## 4 Unified Model of Permanent Deformation  
### Comparative Analysis of Existing Models  
The establishment of a permanent deformation model has always been the focus of many scholars. The most widely used model was proposed by [PERSON] et al. [40], which was defined as:  
\[\epsilon=\alpha_{1}N^{\beta_{1}} \tag{7}\]  
where \(\epsilon\) indicates the permanent deformation, \(N\) represents the numbers of cyclic loading, and \(\alpha_{1}\) and \(\beta_{1}\) are the fitting parameters of this model.  
The permanent deformation model proposed by [PERSON] et al. [35] was defined as:  
\[\epsilon=\alpha_{2}\Big{(}1-e^{-\beta_{2}N}\Big{)} \tag{8}\]  
where \(\alpha_{2}\) and \(\beta_{2}\) are the fitting parameters.  
[PERSON] et al. [41] proposed the following permanent deformation model:  
\[\epsilon=\alpha_{3}+\beta_{3}\ln N \tag{9}\]  
where \(\alpha_{3}\) and \(\beta_{3}\) are the fitting parameters.  
[PERSON] et al. [37] proposed the permanent deformation model as follows:  
\[\epsilon=\frac{N}{\alpha_{4}+\beta_{4}N+\gamma_{4}N^{0.5}} \tag{10}\]  
where \(\alpha_{4}\), \(\beta_{4}\), and \(\gamma_{4}\) are the fitting parameters.  
According to the relationship between \(\epsilon\) and \(N\) presented in Figures 8-10, the permanent deformation of CGSF under cyclic loading is analyzed using a data-fitting method based on the above models in Equations (7)-(10). Analysis shows that fitting parameters of CGSF in Equation (10) cannot be obtained. Hence, Equation (10) is not investigated in this paper. The fitting parameters are displayed in Table 2.  
According to the correlation coefficient (\(R^{2}\)) of fitting parameters in Table 2, the existing permanent deformation model cannot accurately enough describe the permanent deformation of CGSF under cyclic loading, and \(R^{2}\) values are all less than 0.9. The permanent deformation curve can be divided into three types [42]: (1) Stable/attenuation type, where \(\epsilon\) first increases with the increase of \(N\) and then tends to stay at a stable value when \(N\) achieves a larger value. (2) Failure type, where \(\epsilon\) will increase sharply until the specimen fails when \(N\) reaches a bigger value. (3) Criticality type, where \(\epsilon\) distributes between (1) and (2). The existing permanent deformation model can describe the variation of one certain type of the mentioned curves well; however, there is a lack of a unified model of permanent deformation used for different types of permanent deformation curves. Therefore, it is necessary to put forward a unified model which can be generally applicable to different types of permanent deformation based on the mentioned CGSF tests.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Specimen** & **DT01** & **DT02** & **DT03** & **DT04** & **DT05** & **DT06** & **DT07** & **DT08** \\ \hline \(\alpha_{1}\) & 3.9085 & 2.8126 & 3.2501 & 4.9523 & 1.6604 & 12.1707 & 2.5312 & 3.7737 \\ \(\beta_{1}\) & 0.0667 & 0.0550 & 0.0345 & 0.0432 & 0.0693 & 0.0323 & 0.0635 & 0.0482 \\ \(R^{2}\) & 0.8159 & 0.7709 & 0.7050 & 0.7170 & 0.8858 & 0.3876 & 0.8570 & 0.7742 \\ \(\alpha_{2}\) & 7.3600 & 4.7367 & 4.4983 & 7.4476 & 3.2039 & 16.5600 & 4.6217 & 5.9533 \\ \(\beta_{2}\) & 0.0033 & 0.0041 & 0.0102 & 0.0062 & 0.0031 & 0.0051 & 0.0033 & 0.0047 \\ \(R^{2}\) & 0.6947 & 0.7091 & 0.5936 & 0.7042 & 0.6320 & 0.8911 & 0.6038 & 0.6182 \\ \(\alpha_{3}\) & 2.8128 & 2.2945 & 3.0339 & 4.4157 & 1.1584 & 11.2994 & 1.9108 & 3.2612 \\ \(\beta_{3}\) & 0.4808 & 0.2584 & 0.1558 & 0.3216 & 0.2163 & 0.5537 & 0.2866 & 0.2850 \\ \(R^{2}\) & 0.8560 & 0.8095 & 0.7321 & 0.7494 & 0.8980 & 0.4149 & 0.8937 & 0.8091 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Model parameters.