Rigidity transitions in anisotropic networks happen in multiple steps

TL;DR

This paper reveals that rigidity transitions in anisotropic networks occur in multiple steps, with stress percolation happening at different thresholds along different directions, and introduces universal scaling laws for these transitions.

Contribution

It uncovers a multi-step rigidity transition in anisotropic networks and develops universal scaling functions, advancing understanding of mechanical phase transitions in aligned biological materials.

Findings

Rigidity in anisotropic networks occurs in at least two steps.

Stress percolation thresholds differ along various directions.

Universal exponents and scaling functions describe the transition.

Abstract

We study how the rigidity transition in a triangular lattice changes as a function of anisotropy by preferentially filling bonds on the lattice in one direction. We discover that the onset of rigidity in anisotropic spring networks arises in at least two steps, reminiscent of the two-step melting transition in two dimensional crystals. In particular, our simulations demonstrate that the percolation of stress-supporting bonds happens at different critical volume fractions along different directions. By examining each independent component of the elasticity tensor, we determine universal exponents and develop universal scaling functions to analyze isotropic rigidity percolation as a multicritical point. We expect that these results will be important for elucidating the underlying mechanical phase transitions governing the properties of biological materials ranging from the cytoskeletons of cells to the extracellular networks of tissues such as tendon where the networks are often preferentially aligned.