Elastoplastic and Viscoplastic Deformations in Solids and Structures

Abstract

AbstractThis chapter presents a summary of some recent developments in the formulation and the numerical integration of elastoplastic and viscoplastic models of solids and structures. A brief description of the constitutive relations in these models in both the infinitesimal and finite deformation ranges is presented first. A key aspect of the elastoplastic models is the yield condition constraining the stress state to the so‐calledelastic domain, with the viscoplastic models resulting in a regularization of this unilateral constraint. This constrained character of the problem leads naturally to the traditional return‐mapping algorithms used in the numerical integration of the local plastic evolution equations. The resulting discrete equations correspond to the so‐calledclosest‐point projectionequations by which an elastic trial state is projected onto the elastic domain. The mathematical structure of these equations is studied to obtain efficient numerical techniques for their solution. In particular, the rich variational structure of the associated problem is investigated in detail. Primal and dual variational formulations are derived for the elastoplastic and viscoplastic models. Both lead to unilaterally constrained elastoplastic problems reflecting the underlying constrained physical system. Augmented Lagrangian extensions of these formulations resulting in unconstrained forms of the discrete equations, both in their primal and dual versions, are also explored. The different numerical algorithms associated with each of these formulations, referred generically as closest‐point projection (CPPM) algorithms, are presented in detail for a general plastic model, not necessarily associated.