## 2 Teachings of Hooke's Law and the Kirchhoff-Love Theory  
[PERSON]'s law for crystalline graphite, with its hexagonal symmetry, writes [44]  
\[ \begin{pmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{32}\\ \sigma_{13}\\ \sigma_{21}\end{pmatrix} = \begin{pmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\ C_{12}&C_{11}&C_{13}&0&0&0\\ C_{13}&C_{13}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&(C_{11}-C_{22})/2\end{pmatrix}\begin{pmatrix}\epsilon_{11}\\ \epsilon_{22}\\ 2\epsilon_{32}\\ 2\epsilon_{13}\\ 2\epsilon_{21}\end{pmatrix}  \tag{1}\]  
Equation (1) relies on five independent elastic constants. Two sets of experimental values of the elastic constants of graphite are compared in Table 2. The older data set [45] was obtained by ultrasonic, sonic resonance, and static test methods applied to pyrolytic graphite. The more recent data set [20] was deduced from the dispersion curves of acoustic phonons of single-crystalline graphite measured by inelastic X-ray scattering experiments. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \(A\) (N/m) & \(\
u\) & \(C\) (N/m) & \(g\) (N/m) & \(D\) (eV) & \(\
u^{\prime}\) & **Method** \\ Equation (2) & Equation (2) & Equation (4) & Equation (3) & Equation (7) & Equation (7) & \\ \hline
372 & 0.125 & 366 & 163 & & & Elastic constants of graphite [20]: \(A=hC_{11},\
u=C_{12}/C_{11},\) \(C=h(C_{11}-C_{12}/C_{11}),\)\(g=h(C_{11}-C_{22})/2\) with \(h\) the graphite interlayer distance (0.335 nm) \\
368 & & & 149 & 1.75 & HREELS phonon dispersion measurements [21] \\
405 & & & 156 & 1.85 & Phonons dispersion (derived from experimental data) [22] \\
363 & & & 146 & 1.33 & Phonons dispersion (GW-DFT calculations) [23] \\
350 \(\pm\) 65 & & & & radial breathing mode of carbon SWNTs (see text) \\  & 0.412 & 236 & 83 & 0.85 & 1 st generation Brenner potential [24, 25] \\  & & & 0.91 & 2 nd generation Brenner potential [4] \\
289 & 0.397 & 243 & & 1.39 & 2 nd generation Brenner potential [16, 25, 26] \\
355 & 0.15 & & & 1.1 & Monte-Carlo, empirical bond-order potential [3, 27] \\  & 0.22 & 320 & & DFT-derived in-plane force field [28] \\  & 0.26 & 312 & 122 & & AMBER 3D force field [29] \\  & & 410 & & 1.88 & Non-orthogonal tight-binding [30] \\  & 0.24 & 358 & & 1.10 & Non-orthogonal tight-binding [31] \\
407 & 0.281 & & & & Non self-consistent \(ab\)-\(initio\)[32] \\  & & 377 & & & Hartree Fock 6-31G\({}^{\circ}\)[34] \\  & 0.149 & 345 & & DFT Gaussian orbitals [35] \\  & & 370 & & 1.53–1.65 & LDA SIESTA [36] \\  & & & 1.61 & 0.56 & DFT tight-binding [37] \\  & 0.17 & 336 & & & DFT ABINIT code[38] \\
358 & 0.169 & & & & DFT VASP program [39] \\
361 & 0.184 & 350 & & & LDA Vienna VASP [40] \\  & 0.20 & 350 & & & DFT AIMPRO code [41] \\  & & & 95 & & Measurement of torsional modulus [42] \\  & & & 330 \(\pm\) 15 & & & Experimental AFM nano-indentation [43] \\
365 & 0.205 & 350 & 145 & 1.6 & Best fit (see Section 9) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Elastic constants of graphene obtained by various methods. Each line of the table lists parameters obtained by independent measurements or calculations. The authors of Reference [20] conclude that, although poorly defined, the \(C_{13}\) constant is quite close to zero. Adopting this conclusion as a simplifying assumption allows one to separate Equation (1) in two independent blocks governing the in-plane and out-of-plane components, respectively, of the stress and strain tensors. Let us consider a plate of graphite with (0001) free surfaces and arbitrary thickness \(h\) and integrate the in-plane block of Hooke's law over the thickness of the plate. One then obtains constitutive relations between tension components acting on the sides of the plate (\(\gamma=h\sigma\), N/m) and strain components (\(\epsilon\))  
\[ \begin{array}{rcl}\gamma_{11}&=&A(\epsilon_{11}+\
u\epsilon_{22})\\ \gamma_{22}&=&A(\epsilon_{22}+\
u\epsilon_{11})\\ \gamma_{12}&=&A(1-\
u)\epsilon_{12}\end{array}  \tag{2}\]  
where \(A\) is the extensional stiffness (N/m) and \(\
u\) is the in-plane Poisson coefficient.