Reflection of plane waves from the boundary of a pre-stressed compressible elastic half-space

Abstract

The effect of pre-stress on the propagation and reflection of plane waves in an incompressible isotropic elastic half-space has been examined recently by the authors (Ogden & Sotiropoulos, 1997). In the present paper the corresponding analysis for compressible materials is detailed. In the two-dimensional context considered for incompressible materials the (homogeneous) plane waves were necessarily shear waves. By contrast, in the compressible context pure shear waves can propagate only in specific directions in the considered principal plane and, in a general direction, a quasi-shear wave may be accompanied by a quasi-longitudinal wave, as is the case in the anisotropic linear theory. The dependence of the (in-plane) slowness section on the pre-stress (and finite deformation) and on the choice of constitutive law is elucidated. This information is used to determine the reflection coefficients for reflection of either a (quasi-) shear wave or a (quasi-) longitudinal wave from the boundary of the half-space and to characterize the different cases which arise depending on the geometry of the slowness section. The theoretical results are illustrated by numerical calculations for the range of possible types of behaviour with reference forms of strain-energy function and different states of finite deformation and to the question of stability of the half-space.