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Journal of Convex Analysis 30 (2023), No. 1, 081--110
Copyright Heldermann Verlag 2023

Mixed Boundary Conditions as Limits of Dissipative Boundary Conditions in Dynamic Perfect Plasticity

Jean-Francois Babadjian
Lab. de Mathématiques d'Orsay, Université Paris-Saclay, France

Randy Llerena
Research Platform MMM Mathematics - Magnetism - Materials, University of Vienna, Austria

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 by J.-F. Babadjian and V. Crismale [ Dissipative boundary conditions and entropic solutions in dynamical perfect plasticity, J. Math. Pures Appl. (9) 148 (2021) 75--127]. 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.

Keywords: Elasto-plasticity, boundary conditions, convex analysis, functionals of measures, functions of bounded deformation, calculus of variations, dynamic evolution.

MSC: 74C10, 35Q74, 49J45, 49Q20, 35F31.

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