The strain-stiffening critical exponents in polymer networks and their universality

Disordered athermal biopolymer materials, such as collagen networks that constitute a major component in extracellular matrices and various connective tissues, are initially soft and compliant but stiffen dramatically under strain. Such network materials are topologically sub-isostatic and feature strong rigidity scale separation between the bending and stretching response of the constituent polymer fibers. Recently, a comprehensive scaling theory of the athermal strain-stiffening phase transition has been developed, providing predictions for all mean-field critical exponents characterizing the transition in terms of the distance to the critical strain and of the small rigidity scales ratio. Here, we employ large-scale computer simulations, at and away from criticality, to test the analytic predictions. We find that all numerical critical exponents are in quantitative agreement with the analytically predicted ones. Moreover, we find that all predicted mean-field exponents remain valid whether the driving strain is shear, i.e., volume-preserving, or dilation, and independent of the degree of the network's sub-isostaticity, thus establishing the universality of the strain-stiffening phase transition with respect to the symmetry of the driving strain and the network's topology.