## 4 Development of a Damage Constitutive Model for Argillaceous Silitstone
### Determination of Damage Variable  
The effect of water temperature compromises the internal structure of argillaceous siltstone, which is macroscopically reflected as decreases in peak stress, elastic modulus, and related parameters. The elastic modulus, as a representation of the rock's deformation capacity, more effectively captures the extent of damage in argillaceous siltstone. Consequently, the damage variable \(D_{1}\) induced by water temperature can be defined using the elastic modulus as follows:  
\[D_{1}=1-\frac{E_{i}}{E_{0}},(i=1,2,3\dots) \tag{13}\]  
In this formulation, \(E_{i}\) represents the elastic modulus of argillaceous siltstone after exposure to different water temperatures, while \(E_{0}\) indicates the elastic modulus in its original state.  
From the perspectives of statistical damage mechanics and continuum mechanics, rock damage is viewed as a continuous process, with the microelemental damage of rocks following a statistical distribution. Current research frequently models this microelemental damage using a Weibull distribution. [PERSON] et al. suggest that, under external loads, damage only initiates when these loads reach a specific threshold, a stress-strain state known as the damage threshold [25]. This threshold is further proposed to generally correspond to the yield point. Assuming that the damage to argillaceous siltstone under external loading is a function of axial strain \(\varepsilon_{1}\), with a load damage threshold \(\varepsilon_{1d}\), a three-parameter Weibulldistribution is utilized to define the density function for damage beyond the threshold under external load [26]:  
\[\phi(\varepsilon)=\frac{m}{\lambda}(\frac{\varepsilon_{1}-\varepsilon_{1\text{d }}}{\lambda})^{m-1}\text{exp}[-(\frac{\varepsilon_{1}-\varepsilon_{1\text{d }}}{\lambda})^{m}]\ \ (\varepsilon_{1}\geq\varepsilon_{1\text{d}}) \tag{14}\]  
In this expression, \(m\) and \(\lambda\) denote the shape and scale parameters of the Weibull distribution for the microelements of argillaceous siltstone, respectively. Here, \(m\) characterizes the heterogeneity of the material, while \(\lambda\) reflects the rock's plasticity characteristics to some extent. The axial strain \(\varepsilon_{1\text{d}}\) corresponds to the yield point. By integrating this density function, the load-induced damage variable can be obtained.  
The expression for \(D_{2}\) as a function of axial strain \(\varepsilon_{1}\) is as follows:  
\[D_{2}=\begin{cases}0\ \ \ \ \varepsilon_{1}<\varepsilon_{1\text{d}}\\ 1-\text{exp}\Big{[}-\Big{(}\frac{\varepsilon_{1}-\varepsilon_{1\text{d}}}{ \lambda}\Big{)}^{m}\Big{]}\ \ \varepsilon_{1}>\varepsilon_{1\text{d}}\end{cases} \tag{15}\]  
The damage of argillaceous siltstone under the combined effects of water temperature and external load is not simply an additive process; rather, it results from a coupling effect. By substituting Equations (13) and (15) into Equation (10), the expression for the total damage variable \(D_{2}\) of argillaceous siltstone is obtained.  
\[D_{12}=\begin{cases}\ \ \ \ \ \ \ 1-\frac{\varepsilon_{i}}{E_{0}}\ \varepsilon_{1}< \varepsilon_{1\text{d}}\\ 1-\frac{\varepsilon_{i}}{E_{0}}\ \text{exp}\Big{[}-\Big{(}\frac{\varepsilon_{1}- \varepsilon_{1\text{d}}}{\lambda}\Big{)}^{m}\Big{]}\ \varepsilon_{1}>\varepsilon_{1\text{d}}\end{cases} \tag{16}\]