## 3 Numerical Analysis for the LLB Method
### Material Model and State Equation  
A JH-2 constitutive model was utilized to simulate the behavior of rock mass in the dynamic state [23; 24; 25]. This model is widely used to simulate the behavior of brittle materials under dynamic conditions in LS-DYNA. This model is based on the relation of normalized equivalent stress and pressure. The JH-2 model reflects the pressure, damage evolution, fracture, strain-rate-dependent strength, and softening characteristics of the brittle materials [26]. It presents the strength and damage model and EOS, as shown in Figure 6. The strength model (Figure 6a) includes the intact (\(\sigma_{i}^{*}\)), damaged (\(\sigma^{*}\)), and fractured states (\(\sigma_{f}^{*}\)), which are calculated as follows:  
\[\sigma_{i}^{*}=A(P^{*}+T^{*})^{N}\Big{(}1+\mathrm{C}\cdot\ln\hat{\epsilon}^{* }\Big{)}\sigma^{*}=\sigma_{i}^{*}-D\Big{(}\sigma_{i}^{*}-\sigma_{f}^{*}\Big{)} \sigma_{f}^{*}=B(P^{*})^{M}\Big{(}1+\mathrm{C}\cdot\ln\hat{\epsilon}^{*} \Big{)} \tag{1}\]  
where \(\sigma_{i}^{*}\), \(\sigma^{*}\), and \(\sigma_{f}^{*}\) are the normalized intact, damaged, and equivalent stress, respectively; \(A\), \(B\), \(C\), \(M\), and \(N\) are material constants; \(P^{*}\) is the normalized hydrostatic pressure, which is calculated as \(P^{*}=P/P_{HEL}\), where \(P\) and \(P_{HEL}\) are the hydrostatic pressure of the material and hydrostatic pressure at the Hugoniot elastic limit (HEL), respectively; and \(T^{*}\) is the normalized maximum tensile hydrostatic pressure, which can be computed as \(T^{*}=T/P_{HEL}\), where \(T\) is the maximum tensile hydrostatic pressure. The strain rate can be expressed as \(\hat{\epsilon}^{*}=\hat{\epsilon}/\hat{\epsilon}_{0}\), where \(\hat{\epsilon}^{*}\) and \(\hat{\epsilon}_{0}\) are the actual equivalent strain rate and reference strain rate (1.0), respectively. Figure 6: JH-2 material model: (**a**) strength model, (**b**) damage model, and (**c**) EOS [23]. In addition, damage accumulation, which exhibits non-linear behavior due to fracture generation, can be determined from the graph shown in Figure 6b. Depending on the pressure, the material status changes from intact to fractured states due to plastic deformation. In this case, the equivalent plastic strain can be expressed as follows [27]:  
\[\varepsilon_{f}^{p}=D_{1}(P^{*}+T^{*})^{D_{2}} \tag{2}\]  
where \(\varepsilon_{f}^{p}\) is the equivalent plastic strain causing fracture under constant hydrostatic pressure; and \(P\), \(D_{1}\), and \(D_{2}\) are damage constants. The damage is accumulated as the plastic deformation increase as follows:  
\[D=\sum\Delta\varepsilon^{p}\varepsilon_{f}^{p} \tag{3}\]  
where \(\Delta\varepsilon^{p}\) is the equivalent plastic strain increment during an integration cycle. Figure 6c shows the EOS of the JH-2 model, which describes the relationship of hydrostatic pressure and volumetric strain. The hydrostatic pressure before fractures and after damages begin to accumulate can be computed as follows:  
\[P=K_{1}\mu+K_{2}\mu^{2}+K_{3}\mu^{3}\cdots(D=0)P=K_{1}\mu+K_{2}\mu^{2}+K_{3}\mu ^{3}+\Delta P\cdots(0<D<1) \tag{4}\]  
where \(K_{1}\), \(K_{2}\), and \(K_{3}\) are EOS constants, and \(K_{1}\) is the initial bulk modulus of the material. The volumetric strain is \(\mu=\rho/\rho_{0}-1\), where \(\rho\) and \(\rho_{0}\) are the current and initial density, respectively. As damage begins to accumulate in the brittle material, the hydrostatic pressure (\(\Delta P\)) is added to the polynomial EOS. The parameters of granite, which is prevalent in South Korea, were input into the JH-2 model; the input parameters are summarized in Table 1[28]. An element erosion technique was applied to simulate rock failure under blasting. Element deletion was set to initiate when the tensile stress induced by the free face exceeds the maximum tensile strength of the rock material.