Rigidity transitions in anisotropic networks happen in multiple steps

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.