## 2 Establishment of Constitutive Model
In the dynamic loading experiment of SHPB in this paper, the impact loading time is very short. The loading time scale of the low-frequency Maxwell body was between \(10^{0}\) and \(10^{2}\) s. Therefore, the loading time in this experiment is not one order of magnitude as that of the low-frequency Maxwell body, and the former is much smaller than the latter. Therefore, when the dynamic load is completed, the low-frequency Maxwell body is not able to relax. Therefore, the second medium-low-frequency Maxwell body in the ZWT constitutive model can be replaced by a simple spring. The simplified physical model is shown in Figure 2, and the formula of the ZWT constitutive model can be further simplified as:  
\[ \begin{split}\sigma=\sigma_{e}+\sigma_{m1}+\sigma_{m2}& =E_{0}\varepsilon+E_{1}\varepsilon+E_{2}\int_{0}^{t}\frac{\bullet}{ \varepsilon}(\tau)\exp\Bigl{(}-\frac{t-\tau}{\theta_{2}}\Bigr{)}d\end{split}  \tag{3}\]  
Figure 1: The “Zhu–Wang–Tang” constitutive model. As shown in Figure 3, springs \(E_{0}\) and \(E_{1}\) are parallel, and a simple spring \(E_{a}\) can replace the two parallel springs. The equivalent physical model is shown in Figure 3. The ZWT constitutive model is further simplified to:  
\[\sigma=\sigma_{e}+\sigma_{m1}+\sigma_{m2}=E_{a}\varepsilon+E_{2}\int_{0}^{t} \overset{\bullet}{\varepsilon}(\tau)\exp\biggl{(}-\frac{\mathrm{t}-\tau}{ \theta_{2}}\biggr{)}d\tau \tag{4}\]  
(c) Improvement I. The material cannot be completely evenly distributed in the production process of composite concrete. From a microscopic point of view, there are disordered micropores and micro-cracks inside, so the damage factor must be considered when establishing the constitutive model. The distribution of micro-pores and micro-cracks in concrete is very complex and has great discreteness. In order to simplify the model, the continuum damage mechanics method is adopted, that is, the composite concrete is assumed to be a continuous medium. The macroscopic damage \(D\) is introduced to evaluate the damage degree of composite concrete. According to the Lemaitre strain equivalence principle, the basic form of the constitutive model is as follows:  
\[\sigma_{a}=(1-D)\;\sigma_{r} \tag{5}\]  
In this formula, \(\sigma_{a}\) represents effective stress; \(\sigma_{r}\) represents the original stress; and \(D\) represents the damage variable. Strain and strain rate will affect the damage evolution process, and the damage evolution equation is as follows:  
\[D=A\times\left(\frac{\overset{\bullet}{\varepsilon}}{\varepsilon_{0}}\right) ^{a}\varepsilon^{\beta} \tag{6}\]  
Figure 3: Equivalent constitutive model. Figure 2: The “Zhu–Wang–Tang” model simplified. In this formula, \(\mathbf{\tilde{\epsilon}}_{0}=10^{-4}s^{-1}\) represents the reference value of the strain rate; \(\mathbf{\tilde{\epsilon}}\) represents the strain rate; and \(A\), \(\alpha\), and \(\beta\) represent undetermined coefficients respectively. In summary, the Equations (4)-(6) can be obtained as:  
\[\sigma_{a}=(1-D)\sigma_{r}=\left[1-A\left(\frac{\mathbf{\tilde{\epsilon}}}{\mathbf{ \tilde{\epsilon}}_{0}}\right)^{\alpha}\epsilon^{\beta}\right]\left[E_{a} \epsilon+E_{2}\int_{0}^{t}\mathbf{\tilde{\epsilon}}(\tau)\exp\left(-\frac{t-\tau} {\theta_{2}}\right)d\tau\right] \tag{7}\]  
Considering that SHPB is approximately assumed to be constant strain rate, the constitutive model can be rewritten as follows:  
\[\sigma_{a}=(1-D)\sigma_{r}=\left[1-A\left(\frac{\mathbf{\tilde{\epsilon}}}{\mathbf{ \tilde{\epsilon}}_{0}}\right)^{\alpha}\epsilon^{\beta}\right]\left\{E_{a} \epsilon+E_{2}\theta_{2}^{\bullet}\mathbf{\tilde{\epsilon}}\Big{[}1-\exp\left(- \frac{\epsilon}{\mathbf{\tilde{\epsilon}}\theta_{2}}\right)\Big{]}\right\} \tag{8}\]  
Although the improved ZWT constitutive model in this paper considers the damage factor, it still assumes that the material is continuous. In reality, there are a series of uncertainties in the material such as voids, cracks, and in homogeneous weave that may lead to early or delayed arrival of the stress maximum in the stress-strain curve.