## 4 Results of Calculations  
The calculation of elastic constants of graphene by empirical force-field method and _ab-initio_ technique often proceeds via total-energy calculation of deformed graphene and fitting of the results with Equations (5) or (7). The first calculations date back to the period that followed the discovery of carbon nanotubes [24,33]. According to Equation (5) a longitudinal tensile or compressive axial strain \(\epsilon\) increases the energy of a single-walled nanotube by \(\Delta E/S=(C/2)\epsilon^{2}\) where \(C\) is the stiffness constant (Equation (4)) and \(S\) is the lateral surface of the rolled up graphene sheet [30]. At that time, graphene was considered by reference with the limiting case of infinite tube radius. Non-orthogonal tight-binding calculations yielded \(C\) = 410 N/m [30], slightly more than the results of pseudopotential LDA calculations, \(C\) = 370 N/m (60 eV/atom) [36]. Since then, there have been many atomistic calculations specifically devoted to the elastic properties of graphene [28]. The results depend on the specific microscopic model used (see Table 1) and on the type of boundary conditions applied [17]. When calculations are performed on a finite-size graphene sample, the Young modulus is found to be size dependent: below 3 nm, the Young modulus decreases by \(\sim\)10% per each nm reduction [55].  
Total-energy calculations of graphene also gives access to the two-dimensional bulk modulus \(B\) through the relation \(\Delta E=\left(B/2\right)\left(\Delta S/S\right)^{2}\), with \(\Delta S\) the change of the unit-cell area \(S\) obtained by an assumed hydrostatic tension. According to Equation (5), \(B=A(1+\
u)/2\). Published values of \(B\) computed by empirical potentials are 200 N/m, typically [27,28], whereas _ab-initio_ VASP calculations yield \(B\) = 180 N/m [19].  
The in-plane shear stiffness \(g\) can be calculated directly from its definition (Equation (3)). To obtain this parameter, a rectangular unit cell of graphene with zigzag and armchair edges is deformed by in-plane shear tensions applied along the four edges. (Note: The honeycomb structure of graphene hastwo special directions, orthogonal to each other, qualified as zigzag and armchair. These names come from the shape of the chains of first-neighbor C atoms aligned along these directions. In Figure 1, the tension \(\gamma\) is applied on zigzag edges and is parallel to the armchair direction.) The AMBER molecular mechanics model yields \(g\) = 122 N/m [29]. Alternatively, a parabolic fit of the strain energy \(\Delta E/S=(g/2)\,\beta^{2}\)_versus_ the shear angle \(\beta\) can be used to obtain \(g\). First-generation Brenner potential gives \(g\) = 83 N/m [15,25]. Some authors apply a shear along two opposite edges, letting the other two edges free while preventing the free rotation of the graphene sheet [56]. The shear stiffness calculated in this way depends strongly on whether the shear is applied along the zigzag or the armchair edges [28,56].  
The nanotubes are ideal objects to deal with for calculating the bending stiffness \(D\) of graphene. According to Equation (7), rolling up a sheet of graphene for making a nanotube of radius \(R\) costs an elastic energy \(\Delta E/S=(D/2)\,R^{-2}\)[24,33]. This penalty is offset by the covalent bonds closing the lateral surface of the tube [57]. Typically, the value of \(D\) obtained is 1.6 eV (see Table 1).  
It is interesting to observe in Table 1 that the elastic constants of graphene calculated with the widely-used Brenner bond-order potential [25] deviate significantly from tight-binding and DFT data. The contribution of dihedral angles in the bond-order coefficient of the interatomic potential of the second generation [58], not present in the original formulation [59], raises the bending stiffness \(D\) from 0.8 to 1.4 eV [26].