Elasticity of a filamentous kagome lattice

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 4. If lines are treated as semiflexible polymers and crossing points are treated as cross-links, 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) approximate to 0.605 that is the same for all nonzero 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(b), that match simulations with no adjustable parameters near p = 1. The affine response as p -> 1 is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.