Mathematical Problems in Mechanics
Expression of Dirichlet boundary conditions in terms of the Cauchy–Green tensor field
Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 323-327.

In a previous work, it was shown how the Cauchy–Green tensor field C:=ΦTΦW2,s(Ω;S>3), s>3/2, can be considered as the sole unknown in the homogeneous Dirichlet problem of nonlinear elasticity posed over a domain ΩR3, instead of the deformation ΦW3,s(Ω;R3) in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition Φ=Φ0 on a portion Γ0 of the boundary of Ω can be recast, again as boundary conditions on Γ0, but this time expressed only in terms of the new unknown CW2,s(Ω;S>3).

Dans un travail antérieur, on a montré comment le champ C:=ΦTΦW2,s(Ω;S>3), s>3/2, des tenseurs de Cauchy–Green peut être considéré comme la seule inconnue dans le problème de Dirichlet homogène pour lʼélasticité non linéaire posé sur un domaine ΩR3, au lieu de la déformation ΦW3,s(Ω;R3) dans lʼapproche habituelle. Lʼobjet de cette Note est de montrer que la même approche sʼapplique aussi bien au problème de Dirichlet–Neumann. À cette fin, nous montrons comment la condition aux limites Φ=Φ0 sur une portion Γ0 de la frontière de Ω peut être ré-écrite, à nouveau sous forme de conditions aux limites sur Γ0, mais exprimées cette fois uniquement en fonction de la nouvelle inconnue CW2,s(Ω;S>3).

Accepted:
Published online:
DOI: 10.1016/j.crma.2013.05.001
Ciarlet, Philippe G. 1; Mardare, Cristinel 2

1 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
2 Université Pierre-et-Marie-Curie, laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, France
@article{CRMATH_2013__351_7-8_323_0,
     author = {Ciarlet, Philippe G. and Mardare, Cristinel},
     title = {Expression of {Dirichlet} boundary conditions in terms of the {Cauchy{\textendash}Green} tensor field},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {323--327},
     publisher = {Elsevier},
     volume = {351},
     number = {7-8},
     year = {2013},
     doi = {10.1016/j.crma.2013.05.001},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.05.001/}
}
TY  - JOUR
AU  - Ciarlet, Philippe G.
AU  - Mardare, Cristinel
TI  - Expression of Dirichlet boundary conditions in terms of the Cauchy–Green tensor field
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 323
EP  - 327
VL  - 351
IS  - 7-8
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.05.001/
DO  - 10.1016/j.crma.2013.05.001
LA  - en
ID  - CRMATH_2013__351_7-8_323_0
ER  - 
%0 Journal Article
%A Ciarlet, Philippe G.
%A Mardare, Cristinel
%T Expression of Dirichlet boundary conditions in terms of the Cauchy–Green tensor field
%J Comptes Rendus. Mathématique
%D 2013
%P 323-327
%V 351
%N 7-8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.05.001/
%R 10.1016/j.crma.2013.05.001
%G en
%F CRMATH_2013__351_7-8_323_0
Ciarlet, Philippe G.; Mardare, Cristinel. Expression of Dirichlet boundary conditions in terms of the Cauchy–Green tensor field. Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 323-327. doi : 10.1016/j.crma.2013.05.001. http://www.numdam.org/articles/10.1016/j.crma.2013.05.001/

[1] Aubin, T. Some Nonlinear Problems in Riemannian Geometry, Springer, Berlin, 2010

[2] Choquet-Bruhat, Y.; DeWitt-Morette, C.; Dillard-Bleick, M. Analysis, Manifolds, and Physics. Part I. Basics, North-Holland, Amsterdam, 1982

[3] Ciarlet, P.G. An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005

[4] Ciarlet, P.G.; Mardare, C. On rigid and infinitesimal rigid displacements in shell theory, J. Math. Pures Appl., Volume 83 (2004), pp. 1-15

[5] Ciarlet, P.G.; Mardare, C. Existence theorems in intrinsic nonlinear elasticity, J. Math. Pures Appl., Volume 94 (2010), pp. 229-243

[6] Ciarlet, P.G.; Mardare, C. Boundary conditions in nonlinear intrinsic elasticity, J. Math. Pures Appl. (2013) (in press)

[7] Mardare, S. On Pfaff systems with Lp coefficients and their applications in differential geometry, J. Math. Pures Appl., Volume 84 (2005), pp. 1659-1692

Cited by Sources: