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

We derive homogenized von Kármán shell theories starting from three dimensional nonlinear elasticity. The original three dimensional model contains two small parameters: the period of oscillation $\epsilon$ of the material properties and the thickness $h$ of the shell. Depending on the asymptotic ratio of these two parameters, we obtain different asymptotic theories. In the case $h<<\epsilon$ we identify two different asymptotic theories, depending on the ratio of $h$ and $\epsilon^2$. In the case of convex shells we obtain a complete picture in the whole regime $h<<\epsilon$