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

top
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

top

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

top
  1. E. Acerbi and N. Fusco, Semicontinuity results in the calculus of variations. Arch. Rat. Mech. Anal.86 (1984) 125–145.  
  2. M. Bocea and I. Fonseca, Equi-integrability results for 3D-2D dimension reduction problems. ESAIM: COCV7 (2002) 443–470.  
  3. G. Bouchitté, I. Fonseca and M.L. Mascarenhas, Bending moment in membrane theory. J. Elasticity73 (2003) 75–99.  
  4. A. Braides, personal communication.  
  5. A. Braides and A. Defranceschi, Homogenization of multiple integrals. Oxford lectures Ser. Math. Appl. Clarendon Press, Oxford (1998).  
  6. A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J.49 (2000) 1367–1404.  
  7. B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin (1988).  
  8. G. Dal Maso, An introduction to Γ-convergence. Birkhaüser, Boston (1993).  
  9. I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Gauthiers-Villars, Paris (1974).  
  10. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Boca Raton, CRC Press (1992).  
  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.  
  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.  
  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.  
  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.  
  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.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.