Tension-dependent transverse buckles and wrinkles in twisted elastic sheets

We investigate with experiments the twist induced transverse buckling instabilities of an elastic sheet of length $L$, width $W$, and thickness $t$, that is clamped at two opposite ends while held under a tension $T$. Above a critical tension $T_{\lambda }$ and critical twist angle ${\eta }_{tr}$, we find that the sheet buckles with a mode number $n\ge 1$ transverse to the axis of twist. Three distinct buckling regimes characterized as clamp-dominated, bendable, and stiff are identified, by introducing a bendability length $L_B$ and a clamp length $L_C(<L_B)$. In the stiff regime ($L>L_B$), we find that mode $n=1$ develops above ${\eta }_{tr}\equiv {\eta }_S\sim (t\mathrm{/}W)T^{-1\mathrm{/}2}$, independent of $L$. In the bendable regime $L_C<L<L_B$, $n=1$ as well as $n > 1$occurabove$\eta_{tr} \equiv \eta_B \sim \sqrt{t/L}T^{-1/4}$. Here, we find the wavelength ${\lambda }_B\sim \sqrt{Lt}{T}^{-1\mathrm{/}4}$, when $n>1$. These scalings agree with those derived from a covariant form of the F"oppl-von K'arm'an equations, however, we find that the $n=1$ mode also occurs over a surprisingly large range of $L$ in the bendable regime. Finally, in the clamp-dominated regime ($L<L_c$), we find that ${\eta }_{tr}$ is higher compared to ${\eta }_B$ due to additional stiffening induced by the clamped boundary conditions.