Integral Equations for Crack Scattering∗

This chapter is concerned with cracks . Real cracks in solids are complicated: they are thin cavities, their two faces may touch, and the faces may be rough. We consider ideal cracks. By definition, such a crack is modelled by a smooth open surface Ω (such as a disc or a spherical cap); the elastic displacement is discontinuous across Ω, and the traction vanishes on both sides of Ω (so that the crack is seen as a cavity of zero volume). We suppose that we have one crack with a smooth edge, ∂Ω, embedded in an infinite, unbounded, three-dimensional solid. Extensions to multiple cracks, to cracks in two dimensions, to cracks in half-spaces or in bounded domains, or to cracks with less smoothness may be made, with varying degrees of difficulty. For a variety of applications, see the book by Zhang and Gross [16]. In fact, to keep the analysis relatively simple, we shall focus on analogous scalar problems coming from acoustics. Thus, we suppose that Ω is a thin screen in a compressible fluid. The screen is hard (or rigid), which means that we have a Neumann boundary condition. See Section 2 for details. We are interested in scattering time-harmonic waves by the screen. Much is known about how to calculate scattering from objects of non-zero volume [9]. Except in a few special cases (such as scattering by a sphere), it is usual to derive and solve (numerically) a boundary integral equation over the boundary of the scatterer. However, special methods are needed for zero-volume obstacles such as cracks and screens. In particular, the Neumann boundary condition means that it is inevitable that we shall encounter hypersingular boundary integral equations over the screen. These equations can be tackled directly (using boundary elements, perhaps), or they may be recast into other equivalent forms. For example, if the screen is flat, various simplifications can be made. Integral equations can also be used as the basis for various approximation schemes. After formulating our scattering problem in the next section, we give the governing hypersingular integral equation in Section 3. This equation is solved approximately for long waves (low-frequency scattering) in Section 4. The approach used is elevated to a well-known ‘strategy’ in Section 5 prior to further applications.