Mathematical Problems in Mechanics
Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity
Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 329-334.

In a previous work, it was shown how the linearized strain tensor field e:=12(uT+u)L2(Ω) can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain ΩR3, instead of the displacement vector field uH1(Ω) 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 u=0 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 eL2(Ω).

Dans un travail antérieur, on a montré comment le champ e:=12(uT+u)L2(Ω) des tenseurs linéarisés des déformations peut être considéré comme la seule inconnue dans le problème de Neumann pour lʼélasticité linéarisée posé sur un domaine ΩR3, au lieu du champ uH1(Ω) des déplacements 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 u=0 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 eL2(Ω).

Received:
Published online:
DOI: 10.1016/j.crma.2013.04.015
Ciarlet, Philippe 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_329_0,
     author = {Ciarlet, Philippe and Mardare, Cristinel},
     title = {Expression of {Dirichlet} boundary conditions in terms of the strain tensor in linearized elasticity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {329--334},
     publisher = {Elsevier},
     volume = {351},
     number = {7-8},
     year = {2013},
     doi = {10.1016/j.crma.2013.04.015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.04.015/}
}
TY  - JOUR
AU  - Ciarlet, Philippe
AU  - Mardare, Cristinel
TI  - Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 329
EP  - 334
VL  - 351
IS  - 7-8
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.04.015/
DO  - 10.1016/j.crma.2013.04.015
LA  - en
ID  - CRMATH_2013__351_7-8_329_0
ER  - 
%0 Journal Article
%A Ciarlet, Philippe
%A Mardare, Cristinel
%T Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity
%J Comptes Rendus. Mathématique
%D 2013
%P 329-334
%V 351
%N 7-8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.04.015/
%R 10.1016/j.crma.2013.04.015
%G en
%F CRMATH_2013__351_7-8_329_0
Ciarlet, Philippe; Mardare, Cristinel. Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity. Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 329-334. doi : 10.1016/j.crma.2013.04.015. http://www.numdam.org/articles/10.1016/j.crma.2013.04.015/

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

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

[3] Ciarlet, P.G. Linear and Nonlinear Functional Analysis with Applications, SIAM, Providence, 2013

[4] Ciarlet, P.G.; Ciarlet, P. Jr. Another approach to linearized elasticity and a new proof of Kornʼs inequality, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 259-271

[5] P.G. Ciarlet, C. Mardare, Intrinsic formulation of the displacement–traction problem in linearized elasticity, in preparation.

[6] Ciarlet, P.G.; Mardare, S. On Kornʼs inequalities in curvilinear coordinates, Math. Models Methods Appl. Sci., Volume 11 (2001), pp. 1379-1391

[7] Nečas, J. Equations aux Dérivées Partielles, Presses de lʼuniversité de Montréal, 1966

Cited by Sources: