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 non-linear elasticity via homogenization and dimension reduction. We compare effective models obtained by either simultaneously or consecutively passing to the $\mathrm{\Gamma }$-limits as the thickness $h\ll 1$ and the size of the material microstructure $\text{e}\ll 1$ vanish. In the regime $\text{e}\ll h$ 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 $h\ll \text{e}$.