An internal variable theory of elastoplasticity based on the maximum plastic work inequality

Authors:
R. A. Eve, B. D. Reddy and R. T. Rockafellar

Journal:
Quart. Appl. Math. **48** (1990), 59-83

MSC:
Primary 73E99; Secondary 73B30, 73E50

DOI:
https://doi.org/10.1090/qam/1040234

MathSciNet review:
MR1040234

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Abstract | References | Similar Articles | Additional Information

Abstract: The methods of convex analysis are used to explore in greater depth the nature of the evolution equation in internal variable formulations of elastoplasticity. The evolution equation is considered in a form in which the thermodynamic force belongs to a set defined by a multi-valued map . It is shown that the maximum plastic work inequality together with the assumption that is maximal responsive (a term defined in Sec. 4), is necessary and sufficient to give a theory equivalent to that proposed by Moreau. Further consequences are investigated or elucidated, including the relationship between the yield function and the dissipation function; these functions are polars of each other. Examples are given to illustrate the theory.

**[1]**Jean-Pierre Aubin,*Applied functional analysis*, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Translated from the French by Carole Labrousse; With exercises by Bernard Cornet and Jean-Michel Lasry. MR**549483****[2]**P. Carter and J. B. Martin,*Work bounding functions for plastic materials*, J. Appl. Mech.**43**, 434-438 (1976)**[3]**Frank H. Clarke,*Generalized gradients and applications*, Trans. Amer. Math. Soc.**205**(1975), 247–262. MR**0367131**, https://doi.org/10.1090/S0002-9947-1975-0367131-6**[4]**B. D. Coleman and M. E. Gurtin,*Thermodynamics with internal state variables*, Jour. Chem. Phys.**47**, 597-613 (1967)**[5]**R. A. Eve, T. Gültop, and B. D. Reddy,*An internal variable finite strain theory of plasticity*, to appear**[6]**P. Germain, Q. S. Nguyen, and P. Suquet,*Continuum thermodynamics*, J. Appl. Mech.**50**, 1010-1019 (1983)**[7]**A. E. Green and P. M. Naghdi,*A general theory of an elastic-plastic continuum*, Arch. Rational Mech. Anal.**18**(1965), no. 4, 251–281. MR**1553473**, https://doi.org/10.1007/BF00251666**[8]**M. E. Gurtin,*Modern continuum thermodynamics*, Mechanics Today**1**(ed. S. Nemat-Nasser), Pergamon, Oxford, 1974, pp. 168-210.**[9]**Bernard Halphen and Nguyen Quoc Son,*Sur les matériaux standards généralisés*, J. Mécanique**14**(1975), 39–63 (French, with English summary). MR**0416177****[10]**R. Hill,*The essential structure of constitutive laws for metal composites and polycrystals*, Jour. Mech. Phys. Solids**15**, 79-95 (1967)**[11]**R. Hill,*Constitutive dual potentials in classical plasticity*, J. Mech. Phys. Solids**35**(1987), no. 1, 23–33. MR**873171**, https://doi.org/10.1016/0022-5096(87)90025-1**[12]**J. Kestin and J. R. Rice,*Paradoxes in the application of thermodynamics to strained solids*, A Critical Review of Thermodynamics (ed. E. B. Stuart), Mono Book Corp., Baltimore, 1970, pp. 275-298**[13]**S. J. Kim and J. T. Oden,*Generalised flow potentials in finite elastoplasticity*, Int. Jour. Eng. Sci.**22**, 1235-1257 (1984)**[14]**S. J. Kim and J. T. Oden,*Generalized flow potentials in finite elastoplasticity. II. Examples*, Internat. J. Engrg. Sci.**23**(1985), no. 5, 515–530. MR**792727**, https://doi.org/10.1016/0020-7225(85)90061-8**[15]**E. H. Lee,*Elastic-plastic deformation at finite strains*, Jour. Appl. Mech.**36**, 1-6 (1969)**[16]**J. Mandel,*Thermodynamics and plasticity*, Foundations of Continuum Thermodynamics (eds. J. J. Delgado Domingos, M. N. R. Nina, and J. H. Whitelaw), MacMillan Press, London, 1974, pp. 283-304**[17]**J. B. Martin,*An internal variable approach to the formulation of finite element problems in plasticity*, Physical Nonlinearities in Structural Analysis (eds. J. Hult and J. Lemaitre), Springer-Verlag, Berlin, 1981, 165-176**[18]**J. B. Martin and A. Nappi,*An internal variable formulation of perfectly plastic and linear kinematic and isotropic hardening relations with von Mises yield condition*, Meccanica, to appear**[19]**J. B. Martin and B. D. Reddy,*Variational principles and solution algorithms for internal variable formulations of problems in plasticity*, Omaggio a Giulio Ceradini: Note Scientifiche in Occasione del 70 Compleanno (ed. U. Andreaus et. al.), Università di Roma 'La Sapienza' (Roma), 1988, pp. 465-477**[20]**J. J. Moreau,*Sur les lois de frottement, de viscosité et plasticité*, C. R. Acad. Sc.**271**, 608-611 (1970)**[21]**S. Nemat-Nasser,*On finite deformation elasto-plasticity*, Internat. J. Solids and Structures**18**, 857-872 (1982)**[22]**J. T. Oden,*Qualitative Methods in Nonlinear Mechanics*, Prentice-Hall, New Jersey, 1979**[23]**J. R. Rice,*Inelastic constitutive relations for solids: An internal variable theory and its application to metal plasticity*, Jour. Mech. Phys. Solids**19**, 433-455 (1971)**[24]**J. R. Rice,*On the structure of stress-strain relations for time-dependent plastic deformation in metals*, J. Appl. Mech.**137**, 728-737 (1970)**[25]**R. Tyrrell Rockafellar,*Convex analysis*, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR**0274683****[26]**R. T. Rockafellar,*On the maximal monotonicity of subdifferential mappings*, Pacific J. Math.**33**(1970), 209–216. MR**0262827****[27]**J. C. Simo,*A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. I. Continuum formulation*, Comput. Methods Appl. Mech. Engrg.**66**(1988), no. 2, 199–219. MR**927418**, https://doi.org/10.1016/0045-7825(88)90076-X**[28]**Eberhard Zeidler,*Nonlinear functional analysis and its applications. III*, Springer-Verlag, New York, 1985. Variational methods and optimization; Translated from the German by Leo F. Boron. MR**768749**

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Additional Information

DOI:
https://doi.org/10.1090/qam/1040234

Article copyright:
© Copyright 1990
American Mathematical Society