## 4 Model Construction Based on Damage and Residual Stress  
As a material formed by a combination of multiple materials, aluminum oxide fiber concrete is formed by a cement stone which is wrapped and united with other materials by the action of \"glue\". Zhu-Wang-Tang non-linear viscoelastic model (ZWT model) [29] is a constitutive model based on nonlinear theory with a wide application range of strain rate; as shown in Figure 8, the model better describes the behavior of cement-based, rock, and other media in a certain range. The specific expression is as follows:  
\[\left\{ \begin{array}{l}\sigma=\sigma_{fx}+\sigma_{nt1}+\sigma_{nt2}\\ \sigma_{fx}=E_{fx}\varepsilon_{s}(t)+\alpha\varepsilon_{s}(t)^{2}+\beta \varepsilon_{s}(t)^{3}\\ \sigma_{nt1}=E_{nt1}\int_{0}^{t}\frac{\sigma}{\varepsilon_{s}}(t)\exp(\frac{ \tau-t}{\theta_{nt1}})d\tau\\ \sigma_{nt2}=E_{nt2}\int_{0}^{t}\frac{\sigma}{\varepsilon_{s}}(t)\exp(\frac{ \tau-t}{\theta_{nt2}})d\tau\end{array} \right. \tag{5}\]  
In the Formula (5): \(\sigma_{fx}\) represents nonlinear elastomer, the first unit of the elastomer composition model; \(\sigma_{nt1}\sigma_{nt2}\) is a viscoelastic body, which is used to describe the behavior of low and high strain rates, respectively, and constitutes the second and third units of the model. \(E_{fx}\) is the elastic constant; \(E_{nt1}\) and \(E_{nt2}\) are linear elastic moduli; \(\alpha\) and \(\beta\) are material parameters; \(\theta_{nt1}\) and \(\theta_{nt2}\) are the reaction characteristic time of the sticky pot in the viscoelastic body. In this model, \(\alpha\) and \(\beta\) in part \(\sigma_{fx}\) are constant values. In order to solve the problem that the nonlinear elastic part is difficult to calculate and to simplify the difficulty of calculation, the maintenance time of impact compression in this test is between \(10^{-6}\) s and \(10^{-5}\) s, and part \(\alpha\varepsilon_{s}(t)^{2}+\beta\varepsilon_{s}(t)^{3}\) of non-linear elastomer that fails to react in time and the part under the low strain rate are ignored. The final optimized ZWT model is shown in the Figure 9.  
After simplifying and integrating the optimized Formula (5), it is sorted out as follows:  
\[\left\{ \begin{array}{l}\sigma_{xt}=E_{fx}\varepsilon_{s}(t)+E_{nt1} \varepsilon_{s}(t)\\ \sigma_{nt2}=\theta_{nt2}E_{nt2}\vec{\varepsilon}_{s}(t)-\
icefrac{{-\theta_{ nt2}\vec{\varepsilon}_{s}(t)}}{{\sqrt{\theta_{nt2}E_{nt2}\exp\varepsilon_{s}(t)}}}\\ \sigma=\sigma_{xt}+\sigma_{nt2}\end{array} \right. \tag{6}\]  
Based on the experimental analysis, the addition of alumina fiber has a significant strengthening effect on concrete, and its overall stress-strain curve has an obvious tendency of defect damage being suppressed. The defect damage coefficient is incorporated into the model, and the expression of the damage model is as follows:  
\[SS=1-\exp[-\frac{(\varepsilon_{s}(t)-\varepsilon_{th})^{q}}{p}] \tag{7}\]  
In the Formula (7), \(p\) and \(q\) are the parameters to be solved, respectively; \(\varepsilon_{th}\) is the critical strain, when the strain exceeds this range, damage will occur; otherwise, it is assumed that its integrity is good; \(\varepsilon_{th}\) is the final value of the linear elastic stage; 0.8 \(\varepsilon_{p}\) is for easy calculation. After adding the damage coefficient, the original formula becomes:  
\[\sigma_{ss}=(1-SS)\sigma \tag{8}\]  
After impact compression, the stress-strain curve of the specimen at the end of the test does not directly contact with the coordinate axis, so it can be judged that there is still residual stress, so it is expressed as follows:  
\[\sigma_{z}=(1-SS)\sigma+SS\cdot Y_{al} \tag{9}\]  
Figure 8: ZWT model. Figure 9: Concept diagram of optimized model. In Formula (9), \(Y_{al}\) is the part of stress borne by alumina fiber at the damaged site.