Iterative method for the determination of a diffraction field in the interaction of a longitudinal shear wave with a system of cracks

Abstract

UDC 539.3 We consider the problem of determination of a diffraction field that is formed due to the interaction of longitudinal shear waves with a system of cracks arbitrarily located in the infinite body. The initial problem is reduced to a system of singular integro-differential equations. We propose an iterative method of solution of this system, where the zero approximation is the solutions of integral equations for separate cracks. Modern methods of the mechanics of deformable bodies (method of potentials, method of discontinuous solutions, etc.) allow one to easily reduce the problem of determination of a wave field in the body with an arbitrary system of cracks to a system of integral equations. As a matter of fact, the interaction of longitudinal-shear waves (SH-waves) with an arbitrary system of cracks in the infinite elastic body was studied within the method of integral equations in [7, 11]. In [4], an analogous problem for radially positioned cracks was solved within the method of discontinuous solutions. These studies were continued in [3], where an elastic body under conditions of a plane deformation was considered. In these works, it is proposed to solve the systems of integrodifferential equations for the opening displacements of cracks and their derivatives in order to avoid hypersingular integrals. For such systems, an efficient procedure of numerical solution is available. To determine the wave fields in bodies with systems of cracks, the method of boundary elements is used more and more frequently in recent times. For example, this method was applied to the solution of two- and three-dimensional problems in [6, 8] and [9, 10], respectively. However, an analysis of these works shows that the reduction of initial boundary-value problems to boundary integral equations encounters serious difficulties, which consist of the necessity to numerically solve the systems of integral equations of high dimensions. The number of equations of such systems is proportional to the number of cracks, and the kernels of the equations themselves have singularities of high orders. For this reason, all authors restricted themselves to the case of several, most often two, cracks, while performing the numerical calculations of wave fields and stress intensity factors. Therefore, the problem of determination of a stressed state and diffraction fields in bodies with systems of cracks is not completely solved. In the present work, we propose an iterative approach to the determination of a diffraction field and the stress intensity factor in the case of the interaction of an SH-wave with a system of N cracks. As a result, there is no necessity to solve a system of N integro-differential equations, and, at each iteration, N separate equations corresponding to isolated cracks are solved. Statement of the Problem Let N through cracks be located in an elastic body in the state of plane deformation. In the plane Oxy , these cracks occupy intervals 2d k in length, whose middle points have coordinates (a k , b k ) , k = 1, 2, … , N