# A Variable-Order Dynamic Constitutive Model for Clay Based on the Fractional Calculus
## 4 Establishment of Dynamic Constitutive Model  
### Creep Strain of Clay under Long-Term Static Load  
According to the previous analysis, we can draw the conclusion that the rheological strain of clay contains instantaneous strain \(\varepsilon_{e}\) and viscoelastic strain \(\varepsilon_{ve}\), and these can be expressed by the hook body and the variable-order fractional Abel dashpot, respectively. Assuming the stress applied to the sample is equal to \(\sigma_{0}\), i.e., \(\sigma_{0}=(\sigma_{\text{d.max}}+\sigma_{\text{d.min}})/2\), the instantaneous strain can be expressed as  
\[\varepsilon_{e}=\frac{\sigma_{0}}{E_{0}} \tag{15}\]  
where \(E_{0}\) is the elastic modulus of the spring in the hook body. Based on the constitutive model of the variable-order fractional Abel dashpot, the viscoelastic strain expressed as \(\varepsilon_{ve.1}(t)\) for the initial creep stage sand \(\varepsilon_{ve.2}(t)\) for the steady creep stage can be calculated as follows:  
\[\varepsilon_{ve.1}(t)=\frac{\sigma_{0}}{\eta_{1}}\frac{(t-0)^{\alpha_{1}}}{ \Gamma(\alpha_{1}+1)} \tag{16}\]  
\[\varepsilon_{ve.2}(t)=\frac{\sigma_{0}}{\eta_{2}}\frac{(t-t_{1})^{\alpha_{2}}} {\Gamma(\alpha_{2}+1)} \tag{17}\]  
The viscosity coefficient \(\eta_{k}\) and the fractional order \(\alpha_{k}\) can be assigned by a mass of values in the creep process, and the relation between \(\eta_{k}\) and \(\alpha_{k}\) is shown in Table 1. To simplify the final creep constitutive model, two orders were taken out in the initial and steady stages, respectively. If the stress \(\sigma_{0}\) remains unchanged, when the loading duration is in the range of 0 to \(t_{1}\), the creep constitutive model in initial stage is expressed thus:  
\[\varepsilon_{s}(t)=\varepsilon_{e}+\varepsilon_{ve.1}(t)=\frac{\sigma_{0}}{E_ {0}}+\frac{\sigma_{0}}{\eta_{1}}\frac{t^{\alpha_{1}}}{\Gamma(\alpha_{1}+1)}\ \left(0<t\leq t_{1}\right) \tag{18}\]  
Similarly, in the case of steady stage, we have  
\[\varepsilon_{s}(t)=\varepsilon_{ve.2}(t)=\frac{\sigma_{0}}{\eta_{2}}\frac{(t- t_{1})^{\alpha_{2}}}{\Gamma(\alpha_{2}+1)}\ \left(t_{1}<t\leq t_{2}\right) \tag{19}\]  
Whether the clay undergoes the steady stage or the accelerating stage during the loading process mainly depends on the magnitude of \(\sigma_{0}\) and on the clay's material properties; when the static load level exceeds the long-term strength \(\sigma_{s}\), the clay will experiences accelerating creep; when \(\sigma_{0}\) is less than the long-term strength \(\sigma_{s}\), the clay will experience decay and steady creep. The entire creep strain of clay can be portrayed by the model based on the variable-order fractional Abel dashpot, as shown in Figure 9. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Time** & **Fractional Order \(\alpha_{k}\)** & **Viscosity Coefficient \(\eta_{k}\)** \\ \hline \(0<t\leq t_{1}\) & \(\alpha_{1}\) & \(\eta_{1}\) \\ \(t_{1}<t\leq t_{2}\) & \(\alpha_{2}\) & \(\eta_{2}\) \\ \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \(t_{n-1}<t\leq t_{n}\) & \(\alpha_{n}\) & \(\eta_{n}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of fractional order \(\alpha_{k}\) and viscosity coefficient \(\eta_{k}\) versus time. For this viscoplastic body, the stress \(\sigma_{0}\) of the fractional element is given by  
\[\sigma_{0}=\left\{ \begin{aligned} 0&\quad\sigma< \sigma_{\text{s}}\\ \sigma_{\text{s}}&\quad\sigma\geq\sigma_{\text{s}}\end{aligned} \right. \tag{20}\]  
Although the variable-order Abel dashpot produces a better fit of experimental data in the attenuation and steady stages, when compared with the constant-order Abel dashpot, the method of adjusting these parameters in the variable-order Abel dashpot is not optimal for describing the mechanical properties of clay in the accelerating stage. On the one hand, the order of fractional derivative is related to the state of clay; once the order exceeds 1, the underlying physical meaning is hard to explain.