Archive for Rational Mechanics and Analysis

, Volume 180, Issue 2, pp 237–291

Quasistatic Evolution Problems for Linearly Elastic–Perfectly Plastic Materials

  • Gianni Dal Maso
  • Antonio DeSimone
  • Maria Giovanna Mora


The problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate-independent processes. Existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation. This approach provides a new approximation result for the solutions of the quasistatic evolution problem, which are shown to be absolutely continuous in time. Four equivalent formulations of the problem in rate form are derived. A strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain.


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  1. 1.
    Anzellotti, G.: On the extremal stress and displacement in Hencky plasticity. Duke Math. J. 51, 133–147 (1984)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Anzellotti, G., Giaquinta, M.: On the existence of the field of stresses and displacements for an elasto-perfectly plastic body in static equilibrium. J. Math. Pures Appl. 61, 219–244 (1982)MATHMathSciNetGoogle Scholar
  3. 3.
    Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. 2nd rev. ed. Reidel, Dordrecht, 1986Google Scholar
  4. 4.
    Brezis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam-London; American Elsevier, New York, 1973Google Scholar
  5. 5.
    Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458, 299–317 (2002)CrossRefADSMATHMathSciNetGoogle Scholar
  6. 6.
    Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Eve, R.A., Reddy, B.D., Rockafellar, R.T.: An internal variable theory of elastoplasticity based on the maximum plastic work inequality. Quart. Appl. Math. 48, 59–83 (1990)MATHMathSciNetGoogle Scholar
  8. 8.
    Giusti, E.: Minimal surfaces and functions of bounded variation. Birkhäuser, Boston, 1984Google Scholar
  9. 9.
    Goffman, C., Serrin, J.: Sublinear functions of measures and variational integrals. Duke Math. J. 31, 159–178 (1964)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Han, W., Reddy, B.D.: Plasticity. Mathematical Theory and Numerical Analysis. Springer Verlag, Berlin, 1999Google Scholar
  11. 11.
    Hill, R.: The mathematical Theory of Plasticity. Clarendon Press, Oxford, 1950Google Scholar
  12. 12.
    Johnson, C.: Existence Theorems for Plasticity Problems. J. Math. Pures Appl. 55, 431–444 (1976)MATHMathSciNetGoogle Scholar
  13. 13.
    Kohn, R.V., Temam, R.: Dual spaces of stresses and strains, with applications to Hencky plasticity. Appl. Math. Optim. 10, 1–35 (1983)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Lubliner, J.: Plasticity Theory. Macmillan New York, 1990Google Scholar
  15. 15.
    Mainik, A., Mielke, A.: Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differential Equations 22, 73–99 (2005)MATHMathSciNetGoogle Scholar
  16. 16.
    Martin, J.B.: Plasticity. Fundamentals and General Results. MIT Press, Cambridge, 1975Google Scholar
  17. 17.
    Matthies, H., Strang, G., Christiansen, E.: The saddle point of a differential program. Energy Methods in Finite Element Analysis, R. Glowinski, E. Rodin, O.C. Zienkiewicz, (ed.) Wiley, New York, pp. 309–318, 1979Google Scholar
  18. 18.
    Miehe, C.: Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Internat. J. Numer. Methods Engrg. 55, 1285–1322 (2002)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Mielke, A.: Analysis of energetic models for rate-independent materials. Proceedings of the International Congress of Mathematicians, Vol. III, Higher Ed. Press, Beijing, pp. 817–828, 2002Google Scholar
  20. 20.
    Mielke, A.: Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodynamics 15, 351–382 (2003)CrossRefADSMATHMathSciNetGoogle Scholar
  21. 21.
    Mielke, A., Roubíček, T.: A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul. 1, 571–597 (2003)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Mielke, A., Theil, F.: A mathematical model for rate-independent phase transformations with hysteresis. Proceedings of the Workshop on ``Models of Continuum Mechanics in Analysis and Engineering'', H-D. Alber, R. Balean, and R. Farwig, (eds), Shaker-Verlag, pp. 117–129, 1999.Google Scholar
  23. 23.
    Mielke, A., Theil, F., Levitas, V.: A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162, 137–177 (2002)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Ortiz, M., Martin, J.B.: Symmetry preserving return mapping algorithm and incrementally extremal paths: a unification of concepts. Internat. J. Numer. Methods Engrg. 28, 1839–1853 (1989)CrossRefMATHGoogle Scholar
  25. 25.
    Ortiz, M., Stanier, L.: The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Engrg. 171, 419–444 (1999)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, 1970Google Scholar
  27. 27.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York, 1966Google Scholar
  28. 28.
    Suquet, P.: Sur les équations de la plasticité: existence et regularité des solutions. J. Mécanique 20, 3–39 (1981)MATHMathSciNetGoogle Scholar
  29. 29.
    Temam, R.: Mathematical problems in plasticity. Gauthier-Villars, Paris, 1985. Translation of Problèmes mathématiques en plasticité. Gauthier-Villars, Paris, 1983Google Scholar
  30. 30.
    Temam, R., Strang, G.: Duality and relaxation in the variational problem of plasticity. J. Mécanique 19, 493–527 (1980)MATHMathSciNetGoogle Scholar
  31. 31.
    Visintin, A.: Strong convergence results related to strict convexity. Comm. Partial Differential Equations 9, 439–466 (1984)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Gianni Dal Maso
    • 1
  • Antonio DeSimone
    • 1
  • Maria Giovanna Mora
    • 1
  1. 1.Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly

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