## 3 Packing Model and Property Simulation  
### Establishment of Packing Model  
The 2D/3D packing model of resin concrete was established based on relevant knowledge of material meso-structure and meso-mechanics. At least millions of particles exist in a granular pack. Hence, computer performance analysis of powder material is mainly applied to 2D packing models [21]. We consider that 3D model can be obtained by random arbitrary distribution, and then the 2D model is one slice directly captured from the 3D model.  
Based on gradation results that we determine, related settings used to simulate packing model of resin concrete are as follows: According to Japan national standard JIS G5901-1974 [22], the aggregate size distribution is 0.076-0.15 mm, 0.15-0.22 mm, 0.22-0.5 mm, 0.5-0.9 mm, 0.9-1.6 mm, 1.6-2.5 mm and 2.5-4 mm, respectively. According to mass fractions of aggregate, the number of all kinds of aggregate is calculated for precision improvement. Detailed aggregates ratio and particle number of each grade are shown in Table 4. Aggregates are packed from the largest to the smallest size in a cube with an h side and \(h=12\) mm. Pores represent resin binder. In order to improve the reality and accuracy of the packing model, this paper adopts the polygonal aggregates model, not the spherical one. The process of establishing 3D microstructure is as follows: (1) Coordinate of sphere center x(i),y(i),z(i) and radius r(i) of each random aggregate are determined for be completely inside the cube and can be given by in Matlab:  
\[\begin{cases}x\left(i\right)=r\left(i\right)+\left(h-2\times r\left(i\right) \right)\times rand()\\ y\left(i\right)=r\left(i\right)+\left(h-2\times r\left(i\right)\right)\times rand ()\\ z\left(i\right)=r\left(i\right)+\left(h-2\times r\left(i\right)\right)\times rand ()\end{cases} \tag{10}\]  
The generated aggregate is reserved; (2) For newly generated sphere, intersection with other spheres might exist. Hence, interference calculation with generated spheres is carried out in order to determine whether interference exists or not. If it exists, the sphere is deleted; (3) According to the above rules, particles are accumulated grade by grade. For each grade, the volume should meet the need of  
\begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Subject** & **Mass ratio** & **Volume ratio** \\ \hline
1 & Resin binder:quartz & 0.073:0.927 & 0.148:0.852 \\
2 & Resin binder:quartz:ZrW\({}_{2}\)O\({}_{8}\) & 0.066:0.742:0.192 & 0.148:0.767:0.085 \\
3 & Resin binder:quartz:ZrW\({}_{2}\)O\({}_{8}\) & 0.059:0.597:0.344 & 0.148:0.682:0.170 \\ \hline \end{tabular}
\end{table}
Table 2: Comparison of the proportions of three mixing compositions.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**No.**} & \multicolumn{4}{c}{**CTE (\(\times\) 10\({}^{-6}\)/K)**} \\ \cline{2-5}  & **Model A + A** & **Model A + B** & **Model B + A** & **Model B + B** \\ \hline
1 & 9.161 & — & — & 6.598 \\
2 & 8.777 & 3.512 & 6.158 & 2.768 \\
3 & 7.577 & 2.909 & 3.403 & 1.536 \\ \hline \end{tabular}
\end{table}
Table 3: Results of the prediction formula.  
aggregate ratio mentioned above; (4) We take \(n\) (\(n>3\)) random points in one sphere. Then, planes are drawn by using three arbitrary points. After drawing, overlapped planes are deleted by Boolean Operation and polygonal aggregate is obtained. To avoid sharp aggregates, all points should not be placed in the same half spherical region.  
The filling rate of the 3D model is higher than 85% which satisfies the condition of high density packing, and polygonal aggregates occupy the square area of 85.2% in the 2D model. Therefore, the generated 2D/3D models are given in Figure 2.