Mathematics

 In the talk we present the  derivation of  homogenized bending plate models from 3d elasticity by means of Gamma convergence. We obtain different models, depending on the quotient between periodicity of the material and the thickness of the body. Peculiarities arise in the case when dimensional reduction dominates i.e. when the thickness is on the smaller scale than the period of the material, since in that case we have to deal more with the geometric constraint. We are able only partially to solve this regime; namely when $\varepsilon^2 \ll h \ll \varepsilon$, $\varepsilon$ being the period of the material and $h$ being the thickness of the body. We can justify all these models in the stochastic setting by using the stochastic two-scale convergence introduced by Zhikov and Piatnitski. In the simpler case of bending rod and von Karman plate we are able to obtain the model without any periodicity assumption.