Composite Structures 125 (2015) 314–327

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

XFEM analysis of fiber bridging in mixed-mode crack propagationin composites

http://dx.doi.org/10.1016/j.compstruct.2015.02.0020263-8223/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: High Performance Computing Lab, School of CivilEngineering, University of Tehran, Tehran, Iran. Tel.: +98 21 6111 2258; fax: +98 216640 3808.

E-mail address: smoham@ut.ac.ir (S. Mohammadi).

A. Afshar, A. Daneshyar, S. Mohammadi ⇑School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran

a r t i c l e i n f o

Article history:Available online 13 February 2015

Keywords:Fiber bridgingExtended finite element method (XFEM)Mixed-mode fractureArbitrary crack propagation

a b s t r a c t

In this work, the extended finite element method (XFEM) is further generalized to study the fiber bridgingphenomenon in fracture analysis of unidirectional composites. Treating composites as a biphase material,fibers effects on increasing fracture toughness are modeled by non-linear springs integrated in the XFEMformulation of displacement discontinuity. Combining the capability of XFEM in modeling arbitrary crackpath with the traction-separation representation of the fracture process zone in the crack-bridged zonemodel, this formulation allows for simulation of fiber bridging in a crack propagation process without apriori knowledge of the crack path. Moreover, the criterion originally proposed for predicting the crackpropagation path in orthotropic solids is modified to incorporate the effects of fiber orientation. Severalnumerical simulations of mode I and mixed-mode problems are provided, and the corresponding force–displacement curves are analyzed to address the accuracy and efficiency of the proposed method.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Reinforcing a matrix with glass or carbon fibers not onlyimproves the stiffness, but also increases its crack growth resis-tance. When a crack starts to propagate in a fiber reinforced com-posite, the fibers bridge the crack and exert force on the crack face,enhancing the fracture toughness of the composite. Due to theincreasing significance of composite materials in aerospace andautomotive industry, many researchers tried to investigate theeffects of fiber bridging on crack propagation in composites byexperimental and/or numerical methods. The aim of this work isto propose a simple yet powerful formulation for simulation offiber bridging in arbitrary crack propagation in composites.

In unidirectional fiber reinforced composites, many researchershave tried to incorporate the effects of fiber cross-over bridginginto a cohesive zone model by utilizing a traction-separation lawof the fracture process zone [1–4]. Adopting appropriate forms ofthe traction-separation law and accurate calibration of cohesiveparameters have allowed these works to simulate the fiber bridg-ing in the fracture process zone. Nonetheless, these studies arerestricted to discrete interelement approach of the cohesive zonemodel [5–8], in which the crack growth path must be known a

priori. While this assumption appears to be reasonable forinterlaminar fracture, the crack growth path may not be predeter-mined in a general mixed mode fracture problem of composites,and hence a more realistic method is required to allow for arbitrarycrack propagation path. Recently, Rudraraju et al. [9–11] employeda micro–macro decomposition of the displacement field, combinedwith the embedded element discontinuity, to investigate fractureof composite laminates, focusing on determining force–displace-ment curves and analysis of the traction-separation law and thebridging zone. While the embedded discontinuity approach ofthe cohesive zone model [12,13] can be utilized for fracture analy-sis of composites [14,15], its numerical implementation results inan asymmetric stiffness matrix, nonconforming displacementkinematics [16] and may cause stability issues [17]. Therefore, analternative formulation based on the extended finite elementmethod (XFEM) is proposed in this study to account for the effectsof fiber bridging in general arbitrary crack path problems.

XFEM, originally proposed by Belytschko and Black [18], is apowerful numerical technique suited to problems concerning dis-continuities, in which neither singular elements are required toreproduce the crack tip singularity nor element edges should con-form to the crack face in a general crack propagation problem. Forfracture analysis of composites, XFEM has been successfully usedin both static and dynamic problems [19–27]. However, almostall of these studies are based on a linear elastic fracture mechanics,neglecting the nonlinearity effects of fiber bridging in the fractureprocess zone. Such a major drawback is surmounted in this work

Fig. 1. A linear elastic body containing a bridged crack.

Fig. 2. Definition of the distance function and the equivalent domain integral.

A. Afshar et al. / Composite Structures 125 (2015) 314–327 315

by simulating the fiber bridging, while preserving the frameworkof XFEM and its advantages.

In addition to the cohesive zone models, in which the materialis considered to be homogenous, bridged crack models are alsoused to describe the nonlinear fracture process zone [28–37].Bridged crack models are based on the assumption that the mate-rial is multiphase, and hence more than one phenomenon con-tributes to the fracture [38]. Separating the matrix phase failurefrom the fibers pull-out, the bridged crack models seem morepromising for analysis of fracture in composites [39], and thereforethese models are adopted in this study. Though very similar in theformulation, this distinction between the two models results in aremoval of the stress singularity in the cohesive zone approachand a non-vanishing singular field in the bridged crack models.In the latter model, the crack growth criterion is mainly affectedby the matrix phase, and the traction-separation law capturesthe closing forces of fibers on the crack faces. In this work, bothmechanisms are incorporated in the formulation of XFEM usingthe tip and Heaviside enrichments.

Usually, the effects of fiber bridging in increasing the crackgrowth resistance is illustrated by a fracture resistance curve (R-curve). In a crack growth process which includes fiber bridging,the R-curve starts at the fracture initiation toughness, and thenincreases to a plateau. It has been shown that this curve is not acomposite inherent characteristic and depends on the geometryof the specimen [40]. As a result, bridging laws should be used asthe material property representing the fracture resistance. By usingthe bridging laws and the J-integral, many authors studied the fiberbridging phenomenon [41–43], and the same approach is consid-ered here to construct the fracture resistance curve. More detailsare presented in Section 3.

The structure of the paper is as follows: Section 2 provides theconcepts of the LEFM based XFEM formulation of the problem andhow it can be employed to model crack propagation in brittlematerials. Section 3 presents the basics of the bridging law, thesoftening behavior, and evaluation of the fracture toughness forbridged cracks. Numerical evaluation of energy release rate fordetermining the crack growth length and direction is describedin Section 4. Then, in Section 5 the original formulation of XFEMis altered in order to account for the nonlinearity of the fiber bridg-ing phenomenon. Section 6 verifies this formulation by solvingmode I delamination problems and a number of mixed mode frac-ture simulations. Finally, Section 7 completes this paper by dis-cussing the advantages of this formulation over the conventionalXFEM and cohesive zone models.

2. XFEM for brittle fracture

2.1. Displacement approximation

Consider a crack in a fiber reinforced composite, as depicted inFig. 1. Neglecting the effects of fiber bridging, the displacementapproximation of the conventional finite element method isenriched by

uXFEM ¼ uFEM þ ucrack�face þ ucrack�tip ð1Þ

where ucrack�face and ucrack�tip are added in order to model the discon-tinuity in the displacement field and the singularity in the stressfield. The expression for the discontinuity enriched part of the dis-placement field is [44]

ucrack�face ¼Xi2M

NiðxÞHðxÞai ð2Þ

where M is the set of nodes enriched by the Heaviside function. TheHeaviside function for a point x is defined as [44]

HðxÞ ¼þ1 ifðx� �xÞ:en > 0�1 otherwise

�ð3Þ

where x� denotes the coordinate of the closest point of the crack tothe point x (Fig. 2). ai represents additional degrees of freedom usedfor modeling crack face discontinuity. In a similar manner, theexpression for the crack tip enriched part of the displacement fieldis

ucrack�tip ¼Xi2N

NiðxÞXk2F

f k xð Þbik

!ð4Þ

where N is the set of nodes enriched by the crack tip functions, andF is the set of f k crack tip functions. Due to the presence of fibers,the following orthotropic functions are adopted [22]

Fðr; hÞ ¼ffiffiffirp

cosh1

2

� � ffiffiffiffiffiffiffiffiffiffiffig1ðhÞ

p;ffiffiffirp

cosh2

2

� � ffiffiffiffiffiffiffiffiffiffiffig2ðhÞ

p;

�ffiffiffirp

sinh1

2

� � ffiffiffiffiffiffiffiffiffiffiffig1ðhÞ

p;ffiffiffirp

sinh2

2

� � ffiffiffiffiffiffiffiffiffiffiffig2ðhÞ

p �ð5Þ

316 A. Afshar et al. / Composite Structures 125 (2015) 314–327

with

gjðhÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðcosðhÞ þ fjsinðhÞÞ2 þ ðbjsinðhÞÞ2

qðj ¼ 1;2Þ ð6Þ

hkðhÞ ¼ tan�1 bksinðhÞcosðhÞ þ fksinðhÞ

� �ðk ¼ 1; 2Þ ð7Þ

where fi and bi are the real and imaginary parts of the roots of thecharacteristic equation [45]

atip11l

4 � 2atip16l

3 þ 2atip12 þ atip

66

� �l2 � 2atip

26lþ atip22 ¼ 0 ð8Þ

l1 ¼ f1 þ ib1

l2 ¼ f2 þ ib2ð9Þ

where aij are the elements of the compliance tensor a

e ¼ ar ð10Þ

More details of the XFEM implementation are presented inAppendix A.

2.2. Updating degrees of freedom

Due to the movement of the crack tip position, enriched nodesshould be updated accordingly. In each crack propagation step,enrichments associated with the previous tip nodes are changedto the Heaviside type enrichment. As depicted in Fig. 3, eliminationof previous degrees of freedom results in invalid discontinuity inthe crack geometry which generates an out of balance force inthe equilibrium equation. Nevertheless, by solving the equilibriumequation in the updated configuration, the inconsistency in thecrack discontinuity geometry relaxes, and therefore this residualforce vanishes. The detailed algorithm of this procedure in thenth loading increment (Fig. 3) is provided below:

� Solve fext � f int ¼ 0 in configuration (a) using the iterative New-ton–Raphson algorithm� Check the propagation criterion G ¼ Gcr

� Update the crack geometry and advance to configuration (b)

� Calculate ui;Ki and f inti

� Imbalance equation fext � f inti ¼ ri–0

Fig. 3. Updating the enriched nodes and correc

� Solve KiDu ¼ ri for Du

� Update uiþ1 ¼ ui þ Du and f intiþ1

� Check the equilibrium fext � f intiþ1 ¼ 0 and advance to configura-

tion (c)

3. Fiber bridging concept

3.1. Bridging laws

Every model of fiber bridging requires a traction-separationlaw, describing the forces fibers exert on the crack faces. They usu-ally consist of a linearly increasing line that reaches a maximumtraction, and a softening part which could be linear or nonlinear.

Since a bi-linear softening law has been proved to provideacceptable results [46], the same laws are employed in this studyto describe the formulation of spring elements. In addition, tri-lin-ear laws (Fig. 4 for mode I and Fig. 5 for mode II) are also utilized inthis work since these models are proved to improve the accuracy ofthe simulation by incorporating the initial fracture toughness[41,43,47].

3.2. Fracture resistance (the R-curve)

As mentioned earlier, for two failure phenomena are involved inbuilding the R-curve for bridged crack models. A part of the frac-ture resistance is provided by the matrix toughness Gcrack�tip, andanother part results from the traction forces of the fibersGfiber�bridging

GR ¼ Gcrack�tip þ Gfiber�bridging ð15Þ

with

Gfiber�bridging ¼Z d

0tðd0Þ d d0 ð16Þ

where d is the opening of crack face in the position of the fibers. Thefirst term in Eq. (15), Gcrack�tip; is the inherent fracture toughness ofthe matrix, and the second term is the energy dissipated in thefibers. While the matrix fracture toughness is constant, the dissipat-ed energy in the fibers depends on the elongation d of the fibersbridging the crack. The maximum energy absorbed in the fibers

tion of crack discontinuity after relaxation.

Fig. 5. Mode II bi-linear law.

Fig. 4. Mode I bi-linear and tri-linear laws.

A. Afshar et al. / Composite Structures 125 (2015) 314–327 317

equals the area under the traction-separation curve (GIC and GIIC inFigs. 4 and 5). By calculating the area under the traction-separationlaw for fibers that bridge the crack and adding this value to thematrix toughness, the fracture resistance curve can be estimated.

4. Crack propagation scheme

4.1. Basic algorithm

An energy based crack propagation algorithm is adopted in thiswork. In each step of the loading process, the crack propagatesincrementally with small segments, until its energy release ratedrops below the matrix fracture toughness. Afterward, the nextloading step is applied, and this process repeats until the final fail-ure of the specimen. In this manner, the corresponding load–dis-placement and fracture resistance curves of the specimen areobtained. This algorithm further requires a method for calculatingthe energy release rate to evaluate the crack growth length in eachstep of the loading, and a criterion for determination of thepropagation direction.

4.2. Crack growth length

The energy release rate of the crack during the growth isrequired for estimation of the growth length. The energy release

rate could be quantified using the path-independent contour inte-gral. In this work, the equivalent domain integral is utilized to cal-culate the J-integral

J ¼Z

Aðrijui;1 �wd1jÞq;jdA ð17Þ

In the above integral, dij is the Kronecker delta and w is thestrain energy density, defined as

w ¼ 12ðr11e11 þ r22e22 þ 2r12e12Þ ð18Þ

In Eq. (17), A is the interior region of an arbitrary contour C enclos-ing the crack tip (Fig. 2), and q is a smooth function, which equalsunity on the interior boundary of A and zero on the exterior bound-ary, as depicted in Fig 2.

4.3. Crack propagation direction

In the numerical simulations the crack is deemed to propagatein the direction h at which the circumferential stress in maximum.Therefore, the criterion originally proposed by Saouma et al. [48] ismodified by contributing the angle of fiber directions,

MaxRe AðBl1 � Cl2Þ

þ K2K1

Re½AðB� CÞ�cos2ðhþ aÞ þ E1

E2sin2ðhþ aÞ

!ð19Þ

with

A ¼ 1l1 � l2

ð20Þ

B ¼ ðl2 sinðh� hmat þ aÞ þ cosðh� hmat þ aÞÞ32 ð21Þ

C ¼ ðl1 sinðh� hmat þ aÞ þ cosðh� hmat þ aÞÞ32 ð22Þ

where hmat and a are the fibers direction and the crack angle, respec-tively. li are the roots of the characteristic Eq. (8). K1 and K2 aremode I and II stress intensity factors, extracted from the followinginteraction integral,

M ¼Z

Arijuaux

i;1 þ rauxij ui;1 �

12ðrikeaux

ik þ rauxik eikÞd1j

� �q;jdA ð23Þ

To numerically evaluate the domain form of the interactionintegral, the same finite element mesh is employed to calculate dif-ferent terms of Eq. (23) (see Fig. 6). The standard 4� 4 Gauss rule isused for ordinary elements, while the subdomain triangulationtechnique is employed for enriched elements. For the auxiliary dis-placement and stress fields, the asymptotic solution of a crack in anorthotropic medium is employed [44] (see Appendix B).

Fig. 6. Finite element implementation of the interaction integral.

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The real mode I and II stress intensity factors can then be deter-mined by solving the following simultaneous equations obtainedfrom two different assumptions for the auxiliary field,

Ml1 ¼ 2c11KI þ c12KII ðKaux

I ¼ 1 and KauxII ¼ 0Þ

Ml2 ¼ c12KI þ 2c22KII ðKaux

I ¼ 0 and KauxII ¼ 1Þ

(ð24Þ

with

c11 ¼ �a22

2Im

l1 þ l2

l1l2

� �ð25Þ

c12 ¼ �a22

2Im

1l1l2

� �þ a11

2Imðl1l2Þ ð26Þ

c22 ¼a11

2Imðl1 þ l2Þ ð27Þ

where are the elements of the compliance tensor a (Eq. (10)).Following the calculation of KI and KII , the crack growth

direction h can be obtained by maximizing Eq. (19), which dependsto KI and KII , a number of material dependent constants and therelative angles of long fibers and the crack.

Fig. 7. Spring elements located a

As illustrated in the numerical examples, criterion (19) ensuresthat the propagation direction is accurately affected by both thefibers direction and the specimen configuration.

5. Modeling of the fiber bridging zone

In order to simulate the effects of fiber bridging, spring ele-ments are placed between a number of points located at differentsides of the crack in a Heaviside element (Fig. 7). Displacements ofthese points, denoted by uþ and u�, can be written as

uþ ¼X

i

ðNi �ui þ Nið1� HiÞ�aiÞ

u� ¼X

i

ðNi �ui þ Nið�1� HiÞ�aiÞð28Þ

where �ui and �ai are the real and enriched displacements of theHeaviside element, respectively.

Projections of these values onto the spring direction are

uþs ¼X

i

ðNi �ui þ Nið1� HiÞ�aiÞ:~es

u�s ¼X

i

ðNi �ui þ Nið�1� HiÞ�aiÞ:~es

ð29Þ

The displacement of the spring element ubar is approximated bythe finite element shape functions

ubar ¼X2

j¼1

Nbarj

�ubarj ð30Þ

where Nbarj is the shape function of the spring element, �ubar

j is thenodal displacements of the spring, and the vector containing thesevalues are

�ubar ¼�ubar

1

�ubar2

( )¼

u�suþs

� �ð31Þ

The corresponding strain of the spring element is computedfrom

ebar ¼ @ubar

@n¼ Bbar �ubar ð32Þ

where n is the local axis of the spring, and Bbar is the strain–dis-placement matrix for the spring element

Bbar ¼@Nbar

j

@n¼ �1 1½ � ð33Þ

t different sides of the crack.

Fig. 9. Force displacement curve.

A. Afshar et al. / Composite Structures 125 (2015) 314–327 319

Replacing Eq. (31) and Eq. (33) into Eq. (32) results in the fol-lowing equations

ebar ¼ 2X

i

ðNi�aiÞ:~es ð34Þ

Bsi ¼ ½2Ni~es:~eX 2Ni~es:~eY 2Ni~es:~eZ � ð35Þ

Ks ¼Z

XððBsÞTasB

sÞdX ð36Þ

where Bs and Ks are the shape functions gradient and stiffnessmatrix of the spring element in the global coordinates.

Employing the Newton–Raphson iterative solver to determinethe stiffness of the springs ensures that the solutions convergerapidly in all simulations, even with a high number of springs oper-ating at the same time, and no numerical instability is observed.Furthermore, it should be noted that all the extra terms are lumpedinto the XFEM part of the stiffness matrix (Eqs. (35) and (36)).Therefore, once this formulation is incorporated into the stiffnessmatrix of the Heaviside elements, unlike the conventional cohesivezone models or models solely based on nonlinear springs, the fiberbridging effects are automatically simulated without a priorknowledge of the crack propagation path. For a mixed mode prob-lem, the same formulation can be applied to model the mode IIresistance of the fiber bridging. The efficacy of this formulation isexamined by solving several problems in the next section.

6. Numerical examples

6.1. Mode I double cantilever beam

As the first example and in order to validate the proposed for-mulation, the standard double cantilever beam, which has beenwidely analyzed by experimental or numerical approaches, isreconsidered. The configuration is illustrated in Fig. 8 and thematerial properties are [49]

E1 ¼ 130 GPa; E2 ¼ E3 ¼ 8:2 GPa

t12 ¼ t13 ¼ t23 ¼ 0:3

G12 ¼ 4:1 GPa

The tri-linear form of the force-separation law, originally pro-posed by Ref. [49], with the properties presented in Table 1 isassumed.

Fig. 8. Double can

Table 1Force-separation parameters.

P0 (MPa) P1 (MPa) d0 (mm) d1 (mm) d2 (mm) Gcrack-tip (N/mm)

45 1 7.1 � 10�4 7.1 � 10�3 1.28 0.2

The analysis is performed in the plane stress state and with thedisplacement control conditions using 2250 elements. The corre-sponding force-end beam displacement, presented in Fig. 9, showsa good agreement with the reference results. In the first stage ofthe loading, the response is linear, and when the fracture energyreaches the initial fracture toughness, the crack starts to propagate,causing a decrease in the stiffness of the specimen. Due to the fiberbridging effects, the loading capacity increases to the maximumvalue of 65N, and when the displacement of the fibers in the frac-ture process zone reaches its maximum, the loading capacity dropsas a result of the final failure of the specimen. Moreover, the effectsof fiber bridging mechanism in increasing the load carrying capa-city is illustrated in Fig. 9 by modeling the same problem withoutconsidering the effects of fiber bridging. The corresponding R-curvefor the current problem is depicted in Fig. 10, indicating theincreasing trend in the fracture energy until reaching a plateau.In the bridging zone behind the propagating crack tip, fibers exertforces on the crack and increase the fracture toughness. Thebridging length extends until the fracture process zone is fullydeveloped. In the current problem, this length is approximately19 mm The maximum value of 1000 N/mm equals the totalfracture resistance of the matrix and fibers bridging. Clearly, theresults are in good agreement with the reference values [49].

6.2. Eccentric three-point bending specimen

An attempt is now made to extend the mode I crack propaga-tion simulation to a problem with curved crack propagation. How-ever, since no numerical simulation of a fiber bridged crack witharbitrary propagation path in the is available literature, the present

tilever beam.

Fig. 10. Fracture resistance curve.

Fig. 11. Three-point bending specimen.

320 A. Afshar et al. / Composite Structures 125 (2015) 314–327

verification is applied to a specimen with material properties(cohesive strength and fracture toughness) that are irrelevant toconventional fiber reinforced composites. In fact, these materialproperties are more suited to cement based composites insteadof fiber reinforced polymers. With this fact in mind, an isotropicthree-point bending specimen with a 0:4 ðmmÞ eccentric crack isassumed (Fig. 11), and the materials and bi-linear spring propertiesareE ¼ 100 MPa; t ¼ 0

Cohesive Strength ðP0Þ ¼ 0:5 N=mm2

Fracture Toughness ðGcÞ ¼ 0:01 N=mm

This problem was originally analyzed by Mergheim et al. [50]with the finite element method and a discrete damage-type model,and Sun et al. [51] by a cohesive zone model within a mesh-free

Fig. 12. Load–displ

method. The proposed formulation, combined with the criterion(9), is utilized to simulate the crack propagation, and to obtainthe load–displacement curve, as depicted in Fig. 12. The same

acemnt curve.

A. Afshar et al. / Composite Structures 125 (2015) 314–327 321

trend of Fig. 9 in example 5.1 is also observed here. Numericaldeformed configurations, the crack mouth opening, and the crackpath in each stage of the applied displacement are depicted inFig. 13.

Considering the fact that the behavior of springs includes a soft-ening part, a mesh sensitivity analysis of the results seems neces-sary. Therefore, the same problem is analyzed by three differentmeshes, structured and unstructured. Fig. 14 shows that in allcurves, even with a relatively coarse structured mesh, the resultsremain very close and accurate, clearly indicating that the pro-posed formulation does not suffer from any mesh dependency ofthe results.

Fig. 13. Numerical deformed configurations in

Fig. 14. Mesh insensiti

6.3. Mixed-mode orthotropic composite

The proposed approach is now examined for the fracture analy-sis of composites with different fibers orientations (Fig. 15). Bothmode-I and mode-II contributions of the fiber bridging are consid-ered by assuming bi-linear forms of traction-separation law, withparameters presented in Tables 2 and 3, and the problem is solvedin the plane stress state with 4714 elements. The configuration andmaterial properties areE11 ¼ 114:8 GPa; E22 ¼ 11:7 GPa

G12 ¼ 9:66 GPa; t12 ¼ 0:21

different stages of applied displacements.

vity of the results.

Fig. 15. Eccentric three-point bending composite specimen. Fig. 17. Prediction of crack paths for different fiber orientations.

Table 2Mode-I force-separation parameters.

P0 (MPa) d0 (mm) d1 (mm) Gcrack-tip (N/mm)

60 0.000223 0.0223 0.67

Table 3Mode-II force-separation parameters.

P0 (MPa) d0 (mm) d1 (mm) Gcrack-tip (N/mm)

90 0.000371 0.0371 1.67

Fig. 18. Inaccurate prediction of the propagation direction by the original criterionof Saouma [48].

322 A. Afshar et al. / Composite Structures 125 (2015) 314–327

The corresponding force–displacement curves for various fiberangles are depicted in Fig. 16. It is observed that in configurationswith lower fiber angles, the mode-II fracture mechanism becomesmore significant, which provides more resistance to the crackgrowth, and increases the load-carrying capacity.

In order to find the crack propagation path, the modified max-imum hoop stress criterion (Eq. (19)) is employed. According toFig. 17, the crack path in this problem is mainly determined bythe fiber orientation, rather than the loading/geometric configura-tion. Had we used a criterion excluding the effects of fiber orienta-tions, the crack would have propagated towards the concentratedload. It should be noted, however, the fibers angle should be explic-itly included in the propagation criterion, otherwise the propaga-tion direction is incorrectly estimated. For cases where the fiber

Fig. 16. Load –disp

breakage is unlikely to occur (due to higher fracture resistance offibers), it is expected that the crack should propagate in a directiongenerally parallel to fibers orientation. Such a propagation path isalso affected by the loading and boundary condition. The originalcriterion of Saouma [48] for an orthotropic medium, nonetheless,does not incorporate the fibers orientation. To better examine theinaccuracy of the original criterion, this example is analyzed fortwo cases of 30� and 45� fibers with similar loading and boundaryconditions. According to Fig. 18, the predicted propagation path for

lacemnt curve.

Table 4Properties of the fiber reinforced plate.

E11 (GPa) E22 = E33 (GPa) t12 = t23 = t31 G12 (GPa)

33.5 10.23 0.27 4.26

Table 5Mode-I force-separation parameters.

P0 (MPa) d0 (mm) d1 (mm) Gcrack-tip (N/mm)

4 0.00425 0.0425 0.085

Table 6Mode-II force-separation parameters.

P0 (MPa) d0 (mm) d1 (mm) Gcrack-tip (N/mm)

A. Afshar et al. / Composite Structures 125 (2015) 314–327 323

the 30� lamina is located above the lamina with 45� fibers, clearlycontradicting the expectation that the crack tends to propagatenearly parallel to fibers orientation. To improve the predictions, aslightly modified criterion has been proposed, which explicitlyincorporates the fibers orientations. As a result, instead of havingthe absolute value of the crack angle alone, the relative angle ofthe crack and fibers orientation becomes decisive. The predictionsfor crack paths based on the modified criterion are now physicallyfeasible and better consistent with high fiber fracture resistance(Fig. 17).

6.4. Fiber reinforced plate with three holes

The final example consists of a crack in a fiber reinforced platewith three holes (Fig. 19). The mechanical properties of the plate

Fig. 19. Fiber reinforced plate with three holes.

6 0.00708 0.0708 0.2124

Fig. 20. Comparison of predicted crack propagation paths for different fiberorientations.

Fig. 21. Comparison of force–displacement curves for different fiber orientations.

324 A. Afshar et al. / Composite Structures 125 (2015) 314–327

and the traction-separation parameters of the springs are present-ed in Tables 4–6, respectively. The analysis is performed in theplane strain and displacement control state with 4877 elementsfor different fibers orientations. It should also be noted that thesame XFEM mesh has been used for all fibers orientations, irre-spective of the crack propagation path (Fig. 19).

Fig. 20 illustrates the predicted crack propagation paths for dif-ferent fiber orientations based on the modified criterion (19),which accounts for the collective effects of geometry, boundarycondition and fiber orientations. Clearly, the crack tends to growin a direction parallel with fibers, where much less fracture resis-tance is encountered.

Moreover, the force–displacement curves for all fiberorientations are depicted in Fig. 21. Again, in configurations inwhich the mode II fracture becomes more pronounced the load-carrying capacity of the specimen increases significantly. Also,

Fig. 22. Stress contours and the deformed configurations at d ¼ 0:1 mm for the case omagnification factor of 10.

Fig. 22 presents the stress field and numerical deformed configura-tion for the case of hmat ¼ 90� and the applied displacement ofd ¼ 0:1 mm.

Finally, to emphasize the importance of including fiberbridging effects in the formulation, the same problem is solvedand compared with and without the fiber bridging. Fig. 23 pre-sents the force–displacement curve for fiber orientation of 90�.Clearly, neglecting the effects of fiber bridging could erro-neously underestimate the load-carrying capacity of the speci-men, indicating the fact that any accurate simulation offracture in composites must include the bridging effects. Incontrast to the load carrying capacity, neglecting the fiberbridging has a minor effect on the crack propagation path, asdepicted in Fig. 24.

In order to demonstrate the mesh independency of the crackgrowth criterion, the crack path has been determined by

f 90� fiber orientation: (a) rxx (b) ryy (c) rxy (d) deformed configuration with the

Fig. 23. Increase in load-carrying capacity due to the fiber bridging.

Fig. 24. Deviation of the crack path if the fiber bridging is neglected.

Fig. 25. Three meshes used in the mesh sensitivity a

A. Afshar et al. / Composite Structures 125 (2015) 314–327 325

simulating the cases of fiber orientations of 75� and 90� withthree different meshes (Fig. 25). According to the results pre-sented in Fig. 26, by refining the mesh, the crack propagationpaths tend to converge to a final path, while the coarser meshremains slightly deviated from the converged predicted crackpath. This clearly shows that by enriching the displacementwith asymptotic functions and using the interaction integralmethod for deriving fracture parameters, the numerical predic-tions remain insensitive to the size of the mesh and quicklyconverges by refining the mesh; a clear indication of mesh inde-pendency of the results.

7. Conclusion

A new formulation is proposed for fiber bridging and modelingmixed mode crack propagation in fiber reinforced composites with-in the framework of extended finite element method. The formula-tion is proved to be capable of efficiently simulating the increase inload bearing capacity and fracture toughness of fiber reinforcedcomposites in mixed mode crack propagations with arbitrary crackpaths, and improving the accuracy of the linear elastic based XFEMfor modeling nonlinear fracture process zone, while eliminating therequirement of predetermined crack path in cohesive zone models.

nalysis. Number of elements: 2985, 4877, 6058.

Fig. 26. Mesh sensitivity of the predicted crack path: (a) fiber orientation of 75� , (b) fiber orientation of 90�.

326 A. Afshar et al. / Composite Structures 125 (2015) 314–327

Also, the modified maximum hoop stress criterion is introducedand utilized to predict the crack growth direction. Numerical exam-ples of mode I and mixed mode problems are solved, and promisingresults have been obtained, confirming that this simple formulationcan be readily used in accurate numerical simulations of fracture offiber reinforced composites.

Acknowledgments

The authors wish to gratefully acknowledge the technical sup-port of the High Performance Computing Lab, School of Civil Engi-neering, University of Tehran. Also, the financial support of IranNational Science Foundation (INSF) is gratefully acknowledged.

Appendix A.

The finite element formulation of the XFEM is presented here[19]

Krsij ¼

ZXeðBr

i ÞT DBs

i dX ðr; b ¼ u; a; bÞ ðA1Þ

fui ¼

Zt

NiftdCþ

ZXe

NifbdC ðA2Þ

fai ¼

Zt

NiHftdCþZ

XeNiHfbdC ðA3Þ

fbai ¼

Zt

NiFaftdCþZ

XeNiFafbdC ða ¼ 1;2;3; and 4Þ ðA4Þ

In which ft is the external force, fb is the body force, D is theelasticity matrix, Ni are standard shape functions, and Br

i are matri-ces of shape functions derivatives, which in the case of XFEM aredefined as

Bui ¼

Ni;x 00 Ni;y

Ni;y Ni;x

24

35 ðA5Þ

Bai ¼

ðNiðH � HiÞÞ;x 00 ðNiðH � HiÞÞ;y

ðNiðH � HiÞÞ;y ðNiðH � HiÞÞ;x

264

375 ðA6Þ

Bbi ¼ ½Bb1

i Bb2i Bb3

i Bb4i� ðA7Þ

Bbai ¼

ðNiðFa � FaiÞÞ;x 00 ðNiðFa � FaiÞÞ;y

ðNiðFa � FaiÞÞ;yFaðNiðFa � FaiÞÞ;x

264

375 ða ¼ 1;2;3; and 4Þ

ðA8Þ

where H and in above equation are the enrichment functions usedto reproduce the displacement jump across the crack face and stressfields near the crack tip, respectively.

Appendix B.

Orthotropic crack tip fields used in the domain integral eval-uation are presented in this section:

uaux1 ¼ KI

ffiffiffiffiffi2rp

rRe

1l1 � l2

l1p2g2ðhÞ � l2p1g1ðhÞ � �

þ KII

ffiffiffiffiffi2rp

rRe

1l1 � l2

½p2g2ðhÞ � p1g1ðhÞ�� �

ðA1Þ

uaux2 ¼ KI

ffiffiffiffiffi2rp

rRe

1l1 � l2

½l1q2g2ðhÞ � l2q1g1ðhÞ�� �

þ KII

ffiffiffiffiffi2rp

rRe

1l1 � l2

½q2g2ðhÞ � q1g1ðhÞ�� �

ðA2Þ

and the associated asymptotic crack-tip stress fields are defined as

raux11 ¼

KIffiffiffiffiffiffiffiffiffi2prp Re

ltip1 ltip

2

ltip1 � ltip

2

ltip2

gtip2 ðhÞ

� ltip1

gtip1 ðhÞ

" #( )

þ KIIffiffiffiffiffiffiffiffiffi2prp Re

1

ltip1 � ltip

2

ltip2

� �2

gtip2 hð Þ

�ltip

1

� �2

gtip1 ðhÞ

264

375

8><>:

9>=>; ðA3Þ

raux22 ¼

KIffiffiffiffiffiffiffiffiffi2prp Re

1

ltip1 � ltip

2

ltip1

gtip2 ðhÞ

� ltip2

gtip1 ðhÞ

" #( )

þ KIIffiffiffiffiffiffiffiffiffi2prp Re

1

ltip1 � ltip

2

1

gtip2 ðhÞ

� 1

gtip1 ðhÞ

" #( )ðA4Þ

A. Afshar et al. / Composite Structures 125 (2015) 314–327 327

raux12 ¼

KIffiffiffiffiffiffiffiffiffi2prp Re

ltip1 ltip

2

ltip1 � ltip

2

1

gtip1 ðhÞ

� 1

gtip2 ðhÞ

" #( )

þ KIIffiffiffiffiffiffiffiffiffi2prp Re

1

ltip1 � ltip

2

ltip1

gtip1 ðhÞ

� ltip2

gtip2 ðhÞ

" #( )ðA5Þ

with

giðhÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosðhÞ þ lisinðhÞ

qði ¼ 1;2Þ ðA6Þ

pk ¼ a11ðlkÞ2 þ a12 � a16lk ðk ¼ 1;2Þ ðA7Þ

qk ¼ a12lk þa22

lk� a26 ðk ¼ 1;2Þ ðA8Þ

where aij are the elements of the compliance tensor (Eq. (10)), lk

are the roots of the characteristic equation (Eq. (8)), and ðr; hÞ arethe polar coordinates measured from the crack tip (Fig. 1).

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