## 2 New Extended Semi-Weight Function Method for Calculating SIFs under Hydro-Mechanical Coupling
### Derivation of New Stress Function for Hydro-Mechanical Coupling Cracks  
Figure 1 shows an infinite plate with a crack under a uniform hydraulic pressure \(P_{h}\) on the crack surface, where a rectangular coordinate (_xoy_) and a polar coordinate (_ro_; anti-clockwise \(\theta\) is defined as positive) are set at the right tip of the crack with the \(X\)-axis along the direction of the original crack surface.  
According to Williams expansion, the common stress fields at the crack tip of a polar coordinate system are expressed as follows:  
\[\begin{array}{l}\sigma_{r}=-\lambda r^{\lambda-1}[(\lambda+1)(A\sin(\lambda+ 1)\theta+B\cos(\lambda+1)\theta)+(\lambda-3)(Cin(\lambda-1)\theta+D\cos( \lambda-1)\theta)]\\ \sigma_{\theta}=\lambda(\lambda+1)r^{\lambda-1}[A\sin(\lambda+1)\theta+B\cos( \lambda+1)\theta+C\sin(\lambda-1)\theta+D\cos(\lambda-1)\theta]\\ \tau_{r\theta}=-\lambda r^{\lambda-1}[(\lambda+1)(A\cos(\lambda+1)\theta-B \sin(\lambda+1)\theta)+(\lambda-1)(C\cos(\lambda-1)\theta-D\sin(\lambda-1) \theta)]\\ \end{array} \tag{1}\]  
where \(A\), \(B\), \(C\), and \(D\) are undetermined weight functions, and \(\lambda\) is the crack-tip singularity index.  
For single material cracks, the singularity index is usually taken as \(\lambda=0.5\). In this case, by substituting \(\theta=\pi\) and \(\theta=-\pi\) into the second term of Equation (1), the following can be obtained:  
\[\begin{array}{l}\sigma_{\theta}(\theta=\pi)=\frac{3}{4\sqrt{\
u}}(-A-C)\\ \sigma_{\theta}(\theta=-\pi)=\frac{3}{4\sqrt{\
u}}(A+C)\end{array} \tag{2}\]  
Figure 1: Calculation model of hydro-mechanical coupling stress function.  
However, the boundary condition \(\sigma_{\theta}(\theta=\pi)=\sigma_{\theta}(\theta=-\pi)=P_{h}\) is required under hydraulic pressure \(P_{h}\). Obviously, the crack-tip stress function under hydro-mechanical coupling needs to be modified due to the fact that Equation (2) cannot satisfy this boundary condition.  
Assuming a new stress function \(F(r,\,\theta,\,P_{h})\), its relationship with the stress field and displacement field needs to be satisfied:  
\[\left\{\begin{array}{l}\sigma_{r}=\frac{1}{r^{2}}\frac{\partial^{2}F(r,\theta,P_{h})}{\partial\theta^{2}}+\frac{1}{r}\frac{\partial F(r,\theta,P_{h})}{ \partial r}\\ \sigma_{\theta}=\frac{\partial^{2}F(r,\theta,P_{h})}{\partial r^{2}}\\ \tau_{r\theta}=-\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial F (r,\theta,P_{h})}{\partial\theta}\right)\end{array}\right. \tag{3}\]  
\[\left\{\begin{array}{l}2 Gu_{r}=-\frac{\partial F(r,\theta,P_{h})}{\partial r }+\left(1-\frac{v}{1+v}\right)r\frac{\partial\Phi_{1}}{\partial\theta}\\ 2 Gu_{\theta}=-\frac{1}{r}\frac{\partial F(r,\theta,P_{h})}{\partial\theta}+ \left(1-\frac{v}{1+v}\right)r^{2}\frac{\partial\Phi_{1}}{\partial r}\end{array}\right. \tag{4}\]  
where \(G\) is the shearing modulus, \(v\) is [PERSON]'s ratio, and the relationship between \(F(r,\,\theta,\,P_{h})\) and \(\psi_{1}\) can be defined as follows:  
\[\
abla^{2}F(r,\theta,P_{h})=\frac{\partial}{\partial}\left(r\frac{\partial \psi_{1}}{\partial\theta}\right) \tag{5}\]  
The stress function \(F(r,\,\theta,\,P_{h})\) must satisfy the biharmonic equation:  
\[\
abla^{2}\left(\
abla^{2}F(r,\theta,P_{h})\right)=0 \tag{6}\]  
To make the stress field in Equation (3) satisfy the hydro-mechanical coupling crack boundary condition,  
\[\begin{array}{l}\sigma_{\theta}(\theta=\pi)=P_{h},\,\,\,\sigma_{\theta}( \theta=-\pi)=P_{h}\\ \tau_{r\theta}(\theta=\pi)=0,\,\,\,\,\tau_{r\theta}(\theta=-\pi)=0\end{array} \tag{7}\]  
The expression for a new hydro-mechanical coupling crack tip \(F(r,\,\theta,\,P_{h})\) can be obtained by substituting Equation (7) into Equations (3) and (4) and integrating to solve, as follows:  
\[F(r,\theta,P_{h})=r^{\lambda+1}\bigg{[}A\sin(\lambda+1)\theta+B\cos(\lambda+ 1)\theta+C\sin(\lambda-1)\theta+D\cos(\lambda-1)\theta+r^{-\lambda+1}\frac{P_ {h}}{2}\bigg{]} \tag{8}\]