In this chapter we begin to look at the features that characterize elastoplastic materials and at how these physical features are translated into a mathematical theory. The theory has grown steadily during the last century, with the impetus for development coming alternately from physical understanding of such materials and from insight into how the physical attributes might be modeled mathematically. We will eventually arrive, towards the end of this chapter, at a theory that is now regarded as classical and that incorporates all the main features of elastoplasticity. This theory may be further generalized, and placed in a unifying framework, if the ideas and techniques of convex analysis are employed. While the entire theory could easily have been developed ab initio in such a framework, we have chosen instead to focus first on giving a clear outline of the main features of the mathematical theory, without introducing any sophisticated ideas from convex analysis. In this way, we hope that the connection between physical behavior and its mathematical idealization may be more readily seen. Once such a theory is in place, the business of abstraction and generalization, using the tools of convex analysis, may begin.
KeywordsZirconium Manifold Hexagonal Stein Incompressibility
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