## 3 Teachings of Phonon Dispersion Laws  
The first measurements, by high-resolution electron-energy-loss spectroscopy (HREELS), of phonon dispersion curves of graphene on various substrates date back to the years 1980's (see Reference [49] for a review). The elastic properties of graphene can be obtained from phonon dispersion measurements [21]. The dispersion curves of acoustic phonons in graphene are indeed directly related to its elastic constants and to its mass per unit area \(\mu\). The later parameter is a well-defined quantity, which we take as 0.761 mg/m\({}^{2}\) (lattice parameter \(a\) = 0.246 nm). The long-wavelength velocities of the longitudinal acoustic (LA) and transverse acoustic (TA) phonons, whose polarization lies inside the graphene plane, can be derived from Equation (5) as follows. The flat plate is supposed to be deformed by a two-dimensional wave \(\vec{u}\) whose polarization is parallel to the plate. A small perturbation \(\delta\vec{u}\) of the wave changes the strain energy by a small quantity that, to first order in the perturbation, can be identified with a virtual work \(\delta E_{s}=-\int\int\vec{F}\cdot\delta\vec{u}\:dS\). Here, \(\vec{F}\) is the restoring elastic force per area unit. Equation (5) together with Equation (6) cleaned from its nonlinear term yield  
\[\vec{F}=A\vec{\
abla}\vec{\
abla}\cdot\vec{u}-g\vec{\
abla}\times\vec{\
abla} \times\vec{u} \tag{9}\]  
where \(A\) and \(g\) are the extensional stiffness and the in-plane shear stiffness, respectively, introduced above. With that expression, [PERSON]'s law \(\mu dS\:\vec{\vec{u}}=\vec{F}dS\) takes the form of a wave equation for \(\vec{u}\). That equation can further be separated in longitudinal (\(\vec{\
abla}\times\vec{u}=0\)) and transverse (\(\vec{\
abla}\cdot\vec{u}=0\)) components, with respective phase velocities  
\[c_{l}=\sqrt{A/\mu}\:\:\:\text{and}\:\:\:c_{t}=\sqrt{g/\mu} \tag{10}\]  
These in-plane elastic wave velocities can be obtained by measuring the slope of the LA and TA phonons at the \(\Gamma\) point. In complement to the work of Reference [21], based on HREELS measurements of the phonon dispersion in CVD (chemical vapor deposition) graphene on Pt(111), two approaches have been followed hereafter. First, a semi-empirical phonon band structure of graphene [22] based on force constants fitted to experimental inelastic X-ray scattering data of graphite [50]. Second, a theoretical _ab-initio_ phonon band structure of graphene obtained by GW-DFT [23]. The elastic wave velocities obtained by these two approaches are compared in Table 3.  
Out-of plane oscillations of an elastic plate lead to flexural waves. In a similar way as for Equation (9), virtual-work theory can be used to deduce the expression of the elastic forces from the bending energy (Equation (7)) of the plate. For the particular case of a plane wave with small amplitude \(\zeta(x,t)\) traveling along an arbitrary direction \(x\), the flexural-wave equation is  
\[\mu\frac{\partial^{2}\zeta}{\partial t^{2}}+D\frac{\partial^{4}\zeta}{ \partial x^{4}}=0 \tag{11}\]  
which imposes a quadratic dispersion relation \(\omega_{ZA}=\alpha k^{2}\)_versus_ wave vector \(k\), with \(\alpha=\sqrt{D/\mu}\). Parabolic fitting of the dispersion curve of the out-of-plane acoustical (ZA) phonon of graphene close to the \(\Gamma\) point gives the values listed in Table 3 for the band structures referred to above. From the data of Table 3, the elastic parameter \(A\), \(g\), and \(D\) can be obtained. Their values are listed in Table 1.  
The radial breathing mode (RBM) of single-walled carbon nanotubes (SWNT) provides us with important piece of information on the elastic properties of the rolled-up graphene sheet. Its real advantage is that it can be measured accurately in resonant Raman spectroscopy. In cylindrical geometry, the \(x_{1}\) and \(x_{2}\) directions referred to in Equation (5) can be taken along the local circumferential and the axial directions of the tube, respectively. The RBM modifies the radius by a periodic variation \(\delta R\) that