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Rigorous derivation of Föppl's theory for clamped elastic membranes leads to relaxation
Sergio Conti, Francesco Maggi and Stefan Müller
We consider the nonlinear elastic energy of a thin membrane whose boundary is kept fixed, and assume that the energy per unit volume scales as $h^\beta$, with $h$ the film thickness and $\beta\in(0,4)$. We derive, by means of Gamma convergence, a limiting theory for the scaled displacements, which takes a form similar to the one proposed by Föppl in 1907. The difference can be understood as due to the fact that we fully incorporate the possibility of buckling, and hence derive a theory which does not have any resistence to compression. If forces normal to the membrane are included, then our result predicts that the normal displacement scales as the cube root of the force. This scaling depends crucially on the clamped boundary conditions. Indeed, if the boundary is left free then a much softer response is obtained, as was recently shown by Friesecke, James and Müller.