## 6 Buckling Instability  
Buckling will be illustrated here for the case of a graphene ribbon physisorbed on a flat substrate, see Figure 1. The equations will be derived by assuming a double-clamp geometry and uniform loads applied along the width \(l\) of the ribbon, chosen as \(x\) direction. The graphene-substrate interaction potential per unit area is denoted by \(V\). In the continuous medium approximation used throughout this paper, there is no atomic corrugation, with the result that \(V\) is a function of the height \(z\) above the substrate only. Polymers are well beyond the framework of this approach. They present complex, possibly wavy, surface structures that may influence the geometrical shape of the of the adsorbed graphene depending on the interfacial energy [46]. Investigating the mechanical property of a polymer/graphene interface, where the polymer chains can change their geometry to accommodate the deposited graphene layer, requires sophisticated atomistic calculations [66,67]. For small load, the ribbon accumulates strain energy. Once buckling has taken place, strain becomes negligible compared to bending. The total energy per unit length, including the work done by the external tension \(\gamma\)[68, 69], is  
\[W=\int_{0}^{l}[\frac{D}{2}\kappa^{2}+V(z)]\,ds-\gamma\delta l=V(z_{0})l+\frac{1}{ 2}\int_{0}^{l}[D(\frac{d^{2}\zeta}{dx^{2}})^{2}+K\zeta^{2}-\gamma(\frac{d\zeta }{dx})^{2}]\,dx \tag{15}\]  
with \(ds\) a length element along the width of the deformed ribbon, \(\kappa\) is the local curvature (see Equation (7)), and \(\delta l\) is the shortening of the distance between the anchor points of the forces (see Figure 1). The expression in the most right-hand side of Equation (15) is an approximation valid for small values of the deformation \(\zeta=z-z_{0}\) (see Equation (8)), \(z_{0}\) being the equilibrium distance of the graphene plane above the substrate surface, and \(K\) is the second derivative of \(V(z)\) at \(z_{0}\). The deformation \(\zeta\) is the one that minimizes the energy \(W\). The Euler-Lagrange equation derived from the right-hand side of Equation (15) is the fourth-order differential equation  
\[D\frac{d^{4}\zeta}{dx^{4}}+\gamma\frac{d^{2}\zeta}{dx^{2}}+K\zeta=0 \tag{16}\]  
whose boundary conditions are either imposed values of \(\zeta\) and \(d\zeta/dx\) at \(x=0\) and \(x=l\) or periodic boundary conditions for \(\zeta\) and its first three derivatives over the interval \(l\). The critical load is the smallest value of the tension \(\gamma\) for which Equation (16) admits a non trivial solution. In free space (\(K\) = 0), a non-zero solution for the double-clamp geometry exists when \(\gamma\) reaches the critical value \(4\pi^{2}D/l^{2}\)[16, 70]. This solution, \(\zeta(x)=A\,\sin^{2}(\pi x/l)\), has even symmetry with respect to mid width. The shape of the buckled sheet of graphene is illustrated in Figure 1.  
When \(K\
eq 0\), still for the double-clamp geometry of a graphene ribbon of width \(l\) (Figure 1), non-trivial solutions of Equation (16) can only exist for \(\gamma>2\sqrt{DK}\). The differential equation depends on two dimensionless parameters, \(\gamma l^{2}/D\) and \(Kl^{4}/D\). The relationship between these two parameters is displayed in Figure 2, where the two curves correspond to the \"ground state\" solutions of Equation (16) that are symmetric and antisymmetric, respectively, with respect to the mid point between the two edges. In both cases, the substrate interaction stabilizes the ribbon against buckling. Interestingly, the difference between the critical tensions for the even and odd symmetries, which is important in the absence of substrate (\(K\) = 0), rapidly dims as the force constant of the substrate interaction increases. In some ranges of the parameters, the critical load for the antisymmetric solution comes below the one of the symmetric mode. In the abscissa interval \([0.7\times 10^{4}\), \(5\times 10^{4}]\) the average of the two curves is well fitted by the expression \(\gamma l^{2}/D=4.2154\,(Kl^{4}/D)^{0.4384}\).