Spatial heterogeneity in 3D-2D dimensional reduction
ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 1, pp. 139-160.

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 $\Gamma$-convergence of the elastic energy, as the thickness tends to zero.

DOI : https://doi.org/10.1051/cocv:2004031
Classification : 49J45,  74B20,  74G65,  74K15,  74K35
Mots clés : dimension reduction, $\Gamma$-convergence, equi-integrability, quasiconvexity, relaxation
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author = {Babadjian, Jean-Fran\c{c}ois and Francfort, Gilles A.},
title = {Spatial heterogeneity in {3D-2D} dimensional reduction},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {139--160},
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Babadjian, Jean-François; Francfort, Gilles A. Spatial heterogeneity in 3D-2D dimensional reduction. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 1, pp. 139-160. doi : 10.1051/cocv:2004031. http://www.numdam.org/articles/10.1051/cocv:2004031/

[1] E. Acerbi and N. Fusco, Semicontinuity results in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1984) 125-145. | Zbl 0565.49010

[2] M. Bocea and I. Fonseca, Equi-integrability results for 3D-2D dimension reduction problems. ESAIM: COCV 7 (2002) 443-470. | Numdam | Zbl 1044.49010

[3] G. Bouchitté, I. Fonseca and M.L. Mascarenhas, Bending moment in membrane theory. J. Elasticity 73 (2003) 75-99. | Zbl 1059.74034

[4] A. Braides, personal communication.

[5] A. Braides and A. Defranceschi, Homogenization of multiple integrals. Oxford lectures Ser. Math. Appl. Clarendon Press, Oxford (1998). | MR 1684713 | Zbl 0911.49010

[6] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49 (2000) 1367-1404. | Zbl 0987.35020

[7] B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin (1988). | MR 2361288 | Zbl 0703.49001

[8] G. Dal Maso, An introduction to $\Gamma$-convergence. Birkhaüser, Boston (1993). | MR 1201152 | Zbl 0816.49001

[9] I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Gauthiers-Villars, Paris (1974). | MR 463993 | Zbl 0281.49001

[10] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Boca Raton, CRC Press (1992). | MR 1158660 | Zbl 0804.28001

[11] D. Fox, A. Raoult and J.C. Simo, A justification of nonlinear properly invariant plate theories. Arch. Rat. Mech. Anal. 25 (1992) 157-199. | Zbl 0789.73039

[12] 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. | Zbl 1012.74043

[13] G. Friesecke, R.D. James and S. Müller, A Theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure Appl. Math. 55 (2002) 1461-1506. | Zbl 1021.74024

[14] G. Friesecke, R.D. James and S. Müller, The Föppl-von Kármán plate theory as a low energy $\Gamma$-limit of nonlinear elasticity. C.R. Acad. Sci. Paris, Série I 335 (2002) 201-206. | Zbl 1041.74043

[15] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549-578. | Zbl 0847.73025

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