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Disordered fibrous networks are ubiquitous in nature as major structuralcomponents of living cells and tissues. The mechanical stability of networksgenerally depends on the degree of connectivity: only when the average numberof connections between nodes exceeds the isostatic threshold are networksstable (Maxwell, J. C., Philosophical Magazine 27, 294 (1864)). Upon increasingthe connectivity through this point, such networks undergo a mechanical phasetransition from a floppy to a rigid phase. However, even sub-isostatic networksbecome rigid when subjected to sufficiently large deformations. To study thisstrain-controlled transition, we perform a combination of computationalmodeling of fibre networks and experiments on networks of type I collagenfibers, which are crucial for the integrity of biological tissues. We showtheoretically that the development of rigidity is characterized by astrain-controlled continuous phase transition with signatures of criticality.Our experiments demonstrate mechanical properties consistent with our model,including the predicted critical exponents. We show that the nonlinearmechanics of collagen networks can be quantitatively captured by thepredictions of scaling theory for the strain-controlled critical behavior overa wide range of network concentrations and strains up to failure of thematerial.

Strain-controlled criticality governs the nonlinear mechanics of fibre networks