Composites: Part B 40 (2009) 7–16

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Composites: Part B

journal homepage: www.elsevier .com/locate /composi tesb

Developing a stochastic model to predict the strengthand crack path of random composites

Hunain Alkhateb a, Ahmed Al-Ostaz a,*, Khalid I. Alzebdeh b

a Department of Civil Engineering, University of Mississippi, 202 Carrier Hall, University, MS 38655, USAb Department of Mechanical & Industrial Engineering, Sultan Qaboos University, Muscat, Oman

a r t i c l e i n f o

Article history:Received 10 September 2007Received in revised form 20 February 2008Accepted 21 September 2008Available online 30 September 2008

Keywords:A: Computational modellingM: Nano-structuresP: FracturePolymer

1359-8368/$ - see front matter Published by Elsevierdoi:10.1016/j.compositesb.2008.09.001

* Corresponding author. Tel.: +1 662 915 5364.E-mail address: alostaz@olemiss.edu (A. Al-Ostaz)

a b s t r a c t

We characterize fracture and effective stress–strain graphs in 2D random composites subjected to a uni-axial in-plane uniform strain. The fibers are arranged randomly in the matrix. Both fibers and matrix areisotropic and elastic–brittle. We conduct this analysis numerically using a very fine two-dimensional tri-angular spring network and simulate the crack initiation and propagation by sequentially removingbonds which exceed a local fracture criterion. In particular, we focus on effect of geometric randomnesson crack path of random composites. Based on that two stochastic micro-mechanic models are presentedthat can predict with confidence the failure probability of random matrix–inclusion composites.

Published by Elsevier Ltd.

1. Background

Statistical uncertainty in strength properties of engineeringmaterials has been a known problem for decades. Predicting crackpatterns in random composites is controlled by many factors: dis-tribution and shape of inclusions, material constants of matrix andinclusion, boundary conditions at inclusion–matrix interfaces andproximity to the surface.

Numerical approaches usually involve assumptions that thearrangement of inclusions is periodic. However, the distributionof inclusions in composites is in general disordered. Some of theearliest studies on the subject date back to the twenties and thir-ties when Weibull [1] introduced his well known probability fit.However due to the complexity of the problem and lack of ade-quate tools, these derivations had to be based on rather simplistictreatments. Therefore, they are perfectly suited for treatment of asingle crack in a homogenous material, rather than for analysis offields of many crack in heterogeneous microstructures. The lackof adequate tools for treatment of such problems was sorely feltin damage mechanics, which is characterized by interplay of manycracks in disordered materials. The situation has changed over a re-cent decades due to the advent of computational mechanics andpowerful computers, which now begin to permit a’brute force’ sim-ulation of complex fracture events; theses are the tools that werenot available to Weibull and his contemporaries.

Ltd.

.

As far as the computational fracture mechanics is concerned,two principle methods are available: finite element and spring net-works. The second one is borrowed from condensed matter phys-ics; it relies on a regular shape lattice representation of ahomogenous continuum, which avoids any mesh generation (pre-processing) that is a problem with finite elements. This means het-erogeneous media can be easily modeled by assigning the springconstants all over the lattice according to local phase properties.The second major advantage of the spring networks, is the possibil-ity to create cracks by removing the spring bonds without any needfor re meshing that is often required in simulations of crack prop-agation in finite element methods. For slow phenomena one simu-lates the crack evolution in a quasi-static manner, while fordynamic problems one introduces the inertia forces to turn themethod into a variant of molecular dynamics.

Another, important issue in the area of composites materials isthe size/scale effect. It is generally established that the fracturestrength of structural materials is not static, but rather, it decreaseswith increasing size of the specimen. Until recently, it was gener-ally believed that the size effect in structural failure is of statisticalorigin, caused by randomness of material strength [2]. However, ithas been discovered that an important and often dominant size ef-fect can be of purely mechanical and deterministic origin. This ef-fect can be caused by the influence of the stored elastic strainenergy on the nominal strength of the structure. Constitutive mod-eling for the particle size effect on the strength of particulate-rein-forced metal matrix composite was investigated by Zhu et al. [3].The approach is based on a gradient-dependent theory of plasticitythat incorporates strain gradients into the expression of the flow

L

d

Fig. 1. A disordered composite with randomly arranged, non-overlapped inclusionsof diameter d and a square window of observation with a side length of L.

8 H. Alkhateb et al. / Composites: Part B 40 (2009) 7–16

stress of matrix materials, and a finite unit cell technique that isused to calculate the overall flow properties of composites. Itwas found that the strain gradient term introduces a spatial lengthscale in the constitutive equations of composite material and thedependence of the flow stresses on the particle size/spacing couldbe obtained. Moreover, a non-dimensional analysis along with thenumerical result yields an explicit relation for the strain gradientcoefficient in terms of particle size, strain and yield stress. Theauthors of this paper addressed this issue in [4].

Various studies related to the present one have recently beenconducted using similar tools in the fracture/damage mechanicsof composite materials. Several symposia, conferences, journal spe-cial issues and books reflect the activity in this area, e.g. [5–25].The present paper adopts the spring network method for the anal-ysis of fracture characteristics of a two-dimensional matrix–inclu-sion composites with locally isotropic elastic–brittle phases.Results of this research will help in developing a framework of sto-chastic fracture and damage, mechanics for a class of random com-posites in which inclusions are randomly distributed in the matrix,an effort begun in recent decades [26–30]. More specifically crackinitiation and propagation as well as effective responses in two-dimensional brittle matrix are studied for the in-plane elasticityproblem under uniaxial stretching loading.

Al-Ostaz and Jasiuk [31,32] studied the crack initiation andpropagation in elastic–brittle (epoxy) and elasto-plastic (alumi-num) thin sheets containing over thirty randomly arranged circu-lar holes. The specimens were subjected to a uniaxialdisplacement. The authors studied a single configuration of a ran-dom arrangement both numerically, by using a finite elementmethod or a spring network, and experimentally using an Instronmachine and a high speed camera. The goal of this study was topredict numerically the crack path and compare that with thecrack patterns obtained experimentally. In the numerical studyfour fracture criteria were used to identify the location of a crackinitiation, namely the maximum principal stress criterion, the min-imum strain energy criterion, the maximum circumferential stresscriterion, and the minimum thickness criterion. The location of thecrack initiation was found to be the same regardless of the crite-rion. The numerical results using finite element and finite differ-ence [31,32] methods showed different crack paths for the samegeometric arrangement of holes using different failure criteriaand different meshes. The experimental study of crack initiationand propagation in a brittle material (epoxy) with randomly dis-tributed holes showed a catastrophic failure. Similarly, several dif-ferent crack patterns were obtained experimentally by fracturingseveral specimens with the identical hole arrangements. These re-sults emphasized the need for a stochastic fracture model. Themodel need to provide the means to customize the compositesfor specific applications. The major pay-off of this work will be toprovide an understanding of a new class of materials that can thenbe incorporated into various airfield and structural applications.

However, an important and complicating factor in studying thefracture of brittle materials is the probability of failure. Until re-cently, engineers avoided brittle materials in load bearing applica-tions. Advantages in strength, weight or high temperatureresistance would not compensate for main drawback: tendencyto fail unpredictably. In the last decade progress in brittle materialtechnology (ceramics composites, carbon–carbon composites,polymer composites and other) made these composites attractivefor structural applications. However, these can not be used safelyunless one can predict with confidence their probability of failureand statistical strength. A significant portion of the research in thisarea was directed at studying the probabilistic aspects of failure inbrittle materials [33–36]. The research attempted to answer twotypes of questions. First, what is the probability of the elementssudden failure under a given load? That is, how to evaluate the

probability distributions of critical loads, crack lengths, etc. Second,what is the probability of the element’s failure in a given time?That is, how is the probability distribution of time to failure is eval-uated? The crack diffusion theory (CDT) developed by Chudnovskyand Kunin [33] addresses the first question under various condi-tions. It combines continuum mechanics concepts of a crack, stressconcentration, energy release rate, etc. with statistical weakest linkideas in a Griffith type criterion of infinitesimal crack advancethrough a random field of a specific fracture energy. Various prob-abilities associated with crack formation are expected as averages,of functional integrals, over the space of virtual crack paths. CDTintroduces parameters characterizing materials’ resistance to frac-ture with an experimental methodology of their evaluation.

In this paper, we study damage patterns and effective stress–strain responses as a function of several parameters: mismatch ofelastic properties of matrix and inclusions, mismatch of strengthproperties of matrix and inclusions. We display the results on aso called damage plane, that is a plane of both mismatch parame-ters, and we call such results damage maps. Two stochastic modelsare used for predicting crack path and effective stress–strainresponses.

2. Random composite and its spring network representation

2.1. Model of a random composite

In random matrix–inclusion composites there are two aspectsimmediately noted; the spatial disorder and the periodicity onthe scale of window L (Fig. 1). Thus, these two aspects need to beconsidered whenever a simulation of random composite of an infi-nite medium is desired.

As classically defined in mechanics of random media, a randomcomposite is a set B = {B(x); x 2X} of a deterministic mediaB(x)s, where X refers to the total number of subsets B(x) (proba-bility space), and x specifies a realization in which centers of cir-cular inclusions generated within a periodic window size L. Therealizations are generated according to a planner Poisson pointprocess. This Poisson process is subjected to the condition of min-imum separation distance between any pair of generated circularinclusions to ensure that they do not over lap.

Our interest is in the effective response of the random mediumon finite scales with respect to the inclusion size. The effective re-sponse in such a situation will strongly depend on the type ofboundary conditions specified on the edges of a window. For

Table 1Inclusion diameter and scale d for a constant periodicity size L = 1.

Inclusion diameter (d) Scale d

0.4 2.50.2 50.1 10

H. Alkhateb et al. / Composites: Part B 40 (2009) 7–16 9

example, one could consider kinematic (displacement-controlled)boundary conditions which would lead to stable crack propagation,or force (stress-controlled) boundary conditions, which would re-sult in unstable fracture. The drawback of these two types of con-ditions lies in the need for an arbitrary interpretation ormodification of the microstructure: (i) either isolate a test window‘‘at random” from a theoretically infinite disordered composite,which would result in some inclusions being cut at the windowboundaries or (ii) modify the microstructure in the boundary zoneso as to avoid any such non-physical cutting of the inclusions. Toobviate this issue of boundary effects, we introduce a periodicityon the scale L, as shown in Fig. 2. This immediately defines adimensionless parameter d = L/d which specifies the length scalesof material disorder, and thus, the size of a meso-scale element,which we call a statistical volume element (SVE).

In this study, periodic boundary conditions are applied suchthat:

uiðxþ LÞ ¼ uiðxÞ þ e0ijxj

tiðxþ LÞ ¼ �tiðxÞ8x 2 ob ð1Þ

where u = (u1,u2), e0 = (e11,e22) is a given constant strain, andt = (r11n1,r22n2) is the surface traction. Also L = Le, with e being aunit normal, and oBd is the boundary of Bd shown in Fig. 2. Fig. 2shows the situation for e0 = (0,e22), which will henceforth be theloading case of choice in this study. Due to the isotropy of the Pois-

X1

X2

Fig. 2. RE of a periodic window of scale d = L/d for the case of disordered matrix–inclu

Cm

icrε m

crεε

σ

Ci

a

Fig. 3. (a) Elastic–brittle stress–strain curves for matrix an

son point process of inclusions’ placement no generality is lost. Also,it is important to note that due to the introduction of periodicity onthe length scale L, a relatively weak variability will be observed forsmall d compared to what would be obtained under other types ofboundary conditions (kinematic, force, and mixed). For more detailsee Alzebdeh et al. [26].

Three different length scales for a window in which inclusionsare randomly distributed at a constant volume fraction of 40%are considered in this study (Table 1). For all cases, the inclusionhas a size of 5 lattice elements in radius. However in our discussionof this paper we will focus for the case of d = 10.

Both phases of the composite material are taken to be linearelastic homogenous (Fig. 3a). The boundary condition defined byEq. (1) on Bd results in an effective random stiffness tensor Cd(x)with constitutive law stated as

rðxÞ ¼ CdðxÞ:e0 ð2Þ

1Bδ∂ 1Bδ∂

2Bδ∂

2Bδ∂

d

oε

oε

sion composite of periodicity L showing a schematic of the applied loading used.

m

i

C

C

Soft

Weak

Stiff

Strong

Stiff

Weak

mcr

icr

εε

Soft

Strong

b

d inclusion materials; (b) sketch of the damage plane.

10 H. Alkhateb et al. / Composites: Part B 40 (2009) 7–16

The variation in the arrangement of microstructural entities i.e.inclusions distributed in the matrix makes up a realization x inwith a distinct random geometry. This results in fluctuation of frac-ture characteristic of such composites which apparently disap-

Post PeakLinear Elastic

a b

Fig. 4. Effective stress–strain curves: (a) elasti

L

d

Fig. 5. Fine mesh spring network

Fig. 6. Final crack path for six realizations of a two phase co

pears as d ?1. Fracture response in random composites is afunction of the mismatch in elastic constants of the constituents(expressed here as a ratio, Ci/Cm) and the mismatch in strength ofconstituents ðei

cr=emcrÞ. Ci, and Cm are the elastic constants of inclu-

Pre- Peak

c

c–brittle; (b) post-peak; and (c) pre-peak.

2

4 1

1

2

1

6 1

2 32

3

45

6

2

3

4

5 6

1

1= 4 2= 3= 5= 6 1= 2= 4= 6 2 = 5

foe interface springs: kb= (lm/ l km + li / l ki)

-1

where l is spring length

with a zoom in for a unit cell.

mposite with scale d = 10, Ci/Cm = 0.01 and eicr=em

cr ¼ 10.

Fig. 8. Specimen x with a specific realization of fracture path.

H. Alkhateb et al. / Composites: Part B 40 (2009) 7–16 11

sion and matrix respectively, while eicr and em

cr are the elastic strainstrength of inclusion and matrix, respectively. The schematicsketch in Fig. 3b shows different regions as a result of differentcombination of ratios of mismatch in stiffnesses and strengths.

One of the basic questions is: given a composite made of twodifferent elastic–brittle phases, is its effective response also elas-tic–brittle or not? The spring network method provides a conve-nient way to answer this question for a wide range ofparameters. At this stage we note that besides to the elastic re-sponse, two other types of responses are also possible: post-peakand pre-peak depending on whether damage occurs after or priorto the peak rmax in the effective stress–strain curve (Fig. 4).

2.2. Spring network discretization

Spring networks and finite element meshes are common mod-els employed to simulate the crack propagation in 2D elastic–brit-tle composite materials. Alternatively, one can use mesh lessmethods or particle dynamics which also were used in the litera-ture [37–39]. Element breaking rules are established based onthree main criteria: critical energy, critical stress, and critical strainin the bonds (i.e. springs). Regular meshes in spring networks arehighly anisotropic with respect to crack propagation.

In this paper, regular triangular lattices with linear centralsprings (Fig. 5) are adopted. Elastic moduli of individual phasesare mapped into spring stiffness according to the formula.

Cijkl ¼a

2ffiffiffi3p

X6

n¼1

lni ln

j lnk ln

l ð3Þ

where l1 = cosh, l2 = sinh are the direction of the spring, a is springconstant.

The spring network discretization relies on that each circularinclusion diameter is a multiplier of several springs, and the inclu-sion centers are adjusted such that they fall on spring networknodes. The springs stiffness are assigned according to the followingcriteria: if the spring falls within the inclusion boundary it is as-signed a stiffness ki, if it falls with in the matrix boundaries it is as-signed a stiffness km, and any bonding spring connects both phasesit is assigned a stiffness kb. The values of ki and km are calculatedaccording to individual phases elastic properties according to Eq.(4):

ε

σ

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

Fig. 7. Effective stress–strain curves for six realizations of a two phase compositewith scale d = 10, Ci/Cm = 0.01 and ei

cr=emcr ¼ 10.

C1111 ¼ C11 ¼3ffiffiffi3p

8a; C1122 ¼ C12 ¼

ffiffiffi3p

8a; C1212 ¼ C66

¼ffiffiffi3p

8a ð4Þ

For the boundary springs, a weighted average of spring con-stants of the inclusion (ki) and the matrix (km) according to the fol-lowing relation:

kb ¼ ðlm=lkm þ li=lkiÞ�1; l ¼j lb j¼ lm þ li ð5Þ

where lm is the partial length of the boundary spring inside the ma-trix and li is the partial length of the boundary spring inside theinclusion.

The key unknown in spring network method is the displace-ment domain u(i,j) over a given Statistical Volume Element (SVE).For 2D in plane elasticity problems, each node will have two de-grees of freedom (u1,u2). A solution is accomplished by employinga conjugate gradient method with respect to the total energy (sumof the energies stored in all spring bonds) and subjected to theboundary condition in Eq. (1). With u(i,j) solution is in hand, the to-tal potential energy E, stored in the network provides a basis fordetermination of the equivalent effective medium according tothe relation

E ¼ Vd=2 � CdðxÞð:e0Þ2 ð6Þ

where Vd is the volume of Bd(x).Crack evolution is simulated by removal of springs whenever a

failure criterion is met. In this case we use maximum strain crite-rion such that, if the strain of the spring in the matrix reaches acritical strain ðem

crÞ or if the strain in the inclusions reaches a criticalstrain ðei

crÞ then that spring will be removed:

em P emcr and ei P ei

cr ð7Þ

m: refers to matrix; i: refers to inclusion.Damage maps and effective stress–strain curves of a wide range

of stiffness ratios and strength ratios for three different scales arereported in [4]. Typical final crack and effective stress–strain ofsix realizations for the case of stiffness mismatch Ci/Cm = 0.01 andstrength mismatch ei

cr=emcr ¼ 10 and periodicity scale d = 10 are

Table 2Maximum Entropy Formulation Constant A for the case of periodicity scale d = 10.

Ci/Cm

0.1 1 10

ei/em

0.1 NA 401.3314 151.98571 151.9857 23.048510 151.9857 23.0485 22.2045

Table 3Maximum Entropy Formulation Constant k1 for the case of periodicity scale d = 10.

Ci/Cm

0.1 1 10

ei/em

0.1 NA �215.522 �80.75581 �80.7558 �169.383510 �80.7558 �169.3835 �158.3948

Table 4Maximum Entropy Formulation Constant k2 for the case of periodicity scale d = 10.

Ci/Cm

0.1 1 10

ei/em

0.1 NA �189.4784 �70.80871 �70.8087 16.243710 �70.8087 16.2437 �106.8371

Table 5The transition probability density q.

Ci/Cm

0.1 1 10

ei/em

0.1 NA 0.004365 0.011481 0.01148 0.333610 0.01141 0.3336 0.3064

12 H. Alkhateb et al. / Composites: Part B 40 (2009) 7–16

shown in Figs. 6 and 7 respectively which shows the non-unique-ness of crack paths and effective responses due to random distribu-tion of the inclusions.

3. Stochastic modeling

3.1. Stochastic modeling of crack path

In this paper the following formulation is used.For the case of n specimens, consider a specimen x shown in

Fig. 8 with one specific realization of fracture path. It occurred atthe expense of the dissipated energy E(x), which we can get fromthe r–e plots. The fracture path has a length L(x), which may bemeasured (either experimentally or numerically). Thus, we can cal-culate the average dissipated energy per unit length: cavgðxÞ �

EðxÞLðxÞ.

For n specimens we have a set: {cavg(xi); x1, x2, . . . ,xn} each ofwhich carries a probability P(x1) = P(x2) = � � � = P(xn). Thus wecan compute the moments of the (as yet unknown) distributionPðcavgÞ; i:e: : hcavgi; hc2

avgi; hc3avgi; . . ..

If number n is small, then only the first two moments are reli-able (average and standard deviation). We consider these as con-straints in a calculation of P(cavg) according to the MaximumEntropy Formulation, or MEF. (By definition: qðcavgÞ ¼

dPðcavgÞdcavgÞ.

According to MEF, we will have the probability density of cavg inthe following form:

qðcavgÞ ¼1A

exp½�k1:cavg � k2:c2avg� ð8Þ

where A, k1, k2 are to be determined from the three coupledequations:

10

1

0.10

0.10 10 /icrε

Ci /Cm

.011 .012 .013 .014

.6

.5

.4

.3

01

γ avg γ avg

γ avg

γ avg γ avg γ avg

γ avg

ρ

.011 .01 2 .013 .014

.6

.5

.4

.3

01

ρ

.011 .01 2 .013 .014

.6

.5

.4

.3

ρ

.011 .01 2 .013 .014

.6

.5

.4

.3

ρ

.0025 .003

.005

.004

ρ

.006 .008

.013

.012

..011

.01

ρ

.011 .01 2 .013 .014

.5

.4

.3

.01

ρ

1 εcrm

Fig. 9. Relation between energy per unit length cavg and its transition probability density q(cavg) for the case of d = 10.

Table 6Std (cavg).

Ci/Cm

0.1 1 10

ei/em

0.1 NA 0.0002526 0.00079151 0.0007915 0.000764810 0.0007915 0.0007648 0.0008789

H. Alkhateb et al. / Composites: Part B 40 (2009) 7–16 13

AZ 1

0e�k1cavg�k2c2

avg dcavg ¼ 1

hcavgi ¼Z 1

0cavgAe�k1cavg�k2c2

avg dcavg

hc2avgi ¼

Z 1

0c2

avgAe�k1cavg�k2c2avg dcavg

ð9Þ

Clearly, cavg is a positive valued real number. With q(cavg) inhand, we can predict the probability of occurrence of any othercrack path. This derivation-model will be further improved to ac-count for the orientation of cracks with respect to the mean load-ing conditions (i.e. principal stresses/strains) and for the local crackinteractions.

In order to use the Maximum Entropy Formulation, both thecrack lengths and stored strain energy need to be predeterminedexperimentally or numerically. Using spring lattice model de-scribed in Alzebdeh et al. [4], number of broken springs for eachrealization (x) in the space X are calculated. Then the total cracklength is estimated to be equal to the total number of brokenspring � length of each spring. The total energy dissipated to forma full crack path is calculated as the area under stress–strain curve.The average dissipated energy per unit length is cavgðxÞ �

EðxÞLðxÞ.

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.40

0.02

0.04

0.06

0.08

0.1

0 0.2 0.40

0.02

0.04

0.06

0.08

0.1

10

1

0.10

0.10 1

Ci /Cm

ε

ε

0 0.1 0.20

0.1

0.2

0.3

0.4

ε

0

ε

σ

σ σ

σ

0 0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

ε

σ

Fig. 10. Effective stress–strain curves in the damage plane for scale d

The three coupled equations 9 are to be solved to obtain theconstants k1, k2, A. The right hand side of the first equation in Eq.(9) is equal to 1 illustrates an assumption that the we have stablecrack propagation. The second equation in the three coupled equa-tions 9 is the first generated moment function which evaluates themean of the average energy release rate. The third equation is thesecond generated moment function of the energy release rate.Since the number of samples considered in this study is relativelysmall, there is no need to calculate higher order moments.

Calculated values of Maximum Entropy Formulation constantsA, k1 and k2 are shown in Tables 2–4 for range of Ci/Cm and ei

cr=emcr

combinations for the case of periodicity scale d = 10. The calculatedtransition probability densities are shown in Table 5.

The relation between energy per unit length cavg and its transi-tion probability density q(cavg) for the case of d = 10 is shown inFig. 9.

From continuum theories of fracture mechanics, it is wellknown that the mode of the slow crack path in brittle materialsare observed from Markovian stochastic pattern of microscopicjump, followed by a random waiting time, followed by anotherjump, and so on. This is related to energy dissipated at the arrestpoints. One can treat such random jumps as a magnitude withinthe crack diffusion theory, those jumps intercept as a consequenceof local failures ahead of the current crack tip to adopt the Griff-thes–type criterian crack advance. The potential energy releaserate should exceed at acritical value of yavg, the specific fracture en-ergy, to initiate a crack.

For a homogenous material, yavg statistics should be indepen-dent of the position and it we will have a deterministic unique va-lue. On the other hand, for heterogeneous material which, arandom field for yavg is expected and crack growth between anytwo points is not any more a function of the position and the prob-

0.6 0.8

10

0.3 0.4 0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

ε

00

0.1 0.2 0.3 0.4

0.1

0.2

0.3

0.4

ε

σ

σ

/icrε εcrm

= 10 and for the same random arrangement (same realization).

14 H. Alkhateb et al. / Composites: Part B 40 (2009) 7–16

ability of crack growth path criterian is the same for any path, andin additional the probability of any backward jump is always zero.

The transition probability density q satisfies a hyperbolic PDE inprobabolic Kalmogroff equations, q is arrested by an x to X map-ping once it is initiated at x within a certain path. Still there is aprobability that the crack will not be arrested at this particularpath at all, so when q has a small value that indicates that theprobability for a linear crack propagation between the two edgesis very small value. By examining the damage maps in Alzebdehet al. [4], we observe that most of our crack paths are not linearfrom one edge of the specimen to the other. This is in good agree-ment with the small values of q that we have in Table 5. The smallvalues of standard of deviation of the energy release rate yavg (Ta-ble 6) indicates that the average dissipated energy is constant andnot function of the geometry of x.

Fig. 11. (a) Effective r(e) response in a macroscopic, deterministic continuum limit L/drealization r(x,e) of random process r(e); (c) a corresponding realization C(x,e) of theprocess Dr(e). Note that e plays the role of a controllable, time-like parameter of stocha

0 0.1 0.20.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.31.08

1.1

1.12

1.14

1.16

1.18

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

0 0.1 0.2 00.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.40

0.5

1

1.5

10

1

0.10

0.10 1

Ci /Cm

εε

ε

ε ε

cc

c

c c

Fig. 12. Realization stiffness variation C(x,e) of

For the energy release rate cavg standard deviation, Table 6,small values, 10e�3 shows that we have a constant value, and theaverage dissipated energy value, is a constant, and its not a func-tion of the geometry of the (x).

3.2. Stochastic modeling of stress–strain response of randomcomposite

We consider a mesoscopic picture (d = d/L is finite). Whered = inclusion diameter, and L is RVE size. In this case the zigzagcharacter and randomness are present, and softening is tractableas a random jump. The Markov process C(e) is a kinetic equationof damage evolution. Effective stress–strain curves for a singlerealization and statistical scale d = 10 are shown in Fig. 10 for

?1 with local magnification showing an infinitesimal zigzag; (b) general zigzagrandom process C(e); and (d) a corresponding realization Dr(x,e) of the randomstic processes [16].

0 0.1 0.2 0.3 0.41.05

1.1

1.15

1.2

0.3

0 0.1 0.2 0.31.08

1.1

1.12

1.14

1.16

1.18

.3 0.4

10

ε

ε

c

c

/icrε εcrm

the random process C(e) for selected cases.

0 0.1 0.2 0.3 0.4-0.05

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4-0.02

0

0.02

0.04

0.06

0.08

0 0.1 0.2 0.3-0.1

0

0.1

0.2

0.3

0 0.1 0.2 0.3-0.02

0

0.02

0.04

0.06

0.08

0 0.1 0.2 0.3-0.1

0

0.1

0.2

0.3

Δ σΔ σ

Δ σ Δ σ Δ σΔ σΔ σ

0 0.1 0.2 0.3-0.1

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4-0.1

0

0.1

0.2

0.3

0.410

1

0.10

0.10 1 10 /i m

cr crε ε

Ci /Cm

ε ε ε

ε ε

ε

ε

Fig. 13. Realization stress jump Dr(x,e) of the random process Dr(e) for selected cases.

10

1

0.10

0.10 1 10

Ci /Cm

.4

.3

.2

.1

ε

σ

ε

σ

.3

.2

.1

0

.10 . 2

.4

.3

.2

.1

ε

σ

.3 .4

.08

.06

.04

.02

ε

σ

ε

σ

.10 . 2 .3

.10 . 2 .3

.10 . 2 .3 .10 . 2 .3

.3

.2

.1

0

.08

.06

.04

.02

ε

σ

.3

.2

.1

0

ε

σ

.4 .3

.10 . 2 .4 .3

.1 . 2

/icrε εcrm

Fig. 14. Predicted stress–strain curve for the cases shown in Fig. 10.

H. Alkhateb et al. / Composites: Part B 40 (2009) 7–16 15

16 H. Alkhateb et al. / Composites: Part B 40 (2009) 7–16

a range of stiffness and strength ratios. The curves are obtainedusing spring network triangle lattice model [4].

A second stochastic model which was originally proposed byOstoja-Starzweski and Lee [16] is used. A stochastic response lawr(x,e0) where e0 is the applied strain and x indicates randomnessand parametric dependant with respect to X space. Assuming thatthe observed response is some zigzag curve r(x,e0), as shown inFig. 10. Considering e0 as constant for every time step, then the ran-dom process may be expressed as:

rðeÞ � frðx; eÞ;x 2 X; e 2 Tg; T ¼ ½0;1Þ ð10Þ

A more convenient formulation is obtained from this by consid-ering the slope dr/de of all the piecewise linear segments of thezigzag process r(e), whereby we arrive at a random jump processC(e) � {C(x,e); x 2X, e 2 T} with constant states, which has piece-wise constant realization C(x,e). Associated to C(e) there is a ran-dom impulse process Dr(e) � {Dr(x,e); x 2X, e 2 T}. Thus forany realization x 2X we have

rðx; eNÞ ¼XN�1

n¼0

CnðxÞ½enþ1 � en� þ Drðx; enÞ� �

ð11Þ

where, n is the total number of previous jumps.Using the generated zigzag stress–strain curves (e.g. Fig. 8) for

periodicity scale d = 10, the corresponding realization C(x,e) ofthe random process C(e) are shown in Fig. 12, the correspondingrealization Dr(x,e) of the random process Dr(e) are shown inFig. 13. The big jumps in Dr(e) indicates that the failure is domi-nated by an elastic–brittle behavior characterized by one majordrop in the stress values. Predicted stress–strain relations for thecases described in Figs. 10 and 11 utilization the formulation pre-sented by Eq. (11) is shown in Fig. 14.

Now a simple model will result once we introduce a Markovassumption for the process C(e), so that it may be considered aMarkov jump process. Its evolution will be completely determinedby the next-jump probability density function. Such probabilityfunction for a given location of damage will be assessed from thestatistic realizations obtained by Monte Carlo simulations andusing previously obtained experimental data.

4. Conclusion

In this paper two stochastic models were used to study quasi-static crack growth in brittle materials with randomly distributedinclusions. Direction of crack and connectivity of micro cracksare the subject of the authors next publication.

Acknowledgements

The authors wish to acknowledge the partial support for this re-search from US Air Force Grant # F08637-03-C-6006 with a subcontract # S-29000.23 from Applied Research Associates Inc.

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