  Disordered fibrous networks are ubiquitous in nature as major structural
components of living cells and tissues. The mechanical stability of networks
generally depends on the degree of connectivity: only when the average number
of connections between nodes exceeds the isostatic threshold are networks
stable (Maxwell, J. C., Philosophical Magazine 27, 294 (1864)). Upon increasing
the connectivity through this point, such networks undergo a mechanical phase
transition from a floppy to a rigid phase. However, even sub-isostatic networks
become rigid when subjected to sufficiently large deformations. To study this
strain-controlled transition, we perform a combination of computational
modeling of fibre networks and experiments on networks of type I collagen
fibers, which are crucial for the integrity of biological tissues. We show
theoretically that the development of rigidity is characterized by a
strain-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 nonlinear
mechanics of collagen networks can be quantitatively captured by the
predictions of scaling theory for the strain-controlled critical behavior over
a wide range of network concentrations and strains up to failure of the
material.
