## Appendix A Solving Steps of the Interactions in Section 4.3  
**Step (a):**  
For the first iteration process, \(\varphi(z)\) and \(\psi(z)\) can be obtained by the Cauchy integral method. The calculation steps are as follows:  
When \(\sigma_{\rho}=P_{1}\), \(\sigma_{\theta}=\tau_{\rho\theta}=0\), the solutions are indicated by Equation (A1). \[\left\{ \begin{array}{l}\varphi_{1}(z)=\frac{P_{1}}{2}\Big{(}\frac{z}{2}+ \frac{R^{2}}{z}\Big{)}\\ \\ \psi_{1}(z)=-\frac{P_{1}}{2}\Big{(}z+\frac{R^{2}}{z}-\frac{R^{4}}{z^{3}}\Big{)} \end{array} \right.\] (A1)  
When \(\sigma_{\theta}=P_{2}\), \(\sigma_{\rho}=\tau_{\rho\theta}=0\), the solutions are indicated by Equation (A2). \[\left\{ \begin{array}{l}\varphi_{2}(z)=\frac{P_{2}}{2}\Big{(}\frac{z}{2}- \frac{R^{2}}{z}\Big{)}\\ \\ \psi_{2}(z)=\frac{P_{2}}{2}\Big{(}z-\frac{R^{2}}{z}-\frac{R^{4}}{z^{3}}\Big{)} \end{array} \right.\] (A2)  
Therefore, the solution for \(\sigma_{\rho}=P_{1}\), \(\sigma_{\theta}=P_{2}\), \(\tau_{\rho\theta}=0\) can be solved by superimposing Equations (A1) and (A2), which is the functional solution of \(\varphi_{1}^{(1)}(z_{1})\) and \(\psi_{1}^{(1)}(z_{1})\), which are given by  
\[\left\{ \begin{array}{l}\varphi_{1}^{(1)}(z_{1})=\frac{P_{A}z_{1}}{2}-\frac{ P_{B}R_{1}^{2}}{z_{1}}\\ \\ \psi_{1}^{(1)}(z_{1})=P_{B}z_{1}-\frac{P_{A}R_{1}^{2}}{z_{1}}-\frac{P_{B}R_{1}^ {4}}{z_{1}^{3}}\end{array} \right.\] (A3)  
where \(P_{A}\) and \(P_{B}\) are related to \(P_{1}\) and \(P_{2}\), \(P_{A}=(P_{1}+P_{2})/2\), \(P_{B}=(P_{2}-P_{1})/2\), respectively. **Step (b):**  
According to the coordinate transformation formula, the valuable \(z_{1}\) is converted to \(z_{2}\) to obtain the function of the complex stresses \(\varphi_{2}^{(1)}(z_{2})\) and \(\psi_{2}^{(1)}(z_{2})\) around the FT-I section when the AT section is excavated in the coordinate system \(o_{2}x_{2}y_{2}\). \[\left\{ \begin{array}{l}\varphi_{2}^{(1)}(z_{2})=\frac{P_{A}(z_{2}+c)}{2}- \frac{P_{B}R_{1}^{2}}{z_{2}+c}\\ \\ \psi_{2}^{(1)}(z_{2})=P_{B}(z_{2}+c)-\frac{P_{A}R_{1}^{2}}{z_{2}+c}-\frac{P_{B} R_{1}^{4}}{(z_{2}+c)^{3}}\end{array} \right.\] (A4)  
**Step (c):**  
In addition, the excess surface force \(f_{2}^{(1)}\) generated around the FT-I section can be obtained, and follows the form indicated in Equation (A5). \[f_{2}^{(1)}=\varphi_{2}^{(1)}(t_{2})+t_{2}\overline{\varphi_{2}^{(1)}\prime (t_{2})}+\overline{\psi_{2}^{(1)}(t_{2})}\] (A5)  
where \(t_{2}\) represents the point on circle 2. Apply the Cauchy operator to the boundary conditions Equation (A6), as follows:  
\[\frac{1}{2\pi i}{\int_{\rm O1}}\frac{f_{2}^{(1)}}{t_{2}-z_{2}}dt_{2}=\frac{1} {2\pi i}{\int_{\rm O1}}\frac{\varphi_{2}^{(1)}(t_{2})}{t_{2}-z_{2}}dt_{2}+ \frac{1}{2\pi i}{\int_{\rm O1}}\frac{\overline{\varphi_{2}^{(1)}\prime(t_{2})} }{t_{2}-z_{2}}dt_{2}+\frac{1}{2\pi i}{\int_{\rm O1}}\frac{\overline{\psi_{2}^ {(1)}(t_{2})}}{t_{2}-z_{2}}dt_{2}\] (A6)From the Cauchy integral formula, the analytical solution of circle 2, \(\varphi_{2}^{(1)}(z_{2})\), can be obtained by Equation (A7). \[\frac{1}{2\pi i}\int_{\rm O1}\frac{\overline{\psi_{2}^{(1)}(t_{2})}}{t_{2}-z_{2} }dt_{2}=-\varphi_{2}^{(1)}(z_{2})\] (A7)  
To balance the surface force around the FT-I section, the surface force with opposite direction, \(-f_{2}^{(1)}\), needs to be added. Under the balance external forces, the complex stress functions \(\varphi_{2}^{(2)}(z_{2})\) and \(\psi_{2}^{(2)}(z_{2})\) are indicated by Equation (A8), as follows:  
\[\left\{