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 $n \geq 1$ 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 $\eta_{tr} \equiv \eta_S \sim (t/W) T^{-1/2}$, independent of $L$. In the bendable regime $L_{C}<L<L_B$, $n=1$ as well as $n > 1$ occur above $\eta_{tr} \equiv \eta_B \sim \sqrt{t/L}T^{-1/4}$. Here, we find the wavelength $\lambda_B \sim \sqrt{Lt}T^{-1/4}$, 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.