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$%E. Sánchez-Palencia, On the membrane approximation for thin elastic shells in the hyperbolic
case, Rev. Mat. Univ. Complut. Madrid 6 (1993), no. 2, 311–331.%$

On the Membrane Approximation for Thin
Elastic Shells in the Hyperbolic Case

Évariste SÁNCHEZ-PALENCIA

Université Pierre et Marie Curie

4 Place Jussieu

75252 Paris Cedex 05 — France

Revised: March 23, 1992

ABSTRACT

We consider the variational formulation of the problem of elastic shells in the membrane approximation, when the medium surface is hyperbolic. It appears that the corresponding bilinear form behaves as some kind of two-dimensional elasticity without shear rigidity. This amounts to saying that the membrane behaves rather as a net made of elastic strings disposed along the asymptotic curves of the surface than as an elastic two-dimensional medium. The mathematical and physical reasons of this behavior are explained and consequences are thrown concerning the admissible applied forces and the behavior of the solutions. The normal component of the displacement is somewhat non smooth. Our approach gives a description of the problem in somewhat general situations concerning the boundary conditions, whereas the classical approach in terms of a hyperbolic system of total order 4 with 2 double characteristics (the asymptotic lines) only works in the case when the boundary conditions lead to either Cauchy or Goursat problems.

2000 Mathematics Subject Classification: 73K15.