Commutativity and Non-commutativity of Limits in the Nonlinear Bending Theory for Prestrained Microheterogeneous Plates

In this paper we study the derivation of nonlinear bending models for prestrained elastic plates from three-dimensional nonlinear elasticity via homogenization and dimension reduction. We compare effective models obtained by either simultaneously or consecutively passing to the Γ Γ -limits as the thickness h 1 h ≪ 1 and the size of the material microstructure ε 1 ϵ ≪ 1 vanish. In the regime ε h ϵ ≪ h we show that the consecutive and simultaneous limit are equivalent, and also analyze the rate of convergence. In contrast, we observe that there are several different limit models in the case h ε h ≪ ϵ .