## 4 Numerical Simulation
### Determination of Concrete Material Parameters  
The HJC constitutive model is suitable for studying the dynamic mechanical properties of concrete, allowing for accurate characterization of changes in the mechanical properties of concrete under the dynamic impact [32; 33; 34; 35]. It mainly consists of a yield surface equation, a damage evolution equation, and a state equation. (1) Yield surface equation  
The yield surface equation for the HJC constitutive model can be expressed by Equation (4):  
\[\sigma^{*}=\Big{[}A(1-D)+BP^{*N}\Big{]}\Big{(}1+C\ln\dot{\epsilon}^{*}\Big{)} \tag{4}\]  
Figure 17: Uniaxial dynamic impact compression model. where \(A\) is the normalized viscous strength; \(B\) is the normalized pressure hardening coefficient; \(C\) is the influence coefficient of strain rate; \(N\) is the pressure hardening coefficient. \(D\) is the damage factor; \(\sigma^{*}\) and \(P^{*}\) are the normalized equivalent stress and hydrostatic pressure obtained by dividing the actual equivalent stress and the true hydrostatic pressure of the unit by the static compressive strength, respectively; and \(\dot{\bar{\epsilon}}^{*}\) is the normalized true strain rate obtained by dividing the true strain rate \(\dot{\bar{\epsilon}}\) by the reference strain rate \(\dot{\bar{\epsilon}}_{0}\). (2) Damage evolution equation  
The damage evolution equation in the HJC model characterizes the damage based on the equivalent plastic strain and the plastic volume strain accumulation, which can be expressed by Equation (5):  
\[D=\sum\frac{\Delta\varepsilon_{p}+\Delta u_{p}}{D_{1}(P^{*}+T^{*})^{D_{2}}} \tag{5}\]  
where \(D\) is the damage factor; \(D_{1}\) and \(D_{2}\) are the damage coefficients; \(\Delta\varepsilon_{p}\) and \(\Delta u_{p}\) are the equivalent plastic strains and the corresponding volumetric strains, respectively; and \(T^{*}\) is the normalized maximum tensile stress obtained by dividing the maximum tensile strength by the static compressive strength. (3) State equation  
The state equation describes the relationship between hydrostatic pressure \(P\) and volumetric strain \(\mu\) (see Figure 18). The variation can be divided into three stages. The first stage (\(O\) to \(A\)) is the linear elastic stage (\(P<Pc\)). When the pressure is less than or equal to \(Pc\), it satisfies the Equation (6), where \(K\) is the bulk modulus. \[P=K\mu \tag{6}\]  
The second stage (\(A\) to \(B\)) is the compaction transition stage (\(Pc\leqslant P\leqslant P_{l}\)). When the pressure is greater than or equal to \(P_{c}\) or less than or equal to \(P_{l}\), the concrete material is compacted and produces a plastic volumetric strain, which can be expressed by Equation (7):  
\[P=\frac{(P_{l}-P_{c})(\mu-\mu_{c})}{\mu_{l}-\mu_{c}}+P_{c} \tag{7}\]  
The third stage (\(B\) to \(C\)) is the failure stage (\(P>P_{l}\)). When the pressure is greater than \(P_{l}\), the concrete is completely crushed and damaged, satisfying Equations (8) and (9):  
\[P=K_{1}\overline{\mu}+K_{2}\overline{\mu}^{2}+K_{3}\overline{\mu}^{3} \tag{8}\]  
Figure 18: Relationship between hydrostatic pressure and volumetric strain. \[\overline{\mu}=\frac{\mu-\mu_{l}}{1+\mu_{l}} \tag{9}\]  
where \(P_{c}\) and \(\mu_{c}\) are the crushing pressure and volumetric strain; \(P_{l}\) and \(\mu_{l}\) are the locking pressure and locking volumetric strain, respectively; \(K_{1}\), \(K_{2}\) and \(K_{3}\) are the pressure parameters; and \(\overline{\mu}\) is the corrected volumetric strain. The HJC constitutive model consists of 21 parameters, and all of them can be determined by tests or computational equations, except for normalized viscous strength \(A\), normalized pressure hardening coefficient \(B\), strain rate influence coefficient \(C\), and pressure hardening coefficient \(N\).