ENCYCLOPEDIA OF PHYSICS

ENCYCLOPEDIA OF PHYSICS

CHIEF EDITOR

S. FLUGGE

VOLUME VIa/2

MECHANICS OF SOLIDS II

EDITOR

C. TRUESDELL

WITH 25 FIGURES

SPRINGER-VERLAG BERLIN • HEIDELBERG • NEWYORK

1972

HANDBUCH DER PHYSIK

HERAUSGEGEBEN VON

S. FLUGGE

BAND VIa/2

FESTKORPERMECHANIK II

BANDHERAUSGEBER

C.TRUESDELL

MIT 25 FIG U REN

SPRINGER-VERLAG BERLIN • HEIDELBERG • NEW YORK

1972

ISBN 3-540-0SS3S-S Springer-Verlag Berlin Heidelberg New York

ISBN 0-387-0S53S-S Springer-Verlag New York Heidelberg Berlin

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Contents

The Linear Theory of Elasticity. By Morton E. Gurtin, Professor of Mathematics,

Carnegie-Mellon University, Pittsburgh, Pennsylvania (USA). (With 18 Figures) . l

A. Introduction. 1

1. Background. Nature of this treatise. 1

2. Terminology and general scheme of notation. 2

B. Mathematical preliminaries. 5

I. Tensor analysis. 5

3. Points. Vectors. Second-order tensors. 5

4. Scalar fields. Vector fields. Tensor fields.10

II. Elements of potential theory.12

5. The body B. The subsurfaces and of dB.12

6. The divergence theorem. Stokes’ theorem. 16

7. The fundamental lemma. Rellich’s lemma. 19

8. Harmonic and biharmonic fields.20

III. Functions of position and time .24

9. Class 24

10. Convolutions.25

11. Space-time.27

C. Formulation of the linear theory of elasticity.28

I. Kinematics.28

■yl2. Finite deformations. Infinitesimal deformations.28

V 13. Properties of displacement fields. Strain. 31

vl4. Compatibility. 39

II. Balance of momentum. The equations of motion and equilibrium .... 42

15. Balance of momentum. Stress.42

16. Balance of momentum for finite motions. 51

^17. General solutions of the equations of equilibrium.53

t 18. Consequences of the equation of equilibrium. 59

\ 19. Consequences of the equation of motion.64

III. The constitutive relation for linearly elastic materials.67

\o . The elasticity tensor.67

'^21. Material symmetry.69

' 22. Isotropic materials. 74

' 23. The constitutive assumption for finite elasticity .8o

* 24. Work theorems. Stored energy. 81

\25. Strong ellipticity.86

\j26. Anisotropic materials.87

D. Elastostatics.89

I. The fundamental field equations. Elastic states. Work and energy .... 89

27. The fundamental system of field equations.89

28. Elastic states. Work and energy.94

II. The reciprocal theorem. Mean strain theorems.96

29. Mean strain and mean stress theorems. Volume change.96

30. The reciprocal theorem.98

[ Contents.

III. Boundary-value problems. Uniqueness.102

31. The boundary-value problems of elastostatics .102

32. Uniqueness .104

33. Nonexistence.109

\ IV. The variational principles of elastostatics.110

34. Minimum principles.110

35- Some extensions of the fundamental lemma.115

36. Converses to the minimum principles.116

37. Maximum principles .120

38. Variational principles.122

39. Convergence of approximate solutions.125

V. The general boundary-value problem. The contact problem.129

40. Statement of the problem. Uniqueness .129

41. Extension of the minimum principles.130

VI. Homogeneous and isotropic bodies.131

42. Properties of elastic displacement fields.131

43. The mean value theorem .I33

44. Complete solutions of the displacement equation of equilibrium .... 138

VII. The plane problem.I50

45. The associated plane strain and generalized plane stress solutions . . . 150

46. Plane elastic states.154

47- Airy’s solution.156

VIII. Exterior domains.165

48. Representation of elastic displacement fields in a neighborhood of

infinity.165 49. Behavior of elastic states at infinity.I67

50. Extension of the basic theorems in elastostatics to exterior domains . . 169

IX. Basic singular solutions. Concentrated loads. Green’s functions .173

51. Basic singular solutions.173

52. Concentrated loads. The reciprocal theorem.179

53. Integral representation of solutions to concentrated-load problems . . 185

X. Saint-Venant’s principle.190

54. The V. Mises-Sternberg version of Saint-Venant’s principle.190

55. Toupin’s version of Saint-Venant’s principle.196

56. Knowles’ version of Saint-Venant’s principle.200

56a. The Zanaboni-Robinson version of Saint-Venant’s principle.206

XI. Miscellaneous results.207

57. Some further results for homogeneous and isotropic bodies.207

58. Incompressible materials .210

E. Elastodynamics.212

I. The fundamental field equations. Elastic processes. Power and energy.

Reciprocity.212

59. The fundamental system of field equations.212

60. Elastic processes. Power and energy.215

61. Graffi’s reciprocal theorem.218

II. Boundary-initial-value problems. Uniqueness.219

62. The boundary-initial-value problem of elastodynamics.219

63. Uniqueness.222

III. Variational principles.223

64. Some further extensions of the fundamental lemma.223



IV. Homogeneous and isotropic bodies.232

67. Complete solutions of the field equations.232

68. Basic singular solutions.239

69. Love’s integral identity.242

Contents. VII

V. Wave propagation.243

70. The acoustic tensor.243

71. Progressive waves .245

72. Propagating surfaces. Surfaces of discontinuity.248

73. Shock waves. Acceleration waves. Mild discontinuities.253

74. Domain of influence. Uniqueness for infinite regions.257

VI. The free vibration problem.261

75. Basic equations .261

76. Characteristic solutions. Minimum principles.262

77. The minimax principle and its consequences.268

78. Completeness of the characteristic solutions.270

References .273

Linear Thermoelasticity. By Professor Donald E. Carlson, Department of Theoretical

and Applied Mechanics, University of Illinois, Urbana, Illinois (USA).297

A. Introduction.297

1. The nature of this article.297

2. Notation.297

B. The foundations of the linear theory of thermoelasticity.299

3. The basic laws of mechanics and thermodynamics.299

4. Elastic materials. Consequences of the second law.301

5. The principle of material frame-indifference.305

6. Consequences of the heat conduction inequality.307

7. Derivation of the linear theory .307

8. Isotropy.311

C. Equilibrium theory.312

9. Basic equations. Thermoelastic states.312

10. Mean strain and mean stress. Volume change.314

11. The body force analogy.316

12. Special results for homogeneous and isotropic bodies.317

13. The theorem of work and energy. The reciprocal theorem.319

14. The boundary-value problems of the equilibrium theory. Uniqueness .... 320

15. Temperature fields that induce displacement free and stress free states . . . 322


17. The uncoupled-quasi-static theory.325

D. Dynamic theory .326

18. Basic equations. Thermoelastic processes.326

19. Special results for homogeneous and isotropic bodies.327

20. Complete solutions of the field equations.329

21. The theorem of power and energy. The reciprocal theorem . ;.33I

22. The boundary-initial-value problems of the dynamic theory.335

23. Uniqueness.337


25. Progressive waves.342

List of works cited.343

Existence Theorems in Elasticity. By Professor Gabtano Fichbra, University of Rome, Rome (Italy).347

1. Prerequisites and notations.■.348 O

2. The function spaces and .349

3. Elliptic linear systems. Interior regularity.355

4. Results preparatory to the regularization at the boundary.357

5. Strongly elliptic systems.365

6. General existence theorems.368

7. Propagation problems.371

8. Diffusion problems.373

9. Integro-differential equations .373

10. Classical boundary value problems for a scalar 2nd order elliptic operator .... 374

11. Equilibrium of a thin plate .377

VIII Contents.

12. Boundary value problems of equilibrium in linear elasticity.380

13. Equilibrium problems for heterogeneous media.386

Bibliography.388

Boundary Value Problems of Elasticity with Unilateral Constraints. By Professor Gaetano Fichera, University of Rome, Rome (Italy).391

1. Abstract unilateral problems: the symmetric case.391

2. Abstract unilateral problems: the nonsymmetric case.395

3. Unilateral problems for elliptic operators.399

4. General definition for the convex set V.401

5. Unilateral problems for an clastic body.402

6. Other examples of unilateral problems.404 7. Existence theorem for the generalized Signorini problem.407

8. Regularization theorem: interior regularity.408

9. Regularization theorem: regularity near the boundary.411 10. Analysis of the Signorini problem.413

11. Historical and bibliographical remarks concerning Existence Theorems in

Elasticity.418

Bibliography.423

The Theory of Shells and Plates. By P. M. Naghdi, Professor of Engineering Science,

University of California, Berkeley, California (USA). (With 2 Figures).425

A. Introduction.425

1. Preliminary remarks.425

2. Scope and contents.429

3. Notation and a list of symbols used .431

B. Kinematics of shells and plates.438

4. Coordinate systems. Definitions. Preliminary remarks.438

5. Kinematics of shells: I. Direct approach.449

a) General kinematical results .449

/S) Superposed rigid body motions.452

y) Additional kinematics.455

6. Kinematics of shells continued (linear theory): I. Direct approach.456

(5) Linearized kinematics.456

e) A catalogue of linear kinematic measures.458

^) Additional linear kinematic formulae .461

rj) Compatibility equations.463

7. Kinematics of shells: II. Developments from the three-dimensional theory . . 466

a) General kinematical results.466

P) Some results valid in a reference configuration.471

y) Linearized kinematics.473

(5) Approximate linearized kinematic measures .476

e) Other kinematic approximations in the linear theory.477

C. Basic principles for shells and plates.479

8. Basic principles for shells: I. Direct approach.479

a) Conservation laws.479

P) Entropy production.483

y) Invariance conditions.484

d) An alternative statement of the conservation laws.487

e) Conservation laws in terms of field quantities in a reference state .... 490

9. Derivation of the basic field equations for shells: I. Direct approach .... 492

а) General field equations in vector forms .492

/S) Alternative forms of the field equations.498

y) Linearized field equations.500

б) The basic field equations in terms of a reference state.502

10. Derivation of the basic field equations of a restricted theory: I. Direct approach 503

11. Basic field equations for shells: II. Derivation from the three-dimensional

theory.508

a) Some preliminary results.508 /S) Stress-resultants, stress-couples and other resultants for shells .512

y) Developments from the energy equation. Entropy inequalities .515

Contents. IX

12. Basic field equations for shells continued: II. Derivation from the three-

dimensional theory.519

(5) General field equations .519

e) An approximate system of equations of motion.522

Linearized field equations.523

ij) Relationship with results in the classical linear theory of thin shells and

plates.524

12 A. Appendix on the history of derivations of the equations of equilibrium for

shells.527

D. Elastic shells.528

13. Constitutive equations for elastic shells (nonlinear theory); I. Direct approach 528

а) Gener8il considerations. Thermodynamical results.529

/3) Reduction of the constitutive equations under superposed rigid body

motions.534

y) Matericil symmetry restrictions.537

б) Alternative forms of the constitutive equations.540

14. The complete theory. Special results: I. Direct approach.544

а) The boundary-value problem in the general theory.544

P) Constitutive equations in a mechanical theory .544

y) Some special results.546

б) Special theories.546

15. The complete restricted theory: I. Direct approach.549

16. Linear constitutive equations: I. Direct approach.553

a) General considerations.553

P) Explicit results for linear constitutive equations.555

y) A restricted form of the constitutive equations for an isotropic material . 557

(5) Constitutive equations of the restricted linear theory .560

17. The complete theory for thermoelastic shells: II. Derivation from the three-

dimensional theory.561

a) Constitutive equations in terms of two-dimensional variables. Thermo¬

dynamical results.561

P) Summary of the basic equations in a complete theory.565

18. Approximation for thin shells: II. Developments from the three-dimensional

theory.566

a) An approximation procedure.566

P) Approximation in the linear theory.568

19. An alternative approximation procedure in the linear theory: II. Developments

from the three-dimensional theory.569

20. Explicit constitutive equations lor approximate linear theories of plates and

shells: II. Developments from the three-dimensional theory.572

a) Approximate constitutive equations for plates .572

P) The classiccil plate theory. Additional remarks .575

y) Approximate constitutive relations for thin shells .578

(5) Classical shell theory. Additional remarks .580

21. Further remarks on the approximate linear and nonlinear theories developed

from the three-dimensional equations.585

21 A. Appendix on the history of the derivation of linear constitutive equations for

thin elastic shells .589

22. Relationship of results from the three-dimensional theory and the theory of

Cosserat surface.594

E. Linear theory of elastic plates and shells.595

23. The boundary-v8ilue problem in the linear theory .596

a) Elastic plates.596

P) Elastic shells.597

24. Determination of the constitutive coefficients.598

a) The constitutive coefficients for plates.598

P) The constitutive coefficients for shells.606

25. The boundary-value problem of the restricted linear theory.607

26. A uniqueness theorem. Remarks on the general theorems.610

I b Handbuch der Physik, Bd. VIa/2.

X Contents.

F. Appendix: Geometry of a surface and related results .615

A. 1. Geometry of Euclidean space.615

A.2. Some results from the differential geometry of a surface.621

a) Definition of a surface. Preliminaries.621

P) First and second fundamental forms.623

y) Covariant derivatives. The curvature tensor.624

(5) Formulae of Weingartcn and Gauss. Integrability conditions .625

e) Principal curvatures. Lines of curvature.627

A. 3. Geometry of a surface in a Euclidean space covered by normal coordinates . 628

A.4. Physical components of surface tensors in lines of curvature coordinates . . 631

References.633

The Theory of Rods. By Professor Stuart S. Antman, New York University, New York

(USA). (With 5 Figures) .641

A. Introduction.641

1. Definition and purpose of rod theories. Nature of this article.641

2. Notation.642

3. Background .643

B. Formation of rod theories.646

I. Approximation of three-dimensional equations.646

4. Nature of the approximation process.646

5. Representation of position and logarithmic temperature.647

6. Moments of the fundamental equations.649

7. Approximation of the fundamental equations.652

8. Constitutive relations.654

9. Thermo-elastic rods.656

10. Statement of the boundary value problems.658

11. Validity of the projection methods.660

12. History of the use of projection methods for the construction of rod

theories.663

13. Asymptotic methods.664

II. Director theories of rods.665

14. Definition of a Cosserat rod.665

15. Field equations.666

16. Constitutive equations.669

III. Planar problems.670

17. The governing equations.670

18. Boundary conditions.674

C. Problems for nonlinearly elastic rods .676

19. Existence .676

20. Variational formulation of the equilibrium problems.676

21. Statement of theorems.680

22. Proofs of the theorems.682

23. Straight and circular rods.690

24. Uniqueness theorems.692

25. Buckled states.694

26. Integrals of the equilibrium equations. Qualitative behavior of solutions . 696

27. Problems of design .698

28. Dynamical problems.699

References.700

Namenverzeichnis. — Author Index.705

Sachverzeichnis (Deutsch-Englisch).711

Subject Index (English-German).729

The Linear Theory of Elasticity

By

Morton E. Gurtin.

With 18 Figures,

Dedicated to Eli Sternberg.

A. Introduction.

1. Background. Nature of this treatise. Linear elasticity is one of the more successful theories of mathematical physics. Its pragmatic success in describing the small deformations of many materials is imcontested. The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson. The theo¬ retical development of the subject continued at a brisk pace imtil the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff,

Lam6, Saint-Venant, Somigliana, Stokes, and others. These authors established the basic theorems of the theory, namely compatibility, reciprocity, and imiqueness, and deduced important general solutions of the imderlying field equations. In the 20th century the emphasis shifted to the solution of boundary-value prob¬ lems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Ve- nant’s principle, stress functions, variational principles, and imiqueness.

It is the purpose of this treatise to give an exhaustive presentation of the linear theory of elasticity.^ Since this volume contains two articles by Fichera con¬ cerning existence theorems, that subject will not be discussed here.

I have tried to maintain the level of rigor now customary in pure mathematics. However, in order to ease the burden on the reader, many theorems are stated with h3q)otheses more stringent than necessary.

Acknowledgemenl. I would like to acknowledge my debt to my friend and teacher, Eli

Stbrnbbrg, who showed me in his lectures^ that it is possible to present the linear theory in

a concise and rationed form—a form palatable to both engineers and mathematicians. Portions

of this treatise are based on Stbrnbbrg’s unpublished lecture notes; I have tried to indicate

when such is the case. I would like to express my deep gratitude to D. Carlson, G. Fichera,

R. Huilgol, E. Stbrnbbrg, and C. Truesdell for their valuable detailed criticisms of the

manuscript. I would cdso like to thank G. Benthien, W. A. Day, J. Ericksbn, R. Knobs,

M. Oliver, G. de La Penha, T. Ralston, L. Solomon, E. Walsh, L. Wheeler, and

W. Williams for valuable comments, and H. Ziegler for generously sending me a copy of

1 Specific applications are not taken up in this article. They will be treated in a sequel by

L. Solomon, Some Classic Problems of Elasticity, to appear in the Springer Tracts in Natural

Philosophy.

^ At Brown University in 1959-1961.

Handbuch der Physik, Bd, VI a/2. 1

2 M. E. Gurtin: The Linear Theory of Elasticity. Sect. 2.

Prangb’s 1916 Habilitation Dissertation. Most of the historical research for this treatise was

carried out at the Physical Sciences Library of Brown University; without the continued sup¬

port and hospitality of the staff of that great library this research would not have been pos¬

sible. Finally, let me express my gratitude to the U.S. National Science Foundation for their

support through a research grant to Carnegie-Mellon University.

2. Terminology and general scheme of notation. I have departed radically from the customary notation in order to present the theory in what I believe to be a form most easily understood by someone not prejudiced by a past acquaintance with the subject. Direct notation, rather than cêsian or general coordinates, is utilized throughout. I do not use what is commonly called “dyadic notation”; most of the notions used, e.g. vector, linear transformation, tensor product, can be found in a modern text in linear algebra.

General scheme of notation.

Italic boldface minuscules a,b,ii,v vectors and vector fields points of space.

Italic boldface majuscules A,B, (second-order) tensors or tensor fields.

Italic lightface letters A, a, on, 0, scalars or scalar fields.

C, K: fourth-order tensors.

Sans-serif boldface majuscules M, N ... (except C, K): four-tensors or fields with such values.

Sans-serif boldface minuscules u,%,...: four-vectors or four-vector fields.

Italic lightface majuscules B, D, S, ...: regions in euclidean space,

a group of second order tensors.

Script majuscules SA, ... (except : surfaces in euclidean space.

Italic indices i, j, ...: tensorial indices with the range (1, 2, 3).

Greek indices a, /S, ...: tensorial indices with the range (1,2).

Index of frequently used symbols. Only s}mibols used frequently are listed. It has not been possible to adhere rigidly to these notations, so that sometimes within a single section these same letters are used for quantities other than those listed below.

Symbol Name Place of definition or first occurrence

A (m) Acoustic tensor A Beltrami stress function B Body C Elasticity tensor D| Set of points of apphcation of system I of

concentrated loads E Strain tensor O

E Traceless part of strain tensor

E (B) Mean strain ^ Three-dimensional euclidean space

^ X (—00, 00) = space-time

Sect. 2. Terminology and general scheme of notation. 3

S}mibol Name Place of definition or first occurrence

S}mimetry group for the material at x K Kinetic energy K Compliance tensor M Stress-momentum tensor 0 Origin, zero vector, zero tensor P Part of B Q Orthogonal tensor R Plane region S Stress tensor o

S Traceless part of stress tensor

S (B) Mean stress S^, Complementary subsets of SB TJ {E} Strain energy

Total energy tC Vector space associated with ^ ir{i) =^X(-oo, 00)

W Rotation tensor iT Singular surface a Amphtude of wave b Body force c Centroid of B c Speed of propagation, also the constant

• i67lfl{i — v] Cl Irrotational velocity C2 Isochoric velocity e,- Orthonormal basis / Pseudo body force field / System of forces i ^so function with values i{t) —t k Modulus of compression I System of concentrated loads m Direction of propagation n Outward imit normal vector on SB p Pressure p Position vector from the origin 0 Pc Position vector from the centroid c p Admissible process, elastic process r =|as —0| s Surface traction s Prescribed surface traction i Admissible state, elastic state iy [f] Kelvin state corresponding to a concentrated

load IdAy Unit Kelvin state corresponding to the imit load

BiSity Unit doublet states at y

iy Center of compression at y 4y Center of rotation at y parallel to the 2;,-axis

1*

4 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 2.

Symbol Name Place of definition or first occurrence

t Time u Displacement vector u Prescribed displacement on boundary Uq Initial displacement Vq Initial velocity V {B) Volmne of B w Rigid displacement X, y, z Points in space Xi Cartesian components of x z Complex variable 0{<i} Functional in Hellinger-Prange-Reissner principle A{i} Functional in Hu-Washizu principle

{y) Open ball with radius d and center at y 0{i(} Functional in principle of minimum potential

energy !P{<i} Functional in principle of minimum

complementary energy /S Young’s modulus dfj Kronecker’s delta dv{B) Volume change e Internal energy density Bij;, Three-dimensional alternator

Two-dimensional alternator A Lame modulus fi Shear modulus /j.^ Maximum elastic modulus fi„ Minimum elastic modulus V Poisson’s ratio % Point in space-time Q Density q) Scalar field in Boussinesq-Papkovitch-Neuber

solution, Airy stress function, scalar field in Lame solution

ij/ Vector field in Boussinesq-Papkovitch-Neuber solution, vector field in Lam6 solution

to Rotation vector 1 Unit tensor s}mi S}mimetric part of a tensor skw Skew part of a tensor tr Trace of a tensor ® Tensor product of two vectors V Gradient

^(4) Gradient in space-time

V S}mimetric gradient curl Curl div Divergence div(4) Divergence in space-time A Laplacian

Sect. 3. Points. Vectors. Second-order tensors. 5

S}mibol Name Place of definition or first occurrence

Di, D2 Wave operators

[] Jmnp in a function da Element of area dv Element of volume * Convolution # Tensor product convolution

B. Mathematical preliminaries.

I. Tensor analysis.

3. Points. Vectors. Second-order tensors. The space imder consideration is always a three-dimensional euclidean point space S. The term "point” will be reserved for elements of S, the term “vector” for elements of the as¬ sociated vector space 'f. The inner product of two vectors u and v will be designated by m • ». A cartesian coordinate frame consists of an orthonormal basis {ej ={^, eg, eg} together with a point 0 called the origin. The s5mibol p will always denote the position-vector field on S defined by

piye) =* —0.

We assume once and for all that a single, fixed cartesian coordinate frame is given. If M is a vector and a; is a point, then and Xi denote their (cartesian) components:

UfûCi, Xi = {x — Q)-ei.

We will occasionally use indicial notation; thus subscripts are assmned to range over the integers 1, 2, 3, and summation over repeated subscripts is implied:

3

U-VÛiVi=Y,1*i *'»•

We denote the vector product of two vectors u and vhy uxv. In components

where 6^,* is the alternator'.

îjh —

if {i, j, k) is an even permutation of (1, 2, 3)

• — 1 if {i, j, k) is an odd permutation of (1, 2, 3)

0 if {i, j, k) is not a permutation of (1, 2, 3) •

We will frequently use the identity

îjhîpq ^jp^kq ^jq^kp>

where is the Kronecker delta: ft if i=j

to if i^j.


For convenience, we use the term “(second-order) tensor” as a synonym for “linear transformation from Tînto T^”. Thus a tensor S is a linear mapping that assigns to each vector v a vector

u =Sv.

The (cartesian) components S,-^- of S are defined by

so that M = S» is equivalent to Ui = Si,.Vj.

We denote the identity tensor by 1 and the zero tensor by 0;

lv=v and 0» = 0 for every vector v.

Clearly, the components of 1 are 5,-^. The product ST of two tensors is defined by composition;

(ST) (») =S (T(»)) for every vector v; hence

(ST),,.=s,, r,,..

We write S^ for the transpose of S; it is the unique tensor with the following property:

Su ■ V =u ■ V for all vectors u and v.

We calls symmetric ifS =S^, skewitS = —S^. Since it follows that S is symmetric or skew according as or S,-, = — S,-,-. Every tensor S admits the unique decomposition

S = sym S -f skw S,

where s}mi S is s}mimetric and skw S is skew; in fact,

symS = J(S + S^, skwS = J(S-S^.

We call s}mi S the symmetric part and skw S the skew part of S.

There is a one-to-one correspondence between vectors and skew tensors: given any skew tensor W, there exists a imique vector to such that

Wu = (oXu for every vector u; indeed.

We call to the axial vector corresponding to W. Conversely, given a vector to, there exists a unique skew tensor W such that the above relation holds; in fact,

Wij = — eijkCO^-

We write tr S for the trace and det S for the determinant of S. In terms of the components S.-^- of S:

tTS=Sii,

and det S is the determinant of the matrix

•5ii Si 2 Si2

S21 S2 2 S22 ■

,•^31 •^82 •^3 3,


Given any two tensors S and T,

tr(ST)=tr(TS),

det (ST) =(det S) (detT).

A tensor Q is orthogonal provided

The set of all orthogonal tensors forms a group called the orthogonal group; the set of aU orthogonal tensors with positive determinant forms a subgroup called the proper orthogonal group.

If {e,} is an orthonormal basis and

with Q orthogonal, then {e(} is orthonormal. Conversely, if {e^} and {e^} are ortho¬ normal, then there exists a unique orthogonal tensor Q such that ei = Qef. If such is the case, the components w'f of a vector w with respect to {Cj} obey the following law of transformation of coordinates;

Here and Wf are the components of Q and w with respect to {e^}. Likewise, for a tensor S,

The tensor product a ® 6 of two vectors a and b is the tensor that assigns to each vector u the vector a{b ■ u):

In components

and it follows that

{a®b){u) =a{b • u) for every vector u.

(a (8) 6)i,-=

tr(a(g)b) =a • 6.

Further, we conclude from the above definition that the negative of the vector product a X 6 is equal to twice the axial vector corresponding to the skew part of a® b.

The inner product S • T of two second-order tensors is defined by

S.T=tr(S^T)=S,,.7;.,., while _

|s| =ys-s is called the magnitude of S.

Given any tensor S and any pair of vectors a and b,

a • Sb =S ■ (a® b). This identity implies that

=-e^ ■ [(e,-®e.)e;] =e^ ■ [6^(6,. • e,)];

therefore, since {ej is orthonormal,

(e,®e,.). {e^®ei)=di^dji,

8 M. E. Gurtin : The Linear Theory of Elasticity,

and the nine tensors are orthonormal. Moreover,

{Si,- ei®e^) V =e,. Sîv ■ e,-) =6^ S,-,- =S»,

Sect. 3.

and thus S=Sijei®e,-.

Consequently, the nine tensors ei®e,- span the set of all tensors, and we have the following theorem:

, and the

{tensors [9 (1) The set of all ■ ' symmetric tensors 1 is a vector space of dimension ■ 6

[skew tensors b. (9)

6\ tensors • ]/2 sym Bi ® e,-

bJ ]/2skwe,-®ej., i<f

form an orthonormal basis.

We say that a scalar A is a principal or characteristic value of a tensor S if there exists a unit vector n such that

Sn =An,

in which case we call n a.principal direction corresponding to A. The charac¬ teristic space f or S corresponding to 2, is the subspace of consisting of all vectors u that satisfy the equation

Su=2,u.

We now record, without proof, the

(2) Spectral theorem.^ Let S be a symmetric tensor. Then there exists an orthonormal basis n^.n^.n^ and three (not necessarily distinct) principal values Ai, ^2, ^ o/ S such that

Sni—2.ini (no sum) (a) and

3

S=2Ain<®ni. (b) •=i

Conversely, if S admits the representation (b) with {nj orthonormal, then (a) holds. The relation (b) is called a spectral decomposition of S.

(i) If Xi, 2^, and 2^ are distinct, then the characteristic spaces for S are the line spanned by n^, the line spanned by n^, and the line spanned by fig.

(ii) If 2^=2^, then (b) reduces to

Conversely, if (c) holds with 2i4=22', then 2^ and 2^ are the only distinct principal values of S, and S has two distinct characteristic spaces: the line spanned by and the plane perpendicular to .

(iii) 21 = 2.2=2^ = 2 if and only if

S=2l, (d)

in which case the entire vector space is the only characteristic space for S.

r See, e.g., Halmos [1958, 9], § 79.


We say that two tensor S and Q commute if

SQ^QS.

(3) Commutation theorem.^ A symmetric tensor S commutes with an orthogonal tensor Q if and only if Q leaves each characteristic space of S invariant, i.e. if and only if Q maps each characteristic space into itself.

A tensor P is a perpendicular projection if P is symmetric and

P2=P.

If P is a perpendicular projection, then

(l-P)(l-P)=l-2P-fP2 = l-P,

and therefore 1 — P is also a perpendicular projection. Since

(1-P)P=P(1-P)=P-P2 = 0, it follows that

Pm • (1 —P) v—u ■ P(l—P) » = 0;

therefore the range of P is orthogonal to the range of 1 — P.

Two simple examples of perpendicular projections are 1 and 0. Two other examples are n (g)n and 1 —n ®n, where n is a imit vector. The first associates with any vector its component in the direction of n, the second associates with any vector its component in the plane perpendicular to n. The next theorem shows that every perpendicular projection is of one of the above four types.

(4) Structure of perpendicular projections. IfP is a perpendicular pro¬ jection, then P admits one of the following four representations:

1, 0,n®n, 1 —n®n, where n is a unit vector.

Proof. Let A be a principal value and n a principal direction of P. Then Pn—Xn, and hence P^n —XPn, which imphes Pn=XPn. Thus each principal value of P is either zero or one. If all principal values are zero, thenP = 0; if all are unity, then P= 1. If exactly one principal value is unity, then we conclude from the spectral theorem that P=n®n with n a unit vector. If exactly two are unity, then

P=mi®mi-f m2®m2,

where and m^ are the corresponding principal directions, and it is a simple matter to verify that this representation is equivalent to

P=1 —n®n, where n —m^xm^. □

Notice that, in view of (4), the spectral decompositions (a)-(d) in (2) are all expansions in terms of perpendicular projections.

We shall identify fourth-order tensors with linear transformations on the space of all (second-order) tensors. Thus a. fourth-order tensor C[-] is a hnear mapping that assigns to each second-order tensor E a second-order tensor

s = c[£:]. 1 This theorem is a corollary to Theorem 2, § 43 and Theorem 3, § 79 of Halmos

[1958, 9].


The components of C [ • ] axe defined by

so that the relation S = C [JS] becomes

If C'ijhi are the components of C[-] with respect to a second orthonormal basis {ei, e'z, 63} with e,' = 06^, then

îjkl QmiQnj Qpk Qql ^mnpq’

The transpose of a fourth-order tensor C is the unique fourth-order tensor with the following property:

E ■ C[i(^ = C^[JS] • 1? for all (second-order) tensors E and P.

In components CjJjj = Qi,-;- Th® magnitude |C| of C is defined by

|C|=sup{|C[£:]|:|£:|=l}. Clearly,

lc[£:]|g|c||£:|. We will occasionally have to deal separately with the components ^2 in

the %, Xg-plane of a vector u. For this reason, Greek subscripts will always range over the integers 1, 2 and summation from 1 to 2 over repeated Greek subscripts is implied, i.e.,

2

2 J'a- a=l

The two-dimensional alternator will be designated by thus

612 ~ ®21~G 611 =622“ O'

4. Scalar fields. Vector fields. Tensor fields. Let ^ be an open set in S, By a scalar, vector, or tensor field on D we mean a function 99 that assigns to each point xiD, respectively, a scalar, vector, or tensor (p{x). We say that a scalar field 93 is differentiable at x if there exists a vector w with the following property:

—(2f—*)+o(|2f—*1) as y-^x.

If such a vector exists, it is unique; we write to = F 99 (as) and call F 99 (as) the gradient of 99 at as. The partial derivatives of 99 at as, written <pî{x), are defined by

?>,.■(*) = =Vcp{x) ■ e~ [F99(as)],..

More generally, let Whe a finite-dimensional inner product space, and let W be a mapping from D into For example, may be the translation space 'f' corresponding to S or the space of all tensors. Then W is differentiable at aseZ? if there exists a linear function • ] from Tînto such that

'P{y)—'P{^) = '^\.y—^'\+o{\y-x\) as y-^x.

If .5Pexists, it is unique; we write VW{x) and caU VW{x) the gradient of W at as. It is not difficult to show that the hnear transformation VW{x) can be computed using the formula

F'F(®)[o] = -/^ 'F(®-f«o)^,

which holds for every vector

Sect. 4. Scalar fields. Vector fields. Tensor fields. 11

We say that W is of class on D, or smooth on D, if is differentiable at every point of D and VW is continuous on D. The gradient of VW is denoted by W. Continuing in this manner, we say that W is of class on D if it is class and its (AT—1)th gradient is smooth. Clearly, the gradient of a smooth scalar field is a vector field, while the gradient of a smooth vector field is a tensor field. We say that W is of class onB it W is of class on D and for each {0,1, F*”) W has a continuous extension toB. (In this case we also write for the extended function.)

A field W is analytic on B if given any x in B, W can be represented by a power series in some neighborhood of x. Of course, if W is analytic, then is of class C°°.

Let M be a vector field on B, and suppose that u is differentiable at a point xiD. Then the divergence of m at as is the scalar

divM(as) =tr FM(as),

and the partial derivatives of u are defined by

=--|^ = 6,- • l7M(as)e,. = [FM(as)],.,.;

hence div M(as) =Ui^j{x).

The curl of u at as, denoted curl u (as), is defined to be twice the axial vector corresponding to the skew part of FM(as); i.e. curl M(as) is the unique vector with the property:

[FM(as) — FM(as)^]a = (curl u(as)) xa

for every vector a. In components

[curl M(as)],- = e,-,-sM;i,,-(as).-2

We write FM(as) for the symmetric gradient of u:

Vu{x) =s}mi FM(as) =^{Vu{x) + FM(as)^}.

vtn (in.iiA, A (A.

" Ac

Let S be a tensor field on B, and suppose that S is differentiable at as. Then the tensor field is also differentiable at as; the divergence of S at as, written div S(as), is the unique vector with the property:

[div S (*)] . a=div (x) a]

for every fixed vector a. In the same manner, we define the curl of S at as, written curl S(as), to be the unique tensor with the property:

[curl S{x)]a = curl [S^ (as) a]

for every a. The partial derivatives of S are given by

thus [divS{x)]i=Sijj{x),

[curl S {x)]ij = eipkS^k,p (*) •

Let 99 be a differentiable scalar field, and suppose that V(p is differentiable at as. Then we define the Laplacian of 99 at as by

J 99 (as) = div F99 (as).

12 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 5-

We define the Laplacian of a vector field u with analogous properties in the same manner:

Au{x) =div Vu(x). Clearly,

A(p{x)=(pîi(x),

[Au(x)]i=Ui^,.^{x).

Finally, the Laplacian JS(a5) of a sufficiently smooth tensor field S is the unique tensor with the property:

in components [d S(a5)] a = J [S (as) a];

[JS(as)],,. = S,,.,,(as).

Of future use are the following

(1) Identities. Let (p be a scalar field, u a vector field, and S a tensor field, all of class on D. Then

If S is symmetric.

cml F9? = 0, (1)

div curl M = 0, (2)

cml curl M = F div u — Au, (3)

curl Fm = 0, (4)

curl(FM^) = F curl u. (5)

Vu = — VuT =» FFm = 0, (6)

div curl S = cml div S^, (7)

div(ctu:lS)^ = 0, (8)

(curl curl S)^ = curl curl S^, (9)

curl (951) = — [curl (931)] (10)

div(S^M) =M • div S +S • Vu. (11)

tr(curlS)=0, (12)

curl curl S = - JS+2F div S- FF(tr S) +1 [A (tr )S -div div S]. (13)

If S is symmetric and S = 6f — 1 tr 6f, then

curl curls =—J6f + 2F div 6f — l div div 6f. (14)

If S is skew and u is its axial vector, then

curl S = 1 div M —Fm. (15)

II. Elements of potential theory.

5. The body B. The subsurfaces and 5^2 of dB. Given a region D in S,

we write D for its closure, D for its interior, and dD for its boundary. We will consistently write [y) for the open ball in of radius q and center at y.

Sect. 5» The body B, The subsurfaces 6^^ and 6^^ of dB. n Let D be an open region in or in the x^, :af2-plane. We say that D is simply-

connected if every closed curve in D can be continuously deformed to a point without leaving D (i.e., if given any continuous function g: [0, f]->-D satisfying g[0) =g{i) there exists a continuous function /: [0,f] X [0,f]->D and a point yeD such that f(s, 0) =g{s) and f{s, f) =y for every se [0, f ]).

The following definitions will enable us to define concisely the notion of a regular region of space.^ A regular arc is a set « in which, for some orientation of the cartesian coordinate system, admits a representation

a={x-. Xi=f{x-^, %=g(%),

where a<b and / and g are smooth on [a, 6]. A regular curve is a set in con¬ sisting of a finite number of regular arcs arranged in order, and such that the terminal point of each arc (other than the last) is the initial point of the following arc. The arcs have no other points in common, except that the terminal point of the last arc may be the initial point of the first arc, in which case is a closed regular curve. A regular surface element is a (non-empty) set MczS which, for some orientation of the cartesian coordinate system, admits a representation

^={05: %=/(%, x^), (xi, X2)eR},

where 2? is a compact connected region in the %, 8R is a closed regular curve, and / is smooth on R. The union of a finite number of regular surface elements is called a regular surface provided:

(i) the intersection of any two of the elements is either empty, a single point which is a vertex for both, or a single regular arc which is an edge for both;

(ii) thfe intersection of any collection of three or more elements consists at most of vertices;

(iii) any two of the elements are the first and last of a chain such that each has an edge in common with the next;

(iv) all elements having a vertex in common form a chain such that each has an edge, terminating in that vertex, in common with the next; the last may, or may not, have an edge in common with the first.

The term edge here refers to one of the (finite number of) regular arcs com¬ prising the boundary of a regular surface element, while a vertex is a point at which two edges meet. If all the edges of a regular surface belong each to two of its surface elements, the surface is a closed regular surface. Note that a regular surface (and hence a closed regular surface) is necessarily both connected and hounded. Thus, e.g., the boundary of the spherical shell

{®: a^|a5—asol ^/S} (a</S)

is not a regular surface, but rather the rmion of two closed regular stirfaces.

Let J5 be an open set in S. We say that J5 is a bounded regular region if B is the interior of a closed and bounded region B in S whose boundary dB is the union of a finite number of non-intersecting closed regular surfaces. Note that the boundary of a bounded regular region may have comers and edges. We win occasionally deal with infinite domains. We say that J5 is an unboun^d regular region if B is unbounded, but Bn2^(0) is a bounded regular region

1 All of the definitions contained in this paragraph are from Kellogg [1929, i], Chap. IV,

§8.


for all sufficiently large q. Note that the boundary of such a region may extend

“ <“*» « called an

article, unless specified to the contrary, B tvill denote a bounded regular region. We will refer to B as the bo2 A S neSsSlv proper) bounded regular subregion P of B will be referred to as a S ofT ^

We shaU designate the outward unit normal to 8B 3.t x^8B by n(x) If

7 ^3.’. ^ 0'

where is the (relatiye) interior of Note that since is regular where is the (relatiye) closure of « gu ar,

B b' ^ B % unbounded! A partition for B is a finite collection ^1. B2. •••, B^ of mutually disjoint regular subregions of B such that

N

^=u^„. n^l

A function W is piec^ê continuous on B if there exists a partition

on’l? ^ ®uch that for each B„ the restriction of !P to B is bounded

sTz‘"a

™ ^ ^ piecewise regnlar on . Then we write <P^} on ^ if IB (as) =W{x) at each regular point x€S^„.

sati JiedT^ ^ P»-opeHy regular^ proyided the following hypotheses are

(i) B is a bounded regular region;

(ii) for each x^CzdB there exists a neighborhood N ai r nnH q

smooth homeomorphismbofiVoB onto the soM^p?er? ^

{x:xsÔ, *? + xi + xi^l}

such that h mapsiV nSB onto the set {*: X3 = 0, xf+x|^t};

(iii) det Vh is positiye and bounded away from zero.

sfar-shaped if there exists a point x„eB such that the line

So,?rT2“f n“ “t^êcts dB oJy ft x We s^l hout proof the following property of star-shaped regions:

(1) Let B be star-shaped. Then there exists a point x^^B such that

. , (® —»o) ■»(») >0 at every regular point xedB.

1 This definition is due to Fichera [1971, i], § 2.

Sect. 5- The body B. The subsurfaces and of 8B. 15

Let (%, x^, denote cartesian coordinates. We say that B is x,-convex (i fixed) if every line segment parallel to the x^-axis lies entirely in B whenever

its end points belong to B.

Consider the following projections of B :

B\<^~{{x^t X2) • apcB'^,

B^^{x^: x^B),

and for each x^^B^, let

^2)- (^l> ^2>

Note that

2 = U -î 2 (^3) • *s€Bs

(2) Lemma. Let M ^1 and N ^1 be fixed integers, and consider the following

hypotheses on B:

(i) B is Xg-convex.

(ii) 512(^3) simply-connected for each X36B3.

(iii) There exist scalar fields x.^ and x^ of class onB^ and on B^ such that

the curve

^2* ^3)* ^l“^l(^3)* ^2~

is contained in B.

(iv) There exists a scalar field Xg of class onB^g and on B^g such that

the surface ^={(xj, Xg, X3): Xg=Xg(xj^, Xg), (%, ^2)^-®i2}

is contained in B.

Let ^{M, N) denote the set of all scalar fields of class on E and class on B. Then the following two assertions are true'.

16 M.E. Gurtin : The Linear Theory of Elasticity. Sect. 6.

(a) If (i) and (iv) hold, and if N), then there exists a field N)

such that « 3=A.

(b) If (ii) and (iii) hold, and if N) and satisfies

^1,2 ^2,1=0,

then there exists a field (pe^{M, N) such that

Va = <P,a-

Proof. In view of (i)-(iv), the ftinctions

(p{Xj^, X^, X^ ~ f ^2> ^s) dXa,

co(%, x^, f ^(%i X2, ^3) dXg,

where r(Xi, x^, X3) is any smooth path in £12(^3) connecting the points

(%(^3). ^2(%)) (%> ^2). have the desired properties. □

Let S^C dB. We say that B is convex with respect to S/" if the line segment connecting any pair of points x and x oiS/" intersects dB only at x and x. Note that this imphes, in particular, that S/’ is convex, though B need not be.

6. The divergence theorem. Stokes’ theorem.

(1) Divergence theorem. Let B be a hounded or unbounded regular region. Let q> be a scalar field, u a vector field, and T a tensor field, and let <p, u and T be continuous onB, differentiable almost everywhere on B, and of hounded support. Then

J (pn da=fVq) dv, dB B

f u®n da=jVu dv, ZB B

f wnda—f div udv, dB B

f nxu da=f curl udv, dB B

f Tn da =f div T dv, dB B

whenever the integrand on the right is piecewise continuous on E.

Proof. The results concerning 9? and u are classical and will not be verified here.^ To estabhsh the last relation, let 6 be a vector. Then

6 • / Tn da= / 6 • Tn da— j {T'^b) ■ n da dB dB dB

=/div [T^b) dv—f (div T) - b dv =b •/div Tdv, B B B

which imphes the desired result since b is arbitrary. □

1 See, e.g., Kellogg [1929,1}.

Sect. 6. The divergence theorem. Stokes' theorem. 17

(2) Stokes’ theorem. Let ubea smooth vector field and T a smooth tensor field on B. Then given any closed regular surface in E,

f (curltt) -nda — O, sr>

f (curl T)^n da=0. se

Proof. The relation concerning u is well known.^ To establish the second result, let 6 be a vector, and let u = T^b. Then

b •/(curl T)^n da=fb- (curl T)^n da=fn- (curl T)b da sr> £r’ sr>

= fn- curl [T'^b) da—f (curl u) • n da—0, y sr>

which implies the desired result, since b is arbitrary. □

(3) Theorem on irrotational fields. Assume that B is simply-connected.

(i) Let ube a class (N'î) vector field on B that satisfies

curl M = 0.

Then there exists a scalar field <p of class on B such that

u — V(p.

(ii) Let T be a class tensor field on B that satisfies

curl T = 0.

Then there exists a vector field u of class on B such that

T — Vu.

(iii) Let T obey the hypotheses of (ii), and, in addition, assume that

trT = 0.

Then there exists a skew tensor field W of class on B such that

T = cm\W.

Proof. Part (i) is well known.^ Indeed, the function cp defined by

X

(p{x) = f u(y) ■ dy Xo

has the desired properties. Here is a fixed point of B, and the integral is taken along any curve in B connecting aSp and x.

To estabhsh (ii) let ti = Tê,. (a)

Then curl t^ = curl [T'êfj = (curl T) e,- = 0,

and by (i) there exist scalar fields Uf such that

ti = Vu,. (b)

^ See, e.g., Courant [1957, 4], Chap. 5. ^ See, e.g., Phillips [1933, 3], § 45.

Handbuch der Physik, Bd. VI a/2. 2

18 M. E. Gurtin ; The Linear Theory of Elasticity. Sect. 6.

Let u—uê^. Then, since Ui—we^, it follows that

= (c)

and (a)-(c) imply T = Vu.

To prove (iii) let u be the vector field established in (ii), and let W be the negative of the skew tensor field corresponding to u. Then

div M = 0,

since tr T = 0, and we conclude from (15) of (4.1) that

CMilW—Vu. □

(4) Theorem on solenoidal fields. Assume that 8B is of class C®. Let S' denote the set of all fields of class C® on E and class C® on B.

(i) ^ Let uiS' he a vector field, and suppose that

fu-n da=0 se

for every closed regular surface £AcB. Then there exists a vector field w^S' such that

u = ctirli®.

(ii) Let TeS' he a tensor field, and suppose that

fT^nda=0 y

for every closed regular surface £AcB. Then there exists a tensor field such that

T = curl W.

Proof. We omit the proof of (i).® To establish (ii) let

ti = Tei. Then

f ti-n da=ef • j T'^ nda =0, y y

and we conclude from (i) that there exist vector fields such that

Let

and note that for any vector a.

Thus

f; = curlMi,..

Wâ — afWf.

Ta = a,- Te,- = curl (a,- mj,) = curl (W'â) — (curl W) a.

1 The usual statement of this theorem (see, e.g., Phillips [1933, 3], §49, Courant

[1957, 4], pp. 404-406), which asserts that div m = 0 implies u = curl w, is correct only when dB consists of a single closed surface. In this instance div m = 0 is equivalent to the asser¬ tion that

f u-nda = o (•) y

for every closed regular surface y CB. However, if dB consists of two or more closed surfaces (e.g. when B is a spherical shell) there exist divergence-free vector fields in S that do not satisfy (•). Note that by Stokes' theorem, (•) holds whenever u is the curl of a vector field.

® See Lichtenstein [1929, 2], pp. 101-106.

Sect. 7. The fundamental lemma. Rellich’s lemma. 19

and hence T = curl W. □

(5) Properties of the Newtonian potential^ Let q> be a scalar field that is continuous on B and of class [N ^ 1) on B, and let

B

for every xeB. Then f is of class on B and

Ay)=q>.

Further, an analogous result holds for vector and tensor fields.

(6) Properties of the logarithmic potential.^ Let R be a measurable region in the %, x^-plane. Let q> be a scalar field that is continuous on R and of class

[N'^X) on R, and let

“ - s/ for every (x^, X2)^R. Then f is of class on R and

Ay) = q>.

(7) Helmholtz’s theorem. Let u be a vector field that is continuous on B and of class (N'î) on B. Then there exists a scalar field cp and a vector field w, both of class on B, such that

u = V(p-\-cm\w,

divt»=0.

Proof. By (5) there exists a class vector field v on B such that

By (3) of (4.1),

thus if we let

u=Av.

u — V div V —curl curl w;

9? = div*5, M> = —curlw,

the desired conclusion follows. □

7. The fundamental lemma. Rellich’s lemma. For *0 a point of and h a positive scalar, let 0,, {x„) denote the set of aU class C°° scalar fields 95 on with the following properties:

(a) 9? > 0 on the open ball [x^ of radius h and center at x^;

(b) 9? = 0 on — i7j(a5o).

For A > 0 the set 0^ {x„) is not empty. Indeed, the function 9? defined by

10, xiE,,(Xo) belongs to 0i,(x„).

^ See e.g., Kellogg [1929, 11, p. 156; Courant [1962, 5], p. 249. ® See, e.g., Kellogg [1929, 11, P- 174: Courant [1962, 5], p. 2S0.

2*


Let be a subset of dB. We say that a function f on B vanishes near provided there exists a neighborhood N ot such that f=0 on NnE. We are now in a position to estabhsh a version of the fundamental lemma of

the calculus of variations that is sufficiently general for our use.

(1) Fundantental lemma. Let W he a finite-dimensional inner product space. Let w: B-^W be continuous and satisfy

fw-v dv=0 B

for en)ery class C°° function v: B-^W that vanishes near dB. Then

w — Q on E.

Proof. Let Cj, Cg,..., e„ be an orthonormal basis in iP", and let

n e^.

t=i

Assume that for some x^^B and some integer k, (aSj) =|= 0. Then there exists

2X1 h>0 such that S^[x^cB and on S],{x^ or w^<0 on Z),[Xq). Assume, without loss in generality, that the former holds. If we let

v = (pe^,

where (pi0^{x^, then v is of class C°° on E and vanishes near 8B, and by prop¬ erties (a) and (b) of 95,

f w ■ V dv — f (pw ■e^dv = f (pw,^dv>0. B B (aCo)

But by hypothesis, fw-v dv=0.

B

Thus m = 0 on B. But w is continuous on E. Thus m = 0 on J5. □

We now state, without proof,

(2) Rellich’s lemma.^ Let B be properly regular_. Further, let {m„} be a sequence of continuous and piecewise smooth vector fields on B, and suppose that there exist constants and M2 such that

f\u„\^dv^Mi, f\Vu„\^^M2, «=1,2,.... B B

Then there exists a subsequence such that

hm f\Un,—u„Mdv=0. j, k—*-oo £

8. Harmonic and biharmonic fields. Let D be an open region in <f, and let W ho 2. scalar, vector, or tensor field on D. We say that W is harmonic on D if W is of class on D and

AW = 0.

We say that W is biharmonic on D iiW is of class C* on D and

AAWÔ.

We now state, without proof, some well known theorems concerning harmonic and biharmonic fields.

^ Rellich [1930, 4]. See also, Fichera [1971, i]. Theorem 2.IV.

Sect. 8. Harmonic and biharmonic fields. 21

(1) Every harmonic or biharmonic field is analytic.

(2) Hanmck’s convergence theorem.^ Let be harmonic on D for each d (0<d<do), and let

as (5-^0

uniformly on each closed subregion of D. Then W is harmonic on D, and for each fixed integer n,

as (5-^0

uniformly on every closed subregion of D.

(3) Mean value theorem. Let cp and ip be scalar, vector, or tensor fields on D with (p harmonic and ip biharmonic. Then given any ball =2Jg(y) in D :

^ dSp

<p{y)= J<P<iv, (2) Sp

y>iy)+ f 4^ / ^ dSp

'p{y)=-i^ J • (4)'

Xp dSp

We now summarize some results concerning spherical harmonics.® Let [r, 6, y) be spherical coordinates, related to the cartesian coordinates (%, x^, x^) through the mapping

%=r sin 6 cos y, rg —r sin 6 sin y, r3=zcos6,

0^r<oo, O^d^Tt, O^y <27t.

Then the general scalar solid spherical harmonic of degree k admits the representation

ffW(z,e,y)-SW(0.y)z* (^ = 0, ± 1, ± 2....),

where {B, y) is the general surface spherical harmonic of the same degree:

1^1

[B, y) = 2 [®i"* « y + sin n y] (cos B), = 0. »=0

Here designates the associated Legendre function of the first kind, of degree k and order n, while a^\ a'^\ Sjf' (« = 1, 2,..., |^|), for fixed k, are 2^ + 1 arbitrary constants. If k and n are non-negative integers.

d»Pkiit) p _ _1_ dHS^-i)^ 2*A! dS!=

where P^ is the Legendre pol5momial of degree k. Consequently,

(I) = Pk (I). A'”> (f) = 0 {n>k^0).

In addition, we have the recursion relations

Pl«i_i(f)=P*'«)(f),

1 See, for example, Kellogg [1929, 1], p. 249- 2 Nicolesco [1936, 3]. See also Diaz and Payne [1958, 6]. ® See, for example, Poincar^: [1899, 1], Kellogg [1929, 1], and Hobson [1931, 4].


which axe valid for unrestricted k and n. As is evident from the above relations, every surface harmonic of degree —\—k is a surface harmonic of degree k. Next, we recall that any two surface harmonics of distinct degree are orthogonal over the unit sphere r = \:

271 71

[SW, S(“)] = / /S(*)(0, y) SW(0, y) sin 0 0=0 {k^m, -m-i). 0 0

A vector solid spherical harmonic of degree ^ is a vector-valued function with the property that given any vector e, the function e • ft**) is a scalar solid spheric^ harmonic of degree k.

The solid harmonics of degree k axe harmonic ftmctions, which are homo¬ geneous of degree k with respect to the cartesian coordinates (x^, x^, Xg); they axe homogeneous polynomials of degree k in the Xf if ^ is a non-negative integer. Further, if H*** is a scalar solid harmonic of degree k, then is a vector solid harmonic of degree k — i and

p.

where p(x) =x —0. Similarly,

for a vector solid harmonic of degree k. Tinning from solid spherical harmonics to general harmonic functions, we

cite the following theorem, in which 27 (oo) is the following neighborhood of infinity:

27(oo) ={a[::ro<la!—0|<oo}. Here ro> 0 is fixed.

(4) Representation theorem for harmonic fields in a neighborhood of infinity. Let q> be a harmonic scalar field on 27(oo). Then:

(i) <p admits the representation

<p= Ill'll. A——OO

where the H*** are uniquely determined solid spherical harmonics of degree k, and the infinite series is uniformly convergent in every closed subregion of 27(oo);

(ii) (p has derivatives of all orders, series representations of which may be ob¬ tained by performing the corresponding termwise differentiations, the resulting expansions being also uniformly convergent in every closed subregion of 27(oo) ;

(iii) if n is a fixed integer, the three statements

(а) (p{x)=0{r’'-^),

(|8) <p{x)=o{/'),

(y) H(*)=0 for k^n,

are equivalent and imply

(б) Vq)(x)=0{r’'~^).

An analogous result holds for harmonic vector fields on 27(oo).

We now state and prove a theorem which is the counterpart for biharmonic functions of (4).

(5) Representation theorem for biharmonic fields in a neighborhood of infinity.^ Let f be a biharmonic scalar field on 27(oo). Then:

1 PicoNE [1936, 4]. See also Gurtin and Sternberg [1961, 11].

Sect. 8. Harmonic and biharmonic fields. 23

(i) y) admits the representation

y,= 2 2 HW, 00 fts* —00

where G**' and H*** are solid spherical harmonics of degree k, and both infinite series are uniformly convergent in every closed subregion of E{oo);

(ii) ip has derivatives of all orders, series representations of which may be obtained by performing the corresponding termwise differentiations, the resulting expansions being also uniformly convergent in every closed subregion of E(oo);

(iii) if n is a fixed integer, the three statements

(a) ip{x)

(/S) ip {x)ô{r”),

(y) G(*)=H(*-®)=0 for k-^n,

are equivalent and imply

{d) Fv'(®)=0(r”-").

An analogous result holds for biharmonic vector fields on 27(oo).

Proof. We show first that ip admits Ahnansi’s representation

ip = <p-\-r^ 0,

where 9? and 0 are harmonic on E(oo). A completeness proof for this represen¬ tation, apphcable to a region that is star-shaped with respect to 0 in its interior, is indicated by Frank and v. Mises.^ To establish the completeness of this representation in the present circumstances, it is sufficient to exhibit a harmonic function 0 that satisfies

A {ip — r^ 0) =0, or equivalently,

4r -^-}-60=Aip. (a)

Since, by h5q)othesis, A ip is harmonic, we know from (4) that it admits the expansion

00

A ip {r, d,y)= 2 {6, y) r’‘, (b) k=—oo

where the $^'‘^6, y) are surface harmonics of degree k and the infinite series has the convergence properties asserted in parts (i) and (ii) of (4). Now consider 0 defined by

0, y) (d, y) r\ S(*) = .

Clearly, 0 is harmonic. Further, this function also satisfies (a), as is confirmed with the aid of (b).

On appl3dng (i) and (ii) of (4) to q> and 0, (i) and (ii) of the present theorem follow at once. With a view toward estabhshing (iii), we observe that (j8) is immediate from (a). We show next that (j8) imphes (y). Let s(*)(6, y) and (0, y) be the respective surface harmonics corresponding to the solid harmonics G*** and We now multiply the right-hand member of the equation in (i) by r““s(”')(0, y) (fM = 0, ± t, ± 2,...) and integrate termwise over OÔ^n,

^ [1943, P. p. 848.


Bearing in mind the convergence properties asserted in (i) as well as the orthogonality relations, we find that the integral

2n 71

Imn (>■) = ^“” / / V (^. y) s'™’ (®. y) sin 0 0 y, (c) 0 0

for all sufficiently large values of r, is given by

_j_ ytn-n+i |^5(>»)^ sWj _j_ y-H-m+1 5W] _

It follows from (j0) and (c) that I^ni^) tends to zero as r^cx> for fixed m and n; consequently, the same is true of the right-hand member of (d). Noting that the coefficients of the inner products appearing in (d) are four distinct powers of r, we conclude that [$*“>, s*”'] =0 if m—nÔ. Thus $(“'^=0, which implies that (;(>») =0 for m'^n. In a similar manner we verify that for m'^n—2. Hence (j0) is a sufficient condition for (y).

We have to show further that (y) implies (a). This is readily seen to be true by observing that each of the series in (i) represents a field that is harmonic on 2'(oo), and by invoking part (iii) oi (4). Finally, (y) implies (d) because of (ii) of the present theorem and (iii) of (4). Theorem (5) has thus been proved in its entirety. □

The following well-known proposition on divergence-free and curl-free vector fields will be extremely useful.

(6) Let V be a smooth vector field on B that satisfies

div«=0, curl«=0. Then v is harmonic.

Proof. Let £) be an arbitrary open ball in B. By hypothesis and (i) of (6.3), there exists a class scalar field <p on D such that

V = V(p.

Since v is divergence-free, <p is harmonic, and this, in turn, implies that v is harmonic. □

III. Functions of position and time.

9. Class C^’^. By a time-interval T we mean an interval of the form ( —oo,/p), (0, /q). or [0, tg), where /o>0 may be infinity. We frequently deal with functions of position and time having as their domain of definition the cartesian product DxT of a set £) in ^ and a time-interval T. In this physical context x will always stand for a point of D and t will denote the time. If is a function on £) X T, we write *P(‘,t) for the subsidiary mapping of D obtained by holding t fixed and *P(x, •) for the mapping of T obtained by holding x fixed. We say that such a function W is continuous (smooth) in time if W(x, •) is continuous (smooth) on T for each xeD. If DxT is open, we write V^’'^*P(x, t) for the «-th gradient of W with respect to_a5 holding t fixed and (x, t) for the dmvatl^bf

T* with respect to t holding x fixed. Ordinarily, we write W instead of

Let M and N be non-negative integers. We say that W is of class on B X (0, /q) if is continuous on B X (0, /q) ^md the functions

P(’”) we{0,1,..., M}, «e{0, i,..., iV}, w-|-«^max{M, AT},

Sect. 10. Convolutions. 25

ejiist and are continuous on Bx(0,io). We say that ^ is of class on

£?x[0, ô) if ^ is of class on i?x(0,/q) and for each me{0, i,M}, ne{0, i, .... iV}, F*™* IF*"’ has a continuous extension to Bx[0, Iq) (in this case we also write F*™' for the extended function). Finally, we write C^for C^’^.

We say that a function ^ on [0, ig) is piecewise regular if W is piecewise continuous on X [0, /p) and is piecewise regular on for 0 ^<<o•

10. Convolutions. Let 7'=[0,/p) or (—oo, cx)), and let 9? andyi be scalar fields on DXT' that are continuous in time. Then the convolution (p*y) of <p and rp is the function on DxT defined by

t

<p*xp{x, t) = f (p{x, t — r)y){x, r) dr. 0

Let be a function on D X [O, cx>). We say that W has a Laplace transform W if there exists a real number »?p ^0 such that for every »?G[»?p, 00) the integral

_ 00

W(x,r])=fe-’i‘W(x,t)dt 0

converges uniformly on D.

(1) Properties of the convolution. Let cp.tp, and co he scalar fields on £)x[0, /p) that are continuous in time. Then

(i) <p*y) — y)*<p;

(ii) {(p*\p)*u) = (p*[\p*m)=<p*'ip*u)\

(iii) (p*{\p-\-oi)=<p*\p-\-(p*u)',

(iv) for xeD, if (p{x,>)>0 on (0, tg), then

(p*ip{x, •) —0 rp{x, •) =0',

(v) for xeD, if tg — 00, then

(p*y)(x,’)=0 => (p(x,-)=0 or ip(x,’)—0',

(vi) if <p is smooth in time, then

= ^*y)-{- <p{’, 0)y);

(vii) if tg — (xi and cp,rp possess Laplace transforms, then so also does <p*f, and

(p*ip = (pip

on Dx [rig, 00) for some rjgÔ.

Proof. The results (i)-(iii), (v), and (vii) are well known,^ while (vi) follows directly upon differentiating 97*^1. To establish (iv) let (p = (p{x,‘), ‘ip=‘ip{x,>), assume that ip docs not vanish identically on [0, 4), and let

h — inf {f ^ [0, tg) • ip {1)^0} •

Since ip is continuous, there exists a number 4 with 4< ^2< ^0 such that yi > 0 on (4,4) or yi < 0 on (<j, 4)- In either case, since 97 > 0, it follows that

/ 95(4— 0

and we have a contradiction. □

* Properties (i)-{iii) are established in Chap. I of Mikusinski [1959, 10]. Titchmarsh’s theorem (v) is proved in Chap. II of the same book.

26 M. E. Gtjrtin : The Linear Theory of Elasticity. Sect. 10.

Let be a finite-dimensional inner product space, e.g. iV = 'f' ov W = the set of all tensors, and let and !P be -valued fields on DxT that are continuous in time. Then we write 9?*!? for the #^-valued field on DxT defined by

I (p* 'P(x, t)= J(p(x,t — T) ’P {x, t) dr,

0

and * !P for the scalar field on DxT defined by

0*P(x,t)^f0{x.t-r) . P(x,T)dr. 0

Thus if 9? is a scalar field, u and v vector fields, and S and T tensor fields, all defined on DxT, then ^*u is a vector field, <p*S is a tensor field, and u*v and S*T are scodar fields; in components

[(p*u\i = (p*Ui, [9J*S]<j. = 9P*Sj,-,

u*v = Ui*Vi, S*T = Sij*Tij.

Finally, we write S*u for the vector field

S*u{x, t) =f S{x, t — r) u[x, t) dv, 0

then

Properties analogous to (i)-(iv) oi (1) obviously hold for the convolutions defined above.

Let I and k be continuous vector functions on (—00, cx>). Then l#k is for the tensor function on this interval defined by

= / l{t—T)®k(T) dr. 0

In components

For convenience, we write for the function on (—00, 00) defined by

!„(/) =!(/-«).

Further, we let denote the set of all smooth vector functions on (—00, 00) that vanish on (—00, 0).

(2) Let I, k^TT and let a>0. Then

=l#kg^>

la^k—T# k^. Proof. Since

l~k = 0 on (—00,0],

= = on (—00, a],

the above relations hold trivially on (—00, 0]. Thus choose te(0, 00). Then

(a)

l„#k{t) ==/ l{t — r—(x.)^k(r) dr.

Sect. H. Space-time. 27

Thus, letting t=A—a, we arrive at

<+a

la#k{t)— f —A)(g)fc„(A) OC

and, in view of (a),

l„#k{t)=fl(l—X) (8)fca(A) dX = l#k„{t). 0

Next,

ia:#k{t) =i#k^(t) = fi(l — r)(S)k„(T) dr. (b) 0

Since

if we integrate (b) by parts, we arrive at

ia#k(t) = — [l(t — r)®k^(t)]J:o + fl(t — r) ® k„{r) dr. (c) 0

Because of (a), (c) yields the second relation in (2). □

11. Space-time. We call the four-dimensional point space

^W^^X(-oo. oo)

space-time', in our physical context a point ? = (x, consists of a point x of space and a time t. The translation space associated with is

irw='rx(-oo, oo);

elements of will be referred to as four-vectors. As is natural, we write

a • b =a • b -j-x/S

for the inner product of two four-vectors a =(a, a) and b = (6, j8).

A four-tensor is a linear transformation from into Thus M assigns to each four-vector (a, a) a four-vector

(&,|8)=M(a,a).

Given a four-tensor M, there exists a unique tensor M (on "T), unique vectors m, mef, and a unique scalar A such that for every four-vector (a, a).

M[a, 0) = (Ma, m-a),

M(0, a) =(ma, Aa).

Thus

We call the array M{a, a) ={Ma+mx, m ■ a-|-Aa).

M m

m A (a)

the space-time partition of M.

As before, we define the transpose of M to be the unique four-tensor with the property:

Ma • b =o ■ M^ b for all four-vectors a and b.


It is not difficult to show that the space-time partition of M ^ is

m

m A

Let 9? be a smooth scalar field on an open set D in (^’W. Then the gradient in of (p is the four-vector field

F(4) <p^[V(p,f).

Let u = (m, h) be a smooth four-vector field and M a smooth four-tensor field on D (i.e. the values of M are four-tensors). We define the divergences of u and M by the relations:

div(4) u = div

(div(4) M) ■ o =div(4) (Mâ) for every

For future use, we now record the following

(1) Identity. Let M he a smooth four-tensor field on an open set D in and let (a) be the space-time partition of M. Then

Proof. Since

it follows that

diV(4) M = (div M + m, div m-\- X).

a = (M^ a -|-ma, m ■ a -|- Aa),

diV(4)(/Mô) =div(Mâ) -fa div m -f a • m -f ai

= (div M + m, div m -f A) • (a, a),

which implies the desired identity. □

(2) Divergence theorem in space-time. Let D be a hounded regular region in Let u and Mbe a four-vector field and a four-tensor field, respectively, both continuous onD and smooth on D. Then

f u ■ n da = J div(4) u dv, dD D

f Mn da — (diV(4)M dv, dD D

whenever the integrand on the right is continuous on D. Here n is the outward unit normal to dD.

Proof. The first result is simply the classical divergence theorem in the second can be established using the procedure given in the proof of (6.1). □

C. Fotmulation of the linear theory of elasticity.

I. Kinematics.

12. Finite deformations. Infinitesimal deformations. In this section we motivate some of the notions of the infinitesimal theory. When we present the linear theory we wiU give the axioms and definitions that form its foundation; while these are motivated by viewing the linear theory as a first-order approximation to the finite theory, the linear theory itself is independent of these considerations and stands on its own as a completely consistent mathematical theory.

Sect. 12. Finite deformations. Infinitesimal deformations. 29

Consider a body^ identified with the region B it occupies in a fixed reference configuration.2 A deformation^ of B is a smooth homeomorphism x oi B onto a region x{B) in S with det Fx>0. The point x(a;) is the place occupied by the material point x in the deformation x, while

u{x)=x{x)—x (a)

is the displacement of x (see Fig. 2). The tensor fields

F = Vx (b)

and Vu are called, respectively, the deformation gradient and the displacement gradient. By (a) and (b),

Vu=F — 1. (c)

The importance of the concept of strain cannot be established by a study of kinematics alone; its relevance becomes clear when one studies the restrictions placed on constitutive assumptions by material-frame indifference.* Many different measures of strain appear in the literature; however all of the properly invariant choices are, in a certain sense, equivalent.® The most useful for our purposes is the finite strain tensor D defined by

D=^(F‘^F-1). (d)«

Of importance in the linear theory is the infinitesimal strain tensor

by (c), E = ^Vu+Vu'^)-

D =E-\-^Vu^Vu.

(e)

(f)

The infinitesimal theory models physical situations in which the displacement u and the displacement gradient Vu are, in some sense, small. In view of (e) and (f), D and E can be considered functions of Vu. Writing

_ e = |FM|, (g)

1 See, e.g., Truesdell and Noll [196S, 22], § IS. 2 See, e.g., Truesdell and Noll [1965, 22], § 21. ® Truesdell and Toupin [i960, 17'\, §§ 13-58 give a thorough discussion of finite

deformations. * See, e.g., Truesdell and Noll [1965, 22], §§ 19, 29. ® See, e.g., Truesdell and Toupin [i960, if?], § 32.

® The tensor C = fTl? = 2D -)-1 is the right Cauchy-Green strain tensor (Truesdell and Noll [196S, 22], § 23).


we conclude from (f) that to within an error of 0{s^) as sÔ, the finite strain tensor D and the infinitesimal strain tensor E coincide.

A finite rigid deformation is a deformation of the form:

where Xg and tfg are fixed points of and 0 is an orthogonal tensor. In this instance

F=-Q and Fm = 0-1 (h) are constants and

u{x)=Ug + Vu[x — x^, (i)

where Ug=yg—Xg. Moreover, since Q^Q = 1, (d), (e), and (h), imply

D=0,

E = i(0+0^)-l.

Thus in a rigid deformation the finite strain tensor D vanishes, but E does not. For this reason E is not used as a measure of strain in finite elasticity. By (f) and (j),

E = -^Vu'^Vu,

and we conclude from (g) that in a rigid deformation to within an error of 0{s^) the infinitesimal strain tensor vanishes. Thus

Vu — — Vu^+0{s^);

i.e., to within an error of 0(e®) the displacement gradient is skew. This motivates our defining an infinitesimal rigid displacement to be a field of the form

u[x) =iig + W[x —aco] with W skew.

The volume change dF in the deformation x is given by

dV= f dv-fdv. k(B) B

Since the Jacobian of the mapping x is det Fx=detF,

thus

f dv= f det F dv; k{B) b

dV = f (det F—\) dv, B

so that detf — t represents the volume change per unit volume in the deformation. In view of (c), we can consider F a function of Vu. A simple analysis based on (c) and (g) implies that as eÔ

det F =det(l + Vu) +tr Vu+0{s^), or equivalently.

detf—t =divM+0(e®);

thus to within an error of 0(e®) the volume change per unit volume is equal to div u. For this reason we caU the number

dv = / div udv — J u-nda B dB

the infinitesimal volume change.

Sect. 13. Properties of displacement fields. Strain. 31

13. Properties of displacement fields. Strain. In this section we study properties of (infinitesimal) displacement fields. Most of the definitions are motivated by the results of the preceding section.

A displacement field m is a class vector field over B; its value u {x) at a point xeB is the displacement of x. The S5mimetric part

E = J(Fm+Fm^)

of the displacement gradient, Fm, is the (infinitesimal) strain field, and the above equation relating E to m is called the strain-displacement relation. The (infinitesimal) rotation field W is the skew part of Vu, i.e.

W = \{Vu-VvF), while

CO — ^ curl u

is the (infinitesimal) rotation vector. Thus

Fm=E + IF,

and CO is the axial vector of W; i.e. for any vector a,

We caU Wa=caxa.

divM=tr E

the dilatation. The (infinitesimal) volume change dv{P) of a part P ot B due to a continuous displacement field w on 5 is defined by

dv(P) z= f u - n da, dP

and we say that u is isochoric if dv{P)=0 for every P. By the divergence theorem,

dv(P) = f iivudv = f tiEdv; p p

thus u is isochoric if and only if tr E = 0

on B.

An (infinitesimal) rigid displacement field is a displacement field u of the form

M(a:)=Mo + W^[®-®o].

where aij is a point, Mq is a vector, and is a skew tensor. In this, instance

Vu{x) = Wo = -W[ and E{x)=0

for every x in B. Of course, u may also be written in the form

U{X) =M„ + C0o X [x—Xo\,

where cOp is the axial vector of

(1) Characterization of rigid displacements. Let u be a displacement field. Then the following three statements are equivalent.

(i) u is a rigid displacement field.

(ii) The strain field corresponding to u vanishes on B.

(iii) u has the projection property on B: for every pair of points x, yeB

[u{x)-u{y)] ■ [a:-j,]=o.

32 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 13.

Proof. We will show that (i) => (iii) => (ii) => (i). If u is rigid, then

u[x)-u{y)=WQ[x-y] with Wq skew; hence

{x-y) ■ [u{x)-u(y)] = {x-y) ■ [x-y] =0

and (iii) holds. Next, assume that u has the projection property. Then differenti¬ ating the relation (iii) with respect to x, we obtain

Vu{x)'^ [x—yl +u{x) —u{y) =0.

Differentiating this equation with respect to y and evaluating the result at y = x, we arrive at

— Vu(x)'^ — Vu(x) =0,

which implies (ii). Now assume that (ii) holds. By (6) of (4.1), VVu=(S. Thus

U[x)=Uo-\-W^[x-Xo\, - "<^0 0. -6 connot.o

and (ii) implies that Wg is skew. □

An immediate consequence of (1) is the following important result.

(2) Kirchhoff’s theorem^. If two displacement fields u and u' correspond to M'"' the same strain field, then

u=u' -\-w,

where w is a rigid displacement field.

The next theorem shows that (iii) in (1) implies (i), even if we drop the assumption that u be of class C^, but, as we shall see, the proof is much more dif¬ ficult.

'^’>1 (3)^ LetF he a non-coplanar set of points in S', and let u be a vector field on F. A'hen u is a rigid displacement field if and only if u has the projection property.

Proof. Clearly, if u is rigid, it has the projection property. To establish the converse assertion we assume that

[u{x) -uiy)] ■ [x-y) =0

for every pair of points x, yiF. Let siF, and let Fg be the set of all vectors from s to points of F:

Fg={v: V =x —z, xeF).

Let g be the vector field on Fg defined by

g[v)—u{x)—u[z), v=x—z, xeF. (a)

Then g(0) =0, and g has the projection property:

[gr(») —g(w)] ■ [V-w) =0 (b)

for every pair of vectors v, w^Fg. Taking tc =0 we conclude that

g[v) .»=0

1 [1859, 11 2 Cf. Nielsen [I935, 5], Chap. 3.


for every and this fact when combined with (b) implies

g{v)-w^-giw)-v (c) for all V, weF^.

Next, since the points of F do not line in a plane, F^ spans the entire vector space Let Wg be linearly independent vectors in J^, and let g be the function on iT defined by

g(v)=v'g(Wi), (d)

where w* are the components of v relative to Wf, i.e. v=v'Wi. Then g is linear. We now show that g is the restriction of g to Fg. Indeed, (c) and (d) imply that for V, keFg

g(v) .k = v^g{Wf)-k = —v^g{k) ■ Wi = —v ■ g[k)=g{v) k,

and since this relation miist hold for every k^Fg,

g{v)=g{v) forall»eF*. (e)

Since g is linear, we may write g(v)^Wv. (f)

where IT is a tensor. Further, using (c) and (d), it is a simple matter to verify that

g{v) ■w = —g(w) ■ V

for all », Thus W is skew. Finally, (a), (e), and (f) imply

u{x) =u{s) -\-W[x—s] for every xeF. □

A homogeneous displacement field is a displacement field of the form

u{x)—Ua +A[x—*0],

where the point Xq, the vector u^, and the tensor A are independent of x. Such a field is determined, to within a rigid displacement, by the strain field E, which is constant and equal to the symmetric pari of A. If Mq = 0 and A is symmetric, i.e. if

u{x) =E\_x—x^'\,

then u is called a pure strain from x^.

(4) Let u be a homogeneous displacement field. Then u admits the decomposition

u=w-\-u,

where w is a rigid displacement field and u is a pure strain from an arbitrary point x^.

Proof. Let u{x)û^+A[x-y^'\,

E = ^{A+A^). W=l[A-A^). Then

u=w-\-u, where

w{x) =Wo +W[x —x^], MJo =“o +^[*0-yoi >

u{x) =E[x—Xo]. □ Handbuch der Physik, Bd. VI a/2. 3


Let oSq be a given point, and let

Po{x)=x-x^.

Some examples of pure strains from the point Xq are;

(a) simple extension of amount e in the direction n, where |n| =1:

u=e(n • Pq) n,

E = en0n;

(b) uniform dilatation of amount e:

u=epQ

E=el]

(c) simple shear of amount x with respect to the direction pair (m, n), where m and n are perpendicular unit vectors:

u =x[{m ■po)n + (n- p^) m],

E = 2x sym{m0n) =x[m0n+n0m].

Simp I e extension

O-

o- o- o- o- c>-

o-

Uniform dilatation

Simple shear

Fig. 3. Examples of pure strains from a5„.


The displacement fields corresponding to (a), (b), and (c) are shown in Fig. 3. The matrix relative to an orthonormal basis {n, 62,63} of E in the simple extension given in (a) is

e

[E] = 0

0

0

0

0

0

0

0

and the matrix of E relative to an orthonormal basis {m, n, 63} in the simple shear given in (c) is

0 X

[£] = X

0

0

0

0

0

0

Notice that the homogeneous displacement field

u = 2x{m -Po) n

has the same strain as the simple shear defined in (c) and hence differs from it by a rigid displacement field.

For a pure strain the ratio of the volume change dv(B) of the body to its total volume v (B) is tvE. Thus for a simple extension of amount e,

6v(B) v{B)

for a uniform dilatation of amount e,

6v{B)

■ _ v{B) and for any simple shear,

dv{B)

v[B 0.

We call the principal values ei, e^, of a. strain tensor E principal strains. By (3.2), E admits the spectral decomposition

3

»=i

where n,- is a principal direction corresponding to e,-. We now use this decompo¬ sition to show that every pure strain can be accomplished in two ways: by three simple extensions in mutually perpendicular directions; by a uniform dilatation followed by an isochoric pure strain.

(5) Decomposition theorem, for pure strains. Let u be a pure strain from Xq. Then u admits the following two decompositions'.

(i) u=ui+u^+u^,

where tij.Mg, and u^ are simple extensions in mutually perpendicular directions from Xq]

(ii)

where is a uniform dilatation from Xq, while is an isochoric pure strain from Xq.

Proof. Let E be the strain tensor corresponding to u. It follows from the spec¬ tral decomposition for E that

EÊ,+E^+E^,

3'


where Ei = efnf0nf (no sum).

Thus if we let u—EiPo, 1=1,2,3,

where Po{x)=x-Xo,

the decomposition (i) follows. On the other hand, if we let

Md = 5-(trfi)Po.

M„ = [JB-J-(trJB) l]po,

then Uj is a uniform dilatation, is an isochoric pure strain, and

Ua + u^=Epo=u. □

(6) Decomposition theorem for simple shears. Let u he a simple shear of amount x with respect to the direction pair (m, n). Then u admits the decomposition

u=u*

where m* is a simple extension of amount in the direction

^(»n±n).

Proof. Let E be the strain tensor corresponding to u. Then

E =x{m0n -\-n0m),

and it follows that A is a principal value and a a principal vector of E if and only if

x{n • a) m -\-x{m • a) n = Aa.

It is clear from this relation that the principal values of E are 0, -\-x, —x, the corresponding principal directions

mxn, ^(m+n), ^(m-n).

Thus we conclude from (i) oi (5) and its proof that the decomposition given in (6) holds. □

Trivially, every simple shear is isochoric. The next theorem asserts that every isochoric pure strain is the sum of simple shears.

(7) Decomposition theorem for isochoric pure strains. Let u be an iso¬ choric pure strain from x^. Then there exists an orthonormal basis {ni,n2,ng} such that u admits the decomposition

M=tll+tt2+M3,

where Mg, and Mg are simple shears from x^ with respect to the direction pairs (Mg, Mj), (Mg, Ml), and (Mj, Mg), respectively.

Proof.^ Let E be the strain tensor corresponding to m. If E=0 the theorem is trivial; thus assume EÔ. Clearly, it sidfices to show that there exists an

1 This proof was furnished by J. Lew (private communication) in 1968.

Properties of displacement fields. Strain. Sect. 13- 37

orthonormal basis such that the matrix of E relative to this basis has zero entries on its diagonal:

Hi ■ En^—n^ • En^=n^ ■ En^ = 0.

To prove this it is enough to establish the existence of non-zero vectors «, such that

»■ p=o,

« • Eoi=P ■ EP =0;

(a)

(b)

for if {til, Hg, Jig} is an orthonormal basis with

then (b) yields Hi ■ En^=n^ • En^ = o,

and the fact that tr I? = 0 implies

Jig • JEng = 0.

Let m^, mg, in^ be principal directions for E, and let e^, —e^, —e^ be the corre¬ sponding principal strains. Since trJE=0, we may assume, without loss in generality, that

(if two of the principal strains are positive and one negative consider —E rather than E). We shall now seek vectors «, P which satisfy not only (a) and (b), but also the condition

*3 > (^)

where a,- and pf are the components of « and P relative to the orthonormal basis m^,m,2,mg. In view of (c), conditions (a) and (b) reduce to the system:

ai-|-a|—oc| = 0

6^ OCl ^2 6g OC3 = 0 .

It is easily verified that this system has the solution

fig,

a| = «i + ^2.

and since E=^0, the vectors a and p so defined will be real and non-zero. □

We now consider displacement fields that are not necessarily homogeneous.

Given a continuous strain field E on B, we call the symmetric tensor

the mean strain. Clearly, B

dv(B)

v(B) -=tvE(B):

thus the volmne change can be computed once the mean strain is known.


(8) Mean strain theorem. Let uhe a displacement field, let E be the corre¬ sponding strain field, and suppose that E and u are continuous onE. Then the mean

strain E[B) depends only on the boundary values of u and is given by

E{B) = Jsy™ u^nda.

dB

Proof. By the divergence theorem (6.1),

f (u(g)n +n(g)M) = / (Fm + Fm^) dv = 2 f E dv. □ SB B B

The next theorem, which is due to Korn, gives restrictions under which the £2 norm of the gradient of u (i.e. the Dirichlet norm of u) is bounded by a constant times the norm of the strain field.

(9) Korn’s inequality.^ Let ubea class C* displacement field on E, and assume that one of the following two hypotheses hold:

(a) M = 0 on dB;

(/S) B is properly regular, and either u = 0 on a non-empty regular subsurface of dB, or

f Wdv=0, B

where fV is the rotation field corresponding to u.

Then f IFuj^dv^KflEj^dv,

B B

where E is the strain field corresponding to u, and K is a constant depending only on B.

Proof. We will establish Korn’s inequality only for the case in which (a) holds. The proof under (/S), which is quite difficult, can be found in the article by Fi- CHERA. *

Thus assume (a) holds. It follows from the definitions of E and W that

|E|2 = |(|Fm|2 + Fm- Fm^),

|Mrp==i(|FM|2-FM- Fm^); hcncc

lK|2_|Pr|2 = FM-Fu^. (a)

Further, we have the identity

Vu- Vu^ = div [(Fm) u — (div u) u] + (div u)^. (b)

By (a), (b), the divergence theorem, and (a).

which implies

J |JE|* dw — / \ W\^dv= J (div u)® dv, B B B

f\W\^dv^f \E\^dv. B B

(c)

^ Korn [1906, 4], [1908, i], [1909, 3]. Alternative proofs of Korn’s inequality were given by Friedrichs [1947, 7], Eidus [1951, 5] (see also Mikhlin [1952, 2], §§ 40-42), Gobert

[1962, 6], and Fichera [1971, 7]. Numerical values of K for various types of regions can be found in the papers by Bernstein and Toupin [i960, 7], Payne and Weinberger [1961, 77], and Dafermos [1968, 3].

“ [1971, 7], § 12.

Sect. 14. Compatibility. 39

Thus we conclude from (c) and the identity

that \Vu\^=\E\^ + \W\^

f\Vu\^dv^2j\E\^civ; B B

therefore Korn’s inequality holds with K = 2. □

Let M be a displacement field on B. Then u is plane if for some choice of the coordinate frame ^

Ma = 'â. (% > ^2) > *<3=0.

In this instance the domain of u can be identified with the following region R in the arj-plane;

R={(Xi, x^): xaB).

Clearly, the corresponding strain field JE is a function of {x-^, x^ only and satisfies

~ If ”3“ WjJ. a) >

£ig = £33 = £33 = 0.

It is a simple matter to verify that a plane displacement field is rigid if and only if

where a: = (%, x^ and and co are constants. We call a two-dimensional field of this form a plane rigid displacement and a complex function of the form

u(z} =ai-\-ia2~icoz, z = Xi-\-ix2,

a complex rigid displacement. Clearly, a complex function m is a complex rigid displacement if and only if its real and imaginary parts and U2 are the components of a plane rigid displacement.

14. Compatibility. Given an arbitrary strain field E, the strain displacement relation

E —^(Vu + Vu^)

constitutes a linear first-order partial differential equation for the displacement field u. The uniqueness question appropriate to this equation was settled in the last section; we proved that any two solutions differ at most by a rigid displace- m'ent. The question of existence is far less trivial. We will show that a necessary condition for the existence of a displacement field is that the strain field satisfy a certain compatibility relation, and that this relation is also sufficient when the body is simply-connected.

The following proposition, which is of interest in itself, supplies the first step in the derivation of the equation of compatibility.

(1) The strain field E and the rotation vector co corresponding to a class dis¬ placement field satisfy

curl E = Fco.

Proof. If we apply the curl operator to the strain-displacement relation and use (4) and (5) of (4,1), we find that

curl E = icurl {Vn -f Vu^) = iFcurl u = V<o. □

* Recall our agreement that Greek subscripts range over the integers 1, 2.


If we take the curl of the relation m (1) and use (4) of (4.1), we are immedi¬ ately led to the conclusion that a necessary condition for the existence of a dis¬ placement field is that E satisfy the following equation of compatibility :

curl curl £ = 0.

This result is summarized in the first portion of the next theorem; the second portion asserts the sufficiency of the compatibiUty relation when the body is simply-connected.

' (2) Compatibility theorem.^ The strain field E corresponding to a class C® displacement field satisfies the equation of compatihility.

Conversely, IdBbe simply-connected, and let E be a class C^{N^2) symmetric tensor field on B that satisfies the equation of compatibility. Then there exists a dis¬ placement field u of class on B such that E and u satisfy the strain-displace¬ ment relation.

Proof. We have only to establish the converse assertion.® Thus let B be simply-connected, and assume that curl curl £=0. Let

A=cva\E. (a) Then

curl4 = 0, (b)

and since E is symmetric, (a) and (12) of (4.1) imply that

txA = 0. (c)

By (b), (c), and (iii) of (6.3), there exists a class skew tensor field W such that

A = ~cmlW, (d) and (a), (d) imply that

curl (£-f WQ =0;

thus, by (ii) of (6.3), there exists a class vector field u on B such that

E + W=Vu,

and taking the symmetric part of both sides of this equation, we arrive at the strain-displacement relation. □

In components the equation of compatibiUty takes the form

or equivalently. Îmn ^jm,kn b.

2^^12,12 =^11,22 ■i'^22,ll<

^11,23 “ ( ^23,1 ■i'^31,2 ■^^12,3), 1> StC.

Under the hypotheses of (2) it is possible to give an expUcit formiola which may be useful in computing a displacement field u corresponding to the strain

* Some of the basic ideas underlying this theorem are due to Kirchhoff [1859, f], who deduced three of the six equations of compatibility and indicated a procedure for determining the displacement when the strain is known. The complete equation of compatibility was first derived by Saint-Venant [1864, f], who asserted its sufficiency. The first rigorous proof of sufficiency was given by Beltrami [1886, f], [I889, fj. Cf. Boussinesq [1871” f], Kirchhoff [1876, 7], Padova [1889, 2], E. and F. Cosserat [1896, 7], Abraham [1901, 7]. Explicit forms of the equation of compatibility in curvilinear coordinates were given by Od-

QVisT [1937, 4], Bunchikov [1938, 7], and Vlasov [1944, 3].

® This portion of the proof is simply a coordinate-free version of Beltrami’s [1886, 7]

argument.

r,, ' ,!. t\ 11-. -ryyc-'-^* A • I -v

i-

- A. t-' -

Sect. 14. Compatibility. 41

field E. Let £Cq be a fixed point in B. Then for each xeB the line integral

u(x) = fV{y, x) dy, ®0

where Uijiy, X) =Ei,-{y) + {x^-y^) [Eij^^{y)-E^fj{y)],

is independent of the path in B from jCq to x, and the function u so defined is a displacement field corresponding to the strain field E}

An alternative form of the equation of compatibility is given in the following proposition.

(3) Let E he a class symmetric tensor field on B. Then E satisfies the equation of compatibility if and only if

. ZlJB+F|7(trE)-2FdivE=0.

Proof. The proof follows from the identities:

curl curlE = -ZlJB-FF(tr E) + 2FdivE + \[A (tr E) -div div JB], (a)

tr[ZIE + FF(tr E) —2FdivE'\=2\A (tr JE) — div divE]. (b)

Indeed, let curl curl E = 0. Then, if we take the trace of (a), we conclude, with the aid of (b), that

Zl(tr E) — div div JE = 0, (c)

and this result, in view of (a), implies the relation in (3). Conversely, if that relation holds, then (b) implies (c), and (c) implies curl curl JE = 0. □

The following proposition will be quite useful.

(4) Assume that B is simply-connected. Let u he a displacement field, and assume that the corresponding strainJield E is of class on B and of class (Ai ^ 1) on B. Then u is of class on B.

Proof. Let u' be the displacement field generated by E using the procedure given in the proof of (2). By (13.2), u and u' differ by a rigid displacement field; hence'it offices to prove that u' is of class on B. Since curl JE is of class on B, the function W in (d) in the proof of (2) is of class on B, and since

E + W= Fu',

u' is of class on B. □

(5) Compatibility theorem for plane displacements. Assume that R is a simply-connected open region in the %, x^-plane. Let u^ he a class C® field on R, and let

Eo,p=^{Uo,,p+UpJ. (i) Then

2-^12,12 =-^11,22+•^22,11- (ii)

Conversely, let E^^{=E^P} be a class [N^2) field on R that satisfies (ii). Then there exists a class field u^ on R such that (i) holds.

* This explicit solution is due to CesAro [1906, 2]. In this connection, see also Volterra

[1907, 4], SoKOLNiKOFF [1956, 12], Boley and Weiner [I960, S\.

4U M.h,. uurtin: The J-,inear iheory ot Elasticity. beet. 14.

If we take the curl of the relation in (1) and use (4) of (4.1), we are immedi¬ ately led to the conclusion that a necessary condition for the existence of a dis¬ placement field is that E satisfy the following equation of compatibility :

curl curl fi = 0.

This result is summarized in the first portion of the next theorem; the second portion asserts the sufficiency of the compatibility relation when the body is simply-connected.

' (2) Compatibility theorem.^ The strain field E corresponding to a class C® displacement field satisfies the equation of compatibility.

Conversely, let B be simply-connect^, and let E be a class C^{N ^ 2) symmetric tensor field on B that satisfies the equation of compatibility. Then there exists a dis¬ placement field u of class on B such that E and u satisfy the strain-displace¬ ment relation.

Proof. We have only to establish the converse assertion.® Thus let B be simply-connected, and assume that curl curl jE=0. Let

4=curlJS. (a) Then

curl4=0, (b)

and since E is symmetric, (a) and (12) of (4.1) imply that

trJl=0. (c)

By (b), (c), and (iii) of (6.3), there exists a class skew tensor field W such that

4 =-curl W, (d) and (a), (d) imply that

curl(E-fW0=O;

thus, by (ii) of (6.3), there exists a class vector field « on B such that

EpW = Vu,

and taking the symmetric part of both sides of this equation, we arrive at the strain-displacement relation. □

In components the equation of compatibility takes the form

îjk Îmn b, or equivalently,

2^12,18 =^11,22 +^22,ll>

^rl,23'=( ^23,1+-^31,24'-S^12,3),1* 6^.

Under the hypotheses of (2) it is possible to give an explicit formula which may be useful in computing a displacement field u corresponding to the strain

* Some of the basic ideas underlying this theorem are due to Kirchhorr [1859,1], who deduced three of the six equations of compatibility and indicated a procedure for determining the displacement when the strain is known. The complete equation of compatibility was first derived by Saint-Venant [1864, i], who asserted its sufficiency. The first rigorous proof of sufficiency was.given by Beltrami [1886, f], [1889, fj. Cf. Boussinesq [1871, i], Kirchhoee [1876, 1], Padova [1889, 2], E. and F. Cosserat [1896, i], Abraham [1901, i]. Explicit forms of the equation of compatibility in curvilinear coordinates were given by Od-

QvisT [1937, 4], Blinchikov [1938, i], and Vlasov [1944, 3]. ® This portion of the proof is simply a coordinate-free version of Beltrami’s [1886, 7]

argument.

Sect. 15. Balance of momentum. Stress. 43

We assume given on 5 a continuous strictly positive function q called the density ; the mass of any part P of B is then

fe^v. s P ..f

Let (0, fo) denote a fixed interval of time. A motion of the body is a class C* vector field u on Bx (0, tg). The vector u(x, t) is the displacement of ae at time t, while the fields u, u,

E=\(Vu+Vu^),

and E are the velocity, acceleration, strain, and strain-rate. We say that a motion

is admissible if u, u, u, E, and JS are continuous on S X [0, ô)- Given an admis¬ sible motion u and a part P of B,

l(P)==fuQdv p

is the linear momentum of P, and

h(P) =fpxuQ dv ^=^“2 p

is the angular momentum (about the origin 0) of P. Note that, for P fixed, I (P) and h (P) are smooth functions of time on [0, t^; in fact,

i{P}—fugdv, h{P)= f pxii Q dv. p p

A system of forces / for the body is defined by assigning to each (x, t)iBx [0, to) a vector b(x, t) and, for each unit vector n, a vector s„{x, t) such that:

(i) s„ is continuous on S x [0, <o) of class C^’® on B X (0, Q;

(ii) 6 is continuous on B X [0, t^.

We call s„{x,t) the stress vector at {x,t). Let ,5^ be an oriented regular surface in B with unit normal n (Fig. 4). Then s„^^^{x, t) is the force per unit

area at x exerted by the portion of B on the side of toward which n(x) points on the portion of B on the other side; thus

fs„da = f s„(^){x,t) da^ .9^ ^


and

jpxs„da^jp{x)xs„(^)(X, t) da^

represent the total force and moment across The same consideration also apphes when x is located on the boundary of B and n is the outward unit normal to dB at ae; in this case s„{x, t) is called the surface traction at (x, t). The vector 6 {x, t) is the body force at (x, t); it represents the force per unit volume exerted on the point x by bodies exterior to B. The total force f(P) on a part P is the total surface force exerted across dP plus the total body force exerted on P by the external world:

f(P) = /s„da+fb dv. dP p

Analogously, the total moment m{P) on P (about 0) is given by

m{P)=Jpxs„da+Jpxbdv. dP p

An ordered array [tt, /], where u is an admissible motion and / a S5^tem of forces, is called a dynamical process^ if it obeys the following postrdate: for every part P oi B

f{P)=i(P) (mO and

m(P)=h(P). (m2)

These two relations constitute the laws of balance of linear and angular momentum,^ (m^) is the requirement that the total force on P be equal to the rate of change of hnear momentum, (mj) that the total moment be equal to the rate of change of angular momentum. Note that by (m^), the relation (mj) holds for every choice of the origin 0 provided it holds for one such choice. Clearly, (m^) and (mj) can be written in the alternative forms:

Js„da + Jbdv = fuQdv, (mi) dP p p

f pxs„da-i-f pxb dv=f pxiiQ dv. (mi) dP p p

For future use, we note that (mj) is equivalent to the relation:

skw I fp<S)s„da-i-fp<S)b diA =skw / p ®ilQ dv. (m^) lap p J p

The next theorem is one of the major results of continuum mechanics.

(1) Cauchy-Poisson theorem.^ Let u be an admissible motion and ^ a system of forces. Then [«,/] is a dynamical process if and only if the following two conditions are satisfied:

For the results of this section weaker definitions would suffice for an admissible motion, a system of forces, and a dynamical process. The strong definitions given here allow us to use these definitions without change in later sections.

2 Cf. the discussion given by Truesdell and Toupin [i960, J!'], § 196.

® Cauchy [1823, f], [1827, f] proved that balance of linear and angular momentum implies (i) and (ii) in (1), while Poisson [1829,2] established the converse assertion. In a sense, the essential ideas are implied or presumed in memoirs written by Fresnel in 1822 [1868, 7],

but his work rests heavily on the constitutive assumptions of linear elasticity. In a still more limited sense the scalar counterpart of (i) is foreshadowed in a work on heat conduction written by Fourier in 1814 [1822, 7], which is even more involved with special constitutive relations. For a discussion of the history of this theorem, the reader is referred to Truesdell

and Toupin [1960,77]. Gurtin, Mizel, and Williams [1968, 6] established the existence of the stress tensor under somewhat weaker hypotheses.


(1) there exists a class C^’® symmetric tensor field S on Bx(0,to), called the stress field, such that for each unit vector n,

s„=Sn;

(ii) u, S, and b satisfy the egmAion of motion:

div S + 6 =gii.

The proof of this theorem is based on two lemmas. The first is usually referred to as the law of action and reaction.

(2) Cauchy’s reciprocal theorem. Let [m,/] be a dynamical process. Then given any unit vector n,

s„ = s_„.

Proof. Since B is bounded, the properties of q, u, and b imply that

k{t) =sup |6(®, t) —q{x) U{x, i)|

is finite for 0^t<tf,. Thus we conclude from (mi) that

I /s„da\ ^kv(P) (a) \dP I

on [0, Q, where v{P) is the volume of P.

Now choose a point xêB, a time te{0, Q, and a unit vector m. For con¬ venience we shall suppress the argument t in what follows. Let P^ be a rectangular parallelepiped contained in B with center at and sides parallel to m (see Fig. 5).

Suppose further that the top and bottom faces and with exterior nor¬ mals m and — m are squares of length a, and that the height of P, is e*. Let denote the union of the four side faces. Then

and (P,) = e«. a «(«’,) = 4 e®.

where, for any surface 6^, a{S^) denotes the area of By (a) and (c).

(b)

(c)


Further, since S| is continuous on B for each fixed unit vector I, we conclude from (c) that

72- /'s±»t<^«->s±m(»o) and ^ j" s„da~^0 as e-^0. (e)

By (b), (d), and (e),

Sm(»o)+S-m(»o)=0,

which completes the proof, since m and acj are arbitrary. □

(3) Lemnm. Let S be a class C^’® tensor field on Bx (0, <o) S and div S continuous on Bx [0, t^. Then given any part P of B,

f px(Sn) da= f pxdiv S dv-j-2 f adv dp p p

on [0, fo). liohere ts is the axial vector corresponding to the skew part of S.

Proof. Let 3 = f px(Sn) da.

d'P

Then given any vector e,

j • e = / (exp) • Snda= f n • S^(exp) <â=/div{S^(exp)} dv, dP dP p

and hence (It) of (4.1) implies

j-e — f (e xp) • div S + jS-V(e xp) dv. (a) p p

Next, (exp) • div S =e • (pxdiv S),

S-P(exp)=2e-(r;

since e is arbitrary, (a) and (b) imply

3 = fP xdiv S dv-)-2 fa dv. □ p p

Proof of (1). Assume first that (i) and (ii) of (1) hold. Then it follows from the properties of u and s„ that S and div S are continuous on B X [0, tg). Thus given any part P of B, (ii) and the divergence theorem imply (m^). Next, since S is

Fig. 6. Tetrahedron for the case in which fej = Cj, fcj = — Cj, fcs = — 63.

Sect. 1 5. Balance of momentum. Stress. 47

symmetric, its axial vector a vanishes, and (3) together with (i) and (ii) oi (1) imply

Ip X Sfi d d - Ip xdiv S dv = f px(QU — b) dv, dP p p

which is (m^.

Conversely, assume that [u, /] is a dynamical process. Let m be a unit vector and assume that m =|= ±6^ for any base vector of the orthonormal basis {Cj, Cj, 63}. Choose a point xeB, and consider the tetrahedron P{h) in B whose sides 71(h) and 7i{(h) have outward unit normals m and k^, where

ki — — [sgn (Cj ■ m)] (no sum),

and whose faces 7ii(h} intersect at x (see Fig. 6). Let the area of 7z(h) be a(h), so that the area of 71 f (h) is —a(h)m- fe,. Since s„ is continuous on B x (0, tg) for each fixed n, we conclude that

f s^da-^s^(x) n(k)

and, by (2), that

j suida-^~{m-ki) s,^(x) = — (m-ei) Sei(x) (no sum) (a)

as h-^0, where we have suppressed the argument t. On the other hand, it follows from the inequality (a) in the proof of (2) that

Janda-^-O as h-^0. (b)

' dPW

Thus, since 8P(h) is the union of 7t(h) and all three 7t^(h), (a) and (b) imply

(a;) = (m • e^) s*, (x) = [Se, (ae) ® e^] m. (c)

Our derivation of (c) required that »n =)= hold for every i. However, it follows from (2) that the first equation of (c) is also valid when »n = thus (c) holds for every unit vector m. Now let S be the tensor field on B x (0, fj) defined by

Then S = Se,®e,.

s„=Sn (d)

for every unit vector n. Fmrther, S is of class C^’® on B x (0, tg) and continuous on B X [0, tg), because s„ has these properties for every n.

Next, if we apply (mi) to an arbitrary part P (with PcB) and use (d) and the divergence theorem we conclude that

divS + 6=gM. (e)

Thus to complete the proof we have only to show that S is symmetric. By (mi),

(d), and (3), f a dv=0 p

for every part P with FcB. Thus a == 0 on B x (0, tg), which yields the symmetry of S. □


We now give an alternative proof of the existence of the stress tensor.^ Let [tt, /■] be a dynamical process, and let [0, tf,). It suffices to show that the mapping n s„{Xf,, t) is the restriction (to the set of unit vectors) of a linear function on For convenience, we write

s(x,n)=s„{x,t)-,

then for any xeB we can extend the function s(ae, •) to all of as follows:

s (ae, ®) = I®I s (x, Jy), ® 4= 0,

s(x, 0) =0.

Let a be a scalar. If a > 0, then

s(«®) = |«®|s(-“-~)=«|®|s(|||)=«s(®), (b)

where we have omitted the argument x. If a<0, then (b) and Cauchy’s recip¬ rocal theorem (2) yield

s(a®) =s(|a| (—»)) =|a|s(—®) =as(®).

Thus s{x,') is homogeneous.

To show that s(ae, •) is additive we first note that

s (a;, Wi + iCa) = * (®> *®i) + * (®> **2)

whenever and iCg linearly dependent. Suppose then that and are linearly independent. Fix e > 0 and consider n^, the plane through x^ with normal

7t2. the plane through Xq with normal w^', and jtg, the plane through acj -b ew^ with normal w^, where

MJ3 = —(MJ1+MJ2). (c)

Consider the solid s/=sd{E) bounded by these three planes and two planes parallel to both and and a distance d from acj (see Fig. 7). Let e and d be sufficiently small that s/ is a. part of B. Then

(J t=r

'^This proof was furnished by W. Noll (private communication) in 1967.


where is contained in jt,- (« = 1,2, 3), and #2 3-re parallel faces. Moreover,

a3 = 0(e) as e-^0,

v(j^) = ~ \wâ^ = 2d ai — 2d a^,

where is the area of Thus, by the continuity of s„.

c . '“-I /s„ 4. =2 -N- J s (x, + 0 (.) 0jai'

as e->0,

and (a) implies

c= 2 s(®o> *®i)+3(t) as e-^0. t=i

On the other hand, we conclude from estimate (a) in the proof of (2) that

c=0(e) as e->0.

The last two results yield 3

2 s(aeo,MJi) =0; 1=1

since s(ae,;) is homogeneous, this relation and (c) imply that s(aej, •) is additive. Thus s(aej,*) is linear, and, since XffiB was arbitrarily chosen, Noll’s proof is complete.

The field S of the Cauchy-Poisson theorem (1) has the following properties:

(I) S is a symmetric tensor field of class C^’® on 5 X (0, Q '<

(II) S and div S are continuous on B X [0, Q.

The continuity of div S on B X [0, follows from the equation of motion and the continuity of q, U, and b on this set. A field S with properties (I) and (II) will be called a time-dependent admissible stress field. In view of the Cauchy- Poisson theorem, the specification of a dynamical process is equivalent to specifying an ordered array [u, S, 6], where

(a) « is an admissible motion;

(j5) S is a time-dependent admissible stress field;

(y) 6 is a continuous vector field on B X [0, to)

(d) u, S, and 6 satisfy the equation of motion

div S -\-b =QU.

Henceforth, a dynamical process is an ordered array [tt, S, 6] with properties

If tt is independent of time, the equation of motion reduces to the eqtmtion of equilibrium

Handbuch der Physik, Bd. VI a/2.

div S -f 6 =0.

4


Let S be a stress tensor at a point x of the body. If

Sn = sn,

then the scalar s is called a principal stress at x and the unit vector n is called a principal direction (of stress). Since S is symmetric, there exist three mutually perpendicular principal directions n^.n^.n^ and three corresponding principal stresses s^, Sg, Sg.

If a symmetric stress field S is constant, it automatically satisfies the equation of equilibrium (with vanishing body forces). Examples of such stress fields are:

(a) pure tension (or compression) with tensile stress a in the direction n, where |n| = t:

S =an®n-, (b) uniform pressure p:

S = -p\)

(c) pure shear with shear stress r relative to the direction pair m, n, where

m and n are orthogonal unit vectors:

S = 2r sym (m® n).

Pure tension

I i I

i i

t t t t t

Pure sheer

Fig. 8.

Sect. 16. Balance of momentum for finite motions. 51

The matrix of the pure tension (a) relative to an orthonormal basis {n, Cj, eg} is

a 0 0"

[S]= 0 0 0 ,

0 0 0.

and the matrix of the pure shear (c) relative to an orthonormal basis {m, n, Cg} is

0 T O'

[S]= T 0 0 .

0 0 0,

Given a unit vector It, the stress vector for each of the above is:

(a) Sjj=cr(fe-n) n,

(b) s^=-pk,

(c) Sjj = T[(m-fe) n-f (n-fe) m].

These three surface force fields are shown in Fig. 8.

y 16. Balance of momentum for finite motions.^ In this section we show that the definitions and results of the previous section are, in a certain sense, consistent v^jT_the finite thepiy^proyided the deformations are small. We again remind the reader that this section is only to help motivate the infinitesimal theory; the remainder of the article is independent of the considerations given here.

Consider a one-parameter family K(, 0<t<t^, of deformations of B, the time t being the parameter. The point k{x, t)=Kt{x) is the place occupied by x at time t, while

u(x, t) =k{x, t)—X (a)

is the displacement of x at time t.

The linear momentum I (P) of a part P is the same in the finite theory as in the linear theory; i.e.

l{P)=fuQdv, (b) p

where q is the density in the reference configuration. The angular momentum, however, is now equal to

h*(P)—f {k — 0)X'Uq dv. (c) p

By (a), k—u; thus, since «xtt=0,

h*(P) =f {k—0)xuq dv. (d) p

In the finite theory we can specify the surface forces and body forces per unit j area and volume in the deformed configuration or per unit area and volume in ; the reference configuration. The former specification leads to the Cauchy stress tensor; the latter, which is most convenient for our purposes, leads to the Piola- Kirchhoff stress tensor. A system of forces / is defined exactly as in the infini¬ tesimal theory with the quantiti^ s„ and 6 now interpreted as foUows: given any closed regular surface 6^ in B [rather than k (S, i)] with unit outward nor¬ mal n at x^S^, s„(x, t) is the force per unit area in the reference configuration

* See, for example, Truesdeli, and Toupin [i960,17’], §§ 196, 200-210, and Truesdell

and Noll [1965, 22], § 43 A.

4*


exerted at the material point x by the material outside of on the material inside; b(x, t) is the force per unit volume in the reference configuration exerted on the material point x by the external world. The force f(P) on a part P is the same as in the Unear theory;

f(P) = f s„da + Jbdv: dP P

however, the moment in*(P) on P is now given by:

m* (P) = /(k —0) xs„ / (k-—0) x5 ^^7^. dP p

(e)

(f)

In terms of these quantities, the laws of balance of linear and angular momentum take the forms

or equivalently. m* (P) = h* (P),

/ s„da+fbdv=J qu dv. dP

J(K — 0)xs„da+J(K —0) xb dv=f {K — 0)XQUdv. dp p p

Now let

I e= sup \u[x, <)| . ; XiB s

<€(0,<o) __

(g)

(h)

(i)

(i)

By (a),

k{x, t) —0=tt\a5, t) -\-x—0=u{x, t) +p(x),

and if we consider the field k as a function of the field u, then (i) and (j) yield

K —0=2>+O(e) as e->0. (k)

Using (k) and (d), we can write (c), (f), and (h) as follows:

h*{P) =J[jp+0{e)]xuQdv, P +

tn*(P) = /[2>+0(e)] xs„ da+f[pif:0{e)]xbdv, (1)

dP

f [2>+0(e)] xs„ da+ J[p+0{e)] xbdv — J[p+0{e)] XQiidv. (m) dP p p

These relations reduce to the corresponding relations in the linear theory provided we neglect the terms written 0(e).

We can summarize the results estabUshed thus far as foUows: In terms of surface forces and body forces measured per unit area and volume in the reference configuration, the linear momentum of any part P and the force on P are the same as in the linear theory, and (hence) so also is the law of balance of linear momentum. The angular momentum and the moment reduce to the corresponding quantities in the linear theory provided we neglect the terms written 0(e) in (1); therefore when these 1 terms are neglected the law of balance of angular momentum reduces to the corre-1 spending law in the infinitesimal theory.

It is a simple matter to estabUsh the consequences of balance of momentum for the finite theory. In fact, the analog of (15.1) meLy be stated roughly as

General solutions of the equations of equilibrium. Sect. 17. 53

follows: There exists a tensor field S, called the Piola-Kirchhoff stress tensor, such that

s„^Sn,

div S + b =qU,

SF^ = FS^, where

(o)

F = Vk

is the deformation gradient. The Eqs. (n) ajce identical to their counterparts in the infinitesimal theory. However, in the linearytheory the stress field Sis sym¬ metric; in contrast, it is clear from (o) that in the finite theory S will in general be non-symmetric. If we let -j-,

1 d=\Vu\, y (p) then, since X—-

F = l + Vu, (q) (o) imphes

S(l-hO(d))=S^(l-hO(d)).

Thus S is symmetric to within terms of 0(8) as 8^0. This fact further demon¬ strates the consistency of the linear theory within the finite theory.

The Cauchy stress tensor may be defined by the relation

T^(detF)-^SF^. (r)

It follows^ from this definition that for any part P

fSnda = J Tv da, dP dKi (P)

where v is the outward unit normal to d K((P); here ic,(P) is the region of space occupied by P in the deformation Kf, i.e.

Ki(P)={y:y = Ki(x),x€P}.

Thus Tv is the force per unit area in the deformed configuration. Clearly, (r) and (o) imply that T is symmetric. Further, as

it follows from (p) and (q) that detF = f-fO((5),

T = S{\-\-0(8)).

I Thus to within terms of 0(8) the Piola-Kirchhoff stress tensor and the Cauchy stress tensor coincide.

Vciv

>; Vi

vi<

17. General solutions of the equations of equilibrium. In the absence of body forces the equations of equihbrium take the form:

divS=0,

If .4 is a field over B, and if iS? is a differential operator with values that are symmetric tensor fields, then

will be a solution of the equations of equihbrium provided

diviS?(4)=0.

1 See, for example, Truesdell and Noll [1965, 22], § 43 A.


A field A with this property is called a stress function. The first solution of the equations of equihbriumm^nns ofayEessTunction was Airy’s^ two-dimensional solution in terms of a single scalar field. Three-dimensional generalizations of Airy’s stress function were obtained by Maxwell^ and Morera®, who estabUshed two alternative solutions, each involving a triplet of scalar stress functions. Beltrami observed that the solutions due to Maxwell and Morera may be regarded as special cases of

(1) Beltrami’s solution.* Let A be a class C® symmetric tensor field on B, and let

/S=curl curl A. Then

divS=0, S=S'^.

Proof. The symmetry of S follows from (9) of (4.1) and the symmetry of A. On the other hand, by (7) of (4.1),

div S = div curl curl A = curl div (curl A) ^,

and thus we conclude from (8) of (4.1) that

diV)S=0. □

Consider now a fixed cartesian coordinate frame. Then Beltrami’s solution takes the form:

îmn ^jpq ^mp,nq'

Further, letting [A] denote the matrix of A relative to this frame, we have for the following special choices of A:

(a) Airy’s solution.

[A] =

•^11—9’,22> •S'22 — T,ll>

b 0 0

0 0 0

P 0 q>

•S12 = 9’,12> •5i3 — Sj — ^33 — 0,

(b) Maxwell’s solution.

[A] = 0

0

0

^2

0

O'

0

®3.

■^11 —®2,33 ■4"®3,22> •512— ^,1 etc.;

1 Airy [1863, i].

“ Maxwell [1868, 2], [1870,1].

® Morera [1892, 4]. * Beltrami [1892, T\. This solution haa,heeiL.err<mecmsly_attributedjtojrjNiia.,bx.SlBBN-

BERG [i960, 12], to Gwyther ancl FiNzi by Truesdell [1959, 16} and Truesdell and

Toupin [i960,17], and to Blokh by Lur’e [1955, 10]. Related contributions and additional

references can be found in the work of Morera [1892, 5], Klein andWiEGHARDT [1905,1],

Love [1906, 5], Gwyther [1912, 2], [1913, i], Finzi [1934,2], Sobrero [1935, 7]. Kuzmin

[1945,2], Weber [1948,7], Finzi and Pastori [1949,1], Krutkov [1949,5], Peretti

[1949, 5], Blokh [1950, 3], [1961, 2], Morinaga and N6no [1950, 9], Filonenko-Borodich

[1951, 7,8], [1957,5], Arzhanyh [1953, 2], Schaefer [1953,78], [1955,72], [1959,72],

Kroner [1954,77], [1955, 7,8], Langhaar and Stippes [1954,72], Ornstein [1954,77],

Gunther [1954, 7], Marguerre [1955, 77], Dorn and Schild [1956, 7], BrdiCka [1957,

3], Rieder [I960, 9], [1964, 77, 78, 79], Minagawa [1962, 77], [1965, 14], Gurtin [1963, 9],

Kawatate [1963, 16], Pastori [1963, 27], Inov and Vvedenskii [1964, 70], N6no [1964, 75],

Carlson [1966, 6], [1967, 4], Stippes [1966, 25], [1967, 74], Schuler and Fosdick [1967, 73].

Sect. 17. General solutions of the equations of equilibrium. 55

(c) Morera’s solution.

[A] =

0 CO3 CO2

CO3 0 coj

CO2 ft>i 0 J

then

■^11— 2C(>i23> ‘^12 — (*^1,1 *^3,3).3> ®tc.

If we define a tensor field G on B through

G=A-iltTA,

A = G-ltTG,

and it follows from (14) of (4.1) that Beltrami’s solution (1) can be written in the alternative form^

S = —AG +2V div G —1 divdiv G.

By a self-equilibrated stress field we mean a smooth symmetric tensor field S on B whose resultant force and moment vanish on eadi closed regular surface in B :

f Snda — f px(Sn) da=0. se se

Clearly, every self-equilibrated stress field S satisfies

div )S = 0.

On the other hand, if the boundary of B consists of a single closed surface, then any closed regular surface 3^ in B encloses only points of B, and it follows from the divergence theorem that every divergence-free S5mimetric tensor field will be self-equilibrated. The assumption that dB consist of a single closed surface is necessary for the validity of this converse assertion. Indeed, let B be the spherical shell

B={a5:a<la5—a5ol <)3}.

Then the stress field S defined on B by

S(x)= ®(x— Xg) (a =)= 0)

satisfies divS = 0,

but its resultant force on any spherical surface

,5^={a5:|a5-a53| =y} (a<y<)3) is not zero. In fact,

J Snda = — a.

This result when combined with the next theorem shows that the Beltrami solution and hence the solutions of Maxwell and Morera are, in general, not complete', i.e. there exist stress fields that do not admit a representation as a Beltrami solution.

(2)^ Beltrami's solution (1) always yields a self-equilibrated stress field.

1 This version of Beltrami’s solution is due to Schaefer [1953, iS],

“ Rieder [lp6o, 9]. See also Gurtin [1963, 9].


Proof. Let S — curl curl A

with A symmetric and of class C® on IB, and let

W=^cm\ A. (a)

Then by the symmetry of S= curl curl .4 together with Stokes’ theorem (6.2),

J Sn da = f n da = f (curl W)^ nda=0 y y y

for every closed regular surface ^ in B. To see that the resultant moment also vanishes on Sf, consider the identity

2>x(curl W)'^ n = [curl(PIF)]^ n-\-W^ n — {\xW)n, (b)

which holds for any tensor field W and any vector n, where P is the skew tensor corresponding to the position vector p. In view of the symmetry of A, (a) and (12) of (4.1) imply that

tTW=0.

Hence (a), (b), and Stokes’ theorem (6.2) yield

J px (Sn) da = J px (S^ n) da= f px (curl W)'^ n da y y y

= j \cax\{PW)Y n da-\- J (cuxl A)^ n da =Q. □ y y

Thus stress fields that are not self-equilibrated cannot be represented as Beltrami 1 solutions. The next theorem shows that Beltrami’s solution is complete provided 1 we restrict our attention to sdf-equilibrated..g.trfess fields^ In orde^to state this

fSeorem concisely, let ^ denote the set of Si fields of class C® on B amd_class_f7® qn_B.

(3) Completeness of the Beltrami solution.^ Assume that the boundary of B is of class C®. Let Si.^ be a self-equilibrated stress field. Then there exists a symmetric tensor field Ai^ such that

S =curl curl 4.

Proof. Since S is self-equiUbrated (and hence symmetric),

/ nda =0, (a) y

f px{S‘^ n) da=0, (b) y

for every closed regular surface in B. By (a) and (ii) of (6.4), there exists a tensor field such that

)S—curl IP. (c)

1 Gurtin [1963, 9]. Completeness proofs were given previously by Morinaga and N6no

[1950, 9], Ornstein [1954,17], Gunther [1954, 7], and Dorn and Schild [1956, i]. How¬ ever, since the necessary restriction that S be self-equilibrated is absent in all of these in¬ vestigations, they can at most be valid for a region whose boundary consists of a single closed surface. Gurtin’s [1963, 9] proof is based on a minor variant of Gunther’s [1954, 7] argument and is quite cumbersome. The elegant proof given here is due to Carlson [1967, 4] (see also Carlson [1966, 6]).

Sect. 17. General solutions of the equations of equilibrium. 57

It follows from (b), (c), the identity (b) given in the proof of (2), and Stokes’ theorem (6.2) that

J[W-{tTW)iy'nda = 0 se

for every closed regular surface 6^ in B. Thus, by (ii) of (6.4), there exists a tensor field such that

hence (c) imphes

If we let

ir-(trir)l=curlF;

)S = curl curl F + curl [(tr W) 1].

A =sym F,

(d)

then (9) and (fO) of (4.1) yield the relations

sym{curl curl F} = curl curl A,

sym{curl[(tr W) 1]} = 0.

Thus, since )S is symmetric, if we take the syunmetric part of both sides of (d), we arrive at

S = curl curl A. □

If the boundary of B consists of a single closed surface, then every symmetric divergence-free stress field is self-equihbrated, and we have the following direct corollary of (3).

(4) Assume that dB consists of a single class C® closed regular surface. Let be a symmetric tensor field that satisfies .

div)S=0.

Then there exists a symmetric tensor field A^^ such that

S = curl curl A.

The next proposition, which is a direct consequence of the compatibihty theorem (14.2), shows that two tensor fields are Beltrami representations of the same stress field if and only if they differ by a strain field.

(5)^ Let S—curl curl.4, (a)

where A is a class tensor field on B. Further, let

Il=Â+E,

£ = J(Ftt-hFu^),

where u is a class C® vector field on B. Then

S = curl curl W.

(b)

(c)

Conversely, let B be simj/lv-connected. and let A and H be class C® symmetric tensor fields on B that satisfy (a) and (c). Then there exists a class C® vector field u on B such that (b) holds.

It is clear from (5) that to prove the completeness of Maxwell’s solution for situations in which Beltrami’s solution is complete it suffices to estabhsh the

1 Finzi [1934, 2].


existence of a class C® displacement field u that satisfies the three equations

= (*■=)=?)• W On the other hand, the completeness of Morera’s solution will follow if we can

find a class C® solution u of the equations

(no sum). (II)

Here .4 is a given class C® symmetric tensor field on B. Under hypotheses (1) and (iv) of (5.2) with M = 1 and iV=3. as weH as analogous hypotheses obtained from (i) and (iv) by cyclic permutations of the integers (I* 3). the system

, (II) has a solution. Therefore under these hypotheses a stress fteld S^^mtts

, I i « Morera representation whenever U amts « BeUramkrepresenMion. To the author’s knowledge, there is, at present, no general existence theorem

j, ^ a lert that the Beltrami solution is complete if and ordy

if wi re™ Hi attention to self-equilibrated stress fields The next two remits ^ show that itis^sible.to modify_3^ repr^sintonas.Tq_fgnn a j

complet^£;eQ^^^. solution of the equation of equilibrium. \

(6) Beltrami-Schae/er solution.^ Let A be a class C® symmetric tensor

field and h a harmonic field on B, and let

S = curlcurl4+2Ffc —Idivfc.

Then ^ ^ ar divS=0, S=S^.

Proof. The proof follows at once from (1) and the identity

diy[Vh + {Vh)^-ldivh]-=Âh. □

(7) Completeness of the Beltrami-Schaefer solution.^ Let S be a sym¬

metric tensor field that is continuous on B and class C® on B, and suppose that

div )S = 0.

Then there exists a class C® symmetric tensor field A and a harmomc vector field h

on B such that , S = curlcurl4-f-2Ffc —1 divh.

Proof. By (6.5) there exists a class C® symmetric tensor field G on B such

S^-AG.

S = —AG + ^ Fdiv G — 1 div div G — 2 F div G +1 div div G,

that

Thus

and letting 4 = G-ltr G,

h — —diy G,

1 Schaefer [1953. i*]- I Gurtin [1963, 9] introduced the following solution:

S = curl curl ^ +2d P6-FFdiv6,

where A is symmetric and 6 biharmonic, and established its completeness. Shortly tter^fter ScIae^r Sd that the completeness of (*) implies the completeness of the Beltram- Schaefer solution (private communication). The direct completeness proo given a ov p

pears here in print for the first time.

Sect. 18. Consequences of the equation of equilibrium. 59

we see from (•14) of (4,1) that

S = curl curl .4 + 2 Ffc — 1 div h.

Further, the fact that S is divergence-free impUes that h is harmonic. □

Let S satisfy the h5q)otheses of (7), and let

)S, = curlcurl4,

S,^ = 2Vh — l div ft,

where A and h are the fields estabUshed in (7). By (2) is self-equiUbrated, while (6) implies that Sj, is divergence-free and symmetric. Further, since h is harmonic, so is Thus we have the

(8) Decomposition theorem for stress fields.^ Let S satisfy the hypoth¬ eses of (7). Then

S^S,+S„

where is a self-equilibrated stress field, and S,, is a harmonic tensor field that satisfies

dwS,=0, S,=Sf.

Finally, it is clear from the proof of (6) that^

S = 2Vh—l div ft,

furnishes a particular solution of the inhomogeneous equations of equihbrium

divS-f6==0, S = S'^.

'' 18. Consequences of the equation of equilibrium. In this section we study properties of solutions to the equation of equilibrium

div S-\-b—0,

assuming throughout that a continuous body force field b is prescribed on B.

For convenience, we introduce the following two definitions. A vector field u is an admissible displacement field provided

(i) ® u is of class C® on B;

(ii) u and Vu are continuous on E.

On the other hand, an admissible stress field is a symmetric tensor field S with the following properties: ^

(i) S is smooth on B;

(ii) S and div S are continuous on B.

The associated surface force field is then the vector field s defined at every regular point x&dB by

s{x) =S(x) n(x),

where n (x) is the outward unit normal vector to dB at x.

1 Cf. Gurtin [1963, 9], Theorem 4.4. 2 Schaefer [1953, 18}. ® Actually, for tiie results established in this section, this requirement may be replaced

by the assumption that m be smooth on B.


The following lemma will be extremely useful throughout this article.

(1) Lemma. Let S be an admissible stress field and u an admissible displace¬ ment field. Then

f s • u da = f u • div S + f S ■ Vudv. dB B B

Proof. By the divergence theorem, (11) of (4.1), and the symmetry of S,

f u • Sn da= f (Su) ■ nda= f div(Stt) dv— f u - div S + f S ■ Fu dv. dB dB B B B

The symmetry of S also requires that

S- |7tt=S- Ftt^=S- Ftt.

and the proof is complete. □

If )S is a solution of the equation of equiUbrium, then divS = —h, and (1) has the following important corollary.

y (2) Theorem of work expended.^ Let S be an admissible stress field and u an admissible displacement field, and suppose that S satisfies the equation of equi¬ librium. Then

f s ■ uda-i- f b ■udv = f S • E dv, dB B B

where E is the strain field corresponding to u.

This theorem asserts that the work done^ by the surface and body forces over the displacement field u is equal to the work done by the stress field over the

strain field corresponding to u. If m isâ rigid displacement field, then Ftt=0, and the above relation reduces to

/ s ■ uda-(- J b ■ udv =0. dB B

Thus the work expended over a rigid displacement field vanishes.

Our next theorem shows that, conversely, the balance laws for forces and moments are impUed by the requirement that the work done by the surface and body forces be zero over every rigid displacement.

(3) Piola’s theorem.^ Let s be an integrable vector field on dB. Then

/ s -w da-\- J b ■ w dv=0 dB B

1 Cf. Theorem (28.3). “ We follow Love [I927, 3], § 120, in referring to, e.g.,

f 8-u da (a) dB

as the work done by the surface force s over the displacement u. In the dynamical theory the work is given by

/ f Sf-Uf da dt, (b) dB

where po, q] is the interval of time during which the time-dependent surface force acts, and where Uf is the velocity of the time-dependent displacement field m^. Formally, (b) reduces to (a) if we assume that the spatial fields s and m are applied suddenly, i.e., if Sj(a;) = 8(x) h(t—A) and Uf(x)=u(x) h(t—A), where h is the Heaviside unit step function and

(ô> h)- ® PioLA [1833, 2], [1848, T\. See also Truesdell and Toupin [i960,17], § 232.

Sect. l8. Consequences of the equation of equilibrium. 61

for every rigid displacement w if and only if

f s da+ f b dv = 0, dB B

Ip xs da-i- f pxb dv = 0. dB B

Proof. Let

with W„ skew. In view of the identity

a • Wge = Wf, • {a®e),

which holds for aU vectors a and e,

A{w) ^ f s ■ w da-i-1 b ■ w dv dB B

= Uq • j Jsda+ I b dvK + Wô ' | I s®pda + / b®p dv\. Ub b J Ub b >

Thus Alyc) =0 for every rigid displacement w (or equivalently, for every vector u„ and every skew tensor if and only if s and b satisfy balance of forces and, in addition,

skw| jp®sda-\- f p<S)bdv\ =0. l0B B J

This completes the proof, since the above relation is equivalent to balance of moments. □

The theorem of work expended was generalized by Signorini^ in the following manner. •

(4) SignorinVs theorem. Let S be an admissible stress field that satisfies the equation of equilibrium, and let w be a smooth vector field on B. Then

f w<S)sda-\- / w(S)bdv— f {Vw)Sdv. dB B B

Proof. By the divergence theorem,

/ w%s da = / w®{Sn) da= J (div S) + / {Vw) dv. dB dB B B

This completes the proof, since —S and div S — —b. □

Since

Pw = ^{Vw + Vw'^), the symmetry of S yields

tT[(Vw) S]=S ■Vw=S- Vw.

Thus if we take the trace of Signorini’s relation, we arrive at the theorem of work expended.

Given an admissible stress field S, we call the symmetric tensor

B

the mean stress. Here v [B) is the volume of B.

1 SiGNoRiNi [1933, 4]. Actually, Signorini’s theorem is slightly more general.


(5) Mean stress theorem.^ The mean stress corresponding to an admissible stress field that satisfies the equation of equilibrium depends only on the associated surface traction and body force fields and is given by

S(B)= jp®sda-\- jp®hdv .

Us B

Proof. We simply take w in Signorini’s theorem equal to p and use the identity Fjp=l. □

The next three theorems give alternative characterizations of the equations of equihbrium and compatibility.

\(6) Let S be an admissible stress field, and suppose that

J S ■ Vudv — J b ■ u dv B B

/' ■

for every class'vector field non B that vanishes near BB. Then

div S +6=0.

Proof. Let m be a class C°° vector field on B that vanishes near BB. By the

S5mimetry of S, S ■ Vu=S ■ Vu; thus we conclude from (1) that the integral identity in (6) must reduce to

/ (div S + b) -udv =0. B

Since u was arbitrarily chosen, we conclude from the fundamental lemma (7.1) that div S + 6 = 0. □

(7) Let S be an admissible stress field, and suppose that

f S ■ E dv =0 B

for every class C°° symmetric tensor field E on B that vanishes near BB and satisfies the equation of compatibility. Then

divS = 0.

Proof. Let m be a class C” vector field on B that vanishes near BB, and let E be the corresponding strain field. Then E satisfies the equation of compatibility and vanishes near BB; hence, by h3q)othesis,

^ S • Vudv =0, B

and the proof follows from (6). □

^(8) Donati’s theorem.^ Let E be a class symmetric tensor field on B. Furth^, suppose that^

f S ■ Edv=0 B

' Chree [1892, 2], p. 336. See also Signorini [1932, 5].

2 Donati [1890,2], [1894,2]. See also Cotterill [1865, 2], Southwell [1936,5],

[1938,5], Locatelli [1940,5,4], Moriguti [1948,5], Klyushnikov [1954,20], Washizu

[1958, 20], Blokh [1962, 2], Stickforti^[1964, 22], [1965, 27], and Tonti [1967, 16, 27].

® Notice that S-E is continuous on B, since S vanishes near 8B.

Consequences of the equation of equilibrium. Sect. 18. 63

for every class C°° symmetric tensor field S on B that vanishes near dB and satisfies

div S = 0.

Then E satisfies the equation of compaiibility.

Proof. Let ^ be a s5Tnmetric tensor field of class C°° on B that vanishes near dB, and let

S = curl curl A.

Then S vanishes near dB, and by (17.1), S is S5nnmetric and divergence-free. Thus

/ E ■ (curl curl A) dv = 0. B

If we apply the divergence theorem twice and use the fact that A and all of its derivatives vanish on dB, we find that

/ E ■ (curl curl A) dv= ( (curl E)"^ ■ (curl A) dv= f (curl curl E) • Adv. B B B

Thus / (curlcurlE) ■ Adv = Q.

B

Since curl curl E is S5mimetric, and since the above relation must hold for every such function A, it follows from the fundamental lemma (7.1) that E satis¬ fies the equation of compatibility. □

Let s be an integrable vector field whose domain is a regular subsurface of dB, let b be an integrable vector field on B, ai^ let I be a vector field whose domain is a finite point set {?i, ?2, of B. Further, let The function s can be thought of as a surface traction field on 5^, 5 as a body force field, and as a concentrated load applied at We say that s, b, and I are in equilibrium if

f sda+ fbdv+J] l„=0, se B „=i

fpxsda+ f pxbdv+'Z(L-^)xln = ^- Se B n=l

A somewhat more stringent condition is that of astatic equihbriiun. m wMch the system of forces is reqmred to_remain in equHibrium, .when..xQtated..through an arbitrary angle. More precisely, s, b and I are in astatic equilibrium if Qs, Qb. and Ql are in equihbrium for every orthogonal tensor Q, i.e. if

fQsda+ f Qbdv+'Z Ql„ = 0,

^ ^ N (A) Sp-xQsda^- jpxQbdv+ 'Z(l^„-0)xQl„ = 0,

Se B n=l

for every orthogonal Q.

(9) Conditions for astatic equilibrium.^ Let s, b, and I be as defined above. Then s, b, and I are in astatic equilibrium if and only if the following two conditions hold:

(i) s, b, and I are in equilibrium;

(ii) f p®sda+ ( p®bdv-\-Y,(^n — ^)®K = ^■ se B n=l

^ In classical statics, in which the loads are all concentrated, this result has long been known (see, e.g., Routh [I908, 3], p. 313). The extension to distributed loads is due to Berg [1969, 11

64 M.E. Gurtin ; The Linear Theory of Elasticity. Sect. 19.

Proof.^ Assume that s, b, and I are in astatic equilibrium. Then (i) follows from (A) with 0 = 1. Let T denote the left-hand side of (ii). By (Eg), skw T = 0; thus T is S5mimetric. Next, by (A2),

0 = skw| Jp®Qsda + f p®Qbdv+'Z (?„ —0) (glgU; iy' B n»l J

thus, since a(8)0c = (a(8)c)0^ for all vectors a and c, skw(T0^=O. Con¬ sequently,

TQ^ = QT. (a)

T = 2 (b) »=i

be the spectral decomposition of T (in the sense of the spectral theorem (3.2)). By (a) and (b),

»=i »=i

and this relation must hold for every orthogonal tensor Q. Let Q be the ortho¬ gonal tensor with the following properties:

Qni=n^, Qnz=na, Qih^ni. (d)

If we operate with both sides of (c) on 1*3 and use (d) and the fact that nj-n,- = we find that

^2 ^2 “ ^ ^ *

which imphes that Ag = Ag = 0. Similarly, = 0. Thus T = 0 and (ii) holds.

Assume now that (i) and (ii) hold. Let T be as before. Then T = 0 and

0^TQ^= J p®Qsda + f p®Qbdv+'Z (?„-0) (8)0f„. (e) Sr B n=l

If we take the skew part of (e), we are led, at once, to (Ag). On the other hand, (Ai) is an immediate consequence of (E^). □

In view of (9), the mean stress theorem (5) has the following interesting con¬ sequence.

(10)^ The mean stress corresponding to an admissible stress field that satisfies the equation of equilibrium vanishes if and only if the associated surface traction and body force fields are in astatic equilibrium.

19. Consequences of the equation of motion. In this section we study properties of solutions to the equation of motion

div S +b =gu

on a given time interval [0, Q, assuming throughout that a continuous and strictly positive density field q is prescribed on B.

Given a d3mamical process® [u, S, b], we call

K=^ J QU^dv

^ This proof differs from that given by Berg [1969, 1].

2 SiGNORINI [1932, 5].

® See p. 49-

Sect. 19. Consequences of the equation of motion. 65

the kinetic energy and

jS-Edv B

the stress power; here E is the strain field associated with u.

(1) Theorem of power expended.^ Let [u, S, b] be a dynamical process. Then

f s ■uda+ fb-iidv = fS-Edv +K. dB B B

Proof. The proof follows from the theorem of work expended (18.2) with b replaced by b — pit and u replaced by u. □

Theorem (1) asserts that the rate of working of the surface tractions and body forces equals the stress power plus the rate of change of kinetic energy. If

m(-, 0) = u(-, t),

then, by (1), T T T , / f s • ii da dt-{- J J b ■ ii dv dt= J ( S • E dv dt\

0 SB OB OB

i.e., when the initial and terminal velocity fields coincide, the work done by the external forces is equal to the work done by the stress field.

As we shall see later, in a homogeneous deformation of a homogeneous body the stress and strain fields S and E are independent of position. In this instance b=QU and (1) implies

S • E— —I s ■ u da;

SB

i.e., the stress power density is equal to the rate of working of the surface trac¬ tions per unit volume.^

The next .theorem will be of use when we discuss variational principles for elastodynamics; it shows that the equation of motion and the appropriate initial conditions are together equivalent to a single integral-differential equation.

Let i be the function on [0, Q defined by

i{t) =t.

(2)^ Let b be a continuous vector field on Bx [0, t^), let u be an admissible motion, and let S be a time-dependent admissible stress field. Further, let and be vector fields on B, and let / be defined on Bx [0, t^ by

f{x, t)=i*b(x, t)-\-Q{x) [Uo(x) -\-tVa(x)].

Then \u, S, b] is a dynamical process consistent with the initial conditions

if and only if m(.,0)=Mo, m(-,0)=Uo

f *div S -\-f — QU. (a)

1 Stokes [1851, 2], Umov [1874, 2]. Cf. also v. Mises [1909, 7], Truesdell and Toupin

[i960, 77], §217. * Cf. Truesdell and Noll [1965, 22], Eq. (18.7). ® Gurtin [1964, S], Theorem 3.1.

Handbuch der Physik, Bd. VI a/2, 5


Proof. If the equation of motion and the initial conditions hold, then

t

i* [div S + 6] {x, t) =q{x) f (i — r) u(x, r) dr 0

= q(x)u (x, t)-Q (x) (x) + (ae)],

which implies (a). The converse is easily established by reversing the above argument. □

The next lemma, which is a counterpart of (18.1), will be extremely useful in establishing variational principles for elastodynamics.

(3) Lemma. Let w be an admissible motion and S a time-dependent admissible stress field. Then

f s*wda~ f w*div S dv f S*Fw dv. SB B B

Proof. By (18.1), t

f s*w da= f f s{x,t — r)' w{x, r) da dr dB 0 dB

t

= f f [div S(ac, t-r)- w(x,r) -\-S{x, t~r) ■ Vw{x, r)] dv dr 0 B

= f div S*w dvJ S*Vw dv. □ B B

When Mq and Vq are the initial values of u and ii, we call the function / defined in (2) the pseudo body force field. Our next theorem gives an analog of the theorem of work expended (18.2).

(4) Let [m, S, b] be a dynamical process, let f be the corresponding pseudo body force field, and let w be an admissible motion. Then

i* f s*w daf f*wdv=i* f S*Vtc dv J qu*w dv. SB B B B

In particular, when w=u,

i* f s*uda->r J f*u dv =i* J S*E dv f QU*udv. SB B B B

Proof. By (2) and (3),

i* f s*w da — f {f*div S)*w dv +f* / S + Ttu dv SB B B

= J {QU—f)*w dv-\-i* J S*Vw dv. □ B B

(5) Reciprocal theorem.^ Let \u, S, b] and [u, S, b] be dynamical processes,

and let f and f be the corresponding pseudo body force fields. Then

i* f s*uda-\- J f*u dv —i* f S*Edv B B (a)

= J s*u da / f*udv—i* f S*Edv, SB B B

f s*uda-\- f b*udv— f S*E dv f Q{Ug ■ u -\-Vq ■ u) dv SB B B B

= Js*uda-\-Jb*udv — f S*E dv f q{Uq ■ it -\-Vq ■ u) dv. ___SB B B B

1 This theorem, which appears for the first tinie here, is based on an idea of Graffi [1947, 2],

[1954, 6], [1963, S] (see Sect. 61 of this article).

Sect. 20. The elasticity tensor. 67

Thus if Mq = = Mq = = 0, then

f s*uda-{- f b*udv— f S*E dv = f s*uda-\- fb*udv — JS*Edv. (c) SB B B SB B B

Proof. The relation (a) follows from (4) and the fact that

/ gu*u dv = f gu*udv. B B

To verify (b) we note first that (vi) of (10.1) imphes

i*y)—y). (d)

Thus, in view of the definition of /,

f*u=b*u->rg{Uf,-u-\-Vf,-u). (e)

Therefore if we differentiate (a) twice with respect to t using (d), (e), and the

analog of (e) for / and u, we arrive at (b). □

Given a time-dependent admissible stress field S and an admissible motion u, let M denote the four-tensor field with space-time partition

S —gii

0 0 j ’

we call M the associated stress-momentum field. The importance of this field is clear from the following theorem.

(6) Balance of momentum in space-time. Let b he a continuous vector field on Bx [0, t^. Further, let S be a time-dependeni admissible stress field, let u be an admissible motion, and let M he the associated stress-momentum field. Then [u, S, b] is a dynamical process if and only if

f Mn da-{- f b dv=0 SD D

for every regular region D with D in Bx (0, <o)- Biere n is the outward unit normal to dD and

b = (b, 0).

Proof. By the divergence theorem (11.2), the above relation is equivalent to

/ (div(4)M-1-fa) <^5!'=0. D

This equation holds for all regular D if and only if

diV(4)/VI-|-fa=0,

or equivalently, by the identity (11.1),

divS — pii-|-b = 0. □

III. The constitutive relation for linearly elastic materials.

20. The elasticity tensor. The laws of momentum balance hold in every d3mamical process, no matter of what material the body is composed. A specific material is defined by a constitutive relation which places restrictions on the class of processes the body may undergo. Here we will be concerned exclusively

5*


with linearly elastic materials. P'or such materials the stress at any time and point in a d3mamical process is a linear function of the displacement gradient at the same time and place. Moreover, while this function may depend on the point under consideration, it is independent of the time.

Therefore we say that the body is linearly elastic if for each xeB there exists a linear transformation from the space of all tensors into the space of all symmetric tensors such that^

®(3K)=C*[P’M(3e)].

We call Ca, the elasticity tensor for x and the function C on B with values the elasticity field. In general, depends on ac; if, however, and the density g(x) are independent of x, we say that B is homogeneous. We postulate that a rigid displacement field results in zero stress at each point.^ Let IT be an arbitrary skew tensor. Then the field

u{x) =lV[x — Xg]

is a rigid displacement field with Fu(x)=W;

thus we conclude from the above postulate that

Since this result must hold for every xeB and for every skew tensor W,

C*[-4]=Ca,[sym^]

for every tensor A. Therefore is completely determined by its restriction to the space of all S5mimetric tensors. In particular, our original constitutive relation reduces to

S(a!)=Ca, [£(»!)],

where E is the strain field corresponding to Vu. Since the space of all S5munetric tensors has dimension 6, the matrix (relative to any basis) of the restriction of

to this space is 6 X 6. For the moment let us write 0 = 0^,. Consider an orthonormal basis (ej, eg, eg},

and let Cfji,i denote the components of C relative to this basis:

= ■ C[e^®ei].

Then S = C[E] takes the form

Moreover, since the values of C are S5mimetric, and since the value of C on any tensor A equals its value on the S5mimetric part of A,

Cijki = (sym e^ (g) ef) ■ C[sym e* (g) ej; hence

6'* jkl 6'; ikl ^ ijlk'

We call the 36 numbers elasticities.

1 Thus we have made the tacit assumption that the stress is zero when the displacement gradient vanishes; or equivalently, that the residual stress in the body is zero. At firstiight it appears that such an assumption rules out elastic fluids, for which there is always a residual pressure. However, if the residual stress is a constant pressure, and if we interpret S(as) as the actual stress minus the residual stress, then all of our results are valid without change, provided, of course, we interpret the corresponding surface tractions accordingly.

2 Poisson 1829 [1831, 1]. See also Cauchy [1829, f]. This axiom is discussed in detail by Truesdell and Noll [1965, 22], §§ 19, 19A.

Sect. 21. Material symmetry. 69

(1) Properties of the elasticity tensor. For every xeB:

(i) [A] — Ca, [s5nn A] for every tensor A;

(ii) D ■ Ca, [Al = (S57m D) ■ [357111 A] for every pair of tensors A and D;

(iii) = (sym (g) ef) • [sym e^ ®efl, so that

îjkl ^ jikl 6^ t jlk'

Proof. Properties (i) and (iii) have already been established. Property (ii) follows from (i) and the fact that the values of Ca, are symmetric tensors. □

If the elasticity tensor is invertible,^ then its inverse

is called the compliance tensor; it defines a relation

E{x) =K„[S{x)]

between the strain E(x) and the stress S{x) at x.

The following definitions will be important in what follows. Consider a fixed point X of the body, and let C = C^ denote the elasticity tensor for x. We say that C is symmetric if

■ A ■ C[B] . C[A]

for every pair of S57mmetric tensors A and B, po^tivejsemi-definite if

A - C[^]^0

for every symmetric tensor A, and positive definite if

A ■ C[A] > 0

for every non-zero symmetric tensor A. Clearly, C is S5mimetric if and only if its components obey

îjkl 6.ft lijt

SO that in this instance C has 21, rather than 36, distinct elasticities. Note also that C is invertible whenever it is positive definite.

For the remainder of the article we assume given an elasticity fidd C on B.

x21. Material S3mimetry. Assume that the material is linearly elastic in the sense of the stress-strain relation

S = C,„[E]

discussed in the previous section. In order to specify the S57mmetry properties of the material, we introduce the notion of a S5mimetry group. Our first step will be to motivate this concept.

Let ae be a fixed point in B, let u and u* be smooth vector fields whose domain is a neighborhood of x, and suppose that u* is related to u by an orthogonal tensor Q :

u*{y*)=Qu{y) whenever y* — Q{y—x)-\-x. (a)

^ By this statement we mean: "if the restriction of the elasticity tensor to the space of all symmetric tensors is invertible.” The elasticity tensor, when regarded as a linear trans¬ formation on the space of all tensors, cannot be invertible since its value on every skew tensor is zero.


Roughly speaking, (a) asserts that if the point y* is obtained from y by a rotation Q about x, then the displacement u*(y*) is simply the displacement u (y) rotated by Q. If for every such u and u* the corresponding stresses

S = C,[E(a(i)], S*=C,[E*(a(i)] (b) satisfy

S*=QSQ\ (c)

then we call Q a symmetry transformation for the material at x. Condition (c) is slightly more transparent when phrased in terms of the surface force vectors s„=Sn, s^>i=S*n*; indeed, (c) is equivalent to the requirement that

s*» — Qs„ whenever n*=Qn (see Fig. 9).

By (a) Vu*(x) =QVu(x) ,

and hence the corresponding strain tensors are related by

E*ix)^QE{x)Q^. (d)

It follows from (b), (c), and (d) that a sufficient condition for an orthogonal tensor to be a S5mimetry transformation is that

QC^[E]Q^ = C^[QEQ^] (e)

for every S5mimetric tensor E. That this condition is also necessary can be easily verified using displacement fields which are pure strains from x.

Thus we have motivated the following definition the symmetry group %. for the material at x is the set of all orthogonal tensors Q that obey (e) for every S5mimetric tensor E.

Our definition is consistent with the notion of symmetry group (isotropy group) as employed in general non-linear continuum mechanics (see, e.g.,TRUESDEl,L and Noll [1965,22], § 31). Since the symmetry group of a solid relative to an undistorted configuration is a sub¬ group of the orthogonal group, our treatment is sufficiently general to include elastic solids of arbitrary symmetry. \île we do not rule out non-orthpgpnal sjmmet^ trai^formatjons, wp do not study thepi. The treatment of rnore general symmetry ôups is essential to a systematic study of fluid crystals (see, e.g., Truesdell and Noll [1965, 22], § 33 bis.).


To show that the above definition is consistent, i.e. that really is a group, it is sufficient to verify that

Q,

Note first that (e) can be written in the form

C^[E]=Q^C^[QEQ^]Q. (f)

If J? e then, since

(f) applied to the S5nnmetric tensor QÊQ yields

c«[Q'ÊQ] = mQÊQ) Q^]Q = Q^C^[E\ Q.

Thus we conclude from (e) that belongs to Next, if Q,Pe^^, then two applications of (f) yield

C* [£] = C, [PEP^]P =P^Q^C, [QiPEP'^) Q^] QP

^(QPfC^[{QP)E(QP)^]QP-,

thus QPeâ;. Therefore is, indeed, a group.

(1) Let be invertible, and let K^. be the corresponding compliance tensor. Then

for every and every symmetric tensor S.

Proof. Choose S arbitrarily, and let P = Ka,[S]. Then S = Ca,[E] and

QSQ^ = c^[QEQ^y. therefore

QEQ^ = K^[QSQ^]. or equivalently,

QK^[SW = *^.[QSQ^] □

We say that the material at x is isotropic if the symmetry group equals the orthogonal group, anisotropic if is a proper subgroup of the orthogonal group. Clearly, â: always contains the two-element group { — 1,1} as a subgroup. In fact, â: is th® direct product of this two-element group and a group which consists only of proper orthogonal transformations; hence the sjroinet^ of Jdie material is completely châcterized byjhe^qup In particular, the material at sc is isotropic or anisotropic according as equals or is a proper subgroup of the proper orthogonal group. Although there are an infinite number of subgroups of the proper orthogonal group, twelve of them seem to exhaust the kinds of S5mimetries occurring in theories proposed up to now as being appropriate to describe the behavior of real anisotropic elastic materials.

The first eleven of these subgroups of the proper orthogonal group correspond to the thirty-two crystal classes.^ We denote~fhese subgroups by Table 1, which is due to Coleman and Noll,^ lists the generators of each of these groups; in this table (fe, I, m} denotes a right-handed orthonormal basis,

g= j/f(fe4-I-|-m), and is the orthogonal tensor corresponding to a right-

1 The thirty-two crystal classes are discussed by Schoenfliess [1891, 2], Voigt [1910, 2], Flint [1948, 4], Dana and Hurlbut [1959, 2], Smith and Rivlin [1958, 26], Coleman and Noll [1964, 5], Truesdell and Noll [1965, 22], Burckhardt [1966, 5].

2 [1964, 5].


handed rotation through the angle (p,0<(p<2n, about an axis in the direction of the unit vector e.

The last t3rpe of anisotropy, called transverse isotropy (with respect to a direction m), is characterized by the group consisting of 1 and the rotations Rj^, 0< (p<2ji. Transverse isotropy is appropriate to real materials having a laminated or a bundled structure.

A material is called orthotropic if its S5mimetry group contains reflections on three mutually perpendicular planes. Such a triple of reflections is —R%,

—Rf, —ii^.I Since RgRf—R^ and (R^^)^ =R%, it follows that the groups

^3. ^6> ^6> ^7 correspond to orthotropic materials.

Table 1.

Crystal class Group Generators of Order of

Triclinic system all classes 1 1

Monoclinic system all classes Rm 2

Rhombic system all classes -^3 i^.Ri 4

Tetragonal system tetragonal-disphenoidal tetragonal-pyramidal tetragonal-dipyramidal

^4 4

tetragonal-scalenohedral di tetragonal-pyramidal tetragonal-trapezohedral ditetragonal-dipyramidal

R^^.Ru 8

Cubic system tetartoidal diploidal

<^3 Rh. Rl, R^q 12

hexa tetrahedral gyroidal hexoctahedral

Rf", 24

Hexagonal system trigonal-pyramidal rhombohedral

'g’s 3

ditrigonal-pyramidal trigonal-trapezohedral hexagonal-scalenohedral

Rh,Rt 6

trigonal-dipyramidal hexagonal-pyramidal hexagonal-dipyramidal

«’io b|^ 6

ditrigonal-dipyramidal dihexagonal-pyramidal hexagonal-trapezohedral dihexagonal-dipyramidal

Rk.Rt^ 12

We say that the material at x has ^„-symmetry if is a subgroup of We call two groups and equivalent if given any S5mmietric elastkityiensor C^,

( A group is distinct if there is no [m^n) equivalent to It will be clear —->1 from the results of Sect. 26 that ^4, ^8> are distinct;

j and are equivalent; ^10 > ^11 > and are equivalent. By definition, the S5mi- ' metry group ^ is a maximal group of S3mimetry transformations; thus the groups


^6> ^7> ^10> ^12 cannot be symmetry groups, since invariance with respect to one of these, say, implies invariance with respect to another group which contains elements not in

A unit vector e is called an axis of symmetry (for the material at x) if

Qe=e

for some with 0 4= 1- Thus e is an axis of S3mimetry if and only if one of the symmetry transformations is a rotation about the axis spanned by e; i.e., if and only if for some (p {0< (p<2n).

Consider now the stress S due to a uniform dilatation E=el'.

where C = C^,. Then

or

S = C[el],

QSQ^ = C[eQlQ^]=Clel] = S

QS=SQ

for every i.e. S commutes with every Q in But by the commutation theorem (3.3), a S5mimetric tensor S commutes with an orthogonal tensor Q if and only if Q leaves each of the characteristic spaces of S invariant. Thônly subspace of invariant. un,der.all,X)r.tllQgonal, transformations is the full vector space^'ftself. Thus if the material at x is isotropic, then every vector in ^ is a principal direction of S. Therefore it follows from the spectral theorem (3.2) that the response to a uniform dilatation is a uniform pressure, i.e.

C[el] =al.

Now a^s5U]tne _thai^the_material a.t x is anisotropic and again consider the stress S = C[el] due to a uniform dilatation. If the only spaces left invariant by the rotation are the one-dimensional space spanned by e, the two-dimen¬ sional space of all vectors perpendicular to e, and the entire space iC. The rotation Rg leaves invariant, in addition, each one-dimensional space spanned by a vector perpendicular to e. These facts, the spectral theorem (3.2), and the commutation theorem (3.3) enable one to establish easily the results collected in the following table.i

Table 2.

Type of symmetry Restriction on the stress S due to a uniform dilatation E=e\

Triclinic system no restriction

Monoclinic system m is a principal direction of S

Rhombic system k, 1, m are principal directions of

Tetragonal system ('^4, '^5) Hexagonal system ('^g, Transverse isotropy ('û)

S = al -f [Sm® m

Cubic system ('^g, Isotropy

II

Coleman and Noll [1964, 5].


Of future use is the following

^ (2) Identity. Given any two vectors a, b, let

A[a, h) =J(o®6+6®o). Th’Cfi

C[A[m, M)] • A{a, h) =C[A{Qm, Qn)] ■ A{Qa, Qb)

for all vectors a, b, m, n and every ^ =

Proof. Since Q(m®n) Q'^ = {Qm) ®[Qn),

it follows that QA(m,, n) Q'^ =Â(Qm, Qn),

and thus QC[A{m, n)] 0^ = C[A{Qm, Qn)] (a)

for every Next, in view of the identity T ■ [u®v) =u ■ Tv,

{QBQ^).{Qa®Qb)=^Qa.QBQ^QbâBb = B.{a0b),

therefore if we let B = C\A[m,, n)] and use (a) and (ii) of (20.1), we arrive at the desired identity. □

In view of (iii) of (20.1),

Cijki=A{ei, ef) ■ C^[A{e^, e^)],

and we conclude from the identity (2) that for an isotropic material the compo¬ nents of are independent of the orthonormal basis {Cj, e^, Cg}.

22. Isotropic materials. In this section we shall determine the restrictions placed on the constitutive relation

when the material at x is isotropic; i.e. when the symmetry group equals the full orthogonal group. In a later section we will discuss the form of the elasticity tensor for various anisotropic materials.

(1) Theorem on the response of isotropic materials. Assume that the material at x is isotropic. Then:

(i) The response to a uniform dilatation is a uniform pressure, i.e. there exists a constant k, called the modulus of compression, such that

c,[i]=3*i.

(ii) The response to a simple shear is a pure shear', i.e. there exists a constant p, called the shear modulus,^ such that

[s5m {m®n)] = 2p sym (m ® n)

whenever m and n are orthogonal unit vectors.

(iii) IfE, is a traceless symmetric tensor, then

C^[E,] = 2pE,.

Proof. Assertion (i) was established in the previous chapter (see the remarks leading to Table 2).

V To estabhsh (ii) let A{m, n)= sym {m(S)n),

1 The shear modulus is often denoted by G.

Sect. 22. Isotropic materials. 75

where m and n are orthogonal unit vectors, let e=m'Kn, and choose Q such that

Qm = —m, Qn = —n, Qe — e,

i.e. 0 is a rotation about e of 180°. Then, writing C = C^, we conclude from the identity (21.2) with a=m and 6 = e that

C[A{m,, n)] ■ A{m, e) = — C[A{m, n)] • A{m, e), and hence

C{A{m, n)] • A{jn, e) = 0. (%) Similarly,

C{A[m,, n)] • A[n, e) =0.

Next, if we choose Q such that

(aa)

Qm=m, Qn = —n, Qe=e,

and take a—b=m,n, or e, we conclude from (21.2) that

Now let

C[A{m, n)] ■ A{a, a) =0 for o=m,n, ore.

IJb = C[A{m, n)] • A{m, n).

(as)

(b)

In view of (3.1), the six tensors ■|/2.4(m, n), ]/2.4(n, e), y2A{m, e), V2A{m, m), ]/24(n, n), y2.4(e, e) form an orthonormal basis for the space of all s5mimetric tensors. By (a) and (b), the only non-zero component of C\A{m,, n)] relative to this basis is the component ]/2/^ relative to y2.4(m, n); thus

C[A(m, n)] = 2fj,A(m, n). (c)

Now let u and v be orthogonal unit vectors and choose Q such that

Qm—u, Qn—v.

Then (21.2) with a =m, b=n implies

C[A{m,, n)] • A{m,, n) = C[A(u, »)] • A{u, v),

and thus the scalar [j, in (c) is independent of the choice of m and n.

To establish (iii) let Eq be a traceless S5mimetric tensor. By (13.7) there exist scalars Xg and an orthonormal basis Cj, e^, such that

JSo =Xi 4(ei, Ca) +X2 A{e^, Cj) -{-xÂ{e^, Cj);

thus (ii) and the linearity of C imply

[■®'o] — 2/j, Eq. Q


(2) Theorem on the elasticity tensor for isotropic materials. The material at x is isotropic if and only if the elasticity tensor C — C^ has the form^

C[JS] = 2/^JS + A(trJS) 1

for every symmetric tensor E; the scalars ^ (as) and A = A (as) are called the Lame moduli.^ Moreover, p is the shear modulus and X=k~%jx, where k is the modulus of compression.

Proof. We first assume that

C[E] = 2/Ê+A(trJS) 1. Let Q be orthogonal. Then

tv{QEQ^)=tr{Q->^QE)=tvE, and therefore

QC[E} Q^^2/xQEQ^+?. (tr E)Q1Q^

= 2/xQEQ^ +k[tT(QEQ^)]l = C[QEQ^].

Thus the material at x is isotropic.

Conversely, assume that the material at x is isotropic. Let E be an arbitrary S3mimetric tensor. Then E admits the decomposition

JS=JSo + 3-(trE) 1, trEo = 0,

and (i) and (iii) oi (1) yield

C[JS] = 2/^JSo + A:(trJS) 1.

But since Eq=JS —g-(trE) 1, if we let X=k—^/x, we arrive at the desired result. □

Note that the isotropic constitutive relation

S = 2fxE+?.{tTE)l takes the form

Sfj = 2fi Efj + XE),), dij

when referred to cartesian coordinates. When the material is isotropic, (1) implies that C maps multiples of the

identity into multiples of the identity and traceless tensors into traceless tensors. The next proposition shows that those are the only two tensor spaces that are invariant under C. /,

(3) Characteristic values of the elasticity tensor. Assume that the ma- j terial at x is isotropic with 2^4=3^- Then the linear transformation^ C = C^ has \ exactly two characteristic values: 3^ and 2ft. The characteristic space corresponding y to is the set of all tensors of the form a 1, where a is a scalar: the characteristic space \ corresponding to 2ft is the set of all traceless symmetric tensors.

Proof. By (i) and (iii) of (1), 2ft and 3^=2/^ + 3A are characteristic values of C. Further, if C[E]=Ê, then (2) implies that

(2/t-(5)JS + A(trJS) 1=0; (a) thus (for JS 4= 0)

/S=2/t4=»tr JS=0. (b)

' The basic ideas underlying this constitutive relation were presented by Fresnel [1866, t] in 1829. This form of the relation with it = X was derived from a molecnlar model by Navier

[1823, 2], [1827, 2] and Poisson [I829, 2]. The general isotropic relation is due to Cauchy

[1823, 2], [1828, 2], [1829, 2], [1830, 2], Cf. Poisson [1831, 2], 2 Lam£ [1852, 2].

® Here C is considered as a linear transformation on the space of symmetric tensors.


(Note that A=t=0, since 3^4=2/^.) On the other hand, if tr JS4=0, then (a) 3delds

/S = 2/^ + 3A't=>JS = J(tr E) 1. (c)

Therefore the two spaces specified in (3) are characteristic spaces of C. Taking the trace of (a), we see that

(2^-|-3 ^—/^) =

Thus either /S = 2|M+3A or tiE — O. If trJS=0, then (a) implies that p = 2fi.

Thus 2fi are the only .cMractoisMc_yAla^ □ Note that if 2fi='^k, so that A =0, then every S3nnmetric tensor is a charac¬

teristic vector for C.

Assume now that the material at x is isotropic, so that

S = 2/^JS + A(trJS) 1. (I)

trS = (2/^-h3A)trE.

Thus if we assume that 2^-}-3A4=0, then

and we have the following result.

(4) If the material at x is isotropic, and i/ ^4=0, 2ix-\-f)X^0, then C is in¬ vertible, and the compliance tensor K = C~^ is given by

K[S] = 1

2fl s %_

(2/t + 32) (trS)l

for every symmetric tensor S.

Let Cj, Cg, Cg be an orthonormal basis and consider the following special choices for S in the relation E = K[S]:

(a) Pure shear. Let the matrix [S] of S relative to this basis have the form

Then

with

0 T

[S] = T 0

0 0

0

0 •

0.

0 X

[E] = H

0

0

0

o'

0

0

i

i.e. E is a simple shear of amount 1

with respect to the direction pair (Cj, Cg). Conversely, if .E is a simple shear of amount S is a pure shear with shear stress r=2fix.

(b) Uniform pressure. Let S correspond to a uniform pressure of amount p; i.e. let

Then E is a uniform dilatation of amount

e — 1

Tk P


k —-f + A. with

Clearly, p=.-k[tvE).

Conversely, S is a uniform pressure whenever JS is a uniform dilatation.

(c) Pure tension. Let S correspond to a pure tension with tensile stress a in the Cj direction:

Then [E] has the form

with

[S] =

[E] =

0

0

0

0

I

0

e = ja.

and

V

p a= —ve,

o M' (2/i + 3 A)

2(^ + A) ■

The modulus /S is obtained by dividing the tensile stress cr by the longitudinal strain e produced by it. It is known as Young’s modulus.^ The modulus v is the ratio of the lateral contraction to the longitudinal strain of a bar imder pure tension. It is known as Poissons’ ratio.^ Table 3 gives the relations between the various moduli.

o o

If we write E and S for the traceless parts of E and S, i.e.

i=JS-i(trE) 1, S=S-J(trS)l,

then the isotropic constitutive relation (I) is equivalent to the following pair of relations®

S = 2fiE,

tvS^ktiE.

Another important form for the isotropic stress-strain relation is the one taken by the inverted relation (4) when Young’s modulus /S and Poisson’s ratio v are used:

E^ [(l+r)S-r(trS) 1].

^ Young neither defined nor used the modulus named after him. In fact, it was introduced and discussed by Euler in [1780, 1] (cf. Truesdell [I960, 15], [i960, 16], pp. 402-403) and, according to Truesdell [1959, 27], was used by Euler as early as 1727. In most texts Young’s modulus is designated by E.

2 Poisson [I829, 2]. Poisson’s ratio is often denoted by a.

® Stokes [1845. 7] introduced the moduli and k and noted that the response of an

isotropic elastic body is completely determined by its response to shearing and to uniform

compression.


Table 3.

Since an elastic solid should increase its length when pulled, should decrease its volume when acted on by a pressure, and should respond to a positive shearing stress by a positive shearing strain, we would expect that

/S>0, k>0, ^>0.

Also, a pure tensile stress should produce a contraction in the direction per¬ pendicular to it; thus

r>0.

Even though the aboveJnegualities„.ê„ph.ysically weU.mptivated, we will not aSume that they hold, for in many circumstances other (somewhat weaker)

80 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 2js.

assunyjtions arejnore natioral. Further, these vyeaker hypotheses are important to the study of infinitesimaj deformations superposed on finite deformations.

Finally, we remark that Gazis, Tadjbakhsh, and Toupin^ have established a method of determining the isotropic material whose elasticity tensor is nearest (in a sense they make precise) to a given elasticity tensor of arbitrary symmetry.

/ 23. The constitutive assumption for finite elasticity. In this section we will show that the linear constitutive assumption discussed previously is, in a certain sense, consistent with the finite theory of elasticity provided we assume infinitesi¬ mal deformations and a stress-free reference configuration.

In the finite theory the stress S at any given material point as is a function of the deformation gradient F at as:

S=S{F). (a)

The function S is the response function for the material point x.

In view of (o) in Sect. 16, we assume that

S{F)F^=FS(F)^ (b)

for every invertible tensor F; this insures that the values of S are consistent with

the law of balance of angular momentum. In addition, we assume that S is compatible with the principle of material frame indifference; this is the requirement that the constitutive relation (a) be independent of the observer.^ During a change in observer F transforms into QF and S into QS. where Q is the orthogonal

tensor corresponding to this change.® Since S must be invariant under all such changes, it must satisfy^

S{F)=Q^S{QF) (c)

for every orthogonal tensor Q and every F. By the polar decomposition theorem,® F admits the decomposition

F=RU. (d) where R is orthogonal and

U={F'^F)i. (e)

Choosing Q in (c) equal to R^, we arrive at the relation

S = RS{U), or equivalently, using (d).

S=FU-^S{U). (f)

If we define a new function S by the relation

S{D) = U-^S{U),

where D is the finite strain tensor

l)=i(!7®-l)=J(F^F-l),

■X^ [1963, 7]. ® See, e.g., Truesdell and Noll [1965, 22], §§ 19, 19A. ® See, e.g., Truesdell and Noll [196S, 22], § 19. The transformation rule for F is an

immediate consequence of their Eq. (19.2)1. The rule for the Piola-Kirchhoff stress S follows from the law of transformation for the Cauchy stress T [their Eq. (19.2)2] and (r) of Sect. 16 relating S to T and F.

* Cf. Truesdell and Noll [196S, 22], Eq. (143.9). ® See, e.g., Halmos [1958, 9], § 83.

Sect. 24. Work theorems. Stored energy. 81

then (f) reduces to

S==FS{D). (g)

On the other hand, it is not difficult to verify that the constitutive relation (g) is consistent with (c). Further, (g) satisfies (b) if and only if

FS(I)) F^=FS(Df F^, or, since F is invertible,

S{D) = S(D)\ (h)

so that the values of S are S3unmetric tensors.

It is important to note that the concept of strain arises naturally in the study of restrictions placed on constitutive relations by material frame-indiffer¬ ence. It is possible to derive other constitutive relations involving other strain measures; we chose one involving D because it simplifies the discussion of in¬ finitesimal deformations.

Let S be differentiable at 0. Then

S{D)=S{0)+C[D]+o{\D\)

as |D| ->0, where C is a linear transformation which, by (h), maps S5mimetric

tensors into S5mimetric tensors. When F = l, D = 0; thus /S(0) represents the residual stress in the reference configuration. We now assume that this residual stress vanishes'} hence

Let S(l))=C[l)]+o(|l)|).

d = \Vu\.

(i)

Then (c), (e), and (f) of Sect. 12 yield

F = \+0[S),

D +0{d'^) =0{S)

as 5->0, and we conclude from (g) and (i) that

S=C[E] + o{d) as 5^0.

Therefore if we assume that the residual stress in the reference configuration vanishes, then to within terms of o{d) as the stress tensor is a linear function of the in¬ finitesimal strain tensor. Further, since JS = 0 when Ftt is skew, to within terms of 0 (8) the stress tensor vanishes in an infinitesimal rigid displacement.

These remarks should help to motivate the constitutive assumptions underly¬ ing the infinitesimal theory.

24. Work theorems. Stored energy. In this section we establish certain con¬ ditions, involving the notions of work and energy, which are necessary and suffi¬ cient for the elasticity tensor to be S5mimetric and positive definite. Thus we consider a fixed point x of the body, and let C=C^ denote the elasticity tensor for X.

' This assumption, wWle basic to the theory presented here, is not satisfied, e.g., by an elastic fluid.



By a strain path Fe connecting Eq and we mean a one-parameter family E{a.), (Xoââi, of symmetric tensors such that £(•) is continuous and piecewise smooth and

®(*o) ~®o>

We say that is closed if Eg =Ei.

Clearly, the strain path rE determines a one-parameter family of stress tensors

S(a) = C[E(a)], Oo ^ a ^ oti;

the work done along Fe is defined to be the scalar

^{rE)= /S(a) ■ ia.

In view of the discussion following (19.1), we have the following interpretation of w {F^): Consider a homogeneous elastic body with elasticity tensor C; w (/^) is the work done (per unit volume) by the surface tractions in a time-dependent homogeneous deformation with strain E.

7 By a stored energy function ^ we mean a smooth scalar-valued fimction e on the space of symmetric tensors such that e(0) =0 and^

C[E]=VEe(E)

for every symmetric tensor E. An immediate consequence of this definition is the following trivial proposition.

(l)If a stored energy function exists, it is unique, and

w{FE)=e{E0-e{E,)

for every strain path Fe connecting E^ and E^.

The next theorem supplies several alternative criteria for the existence of a stored energy fimction.

^(2) Existence of a stored energy function.^ The following four statements are equivalent.

(i) The work done along each closed strain path is non-negative.

(ii) The work done along each closed strain path vanishes.

(iii) C is symmetric.

(iv) The function s defined on the space of symmetric tensors by

e{E)^\E.C[E-] is a stored energy function.

Proof. We will show that (i) => (ii) => (iii) ^ (iv) ^ (i). Assume that (i) holds. Let Fe be an arbitrary closed path, and let Fe* denote the closed path defined by

E*(a) =jG(aQ-l-oq—a), aoâôq.

* The notion of a stored energy function is due to Green [1839, t], [1841, f]. 2 The gradient P'e6(jE) of s at £ is defined in a manner strictly analogous to the definition

given on p. 10 of the gradient of a scalar field on S. Note that fEsiE) is a symmetric tensor; in components

[|7Bs(£)h,- = 8e(E)

3 Green [1839, f], [1841, i] proved that (iii) O (iv), Udeschini [1943, 3] that (i) O (iv).

Sect. 24. Work theorems. Stored energy. 83

Then a simple computation tells us that

w{rE) = —w{rE.).

But (i) implies that te'(iE) and W(rE*) must both be non-negative. Hence “'(■Tb) =0, which is (ii). To see that (ii) => (iii) let A and B be symmetric tensors, and let Fe be the closed strain path defined by

Then

and

H(a) =(cosa —1) 2I-1-(sina) 0â^27r.

C [E(a)] ■ = (sin a — sin a cos oCj A ■ C [A]

271

-j- (cos* a — cos ol) B ■ C [A]

— (sin* ol) A ■ C [B] -j- (sin a cos ol) B • C [B]

w (Fe) = Jc[£(«)] . doi^n[B.C[A}-A • C[B]).

Hence (ii)=>(iii). Next, define

e(E)=\E-C[B\.

Since the gradient pEe(B) can be computed using the formula^

VEe[B).A = {ê{E + XA)l^^.

it follows that

1 d VEe{E).A^^ {E .C[E]+?iE-C[A]+kA-C [E] +XÂ-C [A]},ô

= {{E.C[A]+A-C[E]}.

Thus (iii) implies VebIE) -AÂ-CIE]

for every symmetric tensor A, which yields (iv). Finally, hy (1), (iv) implies (i). □

If C is symmetric, a stored energy function e exists, and the constitutive equation

S = C[E] may be written in the form

S = VEe(E). Let C be invertible, and let

e{A)=Â-K[A]

for every symmetric tensor A, where K is the compliance tensor. In view of the symmetry of K (which follows from the symmetry of C),

Fsi(S)=K[S],

and the above constitutive relation takes the alternative form

E^FsHS)- Moreover,

i(S)=K[E]-K[C[E]],

1 Cf. p. 10.

6*


and, since K is the inverse of C,

e(S)=e(E).

The next two theorems give a physical motivation for some of our future assumptions concerning the elasticity tensor.

(3) The work done along each strain path starting from E = 0 is non-negative if and only if C is symmetric and positive semi-definite.

Proof. Assume that the work done along each strain path starting from 0 is non-negative. Let Fe be an arbitrary closed strain path starting from and let

E*(<x.) =E{<x.)—Eg, aoâÔi.

Then i®. is a closed strain path starting from 0 and

OCo

Thus, in view of the hnearity of C,

tti

0^ f C[E(«)] . î«-C[Eo] •{£(«!)-£(«o)}.

But E(«.i) =£(ao), since Fe is closed. Thus the work done along each closed strain path is non-negative. Thus we may conclude from (2) that C is symmetric and, in addition, that there exists a stored energy function. Let 2I be a symmetric tensor, and let Fe be the strain path defined by

E{a)=aA, Oâ^l.

Then hj (1) and (iv) of (2),

0^w(rE)=s{A)-s(0)^Â-C[Ay,

hence C is positive semi-definite. Conversely, if C is symmetric, then (1) and (2) imply that given any path Fe connecting 0 and E^,

w{rE) = iE^-C[E^].

Thus if C is positive semi-definite, w (Fe) ^0. □

(4) The work done along each non-closed strain path starting from E = Q is strictly positive if and only if C is symmetric and positive definite.

Proof. Assume that the former assertion holds. Let Fe defined by E(a), otQââi, be an arbitrary closed strain path with

®(«o) ='E(«i) =0-

If E(a) = 0 for all a, then w (Fe) = 0. Assume E(a) =t= 0 for some a, and let

a = sup{a:E(a) =4=0}.

w(Fe) = /c[E(a)]

Oto Oto

dEia.)

da. da..

Then

Work theorems. Stored energy. Sect. 24. 85

Since E{ol) (aoââ*) for a*<a but sufficiently close to ol defines a non-closed strain path starting from 0,

j C[£(«)].îa>0;

otn

hence

w(rE) ^0.

Thus the work done along each strain path starting from 0 is non-negative, and we may conclude from (3) that C is symmetric. The remainder of the argument is the same as the proof of the analogous portions of ^3). □

(5) If the material is isotropic, then C is symmetric,

s{E) =iJL\EY + \-{txEY,

and the following five statements are equivalent:

(i) C is positive definite;

(ii) /4>0, 2/4-1-3A>0;

(iii) /4>0, A>0;

(iv) /4>0, —i<v<^-,

(v) /3>0, —1 <i'<-2-.

Proof. The symmetry of C and the above formula for e{E)=\E ■ C\E'] follow from (22.2). That (i)4=>(iii) is an immediate consequence of (22.3) and the sentence following its proof. Finally, the equivalence of (ii)-(v) follows from the relations given in Table 3 on p. 79. □

Let

E=E + h(iTÊ)l (trE=0). Then

|E|2 = |E|2-|-i(trJS:)2 and

e{E)=f^\EY + ^(trEY.

This alternative form for e jdelds a direct verification of the implication (i)4=>(iii) in (5).

At the end of the last section we gave physically plausible arguments to support the inequalities; ^>0, k>0, /j,>0,v>0. It follows from (5) that these inequalities are more than sufficient to insure positive Hefinifêss. On the other Kan^fhaf C is posifive defmifeTmplies ^>6,k>0,/j,>0, but does not imply v>0.

Assume that C is symmetric and positive definite. Then the characteristic values of C (considered as a linear transformation on the six-dimensional space of all symmetric tensors) are all strictly positive. We call the largest characteristic value the maypimum^pasttc^^^ the smallest the minimum elastic modulus! The following trivial, but useful, proposition is a direct consequence of this definition and (22.3).

^ y (6) Let C be symmetric and positive definite and denote the maximum and mini¬ mum elastic moduli by and respectively. Then

_ l^,n\EY<'E • C[E]

1 Cf. Toupin [1965, 21].


for every symmetric tensor E. If, in addition, the material is isotropic with strictly positive Lame moduli fi and X, then

Pm ~ 2^ “I” 3 ^ > Pm ~ 2^.

Of course, if p>0 and 2.—0, then p„,=pM = 2p', if p>0 and A<0 (but 2p+'i?.>0 so that C is positive definite), then pM = 2p and p„ = 2p-\-'iX.

25. Strong ellipticity. In this section we relate various restrictions on the elasticity tensor C = C^ at a given point ace 5.

We say that C is strongly elliptic if

A ■ C[2l]>0 whenever A is of the form

A=a®b, a=t=0, 6=t=0.

This condition is of importance in discussing uniqueness and also in the study of wave propagation.

By (20.1) A • C [A] — (sym A) ■ C [sym A\;

thus Cpositive definite C strongly elliptic.

The fact that the converse assertion is not true is apparent from the_nextj)ro- positmn.

(1) Suppose that the material is isotropic. Then

C strongly elliptic ^p>0, X-\-2p>0,

— C strongly elliptic <^p<0, X-\-2p<0,

where p and A are the Lame moduli corresponding to C. In terms of Poisson’s ratio v,

C strongly elliptic p>0, vi[\, \],

— C strongly elliptic p<0,vi[\,\].

provided ^

Proof. By (i) of (20.1) and (22.2),

{a®h) • C[a(8)6] =(a(g)6) • C[syma(8)6]

= (a (g) 6) • [a (a (g) 6 + 6 (g) a) + A (a • 6) 1]

=pa^ + (A +p) {a ■ 6)2

=p[a^ b^ — (a- 6)2] + (A + 2p){a ■ 6)2.

Since (a ■ b)^â^b^, p>0 and 2. + 2p>0 imply that C is strongly elliptic. Conversely, if C is strongly eUiptic, then X-\-2p>0 follows by taking a = b, p>0'by taking a orthogonal to 6. The assertion concerning — C follows in the same manner. Finally, the last two restilts are consequences of the relation:

A 2(A+/i) •

V = □

Sect. 26. Anisotropic materials. 87

These results, as well as some of the results established in (24.5), are given in the following table:

Table 4.

Condition Restrictions on the moduli

Positive definite /i>0, 2yti-f 3A>o or M > 0, v£ (— 1, i) or p>o. ve{-i,i)

Strongly elliptic n>0, A-f2yti>0 or M>o, f5[|, 1]

> 26. Anisotropic materials. In this section we state the restrictions that material symmetry places on the elasticity tensor C = C^, at a given point x. Let {e ■} be a given orthonormal basis with e^=e,xe^, and let C,,.*, denote the components of C relative to this basis:

îjki~Cjiki — C{jII, = (sym Ci(g)Cy) • C[sym (g)e,].

^^?^5?*5^ jyF9S&hout_tlus section that a stored energy function exists, so that C is symmetric, thus

îjhl=CkUj-

For the triclinic system (■g’l-symmetry) there are no restrictions placed on C by material symmetry.

For the monoclinic system (■g’a-symmetry) let the group be generated by RJ,. By identity ^21.2),

Cii^, = sym[Qei®Qe,) • C[sym(0e,(g)0e,)]

for every If we take Q = -R^, then

Qei-=ei, 062=62, 063 = -63,

and the above identity imphes

^1128 ^1181 ^2228 ^2281 ^8328 — ^3881 —^2812 ~ ^3112 “ 0 .

In the same manner, results for the other symmetries discussed in Sect. 21 can be deduced.^ We now list these results, which are due to Voigt.^ For con¬ venience, the 21 elasticities will be tabulated as follows:

^1111 ^1122 Cu33 r r ^2222 '-'2233

r '^ZZZZ

r '-'1123 ^1131

r '^1112

C2228 r '^2231

r '^2212

^3828 ^3331 ^8812

^2823 r '^2331 ^2812

^3131 ^8112

^1212

[196oîlf deduced using the results of Smith and Rivlin [1958. 16], SIROTI^

2 Yoj(j.j. [1882, i], [1887,3], [190O, S], [1910, i], § 287. See also Kirchhoff [1876,71 i], Neumann [i885, 3], Love [1927. 3], Auerbach [1927, 1], Geckelke

[1928, 2]. A discussion of methods used to measure the elastic constants of anisotropic materials as well as a tabulation of experimental values of these constants for various materials is given by Hearmon [1946, 2].


Monoclinic system (all classes) generated by (13 elasticities)

^1111 ^1122 Cll83 0 0 C1112

r ^^2222 C2233 0 0 C2212

C3383 0 0 C3312

C2323 C2331 0

C3131 0

C1212

Rhombic system (all classes) generated by , R”^ (9 elasticities)

Ciiii C1122 Cl 133 0 0 0

C2222 C2233 0 0 0

C3333 0 0 0

C2323 0

C3131

0

0

^1212

Tetragonal system (tetragonal-disphenoidal, tetragonal-pyramidal, tetragonal-dipyramidal) generated by (7 elasticities)

Cr

C,

Cl

Cl

C. 3333

0

0

0

c.

0

0

0

0

1112

c, c.

1112

0

0

0

1212

Tetragonal system (tetragonal-scalenohedral, ditetragonal-pyramidal, tetragonal-trapezohedral, ditetragonal-dipyramidal) generated by R^,R^^ (6 elasticities)

Cii Cl

c, Cl

C;

c« 1133

0

0

0

c.

c.

0

0

0

0

2323

0

0

0

0

0

Cl212

Cubic system (tetartoidal, diploidal) generated by

R%,. RZ,, R^q'^. =

(hexatetrahedral, gyroidal, hexoctahedral) generated by R^^, ilef (3 elas¬ ticities)

C, C.

C, 1122 Cll22

Cll22

C„„

C,

0

0

0

2323

0

0

0

0

C,

c,

0

0

0

0

0

2323

Sect. 27. The fundamental system of field equations. 89

Hexagonal system (trigonal-pyramidal, rhombohedral) generated by ^7 elasticities)

^1122 ^1133 ^1123 ^1131 0

Ciiii ^1133 ^1123 ^1131 0

C3333 0 0 0 r '^2323 0 “■^1131

c '^2323 ^1123

Hexagonal system (ditrigonal-pyramidal, trigonal-trapezohedral, hexagonal-scalenohedral) "^9 generated by Ke, (6 elasticities)

Cllll c '^1122 ^1133 ^1123 0 0

^1111 ^1133 ~Qi23 0 0

^3333 0 0 0

^2323 0 0

^2828 ^1128

^1122)

Hexagonal system (trigonal-dipyramidal, hexagonal-pyramidal, hexagonal-dipyramidal) "^10 generated by (ditrigonal-dipyramidal, dihexagonal- pyramidal, hexagondl-trapezohedral, dihexagonal-dipyramidal) generated by

Re^, Re, ', transverse isotropy "^12 generated by R%^, 0<(p<27i (5 elas¬ ticities)

C 1111 ^1122

Ciiii

^138

^1183

^833

0

0

0

^2323

0

0

0

0

^2323

0

0

0

0

0

^(^1111 ^1122)

In view of (21.1), if the elasticity tensor C is invertible, then the components of the compliance tensor K = C~^ will suffer restrictions exactly analogous to the ones tabulated above.

D. Elastostatics.

I. The fundamental field equations. Elastic states. Work and energy.

27. The fundamental system of field equations. The fundamental system of field equations for the time-independent behavior of a linear elastic body consists of the strain-displacement relation

E = Vu = ^(Vu + Fu^),

the stress-strain relation S = C[E].


and the equation of equilibrium^

div S + 6 = 0.

Here, u, E, S, and 6 are the displacement, strain, stress, and body force fields, while C is the elasticity field.

By (i) of (20.1),

C[Fm]=C[Fm];

thus when the displacement field is sufficiently smooth, the above equations imply the displacement equation of equilibrium

div C[Fm] 4-6 = 0.

Conversely, if u satisfies the displacement equation of equilibrium, and if E and S are defined by the strain-displacement and stress-strain relations, then the stress equation of equilibrium is satisfied. In components the displacement equa¬ tion of equilibrium has the form

(Q/a i i), / + — 0 •

We assume for the remainder of this section that the material is isotropic.

Then S — Ip E (tr E') 1,

where p and A are the Lame moduh. Of course, p and A are functions of position x when the body is inhomogeneous. Therefore, since

div {pVu) —pAu + {Vu)Vp,

div {pVu^) —pVdiiYU-\-{Vu)^Vp,

div [X(div u) 1] = A Fdiv u 4- (div u) VX,

the displacement equation of equilibrium takes the form

pAu + {X-\-p) FdivM4-2(FM) Vp + {dlvu) F>l-f6 = 0.

Suppose now that the body is homogeiieous. Then Vp = VX= 0, and the above equation reduces to Navier’s equation^

pAu-\-{X-\-p)VA.iYU-\-b=0, (a) or equivalently,

Au-\-F div M 4- — 6 = 0, \ — 2v fl

in terms of Poisson’s ratio

Next, by (3) of (4.1),

= --

2(k+n)

/\u — Vdiv M — curl curl u.

1 Since the values of Ca,[*] are assumed to be symmetric tensors, the requirement that S = S^ is automatically satisfied.

2 This relation was first derived by Navier [1823, 2], [1827, 2] in 1821. Navier’s work, which is based on a molecular model, is limited to materials for which /j. = X. The general relation involving two elastic constants first appears in the work of Cauchy [1828, i]. Cf. Poisson [1829,2], hAut and Clapeyron [1833,7], Stokes [1845, J], Lam^: [1852, 2],

§26. The analogous relation for cubic symmetry was given by Albrecht [1951,7],


and thus (a) jnelds^ [X + 2[j) Fdivtt—curl curl M+6=0.

Operating on (a), first with the divergence and then with the curl, we find

that (A+2;«)Zl divM = —div6,

fjiA curlu — —curl 6.

If (A + 2;m) +0, ;M+0, and div 6 and curl 6 vanish, then

2ldivM = 0, 2lcurlM=0.

(b)2

Thus, in this instance, the divergence and curl of u are harmonic fields. Trivially,

these relations can be written in the form

div2lM=0, curl2lu = 0; !->

and since a vector field with vanishing divergence and curl is harmonic, we have

the followhg important result: ®

div6=0, curl 6=0 AAu = 0.

Thus if the body force field 6 is divergence-free and curl-free, the displacement field > is biharmonic and hence analytic on B. To prove this we have tacitly assumed that u is of class C*; in Sect. 42 we vnff show that these results remain valid when

u is assumed to be only of class C^. Another interesting consequence of the displacement equation of equilibrium

may be derived as follows: Let

V =/Liu + ^(X+/j,)p div u, p(x)—x — 0.

Then Av Au + div u + \pA divw}.

and (a) and (b) imply

Av — —h — 2 (X-\-zp)

p div 6.

Thus we arrive at the following result of Tedone; * if the body force field vanishes,

then Av = Q.

Assume, for the time being, that the body force field 6 = 0. .îeld u that satisfies . Navier’s equation (a) for ah values of p and A is said to be Since Aiiedd- of this type is independent of the elastic constants p and A, it is a possible dis¬ placement field for all homogeneous and isotropic elastic materials. Clearly, 1* is

universal if and only if® 2Im=0,

V div M=0;

1 Lam^: and Clapeyron [1833, li, Lam^: [1852, 2], § 26. * The first of these relations is due to Cauchy [1828, 7]. See also LamA and Clapeyron

[1833, i], LamA [1852, 2], §27- s LamA [1852, 2], p. 70. ‘ [1903, 4], [1904, 4]. See also Lichtenstein [1924, i]. ® Truesdeix [1966, 28], p. 117.


, i.e., if and only if u is harmonic and has constant dilatation. Important examples ^ of universal displacement fields are furnished by the Saint-Venant torsion

solution.^

If ^=t=0,j'=t=—1, the stress-strain relation may be inverted to give

E= p-(trS) 1 . 2/r i+v ' '

By (14.3) we can write the equation of compatibility in the form

^(E)=AE + VV{tvE)-2VdivE=0. (c) Therefore

0 =^(2^ E) =^(S) - ^^[(tr S) 1]. (d)

But .jS?[(tr S) 1] = d (tr S) 1 -f VV{tT S), (e)

and by (c) with E replaced by S in conjunction with the equation of equihbrium,

^{S)=AS + VV{tTS)+2Vb. (f) Eqs. (d)-(f) imply that

^«) -q+VS) l + 2F5 = 0. (g)

and taking the trace of this equation, we conclude that

;;;;d(trs)=-div6. (h)

Eqs. (g) and (h) yield the stress equation of compatibility :^

AS + —VV(tT S)+2Vb + - - - (div 5) 1 =0,

which in the absence of body forces takes the simple form

^S+-Â„-FF(trS)=0.

On the other hand, if S is a symmetric tensor field on B that satisfies the equation of equilibrium and the stress equation of compatibihty, and if E is defined through the inverted stress-strain relation, then E satisfies the equation of compatibility. Thus if B is simply-connected, there exists a displacement field u that satisfies the strain-displacement relation.

In view of the stress-strain and strain-displacement relations,

S=ij,(Vu-\-Vu’^) -l-A(divM) 1,

and the corresponding surface traction on the boundary 8B is given by

s—Sn=ii{Vu-\-Vu'^) n-l-A(divM) n.

1 See, e.g.. Love [1927, 3], Chap. 14: Sokolnikoff [1956, 12], Chap. 4; Solomon

[1968, 12], Chap. 5. 2 This equation was obtained by Beltrami [1892, 1] for 6==0 and by Donati [1894, 2]

and Michell [1900, 3] for the general case. It is usnally referred to as the Beltrami-Michell equation of compatibility although Donati’s paper appeared six years before Michell’s. The appropriate generalization of this equation for transversely isotropic materials is given by Moisil [1952, 3], for cubic materials by Albrecht [1951, 1],


Let duldnhe defined by

then

8u

8n = [Vu) n;

8u s=2fi g-- +fi(Vu^ — Vu) n + A(divu) n.

But [Vu^ — Vu) n =nX curl u\

thus we have the following interesting formula for the surface traction: ^

s = 2/i ^g~ +^nxcurlM+A(divM) n.

We close this section by recording the strain-displacement relations and the equations of equilibrium in rectangular, cyhndrical, and polar coordinates, using physical components throughout.^

(a) Cartesian coordinates (x,y,z). The strain-displacement relations re¬ duce to

F = ^xx — 8u^

8x yy 8y ’

E = — ^yz 2 __ _1_ 8u^\

2 I dy^ 8x )•

F = ■* Z' 4- « 2\ 8x 8x j’

and the equations of equihbrium are given by

8u, 'y I

8 x 8y I •

dx + 8Sxy

8y + 8Sxz 8z

dSxy

dx + 8Syy

8y + 8Sy^ dz

l^xz_ dx + ^^yz

8y + dz

+ — 0 >

by = 0,

+ ^2=0.

(b) Cylindrical coordinates (r, 0, z). The strain-displacement relations have the form

8u^ 8r~‘

'A' 1 / 8u.

~2

E„„ = 1 8ug _ I F 8Q ^ r • 8z

E„ = -

^re =

F 1 / I 'i F 1 / 8m8 1 8u^ \

E., = , ^82 = -2 [-sT + -- ~90 j,

while the equations of equihbrium reduce to

S I dti-Q ^0

r dd ' dy y

dr J.. bSj± I I _^r . jii. I J =0

^ r 80 ^ 8z ^ r ^ ' ’

8S g 1 8Sgg 8Sg 2 ,

+ 7' 80- + -87-+ 8r

^rz dr

I < 858, r 80

dz

dz + 7 2 +^2—0.

1 Betti [1872,1], § 10, Eq. (53). See also Somigliana [I889, 3], p. 38, Korn [1927, 2],

p. 13, Kupradze [1963, 17, J3], p. 9.

2 See, e.g., Ericksen [i960, 6], §§ 11-13.

94 M.E. Gurtin; The Linear Theory of Elasticity. Sect. 28.

(c) Spherical coordinates [r,B,y). The strain-displacement relations be¬ come

E rr 8u^ 8r~’

E, 1 ^“8 I 3 r 8d ' r

8uv r sin 0 8y

I I Mg cot 0

p if 1 8m, My 0My\

''r 2 i r sin 0 8y r ' 8r ) ’

P _ 1 /1 8m,_^ I _^8 ^

2U' 86 r 8rJ’

1/1 SMy My cot 0 1 SMfl \

2 \ r 80 >■ ysinO 8y /’

and the equations of equilibrium are given by

8^fr I t 8Sfy 1 8S^q I ^yv *^88 4- ■^,0 cot 0 I T _„ 8r ' r sin 0 8y 80 r ' ’

85,y 1 dSyy 1 8SyQ . 3'5,y-|- 2 5y g COt 0 | ^

raiad “Ty V “P" r |-0y—Ug

8^y 1 8Sya 1 8Soa 35,a -KSga — Syy) cot 0 , _

8>' sin 0 8y 80 r "re

28. Elastic states. Work and energy. Throughout this section we assume given a continuous elasticity field C on B. We define the strain energy U{E} corre¬ sponding to a continuous strain field E on 5 by

V{E}=\ fE-C[E\dv. B

Note that when a stored energy function e exists, its integral over B gives the strain energy;

U{E} = Je{E) dv. B

For future use we now record the following

(1) Lemma, Let C be symmetric, and let E and E be continuous symmetric tensor fields on B. Then

U{E +E} = U{E} + U{E} + fE-C[E] dv. B

Proof. Since C is symmetric,

Thus E • C [E] =:E • C [E].

(E +E) ■ C [E +E] =E • C [E] -l-i • C [E] -\-2E-C[E],

which imphes the desired result. □

By an admissible state we mean an ordered array a = [u, E, S] with the following properties:

(i) M is an admissible displacement field

(ii) JS is a continuous S5mimetric tensor field on B;

(iii) S is an admissible stress field.^

1 See p. 59.

Sect. 28. Elastic states. Work and energy. 95

Note that the fields u, E, and S need not be related. Clearly, the set of all admissible states is a vector space provided we define addition and scalar multi¬ plication in the natural manner:

[u, E, S] -I- [it, E, S] = [m -l-M, E -l-E, S -l-S],

a [m, E, S] = [att, «.E, aS].

We say that !) = [«, E,S] is an elastic state^ (on B\ corresponding to the body force field 5 if i) is an admissible statejmd ^

' .....'¥=i(FM + FM^), 1 S = C[E], I

divS-1-5=0. j

The corresponding surface traction s is then defined at every regular point of 8B by

s{x) =S{x) n{x),

where n (a?) is the outward unit normal to dB at x. We call the pair [5, s] the external force system for d.

When we discuss the basic singular solutions of elastostatics, we will fre¬ quently deal with elastic states whose domain of definition is a set of the form D=B—F, where F is a finite subset of B. We say that i = [u, E, S] is an elastic state on D corresponding to 5 if u, E, S,^,^âd_5jyre,iaRctiQhS,0 D arid given any closed regular subregion PcD, the restriction of d to P is an elastic state on P corresponding to the restriction of 5 to P.

When we omit mention of the domain of definition of an elastic state, it will always be understood to he E.

K (2) Principle of superposition for elastic states. If \u, E, S] and [u, E, S]

are elastic states corresponding to the external force systems [5, s] and [5, s],

respectively, and if a and r are scalars, then a,[u, E, S] -|-t[m, E, S] is an elastic

state corresponding to the external force system a [5, s] -1-t[5, s], where

a [5, s] -|-t[5, s] = [a5 -f t5, as -|-ts] .

A direct corollary of (18.2) is the

X (3) Theorem of work and energy.^ Let [u, E, S] be an elastic state correspond¬ ing to the external force system [5, s]. Then

fs-uda + fh-udv = 2U{E}.

The quantity on the left-hand side of this equation is the work done by the external forces; (3) asserts that this work is equal to twice the strain energy.

" (4) Theorem of positive work. Let the elasticity field he positive definite. Then given any elastic state, the work done by the external forces is non-negative and vanishes only when the displacement fidd is rigid.

Proof. If C is positive definite, then U{E}^0, and the work done by the ex¬ ternal forces is non-negative. If U{E}=0, then E- C[E] must vanish on B. Since C is positive definite, this implies E = 0] hence by (13.1) the corresponding displacement field is rigid. □

1 This notion is due to Sternberg and Eubanks [1955, 13]. 2 Lam£ [1852, 21.

96 M. E. Gortin ; The Linear Theory of Elasticity. Sect. 29-

The following direct corollary to (4) will be used in Sect. 32 to establish an important uniqueness theorem for elastostatics.

(5) Let the elasticity field be positive definite. Further, let [u,E,S] be an elastic state corresponding to vanishing body forces, and suppose that the surface traction s satisfies

s-11 = 0 on 8B.

Then u is a rigid displacement field and E = S = 0.

The next theorem, which is extremely important, shows that the norm of the displacement gradient is bounded by the strain energy.

(6) Alternative form of Korn’s inequality^ Assume that C is symmetric, positive definite, and continuous on B. Let u be a class displacement field on B, and assume that either (a) or (/?) of (13.9) holds. Then

f lFul^dv^KgU{E}, B

where E is the associated strain field and is a constant depending only on C and B.

Proof. Let (a?) denote the minimum elastic modulus for 0*. By h3q)othesis, (x) is bounded from below on B. Hence

and we conclude from (24.6) that

\E\^^rE-C^[E]

for every xeB and every symmetric tensor E. This inequality and (13.9) yield the desired result. □

II. The reciprocal theorem. Mean strain theorems.

29. Mean strain and mean stress theorems. Volume change. In this section we estabhsh some results concerning the mean values

B B

of the stress and strain fields when the body B is homogeneous.

j: (1) Second mean stress theorem. The mean stress S corresponding to an elastic state [m, E, S] depends only on the boundary values of u and is given by

S(B) = c| Ju®nda .

^dB

Thus for an isotropic body,

S{B)= J{/a{u®n+n®u)+X(u-n) 1} da.

dB

Proof. Since B is homogeneous, C is independent of x and

dv==cy^^jEdv\=C[E{B)].

B

1 See footnote 1 on p. 38. See also HlavACek and Ne6as [1970, 2].

^ See p. 85.

Sect. 29. Mean strain and mean stress theorems. Volume change. 97

The first of the relations in j is an immediate consequence of this result, the mean strain theorem (13.8), and (i) of (20.1), while the second follows from (22.2). □

The above theorem shows that the mean stress is zero when the surface displacements vanish, independent of the value of the body force field.

'' (2) Second mean strain theorem.^ Suppose that the elasticity tensor C Js invertible with K = C'^. Let \u, E, S] be an elastic state corresponding to the external fdrcFTysl^'J^, 's]'.~Tlien the mean strain depends only on the external force system and is given by

E(B)= k\ I"p®sda+ I*p®bdv

Thus for an isotropic body,

= fp®^da-\- fp®bdv\

^dB B ' ^

Proof. The first result follows from the mean stress theorem (18.5), since

E{B)=K[S(B)]-,

the second result follows from (22.4). □

Note that balance of forces

/ sda-)- J bdv =0 dB li

insures that the above formulae are independent of the choice of the origin 0, while balance of moments in the form

skw I ( p®s da-\- J p®b dv\=0 las B I

insures that the right-hand side of each of the expressions in (2) is symmetric.

(3) Volume change theorem.^ Suppose that the elasticity tensor C is sym¬ metric and invertible. Let [u, E, S] be an elastic state corresponding to the external force system [b, s]. Then the associated volume change dv{B) depends only on the external force system and is given by

( f p®sda-j- f p®bd-A, Us B >

where K = is the compliance tensor. If,m addition, the bo^ is isotropic, then

dv{B) = I f p ■ s da-\- j p • b dv

Us s

where k is the modulus of compression.

1 Betti [1872, 7], § 6 for the case in which 6 = 0. The terms involving the body force were

added by Chree [1892, 2], who claimed that his results were derived independently of

Betti’s. See also Love [1927, 3], § 123; Bland [1953, 5].

2 Betti [1872, J], § 6) Chree [1892, 2].

liandbuch der Physik, Bd. VI a/2. 7


Proof. Let

Given any tensor A,

1)= f P0S daf p(S>b dv. SB B

tvA=^l-A.

Thus the remark made in the paragraph preceding (13.8), (2), and the S3mimetry of K imply that

(5(B) = (B) tr E(B) = tr K[D] =1 • K[D] = K[1] D,

which is the first result. For an isotropic material, (22.4) and Table 3 on p. 79 imply that

hence the second formula for dv{B) follows from the first and the identity

1 • (o® c) =tr(o®c) =o • c. □

Assume that the body is homogeneous and isotropic. Then, as is clear from (3), a necessary and sufficient condition that there be no change in volume is that

/p • s da-\- f p • b dv =0. SB B

Thus a solid sphere or a spherical shell under the action of surface tractions that are tangential to the boundary will not undergo a change in volume.

Recall from Sect. 18 that the force S3^tem [5, s] is in astatic equilibrium if

/ Qsda-\- f Qbdv = 0, SB B

f pxQs da -I- f pxQb dv = 0, SB B

for every orthogonal tensor Q. We have the following interesting consequence of (2) and (18.10).

(4)^ Assume that the elasticity tensor is invertible. Then a necessary and suffi¬ cient condition that the mean strain corresponding to an elastic state vanish is that the external force system be in astatic equilibrium.

It follows as a corollary oi (4) that the volume change vanishes when the external force system is in astatic equilibrium.

30. The reciprocal theorem. Betti’s reciprocal theorem is one of the major results of linear elastostatics. In_essencej.it,,express.esjthe fact that the underlying system of field equations is self-adjoint. Further, it is essential in estabhshing integral representation theoreins for elastostaGcs. In this section we wiU estabhsh Betti’s theorem and" discuss”some oFTts consequences. We assume throughout that C is smooth on B.

(1) Betti’s reciprocal theorem.^ Suppose that the elasticity field C is sym¬

metric. Let [u, E, S] and [it, E, S] be elastic states corresponding to external force

systems [5, s] and [5, s], respectively. Then

f s-iida-j- fb-udv=fs-uda-i- J b - udv = f S-E dv — f S- E dv. dB B dB B B B

1 Berg [1969, 1}. 2 Betti [1872, 7], § 6, [1874, 7]. See also L6vy [i888, 3]. An extension of Betti’s theorem

to include the possibility of dislocations was established by Indenbom [I960, 3]. For homo¬

geneous and isotropic bodies, an interesting generalization of Betti’s theorem was given by

Kupradze [1963, 77, 73], § I.l. See also Beatty [1967, 2],

Sect. 30. The reciprocal theorem. 99

Proof. Since C is symmetric, we conclude from the stress-strain relation that

S-E = C[E]-E^C[E]-E=S-E.

Further, it follows from the theorem of work expended (18.2) that

js-uda-\-jh-udv = f S- E dv, SB B H

j s -uda-)- J b ■ u dv = f S-E dv, SB B B


Betti’s theorem asserts that given two elastic states, the work done by the external forces of the first over the displacements of the second equals the work done by the external forces of the second over the displacements of the first. The next theorem shows that the assumption that C be S3mimetric is necessary for the validity of Betti’s theorem.

(2)^ Betti’s reciprocal theorem is_ false if C is not symmetric.

Proof. If C is not symmetric, then there exist S3mimetric tensors E, E such that

E-C[E]=i=E-C[E]. Let

u=Ep, S = C[£], 6 = -divS,

u=Ep, S = C[E], 6 = -divS.

Then \u, E, S] and [it, E, S] are elastic states corresponding to the body force

fields 5 and 5 and

^ S-Edv^ j S-Edv. □

The following elegant coroUary of Betti’s theorem is due to Shield and Anderson.^

(3) Suppose that C is symmetric and positive semi-definite. Let [u, E, S] and

[ft, E, S] be elastic states corresponding to external force systems [5, s] and [5, s], respectively. Then

U{E} ^ U{E} provided

J s ■ (u — ti) da-{- J b - {u — ii) dvÔ

f (s—s) -uda-)- J (b — b) -udvÔ. SB B

Thus, if and are complementary subsets of dB,

u ~u on

s = 0 on

0

II on B

tt =0 on

s = 5 on ■5^2

5=5 on B

1 Truesdell [1963, 24], See also Burgatti [1931, i], P- 152.

2 [1966, 23].

T


Proof. By the principle of superposition (28.2), [u — u,E — E,S—S] is an

elastic state corresponding to the external forces [5 —5, s —s]. By (28.1) with

E replaced by E — E,

U{E} = U{E} + U{E-E} + fE-(S-S)dv, B

and by Betti’s theorem,

/ E ■ {S—S) dv= f {b — b)-udv-\- f (s —s) • u da B B dB

= f b ■ (ti —u) dv-\- j s • (u—u) da. B dB

0^

Thus, since U{E — E]^0, we have the inequalities

U{E} ^ U{E} + f {b-b)-udv+ f {s-s) • u da, B dB

U{E}Û{E}-{- fb-(u—u)dv+ Js-(u—u)da. B dB

The remainder of the proof follows from these inequalities. □

Theorem (3) may be called a least work principle, since by (28.3)

. I U{E} ^ U{E} if and only if the work done by the external forces corresponding f) to \u, E, iS] is less than or equal to the work done by the external forces of

^]-

(4)^ Suppose that Cis symmetric and positive semi-definite. Let [u, E, S] and

[it, E, iS] be elastic states corresponding to external force systems [5, s] and [5, s], respectively. Let ..., ,5'’^ be complementary regular subsurfaces of dB, and suppose that ism-eadun = 1.2, ...,N, the restriction op u to is a rigi^displacement of Then

>U{E}Û{E}.

s = s on

f {s—s) da— f px(s—s) da = 0

n = i, 2,..., N

b = b on B

Proof. By (3) it is sufficient to show that

/ = / (s—s)•«(!!«+ / {b —b) • udv =0. dB B

Since s = s on and b—b an B,

N

^==24. In= f (s—s)-uda.

Further, as the restriction of u to each 9’„, n—\, 2, ...,N, is rigid, there exist vectors and skew tensors W^,W2,..., such that

Thus u = W„p-\-v„ on y„.

! (s -s) -I^da-pj (s -s) ■W^.pda,

1 Shield and Anderson [1966, 23].

Sect. 30. The reciprocal theorem.

and since v„ and W„ are constant and (s—s) • = r(s_.s) (x)„] this expression reduces to ’

—'^n'J (s s) da 4- IV„ • f (s —s) <S)p da. (a)

But ■

J (s~s) da= f px(s—s) da = 0. (b)

The second relation in (b) implies that the tensor

^ = / {s — s)iS)p da

is S3nnmetric. Thus, since W„ is skew, W„ - A =0, and it follows from (a) and the first of (b) that I„~0 for each n. Therefore 1=0. □

Theorem (29.3) can also be established using Betti's theorem. Indeed the volume change dv{B) is given by

dv{B) = f tr E dv. B

If we assume that B is homogeneous and define

S = l,

E = K[1],

u =Ep,

then [m, E, S] is an elastic state corresponding to vanishing body forces and we conclude from Betti’s theorem (1) that

J S • E dv = J s • u da -j- f b ■ u dv, B dB B

= Js • Ep da-\- f b • Ep dv. B

S-E=z\-E = irE,

a ■ Ep —E • (a 0p) for any vector a,

and E = K[l] is constant and S3mimetric, it follows that

dv{B) ==K[1] • { Jp0sda-\- f p^b dv]. dB B ’

The foUowing generalization of Betti’s theorem holds even when C is nol symmetric. '. “ .. .

(5) Reciprocal theorem. Let [u,E,S] and [u.E.S] be elastic states cor-)

responding to external force systems [5, s] and [5, s], respectively, and let [u, E, S]

correspond to the elasticity field C, [m, E, S] to the elasticity field C = C^. Then ^

J s-uda->r j b-udv= js-uda-\- fb-udv = f S-Edv= f S-Edv dB B SB B B B

Proof. We will prove that S ■ E =S • E; the remainder of the proof is identical

to the proof of Betti’s theorem (1). Since C = C^,


III. Boundary-value problems. Uniqueness.

31. The boundary-value problems of elastostatics. Throughout the following six sections we assume given an elasticity field C on B, body forces 6 on B, surface displacements m on .5^, and surface forces s on .5^, where and are com-

^ IA j)lei^ntary_ regular subsurfaces of dB. Given the above data, the mixed problem of elastostatics^ is to find an elastic state [u, E, S] that corresponds to b and satisfies the displacement condition

u=u on and the traction condition

s=Sn=s on .5^.

We will call such an elastic state a solution of the mixed problem.'^' When is empty, so that = dB, the above boundary conditions reduce to

u=u on dB,

and the associated problem is called the displacement problem.^ If ,9^ = dB, the boundary conditions become

s=s on dB,

and we refer to the resulting problem as the traction problem.*

To avoid repeated regularity assumptions we assume that:

(i) C is smooth on B;

(ii) b is continuous on B;

(iii) M is continuous on ;

(iv) s is piecewise regular on .5^.

1 There are problems of importance in elastostatics not included in this formulation; e.g.,

the contact problem, studied in Sect. 40, in which the normal component of the displacement

and the”iahgehtial component of the traction are prescribed over a portion of the boundary.

Another example is the Signorini problem [1959, in which a portion of the boundary

rests on a rigid, frictionless surface; but is allowed to separate from this surface. Thus s is

perpendicular to and at each point of either

or u-n = 0, s-nÔ,

u-n<o, s-n = o.

A detailed treatment of this difficult problem is given by Fichera [1963, 6] and in an article

following in this volume.

2 Our definition of a solution, since it is based on the notion of an elastic state (§ 28), does!

not cover certain important problems. Indeed, as is well known, there exist genuine mixed '

, problems (.5^ 0^2 4=0) and problems involving bodies with reentrant corners for which the v

corresponding stress field isunbounded. For such problems the classical theorems of elastostatics,

as presented here, are vacuous. Fichera [1971, 1], Theorem 12.IV has shown that in the case of

the genuine mixed problem with boundary, body force field, C°° positive definite and ^

symmetric elasticity field, and null boundary data the displacement field u belongs to (B) a

and is C°° on B (B) is the completion of C' (B) with respect to the norm defined 1 j by iiM|p=iiM|i+iiPsiir where || • II2 is the usual norm.] It is possible to establish minor ' variants of most of our results assuming only that ueW (B), but this would require analytical

machinery much less elementary than that used here. (In this connection see the article by

Fichera [1971, 7].) Finally, Fichera ^1951, 6], [1953, 11] has_shown that for ê genuine

mixed problem u will not be smooth-on B unless the, boundary data obey certain hypotheses.

® Sometimes called the first boundary-value problem. ^

* Sometimes called the second boundary-value problem. '

Sect. 31. The boundary-value problems of elastostatics. 103

Assumptions (ii)-(iv) are necessary for the existence of a solution to the mixed problem. In the definition of a solution [m, JS, S], the requirement that S be admissible^ is redundant; indeed, the required properties of S follow from (i), (ii), the admissibility of u, and the field equations.

By a displacement field corresponding to a solution of the mixed problem we mean a vector field u with the property that there exist fields E, S such that \u, E, S] is a solution of the mixed problem. We define a stress field corresponding to a solution of the mixed problem analogously.

(1) Characterization of the mixed problem in terms of displace¬ ments. Let u be an admissible displacemeM field. Thenu corresponds to a solution of the mixed problem if and only if

div C[Fm]+6 =0 on B,

u — u on

C[Fm] n = s on 9’,^.

Further, when B is homogeneous and isotropic, these relations take the forms'^'

[j,Au-\-{X+[j) Vddv u-\-b=Q on B,

u — u on

fi[Vu-\-Vu^)n+X(dLWu)n = s on 9^.

Proof. That the above relations are necessary follows from the discussion given at the beginning of Sect. 27. To estabhsh sufficiency assume that u satisfies the above relations. Define E through the strain-displacement relation and S through the stress-strain relation. Then, since u is admissible, E is continuousTJn B, and, by assumption (i), S is continuous on B and smooth on B. In addition,

div/S-1-6=0 on B, (a)

Sn=s on 9^.

Further, (a) and assumption (ii) imply that div S is continuous on B; thus S is an admissible stress field. Therefore [u, E, S] meets all the requirements of a solution to the mixed problem.

When B is homogeneous and isotropic, the last three relations are equivalent to the first three, as is clear from Sect. 27. □

In view of the discussion given at the end of Sect. 27, the boundary condition in (1) on .9^ for isotropic bodies may be replaced by

2/.«MX curl M-|-A (div m) n = s on

(2) Characterization of the traction problem in tei'ms of stresses. Suppose that the elasticity field is invertible with class inverse K on B. Let_ B_be_ simply-connected, and let S be an admissible stress field of class on B. Then S corresponds to a solution of the traction problem if and only if

div /S-|-6 =01 \ on B,

curl curl K [S] = 0 J

Sn = s on dB.

1 See p. 59.

2 Lodge [1955, 9] has given conditions under which the basic equations of elastostatics

for an anisotropic material can be transformed into equations formally identical with those

for an isotropic material. See also Kaczkowski [1955, 6].


When B is homogeneous and isotropic, these relations are equivalent to

div /S + 6 = 0

AS+ -^l^ VV(trS)+2Vb + ~/_^ (div 6) 1=0

Sn=s ondB.

Proof. Let

Then E is of class on B and E = K[S].

curl curl E — 0.

Thus, by the compatibility theorem (14.2), there exists a class C® displacement field u on B that satisfies the strain-displacement relation. Moreover, we conclude from (14.4) that u is of class C® on B.

When B is homogeneous and isotropic, the last three relations are equivalent to the first three, as is clear from the remarks given at the end of Sect. 27. □

32. Uniqueness.^ In this section we discuss the uniqueness question appropriate to the fundamental boundary-vcdue problems of elastostatics.

We say that two solutions \u, E, S] and [it, E, S] of the mixed problem are equal modulo a rigid displacement provided

[u, E, S] = [it E, S],

where is a rigid displacement field.

(1) Uniqueness theorem for the mixed problem.^ Suppose that the ^stiQity_ jidd is positive definite. Then any two solutions of the mixed problem are equal modulo a rigid disfdacement. Moreover, if is non-empty, then the mixed problem has at most one solution.

Proof. Let i = [u, E, S] and J = [it, E, SI] be solutions of the mixed problem, and let

[it. £,«]=[. E, S]-[u.E,S].

Then, by the principle of superposition (28.2), [u, E, SI] is an elastic state corre¬ sponding to zero body forces. Moreover,

it = 0 on .5^,

s = 0 on .5^;

thus, since and are complementary subsets of dB,

s -11=0 on dB,

and we conclude from (28.5) that it is rigid. Thus J5;=SI =0, and the two solu¬ tions 6 and 5 are equal modulo a rigid displacement. 6^^, since it is a regular surface, if non-empty must contain at least three non-coUinear points. Thus, in this instance, since it =0 on .5^, it must vanish identically. □

It is clear from the proof ot (1) that the smoothness assumptions tacit in the definition of a solution are more stringent than required. For the purpose of this theorem it is sufficient to assume that a solution [m, E, S] is a triplet of fields

* This subject is discussed at great length in the tract by Knops and Payne [19/1, 2]. 2 Kirchhoff [1859, T, [1876, 7]. See also Clebsch [i862, 7].

Sect. 32. Uniqueness. 105

on B with the following properties: (i) u, E, and S are continuous on B; (ii) u and /S are differentiable on B; (iii) u, E, and S satisfy the field equations and bound¬ ary conditions.

The last theorem implies that uniqueness holds for the displacement and trac¬ tion problems provided the elasticity field is positive definite. When B is homo¬ geneous, we can prove a much stronger uniqueness theorem for the displacement problem.

^ (2) Uniqueness theorem for the displaeement problem.^ Let B be homogeneous, let C he symmetric, and assume that either C or — C is strongly.dlipiic^TTEen the~dtsplacement problem has at most one sdluiion with smooth displacements on B.

Proof. By the principle of superposition, it suffices to show that 6 = 0 on B and

M=0 on dB (a)

imply u = Qon B. Thus assume that the body forces and surface displacements are zero. Then by (28,3)

2U{E} = J E-C[E]dv=0, (b) B

and, since E = Vu, (ii) of (20.1) implies

2U{E} = f Vu-C[Vu]dv. (c) B

Let us extend the definition of u from B to all of S by defining

M = 0 on B. (d)

Then by (a) u is continuous and piecewise smooth on S, and Vu has discontinu¬ ities only on dB. Thus u and Vu are both absolutely and square integrable over S and possess the three-dimensional Fourier transforms a and A defined on S

by

“= [Lff «(y) dvy. (e)

A (sc) = j(y) dVy, (f)

g

where p{x) =x —0. Using (d)-(f) and the divergence theorem, we obtain

A = —ia®p. (g)

‘ E. and F. Cosserat [1898, 1, 2, 3] proved that strong ellipticity implies uniqueness for

the displacerhenFprobremlh‘aTiomogeneous and isotropic body. (See alsoTTredholm [19067 3],

Bo'SST6~X19077 TJ. and Sherman [1938, 2].) This "result cari'also be inferred from the earlier work of Kelvin [1888, 4']. For a homogeneous but possibly anisotropic body, theorem

follows from the work of Browder [1954, 4] and Morrey [1954, 14'\. The proof given here,

which isTaseTon the work of Wan Hove [l947i 9], was Jvirnished by R. A. Toupin (private

communication in 196I). A sHghtiy more general theorem was established by Hill [1957, 8],

[1961, 12, 73], who noted that uniqueness holds provided 17{u}>0 for all class fields u that vanish on 8B but are not identically zero. Hill’s result is not restricted to homogeneous

bodies. Hayes [1966, 9] established a uniqueness theorem for the displacement problem

under a slightly weaker hypothesis on C, which he calls moderate strong ellipticity. It is

not difficult to prove that moderate strong ellipticity and strong ellipticity are equivalent

notions when the body is isotropic. Additional papers concerned with uniqueness for the

displacement problem are Muskhelishvili [1933, 2], Misicu [1953, 16], Ericksen and

Toupin [1956, 3], Ericksen [1957, 5], Duffin and Noll [1958, 8], Gurtin and Stern¬

berg [i960, 7], Gurtin [1963, 10], Hayes [1963, 12], Knops [1964, 73], [1965, 72, 73],

ZoRSKi [1964, 24], Edelstein and Fosdick [1968, if], Mikhlin [1966. 16].


(i)

(j)

Since B is homogeneous, a fundamental theorem of Fourier analysis^ yields

/ Vu • C[Fu] dv^C^.^jJ dv=Ci^„iJ dv =JA-C[A] dv, (h)

where A is the complex conjugate of A.

If we write

A ~Ajf-\-iAi,

tt == t Cij y then (g) implies that

Ai{=fti®p, Ai =

On the other hand, we conclude from the symmetry of C that

A-CIA] =Ak ■ C -j-A^. c [Aj].

By (c). (d), and (h)-(j),

2U{E}=.J (aji^p)-C[aji^p]dv+J (aj^p) ■ C[a,^p] dv. (k)

T identicaUy on B. Then by the uniqueness of

U{E}>o,

ThusV= ?on'? ^ ^ contradiction if - C is strongly elliptic.

Suppose that the material is hmmme^^ndîSQtropic, andth^n^ the__elasticitx i^^nor itsAeAa&A is

Then there exists a body.B..and.an el JioM

^,^0 on dB y and u is not identically zero. " ^

Proof. By (25.1) we have only three cases to consider:

yM=0, A + 2/t==0, {?i+2p) p<0.

In view of (31.1), it is sufficient to exhibit a body B and a non-trivial class displacement field u on B that satisfies

pAuA(}-Ap) FdivM = 0,

M = 0 on dB.

(a)

(b)

fnrrJlr Assume A + 2^-0. Then (a) is satisfied by any displacement field of the

on sLlh^that ^ ^ ^ T of class

^93=0 on dB.

1 See, e.g., Goldberg [1961, 9], Theorem 13E.

MikhliT[^966, B956, 3], Ericksen [1957, 5]. See also

^ Ericksen and Toupin [1956, 3]. See also Hill [1961, 73].

Sect. 32. Uniqueness.

107

We take B equal to the open ball of unit radius centered at the origin 0, and

(p{x)=r^(r — \Y, r = |£c-0l.

(ii)i Assume ^ =o. Then it suffices to find a displacement field u that satisfies

divit=o

“saSfyllg ^ ‘^^hibit a vector field

T X O X- . curlM> = 0 on BB. /jx Let B be as in (i) and take

■w(x) =~.r^[r~\fe,

where e is a unit vector. Then it is easily verified that w satisfies (d)

(ill) 2 Assume (A + 2^) ^ < 0. Let

u^[x) =Us(x) =0.

Then u satisfies (a) everywhere and vanishes on an ellipsoid. □

is^ckaf fmmThTtu°” « necessary /or the validity of theorem (2) R / I X? counterexample of Edelstein and Fosdick ^ tLv "

take B to be the spherical shell mey

5={a;: 7r<la; —0| <27r}

composed of the inhomogeneous but isotropic material defined by

^ = 4 {> + r}' ^ (»') = 72 - 2fi (r).

^ + 2/^>0, the elasticity field is strongly eUiptic on B They show that the displacement field u defined in spherical coordinates by ^

u, (r) = sin r, Me=My=0,

satisfies the displacement equation of equilibrium with zero body forces and vanishes on the boundary of B (i.e. when r = 71 or r = 2 7i).

(4) Uniqueness theorem for the traction problem for star-shaned bodies. Suppose that the body is homogeneous, isotropic, and Itar-shahed and let the Lame moduli p and A satisfy ana let

( e h ' t O n-v<2. ^ (A+ 2^) ^<0.

^ Ericksen and Toupin [1956, 3], See also Mills [1963 201

constant that depend^ on the shape of 4 body ^feetlfo rimL rTo6^ " example by E. and F Cosserat ri oni ?l ^=00 oi « [1961, J3].) A counter- lack of uniqueness for ^ Ji !. i ’ IVo! also Bramble and Payne [1961, 3]) establishes o Ai ^ T> 1, while counterexamples by E and E r'r>c;QT5'PAT*

h Skbrmjx [„38, 8] ha, .horn that uaiquenras hSdi If ,f7

S: “"’•-'—Pto o<


Then any two solutions of the traction problem with class displacements on B are equal modulo a rigid displacement.

Proof. It is sufficient to show that

6=0 on B, s=0 on dB (a)

imply JS = 0 on B. Thus assume (a) holds. Then (24.5) and the theorem of work and energy (28.3) yield

U{E} = f[fz\E\^+ ^ (tr£)^ dv — 0. (b)

(c) Recall Navier’s equation

Zlu+yFdivM = 0,

y = A+A

Since B is star-shaped, it follows from (5.1) that there exists a point in B, which we take to be the origin 0, such that

p-n>0 (d)

at every regular point of dB. Consider the identity

f {Au->ryVd.vwu\-(Vu) pdv = A f s ■ (Vu) pda ^ U{E} J J ^

B dB

f iP-n) \E\^+ ^-‘ -(divuY dB

(e) da.

which can be estabhshed with the aid of the divergence theorem, the stress equation of equilibrium div S = 0, and the stress-strain relation in the form

E + ~^- (divu) 1 S =2/lc

By (a), (b), and (c), (e) reduces to

J (p-n) |R|2-f ^■^--(divM) da =0.

(f)

(g)

Since n • Sn =0 on 8B, (f) implies

(y —1) divM = —2n-Rn.

Thus, f or y =(= 1, (g) becomes

/(p-n) \EY + ~^-{n-EnY dB

(fa =0.

The inequcdity {X + 2fx) fx<0 implies y< — 1; hence

y + 1

y-i >0.

(b)

(i)

Therefore (h) may be rewritten as

f [p.n)\E-[\ + \IÂ±){n-En) n®n ' da=0,

Sect. 33. Nonexistence. 109

and by (d) this implies

£ = + j/-^ {n-En)n®n

In view of (i) and (j),

and hence by (j) n ■ En=0 on dB,

E = 0 on dB.

on dB. (j)

This implies that div m = tr JS = 0 on dB. But div u is harmonic;^ hence div m = 0 on B, and we conclude from (b) that JS = 0. □

33. Nonexistence. For the traction problem a necessary condition for the s^êxistence of a solution is thatTEe external forces be in equilibrium, i.e. that

/s / 6 dv=Q, dB B

f pxsda-i- f pxb dv =0. SB B

A deeper result was established by Ericksen^, who proved that, in general, lack of uniqueness implies lack of existence, or equivalently, that existence implies uniqueness.^

Suppose there were two solutions to a given mixed problem, and that these solutions were not equal modulo a rigid displacement. Then their difference [u, E, /S] would have EÔ, would satisfy

u — 0 on .5^, s=/Sn = 0 on (N)

and would correspond to vanishing body forces. We call an elastic state [u, E, S] with the above properties a non-trivial solution of the mixed problem uMh null data.

(1) Nonexistence theorem.* Let the elasticity field he syiêtric. Assume that there exists a non-trivial solution of the mixed problem withjnuU data. Theri there exists a continuous body force field b on. $, of class on B with the following property, the mixed problem corresponding to this body force field and to the null boundary condition {NXJiasjw solution.. Further, if is empty, then b can be chosen so as to satisfy

fbdv=0, fpxbdv=0. B B

Proof. Let 3 = [u, E, S] be a non-trivial solution of the mixed problem with null data. Then if d = [u, E, S] is any other elastic state with b the corresponding bodjTfoFce field, Betti’s theorem (30.1) implies

s - uda— J s ■ uda-\- f b ■ u dv, B

where we have used the fact that 3 corresponds to null data. NowTchoose b=u, and suppose that d satisfies the null boundary conditions (NL Tên the above relation implies

/ |ii|^dvÔ, _ _ B

’ This will be proved in Sect. 42.

2 Ericksen [1963, 4], [1965, 7].

2 This result is a direct analog of the Fredholm alternative for symmetric linear operators

on an inner product space.

* The basic idea behind this theorem is due to Ericksen [1963, 4], who established a

theorem similar to (1) for the traction problem and asserted an analogous result for the

displacement problem. Cf. Fichera [1971, 1], Theorem 6.IV (with A = 0).

110 M. E. Gurtin; The Linear Theory of Elasticity. Sect. 34

which yields m = 0, and we have a contradiction. Thus there cannot exist an elastic state 4 with the above properties. ^ ^

Now assume that Sî=0. Clearly, if [u, E, S] is a non-trivial solution of the

surface force problem with null data, then [u+w, E, S] also has this property for every rigid displacement field w. Thus to complete the proof it suffices to find a rigid displacement w such that

j {u-\-w) dv = Q, f p'K(u+w) dv =0. (a) B B

Let c denote the centroid of B and I its centroidal inertia tensor; i.e. if

p„(x) =*-c, then

J p„dv = 0, B

'^=/[|Pc|^l-Pc®Pc] Av. B

Let w denote the rigid displacement field defined by

t«=a + «)XPc,

a = — J udv, (0 =PcXii dv.

B B

Then a simple computation based on the identity

/p„x(<axpc) dv=I(a B

yields j (u-\-w) dv = 0, ^p„x[u+w) dv—0,

B B

which is equivalent to (a). □

The non-existence theorem (1) has the following immediate corollary; If the mixed problem with null boundary data has a solution whenever the body force field is sufficiently smooth, then there is at most one solution to the mixed problem.

IV. The variational principles of elastostatics.

34. Minimum principles. In this section we will establish the two classical minimum principles of elastostatics; the principle of minimum potential energy and the principle of minimum complementary energy.^ These principles com¬ pletely characterize the solution of the mixed problem discussed previously.

We assume throughout that the data has properties (i)-(iv) of Sect. 3I, ^d, in addition, that ..

(v) the elasticity field C is symmetric and positive definite on B.

Let E and S be continuous symmetric tensor fields on B. For convenience, we now write Uc{E}, rather than U{E], for the strain energy :

Uc{E} = ^ ^ E ■ CiE] dv. B

‘ The treatment of the minimum principles (1) and (3), which is based on the notion of

an elastic state, was furnished by E. Sternberg (private communication) in 1959.

Sect. 34. Minimum principles. Ill

Further, we define the stress energy Uk{S} by

C/K{S}=i/S-K[5!](fj;, .. B

where K = C"' is the compliance tensor. In view of the remarks made on p. 84,

Uk{S}Ûc{E}

provided /S = C[JS].

By a kinematicallji admissible state we mean an admissible state that satisfies the strain-displacement relation, the stress-strain relation, and the displacement boundary condition. .\ i

(1) Principle of minimum potential energy.^ Let ^ denote the sd of_ all kinematically admissible states, and Id 0 be the functional on ^ defined by

0{6} — Uc{E} — ( b - udv— J s - uda B

for every d = \u, E, S] ejj/. Further, Id i be a solution of the mixed problem. Then

0{6}^0{~6}

for every and equality holds only if d = d modulo a rigid displacement.

Proof. Let d, d fjj/ and define d' = d — d. (a)

Then d' is an admissible state and

J5:'=J(Fm'+Fu'^), (b)

«' = €[£'], (c)

m' = 0 on ,5^1. (d)

Moreover, /S = C[JS], since thus (28.1) and (c) imply

Uc{E} - Uc{E} = Uc{E'} + JS-E'dv. (e) B

Next, if we apply (18.1) to S and u', we conclude, with the aid of (b) and (d), that

f /S • E'dv — f s ■ u' da — f u' ■ div S dv. B B

1 It is somewhat difficult to trace the history of this and other variational and minimum |

principles, since in the older work the nature of the allowed variations was often left to be 1

inferred from the result. The basic.ideas appeay in the work of Green [i839, I], pp. 253-256, ! Haughton [1849, F, P- 152, Kirchhpff.[i850, i], § 1, Kelvin'[i8^3, 3], §§ 61-62, Donati

[1894, 2]. As^roved_tljeoremitapp.ears,t.o.,hay,efirstbeen.givei,byi'liyBil9Q6lii], § 119for the displacemenF^pblem, by Hadamard [-1903,3], §264, Colonnetti [1912,7J, and Prange [1916,

^], PP- 5154 the traction problem, and by Trefftz [1928, 3], § 18 for the mixed problem. An extension to'exterior domains was given byTJuRTiN and SiSftJfhERG [1961, 11], Theo¬

rem 6.5. Variants of this theorem, which are useful in the determination of bounds on ef¬

fective elastic moduli for heterogeneous materials, are contained in the work of Hashin

and Shtrikman [1962, 9], Hill [1963, 73], Hashin [1967, 3], Beran and Molyneux

[1966, 3], and Rubenfeld and Keller [1969, 6].

For the displacement problem, the hypothesis ât .C^ he positive de^nite is not neceasaty I

for the validity of the principle of minlmunr*^tential energy. This fact is clear from the 1-^

work of Hill [1957, 3], Gurtin and Sternberg [i960, 7], and Gurtin [1963, 70]. >

For various related references, see Prager and Synge [1947, 6], Synge [1957, 75],

Mikhlin [1957, 70], and Oravas and McLean [1966, 79].

M. E. Gurtin ; The Linear Theory of Elasticity. Sect. 34. 112

Thus 0{Z}-0{6}=Uc{ty}- j {^xvS + h)-u’ dv+ S {s-s)-u'da, (f)

B

and since <) is a solution of the mixed problem, (f) implies

0{?}-0{4} = C/c{E'}.

Thus, since C is positive definite,

0{i} = 0{i}Ê' =.= £-£=0.

Moreover, since <) and 4 are kinematically admissible, E=E only when <j=3 modulo a rigid displacement. □

In words, the principle of minimum potential energy asserts that the differ¬ ence between the strain energy and the work done by the body forces and pre¬ scribed surface forces assumes a smaller value for the solution of the mixed problem than for any other kinematically admissible state.

The uniqueness theorem (32.1) for the mixed problem follows as a corollary of the principle of minimum potential energy. Indeed, let o and S be two solutions. Then this principle tells us that

0{<i}^0{S}, 0{6}^0{6}. Thus

0{o} = 0m.

and we conclude from (1) that and S' must be equal modulo a rigid displacement.

By a kinematically admissible displacement field we mean an admissible displacement field that satisfies the displacement boundary condition, and for which div C[Vu] is continuous on B. The corresponding strain field is then called a kinematically admissible strain field. If \u, E, S] is a kinematically admissible state, then m is a kinematically admissible displacement field. Con¬ versely, the latter assertion implies the former provided E and S are defined through the strain-displacement and stress-strain relations. In view of these equations, 0{<i} can be written as a functional 0{u} of u:

0{u} = i f Vu ■ C[Fm] dv— f b ■ u dv — f s ■ u da; B B y’,

therefore we have the following partial restatement of (1)

(2) Let u correspond to a solution of the mixed problem. Then

0{u} ^ 0{u}

for every kinematically admissible displacement field u.

By a statically admissible sti^s field we mean an admissible stress field that satisfies the equation of equihbrium and the traction boundary condition.

(3) Principle of minimum complementary energy.^ Let .s/ denote the set of all statically admissible stress fields, and let 0 be the functional on s/ defined by

W{S} =U^{S} -fs-uda _ .5^.

1 The basic ideas appear in the work of Cotterill [1865, 1] and Donati [1890, Z],

[1894, 2]. As a proved theorem it seems to have first been given by Colonnetti [1912,1] and France [1916, 1] for the surface force problem and by Trefftz [1928, 3], § I9 for the

mixed problem. See also Domke [1915, f].

Sect. 34. Minimum principles. 113

for every Let S he a stress field corresponding to a solution of the mixed problem. Then

for every Ssj/, and equality holds only if S=S.

Proof. Let [u, E, S] be a solution of the mixed problem, let and define

S' = S-S. (a) Then S' satisfies

div S' = 0 on B, (b)

s' =S n—0 on 5^2•

Since E = K[S], (a) and an obvious analog of (28.1) imply

«^K{S}-C/K{-S} = C/K{S'} + /S'./<:rft;. (c) H

Further, in view of (b) and (18.1),

f S' ■ E dv = f s' ■ u da. B y’,

Thus, since M = M on ,

W{S}~W{S}Û^{S'}.

Therefore, since K is positive definite, !F{S} ^ and ¥^{S} = 1F{S} only if

S'=S-S=0. □

The principle of minimum complementary energy asserts that of all statically admissible .stress fields, the one belonging to a solution of the mixed problem renders a minimum the difference between the stress energy and the work done over the prescribed displacements.

We shall now use the foregoing minimum principles to establish

'^/(4) Upper and lower bounds for the strain energy.^ Let U be the strain energy associated with a solution of the displacement problem. Assume that the body forces vanish. Then

/ s • H da - 17k{S} ^ 17 ^ Uc{E}, dB

where E is a kinematically admissible strain field, S a statically admissible stress field, and s the corresponding surface traction.

On the other hand, let U ^ the strain energy corresponding to a solution of the traction problem. Then

fb-udvA-fs-u da — Uc{E} ^ 17 ^ U,({S}, B oB

where u is a kinematically admissible displacement field, E the corresponding strain

field, and S a statically admissible stress field.

Proof, Let 6 — [u, E, S] be a solution of the displacement problem with b =0, and let U be the associated strain energy:

U = Uc{E} = U^{S}.

^ Cf. Aymerich [1955. t] who obtains upper and lower bounds on the strain energy by

embedding the body in a larger body.


H4 M.E. Gurtin; The Linear Theory of Elasticity. Sect. 34.

Then, since = 0, we conclude from the principle of minimum potential energy that

U=Ûc{E}Ûc{E}.

On the other hand, since S^ = dB, it follows from the principle of work and energy (28,3) that the last term in the expression for W{S} in (3) is equal to — 2U. Thus

and (3) implies

y/{S} = _C/,

f s-uda — UidS}. 8B

Next, let 4 be a solution of the surface force problem. Then 5? = 0, and the principle of minimum complementary energy 5delds

UÛ^{S}.

Moreover, since S^ = dB, we conclude from (28.3) that the last two terms in the expression for 0{4} in (1) are equal to —2U. Thus

0{4} = -17, and (1) implies

fb-udv-]-fs-uda — Uc{E}. □ B dB

Using (18.2) we can write the inequalities in (4) in the following forms:

fS-{E-^K[S]}dvÛÛc{E}, B

f E ■ {S-^C[E]} dvÛ^ Uk{S}. B

By a statically admissible state we mean an admissible state [u, E, S] with S a statically admissible stress field. For convenience, we now define W on

j;k<AUl statically admissible states <i = [m, JS, <S] by writing W{<i} = W{S}. ‘ Clearly, an admissible state 4 is a solution of the mixed problem if and only

• i if 4 is both kinematically and statically admissible. We shall tacitly use this fact 1 in establishing the next theorem, which furnishes an interesting relation between Uhe functionals 0 and W.

(5)1- et 0 be a solution of the mixed problem. Then

0{4} + !F{4}=O.

Proof. In view of the definition of 0 and W,

0{i} +!F{4} = Uc{E} -\-Uk{S}— fb-udv — fs-u da — fs-u da B y’l se,

= 2Uc{E}— f b ■ u dv — f s ■ uda, B 8B

and the desired result follows from the principle of work and energy (28.3). □

If we assume that the mixed problem has a solution, then the preceding theorem has the foUovnng corollary.

(6) Let 4 and 4 be admissible states with 4 kinematically admissible and 4 stati¬ cally admissible. Then

0{4}+!F{4}Ô.

Sect. 35. Some extensions of the fundamental lemma. 115

Proof. Let o be a solution of the mixed problem. Then the principles of mini¬ mum potential and complementary energy imply that

Thus

35. Some extensions of the fundamental lemma. In this section we shall establish three lemmas whose role in elastostatics is analogous to the role of the fundamental lemma in the calculus of variations. Let 3S(£Co. denote the intersection of dB with the open ball Z,, (apj) of radius h and center at Xq , and let î,{x^ denote the set of functions defined in Sect. 7.

(1) Let W he a finite-dimensional inner product space. Let w: he piecewise regular and satisfy

Jw -v da =0

for every class C°° function v: B-->Wthat vanishes near Then

ic=0 on

Proof. Let e,, Cg,..., e„ be an orthonormal basis for W, and let

n

M)= 2 »=i

Assmne that for some regular interior point x^ of and some k, (Xq) > 0 (say). Then there exists an h>0 such that dB{Xo,h) is contained in .5^ and w^>0 on dB(Xg,h). Let v = (pe^ with (p€^>,,[x^. Then v is of class C°° and vanishes near In addition,

/ w ■ V da>0, •51

which is a contradiction. Thus w{x^ =0 at every regular interior point x^ of .$2• But since w is piecewise regular, this implies = 0 on .5^. □

(2) Let u he a piecewise regular vector field on and suppose that

f (Sn) -udaÔ ■51

for every class C°° symmetric tensor field S on B that vanishes near Then

u = 0 on

Proof. Since S is symmetric,

(Sn) • u=S • (u0n) =S - T,

where T is the symmetric part of u^n :

T = ^(M(g)nH-n(8)M). Therefore

fS-Tda=0

whenever S satisfies the above hypotheses. Consequently, letting iP" in (1) be the space of all symmetric tensors, we conclude from (1) that

T = 0 on

8*


and hence that 2Tn=M+n(u • n) =0 on

Taking the inner product of this relation with n, we find that M • n=0 on 5^, and this fact and the above relation imply m = 0 on □

(3) Let u be a piecewise regular vector field on and suppose that

f u ■ divS da=0

for every class C°° symmetric tensor field S on B. Then

u= 0 on

Proof. Let v be an arbitrary class C°° vector field on B which vanishes near Then by (3.5) there exists a class C°° vector field g on B with the property

that Aq =v.

Let^ S=Pgr + Pgr^-ldivgr.

Then <S is a class C°° symmetric tensor field on B, and hence

/ u ■ divS da=0.

But a simple calculation yields

div<S = ^dgr =». Thus

f u ■ V da=0

for every class C°° vector field » on 5 which vanishes near 6^^, and the desired result follows from (1). □

36. Converses to the minimum principles. In this section we shall use the results just established to prove converses to the principles of minimum potential and complementary energy. We continue to assume that hypotheses (i)-(v) of Sects. 31 and 34 hold. Our first theorem shows that if a kinematically admissible state minimizes the functional 0, then that state is a solution of the mixed problem.

^ (1) Converse of the principle of minimum potential energy. Let i be a kinematically admissible state, and suppose that

for every kinematically admissible state 3. Then 6 is a solution of the mixed problem.

Proof. Let u' be an arbitrary vector field of class C°° on B, and suppose that u' vanishes near Further, let o' = [«', E', S'], where

E' = ^{Vu' + Vu''^),

S'=C[E'].

Then 3 = o + <j' is kinematically admissible, and it is not difficult to verify that (f) in the proof of (34.1) also holds in the present circumstances. Thus, since

0{o}^0m,

0 ^ U(;{E'} — f (div S + b) ■ u' dv-j- f (s — s) -u' da. B

1 Cf. the remarks following (17.8).

Sect. 36. Converses to the minimum principles. 117

Clearly, this relation must hold with u' replaced by aw' and E' by olE'] hence

0 â?Uc{E'} —a / (div <S +b) • u' dv-l-oc f (s — s) • u' da B

for every scalar a, which implies

- / (div S -\-h) • u' dv f (s — s) • w' da=0 B

(a)

for every C°° vector field ti' that vanishes near S/^. If, in addition, u' vanishes near dB, then .,,. „

/ (divS + b) ■ u dv=0, B

and we conclude from (7.1) with tCthat

diV)S + b=0. (b) By (a) and (b), .,

^ ^ ^ ’ /(•** —s) -u' da=0

for every C°° field u' that vanishes near and (35.1) with if ='f~ yields

s = s on Si’^. (c)

Thus 4 is a kinematically admissible state that satisfies (b) and (c); hence o is a solution of the mixed problem. □

Note that (1) does not presuppose the existence of a solution to the rnixed problem. If oneTchows a priori that a solution 4 exists,^ then it follows from (34.1) tliat any kinematically admissible state 4 that minimizes ^ must be equal to 4, and hence must be a solution. Indeed, if 4 minimizes 0, then

0{4}^0{<i}; but (34.1) implies

0{4}^0{4}, so that

0{4}=0{4},

and we conclude from (34.1) that 4=4 (modulo a rigid displacement).

The next theorem yields a converse to the principle of minimum complemen-. tary energy. prove.this theorem we need to assume.that .B is .simple-connected ^ and convex with respect to (when In view of the discussion given in[ the preceding paragraph, if B and the boundary data are such that existence holds for the corresponding mixed problem,^ then this converse follows trivially without the above assumptions concerning B.

(2) Converse of the principle of minimum complementary energy.^ Assume that B is simply-connected and convex with respect to and that K is of class on B. Let S be a statically admissible stress field of class on B, and suppose that ^

'P{S}^W{S}

for every statically admissible stress field S. Then S is a stress field corresponding to a solution of the rnixed problem.

Before proving this theorem we shall establish two subsidiary results; these results are not only basic to the proof of (2), but are also of interest in themselves.

^ Existence theorems for the mixed problem are given, e.g., by Fichera [1971, 1], § 12.

^ The basic ideas behind this theorem are duetto Cotterit.t. [1R6'; 7], Donati [1890, 7],

[1894,2], Domke [1915, ^]. Southwell [I936r3]r[1938, 5], Locatelli [1940,3,4], Dorn and ScHiLD [1956, f]. As a proved theorem it seems to have first been given by Sokolni-

KOFF [1956, 72] for the traction problem and by Gurtin [1963, 77] for the mixed problem.

liS M.E. Gurtin: The Linear Theory of Elasticity. Sect. 36.

(3)^ Let w he a continuous vector field on a regular subsurface S/’ of dB, and let B be convex with respect to Further, suppose that

f {Sn) ■ wda = 0 se

for every class C°° symmetric tensor field S on B that vanishes near dB—£^ and satisfies

div S = 0.

Then w is a rigid displacement field. o

Proof. Let x and x be regular points of £T, and choose a cartesian coordinate system such that the coordinates of x are (0, 0, 0) and those of x are (0, 0, %). Let be the closed disc in the %, A:2-plane with radius e > 0 and center at (0, 0), and let be a class C°° scalar field on the entire x^-plane with the following properties:

(a) /eÔ;

iP) /e=0 outside D,;

(y) ffeda = \.

Such a function is easily constructed using the procedure given in the first para¬ graph of Sect. 7. Clearly, the symmetric tensor field S on E defined by

[S{x)] =

0

0

0

0 0

0 0

0 fs(Xi, ^2).

has zero divergence. Let be the infinite solid circular cylinder whose axis coincides with the %3-axis and whose cross-section is D„. Then the assumed con-

Fig. 10.

^ Gurtin [1963, 11].

Converses to the minimum principles. Sect. 36. „

' ■ ■ 1^9

vexity of B with respect to 5” and the regularity ot 3- imply that tor all sntli-

amtly small . there exist disjoint subregions and 3, of such that (see

Thus S vanishes near dB — ,5^ and

0 = =/ (Sn) -wda^J I n^da+ f f^w,n,da.

Next, by property (y) of /,,

r

(a)

-/ fgda = — 1. .^n

or equivalently that W3(*) —w^{x) =0,

[w(®) -w(x)] ■ [*-*] =0.

^rWâs theprlca'S^f'm ®'£eov2:

SSi □ * >»-* “‘i t-ot we conclude troiVS tha°»l

HM ,s C0.„„„„s B a.i 0,

/ (Sn) ■uda= f S ■ E dv ^ B

iZhfils field Sons that vanishes near dB and

div iS = 0.

Then thêxists an admissible displacement field u such that

I E:=^(Vu + Vu^), {

'i u~u on Sf’.

o" B that

¥

fS-Edv=^0 B

Thus we conclude from Donati’s theorem

theofem ^ of compatibility. In view of the compatibility

^^ftence°fan;^^^^

presenrmore grerS Tie“^s [^956, i]. The


strain-displacf^ment relation. Thus we conclude from the present h5ôtheses and (18.1) that for every which vanishes near dB — S^,

f (Sn)-u'da= f {Sn) ■ u'da= f S ■ E dv= f (Sn) -uda. Se dB B Sf’

Therefore if we let

then w = u — u',

f (Sn) ■ w da—0 se

for every that vanishes near dB ~S^. Thus we may conclude from (S) that w is the restriction_to ^ of a rigid displacement field w, and the displacement field u defined on B through

u =u' +iii

has all of the desired properties. □

We are now in a position to give the

Proof of (2). Let S'e& vanish near 5^. Then )S=<S+<S' is a statically admissible stress field, and

'P{S}^W{S}. (a) Let

E = K[S].

Then (c) in the proof of (34.3) holds in the present circumstances, and we con¬ clude from this result, (a), and the definition of W given in (34.3) that

0^'F{§}-'f^{S} = 17K{S'}-h f S' -Edv- f s' -uda. (b) B

The inequality (b) must hold for every S'e^ that vanishes near If we replace S' in (b) by olS', where a is a scalar, we find that

f S' ■ E dv = f s' ■ ii da B y;

for every such field S'. Thus we conclude from (4) that there exists a vector field u such that \u, E, S] is a solution of the mixed problem. □

<^^37. Maximum principles.^ The principles of minimum potential and complemen¬ tary energy can be used to compute upper bounds for the “energies” (^{0} and lF{tf} corresponding to a solution <1. In this section we shall establish two maximum principles which allow one to compute lower bounds for (^{0} and

We continue to assume that the hypotheses (i)-(v) of Sects. 31 and 34 hold.

As our first step, we extend the domain of the functionals 0 and W to the set of all admissible states in the obvious manner:

0{<,} = Uc{E} —fb-tidv — fs-u da, B y’j

= Uk{S} -fs-uda

for every admissible state <i = [u, E, S].

^ underlying these priiwiples are dijfi_t(3_TREFFTZ [1928, 4], who estab- Uished analogous results for boundary-value problems associated with the equations Au = 0

" ^ând AAu = 0. See also Sokolnikoff [1956,72], § 118 and Mikhlin [1957, JO], §§55, 57, 59.

Sect. 37- Maximum principles. 121

(1) Principle of maximum potential energy.^ Let 6 be a solution of the mixed problem, and let 3 be an elastic state that satisfies

f s ■ (u —u) daÔ,

s = s on Then

and equality holds if and only if 4 = 3 modulo a rigid displacement.

Proof. Let o'= 4 —3.

Then (28.1) and the fact that S = C[E] imply

Uc{E} - Uc{E} = Uc{E'} + fS-E' dv. B

Next, if we apply the theorem of work expended (18.2) to S and u', we find that

f S ■ E' dv — f b ■ u' dv f s ■ u' da. B B dB

Thus, since u = u on and s = s on ,5^,

0{4} - 0{3} = Uc{E'} + f~s-{u-u)da.

This completes the proof, since C is positive definite and the last term non¬ negative. ■ □

It is clear from this proof that (1) continues to hold even when o is required only to be kinematically admissible.

Note that in the principle of minimum potential energy the admissible states were required to satisfy, in essence, only the displacement boundary condition. On the other hand, the “admissible state” 3 in the principle of maximum potential energy is required to satisfy aU of the field equations, the traction boundary condition, and a weak form of the displacement boundary condition.

(2) Principle of maximum complementary energy. Let i be a solution of the mixed problem, and let 3 be an elastic state that satisfies

u = u on

f (s — s) ■ u da'^0.

Then 'F{4}^'F{3},

and equality holds if and only if 4 = 3 modulo a rigid displacement.

Proof. Let 4'= 4 — 3;

then 4' is an elastic state corresponding to zero body forces. As before,

Uk{S} - U„{S} = U„{S'} +f S'. Edv, B

^ COOPERMAN [1952, i].


and by (18.1) and the fact that div S' =0,

fS'-Edv = fs'-uda. B 0B

Thus, since u = ii on £/[ and s = s on ,

= U^{S'] + / (s -s) • u da.

which, in view of our h5T5otheses, implies the desired result. □

Note that (2) remains valid under the weaker hypothesis that i be statically

admissible.

38. Variational principles. The admissible states appropriate to the minimum principles of elastostatics are required to meet certain of the field equations and boundary conditions. In some applications it is advantageous to use variational principles in which the~aHm5siWe states satisfy as few constraInti~as~possT61e. IiTtliis'secfidh we eitablisli itwo such principles.^

We continue to assume that the data has properties (i)-(iv) of Sect. 31, but in place of (v) of Sect. 34 we assume that

(v') the elasticity field C is symmetric.

The two variational principles will be concerned with scalar-valued functionals whose domain of definition is a subset of the set of all admissible states. Let A be such a functional, let

i and 3 be admissible states, (a)

6for every scalar A,

and formally define the notation

dj' A{<i} = /l{<)-|-A3}|;i=o-

Then we vTite dA{i}=0

if dj/t{(i} exists and equals zero for every choice of.ljcoft^tent-wit-h (a).

We begin with a variational principle in which the admissible states are not required to meet any of the field equations, initial conditions, or boundary conditions.

(1) Hu-Washizu principle.^ Let denote the set of all admissible states, and let A be the functional on defined by

A{i) = Uc{E}-jS-Edv- j {divS^b)-udv B B

-|- f s • udaA- f (s — s) -uda _. . ^ -51 -51

* The basic ideas behind tJiese.principles are contained in the work of Born [1906, 1], pp. 91-^, who established analog,ous_results. foj t.he„plane elastica-.Born was cognizant of

the fact thafTilsTesults applied to elasticity theory, as is clear from bis statement (p. 96):

,,Ich will hier bemerken, da6 sich der Vorteil dieser Darstellung eigentlich erst zeigt, wenn

man sie auf die allgemeine Elastizitatstheorie anwendet." See also Oravas and McLean

[1966, 19], pp. 927-929 for a detailed study of the early historical development of these

principles. 2 Hu [1955, 5],tVASHizu [1955,74], [1968,75], § 2.3. See also de Veubeke [1965,S],Tonti

[1967, IS, 77], and HlAvaCek [1967, 6, 7]. An extension valid for discontinuous displacement

and stress fields was given by Prager [1967, 77], and one in which the stress fields of the

admissible states are not required to be symmetric was given by Reissner [1965, 75].

Sect. 38. Variational principles. 123

for every i = [u, E, S] Then bA{i) = Q

at an admissible state i if and only if si is a solution of the mixed problem.

Proof. Let si = [u, E, S] and 5 = [it, E, S] be admissible states. Then si + A 3 cj/ for every scalar A, and in view of (28.1) and the symmetry of C,

Uc{E + ?.E} = Uc{E} + A2 Uc{E} +XfE-C[E]dv. B

Thus, since

(5j/l{si} = -^-^/l{si +A3}|^ô.

it follows that

(57/1{si}=/{(C[£:]-S) ■ E-(div S + b) ■ u-S ■ E -u ■ div §} dv B

+ j s ■ ii da-\- /{s-M + (s — s)-M}(fa.

If we apply (18.1) to S and u, we find that

/ u ■ div S dv= f u -sda— f J(Fu + Fu^) ■ S dv; B SB B

thus

d~,A{i}= S [ClE'l-S) ■ E dv - p [div S d-h) -udv B B

+ / {^(Pm + Fu^) —.E} ■ sS (fw + / (u —m) ■ s tia (a) B 5^,

+ f (s — s) ■ u da.

If si is a solution to the mixed problem, then (a) yields

(5yyl{si}=0 for every 3 (b) which implies

dA{i}=0. (c)

To prove the converse assertion assume that (c), and hence (b), holds. If we choose 3 = [m, 0, 0] and let u vanish near 8B, then it follows from (a) and (b) that

/ (div -S + b) -u dv=0. B

Since this relation must hold for every such u of class C* on B, we conclude from the fundamental lemma (7.1) that diviS + b=0. Next, let 3 = [it, 0,0], but this time require only that u vanish near Then (a) and (b) imply that

f (s — s) ■ u da—0,

and we conclude from (35.1) that s—s on Now, let 3 = [0, E, 0] and suppose

E vanishes near dB. Then by (a) and (b).

/ (C[E]-S)-Edv=0. B

Thus, since C [E] — S and E are symmetric tensor fields, it follows from (7.1) with iP equal to the set of all symmetric tensors that /S = C[E]. In the same

124 M. E. Gurtin; The Linear Theory of Elasticity. Sect. 38.

manner, choosing S = [0, 0, S], where S vanishes near dB, we conclude that

E = ^(Vu-\-Vu'^). Finally, if we drop the requirement that S vanish near dB, we conclude from (a) and (b) that

f (m —u) • {Sn) da =0

for every class symmetric tensor field S on B, and it follows from (35.2) that u=u on Thus i = [u, E, S] is a solution of the mixed problem. □

(2) Hellinger-Prange-Reissner principle.^ Assume that the elasticity field is invertible and that its inverse K is smooth on B. Let- denote the set of all admiîble sta^ that satisfy the strain-displacentent relation, and let © be the func¬ tional on defined by ~~

©{i} = Uk{S} — f S ■ E dv f b ■ udv f s ■ {u—u) da -j- f s ■ u da B B se, y,

for every i = [m, E, Then d©{i}=0

at if and only if i is a solution of the mixed problem.

Proof. Let i = [«, E, S] and ~6 = [u, E, S] be admissible states, and suppose that for every scalar A, or equivalently that <s, Jejs/. Then, in view of the symmetry of C, K is symmetric and

dy 0{o} = / {(K [S]-E)-S-S ■E-(-b-u}dv B

+ / {s ■ (u—u) +s • u) da-\- f s ■ uda. se, se.

If we apply (18.1) to S and u and use the fact that E and u satisfy the strain- displacement relation, we find that

f S ■ E dv = f s ■ uda — f u ■ div S dv, B dB B

thus

d^©{i} = f (K[S]-E)-Sdv-(- f (divS-\-b)-udv 6 B

-j- f (u — u) ■ s da -j- f (s —s) ■ u da. (a)

If <) is a solution of the mixed problem, then, clearly,

d^©{<i}=0 for every (b)

which imphes d©(d) =0. On the other hand, (a), (b), (7.1), (35.1), (35.2), and the fact that i satisfies the strain-displacement relation imply that <) is a solution of the mixed problem. □

Let i = [u, E, /S] be kinematically admissible. Then we conclude from (18.1) that

/ u • div S dv = f s ■ uda-\- f s ■ u da — f S • E dv. B se, B

* The basic idea is contained in the work of Hellinger [1914, 1]. As a proved theorem it was first given by France [1916, 1], pp. 54-57 for the traction problem and Reissner

[1950, 10], [1958, 14], [1961, 75] for the mixed problem. See also RUdiger [I960, 10], [1961, 19], HlAvaCek [1967, 6, 7], Tonti [1967, 16,17], Horak [1968, 5], Solomon [1968,12].

Sect. 39. Convergence of approximate solutions. 125

Thus, in this instance, A{<s}, given by (1), reduces to (?{<)}, where 0 is the func¬ tional of the principle of minimum potential energy (34.1). Also, when 6 is kine¬ matically admissible.

^JS-Edv-=Uc{E} = U„{S}.

and &{i} given by (2) reduces to -0{i}. On the other hand, if 4 is a statically admissible state that satisfies the stress-strain relation, then /!{<)} reduces to

^{**}> where 0 is the functional of the principle of minimum complementary energy (34.3); if ii is a statically admissible state that obeys the strain-displace¬ ment relation, then ©{i} reduces to 0{i}.

Table 5 below compares the two variational principles established here with the minimum principles discussed previously. .~ ..

Table 5.

Principle Field equations satisfied

by the admissible states Boundary conditions satisfied by the

admissible states

Minimum potential energy Strain-displacement stress-strain

Displacement

Minimum complementary energy Stress equation

of equilibrium Traction

Maximum potential energy All Traction, weak form

of displacement Maximum complementary energy All Displacement,

weak form of traction Hu-Washizu None None

Hellinger-Prange-Reissner Strain-displacement None

39. Convergence of approximate solutions. Recall that the functional of the principle of minimum potential energy (34.2) has the form

0{u) = U{u}—j h-udv— ( s-uda, (a) , By, ' '

where

C/{u}=i-/ Fm- C[Fu] dv B

is the strain energy written, for convenience, as a functional of the displacement field.

The standard method of obtaining an approximate solution to the mixed problem is to minimize the functional 0 over a restricted class of functions.i That is, one assumes an approximate solution in the form

N

“v = Mat + Z /»> (b) n=l

where /i,/2, ...,/^ are given functions that vanish on 5^, and is a function

thatjpjxQximates the bound^y data-W pn_^. Of course, the term^^v is omitted when is empty. The constants aj^, a2,..., are then chosen so as to render

a minimum. Indeed, if we write

_ _ aa, ..., a^,)

* This idea appears first in the work of Rayleigh (1877) [1945, S] and Ritz [1908, 2] • the method is usually referred to as the Rayleigh-Kitz method.

^26 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 39.

where Uf^ is given by (b), then (assuming that C is symmetric)

N N

aj, ..., a;v) =« + 2 2 ^mn^m^n+ 2 m, »=1 n=l where

D.nn=JVL-C[Vf„]dv, (C)

a — U{uj^} — J b ■ Uf^dv — f s • u^f, B

< = / Vuj, ■ C[F/„] dv-Jb-f„dv-fs-f„da. (d) B B 4

If is empty, the above relations still hold, but with = 0, so that

a=0,

d„ = - fb-f„dv~ fs-f„da. (e) B SB

If C is positive semi-definite, the matrix [£*„„] will be positive semi-definite, and 0(ai, aj, a^) ^ be a niinimum at ai = ai, .... = if and only a 0.2,, a,Y is a solution of the following system of equations:

2 ^mn “ I» 2, ..., JVj . (f) »=1 ' '

will now estabUsh conditions under which solutions of (f) exist, and under which the resulting approximate solutions converge in energy to the actual solution as 2V->’o6. ''. .. . ..—

We assume for the remainder of this section that C is symmetric and positive definite, and that hypotheses (i)-(iv) of Sect. 3i hold.

We wnte for the set of all continuous and piecewise smooth vector fields on B. -

(1) Existence of approximate solutions. Let ^^be an N-dimensional subspace of with the property that each vanishes on .

(i) If let Un be a given function in , and let be the set of all functions f of the form

/=sr+MAr.

(ii) If = 0, let

Then there exists a function such that

^ 0{v] for every (g)

If ^1 + 0. then Uj^ is unique-, if STi=^, then any two solutions of (g) differ by a rigid displacement. Finally, Uj^ is oîmal in the. following.sense: If u is the displace¬

ment field corresponding to a solution of the mixed problem, then^ ' .

U{u — = inf U{u—v}. (h)

Proof. Let/j^,/2, ■■■>fN f*® a basis for and let and d^ be defined by (e), (d), and (e). Then, clearly, to estabhsh the existence of a solution Uj,] of (g) it

suffices to establish the existence of a solution oti, otg,..., a^v of (f) •

» Cf. Schultz [1969, 7], Theorem 2.4.

Sect. 39. Convergence of approximate solutions. 127

Since C is positive definite, we conclude from (c) that

N

y D a„^0

for any iV-tuple (ai, a2,..., a^v). Assume that

and let

N

m,

/= »=i

Then it follows from (c) and (j) that

f\^f.C[Ff]dv = 0.

(i)

(j)

(k)

and since C is positive definite, this, in turn, implies that / is rigid. Assume first that Then /s=0, since it is rigid and vanishes on Consequently, as /j, /g, ..., /^ is a basis for , this implies that

ai = a2==a;v=0. (1)

Therefore, if ,S^=j=0, then (j) impUes (1), and we conclude from (i) that the matrix ( [D^„] is poative definite i hence in this instancenff^haTa unique solution.

Assume next that = 0. Suppose that

' n*l

then (i) holds, and hence / defined by (k) is rigid. Thus any two solutions of (g) (if they exist) differ by aTigiff displacement Next, by (e) and (k),

Zo^»d„ = -fb-fdv-fs-fda. n=l B dB

and it follows ^ from (18.3) that N

n=l I Thus [d^, d^,.... dj^) is orthogonal to every solution («i, a.2, of the homo- j geneous equatiOB (m),-and. we conclude from the Fredholm alternative that the ( inhomogeneous equation (f) has a solution.

We have only to show that is optimal. To facilitate the proof of this

assertion, we now establish the following

(2) Lemma.^ Let [u, E, SJ be a solution of the mixed problem. Then

0{v} — 0{u] — U{v —u) +s ■ (v ~u) da

for every

Proof. Let ^ u' = v—u, E'=Vu'.

1 Here we tacitly assume that (when =0) the external force system [6, s] is in equi¬ librium.

2 Cf. Tong and Pian [1967, 15], Eq. (2.16).


Then by (28.1), U{v} - U{u} = U{u'} + fS-E’dv.

b

Further, since div S + b = 0, we conclude from (18.2) that

f S ■ E' dv= f s ■ u' da+ f b ■ u' dv. B d'B B

The last three equations and (a) imply the desired result, since u=u on 6^ and s = s on . □

It is of interest to note that the principle of minimum potential energy in the form (34.2) follows as a direct corollary of this lemma. Indeed, if v is kinematically admissible, then » =u on and (2) implies that 0{v]'^ 0{u}.

We are now in a position to complete the proof of (1). Assume that

By hypothesis, every satisfies

V =Uf^ on

Thus, since it follows from (2) and (g) that

U{uj^ — u} = — t?{u} — f s ■ {Uf^ — ii) da

^0{v} — 0{u} — J s ■ (uj^—u) da = U{v—u}

for every Thus i[7{ujv — u} ^ U{v — u},

which implies (h). The proof when ^ = 0 is strictly analogous. □

We now estabhsh the convergence of the approximate solutions established

in (1). Thus for each JV=1, 2, ... let and satisfy the hypotheses of (1). We assume that every sufficiently smooth function that satisfies the displacement boundary condition can be approximated arbitrarily closely in energy by a

sequence of elements of , ^2 < • • • • More precisely, we assume that

(A) given any kinematically admissible displacement field u, there exists a

sequence {»;v} with such that

U{u—Vj^}^0 as Nôo. (n)

(3) Approximation theorem.^ For each N=\,2,... let and satisfy the hypotheses of (1), and assume thptJAfihqld.s. Let {u^y} belTsequence of approx¬ imate solutions, i.e. solutions of (g), and let u he the displacement field correspond¬ ing to a solution of the mixed problem. Then

{/{u—as 00. (o)

1 Most of the ideas underlying this theorem are contained in the following works; Courant

and Hilbert [1953, 5j, pp. 175-176; Mikhlin [1957, JO], pp. 88-95: Friedrichs and Keller

[1965, 5]; Key [1966, 77]; Tong and Pian [1967, 15]. These studies contain general results

on the convergence of the Rayleigh-Ritz and finite element methods.

In the usual applications of the Rayleigh-Ritz procedure

^n(-^N+i- (*)

The abstract formulation given here also includes the finite element method (see, e.g., Zienkie-

wicz and Cheung [1967, 19], Tong and Pian [1967, 75], Zienkiewicz [1970, 4]) for which

(*) is not necessarily satisfied.

Sect. 40. Statement of the problem. Uniqueness. 129

Proof. By (A) there exists a sequence {»jv} with such that (n) holds. On the other hand, (h) and the fact that C is positive definite imply

0^ U{u—Ui^] ^ U{u—Vi^},

and the desired result is an immediate consequence of (n). □

V. The general boundary-value problem. The contact problem.

40. Statement of the problem. Uniqueness. In the contact problem of elasto- staticsi the normal component of the displacement and the tangential component of the surface traction are prescribed over a portion of the boundary:

u-n=u and s —(s-n)n=s on .$3,

where u and s are prescribed with .s tangent to .$3. These relations can also be written in the alternative forms:

{n®n) u=un and (1—n®n)s=s on

We now generalize this boundary condition as follows. We assume given a tensor field P on 8B whose value P{x) at any x<idB is a perpendicular projection. The generalized boundary condition then takes the form:

Pu—u and (1—P)s=s on dB,

where u and s are prescribed vector fields. Since ^

(1-P)P=P(1-P)=0,

u and s must satisfy the consistency condition

(1—P)ii = 0 and P.s = 0 on dB.

At a point x for which P{x) =1 we have u{x) =u{x), and no restriction is placed on s{x). Thus displacements are prescribed over the subset of dB on which P = 1; similarly, surface tractions are prescribed over the subset for which P = 0. The mixed problem therefore corresponds to

P = 1 on P = 0 on .9^,

where 6^ and 6^2 complementary subsets of dB.

If P=n®n at a point, then the normal displacement and the tangential traction are prescribed. Thus the contact problem corresponds to situations in which

P = 1 on P = 0 on i/’2, P=n®n on .9^,

where .9^, -9^2 > 3^® complementary subsets of dB.

In view of the preceding discussion, the general problem may be stated as follows: given an elasticity field C and a body force field b on B together with P, u, and s on 8B; find an elastic state [m, E, S] that corresponds to b and satisfies the generalized boundary condition. We call such a state a solution of the general problem.

* Sometimes called the mixed-mixed problem. ^ See p. 9.

llandbuch der Physik, Bd. VIa/2. 9


(1) Uniqueness theorem, for the general problem.^ Assume that the elas¬ ticity field is positive definite. Then any two solutions of the general problem are equal modulo a rigid displacement.

Proof. Let 3 and 3 be solutions of the general problem, and let

i = [u, E, iS]=3 —3.

Then 4 is an elastic state corresponding to zero body forces. Moreover,

Pu = (1 —P) s = 0 on dB ; thus on dB

S-M = [(l-P+P) u] • [(1-P+P) S] = [(l-P) u] •Ps=[P(l-P) m] • s = 0.

Therefore we conclude from (28.5) that u is rigid and E=S = b. □

Of course, (1) yields, as a corollary, the uniqueness theorem (32.1) for the mixed problem.

41. Extension of the minimum principles. The minimum principles established in Sect. 34 are easily extended to the general problem stated in the previous sec¬ tion.^ For example, if we define a kinematically admissible state to be an admis¬ sible state [u, E, S] that satisfies the strain-displacement relation, the stress- strain relation, and the boundary condition

Pu=u on dB, (a)

then the principle of minimum potential energy (34.1) remains valid for the general problem provided we let

0{<s} = Uc{E} — f b - udv — f s- u da. B dB

In view of the consistency condition and the remarks on p. 9,

s-M = (l-P)s-M = (l-P)s- (1-P)m,

and the integral over dB can also be written in the form:

J(l-~P)s.{l-P)uda. (b) dB

For the mixed problem (P = l on P = 0 on ,9^) the boundary condition (a) reduces to m = m on .9^, and the integral (b) reduces to an integral over . Thus in this instance the extended minimum principle is nothing more than the tradi¬ tional theorem (34.1).

On the other hand, if we define a statically admissible stress field to be an admissible stress field S that satisfies the equation of equilibrium and the boundary condition

(1—P)iSn = s on dB,

then the principle of minimum complementary energy (34.3) remains valid for the general problem® provided we let

<P{S} = i[7K{S}-/s- uda, dB

* Cf. Sternberg and Knowles [1966, 2i']. Other uniqueness theorems which are not special cases of (1) are given by Bramble and Payne [1961, 5] and Knops and Payne

[1971, 2]. ® Cf. Rudiger [I960, 10], Prager [1967, 77], HlAvaCek [1967, 7]. ® An extension of the principle of minimum complementary energy to the mixed-mixed

problem was given by Sternberg and Knowles [1966, 24]. See also HlAvaCek [1967, 7]. The result stated here includes, as a special case, the theorem of Sternberg and Knowles.

Sect. 42. Properties of elastic displacement fields. 131

or equivalently, W{S} = Uk{S} -fPs-Puda.

dB

In a similar manner, the maximum principles discussed in Sect. 37 and the variational principles given in Sect. 38 can be extended to the general problem.

VI. Homogeneous and isotropic bodies.

42. Properties of elastic displacement fields. As we saw in Sect. 27, when the divergence and the cmrl of the body force field vanish, the displacement field u is biharmonic, while div u and curl u are harmonic. To prove this we tacitly assumed that u is of class C*. We now show that this somewhat stringent assump¬ tion is unnecessary; we will estabhsh the above assertions under the assumption that u is of class C®.

By an elastic displacement field corresponding to the body force field b we mean a class vector field u on B that satisfies the displacement equation of equihbrium

Au-\-- - VdivM-f — b = 0. 1 — 2V yM

We assume that Poisson’s ratio v is not equal to f or 1, and that ytt=i= 0.

X (1) Analytidty of elastic displacement fields.^ Let u be an elastic dis¬ placement field corresponding to a smooth body force field b on B that satisfies

divb=0, curlb = 0. Then u is analytic.

Proof. 'The proof is based on the following notion, due to Sobolev® and Friedrichs.® A system of mollifiers is a one-parameter family of functions g"*, (5 > 0, such that for each fixed <3:

(i) ^ is a C°° scalar field oni^'A

(ii) 0 on

(iii) g"* (») = 0 whenever |»| ^ 5;

(iv) / q^dv = \. ■r

An example is furnished by

Q^(v) = U^^~ H<<5 [o,

with Ag chosen such that (iv) is satisfied. Let £> be a regular region, and let / be a continuous scalar, vector, or tensor field on D. Then the system of molli¬ fied functions f, (5 > 0, is defined on S by the transformation

f (*) = /?'*(*-y)/(y) D

Clearly,

(v) each f is a class C®° function on S.

* Friedrichs [1947, 7]. See also Duffin [1956, 2], whose proof we give here. ® [1935, «], [1950, 77]. ® [1939, 2], [1944, 2], [1947, 7]. ^ Recall that •f is the vector space associated with S.

9


By (ii), (iii), and (iv),

!/■’(*)-/(*)! = I / Q\x-y)[f{y)~f{x)]dVy\

^sup{|/(t/) -/(*)!: y^Zaix)},

for sufficiently small d, where Ug (a?) is the open ball with radius d and center at x. Hence ——

''(vi) f f as d uniformly on every closed subregion of D.

Assume now that / is of class on D and, for convenience, that / is scalar¬ valued. Let

={xeD: Us (a?)CZ)}.

In view of the divergence theorem,

V^f^(x) = J (X ~y) f(y) dvy== - J [Fy {x -y)] f (y) dvy D D

= - / e^(^-y) f(y)n(y) day-)- / g^(x-y) Vyfiy) dVy. do D

But (iii) implies that

Q^{x—y)=0 for xeD^ and yedD.

Thus we have the following result:

(vii) F(/^)=(F/)'» on D^. \

Of course, (vii) also holds when / is a vector or tensor field.

We are now in a position to complete the proof of the theorem. Let u be an elastic displacement field; let Z) be a regular region with DcB; let g^ {d> 0) be a system of moUifiers; let and {d> 0) be the systems of moUified functions corresponding to u and 6:

u^{x) = / g^(x—y) u{y) dVy, D

(a?) = f g^{x-y) h [y) dVy. D

Then, by (v), v? and 6'’ are of class C°°, and, by (vii), is an elastic displacement field on corresponding to 6'’. Fmther, 6'’ is divergence-free and cmrl-free on D^. Hence we conclude from the results of Sect. 27 (which hold in the present circum¬ stances) that is biharmonic. Next, let

h =Au. Then (vii) imphes that

= (Auy =A{u^)

on D^. Therefore, since is biharmonic,

Ah^ = 0

on D^, and we conclude from (vi) that

h^-^h

uniformly on every closed subregion of D. The last two results and Harnack’s convergence theorem (8.2) imply that h is harmonic. But

Ah==AAu;

Sect. 43. The mean value theorem. 133

thus to prove that u is biharmonic we have only to show that u is of class C^. But since zlu is equal to a harmonic function, (6.5) and (8.1) imply that u is of class C°° on D. Therefore u is biharmonic and, by (8.1), analytic on D. But D was chosen arbitrarily. Thus u is analytic on B. □

In view of the results given in Sect. 27, we have the following immediate corollary of (1).

(2) Properties of elastic states. Let [m, E, S] be an elastic state correspond¬ ing to a smooth body force field h on B that satisfies

div6=0, 01^16=0.

Then:

(i) u, E, and S are biharmonic;

(ii) div u, curl u, tr E, and tr S are harmonic.

The next proposition will be of futme use.

(8) If u is an elastic displacement field corresponding to null body forces, then so also is u ^ = dufdx^.

Proof. Since m is an elastic displacement field, u is analytic; therefore

Q = [Au + Fdiv m) ^ = zlVdiv{Uj,}.

Thus M is an elastic displacement field. □

Under-certain restrictions on the boimdary of B, Ficherai has estabhshed the following important result: If m is an elastic displacement field that is con¬ tinuous on B and corresponds to zero body forces, then

sup |m| sup |m| , B 8B

where the constant H depends only upon v and the properties of dB. We omit the proof of this result, wliich is quite difficult. Ficheea^ has also established the inequality

/ |s| da^H f |6| dv, dB B

where s is the surface_traction and 6 the body force field of a smqo^ elastic displacement field on B that vanishes on dB. ~~

43. The mean value theorem.® The mean value theorem for harmonic fimctions® asserts that the value of a harmonic fimction at the center of a sphere is equal to the arithmetic mean of its values on the surface of the sphere. In this section we

derive similar..results, for .the displacement,..gtram,-and stress fields-bdonging to ah elastic stat£>Jn particular, we show that the values of these fields at the center of a sphere are,„eq.ual to certjiin weighted averages of the displacement .on the su^ce of the sphera We sdso establish a similar result for the stress field in terms of its surface tractions.

^ [1961, S]. See also Adler [1963, i], [1964, i]. ® Our presentation follows that of Diaz and Payne [1958, 6]. ® See (1) of (8.3).


(1) Mean value theorem.^ Assume that the body is homogeneous and iso¬ tropic with shear modulus /j, and Poisson’s ratio v, and assume that

/M>0, —i<v<i.

' Let [u, E, S] be an elastic sMe corresponding to uro bod^ forces. Then given any ballSg=Sg(y)inB,

= ■ n) n + i'l-Av) u} da, (1) 1671 (2 — 3v)

dSo

8.e^7-lo^ -io./(«(8)«+«(8)u)i« 0i'p

+ 7 / (m • n){5n(g)n —1} ,

dZp

iOv J (u0n -yn0u) da

0i’p

J {u ■ n){}Sn^n—a.l} da ,

S{y) = -- ,4^ —, 4jre^(7 —lov)

a =

0i’p

20v“ — 28v + 7

(2)

(3)

1 —2v

^ + 's7)

s —Sn.

j"{iOv(s0n) -{-7{s-n)[$n®n — l]}da,

aip (4)

Proof.Since the body forces vanish, the displacement equation of equilibrium has the form

zlM+y FdivM = 0, (a) where

(b) ^ \—2v

Let

and notice that r{x)=x—y, r = \r\,

n = on dU„. (c)

Then (a), (b), (c), and the divergence theorem imply that

0 — f [zl M + y Fdiv m] dv

— I «+y(divM) n]da — 2 J [(Fm) r+y(divM) r] dv. (d) dSp ^ p

1 Relation (l) is due to Aquaro [1950, 7], Eq. (13) and Synge [1950, 72], Eq. (2.36); see also Sbrana [1952,4], Mikhlin [1952, 2], pp. 206-207, Garibaldi [1957, ^l. Diaz and Payne [1958, 6, 7], Bramble and Payne [1964, 4]. Relations (2) and (3) are due to Diaz

and Payne [1958, 6], Eqs. (4.11), (4.16). Relation (4) is due to Synge [1950, 72]; see also Diaz and Payne [1958, 6], Eq. (5.21). Diaz and Payne [1958, 6], [1963, 3] and Bramble

and Payne [1964, 4], [1965, 2] have obtained a number of mean value theorems giving the displacement and stress at the center of a sphere in terms of integrals over the surface of various combinations of normal and tangential displacements and tractions. A mean-value theorem for plane stress was given by Foppl [1953, 72].

^ The proof given here is that of Diaz and Payne [1958, 6].

Sect. 43- The mean value theorem. ns

The surface integral in (d) is zero, since

/ [(Pm) n+y (div m) n] da=Q^ f [zIm +y PdivM] dv=0. p i'p

If we use the divergence theorem on the remaining integral in (d), we conclude, with the aid of (c), that

Q f [u +y(M ■ n) n] da— f (3 +y) m (fw =0. dUp Sp

On the other hand, by (4) of the mean value theorem (8.8),

j" udv —\ j" uda

fp ^ aip

(e)

(f)

If we eliminate the integral of u over between (e) and (f), and use (b), we arrive at the first relation in (1).

Our next step is to derive (2). Since u is biharmonic. Pm is biharmonic, and we conclude from (4) of (8.3) that

5_ J Vudv — ^ j Vu da , (g)

where 0< Using the divergence theorem, we can rewrite (g) as follows;

Pm (m) = ,

dSx dSx

If we multiply this relation by and integrate from A = 0 to A = p, we find that

J u0n da — J Vuda

5 Vu{y) = 5 j u^rdv — JPm dv

By the divergence theorem,

/ r^Vudv=Q^Ju0nda — 2 /u^r dv, Ep dEp Ep

and (g), (h), and the strain-displacement relation imply that

^-E(y) = ~ 5 16 JT

7 J (u0r -{-r0u) dv J (m(g)nn (g)m) da

Ep dEp

(h)

(i)

(i)

Next, we will ehminate the integral in (j) over By (42.3), m for each fixed k, satisfies the displacement equation of equilibrium. Thus (e) holds with u replaced by m and we are led to the relation

/' Vudv = -^^ - J \\-\-yn®n\Vu da.

Ex SEx

which can be written, with the aid of the divergence theorem, in the form


If we multiply this relation by A, use the fact that n =r/A on dU^, and integrate from A =0 to A = p, we arrive at

I u®r dv = —- - / [f® 1 +y r®*’] Vu dv. (k) J 3 +y j

Sp Sp

Applying the divergence theorem to the right side of (k), we are led to

J u0r dv =J [u0n-\-y(u ■ n) n^n] da ip ' dSp

— J [2M®r+yr(g)M+y(r • m) 1] dv\.

If we add this equation to its transpose and rearrange terms, the result is

J (M(g)r+r (g)M) (fw

■^p

[“®w+w®w + 2y(M • «) «®»*] 1J {r-u)dv\.

Taking the trace of this equation, we obtain

J r-udv=~ Ju-nda,

and the last two equations yield

J {u0r-{-r0u) dv

y |M(g)n+n(g)M +2y(m • n) ^n(g)n—^ ij da.

If we combine (j) and (1) and use (b), the result is (2).

Since div u is harmonic, (2) of (8.3) implies that

divu{y) = -^^^ J div Ju.nda.

The relation (3) follows from (2), (m), and the stress-strain relation in the form

S = 2pE + (tr E)1 =2^ E+ — 3- (divu) 1

Since [u, E, S] is an elastic state corresponding to zero body forces, the stress field S satisfies the equation of equilibrium

div S = 0 (n)

Sect. 43- The mean value theorem. 137

and the compatibility equation ^

zlS+}jPF(trS)=0.

1 where

Jj: i+v

(O)

(P)

Further, by (42.2), S is biharmonic and tr S harmonic. Thus we may conclude from the mean value theorem (4) of (8,3) that

Sx dSx

By the divergence theorem and (n),

j S(n®n) — J S(n0r) da= J S dv,

(q)

(r) ^x

and (q), (r) imply that

s{y) = Sn J S{n0n) da— J Sda

dSx ftSx

If we multiply this equation by X*, integrate from A = 0 to X=q, and solve for S {y), the result is

5 j S(r0r) dv- J r^Sdv L T’

On the other hand, the divergence theorem and (n) imply

thus

/ S{n0n) da = J S{n^r) da= f S dv-^2 / S(r(g)r) dv, (s) dSp dUp Xp 2'p

=-g^-^5 [7 y S{r0r) dv—Q^ j" S{n0n) da L T’ ft

Further, since

where

it follows that

dSp

S{n0n) — {Sn) (g)n = s(g)n,

s = Sn,

^(*f)=-^gT|7 J S(r0r) dv — Q^ J s0nda

^ ^p dSp

(t)

(U)

To complete the proof we have only to transform the first integral in (u) into terms involving surface tractions. By the divergence theorem and (o),

/ [(FS)n+jj(FtrS)(g)n] da = Q, esx

and using this indentity, (o), and two applications of the divergence theorem, we see that

0 = /r^{AS +JJFF(tr S)] (fw = — 2 / [(FS) r+jj(Ftr S)^r]dv

= — 2 A/ [S +Jj(tr S) n^n] da — J [^S +9j(tr S) 1] dv . ftSx Sx

* See p. 92.

138 Sect. 44. M. E. Gurtin : The Linear Theory of Elasticity.

Taking the trace of (r), we obtain

JtrSdv=?,f S - {n®n)da, Sx 0i-;i

and the last two equations with (r) yield

S + J? (tr S) r (g) r] da =J [')S(r®r)+rj(r-Sr)l]da.

If we integrate this relation from A = 0 to A = g, we arrive at

_/ [r^S+fj(tr S) r (g)r] [3S(r ®)r)+rj{r ■ Sr) 1] dv.

that r. Sr is biharmonic; thus (4) of (8.3) implies

Jr-Srdv= j" snda.

dhp

By the last two relations, (s), and (t),

5 j S{r®r)dv=Q^ j^s®n-'^-{fi.n)\ dZp

In view of the identity

(V)

i^hi I'i, rj + S^^r^ Tj +

and the symmetry of S, the volume integral in the right side of (v) takes the form

J (trS) r®rdv = Q^J {s ■ n) n®n da~ f [S(r®r)+(r ®r) S] dv. (w)

On the other hand, if we multiply (r) by A^ and integrate with respect to A from 0 to Q, we arrive at o

! S{r®r)dv = SdvdX. M ■^p 0 Sx '' '

the side of (x) is symmetric. Thus the volume integral in the right side of (w) is equal to twice the volume integral in the left side of (v), hence (v) and (w) imply that

JS{r®r) dv — ~^^~ n.s(g)n+j;(n(g)n —l) (s • «)

' ai'o ^ '

d Cl, (y)

Eqs. (y), (u), and (p) yield the desired result (4). □

44. Complete solutions of the displacement equation of equilibrium. In thi=

hSim solutions of the displacement equation of equi-

Au+ Fdivit+ ^6=0 1 - 2v ' n

assuming throughout that the body force 6, the shear modulus a, and Poisson's ratio r are prescnbed with ^ 4= 0 and r + *, 1. Recall that a solution of c!as_s

coffesponding to h.

Sect, 44. Complete solutions of the displacement equation of equilibrium.

(1) Boussinesq-Papkovitch-Neuber solution.^ Let

where tp and ip are class fields on B that satisfy

Aip = - I b, Aip=-~p-b.

Then u is an elastic displacement field corresponding to b.

Proof. Clearly,

Au-{- - FdivM 1 — 2v

139

(Pi)^

(P2)

=Aip-\-~^;^~VAivip

Since AV=VA and

it follows that

4 (1 — v)

A{p • ip) =p ■Aip^2 div ip,

A V{p ■ y VA [p • q>-\-cp)

r-hv >^divM=zlip+ ^^^-Fdivv- F(p-Aip+A,p + - 2 div ip).

But

p-Aip = —A^ and Aip = b;

thus the above expression reduces to

- ' 6. n □

<•1, ^.ppcar in the work of BpussiNEgfl. [1878, 2], [1885 71. In fact between the two solutions given by Boussinesq on p. 62 and p. 72’of [1885 71

r? t rotational symmetry The !, *" ,p”. (^2) with 9 = 0 IS due to Fontaneau [1890, 3], [1892, 3]. The Lmplete solu-

I NSrS‘^lToir’'4T ^ Papkovitch [1932, 3. 4]; it »as latwrediscovered by ! ^ According to Papkovitch [1939, 4] (footnote on p. 13o), (P,). (P,) should

be attributed to Grodski, who arrived at the solution in 1928 (unpublished research), How-

ever. It is not clear from the statement of Papkovitch whether or not this work of Grodski

idetp'ôl T’ ^’^^kovitch [1937, 5] claimed that it does not. The com-

sibrect Other f ' ^ “.G^°o®ki's [193S, 7] first published research on the subject. Other refer^ces concerning this and related solutions and not mentioned elsewhere

[1888, 7,2], Tedone [1904, 4], Serini [1919 2] Burgatti

[1924,2], [1948,6], [1950, 73], [1951, 77], Weber [1925 7]

n[1933, 7], [1935, 3], [1955, 77], Biezeno [1934, 7] Sobrero [1934, 6], [1935, 8], Neuber [1935, 4], [1937, 3], Westergaard [1935, 9], Zanaboni [1936

b’LATRiER [1941, 7], Gutman [1947, 3], Shapiro

[1947, 6], Krutkov [1949,3], Blokh [1950, 3], [1958, 2], [1961 2], Sternberg, Eubanks, and Sadowski [1951,9], Brdicka [1953 6] [1957 31 Churikov

19 3. ]. Dzhaneodze [1953, 9], Hu [1953, 73],' [1954, 3], ^chaeee4 [i'95T 73] Sin

Sternberg [1954, 5], Kroner [1954, 77], Ling and Yang [1954 73], Lure [1955, 79], Duffin [1956, 2], Solyanik-Krassa [1957, 73], Solomon [1957 121’

^ a'-f [1959, 76], Bramble and Payne^ [I96L 4], Naghih and Hsu [1961, 76], Aleksandrov and Solovev [1962, 7], [1964 21 Gurtin

[1962, 7], Tang and Sun [1963, 22], Volkov and Komissarova [1963, 25], Herrmann [i964

9], Aleksandrov [1965, 7], Stippes [1966, 25], [1967, 74], Solomon [1968, 721 deLaPenhI and Childs [1969, 3], Youngdahl [1969, 9]. ’

^ Recall that p (as) = as — 0.


As a corollary of this theorem we have the following result.

(2) Assume that the body force h is continuous on B and of class {N^2) on B. Then:

(i) there exists an elastic displacement field u of class on B that corresponds to 6;

(ii) every elastic displacement field that corresponds to h is of class on B.

Proof. Let

' Ann J |as-j;|

y,(x) = b{y)

l®-»l -dv„.

Then it is clear from the properties of the Newtonian potential (6.5) that (p and yy are of class on B and satisfy (Pg); thus the field u defined by (Pj) is a class elastic displacement field corresponding to 6.

To prove (ii) let u be an arbitrary elastic displacement field corresponding to 6. By (i) there exists a class elastic displacement field u that also corresponds to 6. Let

u =u—u.

It suffices to prove that u is of class C^. Clearly, ii is an elastic displacement field corresponding to zero body forces; thus, by (42.1), u is of class C°°. □

For the case in which the body force is the gradient of a potential, i.e.

6 = Fa,

it is possible to construct_ at,niiich_sim,pte-par±icular..5oliition. Indeed, the field

u = V(p

satisfies the displacement equation of equUibrium provided

In the same manner, if

then

zl ffi = — —a. ^ X + 2u

6 = curl T,

is a solution provided u = curl to

I zlt0 =-T.

u

By Helmholtz’s theorem (6.7) we can always decompose the body force as follows:

6 = Fa + curlT.

Thus, combining the previous two displacement fields, we arrive at the solution

u = V(p-\-car\ to.

A(p = — _1_ 1+^

4lto = — 1

— T. U

* Cf. LAMfi [1852, 2], p. 149: Betti [1872, 7], § 5; Kelvin and Tait [1883, 7], p. 283;

Voigt [1900, S'], pp. 417, 432, 439; Tedone and Timpe [1907, 3].

Sect. 44. Complete solutions of the displacement equation of equilibrium. 141

This solution, in general, is not complete. Indeed, consider the case in which a = 0. Then

div u =zl 99 =0;

thus, in this instance, the solution can represent at most divergence-free dis¬ placement fields.

(3) Boussinesq-Somigliana-Galerkin solution.^ Let

u=Ag- Fdivg, (Gi)

where g is a class vector field on B that satisfies

AAg = -\h. (G,)

Then u is an elastic displacement field corresponding to 6.

Proof. Clearly,

I z1m-|-FdivM

\ — 2v

=AAg-—-AVdd\g+ fpdivzlo-—^Pzldivgl.

Since AV—VA and divd =A div, the right-hand side of the above expression reduces to AA g, which by hypothesis is equal to

Our next step is to establish the completeness of the Boussiriesq-Pa-pkP^dtcli- Neuber and Boussinesq- Somigliana- Galerkin representatiojvs,. The notion of com¬ pleteness is oT'ulRnost impirartance to applications. Indeed, it is concerned with the question: can any (solvable) problem be solved using a given representa¬ tion; or, in other words, does every sufficiently smooth solution admit such a representation ?

(4) Completeness of the Boussinesq-Papkovitch-Neuber and Bous¬ sinesq-Somigliana-Galerkin solutions? Let u he an elastic displacement field corresponding to b, and assume that u is continuous on B and of class on B. Then there exists a field g of class C* on B that satisfies (Gj), (G2); and fields q>, ip of class C® on B that satisfy (Pj), (Pj).

Proof. We begin by establishing the completeness of the Boussinesq-Somi¬ gliana-Galerkin solution. Clearly, (Gj) is equivalent to

_ zlg + -^J‘-.-|7divg+^6 = 0,

1 This solution is due to Boussinesq [1885, f], p. 281. It was later rediscovered by SoMi-

GLiANA [1889, 3], [1894, 3] and Galerkin [1930, 2, 3], [1931, <5] and is usually referred to as

Galerkin’s solution, although Tedone and Timpe in their classic article [1907, 3], p. 155

attributed it to Boussinesq. See also Iacovache [1949, 4], Ionescu [1954, 9], Moisil

[1949, 6], [1950, 5], Ionescu-Cazimir [1950, 7], Westergaard [1952, 7], BrdiCka [1954, 3], Bernikov [1966, 4].

2 Papkovitch [1932, 4] noticed that if a particular solution to the equation of equilibrium

exists for arbitrary values of Poisson’s ratio and arbitrary body forces, then the Boussinesq-

Somigliana-Galerkin solution is complete. An alternative completeness proof for (Gj), (Gj)

was furnished by Noll [1957, 47]. The proof given here, which is based on Papkovitch’s

argument, is due to Sternberg and Gurtin [1962, 13].

142 M.E. Gurxin: The Linear Theory of Elasticity. Sect. 44.

where

b — —[lu, v=\—v.

Thus the task of finding a class field g that satisfies (Gj) is reducible to finding a class C* particular solution to the displacement equation of equilibrium corre¬ sponding to given body forces that are continuous on B and of class C* on B. This latter objective is met by (i) of (2). Further, using steps identical to those used in the proof of (3), it is easily verified that (Gj) implies

.d u -j- 1

1 — 2v Vdiv u =AAg.

Thus (Gj) and the displacement equation of equilibrium imply (Gj), and we are entitled to conclude the completeness of the Boussinesq-Somigliana-Galerkin solu¬ tion (Gj), (Gj). Next, following Mindlin,i we define

(p^2div g—p-Ag, xp=Ag.

Then (p and tp are of class on B, and (G2) implies (P2), (Gj) implies (Pj). Thus the Boussinesq-Papkovitch-Neuber solution is also complete. □

In cartesian coordinates the solutions (Pj), (Pg), and (G^), (G2), as well as the formulae for the corresponding stresses, have the following form:

Boussinesq-Papkovitch-Neuber solution

Vi, a

Ui=y)i-

h n ‘

Xjbj

A(\—v)

Sir- 4

Boussinesq-Somigliana-Galerkin solution

n bi

î—Si,jj ’2(1 —v) > *

îj ' /A A~Sj,ikk H ^^ Sk,k 11 Sk,kij} j" •

In most applications of elastostatics the Boussinesq-Papkovitch-Neuber solu¬ tion deserves preference over the Boussinesq-Somigliana-Galerkin solution.^ Indeed, the transformation (Ij) underlying the former is one degree lower than that of (Gi); secondly, (in the absence of body forces) the stress functions involved in the former are harmonic, while those of the latter are biharmonic; and, finally, (Ij), (P2) are conveniently transformed into general orthogonal curvilinear coordi¬ nates, whereas (Gj), (G2) give rise to exceedingly cumbersome forms when referred to curvilinear coordinates, with the exception of cylindrical coordinates.

^ [1936,2]. ^ Cf. Sternberg and Gurtin [1962, 73].


Two other complete solutions of the displacement equation of equihbrium (with 5=0) are 4

u=x— , Fa, 2(1 —v)

Ja=divT, Jt=0;

u=l- 1

curl (o, \ — 2v

A(o=cw:\l, Al = Q.

Returning to the Boussinesq-Papkovitch-Neuber solution, we note that a given equihbrium displacement field-does, not, .uniquely, determineits generating stresFIunctiom^ cp and ig. In fact, it is just this lack of uniqueness which, in certain^cifcJinnstances,' allows the ehmination of cp or one of the components of ip. The next theorem, which is due to Eubanks and Sternberg,^ deals with the possibility of setting y = 0.®

(5) Elimination of the scalar potential^ Let B he star-shaped with respect to the origin 0. Let u satisfy the hypotheses of (4) with 5=0. Further, suppose that —1 < v < i with Av not an integer. Then there exists a harmonic vector field ip on B such that

Proof. By (4) and since 5=0, there exist harmonic fields 99 and ^ such that

u=w- V(p-w-F(p)’

Aip — 0, A q> =0.

Thus we must show that there exists a harmonic field ip that satisfies

OLip — V{p • ip) =a^ — F(p • ^ + 99), where

<x=4(i—v).

By h5rpothesis, we may confine a to the range

2 < a < 8. (a)

^ The first solution, which is apparently due to Freiberger [1949, 2, 3], is simply a variant of the Boussinesq-Papkovitch-Neuber solution; indeed, take r = ti/, 2a. = q>+p • in (Pj), (Pj). The second solution is due to Korn [1915, 2], wlm used jt toTJrove existence fpr the displacement pri-ihlom

^ [1956, 4]. Papkovitch [1932, 3, 4], through an erronoeus argument, was led to claim that 99 may be taken equal to zero without loss in completeness, a claim which is reflected in the title of [1932, 3]. In a subsequent note [1932, 2], Papkovitch refers to this argument as “artificial”. Neuber [1937, 3] asserted that 99 may be set equal to zero without impairing the generality of the solution. In his proof of this statement he takes for granted the existence of a function satisfying a pair of simultaneous partial differential equations. Therefore, as was pointed out by Sokolnikoff [1956, 12], p. 331, Neuber’s argument is inconclusive. Slobodyansky [1954, 19] proved that the stress function 99 may be eliminated, but as was observed by Sokolnikoff [1956, 12], p. 331, Slobodyansky’s proof is limited to a region interior or exterior to a sphere.

® The solution (Pj), (Pj) with 99 = 0 appears in the work of Fontaneau [I89O, 3], [I892, 3], and, according to Papkovitch [1937, 5], [1939, 4], p. 130, also in unpublished research of Grodski.

* This result has been extended by Stippes [1969, 3] to an exterior domain whose comple¬ ment is star-shaped, and to the region between two star-shaped surfaces.


It is sufficient to establish the existence of a harmonic field 0 such that^

p • V0 — OL0 = q). (b)

Indeed, given such a field 0, the function

tp=y}-\-V0 (c) has all of the desired properties.

Let {r, 6, y) be spherical coordinates. The function (p, by hypothesis, is har¬ monic. Hence (p (r, 6, y) admits the following expansion in terms of solid spherical harmonics

OO

<p{r.d,y) = ZS„{d,y)r”, (d) n—0

which is uniformly convergent in any spherical neighborhood of the origin. The functions S„ (6, y) are surface harmonics of degree n.

Next, define a function F through

[a+l] F{r, d,y)=(p (r, 6,y)-j] S„ {6, y) r”, (e)

n—0

where [a -|-1] denotes the largest integer not exceeding a 4-1. Clearly, F is defined and harmonic on B. Further, by (d) and (e),

r~^~'^F{r, Q,y) ^0 as r-Ô. (f)

In spherical coordinates (b) has the form

or by (e) [a+l]

r-+^^{r-0)^F+ n—0

Consider the function 0 defined by [a+l]

0{r, e.y)=G(r,e,y)4-2:-f-i5^. n=0

r

G{r, d,y)==r«f F(i. d. y) d^.

(g)

(h)

Since B is star-shaped with respect to the origin 0, and in view of (a) and (f), the relations (h) define 0 on B. Moreover, 0 is of class on B and satisfies (g), and hence (b).

It remains to be shown that 0 is harmonic on B. To this end, it suffices to show that G is harmonic. In spherical coordinates the Laplacian operator appears as

^=-T 0. ri)+M’ Zl* = cosec 6[sin 6cosec^ 6

(i)

1 Neuber [1937, S] assumed, without proof, the existence of such a <P, in the absence of any restrictions on B and v.


Clearly,

dr ^ I7 + (“ + '*) -^+ “(“ + '*) r

A^G=r« f A^F(i. d^. (i)

and, with the aid of two successive integrations by parts applied to the second of (h), we reach the identity

8F U 8F(|. d.y)

H r (k) a(a+ 1) G = — (a + l) r+r“ j 0

Substitution of (k) in (j), and use of (i), now yield

rMG =r« j AF{1 6, y) d^. 0

But F is harmonic. Hence G and 0 are harmonic on B. □

(6)^ TJieorem jS) is false if the hypothesis thai 4.v not be an integer is omitted

Proof. Suppose that Av is an integer. Then the modulus a=4(l — v) is also an integer with 2 < a < 8. Let S be a closed ball centered at 0, and let u be the elastic displacement field (corresponding to 5=0) defined by

u = -^V<p, (a)

where 99 is the solid spherical harmonic

(p(r,d,y)=i^S^{6,y). (b)

Assume that (5) remains valid. Then there exists a field tp that is harmonic on B and satisfies

IV(p-xp). (c)

By (a) and (c), V(p-xp)—a.y} = Vq>, (d)

and upon operating on (d) with the curl, we conclude that xp is irrotational. Since B is star-shaped and hence simply-connected, there exists an anal5d:ic field 0 on B such that \p = V0-, thus we conclude from (d) that

p-V0 — a.0 — (p-\-<î, (e)

where is a constant. By (e) and (b),

y^{r-0)=r-^S,{e.y)+c^. (f)

The general solution of (f) is

0(r, 0, y)=r^f{Q,y)+ >^(log r) (d, y) 4-. (g)

There is no choice of f{0,y) that leaves 0 without a singularity at the origin. But 0 was required to be anal5d:ic on B. Hence (5) cannot be valid in the present circumstances. □

* Eubanks and Sternberg [1956, 4], See also Slobodyansky [1954,19], Solomon

[1961, 20]. Even though the potential y cannot be eliminated when 4v is an integer, Stippes

[1969, S] has shown that y can be set equal to a solid spherical harmonic of degree 4(1 —v).

Handbuch der Physik, Bd. VI a/2. i 0

146 M.E. Gurtin: The Linear Theory of Elasticity.

Sect. 44.

the deals with

displacement field corres-bondine tn h—n n ^ i ** elastic tiL.. .. j u ''^rrebponaing to 0=0, and assume that u is of class Ci o« R Then u admits the representation (P,) with cp and xp harmonic on B anl^Jl

Proof. By (4) there exist harmonic fields p and y, such that

where u~^~ L (p-^ + ip).

«=4(1—r).

Thus we must establish the existence of harmonic fields <p and tp such that

^W~V(p-ip + 9>)=ocip-F(p-^+p),

¥’3=0.

It suffices to find a harmonic field H that satisfies ^

for then the functions (c)

W = W — VH,

(p=p-\-p-VH~a.H

will be harmonic and wiU satisfy (a) and (b).

is o? clasrSVn^ difficult to verify that w

Then /,3=^3.

(-d/),3 =41^3=0,

and there exists a continuous function h on B,, that is class on B,, and satisfies

^J{Xl.X^,X^)^h(Xi,X^).

^ «■“' C> scalar field R

AK = -h.

Thus if we define H on B through

fd{Xi, x^, Xs) =/(ri, ^2, Xs) +K{xi, x^),

then H is harmonic and satisfies (c). □

Assume now that the body force h_0 on/i 1,,+ ■u in thS-^HSimiiqirSSS^^ .a.giy£n.uiuta.e^

... g^kar, 1 [i956_ ,--

^ Neuber [1934, 41 without restricting B, assumes the existence of such a function.


then (Gj), (Ga) reduces to Love’s solution}

AA3!:=0.

(Li)

(L2)

On the other hand, if we take

ip =kip

in the Boussinesq-Papkovitch-Neuber solution (Pj), (Pj), we arrive at Boussi- nesq’s solution:^

u = kf+ (Bj)

Zl99=0, Aip = 0, (Ba)

where z = 7e • (ae — 0) is the coordinate in the direction of k. Both of these solu¬ tions are extremely useful in applications involving torsionless rotational sym¬ metry.

It is clear from (Lj) and (Bj) that these solutions can at most represent dis¬ placement fields that satisfy®

fe-curlu = 0. (C)

The next theorem shows that under certain hypotheses on B, condition (C) is also sufficient for the completeness of the above solutions.

(8) Completeness of Love’s and Boussinesq’s solutions.* Assume that B satisfies hypotheses (i)-(iv) of (5.2) with M=2 and N=S. Let u he an elastic displacement field corresponding to b=0 that satisfies (C) with k — e^, and sup¬ pose that u is of class C® on B. Then there exist scalar fields 3C, (p, and ip on B that satisfy (L^), (La) and (Bj), (Ba).

Proof. By hypothesis and (42.1), u is of class C°° on B. Since k = e^, (C) yields

^,2 ^2,1

and we conclude from (5.2) with M=2 and N=^ that there exists a scalar field T of class C® on B and C® on B such that

(a)

The displacement equation of equilibrium can be written in the form

(b)

Aus-\-'yds=0, (c) where

6=divu, (d)

y = -—- . (e)

' [1927, 3], p. 276. Cf. Michell [1900, 4], ® [1885, i], p. 62 and p. 72. 5 Noll [1957. 11]- * The portion of this theorem concerning Love’s solution is due to Noll [1957, 11],

Theorem 2. Under the assumption of torsionless rotational symmetry. Love [1927, 3], pp. 274-276 established the completeness of (LJ, (Lg); under the same assumption Eubanks

and Sternberg [1956, 4] gave a completeness proof for (Bj), (Bg) (see (9)).

10’*


Before proceeding any further, recall that

By (a) and (b), —{(%> ^2)' i?3—{xg. assB}.

[Ar+yd]^^ = 0-,

A r +y6 = A,

where ^ = ^(xs) is a class C® function on that is continuous on Bg. Now let q be a solution of

it =-^> fe) and let

x (%, X2, x^ = T , X2, ^3) “1“ g (^3). (h)

Then x is class C® on B and class on B. By (h) and (a),

and since

^ Q —Q,33’

it follows from (f), (g), and (h) that

Ax+yS =0. (j)

If we differentiate (j) with respect to x^ and subtract the result from (c), we arrive at

zI(m3-j< 3) =0. (k) Next, by (d) and (i),

6 = Mj, ~l-*<3^3 ~^,aa “k ^,3 X (^3 ^,3) ,3 > (i)

and (j) and (1) imply

{\-\-y)Ax+y{u2—x^^2=^Q- M In view of (5.2), there exists a solution of

0% = x (n)

that is class on B and C® on B; thus (m) takes the form

[(1+y)zl(P*+y(%—j<,3)],3=0. (o)

Since B is A:3-convex, (o) implies that

g* = (i+v)^0*+y(u3->‘.3) (p)

is independent of Xg. Moreover, g* is continuous on and class C® on B^g- Thus, in view of the properties of the logarithmic potential (6.6), there exists a class function g on B^g such that

S.aa. i*-

If we let 0 (%, X2, X^ — 0* {X-^, X2, X^ g (Xj, X,^ ,

then 0 is of class on B and (p), (q) imply

{\-\-y)A0-\-y{u2 — >cA =0.


Moreover, (n) and (r) yield

Finally, using (i), (s), and (t)

M = +% 63 e„+t<3 63 = Fx + (% —63

= F^,3--y^^ê3,

and letting

L+y 0 y

(t)

we arrive, with the aid of (e), at (Lj). Since (Lj) and the displacement equation of equilibrium imply (Lj), the completeness of Love’s solution is established.^

If we now define ^

(p — ISS-*3 '^,33>

f=^,ss>

then (Bj), (Bj) follow at once from (Lj), (Lj). □

Let {r, y, z) be cylindrical coordinates with the z-axis parallel to k. We say that a displacement field u is torsionless and rotationally symmetric tcith respect to the z-aads if

u,—u,{r,z), My = 0, u^ — u^{r,z).

It is not difficult to show that such displacement fields satisfy (C), and hence, by fS), admit the representation (BJ, (Bj). An alternative proof of this assertion is furnished by the following theorem due to Eubanks and Sternberg.^

(9) Suppose that dB is a surface of revolution with the z-axis the axis of revolu¬ tion. Let u he tofsionTcss anE^rofationally symmetric with respect to the z-axis and satisfy the hypotheses of (4) with 5 = 0. Then there exist scalar fields (p = (p{r, z) and y)=y:(r,z) on B such that (Bj), (Bj) hold. Moreover, the functions q> and ip are single-valued and analytic on any simply-connected plane cross-section of B parallel to the z-axis.

Proof. By (4) there exist harmonic fields 99' and ip' such that (Pj) holds. Let

fr Sip'z __ dip, dr dz ’

/ = - — (r ip) + — + h .y gy VWr) ^ y ^ dy py'z_

dz

Then an elementary computation based on (Pj) and the fact that ip' is harmonic and u torsionless and rotationally S5mimetric confirms that

dz

Thus there exists a field ip such that

, _dip

dr ’

, _dy

dz'

1 The proof up to this point is due to Noll [1957, 11). ^ Cf. the last equation in the proof of (4). 3 [1956, 4).

150 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 45-

In the same manner, let d(p' d

dr dr {f'V'r) +2 8y)z

dr

Then &+ 2“i

8gr 8g, _ dz dr

and there exists a field co such that

_ dm _ dm ir— gz~-gf-

Now let q> =0) —zy)]

then a trivial computation reveals that

u=fe^+ ^ V{q>+Zf). □

VII. The plane problem.'^ 45. The associated plane strain and generalized plane stress solutions. Through¬

out this section B is homogeneous and isotropic^ with Lame moduli [i and A. Let (x^, X2, x^ be a rectangular coordinate system with origin at 0, and let B

be a (not necessarily circular) cylinder cetrtered at J) with generators parallel to

1 Most of this subchapter was taken from the unpublished lecture notes of E. Sternberg

(1959)- Extensive reviews of the subject are given by Teodorescu [1964, 22], and Musk-

HELiSHviLi [1965, 15], See also the eighth chapter (by Barenblatx, Kalandya, and Mand-

zhavidze) of Muskhelishvili’s book [1966, 17], ^ The plane problem for anisotropic media is discussed by Fridman [1950, 5] and Kupradze

[1963, 17, 18], Chap. VIII.

Sect. 45- The associated plane strain and generalized plane stress solutions. 151

the A:3-axis and with end faces at a:3 = ± A (Fig. 11). Let ^ and be complementary subsets of the set

{xedB: a:3=)=±A}.

We assume that ^ and ^ are independent of in the sense that S^^r\ consists at most of line segments parallel to the A:3-axis and of length 2h.

Given elastic constants ^ and 1, body forces 5 on B, surface displacements

M on and surface tractions s on ^2> with b, M,_and s independent q£-^3 and parallel to the , ^^..-plane. the plane problem of elastostatics consists in finding an elastic state \u, E, S] on B that corresponds to the body force field b and satisfies the boundary conditions:

u — u on Sfx, s=.s on

s = 0 when a:3 = d; ^.

Since the surface displacements, surface tractions, and body forces are all in¬ dependent of x^ and parallel to the %, A:2-plane, one might expect that the dis- placemeniLfieU^_will.also have this property. With this in mind, we introduce the foUowing definition.

Let [u, E, S] be an elastic state. Then [u, E, S] is a state of plane strain provided

^2). M3=0.

Here and in what follows Greek subscripts range over the integers (1,2). The above restriction, in conjunction with the strain-displacement and stress-strain relations, implies that E =E{x^, x^ and S =/S(%, X2). Further,

= '2'(*<«,/? >

■^13 “ -^23 ~^3S~^ I

■5i3 ~‘^23 ~ d, S33=AFj(j(,

and the equation of equilibrium takes the form

—d-

A simple calculation based on the above relations yields the important relation

where v is Poisson’s ratio: S33=VS aa»

_ j._

2(A

Thus the complete system of field equations consists of

—d,

supplemented by •^13 “‘^23 “d, S33=vS^^',

while the boundary conditions are

u^=u^ on ^1, = on ^2.

(P,)

(Pf)

(P2)


supplemented by 5i3= 523 = 533 = 0 when x^ = ±h. (Pa*)

Here and the intersections, respectively, of 5^ and with the , ^ta- plane.

The system (Pj), (Pg) constitutes a two-dimensional boundary-value problem. This problem has a rmique solution which we call the plane strain solution asso¬ ciated with the plane problem. As is clear from (Pf) and (P*), this solution

actually solues the plane prohkmL .ouly.-wh&tLJe ==0. ‘’^,.,5ii.+ 5g2=.9.- However, the plane strain solution is an actual solution of the modified problem that results if we replace (Pf) by the mixed-mixed boundary condition:

Si3 = S23 = % = 0 when x^ = ^h.

For future use, we note here that in terms of fx and v the two-dimensional stress-strain relation

=2p E^p -\-X8^p Eyy

can be inverted to give 2p E^p =Su^p V 8^p 5yy.

We turn next to the associated generalized plane stress solution^ which char¬ acterizes the thickness averages of u, E, and S under the approximative assump¬ tion :

533 ^ 0 • (•^)

Thus let [m, .E, jS] (no longer assmned to be a state of plane strain) be a solution of the plane problem and assume that (A) holds. It follows from the symmetry inherent in the plane problem that

^2> ^3) ^2> %)> %(%> ^2i ^3) = %(%> ^2> ^3)'

for otherwise the displacement field u* defined by

(% > ^2 > ^3) “ (% > ^2 > ^3) > (% > ^2 > ^3) ~ *^3 (% > ^2 > ^3)

would generate a second solution of the plane problem, a situation ruled out by the uniqueness theorem of elastostatics (at least when the elasticity tensor is positive definite).

Given a frmction f on B, let f denote its thickness average'.

+A

f{x^,x^—-^ J f{Xi,X2,x^dx^.

--h * . . /

Since the thickness average of an :r3-odd function is zero, and since the %-denv- ative of an :r3-even function is %-odd, it follows that

Further, M3 — £-^3 — £33 — 5^3 — S23 — 0.

E^p=i{u^_p+up_^),

S^p =2/x E^p -\-k 8^p{Eyy -j-£33),

^33 = 2ya £33 -1- X (£yy -b £33).

Thickness averages of solutions to the plane problem under the assumption that S33 vanish were first considered by Filon [1903, 2], who derived the results (a)-(c). See also Filon

[1930, J] and Coker and Filon [1931, 2]. The term “generalized plane stress” is due to Love [1927, 3], § 94. The assumption was studied by Michell [19OO, 3].

Sect. 45- The associated plane strain and generalized plane stress solutions.

In view of (A), ■^33 — 0; thus the above relations imply that

where 2fl X

X “f" 2/ii V

Further, the averaged equation of equilibrium takes the form

(a)

S(tP,p+â—0, (b)

while the boundary conditions become

on‘g’l, = on(c)

The relations (a), (b), and (c) constitute a two dimensional boundary-value prob¬ lem. This problem has a unique solution called the generalized plane stress solution associatedTmth "the plane'proBIein71T6te~ffiatT;he relations (a), (b), and (c), which constitute the generalized plane stress solution, are identical to the relations (P^) and (Pg) of the plane strain solution provided we replace the Lame modulus A by

It must be emphasized that basic to^ derivAtian- of (a) is the. approximative assumption that ^33 vanish.

The stress-strain relation for generalized plane stress is easily inverted; indeed,

2flE^p=S^p VâP^yy)

where - V v =

1 -t-V

is the appropriate “Poisson's ratio” for generalized plane stress. Fig. 12 shows the graph of vjv as a function of v.

Let us agree to caU the elastic constants [x and v positive definite if

p>Q, —i<V<Y-


This definition is motivated by (iv) of (24.5). Note that

— l<l'<i-<=» —CX5<V<-g-, (d)

so that jx, V positive definite does not imply fi, v positive definite. Note, however, that

> The result (d) will be of importance when we discuss the imiqueness issue appro-!: priate to the generalized plane stress solution.

46. Plane elastic states. In this section we study properties of the pla^strain solution associated with the plane problein imder the assumption that B is homogeneous and isotropic with ya=i=0, lÎn addition, we assum^throughgut this section and the next that the body forch-afe zero.

We write R for the intersection of B with the x^, x^-plane. Further, we let and “^2 denote complementary regular subcurves^ of the boundary curve 8R

and write n^ for the components of the outward unit normal to dR. In view of the discussion given in the previous section, to specify a state of

plane strain or generalized plane stress it suffices to specify the plane components and of the displacement and stress fields. This remark should motivate

the following definition. We say that S„^] is a plane elastic state corre¬ sponding to the elastic constants p and v if:

(i) is continuous on R and smooth on R;

(ii) is continuous on R and smooth on R;

(iii) and satisfy the field equations

Since every plane elastic state generates, with the aid of (P*) of Sect. 45, a (three-dimensional) state of plane strain, we have the following direct corollary of (42.1) :

(1) Let S^p] be a plane elastic state corresponding to the elastic constants p and V. Then and S^p are analytic on R.

Each state of plane strain is an actual solution of the modified plane problem for which the boundary conditions at x^ = ^h are replaced by the mixed-mixed conditions

■^13 = ■^23 = % = 0 when % = ± ^ •

Therefore, as a consequence of the uniqueness theorem (40.1), we have the

(2) Uniqueness theorem for the plane problem.^ Let S^p] and

be plane elastic states corresponding to the same elastic constants p and v, with p and v positive definite, and let both states correspond to the same boundary data, i.e.,

u^=u^ o« and S^pnp=S^pnp on'if2.

* and ’€2 are assumed to have properties strictly analogous to those of and given on p. 14.

^ Trivially, this theorem is valid without change when the body forces do not vanish provided, of course, that both states correspond to the same body forces. A computation of the strain energy of a plane elastic state leads, at once, to the conclusion that uniqueness holds under the less restrictive hypotheses: m4= 0. v < -I. In view of (d), this extension is impor¬ tant arit assures uniqueness when the plane states correspond to generalized plane stress solutions and when the elastic constants ^ andv = vj(\-\-v) are such that /i andv are positive definite.

Sect. 46. Plane elastic states. 155

Then

on R,

and hence the two states are equal modulo a plane rigid displacement.

The next theorem shows that for plane states the stress equations of com¬ patibility take a particularly simple form.

(3) Characterization of the stress field. Let \u^, be a plane elastic state corresponding to the elastic constants [x and v. Then satisfies the com¬ patibility relation^

ZlS„„=0. (a) 2

Conversely, let (=S^„) be of class on R and smooth on R, and suppose that satisfies (a) and the equation of equilibrium

— (b)

Assume, in addition, that R is simply-connected. Then there exists a field u^ such that _ S^p] is a plane elastic state corresponding to p and v.

Proof. Let be a plane elastic state. By (1) the strains corre¬ sponding to are analytic; thus we conclude from (14.5) that satisfies the compatibility relation

2^12,12 “-^11,22 "k^22,11 • (c)

The stress-strain relation, in conjunction with (c), yields

2'S'12,12 “'^11,22 "T'^22,11 (d)

while the equation of equilibrium implies

'^12,12“

•^12,12 ~ '^22,22^ hence

25i2,12= ■S'22,22- (®)

Eqs. (d) and (e) imply (a).

On the other hand, let satisfy (a) and (b) and define through the stress- strain relation. Then by reversing the above steps, it is a simple matter to verify that (c) holds. Thus (14.5) implies the existence of a class displacement field

that satisfies the strain-displacement relation, and we conclude from (14.4) that is continuous on R. Thus [m„, has all of the desired properties. □

Theorem (3) remains vahd in the presence of body forces provided (a) and (b) are replaced by

-b&a =0.

The next theorem asserts that the stresses corresponding to a solution, of the tractiofr~problem are independent of the elastic constants^ (provided the body is

^ Lfivy [1898, 7]. 2 Here A is the two-dimensional Laplacian:

dx^ dxl ’

2 This assertion was first made by Lfivy [1898, 7]. However, Lfivy did not require that R be simply-connected, and theorem jf£l_ig_ialae--iLtiiis_restrictipn.. is ..omitted, (cf. (47.5) and j (47,6)). The first correct statement is due to Michell [1900, 3], p. 109. Reference should , also be made to a remark by Maxwell [1870, i], p. 161, who appears to anticipate Levy’s claim, but I am unable to follow Maxwell’s argument.

156 M. E. Gurtin : The Lineax Theory of Elasticity. Sect. 47.

simply-connected). Much of the experimental work in photoelasticity theory is based on this result. ~

(4) Levy’s theorem. Assume that R is sim-ply-connecjed. Let S„^] and

be plane elastic states corresponding to positive definite elastic constants

fx, V and fx, v, respectively. Suppose, further, that and are smooth on ^ and

Sapf^p=Sapnp on dR. Then

onR.

Proof. Clearly, satisfies (a) and (b) of (3). Thus there exists a field

such that is a plane elastic state corresponding to y and v. Therefore

S„^] and correspond to the same elastic constants and the same surface forces. Thus we conclude from the uniqueness theorem (2) with ‘^^=0

and ^0.0 on R. □

47. Airy’s solution. In this section some of the functions we study will be multi-valued. Therefore, for this section only, if a function is single-valued we will say so; if nothing is said the function is allowed to be multi-valued. Thus the asser¬ tion “cp is a fimction of class C^” means that "cp is a possibly multi-valued function of class C^”. However, we will always assume that the displacement and stress fields corresponding to a plane elastic state are single-valued.

We continue to assume that the body forces vanish. Then, as we saw in the previous section, a plane stress field is completely characterized by the following pair of field equations:

The next theorem shows that these equations can be solved with the aid of a single biharmonic function.

(1) Airy’s solution. Let cp be a scalar field of class C® on R, and let

6'22 — 9’,ll> 6|]^2— 9’,12- (a)i Then

6'a,8,/5 = 0. (b) Further,

(c) if and only if cp is biharmonic.^

Conversely, let { = be single-valued and of class (iV^l) on R, and suppose that (b) holds. Then there exists a class C^+2 function cponR that satisfies (a). Moreover, cp is single-valued if R is simply-connected.

Proof. Clearly, (a) implies (b), while cp biharmonic is equivalent to (c). To prove the converse assertion assume that (b) holds. Then

— ( 6^12),2>

_ ( '^12),! = (6'22),2>

* This solution is due to Airy [1863, i]. See also Fox and Southwell [1945, i], Schaerer

[1956, ii], Teodorescu [1958, 19], [1963, 23], Strubecker [1962, 15], Schuler and Fosdick [1967, 13]. A generalization for the dynamic plane problem is given by Radok

[1956, 10]. See also Sneddon [1952, 5], [1958,17], Sveklo [1961, 21]. 2 Maxwell [1870, 1] was the first to notice that cp is biharmonic. See also Voigt [1882, 1],

Ibbetson [1886, 3], [1887, 2], pp. 358-359, Lfivy [1898, 7].

Sect. 47. Airy's solution.

157

and there exists class C^+i functions g and h on R such that

'^11 ^,2* '^12 — ■ -i. ^22 ' '5^12-^ Hence

^.1 ~^,2>

and there exists a class C^+2 function cp such that

S = <P,2> ^ =

Eqs. (d) and (e) imply (a). □

Henceforth by an Airy function we mean a (possibly multi-valuedl to.ct.on y on R with the property that the LociSedTjatf ’

(d)

(e)

■^11 — 9^,22. •S'22 = 9^11, C — ^12 — -V,12

il'cSlfS'. written in

V,Xt>

where is the two-dimensional alternator

(p(x)-^{x)^k^x^ + c,

wJr " T and x={x,. X,). Conversely, we can always add a hnear function of position to an Airy function without changing the associated stresses. Further ^ven a point y = y^) in R we can alwa^choose the hS£ function so that the resulting function \p satisfies

9iy) = (p.Ay)=o.

. Th%(tKO:dimensiond) jtr^t^^^ôr a simply-connected resion con

and the boundary condition

•S'ixp,/}—0, ^dS„„=o,

on dR,

where the s are the prescribed surface tractions. In view of the last theorem this is equivalent to findmg a scalar field (p such that ’

AA 95 = 0,

Cal s^T <P,Xr on dR.

Thus the traction problem reduces to finding a biharmonic function w that satisfies the above boundary conditions. Our next theorem, which gives a phvsical nterpretation of the Airy function. aUows us to write tiie bo^ condS

t Wm,™ ™ ^ ^tateênt oftSs

_ F'={;v:(a); OââJ

^ Finzi [1956, 5].


be a piecewise smooth curve in R parametrized by its arc length a, and let and 4 designate the normal and tangent unit vectors along F, i.e.,

where = dxjda.

(2) Physical interpretation of the Airy function.^ Let (pbean^ and write

(p{a)=-(p{x{a)), {o) = (p,a{x{a))n^{a)

for the values if <p and its normal derivative along the curve F. Assume that

9^(0) =95,„(0) =0. Then

where

(p{a) =m[a),

ll W—

, ( V

m{a) = j e^p[_x^{a')-x^{a)'\sji{a')da',

Proof. By hypothesis

Thus, in view of the identity

I [a) =t^(a) / s^(a') da', 0

■S'a/S ^PrT,)if

■\v

(a)

(b)

the surface force field s^ = can be written in the form

In view of (b),

and thus

—£a.l ^PxT.Xx '^p —âXT.Xx^r-

T.Xr^r — —^Xr ^r>

Therefore, since (p_;i{0) = 0,

where

d

da {<P,x) = —£XzSr-

9.x{o) ^fx{o),

fx{(y) = - J^xxSxia') da'.

On the other hand, since cp (0) = 0,

(p{a) = f ^{a') da' = j (p^^{a')x^{a') da'. 0 0

(C)

(d)

(e)

(f)

If we integrate (f) by parts and use (d), we arrive at

(p{a) =/;, Xf\l- f fx{a') x^{a') da'. 0

1 Michell [1900,3], p. 107 was the first to notice that tp and dtpjdn are essentially determined by the surface forces along F. The relations given here, which follow from a minor modification of Michell’s results, are due to Treeftz [1928, 3]. See also Phillips [1934, 5], SoBRERo [1935, 3], and Finzi [1956, 5].

Sect. 47. Airy’s solution. 159

By (e), /;i(0) =0 and therefore

(p(a) = — f x^{a) exx ^xi^') ^xi^) >

or equivalently (p{a) —m{a).

Next, by definition,

and (d), (e), and (b) imply

|?-(u) = -Z(<r). □

If m is of class C2 on then the results of hold even when V is contained in the boundary of R. If R is simply-connected, its boundary consists ofajn^ closed curve; thus taHn^r=,3S±n^(^.i.Ave_^.JbaUhe

redu£fiS-to finding a biharmonic function. that satisfies, the standard bound¬

ary conditions;^ w = m and — — I on dR, T cn

where m and I are completely determined by the prescribed surface tractions

on dR. If (p and a do not vanish at x(0), we can still apply (2) to the Airy function

^{x) =(p(x) —k^ x^ — c.

with the constants and c chosen so that

^(x(0)) =^,„(a;(0)) =0.

Thus, in this instance, the relations for (p and drpjdn should be replaced by

w — m -j- x^-j-c,

If R is multiply-connected, its boundary 8R is the union of a finite number of closed curves '^2, • • •, In this case it is not possible to adjust the Airy function (without changing the stresses) to make q) and (p „ vanish at one point on each Thus relations of the form (*) will hold on each of course, the

constants will, in general, be different on each

(p=m+k^ x^ + c”’ and + on M).

In the above relations and c” can be set equal to zero on one of the curves as we shall see, the remaining constants can be evaluated using the require¬

ment that the displacements be single-valued. • 4. ^ With a view toward deriving a relation for the displacement field m terms of

the Airy function, we now establish an important representation theorem for the Airy function in terms of a pair of analytic functions of the complex variable

z = xi+ixz.

1 Michell [1900, 3], p. 108.

160 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 47-

(3) GoursaVs theorem.^ Let cp be biharmonic on R. Then there exist analytic complex functions ip and ^ on R such that ^

or equivalently, q>(Xi, X2) =Re{zy>(z) +%{z)},

2(p{Xi, X2)=zy){z) +z^) -\-x{z) -\-%{z).

Conversely, if ip and % are analytic functions and cp is defined on R by the above relation, then cp is biharmonic on R.

Proof.^ Assume that cp is biharmonic on R. Since Acp is harmonic, there exists an analytic function / such that is the real part of /. Let ^ be a solution of the equation

dij) 1

dz 4 (a)

and let

(b)

where and gg ^re real. Then, since is harmonic,

^(<p-igccXj=^9>-ig«,cc- (c)

By the Cauchy-Riemann equations,

gi,i=g2,2> (d) and by (a) and (b),

gi.i=^<Pl (e) thus (c) implies that

^{T-ho^xJ=0. (f) Let

= (g)

In view of (f), is harmonic and hence the real part of an analytic function %. This fact, (g), and (b) imply that

(p{Xi, X2)=Re{zip{z)+x{z)}. (h)

Conversely, let cp be given by (h) with ip and x analytic. Then, by reversing the above steps, it is a simple matter to verify that q> is biharmonic. □

(4) Representation theorem for the displacement fields Let S„^] be a plane elastic state corresponding to the elastic constants p and v {with pA=0). Further, let cp be an Airy function corresponding to and let

4V’(^) =gi{xi. X2) +ig2{xi, X2),

where ip is the analytic complex function established in (3) and is real. Then

= ^ [- <P,<2 + (I

where w^ is a plane rigid displacement.

Conversely, let cp be an Airy function of class on R, let be the associated stresses, and Id u^, defined as above, be single-valued. Then is a plane elastic stale corr^ponding to p and v.

* Goursat [1898, 6].

2 Here Re{ } denotes the real part of the quantity { }, and a bar over a letter designates the complex conjugate.

® This proof is due to Muskhelishvili [1919, f], [1954, 76], § 31.

* Love [1927, <3], § 145. See also Clebsch [i862, i], § 39.

Sect. 47. Airy’s solution.

161

Proof. Using (a) of (1) we can write the displacement-stress relations in the form

In ~v)A(p,

2fiU22= —9’_22-j-(t —v)A(p,

/“ +*<2,i) = — 9^,12-

Thus we conclude from (d) and (e) in the proof of (3) that

2/1 Mj 1 ^ -j- (4

2/1 «<2,2 = —9^,22 + (^ —v)g2y,

= —(t —v) g^+2/xw^,

w.

(a)

hence

where (b)

(c)

eqâttr^® ^ê Cauchy-Riemann

= “'2,2=0.

that ^1.2 S2,l>

“'1,2+“'2,1=0. (d).

Uon fOT ^ displacement; thus (b) is the desired representa-

To prove the converse assertion we simply reverse the steps (a)-(d). □

ti./?"® * theorem yields conditions that must be satisfied by the Airv func- order to insure that the corresponding displacements be single-valLd.

Le, R te

M

m2=l

where each is a closed curve. Let <p be an Airy function of class C=» on R, and let

= ifi' +“'<x-

where w^ts rigid and is the function defined in (4). Then a necessarv anil

“• onRislU.f Jff, f Jn(^<P)dor = 0,

•Tm

% 9') + ^2 9^) / ' da

d dn

do = — 1 —V

{Acp) da = 1 —r

doublSneted rLTn P' ^5-5- [1946,. 4J Eroved that for a pnH^k for+h+Ai^l^S the natural boundary conditions of a variational

TTanabubh aeir'PHyslk, Bd. Vla/aT’ 11


for m = 2,'i,M, where^

/i”*’ = / fip da,

and S^p is the stress field generated hy (p.

Proof. The field wiU be single-valued if and only if

J^-da=0 (m = 2,3....,M),

or, in view of the definition of if and only if

/I*‘'<’=.1-7(“=2.3.«). (a)

It is a simple matter to verify that (c) in the proof of (2) is valid in the present circumstances in which tp and do not necessarily vanish at some point on

Thus (a) can be written in the form

where here and in what follows m is assumed to have the range {2,'},,...,M).

Clearly,

da (C)

on where x(a) ={xi{a), X2(a)), Oââ^,

is the parametrization for . Now let P^ and Pg denote the real and imaginary parts of the complex function / established in the proof of (3) :

f(z)=Pi(Xi, X2)+iP2(^l, X2).

By (a) and (b) in the proof of (3),

(Si -k-igi) — -Pi -\-iP2',

thus

gi,i — Pi> gi.i — Pit

gt,2— Pi) g2,i — Pl>

and by (c) and (d) the relation (b) is equivalent to

j + da = -^ft\

J [-Pi% ~ P2^2] da =

(d)

(e)

Integrating the left-hand side of (e) by parts, we arrive at

[P, r, + P, r,]-? -J^x,^^ + X2^]da = -^^~ /<-),

[Pj X^ Pg X2]aô' J 1% ^2 dP2 da

da -- (f)

1 — V

^ We assume that the parametrization of the are such that = e^pip coincides with the outward unit normal to 8R.

Airy’s solution. Sect. 47. 163

It follows from the definition of the Airy function that =Zl tp is single-valued. Thus for (f) to hold it is sufficient that

/K

/h^‘-

dP^

dP^

da^-- ^ 1 —V

(g)

da da =

i—v

Indeed, the first of (g) asserts that Pg single-valued, and the second and third of (g) are equivalent to (f) when both and P^ are single-valued. Next, by (d) and the definition of ,

[%,2 -co]. (h)

where m = i is constant (since is rigid). To prove that (g) is also necessary for (f) to hold it suffices to show that P^ is single-valued whenever is single-valued. But this fact follows from (h).i Finally, the Cauchy-Riemann equations and the relation imply that

= ^2,1 "I" -^2,2 ^2= -^,2 % d" Pi,i ^2

(i) = ^2 + % = ^ =-^ (Zl9>) .

In view of (i) and since Pi = A(p, the relations (g) reduce to the desired 3M—3

conditions. □

The next result extends (46.4) to multiply-connected regions. It asserts that the stresses corresponding to a solution of the traction problem with null body forces are independent of the elastic constants if the resultant loading on each boundary curve vanishes.

^(6) MichelVs theorem.^ Let R be multiply-connected, and suppose that

M

8R=U‘^«,

where each is a closed curve. Let and be plane elastic states corresponding to positive definite elastic constants [i, v and jx, v, respectively, and

assume that and are smooth on R. Then

provided both states correspond to the same surface tractions and the resultant force on each vanishes, i.e.,

= on dR and

Js^da=0 (m = i,2, ...,M).

1 Since is analytic, uâ is single-valued whenever has this property. “ [1900, 3], p. 109.

11*


Proof. '&Y (1) there exists an Airy function (p generating Moreover,

(p is of class C® on R, since is smooth there. Since [u^, is a plane elastic state, we conclude from (4) that

2/7

with rigid. Moreover, since is single-valued, and since the resultant force on each vanishes, it follows from (5) with f^^ =0 that

J-^{A(p)d(r = 0.

J[x,-g^^(Aq>) + x,f^~(Aq>)^da = 0. (a)

(A<p)-x,-^^-(Aq>)\dcr = 0.

for w = 2, 3, .... M. Now let

““ = 2V ('* •

By (a) and (5), is single-valued, and hence the converse assertion in (4)

implies that [#„, S^,^] is a plane elastic state corresponding to pi and v. Therefore

[w„, S^p] and [w„, S„^] correspond to the same elastic constants and the same surface forces. Thus we conclude from the uniqueness theorem (46.2) that

=^S^p on R. □

(7) Kolosov’s theorem,.^ Let [w„, he a plane elastic state corresponding to the elastic constants pi and v. Then there exist complex analytic functions ip and % on R such that^

Ui{xi,x^)+iu^{xi.x^)= [{')-Av)xp{z)-zfi:^-YW>\A-w{z),

^2)+522(%, ^2) ==4Re{v)'(^)},

Sii(xi, x^ 'S'22(%. ^2) 2* Sj2(%, Xg) = 2|T^ (^)]>

where w is a complex rigid displacement.

Conversely, let rp and % he complex analytic functions on R, and let u^ and S^p he defined hy the above relations. In addition, assume that u^ is single-valued and

and S^p continuous on R. Then [w„, S^p] is an elastic state corresponding to pi and V.

Proof. Let ip be an Airy function generated by S^p. By Goursat’s theorem (3), there exist auMytic complex functions ip and y such that

9>(%. ^2) = Re{i v' W 4-X(^)}• (a)

A simple calculation shows that

_9>,i(%.^2)+«>,2(%.^2)=V'W+îr(^ + FW- (b)

^ Kolosov [1909, 2], [1914, 2], [1935, 2]. A related complex representation theorem was established earlier by Eicon [1903, 2]. See also Muskhelishvili [1932, 7], [1934, 3], [1954, 76], § 32; Stevenson [1943, 2], [1945, 7]; Poritsky [1946, 3]; Sokolnikoff [1956, 72], Chap. 5; Solomon [1968, 72], Chap. 6, § 7.

’’ Here ip' = dipjdz.

Sect. 48. Representation of elastic displacement fields. 165

Thus we conclude from (4) that the desired relation for holds. Further, by (a) of (1).

4-4- . (c)

522 ^ 5i2 = {cp x 4" * 9’,2),1>

and (b), (c) imply the desired results concerning Conversely, let ip and x be complex analytic functions on R, let and be defined by the relations in (7), and let tp be defined by (a) so that (b) holds. Then it is not difficult to verify that u satisfies the relation in (4), and that satisfies (c). Thus are the stresses associated with (p, and we conclude from the converse assertion in (4) that [w„, is an elastic state correspondign to pi and v. □

VIII. Exterior domains.

48. Representation of elastic displacement fields in a neighborhood of infinity. The conventional proofs of the fundamental theorems of elastostatics are con¬ fined in their validity to bounded domains. Of course, they can be trivially generalized to exterior domains provided one is wilUng to lay down sufficiently stringent restrictions on the behavior at infinity of the relevant fields, e.g., that the displacement be 0 (r"i) and the stress 0 (r~^) as r-ôo. However, such a priori assumptions are quite artificial: the rate at which these fields decay is an item of information that one would expect to infer from the solution, rather than a condition to be imposed on a solution in advance.

In this section we shall establish representation theorems for elastostatic fields in the following neighborhood of infinity:

^^(oo) = {ac: >-(,< |a? — 0| < oo},

where >"(, > 0 is fixed. These representation theorems will be used in subsequent sections to infer the rate at which elastic states decay at infinity, and this, in turn, win be utilized to extend the fundamental theorems of elastostatics to ex¬ terior domains.

We assume that the body B=Z{oo) is homogeneous and isotropic, and that the elasticity tensor is positive definite) thus the Lame moduli satisfy

pi'^0, 2,pi 0,

(1) Representation theorem.^ Let u be an elastic displacement field on 2^(oo) corresponding to zero body forces. Then u admits the representation

oo oo

2 2 (a-) k<=—oo k = — oo

where gW and are vector solid spherical harmonics of degree k, and

Both of the series in (a), as well as the series resulting from any finite number of termwise differentiations, are uniformly convergent on every closed subregion of Z(oo).

Proof. First of all, it is not difficult to verify that the denominator in (b) cannot vanish, since the elasticity tensor is positive definite.

^ Kelvin [1863, 5]. See also Gurtin and Sternberg [1961, 11], Theorem 1.2. A generaliza¬ tion of this result to anisotropic bodies is given by B^:zier [1967, d].


In view of (42.2), u is bihaxmonic on 2’(oo). Theorem (8.5) therefore assures the existence of an expansion of the form (a) which has the asserted convergence

properties.

This leaves only (b) to be confirmed. A simple computation shows that

A (r^ Alt*)) = 4 (FAi'^)) p + , and thus the identity

(W*))^) = Âi**), (c)

which is valid for any vector sohd harmonic Ai**), imphes

Zl(r2AiW) = 2(2/fe4-3) Hence

00

4lit = 2 2 (2^ + 3) Alt*). (d) A=—00

Next, by hypothesis and by (42.2),

div4lii=0, curl4lii = 0; thus (d) implies

00 00

2 (2*4-3)divAit*)=0, 2 (2*+ 3) curl Alt*) =0. (e) k=gôo A=—00

But divAit*) is a solid harmonic of degree k — 1. Hence the first of (e), by virtue of the uniqueness of the expansion in (8.5), implies

div6<*) = 0. (f) Similarly,

curl 6t*) = 0,

or equivalently.

E-i*

1-1

T (g)

Consequently, (a), (c), (f), and (g) imply

00

FdivM= 2 [Fdiv^t*) 4-2(/fe + l)6<*)]. (h)

By h}q)othesis. [iAu-\- (A + jw) Fdivii= 0, (i)

and substitution of (d) and (h) in (i) yields

2 {2[(* +1) (A + 3^) + p] + (A + ^) FdivSfW} = 0, k = — oo

which implies the desired result (b). □

The next theorem gives a particular solution of the equations of elastostatics in a neighborhood of infinity.

(2)^ Let b be a smooth vector field on 2(oo), and assume that

div6=0, curl 6=0.

^ Gurtin and Sternberg [I961, ll], Theorem 4.3.

Sect. 49. Behavior of elastic states at infinity. 167

Then:

(i) 6 admits the representation 00

6=2 6W, 00

* where the 6*** are divergence-free and curt-freevector solid spherical harmonics of degree k, and this infinite series, as well as the series resulting from any finite number of termwise differentiations, is uniformly convergent in every closed subregion of 2 {00);

(ii) the same convergence properties apply to the series

u = —

„ 00 r‘ ^ 2 ^ (A + 1 j (A + ’iu)

6W,

and the field u so defined is an elastic displacement field on 2{oo) corresponding to the body force field b;

(iii) for any integer n.

implies b{x) = o(r") as r-^00,

u(x) = 0(r”'^^) as r-^00.

Proof. By (8.6) b is harmonic. Thus (8.4) implies part (i) of this theorem. The uniform convergence of the series of (ii), and of the corresponding series

resulting from successive differentiations, is apparent from the convergence prop¬ erties of the series in (i). Thus u is of class C°° on 2(00). That u satisfies the dis¬ placement equation of equilibrium may be verified by direct substitution. This establishes (ii).

To justify (iii) we note from (i) and (8.4) that b(x) =o(r”) implies b^^^x) =0 for k'^n. Bearing in mind (ii) and appealing once more to (8.4), we see that (iii) holds true. □

49. Behavior of elastic states at infinity. We now use the representation theorems developed in the preceding section to determine the behavior of elastic fields at infinity.

We continue to assume that B =^2{oo) is homogeneous and isotropic and that the Lame moduh obey the inequalities

In addition, we assume that the body force field 6 is smooth on 2^ (00), and that

div6 = 0, curl 6=0.

(1) Behavior of elastic states with vanishing displacements at infinity.^ Let [u, E, S] be an elastic state on 2{00) corresponding to 6, and assume that

u(x)=o(\) t-> as r->oo. Then

u(x)=0 (r"i), S(x)=0 (r"2),

E (x) = 0 {r~^), b{x) — 0 (r~^) as r-^00.

1 Fichera [1950, 4]. See also Gurtin and Sternberg [1961,11], Theorem 5.1. Kupradze

[1963, 17, 18], §111.3 established the estimate u—0{r~^), but only with the additional assumption that dujdr = 0 (r~^). Note that if u — o(l), then div u and curl u are both 0{r~^), a conclusion which Duffin and Noll [1958, S] reached by entirely different means.


Proof. By (42.2) u is biharmonic on 2'(oo). If we apply (8.5) to u, we con¬ clude that u—0{r~^). The remaining assertions then foUow immediately from (iii) of (8.5) in conjunction with the strain-displacement relation, the stress- strain relation, and the equation of equilibrium. □

(2) Behavior of elastic states with vanishing stresses at infinity.^ Let [tt, E, /S] be an elastic state on .S(oo) corresponding to b, and assume that

S{x) = o{\), b(x) = o(r-^) as r-^00. Then

u{x) =w (x) -f- 0 (r“i), S{x) = 0 (r'2),

E(x)=0(r-^), b(x)=0(r-^)

as r->oo, where w is a rigid displacement field.

Proof. In view of the assumed properties of 6, we infer from (48.2) the existence of a particular solution u of the displacement equation of equihbrium that is class on 2^ (00) and satisfies u =0(r~^). Accordingly, by (1), the elastic state associated with u (i.e., generated by the strain-displacement and stress- strain relations) satisfies E=0(r'^ and S=0(r'^), while b must conform to b—0{r~^). We are thus able to exhibit a particular elastic state corresponding to 6 which possesses the requisite orders of magnitude at infinity. Therefore, since the fundamental system of field equations satisfied by elastic states is Mnear, it remains to be demonstrated that (2) is true when b=0.

Suppose now that 6=0. Then, by hypothesis and by (48.1), u admits the representation

u — u-\-u, where

00 00

A=-2

M= 2 +>'^2 k——oo 00

and where the soMd harmonics 6*** obey (b) of (48.1). O /N,'

Next, define fields E and E through

E = l{Vu+Fu'^),

E = i{Vu+Fu% so that

E = E-{-E.

The two infinite series entering the second of (a) each represent a harmonic field on 2^(00). The second of (b) and part (iii) of (8.5) therefore imply

ti = 0(r-i), E = 0(r-^), (d)

while we conclude from the stress-strain relation that

S = 0(r-^).

Consequently, aU we have left to show is that u is rigid; or equivalently that

i = 0. (e)

(a)

(b)

(c)

^ Gurtin and Sternberg [1961, 1T\, Theorem 5.2.

Sect. SO. Extension of the basic theorems. 169

To this end we conclude from (b) in (48.1) that

divAi*** = 0,

and hence reach, with the aid of (a) and (b) above,

o 00 00

trE=2divsfW+2 2 (f) A=0 A=—2

where p(x) =x — Q. Next, since divgf**+^* is a solid harmonic of degree ^4-1,

p • P’divsr<*+®> = (^ + 1) divgf<*+®>, (g)

and since g*®* is a constant, F9<®)=0. (h)

Thus if we eliminate from (f), by recourse to (b) of (48.1), and use (g) and (h), we arrive at

tr£’= y--divQ<*> kh {k-i)a+3fi)+fi

By hypothesis, S =o(l); thus the stress-strain relation implies

E = o(t),

and we conclude from (c) and the second of (d) that

(i)

E = o{i). (i)

Thus if we apply (8.4) to the harmonic function tr E given by (i), bearing in mind that div g*** is a solid harmonic of degree k — i, we find that

div gf*** = 0 for ^ ^ 1,

which, together with (b) of (48.1) and (h), yields

fiW = 0 for k^ — 2. (k)

Inserting (k) in the first of (a), and using (h) and the first of (b), we obtain

o 00 ^

E^Z^gW. (1) ^ = 1

Since Ffif*** is a solid harmonic of degree ^ — t, we conclude from part (iii) of (8.4) that (1) is consistent with (j) only if (e) holds. □

50. Extension of the basic theorems in elastostatics to exterior domains. We now use the results estabhshed in the last section to extend the basic theorems of elastostatics to exterior domains. Thus for this section we drop the requirement that B be bounded and assume instead that B is an exterior domain. We shall continue to assume that B is homogeneous and isotropic with Lame moduli [i and A.

Let 4= [tt, E, S] be an elastic state on B corresponding to the body force field 6, and suppose that the limits

/ = lim / s da, g-*oo sXp

m-- lim / pxsda g-*oo usp


exist, where s=Sn, n is the outward unit normal to 32’^, and Ug is the ball of radius q and center at 0. Then we call / and m the associated force and moment at infinity. Since S and b satisfy balance of forces and moments in every part of B,

f s da-ir J b dv = — f s da, dB Bp dSp

jpy.s da + j pxbdv = -~ fP xs da, dB Bp dZp

for sufficiently large q, where Bg = BnZg} Thus, letting qôo, we see that / and m exist if and only if the integrals

/ 6 dv, f pxbdv B B

exist, in which case f = — f s da — J b dv,

dB B

m = — Jpxs da— f pxb dv. dB B

(1) Extension of Betti’s reciprocal theorem.^ Assume that the Lame

moduli obey p>0, 2jtt + 3 A > 0. Let [tt, E, S] and \u, E, S] be elastic states cor¬

responding to the external force systems [b, s] and [ft, s], respectively, with b and b divergence-free and curl-free and of class on B. Further, assume that the associated

forces f, f and the associated moments m, m at infinity exist. Finally, assume that

u(x) = w(x) 4- o(l),

u(x) =w(x) 4-0(1),

as r-^00, where w and w are the rigid displacements:

w = u^-\-(o^xp,

W = Upp-\-(Op,Xp.

Then

fs-uda-y f b •iidvf • u^-\-m' dB B

= f s ■ uda-i- f b ■ udv 4- / • 4- ^ • cOo, = fS-Edv=fS-Edv. dB B B B

Proof. Applying Betti’s theorem (30.1) to the region Bg=Bnhg, with q large enough so that dBC^g, we find that

/ s ■ iida -\- f b ■ udv-\- f s- uda dB Bp dSp , ,

= f s ■ uda -i- f b • udv f s ■ uda = f S ■ Edv = f S - Edv. dB Bp dXp Bp Bp

Clearly,

/ S'uda = J s- (u — w)da-\- f s ■ wda (b) dXp d2 p d^p

1 Note that 8Bg = 8B u 8£g for q sufficiently large, since B is an exterior domain.

“ This result for the case in which w —w = 0 is due to Gurtin and Sternberg [1961, 11], Theorem 6.1.

Sect. 50. Extension of the basic theorems. 171

and

f s-wda= f s • {u^ 4- co^ xp) da = u^ • J s da +co^ • f pxsda. (c) 0X'p 0i'p dSp dSp

By hypothesis and by (49.1),

u = w0(r~^), u = w-{-0{r~''),

E = 0(r-^), S = 0{r-^), b = 0{r~^), (d)

E = 0{r~^), S = 0{r-^), b = 0(r~^).

Thus (b) and (c) imply f s-uda^f-u^+m-m^ (e)

duS Qôo. Similarly,

^ s-uda^f-u^-irrh-m^ (f)

as p-ôo. The estimates (d) also imply that the volume integrals over in (a) are convergent as p-ôo and tend to the corresponding integrals over B. Thus if we let p-ôo in (a) and use (e) and (f), we arrive at the desired conclusion. □

As a direct corollary of this result we have the

(2) Theorem of work and energy for exterior domains.^ Let [tt, E, S], [b, s], /, m, as well as fi and A obey the same hypotheses as in (1). Then

f s-uda-j- f b ■ udv -j- fu^ + m ■ co^ = 2 Uc{E}. es B

This relation differs from the corresponding result for finite regions by the addition of the term f u^ + m m^, which represents the work done by the force and moment at infinity.

(3) Uniqueness theorem for exterior domains.^ Assume that

P'^0,

Let i = [u, E, S] and 'S = [m, E, S] be elastic states corresponding to the same body force field and the same boundary data in the sense that

u = u on s = s on (a)

Further, assume that either u(x) = u{x) 4-0(1)

or both as 00,

S{x)—S{x) + 0(1) as r-^00

and the corresponding traction fields are statically equivalent on dB, i.e.

fsda = fsda, f pxsda= f pxsda. dB SB dB dB

(b)

(c)

(d) 3

Then <i and % are equal modulo a rigid displacement.

1 Gurtin and Sternberg [I96I, 11], Theorem 6.2, for the case in which tp = 0. 2 Fichera [1950,4], Theorems II, V, and VI for case(b): Gurtin and Sternberg

[1961, 11] for case (c). See also Duffin and Noll [1958, S] and Bezier [1967, 3]. Analogous uniqueness theorems for two-dimensional elastostatics are given by Tiffen [1952, 6] and Muskhelishvili [1954,16], § 40.

® Note that this condition is satisfied automatically when B is finite.


Proof. Let 6'=6—6. It suffices to prove that

jE' = 0. (e)

Clearly, <>' is an elastic state corresponding to zero body forces and vanishing boundary data. Assume (b) holds. Then

u'{x) = o{\) as r->oo,

and (2) (with u’^ — = 0) yields = (f)

which impUes (e). Next assume (c) and (d) hold. Then

S'(a5) = o(l) as r->oo,

and we conclude from (49.2) that

u'{x) — w (x)0 (r~^) as r->oo

with w rigid. Further, by (d),

/'=— fs'da = 0, m' = — Jpxs'da = 0, dB dB

and (2) again implies (f) and hence (e). □

It should be noted that (3) is valid only for unbounded regions with finite boundaries. The uniqueness question associated with problems for general domains whose boundaries extend to infinity is yet to be disposed of satis¬ factorily. ^

We emphasize that theorem f3_), in contrast to (1) and (2), involves no explicit restrictions concerning the body force field. If this field is integrable over B, then (d) can be replaced by the requirement that the forces and moments at infinity coincide:

/=^/, m = m.

For the surface force problem, (d) may be omitted because it is implied by (a). On the other hand, if is not empty, (d) is an independent h5q)othesis. This hypothesis, which requires the prescription of the resultant force and moment on dB (or equivalently, the force and moment at infinity), appears at first sight artificial, since the surface forces over at least a portion of the boundary are not known beforehand. One is therefore led to ask whether (d) is necessary for the truth of (3) when the regularity conditions are taken in the form (c). That this is indeed the case is clear from the next theorem.

(4) Theorem (3) is false if either of the two hypotheses in (d) is omitted.

Proof. For the purpose at hand, let

S={£c:|£c —0|>1},

and assume that dB =STx {■5^2 = 0) in (3). It clearly suffices to exhibit two elastic states, both corresponding to zero body forces, both having strains not identically zero, both satisfying

M = 0 on dB, (a)

/S = o(l) as r->oo, (b)

1 For the special case of the displacement and surface force problems appropriate to the half-space, this question was settled by Turteltaub and Sternberg [1967, 18]. See also Knops [1965, 13].

Sect. 51- Basic singular solutions. 173

with one state obeying j sda = Q, (c)

dB

the other f pxsda=0. (d) ■

dB

To this end, let A 4= 0 and to 4= 0 be given vectors and consider the following two displacement fields 4

«(*)= i-+^6.)-^div(-^-)-A. (e)

«(®) = + 4(4-'5V ® XP- (f)

Both of these displacement fields generate elastic states that correspond to zero body forces and obey (a), (b). Moreover, for (e);

/= Js<^« = A4= 0, m = Jpxsda = 0;

dB dB

while for (f): /=0, m = 87ifi(o. □

The leading two terms in (e) represent the displacement induced in the medium by a rigid translation A apphed to the spherical boundary dB, the body being constrained .against displacements at infinity; the last term in (e) corre¬ sponds to a rigid translation —A of the entire medium. On the other hand, the leading two terms in (f) represent the displacement generated by a rotation to of dB, while the last term corresponds to a rotation of the entire medium.

IX. Basic singular solutions. Concentrated loads. Green’s functions.

51. Basic singular solutions. In this section we study the basic singular solutions of elastostatics, assuming throughout that the body is homogeneous and isotropic, and that /t4=0, v4=i-, 1- We begin with the Kelvin problem,^ which is concerned with a concentrated load apphed at a point of a body occup5dng the entire space S. With a view toward giving a solution of this problem, we introduce the following notation.

Throughout this section y is a. fixed point of S,

r = x — y is the position vector from y, and

r= |r|.

Further, we write Z, for the open ball (y) with radius rj centered at y.

1 Gurtin and Sternberg [1961, H] attribute these examples to R. T. Shield.

^ The solution to this problem was first given by Kelvin [1848, .3] (cf. Somigliana

[1885, 4], Boussinesq [1885, i]). It was derived by Kelvin and Tait [1883, 1], §§ 730, 731

through a limit process that was made explicit by Sternberg and Eubanks [1955, 73] (cf. Love [1927, 3], § 130; Mindlin [1936, i], [1953, 75]). The limit formulation presented here follows closely that adopted by Sternberg and Al-Khozai [1964, 20] and Turteltaub

and Sternberg [1968, 74]. The solution for an anisotropic but homogeneous body was given by Fredholm [1900, 2]. A precise statement of Fredholm’s result is contained in the work of SAenz [1953, 77]. See also Zeilon [1911, 7], Carrier [1944, 7], Lifshic and Rozencveig

[1947, 4], Elliot [1948, 2], Synge [1957, 15], Kroner [1953, 74], Basheleishvili [1957, 7],

Murtazaev [1962, 12], BAzier [1967, 3], Bross [1968, 7], C^kala, DomaiJski, and Milicer-

Gruzewska [1968, 2].


By a sequence of body force fields tending to a concentrated load laty we mean a sequence {6^} of class vector fields on S with the following prop¬ erties :

(i) 6„ = 0on.r-i:i/„;

(ii) J b^dv-^l as m-ôo;

(iii) the sequence is bounded. S'

(1) Limit definition of the solution to Kelvin’s problem.^ Let {6^} be a sequence of body force fields tending to a concentrated load I at y. Then:

(i) For each m there exists a unique elastic state 6^ on S that corresponds to the body force b^ and has uniformly vanishing displacements at infinity.

(ii) The sequence converges {uniformly on closed subsets ofS—{y}) to an elastic state 6^ [i] onS — {y).

(iii) The limit state 6y\l] is independent of the sequence {6^} tending to I and is generated by the displacement field Uy\l\ with values

Uy[l] (®) = 1

cr r(g)r

y2~ + (3-4v)l i,

c= 16:71/1(1 — 1').

We call 6y [i] the Kelvin state corresponding to a concentrated load I at y.

Proof. Let u^ be defined by

where -vf

Wm {X)=0ij dvii, (p„,(x) = -(tj î/m î/m

r(^)-b^(g) \x~g\

dv£, in/j,

(a)

Then, by (6.5) and (44.1), u^ is an equihbrium displacement field corresponding to b^. Moreover, it is easy to verify that

M,„(a5) = o(l) as I®—0|-ôo,

and, by the uniqueness theorem (50.3) for exterior domains, there is no other equihbrium displacement field corresponding to b^ with this property. This establishes (i).

Next, a simple calculation shows that the functions defined on S’—{y} by

q>{x) = 0, xi>{x)= (b)

generate, in the sense of the Boussinesq-Papkovitch-Neuber solution (44.1), the displacement field Uy[l] given in (iii). Thus to complete the proof we have only to show that given any closed region R in (^—{y}, uniformly on R; or equivalently, that (p^-^(p and Wm-^W uniformly on R, and that the first and second gradients of q)^ and xp^ tend to the corresponding gradients of (p and \p uniformly on R. Since the argument in each instance is strictly analogous, we shah merely prove that

^ Sternberg and Eubanks [1955.^3], Theorem 4.2. An analogous theorem for con¬ centrated surface loads was given by Turteltaub and Sternberg [1968, 1^].

Sect. 51. Basic singular solutions. 175

Let ^ be a closed subset of {y}. Then there exists an m^>0 such that

= 0; hence

for a5G.R and

is finite. Thus

1

1

1 ^ 1 \x-^\-\x-y\

1

1

m

for xeR, and Next, by (a) and (b),

|V».(®)-V(®)| = |«| -i/m

-.11 /

1

\^-y\ dvf

b„dv — l

consequently, (c) and (d) imply

|Vm(®)-V(®)1 /iKldv + m^lxi Jb^dv-l îlm îlm

(c)

(d)

for a; 6 i? and w>Wq. This inequality, when combined with (ii) and (iii) in the definition of a sequence of body force fields tending to a concentrated load,

impHes uniformly on R. □

The need for condition (iii) in the definition of a sequence of body force fields tending to a concentrated load was estabhshed by Sternberg and Eubanks,^

who showed by means of a counterexample that conclusions (ii) and (iii) in (1) become invalid if this h5rpothesis is omitted.

We will consistently write

for the Kelvin state corresponding to a concentrated load I at y. Clearly, the mapping <iy[l] is linear on iC.

In view of the stress-strain and strain-displacement relations, the stress field Sy[l] has the form

Sy [*] (®) = -~^ |3+ (1 _ 2,) [r ® I + I ® r - (r ■ f) 1]}.

Let Sy [I] be the unique tensor field with the property that

for every vector v, and let

on 8B. We call Sy [I] the adjoint stress field and Sy [I] the adjoint traction field corresponding to <iy[l]. A simple calculation shows that

S* [f] (X) = - +(i-2v)[l®r-r®l+(r-l) 1]}.

Note that [I] is not symmetric.

^ [1955, 13], Theorem 4.3.


(2) Properties of the Kelvin stated The Kelvin state <iy[l\ has the follow¬ ing properties:

(i) 6y [i] is an elastic state onS— {y} corresponding to zero body forces.

(ii) Uy [1] {x) = 0 [r~^) and Sy [H] (x) = 0 (r~^) as r-^0 and also as r-^00.

(iii) For all r]>0, f Sy[l]da = l, f rxsy[l]da = 0,

where

on dZ^ with n the inward unit normal.

(iv) For every vector v,

l-Uy[V]=VUy[l], ^•S^W='»-S*[ir],

where Sy [tJ] is the traction field on 8B corresponding to 6y [v] and Sy [1] is the adjoint traction field on dB corresponding to 6y\l\.

Proof. Property (i) follows from (4d.l) and the fact that the stress functions (p and ip defined in (b) of the proof of (1) are harmonic on^—{y}. Properties (ii) and the first of (iv) are established by inspection of the fields Uy\l] and

Property (iii) can be established by a direct computation based on the form of An alternative proof proceeds as follows. Let and {6^} be as in (1). Then balance of forces and moments imply that

-fs^da+f b^dv = 0,

— f rxs^da-\- J rxb^dv = 0, d£fj Sfj

where is the traction field on that side of dZ^ facing the point y. If we let m->oo in the above relations and use (ii) ot (1) and properties (i)-(iii) of {b^}, we arrive at the desired relations in (iii).

Finally, to estabhsh the second of (iv) note that, by the symmetry of Sy[l] together with the definitions of Sy [I] and [I],

I • = I • (S^M n) = (S^M l)-n= {S* • n

= v(S*[l]n)=vs*[l]. □

Sternberg and Eubanks ^ have shown that the formulation of Kelvin’s prob¬ lem in terms of (i), (iii), and the portion of (ii) concerning the limit as r->oo, which appears in the hterature,® is incomplete in view of the existence of elastic states on ^~{y} that possess self-equilibrated singularities at y.* In contrast, properties (i), (ii), and the first of (iii) suffice to characterize the Kelvin state uniquely.®

1 Properties (i)-(iii), in the precise form stated here, are taken from Sternberg and Eu¬

banks [1955, 13], Theorem 4.4. Assertion (iii) can be found in Love [1927, 3], § 13I; Lur’e

[1955, 10], § 2.1.

" [1955, 13]. ® See, e.g., Timoshenko and Goodier [1951, 10], § 120.

* E.g. the center of compression defined on p. 179.

® As Sternberg and Eubanks [1955, 13] have remarked, Trefftz [1928, 3], § 32 and

Butty [1946, 1], § 350 approached the Kelvin problem on the basis of properties (i), (ii),

and the first of (iii) but made the erroneous assertion that (ii) is a consequence of the first of

(iii). That this is not true is apparent from the state [I] -(- a where a is a scalar and 5^ is the center of compression defined on p. 179. Indeed, this state obeys the first of (iii)

without conforming to (ii).

Sect. 51- Basic singular solutions. 177

Let Uy[l] be defined on S’—{y} by

so that

r®ei+ei(8)r- _(3_4y)y. 1 j;.

By (42.3), Uy[l] is an elastic displacement field on corresponding to null body forces. Let [H] be the elastic state on <S’—{y} generated by Uy[l]. We call d* [1] the doublet state (corresponding to y, I and e^).^

Note that

< [1] (®) = lim ~ {Uy [f] (as) - Uy [i] (as - Ae^)}.

Thus, since Uy[l\{x-a) = Uy+„[l] (as),

it follows that

%[l\{x) = lim^\uy I I (a5) + Mj,+*e, -y* (*)},

or equivalently that

«»[*] = {“» T * + %+*e, - y * }•

Thus Uy[l] is the hmit as h-^0 of the sum of two displacement fields. These fields are associated with two Kelvin states: the first corresponds to a concen¬ trated load

I apphed at y,

the second corresponds to a concentrated load

— apphed at y-[■he^.

The two load systems are shown in Fig. 13. The above relation also 5delds the important result

Fig. 13-

1 These states were introduced by Love [1927, 3], § 132. See also Sternberg and Eubanks

[1955, 73]. Handbuch der Physik, Bd. VI a/2. 12

178 M. E. Gurtin ; The Linear Theory of Elasticity. Sect. 51-

The next theorem is the analog oi (2) for doublet states; its proof is strictly analogous and can safely be omitted.

(3) Properties of the doublet states,^ The doublet state has the following properties:

(i) ^[l] is an elastic state onS—{y] corresponding to zero body forces.

(ii) (x) =0(r~^) and (*) =0{r^^) as r-^0 and also as r-^cx}.

(iii) For all r]>0,

where

fs\,[l\da=b, f rxs\,[l\da =— e^xl,

on dZ.^ with n the inward unit normal.

(iv) For every vector v,

* ■ [®] = ® •

(v) If 6y [H] is the Kelvin state corresponding to the concentrated load I at y, then

i.e.

Uy [i] = - U* [i] , [l] =

Property (iii) could be anticipated intuitively because of the physical meaning attached to the doublet states (cf. Fig. 13). As is clear from (iii), the singularity of 3y [H] is statically equivalent to a couple or to nuU according as

ejXi=|=0 or e,xl! = 0.

In contrast to properties (i)-(iii) of (2), which characterize the Kelvin state uniquely, (i)-(iii) of (3), clearly, do not supply a unique characterization.^ Indeed,

[I + a e j (no sum) has the same properties as [1].

We call

the unit Kelvin state corresponding to a unit load at y in the direction of e,-. On the other hand, the unit doublet states at y are defined by

By (3) the singularity of 5^ at y is statically equivalent to a couple or to null depending on whether i^j or i=j. We refer to 5^ , i^j, as the unit doublet state with moment e,- X e^- at y, and to dy (no sum) as the unit doublet state corresponding to a force doublet at y parallel to the X{-axis. Clearly,

^y [^] = h ^y ’

and the displacement fields belonging to and are given by

<(*) = -/-[^ + (3-41-) e,j.

' Sternberg and Eubanks [1955, IS], Theorem 5.2. 2 Sternberg and Eubanks [1955, 13], p. 152.

Sect. 52. Concentrated loads. The reciprocal theorem. 179

The state

^ ^

is called the center of compression at y, while

is thecenter of rotation at y parallel to the %^-ajds.^ Using the formula for u^,

it is a simple matter to verify that the displacement fields Uy and Uy associated

with Sy and ly are given by

nUx):

cr^

rxci

Sti

Moreover, it follows from (iii) of fS) that

/ Syda= JrxSyda = 0,

fsida = 0, frxsi,da = ei,

where

Sy = Syn, sj,= S^n,

and n is the inward unit normal to dZ^.

52. Concentrated loads. The reciprocal theorem. By a system of concentrated loads we mean a vector-value(^ function I whose domain Di is a finite set of (boundary or interior) points of B; the vector I (^) is to be interpreted as a concen¬ trated load at If DindB = 0, we say that the concentrated loads are internal. Finally, we write I— I (^) when Df is a singleton

Our next definition is motivated by (51,2) and the ensuing discussion. Let I be a system of concentrated loads. We say that d = \u, E, S] is a singular elastic state corresponding to the external force system [6, s, f] if

(i) d is a (regular) elastic state on 5 — Uj corresponding to the external force system [6, s];

(ii) for each ^zDi, u{x) =0(r~^) and S(x) —0(r~^) as r = |a5 —

(iii) for each ^6Z)|, lim J Snda = l{^),

where S^g(^) —BndZg(^) and n is the inward unit normal to dZg(^).

It follows from (51.2) that for ^zB the Kelvin âte corresponding to a concentrated load f at ^ is a singular elastic state on B corresponding to I and to zero body forces.

By (ii) and (iii) we have:

(iv) for each Di,

lim J pxSnda= {^ — 0)xl(^).

1 These solutions were first introduced by Dougalc [1898, ^]. Cf. Love [1927, 3], § 132;

Lur’e [1955, 10], § 2.3; Sternberg and Eubanks [1955, 13]. 12*


Indeed, by (iii),

/ pxSnda—{^ — Q)xl{!i)= f (x — ^ xS{x)n(x) da^-i-o(i)

as g->0, and by (ii) the integral on the right tends to zero in this limit.

The next theorem shows that for internal concentrated loads in a homo¬ geneous and isotropic body, every singular elastic state admits a representation as a regular elastic state plus a sum of Kelvin states.

(1) Decomposition theorem. Assume that the body is homogeneous and isotropic. Let I be a system of internal concentrated loads, and let 6bea singular elastic state corresponding to I and to zero body forces. Then 6 admits the decomposition

*>=*>«+ 2j

where is a (regular) elastic state corresponding to vanishing body forces, and for each ^^Di, <>5(= [I(^)]) is the Kelvin state corresponding to the concentrated load

m at I

Proof. For each let

with Q sufficiently small that

i(^)CS, i(^)nZ)i=0 o

for all âDi. Then, on 2’(^), u admits the representation^

oo oo

«= 2 2 (a) k——oo k=—oo

where and are vector sohd spherical harmonics of degree k being the origin of the spherical coordinate S3^tem) and

m) ^- -_u+n)-y (A)

Further, by (ii) in the definition of a singular elastic state,^

= 0, kS.-2, (c)

and since g^, is a homogeneous polynomial of degree k, (b) implies

fc^-2) = fi^-i) = 0. (d) For each let

“s = +»'! (e)

on S’—then a simple computation using (b) shows that the state corre¬ sponding to (in the sense of the strain-displacement and stress-strain relations) is an elastic state oni^—{^} corresponding to 6=0. Thus if we let

Mjf = M — 2 «£ (f)

1 It is clear that the representation theorem (48.1) remains valid when the deleted neigh- O

borhood of infinity, 27(00), is replaced by the deleted neighborhood of g, 2{g).

‘ This follows from the analog of (8.5) for 27(^).


on B —Di, then the associated state dj; is an elastic state on B—Di also corre¬ sponding to 6=0. It follows from (a) and (c)-(e) that for each

A=0 A=0

§H=i»

(g)

on .r(ij); thus u^(x) can be continuously extended to x = ri, and the resulting extension is C°° on 21j(ij). Therefore so extended is an elastic state on B corre¬ sponding to zero body forces.

To complete the proof, it clearly suffices to show that is the Kelvin state corresponding to l(^). Since

where the vector a is a constant, (b) and (e) 5deld

(A u^{x) = fx4 f^div ■ - . (h) ' 2(2k + Sn) « '

By (f) and (g), M — Mj is C°° on (^); thus the stress field corresponding to satisfies (iii) in the definition of a singular elastic state:

lim / S^n da=l{^). (i)

An elementary computation based on (h) and (i) shows that

a = -- (2A 5/i)

-m-. I27t /l{X + 2/i)

hence we conclude from (h) and (51.1) that 6^ is the Kelvin state corresponding to l{^) at I □

(2) Balance of forces and moments for singular states. Let I be a system of concefdrated loads, and let 6 be a singular dastic state corresponding to the external force system [6, s, I] with b continuous on B. Then

f sda-\- fb dv-\- 2 m = 0, dB B SeDi

f pxsda+ f pxb dv+ 2 (^ —0) xf(^) = 0. dB B Dj

Here the surface integrals are to be interpreted as Cauchy principal values; i.e.

where

/s<^«=hm Jsda, dB a->-o «’(a)

n=l

{^1 > ■■■> ^n} = Din dB, Q — {Qi, ■ ■ ■, Qn) •

Proof. Let Bi = DinB,

B(d,Q)=B- u U (a)

and choose 5 and q sufficiently small that the closed balls mentioned in (a) are mutually disjoint. (Note that •••> are aU points of dB; while each ^6 Bj is an interior point of B.) Then is a (regular) elastic state on B (5, g); hence

/ s J b dv=0. 0B(<5, a) B(a.e)

(b)


Since b is continuous on B,

f bdv^ f b dv as ^^0, (O-Ô. (c) B{d, q) b

Further, by (a), if d and q are sufficiently small,

/ = / + I / + 2 . (d) SB{d,a) n=i S>’ej{„) SSBiS^ms)

where ^^(x) = Bndi:^(x). (e)

Combining (b) and (d) and letting ^-^0, Q->-0, we conclude, with the aid of (c) and (iii) in the definition of a singular elastic state, that

f s da dB

exists as a Cauchy principal value, and that the first relation in (2) holds. The second relation is derived in exactly the same manner using (iv) in place of

(iii). □

We now give an extension of Betti's reciprocal theorem (30.1) which includes concentrated loads. This result, which is due to Turteltaub and Sternberg,^

will be extremely useful in establishing integral representation theorems for elastic states.

(3) Reciprocal theorem for singular states. Assume that the elasticity

field is symmetric and invertible. Let I and I be systems of concentrated loads with Dg

and Df disjoint. Further, let [u, E, S] and [m, E, S] be singular elastic states corre¬

sponding to the external force systems [5 s, 1] and [5, s, f], respectively, with b

and b continuous on E. Then

f s-uda-j- f b ■ u dv i{^) • u{^) dB B SSDg

= f s-uda-\- f b-udv 2 ^(?) ’«(?) dB B {SDg

= fS-Edv = f S-Edv, B B

where the surface integrals are to be interpreted as Cauchy principal values.

Proof. Let

df (x) = u(x)—u ($) (a)

for each x^B and Then (iii) in the definition of a singular elastic state impUes

/ S'uda— f s-5^da-\-l{^) ■ u{^)o{t) as q->-0 (b)

for every êDg. Since o is an elastic state on E—Dg and 3 an elastic state on B—Dg, and since DgnDg =0, (a), (b), and (ii) in the definition of a singular state yield

/ s •uda^l{^) • u{^), f s-udaÔ (c) _êd) »’ed)

1 [1968, U].


as p-^0 for each Similarly,

J s •uda->-i{^) - u{^), f S'uda->-0 ê(«) ê(«)

as p-^0 for each

Now let

(d)

N

W = 1

Q—{Qi> 6n) >

B(^, 0) = B-U ^ - U n=l

(e)

Then <s and 3 are (regular) elastic states on B{d, q), and Betti’s theorem (30.1) implies

/ s-uda-\- / b-udv= f s-uda-\- f b-udv dB(d,Q) B(d,Q) dB(6,Q) B(6,a)

= / S-Edv= / S-Edv. B(d,Q) B(d,a)

Further, by (e), if d and q are sufficiently small,

f =/ + 2 ^/ + 2 «’(a) n=l S>’gJS„) iSBii

where we have used the notation given in (e) in the proof of (2).

(g)

Since C is invertible on B, we conclude from (ii) that JS is 0 (r as the dis¬ tance r from any of its singularities tends to zero. Thus, since DinDj —0, (ii) im¬

plies that S • .E is also 0 (r'^) near its singularities. Clearly, the same property

holds for S • E. Further, since b and b are continuous on B, b-ii and b • u are 0 (r'l) near their singularities. Thus as d -^0 and q -^0 all of the volume integrals in (f) tend to the corresponding integrals over B. Therefore, if we pass to this limit in (f) and use (g), (c), and (d), we arrive at the desired result. □

(4) Somigliana’s theorem.^ Assume that the body is homogeneous and iso¬ tropic with Lame moduli that satisfy ^ =b 0, 2^ -|- 3 A =b 0. Let [m, E, S] he a singular elastic state corresponding to the external force system [5, s, f]. Then^

«(!/) =/(*€„[«] -s*\u])da-\- Ûyib^dv-^r 2 «»[*(?)](?). dB B

or equivalently,

Ui{y) = I {ui,-s — si,-u)da-\-ft4,-bdv-{- 2 *€^(?) •*(?), dB B_ fSDi

1 SoMiGLiANA [1885, [1886, 4], [1889, 3] for a regular elastic state. See also Dougall

[1898, 4], The extension to singular states is due to Turteltaub and Sternberg [1968, 14], pp. 236-23 7. Somigliana’s theorem was utilized to determine upper and lower bounds on elastic states by Diaz and Greenberg [1948, 1], Synge [1950, 12], Washizu [1953, 20] and Bramble

and Payne [I96I, 6], Kanwal [I969, 4] used Somigliana’s theorem to derive integral equa¬ tions analogous to those used to study Stokes flow problems in hydrodynamics. See also Haiti

and Makan [1971, 3].

2 The properties of the Kelvin state in conjunction with this result furnish an alternative proof of the fact that u is of class C°° on B.


for yeB—Di, where for any vector v,Vy[v] is the displacement field and s*[w] the adjoint traction field corresponding to the Kelvin stated while Uy and s'y are the displacement and traction fields for the unit Kelvin state^ <Sy.

Proof. Let v be an arbitrary vector. Then the reciprocal theorem (3) apphed to the states 6 = [m, E, S] and 6y [w] yields

u{y) ■■»= / {s-u [v]—u-Sy[v])da-\- f b-Uy[v]dv-f- 2 *($) ■«!,[»](?)• SB B «6D|

In view of (iv) of (51.2), this relation implies the first formula in (4). On the other hand, if we take in the above relation, we are led to the second formula. □

As a corollary of SomigUana’s theorem we have the

(5) Integral identity for the displacement gradient.^ Assume that the hypotheses of (4) hold. Then for every y^B—Dg,

-f {u^-s-~sjj-u)da-fv^-bdv-'û^ ($) •!($), SB B S€Di

div u{y) = — f {Uy -s—Sy ■u)da — J Uy-bdv—^ %(?) ‘ *(?); SB B S&Di

i[curl= f (v^-s-si,-u)da+ f^-bdv+Zî^-ti^)’ SB B iSDi

where and s'J are the displacement and traction fields for the unit doublet state*

, Uy and Sy for the center of compression^ ly, and and Sy for the center of

rotation ® i'.

Proof. If we differentiate the second relation in (4) with respect to yp we arrive at ’’

Ui.idi)=/ «- 4 ** ■ “) 4 •

In view of (v) of (51.3) and the definition of unit doublet states given on page 178, this relation imphes the first formula in (5). The second and third formulae are immediate consequences of the first. □

It is a simple matter to extend the mean strain and volume change theorems (29.2) and (29.3) to include concentrated loads. In particular, the formula for the volume change in a homogeneous and isotropic body becomes

dv{B) ■sda-\- p • b dv + ^ — 0) ■

where k is the modulus of compression.

* See p. 174. ^ See p. 178. * The first identity is apparently due to Love [1927, 3], § 169. The formulae for the

divergence and curl of u are due to Betti [I872, 7], §§ 8, 9. See also Dougall [I898, 4], Love [1927, 3], §§ 16O, 162.

* See p. 178. 5 See p. 179- ® See p. 179. ’’ The differentiation under the integral sign is easily justified. Cf. the proof (Kellogg

[1929, 7], p. 151) of the differentiability under the integral sign of Newtonian potentials of volume distributions.

Sect. 53. Integral representation of solutions to concentrated-load problems. 185

Suppose now that .s = 0 on SB, 5=0 on B,

and that the system I consists of a pair of concentrated loads. If we let Bj = , §2}, then it follows from balance of forces and moments (2) that

and there results the following elegant formula for the volume change:

Thus the volume change due to a pair of concentrated loads is independent of the shape or volume of B; it depends only upon the magnitude of the loads, their dist¬ ance apart, and the modulus of compression of the material.

53. Integral representation of solutions to concentrated-load problems. In this section we will use the reciprocal theorem for singular states to establish an integral representation theorem for the mixed problem of elastostatics with concentrated loads included. Thus we assume given a symmetric and invertible elasticity field C on B, surface displacements m on .5^, surface forces s on .5^, a continuous body force field 5 on B, and a system of concentrated loads 1. The generalized mixed problem is to find a singular elastic state [m, E, S] that corre¬ sponds to the body force field 5 and to the system of concentrated loads I, and that satisfies the boundary conditions

M = M on s = s on —

We will call such a singular elastic state a solution of the generalized mixed problem.

In order to introduce the notion of a Green’s state we need the following

(1) Lemma. Let f and m be given vectors. Then there exists a unique rigid displacement field w such that

f wda=f, f PcXw da = m, dB dB

where c is the centroid of dB and Pc{x) =x—c. We call w the rigid field on dB with force f and moment m.

Proof. Suppose such a field w exists and let

Since c is the centroid,

and thus

w = a-\-a)XPc.

J Peda = 0, dB

f w da==aa, dB

(a)

(b)

(c)

where a is the area of dB. Further, (a) and (b) also imply

f PcXwda= f PcX(o)XPc)da = Io), (d) dB dB


where I is the centroidal inertia tensor:

1= S [W^-Pc®Pc]da. (e) dB

Since I is invertible, (c) and (d) imply that if such a rigid field w exists, it is unique and

a= -, (o — I~^m. (f) a ''

Conversely, it is a simple matter to verify that w defined by (a) and (f) has all of the desired properties. □

Let j/e5. We call 3^ = [Uy, E'y, (f = 1, 2, 3) Green’s states at y for the mixed problem provided:

(i) iy is a singular elastic state corresponding to vanishing body forces and to a concentrated load e^ at y;

(ii) if is not empty, then

Uy = 0 on sj,= 0 on

where is the surface traction field of '5^;

(iii) if ,5^1 is empty, then Sy = Wy on dB,

where w^y is the rigid field with force —e, and moment — (1/ — c) xe^.

The above boundary condition insures that balance of forces and moments are satisfied when is empty. Indeed, by (52.2) and (i), the Green’s state ~6y must satisfy

ej = 0, dB

/PcXs^«+ (j/-c)xe,. = 0; dB

that 'iy is consistent with these results follows from (1) and (iii).

For a homogeneous and isotropic body, the Green’s states Sj, may be con¬ structed as follows;

^y~ ^y>

where Oy is the unit Kelvin state corresponding to a concentrated load at y,

and *y is a regular elastic state on B corresponding to zero body forces and chosen so that the boundary conditions (ii) and (iii) for Zy are satisfied.

We say that an integrable vector field u on dB is normalised if

f uda = 0, f p„xuda = 0. dB dB

Given a solution u of the traction problem (.5^ = 0), the field m-|-w with tp rigid is also a solution. By (1) there exists a unique rigid w such that M-f-w is normalized. Thus we may always assume, without loss in generality, that solutions of the traction problem are normalized.


(2) Integral representation theorem.^ Let [u^, {i = \, 2, 3) he Green’s states at y^B for the mixed problem. Further, let \u, E, S] he a solution of the generalized mixed problem with u normalized if ^ is empty. Then for yiDi,

Ui(y) = — f si,-uda-{- f ui,-sda+ f ui^-bdv+'^ m^($) •*($). se, se, B iSDi

Proof. Assume first that ^ is not empty. If we apply the reciprocal theorem (52.3) to the states o and Py and use (i) and (ii) in the definition of Py together with the boundary conditions satisfied by o, we find that

m^-uda+Ui(y) = f s-ui^da + J b-ui^dv+X *(^)' “»(?)- ^1 B f€Dl

which implies the above expression for

If ^ is empty, we conclude from (iii) in the definition of Green’s states that there exist vectors o = and to = co^ such that

s'y = a-\- coxpc on dB.

Thus, since u is normalized,

f Sy' u da = f (a+ coxpc) •uda = a- f uda-\-co-f PcXuda = 0, dB dB dB dB

and the reciprocal theorem again implies the desired result. □

(3) Symmetry of the Green’s states. Let nf^ and Uy be displacement fields corresponding to Green’s states at x, yeB (x4=y) for the mixed problem. Further, if

is empty, assume that ni^ and Uy are both normalized. Then

Proof. We simply apply the integral representation theorem (2) to the state [u, E, S] =34 and use properties (i)-(iii) of Green’s states. □

In the presence of sufficient smoothness one can deduce an integral represen¬ tation theorem for the displacement gradient (and hence also for the strain and stress fields) by differentiating the relations in (2) under the integral sign. An alternative approach is furnished by the

(4) Representation theorem for the displacement gradient. Assume that B is homogeneous and isotropic. Choose y^B and let

* A slightly different version of this theorem for the displacement and surface force prob¬ lems without concentrated loads was given by Lauricella [1895, I], who attributed the method to Volterra (cf. the earlier work of Somigliana discussed in footnote 1 on p. 183). A precise statement of the Lauricella-Volterra theorem is contained in the work of Stern¬

berg and Eubanks [1955, 18]. In [1895, I] and [1955, 18] the equilibration of the concentrated load Cj at y in the Green’s state Zy when .S') is empty is effected through the introduction of a

second internal singularity; the normalization of u is achieved by requiring the displacements and rotations to vanish at the location of this added singularity. The method used here to equilibrate the concentrated load and the normalization procedure for u is due to Bergman

and Schiffer [1953. 8], pp. 223-224 (cf. Turteltaub and Sternberg [1968, 14]). For the surface force problem with concentrated loads this theorem is due to Turteltaub and Stern¬

berg [1968, 14]. See also Dougall [1898, 4], [1904, 1], § 32; Lichtenstein [1924, 1]', Korn

[1927, 2], § 14; Love [1927, 3], § 169; Trefftz [1928, 8], § 49; Arzhanyh [1950, 2], [1951, 2], [1953, 1], [1954, 2]; Radzhabov [1966, 20]; Idenbom and Orlov [1968, 0].


where:

(i) 6'^ are the unit doublet states^ at y;

(ii) are regular elastic states corresponding to zero body forces',

(iii) if ^ is not empty, then

uy = 0 on s'J = 0 on £^2,

where is the displacement field and sj/ the traction field of ;

(iv) if ^ is empty, then = w*J on dB,

where is a rigid field.

Let [u, E, S] be a solution of the generalized mixed problem with u normalized if ^ is empty. Then for y<iDi,

;(J/) = —/s» / u^'sda^ /wy •5(f!7+ 2 •*($). ’ se, se, B s5Di

Proof. By (ii) and the reciprocal theorem (52.3),

f (s • tty — s*;' •u)da+ J b-u^dv+'^l{^)' ($) = 0. (a) 8B B $eD|

.. }|c.. ..

Since — Py, if we add (a) to the right-hand side of the first relation in (52.5), we find that

'î.iiy) =/'(»• wj/' — s^'U)da+ f b- dv+J]l(^)- (5). (b) HB B

It follows from (iii) that when ^ is not empty (b) reduces to the required ex¬ pression for Ui j(y). If ^ is empty, then an argument identical to that utilized in the proof oi (2) can be used to verify that

fs'^-uda = 0; (c) dB

and (b) and (c) 5deld the desired relation for Ui^^(y). □

The rigid field w*J of (iv) is not arbitrary. Indeed, in view of (i) and (ii), must obey balance of forces and moments on B—Sg{y). If we apply these laws and then let p-^0, we conclude, with the aid of (51.3), that

s’J da = 0, dB

f rxs*l da = efXe,. dB

Thus w’J must be the rigid field with force equal to zero and moment equal to X e^-.

The proof of the next theorem, which we omit, is completely analogous to the proof of (4) and is based on the relations for divM(i/) and curlM(i/) given in (52.5).

(5) Representation theorem for the dilatation and rotation.^ Assume that B is homogeneous and isotropic. Choose yeB and let

* Z' = *’ — 6’ Of. — Of. Of.

* See p. i 78. 2 Betti [1872, 1]. See also Love [1927, 3], §§ 16I, 163.


where:

(i) ly is the center of compression at y and the center of rotation at y parallel to the Xf-axisp

(ii) 6y and 6y are regular elastic states corresponding to zero body forces;

(iii) if ^ is not empty, then

Uy — Q on Sj, = 0 on =9^2*

£^ = 0 on 9’y, s» = 0 on

where Uy and Uy are the displacement fields and Sy and the traction fields of iy and iy.

(iv) if ^ is empty, then Sj,= 0 on dB,

siy = w^ on dB, where Wy is a rigid field.

Let [u, E, S] he a solution of the generalized mixed problem with u normalized if ^ is empty. Then for

divM(i/) = — j Sy-uda-\- j Uy- s da-\- ^ Uy-bdv+ 2 ‘ > Se, B {6Dj

J [curl u (y)] f Sy - uda ~ f u^- sda — f Uy-bdv— 2 ^» (^) ‘ * (?) • se, B

As before, the field w'y of (iv) is not arbitrary; it follows from (i), (ii), and the relations on page 179 that Wy must he the rigid field with force equal to zero and moment equal to e,-.

In view of the last theorem, div u and curl u can be computed, at least in principle, from the data. Further, since the displacement equation of equilibrium is equivalent to the relation

A M+ I (1+ ^jpdivM = 0,

the determination of u when div u is known is reduced to a problem in the theory of harmonic functions. Moreover, since the surface force field on dB can be written in the form®

s = 2ix +;MnxcurlM+A(divM)n,

when divM and curlw have been found, the surface value of dufdn is known at points at which the surface force field is prescribed. This method of integration is due to Betti.*

Ericksen’s theorem (33.1) extends, at once, to problems involving concen¬ trated loads.

(6) Assume that there exists a non-trivial solution of the generalized mixed problem with null data (u=0, s =0, b =0,1 =0). Then there exists a continuous body force field b on B of class C® on B with the following property: the regular

* See p. 179. ® See p. 91. ® See p. 93. * [1872, 7]. See also Love [1927, 3], § 159.


mixed problem corresponding to this body force and to null boundary data (m = 0, s = 0) has no solution. If ^ is empty, b can be chosen so as to satisfy

J bdv = f pxbdv — 0. B B

In addition, there exists a system of concentrated loads I such that the generalized mixed problem corresponding to I, vanishing body forces, and null boundary data has no solution. If is empty, I can be chosen so as to satisfy

2*(?) = 2(?-o)x*(a = o. se/j, «e£>,

The proof of this theorem is based on the reciprocal theorem (52.3); it is strictly analogous to the proof of (33.1) and can safely be omitted.

As an immediate corollary of (6) we have the following theorem, due to Turteltaub and Sternberg ; ^ J/ the regular mixed problem with null boundary data has a solution whenever the body force field is sufficiently smooth, then there is at most one solution to the generalized mixed problem.

In the case of internal concentrated loads, a uniqueness theorem can be established that does not presuppose existence.

(7) Uniqueness theorem for problems involving internal concen¬ trated loads. Assume that the body is homogeneous and isotropic with positive definite elasticity tensor. Then any two solutions of the generalized mixed problem with internal concentrated loads are equal modulo a rigid displacement.

Proof. Let 4 denote the difference between two solutions. Then, by the de¬ composition theorem (52.1), a is a regular elastic state corresponding to null data. Thus the uniqueness theorem (32.1) implies that 6 = [w, 0, 0] with tn rigid. □

X. Saint-Venant’s principle.

54. The V. Mises-Sternberg version of Saint-Venant’s principle. Saint-Venant,

in his great memoir^ on torsion and flexure, studied the deformation of a cylindrical body loaded by surface forces distributed over its plane ends. In order to justify using his results as an approximation in situations involving other end loadings, Saint-Venant made a conjecture which may roughly be worded as follows: If two sets of loadings are statically equivalent at each end, then the difference in stress fields is negligible, except possibly near the ends; or equivalently, a system of loads having zero resultant force and moment at each end produces a stress field that is negligible away from the ends. This idea has since been elevated to the status of a general principle bearing the name of Saint-Venant. Its first general statement was given by Boussinesq:® "An equilibrated system of external

\'-^'"t‘ôrces apphed to an elastic body, all of the points of application lying within a s!.'^ given sphere, produces deformations of negligible magnitude at distances from

the sphere which are sufficiently large compared to its radius.” As was first pointed out by v. Mises,* this formulation of Saint-Venant’s

principle, which has since become conventional, is in need of clarification, since

* [1968, 141, Theorem 5.2. Actually, the Turteltaub-Sternberg theorem is slightly

different from the one given above; the idea, however, is the same.

“ [1855, 11 ® [l885, 11, p. 298. “Des forces ext^rieures, qui se font equilibre sur un solide elastique

et dont les points d’application se trouvent tons a Tint6rieur d’une sphere donnê, ne produisent

pas de deformations sensibles a des distances de cette sphere qui sont d’une certaine grandeur

par rapport h son rayon.”

* [1945, 5].

Sect. 54. The V. Mises-Sternberg version of Saint-Venant’s principle. 191

the system of external forces must necessarily be in equilibrium, at least when the body is finite. Further, as was noted by Sternberg,^ in the case of an unbounded body loaded on a finite portionîts smiaMTlTie stresses are arbitrarily / small at sufficiently large distances from the load region.^ I

The first of these criticisms led v. Mises® to suggest the following interpreta¬ tion of the principle: "If the forces acting upon a body are restricted to several, small parts of the surface, each included in a sphere of radius q, then the ... stresses : produced in the interior of the body ... are smaller in order of magnitude when the forces for each single part are in equilibrium than when they are not.”

It is clear that this is the meaning intended by Boussinesq. Indeed, in order to justify the principle, Boussinesq examined the strains at an interior point of an elastic half-space subjected to normal concentrated loads upon its bound¬ ary. Assuming the points of application of the loads to lie within a sphere of radius q, he showed that the order of magnitude of the strains i^g if ê resiânt force is zero, and p® when the resultant moment also vanishes. Various arguments have since been given in support of the principle.*

V. Mises utdized two examples involving tangential as well as normal surface loads to demonstrate that the above version of Saint-Venant’s principle cannot be valid without qualification. The two examples chosen by v. Mises are the three-dimensional problem of the half-space and the plane problem of the circular disk, each under concentrated surface loads.® Guided by these examples, v. Mises ® conjectured a modified Saint-Venant priuciple-which he stated as follows:

“ (a) If a system of loads on an adequately supported body, all applied at surface points within a sphere of diameter g, have the vector sum zero, they produce in an inner point y of the body a strain or stress value s of the order of magnitude g.

(b) If the loads, in addition to having vector sum zero,... form an equilib¬ rium system within the sphere of diameter g, the s-value produced in the point y will, in general, still be of the order of magnitude g.

(c) If the loads, in addition to being an equilibrium system, ... form a system in astatic equilibrium, then the s-value produced in y will be of the order of magnitude g® or smaller. In particular, if loads applied to a small area are parallel to each other and not tangential to the surface and if they form an equilibrium system, they are also in astatic equilibrium and thus lead to an s of the order g®.”

The conjectures of v. Mises concerning concentrated surface loads were analyzed by Sternberg, whose results are also valid for continuously distributed surface loads. In. order to give a concise statement of Sternberg’s theorem, we introduce the following definitions:

We say that , 0 < g ^ go, is a family of load regions on 8B contracting to z^dB if:^

(i) ^g = dBn Hg (z) for every g € (0, go];

*[1954,20]. ® Toupin [1965, 20] gives several counterexamples to the conventional statement of

Saint-Venant’s principle. ® [1945, 5]. * Southwell [1923, 2], Supino [1931, 5], Goodier [1937, 1, 2], [1942, 2], Zanaboni

[1937, 6, 7, S], Locatelli [1940, 2], Dou [1966, 7]. ® Erim [1948, 8] has applied v. Mises’ analysis to the half-plane under concentrated loads.

* [1945, 5]. ’’ Cf. Sternberg and Al-Khozai [1964, 20], Def. 4.1.


(ii) there exists a class mapping x: where ^ is an open region in the plane containing the origin, such that

s = x{0), dx{(x,) dx{<x)

doii d CX2

Note that by (ii) the boundary of B possesses continuous curvatures near s; thus this portion of the boundary is required to exhibit a higher degree of smoothness than that automatically assured by the assumption that 5 be a regular region.

For each ke{i, 2,..., K) let 0<p^po* be a family of load regions on dB contracting to a point Zj^^dB. We say that SJ, 0<p<po> is a family of singular elastic states corresponding to loads on [k) if given

any ee(0, Po):^

(iii) is a singular elasticjtate corresponding to zero body forces, to surface tractions s^, and to a system l^ of concentrated loads f . '

K (iv) the traction field s vanishes on dB—\j ^ (k);

k=l

(V)

(vi) |Sj(a5)| for every xedB and for every êDig, where the constants and are independent of q ;

(vii) and Vu^ admit the representations

dB

= !'%■ “j/'+ 2 Ki^)■ dB

for every yeB, where for each yeB the functions Uy and are continuous on B—{y} and of class on for k—\,2,

Since Di^ is the set of points at which Jhe concentrated loads of the system Ig are acting, (v)"Is"simply the requirement that there be no concentrated loads applied'outside of the K load regions ^g(k).

In view of (iii), (v), and the integral representation theorem (53.2), the first representation formula in (vii) is simply the requirement that the Green’s states for the surface force problem exist. For a homogeneous and isotropic body, it is clear from the discussion on page 186 that such Green’s states will exist if the surface force problem for arbitrary (equihbrated) surface tractions has a solution with sufficiently smooth displacements. In view of the integral representation theorem (53.4), this also insures that the assertion in (vii) concerning Fu^ hold.

For convenience, we adopt the following notation:

De(k) =D,g n^g(k),

+„2,./s (?). ^qW

mg{k)= f pxs da+ 2 (?-0)xIj(?),

Mg{k)^ f p®Sgda+ Z (?-O)0i,(?).

1 Cf. Sternberg and Al-Khozai [1964, 20], Def. 4.2.


so that /j (k) and trig (k) denote the resultant force and resultant moment about the origin of the surface tractions and concentrated loads on ^g{k). We say that the force system is in equilibrium on eqcjh^lggd êgiQtiîS

for k = i, 2,K and 0<g<go- Further, using the terminology introduced in Sect. 18, we say that the force system is in astatic equilibrium on each load region if given any orthogonal tensor Q

^fQSgda+ Z QW = 0. ^Q[k) S^Dgik)

fpxQSgda+ Z {^-0)xQlg{^) = 0. S^Dgik)

Taking Q = l, we see that astatic equilibrium imphes equilibrium. In addition, it follows from theorem (18.9) that the force system is in astatic equilibrium on each load region if and only if

/,(^) = 0, Mg{k) = 0.

for = 1, 2, ..., K and 0 < g < go • Finally, we say that the forces are parallel and non-tqngenUal. taJbe. .boundary if there exist scalar functions q>g on dB aiidT^g on Dig such that for each k,

Sg {x) = (pg{x) a„. Ig (?) = for all xe^g{k), êDg(k), and ge(0, go), where is a unit vector that is not tangent to 8B at .

(1) Sternberg’s theorem,^ Let k = \,2, be K distinct regular points of dB, and for each k let ^g{k), 0<g^go, be a family of load regions on 8B contracting to z,,. Let <ig = [Ug, E^, Sg\, 0<g<go, be a family of singular elastic states corresponding to loads on ^glk). Then given y€B:

(a) Ug{y)=0{Q^), Eg{y)=0{Q^), sJ{y)=0{Q^) as qÔ, where d=0;

(b) d = \ if fg{k) = 0 for k — i, 2,K and 0<Q<Qo',

(c) 6=2 if on each load region the force system is in equilibrium with forces that are parallel and non-tangential to the boundary;

(d) 6=2 if the force system is in astatic equilibrium on each load region.

Moreover, if the concentrated loads all vanish (i.e. 1^ = 0), which will be the case if 6g is regular for each g, then:

(a') 6=2 (in general);

(b') d = 3 if /g(^) = 0 for k=\, 2, ...,K and 0<g<go;

(c') 6 =A if on each load region the force system is in equilibrium with forces that are parallel and non-tangential to the boundary;

(d') 6=A if the force system is in astatic equilibrium on each load region.

Proof. Fix y^B. In view of the strain-displacement and stress-strain relations, it clearly suffices to establish the above order of magnitude estimates for Ug and VUg. Let Ug denote either Ugf{y) or Ug{ j-{y). Then by (vii) Ug admits the representation

Zh{^-9{^). (e) _ SB

‘[1954,20]. See also Schumann [1954, iS], Boley [1958,3], [1960,2], Sternberg

[i960, i2], Sternberg and Al-Khozai [1964, 20], Keller [1965, 11]. The last author discusses the form which the principle might be expected to take in the case of thin bodies.

Handbuch der Physik, Bd. Vla/2. 13


where g denotes either iij, or Since the are all distinct, there exists a gje(0, Qf^) such that the K sets ^g{k) are mutually disjoint for each ge(0, Qi).

Thus, in view of (iv) and (v),

k=l where

Vg(k)= f s^-gda+ 2 *e(§)-Sr(?)-

We now fix the integer k and, for convenience, write

= = D^ = D^{k), Z = Zk. SO that

’^8= J%-9da+ 2 •»(?)• fe)

Let x: be the function specified in (ii). It follows from the properties o

of sb that there exists an open set ^ in ^ containing the origin, a number o ^

^26(0, pi), and a mapping a = (ai,a2) of onto ^ such that a is the inverse o

of the restriction of x to Further, it follows from the properties of x that

a (as) = 0(1® —»|) as x^z. (h)

Since the composite function gox (with values g{x(a,))) is class on if we expand this function in a Taylor series in a about a = 0 and evaluate the result at a = a (®), we arrive at

g(®) =ao + aioli(®)+ a2a2(®) 4-c(®) (i)

for where a^, Oj, and ag are fixed vectors and, in view of (h), c is a class function on that satisfies

c{x) =0{\x — z\^) as x^z. (j)

By (h)-(i) and (vi), 2

f s^-gda = a^- f s^da + Z»?' I + , /3==1

= ao-/s ^f« + 0(p3), (k) ■^8

= 0[Q^),

2 «,(?)• g(?) = ao- 2 m + Z»p- 2 + ?€Dg gSUff /9=1 SeDg

zm+oiQ). (1) geOff

= 0(1)

as p->0. Conclusions (a) and (a') follow from (f), (g), (k)3, and (1)3. Further, it follows from the definition of =f^{k) in conjunction with (f), (g), (kjg, and (1)^ that conclusions (b) and (b') are also valid.

To estabUsh the remaining results we first expand the function ® in a Taylor series about a = 0:

® (a) = s + <*i ai + (fa “2 + ^ (a). (m) where

« = 0 (^ = 1.2), (n)


and his a. class function on ^ that obeys

h{x)=0{\x\^) as a-^0. (o)

Since 2> (as) =as —0, we conclude from (m), (h), and (o) that

2

f p0Sgda = {g — O)0 f Sgda + fx^Sgda + 0(g*), ^=1 (p)

2 (§-O)0f,(?) = (s-O)0 2 !,(?) +2(1^0 2 ?c»e SsDg fi=i seDg

If we assume that the force system is in astatic equilibrium on each load region, then

/,^0, M,s0, (q)

and it follows from (p) and the definition of and that

Zap®ffig = 0{Q^). ,8=1 (r)

= 0(p^) if 1,^0, where

/ê=/«,8Sff'^«+(s)

Each of the relations in (r) may be regarded as a system of inhomogeneous linear equations in the two unlmowns ffig(P — i, 2). It follows from (n) and the restric¬ tions placed on x in (ii) that the coefficient matrix of this system has rank two. Hence (r) implies

hg = 0{Q^), (t)

= 0(p*) if

and this result with (f), (g), (k)^, (1)^, and (s) 5delds conclusions (d) and (d').

We have only to establish (c) and (c'). Thus suppose, in place of (q), that

fg=0, m^Ô. (u)

Clearly, (p) holds in the present circumstances with the operation “ 0 ” replaced by “x”. It follows from this relation in conjunction with (u) and (s) that (r) also holds with “0” replaced by “x”. Thus if we assume that s^{x) =(pg{x)a and =%{^ a on and D^=D^{k), respectively, then

2 xaI/« 4- 2 «,8(?) %(?)} = 0(p^), ,8=1 g€Df >

= 0{Q^) if 1,^0. This, in turn, impHes that

[(di xda) -a] I /a^ 9?^ «-f 2 a,8 (?) % (?)} = 0 (g^), SSDg >

= 0{e^) if (V)

It follows from (n) and (ii) that diXd2 4=0. Therefore, since d^xdg is normal to 8B at g, the requirement that a not be tangent to dB at g imphes (dj xdg) • a 4=0. Thus, in this instance, (v) imphes (t), and, as before, we see that (c) and (c') are valid. □

On the basis of the traditional statement of Saint-Venant’s principle dis¬ cussed at the beginning of this section, one would expect a reduction from d = 1

13*


to b=2 in the order of magnitude estimates whenever the force system is in equilibrium on each load region, even when the surface tractions are not of the special form specified in (c), i.e. parallel and non-tangential to the boundary. That this is not the case is apparent from counterexamples of v. Mises.^

The preceding conclusion has a cmmterpart in the. theory of basic singular states discussed in Sect, 51- Thus consider the unit Kelvin state <>* and the unit doublet state with 0 the point of application in both cases. Both of these states are regular on any infinite region S that does not contain 0. If the bound¬ ary of B is bounded, then

/’=j=0, m‘ = 0,

f’' = 0, mb’4=0 (f=i=/),

/*' = 0, = 0 (f = j),

where f, m* and /b, m*' denote the resultant force and moment alût the origin of the surface forces on dB corresponding to d* and d*', respectively. Con¬ sider the rate of decay of the corresponding stress field at infinity. It,is clear from the results on p. 178 that

S‘ (£c) = 0 (r-2), S' (x) =# 0 (r'*)

S'> (x) = 0 (r-®), Sb (35) =4 0 (r-^)

as r = \x — 0\ôo, regardless of whether i =j or i =j=7- Consequently, whereas the stress field decays more rapidly when the resultant force on dB vanishes than when this is not the case, the additional vanishing of the resultant moment does not produce a further reduction in the order of magnitude of the stresses as r-^cx).*

55. Toupin’s version of Saint-'Venant’s principle. Two alternative versions of Saint-Venant’s principle were estabhshed by Toupin® and KNOWLES.*These authors considered the solution corresponding to a single system of surface loads, and both authors arrived at estimates for the strain energy U(1) of a portion of the body beyond a distance I from the load region.® Knowles’ result, which we will discuss in the next section, apphes to the plane problem, while Toupin’s result is for a semi-infinite cylinder loaded on its end face. ®

(1) Toupin’s theorem.’’ Let B he a homogeneous semi-infinite cylindrical body with end face and assume that the elasticity tensor is symmetric and posi¬ tive definite. Let \u, E, S] be an elastic state corresponding to zero body forces and to a surface force field s that satisfies'.

(i) s=0 on dB — £L’^,

(iii) hm Ju-Snda=0, *->00

‘ [1945. 5]. 2 The conclusions expressed in the last two paragraphs are due to Sternberg and

Al-Khozai [1964,20], pp. 144-145. ® [1965, 21]. * [1966, 12]. ® The idea of estimating the energy U{1] of that portion of a circular cylinder a distance I

from the loaded end appears in the work of Zanaboni [1937, S].

® Knowles and Sternberg [1966, 13] have established a Saint-Venant principle for the torsion of solids of revolution. A result that is valid for more general regions was given by Melnik [1963, 19].

’’ [1965, 21]. See also Toupin [1965, 20].

Sect. 55- Toupin’s version of Saint-Venant’s principle. 197

where is the intersection with B of a plane perpendicular to the axis e of the cylinder and a distance I from 5^. Let Oe^, let

Bi = {xzB\ (£B — 0) • e>/},

and let U(l) denote the strain energy of Bg.

Then

U(l) = k 5 'E-C[E\dv.

U(l)Û{0)e yW (l^t)

for any t > 0, where the decay length y (t) is given by

o * I'

+- e

u B

fnirv e ■on

Here and /<„ are the maximum and minimum elastic moduli, while % (t) is the lowest (non-zero) characteristic value for the free vibrations of a disc with cross section and thickness t. a disc that is composed of the same material and that has its boundary traction-free}

Proof. Choose t,l>0 and let

Then B{l.t)=Bi-Bt + t’

[/(/) = limi jE-C[E]dv <ôo

proyided the limit exists. By (i) and the theorem of work and energy (28.3),

Ju-Snda-{- fu-Snda= / E • C[E] ^t+,

Thus if we let f ^ oo and use (iii) we conclude that U{1) exists and obeys the formula

U{1) = \ ( u-Snda. (a)

Equating to zero the total force and moment on the portion B (0,1) of B be¬ tween ^ and we conclude, with the aid of (i) and (ii), that

Snda= f px{Sn)da = 0. (b)

* Thus we make the tacit assumption that the characteristic value problem stated in Sect. 76 has a solution, at least for the lowest non-zero characteristic value.


Let s' be the vector field on 8B (0,1) defined by

s'=Sn on s'=0 on dB{0,l)

Then (b) implies fs'da= fpxs'da = 0,

dB (0,;) en (o, i)

and we conclude from Piola’s theorem (18.3) with s = s' and 6 = 0 that

/ w • Sn da = 0

for every rigid displacement w. Therefore (a) yields

provided

U{l)==i- fu-Snda

u = u-{-w.

By (c) and the Schwarz inequality,

/ u^da (\Sn\^da. ff’i SXi

(c)

(d)

(e)

If we expand (a ]/«— ]/6)2, where a, h, and a are non-negative scalars with a>0, we arrive at the geometric-arithmetic mean inequality

^ — 2

Applying this inequality to (e) yields

C7(/) /"\Sn\^da-\-^ j"WdA. (f)

'•5^, y, '

Since |n| =1 and C is symmetric,

|Sn|2^|S|2=C[E] •€[£]= £;-C2[£;]. 'VvN

Let and denote the maximum and minimum elastic moduli for C; then

by (24.6) p^\E\^Ê-C[E]^p^\E\K

Further, it follows from the definition of and that and are the maximum and minimum moduli for C^; thus

pl\EYÊ-C^[E]^plt\E\^-

The last three inequalities imply that

\SnY^lxlt\E\^^lx*E-C[E-\. where

Lp* f E-C[E-]da + ~ f u^di fe)

thus (/) implies

Sect. 55- Toupin’s version of Saint-Venant’s principle. 199

Next, choose t>0 and let

UWdA.

If we integrate (g) between I and I-I-i, we arrive at

(Q(l, t) ^ A L^* jE-C [E] dv + } f W

Now let X{t) denote the lowest non-zero characteristic value corresponding to the free vibrations of the cylindrical disc B[l, t). Then^

f Vv - C [Vv] dv

' ' — J dv

for every admissible displacement field v on B{1, t) that satisfies

j v'^dvÔ, fvdv— f p„xvdv = 0.

By use of the same procedure as that used to prove (53.1), it is not difficult to show that theîguîsplacement w in (d) can be chosen so as to satisfy

/ wdv = — f udv, f PcXwdv = — f PcXudv, B(l,t) B{l,t) B{l,l)

where c is the centroid of B{1, t) and Pc{x) =x — c. By (d) when this is the case we have

/ udv— f Pf,xudv = 0; B (l, I) B (l, t)

and (i) and (j) yield

j u^dv ^ - J E - C[E] dv, B(l,t) B(l,t)

I

If we differentiate (h) with respect to I, we arrive at

On the other hand,

j E-C[E]dv = jE-C[E']dv- ^ E • C[E]dv = 2[U{r) - U{l + t)], (m) B(l,{) Bt Bi+,

and (k)-(m) imply

where

* See p. 264.


Clearly, y (t) is a minimum when

Henceforth we assume that a has this value, so that

By(n)

and thus

or

4 Q 0} = -+4)^ 0,

^■/vwQ(4, t) _g*./vwQ(4,t)ô (4^4)

Q{h_.t)

Qih.t)

-V,-h) £ e yi‘)

/

(o) »

Since C is positive definite, U{1) is a non-increasing function of I, and by (h) Q(Ziyis'the mean value of U{1) in the interval [/, therefore

and (o) implies

U(h±i)

U(h) ■

-v,-h) ^ e v(6

Setting 4 = 0 and 1^ = 1 —tin (p), we obtain the desired energy estimate.

(P)

□ In view of (24.6), if the material is homogeneous and isotropic with strictly'

positive Lame moduli fx and X, then the characteristic decay length y (t) has the fonn „ , ,,

2/X + 3X

Y2J11)) ■

Toupin^ also derived an estimate for the magnitude of the strain tensor at interior points of the cylinder. A somewhat stronger estimate was established by Roseman 2 for an isotropic cylinder. Roseman proved that

\S{x)\^^K[U{l-a) - U{l + a)]

for every where iC is a constant that depends on the Lame moduli and the cross-section, while a is a constant that depends only on the cross-section. When

combined with Toupin’s theorem, the above inequality yields pointwise exponential decay for the stress throughout the cylinder.

56. Knowles’ version of Saint-Venant’s principle. Knowles’ version® of Saint- Venant’s principle is concerned with the plane problem of elastostatics for a simply-connected region loaded over a portion of its boundary curve. To state this result concisely we introduce the following notation.

Let R be a simply-connected regular region in the x-^^, ATg-plane, so that its boundary curve is a piecewise smooth siinple closed curve. As indicated in

Fig. 15, = % ^0},

_ = * [1965, 2i]. Toupin’s estimate is based on (h) of Sect. 43, which is due to Diaz and Payne

[1958, 6], ® [1966, 22]. The techniques used by Roseman are based on results of John [1963, 15],

[1965, 10]. ® [1966, 12].

Sect. 56. Knowles’ version of Saint-Venant’s principle. 201

and we choose our coordinates so that the minimum value of on is zero. will represent that portion of the bound^y curve along which surface tractions

are applied. Let a = sup%, b = supx2,

Ri — {xeR\ Xi>l},

£^1 = {x^R\ % = /}.

We assume that for each ê(0, a), the set is the union of a finite number of line segments; we regard .5?, as parametrized by x^.

Since R is simply-connected, (46.3) implies that the stress field corre¬ sponding to a plane elastic state is completely characterized by the field equations

■5a/8,/8~0> •^■5aa=0. (a)

Knowles’ theorem gives a decay estimate for the norm

(b) ' JSl

of such a stress field over i?,.

(1) Knowles’ theorem,^ Let (= he smooth on Rq and of class C* on Rq. Siippose further that satisfies (a) on R^ and

Then for 0 â

where

Sapn0 = O on^^.

l|5|L^y2||S||„/ hi

V

k — 71

(C)

^ [1966, 12'\,

2


With a view toward proving this theorem, we first establish the following

(2) Lemma.^ Let f he continuous onRg. Then

~ Jfda = -ffdx^.

Proof. Consider the difference quotient

F(l + d)-F(l)

for the function F defined by '

F{l) = ffda. St

If is connected, this difference quotient can be written, for sufficiently small d, as an iterated integral, and the limit as d->0 can be computed directly to prove (2). If =5?^ is the union of disjoint hne segments, the difference quotient is written as a sum of iterated integrals, and (2) again follows upon letting d^0. □

Proof of (1). By (47.1) there exists an Airy function (p for on R^, thus

^11— T,2.2.’ ^22 — T,11’ ^12— T,12> (^)

AAcp’Ô. (e)

Since R^ is simply-connected, (p is single-valued; further, it follows from (d), (e), and our hypotheses on that cp is analytic on Rq and of class C® on Rq . More¬ over, in view of (c), the relation (p{a) = m(a) in (47.2), (d) and (e) in the proof of (47.2), and the discussion given in the second paragraph following (47.1), we can assume, without loss in generality, that

9’ = 9’.<x = 0 on <^0. (f) Let

(g) Ri

SO that ||Si,=fW(/J. (h)

Then (d) and (g) imply that

W{l)^J<p,.p<p,.pda. (i)

In view of (e), we can write (i) in the form

W{l) = J [9^,19^,11 — 9^9?,Ill -I- 29^,2 9^,12),! Rt

"1“ (T, 2^,22 TT,22 2 29191^112), 2]

where l€(0, a). Thus (f) and Green’s theorem yield the relation

^(0 = - / (T,i T,n -TT.ni + ^(p.2^,12) sei

and hy (2) this relation, in turn, implies that

fiT,i<P,ii—<P<P,iii + 2(P,2T,i2)d‘i- Ri

1 Knowles [1966, 12], Eq. (2.6).

Sect. 56. Knowles- version of Saint-Venant’s principle.

Hence 203

(j)

since the integral on the right vanishes when / = «. FinaUy, since

/PF(A) dk = -^/ -cp9,ii),ida,

/ W{2.) (91^1 + 9,^2 - cp9i_ii) d. (k)

Next, by (i) and (2),

_ c / 9^.a^ 9^,a^ dX2. /j\

Thus given any scal„ a. the fonun.ae (It) and (1) ™p,y the following .gonfity:

+ix‘jw{X) u I

+ 2 (p%, - 4;^^(9ifi + <p%~(pg,,j] dx.

Letting

(m)

F(l)=W(l)+2xJW{X)dX,

we can write (m) in the form ‘

dF(l) i-

' IT'+ 2x F{1) = - j ([^_ 11 +2xV)"

se,

(n)

(o) + (29’fia 4x29)fi) +{cp%i—Ax‘^q?^ — A)^(pi)]dx^,

after a rearrangement of terms in the integrand on the right side of (m) '

» for which .he right

« a smooth function o-S the closed interval P. U

/r’‘"a (tTSy/r*-*'. (P)^

--- ^0

and Bickley [1955. «!’A*^^ricrpr°oof^S be made to § 0.6 of Temple Polya [1959, 6]. ^ found in Hardy, Littlewood, and


with the minimum taken over all smooth functions f on [1^, that vanish at the end points and have

J dt — i. ^0

We require a slight generalization of (p). If, for fixed l€(0,a), f is a smooth

function (of Xj) on and if ip vanishes at the end points of each line segment

that constitutes .=5?^, then

jip%dx^^~^- jip^dx^. (q)

S?i SPi ^

This inequality foUows upon decomposing the integral on the left in (q) into a sum of integrds over the constituent line segments of applying (p) to each of these, and observing that the length of each line segment is at most h.

Fix f€(0, a). Since vanishes on it vanishes at the end points of the

line segments that form Thus, because of the smoothness of (p, we may take ip in (q) equal to with the result that

J^ p" J<pîdx2. S?i

In view of this inequality, we can write (o) in the form

dF(l)

di Si

Alternatively, we can let ip — (p ^ v^ (q) with the result that

J'dx^ ^ J* ^2 St Si

and this result when substituted into (r) yields

dF(l) n 1-1 ■7x2 \ . ix,2

~dl

+ 2xF{l) J +(P%2- (p% - 4«^ r dx„ (r)

-2-’ +2xF{l)^-+

Si

r dx. (s)

We now assume that

2b ’ (t)

SO that the coefficients of and (p^^ in (s) a-re non-negative. Then we may take ip = (pin (q) and infer from (s) that

^2^- -I- 2xF{l) ^ - [(-J-4x^] -J'- - 4«^j jcp^ dx2.

St

The pol5momial in x in the bracket has one positive zero, namely

K k h ’

where

This value of x satisfies if). We now choose this value of x and conclude that Ff) satisfies the differential inequality

0<l<a. (n)

Sect. 56. Knowles’ version of Saint-Venant’s principle. 205

The inequality (u) implies that

F(/)^F(0)

Since xÔ, (i) and (n) imply that

thus

0^1^ a. (v)

(w)

It remains to compute an upper bound for F{0).

If we insert (n) into the left side of (v), we obtain an inequality which may be written in the form

d

~di a —2x1 fww d;^ ^F{0) 0-^"“

Integrating both sides of this inequality from /=0 to l=a, we find, with the aid of (n), that

6

Therefore a

2hJW{X) dX 0

1

4j< W{0) + 2xj W(X) dk (1

1 — iZflJ-iKa W(0)^W(0),

and this result when inserted into (n) with 1=0 yields the inequality

F(0)^2W(0). (X)

The inequalities (w) and (x) imply that

W(l)^2W(0)e-^’“, O^lâ;

in view of (i), this is the desired result. □

As was noted by Knowles,^ when i? is a rectangular domain subject to self- equilibrated tractions along one edge, the optimum choice of the constant k in the estimate

||S||,^l/2|iS||or^

s approximately 4.2.'^ Thus the universal constant A in (1) appears, in this particular case, to be conservative by a factor of approximately three.

As is clear from the relation on page 152, the siram energy U(l) of is given by

U{1) = { js^pE^pdv

Ri

1 [1966, i2], p. 11. ® See, e.g., Fadle [1940, L], Horvay [1957, 9], and Babu§ka, Rektorys, and VYeiCHLO

[1955, The discussion by BabuSka, Rektorys, and VyCichlo of the exponential decay of stresses in the rectangular case appears to be in error in that the roots of smallest modulus of the equation sin®^ —^^ = 0 are underestimated by a factor of two (cf. Fadle [1940, 7]).®

3 Knowles [1966, 12], p. 11, footnote 1.

206 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 56 a.

If we assume that ^m>0, 0<v<i, then the inequality

(1 2v)

is easily derived using the same procedure as was used to establish (24.6). Thus

and the inequality m (1) can be written in the form

r ** f2

Knowles! also proved that under the hypotheses ot (1) the stresses satisfy the inequality

kXi

(%, ^2)1 ^ ,3 /■ 2U{0) -

-e

where d is the distance from the point [x^, to the boundary of ^ and ^ is defined in (!)■ Note that this inequality is poor at points which arc close to the boundary. ' '

56 a. The Zanaboni-Robinson version of Saint-Venant’s principle. (Added in proof.) This version of the principle is concerned with a one-parameter family 5; (0 < / < 00) of bodies (bounded regular regions) such that

and B.CB, (T</)

(t</)

has a regular interior. We assume given a continuous, symmetric, positive definite elasticity field C on ^

XTBi. 00)

a regular surface such that

6^ C SB I (0 < f < 00),

and integrable surface tractions s on ,5^ such that

J sda = f pxs da=0. Sf Sf

(1) Zanaboni-Robinson theorem.^ For each /e[0, 00) let [m^, be an elastic state on Bj corresponding to the elasticity field C (restricted to BJ, to zero body forces, and to the following surface tractions:

Si=s on ,

S;=0 on dBi—£t’.

Let U{r,l)==^ f EfC[Ef\dv (t<l).

! [1966, 12], Theorem 2. See also Roseman [1967, 12]. 2 The precise version stated here is due to Robinson [1966, 21], § 9.7; the underlying

ideas, however, are contained in the work of Zanaboni [1937, 6, 7, S]. See also Segenreich

[Ipyi, 4], who established interesting corollaries of this theorem.

Sect. 57- Some further results for homogeneous and isotropic bodies. 207

Then lim U(r, l) = 0.

X, /-►OO X<1

For the proof of this theorem we refer the reader to the book by Robinson.^

XI. Miscellaneous results.

57. Some further results for homogeneous and isotropic bodies. In this section we shaU establish some results appropriate to homogeneous and isotropic bodies.

Given a smooth vector field u on B and a closed regular surface 6^ in B, we define the vector field t {u) on .5^ by ^

t(u) = (A +2jj,) (divu) n (curlu) xn,

where n is the unit outward normal to

The first theorem, which is due to Fosdick^, shows that the vector field t(u) when integrated over dB gives the actual load on the boundary.

(1) Let y he a closed regular surface in B. Further, let u he a smooth vector field on E, and let s he the corresponding surface traction field on :

s = ^ (Pm + Pm^) n-\-X (div m) n. Then

f t(u) da = f s da. y y

Proof. Since (curl m) xm = (Pm — Pm^) n,

it follows that s — f (m) = [Pm^ — (div m) 1] m ,

and thus that sda~ J t(u) da = 2fi f [Pm^ —(divM) l]nda.

sf y y

Letting T denote the skew tensor field whose axial vector is m, we conclude from

(15) of {4:.l) that

/ s da — f t(u) da = — 2iA, / nda, y sf se

and the desired result follows from Stokes’ theorem (6.2). □

Note that since dB is the union of closed surfaces ..., .5^, if m is constant on each £F„, then (1) imphes

f t(u) • uda = f s-uda; dB dB

thus, in this instance, “the work done by t(u)” is equal to the work done by s.

(2) Reciprocal theorem.* Let u and u he elastic displacement fields of

class on B corresponding to body force fields b and b, respectively. Then

f t(u) -uda-j- f b • udv = f t{u) • uda J b-udv dB B dB B

~ f [(/l + 2^)(divM) (div m)(curl m) • (curl m)] (fi;.

* [1966, 31], §9.7.

2 Fosdick [1968, 5].

® Sometimes we will take S^=dB. It will be clear from the context when this is the case. ‘ Weyl [1915, 3], p. 6; Fosdick [1968, 5].


Proof Let I{u, u) denote the last term in the above relation. It follows from the identity

/ (“ X curl u) ■nda= f (curl u) ■ (curl u) dv — f u- curl curl udv dB B B

in conjunction with the definition of t(u) that

/ t{u) •uda= f [(A + 2^m) (divtt) u • n +^(MXcurlM) • n] da dB dB

= I{u,u) + / [(A + 2^) P div M — ^ curl curl m] -udv. B

But since u is an elastic displacement field,

(A + 2^) Vdiv u — fi curl curl m + b = O.i Therefore

/ t(u) • uda-\- f b ■ udv = I(u, it), dB B

and, since I{u, u) —I{u, u), the proof is complete. □

If we let u — u and b = b in (2), we arrive at the following result, which is an analog of the theorem of work and energy (28.3).

(3) Kelvin’s theorem.^ Let u be an elastic displacement field of class Côn B corresponding to the body force field b. Then

f t{u) -uda-j- f b •udv = 2G{u}, dB B

where = / [(A+ 2^m) (divM)2-f^|curl m|2] dv.

B

In view of the remark made following the proof of (1), when u is constant on each closed surface comprising dB, the left-hand sides of the relations in (3) and (28.3) coincide; therefore, in this instance, G{u} — U{E}. In particular, it follows that G{m} is equal to the strain energy when the displacement field vanishes on the boundary.

Theorem (3) yields a much simpler proof® for the uniqueness theorem (32.2) when B is homogeneous and isotropic. Indeed, if m = 0 on 05 and b=0 on 5, then Kelvin’s theorem imphes

G {m} = 0.

But if C or — C is strongly eUiptic, we conclude from (25.1) that ^M(A-f-2;M)>0, and hence that

divM = 0, curlM = 0,

which imphes Au — Q. But if u is harmonic and vanishes on dB, then u must also vanish on B, and the proof is complete.

Since B is homogeneous and isotropic, the functional 0 of the principle of minimum potential energy (34.2) has the form

0{u} = ^ {AivuY dv — J b-udv— J s-uda,

B B

1 This identity is derived on pp. 90-91. ® Kelvin [1888, 4]. See also Weyl [1915, 3], p. 6. ® This argument is due to Ericksen and Toupin [1956, 3]. See also Duffin and Noll

[1958, S], Gurtin and Sternberg [i960, 7].

Sect. 57- Some further results for homogeneous anctisotro^cjbodies. 209

as is clear from (24.5). If we confine our attention to the displacement problem, we can establish a minimum principle much simpler in nature than the principle of minimum potential energy.

(4) Minimum principle for the displaeement problem^. Consider the displacement problem for which — Let j?/ he the set of all kinemaiically ad¬ missible displacement fields, and let n be the functional on si defined by

j<{m}=G{m}— Jb-udv B

for every uisd. Then given }{{u} ^x{u}

for all u€ if and only if u is a displacement field corresponding to a solution of the displacement problem.

The proof of (4) follows at once from (34.2) by virtue of the following theorem, which estabhshes the connection between the functionals 0 and x.

(5) ^ Suppose that the hypotheses of (4) hold. Let sd be the set of all kinematically admissible displacement fields and let

/1{m} =x{u) — 0{u] for every uesd. Then

for every u, u'isd; i.e. A{m} is independent of u.

Proof. Let Mgjaf. Since |curl = Vu ■ [Vu— Vu’'),

I Fm + 2 = 2 Pm • ( Pm + , it follows that

A{m}=^ / {{6.ivu)^—Vu-Vu^}dv=ij, f div[M(divM) — (Pm) m] B B

Thus X{u}=fi J rjda, = [M(divM) — (Pm) m] • n.

as

We now complete the argument by showing that rj depends solely on the boundary displacement m, which is common to all members of sd. Let the origin 0 of the cêsian frame be an arbitrary regular point of dB, and choose the frame so that the plane ^3=0 coincides with the tangent plane of dB at 0, the axis pointing in the direction of the inner normal of dB at 0. For this choice of co¬ ordinates.

r\=u^ g«3 dx-^ U2

0^2 (a)

In a neighborhood of 0 the boimdary dB admits the parametrization

% — ^1. ^2 — ^2> ^3 — ^(§)> §—(t*) where

^(0)=0 («=1,2). (c)

Further, if we agree to write m(|) =m(Ii, |2< ^(li. I2)). then, since Mgjaf,

u{x) = tt(§)

1 Love [1906, 5], § 116. See also Gurtin and Sternberg [i960, 7], Theorem 2. 2 Gurtin and Sternberg [I960, 7], Theorem 3.



in the neighborhood under consideration. Thus by (b)-(c)

|“(0)=|“(0) (a=1.2),

and we conclude from (a) that

8SI at 0.

Therefore j; at 0 depends exclusively on the boundary data u. But the point 0 was an arbitrarily chosen regular point of dB. Thus k{u) depends solely on the boundary data u; hence A{m} = A{u'} for every u, □

58. Incompressible materials. An incompressible body is one for which only isochoric motions are possible, or equivalently only motions that satisfy

divM = 0.

For a linearly elastic incompressible body the stress is specified by the strain tensor only up to an arbitrary pressure. Thus

S+;/)l=C[£], where

P = -ktTS

and C = Cjp is a linear transformation from the space of traceless symmetric tensors into itself} The symmetry group for the material at x is the set of all ortho¬ gonal tensors Q such that

QC[E-]Q^=C[QEQ'^]

for every traceless symmetric tensor E. This definition is consistent, since

tr [QEQ'^) = tr {Q'^QE) = tr £ = 0

when Q is orthogonal.

If the material is isotropic, then (iii) of (22.1) is stiU valid; therefore

C[E]=2fiE. and hence

S+pl = 2/j,E.

Thus there is but one elastic constant for an isotropic and incompressible material— the shear modulus p.

The fundamental field equations for the time-independent behavior of a linearly elastic incompressible body are therefore

div M=0,

E^k{Vu + Vu^),

S+pi=ClE-]. p = -ktrS.

div S -f b = 0.

1 We could also begin by assuming that

S-f-^l = C[PM];

the steps leading to the reduction C [Pm] = C [£J would then be exactly the same those given in Sect. 20.

Sect. 58. Incompressible materials. 211

If we combine these relations, we are led to the displacement equation of equi¬ librium

div C[Vu] -Vp-i-b — O,

which for a homogeneous and isotropic body reduces to

fiAu—Vp-{-b~0. Since u is divergence-free,

(b)

Ap= div b; (c)

hence in the absence of body forces the pressure p is harmonic. In view of (a)j and (3) of (4.1), we can also write (b) in the form ' ^

^curlcurlM-|-P^ ~b =0.

On the other hand, if b = 0, then (b) and (c) imply that

A curl M = 0,

AAu = 0.

Since tr£ = 0, the compatibility equation (14.3) has the following form for an incompressible body:

AE-VdivE-{VdivEf =0.

If the body is homogeneous and isotropic, then the same procedure used on p. 92 to derive the stress equation of compatibility now yields

AS-2VVp+2Vb + {divb) 1=0,

which in the absence of body forces has the simple form

AS-2VVp=0.

For an isotropic incompressible material the surface traction on the boundary dB takes the form

s=Sn = —p n -\-iA,{Vu-\-Vu^) n.

Using the steps given on p. 93, we can also write this relation as follows;

s = —pn-{-2/j,-^ -f^nxcurlM.

We define, for an incompressible elastic body, the notion of an elastic state in the same manner as before,^ except that the constitutive relation is replaced by (a)3 and the incompressibility condition (a)j is added. If this is done, the theorem of work and energy (28.3) remains valid for incompressible bodies, where, as before,

U{E}=k S B-C[E\ dv, B

but now, of course, E is required to be traceless. The crucial step in proving this result is to note that if S and E obey (a) 3, then

S-E = -pl-E+E-C[E]=E-C[E],

since 1 • E =tr E =0. Using a similar result, it is a simple matter to verify that Betti’s reciprocal theorem (30.1) also remains valid.

‘p. 95-

14*


If we define the mixed problem as before/ then steps exactly the same as those used to establish (32.1) now }deld the

(1) Uniqueness theorem for an incompressible body. Assume that the body is incompressible and that the elasticity jield C is positive dejinite. Let

6 = [u, E, S] and 3 = [u, E, S] be two solutions of the same mixed problem. Then 6 and 3 are equal modulo a rigid displacement and a uniform pressure, i.e.

[u, E,S'\ = [u-{-w,E,S-ypl],

where w is a rigid displacement field and p is a scalar constant. Moreover, w = 0 if

=b 0, while p =0 if ^2,^ 0-

The principle of minimum potential energy (34.1) remains valid in the pre¬ sent circumstances provided we require that the kinematically admissible states satisfy div m = 0 and change the last few words to read: “ only if i —a modulo a rigid displacement and a uniform pressure.” On the other hand, the principle of minimum complementary energy (34.3) remains valid if we replace C/k{S} by C/k{S 1}, where ^ = — 3- tr S and K is the inverse of C, and if we change

the last few words to read: “only if S =;S modulo a uniform pressure.”

E. Elastodynamics.

I. The fundamental field equations. Elastic processes. Power and energy. Reciprocity.

59. The fundamental system of field equations. The fundamental system of field equations describing the motion of a linear elastic body consists of the strain-displacement relation

E=^(Vu-\-Fu^),

the stress-strain relation S = C[E],

and the equation of motion^ div S -\-b —QU.

Here u, E, S, and 6 are the displacement, strain, stress, and body force fields defined on 5 X (0, ô). where (0, Iq) is a given interval of time, while C and g are the elasticity and density fields over B.

Since C\E]—C[Vu'\, if the displacement field is sufficiently smooth, these equations combine to }deld the displacement equation of motion :

div C[Fm] -|-6 =qu,

which in components has the form

{CijkiUi,j)j-\-bi=QUi.

It is a simple matter to verify that, conversely, if u satisfies the displacement equation of motion, and if E and S are defined by the strain-displacement and stress-strain relations, then the equation of motion is satisfied.

‘ p. 102. ® Since the values of Ca- [•] are assumed to be symmetric tensors, we need not add the

requirement that S — S^.

Sect. 59- The fundamental system of field equations. 213

Assume for the moment that the body is homogeneous and isotropic. Then the procedure used to derive (a) on p. 90 now 3delds Navier’s equation of motion^

Aivu-\-h=QU. (a)

By (3) of (4.1), this equation may be written in the alternative form^

(A + 2/t) V div u—fi curl curl m + b = pit.

If we assume that and q are positive, we can define constants q and

in terms of these constants the above equation becomes

cl V div u — c\ curl curl u + b

= U. Q

(C)

For convenience, we introduce the wave operators and nj, defined by

Then, since Uaf=clAf-f (a = 1,2).

Q -4-

Navier’s equation (a) can be written in the form®

□2M+(cf —c|) P’divM + y =0. (d)

Assume now that b =0. It is clear from (c) and (3) of (4.1) that^

curl M = 0 => M = 0,

div M = 0 □2M = 0.

For this reason, we call the irrotational velocity, C2 the isochoric velocity. Operating on Navier’s equation, first with the divergence and then with the curl, we find that®

divM = 0, (e)

□ 2 curl M = 0.

Thus the dilatation div m satisfies a wave equation with speed c^, while the rotation J curl u satisfies a wave equation with speed C2. Clearly,

--1- = l/7+'?fL = l/2(i-v) . Ca \ fi \ \ — 2v ’

This relation was first derived by Navier [1823, 2], [I827, 2] in 1821. Navier’s theory, which is based on a molecular model, is limited to situations for which fi = X. The general relation involving two elastic constants first appears in the work of Cauchy [1828, T\ \ cf. Poisson [1829, 2], Lam^ and Clapeyron [1833, f], Stokes [1845, 1], Lam^; [1852, 2], § 26.

2 LAMk and Clapeyron [1833, f], Lam^: [1852, 2], § 26. ® Cauchy [1840, f], p. 120. * The first of these results is due to Lam^ [1852, 2], § 61-62; the second to Cauchy

[1840, f], p. 137. ® Cauchy [1828, f], [1840, f], pp. 120, 137- See also Lam^ and Clapeyron [1833, f],

Stokes [i851, 1], Lam^ [1852, 2], § 27.


thus for most materials Ci>C2, so that dilatational waves travel faster than rotational waves.^ In fact, c-^=2c^ when v—^.

Next, if we apply to (a) (with 6=0) and use the first of (e), we arrive at

—pit) =0, or equivalently, 2

□in2M=o. (f)

Eq. (f) and the strain-displacement and stress-strain relations imply that

□i D2E = DiD2S = 0.

Eqs. (e) are counterparts of the harmonicity of div u and curl u in the static theory; (f) is the counterpart of the biharmonicity of u.

We now drop the assumption that B be homogeneous and isotropic. The quantity

e=Ê ■ C[E]-\-^q

represents the total energy per unit volume. If C is symmetric, then the stress- strain and strain-displacement relations imply that

Thus, if we define e=S ■ Fu + qU • u.

q = —Su,

we are led, with the aid of the equation of motion, to the following result:

e-\-6iv q = b •u. (g)

The quantity q is called the energy-flux vector, and (g) expresses a local form of balance of energy.

Next, operating on the equation of motion with the symmetric gradient V and using the strain-displacement relation, we find that

P/divS-l-b

\ Q

If we assume that C is invertible, so that

E.

E = K[S],

where K is the compliance field, we arrive at the stress eqrmtion of motion^

i7p_|±^j=K[S].

If Q is independent of x, this relation takes the form

F(divS-h6)=pK[S].

If, in addition, the material is isotropic, then the formula on page 78 giving E in terms of S yields

K[S]=|[(l+r)S-r(trS) 1],

In seismology waves traveling with speed Cj are called primary waves (P-waves), those with speed Cj are called secondary waves (S-waves).

2 Cauchy [i840, i], p. 137- ® Ignaczak [1963, li] generalizing results of Valcovici [1951, 72] for a homogeneous,

isotropic body and Iacovache [1950,6] for a homogeneous, isotropic, and incompressible body. See also Ignaczak [1959, S], Teodorescu [1965, 19],

Sect. 60. Elastic processes. Power and energy. 215

where /9 is Young’s modulus and v is Poisson’s ratio; thus

/9(PdivS + Fb)=e[(1+5')S:-t^(tr S) 1].

Given initial values and for u and ii, the stress equation of motion implies the fundamental system in the following sense: let S satisfy this equation and the initial conditions

S(.,0)=C[Fmo], ;S(-, 0)=C[Fi5o];

let E be defined by the inverted stress-strain relation; and let u be the unique solution of the equation of motion with the initial values and tjg. Then the strain-displacement relation holds, and hence [u, E, S] satisfies the fundamental system of equations. The fact that in elastodynamics the fundamental system of field equations is equivalent to a single equation for the stress tensor is most interesting, especially when compared to the equilibrium theory where (for a simply-connected body) the fundamental system is equivalent to a pair of equa¬ tions: the stress equations of compatibility and equilibrium.

60. Elastic processes. Power and energy. By an admissible process we mean an ordered array = [u, E, S] with the following properties:

(i) u is an admissible motion

(ii) E is a. continuous symmetric tensor field on B X [0, ;

(iii) S is a time-dependent admissible stress field.^

Clearly, the set of all admissible processes is a vector space provided we define addition and scalar multiplication in the natural manner; i.e. as we did for admissible states.®

We say that p = [m, E, S] is an elastic process (on B) corresponding to the body force 6 if ;e is an admissible process and

£ = 1(Fm + Fm^),

S = C[E],

div S -1-6 =QU.

As before, the surface force field s is then defined at every regular point of

dB X [0, <o) by s{x, t) =S(x, t) n{x),

and we call the pair [6, s] the external force system for p.

If u{-, 0) —u{’, 0) =0

on B, we can extend p smoothly to B x{—oo, t^) by defining

(I)

u — 0 and E = S=0 on Bx(— oo, O).

When (I) holds we call the resulting extended process p an elastic process with a quiescent past.^ Further, to be consistent, we also extend the external force system by assuming that 6 and s vanish for negative time.

1 Cf. p. 43- 2 Cf. p. 49-

® Cf. p. 95.

^ This terminology is due to Wheeler and Sternberg [1968, 76].


When we discuss the basic singular solutions of elastod}mamics, we will be forced to deal with processes whose domain is of the form ^jX[0, oo), where ^ is a point of S and

—{^}.

We say that p = [u, E, S] is an elastic process on corresponding to 6 if u, E, S and b are functions on x [0, oo) and given any closed regular subregion the restriction of p to Px [0, oo) is an elastic process on P corresponding to the restriction of 6 to Px [0, oo).

When we omit mention of the domain of definition of an elastic process, it will always be understood to be Bx [0, tg).

The following simple proposition will be extremely useful.

(1) If [u, E, S] is an elastic process corresponding to the body force field b, then given any t^{Q,Q, [m(*, <), E{•,(), S(>, 4] is an elastic state corresponding to the body force b—qil.

Given an elastic process, we call the function on [0, t^ defined by

the total energy. Here K is the kinetic energy :

K — ^ f qiP dv, B

U is the strain energy:

U = Uc{E}=l S E-C[E] dv. B

If C is symmetric,

if j {E ■ C[E-\WE ■ C[E]} dv = p E • C[E] dv = j S • E dv, B B B

therefore the rate of change of total energy equals the rate of change of kinetic energy plus the stress power} This fact and the theorem of power expended (19.1) to¬ gether yield the

(2) Theorem of power and energy. Assume that C is symmetric. Let [u, E, S] be an elastic process corresponding to the external force system [6, s]. Then

j s-ii da}- f b-udv ='^, BB B

where % is the total energy.

This theorem asserts that the rate at which work is done by the surface and body forces equals the rate of change of total energy. The following proposition, which is an immediate corollary of (2), wiU be useful in establishing uniqueness.

(3) Assume that C is symmetric. Let [u, E, S] be an elastic process corresponding to the external force system [6, s], and suppose that

6=0 on P, s • ti = 0 on 9P.

Then the total energy is constant, i.e.

‘^{t)=‘^{0), 0^t<tg.

1 Cf. p. 65.

Sect. 6o. Elastic processes. Power and energy. 217

Another interesting corollary of (2) may be deduced as follows. By (2)

J / s • u da dt -\- J J b • u dv di =‘^{r) —'^(0). 0 dB OB

The left-hand side of this expression is the work done in the time interval [0, t]. If the initial state [m(*, 0), 0), S(*, 0)] is an unstrained state of rest, i.e. if

i!:(.,0)=0, M(.,0)=0, then

'^(0) = 0.

If, in addition, the elasticity tensor is positive semi-definite and the density positive, then “^(t) ^0 and

T T

/ / s-iida dt-{- f fb-udvdtÔ', 0 dB OB

in words, the work done starting jrom an unstrained rest state is always nonnegative}

The next theorem, which is of interest in itself, forms the basis of a very strong uniqueness theorem for elastod}mamics. For the purpose of this theorem we assume that the underl5dng time-interval [O, t^ is equal to [0, oo).

(4) Brun’s theorem} Assume that C is symmetric. Let [m, £, S] be an elastic process corresponding to the external force system [6, s], and suppose that

M(.,0)=0, il(.,0)=0. Let

P(ql, ^)= f s (x, ql) •u(x, /9) dax J b (x, a.) • u{x, /9) dv^ BB B

for all a, /96[0, oo). Then

U{t) -K{t) = h/[P{i + lt-X)-P{t-h,t+X)]dX (0^t<oo), 0

where U is the strain energy and K the kinetic energy.

Proof. By (18.2) with b replaced by 6(*, a) —pu(', a), s by s(',a), u by m(-, /9), and S by C[£l(-, a)],

P(a, /9) = / {C[E{<x)-]-m+Qu{a)-um dv. (a) B

where, for convenience, we have suppressed the argument x. Next, in view of the symmetry of C and the initial vanishing of the strain field,

t

JJ {C [E(t +X)]-Elt-X)-C [E{t - A)] ■E(t+X)}dXdv

. (b)

= -jj ■j^{C[E{t + X)'\-Elt-X)}dXdv = 2 U{t).

B 0

Similarly, t

fj Q{u(t -\-X) • u(t — X) —U(t — X) ■ ii{t X)} dXdv = —2K(t), (c) BO

and (a), (b), and (c) imply the desired result. □

^ Some related inequalities are given by Martin [1964, 14]. 2 Brun [1965, 4], Eq. (7b); [1969, 2].


61. Graffi’s reciprocal theorem. We shall assume throughout this section that the elasticity field is symmetric.

Given an elastic process p = [u, E, S] corresponding to b, we call the pair

Mo=m(-,0), Vq = u{-,0)

the initial data for p and the function / defined by

/{x, t)î*b(x, t)+Q(x) [Mo(x) +1(*)]

the associated pseudo body force field. Here, as before, i is the function on [0, Q defined by

i (i) = t.

Betti’s reciprocal theorem has the following important counterpart in elasto- d}mamics:

(1) Graffi’s reciprocal theorem.^ Let [u, E, S] and [m, E, S] be elastic

processes corresponding to the external force systems [6, s] and [6, s]. Further, let

[Wfl, iig] and [Mq, iio] be the corresponding initial data, f and f the corresponding pseudo body force fields. Then

i* f s*uda-i- f f*udv=i* f s*u da F f f*udv, (a) 8B B dB B

J S*Edv= f S*Edv, (b) B B

f s*u da F f b*u dv F f u+v^ - u) dv SB B B

= f s*uda-{- j b*udvF- J Q{uQ"U-\-Vg-u) dv. SB B B

Thus if both processes correspond to null initial data, then

f s*uda-\- S ^*udv= f s*uda-\- f b*udv. (d) BB B BB B

Proof. In view of the reciprocal theorem (19.5), it suffices to show that

S*E=S*E. By definition,

S*E{t) = J S{t — r)‘E(r)dr, 0

where, for convenience, we have omitted the dependence of these functions on x. Since C is symmetric.

S{t-r)-E{r)= E{r) ■ C[E{t-r)]=E{t^r) ■ C\_E(r)] =E{t-r)-S(r).

Thus

S*E{t)Ê*S{t)=S*E{t). □

' The result (c) for homogeneous and isotropic bodies is due to Graffi [1939, 3], [1947, 2], [1954, 6], whose proof is based on Betti’s theorem and systematic use of the Laplace transform. Later, Graffi [1963, S] gave a direct proof utilizing the properties of the convolution. Hu [1958, 10] noticed that Graffi's results extend trivially to inhomogeneons and anisotropic bodies; however, Hu gave his results in the Laplace transform domain rather than the time domain. Wheeler and Sternberg [1968, 16], assuming homogeneity and isotropy, gave an extension of Graffi's theorem valid for infinite regions; their result was extended to homogeneous but anisotropic bodies by Wheeler [197O, 3]. Ainola [1966 7], showed that Graffi’s theorem can be derived from a variational principle. Some interesting applications of Graffi’s theorem are given by DiMaggio and Bleich [1959, 4] and Payton [1964, 76].

Sect. 62. The boundary-initial-value problem of elastodynamics. 219

GraffiI also noticed that if both systems correspond to null external forces, then (c) implies

/ Q{Uf,-u-\-VQ’u) dv= f q{Uq-u-\-Vq'U) dv; B B

if, in addition, v„=v„ = 0, then

/ qUq - u dv = J QU^-iidv. B B

This relation can be used to establish the orthogonality of the displacement fields corresponding to distinct modes of free vibration.^

For the remainder of this section let = oo. We say that two external force

systems [b, s] and [b, s] are synchronous if there exists a continuous scalar function g on [0, oo) such that

b {X, t) = bo (x) g{t), b {x, t) = bo (®) g (t)

for all {x, t)eBx [0, oo) and

s {x, t) = So {x) g{t), s {x, t) = So (a;) g {()

for all {x, t)edBx [0, oo).

The next theorem shows that for s5mchronous external force systems a recip¬ rocal theorem holds having the same form as Betti’s theorem in elastostatics.

(2) Reciprocal theorem for synchronous external force systems.^ Let

[u, E, S] and [ii, E, S] be elastic processes corresponding to null initial data and to

synchronous external force systems [b, s] and [b, s]. Then

fs-uda-{-fb-udv= f s-uda-\- J b-udv. dB B dB B

Proof. By hypothesis and (d) of (1), g*(p =0, where

(p= f Sg- u da -j- f b^- u dv — fs^-uda— f b^-u dv, dB B BB B

and we conclude from (v) of (10.1) that either ^ = 0 or 95=0. If ^ = 0 the theorem holds trivially. Thus assume 9? = 0. Then g(p=o, and the desired result follows. □

II. Boundary-initial-value problems. Uniqueness.

62. The boundary-initial-value problem of elastodynamics. Throughout the following three sections we assume given the following data: a time interval [0, <q), an elasticity field C on B, a density field q on B, body forces b on Bx(0, 4). initial displacements u„ on B, initial velocities on B, surface displacements u on 5^x(0, ô). a-nd surface tractions s on .9|x(0, t„). Given the above data, the mixed problem of elastodynamics consists in finding an elastic process [u, E, S] that corresponds to b and satisfies the initial conditions

u(',0)=Ug and u(‘,0)=Vg on B,

* [1947, 2].

2 Cf. (76. 3) ® Graffi [1947, 2].


the displacement condition

u=u on 5^X(0, t„),

and the traction condition

s=s on 5^2X(0, t^).

We call such an elastic process a solution of the mixed problem.^ As before, when 5^ = 95 (,$^ = 0) we refer to the displacement problem, when S^2, = ^B =0) we refer to the traction problem.

To avoid repeated h5rpotheses we assume once and for aU that:

(i) C and q are smooth on 5;

(ii) 6 is continuous on 5 x [0, ô) >

(iii) and are continuous on B ;

(iv) M is continuous on X [0, y i

(v) s is continuous in time and piecewise regular on 5^x[0, 4).

The following proposition is an immediate consequence of the remarks made in Sect. 59: its proof is completely analogous to the proof of (31.1) and can safely be omitted.

(1) Characterisation of the mixed problem in terms of displace¬ ments. Let u be an admissible motion. Then u corresponds to a solution of the mixed problem if and only if

div C[Fm]+6 = pM ow5x(0, 4)<

m(»,0)=Mo and m(*,0)=Uo ow B,

u=u on 5iX(0, 4).

C[Vu'\n = s 5^2 (0, 4) ■

Further, when the body is homogeneous and isotropic, the first and last of these conditions may be replaced by

+ FdivM-|-b=pM on Bx{0,t„),

p{Vu-\- Vu^) n-bA(div u)n—s om 5^2(0. 4)■

For the remainder of this section we assume that

Q>0.

(2) Characterisation of the traction problem in terms of stresses.^

Assume that'.

(a) C is invertible on 5;

(/9) Q, Uq, and are of class on 5;

(y) b is of class on Bx [0, 4):

{6) S is a time-dependent admissible stress fidd^ of class on 5x[0, 4)

with S continuous on Bx [0, 4)-

See footnote 2 on p. 102. ® Ignaczak [1963, 141 the case in which U(| = D(|=0. ® See p. 49.

Sect. 62. The boundary-initial-value problem of elastodynamics. 221

Then S corresponds to a solution of the traction problem if and only if

owBx(0,g,

S(-, 0) =C[Fmo] and ;S(-, 0) =C[l7i5o] B,

s~s on dBx (0, to).

Proof. The necessity of the above relations follows from the discussion given at the end of Sect. 59. The sufficiency also follows from this discussion, but we will give the argument in detail. Let S satisfy the above system of equations. Then, since S is a time-dependent admissible stress field,

(1) S and div S are continuous on F X [0, 4) i a-^d, by (i) and (a),

(2) K=C'i is continuous onH.

We now define u and E through the relations

i * div S U=---,

Q

where / is the pseudo body force field

E=K[S], (a)

/{x, t) = i*b (x, t) + () (x) [Mo (x) + tvg(x)], (b)

i{t) = t.

It then follows that u, S, and b satisfy the equation of motion,^ and that u satisfies the initial conditions. Next, by (ii), (iii), (/9), (y), (d), (1), (2), (a), and (b),

(3) u, u,.u, E, and E are continuous on F X [0, y, while u is of class on B X [0, tff);

(4) E is of class C®' ® on F X [0, <o)< and Fu are continuous on F x [0, fo).

and F(m) = Fm.

To complete the proof we have only to show that the strain-displacement relation holds (for then we could conclude from (3) and (4) that u is an admissible motion). It follows from (4), (a)2, the first relation in (2), and the equation of motion that

e=.k[s]=f (c)

On the other hand, since u satisfies the initial conditions, we conclude from (a)2

and the initial conditions satisfied by S that

£;(.,0)=Fm(.,0), F(.,0) = Fm(.,0); (d)

thus, by (c) and (d), E = Fu. □

An interesting consequence of theorem (62,2) and the uniqueness theorem (63.1) is that a body force field with Vlblg) skew induces zero stresses in the body, i.e,, if dB is traction-free, if u„=v„=0, and if V{blQ) is skew, then S=0 and M—f *6/p is a rigid motion.

The next two theorems give alternative characterizations of the mixed problem in which the initial conditions are incorporated into the field equations. These results will be extremely useful in the derivation of variational principles.

* Cf. the proof of (19,2).


(3) ^ Let p — \u, E, S] be an admissible process. Then p is a solution of the mixed problem if and only if

E = Vu,

S = C[E],

t'itdiv S +/ = û

on Bx [0, <o). u = u on 6^1 x(0, Q,

s = s on (0, Q.

Proof. This theorem is a direct consequence of (19.2). □

(4) Characterisation of the mixed problem in terms of stresses.^ A ssume that hypotheses (a), (/S), and {y)of(2) hold. LetSbea time-dependent admissible

stress field of class C®’® on Bx [0, t^, and suppose that S is continuous on Bx [0, ^q)- Then S corresponds to a solution of the mixed problem if and only if

P I* / J = K [S] on Bx [0, Q,

i*dLvvS=QU—f on 6T.iX{0,Q,

s = s on £A^x (0, ô) •

Proof. Clearly, (l) and (2) in the proof of (2) hold. Assume that S satisfies the above relations. If we define

E=K[S],

then (3) of (2) also holds, and it is not difficult to verify that Vu=E and u = u on 6^1 X (0, ô). It therefore follows from fS) that p= [u, E, S] is a solution of the mixed problem. Conversely, if je is a solution of the mixed problem, then we conclude, with the aid of (3), that the relations in (4) hold. □

63. Uniqueness. In this section we discuss the uniqueness question appropriate to the fundamental initial-boundary-value problems discussed in the preceding section. We assume throughout that the density q is continuous on B.

We will first establish the classical uniqueness theorem of Neumann. We will then give a more general theorem due to Brun; we present Neumann’s theorem separately because its hypotheses are sufficiently general to include most appli¬ cations, and because its proof is quite simple.

(1) Uniqueness theorem for the mixed problem.^ Suppose that the density field is strictly positive and the elasticity field symmetric and positive semi- definite. Then the mixed problem of elastodynamics has at most one solution.

Proof. Let p — [u, E, S] be the difference between two solutions. Then p corresponds to vanishing body forces and satisfies

m(-, 0) =m(", 0) =0 on B, (a)

M = 0 on X (0, ô). s = 0 on .91X (0, tg). (b) By (a) _ ^(•,0)=0, (c)

* Gurtin [1964, S], Theorem 3.2. 2 Gurtin [1964, S], Theorem 3.4. ® Neumann [1885, 3], § 61. An interesting generalization was given by Hill [1967, 9].

Sect. 64. Some further extensions of the fundamental lemma. 223

and (a) and (c) yield <^(0) =0, (d)

where ^ is the total energy.* Moreover, in view of (60.3), (b) and (d) yield

'^(0=0, (e)

By definition is the sum of the kinetic and strain energies, and by hypothesis both of these energies are positive; therefore they must both vanish. In particular.

and, since ^>0,

/ Q dv =0, B

u = 0 on Bx{0, tg).

(f)

This fact, when combined with the first of (a), yields

u=0 on Bx(0, ig),

and the proof is complete. □ Recently Brun has established the following major generalization of Neumann’s

theorem (1)> it is important to note that no assumptions other than symmetry are placed on the elasticity field. We suppose for the purpose of this theorem that

so that the underlying time interval is [0, 00).

(2) Extended uniqueness theorem.^ Suppose that the density field is strictly positive (or negative) and the elasticity field symmetric. Then the mixed problem of elastodynamics has at most one solution.

Proof. As in the proof of (1), let = \u, E, S] be the difference between two solutions. Then p corresponds to vanishing body forces and satisfies (a), (b), and (c) in the proof of (1). Thus we conclude from Brun’s theorem (60.4) that

U{t) —K{t) =0, 0^t<oo. (a)

On the other hand, by (e) in the proof of (1), which also holds in the present circumstances,

U(t)-pK(t) =0, 0^t<oo. (b)

Thus K{t) = 0 and the last few steps in (1) yield the desired conclusion that MÔ. □

III, Variational principles.

64. Some further extensions of the fundamental lemma. We now establish, for functions of position and time, counterparts of the lemmas established in Sect. 35. With this in mind, we introduce the following definition. We say that a function / on Bx[0, ô) vanishes near STCdB if there exists a neighborhood N oi 6^ such that / = 0 on (iVi^ 5) X [0, ^q)-

* See p. 216. 2 Brun [1965, 4], [1969, 2]. This theorem was established independently by Knops and

Payne [1968,11], using an entirely different method. For the displacement problem appropriate to a homogeneous and isotropic body, Gurtin and Sternberg [1961, 10] proved that un¬ iqueness holds provided both wave speeds are positive; i.e. provided q is positive and the elasticity field strongly elliptic. This result was extended to anisotropic, but homogeneous, bodies by Gurtin andToupiN [1965, 9]. Hayes and Knops [1968, 7] showed, without assuming C symmetric, that uniqueness also holds when — C is strongly elliptic. Of course, the extended uniqueness theorem (2) includes as special cases Neumann’s theorem and all of the above results except that of Hayes and Knops. See also Knops and Payne [1971, 2], § 8, for a thorough discussion of uniqueness in elastodynamics.


(ir Let W he a finite-dimensional inner product space. Let w. B x [0, t^ he continuous and satisfy

P w*v[x, t) dv^ = 0 {Q'^t<tPi B

for every class C°° function v: B x [0, tî-^W that vanishes near dB. Then

w = Q on Bx\0, tg).

Proof. Let (t) denote the open interval {x—h, x-{-h), and for Ih{x)C. (0, tP) let (t) be the set of all C°° scalar functions ip on [0, t^} with the properties:

(a) ip>0 on

{b)^=0 on [0, g-AW.

It is clear from the discussion at the beginning of Sect. 7 that (t) is not empty.

Let Cj, e^,..., e„ be an orthonormal basis in W, and let

n

As in the proof of the fundamental lemma (7,1), we assume that for some (acg, t)gBx(0, tPj and some integer k, w^{Xq, t)>0. Then there exists an/(>0

such that Ilh{x^)cB, /i(T)C(0, Q, and ^^>0 on ^'^(aEo) xA(t)- Let

t5(ac, s)=(p{x)ip{s) e^,

where^ (pe0i,{Xo),ipeWi,{t — r), and ti (x-ph, Q. Then i5 is of class C°° on 5 x [0,fo). vanishes near 8B, and

t

f W*v(x, t) dVa;— f f w(x, t — s) •v[x, s) ds dv^. B BO

t — f f t—s) (p{x)ip{s) ds dv^

B 0

— f f t—s) (p(x)f(s) ds dv^>0, 2:/,lXc) l-r-ll

which is a contradiction. Thus M5 =0 on S x [0, <o)- □

The proofs of the next three lemmas can safely be omitted; they are completely analogous, respectively, to the proofs of (35.1), (35.2), and (35.3) a.nd utilize the procedure given in the proof of (1).

(2)^ Let iVhe a finite-dimensional inner product space. Let w. [0, t^} he piecewise regular and continuous in time, and assume that

f w*v(x, t) da^=0 (0^t<tg)

for every class C°° function v: Bx[0, tg)-^W that vanishes near 5^. Then

M5 = 0 on [0, tg).

1 Gurtin [1964, S], Lemma 2.1. ^ 0^ (Xg) is defined in Sect. 7. ® Gurtin [1964, S], Lemma 2.2.


(3) ^ Let u be a vector field ow x [0, t^ that is piecewise regular and continuous in time, and suppose that

f (Sn)*u(x, t) dv^ — 0 (0^t<tg)

for every class C°° symmetric tensor field S on Ex [0, tg) which vanishes near Then

M = 0 OM X [0, ô) •

(4) ^ Let ube a vector field on x [0, t^ that is piecewise regular and continuous in time, and suppose that

P uÂiv S{x,t) dva. = 0 (0î<4)

for every class C°° symmetric tensor field S on Bx [0, t^. Then

M = 0 on i/’^x [0, ô) •

65. Variational principles. In this section we will establish several variational principles for elastodynamics. In addition to hypotheses (i)-(v) ® of Sect. 62 we assume throughout that the elasticity field C is symmetric. We begin by giving Kirchhoff’s generalization of Hamilton’s principle in particle mechanics.

By a kinematically admissible process we mean an admissible process that satisfies the strain-displacement relation, the stress-strain relation, and the displacement boundary condition.

(1) Hamilton-Kirchhoff principle. * Let ^ denote the set of all kinematically admissible processes \u, E, S] that have ii and E continuous on B X [O, t^ and

M(*,0) = a, u{>,tg) = p on B,

where a and are prescribed vector fields. Letrj{’} be the functional on defined by

rj{fi}~ f U{E} dt— f f (i QU^-i'b • u) dv dt— J J s • u da dt 0 OB 0 se,

for every p= \u, E, S] Then

dri{p}^0

at p&j!^ if and only if p satisfies the equation of motion and the traction boundary condition.

Proof. Let p£sd, and let p= [it, E, S] be an admissible process with the follow¬ ing property:

p-\-l.pesd for every scalar A. (a)

1 Gurtin [1964, S], Lemma 2.3. In 1965 HlavACek (private communication) pointed out that the proof given by Gurtin [1964, S] is incorrect and furnished a valid proof.

2 Gurtin [1964, S], Lemma 2.4. ® For the validity of (1) we need the stronger assumption that (ii), (iv), and (v) hold with

the interval [0, tg) replaced by [0, fol- ^ Kirchhoff [1852, I], [18S9, f], [1876, f], § 11. See also Love [I927, 3], § 115; Loca-

TELLi [1945, 3, 4]; Washizu [1957, fS]: Kneshke and ROdiger [1962,10]-, Chen [1964, 6]; Yu [1964,23]; Ben-Amoz [1966,2]: Karnopp [1966,70], [1968,70]; Komkov [1966,75]; XoNTi [1966, 26].



Clearly, (a) holds if and only if p satisfies the strain-displacement relation, the stress-strain relation, the boundary condition

M=0 on ,5^ X [0, ô]. (b) and the end conditions

m(., 0) = m(*, = ® on B. (c) Next,

d^r]{p}=

to to (d) = J J(S ■ E—qu ■ u—b ■ u) dv di— II ^ • w d^ di y

OB 0

where we have used the symmetry of C and the stress-strain relation. By (c)

J QU -u dt= — J Qu-udt, (e) 0 0

and (d), (e), the fact that E and u satisfy the strain-displacement relation, (18,1), and (b) imply

d^ri{fi}= — f f (div S-i-b — gii) ■ u dv+ J J (s — s) • u da dt. (f) OB 0

Clearly, di;{;e}=0if

divS-{-6 = ^M on 5x[0,;o). s = s on y’gX[0, ^q)• (g)

On the other hand, if (f) vanishes for every p consistent with (a), then (f) vanishes for every admissible motion u consistent with (b) and (c), and obvious extensions of (7.1) and (35.1) yield (g). □

The Hamilton-Kirchhoff principle is concerned with the variation of a func¬ tional over a set of processes that assume a given displacement distribution at an initial as weU as at a later time. This type of principle clearly does not characterize the initial-value problem of elastodynamics, since it fails to take into account the initial velocity distribution and presupposes the knowledge of the displace¬ ments at a later time—an item of information not available in advance. We now give several variational principles which do, in fact, characterize this initial-value problem.

We assume for the remainder of this section that the density q is strictly positive. Note that, in view of assumptions (i)-(iii) of Sect. 62, the pseudo body force field

/[x, t)=i*b {x, t)+Q(x) [Mo{x) +tVo{x)], i[t) = t

is continuous on .S X [0, t^.

We begin with a variational principle in which the admissible processes are not required to meet any of the field equations, initial conditions, or boundary conditions; this principle is a counterpart of the Hu-Washizu principle (38.1) in elastostatics.

(2)^ Let denote the set of all admissible processes, and for each to, [0, t^ define the functional /!({•} on by

At{p} = / {î*E*C[E^ QU*u — i*S*E — [f ♦div S -f/]{x, t) dv^ B -f f (i*s*u}(x, t) da^j-j- f {i*(s—s)*u}dai^

1 Gurtin [1964, SJ, Theorem 4.1.


for every p = \u, E, S] e Then

dAt{p}=^Q) {Q)^t<Q

at an admissible process p if and only if p is a solution of the mixed problem.

Proof. Let p = \u, E, S] and p = \_u, E, S] be admissible processes. Then p+^p is an admissible process for every scalar A. Since C is S5nnmetric,

and thus

Let

Then (a) implies

Eit - t) • C [E{x)] = C [E{t - t)] • E{x),

E*C[E]=C[E]*E=E*C[E].

Uf{E} = ^f i*E*C[E](x, t) dv^,.

(a)

U,{E + Ê}=U,{E}+k^ Ut{E} +A / i*E*C[E] (x, t) dv^. (b) B

Thus, since

d~At{p}= At{p + Xp}\xô- it follows that

= f {f =i‘(C[B] —S)*E—i*S*E — (f itdiv S-pf — gu) *u — i*u*div S}{x, t) dv^. B

-f^ {i*s*u}{x, t) daÂ-^ i*(s*u -j- (s —s) *u}(x, t) da^.

If we apply (19.3) to S and u, we find that

thus

f* JMitdiv S dv = i* J s*uda—i* f S*Fu dv, B dB B

^7,AM = fi*{C[E] - S)*E{x, t) dv^ B

— / (fitdiv S+/ —^M)=i<M(ac, t) dv^ B

+ /i*{Vu—E)*S{x, t) dv^ B

+ J i*(u — u)*s (x, t) da^

+ / i*(s — s)*u{x,t) da^^ (0^t<tg).

(c)

If jg is a solution of the mixed problem, then we conclude from (c) and (62.3) that

d~At{p}=0 {0^t<to) for every peji^ (d) which implies

dAM=^0 (0^t<tg). (e)

To prove the converse assertion assume that (e) and hence (d) holds. If we choose p = [it, 0, 0] and let u vanish near dB, then it follows from (c) and (d) that

J {i*div SA-f — Qu)*u(x,t) dv^=0 (0^t<tg). B

15’


Since this relation must hold for every m of class C°° on Bx [0, tg) that vanishes near dB, we conclude from (64.1) that f*div S+/ = gM. Next, by choosing

the forms [0, E, 0] and [0, 0, S], where E and S vanish near dB, we con¬ clude from (c), (d), and (64.1) with W equal to the set of all S5nnmetric tensors that

f*(C[jE] — S) =0, jE)=0.

Thus, by (iv) of (10.1) with (p=i, p satisfies the stress-strain and displacement- strain relations. In view of (62.3), to complete the proof we have only to show that p satisfies the boundary conditions. By (c), (d), (iv) of (10.1), and the results established thus far,

/ (u — u)*(Sn) (x, t) da^+ f (s~s)*u(x, t) da^ = 0 {0^t<L)

for every Therefore, if we take peJi/ equal to [u, 0, 0] and [0, 0, S] and appeal to (64.2) and (64.3), we conclude that p satisfies the boundary condi¬ tions. □

Our next theorem is an analog of the Hellinger-Prange-Reissner principle (38.2).

(3)^ Assume that the elasticity field is invertible. Let j?/ denote the set of all admissible processes that satisfy the strain-displacement relation, and for each

<e(0, ffl) define the functional 0<{'}

©f{p}= J {ii*S*K[S] —i-QU*u — i*S*E (x, t) dv^ B

J {i*s*(u—u)} (x,t) dag,,-\-^{i*s*u}{x, t) da^

for every p — \u, E, S] e jaf. Then

d©({p}=0 i0^t<to)

at p^s/ if and only if p is a solution of the mixed problem.

Proof. Let p and p be admissible processes, and suppose that p-\-'kp^s^ ior every scalar A, or equivalently, that p,p^sd. In view of the S5nnmetry of C, K is S5nnmetric. Thus if we proceed as in the proof of (2), we conclude, with the aid of (19.3), that 2

d;i0i{p}= / f*(K[S]—jE) *S(x, t) dVg,, B

f {i*dnw S+f — Qu)*u{x,t) dVai ^ ^ _ (a) f i*{u—u)*s{x,t) daj. ^ '

-|- f i*(s — s) *u{x, f) da^. (0^t<tg).

If is a solution of the mixed problem, then (a), because of (62.3), yields

d©f{p}=0 (0^t<to). (b)

On the other hand, (a), (b), (64.1), (64.2), (64.3), (iv) of (10.1), and (62.3) imply that is a solution of the mixed problem. □

* Gurtin [1964, S], Theorem 4.2. 2 Cf. the proof of (38.2).

Sect. 65. Variational principles 229

lip = [m, E, S] is a kinematically admissible process, then, by virtue of (19.3), At{p} defined in (2) reduces to ^({p}, where

0Ap}=^ f {i*S*E + QU*u-~2f*u} {x, t) dv^.— f (x, t) da^. B Sf,

Thus we are led to the following:

(4) Analog of the principle of minimum potential energy.^ For each <e[0, (q) let functional defined on the set of all kinematically ad¬ missible processes by the above relation. Then

d0t{p}=o (o^«g at p^sd if and only if pis a solution of the mixed problem.

Proof. Let p = [it, E, S] be an admissible process, and suppose that

p + Xp^sd for every scalar X. (a)

Condition (a) is equivalent to the requirement that p satisfy the strain-displace¬ ment relation, the stress-strain relation, and the boundary condition

it = 0 onyiX[0, <o)- (b)

Next, since the restriction of /!«{•} of (2) to (the present) sd is equal to 0t{’}> we conclude from (c) in the proof ot (2) and the properties of p and p that

h = — / (i*div S -b/—gM) * it (a;, t) dv^ B

Ji*{s—s)*u{x,t) da^^ (0^^<<o) (c)

for every admissible motion it that satisfies (b). If is a solution of the mixed problem, then, by virtue of (62.3), d0({p)=O {O^tKQ. Conversely, if the variation of 0( vanishes at p for all t, then it follows from (c), (64.1), and (64.2) that p is a solution of the mixed problem. □

It is a simple matter to prove counterparts of theorems (34.2) and (57.4) characterizing the displacement field corresponding to a solution of the mixed problem.^ Rather than do this we shall establish a variational principle which involves only the stress field and which has no counterpart in elastostatics.

In addition to the assumptions made previously, we now assume that:®

(A) C is invertible oxiB , q,Uq, and Vq are of class C® onS; 6 is of class C®'® on

S X [0, iff).

(5) First variational principle for the stress field^. Let sd denote the set

of all time-dependent admissible stress fields of class C®' ® ow J5 x (0, Q with S continuous on Ex [0, <o)- Por each te [O, t^) let /){•} be the functional on sd defined by

r,{S} {j'*div S* div S +S*K[S] -2S* Vf) (x, t) dv^

+ .f {(f *s} (®> 0 + I {f'* (s —s) *div S) (x, t) da,g, where

1 Gurtin [1964, S], Theorem 4.3. 2 Cf. Gurtin [1964, S], § 5- ® These assumptions are slightly more stringent than necessary. ^ Gurtin [1964, S], Theorem 6.1.


Then ^r,{S}==0 {0<t<Q

at S if and only if S corresponds to a solution of the mixed problem.

Proof. Let Sfj/, so that S -j-AStsd for every scalar A. Then, since K is sym¬ metric, the divergence theorem and the properties of the convolution yield, after some work.

ds ri{S} = - f {V{i' * div S -h/') - K [S]} *S{x, t) dv^ B

(f'iKdiv S —1) da^.

-f/f'=i‘(s—s)=i‘divS(ac, i) (0^f<fo)

(a)

for every S^sd. If S corresponds to a solution of the mixed problem, then (62.4) implies that

^^^{S}=Q (0^f<g. (b)

On the other hand, (a), (b), the symmetry of S, (64.1), (64.3), (64.4), (iv) of (10.1), and (62.4) imply that S corresponds to a solution of the mixed pro¬ blem. □

Let j/ be as defined in (5). By a dynamically admissible stress field we mean a tensor field Sej/ that satisfies the traction boundary condition. For such a stress field, LJ{S}, defined in (5), reduces to

= i f {f'*div SiKdiv S -{-S=i‘K[S] ~2S*Ff'} (x, t) dv^. B

+ I {(/'—“)*»} (®.

Thus we are led to the following theorem, the proof of which is strictly analogous to that given for (5).

(6) Second variational principle for the stress field. ^ For each te [0, Q let Q,{‘} be defined on the set of dynamically admissible stress fields by the above relation. Then

dQt{S}=Q (0^f<g

at a dynamically admissible stress field S if and only if S corresponds to a solution of the mixed problem.

If we apply (19.3) to the term involving S*Vf', we can rewrite the expression for in the form

=2- / {f'=i<div S=i<div S -fS=i‘K[S] -f 2 div S*/'} {x, t) dv^ B

— / {ftiKs} {x, t) da^ — / {/'*s} [x, t) da^. •5^1 *^2

Of course, the last term may be omitted without destroying the validity of (6), since it is the same for every dynamically admissible S.

66. Minimum principles. For the variational principles established in the preceding section the relevant functionals were stationary, but not necessarily a minimum, at a solution. On the other hand, the principles of minimum potential

' Gurtin [1964, S], Theorem 6.2.

Sect. 66. Minimum principles. 231

energy and minimum complementary energy are true minimum principles. This difference is nontrivial; indeed, the lower bound on the energy provided by a minimum principle is useful in establishing the convergence of ntunerical solutions. In this section we establish minimum principles for the Laplace transforms of the functionals associated with the variational principles (65.4) and (65.6).

We let =00, so that the underlying time interval is [0, 00). In addition to hypotheses (i)-(v) of Sect. 62 we now assume that:

(vi) C is S5nnmetric and positive definite;

(vii) Q>0;

(viii) b, u, and s possess Laplace transforms.

Let denote the set of all kinematically admissible processes = \u, E, S] such that^ fi and Vfi possess Laplace transforms. Then for the function

ii-*

of (65.4) has a Laplace transform

_ 00

0

for some rjo ^ 0. In fact, since the Laplace transform of i {i) = i is 1 Irj^, it follows

from (vii) of (10.1) and the transformed stress-strain relation S — C[E] that

^r,{p} = YUc{E(-,r])} + y{I Q(x)u{x,rjy-f{x,r])-u{x,rj)jdv^ B

— ^2 /S (®. »?)•“(* v) iVo ^rj<oo). (L)

It is interesting to note the similarity between and the functional ^{•} of the principle of minimum potential energy (34.1). This similarity is emphasized by the

(1) Theorem of minimum transformed energy.^ Assume that the elas¬ ticity field is positive semi-definite. Let p(zsî_ be a solution of the mixed problem. Then for each there exists an rjoÔ such that

0^{p}^0r,{p} ir]o^r]<oo).

Proof. Since p, there exists an rjoÔ such that the Laplace trans¬ forms of p, p, Vp, Vp, and 6 exist on B x [rjo, 00). Let

p'=p-p. (a)

Then p' is an admissible process, p' and Vp' possess Laplace transforms, and

E'=:UVu'-VVu'^), (b)

S'^C[E'], (c)

tt' = 0 on . (d)

Moreover S = C[jE], since thus (28.1) and (a) imply

C/c{I} - Uc{E} = Uc{E'} +jS-E'dv. (e) B

^ Here Fje = [Pm, FE, PS]. 2 Benthien and Gurtin [1970, 7].

232 M.E. Gurtin: The Linear Theory of El2isticity. Sect. 67.

Similarly,

/ QU^dv— J QU^dv= / Qu’^dv-\-2 J Qu-u'dv. (f) B B B B

If we apply ) to S and u', we conclude, with the aid of (b) and (d), that

f S-E'dv= f s-u'da—J u'• div S dv. (g) B si B

Further, since is a solution of the mixed problem, if we take the Laplace transform of the third and fifth relations in (62.3), we find that

“2 div S{x, fj) + fix. f]) = Q (x) u (x, fj) (h)

for every {x,f])eBx[r)Q, oo) and

s = l on S^2X[rjo,oo). (i) By (L) and (e)-(i),

—= + y f Q{se)u'ix,rifdv^ (%^??<oo); B

this implies the desired result, since C is positive semi-definite and q positive. □

Note that, since q>0, we can strengthen the inequality in (1) as follows:

whenever m(•,?;) =t= S(..

We now assume that (A) on p. 229 holds, and, in addition, that F6 possesses a Laplace transform. Let denote the set of aU dynamically admissible stress fields S such that S and FS possess Laplace transforms. Then for the

function < £2^ {S} of (65.6) has a Laplace transform £2,, {S} ^ < oo) for some ?;o>0. In fact,

= -Six.f,) ■ vrix,r,)\dv^ B

+ f [/'(*.»7)—«(*.»?)]•»(*,??) (fa* (i]oî]<oo), A

and we are led to the

(2) Minimum principle for the stress field. Let be the stress field

correspondingtoasolutionof the mixed problem. Then for each there exists an ^ 0 such that

Q„{S} ^£i,{S} {rjg^rj<oo).

The proof of this theorem is strictly analogous to that oi (1) and can safely be omitted.

IV. Homogeneous and isotropic bodies.

67. Complete solutions of the field equations. In this section we study certain general solutions of the displacement equation of motion ^

c? Fdiv u — cl curl curl it -I- — = ft, Q

1 Cf. p. 213.

Complete solutions of the field equations. Sect. 67. 233

asstiming throughout that Ci>c|>0 and that bfg admits the Helmholtz de¬ composition ^

Q = — Vx-~ curly, divy = 0,

with X and y of class C*’® on B X (0, tg). A solution u of class on Bx (0, will be referred to as an elastic motion corresponding to 6.

(1) Green-Lam^ solution.^ Let

M= V(p-\- cnx\%p, (Li)

where 99 and y} are class C® fields on Bx (0, <o) satisfy^

□i99 = x, □2V’=y- (Lg)

Then u is an elastic motion corresponding to b.

Proof. By (Li)

cf PdivM —c| curl curl It — fc-t-

= V{clA (p — (p — x} + curl{— c|curl curl yy — ip — y).

In view of (1) and (3) of (4.1),

curl curl curl y} = — curl A yy,

and the desired result follows from (Lj). □

(2) Completeness of the Green-Lam^ solution.^ Let u be an elastic motion of class_C^ corresponding to b, and assume that tt(*, t) and it(*, t) are continuous on B at some time t6(0, <o)- Then there exist class C® fields (p and yy on Bx(0,tg) that satisfy (Lj), (Lj). Moreover,

div^ = 0.

y As is clear from the proof of (6.7), such an expansion is ensured if b/p is continuous on

B X (O, fg) and of clciss C®- ® on B X (0, fg). ® Lam^: [1852, 2], p. 149. The two-dimensional version W2is obtained earlier by Green

[1839, i]. The solution u = V<p, Di <p = x appears in the work of Poisson [1829, 2], p. 404. Related solutions are given by Bondarenko [i960, 4], Kilchevskii and Levchuk [1967, Iff], Chadwick and Trowbridge [1967, 5] show that

u=V(p — curl curl (pip)— curl (pm)

□i9’ = 0. □2V = 0, □201 = 0,

defines an elastic motion corresponding to b = 0. Further, they prove that this solution is complete for regions bounded by concentric spheres.

® The wave operators are defined on p. 213- y Clebsch [1863,2] was the first to assert that the Green-Lam6 solution is complete. Clebsch

also gave a proof which Sternberg [I960, 13] showed is open to serious objection. (Actually, Clebsch’s proof is valid when the boundary of the region consists of a single closed surface.) Another inconclusive proof was given by Kelvin [1904, 5] in 1884. The first general com¬ pleteness proof was furnished by Somigliana [1892, 7]. Completeness proofs were also supplied byTEDONE [1897, 1], Duhem [I898, 5], and Sternberg and Gurtin [1962, 13]. An explicit version of Duhem’s proof was given by Sternberg [I960, 13]. The work of Somigliana,

Tedone, and Duhem appears to be little known; in fact, Sneddon and Berry [1958, 18], p. 109 state that the Green-Lame solution is incomplete if the region occupied by the medium has a boundary. Further, as Sternberg [i960, 13] has remarked, the completeness proof given by Pearson [1959, H] is open to objections.


Proof. 1 If we integrate the equation of motion twice, we arrive at

u{t) = m(t) + m(t) (< — t) + F // [cf divM(A) — pc(A)] dXds

t ^ ^ + curl // [—c|curlM(A) — v(A)] dXds,

X X

(a)

where, for convenience, we have omitted all mention of the variable x. If we apply the Helmholtz resolution (6.7) to u(r) and u(t), we find that

u{r) +u (t) {t — r) = Fa {() + curl P (t), (b)

where a and p are class fields that satisfy

a = 0, P = 0, divp=0.

Thus if we define functions 99 and ^ on B x (0, to) by

<p(t) = a(<) + // [cf divM(A) — pc(A)] dXds, X X

~~ If [cicurl m(A) + v(A)] dXds, X X

then M= F^j + curl^p,

and, since div y=0, (c), (d), and (e) imply

div ip —0. Moreover, by (c), (d), and (e).

(c)

(d)

(e)

(f)

(g)

99 = cf div M —PC, ip— — Czcurlu — y, (h)

and (f), (g), and (l)-(3) of (4.1) yield

divM=Zl99, curlM=—zl^p. (i)

Finally, (h) and (i) imply the desired result (Lj). □

It follows from (d), (e), and (i) that 99 and ip are sufficiently smooth that

□1 F99= Fni99,

□ 2 curl v>=curl n%W>

and (2) has the following corollary:

(3) Poisson’s decomposition theorem.^ Let u satisfy the hypotheses of (2) with body force field 6 = 0. Then u admits the decomposition

u — W2,

where tq and are class fields on Bx (0, <o) that satisfy

□jUi = 0, curltq = 0,

□ 2^2 = 0, divM2=0.

1 SoMiGLiANA [1892, 7]. See also Bishop [1953, 4], 2 Poisson [1829, 3], Although Poisson’s results are based on a molecular model which

)rields fi = 2„ his proof in no way depends on this constraint.

Sect. 67. Complete solutions of the field equations. 235

(4) Cauchy-Kovalevski-Somigliana solution.^ Let

“=ni3+(ci-c?) Fdivsr, (Si)

where g is a class C* vector field on Bx (0, Iq) that satisfies

Da [Hi 3= (S2)

Then u is an elastic motion corresponding to b.

Proof. By (SJ

□ 2M + (cf— d) Vdivu

= Da Di 3 + (ci — c!) Da Fdiv3 + (cf - c|){ni Vdivg + {cl — c|) A Fdiv g}.

In view of (S^) and the identity

Dj-Dif={cl-cl)Af,

the right-hand side of the above relation reduces to —bjQ] thus we conclude from (d) on p. 213 that u is an elastic motion corresponding to 6. □

If g is independent of time, then (Si), (Sj) reduce to

u^clAg + icl — cî Vdrwg,

AAg = - b

QC\C\ '

which is, to within a multiphcative constant, the Boussinesq-Somighana-GaJerkin solution (44,^).

If in the time-dependent Cauchy-Kovalevski-Somighana solution we define ^

^=□13. (p = 2cldiyg~p-y},

then (Sj), (S2) reduce to

»'=w+~^''-v{(p+p-v). Oi<p = -p-niw.

which may be regarded as a dynamic generalization of the Boussinesq-Papko- vitch-Neuber solution (44.1). Since the potentials 95 and y} obey a coupled differential equation, this solution is of no practical interest in elastodynamics.®

We now estabhsh the completeness of the Cauchy-Kovalevski-Somigliana solution. For convenience, we shall assume that the body forces are zero.

(5) Completeness of the Cauchy-Kovalevski-Somigliana solution. Let u be a class C* elastic motion corresponding to zero body forces, and suppose that u{‘, t) and u{-, r) are continuous on B at some t€(0, 4)- Then there exists a class vector field g on B x(0, 4) such that (Si), (Sj) hold.

1 Cauchy [1840, 1], pp. 208-209; Kovalevsky [1885, 2J, p. 269; Somigliana [1892, 7].

This solution was arrived at independently by Iacovache [1949,4]; it was extended to include body forces by Sternberg and Eubanks [1957, IT]- See also Bondarenko [1957, 2], Predeleanu [1958, IS], Teodorescu [i960, 14].

2 Sternberg and Eubanks [1957, 14]. Cf. the remarks on p. 142.

2 Sternberg [i960, IS]. '* Somigliana [1892, 7]. This theorem was arrived at independently by Sternberg and

Eubanks [1957, 14] using an entirely different method of proof and without assuming that b =0. A third completeness proof is contained in the work of Sternberg and Gurtin

[1962, 13].


Proof. By h3ôthesis and (2),

c| Fdivit —c|curlcurlM=u, (a)

M= F99 + curlv>, (b)

□i9’ = 0, 02^ = 0. (c)

Further, it follows from the proof of (2) that (p and \p are of class C®. Substi¬ tuting (b) into (a),

U = c\A curl \p — Fdiv {c% curl y> — cl Vcp), (d)

where we have used the fact that div curl = 0. By (c)

c\A curl \p — curl ip = clA curl ^ — Dj curl v’,

and thus

clA curl \p = Di curl xp. (e) Cl — Cg

Eqs. (d), (e), and the first of (c) imply that

it — -jj ^-y Di [cl curl xp — cl F95) — Fdiv(ci curl v>— c\ Vcp), Cj — Cj

and thus u{() =v{r) +v{r){t — r) -j- □!*(<) — (cf —cl) Fdivft(<),

where / S

*('') = ■^71"// [cicurlv>(il)—cf F99(A)]rfArfs, T T

v = u+ —7a- [cl curl xp-c\ F99]. Cl Cj

In the above equations we have omitted all mention of the variable x.

Next, since the Boussinesq-Somigliana-Galerkin solution ^44.3) is complete, we can expand v[r) — v{-, r) and v{r) —v{‘, r) as follows

V (r) = cfAgi -1- (c| - c|) Fdivcjri,

V (t) = cfzl STa + (cl — c|) Fdiv ,

where g^ and g^ are of class C* on B. Thus if we define

g {X, t)=h {X, t) -t- STi {x) + {t — r) g^ {x),

then g is of class C*, M=niSr + (c|-c|) Fdivg,

and (Sj) holds. Finally, (Sj) and (a) imply (Sj). □

In attempting to judge the comparative merits of the Cauchy-Kovalevski- Somighana solution (Sj), (Sg) and the Green-Lame solution (Lj), (Lj), the follow¬ ing considerations would appear to be pertinent. ^ The former remains complete in the equilibrimn case if the generating potential g {x, t) is taken to be a fimction of position alone. In contrast, the Green-Lame solution has no equilibrium covmterpart in the foregoing sense. This circmnstance reflects the relative economy of the Green-Lam6 solution, which is simpler in structure than the available complete solutions to the equations of equilibrium. Fmther, (S^) contains space

^ These observations are due to Sternberg [I960, IS].

Sect. 67- Complete solutions of the field equations. 237

derivatives of g up to the second order while only first-order derivatives of 99

and y} are seen to enter (Lj). Finally, (Lj), (Lj) are conveniently transformed into general orthogonal curvilinear coordinates, whereas (S^), (S^) give rise to exceedingly cumbersome forms when referred to curvilinear coordinates, with the exception of cyUndrical coordinates. For all of these reasons, the Green-Lam4 solution deserves preference over the Cauchy-Kovalevski-Somigliana solution in appUcations.

In the absence of body forces the Cauchy-Kovalevski-Somigliana stress function satisfies the repeated wave equation

• □2ni3=o. The next result yields an important decomposition for solutions of this equation.

(6) Boggio’s theorem.^ Let g be a vector field of class on Ex (0, Q, and suppose that

□2nisr=o. Then

3 = Si + SI2.

where g^ and g^ are class fields on Bx (0, t^ that satisfy

□lSl = 0. □2S2=0-

Proof. 2 It suffices to establish the existence of a class field gôn Bx (0, t^ with the following properties;

□ iSi = 0, □2(S-Si)=0- (a)

Indeed, once such a g^ has been found, we simply define

S2 = S-Si

Since na9j=clAgi-gi (a = l,2),

it follows that if Oj = 0, then

□2Si= 9i-

Thus (a) is satisfied if

□iSi = 0. Si=/. (b) where

Notice that / is of class on B x (O, t^) and obeys

□i/=0. (c)

Thus to complete the proof it suffices to exhibit a field g^ satisfying (b), sub¬ ject to (c).

Let t6 (0, and consider the function g^ defined by

3i (*.<) = ff f(3e,T)dkds + h{3e)+tq(x)

Boggio [1903, t]. This theorem was established independently by Sternberg and Eubanks [1957, Itl-

2 Sternberg and Eubanks [1957,


for {x, t)e By. (0, Iq), where h and q are as of yet unspecified. Then meets the second of (b). We now show that h and q can be chosen such that □^3^=0. Clearly,

0 = // olAf(x, A) dkds—f(x, t) -{■c\Ah{x) A-tc\Aq[x), X X

and by (c)

clAf==f. Thus

□19i (®. i) = —f{3B,r)+ rf{x, r) —tf{x,T)+ cfA h{x)+t 4 A q (x).

Since / is of class on i? x (0, io)> f{‘> 4 ^.nd /(•, t) are of class on B; hence, in view of (6.5), there exist functions h and q of class C® on B such that

clAh (x) =f[x, r) — rf (x, r),

clAq{x)=f(x,r),

which yields Oj = 0. □

An interesting relation between the Cauchy-Kovalevski-Somighana solution and the Green-Lame solution was estabhshed by Sternberg.^ Assume that the body forces vanish. Let g be the stress function in (Sj), (Sj). Then by (Sj) and Boggio’s theorem, there exist functions g^ and g^ such that

9 = 9i + 92,

By the second of (b). □iSri = o, □232 = 0.

□132=(ci-ci)^32:

thus, using (a) and (b), we can write (Sj) as

M = (Ci - cl) [Ag^-V div (g^ -f 32)], and, since

curl curl 32 = F div ^3 — ^ 32.

(d) can be written in the form

(a)

(b)

(c)

(d)

u = (4 — 4) [Fdiv STi + curl curl g^].

If we define functions 95 and xp by

<P = {cl — Cl) div STi, y, = [4- 4) curl g^, (e)

then (e) takes the form

while (b) yields M = F95-j-curl div VI = 0,

□195 = 0, □2V’ = 0.

(f)

(g)

Eqs. (f) and (g) are identical to (Lj) and (L2). Thus we have Sternberg’s result: The Cauchy-Kovalevski-Somigliana solution reduces to the Green-Lame solution provided the jield g is subjected to the transformation (a), (b), and (e). In view of the completeness of the Cauchy-Kovalevski-Somigliana solution, this reduction theorem furnishes an alternative proof of the completeness of the Green-Lame solution.

5 [I960, 13].

Sect. 68. Basic singular solutions. 239

68. Basic singular solutions. In this section we will discuss the basic singular solutions of elastodynamics, assuming throughout that the body is homogeneous and isotropic with Cj >0, Cj > 0.

Throughout this section y is a fixed point of S,

r=x—y, r = jr|,

and 2^ = 2^ (y) is the open ball with center at y and radius t].

Recall^ that PC is the set of aU smooth vector functions on (—00, 00) that vanish on (—00, 0). For each lePClet Uy{f\ be the vector field on {^~{y})x ( — 00, 00) defined by

l/c,

L 1/c,

Further, let Ey{l} and Sy{l} be defined through the strain-displacement and stress-strain relations, so that

Sy{l}{x,t)=---^-^- -(r-Za) l-r(8)/3-/3(8)r ,

where

/i =/i{0 (*- i) with

l/ca

fi{l}{x, t)=~6cl j U{t-Kr) dX + i2\l{t -rjc^) - -^l(t -rjci)

l/Ci ^

+ 3Lî(t-rlc,)-^î(t-rlc,)\,

I/C2

/al*} (sc, t) = —6c| U(t — Xr) dX-\-2l[t — rjc^ -(-^1 — 4j f(< — r/c^)

l/c. ^

/3{I}(a5, t) = -6c\ JXl{t-Xr) dX+’^lit-rjc^)--^Û(t-rlc^)

l/c, ^

We write Py^} = \nyif),Ey{l}.Syi^}-\

and call tiy{l} the Stokes process^ corresponding to a concentrated load I at y. Clearly, the mapping lt-yjiiy{l} is linear on .5?.

' See p. 26. 2 This solution is due to Stokes [i851, t]. It was deduced by Love [1904, 3], [I927, 3],

§ 212 through a limit process based on a sequence of time-dependent body force fields that approaches a concentrated load (cf. (51,1)). The form of Stokes’ solution presented here is taken from Wheeler and Sternberg [1968, 16). A generalization of Stokes' solution, valid for inhomogeneous media, is contained in the work of Mikhlin [1947, 5]. See also Babich

[1961, i], Gutzwiller [1962, 5], Murtazaev [1962, 72], Nowak [1969, 5].


(1) Let le^fand define on K—{y})x(—oo, oo) by

Sy {1} (*. i) = 4 - r 0/2 -/3 (8) r - (r • /a) l],

where /,=/,{0(*,<) is defined above. Then

Sy{k}*l=S*{lY*k.

Proof. It follows from the definition of/j and (10,2) that

m#k=i#fi{ky. hence

fS}*k = l*fi{k}.

Thus, in view of the definition of Sy{k},

4nr^ Sy{k}*l = (r •/^{fe}) *(r-l) -(r -Mk})*!-(f3{k}*l) r-(r ■ 1)*f3{k}

= (-?-) • Uii^# l)]r-{l#h{^) r-{Uk}*l) r-iUk}# t)r

= • ik#fi{l})]r-{fS}# k) r-{k*f,{l}) r-{k#f3{l}) r

= 4nr^ S^{lY*k. □ Let

on dB x{ — oo, 00), where S*{f} is the tensor field defined in (1). We call s*{l} the adjoint traction field on 8B corresponding to itiy{l}.

(2) Properties of the Stokes process.^ Let Then the Stokes process Py{l} has the following properties:

(i) If I is of class C^, then py{l} is an elastic process on S — {y} mth a quiescent past, and Py{l) corresponds to zero body forces.

(ii) Uy{l} {x,')=0{rY and Sy{l} {x, •) =0(r~^) asr-Ô and also as r-ôo, uniformly on every interval of the form ( —00, <].

(iii) lim / Sy{l}da=l, Um / rxs„{f}da=0,

where ^y{n=Sy{l}n

on dZ^x(—oo, 00) with n the inward unit normal. Moreover, the above limits are attained uniformly on every interval of the form (—00, f].

(iv) For every k^ST,

l*Uy{k} = k*Uy{l}, l*Sy{k} = k*s^{l},

where Sy{k} is the traction field on 8B corresponding to Py{1t\, and s* {1} is the adjoint traction field on 8B corresponding to py{l}.

Proof. Parts (i) and (ii) follow from the assumed properties of I and the for¬ mulae for and Sy{l}.

1 This theorem, with the exception of the second of (iv), is due to Wheeler and Stern¬

berg [1968, 16]. See also Love [1927, 3], who cisserts (iii).

Sect. 68. B2isic singular solutions. 241

Consider now part (iii). A simple computation based on the formula for Sy{l} leads to the result

f Sy{l} (x.r) da^= j 1(t—??/ci)+2i(T—jy/cj)

dS„

+ i (t + — i (t - jy/ca)]. Cl Ca

which imphes the first of (iii), since liS?. The second of (iii) is an immediate consequence of the order of magnitude estimate for Sy{l} in (ii).

The first of property (iv) is easily established with the aid of (10.2). To prove the second, note that hy (1)

I*Sy{k} = 1*{Sy{k} n) =(Sy{k}*l) n

= (S*{l}^*k)-n = k*(S*{l}n)=k*s*{l}. a

Let 16.5? be of class and let Uy{l} be defined on (S’ — {y}) x(—oo, oo) by

(*. 0 = (*. 0 •

= -4^^^ 3(-^^^"-^l-e,.0r-r0e<) J U(t-Xr)dk

+ (1 - e,- (8) r - r 0 e<j 1 (< - rjc^) - 1 (t - r/c^)j

^2 ^2

We write

4{0 = [<W.-E’,{0, «•,{*}], where E'y{l} and S'y{l} are defined through the strain-displacement and stress- strain relations, and call the doublet process (corresponding to y, I,

and e^).

(3) Properties of the doublet processes.^ Let l££S be of class C^. Then the doublet process py{l} has the following properties:

(i) If I is of class C®, then Py{l} is an elastic process on S — {y) with a quies¬ cent past, and Py{t} corresponds to zero body forces.

(ii) (ae, •) =0(r"®) and (ae, •) =0(r"®) as r->0 and also as r->oo, uniformly on every interval of the form (—oo, <].

(iii) lim / sj,{f}rfa=0, lim / ry.^,{l] da =—0^x1, as, as,

where s*{l} = S'{l}n

on with n the inward unit normal.

(iv) If keSS is of class C®, then

k*Uy{l} = l*Uy{k} .

1 Wheeler and Sternberg [1968, i6]. Theorem 3.2.


242 M. E. Gurtin : The Linear Theory of Elasticity. Sect.[69.

(v) If py{l} is the Stokes process corresponding to the concentrated load I at y, then

69. Love’s integral identity. In this section we will establish Love’s theorem; this theorem is a direct analog of Somighana’s theorem (52.4) in elastostatics. We continue to assume that B is homogeneous and isotropic with c^ > 0, Cg > 0.

(1) Lemma.^ Let p — [m, JS, S] he an elastic process with a quiescent past, and let py{t} = Stokes process corresponding to a con¬ centrated load at y^B. Then

hm / s*Uy{l} da=0, 0i,’„ (»)

lim / u*Sy{t} da=l*u(y,’),

where s=Sn, Sy{l}=Sy{l}n

on dZ^{y) x(—oo, oo) with n the inward unit normal.

Proof. Since both states have quiescent pasts, the above relations hold trivially on (—oo, 0]. Thus hold t>0 fixed for the remainder of the argument. Let jS be such that Z0{y)<iB, choose »;6(0,/S], and write S^=Z,^{y). Further, let

V {x, r)=u (x, r)—u {y, r) (a)

for (x, r)€B X [0, oo). Then t

\I{ri)—l*u{y,t)\^ f f Sy{l} (x, t — r) • v{x, r) dr da^

" (b) + / u(y,t — r) • / Sy{t} (x, r) da^ — l(r) dr ,

0

where I[rj)= f u*Sy{l}{x, t) da^..

The second term in the right-hand member of (b) tends to zero with r\, since the limit may be taken under the time-integral^ and because of (iii) in (68.2). Conse¬ quently,

\I{ri)—l*u{y,t)\Ânri^tM^{ri)M^{ri)-\-o{\) as ?y->0, (c) where

Mg (?y) = sup {I»{x, r) |: (a?, t) e X [0, t]},

=sup {\Sy{l} [x, t)| : {x, T)e0i7,x [0, <]},

for >ye(0, jd]. From (a) and the continuity of u on S’x[0, oo) follows

M^{rj)=o{\) as ?y->0.

On the other hand, (ii) of (68.2) 5delds

M^(rj) =0{rj~^) as ?y->0.

The inequality (c), when combined with the above estimates, imphes the second result in (1). A strictly analogous argument 5delds the first result in (1). □

' Wheeler and Sternberg [1968, 16], Lemma 3.1. 2 See Mikusinski [1959, 10], p. 143-

Sect. 70. The acoustic tensor. 243

(2) Love’s integral identity.^ Let p = [u, E, S] he an elastic process on B corresponding to the external force system [5, s], let p have a quiescent past, and suppose that s(x, for every xedB, and that b{x, for every xeB. Then

“(y. •) = / [“»{*}—*»{“}] da-\- j u„{h} dv dB B

for yeB, where, for any le^, Uy{l} is the displacement field and the adjoint traction field corresponding to the Stokes process Py{l}.

Proof.^ By h57pothesis, u is an admissible motion and

m(-,O)=m(*,O)=0 (a)

on B. Thus M(a?, •)£ JS? for every x^B. Therefore, since s(a?, for every xedB, the above expression holds trivially on (—oo, 0].

We now establish its validity on (0, oo). Choose a>0 with 2^{y)CB, and set

Bn = B-I,j(y) (b)

for ?y6(0,a]. Let ieJSfbe of class C^. By (a) and (i) of (68.2), p2j\Apy{l] both correspond to null initial data, while py{l} corresponds to vanishing body forces. Thus we are entitled to apply (d) of Graffi’s reciprocal theorem (61.1) to p and Py{(} on B,! with the result that

/ s*Uy{l'}da-\- f b*Uy{l} dv = J Sy{l}*uda (c) SB„ B„ dB„

on (0, oo). If we let ->0 in (c) and use (b) and (l),vre arrive at the result

/ s*Uy{l} da-{- f b*Uy{l} dv = f Sy{t}*uda-{-l*u{y,-). (d) dB B dB

By (iv) of (68.2) and the properties of the convolution (10.1), (d) can be written in the form

[ /[“»{*}ia + / Uy{b} dv—u{y,’)\*l = 0 (e) B J

on [0, oo). Since ieJSfis an arbitrary class function, (e) and (v) of (10.1) yield the desired result. □

V. Wave propagation.

70. The acoustic tensor. As we will see in the next few sections, the notion of an acoustic tensor is central to the study of wave propagation. In this section we define this tensor and estabhsh some of its properties.

Fix x^B and let C = Ca, be the elasticity tensor and q—q{x)>0 the density at X. Let m be a unit vector. Since C is a linear transformation,

g'l C [a (g) m] m

1 Love [1904, 3]. A similar result was deduced previously by Volterra [1894, 4], but Volterra’s result is confined to two-dimensional elastostatics. See also Somigliana [1906, 6], De Hoop [1958, 4], Wheeler and Sternberg [1968, 46]. None of the above authors utilizes the adjoint traction field, which considerably simplifies Love’s identity. Related integral identities are contained in the work of Arzhanyh [1950, 2], [1951, 2, 3, 4], [1954, 4, 2], [1955, 2, 3], Knopoff [1956, 6], Doyle [1966, 8],

A similar integral identity for the stress field, involving linear combinations of doublet processes, was established by Wheeler and Sternberg [1968, 46], Theorem 3.4.

^ The proof we give here is due to Wheeler and Sternberg [1968, 46], pp. 71, 72. Love [1904, 3] based his proof on Betti’s reciprocal theorem (30.1), treating the inertia term as a body force.

16*


is a linear, vector-valued function of the vector a and hence is represented by the operation of a tensor A{m) acting on a:

A (m) a = p C [a (g) m] m for every vector a,

We call A{m) the acoustic tensor for the direction m. In components

(1) Properties of the acoustic tensor.

(i) A{p%) is symmetric for every m if and only if C is symmetric.

(ii) A{m) is positive definite for every m if and only if C is strongly elliptic.

(iii) A[m) is positive definite for every m whenever C is positive definite.

(iv) Given any Q in the symmetry group

QA[m) Q^=A(Qm) for every m.

Proof. Let C=p'iC. Clearly,

5 • A{m,) a = b’ C[a(g)m] m = (5(g)m) • C[a(g)m]. (a)

Properties (i) and (ii) are direct consequences of (a) and (ii) of (20.1). Further, since C positive definite implies C strongly elliptic, (ii) imphes (iii). Finally, since

(Qa) ® (Qm) ==Q(a®m) ,

it follows from (f) on p. 71 that

Q'Â{Qm) Qa==Q^C{Qfi®Qm] Qm = Q'^C[Q[a®m) Q'^] Qm

= C[a(g)m] m—A[m) a,

which imphes (iv). □ /

(2) Properties of the afcoustie tensor when the material is isotropic.^ Assume that the material at p is isotropic with Lame moduli p and A. Then

A{nf) =cf m(g)m-(-c|(l—m(g)m),

where c^ and c^ are the waie speeds:

.2 _ 2/* 4- A a_p Ox- - g . Ca-.

Further, cf and c| are the principal values of A{m); the line spanned by m is the characteristic space for c\, the plane perpendicular to m the characteristic space for c\.

Proof. By (i) of (20.1) and (22.2),

C[a(g)m] =C[s5mia(g)m] =p(a®m-\-m®a) -f A(a-m) 1; thus

C[a®m] m=pa-\-{^+p) {a-m) m = [p m®m] a

= |jm(1 —m®m) +{A + 2p)m®m] a,

which yields the above formula for A{m). The remainder of the proof follows from the special theorem (3.2) . □

* The form 6f the acoustic tensor for various anisotropic materials was given by Sakadi

[1941,2] and Fedorov [1965,5], §19. Fedorov [1963,5] has established a method of computing the isotropic acoustic tensor that is closest (in a precise sense) to a given anisotropic acoustic tensor.

Sect. 71. Progressive waves. 245

When we wish to make exphcit the dependence of the acoustic tensor A{m) on xi:B, we write A{x, m), i.e.

A(x, m) a = Q-^{x) C^[a®rn\m.

In the following proposition the region B is completely arbitrary.

(3) Let C_and, he bounded on B with C symmetric and strongly elliptic. For each see B and unit vector m, let A {x, m) he the largest characteristic value of the acoustic tensor A{x, m). Then the number %>0 defined by

=sup {A{x, m): xeB, |m| =1} is finite, and

|G-c,[G]|ê(®)4lGp

for every xeE whenever G is of the form G==a®v. We call c^j the maximum speed of propagation corresponding to C and q.

Proof. By hypothesis and (i) and (ii) of (1), A(x, m) is always symmetric and positive definite. Thus A (a?, m) >0 and

V ■ A(x, m) vÂ{x, m) (a)

for every vector v. Thus if the supremuni of A (x, m) is finite, wiQ be well defined. Let e{x, m) with \e{x, m) \ =1 be a characteristic vector corresponding to A (a?, m). Then

A (x, m)~e {x, m) • A(x, m) e {x, m), (b) and since

V ■ A(x, m) » =g"^(a?) {v®m) • C^[v®m'\

|C®1 l»(8)ml®=e-i(a?) |Ca,| |»p

for every vector v, (b) imphes that

A(a?, m)ê-i(a?) |Ca,l. (d)

Since q~^ and C are bounded on B, g'^lCj is bounded, and therefore % is finite. The final inequality in (3) follows from (a), (c), and the definition of %. □

^ 71. Progressive waves. In this section we study plane progressive wave solutions ^ to the displacement equation of motion. These simple solutions are important in that they yield valuable information concerning the propagation characteristics of elastic materials.

We assume that the body is homogeneous and the density strictly positive.

By a progressive wave we mean a function m on ^ X ( — oo, oo) of the form

u(x, t) =aq)(p’m — ct), where:

(i) 9) is a real-valued function of class on (—00, 00) with

ds^~

(a) 2

(ii) a and m are unit vectors called, respectively, the direction of motion and the direction of propagation ;

' For studies of steady waves, of which plane progressive waves are a special case, see SAenz [1953, 77] and Stroh [1962, 14].

2 Recall thatp(x)=x — 0.


(iii) c is a scalar called the velocity of propagation.

We say that the wave is longitudinal if a and m are linearly dependent; transverse if a and m are perpendicular; elastic if u satisfies the displacement equation of motion

Aiv C[Vu]=Qii (b)

on ^x(—oo, oo). Thus an elastic progressive wave is an elastic motion of the form (a) that corresponds to zero body forces.

Let a denote any given constant, and let n^ denote the plane defined by

7it={x\ (* —0) • m—ct =«}.

Then at any given time t the displacement field u is constant on Tif This plane is perpendicular to m and, as a function of t, is moving with velocity c in the direction m. For this reason u is sometimes referred to as a plane progressive wave.

By (a)

where

Thus

Pm = (p’ a®vn,

<p'{x, t) =

<p''{x, t) =

d(p(s) I

ds

d^(p(s) I

ds^ \ss=p'm—ct

div u = (p' a -m,

cm\u = (p' rnxa,

(C)

and, since cp' ^0 (because cp" ^0), it follows that a progressive wave is:

By (c)

longitudinal ■?=> curl m = 0,

transverse <=> div m = 0.

C[Pm] = 99'C[a(g)m],

and, as C is independent of x,

div C[Pm] =99"C[a(g)m] m—Qcp"A{m)a, (d)

where A[m) is the acoustic tensor discussed in the previous section:

Next,

and (d) and (e) yield the

A{m) a = g“^C[a(g)m] m.

Qii = Qc^<p" a, (e)

(1) Propagation condition for progressive waves.^ A necessary and sufficient condition that the progressive wave u defined in (a) be elastic is that the Fresnel-Hadamard propagation condition

A[m) a—câ hold.

1 According to Truesdell and Toupin [i960, 17], § 300, the ideas behind this theorem can be traced back to the work of Fresnel. See also Cauchy [1840, 7], pp. 42.50; Green

[1841, 7]; Musgrave [1954, 75]; Miller and Musgrave [1956, S]; Synge [1956, 73], [1957, 76']; Farnell [1961, 7]\ Musgrave [1961, 75].

Sect. 71. Progressive wave. 247

Thus for a progressive wave to propagate in a direction m its amplitude must be a principal vector of the acoustic tensor A{m) and the square of the velocity of propagation must be the associated characteristic value. If the elasticity tensor C is S5mimetric, then, by (i) of (70.1), A{m) is symmetric, and hence has at least three orthogonal principal directions cq, Og, Og and three associated principal values cf, c|, c| for every m. Further, if C is strongly eUiptic, then A[m) will be positive definite and Cj, Cg, and Cg will be real. Thus, if C is symmetric and, strongly elliptic, there exist, for every direction m, three orthogonal directions of motion and three associated velocities of propagation q, Cg, Cg for progressive waves.

Theorem ah when combined with (70.2), impUes the

(2) Propagation condition for isotropic bodies.^For an isotropic material the progressive wave u defined in (a) will be elastic if and only if either

(i) c^—c\ = —and the wave is longitudinal; or

(ii) c^=c%=filQ and the wave is transverse.

Thus for an isotropic material there are but two types of progressive waves: longitudinal and transverse. One can ask whether or not longitudinal and/or transverse waves exist for anisotropic materials. This question is answered by the

(3) Fedorov-Stippes theorem,.^ There exist longitudinal and transverse elastic progressive waves provided the elasticity tensor is symmetric and strongly elliptic.

Proof. By (1) there exists a longitudinal progressive wave in the direction m provided

A[m)m = lm (a)

for some scalar A>0. Since C is strongly eUiptic, A{m) is positive definite; thus, given any m,

m- A[m)'m>0, A{m) (b)

Consider the function I defined on the srrrface of the unit sphere by

l[m) = A(m) m

\A{m) m| (C)

i By the first of (b), I is continuous, maps the surface of the unit sphere into itself, I and maps no point into its antipode. By a fixed point theorem,® any such map has a fixed point. Thus there exists a unit vector such that l(mf) ='nii, or equivalently, by (c),

iMj = Amj, A = |.4(mi) mi|.

Thus there exists a longitudinal progressive wave in the direction m^. By the spectral theorem (3.2), there exist unit vectors mg and mg such that m^, mg, mg are orthonormal and mg and mg are principal directions for A{mf). Since Alm^

1 Cauchy [1840, 1], pp. 137-142. See also Lam^; [1852, 2], § 59; Weyrauch [1884, 2],

§§ 98-99: Butty [1946, i], §§ 60-62; Pailloux [1956, P], p. 35-

2 This theorem was arrived at independently by Fedorov [1964, 7] and Stirpes [1965, 18'\. See also Truesdell [1966, 27], [1968, IS] and Fedorov [1965, 5], § 18. Kolodner [1966, 14] has extended the Fedorov-Stippes theorem by showing that there exist three distinct directions

along which longitudinal waves propagate. Sadaki [1941, 2] has calculated the actual direc¬

tions along which longitudinal and transverse waves propagate for various crystal classes.

See alsoBRUGGER [1965, 3]. Waterman [1959, IS] has established results for waves that are

nearly longitudinal or nearly transverse.

® See, e.g., Bourgin [1963, 2], Theorem 8.7 on p. 132.


is positive definite, the corresponding characteristic values c| and c| are positive and hence are the squares of positive numbers. Thus there exist progressive waves in the direction iMj whose directions of motion are equal to and m^. But and are orthogonal to m^. Thus these progressive waves are trans¬ verse. □

(4)^ Assume that the elasticity tensor is symmetric and strongly elliptic. Let e he an axis of symmetry for the material.'^ Then there exist longitudinal and transverse elastic progressive waves whose direction of propagation is e.

Proof. Clearly, we have only to show that e is a characteristic vector for A{e). Since e is an axis of symmetry,

Qe=e (a)

for some Q^l in the symmetry group. Thus, by (iv) of (70.1),

QA(e)=A(e)Q,

and we conclude from the commutation theorem (3.3) that Q leaves invariant each characteristic space of .4(e). By (a) Q must be a proper rotation about e through an angle If 4= then the only spaces left invariant by Q are the one-dimensional space spanned by e, the plane perpendicular to and the entire space iC. If = n, then Q leaves invariant, in addition, each one-dimension¬ al space spanned by a vector perpendicular to e. In either event, it is clear that at least one characteristic space of A{e) must contain e. □

It follows from the above proof that if elastic transverse waves can propagate in the direction of e with any amphtude perpendicular to e, and all' of these waves propagate with the same velocity.

72. Propagating surfaces. Surfaces of discontinuity. In this section we define the notion of a propagating singular surface. Roughly speaking, this is a surface, moving with time, across which some kinematic quantity suffers a jump dis¬ continuity.

Recall that ^(4)=^X(-oo, oo), 'rW='rx(-oo, oo).

By a smoothly propagating surface we mean a smootl^toee-diniensional mîfold in with the following property: given any {x, there exists a nSmiar (ft, to iF at [x, t) with ft 4= 0, >i: =|= 0. This assumption imphes the existence of a four-vector field

m = (m, — c)

on iP' that is normal to 'W and has

|m|=1, 00.

We call m the direction of propagation and c the speed of propagation.

For the remainder of this section let be a smoothly propagating surface in Bx(0, Q.

Since is a smooth manifold, and since c is strictly positive, W has the follow¬ ing property: given any point (Xq, there exists a neighborhood i7 of a?#, a neighborhood T of and a smooth scalar field ip on 2 such that

t=ip[x) for all [x,t)iN, N=2xT.

' Fedorov [1965, 5], pp. 92-94.

2 See p. 73.

Sect. 72. Propagating surfaces. Surfaces of discontinuity. 249

In terais of f.

m (x, t) = \Vip(x)\ ’

c {x, t) = \Vy>(x)\

for aU (x,t)iNr,ir.

For t fixed, ^^={xeB: {x,t)iir}

is the smooth surface in B occupied by at time it is not difficult to show thatthevectorm(a?, <)isnormaltoy’, at®. A We

Let r=iy(t): teT} be a smooth curve on of the form y(<) t)- We say that T is a ray if (•) is a solution of the differential equation2

y{t) =c{y[t), t) m{y[t), t).

Thus 1/ (•) is the path traversed by a particle on whose trajectory at any time t is perpendicular to and whose speed is c. Assume, without loss in gener^ty that the ray F passes through the neighborhood N m which the manifold iT is

described by y). Let g denote arc length along F.

f \y{r)\dr. 0

Since |y| never vanishes, a is an invertible function of f. we write t for the inverse

function, i.e. _ ^ t = i(G),

and we define y(a)=y{t{o)),

c(a) -=c{y(a),t(<y)),

Vc{a) = V^c{x.y){x))\^^^

rh{a)=rh{^{a),t{o))-

- '^(lFv(*)|) lFvi(®)|/|»=y(a) ‘

it follows that

4/1 dy'

jg-=m(a) = c-(a) (j^)

it follows that

^^-j = [FFy>(y(G))] ^ = =

Thus we are led to the following differential equation for the ray;

c d<fi c2 Uff / V ± c da^

1 Idc\ dy '42 [daj\da

It foUows from this equation that if the velocity c is constant on iT, then e^^ry ray

function on Sx (0, h). We caU iT a singuMr surface of order

zero with respect to / if / is continuous on S X (0, <«) /_suffers_aj!FFP

discontinuit3ûcross.^. The jump in / across iT is then the function [/] oniT

“"i In the literature it is customary to define a smoothly propagating surface to be a (smooth)

one-parameter famUy o<t<t„,oi smooth surfaces m B. In this instance

•)r = {(aB, i):xesrt, o</</„}.

2 Since y • m = (cm, 1) • (m, -c) =0, this definition is consistent in the sense that the

tangent to B is also tangent to 1^.


defined byi [/] {x, t) = {/ (x,t-\-h)—f{x,t — h)}.

Thus the jump in / is the difference between the values of / just behind and immediately in front of the wave. Since / is continuous on each side of we can evaluate [/] by taking limits along the normal m:

[/] (?) = lim {/ (? - A m) - / (? + A m)} h-^0+

for ?= (x, t)elV.

Let n'^\ be a fixed integer. We call a singular surface of order n with respect to / if:^

(i) / is class on S x (0, t^) and class C" on S x (0, t^—iV;

(ii) the derivatives of / of order n suffer jump discontinuities across if.

The next theorem shows that for a singular srrrface of order 1, the jumps in the space and time derivatives of the associated function are related.

^^(1) Maxwell’s compatibility theorem.^ Let (p be a scalar field, v a vector field, and T a tensor field on Bx (0, t^, and suppose that if is a singular surface of order 1 with respect to <p, v, and T. Then

c[V(p\ = — [cp]m, (1)

c[Vv\ = — [»] ®m. (2)

c [div v\= — [»] • m. (3)

c [curl v\={vi\xm. (4)

c [div T] = — [T] m. (5)

Proof. Choose a point '^îf and a (non-zero) four-vector a such that

a - m{^) =0.

Since if has dimension 3, o is tangent to 'f at ?. Further, since if is smooth, there exists a smooth curve = {z (a): ^ a ^ ag} on if through ? whose tangent at ? is o; i.e. for some ixge(ixi, ag)

where *K)=?. *'(ao)=‘*.

d

d

(a)

For ^îf and any function f on Bx (0, tg), let

/i(?) = lim/(?-Am),

4(?) = lm/(?-f/im), A—>0+

so that

_ _ [/l(?)=/i(?)-4(?). (b)

1 This notation was introduced by Christoffel [1877, 1], § 6.

2 The notion of a singular surface of order « ^0 is due to Duhem [1900, 7] and Hada-

MARD [1901, 5], § 1, [1903, 3], § 75. See also Truesdell and Toupin [i960, 77], § I87.

® Maxwell [1873, 7], § 78, [I88I, 7], § 78a. Cf. also Weingarten [1901, 7], Hadamard

[1903, 3], § 73, Truesdell and Toupin [i960, 77], § 175, Hill [196I, 74].

Sect. 72. Propagating surfaces. Surfaces of discontinuity. 251

Then, by the smoothness properties of (p,

9>i(z(«)) = -^^-Im 9p(z(a)-hni)

= lim^^ (p(z{a.)—hm) da. ^ ^ ' '

= 17(4) (p{z{a.)-hm)-z' (a).

Thus (a) implies^

(*(“)) l«=«. = ^ V>)i (?) • (c)

A similar result holds also for q>2', if we subtract this result from (c) and use (b), we arrive at the relation

i[9>](*(«))la=a.=«-[I7(4)9>](?). (d)

On the other hand, since (p\s continuous across #7

[99] =0; hence (d) imphes

«• [1^(4) 9>1 (?)=0

for every and every a tangent to at Thus [17(4)99] (%) must be parallel to m{%):

[I7(4)9>](?)=A(?)m(?); (e) equivalently,

[V(p\=km, [(p\=—Xc, which imphes (1).

If we apply (1) to the scalar field v • a, where a is an arbitrarily chosen vector, we find that

c[V[v ■ a)\ = — [v ■ a\ m. Thus

c{Vv^\ a = —(m(g) [i?]) a, which 5delds (2).

If we take the trace of (2), we arrive at (3); if we take the skew part, we arrive

at (4).

Finally, (5) foUows from (3) with v =T'â, where a is an arbitrary vector. □

We now sketch an alternative proof of the relation (1) in (1). By a four¬ dimensional counterpart of Stokes’ theorem,

/ skw {(F(4)99) ®n}da = 0 se

for every closed T^^-regular^ h5q)ersurface in Bx{0,to)- The same argument used to derive (c) in the proof of (73.1) here yields

skw {[F(4) 99] (g) m} = 0.

Operating with this equation on m, we are led to the relation (e) in the proof of (1).

The next theorem shows that for a vector field v with a discontinuity of order 1, the jumpsîn Fv and v are completely determined by the jumps in the divergence and curl of^.^

1 Hadamard [1903, 3], § 72. Cf. Lichtenstein [1929, 2], Chap. 1, § 9, and Truesdell

and Toupin [i960, 77], § 174. 2 This term is defined on p. 252.


(2) Weingarten’s theorem.^ Let iL" he a singular surface of order 1 with respect to a vector field v on By. (0, tg), and let

a = [v\. Then

a-m = 0<^ [div®] =0, (1)

axm = 0 [curl®] =0, (2)

a =—cm [div ®] + cm X [curl ®]. (3)

Proof. The results (t) and (2) foUow from (3) and (4) of (1). To establish (3) we note that the identity

a = (a • m) m + (a®m) m — {mâ) m

in conjunction with (2) and (3) of (1) imply

a= —c[div®] m —c[|7®] m + c[F®^] m.

This relation yields (3), since

[Vv~ Vv^l m = [curl ®] xm. □

We wiU occasionally deal with integrals of the form

/ Mn da, 9>

where .5^ is a closed hyrpersurface with outward unit normal field n. If is a singular surface of order zero with respect to M, then M will suffer a jump dis¬ continuity across To insure that the above integral exists and that the value of M in the integrand at a point of discontinuity equals its Hmit from the region interior to y, we give the following definitions.

Let .5^ be a closed regular hyqjersurface in J5x(0, 4). i-e. îs the boundary of a regular region Dy> in J5x(0, <o)- A point \sii^-regular if either or

and there exists an open ball X in with center at \ such that ('#' (see Fig. 16). We say that -regular if the set of points of that are not

-regular has zero area measure.

Fig. 16. and are ItT-regular; ^ is not 1#^-regular.

^ [1901, 7]. Cf. also Truesdell and Toupin [i960, 77], § 175.

Sect. 73. Shock waves. Acceleration waves. Mild discontinuities. 253

Let -l^be a singiilar surface of order zero with respect to a four-tensor field M on J5 X (0, ô). Then the hmit

M(?) = h;m/VI(g

exists for every regular ^€.5^. Further, the integral

/ Mn da

exists whenever .5^ is a closed iT'-regular hypersurface in J5 x (0, 4). When such is the case we define

r J Mnda= f Mnda.

73. Shock waves. Acceleration waves. Mild discontinuities. Throughout this section is a smoothly propagating surface in J5 x (0, with direction of propagation m and speed of propagation c. Fiurther, we assume that the elasticity field C and the density field q are continuous on B, that q is strictly positive, and that the body force field vanishes.

Let nî be a fixed integer. By a wave of order n we mean an ordered array [u, S] with properties (i)-(iii) below.

(i) M is a vector field and S a symmetric tensor field on Bx (0, 4).

(ii) -W \sz. singular surface of order n with respect to u and order n — i with respect to S. ' '

(iii) u and S satisfy balance of hnear momentum in the following form: given any closed ■#"-regular hypersurface ^ in J5x (O, Q.

f Mn da =0, se

where M is the stress momentum field^ cpiresponding tQ.iS and m.

Requirement (iii) is motivated by (19.6). For (iii) may be replaced by the assumption that

divS = pM on Bx{0,t^)—ir. (a)

For « = 1 the discussion given at the end of the last section insures the exist¬ ence of the integral in (iii); in addition, the value of M in the integrand is equal to its limit from the region interior to When « =1, (iii) imphes (a), but the converse is not true. However, in this instance (iii) is equivalent to the following requirement:

(iii') for every one-parameter family (4<<<4) of parts of B,

J {Sn) [x. t) da^ = J (qu) (x, t) dv^

dP, n

whenever the integral on the left and the derivative on the right exist. Here the stress Son dPf is to be interpreted as its limit from the interior of Pf.

Given a wave of order n, we call

_ a = [M] ^ See p. 67.


the amplitude, and we assume, for convenience, that a never vanishes. A wave of order t "is“ômetimes referred to as a stress wave or shock wave] a wave of order 2 is often called an acceleration woweTFor aAhdck wave the amplitude equals the jump in the velocity, while for an acceleration wave a is equal to the jump in the acceleration. We a wa-ve longitudinal or transverse according as a and m are parallel or perpendiculari For a wave of order is a singular

surface of order 1 with respect to ^ m ; thus Weingarten’s theorem (72.2) implies

that a is completely determined by the jumps in the divergence and curl of * m s:

o = — cm [div M ] + cm x [curl u ].

In addition, we conclude from (72.2) that the wave is:

(»-i) longitudinals [curl u ]=0,

{»-!) transverse s [div M ] = 0.

For a wave of order n the time-derivative of S of order « — 1 is allowed to jump across the wave. The next result gives a relation between this jump and the amplitude a.

(1) Balance of momentum at the wave.^ For a wave of order « ^ 1,

(»-i)

[ S ] m= —Qca.

Proof. Let « = 1. Let E be an open ball in J5 x(0, t^) with center on For S sufficiently small, divides S into two regular regions and X' with 'W-regular boundaries. Let be that portionofilinto which m points (seeFig. 17).

1 Although we have assumed that the body force field b = 0, the above result remains

valid if we assume instead that b is of class C”—^ on B x {0, fo). For a shock wave (« = 1) the

relation in (1) has a long history. Hydrodynamical special cases for situations in which

S= —pi were obtained by Stokes [i848, 2], Eq. (3), Riemann [i860, i], § 5, Rankine

[1870,2], §§3-5, Christoffel [1877. f], §1. Hugoniot [1887, i], §140, and Jouguet

[1901, 6]; for general S but linearized acceleration, by Christoffel [1877, 2], § 4; for finite

elastic deformation by ZemplIn [1905, 3], p. 448, [1905, 4], § 3, whose derivation is based

on Hamilton's principle. The general result (for « = 1) is due to Kotchine [1926, 2], Eq. (18).

Cf. also Truesdell and Toupin [I960, 17], § 205, Jeffrey [1964, 11].

Sect. 73. Shock waves. Acceleration waves. Mild discontinuities. 255

By (iii)

/ Mnda= f Mnda= f Mnda=0, dS 02+ 02-

where M is the stress momentum field; thus

/ Mnda-\- f Mnda— f Mn da=0. (a) 02+ 02- 02

The boundary of 2 can be decomposed into a portion contained in 82* and a portion contained in 82-] the remainder of 82* and the remainder of 82- are equal, as sets, to 2r\'W-^ as surfaces,/liowever, they have opposite orientation. Thus (a) reduces to

/ {Iim M(5-)m(?)- lim M(5-)m©}rfa, = 0, (b) 20-)^' ?+-+? 5"-+-?

where 5* is a point of 2*^. In deriving (b) we have used the fact that the value of M in the integral over 82^ is equal to its limit from the interior of X*. Since (b) must hold for every sufficiently small open ball 2 with center on iT, it follows that

[M]m=0, (c)

or equivalently, using the definitions of M and m}

[S] m = —Qc[u\ = —Qca,

which is the desired relation for « = 1

Consider now a wave of order w^2. Choose (x, t) in B x{0, t^)—3^. Then there exists an open baU 2m B centered at x and an open time interval TC ((o>h) containing t such that (2xT)=0. Thus S is a time-dependent admissible stress field and u is an admissible motion on 2xT; (iii) and (19.6) therefore imply

divS=pfi on 2xT.

Since {x, t) in Bx(0, i!#) —3^ was arbitrarily chosen, the above relation holds on BX(0, i!o) — If we differentiate n—2 times with respect to t and take the jump in the resulting equation, we arrive at

(«-2> («) [div S ]=(>[«] =(>a. (d)

(n—2)

Clearly, is a singular surface of order 1 with respect to S ; thus (5) of (72.1) yields

(»-2) (»-l)

c[div S ] = -[ s ]m. (e)

Eqs. (d) and (e) imply the desired result for n^2. □

We say that a wave [m, S] of order nî is elastic if

S=C[Vu].

We now show that for an elastic wave of any order travehng in the direction the square of jhe speed must be a characteristic.Malue .Qf the acoustic, tensor

+4(mL,and the amplitude must he in the corresponding characteristic space.

1 See pp. 67, 248.

2 The proof up to this point is due to Keller [1964, J2].


(2) Propagation theorem for waves of order n.^ Elastic waves of all orders « ^ 1 obey the Fresnel-Hadamard propagation condition

A{m) a = câ.

Proof. Let [u, S] be an elastic wave of order « ^ 1. Since 'W is z. singular

surface of order 1 with respect to ® = * m we can apply (2) of ('72.1) with the result:

c[V u ] — — [w] (a)

Next, (a) and the stress-strain relation imply

C [ ‘“s"'] = C [c [I7‘”m^']] = - C [a ® m]. (b) and since

A[m) a = Q^^Clai^m] m,

(b) and (1) imply the desired result. □

More pedantically, the Fresnel-Hadamard condition reads

A{x, m) a {x, t) = c^ (x, t) a {x, t)

for every (x, t)eiF.

Combining (2) and (70.2), we arrive at the

(3) Propagation theorem for isotropic bodies.^ If the body is isotropic then a wave of order « ^ 1 is either longitudinal, in which case

c^ = c\

or transverse, in which case

c^—c\

Comparing (2) and (3)io (71.1) and (71.2), we see that the laws of propagation, of waves of discontinuity are the same as those cd progressive waves; thus the discussion following (71.1) and (71.2) also holds in the present circumstances.

I In particular, we note thaJiTîs'^junnielnc ahcTstfongl^Ûiptic, there'exisfTfor I every direction m, three associated velocities of propagation for singular surfaces.

Note that the maximum speed of propagation defined in (70.3) is the supremum over all possible velocities of propagation of elastic waves of any order.

For a homogeneous and isotropic body the velocity of propagation c is identi¬ cally constant. Therefore (3) and the remark made on p. 249 yield the following result.

1 This theorem is essentially due to Christoffel [1877, 2], who gave a proof for « = 1.

Hadamard [1903, 3], §§ 260, 267-268, extended Christoffel’s result to acceleration waves (m = 2). That the Fresnel-Hadamard condition should hold for all n is clear from the results of Ericksen [1953,70], §6, and Truesdell [196I, 22], §3, in finite elasticity theory (cf. Trues-

DELL and Noll [1965, 22], § 72). See also Hugoniot [1886, 2]; Duhem [1904, 2], Part 4,

Chap. 1, §5: Love [1927,3], §206-209; Finzi [1942, 7]; Pastori [1949, 7]; Petrashen

[1958, 72]; Buchwald [1959,7]; Truesdell and Toupin [i960,77], § 301; Nariboli [1966, 73].

2 Hugoniot [1886, 2]. Elastic waves of order zero (for which is a singular surface of order zero with respect to the displacement field) in an isotropic but inhomogeneous body are discussed by Gvozdev [1959, 5] using the theory of weak solutions. Gvozdev shows that such waves travel with a velocity equal to Cj or . Other studies concerned with wave propagation in isotropic elastic media are; Levin and Rytov [1956, 7], Thomas [1957, 77], Babich and Alekseev [1958,7], Skuridin and Gvozdev [1958,75], Karal and Keller [1959,9],

Skuridin [1959, 74].

2^ A

e

P e ■

Sect. 74. Domain of influence. Uniqueness for infinite regions. 257

(4) Suppose that the body is homogeneous and isotropic. Then, for a waw of any order n^\,J^âys are â^t lines.

74. Domûn of influence. Uniqueness for infinite regions. In this section we allow J5 to be a bounded or unbounded regular region. We assume that C is positive definite and symmetric, that q is strictly positive, and that C and are continuous and hounded on B. It then follows that the maximum speed of propagation c^^ defined in (70.3) is finite.

Given an elastic process [m, E, S] and a given time <€(0,4). D( denote the set of all cc€J5 such that:

(i) if xeB, then

u [x, 0) =t= 0 or ii {x, O) =t= 0 or b {x, t) =t= 0 for some t€ [O, <];

(ii) if xedB, then

s (x, r) •u{x, t) 4= 0 for some re [0, t].

Here [6, s] is the external force system associated with the process. Roughly

speaking, Df is the support of the initial and boundary data. Consider next the

set of all points of B that can be reached by signals propagating from D( with speeds equal to or less than the maximum speed of propagation c^:

Df={xeB\ (x) =t= 0}.

We caU D( the domain of influence^ of the data at time t. The next theorem, which is the main result of this section, shows that on [o, <] the data has no effect on points outside of

(1) Domain of influence theorem.^ Let \u, E, S] he an elastic process, and let Df he the domain of influence of its data at time t. 2 hen

M=0, E=S=0 on (5~D() X[O, <].

The proof of this theorem is based on the following lemma. In the statement and proof of this lemma e (x, G) is the strain energy density at x corresponding to the (not necessarily symmetric) tensor G :

s(x,G)=^G-C^[G]. (a)

(2) Wheeler-Sternberg lemma.^ Let [m, E, S] he an elastic state corres¬ ponding to the external force system [b, s]. Further, let r: [O, 4) be a continuous

1 Since Df is based on the maximum speed of propagation, it is, actually, an upper bound on the domain of influence. Cf. Duff [I960, 5].

2 This is a minor modification of a theorem due to Wheeler and Sternberg [1968, Iff], Lemma 2.2 and Wheeler [I970, 3], Theorem 1. These authors assume vanishing initial data and do not utilize the notion of a domain of influence. The general idea is due to Zaremba

[1915, 4], who established an analogous result for the wave equation. Zaremba’s scheme was rediscovered independently by Rubinowicz [1920, 2] and Friedrichs and Lewy [1928, i].

See also Courant and Hilbert [1962, 5], pp. 659-661, Bers, John, and Schechter

[1964, 3], Durr [i960, 5].

® Wheeler and Sternberg [1968, Iff], Lemma 2.1 for homogeneous and isotropic media; Wheeler [I97O, 3], Lemma 1, for anisotropic but homogeneous media. Both of the above results assume vanishing initial data.

Handbuch der Physik, Bd. VI a/2, 17


and piecewise smooth field on B with {x^B\ t(cc)>0} finite. Then

r{a!) x(x)

t)-u{x,t) dtda-\- JJ b(x, t) •u{x, t) dt dv

dB 0 BO

== J j- 2^- {x,r{x)) —■u^{x, 0)] — £ {x,G(x)) +£ (cc, Vu{x,r{x)) + G (x)) (b)

B

— £ {x, Vu (x, o))| dv

where G(x)=u(x,r{x))^Vr{x). (c)

Proof of (2). For any function / on J5x[0, t^, let be the function on B defined by

U{x)=f{x,x{x)). (d) Further, let

rix)

k=Su, v(x)=zf k(x,t)dt. (e) 0

Then v is continuous on B and piecewise smooth on B, and we conclude, with the aid of the equation of motion, that

div®(cc) =fc^(cc) • Ft(cc) + J ■ Vu-\-{u-u) ~b-u {x, t) dt. (f)

(f) takes the form £ =S • Fii,

divif(£c) =fci(£c) ■ |7t(cc) +£(», Vu.,{x)) —e{x, Vu{x, 0))

r(x)

+ —u^{x, 0)] — j b •u{x, t) dt.

0

A simple calculation using the symmetry of C yields

k^{x) ■ Fr(x) = Fr(x) ■ (S^(x) ujx)) =S^(x) ■G{x)=G{x) ■ C^[Vu.^{x)]

= s(x, Fu^(x) +6r(cc)) —s(x, Fti,(x)) —e(x, G(x)); thus

div ti(£c) = u^{x) —u^{x, 0)j —e(x, G(x)'j +e{x, Vu.^(x) +G(cc))

x{x)

— s{x,Vu{x,0)) — J b-u{x,t) dt.

Each of ê terms on the right hand side of (g) is a piecewise continuous function of X on B. Moreover, it is clear from (e) and the ^sumed properties of t that v has bounded support. Thus div v is integrable on B, and the divergence theorem together with (g) and the second of (e) imply (b). □

Proof of (1). Let {s, X}e{B—Df)x{0, t) be fixed, let

Q = Brsi:^^^{z), (h)

Sect. 74. Domain of influence. Uniqueness for..^finite regions.

Fig. 18.

and let r : B-^ [0, t) be defined by

/ \ [a-xeQ, r\ t(cc)={ cm' ' (i)

fO, x^Q.

Then t is continuous and piecewise smooth on B, and Vr has a weak point dis¬ continuity at cc = s and a jump discontinuity across Br^dQ. Moreover,

1I7t1 = --- on Q,

_ (j) |7t = 0 on B —Q.

Since X<t, we conclude from (h) and the definition of the domain of influence Df that

By (i) and (k),

Qr^Df = 0;

5=0 on X [0, <],

s-ii=0 on (Qr\dB)x[0,t],

m(», O) =m(*, 0) =0 on Q.

f f b(x, rj) •u(x, tj) dtj dv=0, B 0

r(a!) // s(x,ri)' u(x,ri) dtj da=0.

dB 0

Next, (i), (j), and (c) imply

u(x, r{x))—u{x, 0), Vu{x, t(x)) = Vu(x,o), G(x)=0, xiQ. (n)

By (a), (n), and (1), £(£c, 6r(cc)) =0, xaQ,

(x, t(x)) — {x, 0) = u^(x,r(x)'), ccc-f3,

0, xfiQ, , , , ls{x, Vu(x, r(x))+G(x)), x^Q,

s {x, Vu(x, r (X)) + G{x)) — e (sc, |7m(», 0)) = | ^ I0> .

17*


Clearly, t satisfies the hjôtheses of the Wheeler-Sternberg lemma (2). Therefore we may conclude from (b), (m), and (o) that

J ■^^-u^{x,x{x))—e[x,G(x))-\-e[x,Vu{x,r{x))-{-G{x))]^dv=Q. (p)

a

Since \G{x)\^ = \Vr {x)\^ {x, r{x))

for xe(B —{z}), it follows from (a), (j) and the inequality in (70.3) that

£ {x, G{x)) ^ 1*2 [x, T (»));

thus

^x, r (*)) - £ {x, G{x)) ^ 0 (q)

for xe{B —{s}). Since C is positive definite, the function £ is non-negative, and (q) implies that the integrand in (p) is non-negative on Q — {z]. Thus, since this integrand is also continuous on — {z}, we conclude from (p) that it must vanish on this set. This fact, in conjunction with (q), implies

£ [x, Vu(x, r (»)) -|- G{x)) =0, xe{Q — {s}).

Thus, since C is positive definite, (a) yields

E(sc, t(cc))-fsym 6?(cc) =0, xe{Q — {s]). (r)

By (i)

= (s)

In view of (i) and the continuity of E and u,

E(x, r{x))-Ê{z, X), u{x, r{x))-û{z, X) as x-^z. (t)

If we let X —z -\-8e, where e is a fixed unit vector, and take the limit in (r) as d-^0, we conclude, with the aid of (c), (s), and (t), that

E(z, A) — s)ma {u{z, A) ® e} =0. (u)

Since (u) must hold for every unit vector e, (u) must hold with e replaced by — e; thus

E(z, A)=0.

As (s. A) was arbitrarily chosen in (B—D^) x(0, t), and since E is continuous,

£ = 0 on (E — Df) X [0, t]. (v)

By the definition of

6=0 on (5—D() x[0, f], m(*, 0) =m(*, O) =0 on B—Df] (w)

(v) and the first of (w) together with the stress-strain relation and the equation of motion imply

S = 0, M = 0 on (5 — D() X [0, <]. (x)

The second of (x) and (w) yield

M = 0 on (5 —D() x[0, <),


Sect. 75. Basic equations.

261

W^SMsetiiedbi23^fJ^ e^stablish a uniqueness theorem

to doffiaks:'rnlb“h^^^ this theorem to the uniqueness theorems of Neumann (63.1) and Bron f63 2\ it mirct u remembered that among the assumptions iade iThe le/Jôi tTs section ^âssumptions that C.be£osith5„defimteând symmet^^^^^ ^ be strictly

nf flV t^rem.^ Under the assumptions made at the heginnine of this section the mixed problem of elastodynamics has at most one solutioi. ^

^ —[m, E, S] be the difference between two solutions. Then *> corresponds to vanishing body forces and satisfies ^

«(•, 0) =m(., 0) =0 on B, s-M=o on Bx(0,fo).

o’!TThTSfa'Sw ft the domain of influence on n V ro T Q ^ Ttierefore we conclude from (1) that p vanishes on B X [0, f]. Since t was arbitrarily chosen, the proof is compleT. □

I.^Md^be noted that Wheeler and Sternberg’s uniqueness theorem n)

regiqns ,ô.q :4pdêreSSi^ elastodynamic uniqueness theorem valid for infinite regions mav

procTdmrl-«Tjl^“^‘^-^T*^^ classical energy identity foUowing Nctmann’I procedure (63.1), provided one introduces suitable restrictions on the orders of mîtude of the velocity and stress fields at infinity. The advantage of (3) is that It does not mvpke.-Siict. artificial a priori assumptions. The analogLs

issue m elastostatics, where the equations are elliptic rather

IS considerably more involved. For exterior domains elastostatic

TTn“ect To™ prescriptions at iSy 1

VI. The free vibration problem.

BnriT' equations. The free vibration problem^ is concerned with an elastic body undergoing motions of the form

u(x, t)=u(x) sin (cof+y),

E{x, t) =E[x) sin {cot +y),

S {x, t)~S {x) sin (cof +y), (a)

wJ aT*The^s?Sds "" uull body lorces, then these fields wiU satisfy the fundamental field equations of elasto- ynamics if and only if the amplitudes u{x). E(x). and S(») obey the relations

Since

E = i(Fu + Fu^),

S=C[E],

div S -j-gcoû=0.

C[E]=C[Fm],

1 Wheeler and Sternberg [1968, 16'] for the ca<!P in n.hi..v. -r • u isotropic; Wheeler [1970 3] for the case"^ in whirl. ^ which B is homogeneous and

AIL, i/^^:roZzrzz^T£S’“y


the above relations yield the following equation for the displacement amplitude:

div C[Fm]+50* M =0, (b)

which, for a homogeneous and isotropic body, takes the form

[xA M + (A+/i) FdivM + 5C()*M = 0.

Also, a simple calculation, similar to that used to deduce (e) of Sect. 59, 5delds the relations

cl A divM+C()*divM = 0,

c| A curl M +0)* curl m = 0,

where and c^ are the irrotational and isochoric velocities. Of course, these formulae could also have been derived by substituting the relation for u (x, t) in (a) into (e) on p. 213.

Returning to the general inhomogeneous and anisotropic body, we see that the procedure given on p. 214. here 5delds the following relation for the stress amplitude:

f(-^^) +a)^K[S]=0.

where K = is the compliance tensor.

76. Characteristic solutions. Minimum principles. We assmne that the elasticity field has the following properties:

(i) C is smooth on B;

(ii) C is positive definite and symmetric.

Let 2 = the set of all admissible displacement fields,^

and let ^ be the operator on 2 defined by

=div C[Fm] .

For convenience, we introduce the following notation:

(u,v'} = f u-vdv, B

hi=y<u, M>, <u, w>c = f Fu- C[Fv] dv.

B

Note that, by (ii) of (20.1),

(u, v'}c = / Fu- C[Fv] dv. B

The free vibration problem consists in finding combinations of the frequency co and amplitude u that are possible when a portion of the boundary is clamped and the remainder is free. This should motivate the following definition: By a characteristic solution (for the free vibration problem) we mean an ordered pair [A, m] such that A is a scalar, and

S(’u-\-û — 0, h|| = 'l. M = 0 on .Si^, s=0 on

1 See p. 59.

Sect. 76. Characteristic solutions. Minimum principles. 263

here s is the traction field corresponding to u, i.e.

s = C\yu\n on dB.

We call A a characteristic value, u an associated characteristic displace¬ ment. Given A, the corresponding frequency of vibration is given by

as we will soon see, A is always non-negative. Finally, we say that the entire boundary is clamped ii S^^ — dB (5^2 = 0), free if ,5^ = dB (5^ =0).

(1) Lemma. Assume that:

(i) with û continuous on 5;

(ii) V is continuous on B and piecewise smooth on B, while Vv is piecewise continuous on B.

Then

M, w> = — <M, w>c-t- / s-vda, dB

where s is the traction field corresponding to u.

Proof. Let S=C[Fm], so that s=Sn on dB. Then

(ifw) -v dv= f (div S) -v dv= f s • v da — f S • Vv dv B B dB B

= / s-vda — <u,v'}c. □ dB

(2) Properties of characteristic values.^ solution. Then

A = <m, m>c, so that

AÔ.

Let [A, M] he a characteristic

In fact, A = 0 if and only if u is rigid. Moreover, a characteristic solution [A, u] with A=0 exists only if the entire boundary is free, in which case every rigid dis¬ placement u with ||m1|=1 is a characteristic vector corresponding to A=0.

Proof. Since [A, m] is a characteristic solution,

û-\-ku = 0, ||m|| = 1, (a) and

s • M =0 on dB, (b)

where s is the corresponding traction field. By (a)

<ifM, M> = — A<m, m> = —A;

therefore we conclude from (b) and (1) that

A = <m, m>c.

Moreover, since C is positive definite,

_ <M, M>c^0,

1 Clebsch [1862, J], § 20, generalizing a result of Poisson [1829, 2] for the vibrations of an elastic sphere. See iso Ibbetson [1887, 2], § 256, Ces.^ro [1894, 7], p. 58.

Azhmudinov and Mamatordiev [1967, 7] show that when the boundary is clamped A is given by a certain integral over dB.

264 M.E. Gurtin; The Linear Theory of Elasticity.

Sect. 76.

(a)

(b)

(c)

(d)

(e)

L . a cnaractenstic solution whenever MIS rigid and ))m)) = i. q

(3) OrthogotmlUy of chara^teHstic displacements.^ Let [A «! and\l be charactenstic solutions with A =j= A. Then ^ ^ L * J

<M, m> = <m, m>c = <.^m, m>=o.

Proof. By hypothesis

w “h Am = 0,

^m + Am = 0,

s-M==s-M=o on dB,

where s and s are the traction fields of m and m. By (a) and (b)

m)> 4-A<m, m) = 0,

<i?M, m> + A<m, m>=o.

On the other hand, (c), (1), and the s5Tnmetry of C imply that

By (d) and (e) »> ■

(A —A)<m, m>=0;

hen^ <M, M> =0 if A4=A~. The remaining results foUow from (d) and (e). □

distinc?cSi-aSrisS^SueîetrTwS%?^^^ corresponding to characteristic values are equal. IndeeTff m, « ' '' ""cessarily true if the

" s r “■. ties:, —- - ^ ^ '^th the foUowing proper-

(i) « is continuous on B and piecewise smooth on B ;

(ii) Fv is piecewise continuous on 5- (iii) ||«|| = 1;

(iv) M==0 on

c»«.™cteris«c Le,

Define *^>c^ <1^, i^>c for every veJT.

« oMracM elution, and A, h the ch.,aamîc

2 f on p. 263. See also Poisson [1826 l^.

rant [1920,^]; Cou^'^T fnd HiS'ff^sf; S?p.?39fHu"'[^958,^7jf Con-


Proof. Let a4= it be an arbitrary real number and choose weJT. Then ||Mi+aw||4=0, and

belongs to JT; thus, by hypothesis,

Ai = <Mi, Mi>c ^ <W, t»>c = <«1 +««. +««>C.

and hence Ai<Mi +aw. Ml +aw> ^ <Mi +aw, Mj +aw>c.

If we expand both bilinear forms and use the relations ||i7|| = ||tti||=l and Ai = <Mi, Mi>c, we arrive at

2a[Ai<Mi, w> —<Mi, w>c] +cP[Xi, — <v, w>c] ^0.

Since this inequahty must hold for every a4= ± 1. we must have

Ai<Mi, w> —<Mi, w>c=0. (a)

As w = 0 on , (1) implies that

<Mi, w>c= —<^Mi,»> + / s^-vda,

where Si is the traction field corresponding to itj. In view of the last two relations,

f (ifMi + Aitti) -v dv— / Si'V da=0 B s^,

for every continuous and piecewise smooth vector field v with w = 0 on and ||w|| = l. Since this expression is linear in v, we can drop the requirement that ||i7|| = l. Therefore it follows from (7.1) and (35.1) that

.Sftti + Ai 1*1 = 0 on B,

Si = 0 on 5^2-

Thus, since [Ai, Mi] is a characteristic solution.

If [A, m] is another characteristic solution, then (2) implies that

A = <m, m>c.

Thus we conclude from our present hypotheses that

AiÂ,

and hence Ai is the lowest characteristic value. □

Theorem (4) asserts that the minimum (if it exists) of the functional

<«. «>c (a) under the supplementary condition

||w|i = l (b)

is the lowest characteristic value. We now show that the determination of the M-th characteristic value can be reduced to the variational problem of finding a minimum of the functional (a) under the supplementary condition (b) and the constraint

<w,M,>=0, r = \,2, —


where are the characteristic vectors for the first n — \ charac¬ teristic values.

Let Ml, , ..., M„ be piecewise continuous vector fields on B. Then we write

{Mi,M2, ....mJ-L

for the set of all piecewise continuous vector fields v on B such that

<w,M,>=0, r = \,2,...,n.

(5) Minimum principle.^ Let u^.u^,... belong to and let

f

Mj, ..., M,_i}J-, r'^2.

Assume that, for each r^\, u^^Ct, and

(u,, M,>c ^ <v, V >c for every vCJf,.. Define A, by

A, = <m„m,>c.

Then each [A,, m,] is a characteristic solution and

OÂiÂa^ ••• .

Proof. By (4), [Aj, Mj] is a characteristic solution. We now proceed by in¬ duction. Thus we assume that n'^2 and that [A,, mJ is a characteristic solution for 1 —1. Since

<M„,M,>=0, 1 = 1, 2,.... M —1. (a)

On the other hand, our induction hypothesis and (1) imply that

0 = A,- <M„, Mi> = - <M„, ATUi}

= <M„, M,>c— / Sf-u^da, f = 1, 2,..., M-1. SB

But M„=0 on ,9i and s^ = 0 on ,9^; therefore

<“».“f>c = 0, f = 1, 2,..., M-1. (b)

Now choose and let

w — _1

(M„-Faw),

where a4=±f- Then and steps directly analogous to those used to estabhsh (a) in the proof oi (4) now lead to the result that

for every veJf„.

Let vcJf be arbitrarily chosen. If v lies in the span of Mj, u^, ..., m„_i (a) and (b) imply that

«>c=0.

(c)

, then

(d)

Assume that v does not lie in the span of Mj, Uj, ...,m„_i. Then there exist

scalars /3, 4. â, .• ■, 4-i such that / »-i \

« =/9 -1- Uf (e)

1 See footnote 2 on p. 264.


belongs to Indeed, let = «i>.

and ^since ® 4= — 2 <,• choose ^ such that ||w|l = l. By (c) and (e),

n—1 K <“«. «> - <«». ®>C + 2 h [K <«» .«;>-<**„, Mi>c] = 0.

i = l (f)

Thus we conclude from (a) and (b) that (d) remains valid in the present circum¬ stances. Hence (d) holds for every and, as in the proof of (4), this implies that [A„, M„] is a characteristic solution. □

We now show that the minimum principle (5) characterizes all of the solutions of the free vibration problem provided A„-ôo.i

(6) Completeness of the minimum principle. Let ^1,^2,... and ... obey the hypotheses of the minimum principle (5) and assume, in addition, that

00 as n^ 00. Let [A, m] be a characteristic solution. Then A = A,, for some integer r, and if

^r—l ^ ■ • • = <C ^r+n+1 >

then u is contained in the span of m,, .... In particular, if n=0 so that A, is distinct, then m = ± m,.

Proof. Assume that A is not equal to any of the A/s. By (4), Thus, since A„-> 00, there exists an w ^ 2 such that

^ • (^)

Since [A,, is a characteristic value for every r, we conclude from (3) that

Thus, as the hypotheses of (5) hold.

«>c. or equivalently, by (2),

which contradicts (a). Thus A=A, for some r.

Assume now that A,_i<A,=A,+i= ••• =A,+„<A,+„+i, and let

Since A=j=A,- for f<r, it follows from (3) that

U€{Ui,U^, ..., We decompose u as follows:

M = Mq -f Mx •

where m# belongs to the span of ^ and Indeed, we simply take

(b)

(c)

f+»

«0= 2 <“. “;•>“?•• i=r

To complete the proof it suffices to show that If *to=®> fben u=u^, and it follows from (b) and the fact that that

_ ...,m,+„}-L. (d)

1 See (78.1).


Thus, as before,

'^r+n+l

and we have a contradiction. Thus MQ=i=0. Assume now that let

Mo Uj_-- _MX

iMxi

Since Uq is a linear combination of characteristic displacements corresponding to the same characteristic value A, [A, Mq] is a characteristic solution; this fact and (c) imply that [A, m^] i® ^ characteristic solution. Thus, by (3), is orthogonal to every and hence (d) holds with u replaced by Mj^ . Thus we are again led to a contradiction; consequently

MX=0. □

77. The minimax principle and its consequences. We now estabhsh an alter¬ native characterization of the free vibration problem. We continue to assume that C is symmetric, positive definite, and smooth on B. For convenience, we introduce the following notation;

W — the set of all piecewise continuous vector fields on B,

W" (« times).

Recall that JT is the set of all vector fields on B with properties (i)-(iv) listed on p. 264.

(1) Minimax principle.^ Let [A,,m,] satisfy the hypotheses of (76.5) for \^r-^n-\-\. Define cp on W" by

(p[w^, t«2,..., MJj =inf{<«, «>c: w^,..., m5„}-L}. Then

A„+i = 9’K.“2. ■■•,«») =sup {(p(w^,w^,..., t«„): (M5i,t«2, ..., toJeiT"}.

Proof. By hypothesis,

Am+1 = <pi^> **n+l)’C- (S-)

Thus it suffices to prove that

A„+i^9?(m5i,m52. for every (m5i,m52. •••.*»»)e'5^- (b)

To accomphsh this we will show that for every {w^, w^,..., there exists a w^,..., M5„}J- such that

(p>'^yc^K+ii (c) for then (b) would follow from the inequahty

cp{Wi,w^, ...,M5„)^<«, «>cÂ„+i.

Choose (m5i,w^, ..., Let n+l

(d) i=l

where the a/s are as yet undetermined. Clearly, t«2. w5„}-L if

»-f 1

_ _ _ t = l

1 CouRANT [1920, i]. See also Courant and Hilbert [1953, P* 406.

(e)

Sect. 77. The minimax principle and its consequences. 269

2 a,<M,-,t»,>=0 (r=\,2,...,n). (f) t=i

The system (f) consists of n equations in m + 1 unknowns and can always be solved for ai.aa, ...,a„+i. Moreover, since the system is homogeneous, the re¬ quirement (e) can also be met. We have only to show that the function v so defined satisfies (c). Since ^

(d) implies

Further, (e)

= (no sum) 1,/ = !, 2, ..., m-M,

n+l

<W, W>C= 2 t=l

and the fact that

(g)

imply n+l

2 (h)

and (g), (h) yield the desired inequality (c). □

Let Ai, Aj,... and Ui,u^,... obey the hypotheses of the minimum principle (76.5). We call the ordered array

{Ai, A2,...}

the spectrum corresponding to C, B, ,5^^, and and we say that u^, Mg. is an associated system of normal modes. By (76.6), (78.1), and Footnote 2 on p. 270, the spectrum contains every characteristic value, and hence it is independ¬ ent of the choice oiu^.u^,... (as long as this choice is consistent with (76.5)).

We now consider a second body B composed of an elastic material with

elasticity field C. We will discuss the vibration problem for such a body taking as boundary conditions

M = 0 on and s=0 on 6^^,

where and are complementary subsets of dB. In particular, we compare the spectrum corresponding to this problem with the spectrum for the problem discussed previously (assuming, of course that both spectrums exist).

We assume that C is positive definite, symmetric, and smooth on B. We write

if C^C

E-C^[E]Ê-C,,[E]

for every symmetric tensor E and every xeBr^B.

(2) Comparison theorem.^ Let {Ai,A2, ...} he the spectrum corresponding

to C, B, and A2,...}, the spectrum corresponding to C, B, and Assume that either

1 This follows from (d) and (e) in the proof of (16.3) and the fact that

<,Ui,Ujy = dij {i,i=i, 2, ...,n+\).

2 Cf. CouRANT and Hilbert [1953, S], pp. 407-412.


(i) B = B, C^C; or

(ii) the entire boundary of B is clamped, and

Then BCB, C^C.

Proof. Let JT be as before, and let be the corresponding set for B and .

Assume that (i) holds. Then, since B = B and C .

Further, since C ^ C,

(a)

<17, w>c^<w,w>c. (b)

Let be a fixed integer, let 97 be as defined in (1), and let ^ be defined on -iT" by

^(l»l, 1»2, •••. *«„) =inf {<V, 17>g: 17eJ^r^{l<7^, 1172, ..., icJ-L}.

Then by (a) and (b)

and we conclude from the minimax principle (1) that

The proof under hypothesis (ii) is strictly analogous. □

78. Completeness of the characteristic solutions. In this section we show that the characteristic values A„ become infinite as môo. This imphes, in particular, that each characteristic value has only a finite multiphcity. More important, however, is that the unboundedness of the characteristic values yields the com¬ pleteness of the characteristic displacement fields. Thus we assume that the spectrum {Aj, A2,...} exists; in addition, we suppose that C is positive definite, symmetric, and smooth on B.

(1) Infinite growth of the characteristic values.^ Assume that B is properly regular and that 5î=j=0.^ Then the spectrum {A^, A2,...} has the following property:

lim X„ — oo. n~>oo

Proof. Assume that there exists a real number M such that

for all n. Then by (76.2) <u„,u„yc^M,

(a)

where Ui,u^,... is an associated system of normal modes. Thus, since

_ <«».“»>c = 2 !/{£„},

1 Weyl [1915, 3]. See also Courant and Hilbert [1953, S], p. 412 and Fichera [1971, 1], Theorem 6.V.

2 Note that we do not include the important case in which the entire boundary is free. That the theorem also holds in that instance is clear from the work of Weyl [1915, 3] and Fichera [1971, 7], Theorem 6.V. In fact, Weyl [1915, 3] proves that (for a homogeneous

and isotropic body) A„ goes to infinity as Fichera (private communication, 1971) has remarked that Weyl’s result is easily extended to include inhomogeneous and anisotropic bodies (with symmetric and positive definite C).

Sect. 78. Completeness of the characteristic solutions. 271

where E„ is the strain field corresponding to u„, we conclude from Korn’s in¬ equality (13,9) that there exists a constant K such that

f\Vu„\Uv£K B

for all n. Consequently, since

Kll* = l. (b)

we conclude from Rellich’s lemma (7.2) that there exists a subsequence such that

lim M„*||=0. (c) A-*- oo

But in view of the method of construction oi Ui,U2, ... in (76.5),

— 0, (d) By (b) and (d),

II “ II 11^ II P ,

which contradicts (c). Thus there cannot exist a number M that satisfies (a); hence oo as oo. □

The next theorem shows that every vector field / with property (A) below admits an expansion in terms of normal modes.

(A) / is continuous on B, piecewise smooth on B, and vanishes on while

Vf is piecewise continuous on B.

(2) Completeness of the normal modes.^ Let u^yU^,... he a system of normal modes corresponding to the spectrum Aj,...}, and assume that

lim A„ = oo. n—>-oo

Let f satisfy property (A). Then

lim If— 2 c„M„|| = 0, N-I.OO „=l

where the c„ are the Fourier coefficients

= </.**„> • Moreover, the c„ satisfy the completeness relation

Proof. Let

with

Then, since

it follows that

OO

2 c^«=\\f\r

N

9n—/ 2 n=l

C„ = </,«„> •

oîi»^r=i N

f-lc n=l

(a)

(b)

(c)

1 Cf. CouRANT and Hilbert [1953, S], p. 424.


Thus the series 2 converges and satisfies Bessel’s inequality

»=i

By (a) and (c), to complete the proof it suffices to prove that

iSTivIKO as N->oo.

(d)

(e)

Suppose that for some M,\gi^\=0. Then (c) would imply that Cj^+„ = 0, w = l, 2,.... and hence that ||srM+n|| =0. n = \,2.Thus, in this instance, (e) holds trivially.

Assume now that ||srjv|| never vanishes and let

9n —

By (a), (b), and (f). llsr^f

iV = t,2,...

<Sriv.“«>=0- n = \,2....,N,

and we conclude from the properties of and / that

(f)

(g)

9n^^n+1' (h)

where is defined in (76.5). Hence

or equivalently.

'liV+l ^ ^9n> Sriv)c — ilSvIP ‘\9n> 9n')c,

Since s,s N-^00, to complete the proof it suffices to show that

‘\9n’ 9n'}c remains bounded in this limit. It follows from (76.1), the fact that is a characteristic solution, (g), and

(h) that

<9N’^n'>c=0, n = i,2, Thus, since

<««. «m>c = 4 (no sum),

we conclude from (a) that N N

<f.f>C = <9N+ Z C„M„,Sriv+ Z C„M„>C n=l n=l

Consequently, as ^ 0,

— ^9n>9n')c-^ Z ^^n-

‘\9n> 9N}c^(.f>f}c>

hence <51^, 5riv)c remains bounded as N-^00. □

Under reasonable assumptions on B and on the set of vector fields with property (A) ^s dense (with respect to the norm | • ||) in the space of continuous functions on B. Thus, under such assumptions, (2) will hold for functions / that are continuous on B.

References. 273

References,

{Italic numbers in parentheses following the reference indicate the sections where the work is mentioned.)

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fluidelor viscoase. Bui. St. Acad. R. P. Roraane 6, 555-571 (44) 10. Klyushnikov, V. D.: Derivation of the Beltrami-MicheU equations from a varia¬

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7. Schultz, M. H. : Error bounds on the Rayleigh-Ritz-Galerkin method. J. Math. Anal. Appl. 27, 524-533. (39)

8. Stippes, M. : Completeness of the Papkovitch potentials. Q. Appl. Math. 26, 477- 483. (44)

9. Youngdahl, C. K. : On the completeness of a set of stress functions appropriate to the solution of elasticity problems in general cylindrical coordinates. Int. J. Eng. Sci. 7, 61-79. (44)

1970 1. Benthien, G., and M. E. Gurtin: A principle of minimum transformed energy in elastodynamics. J. Appl. Mech. 31, 1147-1149. (66)

2. HlavACek, I., and J. NeCas: On inequalities of Korn’s type. I. Boundary-value problems for elliptic systems of partial differential equations. II. Applications to linear elasticity. Arch. Rational Mech, Anal. 36, 305-334. (28)

3. Wheeler, L. T. : Some results in the linear dynamical theory of anisotropic elastic solids. Q. Appl. Math. 28, 91-IOI. (61, 74)

4. ZiENKiEwicz, O. C.: The finite element method; from intuition to generality. Appl. Mech. Rev. 23, 249-256. (39)

1971 1. Fichera, G. : Existence theorems in elasticity. Below in Volume VIa/2 of the Hand- buch derPhysik, edited by C. Truesdell. Berlin-Heidelberg-New York: Springer. (5, 7, 13, 31, 33. 36, 78)

2. Knops, R. j., and L. E. Payne: Uniqueness theorems in linear elasticity. In: Sprin¬ ger Tracts in Natural Philosophy (ed. B. D. Coleman), Vol. 19. Berlin-Heidelberg- New York: Springer. (32, 40, 63)

3. Maiti, M., and G. R. Makan: On an integral equation approach to displacement problems of classical elasticity. Q. Appl. Math. To appear. (52)

4. Segenreich, S. a. : On the strain energy distribution in self equilibrated cylinders. 5. Thesis. COPPE, Federal Univ. Rio de Janeiro.

Addendum.

The following additional papers would have been referred to, had I seen them in time.

1956 Teleman, S. : The method of orthogonal projection in the theory of elasticity. Rev. Math. Pures Appl. 1, 49-66.

1957 Roseau, M.: Sur un thdorfeme d’unicitd applicable k certains problfemes de diffraction d’ondes dlastiques. C.R. Acad. Sci., Paris 245, 1780-1782.

1959 Campanato, S.: Sui problemi al contorno per sistemi de equazioni differenziali lineari del tipo dell’elasticita. Ann. Scuola Norm. Pisa (3) 13, 223-258, 275-302.

1961 Baikuziev, K. : Some methods of solving the Cauchy problem of the mathematical theory of elasticity [in Russian]. Izv. Akad. Nauk USSR Ser. Fiz.-Mat. 3-12.

1962 Gegelia, T. G. : Some fundamental boundary-value problems in elasticity theory in space [in Russian]. Akad. Nauk Gruzin. SSSR Trudy Tbiliss. Mat. Inst. Razmadze 28, 53-72. Kupradze, V. D.: Singular integral equations and boundary-value problems of elasticity theory [in Russian]. Tbiliss. Gos. Univ. Trudy Ser. Meh.-Mat. Nauk. 84,

63-75. Rieder, G. : Iterationsverfahren und Operatorgleichungen in der Elastizitatstheorie. Abhandl. Braunschweig. Wiss. Ges. 14, IO9-343.

References. 295

1963 Deev, V. M.: Representation of the general solution of a three-dimensional problem of elasticity theory by particular solutions of Lame’s equations [in Ukrainian], Dopovidi Akad. Nauk. Ukr. RSR 1464-1467. Volkov, S. D., and M. L. Komissarova : Certain representations of the general solutions of the boundary-value problems of elasticity theory [in Russian]. Inzh. Zh. 3, 86-92.

1964 Kurlandzki, J.: Mathematical formulation of a certain method for solving boundary problems of mechanics. Proc. Vibration Problems 5, 117-124.

1965 Niemeyer, H.: Uber die elastischen Eigenschwingungen endlicher Korper. Arch. Rational Mech. Anal. 19, 24-61.

1966 Barr, A. D. S.: An extension of the llu-Washizu variational principle in linear elasti¬ city for dynamic problems. J. Appl. Mech. 88, 465. Mikhlin, S. G. : Numerical Realization of Variational Methods [in Russian]. Moscow. (I have not seen this work.)

1967 CoLAUTTi, M. P.: Sui problemi variazionali di un corpo elastico incompressibile. Mem. Accad. Lincei (8) 8, 291-343. Lai, P. T, : Potentiels elastiques; Tenseurs de Green et de Neumann, J. M6can. 6, 211-242.

1968 Cakala, S., Z. Domaiiski, and H. Milicer-Gruzewska: Construction of the fundamen¬ tal solution of the system of equations of the static theory of elasticity and the solution of the first boundary value problem for a half space [in Polish], Zeszyty Nauk. Politech. Warszaw. Mat. No. 14, 151-164. Knops, R, J., and L. E. Payne; Stability in linear elasticity. Int. J. Solids Structures 4, 1233-1242. Lai, P. T.; Elastic potentials and Green’s tensor. Parti. Memorial de I’ArtiUerie Fran9aise 42, 23-96.

1970 Bufler, H.; Erweiterung des Prinzips der virtuellen Verschiebungen und des Prinzips der virtuellen Krafte. Z. Angew. Math. Mech. 50, 104-108.

1971 HlavAcek, I.: On Reissner’s variational theorem for boundary values in linear elasti¬ city. Apl. Mat. 16, 109-124. Horgan, C. O., and J. K. Knowles; Eigenvalue problems associated with Korn’s inequalities. Arch. Rational Mech. Anal. 40, 384-402. Maisonneuve, O. ; Sur le principe de Saint-Venant. 'I'hese, University de Poitiers.

1972 Duvaut, G., and J. L. Lions; Sur les Inequations en Mycanique et en Physique. Paris; Dunod.

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