## 4 Proposed Material Model and Analysis of the Test Results
### Analysis of the Test Results  
The long-term triaxial creep test results for the dam core specimens were assumed to follow the generalized Kelvin model in the linear form of creep compliance. After using trial and error find the best fit of the test results with reasonable accuracies for a simplified equation, two Kelvin elements were taken for Equations (4) and (5), that is,  
\[\rm D(t)=\frac{1}{E_{g}}+\frac{1}{E_{I}}\bigg{(}1-e^{-\frac{1}{v_{1}}}\bigg{)} +\frac{1}{E_{2}}(1-e^{-\frac{1}{v_{2}}})\]  
Figure 6: Comparison of the long-term axial creep strain curves (solid line) using the material model and the test results for the dam core specimens under various radially confining stress levels and at 5 \({}^{\circ}\)C. (**a**) Confining stress of 0.5 MPa (refer to Figure 1a). (**b**) Confining stress of 0.8 MPa (refer to Figure 1b). (**c**) Confining stress of 1.0 MPa (refer to Figure 1c). Figure 7: Comparison of the long-term axial creep strain curves (solid line) using the material model and the test results for dam core specimens under various radially confining stress levels and at 15 \({}^{\circ}\)C. (**a**) Confining stress of 0.5 MPa (refer to Figure 2a). (**b**) Confining stress of 1.0 MPa (refer to Figure 2b). (**c**) Confining stress of 1.5 MPa (refer to Figure 2c). Figure 8: Comparison of the long-term axial creep strain curves (solid line) using the material model and the test results for dam core specimens under various radially confining stress levels and at 20 \({}^{\circ}\)C. (**a**) Confining stress of 0.5 MPa (refer to Figure 3a). (**b**) Confining stress of 0.8 MPa (refer to Figure 3b). (**c**) Confining stress of 1.0 MPa (refer to Figure 3c). (**d**) Confining stress of 1.5 MPa (refer to Figure 3d). Figure 9: Comparison of the long-term axial creep strain curves (solid line) using the material model and the test results for dam core specimens under various radially confining stress levels and at 25 \({}^{\circ}\)C. (**a**) Confining stress of 0.5 MPa (refer to Figure 4a). (**b**) Confining stress of 1.0 MPa (refer to Figure 4b). Figures 6-10 indicate that the long-term axial creep strain curves derived from the model fit the test results well when the axial strain increased and approached a stable value. Equations (6) and (7) are used for constant temperature conditions. When the viscoelastic parameters are considered as functions of temperature, the equations are expressed as follows:  
\[\epsilon(t,T)=(\sigma_{1}-\sigma_{3})D(t,T) \tag{8}\]  
\[D(t,T)=\frac{1}{E(T)_{g}}+\frac{1}{E(T)_{1}}\bigg{(}1-e^{-\frac{t}{e(1_{1})}} \bigg{)}+\frac{1}{E(T)_{2}}(1-e^{-\frac{t}{e(1_{2})}}) \tag{9}\]  
\[D(T)_{0}=\frac{1}{E(T)_{cs}}=\frac{1}{E(T)_{g}}+\frac{1}{E(T)_{1}}+\frac{1}{E( T)_{2}} \tag{10}\]  
The relationships of the viscoelastic parameters and temperature from Table 2 are shown in Figures 11 and 12. Figure 11: Instantaneous elastic (red circle, Eg) and retardation moduli (black circle, E1; triangle, E2) versus temperature. Figure 10: Comparison of the long-term axial creep strain curve (solid line) using the material model and the test results for dam core specimens under a radial stress level of 0.8 MPa and at 30 \({}^{\circ}\)C.  
Figure 11 indicates that the moduli were reduced with the decrease in temperature. The instantaneous elastic modulus presented an exponential relationship with the temperature, while the retardation modulus presented a linear relationship with the temperature. The curves were scattered with data, especially for the retardation modulus curves, but the trends were clear. The asphalt specimens were drilled from the Quxue Dam and Laojiaoxi Dam cores, and there were unavoidable variations in the asphalt mixes, weather conditions, and compaction procedures during construction. Figure 12 indicates that the temperature had insignificant effects on the retardation time. From the viscoelastic theory, the retardation time is equal to the coefficient of viscosity divided by the retardation modulus (\(\tau_{i}=\frac{\tau_{i}}{E_{i}}\)).