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Engineering Fracture Mechanics xxx (2014) xxx–xxx

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Engineering Fracture Mechanics

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

Mixed mode fracture analysis of adiabatic cracks inhomogeneous and non-homogeneous materials in theframework of partition of unity and the path-independentinteraction integral

http://dx.doi.org/10.1016/j.engfracmech.2014.07.0130013-7944/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +98 21 61112258; fax: +98 21 6640 3808.E-mail address: [email protected] (S. Mohammadi).

Please cite this article in press as: Goli E et al. Mixed mode fracture analysis of adiabatic cracks in homogeneous and non-homogmaterials in the framework of partition of unity and the path-independent interaction integral. Engng Fract Mech (2014), http://dx.10.1016/j.engfracmech.2014.07.013

E. Goli a, H. Bayesteh b, S. Mohammadi b,⇑a Department of Civil Engineering, Sharif University of Technology, Tehran, Iranb High Performance Computing Lab, School of Civil Engineering, University of College of Engineering, University of Tehran, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 October 2013Received in revised form 11 June 2014Accepted 17 July 2014Available online xxxx

Keywords:Interaction integralAdiabatic crackThermo-mechanical loadingExtended finite element method (XFEM)Orthotropic

In this paper, the path independent interaction integral has been implemented in theframework of the extended finite element method for mixed mode adiabatic cracks underthermo-mechanical loadings particularly in orthotropic non-homogenous materials. Themesh insensitivity and increased accuracy due to the thermal and displacement asymptoticanalytical solutions are discussed and the contour independency of the interaction integralis investigated in different examples. Finally, the problem of crack propagation in orthotro-pic FGM materials under the thermal loading is investigated to assess the accuracy androbustness of proposed approach.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

New engineering demands have led to the outburst of novel and advanced tailored materials, covering a wide range oflayered composites and inhomogeneous materials. Generally, homogenous materials do not properly perform under highthermal gradient or certain mechanical loadings. On the other hand, composite materials have shown severe disadvantagesmainly in the form of stress concentration and delamination at the interfaces. Alternatively, functionally graded materials(FGMs) have been designed with continuous variation of material properties to remove the deficiencies related to the exis-tence of interfaces and the inefficient response of homogeneous materials to general thermo-mechanical loadings. In recentyears, FGMs have been widely used in high-tech engineering applications such as thermal barrier coating for space applica-tions [1], piezoelectric and thermoelectric devices [2–6], thermionic converters [7], wear and impact resistant components[8] and biomedical and eco-materials [9,10]. FGMs are produced in both isotropic and orthotropic forms, using some of thefabrication techniques such as the plasma sprayed coating [11].

Bending of orthotropic FGMs beams [12], bimaterial FGMs [13], fracture mechanics of thermal barrier FGM coatings [14],crack propagations in FGMs [15,16] and finally FGMs under the impact loading [17,18] are among the main research topicson the subject. Applications of FGMs to withstand high mechanical and thermal loadings simultaneously are probably themost important issue of FGMs, as they may become extremely vulnerable to crack initiation and propagation. As a result,

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Nomenclature

A surface of the domain surface integralbT vector of additional degrees of freedom corresponding to the crack tip enrichmentsaij contracted notation of the compliance matrixB matrix of shape function derivativesBth matrix of thermal shape functions derivatives

bH additional degrees of freedom corresponding to the Heaviside enrichmentCijkl constitutive tensorE1; E2 Young’s modules with respect to the principal axes of orthotropyF; ðFTÞ set of crack tip enrichment functions for mechanical field (for thermal field)f total force vectorf mech: vector of mechanical forcef th thermal force vectorG12 shear modulusH heaviside functionIm imaginary partJ path-independent J integralJact J integral for the actual fieldJaux J integral for the auxiliary fieldK11 and K22 heat conductivity coefficients along the global directions 1 and 2, respectivelyK diagonal matrix of heat conductivity coefficientsKI;KII mode I and II stress intensity factorsKIn;KIIn normalized stress intensity factorsKx

IC ;KyIC mode I toughness with respect to local coordinates at the crack tip

Khh stress intensity factor in h directionKhhC critical stress intensity factor in h directionM interaction integralN shape functionnj unit outward normal vector to the Cs

Q thermal stiffness matrixq continuous weight function�q prescribed value of heat fluxRe real partr radial direction in polar coordinatesSijmn compliance tensorT temperature fieldT prescribed value of temperatureu displacement fieldw strain energy densitya1;a2;a3 coefficients of thermal expansion in principal orthotropy directionsCT boundary related to the thermal loadingCq boundary related to the prescribed heat fluxDT temperature difference from the reference temperaturedij Kronecker deltaeij components of strainem

ij mechanical part of strainet

ij total strain

ethij thermal part of strain

eact mij mechanical part of strain in actual field

eact thij thermal part of strain in actual field

eact tij total strain in actual field

h angular direction in polar coordinatesh0 angle of crack propagationltip

1 ;ltip2 roots of the characteristic equation at the crack tip location

t Poisson’s ratiorij stress componentsraux

ij auxiliary stress fieldrh hoop stress

2 E. Goli et al. / Engineering Fracture Mechanics xxx (2014) xxx–xxx

Please cite this article in press as: Goli E et al. Mixed mode fracture analysis of adiabatic cracks in homogeneous and non-homogeneousmaterials in the framework of partition of unity and the path-independent interaction integral. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.07.013



x initial crack angle with respect to the material x1 axisu matrix of nodal shape functionsf signed distance functionX domain volume

E. Goli et al. / Engineering Fracture Mechanics xxx (2014) xxx–xxx 3

fracture analysis of FGMs has become one of the most important issues for both the material design stage, and the analysisunder sever thermo-mechanical loadings. Consequently, several numerical studies, as the powerful mean to predict behaviorof FGMs for static [19–27] and dynamic [28–31] crack analyses, have been directed towards this subject in recent years.

A critical point on fracture analysis of FGM problems under thermal loading is the way boundary conditions of the cracksurfaces are defined either as adiabatic or isothermal. The heat equation can be solved without considering the defect forisothermal conditions whereas in adiabatic condition, which is the case in this study, the heat equation should be solvedby considering the temperature discontinuity. In practice, if the air is assumed to exist around the crack surfaces, the adia-batic assumption becomes realistic. Borgi et al. [32,33], and Ding and Li [34], studied FGMs with partial adiabatic cracks insemi-infinite media. Also, Noda and Jin performed a similar study by the adiabatic assumption [35]. Jin and Paulino exam-ined edge cracked FGMs under transient thermal loadings for cracks parallel to the heat flux direction, with practically noeffect on it [36].

There are several methods to evaluate the stress intensity factors in thermal conditions, including, the equivalent domainintegral approach for edge cracked FGMs [37], the interaction energy integral for isotropic non-homogenous materials [38],the Jk-integral in orthotropic FGMs [39], the interaction integral for isotropic and orthotropic FGMs [40,42], and finally,FGMs/homogenous biomaterials with system of cracks based on the singular integral equations [42].

In additional to the conventional finite element method (FEM), several other methods such as the phantom node method[43,44], meshless techniques [45–51] and the extended isogeometric method [52] have been developed in the past decade,but none of them is proved to be as robust and powerful as the extended finite element method (XFEM), as it inherits thesimplicity and robustness of FEM and accuracy and efficiency of meshless methods to capture and simulate general crackpropagation problems.

XFEM is a robust and accurate method for solving discontinuity problems. Reproducing the singular field near a crack tip,avoiding expensive remeshing procedures, extraordinary flexibility for crack propagation problems, simple formulation andindependent definition of crack from the FEM mesh are some of the main advantages of XFEM. Among a large number ofvaluable investigation and developments of this method, various 2D [53–59], 3D [60–63], plate and shell [64–68] implemen-tations can be referenced. For a review on recent developments of XFEM methodology refer to [69].

While XFEM was previously developed to analyze isotropic FGMs under mechanical loadings [70], Bayesteh and Moham-madi [71] extended this method to consider asymptotic enrichments in orthotropic FGMs based on the original homogenoussolutions [72–74]. Soon, the method was extended to consider thermomechanical loadings in orthotropic FGMs [73], andZamani et al. [75] evaluated the stress intensity factors for the homogenous materials under the thermal loading usinghigher order tip enrichments.

While Kim and KC [40] developed the interaction energy integral for the orthotropic FGMs under thermomechanical load-ings in the form of incompatible formulation, they never used it for analysis of mixed mode adiabatic cracks, as they solvedthe heat equation without considering the crack. Also, Hosseini et al. [76] used the same approach within an XFEMframework. To the best knowledge of the authors, the interaction energy integral method has never been used for adiabaticthermomechanical analysis of homogeneous or non-homogeneous orthotropic problems. As a result, the main purpose ofthis research is to develop and implement various forms of the interaction integral method within an XFEM frameworkfor evaluation of mixed mode stress intensity factors of adiabatic crack surfaces in isotropic and orthotropic homogeneousand non-homogenous materials. In addition, the challenging issue of plane strain conditions in thermomechanical problems,due to out-of-plane stress, has been addressed separately.

In the first section, a brief description of XFEM formulation for solving thermo-mechanical problems involving adiabaticcracks is presented. Then, the constitutive relations for orthotropic materials in both plane stress and plane strain states arepresented in Section 3. Section 4 discusses solving the thermal equation, followed by derivation of the interaction integralmethod for thermo-elastic problems in Section 5. Furthermore, this section addresses the extraction of SIFs from theM-integral. The final part comprehensively discusses several stationary and propagation crack problems in orthotropicnon-homogenous materials.

2. The extended finite element method

XFEM is a robust and efficient approach for simulation of discontinuity problems; eliminating the need for remeshing incrack propagation problems, reducing the necessary degrees of freedom and computational costs and providing higher accu-racy. In addition, XFEM allows for reproduction of a singular field by including corresponding analytical solutions into theXFEM approximation using the mathematical framework of the Partition of unity [76].





The displacement field in XFEM is composed of the standard and enrichment parts of the approximation,

Pleasemater10.101

u ¼ uFEM þ uXFEM ¼ uFEM þ utip þ uHeaviside ð1Þ

where uFEM is the displacement field in the standard finite element formulation, utip is the crack-tip enriched part of the dis-placement and uHeaviside allows for the displacement discontinuity inside a finite element by the Heaviside enrichmentfunction.

The Heaviside enrichment part of the displacement can be written as [53],

uHeaviside ¼Xi2NH

NiðxÞHðfÞbHi ð2Þ

where bHi is the vector of additional degrees of freedom and the Heaviside function is defined as,

HðfÞ ¼1 8f > 0�1 8f < 0

�ð3Þ

where f is the signed distance function.The singular stress and strain fields near a crack tip can be reproduced by the crack-tip enrichment approximation of the

displacement field [77],

utip ¼Xi2Ntip

NiðxÞXNTF

j¼1

FjðxÞbTij

!ð4Þ

where NTF is the number of crack tip enrichment functions and bTij are additional degrees of freedom corresponding to the

crack tip enrichments [76].F is the set of crack tip enrichment functions, defined as,

F ¼ffiffiffirp

sinh2;ffiffiffirp

cosh2;ffiffiffirp

sinh2

sin h;ffiffiffirp

cosh2

sin h

� �ð5Þ

for the mechanical tip enrichment functions in the isotropic condition. The following orthotropic enrichment functions arealso used [72],

F r; hð Þ ¼ffiffiffirp

cosh1

2

� � ffiffiffiffiffiffiffiffiffiffiffig1ðhÞ

p;ffiffiffirp

cosh2

2


p;ffiffiffirp

sinh1

2


p;ffiffiffirp

sinh2

2


p� �ð6Þ

with

gjðhÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosðhÞ þ nj sinðhÞ� �2 þ ðbj sinðhÞÞ2

qðj ¼ 1;2Þ ð7Þ

hkðhÞ ¼ tan�1 bk sinðhÞcosðhÞ þ nk sinðhÞ

� �ðk ¼ 1;2Þ ð8Þ

where n and b are defined in Eq. (60).For isotropic thermal conditions, FT is defined as [75],

FT ¼ffiffiffirp

sinh2

� �ð9Þ

which reproduces the singular gradient of the field variable (the heat flux). To the best knowledge of the authors, there is noorthotropic enrichment available for the thermal equation. Therefore, Eq. (9) is adopted for enrichment in the orthotropicthermal equation because it is capable of simulating the discontinuity of thermal field across the crack at the tip element,and reproduces the appropriate

ffiffiffirp

order of singularity for the heat flux at the crack tip.Due to high gradient and existence of singularity near a crack tip the standard Gauss quadrature rule cannot provide accu-

rate solutions for numerical evaluation of the FEM integrals. To enhance the accuracy of numerical integration, the techniquebased on partitioning of the elements into sub-triangles is adopted [77]. Detail of XFEM discretized thermal and mechanicalgoverning equations are presented in subsequent sections.

3. Orthotropic constitutive equations

Orthotropic material properties are defined along the two principal axes of orthotropy, as depicted in Fig. 1. In the pres-ence of thermal loading, the total strain can be divided into two parts, mechanical and thermal strains. The stress–strain rela-tion for an orthotropic material subjected to thermal loading and in plane stress condition can be defined as [37]:

cite this article in press as: Goli E et al. Mixed mode fracture analysis of adiabatic cracks in homogeneous and non-homogeneousials in the framework of partition of unity and the path-independent interaction integral. Engng Fract Mech (2014), http://dx.doi.org/6/j.engfracmech.2014.07.013



Fig. 1. A general problem of crack propagation in an orthotropic medium.


Pleasemater10.101

et11

et22

2et12

8><>:

9>=>; ¼

1=E1 �t12=E1 0�t12=E1 1=E2 0

0 0 1=G12

264

375

r11

r22

r12

8><>:

9>=>;þ

a1DT

a2DT0

8><>:

9>=>; ð10Þ

where et denotes the total strain, r represents the stress components, E1 and E2 are the Young’s modulus with respect to theprincipal axes of orthotropy, G12 is the shear modulus, t12 is the Poisson’s ratio, a1 and a2 are the coefficients of thermalexpansion in principal orthotropy directions and DT is the temperature difference from the reference temperature.

Unlike the plane stress condition, material properties in other directions are involved in the plane strain state,

et11

et22

2et12

8><>:

9>=>; ¼

ð1� t31t13Þ=E1 �ðt12 þ t13t32Þ=E1 0�ðt12 þ t13t32Þ=E1 ð1� t23t32Þ=E2 0

0 0 1=G12

264

375

r11

r22

r12

8><>:

9>=>;þ

ðt31a3 þ a1ÞDT

ðt32a3 þ a2ÞDT0

8><>:

9>=>; ð11Þ

where the following relations should also be satisfied:

E1

E2¼ t12

t21;

E1

E3¼ t13

t31;

E2

E3¼ t23

t32ð12Þ

4. Solution of the thermal equation

The thermal equation for an orthotropic non-homogenous medium under steady state conditions can be written as,

@

@X1k11 X1;X2ð Þ @T X1;X2ð Þ

@X1

þ @

@X2k22 X1;X2ð Þ @T X1;X2ð Þ

@X2

¼ 0 ð13Þ

where T X1;X2ð Þ is the temperature on each point of the domain, and k11 and k22 are the heat conductivity coefficients alongthe global directions 1 and 2, respectively. The temperature and heat flux boundary conditions can be defined as,

T ¼ T on CT ð14Þqn ¼ �q on Cq ð15Þ

where �q is the prescribed value of the heat flux, subscript n shows the normal component of the heat flux q ¼ �krT denotesthe heat flux. k for orthotropic media is a diagonal matrix represented by [76]

k ¼k11 00 k22

ð16Þ

In this study, the thermal equation is solved for orthotropic non-homogenous media which include insulated cracks. As aresult a strong discontinuity should be considered for the temperature field across the crack surface, while the heat fluxremains continuous. Fracture analysis of a domain subjected to thermal loading is more complicated than its correspondingmechanical loadings, due to the existence of singularity in the heat flux field, which increases the degrees of freedom and thecomputational costs. Applying the XFEM discretization procedure on the temperature field results in:





Pleasemater10.101

T ¼ TFEM þ TXFEM ¼Xi2X

NiðxÞTi þXi2Ntip

NiðxÞFTðxÞbTi þ

Xi2NH

NiðxÞHðfÞbHi ð17Þ

where Ni represent the finite element shape functions and Ti, bTi and bH

i are thermal nodal DOFs. The discretized form of theheat equation can then be expressed as

½Q �fTg þ ff thg ¼ 0 ð18Þ

with

½Q � ¼Z

XBT

thKBth dX ð19Þ

ff thg ¼Z

Cq

/T �qdC ð20Þ

where

½/� ¼ ½u1ju2j � � � jun�ui ¼ Ni for the standard nodesui ¼ Ni Ni½HðfÞ � HðfiÞ�½ � for the heaviside enriched nodesui ¼ Ni Ni½FðfÞ � FðfiÞ�½ � for the tip enriched nodes

ð21Þ

and

½Bth� ¼ ½u1;Xju2;Xj � � � jun;X� ð22Þ

ui;X ¼Ni;X1

Ni;X2

for the standard nodes

ui;X ¼Ni;X1 ðNi½HðfÞ � HðfiÞ�Þ;X1

Ni;X2 ðNi½HðfÞ � HðfiÞ�Þ;X2

" #for the Heaviside enriched nodes

ui;X ¼Ni;X1 ðNi½FTðfÞ � FTðfiÞ�Þ;X1

Ni;X2 ðNi½FTðfÞ � FTðfiÞ�Þ;X2

" #for the tip enriched nodes

ð23Þ

The total force vector f in the mechanical equation ½K�fUg ¼ ffg consists of the mechanical and the equivalent thermalforce

f ¼Z

XBT Ceth dXþ f mech: ð24Þ

where the thermal strain eth should now be explicitly defined for plane stress and strain conditions. Also, B is the matrix ofshape function derivatives in relation feg ¼ ½B�fUg.

4.1. Plane stress

The strain energy density for plane stress thermo-elasticity problems can be expressed as [37],

w ¼ 12rijem

ij ði; jÞ ¼ 1;2 where emij ¼ et

ij � ethij ð25Þ

or in the component form [37],

em11 ¼ et

11 � a1DT ¼ 1E1

r11 �m12

E2r22

em21 ¼ em

12 ¼ et12 ¼

12G12

r12

em22 ¼ et

22 � a2DT ¼ 1E2

r22 �m12

E1r11

ð26Þ

Derivatives of the thermal strains ethij;1 are required in all M-integral formulations,

eth11;1 ¼ ða1;1DT þ a1DT ;1Þ; eth

12;1 ¼ eth21;1 ¼ 0; eth

22;1 ¼ ða2;1DT þ a2DT ;1Þeth

11;2 ¼ a1;2DT þ a1DT ;2ð Þ; eth12;2 ¼ eth

21;2 ¼ 0; eth22;2 ¼ ða2;2DT þ a2DT ;2Þ

ð27Þ





4.2. Plane strain

For plane strain thermo-elasticity problems, the strain energy density function is expressed as [37],

Pleasemater10.101

w ¼ 12

r11em11 þ r12em

12 þ r21em21 þ r22em

22 þ r33em33

� �ð28Þ

where

r33 ¼E3

E1m13r11 þ

E3

E2m23r22 � E3a3DT; em

33 ¼ �a3DT ð29Þ

Mechanical strain components for plane strain orthotropic media can be defined as,

em11 ¼ et

11 � ðm31a3 þ a1ÞDT ¼ ð1� m31m13ÞE1

r11 �ðm12 þ m31m32Þ

E1r22

em22 ¼ et

22 � m32a3 þ a2ð ÞDT ¼ �ðm12 � m13m32ÞE1

r11 �ð1� m32m32Þ

E2r22

em21 ¼ em

12 ¼ et12 ¼

12G12

r12

ð30Þ

and derivatives of the thermal strain eact thij;1

�are computed from

eth11;1 ¼ m31;1a3DT þ m31a3;1DT þ m31a3DT ;1 þ a1;1DT þ a1DT ;1


eth12;1 ¼ eth

21;1 ¼ 0



eth12;2 ¼ eth

21;2 ¼ 0

ð31Þ

5. The interaction integral

Beginning with the well-known definition of the J-integral [78],

J ¼ limCs!0

ZCs

wd1j � rijui;1� �

nj dC ð32Þ

where w is the strain energy density, which consists of the stress components and the mechanical strains, Eq. (25).d is the Kronecker delta, u is the displacement field, Cs is a contour around the crack tip and n is the unit outward normal

vector to the Cs. The equivalent domain (EDI) form of the J-integral can be written as [78],

J ¼Z

Arijui;1 �

12rikem

ik

� �d1j

� �q;j dAþ

ZA

rijui;1 �12rikem

ik

� �d1j

� �;j

qdA ð33Þ

where q is an arbitrary but continuous function varying from q ¼ 0 on the outer contour integral to q ¼ 1 at the crack tip.Several types of q-function can be appropriately used for the employed numerical method. The conventional form of this

Fig. 2. Definition of the q-function around the crack tip for the case of (a) mechanical loadings and (b) thermal loadings.





function in a FEM solution is depicted in Fig. 2. In pure mechanical loading (homogeneous properties and without the bodyforce), the second integral in (33) vanishes and only the gradient of q appears in the equation, and the domain surface A canbe reduced into a ring, as depicted in Fig. 2(a). In contrast, for the case of thermal loadings, both terms exist in Eq. (33) andthe domain surface A should include the whole inner surface, as shown in Fig. 2(b).

Instead of direct application of the contour or EDI integrals, evaluation of the stress intensity factors are usually per-formed by the concept of the interaction integral, where two sets of actual and auxiliary displacement/stress/strain fieldsare superimposed. The actual field governs the physical problem and satisfies the equilibrium and compatibility equationsin each point of a general inhomogeneous domain. In contrast, no auxiliary field can be found to satisfy all governing equa-tions (e.g. equilibrium, compatibility and constitutive equations). The general form of the J-integral for the combined set ofactual (act) and auxiliary (aux) fields can be written as [78],

Pleasemater10.101

J ¼ Jact þ Jaux þM ð34Þ

where

Jact ¼Z

Aract

ij uacti;1

�� 1

2ðract

ik eact mik Þd1j

� �q;j dAþ

ZA

ractij uact

i;1

�� 1

2ract

ik eact mik

� �d1j

� �� ;j

qdA ð35Þ

Jaux ¼Z

Araux

ij uauxi;1

�� 1

2raux

ik eauxik

� �d1j

� �q;j dAþ

ZA

rauxij uaux

i;1

�� 1

2ðraux

ik eauxik Þd1j

� �;jqdA ð36Þ

and

M ¼Z

Aract

ij uauxi;1 þ raux

ij uacti;1

�� 1

2ðract

ik eauxik þ raux

ik eact mik Þd1j

� �q;j dA

þZ

Aract

ij uauxi;1 þ raux

ij uacti;1 �

12ðract

ik eauxik þ raux

ik eact mik Þd1j

� �;j

qdA ð37Þ

The term 12 ract

ik eauxik þ raux

ik eact mik

� �can be written as

12

ractik eaux

ik þ rauxik eact m

ik

� �¼ 1

2ract

ik eauxik þ Cikpqeaux

pq eact mik

�¼ ract

ik eauxik ð38Þ

Substitution of (38) into (37) results in:

M ¼Z

Aract

ij uauxi;1 þ raux

ij uacti;1

�� ract

ik eauxik d1j

n oq;jdAþ

ZA

ractij uaux

i;1 þ rauxij uact

i;1 � ractik eaux

ik d1j

n o;jqdA ð39Þ

Eq. (39) is the general form of the M-integral and should be calculated over the surface A. It should be noted that in the M-integral relation, the auxiliary field are usually defined in the crack tip local coordinate system, whereas the actual solution isusually computed in the global coordinate system, which complicates the numerical procedure.

The stress intensity factors can be extracted from the M-integral using the following equations [78,79]

M ¼ 2c11KIKauxI þ c12 KIK

auxII þ Kaux

I KII� �

þ 2c22KIIKauxII ð40Þ

where

c11 ¼ �atip

22

2Im

ltip1 þ ltip

2

ltip1 ltip

2

!ð41Þ

c12 ¼ �atip

22

2Im

1

ltip1 ltip

2

!þ atip

11

2Im ltip

1 ltip2

�ð42Þ

c22 ¼atip

11

2Im ltip

1 þ ltip2

�ð43Þ

where superscript tip shows the value is related to the crack tip position, aij is the components of compliance matrix, Imrefers to the imaginary part of a complex value, and l1 and l2 are the roots of the characteristic Eq. (60). KI and KII stressintensity factors can then be calculated from

MI ¼ 2c11KI þ c12KII

MII ¼ c12KI þ 2c22KII

�ð44Þ

where MI denotes the M-integral for the case of KauxI ¼ 1;Kaux

II ¼ 0 and MII represents the M-integral for the case ofKaux

I ¼ 0;KauxII ¼ 1.





5.1. Incompatible and non-equilibrium formulations

Due to the change of material properties within the M-integral domain in an FGM material, the asymptotic homogeneousorthotropic stress, strain and displacement fields cannot satisfy all equilibrium, consistency and constitutive equationssimultaneously. As a result, a number of alternative approaches have been developed to approximate evaluation of the inter-action integral [78].

5.1.1. Incompatible formulationIn this formulation, the auxiliary fields are selected in a way that satisfies the equilibrium and constitutive equations,

Pleasemater10.101

rauxij;j ¼ 0; raux

ij ¼ Cijrseauxrs ð45Þ

but the compatibility equation is violated [78],

eauxij –

12

uauxi;j þ uaux

j;i

�ð46Þ

Replacing ractik eaux

ik d1j with rauxik eact m

ik d1j in the general form of the M-integral (39) and applying the derivative with respect to jleads to,

M ¼Z

Aract

ij uauxi;1 þ raux

ij uacti;1

�� raux

ik eact mik d1j

n oq;j dA

þZ

Aract

ij;j uauxi;1 þ ract

ij uauxi;1j þ raux

ij;j uacti;1 þ raux

ij uacti;1j

�� raux

ij;1 eact mij þ raux

ij eact mij;1

�n oqdA ð47Þ

Equilibrium of the actual and auxiliary fields requires that ractij;j uaux

i;1 ¼ rauxij;j uact

i;1 ¼ 0. In order to account for the thermaleffects,

rauxij eact m

ij;1 ¼ rauxij eact t

ij;1 � rauxij eact th

ij;1 ð48Þ

Noting that the displacement gradients are conjugate with the total strains, and with the use of the compatibility equa-tion for the actual field,

rauxij uact

i;1j ¼ rauxij uact

i;j1 ¼ rauxij eact t

ij;1 ð49Þ

M ¼Z

Aract

ij uauxi;1 þ raux

ij uacti;1

�� raux

ik eact mik d1j

n oq;j dAþ

ZA

ractij uaux

i;1j � rauxij;1 e

act mij þ raux

ij eact thij;1

n oqdA ð50Þ

Based on the constitutive relation for the mechanical field,

rauxij;1 e

act mij ¼ raux

ij;1 sijmnractmn

� �ð51Þ

and introduction of the auxiliary strain field at the crack tip [38]

eauxij ¼ ðsijmnÞtipr

auxmn ð52Þ

and with the help of ractij uaux

i;1j ¼ ractij eaux

ij;1 ¼ ractij ðsijmnÞtipraux

mn;1

�, the final form of the incompatible M-integral is obtained (in a

form similar to [38] for isotropic problems)

M ¼Z

Aract

ij uauxi;1 þ raux

ij uacti;1

�� raux

ik eact mik d1j

n oq;j dAþ

ZA

ractij sijmn� �

tip � sijmn

h iraux

mn;1 þ rauxij eact th

ij;1

n oqdA ð53Þ

5.1.2. Auxiliary fields for incompatibility formulationWithout the loss of accuracy, the auxiliary field for the present thermo-mechanical M-integral can be considered similar

to the one used for orthotropic domains under mechanical loadings [78],

uaux1 ¼ Kaux

I

ffiffiffiffiffi2rp

rRe

1

ltip1 � ltip

2

ltip1 p2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos hþ ltip

2 sin hq

� ltip2 p1


1 sin hq ( )

þ KauxII

�ffiffiffiffiffi2rp

rRe

1

ltip1 � ltip

2

p2


2 sin hq

� p1


1 sin hq ( )

ð54Þ

uaux2 ¼ Kaux

I

ffiffiffiffiffi2rp

rRe

1

ltip1 � ltip

2

ltip1 q2


2 sin hq

� ltip2 q1


1 sin hq ( )

þ KauxII

�ffiffiffiffiffi2rp

rRe

1

ltip1 � ltip

2

q2


2 sin hq

� q1


1 sin hq ( )

ð55Þ





Pleasemater10.101

raux11 ¼

KauxIffiffiffiffiffiffiffiffiffi2prp Re

ltip1 ltip

2

ltip1 � ltip

2

ltip2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos hþ ltip2 sin h

q � ltip1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos hþ ltip1 sin h

q264

375

8><>:

9>=>;

þ KauxIIffiffiffiffiffiffiffiffiffi2prp Re

1

ltip1 � ltip

2


cos hþ ltip2 sin h


cos hþ ltip1 sin h

q264

375

8><>:

9>=>; ð56Þ

raux22 ¼


1

ltip1 � ltip

2


cos hþ ltip2 sin h


cos hþ ltip1 sin h

q264

375

8><>:

9>=>;


1

ltip1 � ltip

2

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos hþ ltip

2 sin hq � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos hþ ltip1 sin h

q264

375

8><>:

9>=>; ð57Þ

raux12 ¼


ltip1 ltip

2

ltip1 � ltip

2

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos hþ ltip

1 sin hq � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos hþ ltip2 sin h

q264

375

8><>:

9>=>;


1

ltip1 � ltip

2


cos hþ ltip1 sin h


cos hþ ltip2 sin h

q264

375

8><>:

9>=>; ð58Þ

eauxij ¼ Sijklraux

kl ð59Þ

where lk are the roots of the characteristic equation [78],

atip11l

4 � 2atip16l

3 þ 2atip12 þ atip

66

�l2 � 2atip

26lþ atip22 ¼ 0 ð60Þ

The roots lk ¼ nk þ ibk are always complex or purely imaginary in conjugate pairs as l1;l1;l2;l2 and the coefficients pi andqi are defined as [78],

p1 ¼ atip11 ltip

1

�2þ atip

12 � atip16l

tip1 ð61Þ

p2 ¼ atip11 ltip

1

�2þ atip

12 � atip16l

tip2 ð62Þ

q1 ¼ atip12l

tip1 þ

atip12

ltip1

� atip26 ð63Þ

q2 ¼ atip12l

tip2 þ

atip12

ltip2

� atip26 ð64Þ

5.1.3. Non-equilibrium formulationIn an alternative formulation, the auxiliary fields are selected in a way that allows for the equilibrium equation to be vio-

lated but compatibility and constitutive equations are kept authentic [78],

rauxij;j – 0; raux

ij ¼ Cijrseauxrs ; eaux

ij ¼12

uauxi;j þ uaux

j;i

�ð65Þ

Despite the fact that the following formulation can be conceptually found elsewhere [40], they are re-derived in a moreappropriate form. Beginning with applying the derivative with respect to j in (39),

M ¼Z

Aract

ij uauxi;1 þ raux

ij uacti;1

�� ract

ik eauxik d1j

n oq;j dA

þZ

Aract

ij;j uauxi;1 þ ract

ij uauxi;1j þ raux


ij uacti;1j � ract

ik eauxik d1j

� �;j

n oqdA ð66Þ

Noting to the equilibrium of the actual stress field and the constitutive relation,

M ¼Z

Aract

ij uauxi;1 þ raux

ij uacti;1

�� ract

ik eauxik d1j

n oq;j dA

þZ

Aract

ij uauxi;1j þ raux


ij uacti;1j � Cijkl;1eact m

kl eauxij � raux

ij eact mij;1 � ract

ij eauxij;1

n oqdA ð67Þ





Then, using the compatibility equation for auxiliary and actual fields, ractij uaux

i;1j ¼ ractij eaux

ij;1 leads to

Pleasemater10.101

M ¼Z

Aract

ij uauxi;1 þ raux

ij uacti;1

�� ract

ik eauxik d1j

n oq;j dAþ

ZA

rauxij;j uact

i;1 � Cijkl;1eact mkl eaux

ij þ rauxij uact

i;1j � eact mij;1

�n oqdA ð68Þ

The bolded term in Eq. (68) can be written as

rauxij uact

i;1j � eact mij;1

�¼ raux

ij eact tij;1 � eact m

ij;1

�¼ raux

ij eact thij;1 ð69Þ

which leads to [40]

M ¼Z

Aract

ij uauxi;1 þ raux

ij uacti;1

�� ract

ik eauxik d1j

n oq;j dAþ

ZA

rauxij;j uact

i;1 � Cijkl;1eact mkl eaux

ij þ rauxij eact th

ij;1

n oqdA ð70Þ

The auxiliary displacement fields of (54) and (55) are similarly adopted. However, the following auxiliary strain and stressfields have to be used [78]

eaux ¼ ðsymrÞuaux ð71Þraux

ij ¼ Cijkleauxkl ð72Þ

6. Numerical examples

In this section, four different examples are examined in order to assess the accuracy of proposed method for evaluation offracture parameters in thermo-elastic adiabatic crack problems. The examples cover a wide range of problems, includinghomogenous and non-homogenous media in isotropic and orthotropic materials. In addition, pure mode I and II and mixmode problems are solved for both plane-stress and plane strain states. The first three examples are dedicated to evaluationof stress intensity factors for stationary cracks, while the last example is presented to study the crack propagation in anorthotropic FGM medium.

6.1. Isotropic homogenous square plate with a center crack

The first example is an isotropic homogenous square plate with an adiabatic straight central crack, which is subjected tothe thermal loading. Configuration and dimensions of the plane strain plate are shown in Fig. 3. The crack surfaces and theleft and right edges of the plate are adiabatic, which prevent the heat flux crossing. The existing mechanical and thermalboundary conditions, shown in Fig. 3, induce a pure sliding mode (Mode II) for the applied thermal loading. The materialproperties are provided in Table 1.

A structured mesh is used to discretize the model, as typically depicted in Fig. 4. The mesh consists of 2601 elements forthe case of a=L ¼ 0:3.

This problem is solved for different crack lengths and is compared with Refs. [80,45]. Duflot [80] solved this problemusing XFEM with an unstructured mesh of 3632 elements, while Bouhala et al. [45] solved this problem using the meshlessXEFGM approach with 3362 nodes.

Fig. 3. Geometry and boundary conditions of the square plate with an adiabatic centered crack.




Table 1Material properties for the cracked plate.

Young’s modulus (GPa) Poisson’s ratio Coefficient of thermal expansion ð�C�1Þ Thermal conductivity ðW=m �CÞ

174 0.3 1.10E�05 3

Fig. 4. The typical finite element mesh for the square cracked plate.

Fig. 5. The temperature field on the square cracked plate (a/L = 0.3).


At the first step, the 2D heat equation is solved by XFEM and the temperature field is evaluated, as depicted in Fig. 5. Theeffect of adiabatic crack surfaces is clearly observed in Fig. 5 in the form of temperature discontinuity across the crack.

The exaggerated deformed shape of the plate illustrates a pure bending deflection, as depicted in Fig. 6. The continuousvertical deformation of the plate even around the crack shows that the crack only behaves in the pure shearing mode II.

Finally, the stress intensity factors are evaluated for each crack length and are normalized as

Pleasemater10.101

KIIn ¼KII

EaTffiffiffiLp ð73Þ




Fig. 6. Deformed shape of the square cracked plate under the thermal loading (exaggeration factor = 100).

Table 2Normalized stress intensity factor for the square cracked problem.

a=L KIIn

Present study [80] [45]

0.2 0.054 0.054 0.0530.3 0.096 0.095 0.0940.4 0.141 0.141 0.1410.5 0.191 0.190 0.188

2.0E+05

4.0E+05

6.0E+05

8.0E+05

1.0E+06

1.2E+06

1.4E+06

1.6E+06

1.8E+06

2.0E+06

500 5500 10500 15500 20500

Con

ditio

n N

umbe

r

DOFs

Fig. 7. Variation of the condition number versus the number of degrees of freedom.


where E; a; T and L are the Young’s modulus, the coefficient of thermal expansion and the dimension of the problem.Comparison of the normalized SIFs with the reference results [45,80] are depicted in Table 2, which indicates a goodagreement.

The condition number can be considered as an index to study the numerical stability of analysis. As depicted in Fig. 7,relatively low condition numbers are obtained even for very large number of degrees of freedom, an indication of the sta-bility of the numerical solution.

Furthermore, to verify the domain-independency of the J-Integral computation, the stress intensity factors for differentradius of J-integral domain are evaluated and compared. Five different contours, as depicted in Fig. 8, with ratio of r=a (radiusof contour integral/half length of crack) varying from 0.15 to 0.65 with a step of 0.1, are considered. Fig. 8 shows the variationof stress intensity factors versus different values of r=a which reveals that the procedure remains contour independent.

In order to analyze the mesh sensitivity of the results, the problem is solved by several different meshes for the case ofa=L ¼ :4. Fig. 9 compares the normalized SIFs for different meshes and the corresponding level of error. Clearly, for modelswith DOFs larger than 5000, the results converge and remain unchanged.




Fig. 8. Variation of the normalized SIF for different values of r/a.

(a)

(b)

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

500 2000 3500 5000 6500 8000

Nor

mal

ized

SIF

s,

Total DOFs

- 0.01

0.015

0.04

0.065

0.09

0.115

0.14

0.165

0.19

500 2000 3500 5000 6500 8000

Err

or

Total DOFs

Initial mesh

Fig. 9. Effect of mesh on results for the case of a/L = 0.4. (a) Normalized KII and (b) error.


6.2. An edge crack in a functionally graded plate

A cracked functionally graded plate subjected to a thermal loading is considered. The plate is a non-homogenous isotropicmaterial and its properties vary in the X1 direction. Configuration and thermal and mechanical boundary conditions areshown in Fig. 10. The left edge is considered 10 degrees colder than the reference temperature while the right edge is fixedto the reference 1D temperature heat flux, parallel to X1. Top and bottom edges are constrained for displacement. Theseedges and the crack surfaces are assumed adiabatic. The crack length is assumed to vary from 0.2 to 1.6. The material prop-erties are defined by the following hyperbolic-tangent functions,

Pleasemater10.101




Fig. 10. Edge cracked FGM plate.

0.5

1

1.5

2

2.5

3

3.5

-1 -0.5 0 0.5 1

You

ng’s

mod

ulus

and

con

duct

ivity

coe

ffic

ient

(G

Pa, W

/mo C

)

X1

Young's Modulus

Conductivity

Fig. 11. Variations of Young’s modulus and conductivity coefficient of FGM plate.


Pleasemater10.101

EðX1Þ ¼E� þ Eþ

2þ E� � Eþ

2tanhðbX1 þ dÞ

mðX1Þ ¼m� þ mþ

2þ m� � mþ

2tanhðdX1 þ dÞ

aðX1Þ ¼a� þ aþ

2þ a� � aþ

2tanhðdX1 þ dÞ

kðX1Þ ¼k� þ kþ

2þ k� � kþ

2tanhðdX1 þ dÞ

ð74Þ

with

d ¼ 0; L ¼ 4 m; W ¼ 2 m; b ¼ 15; d ¼ 5

E�; Eþ� �

¼ 1;3ð Þ GPa; m�; mþð Þ ¼ 0:3; 0:1ð Þa�;aþð Þ ¼ 0:01;0:03ð Þ ð�CÞ�1

; k�; kþ� �

¼ 1;3ð ÞW=m �C




Fig. 12. Unstructured (a) and structured (b) finite element meshes.

Table 3Normalized stress intensity factors for various crack lengths.

aW Incompatible Non-equilibrium Ref. [41]

KIn Difference (%) KIn Difference (%)

0.2 7.446 0.10 7.439 0.00 7.4390.4 8.054 1.33 7.891 0.71 7.9480.5 5.621 1.73 5.713 0.12 5.7200.6 2.100 1.00 2.065 2.67 2.1210.8 0.962 1.49 0.948 0.00 0.948


where E; m; a; k are the Young’s modulus, the Poisson’s ratio, the coefficient of thermal expansion and the thermal conduc-tivity coefficient. Superscripts ð�;þÞ are related to the left and right edges of the plate.

As illustrated in Fig. 11, properties of this FGM vary very sharply on the vicinity of X1 ¼ 0, so it behaves much similar to abimaterial with an interface at X1 ¼ 0. Therefore, this plate does not hold some conventional practical benefits of FGMs.

Here, this problem is selected in order to verify the formulation of the interaction integral method. The crack is parallel tothe heat flux, so the heat conduction is performed in a one-dimensional pattern.

For all lengths of crack, the domain discretization is performed by a unique unstructured triangular mesh which consistsof 619 nodes, as shown in Fig. 12(a).

The normalized stress intensity factors KIn ¼ KI=E�a�TffiffiffiffiffiffiWp

, evaluated by both incompatible and non-equilibrium formu-lations for different a=w ¼ 0:2� 0:8, are presented in Table 3 and compared with the reference results [41], where KC andKim solved the same problem by the conventional FEM on an unstructured mesh, generated by 1001 nodes. The presentXFEM has used almost half number of nodes. The results of both formulations clearly match the reference results.

Table 4Normalized stress intensity factors for two different meshes in the steepest gradient.

aW Mesh Number of nodes KIn Difference (%)

Present XFEM FEM [41]

0.5 Structured 720 5.735 5.720 0.250.5 Unstructured 619 5.621 1.73




0.5

1

1.5

2

2.5

3

-1 -0.5 0 0.5 1 Y

oung

’s m

odul

us a

nd c

ondu

ctiv

ity

coef

ficie

nt (G

Pa, W

/mo C

) X1 (m)

Young's Modulus

Conductivity

Fig. 13. Variation of Young’s modulus and conductivity coefficient of FGM plate for b ¼ 1 and d ¼ 0:5.

0

1

2

3

4

5

6

7

8

9

0 0.2 0.4 0.6 0.8 1

Nor

mal

ized

SIF

s

a/w

Sharp Variation β=15,δ=5

Smooth Variation β=1,δ=0.5

Fig. 14. Comparison of normalized SIFs for sharp and smooth variations of material properties.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.15 0.25 0.35 0.45 0.55 0.65 0.75

Nor

mal

ized

SIF

s

r/a

a/w=0.8

a/w=0.6

Fig. 15. Normalized stress intensity factors versus ratio of radius of the J-domain to the length of crack for two cases of a=w.


Furthermore, when the crack tip is positioned at the steepest material gradient, a mesh with finer elements around thecrack tip is considered to discretize the domain with 1330 triangular elements and 720 nodes, as shown in Fig. 12(b). Theresults for the two different meshes of Fig. 12 are compared in Table 4.

Fig. 13 shows the smooth variation of Young’s modulus and conductivity coefficient for b ¼ 1; d ¼ 0:5. In order to studythe effect of variation of material properties, a comparison is made for SIFs based on different values of b and d, as presentedin Fig. 14. For smaller values of b and d, the rate of variations in the vicinity of X1 ¼ 0 is smoother than their large values.

Furthermore, Fig. 15 illustrates the contour independency of the J-integral for two cases of crack lengths aw ¼ 0:6;0:8.

6.3. A center crack in a Nickel–Alumina FGM layer

An FGM layer with a center crack is considered. The plane stress orthotropic metal–ceramic FGM consists of nickel as anisotropic metal and Alumina as an orthotropic ceramic. The lower edge of specimen is purely Nickel and the upper edge is




Fig. 16. Center cracked orthotropic FGM layer.


purely Alumina and the material properties vary from Nickel to alumina gradually. Dimensions and thermal/mechanicalboundary conditions are shown in Fig. 16.

The right and left edges of the medium and the crack surfaces are considered insulated. The temperature is fixed to thereference temperature T0 on the bottom edge and to 2T0 on the top edge; then the heat flux induces a 2D temperature dis-tribution. The coordinate origin is inserted in the middle of the lower edge and the crack is placed in the distance h1 from theorigin. In this problem, the interaction integral approach is examined for evaluating the mixed mode stress intensity factorsof a non-homogenous orthotropic FGM problem subjected to the steady state thermal loading. At first, the whole domain isfixed at the reference temperature and the stress in all nodes is equal to zero. Afterwards, the upper edge is fixed at 2T0 andthe heat distributes in the domain in both dimensions and the layer is forced to deform similar to a pure bending. Thethermo-mechanical properties of each node of the domain are represented with the following functions of X2,

Pleasemater10.101

E1ðX1Þ ¼ Em1 þ Ecr

1 � Em1

� � X2

h

� �c

; E2ðX1Þ ¼ Em2 þ Ecr

2 � Em2

� � X2

h

� �c

G12ðX1Þ ¼ Gm12 þ Gcr

12 � Gm12

� � X2

h

� �c

m12ðX1Þ ¼ vm12 þ mcr

12 � mm12

� � X2

h

� �b

; m13ðX1Þ ¼ vm13 þ mcr

13 � mm13

� � X2

h

� �b

m31ðX1Þ ¼ vm31 þ mcr

31 � mm31

� � X2

h

� �b

; m32ðX1Þ ¼ vm32 þ mcr

32 � mm32

� � X2

h

� �b

a1ðX1Þ ¼ am1 þ acr

1 � am1

� � X2

h

� �d1

; a2ðX1Þ ¼ am2 þ acr

2 � am2

� � X2

h

� �d2

a3ðX1Þ ¼ am3 þ acr

3 � am3

� � X2

h

� �d1

k1ðX1Þ ¼ km1 þ kcr

1 � km1

� � X2

h

� �x

; k2ðX1Þ ¼ km2 þ kcr

2 � km2

� � X2

h

� �x

ð75Þ

where superscripts cr and m are related to ceramic at X2 ¼ h and metal X2 ¼ 0, respectively, and

Ecr1 ¼ 90:43 GPa; Ecr

2 ¼ 116:36 GPa; Ecr3 ¼

mcr31

mcr13

Ecr1 ¼ 90:43 GPa

Gcr12 ¼ 38:21 GPa; vcr

12 ¼ 0:22; vcr13 ¼ 0:14; vcr

31 ¼ 0:14; vcr32 ¼ 0:21

kcr1 ¼ 21:25 W=m �C; kcr

2 ¼ 29:82 W=m �C acr1 ¼ 8 10�6

�ð�CÞ�1

acr2 ¼ 7:5 10�6

�ð�CÞ�1

; acr3 ¼ 9 10�6

�ð�CÞ�1

Em1 ¼ Em

2 ¼ Em ¼ 204 GPa; mm12 ¼ mm

13 ¼ mm31 ¼ mm

32 ¼ 0:31

Gm12 ¼ 77:9 GPa; km

1 ¼ km2 ¼ km ¼ 70 W=m �C

am1 ¼ am

2 ¼ am3 ¼ am ¼ 13:3ð10�6Þ ð�CÞ�1

Plasma spray forming can be used to produce Nickel–Alumina FGM [81,82] as an orthotropic elastic, non-homogenousmaterial [39]. The material properties depend on power values of c; b; d1; d2;x. If values 2, 3 and 3 are assumed for c, d1

and d2 respectively, variations of mechanical and thermal properties along the X2 direction can be shown in Figs. 17 and18, respectively.




10

60

110

160

210

260

0 0.2 0.4 0.6 0.8 1 Y

oung

’s a

nd S

hear

mod

ules

(GPa

)

x2/h

E1 E2

G12

Fig. 17. Variations of the Young’s and shear modules in the orthotropic Alumina–Nickel FG layer.

6.5E-06

7.5E-06

8.5E-06

9.5E-06

1.05E-05

1.15E-05

1.25E-05

1.35E-05

1.45E-05

0 0.2 0.4 0.6 0.8 1

ther

mal

exp

ansi

on c

oeff

icie

nts (⁰C

)-1

x2/h

alpha1

alpha2

alpha3

Fig. 18. Variations of the thermal expansion coefficients in the orthotropic Alumina–Nickel FG layer.

Fig. 19. Domain discretization using a structured quadrilateral mesh.

Fig. 20. Temperature field is obtained by solving the 2D heat equation.


A structured 4-node quadrilateral finite element mesh is used to discretize the model. The mesh consists of 8460elements and 8662 nodes, as depicted in Fig. 19 for the case of a=w ¼ 0:1; h=w ¼ 0:4; h1=w ¼ 0:25. This example waspreviously solved by Dag et al. [39] based on the Jk-Integral method with a triangular mesh containing 93,952 elementsand 189,223 nodes; almost 21 times more number of nodes than the present study.

The temperature field is obtained by solving the 2D heat equation, as depicted in Fig. 20. The adiabatic condition on cracksurfaces and other thermal boundary conditions are presented in Fig. 16 in the case of T0 ¼ 1000. Variation of the temper-ature gradient in both global directions dT=dx1; dT=dx2 are shown in Fig. 21, indicating a high gradient variation near thecrack tips.

The predicted stress intensity factors, evaluated by the incompatible form of the interaction integral method and normal-ized with equations (76) and (77), are compared with the reference results by Dag et al. [39], which used the JK -Integralmethod. Table 5 shows the results for the following setting:




Fig. 21. Variation of (a) dT/dx and (b) dT/dy.

Table 5Normalized stress intensity factors for the mixed mode fracture of Alumina–Nickel FG layer.

d2 KIn KIIn

Present study [39] Diff. (%) Present study [39] Diff. (%)8662 nodes 189223 nodes 8662 nodes 189223 nodes

1/3 0.0175 0.0176 0.57 0.1227 0.1197 2.503 0.0204 0.0207 1.45 0.1300 0.1271 2.28


Pleasemater10.101

aw¼ 0:1;

hw¼ 0:4;

h1

w¼ 0:25

c ¼ 2; b ¼ 1:5; d1 ¼ 3; d2 ¼13;3

; x ¼ 4;

with the normalized values

KIn ¼KI

acr1 Ecr

1 T0ffiffiffiffiffiffipap ð76Þ

KIIn ¼KII

acr1 Ecr

1 T0ffiffiffiffiffiffipap ð77Þ

Again, the condition number for the global stiffness matrix, shown in Fig. 22, indicates the stability of the adopted methodeven for very high number of degrees of freedom.

Fig. 23 shows the results of normalized stress intensity factors for different values of a=w. As the crack length increases,the effect of mode II becomes more dominant.

As a parametric study and in order to evaluate the effect of thermal expansion coefficient in both global directions on SIFs,Figs. 24 and 25 illustrate the results while the input data is kept unchanged except for variable d1 and d2. Clearly, by increas-ing d1 and d2, the stress intensity factors and the level of the crack instability are increased.




0.00E+00

1.00E+07

2.00E+07

3.00E+07

4.00E+07

5.00E+07

6.00E+07

7.00E+07

4000 9000 14000 19000 24000

Con

ditio

n N

umbe

r

DOFs

Fig. 22. Condition number of the global stiffness matrix versus DOFs for the Alumina–Nickel FG layer.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3

Nor

mal

ized

SIF

s

a/W

Normalized KI

Normalized KII

Fig. 23. Normalized stress intensity factors for different values of a/w.

0.017

0.0175

0.018

0.0185

0.019

0.0195

0.02

0.0205

0.021

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

Nor

mal

ized

KI

δ1 or δ2

Variation of δ2

Variation of δ1

Fig. 24. Normalized KI for a/w = 0.1 versus variation of d1 and d2.


To examine the effect of radius of J-integral domain, r, variations of the energy release rate, G, versus the ratio of ra is

depicted in Fig. 26 for three cases of a=w ¼ 0:1; a=w ¼ 0:15 and a=w ¼ 0:2, clearly showing the independency of the energyrelease rate with respect to r=a.

Fig. 27 compares the results of normalized SIFs computed by the incompatible and non-equilibrium formulations of theM-integral, as discussed in Section 5. Close results are obtained by both approaches for both stress intensity factors.

6.4. Square plate with a circular hole and an inclined crack

Consider a circular hole with radius of 2.5 mm, placed in the center of a square plate with side length of 20 mm, as shownin Fig. 28. An inclined crack ða ¼ 2:5 mmÞ is induced on the edge of the hole with 30 degrees angle with respect to the hor-izental axis which passes through the center of the circular hole. The temperature in all edges of the square plate is fixed to




0.1

0.105

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

Nor

mal

ized

KII

δ1 or δ2

Variation of δ2

Variation of δ1

Fig. 25. Normalized KII versus variation of d1 and d2.

0

50000

100000

150000

200000

250000

300000

350000

400000

0.1 0.2 0.3 0.4 0.5 0.6 0.7

G (J

/m2 )

r/a

a/w=0.1a/w=0.2a/w=0.15

Fig. 26. Energy release rate versus r/a for three cases of a/w = 0.1, 0.15 and 0.2.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2 0.25 0.3

Nor

mal

ized

SIF

s

a/W

Normalized KI-Incompatible

Normalized KII-Incompatible

Normalized KI Nonequiibrium

Normalized KII Nonequilibrium

Fig. 27. SIFs for incompatible and non-equilibrium formulations.


1000 �C, while it is set to Th ¼ 615 �C for the hole. The problem is considered in the plane stress condition and the upper andlower edges of the square plate are fully constrained. The main aim of this example is to study the crack propagation in thisorthotropic nonhomogenous medium under the mixed mode fracture state. Material properties of the FGM specimen vary byan exponential function as,

Pleasemater10.101

A ¼ A0 � eðbax1Þ ð78Þ

where

A ¼ E1; E2; G12; a1; a2; k1; k2; KIcr

E01 ¼ 114:8 GPa; E0

2 ¼ 11:7 GPa; G012 ¼ 9:66 GPa; v0

12 ¼ 0:21; a01 ¼ 15� 10�6 1=�C

a02 ¼ 10� 10�6 1=�C; k0

1 ¼ 5 W=m �C; k02 ¼ 1 W=m �C; K0

Icr ¼ 2 MPaffiffiffiffiffimp

; ba ¼ 0:05

The domain is discretized by an unstructured quadrilateral 4-node finite element mesh which consists of 7607 elementsand 7799 nodes, as shown in Fig. 29. The predicted temperature field is shown in Fig. 30, which reveals a strong discontinuityin the thermal field.




Fig. 28. Geometry of the square plate with a circular hole and an inclined crack.

Fig. 29. Domain discritization with an unstructured mesh.


Crack propagation angle is determined by the maximum circumferential stress to strength ratio criterion [71,83]. Theangle of propagation h with respect to the crack x1 – axis, is obtained by maximizing Khh=KhhC ,

Pleasemater10.101

Khh

KhhC¼ rh

rmaxh

¼KIRe D ltip

1 B2 � ltip2 B1

�n oþ KIIRe DðB2 � B1Þf g

KXIC � cos2ðh0 þxÞ þ KY

IC � sin2ðh0 þxÞ¼ 1 ð79Þ

where x is the initial crack angle with respect to the material x1 principle axis, as depicted in Fig. 1. ltipi ði ¼ 1;2Þ are the

roots of the characteristic Eq. (60) and coefficients D and B are

D ¼ 1

ltip1 � ltip

2

ð80Þ

Bi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiltip

i sin hþ cos h �3

rð81Þ




Fig. 30. The temperature field for the orthotropic FGM.

Table 6Main fracture mechanics parameters in the beginning of crack propagation.

Angle of propagation (h) KI ðMPaffiffiffiffiffimpÞ KII ðMPa

ffiffiffiffiffimpÞ w ¼ tan�1 KII

KI

�29.64 2.253 �0.109 �2.76


KXIC and KY

IC are the independent mode I toughness with respect to material coordinate systems X and Y, respectively.Table 6 presents the main predicted fracture parameters at the time of first crack propagation. The length of the crack

increment is assumed to be 1 mm. The crack trajectory for successive crack propagation steps is shown in Fig. 31, whileFig. 32 depicts the stress contours for the first step of propagation.

In order to evaluate the effect of crack increment length ðDaÞ on the path of crack propagation, the problem is solved inthree cases of Da ¼ 0:9;1;1:2 and similar results are obtained, as shown in Fig. 33.

Fig. 31. Crack trajectory after 4 steps of crack propagation (ba = 0.05).




Fig. 32. Stress fields after the initial step of crack propagation: (a) r11, (b) r22 and (c) r12.





Fig. 33. Crack trajectory for different values of crack length increments (0.9,1,1.2), (ba = 0.05).


7. Conclusion

The interaction integral method for both incompatible and non-equilibrium formulations has been implemented for adi-abatic cracks in the framework of XFEM. The path independency of the method has been investigated in several examples.Also, the mesh independency of XFEM has been discussed for different plane stress and plane strain formulations in thermo-mechanical conditions. The method can be efficiently adopted to simulate complicated crack propagation thermomechanicalproblems, irrespective of the adopted crack length increment.

Acknowledgements

The authors would like to acknowledge the technical support of the High Performance Computing Lab, School of CivilEngineering, University of Tehran. Furthermore, the financial support of Iran National Science Foundation (INSF) is gratefullyacknowledged.

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