## 4 Discussion
### Constitutive Relation
Under the action of temperature and load, the total damage of rock [45] can be expressed as:  
\[D_{s}=D+D_{T}-DD_{T} \tag{16}\]  
Substitute Formulas (14) and (15) into (16) to get the expression of total damage variable:  
\[D_{s}=1-\frac{E_{T}}{E}\exp[-(\frac{F}{F_{0}})^{m}] \tag{17}\]  
Substituting Formulas (8), (10), and (17) into Formula (9), the constitutive relation [46] of the assembly can be obtained as follows:  
\[\eta\dot{\sigma}+[E_{1}(1-D_{s})+E_{2}(1-D_{s})]\sigma=E_{2}(1-D_{s})\cdot[ \eta\dot{\varepsilon}+E_{1}(1-D_{s})\varepsilon] \tag{18}\]Without considering the damage characteristics, replace the effective elastic modulus \(E\left(1-D_{s}\right)\) with the elastic modulus \(E\) of the damaged body before the damage, and solve the Formula (18):  
\[\sigma=E(1-D)\varepsilon \tag{10}\]0  
When the dynamic load moment t = 0, the sample is subjected to static axial load, that is, \(\sigma_{0}=\sigma_{s}\), \(\epsilon(0)=\epsilon_{0}\). According to Laplace transform principle:  
\[\sigma=E(1-D)\varepsilon \tag{10}\]1  
Substitute Formula (16) into (20) to get:  
\[ \begin{array}{l}\sigma(t)=\frac{E_{T}}{E}\exp[-(\frac{F}{F_{0}})^{m}]( \epsilon_{0}+Ct)E_{2}-\frac{E_{T}}{E}\exp[-(\frac{F}{F_{0}})^{m}]\frac{E_{2}}{ E_{1}+E_{2}}(E_{2}\eta C-\\ E_{2}\epsilon_{0})e^{-\frac{E_{T}}{E}\exp[-(\frac{F}{F_{0}})^{m}](E_{1}+E_{2} )}{\eta}t-\frac{E_{2}}{E_{1}+E_{2}}(Ct-\frac{E_{2}\eta C}{E_{1}+E_{2}}+\frac{ E_{T}}{E}\exp[-(\frac{F}{F_{0}})^{m}]E_{2}\epsilon_{0})\end{array}  \tag{21}\]  
A one-dimensional constitutive equation of fine sandstone in a one-dimensional stress state under the combined influence of temperature and dynamic load is found in Formula (21). When applying the above model, six parameters of \(E_{T}\), \(E_{1}\), \(E_{2}\), \(F_{0}\), and \(m\) need to be determined. The basis for determining the parameters is as follows: According to the previous research results, the static elastic modulus of the coal rock mass can be used to replace the constitutive model's \(E_{1}\), and the value of \(E_{2}\) is comparable to the linear elastic modulus of the coal rock mass's stress-strain curve under various levels of strain. It can be replaced by the dynamic elastic modulus, and the variation range of viscosity coefficient \(\eta\) is generally 0-1000 GPa-s. The strain \(\varepsilon_{r}(t)\), the average strain rate \(C\) and the time \(t_{0}\) when the static load begins to load are measured data. Based on the above model, simulation analysis was conducted on the dynamic stress process of fine sandstone under different temperature conditions. The simulation results are shown in Figure 11.  
Figure 11: _Cont._  
From Table 4, the static elastic modulus of fine sandstone varies from 9.12 to 9.86 GPa, the dynamic elastic modulus varies from 15.95 to 26.77 GPa, and \(F_{0}\) varies from 67.89 to 143.60. The established theoretical model has a good response to the stress-strain curve (before the rebound stage) of fine sandstone and coal samples during the impact process, and cannot explain the stress mutation and post peak rebound phenomenon of damage to fine sandstone and coal samples. This is also a common problem in constructing a one-dimensional coal rock damage impact dynamic model, and some scholars have explored the stress rebound segment using segmented simulation analysis methods.