Elastodynamic fundamental solutions for certain families of 2d inhomogeneous anisotropic domains: basic derivations

The existence of frequency-dependent fundamental solutions for anisotropic, inhomogeneous continua under plane strain conditions is a necessary pre-requisite for studying wave motion, either in geological media or in composites with both depth and direction-dependent material parameters. The path followed herein for recovering such types of solutions is (a) to use a simple algebraic transformation for the displacement vector so as to bring about a governing partial differential equation of motion with constant coefficients, albeit at the cost of introducing a series of constraints on the types of material profiles; (b) to carefully examine these constraints, which reveal a rather rich range of possible variations of the elastic moduli in both vertical and lateral directions; and (c) to use the Radon transformation for handling material anisotropy. Depending on the type of constraints that have been introduced, two basic classes of materials are identified, namely 'Case A' where further restrictions are placed on the elasticity tensor and 'Case B' where further restrictions are placed on the material profile. We note at this point that for isotropic materials, the elasticity tensor constraints correspond to equal Lamé constants or, alternatively, to a fixed Poisson's ratio. The present methodology is quite general and the homogeneous anisotropic medium, as well as the inhomogeneous isotropic one, can both be recovered as special cases from the results given herein.