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Spatial heterogeneity in 3D-2D dimensional reduction

Published online by Cambridge University Press:  15 December 2004

Jean-François Babadjian  and
Gilles A. Francfort
Jean-François Babadjian
Affiliation:
LPMTM, Institut Galilée, Université Paris-Nord, 93430 Villetaneuse, France; jfb@galilee.univ-paris13.fr; francfor@galilee.univ-paris13.fr
Gilles A. Francfort
Affiliation:
LPMTM, Institut Galilée, Université Paris-Nord, 93430 Villetaneuse, France; jfb@galilee.univ-paris13.fr; francfor@galilee.univ-paris13.fr
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Abstract

A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem of Γ-convergence of the elastic energy, as the thickness tends to zero.

Keywords

Dimension reductionΓ-convergenceequi-integrabilityquasiconvexityrelaxation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 1 , January 2005 , pp. 139 - 160
Copyright
© EDP Sciences, SMAI, 2005

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References

Acerbi, E. and Fusco, N., Semicontinuity results in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1984) 125–145. CrossRef
Bocea, M. and Fonseca, I., Equi-integrability results for 3D-2D dimension reduction problems. ESAIM: COCV 7 (2002) 443–470. CrossRef
Bouchitté, G., Fonseca, I. and Mascarenhas, M.L., Bending moment in membrane theory. J. Elasticity 73 (2003) 75–99. CrossRef
A. Braides, personal communication.
A. Braides and A. Defranceschi, Homogenization of multiple integrals. Oxford lectures Ser. Math. Appl. Clarendon Press, Oxford (1998).
Braides, A., Fonseca, I. and Francfort, G., 3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49 (2000) 1367–1404. CrossRef
B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin (1988).
G. Dal Maso, An introduction to Γ-convergence. Birkhaüser, Boston (1993).
I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Gauthiers-Villars, Paris (1974).
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Boca Raton, CRC Press (1992).
Fox, D., Raoult, A. and Simo, J.C., A justification of nonlinear properly invariant plate theories. Arch. Rat. Mech. Anal. 25 (1992) 157–199.
G. Friesecke, R.D. James and S. Müller, Rigorous derivation of nonlinear plate theory and geometric rigidity. C.R. Acad. Sci. Paris, Série I 334 (2001) 173–178.
Friesecke, G., James, R.D. and Müller, S., Theorem, A on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure Appl. Math. 55 (2002) 1461–1506. CrossRef
G. Friesecke, R.D. James and S. Müller, The Föppl-von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity. C.R. Acad. Sci. Paris, Série I 335 (2002) 201–206.
Le Dret, H. and Raoult, A., The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549–578.