Rigorous derivation of Föppl's theory for clamped elastic membranes leads to relaxation
S. Conti, F. Maggi, and S. 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
, with 
the film thickness and
. 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.
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Published in: SIAM J. Math. Anal. 38, pp. 657-680, 2006.
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