Nonlinear Elliptic Systems Arising from plasticity Theory

Abstract

In this chapter (which is a continuation of Chapter 9 and will rely heavily on it) we are interested in specific models of plasticity. We shall need to attach to tensor σ its deviator

$${\sigma _D} = \sigma - 1/nI\;tr\;\sigma ,$$

which has trace 0. We will consider two models of plasticity, the Hencky model, which is a model of perfect plasticity where,

$$|{\sigma _D}|\mu $$

(µ is a given constant), and the Norton—Hoff model, which is an approximation to the Hencky model, where the constraint of perfect plasticity is relaxed with a penalty term. In fact, the Norton—Hoff model will be a particular case of the models considered in Chapter 9, but we shall consider a sequence of these models. Again these models are formulated as variational problems in which the unknown is the stress tensor and the displacement is recovered indirectly. The convergence of the approximation is very natural in the context of variational problems and follows from general penalty methods (see R. Temam [101], G. Duvaut, J.L. Lions [15], J.L. Lions [70], P. Le Tallec [69]).