## 4 Discussion
### Constitutive Relation  
Based on [PERSON]'s effective stress principle, rock mechanical strength theory, and statistical damage theory, the constitutive model of fine sandstone impact dynamics under the coupling of dynamic load and temperature was derived. The assumptions are as follows:  
1. The Nishihara model was used to reflect the elastic-viscoelastic-viscoplastic deformation characteristics of fine sandstone, which is expressed as a composite structure composed of three elements (viscosity mass and damaged body), and its stress process is shown in Figure 10.
2. Micro-elements exhibit linear elastic change in the stress-strain relationship, follow the generalized Hooke's law prior to failure, and have the macroscopic isotropic damage characteristics. 3. The micro-element strength parameters obey [PERSON] probability statistical distribution, and its distribution pattern conforms to the following equation [37; 38]: \[\phi(F)=\frac{m}{F_{0}}(\frac{F}{F_{0}})^{m-1}\exp[-(\frac{F}{F_{0}})^{m}]\] (7) where formula: \(F\) is the distribution variable of fine sandstone strength, and \(m\) and \(F_{0}\) are constants to characterize the materials mechanical properties. 4. The viscous cylinder has no damage characteristics, and the constitutive relationship [39; 40] is: \[\sigma_{b}=\eta d\epsilon_{b}/dt\] (8) In the formula: \(\eta\) is the viscosity parameter of the viscous cylinder. 5. In the constitutive relation modeling, the influence of inertia effect is ignored [41]. The dynamic stress and strain of the damaged body and the viscous cylinder in the combination satisfy the following formulas:  
\[\left. \begin{array}{l}\sigma=\sigma_{s}+\sigma_{i}=\sigma_{a}+\sigma_{b}= \sigma_{c}\\ \epsilon=\epsilon_{a}+\epsilon_{c}\\ \epsilon_{a}=\epsilon_{b}\end{array} \right\} \tag{9}\]  
Figure 10: One-dimensional constitutive model under the coupling action of dynamic load and temperature. According to the strain equivalence principle [42], the constitutive relation of damaged materials can be expressed as:  
\[\sigma=E(1-D)\varepsilon \tag{10}\]  
where \(E\) is the elastic modulus of the fine sandstone when it is lossless, \(D\) is the damage variable, and \(\varepsilon\) is the strain. Drawing from the theories of continuous damage and statistical strength, A novel statistical model was suggested by [PERSON] et al. [43] to characterize the damage of specimens:  
\[\sigma=E\varepsilon(1-\frac{n}{N}) \tag{11}\]  
where \(n\) is the number of micro-units that have been destroyed, \(N\) is the total number of micro-units. Combining Formulas (8) and (9), we can get:  
\[D=\frac{n}{N} \tag{12}\]  
When the fine sandstone is subjected to load, the micro-elements are destroyed, and the number of micro-elements destroyed in any interval \([F,F+dF]\) can be calculated by the probability density function of the strength of the micro-element. That is, when the external load changes from 0 to a certain load \(f\), the number of micro-elements destroyed is: \(N\phi(F)dF\). When the external load changes from 0 to a certain load \(F\), the number of damaged micro-elements is:  
\[n(F)=N\!\int_{0}^{F}\phi(F)dF=N\bigg{\{}1-\exp[-(\frac{F}{F_{0}})^{m}]\bigg{\}} |_{0}^{F} \tag{13}\]  
Substitute Formula (13) into Formula (12):  
\[D=1-\exp[-(\frac{F}{F_{0}})^{m}] \tag{14}\]  
When the dynamic and static load and temperature act together on coal and rock mass, in addition to the dynamic load stress deterioration sample, the temperature rise also increases the instantaneous elastic modulus damage degree. According to the previous research results [44], the instantaneous thermal damage variable \(D_{T}\) can be expressed as:  
\[D_{T}=1-E_{T}/E_{0} \tag{15}\]  
where \(E_{T}\) is the instantaneous elastic modulus of coal and rock at temperature \(T\), and \(E_{0}\) is the instantaneous elastic modulus at normal temperature.