# Study on Rock Damage Mechanism for Lateral Blasting under High In Situ Stresses
## 2 Numerical Modeling
### Constitutive Model Parameters and Validation
#### 2.1.3 Material Identifications of Charge, Air and Stemming  
The charge is modeled by MAT_HIGH_EXPLOSIVE_BURN in LS-DYNA [30]. The JWL(Jones-Wilkins-Lee) EOS are given by:  
\[P=A\bigg{(}1-\frac{\omega}{R_{1}V}\bigg{)}\mathrm{e}^{-R_{1}V}+B\bigg{(}1-\frac {\omega}{R_{2}V}\bigg{)}\mathrm{e}^{-R_{2}V}+\frac{\omega E}{V} \tag{7}\]  
where \(P\) is the pressure, \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(\omega\) are constants, \(V\) is the specific volume, and \(E\) is the internal energy with an initial value of \(E_{0}\).  
In this study, the explosive is a mixture of RDX (Hexogen), PETN, DDNP, et al. The estimation of JWL parameters of explosive is complex and costly [31], so the parameters refer to similar explosive parameters [32]: \(A\) = 524 GPa, \(B\) = 7.68 GPa, \(R_{1}\) = 4.2, \(R_{2}\) = 1.1, \(\omega\) = 0.34, \(E_{0}\) = 8.5 GPa. The charge density is 1.6 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\) and the detonation velocity is 6950 m/s.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Density \(\rho_{0}\) & 2456 kg/m\({}^{3}\) & Compressive strain rate \(\beta_{c}\) & 0.024 \\ Shear modulus \(G\) & 13 GPa & Tensile strain rate \(\beta_{t}\) & 0.029 \\ ONEMPA \({}^{1}\) & 1.0 \(\times\) 10\({}^{6}\) & Pressure influence on plastic flow in tension PTF & 0.001 \\ Eroding plastic strain EPSF & 2.0 & Compressive yield surface \(g_{c}^{*}\) & 0.53 \\ Polynomial EOS \(B_{0}\) & 1.22 & Tensile yield surface \(g_{t}^{*}\) & 0.7 \\ Polynomial EOS \(B_{1}\) & 1.22 & Shear modulus reduction factor \(\xi\) & 0.5 \\ Polynomial EOS \(T_{1}\) & 35 GPa & Damage parameter \(D_{1}\) & 0.04 \\ Failure surface \(A\) & 2.439 & Damage parameter \(D_{2}\) & 1 \\ Failure surface \(N\) & 0.7528 & Minimum damaged residual strain \(\epsilon_{p}^{m}\) & 12 \(\times\) 10\({}^{-3}\) \\ Compressive strength \(f_{c}\) & 48.3 MPa & Residual surface parameter \(A^{f}\) & 1.6 \\ Relative shear strength \(f_{c}^{*}\) & 0.18 & Residual surface parameter \(n^{f}\) & 0.61 \\ Relative tensile strength \(f_{t}^{*}\) & 0.1 & Gruneisen gamma GAMMA & 0 \\ Lode angle \(Q_{0}\) & 0.681 & Hugoniot polynomial coefficient \(A_{1}\) & 3.6 \(\times\) 10\({}^{10}\) \\ Lode angle \(B\) & 0.0105 & Hugoniot polynomial coefficient \(A_{2}\) & 4.04 \(\times\) 10\({}^{10}\) \\ Polynomial EOS \(T_{2}\) & 0 & Hugoniot polynomial coefficient \(A_{3}\) & 0.48 \(\times\) 10\({}^{10}\) \\ Ref. compressive strain rate \(\dot{\epsilon}_{0}^{c}\) & 3.0 \(\times\) 10\({}^{-5}\) & Crush pressure \(P_{el}\) & 16.1 MPa \\ Ref. tensile strain rate \(\dot{\epsilon}_{0}^{t}\) & 3.0 \(\times\) 10\({}^{-6}\) & Compaction pressure \(P_{co}\) & 6 MPa \\ Break compressive strain rate \(\dot{\epsilon}^{c}\) & 3.0 \(\times\) 10\({}^{25}\) & Porosity exponent \(N_{p}\) & 3 \\ Break tensile strain rate \(\dot{\epsilon}^{t}\) & 3.0 \(\times\) 10\({}^{25}\) & Initial porosity \(\alpha_{0}\) & 1 \\ \hline \hline \end{tabular} \({}^{1}\) ONEMPA is the unit conversion factor defining 1 MPa in the pressure units used.  
\end{table}
Table 2: RHT parameters for rock mass.  
The air is modeled by MAT_NULL in LS-DYNA, and the corresponding EOS is given by [30]:  
\[P=C_{0}+C_{1}u+C_{2}u^{2}+C_{3}u^{3}+\Big{(}C_{4}+C_{5}u+C_{6}u^{2}\Big{)}e \tag{8}\]  
where \(C_{0}\), \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), \(C_{5}\) and \(C_{6}\) are polynomial coefficients; \(\mu=\frac{\rho}{\rho_{0}}-1\) is specific volume; \(e\) is the internal energy per volume and has the unit of pressure, Pa. In this study, the air is modeled as an ideal gas by setting \(C_{0}=C_{1}=C_{2}=C_{3}=C_{4}=0\) and \(C_{5}=C_{6}=0.4\), and the initial internal energy per volume is set to 0.25 J/cm\({}^{3}\)[25].  
The stemming is modeled by MAT_SOIL_AND_FOAM in LS-DYNA and its parameters are shown in Table 3.