The effect of pre-stress on the propagation and reflection of plane waves in incompressible elastic solids

This paper illustrates the influence of pre-stress and finite strain on the reflection of homogeneous plane waves from a plane boundary in an incompressible isotropic elastic solid. For a special class of constitutive laws it is shown that an incident SV wave propagating in a principal plane of the finite deformation gives rise to a surface wave in addition to a reflected SV wave for every angle of incidence (except discrete angles for which the surface wave amplitude vanishes). For materials not in this special class and for certain states of deformation and stress, on the other hand, two SV waves are in general reflected when the angle of incidence (measured from parallel to the plane boundary) is less than a critical value. One reflected wave has angle of reflection equal to the angle of incidence. The angle of reflection of the second reflected wave may be greater than, equal to or less than the angle of incidence, but for the second reflected wave to have time-averaged energy flux directed into the material away from the boundary its wave normal is, for certain angles of incidence, necessarily directed out of the material. For angles of incidence greater than the critical angle the situation is similar to that for the special class of materials. This contrasts with the situation in the (compressible) classical theory in which an incident SV wave leads to a surface wave for angles of incidence less than some critical value. The dependence of the reflection coefficients on the angle of incidence depends markedly on the state of deformation and stress, and the range of behaviour is illustrated graphically for a specific material model. The surface wave speed increases indefinitely as the angle of incidence approaches normal incidence while its amplitude approaches zero. This is a consequence of the degeneracy associated with the incompressibility constraint that the P-wave speed is infinite.