Problèmes mathématiques de la mécanique
Modélisation de films courbés non simples de second gradient
[Curved nonsimple grade-two thin films]
Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 343-347.

The behavior of a curved thin film made of a nonsimple grade two material is described by a nonconvex bulk energy depending on the first and second order derivatives of the deformation. We show using Γ-convergence arguments that the quasiminimizers of the three-dimensional energy converge, when the thickness of the curved film vanishes, to the minimizers of an energy which is a function of a two-dimensional deformation and of a Cosserat vector. Part of the energy density is obtained by A-quasiconvexification arguments.

Le comportement d'un film courbé mince composé d'un matériau non simple de second gradient est décrit par une énergie interne non convexe dépendant des dérivées secondes de la déformation. On démontre en utilisant des arguments de Γ-convergence, que lorsque l'épaisseur du film tend vers zéro, les quasiminimiseurs de l'énergie tridimensionnelle convergent vers les minimiseurs d'une énergie dépendant d'une déformation bidimensionnelle et d'un vecteur de Cosserat. Une partie de la densité d'énergie est obtenue par des arguments de A-quasiconvexification.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.01.018
Gargiulo, Giuliano 1; Zappale, Elvira 2; Zorgati, Hamdi 3

1 Dipartimento di Scienze biologiche ed ambientali, Universita' degli Studi del Sannio, 82100 Benevento, Italie
2 DIIMA, Università degli Studi di Salerno, via Ponte Don Melillo, 84084 Fisciano, Italie
3 Faculté des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisie
@article{CRMATH_2007__344_5_343_0,
     author = {Gargiulo, Giuliano and Zappale, Elvira and Zorgati, Hamdi},
     title = {Mod\'elisation de films courb\'es non simples de second gradient},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {343--347},
     publisher = {Elsevier},
     volume = {344},
     number = {5},
     year = {2007},
     doi = {10.1016/j.crma.2007.01.018},
     language = {fr},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.01.018/}
}
TY  - JOUR
AU  - Gargiulo, Giuliano
AU  - Zappale, Elvira
AU  - Zorgati, Hamdi
TI  - Modélisation de films courbés non simples de second gradient
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 343
EP  - 347
VL  - 344
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.01.018/
DO  - 10.1016/j.crma.2007.01.018
LA  - fr
ID  - CRMATH_2007__344_5_343_0
ER  - 
%0 Journal Article
%A Gargiulo, Giuliano
%A Zappale, Elvira
%A Zorgati, Hamdi
%T Modélisation de films courbés non simples de second gradient
%J Comptes Rendus. Mathématique
%D 2007
%P 343-347
%V 344
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.01.018/
%R 10.1016/j.crma.2007.01.018
%G fr
%F CRMATH_2007__344_5_343_0
Gargiulo, Giuliano; Zappale, Elvira; Zorgati, Hamdi. Modélisation de films courbés non simples de second gradient. Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 343-347. doi : 10.1016/j.crma.2007.01.018. http://www.numdam.org/articles/10.1016/j.crma.2007.01.018/

[1] Bhattacharya, K.; James, R.D. A theory of thin films of martensitic materials with applications to microactuators, J. Mech. Phys. Solids, Volume 47 (1999), pp. 531-576

[2] Braides, A.; Fonseca, I.; Leoni, G. A-quasiconvexity: relaxation and homogenization, ESAIM Control Optim. Calc. Var., Volume 5 (2000), pp. 539-577

[3] Ciarletta, M.; Iesan, D. Non-Classical Elastic Solids, Pitman Research Notes in Mathematics Series, vol. 293, Longman Scientific & Technical, Harlow, 1993

[4] Fonseca, I.; Müller, S. A-quasiconvexity, Lower semicontinuity, and Young measures, SIAM J. Math. Anal., Volume 30 (1999) no. 6, pp. 1355-1390

[5] G. Gargiulo, E. Zappale, The energy density of nonsimple materials grade two thin films via a Young measure approach, Boll. UMI. Ser. I, in press

[6] Le Dret, H.; Raoult, A. The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9), Volume 74 (1995) no. 6, pp. 549-578

[7] Le Dret, H.; Raoult, A. The membrane shell model in nonlinear elasticity: A variational asymptotic derivation, J. Nonlinear Sci., Volume 6 (1996), pp. 59-84

[8] Le Dret, H.; Zorgati, H. Films courbés minces martensitiques, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 65-69

[9] Le Dret, H.; Zorgati, H. Asymptotic modeling of thin curved martensitic films, Asymptotic Anal., Volume 48 (2006), pp. 141-171

[10] Santos, P.M.; Zappale, E. Second order analysis for thin structures, Nonlinear Anal., Volume 56 (2004) no. 5, pp. 679-713

[11] Toupin, R.A. Elastic materials with couple-stresses, Arch. Rational Mech. Anal., Volume 11 (1962), pp. 386-414

[12] Toupin, R.A. Theories of elasticity with couple stress, Arch. Rational Mech. Anal., Volume 17 (1964), pp. 85-112

Cited by Sources: