Stress Intensity Factors for a Crack in Bonded Dissimilar Materials Subjected to Various Stresses

The modified complex variable function method with the continuity conditions of the resultant force and displacement function are used to formulate the hypersingular integral equations (HSIE) for an inclined crack and a circular arc crack lies in the upper part of bonded dissimilar materials subjected to various remote stresses. The curve length coordinate method and appropriate quadrature formulas are used to solve numerically the unknown crack opening displacement (COD) function and the traction along the crack as the right hand term of HSIE. The obtained COD is then used to compute the stress intensity factors (SIF), which control the stability behavior of bodies or materials containing cracks or flaws. Numerical results showed the behavior of the nondimensional SIF at the crack tips. It is observed that the nondimensional SIF at the crack tips depend on the various remote stresses, the elastic constants ratio, the crack geometries and the distance between the crack and the boundary.


Introduction
A number of papers have been publish to analyze the behaviour of stress intensity factors (SIF) at the crack tips subjected to specific remote stress for the crack problems in an infinite plane [1,2], finite plane [3,4], half plane [5,6] or bonded dissimilar materials [7][8][9]. The body force method with continuous distributions along cracks were used to find the nondimensional SIF of the crack problems in bonded dissimilar materials subjected to various stress [7]. The Fredholm integral equations with density distributions as undetermined functions were used to calculate the nondimensional SIF for multiple crack problems in bonded dissimilar materials [8]. The logarithmic singular integral equations were used to solve the nondimensional SIF for a circular arc crack lie in the upper part of bonded dissimilar materials [9]. The combinations of Chebyshev polynomials and collocation methods were utilized to solve the nondimensional SIF of a perpendicular crack to the interface of bonded dissimilar materials [10]. The effect of elastic constants ratio to the penny-shaped crack problems in bonded dissimilar materials was determined by reducing the dual integral equations to an infinite system of simultaneous equations [11]. The solidification crack growth patterns in bonded dissimilar materials were predicted by analyzes the nondimensional SIF [12]. The nondimensional SIF of an interface crack in bonded dissimilar materials was evaluated by combining the extended finite element method and a domain independent interaction integral method [13].
The objectives of this paper is to formulate the HSIE and analyze the behavior of nondimensional SIF by using the modified complex variable function method for an inclined and a curved crack lie in the upper part of bonded dissimilar materials subjected to the various remote stresses such as shear stress

Mathematical Formulation
The complex variable function method plays an important role in solving the cracks problem in plane elasticity [14]. In this method, the stress components where N iT + denotes the normal and tangential traction along the crack segment , z z dz + . The complex potentials for a crack L in an infinite plane can be expressed by where g(t) is crack opening displacement (COD) function defined by  , , where subscripts 1 and 2 are the stress component for the upper and lower parts of bonded dissimilar materials, respectively. Those stresses can be expresses as Young's modulus of elasticity for upper and lower parts of bonded dissimilar materials, respectively. For the shear stress 1 1 respectively. The modified complex potentials for the crack lies in the upper part of bonded dissimilar materials are defined as where subscript p and c represent the principal and complementary parts, respectively. Whereas, for the crack lies in the lower part, the complex potentials are represented by Applying (18) into (19) and (20), we can obtained the following complex potentials where L b is boundary of two bonded half materials, S 1 and S 2 are upper and lower parts of bonded dissimilar materials, respectively, and the principle part of complex potentials are referred to an infinite materials. The HSIE for a crack lies in the upper part of bonded dissimilar materials can be obtained by substituted (18) into (5) and applying (6), (7), (21) and (22), then letting point z approaches t 0 on the crack and changing d z dz Note that if G 2 = 0 and G 1 = G 2 , then the HSIE for a crack in bonded dissimilar (25) Here U n (t) is a Chebyshev polynomial of the second kind, define by

Results and Discussions
Stress intensity factor (SIF) at crack tips A j is defined as Consider an inclined crack lie in the upper part of bonded dissimilar materials subjected to various stresses as defined in Figure 2.  Figure 2. Our results are totally agrees with those of Isida and Noguchi [7]. It is observed that the nondimensional SIF at F 1A1 and F 2A1 are equals to negative of F 1A2 and F 2A2 , respectively.  Consider a circular arc crack lie in the upper part of bonded dissimilar materials subjected to various stresses as defined in Figure 5. Figure 6 shows the nondimensional SIF for a circular arc crack lies in the upper part of bonded dissimilar materials subjected to various stresses at crack tip A 1 when h = 0.5R and varies as defined in Figure 5. It is observed that the nondimensional SIF at crack A 1 equal to the SIF A 2 subjected to shear stress , normal stress and mixed stress .
Whereas for tearing stress the nondimensional SIF at crack A 1 equal to the negative SIF at A 2 . The nondimensional SIF increases subjected to shear stress and tearing stress , however SIF decreases subjected to normal stress and mixed stress as increases at crack tip A 1 .

Conclusions
In this paper, the HSIE for an inclined crack and a circular arc crack lies in the upper part of bonded dissimilar materials subjected to various remote stresses such as shear stress From the numerical results we conclude that the nondimensional SIF for a crack lies in the upper part of bonded dissimilar materials depends on the various remote stresses, the elastic constants ratio, the crack geometries and the distance between the crack and the boundary. The close the crack to the boundary the smaller the nondimensional SIF which interprets the weaker of the materials. However the nondimensional SIF is altered for different stresses, elastic constants ratio or crack geometries.