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 $\mu$ 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 $p\to 1$ is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.

arXiv comment15 pages, 10 figures

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