On Nonlinearly Elastic Membranes under Compression

Abstract

The classical equations of a nonlinearly elastic plane membrane made of Saint Venant-Kirchhoff material have been justified by Fox, Raoult and Simo (1993) and Pantz (2000). We show that, under compression, the associated minimization problem admits no solution. The proof is based on a result of non-existence of minimizers of non-convex functionals due to Dacorogna and Marcellini (1995). We generalize the application of their result from plane elasticity to three-dimensional plane membranes.

Keywords

Nonlinear elasticity / Minimization / Quasiconvexity / Quasiconvex envelope / Rank-1-convexity / 49J45 / 73C50 / 74K15

References

[1] Acerbi Arch. Rational Mech. Anal., 1984, 86: 125

[2] Ball J. Funct. Anal., 1984, 58: 225

[3] Ben ESAIM Con- trol Optim. Calc. Var., 2000, 5: 71

[4] Buttazzo, Giuseppe, Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Research Notes in Mathematics Series, Vol. 207, Longman Scientific & Technical, Harlow, 1989.

[5] Ciarlet, P. G., Mathematical Elasticity, Vol. I, Three-Dimensional Elasticity, Studies in Mathematics and Its Applications, Vol. 20, North-Holland Publishing Co., Amsterdam, 1988.

[6] Ciarlet, P. G., Mathematical Elasticity, Vol. II, Theory of Plates, Studies in Mathematics and Its Appli- cations, Vol. 27, North-Holland Publishing Co., Amsterdam, 1997.

[7] Ciarlet, P. G. and Destuynder, P., Approximation of three-dimensional models by two-dimensional models in plate theory, Energy Methods in Finite Element Analysis, Wiley, Chichester, 1979, 33–45.

[8] Ciarlet J. Mécanique, 1979, 18: 315

[9] Coutand J. Elasticity, 1998/1999, 53: 147

[10] Coutand, D., Analyse mathématique de quelques problèmes d'élasticité non linéaire et du calcul des variations, Ph.D thesis, Université Pierre et Marie Curie, Paris, 1999.

[11] Coutand M2AN Math. Model. Numer. Anal., 1999, 33: 1019

[12] Coutand Chin. Ann. Math., 1999, 20B: 279

[13] Dacorogna, B., Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, 78, Springer- Verlag, Berlin, 1989.

[14] Dacorogna Arch. Rational Mech. Anal., 1995, 131: 359

[15] Fox Arch. Rational Mech. Anal., 1993, 124: 157

[16] Genevey Math. Mech. Solids, 1997, 2: 215

[17] Green, A. E. and Zerna, W., Theoretical Elasticity, 2nd edition, Dover Publications Inc., New York, 1992.

[18] Kohn Comm. Pure Appl. Math., 1986, 39: 139

[19] Le Dret C. R. Acad. Sci. Paris Sér. I Math., 1993, 317: 221

[20] Le Dret C. R. Acad. Sci. Paris Sér. I Math., 1994, 318: 93

[21] Le Dret Comm. Appl. Nonlinear Anal., 1994, 1: 85

[22] Le Dret J. Math. Pures Appl., 1995, 74: 549

[23] Le Dret Proc. Roy. Soc. Edinburgh Sect. A, 1995, 125: 1179

[24] Meyers Trans. Amer. Math. Soc., 1965, 119: 125

[25] Morrey Pacific J. Math., 1952, 2: 25

[26] Morrey, C. B. Jr., Multiple integrals in the calculus of variations, Die Grundlehren der Mathematischen Wissenschaften, Band 130, Springer-Verlag, Inc., New York, 1966.

[27] Pantz C. R. Acad. Sci. Paris Sér. I Math., 2000, 331: 171

[28] Pantz, O., Quelques problèmes de modélisation en élasticité non linéaire, Ph.D thesis, Université Pierre et Marie Curie, Paris, 2000.

[29] Raoult Apl. Mat., 1986, 31: 417

[30] Tartar, L., Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., Vol. 39, Pitman, Boston, Mass., 1979, 136–212.

[31] Trabelsi C. R. Math. Acad. Sci. Paris, 2003, 337: 553

[32] Trabelsi, K., Sur la modélisation des plaques minces en élasticité non linéaire, Ph.D thesis, Université Pierre et Marie Curie, Paris, 2004.