# Dynamic Mechanical Characteristics of Horseshoe Tunnel Subjected to Blasting and Confining Pressure
## 2 Materials and Numerical Modeling  
### Materials  
The granite was taken from Shandong, China. The static properties of rock materials were tested in the Laboratory of Explosion at the Beijing Institute of Technology. The static mechanical parameters are listed in Table 1.  
[PERSON] and [PERSON] suggested a constitutive mode (JH model) to analyze the behavior of brittle materials under dynamic loading, and it has been applied as a damage model in ANSYS AUTODYN and LS-DYNA and provided the explicit code for nonlinear dynamics simulation [29; 30; 31]. The Polynomial EOS, JH strength model and JH failure criterion have three parts to describe rock material's response under dynamic loading. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Unit** & **Value** \\ \hline Density & g/cm\({}^{3}\) & 3.02 \\ Poisson’s ratio & & 0.17 \\ Young’s modulus & GPa & 37.61 \\ Tensile strength & MPa & 7.64 \\ Compressive strength & MPa & 86.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Static mechanical parameters of granite. Polynomial EOS is:  
\[P=k_{1}\mu+k_{2}\mu^{2}+k_{3}\mu^{3}+\Delta P\text{ for compression conditions}, \tag{1}\]  
\[P=k_{1}\mu+k_{2}\mu^{2}\text{ for tension conditions}, \tag{2}\]  
where \(\mu\) is the compression, \(\mu=\rho/\rho_{0}-1,\rho\) represents current density, \(\rho_{0}\) is reference density and \(\Delta P\) is an additional pressure. The equivalent stress can be expressed as:  
\[\sigma=\sqrt{3J_{2}}=\left\{\frac{\left(\sigma_{x}-\sigma_{y}\right)^{2}+ \left(\sigma_{x}-\sigma_{z}\right)^{2}+\left(\sigma_{y}-\sigma_{z}\right)^{2} +6\left(\tau_{xy}^{2}+\tau_{xz}^{2}+\tau_{yz}^{2}\right)}{2}\right\}^{\frac{1} {2}}, \tag{3}\]  
where \(\sigma_{x}\), \(\sigma_{y}\) and \(\sigma_{z}\) are the three normal components of the stress tensor, and \(\tau_{xy}\), \(\tau_{xz}\) and \(\tau_{yz}\) are the three shear stresses. The intact strength, fully fractured strength and strength of damaged material are given as follows:  
\[\sigma_{i}^{*}=A(P^{*}+T^{*})^{N}\Big{(}1+Cln\dot{\varepsilon}^{*}\Big{)} \tag{4}\]  
\[\sigma_{f}^{*}=\text{B}(P^{*})^{N}\Big{(}1+Cln\dot{\varepsilon}^{*}\Big{)} \tag{5}\]  
\[\sigma^{*}=\sigma_{i}^{*}-D(\sigma_{i}^{*}-\sigma_{f}^{*}), \tag{6}\]  
where \(A,N,B,M\) and \(C\) are constants, \(\sigma_{i}^{*},~{}\sigma_{f}^{*},~{}\sigma^{*},~{}P^{*},~{}T^{*}\) are intact effective stress, fractured effective stress, current effective stress, pressure and tensile strength, which are normalized by the following expressions:  
\[\sigma_{i}^{*}=\sigma_{i}/\sigma_{\text{HEL}},\sigma_{f}^{*}=\sigma_{f}/\sigma _{\text{HEL}},\sigma^{*}=\sigma_{i}/\sigma_{\text{HEL}},P^{*}=P/P_{\text{HEL} },T^{*}=T/T_{\text{HEL}}, \tag{7}\]  
\(P\) is the current hydrostatic pressure, \(T\) is the maximum tensile hydrostatic pressure, \(\sigma_{\text{HEL}}\) is the effective stress at the Hugoniot elastic limit (HEL), \(P_{\text{HEL}}\) is the associated hydrostatic pressure at the HEL, \(D\) is the damage variable and \(\sigma\) is the effective stress. The equivalent plastic strain to fracture is calculated as follows:  
\[\varepsilon_{p}^{f}=D_{1}(P^{*}+T^{*})^{D_{2}}, \tag{8}\]  
where \(D_{1}\) and \(D_{2}\) is constants. When plastic deformation happens in the material, the damage accumulates, and its value can be obtained from the following:  
\[D=\sum\frac{\Delta\varepsilon_{p}}{\varepsilon_{p}^{f}}, \tag{9}\]  
where \(\Delta\varepsilon_{\text{p}}\) is the increment in the equivalent plastic strain during a calculation cycle.