= Bending and torsion of filament bundles
{title2=$B,C\sim Er^2R^2,\ ErR^3,\ ER^4$}
For independent sliding filaments in a filled bundle, bending energies add and $B\sim NEr^4\sim Er^2R^2$. A coherently bonded one-filament-thick tube has $B\sim ErR^3$, while a bonded filled rod has $B\sim ER^4$. The same torsional scalings apply with the <shear modulus> instead of <Young's modulus> if individual sliding filaments are forced to undergo a common material twist. Completely free filament spin/slip does not define that torsional response: collective helical bending alone begins at fourth order in the twist rate.
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