Rutting Prediction Model of Asphalt Mixture Based on the Triaxial Repeated Load Test

School of Civil Engineering and Transportation, Beijing University of Civil Engineering and Architecture, Beijing 100044, China Beijing Advanced Innovation Center for Future Urban Design, Beijing 100044, China Beijing Collaborative Innovation Center for Energy Conservation & Emission Reduction and Sustainable Urban-Rural Development, Beijing 100044, China National Institute of Clean-and-Low-Carbon Energy (NICE), Beijing 102211, China Guangxi Transportation Research & Consulting Co., Ltd., Guangxi, Nanning 530007, China


Introduction
Establishing the rutting prediction model of asphalt mixture is the mainstream scheme to study the rutting resistance of asphalt pavement. e prediction of the model requires a feasible test method which can accurately reflect the characteristics of asphalt mixture to calibrate the parameters of the model [1][2][3]. Current tests can be roughly divided into two categories. One is the empirical test, such as the rutting test in lab and asphalt pavement analyzer test. At present, the rutting test in lab is one of the main methods to assess the rutting resistance of asphalt mixture in China. However, this kind of test cannot directly reflect the mechanical properties of materials, so the model established by it has certain limitations. Another kind of test is based on mechanical principles, such as the uniaxial creep test and triaxial repeated load test [4][5][6]. Because there is a linear relationship between stress and strain of specimens in this kind of test, mechanical parameters reflecting material properties can be calculated and a complete rutting prediction model can be established. e uniaxial test equipment is simple and easy to implement. However, due to the lack of lateral restraint, the stress state of specimen is different from that of actual road surface [7]. erefore, multiaxial repeated tests, for example, the triaxial repeated load test, were considered as a more suitable choice.
It is proven that the triaxial repeated load test with adjustable lateral confining pressure and dynamic load can better simulate the actual stress state of asphalt pavement [8,9]. e reports from NCHRP (the National Cooperative Highway Research Program) and SHRP (Strategic Highway Research Program) recommend the dynamic modulus test, triaxial static creep test, and triaxial repeated load test to assess the rutting resistance of asphalt mixture [10,11]. G. Cerni put forward a simplified method based on the triaxial repeated load test, which can accurately predict the material characteristics by adjusting the combination of stress and applied stress direction [12]. Park verified the effect of temperature on asphalt mixture by the triaxial repeated load test and predicted the relationship between temperature and rutting by the temperature conversion factor [13]. Zhang et al. studied the triaxial repeated load test and put forward the complete test parameters and method flow [14,15]. us, the triaxial repeated load test is widely considered to be the most accurate and feasible test method to reflect the characteristics of asphalt mixture.
At present, the rutting prediction models of asphalt pavement can be roughly divided into two categories: empirical model and rheological model [16]. Archilla et al. put forward the empirical rutting prediction model based on AASHTO test pavement data. It has strong pertinence, but the cost of such models is high and the overall effect of pavement structure is not considered. erefore, its scope of application is narrow. e rheological model is currently widely used [17]. Blab [23]. us, it is generally believed that the modified Burgers model can accurately reflect the deformation law of asphalt mixture.
On the other hand, in order to solve the energy problem, a large amount of direct coal liquefaction residue (DCLR) is produced every year in China, which can be used as a good modifier of asphalt. At present, there is no efficient method of DCLR utilization, and it is simply being stacked and burned [24]. is will not only cause serious damage to the environment but also be a waste of resources. e previous research of our group shows that DCLR can be developed as a modifier and improve the high temperature performance and water stability of asphalt mixture [25,26]. Unfortunately, there is no research on the rutting model of DCLR.

Objectives and Procedures
e motivation of this study is to establish the rutting prediction model of asphalt mixture based on the triaxial repeated load test.
First, the triaxial repeated load test and multitemperature and load rutting test were conducted on asphalt mixture. e rutting depth of asphalt mixture was simulated based on the existing modified Burgers model and compared with the measured value in the multitemperature and load rutting test.
Second, the quadratic modified Burgers model and parameters were remodified based on the triaxial repeated load test, and the rutting prediction model was reconstructed.
en, the rutting depth based on the quadratic modified Burgers model was simulated and also compared with the measured value in the multitemperature and load rutting test.
Finally, the error ratios and residual sum of squares of simulated and measured rutting depths were calculated and compared to verify the rationality and effectiveness of the abovementioned two rutting prediction models, respectively.

Test Material.
e raw materials included DCLR, four kinds of asphalt binders, and three kinds of aggregates. Additionally, the DCLR, provided by China SHENHUA Co., Ltd., is a solid powder at room temperature and its melting point is 170°C. e four kinds of asphalt binders were SK-90 asphalt, SBS modified asphalt, DCLR, and composite DCLR modified asphalt. Limestone was used for coarse aggregate and fine aggregate, and limestone powder was used for mineral powder. e properties of raw materials were referred to the previous research studies of our group [25,26]. e four kinds of asphalt mixtures included SK-90 asphalt mixture, SBS modified asphalt mixture, DCLR modified asphalt mixture, and composite DCLR modified asphalt mixture. ey had the same gradation (AC-20) and optimum asphalt content (4.3%). eir performances were also referred to the previous research studies of our group [25,26].

Multitemperature and Load Rutting Test.
According to T0719-2011 in JTG E20-2011 specified in China [27], the size of each sample is width 300 mm * length 300 mm * height 50 mm, the test temperature is 60°C, and the test wheel pressure is 0.7 MPa. However, in most areas of China, such as Turpan, Chongqing, Wuhan, and so on, the maximum temperature of asphalt pavement can reach about 70°C, and the wheel load of heavy-duty vehicles and overloaded vehicles can reach 1.0 MPa [28]. In order to consider the influence of the most unfavorable temperature and heavy load conditions on the high temperature characteristics of asphalt pavement, the test conditions were extended according to the specifications as follows: (i) e test temperature is 55°C, 60°C, 65°C, and 70°C (ii) e test wheel pressure is 0.7 MPa, 0.8 MPa, 0.9 MPa, and 1.0 MPa 2 Advances in Civil Engineering e test will stop automatically after 60 mins or the accumulated deformation reaches 25 mm. Each test condition is repeated three times, and the average value is taken if the coefficient of variation is less than 20%. e rutting tester independently developed by the research group is used in the test, as shown in Figure 1.

Triaxial Repeated Load Test.
According to the improvement of T0718-2011 in JTG E20-2011 specified in China [27], the asphalt mixture was formed by rotary compaction, and the cylindrical specimen with diameter of 100 mm and high 160 mm was obtained through core drilling. e cutting machine was used to cut the test piece for ensuring the flatness of both ends of the piece. e height of the test piece is 150 ± 2 mm, and the void ratio is 4.3 ± 0.5%, as shown in Figure 2. e IPC UTM-25 pneumatic servo test instrument is selected for the test, with a total of 12 test conditions, as shown in Figure 3. e test will stop automatically after 10000 cycles of loading or more than 5% permanent deformation of the piece. Each test condition is repeated three times, and the average value is taken if the coefficient of variation is less than 20%. e test conditions are as follows: (i) e preloading is 90 s vertical load, and the load level is 0.01 MPa. (ii) e test temperature is 50°C, 60°C, and 70°C (iii) e test load level is 0.7 MPa, 0.8 MPa, 0.9 MPa, and 1.0 MPa; the lateral pressure is 0.138 MPa; (iv) e loading mode is half sine wave intermittent load (0.1 s loading, 0.9 s unloading)

Existing Modified Burgers Model.
A large number of studies show that the deformation process of asphalt mixture in the triaxial repeated load test is divided into three stages, which are the first stabilization stage, the second migration stage, and the third failure stage [29][30][31]. Since there is no obvious law for the third failure stage, this study only analyzed the data obtained from the stabilization and migration period. e relationships between the permanent deformations and the load numbers under different temperatures and loads are shown in Figure 4. It can be seen that under the same load condition, temperature is positively correlated with the permanent deformation of asphalt mixture. Under the same temperature condition, the load is positively correlated with the permanent deformation degree of asphalt mixture. Under the same load and temperature, the order of permanent deformation relationship of the four asphalt mixtures is as follows: SK-90 asphalt mixture > DCLR modified asphalt mixture > SBS modified asphalt mixture > composite DCLR modified asphalt mixture.
Currently, the existing rheological model which can accurately describe the rutting resistance of asphalt mixture is the Burgers model which is modified by external clay pot [32,33]. e load process strain of modified Burgers model is given as where σ is the applied load; ε is the strain; E1 and E2 are the elastic modulus of the internal and external springs,   Advances in Civil Engineering 3 respectively; η1 and η2 are the viscosity coefficients of the internal and external viscous pots, respectively; t is the time; and A and B are the parameters of the modified Burgers model. Formula (2) shows the unloading process strain: where t 0 is the loading time, and other parameters are the same as formula (1). e existing modified Burgers rheological model and triaxial repeated load test data are combined to build the rutting prediction model. e rutting depth of four kinds of asphalt mixtures under the same conditions as 16 kinds of multitemperature and load rutting tests is calculated, that is, rutting depth under different temperatures (55°C, 60°C, 65°C, and 70°C) and loads (0.7 MPa, 0.8 MPa, 0.9 MPa, and 1.0 MPa). Table 1 compares the predicted rutting depth based on the modified Burgers prediction model and the measured rutting depth using the multitemperature and load rutting test.
It can be seen from Table 1 that the error ratio ranges from 5% to 35%, and the residual sum of squares changes from 5.0% to 8.74%. It is generally believed that the smaller the error rate and the residual sum of squares, the closer the two sets of data are. Numerous studies show that, if the residual sum of squares is less than 5%, it is considered that the correlation between the two sets of data is relatively high. Obviously, the existing modified Burgers model cannot accurately simulate the deformation characteristics of asphalt mixture [34,35]. is is mainly because the viscoelastic parameters of the existing modified Burgers model are calibrated by the uniaxial static creep test, so it is more suitable for rutting prediction under static load but not for dynamic load. Because the dynamic load can simulate the actual road stress state more accurately, therefore, it is necessary to remodify the model to improve the accuracy.

e Quadratic Modified Burgers Model.
Since the viscoelastic parameters of the existing Burgers model are calibrated by the uniaxial static creep test, it is only suitable for the prediction of rutting under static load, but not suitable for simulation under intermittent load. erefore, the remodified existing Burgers model based on the triaxial repeated load test aimed to consider dynamic loads. e main ideas of remodification include the following. First, the independent variable time of the uniaxial static creep test is converted to the number of times of the triaxial repeated load test. Second, the static load is converted into semisinusoidal intermittent load. Finally, the loading strain is converted into loading and unloading strain, and the quadratic modified Burgers model considering dynamic load is obtained.
e steps of remodification are as follows: (1) Strain conversion: since the stress-strain at time t is equal to the sum of all the stress-strain at time from 0 to t and the unloading stress σ at time t is equal to applying a reverse -σ, the sum of the loading and unloading strains is converted into In equation (3), σ 0 is the equivalent static load axial compressive stress level, and other parameters are the same as in formula (1). (2) Load form conversion: the load form of the triaxial load test is half sine wave intermittent load as In equation (4), t is the load period, σ t is the maximum load at time t, and other parameters are the same as in formula (1). According to the principle of stress equivalent impulse, the stress load of half sine wave is transformed into equivalent load as (3) Independent variable conversion: the independent variable in formula (5) is converted from time T to number of actions N by the method of derivative of strain rate, and the strain produced in the i th time is recorded as ε Ni ; then, the permanent strain rate of the N th time is In equation (6), τ � E 2 /η 2 is the parameter of the parallel spring and the glue pot of the modified Burgers model; the other parameters are the same as in formula (1).
By integrating formula (6), the function relationship between permanent deformation and N is obtained, that is, the quadratic modified Burgers model considering dynamic load as the following formula.  Table 2 provides the rheological parameters of four asphalt mixtures. As given in Table 2, the value of determination coefficient R 2 is 0.81-0.99, and the significance P value is 0.00-0.02, indicating that there is a significant linear correlation between the triaxial repeated load test data and the quadratic modified Burgers model. e quadratic modified Burgers model can simulate the relevance between the permanent deformation and the load numbers effectively, and the rheological parameters can also accurately characterize the deformation characteristics of asphalt mixture.

Rutting Prediction
With ABAQUS software, the rheological parameters determined in Table 2 are used as input parameters to build the rutting prediction model of asphalt mixture. Because ABAQUS software cannot directly input the rheological parameters of materials, the Prony series method is used to transform the calibrated rheological parameter A, B, ε 1 、ε 2 , and η 2 into g 1 , g 2 , τ1, and τ2 of shear modulus, so that they can be identified by ABAQUS software. e number of iterations in the transformation process is 2 times, as given in Table 3.
2D models of four kinds of asphalt mixtures are constructed, respectively, to calculate the strain of flat shell elements.
e rutting depth of four kinds of asphalt mixtures under the same conditions as 16 kinds of multitemperature and load rutting tests is calculated. In order to reduce the influence of variables on the simulation results, the dimension of the finite element model is the same as that of the rutting plate in the multitemperature and load rutting test, the size is 300 mm * 5 mm, and Figure 5 shows that the force situation of the simulated rutting prediction model. Note. e error ratio is the ratio of the difference between the predicted and experimental values of rutting depth, and the residual sum of squares is the discretization degree of the fitting curve between the predicted and experimental values of rutting depth.

Comparison on the Two Rutting Prediction
Models. e rutting depths were calculated based on the existing modified Burgers model and quadratic modified Burgers model and compared with the measured rutting depths in the multitemperature and load rutting test, as shown in Figure 6. Table 4 provides the error ratios and residual sums of squares between the estimated and measured rutting depths based on the two models.
From Figure 6 and Table 4, it can be concluded that the error ratio between the predicted rutting depth and the measured rutting depth in the indoor multitemperature and load rutting test is 3-12%, and the residual sum of squares is 0.94-3.10%. Compared with the estimated values based on the existing modified Burgers model, the error ratio and residual sum of squares of rutting depth are reduced by about 70% and 60%, respectively. It shows the accuracy of the results is significantly improved, which can reflect the rutting resistance of asphalt mixture more accurately.

Conclusions
rough the above tests and analysis, some conclusions can be drawn: (i) e error ratio and residual sum of squares of the rutting depth simulated by the existing modified Burgers model between the measured values in the multitemperature and load rutting test is high, which indicates the model is not suitable for predicating the rutting resistance of asphalt mixture (ii) e quadratic modified Burgers model considering dynamic loads was built by the existing modified Burgers model through changes in independent variable, load form, and the transformation of loading and unloading strain. e least square method is used to fit the triaxial repeated load test data with the model, and the R 2 value is greater than 0.8, which means the quadratic modified Burgers model has a good correlation with the triaxial repeated load test. (iii) Compared with the rutting depth predicted by the existing modified Burgers model, the range of the error ratio and the square sum of the residual error of the predicted rutting depth by the quadratic modified Burgers model are reduced by 70% and 60%, respectively. It shows that this model can improve the accuracy and effectiveness.

Data Availability
Previously

Conflicts of Interest
e authors declare that they have no conflicts of interest.