The discussion of dynamic modulus of asphalt mixtures under biaxial stresses

Asphalt pavement in service is subjected to a complicated state of stress, especially in the uphill or starting and braking sections. The normal stress and the shear stress load it together. Only uniaxial vertical load was taken into account in the testing of dynamic modulus when a pavement was designed at present. In this paper, the dynamic response tests were carried out by applying the dynamic normal and shear loads to cylindrical specimens of asphalt mixtures. The dynamic modulus under vertical loading and vertical-horizontal loading were compared to analyze the effect of the combination loading. The result indicated vertical dynamic modulus under biaxial loading was significantly lower than vertical loading, as opposed to dynamic shear modulus. It could be concluded that the properties of asphalt mixture from biaxial stresses state should be paid more attention in the design of pavement, especially in the intersection and uphill sections.


Introduction
The dynamic modulus is an essential parameter used to characterize material properties and is widely employed in pavement design methodologies [1] .Various test methods [2][3] are available for determining the dynamic modulus, such as the application of repeated loads, the utilization of instruments like the Dynamic Shear Rheometer (DSR), Superpave Shear Tester (SST), and Gyratory Testing Machine (GTM), as well as techniques like Dynamic Mechanical Analysis (DMA) and Impact Resonant methods.Among these methods, the repeated load test has gained popularity for its effectiveness in identifying the dynamic modulus.
A lot of research work about the dynamic modulus of asphalt mixture has been carried out in recent years.Cooper et al. [4][5][6] explored the dynamic modulus test method based on indirect tensile test, semi-circular bending test, resonant column test, or DSR complex modulus test by adjusting the testing device or plan.Besides, Zhang et al. [7][8] determined dynamic modulus by using an impact resonance test and investigated various factors that affect the impact resonance test, such as dimensions of the specimens, sensor placement, type of asphalt mixtures, impacting positions, and test temperature.Dynamic modulus test [9] was conducted based on ultrasonic wave propagation as well.
To further study the dynamic performance, it is vital to simulate the loading situation of actual pavement as much as possible.The traditional dynamic modulus test was carried out at different temperatures and frequencies under uniaxial load.However, the dynamic loads applied to the pavement during the driving process contain both vertical vibration force and horizontal shear force, especially in the uphill or starting and braking sections.Thus, there is a certain difference between the actual stress state of asphalt pavement and the laboratory testing.In this paper, both sinusoidal vertical and horizontal shear stresses were applied to specimens by using the self-developed device to consider the impact of acting biaxial stress on the dynamic modulus.

Materials
Three types of asphalt mixtures (AC-13, AC-16, and AC-20) containing asphalt 90# were adopted as the representative material in this paper.The asphalt's properties are itemized in Table 1.Aggregate physical properties are shown in Table 2.The aggregate gradation of asphalt mixtures used in this paper is shown in Table 3.In addition, AC-13, AC-16, and AC-20 optimum asphalt content determined through Marshall testing were 4.7%, 4.5%, and 4.2% by weight of asphalt mixtures, respectively.

Experimental device and methods
The device used in this test was a self-developed LDMTS multi-function dynamic tester developed by ourselves, which was mainly composed of a loading and temperature control system.The tester can provide various common loading waveforms such as sinusoidal, square, and triangular loads and the loading frequency can be adjusted from 0.1 to 10 Hz.Moreover, this tester can apply vertical compressive and horizontal shear dynamic loads to the specimens simultaneously or separately.The specimens were stressed by using haversine without a rest period.The modulus was acquired by calculating the average ratio of maximum stress to maximum strain from collected load-deformation curves of the last five loading periods.To simulate the actual traffic load and ensure the reusability of the specimen, the stress level used in this test was 0.7 MPa, and the vertical and horizontal maximum loads applied to the specimen were 5 KN and 2 KN, respectively.The minimum load was 4% of the maximum load to avoid disengaging and ensure the stability of the specimens during testing [10] .Specimens were tested at six temperatures (−10, 5, 15, 25, 35, and 50) and five loading frequencies (0.1, 0.5, 1, 5, and 10 Hz) under haversine loading.The rest period between each frequency run was at least 2 minutes and did not exceed 30 minutes for any two frequency runs.Laboratory cylindrical specimens of 100 mm were fabricated by a gyratory compactor and cut by a cutting machine.

Dynamic modulus test under uniaxial normal stress
Given that the performance of asphalt and asphalt mixtures is susceptible to the temperature, the dynamic modulus under uniaxial stress and biaxial stress at different temperatures was compared respectively.Firstly, the dynamic modulus test under uniaxial vertical stress at different temperatures was carried out.To reduce the systematic error in the test, the moduli in this paper were all analyzed by using relative values, namely all dynamic modulus and dynamic shear modulus were divided by the maximum value respectively.Based on the results depicted in Figure 1, the dynamic modulus exhibits a significant decrease as the temperature increases.Additionally, the dynamic modulus shows an increasing trend with higher loading frequencies.

Dynamic modulus test under uniaxial shear stress
Similarly, the shear dynamic modulus with three gradations was obtained by applying only direct shear load to the specimens, as shown in Figure 2. From Figure 2, the dynamic shearing moduli of the mixes were found to decrease as the temperature rose and increase as the frequency of loading rose.

Dynamic modulus test under biaxial stresses
The normal stress and shear stress were simultaneously applied to the specimens.Meanwhile, the dynamic modulus and dynamic shear modulus were obtained by using the same method mentioned above, and the results are presented in Figures 3 and 4  Based on the results shown in Figure 5, the vertical dynamic modulus decreases noticeably when a horizontal shear load is applied compared to the case without a shear load.However, this difference gradually diminishes as the temperature increases.One possible explanation for this observation is that the application of shear force on the specimens causes internal particle sliding, resulting in a decrease in friction and cohesion between the aggregates in the asphalt mixture.Consequently, this results in a reduction of the vertical dynamic modulus.It can be predicted that the disparity between the uniaxial dynamic modulus and biaxial dynamic modulus cases will further increase with an increase in shear load.However, it is important to note that the viscosity and the interaction decrease as the temperature rises, eventually reducing the difference between the two cases.
Figure 6 indicates that the dynamic shear modulus increases significantly when vertical compressive stress is added due to the variation of the granular material.

Conclusions
Based on the previous discussion, the following conclusions can be drawn from the conducted dynamic modulus tests on three typical asphalt mixtures (AC-13, AC-16, and AC-20) at a variety of temperatures and load frequencies, under uniaxial and biaxial loading conditions: (1) The vertical and shear dynamic modulus of the asphalt concrete decreases with increasing temperature and increases with increasing loading frequency, regardless of whether uniaxial or biaxial load is applied to the specimens.
(2) The vertical dynamic modulus of asphalt mixtures under biaxial loading is significantly lower than that under vertical loading alone.
(3) The shear dynamic modulus of asphalt mixtures under biaxial loading is significantly higher than that under horizontal shear loading.
The observed changes in the dynamic modulus due to biaxial stresses applied to the actual pavement in service may contribute to the occurrence of distresses such as rutting and slippage in intersections and uphill sections.Therefore, it is crucial to pay closer attention to the dynamic modulus when designing or evaluating pavement for rehabilitation, especially when the pavement is subjected to complex stress conditions.

Figure 3 .Figure 4 .Figure 5 .Figure 6 .
Dynamic modulus (vertical direction) under biaxial stresses.(a) AC-13; (b) AC-16; (c) AC-20.Dynamic shear modulus (horizontal direction) under biaxial stresses.(a) AC-13; (b) AC-16; (c) AC-20.To figure out the different effects of uniaxial loading and biaxial loading on normal and shear modulus, the modulus of the mixtures with different temperatures and the same frequency (1 Hz) under different loading modes were taken for comparison.The comparative results are shown in Figures 5 and 6 respectively, where Label 1 in the figure refers to the uniaxial loading mode, and Label 2 refers to the biaxial loading mode accordingly.Effect of loading mode on dynamic modulus (vertical direction).(a) AC-13; (b) AC-16; (c) AC-20.Effect of loading mode on dynamic shear modulus (horizontal direction).(a) AC-13; (b) AC-16; (c) AC-20.

Table 1 .
Properties of the asphalt.

Table 2 .
Physical properties of the aggregates.

Table 3 .
Passing of the asphalt mixtures with different gradations (%)