Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution

  • Gianni Dal Maso
  • Antonio DeSimone
  • Francesco Solombrino


Cam-Clay plasticity is a well-established model for the description of the mechanics of fine grained soils. As solutions can develop discontinuities in time, a weak notion of solution, in terms of a rescaled time s, has been proposed in Dal Maso, DeSimone and Solombrino (Calc Var Partial Equ 40:125–181, 2011) to give a meaning to this discontinuous evolution. In this paper we first prove that this rescaled evolution satisfies the flow-rule for the rate of plastic strain, in a suitable measure-theoretical sense. In the second part of the paper we consider the behavior of the evolution in terms of the original time variable t. We prove that the unrescaled solution satisfies an energy-dissipation balance and an evolution law for the internal variable, which can be expressed in terms of integrals depending only on the original time. Both these integral identities contain terms concentrated on the jump times, whose size can only be determined by looking at the rescaled formulation.

Mathematics Subject Classification (2000)

74C05 74L10 74G65 49L25 35K55 


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  1. 1.
    Ambrosio L., Kircheim B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318, 527–555 (2000)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Anzellotti G.: On the extremal stress and displacement in Hencky plasticity. Duke Math. J. 51, 133–147 (1984)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Boccardo L., Murat F.: Remarques sur l’homogénéisation de certains problèmes quasi-linéaires. Portugal. Math. 41, 535–562 (1982)MathSciNetMATHGoogle 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 (1973)Google Scholar
  5. 5.
    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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dal Maso G., Solombrino F.: Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case. Netw. Heterog. Media 5, 97–132 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dal Maso G., Toader R.: Quasistatic crack growth in elasto-plastic materials: the two-dimensional case. Arch. Ration. Mech. Anal. 196, 867–906 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    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)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dal Maso G., Demyanov A., DeSimone A.: Quasistatic evolution problems for pressure-sensitive plastic materials. Milan J. Math. 75, 117–134 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dal Maso G., DeSimone A., Solombrino F.: Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling. Calc. Var. Partial Differ. Equ. 40, 125–181 (2011)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHGoogle Scholar
  12. 12.
    Goffman C., Serrin J.: Sublinear functions of measures and variational integrals. Duke Math. J. 31, 159–178 (1964)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Mielke A., Rossi R., Savaré G.: Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dynam. Syst. 25, 585–615 (2009)MATHCrossRefGoogle Scholar
  14. 14.
    Mielke, R., Rossi, G., Savaré, G.: BV solutions and viscosity approximations of rate-independent systems. ESAIM Control Optim. Calc. Var. (2011), to appearGoogle Scholar
  15. 15.
    Moreau, J.J.: Bounded variation in time. In: Moreau J.J., Panagiotopoulos P.D., Strang G. (eds.) Topics in Nonsmooth Mechanics, pp. 1–74. Birkhuser (1988)Google Scholar
  16. 16.
    Rockafellar R. T.: Convex analysis. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  17. 17.
    Roscoe K. H., Burland J. B.: On the generalised stress-strain behaviour of ‘wet’ clay. In: Heyman, J., Leckie, F. A. (eds) Engineering Plasticity, pp. 535–609. Cambridge University Press, Cambridge (1968)Google Scholar
  18. 18.
    Roscoe K. H., Schofield A. N., Wroth C. P.: On the yielding of soils. Géotechnique 8, 22–53 (1958)CrossRefGoogle Scholar
  19. 19.
    Roscoe, K.H., Schofield, A.N.: Mechanical behaviour of an idealised ‘wet clay’. In: Proceedings 2nd European Conference on Soil Mechanics and Foundation Engineering, Wiesbaden, vol. I, pp. 47–54 (1963)Google Scholar
  20. 20.
    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 (2009)Google Scholar
  21. 21.
    Schofield A. N., Wroth C. P.: Critical State Soil Mechanics. McGraw-Hill, London (1968)Google Scholar
  22. 22.
    Solombrino F.: Quasistatic evolution problems for nonhomogeneous elastic-plastic materials. J. Convex Anal. 16, 89–119 (2009)MathSciNetMATHGoogle Scholar
  23. 23.
    Solombrino, F.: Rescaled viscosity solutions of a quasistatic evolution problem in non-associative plasticity. Phd Thesis, SISSA, Trieste (2010)Google Scholar
  24. 24.
    Temam, R.: Mathematical problems in plasticity. Gauthier-Villars, Paris, 1985. Translation of Problèmes mathématiques en plasticité. Gauthier-Villars, Paris (1983)Google Scholar
  25. 25.
    Visintin A.: Strong convergence results related to strict convexity. Comm. Partial Differ. Equ. 9, 439–466 (1984)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Gianni Dal Maso
    • 1
  • Antonio DeSimone
    • 1
  • Francesco Solombrino
    • 2
  1. 1.SISSATriesteItaly
  2. 2.Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of SciencesLinzAustria

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