Conical Defects in Growing Sheets

A growing or shrinking disc will adopt a conical shape, its intrinsic geometry characterized by a surplus angle ${\phi }_e$ at the apex. If growth is slow, the cone will find its equilibrium. Whereas this is trivial if ${\phi }_e\le 0$, the disc can fold into one of a discrete infinite number of states if ${\phi }_e>0$. We construct these states in the regime where bending dominates and determine their energies and how stress is distributed in them. For each state a critical value of ${\phi }_e$ is identified beyond which the cone touches itself. Before this occurs, all states are stable; the ground state has twofold symmetry.

Figure 1

Geometry of the $e$ cone with ${\phi }_e=\frac{2\pi }{9}$.

Figure 2

Paper model (a) and calculated surface shapes for ${\phi }_e=2\pi$ with $n=2$ (b), $n=3$ (c), and $n=4$ (d).

Figure 3

Stress $C_{\parallel }$ as a function of ${\phi }_e$ for $n=2$ (solid line), 3 (short-dashed), 4 (dash-dotted), 5 (long dashed), and 10 (bold solid). The red curve shows $C_{\parallel }$ for the touching twofold. Above $C_{\parallel }=1$ the tension $\tilde{\sigma }$ along the curve $\Gamma$ changes sign.

Figure 4

Scaled bending energy $\tilde{B}$ for $n=2$ (solid line), 3 (short-dashed), 4 (dash-dotted), 5 (long dashed), and 10 (bold solid). The red curve shows $\tilde{B}$ for the touching twofold.