Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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 $ G$. It is shown that the maximum plastic work inequality together with the assumption that $ G$ 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.

References [Enhancements On Off] (What's this?)

  • [1] J. P. Aubin, Applied Functional Analysis, Wiley, New York, 1979 MR 549483
  • [2] P. Carter and J. B. Martin, Work bounding functions for plastic materials, J. Appl. Mech. 43, 434-438 (1976)
  • [3] F. H. Clarke, Generalised gradients and applications, Trans. Amer. Math. Soc. 205, 247-267 (1975) MR 0367131
  • [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, 251-281 (1965) MR 1553473
  • [8] M. E. Gurtin, Modern continuum thermodynamics, Mechanics Today 1 (ed. S. Nemat-Nasser), Pergamon, Oxford, 1974, pp. 168-210.
  • [9] B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, Jour. Méc. 14, 39-63 (1975) 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, Jour. Mech. Phys. Solids 35, 23-33 (1987) MR 873171
  • [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, Generalised flow potentials in finite elastoplasticity: II. Examples, Int. Jour. Eng. Sci. 23, 515-530 (1985) MR 792727
  • [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$ ^{0}$ 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. T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, 1970 MR 0274683
  • [26] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33, 174-222 (1970) MR 0262827
  • [27] J. C. Simo, A framework for finite strain elasto-plasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation, Comput. Methods Appl. Mech. Engrg. 66, 199-219 (1988) MR 927418
  • [28] E. Zeidler, Nonlinear Functional Analysis and its Applications. III: Variational Methods and Optimization, Springer-Verlag, New York, 1985 MR 768749

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DOI: https://doi.org/10.1090/qam/1040234
Article copyright: © Copyright 1990 American Mathematical Society

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