Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-29T09:46:29.626Z Has data issue: false hasContentIssue false

A model problem for boundary layers of thin elastic shells

Published online by Cambridge University Press:  15 April 2002

Philippe Karamian
Affiliation:
Laboratoire de Mécanique, Université de Caen, Boulevard Maréchal Juin, 14032 Caen Cedex, France.
Jacqueline Sanchez-Hubert
Affiliation:
Laboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, France Laboratoire de Mécanique, Université de Caen, Boulevard Maréchal Juin, 14032 Caen Cedex, France.Laboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, FranceLaboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, France. (sanchez@lmm.jussieu.fr)
Évarisite Sanchez Palencia
Affiliation:
Laboratoire de Mécanique, Université de Caen, Boulevard Maréchal Juin, 14032 Caen Cedex, France.Laboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, FranceLaboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, France. (sanchez@lmm.jussieu.fr)
Get access

Abstract

We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theory as the relative thickness ε of the shell tends to zero. For ε = 0 our problem is parabolic, then it is a model of developpable surfaces. Boundary layers along and across the characteristic have very different structure. It also appears internal layers associated with propagations of singularities along the characteristics. The special structure of the limit problem often implies solutions which exhibit distributional singularities along the characteristics. The corresponding layers for small ε have a very large intensity. Layers along the characteristics have a special structure involving subspaces; the corresponding Lagrange multipliers are exhibited. Numerical experiments show the advantage of adaptive meshes in these problems.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Bernadou, Méthodes d'éléments finis pour les problèmes de coques minces. Masson, Paris (1994).
F. Brezzi and F. M. Fortin, Mixed and hybrid finite elements methods. Springer (1991).
Choï, D., Palma, F.J., Sanchez Palencia, É. and Vilari, M.A. no, Remarks on membrane locking in the finite element computation of very thin elastic shells. Math. Modell. Num. Anal. 32 (1998) 131-152.
P.G. Ciarlet, Mathematical elasticity, Vol. III, Theory of shells. North Holland, Amsterdam (to appear).
D. Chapelle and K.J. Bathe, Fundamental considerations for the finite element analysis of shell structures, Computers and Structures 66 (1998) 19-36.
P. Gérard and É. Sanchez Palencia, Sensitivity phenomena for certain thin elastic shells with edges. Math. Meth. Appl. Sci. (to appear).
A.L. Goldenveizer, Theory of elastic thin shells. Pergamon, New York (1962).
Karamian, P., Nouveaux résultats numériques concernant les coques minces hyperboliques inhibées: cas du paraboloïde hyperbolique. C. R. Acad. Sci. Paris Sér. IIb 326 (1998) 755-760.
Karamian, P., Réflexion des singularités dans les coques hyperboliques inhibées. C.R. Acad. Sci. Paris Sér. IIb 326 (1998) 609-614.
P. Karamian, Coques élastiques minces hyperboliques inhibées : calcul du problème limite par éléments finis et non reflexion des singularités. Thèse de l'Universté de Caen (1999).
Leguillon, D., Sanchez-Hubert, J. and Sanchez Palencia, É., Model problem of singular perturbation without limit in the space of finite energy and its computation. C.R. Acad. Sci. Paris Sér. IIb 327 (1999) 485-492.
J.L. Lions and É. Sanchez Palencia, Problèmes sensitifs et coques élastiques minces. in Partial Differential Equations and Functional Analysis, in memory of P. Grisvard (J. Céa, D. Chesnais, G. Geymonat, J.L. Lions Eds.), Birkhauser, Boston (1996) 207-220.
J.L. Lions and É. Sanchez Palencia, Sur quelques espaces de la théorie des coques et la sensitivité, in Homogenization and applications to material sciences, Cioranescu, Damlamian, Doneto Eds., Gakkotosho, Tokyo (1995) 271-278.
A.E.H Love, A treatrise on the mathematical theory of elasticity, Reprinted by Dover, New-York (1944).
Pitkaranta, J. and Sanchez Palencia, É., On the asymptotic behavior of sensitive shells with small thickness. C.R. Acad. Sci. Paris Sér. IIb 325 (1997) 127-134.
H.S. Rutten, Theory and design of shells on the basis of asymptotic analysis. Rutten and Kruisman, Voorburg (1973).
J. Sanchez-Hubert and É. Sanchez Palencia, Introduction aux méthodes asymptotiques et à l'homogénéisation, Masson, Paris (1992).
J. Sanchez-Hubert and É. Sanchez Palencia, Coques élastiques minces. Propriétés asymptotiques. Masson, Paris (1997).
Sanchez-Hubert, J. and Sanchez Palencia, É., Pathological phenomena in computation of thin elastic shells. Transactions Can. Soc. Mech. Engin. 22 (1998) 435-446.
M. Van Dyke, Perturbation methods in fluid mechanics. Academic Press, New-York (1964).