Plasticity and Viscoplasticity

Abstract

The theory of plasticity describes the mechanics of deformation in plastically deforming solids, and, as applied to metals and alloys, it is based on experimental studies of the relations between stresses and strains under simple loading conditions. The theory described here assumes the ideal plastic body for which the Bauschinger effect and size effects are neglected. The theory also is valid only at temperatures for which recovery, creep, and thermal phenomena can be neglected. The basic theory of classical plasticity is described by Hill, and also in References, in addition to the books listed in Chap. 1. A concise description of the general plasticity theory necessary for metal forming is given in the book by Johnson et al.. In this chapter, certain important aspects of the theory are presented in order to elucidate the developments of the finite-element solutions of metal-forming problems discussed in this book. First, various measures of stress and strain are introduced. Then, the governing equations for plastic deformation and principles that are the foundations for the analysis are described. The extension of the theory of plasticity to time-dependent theory of viscoplasticity is outlined in Section 4.8. Particular references are made, in Sections 4.3 through 4.7, to the books by Hill and by Johnson and Mellor, and to the section on general plasticity theory in the book by Johnson et al.. The basic quantities that may be used to describe the mechanics of deformation when a body deforms from one configuration to another under an external load are the stress, strain, and strain-rate. Various measures of these quantities are defined, depending upon how closely formulations represent actual situations. Although it is not possible to provide the complete mathematical formulations in one-dimensional deformation, these measures are introduced for the case of simple uniaxial tension. Consider the uniaxial tension test of a round specimen whose initial length is l0 and cross-sectional area is A0. The specimen is stretched in the axial direction by the force P to the length l and the cross-sectional area A at time t, as shown in Fig. 4.1. The response of the material is recorded as the load-displacement curve, and converted to the stress-strain curve as shown in the figure. The deformation is assumed to be homogeneous until necking begins.