Comptes Rendus
Mathematical Problems in Mechanics
Some unilateral Korn inequalities with application to a contact problem with inclusions
Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 861-865.

In the first part of this note, some unilateral inequalities of the Korn type are established. These inequalities seem to be new.

In the second part, these inequalities are used in an essential way to prove the existence of a solution (which is not necessarily unique) for a unilateral contact problem involving a matrix material with inclusions of various shapes (the conditions depend on the shape of each inclusion in a remarkable way).

Dans une première partie sont démontrées quelques inégalités de Korn unilatérales qui semblent nouvelles.

Ces inégalités sont alors utilisées de façon essentielle dans la démontration de lʼexistence dʼune solution pour un problème de contact unilatéral dans une matrice avec des inclusions de diverses formes (les conditions dépendent de façon remarquable de la forme de chaque inclusion).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.09.006

Alain Damlamian 1

1 Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-94010, Créteil, France
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Alain Damlamian. Some unilateral Korn inequalities with application to a contact problem with inclusions. Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 861-865. doi : 10.1016/j.crma.2012.09.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.09.006/

[1] D. Cioranescu, A. Damlamian, J. Orlik, Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptot. Anal., in press.

[2] G. Duvaut; J.-L. Lions Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques, vol. 21, Dunod, Paris, 1972

[3] G. Fichera Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti Acc. Naz. Lincei (s. VIII), Volume VII (1964)

[4] G. Fichera Elastostatics problems with unilateral constraints, Séminaire Jean Leray, Volume 3 (1966–1967), pp. 64-68

[5] G. Fichera Unilateral constraints in elasticity, Actes Congrès Int. Math. (Nice 1970), vol. 3, Gauthier–Villars, Paris, 1970, pp. 79-84

[6] J. Gobert Une inégalité fondamentale de la théorie de lʼélasticité, Bull. Soc. Roy. Sci. Liège, Volume 31 (1962), pp. 182-191

[7] I. Hlaváček; J. Haslinger; J. Nečas Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, vol. IV, North-Holland, Amsterdam, 1996, pp. 313-485

[8] I. Hlaváček; J. Haslinger; J. Nečas; J. Lovíšek Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences, vol. 66, Springer-Verlag, New York, 1988 (x+275 pp. Translated from Slovak by J. Jarník) (ISBN: 0-387-96597-1)

[9] N. Kikuchi; J.T. Oden Contact Problems in Elasticity, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1988

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