The scattering of elastic waves by cracks in isotropic and anisotropic solids has important applications in various areas of mechanical engineering and geophysics, in particular in ultrasonic nondestructive testing and evaluation. The scattering by cracks can be investigated by integral equation methods, eg, boundary element methods, but here we are particularly concerned with more analytically oriented hypersingular integral equation methods. In these methods, which are only applicable to very simple crack shapes, the unknown crack opening displacement in the integral equation is expanded in a set of Chebyshev functions, or the like, and the integral equation is projected onto the same set of functions. This procedure automatically takes care of the hypersingularity in the integral equation. The methods can be applied to cracks in 2D and 3D, and to isotropic or anisotropic media. The crack can be situated in an unbounded space or in a layered structure, including the case with an interface crack. Also, problems with more than one crack can be treated. We show how the crack scattering procedures can be combined with models for transmitting and receiving ultrasonic probes to yield a complete model of ultrasonic nondestructive testing. We give a few numerical examples showing typical results that can be obtained, also comparing with some experimental results. This review article cites 78 references.
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