Elasticity of Filamentous Kagome Lattice

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 $μ$ and bending modulus $κ$, are used to study the elasticity of this lattice as functions of $p$ and $κ$. At $p=1$, elastic response is purely affine, and the macroscopic elastic modulus $G$ is independent of $κ$. When $κ= 0$, the lattice undergoes a first-order rigidity percolation transition at $p=1$. When $κ> 0$, $G$ 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 $κ$. The effective medium theories predict scaling forms for $G$, which exhibit crossover from bending dominated response at small $κ/μ$ to stretching-dominated response at large $κ/μ$ 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.