  Disordered elastic networks provide a framework for describing a wide variety
of physical systems, ranging from amorphous solids, through polymeric fibrous
materials to confluent cell tissues. In many cases, such networks feature two
widely separated rigidity scales and are nearly floppy, yet they undergo a
dramatic stiffening transition when driven to sufficiently large strains. We
present a complete scaling theory of the critical strain-stiffened state in
terms of the small ratio between the rigidity scales, which is conceptualized
in the framework of a singular perturbation theory. The critical state features
quartic anharmonicity, from which a set of nonlinear scaling relations is
derived. Scaling predictions for the macroscopic elastic modulus beyond the
critical state are derived as well, revealing a previously unidentified
characteristic strain scale. The predictions are quantitatively compared to a
broad range of available numerical data on biopolymer network models and future
research questions are discussed.
