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

• Volume: 33, Issue: 5, page 1019-1032
• ISSN: 0764-583X

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## Abstract

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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 ${ℝ}^{3}$, 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.

## How to cite

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Coutand, Daniel. "Existence of a solution for a nonlinearly elastic plane membrane “under tension”." ESAIM: Mathematical Modelling and Numerical Analysis 33.5 (2010): 1019-1032. <http://eudml.org/doc/197427>.

@article{Coutand2010,
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. },
author = {Coutand, Daniel},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear membrane; implicit function theorem; asymptotic behaviour of solutions of nonlinear systems of partial differential equations.; coercive non-lower semi-continuous energy functional; existence; nonlinear membrane-plate equations},
language = {eng},
month = {3},
number = {5},
pages = {1019-1032},
publisher = {EDP Sciences},
title = {Existence of a solution for a nonlinearly elastic plane membrane “under tension”},
url = {http://eudml.org/doc/197427},
volume = {33},
year = {2010},
}

TY - JOUR
AU - Coutand, Daniel
TI - Existence of a solution for a nonlinearly elastic plane membrane “under tension”
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 33
IS - 5
SP - 1019
EP - 1032
AB - 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.
LA - eng
KW - Nonlinear membrane; implicit function theorem; asymptotic behaviour of solutions of nonlinear systems of partial differential equations.; coercive non-lower semi-continuous energy functional; existence; nonlinear membrane-plate equations
UR - http://eudml.org/doc/197427
ER -

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