## 2 Materials and Methods
### Viscoelastic Model and Its Adaptability  
Since the linear viscoelastic theory has been widely applied in permanent deformation analysis of asphalt pavement [6; 21], this research selected the Burgers model in the simulation, which was found effective in reflecting the instantaneous and lagged elastic strain as well as the viscous flow of asphalt mixture [22]. Its constitutive equation is expressed as follows. \[\epsilon(t)=\sigma_{0}\bigg{[}\frac{t}{\eta_{1}}+\frac{1}{E_{1}}+\frac{1}{E_ {2}}\bigg{(}1-\mathrm{e}^{-\frac{tE_{2}}{\eta_{2}}}\bigg{)}\bigg{]} \tag{1}\]  
where \(\sigma_{0}\) is the original stress; \(\epsilon(t)\) is the creep strain at time \(t\); and \(E_{1}\), \(E_{2}\), \(\eta_{1}\) and \(\eta_{2}\) are the model coefficients which could be obtained by regression on the creep data. According to the creep test, the total deformation-time curve of the specimen was obtained automatically, as is shown in Figure 3. Based on this curve, the total strain-time curve could be obtained by dividing the deformation by specimen height. However, after subtracting the elastic strain from the total strain, the creep strain-time curve during the constant 3600 s loading period was developed as shown in Figure 4. Finally, the parameters of Burgers model (\(E_{1}\), \(\eta_{1}\), \(E_{2}\) and \(\eta_{2}\)) for various mixtures could be acquired by conducting multivariate regression on creep data. Figure 4: Creep strain–time curve of the specimen. Figure 3: Deformation–time curve of the specimen. In ANSYS, the creep model for a linear viscoelastic solid are not represented in Equation (1) but instead in the Prony series representation shown in Equation (2), because the Prony series uses a series of decaying exponentials to reach remarkable computational efficiency and explicit physical basis [23]. \[G(t)=G_{1}[g_{1}\mathrm{e}^{-\frac{t}{1_{1}}}+g_{2}\mathrm{e}^{-\frac{t}{1_{2}} }] \tag{2}\]  
where \(G(t)\) is the shearing modulus at time \(t\), \(G_{1}\) is the initial shearing modulus; \(g_{1}\), \(g_{2}\), \(\tau_{1}\) and \(\tau_{2}\) are the Prony representation coefficients. Since the Burgers model parameters have some connection with the Prony coefficients, these four parameters can be converted into the Prony representation coefficients (\(g_{1}\), \(g_{2}\), \(\tau_{1}\), \(\tau_{2}\)), according to the method in the literature [24]. The fitted Burgers model parameters at 50 \({}^{\circ}\)C and the corresponding converted Prony series coefficients for different asphalt mixtures are shown in Table 1.  
To ensure the included model could characterize the deformation of HMAC adequately, verification of the model adaptability was conducted by modeling the creep test numerically and by comparing the simulation with the test results. Since the creep test took the \(\Phi\)100 mm cylinder, which was 100 mm high as the specimen, a three-dimensional model was constructed accordingly in the verification analysis. For the boundary condition the degree of freedom (DOFs) along the Z-axis on the model bottom was zero and the model was horizontally unconfined. The same loading program to the aforementioned creep test was employed in the simulating. The creep compliance-time curves obtained from both the numerical simulation and the creep test are given in Figure 5. From this figure, a good fitness between them with a relative error no more than 3.2% was observed, indicating that the Burgers model is feasible in evaluating the permanent deformation of HMAC.