Derivation of a homogenized von-Karman shell theory from 3D elasticity

Abstract: We derive the model of homogenized von K\'arm\'an shell theory, starting from three dimensional nonlinear elasticity. The original three dimensional model contains two small parameters: the oscillations of the material $\e$ and the thickness of the shell $h$. Depending on the asymptotic ratio of these two parameters, we obtain different asymptotic theories. In the case $h\ll\e$ we identify two different asymptotic theories, depending on the ratio of $h$ and $\e^2$. In the case of convex shells we obtain a complete picture in the whole regime $h\ll\e$.