Loss of polyconvexity by homogenization: a new example

Original Article

Abstract

This article is devoted to the study of the asymptotic behavior of the zero-energy deformations set of a periodic nonlinear composite material. We approach the problem using two-scale Young measures. We apply our analysis to show that polyconvex energies are not closed with respect to periodic homogenization. The counterexample is obtained through a rank-one laminated structure assembled by mixing two polyconvex functions with P-growth, where p ≥ 2 can be fixed arbitrarily.

Mathematics Subject Classification (2000)

28A20 35B27 35B40 49J45 73B27 

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References

  1. 1.
    Ambrosio L., Fusco N. and Pallara D. (2000). Functions of bounded variation and free problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York Google Scholar
  2. 2.
    Astala K. and Faraco D. (2002). Quasiregular mappings and Young measures. Proc. R. Soc. Edinburgh Sect. A. Math. 132(5): 1045–1056 MATHMathSciNetGoogle Scholar
  3. 3.
    Babadjian, J.-F., Baía, M., Santos, P.M.: Characterization of two-scale gradient Young measures and application to homogenization. Preprint SISSA (2006)Google Scholar
  4. 4.
    Ball, J.M.: A version of the fundamental theorem for Young measures. In: PDEs and continuum models of phase transitions (Nice, 1988), Lecture Notes in Physics, vol. 3445, pp. 207–215. Springer, Berlin (1989)Google Scholar
  5. 5.
    Ball J.M. and James R.D. (1992). Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond A. 338: 389–450 MATHCrossRefGoogle Scholar
  6. 6.
    Ball, J.M.: Some open problems in elasticity. In: Geometry, mechanics, and dynamics, pp. 3–59. Springer, Berlin (2002)Google Scholar
  7. 7.
    Barchiesi, M.: Multiscale homogenization of convex functionals with discontinuous integrand. J. Convex Anal. (To appear)Google Scholar
  8. 8.
    Braides A. (1994). Loss of polyconvexity by homogenization.. Arch. Ration Mech. Anal. 127(2): 183–190 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Braides, A., Defranceschi, A.: Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications, vol. 12. The Clarendon Press Oxford University Press, New York (1998)Google Scholar
  10. 10.
    Braides, A.: A handbook of Γ-convergence. In: Handbook of Differential Equations: stationary partial differential equations, vol. 3, pp. 101–213. Elsevier, Amsterdam (2006)Google Scholar
  11. 11.
    Dacorogna, B.: Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78. Springer, Berlin (1989)Google Scholar
  12. 12.
    Dacorogna, B., Marcellini, P.: Implicit partial differential equations, Progress in Nonlinear Differential Equations and their Applications, vol. 37. Birkhäuser Boston Inc., Boston (1999)Google Scholar
  13. 13.
    Dal Maso, G.: An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser Boston Inc., Boston (1993)Google Scholar
  14. 14.
    Ekeland, I., Témam, R.: Convex analysis and variational problems, Classics in Applied Mathematics, vol. 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999). Translated from the French (English edition)Google Scholar
  15. 15.
    Evans, L.C.: Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74. Published for the Conference Board of the Mathematical Sciences, Washington (1990)Google Scholar
  16. 16.
    Faraco D. and Zhong X. (2003). Quasiconvex functions and Hessian equations. Arch. Ration Mech. Anal. 168(3): 245–252 MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Müller S. (1999). A sharp version of Zhang’s theorem on truncating sequences of gradients. Trans. Am. Math. Soc. 351(11): 4585–4597 MATHCrossRefGoogle Scholar
  18. 18.
    Müller, S.: Variational models for microstructure and phase transitions. In: Calculus of variations and geometric evolution problems (Cetraro, 1996), Lecture Notes in Mathematics, vol. 1713, pp. 85–210. Springer, Berlin (1999)Google Scholar
  19. 19.
    Pedregal P. (2006). Multi-scale Young measures. Trans. Am. Math. Soc. 358(2): 591–602 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Šverák V. (1992). New examples of quasiconvex functions. Arch. Ration Mech. Anal. 119(4): 293–300 CrossRefMATHGoogle Scholar
  21. 21.
    Šverák V. (1993). On Tartar’s conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire 10(4): 405–412 MATHGoogle Scholar
  22. 22.
    Valadier, M.: Young measures. In: Methods of nonconvex analysis (Varenna, 1989), Lecture Notes in Mathematics, vol. 1446, pp. 152–188. Springer, Berlin (1990)Google Scholar
  23. 23.
    Valadier M. (1997). Admissible functions in two-scale convergence. Port. Math. 54(2): 147–164 MATHMathSciNetGoogle Scholar
  24. 24.
    Zhang K. (1992). A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Serie IV 19(3): 313–326 MATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.S.I.S.S.A.TriesteItaly

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