Mode III fracture analysis of a non-homogeneous layer …scientiairanica.sharif.edu/article_4493_1351a76aac1fb...that the SIFs of cracks depend on the crack length, FG parameter, and

Scientia Iranica B (2018) 25(5), 2570{2581

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

Mode III fracture analysis of a non-homogeneous layerbonded to an elastic half-plane weakened by multipleinterface cracks

R. Sourkia, M. Ayatollahia;�, and M.M. Monfaredb

a. Faculty of Engineering, University of Zanjan, Zanjan, P.O. Box 45195-313, Iran.b. Department of Mechanical Engineering, Hashtgerd Branch, Islamic Azad University, P.O. Box 33615-178, Alborz, Iran.

Received 11 November 2016; received in revised form 8 June 2017; accepted 11 September 2017

KEYWORDSInterface cracks;Distributeddislocation technique;Functionally gradedlayer;Volterra type screwdislocation;Stress intensityfactors.

Abstract. In this paper, the mode III crack problem of a non-homogenous layer bonded toan elastic half-plane was considered. It was assumed that the half-plane is homogeneous andthe layer is non-homogeneous where the elastic properties are continuous throughout thelayer. The stress �eld in a non-homogeneous layer and in an elastic half-plane with Volterra-type screw dislocation was obtained. Fourier transforms were applied to the governingequations to derive a system of singular integral equations with simple Cauchy kernel.Then, the integral equations were solved numerically by being converted into a system oflinear algebraic equations and using a collocation technique. The given results demonstratethe e�ects of the non-homogeneity parameters, interaction between the multiple cracks, anddistance of the cracks in the stress intensity factors for gaining better understanding of themechanical behavior of the non-homogenous coating.

© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

Functionally Graded Materials (FGMs) with materialproperties varying continuously possess apparent ad-vantages in maintaining the integrity of the structurewhen subjected to applied thermo-mechanical loads.They are also widely used in engineering �elds asa thermal barrier to prevent chemical corrosion [1].Functionally Graded Materials (FGMs) have beenwidely applied under extreme loading environment,and the fracture mechanics of FGMs have attractedmuch attention [2,3]. Practical application shows that

*. Corresponding author. Tel.: +98 24 33052488;Fax: +98 241 228-3204E-mail addresses: [email protected] (R. Sourki);[email protected] (M. Ayatollahi);mo m [email protected] (M.M. Monfared).

doi: 10.24200/sci.2017.4493

interface cracks constitute a major reason for failuresof structural connection of di�erent materials. Thus,the study of the stress �eld of the interface is playing asigni�cant role in the design of safe layered structures.In addition, due to the nature of the techniques usedin processing, these materials are imperfect. Moreover,by far, the most common forms of such imperfec-tions, which may have the potential of growing intoa macroscopic crack and causing the failure of thestructure eventually, are the surface aws located inthe intersection of the interfaces [4].

In fracture analysis of mediums with interfacialcracks, Erdogan is an expert as one of the pioneeringresearchers; therefore, a concise review regarding hiscontributions in this category is presented herein at�rst. The analytical expression of Stress IntensityFactor (SIFs) in two bonded elastic layers containingcracks perpendicular to the interface was obtainedby Ming-Che and Erdogan [4]. The problem was

R. Sourki et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2570{2581 2571

formulated in terms of integral equations and thesingular behavior of the solution near to and at theends of intersection of the cracks. The solution toan interface crack between two bonded homogeneousand non-homogeneous half planes was presented byDelale and Erdogan [5]. The results led to theconclusion that the singular behavior of stresses inthe nonhomogeneous medium is identical to that ina homogeneous material. In another paper, Erdoganand Wu [6] investigated the inuence of the structureand thickness of the interfacial regions on the strainenergy release rate in bonded isotropic or orthotropicmaterials containing collinear interface cracks. Theyformulated the problem in terms of a system of singularintegral equations of the second kind. The results showthat the e�ects of properties and the relative thicknessof the interfacial region on the stress intensity factorsand strain energy release rate can be highly signi�cant.Erdogan et al. [7] solved the mode III crack problem fortwo bonded homogeneous half planes. The interfacialzone was modelled by a non-homogeneous strip insuch a way that the shear modulus is a continuousfunction throughout the composite medium and hasdiscontinuous derivatives along the boundaries of theinterfacial zone. It was shown that the stresses had thestandard square root singularity.

Besides, in recent years, there have been severalinvestigations regarding mode III interface crack prob-lems as well. For example, Lu et al. [8] investigated theasymmetrical dynamic propagation problems of modeIII interface crack under the condition of point loadsand unit-step loads by the application of the theoryof complex functions. The problem of a functionallygraded coating bonded to an elastic substrate withinterface crack subject to anti-plane loading was ana-lyzed by Ding and Li [9]. In this investigation, theystudied the inuence of the interaction between theperiodic arrays of interface cracks on stress intensityfactors. Chen and Chue [10] dealt with the anti-planeproblems of two bonded FGM strips weakened by aninternal crack normal to the interface. The materialproperties are assumed to vary along the direction ofthe crack lines. The derived system of singular integralequations was solved numerically by Gauss-Chebyshevintegration formula. Chue and Yeh [11] consideredthe anti-plane crack problem of two bonded FGMstrips. Each strip concludes an arbitrarily orientedcrack, and the material properties are assumed to bein exponential forms. Mode III crack cutting theinterface perpendicularly between two dissimilar semi-in�nite magneto-electro-elastic solids was investigatedby Wan et al. [12]. The problem was formulated andsolved based on the complex variable method, andthe analytical solution was then found for the entireplane. Chao and Lu [13] studied the problem of alayered structure demonstrating an arbitrary oriented

crack crossing the interface in anti-plane elasticity. Thedistributed dislocation was used to model a crack, andSIFs for various inclinations with di�erent materialproperties were found. Pan et al. [14] investigated thefracture problem of multiple parallel symmetric andpermeable mode III cracks in a FG piezoelectric mate-rial plane under anti-plane loadings. They have foundthat the SIFs of cracks depend on the crack length,FG parameter, and the distance between the parallelcracks. Ding and Li [15] investigated an anti-planecrack problem of two collinear cracks perpendicular tothe interface of a FG orthotropic layer bonded to anorthotropic homogeneous substrate. Choi [16] providedthe solution to the anti-plane problem of dissimilarnonhomogeneous layers weakened by an embeddedor edge interfacial crack. Ordyan and Petrova [17]presented a solution to the interaction of an interfacecrack with internal cracks in dissimilar materials underanti-plane shear loading. The fracture behavior of apiezoelectro-magneto-elastic medium under anti-planeloading was analyzed by Rogowski [18]. It was foundthat the stress, electric and magnetic �elds exhibitsquare root singularity at the crack tips. Lapusta etal. [19] analyzed an interface crack in a biomaterialpiezoelectric plane under anti-plane mechanical and in-plane electric loadings.

Lu et al. [20] investigated the asymmetrical dy-namic propagation problem on the edges of mode IIIinterface crack subjected to superimposed loads byapplication of the theory of complex variable functions.Hu et al. [21] investigated a moving Dugdale interfacialcrack model between dissimilar magneto-electro-elasticmaterials under anti-plane shear and in-plane electricand magnetic loadings. Monfared and Ayatollahi [22]investigated the dynamic stress intensity factors ofcracked orthotropic half-plane and functionally gradedmaterial coating of a coating-substrate material due tothe action of anti-plane traction on the crack surfaces.The dislocation solution was utilized to derive integralequations for multiple interacting cracks. The transientresponse analysis of a mode III interface crack betweena piezoelectric layer and a FG orthotropic material wascarried out by Shin and Kim [23].

Many researchers have also studied the mixedmode interfacial crack problems in recent years. Forinstance, Yang et al. [24] studied the fracture prob-lems near the interface crack tip of double dissimilarorthotropic composite materials. Through the instru-mentality of complex function method, the singularityexponents were derived and determined. Cheng etal. [25] studied the plane elasticity problem of twobonded dissimilar functionally graded strips containingan interface crack with material properties varyingarbitrarily. The governing equation in terms of Airystress function was formulated, and exact solutionswere obtained for several special variations of material

2572 R. Sourki et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2570{2581

properties in Fourier transformation domain. Thefatigue crack growth of interfacial cracks in bi-layeredmaterials using the extended �nite-element methodwas analysed by Bhattacharya et al. [26]. Ding etal. [27] studied the behavior of an interface crack fora homogeneous orthotropic strip sandwiched betweentwo various FG orthotropic materials subjected to ther-momechanical loading. Crack tip opening displacementand plastic zone size were investigated for a curvedinterface crack between a circular inclusion and anin�nite matrix by Fan et al. [28]. Zhao et al. [29]studied the stress intensity factor in orthotropic bi-material interface cracks adopting the complex functionapproach.

Moreover, there have been several investigationsusing numerical techniques to solve such problems. Forinstance, Bui and Zhang [30] presented a transient dy-namic analysis of stationary cracks in two-dimensionalsolids subjected to coupled electromechanical impactloads using the extended Finite-Element Method (X-FEM). Sharma et al. [31] analyzed a sub-interface crackproblem in piezoelectric biomaterials by the X-FEM.In this investigation, the e�ects of di�erent polingdirections and electromechanical impact loads wereanalyzed. The stress intensity factors were evaluatedby utilizing the asymptotic crack-tip �elds. Liu etal. [32] studied the transient dynamic fracture behav-iors of stationary cracked FG piezoelectric materialsunder impact loading using the X-FEM. The tran-sient thermal shock fracture analysis of functionallygraded piezoelectric materials using the X-FEM wasconducted by Liu et al. [33]. It was found that thee�ect of the thermal shock loading on the dynamicintensity factors was signi�cant. Yu et al. [34] solvedthe problem of interfacial cracks between dissimilarpiezoelectric solids under coupled electromechanicalimpact loadings by the X-FEM. Bui [35] presentedan extension of the extended isogeometric analysis tosimulate two-dimensional fracture mechanics problemsin piezoelectric materials under dynamic and staticcoupled electromechanical loading conditions. Bui etal. [36] developed a dynamic extended Iso Geomet-ric Analysis (XIGA) of transient fracture of crackedmagneto-electroelastic solids under coupled electro-magneto-mechanical loading. Doan et al. [37] presenteda numerical simulation of dynamic crack propagationin functionally graded glass-�lled epoxy beams byutilizing a regularized variation formulation. TheGri�th theory-based hybrid phase �led approach wasused to simulate the dynamic crack growth accurately.Wang and Waisman [38] proposed a set of enrichmentfunctions within the framework of the X-FEM for theanalysis of interface cracks in biomaterials. Itou [39]determined the stress intensity factors for four collinearinterface cracks which are situated at the interfacebetween a nonhomogeneous elastic bonding layer and

one of two dissimilar elastic half-planes. The behaviorof an interface crack in two bonded dissimilar materialssubjected to in-plane loading was studied by Monfaredet al. [40]. The dislocation density on the faces of thecracks was obtained numerically and, then, was usedto calculate the mixed mode stress intensity factors ofmultiple interface cracks.

In this article, we present an analytical methodto study multiple interfacial cracks with arbitraryarrangement under anti-plane shear loading. Thistechnique is e�cient and applicable to modelling theevolution of a developing crack in two or three dimen-sions. Basically, the same strategies will be employedto formulate the solution to three-dimensional crackproblems by dislocation loop instead of a straight linedislocation.

This paper presents the solution to the problemof a graded coating bonded to an elastic isotropic half-plane weakened by multiple interface cracks. The stress�elds in a medium containing Volterra-type screw dis-location are �rst presented (Section 2). The analyticalstudy is based on the use of Fourier transform. In thenext section, the stress analysis of a medium underpoint loading is carried out (Section 3). With respectto the results of Sections 2 and 3, the integral equationsfor multiple interface cracks are derived in terms of thedislocation density function given in Section 4. Severalexamples are solved to demonstrate the applicabilityof the problem presented in Section 5. Numericalexamples are given to show the e�ects of the thicknessof the FGM strips, material properties, and length ofthe crack upon the fracture behavior. Finally, Section 6will conclude the paper.

2. Formulation of the problem

The anti-plane problem under consideration consistsof a graded coating bonded to isotropic semi-in�nitesubstrate with an interface crack (see Figure 1). Thethickness of strip is h in y-direction, where shear mod-ulus, �, varies continuously in the thickness direction.There is a screw dislocation at y = 0 between thegraded coating and elastic substrate. The constitutiverelations for FGM and isotropic material may beexpressed as follows:

Figure 1. Geometry of the problem.


�zx1 = �(y)@w1@x

; �zy1 = �(y)@w1@y

;

�zx2 = �0@w2@x

; �zy2 = �0@w2@y

: (1)

To facilitate the solution of the problem, non-homogeneity of the material may be approximated by:

�(y) = �0e2�y; (2)

where � is a non-homogeneity parameter. In the aboveequations, �0 is the shear module of elasticity. Thestrip is traction free on the boundary. Thus, boundarycondition is as follows:

�zy1(x; h) = 0: (3)

A Volterra-type screw dislocation with Burgers vector;bz, is placed at the origin of coordinates. The conditionillustrating the dislocation is:

w1(x; 0+)� w2(x; 0�) = bzH(x); (4)where H(:) is the Heaviside step function. The condi-tion of self-equilibrium of stress between the strip andhalf-plane implies that:

�zy1(x; 0+) = �zy2(x; 0

�): (5)

The equilibrium equations in terms of displacement willbe written as follows:

@2w@x2

+@2w@y2

+ 2�@w@y

= 0; 0 � y � h;

@2w@x2

+@2w@y2

= 0; y � 0: (6)The solution to Eqs. (6) can be solved by the Fouriertransformation, de�ned as follows:

F (s) =1Z�1

f(x)eisxdx; (7)

and the inversion of the Fourier transform is:

f(x) =1

2�

1Z�1

F (s)e�isxds: (8)

By applying the Fourier transform Eq. (7), the solutionto Eqs. (6) may be expressed as follows:

d2W (s; y)dy2

+ 2�dW (s; y)

dy

� s2W (s; y) = 0; 0 � y � h;d2W (s; y)

dy2� s2W (s; y) = 0; y � 0; (9)

where s is the transform variable, and the solution willlead to the following equations for each region:

W (s; y) =A1(s)e(��)y +A2(s)e(��+�)y

0 � y � h;W (s; y) = B(s)ejsjy; y � 0; (10)

where � =p� 2 + s 2 and unknown functions, A1(s),

A2(s), and B(s) are determined from the boundaryconditions. The Fourier transforms of boundary condi-tions (3) to (5) result in:

dW1(s; h)dy

= 0;

W1(s; 0+)�W2(s; 0�) = bz�� (s) +

is

�;

dW1(s; 0+)dy

=dW2(s; 0�)

dy; (11)

where �(:) is the Dirac delta function. The solution toEq. (10) subject to the above boundary conditions isstraightforward. Hence, unknowns A1(s), A2(s), andB1(s) may be expressed as shown in Box I.

By substituting Eq. (12) into Eq. (10) and ap-plying the Fourier transform inversion formula (8), the

A1(s) = � bzjsj(��(s) + i=s)(� � �)e�h

jsj[(� + �)e��h � (� � �)e�h] + (�2 � �2)(e��h � e�h) ;

A2(s) =bzjsj(��(s) + i=s)(� + �)e��h

jsj[(� + �)e��h � (� � �)e�h] + (�2 � �2)(e��h � e�h) ;

B1(s) =bz(��(s) + i=s)(�2 � �2)(e�h � e��h)

jsj[(� + �)e��h � (� � �)e�h] + (�2 � �2)(e��h � e�h) : (12)

Box I


w1(x; y) = � bz2�Z 1�1jsj(��(s) + i/s)e��y[(� � �)e�(h�y) � (� + �)e��(h�y)]jsj[(� + �)e��h � (� � �)e�h] + (�2 � �2)(e��h � e�h) e�isxds; 0 � y � h;

w2(x; y) =bz2�

Z 1�1

(��(s) + i/s)(�2 � �2)(e�h � e��h) ejsjyjsj[(� + �)e��h � (� � �)e�h] + (�2 � �2)(e��h � e�h)e�isxds; y � 0: (13)

Box II

displacement �eld in the medium can be obtained byEq. (13) as shown in Box II.

By splitting the integrals in Eq. (13) into oddand even parts with respect to parameter s in viewof Eq. (1), the stress �elds are written as follows:

�zy1(x; y) = �bz�0e�y

�

Z 10

s(e�(h�y)�e��(h�y))(�+��s)e��h�(��s)e�h sin(sx)ds;

�zx1(x; y) = �bz�0e�y

�

Z 10

(��)e�(h�y)�(�+�)e��(h�y)(�+�� s)e��h�(��s)e�h cos(sx)ds;

0 � y � h;

�zy2(x; y) = �bz�0�Z 1

0

sesy(e�h � e��h)(�+��s)e��h � (�� s)e�h sin(sx)ds;

�zx2(x; y) = �bz�0�Z 1

0

sesy(e�h�e��h)(�+��s)e��h�(��s)e�h cos(sx)ds;

y � 0: (14)In order to specify the singular nature of the stresscomponents, the asymptotic behavior of integrands inEq. (14) should be examined. Since the integrands arecontinuous functions of s and also �nite at s = 0, thesingularity must occur as s tends to in�nity. The �nalresults may be presented as follows:

�zy1(x; y) =

� �0bze2�y2�

�x

x2+y2+

xx2+(2h�y)2

��0bze�y

�

�Z 1

0

�s(e�he�(�+�)y � e��he�(��)y)(�+��s)e��h�(��s)e�h

� e�sy � e�s(2h�y)2

�sin(sx)ds;

0 � y � h;

�zx1(x; y) =�0bze2�y

2�

�y

x2 + y2+

2h� yx2 + (2h� y)2

�� 0bze�y

�

�Z 1

0

�(��)e�he�(�+�)y�(�+�)e��he�(��)y

(�+��s)e��h�(��s)e�h

+e�sy � e�s(2h�y)

2

�cos(sx)ds;

0 � y � h:

�zy2(x; y) = ��0bz2��

xx2 + y2

�� 0bz

�

Z 10�

s(e�h � e��h)(�+��s)e��h � (��s)e�h �

12

�esy sin(sx)ds;

y � 0;

�zx2(x; y) =�0bz2�

(x

x2 + y2)� �0bz

�

Z 10�

s(e�h � e��h)(�+��s)e��h � (��s)e�h �

12

�esy cos(sx)ds;

y � 0: (15)


It is noteworthy to mention that stress �elds exhibitCauchy singularity at the dislocation position. More-over, the integrands in Eq. (15) decay quite rapidlyas s ! 1, which makes the integrals susceptible tonumerical evaluation.

3. Point load solution

It is assumed that the medium is under an anti-plane point load with magnitude �0 on the boundary,represented by the following:

�zy1(x; h) = �0�(x): (16)

The continuity conditions can be expressed as follows:

w1(x; 0+) = w2(x; 0�);

�zy1(x; 0+) = �zy2(x; 0

�): (17)

The expressions for the displacements and stresses maybe obtained as shown in Box III.

The singular parts of the kernels in Eq. (18) maybe evaluated by separating the leading terms in the

asymptotic analysis for s!1. Thus, after performingthe appropriate asymptotic analysis using a symbolicmanipulator and separating the singular parts of thekernels, we obtain:

�zy1(x; y) =�0e�(h�y)

�

�(h� y)

x2+(h� y)2

+Z 1

0

�(�+��s)e��y � (�� s)e�y(�+��s)e��h � (�� s)e�h

� e�s(h�y)�

cos(sx)ds�; 0 � y � h;

�zx1(x; y) =� �0e�(h�y)�

�x

x2 + (h� y)2

+Z 1

0

�(�+��s)e��y � (�� s)e�y(�+��s)e��h � (�� s)e�h

� e�s(h�y)�

sin(sx)ds�; 0 � y � h;

w1(x; y) =�0

2��0

Z 1�1

e(�+�)(h�y)�2�y[(� + �+ jsj)e2�y � (� � �+ jsj)]s2(e2�h � 1) + jsj[(� + �)� (� � �)e2�h] e�isxds;

0 � y � h;

w2(x; y) =�0��0

Z 1�1

�e(jsj�2�)y+(�+�)h(1� e2�h)(�2 � �2) + jsj[�(1 + e2�h) + �(1� e2�h)]e�isxds;

y � 0;

�zy1(x; y) =�0e�(h�y)

�

Z 10

(� + �� s)e��y � (� � �� s)e�y(� + �� s)e��h � (� � �� s)e�h cos(sx)ds;

0 � y � h;

�zx1(x; y) = ��0e�(h�y)�

Z 10

(� + �� s)e��y � (� � �� s)e�y(� + �� s)e��h � (� � �� s)e�h sin(sx)ds;

0 � y � h;

�zy2(x; y) =2�0�

Z +10

�(s� 2�)e�(h�2y)+sys[(� + �� s)e��h � (� � �� s)e�h] cos(sx)ds;

y � 0;

�zx2(x; y) =� 2�0�Z 1

0

�e�(h�2y)+sy(� + �� s)e��h � (� � �� s)e�h sin(sx)ds;

y � 0: (18)

Box III


�zy2(x; y) =�0e�(h�2y)

�

�(h�y)

x2+(h� y)2

+Z 1

0

�2�(s� 2�)

s[(�+��s)e��h� (��s)e�h]

� e�sh�esy cos(sx)ds

�; y � 0;

�zx2(x; y) =� �0e�(h�2y)�

�x

x2+(h� y)2

+Z 1

0

�2�

(�+��s)e��h � (�� s)e�h

�e�sh�esy sin(sx)ds

�; y � 0; (19)

At the points of application of load, i.e., r ! 0, stress�elds exhibit the familiar Cauchy type singularity, 1=r,where r is the distance from the load.

4. Medium weakened by crack

The dislocation solutions accomplished in the pre-ceding section may be employed to analyze severalarbitrarily cracks located in the interface of two dis-similar mediums. A crack con�guration with respectto Cartesian coordinate x � y may be described in aparametric form as follows:

xi = xi0 + ai �;

yi = yi 0; i = 1; 2; :::; N; �1 � � � 1; (20)where (xi 0; yi 0) is the coordinates of the ith crackcenter. Suppose that the screw dislocation unknowndensity, Bzi(t), is distributed on segment aidt at thesurface of the ith crack, where �1 � t � 1. Coveringthe cracks surfaces by dislocations, the principal su-perposition may be invoked to obtain traction on thegiven crack surface. The anti-plane �eld traction on theface of the ith crack due to the presence of distributionof the above-mentioned dislocation on all N cracks isobtained. The system of a singular equation can bewritten in the following form utilized in a numericalprocedure:

�zy�xi(�); yi(�)

�=

NXj=1

1Z�1

kij(�; t)ajBzj(t)dt;

� 1 � � � 1; i = 1; 2; :::; N: (21)Kernel Kij in integral Eqs. (21) is coe�cient of bz inthe stress components. By virtue of the Buckner'sprinciple [41], the left-hand side of Eqs. (21), afterchanging the sign, is the traction caused by the pointload given in Section 3. The equation for the crack

opening displacement across the ith crack is as follows:

w1i(�)�w2i(�)=Z ��1aiBzi(t)dt; i 2 f1; 2; :::; Ng:

(22)

The displacement �eld is single-valued out of cracksurfaces. Consequently, the dislocation density issubjected to the following closure requirement:Z 1�1Bzi(t) dt = 0; i 2 f1; 2; :::; Ng : (23)

The stress �elds in the neighborhood of crack tipsbehave like 1=

pr, where r is the distance from the

crack tip. Therefore, the dislocation densities are takenas follows:

Bzi(t) =gzi(t)p1� t2 ;

�1 � t � 1; i 2 f1; 2; :::; Ng : (24)Parameters gzi(t) are obtained by solving the systemof Eqs. (21) and (24) using numerical solutions ofintegral equations with Cauchy-type kernel developedby Erdogan et al. [42]. The stress intensity factors willreduce to:

KIIIL =p

24�(yLi) lim

rLi!0w1i(�)� w2i(�)prLi ;

KIIIR =p

24�(yRi) lim

rRi!0w1i(�)� w2i(�)prRi ; (25)

where subscripts L and R designate the left and rightcrack tips, respectively, and the geometry of crackimplies the following:

rLi = [(xi(�)� xi(�1))2 + (yi(�)� yi(�1))2] 12 ;rRi = [(xi(�)� xi(1))2 + (yi(�)� yi(1))2] 12 : (26)

By substituting Eq. (24) intoEq. (22) and the resultantequation into Eq. (25) after utilizing the Taylor seriesexpansion of functions xi(�) and yi(�) in the vicinityof points � = �1, the stress intensity factors can beexpressed as follows:

KIIIL =�(yLi)

2

h[x0i(�1)]2 + [y0i(�1)]2

i 14gzi(�1);

KIIIR = ��(yRi)2h[x0i(1)]2 + [y0i(1)]

2i 1

4gzi(1): (27)

5. Results and discussion

The main results of this study demonstrate that thestress intensity factors have been calculated for dif-


ferent non-homogeneity parameters, crack length, andinteractions between several cracks which are locatedat the interface between the nonhomogeneous layer andthe isotropic half-plane. The above presented methodallows for the consideration of dissimilar materialswith multiple straight cracks subjected to anti-planetractions. The results have been obtained through theinstrumentality of distributed dislocation technique.The validity of the analysis is veri�ed by consideringa few examples, wherein the cracks are under constantnormal traction.

The �rst problem is an interface crack in afunctionally graded coating-substrate structure underthe uniform anti-plane loading. External loading,�zy = �0, is applied on the crack surface. The plots ofdimensionless stress intensity factors for two di�erentnon-homogeneity parameters are drawn in Figure 2.In this case, excellent agreement is observed betweenthe results of this study and those presented by Dingand Li [9]. The second example deals with periodicinterface cracks with three di�erent distances betweencracks centers, as shown in Figure 3. The cracks'faces are subjected to uniform anti-plane shear loads.The results favorably match results of Ding and Li [9].

Figure 2. Comparison of the normalized stress intensityfactors for straight crack in the interface under a uniformload.

Figure 3. Comparison of the normalized stress intensityfactors for periodic cracks in the interface under a uniformload.

The applicability of the methodology developed here isillustrated by solving some new problems.

In all remaining examples, since the medium isunder concentrated anti-plane point loads as speci�edin Eq. (16), the quantity for making stress intensityfactors dimensionless is K0 = �0=

pa where a is the

half length of an embedded crack. In addition, itshould be mentioned that the dimensions of distancebetween multiple cracks are taken c=a = 1:125 as inall numerical results presented in this paper unlessotherwise stated.

The e�ects of crack length and non-homogeneityparameter on the normalized stress intensity factors aregiven in Figure 4. The symmetry of the problem withrespect to y-axis implies that the stress intensity factorsat crack tips are identical. As physically expected,the stress intensity factors increase with the increaseof crack length. Moreover, it can be observed thatthe normalized stress intensity factors increase as thenon-homogeneity parameter rises. The e�ect of non-homogeneity parameter on stress intensity factors isdepicted in Figure 5. It can be seen that the highestvalue of dimensionless stress intensity factors occurswhere the non-homogeneity parameter increases. Ingeneral, the inuence of the coating thickness on the

Figure 4. Normalized stress intensity factors for aninterface crack versus the dimensionless crack length fordi�erent non-homogeneity parameters.

Figure 5. Normalized stress intensity factors for aninterface crack versus the non-homogeneity parameter.


Figure 6. Normalized stress intensity factors for twointerface cracks versus the dimensionless crack length.

Figure 7. Normalized stress intensity factors for twointerface cracks versus the non-homogeneity parameter.

Figure 8. Normalized stress intensity factors for twointerface cracks versus the dimensionless distance betweenthe cracks.

stress intensity factors is as signi�cant as that of thenon-homogeneity parameter. Figure 6 shows the e�ectsof non-homogeneity parameter and cracks' lengthsbetween two interface cracks on the normalized stressintensity factors. It can be seen that an increase in ei-ther �a or a=h causes an increase in the stress intensityfactors. As observed, dimensionless stress intensity fac-tors for the two approaching crack tips change rapidly.

Two equal-length interface cracks are considered,as shown in Figures 7 and 8. The lengths of cracks

Figure 9. Normalized stress intensity factors for threeinterface cracks versus the non-homogeneity parameter.

Figure 10. Normalized stress intensity factors for threeinterface cracks versus the dimensionless distance betweenthe cracks.

Figure 11. Normalized stress intensity factors for threeinterface cracks versus the dimensionless crack lengths.

remain �xed, while the non-homogeneity parameterand crack centers are changing. From Figures 7 and8, we may conclude that the stress intensity factorsof the two interface cracks are mainly a�ected bythe non-homogeneity parameter and interactions ofcracks. According to the given results in Figure 8, itis evident that the normalized stress intensity factorsdecline steadily, while the distances between cracksincrease as expected, due to the dramatic declineof the interaction between cracks. The �nal set ofresults, shown in Figures 9-11, presents stress intensity


Figure 12. Normalized stress distribution around thecrack tips.

factors for three interface cracks with changing non-homogeneity parameter. Figure 11 shows the stressintensity factors of three interface cracks as a functionof the crack lengths. Note that the stress intensityfactors are highly inuenced by the non-homogeneityparameter in which the stress intensity factors go upas non-homogeneity parameter rises.

Last but not least, stress distribution along x-axisis depicted in Figure 12 while �a = 2:0. Singularitiesare, obviously, in the vicinity of the crack tips. It iscrystal clear that the stress values plummet dramati-cally to be equal to the far �eld loadings at the su�cientdistance of the crack tips.

6. Conclusions

The fracture behavior of a non-homogeneous layeralong with a half plane bonded together was investi-gated. It was assumed that the half plane is homoge-neous and the layer is non-homogeneous, in which theelastic properties are continuous throughout the layerand have discontinuous derivatives along the interface.The dislocation solution was obtained in a functionallygraded layer and in half-plane containing Volterra-typescrew dislocation by means of Fourier transformation.The dislocation solution was utilized to derive integralequations for the layer weakened by multiple cracks.The conclusions can be made as follows:

1. The stress intensity factors are highly inuenced bythe non-homogeneity parameter;

2. It was observed that as the crack length increases,the stress intensity factors escalate;

3. For the multiple interface cracks, the stress inten-sity factors are highly a�ected by the cracks' lengthand distance;

4. Negative values for the non-homogeneity parame-ters tend to decrease the stress intensity factor.

5. It was seen that decreasing the value of coating

thickness while keeping � constant tends to increasethe stress intensity factor for negative values ofparameter �.

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Biographies

Reza Sourki is a graduate student in MechanicalEngineering from University of Zanjan, Iran. He hasreceived his MSc and BSc degree (both with honors)

from University of Zanjan and has been awardedas the 1st ranked student. He, also, is a memberof Exceptional Talents at the university as well asNational Elites Foundation of Iran. His current areasof interest include theoretical fracture mechanics andvibrations.

Mojtaba Ayatollahi was born in Zanjan, Iran in1977. He received his PhD degree from the AmirkabirUniversity of Technology in 2007. He joined theDepartment of Mechanical Engineering, University ofZanjan, as an Assistant Professor in 2007; he becamean Associate Professor in 2011 and a Professor in2016. His current research interests include appliedmathematics, elasticity, and theoretical fracture me-chanics. He published over 40 papers in some well-known journals since 2009.

Mojtaba Mahmoudi Monfared received his PhDdegree from the University of Zanjan. He is an Assis-tant Professor of Practice in Mechanical Engineering.His research interests include mixed boundary valueproblems, singular integral equations, solid mechanics,and fracture mechanics.

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Mode III fracture analysis of a non-homogeneous layer …scientiairanica.sharif.edu/article_4493_1351a76aac1fb...that the SIFs of cracks depend on the crack length, FG parameter, and