5 - Integral equations for crack scattering

Summary

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 modeled 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 halfspaces 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 (1998).

After formulating our scattering problem, we give the governing hypersingular integral equation in §5.2. This equation is solved approximately for long waves (low-frequency scattering) in §5.3. The approach used is elevated to a well-known ‘strategy’ in §5.4 prior to further applications. For flat cracks and screens, we can simplify the governing hypersingular integral equation. This is done in §5.5. Alternatively, we can use a direct approach, using Fourier transforms; see §5.6. Methods for solving the resulting equations are discussed in §5.7. The final section is concerned with curved cracks and screens.