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, $K_{\mathrm{tar}}$. We vary the disks’ thickness $t_0$ and measure the dependence of configurations, surface curvature, and energy content on $t_0$. For $K_{\mathrm{tar}}<0$, 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 $t_0\to 0$, 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 $K_{\mathrm{tar}}>0$.

Figure 1

Variation of configurations with thickness. Disks of $K_{\mathrm{tar}}=0.0011{\mathrm{mm}}^{-2}$ (left) and $K_{\mathrm{tar}}=-0.0011{\mathrm{mm}}^{-2}$ (middle and right) of various initial thicknesses ($t_0=0.75$, 0.6, 0.25, 0.19, and 0.125 mm, top to bottom). For $K_{\mathrm{tar}}>0$ the disks keep the same basic shape, a hemisphere, with minor variations along the edge. The disks of $K_{\mathrm{tar}}<0$ undergo a set of bifurcations, in which the number of nodes $n$ increases with decreasing thickness. Surface amplitude of hyperbolic disks (right) shows that each configuration consists of a single wavy mode (the color bar, in mm, is common to all figures).

Figure 2

(a) The number of nodes $n$ as a function of sheet thickness for disks of $K_{\mathrm{tar}}=-0.0011{\mathrm{mm}}^{-2}$. A log-log plot (inset) shows that the data are well described by $n\sim t_0{}^{-0.5}$ (solid line). (b) The amplitude of the waviness $A$ as a function of radius on the buckled disk, for configurations with different number of nodes, $n=2–6$ (open squares down to stars). Inset: Multiplying $A(r)$ by $n$ leads to data convergence that is well described by $n{A}_n(r)=\sqrt{|K_{\mathrm{tar}}|}{r}^2$ (solid line).

Figure 3

Scaling of curvature and energy. (a) The average bending content $⟨\mathit{B}⟩$ as a function of sheet thickness. For $K_{\mathrm{tar}}<0$ (solid circles), $⟨\mathit{B}⟩$ increases sharply with decreasing thickness to values an order of magnitude larger than $|K|$. For $K_{\mathrm{tar}}>0$, $⟨\mathit{B}⟩$ is “saturated” at small thickness. Plotting the same data on a log-log scale (inset) shows that for $K_{\mathrm{tar}}<0$, $⟨\mathit{B}⟩\sim t_0{}^{-1}$ (solid line). (b) The total bending energy in a disk versus thickness. In the disks of $K_{\mathrm{tar}}>0$ (open triangles) the energy decays like $t_0{}^{3\pm 0.1}$ (dotted line), while for $K_{\mathrm{tar}}<0$ (solid circles) the bending energy scales like $t_0{}^{1.9\pm 0.1}$ (solid line).

Figure 4

The average value of the Gaussian curvature (stars) is close to $K_{\mathrm{tar}}$ for all disk thicknesses, indicating that on average all configurations are nearly stretch-free. Estimation of the stretching content by $⟨S⟩\approx ⟨\Delta K^2⟩/n^4$ (triangles) shows a linear (solid line) dependence on the initial thickness.