Spherical-multipole analysis of elastodynamic wave fields

A particular goal of ultrasonic nondestructive testing comprises the identification of cracks, for instance, in welds. Set as an inverse scattering problem for elastic waves in solids it is essentially the crack tip scattering that delivers information about the crack size, which, on the other hand, is the crucial input for fracture mechanical life time estimation of the part under concern. Unfortunately, only ab ovo numerical methods are available to model crack tip scattering quantitatively. Therefore, this paper aims at the step-by-step development of an analytically based method to predict crack tip scattering via translating the spherical multipole expansion of electromagnetic waves scattered by the tip of a perfectly conducting circular cone to elastic wave scattering by a similar geometry as a discontinuity in an otherwise solid material implying the boundary condition of vanishing normal components of the elastodynamic stress tensor and/or, alternatively, vanishing components of the particle displacement vector. Spherical multipoles have been extensively exploited to calculate electromagnetic wave scattering by circular as well as elliptical cones by the first author of the present paper. Basic ingredients are essentially multipole expansions in terms of spherical harmonics for divergence-free N- and M-vector wave functions. They compose dyadic Green functions as well as the electromagnetic field. To translate the above formalism to elastodynamic wave fields the underlying wave equation for the elastodynamic particle displacement has to be separated in terms of vector wave functions. In contrast to the electromagnetic Helmholtz equation this is a Navier-Stokes equation exhibiting two characteristic terms for pressure and shear waves with different wave speeds, the latter one being identical to the electromagnetic Helmholtz equation term, thus the separation in N- and M-functions applies as well. The remaining pressure wave term closely resembles the curl-free electromagnetic L-vector wave function term that is required in source regions, yet disappearing outside the sources. Hence, for elastodynamic wave fields it has to be kept as crucial to represent pressure waves. Therefore, to approach the solution of the problem posed in the introduction the first step must be the appropriate separation of the elastodynamic particle displacement as well as the pertinent second rank Green tensor in terms of elastodynamic L-, N- and M-vector wave functions. Once the representation of the Green function has been obtained we are ready to insert a complex valued source point to produce an elastodynamic Gaussian Beam that can be used as an incident field emanating from an ultrasonic piston radiator (an aperture antenna).