Rigorous derivation of Föppl's theory for clamped elastic membranes leads to relaxation

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^β, with h the film thickness and β ∈ (0, 4). We derive, by means of Γ convergence, a limiting theory for the scaled displacements, which takes a form similar to the one proposed by Foppl in 1907. Our variational approach fully incorporates the possibility of buckling already observed during the derivation of the reduced two-dimensional theory. At variance with Foppl’s, our limiting model is lower semicontinuous and has an energetics that vanishes on all contractions. Therefore buckling does not need to be explicitly resolved when computing with the reduced theory. 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 Mu ̈ller [Arch. Ration. Mech. Anal., 180 (2006), pp. 183–236].