Correlated rigidity percolation in fractal lattices

TL;DR

This paper investigates rigidity percolation in fractal lattices, revealing that fractal networks achieve mechanical stability at much lower densities than regular lattices, with implications for understanding materials like gels and fiber networks.

Contribution

It introduces a study of rigidity percolation in fractal Sierpiński gaskets, showing lower critical volume fractions and developing a simplified model for upper bounds.

Findings

Critical volume fraction is significantly lower in fractal lattices.

Fractal nature does not change the correlation length exponent or fractal dimension.

A simplified model provides an upper bound for the critical packing fraction.

Abstract

Rigidity percolation (RP) is the emergence of mechanical stability in networks. Motivated by the experimentally observed fractal nature of materials like colloidal gels and disordered fiber networks, we study RP in a fractal network. Specifically, we calculate the critical packing fractions of site-diluted lattices of Sierpi\'nski gaskets (SG's) with varying degrees of fractal iteration. Our results suggest that although the correlation length exponent and fractal dimension of the RP of these lattices are identical to that of the regular triangular lattice, the critical volume fraction is dramatically lower due to the fractal nature of the network. Furthermore, we develop a simplified model for an SG lattice based on the fragility analysis of a single SG. This simplified model provides an upper bound for the critical packing fractions of the full fractal lattice, and this upper bound is strictly obeyed by the disorder averaged RP threshold of the fractal lattices. Our results characterize rigidity in ultra-low-density fractal networks.