## 5 Performance Prediction of the Rebuilt Models
### Simulations for Performance Prediction  
Prior to simulations, several micromechanical models were adopted to characterize the contact behavior between coarse aggregates within PFC2D; including the contact-stiffness model and sliding model [38], as the following equation shows. Contact-stiffness model:  
\[ \begin{array}{c}F_{i}^{n}=K^{n}U^{n}\\ \Delta F_{i}^{s}=-k^{s}\Delta U^{s}\end{array}  \tag{13}\]  
where \(F_{i}^{n}\) is the normal force at the i-th contact point; \(\Delta F_{i}^{s}\) is the shear contact force increment at the \(i\)-th contact point; \(K^{n}\) is the normal stiffness at the contact; \(k^{s}\) is the shear stiffness at the contact; \(U^{n}\) is the overlap of the two contact entities along the normal direction; \(\Delta U^{s}\) is the overlap of the two contact entities along the shear direction. Sliding model:  
\[F_{max}^{s}=\mu|F_{i}^{n}| \tag{14}\]  
where \(F_{max}^{s}\) is the maximum allowable shear contact force; \(F_{i}^{n}\) is the normal contact force between two entities; \(\mu\) is the friction coefficient in PFC2D. As shown in Equations (13) and (14), the contact force between two entities (balls) in the normal/shear direction was determined by the contact normal/shear stiffness and their overlap in corresponding directions. As for the normal contact force, the formula is always effective in the simulation, such that the intensity of \(F_{i}^{n}\) depends on the value of \(K^{n}\) and \(U^{n}\) directly. However, the shear contact force was controlled and calculated based on two different modes for different periods. At the beginning, the shear contact force increment depends on \(k^{s}\) and \(\Delta U^{s}\), when the adjacent entities contact each other. As the accumulative shear force reaches the limiting condition, as shown in Equation (14), the calculation process is converted immediately, in which the intensity of the shear contact force (not the shear force increment) is determined by the friction coefficient and absolute value of the normal contact force currently. As shown, it is significantly important to determine the values of \(K^{n}\) and \(k^{s}\) for modeling the micro-contact process in the simulation. According to the PFC2D manual, these two key micro-parameters are calculated by the input value of the balls' normal stiffness \(k_{n}\) and shear stiffness \(k_{s}\), respectively. The appropriate values of \(k_{n}\) and \(k_{s}\) can be obtained through a calibration process during the simulation based on the laboratory results. When calibrating the micro-parameters, the size influences should be taken into consideration. This means that the micro-parameters differed from each other for different sizes of particles and should be determined based on the size, respectively. Thus, the simulations were conducted in two steps. In the first step, the virtual penetration tests of three different compositions were simulated. The three virtual specimens were developed by the proposed algorithm with 4.75-, 9.5- and 13.2-mm particles only, respectively. Then, the calibrated process for the best micro-parameters was done as follows: (1) conduct the laboratory penetration test to get the actual force-displacement curve of the three specimens; (2) conduct the simulations to get the virtual force-displacement curve of the three specimens; (3) obtain the best group of micro-parameters by adjusting their values continuously until the force-displacement curve of the simulation is close to the actual result. By the former process, the calibrated parameters can be obtained as shown in Table 6. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Sieving Size (mm)**} & \multirow{2}{*}{**Particle Density (g/cm\({}^{3}\))**} & **Density of Compacted** & **Void Contents of** \\  & & **Specimen (g/cm\({}^{3}\))** & **Compacted Specimens (\%)** \\ \hline
13.2 & 2.744 & 1.625 & 39.55 \\
9.5 & 2.730 & 1.616 & 39.66 \\
4.75 & 2.729 & 1.594 & 40.70 \\ \hline \end{tabular}
\end{table} 
Table 5: Materials for laboratory penetration tests.