A thermodynamics based damage mechanics model for

International Journal of Solids and Structures 44 (2007) 1099–1114

www.elsevier.com/locate/ijsolstr

A thermodynamics based damage mechanics modelfor particulate composites

Cemal Basaran *, Shihua Nie

Department of Civil, Structural and Environmental Engineering, University at Buffalo, SUNY, Buffalo, NY 14260, United States

Received 14 June 2005; received in revised form 10 February 2006Available online 10 June 2006

Abstract

A micro-mechanical damage model is proposed to predict the overall viscoplastic behavior and damage evolution in aparticle filled polymer matrix composite. Particulate composite consists of polymer matrix, particle fillers, and an interfa-cial transition interphase around the filler particles. Yet the composite is treated as a two distinct phase material, namelythe matrix and the equivalent particle-interface assembly. The CTE mismatch between the matrix and the filler particles isintroduced into the model. A damage evolution function based on irreversible thermodynamics is also introduced into theconstitutive model to describe the degradation of the composite. The efficient general return-mapping algorithm isexploited to implement the proposed unified damage coupled viscoplastic model into finite element formulation. Further-more, the model predictions for uniaxial loading conditions are compared with the experimental data.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

The application and estimation of the effective mechanical properties of random heterogeneous multiphasematerials are of great interest to researchers and engineers in many science and engineering disciplines. Thereare many different methods and tools that can be used to deliver the macroscopic constitutive response of het-erogeneous materials using a local description of the micro-structural behavior (Ju and Lee, 2000, 2001; Juand Sun, 2001; Ju, 1990a,b; Ju and Tseng, 1997; Ju and Chen, 1994a,b; Wong and Ait-Kadi, 1997; Willis,1991; Voyiadjis and Thiarajan, 1996; Sun and Ju, 2001; Ravichandran and Liu, 1995; Kwon and Liu,1998; Ahmed and Jones, 1990; Achenbach and Zhu, 1989. In the development of the homogenization proce-dures for heterogeneous materials, we have to define both the homogenization step itself (from local variablesto overall ones) and the often more complicated localization step (from overall controlled quantities to the cor-responding local ones).

The nature of the bond between particles and the matrix material has a significant effect on the mechanicalbehavior of particulate composites. Most analytical and numerical models assume that the bond between the

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doi:10.1016/j.ijsolstr.2006.06.001

* Corresponding author. Tel.: +1 716 645 2114; fax: +1 716 645 3733.E-mail address: [email protected] (C. Basaran).

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1100 C. Basaran, S. Nie / International Journal of Solids and Structures 44 (2007) 1099–1114

filler and matrix is perfect and can be modeled using the continuity of tractions and displacements across adiscrete interface. However, internal defects and imperfect interfaces are well known to exist in composites.The incorporation of such phenomena into the general theory requires modification and relaxation of the con-tinuity of displacements between the constituents (Nie and Basaran, 2005). The imperfect interface bond maybe due to the compliant interfacial layer known as interphase or interface damage, which may have been cre-ated deliberately by coating the particles. It may also develop during the manufacturing process due to chem-ical reactions between the contacting particles and the matrix material or due to interface damage from cycleloading. Moreover, the strength of the bond at the interface controls the mechanical response and fatigue lifeof the composite (Basaran et al., 2006). By controlling the stress–strain response of the interphase, it is possibleto control the overall behavior of the composite.

Based on the generalizations of the Eshelby method (Eshelby, 1957), a novel micro-mechanical frameworkhas been proposed by Ju and Chen (1994a,b) to investigate the effective mechanical properties of elastic mul-tiphase composites containing many randomly dispersed ellipsoidal inhomogeneities. Within the context ofthe representative volume element (REV), three governing micro-mechanical ensemble-volume averaged fieldequations are presented to relate ensemble-volume averaged stresses, strains, volume fractions, eigenstrains,particle shapes and orientations, as well as elastic properties of constituent phases of linear elastic particulatecomposites. Then, various micro-mechanical models have been developed based on the ensemble-volume aver-aged constitutive equations. For example, in Ju and Tseng (1996), a formulation combining a micro-mechan-ical interaction approach and the continuum plasticity is proposed to predict effective elastoplastic behavior ofa two-phase particulate composite containing many randomly dispersed elastic spherical inhomogeneities.Explicit pairwise inter-particle interactions are considered in both the elastic and plastic responses. Further-more, the ensemble-volume averaging procedure is employed and the formulation is of complete second order.

Constituents of particle filled composites often have very different properties such as the coefficient of ther-mal expansion (CTE), ductility, and elastic modulus. It is well known that within the particulate compositemicrostructure there exists a micro-stress associated with the CTE mismatch between the matrix and the fillerparticles. For a composite made up of lightly cross-linked poly-methyl methacrylate (PMMA) matrix filledwith alumina trihydrate (ATH) particles, these micro-stresses can be imaged using the fact that PMMA is opti-cally birefringent under stresses. The micro-stresses imaged using this technique are indeed due to CTE mis-match because they dissipate as the temperature gets close to the Tg of the PMMA (as shown in Fig. 1). Thesestresses can be quantified by calculation or by direct measurement and are of the order between 15% and 75%of the tensile strength of the composite. Therefore the thermal stress associated with the CTE mismatchbetween the matrix and particles is an important factor for the failure of particulate composites subjectedto the thermo-mechanical loads. However, the original micro-mechanical models of Ju and Chen (1994a,b),Ju and Tseng (1996), do not account for the effects of the CTE mismatch between the filler particles andthe matrix. Lee (1992) analyzed metal matrix composite for strains induced by CTE mismatch. He reportedthat spherical reinforcement particle to be in hydrostatic stress state and remains in the elastic state. Yet,

Fig. 1. Stress fringes when cooling the particle filled composite.

C. Basaran, S. Nie / International Journal of Solids and Structures 44 (2007) 1099–1114 1101

Lee (1992) reported that stresses and strains are the largest and plastic deformation occurs in the matrix adja-cent to the reinforcement particle. Accordingly, he reported, the reinforcement particle/matrix interfacebecomes a potential crack initiation site. Taya (1991) reported strengthening mechanisms of metal matrixcomposites from macro- and nanolevel models. Taya (1991) observed that in both macro- and nanolevel mod-els the residual stresses induced due to CTE mismatch strain, between the filler and the matrix, play an impor-tant role in influencing the yield stress of the composite.

Chatuvedi and Shen (1998) proposed a micro-mechanical characterization for thermal expansion of particlefilled plastic encapsulant material where they accounted for different CTE values of the filler and the matrix.Yet their model is based on theory of elasticity therefore viscoplastic damage modeling is not possible. Chenet al. (1982) studied effect of residual thermal stresses due to CTE mismatch on thermal conductivity of W–Cucomposites, experimentally. They reported that there are significant thermal stresses at the interface due toCTE mismatch. Liu et al. (2005) conducted finite element simulation of the thermal properties of particulateand continuous network-reinforced metal–matrix composites. They calculated effective CTE using FEM dis-cretization. Agrawal et al. (2003) measured thermal residual stresses in two types of co-continuous composites.They used FEM to simulate the state of stress at the interfaces, where CTE mismatch led to tensile thermalstresses at the metal–ceramic interfaces. Kitazono et al. (2001) proposed micro-mechanics models for CTE-mismatch super-plasticity for metal matrix composites and monolithic poly-crystals. Vo et al. (2001) presenteda novel model for predicting the effective CTE of particle filled epoxy-polymer (underfill) composites by con-sidering the effect of an interphase zone surrounding the filler particles in polymer matrix. Xie et al. (2001)measured the CTE values of TiC particulate reinforced ZA43 composites. They reported that thermalstresses developed as a result of CTE mismatch between the reinforcement and the matrix. Elemori et al.(1998) reported observed thermal stresses as a result of CTE mismatch between the reinforcement and thematrix.

It is well known that the interfacial bond properties between the matrix and particles have a significanteffect on the behavior of the composite. In order to directly use the micro-mechanical field equations (Mura,1987) and also to consider the CTE mismatch and interfacial bond properties between the matrix and fillerparticles, the composite is treated as two distinct phases: namely the bulk matrix and the equivalent parti-cle-interface-region assembly. In the proposed model, the interface region is given a thickness and effectiveelastic properties of its own. In this paper, we expand Ju and Chen’s micro-mechanical model to incorporatethe effects of CTE mismatch between the matrix and the filler particles.

2. Micro-mechanical field equations

In order to obtain effective constitutive equations and properties of random heterogeneous composites, onetypically performs the ensemble-volume averaging process within a mesoscopic representative volume element(RVE). The volume-averaged stress tensor is defined as

�r ¼ 1

V

ZV

rðxÞdx ¼ 1

V

ZV 0

rðxÞdxþXn

q¼1

ZV q

rðxÞdx

" #ð1Þ

where V is the volume of a RVE, V0 is the volume of the matrix, Vq is the volume of the qth-phase particles,and n denotes the number of particulate phases of different material properties (excluding the matrix).

Similarly, the volume-averaged strain tensor is defined as follows:

�e ¼ 1

V

ZV

eðxÞdx ¼ 1

V

ZV 0

eðxÞdxþXn

q¼1

ZV q

eðxÞdx

" #� 1

VV 0�e0 þ

Xn

q¼1

V q�eq

" #ð2Þ

According to Eshelby’s equivalence principle, the perturbed strain field e 0(x) induced by inhomogeneitiescan be related to specified eigenstrain e*(x) by replacing the inhomogeneities with the matrix material. Theinhomogeneities may also involve their own eigenstrain caused by phase transformation, precipitation, plasticdeformation, or CTE mismatch between different constituents of the composites. However, it is not essentialto attribute the eigenstrain to any specific source. Then the stress is the sum of the two parts, one caused by theinhomogeneity, and the other by the eigenstress associated with eigenstrain. For the domain of the qth-phase


particles with an elastic stiffness tensor Cq, Mura (1987) and Mura and Tanaka (1973) proposed the followingstress relation:

Cq : e0 þ e0ðxÞ � eTq ðxÞ

h i¼ C0 : e0 þ e0ðxÞ � eT

q ðxÞ � e�qðxÞh i

ð3Þ

where C0 is the stiffness tensor of the matrix and e0 is the uniform elastic strain field induced by far-field loadsfor a homogeneous matrix material only. eT

q is its own eigenstrain associated with the qth particle. e�q is thefictitious equivalent eigenstrain by replacing the qth particle with the matrix material. e 0(x) is the perturbedstrain due to distributed eigenstrain eT

q and e�q associated with all particles in the RVE.The strain at any point within an RVE is decomposed into two parts, the uniform strain and the perturbed

strain due to the distributed eigenstrain. It is emphasized that the eigenstrains e* and eT are non-zero in theparticle domain and zero in the matrix domain, respectively. In particular, in accordance with the Eshelbyprinciple, the perturbed strain field induced by all the distributed eigenstrains e* and eT can be expressed as(Mura, 1987)

e0ðxÞ ¼Z

VGðx� x0Þ : ½eT ðx0Þ þ e�ðx0Þ�dx0 ð4Þ

where x, x 0 2 V, and G is the Green’s function in a linear elastic homogeneous matrix.From Eqs. (3) and (4), we arrive at

�Aq : e�qðxÞ ¼ e0 � eTq ðxÞ þ

ZV

Gðx� x0Þ : ½eT ðx0Þ þ e�ðx0Þ�dx0; x0 2 V ð5Þ

where Aq is given by

Aq ¼ ðCq � C0Þ�1 � C0 ð6Þ
Using the renormalization procedure proposed by (Ju and Chen, 1994a), for spherical inclusions, the three
basic governing micro-mechanical ensemble-volume averaged field equations can be derived as follows:

�r ¼ C0 : �e�Xn

q¼1

/q �e�q þ �eTq

� �" #ð7Þ

�e ¼ e0 þXn

q¼1

/qS : �eTq þ �e�q

� �ð8Þ

� ðAq þ SÞ : �e�q ¼ e0 � eT þ S : �eTq þ �e0pq ð9Þ

where S is the Eshelby tensor, and �e0pq represents the inter-particle interaction effects. In order to actually solveEqs. (7)–(9) and to obtain effective elastic properties of composites, it is essential to express the qth-phase aver-age eigenstrain �e�q in terms of the average strain �e. Namely, one has to solve the integral Eq. (5) exactly for eachphase, which requires details of the random microstructure.

3. Effective thermo-mechanical properties of composite sphere assemblage

In many particulate composites, a thin layer of some other elastic phase intervenes between a particle andthe matrix. The imperfect interface bond may be due to the very compliant thin interfacial layer that isassumed to have perfect boundary conditions with the matrix and the particle. This defines a three-phase com-posite that includes particles, thin interphase, and the matrix as shown in Fig. 2. Once the effective mechanicaland thermal properties of the inner composite sphere assemblage (CSA), which consists of the particle andinterphase layer of thickness d, are found, the composite models used for the perfect interface compositemodel can be readily applied to this case of imperfect interface conditions.

The composite spheres assemblage (CSA) model was first proposed by Kerner (1956) and Van der Poel(1958) as shown in Fig. 3. Smith (1974, 1975), Christensen and Lo (1979) and Hashin and his co-workers(1962, 1963, 1968, 1990, and 1991a,b) eventually improved on the CSA model. The CSA assumes that the par-ticles are spherical and, moreover, that the action on the particle is transmitted through a spherical interphase

Fig. 3. Schematic illustration of composite spherical assemblage (CSA).

Fig. 2. Three-phase composite system.


shell. The overall macro-behavior is assumed to be isotropic and is thus characterized by two effective moduli:the bulk modulus k and the shear modulus l. In this section, I summarized the theoretical solution for effectivethermo-mechanical properties of the CSA consisting of an elastic spherical particle and an elastic interphaselayer with a thickness of d. In the following formulae, k represents the bulk modulus, l represents shearmodulus, a represents the coefficient of thermal expansion. The subscripts i, f and m refer to the interphase,filler and matrix, respectively.

3.1. Effective bulk modulus

The effective bulk modulus k1 for the CSA as obtained by Hashin (1962) is given as

k1 ¼ ki þ/

1kf�kiþ 3ð1�/Þ

3kiþ4li

ð10Þ

where

/ ¼ rr þ d

� �3

ð11Þ

r is the radius of the filler particle and d is the thickness of interphase.

3.2. Effective coefficient of thermal expansion (ECTE)

The effective coefficient of thermal expansion a1 for the CSA as obtained by Levin (1967) is given as

a1 ¼ ai þaf � ai

1kf� 1

ki

1

k1

� 1

ki

� �ð12Þ

3.3. Effective shear modulus

Based on the generalized self-consistent scheme (GSCS) model proposed by (Hashin, 1962), Christensenand Lo (1979) have given the condition for determining the effective shear modulus of CSA as follows:


Al1

li

� �2

þ Bl1

li

� �þ D ¼ 0 ð13Þ

where

A ¼ 8½lf=li � 1�ð4� 5viÞg1/10=3 � 2½63ðlf=li � 1Þg2 þ 2g1g3�/7=3

þ 252½lf=li � 1�g2/5=3 � 50½lf=li � 1� 7� 12vi þ 8v2

i

� �g2/þ 4ð7� 10viÞg2g3 ð14Þ

B ¼ �4½lf=li � 1�ð1� 5viÞg1/10=3 þ 4½63ðlf=li � 1Þg2 þ 2g1g3�/7=3

� 504½lf=li � 1�g2/5=3 þ 150½lf=li � 1�ð3� viÞvig2/þ 3ð15vi � 7Þg2g3 ð15Þ

D ¼ 4½lf=li � 1�ð5vi � 7Þg1/10=3 � 2½63ðlf=li � 1Þg2 þ 2g1g3�/7=3

þ 252½lf=li � 1�g2/5=3 þ 25½lf=li � 1� v2

i � 7� �

g2/� ð7þ 5viÞg2g3 ð16Þ
with
g1 ¼ ½lf=li � 1�ð49� 50vf viÞ þ 35ðlf=liÞðvf � 2viÞ þ 35ð2vf � viÞ ð17Þg2 ¼ 5vf ½lf=li � 8� þ 7ðlf=li þ 4Þ ð18Þg3 ¼ ðlf=liÞð8� 10viÞ þ ð7� 5viÞ ð19Þ

/ is given in Eq. (11).

4. Viscoplastic behavior of two-phase composites

Let us consider a perfectly bonded two-phase composite consisting of a viscoplastic matrix (phase 0) withan elastic bulk modulus k0 and elastic shear modulus l0, and randomly dispersed elastic spherical particle-interphase assembly (phase 1) with effective bulk modulus k1 and effective shear modulus l1. The effective elas-tic moduli of the two-phase composite by neglecting the inter-particle interaction effects has been derived by Juand Chen (1994a) using the field equations given above without the CTE mismatch between the filler particlesand the matrix:

k ¼ k0 1þ 3ð1� v0Þðk1 � k0Þ/3ð1� v0Þk0 þ ð1� /Þð1þ v0Þðk1 � k0Þ

� ð20Þ

l ¼ l0 1þ 15ð1� v0Þðl1 � l0Þ/15ð1� v0Þl0 þ ð1� /Þð8� 10v0Þðl1 � l0Þ

� ð21Þ

where / is the particle volume fraction, v0 is the Poisson’s ratio of the matrix.For simplicity, the Von Mises yield criterion is assumed for the matrix. Accordingly, at any matrix material

point, the stress r and the equivalent plastic strain a must satisfy the following yield criterion:

F ðr; aÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir : Id : r

p�

ffiffiffi2

3

rryðaÞ ð22Þ

where ry(a) is the isotropic hardening function of the matrix-only material, and Id denotes the deviatoric partof the fourth rank identity tensor.

In order to solve for the elastoplastic response exactly, the stress at any local point has to be determined andused to determine the plastic response through the local yield criterion. This approach is generally infeasibledue to the complexity of accounting for all possible configurations statistically and micro-structurally. There-fore, a framework in which an ensemble averaged yield criterion is constructed for the entire composite is usedinstead (Ju and Tseng, 1996). The averaged current stress norm in a matrix point x can be derived as

hHimðxÞ ¼ ð�r� �rT Þ : T : ð�r� �rT Þ ð23ÞrT ¼ AC1 : eT ð24Þ

where eT is the uniform eigenstrain associated with CTE mismatch between matrix and particles, and the com-ponents of T are given as

T ijkl ¼ T 1dijdkl þ T 2 dikdjl þ dildjk

� �ð25Þ


with

3T 1 þ 2T 2 ¼200/ð1� 2v0Þ2

ð3aþ 2bÞ2ða/þ 1Þ2ð26Þ

T 2 ¼12þ ð23� 50v0 þ 35v2

0Þ/b2

ðb/þ 1Þ2ð27Þ

a ¼ 2ð5v0 � 1Þ þ 10ð1� v0Þk0

k1 � k0

� l0

l1 � l0

� �ð28Þ

b ¼ 2ð4� 5v0Þ þ 15ð1� v0Þl0

l1 � l0

ð29Þ

a ¼ 20ð1� 2v0Þ3aþ 2b

ð30Þ

b ¼ ð7� 5v0Þb

ð31Þ

In this work it is assumed that viscoplastic yielding and flow occur only in the matrix, and the matrix solelydetermines the viscoplastic behavior of the composite. The magnitude of the current equivalent stress normof the matrix can be utilized to determine the possible viscoplastic behavior in the composite. When theensemble-volume averaged current stress norm in the matrix reaches a certain level, the composite undergoesviscoplastic flow. The effective yield function for the composite in the presence of CTE mismatch inducedstresses can be written as

f ð�r; �aÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�r� �rT Þ : T : ð�r� �rT Þ

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT 1 þ 2T 2

p�ryð�aÞ ð32Þ

where �ryð�aÞ is the isotropic hardening function of the composite materials, �a is the equivalent viscoplasticstrain that defines isotropic hardening of the yield surface of the composites, and

_�a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT 1 þ 2T 2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_�evp : T�1 : _�evp

pð33Þ

The factors in the effective yield and effective plastic strain increment equations are chosen so that the effec-tive stress and effective plastic strain increments are equal to the uniaxial stress and uniaxial plastic strainincrement in a tensile test. It should be noted that the effective yield function is pressure dependent now (asa result of accounting for CTE mismatch) and not of the Von-Mises type any more. Therefore, the particleshave significant effect on the viscoplastic behavior of the matrix.

In order to simulate damage evolution behavior of composite materials, there is a need for introduction of adamage parameter in the above proposed constitutive model. Damage mechanics provides us a basic frame-work to develop damage evolution models (Lemaitre, 1996). According to the strain equivalence principle, theeffective damage coupled yield function for an isotropic damage parameter D can be written as follows:

f ð�r; �aÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�r

1� D� �rT

� �: T :

�r

1� D� �rT

� �s�


p�ryð�aÞ ð34Þ

where D is the damage parameter. It is obvious that the damage increases the equivalent stress norm of thecomposite, which tends to amplify the viscoplastic flow of the composite. For simplicity, the Perzyna-typeviscoplasticity model is employed to characterize the rate dependency (viscosity) behavior of the matrix. Itis assumed that viscoplastic flow that takes place in the matrix follows the Perzyna model, where creep strainrate is proportional to over-stress and viscosity coefficient of the material. Therefore, the effective ensemble-volume averaged plastic strain rate for the composite can be expressed as

_evp ¼ cofor¼ 1

1� Dcn ð35Þ


where

n ¼T �r

1�D� �rT� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�r

1�D� �rT� �

: T : �r1�D� �rT� �q ð36Þ

and c denotes the plastic consistency parameter

c ¼ hf ig¼ hf i

2lsð37Þ

gis viscosity coefficient, s is relaxation time. And the equivalent plastic strain is defined as

_a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT 1 þ 2T 2

p c1� D

ð38Þ

5. Damage evolution function

In solids, the entropy production is a non-negative intrinsic quantity based on irreversible thermody-namics and thus can serve as a basis for the systematic description of the irreversible processes occurringin a solid. Isotropic damage evolution function based on entropy production was proposed and experi-mentally validated by Basaran and Yan (1998), Basaran and Tang (2002), Tang and Basaran (2003),Basaran et al. (2003), Basaran and Nie (2004) and Gomez and Basaran (2005). The damage evolutionfunction is given by

D ¼ Dcr 1� e�msR Ds

� �ð39Þ

where Dcr is a proportionally critical damage coefficient, R is the gas constant, and Ds is the internal entropyproduction and can be calculated as follows:

Ds ¼Z t

t0

�r : _�evp

Tqdt þ

Z t0

t

k

T 2qjgradT j2

� �dt þ

Z t

t0

rT

dt ð40Þ

where q is the density, T is temperature, k is the thermal conductivity of composite, r is the distributed internalheat production rate.

6. Computational integration algorithms

In this section, we employ the general return-mapping algorithm to solve for the unified damaged coupledviscoplastic corrector problem proposed above. The general return-mapping algorithm was proposed by Simoand Taylor (1985) for rate independent elastoplasticity and summarized in details by Simo and Hughes (1998).In order to minimize the confusion, the symbol D is used to denote an increment over a time step, or an incre-ment between successive iterations. For the rate-of-slip c, we adhere to the conventions: Dc = cDt denotes theincrement of c over a time step, and D2c denotes the increment of Dc between iterations.

Let C be the elastic stiffness matrix of the composite, then Hooke’s law yields

rnþ1 ¼ ð1� DÞC : enþ1 � evpnþ1

� �ð41Þ

Since en+1 is fixed during the return-mapping stage, it follows that

Devp ¼ � 1

1� DC�1Dr ð42Þ

From Eqs. (39) and (40), we have

DD ¼ � 1

1� DDcrms

TqRexp �ms

RDs

� �rC�1Dr ð43Þ


From Eqs. (35) and (38) we also have

evpnþ1 ¼ evp

n þ1

1� DDcn ð44Þ

anþ1 ¼ an þ1

1� D


pDc ð45Þ

Let us define the residual functions as follows:

Rc ¼2lsDc

Dt� hf i ð46Þ

Re ¼ �evpnþ1 þ evp

n þ1

1� DDcn ð47Þ

Ra ¼ �anþ1 þ an þ1

1� D


pDc ð48Þ

If we linearize these residual functions and solve them, we obtain

D2cnþ1 ¼�ð1� DÞðRc þ


pKRaÞ � n0 : E : Re

ðT 1 þ 2T 2ÞK þ 2ð1�DÞlsDt þ 1

1�D n0 : E : nð49Þ

D�rnþ1 ¼ �E Re þD2cnþ1

1� Dn

� �ð50Þ

D�evpnþ1 ¼ �

1

1� DC�1D�rnþ1 ð51Þ

D2snþ1 ¼�rnþ1D�enþ1

Tqð52Þ

DDnþ1 ¼Dcrms

TqRexp �ms

RDsnþ1

� �ð�rnþ1D�enþ1Þ ð53Þ

D�anþ1 ¼ Ra þ1

1� D


pD2cnþ1 þ

1

ð1� DÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT 1 þ 2T 2

pDcnþ1DDnþ1 ð54Þ

where

E�1 ¼ 1

1� DC�1 þ DcN

1� D� DcrmsDc

ð1� DÞ2T qRexp �ms

RDs

� �nþ 1

1� DN�r

� �� ð�rC�1Þ

" #

N ¼ T� n� nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�r

1�D� �rT� �

: T : �r1�D� �rT� �q

n0 ¼ n� Dcrms

ð1� DÞ2TqRexp �ms

RDs

� �n�r� ðT 1 þ 2T 2ÞKDc� �

ð�rC�1Þ

K ¼ o�ry

o�a is the isotropic hardening modulus.

We then update the viscoplastic strain, consistency parameter, stress, damage parameter and entropyproduction and effective equivalent plastic strain at each iteration and iterate until the convergence tolerancesare achieved. An important advantage of this algorithm lies in the fact that it can be exactly linearized inclosed form. This leads to the notion of consistent elastoplastic tangent moduli, which is derived for thecomposite as

C� ¼ E�

� E�Wn� n�E�

2lð1�DÞ2 sDtþðT 1þ 2T 2Þð1�DÞK þ n� : E�W : n� Dcrms

ð1�DÞT qR exp �ms

R Ds� �

½nr�ðT 1þ 2T 2ÞKDc�rWn

ð55Þ


where

n� ¼ nþ Dcrms

ð1� DÞ3TqRexp �ms

RDsnþ1

� �½nr� ðT 1 þ 2T 2ÞKDcnþ1�ðr : WNÞ

E��1 ¼ 1

1� DC�1 þ Dcnþ1

1� DWN

�

W�1 ¼ 1� DcrmsDcnþ1

ð1� DÞ2T qRexp �ms

RDsnþ1

� �nþ 1

1� DN�r

� �� r

7. Numerical simulations and experimental comparison

The proposed unified damage coupled micro-mechanical model is implemented into ABAQUS throughuser defined subroutine UMAT. The influences of interfacial bond and CTE mismatch (between the matrixand the filler particles) are studied under uniaxial loading conditions. The finite element simulation resultsare compared with the corresponding strain-controlled monotonic tensile tests on particulate composite pre-pared using lightly cross-linked PMMA filled with ATH particles. Details of this testing procedure is given inBasaran et al. (submitted for publication).

The geometry of the dog boned shape specimen prepared according to ASTM D 638-98 used in the uniaxialtests is shown in Fig. 4. Owing to the symmetry, only one half of the gauge length was used for finite elementsimulation.

The Ramberg–Osgood plastic model was used to model the monotonic tensile plastic response, which isreasonably accurate for tensile behavior:

�ry ¼ �r0 þ Kan ð56Þ
where the parameters are determined from the experimental data as
�r0 ¼ �0:1791T þ 19:6; T 6 100 �C

K ¼ �8:5881 10�4T 3 þ 9:1312 10�2T 2 � 4:8155T þ 424:9; T 6 100 �C

n ¼ 4:098 10�7T 3 � 4:476 10�5T 2 þ 2:33 10�3T þ 0:3542; T 6 100 �C

Viscosity relaxation time s and the coefficient of thermal expansion (CTE) of the matrix are determinedfrom the test data. They are as follows:

s ¼ 1:12406 10�6T 3 � 1:67823 10�4T 2 þ 7:91134 10�3T þ 7:35 10�3; T 6 90 �C

�2:6348 10�6T 4 � 1:08452 10�3T 3 � 1:64629 10�1T 2 þ 10:9673T � 271:11; T P 90 �C

(

Cte ¼ 3:035 10�7T þ 2:347 10�5; T 6 90 �C

1:0992 10�6T � 5:012 10�5; T P 90 �C

(

The material parameters needed for the proposed unified damage coupled viscoplastic micro-mechanicsmodel are listed as follows:

the average specific mass of composite ms = 85 g/mole;density of composite q = 1750 kg/m3;

Fig. 4. Uniaxial tensile test specimen geometry (mm).

Figdat

Figdat


gas constant R = 8.3145 J/mole/K;volume fraction of filler particle / = 0.48;Poisson’s ratio of filler particle mp = 0.24;elastic modulus of filler particle Ep = 70,000 MPa;coefficient of thermal expansion of filler particle Cte = 1.47 · 10�6;Poisson ratio of the matrix PMMA mm = 0.31;elastic modulus of the matrix PMMA Em = �23.36T + 4123.8 MPa.

. 5. Comparison of stress–strain relationship among damage coupled plastic model Ramberg–Osgood plasticity model and experimenta at 24 �C.

. 6. Comparison of stress–strain relationship among damage coupled plastic model Ramberg–Osgood plasticity model and experimenta at 75 �C.


For the interface region between the particle and the matrix, Young’s modulus and interface thickness areadjustable parameters according to the new proposed model. We assume that the Poisson’s ratio and CTE forthe interface are the same as the matrix PMMA.

The stress–strain relationship obtained from the unified damage coupled plastic model is compared with theRamberg–Osgood plastic model and also compared with experimental data as shown in Figs. 5 and 6 at 24 �Cand 75 �C, respectively. For simulations, 75 �C is used as the highest temperature because this particular mate-rial is not designed for or expected to experience much higher temperatures in its service life. The simulationresults are obtained based on the assumption that the thickness of interface region is 1% of the radius of a fillerparticle and the elastic modulus of interface region is 50% of that of the matrix PMMA. It is seen that thedamage effects must be accounted to be able to simulate the material behavior properly.

The influence of CTE mismatch between the filler particle and matrix on the overall uniaxial behavior of thecomposite is shown in Figs. 7 and 8 at 24 �C and 75 �C, respectively. Tc is the temperature that the residual

Fig. 7. The effect of CTE mismatch between matrix and particle on the overall stress–strain relation of particulate composite at 24 �C.

Fig. 8. The effect of CTE mismatch between matrix and particle on the overall stress–strain relation of particulate composite at 75 �C.


stresses are set in the microstructure due to the CTE mismatch during cooling the composite from its curingtemperature. These simulation results are obtained based on the assumption that the thickness of interface is1% of the radius of a particle and the elastic modulus of interphase is 50% of that of PMMA.

The influence of stiffness of the interphase around the filler particle on the overall uniaxial behavior of thecomposite is shown in Figs. 9 and 10 at 24 �C and 75 �C, respectively. These results are obtained by onlychanging Young’s modulus of the interphase as the percent of that of matrix and keeping other parametersfixed (Tc = 100 �C and the thickness of interphase is 1% of the radius of particle). As shown in Figs. 9 and10, when the interphase between the filler and matrix becomes more flexible (smaller elastic modulus) andstress in the matrix is reduced, which is in agreement with traditional weighted average stress equations forcomposites. As the interface gets more flexible, there is less contribution from the filler to the overall modulus,because the filler is a much stronger material then the matrix.

The influence of thickness of the interphase around the particle on the overall uniaxial behavior of the com-posite is shown in Figs. 11 and 12 at 24 �C and 75 �C, respectively. These results are obtained by only changing

Fig. 9. The effect of the stiffness of the interphase around particle on the overall stress–strain relation of particulate composite at 24 �C.

Fig. 10. The effect of stiffness of interphase around particle on the overall stress–strain relation of particulate composite at 75 �C.

Fig. 11. The effect of thickness of interphase around particle on the overall stress–strain relation of particulate composite at 24 �C.

Fig. 12. The effect of thickness of interphase around particle on the overall stress–strain relation of particulate composite at 75 �C.


the thickness of the interphase as a percent of the radius of the particle and keeping all other parameters fixed(Tc = 100 �C and the elastic modulus of interphase is about 50% of that of PMMA). The results indicate that,as the interphase between the matrix and the filler gets thicker, the interface becomes softer. A more flexibleinterface makes the composite more flexible (meaning smaller stress level for the same strain). Having a moreflexible interface also makes the composite more ductile, similar to the ductile matrix.

8. Conclusions

A unified damage coupled viscoplastic micro-mechanics constitutive model is proposed for a particulatecomposite. The proposed model accounts for the CTE mismatch between the filler and the matrix. Moreover,the model accounts for the stiffness of the interface region between the filler particles and the matrix. Theinterface is modeled as a layer with thickness and elastic properties. Then, the model is unified with a thermo-


dynamics based damage mechanics formulation. Comparison with test data indicates that the model is stableand predicts the test data well.

Acknowledgements

This project is sponsored by DuPont Surfaces Corp. Yerkes R&D Lab under the supervision of Dr. ClydeHutchins. Generous help and guidance received from Dr. Hutchins is gratefully acknowledged.

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A thermodynamics based damage mechanics model for