Time-Independent Plasticity Based on Thermodynamic Equilibrium and Its Stability

Abstract

Within the thermodynamic framework with internal variables by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19(6), pp. 433–455), Yang et al. (2014, “Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional,” ASME J. Eng. Mater. Technol., 136(2), p. 021001) established a model of time-independent plasticity of three states. In this model, equilibrium states are the states with vanishing thermodynamic forces conjugate to the internal variables, and correspond to critical points of the free energy or its complementary energy functions. Then, the conjugate forces play a role of yield functions and further lead to the consistency conditions. The model is further elaborated in this paper and extended to nonisothermal processes. It is shown that the incremental stress–strain relations are fully determined by the local curvature of the free energy or its complementary energy functions at the critical points, described by the Hessian matrices. It is further shown that the extended model can be well reformulated based on the intrinsic time in the sense of Valanis (1971, “A Theory of Viscoplasticity Without a Yield Surface, Part I. General Theory,” Arch. Mech., 23(4), pp. 517–533; 1975, “On the Foundations of the Endochronic Theory of Viscoplasticity,” Arch. Mech., 27(5–6), pp. 857–868), by taking the intrinsic time as the accumulated length of the variation of the internal variables during inelastic processes. It is revealed within this framework that the stability condition of equilibrium directly leads to Drucker (1951, “A More Fundamental Approach to Stress–Strain Relations,” First U.S. National Congress of Applied Mechanics, pp. 487–497) and Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech., 25(2), pp. 746–750) inequalities, by introducing the consistency condition into the work of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25(3), pp. 448–461). Generalized inequalities of Drucker (1951, “A More Fundamental Approach to Stress–Strain Relations,” First U.S. National Congress of Applied Mechanics, pp. 487–497) and Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech., 25(2), pp. 746–750) for nonisothermal processes are established straightforwardly based on the connection.