Effective Medium Theory of Filamentous Triangular Lattice

TL;DR

This paper develops an effective medium theory for a filamentous triangular lattice that incorporates both bending and stretching forces, analyzing its mechanical response and rigidity thresholds across different elastic regimes.

Contribution

It introduces a comprehensive effective medium theory including bending and stretching, and identifies a universal rigidity threshold in filamentous lattices.

Findings

Identifies a rigidity threshold $p_b$ independent of bending rigidity.

Characterizes elastic regimes controlled by central-force percolation.

Provides a framework for understanding mechanical response in filamentous networks.

Abstract

We present an effective medium theory that includes bending as well as stretching forces, and we use it to calculate mechanical response of a diluted filamentous triangular lattice. In this lattice, bonds are central-force springs, and there are bending forces between neighboring bonds on the same filament. We investigate the diluted lattice in which each bond is present with a probability $p$. We find a rigidity threshold $p_b$ which has the same value for all positive bending rigidity and a crossover characterizing bending-, stretching-, and bend-stretch coupled elastic regimes controlled by the central-force rigidity percolation point at $p_{\textrm{CF}} \simeq 2/3$ of the lattice when fiber bending rigidity vanishes.