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53 (a) Stress history (b) Strain history Figure 5.17. Typical stress/strain history for constant amplitude uniaxial fatigue test. 2 Specimen #15 Specimen #14 1.5 Monotonic 1 C 0.5 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 S Figure 5.18. Comparison monotonic and cyclic curves for PG 67-22 optimum. during the tests until failure, at which point the mean strain fSp 0 2 = (23) p ( 0.125 IC11C12 ) N ( E ) drops. 2 Figure 5.18 shows the characteristic curves constructed from both the fatigue and monotonic tests for the PG 67-22 2 f [ e - kS - 1] optimum mixture. Previous research has shown better agree- 0 2 = (24) I ( -k ) N ( E ) 1+ 2 ment between samples and between monotonic and cyclic tests (32). where, 0 is the strain level required to sustain N number of load Evaluation of Endurance Limit repetitions, Prediction from Characteristic S is the damage parameter value at failure (measured from Damage Curves the damage characteristic curve for the mixture at the point where C = 0.3, identified in previous research [32]), Once dynamic modulus, initial stiffness, testing frequency, I is the initial pseudo stiffness, and damage curve coefficients are known, the strain level |E| is the dynamic modulus at testing frequency ( f ), and required to sustain any number of design load repetitions can is the material constant, p = 1 + (1 - C12). be predicted. Equations 23 and 24 can be used to find required strain level with generalized power law and exponential mod- Plots of strain level (0) versus design load repetitions (N) els, respectively (60, 61). obtained for individual asphalt mixtures are presented in

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54 1.0E-02 Exponential model Generalized power model 1.0E-03 Strain 1.0E-04 1.0E-05 1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08 Number of repetitions Figure 5.19. Plot of strain level versus load repetitions for PG 67-22 optimum. Figures 5.19 through 5.22. A comparison of the relations sented graphically in Figure 5.24. The values obtained using of all mixtures together is presented Figures 5.23(a) and the exponential model are much lower than those obtained (b) for the generalized power law and exponential models, from the generalized power law model. There is more confi- respectively. dence in the values from the generalized power law because The strain levels required to sustain 50 million cycles of this function fits the C-versus-S data for these mixtures bet- repetitions for all mixtures are shown in Table 5.3 and pre- ter than the exponential function. 1.0E-02 Exponential model Generalized power model 1.0E-03 Strain 1.0E-04 1.0E-05 1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08 Number of repetitions Figure 5.20. Plot of strain level versus load repetitions for PG 76-22 optimum. 1.0E-02 Exponential model Generalized power model 1.0E-03 Strain 1.0E-04 1.0E-05 1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08 Number of repetitions Figure 5.21. Plot of strain level versus load repetitions for PG 67-22 at optimum plus.

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55 1.0E-02 Exponential model Generalized power model 1.0E-03 Strain 1.0E-04 1.0E-05 1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08 Number of repetitions Figure 5.22. Plot of strain level versus load repetitions for PG 76-22 at optimum plus. 1.0E-03 Strain 1.0E-04 1.0E-05 1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08 Number of repetitions 67-22 @ Opt. 76-22 @ Opt. 67-22 @ Opt. + 76-22 @ Opt. + (a) Generalized power models 1.0E-02 Strain 1.0E-03 1.0E-04 1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08 Number of repetitions 67-22 @ Opt. 76-22 @ Opt 67-22 @ Opt. + 76-22 @ Opt+ (b) Negative exponential models Figure 5.23. Plot of strain level versus load repetitions LVDTs for all mixtures.

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56 Table 5.3. Computed critical micro-strain to sustain of the material. The hysteresis loop collapses when the pseudo 50 million cycles. strain is plotted versus the stress, as seen in Figure 5.27. If dam- age occurs in the material, a hysteresis loop will appear in the Damage Asphalt Mixture Grade Characteristic 67-22 76-22 67-22 76-22 stress-pseudo strain plot. Curve Form Optimum Optimum Optimum+ Optimum+ The analysis of the increasing strain amplitude fatigue tests Exponential model 96 70 64 47 involves calculating the pseudo strain and then plotting Generalized pseudo strain versus stress at each strain amplitude to deter- 261 197 194 164 power model mine the strain level at which loops begin to appear. The pres- ence of a loop in the stress-pseudo strain plot indicates that damage is occurring in the specimen. Figures 5.28 and 5.29 Increasing Strain Amplitude Test show the stress versus pseudo strain plots for PG 67-22 op- This test consists of applying blocks of haversine loading to timum plus and PG 76-22 optimum plus specimens, respec- a uniaxial test specimen. Initially, a relatively low strain am- tively. For the PG 67-22 optimum plus specimen, it is clear plitude that is thought to be below the fatigue endurance limit that there is no loop at the lowest strain amplitude, a loop is applied. Typically, this is close to the same amplitude at appears to just be forming at the middle strain amplitude and which dynamic modulus tests are performed. Approximately a loop is definitely apparent at the highest strain amplitude. 10,000 cycles are applied at this amplitude to allow the speci- These two figures indicate that the fatigue endurance limit men to reach steady-state response. The applied strain ampli- (strain level below which no damage is occurring) for this tude is then increased and 10,000 more cycles are applied. This specimen is around 150 ms. For the PG 76-22 optimum plus procedure is continued until the specimen fails. Figures 5.25(a) specimen shown in Figure 5.29, the loop appears in the sec- and (b) show a typical load and strain history, respectively, ond load level, which is 245 ms. Hence, the fatigue endurance recorded during an increasing amplitude fatigue test. It should limit is somewhere between 93 and 245 ms. Table 5.4 sum- be noted that, for this project, the crosshead displacement marizes the results of the increasing amplitude fatigue tests was controlled while the on-specimen LVDT measurements for all specimens tested. It should be noted that only three were used for strain analysis. Machine compliance made it mixtures were subject to this test method. difficult to precisely control the strain amplitude in the spec- The third column in Table 5.4 shows the bounds of the imen at the different levels. strain level at which loop formation was observed. For sev- eral specimens, the first level tested resulted in loop forma- tion, so only an upper bound is reported. The specimen Concept of Pseudo Strain stiffness at 50 cycles and air void content are also shown in The use of pseudo strain instead of engineering strain in Table 5.4. From this information, it is evident that there are constitutive analysis removes the hysteretic effect of vis- two groups of specimens for the 67-22 optimum plus and coelasticity. For example, a plot of stress versus strain data 76-22 optimum plus mixtures. The 67-22 optimum plus Spec- obtained from a dynamic modulus test produces a hysteresis imens 1 and 2, and the 76-22 optimum plus Specimens 1 loop, as shown in Figure 5.26. The load levels applied during a and 3, have lower air void contents and correspondingly higher dynamic modulus test are low enough not to induce damage, stiffnesses and strain range at which loop formation occurs so the hysteresis loop is purely due to the viscoelastic response than the other specimens. Using engineering judgment, the 0.0003 Exponential Model Generalized Power Model 0.0002 Strain 0.0001 0.0000 67-22 @ 76-22 @ 67-22 @ 76-22 Optimum+ Optimum Optimum Optimum+ Asphalt concrete mix Figure 5.24. Comparison of critical strain to sustain 50 million cycles for all mixtures.

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57 Stress, MPa Load Pseudo Strain, m/m Time (a) Load History Figure 5.27. Stress versus pseudo strain plot for several cycles of loading. LVDT reading 4 Stress, MPa Time (b) Strain History Pseudo Strain Figure 5.25. Typical stress/strain history for increasing Figure 5.28. Stress versus pseudo strain plots for amplitude uniaxial fatigue test. PG 67-22 optimum plus. Stress, MPa Stress (MPa) Strain, m/m Figure 5.26. Stress versus strain plot for Pseudo strain several cycles of loading. Figure 5.29. Stress versus pseudo strain plots for PG 76-22 optimum plus.

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58 Table 5.4. Summary of increasing amplitude fatigue test data. Loop Estimated Specimen Stiffness @ 50th Air Voids, Mixture Formation Endurance Number Cycle (MPa) % Strain Range Limit 15 <1811 8955 6.9 67-22 Opt 17 107