Variational Approaches to Fracture

  • Gianpietro Del Piero
Conference paper
Part of the Springer Optimization and Its Applications book series (SOIA, volume 33)


Fracture Mechanics is largely based on an assumption of A.A. Griffith, according to which the evolution of a crack in a fractured body is governed by the competition between the elastic strain energy and the energy spent in the crack growth. In the recent variational formulation of Francfort and Marigo, some analytical difficulties due to the presence of discontinuous displacements are avoided using the Г-convergence theory. In it, the problem is approximated by a sequence of more regular problems, formulated in Sobolev spaces, and therefore solvable with standard finite elements techniques. A further regularization introduced by G.I. Barenblatt allows to solve the problem of determining the fracture onset in an initially unfractured body.

In this communication I discuss the role plaid by local energy minimizers in the post-fracture evolution. I show that a dissipation inequality leads to the formulation of an incremental problem, which determines a quasi-static evolution along a path made of local energy minimizers. The introduction of a dissipation inequality provides different responses at loading and unloading, in accordance with experimental evidence. Finally, I briefly discuss the possibility of bulk regularization, based on the assumption that the fracture energy be diffused in a three-dimensional region, instead of being concentrated on a singular surface.


Fracture Energy Elastic Strain Energy Discontinuous Displacement Half Line Fracture Problem 
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This work was supported by the Research Project Mathematical Models for Materials Science - PRIN 2005 of the Italian Ministry for University.


  1. 1.
    Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Math. Monogr., Oxford University Press, New York (2000)Google Scholar
  2. 2.
    Ambrosio L., Tortorelli M.V.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. 6-B, 105–123 (1992)MATHMathSciNetGoogle Scholar
  3. 3.
    Barenblatt G.I.: The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 7, 55–129 (1962)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bažant Z.P., Chen E.P.: Scaling of structural failure. Appl. Mech. Review 50, 593–627 (1997)CrossRefGoogle Scholar
  5. 5.
    Braides A.: G-Convergence for Beginners. Oxford University Press, Oxford (2002)CrossRefGoogle Scholar
  6. 6.
    Bourdin B., Francfort G.A., Marigo J.-J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797–826 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chambolle A.: An approximation result for special functions with bounded deformation. J. Math. Pures Appl., IX Sér. 83, 929–954 (2004)MathSciNetGoogle Scholar
  8. 8.
    Chambolle A., Giacomini A., Ponsiglione M.: Crack initiation in brittle materials. Arch. Rational Mech. Analysis, 188, 309–349(2008)Google Scholar
  9. 9.
    Ciarlet P.G.: Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity. North-Holland, Amsterdam (1987)Google Scholar
  10. 10.
    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
  11. 11.
    De Giorgi E., Franzoni T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. 58, 842–850 (1975)MATHMathSciNetGoogle Scholar
  12. 12.
    De Giorgi E., Ambrosio L.: Un nuovo funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei,Rend. Cl. Sci. Fis. Mat. Natur. 82, 199–210 (1988)MATHMathSciNetGoogle Scholar
  13. 13.
    Del Piero G.: One-dimensional ductile-brittle transition, yielding, and structured deformations. In: P. Argoul et al. (eds.) Variations of Domains and Free-Boundary Problems in Solid Mechanics, 203–210, Kluwer, Dordrecht (1999)Google Scholar
  14. 14.
    Del Piero G.: The energy of a one-dimensional structured deformation. Math. Mech. Solids 6, 387–408 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Del Piero G.: Interface energies and structured deformations in plasticity. In: G. Dal Maso et al. (eds.) Variational Methods for Discontinuous Structures, 103–116, Birkhäuser, Basel (2002)Google Scholar
  16. 16.
    Del Piero G.: Foundations of the theory of structured deformations. In: Del Piero G., Owen D.R. (eds.) Multiscale Modeling in Continuum Mechanics and Structured Deformations. CISM Courses and Lectures n. 447 (2004)Google Scholar
  17. 17.
    Del Piero G., March R., Lancioni G.: A variational model for fracture mechanics: Numerical experiments. J. Mech. Phys. Solids 55, 2513–2537 (2007)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Del Piero G., Truskinovsky L.: Elastic bars with Cohesive energy. Cont. Mech. thermodynamics, in press(DOI 10.1007/S00161-009-0101-9)(2009)Google Scholar
  19. 19.
    Francfort G.A., Marigo J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fusco N., Leone C., March R., Verde A.: A lower semicontinuity result for polyconvex functionals in SBV, Proc. R. Soc. Edinb. A, 136, 321–336 (2006)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Griffith A.A.: The phenomenon of rupture and flow in solids. Phil. Trans. Roy. Soc. London A, 221, 163–198 (1920)Google Scholar
  22. 22.
    Hill R.: The Mathematical Theory of Plasticity. Oxford University Press (1950). Reprinted in: Oxford Classic Series, Clarendon Press, Oxford (1998)MATHGoogle Scholar
  23. 23.
    Mumford D., Shah J.: Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 577–685 (1989)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Vol’pert A.I., Hudjaev S.I.: Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Niyhoff, Dordrecht (1985)MATHGoogle Scholar

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© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Dipartimento di IngegneriaUniversitàdi FerraraFerraraItaly

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