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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:

Rectangular supercell

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

Energy vs strain

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: Strain energy vs x/y uniaxial strain

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
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  • $\begingroup$ What does this symbol : Ꜫ mean? $\endgroup$ Aug 19, 2023 at 15:25

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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.

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