## 3 Solution Procedure of Mathematical Physics Equations  
### Strain Distribution Law of Collision Interface  
#### Dynamic Equation of Particles of Collision Interface  
(i)   Dynamic equation of arbitrary particles in collision interface  
For any moment, in the collision interface, \(u_{0}\) is independent of \(x,y\), but only related to \(z\) and \(t\), that is:  
\[\left\{ \begin{array}{l}\frac{\partial u_{0}}{\partial x}=\frac{\partial \delta}{\partial x}=0\\ \frac{\partial^{2}u_{0}}{\partial x^{2}}=\frac{\partial^{2}\delta}{\partial x^ {2}}=0\\ \frac{\partial u_{0}}{\partial y}=\frac{\partial\delta}{\partial y}=0\\ \frac{\partial^{2}u_{0}}{\partial y^{2}}=\frac{\partial^{2}\delta}{\partial y^ {2}}=0\end{array} \right. \tag{63}\]  
Therefore, substituting Equations (5) and (6) into Equation (49):  
\[\frac{\partial^{2}(u+\delta)}{\partial t^{2}}\big{|}_{t\geq t^{\prime}}=\frac{ \partial^{2}u_{0}}{\partial t^{2}}\big{|}_{t\geq t^{\prime}}=c^{2}\bigg{[} \eta\frac{\partial^{2}(u_{0}+\delta-\omega)}{\partial z^{2}}-\frac{\partial^{2} \omega}{\partial x^{2}}-\frac{\partial^{2}\omega}{\partial y^{2}}\bigg{]} \tag{64}\]Considering  
\[ \begin{array}{c}\frac{\partial\omega}{\partial x}=\frac{x}{\sqrt{R^{2}-x^{2}-y^{2 }}}\\ \frac{\partial^{2}\omega}{\partial x^{2}}=\frac{1}{\sqrt{R^{2}-x^{2}-y^{2}}}+ \frac{x^{2}}{\left(R^{2}-x^{2}-y^{2}\right)^{\frac{3}{2}}}\\ \frac{\partial^{2}\omega}{\partial y^{2}}=\frac{1}{\sqrt{R^{2}-x^{2}-y^{2}}}+ \frac{y^{2}}{\left(R^{2}-x^{2}-y^{2}\right)^{\frac{3}{2}}}\end{array}  \tag{65}\]  
Substituting Equation (65) into Equation (64) and setting \(u+\delta=u_{\rm T}\), the dynamic equation of any particle at the collision interface can be further expressed as:  
\[\frac{\partial^{2}u_{0}}{\partial t^{2}}=c^{2}\left\{\eta\frac{\partial^{2}u_ {\rm T}}{\partial z^{2}}-\frac{1}{R-(u_{0}+\delta)+u_{\rm T}}-\frac{R^{2}}{ \left[R-(u_{0}+\delta)+u_{\rm T}\right]^{3}}\right\} \tag{66}\]  
(ii) Dynamic equation of center point of collision interface  
When \(x=y=0\), the dynamic equation at the center of the collision interface is:  
\[\frac{\partial^{2}u_{0}}{\partial t^{2}}=c^{2}\left[\eta\frac{\partial^{2}(u_ {0}+\delta-\omega)}{\partial z^{2}}\right|_{\omega=0}-\frac{2}{R}\right] \tag{67}\]  
At the center point, \(\omega\equiv 0\), so:  
\[\eta\frac{\partial^{2}(u_{0}+\delta-\omega)}{\partial z^{2}}=\eta\frac{ \partial^{2}(u_{0}+\delta)}{\partial z^{2}} \tag{68}\]  
Therefore, the dynamic equation of the center point can be expressed as:  
\[\frac{\partial^{2}u_{0}}{\partial t^{2}}=c^{2}\left[\eta\frac{\partial^{2}(u_ {0}+\delta)}{\partial z^{2}}-\frac{2}{R}\right] \tag{69}\]  
According to the initial value conditions, it can be known:  
\[g=c^{2}\left[\frac{\partial^{2}(u_{0}+\delta)}{\partial z^{2}}\right|_{u_{0} =0}-\frac{2}{R}\right] \tag{70}\]  
(iii) Dynamic equation of boundary point of collision interface  
When \(x^{2}+y^{2}=2 Ru_{0}-{u_{0}}^{2}\), it is the dynamic equation of the collision boundary point particle, and \(\omega=u_{0}\):  
\[\frac{\partial^{2}u_{0}}{\partial t^{2}}=c^{2}\left[\eta\frac{\partial^{2}(u _{0}+\delta-\omega)}{\partial z^{2}}\right|_{\omega=u_{0}}-\frac{1}{R-u_{0}} -\frac{R^{2}}{\left(R-u_{0}\right)^{3}}\right] \tag{71}\]  
At any time \(t\), the collision depth of the center point \(u_{0}\) and the velocity and acceleration of each particle point on the interface is consistent, and the dynamic equation of the center point, boundary point, and particle point at any point can be formed:  
\[\left\{