= Finite-time extension of a quadratically thickened sheet
{title2=$L\sim\frac{\pi}{|k|}\sqrt{\frac{\mu H}{F}}(t^*-t)^{-1/2},\quad t^*=4\mu H/F$}
For initial thickness $H(1+k^2z_0^2)$ and constant positive force, the material-map integral is dominated by a shrinking region around the minimum thickness as $\Delta\uparrow H$. With $k\ne0$, the integral of $1/[H-\Delta+Hk^2z_0^2]$ 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 $L=HL_0/(H-\Delta)$, with the stronger inverse-time divergence $L\sim4\mu HL_0/[F(t^*-t)]$. These are ideal slender-sheet predictions up to their singular limit.
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