Experimental Study of Shape Transitions and Energy Scaling in Thin Non-Euclidean Plates

We present the first quantitative measurements of shape and energy variation in non-Euclidean plates. Using environmentally responsive gel, we construct non-Euclidean disks of constant imposed Gaussian curvature, K-tar. We vary the disks' thickness t(0) and measure the dependence of configurations, surface curvature, and energy content on t(0). For K-tar < 0, configurations are of a single wavy mode and undergo a set of bifurcations that leads to their refinement with decreasing thickness. This leads to sharp increase in the amount of surface bending as t(0) -> 0, and to a slow decay of both bending and stretching energies. Both vary like t(0)(2), compared with t(0)(3) of the bending energy in disks with K-tar > 0.