# Fatigue Performance and Model of Polyacrylonitrile Fiber Reinforced Asphalt Mixture
## 1 Introduction
For the past several decades, significant research efforts have been devoted to the selection of an appropriate laboratory test procedure and developing reliable fatigue models that could be used to predict the fatigue life of HMA pavements. [PERSON] and [PERSON] [17] suggested that the controlled-strain mode of loading be approached by asphalt pavements when the thickness of the HMA layer is less than 50 mm (except when the HMA layer is extremely stiff or the base is very weak). The researchers of the University of California Berkeley recommended that the repeated flexural beam (three point loading) fatigue test method under the controlled-strain mode of loading be a relatively simple test, which can best simulate field conditions, compared with various laboratory fatigue tests under the Strategic Highway Research Program (SHRP) Project [18]. [PERSON] et al. [19] conducted tests to evaluate the fatigue characteristics of HMA, designed using the Superpave volumetric method and Nevada's Iveveem method. Their tests used the flexural beam fatigue test under the strain-controlled mode of loading. The influence of long-term oven aging on the fatigue performance was also evaluated. [PERSON] et al. [20] developed a new fatigue response model based on the dissipated energy equation and the equation correlating the cumulative energy and fatigue life under the same strain level. [PERSON] and [PERSON] [21] conducted five kinds of asphalt mixtures' fatigue tests on the control strain mode. They concluded that asphalt mixtures' logarithm fatigue life follows a normal distribution at the same strain level. Meanwhile, the strain level has a linear relationship with the logarithm fatigue life. In the fatigue testing of the control strain mode, when asphalt mixtures modulus is much lower, its fatigue life is much longer. Based on the phenomenological approach, many fatigue response models have been generalized. For many years, the most commonly accepted model of the fatigue life for a given asphalt mixture can be characterized by the following relationships:  
\[N_{f}=k_{1}\bigg{(}\frac{1}{\varepsilon_{\mathrm{t}}}\bigg{)}^{k_{2}}, \tag{1}\]\[\text{or}\qquad\qquad N_{f}=k_{1}\bigg{(}\frac{1}{\sigma_{t}}\bigg{)}^{k_{2}}, \tag{2}\]  
where, \(N_{f}\) is the fatigue life; \(\epsilon_{t}\) or \(\sigma_{t}\) is the applied tensile strain or stress, respectively; and \(k_{1}\) and \(k_{2}\) are experimental coefficients. These relationships were consistently confirmed in the SHRP project for the ranges of stresses and strains to which laboratory specimens were subjected [18]. [PERSON] et al. [22] suggested introducing the mixture stiffness term to the original form so that the original effects of asphalt mixture could be taken into consideration:  
\[N_{f}=k_{1}\bigg{(}\frac{1}{\epsilon_{t}}\bigg{)}^{k_{2}}\bigg{(}\frac{1}{S_{0 }}\bigg{)}^{k_{3}}, \tag{3}\]  
where, \(N_{f}\) is the fatigue life; \(\epsilon_{t}\) is the applied tensile strain; \(S_{0}\) is the mixture initial flexural stiffness; and \(k_{1}\), \(k_{2}\), and \(k_{3}\) are experimental coefficients. Prior to [PERSON]'s contribution, [PERSON] and [PERSON] [23] suggested introducing the effect of the volumetric asphalt content (\(Va\)) and the air void (\(AV\)) content to the fatigue performance model of HMA mixtures:  
\[N_{f}=k_{1}\bigg{(}\frac{1}{\epsilon_{t}}\bigg{)}^{k_{2}}\bigg{(}\frac{1}{S_{ 0}}\bigg{)}^{k_{3}}\bigg{(}\frac{V_{a}}{V_{a}+AV}\bigg{)}^{k_{4}}, \tag{4}\]  
where, \(N_{f}\) is the fatigue life; \(\epsilon_{t}\) is the applied tensile strain; \(S_{0}\) is the mixture initial flexural stiffness; \(Va\) is the asphalt content; \(AV\) is the air void content; and \(k_{1}\), \(k_{2}\), \(k_{3}\), and \(k_{4}\) are experimental coefficients. For asphalt mixture, the concept of an improved performance with a reduced air void content and/or increased asphalt content has often been expressed in terms of voids filled with asphalt (\(VFA\)).