How can I calculate the in-plane stiffness and Poisson’s ratio of a graphene monolayer?

Question

I used a 2x2x1 rectangular supercell of graphene to calculate the elastic properties (in-plane stiffness and Poisson’s ratio). The in-plane stiffness can be given by the following equation, $$C^2D= \frac{1}{A_0} \frac{\partial^2 E_S}{\partial \epsilon ^2}$$ where $A_0$ is the equilibrium area of the system, $E_S$ corresponds to the energy difference between the total energy of the strained system from the unstrained system (i.e., at zero strain applied), and $\epsilon$ is the uniaxial strain ($\epsilon = \frac{a − a_0}{a_0}$, $a$ and $a_0$ being the lattice constant before and after deformation, respectively).

The Poisson ratio, which is the ratio of the transverse strain to the axial strain can be defined straightforwardly as $$\upsilon = \frac{-\epsilon_\text{transverse}}{\epsilon_\text{axial}} = \frac{-\epsilon_{yx}}{\epsilon_x}$$
The model developed by Topsakal et al. (Ref. 1) was applied to calculate the elastic stiffness. A rectangular supercell with x and y axes along the armchair and zigzag directions, respectively, was used (as shown in Fig. below:

to calculate the elastic constant of the monolayer. The strains in two directions perpendicular to each other, Ꜫx (along the armchair direction) and Ꜫy (along the zigzag direction) were varied from -2.0% to 2.0% in an increment of 0.5%. so I plotted the energy vs. strain as shown below

I used this formula to calculate the elastic constant C11= (2/A0) (Es/Ꜫ2) C11= 2(231.46)/A0 C11= 350 N/m Now the question is how can calculate the in-plane stiffness and Poison’s ratio? For more information, most researchers use this formula to calculate in-plane stiffness E = b1Ꜫx2 + b2Ꜫy2 + b3ꜪxꜪy and they have this diagram as shown below:

So how can I get this formula and the diagram?

References

1. Topsakal, M.; Cahangirov, S.; Ciraci, S. The Response of Mechanical and Electronic Properties of Graphane to the Elastic Strain. Applied Physics Letters, 2010, 96. DOI

Answer 1

If you already have the elastic constants $C_{ij}$ (in Voigt notation), you should be able to use the following equations for the general angle-dependent in-plane Young's modulus $E(\theta)$ and Poisson's ratio $\nu(\theta)$. They were obtained from J. Phys. Chem. C 2021, 125, 36, 19666–19672.