## 2 Method of Calculation
### Stress Field
#### 2.1.3 Continuity of Boundary Displacements  
To solve for the coefficient \(x\) in Equations (3) and (8), it is necessary to solve for the displacements of the circular perforated plate and the circular body separately so that their radial displacements at the junction are equal. The first type of problem is analyzed.  
The stress component is initially incorporated into the physical equation in order to derive an expression for the strain component. The physical equations in elasticity are shown below:  
\[\left\{\begin{array}{l}\varepsilon_{\rho}=\frac{1}{E}(\sigma_{\rho}-u\sigma_{ \theta})\\ \varepsilon_{\theta}=\frac{1}{E}(\sigma_{\theta}-u\sigma_{\rho})\\ \gamma_{\rho\theta}=\frac{2(1+\
u)}{E}\tau_{\rho\theta}\end{array}\right. \tag{9}\]  
The differential expression for displacement can be obtained by substituting the strain component obtained from Equation (9) into the geometric equations. The geometric equations in elasticity are shown below:  
\[\left\{\begin{array}{l}\varepsilon_{\rho}=\frac{\partial u_{\rho}}{\partial\rho} \\ \varepsilon_{\theta}=\frac{u_{\rho}}{\rho}+\frac{1}{\rho}\frac{\partial u_{ \theta}}{\partial\theta}\\ \gamma_{\rho\theta}=\frac{1}{\rho}\frac{\partial u_{\rho}}{\partial\theta}+ \frac{\partial u_{\theta}}{\partial\rho}-\frac{u_{\theta}}{\rho}\end{array}\right. \tag{10}\]  
An expression for the displacement of the circular orifice plate can be obtained by integrating the first two terms of Equation (10).  
\[\left\{\begin{array}{l}u_{\rho 1}=\frac{(1+\lambda)}{2E_{1}}q\Big{[}-\frac{R^{2} }{\rho^{2}}(1-x)(1+v_{1})-\rho(1-v_{1})\Big{]}+f_{1}(\theta)\\ u_{\theta 1}=-\widetilde{f}_{1}(\theta)+f_{1}(\rho)\end{array}\right. \tag{11}\]  
Inserting Equation (11) into the third equation of Equation (10) yields the new equation.  
\[{f^{\prime}}_{1}(\theta)+\widetilde{f}_{1}(\theta)+\rho{f^{\prime}}_{1}(\rho) -f_{1}(\rho)=0 \tag{12}\]  
Separating the variables gives the following equation:  
\[\left\{\begin{array}{l}f_{1}(\theta)=a_{1}\sin\theta+b_{1}\cos\theta\\ f_{1}(\rho)=c_{1}\rho\end{array}\right. \tag{13}\]  
By substituting Equation (13) into Equation (11) and simplifying, an expression for the displacement of the circular orifice plate can be obtained.  
\[\left\{\begin{array}{l}u_{\rho 1}=\frac{(1+\lambda)}{2E_{1}}  
\[x=\frac{2E_{2}}{E_{1}(1-v_{2})+E_{2}(1+v_{1})} \tag{17}\]  
The expression for the stress field of the model is obtained.  
\[\left\{\begin{array}{l}\sigma_{\rho}=\sigma_{\rho 1}+\sigma_{\rho 2}\\ \sigma_{\theta}=\sigma_{\theta 1}+\sigma_{\theta 2}\\ \tau_{\rho\theta}=\tau_{\rho\theta 2}\end{array}\right. \tag{18}\]  
The following three special cases of the model can be derived from Equation (17):  
\[\left\{\begin{array}{ll}E_{2}\to 0&x\to 0\\ E_{1}=E_{2}&v_{1}=v_{2}&x=1\\ E_{2}\rightarrow\infty&x=\frac{2}{1+v_{1}}\in(\frac{4}{3},2)\end{array}\right. \tag{19}\]  
If the modulus of elasticity of ice is significantly lower than that of rock, the model can be considered a problem of stress concentration at the edge of a hole in a circular porous plate. When the modulus of elasticity of ice and rock is the same, it confirms the homogeneity of the model. If the modulus of elasticity of the ice is much higher than that of the rock, the ice can be considered a rigid body in comparison to the rock. As a result, displacement of the pore edge occurs.