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A Density Result in GSBDp with Applications to the Approximation of Brittle Fracture Energies

  • Antonin Chambolle
  • Vito Crismale
Article

Abstract

We prove that any function in \({GSBD^p(\Omega)}\), with \({\Omega}\) a n-dimensional open bounded set with finite perimeter, is approximated by functions \({u_k\in SBV(\Omega; \mathbb{R}^{n})\cap L^\infty(\Omega; \mathbb{R}^{n})}\) whose jump is a finite union of C1 hypersurfaces. The approximation takes place in the sense of Griffith-type energies \({\int_\Omega W(e(u)) {\rm dx} +\mathcal{H}^{n-1}(J_u)}\), e(u) and Ju being the approximate symmetric gradient and the jump set of u, and W a nonnegative function with p-growth, p > 1. The difference between uk and u is small in Lp outside a sequence of sets \({E_k\subset \Omega}\) whose measure tends to 0 and if \({|u|^r \in L^1(\Omega)}\) with \({r\in (0,p]}\), then \({|u_k-u|^r \to 0}\) in \({L^1(\Omega)}\). Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce \({\Gamma}\)-convergence approximation à la Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are a priori not even in \({L^1(\Omega; \mathbb{R}^{n})}\).

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Acknowledgments

V. Crismale has been supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and is currently funded by the Marie Skłodowska-Curie Standard European Fellowship No.793018. The authors wish to thank the anonymous referees for their valuable comments.

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Conflict of interest

The authors declare no competing financial interest.

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References

  1. 1.
    Amar, M., De Cicco, V.: A new approximation result for BV-functions. C. R. Math. Acad. Sci. Paris 340, 735–738 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L.: Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111, 291–322 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Coscia, A., Dal Maso, G.: Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139, 201–238 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)zbMATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 7(6), 105–123 (1992)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Babadjian, J.-F.: Traces of functions of bounded deformation. Indiana Univ. Math. J. 64, 1271–1290 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Babadjian, J.-F., Giacomini, A.: Existence of strong solutions for quasi-static evolution in brittle fracture. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13, 925–974 (2014)Google Scholar
  9. 9.
    Bellettini, G., Coscia, A., Dal Maso, G.: Compactness and lower semicontinuity properties in \({\rm SBD}(\Omega )\). Math. Z. 228, 337–351 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourdin, B.: Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound. 9, 411–430 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bourdin, B., Francfort, G.A., Marigo, J.-J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797–826 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Braides, A., Chiadò Piat, V.: Integral representation results for functionals defined on \({\rm SBV}(\Omega; {\bf R}^m)\). J. Math. Pures Appl. 9(75), 595–626 (1996)zbMATHGoogle Scholar
  13. 13.
    Braides, A., Conti, S., Garroni, A.: Density of polyhedral partitions. Calc. Var. Partial Differ. Equ. 56, pp. Art. 28, 10. (2017)Google Scholar
  14. 14.
    Burke, S., Ortner, C., Süli, E.: An adaptive finite element approximation of a generalized Ambrosio-Tortorelli functional. Math. Models Methods Appl. Sci. 23, 1663–1697 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Buttazzo, G.: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, vol. 207 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989Google Scholar
  16. 16.
    Cagnetti, F., Toader, R.: Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach. ESAIM Control Optim. Calc. Var. 17, 1–27 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Caroccia, M., Van Goethem, N.: Damage-driven fracture with low-order potentials: asymptotic behavior and applications (2018). Preprint arXiv:1712.08556
  18. 18.
    Chambolle, A.: A density result in two-dimensional linearized elasticity, and applications. Arch. Ration. Mech. Anal. 167, 211–233 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chambolle, A.: An approximation result for special functions with bounded deformation. J. Math. Pures Appl. 9(83), 929–954 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chambolle, A.: Addendum to: ``An approximation result for special functions with bounded deformation'' [J. Math. Pures Appl. (9) 83(7), 929–954 (2004). mr2074682]. J. Math. Pures Appl. (9) 84, 137–145. (2005)Google Scholar
  21. 21.
    Chambolle, A., Conti, S., Francfort, G.: Korn-Poincaré inequalities for functions with a small jump set. Indiana Univ. Math. J. 65, 1373–1399 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chambolle, A., Conti, S., Francfort, G.A.: Approximation of a brittle fracture energy with a constraint of non-interpenetration. Arch. Ration. Mech. Anal. 228, 867–889 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Chambolle, A., Conti, S., Iurlano, F.: Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy (2017). Preprint arXiv:1710.01929 (Accepted for publication on J. Math. Pures Appl)
  24. 24.
    Chambolle, A., Crismale, V.: Phase-field approximation of some fracture energies of cohesive type. Preprint arXiv:1812.05301
  25. 25.
    Chambolle, A., Crismale, V.: Existence of strong solutions to the Dirichlet problem for Griffith energy (2018). Preprint arXiv:1811.07147
  26. 26.
    Chambolle, A., Crismale, V.: A density result in \(GSBD^p\) with applications to the approximation of brittle fracture energies (2018). Preprint arXiv:1802.03302 (to appear on J. Eur. Math. Soc.)
  27. 27.
    Conti, S., Focardi, M., Iurlano, F.: Which special functions of bounded deformation have bounded variation? Proc. R. Soc. Edinb. Sect. A 148, 33–50 (2018)Google Scholar
  28. 28.
    Conti, S., Focardi, M., Iurlano, F.: Integral representation for functionals defined on \(SBD^p\) in dimension two. Arch. Ration. Mech. Anal. 223, 1337–1374 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Conti, S., Focardi, M., Iurlano, F.: Existence of strong minimizers for the Griffith static fracture model in dimension two. Ann. Inst. H. Poincaré Anal. Non Linéaire.  https://doi.org/10.1016/j.anihpc.2018.06.003 (in press)
  30. 30.
    Conti, S., Focardi, M., Iurlano, F.: Approximation of fracture energies with \(p\)-growth via piecewise affine finite elements. ESAIM Control Optim. Calc. Var.  https://doi.org/10.1051/cocv/2018021 (in press)
  31. 31.
    Cortesani, G., Toader, R.: A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38, 585–604 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Crismale, V.: On the approximation of \(SBD\) functions and some applications (2018). Preprint arXiv:1806.03076
  33. 33.
    Crismale, V., Lazzaroni, G., Orlando, G.: Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue. Math. Models Methods Appl. Sci. 28, 1371–1412 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Dal Maso, G.: An introduction to \(\Gamma \)-convergence, vol. 8 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston (1993)Google Scholar
  35. 35.
    Dal Maso, G.: Generalised functions of bounded deformation. J. Eur. Math. Soc. (JEMS) 15, 1943–1997 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    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)Google Scholar
  38. 38.
    Dal Maso, G., Toader, R.: A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162, 101–135 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Dal Maso, G., Zanini, C.: Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. Roy. Soc. Edinburgh Sect. A 137, 253–279 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    De Giorgi, E., Ambrosio, L.: New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82, 199–210 (1988)Google Scholar
  41. 41.
    De Giorgi, E., Carriero, M., Leaci, A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. Anal. 108, 195–218 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    de Philippis, G., Fusco, N., Pratelli, A.: On the approximation of SBV functions. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28, 369–413 (2017)Google Scholar
  43. 43.
    Dibos, F., Séré, E.: An approximation result for the minimizers of the Mumford-Shah functional. Boll. Un. Mat. Ital. A 7(11), 149–162 (1997)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, vol. 85. Cambridge University Press, Cambridge (1986)Google Scholar
  45. 45.
    Federer, H.: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153. New York Inc., New York, Springer-Verlag (1969)Google Scholar
  46. 46.
    Focardi, M., Iurlano, F.: Asymptotic analysis of Ambrosio-Tortorelli energies in linearized elasticity. SIAM J. Math. Anal. 46, 2936–2955 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Francfort, G.A., Larsen, C.J.: Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. 56, 1465–1500 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Francfort, G.A., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Friedrich, M.: A derivation of linearized Griffith energies from nonlinear models. Arch. Ration. Mech. Anal. 225, 425–467 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Friedrich, M.: A Piecewise Korn Inequality in SBD and Applications to Embedding and Density Results. SIAM J. Math. Anal. 50, 3842–3918 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Friedrich, M.: A compactness result in \(GSBV^p\) and applications to \(\Gamma \)-convergence for free discontinuity problems (2018). Preprint arXiv:1807.03647
  52. 52.
    Friedrich, M., Solombrino, F.: Quasistatic crack growth in 2d-linearized elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 35, 27–64 (2018)Google Scholar
  53. 53.
    Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Ser. A 221, 163–198 (1920)Google Scholar
  54. 54.
    Hutchinson, J.W.: A Course on Nonlinear Fracture Mechanics. Technical University of Denmark, Kongens Lyngby, Department of Solid Mechanics (1989)Google Scholar
  55. 55.
    Iurlano, F.: A density result for GSBD and its application to the approximation of brittle fracture energies. Calc. Var. Partial Differ. Equ. 51, 315–342 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Kohn, R., Temam, R.: Dual spaces of stresses and strains, with applications to Hencky plasticity. Appl. Math. Optim. 10, 1–35 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Lazzaroni, G.: Quasistatic crack growth in finite elasticity with Lipschitz data. Ann. Mat. Pura Appl. 4(190), 165–194 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Mumford, D., Shah, J.: Boundary detection by minimizing functionals. In: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, San Francisco (1985)Google Scholar
  59. 59.
    Negri, M.: A finite element approximation of the Griffith's model in fracture mechanics. Numer. Math. 95, 653–687 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Negri, M.: A non-local approximation of free discontinuity problems in \(SBV\) and \(SBD\). Calc. Var. Partial Differential Equations 25, 33–62 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Nitsche, J.A.: On Korn's second inequality. RAIRO Anal. Numér. 15, 237–248 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Suquet, P.-M.: Sur les équations de la plasticité: existence et régularité des solutions. J. Mécanique 20, 3–39 (1981)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Temam, R.: Mathematical Problems in Plasticity, Gauthier-Villars, Paris, 1985. Translation of Problèmes mathématiques en plasticité. Gauthier-Villars, Paris (1983)Google Scholar
  64. 64.
    Temam, R., Strang, G.: Duality and relaxation in the variational problem of plasticity. J. Mécanique 19, 493–527 (1980)MathSciNetzbMATHGoogle Scholar
  65. 65.
    White, B.: The deformation theorem for flat chains. Acta Math. 183, 255–271 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CMAP, École Polytechnique, CNRSPalaiseau CedexFrance

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