# New Perspectives on Excavation Disturbance Zones: Main Driving Forces
## 2 Numerical Methodology
### Thermo-Mechanical Coupling Model with Damage Evolution  
In the case of thermal transients, thermodynamic equilibrium in the material domain conforms to [PERSON]'s law [47; 48]. Therefore, the temperature distribution in the rock can be expressed as follows:  
\[\frac{\partial}{\partial x}\bigg{(}x_{x}\frac{\partial T}{\partial x}\bigg{)}+ \frac{\partial}{\partial y}\bigg{(}x_{y}\frac{\partial T}{\partial y}\bigg{)} +Q=\rho c\frac{\partial T}{\partial t} \tag{9}\]  
where \(x\) represents the thermal conductivity, subscripts \(x\) and \(y\) represent the coordinates, \(T\) represents the temperature, \(Q\) is the heat source, and \(\rho\) and \(c\) denote the density and specific heat of the material, respectively. The description of rock damage in the RFPA\({}^{\text{2D}}\)-thermal code is the same as that in the RFPA\({}^{\text{2D}}\)-humid code, i.e., the damage variable can be expressed using Equation (4) or Equation (5). Heat conduction forms a temperature stress field. The external load causes rock damage and changes the mechanical properties of the rock, thereby influencing heat conduction. Coupling can be observed among the temperature, stress, and damage, which can be mainly described with respect to the thermal conductivity, damage variable, and temperature. Furthermore, a correlation can be observed between these three parameters, which can be expressed in different ways depending on the type of rock damage [36]. When tensile damage occurs in a rock, the relationship between the three parameters is expressed as Equation (10); conversely, when shear damage occurs in a rock, the relationship between them is expressed as in Equation (11). \[\kappa= \begin{cases}\kappa_{0}&d=0\\ \frac{\kappa_{0}}{0.99+T(a-b/\kappa_{0})}&0<d<1\\ \xi_{1}\,\lambda_{0}&d=1\end{cases}  \tag{10}\]  
Figure 2: Comparisons of the deformation of the surrounding rock between field measurements and numerical simulations under a high-humidity environment [8]. \[\kappa=\left\{ \begin{array}{ll}\kappa_{0}&d=0\\ \frac{\kappa_{0}}{0.99+T(a-b/\kappa_{0})}&d>0\end{array} \right. \tag{11}\]  
where \(\kappa_{0}\) is the initial thermal conductivity; and \(a\) and \(b\) are empirical parameters, such as the experimental results of several lithologies such as crystalline rock and sedimentary rock, as reported in the literature [49]. Moreover, \(\xi_{1}\) (\(\xi_{1}<1\)) represents the thermal conductivity mutation coefficient. The influence of heat on a rock mass is similar to that of humidity on a rock mass; therefore, the constitutive law of rock mass with thermal effect can be expressed as follows [36]:  
\[\sigma_{ij}=D_{ijkl}\epsilon_{kl}-(3\lambda+2\mu)\beta\delta_{ij}T \tag{12}\]  
where \(D_{ijkl}\) is the elasticity tensor, \(\beta\) is the thermal expansion coefficient, and \(\delta_{ij}\) is the Kronecker constant. The variation in the specific heat of an element in the model can be expressed as follows [36]:  
\[c(T)=c_{0}(1+\psi T) \tag{13}\]  
where \(c_{0}\) is the specific heat at \(0\)\({}^{\circ}\)C and \(\psi\) is the impact factor, which is generally \(3\times 10^{-3}/^{\circ}\)C.  
The expressions above constitute the thermo-mechanical coupling model with damage evolution, which is the core part of the RFPA2D-thermal code. To verify the accuracy of the RFPA2D-thermal code, [PERSON] et al. compared the results of theoretical analysis with those of numerical calculation [37]. First, as shown in Figure 3, a calculation model consisting of two concentric circles was established. In the model, a and b represent the radii of the embedded particles and matrix, respectively, and g and m represent the embedded particles and matrix, respectively. It should be noted that the application of an initial temperature to the model would lead to thermal stress due to the different thermal expansion coefficients of the internal and external media, which would then induce continuous changes in the thermal stress of the model due to uniform heating.