Gerry R. Wickham
Dept. of Mathematics and Statistics, Brunel Univ., Uxbridge, Middlesex UB8 3PH, UK
Patricia A. Lewis
Bolton Inst., Bolton BL3, UK
The diffraction of sound by a semi-infinite planar crack arbitrarily oriented in a homogeneous anisotropic linearly elastic solid is considered. The problem is formulated exactly as an integrodifferential 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 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. However, by exploiting the fact that the diffraction coefficient is independent of frequency, it is possible to develop a novel numerical scheme for its evaluation. The exact diffracted field is then analytically expressed in terms of these coefficients. It is shown how the rich qualitative geometrical structure of the diffracted field may be obtained and how the solution reproduces known results for special material symmetries.