kinked cracks at the interface of two bonded anisotropic ... · functions. by using the singular...

−75−

Research Reports of Nagaoka National College of Tecnology, Vol.50, 2014

Toshimi KONDO1, Kyosuke OHASHI2, Takahiko KURAHASHI3 and Toru SASAKI1

1Department of Mechanical Engineering, Nagaoka National College of Technology 2 Nanotemu Corporations, Nagaoka Japan

3Department of Mechanical Engineering, Nagaoka University of Technology

Key Words: kinked crack, generalized stress intensity factors, longitudinal shear, fracture mechanics

1. Introduction Kinked cracks or deflected cracks are frequently

observed in the fracture of brittle materials under non-uniform loadings or non-homogeneity of material properties. It is important in achievement of high fracture toughness in many brittle materials, in particular composite materials, to make clear the fracture behavior of kinked cracks at the interface of these materials. Primary concerns of this subject are to calculate the stress intensity factors at the crack tips and to clarify the behavior of crack propagation of the kinked cracks.

A number of papers in this area has been published in isotropic materials1)~14) so far, where fracture parameters and fracture criteria were

studied. However, much less work has been done on the area for anisotropic materials15)~19), particularly bonded anisotropic materials. In the previous papers20)~23), the author has presented the analysis method of cracks and analyzed bonded anisotropic media with boundary cracks, taking account of the stress singularity of the vertex of the kinked crack; a crack terminating at the interface in arbitrary crack angle 20), a crack kinked at and going through the interface 21), a crack kinked at the interface and going along the interface22), a crack kinked at the interface back into the first medium23), and clarified the influence of the various factors on the stress intensity factors at the kinked crack tips.

In the present paper, we deal with the singular stress fields at the vertex of the kinked crack back into

Kinked Cracks at the Interface of Two Bonded Anisotropic Elastic Media under Antiplane Shear -Generalized Stress Intensity Factors at the Kinked Crack Vertex-

The kinked crack, which is reflecting back into the first medium at the interface of two bonded anisotropic media under antiplane shear loading, is analyzed by using the singular point method of complex potential functions. The crack is modeled by means of continuous distributions of dislocations, and a system of singular integral equations with generalized Cauchy kernels is obtained. The vertex of the kinked crack has the different stress singularity from that of the tips of the kinked crack. Taking account of this stress singularity of the vertex as well as the stress singularity of the kinked crack tips, the singular integral equations are solved numerically. Numerical results of the generalized stress intensity factors of the vertex of the kinked crack are shown in figures, and the influences of anisotropy of the media, the kinked crack angle, and the ratio of crack length of the kinked crack are clarified.

Paper

−76− −77−

Toshimi KONDO, Kyosuke OHASHI, Takahiko KURAHASHI and Toru SASAKI

Fig.1 A crack kinked at the interface of bonded anisotropic media under longitudinal shear loadings and subsidiary coordinate systems.

y

w2

y1 x, x*

x1

x2

y2

L1 L2

I

II

B

O A

y*

h1

h2

L

a1 a2

○ ●

○ ●

○ ×

○ ×

○ ×

○ ×

○ ●

○ ●

○ ●

○ ● ○ ● ○ ● ○ ●

○ × ○ × ○ ×

○ ×

○ ×

the first medium as shown in Fig.1. This problem is closely related to, for example, the design of the interface between fiber and matrix in fiber reinforced composites where it is desired that any matrix crack approaching a fiber kinks at the interface, thereby allowing the fiber survive. The solution method of this paper are founded on the previous paper23), based on two dimensional, anisotropic elasticity of complex potential functions. By using the singular point method, the problem is reduced to solving a system of singular integral equations with generalized Cauchy kernels. The singular stress fields near the vertex of the kinked crack are obtained by the function theoretic method24). Solving the singular integral equation numerically, the generalized stress intensity factor at the kinked vertex is calculated and clarified the influences of anisotropy of bonded media, kinked angles, and the ratio of the kinked crack lengths.

2. Statement of the problem and basic equations

As shown in Fig.1, a crack kinks at the interface and goes back to the first medium with arbitrary angles of . The medium is made of anisotropic two semi-infinite spaces I and II with different elastic constants. These semi- infinite spaces are perfectly bonded together

along the common surface L. A rectangular coordinate system (x, y) is located on the bonded surface of the media. The subsidiary rectangular coordinate systems (xj, yj)(j=1,2) are also located on the crack L1 of length 2a1, and the crack L2 of length 2a2 as shown in Fig.1. The cracks Lj(j=1,2) make angles j(j=1,2) with the x axis, as shown in Fig.1. The distances between the center of the crack Lj(j=1,2) and the interface are hj(j=1,2). The media are subjected to antiplane shear stresses j(j=0,1,2) at infinity. The three complex variables referred to the coordinates )y,x( and (xj, yj,)(j=1,2) are defined as z=x+iy, zj=xj+iyj(j=1,2). In the following analysis, we employ the subscript j (=1, 2) for the quantities referred to the coordinate systems (xj, yj,)(j=1,2), unless stated otherwise. The Greek numerals I and II are used to denote the quantities associated with the lower and upper half-spaces, respectively. The stress components m

zymzx , σσ and the displace-

ment mu ( m=I , II) with respect to the rectangular coordinate system (x,y) are expressed by the two complex potential functions mφ (m=I,II) as follows23):

)()1/(k)()1/(ki mmm

m2mmm

m1

mzy

mzx ζφγζφγσσ

(1) )()(x/u mmmm

m (2) where

m11m

m12

m2

m12m

m11

m1 kkk,kkk (3)

m44

m55

m12

m22

m45

m44

m55

m11 CCk,kiC2CCk (4)

and Cst (s,t=4,5) are elastic constants for antiplane shear loading which is characterized by the symmetry with respect to the x, y plane. The characteristic value of m which satisfies the condition |m|<1 is obtained by the quadratic equation:

0kk2k m11m

m12

2m

m22 γγ (5)

The relation between the complex potential mj(m) with respect to the coordinate systems (xj, yj) in the

lower half space and the complex potential mj(m) are, if we write down only the potentials in the lower half space, as follows:

−76− −77−

Kinked Cracks at the Interface of Two Bonded Anisotropic Elastic Media under Antiplane Shear−Generalized Stress Intensity Factors at the Kinked Crack Vertex−

)(')1()Γ1(e)('Φ I1I

1

1i11I

1 ζφγ

η α

, (6)

)(')1()Γ1(

e)('Φ I1II

IIi21I

2 ζφγ

η α

, (7)

where the relation between mζ m=I,II) and jη (j=1,2) is

)1()Γ1(

edm

jij

mjm

j

γηζ α

(8)

and

m

mm 1

zzγγ

ζ

(9)

j

jjjj Γ1

zΓz

η (10)

)1()1(

ihdm

m1

m1 γ

γ

(11)

)1()1(

ihwdm

m22

m2 γ

γ

(12)

ji2mj eΓ αγ (13)

The constants kjm,, which concern with the

rectangular coordinates (x,y), are connected with Kj

m related to the rectangular coordinates (xj,yj) (j=1,2) as

1i2m1

m1 eKk α , m

2m2 Kk (14)

The second equation shows invariant with rotation of the coordinate. The boundary conditions of this problem can be written in the forms: (a)Along the interface of the media (y=0)

IIIIIzy

Izy uu, (15)

(b)Along the cracks Lj (yj=0, |xj|<aj, j=1,2) 0I

y,z jjσ (16)

(c)At infinity )( m

0IIzy

Izy2

IIzx1

Izx ,,

(17) 3. Derivation of singular integral equations with generalized Cauchy kernels

To apply the method of continuous distributions of the screw dislocations along the kinked crack, it is necessary to obtain the fundamental complex potentials of two screw dislocations existing at the two points (xs0*, ys0*) (s=1,2) in the lower half space of the media. This potentials, which satisfies the boundary

conditions given by Eq.(15), can be obtained by the use of the “Riemann-Hilbert boundary value problem” as follows:

0II

22

0II

21

0I

12

0I

11 bi2

Abi2

Abi2

Abi2

A)('ζζπζζπζζπζζπ

ζφ

ⅠⅠⅠⅠ

ⅠⅠ

0IIII

2

0III

13 bbi2

A)('ζζζζπ

ζφ ⅠⅠ

where )1()zz( m

*0sm

*0s0m γγζ

*0s

*0s

*0s iyxz (s=1, 2) (20)

and bj(j=1,2) are the magnitude of the Burgers vectors of the screw dislocations, and Aj(j=1,2,3) are constants including the elastic constants and given by Appendix A, and we assumed orthotropic elastic media in the present analysis, and so forth. In this case, it should be noted that the material constants km, kjm and m reduce to be real. By using the fundamental complex potentials of Eqs.(18) and (19), and distributing the dislocations on the two cracks L1 and L2, we assume the required complex potentials in the forms:

2

2

1

1

1

1

a

a 211422

223

a

a 111211

111

a

a 111

11*111

dsaBas

sbi4

B

dsaBas

sbi4

B

dss

sbi4

1T'Φ

ηπ

ηπ

ηπηⅠ

2

2

a

a 211622

225 dsaBas

sbi4

Bηπ

(21)

2

2

1

1

1

1

2

2

a

a 222622

225

a

a 122411

113

a

a 122211

111

a

a 222

22*222

dsaCas

sbi4

C

dsaCas

sbi4

C

dsaCas

sbi4

C

dss

sbi4

1T'Φ

ηπ

ηπ

ηπ

ηπηⅠ

where Bj (j=1,…,6) and Cj (j=1,…,6) are constants and given in Appendix B, and *

1T and *2T corre-

sponding to far stress fields of the present media are given by

2i

01i

1*1 DeiDeT 11 (23)

2i

01i

1*2 DeiDeT 22 (24)

(18),(19)

(22)

−78− −79−


where Dj (j=1,2) are the constants and given by Appendix C. In derivation of Eqs. (21) and (22), Eqs. (8), (11), (12) and transformation formulas Eqs. (6) and (7) of complex potentials are used, in addition the conditions of continuity of two cracks and geometry of crack configuration are also used, namely:

22112 cosacosaw jjj sinah α )2,1j( (25)

,ihwesz,ihesz 22i

2*201

i1

*10

21 αα (26)

where sj (j=1,2) denote the points on the crack Li(i=1,2). A relation between the far field stresses j(j=1,2) is given by

)kk/()1()kk/()1( II2

II1II2

I2

I1I1 γτγτ (27)

Equation (27) can be derived from the condition of the continuity of displacement at the bonding interface similar to the case of plane problem of dissimilar media. The complex potentials Eqs.(21) and (22) of the present problem satisfy the boundary conditions of Eqs.(15) and (17) automatically. Then, to satisfy the remaining boundary condition Eq.(16), a system of singular integral equations with generalized Cauchy kernels for the density functions bj(sj) must hold as follows :

*1

1

1 2222113

1

1 2222112

1

1 11111111

1

111

11

CdSSBS,XM1

dSSBS,XM1

dSSBS,XM1dSXS

)S(B1

π

π

ππ

*2

1

1 1111223

1

1 1111222

1

1 2222221

1

1 222

22

CdSSBS,XM1

dSSBS,XM1

dSSBS,XM1dSXS

SB1

π

π

ππ

where the kernels Mij (X, S) and the constants Ci* (i=1, 2) are given in Appendix D. In the integral equations (28) and (29), the following non-dimensional quantities are introduced:

)2,1j(,)s(b)S(B

,aaR,asS,axX

jjjj

12jjjjjj

(30)

The single-valuedness of the displacement yields:

0dS)S(BRdS)S(B 221

1

1

1 2111 (31)

4. The characteristic equation at the vertex of the kinked crack To obtain the characteristic equation for determination of the stress singularity of the kinked vertex, we use the function-theoretic method24). Taking account of the known order of stress singularity of the crack tips -1/2 and the unknown order of stress singularity ω at the vertex of the kinked crack, we assume the density functions Bj(Sj)(j=1,2) as follows:

21

11i

11

21111111

1S1SeSG

S1S1SGSB

ωωπ

ω

for 1S1,01 1 (32)

1S1SeSG

S1S1SGSB

221

22i

22

221

22222

for 1S1,01 2 (33) where )2,1j()S(G jj are the bounded functions in the closed interval |Sj|<1. Since integral equations of Eqs.(28) and (29) have the generalized Cauchy kernels, and after introducing the Eqs.(32) and (33) into the integral Eqs. (28) and) (29), it must look into the singular behavior of the integrals at the upper and lower points

1Sj (see Appendix E) . By using these singular behavior of integrals, and by the condition due to )2,1j()S(G jj having the non-zero values, we obtain the characteristic equation for determination of the order of stress singularity ω of the kinked vertex in the form of Eq. (34) (see next page). Equation (34) shows that the stress singularity ω depends on the elastic constants as well as the kinked angles )2,1j(j α . It should be noted that Eq. (34) has multiple solutions of order ω, which correspond to those of two bonded wedge problems at the kink A and B in Fig.1 for arbitrary angles of )2,1j(j α , where the stresses are generally singular except for the particular cases. For the nonhomogeneous isotropic materials, Eq.(34) reduces to Eq.(35),

(28)

(29)

−78− −79−


namely, if we put 0kk II1

I1 , I

I2 G2k ,

IIII2 G2k , II Gk , IIII Gk , 0m and

III G/GΓ (G is the shear elastic constant) in Eq.(34). Numerical results of Eq.(35) coincides with the past investigation25).

0)1)(cos(

)Γ1/()Γ1()1)(cos()1(2cos)Γ1/()Γ1(cos

)1(2cos)Γ1/()Γ1(cos

221

21

2

1

ωαα

ωααωαωπωπ

ωαωπωπ

0)1)((coscos 21

22

5. Analysis of interface stresses and generalized stress intensity factors at the vertex of the kinked crack

First, we analyze the singular stress field at the kinked point A shown in Fig.1. This potential which is in the upper medium (II), will be obtained from the Eq.(19), taking account of the distributing the dislocations on the cracks and the far stress field as:

2

1

1ii2

22ii

1

1

1ii1

11ii

*2

*

dSZ

R1

ee11S

SBee

1kki2

k

dSZ

ee11S

SBee

1kki2

k

T)Z('

22

22

11

11

Ⅱ

Ⅰ

ⅠⅠ

Ⅰ

ⅡⅠ

Ⅰ

Ⅱ

Ⅰ

ⅠⅠ

Ⅰ

ⅡⅠ

Ⅰ

ⅡⅡ

αα

αα

αα

αα

γγγ

γπ

γγγ

γπ

φ

(37) where the following non-dimensional notations were used:

Especially, when 1 , namely, for the case of homogeneous isotropic elastic medium, Eq.(35) are reduced Eq.(36). Equation (36), of course, includes the order of stress singularities of the points A and B as shown in Fig.1.

(34)

,a/sS,a/sS,y11ix

a1Z 222111

**

1

Ⅱ

ⅡⅡ γ

γ

jjjj12 sbSB,a/aR (38)

By using the asymptotic behavior of the integral (37) as 0ZfI (see Appendix E), we obtain the complex potential of the upper half space, as follows:

ω

ααω

ω

ωπ

αα

ω

ααω

αα

γγ

ωπγγ

π

γγ

ωπγγ

πζφ

22

22

11

11

ii2

i21

ii

ii1

21

ii*

2**

ee1Z

R11G

sine2

ee1

kki2k

ee1Z1G

sin2

ee1

kki2kT)('

Ⅰ

ⅠⅡ

Ⅰ

Ⅰ

ⅡⅠ

Ⅰ

Ⅰ

ⅠⅡ

Ⅰ

Ⅰ

ⅡⅠ

ⅠⅡⅡ

where, IIZ , since 0y* on the interface, is given by

*

1

***

1

Xaxy

11

ixa1Z

Ⅱ

ⅡⅡ γ

γ (40)

Thus, the interface stresses near the kinked vertex in the upper half space with respect to the coordinates (x*, y*) are given by

ωπγ

γσ

sinkk21k2iT

1kkRe

2/1*

2

II2

II1

xz **

ⅡⅠ

ⅠⅠ

ⅡⅡ

(35)

(39)

0

eeee

eekkekekkk

Reeeee

eeekek

Re

eeee

eekkekekkk

Reee

eeee

ekekRe

eeee

eeekk

ekekkkRecos

e1kek

Re

eeee

eeekk

ekekkkRecos

e1kek

Re

11

22

11

22

11

22

11

22

22

11

22

11

22

11

22

11

22

22

22

22

2

2

11

11

11

11

1

1

ii

ii

ii

iI2

iI1

ii

ii

ii

iI2

iI1

ii

ii

ii

iI2

iI1

ii

ii

ii

iI2

iI1

ii

iii

ii

iI2

iI1

i2

I2

i2I1

ii

iii

ii

iI2

iI1

i2

I2

i2I1

ω

αα

αα

αα

ααω

αα

αα

αα

αα

ω

αα

αα

αα

ααω

αα

αα

αα

αα

ω

αα

ααωπ

αα

αα

α

α

ω

αα

ααωπ

αα

αα

α

α

γγ

γγγ

γ

γγ

γγγ

γ

γγ

γωπ

γ

γγ

γωπ

γ

Ⅰ

Ⅰ

ⅠⅡⅠ

ⅡⅠ

Ⅰ

Ⅰ

Ⅰ

Ⅰ

Ⅰ

ⅠⅡⅠ

ⅡⅠ

Ⅰ

Ⅰ

Ⅰ

Ⅰ

Ⅰ

ⅠⅡⅠ

ⅡⅠ

Ⅰ

Ⅰ

Ⅰ

ⅠⅡⅠ

ⅡⅠ

Ⅰ

(36)

−80− −81−


1

ii1*

11 ee11GX

ω

αα

ω

γⅠ

1

iii

2 22 ee1

R1e1G

ω

ααωωπ

γⅠ

1

iii

2

1

ii1*

12/1*

2

II2

II1

yz

22

11

**

ee1

R1e1G

ee11GX

sinkk21k2iT

1kkRe

ω

ααωωπ

ω

αα

ω

ω

γ

γ

ωπγ

γσ

Ⅰ

Ⅰ

ⅡⅠ

ⅠⅠ

ⅡⅡ

Next, we will proceed to obtain the expression of the generalized stress intensity factor of the kinked vertex. The generalized stress intensity factor *

3k at the kinked point O3 in Fig.2, is defined in regard to the direction of the coordinate ,3x :

*3

0yz0r

kr2lim33

θ

ωσ (43)

where 33yzσ can be obtained from the complex

potential with respect to the coordinates (x3*,y3*) using the coordinate transformation formula. Thus, the complex potentials with respect to the coordinates )y,x( 33 can be written as follows:

ω

αα

θω

θαα

γγγη

ωπ

γγγη

ⅡⅠ

Ⅰ

ⅡⅠ

Ⅰ

ⅡⅠ

ⅠⅡ

1eeeΓ11)1(G

sin2

e1

Γ1ee

1kki2

k'Φ

11

11

ii

i3

31

21

i3ii3

*

ω

αα

θωπω

θαα

γγγη

ωπγγγ

π

ⅡⅠ

Ⅰ

ⅡⅠ

Ⅰ

ⅡⅠ

Ⅰ

1eeeΓ11

R1e)1(G

sin2e

1Γ1

ee1

kki2k

22

22

ii

i3i

32

21

i3ii

where 133333 a/)Γ1/()Γ1(iyx η

and 3 is a characteristic constant with respect to coordinates (x3,y3). Therefore, the singular stress fields near the point O3 are written as

ωωαα

ωπ

ωαα

ω

ω

ωωθθθθ

γγη

γγγ

ωπσ

R1

eee)1(G

ee)1(G

11ee

sinkk22ekekikRe

1ii

i2

1ii1*

3

1

1ii21

iII2

iII1

xz

2211

33

ⅡⅡ

Ⅱ

ⅠⅡ

ⅡⅠ

ⅠⅡ

ωωαα

ωπ

ωαα

ω

ω

ωω

γγη

γγγ

ωπσ

R1

eee)1(G

ee)1(G

11ee

sinkk22ekekikIm

1ii

i2

1ii1*

3

1f

1f

if

i21

i2

i1

yz

2211

33

ⅡⅡ

Ⅱ

ⅠⅡ

ⅡⅠ

ⅡⅡⅠⅡ

From Eqs.(43) and (46), the generalized stress intensity factor then is

ωωαα

ωπ

ωαα

ω

ω

ωωθθθθ

γγ

γγγ

ωπ

R1

ee

e)1(G

ee

)1(Ga

11ee

sinkk2ekekk

Rek

1ii

i2

1ii

11

1

1i1iII2

iII1*

3

2211ⅡⅡ

Ⅱ

ⅠⅡ

ⅡⅠ

Ⅰ

(47) In Eqs.(45), (46) and (47), the values of )1(G1 and

)1(G2 are obtained from the solution of the singular integral equations (28) and (29). 6. Numerical solution of the singular integral equations and the generalized stress intensity factors 6.1 Numerical solution of the singular integral

equations. For the numerical evaluation of the singular

integral equations, we use Gauss-Jacobi type integration formula26):

1

1

n

1kk t,xGWdtt1t1t,xG βα (48)

(41)

(45)

(46)

(42)

(44)

Fig.2 Coordinate systems to decide the generalized stress intensity factor at the kinked vertex.

yf y*

x

0*

-a1

1

2

a2

x2

y2

O2

x

1 y1

x*, xf

x3

y3 P

2

21

2

21

Of

O1

O3 II

0*

I

0*

r

0*

−80− −81−


where n,,1k,0)t(P k

,n βα (49)

1,Re1 βα )(P ,n βα are Jacobi polynomials and the

weighting constants are given by

where (･) denotes Gamma function. Then, the integral equations (28) and (29) may now be expressed as

*1k2'j113k2'j112

k2

1n

1kk22

1n

1kk1j111

j1k1

k1k11

CS,XMS,XMW

SGS,XMXS

1WSG21

π

π

*2k1j223k2j222

k1

1n

1kk11

1n

1kk2j221

j2k2

k2k22

CS,XMS,XMW

SGS,XMXS

1WSG12

π

π

21 ,,2,1':,,2,1 njnj (52)

where

1n,

21,WM,,W 1k1k1 ωλδ

1n,,2,1k,0SP 1k121

,

1n1

ω

1j121

,1

n n,,2,1j,0XP1

ω

1n,,

21WM,,W 2k2k2 ωλδ

1n,,2,1k,0SP 2k2

,21

1n 2

ω

)n,,2,1j(,0XP 2j1

1,21

n2

ω

On the other hand, the single-valuedness of the displacement Eq.(31) is

0SGWRSGW1n

1kk22k2

1n

1kk11k1

21

(54)

Equations (51) and (52) include )2nn( 21 unknown constants, and number of the algebraic equations is (n1+n2), and it is necessary more two equations. One of these is the single-valued

condition Eq.(54), and the other equation is as follows:

ⅡⅠ

ⅡⅠ

Ⅰ kkkkRecos

e1kekRe

1

1

i2

I2

i2I1 ωπ

γ α

α

0)1(Geeee

R1

eekkekekkkRe

eeee

R1

eeekekRe

)1(Gee

eeeeeekek

2ii

ii

ii

iII1

iI1

ii

ii

ii

iII1

iI1

1ii

iii

ii

iI2

iI1

22

11

22

11

22

11

22

11

11

11

11

11

ω

αα

αα

ω

αα

αα

ω

αα

αα

ωαα

αα

ω

αα

ααωπ

αα

αα

γγ

γ

γγ

γ

γγ

γ

Ⅰ

Ⅰ

ⅠⅡⅠ

ⅡⅠ

Ⅰ

Ⅰ

Ⅰ

Ⅰ

Ⅰ

Ⅰ

(55)

Equation (55) can be obtained from the asymptotic behavior of the integral equation of Eq.(28). Equation (55) contains two new unknown constants of G1(1) and G2(－ 1), and the additional two equations to be determine these constants are

1n

1kk1111

1

SG)1(R)1(G (56)

1n

1kk2222

2

SG)1(R)1(G (57)

where

1n

kk1'k 'k1k1

'k1

11n,1k21k,1k21k,1k21,1k2

11n,11k,11k,1111

1

SSSx

SSSSSSSSSxSxSxSx

)x(R

1n

kk1'k 'k2k2

'k2

1n,2k21k,2k21k,2k221k1

11n,21k,21k,2212

2

2

SSSx

SSSSSSSSSxSxSxSx

)x(R

These equations are well known as the Lagrange’s interpolation formula. Thus, all of the unknown constants )4nn( 21 will be determined by the

)4nn( 21 equations, and the generalized stress intensity factors at the vertex may be determined from the Eq.(47). 6.2 Non-dimensional, generalized stress

intensity factors

The generalized stress intensity factor defined

(51)

(50)

lk

,1M

lklk

,1M

lk

SPdS

dSP

21MΓ1MΓ1M!1M

1MΓ2M2M,,W

λδλδ

λδλλδλδ

δλδλδ

(53)

(58)

(59)

−82− −83−


Eq.(43) is normalized as, if only 1 acts on the media:

ωωαα

ωπ

ωαα

ω

ωωθθθθ

γγ

γγγ

ωπ

R1

ee

e)1(G

ee

)1(G

11ee

sinkk2ekekk

Re

'kkK

1ii

i*2

1ii

*1

1

1iiiII2

iII1

3

*3*

3

2211ⅡⅡ

Ⅱ

ⅠⅡ

ⅡⅠ

Ⅰ

(60) where

ωω αατ 21113 cosR1a'k (61)

and corresponds to the entire crack length of projection of kinked crack length to the extension of main crack length, and

ω

ω

αατ

αατ

211

2*2

211

1*1

cosR1)1(G

)1(G

cosR1)1(G

)1(G (62)

Also, for the case of 0 only, we obtain

ωωαα

ωπ

ωαα

ω

ωωθθθθ

γγ

γγγ

ωπ

R1

eee)1(G

ee)1(G

11ee

sinkk2ekekkRe

''kkK

1ii

i**2

1ii

**1

1f

1iiiII2

iII1

3

*3**

3

2211ⅡⅡ

Ⅱ

ⅠⅡ

ⅡⅠ

Ⅰ

(63) where

ωω αατ 21103 cosR1a''k (64) and

ω

ω

αατ

αατ

210

2**2

210

1**1

cosR1)1(G

)1(G

cosR1)1(G

)1(G

The values of Eqs.(62) and (65) can be obtained from Eqs.(51) and (52) which are divided by the Eqs.(61) and (64), respectively. 7. Numerical results of the generalized stress intensity factors and discussion

Numerical calculations of the generalized stress intensity factors of the kinked vertex were carried out on the bonded isotropic media as well as the bonded anisotropic media. The elastic constants and their symbols of bonded media used in the numerical calculations are shown in from Table 1 to Table 4, in which the symbol A denotes an isotropic medium, the symbols B and C orthotropic media. Table 2 and Table 4 show the combinations of those elastic constants. Note that the A/A is the case of homogeneous isotropic medium, and C/A and B/A are the cases of isotropic medium in the lower half space bonded to the orthotropic medium in the upper half space. To check the accuracy of the numerical results, numerical calculations of the generalized stress intensity factor K3* for various values of n1(=n2) were performed for the case of isotropic, 0=0,

=45° and R=0.1, as shown in Fig.3. The

difference between value of n1=50(=n2) and n1=300(=n2) is 0.1%, and we performed as n1=n2=300 for all the numerical calculations. First, for the two bonded isotropic materials with different elastic constants, the generalized stress intensity factor K3*, which is normalized as Eq.(60), as a function of the of crack length R(=a2/a1), is plotted in Fig.4, for the case of 0=0,1≠0,2≠0 at

C55 C45 C44

A1 1 0 1 A2 2 0 2 A0.5 0.5 0 0.5

Table 1 Symbols of elastic constants for isotropic media.

Lower elastic Upper elastic Symbols constants constants

A1 A1 A1/A1 A2 A1 A2/A1 A0.5 A1 A0.5/A1

Table 2 Symbols of bonded isotropic media. (53)

different values of the crack angles 1(=2)= 15°,30°and 45°. A homogeneous isotropic body is also depicted in the figure for comparison between them. The stress intensity factor K3* for each given conditions is nearly constant for the range of R>0.3, and decreases rapidly (absolute values increase) as R decreases in the range of R<0.2. Figure 5 shows the generalized stress intensity

factor K3**, which is normalized as Eq.(61), for the case of 0≠0,1=0,2=0. The other conditions

are the same as Fig.4. At the value of R=1, K3** becomes zero due to their symmetry of kinked crack, as would be expected. The generalized stress intensity factor K3** increases rapidly as R decreases, and the influences of elastic constants and crack angles on K3** also increase rapidly. Figure 6 shows the relation between K3* and R in

the case of anisotropic bonded materials for 0=0,1

≠0 and 2≠0, at different values of the crack angles 1(=2)=15° ,30°and 45° , where the elastic constant in the lower half space is fixed with

C55 C45 C44

A 1 0 1 B 1 0 2 C 1 0 0.5

Table 3 Symbols of elastic constants for anisotropic media.

0 100 200 300 400 500-1.6050

-1.6045

-1.6040

-1.6035

-1.6030

-1.6025

-1.6020 45,0,0,0 21210

11 A/A,1.0R

*K3

)n(n 21 Fig.3 Effect of n1 on the stress-intensity factor K

3* in the

case of isotropic medium, 0=0, and

=45°,R=0.1.

.

0.0 0.2 0.4 0.6 0.8 1.0-5

-4

-3

-2

-1

0

A1/A1

A2/A1

A0.5/A1

*K3

0,0,0 210

1521 3021

4521

)a/a(R 12

Fig.4 Effects of isotropy,

and R on the stress-

intensity factor K3* in the case of

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

A1/A1

A2/A1

A0.5/A1

**K 3

)a/a(R 12

0,0,0 210

1521

3021

4521

Fig.5 Effects of isotropy, 1=

2 and R on the stress-

intensity factor K3

** in the case of 1=

2 =0.

Table 4 Symbols of bonded anisotropic media.

Lower elastic Upper elastic Symbols constants constants

A A A/A B A B/A C A C/A

−82− −83−


an isotropic material to make clear the influence of anisotropy. It is found from the figure that the normalized, generalized stress intensity factor K3* is considerably influenced by anisotropy for all the range of R. Figure 7 shows the generalized stress intensity factor K3** , as a function of R for the case of 0

≠0,1=0 and 2=0. The other conditions are the same as the case of Fig.6. It can be seen from the figure that anisotropy of the materials has significant effects on the generalized stress

intensity factor K3**, as decreasing R. This phenomenon is similar to the case of the Fig.5. 8. Conclusion The singular point method connected with the two dimensional anisotropic elasticity of complex potential functions has been developed for the singular stress analysis of the vertex of the kinked crack at the interface of two bonded anisotropic media under antiplane shear loadings. The problem was reduced to solving a system of the singular integral equations with generalized Cauchy kernels. The singular stress field near the vertex of the kinked crack was obtained with the generalized stress intensity factor. Solving the singular integral equations numerically, the generalized stress intensity factor at the vertex of the kinked crack was calculated, and it was found that the influences of anisotropy of the bonded media, kinked angles and the crack length ratio of the kinked crack on the generalized stress intensity factor were considerable large. Acknowledgement: The author wishes to thank Prof. Emeritus H. Sekine of Tohoku University for his kind advice and helpful discussions. Appendix A

)kk(kA

kkkkkkkA

kkkA

IIII3

IIIII2

I1

ⅠⅠⅠⅠ

ⅠⅠ

)1(2)kk(k mmII

mIm γ

B

)ee()ee(B

eekkeΓ1kkB1111

111

iiii2

iii11

αααα

ααα

γγ

γ

ⅠⅠ

ⅠⅡⅠⅡⅠ

)ee()ee(B

eeeΓ1B2211

221

iiii4

iii13

αααα

ααα

γγ

γ

ⅠⅠ

Ⅰ

)ee()ee(B

eekkeΓ1kkB2211

221

iiii6

iii15

αααα

ααα

γγ

γ

ⅠⅠ

ⅠⅡⅠⅡⅠ

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

A/AB/AC/A

**K 3

0,0,0 210

1521

3021

4521

)a/a(R 12

Fig.7 Effect of anisotropic,

and R on the stress-

intensity factor K3

** in the case of

0.0 0.2 0.4 0.6 0.8 1.0-2.2

-1.8

-1.4

-1.0

A/AB/AC/A

*K3

)a/a(R 12

0,0,0 210

3021

4521

1521

Fig.6 Effects of anisotropy, 1=2 and R on the stress- intensity factor K3* in the case of 0=0.

(a1)

(a2)

−84− −85−


−84− −85−


(a7)

)ee()ee(C

eeeΓ1C1122

112

iiii2

iii21

αααα

ααα

γγ

γ

ⅠⅠ

Ⅰ

)ee()ee(C

eekkeΓ1kkC1122

112

iiii4

iii23

αααα

ααα

γγ

γ

ⅠⅠ

ⅠⅡⅠⅡⅠ

)ee()ee(C

eekkeΓ1kkC2222

222

iiii6

iii25

αααα

ααα

γγ

γ

ⅠⅠ

ⅠⅡⅠⅡⅠ

(a3) C

)kk(Γ1D

)kk(Γ1DI2

I112

I2

I111

(a4)

D

11211

111111 1XB1SFReES,XM

11422

112112 R/1XB1SFReES,XM

11623

112113 R/1XB1SFReES,XM

12624

122221 1XC1SFReES,XM

12215

121222 1XRC1SFReES,XM

12416

121223 1XC1SFReES,XM

(a5) where

11 i2I2

i2I11 e12)kek(ReE αα γ Ⅰ

22 i2I

2i2I

12 e12)kek(ReE αα γⅠ 11

11

ii

iI2

iI11

eekk2

ekekkkFαα

αα

γⅠⅡⅠ

ⅡⅠ

2211 iiiI2

iI12 ee2ekekF αααα γⅠ

22

11

ii

iI2

iI13

eekk2

ekekkkFαα

αα

γ

ⅠⅡⅠ

ⅡⅠ

22

22

ii

iI2

iI14

eekk2

ekekkkFαα

αα

γ

ⅠⅡⅠ

ⅡⅠ

1122 iiiI2

iI15 ee2ekekF αααα γ Ⅰ

11

22

ii

iI2

iI16

eekk2

ekekkkFαα

αα

γⅠⅡⅠ

ⅡⅠ

(a6) and

11

11

iI2

iI1

I2

I10

iI2

iI1

I2

I11

1

12I2

2I1

*1

ekekkkRe

ekekkkIm

E/kk2C

αα

αα

τ

τ

22

22

iI2

iI1

I2

I10

iI2

iI1

I2

I11

2

12I2

2I1

*2

ekekkkRe

ekekkkIm

E/kk2C

αα

αα

τ

τ

E

For example,

)1(G)x1(cot2dSXS

S1S1SG111

2/11

1

1 11

21

1111 ωω

ωππ

ω

ααω

αα

ω

γγ

ωπ

γγπ

11

11

ii1

21

1

1

1ii1

21

1111

ee1Z1G

sin2

dSZ

ee11S

S1S1SG1

Ⅰ

ⅠⅡ

Ⅱ

Ⅰ

Ⅰ

ω

ααω

ω

ωπ

αα

ωπ

γγ

ωπ

γγπ

22

22

ii2

i21

2

1

1 fii2

22/1

22/i

22

ee1Z

R11G

sine2

dSZ

R1

ee11S

)1S()1S(e)S(G1

Ⅰ

ⅠⅡ

Ⅱ

Ⅰ

Ⅰ

(a8) The other asymptotic formulas of the Cauchy integrals are obtained similar to the Eqs.(a8), and they are omitted. References 1) F. Erdogan and V. Biricikoglu: Two bonded half planes with

a crack going through the interface, International Journal of Engineering Science., Vol.11, pp.745-766, 1973.

2) F. Erdogan and T. S. Cook: Antiplane shear crack terminating at and going through a bimaterial interface, Internationa Journal of Fracture, Vol.10, pp.227-240, 1974.

3) J. G. Goree and W. A. Venezia: Bonded elastic half-planes with an interface crack and a perpendicular intersecting crack that extends into the adjacent material, International Journal of Engineering Science, Vol.15, Part I, pp.1-17; Part II, pp.19-27,1977.

4) H. Hayashi and S. Nemat-Nasser: On branched interface cracks, ASME Journal Applied .Mechanics, Vol.48, pp.529-533, 1981.

5) M-C. Lu and F. Erdogan: Stress intensity factors in two bonded elastic layers containing cracks perpendicular to and on the interface, Engineering Fracture Mechanics, Vol.18, Part I, pp.491-528, 1983.

6) M-Y. He and J. W. Hutchinson: Crack deflection at an iterface between dissimilar elastic materials, International Journal of Solids and Structures, Vol.25, pp. 1053-1067, 1989.

7) M-Y. He and J.W. Hutchinson: Kinking of a crack out of

an interface, ASME Journal Applied .Mechanics, Vol.56, pp.270-278, 1989.

8) K. Kasano, K. Shimoyama and H. Matsumoto: Application of a singular integral equation method to a crack branching at an interface, JSME International Journal, Ser.1, Vol.33, pp.431-438, 1990.

9) D. J. Mukai, R. Ballarini and G. R. Miller: Analysis of branched interface cracks, ASME Journal Applied Mechanics, Vol.57, pp.887-893, 1990.

10) R. Yuuki and J-Q. Xu: Fracture criteria on kinking of a crack out of the interface in dissimilar materials, Trans

−86− −87−


JSME Series (A), Vol.56, pp.1945-1951, 1990. 11) S. R. Choi, K. S. Lee and Y. Y. Earmme: Analysis of a

kinked interfacial crack under out-of-plane shear, ASME Journal Applied .Mechanics, Vol.61, pp38-44, 1994.

12) W. Lee, Y-H. Yoo and H. Shin: Reconsideration of crack deflection at planar interfaces in layered systems, Composites Science and Technology, Vol.64, pp.2415- 2423, 2004.

13) A. Azhdari and S. Nemat-Nasser: Energy-release rate and crack kinking in anisotropic brittle solids, Journal of Mechanics and Physics of Solids, Vol.44, pp.929-951, 1996.

14) M. Beghini, M. Benedetti, V. Fontanari and D. Mondelli: Stress intensity factors of inclined kinked edge cracks: A simplified approach, Engineering fracture mechanics, Vol.81, pp.120-129,2012.

15) M. Obata, S. Nemat-Nasser and Y. Goto: Branched cracks in anisotropic elastic solids, ASME Journal Applied Mechanics, Vol.56, pp.858-864, 1989.

16) A. Azhdari and S. Nemat-Nasser: Hoop stress intensity factor and crack kinking in anisotropic brittle solids, International Journal of Solids and Structures, Vol.33, pp929-951, 1991.

17) C. Blanco, J. M. Martinez-Esnaola and C. Atkinson: Kinked cracks in an anisotropic elastic materials，International Journal of fracture,Vol.93,p.87-407, 998.

18) Q-H. Qin and X. Zhang: Crack deflection at an interface between dissimilar piezoelectric materials, International Journal of Fracture, Vol.102, pp.355-370, 2000.

19) H. G. Beam, J. W. Lee and C. B. Cui: Analysis of a antiplane deformation, Journal of Mechanical Science and kinked crack in an anisotropic material under Technology, Vol.26,pp.411-419,2012.

20) T. Kondo: Singular stresses at the tips of a crack terminating at the interface of two bonded anisotropic

Media subjected to longitudinal shear loading, Engineering Fracture Mechanics, Vol.42, pp.445-451, 1992.

21) T. Kondo and H. Sekine,H: The crack kinking at and going through the bonded interface of two semi-infinite anisotropic media under longitudinal shear, International Journal of Engineering Science, Vol.29, pp.797-802,1991.

22) T. Kondo and H. Sekine: Anti-plane shear of kinked interface crack in bonded dissimilar anisotropic solid, Theoretical and Applied Fracture Mechanics, Vol.18,pp. 273-282,1993.

23) T. Kondo, M. Kobayashi and H. Sekine: A crack kinked at the interface of bonded anisotropic elastic media under longitudinal shear, International Journal of the Society of Materials Engineering Resources, Vol.10, pp.158-164, 2002.

24) F.Erdogan: Mixed boundary value problems in mechanics, In; Mechanics Today, Edited by S. Nemat-Nasser, Oxford, 1978.

25) T.Kondo, M.Tsuji, K.Hatori and M. Kobayashi: Stress singularities at the apex of sharp notch terminating at the interface of bonded anisotropic bodies subjected to longitudinal shear loadings, Research Reports of Nagaoka National College of Technology, Vol.26, pp. 111-126, 1990.

26) A.H.Stroud and D.Secrest: Gaussian Quadrature formulas, Prentice-Hall, New York, 1966.

(Received September 30, 2014)

kinked cracks at the interface of two bonded anisotropic ... · functions. by using the singular...

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