Fig. 4.

Overcurvature of an open torus. (A) A circular annulus cut and sealed (Upper) curves more on inflation (Lower) and exhibits an excess angle $\mathrm{\Delta }\alpha$. (B) Dimensionless perimeter difference $u*$ as a function of the curvilinear coordinate $s*$ for various overcurvature strains ${ϵ}_{\alpha }$ in the case $R\gg w$ (circular section on inflation). $u*=0$ in the flat state. (Green line) On inflation with ${ϵ}_{\alpha }=0$. (Red line) For $\mathrm{\Delta }{\alpha }_{tens}\simeq 137^{○}$ (the red cross indicates the abscissa $s_{tens}*$ under tension). (Dashed blue line) For ${ϵ}_{\alpha }=0.5$ (this solution is not physically relevant, since it implies azimuthal extension). (C) Experimental and theoretical (dashed line) target curvature change as a function of the ratio $R/w$. Triangles indicate experiments with 16-$\mathrm{\mu }$m sheets of polypropylene; diamonds indicate experiments with 4-$\mathrm{\mu }$m sheets of polyethylene. (D) Wrinkles are absent along a band of finite width, which is highlighted in blue. The red line corresponds to the theoretically calculated profile (within the limit of inextensibility), and the red crosses mark the positions of the tensions lines for $R/w=3$.