[Un modèle non linéaire de coques de type Koiter]
Nous définissons un nouveau modèle bidimensionnel non linéaire de coques « de type Koiter » qui peut être utilisé pour la modélisation de tout type de coque et de conditions aux limites et pour lequel nous établissons un théorème d'existence. Ce modèle utilise une densité d'énergie de type Ogden satisfaisant toutes les hypothèses du théorème d'existence fondamental de John Ball en élasticité tridimensionnelle non linéaire et qui est adaptée ici à la modélisation des coques non linéairement élastiques minces au moyen de déformations particulières, qui sont quadratiques en la variable transverse.
We define a new two-dimensional nonlinear shell model “of Koiter's type” that can be used for the modeling of any type of shell and boundary conditions and for which we establish an existence theorem. The model uses a specific three-dimensional stored energy function of Ogden's type that satisfies all the assumptions of John Ball's fundamental existence theorem in three-dimensional nonlinear elasticity and that is adapted here to the modeling of thin nonlinearly elastic shells by means of specific deformations that are quadratic with respect to the transverse variable.
Accepté le :
Publié le :
@article{CRMATH_2018__356_2_227_0,
author = {Ciarlet, Philippe G. and Mardare, Cristinel},
title = {A nonlinear shell model of {Koiter's} type},
journal = {Comptes Rendus. Math\'ematique},
pages = {227--234},
publisher = {Elsevier},
volume = {356},
number = {2},
year = {2018},
doi = {10.1016/j.crma.2017.12.005},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2017.12.005/}
}
TY - JOUR AU - Ciarlet, Philippe G. AU - Mardare, Cristinel TI - A nonlinear shell model of Koiter's type JO - Comptes Rendus. Mathématique PY - 2018 SP - 227 EP - 234 VL - 356 IS - 2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2017.12.005/ DO - 10.1016/j.crma.2017.12.005 LA - en ID - CRMATH_2018__356_2_227_0 ER -
%0 Journal Article %A Ciarlet, Philippe G. %A Mardare, Cristinel %T A nonlinear shell model of Koiter's type %J Comptes Rendus. Mathématique %D 2018 %P 227-234 %V 356 %N 2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2017.12.005/ %R 10.1016/j.crma.2017.12.005 %G en %F CRMATH_2018__356_2_227_0
Ciarlet, Philippe G.; Mardare, Cristinel. A nonlinear shell model of Koiter's type. Comptes Rendus. Mathématique, Tome 356 (2018) no. 2, pp. 227-234. doi : 10.1016/j.crma.2017.12.005. http://archive.numdam.org/articles/10.1016/j.crma.2017.12.005/
[1] Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1977), pp. 337-403
[2] Existence theorem for a nonlinear elliptic shell model, J. Elliptic Parabolic Equ., Volume 1 (2015), pp. 31-48
[3] Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988
[4] Mathematical Elasticity, Volume III: Theory of Shells, North-Holland, Amsterdam, 2000
[5] An Introduction to Differential Geometry with Applications to Geometry, Springer, Dordrecht, The Netherlands, 2005
[6] An existence theorem for nonlinearly elastic “flexural” shells, J. Elast., Volume 50 (1998), pp. 261-277
[7] Sur les lois de comportement en élasticité non linéaire compressible, C. R. Acad. Sci. Paris, Ser. II, Volume 295 (1982), pp. 423-426
[8] A mathematical model of Koiter's type for a nonlinearly elastic “almost spherical” shell, C. R. Acad. Sci. Paris, Ser. I, Volume 354 (2016), pp. 1241-1247
[9] P.G. Ciarlet, C. Mardare, An existence theorem for a two-dimensional nonlinear shell model of Koiter's type, in preparation.
[10] Derivation of nonlinear bending theory for shells from three dimensional nonlinear elasticity by Gamma-convergence, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003), pp. 697-702
[11] Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Commun. Pure Appl. Math., Volume 18 (1965), pp. 235-267
[12] Refined interior equations for thin elastic shells, Commun. Pure Appl. Math., Volume 24 (1971), pp. 583-615
[13] On the nonlinear theory of thin elastic shells, Proc. K. Ned. Akad. Wet., Ser. B, Phys. Sci., Volume 69 (1966), pp. 1-54
[14] The membrane shell model in nonlinear elasticity: a variational asymptotic derivation, J. Nonlinear Sci., Volume 6 (1996), pp. 59-84
[15] Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris, 1967 (English translation: Direct Methods in the Theory of Elliptic Equations, 2012, Springer)
Cité par Sources :
