## 8 Nonlocal, Nonlinear and Temperature Effects  
When the deformations vary on a length scale that is not much larger than atomic distances, it is possible to extend the domain of validity of elasticity by turning to a nonlocal response of the material [82]. In the particular case of graphene, nonlocal elasticity consists chiefly in replacing each component of the algebraic [PERSON]'s law (Equation (2)) by a differential equation of the kind  
\[(1-\xi^{2}\Delta)\gamma_{11}=A\left(\epsilon_{11}+\
u\epsilon_{22}\right) \tag{20}\]  
(\(\Delta\) is the Laplacian) and similar equations for \(\gamma_{12}\) and \(\gamma_{22}\). In that relation, \(\xi\) is a non-locality parameter, which is of the order of the interatomic distances. For an unbounded graphene plane, a formal solution of Equation (20) writes  
\[\gamma_{11}(x_{1},x_{2})=\frac{1}{2\pi\xi^{2}}\,\int\int A\left[\epsilon_{11}( x_{1}^{\prime},x_{2}^{\prime})+\
u\epsilon_{22}(x_{1}^{\prime},x_{2}^{\prime}) \right]\,\mbox{K}_{0}(\sqrt{(x_{1}^{\prime}-x_{1})^{2}+(x_{2}^{\prime}-x_{2}) ^{2}}/\xi)\,dx_{1}^{\prime}dx_{2}^{\prime} \tag{21}\]  
Figure 4: Total energy per depth unit above that of flat graphene of a scrolled sheet such as displayed in Figure 3 as a function of the length of the overlapping part. The two curves correspond to a sheet of length \(l\) of 10 and 17 nm, respectively. where K\({}_{0}\) is the modified Bessel function of the second kind and order 0. In nonlocal elasticity, the tension at the reference point \((x_{1},x_{2})\) depends on the strain field everywhere, the dependency decreasing exponentially with the distance to the reference point measured in units of \(\xi\). When this theory is used to study in-plane elastic waves in graphene, the effect of non locality on [PERSON]'s law consists in the substitution of \((1-\xi^{2}\Delta)\ddot{\vec{u}}\) for the acceleration \(\ddot{\vec{u}}\), without modification of the expression of the restoring elastic force (Equation (9)) [83]. As the result, the dispersion relation for plane waves now takes the form  
\[\omega=\frac{ck}{\sqrt{1+\xi^{2}k^{2}}} \tag{22}\]  
with \(k\) the modulus of the two dimensional wave-vector, where the velocity \(c\) denotes \(c_{l}\) or \(c_{t}\) for the longitudinal or transverse modes, respectively, with expressions given by Equation (10). The dispersion relation deviates from the strict linear form. When \(\xi k\) is small, \(\omega\approx ck(1-\xi^{2}k^{2}/2)\). This behavior is qualitatively correct in the sense that the LA and TA modes of graphene now have a negative dispersion (the phase velocity decreases with increasing \(k\)), as indeed observed experimentally [21] and by calculations [23]. If Equation (22) had to reproduce the experimentally-measured frequencies of the TA or LA phonons at the \(K\) point of the first Brillouin zone, \(\xi\) would take the values 0.044 or 0.077 nm, respectively. Should one want to fit the TA and LA frequencies at the \(M\) point, other values of \(\xi\) would be obtained. Not only the non-locality parameter depends on the wave polarization, also does it depend on the direction of the wave vector. In nonlocal elasticity, the equation for flexural waves is transformed along the same way as the one governing in-plane vibrations: the acceleration term in Equation (11) is replaced by \(\mu(1-\xi^{2}\
abla^{2})\ddot{\zeta}\)[84]. Accordingly, the dispersion relation of the out-of-plane wave now becomes \(\omega_{ZA}=\alpha k^{2}/\sqrt{1+\xi^{2}k^{2}}\) with \(\alpha=\sqrt{D/\mu}\) as before. A best fit over the first half of the \(\Gamma K\) line of the ZA phonon branch obtained by GW calculations [23] gives \(\xi\) = 0.052 nm. This value is intermediate between the values of \(\xi\) deduced from the TA and LA phonon dispersion. In any case, \(\xi\) is a small fraction of the CC distance, which raises question on its exact physical meaning.