[Le problème en déplacement pur en élasticité non linéaire tri-dimensionnelle : Formulation intrinsèque et théorèmes d'existence]
Dans cette Note, les équations de l'élasticité non linéaire tri-dimensionnelle correspondant au problème en déplacement pur sont ré-écrites, soit comme un problème aux limites, soit comme un problème de minimisation, l'inconnue étant dans les deux cas le tenseur des déformations de Cauchy–Green, au lieu de la déformation comme il est usuel. On montre ensuite que l'un et l'autre problème ont au moins une solution si les forces sont suffisamment petites et si la densité d'énergie satisfait certaines hypothèses naturelles. Le second problème constitue un exemple de problème de minimisation d'une fonctionnelle non coercive sur une variété de Banach.
In this Note, the equations of nonlinear three-dimensional elasticity corresponding to the pure displacement problem are recast either as a boundary value problem, or as a minimization problem, where the unknown is in both cases the Cauchy–Green strain tensor, instead of the deformation as is customary. We then show that either problem possesses a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. The second problem provides an example of a minimization problem for a non-coercive functional over a Banach manifold.
Publié le :
@article{CRMATH_2009__347_11-12_677_0,
author = {Ciarlet, Philippe G. and Mardare, Cristinel},
title = {The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems},
journal = {Comptes Rendus. Math\'ematique},
pages = {677--683},
publisher = {Elsevier},
volume = {347},
number = {11-12},
year = {2009},
doi = {10.1016/j.crma.2009.03.020},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.03.020/}
}
TY - JOUR AU - Ciarlet, Philippe G. AU - Mardare, Cristinel TI - The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems JO - Comptes Rendus. Mathématique PY - 2009 SP - 677 EP - 683 VL - 347 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.03.020/ DO - 10.1016/j.crma.2009.03.020 LA - en ID - CRMATH_2009__347_11-12_677_0 ER -
%0 Journal Article %A Ciarlet, Philippe G. %A Mardare, Cristinel %T The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems %J Comptes Rendus. Mathématique %D 2009 %P 677-683 %V 347 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.03.020/ %R 10.1016/j.crma.2009.03.020 %G en %F CRMATH_2009__347_11-12_677_0
Ciarlet, Philippe G.; Mardare, Cristinel. The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems. Comptes Rendus. Mathématique, Tome 347 (2009) no. 11-12, pp. 677-683. doi : 10.1016/j.crma.2009.03.020. http://www.numdam.org/articles/10.1016/j.crma.2009.03.020/
[1] Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York, 1988
[2] Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of non-linearly elastic rods and shells, Arch. Ration. Mech. Anal., Volume 61 (1976), pp. 307-351
[3] Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1977), pp. 337-403
[4] -Quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., Volume 58 (1984), pp. 225-253
[5] Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988
[6] Mathematical Elasticity, Volume III: Theory of Shells, North-Holland, Amsterdam, 2000
[7] Another approach to linearized elasticity and a new proof of Korn's inequality, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 259-271
[8] On constitutive equations in compressible nonlinear elasticity, C. R. Acad. Sci. Paris, Sér. II, Volume 295 (1982), pp. 423-426
[9] Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl., Volume 83 (2004), pp. 811-843
[10] P.G. Ciarlet, C. Mardare, Existence theorems in intrinsic nonlinear elasticity, in preparation
[11] P.G. Ciarlet, C. Mardare, The pure displacement problem in intrinsic linearized elasticity, in preparation
[12] -regularity of a manifold as a function of its metric tensor, Anal. Appl., Volume 4 (2006), pp. 19-30
[13] On isometric immersions of a Riemannian space with little regularity, Anal. Appl., Volume 2 (2004), pp. 193-226
[14] On Pfaff systems with coefficients and their applications in differential geometry, J. Math. Pures Appl., Volume 84 (2005), pp. 1659-1692
[15] On systems of first order linear partial differential equations with coefficients, Adv. Differential Equations, Volume 12 (2007), pp. 301-360
[16] Energy minimizers in nonlinear elastostatics and the implicit function theorem, Arch. Ration. Mech. Anal., Volume 114 (1991), pp. 95-117
Cité par Sources :
