# Calibration Method of PFC3D Micro-Parameters under Impact Load
## 2 Laboratory Rock Test
### SHPB Test and Results  
A uniaxial impact compression test was conducted on limestone samples using the SHPB testing system from Shaanxi Coal Chemical Group Co., Ltd. in Xi'an, China. The shock wave of the limestone samples was loaded at a driving pressure of 0.2 MPa. The test system and limestone samples in the clamping state are shown in Figures 6 and 7. The length of the absorption bar is 1.5 m, the length of the incident bar and transmission bar is 3.0 m, and the diameter is 50 mm. The bars are made of high-strength alloy steel, 48 CrMoA, with a density of 7.8 g/cm\({}^{3}\), an elastic modulus of 206 GPa, a yield strength of 835 MPa, and a P-wave velocity of 5172 m/s. The three-wave method was used to process the test data. In the SHPB test, incident energy \(W_{t}\), transmitted energy \(W_{t}\), and reflected energy \(W_{r}\) received by the specimen can be calculated as follows [25]:  
\[W_{i}(t)=AEC_{0}\int_{0}^{t}\epsilon_{i}^{2}(t)dt \tag{1}\]  
\[W_{r}(t)=AEC_{0}\int_{0}^{t}\epsilon_{r}^{2}(t)dt \tag{2}\]  
\[W_{t}(t)=AEC_{0}\int_{0}^{t}\epsilon_{t}^{2}(t)dt \tag{3}\]  
where \(A\) is the cross-sectional area of the bar, m\({}^{2}\); \(E\) is the elastic modulus of the bar material; \(C_{0}\) is the longitudinal wave velocity of the bar, m/s; \(\epsilon_{i}(t)\), \(\epsilon_{r}(t)\), and \(\epsilon_{t}(t)\) are the strain of the incident wave, reflected wave, and transmitted wave on the pressure bar at time, \(t\) respectively. Figure 8 shows the original waveform of three waves at the impact pressure of 0.2 MPa. According to the one-dimensional stress wave theory and the assumption of uniformity introduced into the specimen, the average strain rate, average strain, and average stress on the two end faces of the specimen in the testing process are as follows [26]:  
\[\left\{ \begin{array}{l}\overset{\bullet}{\epsilon}(t)=\frac{C_{0}}{L}( \epsilon_{1}(t)-\epsilon_{r}(t)-\epsilon_{t}(t))\\ \epsilon(t)=\frac{C_{0}}{L_{s}}\int_{0}^{t}|\epsilon_{i}(t)-\epsilon_{r}(t)- \epsilon_{t}(t)|dt\\ \sigma(t)=\frac{A\epsilon}{2A_{s}}(\epsilon_{i}(t)+\epsilon_{r}(t)+\epsilon_{ t}(t))\end{array} \right. \tag{4}\]  
where \(l_{s}\) and \(A_{s}\) are the length and cross-sectional area of the rock specimen. Figure 9 shows the deviatoric stress-strain curve when the impact pressure is 0.2 MPa, after calculation by the three-wave method. The dynamic peak strength and dynamic elastic modulus of the limestone sample are 84.29 MPa and 34.14 GPa, respectively. Figure 8: Typical three-wave original waveform. Figure 7: A sample of limestone in a clamped state. The failure pattern of the limestone sample when the impact pressure is 0.2 MPa is shown in Figure 10. After impact damage occurred, the limestone samples were screened and weighed using a set of sizing screens and an electronic balance [27]. A set of sizing screens selected for the test is shown in Figure 11. The pore sizes, from small to large, are 1.0 mm, 2.0 mm, 5.0 mm, 10.0 mm, and 15.0 mm. Six groups of limestone fragments, with particle sizes ranging from 15.0-50.0 mm, 10.0-15.0 mm, 5.0-10.0 mm, 2.0-5.0 mm, 1.0-2.0 mm, and 0.0-1.0 mm, were obtained through screening, as shown in Figure 12. In order of particles sizes, the range obtained was i = 1, 2, 3...6. After the screening, the percentage of the mass of fragments in each group and the total mass of the sample was counted. The method of obtaining these results is as follows: firstly, a set of sizing screens was used to screen the broken fragments; secondly, an electronic balance was used to obtain the mass m\({}_{i}\) of each group of particle sizes; finally, the mass percentage m\({}_{pi}\) of each group of particle sizes was obtained by using the ratio of the mass m of each group of particle sizes to the total mass m of the sample. The mass percentage of the maximum particle size group was used as the evaluation index of the damage degree. After weighing, each group's mass m\({}_{1}\) to m\({}_{6}\) was 103.94 g, 22.50 g, 3.72 g, 1.72 g, 0.38 g, and 0.28 g, respectively.