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Research Papers

Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional

[+] Author and Article Information
Q. Yang

State Key Laboratory of Hydroscience
and Engineering,
Tsinghua University,
Beijing 100084, China
e-mail: yangq@tsinghua.edu.cn

Y. R. Liu

State Key Laboratory of Hydroscience
and Engineering,
Tsinghua University,
Beijing 100084, China

X. Q. Feng, S. W. Yu

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China

If ξ is viewed as one set of generalized coordinates for a system, as done by Yang et al. [23] and Yang et al. [24], f is then the generalized forces conjugate to ξ, and the condition f = 0 is just the equilibrium condition of the system from the point of view of Lagrangian mechanics, see, e.g., Goldstein et al. [35].

With the existence of the free energy function, the relaxing process from State II to State III will follow a path with minimum dissipated energy [23], and the orthogonality condition of Ziegler [11] is related to the asymptotic stability of the equilibrium state III [25].

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received February 4, 2012; final manuscript received December 4, 2013; published online January 17, 2014. Assoc. Editor: Joost Vlassak.

J. Eng. Mater. Technol 136(2), 021001 (Jan 17, 2014) (9 pages) Paper No: MATS-12-1022; doi: 10.1115/1.4026232 History: Received February 04, 2012; Revised December 04, 2013

In this paper, time-independent plasticity is addressed 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, pp. 433–455). It is shown in this paper that the existence of a free energy function along with thermodynamic equilibrium conditions directly leads to associated flow rules. The time-independent inelastic behaviors can be fully determined by the Hessian matrix at the nondegenerate critical point of the free energy function. The normality rule of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25, pp. 448–461) or the Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech. 25, pp. 746–750) postulate is just a stability requirement of the thermodynamic equilibrium. The existence of a free energy functional which is not a direct function of the internal variables, along with thermodynamic equilibrium conditions also leads to associated flow rules. The time-independent inelastic behaviors with the free energy functional can be fully determined by the quasi Hessian matrix at the quasi critical point of the free energy functional. With the free energy functional, the thermodynamic forces conjugate to the internal variables are nonconservative and are constructed based on Darboux theorem. Based on the constructed nonconservative forces, it is shown that there may exist several possible thermodynamic equilibrium mechanisms for the thermodynamic system of the material sample. Therefore, the associated flow rules based on free energy functionals may degenerate into nonassociated flow rules. The symmetry of the conjugate forces plays a central role for the characteristics of time-independent plasticity.

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