Int J Fract (2006) 141:11–25DOI 10.1007/s10704-006-0047-x

ORIGINAL ARTICLE

Modeling of crack growth through particulate clustersin brittle matrix by symmetric-Galerkin boundary elementmethod

R. Kitey · A.-V. Phan · H. V. Tippur · T. Kaplan

Received: 29 September 2005 / Accepted: 15 February 2006© Springer Science+Business Media B.V. 2006

Abstract The interaction of a crack with perfectlybonded rigid isolated inclusions and clusters of inclu-sions in a brittle matrix is investigated using numericalsimulations. Of particular interest is the role inclusionsplay on crack paths, stress intensity factors (SIFs) andthe energy release rates with potential implications tothe fracture behavior of particulate composites. The ef-fects of particle size and eccentricity relative to the ini-tial crack orientation are examined first as a precursorto the study of particle clusters. Simulations are accom-plished using a new quasi-static crack-growth predic-tion tool based on the symmetric-Galerkin boundaryelement method, a modified quarter-point crack-tip ele-ment, the displacement correlation technique for evalu-ating SIFs, and the maximum principal stress criterionfor crack-growth direction prediction. The numericalsimulations demonstrate a complex interplay of crack-tip

R. Kitey · H. V. TippurDepartment of Mechanical Engineering,Auburn University,Auburn, AL 36849, USA

A.-V. Phan (B)Department of Mechanical Engineering,University of South Alabama,Mobile, AL 36688, USAe-mail: vphan@jaguar1.usouthal.edu

T. KaplanComputer Science and Mathematics Division,Oak Ridge National Laboratory,Oak Ridge, TN 37831, USA

shielding and amplification mechanisms leading to sig-nificant toughening of the material.

Keywords Crack growth · Crack-inclusioninteraction · Crack deflection · Particulatecomposites · Matrix toughening ·Symmetric-Galerkin boundary element method

1 Introduction

Particle filled polymers are useful in a variety of engi-neering applications—as light-weight structural com-posites, as conducting polymers, as potting compoundsin electronic packages, as syntactic foams for civil andmarine structures, surface coats/paints for thermal insu-lation, to name a few. Higher stiffness and fracturetoughness compared to unfilled (neat) matrix materialare some of the positive mechanical attributes derivedby reinforcing a matrix with stiff fillers. A favorableincrease in failure properties occurs in the presenceof secondary phases due to crack tip shielding, crackdeflection and twisting, and crack tip blunting mecha-nisms. Filler particle size, shape and volume fraction,the properties of constituent phases and filler–matrixinterface strength influence the toughening mechanismsand in turn the failure properties. The experimentalresults of particle size and filler–matrix adhesion ef-fects on fracture behavior of heterogeneous media isa motivating factor for the current study. Spanoudakisand Young (1984a, b) observed a decrease in fracture

12 R. Kitey et al.

toughness with increasing particle size for particles ofsizes 4–62µm in diameter at low volume fractions.Unlike the monotonic trend in fracture toughness withparticle size, a minimum energy release rate (ERR) wasobserved for 47µm particles. They also suggested thatpoor inclusion–matrix bonding causes an increase inERR. Crack pinning followed by crack tip blunting wasanticipated to be the potential toughening mechanism.Moloney et al. (1987) reported that for the particle sizesranging from 60 to 300µm, fracture toughness has neg-ligible variation but weaker filler–matrix interface doesincrease the fracture toughness due to possible cracktip blunting. On the contrary, Nakamura et al. (1999),Nakamura and Yamaguchi (1992) noticed prominentincrease in fracture toughness with increasing parti-cle size for the sizes ranging from 2 to 42µm. Theincreased crack deflection, interfacial debonding andparticle fracture were suggested as reasons for the in-creased fracture toughness. They also concluded thatfracture toughness is unaffected by filler–matrix adhe-sion strength. The contrasting results among previousstudies, possibly due to the narrow range of particlesizes were addressed by Kitey and Tippur (2005a, b)recently. They observed a non-monotonic bell shapedvariation in fracture toughness for 7–200µm particlesizes. They observed localized crack tip blunting andcrack front trapping as potential toughening mecha-nisms when filler particles were weakly bonded to thematrix. For strongly bonded particles, on the other hand,fracture toughness values were consistently lower com-pared to the weakly bonded components. They haveattributed this difference to the crack front gettingtrapped at weak filler–matrix interfaces causing local-ized blunting and crack growth retardation. The weakerinterfaces also act as distributed attractors driving thecrack into modes II and III conditions in addition tothe dominant mode I, thereby increasing the crack pathtortuosity.

A few previous works have tried to explain toughen-ing mechanisms using analytical and numerical stud-ies. Atkinson (1972) calculated the stress field around acrack tip for a symmetrically located crack in the pres-ence of an inclusion. He concluded that inverse

√r sin-

gularity exists for crack positions up to a distance veryclose to the inclusion until the crack is not touchingthe inclusion. In a similar investigation Erdogan et al.(1974) calculated the generalized stress field aroundthe crack tip and stress intensity factors (SIFs) usingGreen’s function for an arbitrarily oriented crack with

respect to a circular inclusion. A model predicting anincrease in fracture toughness due to crack deflectionand twisting around a secondary phase filler is pro-posed by Faber and Evans (1983). They have shownthe dependence of fracture toughness on morphologyand volume fraction of the secondary phase, while thefracture toughness is shown to be invariant relative toparticle size.

There have been several analytical and numericalstudies on the topic. Analytical approaches have beendeveloped for problems concerning cracks along theinterface of various types of inclusions (e.g., Tamate1968; Sendeckyj 1974; Hwu et al. 1995). Numericaltechniques, including finite element method (e.g., [Liand Chudnovsky 1993a,b Ferber et al. 1993; Lipetzkyand Schmauder 1994; Lipetzky and Knesl 1995; Haddiand Weichert 1998]) and boundary element method(BEM) (e.g., [Bush 1997; Knight et al. 2002; Wanget al. 1998]), have been employed to investigate thistype of fracture behavior under static loading condi-tions. Recently, the dynamic response of the interactionbetween a crack and an inclusion using the time-domainBEM has been studied by Lei et al. (2005). Note that thedual BEM (DBEM) is used in Bush (1997) and Knightet al. (2002) while the sub-domain BEM technique isadopted in Wang et al. (1998) and Lei et al. (2005). Liand Chudnovsky (1993) have numerically shown thatwhen a crack approaches a rigid inclusion in a relativelycompliant matrix, the crack tip is shielded from the far-field stresses. This decreases stress intensification at thecrack tip and hence lowers the ERR. They also showthat typically, a crack tip gets shielded when the crackapproaches an inclusion while stresses are amplifiedwhen the crack propagates away (departs) from the par-ticle. Bush (1997) has investigated the effect of singleinclusion and an inclusion cluster on crack trajectoryand energy release rate. He considered two differenttypes of inclusions in his simulations; ones which wereperfectly bonded to the matrix and the others with inter-face flaws or partially bonded to the matrix. He showedthat pre-existing flaws on inclusion–matrix interface at-tract the propagating crack. The crack deflection aroundan inclusion is noticed when the crack tip is one radiusaway from the inclusion. Knight et al. (2002) examinedcrack deflection/attraction mechanisms in a crack–par-ticle interaction investigation by performing a series ofparametric studies for different Young’s modulus andPoisson’s ratio mismatches. For the inclusions with andwithout interphase regions between filler and matrix,

Modeling of crack growth by symmetric-Galerkin boundary element method 13

Poisson’s ratio was shown to significantly affect cracktrajectories.

The key feature of the BEM approach is that only theboundary of the domain is discretized and only bound-ary quantities are determined. It is generally recognizedthat the BEM is particularly well suited for linear elasticfracture mechanics, as the method is known to providemore accurate results for stress and there is no needfor re-meshing the boundary during the modeling ofcrack propagation. Recently, the symmetric-Galerkinboundary element method (SGBEM) has been devel-oped (e.g., Bonnet 1995; Bonnet et al. 1998) and hasshown several key advantages in fracture applications:(a) as the name implies, SGBEM formulation results ina symmetric coefficient matrix; (b) the presence of bothdisplacement and traction BIE enables fracture prob-lems to be solved without using artificial sub-domains;(c) unlike BEM or DBEM, there is no smoothnessrequirement on the displacement (e.g., Gray 1991; Mar-tin and Rizzo 1996) in order to evaluate the hypersingu-lar integral; thus, standard continuous and quarter-point(QP) elements can be employed.

In both finite and boundary element fracture mod-eling, the standard approach consists of incorporatingthe

√r displacement behavior at the crack tip by means

of the QP element (Henshell and Shaw 1975; Barsoum1976), where r is the radial distance from the cracktip. Recently, Gray and Paulino (1998) have provedthat, for an arbitrary crack geometry, a constraint ex-ists in the series expansion of the crack opening dis-placement (COD) at the tip. As discussed in Gray andPaulino (1998), the QP element in general fails to sat-isfy this constraint, and this has led to the developmentof an improved modified quarter-point (MQP) element(Gray et al. 2003). It was demonstrated in Gray et al.(2003) that the accuracy of the computed SIFs usingthe Displacement Correlation Technique (DCT) canbe significantly improved by incorporating this MQPelement into the SGBEM. This suggests that stressmethods such as the maximum principal stress crite-rion (MPSC) (Erdogan and Sih, 1963) can effectivelybe used as the criteria for crack-growth direction in aquasi-static crack-growth prediction tool made up fromthe SGBEM, the MQP, and the DCT.

The focus of this paper is on simulations of crack-growth through particulate polymer matrix composites.Of interest is the study of the effects of inclusion size,eccentricity and crack tip shielding on crack path andSIF, and the effects of particle arrangement and volume

fraction on the ERR of a brittle particulate compositesuch as glass-filled epoxy. The simulations are carriedout using the crack-growth prediction tool mentionedabove. The performance of the tool is bench markedby running crack growth simulations in an unfilled orneat matrix and comparing the obtained SIF variationwith an available analytical solution. It is shown fromthis work that, even with a simple local SIF methodsuch as the DCT, and a simple stress method for crackgrowth direction, the proposed numerical tool is ableto capture complex interplay of crack–tip shielding andamplification mechanisms for various crack–inclusioninteraction situations.

2 A Crack-growth prediction tool

2.1 SGBEM fracture analysis

This section provides a very brief review of the SGBEMand its application to fracture. The reader is asked toconsult the cited references for further details.

Consider a domain B containing a crack�c = �+c +

�−c on which only tractions are prescribed and τ+

c =−τ−

c (Fig. 1). The boundary integral equation (BIE)without body forces for linear elasticity is given byRizzo (1967). For a source point P interior to the do-main, if the displacements u+

c and u−c are replaced

by the COD �uc = u+c − u−

c on �+c , and note that

τ+c + τ−

c = 0, the BIE is written as

U(P) = uk(P)

−∫�b

[Ukj (P, Q) τ j (Q)− Tkj (P, Q) u j (Q)

]d Q

+∫�+

c

Tk j (P, Q)�u j (Q) d Q = 0, (1)

τ

Γc-

Γc+

n+B

Γ

n-P

b

x2

x1

Fig. 1 A body B containing a crack

14 R. Kitey et al.

where Q is a field point, τ j and u j are traction anddisplacement vectors, Ukj and Tkj are the Kelvin ker-nel tensors or fundamental solutions, �b denotes theouter boundary of the domain, and d Q is an infinitesi-mal boundary length (for 2-D) or boundary surface (for3-D cases).

It can be shown that the limit of the integral in Eq.(1) as P approaches the boundary exists. From now on,for P ∈ � = �b + �c, the BIE is understood in thislimiting sense.

As P is off the boundary, the kernel functions arenot singular and it is permissible to differentiate Eq.(1) with respect to P , yielding the hypersingular BIE(HBIE) for displacement gradient. Substitution of thisgradient into Hooke’s law gives the following HBIE forboundary stresses:

S(P) = σk�(P)

−∫�b

[Dkj�(P, Q)τ j (Q)− Skj�(P, Q)u j (Q)

]d Q

+∫�+

c

Sk j�(P, Q) �u j (Q) d Q = 0 . (2)

Expressions for the kernel tensors Ukj , Tkj , Dkj�

and Skj� can be found in Bonnet (1995).The Galerkin boundary integral formulation is ob-

tained by taking the shape functions ψm employed inapproximating the boundary tractions and displacementsas weighting functions for the Eqs. (1) and (2). Thus,∫�

ψm(P)U(P) d P = 0 , (3)∫�

ψm(P)S(P) d P = 0 . (4)

A symmetric coefficient matrix, and hence a sym-metric-Galerkin approximation, is obtained by employ-ing Eq. (3) on the boundary �(u) where displacementsubv are prescribed, and similarly Eq. (4) is employed onthe boundary �(τ ) with prescribed tractions τbv . Notethat� = �(u)+�(τ ). The additional boundary integra-tion is the key to obtain a symmetric coefficient matrix,as this ensures that the source point P and field pointQ are treated in the same manner.

2.2 Modified quarter-point element

The 2-D QP element is based upon the three-equidistant-noded quadratic element. For t ∈ [0, 1],the shape functions for this element are given by

ψ1(t) = (1 − t)(1 − 2t) ,

ψ2(t) = 4t (1 − t) , (5)

ψ3(t) = t (2t − 1) .

As mentioned in the Introduction, this QP element failsto satisfy a constraint in the series expansion of theCOD at the crack tip (Gray and Paulino, 1998). A rem-edy was presented in Gray et al. (2003) and this resultsin the following MQP shape functions (ψ̂1(t) is notrequired as �u = 0 at the crack tip):

ψ̂2(t) = − 83 (t

3 − t) ,

(6)ψ̂3(t) = 13 (4t3 − t) .

It can be observed that the modified shape functions(6) still satisfy the Kronecker delta property ψ̂i (t j ) =δi j . This new approximation is only applied to the COD,as we must use the standard shape functions (5) for therepresentation of the crack tip geometry. This ensuresthat the property t ∼ √

r remains.

2.3 Stress intensity factors by the DCT

There are several approaches for numerically evalu-ating SIFs. Among these approaches, the DCT basedupon the COD in the vicinity of the crack tip is a verysimple method. In case of using the MQP element, theexpressions for SIFs by means of the DCT are given by(Gray et al., 2003)

KI = µ

3(κ + 1)

√2π

L

(8�u(2)2 −�u(3)2

),

(7)KII = µ

3(κ + 1)

√2π

L

(8�u(2)1 −�u(3)1

).

where µ is the shear modulus, ν is Poisson’s ratio, and

κ = 3 − 4ν (plane strain) ,

(8)κ = 3 − ν

1 + ν(plane stress) .

Since SIFs are directly given in terms of the nodalvalues of the COD at the crack tip element, the useof MQP element herein is highly beneficial as it wouldenhance the accuracy of the nodal CODs, thus the accu-racy of the obtained SIFs. Further, energy release rateG is calculated using SIFs by,

G = κ + 1

8µ(K 2

I + K 2II). (9)

Modeling of crack growth by symmetric-Galerkin boundary element method 15

2.4 Maximum principal stress criterion

There are several criteria for predicting crack growthdirection (e.g., Erdogan and Sih, 1963; Sih, 1974). TheMPSC proposed by Erdogan and Sih (1963) is adoptedherein. According to this criterion, the crack growthdirection θc is perpendicular to that of the maximumprincipal stress, and the crack will propagate when KI

exceeds the fracture toughness KIc of the material.Since in the direction θc, σrθ = 0, i.e.

σrθ = KI sin θc + KII(3 cos θc − 1) = 0 (10)

thus,

θc = 2 tan−1

⎛⎝ KI

4KII± 1

4

√(KI

KII

)2

+ 8

⎞⎠ . (11)

A quasi-static crack-growth prediction tool basedupon the SGBEM, the MQP crack-tip element, the DCTand the MPSC is employed for the simulations con-ducted in this work. By incorporating the MQP shapefunctions (6) into a SGBEM code, accurate CODs �uat the mid and end nodes of any crack-tip element maybe obtained (see Eqs. (1) and (2)). The SIFs at the cracktip of interest will then be evaluated by substituting therelated CODs in the DCT formulas (7). If KI ≥ KIc thecrack-growth direction θc of the crack tip in questionis determined by Eq. (11), and an infinitesimal crackincrement is added in that direction during the simu-lation. As a result, remeshing a propagating crack isstraightforward. Only a new crack-tip element needsto be added in the crack-growth direction θc while theprevious element discretization remains unchanged. Atissue is obviously the size of the new crack-tip ele-ment. Smaller sizes, although leading to more time-consuming simulations, are expected to result in moreaccurate results.

3 Simulation results and discussion

To investigate the effects of particle arrangement andvolume fraction on crack and particle–cluster interac-tion, a plane stress situation involving a cracked sheetof dimensions 140 mm× 40 mm× 8 mm is considered1.The material properties of the matrix and inclusions

1 The dimensions are chosen with future experimental simula-tions involving optical interferometry in mind (Kitey and Tippur,2005a)

are matched with those for epoxy and glass, respec-tively, with an elastic modulus (E) 3.2 GPa and Pois-son’s ratio (ν) 0.3 for epoxy and E = 70 GPa, ν =0.3 for glass. The loading configuration considered inall simulations is one of 3-point bending of an edgecracked sheet/beam. A crack is located at the edge of thesheet opposite to the loading point and along the load-ing axis. The line joining the initial crack tip and load-ing point is referred to as line-of-symmetry throughoutthe article.

3.1 Benchmarking and mesh convergence

To ensure a high degree of accuracy of SIFs calculatedby BE simulations, first a single edge notched geometry(see inset in Fig. 2(a)) of neat matrix material withoutany inclusions is carried out. The geometry is a sym-metrically loaded 3-point bend configuration. The ini-tial crack length is chosen such that a/W = 0.25 whereW is the total width of the chosen geometry. In thesimulations, equal length boundary elements are usedto represent specimen edges including the initial crack.This length is determined first by performing a con-vergence study. The SIF results for different elementlengths varying from W/10 to W/40 as a function ofcrack length are determined for a symmetrically loaded3-point bend configuration using [Anderson 1995],

KI = 3 P LBW 2

√a

2(1 + 2 a

W

) (1 − 2 a

W

)3/2

[1.99 − a

W

(1 − a

W

)

×{

2.15 − 3.93a

W+ 2.7

( a

W

)2} ]. (12)

Non-dimensional SIF (= KIB

√W

P ) values fromSGBEM are compared with analytical results in Fig.2(a) for different a/W ratios. With a decrease in elementlength accuracy of SIF calculations increase. Good agree-ment between calculated KI values using SGBEM andanalytical results are evident when an element length ofW/40 is used. For this choice, a high degree of accuracycan be expected up to a/W = 0.75 with an error lessthan 2.5%. Beyond a/W = 0.75, however, the accuracyreduces due to the dominance of loading point stresseson the crack tip stress field.

Next, the crack tip increment length in the directionof propagation is determined separately to match exper-imentally determined crack paths in a mixed-mode

16 R. Kitey et al.

(a)

(b)

Fig. 2 (a) Comparison between the analytical results andSGBEM results for various boundary element lengths, (b) Crackpropagation for mixed-mode loading; crack trajectory from BEMis superimposed on experimentally obtained crack path for aneccentrically applied load at the distance L/4 from the line-of-symmetry

loading situation. The crack geometry used in the simu-lations is the same as shown earlier for the mode-I case(Fig. 2(a)), except for the location of the applied load.The loading point is at a distance of L/4 from the line-of-symmetry, where L is the span. Various crack tip incre-ments were used before choosing�a/W = 6.25×10−3

in the simulations. The loading configuration, speci-men geometry and the fractured epoxy specimen froman experiment are shown in Fig. 2(b). The crack trajec-tory from BE analysis is superimposed on experimen-tally obtained crack path. Excellent agreement betweencrack paths shows the accuracy of SGBEM in capturingmode-mixity using the chosen crack tip increment.

3.2 Crack tip shielding

Next, crack growth is simulated in the presence of arigid inclusion in the crack path. When inclusions areperfectly bonded to the matrix material, crack deflec-tion is a prominent toughening mechanism. The crackdeflection depends on inclusion size, inclusion eccen-tricity with respect to the crack orientation, the numberof inclusions surrounding the crack tip, etc. As men-tioned earlier, a rigid inclusion in a relatively compliantmatrix shields the crack tip as it is approached. How-ever, the shielding depends on the location of the inclu-sion with respect to the initial crack orientation. Also,the particle size, particle eccentricity, the number ofparticles and arrangement of particles around the cracktip are some of the other parameters which affect thedegree of shielding.

First a case when a cylindrical inclusion of diam-eter d = W/10 is symmetrically located in front of acrack tip is considered. In this, the inclusion is locatedin front of the initial crack tip at p/W = 0.65, wherep is the distance of the center of the inclusion in thex-direction (see Fig. 3(a)). The plot in Fig. 3(b) showsthe variation of nondimensional KI with crack lengtha/W . For comparison the KI variation for the case ofthe neat matrix is also shown. A sharp decrease in SIFcan be noticed from the plot as the crack approaches theinclusion. The inclusion in front of the crack shows anegligible effect on KI until the crack reaches a lengthof approximately a/W = 0.45. This suggests that thepresence of rigid inclusion in front of the crack is feltonly when the tip is at the distance of ≈ 1d from theinclusion. When the crack is d/4 away from the inclu-sion, KI decreases substantially. The KI approacheszero as the crack propagates further and reaches theinclusion.

The accuracy of ERR calculations were validatedagainst the results reported in Ref. [Bush 1997]. Thegeometry and loading configuration (see Figs. 3 and 7in [Bush 1997]) is shown in Fig. 4(a). The inclusionradius r is considered as L/20, where L is the lengthof the planar body with an edge crack. The ratio of theelastic modulii E p/Em is varied from 2 to 8, where thesubscripts p and m correspond to the inclusion and thematrix, respectively. The Poisson’s ratio for the inclu-sion and matrix are considered as 0.33 and 0.17, respec-tively. In Fig. 4(b) nondimensional ERR, G/G0, is plot-ted with varying nondimensional crack lengths, whereG0 is the ERR for the matrix. The crack tip shielding

Modeling of crack growth by symmetric-Galerkin boundary element method 17

(a)

Fig. 3 Crack tip shielding by a rigid inclusion in front of a mode-I crack: (a) Loading configuration; (b) Comparison between SIFvariations in the presence and absence of inclusion

increases (i.e., G/G0 decreases) with increasing E p/Em .The crack senses the rigid particle in front of it at leastfrom a distance of ≈ 5 radii. The observed G/G0 varia-tions for various E p/Em are same as the ones reportedby Bush 1997) and hence suggests the accuracy of ERRcalculations from SGBEM in the presence of second-ary phases. Minor differences in ERR value, when thecrack reaches the particle can be attributed to the differ-ent methods of ERR calculation (DCT method in cur-rent study).

The crack tip shielding and amplification effects arefurther explored by simulating a mode-I problem with apair of inclusions ahead of a propagating crack. A sym-metrically located pair of inclusions with respect to thecrack orientation, as shown in Fig. 5, leads to crackgrowth under mode-I conditions. Here the inclusionsare located at p/W = 0.5 or, W/4 away in the x-direc-tion from the initial crack tip. The separation distance‘s’ between inclusions is varied from d/8 to 2d, whered = W/10. Figure 5 shows the variation of G/G0 witha/W .

Denoting the inclusion location (the orientation ofthe center of the inclusion(s)) in the x-direction as p/W ,

Fig. 4 Validation of ERR calculation from SGBEM in the pres-ence of secondary phase filler, (a) Problem geometry and loadingconfiguration, (b) Variation of non-dimensional ERR for variousE p/Em , νp = 0.17, νm = 0.33

Fig. 5 Crack tip shielding and amplification effects due to a pairof symmetrically situated inclusions in the crack path

it is evident that as the crack length approaches p/W ,crack tip shielding effects begin to manifest as decreas-ing ERR (G/G0 < 1) values with a/W . It can be noticedthat maximum shielding occurs when the crack tip isapproximately d/2 in front of the center of the inclusion.

18 R. Kitey et al.

(a) (b)

(d)(c)

Fig. 6 The effect of eccentrically situated rigid inclusions rel-ative to the initial crack on crack deflection and energy releaserate; (a) Crack deflection with varying inclusion eccentricity; theinclusion location is shown with respect to initial crack orienta-tion; (b) Variation of maximum crack deflection from the line-

of-symmetry with inclusion eccentricity for inclusion diameterd = W/10; (c) Variation in nondimensional ERR with a/W ; (d)Crack propagation around a rigid inclusion from SGBEM simu-lation

Further crack propagation decreases shielding. That is,the nondimensional ERR reaches a value of unity whenthe propagating crack length a/W reaches p/W . As thecrack propagates beyond p/W , an amplification of ERR(G/G0 > 1) values can be seen. The maximum ampli-fication is seen when the crack tip is at a distance of d/2away from the center of the inclusion. The amplifica-tion begins to vanish as the crack propagates away fromthis location. The above crack shielding/amplificationeffects are similar to the ones reported in Refs. [Li andChudnovsky 1993; Bush 1997].

From the plots in Fig. 5 it can also be seen that theshielding/amplification effect increases as the inclusionseparation distance decreases. The effects are quite evi-dent even when the inclusions are 2d apart. Interest-ingly, the magnitude of maximum shielding is always

greater than the magnitude of maximum amplificationfor a given value of s. If one were to associate the de-crease in inclusion separation as a measure of increasein filler volume fraction, these results suggest that evenat low volume fractions shielding effects are significantand the effect increases as the filler volume fractionincreases.

3.3 Particle eccentricity effect

Next, interaction between a crack and an isolated inclu-sion located eccentrically to the initial crack isinvestigated. Of particular interest is the role inclusioneccentricity plays on crack deflection and the result-ing ERR variation during crack growth. As shown in

Modeling of crack growth by symmetric-Galerkin boundary element method 19

Fig. 6(a), the center of the inclusion is located W/4 awayfrom the initial crack tip in the x-direction. The inclu-sion eccentricity ‘e’ (the distance between the centerof the inclusion and the line-of-symmetry) is variedfrom d/4 to 2d while the load is applied symmetricallyin a 3-point bend configuration as before. The crackpaths around inclusions for different values of eccen-tricity are shown in Fig. 6(a). In these simulations, thecrack propagates nearly in a mode-I fashion until thetip reaches a/W ≈ 0.45 or ≈ 1d in front of the centerof the inclusion. As the crack propagates further, theangle of deflection2 increases. A sharp increase in theangle of deflection can be noticed from a/W ≈ 0.45onwards, when the crack is d/2 behind the the centerof the inclusion. The angle of deflection attains a max-imum value at a/W ≈ 0.475. This is followed by adecrease in the angle of deflection with the crack stillpropagating away from the line-of-symmetry. Whenthe crack travels ≈ d/2 away from the particle locationp/W , the crack begins to propagate towards the line-of-symmetry with a relatively small angle of deflection.Plots suggest that the crack deflection increases as theeccentricity decreases. A symmetrically oriented inclu-sion (e = 0) can be viewed as a limiting case when thecrack gets stalled at the inclusion–matrix interface (seeFig. 3(b)). The variation of maximum crack deflectionwith inclusion eccentricity is shown in Fig. 6(b). Fromthe plot it appears that the maximum crack deflectionincreases exponentially with decreasing eccentricity.

The effect of inclusion eccentricity on ERR is pre-sented in Fig. 6(c). The figure shows the variation ofnondimensional ERR, G/G0, with a/W . It can be seenfrom the plots that ERR decreases as the crack ap-proaches the inclusion. The minimum value of ERRoccurs when a/W ≈ 0.45 or when the crack tip islocated ≈ d/2 behind the the center of the inclusion.With further crack propagation, ERR values increaseand approach unity as a/W reaches p/W . An ampli-fication in SIF and hence ERR can be noticed as thecrack starts propagating away (recede) from the inclu-sion. This relatively small increase in ERR continuesuntil crack propagates a distance of approximately d/2away from the inclusion. With further crack propaga-tion, G/G0 decreases and asymptotically approachesunity. Similar to the effects of eccentricity on crack

2 Angle of deflection is the absolute angle between the line-of-symmetry (initial crack orientation) and the tangent to thecrack path at any instant.

deflection, G/G0 increases as the inclusion eccentricitye decreases. The maximum effect can be seen for zeroeccentricity (also shown in Fig. 3(b)).

It should be be emphasized here that the crack growthand ERR calculations are based on propagation occur-ring in the matrix material at all times. If the crack wereto enter the matrix–inclusion interfacial region, cracktip fields corresponding to dissimilar material inter-faces will have to be used [He and Hutchinson 1989].Simulations reported here correspond to the crackgrowth occurring close to an interface yet in the matrixonly as a sub-interfacial crack. An example of the samefor the case of e = d/4 is shown in Fig. 6(d).

For the sake of completeness, the combined effectsof shielding and eccentricity are studied next using apair of inclusions located eccentrically with respect tothe line-of-symmetry (see Fig. 7). The separation dis-tance ‘s’ between the inclusions is kept constant (s =d/2) while the eccentricity of the inclusion nearest tothe line-of-symmetry is varied from d/2 to 3d/4. Thevariation of crack path for various eccentricities are notshown here for brevity which showed similar eccentric-ity effect as for the case of single inclusion (Fig. 6(a))with an exception of relatively small crack deflections.The variation of G/G0 with a/W is plotted in Fig. 7.Plots overlap on each other in most part except whenthe crack propagates between a/W ≈ 0.4 and 0.5 (i.e.,from a distance ≈ 1d in front of the particle center) forvarious eccentricities. Small but monotonic variation inminimum value of G/G0 can be noticed with increasing

Fig. 7 Variation of G/G0 with a/W in the presence of eccen-trically located pair of inclusions showing the combined effectof shielding and inclusion eccentricity (Note: Eccentricity of theinclusion-pair is defined for the nearest inclusion center relativeto the crack)

20 R. Kitey et al.

inclusion eccentricity. The lowest magnitude of G/G0

(at a/W ≈ 0.45) decreases with decrease in eccentric-ity, the minimum being for the case of symmetricallylocated pair of particles. Interestingly the eccentricityof a pair of particles is noticed to be playing negligiblerole on amplification.

3.4 Particle size effect

In this part the effect of inclusion size on crack deflec-tion and ERR is investigated. An inclusion is located ata distance W/4 away from the initial crack tip with aneccentricity of d/2 (see Fig. 8(a)). The inclusion diam-eter is varied from W/40 to W/5. The crack paths forvarious sizes of inclusions at p/W = 0.5 is shown inFig. 8(a). Again, the load is applied symmetrically in a3-point bend configuration. Evidently, the crack prop-agates in a mode-I fashion until the inclusion is ≈ 1din front of the crack tip. As the distance between theinclusion and the crack tip decreases, crack deflectionincreases. The angle of deflection attains a maximumvalue when the crack tip is at a distance of ≈ d/4 fromthe particle center. Interestingly, the crack deflectionkeeps increasing even after the crack tip grows pastthe particle centerline. Maximum crack deflection oc-curs when the crack travels across the inclusion andpropagates d/2 away from the inclusion center. Withfurther crack propagation a slight reversal of angle ofdeflection can be seen and the crack travels towards theline-of-symmetry with a relatively small angle. Also itcan be noticed that the presence of a larger inclusionis felt earlier by the crack tip. The amount of crackdeflection increases with the increase in inclusion size.Figure 8(b) shows the variation of maximum deflec-tion from the line-of-symmetry with inclusion diam-eter. For an eccentricity of d/2, plot suggests that themaximum deflection varies monotonically (with slightnon-linearity) with inclusion size.

The inclusion size effect on ERR is shown in Fig.8(c). Plots show the variation of G/G0 with a/W for var-ious inclusion sizes. The size effect in terms of decreas-ing ERR is evident for all inclusion sizes. As one wouldexpect, the effect can be noticed much earlier as the par-ticle size increases. The largest among the chosen inclu-sions shows the lowest non-dimensional ERR, while incase of smallest inclusion the crack propagates withG/G0 ≈ 1 until a length of a/W ≈ 0.40. For all inclu-sion sizes G/G0 decreases when the crack propagates

Fig. 8 Role of particle size on crack deflection and ERR: (a)crack deflection in the presence of inclusions of various diame-ters for a fixed eccentricity e = d/2; (b) variation of maximumcrack deflection from the line-of-symmetry with inclusion diam-eter; (c) variation in non-dimensional ERR with a/W

towards the inclusion, which attains a minimum whenthe crack tip is at a distance of d/2 from the inclusioncenterline. This is similar to the results obtained incase of crack-tip shielding and inclusion eccentricityeffects. With further crack propagation G/G0 increases

Modeling of crack growth by symmetric-Galerkin boundary element method 21

and reaches a maximum just before it travels d/2 acrossthe center of the inclusion. This is followed by a mono-tonic but gradual decrease in G/G0. Similar to the pre-vious results, for all inclusion sizes greater shieldingoccurs while the crack is traveling towards the inclu-sion compared to the amplification effect while reced-ing from the inclusion. Also from the plots it can benoticed that these effects increase with an increase ininclusion size.

3.5 Crack propagation through particle clusters

Next, simulations of crack growth through idealizedclusters of particles in a brittle and compliant matrixis undertaken. In a typical particulate composite sec-ondary phase is randomly distributed in a matrix at aknown volume fraction. In the present work, this situ-ation is approximated using a particle–cluster. Particledistribution, particle size, inter particle distance (clus-ter radius), and cluster orientation relative to the initialcrack orientation, are some of the parameters which canbe used to characterize a cluster. In this study, a six-particle-cluster (see Fig. 9(a)) with a centrally locatedparticle surrounded by five others at the corners of auniform pentagon is used. Unlike square, hexagonal oroctagonal arrangements of particles, this pattern cap-tures randomness of fillers in a matrix to a greater de-gree while being characterized by a few simple param-eters. In the current study, a single cluster is positionedahead of a crack tip in the loading geometry used ear-lier. The cluster geometry for a chosen particle diameterd is defined in terms of (i) distance between particleson the periphery and the center one (cluster radius R)and (ii) the smallest angle (θ ) of the surrounding parti-cles from the line-of-symmetry, as shown in the figure.Clearly, the crack particle cluster interaction and itseffects on crack path and ERR depend on the clusterorientation θ and a measure of volume fraction, whichare investigated next.

3.5.1 Effect of cluster orientation

As the propagating crack negotiates various membersof a particle-cluster, the crack path become tortuouscausing greater energy dissipation and higher overallfracture toughness compared to a neat matrix. In thiscontext, it is interesting to address the effect of parti-

Fig. 9 Interaction between a crack and a particle cluster witha pentagonal arrangement: (a) cluster orientation is defined interms of θ . Crack path in the presence of particle cluster is shown:(b) crack deflection in the presence of particle cluster of variousorientations; (c) variation of ERR with a/W showing the effectof cluster orientation

cle arrangement such as cluster orientation relative tothe initial crack impingement. Accordingly, the role ofangular parameter θ (see Fig. 9(a)) on crack growthbehavior through a particle cluster is studied next. Inthese simulations, the center of the cluster is consid-ered to be located at a distance of C/W ≈ 0.4 where Cis the distance of the center of the inclusion from the

22 R. Kitey et al.

initial crack tip. The cluster radius is considered as 2d,d being the inclusion diameter. The angle θ is variedfrom 21◦ to 33◦ for the pentagonal arrangement usedhere. Figure 9(b) shows the crack propagation for var-ious cluster orientation angles. Evidently, significantcrack path deflections within the cluster can be noticedfrom the figure. The crack deflection from the first par-ticle influences the subsequent growth towards the sec-ond and so on. The crack, already deflected away fromthe line-of-symmetry of the cluster at the first inter-action, undergoes further deflection due to the centralparticle of the cluster. As the crack propagates furtherand away from the central particle, the effect of nextneighboring particles can be noticed. The differences incrack paths are greatest when the crack interacts withthe first particle of the cluster. That is, crack deflec-tion increases monotonically with increasing angularparameter θ . The opposite trend is evident when thecrack reaches the central particle—the crack deflectionat the central particle of the cluster decreases with θ .Once the crack recedes from the central particle, allcrack paths essentially coincide as the crack leaves thecluster. These observations suggest that, the net effectof the angular parameter θ on cumulative crack deflec-tion is small even though crack paths within the clusterare significantly different.

The above observations can be further quantified interms of ERR for each crack path corresponding todifferent values of θ . The effect of the change in θon ERR is shown in Fig. 9(c). Nondimensional ERRis plotted against a/W for various cluster orientations.Plots show significant variation in G/G0 as the crackpropagates and interacts with surrounding particles.Interestingly, ERR values for all cluster orientationsare lower when compared to the case of unfilled (neat)matrix material. This suggests lowering of crack tipstress intensification as various elements of the clusterare encountered by the crack resulting in an overall in-crease in fracture toughness of the material. The clusterorientation affects crack propagation and hence G/G0

differently. As can be seen from the plots that for 300

orientation, the first particle interaction gives lowestG/G0, while for the case of 210 orientation the centerparticle in the cluster affects G/G0 the most. For otherorientations, the values of G/G0 are bounded by thevalues for these two cases. In order to estimate whichcluster orientation dissipates the most energy, averagevalue of nondimensional ERR (ERRa) for each case isdetermined. Here, ERRa is defined as,

ERRa = 1

(a/W )

∫G

G0d

( a

W

). (13)

The ERRa for the crack propagation between a/W =0.25 and 0.70 has been evaluated. For the chosen clusterorientations 21◦, 24◦, 27◦ and 30◦, the ERRa values are0.83, 0.83, 0.84 and 0.80, respectively. Thus, it can beagain be concluded that the effect of cluster orientationhas negligible effect on energy dissipation.

3.5.2 Effect of cluster volume fraction

The effect of particle volume fraction on crack growthis considered next. This can be done by changing ei-ther the cluster radius (R) and keeping the particle size(d) constant or by changing the particle size and keep-ing the cluster radius constant. To avoid particle sizeeffects, an investigation is performed for various clus-ter radii with a fixed particle size.3 It has already beenshown in the previous section that cluster orientationhas only a small effect on ERR. Therefore the clusterorientation is chosen such that the central particle ofthe cluster is located symmetrically at C/W ≈ 0.4 rel-ative to the initial crack tip and the particle nearest tothe initial crack tip has d/3 eccentricity (e) as shownin Fig. 9(a). Defining a control volume to find volumefraction is not straight forward for the chosen pentago-nal cluster arrangement because it cannot be replicatedsymmetrically in all directions. Therefore instead ofvolume fraction, a parameter ‘area ratio’ is defined forquantification purpose. The area ratio, AR, is definedas the ratio of the total area occupied by the particlesinside the pentagon to the total area of the pentagonitself.

Figure 10(a) shows crack deflections in the presenceof particle cluster for 10% to 25% area ratios. Similarbut distinct crack paths can be seen for various val-ues of AR. As with the cluster orientation study, crackpaths show dependency on the value of AR. Earliercrack deflection can be noticed for the case of 10% ARdue the presence of a particle much closer to the initialcrack tip when compared to other AR values. With anincrease in AR, the crack deflections occur at differ-ent a/W values sequentially. The first interaction is the

3 CW = 3

40 in this case. A different particle diameter is chosencompared to the previous computations so that simulations canstill be performed for lower volume fractions without the clustergeometry interfering with the initial crack tip.

Modeling of crack growth by symmetric-Galerkin boundary element method 23

Fig. 10 Interaction of a crack with a particle cluster to study theeffect of volume fraction: (a) crack deflection for various volumeratios; volume ratio is changed by expanding the cluster radiusR shown in Fig. 9(a); (b) variation of energy release rate withcrack growth

only dominant distinguishing feature among the differ-ent crack paths. The paths tend to merge while propa-gating around the central inclusion. That is, the cracktrajectory is essentially same between a/W ≈ 0.54 and0.60 in the figure. With further crack propagation thepaths show variation with AR. The distinct crack pathsbeyond a/W ≈ 0.60 are due to the difference in sur-rounding inclusion locations. Interestingly, when com-pared to the crack deflection in the presence of a singleparticle (Figs. 6(a), 8(a)), prominent and higher crackdeflection can be noticed for the cluster for all valuesof AR.

Figure 10(b) shows the variation of nondimension-al ERR with a/W for various area ratios. A combinedeffect of crack tip shielding and amplification, parti-cle size effect and particle eccentricity with respect topropagating crack can be seen in terms of distinct G/G0

variation for different ARs. For lower AR values the

decrease in ERR can be noticed earlier due to the prox-imity of initial crack tip to the nearest particle. But thisalso gives rise to an early amplification as the crackpropagates away from the particle. The amplificationeffect is more prominent for lower AR values due toits larger inter-particle separation distance. In case ofhigher ARs, however, the amplification effect is rela-tively suppressed due to the proximity of next neighbor-ing particle. When a crack negotiates the first periph-eral particle of the cluster, the lowest G/G0 occurs atdifferent a/W values depending upon the particle loca-tion with respect to the initial crack tip. Even thoughthe plots show distinct variations of G/G0 for variousARs when a crack propagates around its first encoun-ter, they all tend to follow the same path as the centralparticle is approached. For all values of AR, nearly thesame G/G0 variation between a/W = 0.5 and 0.58 isevident. Further crack propagation shows the effect ofnext neighboring particle effect on G/G0, where againlowest G/G0 occurs at different a/W depending uponthe particle location. This is followed by some increasein G/G0 similar to the amplification effect, and then amonotonic but gradual decrease in G/G0.

Here again the area ratio effect is quantified usingaverage ERR, ERRa , defined earlier in Eq. (13). Acrack propagates different distances within the parti-cle cluster due to different cluster radius for each AR.Hence, the same crack propagation length can not beused to evaluate ERRa . Subsequently, a crack propaga-tion length of 2R within the particle cluster is consid-ered for comparison. For 10%, 15%, 20% and 25% ARvalues, the ERRa values have been calculated as 0.88,0.86, 0.85 and 0.83 respectively. A monotonic decreasein ERRa with increase in AR can be noticed. This inturn suggests that the material becomes relatively moreresistant to crack propagation with increase in volumefraction of secondary phase rigid fillers.

4 Concluding remarks

An SGBEM-based tool is developed to simulate crackgrowth in a heterogeneous brittle material system. Sim-ulations are performed to study crack paths and tocompute SIFs under plane stress conditions. The quasi-static crack-growth prediction tool is first bench markedand simulations are performed to examine the particle

24 R. Kitey et al.

size, eccentricity and the shielding effects on crackgrowth in the presence of isolated inclusions. In thesesimulations the secondary phases are assumed to beperfectly bonded to a brittle and relatively compliantmatrix. Subsequently, crack growth through idealizedparticle clusters of uniform pentagonal arrangement arestudied. The results are summarized as follows:

• A propagating crack is affected by a rigid inclusionin terms of energy release rate ahead of visible crackdeflection. That is, ERR values are affected whenthe crack is a few inclusion diameters (d) away fromit. On the other hand crack deflection is noticeableonly when the crack is ≤ 3d/2 from the center of theinclusion while approaching it. For both symmetricand eccentric inclusions (with respect to the initialcrack orientation), the energy release rate is mini-mum when the crack tip is at a distance of ≈ d/2from the center of the inclusion. As eccentricityincreases, the crack deflection decreases while theenergy release rate increases.

• Crack tip shielding and amplification increase withdecrease in inter particle separation distance in aparticle-pair arranged symmetrically relative to thecrack. For an eccentrically arranged particle-pair,on the other hand, the crack tip shielding is greaterwhen compared to the symmetric case. However,eccentricity has negligible effect on amplification.In general, shielding effects are greater than ampli-fication effects when a crack propagates around aparticle.

• Crack deflection increases and the energy releaserate decreases with increase in particle size whenthe size of the reinforcement is in the range d/W =2−20%. A propagating crack is influenced by alarger particle in front much earlier than the smallerones. Yet, the crack deflection is relatively unaf-fected by the inclusion when the tip is at a distanceof ≥3d/2 from the center of the inclusion.

• A crack propagating through a particle cluster showsdistinct crack trajectories for various cluster orien-tations. The cluster orientation, however, has neg-ligible effect on overall energy dissipation.

• The crack path depends upon the cluster area ratio,AR. The total energy dissipation decreases with in-crease in cluster AR. This suggests that the materialbecomes more resistant to the crack propagation forhigher area ratios.

Acknowledgements This research was supported in part by theU.S. Army Research Office Grant #W911 NF-04-1-0257 to HVTand A-VP, and by the Applied Mathematical Sciences ResearchProgram of the Office of Mathematical, Information, and Com-putational Sciences, the U.S. Department of Energy under con-tract DE-AC05-00OR22725 with UT-Battelle, LLC., to TK. Theauthors would also like to thank Dr. Len Gray (ORNL) for hiscontribution to the SGBEM codes used in this work.

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