# Numerical Studies on Rockbursts in Tunnels with High In Situ Stresses and Geothermally Rich Areas
## 2 Theoretical Studies on Failure Caused by Decreasing Temperature  
After a tunnel is excavated in a geothermally rich area, the temperature of the tunnel surface will decrease over time according to the ventilation of the tunnel, and then the thermal gradient stress will be generated gradually. The heat conduction equation in the surrounding rock is shown in Equation (1). The adiabatic boundary condition is shown in Equation (2). The boundary condition for convective heat transfer on the tunnel surface is shown in Equation (3) [18]. \[\rho c\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\bigg{(}k_{x} \frac{\partial T}{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(}k_{y} \frac{\partial T}{\partial y}\bigg{)}+\frac{\partial}{\partial z}\bigg{(}k_{z} \frac{\partial T}{\partial z}\bigg{)}+Q \tag{1}\]  
Figure 1: Plate distribution along the Sichuan–Tibet railway. \[-k_{n}\frac{\partial T}{\partial n} =0 \tag{2}\] \[k_{n}\frac{\partial T}{\partial n} =h\left(T_{s}-T\right) \tag{3}\]  
where \(k_{x},k_{y},\text{ and }k_{z}\) are the thermal conductivities in the \(x,y,\) and \(z\) directions, respectively, \(\rho\) is the density, \(C\) is the specific heat, \(Q\) is the heat generation rate, \(h\) is the convective heat transfer coefficient, \(T_{s}\) is the fluid temperature, and \(T\) is the temperature field function. The constitutive equations of the surrounding rock are as follows:  
\[\sigma_{1} =\lambda e+2G\epsilon_{1}-\beta\Delta T \tag{4}\] \[\sigma_{2} =\lambda e+2G\epsilon_{2}-\beta\Delta T\] (5) \[\sigma_{3} =\lambda e+2G\epsilon_{3}-\beta\Delta T \tag{6}\]  
where \(\sigma_{1},\)\(\sigma_{2}\) and \(\sigma_{3}\) are the three effective principal stresses; \(\epsilon_{1},\)\(\epsilon_{2}\) and \(\epsilon_{3}\) are the three principal strains; \(\lambda\) is the Lame constant; \(G\) is the shear modulus; and \(E\) is the elastic modulus; and \(\mu\) is Poisson's ratio. \(\beta=(3\lambda+2G)\)\(\alpha\), where \(\alpha\) is the coefficient of thermal expansion. \(\Delta T\) is the change in temperature, which is written as \(\Delta T=T-T_{0}\) (\(T_{0}\) is the initial temperature, which may be positive or negative). The Mohr circle equation at a point can be written as follows:  
\[\left(\sigma_{\alpha}-\frac{\sigma_{1}+\sigma_{3}}{2}\right)^{2}+\tau_{\alpha }{}^{2}=(\frac{\sigma_{1}-\sigma_{3}}{2})^{2} \tag{7}\]  
where \(\sigma_{\alpha}\) and \(\tau_{\alpha}\) are the normal and shear stresses on the failure plane, respectively. Once the highly geothermal surrounding rock is excavated and starts to cool, the effective principal stresses \(\sigma_{1},\)\(\sigma_{2}\) and \(\sigma_{3}\) can be written as follows:  
\[\sigma_{1} =\sigma_{1}^{\prime}-\beta\Delta T \tag{8}\] \[\sigma_{2} =\sigma_{2}^{\prime}-\beta\Delta T\] (9) Figure 1:0  
where \(\sigma_{1}^{\prime},\)\(\sigma_{2}^{\prime},\) and \(\sigma_{3}^{\prime}\) are the three principal stresses before cooling. Substituting Equations (8) and (10) into Equation (7), as the following can be obtained:  
Figure 1:1  
The Mohr-Coulomb criterion is used to judge whether rock failure occurs [19]. Figure 2 shows how the Mohr circle moves after cooling (only \(\Delta T\) is variable) in Equation (11). This indicates that in cooling (\(t_{0}\to t_{1}\)), the Mohr circle moves and approaches the Mohr-Coulomb failure curve until it intersects with the curve, which is the point of rock failure. Meanwhile, when the deviatoric stress (\(\sigma_{1}-\sigma_{3}\)) is greater, the Mohr circle intersects the Mohr-Coulomb failure curve in the compressive area after cooling, and shear failure occurs in the rock (see Figure 2a). Similarly, when the deviatoric stress (\(\sigma_{1}-\sigma_{3}\)) is smaller, the Mohr circle intersects the Mohr-Coulomb failure curve in the tensile area after cooling, and tensile failure occurs in the rock (see Figure 2b). The results show that the forms of rock failure caused by cooling are different under different deviatoric stresses.