Scaling theory of critical strain-stiffening in disordered elastic networks

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.