# Investigation of Rock Joint and Fracture Influence on Delayed Blasting Performance
## 5 Analysis of Blasting Simulation  
Based on field conditions, pore spacing and lithology, the fluid-structure coupling of the explosive, and the Lagrange algorithm are assumed and used for the ore rock and the blockage simulation. Yield occurs when the average stress \(\sigma\)m at a point in the rock mass exceeds its yield strength \(\sigma\)y (that is, it begins to enter the plastic stage); for high-speed impact processes, such as explosions, the strain rate effect of rock mass is significant. The rock's constitutive model adopts the bilinear kinematic hardening model (plastic kinematic) [27; 28; 29], and the specific expression is as follows:  
\[\sigma_{\rm Y}=\left[1+\left(\frac{\dot{\varepsilon}}{C}\right)^{\frac{1}{P} }\right](\sigma_{0}+\beta E_{\rm P}\varepsilon_{eff}^{p})\]  
\[\varepsilon_{eff}^{p}=\int\limits_{0}^{t}\left(\frac{2}{\xi_{ij}^{p} \varepsilon_{ij}^{p}}\right)^{\
icefrac{{1}}{{2}}}\!\!\!\!dt\]  
where \(\sigma_{0}\) is the initial yield strength; \(C\) and \(P\) are constants related to material properties and \(C\) = 35, \(P\) = 3; \(\dot{\varepsilon}\) is the strain rate; \(\beta\) is an adjustable parameter, \(\beta\) = 1; \(E_{\rm P}\) is the plastic hardening modulus, 23.7 MPa; \(\varepsilon_{\rm ij}^{\rm P}\) is the plastic strain rate; and \(\varepsilon_{\rm eff}^{p}\) is the equivalent plastic strain. Through the analysis of the above two formulas, when the detonation wave generated by the explosion acts on the rock mass and the Mises stress \(\sigma_{\rm vm}\) of the rock in the near explosion source area exceeds the set dynamic compressive strength \(\sigma_{\rm cd}\), the point fails. In the far zone of the explosion source, when the calculated tensile stress \(\sigma_{\rm t}\) of the point in the rock mass is greater than the set dynamic tensile strength \(\sigma_{\rm td}\), the point fails, and in the near zone of the explosion source, the calculated cumulative plastic strain \(\varepsilon\)p of the point in the rock mass exceeds the set failure strain \(\varepsilon_{\rm pf}\), the point fails. In short, the possibility of rock mass failure increases, which is convenient to more realistically simulate the dynamic force of rock mass after the detonation wave and symbiotic gas. Based on the existing reference manual \"Engineering Geology Handbook\" and an engineering geological survey report, the relevant strain rate parameters are selected, and it is believed that the dynamic compressive strength of the rock will be significantly enhanced under high loading strain rate. The conversion relationship between the dynamic compressive strength \(\sigma_{\rm cd}\) and the static compressive strength \(\sigma_{\rm c}\) is as follows. \[\sigma_{\rm cd}=\sigma_{\rm c}\varepsilon^{1/3} \tag{14}\]  
The same material model is used for the stemming and hosting rock [30], in which the specific parameter settings are shown in Tables 1 and 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Density (g/cm\({}^{3}\))**} & **Young’s** & \multirow{2}{*}{**Poisson’s Ratio**} & **Compressive** & **Tangent** \\  & **Modulus (MPa)** & & **Strength (MPa)** & **Modulus (MPa)** \\ \hline
1.85 & 1.2 & 0.38 & 0.8 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Stemming Material Properties. The intervals are filled with air to a density of 1.29 g/L. On the other hand, other parameters are defaulted, and the state equation is expressed as the following: When \(\mu<0\), \(C_{2}\mu^{2}\) and \(C_{6}\mu^{3}\) are 0; \(C_{0}=C_{1}=C_{2}=C_{3}=C_{6}=0\), \(C_{4}=C_{5}=\gamma-1\), where \(\gamma\) is the ratio of two pressures and volume specific heat.