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 Γ-convergence of the elastic energy, as the thickness tends to zero.

DOI : 10.1051/cocv:2004031
Classification : 49J45, 74B20, 74G65, 74K15, 74K35
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://archive.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://archive.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://archive.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://archive.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

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

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

[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 | Zbl

[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

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

[8] G. Dal Maso, An introduction to Γ-convergence. Birkhaüser, Boston (1993). | MR | Zbl

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

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

[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

[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

[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

[14] 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. | Zbl

[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

Cité par Sources :