Orthogonal projections onto convex sets and the application to problems in plasticity

We review the classical theory of static and quasi-static plasticity in an abstract framework of convex analysis. It can be shown that the extended use of orthogonal projections onto closed convex sets simplifies the analysis substantially. We present new characterizations of the primal problem in plasticity. The abstract setting is applied to problems in perfect plasticity and to plasticity with nonlinear hardening. Furthermore, our approach leads to a better understanding of the radial return algorithm which is commonly used in computational plasticity. We study the fundamental role of orthogonal projections in plasticity in an abstract framework of convex analysis. This is the key to the mathematical understanding of the radial return algorithm and the consistent tangent operator introduced by SIMO-TAYLOR [16]. In this abstract framework, the basic results in TEMAM [17] for static perfect plasticity become clearer and can be applied to computational plasticity, and we derive a new characterization of the primal problem. We obtain the quasi-static formulation of plasticity as the limit of increments, where every time increment corresponds to a static problem. Then, the quasi-static problem is studied as a dissipative evolution equation. This is a unified and simplified framework which can be applied to more general problems such as the coupling of plasticity and friction or contact. In addition, this presentation allows a simple analysis of the limit problem. The analytical core of the quasi-static analysis is the observation that the resolvent of the subdifferential of the indicator function of a closed convex set coincides with the orthogonal projection. The main difference between our proof and the arguments given in DUVAUT-LIONS [6] is the consequent use of the orthogonal projection and its properties. Furthermore, our approach formalizes quasi-static plasticity in the context of maximal monotone operators as it is considered for the dynamic case in ALBER [1]; in particular, this makes the theory of the transformation of internal variables available to the quasi-static setting. 1991 Mathematics Subject Classification. 73E50.