Cohesive modeling of dewetting in particulate composites ...kmatous/Papers/MEMA_MM.pdf · This paper focuses on the modeling of particle debonding (sometimes referred to as dewetting)in

Mechanics of Materials 39 (2007) 580–595

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Cohesive modeling of dewetting in particulate composites:micromechanics vs. multiscale finite element analysis

H.M. Inglis a, P.H. Geubelle b,*, K. Matous b,c, H. Tan a, Y. Huang a

a Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W,

Green Street, Urbana, IL 61801, USAb Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 South Wright Street, Urbana, IL 61801, USA

c Center for Simulation of Advanced Rockets, University of Illinois at Urbana-Champaign,

1304 West Springfield Avenue, Urbana, IL 61801, USA

Received 23 December 2005

Abstract

The effect of damage due to particle debonding on the constitutive response of highly filled composites is investigatedusing two multiscale homogenization schemes: one based on a closed-form micromechanics solution, and the other on thefinite element implementation of the mathematical theory of homogenization. In both cases, the particle debonding processis modeled using a bilinear cohesive law which relates cohesive tractions to displacement jumps along the particle–matrixinterface. The analysis is performed in plane strain with linear kinematics. A detailed comparative assessment between thetwo homogenization schemes is presented, with emphasis on the effect of volume fraction, particle size and particle-to-par-ticle interaction.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Particle-reinforced composites; Debonding; Microstructure; Damage mechanics; Micromechanics; Homogenization

1. Introduction

This paper focuses on the modeling of particledebonding (sometimes referred to as dewetting) inparticulate composites, with special emphasis onsystems characterized by high reinforcement volumefraction and substantial modulus mismatch betweenconstituents. Examples include solid propellants(SP), energetic materials and other reinforcedelastomers, which are typically composed of stiff

0167-6636/$ - see front matter � 2006 Elsevier Ltd. All rights reserved

doi:10.1016/j.mechmat.2006.08.008

* Corresponding author.E-mail address: [email protected] (P.H. Geubelle).

particles (such as ammonium perchlorate and alu-minum for SP) bonded together in a very compliantelastomeric matrix. The particles in these materialsrange from a few micrometers to several hundredmicrometers in diameter, with a total volume frac-tion as high as 75%.

Bencher et al. (1995) and Ide and Ho (1999) havestudied fracture processes in solid propellants at lowstrain rates (�10�3 s�1) and temperatures rangingfrom �55 to 70 �C. They report a damage processwhich begins with particle dewetting and formationof matrix fibrils ahead of the crack tip followed byvoid coalescence and fibril rupture leading to crackadvancement. Temperature decrease results in

.

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H.M. Inglis et al. / Mechanics of Materials 39 (2007) 580–595 581

strengthening of the elastomeric matrix andincreased occurrence of particle debonding ratherthan matrix tearing, leading to an increase in thesize of the process zone and higher fracturetoughness.

In experiments on an inert comparison material,Trumel et al. (2001) also noted particle debondingas the origin of the damage process in uniaxialquasi-static tests. This is in contrast to the behaviorunder high strain rates (103 s�1), in which particlefracture is the dominant failure mode, and highpressures (�500 MPa), in which grain plasticity isobserved. Rae et al. (2002a,b) performed experi-ments on high explosive, another closely packedparticulate composite system. Their results showclearly that the damage process is dominated by fail-ure of the particle–matrix interface, with failure ini-tiating at multiple sites perpendicular to the appliedloading direction, especially along the boundaries oflarge particles.

Failure in energetic materials is thus dominatedby processes which occur at a scale of several hun-dred microns, orders of magnitude smaller thanthe scales of the structural components in whichthey are commonly used. It is prohibitively expen-sive to model a typical component with sufficientresolution to capture the failure processes, hence itis customary to model different scales separatelyand introduce simplifying assumptions to bridgethem. The most common assumption is that it ispossible to represent the complexity of the micro-structure through some average material character-ization at the macroscale, a process of ‘‘smearing’’or homogenization of the microstructural response.We consider hereafter two complementary homoge-nization approaches—an analytical method basedon micromechanics, and a numerical scheme basedon the mathematical theory of homogenization(MTH).

Micromechanics relates microstructural phenom-ena such as heterogeneities to the mechanics ofmaterial behavior. Building on Eshelby’s detailedanalytical solution of the problem of an ellipsoidalinclusion in an infinite matrix (Eshelby, 1957), themethod is extended by considering boundary condi-tions which account for some interactions betweenparticles. In the Mori–Tanaka method (Mori andTanaka, 1973), the inclusion is subjected to the localstrain in the surrounding matrix, while the self-consistent method, attributed to Hill (1965) andBudiansky (1965), embeds the particle in a materialwith the properties of the homogenized composite.

The Hashin–Shtrikman bounds (Hashin and Shtrik-man, 1962) are computed by comparing the com-posite to a reference material, with stiffness eithergreater than any element in the composite (upperbound) or less than every element (lower bound).Similar bounds on material properties are obtainednumerically by imposing either traction or displace-ment on the boundaries of a representative volumeelement (RVE). In recent work, Kanit et al. (2003)compare such results to those obtained when theunit cell is assumed to be the base cell of a periodicmicrostructure and periodic boundary conditionsare imposed.

Homogenization methods based on asymptoticexpansions of solution fields were presented by Ben-soussan et al. (1978) and Sanchez-Palencia (1980) tostudy periodic heterogeneous media. As outlined byGuedes and Kikuchi (1990), the mathematical the-ory of homogenization (MTH) provides a powerfulframework for linking multiple length scales, wellsuited to computation. This approach has been usedextensively by various research groups; see, forexample, Ghosh et al. (1995); Raghavan et al.(2001); Fish et al. (1997).

When an analytical solution is available, amicromechanics model is substantially cheaper thana numerical solution. However, the assumptionsrequired for an analytical solution to be tractabletend to reduce the accuracy of the model. In thiswork, we are interested in evaluating the extent towhich the simplified model still provides valid mate-rial representation, or, at the least, useful physicalinsights in the case of particle dewetting.

The micromechanics model can be validatedthrough comparison with experimental results.However, when the analytical and empirical resultsdiffer, to what do we attribute the error? The errormay be a consequence of material assumptions(i.e., linear elasticity, small strain) or of simplifica-tions used to arrive at the micromechanics model(i.e., spherical particles, hydrostatic stress, uniformfields in phases). Often, due to the limitations ofexperimentation, only a single macroscopic compar-ison can be made quantitatively, while microscopiccomparison is qualitative at best. Hence comparisonwith experimental data is not always sufficient toidentify errors in the model and consequentlyimprove it. There is a role for solution verificationby comparison with a reference numerical solution.In this paper, we therefore perform such a com-parison between a micromechanics model and anumerical MTH scheme that incorporates the same

582 H.M. Inglis et al. / Mechanics of Materials 39 (2007) 580–595

material and failure models, but accounts for heter-ogeneous fields and particle interactions and allowsfor complex loading paths.

The micromechanics model follows that used byTan et al. (2005a, 2006) to investigate the debondingof spherical particles under remote hydrostatictension. Debonding at the particle–matrix inter-face was modeled through the use of a bilinear cohe-sive law. Beginning with a dilute homogenizationassumption (Tan et al., 2006), the model was subse-quently extended to a Mori–Tanaka homogeniza-tion procedure (Tan et al., 2005a), which is validfor a larger range of volume fractions. Althoughbased on highly simplifying assumptions (linearkinematics and material response), the microme-chanics models captured the size effect observedexperimentally, with large particles debondingbefore small particles. The authors identified a crit-ical particle radius acr above which the compositesoftens rather than hardens after debonding, caus-ing the interface to undergo continuous debondingeven under static load.

The other homogenization approach consideredin this paper, the MTH-based finite element scheme,served as the foundation of the analysis performedby Matous and Geubelle (2006) in a finite strainsetting with cohesive zone modelling of interfacefailure. The finite element formulation was stabi-lized to handle the near-incompressible behaviorof the matrix. Simulations of one- and four-particleunit cells of an idealized solid propellant allowedmicroscale failure processes to be linked with mac-roscopic stress–strain curves, including an examplein which load path bifurcation occurs. The resultspresented in Matous and Geubelle (2006) are notpredictive, as the size and simplicity of the unitcell adopted in that study do not approach the sta-tistical representativity required of an RVE exceptfor very organized microstructures. Increasing prob-lem size would substantially increase the computa-tional costs. In order to further study the physicsof particle debonding in closely packed materialswith much larger unit cells, a simpler computationalmodel, based on assumptions of linear kinematicsand plane strain, is developed in this paper. Thissimpler model provides a useful tool for investiga-tion of multiple physical phenomena, while notattempting to approach the accuracy of the abovework.

The objective of this paper is thus to perform adetailed comparative study between these two mod-eling approaches to assess the range of validity of

the micromechanics model in the cohesive modelingof dewetting in particulate composites. As it was thecase in the micromechanics study, this comparativeanalysis relies on linear kinematics (small strain),linear elasticity and equibiaxial loading assump-tions, although the numerical homogenizationscheme can readily incorporate finite kinematics,nonlinear material response and more complexloading cases (Matous and Geubelle, 2006). Thecomparison is performed under the plane strainassumption, in which the inclusions are actuallyfibers rather than spheres, although we will continueto refer to ‘‘particles’’ and ‘‘particulate composites’’.Both approaches use identical constitutive modelsfor the constituents and their interface.

Section 2 sets out the general problem to besolved, and defines the cohesive law and materialproperties which will be used. In Section 3, wedevelop the plane strain version of the microme-chanics model introduced by Tan et al. (2005a,2006) and present the key results of this homogeniza-tion scheme. Section 4 is devoted to the presentation,implementation and application of the MTH-basedfinite element method. Simulated damage evolutionsobtained on a variety of loading conditions and theireffect on the macroscopic constitutive response ofthe composite are also presented. However, the com-parison between these two homogenization schemes,discussed in Section 5, is limited to the equibiaxialloading case. In Section 6, we evaluate the perfor-mance of the micromechanics model, and presentconclusions.

2. Problem description

The problem of interest consists of a heteroge-neous material composed of particles XpI embeddedin a matrix Xm and subjected to a macroscopic stress�r or strain �e. The volume fraction of particle I isgiven by fI ¼ XpI=X, satisfying

PI fI ¼ f . Super-

scripts m, pI and int denote the matrix, particle I

and interface, respectively. The composite experi-ences damage through debonding of the particle–matrix interface, denoted SpI . We aim to determinethe macroscopic stress–strain response of the com-posite during damage evolution.

The debonding of the particle from the surround-ing matrix is modeled through a traction–separationlaw at the cohesive interface. Tan et al. (2005b) pres-ent experimental results justifying the use of a bilin-ear cohesive law, specified in terms of interfaceproperties,

Δnc

σmax

σmaxSinit

k~σ

kσ

γ int

σint

Δnc

ur

1 1

(1–S )init

Fig. 1. Bilinear cohesive law for opening in the normal direction.The unloading path is always directed towards the origin.

Table 1Cohesive interface properties

Property Symbol Value

Interface strength (MPa)

Normal rmax 0.02Tangential smax 0.02

Critical opening displacement (lm)

Normal Dnc 1.0Tangential Dtc 1.0

Linear modulus (MPa/lm) kr 1.00Softening modulus (MPa/lm) ~kr 0.02

Initial damage (–) Sinit 0.98Interface energy (MPa lm) cint 0.01


rint¼

krsurt for surtrmax

kr;

1þ ~krkr

� �rmax�~krsurt for rmax

krsurt< rmax

1krþ 1

~kr

� �;

0 for surt> rmax1

krþ 1

~kr

� �8>>><>>>:

ð1Þ

where rmax, kr and ~kr respectively represent theinterface strength, modulus and softening modulusof the interface, and surb is the interface opening(or displacement jump) in the radial direction, mak-ing use of the discontinuity notation s• b. Since theloading condition is hydrostatic, they consider onlynormal extension and do not account for tangentialopening or for normal compression.

The bilinear irreversible cohesive law used in thenumerical simulations presented in Section 4 is thatdescribed by Geubelle and Baylor (1998), which, ateach point along the interface, relates the normal(Tn) and tangential (Tt) cohesive tractions to thenormal (Dn) and tangential (Dt) displacement jumpsthrough

T n ¼rmax

Sinit

S1�S

DnDnc

for Dn P 0;

rmax

1�Sinit

DnDnc

for Dn < 0;

(ð2Þ

T t ¼smax

Sinit

S1� S

DtDtc

: ð3Þ

The coupling between normal and tangential failureis achieved through an interface damage parameterS, which degrades from its initial value Sinit, chosenclose to unity, to 0 with increased interface openingaccording to

S ¼ hh1� keDk2ii ¼ hh1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDnDnc

� �2

þ DtDtc

� �2s

iið4Þ

when Dn is positive. In the case of compression(Dn < 0), no further accumulation of damage is al-lowed. In the above expression, hhaii = a if a > 0and hhaii = 0 otherwise. S represents the remainingcapacity of the local interface to sustain tractions.The parameters rmax and smax entering (2) and (3) de-note the tensile and shear interface strengths respec-tively, while Dnc and Dtc represent the critical normaland tangential values of the opening displacementsbeyond which complete failure is assumed.

As shown in Fig. 1, the cohesive laws used in themicromechanics model and in the numerical simula-tions are identical for the normal opening directionin the absence of shear failure, when cohesive stiff-nesses are related through

kr ¼rmax

ð1� SinitÞDnc

; ð5Þ

~kr ¼rmax

SinitDnc

; ð6Þ

and the interface fracture toughness, (i.e., the areaunder the traction-separation curve) is given by

cint ¼ r2max

2

1

krþ 1

~kr

� �¼ 1

2rmaxDnc: ð7Þ

The interface properties, given in Table 1, and vol-umetric material properties, given in Table 2, havebeen chosen to ensure that failure occurs withinthe limit of small strain, as the linear kinematicsassumption is invoked in both the micromechanicsand MTH models. This is acceptable, since we are

Table 2Volumetric material properties

Constituent E (MPa) m

Particles 150 0.3Binder 1 0.4


interested in comparing two models using identicalmaterial properties. Throughout the paper, all re-sults are normalized with respect to the chosenproperties. Note the large mismatch between thematerial properties of the matrix and the particles.

3. Micromechanics model

The micromechanics model with debonding pre-sented by Tan et al. (2005a, 2006) is reformulatedhere for plane strain conditions. Only the key planestrain results are summarized hereafter: for moredetail, the reader is referred to the above references.In this section and the remainder of this paper,uppercase subscripts denote the particle numberand are not summed over unless explicitly specified,while lowercase indices follow conventional summa-tion rules and in this plane strain setting take thevalues 1 and 2.

To find a relationship between macroscopicstrain, �e, and macroscopic stress, �r, we begin withthe following expression for the macroscopic straintaken from Benveniste and Aboudi (1984):

�e ¼Mm : �rþX

I

fIfðMpI �MmÞ : hrpiI þ heintiIg;

ð8Þ

where M is the material compliance tensor, hrpiIrepresents the stress average over particle I, andheintiI represents the strain average over interfacesaround particle I,

heintiI ¼1

2XpI

ZSpI

ðsut� nþ n� sutÞdA; ð9Þ

where n is the normal to the cohesive interface. Inother words, the macroscopic (or average) straincan be considered as the strain due to the averagestress applied to the matrix material, with perturba-tions due to the stress in each particle and due to thedisplacement discontinuity at the damaged particle–matrix interfaces. Note that this relation is generaland only assumes linearity of the matrix and particleconstitutive laws.

In order to find the macroscopic stress–strainrelationship, we need to investigate the stress within

particles and the strain across particle interfaces.We make simplifying assumptions in order to pro-ceed with an analytical solution: isotropy of parti-cles and matrix, equibiaxial applied strain �e andcylindrical particles of radius aI. These assumptionsreduce the problem to an axisymmetric one andensure uniform stress fields in the particles. Theaverage stress in a particle is then given by

1

2hrp

kkiI ¼ rintI ; ð10Þ

while the interface strain (9) reduces to

heintkk iI ¼

2hsurtiIaI

: ð11Þ

The relationship between interface traction and dis-placement jump is the cohesive interface law (1). Therelationship between displacement jump and macro-scopic stress requires a further assumption aboutthe stress in the matrix surrounding the particle.The dilute assumption considers the particle to beembedded in an infinite matrix subjected to remoteequibiaxial stress �r. The Mori–Tanaka assumption,by contrast, considers a particle embedded in thelocal matrix and subjected to stress rm, with themacroscopic stress balanced by

�r ¼ ð1� f Þrm þX

I

fIhrpiI : ð12Þ

It is clear that, for small f, the Mori–Tanaka and thedilute assumptions converge, as we would expect.The two assumptions are represented schematicallyin Fig. 2.

The displacement discontinuity is readily deter-mined (Timoshenko and Goodier, 1970) to be

surt¼ aI2ð1� mmÞð1þ mmÞrE�ð1þ mmÞrint

Em � rint

2KpIpl�e

" #;

ð13Þ

where Kpl�e is the plane strain bulk modulusexpressed in terms of the Young’s modulus E andPoisson’s ratio m as

Kpl�e ¼E

2ð1� 2mÞð1þ mÞ ; ð14Þ

and rE denotes the macroscopic applied equibiaxialstress,

rE ¼�r; for the dilute model;

rm; for the Mori–Tanaka model:

�

σm

σσ

σmσp

σp5 σp

4

σp1

σp3

σp6

σp2

σ

Fig. 2. Schematic comparison of dilute (left) and Mori–Tanaka (right) assumptions under conditions of remote equibiaxial stress.

σ maxσ

Nor

mal

ized

mac

rosc

opic

str

ess,

εMacroscopic strain, [%]

Fully damagedTransitionalUndamaged

Dilute

Mori–Tanaka

1.00

0.75

0.05

0.25

0.00

1.25

0.0 0.4 0.8 1.2 1.6

Fig. 3. Comparison between dilute and Mori–Tanaka solutions;f = 0.2, a = 40 lm.


We use (13) to rewrite the cohesive law (1) as

rint ¼aIrE; undamaged

1þ a0IaI

� �rmax � a0Ir

E; transition

0; fully damaged

8><>:ð15Þ

where aI and a0I are functions of the materialproperties:

aI ¼2ð1� mmÞð1þ mmÞ

Em

kraIþ Em

2KpIpl�e

þ 1þ mm; ð16Þ

a0I ¼�2ð1� mmÞð1þ mmÞ� Em

~kraIþ Em

2KpIpl�e

þ 1þ mm: ð17Þ

The macroscopic strain in the composite can now befound by substituting (10) through (13) in (8) for thedilute and Mori–Tanaka cases, respectively:

�eD ¼2ð1þ mmÞð1� mmÞEm

1� 2mm

2ð1� mmÞþ f� �

�r�X

I

fIrintI

" #;

ð18Þ

�eMT ¼2ð1þ mmÞð1� mmÞEmð1� f Þ

1� 2mmþ f2ð1� mmÞ �r�

XI

fIrintI

" #;

ð19Þ

where rintI is related to �r through (15). The dilute

and Mori–Tanaka macroscopic stress–strain curvesobtained for the volumetric and cohesive propertiesdescribed in Section 2 and for a relatively small vol-ume fraction (f = 0.2) of 40 lm radius particles areshown in Fig. 3. For a single particle size (I = 1)and the equibiaxial loading case, the constitutive re-

sponse is trilinear, with the undamaged and fullydamaged portions of the curve passing throughthe origin (as indicated by the dotted lines inFig. 3). The fully damaged solution, for which allinterfaces have debonded (i.e., rint

I ¼ 0 : 8I), corre-sponds to the response of a linearly elastic materialwith a void volume fraction f (i.e. porous medium).The sign of the coefficient a0I defined in (17) deter-mines the slope of the transitional stage of the mac-roscopic stress–strain curve.

The predicted undamaged response and onset ofthe transition between undamaged and fully dam-aged states are, for this value of f, very similarbetween the dilute and the Mori–Tanaka models.However, as is apparent in Fig. 3 even for this rela-tively low volume fraction, substantial differences


σ σ max

Nor

mal

ized

mac

rosc

opic

str

ess,

large particlesDamage of

Damage of small particles

small particlesFailure of

large particlesFailure of

1.00

0.75

0.50

0.25

0.00 0.0 0.5 1.0 1.5 2.0

Fig. 5. Macroscopic stress strain-curve for a material with twodifferent particle sizes, a1 = 50 lm, f1 = 0.26, a2 = 20 lm,f2 = 0.18.

σ maxσ

Nor

mal

ized

mac

rosc

opic

str

ess,

μma,20304050


0.00

0.50

0.25

0.75

1.25

1.00

1.50

0.000 0.005 0.010 0.015 0.020

Fig. 6. Effect of particle radius: Mori–Tanaka solution, with


exist between the two micromechanics models in thetransition and fully damaged responses. Since weare primarily interested in composite systems withmoderate to high particle volume fractions, we willfocus the remainder of this discussion on theMori–Tanaka model.

The effect of the volume fraction on the �r–�ecurve is shown in Fig. 4. As expected, due to thestiffness mismatch between the particles and thematrix, an increasing value of f leads to a stiffeningundamaged response and a more compliant fullydamaged response. Note that the maximum theo-retical packing density of circles of a single size ina plane is f � 0.91, in contrast to the maximumfor spheres, which is f � 0.78. In practice, the highvolume fractions in energetic materials are achievedby using a distribution of particle sizes. When thereis more than one particle size, as in Fig. 5, the mac-roscopic stress–strain curve becomes more complex,with critical points corresponding to damage initia-tion and complete failure for each of the phases,starting with the larger particles, as observedexperimentally.

Fig. 6 shows the effect of the particle radius on themacroscopic stress–strain curve for a fixed volumefraction. Increasing the radius decreases the slopeof the transitional stage, which suggests the existenceof a critical radius, acr, below which the compositehardens in the transitional phase, and above whichthe material softens. Tan et al. (2006, 2005a) differ-entiate between stable and unstable paths, with cat-astrophic debonding possible under load control forparticles larger than the critical particle size. How-ever, as we will see in Section 5, even the mono-

Nor

mal

ized

mac

rosc

opic

str

ess,


f=0.6

f=0.8

f=0.4

f=0.2σ maxσ

0.00

0.25

0.50

0.75

1.00

1.25

1.75

1.50

0.0 1.0 2.0 3.0

Fig. 4. Effect of volume fraction on the macroscopic stress–straincurve: Mori–Tanaka solution, a = 40 lm.

volume fraction f = 0.2.

tonically increasing macroscopic responses showinstability under equibiaxial loading.

4. MTH-based multiscale finite element method

Consider the multiscale structural problemshown in Fig. 7. The macroscopic scale x is relatedto the microscopic scale y through an asymptoticscaling parameter n, as y = x/n. The macroscopicdomain X with boundary C is subject to mixedboundary conditions. Tractions are applied to Ct

and displacements are imposed on Cd, with C =Ct [ Cd. No body forces are considered inthis work. The heterogeneous microscopic domainH is assumed to be periodic and contains fibers

Γt

Γd

σ

ε

Γc

1

2

1

2

Ω

Microscopic scale

Macroscopic scaley

x

y = xξ

Θ

t

Fig. 7. Multiscale model. The deformed contours drawn on theRVE (dashed curves) emphasize the periodic nature of themicroscale solution and the displacement jump along the cohesivefiber–matrix interface.


embedded in a surrounding matrix. The progressivefailure of the fiber–matrix interfaces, denoted by Cc,is characterized by the cohesive law given in (2)–(4).

We define an asymptotic expansion of the dis-placement field,

uðx; yÞ � uð0Þðx; yÞ þ n1uð1Þðx; yÞ þ n2uð2Þðx; yÞ þ � � � ;ð20Þ

where superscripts in parentheses (0),(1),. . . indicatethe asymptotic order. Next, we use the differentia-tion operator

o/ðx; yÞox

¼ o/oxþ 1

no/oy

ð21Þ

to derive the asymptotic expansion of the displace-ment gradient

oui

oxj� ouð0Þi

oxjþ 1

nouð0Þi

oyj

þ nouð1Þi

oxjþ ouð1Þi

oyj

þ n2 ouð2Þi

oxj

þ nouð2Þi

oyj

þ � � � ð22Þ

The strain, defined to be the symmetric part of thestrain operator in the small strain setting, is thusgiven by

eij � n�1 oSuð0Þi

oyj

!þ n0 oSuð0Þi

oxjþ oSuð1Þi

oyj

!

þ n1 oSuð1Þi

oxjþ oSuð2Þi

oyj

!þ � � � ; ð23Þ

� n�1eð�1Þij þ n0eð0Þij þ n1eð1Þij þ � � � ; ð24Þ

where we have introduced the symmetric gradientoperator

oS�i

oxj¼ 1

2

o�i

oxjþ o�j

oxi

� �: ð25Þ

In terms of the material stiffness tensor, E, the exter-nal tractions applied on Ct, t, and the cohesive trac-tions present on Cc, T, the principle of virtual work(PVW) for this problem isZ

XEijklekl

oSvi

oxjdX�

ZCt

tivi dS þZ

Cc

T isvitdS ¼ 0;

ð26Þfor all admissible displacements v satisfying

v 2 ½H 1�2; v ¼ 0 on Cd; ð27Þwhere [H1]2 is the Sobolev space for the 2-D prob-lem. Expanding (26) and grouping by powers of nleads to

1

n2

ZX

EijkloSuð0Þk

oyl

oSvi

oyj

dX¼ 0; ð28Þ

1

n

ZX

EijkloSuð0Þk

oxlþ oSuð1Þk

oyl

!oSvi

oyj

þ oSuð0Þk

oyl

oSvi

oxj

" #dX

þZ

Cc

T isvitdS ¼ 0; ð29ÞZX

Eijklo

Suð1Þk

oxlþ o

Suð2Þk

oyl

!o

Svi

oyj

þ oSuð0Þk

oxlþ o

Suð1Þk

oyl

!o

Svi

oxj

" #dX

¼Z

Ct

tivi dS: ð30Þ

We make use of the integration operators for y-peri-odic functions

limn!0þ

ZX

/ðx; yÞdX ¼ 1

jHj

ZX

ZH

/ðyÞdHdX; ð31Þ

limn!0þ

nZ

C/ðx; yÞdA ¼ 1

jHj

ZX

ZoH

/ðyÞdAH dX; ð32Þ

in evaluating (28) and (29). Eq. (30) represents equi-librium at the macroscale and is used to fully couplethe macro- and microscale solutions. However, thisrelation is not used in this work, as we assume amacroscopic strain history �e and extract from theperiodic RVE the effect of the microscale damageevolution on the macroscopic stress �r.

As described in detail by Guedes and Kikuchi(1990), from (28) we establish that u(0) depends onlyon the macroscale,

uð0Þðx; yÞ ¼ uð0ÞðxÞ ð33Þwith no y-dependence, and is hence a continuousfield. From (33) and (29), we obtain an expressionof equilibrium at the microscale


1

jHj

ZH

EijkloSuð0Þk ðxÞ

oxlþ oSuð1Þk

oyl

!oSvi

oyj

" #dH

þ 1

jHj

ZCc

T isvitdS ¼ 0; ð34Þ

for all admissible displacements v satisfying

v 2 ½H 1�2; vðyÞ is Y -periodic on oH: ð35ÞDefining the macroscopic strain, �e ¼ oS uð0Þ

ox, and the

fluctuating strain, ~e ¼ oS uð1Þ

oy, and considering the

macroscopic strain as having the effect of a loadingterm at the microscale, we can rearrange (34) in thefollowing form:

1

jHj

ZH

Eijkl~eklo

SviðyÞoyj

dHþ 1

jHj

ZCc

T isvitdS

¼ � 1

jHj

ZH

EijkloSviðyÞ

oyj

dH�ekl; ð36Þ

which serves as the basis for the finite element solu-tion for the discontinuous displacement field u(1). Inthe following, we refer to u(0) as the macroscopic dis-

Fig. 8. (a) Typical RVE mesh, incorporating 3- and 4-noded elementsdistribution.

placement, and u(1) as the fluctuating displacement.The macroscopic stress �r is given by the volumeaverage of the local stress field. Note that the mac-roscopic stress and strain defined in this sectionhave the same meaning as those defined in Section 3.

A sample representative volume element (RVE)of a simulated particulate composite containing 50circular particles is shown in Fig. 8(a). The RVEis square, with edges of length 693.5 lm. The parti-cle radius distribution is bimodal, as shown inFig. 8(b), with peaks at a = 50 and 30 lm, and withvolume fractions of 0.26 and 0.18, respectively. Thetotal particle volume fraction is thus 0.44. The par-ticles and matrix are discretized with 3- and 4-nodedelements using the T3D meshing tool (Rypl andBittnar, 2002), which creates a periodic mesh. Theparticle–matrix interfaces are modeled with 4-nodedcohesive elements. To enforce periodicity, corre-sponding nodes on opposite edges are assigned thesame equation number. The nonlinear system ofequations resulting from (36) is solved using thestiffness matrix from the previous load step, with

, with refinement in areas of high heterogeneity. (b) Particle size


an adaptive load stepping scheme to ensure accu-racy and efficiency of computational effort. Themesh is checked to ensure that there are sufficientcohesive elements within the cohesive zone. Moredetails on the packing and discretization methodol-ogies can be found in Matous et al. (2006).

Fig. 9 shows results from this RVE under animposed macroscopic equibiaxial strain, �e ¼ ð�e11;�e22;�e12Þ ¼ ð0:0045; 0:0045; 0:0000Þ. The macroscopicstress–strain (left axis) and cohesive damage evolu-tion (right axis) are plotted on the same graph toallow for a direct correlation between microscaledamage and macroscopic constitutive response.The macroscopic stresses �r11 and �r22 are normalized

Fig. 9. (Top) MTH prediction of macroscopic stress–strain and damagestress distributions plotted on the deformed shape (displacements magndenoted by open circles in the top figure.

by the critical opening stress rmax. Below the graph,a sequence of von Mises stress fields, rvm, are plot-ted on the deformed shape, with displacements mag-nified 10 times. The gray scale maps have beennormalized and the range limited to allow clear dis-cernment of variations in the fields. For this reason,the signal saturates in regions of particularly highstresses.

The dash-dotted curve corresponds to the evolu-tion of the fraction of cohesive elements which aredamaged, i.e., those which are in the downward por-tion of the cohesive curve in Fig. 1, while the dashedand dotted curves denote the fraction of fully failedcohesive elements on the boundaries of large and

evolution curves for equibiaxial strain case. (Bottom) Von Misesified by 10) of the RVE at the four stages in the damage process


small particles, respectively. As apparent in Fig. 9, asubstantial fraction (about 40%) of the particle–matrix interfaces is damaged before we can discerna deviation from linearity in the stress–strain curves.Under the highly unstable macroscopic equibiaxialstrain, the damage evolution curve rises rapidly upto the point at which the first interfaces fail com-pletely leading to a sudden dropoff in macroscopicstresses, as indicated by (a). Failure commences atthe large particles and in regions of locally high vol-ume fraction and is a sudden event. The initial inter-facial failure drives a localization process, visible in(b), resulting in a loss of stress equibiaxiality, with�r11 < �r22 since the system becomes more compliantin the direction normal to the localization band.Localization continues until all the particles acrossthe height of the RVE have failed, at (c). At this

Fig. 10. Uniaxial macroscopic strain: (a) Macroscopic stress–strain curve, including damage evolution; (b) Von Mises stressdistribution at �e11 ¼ 0:6%. The displacements have been magni-fied by 10 to emphasize the interface failure.

point, we note that, in reality, the small remainingligaments of matrix between the decohered particleswould tear, causing complete failure of the compo-nent. However, since matrix tearing is not consid-ered in this study, continuing loading causes a newprocess of localization to initiate perpendicular tothe original localization direction, as shown in (d),resulting in a return to near equibiaxiality of themacroscopic stresses. Throughout this sequence,the periodicity of the domain is evident, with dis-placements and stresses at opposite boundaries ofthe RVE being identical. Failure occurs preferen-tially at the interfaces of large particles, in agree-ment with experimental observations.

Fig. 10 presents the macro- and micro-scaleresults associated with �e ¼ ð0:006; 0; 0Þ. The uniaxialstate of macroscopic strain leads to a more stable

Fig. 11. Macroscopic shear stress state: (a) Macroscopic stress–strain curve, including damage evolution; (b) Von Mises stressdistribution at �e11 ¼ 2:3%. Displacements magnified by 10.


macroscopic solution. In this case, failure initiatesat large particles, with opening normal to theapplied loading direction. Due to the lower levelof stress biaxiality, the failure process is less sudden,although localization of damage still appears in thedirection perpendicular to the applied macroscopicstrain. Note the presence of stress concentrationsin the vicinity of interfacial crack tips.

The solution is quite different in the shear loadingcase (Fig. 11), for a macroscopic strain of �e ¼ð0; 0; 0:023Þ. The maximum principal stress is at a45� angle to the RVE axes, and the openings areagain normal to the applied load. The large negativeprincipal stress results in considerably less failure ofthe cohesive interfaces than in the two other loadingcases, and prevents crack propagation along theinterfaces. Instead of the damage localizationobserved in the other two loading cases, we observemuch more distributed damage in the RVE, whichleads to a very limited deviation from linearity inthe macroscopic �r12–�e12 curve (Fig. 11(a)). As inthe uniaxial strain case, damage initiates aroundthe larger particles. Some of the cohesive failure isin rotation, particularly evident in those particleswhich are on the periodic boundaries.

5. Comparative assessment

In order to compare the micromechanics andMTH-based finite element (MTHFE) models, weconsider first a single particle unit cell subjected toequibiaxial loading. Periodic boundary conditionsimpose a regular stacking of unit cells, thus the sin-gle particle cell represents a regular array of parti-

σ maxσ

Nor

mal

ized

mac

rosc

opic

str

ess,


a = μmf =

200.2

Mori–Tanaka modelNumerical simulation

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5 2.0

a b

Fig. 12. Comparison between micromechanics a

cles. The volume fraction can be easily controlledby changing the size of the unit cell while keepingthe particle size constant. Fig. 12 shows the resultsof this comparison for low volume fraction(f = 0.2) for particles smaller (a) and larger (b) thanthe aforementioned critical particle size, acr = 27 lmfor this material system. In all the comparativecurves, dashed lines represent the micromechanicsprediction, and solid curves are used for the resultsof the MTHFE simulation.

As apparent in Fig. 12 the Mori–Tanaka solutioncaptures the essential features of the MTHFE result,with the initial slope and damage initiation point ingood agreement between the two, and the final slopeof the MTHFE tending toward that predicted bythe Mori–Tanaka model. However, the two solu-tions differ substantially in the transition phase.Although the MTHFE curve initially follows theslope of the micromechanics prediction, there is aninstability present in the solution regardless ofwhether the predicted material response is mono-tonic or not. This instability can be related to vari-ations in stress concentration around the particleboundary, a consequence of the periodic arrange-ment of particles, where points at the top, bottom,left and right of the particle are the closest to aneighboring particle. Slight differences in cohesivetractions experienced at different points of the parti-cle–matrix interfaces, coupled with randomness inthe mesh (analogous to randomness in the particleshape or local variations in material properties ina real material) cause the problem to lose axisymme-try. The instability of the loading conditions and thesoftening cohesive law create a system which drives

a = 40μmf = 0.2

σ maxσ

Nor

mal

ized

mac

rosc

opic

str

ess,



1.00

1.25

0.05

0.25

0.00 0.0 0.4 0.8 1.2 1.5

0.75

nd MTH solutions (a) a < acr, (b) a > acr.

σ maxσ

Nor

mal

ized

mac

rosc

opic

str

ess,

f = 0.32a = μm30


Mori–TanakaMTHFE: 18 particlesMTHFE: 1 particle

1.0

0.8

0.6

0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0

10000

15000

Fig. 13. Comparison between micromechanics and MTH solu-tions for a unit cell containing 18 particles. The failure of eachparticle is a discrete event.


rapid failure of the portion of the interface whichwas initially perturbed. We see that, while theMori–Tanaka solution assumes that failure occursaxisymmetrically and uniformly, MTHFE allowsfailure to occur non-uniformly, through crack initi-ation at a random location, followed by crack prop-agation around the interface. The interface neverfails completely, hence the final slope of theMTHFE solution does not decline to that predictedby micromechanics. Once a considerable portion ofthe interface has failed, little driving force remainsto debond the last few elements.

To investigate the effect of non-regular particledistribution, we perform the same comparison ona periodic unit cell containing 18 particles in a ran-dom array, shown in Fig. 13. (This locally randomparticle arrangement is, however, repeated throughperiodicity.) For reference, the solution for a single

σ max

σN

orm

aliz

ed m

acro

scop

ic s

tres

s,


σ maxσ

Nor

mal

ized

mac

rosc

opic

str

ess,

a = 40μmf = 0.7

σ11σ22

σ max

σN

orm

aliz

ed m

acro

scop

ic s

tres

s,

a = 40μmf = 0.7

σ maxσ

Nor

mal

ized

mac

rosc

opic

str

ess,


εMacroscopic strain, [%] εMacroscopic strain, [%]

a = 40μmf = 0.3


Mori–Tanaka modelNumericalNumerical


a = 40μmf = 0.5


1.00

1.25

0.05

0.25

0.00 0.0 0.5 1.0 1.5 2.0

0.75

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0

0.5 0.4 0.3 0.2 0.1 0.0 2.5 2.0 1.5 1.0 0.5 0.0

a b

c d

Fig. 14. Comparison between micromechanics and MTH solutions for a = 40 lm, with varying volume fraction: (a) f = 0.3, (b) f = 0.5, (c)f = 0.7, (d) is a magnification of the first part of the curve in (c).

σ maxσ

Nor

mal

ized

mac

rosc

opic

str

ess,

σ m

axσN

orm

aliz

ed m

acro

scop

ic s

tres

s,




1 2a =f = 0.23 0.16f =21

a = μm30 40μm

1 a =f = f =21

a 2 = μm μm40 10


16 particles, 2 sizes

0.11 0.25

18 particles, 2 sizes

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5 2.0

a

b

Fig. 15. Comparison between micromechanics and MTH solu-tions for two different particle sizes: (a) shows a system in whichthe particles are of the same order of magnitude. In (b) thedifference in size is more marked. The small particles have not yetfailed at �e ¼ 0:02, hence the MTHFE is still below the Mori–Tanaka prediction.


particle with the same volume fraction and particlesize is plotted on the same graph, with a dottedcurve. While the beginning of instability occurs atalmost the same point for both simulations, thedropoff is much more sudden for the multi-particlesystem than it is for a single particle, due to particleinteraction and localization. As failure initiates inone particle, that drives failure for all the particlesacross the height or width of the cell. The failureprocess is discrete, as an ongoing failure processrelieves stresses elsewhere in the RVE, delaying thefailure of other particles.

Thus far, we have considered only microstruc-tures with low volume fractions. Fig. 14 showsresults for volume fractions as high as f = 0.7 (closeto the maximum possible for a rectangular array).As seen earlier, for low volume fraction, the modelsare in good agreement about key features of thesolution. As the volume fraction increases, the levelof agreement between the two models in the transi-tion region decreases. The difference is particularlyobvious in Fig. 14(c) and (d). This is expected sincethe Mori–Tanaka assumption is known to be validonly for low to moderate volume fractions. TheMori–Tanaka model also underpredicts the initialslope slightly. The sharp damage initiation point,a consequence of the bilinear cohesive law, ispresent in the solutions at low volume fraction,but becomes less marked as f increases. The increas-ing packing density results in an increase in stressconcentration at localized points on the particleboundary, causing damage to begin earlier thanpredicted. The transitional phase does not displaythe same sudden dropoff as observed for lower vol-ume fraction and the slope is significantly more neg-ative in the entire region, since failure is notoccurring at the same rate everywhere around theboundary.

Real material systems have multiple particlesizes. Fig. 15 shows the results of a simulation withtwo different particle sizes. For particles of similarsizes, shown in (a), stress concentrations and locali-zation processes play a significant role. The failureprocesses for large and small particles are not inde-pendent, as predicted by the micromechanics solu-tion. However, once the difference between particlesizes becomes significant, as in (b), the effect of thesmaller particles on the failure process is reducedsubstantially. The smaller particles serve merely tostiffen the binder and to trigger damage nucleationthrough local stress concentration. This is evidentin both the MTHFE and micromechanics results,

indicating that the micromechanics model, whilenot fully capturing the failure process, can nonethe-less give useful information about which particlesneed to be modeled.

6. Conclusions

We have performed a detailed comparativeassessment of micromechanics and finite elementbased homogenization schemes for the problem ofdebonding damage in a plane strain particulatecomposite system. Special emphasis has been placedon the ability of the two schemes to capture particle-to-particle interactions and the effect of dissimilarparticle sizes.


The plane strain micromechanics model devel-oped by Tan et al. (2005a) is effective at capturingkey features of the macroscopic stress–strainresponse for minimal computational effort. Theshortcomings of this model are that it cannot cap-ture the instability inherent in the system or the het-erogeneous stress and strain fields. In a highly filledcomposite, interactions between particles are a sig-nificant contributor to failure through local stressconcentrations and the occurrence of localization.The micromechanics model does not capture theseinteractions during the failure process, and henceceases to be predictive under high volume fraction,f > 0.5, when stress concentrations begin to play asignificant role in the solution, or when the particledistribution is random, resulting in localization.Both models demonstrate that, for large differencesin particle diameters, it is unnecessary to model thedebonding of smaller particles, but it is sufficient torepresent their contribution to damage nucleationand to the stiffness of the matrix.

The ability of the MTHFE code to function as adirect numerical simulation for validation of simplermodels has been demonstrated. The MTH-basedcode has an ability to capture a richness of detailsabout the physical response of the system. Themethod is capable of solving more complex loadingcases, and can be extended to include different mate-rial models and nonlinear kinematics, which is ofparticular interest in the modeling of damage inenergetic materials.

Acknowledgements

This work was supported by the Center forSimulation of Advanced Rockets (CSAR) underContract number B341494 by the US Departmentof Energy. K. Matous and P.H. Geubelle alsoacknowledge support from ATK/Thiokol (ProgramManagers, J. Thompson and Dr. I. L. Davis). H.Tan and Y. Huang acknowledge additional supportfrom ONR Composites for Marine Structures Pro-gram (Grant N00014-01-1-0205, Program ManagerDr. Y.D.S. Rajapakse).

References

Bencher, C.D., Dauskardt, R.H., Ritchie, R.O., 1995. Micro-structural damage and fracture processes in a composite solidrocket propellant. Journal of Spacecraft and Rockets 32 (2),328–334.

Bensoussan, A., Lions, J.L., Papanicolaou, G., 1978. AsymptoticAnalysis for Periodic Structures. North-Holland.

Benveniste, Y., Aboudi, J., 1984. A continuum model for fiberreinforced materials with debonding. International Journal ofSolids and Structures 20, 935–951.

Budiansky, B., 1965. On the elastic moduli of some heteroge-neous materials. Journal of the Mechanics and Physics ofSolids 13, 223–227.

Eshelby, J.D., 1957. The determination of the elastic field of anellipsoidal inclusion, and related problems. Proceedings of theRoyal Society of London A 241, 376–396.

Fish, J., Shek, K., Pandheeradi, M., Shephard, M.S., 1997.Computational plasticity for composite structures based onmathematical homogenization: theory and practice. Com-puter Methods in Applied Mechanics and Engineering 148,53–73.

Geubelle, P.H., Baylor, J.S., 1998. Impact-induced delaminationof composites: a 2D simulation. Composites Part B 29B,589–602.

Ghosh, S., Lee, K., Moorthy, S., 1995. Multiple scale analysis ofheterogeneous elastic structures using homogenization theoryand Voronoi cell finite element method. International Journalof Solids and Structures 32 (1), 27–62.

Guedes, J.M., Kikuchi, N., 1990. Preprocessing and postpro-cessing for materials based on the homogenization methodwith adaptive finite element methods. Computer Methods inApplied Mechanics and Engineering 83, 143–198.

Hashin, Z., Shtrikman, S., 1962. A variational approach to thetheory of the elastic behaviour of polycrystals. Journal of theMechanics and Physics of Solids.

Hill, R., 1965. A self-consistent mechanics of composite materi-als. Journal of the Mechanics and Physics of Solids 13, 213–222.

Ide, K.M., Ho, S.-Y., 1999. Fracture behaviour of acceleratedaged solid rocket propellants. Journal of Materials Science 34,4209–4218.

Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D., 2003.Determination of the size of the representative volumeelement for random composites: statistical and numericalapproach. International Journal of Solids and Structures 40,3647–3679.

Matous, K., Geubelle, P., 2006. Multiscale modeling of particledebonding in reinforced elastomers subjected to finite defor-mations. International Journal for Numerical Methods inEngineering 65, 190–223.

Matous, K., Inglis, H.M., Gu, X., Rypl, D., Jackson, T.L.,Geubelle, P.H., 2006. Multiscale modeling of solid propellants:From particle packing to grain failure. Composites Science andTechnology, doi:10.1016/j.compscitech.2006.06.017.

Mori, T., Tanaka, K., 1973. Average stress in matrix and averageelastic energy of materials with misfitting inclusions. ActaMetallurgica 21, 571–574.

Rae, P.J., Goldrein, H.T., Palmer, S.J.P., Field, J.E., Lewis, A.L.,2002a. Quasi-static studies of the deformation and failure ofb-HMX based polymer bonded explosives. Proceedings of theRoyal Society of London A 458, 743–762.

Rae, P.J., Palmer, S.J.P., Goldrein, H.T., Field, J.E., Lewis, A.L.,2002b. Quasi-static studies of the deformation and failure ofPBX 9501. Proceedings of the Royal Society of London A458, 2227–2242.

Raghavan, P., Moorthy, S., Ghosh, S., Pagano, N., 2001.Revisiting the composite laminate problem with an adaptivemulti-level computational model. Composites Science andTechnology 61, 1017–1040.

http://dx.doi.org/10.1016/j.compscitech.2006.06.017


Rypl, D., Bittnar, Z., 2002. Hybrid method for generation ofquadrilateral meshes. Engineering Mechanics 9, 49–64.

Sanchez-Palencia, E., 1980. Non-Homogeneous Media andVibration Theory. Lecture Notes in Physics No. 127.Springer-Verlag.

Tan, H., Huang, Y., Liu, C., Geubelle, P.H., 2005a. The Mori–Tanaka method for composite materials with nonlinearinterface debonding. International Journal of Plasticity 21,1890–1918.

Tan, H., Liu, C., Huang, Y., Geubelle, P.H., 2005b. The cohesivelaw for the particle/matrix interfaces in high explosives.Journal of the Mechanics and Physics of Solids 53, 1892–1917.

Tan, H., Huang, Y., Liu, C., Geubelle, P.H., 2006. The effect ofnonlinear interface debonding on the constitutive model ofcomposite materials. International Journal for MultiscaleComputational Engineering 4, 147–167.

Timoshenko, S.P., Goodier, J.N., 1970. Theory of Elasticity,Third ed. McGraw-Hill.

Trumel, H., Dragon, A., Fanget, A., Lambert, P., 2001. Aconstitutive model for the dynamic and high-pressure behav-iour of a propellant-like material: Part I: Experimentalbackground and general structure of the model. InternationalJournal for Numerical and Analytical Methods in Geome-chanics 25, 551–579.

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