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
Full-text PDF Free Access
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.
- 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
P. Carter and J. B. Martin, Work bounding functions for plastic materials, J. Appl. Mech. 43, 434–438 (1976)
- Frank H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262. MR 367131, DOI https://doi.org/10.1090/S0002-9947-1975-0367131-6
B. D. Coleman and M. E. Gurtin, Thermodynamics with internal state variables, Jour. Chem. Phys. 47, 597–613 (1967)
R. A. Eve, T. Gültop, and B. D. Reddy, An internal variable finite strain theory of plasticity, to appear
P. Germain, Q. S. Nguyen, and P. Suquet, Continuum thermodynamics, J. Appl. Mech. 50, 1010–1019 (1983)
- 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, DOI https://doi.org/10.1007/BF00251666
M. E. Gurtin, Modern continuum thermodynamics, Mechanics Today 1 (ed. S. Nemat-Nasser), Pergamon, Oxford, 1974, pp. 168–210.
- Bernard Halphen and Nguyen Quoc Son, Sur les matériaux standards généralisés, J. Mécanique 14 (1975), 39–63 (French, with English summary). MR 416177
R. Hill, The essential structure of constitutive laws for metal composites and polycrystals, Jour. Mech. Phys. Solids 15, 79–95 (1967)
- R. Hill, Constitutive dual potentials in classical plasticity, J. Mech. Phys. Solids 35 (1987), no. 1, 23–33. MR 873171, DOI https://doi.org/10.1016/0022-5096%2887%2990025-1
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
S. J. Kim and J. T. Oden, Generalised flow potentials in finite elastoplasticity, Int. Jour. Eng. Sci. 22, 1235–1257 (1984)
- 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, DOI https://doi.org/10.1016/0020-7225%2885%2990061-8
E. H. Lee, Elastic-plastic deformation at finite strains, Jour. Appl. Mech. 36, 1–6 (1969)
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
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
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
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
J. J. Moreau, Sur les lois de frottement, de viscosité et plasticité, C. R. Acad. Sc. 271, 608–611 (1970)
S. Nemat-Nasser, On finite deformation elasto-plasticity, Internat. J. Solids and Structures 18, 857–872 (1982)
J. T. Oden, Qualitative Methods in Nonlinear Mechanics, Prentice-Hall, New Jersey, 1979
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)
J. R. Rice, On the structure of stress-strain relations for time-dependent plastic deformation in metals, J. Appl. Mech. 137, 728–737 (1970)
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216. MR 262827
- 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, DOI https://doi.org/10.1016/0045-7825%2888%2990076-X
- 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
J. P. Aubin, Applied Functional Analysis, Wiley, New York, 1979
P. Carter and J. B. Martin, Work bounding functions for plastic materials, J. Appl. Mech. 43, 434–438 (1976)
F. H. Clarke, Generalised gradients and applications, Trans. Amer. Math. Soc. 205, 247–267 (1975)
B. D. Coleman and M. E. Gurtin, Thermodynamics with internal state variables, Jour. Chem. Phys. 47, 597–613 (1967)
R. A. Eve, T. Gültop, and B. D. Reddy, An internal variable finite strain theory of plasticity, to appear
P. Germain, Q. S. Nguyen, and P. Suquet, Continuum thermodynamics, J. Appl. Mech. 50, 1010–1019 (1983)
A. E. Green and P. M. Naghdi, A general theory of an elastic-plastic continuum, Arch. Rational Mech. Anal. 18, 251–281 (1965)
M. E. Gurtin, Modern continuum thermodynamics, Mechanics Today 1 (ed. S. Nemat-Nasser), Pergamon, Oxford, 1974, pp. 168–210.
B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, Jour. Méc. 14, 39–63 (1975)
R. Hill, The essential structure of constitutive laws for metal composites and polycrystals, Jour. Mech. Phys. Solids 15, 79–95 (1967)
R. Hill, Constitutive dual potentials in classical plasticity, Jour. Mech. Phys. Solids 35, 23–33 (1987)
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
S. J. Kim and J. T. Oden, Generalised flow potentials in finite elastoplasticity, Int. Jour. Eng. Sci. 22, 1235–1257 (1984)
S. J. Kim and J. T. Oden, Generalised flow potentials in finite elastoplasticity: II. Examples, Int. Jour. Eng. Sci. 23, 515–530 (1985)
E. H. Lee, Elastic-plastic deformation at finite strains, Jour. Appl. Mech. 36, 1–6 (1969)
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
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
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
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
J. J. Moreau, Sur les lois de frottement, de viscosité et plasticité, C. R. Acad. Sc. 271, 608–611 (1970)
S. Nemat-Nasser, On finite deformation elasto-plasticity, Internat. J. Solids and Structures 18, 857–872 (1982)
J. T. Oden, Qualitative Methods in Nonlinear Mechanics, Prentice-Hall, New Jersey, 1979
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)
J. R. Rice, On the structure of stress-strain relations for time-dependent plastic deformation in metals, J. Appl. Mech. 137, 728–737 (1970)
R. T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, 1970
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33, 174–222 (1970)
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)
E. Zeidler, Nonlinear Functional Analysis and its Applications. III: Variational Methods and Optimization, Springer-Verlag, New York, 1985
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73E99,
73B30,
73E50
Retrieve articles in all journals
with MSC:
73E99,
73B30,
73E50
Additional Information
Article copyright:
© Copyright 1990
American Mathematical Society
