Non-smoothness in the asymptotics of thin shells and propagation of singularities. Hyperbolic case

Philippe Karamian; Jacqueline Sanchez-Hubert; Évariste Sanchez Palencia

International Journal of Applied Mathematics and Computer Science (2002)

  • Volume: 12, Issue: 1, page 81-90
  • ISSN: 1641-876X

Abstract

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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.

How to cite

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Karamian, Philippe, Sanchez-Hubert, Jacqueline, and 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 12.1 (2002): 81-90. <http://eudml.org/doc/207571>.

@article{Karamian2002,
abstract = {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.},
author = {Karamian, Philippe, Sanchez-Hubert, Jacqueline, Sanchez Palencia, Évariste},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {hyperbolic systems; propagation of singularities; shells; thin shells; singular perturbation; variational problem; asymptotic expansions; geometric rigidity},
language = {eng},
number = {1},
pages = {81-90},
title = {Non-smoothness in the asymptotics of thin shells and propagation of singularities. Hyperbolic case},
url = {http://eudml.org/doc/207571},
volume = {12},
year = {2002},
}

TY - JOUR
AU - Karamian, Philippe
AU - Sanchez-Hubert, Jacqueline
AU - Sanchez Palencia, Évariste
TI - Non-smoothness in the asymptotics of thin shells and propagation of singularities. Hyperbolic case
JO - International Journal of Applied Mathematics and Computer Science
PY - 2002
VL - 12
IS - 1
SP - 81
EP - 90
AB - 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.
LA - eng
KW - hyperbolic systems; propagation of singularities; shells; thin shells; singular perturbation; variational problem; asymptotic expansions; geometric rigidity
UR - http://eudml.org/doc/207571
ER -

References

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  6. 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. Zbl0989.74047
  7. Goldenveizer A. L. (1962): Theory of Thin Elastic Shells. - New York: Pergamon. Zbl0145.45504
  8. 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. Zbl1004.74050
  9. 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. 
  10. 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. 
  11. 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. 
  12. 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. Zbl1006.74064
  13. 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. Zbl0932.35064
  14. Lions J.L. (1973): Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal. - Berlin: Springer. Zbl0268.49001
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  16. Sanchez-Hubert J. and Sanchez Palencia É. (1989), Vibration and Coupling of Continuous Systems. Asymptotic Methods. -Berlin: Springer. Zbl0698.70003
  17. 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. 
  18. 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. Zbl1079.35010
  19. 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. Zbl1128.74337
  20. Sanchez-Hubert J. and Sanchez Palencia É (1997): Coques Elastiques Minces. Proprietes Asymptotiques. - Paris: Masson. 
  21. 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. Zbl0958.35008
  22. 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. Zbl1059.74037

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