# Oscillatory Singularity Behaviors Near Interface Crack Tip for Mode II of Orthotropic Bimaterial

Authors: Xiaomei Yang; Weiyang Yang; Junlin Li; Xuexia Zhang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The fracture behaviors near the interface crack tip for mode II of orthotropic bimaterial are discussed. The oscillatory singularity fields are researched. The stress functions are chosen which contain twelve undetermined coefficients and an unknown singularity exponent. Based on the boundary conditions and linear independence, the system of twelve nonhomogeneous linear equations is derived. According to the condition for the system of nonhomogeneous linear equations which has a solution, the singularity exponent is determined. Total coefficients are found by means of successive elimination of the unknowns. The theoretical formulae of stress intensity factors and analytic solutions of stress field near the interface crack tip are obtained. The crack tip field is shown by figures.


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## Body

## 1. Introduction

Many researchers have studied the singularity behavior near interface crack tip for isotropic, orthotropic, or anisotropic bimaterial. The method of eigenfunction expansions is used by Williams [1]. The complex function method is developed by Rice and Sih [2]. Erdogan [3] and England [4], present and research various interface crack problems. These academic authorities play a leading role in fracture mechanics for interface crack, and they have made the tremendous contribution. The subsequent papers are published one after another over several decades [5–16]. In this paper, the solution method is proposed to research the singularity behavior near interface crack tip for mode II of orthotropic bimaterial.It can be seen that the following differences exist between this method and previous methods by comparison.(1)
The stress function contains twelve undetermined coefficients, rather than eight.(2)
The system of twelve nonhomogeneous linear equations is deduced based on the boundary conditions, rather than the system of eight homogeneous linear equations.(3)
The characteristic equation is found by using the condition for the system of nonhomogeneous linear equations that  possess a solution, rather than being based on the condition for the system of homogeneous linear equations which has a nontrivial solution.(4)
In order to determine total coefficients, we only need to solve the system of nonhomogeneous linear equations, rather than to solve the system of homogeneous linear equations first, and then use the load conditions at infinity.(5)
The stress intensity factors are defined by right-hand limit and left-hand limit rather than by limit.(6)
The oscillatory singularity fields near interface crack tip of three orthotropic bimaterial are illustrated by the  help of two tables and seven figures.


## 2. Mechanical Model

The planey>0 is the upper orthotropic material (j=1), and its elastic constants are E11,E12,ν11 and μ1. The plane y<0 is the lower orthotropic material (j=2), and its elastic constants are E21,E22,ν21, and μ2. y=0, |x|<a is the crack surfaces, and y=0, |x|>a is the bonded interface of two dissimilar materials.The stress functionsUj(x,y), (j=1,2) satisfy the governing equations [16–20]:
(1)(b22)j∂4Uj∂x4+[2(b12)j+(b66)j]∂4Uj∂x2∂y2+(b11)j∂4Uj∂y4=0,(j=1,2).The boundary conditions of the interface crack for mode II are as follows:(2)y=0,|x|<a:(σy)1=(σy)2=0,(τxy)1=(τxy)2=0,y=0,|x|>a:(σy)1=(σy)2,(τxy)1=(τxy)2,(3)(u)1=(u)2,(v)1=(v)2,(4)|y|⟶+∞:(σy)1=(σy)2=0,(τxy)1=(τxy)2=τ.The geometric and load conditions for the investigated problem are given as shown in Figure1.Interface crack for mode II of orthotropic bimaterial.
(a)
Geometric state
(b)
Shear load


## 3. Stress Function

The relationships between the stresses and the stress functions can be obtained as(5a)(σx)j=∂2Uj∂y2=∑k=12Re{[(Ajk,λ+iCjk,λ)-i(Bjk,λ+iCjk,λ)]×(-βjk2)Ujk,λ(zjk)},(5b)(σy)j=∂2Uj∂x2=∑k=12Re{[(Ajk,λ+iCjk,λ)-i(Bjk,λ+iCjk,λ)]Ujk,λ(zjk)},(5c)(τxy)j=-∂2Uj∂x∂y=-∑k=12Re{[(Ajk,λ+iCjk,λ)-i(Bjk,λ+iCjk,λ)]mmmmm×(iβjk)Ujk,λ(zjk)},(j=1,2)in which the stress functions Uj of right side of (5a)–(5c) contain twelve undetermined coefficients Ajk,λ, Bjk,λ, Cjk,λ(j,k=1,2) and an unknown singularity exponentλ.Considering the boundary conditions (2), (3), and (4), the functions Ujk,λ(zjk) of right side of (5a)–(5c) can be chosen as
(6)Ujk,λ(zjk)=τ(1-a2zjk2)λ=τzjk-2λ(zjk2-a2)-λ,(j,k=1,2).
in which
(7)zjk=x+sjky=x+iβjky,(j,k=1,2)(8)zjk-a=x-a+iβjky=r(cosθ+iβjksinθ),(j,k=1,2).From (5a)–(5c) and (6), an unknown number λ is called singularity exponent, and sometimes it is also called eigenvalue.


## 4. Singularity Exponent

If the singularity exponentλ is the complex number, the stress near the interface crack tip shows oscillatory singularity. We let [1–16]
(9)λ=-12+iε.Substituting (9) into (6), it can be found as
(10)Ujk,ε(zjk)=τ(1-a2zjk2)-(1/2)+iε=τzjk1-i2ε(zjk2-a2)(1/2)-iε,(j,k=1,2).Equations (7) and (8) are substituted into (10), and we have
(11)Ujk,ε(zjk)|y=0,|x|<a=e(-1)jεπτxa2-x2(-1)j-1×[sin(2εlna2-x2x)-icos(2εlna2-x2x)]
at the crack surfaces.Substituting (11), (5b), and (5c) into the boundary condition (2), by the help of the functions cos(2εln((a2-x2)/x)) and sin(2εln((a2-x2)/x)) which are linearly independent, the four homogeneous linear equations can be deduced as(12a)A11,ε+A12,ε-B11,ε-B12,ε+2C11,ε+2C12,ε=0,(12b)-A21,ε-A22,ε+B21,ε+B22,ε-2C21,ε-2C22,ε=0,(12c)-β11A11,ε-β12A12,ε-β11B11,ε-β12B12,ε=0,(12d)β21A21,ε+β22A22,ε+β21B21,ε+β22B22,ε=0.Similarly, we can find other eight linear equations(12e) to (12l) which are omitted here. By (τxy)1=(τxy)2=τ in condition (4), two nonhomogeneous linear equations are contained in the eight linear equations.The system of nonhomogeneous linear equations (12a) to (12l) is solved by eliminating the unknowns. To let the rank of the coefficient matrix Aλ for the system (12a)–(12l) and the rank of the augmented matrix A-λ=(Aλ,bII) be equal, the rank must be 8; that is, rank(Aλ)=rank(A-λ)=8. For this reason [21], the numberεmust satisfy the following characteristic equation:
(13)(e12-f12)f12-4g12h12ε2=0
in which bimaterial parameters are given as (14a)e12=1μ1-1μ2,f12=β11β12-ν11E11-β21β22-ν21E21,(14b)g12=β11+β12E11+β21+β22E21,h12=β11+β12E11β11β12+β21+β22E21β21β22.The singularity exponentλ of (9) is a complex number, so ε≠0 in (13), and e12≠0,f12≠0,g12>0,h12>0 in (14a) and (14b) for determinate bimaterial. To the equation (13) has a solution,  the discrimination rule can be deduced as (*)
D12: ife12 and f12 in (14a) satisfy e12>0 and f12>0, then the negative root of (13) is the bielastic constant ε.Substitutingεsolved by (13) into (9), the complex singularity exponent λ can be obtained.The system of eight equations which remained through sequence elimination is solved by means of the inverse sequence backsubstitution. We found all coefficients as follows:(15a)Ajk,ε+Cjk,ε=(-1)kf12+2βjk*g12ε(βj2-βj1)f12,(15b)Bjk,ε-Cjk,ε=(-1)k-1f12-2βjk*g12ε(βj2-βj1)f12,(j,k=1,2).in which k=1, k*=2; k=2, k*=1. By (15a) and (15b), the stress functionsUjof the right side of (5a)–(5c) contain practically eight coefficients.


## 5. Stress Intensity Factors

Considering the stress expressions (5b) and (5c) and also the load condition in (4), the stress intensity factors are defined as(16a)(k2+)j=-∑k=12limzjk→a+Re{(2π|zjk-a|)(1/2)-iε×[(Ajk,ε+Cjk,ε)-i(Bjk,ε-Cjk,ε)]×(iβjk)Ujk,ε(zjk)(2π|zjk-a|)(2π|zjk-a|)(1/2)-iε},(16b)(k2-)j=-∑k=12limzjk→a-Re{(2π|zjk-a|)(1/2)-iε×[(Ajk,ε+Cjk,ε)-i(Bjk,ε-Cjk,ε)]×(-1)j-1(2ε-i)(iβjk)×Ujk,ε(zjk)(2π|zjk-a|)1/2-iε},(j=1,2),(16c)k2=k2+-k2-2=12∑j=12(k2+)j2-12∑j=12(k2-)j2.(17a)
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In order to express the change process ofzjk→a-completely, it is necessary that the factor of (11): e(-1)jεπ(-1)j-1[sin(2εln((a2-x2)/x))-icos(2εln((a2-x2)/x))]=(-1)j-1(2ε-i),(ε→0) is substituted into the expressions (16b) and (17b). At the same time, the minus is used in (16c) because the load condition is given in (4).Substituting (10), (7), (8), (15a), and (15b) into (16a)–(16c), (17a)–(17c), and (18), the stress intensity factors of interface crack for mode II can be deduced as
(19)k2=τ(πa)1/2p(ε),k1=-τ(πa)1/22g12εf12q(ε),k=k2+ik1
in which(20a)p(ε)=12{[1+(1+2ε)chεπ]cos(εlnπa)-[1+(1-2ε)chεπ]sin(εlnπa)},(20b)q(ε)=12{[1-(1+2ε)chεπ]cos(εlnπa)-[1-(1-2ε)chεπ]sin(εlnπa)}.


## 6. Stress

From (10), (8), and (7), it can be found as follows that
(21)limzjk→a{[2π(zjk-a)](1/2)-iεUjk,ε(zjk)}=τ(πa)(1/2)-iε,(j,k=1,2)
in the vicinity of the crack tip.By (21), it can be known that
(22)Ujk,ε(zjk)=τ(πa)(1/2)-iε(2π)(1/2)-iε1(zjk-a)(1/2)-iε,(zjk⟶a;j,k=1,2).Substituting (22), (15a), (15b), (19), and (8) into (5a)–(5c), the stresses in the vicinity of the interface crack tip(zjk→a,r→0;j,k=1,2)for mode II of orthotropic bimaterial can be expressed as follows:(23a)(σx)j=k2(2πr)1/22(βj2-βj1)p(ε)×{-Imβj22(cosθ+iβj2sinθ)(1/2)-iε][cos(π4-εlnπa)·cos(εln2πr)-sin(π4-εlnπa)sin(εln2πr)]·[Reβj12(cosθ+iβj1sinθ)(1/2)-iε-Reβj22(cosθ+iβj2sinθ)(1/2)-iε]-[cos(π4+εlnπa)cos(εln2πr)+sin(π4+εlnπa)sin(εln2πr)]·[Imβj12(cosθ+iβj1sinθ)(1/2)-iε-Imβj22(cosθ+iβj2sinθ)(1/2)-iε]}+k1(2πr)1/22βj1βj2(βj2-βj1)q(ε)×{-Imβj1(cosθ+iβj1sinθ)(1/2)-iε][cos(π4+εlnπa)·cos(εln2πr)+sin(π4+εlnπa)sin(εln2πr)]·[Reβj2(cosθ+iβj2sinθ)(1/2)-iε-Reβj1(cosθ+iβj1sinθ)(1/2)-iε]+[cos(π4-εlnπa)cos(εln2πr)-sin(π4-εlnπa)sin(εln2πr)]·[Imβj2(cosθ+iβj2sinθ)(1/2)-iε-Imβj1(cosθ+iβj1sinθ)(1/2)-iε]},(23b)(σy)j=k2(2πr)1/22(βj2-βj1)p(ε)×{-Im1(cosθ+iβj1sinθ)(1/2)-iε][cos(π4-εlnπa)·cos(εln2πr)-sin(π4-εlnπa)sin(εln2πr)]·[Re1(cosθ+iβj2sinθ)(1/2)-iε-Re1(cosθ+iβj1sinθ)(1/2)-iε]-[cos(π4+εlnπa)cos(εln2πr)+sin(π4+εlnπa)sin(εln2πr)]·[Im1(cosθ+iβj2sinθ)(1/2)-iε-Im1(cosθ+iβj1sinθ)(1/2)-iε][cos(π4-εlnπa)·cos(εln2πr)-Re1(cosθ+iβj1sinθ)(1/2)-iε]}+k1(2πr)1/22(βj2-βj1)q(ε)×{[cos(π4-εlnπa)·cos(εln2πr)-Im1(cosθ+iβj1sinθ)(1/2)-iε][cos(π4-εlnπa)·cos(εln2πr)[cos(π4+εlnπa)·cos(εln2πr)+sin(π4+εlnπa)sin(εln2πr)]·[Reβj2(cosθ+iβj1sinθ)(1/2)-iε-Reβj1(cosθ+iβj2sinθ)(1/2)-iε]+[cos(π4-εlnπa)cos(εln2πr)-sin(π4-εlnπa)sin(εln2πr)]·[Imβj2(cosθ+iβj1sinθ)(1/2)-iε-Imβj1(cosθ+iβj2sinθ)(1/2)-iε]},(23c)(τxy)j=k2(2πr)1/22(βj2-βj1)p(ε)×{-Imβj1(cosθ+iβj1sinθ)(1/2)-iε][cos(π4+εlnπa)·cos(εln2πr)+sin(π4+εlnπa)sin(εln2πr)]·[Reβj2(cosθ+iβj2sinθ)(1/2)-iε-Reβj1(cosθ+iβj1sinθ)(1/2)-iε]+[cos(π4-εlnπa)cos(εln2πr)-sin(π4-εlnπa)sin(εln2πr)]·[Imβj2(cosθ+iβj2sinθ)(1/2)-iε-Imβj1(cosθ+iβj1sinθ)(1/2)-iε][cos(π4+εlnπa)·cos(εln2πr)-Reβj1(cosθ+iβj1sinθ)(1/2)-iε]}+k1(2πr)1/22βj1βj2(βj2-βj1)q(ε)×{-Im1(cosθ+iβj1sinθ)(1/2)-iε][cos(π4-εlnπa)·cos(εln2πr)-sin(π4-εlnπa)sin(εln2πr)]·[Re1(cosθ+iβj2sinθ)(1/2)-iε-Re1(cosθ+iβj1sinθ)(1/2)-iε]-[cos(π4+εlnπa)cos(εln2πr)+sin(π4+εlnπa)sin(εln2πr)]·[Im1(cosθ+iβj2sinθ)(1/2)-iε-Im1(cosθ+iβj1sinθ)(1/2)-iε]}in whichp(ε)andq(ε)are given by (20a) and (20b).Note that (23a)–(23c) of the stress field contain the following terms:
(24)((σx)j(σy)j(τxy)j)⊃1(2πr)1/2(cos(εln2πr)sin(εln2πr)).
The right side of (24) tends to change the sign rapidly at the crack tip (r→0), and therefore the stresses possess an oscillatory character.


## 7. Oscillatory Field

The test and calculus results [16, 22–24] for the mechanicals properties of three orthotropic bimaterial are shown in Tables 1 and 2.Table 1
Mechanical properties of each orthotropic material.
Materials
E
j
1/GPa
E
j
2/GPa
ν
j
μ
j/GPa
Δ
j
β
j
1
β
j
2
A
j
=
1
12.45
11.56
0.33
4.35
0.54
0.85
1.21
j
=
2
201
12.5
0.21
6.69
813.31
0.74
5.39
B
j
=
1
181
10.3
0.28
7.17
539.01
0.86
4.89
j
=
2
40.06
38.26
0.26
11.24
5.07
0.63
1.63
C
j
=
1
18.3
15.51
0.32
5.87
1.42
0.80
1.35
j
=
2
135
10.3
0.21
6.6
348.96
0.82
4.40Table 2
Mechanical properties of three orthotropic bimaterials.
Bimaterial
e
12
f
12
g
12
h
12
ε
λ
A
0.0804
0.0379
0.1966
0.2948
−0.0834
-
0.5 -0.0834i
B
0.0505
0.0026
0.0881
0.1908
−0.0427
-
0.5 -0.0427i
C
0.0188
0.0166
0.1565
0.2681
−0.0149
-
0.5 -0.0149iFrom (19), the normalized stress intensity factorsk2/τandk1/τdepend on the length of crackaand the bielastic constantε. The variations of the stress intensity factors versus crack length are plotted in Figure 2. The variations of the stress intensity factors with bielastic constant are illustrated in Figure 3. The minute variations of factork1/τcan be observed by the inner small figure of Figure 3. The factorsk2/τandk1/τincrease almost linearly when the lengthaor the constantεincreases as shown in Figures 2 and 3.Figure 2
Normalized stress intensity factors as a function ofa.Figure 3
Normalized stress intensity factors as a function ofε when a=1.By (23a)–(23c), the normalized stressesσx/τ, σy/τ, andτxy/τdepend on the polar angle θ and the polar radiusr. The variations of the normalized stresses for the above three bimaterials with respect to polar angleθare plotted in Figure 4. Figure 4 shows that three stresses have alternately the increase and decrease with θ increase, their maximum values can be always reached at both sides of the crack.Figure 4
Normalized stresses versusθ for r/a=0.0001.Figures5 and 6 show the variations of the normalized stresses for the above three bimaterials with respect to polar radiusrforθ=±45∘andθ=±60∘. Two figures mean that the stressesσx/τ, σy/τ, andτxy/τare the monotonic decreasing functions ofr/aon the two half-planes.Figure 5
Normalized stresses versusr for θ=±45∘.Figure 6
Normalized stresses versusr for θ=±60∘.Figures7 and 8 show the variations of the normalized logarithmic stresses with respect to the polar radiusrforθ=±45∘andθ=±60∘. It can be found that the stress distribution in Figures 7 and 8 is not straight line and does not parallel to each other. Such distributions mean oscillatory singularity state.Figure 7
Normalized stresses versusr for θ=±45∘.Figure 8
Normalized stresses versusr for θ=±60∘.


## 8. Conclusion

From the above derivation, the following results are very significant.(1)
New stress functions are chosen.(2)
The system of twelve nonhomogeneous linear equations is derived.(3)
The characteristic equation can be given.(4)
By the help of the distinction rule(*), the complex singularity exponentλcan be found.(5)
Total coefficients are determined.(6)
For the oscillatory field, the theoretical formulae of stress intensity factors and the analytic solutions for stresses near interface crack tip of mode II are yielded, respectively. The stress intensity factors and the stresses show mixed crack characteristics.(7)
The crack tip fields for oscillatory singularities are illustrated in Figures2 to 8.



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*Source: 716768-2013-07-01.xml*