3.1 Failure mode of deformation superposition effect
Previous studies showed that the shear failure mode of the asphalt mixture in the uniaxial penetration test is similar to that of pavement core [34]. On the basis of the method mentioned in Sect. 2.2, uniaxial and four-point penetration tests were conducted for SMA-13 with a temperature of 60 ℃ and a loading rate of 1 mm/min. This process was performed to obtain and compare the failure mode of the asphalt mixture. The penetration curve is shown in Fig. 7.
As shown in Fig. 7, the load growth rate under uniaxial and four-point penetration of the SMA-13 mixture is 4.33 and 2.50 MPa/mm respectively. This finding indicates that resisting deformation under a four-point penetration load is more difficult for the asphalt mixture.
A direct result is shown in Fig. 8, which illustrates the crack patterns of asphalt mixture under uniaxial and four-point loads.
Under uniaxial penetration load, the cracking path is from the center to the edge of the specimen, indicating that the edge cracking is the main reason for the strength loss of the mixture. Under four-point penetration load, the crack path is the edge of the specimen and the connecting line of the four penetration points, indicating that the deformation is caused by the joint action of four penetration shafts. The deformation appears as a superposition phenomenon. This condition results in considerable cracking of the sample. Thus, the resistance to deformation decreases, and the load growth rate is small. Specifically, Δ4 is less than Δ1, and the superposition coefficient is less than 1.
3.2 Effect of the wheel group on deformation superposition
The impact of wheel spacing and the number of wheels on the superposition effect in asphalt mixtures is quantitatively discussed to investigate the external cause of the deformation superposition in asphalt mixtures.
3.2.1 Effect of wheel spacing
Wheel spacing is regarded as the key factor affecting the deformation behavior of asphalt mixture. This section quantificationally examines the influence of wheel spacing on the deformation superposition effect of the SMA-13 mixture by using the aforementioned test method. The four-point penetration tests with the wheel spacing of 54, 60, and 65 mm were conducted, the loading rate was 1 mm/min, and the test temperature was 60 ℃. The results from the various wheel spacings exhibit the variation of superposition parameters, as shown in Figs. 9 − 10.
Figure 9 presents the penetration load growth rate in uniaxial and four-point penetration tests. On the whole, the penetration load growth rate generated by the uniaxial penetration is greater than that by the four-point. The result of uniaxial penetration is 1.1 MPa/mm, whereas the maximum value of four-point penetration is 0.65 MPa/mm, which is only 59% of the former. This finding confirms the analysis in Sect. 3.1 and indicates that the asphalt mixture is prone to deform under the four-point penetration load. Compared with the result under various four-point loads, the penetration load growth rate increases with wheel spacing. This phenomenon demonstrates the important effect of wheel spacing on superimposed deformation.
A direct explanation of the influence of wheel spacing on deformation superposition is illustrated in Fig. 10. On the basis of the test result in Fig. 9, the deformation superposition effect coefficient is calculated by using Eq. (1). Specifically, the superposition effect coefficient gradually increases with the rise of wheel spacing. For instance, the coefficient is 1.2 times higher for spacing of 54 mm compared with that for spacing of 65 mm. The increase in the superposition effect coefficient suggests that the resistance to superposition deformation of asphalt mixture is worse. This result is attributed to the increase in wheel spacing.
3.2.2 Effect of the number of wheels
The preliminary analysis in Sect. 3.1 showed that the number of wheels is another important factor for the deformation superposition of the asphalt mixture. In this section, the deformation superposition effect under two-point, four-point, and six-point penetration loads was investigated. The test temperature was 60 ℃, the loading rate was 1mm/min, and the wheel spacing was 65mm. The results from the various number of wheels exhibit the variation of superposition parameters, as shown in Figs. 11 − 12.
In Fig. 11, the penetration load growth rate is smaller under multi-point load compared with that under uniaxial load. This condition is the same as the previous analysis in Sect. 3.2.1. Figure 10 displays that the penetration load growth rate decreases with the number of wheels. The penetration load growth rate under uniaxial and six-point loads is 1.1 and 0.4 MPa/mm, and the latter decreases by 63.6%. This finding shows that the number of wheels is an important parameter affecting the superimposed deformation of asphalt mixture, because it has the potential to increase the stress transferring path in the internal structure.
The deformation superposition effect coefficient varies with the number of wheels, as illustrated in Fig. 12. The greater the number of wheels, the smaller the superposition effect coefficient. The coefficient decreases from 0.66 in the case of two-point loading to 0.38 in the case of six-point loading, and the latter decreases by 42.9%. This finding shows that the deformation superposition effect becomes significant with the increase of the number of wheels. The resistance of asphalt mixture to deformation is reduced. Compared with light vehicles, we should pay attention to the multi-wheel heavy vehicles, which aggravate the possibility of superimposed deformation damage.
3.3 Micromechanical characteristics of deformation superposition
In Sect. 3.2, two factors causing the deformation superposition of asphalt mixture, namely, wheel spacing and the number of wheels, are discussed. They are the external reasons that cause the deformation superposition of the asphalt mixture. In the process of deformation, the variation of the micromechanical characteristics of asphalt mixture under multi-wheel load is still unclear. The importance of understanding the micromechanical characteristics is that it can guide asphalt mixture design for anti-superimposed deformation properties. Therefore, this section aims to clarify the micromechanical characteristics of deformation superposition behavior under different wheel group conditions based on DEM.
3.3.1 Wheel spacing
In this section, the uniaxial and the four-point penetration tests with wheel spacings of 20, 50, and 65mm were simulated. The penetration curves are shown in Fig. 13. The black curve describes the result in the uniaxial penetration test, and the red, green, and purple curves describe the result under the wheel spacings of 20, 50, and 65 mm.
As shown in Fig. 13, a linear relationship is found between penetration strength and depth, and the material mainly shows linear elastic characteristics at the initial stage of simulated loading. The mixture was destroyed until the load reached its peak, and the bearing capacity declined rapidly. The slope of the linear segment represents the penetration load growth rate. The slope and peak load are larger in uniaxial tests compared with that in multi-point tests. For instance, the former is 0.13 and 0.65 MPa, whereas, the maximum values of the latter are 0.12 and 0.35 MPa. This finding indicates that the asphalt mixture consistently has low resistance to multi-wheel action. The whole simulation loading process is consistent with the laboratory test, thereby reflecting the reliability and accuracy of the simulation. For multi-point simulation tests, the penetration load growth rate gradually increases with the increase in wheel spacing. The results agree with the reported laboratory test in Sect. 3.2.1.
The superposition effect coefficient obtained from the skeleton model is shown in Fig. 14. The values of the superposition effect coefficient are 0.43, 0.55, and 0.77 for the wheel spacing of 20, 50, and 65 mm, respectively. This finding demonstrates that the model deformation under four-point loads is superimposed compared with that under uniaxial loads. The superposition coefficient increases with the decrease in wheel spacing. This result is the same as that obtained in Sect. 3.2.1.
Figure 15 shows that the number of tensile chains varies with depth. In the penetration process, the number of force chains gradually decreases. This phenomenon suggests that the strength of physical adhesion between the asphalt and the aggregate is reduced. As shown in Fig. 16, the force chains at the penetration depth of 4mm are reduced by 272, 260, and 249 for the wheel spacing of 20, 50, and 65 mm compared with that at 1 mm. The closer the wheel spacing indicates the decrease in the number of tensile chains. This condition indicates further superimposed deformation and damage to the asphalt mixture.
The previous consequence can be illustrated by the force chain evolution in Fig. 17. The lines in the figure represent the contact force between the skeleton particles, and the blue and red lines denote the pressure and tensile chain, respectively.
During the whole loading process, the change in the pressure chain is insignificant. On the contrary, the tensile chain decreases with the penetration depth. At the 1 mm penetration depth, the number of tensile chains reaches a maximum. At the 4 mm depth, a large number of tensile chains are broken. Under the four-point load, cracks are generated from the center to the edge of the model, and deformation superposition is destructive.
With the increase in wheel spacing, the number of tensile chains decreases significantly, and the skeleton model is close to the overall loading state. For the wheel spacing of 50 mm, part of the tensile chain breaks at the edge of the model. For the wheel spacing of 65 mm, only part of the force chain under the penetration shaft breaks. Under the four-point load, the number of broken tensile chains increases with the decrease in wheel spacing, and the superposition effect is significant.
3.3.2 Number of wheels
In this section, the uniaxial, two-point, four-point, six-point, and eight-point penetration tests were simulated. The penetration curves are shown in Fig. 18, and the superposition coefficient obtained from the skeleton model is shown in Fig. 18.
As shown in Fig. 18, the multi-point action has a significant effect on the deformation superposition behavior of the asphalt mixtures. This condition is consistent with the previous analysis in Sect. 3.3.2. With the increase in the number of penetration shafts, the peak load and penetration load growth rate of the model show a decreasing trend. Under the two-point, four-point, six-point, and eight-point loads, the former is 0.31, 0.26, 0.22, and 0.16 MPa, and the latter is 0.078, 0.065, 0.073 and 0.053 MPa/mm, respectively. These results provide important insights into the superimposed deformation damage under multi-wheel load. As shown in Fig. 19, the superposition coefficient is 0.59, 0.52, 0.49, and 0.41 in the aforementioned loading conditions. These results support the idea of the superposition effect being enhanced by multi-wheel load. This condition is consistent with the conclusion obtained in Sect. 3.2.2.
The results in Fig. 20 show the changes in internal tensile chains of the asphalt mixture during the deformation superposition. A significant reduction is found in tensile chains when the number of loads increased. As shown in Fig. 21, under the two-point, four-point, six-point, and eight-point loads, the number of force chains decreases at 4 mm is 203, 250, 307, and 337, respectively, compared with the peak value of tensile chains at the depth of 1mm. With the increase in the number of wheels, the number of tensile chains decreases significantly. This finding indicates that the more the number of wheels, the more serious the deformation and damage of asphalt mixture under multi-wheel load. This result is consistent with the conclusion of the laboratory experiment above.
Figure 22 illustrates the variation of the force chain under the different number of wheels. The meaning of the lines in the figure is the same as in Sect. 3.3.1.
At the same penetration depth, the high number of wheels corresponds to the low number of the tensile chain. The previous analysis in 3.2.2 indicates that the superposition coefficient of the skeleton model is smaller, the deformation superposition effect is more significant, and asphalt mixtures have weak resistance to deformation.
Specifically, for the uniaxial penetration test, the number of tensile chains is 1006 at a penetration depth of 1 mm, and the skeleton model has strong resistance to deformation superposition. This condition is the reason that the shear stress of the whole skeleton structure points to the external, and the full skeleton structure shares shear deformation. Thus, tensile chains are found. When the penetration depth reaches 4 mm, the number of tensile chains in the skeleton model decreases remarkably. Under the eight-point load, the number of tensile chains of the skeleton model is 893 at the depth of 1 mm, whereas the number of tensile chains of the skeleton model rapidly decreases at the depth of 4 mm. Affected by the superposition effect, the tensile chains of the skeleton model near the penetration shaft decrease, and the bearing capacity of the structure decreases rapidly. Under the eight-point load, the number of tension chains at different penetration depths is the minimum, and the asphalt mixture has poor resistance to structure deformation.