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Existence of a solution for a nonlinearly elastic plane membrane “under tension”

Published online by Cambridge University Press:  15 August 2002

Daniel Coutand*
Affiliation:
Université Pierre et Marie Curie, Laboratoire d'Analyse Numérique, 4 Place Jussieu, 75252 Paris, France. coutand@ann.jussieu.fr.
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Abstract

A justification of the two-dimensional nonlinear “membrane” equations for a plate made of a Saint Venant-Kirchhoff material has been given by Fox et al. [9] by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. This model, which retains the material-frame indifference of the original three dimensional problem in the sense that its energy density is invariant under the rotations of ${\mathbb{R}}^3$, is equivalent to finding the critical points of a functional whose nonlinear part depends on the first fundamental form of the unknown deformed surface. We establish here an existence result for these equations in the case of the membrane submitted to a boundary condition of “tension”, and we show that the solution found in our analysis is injective and is the unique minimizer of the nonlinear membrane functional, which is not sequentially weakly lower semi-continuous. We also analyze the behaviour of the membrane when the “tension” goes to infinity and we conclude that a “well-extended” membrane may undergo large loadings.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 1999

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