  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.
