## 3 Results and Discussion
### Law of Energy Evolution
Subsequent to the peak elastic strain energy, the elastic strain energy of the reference group coal-gangue concrete diminishes rapidly, indicating rapid crack penetration and a conspicuous brittle failure. The introduction of fibre, particularly the inclusion of a hybrid fibre, reduces the rate at which elastic strain energy decreases and enhances the characteristics of ductile failure. This improvement is more pronounced with higher hybrid-fibre volume content. This characteristic is attributed to the crack-bridging  
Figure 8: Energy-evolution characteristics of HBPRGC. action of BF and PF, which delays crack propagation and penetration speed, ultimately enhancing the ductile failure characteristics of coal-gangue concrete. A comparison between Figures 9 and 10 reveals that as the matrix strength increases, the peak elastic strain energy increases from 0.020 J\(\cdot\)cm\({}^{-3}\)-0.025 J\(\cdot\)cm\({}^{-3}\) to 0.028 J\(\cdot\)cm\({}^{-3}\)-0.032 J\(\cdot\)cm\({}^{-3}\), the total strain energy and the dissipation energy also show a gradually increasing trend. Simultaneously, the impacts of fibre type and volume content on the energy-evolution characteristics of coal-gangue concrete diminish. This observation suggests that heightened matrix strength attenuates the influence of the fibre on the performance of coal-gangue concrete, hybrid fibres are more effective in enhancing coal-gangue concrete with low matrix strength. The energy conversion rates of HBPRGC are illustrated in Figure 11; \(\alpha=Ue/U\) is defined as elastic strain energy conversion in HBPRGC, and \(\beta=Ud/U\) is defined as dissipative energy conversion. Figure 11 demonstrates that in the closing stage of primary cracks (stage I) when the matrix strength remains constant, the elastic strain energy conversion in HBPRGC decreases to \(\alpha<1\), while the dissipative energy conversion increases to \(\beta>0.1\). This phenomenon is mainly attributed to the energy consumed by primary crack closure due to friction. During the stage of elastic deformation (stage II), the addition of fibre leads to increased internal interfaces within coal-gangue concrete. Consequently, loading induces dislocation and interface friction, resulting in increased energy consumption. Hence,  
Figure 9: Energy-evolution law of HBPRGC with matrix strength grade of C20. (**a**) Elastic strain energy; (**b**) dissipated energy; (**c**) total strain energy. the addition of fibre reduces the elastic strain energy conversion rate and augments the dissipative energy-conversion rate. In the phase of rapid crack expansion (stage III), the rate at which the elastic strain energy conversion diminishes. At the same time, the rate at which the dissipative energy conversion accelerates in coal-gangue concrete is mainly due to the crack-bridging effect of fibre. The crack-bridging effect of fibre is helpful for consuming more energy and preventing crack expansion in the crack-expansion stage of concrete. Thus, coal-gangue concrete doped fibres can absorb more energy for the same state of crack development, which means the strength and ductility of coal-gangue concrete all have improved significantly. As the crack-penetration stage is reached (stage IV), cracks tend to propagate through the material. The crack-limiting effect of fibre becomes more pronounced, reducing the rate of crack penetration in coal-gangue concrete. Consequently, the rate of elastic strain energy conversion notably decreases and the rate of dissipative energy conversion significantly increases, enhancing the ductile deformation of the material. Furthermore, with increased matrix strength, the concentration of elastic strain energy conversion and the curve of dissipative energy conversion in HBPRGC increase. This observation underscores that higher matrix strength weakens the influence of fibre on coal-gangue concrete properties. Figure 10: Energy-evolution law of HBPRGC with matrix strength grade of C30. (**a**) Elastic strain energy; (**b**) dissipated energy; (**c**) total strain energy.