Asphalt mixture is a typical viscoelastic-plastic material. It shows different characteristics in different temperature ranges. It has linear elasticity at low temperatures and small deformation, viscoelasticity at high temperatures and large deformation, and viscoelasticity at temperature transition ranges. Therefore, the appropriate viscoelastic mechanical model should be selected for in-depth study to obtain the viscoelastic parameters of asphalt mixture.

The Kelvin model can describe the creep behavior but cannot reflect the phenomenon of instantaneous elastic deformation and stress relaxation. The mechanical behavior is obviously inconsistent with the viscoelastic deformation behavior of asphalt mixture. The Maxwell model can reflect the elastic and viscous characteristics, but the characteristics of the viscous fluid are closer to the fluid in nature, which is far from the deformation characteristics of the typical viscoelastic solid material (asphalt mixture).

As shown in Figure 1, the Burgers model is composed of the Kelvin and Maxwell models in series, which can better describe the mechanical behavior of asphalt mixture in a certain temperature range. By solving the viscoelastic parameters of the mechanical elements in the model, the constitutive equation can predict the stress-strain relationship of the material. This model can reflect instantaneous elastic deformation, viscoelastic deformation, and viscous flow deformation. However, the viscous flow deformation is expressed as a linear function of loading time, and the actual viscosity increases as the loading increases. Therefore, the increment of permanent deformation decreases with the time. Therefore, the model cannot well reflect the deformation characteristics of asphalt mixture in the whole loading and unloading process.

In Figure 2, an improved Burgers model, namely, the modified Burgers model, is proposed in this study; the “consolidation effect” of asphalt mixture can be demonstrated by this, and better describe the high temperature deformation law of asphalt mixture (Xu, 1992).

The model extends the external sticky pot element that characterizes the viscous flow deformation characteristics of the material in the Burgers model to a generalized sticky pot, and its viscosity is expressed as: η 1 t = A e B t where η 1 t is the viscosity of the external sticky pot element in the model and A and B are the viscosity parameters of the external sticky pot components. t is the loading time.

Zhang Yuqing (Zhang and Xiao-ming, 2008) derived the functional expression of the permanent strain under repeated loading and obtained the functional relationship between ε P N   and N : ε P N = σ 0 1 − e − B T N A B 2 T e B t 0 − 1 + σ 0 e τ t 0 − 1 E 2 τ T 1 − e − τ T N (1) where ε P N is the cumulative permanent strain after the Nth load; σ 0 is the equivalent static load axial compressive stress level; A and B are the parameters of the improved sticky pot; τ = E 2 η 2 is the parameters of parallel spring and sticky pot; t 0 is the action time of the load in a period; and T is the loading cycle.

The test data cannot be directly fitted using Formula 1. Therefore, it is also necessary to substitute the half-sine wave load used in the repeated load permanent deformation test into Formula 1, including the loading time t 0 = 0.1 s , the intermittent time t j = 0.9 s , and the total period T = t 0 + t j = 1 s . Then, the viscoelastic mechanical model shown in Formula 2 can be obtained, which describes the permanent deformation of repeated load. ε P N = 2 σ 1 − e − B N A B 2 π e 0.1 B − 1 + 2 σ e 0.1 τ − 1 E 2 τ π 1 − e − τ N , (2) where σ is the maximum stress of the axial compressive stress peak and N is the number of loading. Formula 2 is the final fitting formula, and the values of the model parameters A , B , E 2 , and   η 2 need to be determined by the data fitting.

In this study, asphalt mixtures AC-13, AC-20, and AC-25 widely used in China’s expressway asphalt pavement were adopted. In all of them, the asphalt is 70 # road petroleum asphalt and the aggregate is limestone. The gradation of these three asphalt mixtures is shown in Table 1.

The optimum asphalt-aggregate ratio of the asphalt mixture was determined by the Marshall test. The optimum asphalt-aggregate ratios of AC-13, AC-20, and AC-25 are 5.0%, 3.8%, and 3.6%, respectively.

With the optimal asphalt-aggregate ratio, the specimen was molded using the rotary compaction method. After demoulding, the molded specimen was 170 mm in height and 100 mm in diameter. With its two ends cut, a specimen with a height of 150 mm and a diameter of 100 mm was obtained, as shown in Figure 3.

In this paper, a simple performance testing machine (SPT) was used to test the permanent deformation of AC-13, AC-20, and AC-25 with lateral repeated loads. The confining pressure was 138 Kpa, and it remained constant throughout the whole testing process. In addition, the stress level was 0.7 Mpa; the loading waveform was half-sine wave intermittent load; the loading time was 0.1 s; the intermittent time was 0.9 s; the temperatures were 40°C and 60°C; and the number of specimens in each group of parallel tests was three. The experimental device is shown in Figure 4.

The test results of AC-13, AC-20, and AC-25 at the same stress level and different temperatures are shown in Figure 5. Additionally, the test results of the same mixture under the above conditions are displayed in Figures 6, 7.

Figure 5 reveals that the permanent deformation of asphalt mixtures escalates as the number of loading cycles amplifies. Additionally, it demonstrates that asphalt mixtures with different gradations exhibit two distinct stages of permanent deformation growth, that is, the rapid growth stage of strain and the relatively stable stage of strain growth.

By observing Figure 6, we can ascertain that the stress level remains consistent and the loading cycle remains constant; the permanent deformation of asphalt mixtures continues to increase as the temperature increases, indicating that the increase of temperature has a great influence on the high temperature deformation resistance of asphalt mixtures.

The multi-parameter regression analysis of the permanent strain change curves of asphalt mixtures at 40°C and 60°C with the number of loading cycles was carried out to obtain the mechanical model parameters A , B , E 2 , and  η 2 , as shown in Table 2. The fitting results show that the parameters A , B , E 2 , and  η 2 of the same asphalt mixture decrease as the temperature increases, indicating that a high temperature leads to large irreversible deformation of the specimen.

To verify the applicability of the model, the repeated load permanent deformation test of AC-13 at 30°C, 50°C, and 0.7 Mpa was conducted, and the test results are shown in Figure 7. It can be seen that the red curve is the measured value of the test, the blue curve is its fitting value, and the correlation coefficient is 0.98499. It is demonstrated that the permanent strain mechanical model equation derived from viscoelastic mechanics under repeated load can accurately characterize the variation law of permanent deformation of asphalt mixtures in the rapid growth stage and relatively stable stage of strain.

In this paper, the representative values of the dynamic modulus of each structural layer were selected; the surface courses of AC-13, AC-20, and AC-25 were chosen, respectively. Considering its viscoelastic properties, the structural damping coefficient 0.05 of the base course and the soil base was taken.

In ABAQUS, the parameters of viscoelastic material were generally defined by inputting the Prony series parameters of the shear relaxation modulus. Based on the convolution relationship between the creep compliance and the relaxation modulus, the modified Burgers model is used to determine the parameters of the viscoelastic material. The viscoelastic parameters of the modified Burgers model were transformed into the Prony series form of the shear relaxation modulus by the Laplace transform, as shown in Table 3.

Before modeling, the pavement structure was simplified. According to the current specification requirements, the structural layers were regarded as completely continuous, and infinite elements were added around the model to reflect the non-reflective boundary conditions. The road surface model is shown in Figure 8. It can be seen that both the length and width of the model are 3 m, and the depth direction regarding the structure of the pavement is 2 m. In this study, the dual-wheel spacing is 0.320 m and the single-wheel grounding width is 0.223 m; therefore, the total loading width is 0.543 m. The pavement model unit type adopts C3D8R, which reduces the amount of calculation by refining the finite element mesh in the loading area along the wheel path. It is 23.4 cm long and 32 cm wide (middle) at the place where the loading needs to be analyzed. The mesh is fine, and the mesh density is reduced in other parts to locally refine the mesh. The minimum size is 0.02 m and the maximum size is 0.2 m. The central part of the grid is 19 according to the number of transverse roads, 30 along the driving direction(y-axis), and the surrounding grids are encrypted toward the middle.

The technical route of finite element modeling is shown in Figure 9.

The boundary conditions of the model are set, the bottom is fixed, and the four sides of the model are symmetrically constrained. According to the current specification requirements, each structural layer is considered to be completely continuous, and infinite elements are added around the model to reflect the non-reflective boundary conditions.

In order to ensure continuous and in-depth study, the moving load in this paper uses the tire load in Zhang (2020b). Moreover, DLOAD was adopted to realize the pressure loading on the user defined surface, and UTRACLOAD was utilized to achieve the arbitrary distributed force loading on the user defined surface. The relationship among the distributed force coordinate and time was provided by the DLOAD and UTRACLOAD subroutines to simulate the application of non-uniform moving load.

In this study, the x-axis, y-axis, and z-axis were the transverse, driving, and depth directions of the pavement structure, respectively. The moving load working conditions were divided into four types: cornering, driving, braking, and free rolling working conditions. The cornering angle of the cornering working condition was 8°; the angular velocity of the driving and braking working conditions was 46.0588 rad/s and 40.0588 rad/s, respectively; and the free rolling speed was 80 km/h.

In this study, research was focused on the analysis of the stress and strain between the dual-wheel and road surface and between the road structure layers under different load working conditions. Figure 10 presents the schematic diagram illustrating the calculation and analysis points, where A is the center line of dual wheels, B is the inner side line of the wheel edge, C is the center line of the single wheel, D is the outer side line of the wheel edge, and E is a calculation point line a distance away from the outer side of the wheel edge. The actual distance can be determined according to the distribution of stress and strain.

From Figures 11 and 12, it can be seen that:

(1) Except for the maximum principal strain of the cornering working condition being biased toward the turning side, the maximum principal strain and stress distributions of the other working conditions show the same law, which have obvious symmetry. The stress distribution of the free rolling condition shows uniformity, which is roughly symmetrical along the center line of the two wheels, and is roughly symmetrical along the transverse axis of the tire center.

From the analysis of the stress and strain of the pavement, it was found that the maximum strain does not necessarily appear at the position where the maximum three-way stress is generated.

In Figures 13 and 14, it is shown that:

(1) Except for that of the cornering working condition, there is clear symmetry in the distribution of the maximum principal stress and strain for the other three load working conditions. The cornering and driving working conditions produce a large principal stress on the pavement.

As shown in Figures 15 and 16, it can be found that:

(1) The position of the maximum transverse tensile strain value is different from that of the maximum transverse tensile stress value. The maximum value for transverse tensile stress is situated at the midpoint of the subgrade and dual wheels, and the maximum transverse strain value is distributed at the outer edge line of the wheel or a distance away from it.

It can be seen from Figures 17 and 18 that:

(1) The driving working condition produces larger longitudinal tensile stress and strain values on the pavement layer, which is differently distributed. The maximum longitudinal tensile stress and strain values are distributed at the center line of the left wheel and single wheel on the AC-13 pavement layer, respectively.

From Figures 19 and 20, it is demonstrated that:

(1) The shear stress distribution of the driving and braking working conditions is roughly anti-symmetrical except for that of the cornering working condition; however, there is no shear stress distribution in the free rolling working condition of the section in the middle of the area where the wheel touches the road surface.