Elasticity of randomly diluted honeycomb and diamond lattices with bending forces

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.