Laboratory Validation of an Endurance Limit for Asphalt Pavements (2013)

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13 The materials used in the uniaxial fatigue test were the same as those of the beam fatigue test, except that only one asphalt binder (PG 64-22) was used instead of three. The same factors and levels as in the beam fatigue experiment were used. The similarity of materials and factors between the uniaxial and the beam fatigue tests allows for direct comparisons of the results of the two test methods. Gyratory specimens were prepared with a 6-in. diameter and a 6.7-in. height using the Superpave gyra- tory compactor. Four-in. diameter specimens were cored from the gyratory compacted specimens and cut to a 6-in. height for complex modulus testing or cored to a 3-in. diameter and cut to a 6-in. height for the uniaxial test. Air voids were measured using the saturated surface-dry procedure (AASHTO T166, Method A). Any specimen with air voids deviating by greater than 1 percent from the target value of 7 percent was rejected. Continuum Damage Approach The Continuum Damage Mechanics (CDM) analysis approach was developed at North Carolina State University and Texas A&M University. This approach utilizes the visco- elastic correspondence principle and Work Potential Theory (WPT) described by Schapery (29) to remove viscous effects in monitoring changes in pseudo stiffness in repeated uniaxial tensile tests. Physical variables are replaced by pseudo variables based on the extended elastic-viscoelastic correspondence principle to transform a viscoelastic (linear or nonlinear) problem to an elastic case. One of the viscoelastic properties required to apply the CDM approach is the relaxation modulus. The relaxation modulus is defined as the stress response of a viscoelastic mate- rial due to a unit step of strain input. The relaxation modulus can be calculated as the time-dependent stress divided by the initial applied strain as shown by Equation 4: (4)E t t o ( ) ( )= σ ε where E(t) is the relaxation modulus at time t, s(t) is the stress at time t, and eo is the initial applied strain. The calculation of the pseudo stiffness (PS) requires the calculation of pseudo strain (eR). The pseudo strain can be calculated rigorously using Equation 5, where e is the mea- sured strain, E(t) is the linear viscoelastic relaxation modulus and ER is the reference modulus (typically taken as 1) used for dimensional compatibility (29). 1 (5) 0E E t d d dR R t ∫ ( )ε = − τ ετ τ Kim et al. (30) proposed a simplified approach for the steady-state assumption to calculate the pseudo strain as shown in Equation 6. This equation is based on the assump- tion that fatigue damage accumulates only under the tensile loading condition, represented by the pseudo strain tension amplitude, eR0,ta. In such conditions, the pseudo strain can be rigorously computed as the product of strain and dynamic modulus, |E*|LVE (at temperature and frequency matching with the test under investigation). 1 1 2 * (6)0, 0, E EtaR i R pp i LVE( )( )( )ε = ⋅ β + ε ⋅ where b is a factor used to quantify the duration that a given stress cycle is tensile (1 means always tensile, 0 means fully reversed loading and -1 means always compressive), and e0,pp stands for peak-to-peak strain amplitude. Underwood et al. (31) suggested a simplified approach to calculate the pseudo strain. For the first loading path where damage growth may be significant, the rigorous calculation (31) is used; for the remaining cycles, the simplified calcula- tion is used. Once the pseudo strain is calculated, the PS is also calcu- lated through Equation 7 using the pseudo strain as defined in Equations 5 and 6. C H A P T E R 3 Developing of Endurance Limit Model Based on Uniaxial Fatigue Tests

14 PS DMR first cycle DMR rest of cycles R ta R = × × σ ε σ ε0,      ( )7 where the DMR is the dynamic modulus ratio to account for specimen-to-specimen variability (32) and is defined as shown in Equation 8. In this equation |E*|LVE is the linear viscoelastic dynamic modulus of the material at the par- ticular temperature and frequency of the test and it can be determined from the |E*| master curve. |E*|fp is the finger- print dynamic modulus that is measured from a fingerprint experiment performed before the uniaxial fatigue test. * * (8)DMR E E fp LVE = For the purpose of developing the Pseudo Stiffness Ratio (PSR) model using the continuum damage approach and predicting the endurance limit, two tests were performed: the complex modulus test and the uniaxial tension-compression fatigue test. Complex Modulus Testing The main objective of the complex modulus test was to construct the dynamic modulus master and phase angle mas- ter curves to estimate the relaxation moduli required in the analysis of the viscoelastic and continuum damage model. Four asphalt mixtures were tested in this part of the study representing combinations of two levels of asphalt content and two levels of air voids (4.2%AC-4.5%AV, 4.2%AC- 9.5%AV, 5.2%AC-4.5%AV, and 5.2%AC-9.5%AV). Complex modulus (E*) tests were performed accord- ing to AASHTO TP 62-07 using a servohydraulic testing machine. Two replicates were tested for each factor combi- nation. E* tests were conducted on each specimen at a full sweep of loading frequencies (25, 10, 5, 1, 0.5 and 0.1 Hz) and temperatures (-10, 4.4, 21.1, 37.8 and 54.4°C). Using the E* test results, i.e., the dynamic modulus, |E*|, and the phase angle, f, master curves were constructed for the four mixtures using the time-temperature superposition prin- ciples (Figure 14). It can be observed from Figure 14 that the |E*| values were more significantly affected by varying the air voids from 4.5 to 9.5% than by varying the asphalt content from 4.2 to 5.2%. Uniaxial Fatigue Testing In order to determine the strain values to be used in the uniaxial fatigue test, a pilot experiment was performed to establish the log Nf-log et relationships for the four mix- tures at three selected temperatures (40, 70, 100°F). These fatigue relationships were then used to determine the three tensile strain values used in the uniaxial fatigue experi- ments. The criterion for selecting the three tensile strain values at each temperature was to reach a fatigue life of 5,000, 20,000, and 100,000 cycles at the high, medium, and low tensile strain values, respectively. To establish a single fatigue relationship, four uniaxial tension-compression fatigue tests were conducted at different strain levels, which required 12 tests for one mixture at three temperatures (40, 70, and 100°F) or 48 tests for the four asphalt concrete mix- tures. These fatigue relationships were then used to deter- mine the tensile strain values for each mixture at the three temperatures. Figure 14. (a) Dynamic modulus master curves and (b) phase angle master curves. 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 -10 -5 0 5 10 D yn a m ic M od u lu s, ps i Log Reduced Frequency, Hz 4.2%AC, 4.5%Va 5.2%AC, 4.5%Va 4.2%AC, 9.5%Va 5.2%AC, 9.5%Va (a) (b) 0 10 20 30 40 -10 -5 0 5 10 Ph as e An gl e, D eg re es Log Reduced Frequency, Hz 4.2%AC, 4.5%Va 5.2%AC, 4.5%Va 4.2%AC, 9.5%Va 5.2%AC, 9.5%Va

15 Figure 15. (a) Uniaxial fatigue test setup and (b) Gluing jig. (b)(a) Waveform Selection Similar to the beam fatigue test, the uniaxial fatigue test can be performed using a haversine (pull-pull) or a sinusoi- dal (pull-push) waveform. The rationale used in the beam fatigue test was also used in the uniaxial fatigue test, in which it was concluded that a haversine waveform cannot be main- tained during the test on a viscoelastic material such as HMA. If a haversine tensile waveform is used, the gauge length of the specimen will change in such a way that the haversine waveform will change to sinusoidal after a few loading cycles. This would result in misleading results because of the incon- sistency between the assumed condition (tension only with a peak-to-peak value) and the actual test conditions (tension and compression with a half peak-to-peak value). Therefore, a sinusoidal (pull-push or tension-compression) waveform was used in the uniaxial fatigue test. Experimental Design The uniaxial fatigue tests were conducted using the test protocol developed in Appendix 2 of this report. Figure 15(a) shows the system setup used to conduct the uniaxial fatigue test. The vertical deformation is measured with three spring- loaded on-specimen linear variable differential transformers (LVDTs) spaced 120 degrees apart. The LVDTs are attached to the specimen using parallel brass studs to secure the LVDTs in place. Three pairs of studs are glued on the surface of the specimen with gauge lengths of 4 in. Because of the difference between the actuator displacement and specimen displace- ment, on-specimen LVDTs had to be used considering the small displacement measurements used in this test. After many trials and several gluing procedures, it was found that the uniaxial test is much more cumbersome and time consuming than the beam fatigue test. Several issues had to be addressed in order to perform successful tests. For example, the specimen has to be glued to the platens in order to subject the specimen to tension. The proper glue has to be selected in order to ensure that failure occurs in the specimen, and not at the interface between the specimen and the platen. Another important issue is specimen alignment. If there is any small eccentricity in the specimen, the specimen may break after a few loading cycles or may break too close to one of the platens and outside the gauge length. A special gluing jig was manufactured in order to carefully center the specimen when it is glued into the end platens as shown in Figure 15(b). The uniaxial fatigue tests were conducted by controlling the on-specimen strain using a sinusoidal strain waveform (tension-compression). Similar to the beam fatigue experi- ment, some tests were performed without rest periods with a frequency of 10 Hz, while others used a 0.1s sinusoidal load- ing followed by a rest period. Fatigue failure was defined at 50% reduction of the initial stiffness. As a part of the uniaxial test protocol, two special uniaxial fatigue software programs were developed by the IPC Com- pany to conduct uniaxial fatigue tests without and with rest periods. The uniaxial fatigue test without rest period was conducted until fatigue failure occurred as discussed subse- quently. However, all the tests with rest period were stopped at 20,000 cycles due to time limitations of the overall study. In order to reduce the number of tests and at the same time determine important effects of all variables, a five-factor frac- tional factorial statistical design was used considering the effect of all five factors, two-factor interactions, and three-factor inter- actions (27, 28). Similar to the beam fatigue test, two or three replicates were tested for each factor combination. An analysis of the results from 132 tests was used to develop the model. Model Development Similar to the beam fatigue test, several initial regression models were tried. Originally, a regression model was devel- oped to relate the PSR to all factors used in the study, which are binder content, air voids, temperature, applied tensile strain, rest period, and number of loading cycles. Later, the model was simplified by replacing the binder content, air voids, and temperature with the initial stiffness of the mixture. The inclusion of the rest period decreases the stiffness deterioration through partial or full healing of fatigue dam- age. That is, the stiffness tends to deteriorate at a slower rate compared to the test without rest period. In this study, the PS values at different loading cycles were determined using the improved calculation methods developed by Underwood et al. (31). Because the PS varies between replicates, the PSR

16 was used, which is the PS value at any cycle (PSn) normalized to the initial stiffness (PSo). Figure 16 shows the relationship of PSR versus time for two tests conducted using 0 and 5s rest periods and a 310 peak-to-peak microstrain at 70°F. In this part of the study, the PSR parameter was used to define the effect of rest period. For the test with rest period, decreasing the strain level or increasing the rest period would decrease the net fatigue damage and increase the PSR over time. If the test was conducted at specific tensile strain and rest period so that the fatigue damage created during loading is equal to the healing occurring during the rest period, the tensile strain is the endurance limit. This means that the PSR is equal to 1.0 at all loading cycles as demonstrated in Figure 17. To determine the tensile strain value that represents the endurance limit when the net damage is zero (PSR = 1.0), the tensile strain versus the PSR at different temperatures and at a certain number of loading cycles is required. The endur- ance limit is calculated as the tensile strain value when this relationship intersects with PSR of 1.0 as shown in Figure 18. The test results from all uniaxial fatigue experiments were combined together to develop the PSR model for the PG 64-22 mixture. For each test with rest period, four PSR val- ues were measured at different N values on the PSR-N curve (Figure 16) to represent the nonlinear change of PSR over time. Only the PSR values at Nf were considered for the tests without rest period. A total number of 161 test results and 385 data points were used in the model development. The PSR model included the main five factors plus one addi- tional factor, which is the value of N where the PSR was measured. A nonlinear optimization analysis was used considering one- and two-factor interactions in the statisti- cal model. A powerful nonlinear optimization technique that uses innovative genetic algorithm (GA) technology was uti- lized to provide an accurate optimization solution. Evolver®, a GA technology-based software that is well-suited to find the best overall answer by exploring the entire universe of possible answers, was used in this part of the study to develop the PSR model (33). The optimization technique requires the main form of the regression model as an input. To construct a rational model, the relationship between the PSR and each factor was investigated by following an iteration process. It was found that there is a need for a logarithmic transformation for strain and number of loading cycles, while the second degree polynomial func- tion was proper for temperature. For the rest period, a special function was used to fit its relationship with the PSR. The lab- oratory tests showed that increasing the rest period increases the PSR, indicating more healing. The rate of increase of PSR decreases as the rest period increases up to a certain threshold value above which there is no more PSR increase as shown in Figure 19. Using the tangent hyperbolic (Tanh) function to fit the PSR and rest period relationship, the threshold rest period can be found. The shape and form of the tangent hyperbolic 0 0.2 0.4 0.6 0.8 1 1.2 0 5000 10000 15000 20000 25000 Ps eu do S tif fn es s R at io Number of Loading Cycles Test with Rest Period Test without Rest Period Figure 16. Pseudo Stiffness Ratio (PSR) versus time relationship for tests with and without rest periods. Figure 17. Effect of strain and rest period of the PSR as a function of the loading cycles. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 5000 10000 15000 20000 Ps e u do St iff n e ss R at io Loading Cycles, N 5.2%AC, 9.5%Va, 113 µε, 5 sec RP 5.2%AC, 9.5%Va, 155 µε, 5 sec RP 5.2%AC, 9.5%Va, 125 µε, 0 sec RP 5.2%AC, 9.5%Va, 155 µε, 0 sec RP Tests with RP Tests without RP Decreasing strain Decreasing strain At Endurance Limit, SR = 1.0 for all values of N Figure 18. Determination of endurance limit at each temperature using PSR parameter. 40°F 70°F 100°F 0 1 Tensile Strain (Full Healing) (No Healing) PS R Endurance Limits

17 function to fit the PSR and rest period relationship are pre- sented in Figure 19. In order to obtain a non-biased regression model, the sum of errors was set to zero. The model was further improved by removing the outlier data points using the method suggested by Montgomery (28). The analysis was then repeated based on the remaining 383 data points and the regression model shown in Equation 9 was obtained. PSR 0.459539 0.090917 log E – 0.104389 log 0.417028 Tanh 0.875884 RP 0.238893 log N 0.120018 log E log 0.041502 log E log N 0.077377 log log N (9) o t o t t o            ( ) = − ε + + + ε − − ε where, PSR = pseudo stiffness ratio Eo = initial flexural stiffness (ksi) et = applied tensile microstrain (the tensile portion of the tension-compression loading cycle, or half peak- to-peak) (10-6 in./in.) RP = rest period (seconds) N = number of loading cycles The adjusted R2 value of the model was 0.951. Figure 20 presents the measured versus predicted PSR, which indicates accurate prediction. Similar to Figure 9, the two clusters of data points represent tests with and without rest periods. For all the cases presented, it is clear that the PSR increases as the rest period increases up to a certain threshold value, after which the PSR is constant. The threshold rest period val- ues for all the cases were around 3s for a loading time of 0.1s. To investigate the effect of N on the PSR and consequently on the endurance limit, the PSR versus the tensile strain relationships were investigated at different RP, N values of 25,000, 50,000, 100,000, and 200,000 loading cycles, and temperatures of 40, 70, and 100°F for the four asphalt mix- tures. Unlike the results of the beam fatigue study, the results showed that PSR-tensile strain relationships are not parallel at different N values. Estimation of Endurance Limit Based on Uniaxial Fatigue Testing Model The PSR model was used to predict the PSR values (when PSR is 1.0) at different tensile strain values for each mixture type at different temperatures. The endurance limit was esti- mated by plotting PSR versus strain at rest periods of 1, 2, 5, 10, and 20s. In each case, the endurance limit was obtained as the strain corresponding to a PSR value of 1.0, indicating complete healing during the rest period. Figure 21 demon- strates examples of stiffness ratio versus strain for rest periods of 1 and 5s. Figure 22 and Table 5 demonstrate the endurance limit val- ues for different stiffness and rest period values at N = 20,000 cycles. It can be observed that the mixtures with higher stiff- ness showed lower endurance limit as expected. In addition, the endurance limit values were stable after 5s of rest period. The threshold rest period occurred at about 3s. The number of loading cycles, N, has a slight effect on the endurance limit values, where higher N showed slightly higher endurance limit values. Figure 23 shows the relation between the PSR and the ten- sile strain at different numbers of loading cycle. The figure shows that the number of loading cycles has little effect on the PSR value. The endurance limit was calculated at 20,000 cycles for the uniaxial model. Figure 19. Effect of rest period on PSR. 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 6 7 8 9 10 11 PS R RP, Second PSR = a + b * Tanh (c * RP) a, b, and c are regression coefficients Figure 20. Measured versus predicted PSR for the uniaxial fatigue test model. y = 0.9501x + 0.0411 R² = 0.9511 Se/Sy = 0.2253 N = 383 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 M e a su re d PS R Predicted PSR

18 Rest Period (seconds) Stiffness (ksi) Predicted EL (µ ) EL Range (µ ) 1 2,000 1 1-25 1,000 1 200 10 50 25 2 2,000 1 1-65 1,000 9 200 42 50 65 5 2,000 3 3-82 1,000 18 200 59 50 82 10 2,000 3 3-82 1,000 18 200 59 50 82 20 2,000 3 3-82 1,000 18 200 59 50 82 Table 5. Endurance Limits (EL) predicted from the uniaxial fatigue test model. Figure 21. PSR versus tensile strain at different initial stiffness values for 1-second and 5-second rest periods. 0.5 0.6 0.7 0.8 0.9 1 1 10 100 1000 PS R Microstrain Eo = 2000 ksi Eo = 1000 ksi Eo = 200 ksi Eo = 50 ksi N = 20,000 RP = 1.0 sec. 0.5 0.6 0.7 0.8 0.9 1 1 10 100 1000 PS R Microstrain Eo = 2000 ksi Eo = 1000 ksi Eo = 200 ksi Eo = 50 ksi N = 20,000 RP = 5.0 sec. Figure 22. Endurance limit values at different rest periods and stiffness values using uniaxial fatigue test model (N  20,000). 0 20 40 60 80 100 1 2 5 10 20 En du ra n ce Li m it, µε Rest Period, seconds Eo=2000 ksi Eo=1000 ksi Eo=200 ksi Eo=50 ksi

19 Figure 23. Effect of N on PSR at different initial stiffness and RP values. 0 0.2 0.4 0.6 0.8 1 000100101 PS R Microstrain N = 25,000 N = 50,000 N = 100,000 N = 200,000 Eo = 500 ksi RP = 1.0 sec 0 0.2 0.4 0.6 0.8 1 000100101 PS R Microstrain N = 25,000 N = 50,000 N = 100,000 N = 200,000 Eo = 100 ksi RP = 1.0 sec 0 0.2 0.4 0.6 0.8 1 000100101 PS R Microstrain N = 25,000 N = 50,000 N = 100,000 N = 200,000 Eo = 500 ksi RP = 10.0 sec 0 0.2 0.4 0.6 0.8 1 000100101 PS R Microstrain N = 25,000 N = 50,000 N = 100,000 N = 200,000 Eo = 100 ksi RP = 10.0 sec