Elasticity of Filamentous Kagome Lattice

TL;DR

This paper investigates the elastic properties of a diluted filamentous kagome lattice, revealing a continuous rigidity transition at a critical bond probability and demonstrating a crossover from bending to stretching dominated elasticity.

Contribution

It introduces a model for filamentous networks using a diluted kagome lattice and combines effective medium theories with simulations to analyze elasticity and rigidity transitions.

Findings

Rigidity percolation transition at p ≈ 0.605 for non-zero bending stiffness

Crossover from bending to stretching dominated response near p=1 and p=p_b

Effective medium theories accurately predict scaling forms of elastic modulus

Abstract

The diluted kagome lattice, in which bonds are randomly removed with probability $1-p$, consists of straight lines that intersect at points with a maximum coordination number of four. If lines are treated as semi-flexible polymers and crossing points are treated as crosslinks, this lattice provides a simple model for two-dimensional filamentous networks. Lattice-based effective medium theories and numerical simulations for filaments modeled as elastic rods, with stretching modulus $\mu$ and bending modulus $\kappa$, are used to study the elasticity of this lattice as functions of $p$ and $\kappa$. At $p=1$, elastic response is purely affine, and the macroscopic elastic modulus $G$ is independent of $\kappa$. When $\kappa = 0$, the lattice undergoes a first-order rigidity percolation transition at $p=1$. When $\kappa > 0$, $G$ decreases continuously as $p$ decreases below one, reaching zero at a continuous rigidity percolation transition at $p=p_b \approx 0.605$ that is the same for all non-zero values of $\kappa$. The effective medium theories predict scaling forms for $G$, which exhibit crossover from bending dominated response at small $\kappa/\mu$ to stretching-dominated response at large $\kappa/\mu$ near both $p=1$ and $p=p_b$, that match simulations with no adjustable parameters near $p=1$. The affine response as $p\rightarrow 1$ is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.