Scale-invariant initial value problems in one-dimensional dynamic elastoplasticity, with consequences for multidimensional nonassociative plasticity

Abstract

This paper solves a class of one-dimensional, dynamic elastoplasticity problems for equations which describe the longitudinal motion of a rod. The initial conditionsU(x, 0)are continuous and piecewise linear, the derivative ∂U/∂x(x, 0) having just one jump atx= 0. Both the equations and the initial data are invariant under the scalingŨ(x, t) = α−1U(αx, αt), where α > 0; hence the termscale-invariant. Both in underlying motivation and in solution, this problem is highly analogous to the Riemann problem from gas dynamics. These ideas are applied to the Sandler–Rubin example of non-unique solutions in dynamic plasticity with a nonassociative flow rule. We introduce an entropy condition that re-establishes uniqueness, but we also exhibit problems regarding existence.