# Safety Performance Evaluation of Ultra-High-Performance Concrete Semi-Assembled Barrier
## 3 Collision Simulation System
### Introduction of the HJC Constitutive Model  
The HJC model was used in this study since it is suitable for concrete subjected to large strains, high strain rates, and high pressures [31] and is widely used by researchers. [PERSON] et al. [31] proposed a value method to determine the original values of basic, damage pressure, and strength parameters in the HJC model of NC by combining experimental data with curve fitting methods. For UHPC, these parameters in the HJC model were calibrated based on mechanical property tests and recommended values found in the related literature [32; 33; 34; 35]. 1. Basic physical parameters: \(f_{c}^{\prime}\), \(T\), \(G\), \(K\)  
The uniaxial compressive strength \(f_{c}^{\prime}\) and tensile strength \(T\) of the UHPC are presented in the test results in Section 2. \(f_{c}^{\prime}=142.48\) MPa, \(T\) = 9.8 MPa, \(E\) = 45.7 GPa, [PERSON]'s ratio \(v\) = 0.2, and \(\rho\) = 2.54 g/cm\({}^{3}\) were used in this study. The shear modulus \(G\) and bulk modulus \(K\) were determined using the following formulae:  
\[G=\frac{E}{1+2v} \tag{1}\]  
\[K=\frac{E}{3(1-2v)} \tag{2}\]  
The calculation can be obtained as follows: \(G\) = 32.64 GPa; \(K\) = 25.38 GPa. 2. Strength parameters: \(A\), \(B\), \(N\), \(C\), \(SF_{max}\)  
\(B\) and \(N\) are the fracture strength of the material at the normalized strain rate \(\dot{\epsilon}^{*}\) = 1.0 s\({}^{-1}\), which can be determined by the conventional triaxial test of UHPC. Ignoringthe influence of damage parameter \(D\) in the elastic stage, the yield surface equation of can be simplified to  
\[\sigma^{*}=A+BP^{*N} \tag{3}\]  
In Equation (3),  
\[\sigma^{*}=\frac{\sigma_{1}+\sigma_{3}}{f_{c}^{\prime}} \tag{4}\]  
\[P^{*}=\frac{\sigma_{1}+2\sigma_{3}}{3f_{c}^{\prime}} \tag{5}\]  
\(\sigma_{1}\), \(\sigma_{3}\) are the axial and radial pressures of the conventional triaxial test in Section 2, respectively. [PERSON] et al. (2019) derived the dimensionless cohesive strength \(A\) based on the interrelationship between the Mises criterion and the Mohr-Coulomb criterion in the \(\pi\)-plane in plastic yield surface theory:  
\[A=\frac{1}{2\sqrt{K}} \tag{6}\]  
where \(K\) is the slope of the \(\sigma_{1}-\sigma_{3}\) linear relationship curve obtained from the conventional triaxial test, which can be obtained by linear fitting. \(K=4.00\), \(A=0.25\). Curve fitting using the triaxial test results and Equation (3) yields \(B=2.49\), \(N=1.08\). Owing to the lack of experimental data, this study takes the initial values of \(C=0.007\) and \(SF_{max}=7.0\), as recommended by [PERSON] et al. (2019). 3. Equation of State (EOS) parameters: \(P_{crush}\), \(\mu_{crush}\), \(P_{lock}\), \(\mu_{lock}\), \(K_{1}\), \(K_{2}\), and \(K_{3}\)  
The parameters of the EOS were mainly determined based on Hugoniot test data. Owing to the lack of experimental data, the original values proposed by [PERSON] et al. (2019) were used in this study, \(K_{1}=85\) GPa, \(K_{2}=-171\) GPa, \(K_{3}=208\) GPa. \(P_{crush}\), \(\mu_{crush}\), can be determined according to the following equations:  
\[P_{crush}=f_{c}^{\prime}/3 \tag{7}\]  
\[\mu_{crush}=P_{crush}/K \tag{8}\]  
Currently, there are few experimental studies on the \(P_{lock}\) of UHPC. The values of \(P_{lock}\) for different strengths were obtained by interpolation as in Equation (9) and substituting \(f_{c}^{{}^{\prime}}=142.48\) MPa can be obtained by substituting \(P_{lock}=1.00\) GPa ([PERSON] et al., 2019; [PERSON] et al., 2020). \(\mu_{lock}\) can be calculated according to Equation (10), where \(\rho_{0}\) is the original density of concrete, \(\rho_{grain}\) is the density of concrete after the pores in the concrete are compacted. In this study, the porosity of UHPC is taken to be 6.35% ([PERSON] et al., 2019), that is, \(1-\rho_{0}/\rho_{grain}=6.35\%\), \(\mu_{lock}=\rho_{grain}/\rho_{0}-1=0.068\). \[P_{lock}= \begin{cases}0.8+(f_{c}^{\prime}-48)/460&f_{c}^{\prime}<140\\ 1.0+(f_{c}^{\prime}-140)/460&f_{c}^{\prime}<140\end{cases}