\sqrt{G}$ in the variation of $\varepsilon_\mathrm{bend}$. The $\varepsilon_\mathrm{bend}$ dependence of the BP frequency is determined by the shear modulus, which is a macroscopic quantity describing the entire system in an averaged manner. It is noted that the recent study [@Baggioli_2019] predicts $\omega_\mathrm{BP} \propto \sqrt{G}$ from the phonon Green’s function with diffusive damping. It might be interesting to study effects of $\varepsilon_\mathrm{bend}$ on phonon transport and the phonon’s Green function.

According to the heterogeneous elasticity theory [@Schirmacher_2006; @Schirmacher_2007; @Schirmacher_2015], the spatial fluctuations of the local shear modulus $\delta G$ control nature of the BP [^2]. The collapse of $g(\omega)/( \omega^2 A_\mathrm{D})$ as a function of $\omega/\omega_\mathrm{D}$ indicates that the shear modulus fluctuations relative to the macroscopic value, $\delta G/G$, are constant for all the cases of different bending rigidities. Therefore, the results of this study can be explained as follows. The increase in bending rigidity does not affect the shear modulus fluctuations (relative to the macroscopic moduli) but only affects the macroscopic shear modulus, which leads to the collapse of the scaled vDOS.

### Participation ratio

To further study the vibrational eigenstates, the participation ratio $P^k$ that measures the extent of localization of the eigenmodes $k$ is calculated as follows [@Schober_1996; 