  We propose a nonlinear extension of the standard tube model for semidilute
solutions of freely-sliding semiflexible polymers. Non-affine filament
deformations at the entanglement scale, the renormalisation of direct
interactions by thermal fluctuations, and the geometry of large deformations
are systematically taken into account. The stiffening response predicted for
athermal solutions of stiff rods is found to be thermally suppressed. Instead,
we obtain a broad linear response regime, supporting the interpretation of
shear stiffening at finite frequencies in polymerised actin solutions as
indicative of coupling to longitudinal modes. We observe a destabilizing effect
of large strains (about 100%), suggesting shear banding as a plausible
explanation for the widely observed catastrophic collapse of in-vitro
biopolymer solutions, usually attributed to network damage. In combination with
friction-type interactions, our analysis provides an analytically tractable
framework to address the nonlinear viscoplasticity of biological tissue on a
molecular basis.
