Boundary Point Method applied for calculating elastic strain and stress in bodies with cracks

Abstract

Summary A new numerical method for the solution of integral equations of the theory of elasticity for bodies with cracks is developed. The method is based on a class of Gaussian approximating functions that simplify essentially the construction of the final matrix of the linear algebraic system of the discretized problem. The results of the application of the method to some plane problems of elasticity were compared with the exact solutions and some other numerical solutions that exist in literature. Introduction In this Study we develop a new numerical method for the solution of the inte- gral equations of the second boundary value problem of elasticity for bodies with cracks. The method is based on a class of Gaussian approximating that are Gaus- sian type functions concentrated in tangent planes to the surface of the body at some number of surface nodes. The idea to use these functions for the solution of a wide class of the integral equations of physics belongs to V. Maz'ya. The theory of approximation Gaussian functions was developed in the works of V. Maz'ya(1,2). These functions were used for the solution of a static problem of elasticity for an infinite medium with thin inclusions and cracks. In this work we present some numerical results for elasticity problems in bodies with cracks. The second boundary value problem of elasticity for bodies with cracks Let us consider the numerical solution of the elasticity problem for a straight crack that occupies an interval (|x| < l, y = 0) in an infinite plane. The plane is subjected to stress field σ0(x) at infinity. The integral equation of this problem takes form: � Γ n(x) ·(S(x −x � ) ·n(x � )) ·b(x � )dΓ � = −n(x) · σ0, for x∈ Γ (1)