Interfacial waves and deformations in pre-stressed elastic media

Abstract

The propagation of interfacial (Stoneley) waves along the boundary between two half-spaces of pre-stressed incompressible isotropic elastic material is examined. The underlying deformation in each half-space corresponds to a pure homogeneous strain with one principal axis of strain normal to the interface and the others having a common orientation. The secular equation governing the wave speed for propagation along a principal axis is obtained in respect of general strain-energy functions. Detailed analysis of the secular equation reveals general sufficient conditions for the existence of a wave and, in particular cases, necessary and sufficient conditions for the existence of a unique interfacial wave. It is also shown that when an interfacial wave exists its speed is greater than that of the least of the Rayleigh wave speeds for the separate half-spaces, paralleling a result from the linear theory. For the special case of quasi-static interfacial deformations (corresponding to vanishing wave speed) an existence criterion is found; moreover, it is shown that inequalities that exclude surface deformations in each half-space also exclude interfacial deformations. Dependence of the above results on the underlying homogeneous deformations and on material parameters is illustrated by numerical results for the neo-hookean strain-energy function.