Lecture Notes 2011 02

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Lecture Notes

Effective Properties of Micro-HeterogeneousMaterials

Dr.-Ing. Daniel BalzaniInstitut fur Mechanik

Universitat Duisburg-EssenFakultat fur Ingenieurwissenschaften

Abteilung Bauwissenschaften

Summer term 2011

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Effective Properties, summer term 2011, c Daniel Balzani 1

1 Introduction

The design and modeling of micro-heterogeneous materials is more and more based onthe computer simulation of how the individual components will behave along the processchain. As an example, the study of how metals react to rolling, deep drawing and weldingand how the final components mechanical properties will be, plays the major role in caseof car body parts. Today numerical methods and material models are able to simulatemany materials, such as aluminum and conventional steel, appropriately. Unfortunately,some other materials as e.g. high-strength steels, fiber-matrix composites, filled polymers,etc. are not standard to be simulated due to their special properties, especially at smallerscales. In order to overcome these difficulties such materials have to be analized circum-stantially. In this context the properties of the individual components or crystallites arecomputed and results herefrom are used to conclude in the behavior of the entire ma-terial. This problem of scales turns out to be one of the most challenging tasks withrespect to micromechanics. Depending on the type of micro-heterogeneity of materials

different lengthscales have to be considered: the macroscale (measured in e.g. meters),the mesoscale (measured in millimeters), the microscale (measured in micrometers) andthe nanoscale (measured in nanometers). The macroscale characterizes systems and struc-tures as e.g. a cantilever beam or a pillar, whereas the mesoscale usually describes thematerial at the level of e.g. inclusions in a matrix. At the microscale we differentiate be-tween individual grains and crystals and the nanoscale considers individual molecules oreven atoms. However, if only two scales are to be analyzed then usually these two scalesare referred to as micro- and macroscale. The main idea behind micromechanics is thatthe real micro-heterogeneous material can be treated as a homogenized one at a largerscale.

A variety of materials is obviously micro-heterogeneous, as for instance concrete due tothe existence of supplements embedded in a cementoid matrix phase. A further exam-ple is wood where a kind of microstructure composed of rays and longitudinal cells isalready visible to the naked eye. But there exist many further materials where the micro-heterogeneity is not as obvious. One example is high-strength steel, whose advantageouselasto-plastic behavior is mainly governed by a complex microstructure at the meso- andat the microscale.

For the task of scale transition there exist a variety of approaches. One (self-consistent)approach observes a single test crystallite in an environment matching the mean propertiesof all other crystallites. Another possibility is to understand metal properties by use ofimproved Taylor models. For the modeling of the macroscopic behavior in the framework ofthe continuum mechanics it is clearly stated in [2], that hereby problems can be calculatedif the properties of the mesoconstituents are given. Conversely, the macroscopic behaviorcan be investigated to determine some of the properties of the microconstituents. Due tothe nature of continuum mechanics it is however not possible to determine these propertiesby fundamental physical and chemical information; there are still experiments necessary.

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As an example continuum mechanics is not able to predict the yield strength from carboncontent in the ferrite of pearlitic or spheroidized steel without any tests.

One important application area of micro-macro modeling results from the fact that crys-tallographic orientation has been observed to have significant effect in most aluminumalloys on crack initiation, which is quite important in the context of probability-of-failureestimate analysis. In [11] an elasto-viscoplastic model and corresponding Finite-Elementimplementation is presented for Al 7075-T651, based on microstructural information. In[15] a dynamic explicit Finite Element Analysis is introduced by using a crystallographichomogenization method to estimate the polycrystalline sheet metal formability. There,the velocity in the homogenized macro- and the micro crystal structure is consideredadditionally. A further step not only considering the micro- and macroscale is realizedin [5], where even the smallest scale at the molecular level is taken into account by us-age of quantum physics. Furthermore, fine scale deformations are described by a particledynamics method extending the molecular dynamics to multi-atom aggregates.

Due to the fact that the consideration of both the micro- as well as the macroscale in

one calculation, is computationally very expensive, parallel computing algorithms seemto become more important especially when large macrostructures have to be analyzed. In[8] a component template library is used which is suitable to adapt for the parallelizationeasily. Another example for a realization of parallel computing in the context of Finite-Element problems at different scales synchronously is given in [10].

Literature Recommendation With respect to an introduction to the homogenizationand localization theory in the context of small strains we refer to [4]. Another importantreference is [17], where also the issue of computational aspects is discussed.Other references used in these lecture notes are [1], [16], [2], [3], [5], [6], [7], [8], [9], [10],[11], [12], [13], [14], [15].

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2 Defects and Fundamental Solutions

In elastic materials defects are characterized by inhomogeneous stress or strain fields.Basically, defects can be subdivided into the two main groups

Defects that are themselves source of eigenstrains or eigenstresses (dislocations,inclusions)

Defects that arise due to the effect of external loadings (inhomogeneous materials,wholes, cracks).

In the case of material inhomogeneities it is reasonable to decouple the stress and strainfields into a homogeneous part , of the virtually undisturbed material (without defects),and a fluctuation part , , which is also referred to as eigenstrain or eigenstress, thenwe obtain

= + and = + . (1)

2.1 Eigenstrains or Eigenstresses

2.1.1 Dilatation Centers The idealization of a punctiform region which is character-ized by a radial expansion (eigenstrain) is referred to as dilatation center. In an isotropicmaterial radial-symmetric strain- and stress fields with tension in circumferential directionand compression in radial direction are obtained. As an example we refer to a DP-steel,where due to the production process an austenitic inclusion transforms to martensite re-sulting in a volumetric jump. A dilatation center can be interpreted as a spherical region

with radius a where a pressure p exists, cf. Fig. 1.

pr

a

Figure 1:Idealization of a dilatation center.

For this idealization an analytical solution can be derived in spherical coordinates (r,,)for the displacements

ur = pa3

4r2, u = u = 0 (2)

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and for the stresses

r = pa3

r3, = = p

a3

2r3, r = r = = 0 . (3)

Remark: a dilatation center can be used as a simple model for the description of aninter-atomic-lattice atom (punctiform defect).

2.1.2 Straight Edge and Screw Dislocations Dislocations are linear defects incrystalline solids, which can be continuum-mechanically characterized by a constant jumpb (Burgers vector), cf. Fig. 2 where x3 is the dislocation line.

a) b)Figure 2: Schematic illustration of an a) edge dislocation and b) screw dislocation, takenfrom [4].

Let us now distinguish between the two dislocation types

Edge dislocations, where the Burgers vector is transverse to the dislocation line, and Screw dislocations, where the Burgers vector follows the dislocation line.

For a straight edge dislocation following Fig. 2a we are able to provide analytical equationswith b = |b| and r2 = x21 + x22 for the displacements

u1 =D

2

2(1 ) + x1x2

r2

and u2 =

D

2

(1 2)lnr + x

22

r2

, (4)

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wherein the abbreviation D = b/2(1 ) is used, and for the stresses

11 = Dx23x21 + x

22

r4, 12 = Dx1

x21 + x22

r4and 22 = Dx2

x21 x22r4

. (5)

More simply the displacement- and stress fields can be analytically computed for thescrew dislocations by

u3 =b

2, 13 = b

2

x2r2

and 23 =b

2

x1r2

. (6)

2.1.3 Inclusions Here, we focus on spatial distributions of eigenstrains t(x), whichresult e.g. from phase transformations in solids, where the distances between the atomschange and rearrange the lattice. Since these eigenstrains are not associated with stressesthey are also referred to as stress-free transformation strains. Other examples for sucheigenstrains are all eigenstrains existing in stress-free states as e.g. thermal or plastic

strains. In the context of small strains the total strains and resulting stresses are additivelyobtained by

= e + t and = C : ( t) . (7)Herein, the elastic strains and the elastic tangent moduli are denoted by e and C. Ifonly a certain area is governed by such eigenstrains t = 0 then this area is referredto as inclusion and the surrounding region with t = 0 as matrix. Usually, thesetwo regions are associated with the same elasticity, otherwise it would be referred to asinhomogeneity. It is remarked that often we speak of inclusions even if different materialproperties exist if there is no danger of confusion.

Please note that in general it is not possible to derive analytical solutions for the distri-

butions of stress or total strains. However, there exist closed-form solutions for severalspecial cases which play an important role in the field of multiscale mechanics and whichare discussed in the following sections.

2.1.4 Eshelby-Solution Here, we focus on an ellipsoidal inclusion which is locatedin an infinite matrix. With the semi axis ai the geometry of the ellipsoid is described by

x1a1

2+

x2a2

2+

x3a3

2 1 , (8)

where the semi axis ai coincide with the xi-axis. If a constant eigenstrain t

= const isacting on the ellipsoidal inclusion, then an interesting finding of J.D. Eshelby (1916-1981)can be proven that also the total strains are constant inside the inclusion . These canbe computed in terms of the Ehelby-tensor S based on the linear relationship

= S : t = const in or ij = Sijkltkl in . (9)

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We conclude with (7) that the stresses have to be constant in the inclusion , too. Thesecan then be computed by

= C : (S I) : t = const in , (10)

wherein I characterizes the fourth-order identity tensor

Inmkl =12(mknl + mlnk) . (11)

The symmetry with respect to the first and second pair of indices does not hold (Sijkl =Sklij) although the symmetries Sijkl = Sjikl = Sijlk hold. For isotropic materials thecomponents ofS depend only on the

the Poissons ratio ,

the semi axis ai and

their orientation in the xi-coordinate system.

Interestingly, the coefficients do not depend on the elasticity modulus. Due to the com-plexity of the individual components we refer to [14] and focus on some more simplespecial cases. However, we conclude several properties:

the strain- and stress fields are not constant in the matrix,

they behave indirectly proportional to the cubic distance from the inclusion, i.e., r3 for r ,

the result of Eshelby holds generally also for anisotropic materials

closed-form representations of the components of S and strain- and stress fields inthe matrix are only possible for isotropic materials .

Let us now consider two simple special cases. If we consider a cylinder of infinite length asthe inclusion geometry, i.e. a3 , as shown in Fig. 3, then we are able to write downa two-dimensional analytical solution.

In this case the matrix strain- and stress fields in the x1 x2-plane behave indirectlyproportional to the square distance from the inclusion (, r2 for r ). The

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a1

a2

x1

x2

x3Figure 3: Schematic illustration of cylinder-shaped inclusion.

non-vanishing components ofS can then be written down for isotropic materials by

S1111 =1

2(1 ) a22 + 2a1a2

(a1 + a2)2

+ (1

2)

a2

a1 + a2S2222 =

1

2(1 )

a21 + 2a1a2(a1 + a2)2

+ (1 2) a1a1 + a2

S1122 =1

2(1 )

a22(a1 + a2)2

(1 2) a2a1 + a2

S2211 =1

2(1 )

a21(a1 + a2)2

(1 2) a1a1 + a2

S1212 =1

2(1

)

a21 + a

22

2(a1 + a2)2+

1 22

S1133 =

2(1 )

2a2a1 + a2

S2233 =

2(1 )

2a1a1 + a2

S1313 =a2

2(a1 + a2)

S2323 =a1

2(a1 + a2).

(12)

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Example:

Consider a cylinder-shaped inclusion with the semi-axis a1 = 10 and a2 = 5 length unitsand the Poissons ratio = 0.2. Then apply heating to the inclusion of with a thermalexpansion coefficient of k in the inclusion and compute the total normal strains into thelong semi-axis

11. Assume that only strains in the x

1-x

2-plane differ from zero.

The resulting coefficients of the eigenstrain tensor in the inclusion are then computed by

tij = k ij . (13)

By taking into account the general relationship ij = Sijkltkl the total strains can be

computed by

11 = S1111t11 + S1122

t22 + S1133

t33 = (S1111 + S1122 + S1133) k . (14)

With 1/(2 2) = 0.625 we evaluate the three coefficients of the Eshelby tensor

S1111 = 0.625

52 + 2 10 5

(10 + 5)2+ (1 2 0.2) 5

10 + 5

= 0.472

S1122 = 0.625

52

(10 + 5)2 (1 2 0.2) 5

10 + 5

= 0.0556

S1133 = 0.625 0.2 2 510 + 5

= 0.0833 ,

(15)

and obtain therewith the result

11 = (0.472

0.0556 + 0.0833) k = 0.4997 k . (16)

Let us now consider a spherical inclusion where ai = 1, then we obtain a geometricalisotropy and the dependency of the orientation in the xi-coordinate system vanishes. Inthis case the Eshelby tensor reduces to

S = 1 1 + (I 131 1) or Sijkl = 13ijkl + (Iijkl 13ijkl) , (17)wherein we have the abbreviations

=1 +

3(1

)

and =2(4 5)15(1

)

. (18)

Please note, that 1 denotes the second-order identity tensor. The interpretation of thesetwo parameters becomes clear when taking into account the volumetric- and deviatoricsplit which ends up in

ii = tii and dev[] = dev[

t] . (19)

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If we apply a thermal expansion in this spherical inclusion, we are able to provide ana-lytical formulae in polar-coordinates for the strains in the inclusion

r = = =1 +

3(1

)

k (20)

and outside of the inclusion

r = 2 1 + 3(1 )

ar

3k and = =

1 +

3(1 )a

r

3k . (21)

2.2 Inhomogeneities

With contrast to the case of eigenstrains we concentrate now on micro-heterogeneousmaterials. This means that the material properties vary with the position in the body.

2.2.1 Concept of Equivalent Eigenstrains Main idea: find an equivalent eigen-strain in a homogeneous replacement material which represents the inhomogeneity, inorder to be able to apply the Eshelby results.

The body V with the elasticity tensor C := C(x) is considered, where an external dis-placement u is applied at the boundary V, cf. Fig. 4.

a) b) c) d)

Figure 4: a) Heterogeneous material, b) homogeneous replacement material, c) equivalenteigenstrains and d) homogenized initial problem, taken from [4].

By neglecting body forces the boundary value problem is described by

div[] = 0 with u|V = u , (22)

wherein the general constitutive law for elasticity is considered

= C(x) : . (23)

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Now we take into account the same physical body with the same boundary conditionsmade of a homogeneous replacement material with the constant elastic properties C0,cf. Fig. 4b. Then we are able to write down the boundary value problem

div[0] = 0 with u0|V = u , (24)where we consider the constitutive law

0 = C0 : 0 . (25)

If we compute the differences of the kinematic quantities

u = u u0 and = 0 , (26)then we are also able to compute the difference of the stresses by

= 0

= C(x) : C0 : ( )= C0 : C0 : + C(x) : = C0 : + [C(x) C0] : = C0 : { + (C0)1[C(x) C0] : } .

(27)

For the case of Fig. 4c, where we focus on the differences of the fields, we write down theassociated boundary value problem

div[] = 0 with u|V = 0 . (28)Therein, the constitutive law reads

= C0 : { } (29)with the equivalent eigenstrains

= (C0)1 : [C(x) C0] : . (30)Due to the formal analogy with (72) we notice that in this case the boundary value problemdescribes a homogeneous material with the elasticity tensor C0, where an eigenstrain field is applied and no diplacements occur at the boundary V. Therewith the complex

heterogeneous problem shown in Fig. 4a can be reduced to the less complex problemdepicted in Fig. 4d, where we only treat a homogeneous material with a distributionof eigenstrains. Now we are generally able to use the Eshelby results also for materialinhomogeneities. The in Equation (30) included quantity

(x) = [C(x) C0] : (31)

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is referred to as stress polarization. It describes the deviation of the true stresses =C : from the stresses that would result from the true strains applied only to thehomogeneous replacement material.

If a true eigenstrain field t exists in addition to the material inhomogeneities, then theabove described proceeding leads to the equivalent eigenstrains

= (C0)1 : [(C(x) C0) : C(x) : t] . (32)

2.2.2 Ellipsoidal Inhomogeneities As an important special case we consider an el-lipsoidal inhomogeneity in an infinite matrix. The elasticity tensors of the inhomogeneityand the matrix are denoted by CI and CM. We chose the matrix material as a replacementmaterial and obtain C0 = CM. Here, we consider a given strain field

0 = const acting atan infinite distance from the inhomogeneity. Now we use = + 0 and the finding fromEquation (30), then we obtain the equivalent eigenstrains

(x) = C1

M : [C

I CM] : ((x) + 0

) . (33)We know that the equivalent eigenstrains should be zero = 0 outside of the inhomo-geneity. Thus, by using the Eshelby result we find that

= S : = const . (34)

We insert (34) into (33) and solve the resulting equation with respect to the equivalenteigenstrains; then we obtain the expression

= [S + (CI CM)1 : CM]1 : 0 in . (35)If we take again into account = 0 + , then we are able to compute the total strains in

the inhomogeneity by = AI :

0 = const . (36)

Herein, the so-called influence tensor is identified by

AI = [I + S : C

1M : (CI CM)]1 (37)

and describes the relation between the strains in and the external strain field 0.Therewith, and by using the relation

0 = C1M : 0 (38)

we are able to compute the stresses

= CI :

= CI : AI :

0

= CI : AI : C

1M :

0 ,

(39)

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as a function of an external stress field acting at an infinite distance from the inhomo-geneity. Please note, that the stresses are also constant in .

Example:

As an example we focus on a spherical isotropic inhomogeneity, which is located in an

isotropic matrix and concentrate on the hydrostatic part. This means that for = 1/3we only need to take into account

{CI}1111 and {CM}1111 {S}1111 = = 23 , see Eq. (18) ,

wherein KI and KM denote the compression moduli for the inhomogeneity and the matrix.Then we are able to compute the stresses

ii = {CI}1111 {AI }1111 {C

1M}1111,

0ii

= 3 KI

1 +

3 KI 3 KM3 KM

11

3 KM0ii

=KIKM

2

3

KIKM

+1

3

10ii

=KIKM

3

2

KI

KM+

2

9

10ii .

(40)

If we analyze the case of a hard inhomogeneity, i.e. KI

KM, then we are able to

neglect the constant 2/9 and obtain for the hydrostatic stresses

ii 32

KIKM

KMKI

0ii = 1.5 0ii . (41)

In case of a soft inhomogeneity, i.e. KI KM, we find that the hydrostatic stresses iiare much smaller than the applied hydrostatic stresses 0ii

ii KIKM

3

2

9

20ii 0ii . (42)

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2.2.3 Cavities and Cracks A special case of inhomogeneities are given by cavatiesand cracks since there the elasticity can be interpreted as zero. For a graphical illustrationof different types of cavaties and cracks see Fig. 5.

a) b) c)

Figure 5: a) Voids, b) straight cracks and c) penny-shaped crack, taken from [4]).

Voids in 2D

If we apply an external stress field 0 at an infinite distance of a spherically-shaped voidof radius a embedded in a plate of infinite size (Fig. 5a), the displacements at the voidboundary (r = a) can be computed by

ur(a, ) =a

E

011(3cos

2 sin2) + 022(3sin2 cos2) + 8012sincos

(43)

for the radial part and for the circumferential part by

u(a, ) = 4 aE011sincos + 022sincos + 012(cos2 sin2) . (44)

Straight Crack in 2D

Let us consider a straight crack of length 2a which is located in a plate of infinite size andapply an external stress at an infinite distance 0 under plain strain conditions (Fig. 5)b,then we observe a jump of the displacements u. These can be computed by

ui(x1) =40i2

E

a2 x21 for i = 1, 2 . (45)

Penny-Shaped Crack in 3DThe displacement jumps in the x1 x2-plane of a spherically shaped crack of radius awhose normal coincides with the x3-coordinate axis (Fig. 5c) can be described by

ui(r) =16(1 2)E(2 )

0i3

a2 r2 for i = 1, 2 . (46)

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The jump in x3-direction can be computed by

u3(r) =8(1 2)

E033

a2 r2 , (47)

wherein we used the abbreviation r = x21 + x22.

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3 Theoretical Concepts with Respect to Homogenization

Figure 6: The idea of homogenization, taken from [4].

3.1 Representative Volume Element (RVE)

Considering micro-heterogeneous materials the continuum mechanical properties at themacroscale are characterized by the geometry and by the properties of the particular con-

stituents at the microscale. For the description of such materials we apply the conceptof representative volume elements (RVE), see e.g. [7] and [6]. We define a partial volumeof the material, which is macroscopically considered to be statistically homogeneous, asthe representative volume element. We refer a partial volume of a microstucture to asstatistically homogeneous, if each arbitrary section of the microstructure with equal di-mensions as the partial volume leads to the same macroscopic quantities. This inducesthat the choice of an RVE is not unique and it should be noted that some geometricstructures are more applicable for the implementation than others. In general an RVEat the microscale has a complex structure, which is characterized by a large amountof micro-heterogeneities, as e.g. inclusions, phase interfaces between the particular con-stituents, cracks or cavities. An important requirement for the application of the conceptof representative volume elements is the existence of two length scales: the length scaleof the macrostructure, which defines the infinitesimal vicinity, and the length scale of themicrostructure, which is characterized by the smallest significant dimension of the micro-heterogeneities. Denoting a typical characteristic length at the macroscale by L and atthe microscale by L, then we require L/l 1 to hold. Therefore, the ratio of lengthscales

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is important and not absolute values of the lengthscales.In order to obtain a volume element, which has a representative character, the represen-tative volume element (RVE) should be much larger than the characteristic size of aninhomogeneity. Otherwise it may happen, that the RVE consisted only of the materialof the matrix or of the inhomogeneity, which would obviously lead to unreasonable ef-

fective properties. Therefore, we conclude that the RVE should satisfy the geometricalrequirement

l d L . (48)There exist various definitions for a RVE, which show that a unique mathematically exactdefinition seems not to be possible. Some important definitions are provided below:

Hill (1963): Overall moduli have to be independent of the surface value of tractionand displacement, so long as these values are macroscopically uniform

Hashin (1983): RVE should be large enough to contain sufficient microstructuralinformation, but much smaller than macroscopic body

Drugan and Willis (1996): RVE is smallest volume element for which the overalleffective modulus is sufficiently accurate to represent the mean constitutive response

Ostoja-Starzewski (2001): RVE is i) unit cell of periodic microstructure, ii)volume possessing statistically homogeneous and ergodic properties

Stroeven, Askes, and Sluis (2002): Determination of RVE size is not straight-forward! It depends on the material under consideration and on the structure sen-sitivity of the physical quantity that is measured

3.2 Concept of an Ensemble

Another approach for the description of the macroscopic material behavior of micro-heterogeneous materials is based on statistic considerations of the microstructure, seethe basic literature in e.g. [1] and [9]. Let S be the collection of samples of randommicrostructures and p the probability density of an individual sample in S, then theensemble average of a material response F is

F(x) =S

F(x, )p() d .

If the number of samples is sufficiently high, then the material response can be interpretedas representative at the macroscale and thus as a suitable effective property F = F.

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3.3 Ergodic Hypothesis

The transition from ensemble average computations to the above defined representativevolume elements is enabled by further physical assumptions strongly associated with theergodic hypothesis. In this contex the ensemble average is replaced by simple volumetric

averages over one RVE. Basically, the ergodic hypothesis can be rephrased in words byAll states available to an ensemble are also available to every sample in the ensemble.

The main message behind is that if one considers only one particular sample which issufficiently large and computes the volume average of the material response

F(x, ) = 1|V|V

F(x + y, ) dy ,

then we are independent of the particular sample. This means that the volume average isidentical to the ensemble average

F(x, )

= F(x, ) for

|V

| and a suitable effective material response is obtained by computing the volumetric averageF = F of a sufficiently large volume.Remark: If periodic composites are considered this is automatically satisfied for the peri-odic unitcell Y

lim|V|

1

|V|V

F(x + y, ) dy =1

|Y|Y

F(x + y, ) dy .

3.4 Statistical Homogeneity

A material is statistically homogeneous, if the ensemble average of a material responseF(x1,...xn) is invariant with respect to translation

F(x1,...xn) = F(x1 y,...xn y) for arbitrary yand if the translation is y = x1, then we obtain the alternative representation

F(x1,...xn) = F(x12,...x1n) with xij = xj xi .

3.5 Statistical Isotropy

A material is statistically isotropic, if the ensemble average of a material response Fis invariant with respect to translation and rotation. In this case the ensemble averagedepends only on absolute values of the vectors xij

F(x12,...x1n) = F(rij)

for arbitrary rij = ||xij||, i = 1, ...n, j = (i + 1),...n.

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3.6 Notation

Due to the importance of the ensemble average on the effective properties of a material atthe macroscale, we denote quantities at the macroscale by ()macro = () if the separationof scales is considered. Exemplarily, the stresses and strains at the macroscale are denoted

by and .

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4 Homogenization of Linear Elastic Materials

4.1 Effective Stresses and Strains

It can be shown that the average of traction forces acting on the boundary of a RVE is

the effective stress at the macroscale

=1

|V|V

t x da . (49)

For the strains we obtain analogously the macroscopic effective representation

=1

|V|V

sym[u n] da . (50)

Let us take into account the property

(xijk), k = xi,kjk + xijk,k

div[x

] = + x

div (51)

and include the balance of linear momentum, then we obtain

div[x ] = + x (b u) . (52)Neglecting body forces and acceleration terms one obtains

div[x ] = . (53)Then an alternative representation for the effective stresses is possible by applying theGauss theorem

=1

|V| Vt

x da

=1

|V|V

div[x ] dv

=1

|V|V

dv ,

(54)

which represents the volumetric average over the microscopic stresses. If we do not neglectbody forces and accelerations and by assuming integrability of the fields we arrive at

=1

|V| Vt

xda

=1

|V|V

div[x ] dv

=1

|V|V

[ + x (b u)] dv .

(55)

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For the macroscopic strains we obtain in an analogous manner

=1

|V|V

sym[u n] da

= 1|V| Vgradsymu dv=

1

|V|V

dv ,

(56)

which represents the volumetric average of the microscopic strains. Please note that inthis representation no cavaties or cracks are taken into account.

In order to obtain a general way to derive the macroscopic stress and strain quantities,we have to take into account cavities and cracks. For this purpose we start from a rep-resentative microstructure V with the boundary V. Furthermore, the microstructure ischaracterized by cavities with the boundary S and singular areas S. Such a singular areaS splits the vicinity NX of a point x V into the sections Nx+ and Nx.

n+

V

S

n

N

n

V

S

Figure 7: Microscopic body V with cavity and cavity boundary S and singular area S.The normal n of the singular area S aims from Nx into the direction of Nx+ , i.e. n =n = n+.

If the divergence theorem is applied attention has to be paid to the fact this theorem isonly applicable in sections where the considered quantity is smooth. As an example, ifsingular areas exist, then the partial integration of the gradient of a vector field y overthe volume of V yields

Vgrad

ydv = Vy n da S [[y]] n da , (57)

with the jump [[y]] := y+ y. Herein, y+ and y denote the thresholds from the rightand from the left ofy at the singular area S, i.e.

y+ := limx+xs

y(x+) and y := limxxs

y(x) (58)

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with x+ Nx+ and x Nx. For a general representation we consider first the twotensor fields K and G in V with

div[K] = 0 in V

[[K]]n = 0 on S

Kn = 0 on S

G = grad[y]

, (59)

wherein we have the smooth function y and piecewise smooth G. Then we obtain thevolumetric average ofKG by

KG := 1|V|V

KG dv =1

|V|V

(Kn) y da . (60)

For the representation of the macroscopic Cauchy stress tensor we set K = and

G = grad[x] = 1 and obtain

div[] = 0 in V

[[]]n = 0 on S

n = 0 on S

. (61)

Inserting this in (60) and taking into account the Cauchy theorem t = n we receive theexpressions for the macroscopic stresses

:=

=

1

|V

| V(n)

xda =

1

|V

| Vt

xda =

1

|V

| V dv . (62)

Hereby, it can be seen that the macroscopic stress tensor can be either computed by thevolumetric average over V or it can also be defined by boundary tractions.By setting K= 1 and G = = symu we obtain for the average of the strain tensor

= 1|V|

V

sym[u n] da +S

sym[u n] da S

sym[[[u]] n] da

. (63)

In the absence of singularities the jump of the displacement field is

[[u]] = 0 on S (64)

and the last term in (63) vanishes. This means that by solving with respect to themacroscopic strains are given by

:=1

|V|V

sym[u n] da = 1|V|S

sym[u n] da . (65)

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Table 1: Definition of macroscopic strains and stresses.

Strain tensor =1

|V

|

V

dv S

sym[un] da

=1

|V|V

sym[u n] da

Stress tensor =1

|V|V

dv

=1

|V|V

t xda

Therefore, is only defined as the volumetric average over the representative microstruc-ture V if no cavities and cracks occur. Thus, we conclude that for the general case themacroscopic quantities are not governed by volumetric averaging. In Table 1 the importantmacroscopic quantities are summarized.

In many cases the volume V consists of n partial volumes V( = 1...n) with the volumefractions

c = V/V andn

=1

c = 1 , (66)

wherein the elastic constant material properties C are observed. In this case the mi-crostructure is said to consist of discrete phases and the macroscopic stresses and strainscan be computed by

= =n

=1

c and = =n

=1

c . (67)

Herein, the phase averages are given by

= 1|V|V

dv and = 1|V|V

dv . (68)

Inside the discrete phases we have the relations

= C : in V , (69)since the elastic properties do not vary within one discrete phase.

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4.2 Effective Elasticity Tensor and Hill Condition

The effective coefficients of the elasticity tensor are defined by

= = C(x) : (x) = C : . (70)

The Hill condition (1963) states that the average microscopic strain energy density shouldbe equal to the macroscopic strain energy density

= or (C : ) = C : . (71)Now we consider the fluctuations of the stresses and strains

= and = . (72)By inserting the relations for the macroscopic stress and strain quantities we find thatthe volumetric averages of the fluctuations have to vanish. For this purpose we focus onthe calculation rule for averages X+ Y = X + Y and obtain

= = = 0 = = = 0 .

(73)

It is emphasized that this holds for the case when the macroscopic strains can be rep-resented by the volumetric averages of the microscopic strains, which is considered here.Then we obtain an additional relation for the volumetric average of the fluctuation strainenergy density

=

( + ) ( + )

= + + +

= + .

(74)

By inserting the Hill condition (71) we find that

= 0 . (75)By using the Gauss theorem and the equilibrium equation div = 0 we are able toreformulate the Hill condition in terms of quantities that are expressed at the boundaryof the RVE

1

|V|V

(u x) w

(t n) t

da = 0 (76)

with the fluctuations of the displacements w and the fluctiations of the tractions t. In thisform we observe that the fluctiation terms vanish energetically at the boundary and havetherefore no influence. This means that the Hill condition can be interpreted in the sensethat the fluctuating fields at the boundary of a heterogeneous material are energeticallyequivalent to their volumetric averages. As already mentioned, this can of course only beexpected if the considered RVE is sufficiently large.

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4.3 Average Strain- and Average Stress Theorem

For the computation of the locally distributed stress and strain fields (x) and (x) ina given volume V at the microscale, we have to solve the microscopic boundary valueproblem

div[] = 0 , (77)

with suitable boundary conditions. The main goal is to replace the real heterogeneousvolume by a homogeneous (effective) material which represents a point at the macroscaleand which only notices homogeneous strains and stresses. Enabled by the Hill-conditionwe apply homogeneous strain states 0 or stress states 0 at the boundary and end up intwo different types of boundary conditions:

a) Linear displacements: first we apply a uniform strain field at the boundary whichthen leads to linear displacements at the boundary, i.e.

u = 0 x on V with 0 = const . (78)

By applying the Gauss-theorem we obtain the propertyV

x n da =V

gradx dv = |V|1 (79)

and find that the macroscopic strain is equal to the homogeneous strain at theboundary, which is constant over the surface, i.e.

=

1

|V| Vu n da =1

|V| V(0 x

) n

da =

1

|V|0

|V| 1 =0

. (80)

b) Uniform stresses: second, a uniform stress field is applied at the boundary andwe obtain

t = 0n on V with 0 = const . (81)

Here, we find analogously that the macroscopic stress is equal to the homogeneousstress at the boundary

=1

|V

|

V

t x da = 1

|V

|

V

(0n) x da = 1

|V

|0 |V| 1T = 0 . (82)

Since for many cases the macroscopic strains and stresses are given by the volumetricaverages, the relations (80) and (82) are referred to as average strain theorem andaverage stress theorem. Taking a look on (76) again we notice that the Hill criterionis satisfied independently by each of the boundary conditions a) and b). In addition, we

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conclude that therefore the Hill condition can be generalized to the case of independentstress (1) and strain fields (2)

(1) (2) = (1) (2) . (83)Due to the fact that in the case of linear elasticity the solutions of the boundary valueproblems are unique and independent from the history, the total strains and stresses canbe computed by

a) (x) = L(x) : 0 for u = 0x on V

b) (x) = L(x) : 0 for t = 0n on V .

(84)

Herein, the localization or influence tensors L and L represent the complete solutionof the boundary value problem and depend on the microstructure in the whole volumeV. By taking into account the average strain theorem and computing the volume average

on both sides of (841) we find that

(x) = L(x) : 0 0 = L : 0 L = I .

(85)

Analogously, we obtain for the stresses when focussing on the average stress theorem

L = I . (86)For the effective elasticity tensor the relation

C : = = = C : , (87)holds, thus, we are able to transform this equation by using relation (841) to

C : = C : L : 0 = C : L : . (88)This leads to an expression for the effective elasticity tensor as a result of boundarycondition a)

C(a)

= C : L . (89)By applying boundary condition b) we transform analogously

C1

: = C1 : L : 0 = C1 : L : , (90)and the associated effective elasticity tensor is given by

C(b)

= C1 : L1 . (91)

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If we insert these results into the representation of the Hill condition (71), then we obtain

(C : ) = C : (L : 0) (C : L : 0) = 0 (C : 0) C(a) = (L)T : C : L

(92)

and analogously for the boundary conditions b)

C(b)

= (L)T : C1 : L1 . (93)

From these expressions we directly notice the symmetries of the effective elasticity tensorswith respect to the first and second pair of indices

C(a)

ijkl = C(a)

klij and C(b)

ijkl = C(b)

klij . (94)

It should be noted that the two different effective moduli C(a)

and C(b)

can be generallycomputed for arbitrary heterogeneous volumes, although they depend on the type ofboundary condition. Therefore, strictly speaking, these elasticity moduli are no effectiveproperties since they can be also computed for a volume which does not satisfy someconditions defining a reasonable RVE. Such a definition could be that if

C(a)

= C(b)

= C (95)

holds for a considered volume V, then this volume can be interpreted as a RVE. In thiscase C characterizes the unique effective elastic properties even if larger volumes are

considered that contain V.

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References

[1] M. Beran. Statistical continuum theories. Wiley, 1968.

[2] D.C. Drucker. Material response and continuum relations; or from microscales to

macroscales. Journal of Engineering Materials and Technology, 106:286289, 1984.

[3] M.G.D. Geers, V. Kouznetsova, and W.A.M. Brekelmans. Multi-scale first-order andsecond-order computational homogenization of microstructures towards continua. In-ternational Journal for Multiscale Computational Engineering, 1:371386, 2003.

[4] D. Gross and T. Seelig. Fracture Mechanics. Springer, 2006.

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[6] Z. Hashin. Analysis of composite materials - a survey. Journal of Applied Mechanics,50:481505, 1983.

[7] R. Hill. Elastic properties of reinforced solids: some theoretical principles. Journalof the Mechanics and Physics of Solids, 11:357372, 1963.

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[9] E. Kroner. Statistical continuum mechanics. In CISM Courses and Lectures, vol-ume 92. Springer-Verlag, Wien, New-York, 1971.

[10] H. Kuramae, K. Okada, M. Yamamoto, M. Tsuchimura, H. Sakamoto, and E. Nakam-chi. Parallel performance evaluation of multi-scale finite element analysis based oncrystallographic homogenization method. In D.R.J. Owen, E. Onate, and B. Suarez,editors, Computational Plasticity / COMPLAS VIII. CIMNE, Barcelona, 2005.

[11] D.J. Littlewood and A.M. Maniatty. Multiscale modeling of crystal plasticity in

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[12] C. Miehe, J. Schotte, and J. Schroder. Computational micro-macro-transitions andoverall moduli in the analysis of polycrystals at large strains. Computational Mate-rials Science, 16:372382, 1999.

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[13] C. Miehe, J. Schroder, and C. G. Bayreuther. Homogenization analysis of linear com-posite materials on discretized fluctuations on the micro-structure. Acta Mechanica,155:116, 2002.

[14] T. Mura. Micromechanics of Defects in Solids. Martinus Nijhoff Publishers, Dor-

drecht, 1982.

[15] E. Nakamachi, H. Kuramae, Y. Nakamura, and H. Sakamoto. Crystallographic ho-mogenization elastoplastic finite element analyses of polycrystal sheet material. InD.R.J. Owen, E. Onate, and B. Suarez, editors, Computational Plasticity / COM-PLAS VIII. CIMNE, Barcelona, 2005.

[16] R.J.M. Smit, W.A.M. Brekelmans, and H.E.H. Meijer. Prediction of the mechanicalbehaviour of nonlinear heterogeneous systems by multi-level finite element modelling.Computer Methods in Applied Mechanics and Engineering, 155:181192, 1998.

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