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Fatigue Resistance (3840). Oxidation, both during refining and during age
hardening, can dramatically increase the value of R (38, 40).
Continuum Damage Approach to The primary objective of NCHRP Projects 9-25 and 9-31 was
Fatigue Phenomena in HMA to develop relationships among mixture compositional char-
During NCHRP Projects 9-25 and 9-31, a practical acteristics and HMA performance. Although R is not a mix-
approach was developed by applying continuum damage the- ture compositional characteristic, it must be included in
ory to characterizing and analyzing the fatigue response of fatigue equations to ensure that the models are valid and
HMA. The following discussion is a summary of this method accurate. It is not recommended that the R value be con-
of analysis and is a relatively minor extension of previous trolled as part of the mix design process; control of R must be
work on continuum damage theory done by other researchers done through the binder specification and should be the topic
(16, 3337). The following equation for fatigue life was of other research projects. However, it must be emphasized
derived based upon continuum damage theory and an expo- that none of the fatigue models presented here involve inter-
nential damage rate: actions between R and other mixture characteristics. There-
fore, even though variations in R are not directly addressed in
2 f (C - - 1) this report, the effects of changes in mixture composition on
N= (3) fatigue resistance are still accurate and valid.
( -C 2 ) 0
1+
2 2
E * LVE
Another important finding in analyzing the NCHRP Proj-
where ects 9-25 and 9-31 fatigue data was that the complex modu-
N = fatigue cycles; lus in tension/compression, as determined at the start of the
= a material constant for viscoelastic material; uniaxial fatigue tests, was significantly lower than that meas-
f = loading frequency, Hz; ured in dynamic compression. Based upon comparing the
C = damage ratio (damaged/undamaged modulus); measurements made during these fatigue tests and dynamic
C2 = continuum damage fatigue constant; compression values predicted using the Hirsch equation, the
0 = applied strain amplitude (1/2 of peak-to-peak strain); following empirical equation was developed for estimating
and tension/compression modulus values (|E*|TC) from |E*| values
|E*|LVE = linear viscoelastic (LVE) complex modulus. determined for dynamic compression:
To apply Equation 3 (and related functions) to fatigue phe- |E*|TC = 0.000209|E*|1.57 (5)
nomena in HMA, the values of and C2 must be known
along with the values for frequency and modulus. Analysis of It was found that tension/compression modulus values com-
uniaxial fatigue data gathered during NCHRP Projects 9-25 puted using this equation agree well with flexural stiffness
and 9-31 lead to the following empirical equation for esti- values predicted with Bonnaure's equation and as measured
mating C2 as a function of |E*|, VFA, and the rheological index during SHRP. Figure 9 is a comparison of tension/compression
of the binder, R: modulus values predicted using the Hirsch equation and Equa-
tion 5 with measured flexural stiffness values (|S*|) measured
-0.629
C2 = -30.1 E * LVE VFA -0.576 R -1.54 (4) during SHRP as part of the SHRP fatigue tests (15).
The rheological index R of the binder is a constant in 6.5
Log T/C |E*|, Pred., psi
the ChristensenAnderson and ChristensenAnderson
Marasteanu (CAM) models for complex modulus and phase
6.0
angle of asphalt binders (38, 39). This constant is directly
related to the dispersion of relaxation times for the binder,
and so it increases as the width of the relaxation spectrum 5.5 2
R = 81%
increases. In the literature of paving technology the width of
the relaxation spectrum is often referred to as "rheologic 5.0
type" and was traditionally characterized by empirical con- 5.0 5.5 6.0 6.5
stants such as the penetration index (PI) and penetration- Log Flexural |E*|, Measured, psi
viscosity number (PVN) (40). The value of R relates well to Figure 9. Predicted Complex Modulus in
these older indexes but is more rational and more exact in the Tension/Compression Compared with Measured
way it characterizes the flow properties of asphalt binders (38, Flexural Complex Modulus for SHRP Mixtures
40). Values for R typically range from about 1.2 to more than (Bars d2s Confidence Limits, Solid Line Regres-
3.0, with a typical value for an unaged binder being about 2.0 sion Function, and Dashed Line Equality).

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Equations 3 and 4 potentially can be used to evaluate the process only estimated modulus values are available. In Equa-
effect of changes in mixture composition on fatigue resistance. tion 6, the coefficients to the log of the predictor variables
However, because the fatigue testing upon which Equation 4 correspond to exponents for these terms; according to con-
was based was somewhat limited and because of the newness tinuum damage theory, the exponent for the strain term is
of continuum damage theory, further verification of these /2. The model represented by Equation 6 was reasonably
results is desirable prior to application to HMA mix design and accurate, with an r2 value of 84% (adjusted for degrees of free-
analysis. This verification was done by applying continuum dom) and gave a value for D = /2 = 1.74, slightly lower than
damage theory to flexural fatigue data gathered during SHRP the value of 2.00 that was used in analyzing the uniaxial
(15). These data included the results of 185 tests on mixtures fatigue data generated during NCHRP Projects 9-25 and
made using eight different asphalt binders and three aggre- 9-31. The next step in analyzing the SHRP data was to calcu-
gates. Properties of this data set are summarized in Table 3. late the value of C2 for each test, using a rearranged form of
Application of continuum damage theory to flexural fatigue Equation 3 and a value for the terminal damage ratio C of
data in part is dependent upon the finding that at completion 0.136 and = 1.74:
1
of a flexural fatigue test, when the flexural stiffness has
decreased by 50%, the extreme fiber damage will be constant
-C 2 =
(
2 f C - - 1 1+
)
at about 86.4%, corresponding to a damage ratio of 0.136. N 2 E * 2 (7)
In analyzing the SHRP fatigue data, the data was first ana- 0
LVE
lyzed statistically using a model of the following form:
Uniaxial fatigue test data collected during NCHRP Projects
9-25 and 9-31 were re-analyzed using = 1.74 rather than
log N = A + B log E * TC + D log ( 0 ) + E log (VFA )
= 2.00 (as was done in the initial analysis of these data).
+ F log ( R ) + error (6) Then, C2 values and related information for both the SHRP
data and for the NCHRP Projects 9-25 and 9-31 data were
In analysis of the SHRP data, the value of the complex combined and analyzed statistically. The best model for esti-
modulus in compression was estimated from the Hirsch mating C2 for this combined data set was somewhat different
model and then converted to a tension/compression value from that given earlier as Equation 4:
using Equation 5. This approach was taken, rather than using
-0.780
measured flexural modulus values, because in the mix design C2 = -466VBEdesign
-0.612 -0.380
N design -8.88
DrelativeR
-1.22
E * LVE (8)
Table 3. Summary of properties of data used in developing fatigue model.
Average
Property Value Minimum Maximum
Total number of tests 200
Number of uniaxial tests (n = 4 analyzed together) 43
Number of replicated flexural tests (n = 2, analyzed 61
separately)
Number of non-replicated flexural tests 35
Mix design methods Superpave, Marshall
Aggregate types Greywacke gravel, low absorption limestone,
limestone (2 sources), granite, gravel
Binder types SHRP core asphalts--eight binders of widely
varying rheology and grade; one SHRP asphalt
modified with three modifiers; NCHRP 9-25/31
binders: PG 58-28, PG 64-22, PG 76-22, all
unmodified
Estimated compaction (Ndesign), gyrations 76 29 125
Air void content, Vol. % 5.1 0.8 8.8
Voids in mineral aggregate, Vol. % 16.5 11.5 21.5
Effective asphalt binder content, Vol. % 11.3 6.1 16.4
Voids filled with asphalt binder, % 69.2 42.4 94.3
Test temperature, °C 19 4 25
Test frequency, Hz 10
Applied strain, × 106 (uniaxial tests) 50 to 100 at 4 °C and 100 to 200 at 20 °C
Applied strain, × 106 (flexural tests) 339 200 1,200
Initial |E*| uniaxial at 20 °C, uniaxial tests, GPa 5.76 1.24 9.52
Initial flexural stiffness, GPa 4.53 1.02 11.35
Cycles to failure (50 % stiffness lost) 119,000 10,000 685,000

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where using Equations 3 and 8, which can be combined to form a
VBEdesign = the effective asphalt binder content at the design single function for predicting fatigue life to a given damage
compaction level, in volume %; ratio C:
Drelative = the bulk density relative to the design bulk
density; 1.05 × 10-6 fVBEdesign
1.677 1.041
N design 24.34
Drelative R
3.335
(C -1.74 - 1)
N= (9)
R = the rheological index of the binder; and ( 2 )0
3.48 1.342
E * LVE
|E*| = in lb/in2.
where the damage ratio at the end of the test (when the flex-
The inclusion of Ndesign in this model significantly improved ural stiffness falls to 50% of its initial value) is 0.136 and the
its accuracy, but was complicated because of the different modulus values are as predicted using the Hirsch model
compaction methods used in the two data sets--the SHRP corrected for tension/compression loading using Equa-
mix designs were prepared using Marshall compaction, while tion 5. Figure 11 shows the cycles to failure predicted using
the NCHRP Projects 9-25 and 9-31 mix designs were pre- Equation 9 and as measured during SHRP. Each point repre-
pared using gyratory compaction. Two different levels of sents the average of two tests and includes d2s confidence lim-
design compaction were used in the SHRP mix designs: its, which represent a 95% prediction limit for the difference
50 blows and 75 blows (15). In the model represented by between two independent observations. If most of these con-
Equation 8, the number of gyrations for each compaction fidence limits include the line of equality, it indicates that the
level for the SHRP mix designs were included as predictor predictions agree well with the experimental value. In this
variables; the analysis indicated that the equivalent number case, 57 of 61 data points agree to within the d2s limits, indi-
of gyrations for 50-blow Marshall compaction was 73 and for cating exceptionally good agreement between the predicted
75-blow Marshall compaction was 92. Equation 8 differs from values and the observed fatigue limits. If two independent sets
Equation 4 in the use of effective binder content rather than of fatigue measurements were compared, it would be
VFA. In analyzing the combined set of fatigue data, it was expected that 58 of the 61 data points would agree to within
found that VFA could be used as an effective predictor if it is the d2s limits, so the predicted values appear to be almost
adjusted to 4% air voids. However, this is a cumbersome cal- interchangeable with experimentally determined values.
culation and in fact provides essentially the same information
as VBE. VBE has the additional advantage that it is nearly
independent of changes in design air void content for the Effect of Mixture Composition
range from 3% to 5%. The model represented by Equation 8 on In Situ Fatigue Resistance
was very effective, with an r2 value of 89% (adjusted for Some discussion of the relationships among fatigue resist-
degrees of freedom). The results are shown graphically in Fig- ance and binder content, design compaction level, and field
ure 10, which shows predicted and observed values for C2 for compaction is useful at this point to illustrate the practical
both data sets. implications of Equation 9. Many pavement engineers and
As a final check on the accuracy of this analysis, the cycles technicians assume that lower values of Ndesign automatically
to failure for the SHRP flexural fatigue data were predicted result in higher binder contents, so lowering Ndesign will
improve fatigue resistance because increased binder content
0.00100 will improve fatigue life. This is true if Ndesign is changed
SHRP Flexural Data
NCHRP 9-25/31 Uniaxial Data
Equality 6.0
Predicted C 2
Log Predicted N f
5.5
5.0
4.5
2
4.0 R = 91%
0.00010 3.5
0.00010 0.00100 3.5 4.0 4.5 5.0 5.5 6.0
Observed C2 Log Measured Cycles to Failure
Figure 10. Predicted and Observed Values for Contin- Figure 11. Predicted and Measured Log Cycles
uum Damage Fatigue Constant C2 for SHRP Flexural to Failure for SHRP Flexural Fatigue Data, with
Fatigue Data and NCHRP Uniaxial Fatigue Data. d2s Confidence Limits.

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without modifying the given aggregate blend, but it is not true · Two binder grades: PG 64-22 and PG 76-22;
when a range of aggregates and mixtures are considered. Fur- · Design VMA ranging from 13 to 16;
thermore, there is no reason to believe that if Ndesign require- · Ndesign of 75; and
ments are changed, materials suppliers will not change · Design air voids of 3%, 4%, and 5%.
aggregate gradations for their mixtures. In fact, in cases where
asphalt binder is not paid as a separate item, there is strong It should be pointed out that the level of Ndesign was not var-
economic incentive to modify aggregate gradations to obtain ied since Equation 9 (and any similar fatigue equation derived
the minimum binder content that will not incur a penalty. from this analysis) predicts that fatigue life will increase with
Because changes in volumetric requirements cannot possibly increasing values of Ndesign and will decrease with lower values
include a requirement that Ndesign should be changed without of Ndesign, all else being equal. There is no mechanism for
any modification in aggregate gradation or sources, there is changing this relationship in the field. Mixture modulus val-
no basis for suggesting that implementing lower Ndesign values ues were estimated using the Hirsch model. Binder R values
will result in increased binder contents and improved fatigue were 1.70 for the PG 64-22 and 2.17 for the PG 76-22; these
resistance. Decreasing Ndesign may improve the ease with which and other binder properties were taken from actual materials
a mixture can be compacted in the field, but this will not nec- tested in Advanced Asphalt Technologies' laboratory. As dis-
essarily mean that field compaction will be improved since cussed above, the statistical analyses showed no interaction
there is an economic incentive to compact a pavement only to between mix composition and binder R value. Therefore,
the highest air void content that will not incur a significant using typical values for R should not affect the sensitivity of
penalty. In summary, all else being equal, increasing Ndesign will this analysis to changes in mix composition. Although an
improve fatigue resistance, and decreasing it will do the oppo- analysis of the affect of changes in R on fatigue resistance
site. If an agency feels that higher binder contents and lower might be enlightening, it is clearly outside the scope of
in-place air voids are needed to improve fatigue resistance, NCHRP Projects 9-25 and 9-31 and is not included in this
higher minimum binder contents (higher minimum VMA at report. The subgrade modulus values were allowed to vary
a given design air void level) and improved field compaction according to the time of year, using the same values incorpo-
requirements should be specified, perhaps in combination rated into the 1991 edition of the Asphalt Institute's Mix
with lower Ndesign values if it is felt that this latter change will Design Methods for Asphalt Concrete and Other Hot-Mix Types
help materials suppliers and contractors deal with the first (42) for the thickness design of flexible pavements as reported
two changes. by Huang (41). The results of this analysis were then compiled
Fatigue resistance in situ involves more than the inherent in terms of relative fatigue life--in this case, fatigue life
fatigue resistance of the mixture because mixture stiffness will expressed as a fraction of that for a design VMA of 15%, design
affect the magnitude of strains resulting from traffic loading, air voids of 4%. These relative fatigue life values were then
in addition to affecting the resulting rate of damage as pre- summarized statistically using means and standard deviations.
dicted by Equation 9 and similar functions. Therefore, in Plots were prepared showing average changes and d2s confi-
order to evaluate the overall relationships among volumetric dence limits in relative fatigue life with design VMA, design air
composition and fatigue resistance, a simplified evaluation of voids, and in-place air voids. The results are shown in Figures
field fatigue resistance was performed. The general approach 12 through 15. In Figure 12, the effect of changes in design air
involved using the Illipave algorithms, as described by Huang
(41), to estimate tensile strains at the bottom of the bound
materials in various pavement structures; Equation 9 was 1.5
then used to determine the pavement fatigue life using a ter-
minal damage ratio of 0.20. A variety of climates, pavement
Relative Nf
1.0
structures, and mixture compositions were considered:
5% design / 7% in-place
0.5
· Two climates: New York state and South Carolina; 4% design / 7% in-place
· Two pavement structures: 100-mm bound material over 3% design / 7% in-place
150-mm granular subbase and 200-mm bound material 0.0
over 300-mm granular subbase; 12 13 14 15 16 17 18
· Three average subgrade stiffness conditions: weak, moder- Design VMA, Vol. %
ate, and stiff; Figure 12. Effect of Design Air Voids and
· Four different times of year: mid-winter (January), early Design VMA on Relative In-Situ Fatigue Life,
spring (March/April), spring thaw (April/May), and late In-Place Air Voids Constant at 7% (Errors Bars
summer (August); 2s Confidence Limits).

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1.5 3.0
5% design voids
Relative Nf
Relative N f
1.0 2.0 4% design voids
3% design voids
5% design / 8% in-place 1.0
0.5
4% design / 7% in-place
3% design / 6% in-place
0.0 0.0
12 13 14 15 16 17 18 4 6 8 10 12 14
Design VMA, Vol. % In-Place Voids, Vol. %
Figure 13. Effect of Design Air Voids and Design Figure 15. Effect of In-Place Air Voids and
VMA on Relative In-Situ Fatigue Life, In-Place Design Air Voids on Relative In-Situ Fatigue
Air Voids of 6%, 7%, and 8% for Design Air Life (Errors Bars 2s Confidence Limits).
Voids of 3%, 4%, and 5%, Respectively (Errors
Bars 2s Confidence Limits).
To illustrate the good relationship between effective
binder content and fatigue resistance, Figure 14 was con-
voids and design VMA are shown at a constant in-place air
structed. This plot is nearly identical to Figure 13, but the
void level of 7%. For every 1% increase in VMA, the fatigue life
horizontal axis is VBE rather than VMA. There is an excel-
increases from about 13% to 21% (typically about 16%). For
lent, nearly linear relationship between relative fatigue life
every 1% increase in design air void content, the fatigue life
and effective binder content--even though this figure was
increases about 7% to 14% (typically about 10%). This later
generated using different pavement structures, climates, and
finding may at first seem counter-intuitive, but it must be
times of the year. For every 1% increase in VBE, there is typ-
remembered that the analysis summarized in Figure 12 was
ically a 13% to 15% increase in relative fatigue life. There-
generated assuming constant in-place air voids--therefore,
fore, to control the fatigue resistance of HMA, VBE should
increasing design air voids mostly has the effect of increasing
be specified. Alternately, asphalt binder content by weight
the compaction effort during construction. For comparison,
can be specified as a function of aggregate specific gravity,
Figure 13 represents an analysis in which in-place air voids are
but this is a somewhat more cumbersome approach. Fur-
allowed to vary with design air voids--that is, in-place air
thermore, it is clear that if in-place air voids are allowed to
voids were assumed to be 6%, 7%, and 8% for design air voids
vary with design air voids, there is little net affect on fatigue
of 3%, 4%, and 5%, respectively. In this case, the advantage of
resistance. As discussed previously, the same situation exists
using higher air voids disappears--and, in fact, it appears to
for rut resistance--that is, changing in-place air voids
become a disadvantage in that it significantly decreases fatigue
simultaneously with design air voids has little net effect on
life. However, this is only because as design air voids change at
rut resistance.
constant VMA, it directly affects VBE--as air voids increase at
Although there appears to be some advantage to linking
constant VMA, VBE decreases.
design and in-place air voids, such an approach would be
impractical since in most paving projects the in-place air
1.5 voids cannot be predicted with any certainty. Paving engi-
neers and technicians should nevertheless understand the
relationship among design air voids, in-place air voids, and
Relative Nf
1.0
performance:
5% design / 8% in-place
0.5
4% design / 7% in-place · At a constant level of in-place air voids, increasing design
3% design / 6% in-place air voids will improve performance because it forces more
0.0 compaction energy to be used during construction.
7 8 9 10 11 12 13 14 15 16 · At a constant level of design air voids, increasing field air voids
Design VBE, Vol. % will decrease performance because it will result in less com-
Figure 14. Effect of Design Air Voids and Design paction energy being used during construction (it will also
VBE on Relative In-Situ Fatigue Life, In-Place Air increase the permeability of the pavement, potentially
Voids of 6%, 7%, and 8% for Design Air Voids of decreasing resistance to age hardening and moisture damage).
3%, 4%, and 5%, Respectively (Errors Bars 2s · If design air voids and in-place air voids vary in a similar
Confidence Limits). way, there will be little effect on performance.