Unification of the mathematical model of elastic perfectly plastic solids: a convex analysis approach

TL;DR

This paper introduces a unified convex analysis framework for elastic perfectly plastic solids, providing a clearer mathematical formulation of constitutive laws and simplifying the flow evolution equations.

Contribution

It presents a novel convex analysis approach that unifies the mathematical modeling of elastic-perfectly plastic materials, including practical yield criteria.

Findings

Orthogonal decomposition of strain rates with respect to yield cones

Stress rate as projection on the tangent cone

Simplified flow evolution equations for elasto-plastic models

Abstract

A new mathematical formulation for the constitutive laws governing elastic perfectly plastic materials is proposed here. In particular, it is shown that the elastic strain rate and the plastic strain rate form an orthogonal decomposition with respect to the tangent cone and the normal cone of the yield domain. It is also shown that the stress rate can be seen as the projection on the tangent cone of the elastic stress tensor. This approach leads to a coherent mathematical formulation of the elasto-plastic laws and simplifies the resulting system for the associated flow evolution equations. The cases of one or two yields functions are treated in detail. The practical examples of the von Mises and Tresca yield criteria are worked out in detail to demonstrate the usefulness of the new formalism in applications.