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(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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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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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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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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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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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