An explicit expression for the surface-impedance matrix of a generally anisotropic incompressible elastic material in a state of plane strain

Surface and interfacial impedance matrices play an important role in the construction of Green's functions, the analysis of surface and interfacial waves and the stability assessment of pre-stressed half-spaces or joined half-spaces. This paper studies these matrices for generally anisotropic pre-stressed incompressible elastic materials. It is shown that the surface-impedance matrix satisfies a simple matrix equation which, for plane-strain deformations, can be solved exactly. As a result, explicit secular equations for surface and interfacial wave speeds and explicit wrinkling/buckling conditions for pre-stressed half-spaces and joined half-spaces are obtained. It is also shown that the plane-strain surface-wave problem is mathematically identical to the edge-wave problem for thin elastic plates. Thus, the uniqueness of surface-wave speed is settled by drawing upon a recent proof of the uniqueness of edge-wave speed. Examples are used to show that it is straightforward to solve the secular equations based on the given formulae either exactly (where possible) or numerically.