## 6 Buckling Instability
In other words the critical bucking tension becomes almost independent of the ribbon width \(l\), at least in this range of parameter. As shown in Figure 2, the curve \(\gamma=2\sqrt{DK}\) underestimates by \(\sim\)10% the critical load and is indeed independent on the ribbon width. To estimate the force constant \(K\) of the interaction potential, one may again have recourse to phonon dispersion curves. When the graphene plane is bound to a medium by an assumed harmonic potential, the flexural wave equation introduced here above for a free-standing plate (Equation (11)) is modified in  
\[\mu\frac{\partial^{2}\zeta}{\partial t^{2}}+D\frac{\partial^{4}\zeta}{ \partial x^{4}}+K\zeta=0 \tag{17}\]  
The effects of the restoring force \(-K\zeta\) imposed by the substrate is to shift the square of the flexural wave frequency \(\omega^{2}\) by the quantity \(K/\mu\). In other words, the frequency of the ZA phonon of graphene at the \(\Gamma\)point is raised from 0 to \(\sqrt{K/\mu}\) when it is adsorbed on a substrate. _Ab-initio_ calculations of the phonon band structure of graphene bound to several metallic surfaces are available [71,72], making it possible to evaluate the force constant \(K\). Results are given in Table 4. The specificity of Cu as substrate for CVD growth of graphene is immediately perceived from the low value of \(K\) it leads to. Not only is the force constant weak for graphene on Cu(111), but also the adhesion energy is small: 68 meV/atom according to recent RPA calculations [73] or 120 meV/atom according to nano-mechanical measurement [74]. The force constant for the interaction of graphene on SiO\({}_{2}\) has been deduced from semi-empirical energy calculations of the interface adhesion energy of graphene and few-layer graphite [75]. For graphene on graphite and for graphene on h-BN, the force constant \(K\) listed in Table 4 has been obtained by the method described in the Appendix. **Figure 2.** The homogeneous differential equation (Equation (16)) has non trivial solutions for values of the parameters outside the convex side of the curve \(\gamma=2\sqrt{DK}\) (dotted line). The figure illustrates the relationship between the equation parameters for the fundamental bucking modes of even (solid-line curve) and odd (dashed-line curve) symmetries when the graphene ribbon is clamped along its two edges distant of \(l\). The value of the critical tension at \(K\) = 0 are \(39.48\,D/l^{2}\) and \(80.76\,D/l^{2}\) for the symmetric and antisymmetric modes, respectively. Under special preparation techniques, such as liquid exfoliation of graphite, graphene can be strained. A compressive strain may lead to a periodic buckling of the sheet, which will therefore present a regular pattern of ripples. Equation (16) possesses periodic solutions of the type \(\zeta=\alpha\cos(kx)+\beta\sin(kx)\) provided \(\gamma=Dk^{2}+K/k^{2}\). \(\gamma\) is the uniaxial tension in the sheet just before it buckles. The minimum of the last expression, realized for \(k=\sqrt[4]{K/D}\), is the critical load for rippling, \(\gamma_{cr}=2\sqrt{DK}\). This parameter is listed in the fourth column of Table 4. The wavelength of the rippled solution is given in the fifth column of the same table. Just before buckling, the strain of the graphene sheet can be estimated by [PERSON]'s law (Equation (4)). The critical strain is therefore \(\epsilon_{cr}=2\sqrt{DK}/C\). This parameter is listed in the last column of Table 4. Remarks that \(\epsilon_{cr}\) vanishes at the limit \(K\to 0\), which agrees with the instability of a self-supported 2D crystal [1]. It is interesting to point out that periodic, short-wavelength buckling have been observed by high-resolution transmission electron microscopy (TEM) in the top layer of a bi-layer graphite (\(\lambda\) = 0.64 nm) and in a folded graphene sheet near the folding edge (\(\lambda\) = 0.36 nm) [76]. The corresponding observed strains were 0.047 and 0.14, respectively. These values lie above the critical strain predicted for rippling graphene on graphite, 0.03 (see Table 4).