Work Conditions and Energy Functions for Ideal Elastic-Plastic Materials

Abstract

The paper deals with the cyclic second law and Il’yushin’s condition for isothermal, ideal, isotropic, elastic-plastic materials at large deformations. The second law is equivalent to the existence of the elastic potential and the nonnegativity of plastic power. The material admits infinitely many free energies: The set of all energy functions is described in terms of a dissipation function. Its convexification provides the optimal lower bound for plastic work; it also figures in the maximal and minimal energies. Il’yushin’s condition is equivalent to the existence of the elastic potential and a new condition that is stronger than the normality of the plastic stretching and the convexity of the stress range. Il’yushin’s condition is also equivalent to the existence of a new kind of energy function called “the initial and final extended energy functions.” Materials of type C are introduced for which the initial extended energy function has additional convexity properties. It can be viewed as a stored energy of a Hencky hyperelastic material associated with the elastic-plastic material.