## 3 Results and Discussion
### Law of Energy Evolution
As a result, it can be obtained, that is:  
\[d\sigma/d\varepsilon=E\exp\left[-(\varepsilon_{c}/F_{0})^{m}\right]\left[1-m( \varepsilon_{c}/F_{0})^{m}\right]=0 \tag{8}\]  
As \(E\
eq 0\), \(\exp\left[-(\varepsilon_{c}/F_{0})^{m}\right]\
eq 0\), \(1-m(\varepsilon_{c}/F_{0})^{m}=0\), so \(m(\varepsilon_{c}/F_{0})^{m}=1\), As a result, it can be obtained, that is:  
\[F_{0}=\varepsilon_{c}/m^{-\frac{1}{m}} \tag{9}\]  
Formula (9) is substituted for (6) and, considering the condition 3\(\varepsilon=\varepsilon_{c},\;\sigma=\sigma_{c}\), the final result can be obtained that:  
\[m=1/\ln(E\varepsilon_{c}/\sigma_{c}) \tag{10}\]  
Therefore, the constitutive relation and damage of fiber-doped coal-gangue concrete under uniaxial compression can be expressed as:  
\[\left\{ \begin{array}{l}\sigma=E\varepsilon\exp\left[-\left(\frac{\varepsilon }{\varepsilon_{c}}m^{-\frac{1}{m}}\right)^{\frac{1}{m(\varepsilon_{c})}} \right]\\ D=1-\exp\left[-\left(\frac{\varepsilon}{\varepsilon_{c}}m^{-\frac{1}{m}} \right)^{\frac{1}{m(\varepsilon_{c})}}\right]\end{array} \right. \tag{11}\]  
Formula (11) reveals that the damage constitutive equation and damage evolution equation of fibre-doped coal-gangue concrete depend on the peak stress, peak strain, and initial elastic modulus of the material. Measuring these three values is sufficient to obtain the damage stress-strain curve and damage evolution curve of fibre-doped coal-gangue concrete. Based on the peak stress, peak strain, and initial elastic modulus obtained from the test, we derive the damage constitutive equation and damage evolution equation for fibre-doped coal-gangue concrete under uniaxial compression. The measured values of the model parameters are provided in Table 6, while the damage constitutive equation and damage evolution equation can be found in Table 7.  
Figure 12 demonstrates that the chosen damage constitutive model aligns well with the test results. The model curve closely matches the test curve before reaching the peak stress, but deviations occur after the peak stress. Similarly, the prereverse bending point of the stress-strain curve, based on the selected constitutive model, exhibits a strong alignment with the test curve, and the stress values decline rapidly after the reverse bending point in the descending section. The test curve shows a drop in stress when strain increases to a certain extent; the stress tends towards zero, yet the rate of decline is noticeably slower than that of the model curve. This slow rate can be attributed to a critical damage threshold in the descending section of the curve. Once the damage value of coal-gangue concrete surpasses this threshold, the material properties rapidly deteriorate, causing a substantial increase in damage and a sharp reduction in the load-bearing capacity of fibre-doped coal-gangue concrete. The damage evolution characteristics of HBPRGC are illustrated in Figure 13 below. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Serial Number** & \(\mathbf{F_{0}}\) & \(\mathbf{m}\) & **Initial Modulus of Elasticity/MPa** & **Peak Strain/\%** & **Peak Stress/MPa** \\ \hline OC-30-0.1 & 0.003417 & 2.2311 & 13,195 & 2.31 & 30.48 \\ PC-30-0.1 & 0.003198 & 2.2480 & 12,456 & 2.52 & 31.39 \\ BC-30-0.1 & 0.002988 & 1.8519 & 12,759 & 2.66 & 33.94 \\ HBPRGC-30-0.1 & 0.003915 & 1.8434 & 11,381 & 2.83 & 32.21 \\ HBPRGC-30-0.2 & 0.004401 & 2.0044 & 8945 & 2.96 & 26.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Damage constitutive model parameters under uniaxial compression. Figure 12: _Cont._