Spatial heterogeneity in 3D-2D dimensional reduction

Jean-François Babadjian; Gilles A. Francfort

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 (2010): 139-160. <http://eudml.org/doc/90754>.

@article{Babadjian2010,
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. },
author = {Babadjian, Jean-François, Francfort, Gilles A.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Dimension reduction; Γ-convergence; equi-integrability; quasiconvexity; relaxation.; dimension reduction; -convergence; relaxation},
language = {eng},
month = {3},
number = {1},
pages = {139-160},
publisher = {EDP Sciences},
title = {Spatial heterogeneity in 3D-2D dimensional reduction},
url = {http://eudml.org/doc/90754},
volume = {11},
year = {2010},
}

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
DA - 2010/3//
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 Γ-convergence of the elastic energy, as the thickness tends to zero.
LA - eng
KW - Dimension reduction; Γ-convergence; equi-integrability; quasiconvexity; relaxation.; dimension reduction; -convergence; relaxation
UR - http://eudml.org/doc/90754
ER -

References

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  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. G. Friesecke, R.D. James and S. Müller, Rigorous derivation of nonlinear plate theory and geometric rigidity. C.R. Acad. Sci. Paris, Série I334 (2001) 173–178.  Zbl1012.74043
  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.  Zbl1021.74024
  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 I335 (2002) 201–206.  Zbl1041.74043
  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.  Zbl0847.73025

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