## 2 Improved Gene Expression Programming for Determining a Creep Model
### Validation of Determining Creep Models Based on Improved Gene Expression Programming
#### 2.3.2 The Establishment of the Creep Model of Mine Rock  
Moreover, an extra creep body can be added to the proposed method. For example, [PERSON] et al. [42] proposed a creep yield joint (CYJ) body; based on the definition of the CYJ creep body, its constitutive model can be described as follows:  
\[\left\{ \begin{array}{l}\dot{\epsilon}=\frac{\sigma}{\eta_{\mathrm{c}}}, \epsilon\leq\epsilon_{\mathrm{c}}\\ \dot{\epsilon}=\frac{\sigma}{\eta_{\mathrm{c}}}(\frac{\epsilon}{\epsilon_{ \mathrm{c}}})^{\eta},\epsilon>\epsilon_{\mathrm{c}}\end{array} \right. \tag{13}\]  
The creep characteristics of the CYJ creep body were determined by three parameters: the fitting parameter \(n\), the viscosity parameter \(\eta_{\mathrm{c}}\), and the strain parameter \(\epsilon_{\mathrm{c}}\). Equation (13) can be written as follows:  
\[\left\{ \begin{array}{l}\epsilon=\frac{\sigma}{\eta_{\mathrm{c}}}t,t\leq t_{ \mathrm{c}}\\ \epsilon=\epsilon_{\mathrm{c}}[\frac{(1-n)\sigma}{\eta\epsilon_{\mathrm{c}}}t +n]^{\frac{1}{1+n}},t>t_{\mathrm{c}}\end{array} \right. \tag{14}\]  
where \(t_{\mathrm{c}}=\frac{t_{\mathrm{c}}\epsilon_{\mathrm{c}}}{\sigma}\). To further verify the validity of the proposed method, another example was provided. [PERSON] et al. [10] studied the creep behaviour of the rock from the Jingchuang mine. Loading and unloading creep experiments under uniaxial compression were conducted. The uniaxial compressive strength of the rock is 44.5 MPa. The stress level contains five levels, namely, 8.089 MPa, 12.782 MPa, 17.473 MPa, 23.565 MPa, and 33.807 MPa, and the loading time determines the strain variation. When the axial strain variation is less than 0.001 mm/h, the strain tends to stabilize, and then the stress is unloaded to 0 MPa for 24 h; then, the next stress level begins. In the experiments, the loading and unloading speed was 0.5 MPa/s. based on the experiments. The strain-time curve was obtained, which is illustrated in Figure 11.  
In this example, the CYJ model was added to the function set: {N Y H C Y} (N: viscous creep body, Y: plastic creep body, H: elastic creep body, and C Y: CYJ creep body). By using the improved gene expression programming method, the creep chromosome model was obtained:  
\[\underbrace{+\ ||+\mathrm{N}\,\mathrm{H}\,\mathrm{H}\,||\ \mathrm{Y}\,\mathrm{C}_{ \mathrm{YJ}}+||\ \mathrm{N}+\mathrm{C}_{\mathrm{YJ}}\ \mathrm{YH}\cdot\cdot\cdot\cdot\cdot\mathrm{Y}}_{436} \tag{15}\]  
Figure 11: Strain–time curves from the creep experiments [10]. Then, Equation (15) can be decoded as a tree (Figure 12):  
The subsequent creep model is illustrated in Figure 13.  
The constitutive equation for the creep model (Figure 13) can be expressed as follows:  
\[ \begin{cases}\epsilon=\frac{\varphi}{k_{1}}+\frac{\varphi}{k_{2}}(1-e^{-\frac{k _{2}}{k_{1}}t})&(\sigma<\sigma_{\text{s}})\\ \epsilon=\frac{\varphi}{k_{1}}+\frac{\varphi}{k_{2}}(1-e^{-\frac{k_{2}}{k_{1} }t})+\frac{\sigma-\sigma_{\text{s}}}{\eta_{\text{c}}}t&(\sigma\geq\sigma_{ \text{s}},t\leq t_{\text{c}})\\ \epsilon=\frac{\varphi}{k_{1}}+\frac{\varphi}{k_{2}}(1-e^{-\frac{k_{2}}{k_{1} }t})+\epsilon_{\text{c}}[\frac{(1-n)(\sigma-\sigma_{\text{s}})}{\eta_{\text{c }}\varepsilon_{\text{c}}}+n]^{\frac{1}{1+n}}&(\sigma\geq\sigma_{\text{s}},t>t _{\text{c}})\\ t_{\text{c}}=\frac{\eta_{\text{c}}\varepsilon_{\text{c}}}{\sigma-\sigma_{ \text{s}}}&\end{cases}  \tag{16}\]  
The fitting results are shown in Figure 13, and the basic parameters of the creep model are listed in Table 3.