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

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Abstract This is the second of a two part paper dealing with the inelastic response of materials. Part I (see Rajagopal and Srinivasa, 1998 International Journal of Plasticity 14 , 945–967) dealt with the structure of the constitutive equations for the elastic response of a material with multiple natural configurations. We now focus attention on the evolution of the natural configurations. We introduce two functions-the Helmholtz potential and the rate of dissipation function—representing the rate of conversion of mechanical work into heat. Motivated by, and generalizing the work of Ziegler (1963) in Progress in Solid Mechanics , Vol. 4, North-Holland, Amsterdam/New York; (1983) An Introduction to Thermodynamics, North-Holland, Amsterdam/New York) we then assume that the evolution of the natural configurations occurs in such a way that the rate of dissipation is maximized. This maximization is subject to the constraint that the rate of dissipation is equal to the difference 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 constitutive equations for the stress response and the evolution of the natural configurations from these two scalar functions. Of course, the maximum rate of dissipation criterion that is stated here is only an assumption that holds 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 that the resulting constitutive equations allow for response with and without yielding behavior and obtain a generalization of the normality and convexity conditions. We also show that, in the limit of quasistatic deformations, if one considers materials that possess yielding behavior, then the constitutive equations reduce to those corresponding to the strain space formulation of the rate independent theory of plasticity (see e.g. Naghdi (1990) Journal of Applied Mathematics and Physics A345 , 425–458.). Moreover, in this limit, the maximum rate of dissipation criterion, as stated here, is equivalent to the work inequality of Naghdi and Trapp (1975) Quartely Journal of Mechanics and Applied Mathematics 28 , 25–46). The main results together with an illustrative example are presented.