Summary
We discuss the derivation of two-dimensional models for thin elastic sheets as Γ-limits of three-dimensional nonlinear elasticity. We briefly review recent results and present an extension of the derivation of a membrane theory, first obtained by LeDret and Raoult in 1993, to the case of incompressible materials. The main difficulty is the construction of a recovery sequence which satisfies pointwise the nonlinear constraint of incompressibility.
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References
J. Adams, S. Conti, and A. DeSimone. Soft elasticity and microstructure in smectic C elastomers. Preprint, 2006.
E. Acerbi and N. Fusco. Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal., 86, 125–145, 1984.
H. Ben Belgacem, S. Conti, A. DeSimone, and S. Müller. Rigorous bounds for the Föppl-von Kármán theory of isotropically compressed plates. J. Nonlinear Sci., 10, 661–683, 2000.
H. Ben Belgacem, S. Conti, A. DeSimone, and S. Müller. Energy scaling of compressed elastic films. Arch. Rat. Mech. Anal., 164, 1–37, 2002.
H. B. Belgacem. Modélisation de structures minces en élasticité non linéaire. PhD thesis, Univ. Paris 6, 1996.
H. B. Belgacem. Une méthode de Γ-convergence pour un modèle de membrane non linéaire. C. R. Acad. Sci. Paris, 323, 845–849, 1996.
J. M. Ball and F. Murat. W 1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal., 58, 225–253, 1984.
A. Braides. Γ-convergence for beginners. Oxford University Press, Oxford, 2002.
S. Conti and G. Dolzmann. Derivation of a plate theory for incompressible materials. Preprint, 2005.
S. Conti, A. DeSimone, and G. Dolzmann. Semi-soft elasticity and director reorientation in stretched sheets of nematic elastomers. Phys. Rev. E, 66, 061710, 2002.
P. G. Ciarlet. Theory of plates, volume II of Mathematical elasticity. Elsevier, Amsterdam, 1997.
S. Conti and F. Maggi. Confining thin elastic sheets and folding paper. Preprint, 2005.
S. Conti, F. Maggi, and S. Müller. Rigorous derivation of Föppl’s theory for clamped elastic membranes leads to relaxation. Preprint, to appear in SIAM J. Math. Anal..
S. Conti. Low-energy deformations of thin elastic sheets: isometric embeddings and branching patterns. Habilitation thesis, Universität Leipzig, 2003.
B. Dacorogna. Direct methods in the calculus of variations. Springer-Verlag, New York, 1989.
G. Dal Maso. An introduction to Γ-convergence. Birkhäuser, Boston, 1993.
A. DeSimone and G. Dolzmann. Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies. Arch. Rat. Mech. Anal., 161, 181–204, 2002.
G. Friesecke, R. 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, 1461–1506, 2002.
I. Fonseca. The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. pures et appl., 67, 175–195, 1988.
E. De Giorgi and T. Franzoni. Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. (8), 58, 842–850, 1975.
W. Jin and P. Sternberg. Energy estimates of the von Kármán model of thin-film blistering. J. Math. Phys., 42, 192–199, 2001.
W. Jin and P. Sternberg. In-plane displacements in thin-film blistering. Proc. R. Soc. Edin. A, 132A, 911–930, 2002.
N. Kuiper. On C 1 isometric imbeddings I and II. Proc. Kon. Acad. Wet. Amsterdam A, 58, 545–556 and 683–689, 1955.
H. LeDret and A. Raoult. Le modèle de membrane nonlinéaire comme limite variationelle de l’élasticité non linéaire tridimensionelle. C. R. Acad. Sci. Paris, 317, 221–226, 1993.
H. LeDret and A. Raoult. The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl., 73, 549–578, 1995.
S. Müller. Variational models for microstructure and phase transitions. In F. Bethuel et al., editors, in: Calculus of variations and geometric evolution problems, Springer Lecture Notes in Math. 1713, pages 85–210. Springer-Verlag, 1999.
J. Nash. C 1 isometric imbeddings. Ann. Math., 60, 383–396, 1954.
O. Pantz. On the justification of the nonlinear inextensional plate model. Arch. Rat. Mech. Anal., 167, 179–209, 2003.
K. Trabelsi. Incompressible nonlinearly elastic thin membranes. C. R. Acad. Sci. Paris, Ser. I, 340, 75–80, 2005.
K. Trabelsi. Modeling of a nonlinear membrane plate for incompressible materials via Gamma-convergence. To appear in Anal. Appl. (Singap.).
H. Whitney. The general type of singularity of a set of 2n-1 smooth functions of n variables. Duke Math. J., 10, 161–172, 1943.
H. Whitney. The self-intersections of a smooth n-manifold in 2n-space. Ann. Math., 45, 220–246, 1944.
H. Whitney. The singularities of a smooth n-manifold in 2n-1-space. Ann. Math., 45, 247–293, 1944.
M. Warner and E. M. Terentjev. Nematic elastomers — a new state of matter? Prog. Polym. Sci., 21, 853–891, 1996.
M. Warner and E. M. Terentjev. Liquid Crystal Elastomers. Oxford Univ. Press, 2003.
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Conti, S., Dolzmann, G. (2006). Derivation of Elastic Theories for Thin Sheets and the Constraint of Incompressibility. In: Mielke, A. (eds) Analysis, Modeling and Simulation of Multiscale Problems. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35657-6_9
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DOI: https://doi.org/10.1007/3-540-35657-6_9
Publisher Name: Springer, Berlin, Heidelberg
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