A time-harmonic fundamental solution for a class of inhomogeneous transversely isotropic media

We present here some numerical results for a two-dimensional fundamental solution in a class of inhomogeneous transversely isotropic media. The inhomogeneity is assumed to be the same not only for the stiffnesses, but also for the density. The derivation, which is based on a previous work by Rangelov et al. [T.V. Rangelov et al., Elastodynamic fundamental solutions for certain families of 2d inhomogeneous anisotropic domains: basic derivations, Eur. J. Mech. A Solids 24 (2005) 820–836], is accomplished in terms of the Radon transform and numerical integration procedures. The time-harmonic fundamental solution reveals its non-wave nature for sources with lower frequencies than the critical one. We identify a subcase of the fundamental solution which is amenable to numerical evaluation, requiring however additional constraints with respect to the elasticity constants. The general fundamental solution reveals a more complicated structure, with additional effects in comparison to the above mentioned subcase, as e.g. the loss of symmetry of the Green’s Tensor.