## 4 The Inversion of HJC Constitutive Parameters for Hard Rock
### Numerical Simulation of Simulated Reservoir Penetrating Hard Rock Target
#### 4.1.2 Material Model and Parameters  
The material model used for the explosive, shell case, and air is the same as above, and the model parameters of the powder liner are selected after inversion. Furthermore, both the gun holster and casing are composed of steel 45, and its model and parameters are the same as above. Therefore, these material models and parameters will not be reiterated in this section. This section mainly introduces the constitutive models and parameters of hard rock, water, and cement ring materials not mentioned above. (1) Hard rock Based on previous studies, the HJC model is utilized to characterize hard rock in this study [28]. The HJC model, proposed by [PERSON] et al. [29], serves as a constitutive model for concrete under conditions of large strain and a high strain rate, enabling a more comprehensive depiction of its mechanical properties during explosion, impact, and penetration. This model effectively accounts for the accumulation of compressive damage while also capturing the influence of strain rate, compressive strength, and confining pressure on material behavior. Consequently, it provides a more accurate representation of dynamic damage occurring in materials subjected to significant deformation and high strain rates. In this paper, the HJC constitutive model considering strain rate and damage is described as follows:  
Strength model equation:  
\[\sigma^{*}=[A_{1}(1-D)+B_{1}P^{*N_{1}}][1+C_{7}\ln\varepsilon^{*}] \tag{9}\]  
In the equation, \(\sigma^{*}=\sigma/f_{c}\) is the characteristic equivalent stress, where \(\sigma\) is the actual equivalent stress, and \(f_{c}\) is the quasi-static uniaxial compressive strength of the material; \(P^{*}=P/f_{c}\) is the characteristic hydrostatic pressure, where \(P\) is the actual pressure borne by the material; \(\varepsilon^{*}=\varepsilon/\varepsilon_{0}\) is the characteristic strain rate, where \(\varepsilon\) is the actual strain rate of the material, and \(\varepsilon_{0}\) is the reference strain rate; \(D\) is the damage parameter; and \(A_{1}\), \(B_{1}\), \(N_{1}\), \(C_{7}\), and \(S_{\max}\) are material constants, Among them, \(A_{1}\) is the standardized cohesion strength, \(B_{1}\) is the standardized pressure hardening coefficient, \(N_{1}\) is the pressure hardening index, \(C_{7}\) is the strain rate coefficient, and \(S_{\max}\) is the characteristic ultimate strength of the material. Damage evolution equation:  
\[\mathrm{D}=\sum\frac{\Delta\varepsilon_{p}+\Delta\mu_{p}}{e_{p}^{f}+\mu_{p}^{ f}} \tag{10}\]  
Figure 14: Mesh sensitivity analysis of the model. In the equation, \(e_{p}^{f}+\mu_{p}^{f}\) represents the plastic strain when the material breaks under normal pressure; \(e_{p}^{f}+\mu_{p}^{f}=D_{1}(p*+T*)^{D_{2}}\), where \(\Delta\epsilon_{p}\) and \(\Delta\mu_{p}\) represent the equivalent plastic strain and plastic volume strain of the material in a calculation cycle; \(T*\) represents the maximum characteristic hydrostatic tension that the material can withstand; and \(D_{1}\) and \(D_{2}\) are damage constants. In the process of calculation with the LS-DYNA program, it is found that the FS failure parameter of the HJC constitutive model cannot effectively control the unit failure. The \"MAT_ADDE REROSION\" erosion failure criterion is therefore introduced to regulate unit failure. This paper adopts the maximum principal strain failure criterion (\(\epsilon_{\text{max}}\)), which is currently the most widely employed criterion for rock materials [30]. Normalized parameters \(A_{1}\), \(B_{1}\), \(N_{1}\), and \(C_{7}\), damage parameter \(D_{1}\), uniaxial tensile mild \(f_{c}\), and maximum failure principal strain \(\epsilon_{\text{max}}\) were selected as design variables based on previous studies [31; 32; 33; 34; 35]. The values of key constitutive parameters of hard rock are shown in Table 6.  
(2) Water layer and cement ring  
In this study, the wellbore pressure is considered in the numerical simulation of simulated reservoir penetrating hard rock targets.