Spatial heterogeneity in 3D-2D dimensional reduction

Jean-François Babadjian; Gilles A. Francfort

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 1, page 139-160
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Babadjian, Jean-François, and Francfort, Gilles A.. "Spatial heterogeneity in 3D-2D dimensional reduction." ESAIM: Control, Optimisation and Calculus of Variations 11.1 (2005): 139-160. <http://eudml.org/doc/245696>.

@article{Babadjian2005,
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 $\Gamma $-convergence of the elastic energy, as the thickness tends to zero.},
author = {Babadjian, Jean-François, Francfort, Gilles A.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {dimension reduction; $\Gamma $-convergence; equi-integrability; quasiconvexity; relaxation; -convergence},
language = {eng},
number = {1},
pages = {139-160},
publisher = {EDP-Sciences},
title = {Spatial heterogeneity in 3D-2D dimensional reduction},
url = {http://eudml.org/doc/245696},
volume = {11},
year = {2005},
}

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
PB - EDP-Sciences
VL - 11
IS - 1
SP - 139
EP - 160
AB - 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.
LA - eng
KW - dimension reduction; $\Gamma $-convergence; equi-integrability; quasiconvexity; relaxation; -convergence
UR - http://eudml.org/doc/245696
ER -

References

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  3. [3] G. Bouchitté, I. Fonseca and M.L. Mascarenhas, Bending moment in membrane theory. J. Elasticity 73 (2003) 75–99. Zbl1059.74034
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  6. [6] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49 (2000) 1367–1404. Zbl0987.35020
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  10. [10] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Boca Raton, CRC Press (1992). Zbl0804.28001MR1158660
  11. [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. Zbl0789.73039
  12. [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. Zbl1012.74043
  13. [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. Zbl1021.74024
  14. [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. Zbl1041.74043
  15. [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. Zbl0847.73025

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