Polymerization-driven cantilever postbuckling ($F\simeq {\pi }^{2}B[1-3(\ell -d)/(2d)]/(4d^2)$)

A clamped filament touching a laterally sliding, moment-free wall has fixed projected gap $d$. For an inextensible small-amplitude elastica, parameterize its tip angle by $k=\sin ({\theta }_{\mathrm{tip}}/2)$. Then $\ell =\sqrt{B/F}K(k)$ and $d=\sqrt{B/F}[2E(k)-K(k)]$, where $K,E$ are complete elliptic integrals. Their small-$k$ expansions give the displayed decrease of wall force with excess contour length. It concerns the weakly buckled branch, before overhang or additional wall contact changes the boundary conditions.