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

Published online by Cambridge University Press:  16 July 2009

David G. Schaeffer  and
Michael Shearer
David G. Schaeffer
Affiliation:
Department of Mathematics, Duke University, NC 27706, USA
Michael Shearer
Affiliation:
Department of Mathematics, North Carolina State University, NC 27607, USA
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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.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 3 , September 1992 , pp. 225 - 254
Copyright
Copyright © Cambridge University Press 1992

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