Advertisement

Archive for Rational Mechanics and Analysis

, Volume 197, Issue 2, pp 619–655 | Cite as

Invertibility and Weak Continuity of the Determinant for the Modelling of Cavitation and Fracture in Nonlinear Elasticity

  • Duvan HenaoEmail author
  • Carlos Mora-Corral
Article

Abstract

In this paper, we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture. The main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface. Accordingly, we define a functional that measures the area of the created surface. This functional has relationships with the theory of Cartesian currents. We show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient, and that the weak limit of one-to-one almost everywhere deformations is also one-to-one almost everywhere. We then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy, taking into account orientation-preserving and non-interpenetration conditions.

Keywords

Cavitation Elastic Energy Nonlinear Elasticity Lipschitz Boundary Existence Theory 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosio L.: A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B 3(7), 857–881 (1989)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Ambrosio L.: Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111, 291–322 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ambrosio L.: On the lower semicontinuity of quasiconvex integrals in SBV(Ω, R k). Nonlinear Anal. 23, 405–425 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ambrosio L., Braides A.: Energies in SBV and variational models in fracture mechanics. Homogenization and Applications to Material Sciences (Nice, 1995), Gakkōtosho, Tokyo, pp. 1–22, 1995Google Scholar
  5. 5.
    Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000)zbMATHGoogle Scholar
  6. 6.
    Ball J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1976/1977)CrossRefGoogle Scholar
  7. 7.
    Ball J.M.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinb. Sect. A 88, 315–328 (1981)zbMATHGoogle Scholar
  8. 8.
    Ball J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. Ser. A 306, 557–611 (1982)zbMATHCrossRefADSGoogle Scholar
  9. 9.
    Ball J.M., Currie J.C., Olver P.J.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41, 135–174 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ball J.M., Murat F.: W 1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58, 225–253 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bourdin B., Francfort G.A., Marigo J.-J.: The variational approach to fracture. J. Elast. 91, 5–148 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chambolle A., Giacomini A., Ponsiglione M.: Crack initiation in brittle materials. Arch. Ration. Mech. Anal. 188, 309–349 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ciarlet P.G., Nečas J.: Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97, 171–188 (1987)CrossRefGoogle Scholar
  14. 14.
    Conti S., De Lellis C.: Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5(2), 521–549 (2003)MathSciNetGoogle Scholar
  15. 15.
    Dal Maso G., Francfort G.A., Toader R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Dal Maso G., Lazzaroni G.: Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. H. Poincaré Anal. Non Linéaire (2009, in press)Google Scholar
  17. 17.
    Dal Maso G., Toader R.: A model for the quasi-static growth of brittle fractures based on local minimization. Math. Models Methods Appl. Sci. 12, 1773–1799 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Federer H.: Geometric Measure Theory. Springer, New York (1969)zbMATHGoogle Scholar
  20. 20.
    Federer H., Fleming W.H.: Normal and integral currents. Ann. Math. (2) 72, 458–520 (1960)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Francfort G.A., Larsen C.J.: Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. 56, 1465–1500 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Francfort G.A., Marigo J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Fusco N., Leone C., Verde A., March R.: A lower semi-continuity result for polyconvex functionals in SBV. Proc. R. Soc. Edinb. Sect. A 136, 321–336 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Gent A.: Cavitation in rubber: a cautionary tale. Rubber Chem. Tech. 63, G49–G53 (1991)Google Scholar
  25. 25.
    Gent A., Wang C.: Fracture mechanics and cavitation in rubber-like solids. J. Mater. Sci. 26, 3392–3395 (1991)CrossRefADSGoogle Scholar
  26. 26.
    Giacomini A., Ponsiglione M.: Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials. Proc. R. Soc. Edinb. Sect. A 138, 1019–1041 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Giaquinta M., Modica G., Souček J.: Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 106, 97–159 (1989)zbMATHCrossRefGoogle Scholar
  28. 28.
    Giaquinta M., Modica G., Souček J.: Cartesian Currents in the Calculus of Variations. I. Springer, Berlin (1998)zbMATHGoogle Scholar
  29. 29.
    Giaquinta M., Modica G., Souček J.: Cartesian Currents in the Calculus of Variations II. Springer, Berlin (1998)zbMATHGoogle Scholar
  30. 30.
    Henao D., Mora-Corral C.: Fracture surfaces and the regularity of inverses for BV deformations. Preprint available at http://www.bcamath.org/documentos/archivos/publicaciones/HenaoMora2.pdf
  31. 31.
    Mucci D.: Fractures and vector valued maps. Calc. Var. Partial Differ. Equ. 22, 391–420 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Müller S.: Weak continuity of determinants and nonlinear elasticity. C. R. Acad. Sci. Paris Sér. I Math. 307, 501–506 (1988)zbMATHGoogle Scholar
  33. 33.
    Müller S.: Det=det. A remark on the distributional determinant.. C. R. Acad. Sci. Paris Sér. I Math. 311, 13–17 (1990)zbMATHGoogle Scholar
  34. 34.
    Müller S., Spector S.J.: An existence theory for nonlinear elasticity that allows for cavitation. Arch. Ration. Mech. Anal. 131, 1–66 (1995)zbMATHCrossRefGoogle Scholar
  35. 35.
    Müller S., Tang Q., Yan B.S.: On a new class of elastic deformations not allowing for cavitation. Ann. Inst. H. Poincaré Anal. Non Linéaire 11, 217–243 (1994)zbMATHGoogle Scholar
  36. 36.
    Niven I., Zuckerman H.S.: An Introduction to the Theory of Numbers. John, New York (1960)zbMATHGoogle Scholar
  37. 37.
    Petrinic N., Curiel Sosa J.L., Siviour C.R., Elliott B.C.F.: Improved predictive modelling of strain localisation and ductile fracture in a Ti-6Al-4V alloy subjected to impact loading. J. Phys. IV 134, 147–155 (2006)CrossRefGoogle Scholar
  38. 38.
    Petrinic N., Siviour C.R., Curiel Sosa, J.L., Elliott B.C.F.: On competing volumetric and deviatoric damage mechanisms in simulation of ductile fracture in Ti64 alloy at high rates of strain. Proceedings of Euromech EMMC-10 Conference “Multi-phase and multi-component materials under dynamic loading”, Kazimierz Dolny, 2007, pp. 467–478Google Scholar
  39. 39.
    Rudin W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974)zbMATHGoogle Scholar
  40. 40.
    Sivaloganathan J., Spector S.J.: On the existence of minimizers with prescribed singular points in nonlinear elasticity. J. Elast. 59, 83–113 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Tang Q.: Almost-everywhere injectivity in nonlinear elasticity. Proc. R. Soc. Edinb. Sect. A 109, 79–95 (1988)zbMATHGoogle Scholar
  42. 42.
    Williams M., Schapery R.: Spherical flaw instability in hydrostatic tension. Int. J. Fract. Mech. 1, 64–71 (1965)CrossRefGoogle Scholar
  43. 43.
    Ziemer W.P.: Weakly Differentiable Functions. Springer, New York (1989)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParis Cedex 05France
  2. 2.Basque Center for Applied Mathematics (BCAM)Parque Tecnológico de BizkaiaDerio (Vizcaya)Spain

Personalised recommendations