A homogenization bending shell theory for multiscale materials from $3D$ nonlinear elasticity

We derive homogenized bending shell theories starting from three dimensional nonlinear elasticity. The original three dimensional model contains three small parameters: the two homogenization scales $\epsilon$ and ${\epsilon }^2$ of the material properties and the thickness $h$ of the shell. Depending on the asymptotic ratio of these three parameters, we obtain different asymptotic theories.