Multiscale modeling of solid propellants: From particle ...kmatous/Papers/CST_PPF.pdfMultiscale modeling of solid propellants: From particle packing to failure ... [14,15] in their

COMPOSITES

www.elsevier.com/locate/compscitech

Composites Science and Technology 67 (2007) 1694–1708

SCIENCE ANDTECHNOLOGY

Multiscale modeling of solid propellants: From particle packingto failure

K. Matous a,c,*,1, H.M. Inglis a,2, X. Gu a,2, D. Rypl b,3, T.L. Jackson a,2, P.H. Geubelle c,1

a Center for Simulation of Advanced Rockets, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USAb Department of Structural Mechanics, Czech Technical University in Prague, Prague 160 00 P6, Czech Republic

c Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Received 17 January 2006; received in revised form 24 May 2006; accepted 29 June 2006Available online 1 September 2006

Abstract

We present a theoretical and computational framework for modeling the multiscale constitutive behavior of highly filled elastomers,such as solid propellants and other energetic materials. Special emphasis is placed on the effect of the particle debonding or dewettingprocess taking place at the microscale and on the macroscopic constitutive response. The microscale is characterized by a periodic unitcell, which contains a set of hard particles (such as ammonium perchlorate for AP-based propellants) dispersed in an elastomeric binder.The unit cell is created using a packing algorithm that treats the particles as spheres or discs, enabling us to generate packs which matchthe size distribution and volume fraction of actual propellants. A novel technique is introduced to characterize the pack geometry in away suitable for meshing, allowing for the creation of high-quality periodic meshes with refinement zones in the regions of interest. Theproposed numerical multiscale framework, based on the mathematical theory of homogenization, is capable of predicting the complex,heterogeneous stress and strain fields associated, at the microscale, with the nucleation and propagation of damage along the particle–matrix interface, as well as the macroscopic response and mechanical properties of the damaged continuum. Examples involving simpleunit cells are presented to illustrate the multiscale algorithm and demonstrate the complexity of the underlying physical processes.� 2006 Elsevier Ltd. All rights reserved.

Keywords: A. Particle-reinforced composites; B. Debonding; B. Microstructure; C. Damage mechanics; Mathematical homogenization

1. Introduction

Particulate elastomeric matrix composites are usedtoday in a variety of applications such as solid rocket pro-

0266-3538/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2006.06.017

* Corresponding author. Address: Center for Simulation of AdvancedRockets, University of Illinois at Urbana-Champaign, Urbana, IL 61801,USA. Tel.: +1 217 333 8448; fax: +1 217 333 8497.

E-mail addresses: [email protected] (K. Matous), [email protected](H.M. Inglis), [email protected] (X. Gu), [email protected] (D. Rypl),[email protected] (T.L. Jackson), [email protected] (P.H. Geubelle).

1 Supported by the US Department of Energy under contract number

2 Supported by the US Department of Energy under contract numberB341494.

3 Supported by the Grant Agency of the Czech Republic under contractnumber GACR 103/05/2315.

pellants, energetic materials and automobile tires. The verycomplex mechanical behavior of these materials is drivenby the complicated microstructure and physical processesoccurring in the body at multiple length scales undermechanical loading. A key source of such processesappears to be associated with microcracks initiating andgrowing along the matrix–particle interface and with thenucleation and coalescence of voids in the matrix.

How to model these complex materials and damage pro-cesses has been a long standing quest. Ha et al. [1] andSimo [2] studied viscoelasticity with growing damage;Mullins [3] discovered the Mullins effect, Farris [4] studiedvacuole formation and growth and Christensen et al. [5]proposed models for stiffness reduction in coated particlesand/or fibers. Interface friction or sliding theories have alsobeen investigated by Hutchinson et al. [6]; Hashin [7]

mailto:[email protected]






K. Matous et al. / Composites Science and Technology 67 (2007) 1694–1708 1695

provided analytical expressions of local and overall stressand strain fields as functions of the phase moduli andinterfacial properties. Another group of models publishedin recent years devoted to progressive interfacial decohe-sion and/or progressive volumetric damage has been pro-posed by Matous [8], Krajcinovic [9] and Chaboche et al.[10]. Zhong and Knauss employed a cohesive model toresolve the debonding of rigid particles from an elastomericmatrix [11,12]. Also, three-dimensional periodic cells wererecently used by Michel et al. [13] and Llorca and Segurado[14,15] in their study of elastic and elasto-plastic particulatecomposites. In their work, Llorca and Segurado consideredsimple periodic boundary conditions without interfacialdamage and no mathematical theory of homogenization(MTH) was employed. Moreover, difficulties with meshgeneration were discussed. Three faces of a cube weremeshed with quadratic triangles, and the meshes were man-ually copied to the opposite sides. Such an approach isbypassed here and novel meshing of periodic structures isproposed as part of an automated numerical frameworkto model particulate composites from packing to failure.Note that the emphasis of this paper is not to investigatethe size of the representative volume element. Instead, weinvestigate hereafter the damage response of a periodic unitcell (PUC).

The mathematical/numerical framework described inthis paper consists of several fully automated components.First, a packing algorithm developed in [16,17] is used tocreate the characteristic pack matching the particledistribution and volume fraction. Next, a pre-processingprocedure is applied to create a geometric model withrefinement zones in the regions of interest. Once thepre-processing is performed and the geometric model iscreated, we employ the T3d mesh generator developedby Rypl [18,19] to discretize a periodic unit cell. T3d isbased on an advancing front technique and can meshcomplex two-dimensional (2D) and three-dimensional(3D) domains into triangular and tetrahedral meshes ofa high quality. Moreover, mixed meshes consisting ofquadrilaterals and triangles are allowed for 2D domains.This feature is of great interest, since meshes composedof triangles are known to be subject to locking for nearlyincompressible materials. In this work, with examplesdedicated primarily to the 2D case, the B (Bbar) proce-dure [20] is used to eliminate the pressure instabilitiesassociated with material incompressibility. In particular,Q1/P0 elements, where the displacement field is approxi-mated by a bilinear function and the pressure is constantover the element, are used and despite the fact that thiselement violates the Babuska-Brezzi stability conditions,an optimal convergence rate can be proven under suitableassumptions (see [20] for more details). A novel stabilizedfinite element framework was used for 3D finite strainanalysis by Matous and Geubelle [21]. The mechanicalbehavior of the interface between particles and a blend(composed of the elastomeric matrix and the particlestoo small to be incorporated in the unit cell) is modeled

by a cohesive law, which is embedded in the finite elementimplementation of the mathematical theory ofhomogenization.

The mathematical theory of homogenization, includingthe cohesive model, is described in Section 3. This theoryis very popular in computational multiscale modeling andhas been used by several researchers [22,23]. The finitestrain counterpart of this theory was described in [21].Next, various 2D examples involving the multiaxial load-ing of an idealized propellant are presented, showing theability of the proposed multiscale scheme to relate the dam-age processes taking place at the microscale to the macro-scopic constitutive relations. As illustrated in theseexamples, one of the key aspects of the proposed schemeis its ability to capture the complex interactions betweenthe closely packed particles.

The symbolic notation adopted herein uses upper caseboldface italic and lower case boldface Greek letters, e.g.,P and r for second-order tensors. The trace of the sec-ond-order tensor is denoted as tr(A), and the tensor oper-ations between two second-order tensors S and E areindicated as SE for a contraction of tensors (a second-ordertensor) or S:E for the scalar product (a double contrac-tion). A quantity marked by an underline, �, denotes thequantity � at the macroscale. Other notational conventionsadopted in this paper are introduced as they are used.

2. Packing algorithm, pre-processing and meshing

procedures

To achieve an accurate description of the damageevolution in heterogeneous materials, it is often essentialto perform the microscale analysis on a representativevolume element that adequately captures the heteroge-neous microstructure. This computational description ofthe representative volume element is particularly chal-lenging in the case of highly packed particulate compos-ites, such as solid propellants, due to the geometricalconstraints associated with the small inter-particle dis-tances and with the range of particle sizes needed toachieve the high particle volume fraction. To that effect,we have developed a fully automated process thatinvolves a dynamic packing algorithm used to create acomputational description of the embedded particles, apre-processing procedure to eliminate the singularitiesinherent in contacting particles, and a very general mesh-ing tool to generate a high-quality periodic finite elementdiscretization of the periodic unit cell. These three pre-processing tools are described next.

2.1. Rocpack – a dynamic packing algorithm

The heterogeneous propellants of interest consist ofammonium perchlorate oxidizer (AP) and aluminum(Al) particles embedded in a fuel polymeric binder. Typi-cal loadings are 80–87% by weight AP if no aluminum ispresent; 68–76% by weight AP and 10–18% by weight Al

1696 K. Matous et al. / Composites Science and Technology 67 (2007) 1694–1708

otherwise. It has long been recognized that various char-acteristics, such as the burning rate or the response todynamic loading, of a heterogeneous propellant are influ-enced by the propellant morphology, by the size and sizedistribution of the AP particles. It follows that any seri-ous attempt to describe the characteristics of heteroge-neous propellants must include a packing algorithm, astrategy for defining and constructing a model propellant.Lattice (or crystal) packs are easily defined, but do notreflect the random nature of a true propellant. For this,a random packing algorithm must be used, and a suitableone, called Rocpack, is defined and discussed in [16,17].This algorithm is dynamic in nature and can closely packspheres or discs of arbitrary size. The algorithm hasrecently been extended to ellipsoids to study non-isotropicpacks [24]. For completeness, we briefly describe the algo-rithm here.

The algorithm begins with an infinite computationaldomain defined by the periodic continuation of a cube,and points are randomly assigned to this domain at timet ¼ 0 with random velocities. For t > 0, the points becomespheres that grow linearly in time with a distribution ofrates that defines the final distribution of AP and Al diam-eters. Collisions between the particles are dealt with usingstraightforward but non-classical dynamics, and this pre-vents overlapping. The pack is defined by stopping the cal-culation at some assigned volume fraction (defined as thetotal volume of the spheres to the volume of the cube),or after the interval between collisions becomes too smallto continue. The output is a random distribution of sphereswith centers Yi and radii ri, where subscript ‘‘i’’ representsthe particle index.

We use the data of McGeary [25], who packed bimodalsteel shot in a cylindrical container, to validate the pack-ing code. Fig. 1 shows comparisons between the experi-mental data and the numerical data for two bimodal

60

65

70

75

80

Volu

me

Fra

ctio

n (

%)

0 20 40 60 80 10060

65

70

75

80

Fine Mode (%)

Fig. 1. Volume fraction for a coarse to fine ratio of 3.44 (top) and 4.77(bottom) bimodal mix; numerical (dash; box) and experimental (solid;circle). The fine particles have a single diameter of 360 lm.

mixes. In each case the fine particles have a single diam-eter of 360 lm, and the ratio of coarse to fine particlesis 3.44 (top panel) and 4.77 (bottom panel). In each panelthe percent of fine particle is varied from zero (all coarse;monomodal) to 100 (all fine; monomodal). Note the excel-lent agreement between experimental data and numericalresults. Also note that, for the 3.44 ratio case, we wereable to generate slightly higher volume fractions than thatobtained experimentally in the range of 10–30% finemode. The slight over prediction in packing fraction isdue to the difficulty in packing bimodal steel and compar-ison with experimental data is only informative. We donot have detailed information about experiments (numberof tests, error bars, etc.)

Packs generated in this fashion can be designed to cap-ture the volume fractions of various sized particles inexperimental or industrial packs. Fig. 2 shows an exampleof a periodic pack generated with 10,000 particles per cube,colored according to particle diameter. The binder fills thespace between particles. This pack mimics a propellantbuilt and burnt by Miller [26].

From packs as described above, a periodic unit cell ofmuch smaller size can be derived. As shown in [27], theperiodic unit cell can possesses the same material statisticsas the original characteristic pack. Hence, the computergrown pack obtained by particle packing is used as a digitalrepresentation of a microcontinuum.

2.2. Pre-processing procedure

Although the output from the packing algorithmdescribed above is geometrically very simple, it is unsuit-able to be used directly in a numerical framework as con-tacting particles would lead to singularities. Moreover, asindicated earlier, the proximity of adjacent, closelypacked particles poses some important challenges in the

Fig. 2. A typical randomly generated propellant pack with 10,000 spheres,colored according to particle diameter.

4 Note that T3d is based on a Bezier representation of curves andsurfaces. Thus, a surface is defined by a list of clockwise or counterclock-wise oriented boundary curves each of which is in turn topologicallydefined by a starting and an ending vertex.


finite element discretization of the periodic unit cell,because very small elements need to be used in someregions of the cell. A uniform finite element discretizationwould result in a prohibitively large number of degrees offreedom. Therefore, a more sophisticated approach needsto be adopted to mesh a unit cell. In the present work,the geometry is characterized by rational Bezier curvesand/or surfaces, and the general meshing tool T3d

developed by Rypl [18,19] is employed to discretize themicrostructure as described below. This meshing frame-work can automatically provide non-uniform 2D–3Dmeshes with refinement zones in regions of interest, suchas particles in close proximity, where the stress concentra-tion will be pronounced and a fine mesh is required. Italso allows for the generation of periodic meshes neededto enforce the periodicity of the microscopic solutionsobtained as part of the mathematical theory ofhomogenization.

A pre-processing algorithm is developed to create thegeometric model from the Rocpack output: shrink particlesin contact with other particles and/or the reference box andcompute the refinement parameters. The separation dis-tance between particles i and j is computed simply as

xij ¼ kY i � Y jk � ðri þ rjÞ: ð1Þ

Whenever xij < a, the radii of the corresponding particles i

and j are modified according to

rnewi ¼ ri � ða� xijÞ

ri

ri þ rj;

rnewj ¼ rj � ða� xijÞ

rj

ri þ rj:

ð2Þ

Here, a represents a user-defined minimum separation be-tween the surfaces of adjacent particles.

The volume change associated with particle shrinkingis added to the blend, which consists of a binder andsmall particles not accounted for in the packing proce-dure. Then, the spatially-dependent refinement factorgij 2 (0,1), which modifies the user-prescribed element sizeh, is given by

gij ¼0:1þ 0:9

b�a ðxij � aÞ if a 6 xij 6 b;

1:0 if xij P b;

(ð3Þ

where b is the maximum distance influenced by refinement(second user-defined parameter). Thus, the geometricmodel and the resulting mesh depend on two parametersa and b, and the default mesh size h, respectively. Allparameters a, b, h must be suitably chosen; therefore, thepre-processing and meshing procedures require some trialand error to determine the minimum and maximum parti-cle radius, mesh density, etc.

2.3. Meshing tool – periodic mesh

One possible choice for prescribing the periodicboundary conditions required in the mathematical theory

of homogenization is to assign the same deformationidentification numbers to the corresponding degrees offreedom related to nodes on opposite sides of the unitcell. Thus, the periodic constraints are imposed directlyon the discrete unknowns in the finite element solutionprocedure. We have thus modified T3d to discretize pairsof model surfaces on opposite sides of the PUC by topo-logically and geometrically identical meshes. However,this can only be successfully accomplished under certainconditions. First, the opposite surfaces must be geometri-cally identical. Second, they must be topologically identi-cal, i.e., the ordering of curves bounding the surfaces andtheir orientation with respect to the surface normal mustbe exactly the same.4 Both conditions are directly satis-fied by the adopted pre-processing procedure.

The discretization engine T3d is based on the advanc-ing front technique and can provide high-quality unstruc-tured surface and solid meshes. Assuming that theopposite surfaces are topologically and geometricallyidentical, one may get the impression that exactly thesame meshes must be produced if the advancing fronton both surfaces is initialized identically. But this is trueonly if the same mesh density control is used and if pre-cise arithmetics without round-off error is considered.However, neither of these two conditions are usually sat-isfied. Therefore, a different approach based on mirroring

surface meshes on opposite sides of the PUC has beenadopted. This approach requires that there is a parametricspace associated with each of the mirrored surfaces (mir-ror master surface on one side and mirror slave surface onthe opposite side of the PUC) and that the mappingbetween the parametric and real spaces is identical forboth of them. This is again achieved by the pre-processingprocedure that sets up the pairs of opposite mirroredsurfaces.

The mirror master surface is discretized in the stan-dard way using the advancing front technique [28]. Onthe mirror slave surface, nodes with the same parametriccoordinates as their counterparts on the mirror mastersurface are created (Fig. 3). Their real coordinates aregiven by the mapping applied to the geometry of themirror slave surface. Each of the nodes on the mirrormaster surface temporarily stores a pointer to its coun-terpart on the mirror slave surface. Similar pointers arealso established on curves bounding the mirror mastersurface. The final mesh on the mirror slave surface isthen generated by traversing the elements on the mirrormaster surface and creating for each of them a new ele-ment on the mirror slave surface using the nodes beingstored as pointers at the nodes of the original elementon the mirror master surface (Fig. 3). After all elementson a particular mirror slave surface are formed, the

Fig. 4. 2D cell geometry of an idealized bimodal propellant PUC.

real

spa

cepa

ram

etri

c sp

ace

mirror master mirror slave

Fig. 3. Mirroring of surface mesh. The arrows visualize the flow ofinformation to create nodes (empty arrows) and to form elements (fullarrows) on the mirror slave surface.


pointers to slave nodes on the mirror master surface andits boundary curves are discarded. In this way, topolog-ically identical meshes of the same geometry are pro-duced for all pairs of mirrored surfaces. Note that thediscretization of boundary curves of mirrored surfaces,which actually precedes the surface discretization, is per-formed in a similar manner. In this case, however, themirroring of appropriate pairs of curves (derived fromthe mirroring of opposite surfaces) is performed auto-matically by T3d.

A similar meshing strategy for PUCs was employed byWentorf et al. [29]. In our framework, however, discreti-zation is performed directly in the real space and theparametric space is used only for the mirroring of nodes.This concept avoids the demanding generation of aniso-

Fig. 5. Bezier curve representation (left) and refinem

tropic meshes in the parametric space, where distortionand stretching induced by the mapping may generallyoccur.

As mentioned in Section 1, the proposed numericalframework is based on a finite element scheme in whichconventional (volumetric) elements are used to describethe response of the particles and the binder, while interfa-cial (cohesive) elements are introduced along the particle/matrix interface to model progressive interfacial failureduring the debonding (or dewetting) process. Therefore,construction of cohesive elements is also necessary. Thisis enabled in the T3d framework by creating two identicalcurves in 2D and/or surfaces in 3D along which the cohe-sive elements are inserted. The same mirroring procedureas for the periodic sides of the unit cell is used to createtwo overlapping identical curve/surface meshes.

ent octree (right) of 2D PUC shown in Fig. 4.

Fig. 6. High-quality mixed periodic mesh, composed of quadrilaterals and triangles.

Table 1Finite element discretization for the 2D unit cell

nn neQ neT nce

Periodic mesh 7896 6146 184 1574

Fig. 7. An idealized bimodal propellant. 3D cell geometry and hig


2.4. Examples – packing/pre-processing/meshing

We next demonstrate the packing/pre-processing/mesh-ing procedures on an idealized propellant, with bimodal

h-quality triangular (surface) and tetrahedral (volume) meshes.

Y

X1

X2

X3

ς

ςY = X /

Macroscale

Y

PUCsY

zoom

ΩX1

Y 2

3

Θ Θ Θ

ΘΘ

t

u

Microscale Y periodic

Fig. 8. Microscopic and macroscopic structures.


particle distribution in a 2D periodic unit cell. Note thatboth packing and meshing can be executed without restric-tions in 3D with features such as particle shrinking andnon-uniform mesh size control fully operational. More-over, the periodic boundary meshes for sides with cut par-ticles would be identical to the 2D meshes presented here,since the only allowable geometrical entity after sphere cut-ting is a circle. A sphere in contact with the boundary boxwould be scaled down according to Eq. (2).

Fig. 4 shows an idealized 2D pack obtained from thepacking algorithm (Rocpack). The pack is composed of abimodal distribution of particle sizes and involves some con-tacting particles. After executing the pre-processingalgorithm described above, we obtain a geometrical descrip-tion based on rational Bezier curves/surfaces, where parti-cles in contact with other particles and/or the referencebox are modified according to Eq. (2) and a refinementparameter is set by Eq. (3). The 2D unit cell of dimensions693 · 693 lm contains a bimodal pack, with diameters100 lm (30% by volume of total particles) and 60 lm (70%by volume of total particles). The total particle volume frac-tion is 0.433. The original volume fraction of particles beforeparticle shrinking was 0.451. The particle contraction hastherefore led to a 3.9% change in the particle volume frac-tion. The error in converting the rational Bezier curves/sur-faces to the finite element mesh in terms of volume fractionof particles is neglected due to high mesh refinement.Fig. 5 shows the resulting geometry and the refinementoctree that governs the element size in the meshing proce-dure. The non-uniform mesh generated is displayed inFig. 6. The characteristics of the 2D periodic mesh are listedin Table 1, where nn is the number of finite element nodes,neQ represents the number of volumetric quadrilateral ele-ments, neT denotes the number of volumetric triangular ele-ments, and nce is the number of cohesive elements.

An idealized 3D pack is obtained in a similar manner, asshown in Fig. 7. The unit cell consists of 100 reinforcingparticles with a bimodal particle size distribution and themesh contains 1,072,210 elements and 190,311 nodes.

3. Multiscale formulation of the damage response

Here, we describe the mathematical theory of homogeni-zation and some aspects of its finite element implementa-tion as used in this work to relate the complex damageprocesses taking place at the microscale to the homoge-nized macroscopic response of the reinforced elastomer.The formulation includes the cohesive modeling of parti-cle–matrix debonding, but is limited to the small strainregime. The formulation presented hereafter is fully 3D,although the illustrative examples described in Section 4are solved in a 2D (plane strain) setting.

3.1. Governing equations

Let us consider a periodic (Y-periodic) composite mate-rial with the microstructural period defined by the PUC

created in Section 2 and denoted by H, as illustrated inFig. 8. Next, consider X 2 R3 to be the position of aparticle in the reference macroscopic configurationX � R3 in the Cartesian co-ordinate system and Y = X/1to be the microscopic position vector in H � R3 in theCartesian co-ordinate system. Hence, 1 denotes a verysmall positive number that, roughly speaking, correspondsto the size of the microstructure. X is commonly referred toas the slow variable while Y is called the fast variable. Theheterogeneity is enclosed by a cohesive surface S, and weorient the cohesive surface by a unit normal N. For anymacroscopically periodic function / we have /ðX ;YÞ ¼/ðX ;Y þ k bY Þ in which vector bY is the basic period ofthe microstructure and k is a 3 · 3 diagonal matrix withinteger components. Adopting the classical nomenclature,any locally Y-periodic function / can be represented as

/1ðXÞ � /ðX ;YðXÞÞ; ð4Þwhere the superscript ‘‘1’’ denotes a Y-periodic function /.

The corresponding boundary value problem at the micro-scale, including the cohesive description of the interface, isdetermined by the following set of governing equations:

divðr1Þ þ f 1 ¼ 0 in X1;

r1 ¼ L�1 in X1;

�1 ¼ rsu1 in X1;

r1 �N ¼ t on Ct;

u1 ¼ �u on Cu;

br1 �Ne � bt1e ¼ 0 on S1;

ð5Þ

where div(•) is the divergence operator, $s represents thesymmetric gradient operator, and r1 and �1 are stress andstrain tensors, respectively. L(X,Y) denotes the spatiallydependent symmetric material stiffness tensor, f(X,Y) de-notes the body forces and t(X) represents the prescribedmacroscopic tractions on the boundary Ct. We also con-sider Dirichlet boundary conditions �u on Cu, such thatC = Ct [ Cu. Moreover, the symbol b•e denotes the jumpof a quantity • across the cohesive surface. The cohesive


law used in this work to relate the cohesive traction actingalong S1 to the associated displacement jump vector bu1e isdescribed later in this section.

3.2. Mathematical theory of homogenization

We start by approximating the displacement fieldu1(X) = u(X,Y) in terms of the two-scale asymptotic expan-sion on X · H as

uðX ;YÞ � 0uðX ;YÞ þ 11uðX ;YÞ þ � � � ; ð6Þwhere left superscripts ‘‘0,1, . . .’’ represent the asymptoticorder. Employing the indirect macroscopic spatial deriva-tives, rs

X ¼ rsX þ 1

1rsY , the strain tensor reads

�ðX ;YÞ � 1

1rs

Y0uþ 10 rs

X0uþrs

Y1u

� �þ � � � ; ð7Þ

yielding strain tensors of various orders

ð�1Þ� ¼ rsY

0u; 0� ¼ rsX

0uþrsY

1u� �

; ð8Þ

where rsX and rs

Y , respectively, denote the symmetric partof the gradient with respect to X and Y coordinates. In thissmall strain linearly elastic setting, the stress and strain ten-sors for different orders of 1 are related by the constitutiveequations

ð�1Þr ¼ Lð�1Þ�; 0r ¼ L0�; ð9Þand the resulting asymptotic expansion of the stress yields

rðX ;YÞ � 1

1ð�1Þrþ 0rþ � � � ð10Þ

Applying standard variational methods, the principle ofvirtual work readsZ

X1r1 :rs dudXþ

ZS1

t1 � bduedS�Z

X1f 1 � dudX�

ZCt

t � dudA¼ 0

ð11Þfor all admissible variations du satisfying

du 2 ½H 1ðX1Þ�N; du ¼ 0 on Cu; ð12Þwith N being the space dimension and H1 representing theSobolev space.

Introducing the asymptotic expansions (6) and (10) into(11), taking the limit when 1! 0+ and making use of theindirect spatial derivatives, the principle of virtual work(11) holds in terms of the same powers of 1 as

Oð1�2Þ :1

jHj

ZX

ZH

ð�1Þr : rsY dudHdX ¼ 0;

Oð1�1Þ :1

jHj

ZX

ZH

ð�1Þr : rsX duþ 0r : rs

Y du� �

dHdX

þ 1

jHj

ZX

ZS

t � bduedSdX ¼ 0;

Oð10Þ :1

jHj

ZX

ZH

0r : rsX dudHdX� 1

jHj

ZX

ZH

f � dudHdX

�Z

Ct

t � dudA ¼ 0 ð13Þ

for "du 2 VX·H, where

V X�H ¼ fduðX ;YÞjduðX ;YÞ 2 H 1ðXÞ � L2ðX; V HÞ;duð�;YÞ is Y -periodic; dujCu

¼ 0g;V X ¼ fduðXÞjduðXÞ 2 H 1ðXÞ; dujCu

¼ 0g;V H ¼ fduðYÞjduðYÞ 2 H 1ðHÞ; duðYÞ is Y -periodicg;V H ¼ V H=R:

ð14ÞHere, the V H is a quotient space, the uniqueness of 1umeans that 1u(X,Y) is uniquely defined up to the additionof an arbitrary function of X. For more information onthe space selection please see [30,31]. The following integra-tion rules have been applied to integrate any Y-periodic

function • [30,23]:

lim1!0þ

ZX1ðX=1ÞdX! 1

jHj

ZX

ZHðYÞdHdX;

lim1!0þ

1Z

S1ðX=1ÞdS ! 1

jHj

ZX

ZSðYÞdS dX:

ð15Þ

Let us first consider the Oð1�2Þ Eq. (131) and choose du =du(Y), i.e., du 2 VH. Then, integrating by parts, applyingthe divergence theorem and noting that the terms on theopposite faces of the unit cell cancel due to the periodicitycondition, one has

rsY � ð�1Þr ¼ 0 and 0u ¼ 0uðXÞ ) ð�1Þ� ¼ ð�1Þr ¼ 0:

ð16ÞUsing the results from Eq. (16), the Oð1�1Þ equation withdu = d1u 2 VH yieldsZ

X

1

jHj

ZH

0r : rsY d1u dHþ 1

jHj

ZS

t � bd1uedS� �

dX ¼ 0;

ð17Þwhere

1u is Y -periodic on CH and d1u 2 H 1ðHÞ;d1u is Y -periodic on CH: ð18Þ

In principle, Eq. (17) represents the weak form of the equi-librium on the microscale H for purely kinematic boundaryconditions. Moreover, to satisfy the equilibrium with theassumed neighboring PUCs, the stress on the boundaryCH will also be periodic.

Finally, the Oð10Þ equation with du = d0u 2 VX denotesthe weak form of the equilibrium equation on themacroscaleZ

Xr : rd0u dX�

ZX

f � d0u dX�Z

Ct

t � d0udA ¼ 0; ð19Þ

where the macroscopic stress tensor and body force vectorare given by

r ¼ 1

jHj

ZH

0rdH; f ¼ 1

jHj

ZH

f dH; ð20Þ

with


0u ¼ �u on Cu and d0u 2 H 1ðXÞ; d0u ¼ 0 on Cu: ð21ÞNote that to apply the boundary conditions properly,since the periodicity breaks down close to oX1, one wouldneed to introduce special boundary layer terms as sug-gested in [31] and solve additional PDEs on a half spacewith exponentially decaying coefficients. Here, we assumethat the boundary layer does not influence the solutionand that prescribed displacements are a macroscopicquantity only.

With the aid of Eqs. (8), (9) and (20), the first term inEq. (19) can be written asZ

Xr : rd0udX

¼Z

X

1

jHj

ZH

L rSX

0u|fflffl{zfflffl}�

þrSY

1u|ffl{zffl}~�

0B@1CAdH

264375 : rd0u

8><>:9>=>;dX;

ð22Þwhere �ðXÞ ¼ rS

X0u denotes the macroscopic strain ten-

sor (recall that 0u = 0u(X)), and ~�ðX ;YÞ ¼ rSY

1u repre-sents the microscopic fluctuation strain. Usingseparation of variables and introducing the fourth-ordermechanical damage tensor, A, the asymptotic expansionof the strain (7) can be written in terms of the macro-scopic strain � as

� ¼ ½I þG�� and ~�ðX ;YÞ ¼ GðYÞ�ðXÞ; ð23Þwhere A ¼ ½I þG� and I denotes the fourth-order identitytensor, while GðYÞ is the Y-periodic damage polarizationfunction. Note that the mechanical damage tensor A con-tains both the elastic concentration function, which arisesdue to the material heterogeneity, and the damage transfor-mation function, which originates due to the gradual deb-onding along the particle matrix interface and is relatedto the displacement jump introduced in Section 3.3. Sepa-ration of these effects is non-trivial and would require theintroduction of a certain eigenstrain field. Therefore, wehave combined both effects into one tensor, which is hand-ily numerically computed.

Substituting (23) back into (22), we arrive atZX

r : rd0udX ¼Z

X½L�� : rd0udX; ð24Þ

where the instantaneous macroscopic stiffness tensor is gi-ven by

L � 1

jHj

ZH

LA dH; ð25Þ

and the macroscopic stress–strain law yields

r ¼L�: ð26ÞNote that the instantaneous macroscopic stiffness tensoris influenced by inhomogeneous material variations andthe cohesive interface. Also, it can be proven that Lpossesses the same minor and major symmetries as L[32].

3.3. Irreversible cohesive law

Cohesive models are popular in computational mechan-ics and have been extensively used in many different con-texts [33–35]. The cohesive law adopted in this work tomodel the progressive failure of the particle–binder inter-face is the simple bilinear law used by Geubelle and Baylor[36], in which the normal and tangential components of thecohesive tractions, tn and tt, are related to the normal andtangential opening displacements, vn and vt, through

tn ¼rc

cin

c1�c

vn

vnc; vn P 0;

sc

1�cin

vn

vnc; vn < 0;

(ð27Þ

tt ¼sc

cin

c1� c

vt

vtc

; ð28Þ

where

vn ¼ b1ue �N ; vt ¼ ð1�N NÞb1ue: ð29ÞRecall that b1ue denotes the jump of the fluctuating dis-placement across the cohesive surface and N representsthe normal to the cohesive surface. The traction vector tintroduced in Eq. (5) is, as expected, constructed ast = RTtl, where tl = (tt1, tt2, tn)T and R defines the orthogo-nal transformation from the global reference frame to theelement specific (local) coordinate system. Standard iso-parametric element procedure yields the required rotationmatrix R. The parameters rc and sc denote the tensileand shear interface strengths, respectively, while vnc

andvtc

represent the normal and tangential critical opening dis-placements. The normal and tangential directions are cou-pled through the damage parameter, c, defined by

c ¼ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivn

vnc

�2

þ jvtjvtc

�2s* +

8vn P 0; ð30Þ

with h•i denoting McAuley brackets. As apparent in Eqs.(27) and (28), a decreasing value of c leads to an increas-ingly compliant cohesive interface. c is initially set to a va-lue cin, taken to be 0.98. Once c < cin, the traction carriedby the interface begins to reduce and the interface is consid-ered to be locally damaged. The interface continues to de-grade with decreasing c and increasing separation of theinterface until c = 0. At this point, the local interface isconsidered to be completely failed. In the case of compres-sion (vn < 0), no damage is accumulated. The cohesivemodel described above is irreversible; c is only allowed todecrease monotonically so that, even if the interface is un-loaded, it retains the current state of damage and unload-ing is towards the origin. The bilinear law is representedgraphically in Fig. 9.

3.4. Numerical implementation

Euler–Lagrange equations on the microscale Oð1�1Þ(132) and on the macroscale, Oð10Þ (133), are solved witha conventional finite element scheme. A B procedure is

τ

n

ncn

tt t

ccσ τ

c

χχ χtc

χt

Fig. 9. Bilinear cohesive law for normal separation (left) and tangentialfailure (right). The dashed line indicates an unloading path prior tocomplete failure. The normal and shear failures are coupled through thedamage parameter c, defined in Eq. (30).

Table 2Material properties

Constituent E (MPa) m

Particles 32,450 0.1433Binder 7.39 0.4991

Table 3Cohesive interface properties

Direction Interface strength (MPa) Critical openingdisplacement (lm)

Normal rc = 0.05 vnc¼ 0:75

Tangential sc = 0.05 vtc¼ 0:75


used to eliminate pressure oscillations and create a stable2D finite element framework for nearly incompressiblematerials. Cohesive elements are inserted at the particle–matrix interfaces and a three-point integration rule is used,with the local damage parameter c stored as a state variableat each integration point. The choice of cin < 1 introduces asmall amount of compliance into the undamaged cohesiveinterface, eliminating the numerical instabilities associatedwith rigid elements.

The numerical implementation uses a semi-implicit solu-tion procedure with adaptive load stepping to ensure effi-ciency of computational effort. When no damage isaccumulated, the model is linear, allowing for larger loadsteps. As damage increases, however, smaller load stepsare required in order to capture accurately the physics ofthe failure process, suggesting the introduction of an adap-tive load-stepping scheme. The maximum increment ofdamage accumulated at any integration point in a load stepis used as an error indicator to determine whether the load-step size needs to be reduced or increased. If the accumu-lated damage c is greater than a certain threshold, cmax,the load-step size is reduced and the current load step isrestarted. This is similar to a predictor–corrector method.If the damage accumulated is less than cmin, the load-stepsize for future load steps is increased. Typically, the valueused for cmin is half that for cmax, which is in the rangeof 0.01–0.001.

4. Examples

To illustrate the proposed multiscale numerical frame-work, we analyze the damage evolution in the model 2Dparticulate composite system shown earlier. To analyzemore representative unit cells, high-performance parallelcomputing is necessary due to the size and complexity ofthe finite element model. Finite element meshes are createdas described in Section 2. All constituents are assumed tobe isotropic linearly elastic solids. The elastic material char-acteristics (stiffness E and Poisson’s ratio m) of the particlesand of the homogenized blend (consisting of the binder andsmall particles) are listed in Table 2. Note the very high

stiffness mismatch between the particle and the blend andthe near incompressibility of the blend, as quantified bythe Poisson’s ratio approaching 1/2. The interface proper-ties, given in Table 3, have been chosen to ensure that fail-ure occurs within the limit of small strains. The stresses inthe surrounding material are influenced by the strength ofthe cohesive interface. Therefore, low macroscopic stressesare observed in the results presented below.

Instead of solving the macroscopic boundary valueproblem explicitly, Oð10Þ Eq. (132), we impose hereafter apredefined deformation path of a macroscale materialpoint as in the microhistory recovery procedure proposedby Fish et al. [37]. Three loading cases are considered here,characterized by the following macroscopic strains, �:

�A ¼0:01 0:00

0:00 �0:01

� �; �B ¼

0:01000 0:00000

0:00000 �0:00982

� �;

�C ¼0:00 0:02

0:02 0:00

� �:

ð31ÞThe third row and column of the macroscopic strain ten-sors are not shown as these are zero under plane strain con-ditions. Cases A and C are volume preserving macroscopicstrains and are equivalent in terms of principal stresses.Case B is a quasi-uniaxial stress, which is achieved byimposing a strain which is a slight deviation from load caseA in order to study the dilatational stress component (pres-sure p = 1/3tr(r)) resulting from the attempt to force anearly incompressible cell to deform in a non-volume pre-serving manner.

4.1. Load cases A and B

Fig. 10 shows the macroscopic stress–strain curve (leftaxis) and evolution of interfacial damage (right axis) forthe volume preserving tension case (load case A). The solidline denotes macroscopic stress in the direction of appliedtension, r11. The dash-dotted curve corresponds to the evo-lution of cohesive interface elements which have begun toexperience damage, defined as elements for which c < cin.

Damaged interface

σ11

ε11Macroscopic strain,

Mac

rosc

opic

str

ess

[MPa

]

0.4 0.06

0.05

0.04

0.03

0.02

0.01

0.00

Frac

tion

of c

ohes

ive

inte

rfac

e

0.3

0.2

0.1

0.0

0.000 0.002 0.004 0.006 0.008 0.010

Failed interface large particlesFailed interface small particles

Fig. 10. Load case A – Macroscopic stress–strain and damage curves. Theinterface is considered to be damaged when c < cin and failed when c = 0.

0.5

0.0

0.1

0.2

0.3

0.4Damaged interface

ε11

σ11

0.25

0.20

0.15

0.10

0.05

0.00

Mac

rosc

opic

str

ess

[MPa

]

Frac

tion

of c

ohes

ive

inte

rfac

e

0.000 0.002 0.004 0.006 0.008 0.010

Macroscopic strain,


Fig. 11. Load case B – Macroscopic stress–strain and damage curves.

11σA

σ22A

σ33A

11σB

σ22B

σ33B

ε11Macroscopic strain,

0.00

0.05

0.05

0.10

0.10

0.15

0.15

0.20

0.20

0.25

0.000 0.002 0.004 0.006 0.008 0.010

Mac

rosc

opic

str

ess

[MPa

]

Fig. 12. Macroscopic stress–strain curves — Comparison between volumepreserving (case A) and quasi-uniaxial loading (case B). The largedifference in stress response with only a small difference in appliedmacroscopic strain is explained by the increased dilatational stress for loadcase B, which is not volume preserving.


The length of damaged cohesive interface is divided by thetotal interface length to obtain the damaged fraction. Thisrepresents the portion of the interface which is in the cohe-sive zone (softening part of the traction–separation curve).The dashed and dotted curves indicate the fraction of fullyfailed cohesive elements on the boundaries of large andsmall particles, respectively, where fully failed elementsare those for which c = 0.

It is clear that the initial deviation from linearity of thematerial response is a result of the onset of damage of thecohesive elements. This early damage is fairly evenly dis-tributed throughout the microstructure, with no significantdifference between interfaces associated with large andsmall particles. The locations at which damage nucleatesdepend on local stress concentrations, but not on the sizeof the particles. For this reason, only a single curve is plot-ted, representing the damage fraction over all cohesiveinterfaces. Damage of the cohesive interface saturates atthe point at which complete failure begins to be observed.

In contrast to damage nucleation, the failure of theinterfaces shows a marked size effect. As often observedexperimentally, most of the failure occurs on large particleinterfaces, and very little on interfaces associated withsmall particles. The failure process is discrete, with only afew particles failing at a time. Once failure initiates on aparticle, the increased compliance of the failing interfacecauses load redistribution such that failure does not, inmany cases, initiate elsewhere until it is complete for thatparticle.

The imposed strain for load case B, �B, was chosen bytrial and error such that r22 � 0 (uniaxial tension loadingcase). The damage and failure responses, shown inFig. 11, are very similar to those for load case A. However,the macroscopic stress–strain response is markedly differ-ent. Due to the quasi-incompressibility of the binder andthe quasi-rigidity of the embedded particles, there is a sig-nificant change in the state of stress for load case B, wherethe imposed strain is not volume preserving, resulting in a

rapid increase in the dilatational component of the stress(pressure). The large difference in macroscopic responsebetween load cases A and B is clearly seen in Fig. 12.For both cases, although 40% of the interface has begunto degrade, the macroscopic stress continues to growmonotonically. One might assume that stress softeningwould dominate the behavior once such a large percentageof the interface has degraded. This weak non-linearity isassociated in part with the constrained loading cases con-sidered. The compression of the nearly incompressibleand compliant blend against the hard particles also servesto prevent the propagation of the debonding cracks.

If we exclude dilatational stress from our analysis byconsidering a deviatoric measure of the stress field, suchas the von Mises stress, we find that load cases A and Bare nearly identical. The von Mises stress distribution forload case A is shown on the deformed mesh in Fig. 13.

[MPa]0.60

0.45

0.30

0.15

0.00

1

2

34

Fig. 13. Load case A – von Mises stress distribution plotted on thedeformed configuration with displacements magnified 10 times. Thenumbers refer to comments in the text.


The deformation is magnified 10 times, so that the openingof cohesive interfaces is easily visible. Void opening isaligned with applied load, i.e., on the left and right edgesof the particles (box 1), however, in box 2, we note that astress concentration due to particle proximity causes thevoids to open on a slightly different face. Stress concentra-tions at the crack tips can be seen, in box 3 particularly.Note the heterogeneous stress distribution in the matrixmaterial and the periodicity of the boundaries.

As expected from the macroscopic response, failureoccurs predominantly on large particles. Small particles failprimarily when they are in an area of local stress concentra-tion, e.g., close to another particle. Examples of such fail-ures can be found in Fig. 13, with particle pairs failing intension (box 1) or in shear (box 4).

If we compare results from the same load case appliedto different unit cells (Fig. 14), we see a minor deviation inthe macroscopic stress. Since the emphasis of this paper isnot to investigate the size of the representative volumeelement, we make no claim on representativeness of thiscomparison. One would need to compute the bounds onthe solution with respect to the cell size to determine

(a)

(b)

11

Mac

rosc

opic

str

ess,

[M

Pa]

σ 11

0.00

0.01

0.02

0.03

0.05

0.04

0.06

0.000 0.002 0.004 0.006 0.008 0.010

εMacroscopic strain,

Fig. 14. Load case A – Macroscopic stress–strain curves obtained for twodifferent periodic unit cells.

the representativeness of these results. Unit cells (a) and(b) are identical in terms of geometric statistics, but differin the arrangement of particles within the domain. In par-ticular, cell (b) is obtained by rotating cell (a) by 90�, butthe finite element meshes are identical for both cells toeliminate the numerical errors in the comparison. The ini-tial (undamaged) stiffness is in agreement, but after onsetof damage, cell (a) is seen to be more compliant than cell(b). The reason for this difference might be explained bycomparing the evolution of damage for the two cases(Fig. 15). The curves shown on the right two figuresdenote the percentile of degraded interface for a particularvalue of c defined by (30). No curve is plotted forc = 0.98, as the fraction of cohesive elements withc 6 0.98 is identically 1 at all times. The c = 0.95 curverepresents the fraction of the cohesive interface whichhas experienced any damage, equivalent to the dash-dot-ted line in Fig. 10. The c = 0.0 curve represents the frac-tion of the cohesive interface which has completely failed,equivalent to the average of the dashed and dotted curvesin Fig. 10. The spacing between contours indicates howrapidly damage progresses. It is clear that damaged ele-ments proceed to failure more suddenly in cell (a) thanin cell (b). A higher fraction of the cohesive interfaceexperiences the onset of damage in cell (b) than in cell(a), but a lower fraction of the cohesive interface experi-ences complete failure. This lower fraction of failed inter-face suggest increased stiffness of the macroscopic stress–strain response for cell (b). Considering the deformedshapes, we can see that more of the damaged interfacesin cell (a) have opened up to form voids than those in cell(b). Since the microscopic damage processes for both cellsare markedly different, but macroscopic stress–straincurves disagree only slightly, one might assume that thecells are representative. Although, a more detailed investi-gation is needed to assess the representativeness of thesecells.

4.2. Load case C

The macroscopic response for load case C, a volume-preserving imposed shear state of macroscopic strain, isshown in Fig. 16, with the corresponding von Mises stressdistribution presented in Fig. 17. Again, the periodicity ofthe computed fields is evident. The fluctuating displace-ments can also be distinguished by observing the heteroge-neity of the boundary deformations. The von Mises stressdistribution is heterogeneous, showing the effects of multi-ple stress concentrations. Void opening is aligned with theprincipal tensile stress direction of 45�.

Particle pairs separate in opening (box 1) or in shearing(boxes 2 and 3). In box 1, interfacial failure for the particlepair initiates in alignment with the highest stress concentra-tion. Failure is often associated with the rotation of theparticle, resisted by the surrounding matrix. This tendencyto rotation is apparent for boundary particles 4 and 5. Forthis reason, voids do not open as much as for load case A.

[MPa]0.60

0.45

0.15

0.30

[MPa]0.60

0.45

0.30

0.15

11

Frac

tion

of c

ohes

ive

inte

rfac

e

11

Frac

tion

of c

ohes

ive

inte

rfac

e

0.95

0.75

0.50

0.250.00

0.95

0.75

0.50

0.250.00

0.00

a

b

0.00

εMacroscopic strain,

0.2

0.4

0.01 0.0

0.00

ε

0.4

0.2

0.0 0.01 0.00

Macroscopic strain,

Fig. 15. Load case A – Details of the microscopic damage for the two periodic unit cells whose macroscopic response is presented in Fig. 14. von Misesstress distribution for �11 = 0.01 (left); Damage evolution curves for 20 values of the damage parameter c (right).

Damaged interface

0.48

0.12

0.24

0.36

0.00

12

12

0.10

0.08

0.06

0.04

0.02

0.12

Frac

tion

of c

ohes

ive

inte

rfac

e

0.00

Mac

rosc

opic

str

ess

[MPa

]

Macroscopic strain, 0.000 0.004 0.008 0.012 0.016 0.020


Fig. 16. Load case C – Macroscopic stress–strain and damage curves.

[MPa]

0.0

0.2

0.4

0.6

0.8

3

4

4

1

2

5

5

Fig. 17. Load case C – von Mises stress distribution plotted on adeformed mesh with displacements magnified 10 times. Numbers 1–5 referto features discussed in the text.


It was noted that volume preserving tension and shearare identical in terms of principal stresses. This is clear ifwe compare the von Mises stress fields for load cases Aand C. Rotating Fig. 17 so that the principal stresses arealigned with those in Fig. 13, we see bands of increasedstress running along the major tension axis, and maximumopening on the transverse edges of the particles.

The similarity between load cases A and C is further evi-dent through comparison between Figs. 16 and 10. The

damage fraction rises early, and then levels off to form aplateau, a significant feature in both curves, indicating thatdamage propagation stops due to the transverse compres-sive effect. At this point, interfaces fail completely, as evi-dent in the failure fraction curves. This late opening ofvoids, however, may be due to the choice of cohesive inter-face properties, and in particular, the choice of high critical


opening displacements. In order to simulate a failureprocess more representative of actual solid propellantbehavior, we would need to use a non-linear kinematic for-mulation. The monotonically increasing stress–strain curveis, once again, associated with the near incompressibility ofthe blend.

5. Conclusions

A fully automated mathematical/numerical frameworkfor multiscale modeling of heterogeneous propellants fromparticle packing to failure has been presented. The micro-scale description is based on a periodic unit cell consistingof particles dispersed in a blend and incorporates the localnon-homogeneous stress and deformation fields present inthe unit cell during the failure of the particle/matrix inter-faces. A packing algorithm, treating the embedded particlesas spheres or discs, is used to generate packs which matchthe size distribution and volume fraction of actual propel-lants. Moreover, a sophisticated pre-processing tool hasbeen developed to generate a geometric model based onBezier curves and/or surfaces. This geometric model is thenused in a general meshing tool, T3d, to create high-qualityperiodic meshes. Since the identical meshing of the periodicentities using the advancing front technique is not usuallyviable, a different approach based on mirroring has beenadopted. Next, the mathematical theory of homogenizationbased on the asymptotic expansion of the displacement,strain and stress fields has been derived and used in model-ing debonding (or dewetting) damage evolution in rein-forced elastomers.

Various examples involving 2D unit cells and macro-scopic deformation histories of an idealized solid propel-lant have been considered to study the link between thefailure process taking place at the particle size scale andits effect on the macroscopic stress–strain curves and theevolution of void volume. The emphasis of this workhas been to develop a damage analysis tool at multiplescales from particle packing to failure. Further researchwill involve the inclusion of large deformations, a morecomplex, rate-dependent description of the binder and amatrix tearing model needed to capture the initiationand propagation of cracks in the solid propellant duringvoid coalescence. Moreover, the size of the representativevolume element in the presence of damage needs to beinvestigated.

Acknowledgements

The work of K. Matous, H.M. Inglis, X. Gu, T.L. Jack-son and P.H. Geubelle was supported by the Center forSimulation of Advanced Rockets (CSAR) under contractnumber B341494 by the US Department of Energy as apart of its Advanced Simulation and Computing program(ASC). K. Matous and P.H. Geubelle also acknowledgesupport from ATK/Thiokol, with J. Thompson and Dr.I.L. Davis as a program monitors. The work of Dr. Rypl

was supported by the Grant Agency of the Czech Republicunder contract number GACR 103/05/2315.

References

[1] Ha K, Schapery RA. A three-dimensional viscoelastic constitu-tive model for particulate composites with growing damage andits experimental validation. Int J Solids Struct 1998;26(27):3497–517.

[2] Simo JC. On a fully three-dimensional finite-strain viscoelasticdamage model: formulation and computational aspects. Comp MethAppl Mech Engng 1987;60:153–73.

[3] Mullins L. Softening of rubber by deformation. Rubber Chem. Tech.1969;42:339–61.

[4] Farris RJ. The character of the stress–strain function for highly filledelastomers. Trans Soc Rheol 1968;12:303–14.

[5] Christensen RN, Lo KH. Solutions for effective properties ofcomposite materials. J Mech Phys Solids 1979;27:315–30.

[6] Hutchinson JW, Jensen HM. Models of fiber debonding and pulloutin brittle composites with friction. Mech Mater 1990;9:335–442.

[7] Hashin Z. The spherical inclusion with imperfect interface. ASME JAppl Mech 1991;58:444–9.

[8] Matous K. Damage evolution in particulate composite materials. IntJ Solids Struct 2003;40:1489–503.

[9] Krajcinovic D. Damage mechanics. Amsterdam: Elsevier Science;1996.

[10] Chaboche JL, Kruch S, Maire JF, Poittier T. Towards a microme-chanics based inelastic and damage modeling of composites. Int JPlasticity 2001;17:411–39.

[11] Zhong AX, Knauss WG. Analysis of interfacial failure in particle-filled elastomers. J Engng Mater Technol 1997;119:198–204.

[12] Zhong AX, Knauss WG. Effects of particle interaction and sizevariation on damage evolution in filled elastomers. Mech ComposMater Struct 2000;7:35–53.

[13] Michel JC, Moulinec H, Suquet P. Effective properties of compositematerials with periodic microstructure: a computational approach.Comp Meth Appl Mech Engng 1999;172:109–43.

[14] Segurado J, Llorca J. A numerical approximation to the elasticproperties of sphere-reinforced composites. J Mech Phys Solids2002;50:2107–21.

[15] Llorca J, Segurado J. Three-dimensional multiparticle cell simulationsof deformation and damage in sphere-reinforced composites. MaterSci Engng A 2004;365:67–274.

[16] Knott GM, Jackson TL, Buckmaster J. The random packing ofheterogeneous propellants. AIAA J 2001;39(4):678–86.

[17] Kochevets S, Buckmaster J, Jackson TL, Hegab A. Randompropellant packs and the flames they support. AIAA J Propul Power2001;17(4):883–91.

[18] Rypl D. Sequential and parallel generation of unstructured 3Dmeshes. PhD thesis. CTU Reports, vol. 3, no. 2. Prague: CzechTechnical University; 1998. T3d page. Available from: http://mech.fsv.cvut.cz/~dr/t3d.html.

[19] Rypl D, Bittnar Z. Hybrid method for generation of quadrilateralmeshes. Engng Mech 2002;9(1/2):49–64.

[20] Hughes TJR. The finite element method, linear static and dynamicfinite element analysis. New Jersey: Prentice-Hall, Inc. A Division ofSimon & Schuster; 1987.

[21] Matous K, Geubelle PH. Multiscale modeling of particle debondingin reinforced elastomers subjected to finite deformations. Int J NumerMeth Engng 2006;65:190–223.

[22] Fish J, Yu Q. Multiscale damage modeling for composite materials:theory and computational framework. Int J Numer Meth Engng2001;52(1–2):161–92.

[23] Guedes JM, Kikuchi N. Preprocessing and postprocessing formaterials based on the homogenization method with adaptive finiteelement methods. Comp Meth Appl Mech Engng 1990;83(1–2):143–98.

http://mech.fsv.cvut.cz/~dr/t3d.html

http://mech.fsv.cvut.cz/~dr/t3d.html


[24] Wang X, Buckmaster J, Jackson TL. Burning of ammonium-perchlorate ellipses and spheroids in fuel binder. AIAA J PropulPower 2006;22(4):764–8.

[25] McGeary RK. Mechanical packing of spherical particles. J AmCeram Soc 1961;44(10):513–22.

[26] Miller RR. Effects of particle size on reduced smoke propellantballistics. Paper No. 82-1096. AIAA; 1982.

[27] Zeman J, Sejnoha M. Numerical evaluation of effective elasticproperties of graphite fiber tow impregnated by polymer matrix. JMech Phys Solids 2001;49:69–90.

[28] Bittnar Z, Rypl D. Direct triangulation of 3D surfaces usingadvancing front technique. In: Proceedings of the second ECCOMASconference on numerical methods in engineering. Paris, France:Wiley; 1996. p. 86–90.

[29] Wentorf R, Collar R, Shephard MS, Fish J. Automated modeling forcomplex woven mesostructures. Comput Meth Appl Mech Engng1999;172(1–4):273–91.

[30] Bensoussan A, Lions JL, Papanicolaou G. Asymptotic analysis forperiodic structures. New York: North-Holland; 1978.

[31] Lions JL. Some methods in the mathematical analysis of systems andtheir control. New York: Gordon and Breach; 1981.

[32] Duvaut G. Homogeneization et materiaux composites. In: Ciarlet PG,Rouseau M, editors. Trends and applications of pure mathematics tomechanics. Lecture notes in physics, vol. 195. Berlin: Springer; 1984.

[33] Ortiz M, Pandolfi A. Finite-deformation irreversible cohesive elementfor three-dimensional. crack-propagation analysis. Int J Numer MethEngng 1999;44:1267–82.

[34] Roy YA, Dodds RH. Simulation of ductile crack growth in thinaluminum panels using 3-D surface cohesive elements. Int J Fracture2001;110:21–45.

[35] Xu XP, Needleman A. A numerical study of dynamic crack growth inbrittle solids. Int J Solids Struct 1994;84:769–88.

[36] Geubelle PH, Baylor JS. Impact-induced delamination of composites:a 2D simulation. Composites B 1998;29:589–602.

[37] Fish J, Shek K, Pandheeradi M, Shephard MS. Computationalplasticity for composite structures based on mathematical homoge-nization: theory and practice. Comp Meth Appl Mech Engng1997;148:53–73.

Documents

Multiscale modeling of solid propellants: From particle ...kmatous/Papers/CST_PPF.pdfMultiscale modeling of solid propellants: From particle packing to failure ... [14,15] in their