Understanding Mechanisms Leading to Asphalt Binder Fatigue

Transcription

Understanding Mechanisms Leading to Asphalt
Binder Fatigue
By:
Cassie Hintz
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
(Civil & Environmental Engineering)
University of Wisconsin – Madison
2012
Date of Final Oral Examination: 06/25/2012
The dissertation is approved by the following members of the Final Oral Committee:
Hussain U. Bahia, Professor, Civil and Environmental Engineering
Steven M. Cramer, Professor, Civil and Environmental Engineering
Tuncer B. Edil, Professor, Civil and Environmental Engineering
Dante O. Fratta, Associate Professor, Civil and Environmental Engineering
Robert E. Rowlands, Professor, Mechanical Engineering
© Copyright by Cassie Hintz, 2012
All rights reserved
i
Table of Contents
List of Figures ................................................................................................................................ iii
List of Tables .................................................................................................................................. v
Acknowledgements ........................................................................................................................ vi
Abstract ......................................................................................................................................... vii
1
2
Introduction ............................................................................................................................. 1
1.1
Background ...................................................................................................................... 1
1.2
Problem Statement ........................................................................................................... 2
1.3
Hypothesis ........................................................................................................................ 2
1.4
Objectives ......................................................................................................................... 3
1.5
Outline .............................................................................................................................. 3
Literature Review.................................................................................................................... 5
2.1
Asphalt Pavement Fatigue ................................................................................................ 5
2.2
Asphalt Binder Fatigue Testing........................................................................................ 6
2.3 Use of Viscoelastic Continuum Damage (VECD) Mechanics for Fatigue
Characterization ........................................................................................................................ 10
3
2.3.1
Viscoelastic Continuum Damage Mechanics (VECD) ........................................... 10
2.3.2
Application of VECD to Asphalt Mixture Fatigue ................................................. 13
2.3.3
Application of VECD to Asphalt Binder Fatigue ................................................... 15
2.4
Damage........................................................................................................................... 17
2.5
Thixotropy ...................................................................................................................... 18
2.6
Fracture under Torsional Loading .................................................................................. 20
2.6.1
Edge fracture in fatigue testing of cylindrical specimens under torsion ................. 20
2.6.2
Mechanism of Fracture under Torsion .................................................................... 22
2.6.3
Linear Elastic Fracture Mechanics of Edge Fracture .............................................. 24
2.6.4
Incorporation of Viscoelasticity.............................................................................. 30
2.6.5
Nonlinearity ............................................................................................................ 31
2.7
Fracture Based Fatigue Models ...................................................................................... 34
2.8
Time-Temperature Superposition .................................................................................. 35
Materials and Methods .......................................................................................................... 38
3.1
Verification of Hypothesis ............................................................................................. 39
3.2
Model Development ....................................................................................................... 41
ii
3.3
4
5
Verification of Hypothesis .................................................................................................... 46
4.1
Experimental Verification of Hypothesis ....................................................................... 46
4.2
Determination of Fracture Mode .................................................................................... 59
4.3
Energy Release Rate Calculation ................................................................................... 65
Fatigue Model Development and Results ............................................................................. 72
5.1
Frequency Sweep Results............................................................................................... 72
5.2
Time Sweep Results and Analysis ................................................................................. 74
5.2.1
Verification of Energy Release Rate Solution ........................................................ 74
5.2.2
Development of Fatigue Failure Criteria ................................................................ 79
5.2.3
Life
Effect of Binder Modification, Temperature, and Loading Amplitude on Fatigue
80
5.2.4
Separating Crack Initiation and Propagation .......................................................... 81
5.3
6
7
Validation with Mixture Fatigue Data ........................................................................... 44
Model development ........................................................................................................ 88
5.3.1
Crack Propagation Model ....................................................................................... 88
5.3.2
Crack Initiation Model ............................................................................................ 95
5.3.3
Comprehensive Fatigue Model ............................................................................... 98
5.3.4
Application of Time-Temperature Superposition to Fatigue Crack Propagation . 103
5.3.5
Summary ............................................................................................................... 111
5.4
Model Validation.......................................................................................................... 111
5.5
Application of Fracture Analysis to Linear Amplitude Sweep (LAS) Test ................. 114
5.5.1
Evaluation of the LAS Test as a Fatigue Test....................................................... 114
5.5.2
Modification of the LAS Procedure...................................................................... 117
5.5.3
Summary ............................................................................................................... 125
Conclusions and Recommendations ................................................................................... 126
6.1
Conclusions .................................................................................................................. 126
6.2
Recommendations ........................................................................................................ 128
References ........................................................................................................................... 131
iii
List of Figures
Figure 1. Dynamic Shear Rheometer (DSR). ................................................................................. 7
Figure 2. Depiction of the Linear Amplitude Sweep (LAS) loading sequence. ............................. 9
Figure 3. Fatigue law. ................................................................................................................... 10
Figure 4. Edge fracture in DSR testing. ........................................................................................ 21
Figure 5. Fracture under torsion in (a) Mode III and (b) Mode I.................................................. 23
Figure 6. Fracture Morphology of Steel. Source: Tschegg et al. 1983. ........................................ 24
Figure 7. Fracture Morphology of Poly-dimethylsiloxane. Source: Keller et al. 2008. ............... 24
Figure 8. Edge fracture schematic. ............................................................................................... 25
Figure 9. Energy release rate as a function of crack length ( (Fleischman, Kerchman and Ebbott
2001). ............................................................................................................................................ 27
Figure 10. Depiction of (a) linear elastic behavior and (b) linear viscoelastic behavior. ............. 32
Figure 11. Depiction of ΔK. ......................................................................................................... 34
Figure 12. Mode I crack growth. .................................................................................................. 35
Figure 13. (a) Original frequency sweep data at multiple temperatures, (b) Shifted frequency data
to a reference temperature. ............................................................................................................ 37
Figure 14. Depiction of time sweep loading and response. .......................................................... 39
Figure 15. Procedure for attaining image of fractured surface: (a) paint application and (b)
sample after spindle detachment. .................................................................................................. 40
Figure 16. Normalized torque results for Flint Hills binder. ........................................................ 48
Figure 17. Normalized torque results for the Flint Hills binder, based on averaged data. ........... 48
Figure 18. Normalized torque results for Nustar + 2% SBS binder. ............................................ 49
Figure 19. Normalized torque results for the Nustar + 2%SBS binder, based on averaged data. 49
Figure 20. Fractured sample with paint application...................................................................... 50
Figure 21. Microscope image of fractured sample with paint application #1. ............................. 51
Figure 22. Microscope image of fractured sample with paint application #2. ............................. 51
Figure 23. Image processing steps. ............................................................................................... 54
Figure 24. Visualization of edge fracture of Flint Hills binder at (a) 30, (b) 40, and (c) 55 minutes
of loading. ..................................................................................................................................... 55
Figure 25. Visualization of edge fracture of Nustar + 2% SBS binder at (a) 40, (b) 50, (c) 60, and
(d) 80 minutes of loading. ............................................................................................................. 56
Figure 26. Comparison between crack lengths predicted from torque measurements and
estimated from digital image analysis for Flints Hills binder. ...................................................... 58
estimated from digital image analysis for Nustar + 2% SBS binder. .......................................... 59
Figure 28. Schematic of FEM of DSR. ......................................................................................... 60
Figure 29. Von Mises stress distribution. ..................................................................................... 61
Figure 30. Normal stress distribution............................................................................................ 62
Figure 31. Radial distribution of shear stress................................................................................ 62
Figure 32. Fatigue fractured samples at (a) 6.5MPa iso-stiffness condition, (b) 30MPa isostiffness condition. ........................................................................................................................ 63
Figure 33. Predicted crack growth rate as a function of number of loading cycles for Flint Hills
binder. ........................................................................................................................................... 64
Figure 34. Predicted crack growth rate as a function of number of loading cycles for Nustar + 2%
SBS binder. ................................................................................................................................... 65
iv
Figure 35. Energy release rate calculated using Equation (40) and crack growth rate as a function
of number of loading cycles for Flint Hills Binder. ...................................................................... 66
Figure 36. Energy release rate calculated using Equation (40) and crack growth rate as a function
of number of loading cycles for Nustar + 2% SBS binder. .......................................................... 67
Figure 37. Comparison between measured radii and predicted radii from torque measurements.68
Figure 38. Comparison between measured radius with correction factor and predicted radius
from torque measurements. ........................................................................................................... 68
Figure 39. Energy release rate and crack growth rate as a function of number of loading cycles
for Flint Hills Binder..................................................................................................................... 69
Figure 40. Energy release rate and crack growth rate as a function of number of loading cycles
for Nustar + 2% SBS binder. ........................................................................................................ 70
Figure 41. Total harmonic distortion. ........................................................................................... 71
Figure 42. |G*| master curves. ...................................................................................................... 73
Figure 43. Phase angle master curves. .......................................................................................... 73
Figure 44. Depiction of typical trend of energy release rate versus crack length with c equal to
0.1.................................................................................................................................................. 75
Figure 45. Comparison of energy release and crack growth rates for Nustar neat at undamaged
strain amplitudes of (a) 2%, (b) 3%, and (c) 3.5% at 6.5MPa iso-stiffness condition, and at 2%
undamaged strain at (d) 2MPa and (e) 30MPa iso-stiffness conditions. ...................................... 78
Figure 46. Crack growth rate as crack length (a) function of a when sample height = 3mm. ...... 79
Figure 47. Fatigue lives at undamaged strain amplitudes of 2% and 3%. .................................... 81
Figure 48. Flint Hills neat: Crack growth rate versus energy release. .......................................... 82
Figure 49. Flint Hills + SBSX: Crack growth rate versus energy release. ................................... 83
Figure 50. Flint Hills + SBSL: Crack growth rate versus energy release. .................................... 83
Figure 51. Flint Hills + PE: Crack growth rate versus energy release. ......................................... 84
Figure 52. Nustar Neat: Crack growth rate versus energy release. ............................................... 84
Figure 53. Nustar + SBSX: Crack growth rate versus energy release. ......................................... 85
Figure 54. Relative contributions of number of cycles of crack initiation (Ni) and crack
propagation (Np) on total fatigue life (Nt) at 2MPa iso-stiffness condition. ................................ 87
propagation (Np) on total fatigue life (Nt) at 6.5MPa iso-stiffness condition. ............................. 87
Figure 56. Flint Hills Neat: Crack propagation results. ................................................................ 88
Figure 57. Flint Hills + SBSX: Crack propagation results. .......................................................... 89
Figure 58. Flint Hills + SBSL: Crack propagation results. ........................................................... 89
Figure 59. Flint Hills + PE: Crack propagation results................................................................. 90
Figure 60. Nustar Neat: Crack propagation results. ...................................................................... 90
Figure 61. Nustar + SBSX: Crack propagation results. ................................................................ 91
Figure 62. k as a function of temperature and binder type............................................................ 92
Figure 63. α as a function of temperature and binder type. .......................................................... 93
Figure 64. Comparison of measured and predicted Ni. ................................................................ 96
Figure 65. Effect of binder type and temperature on β. ................................................................ 97
Figure 66. Comparison between measured and predicted fatigue lives (Nf). ............................... 99
Figure 67. Effect of temperature and binder type on fatigue law parameter A. .......................... 103
Figure 68. Crack propagation master curve. ............................................................................... 104
Figure 69. Flint Hills Neat: Fatigue crack propagation master curve. ........................................ 105
Figure 70. Flint Hills + SBSX: Fatigue crack propagation master curve. .................................. 106
v
Figure 71. Flint Hills + SBSL: Fatigue crack propagation master curve.................................... 106
Figure 72. Flint Hills + PE: Fatigue crack propagation master curve. ....................................... 107
Figure 73. Nustar Neat: Fatigue crack propagation master curve. ............................................. 107
Figure 74. Nustar + SBSX: Fatigue crack propagation master curve. ........................................ 108
Figure 75. Comparison between measured k value and predicted k values using time-temperature
shift factors.................................................................................................................................. 109
Figure 76. Fatigue crack growth master curve including crack initiation for Flint Hills + PE. . 110
Figure 77. Mixture validation fatigue crack propagation curves. ............................................... 113
Figure 78. Sample after LAS test. ............................................................................................... 115
Figure 79. Trends in crack growth rate and energy release rate using current LAS test. ........... 116
Figure 80. Modified LAS loading schematic. ............................................................................. 118
Figure 81. Trends in crack growth rate and energy release rate using modified LAS test. ........ 119
Figure 82. Crack growth rate as a function of crack length under modified LAS testing. ......... 120
Figure 83. Crack growth rate versus energy release rate under modified LAS testing. ............. 121
Figure 84. Illustration of crack length at failure, af . ................................................................... 122
Figure 85. Trends in torque and crack growth rate for Flint Hills Neat binder. ......................... 122
Figure 86. Trends in torque and crack growth rate for Flint Hills + SBSX binder. ................... 123
Figure 87. Evaluation of crack length at failure as failure parameter. ........................................ 124
Figure 88. Evaluation of effective strain at failure as failure parameter. ................................... 124
List of Tables
Table 1. Experimental plan for verification of hypothesis. .......................................................... 40
Table 2. Experimental Plan for Model Development. .................................................................. 44
Table 3. Experimental Plan for Validation ................................................................................... 45
Table 4. Comparison between Measured and Torque Predicted Sample Radii for Flint Hills
Binder............................................................................................................................................ 57
Table 5. Comparison between Measured and Torque Predicted Sample Radii for Nustar + 2%
SBS Binder.................................................................................................................................... 58
Table 6. Test Temperatures........................................................................................................... 74
Table 7. Crack Propagation Curve Fit Results.............................................................................. 91
Table 8. Crack Initiation Curve Fit Results. ................................................................................. 97
Table 9. Comparison between Measured and Predicted Fatigue Lives at 2MPa Iso-stiffness
Condition..................................................................................................................................... 100
Table 10. Comparison between Measured and Predicted Fatigue Lives at 6.5MPa Iso-stiffness
Condition..................................................................................................................................... 101
Condition..................................................................................................................................... 101
Table 12. Summary of Fatigue Law Parameters A and B .......................................................... 102
Table 13. Mixture Validation Results ......................................................................................... 113
vi
Acknowledgements
I would like to sincerely thank those who have contributed to the completion of this
research. First, I want to thank my advisor, Dr. Hussain Bahia for his guidance and support. Dr.
Bahia has both encouraged and challenged me throughout the course of my studies. I sincerely
appreciate his technical and professional advice. My interactions with Dr. Bahia have opened
career opportunities for which I will always be grateful.
I sincerely appreciate the willingness of Dr. Steven Cramer, Dr. Tuncer Edil, Dr. Dante
Fratta, and Dr. Robert Rowlands to serve as committee members. Their insight and criticisms
have contributed greatly to this work.
The support and friendship of my colleagues at the Modified Asphalt Research Center
(MARC) is also greatly appreciated. I am thankful to have been a part of such a dynamic and
diverse group. I would like to offer special thanks to Dr. Raul Velasquez for his input and
technical guidance.
Lastly, I would like to thank my family and friends whose patience and support have been
instrumental in my success.
vii
Abstract
Fatigue cracking is one of the primary modes of failure in asphalt pavements. Asphalt
pavements are a composite materials consisting of aggregate, asphalt binder, and air voids.
Cracking typically occurs either within the asphalt binder or at the asphalt binder - aggregate
interface. Thus, asphalt binder fatigue resistance plays a critical role in determining overall
pavement fatigue performance. One procedure commonly used to characterize asphalt binder
fatigue resistance is the time sweep test, which consists of repeated cyclic loading in the
Dynamic Shear Rheometer (DSR). In the DSR, cyclic torsion is applied to a cylindrical asphalt
binder specimen inserted between two parallel plates. Generally, apparent changes in material
properties with respect to number of loading cycles are used to define fatigue failure of the
asphalt binder. Results of this test have been shown to correlate well with asphalt mixture fatigue
performance. However, the mechanisms responsible for changes in material properties during
fatigue testing in the DSR were previously not well understood.
Results in this study demonstrate that fracture can explain the changes in loading
resistance during fatigue testing in the DSR. Under cyclic torsional loading of cylindrical
specimens, macro fracture is shown to manifest in the form of edge fracture. Edge fracture is a
circumferential crack starting at the periphery of a cylindrical sample that propagates inward as
loading is applied, reducing the effective sample size. Digital visualization of the fractured
specimens allowed for determination of the fractured and intact area (i.e., effective area).
Predictions of fracture propagation based on measurements of loading resistance and fracture
mechanics concepts agreed favorably with actual measurements based on visualization.
Furthermore, the fracture morphology and progression of crack growth of asphalt binders under
viii
fatigue loading in the DSR matched those observed for other materials under similar loading
conditions.
Based on these results, fracture mechanics concepts can be used to improve current
analyses of DSR fatigue test results. A fracture based analysis framework that considers crack
initiation and propagation is presented. The proposed model allows predicting fatigue life at any
loading amplitude using the results of a single fatigue test. Additionally, it is demonstrated that
time-temperature superposition is applicable to fatigue crack propagation of asphalt binders,
allowing for efficient prediction of fatigue performance at multiple temperatures. The model is
validated using a comparison between asphalt mixture and binder fatigue test results.
1
1
1.1
Introduction
Background
Fatigue cracking is one of the primary modes of failure in asphalt pavements. Cracking
typically occurs either within the asphalt binder or at the asphalt binder - aggregate interface.
Thus, asphalt binder plays a critical role in overall pavement fatigue performance.
One of the most widely used procedures to evaluate cohesive asphalt binder fatigue
resistance is to apply repeated cyclic loading in the Dynamic Shear Rheometer (DSR). This test
is typically referred to as the time sweep. Results of time sweep testing of asphalt binder have
been shown to correlate well with asphalt mixture fatigue performance, indicating the procedure
captures asphalt binder contribution to mixture fatigue (Bahia, et al. 2001).
Generally, apparent changes in material properties (e.g., |G*|, |G*|·sin) with number of
cycles of loading are used to define fatigue failure of the asphalt binder. However, the
mechanisms responsible for changes in material properties during fatigue testing in the DSR are
not well understood. Fatigue testing of asphalt binders is conducted at intermediate temperatures.
For this range of temperatures, asphalt binders are relatively soft and exhibit viscoelastic
behavior. As such, multiple phenomena could potentially contribute to changes in material
properties (e.g., complex moduli) during fatigue testing of asphalt binders, including thixotropy,
micro-defect/void formation, and macro fracture. Thixotropy is the time-dependent decrease in
viscosity under loading, which recovers with rest (Mewis and Wagner 2009). Note that
thixotropy does not lead to permanent changes in material properties and is thus, not actually a
damage mechanism. Several researchers have attributed the changes in asphalt binder material
properties during loading to thixotropy (Shan, et al. 2010); (Soltani and Anderson 2005).
2
Fracture is also a possible source for apparent changes in material properties. Under cyclic
torsional loading of cylindrical specimens, cracking typically manifests as edge fracture, which is
circumferential crack that initiates at the periphery of a sample and propagates inward
(Aboutorabi, Ebbot and Gent 1998).
1.2
Problem Statement
There is a lack of an understanding of the mechanisms of fatigue damage in asphalt
binders. While it has previously been demonstrated that asphalt binder fatigue test results from
the DSR correlate well with asphalt mixture fatigue performance, the mechanisms responsible
for changes in material properties during DSR testing are not well understood.
Multiple phenomena potentially contribute to changes in material properties under cyclic
loading, including thixotropy, micro void/defect formation, and fracture. A fundamental
understanding of the damage mechanism(s) is needed to determine if the DSR is an appropriate
method for fatigue characterization. There is currently concern that fatigue characterization
methods for asphalt binders are empirically based and rely on linear viscoelastic properties.
Understanding the damage mechanism(s) leading to fatigue will facilitate the improvement of
current analysis methods for mechanistic fatigue characterization.
1.3
Hypothesis
It is hypothesized that changes in sample geometry due to cracking, rather than changes
in viscoelastic properties, are responsible for the apparent changes in load carrying capacity
during fatigue testing of asphalt binders in the DSR. If this hypothesis is confirmed, fracture
mechanics concepts can be applied for estimation of fatigue damage resistance of asphalt binders
under different loading, climate, and pavement structure conditions.
3
1.4
Objectives
The objectives of this research are to:
– Develop an understanding of the damage mechanisms in asphalt binder during fatigue
testing in the DSR.
– Based on the mechanisms identified, propose a testing and analysis framework to
improve characterization of asphalt binder fatigue resistance.
– Evaluate the Linear Amplitude Sweep (LAS) Test proposed for AASHTO specification,
and propose modification if needed.
1.5
Outline
This thesis includes six chapters:
– Chapter 1: Introduction
The introduction includes background information on asphalt binder fatigue, a problem
statement, hypothesis, and objectives of this research.

Chapter 2: Literature Review
The literature review discusses topics related to asphalt binder and mixture fatigue,
viscoelastic continuum damage mechanics, thixotropy, fracture based fatigue modeling,
and characteristics of fracture under cyclic torsional loading.

Chapter 3: Materials and Methods
This chapter presents the experimental methods and materials used to accomplish the
objectives of this research.

Chapter 4: Verification of Hypothesis
This chapter presents experimental verification of the hypothesis that fracture is the
source of fatigue damage during time sweep testing of asphalt binders in the DSR.
4
Several measures were taken to verify this hypothesis including a comparison between
predicted crack lengths based on torque and direct measurements of crack length, Finite
Element Modeling (FEM), and a comparison between the fracture morphology and crack
growth trends in asphalt binders and other materials under similar loading conditions.

Chapter 5: Model Development
This chapter details the development and validation of a fracture based analysis
framework that allows for prediction of fatigue life at any loading amplitude from a
single fatigue test. The model accounts for both crack initiation and propagation. It is
demonstrated that time-temperature superposition is applicable to the fatigue crack
propagation model, allowing for efficient fatigue characterization at multiple
temperatures. The model is validated through a comparison with asphalt mixture fatigue
data. Additionally, based on the findings of this study, modifications to the Linear
Amplitude Sweep (LAS) test, an accelerated damage tolerance test, are proposed along
with an accompanying analysis framework.

Chapter 6: Conclusions and Recommendations
This chapter summarizes the major findings and contributions of this work. Additionally,
recommendations for future research are presented.
5
2
2.1
Literature Review
Asphalt Pavement Fatigue
Fatigue cracking in asphalt pavements is caused by repetitive traffic loading. There are
two mechanisms of fatigue cracking in asphalt pavements: top-down and bottom-up (Huang
2004). Bottom-up cracks initiate at the bottom of an asphalt surface or asphaltic base layer as a
result of high strains associated with flexure and propagate to the surface. Bottom-up cracking is
the most common form of fatigue cracking. Top-down cracks initiate at the surface under the
wheel path and propagate downward. Top-down cracking manifests as longitudinal cracks in the
wheel path whereas bottom-up cracking leads to an inter-connected network of cracks, often
referred to as alligator cracking.
Pavements are thought to be most susceptible to fatigue damage during spring. During
thawing of snow and ice in the spring, granular layers underlying the asphalt surface become
saturated. Saturation reduces load carrying capacity, leading to high stresses and strains in the
asphalt layer(s) (Huang 2004). Therefore, fatigue testing of asphalt pavements is typically
conducted at intermediate temperatures corresponding to spring conditions.
Fatigue testing of asphalt mixtures typically consists of repeated cyclic loading with
either constant load or constant displacement amplitude. Multiple testing configurations are
currently used for fatigue testing of asphalt mixtures. Traditionally, fatigue testing of asphalt
mixtures was conducted using two or four point bending of rectangular asphalt concrete samples
(Monismith, et al. 1970). More recently, uniaxial loading of cylindrical asphalt mixture samples
has gained increased interest because the uniform stress state on cross sections simplifies
calculations used in analyses and modeling (Daniel and Kim 2002); (Kutay, Gibson and
Youtcheff 2008).
6
2.2
Asphalt Binder Fatigue Testing
Asphalt binders exhibit viscoelastic behavior which is time, rate of loading, and
temperature dependent. (Lakes 2009). At short loading times and low temperatures, asphalts
behave elastically. Elastic behavior is observed when there is insufficient time for rearrangement
of molecular structure (Lakes 2009). Similarly, at low temperatures molecular mobility is low,
giving rise to elastic behavior. However, when subjected to high temperatures and/or long
loading times (and low rates of loading), asphalts exhibits more viscous behavior. When given
time, flow occurs due to molecular rearrangements and viscous behavior is observed (Lakes
2009). Likewise, at high temperatures molecular mobility increases, giving rise to viscous
behavior.
The Dynamic Shear Rheometer (DSR) is a common device used to evaluate asphalt
binder properties, including fatigue resistance. In DSR testing of asphalt binders, a cylindrical
specimen is placed between two parallel plates as depicted in Figure 1. The top plate rotates
while the bottom plate is fixed, effectively applying torsion to the asphalt binder specimen. The
DSR is able to directly apply and measure torque (T) and deflection angle (φ). Shear stresses (τ)
and shear strains (γ) are calculated and reported by the instrument using the following equations:
(1)
(2)
where r is the sample radius and h is the sample height. Note that the shear stress and shear strain
are not uniform in DSR testing. Rather, stress and strain are maximum at the sample periphery
7
and zero at the sample center. The maximum stress and strain at the edge of the sample is
normally reported in DSR testing, assuming a fixed sample radius and height.
Figure 1. Dynamic Shear Rheometer (DSR).
Rheometers are designed to test materials with a fluid or soft consistency which tend to
flow under their own self weight (Lakes 2009), making it a good instrument for measuring the
fatigue resistance of binders at the intermediate temperatures used for fatigue testing.
Additionally, DSRs allow for fine control and measurement of torque, displacement, and
temperature. The ability to precisely measure and control temperature, torque and deflection
angle is critical when testing asphalt binders due their high temperature and loading
amplitude/rate sensitivity.
The current Superpave specification for characterizing asphalt binder fatigue resistance
was developed under the assumption that a softer, more elastic binder will reduce fatigue
cracking. The specification places a maximum limit of 5,000kPa on |G*|·sinδ, (where |G*| is the
complex shear modulus and δ is the phase angle), measured at a strain amplitude of 1% and
frequency of 10 rad/sec in the DSR. This approach lacks the ability to characterize actual damage
resistance (Bahia, et al. 2001); (Bahia, et al. 2002). Furthermore, this specification does not
8
account for pavement structure or traffic loading as the measurement is made at one, low strain
level over very few cycles of loading. When this specification was developed, it was speculated
that binder in pavements function mostly in the linear viscoelastic regime. However, subsequent
research has proven this is not the case (Bahia, et al. 2001).
To solve the limitations of the current specification, the time sweep test procedure was
proposed in National Cooperative Highway Research Program (NCHRP) Project 9-10 (Bahia, et
al. 2001). The time sweep is a test method that consists of applying repeated sinusoidal loading
at fixed amplitude to an asphalt binder specimen in the DSR. Generally, in time sweep testing
geometry consists of 8mm diameter parallel plates with a fixed gap of 2mm. Changes in |G*| and
δ with number of loading cycles are used to determine fatigue life. The procedure allows for
selection of load amplitude, thus considering pavement structure and traffic loading.
Furthermore, results of time sweep testing have been shown to correlate to mixture fatigue
results (R2 = 0.84), indicating the time sweep captures binder contribution to mixture fatigue
(Bahia, et al. 2001).
Recently, the Linear Amplitude Sweep (LAS) test has been proposed as a surrogate to the
time sweep (Johnson 2010). The LAS test is similar to the time sweep in that it consists of cyclic
loading in the DSR and utilizes the same testing geometry. However, in the LAS test, loading
amplitudes are systematically increased to accelerate damage. Additionally, the LAS test
includes a frequency sweep test prior to the amplitude sweep to obtain an undamaged material
response used in viscoelastic continuum damage modeling. The frequency sweep is conducted at
using a small loading amplitude of 0.1% strain to avoid inducing damage. Figure 2 provides a
schematic of the amplitude sweep loading. Loading begins with 100 cycles of sinusoidal loading
at 0.1% and 10 Hz frequency. Loading proceeds in 1% strain step increments. Each strain step
9
consists of 100 cycles of loading at 10 Hz frequency. Hintz et al. (2011) recommended using
loading steps spanning the range of 1% and 30% applied strain to evaluate damage evolution.
Total testing time, including thermal equilibration, is approximately 30 minutes.
Applied Strain, %
35
30
25
20
15
10
5
0
0
500
1000
1500
2000
2500
3000
3500
Loading Cycles
Figure 2. Depiction of the Linear Amplitude Sweep (LAS) loading sequence.
Dynamic Mechanical Analysis (DMA) has also been proposed for characterizing asphalt
binder fatigue behavior (Kim, Little and Lytton 2002). In DMA fatigue testing, a cylindrical
sample composed of a sand-asphalt binder mix is subjected to torsional, oscillatory loading.
Changes in material properties with number of loading cycles are used to quantify fatigue life
similar to testing and analysis of time sweeps. Kim et al. (Kim, Little and Lytton 2002)
hypothesized that DMA allows for accurate fatigue characterization because there are thin films
of asphalt binder within the sand mix which are similar to those in asphalt-aggregate mixtures.
10
2.3
Use of Viscoelastic Continuum Damage (VECD) Mechanics for Fatigue
Characterization
2.3.1
Viscoelastic Continuum Damage Mechanics (VECD)
For comprehensive fatigue characterization of asphalt materials, a model relating applied
load and fatigue life is often derived. Obtaining such a model allows for consideration of
pavement structure (i.e., applied load amplitude) and traffic (i.e., number of cycles to failure).
Asphalt mixtures and binder demonstrate a well-defined relationship between load amplitude and
fatigue life (Nf) as follows (Monismith, et al. 1970):
Nf = A · (Load Amplitude)B
(3)
where A and B are material dependent parameters. A schematic of Equation (3) is provided in
Figure 3.
Number of Cycles to Failure
1.0E+12
A
1.0E+11
1.0E+10
Nf = A·γB
B
1.0E+09
1.0E+08
1.0E+07
1.0E+06
1.0E+05
1.0E+04
1
10
Applied Strain (%)
Figure 3. Fatigue law.
Obtaining the fatigue law parameters in Equation (3) directly requires conducting tests at
multiple load amplitudes, which is time consuming and hence, impractical. This has motivated
11
researchers to utilize viscoelastic continuum damage (VECD) mechanics to model the fatigue
behavior of asphalt mixtures (Park, Kim and Schapery 1996); (Lee and Kim 1998); (Daniel and
Kim 2002); (Kutay, Gibson and Youtcheff 2008) and more recently, asphalt binders (Johnson
2010); (Hintz, et al. 2011). The primary benefit of VECD analysis is that it allows for prediction
of fatigue life at any loading amplitude from a single test. VECD analysis frameworks used to
model fatigue behavior of asphalt are derived based on the work of Schapery (R. Schapery
1975); (R. Schapery 1984); (R. Schapery 1990).
Schapery (R. Schapery 1990) developed a theory to describe the behavior of elastic
materials with growing damage based on the thermodynamics of irreversible processes as
follows:
(4)
where
is the strain energy density, which is a function of strain and D (i.e.,
internal state variable used to describe amount of work required to produce a given extent of
damage), and
) is the dissipated energy due to structural changes.
The theory was extended to viscoelastic materials through the use of Schapery's concept
of "pseudo-strain". Schapery (1984) postulated that constitutive equations for certain viscoelastic
materials are identical to those for the elastic case but stresses and strains are not necessarily
physical quantities. Rather, they are “pseudo” variables. Pseudo-strain (
) is defined as follows:
12
∫
(5)
where
is a reference modulus of arbitrary value, typically selected to be one. E(t) is the
relaxation modulus, t is time and
is a time variable of integration. Linear viscoelastic stress is
defined as:
∫
(6)
Hence,
(7)
If
, then pseudo-strain is equal to the linear viscoelastic shear stress. Equation (7) has the
form of the elastic stress-strain equation given in Hooke's law even though it is a viscoelastic
stress-strain equation. Thus, use of pseudo-strain eliminates hysteresis (i.e., differences in
loading and unloading paths) when the material is undamaged.
Additionally, to apply the work potential theory to viscoelastic materials (Park, Kim and
Schapery 1996) demonstrated the growth of D is rate dependent for most viscoelastic materials.
As such, (Park, Kim and Schapery 1996) proposed the following evolution law:
(
)
(8)
13
where
is the pseudo strain energy density function, and
is a material-
dependent constant.
While VECD analysis is based on continuum damage mechanics, the parameter α in
Equation (8) is typically defined a priori, based on fracture mechanics. Schapery (R. Schapery
1975) suggested that α can be calculated based on crack growth theory for viscoelastic media.
Schapery demonstrated that for Mode I fracture (i.e., opening), viscoelastic media obey a power
law relationship between crack growth rate (da/dN) and energy release rate (Gf):
(9)
where α is defined based on the maximum slope, m, of the log J(t) (creep) or log E(t) (relaxation)
curve. Schapery (R. Schapery 1990) demonstrated that if a material's fracture energy and fracture
stress are constant, α=1+1/m whereas if the fracture energy and fracture process zone size are
constant, α=1/m.
2.3.2
Application of VECD to Asphalt Mixture Fatigue
Researchers have applied VECD mechanics to cyclic fatigue testing of asphalt mixtures
to model fatigue damage evolution. Pseudo-strain energy density can be used to quantify work
performed (Daniel and Kim 2002); (Kutay, Gibson and Youtcheff 2008):
(10)
where
is the peak pseudo-strain in a given stress-pseudo strain cycle. The parameter C is a
normalized material integrity parameter defined as (Daniel and Kim 2002) (Kutay, Gibson and
Youtcheff 2008):
14
(11)
where
is the peak stress in cycle N and I is defined as:
(12)
where |E*|initial is the initial, undamaged dynamic modulus determined from the fatigue test and
|E*|LVE is the linear viscoelastic dynamic modulus at a reference temperature. To derive damage
as a function of time, the following chain rule is used (Park, Kim and Schapery 1996):
(13)
Equation (8) can be combined with Equation (13) to allow for use of numerical integration to
derive damage as a function of time (Park, Kim and Schapery 1996):
∑[
]
(14)
Typically, α has been defined as 1+1/m for displacement controlled tests and 1/m for load
controlled tests in cyclic loading based on the findings of (Lee and Kim 1998). The parameter m
is the slope of the log E(t) vs. time curve. Lee and Kim found that these definitions eliminated
the loading amplitude dependence of the damage curve (i.e., the C(D) curve). However, in later
work, (Kutay, Gibson and Youtcheff 2008) found that use of α =1/m for both stress and strain
controlled fatigue tests unified all damage curves (i.e., from both load and displacement
15
controlled loading). Thus, there is discrepancy in the literature as to which definition of α should
be used. Therefore, there appears to be a lack of theoretical basis in defining α.
2.3.3
Application of VECD to Asphalt Binder Fatigue
Johnson (Johnson 2010) applied VECD concepts developed by (Kim, Little and Lytton
2006) to sand asphalt mixtures tested using DMA and asphalt binders tested in the DSR. In
Johnson’s approach, instead of using pseudo strain energy density, dissipated energy is used to
quantify work performed:
(15)
where ID is the initial undamaged dynamic shear modulus divided by a modulus of one, 0 is the
applied shear strain amplitude, |G*| is complex shear modulus and  is the phase angle.
Using Equation (15) and Equation (8), damage is calculated based on changes in
|G*|·sinδ with respect to time. Thus, |G*|·sinδ is used to define the material integrity in asphalt
binders. Correspondingly, damage is calculated as follows (Kim, Little and Lytton 2006):
∑[
]
(16)
In order to derive the relationship between number of cycles to failure and applied load
amplitude, in both asphalt binders and mixtures, a model is typically fit to the material integrity
versus damage plot. As discussed, in mixtures C is the material integrity parameter whereas in
binders, |G*|·sin δ is used as the material integrity parameter. One of the most commonly used
models relating damage and material integrity is a power law (Kim, Little and Lytton 2006);
(Johnson 2010); (Hintz, et al. 2011):
16
(17)
where C0, C1, and C2 are model coefficients. To derive the fatigue law, the derivative of Equation
(17) is then taken with respect to damage and substituted into Equation (8). The derivative of
Equation (17) is:
(18)
Equation (18) is substituted into Equation (8) and integration is performed to derive the
relationship between D and time:
(19)
where k = 1 + (1 – C2). Equation (19) can be re-written in terms of number of cycles to failure,
Nf, as follows:
(
)
(20)
where f is the loading frequency (Hz) and Df is the damage at failure. Equation (20) can be rewritten by grouping terms as follows:
(21)
where:
(
)
(22)
17
B = -2
(23)
Thus, the number of cycles to failure can be calculated at any strain amplitude from a
single test. Note that Equation (21) has the same general form as Equation (3). Similar
manipulations can be made to derive the relationship between number of cycles to failure and
applied load when using pseudo strain energy density (i.e., for asphalt mixtures).
Derivation of Equation (21) requires a failure definition. Johnson (Johnson 2010)
recommended defining failure as a 35% reduction in |G*|·sinδ because this gave the highest
correlation between VECD predictions of fatigue life from the time sweep (true fatigue test) and
LAS (accelerated, surrogate test).
Johnson (Johnson 2010) defined α as 1+1/m where m is the slope of the log-log plot of
storage modulus (i.e., |G*|·cosδ) versus frequency. Approximate inter-conversions proposed by
(Schapery and Park 1999) from dynamic loading to the relaxation domain rely on the
relationship between storage modulus and frequency. Johnson (Johnson 2010) and Hintz et al.
(Hintz, et al. 2011) demonstrated that α determined using storage modulus and relaxation
modulus are statistically equivalent.
2.4
Damage
Damage analysis is often based on the concept of effective stress (Lemaitre and Desmarat
2005). Damage is assumed to cause a decrease in the surface area over which stresses are
applied. This decrease in effective area occurs because the surface area which is stressed
decreases due to the formation and growth of defects and voids. This is the basic concept used in
continuum damage modeling. In fatigue testing of asphalt mixtures, it is reasonable to assume
repeated loading leads to the formation of micro-voids and defects. However, the cause for
changes in material response during DSR fatigue testing of asphalt binders are less clear.
18
2.5
Thixotropy
Thixotropy is defined as "the continuous decrease of viscosity with time when flow is
applied to a sample that has been previously at rest and the subsequent recovery of viscosity in
time when flow is discontinued" (Mewis and Wagner 2009). Thixotropy is a form of structural
nonlinearity and thus, not a form of damage. Structural nonlinearity is associated with reversible
micro-structural changes (Malkin 1995). Structural changes can include changes in "regular"
structure or configuration and "rupture" of inner micro-structure. Structural nonlinearity is
possible either if the material is being loaded outside the linear viscoelastic regime or when
continuous, reversible changes evolve over time (i.e., thixotropy).
The changes in microstructure leading to thixotropic behavior are not well understood.
Generally, thixotropy is thought to be associated with relatively weak attractive forces. There are
two competing mechanisms during flow: break-down due to stresses and build-up due to random
collisions between particles and Brownian motion (Barnes 1997). Brownian motion is the
random thermal motion of molecules. When molecules collide due to this random motion, they
can attach to each other, given the necessary attractive forces, forming flocs. During loading,
stresses can cause flocs to break apart over time, leading to thixotropic behavior. When loading
is ceased, Brownian motion dominates and molecules will collide and flocs will be formed.
Recovery / formation of flocs can take significantly longer than break-down (Barnes 1997). Both
increased shear rate and loading time increase breakage (Mewis and Wagner 2009).
Complex moduli tests, such as the time sweep, can be used to evaluate thixotropy (Mewis
and Wagner 2009). Repeated cyclic loading in a nondestructive manner allows for monitoring
the breakdown of microstructure during loading and recovery with rest. Typically, a load
sufficiently small to avoid disruption of recovery is applied during the rest phase to allow for
19
monitoring the rate of recovery. When dynamic modulus testing is used, the dynamic viscosity,
|η*|, or |G*| is typically used to evaluate thixotropy. The complex modulus is related to |η*| as
follows:
(24)
where |η*| is related to viscosity through the Cox-Merz relationship (Cox and Merz 1958):
̇
(25)
Several researchers have considered thixotropy when studying asphalt fatigue and
healing. Soltani and Anderson (Soltani and Anderson 2005) conducted uniaxial push-pull fatigue
tests on asphalt mixtures and concluded that the decrease in modulus during loading and
recovery of modulus with rest is largely due to thixotropy. Di Benedetto et al. (Di Benedetto,
Nguyen and Sauzeat 2011) attributed a sharp initial decrease in modulus observed at the
beginning of fatigue testing of asphalt mixtures to thixotropy.
Shan et al. (Shan, et al. 2010) evaluated asphalt binder fatigue using stress-controlled
time sweeps with rest. Rest periods were inserted at various percent reductions in |G*| to study
healing and thixotropy. Thixotropy was characterized using flow tests with step-wise increases in
shear rate. The time required to reach a steady state viscosity at different shear rates was used to
characterize thixotropy. Based on a comparison of step-wise shear test results and fatigue tests, it
was concluded part of the modulus reduction in fatigue testing in the DSR is associated with
thixotropy.
20
2.6
Fracture under Torsional Loading
2.6.1
Edge fracture in fatigue testing of cylindrical specimens under torsion
Generally, in cyclic torsional testing of cylindrical specimens, fracture manifests in the
form of an “edge fracture.” Edge fracture is a circumferential crack or indentation that forms at
the periphery of a cylindrical sample (Aboutorabi, Ebbot and Gent 1998); (Keentok and Xue
1999).
Edge fracture in DSR testing can be a result of flow instability caused by the
development of normal stresses (Keentok and Xue 1999) or fracture. When edge fracture is
caused by flow instability, it is more properly termed "edge flow" as it is not actually a fracture
phenomenon. Previous researchers have attributed edge fracture in DSR testing of asphalt
binders to flow instability (Anderson, et al. 2001). However, these researchers did not consider
normal stresses. Bahia et al. (Bahia, et al. 2011) investigated the normal stresses during fatigue
testing in the DSR for a wide range of binders and testing conditions. No significant normal
stresses were observed and the normal stresses actually decreased with increasing number of
cycles. This indicates that the edge fracture observed during fatigue testing in the DSR is actually
a fracture phenomenon and not a result of flow instability.
A depiction of edge fracture is provided in Figure 4. As the crack propagates, the area
which is sheared is decreased. This follows the definition of damage previously discussed: a
decrease over the effective area in which loads are distributed. Hence, edge fracture associated
with fatigue testing of asphalt binders in the DSR is a valid form of damage.
21
Figure 4. Edge fracture in DSR testing.
Mattes et al. (Mattes, Vogt and Friedrich 2008) studied the edge fracture of polystyrene
melts during oscillatory DSR testing. Time sweeps were conducted over a wide range of strains
and frequencies. A significant decrease in complex modulus with respect to number of loading
cycles was observed in samples with edge fracture.
The DSR is only able to control/measure torque and deflection angle directly. Thus,
stresses and strains are computed assuming a constant sample radius (i.e., R in Figure 4).
Therefore, calculations are erroneous in the presence of fracture. Thus, under controlled
deflection angle amplitude testing, an observed decrease in complex modulus reflects a change in
torque only.
By assuming all changes in torque were due to edge fracture, Mattes et al. (Mattes, Vogt
and Friedrich 2008) calculated the theoretical sample radius as a function of number of cycles of
loading. Predictions of edge fracture from torque measurement were compared with visual
observations of edge fracture propagation. At various loading times and recovery times, tests
were terminated and samples were chilled. Chilling the samples allowed for easy sample
extraction. Intact radii (i.e., r(t)) of extracted samples were measured. The authors concluded
22
edge fracture was the primary cause for modulus reduction based on comparison of r(t) and
torque measurements.
Mattes et al. (Mattes, Vogt and Friedrich 2008) also studied the effect of rest on edge
fracture. Measurements and visual observations demonstrated the edge crack closes (heals) with
rest. Bubbles occurred during healing of the edge fracture, which prevented full recovery.
Asphalt binders are also known to possess self-healing capabilities (Carpenter and Shen 2006);
(Bommavaram, Bhasin and Little 2009). Hence, recovery attributed to thixotropy could actually
be a result of fracture healing.
2.6.2
Mechanism of Fracture under Torsion
Crack initiation and propagation under torsional loading depends on whether the crack is
forming in Mode I (tensile opening) or Mode III (anti-plane shear). Under pure torsion, if
fracture occurs due to tension, cracking will occur +/- 45 degrees relative to the specimen axis
(i.e., along principal plane under pure torsion loading) whereas in Mode III fracture, cracking
will occur along axial or radial shear planes as depicted in Figure 5. Hence, one can infer the
mode of crack initiation and propagation based on the orientation of crack growth.
Fatigue crack growth in Mode III is generally significantly slower than in Mode I.
Several researchers have tried to explain this phenomenon. The primary reason reported in the
literature is based on the depth to length ratio of cracks formed in Mode I versus Mode III (or II)
(Doquet 1997). Fracture under tensile (Mode I) loading will form perpendicular to the direction
of the stress driving fracture. In Mode III, fracture will occur parallel to the maximum shear
stress. This leads to the depth to length ratio of cracks under Mode I to be much greater than
Mode III, which leads to crack face surface interaction in Mode III, delaying crack growth.
23
Fracture
(a)
Fracture
(b)
Figure 5. Fracture under torsion in (a) Mode III and (b) Mode I
The fracture morphology of cylindrical specimens with an edge crack subjected to torsion
has been studied in steel (Tschegg, Ritchie and McClintock 1983) and polymeric materials
(Keller, White and Sottos 2008). Experimental observations of torsional fatigue fracture in
notched steel specimens demonstrate that as fracture initiates in Mode III and as the fracture
propagates, there is increased surface interaction between fracture planes. Typical fatigue
fracture surfaces of torsionally fatigued steel and poly-dimethylsiloxane (an elastomeric
polymer) are shown in Figure 6 and Figure 7, respectively. The initial fracture surface,
corresponding to the outer edge of the sample, appears macroscopically flat (pure Mode III). As
the crack propagates inward, a rough fracture surface develops, typically referred to as "factory
roof" morphology (Tschegg, Ritchie and McClintock 1983). Factory rough morphology consists
of radial peaks and valleys, which leads to surface interactions between crack faces. This surface
interaction creates radial crack branches which leads to combined Mode I and Mode III fracture
and consequently slows crack growth (Tschegg, Ritchie and McClintock 1983).
24
Figure 6. Fracture Morphology of Steel. Source: Tschegg et al. 1983.
Figure 7. Fracture Morphology of Poly-dimethylsiloxane. Source: Keller et al. 2008.
2.6.3
Linear Elastic Fracture Mechanics of Edge Fracture
Based on linear elastic fracture mechanics, equations for energy release rate have been
derived for edge fracture of a cylindrical disk under pure torsion assuming Mode III fracture
(Aboutorabi, Ebbot and Gent 1998), (De and Gent 1998), (Gent and Yeoh 2003), (Fleischman,
Kerchman and Ebbott 2001). Figure 8 provides a depiction of edge fracture where ri is the initial
sample radius, r is the intact sample radius, and a is the crack length. Crack propagation
effectively reduces the sample radius.
25
Figure 8. Edge fracture schematic.
Gent et al. (Aboutorabi, Ebbot and Gent 1998); (Gent and Yeoh 2003) determined the
energy release rate, of a cylindrical disk subjected to rotation by an angle, φ.
Energy stored elastically for a cylindrical disk under torsion loading is:
(26)
where r is the sample radius, φ is the defection angle, G is the shear modulus, and h is the sample
height. Note that Equation (26) is derived by integrating over the volume of the sample and
therefore accounts for the radial gradients in stress and strain. The energy release rate is the rate
of strain energy release as the crack propagates inward:
(
)
(27)
The subscript φ indicates controlled displacement conditions. A is the surface area created by the
crack. Equation (27) can be rewritten as:
(28)
26
The new surface area created by the edge fracture is:
(29)
Combining Equations (26), (28), and (29) gives the following solution for energy release rate
(Gf):
(30)
where ri is the initial radius and a is the crack length. Note that Equation (30) indicates Gf
decreases with increasing crack depth, which does not agree with experimental observations.
Aboutorabi et al. (Aboutorabi, Ebbot and Gent 1998), demonstrated that initially both
crack growth rate and energy release rate increase as edge fracture progresses. Gent and Yeoh
(Gent and Yeoh 2003) called this period of crack growth "shallow" crack growth. Once a critical
crack length is exceeded, both energy release rate and crack growth rate decrease with increasing
crack length, which Gent and Yeoh referred to as "deep" crack growth (Gent and Yeoh 2003).
The stages of crack growth are depicted in Figure 9, which shows energy release rate as a
function of crack length for rubber under cyclic torsion.
27
Figure 9. Energy release rate as a function of crack length (Fleischman, Kerchman and
Ebbott 2001).
There are two reasons for these trends in crack growth rate. The first concerns the energy
source driving fracture and the second involves fracture surface interactions. Initially, the energy
driving crack growth is derived from the material near the edge in the immediate vicinity of the
crack (Gent and Yeoh 2003). As the crack grows, the available energy source increases (and
consequently crack growth rate increases) until limited by the physical boundaries of the
specimen. At this point, the crack becomes “deep” and energy for crack growth is derived from
interior of specimen (i.e., Equation (31)). The interior specimen size decreases with increasing
crack depth. Hence, energy release rate and crack growth rate decrease with increasing crack
depth. Additionally, as the crack becomes deep interaction between fractured surfaces increase,
which reduces the crack growth rate (Tschegg, Ritchie and McClintock 1983).
Fleischman et al. resolved the contradiction between Equation (31) and experimental
observations (Fleischman, Kerchman and Ebbott 2001). Based on comparison of torque and
fracture length (i.e., a), Fleischman et al. found that the fractured portion of the disk carries some
28
load. Based on this finding, which was also noted by Aboutorabi et al. (Aboutorabi, Ebbot and
Gent 1998) but not accounted for in analyses, Fleischman et al. (Fleischman, Kerchman and
Ebbott 2001) proposed an adjustment factor, δc to account for the stress carried by the fractured
edge. The correction factor is defined as follows:
(31)
where c is a constant. The correction factor increases the effective radius of the disk. That is,
(32)
Note that the exponential model is used because as crack length approaches zero, so should δc.
Otherwise, the effective radius would exceed the actual sample radius. Gent and Yeoh (Gent and
Yeoh 2003) also found that for shallow cracks, crack length can be estimated using the torque
measurements alone since the correction factor is small.
Substituting the effective radius in place of the actual sample radius into Equation (26)
yields the following equation for energy release rate:
(33)
Equation (33) was found to compare favorably with experimental results in rubber (Fleischman,
Kerchman and Ebbott 2001) and elastomeric polymers (Keller, White and Sottos 2008).
The load carrying capacity of the fractured sample edge implies the front of the crack is
acting as a process zone rather than fully fractured, which could be influenced by crack face
interactions and plasticity. Typically, a process zone is defined as a region in front of a crack tip
29
that carries less stress than if undamaged (T. Anderson 2005). The concept of a process zone is
often incorporated into elastic-plastic fracture mechanics using similar to the concept to the
correction factor applied to edge fracture under torsional loading. The Irwin approach (T.
Anderson 2005) is used to account for crack tip plasticity by defining an effective crack length
(aeff), which is slightly longer than the actual crack length (a) by introducing a plastic zone
correction factor (ry) as follows:
(34)
Additionally, the Mode III stress intensity factor, KIII for an infinitely long bar with an edge
crack under torsion includes a correction factor:
[
]
(35)
where T is the torque and α is a numerical (i.e., correction) factor defined as:
[
]
Note that the stress intensity factor, K, and energy release rate, Gf , are related as follows for
linear elastic materials (T. Anderson 2005):
(36)
where KI is the stress intensity factor for mode I loading (i.e., crack opening), KII is the stress
intensity factor for mode II loading (i.e., in-plane shear), and KIII is the stress intensity factor for
30
Mode III loading (i.e., out of plane shear). The parameter G is the shear modulus and E' is
defined as E for plane stress problems and E/(1-ν2) for plane strain problems where E is Young's
modulus. Making use of Equation (35) and noting the following equation for torsional stiffness
(T/φ), one can derive the energy release rate as shown in Equation (38).
(37)
(38)
Note that Equation (38) follows a similar trend with crack length as Equation (33). Both
equations follow general experimental trends: an initial increase in energy release rate with
increasing crack length but as the crack becomes deep, the energy release rate decreases. Gent
and Yeoh (Gent and Yeoh 2003) found that Equation (38) was a reasonable solution for
relatively thin rubber disks with an edge crack, despite being derived for an infinitely long
specimen. It is also important to note that edge fracture has been used to study fatigue
phenomenon in rubber (Aboutorabi, Ebbot and Gent 1998); (Fleischman, Kerchman and Ebbott
2001) and self-healing elastomers (Keller, White and Sottos 2008). However, these studies did
not use the DSR for testing.
2.6.4
Incorporation of Viscoelasticity
Asphalt binders exhibit viscoelastic behavior and thus, use of linear elastic fracture
mechanics is questionable. Although there are many examples of using elastic fracture
mechanics to characterize polymers (viscoelastic materials), it is recognized that applicability is
limited and that used is driven by practical rather than fundamental reasons (T. Anderson 2005).
31
Edge fracture presents a unique case in which the elastic-viscoelastic correspondence
principle can be effectively applied to determine the quasi-linear viscoelastic fracture mechanics
solution based on the elastic case. The correspondence principle, applied to dynamic loading,
states that the "linear viscoelastic solution to a problem can be determined from the linear elastic
solution by replacing all material properties which depend on frequency by their Fourier
transform", given boundary conditions do not change with time (Lakes 2009). This principle
leads to simply replacing elastic constants with the complex dynamic viscoelastic functions.
Unlike most fracture problems, the correspondence principle can be applied to the energy release
rate solution for edge fracture because the effective boundary conditions do not change with time
(i.e., the effective sample size simply reduces). Hence, the quasi-linear viscoelastic fracture
energy solutions for edge fracture energy release rate can be written by simply replacing the
shear modulus (G) for the elastic solution shown in Equations (33) and (38) with the complex
shear modulus as a function of frequency (|G*|(ω)) as shown in the following equations,
respectively:
(39)
(40)
2.6.5
Nonlinearity
If a material is behaving linear elastically under dynamic loading, the plot of stress versus
strain should appear as a straight line as depicted in Figure 10(a). For linear viscoelastic
32
materials under sinusoidal loading, the plot of stress versus strain is elliptical due to a phase lag
between the loading input and response as depicted in Figure 10(b).
(a)
(b)
Figure 10. Depiction of (a) linear elastic behavior and (b) linear viscoelastic behavior.
In the nonlinear regime under cyclic loading, the response to sinusoidal strain loading
becomes distorted and hence, the plot of stress versus strain becomes non-elliptical (Lakes
2009). To adequately represent nonlinear viscoelastic response, a Fourier series can be used to
represent the oscillatory stress response (Wilhelm 2002). A Fourier transformation represents the
periodic contributions to a time dependent signal. The Fourier transformation provides the
amplitudes and phases (i.e., real and imaginary components) of the contributions from harmonics
of the input frequency. Fourier transformation, S(t), applied to a time signal, s(t), is defined as
follows:
∫
(41)
33
The Fourier transform generates real, R(ω), and imaginary, I(ω), components of the
contributions from frequencies. These components can be used to determine the magnitude m(ω)
= (R2 + I2)0.5 and phase shift φ(ω) = tan-1(I(ω)/R(ω)) of the contributions of higher harmonics.
Typically, the relative intensity of harmonic contributions is used to quantify harmonic distortion
where the relative intensity In/I1 is obtained by normalizing the magnitude of the nth harmonic to
the first harmonic (i.e., n = 1) (Kyun, et al. 2011). The total harmonic distortion can be
quantified as the sum of all In divided by I1.
The Fourier series representation of the nonlinear stress (τ) response to a sinusoidal strain
input can be described as follows:
∑
(42)
where an represents of magnitude of contribution from the nth harmonic and ωn is the angular
frequency of loading input. Note that this framework implies there can be nonlinearity (i.e.,
distortion of the hysteresis) even if the magnitude of the complex modulus does not change with
loading amplitude.
In studying asphalt binders, the most common method used to characterize nonlinearity
under dynamic loading is to observe the trend in complex modulus with loading amplitude under
dynamic loading rather than studying hysteretic behavior. Typically, the asphalt binder is thought
to be within the linear viscoelastic regime if the complex modulus until a loading amplitude is
reached where the complex modulus is equal to 95% of the complex modulus at lower loading
amplitudes (Airy, Rahimzadeh, and Collop 2002).
34
2.7
Fracture Based Fatigue Models
Equation (9) represents the fracture based fatigue model proposed by Schapery (R.
Schapery 1975). Note that Equation (9) resembles Paris' Law, a model, which relates crack
growth rate to ΔK (the range of stress intensity factors over a single cycle of loading) (T.
Anderson 2005):
(43)
where ΔK = Kmax - Kmin. Figure 11 provides a depiction of Kmax and Kmin.
Figure 11. Depiction of ΔK.
Paris' law is only applicable to stable crack growth (T. Anderson 2005). There are
typically three regimes of Mode I fatigue fracture: crack initiation, steady crack growth, and
unsteady crack growth (T. Anderson 2005). These stages are depicted in Figure 12. Stage I is
crack initiation, which typically corresponds to slow crack growth. Stage II is stable crack
growth, corresponding to crack propagation where crack growth rate increases steadily. In Stage
III, often assumed to correspond to failure, corresponds to unsteady crack growth where crack
growth rate increases rapidly.
35
Figure 12. Mode I crack growth.
Total fatigue life, NT, can be defined as (Hertzberg 1989):
(44)
where Ni defines the number of cycles for crack initiation (i.e., Stage I) and corresponds to the
number of cycles to develop a crack of specified size and Np corresponds to the number of cycles
of crack propagation until a specified crack size at failure (i.e., Stage II) is reached. No precise
definition of crack initiation exists due to the complex geometry dependence of crack initiation
(Hertzberg 1989).
2.8
Time-Temperature Superposition
The principle of time-temperature superposition states that the change in a material
property (e.g., |G*|) with respect to temperature is equivalent to a horizontal shift on the log time
or log frequency scale (Lakes 2009). This principle implies that increasing temperature is
equivalent to increasing loading time (or decreasing loading frequency). Similarly, decreasing
temperature is equivalent to decreasing loading time (or increasing loading frequency). Under
the principle of time-temperature superposition, the concept of reduced time (or reduced
36
frequency) can be used to form a master curve for a given temperature. A master curve consists
of shifting data from different testing temperature horizontally on the log time or log frequency
scale (to form a "reduced" time or frequency scale) to align with the data at a reference
temperature. The concept of a master curve is illustrated in Figure 13.
The reduced time is defined using a time-temperature shift factor (aT):
(45)
The Williams-Landel-Ferry equation (Ferry 1980) can be used to determine the time-temperature
shift factors at any temperature for a given reference temperature:
(46)
Where T is the temperature of interest, T0 is the reference temperature, and c1, c2 are curve fit
coefficients.
The concept of time-temperature superposition was developed for characterization of
viscoelastic materials in the linear viscoelastic regime. However, researchers have demonstrated
that time-temperature superposition remains valid for asphalt mixtures with growing microdamage (Daniel and Kim 2002) and macro-crack propagation (Seo 2003). This is an important
concept as it allows for predicting fatigue performance at any temperature using fatigue test data
at a single, reference temperature coupled with corresponding time-temperature shift factor for
the temperature of interest.
37
1.0E+08
Reference
Temperature
|G*| (Pa)
1.0E+07
1.0E+06
1.0E+05
10C
20C
25C
30C
40C
1.0E+04
1.0E+03
1.0E-01
1.0E+00
1.0E+01
1.0E+02
Frequency (Hz)
(a)
1.0E+08
1.0E+07
|G*| (Pa)
1.0E+06
1.0E+05
10C
20C
25C
30C
1.0E+04
1.0E+03
1.0E-03
1.0E-01
1.0E+01
Reduced Frequency (Hz)
1.0E+03
(b)
Figure 13. (a) Original frequency sweep data at multiple temperatures, (b) Shifted
frequency data to a reference temperature.
38
3
Materials and Methods
Time sweep testing (i.e., repeated cyclic loading at constant loading amplitude) in the DSR
was the primary experimental method used in this study. For all tests (except where otherwise
specified), 8mm diameter parallel plate geometry with a fixed 2mm sample height was used.
Controlled deflection angle tests rather than controlled torque tests were used because failure
occurs at a slower rate in the displacement controlled mode and there is no accumulated
deformation. For all tests, a constant loading frequency of 10Hz was used. Note that if fracture is
responsible for fatigue damage, it is extremely difficult to conduct a strain-controlled or stresscontrolled test because the sample radius is not constant. A depiction of deflection angle
controlled time sweep loading and response is provided in Figure 14. Prior to loading, samples
were allowed to thermally equilibrate at the test temperature for 15 minutes. A minimum of two
replicates for each test was conducted. In cases where repeatable results are not obtained, a third
replicate was tested.
Additionally, all asphalt binders were subjected to short-term aging using the Rolling Thin
Film Oven (RTFO) following the American Association of State Highway and Transportation
Officials (AASHTO) T 240-09, “Effect of Heat and Air on Rolling Film of Asphalt” (AASHTO
2009).
39
Figure 14. Depiction of time sweep loading and response.
3.1
Verification of Hypothesis
To determine if fracture is responsible for fatigue damage in the DSR, numerous tests
were conducted under controlled deflection angle conditions and terminated at different loading
durations in order to attain visual measurements of crack length at various stages of the fracture
process. Following each test, samples were chilled to 0°C and yellow paint was applied around
the sample perimeter as depicted in Figure 15(a). The paint used was an acrylic polymer
emulsion, produced by Liquitex®. Additional dyes/paints specifically developed for crack
detection were tried with no success. Due to the dark color of asphalt, an opaque paint/dye was
needed for visibility on the crack in the asphalt sample. The yellow paint used in this study was
fluid at room temperature, which allowed the paint to seep into cracks. Following paint
application, the paint was allowed to dry and then the upper portion of the sample adhering to the
DSR spindle is detached from the lower part of the sample adhering to the lower plate. A
depiction of a sample after spindle removal is provided in Figure 15(b). The paint was used to
enhance contrast between the fractured and intact portions of samples. After detachment of the
40
spindle, samples were photographed and digital imaging analysis was used to determine the
change in effective sample area due to fracture.
(a)
(b)
Figure 15. Procedure for attaining image of fractured surface: (a) paint application and (b)
sample after spindle detachment.
Two binders were tested using this procedure: one neat (i.e., unmodified) and one binder
with cross linked sytrene-butadiene-styrene (SBS) polymer modification. Tests were terminated
in five minute intervals beginning once a significant decrease in torque was achieved and once
torque measurements became relatively constant with time. Table 1 provides a summary of the
experimental plan for hypothesis verification. Note that tests are specified in terms of initial,
undamaged strain amplitude instead of the deflection amplitude of testing.
Table 1. Experimental Plan for Verification of Hypothesis
Asphalt Binder
Flint Hills (neat)
Nustar + 2% SBS
(modified)
Temperature
(°C)
25
25
Initial Strain
(%)
4
4
Times of Test
Termination (min)
25, 30, 35, 40, 45,50, 55
35, 40, 45, 50, 55, 60,
65, 70, 75, 80
41
3.2
Model Development
Based on the existing literature on fatigue cracking of elastic and viscoelastic materials, an
analysis framework based on fracture principles is presented. The goal is to propose a
mechanistic-empirical model that has similar capabilities to the Viscoelastic Continuum Damage
(VECD) modeling approach.
In order to develop a model for comprehensive fatigue characterization, two base binders
(i.e., Flint Hills and Nustar) with different types of modification were evaluated using numerous
fatigue tests including time sweep. Two modifiers were used: polyethylene (PE), a plastomeric
modifier and styrene-butadiene-styrene (SBS), an elastomeric modifier. SBS was used in two
forms: cross linked and linear. All binders were evaluated using time sweep tests at a minimum
of two temperatures. To eliminate effects of initial stiffness, iso-stiffness conditions of |G*|·sin δ
at 10Hz loading equal to 30MPa, 6.5 MPa and 2 MPa were used to determine test temperatures.
Frequency sweep tests were used to determine the temperatures corresponding to the isostiffness conditions for the individual binders. The frequency sweep test consists of sinusoidal
loading over a range of frequencies and temperatures in the DSR at small loading amplitude,
(assumed to be in the linear viscoelastic regime). In this study, a 0.1% strain amplitude was
applied using a frequency sweep over the range of 0.1 through 30 Hz (determined based upon
machine limitations). Frequency sweeps were conducted over a range of temperatures (10°C,
20°C, 25°C, 30°C, and 40°C). Frequency sweep data was used to produce a "master curve" using
time-temperature superposition (Ferry 1980) .
Master curves were constructed using a two-step process. First, a phenomenological
model relating complex modulus to frequency of loading proposed by Bahia et al. (Bahia, et al.
2001) was fit to the data:
42
[
]
(47)
where f’ = reduced frequency (Hz),
|G*|e = |G*| as f’ → 0: equilibrium dynamic modulus, which is equal to zero for asphalt binders,
|G*|g = |G*| as f’ → ∞: glassy dynamic modulus, equal to 1 GPa for asphalt binder;
fc = location parameter with dimensions of frequency, and
k, me = shape parameters, dimensionless.
Next, the relationship between phase angle () and frequency was modeled by (Bahia et
al. 2001):
{
[
⁄
] }
(48)
where m = phase angle at fd, the value at inflection, fd = location parameter with dimensions of
frequency, Rd, md = shape parameters, dimensionless, and I = 0 if f’ > fd, 1 if f’ ≤ fd.
Then, Equation (45) was fit to the data to allow for prediction of the time-temperature
shift factor at any temperature. The frequency sweep testing and analysis allows for prediction of
complex modulus and phase angle at any temperature and loading frequency, hence allowing for
prediction of the temperature where iso-stiffness conditions are met (e.g., temperature where
|G*|·sinδ at 10Hz frequency = 6.5MPa).
Table 2 presents the experimental plan associated with model development. At the 6.5
and 2MPa iso-stiffness conditions, asphalt binders were tested using deflection angle controlled
time sweeps at multiple initial, undamaged strain amplitudes (i.e., deflection angle amplitudes).
Additionally, two binders were tested at a 30MPa iso-stiffness condition at a single loading
43
amplitude. Generally, binders were tested at three loading amplitudes at the 6.5MPa iso-stiffness
condition and two loading amplitudes at the 2MPa iso-stiffness condition. The Flint Hills with
cross linked SBS modification exhibited extreme fatigue performance and thus, was tested at
different loading amplitudes than the other binders at the 2MPa iso-stiffness condition to produce
failure in a reasonable time. Additionally, this binder was only tested at two loading amplitudes
at the 6.5MPa iso-stiffness conditions due to the long testing times associated with this binder.
As will be demonstrated in proceeding analysis, two loading amplitudes of testing is more than
sufficient and hence, it is not problematic that this binder was only tested at two loading
amplitudes at the 6.5MPa iso-stiffness condition. Two binders were also be tested with a 3mm
sample height at the 6.5MPa iso-stiffness condition to determine the significance of testing
geometry on results. Additionally, a subset of the binders was tested using the Linear Amplitude
Sweep (LAS) test at the 6.5 MPa iso-stiffness condition. Note that the analysis of LAS results are
more complex because deflection angle amplitude changes systematically with time. Hence, the
applicability of findings for constant deflection angle tests to the LAS is evaluated in this study.
Additionally, all binders were tested using modified LAS test, which is proposed and described
in Chapter 6.
44
Table 2. Experimental Plan for Model Development
Undamaged Strain (%)
Base
Binder
Modification
|G*|·sinδ=6.5MPa |G*|·sinδ=2MPa |G*|·sinδ=30MPa
none
2% SBS
(x-linked)
Flint Hills
(PG 64-22) 2% SBS
(linear)
2%
Polyethylene
Nustar
(PG 64-22)
none
LAS*
Modified
Las*
X
X
2, 3, 3.5
2 ,3
2, 3
3, 5
X
2, 3, 4
2, 3
X
2, 3, 4
2, 3
2, 3, 3.5
2, 3
2% SBS
2, 3, 4
(x-linked)
* tested at 6.5MPa iso-stiffness condition
3.3
3mm
Gap*
3
2, 3
2
X
2
X
3
X
Validation with Mixture Fatigue Data
The developed model is validated using previously collected laboratory mixture fatigue
data from uniaxial push-pull testing. Asphalt binders were tested at the same temperatures at the
temperature of mixture fatigue testing. The loading amplitude of testing was selected such that
failure would occur in a reasonable time. Target test time was one hour. Table 3 summarizes the
experimental plan associated with the validation effort.
X
X
45
Table 3. Experimental Plan for Validation
Asphalt Binder
Laboratory Mix
Fatigue Data
Testing
Temp
(°C)
Undamaged
Strain (%)
PG 64-28
Uniaxial Push-Pull
20°C
20
3
PG 64-28
+Polyphosphoric
Acid (PPA)
Uniaxial Push-Pull
20°C
20
3
20
12
20
4
20
4
PG 64-34
PG 76-22
PG 64-28 + 2%
Latex Rubber
Uniaxial Push-Pull
20°C
Uniaxial Push-Pull
20°C
Uniaxial Push-Pull
20°C
46
4
Verification of Hypothesis
This chapter presents experimental verification of the hypothesis that fracture is the
source of fatigue damage during time sweep testing of asphalt binders in the DSR. The first step
to verify this hypothesis included an experimental study in which torque measurements in the
DSR were to predict crack length. These crack length predictions based on relative changes in
loading resistance were compared with crack lengths measured from direct imaging of the
samples following testing. Next, the mode of fracture was evaluated through Finite Element
Modeling (FEM) of the DSR. This was followed by evaluation of fracture morphology, and
observations in crack growth trends.
4.1
Experimental Verification of Hypothesis
The hypothesis that edge fracture is the primary mechanism of material degradation in
fatigue testing of asphalt binders in the DSR was tested by using constant deflection angle testing
and comparing predictions of crack length based on torque measurements with direct
measurements of intact sample radii obtained from image analysis. A similar analysis could be
conducted by comparing the change in deflection angle with visual observations in a constant
torque amplitude test. However, failure occurs rapidly in controlled torque tests. Therefore,
controlled deflection angle tests were used in this study because failure occurs at a slower rate,
and there is no accumulated deformation as the deflection is fully reversed. By combining
Equations (1) and (2), it can be seen that complex modulus is calculated based on peak torque
and peak deflection angle measurements as follows:
(49)
47
Rearranging Equation (49) in terms of sample radius gives:
(50)
Equation (50) can be used to calculate the theoretical intact sample radius (r) for a deflection
angle controlled test based on changes in torque measurements, assuming fracture leads to a
reduction in torque and |G*| is constant at the value measured for the undamaged binder. Note
that initially sample radius and height are 4mm and 2mm, respectively. The DSR is programmed
to keep the gap between the upper and lower plate constant so sample height can be assumed
constant. Sample radius, however, effectively decreases due to the propagation of a
circumferential crack originating at the outside edge of the sample.
Two binders were studied to verify the hypothesis that fracture rather than change in
|G*| is responsible for apparent changes in modulus during time sweep testing: Flint Hills Neat,
an unmodified binder asphalt and Nustar + SBS, a polymer modified asphalt. Torque values with
normalization by the initial torque for the Flint Hills binder versus number of loading cycles at
all times of test termination are presented in Figure 16. Results demonstrate a reduction in torque
of roughly 75% at the maximum loading time of 55 minutes. Figure 17 shows the average results
based on all tests for the Flint Hills binder. It can be seen that there is a significant change in
torque between each time interval at which the tests were aborted.
Normalized Torque (T/Tinitial)
48
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20min
25min
30min
35min
40min
45min
50min
55min
0
5,000
10,000 15,000 20,000 25,000
Number of Loading Cycles
30,000
35,000
Figure 16. Normalized torque results for Flint Hills binder.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20min
25min
30min
35min
40min
45min
50min
55min
0
5,000
10,000 15,000 20,000 25,000
30,000
35,000
Figure 17. Normalized torque results for the Flint Hills binder, based on averaged data.
Figure 18 presents the normalized torque results of Nustar +2% SBS for tests conducted
at various loading times. Roughly 75% reduction in torque is reached at the maximum loading
49
time of 80 minutes. It can be seen that the Nustar + 2% binder exhibits higher fatigue resistance
than the Flint Hills binder as the torque decreases at a slower rate. Figure 19 shows the average
results of normalized torque at all times of test abortion.
1.0
0.9
35min
40min
45min
50min
55min
60min
65min
70min
75min
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
10,000
20,000 30,000 40,000
50,000
60,000
Figure 18. Normalized torque results for Nustar + 2% SBS binder.
1.00
0.90
35min
40min
45min
50min
55min
60min
65min
70min
75min
80min
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
10,000
20,000 30,000 40,000 50,000
60,000
Figure 19. Normalized torque results for the Nustar + 2%SBS binder, based on averaged
data.
50
Equation (50) was used to predict the undamaged sample radius based on changes in
torque with number of cycle. Additionally, samples were photographed following termination of
tests for visual measurement of the crack length. To prepare samples for photographing, paint
was applied to the outer edge of samples immediately following removal of loading. The paint is
expected to seep into the cracked portion of the sample only, which allows for contrast in color
between the fractured and intact portions of the sample. The image captured allowed for easy
detection of intact and fractured portions of samples using digital image analysis as shown in
Figure 20. The expectation that paint would not seep in asphalt material was verified by
submerging a binder sample in paint for 30 minutes at the testing temperature. Following
soaking in paint, samples were cut to see if the paint diffused into the asphalt. No diffusion was
observed, which confirm that paint cannot seep into binders. It is also believed that due to very
narrow crack depth at the end of the crack, capillary action forces the paint to reach very close to
the actual end of the crack.
Figure 20. Fractured sample with paint application.
Microscope imaging of fractured samples after paint application was conducted to verify
the paint did not diffuse into the asphalt binder and that the paint reached the crack tip. Two
51
microscope images, taken using an optical microscope at differing magnification levels, are
shown in Figure 21 and Figure 22. At the inner edge of the paint, the fracture front can be seen.
The fracture morphology at the crack front is characterized by radial peaks and valleys. It can be
seen that these peaks contain yellow paint but there is no paint in front of these features. Thus,
the microscope images demonstrate the paint reaches the crack tip and does not diffuse within
the asphalt.
Figure 21. Microscope image of fractured sample with paint application #1.
Figure 22. Microscope image of fractured sample with paint application #2.
52
Images of samples after testing were processed using MATLABTM. A program was
written using built-in MATLABTM image analysis functions in order to detect the fractured and
intact portions of the sample. Based on Gonzalez et al. (Gonzalez, Woods and Eddins 2004), the
most appropriate image processing functions were identified. Upon identification of the suitable
image filters, a variety of functions and sequences of these functions were tried to determine the
optimal combination and sequence for fracture identification.
Figure 23 provides a depiction of the effects of the image filters used. Using the program
developed, the image is first converted from a color image to a grayscale image. Next, a median
filter is applied. Median filtering adjusts the grayscale value of a given pixel to the median value
of pixels within a specified radius of the pixel of interest. After the median filter is applied,
Hmax filtering is used to reduce the variation in grayscale intensity. The Hmax filter effectively
converts the fractured portion of the sample to a single grayscale intensity. Next, to enhance
contrast between the fractured edge, intact sample, and background, histogram equalization is
applied. Histogram equalization enhances contrast by expanding the range of pixel intensities of
the image to cover the full range of possible grayscale intensities (i.e., 0 to 256). Then, the image
is converted to a binary image (i.e., black and white image). The binary image is processed to
isolate the edge fracture only. First, the binary image is "closed". Closing first dilates the image
and then erodes it using the same size element for both processes. Dilation thickens objects by a
specified number / sequence of pixels whereas erosion thins objects by a specified number /
sequence of pixels. When these two operations are combined as in closing, small objects
connected to larger objects become disconnected. After closing, the clear border function is used
to clear any white objects connected to the image border. Lastly, the area open function is used
to remove any objects smaller than a specified number of pixels. Once the fractured portion of
53
the sample is detected, the outer area including cracked and uncracked portions of the sample
(i.e., initial sample area) and intact sample area (i.e., black area inside white fractured portion)
are calculated in terms of number of pixels. Using the areas, measured in pixels, radii are
calculated assuming the area is circular (i.e., assuming area equals πr2). Noting that the initial
sample radius equals 4mm, one can convert the radii measured in pixels to millimeters. The
difference between the radius calculated for the initial and intact radius is used to estimate crack
length. Thus, crack length is taken to be the difference between the initial an intact sample
radius.
54
Figure 23. Image processing steps.
55
Figure 24 depicts the progression of edge fracture based on visual observation for the
Flint Hills binder. It can be seen that as loading time increases, the intact sample size decreases.
Visual observations of edge fracture in the Nustar + 2%SBS binder at different loading times are
shown in Figure 25. As with the Flint Hills binder, visual observations match expected trends:
fracture progresses as loading time increases and the intact sample area decreases.
(a)
(b)
(c)
Figure 24. Visualization of edge fracture of Flint Hills binder at (a) 30, (b) 40, and (c) 55
minutes of loading.
56
(a)
(b)
(c)
(d)
Figure 25. Visualization of edge fracture of Nustar + 2% SBS binder at (a) 40, (b) 50, (c)
60, and (d) 80 minutes of loading.
Crack lengths determined from image analysis were compared with crack lengths
predicted from torque measurements. The comparison between measured and predicted crack
lengths for the Flint Hills binder and Nustar + 2% SBS are presented in Table 4 and Table 5,
respectively. Note that torque predicted values were made based on the results of the same test
that the images were taken (i.e., comparison at 25 minutes of loading corresponds to torque
predictions from the test that was terminated at 25 minutes loading time). Note also that the first
data point for each binder corresponding to 20 minutes of loading for the Flint Hills binder and
35 minutes for the Nustar + 2% SBS binder had to be excluded. Sample detachment from the
spindle without disturbing fractured portion of the sample was not possible at these loading times
57
because edge fracture had not propagated substantially. The crack lengths predicted from torque
measurements, based on averaging all test data and visualization of edge fracture (i.e.,
"measured"), for the Flint Hills binder and Nustar + 2% SBS are shown in Figure 26 and Figure
27, respectively. Note that all data series except for "measured" correspond to predictions based
on torque measurements. Results indicate good agreement between visual observations and
torque predictions of crack length. Furthermore, results match the trends reported in the literature
(Fleischman, Kerchman and Ebbott 2001): the torque predicted values of crack length are
slightly less than the measured fracture, particularly at higher crack lengths. This is thought to be
a result of the fractured edge carrying a small portion of the load due to interaction between
crack faces. Therefore, the comparison between crack length measurements using digital image
analysis and predictions based on torque measurements indicates fracture is responsible for the
apparent changes in complex modulus during fatigue testing of asphalt binders in the DSR.
Table 4. Comparison between Effective Sample Radii Estimated from Images and
Predicted from Torque measurements for Flint Hills Binder
Loading Time
(min)
25
30
35
40
50
55
60
Intact Radius (mm)
Predicted from
Measured
Torque
3.44
3.41
3.35
3.40
3.22
3.24
2.90
3.07
2.78
2.87
2.62
2.77
2.61
2.72
58
Table 5. Comparison between Effective Sample Radii Estimated from Images and
Predicted from Torque measurements for Nustar + 2% SBS Binder
Intact Radius (mm)
Loading Time
(min)
Measured
Predicted from
Torque
40
3.54
3.58
45
3.37
3.48
50
3.30
3.39
55
3.16
3.20
60
3.10
3.14
65
2.98
2.98
70
2.92
2.96
75
2.86
2.94
80
2.86
2.94
Crack Length(mm)
1.6
1.4
20min
1.2
25min
1
30min
35min
0.8
40min
0.6
45min
0.4
50min
0.2
55min
Measured
0
0
5000
10000 15000 20000 25000
30000
35000
estimated from digital image analysis for Flints Hills binder.
59
1.20
35min
40min
Crack Length (mm)
1.00
45min
0.80
50min
55min
0.60
60min
65min
0.40
70min
75min
0.20
80min
Measured
0.00
0
10,000
20,000 30,000 40,000
50,000
60,000
estimated from digital image analysis for Nustar + 2% SBS binder.
4.2
Determination of Fracture Mode
To better understand asphalt binder fracture during DSR fatigue testing a Finite Element
Modeling (FEM) simulation of the DSR was conducted to evaluate the stress state of the binder
in the DSR (shear versus normal stress). In addition, the fracture morphology of specimens after
testing was evaluated using high resolution images for select samples.
The FEM of the DSR was constructed by modeling asphalt binder as a linear viscoelastic
materials and the metal spindle as a rigid body. Typical viscoelastic asphalt binder properties
were input into the model. The asphalt binder specimen and rigid body were considered fully
bounded, which forces the top of the binder specimen to be fully fixed to the rigid body. The
bottom of the asphalt sample was fully fixed. A schematic of the DSR FEM is provided in Figure
28.
60
Figure 28. Schematic of FEM of DSR.
The load applied consisted of a small ramped displacement (0.01 radians) applied over
0.1 seconds. The distribution of Von Mises stresses is shown in Figure 29. It can be seen that the
stresses increase radially from zero at the center to a maximum at the specimen edge, as
expected. Figure 30 provides the distribution of normal stress in the specimen. It can be seen
that normal stresses are insignificant, (the maximum normal stress is 24.5Pa whereas the
maximum Von Mises stress is 23,450Pa). Thus, the shear stress can practically be considered
equivalent to the Von Mises stress, which varies linearly from zero at the sample center to
maximum at the sample edge as shown in Figure 31. It can be concluded that the DSR at the
temperatures used in the test (relatively high modulus) does not impose any important normal
stresses perpendicular to torsional loading when typical binder properties are used. Thus, it is
61
expected that fracture will propagate either in Mode III, parallel to the torsional loading or in
Mode I, at a 45⁰ angle relative to the torsional loading is expected.
Figure 29. Von Mises stress distribution.
62
Figure 30. Normal stress distribution.
25,000
y = 23035x
R² = 0.9989
Shear Stress (Pa)
20,000
15,000
10,000
5,000
0
0
0.2
0.4
0.6
0.8
r/rmax
Figure 31. Radial distribution of shear stress.
1
63
Photographs of fractured samples at the 6.5MPa and 30MPa iso-stiffness conditions of
testing are provided in Figure 32. The fracture morphology matches experimental observations in
steel, rubber, and elastomers under torsional fatigue loading where crack growth originates in
Mode III and radial peaks and valleys (“factory roof”) morphology develops due to surface
interactions as shown in many studies (Fleischman, Kerchman and Ebbott 2001) (Tschegg,
Ritchie and McClintock 1983) (Keller, White and Sottos 2008). Examples are shown in Figure 6
and Figure 7. It can be observed in Figure 32 that fracture does not occur at a 45⁰ angle relative
to specimen axis, indicating that tension is not the main cause of fracture and hence fracture can
be considered to occur in Mode III. Furthermore, it can be seen that the density of radial fracture
lines is much greater when the binder is softer, which matches trends observed in steel (Tschegg,
Ritchie and McClintock 1983).
(a)
(b)
Figure 32. Fatigue fractured samples at (a) 6.5MPa iso-stiffness condition, (b) 30MPa isostiffness condition.
Crack growth rate (da/dN) with respect to crack length, based on torque predictions of
crack length, for the Flint Hills Neat and Nustar + 2% SBS binders are shown in Figure 33 and
64
34, respectively. The results shown indicate initially there is a decrease in da/dN with increasing
crack length (or number of loading cycles). This contradicts the crack growth trends for torsional
Mode III fracture reported in the literature, which demonstrate an increase in the rate of energy
release and hence, crack propagation with increasing crack length for shallow cracks. However,
the results reported in the literature are based on notched specimens. As previously discussed,
crack growth can be divided into two phases: crack initiation and crack propagation. Energy
release rate calculations may be invalid during crack initiation due to complex geometry
dependence of initiation (Hertzberg 1989), which could explain the decreasing magnitude of
da/dN during the initial portion of crack growth. Excluding initiation, results follow the
observations by Gent et al. (Gent and Yeoh 2003) who reported that for rubber disks initially,
there is an increase in crack growth rate with number of cycles, assumed to be "shallow" crack
growth. As loading progresses, the crack becomes "deep" and there is a decrease in the crack
growth rate. These results provide promising evidence that edge fracture is responsible for
fatigue in asphalt binders tested in the DSR.
da/dN (mm/cycle)
0.00008
0.00006
0.00004
0.00002
0
0
0.2
0.4
0.6
0.8
a (mm)
1
1.2
1.4
Figure 33. Predicted crack growth rate as a function of number of loading cycles for Flint
Hills binder.
65
da/dN (mm/cycle)
0.00005
0.00004
0.00003
0.00002
0.00001
0
0
0.2
0.4
0.6
0.8
1
1.2
a (mm)
Figure 34. Predicted crack growth rate as a function of number of loading cycles for Nustar
+ 2% SBS binder.
4.3
Energy Release Rate Calculation
The applicability of the two solutions presented in Chapter 3for energy release rate for
growth of an edge crack under torsion was evaluated. The first, Equation (40) is based on an
infinitely long specimen so its applicability to the DSR is unknown. However, it has been found
to be applicable to thin rubber disks (Gent and Yeoh 2003). The second, Equation (39) requires
determination of the adjustment coefficient c using experimental data.
A comparison between trends in crack growth rate and energy release rate calculated
using Equation (40) with number of loading cycles for the Flint Hills neat and Nustar + SBS
binders, (excluding initial decrease in da/dN), are presented in Figure 35 and Figure 36. Note that
crack lengths were estimated based on torque measurements in these calculations. If the rate of
energy release is decreasing, the crack growth cannot increase. Thus, the energy release rate and
crack propagation rate should peak at roughly the same point in time. As evident by Figures
66
Figure 35 and Figure 36, the energy release and crack propagation rates peak at different number
of loading cycles. Hence, Equation (40) is not applicable to fatigue crack growth of asphalt
binders in the DSR.
7.0E+03
0.00009
0.00008
6.0E+03
Gf (Pa·mm)
5.0E+03
0.00006
4.0E+03
0.00005
3.0E+03
0.00004
2.0E+03
Gf
da/dN
1.0E+03
0.00003
da/dN (mm/cycle)
0.00007
0.00002
0.00001
0.0E+00
0
10000
20000
30000
0
40000
Figure 35. Energy release rate calculated using Equation (40) and crack growth rate as a
function of number of loading cycles for Flint Hills Binder.
67
6.00E+03
0.000045
0.00004
0.000035
4.00E+03
0.00003
0.000025
3.00E+03
0.00002
2.00E+03
0.000015
1.00E+03
Gf
0.00001
da/dN
0.000005
0.00E+00
0
10000
20000 30000 40000 50000
da/dN (mm/cycle)
Gf (Pa·mm)
5.00E+03
0
60000
Figure 36. Energy release rate calculated using Equation (40) and crack growth rate as a
function of number of loading cycles for Nustar + 2% SBS binder.
Based on the comparison of intact radii estimated from imaging and those predicted
based on torque measurements for both the Flint Hills and Nustar + 2% SBS binders, the
parameter c in Equation (31) was determined by minimizing the squared error between measured
radii + δc and radii predicted based on torque measurements. Note that the discrepancy between
torque predicted value of sample radius is the effective radius. Hence, the discrepancy between
the measured radius and torque predicted radius at any crack length is equal to δc. It was
determined that same trend between predicted and measured radii is found for both binders as
shown in Figure 37. Therefore, it is speculated that c is independent of binder type. A single
value of 0.1 mm for the parameter c was fit to the data for both binders evaluated. Results with c
equal to 0.1mm appear to show a very good fit as shown in Figure 38.
68
Measured Radius (mm)
4
3.5
3
2.5
Flint Hills
2
2
2.5
3
3.5
4
Radius Predicted from Torque Measurements (mm)
Mesured Radius + Correction (mm)
Figure 37. Comparison between measured radii and predicted radii from torque
measurements.
4
3.5
3
2.5
Flint Hills
Nustar + 2% SBS
Equality Line
2
2
2.5
3
3.5
4
Predicted Radius from Torque Measurements (mm)
Figure 38. Comparison between measured radius with correction factor and predicted
radius from torque measurements.
69
Once the estimate of c was determined, the energy release rate, Gf, was calculated using
Equation (69). The evolution of energy release rate and crack growth rate predicted based on
torque measurements as a function of number of loading cycles were compared for both the Flint
Hills binder (Figure 39) and Nustar + 2% SBS binder (Figure 40). It can be seen that both the
energy release and crack growth rates peak at approximately the same point for both binders
tested. This provides promising evidence the c factor and energy release rate are sufficient for
describing edge fracture of asphalt binders in the DSR. Based on these results and those reported
in the literature (Fleischman, Kerchman and Ebbott 2001), it is expected that c will depend on
sample geometry but not asphalt binder type.
14,000
0.00009
0.00008
12,000
0.00007
0.00006
8,000
0.00005
6,000
0.00004
4,000
0.00003
Gf
da/dN
2,000
da/dN
Gf (Pa·mm)
10,000
0.00002
0.00001
0
0
5,000
10,000 15,000 20,000 25,000
30,000
0
35,000
Figure 39. Energy release rate and crack growth rate as a function of number of loading
cycles for Flint Hills Binder.
70
12,000
0.000045
0.00004
0.000035
8,000
0.00003
0.000025
6,000
0.00002
4,000
da/dN
Gf (Pa·mm)
10,000
0.000015
0.00001
Gf
2,000
da/dN
0
0
10,000
20,000 30,000 40,000 50,000
0.000005
0
60,000
Figure 40. Energy release rate and crack growth rate as a function of number of loading
cycles for Nustar + 2% SBS binder.
As mentioned in the literature review section, there is a possibility of confounding effects
of nonlinearity in the viscoelastic fracture analysis. To rule out such interferences, the effect of
nonlinearity was evaluated by computing the total harmonic distortion based on a Fourier
transform analysis as discussed in the literature review section 6.2.5. Examples of results for the
binders tested are presented in Figure 41 as a function of number of cycles. It can be seen that
harmonic distortions are on the order of 10-3, which is very small. Therefore, it can be concluded
that nonlinearity effects are insignificant and that the assumption of linearity of binder behavior
for the conditions used in the testing can be used successfully for the analysis method.
Total Harmonic Distortion (∑In/I1)
71
0.002
0.0015
0.001
0.0005
Flint Hills
Nustar + 2% SBS
0
0
10,000
20,000
30,000
Figure 41. Total harmonic distortion.
40,000
50,000
72
5
Fatigue Model Development and Results
This chapter documents the development of a fatigue analysis framework applicable to the
time sweep test based on the findings presented in Chapter 4. The model is developed based on
test results of six binders and is validated using asphalt mixture fatigue test results. Additionally,
the applicability of time-temperature superposition to fatigue crack growth in asphalt binders is
investigated. The implications of findings from time sweep testing were applied to the Linear
Amplitude Sweep (LAS) test, an accelerated damage test currently proposed as a specification
standard for asphalt binders, are evaluated and a modified standard testing procedure and
analysis framework are proposed.
5.1
Frequency Sweep Results
Complex modulus and phase angle master curves generated from frequency sweep data
are presented in Figure 42 and
Figure
43,
respectively.
Testing
temperatures
corresponding to the selected iso-stiffness conditions determined from the master curves are
presented in Table 6. These temperatures were used for fatigue testing (time sweep and LAS). It
can be seen in Table 6 that testing temperatures spanned a range of approximately 10°C to 30°C.
73
1.0E+09
1.0E+08
1.0E+07
|G*| (Pa)
1.0E+06
1.0E+05
FH Neat
Nustar
PE
FH+CBE
FH+SBSL
FH+SBSX
Nustar+SBSX
1.0E+04
1.0E+03
1.0E+02
1.0E+01
1.0E+00
1.0E-07 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05 1.0E+07
Figure 42. |G*| master curves.
Phase Angle (°)
100
10
FH Neat
Nustar
FH+CBE
PE
FH+SBSL
FH+SBSX
Nustar+SBSX
1
1.0E-07 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05 1.0E+07
Figure 43. Phase angle master curves.
74
Table 6. Test Temperatures
Base
Binder
Flint Hills
(PG 64-22)
Nustar
(PG 64-22)
5.2
5.2.1
Modification
none
2% SBS
(x-linked)
2% SBS (linear)
2% Polyethylene
(PE)
none
2% SBS
(x-linked)
Test Temperature
(|G*|·sinδ = 6.5 MPa)
(°C)
23.5
Test Temperature
(|G*|·sinδ = 2MPa)
(°C)
32.3
Test Temperature
(|G*|·sinδ = 30MPa)
(°C)
-------
21.8
23.0
31.2
31.9
-------------
23.5
20.0
32.8
28.5
11.6
13.0
23.0
32.5
-------
Time Sweep Results and Analysis
Verification of Energy Release Rate Solution
The relationship between the crack growth rate (da/dN) and energy release rate (Gf ) was
investigated for use as the basis for a fatigue model. Prior to further analysis, it is important to
evaluate use of a constant parameter c in Equation (39) for all binders and testing conditions. The
parameter c was established based on the results of two binders at a single deflection angle
amplitude (0.02rad) and temperature of testing (25°C). The energy release rate should peak at the
same crack length as the crack propagation rate (i.e., da/dN). It can be seen that given a constant
c value, the energy release rate, (i.e., Equation (39)), peaks at a specific crack length regardless
of other parameters. With c equal to 0.1mm, the peak energy release rate occurs at a crack length
of approximately 0.3mm, which is illustrated in Figure 44, in which |G*|, φ, h, were set to unity.
This represents the crack length where crack growth transitions from shallow to deep crack
growth. To verify the parameter c is independent of binder type and testing conditions, the crack
75
growth rates and corresponding energy release rates were compared for the different binders and
testing conditions used in this study.
16
14
Gf (Pa-mm)
12
10
8
6
4
2
0
0.00
0.10
0.20
0.30
0.40
0.50 0.60
a (mm)
0.70
0.80
0.90
1.00
Figure 44. Depiction of typical trend of energy release rate versus crack length with c equal
to 0.1.
Crack propagation rates were compared with energy release rates for various
combinations of binders and the deflection angle amplitude as shown in Figure 45. Good
agreement between the crack length at peak crack growth rate and peak energy release rate is
observed. This is illustrated in Figure 45 which presents data for the Nustar neat binder, tested at
multiple loading amplitudes and temperatures (i.e., stiffnesses). Note that loading amplitudes are
specified in terms of the initial, undamaged strain, referred to as γi in proceeding analyses.
Trends shown in Figure 45 demonstrate that the peak rate of crack growth occurs at a crack
length of roughly 0.3mm. These results provide promising evidence that c is independent of
testing conditions and binder type. Based on this analysis a constant value of c is recommended
for all binder fracture analyses.
76
3E-05
3,000
2E-05
2,000
1E-05
1,000
Gf
da/dN
0
0.00
0.10
0.20
0.30
a (mm)
0.40
da/dN (mm/cycle)
Gf (Pa-mm)
4,000
0
0.50
8,000
8E-05
6,000
6E-05
4,000
4E-05
2,000
0
0.00
Gf
da/dN
2E-05
0
0.20
0.40
a (mm)
(b)
0.60
da/dN (mm/cycle)
Gf (Pa-mm)
(a)
1.E+04
0.00012
8.E+03
0.0001
8E-05
6.E+03
6E-05
4.E+03
Gf
2.E+03
da/dN
0.E+00
0.00
0.20
0.40
a (mm)
0.60
4E-05
2E-05
da/dN (mm/cycle)
Gf (Pa-mm)
77
0
0.80
1,800
1,600
1,400
1,200
1,000
800
600
400
200
0
0.00
0.00003
0.000025
0.00002
da/dN (mm/cycle)
Gf (Pa-mm)
(c)
0.000015
Gf
da/dN
0.20
0.40
a (mm)
(d)
0.60
0.00001
0.000005
0
0.80
20,000
18,000
16,000
14,000
12,000
10,000
8,000
6,000
4,000
2,000
0
0.00
0.00007
0.00006
da/dN (mm/cycle)
Gf (Pa-mm)
78
0.00005
0.00004
0.00003
Gf
0.00002
da/dN
0.20
0.40
0.60
a (mm)
0.00001
0.80
0
1.00
(e)
Figure 45. Comparison of energy release and crack growth rates for Nustar neat at
undamaged strain amplitudes of (a) 2%, (b) 3%, and (c) 3.5% at 6.5MPa iso-stiffness
condition, and at 2% undamaged strain at (d) 2MPa and (e) 30MPa iso-stiffness conditions.
Fracture is dependent on sample size and geometry when there are significant effects of a
fracture process zone (T. Anderson 2005). Thus, it is expected that the correction factor, δc in the
energy release rate calculation, Equation (39) is dependent on the sample geometry. In order to
verify this hypothesis, two binders (Flint Hills Neat and Nustar + SBSX) were tested using an
initial, undamaged strain of 3% using a 3mm as opposed to 2mm sample height. Crack lengths
were approximated based on torque measurements. Figure 46 presents the crack growth rates as
a function of crack length for the Flint Hills Neat and Nustar + SBSX binders with a sample
height of 3mm. Results demonstrate that when the sample height is 3mm the crack growth rate
peaks at a crack length of approximately 0.6mm. In order for the energy release and crack
propagation rate to peak at the same crack length, the correction factor for a 3mm sample height
would be approximately 0.25. Recall that when the sample height is 2mm, the crack growth rate
peaks at a crack length of approximately 0.3mm and the corresponding correction factor is 0.1.
79
Hence, there is a significant geometry effect on the correction factor. Thus, the energy release
rate formula is considered specific to the testing geometry. These trends are in agreement with
studies on rubber under Mode III fracture (Aboutorabi, Ebbot and Gent 1998), which
demonstrated that the correction factor increases as the sample height increases. However, this
geometry effect is not an issue if the standard testing geometry of an 8mm diameter with 2mm
height is used.
0.00008
da/dN (mm/cycle)
0.00007
0.00006
0.00005
0.00004
0.00003
0.00002
Nustar + SBSX
0.00001
FH Neat
0
0
0.2
0.4
0.6
a (mm)
0.8
1
1.2
Figure 46. Crack growth rate as crack length (a) function of a when sample height = 3mm.
5.2.2
Development of Fatigue Failure Criteria
The transition from a shallow to deep crack, which is assumed to occur prior to the peak
in energy release rate, is found to occur at a crack length of 0.3 mm for the standard testing
geometry (i.e., 8mm diameter samples with 2mm height. Beyond the peak energy release rate,
crack surface interactions can become significant, leading to mixed mode fracture complicating
analysis. Additionally, the transition from shallow to deep crack growth is a marked transition in
80
crack growth, and hence, is a reasonable and meaningful failure criterion. Thus, in the
proceeding analysis failure will be defined as the peak in the energy release rate. Note that
typical fatigue analyses rely on arbitrary failure definitions and hence, the proposed failure
criterion is advantageous over traditional analyses as it marks a distinct benchmark in material
damage resistance.
5.2.3
Effect of Binder Modification, Temperature, and Loading Amplitude on Fatigue Life
The effect of different factors evaluated in this study, including asphalt binder
modification, temperature, and loading amplitude on fatigue life for the binders tested is
illustrated in Figure 47. Note that fatigue lives were calculated as the number of cycles to the
transition from a shallow to a deep crack as described in the previous section. Also note that only
two binders were tested at the 30MPa iso-stiffness condition and at this condition, testing was
only conducted at an undamaged strain amplitude of 2%. Note that in all proceeding analyses,
loading amplitudes are specified in terms of initial, undamaged strain amplitude rather than
deflection angle amplitude.
The results presented in Figure 47 highlight the ability of the time sweep test to
discriminate fatigue performance of different binder types. Results clearly demonstrate that
polymer modification improves fatigue performance. It can be seen that the two neat binders
have similar fatigue performance, whereas the modified binders have considerably higher fatigue
resistance. The Flint Hills binder modified with cross-linked SBS (SBSX) demonstrates superior
performance over the other binders. Additionally, results demonstrate temperature sensitivity of
fatigue life is binder dependent and it is clear that increasing stiffness from 6.5MPa to 30MPa
has a much greater effect on fatigue life than increasing stiffness from 2MPa to 6.5MPa.
81
250,000
Nustar Neat 30MPa
Nustar Neat 6.5MPa
Number of Cycels to Failure
Nustar Neat 2MPa
200,000
Flint Hills Neat 6.5 MPa
Flint Hills Neat 2MPa
Flint Hills + SBSL 6.5MPa
150,000
Flint Hills Neat + SBSL 2MPa
Flint Hills + PE 30MPa
Flint Hills + PE 6.5MPa
100,000
Flint Hills + PE 2MPa
Flint Hills + SBSX 6.5MPa
Flint Hills + SBSX 2MPa*
50,000
Nustar SBSX 6.5MPa
Nustar + SBSX 2MPa
0
2%
3%
Initial, Undamaged Strain
*number of cycles predicted at 2%, using Equation (57) to be presented later.
Figure 47. Fatigue lives at undamaged strain amplitudes of 2% and 3%.
5.2.4
Separating Crack Initiation and Propagation
Distinguishing between crack initiation and propagation was accomplished through
investigation of the relationship between crack growth and energy release rates. Schapery’s (R.
Schapery 1990) work in studying fracture propagation of viscoelastic media was the basis for
using the relationship between crack growth rate and energy release rate to discriminate between
crack initiation and propagation. Schapery demonstrated that under Mode I loading, viscoelastic
media obey a power law relationship between crack growth and energy release rate (i.e.,
Equation (9)). Note that since Equation (9) was originally proposed for Mode I crack growth its
applicability to Mode III needed to be investigated for application to the DSR. If the edge
fracture propagation follows Schapery's model, Equation (9), for shallow crack growth (i.e.,
82
crack lengths up to 0.3 mm), the plot of crack growth rate versus energy release rate should
appear linear on a log-log scale and also should be independent of initial strain (deflection angle
amplitude of testing). Results at different initial strain amplitudes were compared to determine if
initial strain amplitude affects the relationship between crack growth and energy release rates.
Crack growth rate as a function of energy release rate for the six binders tested at all testing
temperatures and loading amplitudes, (specified in terms of initial, undamaged strain), are
plotted in Figure 48 through Figure 53.
0.001
da/dN (mm/cycle)
2% Rep.1
2% Rep.2
0.0001
2MPa
3% Rep. 1
3% Rep.2
0.00001
2% Rep. 1
0.000001
2% Rep. 2
3% Rep. 1
0.0000001
1
10
100
Gf (Pa-mm)
1,000
10,000
Figure 48. Flint Hills neat: Crack growth rate versus energy release.
6.5MPa
83
0.001
3% Rep. 1
3% Rep. 2
5% Rep. 1
5% Rep. 2
2% Rep.1
2% Rep.2
3% Rep.1
3% Rep.2
da/dN (mm/cycle)
0.0001
0.00001
0.000001
2 MPa
6.5 MPa
0.0000001
1
10
100
1,000
10,000
Gf (Pa-mm)
Figure 49. Flint Hills + SBSX: Crack growth rate versus energy release.
0.001
2% Rep. 1
2% Rep. 2
0.0001
3% Rep. 1
2 MPa
da/dN (mm/cycle)
3% Rep. 2
2% Rep. 1
0.00001
2% Rep. 2
3% Rep. 1
0.000001
3% Rep. 2
4% Rep. 1
4% Rep. 2
0.0000001
1
10
100
Gf
1,000
10,000
Figure 50. Flint Hills + SBSL: Crack growth rate versus energy release.
6.5 MPa
84
0.001
da/dN (mm/cycle)
0.0001
0.00001
0.000001
0.0000001
1
10
100
1,000
Gf (Pa-m)
10,000 100,000
2% Rep. 1
2% Rep. 2
3% Rep. 1
3% Rep. 2
2% Rep. 1
2% Rep. 2
3% Rep. 1
3% Rep. 2
4% Rep. 1
4% Rep. 2
2% Rep.1
2% Rep.2
2 MPa
6.5 MPa
30 MPa
Figure 51. Flint Hills + PE: Crack growth rate versus energy release.
2% Rep.1
0.001
2% Rep.2
3% Rep.1
0.0001
2MPa
da/dN (mm/cycle)
3% Rep.2
2% Rep. 1
0.00001
2% Rep. 2
3% Rep. 1
0.000001
3% Rep. 2
6.5MPa
3.5% Rep. 1
0.0000001
3.5% Rep. 2
1E-08
1
10
100
1,000
Gf (Pa-mm)
10,000
2% Rep. 1
100,000 2% Rep. 2
Figure 52. Nustar Neat: Crack growth rate versus energy release.
30MPa
85
0.001
2% Rep.1
2% Rep.2
0.0001
3% Rep.1
2MPa
da/dN (mm/cycle)
3% Rep.2
2% Rep.1
0.00001
2% Rep.2
3% Rep.1
0.000001
3% Rep.2
6.5MPa
4% Rep.1
4% Rep.2
0.0000001
1
10
100
Gf (Pa-mm)
1,000
10,000
Figure 53. Nustar + SBSX: Crack growth rate versus energy release.
Results in Figures 48 through 53 demonstrate that the relationship between crack growth
rate and energy release rate follows a power law relationship that is independent of the loading
amplitude of testing with the exception of data corresponding to low energy release/crack growth
rates, which correspond to very short crack lengths. For very small crack lengths, there is no
consistent trend in the relationship between energy release and crack growth rates. For all
binders tested, removing data corresponding to small crack lengths less than 0.05 mm leads to
agreement between Equation (9) and test data. Thus, crack propagation is assumed to begin at a
crack length of 0.05mm for the sample geometry of 2mm height and 4mm initial radius. Note
that 0.05 mm is considered an extremely small crack length as this crack size cannot be detected
using visual observations.
86
Also, Figures 48 through 53 demonstrate for a given crack growth rate, the energy release
rate increases with decreasing temperature (primarily as a result of a decrease in modulus with
increasing temperature). It is expected that there will be differences in the relationship between
crack growth rate and energy release rate at different temperatures due to viscoelastic nature of
asphalt. Hence, changes in temperature (or loading rate) are expected to affect crack growth rate.
The significance of crack initiation in the total fatigue life was also investigated. It was
found that the significance of crack initiation is highly dependent on temperature. A comparison
between the number of cycles of crack initation (Ni) and crack propagation (Np) on total fatigue
life (Nt) at the 2MPa and 6.5MPa iso-stiffness conditions for an undamaged strain amplitude of
3% are provided in Figures Figure 54Figure 55, respectively. These results demonstrate the
relative contribution of crack initiation increases as temperature increases (stiffness decreases).
At the 6.5MPa iso-stiffness condition, the number of cycles of crack initiation only accounts for
roughly 20% of the total fatigue life. However, the relative contribution of initiation at the 2MPa
iso-stiffness condition is significant and highly binder dependent. Based on these results, it is
important to consider crack initiation in characterization of fatigue damage in the DSR.
87
70,000
Nt
60,000
Np
50,000
Ni
40,000
30,000
20,000
10,000
0
Nustar Neat
FH Neat
Nustar+SBSX FH+SBSX
FH+SBSL
FH+CBE
propagation (Np) on total fatigue life (Nt) at 2MPa iso-stiffness condition.
60,000
Nt
50,000
Np
Ni
40,000
30,000
20,000
10,000
0
Nustar Neat
FH Neat
Nustar+SBSX FH+SBSX
FH+SBSL
FH+CBE
propagation (Np) on total fatigue life (Nt) at 6.5MPa iso-stiffness condition.
88
5.3
Model development
5.3.1
Crack Propagation Model
The results of fitting the power law crack propagation model (i.e., Equation (9)) to the
time sweep results are shown in Figures Figure 56 through Figure 61. Curve fitting was
conducted excluding crack initiation data (i.e., data corresponding to crack lengths less than
0.05mm). Curve fit parameters were determined using least squares regression. One curve was fit
to each temperature, corresponding to multiple loading amplitudes for each curve fit. The results
demonstrate very good agreement between the model predictions and measured crack growth
rates (R2 values ranging from 0.86 to 0.99).
0.001
2% Rep.1
da/dN (mm/cycle)
2% Rep.2
3% Rep.1
0.0001
2MPa
3% Rep.2
Fit
2% Rep. 1
0.00001
2% Rep. 2
3% Rep. 1
3% Rep. 2
3.5% Rep. 1
0.000001
1
10
100
1,000
Gf (Pa-mm)
10,000 100,000
3.50% Rep. 2
Fit
Figure 56. Flint Hills Neat: Crack propagation results.
6.5MPa
89
0.001
3% Rep. 1
3% Rep. 2
5% Rep. 1
5% Rep. 2
Fit
2% Rep.1
2% Rep.2
3% Rep.1
3% Rep.2
Fit
da/dN (mm/cycle)
0.0001
0.00001
0.000001
0.0000001
1
10
100
1,000
2 MPa
6.5 MPa
10,000 100,000
Gf (Pa-mm)
Figure 57. Flint Hills + SBSX: Crack propagation results.
0.001
2% Rep. 1
2% Rep. 2
3% Rep. 1
3% Rep. 2
4% Rep. 1
4% Rep. 2
Fit
2% Rep. 1
2% Rep. 2
3% Rep. 1
3% Rep. 2
Fit
da/dN (mm/cycle)
0.0001
0.00001
0.000001
0.0000001
1
10
100
Gf 1,000
10,000
100,000
Figure 58. Flint Hills + SBSL: Crack propagation results.
2 MPa
6.5 MPa
90
da/dN (mm/cycle)
0.001
0.0001
0.00001
0.000001
0.0000001
1
10
100
1,000
Gf (Pa-mm)
10,000
100,000
2% Rep. 1
2% Rep. 2
3% Rep. 1
3% Rep. 2
Fit
2% Rep. 1
2% Rep. 2
3% Rep. 1
3% Rep. 2
4% Rep. 1
4% Rep. 2
Fit
2% Rep.1
2% Rep.2
Fit
2MPa
6.5MPa
30MPa
Figure 59. Flint Hills + PE: Crack propagation results.
da/dN (mm/cycle)
0.001
2% Rep.1
2% Rep.2
3% Rep.1
3% Rep.
Fit
2% Rep. 1
2% Rep. 2
3% Rep. 1
3% Rep. 2
3.5% Rep. 1
3.5% Rep. 2
Fit
2% Rep. 1
2% Rep. 2
Fit
0.0001
0.00001
0.000001
1
10
100
1,000
Gf (Pa-mm)
10,000
100,000
Figure 60. Nustar Neat: Crack propagation results.
2MPa
6.5MPa
30MPa
91
0.001
2% Rep.1
2% Rep.2
3% Rep.1
3% Rep.2
Fit
2% Rep.1
2% Rep.2
3% Rep.1
3% Rep.2
4% Rep.1
4% Rep.2
Fit
da/dN (mm/cycle)
0.0001
0.00001
0.000001
2MPa
6.5MPa
0.0000001
1
10
100
1,000
Gf (Pa-mm)
10,000
100,000
Figure 61. Nustar + SBSX: Crack propagation results.
A summary of the k and  values based on curve fitting of Equation (9) to crack
propagation data are listed in Table 7 and shown in and FiguresFigure 62Figure 63. Results
indicate that temperature leads to a shift of the fatigue propagation curve (i.e., significant
changes in k parameter), but not a change in slope (i.e., only minor changes in the  parameter).
This is illustrated in Figure 62 Figure 63, which show model parameters k and α as a function of
binder type and temperature. It can be seen that α is relatively constant with temperature whereas
k decreases with increasing stiffness (decreasing temperature). This implies increasing or
decreasing temperature leads to a vertical shift in the crack propagation curve. Note that there is
a significant difference between α at the 30MPa and other iso-stiffness conditions for the PE
modified binder, which could imply a brittle transition, increasing rate of crack propagation or
partial adhesive failure between the plates and the binder specimen.
92
Table 7. Crack Propagation Curve Fit Results
Binder
Flint Hills Neat
Flint Hills + 2% SBS X
Flint Hills + 2% SBS L
Flint Hills + 2% PE
Nustar Neat
Nustar + 2% SBS X
|G*|·sinδ
6.5 MPa
2 MPa
6.5 MPa
2 MPa
6.5 MPa
2 MPa
6.5 MPa
2 MPa
30 MPa
6.5 MPa
2 MPa
30 Mpa
6.5 MPa
2 MPa
k
8.94E-10
3.99E-09
8.73E-11
3.044E-10
4.76E-10
2.87E-09
1.62E-10
8.91E-10
1.50E-13
2.76E-10
1.70E-09
1.18E-10
4.77E-10
2.11E-09
α
1.274
1.262
1.310
1.383
1.208
1.170
1.388
1.341
1.956
1.402
1.393
1.459
1.255
1.229
R2
0.99
0.99
0.86
0.94
0.92
0.98
0.94
0.83
0.92
0.99
0.94
0.99
0.95
0.93
4.5E-09
4.0E-09
2MPa
3.5E-09
6.5MPa
3.0E-09
30MPa
k
2.5E-09
2.0E-09
1.5E-09
1.0E-09
5.0E-10
0.0E+00
Flint Hills
Neat
Flint Hills +
2% SBS X
Flint Hills +
2% SBS L
Flint Hills +
2% PE
Nustar Neat
Nustar + 2%
SBS X
Figure 62. k as a function of temperature and binder type.
93
2.5
2MPa
2.0
6.5MPa
α
30MPa
1.5
1.0
0.5
0.0
Flint Hills Neat Flint Hills +
2% SBS X
Flint Hills +
2% SBS L
Flint Hills +
2% PE
Nustar Neat
Nustar + 2%
SBS X
Figure 63. α as a function of temperature and binder type.
Based on the results presented, it is proposed that fatigue life can be represented as the
summation of the number of cycles of crack initiation and number of cycles of crack propagation
to failure (i.e., Equation (44)). It is important to note that since it is very difficult to conduct a
true strain controlled fatigue test in the DSR, a fatigue model in terms of a constant strain
amplitude cannot be developed. However, a fatigue model can be developed in terms of the
initial, undamaged strain amplitude, γi. Number of cycles of crack propagation to failure can also
be represented in terms of initial strain amplitude. Energy release rate, (i.e., Equation (39)) can
be represented in terms of initial strain as follows:
(51)
Substituting Equation (51) into Schapery's fracture propagation model, Equation (9), and solving
for N yields:
94
(
)
∫
(
(
))
(
)
(52)
where Np is the number of cycles of crack propagation to failure, ai is crack length at end of
initiation (i.e., 0.05 mm) and af is the crack length at failure. It is recommended that failure be
defined as the transitions from shallow to deep crack growth as previously discussed, which has
been found to occur at a crack length of 0.3mm (i.e., crack length at peak energy release rate).
Note that typical asphalt binder and mixture fatigue failure criteria are based on an arbitrary
reduction in effective modulus. Hence, the proposed failure criterion is advantageous over
current practices as it is not arbitrary. Note that the integral portion of Equation (52) is
independent of undamaged strain amplitude. Assuming, |G*|, k, ri, and h are all independent of
loading amplitude, Equation (52) can be re-written as:
(53)
where
(
)
∫
(
(
))
(
)
(54)
(55)
Thus, the number of cycles to crack propagation follows a power law relationship with initial
strain amplitude, with exponent -2α.
95
5.3.2
Crack Initiation Model
A model for crack initiation could not be developed using the methodology used for
propagation because there is no consistent trend in the relationship between energy release and
crack growth rate during crack initiation. Thus, a simpler approach was taken for characterizing
the loading amplitude dependence of crack initiation. Based on the crack propagation model
presented in the previous section (Equation (53)), it was hypothesized that the number cycles of
crack initiation (Ni) and the initial, undamaged strain also follows a power law relationship with
the same exponent, -2·α from the crack propagation model (i.e., Equation (53)), as shown in
Equation (56).
(56)
Using an exponent of -2·α determined from crack propagation data, the coefficient β was
determined through least squares fitting to crack initiation data for the binders tested, using the
criterion that crack initiation ends at a crack length of 0.05 mm. A comparison of measured Ni
and predicted Ni using Equation (56) is provided in Figure 64. Results demonstrate good
agreement between model predictions and measured values (R2 = 0.95), indicating the
assumption that number of cycles of crack initiation and crack propagation follow the same
loading amplitude dependency is valid.
96
100,000
Predicted Ni
80,000
60,000
Equality Line
R2 = 0.95
40,000
20,000
0
0
20,000
40,000
60,000
Measured Ni
80,000
100,000
Figure 64. Comparison of measured and predicted Ni.
A summary of β coefficients for the crack initiation model and corresponding R2 values
are shown in Table 8. Results demonstrate reasonable agreement between measurements and
predictions using a power law exponent of -2·α. The parameter β as a function of temperature
and binder type is shown graphically in Figure 65. It can be seen that the parameter β increases
as stiffness decreases (temperature increases). A higher β indicates higher number of cycles to
crack initiation for a given α. Results indicate in some cases asphalt binder modification
significantly increases β indicating the modifier delays crack initiation. In other cases, binder
modification does not affect β substantially, indicating the benefit of modification is reflected in
the fatigue crack propagation performance.
97
Table 8. Crack Initiation Curve Fit Results.
Binder
Flint Hills Neat
Flint Hills + 2% PE
Nustar Neat
Nustar + 2% SBS X
|G*|·sinδ
6.5 MPa
2 MPa
6.5 MPa
2 MPa
6.5 MPa
2 MPa
6.5 MPa
2 MPa
30 MPa
6.5 MPa
2 MPa
30 Mpa
6.5 MPa
2 MPa
β
R2
0.500
1.046
0.600
1.113
4.049
5.449
0.293
0.672
0.0001
0.129
0.405
0.124
2.229
4.917
0.961
0.983
0.856
0.899
0.590
0.932
0.801
0.990
n/a
0.962
0.957
n/a
0.879
0.918
6.0
5.0
β
4.0
2MPa
3.0
6.5MPa
2.0
30MPa
1.0
0.0
Flint Hills
Neat
Flint Hills + Flint Hills + Flint Hills + Nustar Neat Nustar + 2%
2% SBS X 2% SBS L
2% PE
SBS X
Figure 65. Effect of binder type and temperature on β.
98
5.3.3
Comprehensive Fatigue Model
Combining Equation (56) and (52) with input of initial sample dimensions, c value of 0.1,
and values of ai and af yields the following relationship between total number of cycles to failure
and undamaged strain amplitude:
(57)
where
∫
(
)
(58)
and B is defined as in Equation (55).
All coefficients needed to calculate A and B in Equation (57) can theoretically be derived
from a single test. This is possible because parameters α and k are determined from the
relationship between crack propagation and energy release rates, which is independent of initial
strain. Additionally, with α known and the number of cycles until completion of crack initiation,
Equation (56) can be used to determine β. The proposed analysis framework is advantageous
over current VECD framework because no parameters need to be assumed a priori as with the
parameter α in Equation (8), which is estimated from a frequency sweep in the linear viscoelastic range.
There is no closed form solution to the integral in Equation (58). Thus, it must be
evaluated numerically. In this study, Simpson's rule was used for numerical integration.
Simpson's rule uses a piecewise second order polynomial to approximate an integral (Atkinson
1989). Simpsons rule applied to a function, f(x) over the interval (a,b), by dividing the interval
into n pieces at an interval of Δx can be represented as follows:
99
∫
[
]
(59)
Equation (57) was validated using the results of the six binders tested where model
parameters were determined using all data (i.e., at all deflection amplitudes). This comparison
was conducted to validate the proposed model and to verify that k and α are independent of
loading amplitude. Model predictions of fatigue lives (Nf) were compared with direct
measurements. Results are provided in Figure 66 and Table 9 through 11. Results indicate good
agreement between measured and predicted values (R2 = 0.98). Thus, it can be concluded that
Equation (57) is a sufficient model and that fatigue life can be predicted at any loading amplitude
from a single test.
250,000
Predicted Nf
200,000
150,000
Equality Line
R2 = 0.98
100,000
30MPa
6.5MPa
2MPa
Equality Line
50,000
0
0
50,000
100,000
150,000
Measured Nf
200,000
250,000
Figure 66. Comparison between measured and predicted fatigue lives (Nf
100
Condition
Binder
Nustar Neat
Flint Hills Neat
Nustar + 2% SBS X
Flint Hills + 2% PE
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
Initial Strain
2%
3%
5%
49,500
2,546
47,472
46,950
212
46,141
125,250
15,910
119,810
96,450
1,061
88,170
75,000
0
80,781
16,650
636
15,336
15,300
0
16,583
34,050
2,333
44,239
64,650
8,273
62,451
30,000
2,546
34,140
22,800
1,697
27,231
15,450
1,061
15,196
101
Table 10. Comparison between Measured and Predicted Fatigue Lives at 6.5MPa Isostiffness Condition
Binder
Nustar Neat
Flint Hills Neat
Nustar + 2% SBS X
Flint Hills + 2% PE
2%
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
Average
Standard Deviation
Predicted
41,250
636
40,718
40,500
424
40,999
79,650
2,758
78,579
189,900
27,789
167,898
101,850
212
97,113
78,300
1,061
82,675
Initial Strain
3%
3.5%
15,450
636
13,058
13,800
352
14,595
28,500
424
28,394
49,950
3,606
36,471
33,450
212
36,471
26,700
1,018
26,820
4%
10,350
212
8,474
10,800
600
9,855
12,750
212
13,790
14,700
849
18,204
17,100
212
12,066
Table 11. Comparison between Measured and Predicted Fatigue Lives at 30MPa Isostiffness Condition
Binder
Average
Standard Deviation
Predicted
Nustar Neat
Average
Standard Deviation
Flint Hills + 2% PE Predicted
Initial Strain
2%
23,200
400
23,153
34,350
3,180
43,273
5%
102
Table 12 provides a summary of the fatigue law parameters A and B in Equation (57).
Additionally, a graph depicting trends in fatigue law parameter A with temperature and binder
type is provided in Figure 67. As previously discussed, α is more or less constant with
temperature. Thus, fatigue law parameter B, equal to -2·α, is also relatively insensitive to
temperature. However, fatigue law parameter A is temperature dependent. As illustrated in
Figure 67, with the exception of the Flint Hills + SBSX binder, the parameter A increases with
increasing temperature (decreasing stiffness). Thus, given that B is relatively constant with
temperature, an increasing A with increasing temperature implies increasing temperature has a
positive effect on fatigue performance, which is a logical trend. The Flint Hills + SBSX binder
exhibited extreme fatigue performance, which is possibly causing the trends observed. It is
speculated that even for this binder, A values would show a decreasing trend with temperature if
temperature was decreased significantly.
Table 12. Summary of Fatigue Law Parameters A and B
Nustar Neat
Flint Hills Neat
Nustar + 2% SBS X
Flint Hills + 2% PE
30 MPa
6.5 MPa
2 MPa
6.5 MPa
2 MPa
6.5 MPa
2 MPa
6.5 MPa
2 MPa
6.5 MPa
2 MPa
30 MPa
6.5 MPa
2 MPa
A
0.155
0.699
0.875
1.927
2.377
4.266
8.013
5.524
3.820
7.648
9.326
0.001
1.585
2.244
B
-2.918
-2.805
-2.787
-2.547
-2.524
-2.511
-2.457
-2.639
-2.767
-2.415
-2.340
-4.559
-2.777
-2.682
103
2MPa
A
10.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
6.5MPa
30MPa
Flint Hills
Neat
Flint Hills + Flint Hills + Flint Hills + Nustar Neat Nustar + 2%
2% SBS X 2% SBS L
2% PE
SBS X
Figure 67. Effect of temperature and binder type on fatigue law parameter A.
5.3.4
Application of Time-Temperature Superposition to Fatigue Crack Propagation
The applicability of time-temperature superposition to fatigue crack propagation in
asphalt binders is evaluated in this section. In order to apply time-temperature superposition to
fatigue crack propagation in asphalt binders, the concept of a “reduced crack growth rate” is
introduced, analogous to the concept of reduced time and frequency (Seo 2003). A reduced crack
growth rate can be derived as follows:
(60)
where ξ is the reduced time and aT is the time-temperature shift factor defined in Equation (45).
Since,
(61)
104
The reduced crack growth rate can be written as:
(62)
where Nξ is the reduced number of loading cycles. Hence, in the case of fatigue crack
propagation, a master curve is constructed by shifting the da/dN versus Gf curve vertically on a
log da/dN scale as illustrated in Figure 68 where T1, Tref, T2 are temperatures where T1 > Tref
> T2.
0.001
da/dN (mm/cycle)
0.0001
0.00001
0.000001
T1
Tref
T2
Master Curve
0.0000001
1
10
100
1,000
Gf (Pa-mm)
10,000
100,000
Figure 68. Crack propagation master curve.
The time-temperature shift factors determined from the frequency sweep (i.e., linear
viscoelastic) data were applied to shift fatigue crack propagation curves (da/dN versus Gf) to a
reference temperature, selected to be the temperature where the 6.5MPa iso-stiffness condition is
105
met. Results are presented in Figure 69 through Figure 74 for the six binders included in this
study. The “master curves” correspond to the fatigue crack propagation model (i.e., Equation (9))
curve fit parameters derived at the 6.5MPa iso-stiffness condition. Results demonstrate timetemperature superposition applies to fatigue crack growth in asphalt binders. There is generally
good agreement between the curve fit derived from data at the reference temperature and the
shifted crack propagation data corresponding to results at alternative temperatures. There is poor
agreement between model predictions and fatigue crack propagation data at the 30MPa isostiffness condition for the PE modified binder (Figure 72). As previously discussed, the results at
the 30MPa iso-stiffness condition for the PE modified binder demonstrates unique behavior,
which is thought to potentially be related to either a brittle transition or a partial loss of adhesion
between DSR plates and the binder specimen.
Reduced da/dN (mm/cycle)
0.001
0.0001
R2 = 0.99
0.00001
32.3C
23.5C
0.000001
Mater Curve
Reference Temperature = 23.5C
0.0000001
1
10
100
Gf (Pa-mm)
1,000
10,000
Figure 69. Flint Hills Neat: Fatigue crack propagation master curve.
106
0.001
0.0001
R2 = 0.93
0.00001
31.2C
21.8C
Master Curve
Reference Temperature = 21.8C
0.000001
0.0000001
1
10
100
1,000
10,000
Gf (Pa-mm)
Figure 70. Flint Hills + SBSX: Fatigue crack propagation master curve.
0.001
0.0001
R2 = 0.93
0.00001
23C
32.8C
Master Curve
0.000001
Reference Temperature = 23C
0.0000001
1
10
100
1,000
Gf (Pa-mm)
10,000
Figure 71. Flint Hills + SBSL: Fatigue crack propagation master curve.
100,000
107
0.001
0.0001
R2 = 0.55
32.8C
0.00001
23.5C
11.6C
0.000001
Master Curve
Reference Temperature = 23.5⁰C
0.0000001
1
10
100
1,000
Gf (Pa-mm)
10,000
100,000
Figure 72. Flint Hills + PE: Fatigue crack propagation master curve.
0.001
R2 = 0.93
0.0001
0.00001
28C
20C
13C
Master Curve
0.000001
Reference Temperature = 20⁰C
0.0000001
1
10
100
1,000
Gf (Pa-mm)
10,000
100,000
Figure 73. Nustar Neat: Fatigue crack propagation master curve.
108
0.001
0.0001
R2 = 0.95
0.00001
23C
32.5C
0.000001
Master Curve
Reference Temperature = 23C
0.0000001
1
10
100
1,000
10,000
100,000
Gf (Pa-mm)
Figure 74. Nustar + SBSX: Fatigue crack propagation master curve.
Based on results presented, it is apparent that the parameter α in Equation (9) can be
assumed to be independent of testing temperature. Furthermore, results suggest the parameter k
can be determined at any temperature using fatigue test results at a single testing temperature and
time-temperature shift factors determined from linear viscoelastic characterization. Following
this logic, k can be calculated as follows:
(63)
where kT is the model parameter at the temperature of interest, kTref is the value of k at the
reference temperature and aT is the time-temperature shift factor. To validate the applicability of
time-temperature superposition to the fatigue crack propagation model parameter k, the k values
determined directly at the 2MPa and 30MPa iso-stiffness testing conditions were compared to
109
those predicted using Equation (63) with input of the k values determined at the 6.5MPa isostiffness coupled with time-temperature shift factors. The comparison between measured and
predicted k values is shown in Figure 75. Good agreement (R2 = 0.91) is found between
predicted and measured values, indicating one can use results of a single fatigue test along with
time temperature shift factors to determine the fatigue crack propagation behavior at any
temperature and loading amplitude.
5E-09
Predicted k
4E-09
3E-09
Equality Line
R2 = 0.91
2E-09
1E-09
0
0
1E-09
2E-09
3E-09
Measured k
4E-09
5E-09
Figure 75. Comparison between measured k value and predicted k values using timetemperature shift factors.
Crack initiation is complex and does not represent a steady state of crack growth
(Hertzberg 1989). Thus, time-temperature superposition is not applicable to fatigue crack
initiation of asphalt binders. As shown in Figures 48 through 53, crack initiation does not follow
any defined trend with respect to energy release rate and hence, it is not possible to apply timetemperature superposition to attain an overall fatigue master curve. This is demonstrated in
110
Figure 76, which shows the fatigue master curve for the Flints Hills + PE binder. Note similar
trends were observed for the other binders evaluated in this study.
0.001
2% Rep. 1
2% Rep. 2
0.0001
3% Rep. 1
3% Rep. 2
2% Rep. 1
0.00001
2% Rep. 2
3% Rep. 1
3% Rep. 2
0.000001
4% Rep. 1
4% Rep. 2
0.0000001
1
10
100
1,000
Gf (Pa-mm)
10,000
100,000
2% Rep.1
2% Rep.2
Figure 76. Fatigue crack growth master curve including crack initiation for Flint Hills +
PE.
While time-temperature superposition is not applicable to fatigue crack initiation, one can
still efficiently predict the fatigue law parameters in Equation (57) at multiple temperatures. A
fatigue test at a single temperature must be conducted to failure and short fatigue tests must be
conducted at the temperature(s) of interest to estimate initiation trends. These short fatigue tests
require loading until the start of crack propagation in order to derive the parameter β in Equation
(56), used for characterizing the number of cycles of crack initiation. The parameters k and α,
describing crack propagation can be determined at any temperature using fatigue test data at a
single temperature along with time-temperature shift factors as previously discussed.
111
5.3.5
Summary
In summary, an analysis framework was developed based on fracture mechanics concepts
that allows for prediction of fatigue life at any loading amplitude from the results of a single time
sweep test. The proposed analysis framework takes into account both crack initiation and
propagation. Additionally, it was demonstrated that time-temperature superposition is applicable
to asphalt binder fatigue crack propagation, allowing for efficient prediction of fatigue life at
multiple temperatures. Time-temperature superposition could not be applied to crack initiation
and thus, fatigue life prediction at multiple temperatures requires short fatigue tests at all
temperatures of interest to estimate the initiation stage.
5.4
Model Validation
The model presented in section 5.3 was validated by comparing the binder fatigue
performance under time sweep testing with laboratory mixture fatigue performance. The mixture
data used in the validation effort was collected by the University of Massachusetts – Dartmouth.
The asphalt mixture specimens consisted of a 9.5mm Superpave coarse gradation mixed with
various binders as specified in Table 3. Mixture samples were mixed and compacted using the
Superpave Gyratory Compactor according to AASHTO TP 62 (AASHTO 2007). After
compaction, the mixture samples were cored to produce cylindrical specimens 150mm tall with
100mm diameter.
Asphalt mixture fatigue testing consisted of uniaxial, fully reversed (tension and
compression) sinusoidal loading at frequency of 10Hz using a constant displacement amplitude.
The testing temperature for both asphalt binders and mixtures was 20⁰C. Asphalt mixture fatigue
data was analyzed using the VECD analysis framework proposed by Kutay et al. (Kutay, Gibson
and Youtcheff 2008). Asphalt binders were tested using the time sweep. The undamaged strain
112
amplitudes of testing were selected based on a target failure time of one hour. Note that the
loading amplitude of testing is insignificant because the proposed analysis framework allows for
prediction of fatigue life at any loading amplitude from a single fatigue test. The data was
analyzed using the proposed analysis framework in order to derive the fatigue law parameters A
and B in Equation (57). The fatigue law parameters were used to estimate the fatigue life of the
individual binders at the same loading amplitude to allow for ranking the relative fatigue
performance of the binders. The loading amplitude selected for comparison with mixture
performance was an initial, undamaged strain of 3%. Relative rankings of mixture fatigue
performance were based on the results presented by Mogawer et al. (Mogawer, et al. 2011). The
fatigue crack propagation curves (da/dN versus Gf) are provided in Figure 77. It can be seen that
these binders exhibit very different trends in the relationship between da/dN and Gf, both with
respect to the slope and intercept. This is reflected in the fatigue law parameters, which are
presented in Table 10. The parameters A and B vary considerably among binder types. Table 10
also includes the comparison between the ranking of binder fatigue performance, based on an
undamaged strain amplitude of 3%, and mixture fatigue performance. The rankings of binder and
mixture performance are in agreement. (Note that the PG 64-28 and the PG 64-28 + 2% Latex
binders have very similar fatigue lives at 3% undamaged strain). Thus, these results indicate the
time sweep test coupled with the proposed analysis framework can adequately quantify the
relative performance of asphalt binders within an asphalt mixture.
113
0.001
64-28 Rep.1
64-28 Rep. 2
0.0001
da/dN (mm/cycle)
64-28+2% Latex Rep.1
64-28+2% Latex Rep.2
0.00001
64-28+PPA Rep.1
64-28+PPA Rep. 2
64-34 Rep.1
0.000001
64-34 Rep.2
76-22 Rep.1
76-22 Rep.2
0.0000001
100
1000
10000
Gf (Pa-mm)
100000
Figure 77. Mixture validation fatigue crack propagation curves.
Table 13. Mixture Validation Results
Asphalt Binder
PG 64-28
PG 64-28 +
Polyphosphoric Acid
(PPA)
PG 64-34
PG 76-22
A
6.14E-05
B
-5.811
Nf
(γi = 3%)
43,411
Mixture
Ranking
D
Binder
Ranking
D
1.51E-01
8.70E+01
2.21E+02
-3.536
-2.430
-1.603
36,593
436,246
60,877
E
A
B
E
A
B
PG 64-28 + 2% Latex
Rubber
1.54E+01
-2.265
43,456
C
C
114
5.5
Application of Fracture Analysis to Linear Amplitude Sweep (LAS) Test
5.5.1
Evaluation of the LAS Test as a Fatigue Test
The linear amplitude sweep (LAS) test (Johnson 2010) is currently being proposed for
fatigue specification of asphalt binders. It is an accelerated test that uses systematically
increasing loading amplitudes to increase the rate of damage. The LAS test was introduced as a
practical surrogate to the time sweep due to the difficulty of estimating the time required for the
time sweep test to reach fracture stage. A schematic depicting the loading sequence is provided
in Figure 2. Note that the test is specified in terms of strain amplitudes, assuming the sample
geometry is constant. Hence, actual strain amplitudes differ from these specified amplitudes as
the sample becomes damaged since the DSR controls deflection angle, not strain. The test is
conducted in the DSR using the same testing geometry as the time sweep test.
VECD has been proposed for LAS data analysis (Johnson 2010). However, the findings of
this study suggest the LAS test results should be analyzed using a fracture based model since it is
known that macro fracture occurs during testing. This was validated through observation of
samples following LAS testing. Figure 78 shows a binder sample after LAS testing. Fracture
morphology shows a smooth outer edge with radially lines towards the crack tip, consistent with
fracture morphology observed under true (constant amplitude) fatigue testing. The applicability
of the analysis framework developed for the time sweep test to the LAS test is investigated in
this section.
115
Figure 78. Sample after LAS test.
In the LAS test, both torque and deflection angle amplitude change with number of loading
cycles. In the previous analysis of the time sweep, only torque amplitude was changing with
number of loading cycles as the deflection angle amplitude was fixed. Thus, in utilizing Equation
(50) to predict changes in sample radius, which relates sample radius to torque, deflection angle,
|G*|, and sample height, the relative change in the torsional stiffness, equal to torque / deflection
angle amplitude was used to predict the relative change in effective sample radius. This analysis
assumes all changes in torsional stiffness are related to crack growth and neglects any effect of
nonlinearity. The basis for this assumption is that the accelerated loading procedure induces
rapid crack growth which makes nonlinearity effects small in comparison to crack growth.
Energy release rates were calculated for the LAS test using the same equation as utilized for the
time sweep, (Equation (39)).
The trends in crack growth rate and energy release rate were evaluated using the current
LAS procedure, which consists of incremental increases loading amplitudes every 100 loading
cycles. Similar trends were observed for all binders. Figure 79 depicts the evolution of crack
growth and energy release for the Flint Hills neat binder. Figure 79 demonstrates the crack
growth and energy release rates follow very different trends than what was observed for the time
116
sweep. It can be seen that within each strain step, the crack growth rate decreases. Under time
sweep loading of asphalt binders, initially the crack growth rate increases (corresponding to
shallow crack growth) and after a critical crack length is reached the crack growth rate decreases
(corresponding to deep crack growth). Between each strain step, this general trend is observed.
However, the decreasing rate of crack growth within each strain step is not indicative of fatigue
crack propagation. The decrease in crack growth rate within each strain step could be indicative
of “crack tip conditioning”, which is observed in notched specimens (Keller, White and Sottos
2008). Crack tip conditioning is the period in which a crack in front of a notch reaches steady
state crack growth following the start of load application. In the LAS test, each time the loading
amplitude is increased a response indicative of crack tip conditioning ensues, indicating
increasing the loading amplitude is analogous to initial loading of a notched specimen. Based on
these results, the LAS test is considered a very complex procedure that cannot be analyzed with
the analysis framework developed for the time sweep in this study.
Gf
da/dN
Gf (Pa-mm)
2.00E+02
0.00012
0.0001
0.00008
1.50E+02
0.00006
1.00E+02
0.00004
5.00E+01
0.00E+00
0.00
da/dN (mm/cycle)
2.50E+02
0.00002
0.50
1.00
1.50
a (mm)
2.00
2.50
0
3.00
Figure 79. Trends in crack growth rate and energy release rate using current LAS test.
117
5.5.2
Modification of the LAS Procedure
While the current LAS test is not a simple fatigue test and is problematic due to the
relatively large and abrupt changes in loading amplitude, it is successful at producing damage at
an accelerated rate and could be useful as an index test. Such an index test can be considered a
“damage tolerance test” that uses accelerated loading to measure the relative damage resistance
of asphalt binders However, first there is a need to modify the procedure to eliminate the effects
of crack tip conditioning when increasing loading amplitudes. To eliminate the effect of the
crack tip conditioning, it is recommended that instead of using step-wise increases in loading
amplitude that the loading amplitude be increased a very small amount in each successive
loading cycle. This revised loading schematic, specified in terms of deflection angle amplitude is
presented in Figure 80. Note that the revised loading consists of the same total number of loading
cycles (i.e., testing time) and the same range of loading amplitudes as the current LAS test. Also,
it is noteworthy to mention that DSRs are not designed for significant, abrupt changes loading
amplitude. Thus, the proposed modified procedure will be more compliant with the capabilities
of typical rheometers whereas the current procedure can be problematic when using non-research
grade rheometers.
118
0.16
0.14
ϕ
0.12
0.1
0.08
0.06
0.04
0.02
0
0
500
1000
1500
2000
2500
3000
3500
Figure 80. Modified LAS loading schematic.
The trends in energy release and crack growth rates under the modified loading scheme
are depicted in Figure 81. The results presented in Figure 81 correspond to the Nustar Neat
binder. Similar trends were observed for the other binders tested. It can be seen that using the
revised loading with increasing loading amplitudes in very small increments leads to much
smoother crack growth rate and energy release rate curves. Also, note that the energy release and
crack growth rates peak at roughly the same point, indicating the energy release rate equation
used for the time sweep is also applicable to the amplitude sweep.
119
0.00014
Gf
1.40E+05
da/dN
Gf (Pa-mm)
1.20E+05
0.00012
0.0001
1.00E+05
0.00008
8.00E+04
0.00006
6.00E+04
0.00004
4.00E+04
0.00002
2.00E+04
0.00E+00
0.00
da/dN (mm/cycle)
1.60E+05
1.00
2.00
a (mm)
3.00
0
4.00
Figure 81. Trends in crack growth rate and energy release rate using modified LAS test.
Figure 82 shows crack growth rate as a function of crack length for all binders tested (at
the 6.5MPa iso-stiffness condition) under the modified LAS procedure. It can be seen that crack
growth rate curves follow somewhat similar trends as the curves observed for time sweep tests
(see Figure 45) with the exception of the initial crack growth where there is small local peak
followed by a very rapid increase in crack growth rate to the absolute peak.
120
1.40E-04
da/dN (mm/cycle)
1.20E-04
FH Neat
FH + SBSX
FH + SBSL
FH + PE
Nustar
Nustar + SBSX
1.00E-04
8.00E-05
6.00E-05
4.00E-05
2.00E-05
0.00E+00
0.00
1.00
2.00
a (mm)
3.00
4.00
Figure 82. Crack growth rate as a function of crack length under modified LAS testing.
The relationship between energy release rate and crack growth rate, prior to the peak
crack growth rate, under the modified LAS testing procedure for the asphalt binders tested is
shown in Figure 83. It can be seen that the curves do not match the trends seen in the time sweep
test. Hence, the fatigue analysis framework proposed for the time sweep test is not applicable to
the modified LAS. It can be seen in Figure 83 that initially crack growth rate increases rapidly
and then becomes fairly constant. Then, at a critical energy release rate that varies for the
individual binders, crack growth rate increases very rapidly. The crack length and deflection
angle amplitude at which crack growth becomes rapid varies among binders, which explains the
variation in energy release rates where the crack growth rate among binders.
121
0.00012
0.0001
da/dN
0.00008
0.00006
0.00004
0.00002
FH Neat
FH + SBSX
FH + SBSL
FH + PE
Nustar Neat
Nustar + SBS
0
0.00E+00 5.00E+04 1.00E+05 1.50E+05 2.00E+05 2.50E+05 3.00E+05
Gf (Pa-mm)
Figure 83. Crack growth rate versus energy release rate under modified LAS testing.
A practical analysis method to rank the relative damage tolerance of asphalt binders using
the modified LAS test can be proposed for ranking of binders. The failure criterion proposed for
analyzing the modified LAS is based on the local minimum in crack growth rate prior to the
rapid increase in crack growth rate (illustrated in Figure 84). This is a logical failure criterion as
it indicates the start of un-steady, rapid failure. Additionally, it coincides roughly with the start of
a rapid decrease in loading resistance (i.e., torque). This is illustrated in Figure 85 and Figure 86,
which show a comparison of trends in crack growth rate and torque with deflection angle
amplitude for the Flint Hills neat and Flint Hills + SBSX binders. (Note that similar trends were
observed for the other binders tested). It can be seen that initially there is an increase in torque
(i.e., loading resistance) with increasing deflection angle amplitude. During this period, crack
growth is steady. However, once the torque begins to decrease, the crack growth rate increases
122
rapidly. Also noteworthy, Figure 85 and Figure 86 demonstrate the deflection angle at the peak
torque for both the neat and SBS modified binders are very similar. However, the breadth of the
peak in torque is much greater for the modified asphalt, clearly showing the benefit of
modification on damage tolerance.
1.40E-04
1.20E-04
da/dN (mm/cycle)
1.00E-04
8.00E-05
6.00E-05
4.00E-05
2.00E-05
0.00E+00
0.00
af
1.00
2.00
a (mm)
3.00
4.00
Figure 84. Illustration of crack length at failure, af .
Torque (mNm)
50
40
Torque
da/dN
30
20
10
0
0
0.05
ϕ
0.1
0.0001
0.00009
0.00008
0.00007
0.00006
0.00005
0.00004
0.00003
0.00002
0.00001
0
0.15
Figure 85. Trends in torque and crack growth rate for Flint Hills Neat binder.
da/dN (mm/cycle)
60
123
60
0.000045
0.00004
0.000035
40
0.00003
da/dN (mm/cycle)
Torque (mNm)
50
0.000025
30
0.00002
20
0.000015
torque
10
da/dN
0
0
0.05
ϕ
0.1
0.00001
0.000005
0
0.15
Figure 86. Trends in torque and crack growth rate for Flint Hills + SBSX binder.
Two parameters at the failure point were evaluated for ranking the relative damage
tolerance of asphalt binders. The first parameter is the crack length at failure (af) and the second
is the “effective” strain at failure (γf). Effective strain is the strain calculated assuming initial
geometry and the corresponding deflection angle amplitude. Two considerations were made in
evaluating the parameters: sensitivity to binder type and if ranking of materials was equivalent to
the time sweep test (TS). Figure 87 and Figure 88 display a comparison between the number of
cycles to failure (Nf) at 2% undamaged, initial strain from the TS and the LAS parameters
evaluated: crack length at failure and effective strain at failure, respectively. The comparison
between crack length at failure from the modified LAS test and the number of cycles to failure
from the TS reveals generally good agreement between results, indicating crack length at failure
provides a good indication of damage resistance. However, the effective strain at failure from the
LAS test and the number of cycles to failure from the time sweep test exhibits some discrepancy
in ranking materials. Hence, it is proposed that the crack length at failure be selected as the
124
parameter to rank the relative damage tolerance of asphalt binders using the modified LAS test.
A higher crack length at failure is desirable as this indicates the binder can withstand a greater
crack length prior to rapid crack propagation.
200,000
1.2
Nf (TS)
180,000
1.1
af (LAS)
160,000
1
140,000
Nf
0.8
100,000
0.7
80,000
0.6
60,000
40,000
0.5
20,000
0.4
0
af (mm)
0.9
120,000
FH Neat
Nustar Neat
FH + PE
Nustar + SBS FH + SBSL
FH + SBSX
0.3
Figure 87. Evaluation of crack length at failure as failure parameter.
180,000
160,000
18
Nf (TS)
Failure strain
16
140,000
14
Nf
120,000
100,000
12
80,000
10
60,000
40,000
8
20,000
0
FH Neat
Nustar Neat
FH + PE
Nustar + SBS
FH + SBSL
FH + SBSX
Figure 88. Evaluation of effective strain at failure as failure parameter.
6
γf (%)
200,000
125
5.5.3
Summary
In summary, the current approach of analyzing LAS test using a VECD framework is not
optimum because continuum damage mechanics assumes damage accumulation is on a microscale (e.g., micro crack and micro void formation) whereas there is macro-crack propagation in
cyclic DSR testing. The current LAS procedure is problematic because it contains relatively
large increments in loading amplitude, which causes crack tip condition upon changing each
strain step. Additionally, the procedure is difficult to conduct in some rheometers. Therefore, it is
recommended that the LAS procedure be changed from a step amplitude sweep to a continuous
amplitude sweep in which the deflection angle amplitude is increased a very small amount every
cycle, which eliminates the problem of crack tip conditioning associated with abrupt changes in
loading amplitude. The proposed modified procedure utilizes the same range of loading
amplitudes as the current LAS and takes the same amount of time. It is also recommended that
the LAS test be considered a “damage tolerance” test rather than a fatigue test because there are
different trends in the crack propagation of binders under the true fatigue test (time sweep) and
the accelerated procedure (modified LAS). The crack length at failure described previously is
suggested as the parameter to rank the relative damage tolerance of asphalt binders. This
parameter was found to be sensitive to binder type and provides a similar ranking of materials as
the time sweep test. However, for true fatigue characterization it is recommended that a time
sweep test is conducted and analyzed using the rigorous fracture based analysis framework
presented in section 5.3.
126
6
Conclusions and Recommendations
6.1
Conclusions
The following points summarize the major findings of this study:

It is demonstrated in this study that fracture, leading to a reduction in effective sample
size, is the main cause of fatigue damage during time sweep testing of asphalt binders in
the dynamic shear rheometer (DSR). Visual observations of fracture morphology and
trends in fatigue crack growth rate of asphalt binders show similar trends to other
materials under similar loading conditions, validating the damage mechanism. Based on
these findings, the time sweep test is recommended for fatigue characterization of asphalt
binders.

An analysis framework based on fracture mechanics concepts was developed for fatigue
characterization of asphalt binders using the time sweep test. The model includes two
components representing crack initiation and propagation. The framework developed
allows for predicting fatigue life at any loading amplitude from a single fatigue test
without any assumption of model parameters.

Fatigue life of asphalt binders in this study is defined as the number of cycles at which
there is a transition from shallow to deep crack growth, corresponding to the peak in
crack growth rate. Although the definition is phenomenological, it is considered easier to
quantify and a better approach than arbitrarily selecting a percent reduction in effective
stiffness as fatigue failure.
127

Time-temperature shift factors calculated from linear viscoelasticity were successfully
applied to fatigue crack propagation within the range of temperatures evaluated in this
study (i.e., 10⁰C to 30⁰C). This allows for efficient fatigue characterization at multiple
temperatures. Time-temperature superposition, however, could not be applied to crack
initiation. Thus, fatigue testing should be conducted at all target temperatures for
estimation of crack initiation behavior. Future research is needed to evaluate the
applicability of time-temperature superposition to temperatures outside of the range
evaluated in this study.

The proposed analysis framework was validated using a comparison between asphalt
mixture and binder fatigue lives. Asphalt mixture fatigue lives were determined using
viscoelastic continuum damage (VECD) analysis of uniaxial push-pull fatigue. Asphalt
binder fatigue lives were estimated using the time sweep test and fracture based analysis
framework developed in this study. The ranking of asphalt mixtures and binders were
consistent, validating the use of the time sweep test and accompanying analysis
framework for fatigue characterization of asphalt binders.

Based on the findings of this study, several modifications to the Linear Amplitude Sweep
(LAS) test are proposed. Results demonstrate that while the LAS test induces damage to
asphalt binders, the proposed fatigue fracture mechanics based analysis proposed for the
time sweep could not be applied to LAS test results. Thus, it is recommended that the
LAS test be referred to as a damage tolerance test. Second, it is recommended that the
loading sequence of the LAS test be modified to include small increments in loading
128
amplitude every cycle (i.e., continuous rather step-wise amplitude sweep). This resolves
issues regarding compliance with DSR capabilities and eliminates crack tip conditioning
which occurs each time a load is incremented abruptly. A simple means to rank the
relative damage tolerance of asphalt binders using trends in crack growth rate is
presented. Based on the binders evaluated in this study, the simple analysis of the
modified LAS test provides similar ranking asphalt binders as the rigorous fracture
mechanics based analysis of time sweep, indicating the LAS test can provide an
indication of fatigue resistance of binders.
6.2
Recommendations
The results of this study provide a significant contribution to the understanding and
characterization of asphalt binder fatigue. However, there are areas that merit further
investigation. The following provides recommendation for future research based on the results of
this study.

Asphalt pavements have the ability to self-heal during rest periods in traffic and hence,
reverse damage due to fatigue (Kim, Little and Lytton 2003). The intrinsic self-healing
properties of asphalt binder are responsible for damage reversal in pavements. The
proposed testing and analysis method does not allow for consideration of healing.
However, the discovery that fracture is the source of damage in asphalt binders during
fatigue loading in the DSR allows for a better understanding of healing observed when
rest is given to a binder sample during DSR testing. Thus, future research should aim at
better understanding the healing mechanism of crack closure in asphalt binders and
should incorporating consideration of healing into a fatigue model.
129

In this study, it is demonstrated that asphalt binders experience three distinct stages in
crack growth during time sweep testing in the DSR: crack initiation, shallow crack
propagation, and deep crack propagation. Development of a rigorous analysis framework
to describe shallow crack propagation was the primary focus in this study. Future
research is needed to better describe and model crack initiation and deep crack growth for
comprehensive fatigue crack growth characterization.

This study focused on Mode III cohesive fatigue fracture of asphalt binders, which was
motivated by the wide spread availability and use of Dynamic Shear Rheometers (DSR)
and its demonstrated ability to be able discriminate between the relative fatigue resistance
of asphalt binders. Presently, test methods for evaluating asphalt binder fatigue fracture
under other modes of loading are relatively limited. Future research is needed to evaluate
asphalt binders under different fracture modes (i.e., tensile and shear) and to assess
adhesive fatigue failure to better understand and characterize fatigue in asphalt
pavements. Additionally, research is needed to obtain a better understanding of the
distribution of stresses and strains in the binder phase of asphalt mixtures with respect to
magnitude, loading rate, and type of load (i.e., compressive, tensile, shear). The
recommended research will allow for improved simulation of field conditions during
testing and analysis/modeling.

This study focused on developing a better understanding and means for characterizing
asphalt binder fatigue. However, asphalt pavements are comprised of a composite
composed of asphalt binder, aggregates of different size scale, and air voids. Future
130
research involving multi-scale modeling is needed to better understand fatigue at
differing length scales and to better understand the interactions between mixture
constituents. Multi-scale modeling for improved understanding and characterization of
fatigue could potentially lead to improved material selection, mixture design, and
consequently pavement performance.
131
7
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Understanding Mechanisms Leading to Asphalt Binder Fatigue

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