Micromechanical modeling of finite viscoelastic multiphase composites

Z. angew. Math. Phys. 51 (2000) 114–1340044-2275/00/010114-21 $ 1.50+0.20/0c© 2000 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Micromechanical modeling of finite viscoelastic multiphasecomposites

Jacob Aboudi

Abstract. A three-dimensional finite viscoelastic constitutive relation for monolithic materialsis incorporated with a micromechanical multiphase short-fiber composite model. As a resulta micromechanical modeling of viscoelastic multiphase composites in which the constituentsexhibit, in general, finite viscoelastic deformation is established. The micromechanical analysisprovides closed-form representations for the instantaneous concentration tensors as well as forthe current effective tangent tensor. Results exhibit, in particular, the hysteretic response offinite strain viscoelastic composites that are subjected to cyclic loadings.

Keywords. Finite viscoelasticity, finite elasticity, micromechanics, composite materials, porousmaterials.

1. Introduction

The modeling of finite viscoelastic materials has been discussed in monographs byLockett [10], Christensen [6] and Drozdov [7] for example. A convenient three-dimensional finite-strain viscoelastic model, which is motivated by the linear the-ory of viscoelasticity, has been developed by Simo [14]. This model recovers thestrain energy for finite elasticity (e.g. Blatz and Ko [4]) of rubber elasticity) ofthe material for very fast or very slow processes. As was indicated by Simo [14],this constitutive model is particularly well suited for large-scale computation. Ac-cording to Simo’s model, a finite viscoelastic isotropic material is specified by theparameters of the strain energy (that characterizes the short-time elastic behav-ior), relaxation times (which are involved in the chosen kernel that describes theviscoelastic response) and weighting factors. This model is very general since theconvolution integral that appears in this representation may involve several re-laxation times, a continuous spectrum of relaxation times, fractional derivativesor power type of kernels. Another formulation and implementation of finite vis-coelasticity which is based on the generalized Maxwell model has been recentlyproposed by Kaliske and Rothert [9].

Although Simo’s model can be employed to model the finite viscoelastic be-havior of anisotropic materials, the number of parameters involved would be ex-

Vol. 51 (2000) Micromechanical modeling of multiphase composites 115

ceedingly large and their determination might be formidable. Composite materialsare in general anisotropic materials. It is well known that the determination ofthe overall behavior of composites can be achieved by following a micromechan-ical analysis that takes into account the interaction between the various phases.This circumvents the necessity of determining the model parameters for every fibervolume fraction, fiber shape and architecture, which is obviously a considerableadvantage. In the present paper we present a micromechanical modeling of fi-nite viscoelastic multiphase composites with short fibers or inclusions. The finitestrain viscoelastic behavior of the composite results from the existence of one ormore phases that are nonlinearly viscoelastic materials. The constitutive law of thefinite viscoelastic phases is based on Simo’s three dimensional convenient represen-tation. The micromechanical analysis is based on the generalized method of cells(GMC) which for elastic composite materials that involve infinitesimal and largestrains has been reported in Aboudi [1] and Aboudi and Arnold [3], respectively.The method is an analytical model that satisfies the compatibility of displace-ments and equilibrium of tractions at the various interfaces in an average sense,and along the boundaries of the repeating volume element. It provides analyticalexpressions for the effective properties and the three-dimensional constitutive be-havior of such composites. Consequently, any type of simple or combined loading(multiaxial state of stress) can be applied irrespective of whether symmetry existsor not, as well as without resorting to different boundary condition applicationstrategies. This is particularly advantageous when analyzing realistic compositestructural components, given that different loading conditions exist throughout thestructure, which in turn necessitates the repeated application of the establishedmicromechanical equations at each of these locations. For an update review, verac-ity and implementation of the method by various researchers the reader is referredto Aboudi [2].

The present paper offers in the framework of GMC a micromechanical analysis,based on the tangent approach in conjunction with an incremental procedure, thatcan predict the overall behavior of nonlinearly viscoelastic multiphase composites.Every constituent in such a composite can be assumed, in general, to behave as anonlinearly viscoelastic material with large deformation. As a result of the tangentapproach, it is necessary to solve at each time increment and a specific type ofloading a system of linear algebraic equations. Furthermore, in the course of themicromechanical analysis, the instantaneous concentration tensor (which relatesthe local deformation gradient to the global one), is established. In addition, theeffective current tangent tensor (which relates the average stress increment to theaverage deformation gradient increment), is determined. Both the concentrationand tangent tensors are given at each time increment in a closed-form manner interms of the geometrical dimensions and material properties.

Results are given for a finite viscoelastic matrix whose short-term behavior isdescribed by the Blatz and Ko [4] strain energy which is applicable for polyurethanecompressible rubbers. The finite viscoelastic effects of porous materials that in-

116 J. Aboudi ZAMP

volve this type of a matrix are shown. Another set of results are given for a contin-uous reinforced finite viscoelastic composite. Here the finite viscoelastic matrix ischaracterized for short-terms by the Murnaghan [12] finite elastic representation.The overall viscoelastic effect on the composite response in the longitudinal andtransverse directions are given.

2. Material representation: nonlinearly elastic materials

The present approach is based on the representation of the finite elastic materi-al in terms of its current deformation-dependent tangent tensor that relates theincrements of the stress and deformation tensors. For an isotropic nonlinearlycompressible elastic material, the strain energy is given in terms of the invariantsof the Cauchy-Green deformation tensor C. Thus the strain energy of the materialcan be represented in terms of the three invariants I1, I2, I3 of the Cauchy-Greendeformation tensor in the form:

W = W (I1, I2, I3) (1)

Let F denote the deformation gradient. The Cauchy-Green deformation tensor Cis given by

C = FTF (2)

where the superscript T denotes the transverse operation, and the invariants of Care

I1 = tr C = Cii

I2 =12

( tr2 C− tr C2)

I3 = det C

(3)

It should be noted that in this paper, the summation convention is implied forrepeating Latin indices. The Cauchy-Green strain tensor E is given by

E =12

(C− I) (4)

where I is the unit matrix.Let us denote by S the 2nd (symmetric) Piola-Kirchhoff tensor. It follows that

Sij =∂W

∂Cij+∂W

∂Cji=

∂W

∂Eij(5)

Furthermore,

Sij = 2∂W

∂Ip

∂Ip∂Cij

(6)


where∂I1∂Cij

= δij

∂I2∂Cij

= I1δij − Cij

∂I3∂Cij

= I2δij − I1Cij + CikCkj

with δij being the Kronecker delta. Thus the requested expression of S for a givenmaterial can be readily determined for a given state of deformation.

Let us represent the constitutive law of the nonlinearly elastic material in theincremental form:

∆S =12D∆C (7)

where D denotes the instantaneous mechanical tangent tensor.The tangent tensor D of the material at the current instant of loading is de-

termined from

Dijkl =∂Sij∂Ckl

+∂Sij∂Clk

= 4∂2W

∂Cij∂Ckl(8)

In establishing the requested tangent modulus D for a given material, the followingrelations have to be used:

∂2I1∂Cij∂Ckl

= 0

∂2I2∂Cij∂Ckl

= δijδkl − I(4)ijkl

∂2I3∂Cij∂Ckl

= (I1δkl − Ckl)δij − Cijδkl − I1I(4)ijkl

+12

(δikCjl + δilCjk)

+12

(Cikδjl + Cilδjk) (9)

andI

(4)ijkl =

12

(δik δjl + δil δjk)

Two examples for strain energy, that will be utilized in the implementation ofthe proposed theory, are given below.

(a) The Blatz and Ko compressible material (Blatz and Ko [4]) :Blatz and Ko proposed a strain energy form that is valid for for compressible

polyurethane rubbers which is given by

W =µ

2(I1 +

1− 2νν

I− ν

1−2ν3 − 1 + ν

ν) (10)

118 J. Aboudi ZAMP

where µ and ν are material constants.(b) The nonlinear material representation of Murnaghan [12]:According to this representation, the nonlinearly elastic material has the form

W =λ+ 2µ

2K2

1 − 2µ K2 +l + 2m

3K3

1 − 2m K1K2 + n K3 (11)

where K1,K2,K3 are the invariants of the large Green strain tensor E, and λ, µ,l,m, n are material constants. These invariants can be related to the invariantsI1, I2, I3 of C as follows

K1 =12

(I1 − 3)

K2 =14

(−2I1 + I2 + 3)

K3 =12

(I1 − I2 + I3 − 1)

It should be remarked that both Blatz & Ko and Murnaghan strain ener-gies couple the bulk (dilatational) and deviatoric (distortional) responses over anyrange of deformation. Since some viscoelastic materials exhibit viscoelastic effectsin the deviatoric response only, a decoupling between these two types of responsesis necessary. To this end, let us decompose F and C as follows (Ogden [13])

F = J1/3F, C = J2/3C (12)

where J = det F =√I3 with det F = 1, so that F and C are volume preserving

deformation gradient and Cauchy-Green tensors, respectively. The strain energyfor finite isotropic materials can be represented in the following form

W = W (I1, I2, J) (13)

whereI1 = I1I

−1/33 , I2 = I2I

−2/33 (14)

These invariants of C are insensitive to dilatational deformations as the lattersare completely represented by the dependence on J only. The stress tensor S canbe determined from (13) by using eqn.(5) and taking into account that ∂J/∂C =JC−1/2, and ∂C/∂C = J−2/3[I⊗ I− C⊗ C−1/3]. In the resulting expression forthe stress, the dependence of W on I1 and I2 yields the stress deviator, and itsdependence on J yields the hydrostatic stress.

An example for such a strain energy, that characterizes a compressible rubber-like material, is provided by Sussman and Bathe [15] who generalized the Mooney-Rivlin description to a compressible material. The resulting expression is

W = C1(I1 − 3) + C2(I2 − 3) +κ

2(J − 1)2 (15)


where κ is the bulk modulus and C1,C2 are material parameters. The second Piola-Kirchhoff stress tensor that corresponds to this strain energy can be determinedfrom (5), and is given by Sussman and Bathe [15]. This expression of the stresstensor separates the volumetric from the deviatoric effects. These authors providealso the tangent tensor of this material.

Finally, as the micromechanical analysis described in the following sectionsutilizes the first Piola-Kirchhoff stress tensor T, instead of the second, we needto relate the displacement gradient increment to the corresponding first Piola-Kirchhoff stress increment. To this end, we can use eqn.(2) in order to rewrite (7),and, after some manipulations, obtain the following:

∆S = Q ∆F (16)

whereQijkl =

12

[Dijlp Fkp +Dijpl Fkp] (17)

Given the fact that the relationship between the first and second Piola-Kirchhoffstresses is defined as

T = S FT (18)

it follows, then, after some manipulations, that the desired incremental constitutiverelation for the considered nonlinear elastic material is:

∆T = R ∆F (19)

where the current mechanical tensor R in (19) is given by

Rijkl = QipklFjp + Sil δjk (20)

It is readily observed that the determination of R at a given state of deformationF depends on the knowledge of S and D.

3. Material representation: nonlinearly viscoelastic materials

The finite viscoelastic representation of the material is based on Simo [14] gener-alization of the infinitesimal viscoelastic behavior to large strain and was imple-mented by MARC [11]. The energy functional is taken as

W (E,H(n)) = W∞(E) +N∑n=1

H(n)E (21)

where W∞ is the elastic strain energy for long-term deformations, and H(n) is aset of N internal variables. Using this energy representation in (5), the followingexpression for the second Piola-Kirchhoff stress tensor is obtained.

S =∂W∞(E)

∂E+

N∑n=1

H(n) (22)

120 J. Aboudi ZAMP

Since the long-term contribution can be related to the short-term one, it can beconcluded that this model is based on the additive split of the stress tensor intoinitial and nonequilibrium parts.

Let the strain energy be expressed by the following series expansion

W = W∞ +N∑n=1

W (n) exp[−t/λ(n)] (23)

where λ(n) are relaxation times.Motivated by the similarity of (22) and the equations of small strain viscoelas-

ticity that correspond to the generalized Maxwell model, the internal variablesH(n) at time t can be expressed in terms of convolution integrals, Simo [14],

H(n)(t) =∫ t

0S(n)(τ) exp[−(t− τ)/λ(n)] dτ (24)

where S(n) are internal stresses obtained from the energy functions

S(n) =∂W (n)

∂E(25)

and dot denotes a time derivative.Next, the following simplification is introduced. It is assumed that in the

energy expansion (23) each term W (n) is just a scalar multiplier of W (0), namelyW (n) = δ(n)W (0), where W (0) is the short-term elastic energy given, for example,by (10). The use of this assumption in (23) provides

W∞ = W (0) [1−N∑n=1

δ(n)] (26)

Consequently, the second Piola-Kirchhoff stress tensor can be readily deter-mined from (22) and (26) as

S(t) = S∞(t) +N∑n=1

H(n)(t) (27)

where

S∞(t) =∂W∞

∂E= [1−

N∑n=1

δ(n)]∂W (0)

∂E(28)

and

H(n)(t) =∫ t

0δ(n)S(0)(τ) exp[−(t− τ)/λ(n)] dτ (29)


with S(0) = ∂W (0)/∂E. As it was indicated by Simo [14], the exponential termsin the kernels can be replaced by continuous spectrum or by fractional derivatives.Alternatively, these kernels can be replaced by power type of kernels.

Relations (27)-(29) form the constitutive law of the a finite viscoelastic con-stituent which is just one phase of the nonlinearly viscoelastic composite. Theviscoelastic constituent is characterized by the proper choice of the short termstrain energy function W (0), the relaxation times λ(n) and the weighting factorsδ(n). It should be noted that finite elasticity is recovered for very slow and veryfast processes.

The above constitutive relations cannot be used in the micromechanical for-mulation since they are not given in an incremental form. In order to cast theseequations in incremental form, let us divide the total time interval into subinter-vals [tm−1, tm] with a time step ∆tm = tm − tm−1. It can be easily shown thatthe following approximation can be established for the internal variables H(n)

H(n)(tm) = exp[−∆tmλ(n)

] H(n)(tm−1) + δ(n) ∆S(0)

∆tmλ(n) [1− exp(−∆tm

λ(n))] (30)

Thus the following expression for the increment of H(n)(tm) can be obtained

∆H(n)(tm) = β(n)δ(n)∆S(0)(tm)− α(n)H(n)(tm−1) (31)

where α(n) = 1 − exp(−∆tmλ(n) ), β(n) = α(n)λ(n)/∆tm. This recursive relation is

used to update the internal variables at every increment.By using eqns.(27)-(28) the following requested incremental expression for the

second Piola-Kirchhoff stress tensor is obtained

∆S(n)(tm) = [1−N∑n=1

(1− β(n))δ(n)] ∆S(0)(tm)−N∑n=1

α(n)H(n)(tm−1) (32)

Notice that in both eqns. (31) and (32), ∆S(0)(tm) is determined at any instantfrom the deformation gradient increment by (16).

The increment of the first Piola-Kirchhoff stress tensor can be obtained from(18) which, by using (32), (22) and (28), provides

∆T = [ξ∆S(0) −N∑n=1

α(n)H(n)(tm−1)] FT + [ηS(0) +N∑n=1

H(n)] ∆FT (33)

where ξ = 1−∑Nn=1(1− β(n))δ(n) and η = 1−

∑Nn=1 δ

(n).By using (16) this increment can be rewritten as

∆Tij = [ξQikrsFjk + ηδjrS(0)is ] ∆Frs +

N∑n=1

H(n)ik ∆Fjk −

N∑n=1

α(n)H(n)ik (tm−1)Fjk

(34)

122 J. Aboudi ZAMP

so that

∆Tij = [ξQikrsFjk+ηδjrS(0)is +δjr

N∑n=1

H(n)is ] ∆Frs−

N∑n=1

α(n)H(n)ik (tm−1)Fjk (35)

Consequently the final form of the first Piola-Kirchhoff stress tensor increment,which is expressed in terms of the deformation gradient increment and deformationhistory, is

∆T = V ∆F−Y (36)

where the viscoelastic tangent tensor V is given by

Vijrs = ξQikrsFjk + ηδjrS(0)is + δjr

N∑n=1

H(n)is (37)

and

Y =N∑n=1

α(n)H(n)(tm−1) FT (38)

accounts for the history of deformation.In the special case of a finite elastic material δ(n) = 0, ξ = η = 1, and α(n) = 0

so that eqn.(36) reduces to (19), with equal tangent tensors namely, V = R

4. Micromechanical model description

Consider a multiphase composite material in which some or all phases are modeledas nonlinearly viscoelastic materials. It is assumed that the composite possessesa periodic structure such that a repeating cell can be defined. In Fig. 1, sucha repeating cell is shown which consists of NαNβNγ rectangular parallelepipedsubcells. The volume of each subcell is dαhβlγ , where α, β, γ are running indicesα = 1, · · · , Nα; β = 1, · · · , Nβ; γ = 1, · · · , Nγ in the three orthogonal directions,respectively. The volume of the repeating cell is dhl where

d =Nα∑α=1

dα , h =Nβ∑β=1

hβ , l =Nγ∑γ=1

lγ (39)

Any subcell can be filled in general by nonlinearly viscoelastic materials. Non-linear viscoelastic unidirectional long-fiber composites, short-fiber composites, non-linear porous materials, and laminated materials are obtained by a proper selectionof the geometrical dimensions of the subcells and with an appropriate material fill-ings.


x1

x2

x3

l 1

l 2

l

l 3

l N g

h

d

g=1

g=2

g=3

g=N g

……

h1

h1

h1

hNb

dNa

d3

d2

d1

b=1

a=1

a=2

a=3

b=2 b=3 b=Nb

a=Na

…

…

…

…

Figure 1.A repeating cell in GMC consisting of Nα,Nβ and Nγ subcells in the 1,2 and 3 directions,respectively.

The following formulation is based on a Lagrangian description of the motionof the composite. To this end, let X denote the position of a material point in theundeformed configuration at time t=0. The location of this point in the deformedconfiguration is denoted by x. This current position is given by

x = X + u(X, t) (40)

where u denotes the displacement vector.The micromechanical model employs a first order expansion of the displacement

increment in the subcell (αβγ) in terms of the local coordinates (X(α)1 , X

(β)2 , X

(γ)3 )

located at the center of the subcell.

∆u(αβγ) = ∆w(αβγ) + X(α)1 ∆φ(αβγ) + X

(β)2 ∆χ(αβγ) + X

(γ)3 ∆ψ(αβγ) (41)

The increment of the deformation gradient in the subcell is given according to(41) by

∆F(αβγ) =

∆φ(αβγ)1 ∆χ(αβγ)

1 ∆ψ(αβγ)1


2 ∆ψ(αβγ)2


3 ∆ψ(αβγ)3

(42)

124 J. Aboudi ZAMP

The average deformation gradient in the entire repeating cell is given by

∆F =1dhl

Nα∑α=1

Nβ∑β=1

Nγ∑γ=1

dα hβ lγ ∆F(αβγ) (43)

Similarly, let the increment of the first Piola-Kirchhoff stress tensor in thesubcell (αβγ) be denoted by ∆T(αβγ). The average increment of this stress tensoris given by

∆T =1dhl

Nα∑α=1

Nβ∑β=1

Nγ∑γ=1

dα hβ lγ ∆T(αβγ) (44)

The micromechanical analysis in the next section establishes closed-form ex-pressions for the instantaneous concentration tensor that relates the increment ofthe deformation gradient ∆F(αβγ) in the subcell (αβγ), to the average (global) in-crement of the deformation gradient ∆F. The derivation of such relations betweenlocal and global quantities is referred to as localization. The subsequent use ofthese localization relationships establishes a relationship between the increment ofthe global stress tensor ∆T and the increments of the global deformation gradient∆F.

5. Micromechanical analysis

The micromechanical analysis is based on the satisfaction of the equilibriumequations in the subcell, and the fulfillment of the continuity of displacement andtraction increments at the interfaces between the subcells in the repeating cells, andbetween neighboring cells. A complete description of the micromechanical analysisfor elastic materials subjected to finite deformation can be found in Aboudi andArnold [3]. For a detailed explanation of the two-dimensional infinitesimal casewith 2 by 2 subcells see a recent text book by Herakovich [8].

Following the analysis which was previously presented by Aboudi and Arnold[3] for finite elastic composite materials, the following localization relation can bereadily established

∆F(αβγ) = A(αβγ) ∆F + AV (αβγ)Ys (45)

Equation (45) expresses the deformation gradient increment in the subcell (αβγ)in terms of the applied average (macro) deformation gradient increment ∆F, viathe current concentration tensor A(αβγ) and the history of deformation. The lat-ter is accounted to via Ys which involves the microscopic viscoelastic terms in allsubcells. In eqn.(45), AV (αβγ) denotes the resulting viscoelastic concentration ten-sor that is operating on Ys. Both concentration tensors are given by closed-form


complicated expressions that involve the geometric dimensions of the repeatingvolume element and the properties of materials which occupy the subcells.

Substitution of eqn.(45) into relation (36), that governs the finite viscoelasticbehavior of the material which fills subcell (αβγ), provides

∆T(αβγ) = V(αβγ)[ A(αβγ) ∆F + AV (αβγ)Ys]−Y(αβγ) (46)

Consequently, in conjunction with eqn.(44), the following overall (macroscopic)nonlinear viscoelastic constitutive law that governs the average behavior of themultiphase composite which consists of nonlinearly viscoelastic phases can be es-tablished

∆T = V∗ ∆F− Y (47)

where the current effective tangent tensor, that relates the average first Piola-Kirchhoff increment to the applied average deformation gradient increment, isgiven in a closed-form manner by

V∗ =1dhl

Nα∑α=1

Nβ∑β=1

Nγ∑γ=1

dα hβ lγ V(αβγ) A(αβγ) (48)

The overall viscoelastic contribution that accounts for the history of deformationis given by

Y = − 1dhl

Nα∑α=1

Nβ∑β=1

Nγ∑γ=1

dα hβ lγ [V(αβγ)AV (αβγ) Ys −Y(αβγ)] (49)

Once V∗ and Y have been determined at the current stage of deformation,one can obtain the current average stress tensor T from the computed stress atthe previous stage T|previous according to T = T|previous + ∆T. The currentlocal F(αβγ) and average F deformation gradients can be determined in the samemanner.

The derived constitutive law, eqns.(47), that governs the overall behavior ofthe nonlinear multiphase viscoelastic composite, has the advantage that it can bereadily utilized irrespective whether symmetry exists or not, as well as withoutresorting to different boundary condition application strategies as in the case ofthe finite element unit cell procedure. Furthermore, the availability of an analyt-ical expression representing the macro response of the composite is particularlyimportant when analyzing realistic structural components, since different loadingconditions exist throughout the structure, thus necessitating the application of themacromechanical equations repeatedly at these locations.

126 J. Aboudi ZAMP

6. Computational procedure

The proposed approach for establishing the response of a nonlinearly viscoelasticmultiphase composite is based on an incremental procedure in time. At time t = 0the deformation gradient F(αβγ) in any subcell is equal to the unit matrix I, sothat its increment is equal to zero. In order to initiate the procedure a very smallvalue is assigned to F(αβγ) = I, e.g. F(αβγ) = (1 + 10−14)I (say). With thisinitial value we can determine the viscoelastic tangent tensors V(αβγ) and tensorsY(αβγ) that account for the history of deformation by employing eqns(37) and(38), respectively. With these known values of V(αβγ) and Y(αβγ) in all subcells,we proceed with the micromechanical analysis to obtain the concentration tensorsA(αβγ) and AV (αβγ). Consequently, we can use eqns(48) and (49) to establish thecurrent effective tangent tensor V∗ and the overall viscoelastic contribution thataccounts for the history of the composite deformation Y.

The applied increment of the global deformation gradient ∆F is determinedin accordance with the prescribed type of loading. For a uniaxial deformationin the 1-direction, for example, ∆F11 is known from the specific applied time-dependent loading, while all other components of ∆F are zero. Hence we canproceed by computing the resulting stress increments ∆T using eqn(47). Once ∆Fand ∆T have been determined, we can compute ∆F(αβγ) from eqn(45), F(αβγ)

from F(αβγ) = F(αβγ)|previous + ∆F(αβγ), T from T = T|previous + ∆T, and Ffrom F = F|previous + ∆F, and the incremental procedure can be continued.

Suppose, on the other hand, that a uniaxial stress loading in the 1-direction(say) is applied, so that ∆F11 is known, but all other ∆Fij are unknown. Here,On the other hand, all components of ∆Tij are zero for i + j 6= 2. In this casean iterative procedure is employed to determine the unknown components ∆Fij .This iterative procedure is necessary since the relationship between ∆Tij and ∆Fijis not linear. For sufficiently small increments of loading the iterations rapidlyconverge, yielding very small values of ∆Tij for i+ j 6= 2, and the aforementionedprocedure is continued.

7. Applications

7.1 Monolithic finite viscoelastic material

The derived micromechanical equations can be implemented to study the behaviorof a finite viscoelastic composite. To this end, consider a polyurethane matrixwhose short term strain energy function W (0) is described by Blatz and Ko [4] asis given by eqn.(10) with µ = 0.23 MPa and ν = 0.25. Let us characterize theviscoelastic behavior of this material by N = 1, δ(1) = 0.6 and λ(1) = 0.25 and 1s.


0.6 0.8 1.0 1.2 1.4

–400

–200

200

400

T(k

Pa)

11

L1

d(I)=0

0.6 0.8 1.0 1.2 1.4

–400

–200

200

400

T(k

Pa)

11

L1

d

l

(I)

(I)

=0.6

=0.25s

0.6 0.8 1.0 1.2 1.4

–400

–200

200

400

T(k

Pa)

11

L1

d

l

(I)

(I)

=0.6

=1s

Figure 2.Stress-stretch response of monolithic finite elastic and viscoelastic materials subjected to uniaxialstress loading (described by eqn.(50)) in the 1-direction.

It should be noted that from eqn.(22) this characterization implies that

δ(1) =W (1)

W (0)=W (0) −W∞

W (0)= 0.6

so that the long-time strain energy is 0.4 of the short-time one.Fig. 2 displays the response of this monolithic material when it is subjected to a

uniaxial sinusoidal stress loading in which the average stretch Λ1 in the 1-directionis given by

Λ1(t) = A sinπt

2t0(50)

where A and t0 are amplitude factor and time duration, chosen as A = 0.4 andt0 = 0.25s (it should be mentioned that for a monolithic material the averagestretch is just the imposed stretch). The above stretch function describes a loadingfollowed by unloading, followed by a stretching in the negative direction and so on.For the elastic case (labeled by δ(1) = 0) the loading and unloading portions of

128 J. Aboudi ZAMP

d

l

(I)

(I)

=0.6

=0.25s

d(I) =0

1 1.1 1.2 1.3 1.4

T(k

Pa)

11

L1

–50

0

50

100

150

200

Figure 3.Stress-stretch response of monolithic finite elastic and viscoelastic materials subjected to uniaxialstress loading (described by eqn.(51)) in the 1-direction.

d

l

(I)

(I)

=0.6

=0.25sd(I)=0

0.6 0.8 1.0 1.2 1.4

–400

–200

200

400

T(k

Pa)

11

L1

Figure 4.Average stress-stretch response of finite elastic and viscoelastic porous materials subjected to auniaxial stress loading (described by eqn.(50)) in the 1-direction. The amount of porosity is 40percent.

the response coincide as expected. For finite viscoelasticity, the response is shownfor δ(1) = 0.6 and two values of relaxation constants: λ(1) = 0.25 and 1s. Theappreciable effect of the viscoelastic mechanism is well observed, and is obviouslymore significant as the relaxation constant decreases. Of particular interest is thepermanent deformation obtained at zero stress after unloading of the material. Itshould be mentioned that for a relaxation constant that approaches zero or infinity,finite elastic behavior would be recovered.

The behavior of this finite viscoelastic material is further illustrated by Fig. 3


d

l

(I)

(I)

=0.6

=0.25sd(I)=0

0.6 0.8 1.0 1.2 1.4

–400

–200

200

400

T(k

Pa)

11

L1

Figure 5.Average stress-stretch response of finite elastic and viscoelastic porous materials subjected to auniform dilatational loading described by eqn.(50). The amount of porosity is 40 percent.

–0.4

–0.2

0 0.2 0.4

–400

–200

200

400

T –F23 23

T –F23 23

(kP

a)

d

l1

1

=0.6

=0.25s

–0.4 –0.2 0 0.2 0.4

0.8

0.7

0.6

0.9

1.

1.1

J–F23

J–F23

d

l1

1

=0.6

=0.25s

(a) (b)

Figure 6.(a) Stress-shear deformation gradient response of a finite viscoelastic monolithic material, andthe corresponding average stress-shear deformation gradient response of the porous materialsubjected to a shear loading described by eqn.(52). (b) Dilatation-shear deformation gradientresponse of a finite viscoelastic monolithic material, and the corresponding average dilatational-shear deformation gradient response of the porous material subjected to a shear loading describedby eqn.(52). The amount of porosity is 40 percent.

which exhibits its response when it is subjected to increasing-decreasing positiveuniaxial stress loading which is characterized by

Λ1(t) = A sin2 πt

2t0(51)

with A = 0.4 and t0 = 0.25s, δ(1) = 0.6 and λ(1) = 0.25. The converging responseas the number of cycles increases is well observed.

130 J. Aboudi ZAMP

–0.4

–0.2

0 0.2 0.4

–200

–100

100

200

T(k

Pa)

23

F23

d

l

(1)

(1)=0.6

=0.25s

d

l

(1)

(1)=0.6

=0.25s

d(1)

=0d

(1)=0

–0.4 –0.2 0 0.2 0.4

0.8

0.7

0.6

0.9

1.

1.1

F23

J

(a) (b)

Figure 7.(a) Average stress-shear deformation gradient response of finite viscoelastic and elastic porousmaterials subjected to a shear loading described by eqn.(52). (b) Average dilatation-shear de-formation gradient response of a finite viscoelastic and elastic porous materials subjected to ashear loading described by eqn.(52). The amount of porosity is 40 percent.

d(I)=0

0.96 0.98 1 1.02 1.04

T(G

Pa)

11

L1

–15

–10

–5

5

10

15

d

l

(I)

(I)

=0.6

=0.25s

0.96 0.98 1 1.02 1.04

T(G

Pa)

11

L1

–15

–10

–5

5

10

15

Figure 8.Stress-stretch response of the aluminum 8091 alloy subjected to uniaxial stretch loading (de-scribed by eqn.(50)) in the 1-direction.

7.2 Finite viscoelastic porous material

So far the micromechanical model has not activated as we considered monolithicmaterials only. Let us consider a porous material in which the solid matrix is iden-tical with aforementioned finite viscoelastic material (whose response was shownin Figs. 2 and 3). In Figs. 4 and 5 the response to uniaxial stress in the 1-directionand uniform dilatational loading, respectively, of the porous material when it issubjected to cyclic pulse (50) are shown for finite strain elastic and viscoelastic


d(I)=0

0.96 0.98 1 1.02 1.04

T(G

Pa)

11

L1

–15

–10

–5

5

10

15

d

l

(I)

(I)

=0.6

=0.25s

0.96 0.98 1 1.02 1.04

T(G

Pa)

11

L1

–15

–10

–5

5

10

15

Figure 9.Average stress-stretch response of the SiC/Al composite system subjected to uniaxial stretchloading (described by eqn.(50)) in the axial 1-direction (parallel to the fibers). The fiber volumeratio is 40 percent.

d(I)=0

0.96 0.98 1 1.02 1.04

T(G

Pa)

22

L2

–15

–10

–5

5

10

15

d

l

(I)

(I)

=0.6

=0.25s

0.96 0.98 1 1.02 1.04

T(G

Pa)

22

L2

–15

–10

–5

5

10

15

Figure 10.Average stress-stretch response of the SiC/Al composite system subjected to uniaxial stretchloading (described by eqn.(50)) in the transverse 2-direction (perpendicular to the fibers). Thefiber volume ratio is 40 percent.

matrices. The effect of porosity and viscosity are well exhibited in these figures.It is also interesting to study the behavior of the porous material when it is

subjected to shear loading. To this end consider the following type of applied sheardeformation gradient

F23(t) = F32(t) = A sinπt

2t0(52)

while keeping all stress components in the other directions to be zero. Fig. 6(a)exhibits the effects of porosity for this type of loading with A = 0.4 and t0 =0.25s. This figure displays the response (i.e. T23 vs F23) of the solid monolithic

132 J. Aboudi ZAMP

viscoelastic material and its response (i.e. T23 vs F23) when 40 percent porosityis assumed to exist.

Due to the nonlinearity of the material Kelvin effect is obtained according towhich an applied shear generates dilatational effect. This effect is shown in Fig.6(b) where the average dilatation

J(t) = Λ1(t)Λ2(t)Λ3(t) (53)

is shown for the monolithic viscoelastic (i.e. J vs F23) and porous materials (i.e.J vs F23) subjected to shear loading (52).

By contrasting the response of the viscoelastic porous material with the corre-sponding elastic (i.e. δ(1) = 0) porous material, the effect of viscoelasticity can bedetected. Such a comparison is given by Fig. 7 where the response of the material,and the resulting dilatational behavior are shown for the shear loading given by(52).

7.3 Finite viscoelastic continuous reinforced composite

Next consider an aluminum alloy 8091 reinforced by SiC long fibers. Both mate-rials are modeled according to Chen and Jiang [5] by Murnaghan’s strain energyrepresentation (see eqn.(11)). The material constants of the phases are given inTable 1 (Chen and Jiang [5]). In the present investigation the SiC fiber is assumedto be nonlinearly elastic, whereas the aluminum alloy is taken as nonlinearly vis-coelastic with the associated viscoelastic parameters δ(1) = 0.6 and λ(1) = 0.25s.

Table 1. Material constants for SiC/Al 8091 system

Constituent λ(GPa) µ(GPa) l(GPa) m(GPa) n(GPa)

SiC 97.66 188 -82.1 -310 -683Al 8091 44.93 31 -218 -378 -435

In Fig. 8 the response of the monolithic aluminum alloy is shown under uniaxialstretching (i.e., Λ1 is given by eqn.(50), whereas Λ2 = Λ3 = 1). The figure exhibitsthe response of the material assuming both finite elastic and viscoelastic behavior.It can be seen that the hysteretic response of the viscoelastic material and theaforementioned applied loading converges rapidly after the first several few cycles.

Next consider the continuous fiber reinforced system SiC/Al with a fiber vol-ume fraction of 0.4. Fig. 9 presents the composite behavior for uniaxial stretchloading (50) in the fibers direction (i.e. in the 1-direction). In order to illustratethe viscous effect of the aluminum alloy matrix, both finite elastic and finite vis-coelastic behaviors are shown in figure. Similarly, Fig. 10 exhibits the responseof the composite to transverse loading (i.e. uniaxial stretching in the 2-direction).Both figures clearly show the nonlinearity and directionally effects of the compos-ite. It should be noted that under tensile stretching the composite system exhibits


at certain stage of loading an instability in the sense that the stress-deformationcurve starts to decrease. Consequently, the computed responses in Fig. 10 havebeen stopped at this stage of loading.

8. Conclusions

By incorporating Simo [14] finite viscoelastic model with the micromechanicalmethod, referred to as GMC, a convenient and powerful approach for the predic-tion of the overall response of finite viscoelastic multiphase composites with shortfibers (or inclusions) has been established. The predicted response is based ofthe properties of the individual constituents which, in general, can be assumedto behave as finite viscoelastic materials. The single phase is characterized bythe short-term strain energy function (e.g. Blatz and Ko [4], Murnaghan [12],compressible Mooney-Rivlin material description (Sussman and Bathe [15]) or Og-den material description (Ogden [13]), relaxation times and weight factors. Thederivation is based on the tangential formulation in conjunction with the history ofdeformation, and provides closed-form expressions for the concentration matrix,the effective tangent tensor of the composite, and the overall contribution thataccounts for the history of deformation of the entire composite. The correspond-ing overall response of finite elastic composites is obtained as a special case byneglecting the viscoelastic effects. The derived multiaxial constitutive equation ofthe composite can be readily employed to analyze finite viscoelastic structures.

Acknowledgment

The author gratefully acknowledges the support of the Diane and Arthur Belferchair of Mechanics and Biomechanics.

References

[1] Aboudi, J., Micromechanical analysis of thermoinelastic multiphase short-fiber composites.Composites Eng. 5 (1995), 839-850.

[2] Aboudi, J., Micromechanical analysis of composites by the method of cells - Update. Appl.Mech. Rev. 49 (1996), 83-91.

[3] Aboudi, J. and Arnold, S.M., Micromechanical Modeling of the finite deformation of ther-moelastic multiphase composites. NASA Technical Memorandum 107531, 1997. To appearalso in Math. & Mech. of Solids (1999).

[4] Blatz, P. J. and Ko, W. L., Application of finite elastic theory to the deformation of rubberymaterials. Trans. Soc. Rheology 6 (1962), 223-251.

[5] Chen, Y. C. and Jiang, X., Nonlinear elastic properties of particulate composites. J. Mech.Phys. Solids 41 (1993), 1177-1190.

134 J. Aboudi ZAMP

[6] Christensen R. M., Theory of Viscoelasticity . Academic Press, New York 1982.[7] Drozdov, A. D., Finite Elasticity and Viscoelasticity , World Scientific, Singapore 1996[8] Herakovich, C. T., Mechanics of Fibrous Composites. Wiley, New York 1998.[9] Kaliske, M. and Rothert, H., Formulation and implementation of three-dimensional vis-

coelasticity. Computational Mech. 19 (1997), 228-239.[10] Lockett, F. J., Nonlinear Viscoelastic Solids. Academic Press, London 1972.[11] MARC Analysis Research Corporation, volume A, Revision K6, 1994.[12] Murnaghan, F. D., Finite Deformation of an Elastic Solid . Dover, New York 1967.[13] Ogden, R. W., Non-Linear Elastic Deformations. Ellis Horwood, Chichester 1984.[14] Simo, J. C., On a fully three-dimensional finite-strain viscoelastic damage model: refor-

mulation and computational aspects. Computer Meth. Appl. Mech. & Eng. 60 (1987),153-173.

[15] Sussman, T. and Bathe, K. J., A finite element formulation for nonlinear incompressibleelastic and inelastic analysis. Computers & Structures 26 (1987), 357-409.

Jacob AboudiDepartment of Solid Mechanics, Materials & StructuresFaculty of EngineeringTel-Aviv UniversityRamat-Aviv 69978, Israel(e-mail: [email protected])

(Received: June 15, 1998; revised: December 1, 1998)

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Micromechanical modeling of finite viscoelastic multiphase composites