## 3 Results and Discussion
The increase in both total strain energy and dissipative energy accelerates notably at a fibre volume content of 0.1%; HBPRGC-20-0.1 and HBPRGC-30-0.1 have elevations of 53.66% and 51.45% in total strain energy and 54.11% and 50% in dissipative energy, respectively. When the volume content of hybrid fibres reaches 0.2%, HBPRGC-20-0.2 and HBPRGC-30-0.2 have elevations of 31.95% and 30.32% in total strain energy and \(-\)3.46% and 28.71% in dissipative energy, respectively. These values are second to coal-gangue concrete with a mixed fibre volume of 0.1%. Therefore, 0.1% also seems to be the optimum hybrid-fibre volume content for increasing both the total strain energy and dissipative energy of coal-gangue concrete during its uniaxial compression. ### Law of Energy Evolution  
The nonlinearity of concrete materials arises from two primary energy-dissipation mechanisms: microcrack development and plastic yielding. Plastic yielding results in irreversible plastic deformation, while microcrack propagation reduces stiffness and causes strain softening in concrete. To capture the irreversible damage evolution process of materials, global scholars [48,49] have extended constitutive models using damage variables within the framework of continuum mechanics. [PERSON], a renowned French scholar, introduced the strain equivalent principle, which states that the strain constitutive relationships of damaged materials under uniaxial stress can be derived from the constitutive equations of undamaged materials. This derivation involves replacing the nominal stress in the constitutive relationships of undamaged materials with the effective stress after damage occurs. This hypothesis allows us to establish constitutive equations for damaged materials:  
\[\varepsilon=\sigma_{n}/E \tag{1}\]  
The relationship between effective stress and Cauchy stress is as follows:  
\[\sigma_{n}=\sigma/1-D \tag{2}\]  
Combining Equations (1) and (2):  
\[\sigma=E(1-D)\varepsilon \tag{3}\]  
Figure 11: Energy-conversion ratio of HBPRGC. The meso-heterogeneous mechanical properties and damage of quasi-brittle materials, like concrete and rock, obey Weibull distribution, so the damage variable can be further expressed as:  
\[D=1-\exp\left[-(\varepsilon/F_{0})^{m}\right] \tag{4}\]  
Therefore,  
\[\sigma=E\varepsilon\exp\left[-(\varepsilon/F_{0})^{m}\right] \tag{5}\]  
In the formula, \(\sigma_{n}\) is the effective stress, \(\sigma\) is the Cauchy stress, \(D\) is the damage variable, \(E\) is the elastic modulus of the undamaged material, \(\varepsilon\) is strain, and \(F_{0}\) and \(m\) are the scale parameters and shape parameters, respectively. According to the characteristic that the stress-strain curve is divided into four segments, the above Formula (5) needs to satisfy the following four corresponding boundary conditions:  
\[\left\{ \begin{array}{l}\vartriangle\sigma=0,\;\varepsilon=0\\ \vartriangle\sigma=0,\;d\sigma/d\varepsilon=E\\ \vartriangle\varepsilon=\varepsilon_{c},\;\sigma=\sigma_{c}\\ \vartriangle\varepsilon=\varepsilon_{c},\;d\sigma/d\varepsilon=0\end{array} \right. \tag{6}\]  
In the formula, \(\sigma_{c}\) and \(\varepsilon_{c}\) are the peak stress and strain, and \(E\) is the elastic modulus of the undamaged material. Equation (6) can be obtained after derivation:  
\[d\sigma/d\varepsilon=E\exp\left[-(\varepsilon/F_{0})^{m}\right]\left[1-m( \varepsilon/F_{0})^{m}\right] \tag{7}\]  
Conditions 1 and 2 are substituted (5), and conditions 2 and 4 are substituted (7), respectively.