Shear waves in hardening rigidly plastic bodies

Abstract

The dynamic flow of a hardening rigidly plastic medium with a flow law related to the Mises or Tresca yield condition is considered. The Odqvist parameter is taken as the hardening parameter (the Odqvist parameter is related to the specific plastic work w in the Mises theory by the dependence dw = kdχ where k is the shear yield limit). It is shown that in an arbitrary continuous medium the surface of weak velocity discontinuity ( S) at each point should be tangent to the principal direction of the strain rate tensor, if the principal directions of this tensor are continuous. The flow law imposes new constraints on S. Thus, weak velocity discontinuities can only be on the maximal shear surface in the Mises theory. As an illustration, the problem of plastic deformation propagation in a half-plane is considered when the velocity is given on an edge, where the solution shows that weak discontinuities can be caused by the boundary conditions. It turns out also that a continuous solution or one containing only weak velocity discontinuities is not always possible. In this connection, the problem of the structure of a strong velocity discontinuity (shock) is solved with viscosity taken into account. Existence conditions for the shock, and an equation governing its propagation velocity are obtained. The results obtained are applied to the problem of strain propagation in rigidly plastic bodies. Among the large quantity of papers on the wave theory of plastic media, we note /1–3/ as being closest to the theme of the present paper.