Pattern selection and multiscale behavior in metrically discontinuous non-Euclidean plates

We study equilibrium configurations of non-Euclidean plates, in which the reference metric is uniaxially periodic. This work is motivated by recent experiments on thin sheets of composite thermally responsive gels [1]. Such sheets bend perpendicularly to the periodic axis in order to alleviate the metric discrepancy. For abruptly varying metrics, we identify multiple scaling regimes with different power law dependences of the elastic energy $\En$ and the axial curvature $\kappa$ on the sheet's thickness $h$. In the $h\to0$ limit the equilibrium configuration tends to an isometric embedding of the reference metric, and $\En\sim h^2$. Two intermediate asymptotic regimes emerge in between the buckling threshold and the $h\to0$ limit, in which the energy scales either like $h^{4/5}$ or like $h^{2/3}$. We believe that this system exemplifies a much more general phenomenon, in which the thickness of the sheet induces a cutoff length scale below which finer structures of the metric cannot be observed. When the reference metric consists of several separated length scales, a decrease of the sheet's thickness results in a sequence of conformational changes, as finer properties of the reference metric are revealed.