# A Dynamic Damage Constitutive Model of Rock-like Materials Based on Elastic Tensile Strain
## 2 The Construction of the HK Model
### Simple Loading Numerical Calculations  
#### 3.2.1 Numerical Calculation of a Single-Element Model  
The numerical calculation of a single-element model is hypothetical. A single-element model is established, and numerical simulations are performed using the HK and HK\({}_{0}\) models for pre-compression followed by tensile loading, uniaxial and triaxial equal loading, as well as tensile loading at different strain rates. The material parameters are detailed in Table 1, with corresponding results presented in Figure 1. In Figure 1a, the initiation of damage occurs earlier in the HK model as compared to the HK\({}_{0}\) model. The hydrostatic tensile strength calculated by the HK model aligns with the input value of parameter T, while that computed by the HK\({}_{0}\) model significantly exceeds the specified strength. In Figure 1b, the uniaxial and triaxial tensile strengths determined by the HK model are identical, whereas substantial disparities exist in those calculated by the HK\({}_{0}\) model. In Figure 1c, under loading conditions with a strain rate greater than the reference, both material strengths computed by the HK and HK\({}_{0}\) models exhibit rapid escalation with an increasing strain rate. Conversely, under loading conditions with a strain rate lower than the reference, while material strength remains constant for the HK model, it declines sharply for the HK\({}_{0}\) model.  
The computational results indicate the following: (1) When subjected to plastic compression followed by reverse stretching, the HK model effectively accounts for the influence of accumulated plasticity on damage calculation. Additionally, it synchronously calculates tensile damage and stress without encountering situations where the evolution relationship of damage fails to cover tensile stress as observed in the HK\({}_{0}\) model calculations, as shown in Figure 1a. (2) In comparison to the tensile strength predicted by the HK\({}_{0}\) model for uniaxial/triaxial tensile tests, the tensile strength predicted by the HK model is found to be consistent, aligning with established experimental findings, as shown in Figure 1b. (3) The HK model can reasonably describe the strain rate effect of the segmented form  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(G\)/GPa & \(f_{\text{C}}\)/MPa & \(T\)/MPa & \(C\) & \(A\) & \(B\) & \(n\) & \(S_{\text{max}}\) \\ \hline
17.3 & 131.2 & 11.4 & 0.007 & 0.54 & 1.53 & 0.82 & 10 \\ \hline \(\epsilon_{f}^{\text{min}}\) & \(D_{1}\) & \(D_{2}\) & \(P_{\text{C}}\)/MPa & \(U_{\text{C}}\) & \(P_{\text{L}}\)/MPa & \(U_{\text{L}}\) & \(K_{1}\)/GPa \\
0.02 & 0.04 & 1.0 & 92.4 & 0.004 & 800 & 0.1 & 85 \\ \hline \(K_{2}\)/GPa & \(K_{3}\)/GPa & \(K_{\text{IC}}\)/(MN\(\cdot\)cm\({}^{-3/2}\)) \(k\)/(cm\({}^{-3}\)) & \(m\) & & & \\ \(-171\) & 208 & \(2.747\times 10^{-4}\) & \(3.0\times 10^{14}\) & 6 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of high-strength steel fiber concrete [7; 10; 11; 12].  
Figure 1: Numerical calculation results of two models under different loading methods: (**a**) the evolution curve of hydrostatic pressure and damage under compress then tensile loading with volumetric strain (the tensile loading stage); (**b**) stress and damage evolution curves under uniaxial/triaxial tensile loading; (**c**) the curve illustrating the variation in tensile strength with strain rate.  
of the material, and the tensile strength obtained under low strain rate loading is more reasonable than that obtained using the HK\({}_{0}\) model, as shown in Figure 1c.