# Dynamic Response and Rock Damage of Different Shapes of Cavities under Blasting Loads
## 4 Numerical Simulation Study on the Response and Fracture Characteristics of Cavity Blasting
### Three-Dimensional Numerical Modeling and Damage Characteristics of Cavities  
Based on the stress distribution characteristics of cavities under blasting, a three-dimensional elastic-plastic model was constructed to further study the destructive characteristics of cavities under blasting. Some scholars have employed the MAT_PLASTIC_KINEMATIC model for simulating the dynamic damage of rock due to its computational efficiency and alignment with the dynamic rock damage criteria [29,30]. The rock material is primarily defined using the material keyword MAT_PLASTIC_KINEMATIC (kinematic plasticity model material). This material's constitutive model can reflect the elastic and plastic failure processes of soils, rocks, concrete, and other media under the action of explosive blasting. According to the mechanical testing results of cement mortar specimens from similar model tests, the specific parameters used are listed in Table 3.  
The keyword MAT_ADD_EROSION can define various parameters of material failure, such as compression stress, tensile stress, equivalent stress, strain, and failure time. For the MAT_PLASTIC_KINEMATIC material, therefore, it is only necessary to use this keyword to define the material's tensile strength [31]. In this simulation, any model that defines the tensile strength parameter with MAT_ADD_EROSION is set to \(-0.47\) MPa (the tensile stress value for MAT_ADD_EROSION is labeled as a negative value). Additionally, the MAT_HIGH_EXPLOSIVE_BURN material is used to define high-energy explosives and void material. This paper uses No. 2 rock emulsion explosives with the same explosive performance as the similar model test, with a density of 1050 kg/m\({}^{3}\) and a detonation velocity of 4200 m/s. Besides material parameters, in ANSYS/LS-DYNA, it is also necessary to set the \"EOS_JWL equation of state parameters [32]. The EOS_JWL equation of state is as follows:  
\[P=A[1-\frac{\omega}{R_{1}V}]e^{-R_{1}V}+B[1-\frac{\omega}{R_{2}V}]e^{-R_{2}V}+ \frac{\omega E}{V} \tag{2}\]  
where \(A\), \(B\), \(R_{1}\), \(R_{2}\), \(\omega\) are parameters related to explosives determined by experiments, \(V_{0}\) is relative specific volume, \(E_{0}\) is initial specific energy. The specific parameters of the equation of state of the explosive are shown in Table 4.  
The three-dimensional numerical modeling employs the Lagrange (Lagrange) method with shared nodes, which requires not only the definition of explosive and rock materials but also the additional definition of void material. The ALE12 algorithm for \"SECTION_SOLID is used to calculate the void material, which provides the necessary volume for the expansion of the explosive, thereby preventing negative volume errors during the simulation. The void material is co-nodally coupled with the rock mass without requiring additional special boundaries, ensuring that the transmission of the blasting stress wave remains unaffected. The void material (void) is defined using the same material parameters as the explosive material. The three-dimensional numerical model is shown in Figure 11. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**A** & **B** & **R\({}_{1}\)** & **R\({}_{2}\)** & **OMEG** & **F\({}_{0}\)** & **V\({}_{0}\)** \\ \hline
2.144 \(\times\) 10\({}^{11}\) & 0.182 \(\times\) 10\({}^{9}\) & 4.2 & 0.9 & 0.15 & 4.192 \(\times\) 10\({}^{9}\) & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the explosive equation of state. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Density** & **Modulus of** & **Poisson’s** & **Yield** & **Tensile** & **Failure** \\
**kg/m\({}^{3}\)** & **Elasticity/GPa** & **Ratio** & **Strength/MPa** & **Strength/MPa** & **Strain** \\ \hline
2000 & 2.60 & 0.23 & 4.65 & 0.47 & 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material parameter settings. Consistent with similar model tests, the model is rectangular in structure with dimensions of 60 cm \(\times\) 40 cm \(\times\) 30 cm, and the total number of elements is approximately 640,000.