# Stress Analysis and Spalling Failure Simulation on Surrounding Rock of Deep Arch Tunnel
## 2 Stress Analysis of Surrounding Rock  
### Complex Functions for Stress Distribution  
For simplicity, the deep subsurface rock mass is assumed as a homogeneous, isotropic, and linearly elastic material. For the burial depths much greater than the cross-sectional dimensions of openings, rock mass can be considered as an infinite plane containing holes (z-plane) under plane strain conditions. According to [PERSON]'s theory [59], the stress component on the z-plane (\(\sigma_{z},\sigma_{y},\tau_{zy}\)) in the right-angle coordinate system can be expressed as follows:  
\[\left\{ \begin{aligned} \sigma_{z}+\sigma_{y}&=4 \text{Re}[\sigma_{1}^{\prime}(z)]\\ \sigma_{z}-\sigma_{y}+2i\tau_{zy}&=2[\varpi_{1}^{ \prime}(z)+\psi_{1}^{\prime}(z)]\end{aligned} \right. \tag{1}\]  
where \(\varphi_{1}(z)\) and \(\psi_{1}(z)\) are two complex stress functions of the complex variable z. The stress component (\(\sigma_{\rho},\ \sigma_{\theta},\ \tau_{r\theta}\)) in polar coordinates can be obtained by [25]  
\[\left\{ \begin{aligned} \sigma_{\rho}+\sigma_{\theta}& =\sigma_{z}+\sigma_{y}\\ \sigma_{\rho}-\sigma_{\theta}+2i\tau_{r\theta}&=( \sigma_{z}-\sigma_{y}+2i\tau_{zy})e^{2i\theta}\end{aligned} \right. \tag{2}\]  
where \(\sigma_{\theta}\sigma_{\rho}\), and \(\tau_{r\theta}\) are the tangential stress, radial stress, and shear stress of surrounding rock, respectively. According to the Riemann mapping theorem, any single connected domain with multiple boundary points in the z-plane can usually be mapped to a unit circle on the complex plane via a mapping function. Substituting Equation (2) into Equation (1), the stress component can be expressed as follows:  
\[\left\{ \begin{aligned} \sigma_{\rho}+\sigma_{\theta}& =4\text{Re}[\Phi(\zeta)]\\ \sigma_{\theta}-\sigma_{\rho}+2i\tau_{r\theta}&=\frac{2 \zeta^{2}}{\rho^{2}\omega^{\prime}(\xi)}[\varpi(\zeta)\Phi^{\prime}(\zeta)+ \omega^{\prime}(\zeta)\Psi(\zeta)]\end{aligned} \right. \tag{3}\]  
where \(\zeta\) is the coordinate of a given point on the boundary of the complex plane mapping, Re is the real part of the complex number, \(i\) is the imaginary unit, \(\omega(\zeta)\) is the mapping function, and \(\Phi(\zeta)\), and \(\Psi(\zeta)\) are two complex potential functions, which can be expressed by the following:  
\[\Phi(\zeta)=\varphi_{1}^{\prime}(z)=\frac{\phi^{\prime}(\zeta)}{\omega^{\prime }(\zeta)},\ \Psi(\zeta)=\psi_{1}^{\prime}(z)=\frac{\psi^{\prime}(\zeta)}{\omega^{\prime}( \zeta)} \tag{4}\]  
The two complex stress functions \(\varphi(\zeta)\) and \(\psi(\zeta)\) in the \(\zeta\)-plane can be expressed as [60]  
\[\left\{ \begin{aligned} \varphi(\zeta)&=B\omega(\zeta)+ \varphi_{0}(\zeta)\\ \psi(\zeta)&=(B^{\prime}+iC^{\prime})\omega(\zeta)+ \psi_{0}(\zeta)\end{aligned} \right. \tag{5}\]The constants \(B\), \(B^{\prime}\), \(C^{\prime}\) reflect the stress conditions of far-field surrounding rock and can be derived using the following equation:  
\[B=\frac{\sigma_{y}^{\infty}+\sigma_{z}^{\infty}}{4},\ B^{\prime}=\frac{\sigma_{ z}^{\infty}-\sigma_{y}^{\infty}}{2},\ C^{\prime}=\tau_{zy}^{\infty} \tag{6}\]  
Let \(\sigma_{z}^{\infty}=p\), the coefficient of lateral pressure \(\lambda=\frac{\sigma_{y}^{\infty}}{\sigma_{z}^{\infty}}\), \(\tau_{zy}^{\infty}=0\).