## 2 New Extended Semi-Weight Function Method for Calculating SIFs under Hydro-Mechanical Coupling
### Calculation of Hydro-Mechanical Coupling Stress-Intensity Factors  
To calculate hydro-mechanical coupling SIFs (Figure 2) using the modified semi-weight function method, an arbitrary closed region \(\Omega\) for the reciprocal theorem should be selected that contains the crack surface (\(\Phi_{1}\) and \(\Phi_{2}\)) and internal and external circles \(\Gamma_{1}\) and \(\Gamma_{2}\) (with the same center of coordinate origin and radii \(r_{1}\) and \(r_{2}\)). Since the existing modified semi-weight function method is applicable mainly to free surface conditions (\(p=0\) and \(p^{\prime}=0\) on the integration paths \(\Phi_{1}\) and \(\Phi_{2}\)), the reciprocal theorem for the work integral formula can be simplified as follows:  
\[\int\limits_{\Gamma_{1}}p^{\prime}_{i}u_{i}-p_{i}u^{\prime}_{i}\mathrm{d}s= \int\limits_{\Gamma_{2}}p_{i}u^{\prime}_{i}-p^{\prime}_{i}u_{i}\mathrm{d}s \tag{21}\]  
where \(p\) and \(u\) and \(p^{\prime}\) and \(u^{\prime}\) are the stress and displacement expressed by the weight function (without superscript \"'\") and the semi-weight function (with superscript \"'\"), respectively. Figure 2: Integral path. It can be seen that Equation (21) does not adhere to the case of hydro-mechanical coupling loading cracks (\(p\
eq 0\), \(p^{\prime}\
eq 0\) and the path integrals on \(\Phi_{1}\) and \(\Phi_{2}\) cannot be neglected). Therefore, Equation (21) needs to be extended and expressed in a polar coordinate form, which can be obtained as follows:  
\[ \begin{array}{l}\int\limits_{\Gamma_{1}}\big{(}\sigma^{\prime}_{r}u_{r}+ \tau^{\prime}_{r\theta}u_{\theta}-\sigma_{r}u^{\prime}_{r}-\tau_{r\theta}u^{ \prime}_{\theta}\big{)}\mathrm{d}s+\int\limits_{\Phi_{1}+\Phi_{2}}\big{(} \sigma^{\prime}_{\theta}u_{\theta}+\tau^{\prime}_{r\theta}u_{r}-\sigma_{ \theta}u^{\prime}_{\theta}-\tau_{r\theta}u^{\prime}_{r}\big{)}\mathrm{d}s\\ =\int\limits_{\Gamma_{2}}\big{(}\sigma^{\prime}_{r}u_{r}+\tau^{\prime}_{r \theta}u_{\theta}-\sigma_{r}u^{\prime}_{r}-\tau_{r\theta}u^{\prime}_{\theta} \big{)}\mathrm{d}s\end{array}  \tag{22}\]  
For the left side of Equation (22) (i.e., integral path \(\Gamma_{1}\), \(\Phi_{1}\), and \(\Phi_{2}\)), by substituting \(\sigma^{\prime}_{r}\), \(\sigma^{\prime}_{\theta}\), \(\tau^{\prime}_{r\theta}\), \(u^{\prime}_{r}\), and \(u^{\prime}_{\theta}\), represented by the semi-weight function, and \(\sigma_{r}\), \(\sigma_{\theta}\), \(\tau_{r\theta}\), \(u_{r}\), and \(u_{\theta}\), represented by the weight function, the following can be obtained:  
\[ \begin{array}{l}\int\limits_{\Gamma_{1}}\big{(}\sigma^{\prime}_{r}u_{r}+ \tau^{\prime}_{r\theta}u_{\theta}-\sigma_{r}u^{\prime}_{r}-\tau_{r\theta}u^{ \prime}_{\theta}\big{)}\mathrm{d}s+\int\limits_{\Phi_{1}+\Phi_{2}}\big{(} \sigma^{\prime}_{\theta}u_{\theta}+\tau^{\prime}_{r\theta}u_{r}-\sigma_{ \theta}u^{\prime}_{\theta}-\tau_{r\theta}u^{\prime}_{r}\big{)}\mathrm{d}s\\ =\delta_{1}Q_{1}K_{1}+\delta_{2}Q_{2}K_{2}+\delta_{3}K_{1}P_{H}\tau_{2}^{1.5}+ \delta_{4}Q_{1}P_{H}\tau_{2}^{0.5}\end{array}  \tag{23}\]  
where \(\delta_{i}\) (\(i=1,2,3,4\)) represents the integration coefficients related to the material parameter (\(G\), \(\kappa\)); \(P_{h}\) is the hydraulic pressure of the crack surface; and \(r_{2}\) is the radius of the external integrate path \(\Gamma_{2}\). The calculation of the undetermined SIFs can be simplified by arbitrarily selecting two groups with different \(Q_{1}\) and \(Q_{2}\):  
\[\left\{ \begin{array}{l}Q_{1}=\frac{1}{s_{1}}\\ Q_{2}=0\end{array} \right.\text{ and }\left\{ \begin{array}{l}Q_{1}=0\\ Q_{2}=\frac{1}{s_{2}}\end{array} \right.