We consider the limit behaviour of elastic shells when the relative thickness tends to zero. We address the case when the middle surface has principal curvatures of opposite signs and the boundary conditions ensure the geometrical rigidity. The limit problem is hyperbolic, but enjoys peculiarities which imply singularities of unusual intensity. We study these singularities and their propagation for several cases of loading, giving a somewhat complete description of the solution.
@article{bwmeta1.element.bwnjournal-article-amcv12i1p81bwm,
author = {Karamian, Philippe and Sanchez-Hubert, Jacqueline and Sanchez Palencia, \'Evariste},
title = {Non-smoothness in the asymptotics of thin shells and propagation of singularities. Hyperbolic case},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {12},
year = {2002},
pages = {81-90},
zbl = {1023.74030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv12i1p81bwm}
}
Karamian, Philippe; Sanchez-Hubert, Jacqueline; Sanchez Palencia, Évariste. Non-smoothness in the asymptotics of thin shells and propagation of singularities. Hyperbolic case. International Journal of Applied Mathematics and Computer Science, Tome 12 (2002) pp. 81-90. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv12i1p81bwm/
[000] Bernadou M. (1994): Methodes d'Elements Finis pour les Problèmes de Coques Minces. - Paris: Masson. | Zbl 0355.65082
[001] Chapelle D. and Bathe K.J. (1998): Fundamental considerations for thefinite element analysis of shell structures. - Comp. Struct., Vol. 66, No. 1, pp. 19-36. | Zbl 0934.74073
[002] Ciarlet P.G. (2000): Mathematical Elasticity, Vol. III, Theory of Shells. - Amsterdam: North Holland. | Zbl 0953.74004
[003] Egorov Yu.V. and Shubin M.A. (1992): Linear partial differential equations. Foundations of the classical theory, In: Encyclopaedia of Mathematical Sciences, Vol. 30 (Part. Diff. Eqs. I). - New York: Springer, pp. 345-375.
[004] Gerard P. (1988): Solutions conormales analytiquesd'equations hyperboliques non lineaires. - Comm. Part. Diff. Eqns., Vol. 13, No. 3, pp. 345-375. | Zbl 0639.35054
[005] Gerard P. and Sanchez Palencia É. (2000): Sensitivity phenomena for certain thin elastic shells with edges. - Math. Meth. Appl.Sci., Vol.23, No. 4, pp. 379-399. | Zbl 0989.74047
[006] Goldenveizer A. L. (1962): Theory of Thin Elastic Shells. - New York: Pergamon. | Zbl 0145.45504
[007] Karamian P., Sanchez-Hubert J. and Sanchez Palencia É. (2000): Amodel problem for boundary layers of thin elastic shells. - Math. Modell.Num. Anal., Vol. 34, No. 1, pp. 1-30. | Zbl 1004.74050
[008] Karamian P. (1998a): Nouveaux resultats numeriques concernant les coques minces hyperboliques inhibees: Cas du paraboloide hyperbolique. - Compt. Rend. Acad. Sci., Paris, Serie IIb, Vol. 326, No. 11, pp. 755-760.
[009] Karamian P. (1998b): Reflexion des singularites dans les coques hyperboliques inhibees. - Compt. Rend. Acad. Sci., Paris, Serie IIb, Vol. 326, No. 1, pp. 609-614.
[010] Karamian P. (1999) Coques elastiques minces hyperboliques inhibees: calcul du problème limite par elements finis et non reflexion des singularites. - Ph. D. thesis, Universite de Caen.
[011] Karamian P. and Sanchez-Hubert J. (2002): Boundary layers in thin elastic shells with developable middle surface. - Euro. J. Mech. Asolids, Vol. 21, No. 1, pp. 13-47. | Zbl 1006.74064
[012] Leguillon D., Sanchez-Hubert J. and Sanchez Palencia É. (1999): Model problem of singular perturbation without limit in the space off inite energy and its computation. - C. R. Acad. Sci. Paris, Serie IIb, Vol. 327, No. 5, pp. 485-492. | Zbl 0932.35064
[013] Lions J.L. (1973): Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal. - Berlin: Springer. | Zbl 0268.49001
[014] Pitkaranta J., Matache A.M. and Schwab C. (1998): Fourier mode analysis of layers in shallow shell deformation. - Res. Rep., No. 98-18, seminar fur Angewandte Mathematik, Technische Hochschule Zurich. | Zbl 0983.74039
[015] Sanchez-Hubert J. and Sanchez Palencia É. (1989), Vibration and Coupling of Continuous Systems. Asymptotic Methods. -Berlin: Springer. | Zbl 0698.70003
[016] Sanchez-Hubert J. and Sanchez Palencia É. (1998): Pathological phenomena in computation of thin elastic shells. - Trans. Can. Mech. Eng., Vol. 22, No. 4B, pp. 435-446.
[017] Sanchez-Hubert J. and Sanchez Palencia É. (2001a): Singular perturbations with non-smooth limit and finite element approximation of layers for model problems of shells, In: Partial Differential Equations in Multistructures (F. Ali Mehmeti, J. von Below and S. Nicaise, Eds.). - New York: Dekker. | Zbl 1079.35010
[018] Sanchez-Hubert J. and Sanchez Palencia É. (2001b): An isotropic finite element estimates and local locking for shells: parabolic case. - Compt. Rend. Acad. Sci., Paris, Serie IIb, Vol. 329, No. 2, pp.153-159. | Zbl 1128.74337
[019] Sanchez-Hubert J. and Sanchez Palencia É (1997): Coques Elastiques Minces. Proprietes Asymptotiques. - Paris: Masson.
[020] Sanchez Palencia É. (2000): On a singular perturbation going out of the energy space. - J. Math. Pures Appl., Vol. 79, No. 8, pp. 591-602. | Zbl 0958.35008
[021] Sanchez Palencia É. (2001) New cases of propagation of singularities along characteristic boundaries for model problems of shell theory. - Compt. Rend. Acad. Sci., Paris, Serie IIb, Vol. 329, No. 5, pp. 315-321. | Zbl 1059.74037