Scattering of Acoustic and Elastic Waves by Cracklike Objects: The Role of Hypersingular Integrals

Abstract

Cracks in NDE are often modeled as slender - shaped voids with little enclosed volume, or as cracks with asperities, or thin slits cut into the surface of a solid, or as touching surfaces with zero enclosed volume, i.e. a ‘mathematical’ crack (see Fig. 1). Any of these models, among others, may be employed depending on a variety of factors. Although the models with nonzero enclosed volume usually represent reality better, the ‘mathematical’ model is used very often because of its simplicity and utility, despite certain analytical and numerical difficulties associated with the zero volume aspect. Nevertheless, the more realistic model presents difficulties because of the thinness of the shapes enclosed by the crack surfaces. When such models are used in computations, difficulties with at least the following two features arise: (i) poor conditioning of the final system of equations and (ii) numerical inaccuracy, Both features are due to the proximity of the crack surfaces to each other. This paper demonstrates how a combination of conventional and hypersingular boundary integral equations provides a formulation for scattering of waves from thin - body shapes which is free of the difficulties (i) and (ii). The methodology should be valuable in solving the rough crack and partially - closed crack, as well as the incompletely bonded crack or thin - body inclusion problem. Numerical results are given in this paper for scattering of acoustic waves from certain thin cracklike shapes and data are compared in the near and far field with data from a mathematical crack model. The vector counterpart of such problems, i.e. scattering of elastic waves from cracklike objects, is part of our ongoing research and will be discussed in a future paper.