Elasticity of randomly diluted honeycomb and diamond lattices with bending forces

TL;DR

This study investigates how weak bond-bending forces affect the rigidity transition in honeycomb and diamond lattices, using simulations and theory to understand elastic properties near the percolation threshold.

Contribution

It introduces a rotationally invariant bending potential that isolates bending effects, providing new scaling functions and demonstrating agreement between theory and simulations.

Findings

Bulk modulus is independent of bending stiffness $$.

Scaling functions describe elastic moduli near the percolation threshold.

Good quantitative agreement between theory and simulations for high bond occupation probabilities.

Abstract

We use numerical simulations and an effective-medium theory to study the rigidity percolation transition of the honeycomb and diamond lattices when weak bond-bending forces are included. We use a rotationally invariant bond-bending potential, which, in contrast to the Keating potential, does not involve any stretching. As a result, the bulk modulus does not depend on the bending stiffness $\kappa$. We obtain scaling functions for the behavior of some elastic moduli in the limits of small $\Delta \mathcal{P} = 1 - \mathcal{P}$, and small $\delta \mathcal{P} = \mathcal{P} - \mathcal{P}_c$, where $\mathcal{P}$ is an occupation probability of each bond, and $\mathcal{P}_c$ is the critical probability at which rigidity percolation occurs. We find good quantitative agreement between effective-medium theory and simulations for both lattices for $\mathcal{P}$ close to one.