Abstract

We give an overview of relaxation and 3d-2d passage theorems in hyperelasticity in the framework of the multidimensional calculus of variations. We give several improvements of the proofs and we introduce the concept of p-ample integrand in showing its interest with respect to determinant type constraints. Some open questions are addressed.

Author information

Contact details are reproduced from the original publication and may be historical.

Omar Anza Hafsa

Université de Nîmes, Laboratoire MIPA, Site des Carmes, Place Gabriel Péri, 30021 Nîmes, France
and: LMGC, UMR-CNRS 5508, Place Eugène Bataillon, 34095 Montpellier, France

omar.anza-hafsa@unimes.fr

Keywords

  • Calculus of variations
  • integral representation
  • relaxation
  • 3d-2d passage
  • Gamma(pi)-convergence
  • determinant type constraints
  • hyperelasticity

References

  1. E. Acerbi, G. Buttazzo, D. Percivale: A variational definition of the strain energy for an elastic string, J. Elasticity 25(2) (1991) 137–148.https://doi.org/10.1007/BF00042462Google Scholar
  2. G. Anzellotti, S. Baldo, D. Percivale: Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity, Asymptotic Anal. 9(1) (1994) 61–100.https://doi.org/10.3233/ASY-1994-9105Google Scholar
  3. J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Systems & Control: Foundations & Applications 2, Birkhäuser, Boston (1990).Google Scholar
  4. O. Anza Hafsa, J.-P. Mandallena: Interchange of infimum and integral, Calc. Var. Partial Differ. Equ. 18(4) (2003) 433–449.https://doi.org/10.1007/s00526-003-0211-3Google Scholar
  5. O. Anza Hafsa, J. -P. Mandallena: The nonlinear membrane energy: variational derivation under the constraint “det∇u ≠ 0”, J. Math. Pures Appl., IX. Sér. 86(2) (2006) 100–115.https://doi.org/10.1016/j.matpur.2006.01.004Google Scholar
  6. O. Anza Hafsa, J.-P. Mandallena: Relaxation of variational problems in two-dimensional nonlinear elasticity, Ann. Mat. Pura Appl. (4) 186(1) (2007) 187–198.https://doi.org/10.1007/s10231-005-0177-1Google Scholar
  7. O. Anza Hafsa, J.-P. Mandallena: Relaxation theorems in nonlinear elasticity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25(1) (2008) 135–148.https://doi.org/10.1016/j.anihpc.2006.11.005Google Scholar
  8. O. Anza Hafsa, J.-P. Mandallena: The nonlinear membrane energy: variational derivation under the constraint “det∇u > 0”, Bull. Sci. Math. 132(4) (2008) 272–291.https://doi.org/10.1016/j.bulsci.2007.05.004Google Scholar
  9. O. Anza Hafsa, J.-P. Mandallena: Relaxation et passage 3d-2d avec contraintes de type déterminant, ArXiv:math.AP/0901.3688 (2009).Google Scholar
  10. O. Anza Hafsa, J.-P. Mandallena: Relaxation and 3d-2d passage 3d-2d with determinant type constraints: an outline, ArXiv:math.AP/0906.5262 (2009).Google Scholar
  11. J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63(4) (1977) 337–403.https://doi.org/10.1007/BF00279992Google Scholar
  12. H. Ben Belgacem: Modélisation de Structures Minces en Élasticité Non Linéaire, PhD Thesis, Université Pierre et Marie Curie, Paris (1996).Google Scholar
  13. H. Ben Belgacem: Une méthode de Γ-convergence pour un modèle de membrane non linéaire, C. R. Acad. Sci., Paris, Sér. I 324(7) (1997) 845–849.https://doi.org/10.1016/S0764-4442(97)86956-XGoogle Scholar
  14. H. Ben Belgacem: Relaxation of singular functionals defined on Sobolev spaces, ESAIM, Control Optim. Calc. Var. 5 (2000) 71–85.https://doi.org/10.1051/cocv:2000102Google Scholar
  15. P. Celada, S. Perrotta: Functions with prescribed singular values of the gradient, NoDEA, Nonlinear Differ. Equ. Appl. 5(3) (1998) 383–396.https://doi.org/10.1007/s000300050052Google Scholar
  16. B. Dacorogna: Quasiconvexity and relaxation of nonconvex problems in the calculus of variations, J. Funct. Anal. 46(1) (1982) 102–118.https://doi.org/10.1016/0022-1236(82)90046-5Google Scholar
  17. E. De Giorgi: Sulla convergenza di alcune successioni d’integrali del tipo dell’area, Rend. Mat., VI. Ser. 8 (1975) 277–294.Google Scholar
  18. E. De Giorgi, T. Franzoni: Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 58(6) (1975) 842–850.Google Scholar
  19. B. Dacorogna, A. M. Ribeiro: Existence of solutions for some implicit partial differential equations and applications to variational integrals involving quasi-affine functions, Proc. R. Soc. Edinb., Sect. A, Math. 134(5) (2004) 907–921.https://doi.org/10.1017/S0308210500003541Google Scholar
  20. G. Friesecke, R. D. James, S. Müller: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Commun. Pure Appl. Math. 55(11) (2002) 1461–1506.https://doi.org/10.1002/cpa.10048Google Scholar
  21. I. Fonseca: The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. Pures Appl., IX. Sér. 67(2) (1988) 175–195.Google Scholar
  22. M. L. Gromov, Ja. M. Èliašberg: Construction of nonsingular isoperimetric films, Trudy Mat. Inst. Steklov. 116 (1971) 18–33, 235 (in Russian); Proc. Steklov Inst. Math. 116 (1971) 13–28 (in English).Google Scholar
  23. M. Gromov: Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete 3(9) [Results in Mathematics and Related Areas 3(9)], Springer, Berlin (1986).Google Scholar
  24. R. V. Kohn, G. Strang: Optimal design and relaxation of variational problems. II, Commun. Pure Appl. Math. 39(2) (1986) 139–182.https://doi.org/10.1002/cpa.3160390202Google Scholar
  25. H. Le Dret, A. Raoult: Le modèle de membrane non linéaire comme limite variationnelle de l’élasticité non linéaire tridimensionnelle, C. R. Acad. Sci., Paris, Sér. I 317(2) (1993) 221–226.Google Scholar
  26. H. Le Dret, A. Raoult: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., IX. Sér. 74(6) (1995) 549–578.Google Scholar
  27. D. Percivale: The variational method for tensile structures, Preprint 16, Dipartimento di Matematica Politecnico di Torino (1991).Google Scholar

Suggested citation

O. Anza Hafsa, J.-P. Mandallena. “Relaxation and 3d-2d Passage Theorems in Hyperelasticity.” Journal of Convex Analysis 19 (2012), No. 3, 759–794. https://doi.org/10.68381/jca19040

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.