# A Creep Model of Asphal Mixture Based on Variable Order Fractional Derivative
## 5 Model Verification
### Results and Disscussions  
In order to distinguish the linear/nonlinear characteristics of the creep behavior, the creep strain is divided by the corresponding applied stress to obtain the creep compliance curve, as shown in Figure 8. It can be seen that the creep compliance curves for stresses of 0.7 MPa, 0.9 MPa and 1.1 MPa are almost coincident, indicating that the creep compliances at these stress levels are stress independent, and the asphalt mixture exhibits linear viscoelastic creep behavior; when the stresses are higher than 1.1 MPa, the creep compliance curves depend on the stress level, and the asphalt mixture shows nonlinear characteristics. Assuming that the nonlinear creep is caused by damage evolution, the creep damage initiation stress can be taken as \(\sigma_{\rm d}=1.1\) MPa. Figure 9 shows the creep rate curve and creep acceleration curve at various stress levels, and it can be seen that the primary creep stages under all stresses are completed at \(t_{1}=300\) s. While, for the stresses of 1.5 MPa and 1.7 MPa, the creep acceleration phases start at about \(t_{2}=1250\) s and \(t_{2}=620\) s, respectively. [PERSON] et al. [23] conducted uniaxial compression tests on AC-13C asphalt mixture, and obtained the compression strength ranging 3.70 MPa \(\sim\) 4.38 MPa. In this study, the compression strength is set to be the average value, i.e., \(\sigma_{\rm f}=4.0\) MPa. As shown in Figure 10, using Equation (16) to fit the obtained damage incubation time data, we obtained \(A=45\) s, \(n=1.95\). Therefore, the relationship between creep damage incubation time and the creep stress obeys the equation below:  
\[t_{2}=45\cdot\left(\frac{4.0-\sigma}{\sigma-1.1}\right)^{1.95},\ (1.1\ {\rm MPa}< \sigma\leq 4.0\ {\rm MPa}) \tag{23}\]  
Figure 8: Creep compliance curves of asphalt mixture under various compressive stress levels. Figure 9: Creep rate curves (a) and creep acceleration curves (b) of asphalt mixture under various compressive stress levels. It can be calculated that \(t_{2}=7200\) s when \(\sigma=1.3\) MPa, meaning that there is no damage within the test duration of 6000 s for the creep under 1.3 MPa, and, thus, the experimental creep curve only includes the primary stage and steady stage, as it behaves in a manner for the creep under stresses lower than 1.1 MPa because of no damage. By means of the Levenberg-Marquardt optimization algorithm in 1 stop, nonlinear fitting of the test data in the primary stage, the steady stage and the tertiary stage for various stress levels were completed by use of Equation (22), and the corresponding parameters were determined, as listed in Table 2. It can be seen that in the stress range considered, the model parameters of the primary stage creep are the same constants, regardless of the applied stress, which indicates that when the strain is small, the asphalt mixture behaves linearly viscoelastic; under the same applied stress, the fractional differential order for the steady creep stage is greater than that for the primary stage creep, i.e., \(\alpha_{2}>\alpha_{1}\). Moreover, in the steady creep stage, the fractional differential order \(\alpha_{2}\) increases with increasing stress level, which demonstrates that the viscoelastic behavior of the material is more inclined to viscous flow when the applied stress increases, indicating the shortening of the characteristic time and the increase of the creep rate. In the tertiary stage, the parameter \(\omega\) increases with the applied stress level, which is consistent with the fact that the creep damage rate increases with stress level. Figure 11 shows the comparison between the model and the test results. It can be seen that the model is in good agreement with the test results, which verifies that the model can effectively describe the whole primary-steady-tertiary creep process of the asphalt mixture under different stress levels.