Skip to main content
SpringerLink
Log in

A Quasistatic Evolution Model for the Interaction Between Fracture and Damage

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

This paper is devoted to the investigation of a quasistatic evolution model for a continuum which undergoes damage and possibly fracture. In both cases, the model appears to be ill posed so that it is necessary to introduce a relaxed variational evolution preserving the irreversibility of the process, the minimality at each time, and the energy balance. From a mechanical point of view, it turns out that the material prefers to form microstructures through the creation of fine mixtures between the damaged and healthy parts of the medium. The brutal character of the damage process is then replaced by a progressive one, where the original damage internal variable, that is, the characteristic function of the damaged part, is replaced by the local volume fraction. The analysis rests on a locality property for mixtures which enables us to use an alternative formula for the lower semicontinuous envelope of the elastic energy in terms of the G-closure set.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Allaire, G.: Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, Vol. 146. Springer, Berlin, 2002

  3. Allaire, G., Aubry, S., Jouve, F.: Simulation numérique de l’endommagement à l’aide du modèle Francfort-Marigo. Actes du 29ème congrès d’analyse numérique, ESAIM Proceedings, Vol. 3, 1–9, 1998

  4. Allaire, G., Jouve, F., Van Goethem, N.: A level set method for the numerical simulation of damage evolution. Proceedings of ICIAM 2007 Zürich (Eds. R. Jeltsch, G. Wanner). EMS, Zürich, 3–22, 2009

  5. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, 2000

  6. Babadjian J.-F.: Quasistatic evolution of a brittle thin film. Calc. Var. PDEs 26, 69–118 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Babadjian J.-F., Barchiesi M.: A variational approach to the local character of G-closure: the convex case. Ann. I. H. Poincaré 26, 351–373 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Bouchitté G., Fonseca I., Leoni G., Mascarenhas L.: A global method for relaxation in W 1,p and in SBV p. Arch. Rational Mech. Anal. 165, 187–242 (2002)

    Article  ADS  MATH  Google Scholar 

  9. Bourdin B., Francfort G.A., Marigo J.-J.: The Variational Approach to Fracture. Springer, New York (2008)

    Book  Google Scholar 

  10. Braides A.: Homogenization of some almost periodic coercive functionals. Rend. Accad. Naz. Sci. Mem. Mat. 9(5), 313–321 (1985)

    MathSciNet  MATH  Google Scholar 

  11. Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals, Vol. 12. The Clarendon Press, Oxford University Press, New York, 1998

  12. Buttazzo G., Dal Maso G.: Integral representation and relaxation of local functionals. Nonlinear Anal. 9, 515–532 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Conti S., Theil F.: Single-slip elasto-plastic microstructures. Arch. Rational Mech. Anal. 178, 125–148 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Dacorogna B.: Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin (1989)

    MATH  Google Scholar 

  15. Dal Maso G.: An Introduction to Γ-Convergence. Birkhaüser, Boston (1993)

    Book  Google Scholar 

  16. Dal Maso G., De Simone A., Mora M.G.: Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Rational Mech. Anal. 180, 237–291 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Dal Maso G., Francfort G.A., Toader R.: Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176, 165–225 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Dal Maso G., Toader R.: A model for the quasistatic growth of brittle fractures: existence and approximation results. Arch. Rational Mech. Anal. 162, 101–135 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Dal Maso G., Toader R.: Quasistatic crack growth in elasto-plastic materials: the two-dimensional case. Arch. Rational Mech. Anal. 196(3), 867–906 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Ekeland I., Temam R.: Analyse convexe et problèmes variationnels. Dunod, Gauthier-Villars, Paris (1974)

    MATH  Google Scholar 

  21. Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  22. Fonseca I., Francfort G.A.: Relaxation in BV versus quasiconvexification in W 1,p; a model for the interaction between fracture and damage. Calc. Var. PDEs 3, 407–446 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fonseca I., Fusco N.: Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24(3), 463–499 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Fonseca I., Fusco N., Marcellini P.: An existence result for a nonconvex variational problem via regularity. ESAIM Control Optim. Calc. Var. 7, 69–95 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: L p Spaces. Springer Monographs in Mathematics. Springer, New York, 2007

  26. Fonseca I., Müller S., Pedregal P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29, 736–756 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Francfort G.A., Garroni A.: A variational view of partial brittle damage evolution. Arch. Rational Mech. Anal. 182, 125–152 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Francfort G.A., Larsen G.J.: Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. 56, 1465–1500 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Francfort G.A., Marigo J.-J.: Stable damage evolution in a brittle continuous medium. Eur. J. Mech. A Solids 12, 149–189 (1993)

    MathSciNet  MATH  Google Scholar 

  30. Francfort G.A., Marigo J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Garroni A., Larsen C.: Threshold-based quasi-static brittle damage evolution. Arch. Rational Mech. Anal. 194, 585–609 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Giacomini A., Ponsiglione M.: A Γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications. Arch. Rational Mech. Anal. 180, 399–447 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies, Vol. 105. Princeton University Press, Princeton, 1983

  34. Giusti E.: Direct Methods in the Calculus of Variations. World Scientific, River Edge (2003)

    Book  MATH  Google Scholar 

  35. Mainik A., Mielke A.: Existence results for energetic models for rate-independent systems. Calc. Var. PDEs 22, 73–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mielke, A.: Evolution of Rate-Independent Systems. Evolutionary Equations, Vol. II (Eds. C.M. Dafermos, E. Feireisl). Handbook of Differential Equations. Elsevier/North-Holland, Amsterdam, 461–559, 2005

  37. Mielke A.: Existence of minimizers in incremental elasto-plasticity with finite strains. SIAM J. Math. Anal. 36, 384–404 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mielke A., Roubicek T., Stefanelli U.: Γ-limits and relaxations for rate-independent evolutionary problems. Calc. Var. PDEs 31, 387–416 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Müller S.: Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rational Mech. Anal. 99, 189–212 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Raitums U.: On the local representation of G-closure. Arch. Rational Mech. Anal. 158, 213–234 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire J.-L. Lions, CNRS, Université Pierre et Marie Curie – Paris 6, UMR 7598, 75005, Paris, France

    Jean-François Babadjian

Authors
  1. Jean-François Babadjian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-François Babadjian.

Additional information

Communicated by G. Dal Maso

Rights and permissions

Reprints and permissions

About this article

Cite this article

Babadjian, JF. A Quasistatic Evolution Model for the Interaction Between Fracture and Damage. Arch Rational Mech Anal 200, 945–1002 (2011). https://doi.org/10.1007/s00205-010-0379-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-010-0379-6

Keywords

Search

Navigation