Reflection and Refraction of Weak Elastic-Plastic Waves

Abstract

The method of singular surfaces is used to obtain expressions for the amplitudes of weak discontinuities reflected from or transmitted across interfaces between solids of dissimilar elastic-plastic properties. Here weak discontinuities are taken to mean discontinuities in derivatives of stress, strain, and velocity components. These discontinuities occur across singular surfaces which propagate at characteristic wave speeds and are referred to as weak waves. Analogous to elastic wave propagation results, two reflected and two refracted fronts satisfy stress and velocity continuity conditions in media prestressed into the plastic range. However, the speeds of these fronts are generally less than the elastic dilatational and shear wave speeds, and the amplitudes of the reflected and refracted discontinuities can differ dramatically from their elastic counterparts. Numerical examples are considered in which weak waves are reflected from rigid and stress-free surfaces. The medium through which the waves pass is prestressed in the direction parallel to the reflecting surface. Results are presented which show the dependence of the reflected velocity and stress discontinuity amplitudes on the angle of incidence of the oncoming wave. As compared to elastic wave propagation, the presence of plastic deformation reduces the amplitude of the reflected front which travels at the speed of the incident front and raises the amplitude of the other reflected front. The most pronounced effect of plastic deformation is found when the incident front travels at the slow wave speed (SV-type wave). In this case, the critical angle of incidence (beyond which reflected weak waves alone cannot satisfy the boundary conditions) decreases to 22.5 deg from the elastic value of 30 deg when Poisson’s ratio is 1/3. It is conjectured that elastic-plastic surface waves may be needed to satisfy the interface conditions at incidence angles beyond this critical value.