On the reflection of plane waves at the boundary of an elastic half-space subject to simple shear

Abstract

This paper is concerned with the effect of finite strain principal axis orientation on the reflection from a plane boundary of infinitesimal plane waves propagating in a half-space of incompressible isotropic elastic material. Attention is focussed on waves propagating in a principal plane of the deformation corresponding to simple shear. For a special class of constitutive laws it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave for every angle of incidence although its amplitude may vanish at certain discrete angles depending on the state of stress and deformation. Unlike the situation in which the underlying deformation is a pure homogeneous strain, however, the amplitude ratio of the reflected (plane harmonic) wave does not in general have unit magnitude, but its magnitude is independent of the pre-stress. Moreover, the angle of reflection differs from the angle of incidence. For materials not in this special class, on the other hand, it is shown that two plane harmonic waves may be reflected when the angle of incidence lies within certain ranges of values (which depend on the shear deformation). Outside this range there is in general a single reflected wave, and a surface wave is generated. For each class of materials the dependence of the reflected and surface wave behaviour on the angle of incidence, the magnitude of the shear strain and the state of stress is illustrated graphically.