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

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.


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][6][7][8][9][10][11][12][13][14][15][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.

Mechanical Model
The plane  > 0 is the upper orthotropic material ( = 1), and its elastic constants are  11 ,   The stress functions   (, ), ( = 1, 2) satisfy the governing equations [16][17][18][19][20]: The boundary conditions of the interface crack for mode II are as follows: → +∞ : The geometric and load conditions for the investigated problem are given as shown in Figure 1.

Stress Function
The relationships between the stresses and the stress functions can be obtained as in which the stress functions   of right side of (5a)-(5c) contain twelve undetermined coefficients  , ,  , ,  , (,  = 1, 2) and an unknown singularity exponent .
Substituting  solved by ( 13) into (9), the complex singularity exponent  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: in which  = 1,  * = 2;  = 2,  * = 1.By (15a) and (15b), the stress functions   of the right side of (5a)-(5c) contain practically eight coefficients.

Stress Intensity Factors
Considering the stress expressions (5b) and (5c) and also the load condition in (4), the stress intensity factors are defined as In order to express the change process of   →  − completely, it is necessary that the factor of (11): ) 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).

Oscillatory Field
The test and calculus results [16,[22][23][24] for the mechanicals properties of three orthotropic bimaterial are shown in Tables 1 and 2. From (19), the normalized stress intensity factors  2 / and  1 / depend on the length of crack  and the bielastic constant .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 factor  1 / can be observed by the inner small figure of Figure 3.The factors  2 / and  1 / increase almost linearly when the length  or the constant  increases as shown in Figures 2 and 3.
By (23a)-(23c), the normalized stresses   /,   /, and   / depend on the polar angle  and the polar radius .The variations of the normalized stresses for the above three bimaterials with respect to polar angle  are plotted in Figure 4. Figure 4 shows that three stresses have alternately the increase and decrease with  increase, their maximum values can be always reached at both sides of the crack.
Figures 5 and 6 show the variations of the normalized stresses for the above three bimaterials with respect to polar radius  for  = ±45 ∘ and  = ±60 ∘ .Two figures mean that the stresses   /,   /, and   / are the monotonic decreasing functions of / on the two half-planes.Figures 7 and 8 show the variations of the normalized logarithmic stresses with respect to the polar radius  for  = ±45 ∘ and  = ±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.

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 exponent  can be found.

Acknowledgment
This work was supported by the Natural Science Foundation of Shanxi province (no.2011011021-3), and the doctoral fund of Taiyuan University of Science and Technology (no.20102028).

Figure 1 :
Figure 1: Interface crack for mode II of orthotropic bimaterial.

Figure 2 :
Figure 2: Normalized stress intensity factors as a function of .

Figure 3 :
Figure 3: Normalized stress intensity factors as a function of  when  = 1.

Table 1 :
Mechanical properties of each orthotropic material.