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Publication 99-CNA-26

On the Inadequacy of the Scaling of Linear Elasticity for 3d-2d Asymptotics in a Nonlinear Setting

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

and

Gilles Francfort
L.P.M.T.M.
Universite Paris-Nord
93430 Villetaneuse, France

Abstract: Rescaling of independent and/or dependent variables is the usual first step when performing a 3D-2D asymptotic analysis of elastic equilibrium for an $\varepsilon$-thin three-dimensional domain. The direction transverse to the thickness of the domain is dilated by $\frac{1}{\varepsilon}$ in the linearized setting, as well as in its nonlinear analogue. The dependent variables (i.e., the components of the displacement field) are however left untouched in the nonlinear setting, while the third component is contracted by a factor $\varepsilon$ in the linearized setting. We investigate the consequences of adopting the contrary scaling of the dependent variables in both settings and evidence a striking difference at first order in $\varepsilon$: linearized elasticity is only affected through the kinematics of the limit fields on the plate (the resulting 2d-domain), while nonlinear elasticity loses its structure because the resulting plate energy depends on the imposed lateral boundary conditions. Therefore, there is no limit model behavior under such a scaling, at least at first order.

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