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Fracture

  • B. Bourdin
  • G. A. Francfort
Part of the CISM Courses and Lectures book series (CISM, volume 535)

Abstract

These notes begin with a review of the mainstream theory of brittle fracture, as it has emerged from the works of Griffith and Irwin. We propose a re-formulation of that theory within the confines of the calculus of variations, focussing on crack path prediction. We then illustrate the various possible minimality criteria in a simple 1d-case as well as in a tearing experiment and discuss in some details the only complete mathematical formulation so far, that is that where global minimality for the total energy holds at each time. Next we focus on the numerical treatment of crack evolution and detail crack regularization which turns out to be a good approximation from the standpoint of crack propagation. This leads to a discussion of the computation of minimizing states for a non-convex functional. We illustrate the computational issues with a detailed investigation of the tearing experiment.

Keywords

Crack Length Crack Path Bulk Energy Dissipation Potential Elastic Energy Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. G. Alberti. Variational models for phase transitions, an approach via Γ-convergence. In G. Buttazzo, editor, Calculus of Variations and Partial Differential Equations, pages 95–114. Springer-Verlag, 2000.Google Scholar
  2. L. Ambrosio. Existence theory for a new class of variational problems. Arch. Ration. Mech. An., 111:291–322, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  3. L. Ambrosio. On the lower semicontinuity of quasiconvex integrals in SBV(Ω,R k). Nonlinear Anal.-Theor., 23(3):405–425, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  4. L. Ambrosio and V.M. Tortorelli. Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commun. Pur. Appl. Math., 43(8):999–1036, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  5. L. Ambrosio and V.M. Tortorelli. On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992.MathSciNetzbMATHGoogle Scholar
  6. L. Ambrosio, N. Fusco, and D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, 2000.Google Scholar
  7. J. M. Ball and F. Murat. W 1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal., 58(3):225–253, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  8. G. Bellettini and A. Coscia. Discrete approximation of a free discontinuity problem. Numer. Func. Anal. opt., 15(3–4):201–224, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  9. B. Bourdin. Une méthode variationnelle en mécanique de la rupture. Théorie et applications numériques. Thèse de doctorat, Université Paris-Nord, 1998.Google Scholar
  10. B. Bourdin. Image segmentation with a finite element method. ESAIM Math. Model. Numer. Anal., 33(2):229–244, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  11. B. Bourdin. Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound., 9(3):411–430, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  12. B. Bourdin and A. Chambolle. Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math., 85(4): 609–646, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  13. B. Bourdin, G. A. Francfort, and J.-J. Marigo. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids, 48:797–826, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  14. B. Bourdin, G. A. Francfort, and J.-J. Marigo. The variational approach to fracture. Springer, New York, 2008. Reprinted from J. Elasticity 91 (2008), no. 1–3, With a foreword by Roger Fosdick.CrossRefGoogle Scholar
  15. A. Braides. Γ-convergence for Beginners, volume 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2002.Google Scholar
  16. D. Bucur and N. Varchon. Boundary variation for a Neumann problem. Ann. Scuola Norm.-Sci., 29(4):807–821, 2000.MathSciNetGoogle Scholar
  17. S. Burke, C. Ortner, and E. Süli. An adaptive finite element approximation of a variational model of brittle fracture. SIAM J. Numer. Anal., 48(3): 980–1012, 2010.Google Scholar
  18. A. Chambolle. A density result in two-dimensional linearized elasticity, and applications. Arch. Ration. Mech. An., 167(3):211–233, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  19. A. Chambolle. An approximation result for special functions with bounded variations. J. Math. Pure Appl., 83:929–954, 2004.MathSciNetCrossRefGoogle Scholar
  20. A. Chambolle. Addendum to “an approximation result for special functions with bounded deformation” [j. math. pures appl. (9) 83 (7) (2004) 929–954]: the n-dimensional case. J. Math. Pure Appl., 84:137–145, 2005.MathSciNetCrossRefGoogle Scholar
  21. A. Chambolle and F. Doveri. Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets. Commun. Part. Diff. Eq., 22(5–6):811–840, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  22. A. Chambolle, A. Giacomini, and M. Ponsiglione. Crack initiation in brittle materials. Arch. Ration. Mech. Anal., 188(2):309–349, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  23. B. Dacorogna. Direct Methods in the Calculus of Variations. Springer Verlag, Berlin, Heidelberg, 1989.zbMATHGoogle Scholar
  24. G. Dal Maso. An introduction to Γ-convergence. Birkhäuser, Boston, 1993.Google Scholar
  25. G. Dal Maso and G. Lazzaroni. Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. H. Poincaré Anal. Non Linéaire, 27(1):257–290, 2010.zbMATHCrossRefGoogle Scholar
  26. G. Dal Maso and R. Toader. A model for the quasistatic growth of brittle fractures: Existence and approximation results. Arch. Ration. Mech. An., 162:101–135, 2002.zbMATHCrossRefGoogle Scholar
  27. G. Dal Maso, G. A. Francfort, and R. Toader. Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. An., 176(2):165–225, 2005.zbMATHCrossRefGoogle Scholar
  28. G. Dal Maso, A. Giacomini, and M. Ponsiglione. A variational model for quasistatic crack growth in nonlinear elasticity: some qualitative properties of the solutions. Boll. Unione Mat. Ital. (9), 2(2):371–390, 2009.MathSciNetzbMATHGoogle Scholar
  29. E. De Giorgi, M. Carriero, and A. Leaci. Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. An., 108:195–218, 1989.zbMATHCrossRefGoogle Scholar
  30. P. Destuynder and M. Djaoua. Sur une interprétation mathématique de l’intégrale de rice en théorie de la rupture fragile. Math. Met. Appl. Sc, 3:70–87, 1981.MathSciNetzbMATHCrossRefGoogle Scholar
  31. L.C. Evans and R.F. Gariepy. Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, 1992.zbMATHGoogle Scholar
  32. H. Federer. Geometric measure theory. SpringerVerlag, New York, 1969.zbMATHGoogle Scholar
  33. I. Fonseca and N. Fusco. Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24(3):463–499, 1997.MathSciNetzbMATHGoogle Scholar
  34. G.A. Francfort and C. Larsen. Existence and convergence for quasistatic evolution in brittle fracture. Commun. Pur. Appl. Math., 56(10):1465–1500, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  35. G.A. Francfort, N. Le, and S. Serfaty. Critical points of ambrosio-tortorelli converge to critical points of mumford-shah in the one-dimensional dirichlet case. ESAIM Control Optim. Calc. Var., 15(3):576–598, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  36. A. Giacomini. Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Dif., 22(2):129–172, 2005.MathSciNetzbMATHGoogle Scholar
  37. A. Giacomini and M. Ponsiglione. A discontinuous finite element approximation of quasi-static growth of brittle fractures. Numer. Func. Anal. Opt., 24(78): 813–850, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  38. A. Giacomini and M. Ponsiglione. Discontinuous finite element approximation of quasistatic crack growth in nonlinear elasticity. Math. Mod. Meth. Appl. S., 16(1):77–118, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  39. A.A. Griffith. The phenomena of rupture and flow in solids. Philos. T. Roy. Soc. A, CCXXIA: 163–198, 1920.Google Scholar
  40. M.E. Gurtin. Configurational forces as basic concepts of continuum physics, volume 137 of Applied Mathematical Sciences. Springer-Verlag, New York, 2000.Google Scholar
  41. H. Hahn. Über Annäherung an Lebesgue’sche integrale durch Riemann’sche summen. Sitzungsber. Math. Phys. Kl. K. Akad. Wiss. Wien, 123:713–743, 1914.Google Scholar
  42. R.V. Kohn and P. Sternberg. Local minimizers and singular perturbations. Proc. Roy. Soc. Edin. A, 111(A):69–84, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  43. F. Leonetti and F. Siepe. Maximum principle for vector valued minimizers. J. Convex Anal., 12(2):267–278, 2005.MathSciNetzbMATHGoogle Scholar
  44. D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pur. Appl. Math., XLII:577–685, 1989.MathSciNetCrossRefGoogle Scholar
  45. F. Murat. The Neumann sieve. In Nonlinear variational problems (Isola d’Elba, 1983), volume 127 of Res. Notes in Math., pages 24–32. Pitman, Boston, MA, 1985.Google Scholar
  46. M. Negri. The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional. Numer. Func. Anal. opt., 20(9–10):957–982, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  47. M. Negri. A finite element approximation of the Griffith model in fracture mechanics. Numer. Math., 95:653–687, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  48. M. Negri and M. Paolini. Numerical minimization of the Mumford-Shah functional. Calcolo, 38(2):67–84, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  49. C. A. Rogers. Hausdorff measures. Cambridge University Press, London, 1970.zbMATHGoogle Scholar

Copyright information

© CISM, Udine 2011

Authors and Affiliations

  • B. Bourdin
    • 1
  • G. A. Francfort
    • 2
  1. 1.Louisiana State UniversityBaton RougeUSA
  2. 2.Université Paris-NordVilletaneuseFrance

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