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 -convergence of the elastic energy, as the thickness tends to zero.
Mots-clés : dimension reduction, $\Gamma $-convergence, equi-integrability, quasiconvexity, relaxation
@article{COCV_2005__11_1_139_0,
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},
publisher = {EDP-Sciences},
volume = {11},
number = {1},
year = {2005},
doi = {10.1051/cocv:2004031},
mrnumber = {2110618},
zbl = {1085.49015},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2004031/}
}
TY - JOUR AU - Babadjian, Jean-François AU - Francfort, Gilles A. TI - Spatial heterogeneity in 3D-2D dimensional reduction JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2005 SP - 139 EP - 160 VL - 11 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2004031/ DO - 10.1051/cocv:2004031 LA - en ID - COCV_2005__11_1_139_0 ER -
%0 Journal Article %A Babadjian, Jean-François %A Francfort, Gilles A. %T Spatial heterogeneity in 3D-2D dimensional reduction %J ESAIM: Control, Optimisation and Calculus of Variations %D 2005 %P 139-160 %V 11 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2004031/ %R 10.1051/cocv:2004031 %G en %F COCV_2005__11_1_139_0
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] and, Semicontinuity results in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1984) 125-145. | Zbl
[2] and, Equi-integrability results for 3D-2D dimension reduction problems. ESAIM: COCV 7 (2002) 443-470. | Numdam | Zbl
[3] , and, Bending moment in membrane theory. J. Elasticity 73 (2003) 75-99. | Zbl
[4] , personal communication.
[5] and, Homogenization of multiple integrals. Oxford lectures Ser. Math. Appl. Clarendon Press, Oxford (1998). | MR | Zbl
[6] , and, 3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49 (2000) 1367-1404. | Zbl
[7] , Direct methods in the calculus of variations. Springer-Verlag, Berlin (1988). | MR | Zbl
[8] , An introduction to -convergence. Birkhaüser, Boston (1993). | MR | Zbl
[9] and, Analyse convexe et problèmes variationnels. Dunod, Gauthiers-Villars, Paris (1974). | MR | Zbl
[10] and, Measure theory and fine properties of functions, Boca Raton, CRC Press (1992). | MR | Zbl
[11] , and, A justification of nonlinear properly invariant plate theories. Arch. Rat. Mech. Anal. 25 (1992) 157-199. | Zbl
[12] , and, Rigorous derivation of nonlinear plate theory and geometric rigidity. C.R. Acad. Sci. Paris, Série I 334 (2001) 173-178. | Zbl
[13] , and, 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
[14] , and, The Föppl-von Kármán plate theory as a low energy -limit of nonlinear elasticity. C.R. Acad. Sci. Paris, Série I 335 (2002) 201-206. | Zbl
[15] and, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549-578. | Zbl
Cité par Sources :
