Surface and interfacial waves and deformations in pre-stressed elastic materials
This thesis is concerned with the effect of pre-stress on the propagation of surface and interfacial waves in elastic materials. Following a review of the classical theory of Rayleigh and Stoneley waves for linear elastic materials we consider first the propagation of infinitesimal surface waves on a half-space of incompressible material subject to a general pure homogeneous pre-stress; the secular equation for propagation along a principal axis of the pre-stess is obtained for a general strain-energy function, and conditions which ensure stability of the underlying pre-stress are derived; the influence of the pre-stress on the existence of surface waves is examined, and the secular equation is analysed in detail for particular deformations and, for a number of specific forms of strain-energy function, numerical results are used to illustrate the dependence of the wave speed on the pre-stress. Necessary and sufficient conditions for the existence of a unique surface wave are obtained. Corresponding results for a compressible material are also derived. The propagation of (Stoneley) interfacial waves along the boundary between two half-spaces of pre-stressed incompressible isotropic elastic material is then 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 material.