Finite-time extension of a quadratically thickened sheet ($L\sim \frac{\pi }{|k|}\sqrt{\frac{\mu H}{F}}(t^{\ast }-t{)}^{-1/2},t^{\ast }=4\mu H/F$)

For initial thickness $H(1+k^2{z}_0^2)$ and constant positive force, the material-map integral is dominated by a shrinking region around the minimum thickness as $\Delta \uparrow H$. With $k\ne 0$, the integral of $1/[H-\Delta +H{k}^2{z}_0^2]$ gives the displayed inverse-square-root divergence, whose coefficient is independent of the initial half-length. If $k=0$, every material point thins equally and $L=H{L}_0/(H-\Delta )$, with the stronger inverse-time divergence $L\sim 4\mu H{L}_0/[F(t^{\ast }-t)]$. These are ideal slender-sheet predictions up to their singular limit.