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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 721656 14 pageshttpdxdoiorg1011552013721656

Research ArticleTwo-Dimensional Stress Intensity Factor Analysis of Cracks inAnisotropic Bimaterial

Chia-Huei Tu1 Jia-Jyun Dong23 Chao-Shi Chen1

Chien-Chung Ke4 Jyun-Yong Jhan1 and Hsien Jui Yu1

1 Department of Resources Engineering National Cheng Kung University No 1 University Road Tainan City 70101 Taiwan2Graduate Institute of Applied Geology National Central University No 300 Jhongda Road Jhongli City 32001 Taiwan3 Graduate Institute of Geophysics National Central University No 300 Jhongda Road Jhongli City 32001 Taiwan4Geotechnical Engineering Research Center Sinotech Engineering Consultants Inc Taipei 110 Taiwan

Correspondence should be addressed to Chao-Shi Chen chencsmailnckuedutw

Received 11 June 2012 Revised 3 November 2012 Accepted 9 November 2012

Academic Editor Giuseppe Carbone

Copyright copy 2013 Chia-Huei Tu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a 2D numerical technique based on the boundary element method (BEM) for the analysis of linear elastic frac-ture mechanics (LEFM) problems on stress intensity factors (SIFs) involving anisotropic bimaterials The most outstanding featureof this analysis is that it is a singledomain method yet it is very accurate efficient and versatile (ie the material properties of themedium can be anisotropic as well as isotropic) A computer program using the BEM formula translation (FORTRAN 90) codewas developed to effectively calculate the stress intensity factors (SIFs) in an anisotropic bi-material This BEM program has beenverified and showed good accuracy compared with the previous studies Numerical examples of stress intensity factor calculationfor a straight crack with various locations in both finite and infinite bimaterials are presented It was found that very accurate resultscan be obtained using the proposed method even with relatively simple discretization The results of the numerical analysis alsoshow that material anisotropy can greatly affect the stress intensity factor

1 Introduction

The problem of cracks between two dissimilar materials hasbeen widely studied over the past several decades stemmingmainly from the desire to understand the failure modesof composites including structures rocks and concreteWilliams [1] presented the first study of the plane problemof cracks between dissimilar isotropic materials Williamsshowed that stresses possess the singularity of 119903minus12plusmn119894120576 where119903 is the radius distance from the crack tip and 120576 is a bi-materialconstant England [2] investigated the problem of finitecracks between dissimilar isotropicmaterials Rice and Sih [3]studied similar problems and derived the expressions of thestress fields near crack tips Rice [4] reexamined the elasticfracture mechanics concepts of the isotropic interfacial crackand introduced an intrinsic material length scale so that thedefinition of the stress intensity factors (SIFs) possessed thesame physical significance as those for homogeneous cracks

Clements [5] and Willis [6] extended the problem studiedby England [2] for dissimilar anisotropic materials They alsoshowed the oscillatory behavior of the stresses and the phe-nomenon of interpenetration of the crack face near the cracktip in anisotropic interface cracks Wu [7] extended the prob-lem studied by Rice [4] for anisotropic bi-material cracksRecently many authors [7ndash10] have conducted studies oninterfacial cracks in anisotropic materials and different def-initions for the stress intensity factor existence However thedata on different crack locations crack lengths and degree ofmaterial anisotropy are scarce in the literature

Recently several BEMs have been proposed for the studyof cracked media [11ndash13] It involves two sets of boundaryintegral equations and is in general superior to the afore-mentioned BEMs Consequently general mixed mode crackproblems can be solved in a BEM formulation The single-domain analysis can eliminate remeshing problems whichare typical of the FEM and the subregional BEM

2 Mathematical Problems in Engineering

In this paper we develop a technique for single-domainBEM formulation in which neither the artificial boundarynor the discretization along the uncracked interface is nec-essary Chen et al [14] presented this single-domain BEMformulation for homogeneous materials We combined itwith Greenrsquos functions of bi-materials [15] to extend it toanisotropic bi-materials The BEM formulation is such thatthe displacement integral equation is applied only to the outerboundary and the traction integral equation is applied only toone side of the cracked surface A decoupling technique canbe used to determine mixed mode SIFs based on the relativedisplacement at the crack tip Numerical cases for mixedmode SIFs for anisotropic bi-materials with different cracklocations and degrees of material anisotropy are presentedThe numerical results obtained using the BEM formulationare verified by several previous studies [8 16ndash18]

2 Methodology

21 Basic Equations for Anisotropic Elasticity For linearelastic homogeneous and anisotropic material the stressand displacement fields can be formulated in terms of twoanalytical functions 120601

119896

(119911119896

) of the complex variables 119911119896

=

119909 + 120583119896

119910 (119896 = 1 2) where 120583119896

is the root of the followingcharacteristic equation [19]

11988611

1205834

minus 211988616

1205833

+ (211988612

+ 11988666

) 1205832

minus 211988626

120583 + 11988622

= 0 (1)

where the coefficient 119886119894119895

is the compliance component cal-culated using the 119909-119910 coordinate system The detailed rela-tionships of these components with material elasticity can befound in Chen et al [14] If the root 120583

119895

of (1) is assumed to bedistinct the general expressions for the stress and displace-ments are

120590119909

= 2Re [12058321

1206011015840

1

(1199111

) + 1205832

2

1206011015840

2

(1199112

)]

120590119910

= 2Re [12060110158401

(1199111

) + 1206011015840

2

(1199112

)]

120591119909119910

= minus2Re [1205831

1206011015840

1

(1199111

) + 1205832

1206011015840

2

(1199112

)]

119906 = 2Re [11986011

1206011

(1199111

) + 11986012

1206012

(1199112

)]

120592 = 2Re [11986021

1206011

(1199111

) + 11986022

1206012

(1199112

)]

(2)

where the elements of the complex matrices 119860 are

1198601119895

= 11988611

1205832

119895

+ 11988622

minus 11988616

120583119895

1198602119895

= 11988612

120583119895

+

11988622

120583119895

minus 11988626

(119895 = 1 2)

(3)

The traction components in the 119909 and 119910 directions are

119879119909

= 2Re [1205831

1206011

(1199111

) + 1205832

1206012

(1199112

)]

119879119910

= minus2Re [1206011

(1199111

) + 1206012

(1199112

)]

(4)

Here the complex analytical functions 120601119894

(119911119894

) can in generalexpress (2) and (4) as follows [19ndash21]

119906119894

= 2Re [1198601119894

1206011

(1199111

) + 1198602119894

1206012

(1199112

)]

119879119894

= minus2Re [1198611119894

1206011

(1199111

) + 1198612119894

1206012

(1199112

)]

1205902119894

= 2Re [1198611119894

1206011015840

1

(1199111

) + 1198612119894

1206011015840

2

(1199112

)]

1205901119894

= minus2Re [1198611119894

1206011015840

1

(1199111

) + 1198612119894

1206011015840

2

(1199112

)] (119894 = 1 2)

(5)

where 119911119895

= 119909 + 120583119895

119910 Re denotes the real part of a complexvariable or function a prime denotes the derivative the com-plex number 120583

119895

(119895 = 1 2) the elements of complex matrices119860 are defined in (3) and the elements of complex matrices 119861can be defined as

119861119894119895

= [

minus1205831

minus1205832

1 1] (6)

Assume that the medium is composed of two joined dis-similar anisotropic and elastic half-planes Let the interface bealong the 119909-axis and let the upper (119910 gt 0) and lower (119910 lt 0)half-planes be occupied by materials 1 and 2 respectively(as shown in Figure 1)

Considering a concentrated force acting at the sourcepoint (1199090 1199100) in material 2(1199100 lt 0) we express the complexvector function as [20]

120601 (119911) =

1206011(119911) 119911 isin (1)

1206012(119911) + 120601

0

2 (119911) 119911 isin (2)

(7)

where the vector function is

120601 (119911) = [1206011

(119911) 1206012

(119911)]119879

(8)

where the argument has the generic form 119911 = 119909 + 120583119910In (7) 12060102 is a singular solution corresponding to a point

force acting at the point (1199090 1199100) on an anisotropic infiniteplane with the elastic properties of material 2 This singularsolution can be expressed as [20 21]

1206010

119896

(119911119896

) = minus

1

2120587

[1198671198961

1198751

ln (119911119896

minus 119904119896

) + 1198671198962

1198752

ln (119911119896

minus 119904119896

)]

(9)

where 119904119896

= 1199090

+ 120583119896

1199100 119875119896

is the magnitude of the point forcein the 119896-direction and

119867 = 119860minus1

(119884minus1

+ 119884

minus1

)

minus1

119884 = 119894119860119861minus1

(10)

where 119894 = radicminus1 the overbar indicates the complex conju-gate and superscript minus1 indicates the matrix inverse Twounknown vector functions 1206011(119911) and 1206012(119911) are solved in(7)The former is analytic inmaterial 1 and the latter is ana-lytic inmaterial 2These expressions can be found by requir-ing the continuity of the resultant traction and displacement

Mathematical Problems in Engineering 3

Interface

0

Crack tip

Counterclockwise tobuild the domain

Material 1

Material 2

Γ119861 Displacement equationΓ119862 Traction equation

119910

119909Γ119862+

Γ119862minus 1198649984002

1198649984001

1198642

1198641119899119898

Γ119861

1205951

minus1205952

Figure 1 Definition of the coordinate systems in an anisotropic bi-material

across the interface in addition to the standard analyticcontinuation arguments Substituting (9) for (7) we obtain

1206011(119885) = 119861

minus1

1 (1198841 + 1198842)minus1

(1198842 + 1198842) 11986121206010

2 (119911)

119911 isin 1

1206012(119885) = 119861

minus1

2 (1198841 + 1198842)minus1

(1198842 minus 1198841) 11986121206010

2 (119911)

119911 isin 2

(11)

Therefore the complex vector functions can be expressed as

120601 (119911) =

119861minus1

1 (1198841 + 1198842)minus1

times (1198842 + 1198842) 11986121206010

2 (119911) 119911 isin 1119861minus1

2 (1198841 + 1198842)minus1

(1198842 minus 1198841)

times11986121206010

2 (119911) + 1206010

2 (119911) 119911 isin 2

(12)

In (12) subscripts 1 and 2 are used to denote that thecorrespondingmatrix or vector is for material 1 (119910 gt 0) andmaterial 2 (119910 lt 0) respectively

Similarly for a point force in material 1 (1199100 gt 0) thecomplex vector functions can be expressed as

120601 (119911) =

119861minus1

1 (1198842 + 1198841)minus1

times (1198841 minus 1198842) 11986111206010

2 (119911) + 1206010

1 (119911) 119911 isin 1

119861minus1

2 (1198842 + 1198841)minus1

times (1198841 + 1198841) 11986111206010

1 (119911) 119911 isin 2(13)

where vector function 12060101 is the infinite-plane solution givenin (9) but with the elastic properties of material 1

The Greenrsquos functions of the displacement and tractioncan be obtained by substituting the complex vector functionsin (12) and (13) for (5) Here 119880lowast

119896119897

is the Greenrsquos function fordisplacement and 119879

lowast

119896119897

is the Greenrsquos function for tractionSuperscripts and subscripts 1 and 2 are used to denotethe corresponding quantities for materials 1 (119910 gt 0) and2 (119910 lt 0) respectively [15]

(i) For source point (119904) and field point (119911) on material1 (119910 gt 0)

119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198601119897119895

[ ln (1199111119895

minus 1199041119895

)1198671119895119896

+

2

sum

119894=1

11988211

119895119894

ln (1199111119895

minus 1199041119894

)119867

1119894119896

]

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198611119897119895

[

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199041119895

1198671119895119896

+

2

sum

119894=1

11988211

119895119894

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199041119894

119867

1119894119896

]

(14)

where the matrix119867 is defined in (10) with the aniso-tropic elastic properties of material 1 and

11988211

= 119861minus1

1 (1198841 + 1198842)minus1

(1198841 minus 1198842) 1198611 (15)

(ii) For source point (119904) and field point (119911) on material1 (119910 gt 0) and field point (119911) on material 2 (119910 lt 0)

119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198602119897119895

[

2

sum

119894=1

11988212

119895119894

ln (1199112119895

minus 1199041119894

)1198671119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198612119897119895

[

2

sum

119894=1

11988212

119895119894

1205832119895

119899119909

minus 119899119910

1199112119895

minus 1199041119894

119867

1119894119896

]

(16)

with

11988212

= 119861minus1

2 (1198842 + 1198841)minus1

(1198841 minus 1198841) 1198611 (17)

(iii) For source point (119904) and field point (119911) on material2 (119910 lt 0)

119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198602119897119895

[ ln (1199112119895

minus 1199042119895

)1198672119895119896

+

2

sum

119894=1

11988222

119895119894

ln (1199112119895

minus 1199042119894

)119867

2119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198612119897119895

[

1205832119895

119899119909

minus 119899119910

1199112119895

minus 1199042119895

1198672119895119896

+

2

sum

119894=1

11988222

119895119894

1205832119895

119899119909

minus 119899119910

1199112119895

minus 1199042119894

119867

2119894119896

]

]

(18)

wherematrix119867 is defined in (10) with the anisotropicelastic properties of material 2 and

11988222

= 119861minus1

2 (1198842 + 1198841)minus1

(1198842 minus 1198841) 1198612 (19)


(iv) For source point (119904) and field point (119911) on materials2 (119910 lt 0) and field point (119911) on material 1 (119910 gt 0)

119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198601119897119895

[

2

sum

119894=1

11988221

119895119894

ln (1199111119895

minus 1199042119894

)1198672119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198611119897119895

[

2

sum

119894=1

11988221

119895119894

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199042119894

119867

2119894119896

]

(20)

with

11988221

= 119861minus1

1 (1198841 + 1198842)minus1

(1198842 minus 1198842) 1198612 (21)

It should be noted that these Greenrsquos functions can beused to solve both plane stress and plane strain problemsfor anisotropic bi-materials Although the isotropic solutioncannot be analytically reduced from these Greenrsquos functionsone can numerically approximate it by selecting a very weakanisotropic (or nearly isotropic) medium [22 23]

22 Single-Domain Boundary Integral Equations In this sec-tion we present single-domain boundary integral equationsin which neither the artificial boundary nor the discretizationalong the uncracked interface is necessary This single-domain boundary integral equation was used recently byChen et al [14] for homogeneous materials and it is nowextended to anisotropic bi-materials

For a point 1199040119896119861

on the uncracked boundary the displace-ment integral equation applied to the outer boundary resultsin the following form (1199040

119896119861

isin Γ119861

only as shown in Figure 1)

119887119894119895

(1199040

119896119861

) 119906119895

(1199040

119896119861

)

+ int

Γ119861

119879lowast

119894119895

(119911119896119861

1199040

119896119861

) 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ int

Γ119862

119879lowast

119894119895

(119911119896119862

1199040

119896119861

) [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= int

Γ119861

119880lowast

119894119895

(119911119896119861

1199040

119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(22)

where 119894 119895 119896 = 1 2 119879lowast

119894119895

and 119880lowast119894119895

are the Greenrsquos tractions anddisplacements given in (14) (16) (18) and (20) 119906

119895

and 119905119895

are the boundary displacements and tractions respectivelyand 119911

119896

and 1199040

119896119861

are the field points and the source pointson boundary Γ of the domain respectively Γ

119862

has the sameoutwardnormal asΓ

119862+

119887119894119895

are coefficients that dependonly onthe local geometry of the uncracked boundary at point 1199040

119896119861

and are equal to 120575119894119895

2 for a smooth boundaryHere subscripts119861 and 119862 denote the outer boundary and the cracked surfacerespectively In deriving (22) we have assumed that the trac-tions on the two faces of a crack are equal and opposite Weemphasize here that since the bi-material Greenrsquos functionsare included in (22) discretization along the interface can

be avoided with the exception of the interfacial crack partwhich is treated by the traction integral equation presentedin the following

It should be noted that all the terms on the right-handside of (22) have weak singularities and thus are integrableAlthough the second term on the left-hand side of (22) hasa strong singularity it can be treated using the rigid-bodymotion method

The traction integral equation (for 1199040119896

being a smoothpoint on the crack) applied to one side of the cracked surfaceis (1199040119896119862

isin Γ119862+

only)

05119905119895

(1199040

119896119862

) + 119899119898

(1199040

119896119862

) int

ΓΒ

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119861

)

times 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ 119899119898

(1199040

119896119862

)int

Γ119862

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119862

)

times [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= 119899119898

(1199040

119896119862

)int

Γ119861

119862lim 119896119880lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(23)

where 119899119898

is the unit outward normal at cracked surface 1199040119896119862

and 119862lim 119896 is the fourth-order stiffness tensorEquations (22) and (23) form a pair of boundary integral

equations [14 21 24] and can be used to calculate SIFs inanisotropic bi-materialsThemain feature of the BEM formu-lation is that it is a single-domain formulation with the dis-placement integral equation (22) being collocated only on theuncracked boundary and the traction integral equation (23)only on one side of the crack surface For problems withoutcracks one only needs (22) with the integral on the crackedsurface being discarded Equation (22) then reduces to thewell-known displacement integral on the uncracked bound-ary

The internal stresses 120590(119911119896

) are determined using the fol-lowing expression

120590lm (119911119896) + intΓΒ

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ int

Γ119862

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119862

)

times [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= int

Γ119861

119862lim 119896119880lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(24)

For given solutions (related to the body force of gravityrotational forces and the far-field stresses) the boundaryintegral equations (22) and (23) can be discretized andsolved numerically for the unknownboundary displacements(or displacement discontinuities on the cracked surface)and tractions In solving these equations the hypersingularintegral term in (23) can be handled using an accurate and


efficient Gauss quadrature formula which is similar to thetraditional weighted Gauss quadrature but has a differentweight [25]

23 Evaluation of Stress Intensity Factor In fracture mechan-ics analysis especially in the calculation of SIFs one needsto know the asymptotic behavior of the displacements andstresses near the crack tip In our BEM analysis of SIFswe propose to use the extrapolation method of crack-tip displacement We therefore need to know the exactasymptotic behavior of the relative crack displacement (RCD)behind the crack tip The form of the asymptotic expressiondepends on the location of the crack tip In this study twocases are discussed a crack tip in homogeneous material andan interfacial crack tip

(i) Crack Tip in Homogeneous Material Assume that thecrack tip is in material 1 The asymptotic behavior of therelative displacement at a distance 119903 behind the crack tip canbe expressed in terms of the three SIFs as [17]

Δ119906 (119903) = 2radic2119903

120587

Re (1198841)119870 (25)

where 119870 = [119870II 119870I 119870III]119879 is the SIF vector and 119884 is a matrix

with elements related to the anisotropic properties inmaterial1 as defined in (10)

In order to calculate the square-root characteristic of theRCD near the crack tip we constructed the following newcrack tip element with the tip at 120585 = minus1 (as shown inFigure 2(e))

Δ119906119894

=

3

sum

119896=1

119891119896

Δ119906119896

119894

(26)

where subscript 119894 denotes the RCD component and super-script 119896 (1 2 3) denotes the RCDs at nodes 120585 = minus23 0 23

(as shown in Figure 2(b)) respectively The shape functions119891119896

are those introduced by [25]

1198911

=

3radic3

8

radic120585 + 1 [5 minus 8 (120585 + 1) + 3(120585 + 1)2

]

1198912

=

1

4

radic120585 + 1 [minus5 + 18 (120585 + 1) + 9(120585 + 1)2

]

1198913

=

3radic3

8radic5

radic120585 + 1 [1 minus 4 (120585 + 1) + 3(120585 + 1)2

]

(27)

In this case the relation of the RCDs at a distance 119903 behindthe crack tip and the SIFs can be found as [26ndash28]

Δ1199061

= 2radic2119903

120587

(11986711

119870I + 11986712119870II)

Δ1199062

= 2radic2119903

120587

(11986721

119870I + 11986722119870II)

(28)

where

11986711

= Im(

1205832

11987511

minus 1205832

11987512

1205831

minus 1205832

)

11986712

= Im(

11987511

minus 11987512

1205831

minus 1205832

)

11986721

= Im(

1205832

11987521

minus 1205832

11987522

1205831

minus 1205832

)

11986722

= Im(

11987521

minus 11987522

1205831

minus 1205832

)

(29)

Substituting the RCDs into (26) and (28) we may obtaina set of algebraic equations with which the SIFs 119870I and 119870IIcan be solved

(ii) Interfacial Crack Tip In this case the relative crackdisplacements at a distance 119903 behind the interfacial crack tipcan be expressed in terms of the three SIFs as [29]

Δ119906 (119903) = (

3

sum

119895=1

119888119895

119863119876119895

119890minus120587120575119895

119903(12)+120575119895

)119870 (30)

where 119888119895

120575119895

119876119895

and119863 are the relative parameters inmaterials1 and 2 Utilizing (10) we defined the matrix of a materialas

1198841 + 1198842 = 119863 minus 119894119881 (31)

where119863 and 119881 are two real matrices These two matrices areused to define matrix 119875 as

119875 = minus119863minus1

119881 (32)

The characteristic 120573 relative to material is as follows

120573 = radicminus

1

2

tr (1198752) (33)

We use the characteristic 120573 to define oscillation index 120576 as

120576 =

1

2120587

ln1 + 120573

1 minus 120573

=

1

120587

tanhminus1120573

1205751

= 0 1205752

= 120576 1205753

= minus120576

1198761

= 1198752

+ 1205732

119868 1198762

= 119875 (119875 minus 119894120573119868)

1198763

= 119875 (119875 + 119894120573119868)

(34)

where 119868 is a 3 times 3 identity matrixThe relationship between characteristic 120573 and oscillation

index 120576 is used to define constant 119888119895

as

1198881

=

2

radic21205871205732

1198882

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 + 2119894120576) 1205732 cosh (120587120576)

1198883

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 minus 2119894120576) 1205732 cosh (120587120576)

(35)

where 119889 is the characteristic distance along thematerial inter-face to the crack tip


1

2

3

31 2ξ

120585 = minus23 120585 = 0 120585 = 1

119909

119910

(a) Discontinuous quadratic element of Type I

1

2

3

31 2

120585 = minus23 120585 = 0 120585 = 23

119909

119910

120585

(b) Crack surface quadratic element of Type II

1

2

3

31 2

120585 = minus1 120585 = 0 120585 = 23

120585

119909

119910

(c) Discontinuous quadratic element of Type III

1

2

3

31 2119909

119910

120585

120585 = minus1 120585 = 0 120585 = 1

(d) Discontinuous quadratic element of Type IV

Crack tip position1

2

3

31 2

120585 = minus1 120585 = minus23 120585 = 0 120585 = 23

120585

119909

119910

(e) Crack tip quadratic element of Type V

Figure 2 Three-geometric-node quadratic elements used to approximate the uncracked boundary and the crack for two-dimensionalproblems


Comparing (25) with (30) we noticed that while therelative crack displacement behaves as a square root for acrack tip in a homogeneousmedium an interfacial crack-tiprsquosbehavior is 119903(12)+119894120575 a square root featuremultiplied by a weakoscillation

Equation (30) can be written in the following form whichis more convenient for the current numerical applications

Δ119906 (119903) = radic2119903

120587

119872(

119903

119889

)119870 (36)

where119889 is the characteristic length and119872 is amatrix functionexpressed as

119872(119909) =

119863

1205732

(1198752

+ 1205732

119868)

minus ([cos (120576 ln119909) + 2120576 sin (120576 ln119909)] 1198752

+120573 [sin (120576 ln119909) minus 2120576 cos (120576 ln119909)] 119875)

times ((1 + 41205762

) cosh (120587120576))minus1

(37)

Again in order to capture the square root and weakoscillatory behavior we constructed a crack-tip element withthe tip at 120585 = minus1 (as shown in Figure 2(e)) in terms of whichthe relative crack displacement is expressed as

Δ119906 (119903) = 119872(

119903

119889

)

[

[

[

[

[

[

1198911

Δ1199061

1

1198912

Δ1199062

1

1198913

Δ1199063

1

1198911

Δ1199061

2

1198912

Δ1199062

2

1198913

Δ1199063

2

1198911

Δ1199061

3

1198912

Δ1199061

3

1198913

Δ1199063

3

]

]

]

]

]

]

(38)

24 Construction of the Numerical Model Quadratic ele-ments are often used to model problems with curvilineargeometry Three-geometric-node quadratic elements can besubdivided into five types as shown in Figure 2 during anal-ysis For these elements both the geometry and the boundaryquantities are approximated by intrinsic coordinate 120585 andshape function 119891

119896

is obtained from (27) Generally the threegeometric nodes of an element are used as collocation pointsSuch an element is referred to as a continuous quadraticelement of Type IV (as shown in Figure 2(d)) To modelthe corner points or the points where there is a change inthe boundary conditions one-sided discontinuous quadraticelements (commonly referred to as partially discontinuouselements) are used These elements can be one of four typesdepending on which extreme collocation point has beenshifted inside the element to model the geometric or physicalsingularity If the third collocation node is shifted inside thenthe resulting element is referred to as a partially discontin-uous quadratic element of Type I (as shown in Figure 2(a))If the first node is shifted inside then the element is calleda partially discontinuous quadratic element of Type III (asshown in Figure 2(c)) Onemay opt for moving both extremenodes inside which results in discontinuous elements ofTypeII and Type V (to model the cracked surface or the points

where there is in the crack tip positions) as shown in Figures2(b) and 2(e) respectively In the graphical representation ofthese elements we use an ldquotimesrdquo for a geometric node a ldquo ⃝rdquo fora collocation node and a ldquolozrdquo for the crack tip position

3 Verification of the BEM Program

The Greenrsquos functions and the particular crack-tip elementswere incorporated into the boundary integral equations andthe results were programmed using FORTRAN code Inthis section the following numerical examples are presentedto verify the formulation and to show the efficiency andversatility of the present BEM method for problems relatedto fractures in bi-material

31 Horizontal Crack in Material 1 A horizontal crackunder uniform pressure 119875 is shown in Figure 3 The crackwhose length is 2119886 is located distance 119889 from the interfaceThe Poissonrsquos ratios for materials 1 and 2 are ]1 = ]2 =03 and the ratio of the shear modulus 11986621198661 varies Aplane stress condition is assumed In order to calculate theSIFs at crack tip A or B 20 quadratic elements were used todiscretize the crack surfaceThe results are given in Table 1 forvarious values of the shear modulus ratioThey are comparedto the results of Isida andNoguchi [16] who used a body forceintegral equation method and those of Ryoji and Sang-Bong[17] who used a multidomain BEM formulation It can beseen in this table that the results are in agreement

32 Interfacial Crack in an Anisotropic Bi-Material Plate Thefollowing example is used for comparison with results fromthe literature in order to demonstrate the accuracy of theBEM approach for an interfacial crack in an anisotropic bi-material plate The geometry is that of a rectangular plate asshown in Figure 4 For the comparison the crack length is2119886 ℎ = 2119908 119886119908 = 04 and static tensile loading 120590 is appliedto the upper and the lower boundaries of the plate

A plane stress condition is assumed The anisotropicelastic properties for materials 1 and 2 are given inTable 2 The normalized complex stress intensity factors atcrack tips A and B are listed in Table 3 together with thosefrom Sang-Bong et al [18] who used a multidomain BEMformulation and the results from Wunsche et al [8] for afinite body The outer boundary and interfacial crack surfacewere discretized with 20 continuous and 20 discontinuousquadratic elements respectively Table 3 shows that the BEMapproach produces results that agree well with those obtainedby other researchers

4 Numerical Analysis

After its accuracywas checked the BEMprogramwas appliedto more challenging problems involving an anisotropic bi-material to determine the SIFs of crack tips A generalizedplane stress condition was assumed in all the problems Inthe analysis zero traction on the crack surface was assumedHere119865I and119865II are the normalized SIFs for119865I gt 0 the failuremechanism is tensile fracture and for 119865II gt 0 the crack is


Table 1 Comparison of SIFs (horizontal crack)

119866(2)

119866(1)

1198892119886 Proposed approach Isida and Noguchi [16] Diff () Ryoji and Sang-Bong [17] Diff ()119870I (119901radic120587119886) of tip A (or B)

025 005 1476 1468 minus057 1468 minus054025 05 1198 1197 minus009 1197 minus01220 005 0871 0872 014 0869 minus01720 05 0936 0935 minus006 0934 minus016

119870II (119901radic120587119886) of tip A (or B)

025 005 0285 0286 035 0292 250025 05 0071 0071 070 0072 16720 005 minus0088 minus0087 minus138 minus0085 minus40120 05 minus0023 minus0024 250 minus0023 minus354

Table 2 Elastic properties for materials 1 and 2

Material 119864 (MPa) 1198641015840 (MPa) ]1015840 119866

1015840

Material 1 100 50 03 10009Material 2(i) 100 45 03 9525(ii) 100 40 03 9010(iii) 100 30 03 7860(iv) 100 10 03 4630

subjected to an anticlockwise shear force For a 2D problemunder mixed mode loading the failure mechanism may bedefined by the SIFs of crack tips (tensile fracture 119865I119865II gt 1and shear fracture 119865I119865II lt 1 resp)

41 Effect of the Degree of Anisotropy on the Stress IntensityFactor In this chapter we analyze the SIFs for an aniso-tropic bi-material with various anisotropic directions 120595 anddegrees of material anisotropy using the BEM program Thenormalized SIFs defined as 119865I and 119865II are equal to

119865I =119870I1198700

119865II =119870II1198700

(39)

with

1198700

= 120590radic120587119886 (40)

In order to evaluate the influence of bi-material aniso-tropy on the SIFs we considered the following three cases(i) a horizontal crack within material 1 (ii) a horizontalcrack within material 2 and (iii) a crack at the interface ofthe materials The infinite bi-material was subjected to far-field vertical tensile stresses as shown in Figure 5 The crackwhich had a length of 2119886 was located at distances 119889119886 = 1

(material 1)minus1 (material 2) and 119889119886 = 0 (interfacial crack)from the interface respectively Material 1 was transverselyisotropic with the plane of transverse isotropy inclined atangle 1205951 with respect to the 119909-axis Material 1 had fiveindependent elastic constants (1198641 119864

1015840

1 ]1 ]1015840

1 and 1198661) inthe local coordinate system 119864119894 and 119864

1015840

119894 are the Youngrsquosmoduli in the plane of transverse isotropy and in a direction

InterfaceMaterial 1Material 2

2119886

119875

119875

119889

119909

119910

A B

Figure 3 Horizontal crack under uniform pressure in material 1of an infinite bi-material

Interface

119908 119908

ℎ

ℎ

119910

119909

2119886

1198649984002

1198649984001

1198642

1198641

Material 1Material 2

120590

120590

119860 B

Figure 4 Interfacial crack within a finite rectangular plate of a bi-material

normal to it respectively ]119894 and ]1015840119894 are the Poissonrsquos ratiosthat characterize the lateral strain response in the plane oftransverse isotropy to a stress acting parallel and normal toit respectively and 1198661015840119894 is the shear modulus in the planes


Table 3 Comparison of the normalized complex SIFs for finite anisotropic problem (interfacial crack)

Material 21198641015840

21198642

|119870|(120590radic120587119886) of tip A (or B)Sang-Bong et al [18] Wunsche et al [8] Diff () Proposed approach Diff ()

(i) 045 1317 1312 038 13132 029(ii) 040 1337 1333 030 13351 014(iii) 030 1392 1386 043 13922 minus001(iv) 010 1697 1689 047 16968 minus006lowast

|119870| =radic119870

2

I + 1198702

II

Far-field stresses

Isotropic material

Case I


2119886

119909

119910

1198649984001 1198641

1205951

119889

(a) Horizontal crack within material 1

Far-field stresses

Isotropic material


Case II

119909

119910

1198649984001

1205951

119889

2119886

(b) Horizontal crack within material 2

Far-field stresses

Isotropic material

Case III


119909

119910

1205951

1198649984001 1198641

2119886

(c) Interfacial crack within bi-material

Figure 5 Horizontal cracks in an infinite bi-material under far-field stresses

normal to the plane of transverse isotropy For material 21198642119864

1015840

2 = 1 11986421198661015840

2 = 25 and ]2 = ]10158402 = 025 Inthis problem the anisotropic direction (120595)withinmaterial 1varies from 0

∘ to 180∘For all cases three sets of dimensionless elastic constants

are considered They are defined in Table 4 Here 11986411198642which is the ratio of the Youngrsquos modulus of material 1 tothe Youngrsquos modulus of material 2 equals one In order tocalculate the SIFs at the crack tip 20 quadratic elements wereused to discretize the crack surfaceThe numerical results areplotted in Figures 7 to 9 for cases I II and III respectivelyThe results for the isotropic case (1198641119864

1015840

1 = 1 11986411198661015840

1 = 25

Table 4 Sets of dimensionless elastic constants (]1 = 025)

11986411198641015840

1 11986411198661015840

1 ]1015840113 12 1 2 3 25 025lowast

Subscript 1 is applied in material 1

and ]10158401 = 025) are shown as solid lines in the figures forcomparison

42 An Inclined Crack Situated Near the Interfacial CrackThis case treats interaction between the interfacial crack



Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A

Figure 6 A crack situated near the interfacial crack within infinite bi-materials

09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)

Orientation 1205951 (degrees)

11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)

Figure 7 Variation of normalized SIFs of the crack tip in material 1 with the anisotropic directions 1205951 for case I (11986411198661015840

1 = 25 V10158401 =

025 11986411198642 = 1)

and an inclined crack with an angle 120573 The ratio 119889119905

119886 =

025 respectively The same consider an infinite plate of bi-materials having the elastic constants ratio 1198641119864

1015840

1 = 1

11986411198661015840

1 = 04 and ]10158401 = 02499 for material 1 and1198642119864

1015840

2 = 13 11986421198661015840

2 = 04 and ]10158402 = 02499 for material2 The Poisson ratios are ]1 = ]2 = 025 material orienta-tion 1205951 = 1205952 = 0

∘ and the Young modulus ratio 11986411198642 =15 One considers an interfacial crack of length 2119886 and ahorizontal crack situated near the interfacial crack with ahThe ratio 119886119886

ℎ

is equal to 1The longitudinal distance119889119897

119886 = 1and the plate is subjected to a far-field vertical tensile stressesas shown in Figure 6 For each crack surface is meshedby 20 quadratic elements The plane stress conditions weresupposed Calculations were carried out by our BEM code

Figures 10 and 11 represent the variations of normalized SIFs(119865I and 119865II) of interfacial crack and inclined crack accordingto the inclined angle 120573 respectively

43 Discussion An analysis of Figures 7 to 9 reveals thefollowing major trends

(i) For the SIFs in mode I the orientation of the planesof material anisotropy with respect to the horizontalplane has a strong influence on the value of the SIFsfor case I and case II The influence is small for caseIII This means that the effects of mode I on the crackin the homogeneous material are greater than thoseof the interface


0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)

Figure 8 Variation of Normalized SIFs of crack tip in material 2 with the anisotropic directions 1205951 for case II (11986411198661015840

1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)

Figure 9 Variation of normalized SIFs of interfacial crack tip with the anisotropic directions 1205951 for case III (11986411198661015840

1 = 25 ]10158401 =

025 11986411198642 = 1)

(ii) For all cases the variation of the SIFs with the aniso-tropic direction 1205951 is symmetric with respect to1205951 = 90

∘ However this type of symmetry was notobserved for the SIFs (mode II) in case 1 and case 2(Figures 7(b) and 8(b))

(iii) For the interfacial crack (case III) the anisotropicdirection 1205951 has a strong influence on the value ofthe SIFs for mode II (Figure 9(b)) The influence issmall for mode I It should be noted that the effects of1198641119864

1015840

1 on the SIFs of mode II are greater than thoseof mode I

(iv) There is a greater variation in the SIFs with theanisotropic direction 1205951 for bi-materials with a highdegree of anisotropy (1198641119864

1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840

1 = 13) indicates that the max-imum values of the SIFs (mode I) occur when thefar-field stresses are perpendicular to the plane oftransverse isotropy (ie 1205951 = 0

∘ or 180∘) for case1 (crack within material 1) Figure 9(b) (1198641119864

1015840

1 =13) shows that the minimum values of the SIFs(mode II) occur when the anisotropic direction angleis about 90∘ for case 3 (interfacial crack)


07075

08085

09095

1105

11115

12125

13135

Tip ATip B

0 15 30 45 60 75 90 105 120 135 150 165 180Inclined angle 120573 (deg)

Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)

Figure 10 Variation of normalized SIFs of an interfacial crack with an inclined crack angle 120573

01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

Inclined angle 120573 (deg)

minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)

Figure 11 Variation of normalized SIFs of an inclined crack with the inclined angle 120573

Analysis of Figures 10 and 11 reveals that severalmajortrends can be underlined as follows

(vi) Figure 10(a) presents the SIFs (mode I) variation asan inclined angle 120573 of the inclined crack for differentvalues of the interfacial crack It is shown that theincrease in the inclined angle120573 causes a slight increasein SIFs (mode I) until reaching a value at 120573 = 90

∘From these angles the tip D meets at the interfaceof bi-materials and the normalized SIFs start animmediate sharp increasing The rise continued untilit reached 120573 = 105

∘ The maximum of SIFs (modeI) is reached for this last angle In this positionthe distance between tip D and the interfacial crackis minimal which increases the stress interactionbetween the two crack tips For 120573 gt 120

∘ the SIFsstart decreasing and stabilizing when approaching180∘

(vii) The same phenomenon was observed in Figure 10(b)It is shown that the increase in the inclined angle 120573causes the reduction in SIFs (mode II) until reachinga value at 120573 = 100

∘ and starts an immediate sharpincreasingThemaximumof SIFs (mode II) is reachedfor120573 = 110∘ In this position the distance between tipD and the interfacial crack isminimal which increasesthe stress interaction between the two crack tips For120573 gt 120

∘ the SIFs start a slight increasing andstabilizing when approaching 180∘

(viii) Figure 11 illustrates respectively the variations of thenormalized SIFs of tip C and D according to inclinedangle 120573 For Figure 11(a) between 0

∘ and 90∘ one

observes thatmode I of the normalized SIFs decreaseswith 120573Theminimal value is reached at120573 = 90∘ From90∘ the two normalized SIFs increase until 138∘ and

the normalized SIFs at the tip D reaches a maximum


Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1

Figure 12 Variation of the ratio of 119865I119865II of an inclined crack withthe inclined angle 120573

value In the interval of 90∘ at 138∘ the normalizedSIFs of tip D are higher than those of tip C becausethe tip D approaches the tip of the interfacial crackBetween 138∘ and 180∘ tip D moves away from thetip A of interfacial crack

(ix) For Figure 11(b) the normalized SIFs (mode II) ofthe two tips grow in interval of the angles rangingbetween 0

∘ and 35∘ and decrease between 35

∘ and80∘ In the interval of 0∘ at 32∘ the normalized SIFs

of tip C are slightly higher than those of tip D andthe inverse phenomenon is noted for angles rangingbetween 32∘ and 100∘These results permit to note onone hand that the orientation of the inclined crack hasinfluence on the variations of 119865I and a weak influenceon the variations of 119865II and in another hand whenthe angle of this orientation increases the normalizedSIFs of mode I increase and automatically generatethe reduction in mode II SIFs For homogeneous andisotropic materials the two SIFs for inclined crackare nulls at 120573 = 90

∘ The presence of the bi-materialinterface gives 119865I different of zero whatever the valueof 120573 and 119865II = 0 at 120573 = 80

∘

(x) The ratio of the normalized SIFs (119865I119865II) is concernedFigure 12 shows that the fracture of shear mode(119865I119865II lt 1) occurs to the inclined crack angle isbetween 30

∘ and 150∘ (tip C) For tip D the shear

mode (119865I119865II lt 1) occurs to the 120573 is between 50∘ and95∘ and between 105∘ and 120∘ respectively And the

openingmode (119865I119865II gt 1) occurs to the120573 is between95∘ and 105∘ In this position the distance between tip

D and tip A of the interfacial crack is minimal whichincreases the stress interaction between the two cracktips

5 Conclusions

In this study we developed a single-domain BEM for-mulation in which neither the artificial boundary nor thediscretization along the uncracked interface is necessaryFedelinski [11] presented the single-domain BEM formula-tion for homogeneous materials We combined Chenrsquos for-mulation with the Greenrsquos functions of bi-materials derivedby Ke et al [12] to extend it to anisotropic bi-materials Themajor achievements of the research work are summarized inthe following

(i) A decoupling technique was used to determine theSIFs of the mixed mode and the oscillation on theinterfacial crack based on the relative displacementsat the crack tip Five types of three-node quadraticelements were utilized to approximate the crack tipand the outer boundary A crack surface and aninterfacial crack surface were evaluated using theasymptotical relation between the SIFs Since theinterfacial crack has an oscillation singular behaviorwe used a special crack-tip element [26] to capturethis behavior

(ii) Calculation of the SIFs was conducted for severalsituations including cracks along and away from theinterface Numerical results show that the proposedmethod is very accurate even with relatively coarsemesh discretization In addition the use of proposedBEM program is also very fast The calculation timefor each samplewas typically 10 s on a PCwith an IntelCore i7-2600 CPU at 34GHz and 4GB of RAM

(iii) The study of the fracture behavior of a crack is ofpractical importance due to the increasing applicationof anisotropic bi-material Therefore the fracturemechanisms for anisotropic bi-material need to beinvestigated

Acknowledgments

The authors would like to thank the National Science Councilof the Republic of China Taiwan for financially support-ing this research under Contract no NSC-100-2221-E-006-220-MY3 They also would like to thank two anonymousreviewers for their very constructive comments that greatlyimproved the paper

References

[1] M L Williams ldquoThe stresses around a fault or crack in dissim-ilar mediardquo Bulletin of the Seismological Society of America vol49 pp 199ndash204 1959

[2] A H England ldquoA crack between dissimilar mediardquo Journal ofApplied Mechanics vol 32 pp 400ndash402 1965

[3] J R Rice and G C Sih ldquoPlane problems of cracks in dissimilarmediardquo Journal of Applied Mechanics vol 32 pp 418ndash423 1965

[4] J R Rice ldquoElastic fracture mechanics concepts for interfacialcrackrdquo Journal of Applied Mechanics vol 55 pp 98ndash103 1988


[5] D L Clements ldquoA crack between dissimilar anisotropic mediardquoInternational Journal of Engineering Science vol 9 pp 257ndash2651971

[6] J RWillis ldquoFracturemechanics of interfacial cracksrdquo Journal ofthe Mechanics and Physics of Solids vol 19 no 6 pp 353ndash3681971

[7] K C Wu ldquoStress intensity factor and energy release rate forinterfacial cracks between dissimilar anisotropic materialsrdquoJournal of Applied Mechanics vol 57 pp 882ndash886 1990

[8] M Wunsche C Zhang J Sladek V Sladek S Hirose and MKuna ldquoTransient dynamic analysis of interface cracks in layeredanisotropic solids under impact loadingrdquo International Journalof Fracture vol 157 no 1-2 pp 131ndash147 2009

[9] J Y Jhan C S Chen C H Tu and C C Ke ldquoFracture mechan-ics analysis of interfacial crack in anisotropic bi-materialsrdquo KeyEngineering Materials vol 467ndash469 pp 1044ndash1049 2011

[10] J Lei and C Zhang ldquoTime-domain BEM for transient inter-facial crack problems in anisotropic piezoelectric bi-materialsrdquoInternational Journal of Fracture vol 174 pp 163ndash175 2012

[11] P Fedelinski ldquoComputer modelling and analysis of microstruc-tures with fibres and cracksrdquo Journal of Achievements in Materi-als and Manufacturing Engineering vol 54 no 2 pp 242ndash2492012

[12] C C Ke C L Kuo S M Hsu S C Liu and C S ChenldquoTwo-dimensional fracture mechanics analysis using a single-domain boundary element methodrdquoMathematical Problems inEngineering vol 2012 Article ID 581493 26 pages 2012

[13] J Xiao W Ye Y Cai and J Zhang ldquoPrecorrected FFT accel-erated BEM for large-scale transient elastodynamic analysisusing frequency-domain approachrdquo International Journal forNumerical Methods in Engineering vol 90 no 1 pp 116ndash1342012

[14] C S Chen E Pan and B Amadei ldquoFracturemechanics analysisof cracked discs of anisotropic rock using the boundary elementmethodrdquo International Journal of Rock Mechanics and MiningSciences vol 35 no 2 pp 195ndash218 1998

[15] E Pan and B Amadei ldquoBoundary element analysis of fracturemechanics in anisotropic bimaterialsrdquo Engineering Analysiswith Boundary Elements vol 23 no 8 pp 683ndash691 1999

[16] M Isida and H Noguchi ldquoPlane elastostatic problems ofbonded dissimilar materials with an interface crack and arbi-trarily located cracksrdquo Transactions of the Japan Society ofMechanical Engineering vol 49 pp 137ndash146 1983 (Japanese)

[17] Y Ryoji andC Sang-Bong ldquoEfficient boundary element analysisof stress intensity factors for interface cracks in dissimilarmaterialsrdquo Engineering Fracture Mechanics vol 34 no 1 pp179ndash188 1989

[18] C Sang-Bong L R Kab C S Yong and Y Ryoji ldquoDetermina-tion of stress intensity factors and boundary element analysis forinterface cracks in dissimilar anisotropicmaterialsrdquoEngineeringFracture Mechanics vol 43 no 4 pp 603ndash614 1992

[19] S G Lekhnitskii Theory of Elasticity of an Anisotropic ElasticBody Translated by P Fern Edited by Julius J BrandstatterHolden-Day San Francisco Calif USA 1963

[20] Z Suo ldquoSingularities interfaces and cracks in dissimilar aniso-tropicmediardquo Proceedings of the Royal Society London Series Avol 427 no 1873 pp 331ndash358 1990

[21] T C T TingAnisotropic Elasticity vol 45 ofOxford EngineeringScience Series Oxford University Press New York NY USA1996

[22] E Pan and B Amadei ldquoFracture mechanics analysis of cracked2-D anisotropic media with a new formulation of the boundaryelement methodrdquo International Journal of Fracture vol 77 no2 pp 161ndash174 1996

[23] P SolleroMH Aliabadi andD P Rooke ldquoAnisotropic analysisof cracks emanating from circular holes in composite laminatesusing the boundary element methodrdquo Engineering FractureMechanics vol 49 no 2 pp 213ndash224 1994

[24] S L Crouch and A M Starfield Boundary Element Methods inSolid Mechanics George Allen amp Unwin London UK 1983

[25] G Tsamasphyros and G Dimou ldquoGauss quadrature rulesfor finite part integralsrdquo International Journal for NumericalMethods in Engineering vol 30 no 1 pp 13ndash26 1990

[26] E Pan ldquoA general boundary element analysis of 2-D linearelastic fracture mechanicsrdquo International Journal of Fracturevol 88 no 1 pp 41ndash59 1997

[27] G C Sih P C Paris and G R Irwin ldquoOn cracks in rectilinearlyanisotropic bodiesrdquo International Journal of Fracture vol 3 pp189ndash203 1965

[28] P Sollero and M H Aliabadi ldquoFracture mechanics analysis ofanisotropic plates by the boundary element methodrdquo Interna-tional Journal of Fracture vol 64 no 4 pp 269ndash284 1993

[29] H Gao M Abbudi and D M Barnett ldquoInterfacial crack-tipfield in anisotropic elastic solidsrdquo Journal of the Mechanics andPhysics of Solids vol 40 no 2 pp 393ndash416 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In this paper we develop a technique for single-domainBEM formulation in which neither the artificial boundarynor the discretization along the uncracked interface is nec-essary Chen et al [14] presented this single-domain BEMformulation for homogeneous materials We combined itwith Greenrsquos functions of bi-materials [15] to extend it toanisotropic bi-materials The BEM formulation is such thatthe displacement integral equation is applied only to the outerboundary and the traction integral equation is applied only toone side of the cracked surface A decoupling technique canbe used to determine mixed mode SIFs based on the relativedisplacement at the crack tip Numerical cases for mixedmode SIFs for anisotropic bi-materials with different cracklocations and degrees of material anisotropy are presentedThe numerical results obtained using the BEM formulationare verified by several previous studies [8 16ndash18]

2 Methodology

21 Basic Equations for Anisotropic Elasticity For linearelastic homogeneous and anisotropic material the stressand displacement fields can be formulated in terms of twoanalytical functions 120601

119896

(119911119896

) of the complex variables 119911119896

=

119909 + 120583119896

119910 (119896 = 1 2) where 120583119896

is the root of the followingcharacteristic equation [19]

11988611

1205834

minus 211988616

1205833

+ (211988612

+ 11988666

) 1205832

minus 211988626

120583 + 11988622

= 0 (1)

where the coefficient 119886119894119895

is the compliance component cal-culated using the 119909-119910 coordinate system The detailed rela-tionships of these components with material elasticity can befound in Chen et al [14] If the root 120583

119895

of (1) is assumed to bedistinct the general expressions for the stress and displace-ments are

120590119909

= 2Re [12058321

1206011015840

1

(1199111

) + 1205832

2

1206011015840

2

(1199112

)]

120590119910

= 2Re [12060110158401

(1199111

) + 1206011015840

2

(1199112

)]

120591119909119910

= minus2Re [1205831

1206011015840

1

(1199111

) + 1205832

1206011015840

2

(1199112

)]

119906 = 2Re [11986011

1206011

(1199111

) + 11986012

1206012

(1199112

)]

120592 = 2Re [11986021

1206011

(1199111

) + 11986022

1206012

(1199112

)]

(2)

where the elements of the complex matrices 119860 are

1198601119895

= 11988611

1205832

119895

+ 11988622

minus 11988616

120583119895

1198602119895

= 11988612

120583119895

+

11988622

120583119895

minus 11988626

(119895 = 1 2)

(3)

The traction components in the 119909 and 119910 directions are

119879119909

= 2Re [1205831

1206011

(1199111

) + 1205832

1206012

(1199112

)]

119879119910

= minus2Re [1206011

(1199111

) + 1206012

(1199112

)]

(4)

Here the complex analytical functions 120601119894

(119911119894

) can in generalexpress (2) and (4) as follows [19ndash21]

119906119894

= 2Re [1198601119894

1206011

(1199111

) + 1198602119894

1206012

(1199112

)]

119879119894

= minus2Re [1198611119894

1206011

(1199111

) + 1198612119894

1206012

(1199112

)]

1205902119894

= 2Re [1198611119894

1206011015840

1

(1199111

) + 1198612119894

1206011015840

2

(1199112

)]

1205901119894

= minus2Re [1198611119894

1206011015840

1

(1199111

) + 1198612119894

1206011015840

2

(1199112

)] (119894 = 1 2)

(5)

where 119911119895

= 119909 + 120583119895

119910 Re denotes the real part of a complexvariable or function a prime denotes the derivative the com-plex number 120583

119895

(119895 = 1 2) the elements of complex matrices119860 are defined in (3) and the elements of complex matrices 119861can be defined as

119861119894119895

= [

minus1205831

minus1205832

1 1] (6)

Assume that the medium is composed of two joined dis-similar anisotropic and elastic half-planes Let the interface bealong the 119909-axis and let the upper (119910 gt 0) and lower (119910 lt 0)half-planes be occupied by materials 1 and 2 respectively(as shown in Figure 1)

Considering a concentrated force acting at the sourcepoint (1199090 1199100) in material 2(1199100 lt 0) we express the complexvector function as [20]

120601 (119911) =

1206011(119911) 119911 isin (1)

1206012(119911) + 120601

0

2 (119911) 119911 isin (2)

(7)

where the vector function is

120601 (119911) = [1206011

(119911) 1206012

(119911)]119879

(8)

where the argument has the generic form 119911 = 119909 + 120583119910In (7) 12060102 is a singular solution corresponding to a point

force acting at the point (1199090 1199100) on an anisotropic infiniteplane with the elastic properties of material 2 This singularsolution can be expressed as [20 21]

1206010

119896

(119911119896

) = minus

1

2120587

[1198671198961

1198751

ln (119911119896

minus 119904119896

) + 1198671198962

1198752

ln (119911119896

minus 119904119896

)]

(9)

where 119904119896

= 1199090

+ 120583119896

1199100 119875119896

is the magnitude of the point forcein the 119896-direction and

119867 = 119860minus1

(119884minus1

+ 119884

minus1

)

minus1

119884 = 119894119860119861minus1

(10)

where 119894 = radicminus1 the overbar indicates the complex conju-gate and superscript minus1 indicates the matrix inverse Twounknown vector functions 1206011(119911) and 1206012(119911) are solved in(7)The former is analytic inmaterial 1 and the latter is ana-lytic inmaterial 2These expressions can be found by requir-ing the continuity of the resultant traction and displacement


Interface

0

Crack tip


Material 1

Material 2


119910

119909Γ119862+

Γ119862minus 1198649984002

1198649984001

1198642

1198641119899119898

Γ119861

1205951

minus1205952



1206011(119885) = 119861

minus1

1 (1198841 + 1198842)minus1

(1198842 + 1198842) 11986121206010

2 (119911)

119911 isin 1

1206012(119885) = 119861

minus1

2 (1198841 + 1198842)minus1

(1198842 minus 1198841) 11986121206010

2 (119911)

119911 isin 2

(11)


120601 (119911) =

119861minus1

1 (1198841 + 1198842)minus1

times (1198842 + 1198842) 11986121206010

2 (119911) 119911 isin 1119861minus1

2 (1198841 + 1198842)minus1

(1198842 minus 1198841)

times11986121206010

2 (119911) + 1206010

2 (119911) 119911 isin 2

(12)



120601 (119911) =

119861minus1

1 (1198842 + 1198841)minus1

times (1198841 minus 1198842) 11986111206010

2 (119911) + 1206010

1 (119911) 119911 isin 1

119861minus1

2 (1198842 + 1198841)minus1

times (1198841 + 1198841) 11986111206010

1 (119911) 119911 isin 2(13)



119896119897


lowast

119896119897



119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198601119897119895

[ ln (1199111119895

minus 1199041119895

)1198671119895119896

+

2

sum

119894=1

11988211

119895119894

ln (1199111119895

minus 1199041119894

)119867

1119894119896

]

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198611119897119895

[

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199041119895

1198671119895119896

+

2

sum

119894=1

11988211

119895119894

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199041119894

119867

1119894119896

]

(14)


11988211

= 119861minus1

1 (1198841 + 1198842)minus1

(1198841 minus 1198842) 1198611 (15)


119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198602119897119895

[

2

sum

119894=1

11988212

119895119894

ln (1199112119895

minus 1199041119894

)1198671119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198612119897119895

[

2

sum

119894=1

11988212

119895119894

1205832119895

119899119909

minus 119899119910

1199112119895

minus 1199041119894

119867

1119894119896

]

(16)

with

11988212

= 119861minus1

2 (1198842 + 1198841)minus1

(1198841 minus 1198841) 1198611 (17)


119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198602119897119895

[ ln (1199112119895

minus 1199042119895

)1198672119895119896

+

2

sum

119894=1

11988222

119895119894

ln (1199112119895

minus 1199042119894

)119867

2119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198612119897119895

[

1205832119895

119899119909

minus 119899119910

1199112119895

minus 1199042119895

1198672119895119896

+

2

sum

119894=1

11988222

119895119894

1205832119895

119899119909

minus 119899119910

1199112119895

minus 1199042119894

119867

2119894119896

]

]

(18)


11988222

= 119861minus1

2 (1198842 + 1198841)minus1

(1198842 minus 1198841) 1198612 (19)



119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198601119897119895

[

2

sum

119894=1

11988221

119895119894

ln (1199111119895

minus 1199042119894

)1198672119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198611119897119895

[

2

sum

119894=1

11988221

119895119894

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199042119894

119867

2119894119896

]

(20)

with

11988221

= 119861minus1

1 (1198841 + 1198842)minus1

(1198842 minus 1198842) 1198612 (21)



For a point 1199040119896119861


119896119861

isin Γ119861


119887119894119895

(1199040

119896119861

) 119906119895

(1199040

119896119861

)

+ int

Γ119861

119879lowast

119894119895

(119911119896119861

1199040

119896119861

) 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ int

Γ119862

119879lowast

119894119895

(119911119896119862

1199040

119896119861

) [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= int

Γ119861

119880lowast

119894119895

(119911119896119861

1199040

119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(22)

where 119894 119895 119896 = 1 2 119879lowast

119894119895

and 119880lowast119894119895


119895

and 119905119895


119896

and 1199040

119896119861


119862


119862+

119887119894119895


119896119861

and are equal to 120575119894119895






isin Γ119862+

only)

05119905119895

(1199040

119896119862

) + 119899119898

(1199040

119896119862

) int

ΓΒ

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119861

)

times 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ 119899119898

(1199040

119896119862

)int

Γ119862

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119862

)

times [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= 119899119898

(1199040

119896119862

)int

Γ119861

119862lim 119896119880lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(23)

where 119899119898






120590lm (119911119896) + intΓΒ

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ int

Γ119862

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119862

)

times [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= int

Γ119861

119862lim 119896119880lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(24)






Δ119906 (119903) = 2radic2119903

120587

Re (1198841)119870 (25)




Δ119906119894

=

3

sum

119896=1

119891119896

Δ119906119896

119894

(26)




1198911

=

3radic3

8

radic120585 + 1 [5 minus 8 (120585 + 1) + 3(120585 + 1)2

]

1198912

=

1

4

radic120585 + 1 [minus5 + 18 (120585 + 1) + 9(120585 + 1)2

]

1198913

=

3radic3

8radic5

radic120585 + 1 [1 minus 4 (120585 + 1) + 3(120585 + 1)2

]

(27)


Δ1199061

= 2radic2119903

120587

(11986711

119870I + 11986712119870II)

Δ1199062

= 2radic2119903

120587

(11986721

119870I + 11986722119870II)

(28)

where

11986711

= Im(

1205832

11987511

minus 1205832

11987512

1205831

minus 1205832

)

11986712

= Im(

11987511

minus 11987512

1205831

minus 1205832

)

11986721

= Im(

1205832

11987521

minus 1205832

11987522

1205831

minus 1205832

)

11986722

= Im(

11987521

minus 11987522

1205831

minus 1205832

)

(29)



Δ119906 (119903) = (

3

sum

119895=1

119888119895

119863119876119895

119890minus120587120575119895

119903(12)+120575119895

)119870 (30)

where 119888119895

120575119895

119876119895


1198841 + 1198842 = 119863 minus 119894119881 (31)


119875 = minus119863minus1

119881 (32)


120573 = radicminus

1

2

tr (1198752) (33)


120576 =

1

2120587

ln1 + 120573

1 minus 120573

=

1

120587

tanhminus1120573

1205751

= 0 1205752

= 120576 1205753

= minus120576

1198761

= 1198752

+ 1205732

119868 1198762

= 119875 (119875 minus 119894120573119868)

1198763

= 119875 (119875 + 119894120573119868)

(34)



as

1198881

=

2

radic21205871205732

1198882

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 + 2119894120576) 1205732 cosh (120587120576)

1198883

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 minus 2119894120576) 1205732 cosh (120587120576)

(35)



1

2

3

31 2ξ

120585 = minus23 120585 = 0 120585 = 1

119909

119910


1

2

3

31 2

120585 = minus23 120585 = 0 120585 = 23

119909

119910

120585


1

2

3

31 2

120585 = minus1 120585 = 0 120585 = 23

120585

119909

119910


1

2

3

31 2119909

119910

120585

120585 = minus1 120585 = 0 120585 = 1


Crack tip position1

2

3

31 2

120585 = minus1 120585 = minus23 120585 = 0 120585 = 23

120585

119909

119910






Δ119906 (119903) = radic2119903

120587

119872(

119903

119889

)119870 (36)


119872(119909) =

119863

1205732

(1198752

+ 1205732

119868)



times ((1 + 41205762

) cosh (120587120576))minus1

(37)


Δ119906 (119903) = 119872(

119903

119889

)

[

[

[

[

[

[

1198911

Δ1199061

1

1198912

Δ1199062

1

1198913

Δ1199063

1

1198911

Δ1199061

2

1198912

Δ1199062

2

1198913

Δ1199063

2

1198911

Δ1199061

3

1198912

Δ1199061

3

1198913

Δ1199063

3

]

]

]

]

]

]

(38)


119896












119866(2)

119866(1)






Material 119864 (MPa) 1198641015840 (MPa) ]1015840 119866

1015840




119865I =119870I1198700

119865II =119870II1198700

(39)

with

1198700

= 120590radic120587119886 (40)



1015840

1 ]1 ]1015840


1015840



2119886

119875

119875

119889

119909

119910

A B


Interface

119908 119908

ℎ

ℎ

119910

119909

2119886

1198649984002

1198649984001

1198642

1198641


120590

120590

119860 B





Material 21198641015840

21198642



|119870| =radic119870

2

I + 1198702

II

Far-field stresses

Isotropic material

Case I


2119886

119909

119910

1198649984001 1198641

1205951

119889


Far-field stresses

Isotropic material


Case II

119909

119910

1198649984001

1205951

119889

2119886


Far-field stresses

Isotropic material

Case III


119909

119910

1205951

1198649984001 1198641

2119886




1015840

2 = 1 11986421198661015840




1015840

1 = 1 11986411198661015840

1 = 25


11986411198641015840

1 11986411198661015840

1 ]1015840113 12 1 2 3 25 025lowast






Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A


09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 V10158401 =

025 11986411198642 = 1)


119886 =


1015840

1 = 1

11986411198661015840


1015840

2 = 13 11986421198661015840



ℎ







0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Interface

0

Crack tip


Material 1

Material 2


119910

119909Γ119862+

Γ119862minus 1198649984002

1198649984001

1198642

1198641119899119898

Γ119861

1205951

minus1205952



1206011(119885) = 119861

minus1

1 (1198841 + 1198842)minus1

(1198842 + 1198842) 11986121206010

2 (119911)

119911 isin 1

1206012(119885) = 119861

minus1

2 (1198841 + 1198842)minus1

(1198842 minus 1198841) 11986121206010

2 (119911)

119911 isin 2

(11)


120601 (119911) =

119861minus1

1 (1198841 + 1198842)minus1

times (1198842 + 1198842) 11986121206010

2 (119911) 119911 isin 1119861minus1

2 (1198841 + 1198842)minus1

(1198842 minus 1198841)

times11986121206010

2 (119911) + 1206010

2 (119911) 119911 isin 2

(12)



120601 (119911) =

119861minus1

1 (1198842 + 1198841)minus1

times (1198841 minus 1198842) 11986111206010

2 (119911) + 1206010

1 (119911) 119911 isin 1

119861minus1

2 (1198842 + 1198841)minus1

times (1198841 + 1198841) 11986111206010

1 (119911) 119911 isin 2(13)



119896119897


lowast

119896119897



119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198601119897119895

[ ln (1199111119895

minus 1199041119895

)1198671119895119896

+

2

sum

119894=1

11988211

119895119894

ln (1199111119895

minus 1199041119894

)119867

1119894119896

]

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198611119897119895

[

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199041119895

1198671119895119896

+

2

sum

119894=1

11988211

119895119894

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199041119894

119867

1119894119896

]

(14)


11988211

= 119861minus1

1 (1198841 + 1198842)minus1

(1198841 minus 1198842) 1198611 (15)


119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198602119897119895

[

2

sum

119894=1

11988212

119895119894

ln (1199112119895

minus 1199041119894

)1198671119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198612119897119895

[

2

sum

119894=1

11988212

119895119894

1205832119895

119899119909

minus 119899119910

1199112119895

minus 1199041119894

119867

1119894119896

]

(16)

with

11988212

= 119861minus1

2 (1198842 + 1198841)minus1

(1198841 minus 1198841) 1198611 (17)


119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198602119897119895

[ ln (1199112119895

minus 1199042119895

)1198672119895119896

+

2

sum

119894=1

11988222

119895119894

ln (1199112119895

minus 1199042119894

)119867

2119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198612119897119895

[

1205832119895

119899119909

minus 119899119910

1199112119895

minus 1199042119895

1198672119895119896

+

2

sum

119894=1

11988222

119895119894

1205832119895

119899119909

minus 119899119910

1199112119895

minus 1199042119894

119867

2119894119896

]

]

(18)


11988222

= 119861minus1

2 (1198842 + 1198841)minus1

(1198842 minus 1198841) 1198612 (19)



119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198601119897119895

[

2

sum

119894=1

11988221

119895119894

ln (1199111119895

minus 1199042119894

)1198672119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198611119897119895

[

2

sum

119894=1

11988221

119895119894

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199042119894

119867

2119894119896

]

(20)

with

11988221

= 119861minus1

1 (1198841 + 1198842)minus1

(1198842 minus 1198842) 1198612 (21)



For a point 1199040119896119861


119896119861

isin Γ119861


119887119894119895

(1199040

119896119861

) 119906119895

(1199040

119896119861

)

+ int

Γ119861

119879lowast

119894119895

(119911119896119861

1199040

119896119861

) 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ int

Γ119862

119879lowast

119894119895

(119911119896119862

1199040

119896119861

) [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= int

Γ119861

119880lowast

119894119895

(119911119896119861

1199040

119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(22)

where 119894 119895 119896 = 1 2 119879lowast

119894119895

and 119880lowast119894119895


119895

and 119905119895


119896

and 1199040

119896119861


119862


119862+

119887119894119895


119896119861

and are equal to 120575119894119895






isin Γ119862+

only)

05119905119895

(1199040

119896119862

) + 119899119898

(1199040

119896119862

) int

ΓΒ

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119861

)

times 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ 119899119898

(1199040

119896119862

)int

Γ119862

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119862

)

times [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= 119899119898

(1199040

119896119862

)int

Γ119861

119862lim 119896119880lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(23)

where 119899119898






120590lm (119911119896) + intΓΒ

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ int

Γ119862

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119862

)

times [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= int

Γ119861

119862lim 119896119880lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(24)






Δ119906 (119903) = 2radic2119903

120587

Re (1198841)119870 (25)




Δ119906119894

=

3

sum

119896=1

119891119896

Δ119906119896

119894

(26)




1198911

=

3radic3

8

radic120585 + 1 [5 minus 8 (120585 + 1) + 3(120585 + 1)2

]

1198912

=

1

4

radic120585 + 1 [minus5 + 18 (120585 + 1) + 9(120585 + 1)2

]

1198913

=

3radic3

8radic5

radic120585 + 1 [1 minus 4 (120585 + 1) + 3(120585 + 1)2

]

(27)


Δ1199061

= 2radic2119903

120587

(11986711

119870I + 11986712119870II)

Δ1199062

= 2radic2119903

120587

(11986721

119870I + 11986722119870II)

(28)

where

11986711

= Im(

1205832

11987511

minus 1205832

11987512

1205831

minus 1205832

)

11986712

= Im(

11987511

minus 11987512

1205831

minus 1205832

)

11986721

= Im(

1205832

11987521

minus 1205832

11987522

1205831

minus 1205832

)

11986722

= Im(

11987521

minus 11987522

1205831

minus 1205832

)

(29)



Δ119906 (119903) = (

3

sum

119895=1

119888119895

119863119876119895

119890minus120587120575119895

119903(12)+120575119895

)119870 (30)

where 119888119895

120575119895

119876119895


1198841 + 1198842 = 119863 minus 119894119881 (31)


119875 = minus119863minus1

119881 (32)


120573 = radicminus

1

2

tr (1198752) (33)


120576 =

1

2120587

ln1 + 120573

1 minus 120573

=

1

120587

tanhminus1120573

1205751

= 0 1205752

= 120576 1205753

= minus120576

1198761

= 1198752

+ 1205732

119868 1198762

= 119875 (119875 minus 119894120573119868)

1198763

= 119875 (119875 + 119894120573119868)

(34)



as

1198881

=

2

radic21205871205732

1198882

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 + 2119894120576) 1205732 cosh (120587120576)

1198883

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 minus 2119894120576) 1205732 cosh (120587120576)

(35)



1

2

3

31 2ξ

120585 = minus23 120585 = 0 120585 = 1

119909

119910


1

2

3

31 2

120585 = minus23 120585 = 0 120585 = 23

119909

119910

120585


1

2

3

31 2

120585 = minus1 120585 = 0 120585 = 23

120585

119909

119910


1

2

3

31 2119909

119910

120585

120585 = minus1 120585 = 0 120585 = 1


Crack tip position1

2

3

31 2

120585 = minus1 120585 = minus23 120585 = 0 120585 = 23

120585

119909

119910






Δ119906 (119903) = radic2119903

120587

119872(

119903

119889

)119870 (36)


119872(119909) =

119863

1205732

(1198752

+ 1205732

119868)



times ((1 + 41205762

) cosh (120587120576))minus1

(37)


Δ119906 (119903) = 119872(

119903

119889

)

[

[

[

[

[

[

1198911

Δ1199061

1

1198912

Δ1199062

1

1198913

Δ1199063

1

1198911

Δ1199061

2

1198912

Δ1199062

2

1198913

Δ1199063

2

1198911

Δ1199061

3

1198912

Δ1199061

3

1198913

Δ1199063

3

]

]

]

]

]

]

(38)


119896












119866(2)

119866(1)






Material 119864 (MPa) 1198641015840 (MPa) ]1015840 119866

1015840




119865I =119870I1198700

119865II =119870II1198700

(39)

with

1198700

= 120590radic120587119886 (40)



1015840

1 ]1 ]1015840


1015840



2119886

119875

119875

119889

119909

119910

A B


Interface

119908 119908

ℎ

ℎ

119910

119909

2119886

1198649984002

1198649984001

1198642

1198641


120590

120590

119860 B





Material 21198641015840

21198642



|119870| =radic119870

2

I + 1198702

II

Far-field stresses

Isotropic material

Case I


2119886

119909

119910

1198649984001 1198641

1205951

119889


Far-field stresses

Isotropic material


Case II

119909

119910

1198649984001

1205951

119889

2119886


Far-field stresses

Isotropic material

Case III


119909

119910

1205951

1198649984001 1198641

2119886




1015840

2 = 1 11986421198661015840




1015840

1 = 1 11986411198661015840

1 = 25


11986411198641015840

1 11986411198661015840

1 ]1015840113 12 1 2 3 25 025lowast






Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A


09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 V10158401 =

025 11986411198642 = 1)


119886 =


1015840

1 = 1

11986411198661015840


1015840

2 = 13 11986421198661015840



ℎ







0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880lowast

119896119897

=

minus1

120587

Re

2

sum

119895=1

1198601119897119895

[

2

sum

119894=1

11988221

119895119894

ln (1199111119895

minus 1199042119894

)1198672119894119896

]

119879lowast

119896119897

=

1

120587

Re

2

sum

119895=1

1198611119897119895

[

2

sum

119894=1

11988221

119895119894

1205831119895

119899119909

minus 119899119910

1199111119895

minus 1199042119894

119867

2119894119896

]

(20)

with

11988221

= 119861minus1

1 (1198841 + 1198842)minus1

(1198842 minus 1198842) 1198612 (21)



For a point 1199040119896119861


119896119861

isin Γ119861


119887119894119895

(1199040

119896119861

) 119906119895

(1199040

119896119861

)

+ int

Γ119861

119879lowast

119894119895

(119911119896119861

1199040

119896119861

) 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ int

Γ119862

119879lowast

119894119895

(119911119896119862

1199040

119896119861

) [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= int

Γ119861

119880lowast

119894119895

(119911119896119861

1199040

119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(22)

where 119894 119895 119896 = 1 2 119879lowast

119894119895

and 119880lowast119894119895


119895

and 119905119895


119896

and 1199040

119896119861


119862


119862+

119887119894119895


119896119861

and are equal to 120575119894119895






isin Γ119862+

only)

05119905119895

(1199040

119896119862

) + 119899119898

(1199040

119896119862

) int

ΓΒ

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119861

)

times 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ 119899119898

(1199040

119896119862

)int

Γ119862

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119862

)

times [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= 119899119898

(1199040

119896119862

)int

Γ119861

119862lim 119896119880lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(23)

where 119899119898






120590lm (119911119896) + intΓΒ

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119906119895

(119911119896119861

) 119889Γ (119911119896119861

)

+ int

Γ119862

119862lim 119896119879lowast

119894119895119896

(1199040

119896119862

119911119896119862

)

times [119906119895

(119911119896119862+

) minus 119906119895

(119911119896119862minus

)] 119889Γ (119911119896119862

)

= int

Γ119861

119862lim 119896119880lowast

119894119895119896

(1199040

119896119862

119911119896119861

) 119905119895

(119911119896119861

) 119889Γ (119911119896119861

)

(24)






Δ119906 (119903) = 2radic2119903

120587

Re (1198841)119870 (25)




Δ119906119894

=

3

sum

119896=1

119891119896

Δ119906119896

119894

(26)




1198911

=

3radic3

8

radic120585 + 1 [5 minus 8 (120585 + 1) + 3(120585 + 1)2

]

1198912

=

1

4

radic120585 + 1 [minus5 + 18 (120585 + 1) + 9(120585 + 1)2

]

1198913

=

3radic3

8radic5

radic120585 + 1 [1 minus 4 (120585 + 1) + 3(120585 + 1)2

]

(27)


Δ1199061

= 2radic2119903

120587

(11986711

119870I + 11986712119870II)

Δ1199062

= 2radic2119903

120587

(11986721

119870I + 11986722119870II)

(28)

where

11986711

= Im(

1205832

11987511

minus 1205832

11987512

1205831

minus 1205832

)

11986712

= Im(

11987511

minus 11987512

1205831

minus 1205832

)

11986721

= Im(

1205832

11987521

minus 1205832

11987522

1205831

minus 1205832

)

11986722

= Im(

11987521

minus 11987522

1205831

minus 1205832

)

(29)



Δ119906 (119903) = (

3

sum

119895=1

119888119895

119863119876119895

119890minus120587120575119895

119903(12)+120575119895

)119870 (30)

where 119888119895

120575119895

119876119895


1198841 + 1198842 = 119863 minus 119894119881 (31)


119875 = minus119863minus1

119881 (32)


120573 = radicminus

1

2

tr (1198752) (33)


120576 =

1

2120587

ln1 + 120573

1 minus 120573

=

1

120587

tanhminus1120573

1205751

= 0 1205752

= 120576 1205753

= minus120576

1198761

= 1198752

+ 1205732

119868 1198762

= 119875 (119875 minus 119894120573119868)

1198763

= 119875 (119875 + 119894120573119868)

(34)



as

1198881

=

2

radic21205871205732

1198882

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 + 2119894120576) 1205732 cosh (120587120576)

1198883

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 minus 2119894120576) 1205732 cosh (120587120576)

(35)



1

2

3

31 2ξ

120585 = minus23 120585 = 0 120585 = 1

119909

119910


1

2

3

31 2

120585 = minus23 120585 = 0 120585 = 23

119909

119910

120585


1

2

3

31 2

120585 = minus1 120585 = 0 120585 = 23

120585

119909

119910


1

2

3

31 2119909

119910

120585

120585 = minus1 120585 = 0 120585 = 1


Crack tip position1

2

3

31 2

120585 = minus1 120585 = minus23 120585 = 0 120585 = 23

120585

119909

119910






Δ119906 (119903) = radic2119903

120587

119872(

119903

119889

)119870 (36)


119872(119909) =

119863

1205732

(1198752

+ 1205732

119868)



times ((1 + 41205762

) cosh (120587120576))minus1

(37)


Δ119906 (119903) = 119872(

119903

119889

)

[

[

[

[

[

[

1198911

Δ1199061

1

1198912

Δ1199062

1

1198913

Δ1199063

1

1198911

Δ1199061

2

1198912

Δ1199062

2

1198913

Δ1199063

2

1198911

Δ1199061

3

1198912

Δ1199061

3

1198913

Δ1199063

3

]

]

]

]

]

]

(38)


119896












119866(2)

119866(1)






Material 119864 (MPa) 1198641015840 (MPa) ]1015840 119866

1015840




119865I =119870I1198700

119865II =119870II1198700

(39)

with

1198700

= 120590radic120587119886 (40)



1015840

1 ]1 ]1015840


1015840



2119886

119875

119875

119889

119909

119910

A B


Interface

119908 119908

ℎ

ℎ

119910

119909

2119886

1198649984002

1198649984001

1198642

1198641


120590

120590

119860 B





Material 21198641015840

21198642



|119870| =radic119870

2

I + 1198702

II

Far-field stresses

Isotropic material

Case I


2119886

119909

119910

1198649984001 1198641

1205951

119889


Far-field stresses

Isotropic material


Case II

119909

119910

1198649984001

1205951

119889

2119886


Far-field stresses

Isotropic material

Case III


119909

119910

1205951

1198649984001 1198641

2119886




1015840

2 = 1 11986421198661015840




1015840

1 = 1 11986411198661015840

1 = 25


11986411198641015840

1 11986411198661015840

1 ]1015840113 12 1 2 3 25 025lowast






Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A


09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 V10158401 =

025 11986411198642 = 1)


119886 =


1015840

1 = 1

11986411198661015840


1015840

2 = 13 11986421198661015840



ℎ







0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Δ119906 (119903) = 2radic2119903

120587

Re (1198841)119870 (25)




Δ119906119894

=

3

sum

119896=1

119891119896

Δ119906119896

119894

(26)




1198911

=

3radic3

8

radic120585 + 1 [5 minus 8 (120585 + 1) + 3(120585 + 1)2

]

1198912

=

1

4

radic120585 + 1 [minus5 + 18 (120585 + 1) + 9(120585 + 1)2

]

1198913

=

3radic3

8radic5

radic120585 + 1 [1 minus 4 (120585 + 1) + 3(120585 + 1)2

]

(27)


Δ1199061

= 2radic2119903

120587

(11986711

119870I + 11986712119870II)

Δ1199062

= 2radic2119903

120587

(11986721

119870I + 11986722119870II)

(28)

where

11986711

= Im(

1205832

11987511

minus 1205832

11987512

1205831

minus 1205832

)

11986712

= Im(

11987511

minus 11987512

1205831

minus 1205832

)

11986721

= Im(

1205832

11987521

minus 1205832

11987522

1205831

minus 1205832

)

11986722

= Im(

11987521

minus 11987522

1205831

minus 1205832

)

(29)



Δ119906 (119903) = (

3

sum

119895=1

119888119895

119863119876119895

119890minus120587120575119895

119903(12)+120575119895

)119870 (30)

where 119888119895

120575119895

119876119895


1198841 + 1198842 = 119863 minus 119894119881 (31)


119875 = minus119863minus1

119881 (32)


120573 = radicminus

1

2

tr (1198752) (33)


120576 =

1

2120587

ln1 + 120573

1 minus 120573

=

1

120587

tanhminus1120573

1205751

= 0 1205752

= 120576 1205753

= minus120576

1198761

= 1198752

+ 1205732

119868 1198762

= 119875 (119875 minus 119894120573119868)

1198763

= 119875 (119875 + 119894120573119868)

(34)



as

1198881

=

2

radic21205871205732

1198882

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 + 2119894120576) 1205732 cosh (120587120576)

1198883

=

minus119890minus120587120576

119889119894120576

radic2120587 (1 minus 2119894120576) 1205732 cosh (120587120576)

(35)



1

2

3

31 2ξ

120585 = minus23 120585 = 0 120585 = 1

119909

119910


1

2

3

31 2

120585 = minus23 120585 = 0 120585 = 23

119909

119910

120585


1

2

3

31 2

120585 = minus1 120585 = 0 120585 = 23

120585

119909

119910


1

2

3

31 2119909

119910

120585

120585 = minus1 120585 = 0 120585 = 1


Crack tip position1

2

3

31 2

120585 = minus1 120585 = minus23 120585 = 0 120585 = 23

120585

119909

119910






Δ119906 (119903) = radic2119903

120587

119872(

119903

119889

)119870 (36)


119872(119909) =

119863

1205732

(1198752

+ 1205732

119868)



times ((1 + 41205762

) cosh (120587120576))minus1

(37)


Δ119906 (119903) = 119872(

119903

119889

)

[

[

[

[

[

[

1198911

Δ1199061

1

1198912

Δ1199062

1

1198913

Δ1199063

1

1198911

Δ1199061

2

1198912

Δ1199062

2

1198913

Δ1199063

2

1198911

Δ1199061

3

1198912

Δ1199061

3

1198913

Δ1199063

3

]

]

]

]

]

]

(38)


119896












119866(2)

119866(1)






Material 119864 (MPa) 1198641015840 (MPa) ]1015840 119866

1015840




119865I =119870I1198700

119865II =119870II1198700

(39)

with

1198700

= 120590radic120587119886 (40)



1015840

1 ]1 ]1015840


1015840



2119886

119875

119875

119889

119909

119910

A B


Interface

119908 119908

ℎ

ℎ

119910

119909

2119886

1198649984002

1198649984001

1198642

1198641


120590

120590

119860 B





Material 21198641015840

21198642



|119870| =radic119870

2

I + 1198702

II

Far-field stresses

Isotropic material

Case I


2119886

119909

119910

1198649984001 1198641

1205951

119889


Far-field stresses

Isotropic material


Case II

119909

119910

1198649984001

1205951

119889

2119886


Far-field stresses

Isotropic material

Case III


119909

119910

1205951

1198649984001 1198641

2119886




1015840

2 = 1 11986421198661015840




1015840

1 = 1 11986411198661015840

1 = 25


11986411198641015840

1 11986411198661015840

1 ]1015840113 12 1 2 3 25 025lowast






Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A


09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 V10158401 =

025 11986411198642 = 1)


119886 =


1015840

1 = 1

11986411198661015840


1015840

2 = 13 11986421198661015840



ℎ







0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

3

31 2ξ

120585 = minus23 120585 = 0 120585 = 1

119909

119910


1

2

3

31 2

120585 = minus23 120585 = 0 120585 = 23

119909

119910

120585


1

2

3

31 2

120585 = minus1 120585 = 0 120585 = 23

120585

119909

119910


1

2

3

31 2119909

119910

120585

120585 = minus1 120585 = 0 120585 = 1


Crack tip position1

2

3

31 2

120585 = minus1 120585 = minus23 120585 = 0 120585 = 23

120585

119909

119910






Δ119906 (119903) = radic2119903

120587

119872(

119903

119889

)119870 (36)


119872(119909) =

119863

1205732

(1198752

+ 1205732

119868)



times ((1 + 41205762

) cosh (120587120576))minus1

(37)


Δ119906 (119903) = 119872(

119903

119889

)

[

[

[

[

[

[

1198911

Δ1199061

1

1198912

Δ1199062

1

1198913

Δ1199063

1

1198911

Δ1199061

2

1198912

Δ1199062

2

1198913

Δ1199063

2

1198911

Δ1199061

3

1198912

Δ1199061

3

1198913

Δ1199063

3

]

]

]

]

]

]

(38)


119896












119866(2)

119866(1)






Material 119864 (MPa) 1198641015840 (MPa) ]1015840 119866

1015840




119865I =119870I1198700

119865II =119870II1198700

(39)

with

1198700

= 120590radic120587119886 (40)



1015840

1 ]1 ]1015840


1015840



2119886

119875

119875

119889

119909

119910

A B


Interface

119908 119908

ℎ

ℎ

119910

119909

2119886

1198649984002

1198649984001

1198642

1198641


120590

120590

119860 B





Material 21198641015840

21198642



|119870| =radic119870

2

I + 1198702

II

Far-field stresses

Isotropic material

Case I


2119886

119909

119910

1198649984001 1198641

1205951

119889


Far-field stresses

Isotropic material


Case II

119909

119910

1198649984001

1205951

119889

2119886


Far-field stresses

Isotropic material

Case III


119909

119910

1205951

1198649984001 1198641

2119886




1015840

2 = 1 11986421198661015840




1015840

1 = 1 11986411198661015840

1 = 25


11986411198641015840

1 11986411198661015840

1 ]1015840113 12 1 2 3 25 025lowast






Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A


09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 V10158401 =

025 11986411198642 = 1)


119886 =


1015840

1 = 1

11986411198661015840


1015840

2 = 13 11986421198661015840



ℎ







0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Δ119906 (119903) = radic2119903

120587

119872(

119903

119889

)119870 (36)


119872(119909) =

119863

1205732

(1198752

+ 1205732

119868)



times ((1 + 41205762

) cosh (120587120576))minus1

(37)


Δ119906 (119903) = 119872(

119903

119889

)

[

[

[

[

[

[

1198911

Δ1199061

1

1198912

Δ1199062

1

1198913

Δ1199063

1

1198911

Δ1199061

2

1198912

Δ1199062

2

1198913

Δ1199063

2

1198911

Δ1199061

3

1198912

Δ1199061

3

1198913

Δ1199063

3

]

]

]

]

]

]

(38)


119896












119866(2)

119866(1)






Material 119864 (MPa) 1198641015840 (MPa) ]1015840 119866

1015840




119865I =119870I1198700

119865II =119870II1198700

(39)

with

1198700

= 120590radic120587119886 (40)



1015840

1 ]1 ]1015840


1015840



2119886

119875

119875

119889

119909

119910

A B


Interface

119908 119908

ℎ

ℎ

119910

119909

2119886

1198649984002

1198649984001

1198642

1198641


120590

120590

119860 B





Material 21198641015840

21198642



|119870| =radic119870

2

I + 1198702

II

Far-field stresses

Isotropic material

Case I


2119886

119909

119910

1198649984001 1198641

1205951

119889


Far-field stresses

Isotropic material


Case II

119909

119910

1198649984001

1205951

119889

2119886


Far-field stresses

Isotropic material

Case III


119909

119910

1205951

1198649984001 1198641

2119886




1015840

2 = 1 11986421198661015840




1015840

1 = 1 11986411198661015840

1 = 25


11986411198641015840

1 11986411198661015840

1 ]1015840113 12 1 2 3 25 025lowast






Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A


09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 V10158401 =

025 11986411198642 = 1)


119886 =


1015840

1 = 1

11986411198661015840


1015840

2 = 13 11986421198661015840



ℎ







0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119866(2)

119866(1)






Material 119864 (MPa) 1198641015840 (MPa) ]1015840 119866

1015840




119865I =119870I1198700

119865II =119870II1198700

(39)

with

1198700

= 120590radic120587119886 (40)



1015840

1 ]1 ]1015840


1015840



2119886

119875

119875

119889

119909

119910

A B


Interface

119908 119908

ℎ

ℎ

119910

119909

2119886

1198649984002

1198649984001

1198642

1198641


120590

120590

119860 B





Material 21198641015840

21198642



|119870| =radic119870

2

I + 1198702

II

Far-field stresses

Isotropic material

Case I


2119886

119909

119910

1198649984001 1198641

1205951

119889


Far-field stresses

Isotropic material


Case II

119909

119910

1198649984001

1205951

119889

2119886


Far-field stresses

Isotropic material

Case III


119909

119910

1205951

1198649984001 1198641

2119886




1015840

2 = 1 11986421198661015840




1015840

1 = 1 11986411198661015840

1 = 25


11986411198641015840

1 11986411198661015840

1 ]1015840113 12 1 2 3 25 025lowast






Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A


09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 V10158401 =

025 11986411198642 = 1)


119886 =


1015840

1 = 1

11986411198661015840


1015840

2 = 13 11986421198661015840



ℎ







0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Material 21198641015840

21198642



|119870| =radic119870

2

I + 1198702

II

Far-field stresses

Isotropic material

Case I


2119886

119909

119910

1198649984001 1198641

1205951

119889


Far-field stresses

Isotropic material


Case II

119909

119910

1198649984001

1205951

119889

2119886


Far-field stresses

Isotropic material

Case III


119909

119910

1205951

1198649984001 1198641

2119886




1015840

2 = 1 11986421198661015840




1015840

1 = 1 11986411198661015840

1 = 25


11986411198641015840

1 11986411198661015840

1 ]1015840113 12 1 2 3 25 025lowast






Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A


09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 V10158401 =

025 11986411198642 = 1)


119886 =


1015840

1 = 1

11986411198661015840


1015840

2 = 13 11986421198661015840



ℎ







0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Far-field stresses

B

C

1198649984001

1198649984002

1198641

11986421205952 = 0∘

1205951 = 0∘

2119886

2119886ℎ

119910

119909

119889119905

1198891120573

A


09

095

1

105

11

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus005

minus01


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 V10158401 =

025 11986411198642 = 1)


119886 =


1015840

1 = 1

11986411198661015840


1015840

2 = 13 11986421198661015840



ℎ







0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 15 30 45 60 75 90 105 120 135 150 165 18009

095

1

105

11N

orm

aliz

ed S

IFs (

mod

e I)

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(a)

0

005

01

0 15 30 45 60 75 90 105 120 135 150 165 180

Nor

mal

ized

SIF

s (m

ode I

I)

minus01

minus005

11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)


(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)

15 30 45 60 75 90 105 120 135 150 165 18009

095

105

11

0

1

Nor

mal

ized

SIF

s (m

ode I

)


11986411198649984001 = 2

11986411198649984001 = 3

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic)

(a)

0

005

01

15 30 45 60 75 90 105 120 135 150 165 1800

minus005

minus01

Nor

mal

ized

SIF

s (m

ode I

I)


11986411198649984001 = 2

11986411198649984001 = 13

11986411198649984001 = 12

11986411198649984001 = 1 (isotropic) 1198641119864

9984001 = 3

(b)


1 = 25 ]10158401 =

025 11986411198642 = 1)




1015840



1015840

1 = 3 or 13)

(v) Figure 7(a) (11986411198641015840



1015840



07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










07075

08085

09095

1105

11115

12125

13135

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

)

(a)

01015

02025

03035

04045

05055

06

Tip ATip B


Nor

mal

ized

SIF

s (m

ode I

I)

(b)


01

06

11

16

Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180


minus04

Nor

mal

ized

SIF

s (m

ode I

)

(a)

02

07

Tip CTip D


minus08

minus03

minus18

minus13Nor

mal

ized

SIF

s (m

ode I

I)

(b)











∘ and 90∘ one




Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Tip CTip D

0 15 30 45 60 75 90 105 120 135 150 165 180

0

05

1

15

2

25

3

Ratio

of n

orm

aliz

ed S

IFs (119865

I119865II

)


minus05

minus1








∘






5 Conclusions





Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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