Topological mechanics in quasicrystals

We study topological mechanics in two-dimensional quasicrystalline parallelogram tilings. Topological mechanics has been studied intensively in periodic lattices in the past a few years, leading to the discovery of topologically protected boundary floppy modes in Maxwell lattices. In this paper we extend this concept to quasicrystalline parallelogram tillings and we use the Penrose tiling as our example to demonstrate how these topological boundary floppy modes arise with a small geometric perturbation to the tiling. The same construction can also be applied to disordered parallelogram tilings to generate topological boundary floppy modes. We prove the existence of these topological boundary floppy modes using a duality theorem which relates floppy modes and states of self stress in parallelogram tilings and fiber networks, which are Maxwell reciprocal diagrams to one another. We find that, due to the unusual rotational symmetry of quasicrystals, the resulting topological polarization can exhibit orientations not allowed in periodic lattices. Our result reveals new physics about the interplay between topological states and quasicrystalline order, and leads to novel designs of quasicrystalline topological mechanical metamaterials.Comment: 16 pages, 8 figure

Elasticity of 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 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.Comment: 15 pages, 10 figure