## 4 Properties of GR
### Mechanical Properties  
The two main parameters for elastic properties in 2D-type materials are in-plane stiffness and Poisson's ratio. They are represented by C and v, respectively, and by utilizing the below-mentioned equation. The value of C can be calculated. \[C=(\frac{1}{S_{0}})\bigg{[}\frac{\partial^{2}E}{\partial e^{2}}\bigg{]} \tag{1}\]  
Figure 15: The structures of \(\alpha\)-, \(\beta\)-, and \(\gamma\)-graphynes include: (**a**) 6,6,12-graphyne, (**b**) 6,6,14-graphyne, (**c**) 14,14,14-graphyne, (**d**) 12,12,12-graphyne, (**e**) 14,14,18-graphyne (also referred to as \(\beta\)-graphyne), and (**f**) 18,18,18-graphyne (commonly known as \(\alpha\)-graphyne). The figure is adopted from Ref. [119] © 2015 ACS. \(\text{S}_{0}\), \(\varepsilon\), and E represent the equilibrium area, uniaxial strain, and total energy of the system [87]. This explains how much a 2D material can tend along an axis and the value of deformation in the direction of the axis. \(v\) shows the relationship between transverse strain and uniaxial strain and is represented by  
\[\
u=-\frac{\varepsilon_{trans}}{\varepsilon_{axial}} \tag{2}\]  
For the hexagonal structure, the isotropic parameters are used, but for the structures 6,6,12-GR, the anisotropic parameters have been adopted. GR has a smaller stiffness with respect to G, see Table 1.  
This can be explained with two terms: (1) In G, every C atom has a coordination number \(\text{N}_{\text{C}}\) of 3, and in GR, acetylenic linkage reduces the coordination number \(\text{N}_{\text{C}}\). Due to a small coordination number, a smaller number of atoms are present in the GR. (2) GR has a smaller atomic mass density in comparison with G. The calculation of stiffness of the  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Name**} & **Ultimate Strength** & **Ultimate Strength** & **Poisson’s** & **Ratio \(\text{v}_{\text{x}}\)** & **Poisson’s** & **In-Plane** & **In-Plane** & **Ultimate** & **Ultimate** \\  & **\(\sigma_{\text{x}}\) (N/m)** & **\(\sigma_{\text{y}}\) (N/m)** & **Ratio \(\text{v}_{\text{x}}\)** & **Ratio \(\text{v}_{\text{y}}\)** & **Stiffness** & **Stiffness** & **Stiffness** & **Stain \(\varepsilon_{\text{x}}\)** & **Stain \(\varepsilon_{\text{y}}\)** \\  & & & & & & & & & \\ \hline \multirow{2}{*}{Graphyne} & 17.84 [121] & 18.83 [121] & 0.417 [56], & \multirow{2}{*}{0.42 [123]} & 166 [56], & 224.0 [115], & \multirow{2}{*}{20 [121]} & \multirow{2}{*}{20 [121]} \\  & & & 0.416 [122] & & & & & & \\  & & & 0.429 [121], & & & & & & \\  & & & 0.38 [124] & 166.3 [122], & 162.5 [123], & \multirow{2}{*}{11.2 [117]} & 14.8 [117] \\  & & & 0.42 [123] & & & & & & \\  & & & 0.39 [124] & & & & & & \\  & & & 0.40 [124] & & & & & & \\  & & & 0.40 [125] & & & & & & \\ \hline \multirow{2}{*}{Graphadiyne} & 10.71 [116] & 13.54 [116] & 0.446 [122], & \multirow{2}{*}{0.40 [124]} & 123.1 [122], & 185.2 [116], & \multirow{2}{*}{6.3 [116]} & \multirow{2}{*}{8.0 [116]} \\  & & & 0.4 [124] & & & & & & \\  & & 0.4 [124] & & & & & & \\  & & & 0.42 [125] & & & & & & \\  & & & 0.453 & & & & & & \\  & & & & & & & & & \\  & & & & & & & & \\  & & & & & & & & & \\  & & & & & & & & & \\GR-\(n\) structure was proposed by Buehler et al., and a linear model scaling was proposed in the study [116]. They proposed a formula [116]. \[C_{n}=C_{1}(\frac{a_{1}}{a_{n}}) \tag{3}\]  
where C\({}_{n}\) is utilized to represent GR in-line stiffness and a\({}_{n}\) is showing lattice constant. In another study, Fonseca proposed that the mass density is proportional to in-plane stiffness C-\(\rho^{p}\)[124]. The Poisson's ratio for GR is from 0.39 to 0.8, and these values are much higher in comparison with G. The upper bound value of 2D materials is 1.0 [131], and it means that the material cannot change its area under uniaxial strain. The value of 0.87 for \(\alpha\)-graphyne may result in the ability to preserve its sheet area.