Existence and uniqueness of solutions to uni-axial elastic-plastic wave interactions

Abstract

This paper treats the propagation of stress waves through an elastic-plastic medium on the assumption of uni-axial displacement. With the further simplification to a piecewise linear stress-strain curve in terms of engineering stress and strain, wave equations are obtained for the longitudinal stress in both elastic and plastic regions, each with a distinct constant Lagrangian wave speed. The stress distribution in any region is then simply expressed in terms of two wave functions. In a general motion the medium will be divided into a sequence of alternating elastic and plastic regions separated by moving interfaces. A detailed analysis is presented for a single-interface wave interaction under general initial conditions, namely, continuous initial waves in the two directions in both elastic and plastic regions with a non-uniform yield stress in the elastic region. For different sets of initial conditions six distinct types of solution are shown to exist, and these are classified according to the direction and speed of the interface. In particular, two types involve interface speeds in excess of the elastic wave speed, not, to the authors’ knowledge, demonstrated in previous plastic wave treatments, noting the absence of possible shock formation for the present linearized stress-strain laws. Further, it is shown that stress discontinuities cannot form at the interface (or elsewhere) from initially continuous stress profiles. Associated with the different types of solution are four distinct sets of interface conditions so that there is no common form for the interaction solution. Each of the six types of solution is shown to be consistent with the elastic-plastic model only under a restricted set of initial conditions, and these sets are found to be mutually exclusive for the six types, thus deciding a unique choice for the type of single-interface solution. The six sets, however, are not inclusive of all possible initial conditions, indicating a need for multi-interface solutions in the exceptional situations. Multi-interface solutions may be possible even in the non-exceptional situations, but this possibility is felt to be unlikely. Finally, it can be noted that the analysis dealing with validity of solution is, for most cases, only local in that it applies in some small neighbourhood of the current point on the interface path, being based on expansions about this point. The results of such local analysis will therefore extend to the case of non-uniform wave speeds arising from non-linear stress-strain laws, provided that no shock is formed in the neighbourhood, but a global solution can no longer be expressed simply in terms of wave functions.