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# (a) An inclined crack in upper position of a curved crack (Mode I). (b) An inclined crack in upper position of a curved crack (Mode II). (c) An inclined crack in upper position of a curved crack (Mode III). (d) An inclined crack is located below the curved crack (Mix Mode). (e) An inclined crack is located on the right position of the curved crack (Mix Mode).

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The interaction between the inclined and curved cracks is studied. Using the complex variable function method, the formulation in hypersingular integral equations is obtained. The curved length coordinate method and suitable quadrature rule are used to solve the integral equations numerically for the unknown function, which are later used to evalua...

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## Citations

... The curve coordinate method provides an effective way to mapped the configurations of the cracks into a straight line on a real axis that require less collocation points, therefore give faster convergence. This approach was suggested to solve a singular integral equation (SIE) for curved crack problems in plane elasticity by Chen [1,2], and hypersingular integral equation (HSIE) for multiple curved cracks problem in plane elasticity by Nik Long and Eshkuvatov [3], also investigated for the interaction between curved and inclined cracks [4,5]. Recently, Rafar et al. used it with HSIE to study the behavior of the stress intensity factor for multiple inclined or curved cracks in circular positions in plane elasticity [6]. ...

In this paper, the problem of arc cracks that lie in the boundary of half circle in an elastic half plane is investigated. The complex potential variables with free traction boundary condition is used to formulate the problem into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function with the help of curve length coordinate method. The numerical results have shown that our results are in good agreement with the previous works. Stress intensity factors for different cracks position are presented.

... They focused their works either in plane [1][2][3][4][5] or half plane [6][7][8][9] elasticity. For plane elasticity, Fredholm type integral equations for multiple cracks problems were proposed in [10] , weakly singular integral equations (WSIEs) [11] , Cauchy type singular integral equations (SIEs) [12,13] , and hypersingular integral equations [14][15][16][17] were used for solving various cracks problems. ...

Modified complex potential with free traction boundary condition is used to formulate the curved crack problem in a half plane elasticity into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function. Numerical examples exhibit the stress intensity factor increases as the cracks getting close to each other, and close to the boundary of the half plane.

... For example, Chen [18] solved the hypersingular integral equation in a closed form which the unknowns are approximated by a weight function multiplied by a polynomial. In addition, Nik Long and Eshkuvatov [3], Aridi et al. [20] and Nik Long et al. [5] used the complex variable function method to formulate the hypersingular integral equation. The curved length coordinate method is then used to solve the hypersingular integral equation numerically. ...

... The first type is SIE with crack opening displacement function (COD) [1], the second type is SIE with the dislocation distribution function [2,3], the third is weakly singular integral equation with logarithmic kernel [4,5], and the fourth type is HSIE [6,7,8,9]. A curve length method was developed to solve the integral equations for the curved crack problems numerically for infinite plate [10,11], and for elastic half plane [7]. This method is useful to transform the integral defined on the curve into the one on the real axis. ...

The multiple cracks problem in an elastic half-plane is formulated into singular integral equation using the modified complex potential with free traction boundary condition. A system of singular integral equations is obtained with the distribution dislocation function as unknown, and the traction applied on the crack faces as the right hand terms. With the help of the curved length coordinate method and Gauss quadrature rule, the resulting system is solved numerically. The stress intensity factor (SIF) can be obtained from the unknown coefficients. Numerical examples exhibit that our results are in good agreement with the previous works, and it is found that the SIF increase as the cracks approaches the boundary of half plane.

The various mode of stresses for the interaction between two inclined cracks in the upper part of bonded two half planes which are normal stress (Mode I), shear stress (Mode II), tearing stress (Mode III) and mixed stress was studied. For this problem, the modified complex potentials (MCPs) method was used to develop the new system of hypersingular integral equations (HSIEs) by applying the conditions for continuity of resultant force and displacement functions with the unknown variable of crack opening displacement (COD) function and the right hand terms are the tractions along the crack. The curve length coordinate method and Gauss quadrature rules were used to solve numerically the obtained HSIEs to compute the stress intensity factors (SIFs) in order to determine the strength of the materials containing cracks. Numerical solutions presented the characteristic of nondimensional SIFs at the cracks tips. It is obtained that the various stresses and the elastic constants ratio are influences to the value of nondimensional SIFs at the crack tips.

This paper deals with the interaction between two inclined cracks in the upper part of bonded dissimilar materials subjected to various stresses which is normal stress (Mode I), shear stress (Mode II), tearing stress (Mode III) and mixed stress. This problem is formulated into hypersingular integral equations (HSIE) by using modified complex potentials (MCP) with the help of continuity conditions of the resultant force and displacement functions where the unknown is the crack opening displacement (COD) function and the tractions along the crack as the right hand terms. Then, the curve length coordinate method and appropriate quadrature formulas are used to solve numerically the obtained HSIE to compute the stress intensity factors (SIF) in order to determine the stability behavior of materials containing cracks. Numerical results showed the behavior of the nondimensionalSIF at the cracks tips. It is observed that the various stresses and the elastic constants ratio are influences to the value of nondimensional SIF at the crack tips.

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.

This paper deals with the multiple inclined or circular arc cracks in the upper half of bonded dissimilar materials subjected to shear stress. Using the complex variable function
method, and with the help of the continuity conditions of the traction and displacement, the problem is formulated into the hypersingular integral equation (HSIE) with the crack
opening displacement function as the unknown and the tractions along the crack as the right term. The obtained HSIE are solved numerically by utilising the appropriate quadrature formulas. Numerical results for multiple inclined or circular arc cracks problems in the upper half of bonded dissimilar materials are presented. It is found that the nondimensional stress intensity factors at the crack tips strongly depends on the elastic constants ratio, crack geometries, the distance between each crack and the distance between the crack and boundary.

A system of hypersingular integral equation for the multiple straight cracks in a circular position in plane elasticity is formulated and presented. The center of the cracks are placed at the edge of a circle with radius R. The second crack in this problem is located in a different position based on the varying angle, θ. The sraight cracks problem is reduced to a system of hypersingular integral equations by using the method of complex potential. With the help of particular quadrature rules, the unknown coefficients are solved numerically from the resulting system of hypersingular integral equations. The obtained unknown coefficients are then can be used for determining the stress intensity factor (SIF). To prove that the suggested method can be used to solve more complicated model cases of the cracks in circular positions, examples are given to demonstrate the behaviour of SIF for different crack positions.