*4.3. Determination of the Values of Parameters* ε*<sup>f</sup>* min*, D*<sup>1</sup> *and D*<sup>2</sup>

Damage constant ε*<sup>f</sup>* min is the plastic strain at the moment when the minimum strength fracture of the material is achieved. The acquisition method is given in the literature [15] (Figure 7). Determination of the value of parameter ε*<sup>f</sup>* min required a uniaxial compression cycle loading experiment, and a hypothetical failure interface was defined based on the curve drawn by the experimental results. The failure interface revealed that when the axial strain reached the intersection of the interface and the strain axis, the sample lost its strength completely, and the strain value was equal to the value of ε*<sup>f</sup>* min. When the equivalent fracture strain was achieved, *P*<sup>∗</sup> = 1/6 and *T*<sup>∗</sup> could be calculated as *T*<sup>∗</sup> = *T*/ *fC*. If ε*<sup>f</sup>* min ≤ *D*1(*P*<sup>∗</sup> + *T*<sup>∗</sup> ) *<sup>D</sup>*<sup>2</sup> , then *<sup>D</sup>*<sup>1</sup> = <sup>ε</sup>*<sup>f</sup>* min/(1/6 + *<sup>T</sup>*<sup>∗</sup> ). Due to the lack of real data, we assumed that *D*<sup>2</sup> = 1.0.

**Figure 7.** The acquisition method for parameter ε*<sup>f</sup>* min.

In this test, cylindrical samples with a diameter of 50 mm and length of 100 mm were used. During the loading process, the briquette samples were first loaded to 70% of their uniaxial compressive strength and then unloaded to zero at the same rate. Then, loading was repeated and its intensity was decreased by 10% at each cycle until the sample was destroyed. Uniaxial cyclic loading experiments were repeated for 5 test specimens. Due to operational error, only two sets of effective data were obtained. According to the experimental data, the stress–strain curve of the uniaxial compression cycle loading experiment was plotted, as shown in Figure 8.

**Figure 8.** Experimental stress–strain curve of uniaxial cyclic loading.

A hypothetical failure interface was obtained on the stress–strain curve, and its intersection with the strain axis was considered to be the value of ε*<sup>f</sup>* min. The parameter values obtained by the above method are summarized in Table 6.


According to the results presented in Table 6, the damage parameter ε*<sup>f</sup>* min takes the average of the values obtained for the two sets of samples, which is 0.025. *D*<sup>1</sup> = 0.0131 and *D*<sup>2</sup> = 1.0 were obtained according to the method described above.

#### *4.4. Determination of the Values of Parameters B and N*

Normalized pressure hardening coefficient *B* and pressure hardening index *N* were obtained by triaxial compression experiments. In these experiments, the confining pressure was set at σ<sup>2</sup> = σ3. First, the three axes coordinately load to the values of the confining pressure, and then the confining pressure was kept constant at σ<sup>2</sup> = σ3. In the next stage, a load was applied along the axial direction until the sample failed, and the peak value of axial stress σ<sup>1</sup> was recorded. The calculation of the normalized pressure hardening coefficient *B* and pressure hardening index *N* required hydrostatic pressure *P* and principal stress difference Δσ, respectively. The hydrostatic pressure *P* and principal stress difference Δσ were calculated as *P* = (σ + 2σ3)/3 and Δσ = σ<sup>1</sup> − σ3, respectively. A series of values (*P*<sup>∗</sup> , σ<sup>∗</sup> ) were obtained by normalizing the values of (*P*, Δσ) according to the equations *P*<sup>∗</sup> = *P*/ *fc* and σ<sup>∗</sup> = Δσ/ *fC*. The obtained values were fitted by equation σ<sup>∗</sup> = *BP*∗*<sup>N</sup>* to obtain the values of *B* and *N*.

A triaxial compression test was performed using confining pressure at the rate of 0.02 MPa/s. After a stabilization period, the axial load was applied at the rate of 0.1 mm/min until the sample failed. Since the strength of the briquette sample was not high, excessively high confining pressures could break the sample. Therefore, confining pressure gradients were set at 1, 2, 3, and 4 MPa. The principal stress difference–axial strain curves of the samples using the experimental data obtained under different confining pressures are shown in Figure 9.

**Figure 9.** Curves of different principal stresses at different confining pressures.

The value of hydrostatic pressure *P* was calculated by equation *P* = (σ + 2σ3)/3, and a series of (*P*, Δσ) values were obtained. The obtained main stress difference Δσ and hydrostatic pressure *P* were normalized to obtain a series of values (*P*<sup>∗</sup> , σ<sup>∗</sup> ) (Table 7).

**Table 7.** Normalized principal stress difference σ<sup>∗</sup> and hydrostatic pressure *P*<sup>∗</sup>

.


Using the data presented in Table 7, fitting was performed by equation σ<sup>∗</sup> = *BP*∗*N*, and the fitting curve was drawn as shown in Figure 10. The values of *B* and *N* were 1.86 and 0.75, respectively.

**Figure 10.** Curve of *B* and *N* fitting values.

#### *4.5. Determination of the Values of the Remaining Parameters*

The average density of the coal samples ρ0, the elastic modulus *E*, Poisson's ratio ν, volume pressure at crushing point *PC*, and volumetric strain at crushing point μ*<sup>C</sup>* were mainly derived indirectly using different equations. Based on the data shown in Table 2, the average density ρ<sup>0</sup> of the briquette samples was 1.228 g/cm3. According to the data summarized in Table 4, the mean value of the three samples was taken as the elastic modulus and Poisson's ratio of the briquette samples, which were 101.56 MPa and 0.36, respectively. Shear modulus G was calculated using the equation *G* = *E*/2(1 + ν) to be 37.34 MPa. Based on previous literature, the value of *P* was calculated to be 0.78 MPa using the equation *PC* = *fC*/3 [14], where uniaxial compressive strength *fc* was considered to be 2.33 MPa. Volumetric strain at crushing point μ*<sup>C</sup>* was obtained to be 0.0064 using the equation μ*<sup>C</sup>* = *PC*/*K*, where the bulk modulus *K* was calculated according to the equation *K* = *E*/3(1 − 2ν), where the elastic modulus *E* and Poisson's ratio ν were assumed to be 101.56 MPa and 0.36, respectively.

The 11 parameters in the HJC constitutive model for coal were determined by experimental measurements and different equations. Pressure at compaction point *P*1, volume strain at compaction point μ1, pressure constants *k*1, *k*2, and *k*3, normalized cohesive strength A, strain rate coefficient *C*, maximum normalized intensity *S*max, and some other parameters were determined using a flying impact test [28]. The remaining parameters were not accessible due to their low sensitivity and limited experimental conditions. Therefore, their values were considered to be similar to those provided in [20]. Thus, the values of all parameters in the HJC constitutive model were determined, and the results are shown in Table 8.

**Table 8.** Coal HJC constitutive model parameter values.


#### **5. SHPB Experiment and Numerical Simulation Analysis**

#### *5.1. SHPB Experimental Device*

Dynamic impact tests on coal samples were carried out using an SHPB test device at China University of Mining and Technology (Beijing). The SHPB device consisted of a striking rod (bullet), an incident rod, and a projection rod. As shown in Figure 11, the bullet was a 540-mm long heavy double hammer-spun cone with a cone ratio of 310:100:130 [29]. As shown in Figure 12, the experimental and projection rods had a diameter of 75 mm and length of 2000 mm, and the tested coal sample was sandwiched between incident and projection rods. The SHPB experiment used coal briquettes with a diameter of 50 mm and length of 25 mm.

**Figure 11.** Spinning cone bullet.

**Figure 12.** 50 mm split Hopkinson pressure bar (SHPB) experimental device.
