1. Introduction
In recent years, more and more asphalt pavements have been built, but the qualities of part of them are not guaranteed. The asphalt pavement, subjected to repeated actions of different kinds of loads, is prone to fatigue failure, because of the strength reduction and the fatigue damage of materials [
1]. The strength of asphalt mixture decreases gradually with the increase of the cyclic loading times, and the attenuated material strength can be used to define the damage. Some damage models were proposed to study the fatigue performance of asphalt mixture based on the change laws of the strength reduction [
2,
3]. The bearing capacity of asphalt mixture decreases with the increase of damage. The strength reduction will lead to the damage and the damage evolution, and the damage evolution accelerates the strength reduction [
4,
5]. When the damage evolves to the threshold of the failure, the bearing capacity will be less than the applied load, and the fatigue failure will occur to the material structure [
6,
7].
Many researches have been carried out to study the influence of experimental factors on the fatigue resistance of asphalt mixture. Furthermore, many mathematical models (such as the mathematical models of recovery ratio of elastic deformation, permanent deformation and permanent deformation ratio, etc.) were proposed to study the fatigue resistance of asphalt mixture, and it was found that the permanent deformation ratio reflects the damage evolution speed and can be well used to predict the fatigue life [
8]. Logarithmic fatigue life linearly decreases with the increase of stress-strength ratio, and the fatigue life at low frequency is much less than that at high frequency [
9]. There are many researches about the effects of the asphalt type on the fatigue life were carried out by experts. Three kinds of asphalt were used to study the effects of the asphalt type on the fatigue life through the fatigue test, and the results show that the rubber asphalt can well improve the fatigue performance of asphalt mixture compared with the base asphalt, styrene butadiene styrene (SBS) modified asphalt [
9]. When the bitumen aggregate ratio is between 7.5% and 9%, the fatigue life of the rubber asphalt mixture increases with the increase of asphalt content [
10]. The fatigue life of the rubber asphalt mixture increases with the decrease of the air voids in the reasonable range [
11]. The high fatigue life of the rubber asphalt mixture was found to be closely related to the gradation [
12]. Three kinds of gradation were used to study the effects of the asphalt mixture type on the fatigue life through the fatigue test, and the results of the research show that when the stress-strength ratio is relatively low (0.3–0.5), the gradation has a significant effect on the fatigue life of the asphalt mixture, and the increasing sequence of the fatigue life of different asphalt mixture types corresponds to the asphalt mixture types of AC-13, stone mastic asphalt (SMA), and gap gradation, respectively, which was caused by the reason that the internal structure and the air voids are different when the gradation changes. However, when the stress-strength ratio is relatively high (0.6–0.8), the gradation has no obvious effect on the fatigue life of the asphalt mixture [
9].
The energy (damage) approach is widely used in the fatigue-life prediction, because of its simple principle and convenient operation [
13,
14,
15,
16]. The energy dissipation law of asphalt mixture was studied and the viewpoint that the dissipated strain energy (DSE) of material is related to their fatigue properties was proposed by Heukelom [
17]. The total amount of DSE before the fracture of specimen was analyzed and the viewpoint that the maximum number of loading cycles can be affected by the cumulative DSE was proposed by Chomton and Valayer [
18]. The viewpoint that there is a power function relation between the fatigue life and the cumulative DSE of the material structure before its fatigue failure was proposed based on the assumption that all the DSE before failure can cause the damage of material by Van Dijk [
19]. The conclusion that temperature and loading mode can affect the mathematical relationship between the fatigue life and cumulative DSE of asphalt mixture was proved by Tayebali [
20]. The damage evolution process was studied from the angle of energy by using the change laws of DSE and recoverable strain energy based on the different viscoelastic properties of asphalt mixture in the tensile and compressive portion by Xue Luo [
21].
Most of the above researches on the fatigue-life prediction for asphalt mixture, based on the assumption that all the DSE before failure can cause the damage of material, cannot accurately reveal the damage evolution mechanism. In fact, only partial DSE can cause the fatigue damage to asphalt mixture, and the change of DSE is the real cause of material damage [
22,
23,
24,
25,
26]. Therefore, the damage variable, defined by the cumulative total amount of dissipative energy before the fatigue failure, cannot accurately reveal the damage evolution process of asphalt mixture. The fatigue equation based on the damage variable cannot be used to correctly predict the fatigue life before fatigue failure.
In order to accurately reveal the damage evolution process and accurately predict the fatigue life of asphalt mixture, three-point bending fatigue tests were carried out. The conclusion that the DSE becomes smaller, with the increase of fatigue life, was obtained by analyzing the influences of temperature, stress ratio and rate on the calculation results of DSE. The mathematical models of DSE and dissipated strain energy ratio (DSER) was established based on the change laws of the DSE. The conclusion that the fatigue life decreases with the increase of DSER was obtained by defining the damage variable using DSER and studying the damage evolution process. The fatigue equation, established based on the relation of the fatigue life to the plateau value (PV) of DSER, was used to predict the fatigue life, and the results show that it can accurately predict the fatigue life of asphalt mixture.
The applied main research methodology was that the fatigue test was used for the theoretical analysis of the relation of the fatigue life to the PV, and the fatigue equation was established based on the relation.
4. Construction of DSER Mathematical Model
The DSE is gradually accumulated during the damage evolution. At present, the study of fatigue damage evolution based on DSE is mostly based on the assumption that all the DSE causes damage to asphalt mixture. The assumption has been proved to be inaccurate and damage to the asphalt mixture will be caused by the change of DSE [
31]. The concept of a dissipated strain energy ratio (DSER) was proposed based on the absolute value of the change in dissipated strain energy between load cycle
N and load cycle
N + 1 divided by the dissipated strain energy in load cycle
N [
22,
32]:
where DSE
N+1 and DSE
N are the DSE of the loading cycle
N + 1 and
N. The DSER represents the percentage of dissipated energy causing damage to the material.
The
of the loading cycle
N is a fixed value under certain conditions, therefore, the proportion of the
to the DSE
N is reflected by the DSER that is only related to the damage deformation [
22,
23]. Mastering the change law of DSER can provides a theoretical basis for the study of damage evolution. In order to accurately describe the change law of DSER in the process of damage evolution, the DSER mathematical model of the three-point bending fatigue test, obtained by substituting the Equation (2) into the Equation (3), is shown in Equation (4). The DSER results of the mathematical model were used to compare with that obtained by tests, and the comparison results are shown in
Figure 9, from which it can be seen that the mathematical model well predict the overall change trend of DSER.