# A Modified Plastic Burgers Model for the Non-Decaying Creep of Frozen Soils
## 4 Parameter Calibration Method  
### Plastic Burgers Model Deformation Parameters  
As expressed in Formula (1), the sum of the Maxwell, Kelvin and plastic Burgers model strains enables the description of the decaying creep process. Correspondingly, the deformation parameters (\(G_{\text{M}}\), \(G_{\text{K}}\), \(H_{\text{M}}\) and \(H_{\text{K}}\)) can be determined on the typical creep \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Temperature/\({}^{\circ}\)C** & & & & & **Axial Stress/MPa** & & & \\ \hline \(-5.0\) & 1.13 & 1.23 & 1.35 & 1.42 & 1.45 & 1.52 & 1.67 & 1.95 \\ \(-10.0\) & 1.99 & 2.20 & 2.28 & 2.44 & 2.58 & 3.00 & 3.47 & 3.97 \\ \(-15.0\) & 2.69 & 2.85 & 3.00 & 3.10 & 3.40 & 3.79 & 4.19 & 5.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The applied axial stresses of the triaxial creep tests. Figure 4: Strength envelopes of the tested soil at different temperatures. Figure 3: Grain size distribution of the tested silty clay. curve during the decaying stage. The tested creep curve under a temperature of \(-5.0\)\({}^{\circ}\)C and with an axial stress of 1.52 MPa was used to demonstrate the determination of each deformation parameter (Figure 5). According to Formula (1), the creep strain without the plastic part is expressed as  
\[\varepsilon_{ij}=\frac{\sigma_{ij}}{2G_{M}}+\frac{\sigma_{ij}}{2H_{M}}t+\frac{ \sigma_{ij}}{2G_{k}}[1-\exp(-\frac{G_{k}}{H_{k}}t)] \tag{9}\]  
When time \(t=0\), Formula (9) becomes  
\[\varepsilon_{t_{0}}=\frac{\sigma_{ij}}{2G_{M}} \tag{10}\]  
where \(\varepsilon_{t_{0}}\) is the instantaneous strain after overloading is applied, and the Maxwell elastic shear modulus \(G_{M}\) is calculated with Formula (10) as  
\[G_{M}=\frac{\sigma_{ij}}{2\varepsilon_{t_{0}}} \tag{11}\]  
As for the Kelvin strain, its development tendency is dependent upon the law of an exponential function (Formula (9)). Theoretically, when the time \(t\) approaches infinity, the exponential function part of the Kelvin strain will approach zero. For a practical creep curve, as shown in Figure 5, when the time is larger than a time point \(t_{1}\), the exponentially functioning part of the Kelvin strain will decrease to be about zero. Here, \(t_{1}\) is the intersection time of the creep strain rate's non-linearly decreasing section and the straight section with an essentially constant creep strain rate. And thus, the creep strain (\(\varepsilon_{t1+}\)) after time \(t_{1}\) can be expressed as  
\[\varepsilon_{t1+}=\frac{\sigma_{ij}}{2G_{M}}+\frac{\sigma_{ij}}{2H_{M}}t+\frac {\sigma_{ij}}{2G_{k}} \tag{12}\]  
Formula (12) is the expression of the straight section on the creep curve, and the intersection \(\varepsilon_{t1}\) (Figure 5) between the straight section extension line and the strain axis is  
\[\varepsilon_{t1}=\frac{\sigma_{ij}}{2G_{M}}+\frac{\sigma_{ij}}{2G_{k}} \tag{13}\]  
With Formula (13), the Kelvin elastic shear modulus \(G_{k}\) is calculated as  
\[G_{k}=\frac{\sigma_{ij}G_{M}}{2\varepsilon_{t1}G_{M}-\sigma_{ij}} \tag{14}\]  
Figure 5: Creep curve of the tested silty clay under a temperature of \(-5.0\)\({}^{\circ}\)C and with an axial stress of 1.52 MPa. In the second term on the right side of Formula (12), the \(\sigma_{\rm ij}/2H_{\rm M}\) is the slope of the straight section (Figure 5). Apparently, the Maxwell viscosity coefficient \(H_{\rm M}\) is calculated as  
Figure 4:0  
where \(\Delta\varepsilon_{ij}\) is the incremental strain after a time step \(\Delta t\). After \(G_{\rm M}\), \(G_{\rm K}\) and \(H_{\rm M}\) are determined with Formulas (11), (14) and (15), respectively, the Kelvin viscosity coefficient \(H_{\rm K}\) can be calculated with a strain value on the creep strain curve between \(t_{0}\) and \(t_{1}\) (Figure 5). According to Formula (9), the Kelvin viscosity coefficient \(H_{\rm K}\) is calculated as  
Figure 4:1  
According to Formulas (11), (14)-(16), the plastic Burgers model deformation parameters were determined with the test results and are listed in Figure 4:2