# Effect of Gradation Segregation on Mechanical Properties of an Asphalt Mixture
## 2 DEM Model Development
### Model Development  
To simplify the DEM model, aggregates in asphalt mixture are modeled as spheres with different diameters. In addition, it is assumed that the density of all the aggregates is same and the surface area of aggregates passing sieve is uniformly distributed. Therefore, the mass ratio of aggregate with different diameters equals their area percentage per unit area. The number of aggregate particles in asphalt mixture can be approximately calculated through Equation (2) to Equation (3):  
\[A_{i}=A\times(1-\theta)\times P_{i} \tag{2}\]  
\[n_{i}=\frac{A_{i}}{\pi\cdot(r_{\min}^{2}+r_{\max}^{2})/2} \tag{3}\]  
where,  
\(A=\) total area of the designed model;  
\(\Theta=\) voids in the mineral aggregate;  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **High-Level Segregation** & **Medium-Level Segregation** & **Low-Level Segregation** \\ \hline Coarse gradation & HC & MC & LC \\ Fine gradation & HF & MF & LF \\ \hline \hline \end{tabular}
\end{table}
Table 1: Label of aggregate gradations.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Sieve Size/mm**} & \multicolumn{6}{c}{**Passing Ratio (\%) for Different Sieving Size (mm)**} \\ \cline{2-9}  & **HF** & **MF** & **LF** & **D** & **LC** & **MC** & **HC** \\ \hline
26.5 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
19 & 100 & 96 & 94 & 92 & 91 & 88 & 83  
\(P_{i}\) = percentage of aggregate remaining on the \(i\)-th sieve;  
\(A_{i}\) = total area of aggregate remaining on the \(i\)-th sieve;  
\(n_{i}\) = total number of the aggregate remaining on the \(i\)-th sieve;  
\(r_{\text{max}}\) = maximum radius of the spheres  
\(r_{\text{min}}\) = minimum radius of the spheres.  
To calculate the number of aggregates with different diameters, the air voids in mineral aggregate (VMA) should be calculated in advance. In this study, VMAs were calculated by means of trial-calculating. Specifically, let the model area equal the total area of aggregates. The amounts of aggregate with different diameters can be estimated. Then, develop the DEM model accordingly followed by calculating the VMA of aggregates within the built DEM model. For each gradation, the VMA calculation trial-calculating process was repeated 10 times to minimize errors. The trail-calculated VMAs are listed in Table 3.  
Once the VMA was obtained, the amounts of aggregate spheres can be calculated according to Equations (2) and (3). However, due to the randomness of the generated particle area, in order to ensure that the total area of each aggregate gradations is consistent with the design value, after generating the aggregate particles for each gradation according to the calculated results, it is necessary to count the total area of all particles. Compared with the design value, if the difference between the two is greater than \(\pi\cdot(r_{\text{min}}^{2}+r_{\text{max}}^{2})/2\), the result is modified by randomly adding or deleting one particle, and then repeat the above process until the area of each grade of aggregate particles meets the design requirement. Based on this, 2D DEM model is built.  
Figure 1 shows the 2D DEM model development process. A \(150\times 100\) mm size model was built firstly by generating four rigid walls. Then, loose aggregates were generated uniformly in the certain space surrounded by the four walls. A certain speed was then given to the upper and lower walls for compacting the loose aggregates. Finally, to avoid the aggregates being excessively compacted, the upper wall moves up a small distance, and then the PFC \"solve\" command was used to delete the imbalance force in the model. Repeat this step until the stress on the upper wall is 0. The heights of the models can be found in Table 3. As excepted the heights of the models are basically consistent and this meets the research expectation.