# Energy Generation and Attenuation of Blast-Induced Seismic Waves under In Situ Stress Conditions
## 3 Numerical Study of Blasting Seismic Energy Generation and Attenuation under In Situ Stress
### Numerical Model and Simulation Method
#### 3.2.2 Rock and Explosive Models  
It is generally accepted that under blast loading, the rock in the crushed zone is mainly subjected to compressive-shear failure, and the rock in the cracked zone mainly suffers tensile failure [34]. The Riedel-Hiermaier-Thoma (RHT) constitutive model developed by [PERSON] et al. [49] takes compression, tension, and strain rate effects into account together and is widely used in modeling rock fracture caused by blasting. Herein, the rock behavior induced by blasting under in situ stress is also simulated by using the RHT model. In this model, the change of the material strength under loading is divided into three phases, namely the elastic phase, the linear strengthening phase, and the damage softening phase, as depicted in Figure 3. The three phases are distinguished by using three stress limit surfaces, namely the initial elastic yield surface \(Y_{\text{elastic}}\), the failure surface \(Y_{\text{failure}}\) and the residual friction surface \(Y_{\text{residual}}\). A more detailed description of the RHT constitutive model is presented in [PERSON] et al. [49] and Borrvall and Riedel [50].  
Figure 2: Schematic diagram of the coupled SPH-FEM method.  
In the damage softening phase, a parameterized damage model driven by plastic strain governs the evolution of material damage. The damage variable \(D\) is defined as:  
\[D=\sum\frac{\Delta\varepsilon_{\text{p}}}{\varepsilon_{\text{p}}^{\text{f}}},\;0 \leq D\leq 1 \tag{8}\]  
where \(\Delta\varepsilon_{\text{p}}\) represents the increment of the equivalent plastic strain, \(\varepsilon_{\text{p}}^{\text{f}}\) denotes the equivalent plastic strain of material failure under a constant pressure \(p\). The failure strain \(\varepsilon_{\text{p}}^{\text{f}}\) is given by:  
\[\varepsilon_{\text{p}}^{\text{f}}=D_{1}(p^{*}-p_{\text{s}}^{*})^{D_{2}}\geq \varepsilon_{\text{f, min}} \tag{9}\]  
where \(p^{*}\) is the pressure normalized by \(f_{c}\), \(p^{*}=p/f_{c}\), in which \(f_{c}\) is the uniaxial compressive strength; \(p_{\text{s}}^{*}=f_{\text{f}}/f_{c}\), in which \(f_{\text{t}}\) is the uniaxial tensile strength; \(\varepsilon_{\text{f, min}}\) denotes the minimum strain to reach material failure; \(D_{1}\) and \(D_{2}\) are damage constants.  
The post-failure stress limit surface for a given damage level is obtained by interpolating between the failure surface \(Y_{\text{failure}}\) and the residual friction surface \(Y_{\text{residual}}\). Then the equivalent strength in the damage phase \(\sigma_{\text{d}}\) is expressed as:  
\[\sigma_{\text{d}}=(1-D)\sigma_{\text{f}}+D\sigma_{\text{r}} \tag{10}\]  
where \(\sigma_{\text{f}}\) represents the equivalent failure surface strength, and \(\sigma_{\text{r}}\) represents the equivalent residual strength.  
The damaged shear modulus \(G_{\text{D}}\) is given through the following Equation:  
\[G_{\text{D}}=(1-D)G_{0}+DG_{\text{r}} \tag{11}\]  
where \(G_{0}\) denotes the undamaged shear modulus, and \(G_{\text{r}}\) denotes the residual value.  
The explosive detonation process is simulated by using the Jones-Wilkins-Lee (JWL) equation of state (EOS). The JWL EOS defines the detonation pressure as a function of the volume and energy of detonation products,  
\[P=C_{1}\bigg{(}1-\frac{\omega}{R_{1}V}\bigg{)}e^{-R_{1}V}+C_{2}\bigg{(}1-\frac {\omega}{R_{2}V}\bigg{)}e^{-R_{2}V}+\frac{\omega E_{0}}{V} \tag{12}\]  
where \(P\) denotes the detonation pressure, \(V\) is the relative volume of detonation products, \(E_{0}\) is the initial special internal energy, and \(C_{1}\), \(C_{2}\), \(R_{1}\), \(R_{2}\), and \(\omega\) are explosive constants.