## 4 Factors Affecting Mechanical Properties of Porous Materials
### Circular Pores
#### 4.1.3 Porosity  
Based on the given model, the number of pores in each group is controlled to remain unchanged at 10. The models of 5%, 10%, 15%, and 20% porosity are shown in Figure 10.  
According to Table 3 and Figure 11, Poisson's ratio and equivalent modulus change strongly and have obvious rules. The equivalent modulus decreases and its variance  
Figure 10: Change in model porosity. Figure 9: (a) Influence of pores number on equivalent modulus of porous materials. (b) Influence of pores number on Poisson’s ratio of porous materials. (c) Influence of pores number on major principal stress of porous materials. increases with the increase in porosity. It shows that the results of equivalent modulus fluctuate increasingly. With the increase in porosity, Poisson's ratio gradually increases, and its variance also increases. Reference [26] also concluded that porosity changes. In particular, Poisson's ratio also has a corresponding quantitative relationship: for every 5% increase in porosity, the variance increases by about 5%. Therefore, it is verified that when the porosity increases by 2%, whether the value of variance increases by 2%, it can be found that the results also have a corresponding quantitative relationship. From Figure 11c, it can be found that the major principal stress does not change significantly with the increase in porosity, and its variance does not change much. The Poisson's ratio and equivalent modulus of porous materials are clearly affected by changes in porosity, but the major principal stress of porous materials is unaffected. Figure 11: (**a**) Influence of porosity on equivalent modulus of porous materials. (**b**) Influence of porosity on Poisson’s ratio of porous materials. (**c**) Influence of porosity on major principal stress of porous materials. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Porosity**} & **Equivalent** & **Variance of** & & **Variance of** & **First Principal** & **Variance of The** \\  & **Modulus** & **Equivalent** & **Poisson’s Ratio** & **Poisson’s Ratio** & **Stress (_Mpa_)** & **First Principal** \\  & **(_MPa_)** & **Modulus** & & & **Stress (_Mpa_)** & **Stress** \\ \hline
5\% & 19473.01 & 134.67 & 0.220 & 0.005 & 7.67 & 1.12 \\
10\% & 16864.78 & 303.17 & 0.238 & 0.010 & 7.72 & 1.09 \\
15\% & 14605.79 & 393.03 & 0.253 & 0.015 & 7.70 & 1.01 \\
20\% & 12648.98 & 489.04 & 0.267 & 0.020 & 7.64 & 1.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of mechanical properties of porous materials with different porosity. In order to adapt to a variety of materials, three materials with an initial modulus of 1.67 _GPa_, 3.4 _GPa_, 4.2 _GPa_, and Poisson's ratio of 0.2 are selected to generate a large number of models for calculation. Studying the law of calculation results yielded the following fitting formulas for mechanical properties:  
\[M_{P}^{*}=M_{0}\times 10^{-4}(28702.9P^{2}-27185.01K+9945.42) \tag{3}\]  
\[S_{P}^{*}=0.6594(M_{0}\times 10^{-4})^{2}-0.0335(M_{0}\times 10^{-4})+4.4922 \tag{4}\]  
where \(M_{0}\) represents the initial modulus of the material, \(M_{P}^{*}\) represents the calculated modulus with the change in porosity and its unit is _MPa_, \(P\) represents porosity and \(S_{P}^{*}\) represents the calculated major principal stress and its unit is _MPa_. The first principal stress is independent of the porosity but will change with the initial elastic modulus of the material. The equivalent modulus of 8.5% porosity silica gel material is estimated. According to the data, the elastic modulus of the silica gel matrix is 1.2 _GPa_, Poisson's ratio is 0.48. Table 4 shows the numerical comparison between the fitting formula and the Monte Carlo simulation. It is evident that the fitting formula has a small error. Researchers can use the fitting formula to estimate the modulus and maximum stress of materials accurately. Note that when an initial Poisson's ratio is given, the initial elastic modulus is changed.