# Study on Rock Damage Mechanism for Lateral Blasting under High In Situ Stresses
## 2 Numerical Modeling
### Constitutive Model Parameters and Validation
#### 2.1.2 RHT Material Model for Rock  
In this study, the dynamic response and damage process of rock mass were simulated by [PERSON], [PERSON], and [PERSON] (RHT) in LS-DYNA, which were widely used in the numerical simulation of concrete and rock [3; 20; 21; 22; 23; 24; 25]. The damage degree of the RHT material model is given by \(D=\sum\frac{\Delta e^{p}}{\varepsilon^{f}}\), in which \(\Delta e^{p}\) is the accumulated plastic strain and \(\varepsilon^{f}\) is the failure strain. The detailed introduction of the RHT constitutive model can be  
Figure 1: Single-hole crater blasting model and the comparison of explosion craters: (**a**) physical model; (**b**) numerical model; (**c**) the comparison of experimental and simulation results. found in [20]. Based on the tested mechanical parameter, the following sections obtain the other material parameters through empirical formulas or related literatures. 1. Strain rate parameters  
The effect of strain rate on the rock strength is clear. The strain rate strength factor \(F_{r}\big{(}\dot{\varepsilon}_{p}\big{)}\) is expressed as [21]:  
\[F_{r}\big{(}\dot{\varepsilon}_{p}\big{)}=\left\{ \begin{array}{ll}\big{(}\dot{ \varepsilon}_{p}/\varepsilon_{0}^{\epsilon}\big{)}^{\beta_{c}}&P\geq f_{c}/3\\ \frac{P_{r}+f_{c}/3}{f_{c}/3+f_{c}/3}\big{(}\dot{\varepsilon}_{p}/\varepsilon_ {0}^{\epsilon}\big{)}^{\beta_{c}}-\frac{P_{r}-f_{c}/3}{f_{c}/3+f_{c}/3}\big{(} \dot{\varepsilon}_{p}/\varepsilon_{0}^{\epsilon}\big{)}^{\beta_{\tilde{r}}}&-f_ {t}/3<P<f_{c}/3\\ \big{(}\dot{\varepsilon}_{p}/\varepsilon_{0}^{\epsilon}\big{)}^{\beta_{\tilde{ r}}}&P\leq-f_{t}/3\end{array} \right. \tag{1}\]  
where \(\dot{\varepsilon}_{p}\) is the strain rate; \(\dot{\varepsilon}_{0}^{\epsilon}\) and \(\dot{\varepsilon}_{0}^{t}\) are the compressive and tensile reference strain rates, which are \(3\times 10^{-5}\) s\({}^{-1}\) and \(3\times 10^{-6}\) s\({}^{-1}\), respectively; \(P\) is the hydrostatic pressure; \(f_{c}\) and \(f_{t}\) are the uniaxial strengths in compression and tension. The strain rates in compression (\(\beta_{c}\)) and in tension (\(\beta_{t}\)) are constant for the material and can be calculated by  
\[\beta_{c}=\frac{4}{20+3f_{c}},\ \beta_{t}=\frac{2}{20+f_{c}} \tag{2}\]  
where the unit is MPa, and then \(\beta_{c}\) and \(\beta_{t}\) are determined as 0.024 and 0.029. 2. Failure surface parameters  
\[A\ \text{and}\ N\ \text{are}\ \text{the}\ \text{constants}\ \text{the}\ \text{constants}\ \text{the}\ \text{failure}\ \text{surface}\ \text{and}\ \text{can}\ \text{be}\ \text{obtained by:} \tag{3}\]  
where \(\sigma_{\tilde{t}}^{*}(P^{*},F_{r})\) is the normalized strength and can be calculated by \(\sigma_{\tilde{t}}^{*}=\frac{\sigma_{\tilde{t}}}{f_{c}}\); \(P^{*}\) is the normalized hydrostatic pressure and can be calculated by \(P^{*}=\frac{P}{f_{c}}\); \(F_{r}\) is the strain rate strength factor. When the rock is in a quasi-static state, \(\dot{\varepsilon}_{p}=3.0\times 10^{-5}\) s\({}^{-1}\), and then \(F_{r}=1\) can be calculated by Equation (1). The rock strength under different confining pressures can be calculated by empirical equation of Hoek-Brown [26] and the fitting equations for rock material are as follows [27]:  
\[\sigma_{1}=\sigma_{3}+48.3\Big{(}24\frac{\sigma_{3}}{48.3}+1\Big{)}^{1/2} \tag{4}\]  
The axial stress at failure (\(\sigma_{1}\)) under different confining pressures (\(\sigma_{2}=\sigma_{3}\)) can be calculated by Equation (4). The results are shown in Table 1. Subsequently, the pressure \(P=\frac{1}{3}(\sigma_{1}+\sigma_{2}+\sigma_{3})\), equivalent stress at failure \(\sigma_{\tilde{t}}=\sqrt{\frac{1}{2}\Big{[}(\sigma_{1}-\sigma_{2})^{2}+(\sigma _{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\Big{]}}\), \(P^{*}\), and \(\sigma_{t}^{*}\) can be obtained and are shown in Table 1. Based on Equation (3), \(A=2.439\) and \(N=0.7528\) can be obtained by substituting the values of \(P^{*}\) and \(\sigma_{\tilde{t}}^{*}\) corresponding to the confining pressures of 5 MPa and 30 MPa.