Archive for Rational Mechanics and Analysis

, Volume 215, Issue 3, pp 831–866 | Cite as

Quasi-Static Brittle Damage Evolution in Elastic Materials with Multiple Damaged States

Article

Abstract

We present energetic and strain-threshold models for the quasi-static evolution of brutal brittle damage for geometrically-linear elastic materials. By allowing for anisotropic elastic moduli and multiple damaged states we present the issues for the first time in a truly elastic setting, and show that the threshold methods developed in (Garroni, A., Larsen, C. J., Threshold-based quasi-static brittle damage evolution, Archive for Rational Mechanics and Analysis 194 (2), 585–609, 2009) extend naturally to elastic materials with non-interacting damage. We show the existence of solutions and that energetic evolutions are also threshold evolutions.

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References

  1. 1.
    Chenchiah I.V., Bhattacharya K.: The relaxation of two-well energies with possibly unequal moduli. Arch. Ration. Mech. Anal. 187(3), 409–479 (2008)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Cherkaev A.: Variational Methods For Structural Optimization. Applied Mathematical Sciences, vol. 140. Springer, New York, 2000Google Scholar
  3. 3.
    Chenchiah I.V., Rieger M.O., Zimmer J.: Gradient flows in asymmetric metric spaces. Nonlinear Anal. 71(11), 5820–5834 (2009)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Francfort G.A., Garroni A.: A variational view of partial brittle damage evolution. Arch. Ration. Mech. Anal. 182(1), 125–152 (2006)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Francfort G.A., Marigo J.-J.: Stable damage evolution in a brittle continuous medium. Eur. J. Mech. A Solids 12(2), 149–189 (1993)MATHMathSciNetGoogle Scholar
  6. 6.
    Francfort G.A., Marigo J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids. 46(8), 1319–1342 (1998)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Francfort G.A., Mielke A.: Existence results for a class of rate-independent material models with nonconvex elastic energies. Journal für die Reine und Angewandte Mathematik 595, 55–91 (2006)MATHMathSciNetGoogle Scholar
  8. 8.
    Fonseca, I., Müller, S., Pedregal, P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29, 736–756 (1998)Google Scholar
  9. 9.
    Garroni A., Larsen C.J.: Threshold-based quasi-static brittle damage evolution. Arch Ration. Mech. Anal. 194, 585–609 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Han, W., Daya Reddy, B.: Plasticity: Mathematical theory and numerical analysis. 2nd edn., Interdisciplinary applied mathematics, vol. 9, Springer, New York, 1999Google Scholar
  11. 11.
    Larsen C.J.: Regularity of components in optimal design problems with perimeter penalization. Calc Var. Part. Differ Equ. 16(1), 17–29 (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Mielke A.: Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15(4), 351–382 (2003)ADSCrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Mielke, A.: Existence of minimizers in incremental elasto-plasticity with finite strains. SIAM J. Math. Anal. 36(2), 384–404 (2004) (electronic)Google Scholar
  14. 14.
    Milton, G.W.: The theory of composites. Cambridge monographs on applied and computational mathematics, vol. 6. Cambridge University Press, Cambridge, 2002Google Scholar
  15. 15.
    Mainik A., Mielke A.: Existence results for energetic models for rate-independent systems. Calc. Var. Part. Differ. Equ. 22(1), 73–99 (2005)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Isaac Vikram Chenchiah
    • 1
  • Christopher J. Larsen
    • 2
  1. 1.School of MathematicsUniversity of Bristol, University WalkBristolUK
  2. 2.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA

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