  Disordered biopolymer gels have striking mechanical properties including
strong nonlinearities. In the case of athermal gels (such as collagen-I) the
nonlinearity has long been associated with a crossover from a bending dominated
to a stretching dominated regime of elasticity. The physics of this crossover
is related to the existence of a central-force isostatic point and to the fact
that for most gels the bending modulus is small. This crossover induces scaling
behavior for the elastic moduli. In particular, for linear elasticity such a
scaling law has been demonstrated [Broedersz et al. Nature Physics, 2011 7,
983]. In this work we generalize the scaling to the nonlinear regime with a
two-parameter scaling law involving three critical exponents. We test the
scaling law numerically for two disordered lattice models, and find a good
scaling collapse for the shear modulus in both the linear and nonlinear
regimes. We compute all the critical exponents for the two lattice models and
discuss the applicability of our results to real systems.
