Nonlinear Plasticity

Abstract

In this chapter we discuss a general class of materials with memory, the plastic materials, which are useful in describing the behavior of metals. Our purpose is to present the basic theory, in which some concepts of Chap. XII are further developed and illustrated, and in which the theory of elasticity plays a central role, in as simple a context as is compatible with the underlying physics. The exposition is simpler than that of most treatments because we consistently use internal variables in the material formulation, which obviates the need for a complicated treatment of frame-indifference, as is necessary in the spatial treatment of theories involving stress rates. To illustrate the nature of such theories, we treat a model with a lot of physical structure in Secs. 2 and 3. In Sec. 4 we show how to formulate a natural numerical approach for the solution of a particular dynamical problem. In Sec. 5 we give a general formulation of antiplane problems, whose degeneracies illuminate subtle difficulties with the concept of permanent plastic deformation. We conclude this chapter with a brief discussion of discrete models, which are used to motivate various theories.