  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.
