# Fatigue Life Prediction of High Modulus Asphalt Concrete Based on the Local Stress-Strain Method
## 4 True Cyclic Stress-Strain Curve  
The cyclic stress-strain curve, which is made by the vertices of the stable hysteresis loop at different strain levels, is a smooth curve. The following methods were used to determine the cyclic stress-strain curves:  
1. Multistage test method. Cyclic loading under several strain amplitude levels was conducted on the specimen, each cycle was loaded a number of times to achieve stability, and a smooth curve was drawn using the overlapping and stable hysteresis loop through the top. This loop was named the cyclic stress-strain curve. The method required few specimens and had a quick measurement speed; however, this method has low precision and it easily produces fatigue damage. 2. Relegation-enhancement test method. When the specimen was loaded, the strain amplitude was gradually relegated and then enhanced. The other steps of this method are similar to the multistage test method
3. Stretching after cycle stability method. The specimen underwent a series of strain decreases and increases, reducing the level of strain amplitude after cycle stability was reached. A specimen was then loaded to tensile failure to determine the stress-strain curve. Results of this method are in good agreement with those in the above two methods, however, it has low precision. 4. Multistage multi-sample method. Multiple experiments controlled by a constant strain were conducted on specimens in multistage strain stages in order to obtain a stable hysteresis loop, then the top of the hysteresis loop was connected and a smooth cyclic stress-strain curve was obtained. This method can accurately reflect the cycle stress-strain properties of materials with high precision, but it requires more specimens and a longer test period than the other methods. The strength coefficient and cyclic strain hardening index are calculated using the cyclic stress-strain curve [22; 23]: \[\frac{\Delta\varepsilon}{2}=\frac{\Delta\sigma}{2E}+\left(\frac{\Delta\sigma}{ 2K^{\prime}}\right)^{\frac{1}{\alpha^{\prime}}}\] (2) where: \(\Delta\varepsilon=\) cyclic strain amplitude, \(\Delta\sigma=\) cyclic stress amplitude. The following equation can be obtained using the plastic component calculation:  
\[\varepsilon=\frac{\sigma}{E}+\left(\frac{\sigma}{K^{\prime}}\right)^{\frac{1}{n^{ \prime}}} \tag{3}\]  
where: \(\sigma\) = stress, \(\varepsilon\) = strain, \(n^{\prime}\) = Cyclic strain hardening exponent, \(K^{\prime}\) = Cyclic stress hardening exponent. The strain fatigue curve shows that the values of the hysteresis loop apex were calculated when fatigue life reached 50% of the specimen at strain levels ranging from 0.3 to 0.6 in order to determine the true strain and true stress amplitude at different strain levels and the plastic strain component (see Table 2). Table 2 shows the different fitting algorithms that were adopted to fit Equation (3). The fitting parameters \(K^{\prime}\) = 26.847 and \(n^{\prime}\) = 0.725 were then obtained and used in the following formula that predicts the strain of HMAC in the strain-controlled mode:  
\[\varepsilon=\frac{\sigma}{E}+\left(\frac{\sigma}{26.847}\right)^{1.3793} \tag{4}\]  
Stress can be obtained using Equation (4) with strains ranging from 0.0001-0.001; the results are presented in Figure 6.  
The parameters \(K\), \(n\), and the HMAC monotone curve are compared, respectively, with the parameters \(K^{\prime}\), \(n^{\prime}\), and the cyclic curves. The cyclic curves are far higher than the monotone curve. Based on the position of the curve, HMAC was determined to be a cyclic hardening material [24], indicating that the stress required for producing the same strain during cyclic loading is higher than when under a static load. However, the hardening phenomenon gradually disappears when the cycling times increase. Figure 7 shows the relationship between stress and time in the cyclic stress strain test with constant strain controlled. The results show that stress fluctuates as the length of time increases, whereas the