Free-end curvature boundary layer ($\ell =\sqrt{B/F}$)

Moment-free ends require ${\zeta }^{\prime \prime }={\kappa }_0$ there, so a highly stretched finite filament can retain curvature in boundary layers of width $\ell =\sqrt{B/F}$. For constant intrinsic curvature $\kappa$, zero transverse end forces give slope ${\zeta }^{\prime }(x)=\kappa \ell \sinh [(x-L/2)/\ell ]/\cosh [L/(2\ell )]$. Direct integration yieldsIts large-force deficit is ${\kappa }^2{\ell }^3/2$, of order $F^{-3/2}$, even though the smooth bulk curvature derivative vanishes. This shows why endpoint assumptions matter in force-extension of an intrinsically curved filament.