Comptes Rendus
no. 6
Mathematical problems in mechanics
Thin layer approximations in mechanical structures: The Dirichlet boundary condition case
[Approximations de couche mince dans les structures mécaniques : le cas de la condition aux limites de Dirichlet]

Présenté par : Philippe G. Ciarlet

Frédérique Le Louër1
1 Sorbonne Universités, Université de technologie de Compiègne, LMAC EA2222 Laboratoire de mathématiques appliquées de Compiègne, CS 60 319, 60203 Compiègne cedex, France
Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 576-581.

Cette note concerne un problème de transmission dans une structure mécanique contenant une couche d'épaisseur mince ε>0. Nous construisons un développement asymptotique de la solution lorsque ε→0 qui permet de remplacer la couche mince par une condition aux limites approchées et nous en déduisons des estimations d'erreurs optimales. Nous présentons de nouveaux résultats lorsqu'une condition de Dirichlet est imposée sur la frontière interne de la couche mince, tandis que le cas d'une condition de Neumann est étudié dans F. Caubet, D. Kateb, F. Le Louër, J. Elast. 136 (1) (2019) 17–53.

We consider the solution to a transmission problem at a thin layer interface of thickness ε>0 in a mechanical structure. We build a multi-scale expansion for that solution as ε→0, which enables to replace the thin layer with an improved boundary condition and leads to optimal estimates for the remainders. This short note presents new results when a Dirichlet condition is imposed on the internal boundary of the thin layer and is the counterpart of F. Caubet, D. Kateb, F. Le Louër, J. Elast. 136 (1) (2019) 17–53, where the Neumann case was considered.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.06.001
Frédérique Le Louër 1

1 Sorbonne Universités, Université de technologie de Compiègne, LMAC EA2222 Laboratoire de mathématiques appliquées de Compiègne, CS 60 319, 60203 Compiègne cedex, France 
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     title = {Thin layer approximations in mechanical structures: {The} {Dirichlet} boundary condition case},
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     pages = {576--581},
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     doi = {10.1016/j.crma.2019.06.001},
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Frédérique Le Louër. Thin layer approximations in mechanical structures: The Dirichlet boundary condition case. Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 576-581. doi : 10.1016/j.crma.2019.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.06.001/

[1] F. Cakoni; I. de Teresa; H. Haddar; P. Monk Nondestructive testing of the delaminated interface between two materials, SIAM J. Appl. Math., Volume 76 (2016) no. 6, pp. 2306-2332

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[7] J.-C. Nédélec Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Applied Mathematical Sciences, vol. 144, Springer-Verlag, New York, 2001

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