## 2 Teachings of Hooke's Law and the Kirchhoff-Love Theory
These expressions are valid in any orthogonal basis set tangent to the plate. For some applications, it will be useful to cast the third relation of Equation (2) in the form  
\[\gamma_{12}=2g\epsilon_{12}\text{ with }g=A(1-\
u)/2 \tag{3}\]  
where \(g\) is the in-plane shear stiffness. Equation (2) is the basis of Kirchhoff-Love linear theory of elastic isotropic plates (see e.g., Reference [39]). When the plate experiences a uniaxial tension \(\gamma\) (say \(\gamma_{11}=\gamma\), \(\gamma_{22}=0\)), the component \(\epsilon\) of the strain along the stress direction fulfills the elementary [PERSON]'s law  
\[\gamma=C\epsilon\text{ with }C=A(1-\
u^{2}) \tag{4}\]  
where \(C\) (N/m), which will be called in-plane stiffness constant, plays the same role as the Young modulus does for a macroscopic 3D object. In some cases, it is more easy to measure or to calculate \(C\) than \(A\). In Kirchhoff-Love theory, the elastic strain energy accumulated in the plate subject to in-plane deformation is [46]  
\[E_{s}=\frac{A}{2}\int\int[(\epsilon_{11}+\epsilon_{22})^{2}+2(1-\
u)(\epsilon _{12}^{2}-\epsilon_{11}\epsilon_{22})]\,dS \tag{5}\]  
knowing that the strain tensor is related to the in-plane displacement field \(\vec{u}\) by the usual definition that, in Cartesian coordinates, writes [16]  
\[\epsilon_{ij}=\frac{1}{2}(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}+\sum_{k=1}^{2}\frac{\partial u_{k}}{\partial x _{i}}\frac{\partial u_{k}}{\partial x_{j}}) \tag{6}\] \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**References** & \(C_{11}\) & \(C_{12}\) & \(C_{13}\) & \(C_{33}\) & \(C_{44}\) & \(C_{66}\) \\ \hline Reference [45] & 1060(20) & 180(20) & 15(5) & 36.5(10) & 0.18–0.35 & 440(40) \\ Reference [20] & 1109(16) & 139(36) & 0(3) & 38.7(7) & 5.0(3) & 485(10) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental values of the elastic constants of graphite (all data in GPa). The last term may be neglected when \(\vec{u}\) varies slowly with coordinates. The bending energy takes a similar expression as the strain energy [47]  
\[E_{b}=\frac{D}{2}\int\int[(\kappa_{11}+\kappa_{22})^{2}+2(1-\
u^{\prime})(\kappa_ {12}^{2}-\kappa_{11}\kappa_{22})]\,dS \tag{7}\]  
where \(D\) is the bending stiffness or flexural rigidity (J) of the plate, \(\kappa\) is the two-dimensional curvature tensor, and \(\
u^{\prime}\) is the same Poisson coefficient \(\
u\) as in Equation (5) (see below). For small deformations,  
\[\kappa_{ij}=\frac{\partial^{2}\zeta}{\partial x_{i}\partial x_{j}} \tag{8}\]  
where \(\zeta\) is the mid-surface displacement of the plate with respect to its planar equilibrium shape. Equations (5) and (7) could simply follow by assuming that the energy is a quadratic form of the deformation (linear-response theory) that must involve invariant scalar expressions of the rank-2 tensors \(\gamma\) and \(\kappa\) due to the assumed in-plane isotropy. The hypothesis will now be made that Equations (5)-(7) remain valid for one-atom thick graphene, provided suitable elastic constants are used. From now on, \(A\), \(D\), \(\
u\), and \(\
u^{\prime}\) will be assumed to be independent elastic constants, which they are not in the traditional plate theory. In the way this theory was derived, \(A=hC_{11}\), \(\
u=C_{12}/C_{11}=\
u^{\prime}\), and \(D=Ah^{2}/12\). For graphene, one cannot rely on expressions that involve the thickness \(h\), which is ill defined (see Table 2 in Reference [48]). More precisely stated [47], the thickness for extensional deformation of graphene (mostly controlled by the response of the \(\sigma\) bonds) cannot be the same as the one for bending (the flexural rigidity comes chiefly from the \(\pi\) orbitals). Extending the reasoning a step further, there is no reason on why the same Poisson coefficient should take part in both Equations (5) and (7) for graphene. This is the reason why Equation (7) was written with \(\
u^{\prime}\) instead of \(\
u\).