A density result for GSBD and its application to the approximation of brittle fracture energies

Article

Abstract

We present an approximation result for functions \(u:\Omega \rightarrow \mathbb {R}^n\) belonging to the space \(GSBD(\Omega )\cap L^2(\Omega ,{{\mathbb R}}^n)\) with \(e(u)\) square integrable and \(\fancyscript{H}^{n-1}(J_u)\) finite. The approximating functions \(u_k\) are piecewise continuous functions such that \(u_k\rightarrow u\) in \(L^2(\Omega ,\mathbb {R}^n)\), \(e(u_k)\rightarrow e(u)\) in \(L^2(\Omega ,\mathbb {M}^{n{\times }n}_{sym})\), \(\fancyscript{H}^{n-1}(J_{u_k}\triangle J_u)\rightarrow 0\), and \(\int _{J_{u_k}\cup J_u}|u_k^\pm -u^\pm |\wedge 1d{\fancyscript{H}}^{n-1}\rightarrow 0\).  As an application, we provide the extension to the vector-valued case of the \(\Gamma \)-convergence result in \(GSBV(\Omega )\) proved by Ambrosio and Tortorelli (Commun Pure Appl Math 43:999–1036, 1990; Boll. Un. Mat. Ital. B (7) 6:105–123, 1992).

Mathematics Subject Classification (2000)

49Q20 49J45 26B30 74R10 35A35 

References

  1. 1.
    Alberti, G., Bouchitté, G., Seppecher, P.: Phase transition with the line-tension effect. Arch. Rational Mech. Anal. 144, 1–46 (1998)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Coscia, A., Dal Maso, G.: Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal. 139, 201–238 (1997)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, New York (2000)MATHGoogle Scholar
  4. 4.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma \)-convergence. Commun. Pure Appl. Math. 43, 999–1036 (1990)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) 6, 105–123 (1992)Google Scholar
  6. 6.
    Bellettini, G., Coscia, A., Dal Maso, G.: Special Functions of Bounded Deformation. Preprint SISSA, Trieste (1995)Google Scholar
  7. 7.
    Bellettini, G., Coscia, A., Dal Maso, G.: Compactness and lower semicontinuity properties in \({\rm SBD}(\Omega )\). Math. Z. 228, 337–351 (1998)Google Scholar
  8. 8.
    Bourdin, B.: Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound. 9, 411–430 (2007)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bourdin, B., Francfort, G.A., Marigo, J.-J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797–826 (2000)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Bourdin, B., Francfort, G.A., Marigo, J.-J.: The variational approach to fracture. J. Elast. 91, 5–148 (2008)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Buttazzo, G.: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation. Pitman Research Notes in Mathematics Series, vol. 203. Longman Scientific & Technical, Harlow (1989)Google Scholar
  12. 12.
    Chambolle, A.: An approximation result for special functions with bounded deformation. J. Math. Pures Appl. (9) 83, 929–954 (2004)Google Scholar
  13. 13.
    Chambolle, A.: Addendum to: “An approximation result for special functions with bounded deformation” [J. Math. Pures Appl. (9) 83 (2004), no. 7, 929–954; MR2074682]. J. Math. Pures Appl. (9) 84, 137–145 (2005)Google Scholar
  14. 14.
    Cortesani, G.: Strong approximation of GSBV functions by piecewise smooth functions. Ann. Univ. Ferrara Sez. VII (N.S.) 43, 27–49 (1997)Google Scholar
  15. 15.
    Cortesani, G., Toader, R.: A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38, 585–604 (1999)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser, Boston (1993)Google Scholar
  17. 17.
    Dal Maso, G.: Generalised functions of bounded deformation. J. Eur. Math. Soc. (JEMS) 15(5), 1943–1997 (2013)Google Scholar
  18. 18.
    Dal Maso, G., Iurlano, F.: Fracture models as \(\Gamma \)-limits of damage models. Commun. Pure. Appl. Anal. 12, 1657–1686 (2013)Google Scholar
  19. 19.
    Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)Google Scholar
  20. 20.
    Francfort, G.A., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Francfort, G.A., Marigo, J.-J.: Cracks in fracture mechanics: a time indexed family of energy minimizers. Variations of domain and free-boundary problems in solid mechanics (Paris, 1997), Solid Mech. Appl. vol. 66. Kluwer, Dordrecht, pp. 197–202 (1999)Google Scholar
  22. 22.
    Iurlano, F.: Fracture and plastic models as \(\Gamma \)-limits of damage models under different regimes. Adv. Calc. Var. 6, 165–189 (2013)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Temam, R.: Problèmes mathématiques en plasticité. Méthodes Mathématiques de l’Informatique 12, Gauthier-Villars, Montrouge (1983)Google Scholar
  24. 24.
    Temam, R., Strang, G.: Functions of bounded deformation. Arch. Rational Mech. Anal. 75, 7–21 (1980/81)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.SISSATriesteItaly

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