## 4 Model Construction Based on Damage and Residual Stress
By combining the above types:  
\[\sigma_{z}(t)=\exp[-\frac{(\epsilon_{s}(t)-\epsilon_{th})^{q}}{p}]*[(E_{fx}+E_{nt1 })\epsilon_{s}(t)+\theta_{nt2}E_{nt2}\epsilon_{s}(t)-\theta_{nt2}\epsilon_{s}^{ \bullet}(t)\sqrt{\theta_{nt2}E_{nt2}\exp\varepsilon_{s}(t)}]+\{1-\exp[-\frac{ (\epsilon_{s}(t)-\epsilon_{th})^{q}}{p}]Y_{al}\} \tag{10}\]  
In Formula (10), the sum values of \(E_{fx}\) and \(E_{nt1}\) are substituted into the overall calculation, and \(E_{fx}+E_{nt1}=E_{h}\) is given to facilitate calculation. Therefore, this model needs to be fitted with a total of 6 unknowns, which are \(p\), \(q\), \(E_{h}\), \(E_{nt2}\), \(\theta_{nt2}\), \(Y_{al}\), respectively. The statistical analysis of optimization model parameters under different strain rates and alumina fiber contents is shown in Table 4. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Testing Program** & \(\tilde{\mathbf{\epsilon}}\) & \(\epsilon_{th}\) & \(p\) & \(q\) & \(E_{h}\) & \(E_{nt2}\) & \(\theta_{nt2}\) & \(Y_{al}\) \\ \hline \multirow{8}{*}{PCCY} & 60.20 & 5.52 & 5.68 & 0.92 & 107.28 & \(-90.05\) & \(-404.64\) & 79.40 \\ \cline{2-10}  & 69.55 & 6.98 & 3.40 & 1.06 & 723.76 & 21.05 & 3.92 & 10.32 \\ \cline{2-10}  & 83.98 & 7.92 & 5.22 & 1.14 & 74.25 & \(-76.25\) & 16.78 & 10.44 \\ \cline{2-10}  & 97.60 & 6.88 & 3.02 & 0.92 & 0.09 & \(-1.90\) & \(-0.01\) & 30.95 \\ \cline{2-10}  & 115.13 & 6.98 & 12.31 & 0.97 & 617.86 & 33.92 & 90.27 & 126.49 \\ \hline \multirow{8}{*}{1 ACCY} & 70.67 & 5.47 & 3.23 & 1.00 & 195.23 & \(-98.10\) & 24.87 & 44.24 \\ \cline{2-10}  & 81.67 & 6.94 & 4.85 & 1.08 & 965.22 & 64.30 & \(-84.73\) & 31.38 \\ \cline{2-10}  & 91.05 & 8.63 & 3.92 & 0.83 & \(-2.82\) & 17.32 & 0.02 & 4.11 \\ \cline{2-10}  & 98.87 & 7.97 & 14.07 & 0.83 & 792.60 & 9.48 & 9.09 & 86.47 \\ \cline{2-10}  & 114.79 & 8.54 & 13.32 & 1.65 & 900.00 & \(-3.19\) & \(-87.47\) & \(-10.23\) \\ \hline \multirow{8}{*}{2 ACCY} & 66.23 & 6.12 & 6.34 & 1.12 & 36.16 & \(-17.33\) & \(-151.36\) & 105.51 \\ \cline{2-10}  & 83.87 & 6.71 & 43.73 & 1.75 & 434.89 & 51.19 & 95.52 & \(-28.76\) \\ \cline{2-10}  & 93.88 & 6.81 & 14.14 & 1.75 & 157.26 & 40.58 & \(-7.53\) & \(-18.96\) \\ \cline{2-10}  & 98.57 & 7.90 & 4.97 & 1.60 & 137.22 & \(-19.68\) & \(-118.58\) & \(-15.71\) \\ \cline{2-10}  & 113.97 & 7.30 & 4.43 & 0.97 & 95.52 & 94.69 & \(-20.94\) & 12.13 \\ \hline \multirow{8}{*}{3 ACCY} & 67.16 & 6.14 & 3.54 & 1.12 & 110.50 & 7.82 & \(-94.71\) & \(-0.28\) \\ \cline{2-10}  & 87.76 & 6.66 & 3.53 & 1.12 & 300.20 & \(-2.86\) & 88.19 & \(-0.29\) \\ \cline{2-10}  & 98.86 & 7.19 & 4.16 & 1.05 & 577.63 & 4.95 & \(-58.28\) & \(-0.32\) \\ \cline{2-10}  & 110.66 & 8.03 & 5.17 & 1.09 & 326.62 & 23.67 & \(-73.23\) & \(-0.46\) \\ \cline{2-10}  & 116.98 & 8.56 & 5.35 & 1.07 & 262.97 & \(-66.73\) & 126.69 & \(-0.41\) \\ \hline \multirow{8}{*}{4 ACCY} & 63.51 & 5.02 & 3.24 & 1.59 & 403.80 & 418.67 & \(-51.08\) & 15.19 \\ \cline{2-10}  & 84.80 & 5.50 & 0.65 & 0.71 & 505.23 & \(-92.35\) & 4.05 & 84.05 \\ \cline{2-10}  & 90.03 & 5.89 & 8.70 & 0.99 & 732.38 & 47.14 & 102.07 & 31.47 \\ \cline{2-10}  & 101.11 & 6.27 & 4.23 & 1.03 & 450.45 & 52.27 & 77.78 & \(-0.21\) \\ \cline{2-10}  & 106.54 & 6.54 & 14.88 & 1.54 & 110.42 & 7.03 & \(-18.54\) & \(-13.12\) \\ \hline \multirow{8}{*}{5 ACCY} & 59.66 & 3.72 & 8.59 & 1.43 & 327.36 & 49.64 & 164.21 & 2.15 \\ \cline{2-10}  & 69.04 & 4.32 & 6.42 & 1.28 & 184.68 & 55.71 & 63.99 & 5.57 \\ \cline{2-10}  & 85.78 & 5.09 & 2.95 & 1.03 & 442.97 & 42.03 & \(-63.40\) & \(-0.10\) \\ \cline{2-10}  & 95.90 & 5.35 & 3.09 & 0.96 & 125.43 & 56.97 & 29.64 & \(-0.07\) \\ \cline{2-10}  & 104.08 & 6.04 & 4.15 & 0.98 & 276.95 & \(-73.65\) & 82.51 & 0.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitting results of impact compression test. So far, the prediction model of aluminum oxide fiber-reinforced concrete under different strain rates has been constructed. The compressive strength obtained by the prediction model is compared with the actual measured compressive strength (as shown in Figure 10).