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Spatial regularity for general yield criteria in dynamic and quasi-static perfect plasticity

Jean-François Babadjian, Alessandro Giacomini, Maria Giovanna Mora

Abstract

This work addresses the question of regularity of solutions to evolutionary (quasi-static and dynamic) perfect plasticity models. Under the assumption that the elasticity set is a compact convex subset of deviatoric matrices, with $C^2$ boundary and positive definite second fundamental form, it is proved that the Cauchy stress admits spatial partial derivatives that are locally square integrable. In the dynamic case, a similar regularity result is established for the velocity as well. In the latter case, one-dimensional counterexamples show that, although solutions are Sobolev in the interior of the domain, singularities may appear at the boundary and the Dirichlet condition may fail to be attained.

Spatial regularity for general yield criteria in dynamic and quasi-static perfect plasticity

Abstract

This work addresses the question of regularity of solutions to evolutionary (quasi-static and dynamic) perfect plasticity models. Under the assumption that the elasticity set is a compact convex subset of deviatoric matrices, with boundary and positive definite second fundamental form, it is proved that the Cauchy stress admits spatial partial derivatives that are locally square integrable. In the dynamic case, a similar regularity result is established for the velocity as well. In the latter case, one-dimensional counterexamples show that, although solutions are Sobolev in the interior of the domain, singularities may appear at the boundary and the Dirichlet condition may fail to be attained.

Paper Structure

This paper contains 19 sections, 11 theorems, 297 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Linear algebra
  5. Measures
  6. Functional spaces
  7. Banach space-valued functions
  8. Spaces of admissible fields and duality between stress and plastic strain
  9. Some results concerning the distance function from a convex set
  10. The dynamical model
  11. Setting of the problem
  12. Norton-Hoff in dynamical perfect plasticity
  13. Existence of a strong solution in dynamical perfect plasticity
  14. The quasi-static model
  15. Norton-Hoff approximation
  16. ...and 4 more sections

Key Result

Theorem 2.2

Let $K\subseteq \mathbb{R}^N$ be a closed convex set satisfying eq:Kballs. The following properties hold true.

Figures (1)

  • Figure 1: The elastic region (in light blue) and the plastic region (in light red) in the example of Section \ref{['subsec52']}.

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Theorem 3.1: Norton-Hoff approximation
  • proof
  • Remark 3.2
  • Theorem 3.3: Higher spatial regularity
  • proof
  • Theorem 3.4: Strong evolutions in dynamic perfect plasticity
  • ...and 14 more