# Modeling of Fatigue-Strength Development
## 2 Materials and Methods
### Fatigue Testing and Data Analysis  
Fatigue tests on the cold-emulsion reclaimed asphalt pavement beams was conducted in accordance with procedures set out in BS EN 12697-24 (E) Annex D [69]. Flexural tests were conducted on a four-point bending (4 PB) test jig enclosed in a universal testing machine, Dynapave UTM 30, developed and supplied by Industrial Process Controls Ltd., Italy. The test setup is composed of a loading frame, an environmental chamber, a control panel, a hydraulic power supply unit and a personal computer equipped with automated control and data acquisition system (CDAS) that collects and stores information in American Standard Code for Information Interchange (ASCII), Comma Separated Variable (CSV) format. Six specimens for each cure duration, within each cure temperature, were tested in controlled-strain set up at 5 \({}^{\circ}\)C, 10 Hz, and strain levels spread at intervals of 25 \(\upmu\)m/m between 125 \(\upmu\)m/m and 250 \(\upmu\)m/m. Upon completion of each test, data collected and stored by the control and data acquisition system (CDAS) was exported and saved in notepad form. Figure 2 depicts the setup of equipment used in the four-point bending fatigue test.  
For purposes of data analysis, fatigue test results were imported from notepad to excel as semicolon delimited data. The number of load cycles to failure, N\({}_{\text{f}}\), at a particular strain level was determined by plotting the complex stiffness modulus, E*, against the number of load applications, N, for each of the 168 tested specimens. A third-order polynomial equation was fitted to the data to obtain the number of load cycles, N\({}_{\text{f}}\), at which the stiffness of the specimen had reduced to 50% of the initial value. Typically, a material under cyclic loading accumulates damage and experiences a reduction in stiffness as the test progresses. According to BS EN 12697-24 (E) Annex D [69], the initial value of complex stiffness modulus is taken as that of the 100 th cycle, to allow the specimen to \"bed-in\" in the loading jig. Figure 3 shows how the stiffness of an asphalt specimen evolves during a flexural fatigue test. In Figure 3, E*\({}_{100}\) denotes initial complex stiffness modulus of the specimen measured at the 100 th load cycle, E*\({}_{\text{f}}\) denotes the complex stiffness modulus at failure and N\({}_{\text{f}}\) denotes the number of load cycles at failure.  
Figure 3: Stiffness evolution in an asphalt fatigue test.  
Figure 2: 30 kN Universal Testing Machine (UTM 30-IPC Global).  
Upon completion of fatigue tests for each cure duration, within each cure temperature, the number of load cycles to failure were plotted against the test strain levels and fitted to Equation (14) using linear regression approach. Equation (14) was proposed by [PERSON] et al. [70] as a suitable predictive model for asphalt fatigue strength at a loading frequency of 10 Hz in controlled-strain test mode. Based on Miner's law of cumulative pavement damage, Equation (14) assumes logarithmic linearity between the test strains and the number of load applications to failure [54]. This linearity has been found to hold best at temperatures below 5 \({}^{\circ}\)C, when asphalt behaves in an almost linear-elastic manner  
\[\ln\left(\text{N}_{\text{f}}\right)=\text{a}+\text{b}\ln\left(\varepsilon\right) \tag{14}\]  
where,  
\(\text{N}_{\text{f}}=\text{Number of load cycles to failure}\)  
\(\varepsilon=\text{Strain levels in (\mu\text{m/m})}\)  
\(\text{a}=\text{Intercept of the regression curve}\)  
\(\text{b}=\text{Slope of the regression curve}\).