# Study on Creep Behavior of Silty Clay Based on Fractal Derivative
## 3 Fractal Creep Model
### Fractal Burgers Model  
The traditional Burgers model is composed of a Maxwell model and a Kelvin model in series, as shown in Figure 12a. The creep modulus of the traditional Burgers model is expressed as  
\[J(t)=\frac{1}{E_{m}}+\frac{1}{\eta_{m}}t+\frac{1}{E_{k}}\bigg{(}1-e^{-\frac{E_{k} }{\eta_{k}}t}\bigg{)} \tag{2}\]  
where \(\sigma\) and \(\varepsilon\) are the stress and strain, \(E\) and \(\eta\) are [PERSON]'s modulus and the viscosity coefficient, and the subscripts \(m\) and \(k\) represent Maxwell and Kelvin elements in a fractal form, respectively. Since the typical traditional Burgers model is unable to exactly fit the initial stage of viscoelastic creep, the fractional Burgers model replacing the Newton dashpot with the Abel dashpot was proposed, as shown in Figure 12b. The [PERSON] fractional derivative theory can be expressed as [42]  
\[\frac{d^{\beta}f(t)}{dt^{\beta}}=\frac{1}{\Gamma(n+1-\beta)}\int\frac{f^{(n+1)} (\tau)}{(t-\tau)^{(\beta-n)}}d\tau,n-1\leq\beta<n \tag{3}\]  
where \(\beta\) is the order of time fractal derivative, \(n\) represents a rational number, \(\Gamma(\cdot)\) denotes the gamma function. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Hooke Sping** & **Newton Dashpot** & **Abel Dashpot** & **Fractal Dashpot** \\ \hline Symbol & \(\sqrt[]{\text{N}}\) & & & \\ \hline Constitutive & & & & \\ modelRelations & & & & \\ \hline Relationship of & & & & \\ stress and strain & & & & \\ \hline The order value & \(\alpha=\beta=0\) & \(\alpha=\beta=1\) & \(0\leq\alpha=\beta\leq 1\) & \(\alpha=1,0\leq\beta\leq 1\) \\ \hline Creep model & traditional model & traditional model & fractional model & fractal model \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of different component models. Figure 11: Sensitivity of the creep strain to the derivative order and viscosity coefficient: (a) \(\sigma=\)100 kPa and \(\eta=\) 40 kPa-h\({}^{\beta}\); (b) \(\sigma=\) 100 kPa and \(\beta=\) 0.4. The fractional Burgers creep modulus based on [PERSON] fractional derivative theory can be expressed as [43]  
\[J(t)=\frac{1}{E_{m}}+\frac{1}{\eta_{m}}\frac{t^{\beta}}{\Gamma(1+\beta)}+\frac{1} {E_{k}}\Bigg{[}1-E_{\beta}\Bigg{(}-\Big{(}\frac{t}{\eta_{k}}\Big{)}^{\beta} \Bigg{)}\Bigg{]},0\leq\beta\leq 1 \tag{4}\]  
where \(E_{\beta}(\cdot)\) is a single-parameter Mittag-Leffler function proposed as [44]  
\[E_{\beta}(x)=\sum_{n=0}^{\infty}\frac{x^{n}}{\Gamma(1+\beta n)},\beta>0 \tag{5}\]  
This fractional derivative creep model can accurately fit the initial stage of viscoelastic creep and has been widely promoted and applied. However, it has the defect of the complex solution process of the fractional derivative convolution integral [45]. In this paper, we attempt to construct the fractal Burgers model by replacing the Abel dashpot with a fractal dashpot, as shown in Figure 12c. Under the premise of ensuring the accuracy of the simulated creep behavior, the calculation efficiency is improved by avoiding the local operator of convolutional integration. According to the component element rules between Maxwell and Kelvin, the stress-strain relationship of the fractal Burgers model can be described as  
\[\left\{ \begin{array}{l}\varepsilon=\varepsilon_{m}+\varepsilon_{k}\\ \sigma=\sigma_{m}=\sigma_{k}\\ \frac{\sigma_{m}}{\eta_{m}}+\frac{1}{E_{m}}\frac{d\sigma_{m}}{dt^{\beta}}= \frac{d\varepsilon_{m}}{dt^{\beta}}\\ \frac{\sigma_{k}}{\eta_{k}}=\frac{E_{k}}{\eta_{k}}\varepsilon_{k}+\frac{d \varepsilon_{k}}{dt^{\beta}}\end{array} \right. \tag{6}\]  
The constitutive relation is  
\[\sigma+\bigg{(}\frac{\eta_{m}+\eta_{k}}{E_{k}}+\frac{\eta_{m}}{E_{m}}\bigg{)} \frac{d\sigma}{dt^{\beta}}+\frac{\eta_{m}\eta_{k}}{E_{m}E_{k}}\frac{d}{dt^{ \beta}}(\frac{d\sigma}{dt^{\beta}})= \eta_{m}\frac{d\varepsilon}{dt^{\beta}}+\frac{\eta_{m}\eta_{k}}{E_{m}} \frac{d}{dt^{\beta}}(\frac{d\varepsilon}{dt^{\beta}}) \tag{7}\]  
Figure 12: Schematic of different Burgers models: (**a**) Burgers model; (**b**) fractional Burgers model; (**c**) fractal Burgers model.