Scaling theory of critical strain-stiffening in athermal biopolymer networks

Athermal biopolymer networks are disordered fibrous biomaterials abundant in living cells and tissues that feature strong rigidity scale separation between the bending and stretching response of the constituent fibers. Such networks -- that are generically underconstrained in terms of their degree of connectivity -- undergo a dramatic macroscopic stiffening transition when subjected to sufficiently large external strains, which in turn plays major roles in determining the mechanical stability and functionality of living systems. We present a complete scaling theory of the critical strain-stiffened state in terms of the small ratio between fiber bending and stretching/compression rigidities. We show that the small bending forces may be viewed as an isotropic singular perturbation applied to the stiff anisotropic backbone corresponding to fibers' stretching/compression. The critical state features quartic anharmonicity, from which a set of nonlinear scaling relations for various fundamental biophysical quantities are derived. These results, which are validated by highly accurate numerical simulations, are then used to derive scaling predictions for the macroscopic elastic modulus beyond the critical state, revealing a previously unidentified characteristic strain scale. We thus provide a comprehensive understanding of the strain-stiffening transition in athermal biopolymer networks.