# Experimental Study on the Dynamic Characteristics of Fractured Coal Under Cumulative Impact
## 3 Results and Analysis
### Energy Dissipation Analysis  
Macroscopic material failure is fundamentally an energy-driven process from micro-deformation to fragmentation. Based on the recorded incident strain \(\epsilon_{\mathrm{i}}\), reflected strain \(\epsilon_{\mathrm{r}}\) and transmitted strain \(\epsilon_{\mathrm{t}}\) from SHPB tests, the incident energy \(E_{\mathrm{i}}\), reflected energy \(E_{\mathrm{r}}\) and transmitted energy \(E_{\mathrm{t}}\) can be calculated using Formulae (7)-(9) [45]:  
\[E_{\mathrm{i}}=A\cdot E\cdot v\int_{0}^{t}\epsilon_{\mathrm{i}}^{2}\mathrm{d}t \tag{7}\]  
\[E_{\mathrm{r}}=A\cdot E\cdot v\int_{0}^{t}\epsilon_{\mathrm{r}}^{2}\mathrm{d}t \tag{8}\]  
Figure 8: Morphological diagram of crushed coal samples under cumulative impact. (**a**) No fracture-impact effect-1; (**b**) Fracture angle 30\({}^{\circ}\)-impact effect-1; (**c**) Fracture angle 60\({}^{\circ}\)-impact effect-1; (**d**) Fracture angle 90\({}^{\circ}\)-impact effect-1; (**e**) No fracture-impact effect-3; (**f**) Fracture angle 30\({}^{\circ}\)-impact effect-3; (**g**) Fracture angle 60\({}^{\circ}\)-impact effect-3; (**h**) Fracture angle 90\({}^{\circ}\)-impact effect-3; (**i**) No fracture-impact effect-5; (**j**) Fracture angle 30\({}^{\circ}\)-impact effect-5; (**k**) Fracture angle 60\({}^{\circ}\)-impact effect-5; (**l**) Fracture angle 90\({}^{\circ}\)-impact effect-5.  
\[E_{\rm t}=A\cdot E\cdot v\int_{0}^{t}\varepsilon_{\rm t}^{2}{\rm d}t \tag{9}\]  
The dissipated energy \(E_{\rm d}\) during dynamic failure of fractured coal samples can be derived from the law of energy conservation as Formula (10) [47; 48]:  
\[E_{\rm d}=E_{\rm i}-E_{\rm r}-E_{\rm t} \tag{10}\]  
Energy dissipation constitutes the fundamental mechanism of material failure. The dissipated energy comprises three components: fracture energy, kinetic energy of the ejected fragments and other auxiliary losses. The fracture energy is the energy used to produce cracks and crack propagation of materials, which is the energy absorbed by material damage in a strict sense. Fracture energy, accounting for over 90% of dissipated energy, strictly represents the energy absorbed during material disintegration. Neglecting fragment kinetic energy and auxiliary losses, fracture energy can be approximated as the total dissipated energy.  
Under constant impact pressure, the incident energy remained consistent across tests. The energy dissipation ratio is defined as Formula (11):  
\[N=E_{\rm d}/E_{\rm i} \tag{11}\]  
The energy dissipation ratio variations in fractured coal samples under cumulative impacts are analyzed in Figure 9. Prefractured specimens exhibit lower energy dissipation rates compared to intact samples, though the influence of fracture inclination angles on the dissipation rates is statistically insignificant. Furthermore, as the number of impact cycles increases, the energy dissipation rate of coal samples continuously decreases. This occurs because repeated impacts inevitably increase the quantity of micro-fractures within the coal matrix. Consequently, the material's compactness and homogeneity diminish, while internal damage accumulation intensifies. This structural degradation reduces the energy required to induce failure in fractured coal samples under dynamic impact loading. These observations substantiate the notion that cumulative impact loading induces significant damage deterioration effects in fractured coal specimens.  
Figure 9: Energy dissipation rate of fractured coal samples under cumulative impact.