## 4 Factors Affecting Mechanical Properties of Porous Materials
### Elliptical Pore
#### 4.2.2 Elliptical Shape Parameters  
On the basis of the given model and the constant control conditions in this part (porosity = 10%), the elliptical angle (\(\theta\) = 90\({}^{\circ}\)) is controlled to remain unchanged, and \(K\) changes from 0.4 to 0.5 to 0.6 to 2.5. Similarly, this section also provides the fitting function of parameter \(K\). It can be seen from Figure 14a and Table 7 that with the increase in \(K\), the equivalent modulus increases, and the value of each increase gradually decreases. The increasing trend is increasingly gentle. With the increase in \(K\), the equivalent modulus of the model should gradually increase and then tend to stabilize. The variation gradually reduces, and the magnitude of the decline increases, indicating that the fluctuation is becoming increasingly smaller. It demonstrates that the equivalent modulus calculation findings are becoming increasingly stable. Poisson's ratio also gradually increases with the increase in \(K\), and the increasing trend gradually decreases. Its variance increases first and then decreases and then gradually tends to be stable, and there is a maximum value of variance. The model's major principal stress gradually decreases as \(K\) increases, as shown in Figure 14c. The variance is gradually reduced as well. Figure 13: (**a**) Influence of elliptical angle on equivalent modulus of porous materials. (**b**) Influence of elliptical angle on Poisson’s ratio of porous materials. (**c**) Influence of elliptical angle on major principal stress of porous materials. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(K\)} & \multicolumn{2}{c}{**Equivalent**} & \multicolumn{2}{c}{**Variance of**} & \multirow{2}{*}{**Poisson’s Ratio**} & \multicolumn{2}{c}{**Variance of**} & \multirow{2}{*}{**First Principal**} & \multicolumn{1}{c}{**Variance of the**} \\  & **Modulus** & **Equivalent** & & & **Poisson’s Ratio** & **Stress (_Mpa_)** & **First Principal** \\  & _(Mpa_)_ & **Modulus** & & & & **Stress (_Mpa_)** & **Stress** \\ \hline
0.4 & 12939.89 & 381.79 & 0.184 & 0.006 & 12.57 & 0.18 \\
0.5 & 14139.62 & 359.53 & 0.200 & 0.006 & 12.07 & 0.20 \\
0.6 & 14976.78 & 357.76 & 0.210 & 0.007 & 11.087 & 0.21 \\
0.7 & 15632.15 & 360.72 & 0.219 & 0.008 & 10.22 & 0.22 \\
1.0 & 16867.24 & 295.42 & 0.237 & 0.010 & 8.58 & 0.24 \\
1.2 & 17467.59 & 228.15 & 0.244 & 0.010 & 7.14 & 0.24 \\
1.6 & 18173.00 & 153.97 & 0.253 & 0.010 & 5.85 & 0.25 \\
2.0 & 18610.51 & 122.44 & 0.258 & 0.009 & 5.16 & 0.26 \\
2.5 & 18962.58 & 99.48 & 0.264 & 0.009 & 4.62 & 0.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The values of mechanical properties of porous materials with different elliptical shape parameters. Figure 14: (**a**) Influence of \(K\) on equivalent modulus of porous materials. (**b**) Influence of \(K\) on Poisson’s ratio of porous materials. (**c**) Influence of \(K\) on major principal stress of porous materials. Various material properties are given by the fitting formula of mechanical properties of porous materials, which is applicable to the change in elliptical shape parameters:  
\[M_{K}^{*}=M_{0}\times 10^{-4}((-863.77)K^{2}+3679.23K+4584.13) \tag{7}\]  
\[S_{K}^{*}=M_{0}\times 10^{-4}(1.03K^{2}-4.78K+7.3) \tag{8}\]  
where \(M_{K}^{*}\) represents the calculated modulus with the change in the elliptical shape parameters. \(K\) represents the elliptical shape parameters and \(S_{K}^{*}\) represents the calculated major principal stress with the change in the elliptical shape parameters. Similarly, silica gel materials with the same parameters were selected for comparison. At this point, the shape parameter is set to 0.84. The values calculated by the Monte Carlo method and fitting formula were compared. As it can be seen in Table 8, the fitting formula calculation results are basically consistent with the Monte Carlo simulation results.