Calculation of diffraction coefficients for a semi-infinite crack embedded in an infinite anisotropic linearly elastic body

Abstract

In this paper we consider the diffraction of sound by a semi–infinite planar crack arbitrarily oriented in a homogeneous anisotropic linearly elastic solid. The problem is formulated exactly as an integro–differential equation with a difference kernel which is the stress tensor corresponding to the fundamental point force solution for the uncracked solid. The solution of this equation is the crack opening displacement (COD) induced by the incident field, and this may be expressed in terms of the Wiener–Hopf factors of the Fourier symbol of the kernel. The quantity of physical interest is the so–called diffraction coefficient which is proportional to the Fourier transform of the COD and relates the vector amplitude on an incident ray to the crack edge to the amplitudes on the diffracted rays. In the general case, the necessary Wiener–Hopf factorization cannot be found explicitly, but it is possible to construct an iterative procedure which enables a numerical evaluation. In this, the first of two papers, we give a general formal solution of the diffraction problem and show how it reproduces known results with special material symmetries. In the second paper we shall discuss the numerical procedure in detail and provide some new tabulations of diffraction coefficients.