## 2 Characterization of Damage Model and Hydro-Mechanical Properties Model  
### Damage Model in Rock creep Process  
The degree of rock damage is not only related to the stress level, but also to the internal structure of the rock, which continuously deteriorates under the action of time-dependent damage. The stress level includes its stress state, permeability characteristics, chemical corrosion and temperature changes. The internal structure mainly involves the friction of small particles within the rock, the generation and expansion of internal microcracks. Itis necessary to establish a criterion for the evolution of time-dependent damage, which is used to describe the time-dependent damage law in the creep process of rocks. Here, the internal variable of time-dependent damage is introduced to describe the characteristics of time-dependent damage. Internal variable \(\zeta\) is used to describe the time-dependent damage of rocks, which can explain the relative damage changes of micro contact surfaces between solid particles per unit volume unit of rock under stress. It can be seen as a time-dependent function describing the plastic deformation of rocks, which can be expressed as Equation (1):  
\[\zeta(t)=\zeta(\epsilon^{p},t) \tag{1}\]  
where \(\epsilon^{p}=\) plastic deformation and \(t=\) time. It can be seen from the creep characteristics of rock in the creep mechanical property test that with the increase of time, the time-dependent damage variable \(\zeta\) will tend towards a state of equilibrium. Set the time-dependent damage variable function in this state to \(\overline{\zeta}\), which represents an equilibrium state of the internal microstructure of the rock, that is, when time approaches infinity, \(\zeta\) will tend towards \(\overline{\zeta}\). Assuming that the damage evolution of the microstructure inside the rock can be described by the deviation of the damage variable from the equilibrium state, then this feature can be achieved using \((\overline{\zeta}-\zeta)\). A simple linear model can effectively describe the time-dependent damage characteristics of the internal microstructure of rocks over time, as shown in Equation (2). \[\frac{\partial\zeta(t)}{\partial t}=\gamma\big{(}\overline{\zeta}-\zeta\big{)} \tag{2}\]  
In the equation, \(\gamma\) represents the experimental constant that ignores external factors such as temperature changes and chemical corrosion and can be taken as a certain value. The evolution characteristics of time-dependent damage are mainly determined by the equilibrium state function \(\overline{\zeta}\) of time-dependent damage variables, which depends on the magnitude of stress levels and ranges from 0 to 1. The range of values for the time-dependent damage variable \(\zeta\) is between 0 and, when time \(t=0\), \(\zeta=0\); when \(t\) approaches infinity, \(\zeta=\overline{\zeta}\). According to the mathematical characteristics of the time-dependent damage variable, the time-dependent damage characteristics can be represented by Equation (3). \[\overline{\zeta}-\zeta=\overline{\zeta}e^{-\gamma t} \tag{3}\]  
In the equation, when time \(t=0\), both sides are equal to \(\overline{\zeta}\); when \(t\) approaches infinity, \(e^{-\gamma t}\) tends to 0, and both sides of the equation are equal to 0. At the same time, the exponential function represents the memory effect of the previous loading process, which can better express the change characteristics of time-dependent damage at each stage. In the establishment of constitutive models for rock creep damage, it is not only necessary to consider the time-dependent damage characteristics during the creep process, but also the instantaneous damage \(d\) during the instantaneous loading stage. [PERSON] [24] proposed a damage criterion to characterize the transient loading damage evolution of rock.