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] Stuart S. Antman and William G. Szymczak, Nonlinear elastoplastic waves, Current progress in hyperbolic systems: Riemann problems and computations (Brunswick, ME, 1988) Contemp. Math., vol. 100, Amer. Math. Soc., Providence, RI, 1989, pp. 27–54. MR 1033507, https://doi.org/10.1090/conm/100/1033507
  • [2] John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 0475169, https://doi.org/10.1007/BF00279992
  • [3] Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
  • [4] Philippe G. Ciarlet and Giuseppe Geymonat, Sur les lois de comportement en élasticité non linéaire compressible, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 295 (1982), no. 4, 423–426 (French, with English summary). 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 (1990), no. 1, 59–83. MR 1040234, https://doi.org/10.1090/qam/1040234
  • [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] Bernard Halphen and Nguyen Quoc Son, Sur les matériaux standards généralisés, J. Mécanique 14 (1975), 39–63 (French, with English summary). 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 (1984), no. 11-12, 1235–1257. MR 769970, https://doi.org/10.1016/0020-7225(84)90019-3
  • [15] 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
  • [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 (1985), no. 3, 277–302. MR 820832, https://doi.org/10.1016/0045-7825(85)90119-7
  • [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, Internat. J. Numer. Methods Engrg. 29 (1990), no. 3, 483–514. MR 1040734, https://doi.org/10.1002/nme.1620290304
  • [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 (1988), no. 1, 101–117. MR 918319, https://doi.org/10.1007/BF01401024
  • [26] B. D. Reddy and F. Tomarelli, The obstacle problem for an elastoplastic body, Appl. Math. Optim. 21 (1990), no. 1, 89–110. MR 1014947, https://doi.org/10.1007/BF01445159
  • [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. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
  • [29] 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
  • [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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