## 2 Improved Gene Expression Programming for Determining a Creep Model
### Validation of Determining Creep Models Based on Improved Gene Expression Programming
#### 2.3.1 The Construction of the Creep Model of Silstone  
In the first example, the siltstone from the Niumasi Shuitou Coal Mine in the Hunan province of China was taken as an example to illustrate how to obtain a creep model based on the improved gene expression programming. The basic parameters of the siltstone are listed in Table 1.  
In our experiments, the sample was under the uniaxial compressive loading state, and four stress levels were utilized: 21 MPa, 26 MPa, 31 MPa, and 36 MPa. The time for every stress level was 20 h; after 20 h, the stress was loaded to the next stress level at a speed of 1 MPa/s. The strain-time experimental data, which are displayed in Figure 6, were subsequently obtained.  
For convenience in obtaining the creep model of the siltstone, the experimental data were divided into four parts when the stress levels were 21 MPa, 26 MPa, 31 MPa, and 36 MPa, as shown in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Uniaxial Compressive Strength (MPa)** & **Poisson’s Ratio** &
\begin{tabular}{c} **Young’s Modulus** \\ **(GPa)** \\ \end{tabular} & **Friction Angle (\({}^{\circ}\))** & **Cohesion (MPa)** \\ \hline
48.44 & 0.41 & 42 & 15.8 & 18.37 \\ \hline \end{tabular}
\end{table}
Table 1: Basic mechanical parameters of the siltstone.  
Figure 6: Creep experimental data of siltstone under uniaxial compression.  
In this example, the function set is \(\{\text{N}\,\text{Y}\,\text{H}\}\) (N: viscous creep body, Y: plastic creep body, H: elastic creep body), and the function set is \(\{+||\}\) (+: connection in series and \(\ ||\): parallel connections), the head of a chromosome is 15, and the maximum argument of all functions in head \(n\) is 30. Based on Equation (3), the tail length of a chromosome is 436.  
Based on the improved gene expression programming, the creep constitutive model was obtained, and an improved GEP chromosome was obtained:  
\[\underbrace{++||\,\text{N}\,\text{H}\,\text{N}\,\text{H}+||\,\,||\,\text{H}\, \text{Y}\,\text{Y}\,\text{N}+}\underbrace{\text{N}\cdot\cdot\cdot\cdot\cdot \text{Y}}_{436} \tag{11}\]  
The expression tree can be expressed as follows (Figure 8):  
From the analysis of Figure 8, it should be noted that the yield creep body (Y) did not appear, which means that the yield creep body was eliminated in the procedure of the improved GEP. The expression tree (Figure 8) can be decoded as a creep constitutive model, which is illustrated as follows (Figure 9).  
Figure 8: The expression tree based on the chromosome code Equation (11).  
Figure 7: Experimental data of the creep experiments when the stress level is 21 MPa, 26 MPa, 31 MPa, and 36 MPa.  
The corresponding constitutive equation was as follows.  
\[\varepsilon(t)=\frac{\sigma_{0}}{k_{1}}+\frac{\sigma_{0}}{\eta_{1}}t+\frac{ \sigma_{0}}{k_{2}}(1-e^{-\frac{k_{2}}{\eta_{2}}t}) \tag{12}\]  
The fitting results are displayed in Table 2 and Figure 10.  
An analysis of the data in Table 2 and Figure 10 reveals that the theoretical curves of the creep model obtained via improved gene expression programming are in good agreement with the experimental data, with a correlation coefficient (R\({}^{2}\)) above 0.93. The validity of the improved GEP for establishing the creep model was verified.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Stress Level (MPa)** & \(k_{1}\) **(MPa)** & \(\eta_{1}\) **(MPa \(\cdot\) h)** & \(k_{2}\) **(MPa)** & \(\eta_{2}\) **(MPa \(\cdot\) h)** & \(\mathbf{R}^{2}\) \\ \hline
21 MPa & 34,711.05 & 1,100,846.61 & 99,293.74 & 155,980.92 & 0.93 \\
26 MPa & 13,047.79 & 123,918,901.65 & 90,361.39 & 375,980.95 & 0.98 \\
31 MPa & 9953.07 & 3,908,880.31 & 188,253.59 & 225,143.76 & 0.94 \\
36 MPa & 5328.21 & 1,029,447.11 & 40,585.21 & 98,900.92 & 0.99 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The parameters of the creep body in the creep model.  
Figure 10: The fitting result.  
Figure 9: Creep model obtained via improved GEP (Burgers model).