# Study on Dynamic Strain Distribution in Rock Broken by Abrasive Slurry Jet
## 4 Theoretical Analysis for the Dynamic Strain Distribution in Rock Broken by ASJ  
In this section, theoretical analysis for the dynamic strain distribution in rock broken by ASJ was performed, obtaining the propagation regularity of strain within the rock. With the impacting load of ASJ approximately acting at a point on the rock surface, the stress wave produced by ASJ impacting will be propagated within the rock in form of spherical wave [18].  
As shown in Figure 8, the stress wave propagates within the rock in the form of a spherical wave, whose state parameters are related to the sphere radius (\(r\)) and time (\(t\)), and not related to the angles (\(\theta\) and \(\varphi\)) in polar coordinates.  
\[\begin{array}{l}\varepsilon_{r}(r,t)=\frac{\partial u(r,t)}{\partial r}\\ \varepsilon_{\theta}(r,t)=\frac{u(r,t)}{r}\end{array} \tag{1}\]  
where, \(u(r,t)\) is the rock radial displacement, \(\varepsilon_{r}(r,t)\) is the rock radial strain, and \(\varepsilon_{\theta}(r,t)\) is the rock tangential strain.  
Figure 6: Filter processing of the rock dynamic strain data.  
Figure 7: Fast Fourier Transformation (FFT) of rock dynamic strain data.  
Assuming that rock is a standard elastomer with no initial crack, its constitutive equations can be expressed as follows  
\[\begin{array}{c}\sigma_{r}=\frac{E}{(1+\
u_{2})(1-2\
u_{2})}[(1+\
u_{2})\varepsilon _{r}+2\
u_{2}\varepsilon_{\theta}]\\ \sigma_{\theta}=\frac{E}{(1+\
u_{2})(1-2\
u_{2})}[\
u_{2}\varepsilon_{r}+ \varepsilon_{\theta}]\end{array} \tag{2}\]  
where, \(\
u_{2}\) is the Poisson ratio of the rock, and \(E\) is the elastic modulus of the rock.  
Based on the elastokinetics, the wave equation of the spherical wave in the rock can be expressed as follows [19]  
\[\frac{\partial^{2}(r\phi)}{\partial r^{2}}-\frac{1}{c^{2}}\frac{\partial^{2} (r\phi)}{\partial t^{2}}=0 \tag{3}\]  
where, \(\phi\) is a potential function of the rock radial displacement. \(c\) is the velocity of the spherical wave in the rock, which is related to rock density \(\rho_{0}\).  
\[u=\frac{\partial\phi(r,t)}{\partial r} \tag{4}\]  
\[c=\sqrt{\frac{E(1-\
u_{2})}{\rho_{0}(1+\
u_{2})(1-2\
u_{2})}} \tag{5}\]  
Solving the simultaneous equations (the five equations above), the paper can obtain the wave equation of stress wave in the rock impacted by ASJ. Outside the jet area, rock stress caused by the slurry jet impacting is much larger than that by the abrasive particles impacting [20]. Therefore, with the impacting of abrasive particles ignored, the final stress wave equation was derived.  
\[\phi(r,t)=\frac{p_{0}d^{3}}{4 rG}[1-e^{-\beta\tau}[\cos(\alpha\tau)+\frac{\beta }{\alpha}\sin(\alpha\tau)]] \tag{6}\]  
where, \(\tau=t-(r-d)/c\), \(\alpha=\frac{2\
u_{2}}{d}\sqrt{\frac{2}{\rho_{0}(1-\
u_{2})}}\), \(\beta=\frac{2}{d}\sqrt{\frac{(1+\
u_{2})(1-2\
u_{2})}{\rho_{0}(1-\
u_{2})strain will increase to a maximum value with the increase of time, which can be regarded as a process of pressure increasing within the rock caused by ASJ impacting. Moreover, the maximum value of rock strain is proportional to jet pressure, while it is inversely proportional to the square of the propagation distance, which is in accordance with the experimental results. Figure 10 also shows the time for rock strain to increase from zero to the maximum value is not affected very much by the jet pressure and propagation distance, which is related to the rock's physical properties only.