Mutiple curved crack problems in antiplane elasticity for circular region with traction free boundary

The multiple curved cracks in a circular region problem in antiplane elasticity is formulated in terms of hypersingular integral equation in conjunction with the complex variable function method. The obtained hypersingular integral equations are solved numerically using the curve length coordinate method, where the curved crack configurations are mapped on the real axes s with intervals( , ) 1, 2,..., . i i − = a a n . Suitable scheme is used for the determination of the unknown functions. For numerical purposes only a particular case of doubly circular arc cracks is considered, and it is found that the stress intensity factors (SIFs) are higher as the cracks become closer to the circular boundary

Stress intensity factor for the interaction between a straight crack and a curved crack in plane elasticity

Formulation in terms of hypersingular integral equations for the interaction between straight and curved cracks in plane elasticity is obtained using the complex variable functions method. The curved length coordinate method and a suitable numerical scheme are used to solve such integrals numerically for the unknown function, which are later used to find the stress intensity factor, SIF

An approximate solution of two dimensional nonlinear Volterra integral equation using Newton-Kantorovich method

This paper studies the method for establishing an approximate solution of nonlinear two dimensional Volterra integral equations (NLTD-VIE). The Newton-Kantorovich (NK) suppositions are employed to modify NLTD-VIE to the sequence of linear two dimensional Volterra integral equation (LTD-VIE). The proper-ties of the two dimensional Gauss-Legenre (GL) quadrature fromula are used to abridge the sequence of LTD-VIE to the solution of the linear algebraic system. The existence and uniqueness of the approximate solution is demonstrated, and an illustrative example is provided to show the precision and authenticity of the method

Solving system of nonlinear integral equations by Newton-Kantorovich method

Newton-Kantorovich method is applied to obtain an approximate solution for a system of nonlinear Volterra integral equations which describes a large class of problems in ecology, economics, medicine and other fields. The system of nonlinear integral equations is reduced to find the roots of nonlinear integral operator. This nonlinear integral operator is solved by the Newton-Kantorovich method with initial guess and this procedure is continued by iteration method to find the unknown functions. Finally, numerical examples are provided to show the validity and the efficiency of the method presented

Interaction between inclined and curved cracks problem in plane elasticity

Interaction between inclined and curved cracks are studied and the hypersingular integral equations for the problem in plane elasticity are obtained using the complex variable functions method. The curved length coordinate technique and a suitable quadrature rule are used to solve the algebraic equations numerically for the unknown function, which are later used to find the stress intensity factor (SIF)

Formulation for multiple curved crack problem in a finite plate

The formulation for the curved crack in a finite plate is established. The technique is the curved crack in a finite plate is divided into two sub-problems i.e. the curved crack problem in an infinite plate and the finite plate without crack. For the first problem, the curved problem is formulated into Fredholm integral equation, where as for the second problem the complex boundary integral equations based on complex variables are considered. The solution of the coupled boundary integral equations gives the solution on the domain of the boundary

On the semi-bounded solution of cauchy type singular integral equations of the first kind.

This paper presents an efficient approximate method to obtain a numerical solution, which is bounded at the end point x = −1, for Cauchy type singular integral equations of the first kind on the interval [−1,1]. The solution is derived by approximating the unknown density function using the weighted Chebyshev polynomials of the third kind, and then computing the Cauchy singular integral which is obtained analytically. The known force function is interpolated using the Chebyshev polynomials of the fourth kind. The exactness of this approximate method is shown for characteristic equation when the force function is a cubic. Particular result is also given to show the exactness of this method

Formulation for multiple curved and kinked cracks problems in antiplane elasticity

Formulation in term of hypersingular integral equation for multiple curved and kinked cracks in antiplane elasticity are obtained using the complex variable function. The curved length coordinate technique and a suitable numerical scheme for such an integral are developed to solve numerically for the unknown function, which are later used to find the stress intensity factor, SIF

Stress intensity factor for multiple cracks in half plane elasticity

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

Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods