Abstract

In a previous article of the authors [J. Elasticity 73 (2004) 75–99] a model of nonlinear membrane was studied, where the external surface loading induces a density of bending moment. Due to the special form of the applied surface forces, the emerging Cosserat vector, resulting from the 3D-2D dimension reduction, was restricted to a class of two dimensional functions. In this paper the full 3D dependence of the Cosserat vector is analyzed via Γ-convergence techniques.

Author information

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

Guy Bouchitté

Dép. de Mathématiques, Université du Sud-Toulon-Var, 83957 La Garde, France

bouchitte@univ-tln.fr

Irene Fonseca

Dept. of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.

fonseca@andrew.cmu.edu

M. Luísa Mascarenhas

Dep. de Matemática, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal

mascar@fct.unl.pt

Keywords

  • Dimension reduction
  • Gamma-convergence
  • relaxation
  • quasiconvexity
  • bending effect

Mathematics Subject Classification

35E99, 35M10, 49J45, 74B20, 74K15, 74K20, 74K35

References

  1. E. Acerbi, N. Fusco: Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984) 125–145.https://doi.org/10.1007/BF00275731Google Scholar
  2. G. Bouchitté, I. Fonseca, M. L. Mascarenhas: Bending moment in membrane theory, J. Elasticity 73(1–3) (2004) 75–99.https://doi.org/10.1023/B:ELAS.0000029996.20973.92Google Scholar
  3. B. Dacorogna: Direct Methods in the Calculus of Variations. 2nd Ed., Springer, Berlin (2008).https://doi.org/10.1142/p616Google Scholar
  4. I. Ekeland, R. Témam: Convex Analysis and Variational Problems, SIAM, Philadelphia (1999).https://doi.org/10.1137/1.9781611971088Google Scholar
  5. I. Fonseca, D. Kinderlehrer, P. Pedregal: Energy functionals depending on elastic strain and chemical composition, Calc. Var. Partial Differ. Equ. 2(3) (1994) 283–313.https://doi.org/10.1007/BF01235532Google Scholar
  6. I. Fonseca, S. Müller, P. Pedregal: Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal. 29(3) (1998) 736–756.https://doi.org/10.1137/S0036141096306534Google Scholar
  7. H. Le Dret, A. Raoult: Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results, Arch. Ration. Mech. Anal. 154 (2000) 101–134.https://doi.org/10.1007/s002050000100Google Scholar
  8. P. Marcellini: Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscr. Math. 51(1–3) (1985) 1–28.https://doi.org/10.1007/BF01168345Google Scholar

Suggested citation

G. Bouchitté, I. Fonseca, M. L. Mascarenhas. “The Cosserat Vector in Membrane Theory: a Variational Approach.” Journal of Convex Analysis 16 (2009), No. 2, 351–365. https://doi.org/10.68381/jca16018

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.