## 4 Factors Affecting Mechanical Properties of Porous Materials
### Elliptical Pore
#### 4.2.1 Angle of Elliptical Pore  
When the given model and the control conditions of this part are constant (porosity = 10%), the elliptic parameter \(K\) = 0.5 is controlled to remain unchanged. The angle changes regularly from 0\({}^{\circ}\) to 90\({}^{\circ}\)and the results are as follows:  
Table 5 shows that the equivalent modulus decreases with increasing angle. The variance of equivalent modulus also increases gradually with angle. The larger the modulus calculation result deviates from the expected value, the flatter the curve is shown in Figure 9a. The Poisson's ratio decreases with the increase in the angle from 0\({}^{\circ}\) to 90\({}^{\circ}\). The variance of the Poisson's ratio increases first and then decreases (shown in Figure 13b that the curve becomes flat first and then becomes thin and high). The decreased amplitude is larger than the increased amplitude. The major principal stress increases with the increase in angle, and its variance shows an increasing trend. However, when the angle is larger, the increasing trend is not obvious.  
The elliptical angle has a significant effect on the mechanical properties of porous materials. The fitting formula for mechanical properties of porous materials is also given, which can be applied to a variety of materials.  
\[M_{\theta}^{*}=M_{0}\times 10^{-4}(7484.3+512.4 cos(0.034426\theta)-26.654 sin(0.03 4426\theta)) \tag{5}\]  
\[S_{\theta}^{*}=M_{0}*10^{-4}(3.6397-0.852 cos(0.033773\theta)+0.005638 sin(0.03 3773\theta)) \tag{6}\]  
where \(M_{\theta}^{*}\) represents the calculated modulus with the change in the ellipse angle and its unit is \(MPa\). \(\theta\) represents the angle and \(S_{\theta}^{*}\) represents the calculated major principal stress with the change in the ellipse angle and its unit is \(MPa\). Based on the fitting formula and Monte Carlo simulation, the values of equivalent modulus and first principal stress were obtained for the silica gel material with an elliptical angle of 37.4\({}^{\circ}\). Table 6 compares the values calculated by the Monte Carlo method and fitting formula.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Equivalent Modulus** & **First Principal Stress** & **Error of Equivalent** & **Error of First** \\  & **(_Mpa_)** & **(_Mpa_)** & **Modulus** & **Principal Stress** \\ \hline Monte Carlo simulation value & 9099.16 & 4.13 & & \\ Calculation value of fitting formula & 9126.43 & 4.058 & 0.30\% & 1.77\% \\ \hline \end{tabular}
\end{table}
Table 6: \(\theta=37.2^{\circ}\), Comparison of calculation results between Monte Carlo simulation and fitting formula.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{\(\theta\)} & **Equivalent** & **Variance of** & & & \\  & **Modulus** & **Equivalent** & **Poisson’s Ratio** & **Variance of** & **First Principal** & **Variance of the First** \\  & **(_Mpa_)** & **Modulus** & & & **Poisson’s Ratio** & **Stress (_Mpa_)** & **Principal Stress** \\ \hline
0\({}^{\circ}\) & 18621.92 & 119.82 & 0.258 & 0.009 & 5.14 & 0.28 \\
15\({}^{\circ}\) & 18286.02 & 142.53 & 0.253 & 0.010 & 5.63 & 0.40 \\
30\({}^{\circ}\) & 17395.98 & 194.33 & 0.241 & 0.011 & 6.99 & 0.65 \\
45\({}^{\circ}\) & 16216.89 & 250.88 & 0.226 & 0.011 & 8.72 & 1.03 \\
60\({}^{\circ}\) & 15146.41 & 327.97 & 0.212 & 0.009 & 10.08 & 1.08 \\
75\({}^{\circ}\) & 14372.00 & 361.64 & 0.203 & 0.008 & 11.22 & 1.33 \\
90\({}^{\circ}\) & 14114.00 & 394.93 & 0.200 & 0.007 & 12.10 & 1.31 \\ \hline \end{tabular}
\end{table}
Table 5: Values of mechanical properties of porous materials at different angles.