Some implications of work hardening and ideal plasticity
1. Summary. The purpose of this note is to point out the severe restriction imposed on possible stress-strain relations by a mathematical formulation of the concepts of work hardening and ideal plasticity. Using this condition, an algebraic derivation and extension is given of Prager's extension1 of the Mises plastic potential function. A brief discussion is also given of the meaning of stability of plastic deformation as contrasted with stability of non-conservative systems in general. 2. Introduction. A mathematical theory of plasticity must of necessity be based upon simplifying assumptions. For example, a good first approximation may be obtained by ignoring time effects in most structural metals at room temperature under isothermal conditions. A stress-strain diagram, therefore, represents a succession of states of static equilibrium. Furthermore, it is an experimental fact that metals generally work harden; then each equilibrium state is said to be stable. The phenomenon of the upper yield point, exhibited by a few metals, notably mild steel, is an important exception. Another exception is the region beyond the ultimate strength on a nominal tensile stress-strain plot when necking occurs, Fig. 1. However, this real instability is in terms of force; the actual stress continues to increase until failure occurs.