Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



An internal variable finite-strain theory of plasticity within the framework of convex analysis

Authors: R. A. Eve, T. Gültop and B. D. Reddy
Journal: Quart. Appl. Math. 48 (1990), 625-643
MSC: Primary 73G20; Secondary 73E05, 73S10
DOI: https://doi.org/10.1090/qam/1079910
MathSciNet review: MR1079910
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Abstract | References | Similar Articles | Additional Information

Abstract: An internal variable constitutive theory for elastic-plastic materials undergoing finite strains is presented. The theory is based on a corresponding study in the context of small strains [6], and has the following features: first, with a view to embracing the classical notions of convex yield surfaces and the normality law, the evolution law is developed within the framework of nonsmooth convex analysis, which proves to be a powerful unifying tool; secondly, the special case of elastic materials is recovered from the theory in a natural manner. After presentation of the theory a concrete example is discussed in detail.

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  • [1] S. S. Antman and W. G. Szymczak, Nonlinear elastoplastic waves, Contemp. Math. 100, 27-54 (1989) MR 1033507
  • [2] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337-406 (1977) MR 0475169
  • [3] P. G. Ciarlet, Mathematical Elasticity, Volume I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988 MR 936420
  • [4] P. G. Ciarlet and G. Geymonat, Sur les lois de comportement en élasticité nonlinéaire compressible, C. R. Acad. Sci. Paris Sér. II 295, 423-426 (1982) MR 695540
  • [5] B. D. Coleman and M. E. Gurtin, Thermodynamics with internal state variables, J. Chem. Phys. 47, 597-613 (1967)
  • [6] R. A. Eve, B. D. Reddy, and R. T. Rockafellar, An internal variable theory of elastoplasticity based on the maximum plastic work inequality, Quart. Appl. Math. 48, 59-83 (1990) MR 1040234
  • [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] B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, J. Méc. 14, 39-63 (1975) MR 0416177
  • [9] R. Hill, The essential structure of constitutive laws for metal composites and polycrystals, J. Mech. Phys. Solids 15, 79-95 (1967)
  • [10] R. Hill, On constitutive inequalities for simple materials. II, J. Mech. Phys. Solids 16, 315-322 (1968)
  • [11] R. Hill and J. R. Rice, Elastic potentials and the structure of inelastic constitutive laws, SIAM J. Appl. Math. 25, 448-461 (1973)
  • [12] J. W. Hutchinson, Finite strain analysis of elastic-plastic solids and structures, Numerical Solution of Nonlinear Structural Problems (ed. R. F. Hartung), Amer. Soc. Mech. Eng., New York, pp. 17-29 1973
  • [13] 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
  • [14] S. J. Kim and J. T. Oden, Generalized potentials in finite elastoplasticity, Internat. J. Engrg. Sci. 22, 1235-1257 (1984) MR 769970
  • [15] S. J. Kim and J. T. Oden. Generalized potentials in finite elastoplasticity, II. Example, Internat. J. Engrg. Sci. 23, 510-530 (1985) MR 792727
  • [16] S. J. Kim and J. T. Oden, Finite element analysis of a class of problems in finite elastoplasticity based on the thermodynamical theory of materials of type N, Comput. Methods Appl. Mech. Engrg. 53, 277-302 (1985) MR 820832
  • [17] E. H. Lee, Elastic-plastic deformations at finite strains, J. Appl. Mech. 36, 1-6 (1969)
  • [18] J. Mandel, Thermodynamics and plasticity, Foundations of Continuum Thermodynamics (ed. J. J. Delgado Domingos, M. N. R. Nina, and J. H. Whitelaw), Macmillan, 1974, pp. 283-304
  • [19] J. B. Martin, An internal variable approach to the formulation of finite element problems in plasticity, Physical Nonlinearities in Structural Analysis (ed. J. Hult and J. Lemaitre), Springer, Berlin, 1981, pp. 165-176
  • [20] 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^ \circ }$ Compleanno (ed. U. Andreaus et al.), Università di Roma 'La Sapienza', Roma, 1988, pp. 465-477
  • [21] J. B. Martin, B. D. Reddy, T. B. Griffin, and W. W. Bird, Application of mathematical programming concepts to incremental elastic-plastic analysis, Engrg. Struct. 9, 171-176 (1987)
  • [22] B. Moran, M. Ortiz, and C. F. Shih, Formulation of implicit finite element methods for multiplicative finite deformation plasticity, Technical Report, Division of Engineering, Brown Univ., 1989 MR 1040734
  • [23] J. J. Moreau, Sur les lois de frottement, de viscosité et plasticité, C. R. Acad. Sci 271, 608-611 (1970)
  • [24] A. Needleman, On finite element formulations for large elastic-plastic deformations, Comp. Struct. 20, 247-257 (1985)
  • [25] B. D. Reddy and T. B. Griffin, Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem, Numer. Math. 52, 101-117 (1988) MR 918319
  • [26] B. D. Reddy and F. Tomarelli, The obstacle problem for an elastic-plastic body, Appl. Math. Optim. 21, 89-110 (1990) MR 1014947
  • [27] J. R. Rice, Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity, J. Mech. Phys. Solids 19, 433-455 (1971)
  • [28] R. T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, 1970 MR 0274683
  • [29] J. C. Simo, A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation, Comput. Methods Appl. Mech. Engrg. 66, 199-219 (1988) MR 927418
  • [30] J. C. Simo and M. Ortiz, A unified approach to finite deformation elastoplasticity based on the use of hyperelastic constitutive equations, Comput. Methods Appl. Mech. Engrg. 49, 222-235 (1985)

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DOI: https://doi.org/10.1090/qam/1079910
Article copyright: © Copyright 1990 American Mathematical Society

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