Annali di Matematica Pura ed Applicata

, Volume 190, Issue 1, pp 165–194 | Cite as

Quasistatic crack growth in finite elasticity with Lipschitz data

Article

Abstract

We extend the recent existence result of Dal Maso and Lazzaroni (Ann Inst H Poincaré Anal Non Linéaire 27:257–290, 2010) for quasistatic evolutions of cracks in finite elasticity, allowing for boundary conditions and external forces with discontinuous first derivatives.

Keywords

Variational models Energy minimization Free-discontinuity problems Polyconvexity Quasistatic evolution Rate-independent processes Brittle fracture Crack propagation Griffith’s criterion Finite elasticity Non-interpenetration 

Mathematics Subject Classification (2000)

35R35 74R10 74B20 49J45 49Q20 35A35 

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References

  1. 1.
    Amar, M., De Cicco, V., Fusco, N.: Lower semicontinuity results for free discontinuity energies. To appear on Math. Models Methods Appl. Sci.Google Scholar
  2. 2.
    Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (2000)Google Scholar
  3. 3.
    Aronszajn N.: Differentiability of Lipschitzian mappings between Banach spaces. Stud. Math. 57, 147–190 (1976)MATHMathSciNetGoogle Scholar
  4. 4.
    Ball J.M.: Some open problems in elasticity. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, New York (2002)CrossRefGoogle Scholar
  5. 5.
    Bourdin B., Francfort G.A., Marigo J.-J.: The variational approach to fracture. J. Elastic. 91, 5–148 (2008)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies vol. 5, Notas de Matemática vol. 50, North-Holland, Amsterdam-London, American Elsevier, New York (1973)Google Scholar
  7. 7.
    Chambolle A.: A density result in two-dimensional linearized elasticity, and applications. Arch. Rational Mech. Anal. 167, 211–233 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ciarlet P.G., Nečas J.: Injectivity and self–contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97, 171–188 (1987)CrossRefGoogle Scholar
  9. 9.
    Dal Maso G., Francfort G.A., Toader R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dal Maso G., Lazzaroni G.: Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 257–290 (2010)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dal Maso, G., Lazzaroni, G.: Crack growth with non-interpenetration: a simplified proof for the pure Neumann problem. Preprint SISSA 70/2009/M (http://cvgmt.sns.it/)
  12. 12.
    Dal Maso G., Toader R.: A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Rational Mech. Anal. 162, 101–135 (2002)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dunford, N., Schwartz, J.T.: Linear operators—Part I: general theory. With the assistance of William G. Bade and Robert G. Bartle. Reprint of the 1958 original. A Wiley-Interscience Publication, Wiley Classics Library, John Wiley & Sons, Inc., New York (1988)Google Scholar
  14. 14.
    Francfort G.A., Larsen C.J.: Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. 56, 1465–1500 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Francfort G.A., Marigo J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Francfort G.A., Mielke A.: Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595, 55–91 (2006)MATHMathSciNetGoogle Scholar
  17. 17.
    Fusco N., Leone C., March R., Verde A.: A lower semi-continuity result for polyconvex unctionals in SBV. Proc. Roy. Soc. Edinb. Sect. A 136, 321–336 (2006)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Giacomini A., Ponsiglione M.: Non interpenetration of matter for SBV-deformations of hyperelastic brittle materials. Proc. Roy. Soc. Edinb. Sect. A 138, 1019–1041 (2008)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Griffith A.A.: The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. Lond. A 221, 163–198 (1920)CrossRefGoogle Scholar
  20. 20.
    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
  21. 21.
    Mielke A.: Evolution of rate-independent systems. In: Dafermos, C.M., Feireisl, E. (eds) Evolutionary Equations. Handbook of Differential Equations, pp. 461–559. Elsevier/North-Holland, Amsterdam (2005)Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2010

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.Institut d’AlembertUniversité Paris 6 “Pierre et Marie Curie”Paris Cedex 05France

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