Mathematical Modeling of Plasticity in Metals

We present a short introduction to continuum models for the plastic ow of metals. Our emphasis is on the physical principles underlying these models, the nature and validity of approximations involved, and the mathematical structure of the ow equations. Using the framework developed, we derive a simple, but realistic, model describing one-dimensional plastic ow. 1. Introduction When a piece of metal is subjected to stress, it responds by deforming. If only small stresses are applied, then the metal returns to its original shape when stress is relieved. In this regime, the metal is elastic. If, however, the stress exceeds a threshold, the yield stress, then the metal suuers permanent deformation. This is plastic behavior. Models of plasticity have been investigated by many scientists for many years. Early work assumed the deformations to be small; this approximation is often quite good, and it leads to some simpliication. A primary focus was static solutions and linearized dynamics. More recent work has addressed the problems posed by large deformations and nonlinear dynamics, but much remains to be understood. Physically realistic models of dynamic plasticity are necessarily complicated. The kine-matics of a three-dimensional continuum, the thermodynamics of materials, and the physics of microscopic defects all enter the description of plastic phenomena. Simpliied versions of such models are, of course, valuable for understanding the mathematical features of plastic ow. It is essential, however, for these simpliied models to be faithful to the physics. The purpose of this paper is to derive such a simpliied model, paying close attention to the nature and validity of the approximations made. Section 2 contains an overview of continuum models of plasticity. We discuss the governing conservation laws, measures of elastic and plastic strain, the form of constitutive relations dictated by thermodynamics, and the phenomenon of yielding. More speciic constitutive models are considered in Sec. 3. The elastic response of a metal is isotropic, and the shear strain that it can support is tiny; these properties allow us to formulate a constitutive model with a manageable number of empirical parameters. In Sec. 4, we specialize the general three-dimensional ow equations to one-dimensional ow with uniaxial strain. This ow connguration is used in the experimental determination of material properties under conditions of high pressure and high strain rate. The governing equations resemble those for reactive gas dynamics, but there is an important diierence, as we discuss. We conclude by brieey describing …