Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling

  • Gianni Dal Maso
  • Antonio DeSimone
  • Francesco Solombrino
Article
  • 101 Downloads

Abstract

Cam-Clay nonassociative plasticity exhibits both hardening and softening behaviour, depending on the loading. For many initial data the classical formulation of the quasistatic evolution problem has no smooth solution. We propose here a notion of generalized solution, based on a viscoplastic approximation. To study the limit of the viscoplastic evolutions we rescale time, in such a way that the plastic strain is uniformly Lipschitz with respect to the rescaled time. The limit of these rescaled solutions, as the viscosity parameter tends to zero, is characterized through an energy-dissipation balance, that can be written in a natural way using the rescaled time. As shown in Dal Maso and DeSimone (Math Models Methods Appl Sci 19:1–69, 2009) and Dal Maso and Solombrino (Netw Heterog Media 5:97–132, 2010), the proposed solution may be discontinuous with respect to the original time. Our formulation allows us to compute the amount of viscous dissipation occurring instantaneously at each discontinuity time.

Mathematics Subject Classification (2000)

74C05 74L10 74G65 49L25 35K55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boccardo L., Murat F.: Remarques sur l’homogénéisation de certains problèmes quasi-linéaires. Portugal. Math. 41, 535–562 (1982)MATHMathSciNetGoogle Scholar
  2. 2.
    Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland/American Elsevier, Amsterdam/New York (1973)Google Scholar
  3. 3.
    Buttazzo G.: Semicontinuity, Relaxation and Integral Representation Problems in the Calculus of Variations. Pitman Research Notes in Mathematics Series. Longman, Harlow (1989)Google Scholar
  4. 4.
    Dal Maso G., DeSimone A., Mora M.G.: Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180, 237–291 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dal Maso G., Demyanov A., DeSimone A.: Quasistatic evolution problems for pressure-sensitive plastic materials. Milan J. Math. 75, 117–134 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dal Maso G., DeSimone A.: Quasistatic evolution problems for Cam-Clay plasticity: examples of spatially homogeneous solutions. Math. Models Methods Appl. Sci. 19, 1–69 (2009)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Dal Maso G., Solombrino F.: Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case. Netw. Heterog. Media 5, 97–132 (2010)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Dal Maso, G., DeSimone, A., Solombrino, F.: Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution (in preparation)Google Scholar
  9. 9.
    Doob J.L.: Stochastic Processes. Wiley, New York (1953)MATHGoogle Scholar
  10. 10.
    Duvaut G., Lions J.-L.: Inequalities in Mechanics and Physics. Grundlehren der Mathematischen Wissenschaften, vol. 219. Springer, Berlin (1976)Google Scholar
  11. 11.
    Efendiev M., Mielke A.: On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13, 151–167 (2006)MATHMathSciNetGoogle Scholar
  12. 12.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976). Translation of Analyse convexe et problèmes variationnels. Dunod, Paris (1972)Google Scholar
  13. 13.
    Gagliardo E.: Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova 27, 284–305 (1957)MATHMathSciNetGoogle Scholar
  14. 14.
    Goffman C., Serrin J.: Sublinear functions of measures and variational integrals. Duke Math. J. 31, 159–178 (1964)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hahn H.: Über Annäherung an Lebesgue’sche Integrale durch Riemann’sche Summen. Sitzungsber. Math. Phys. Kl. K. Akad. Wiss. Wien 123, 713–743 (1914)Google Scholar
  16. 16.
    Henstock R.: A Riemann-type integral of Lebesgue power. Can. J. Math. 20, 79–87 (1968)MATHMathSciNetGoogle Scholar
  17. 17.
    Hughes T.J.R., Taylor R.: Unconditionally stable algorithms for quasi-static elasto-viscoplastic finite element analysis. Comput. Struct. 8, 169–173 (1978)MATHCrossRefGoogle Scholar
  18. 18.
    Matthies, H., Strang, G., Christiansen, E.: The saddle point of a differential program. In: Glowinski, R., Rodin, E., Zienkiewicz, O.C. (eds.) Energy Methods in Finite Element Analysis, pp. 309–318. Wiley, New York (1979)Google Scholar
  19. 19.
    Mawhin, J.: Analyse. Fondements, techniques, évolution. Second edn. Accès Sciences. De Boeck Université, Brussels (1997)Google Scholar
  20. 20.
    Mielke A., Rossi R., Savaré G.: Modeling solutions with jumps for rate-independent systems on metric spaces. Discret. Cont. Dyn. Syst. 25, 585–615 (2009)MATHCrossRefGoogle Scholar
  21. 21.
    Mielke, A., Rossi, R., Savaré, G.: BV-solutions of the viscosity approximation of rate-independent systems. WIAS Preprint, Berlin (2009)Google Scholar
  22. 22.
    Moreau, J.J.: Champs et distributiond de tenseurs déformation sur un ouvert de connexité quelconque. Séminaire d’analyse convexe, vol. 6. Université de Montpellier, Montpellier (1976)Google Scholar
  23. 23.
    Ortiz, M.: Topics in constitutive theory for inelastic solids. PhD thesis, University of California, Berkeley (1981)Google Scholar
  24. 24.
    Perzyna P.: Thermodynamic theory of viscoplasticity. Adv. Appl. Mech. 11, 313–354 (1971)CrossRefGoogle Scholar
  25. 25.
    Phillips R.B.: Crystals, Defects and Microstructures. Modeling Across Scales. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  26. 26.
    Reshetnyak Y.G.: Weak convergence of completely additive vector functions on a set. Sib. Math. J. 9, 1039–1045 (1968)MATHGoogle Scholar
  27. 27.
    Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  28. 28.
    Rossi, R.: Interazione di norme L 2 e L 1 in evoluzioni rate-independent. Lecture given at the “XIX Convegno Nazionale di Calcolo delle Variazioni”, Levico (Trento), February 8–13, 2009Google Scholar
  29. 29.
    Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1966)MATHGoogle Scholar
  30. 30.
    Saks S.: Sur les fonctions d’intervalle. Fundam. Math. 10, 211–224 (1927)MATHGoogle Scholar
  31. 31.
    Solombrino F.: Quasistatic evolution problems for nonhomogeneous elastic-plastic materials. J. Convex Anal. 16, 89–119 (2009)MATHMathSciNetGoogle Scholar
  32. 32.
    Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  33. 33.
    Suquet P.: Sur les équations de la plasticité: existence et regularité des solutions. J. Mécanique 20, 3–39 (1981)MATHMathSciNetGoogle Scholar
  34. 34.
    Temam R.: Mathematical Problems in Plasticity. Gauthier-Villars, Paris (1985). Translation of Problèmes mathématiques en plasticité. Gauthier-Villars, Paris (1983)Google Scholar
  35. 35.
    Temam R., Strang G.: Duality and relaxation in the variational problem of plasticity. J. Mécanique 19, 493–527 (1980)MATHMathSciNetGoogle Scholar
  36. 36.
    Truskinovsky, L.: Quasi-static deformation of a system with nonconvex energy from a perspective of dynamics. Lecture given at the workshop “Variational Problems in Materials Science”, SISSA, Trieste, September 6–10, 2004Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Gianni Dal Maso
    • 1
  • Antonio DeSimone
    • 1
  • Francesco Solombrino
    • 1
  1. 1.SISSATriesteItaly

Personalised recommendations