Mixed boundary conditions as limits of dissipative boundary conditions in dynamic perfect plasticity

TL;DR

This paper investigates the mathematical well-posedness of a dynamic perfect plasticity model with mixed boundary conditions, using asymptotic analysis and measure theoretic duality to extend results to more general boundary scenarios.

Contribution

It extends the analysis of perfect plasticity models to mixed boundary conditions, including unbounded deviatoric stresses, and provides new theoretical insights in this complex setting.

Findings

Complete solutions for Dirichlet and Neumann boundary conditions.

Partial results for mixed boundary conditions in 2D and 3D.

Extension of measure theoretic duality to unbounded stresses.

Abstract

This paper addresses the well posedness of a dynamical model of perfect plasticity with mixed boundary conditions for general closed and convex elasticity sets. The proof relies on an asymptotic analysis of the solution of a perfect plasticity model with relaxed dissipative boundary conditions obtained in [7]. One of the main issues consists in extending the measure theoretic duality pairing between stresses and plastic strains, as well as a convexity inequality to a more general context where deviatoric stresses are not necessarily bounded. Complete answers are given in the pure Dirichlet and pure Neumann cases. For general mixed boundary conditions, partial answers are given in dimension $2$ and $3$ under additional geometric hypothesis on the elasticity set and the reference configuration.