The main concern here is the analysis of plastic deformation processes in the warm and hot forming regimes. When deformation takes place at high temperatures, material properties can vary considerably with temperature. Heat is generated during a metal-forming process, and if dies are at a considerably lower temperature than the workpiece, the heat loss by conduction to the dies and by radiation and convection to the environment can result in severe temperature gradients within the workpiece. Thus, the consideration of temperature effects in the analysis of metal-forming problems is very important. Furthermore, at elevated temperatures, plastic deformation can induce phase transformations and alterations in grain structures that, in turn, can modify the flow stress of the workpiece material as well as other mechanical properties. Since materials at elevated temperatures are usually rate-sensitive, a complete analysis of hot forming requires two considerations—the effect of the rate-sensitivity of materials and the coupling of the metal flow and heat transfer analyses. A material behavior that exhibits rate sensitivity is called viscoplastic. A theory that deals with viscoplasticity was described in Chap. 4. It was shown that the governing equations for deformation of viscoplastic materials are formally identical to those of plastic materials, except that the effective stress is a function of strain, strain-rate, and temperature. The application of the finite-element method to the analysis of metal-forming processes using rigid-plastic materials leads to a simple extension of the method to rigid-viscoplastic materials. The importance of temperature calculations during a metal-forming process has been recognized for a long time. Until recently, the majority of the work had been based on procedures that uncouple the problem of heat transfer from the metal deformation problem. Several researchers have used the following approach. They determined the flow velocity fields in the problem either experimentally or by calculations, and they then used these fields to calculate heat generation. Examples of this approach are the works of Johnson and Kudo on extrusion, and of Tay et al. on machining. Another approach uses Bishop’s numerical method in which heat generation and transportation are considered to occur instantaneously for each time-step with conduction taking place during the time-step.
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.
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