## 4 Validation of the Viscoelastic Constitutive Model with Damage
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \(\dot{\epsilon}\) & \(A\) & \(\alpha\) & \(\beta\) & \(E_{a}\) & \(E_{2}\) & \(\theta_{2}\) & \(\delta\) & \(R^{2}\) \\ \hline \multirow{4}{*}{Coal rock} & \(120\) & \(3\) & \(-0.6\) & \(-1\) & \(-1.77042\times 10^{10}\) & \(1.77042\times 10^{10}\) & \(3\) & \(-0.0002\) & \(0.9648\) \\  & \(145\) & \(3\) & \(-0.59\) & \(-1\) & \(-2.76368\times 10^{10}\) & \(2.76368\times 10^{10}\) & \(3\) & \(-0.0002\) & \(0.9401\) \\  & \(186\) & \(3\) & \(-0.545\) & \(-1\) & \(-2.1608\times 10^{10}\) & \(2.1608\times 10^{10}\) & \(3\) & \(-0.0002\) & \(0.9584\) \\  & \(212\) & \(3\) & \(-0.51\) & \(-1\) & \(-1.5297\times 10^{10}\) & \(1.5297\times 10^{10}\) & \(3\) & \(-0.0002\) & \(0.9739\) \\ \hline \multirow{4}{*}{Foamed concrete} & \(156.47\) & \(3\) & \(0.80\) & \(0.30\) & \(3005.86\) & \(-3005.93\) & \(3\) & \(0.0002\) & \(0.9953\) \\  & \(235.29\) & \(3\) & \(0.68\) & \(0.20\) & \(12,479.65\) & \(-12,479.82\) & \(3\) & \(0.0002\) & \(0.9945\) \\  & \(254.12\) & \(3\) & \(0.64\) & \(0.30\) & \(14,062.01\) & \(-14,062.27\) & \(3\) & \(-0.0002\) & \(0.9946\) \\  & \(298.82\) & \(3\) & \(0.50\) & \(0.25\) & \(88,387.15\) & \(-88,388.46\) & \(3\) & \(0.0002\) & \(0.9946\) \\  & \(342.94\) & \(3\) & \(0.70\) & \(0.35\) & \(21,752.35\) & \(-21,752.57\) & \(3\) & \(-0.0002\) & \(0.9881\) \\ \hline Steel fiber-reinforced concrete & \(38.24\) & \(3\) & \(-0.485\) & \(-1\) & \(-2.58732\times 10^{8}\) & \(2.58731\times 10^{8}\) & \(3\) & \(-0.0002\) & \(0.9625\) \\  & \(120.13\) & \(3\) & \(-0.14\) & \(-0.2\) & \(3.84645\times 10^{8}\) & \(-3.84677\times 10^{8}\) & \(3\) & \(-0.0002\) & \(0.9712\) \\ \hline \multirow{4}{*}{Granite} & \(70.99\) & \(3\) & \(-0.27\) & \(-0.47\) & \(-2.77818\times 10^{9}\) & \(2.77818\times 10^{9}\) & \(3\) & \(-0.0002\) & \(0.9583\) \\  & \(246.93\) & \(3\) & \(0.02\) & \(0.33\) & \(2.18859\times 10^{10}\) & \(-2.18859\times 10^{10}\) & \(3\) & \(-0.0002\) & \(0.9943\) \\ \hline Lightweight & \(64\) & \(3\) & \(-0.0026\) & \(-0.7\) & \(155,087.9264\) & \(-155,093.636\) & \(3\) & \(-0.0002\) & \(0.9078\) \\ foamed concrete & \(266\) & \(3\) & \(-0.34\) & \(-0.7\) & \(5.30233\times 10^{7}\) & \(-5.30245\times 10^{7}\) & \(3\) & \(-0.0002\) & \(0.9105\) \\ \hline EPS concrete & \(51.12\) & \(3\) & \(-0.077\) & \(0.06\) & \(-2.97949\times 10^{9}\) & \(2.9796\times 10^{9}\) & \(3\) & \(-0.0002\) & \(0.9776\) \\  & \(74.29\) & \(3\) & \(0.1\) & \(0.5\) & \(-1.00786\times 10^{9}\) & \(1.00788\times 10^{9}\) & \(3\) & \(-0.0002\) & \(0.9838\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitted curve parameters. the constitutive equation established in this paper has wide applicability. 1 stOpt is a mathematical optimization analysis synthesis tool developed by 7D-Soft High Technology Inc. in China, which is widely used in the fields of nonlinear regression, curve fitting, and nonlinear complex engineering mode parameter evaluation solutions, etc. The data in the references in this paper were extracted by GetData. The parameter A is a constant in the principal structure model established in this paper, and determines the magnitudes of the parameters \(E_{a}\) and \(E_{2}\). The parameter alpha and the parameter beta jointly determine the trend of the whole curve. The parameter alpha has a greater influence on the peak strain and peak stress of the curve. The parameter beta has a greater influence on the direction of the opening of the curve. The parameter \(E_{a}\) is determined by the rising section of the curve, and \(E_{2}\) is determined by the falling section of the curve. The intrinsic constitutive model established in this paper also has certain defects, for example, the model was only able to describe the large trend of stress-strain, and it could not accurately describe the curve wavy fluctuation. The description of material strength and weakness by the intrinsic constitutive model is described by \(E_{a}\) and \(E_{2}\).