## 8 Nonlocal, Nonlinear and Temperature Effects
Within nonlocal elasticity, one-dimensional out-of-plane deformations of graphene bound to a substrate (Equation (16)) are governed by the following equation [85, 86]  
\[D\frac{d^{4}\zeta}{dx^{4}}+\gamma\,(\frac{d^{2}\zeta}{dx^{2}}-\xi^{2}\frac{d^ {4}\zeta}{dx^{4}})+K\,(\zeta-\xi^{2}\frac{d^{2}\zeta}{dx^{2}})=0 \tag{23}\]  
Numerical calculations show that the buckling load is reduced compared to the predictions of local elasticity [86, 87]. However, this reduction is significant only for graphene ribbons with small width, of the order of \(\xi\). This conclusion can readily be substantiated for the case of a self-supported graphene strip, by setting \(K\)=0 in the above equation. For a given applied compressive tension \(\gamma\), the role of the non-locality parameter is to weaken the bending stiffness to \(D-\xi^{2}\gamma\). Accordingly, the critical load triggering the buckling mode illustrated in Figure 1 is \(\gamma_{cr}=4\pi^{2}D/(l^{2}+4\pi^{2}\xi^{2})\). Non locality has little effects as soon as \(l\gg 2\pi\xi\approx 0.3\) nm. The stability of physisorbed graphene is now briefly revisited in the light of nonlocal elasticity. A rippled solution of Equation (23) with wave vector \(k\) requires \(\gamma=Dk^{2}/(1+\xi^{2}k^{2})+K/k^{2}\). The minimum of that expression is \(\gamma_{cr}=2\sqrt{KD}-K\xi^{2}\) if \(\xi^{2}<\sqrt{D/K}\) or \(\gamma_{cr}=D/\xi^{2}\) if \(\xi^{2}>\sqrt{D/K}\). The latter condition could possibly be fulfilled for the case of Co and Ni substrates, not for the other cases listed in Table 4. For Cu and h-BN, a non-locality parameter \(\xi\) = 0.05 nm reduces the critical strain given in the last column of Table 4 by \(\sim 5\times 10^{-4}\), which is negligible. Besides non-locality, nonlinear contributions may be worth accounting for when the strain exceeds a few percents, which is not unrealistic in the case of graphene [62]. First of all, the last term in Equation (6) must be kept when dealing with a rapidly varying in-plane displacement field \(vectu\). For strong bending, the simplified expression \(\partial^{2}\zeta/\partial x_{i}\partial x_{j}\) of the curvature tensor \(\kappa_{ij}\) is no longer valid [16]. Second, nonlinear [PERSON]'s law should be used. For uniaxial tension, in particular, the elementary Hooke's law (Equation (4)) transforms into  
\[\gamma=C\epsilon+C_{2}\epsilon^{2}\text{ (plus higher order terms)} \tag{24}\]  
The nonlinear response coefficient \(C_{2}\) depend on elements of the second- and third-order elastic-constant tensors generated by Taylor expansion of the elastic energy _versus_ strain [39]. Equation (24) is slightly anisotropic in the sense that the value of \(C_{2}\) depends on the direction of the applied tension \(\gamma\) (\(C_{2}\) = \(-\)670 N/m and \(-\)700 N/m in the zigzag and armchair directions, respectively, according to _ab-initio_ calculations of Reference [28]). It is worth to mention, however, that in nanoindentation experiments performed on graphene, the nonlinear coefficient \(C_{2}\) plays a role in such a small fraction of the deformed graphene sheet that its overall effect can be ignored [62]. Temperature affects the elastic properties of any material. In graphene, thermal fluctuations of the out-of-plane displacements are important, at least in self-supported geometry [4]. In the harmonic approximation, the mean square displacement normal to the plane is [3,55]  
\[\langle\zeta^{2}\rangle\approx\frac{k_{B}T}{D}\frac{S}{30\pi^{2}} \tag{25}\]  
where \(k_{B}\) is the Boltzmann constant and \(S\) is the area of the sample. The root mean square displacement increases therefore like \(\sqrt{S}\) and may become important in a large free-standing sample. In direct connection with the previous Section, atomistic simulations show that the transformation of a graphene nanoribbon into a nanoscroll may happen spontaneously at room temperature when the aspect ratio \(l/W\) is large, thanks to large-amplitude thermal fluctuations [4].