Archive for Rational Mechanics and Analysis

, Volume 194, Issue 2, pp 585–609 | Cite as

Threshold-based Quasi-static Brittle Damage Evolution

Article

Abstract

We introduce models for static and quasi-static damage in elastic materials, based on a strain threshold, and then investigate the relationship between these threshold models and the energy-based models introduced in Francfort and Marigo (Eur J Mech A Solids 12:149–189, 1993) and Francfort and Garroni (Ration Mech Anal 182(1):125–152, 2006). A somewhat surprising result is that, while classical solutions for the energy models are also threshold solutions, this is shown not to be the case for nonclassical solutions, that is, solutions with microstructure. A new and arguably more physical definition of solutions with microstructure for the energy-based model is then given, in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors, rather than by the tensors themselves. We prove existence for this energy-based problem, and show that these solutions are also threshold solutions. A by-product of this analysis is that all local minimizers, in both the classical setting and for the new microstructure definition, are also global minimizers.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86, 125–145 (1984)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Allaire G., Kohn R.V.: Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials. Q. Appl. Math. 51, 643–674 (1993)MATHMathSciNetGoogle Scholar
  3. 3.
    Ball J.M., James R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100(1), 13–52 (1987)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ball J.M., Murat F.: W 1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58(3), 225–253 (1984)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Conti S., Theil F.: Single-slip elasto-plastic microstructures. Arch. Ration. Mech. Anal. 178(1), 125–148 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dal Maso G., De Simone A., Mora M.G.: Quasistatic evolution problems for linearly elastic—perfectly plastic materials. Arch. Ration. Mech. Anal. 180(2), 237–291 (2006)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dal Maso, G., Kohn, R.V.: The Local Character of G-closure (unpublished)Google Scholar
  8. 8.
    Dal Maso G., Francfort G., Toader R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dal Maso G., Toader R.: A model for quasi–static growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162, 101–135 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    De Giorgi E., Spagnolo S.: Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital. 4, 391–411 (1973)MathSciNetGoogle Scholar
  11. 11.
    Fonseca I., Müller S., Pedregal P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29(3), 736–756 (1998)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Francfort G.A., Garroni A.: A variational view of partial brittle damage evolution. Arch. Ration. Mech. Anal. 182(1), 125–152 (2006)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Francfort G.A., Larsen C.J.: Existence and convergence for quasi-static evolution of brittle fracture. Comm. Pure Appl. Math. 56, 1495–1500 (2003)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Francfort G.A., Marigo J.-J.: Stable damage evolution in a brittle continuous medium. Eur. J. Mech. A Solids 12, 149–189 (1993)MATHMathSciNetGoogle Scholar
  15. 15.
    Francfort G., Marigo J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Francfort, G., Mielke, A.: Existence results for a class of rate independent materials with non-convex elastic energies (to appear)Google Scholar
  17. 17.
    Larsen C.J.: Regularity of components in optimal design problems with perimeter penalization. Calc. Var. Partial Differ. Equ. 16, 17–29 (2003)MATHCrossRefGoogle Scholar
  18. 18.
    Mainik A., Mielke A.: Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differ. Equ. 22(1), 73–99 (2005)MATHMathSciNetGoogle Scholar
  19. 19.
    Mielke A.: Energetic formulation of multiplicative elasto–plasticity using dissipation distances. Cont. Mech. Thermodyn. 15, 351–382 (2003)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Mielke A.: Existence of minimizers in incremental elasto–plasticity with finite strains. SIAM J. Math. Anal. 36, 384–404 (2004)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Mielke A., Theil F.: On rate–independent hysteresis models. Nonl. Diff. Equ. Appl. (NoDEA) 11, 151–189 (2004)MATHMathSciNetGoogle Scholar
  22. 22.
    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)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Murat, F., Tartar, L.: Calcul des variations et homogénéisation. In: Les méthodes de l’homogénéisation: théorie et applications en physique, pp. 319–369, Eyrolles, 1997. Collection Etudes et Recherches EDFGoogle Scholar
  24. 24.
    Murat, F., Tartar, L.: H-convergence. In A Cherkaev and R.V. Kohn, editors, Topics in the mathematical modelling of composite materials, pp. 21–43. Birkhäuser, Boston, 1997. Progress in Nonlinear Differential Equations and Their Applications, 31Google Scholar
  25. 25.
    Ortiz M., Repetto E.: Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47(2), 397–462 (1999)MATHCrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Raitums U.: On the local representation of G-closure. Arch. Ration. Mech. Anal. 158(3), 213–234 (2001)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics “G. Castelnuovo”Università di Roma “La Sapienza”RomeItaly
  2. 2.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA

Personalised recommendations