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 conditions U(x, 0) are continuous and piecewise linear, the derivative ∂U/∂x(x, 0) having just one jump at x = 0. Both the equations and the initial data are invariant under the scaling Ũ(x, t) = α−1U(αx, αt), where α > 0; hence the term scale-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.

Numerical computations for antiplane shear in a granular flow model

“…Scale-invariant problems. Scale-invariant problems studied by Schaeffer and Shearer [10] consist of system (1) with a = 0 and initial data that is continuous and piecewise linear. Schaeffer and Shearer construct piecewise linear solutions of this problem.…”

“…Relation (10) holds also for left-waves. The assumptions on H(y) imply that /?>VA ^ 0 and therefore the left-and right-wave families are genuinely nonlinear.…”

“…The first three cases are applicable to granular flow [9], [10]. If H(y) is given by (3-iv), then system (1) with a -0 reduces to a model for longitudinal deformation discussed by Lee [7], In this setting, the variables of the system are velocity and stress in the x-direction.…”

A mathematical model for localization in granular flow

“…which exceeds ce if the inner products ( j , £> and (JUt, £> have opposite signs. The Sandler-Rubin example was further analysed in (Schaeffer & Shearer 1991).I t turns out th a t the non-uniqueness problem can be eliminated by imposing an appropriate Lax-type (Lax 1957) entropy condition a t elastic-plastic fronts, but th a t there are still continuous, piecewise smooth initial conditions for which, even in the small, the equations have no continuous solution; i.e. existence is problematic.…”

Stability of Shear Bands in an Elastoplastic Model for Granular Flow: The Role of Discreteness

“…In particular, the strength of the shear band varies periodically. Elastic waves and elastic-plastic interfaces [8] propagate through the sample, and changes in the strength of the shear band are triggered by their arrival.…”