Mechanics of the inelastic behavior of materials. Part II: Inelastic response

This is the second of a two part paper dealing with the inelastic response of materials. Part I (see Rajagopal and Srinivasa (1998) International Jour nal of Plasticity 14, 945-967) dealt with the structure of the constitutive equations for the elastic response of a material with multiple natural con figurations. We now focus attention on the evolution of the natural configu rations. We introduce two functions-the Helmholtz potential and the rate of dissipation function-representing the rate of conversion of mechanical wor k into heat. Motivated by, and generalizing the work of Ziegler (1963) in P rogress in Solid Mechanics, Vol. 4, North-Holland, Amsterdam/New York; (198 3) An Introduction to Thermodynamics, North-Holland, Amsterdam/New York) we then assume that the evolution of the natural configurations occurs in suc h a way that the rate of dissipation is maximized. This maximization is sub ject to the constraint that the rate of dissipation is equal to the differe nce between the rate of mechanical working and the rate of increase of the Helmholtz potential per unit volume. This then allows us to derive the cons titutive equations for the stress response and the evolution of the natural configurations from these two scalar functions. Of course, the maximum rat e of dissipation criterion that is stated here is only an assumption that h olds for a certain class of materials under consideration. Our quest is to see whether such an assumption gives reasonable results. In the process, we hope to gain insight into the nature of such materials. We demonstrate tha t the resulting constitutive equations allow for response with and without yielding behavior and obtain a generalization of the normality and convexit y conditions. We also show that, in the limit of quasistatic deformations, if one considers materials that possess yielding behavior, then the constit utive equations reduce to those corresponding to the strain space formulati on of the rate independent theory of plasticity (see e.g. Naghdi (1990) Jou rnal of Applied Mathematics and Physics A345, 425-458.). Moreover, in this limit, the maximum rate of dissipation criterion, as stated here, is equiva lent to the work inequality of Naghdi and Trapp (1975) Quartely Journal of Mechanics and Applied Mathematics 28, 25-46). The main results together wit h an illustrative example are presented. (C) 1998 Elsevier Science Ltd. All rights reserved.