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Although a lot of crack problems in bi-materials plate were previously treated, few solutions are available under mechanical loadings, arbitrary crack lengths and material combinations. In this paper the dimensionless stress intensity factors (SIFs) of two slanted cracks in the upper plate of bi-materials are considered under mechanical loadings with varying the crack length and material combinations systematically. In order to calculate the dimensionless SIFs accurately, the hypersingular integral equations (HSIEs) was formulated by using the modified complex potentials (MCP) function. The details numerical results of the dimensionless SIFs are given in tabular form and graphical presentations. Comparisons with the existing exact solutions show that the numerical results in this paper have high accuracy. Our results are described with clarifying the effect of the mechanical loadings, bi-elastic constant ratio and element size of cracks on the dimensionless SIFs.

In this note, the problems of solvability and construction of solutions for a nonlinear Fredholm one-order integro-differential equation with degenerate kernel and nonlinear maxima are considered. Using the method of degenerate kernel combined with the method of regularization, we obtain an implicit the first-order functional-differential equation with the nonlinear maxima. Initial boundary conditions are used to ensure the solution uniqueness. In order to use the method of a successive approximations and prove the one value solvability, the obtained implicit functional-differential equation is transformed to the nonlinear Volterra type integro-differential equation with the nonlinear maxima.

In this paper, an unequal arms branch crack problem in a plane elasticity is treated. Using distribution dislocation function and complex variable potential method, the problem is formulated into a singular integral equation. The appropriate integration scheme, in which a point dislocation is set at the origin and the distribution dislocation, is applied through all arms of the branch crack to solve the obtained singular integral equations numerically. Stability, convergence, the order of convergence, and the error term of the solution are analyzed. Some numerical examples are examined to describe the behavior of stress intensity factors at the endpoints of each branch crack.

This study will introduce a new differentiation operator, the Hilfer fractional-fractal derivative (H-FFD). The new proposed derivative aims to attract more non-local problems that show with the same time fractal behaviors. For numerical settlement of initial value problems, we use the shifted Legendre operational matrix. The main advantage of this method is that it reduces both linear and non-linear problems alike in solving the problem into a system of linear and non-linear algebraic equations. In addition, the numerical approximation of this new operator also offers some applications to systems of linear and non-linear problems.

The new system of hypersingular integral equations (HSIEs) for the thermally insulated inclined cracks and thermally insulated circular arc cracks subjected to remote shear stress in bonded dissimilar materials was formulated by using the modified complex potentials (MCPs) function method with the continuity conditions of the resultant force, displacement and heat conduction functions. This new system of HSIEs is derived from the elasticity problem and heat conduction problem by using crack opening displacement (COD) function and temperature jump along the crack faces. The appropriate quadrature formulas were used to solve numerically the new system of HSIEs for the unknown COD function and the known traction along the crack as the right hand term. Numerical solutions for the value of nondimensional stress intensity factors (SIFs) at all the cracks tips are illustrated. The comparison of nondimensional SIFs for the cracks with and without thermal is also illustrated.

The new hypersingular integral equations (HSIEs) for the multiple cracks problems in both upper and lower parts of the bonded dissimilar materials are formulated using the modified complex potential method, and with the help of the continuity conditions of the resultant force function and displacement. The crack opening displacement is the unknown and the traction along the crack as the right term of the equations. The appropriate quadrature formulas are used in solving the obtained HSIEs for the unknown coefficients. Numerical results for the multiple inclined or circular arc cracks subjected to the remote shear stress are presented.

In this note, we consider a general 2 × 2 system of nonlinear Volterra type integral equations. The modified Newton method (modified NM) is used to reduce the nonlinear problems into 2 × 2 linear system of algebraic integral equations of Volterra type. The latter equation is solved by discretization method. Nystrom method with Gauss–Legendre quadrature is applied for the kernel integrals and Newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points. Existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained. Finally, numerical examples are provided to show the validity and efficiency of the method presented. Numerical results reveal that the proposed methods is efficient and accurate. Comparisons with other methods for the same problem are also presented.

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.

The modified complex variable function method with the continuity conditions of the resultant force, displacement and heat conduction functions are used to formulate the hypersingular integral equation (HSIE) for a thermally insulated circular arc crack or a thermally insulated inclined crack in the upper part of bonded dissimilar materials subjected to remote stress. The HSIE is solved numerically for the unknown crack opening displacement function and the known traction along the crack as the right hand term using the appropriate quadrature formulas. Numerical results showed the behavior of the nondimensional stress intensity factor (SIF) at all crack tips. The nondimensional SIF depends on the crack geometries, the distance between crack and the boundary, the elastic constants ratio, the heat conductivity ratio and the thermal expansion coefficients ratio.