## 4 Discussion
### Constructing Constitutive Model  
The damage variable, as a parameter reflecting the degree of internal damage of rock materials, is closely related to the strain equivalence assumption [29]. Therefore, a damaging constitutive model is constructed based on damage variables and the assumption of strain equivalence [30]:  
\[\sigma=(1-D)E\varepsilon \tag{5}\]  
where \(D\) is the damage variable.  
Figure 5: Relationship between energy and failure mechanism: (**a**) 1 mm/min; (**b**) 3 mm/min; (**c**) 5 mm/min; (**d**) 7 mm/min.  
The damage behavior of the microunits inside the rock is random. In this paper, the Weibull distribution function [31] is introduced to quantitatively describe the damage behavior of internal microunits in rock samples [32]:  
\[f(\epsilon)=\lambda k(\lambda\epsilon)^{k-1}\mathrm{e}^{-(\lambda\epsilon)^{k}} \tag{6}\]  
where \(\lambda\), \(k\) are the parameters of the Weibull distribution.  
Under different strain rates, the greater the loading stress, the more internal failure units there are in the rock sample, and the more severe the damage to the rock sample. Based on this, the damage variable can be expressed as [33]  
\[D=\frac{N}{N_{P}} \tag{7}\]  
where \(N\) is the destruction of microunits; \(N_{P}\) is the total number of microunits.  
In combination with Equations (6) and (7), the destruction of the microunit is further expressed as  
\[N=\int_{0}^{\epsilon}N_{P}f(x)dx \tag{8}\]  
\[N=N_{P}\Big{[}1-\mathrm{e}^{-(\lambda\epsilon)^{k}}\Big{]} \tag{9}\]  
Substitute Equation (9) into Equation (8), and the damage variable is further expressed as  
\[D=1-\mathrm{e}^{-(\lambda\epsilon)^{k}} \tag{10}\]  
Substituting Equation (10) into Equation (5), the damaging constitutive model is further expressed as [28]  
\[\sigma=E\mathrm{e}^{-(\lambda\epsilon)^{k}} \tag{11}\]  
Dissipative strain energy refers to the energy that cannot be recovered during the deformation process of the rock sample. Under different strain rates, damage accumulation will release elastic strain energy and convert it into dissipated strain energy. Meanwhile, an increase in dissipated energy will accelerate the accumulation of damage. Therefore, the damage variable based on cumulative dissipative strain energy is expressed as  
\[D=\frac{U}{U_{P}} \tag{12}\]  
where \(U\) is the cumulative dissipated strain energy at a certain moment; \(U_{P}\) is the cumulative dissipated strain energy at the peak stress.  
The unknown parameters \(k\) and \(\lambda\) in Equation (10) can be determined by fitting the energy damage variable. Introduce the control parameter (\(\omega\)), which is expressed as [28]  
\[W=A\epsilon^{k} \tag{13}\]  
\[\rho=\frac{1}{\epsilon_{n}} \tag{14}\]  
where \(\epsilon_{n}\) is peak strain corresponding to peak stress.  
In combination with Equations (10), (13), and (14), the constitutive model of energy damage is expressed as  
\[\sigma=E\epsilon\cdot\mathrm{e}^{-(\rho\lambda\epsilon)^{k}} \tag{15}\]To accurately characterize the compaction and hardening characteristics of rock samples, [PERSON] et al. (2015) proposed the compaction coefficient. The compaction coefficient (\(\eta\)) is expressed as  
\[\eta=\log_{m_{1}}\left[\frac{(m_{2}-1)\varepsilon}{\varepsilon_{m}}+1\right] \tag{16}\]  
where \(\varepsilon_{m}\) is the yield strain; \(m_{1}\) and \(m_{2}\) are the experimental coefficients.  
Substituting Equation (16) into Equation (15), the constitutive model of energy damage can be obtained:  
\[\sigma=\mu\log_{m_{1}}\left[\frac{(m_{2}-1)\varepsilon}{\varepsilon_{m}}+1 \right]E\varepsilon\cdot\mathrm{e}^{-(\rho\lambda\varepsilon)^{k}} \tag{17}\]  
where \(\mu\) is the correction coefficient.