Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

Development of a Multiaxial

Viscoelastoplastic Continuum

Damage Model for Asphalt Mixtures

Publication No. FHWA-HRT-08-073 SEPTEMBER 2009
FOREWORD 

The constitutive modeling of asphalt concrete behavior is a topic that has gained national 

importance in the past few years. Such modeling efforts have the explicit goal of providing for 

better design and analysis of asphalt pavement structures to resist failure and/or better predict 

when failure will occur. These efforts should thus provide the tools necessary to better utilize 

available resources and/or to gain maximum results from limited resources. One such modeling 

effort that encompasses the two main forms of pavement distress, cracking and permanent 

deformation, is the multiaxial viscoelastoplastic continuum damage (MVEPCD) model and finite 

element package, finite element program (FEP++). The MVEPCD model combines elements of 

viscoelasticity, continuum damage mechanics, and viscoplasticity to model the material behavior, 

and FEP++ is used to model the interaction of material and structure. 

 

The MVEPCD model has been characterized and verified using asphalt concrete mixtures tested 

at the Federal Highway Administration’s Accelerated Load Facility in McLean, VA. A novel 

approach to modeling this process is suggested and verified in this work. In light of practical 

concerns related to constant rate tests using the Asphalt Mixture Performance Tester and due to 

the complexities of performing true time-dependent analysis of cyclic fatigue tests, a refined and 

simplified viscoelastoplastic continuum damage model is presented. A robust FEP++ has been 

developed to account for the effects of loading and boundary conditions. Analysis can be 

performed in either two-dimensional or three-dimensional configurations. The resulting 

predictions are deemed reasonable and, thus, a reliable simulation of pavement response. 

 

Cheryl Allen Richter 

Acting Director, Office of Infrastructure 

Research and Development 

Notice 

This document is disseminated under the sponsorship of the U.S. Department of Transportation 

in the interest of information exchange. The U.S. Government assumes no liability for the use of 

the information contained in this document. 

The U.S. Government does not endorse products or manufacturers. Trademarks or 

manufacturers’ names appear in this report only because they are considered essential to the 

objective of the document. 

Quality Assurance Statement 

The Federal Highway Administration (FHWA) provides high-quality information to serve 

Government, industry, and the public in a manner that promotes public understanding. Standards 

and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its 

information. FHWA periodically reviews quality issues and adjusts its programs and processes to 

ensure continuous quality improvement. 
TECHNICAL REPORT DOCUMENTATION PAGE 

1. Report No. 

FHWA-HRT-08-073 

2. Government Accession No. 3. Recipient’s Catalog No. 

4. Title and Subtitle 

Development of a Multiaxial Viscoelastoplastic Continuum Damage 

Model for Asphalt Mixtures

5. Report Date 

September 2009 

6. Performing Organization Code 

7. Author(s) 

Y. Richard Kim, Ph.D. P.E., M.N. Guddati, Ph.D., B.S. Underwood, 

T.Y. Yun, V. Subramanian, S. Savadatti 

8. Performing Organization Report No. 

9. Performing Organization Name and Address 

North Carolina State University 

Department of Civil, Construction, and Environmental Engineering 

Campus Box 7908 

Raleigh, NC 27695 

10. Work Unit No. (TRAIS) 

11. Contract or Grant No. 

DTFH61-05-H-00019 

12. Sponsoring Agency Name and Address 

Office of Research and Technology Services 

Federal Highway Administration 

6300 Georgetown Pike 

McLean, VA 22101-2296 

13. Type of Report and Period 

Covered 

14. Sponsoring Agency Code 

FHWA 

15. Supplementary Notes 

Project performed under FHWA DTFH61-05-H-00019; FHWA AOTR: Katherine Petros 

16. Abstract 

This report highlights findings from the FHWA DTFH61-05-H-00019 project, which focused on the 

development of the multiaxial viscoelastoplastic continuum damage model for asphalt concrete in both 

compression and tension. Asphalt concrete pavement, one of the largest infrastructure components in the United 

States, is a complex system that involves multiple layers of different materials, various combinations of irregular 

traffic loading, and various environmental conditions. The performance of this structure is closely related to the 

performance of asphalt concrete. To predict the performance of asphalt concrete with reasonable accuracy, a 

better understanding of its deformation behavior under realistic conditions is urgently needed. Over the past 

decade, the authors have been successful in developing uniaxial material models that can accurately capture 

various critical phenomena such as microcrack-induced damage that is critical in fatigue modeling, strain-rate 

temperature interdependence, and viscoplastic flow, which is critical for high temperature modeling. The 

resulting model is termed the viscoelastoplastic continuum damage model. However, to consider the complicated 

nature of in-service stress states, a multidimensional model is needed. To predict the performance of the real 

pavement structures, it is also important to incorporate the material model in a pavement model that considers the 

vehicle and climatic loads as well as the boundary conditions; the in-house finite element package has been 

developed for this purpose. 

17. Key Words 

Constitutive modeling, Asphalt concrete, Viscoelastic, 

Viscoplastic, Multiaxial, VECD, Damage mechanics 

18. Distribution Statement 

No restrictions. This document is available through the 

National Technical Information Service, Springfield, 

VA 22161. 

19. Security Classif. (of this report) 

Unclassified 

20. Security Classif. (of this page) 

Unclassified 

21. No. of Pages 

264 

22. Price 

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized 
ii 
iii

TABLE OF CONTENTS 

EXECUTIVE SUMMARY .......................................................................................................... 1 

CHAPTER 1. INTRODUCTION ................................................................................................ 3

1.1. PROBLEM STATEMENT ............................................................................................. 3

1.2. OBJECTIVES .................................................................................................................. 4

1.3. RESEARCH SCOPE ....................................................................................................... 4

CHAPTER 2. CONSTITUTIVE MODELING WITH A MULTIAXIAL 

VISCOELASTOPLASTIC CONTINUUM DAMAGE MODEL ............................................ 5

2.1. HISTORICAL PERSPECTIVE ..................................................................................... 5

2.2. MODELING APPROACH ............................................................................................. 5

2.2.1. Linear Viscoelasticity .............................................................................................. 6 

2.2.2. Correspondence Principle ........................................................................................ 9 

2.2.3. Continuum Damage ............................................................................................... 12 

2.2.4. Viscoelastic Continuum Damage Theory .............................................................. 12 

2.3. MULTIAXIAL VISCOELASTIC CONTINUUM DAMAGE MODEL .................. 17

2.3.1. Review of Mechanistic Principles ......................................................................... 17 

2.3.2. Application of Schapery’s Work Potential Theory ............................................... 21 

2.4. THEORY OF VISCOPLASTICITY............................................................................ 24

2.4.1. Flow Rule .............................................................................................................. 25 

2.4.2. Yield Criterion ....................................................................................................... 26 

2.5. VISCOPLASTIC MODELS ......................................................................................... 28

2.5.1. Simple Strain-Hardening Model ........................................................................... 28 

2.5.2. HISS-Perzyna Model ............................................................................................. 30 

2.5.3. Unified Model ....................................................................................................... 32 

2.6. VISCOELASTOPLASTIC CONTINUUM DAMAGE MODEL ............................. 35

CHAPTER 3. SPECIMEN FABRICATION AND TESTING PROTOCOLS ..................... 37

3.1. MATERIALS ................................................................................................................. 37

3.2. SPECIMEN FABRICATION ....................................................................................... 39

3.3. TEST SETUP ................................................................................................................. 39

3.4. TEST PROTOCOLS ..................................................................................................... 40

3.4.1. Complex Modulus Test ......................................................................................... 41 

3.4.2. Constant Crosshead Rate Tests ............................................................................. 41 

3.4.3. Repetitive Creep and Recovery Tests.................................................................... 43 

3.5. DATA ANALYSIS ......................................................................................................... 48

CHAPTER 4. MVEPCD CHARACTERIZATION AND VERIFICATION IN 

TENSION..................................................................................................................................... 49

4.1. INTRODUCTION.......................................................................................................... 49

4.2. MVECD CHARACTERIZATION IN TENSION ...................................................... 50

4.2.1. Linear Viscoelastic Characterization ..................................................................... 50 

4.2.2. Modeling Stress State Dependence of the |E*| ...................................................... 59 

4.2.3. MVECD Damage Function Characterization ........................................................ 60 

4.3. VISCOPLASTIC MODELING OF ASPHALT CONCRETE IN TENSION ......... 78

4.4. MVEPCD MODEL VERIFICATION ......................................................................... 79
iv

4.5. t-TS WITH GROWING DAMAGE IN TENSION .................................................... 81

4.6. ENGINEERING BEHAVIOR IN CONFINED TENSION ....................................... 87

4.7. FATIGUE ANALYSIS WITH VECD MODEL ......................................................... 89

4.7.1. VECD Integrated Formulation Summary .............................................................. 89 

4.7.2. Formulation of Lee, Daniel, and Kim ................................................................... 90 

4.8. SIMPLIFIED ANALYSIS OF CYCLIC FATIGUE DATA ...................................... 92

4.8.1. Correction Factor in Formulation of Lee, Daniel, and Kim .................................. 92 

4.8.2. Rigorous Analysis of Pseudo Strain History Effects ............................................. 94 

4.8.3. Discussion of Simplified Model Approach ......................................................... 101 

4.9. REFINED DERIVATION OF SIMPLIFIED FATIGUE MODEL ........................ 102

4.9.1. Identification of Tensile Loading Time ............................................................... 103 

4.9.2. Defining Alpha .................................................................................................... 104 

4.9.3. Failure in Cyclic Mode of Loading ..................................................................... 104 

4.9.4. Refined Simplified Model Derivation ................................................................. 105 

4.9.5. Model Application ............................................................................................... 107 

4.9.6. Simplifications to Refined Model ....................................................................... 110 

4.10. SUMMARY ................................................................................................................ 113

CHAPTER 5. MVEPCD CHARACTERIZATION AND VERIFICATION IN 

COMPRESSION ....................................................................................................................... 115

5.1. ENGINEERING BEHAVIOR OF ASPHALT CONCRETE IN COMPRESSION ... 115

5.1.1. Constant Rate Compression ................................................................................ 115 

5.1.2. Repetitive Creep and Recovery Test ................................................................... 119 

5.1.3. t-TS with Growing Damage in Compression ...................................................... 125 

5.2. MVECD CHARACTERIZATION IN COMPRESSION ........................................ 136

5.2.1. Linear Viscoelastic Characterization ................................................................... 136 

5.2.2. Comparison of Zero-Mean and Zero-Maximum Deviatoric Stress Results ........ 140 

5.2.3. MVECD Damage Function Characterization ...................................................... 143 

5.2.4. Comparison of MVECD Behavior in Compression and Tension ....................... 146 

5.3. VISCOPLASTIC MODELING OF ASPHALT CONCRETE IN 

COMPRESSION ................................................................................................................. 149

5.3.1. A Phenomenological Model Considering Pulse Time Effects ............................ 149 

5.3.2. HISS-Perzyna Model ........................................................................................... 158 

5.3.3. Development of a Viscoplastic Model Using Rate-Dependent Yield Stress ...... 166 

5.4. CHARACTERIZATION AND VERIFICATION OF THE VISCOPLASTIC 

MODEL ............................................................................................................................... 172

5.4.1. Calibration ........................................................................................................... 172 

5.4.2. Verification .......................................................................................................... 175 

CHAPTER 6. ENHANCEMENT OF VEPCD-FEP++ FOR PAVEMENT 

MODELING .............................................................................................................................. 181

6.1. DAMAGE IN VISCOELASTIC MATERIALS ....................................................... 181

6.1.1. The Original Model ............................................................................................. 181 

6.1.2. Upgraded Constitutive Model ............................................................................. 184 

6.1.3. Damage Model .................................................................................................... 186 

6.1.4. Finite Element Implementation ........................................................................... 187 

6.1.5. Verification .......................................................................................................... 188 

6.1.6. Implementation of Temperature Dependency ..................................................... 190 
v

6.2. REDESIGN OF FEP++ ............................................................................................... 192

6.2.1. Domain Module ................................................................................................... 192 

6.2.2. Analysis Module .................................................................................................. 195 

6.3. PREPROCESSOR ....................................................................................................... 196

6.3.1. Input Stage ........................................................................................................... 200 

6.3.2. Analysis Stage ..................................................................................................... 201 

6.3.3. Postprocessing Stage ........................................................................................... 203 

6.4. POSTPROCESSOR ..................................................................................................... 203

CHAPTER 7. A SAMPLE 3D PAVEMENT SIMULATION USING FEP++ .................... 205

7.1. THE INPUT DATA ..................................................................................................... 205

7.2. EFFECT OF TEMPERATURE ................................................................................. 205

7.3. EFFECT OF MATERIAL .......................................................................................... 210

7.4. EFFECT OF WHEEL SPEED ................................................................................... 213

CHAPTER 8. CONCLUSIONS AND FUTURE WORK ..................................................... 219

8.1. CONCLUSIONS .......................................................................................................... 219

8.2. FUTURE WORK ......................................................................................................... 220

APPENDIX. USER MANUAL FOR FEP++ .......................................................................... 223

A.1. INSTALLATION OVERVIEW ................................................................................ 223

A.2. LIST OF INPUT DIALOGS ...................................................................................... 223

A.2.1. General Information Dialog................................................................................ 223 

A.2.2. Material Properties Dialog ................................................................................. 224 

A.2.3. Elastic Material Properties Dialog ...................................................................... 226 

A.2.4. Viscoelastic Material Properties Dialog ............................................................. 227 

A.2.5. Prony Coefficients Dialog .................................................................................. 228 

A.2.6. Layer Properties Dialog ...................................................................................... 230 

A.2.7. Mesh Parameters Dialog ..................................................................................... 231 

A.2.8. Load Properties Dialog ....................................................................................... 233 

A.2.9. Analysis Parameters Dialog ............................................................................... 235 

A.2.10. Summary Dialog ............................................................................................... 237 

A.2.11. Generating an Input File ................................................................................... 238 

A.2.12. Running Analysis ............................................................................................. 238

REFERENCES .......................................................................................................................... 241 
vi

LIST OF FIGURES 

Figure 1. Graph. Schematic representation of dynamic modulus shifting process with 

unshifted data ................................................................................................................................ 10 

Figure 2. Graph. Schematic representation of dynamic modulus shifting process with shifted 

data ................................................................................................................................................ 10 

Figure 3. Graph. Schematic representation of dynamic modulus shifting process with 

time-temperature shift factor ......................................................................................................... 11 

Figure 4. Graph. Constant crosshead test results in stress-strain space ........................................ 11 

Figure 5. Graph. Constant crosshead test results in stress-pseudo strain space ............................ 12 

Figure 6. Graph. Comparison of refined and approximate damage calculation techniques ......... 16 

Figure 7. Graph. Comparison of refined and approximate damage characteristic relationship ... 16 

Figure 8. Illustration. Mechanical analog for the viscoplastic model ........................................... 26 

Figure 9. Illustration. Isotropic hardening diagram ...................................................................... 28 

Figure 10. Illustration. Kinematic hardening diagram .................................................................. 28 

Figure 11. Illustration. Strain decomposition from creep and recovery testing ............................ 30 

Figure 12. Illustration. Typical yield surface of HISS model ....................................................... 31 

Figure 13. Graph. Mixture gradation chart ................................................................................... 38 

Figure 14. Graph. Comparison of test lane and laboratory gradations ......................................... 39 

Figure 15. Graph. Stress history of VL testing (unconfined and 140 kPa confinement VL) ....... 45 

Figure 16. Graph. Stress history of VL testing (500 kPa confinement VL) ................................. 45 

Figure 17. Graph. Stress history of VLT testing (140 kPa confinement) ..................................... 47 

Figure 18. Graph. Stress history of VLT testing (500 kPa confinement) ..................................... 48 

Figure 19. Graph. Effect of 500 kPa confining pressure on the dynamic modulus in semi-log 

space .............................................................................................................................................. 55 

Figure 20. Graph. Effect of 500 kPa confining pressure on the dynamic modulus in log-log 

space .............................................................................................................................................. 56 

Figure 21. Graph. Effect of 500 kPa confining pressure on observed elasticity in the Control 

mixture .......................................................................................................................................... 56 

Figure 22. Graph. Effect of 500 kPa confining pressure on the log shift factor function ............. 57 

Figure 23. Graph. Effect of confining pressure on the relaxation spectrum ................................. 57 

Figure 24. Graph. Use of the uniaxial relaxation spectrum for multiaxial test results ................. 58 

Figure 25. Graph. Effect of performing confined temperature/frequency sweep testing on the 

unconfined dynamic modulus ....................................................................................................... 58 

Figure 26. Graph. Multiaxial equilibrium characterization results ............................................... 59 

Figure 27. Graph. Multiaxial dynamic modulus model strength .................................................. 60 

Figure 28. Graph. C11 versus S for tension for Control-2006 mixture (5 °C reference) ............... 61 

Figure 29. Graph. Effect of time-dependent Poisson’s ratio on C12 calculation ........................... 62 

Figure 30. Graph. C12 characteristic curve for Control-2006 mixture .......................................... 63 

Figure 31. Graph. C12 characteristic curve for Control-2006 mixture in semi-logarithmic 

space .............................................................................................................................................. 64 

Figure 32. Graph. Effect of different time-independent Poisson’s ratio values on radial strain 

predictions ..................................................................................................................................... 65 

Figure 33. Graph. C12 versus S for tensile loading for Control-2006 mixture (5 °C reference) ... 66 

Figure 34. Graph. S as a function of reduced time calculated by three different methodologies 

 in arithmetic space for 5-1-T ........................................................................................................ 70 
vii

Figure 35. Graph. S as a function of reduced time calculated by three different methodologies 

in arithmetic space for 5-3-T ......................................................................................................... 70 

Figure 36. Graph. S as a function of reduced time calculated by three different methodologies 

in arithmetic space for 5-4-T ......................................................................................................... 71 

Figure 37. Graph. S as a function of reduced time calculated by three different methodologies 

in arithmetic space for 5-5-T ......................................................................................................... 71 

Figure 38. Graph. S as a function of reduced time calculated by three different methodologies 

in logarithmic space for 5-1-T ...................................................................................................... 72 

Figure 39. Graph. S as a function of reduced time calculated by three different methodologies 

in logarithmic space for 5-3-T ...................................................................................................... 72 

Figure 40. Graph. S as a function of reduced time calculated by three different methodologies 

in logarithmic space for 5-4-T ...................................................................................................... 73 

Figure 41. Graph. S as a function of reduced time calculated by three different methodologies 

in logarithmic space for 5-5-T ...................................................................................................... 73 

Figure 42. Graph. Predicted and measured stress as a function of reduced time for different S

calculation methodologies for 5-1-T ............................................................................................. 74 

Figure 43. Graph. Predicted and measured stress as a function of reduced time for different S

calculation methodologies for 5-3-T ............................................................................................. 74 

Figure 44. Graph. Predicted and measured stress as a function of reduced time for different S

calculation methodologies for 5-4-T ............................................................................................. 75 

Figure 45. Graph. Predicted and measured stress as a function of reduced time for different S

calculation methodologies for 5-5-T ............................................................................................. 75 

Figure 46. Graph. C22 as a function of S calculated by different methodologies for 5-1-T .......... 76 

Figure 47. Graph. C22 as a function of S calculated by different methodologies for 5-3-T .......... 76 

Figure 48. Graph. C22 as a function of S calculated by different methodologies for 5-4-T .......... 77 

Figure 49. Graph. C22 as a function of S calculated by different methodologies for 5-5-T .......... 77 

Figure 50. Graph. Representative C22 versus S for tension using optimization methodology 

(5 °C reference) ............................................................................................................................. 78 

Figure 51. Graph. Effect of confining pressure on coefficient Y .................................................. 79 

Figure 52. Graph. Results of MVEPCD model simple verification for constant crosshead rate 

tests under 500 kPa confinement at 25 °C .................................................................................... 80 

Figure 53. Graph. Results of MVEPCD model verification for constant crosshead rate tests 

under 250 kPa confinement at 40 °C ............................................................................................ 80 

Figure 54. Graph. Results of MVEPCD model verification for constant crosshead rate tests 

under 250 kPa confinement at 5 °C .............................................................................................. 81 

Figure 55. Illustration. Schematic representation of first step for verifying the t-TS principle 

under growing damage, finding stress for different tests at a consistent strain level ................... 82 

Figure 56. Illustration. Schematic representation of the second step for verifying the t-TS 

principle under growing damage, finding reduced time for the stress found in the first step for 

each test ......................................................................................................................................... 82 

Figure 57. Illustration. Schematic representation of the third step for verifying the t-TS 

principle under growing damage, plotting the stress from the first step against the reduced 

time from the second step ............................................................................................................. 83 

Figure 58. Graph. Strain levels examined for verifying the t-TS principle under growing 

damage and confining pressure for the Control-2006 mixture ..................................................... 84 
viii

Figure 59. Graph. t-TS with growing damage under confinement verification at a 0.0001ε 

level ............................................................................................................................................... 84 

Figure 60. Graph. t-TS with growing damage under confinement verification at a 0.0005ε 

level ............................................................................................................................................... 85 

Figure 61. Graph. t-TS with growing damage under confinement verification at a 0.001ε 

level ............................................................................................................................................... 85 

Figure 62. Graph. t-TS with growing damage under confinement verification at a 0.0022ε 

level ............................................................................................................................................... 86 

Figure 63. Graph. t-TS with growing damage under confinement verification at a 0.004ε 

level ............................................................................................................................................... 86 

Figure 64. Graph. t-TS with growing damage under confinement verification at a 0.007ε 

level ............................................................................................................................................... 87 

Figure 65. Graph. Effect of 500 kPa confining pressure on strength mastercurves ..................... 88 

Figure 66. Graph. Effect of 500 kPa confining pressure on ductility in constant crosshead 

rate tests ........................................................................................................................................ 88 

Figure 67. Illustration. A schematic representation of the concept of average dC/dt ................... 93 

Figure 68. Illustration. A schematic representation of the effect of the M factor on dC/dt 

used in calculation ......................................................................................................................... 93 

Figure 69. Illustration. Mathematical equivalence of the formulation used by Lee, Daniel, 

and Kim ......................................................................................................................................... 94 

Figure 70. Illustration. Schematic representation of assumptions made for controlled stress 

cyclic loading to develop Q and M factors ................................................................................... 96 

Figure 71. Illustration. Schematic representation of assumptions made for controlled 

crosshead cyclic loading to develop Q and M factors ................................................................... 97 

Figure 72. Graph. Damage characteristic comparison, cyclic to monotonic using equation 127 

Control mixture ........................................................................................................................... 100 

Figure 73. Graph. Damage characteristic comparison, cyclic to monotonic using equation 127 

CRTB mixture ............................................................................................................................. 100 

Figure 74. Graph. Damage characteristic comparison, cyclic to monotonic using equation 127 

SBS mixture ................................................................................................................................ 101 

Figure 75. Graph. Damage characteristic comparison, cyclic to monotonic using equation 127 

Terpolymer mixture .................................................................................................................... 101 

Figure 76. Graph. Damage characteristic comparison, cyclic to monotonic using refined 

model Control mixture ................................................................................................................ 108 

Figure 77. Graph. Damage characteristic comparison, cyclic to monotonic using refined 

model CRTB mixture .................................................................................................................. 108 

Figure 78. Graph. Damage characteristic comparison, cyclic to monotonic using refined 

model SBS mixture ..................................................................................................................... 109 

Figure 79. Graph. Damage characteristic comparison, cyclic to monotonic using refined 

model Terpolymer mixture ......................................................................................................... 109 

Figure 80. Graph. Damage characteristic comparison, cyclic to monotonic using refined 

simplified model Control mixture ............................................................................................... 110 

Figure 81. Graph. Damage characteristic comparison, cyclic to monotonic using refined 

simplified model CRTB mixture ................................................................................................ 111 

Figure 82. Graph. Damage characteristic comparison, cyclic to monotonic using refined 

simplified model SBS mixture .................................................................................................... 111 
ix

Figure 83. Graph. Damage characteristic comparison, cyclic to monotonic using refined 

simplified model Terpolymer mixture ........................................................................................ 112 

Figure 84. Graph. Damage characteristic comparison, cyclic to monotonic using refined 

simplified model 9.5-mm Fine mixture ...................................................................................... 112

Figure 85. Graph. Stress-strain curves for unconfined constant strain-rate tests ........................ 116 

Figure 86. Graph. Stress-strain curves for 500 kPa confinement constant strain-rate tests ........ 116 

Figure 87. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests 

for 5 °C ........................................................................................................................................ 117 

Figure 88. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests 

for 25 °C ...................................................................................................................................... 117 

Figure 89. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests 

for 40 °C ...................................................................................................................................... 118 

Figure 90. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests 

for 55 °C ...................................................................................................................................... 118 

Figure 91. Graph. Viscoplastic strain versus cumulative loading time (unconfined VL) .......... 119 

Figure 92. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement 

VL) .............................................................................................................................................. 120 

Figure 93. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

VL) .............................................................................................................................................. 120 

Figure 94. Graph. Viscoplastic strain versus cumulative loading time (unconfined VT 

testing)......................................................................................................................................... 122 

Figure 95. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

for VT and RVT testing) ............................................................................................................. 122 

Figure 96. Graph. Incremental viscoplastic strain rate versus viscoplastic strain (500 kPa 

confinement, 1,600 kPa deviatoric) ............................................................................................ 123 

Figure 97. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

CLT) ............................................................................................................................................ 124 

Figure 98. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement 

VT) .............................................................................................................................................. 124 

Figure 99. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.0001 strain level under uniaxial conditions ...................... 126 

Figure 100. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.0005 strain level under uniaxial conditions ...................... 127 

Figure 101. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.001 strain level under uniaxial conditions ........................ 127 

Figure 102. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.003 strain level under uniaxial conditions ........................ 128 

Figure 103. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.005 strain level under uniaxial conditions ........................ 128 

Figure 104. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.01 strain level under uniaxial conditions .......................... 129 

Figure 105. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.015 strain level under uniaxial conditions ........................ 129 

Figure 106. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.02 strain level under uniaxial conditions .......................... 130 
x

Figure 107. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.0001 strain level under 500 kPa conditions ...................... 130 

Figure 108. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.0005 strain level under 500 kPa conditions ...................... 131 

Figure 109. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.001 strain level under 500 kPa conditions ........................ 131 

Figure 110. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.003 strain level under 500 kPa conditions ........................ 132 

Figure 111. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.005 strain level under 500 kPa conditions ........................ 132 

Figure 112. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.01 strain level under 500 kPa conditions .......................... 133 

Figure 113. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.015 strain level under 500 kPa conditions ........................ 133 

Figure 114. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.02 strain level under 500 kPa conditions .......................... 134 

Figure 115. Graph. Stress mastercurves for the Control mixture under uniaxial conditions ...... 134 

Figure 116. Graph. Stress mastercurves for the Control mixture under triaxial conditions 

(500 kPa) ..................................................................................................................................... 135 

Figure 117. Graph. Variation of strain rate during unloading .................................................... 135 

Figure 118. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement 

VT at 40 and 55 °C) .................................................................................................................... 136 

Figure 119. Graph. Confining stress effect on the relaxation modulus ...................................... 140 

Figure 120. Graph. Comparison of zero-mean and zero-maximum deviatoric stress dynamic 

modulus mastercurves in semi-logarithmic scale ....................................................................... 141 

Figure 121. Graph. Comparison of zero-mean and zero-maximum deviatoric stress dynamic 

modulus mastercurves in logarithmic scale ................................................................................ 141 

Figure 122. Graph. Effect of test method on shift factor functions ............................................ 142 

Figure 123. Graph. Comparison of zero-mean and zero-maximum deviatoric stress phase 

angle mastercurves ...................................................................................................................... 142 

Figure 124. Graph. Application of stress state-dependent model to zero-maximum deviatoric 

stress tests.................................................................................................................................... 143 

Figure 125. Diagram. MVECD model characterization ............................................................. 144 

Figure 126. Graph. C11 versus S for compression for Control mixture (5 °C reference)............ 144 

Figure 127. Graph. C12 versus S for compression for Control mixture (5 °C reference)............ 145 

Figure 128. Graph. C22 versus S for compression for Control mixture (5 °C reference)............ 146 

Figure 129. Graph. Comparison of tension and compression of C11 damage function .............. 147 

Figure 130. Graph. Comparison of tension and compression of C12 damage function .............. 148 

Figure 131. Graph. Comparison of tension and compression of C22 damage function .............. 148 

Figure 132. Graph. Incremental viscoplastic strain rate versus viscoplastic strain (500 kPa 

confinement, 2,000 kPa) ............................................................................................................. 151 

Figure 133. Graph. Determined fitting results and coefficients of function a(tp) ....................... 152 

Figure 134. Graph. Determined fitting results and coefficients of function D(tp σ ) ................. 153 

Figure 135. Graph. VT predictions ............................................................................................. 153 

Figure 136. Graph. VL predictions ............................................................................................. 154 

Figure 137. Graph. CLT predictions (2.0 MPa deviatoric stress—0.1-s pulse time) ................. 155 
xi

Figure 138. Graph. CLT predictions (2.0 MPa deviatoric stress—0.4-s pulse time) ................. 155 

Figure 139. Graph. CLT predictions (2.0 MPa deviatoric stress—1.6-s pulse time) ................. 156 

Figure 140. Graph. CLT predictions (2.0 MPa deviatoric stress—6.4-s pulse time) ................. 156 

Figure 141. Graph. CLT predictions (1.8 MPa deviatoric stress—1.6-s pulse time) ................. 157 

Figure 142. Graph. CLT predictions (2.2 MPa deviatoric stress—1.6-s pulse time) ................. 157 

Figure 143. Graph. Compressive and tensile peak stress in SQRT(J2) - I1 space ....................... 159 

Figure 144. Graph. Determined γ 0 parameter function .............................................................. 160 

Figure 145. Graph. Determined R parameter function ............................................................... 161 

Figure 146. Graph. Determined n parameter function ................................................................ 161 

Figure 147. Graph. Determined α 0 parameter function ............................................................. 162 

Figure 148. Graph. Rate-dependent initial yield surface ............................................................ 162 

Figure 149. Graph. Variation of α for 1,800 kPa CLT loading (500 kPa confinement) ........... 164 

Figure 150. Graph. Viscoplastic strain predictions for VT tests (500 kPa confinement) ........... 164 

Figure 151. Graph. Viscoplastic strain predictions for CLT tests (500 kPa confinement) ......... 165 

Figure 152. Graph. Viscoplastic strain predictions for RVT tests (500 kPa confinement) ........ 165 

Figure 153. Illustration. Variation of yield stress (Standard Linear Solid model) ..................... 167 

Figure 154. Graph. Stress histories for rest period analysis ....................................................... 169 

Figure 155. Graph. Yield stress versus cumulative loading time (rest period analysis) ............. 170 

Figure 156. Graph. Viscoplastic strain versus cumulative loading time (rest period analysis) .. 170 

Figure 157. Graph. Stress history for loading time analysis ....................................................... 171 

Figure 158. Graph. Viscoplastic strain versus cumulative loading time (loading time 

analysis) ...................................................................................................................................... 172 

Figure 159. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement 

VT) .............................................................................................................................................. 173 

Figure 160. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement 

VL) .............................................................................................................................................. 173 

Figure 161. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

VT) .............................................................................................................................................. 174 

Figure 162. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

VL) .............................................................................................................................................. 174 

Figure 163. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement 

VT) .............................................................................................................................................. 175 

Figure 164. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement 

VLT) ........................................................................................................................................... 176 

Figure 165. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement 

VT) .............................................................................................................................................. 176 

Figure 166. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement 

VT + flow) .................................................................................................................................. 177 

Figure 167. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

1,600 deviatoric VT) ................................................................................................................... 177 

Figure 168. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

2,000 deviatoric VT) ................................................................................................................... 178 

Figure 169. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

0.4 s CLT) ................................................................................................................................... 178 

Figure 170. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

1.6 s CLT) ................................................................................................................................... 179 
xii

Figure 171. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

6.4 s CLT) ................................................................................................................................... 179 

Figure 172. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 

VLT) ........................................................................................................................................... 180 

Figure 173. Graph. Damage characteristic relationship used in the finite element 

implementation of one-dimensional VECD model..................................................................... 183 

Figure 174. Graph. Layout of the numerical experiment specimen ........................................... 183 

Figure 175. Graph. Effect of continuum damage evolution on the vertical displacement of 

the test specimen at point A in the test simulation ...................................................................... 184 

Figure 176. Graph. Verification of strain prediction for monotonic uniaxial test using the 

new continuum damage material model ..................................................................................... 189 

Figure 177. Graph. Evolution of damage parameter, S, for the monotonic test ......................... 190 

Figure 178. Graph. Plot of function C(S) with time ................................................................... 190 

Figure 179. Diagram. Domain module ....................................................................................... 192 

Figure 180. Illustration. Infinite elastic layer on a rigid base ..................................................... 193 

Figure 181. Illustration. Finite element mesh required to model the physical problem with 

1-percent accuracy without special elements .............................................................................. 194 

Figure 182. Illustration. Finite element mesh required to model the physical problem with 

1-percent accuracy with special elements ................................................................................... 194 

Figure 183. Diagram. Analysis module ...................................................................................... 195

Figure 184. Screen capture. Main window of the preprocessor ................................................. 196 

Figure 185. Screen capture. Control panel .................................................................................. 197 

Figure 186. Screen capture. Mesh discretization ........................................................................ 198 

Figure 187. Screen capture. Zoom operation on a mesh ............................................................. 199 

Figure 188. Diagram. Stages involved in an FEP++ analysis .................................................... 200 

Figure 189. Screen capture. Sample data entry window ............................................................. 201 

Figure 190. Screen capture. Error dialog for a semantic error .................................................... 202 

Figure 191. Screen capture. Analysis run-time window ............................................................. 203 

Figure 192. Graph. Temperature variations used for simulations .............................................. 206 

Figure 193. Illustration. Vertical strains in winter ...................................................................... 207 

Figure 194. Illustration. Vertical strains in summer ................................................................... 207 

Figure 195. Illustration. Longitudinal strains in winter .............................................................. 208 

Figure 196. Illustration. Longitudinal strains in summer ........................................................... 208 

Figure 197. Illustration. Transverse strains in winter ................................................................. 209 

Figure 198. Illustration. Transverse strains in summer ............................................................... 209 

Figure 199. Illustration. Vertical strains for Control mixture ..................................................... 210 

Figure 200. Illustration. Vertical strains for SBS mixture .......................................................... 211 

Figure 201. Illustration. Longitudinal strains for Control mixture ............................................. 211 

Figure 202. Illustration. Longitudinal strains for SBS mixture .................................................. 212 

Figure 203. Illustration. Transverse strains for Control mixture ................................................ 212 

Figure 204. Illustration. Transverse strains for SBS mixture ..................................................... 213 

Figure 205. Illustration. Vertical strains for a wheel speed of 13.41 m/s ................................... 214 

Figure 206. Illustration. Vertical strains for a wheel speed of 26.82 m/s ................................... 215 

Figure 207. Illustration. Longitudinal strains for a wheel speed of 13.41 m/s ........................... 215 

Figure 208. Illustration. Longitudinal strains for a wheel speed of 26.82 m/s ........................... 216 

Figure 209. Illustration. Transverse strains for a wheel speed of 13.41 m/s .............................. 216 
xiii

Figure 210. Illustration. Transverse strains for a wheel speed of 26.82 m/s .............................. 217 

Figure 211. Screen capture. General Information dialog ............................................................ 224 

Figure 212. Screen capture. Material Properties dialog .............................................................. 225 

Figure 213. Screen capture. Elastic material properties dialog .................................................. 226 

Figure 214. Screen capture. Viscoelastic material properties dialog .......................................... 227 

Figure 215. Screen capture. Prony Coefficients dialog .............................................................. 229 

Figure 216. Screen capture. Layer properties dialog .................................................................. 230 

Figure 217. Screen capture. Mesh properties dialog ................................................................... 232 

Figure 218. Screen capture. Load properties dialog ................................................................... 234 

Figure 219. Screen capture. Analysis Parameters dialog ............................................................ 236 

Figure 220. Screen capture. Summary dialog ............................................................................. 238 

Figure 221. Screen capture. FEP++ analysis in progress ........................................................... 239 
xiv

LIST OF TABLES 

Table 1. Relevant asphalt binder information ............................................................................... 37

Table 2. Summary of constructed lanes’ air void and asphalt content ......................................... 38 

Table 3. Controlled crosshead testing matrix for Control-2006 in tension ................................... 42 

Table 4. Controlled crosshead testing matrix for Control in compression ................................... 43 

Table 5. Creep and recovery testing matrix for Control mixture in compression ........................ 44 

Table 6. Test conditions for the VT and RVT tests ...................................................................... 46 

Table 7. Loading times for VLT test ............................................................................................ 48 

Table 8. Linear viscoelastic characterization and variation for Control-2006 in unconfined 

state at selected frequencies and temperatures .............................................................................. 51 

Table 9. Linear viscoelastic characterization and variation for Control-2006 in 250 kPa 

confined state at selected frequencies and temperatures ............................................................... 52 

Table 10. Linear viscoelastic characterization and variation for Control-2006 in 500 kPa 

confined state at selected frequencies and temperatures ............................................................... 53 

Table 11. Effect of confining pressure on shift factor function coefficients for Control-2006 

mixture .......................................................................................................................................... 53 

Table 12. Cyclic tests performed .................................................................................................. 99 

Table 13. Summary of cyclic correction factors ........................................................................... 99 

Table 14. Summary of α values for refined model ..................................................................... 104 

Table 15. Cyclic tests performed for 9.5-mm Fine mixture ....................................................... 113 

Table 16. Linear viscoelastic characterization and variation for the Control mixture in 

unconfined compression state at selected frequencies and temperatures ................................... 137 

Table 17. Linear viscoelastic characterization and variation for the Control mixture in 

140 kPa confined compression state at selected frequencies and temperatures ......................... 138 

Table 18. Linear viscoelastic characterization and variation for the Control mixture in 

500 kPa confined compression state at selected frequencies and temperatures ......................... 139 

Table 19. Effect of confining pressure on shift factor function coefficients for the Control 

mixture in compression state ...................................................................................................... 139 

Table 20. Delft material model coefficients functions ................................................................ 159 

Table 21. Material coefficients used for the developed model analysis ..................................... 169 

Table 22. Compression viscoplastic material model coefficients ............................................... 172 

Table 23. Properties of pavement ............................................................................................... 205 

Table 24. Properties of moving wheel load ................................................................................ 205 
xv

LIST OF ACRONYMS AND SYMBOLS 

Abbreviations 

2D Two-dimensional 

3D Three-dimensional 

ALF Accelerated Load Facility 

AMPT Asphalt Mixture Performance Tester 

CLT Constant loading level and time test 

CR-TB Crumb Rubber Terminal Blend 

CS Controlled stress 

CX Controlled crosshead 

EICM Enhanced Integrated Climatic Model 

FEM Finite element method 

FEP++ Finite element package (proper name) 

FHWA Federal Highway Administration 

GUI Graphical user interface 

HISS Hierarchical single surface 

HMA Hot mix asphalt 

IDT Indirect tension test 

LVDT Linear variable displacement transducers 

LVE Linear viscoelastic 

MVECD Multiaxial viscoelastic continuum damage 

MVEPCD Multiaxial viscoelastoplastic continuum damage 

NCHRP National Cooperative Highway Research Program 

NCSU North Carolina State University 

NMSA Nominal maximum size aggregate 

RVT Reversed variable loading time test 

SBS Styrene-butadiene-styrene 

SHRP Strategic Highway Research Program 

SPT Simple Performance Tester 

t-TS Time-temperature superposition 

TFHRC Turner-Fairbank Highway Research Center 

TRS Thermorheologically simple 

UVECD Uniaxial viscoelastic continuum damage 

UVEPCD Uniaxial viscoelastoplastic continuum damage 

VECD Viscoelastic continuum damage 

VECD-FEP++ Viscoelastic continuum damage model included in finite element package 

VEPCD Viscoelastoplastic continuum damage 

VP Viscoplastic 

VL Variable loading level test 

VLT Variable loading level and time test 

VT Variable loading time test 

VTK Visualization Toolkit 
xvi

Symbols 

e

ij ε elastic strain 

p

ij ε plastic strain 

c

ij ε creep strain 

vp

ij ε viscoplastic strain 

ve

ij ε elastic plus linear viscoelastic strain due to damage 

ve ε elastic plus linear viscoelastic strain due to damage 

Rf reduced physical frequency (Hz) 

ωR reduced angular frequency (=2πfRΔ t) 

T a time-temperature shift factor at specific temperature, T 

π 3.141593 

σ stress 

ε strain 

ε x strain in direction x

με microstrain 

E( )t relaxation modulus 

D t( ) creep compliance 

τ dummy integration variable 

ρi relaxation time (fitting coefficient) 

j τ retardation time (fitting coefficient) 

E∞ relaxation modulus at infinite time 

Ei Prony coefficient for relaxation modulus 

D

g glassy compliance 

Dj Prony coefficient for creep compliance 

E ' storage modulus 

* E dynamic modulus 

φ phase angle 

t time 

ξ reduced time 

∂ partial derivative 

ΔX delta, finite difference in X 

δ X delta, finite difference in X 

R ε pseudo strain 

ER reference modulus 

C normalized pseudo secant modulus 

I normalization parameter 

S damage 

W strain energy density 
xvii

R Wd pseudo energy density 

α viscoelastic damage growth rate 

Cijkl general stiffness matrix 

ν Poisson’s ratio 

E elastic elongational modulus 

G elastic shear modulus 

Zij stiffness matrix for transversely isotropic material 

E3 modulus along axis of symmetry 

E modulus perpendicular to axis of symmetry 

ν 3132 Poisson’s ratio between axis of symmetry and perpendicular plane 

ν 1323 Poisson’s ratio between perpendicular plane and axis of symmetry 

ν 12 Poisson’s ratio on perpendicular plane 

Υ Poisson’s ratio term 

ij S compliance matrix for transversely isotropic material 

Aij alterative stiffness matrix terms for transversely isotropic material 

v e volumetric strain (dilation) 

3 e major deviatoric strain 

2 e strain difference 

p pressure 

C11 first material integrity term 

C12 second material integrity term 

C22 third material integrity term 

υ dilation 

R υ pseudo dilation 

Qi generalized loads 

i q generalized displacements 

Ue complementary strain energy 

vp ε& viscoplastic strain rate 

λ positive scalar factor 

g plastic potential function (general viscoplasticity) 

Φ overstress function 

η viscosity 

Γ fluidity 

1I first stress invariant 

2 J second deviatoric stress invariant 

3 J third deviatoric stress invariant 

ij s deviatoric stress 

ij δ Kroneker delta 
xviii

κ isotropic hardening parameter 

αij viscoplastic kinematic hardening parameter 

g stress function (strain hardening model specific) 

Pa atmospheric pressure 

R tensile strength parameter 

n yield stress shape parameter 

β parameter determining shape of yield stress in deviatoric stress space in ε& inelastic strain rate 

k isotropic hardening function 

α viscoplastic back stress parameter (kinematic hardening) 

D drag stress 

R isotropic hardening function 

H kinematic hardening function 

G back stress (viscoplastic model) 

ξ viscoplastic strain trajectory

θ angular direction in axisymmetric coordinate system 

α1 time-temperature shift factor function coefficient 1 

α2 time-temperature shift factor function coefficient 2 

α3 time-temperature shift factor function coefficient 3 

θ bulk stress

R

pressure ε pseudo strain due to pressure 

R υ pressure pseudo dilation due to pressure 

R

e ε effective pseudo strain 

R

s ε permanent pseudo strain 

R

m ε total maximum pseudo strain 

R

me ε total effective pseudo strain at peak of loading 

M time change correction factor 

p t pulse time 

p ξ reduced pulse time 

N number of points in a calculation 

F pseudo strain slope function 

G pseudo strain hysterisis function 

H healing function 

ˆ ( ) R ε t analytical expression for pseudo strain as a function of time 

Q pseudo strain shape factor 

Z combined pseudo strain shape and pseudo stiffness time factor 

0,

R

ta ε pseudo strain tension amplitude only 

σ 0,ta stress tension amplitude only 
xix

σ pp peak-to-peak stress magnitude 

β factor quantifying time under tensile loading 

σ peak maximum value of stress in a cycle 

σ valley minimum value of stress in a cycle 

i ξ reduced time within a cycle when tension loading begins 

f ξ reduced time within a cycle when tension loading ends 

Rk form adjustment factor for characterization 

K1 form adjustment factor for prediction 

0,ta ε strain tension amplitude only 

( ) p a t phenomenological viscoplastic model slope pulse time function 

D phenomenological viscoplastic model intercept function 

γ viscoplastic softening parameter 

1,dilation I first stress invariant value at the beginning of dilation 

2,dilation J second deviatoric stress invariant value at the beginning of dilation 

reduced ε& reduced strain rate 

D viscosity parameter (final viscoplastic model) 

G orientation-dependent isotropic hardening function 

n 1 G + value of hardening function at next time step 

n G value of hardening function at current time step 

0

n ζ value of elastic state variable at current time step 

1

0

n ζ + value of elastic state variable at next time step

n

i ζ value of state variable for element i at current time step 

n 1

i ζ + value of state variable for element i at next time step 

1

EXECUTIVE SUMMARY 

This report highlights findings from the FHWA DTFH61-05-H-00019 project, which focused on 

the development of the multiaxial viscoelastoplastic continuum damage (MVEPCD) model for 

asphalt concrete in both compression and tension. The MVEPCD model and finite element 

package (FEP++) have been developed for this purpose. The MVEPCD model combines 

elements of viscoelasticity, continuum damage mechanics, and viscoplasticity to model the 

material behavior. FEP++ is used to model the interaction of material and structure. 

An introduction that outlines the research objectives and scope of the project is given in 

chapter 1 of this report. Chapter 2 provides a practical review of the underlying theories for the 

MVEPCD material model. Particular attention is paid to the concept of continuum damage 

mechanics and their implementation with viscoelastic materials. In addition, methodologies used 

to characterize the MVEPCD model are outlined in chapter 2. Details regarding specimen 

fabrication and testing protocols as well as background for the materials tested are in chapter 3. 

Experimental results are categorized by loading direction, tension, or compression in chapter 4 

and chapter 5, respectively. Specific comparisons of the effects of direction of loading are given 

in subsections 5.2.2 and 5.2.4. Each chapter provides the procedure for characterizing the 

viscoelastoplastic continuum damage (VEPCD), beginning with the linear viscoelastic 

characterization, proceeding to the viscoelastic damage characterization and viscoplastic 

characterization, and ending with the validation of the models. In addition, there is some 

discussion about the engineering properties of the materials in compression and tension, as well 

as verification of the time-temperature superposition (t-TS) principle with growing damage. 

Chapter 4 discusses the use of the VEPCD model for fatigue predictions. Because the rutting 

distress, a permanent deformation phenomenon, is related primarily to the compressive behavior 

of asphalt concrete, chapter 5 focuses on the viscoplastic behavior of the material in compression. 

Chapter 6 presents the research efforts to enhance the finite element program with the 

viscoelastic continuum damage (VECD) model and the viscoelastic continuum damage model 

included in finite element package (VECD-FEP++). The resulting program is then used in 

chapter 7 to perform the three-dimensional (3D) finite element analysis to study the effects of 

temperature, material type, and vehicle speed on pavement responses. Chapter 8 summarizes the 

conclusions from both the experimental and computational work for this project. The future 

direction of the research at hand is provided at the end of chapter 8 as well.

3

CHAPTER 1. INTRODUCTION 

1.1. PROBLEM STATEMENT 

Asphalt concrete pavement, one of the largest infrastructure components in the United States, is a 

complex system that involves multiple layers of different materials, various combinations of 

irregular traffic loading, and various environmental conditions. Therefore, a realistic prediction 

of the long-term service life of asphalt pavements is one of the most challenging tasks for 

pavement engineers. The performance of asphalt concrete pavements is closely related to the 

performance of asphalt concrete. In order to predict the performance of asphalt concrete with 

reasonable accuracy, a better understanding of its deformation behavior under realistic conditions 

is urgently needed. 

Asphalt concrete is a viscoelastic particulate composite that consists of aggregate particles and 

an asphalt binder matrix. When the asphalt-aggregate composite is subjected to repeated traffic 

loading at low temperatures, distributed microstructural damage occurs primarily in the forms of 

microcrack nucleation and growth due to the embrittled binder and high-stress concentrations 

along the aggregate-binder interfaces. Therefore, the role of the binder and the variables that 

influence the properties of the binder (e.g., aging, adhesion, etc.) become important to the study 

of this type of damage. At high temperatures, the asphalt binder becomes too soft to carry the 

load, and thus, the principal type of damage is permanent deformation due to volume change 

(i.e., densification) and rearrangement of aggregate particles. Therefore, a reliable performance 

prediction model should account for the effects of various constitutive factors that affect the 

aggregate-binder and aggregate-aggregate interactions. 

With the goal of accurate pavement performance evaluation, researchers at North Carolina State 

University (NCSU) have been developing advanced models for asphalt concrete under complex 

loading conditions. Over the past decade, they have successfully developed material models that 

can accurately capture various critical phenomena such as microcrack-induced damage, which is 

critical in fatigue modeling, strain-rate temperature interdependence, and viscoplastic flow, 

which is critical for high-temperature modeling. The resulting model is termed the viscoelastic 

continuum damage (VEPCD) model. While the initial development of the VEPCD model 

focused on uniaxial tension behavior, the accurate performance prediction of an asphalt mixture 

in a pavement structure requires a multidimensional model. 

To predict the performance of real pavement structures, it is also important to incorporate the 

material model in a pavement model that considers the vehicle and climatic loads as well as the 

boundary conditions. The finite element method is best suited for this purpose due to the 

nonlinear material behavior. The group has an in-house finite element code (FEP++) which can 

analyze general nonlinear dynamical systems. FEP++ is a research code and requires several 

modifications in order to be used for routine pavement modeling. 
4

1.2. OBJECTIVES 

The long-term goal of the asphalt pavement modeling research at NCSU is to develop a 

mechanistic asphalt pavement performance prediction methodology that can be used by State 

highway agencies. This research focuses on the following objectives to accomplish this goal: 

1. To develop a multiaxial viscoelastoplastic continuum damage (MVEPCD) model for 

asphalt concrete in both compression and tension. 

2. To enhance FEP++ so that nonlinear analysis of pavements can be easily conducted. 

1.3. RESEARCH SCOPE 

This research includes four of the mixtures used in the Federal Highway Administration’s 

(FHWA) Accelerated Load Facility (ALF) current study that is funded through the pooled-fund 

study, TPF-5(019). Three of the mixtures contain asphalt binders modified with polymeric 

additives; the remaining mixture contains an unmodified asphalt binder. Material behavior under 

both tension and compression is addressed in this research. For the current research in multiaxial 

model development, only the unmodified mixture is being used. 
5

CHAPTER 2. CONSTITUTIVE MODELING WITH A MULTIAXIAL 

VISCOELASTOPLASTIC CONTINUUM DAMAGE MODEL 

2.1. HISTORICAL PERSPECTIVE 

The foundation for the work presented in this report was laid by researchers at Texas A&M 

University almost two decades ago.(1) These researchers successfully applied Schapery’s 

nonlinear viscoelastic constitutive theory for materials with distributed damage to describe the 

behavior of sand asphalt under controlled strain cyclic loading. Later research shows that this 

theory can also describe the behavior of asphalt concrete under both controlled stress and 

controlled strain cyclic loading.(2–4) Later research shows that the damage characteristics of a 

material are independent of the mode of loading and can be determined using simpler tests.(5)

Further, verification of the t-TS principle at high levels of damage is an equally significant 

discovery.(6) These two findings significantly reduce the required testing protocol while 

simultaneously extending the realm of application for the model. Work by Chehab et al. utilizes 

Schapery’s theory, along with strain decomposition, to account for both viscoelastic and 

viscoplastic strains. (See references 7 through 11.) The most recent work applies this theory to 

mixtures tested at the FHWA ALF in Mclean, VA,(12) and successfully demonstrates the 

application of the modeling principles to both modified and unmodified asphalt concrete 

mixtures. This work is the basis for the current research. 

2.2. MODELING APPROACH 

The MVEPCD model is a theoretical and phenomenological extension of the uniaxial VEPCD 

model presented in previous work and briefly reviewed here. These modeling approaches share 

many of the same principles, such as linear viscoelasticity, elastic-viscoelastic correspondence 

principle, continuum damage, and strain-hardening plasticity. A complete review of these 

principles in the uniaxial sense is given elsewhere, and a brief review is given in the following 

sections.(13) For a more rigorous treatment of the subject, the reader is referred to previous work 

and to the work of Schapery for linear viscoelasticity and continuum damage mechanics. (See 

references 1 through 10.) For a detailed review of strain-hardening viscoplasticity, the reader is 

directed to the work of others.(7,10,14) 

Linear viscoelastic materials exhibit time- and temperature-dependent properties that make them 

dependent upon the history of loading, unlike elastic materials. This issue complicates the 

analysis since many continuum theories have been developed assuming an elastic material. In 

this report, the elastic-viscoelastic correspondence principle is used so that these elastic theories 

can be applied to the viscoelastic asphalt concrete mixtures. The correspondence principle uses 

pseudo strains, a quantity calculated from the actual time-dependent strains, in place of the actual 

strains. It is a general theory in so much that for purely elastic materials, the pseudo strains may 

be equal to the physical strains. Many damage theories exist, and in this report Schapery’s theory 

based on energy principles is used. For this theory, consideration is given for both the timedependent nature of energy availability as well as the time dependence of damage resistance. 

These arguments are considered within a thermodynamic framework. Finally, strain-hardening 

plasticity is found to be simple to model and commonly observed viscoplastic phenomenon. 
6

A well-known assumption, founded in the theory of plasticity, is that total strain can be 

decomposed into elastic and plastic strains. Likewise, total strain can be decomposed into elastic 

strain, plastic strain, and creep strain, according to the theory of viscoplasticity, to account for 

the rate-dependent plastic strain of materials, as shown equation 1. In some studies that involve 

such rate-dependent materials, the rate-independent plastic strain and the rate-dependent creep 

strain are defined as viscoplastic strain because it is difficult to distinguish the plastic 

deformation from creep deformation. The theoretical background for the viscoplastic strain 

concept was first discussed by Perzyna, and since then more complicated models have been 

developed to explain the behavior of a material due to plasticity-creep interaction.(15)

total e p c e vp

ij ij ij ij ij ij ε = εεεεε ++=+ (1) 

Where: e

ij ε = Elastic strain. 

p

ij ε = Plastic strain. 

c

ij ε = Creep strain. 

vp

ij ε = Viscoplastic strain. 

From a similar perspective, Schapery suggests that total strain may be separated into several 

components, such as elastic, viscoelastic, and viscoplastic strains.(10) For the MVEPCD model, 

elastic, linear viscoelastic, and strains due to microcracking damage are combined into a single 

term, and plastic and viscoplastic strains are combined into another, as shown in equation 2. 

total ve vp

ij ij ij ε = ε ε + (2) 

Where: ve

ij ε = Elastic plus linear viscoelastic strain due to damage. 

vp

ij ε = Viscoplastic strain. 

2.2.1. Linear Viscoelasticity 

Linear viscoelastic (LVE) materials exhibit time- and temperature-dependent behavior—the 

current response is dependent on both the current input and all past input (i.e., the input history). 

By contrast, the response of an elastic material is only dependent on the current input. 

Constitutive relationships for LVE materials are typically expressed in the convolution integral 

form, as follows: 

( )

0

t d E t d

d

ε

σ τ τ

τ = − ∫ (3) 

( )

0

t d D t d

d

σ

ε τ τ

τ = − ∫ (4) 
7

Where: 

E(t) and D(t) are the relaxation modulus and creep compliance, respectively. In this report, the 

analytical forms for these functions are given by the common Prony series formulation shown in 

equation 5 and equation 6. Note that the characterization of LVE behavior is undertaken by 

performing temperature and frequency sweep tests which are then processed to obtain the 

coefficients for these functions.(13) 

( )

1

i

t m

i

i

E t E Ee

ρ −

∞

=

= +∑ (5) 

( )

1

1 j

n t

g j

j

Dt D D e τ −

=

⎛ ⎞

=+ − ⎜ ⎟ ⎝ ⎠ ∑ (6) 

2.2.1.1. Linear Viscoelastic Interconversion 

The unit response functions presented in equation 3 and equation 4 are often measured in the 

frequency domain using the complex modulus test because it is often difficult to obtain 

measurements in the time domain due to limitations of the testing machine’s capacity for testing 

time. The complex modulus provides the constitutive relationship between the stress and strain 

amplitudes of a material loaded in a steady-state sinusoidal manner. Then, the storage modulus 

can be determined from the complex modulus and converted to a time-dependent property, such 

as E(t) and D(t), according to the theory of linear viscoelasticity. When the storage modulus is 

expressed in terms of reduced angular frequency,ωR , as shown in equation 7, it can be expressed 

using the Prony series representation given in equation 8.(16,17)

( ) ( ) ( ( )) * ' *sin E E ω ω φω RR R = (7) 

( )

2

2 2

1

' 1

m

R ii

R

i R i

E E E ω ρ ω

ω ρ ∞

=

= +

+ ∑ (8) 

Where: 

E∞ = Elastic modulus. 

ωR = Angular frequency (=2π fRΔt). 

Δt = Time lag between stress and strain. 

Ei = Modulus of i

th Maxwell element (fitting coefficient). 

ρi = Relaxation time (fitting coefficient). 

The coefficients determined from this process are then used with equation 5 to find the relaxation 

modulus. Using the theory of viscoelasticity, the exact relationship between the creep 

compliance and relaxation modulus is given by equation 9. 

( ) ( )

0

1

t dD

Et d

d

τ

τ τ

τ − = ∫ (9) 
8

If the creep compliance, written in terms of the Prony representation (equation 10), is substituted 

into equation 9 along with equation 5 and simplified, then the result can be expressed as a linear 

algebraic system, shown as equation 11. The coefficients {D} are solved by any proper 

numerical method. 

( )

1

1 j

n t

g j

j

Dt D D e τ −

=

⎛ ⎞ =+ − ⎜ ⎟ ⎝ ⎠ ∑ (10) 

[A]{D B } = [ ] (11) 

Where: 

[ ]

1 1

1 m j j

t t t M N

m m

j m m j

E A e e E e ρ ρ τ τ

ρ τ

− − −

∞

= =

⎡ ⎛ ⎞⎛ ⎞⎤

= − ⎢ ⎜ ⎟⎜ ⎟ + − ⎥ ⎢ − ⎥ ⎣ ⎝ ⎠⎝ ⎠⎦

∑ ∑

 (12) 

{D} = Dj (13) 

[ ]

1

1

1 1 m

t N

N m

m

m

m

B E E e

E E

ρ

−

∞

=

∞

=

⎛ ⎞

=− + ⎜ ⎟

+ ⎝ ⎠

∑

∑ (14) 

Once the coefficients Dj are determined, they are substituted into equation 10 to find the creep 

compliance. 

2.2.1.2. Linear Viscoelastic Characterization Methodology 

Numerous research efforts by the authors have led to a methodology for assessing and analyzing 

LVE properties through the dynamic modulus. Explicit details of the experimental method are 

given in subsection 3.4.1; however, it should be known that the test applies cyclic sinusoidal 

loading at several combinations of frequency and temperature either with or without confining 

pressure. Load and axial deformations measured at four locations separated by 90-degree 

intervals are recorded for each combination of frequency and temperature. From these 

measurements, stresses and strains are calculated based on the specimen area and gauge length of 

the deformation measurements, respectively. The analysis procedure is the same for both 

confined and unconfined tests and is given in detail elsewhere.(13,18) 

Asphalt concrete in the LVE range is known to be thermorheologically simple (TRS) and, as 

such, the effects of time and temperature can be combined into a joint parameter, reduced 

time/frequency, through the time-temperature shift factor (aT) using equation 15. 

R T f = f a × (15) 
9

In practical terms, this behavior allows for the horizontal shifting of the processed data to form a 

single curve, the mastercurve, for describing the constitutive behavior of asphalt concrete. The 

amount of horizontal shift is known as the t-TS shift factor, and the relationship between this 

factor and temperature is known as the t-TS shift factor function. When combined with the 

mastercurve, these two functions allow for the prediction of the LVE behavior over a wide range 

of conditions. The process is shown schematically in figure 1 through figure 3. 

2.2.2. Correspondence Principle 

The damage theory used in this research, which was originally developed for elastic materials, is 

generalized for viscoelastic materials using the elastic-viscoelastic correspondence principle.(8) In 

short, this principle states that viscoelastic problems can be solved with elastic solutions when 

physical strains are replaced by pseudo strains. 

( )

0

1 t

R

R

d E t d

E d

ε

ε τ τ

τ = − ∫ (16) 

Where: 

ER is a particular reference modulus, typically taken as one. Using pseudo strain in place of 

physical strain, the uniaxial constitutive relationship presented in equation 3 can be rewritten as 

follows: 

R σ ER = ε (17) 

As equation 17 shows, for the uniaxial condition, a form corresponding to that of a linear elastic 

material (Hooke’s Law) is taken when strains are replaced by pseudo strains. Additional 

theoretical details of this concept can be found elsewhere.(8,19,20) In a practical sense, pseudo 

strains are simply the LVE stress response to a particular strain input. The most important effect 

of pseudo strains is seen when plotting with stress, as the time effects are removed from the 

resulting graph. This property allows the direct quantification of damage independent of any time 

effects. The results of two typical monotonic tests are presented in figure 4 and figure 5 in both 

stress strain space and stress-pseudo strain space. The behavior during initial loading is shown as 

an inset in these figures. In stress-strain space, as seen in figure 4, nonlinearity appears in the 

initial stage of loading, which might suggest that damage commences from the outset. However, 

the nonlinearity in this zone is related only to the time effects of the material. When these time 

effects are removed, as seen in figure 5, it is clear that damage does not commence at the outset 

of loading and does not begin until the stress level reaches approximately 500 kPa. 
10

10

100

1000

10000

100000

1.E-08 1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04

Frequency (Hz)

|E*| (MPa)

54°C

40°C

20°C

 5°C

-10°C

Mastercurve

Figure 1. Graph. Schematic representation of dynamic modulus shifting process with 

unshifted data. 

10

100

1000

10000

100000

1.E-08 1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04

Reduced Frequency (Hz)

|E*| (MPa)

54°C

40°C

20°C

5°C

-10°C

Mastercurve

Figure 2. Graph. Schematic representation of dynamic modulus shifting process with 

shifted data. 
11

-6

-5

-4

-3

-2

-1

0

1

2

3

-15 5 25 45 65

Temperature (C)

log a

T

Figure 3. Graph. Schematic representation of dynamic modulus shifting process with 

time-temperature shift factor. 

0

1000

2000

3000

4000

0.E+00 2.E-03 4.E-03 6.E-03

Strain

Stress (kPa)

0

500

1000

0E+00 1E-04 2E-04

Strain

Stress (kPa)

Figure 4. Graph. Constant crosshead test results in stress-strain space. 

Initial 

Stiffness 
12

0

1000

2000

3000

4000

0 5000 10000 15000 20000 25000

Pseudo Strain

Stress (kPa)

0

500

1000

0 500 1000

Pseudo Strain

Stress (kPa)

Figure 5. Graph. Constant crosshead test results in stress-pseudo strain space. 

2.2.3. Continuum Damage 

On the simplest level, continuum damage mechanics considers a damaged body with some 

stiffness as an undamaged body with reduced stiffness. Continuum damage theories thus attempt 

to quantify two values: damage and effective stiffness. Further, these theories ignore specific 

microscale behaviors and instead characterize a material using macroscale observations (i.e., the 

net effect of microstructural changes on observable properties). The most convenient method to 

assess the effective stiffness in the macro sense is the instantaneous secant modulus. On the other 

hand, damage is oftentimes more difficult to quantify and generally relies on macroscale 

measurements combined with rigorous theoretical considerations. For the model at hand, 

Schapery’s work potential theory, based on thermodynamic principles, is appropriate for the 

purpose of quantifying damage. Within Schapery’s theory, damage is quantified by an internal 

state variable, S, that accounts for microstructural changes in the material. (See references 8 

through 11.) 

2.2.4. Viscoelastic Continuum Damage Theory 

The uniaxial viscoelastic continuum damage (UVECD) model combines elements from the 

preceding sections to arrive at the constitutive relationship. From continuum damage, the 

stiffness reduction is defined by the pseudo secant modulus (pseudo stiffness). This quantity is 

typically normalized for specimen-to-specimen variability by the factor I and denoted as C. 

C R I

σ

ε = × (18) 

Initial Pseudo 

Stiffness 
13

The relationship between damage, S, and the normalized pseudo secant modulus, C, is known as 

the damage characteristic relationship and is a material function independent of loading 

conditions.(19) With these considerations, the nonlinear constitutive relationships used for this 

research are given by equation 19 for stresses and equation 20 for strains. 

( ) R σ = C S ε (19) 

( ) ( )

0

ve R

d

C S

E D t d

d

ξ

σ

ε τ τ

τ

⎛ ⎞ ⎜ ⎟

⎝ ⎠ = − ∫ (20) 

In comparing equation 20 to equation 4, a striking similarity is observed. Equation 4 is the 

constitutive relationship for linear viscoelasticity with a stress input. The modeling approach 

uses the given input (σ ) to determine the input if no damage has occurred ( R ε ) and then utilizes 

the LVE constitutive relationships to find the response. Equation 20 is solved in this research 

using the same state variable type of approach used for solving equation 16. These formulations 

are presented elsewhere.(13) 

2.2.4.1. Refinement of Damage Characteristic Relationship 

The work potential theory specifies an internal state variable, S, to quantify damage. This 

internal state variable quantifies any microstructural changes that result in the observed stiffness 

reduction. For asphalt concrete in tension, this variable is related primarily to the microcracking 

phenomenon. This report highlights only components new to this research, as significant 

theoretical work has been done by others. (See references 2, 7, 20, and 21.) 

The derivation of the UVECD model begins with an assumption of damage behavior 

(equation 21) or damage evolution law. 

R dS Wd

dt S

α

⎛ ⎞ ∂ = −⎜ ⎟ ∂ ⎝ ⎠ (21) 

The method used to solve the damage evolution law is a matter of preference; therefore, two 

different solutions are proposed for solving equation 21. The first transforms the original form of 

the equation into an integral form, assumes α >> 1, and defines a new parameter, ˆ

S.

(21)

Equation 22 presents this method in discrete form. 

1

1 1 1 ˆ S S 1 α

α

⎡ ⎤ + ⎛ ⎞ = + ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ (22) 
14

Where: 

ˆ

S is given by equation 22, as follows: 

( ) ( )

1 2

1

1 ˆ ˆ

2

R

i i i i

SS C tα ε + =− Δ (23) 

The second possible solution shown in equation 24 utilizes the chain rule and makes no 

assumption regardingα.

(3) Both methods have been successfully applied in asphalt concrete 

research.(7,20,21) 

( ) ( )

1 2 1

1

1

1

2

R

i i i i

SS C t

α

α α ε + + +

⎡ ⎤ = +− Δ Δ ⎢ ⎥ ⎣ ⎦ (24) 

To reconcile the approximations of these methods, an iterative refinement technique is 

incorporated into this research. In short, this method assumes that the change in material integrity 

is sufficiently small over some discrete time step. The rate of change of the material integrity 

with respect to damage is determined at a point near the current value of damage, Si + δ S, where 

the extrapolation error is minimized. 

This refinement process begins with an initial calculation of S by either of the approximate 

methods, both of which require results from constant crosshead rate tests for the stress-pseudo 

strain relationship. The initial S values are plotted with the pseudo stiffness values, C, to obtain 

the damage characteristic curve. This relationship is then fitted to some analytical form, as 

shown in equation 25. 

b aS C e = (25) 

Returning to equation 21 and noting that the increments of time are generally small, the rate of 

change in damage can be written as follows: 

dS S

dt t

Δ = Δ (26) 

By substituting this expression into equation 21 and rearranging and writing in the discrete form, 

equation 27 is established: 

1

R

i

i i

W SSt

S

α

δ

δ +

⎛ ⎞ = +Δ −⎜ ⎟ ⎝ ⎠ (27) 

For the uniaxial case, the work function, WR

, is given by equation 28.(19) 

( ) 1

2

R R W CS d = ε (28) 
15

By substituting equation 28 into equation 27 and then simplifying, equation 29 is reached. 

( )2 ( )

1

1

2

R i

i i

C

SSt

S

α

δ

ε

δ +

⎛ ⎞

= +Δ −⎜ ⎟

⎝ ⎠ (29) 

In equation 29, it is assumed that before loading occurs, S and C are zero and 1, respectively. 

Furthermore, δ S must be specified and should be significantly less than the change in damage 

over a time step (typically, 0.1 is used). After calculating the value of damage, Si, and the 

incremental damage, Si +δ S, at a given time step, the corresponding values of C are found by 

equation 25. The difference between these values (δ C) is then used to calculate damage at the 

next time step. The process is repeated until all data points are processed. 

After completing this first iteration, the new values of S are plotted against the original pseudo 

stiffness values, and a new analytical relationship is found. The entire process is repeated until 

the change in successive iterations is small. In this research, eight such iterations were performed, 

but very little improvement was made after the third or fourth iteration. 

Figure 6 and figure 7 present the initial S calculated by both approximate techniques along with 

results from the refinement process. From these figures, it is seen that the refinement process 

results in S values that fall between the two approximate methods. In these figures, the seed 

values for the refinement process are obtained by the chain rule method. However, trials show 

that regardless of the method used to find the seed values, iterations collapse to the same curve. 

Details regarding this refinement process can be found elsewhere.(22)

The final refined values of S are plotted with C to obtain the true damage characteristic curve. As 

noted before, the benefit of this curve is its mode-of-loading independence; the curve represents 

a fundamental behavior of the material. From equation 16 and equation 19, if this fundamental 

relationship is known and the relaxation modulus is given, then the stresses can be directly 

calculated. Conversely, if this fundamental relationship is known and the creep compliance is 

given, the strain response can be directly calculated for any stress input using equation 20. 
16

0

20000

40000

60000

80000

100000

0 25 50 75 100 125

Reduced Time (s)

S

Park's Method

Refined Values

Chain Rule

Transportation Research Board, National Academy of Science. From Appendixes to National Cooperative 

Highway Research Program (NCHRP) Report 547: Sample Performance Tests and Advanced Materials 

Characterization Models (2005)

Figure 6. Graph. Comparison of refined and approximate damage calculation techniques.(22)

0.2

0.4

0.6

0.8

1

0 10000 20000 30000 40000 50000 60000

S

C

Park's Method

Refined Values

Chain Rule Method

Transportation Research Board, National Academy of Science. From Appendixes to NCHRP Report 547: 

Sample Performance Tests and Advanced Materials Characterization Models (2005)

Figure 7. Graph. Comparison of refined and approximate damage characteristic relationship.(22)
17

2.2.4.2. Calculation of Damage 

In a typical application, the user wants to predict strain from only stress and time using 

equation 20. However, from the preceding discussion, the user must know the strain to compute 

S and C. To bypass this requirement, it is first assumed that S and C are initially zero and 1, 

respectively. Then, at some incremental amount of damage (δ S), C can be computed by the 

functional relationship found through the refinement process. Finally, using the relationship in 

equation 19, equation 29 can be rewritten as follows: 

( ) 2

1

1

2

i i

i i

i

C

SSt

C S

α

σ δ

δ +

⎛ ⎞ ⎛ ⎞ = +Δ −⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (30) 

Through equation 30, damage may be incrementally calculated using a technique similar to that 

used to refine S using only stress and time. 

2.2.4.3. Viscoelastic Damage Characterization Methodology 

Viscoelastic damage characterization refers to the development of the characteristic damage 

relationship (i.e., the C versus S relationship). Although such characterization can be performed 

under any loading condition, the simplest method is the constant crosshead rate test.(20) For 

MVEPCD characterization, both confined and unconfined constant crosshead rate tests are 

performed. The only requirement for this test is that viscoelastic damage mechanisms dominate 

the material behavior, such as when the material is at a low temperature or when it is loaded at a 

very fast rate. In this study, characterization is performed at 5 °C. This temperature is convenient 

because it allows for moderate strain rates and does not require consideration of any dynamic 

effects that might be related to extremely high strain rates. In practice, at least three different 

rates are tested at 5 °C, and if the resulting C versus S curves collapse, it is known that 

viscoelastic damage mechanisms dominate. The appropriate strain rates are material-dependent. 

For both VEPCD and MVEPCD characterization, the first step is the calculation of the pseudo 

strains by equation 16 for the constant rate tests. These values are used along with equation 18 

and equation 24 to calculate the initial damage, S, values. The relationship between C and these 

initial S values are fitted to equation 25 and refined through the aforementioned procedure. The 

true C versus S relationship is obtained by refitting equation 25 to these refined S values. Results 

from the tests at different rates are finally averaged to obtain the C versus S relationship for 

modeling. Typically, each test is sampled at different intervals, which complicates this averaging 

process. To overcome this difficulty, a common list of C values is compiled, and the 

corresponding S values are interpolated from the respective test. Averaging can then be easily 

performed because each test has common C values. 

2.3. MULTIAXIAL VISCOELASTIC CONTINUUM DAMAGE MODEL 

2.3.1. Review of Mechanistic Principles 

The arguments presented in this section focus on general stress-strain relationships. For 

notational simplicity, the derivations in this section are given for linear elasticity; however, the 
18

arguments remain valid for viscoelastic materials through the elastic-viscoelastic correspondence 

principle. Much information about this principle has been provided elsewhere; the form given by 

Schapery is adopted here.(8) 

For linear elastic materials, the most general constitutive relationship is given by the following: 

σ ij ijkl kl = C ε (31) 

Cijkl is a fourth-order tensor with 81 constants. Through symmetry arguments, this number is 

reduced to 21. In matrix form, equation 31 can be viewed as equation 32. 

11 12 13 14 15 16

12 22 23 24 25 26

13 23 33 34 35 36

14 24 25 44 45 46

15 25 35 45 55 56

16 26 36 46 56 66

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦

C

 (32) 

If further assumptions such as orthotropic isotropy, transverse isotropy, and cubical isotropy are 

assumed, these numbers reduce further. The simplest of these cases is full isotropy, which 

assumes no directional dependence on the material properties. In this case, two material 

constants are necessary to describe the constitutive relationship between stress and strain. In 

general terms, these are called Lame’s constants and together define the more commonly 

recognized fundamental parameters: Poisson’s ratio (ν ), shear modulus (G), Young’s 

modulus (E), and bulk modulus (K). In terms of compliances, these constants define the shear 

compliance (J), longitudinal compliance (D), and bulk compliance (B). 

For a general isotropic case, it can be shown that C11, C22, and C33 are equal; C12, C13, and C23 are 

equal; and C44, C55, and C66 equal either half of the shear modulus or just the shear modulus 

depending on the definition used for shear strain. All other terms become zero. Written in full, 

assuming the shear strain in equation 31 is given as the total angle form, the stiffness matrix for 

the general isotropic case becomes equation 33. 

( )( )

( )

( )

( )

( )

( )

( )

1 0 0 0

1 000

1 000

1 2

000 0 0

2 1 12

1 2

000 0 0

2

1 2

000 0 0

2

E

νν ν

ν νν

νν ν

ν

ν ν

ν

ν

⎡ − ⎤ ⎢ ⎥ −

−

−

= ⎢ ⎥

+ − ⎢ ⎥ −

−

⎣ ⎦

C

 (33) 
19

It is seen from equation 31 that the use of equation 33 recovers the common form of Hooke’s 

law.(23) For the problem at hand, however, isotropy is of less concern than transverse isotropy. 

For this case, it can be shown that the stiffness matrix is given by equation 34. A change in 

nomenclature from Cij to Zij for general stiffness is made to avoid confusion with later 

terminology. Also, axis three is assumed to be the axis of symmetry. 

11 12 13

11 13

33

44

44

66

000

000

000

0 0

0

ZZZ

Z Z

Z

sym Z

Z

Z

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦

Z

 (34) 

For isotropy, it is seen that the stiffness matrix terms are related to the engineering parameters, 

Young’s modulus and Poisson’s ratio. Using similar terminology, transverse isotropy can be 

considered to have two Young’s moduli and three Poisson’s ratios. The nomenclature used in 

this report for these parameters is as follows: 

1 2

3

12

11 22

3132

33 33

33 33

1323

11 22

13 23

Stiffness on isotropy plane.

Stiffness along axis of symmetry.

Poisson's ratio on isotropy plane.

.

.

Shear modulus between transverse plane

EE E

E

G G

ν

ε ε

ν

ε ε

ε ε

ν

ε ε

===

=

=

=− =−

=− =−

= =

12

and axis of symmetry.

G = Shear modulus on transverse isotropy plane. (35) 

Where: 

ν 1323 is not equal to ν 3132; however, for symmetry, ν 3132/E3 = ν 1323/E. 
20

The relationships between Zij and the values defined in equation 35 are as follows: 

( )

( )

( )

( )( )

11 1323 3132

33 3 12

12 12 1323 3132

13 3132 12 3132 3 1323 12 1323

44 23 13

66 12

2

12 3132 1323 12 3132 1323

1

1

1

12 2

Z E

Z E

Z E

Z E E

ZGG

Z G

ν ν

ν

ν νν

ν νν ν νν

ν ν ν νν ν

=− Υ

= −Υ

=+ Υ

= + Υ= + Υ

= =

=

Υ = −− − (36) 

Or, solving for the engineering parameters, the relationships are as follows: 

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

2 222

11 33 13 12 13 11 33 12

2

11 33 13

2 222

11 33 13 12 13 11 33 12

3 2 2

11 12

2

12 33 13

12 2

11 33 13

13 11 12 13

3132 2 2

11 12

13 11 12 13

1323 2

11 33 13

23 13 44

12 66

2* 2*

2* 2*

Z Z Z Z Z Z ZZ

E

ZZ Z

Z Z Z Z Z Z ZZ

E

Z Z

ZZ Z

ZZ Z

ZZ ZZ

Z Z

ZZ ZZ

ZZ Z

GGZ

G Z

ν

ν

ν

+−− = −

+−− = −

− = −

− = −

− = −

= =

= (37) 
21

If the compliance matrix with generalized compliances, Sij, is considered, then the resulting 

relationships are as shown in equation 38. 

12 3132

3

12 3132

3

1323 1323

3

23

23

12

1 000

1 000

1 000

0 0 0 00 1

0 0 00 0 1

1 0 0 0 00

E EE

EE E

E EE

G

G

G

ν ν

ν ν

ν ν

⎡ ⎤ − −

⎢ ⎥

− −

− −

=

⎣ ⎦

S

 (38) 

The compliance matrix is considered here for completeness. Also, the constitutive relationships 

are more compact and easier to present. 

2.3.2. Application of Schapery’s Work Potential Theory 

Schapery’s work potential theory model contains parameters that characterize the material 

integrity with damage growth.(22,11) To clarify the issue of the physical significance of these 

parameters, a step-by-step process is followed to link the damage functions to the engineering 

parameters defined in equation 37. 

In terms of principal strains, the strain energy density function as defined by the terms found in 

equation 34 is shown in equation 39: 

222

11 1 11 2 33 3 12 1 2 13 1 3 23 2 3

1 2* 2* 2*

2

WZ Z Z Z Z Z = +++ + + ⎡ ⎤ ε ε ε εε εε εε ⎣ ⎦ (39) 

Schapery shows that a more convenient way to express the strain energy density function for 

transversely isotropic materials is as follows: 

2 2 2

11 22 3 12 3 66 2

1 2 *

2

W A e A e A ee A e v v = ++ + ⎡ ⎤ ⎣ ⎦ (40) 

Where: 

123 3 3 2 21 , 3 and v v e ee e =++ =− =− ε εε ε ε ε (41) 
22

The stiffness values, Aij, in equation 40 can be damage dependent for both the elastic and 

viscoelastic case. Through an expansion of equation 40, the Aij values correspond to the more 

recognizable forms in equation 39, thusly: 

11 22 11 22 12 66

33 11 22 12

12 11 22 12 66

23 13 11 22 12

1 2

9 3

4 4

9 3

1 2

9 3

2 1

9 3

Z Z A A AA

ZA A A

Z A A AA

ZZA A A

= =+ − +

=+ +

=+ − −

==− + (42) 

Schapery presents the energy density function as a dual energy density function so that the 

damage characteristics of a cylindrical body subjected to confining pressure, p, and axial 

deformation can be more easily characterized. This dual energy density function, as derived by 

Schapery, is presented in equation 43. To arrive at the relationship presented in equation 43, 

Schapery assumes that work potential exists and uses the principle of virtual work on a 

cylindrical specimen subjected to axial loading and confining pressure.(11) Then, he defines a 

general work potential theory and expands it into a power series, canceling terms to satisfy the 

initial conditions. The relationships between the Aij terms and the damage-dependent variables, 

Cij, are presented in equation 44. These relationships must be satisfied for consistency between 

equation 40 and equation 43. The Cij terms in equation 43 and equation 44 are not the same as 

those defined in equation 32. 

2 2

11 12 22

1 1

2 2 W C C p Cp d = ++ ε ε (43) 

( )2

12

11 11

22

11 12 12

12

22

2

12

22 11

22

44 66 0

1 3

9

1

3 3

C

A C

C

CC C A

C

C A C

C

AAG

⎡ − ⎤ = − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

⎛ ⎞ =+ − ⎜ ⎟ ⎝ ⎠

= −

= ≅ (44) 
23

Making substitutions and canceling terms, the engineering parameters in equation 37 are 

expressed as a function of Cij: 

( )

( )

( )

( )

( )

( )

3 11

0 11

2

11 0 12 12 22 11

3132 12

12 0

1323 2

11 0 12 12 22 11

2

11 0 12 12 22 11

12 2

11 0 12 12 22 11

4

1 2

1 1

2

2 1

1 2

1 2

1 2

E C

G C E

C G C C CC

C

C G

C G C C CC

C G C C CC

C G C C CC

ν

ν

ν

=

= + +− −

= −

− = + +− −

− +− − = + +− − (45) 

For the compliance matrix, the following can be seen: 

( ) ( )

( ) ( )

2 2

11 0 12 12 22 11 11 0 12 12 22 11 12

0 11 0 11 11

2 2

11 0 12 12 22 11 11 0 12 12 22 11 12

0 11 0 11 11

12 12

11 11 11

0

0

1 2 1 2 (1 ) 000

4 42

1 2 1 2 (1 ) 000

4 42

(1 ) (1 ) 1 000

2 2

0 00 1 0 0

0 00 0 1

C G C C CC C G C C CC C

G C G C C

C G C C CC C G C C CC C

G C G C C

C C

C C C

G

G

+ + − − −+ + − − − −

−+ + − − + + − − − −

− − − − S =

0

0

1 0 0 0 00 G

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦(46) 

It is worthwhile to consider the definitions of stress and strain used in equation 43. Stress is 

defined as the stress above stress due to pressure (i.e., the deviator stress). According to the 

definition of strain given in the formulation of the work potential theory presented in the Park 

dissertation, strain includes all changes in length, even those associated with volume change.(24)

In an earlier paper, strain is defined clearly as the change in axial displacement due to load 

divided by the initial length.(11) To clarify the confusion, virtual work is defined by the 

following: 

( )

1

N

i i

i

δ Work Q dq == ∑ (47) 

In equation 47, Qi are the generalized loads, and qi are the generalized displacements. Whether 

the strain is defined as the total strain, the evaluated value of equation 47 does not change 

because the derivative of the strain is independent of these two definitions. Still, the issue does 

present some consequences in the characterization stage, and so, equation 40 and the 
24

relationships in equation 44 will be used to clarify the issue. For a transversely isotropic axial 

specimen subjected to confining pressure and axial elongation along the axis of symmetry, the 

following can be shown: 

3 3 11 12 ( ) 22 1 11 12 ( 22 )

3

44 12 2 39 39

W

σ ε A AA AAA ε

ε

∂

== + + + + −

∂ (48) 

Using the relationships in equation 44, equation 48 can be recast with υ defined the same way as 

ev from equation 40 as the following: 

( )

[ ] 12

3 3 11 12 3

22

C 1

C C

C

σ ε υε − =+ − (49) 

From equation 43, C22 and C12 are related by the following: 

12

22

C C

p

υ − ε = (50) 

If it is assumed that the strain in equation 50 is the total strain,ε 3, and that equation 50 is 

substituted into equation 49 and rearranged, then the following occurs: 

3 3 11 12 σ + p = + ε C Cp (51) 

Equation 51 is compared to the similar form obtained from equation 43 to achieve equation 52: 

11 12

Wd σ ε C Cp

ε

∂

== +

∂ (52) 

Observing that the sign convention for pressure is opposite that of stress above pressure, a 

comparison of equation 51 and equation 52 indicates that the stress in equation 43 is the deviator 

stress, and the strain is the total strain. 

2.4. THEORY OF VISCOPLASTICITY 

The hot mix asphalt (HMA) mixture is a pavement material that exhibits both viscoelastic and 

viscoplastic behavior and therefore shows complicated rate-dependent behavior for repetitive 

traffic loadings. Rutting, one of the major distress types in HMA pavements, is directly related to 

the rate-dependent permanent deformation behavior of HMA. To predict the rutting performance 

of HMA, much effort has been made to develop constitutive models capable of describing the 

rate-dependent permanent strain development in HMA. Some researchers suggest a viscoplastic 

model with strain-hardening features.(25) As the simplest model, it is also able to describe 

monotonic behavior in tension, as shown by researchers.(7,10) Others have presented a 

viscoplastic model for HMA that incorporates Perzyna’s flow rule with Desai’s yield 

function.(26,27) While still others suggest a simplified hierarchical single surface (HISS)-Perzyna 

model that shows a reasonable viscoplastic strain prediction.(28) However, the difficulty in 

developing a constitutive model for HMA is that hardening rules based on the behavior of metals 
25

or soils are not necessarily appropriate to describe both viscoplastic and viscoelastic behaviors of 

HMA. That is, models developed for metals and soils describe only elastic recovery during 

unloading, whereas HMA shows nonlinear strain recovery during unloading due to the 

viscoelastic property of the material. An important observation supporting this phenomenon was 

made by Saadeh.(29) 

2.4.1. Flow Rule 

The general concepts behind the constitutive equations for plastic deformation were proposed by 

Von Mises based on the theory of elasticity.(30) As such, the strain tensor is related to the stress 

through an elastic potential function, the complementary strain energy, Ue. 

e e

ij

ij

U

ε

σ

∂ = ∂ (53) 

Where: 

ε ij

e = Elastic strain tensor. 

Ue = Elastic complementary strain energy. 

σ ij = Stress tensor. 

The plasticity theory based on the above flow rule is called the plastic potential theory. When the 

state of stress reaches the yield criterion, f, plastic strain develops; this mechanism is called 

plastic flow. To generalize this concept to the plasticity theory, Von Mises proposes that a plastic 

potential function, g(σ ij), exists.(30) The plastic strain rate, dε ij

p

, can then be derived from the 

following flow rule: 

p

ij

ij

g dε λ

σ

∂ = ∂ (54) 

Where: 

dε ij

p

 = Plastic strain rate. 

λ = Positive scalar factor. 

ij

g

σ

∂

∂ = Gradient of the plastic potential, g(σ ij). 

In equation 54, λ is a proportional positive scalar factor that can be determined from the yield 

criterion. For some materials, the plastic potential function, g, and the yield function, f, can be 

assumed to be the same. These kinds of materials are considered to follow the associative flow 

rule of plasticity. In other words, the normality rule for this material is associated with the yield 

criterion, f. However, for other materials, the plastic potential function, g, and the yield function, 

f, are different. These materials are considered to follow the nonassociative flow rule of plasticity, 

and the flow rule of the material is derived from a plastic potential, g. In this case, ∂ g/ ∂σ ij is not 

proportional to ∂ f/ ∂σ ij, and therefore, the plastic strain direction is not normal to the yield 

surface, f. Basically, the viscoplastic flow rule takes a similar form to the plastic flow rule except 
26

that it takes advantage of the over-stress function as a replacement forλ , as shown in 

equation 55. 

0

vp

ij

ij ij ij

f f f ε λ

σ ησ σ

∂ Φ∂ ∂ = = = ΓΦ

∂ ∂ ∂ & (55) 

Where: 

Φ = Overstress function. 

η0 = Viscosity. 

Figure 8 represents the viscoplastic flow rule using a mechanical analog, which combines a 

dashpot and a slip element in parallel.(31) In this case, the overstress function in equation 55 is 

represented by the difference from the applied stress (σ ) and the yield stress (G) to the viscosity 

(η 0) in this model. 

η0 G

σ

Figure 8. Illustration. Mechanical analog for the viscoplastic model. 

2.4.2. Yield Criterion 

In conventional viscoplasticity, the elastic limit of the material can be defined by a surface-instress space. Mathematically, the yield surface for general anisotropic materials is expressed as 

equation 56. 

( ) 0 ij f σ = (56) 

For an isotropic material, the orientation of the principal axis is immaterial, and the principal 

stresses, σ 11, σ 22, and σ 33, are sufficient to describe the state of stress. The principal stresses 

form the integrity basis; it is common to use I1, J2, and J3 as the integrity basis. Therefore, the 

yield function becomes equation 57 for the isotropic material. 

f fIJJ (σσσ 11 22 33 , , ,, 0 ) = ( 1 2 3 ) = (57) 
27

Where: 

I1 = σ 11+ σ 22+σ 33 (58) 

J2 = 

1

2 ij ij s s (59) 

J3 = 

1

3 ij jk ki ss s (60) 

sij = 

1

3

σ ij kk ij − σ δ (61) 

δ ij = Kronecker delta. 

Physically, I1 represents the hydrostatic pressure, and J2 represents the distortional energy in the 

material; no clear physical meaning is related to J3. Generally, yield criteria can be classified into 

two subgroups according to their dependence on hydrostatic stress. The isotropic hardening 

model is the simplest hardening model and is based on the assumption that the yield surface 

expands isotropically as the plastic strain develops. The typical isotropic hardening model is 

presented in equation 62 and figure 9. Because the loading surface expands uniformly, it cannot 

account for the Bauschinger effect observed in various materials, which describes the reduction 

of compressive yield strength due to a previously applied tensile stress, or vice versa. Therefore, 

using only the isotropic hardening model frequently limits the characterization of the material 

behavior when both tension and compression loads are applied. 

f K fs (σ κ ij , 0 ) = ( ij) − = (62) 

Where: 

sij = Deviatoric stresses. 

κ = Isotropic hardening parameter. 

The kinematic hardening model assumes that during plastic deformation the yield surface 

translates as a rigid body in the stress space and has the same shape and size as the initial yield 

surface. The kinematic hardening model is represented by equation 63 and figure 10. 

f a fs (σ α ij , 0 ) = ( ij ij − =) (63) 

Where: 

sij = Deviatoric stresses. 

α ij = Kinematic hardening parameters (i.e., coordinates of the center of the yield 

 surface in the deviatoric stress space). 

Equation 64 and equation 65 conceptually represent the classic kinematic hardening rules 

suggested by Prager, Armstrong-Frederick, and Chaboche, respectively.(32–34) The kinematic 

hardening rate is a function of the viscoplastic strain rate in these equations, and the hardening 

rate is always zero when there is no change in viscoplastic strain. Therefore, when a material is 

subjected to one-directional loading, such as a constant strain-rate test and a repetitive creep and 

recovery test, the yield stress increases only in the direction of the developed viscoplastic strain 

and never decreases. 
28

( ) vp α& & = g ε (64) 

( , ) vp α& & = g ε α (65) 

Where: 

α = Kinematic hardening stress. 

vp ε& = Viscoplastic strain rate. 

σ1 

σ2 

Initial yield 

surface 

Subsequent yield surface 

Figure 9. Illustration. Isotropic hardening diagram. 

σ1 

σ2 

Initial yield 

surface 

Subsequent yield surface

(b) 

Figure 10. Illustration. Kinematic hardening diagram. 

2.5. VISCOPLASTIC MODELS 

2.5.1. Simple Strain-Hardening Model 

A simple strain-hardening model has been suggested and is shown in equation 66, which assumes 

that viscosity obeys a power law in viscoplasticity. Researchers have shown that the model is 

applicable to monotonic behavior in tension.(7,10,25) 
29

( )

( ) vp

vp

g σ

ε

η ε & = (66) 

Where: 

g(σ ) = Stress function. 

η = Viscosity. 

Assuming that η is a power law in the viscoplastic strain, equation 67 becomes the following: 

( )

vp p

vp

g

A

σ

ε

ε & = (67) 

Where: 

A and p are model coefficients. Rearranging and integrating yield the following: 

( ) p

vp vp

g dt d

A

σ

ε ε = (68) 

( ) 1

0

1 t

p

vp

p g dt A

ε σ + + = ∫ (69) 

Raising both sides of equation 69 to the 1/(p + 1) power yields the following: 

( )

1 1

1 1

0

1 t p p

vp

p g dt A

ε σ + + + ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∫ (70) 

Letting g(σ ) = Bσ 1

q

 and coupling coefficients A and B into coefficient Y, equation 70 becomes 

the following: 

1 1

1 1

0

1 t p p q

vp

p dt

Y

ε σ + + + ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∫ (71) 

In the current work, the coefficients, p, q, and Y, are pressure-dependent quantities. 

Typically, viscoplastic models are characterized using creep and recovery tests. These tests allow 

relatively easy separation of the viscoplastic and viscoelastic components, as shown in figure 11. 

However, it is difficult (if not impossible in some machines) to maintain zero load during the 

recovery period of the creep and recovery test in tension. Therefore, in tension, viscoplastic 

characterization uses constant rate tests in which the VECD model is used to first predict the 

viscoelastic strains. These viscoelastic strains are then subtracted from the measured strains to 

provide the viscoplastic strains that are needed for curve fitting to equation 71. 
30

The advantage of this model is that it is easy to implement and does not consume much 

computational time. This model’s ability to predict the HMA behavior under complex loading 

histories in tension is reported by Underwood et al.(12) However, this model’s major weakness is 

its one-dimensional nature, which is not sufficient to describe the behavior of HMA in 

pavements. This deficiency is particularly troublesome in compression where the confinement is 

known to play a major role in permanent deformation behavior of HMA. In the following 

subsections, more general viscoplastic models are presented. 

Viscoelastic 

Plastic

Elastic

Elastic

Viscoelastic

+ 

Viscoplastic 

Time 

Axial Strain 

Figure 11. Illustration. Strain decomposition from creep and recovery testing. 

2.5.2. HISS-Perzyna Model 

The HISS plasticity model provides a general formulation for the elastoplastic characterization of 

material behavior. This model, which is potentially able to incorporate isotropic and kinematic 

hardening and associated and nonassociated plasticity characterizations, can be used to represent 

a material response based on the continuum plasticity theory. Therefore, the HISS model allows 

the selection of a more appropriate derivative model for a given material in a specific 

engineering application. Various well-known plasticity models, such as the Von Mises, MohrCoulomb, Drucker-Prager, continuous yielding critical-state, and capped models, can be derived 

as special cases of the HISS model.(30,32,35) 

2.5.2.1. HISS Model Implemented by Delft University of Technology 

Equation 72 represents the HISS criterion.(35) In the criterion, n and α determine the shape of the 

yield stress in deviatoric-hydrostatic stress space, and γ represents ultimate yield stress. R

represents the cohesion of the material and determines the position of the yield stress with 

respect to the hydrostatic stress axis. Because the yield stress of the HISS criterion varies 

depending on the first stress invariant, I1, the model exhibits a spindle shape of its yield surface 
31

when the shape of the yield surface is assumed to be circular (β =0) in the deviatoric stress space, 

as shown in figure 12. 

( )

2

1 1

2

2

3

3

2

2

0

3 3 1

2

n

a a

a

IR IR

P P J f P

J

J

γ α

β

⎡ ⎤ ⎛ ⎞⎛ ⎞ − − ⎢ ⎥ ⎜ ⎟⎜ ⎟ −

⎝ ⎠⎝ ⎠ ⎣ ⎦ = − =

⎛ ⎞ ⎜ ⎟ − ⎜ ⎟

⎝ ⎠ (72) 

Where: 

γ = Softening parameter. 

α = Hardening parameter. 

R = Tensile strength of material when deviatoric stress is zero. 

n = Parameter determining shape of yield stress. 

β = Parameter determining shape of yield stress in deviatoric stress space. 

Pa = Atmosphere pressure. 

σ1

σ2

σ3 σ1= σ2= σ3

Figure 12. Illustration. Typical yield surface of HISS model. 

Researchers using this model suggest an HISS criterion whose parameters are the strain ratedependent functions for a given HMA mixture.(36) For the characterization, a series of constant 

strain-rate tests in tension and in compression are performed at several strain rates and 

temperatures. Then, predictions are made for indirect tension (IDT) specimens subjected to 

constant strain-rate loading. Because this model requires constant strain-rate testing using several 

different strain rates and temperatures for characterization, it is important to determine the 

appropriate range of the strain rates and temperatures to minimize the amount of experimental 

effort. Although it appears that the model can successfully explain the viscoplastic behavior 

when subjected to monotonic loading, numerical issues related to parameter determination and 

prediction still remain. In addition, because the model is characterized using constant strain-rate 

tests, it cannot fully explain the behavior of HMA under discontinuous loading, such as repetitive 

creep and recovery loading. 
32

2.5.2.2. HISS Model Implemented by the University of Maryland 

A viscoplastic model based on the simplified HISS model and using Perzyna’s flow rule has also 

been suggested.(28) Repetitive creep and recovery tests are used for both calibration and 

prediction, and numerical optimization techniques are adapted for calibrations, unlike Erkens’ 

model.(36) As shown in equation 73, γ , which represents both the ultimate yield stress and the 

softening of the material, is considered a constant, and R is a function of the viscoplastic strain. 

In Erkens’ study, however, γ and R are functions of the viscoplastic strain rate and 

temperature.(36)

( ) ( ) () () ( ) 2

2 1 1

n

FJ IR IR γ ξ αξ ξ =− − − − ⎡ ⎤ ⎣ ⎦ (73) 

Where: 

ξ = Viscoplastic strain trajectory. 

γ = Softening parameter (constant). 

α = Hardening parameter (function of viscoplastic strain). 

R = Tensile strength of material when the deviatoric stress is zero. 

The main contribution of this research is to apply the t-TS principle to a conventional 

viscoplastic constitutive model and confirm the validity of the superposition principle. 

2.5.3. Unified Model 

In theory of viscoplasticity, the term unified constitutive model refers to models that describe the 

rate-dependent viscoplastic strain for steel or polymer. However, in other disciplines, the unified 

constitutive model is used to represent not only viscoplastic strain but also viscoelastic strain. 

Because this type of model takes a more flexible form of hardening equations than other 

viscoplastic models, it is worth reviewing the approaches used in the unified model. 

2.5.3.1. Linear Kinematic Hardening Model 

The simplest unified constitutive model is the linear kinematic hardening model shown in 

equation 74 and equation 75.(37) The inelastic strain rate has a linear relationship with the 

overstress, σ - α . The back stress rate is a function of the back stress in the previous time step 

and the strain rate in the current time step. Because the model is designed to explain inelastic 

strain, which is the summation of viscoelastic strain, plastic strain, and viscoplastic strain, the 

inelastic strain rate can be negative depending on the stress history. The back stress rate,α& , is 

another representation of the Maxwell model of mechanical analog.(50)
33

( ) in ε& = C σ α− (74) 

in α& = − Aε αB (75) 

Where: 

in ε& = Inelastic strain. 

α = Back stress. 

A,B,C = Material constants. 

2.5.3.2. Chaboche Model 

In general, the viscoplastic strain of a material obeys a power law, and the hardening of the 

material can be represented by kinematic and isotropic hardening, as shown in equation 76. 

( ) ( ) , ,

n

vp

ij

ij ij

F F F k

k

D

σ α

ε σα

σ σ

∂ − − ∂ = Φ = ∂ ∂ & (76) 

Where: 

Φ = Magnitude of strain rate. 

α = Kinematic hardening function (back stress function). 

k = Isotropic hardening function. 

The Chaboche model is a viscoplastic model consisting of the above flow rule and hardening 

function represented by a summation form of back stress.(34) By using decomposed back stress, 

the model is capable of describing nonlinear hardening with enhanced accuracy for a wider range 

of viscoplastic strain. Equation 77 shows the back stress function in the Chaboche model; it 

becomes the Armstrong-Frederick model when n = 1. 

( )

1

n

i vp

i

α α ε

=

= ∑ (77) 

Where: 

α i(ε vp) = i

th back stress, which is a function of the viscoplastic strain. 

2.5.3.3. Krempl and Ho Models 

As an advanced and recent form of the linear kinematic hardening model, some researchers have 

proposed a viscoplastic model based on the overstress concept.(38,39) The constitutive model also 

includes a description of the time-dependent recoverable strain of the materials. It begins with 

the assumption that the coefficients of viscoelasticity are nonlinear functions. Equation 78 is the 

differential equation used to derive the model.
34

mg g kg ( ) σ − ×+ = + − × (ε ε εσ σ ε σ ) & ( ) ( ( )) & (78) 

Where: 

m(σ -g(ε )) = Positive, bounded and even function. 

k(σ -g(ε )) = Positive, bounded and even function. 

g(ε ) = Odd function of strain. 

σ -g(ε ) = Overstress. 

When the functions m(σ – g(ε )) and k(σ – g(ε )) become constant and when g(ε ) is linear in ε , 

equation 78 reduces to the differential equation of a standard linear solid model that represents 

viscoelastic behavior. The equation can then be expanded to the regular convolution integrals of 

linear viscoelasticity. Because equation 79 holds true for slow loading, the relationship between 

the strain rate and stress can be expressed as equation 80, the simplest form of the viscoplastic 

model.(40)

( ( ))

( ) ( )

m g E

k g

σ ε

σ ε

− = − (79) 

Where: 

E = Material elastic modulus. 

( )

( ) ( )

g

E Ek g

σ σ ε

ε

σ ε

− = +

× − & (80) 

More complicated model forms are available for describing hardening, dynamic and static 

softening, and relaxing, as well as negative and positive rate sensitivities. However, as shown in 

equation 80, Krempl’s constitutive models have neither loading or unloading conditions nor the 

concept of yield stress because they are derived from the concept of general viscoelasticity. 

Therefore, it is not appropriate to describe only the viscoplastic behavior of HMA, even though 

the model gives an idea of the rate dependency of viscoplastic properties under unloading 

conditions. This potential shortcoming is overcome by introducing Macauley’s bracket into the 

second term of equation 80.(41) Because the model is based on equation 80, the back stress 

function is related to the kinematic stress function, as shown in equation 83. 

( ) m

in HR G

B

D G

σ σ

ε

σ

−− − = − & (81) 

Where: 

D = Drag stress. 

R = Isotropic hardening function. 

H = Kinematic stress function ( in H E = ε & & ) (82)

G = Back stress. 
35

( ) in G H in G H

R ψε ε

⎡ ⎤ − = ⎢ ⎥ − +

⎣ ⎦

& & & & (83) 

Where: 

B, m, and ψ = Material constants characterized from experimental results. 

In contrast to Krempl’s model, this model does not allow a change in the material state during 

unloading because the state of the material is a function only of the viscoplastic strain rate. 

2.6. VISCOELASTOPLASTIC CONTINUUM DAMAGE MODEL 

In the previous sections, the viscoelastic and viscoplastic models are developed separately. It has 

been shown by using a thermodynamic formulation that the total strain is the sum of the 

viscoelastic and viscoplastic components even in the non-small strain region.(10) Thus, the 

viscoelastic and viscoplastic models can now be integrated to form the VEPCD model in which 

the predicted viscoelastic and viscoplastic responses are combined to obtain the total response 

for a given stress history. 

The resulting equation from combining equation 20 and equation 71 predicts the total strain 

history for a general loading history at a reference temperature. 

( )

( )

( )

12

1 1 1 1 11

0 0

1 ' ' '

p p

q

T R

C Sp d

C S p E D d d

d Y

ξ ξ

σ

ε ξξ ξ σ ξ ξ

+ +

⎛ ⎞ − ⎜ ⎟

+ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ = − + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∫ ∫ (84) 

Equation 84 is presented in terms of reduced time, (ξ ), due to the verification of the t-TS 

principle with growing damage. Stated simply, this principle allows the use of time-temperature 

shift factors determined from LVE characterization to combine the effects of time and 

temperature at higher levels of damage. This simplification significantly reduces the required 

testing protocols while simultaneously expanding the realm of application for the model. 

Section 4.5 and subsection 5.1.3 present the methodology and experimental work used to verify 

this principle for both tension and compression modes of loading, respectively. 

37

CHAPTER 3. SPECIMEN FABRICATION AND TESTING PROTOCOLS 

3.1. MATERIALS 

Component materials used in this study were obtained from FHWA’s Turner-Fairbank Highway 

Research Center (TFHRC). Two options for materials acquisition were available to the 

researchers at the outset of the project. The first option was to acquire already mixed asphalt 

concrete that was available and used for construction of the ALF testing lanes. The second option 

was to have the component materials sampled and transported to NCSU for fabrication of 

appropriate asphalt concrete mixtures. Based on the availability and desired consistency of the 

materials, the latter option was chosen, and component materials were obtained from TFHRC. 

Four stockpiles, #68, #78, #10, and Sand (FHWA Designation B-6265, B-6264, B-6306, and 

B-6263, respectively), were sampled and transported to NCSU for fabrication. For the ALF lanes, 

hydrated lime was first mixed with the #10 stockpile before mixing it with the asphalt binder. 

The decision was made to not use this stockpile for fabrication; instead, hydrated lime was added 

separately on a specimen by specimen basis at NCSU. In addition to acquiring aggregates, four 

asphalt binders, PG 70-22 (B-6298), Crumb Rubber Terminal Blend (B-6286), StyreneButadiene-Styrene (B-6295), and Ethylene Terpolymer (B-6289), were also acquired. 

The aggregate structure for each of the mixtures was constant and was a coarse 12.5-mm 

nominal maximum size aggregate (NMSA) mixture comprised of 18.0 percent #68 stone, 

36.4 percent #78 stone, 17.1 percent #10 stone, 27.5 percent sand, and 1.0 percent lime. For the 

multiaxial tension work, a slightly modified mixture was used. This mixture contained the same 

binder and gradation as the Control mixture; however, the #78 aggregate for this mixture was 

obtained almost 4 years after the original aggregate was obtained. To reflect the later date of 

aggregate acquisition, this mixture is referred to as Control-2006. The blended gradation is 

shown in figure 13. Four asphalt binders, three polymer-modified and one unmodified, of similar 

performance grade were used for this study. Table 1 summarizes the relevant information for 

each of these binders. The asphalt content for each mixture was set at 5.3 percent by total 

mixture mass based on the mix design values. 

Table 1. Relevant asphalt binder information. 

Binder Designation PG Grade Cont. PG Grade

Unmodified Control/Control-2006 70–22 72–23

Crumb Rubber Term. Blend CR-TB 76–28 79–28

Styrene-Butadiene-Styrene SBS 70–28 74–28

Ethylene Terpolymer Terpolymer 70–28 74–31
38

0

20

40

60

80

100

01234

Sieve Size .45 Power

% Passing

Control Points

Restricted Zone

Mixture Gradation

Figure 13. Graph. Mixture gradation chart. 

The laboratory materials were similar to those found in the field with the following exceptions: 

• Differences between the laboratory air voids and the constructed air voids. 

• Differences in internal structures (not quantified) between the compaction techniques 

used—gyratory in the lab and roller in the field. 

• Small differences in the mixture gradations due to construction factors. 

The actual field mixture conditions are summarized in table 2 and figure 14. 

Table 2. Summary of constructed lanes’ air void and asphalt content. 

Mixture 

Laboratory Test Lanes 

Percent 

Air Voids

Percent 

Asphalt1

Percent 

Air Voids

Percent 

Asphalt1

Ignition 

Percent 

Asphalt1

Nuclear 

Control 

4.0 5.3 

7.8 5.23 4.82 

CR-TB 6.8 5.48 5.09 

SBS 6.0 5.28 5.05 

Terpolymer 6.5 5.52 5.25 1

 By percent mass. 
39

0

20

40

60

80

100

01234

Sieve Size .45 Power

% Passing

Laboratory

Control Lane

CRTB Lane

SBS lane

Terpolymer Lane

Figure 14. Graph. Comparison of test lane and laboratory gradations. 

3.2. SPECIMEN FABRICATION 

All specimens were compacted by the Superpave Gyratory Compactor to a height of 178 mm 

and a diameter of 150 mm. To obtain specimens of uniform air void distribution, these samples 

were cored and cut to a height of 150 mm with a diameter of either 75 mm for tension testing or 

a diameter of 100 mm for compression testing. Details can be found elsewhere.(19,20)

After obtaining specimens of the appropriate dimensions, air void measurements were taken via 

the CoreLok method, and specimens were stored until testing. The air voids for all tests in this 

study were between 3.5 and 4.5 percent. During storage, specimens were sealed in bags and 

placed in an unlit cabinet to reduce aging effects. Furthermore, no test specimens were stored for 

longer than 2 weeks before testing. 

3.3. TEST SETUP 

A closed-loop servo-hydraulic loading frame was used for all the tests. Depending on the nature 

of the test, either an 8.9- or a 25-kN load cell was used. An environmental chamber, equipped 

with liquid nitrogen coolant and a feedback system, was used to control and maintain the test 

temperature. 

Measurements of axial and, in some instances, radial deformations were taken during loading. 

Axial measurements were taken at 90-degree intervals over the middle 100 mm of the specimen 

with loose-core linear variable displacement transducers (LVDTs). When taken, radial 

deformations were measured at 90-degree intervals with spring-loaded LVDTs. These 

measurements were taken on the central plane of the specimen. Load, crosshead movement, 

triaxial cell pressure (when appropriate) and deformation data for the specimen were acquired 

with National Instruments® hardware and collected with LabViewTM software. 
40

Prior to all tension testing, steel end plates were glued to the specimen with Devcon® steel putty. 

Extreme care was taken to completely clean both the end plates and the specimen ends before 

each application. It was observed early in the testing that failure to do so could result in the 

premature failure of the test. To ensure that the specimens were properly aligned, a special 

gluing jig was employed to ensure that the end plates were parallel, minimizing any eccentricity 

that might occur during the test. The triaxial tension end-plates had holes drilled specifically so 

that after the end plate glue had cured, a drill could create another hole through the glue at the 

center of the bottom end of the specimens to allow them to drain excess pore pressure during testing. 

For the uniaxial constant crosshead rate compression tests, a circular hole that was 3 mm in 

depth and 14 mm in diameter was made at the center of the top surface of the specimen. The top 

plate had a key with the same dimensions in its center. The key and the hole were used to prevent 

the specimen from sliding during the test. For the creep and recovery test specimens, an 

aluminum end plate was used to minimize the creep due to the weight of the end plate. A 

0.3048-mm thick rubber membrane with a 100-mm diameter and lubricant were used to avoid 

the end effect caused by friction between the end plate and the surface of the specimen. 

Tests performed under confining pressure were the most difficult to prepare. The preparation 

protocols for confined tests are similar to those of an unconfined setup, except that before testing 

the specimen is encased in a latex membrane. To ensure proper drainage during the compression 

tests, the bottom lubricated membrane was punctured with a hole approximately 25 mm in 

diameter. This drainage ensured that no excess pore pressure developed during testing. 

Following the protocol presented elsewhere, the specimen was encased in a latex membrane after 

attaching LVDT mounting studs.(42) Early observations showed that extreme care had to be taken 

when attaching LVDTs and that failure to remove all air pockets between the membrane and 

specimen, particularly around the LVDT studs, resulted in unusual and inconsistent results. In 

this study, a hole in the membrane approximately 2.5 mm in diameter was punctured at the 

center of each LVDT mounting stud. This hole was then stretched around the LVDT studs, and 

all surrounding air pockets were removed. Studs were mounted and prepared similarly for radial 

measurements. Mounts were prepared for radial measurements instead of directly measuring the 

membrane-encased specimen because it was found that deformation of the membrane was 

significant during pressurization. Finally, after all LVDT mounting brackets were attached, 

acrylic latex caulk was used to seal the areas around the mounts. 

3.4. TEST PROTOCOLS 

The laboratory testing was divided into four phases. The objective of the first phase was to 

determine the temperature and loading frequency dependence of the four mixtures. Temperature 

and frequency sweep complex modulus testing, either under confinement or not, were performed 

during this phase. 

The second phase was intended to verify the t-TS principle with growing damage under 

confinement. This phase was important because the positive verification of the t-TS principle 

with growing damage under confinement significantly reduced the testing program for 

MVEPCD model development. Due to the limitation of time and resources, only the constant 

crosshead rate tests were performed for validation of the t-TS principle. 
41

The third phase of laboratory testing was designed to determine the MVEPCD model 

coefficients of the mixtures. The testing program required for this phase consisted of the 

following: 

• Constant crosshead rate tests at low temperatures and fast loading rates under both 

confined and unconfined conditions for the development of the viscoelastic continuum 

damage model. 

• Constant crosshead rate tests at high temperatures with a range of loading rates or creep 

and recovery tests for the development of the viscoplastic model. The creep and recovery 

tests required a constant loading time with varying load amplitudes and constant loading 

amplitude with varying loading times. 

The final phase of the experimental program was designed to validate the developed model. In 

this phase, constant rate tests under conditions not used in model development were used. 

The four phases of testing applied both to tension and compression testing. For the tension 

testing, 75-mm diameter, 150-mm tall specimens were used, whereas 100-mm diameter, 150-mm 

tall specimens were used for the compression testing. 

3.4.1. Complex Modulus Test 

The complex modulus test was performed in the load-controlled mode in axial tensioncompression (zero-mean deviatoric stress) or in compression only (zero-maximum deviatoric 

stress) according to American Association of State Highway and Transportation Officials 

(AASHTO) TP62-03. For the zero-mean deviatoric stress condition, confining pressures of 

0, 250, and 500 kPa were tested, whereas the zero-maximum deviatoric stress tests were 

performed for confining pressures of 0, 140, and 500 kPa. Load levels for these tests were 

determined by a trial and error process so that the resulting strain amplitudes were between 

50 and 70 με . Based on the work of other researchers, it was assumed that this criterion ensured 

an accurate viscoelastic characterization.(7,42) Under both confined and unconfined conditions, 

tests were performed at five temperatures (-10, 5, 20, 40, and 54 °C) and six frequencies (25, 10, 

5, 1, 0.5, and 0.1 Hz) or in compression at four temperatures (-10, 10, 35, and 54 °C) and eight 

frequencies (25, 10, 5, 1, 0.5, 0.1, 0.05, and 0.01 Hz). 

3.4.2. Constant Crosshead Rate Tests 

Constant crosshead rate tests were performed with the application of a constant rate of 

deformation over the complete loading train. Because each component of the loading train 

(machine ram, load cell, etc.) deformed slightly, the on-specimen displacement rate or strain rate 

was not constant.(20) Prior to the confined tension test, a constant confining pressure was applied 

for 2 hours before testing. This 2-hour period ensured that bulk creep could not affect the 

analysis. To simplify the discussion of these results for the ready, a test identification (ID) 

system (e.g., 55-1, 55-2, etc.) has been used to describe the data for the remainder of this report. 

The first number in the test ID is the test temperature, and the second number indicates the 

ranking of the strain rate, with 1 being the fastest. For tension tests, the test ID is followed by a 

“T” (e.g., 55-1-T, 55-2-T). 
42

Tests in tension were performed at three temperatures (5, 25, and 40 °C) and at multiple rates. 

These tests were used for the following: 

• Viscoelastic (Multiaxial viscoelastic continuum damage (MVECD)) damage 

characterization (5 °C tests). 

• Viscoplastic (VP) characterization (40 °C tests). 

• MVEPCD model validation. 

• t-TS principle with growing damage verification. 

Table 3. Controlled crosshead testing matrix for Control-2006 in tension. 

Test ID

Confining 

Pressure (kPa) Temp. 

(°C) 

Crosshead 

Strain 

Rate 

Purpose 

0 500 

5-1-T X X 

5 

5.50E-05 

MVECD 

characterization 

5-2-T X X 3.00E-05 

5-3-T X X 2.20E-05 

5-4-T X 2.15E-05 

5-5-T X 1.50E-05 

5-6-T X 1.05E-05 

MVEPCD and 

t-TS verification 

25-1-T X 

25 

5.00E-04 

25-2-T X 1.50E-03 

25-3-T X 4.50E-03 

25-4-T X 1.35E-02 

40-1-T X X 

40 

3.00E-02 

40-2-T X X 1.00E-02 

VP 

characterization 

40-3-T X X 3.00E-03 

40-4-T X X 1.00E-03 

40-5-T X X 3.00E-04 

 

Table 3 presents the strain rates calculated from the crosshead displacement rates over the 

150-mm specimen height that were used for each phase. Tension tests were performed at 5, 25, 

and 40 °C at 0 and 500 kPa. 
43

Tests in compression were performed at 5, 25, 40, and 55 °C at 0 and 500 kPa. For compression 

modeling purposes, these tests were used for the following: 

• Viscoelastic damage characterization (5 °C tests). 

• t-TS principle with growing damage verification. 

To conserve materials while still covering a wide range of material behavior, it was decided that 

the best course of action would be to conduct tests at several different combinations of 

temperatures and rates with one replicate test for each combination. The strain rates used for 

testing at each temperature were selected so that an overlap would occur in the reduced strain 

rates between adjacent temperatures. The relevant information for the compression tests is 

presented in table 4. 

Table 4. Controlled crosshead testing matrix for Control in compression. 

Test ID 

Confining 

Pressure (kPa) Temp. 

(°C) 

Crosshead 

Strain 

Rate 

Purpose 

0 500 

5-1 X X 

5 

5.75E-05 

MVECD 

characterization 

5-2 X X 3.83E-05 

5-3 X X 1.92E-05 

5-4 X X 9.60E-06 

25-1 X X 

25 

1.35E-02 

t-TS verification 

25-2 X X 4.50E-03 

25-3 X X 1.50E-03 

25-4 X X 5.00E-04 

40-1 X X 

40 

3.01E-02 

40-2 X X 1.00E-02 

40-3 X X 3.00E-03 

40-4 X X 1.00E-03 

55-1 X X 

55 

2.99E-02 

55-2 X X 1.00E-02 

55-3 X 3.00E-03 

55-4 X 1.00E-03 

3.4.3. Repetitive Creep and Recovery Tests 

Repetitive creep and recovery tests were conducted in uniaxial and multiaxial compression up to 

failure at 55 °C. These tests consisted of the repeated application of creep and recovery cycles. 

For characterization, two different conditions were applied: one with the loading time fixed and 

the stress magnitude increasing in each cycle (i.e., a variable loading (VL) test) and the other 

with a fixed-stress level and increasing loading time as the loading cycle increases (i.e., a 

variable time (VT) test). To gain insight into the material behavior with respect to sequence of 
44

loading, a VT test was performed in the reverse order of the characterization VT test (i.e., a 

reverse VT (RVT)). For model verification, three different conditions were used: 

1. A test with constant load and constant time (CLT). 

2. A variable load and variable time test (VLT). 

3. A VT test that was followed by a flow number test. 

A summary of the test methods and the confining pressures used with each is shown in table 5. 

The following sections provide a brief summary of each test method used. 

3.4.3.1. VL Test 

The repetitive creep and recovery test with a variable load level was performed for the 

calibration of the developed viscoplastic model. In this VL test, the loading time remained 

constant until the end of the test, whereas the load level was varied. One loading block consisted 

of eight increasing loading pulses. The deviatoric stress of the first loading in the loading group 

was the same as that of the third loading in the preceding loading group. For testing with an 

unconfined and confining pressure of 140 kPa, the first deviatoric stress was 137.9 kPa and was 

stepped by 1.2(n-1) as the number of loads, n, increased, as shown in figure 15. The confined 

testing began with a deviatoric stress of 300 kPa for the 500 kPa confining pressure test, as 

shown in figure 16. For both cases, the loading time was set to 0.4 s, and the loading was 

repeated until the specimens failed. 

Table 5. Creep and recovery testing matrix for Control mixture in compression. 

Confining Pressure (kPa) 

0 140 500 

VT X X X 

RVT X X 

VL X X X 

VLT X X 

CLT 

0.4 s X 

1.6 s X 

6.4 s X 

VT + Flow Number X 
45

0

500

1000

1500

2000

2500

3000

3500

1 6 11 16 21 26 31 36 41 46

Load Number

Stress (kPa)

Figure 15. Graph. Stress history of VL testing (unconfined and 140 kPa confinement VL). 

0

1000

2000

3000

4000

5000

1 6 11 16 21 26 31 36

Load Number

Stress (kPa)

Figure 16. Graph. Stress history of VL testing (500 kPa confinement VL). 
46

3.4.3.2. VT and RVT Test 

The repetitive creep and recovery tests with variable loading times were performed to achieve 

three major goals: 

1. Identify the effect of loading time (pulse time) on viscoplastic strain development. 

2. Verify the t-TS principle. 

3. Characterize the viscoplastic model. 

For the VT tests, the level of deviatoric stress remained the same until the end of the test, but the 

duration of the load was varied. As with the VL testing, these tests consisted of the repeated 

application of load pulses in groups or blocks. The first loading block started with a 0.05-s 

loading pulse followed by 200 s of rest. Longer loading pulses with 200 s of rest were applied 

subsequently until the loading block ended, as shown in table 6. This loading block was repeated 

depending on the deviatoric stress and confining pressure of the test. 

Table 6. Test conditions for the VT and RVT tests. 

Confining 

Pressure 

(kPa) 

Deviatoric 

Stress 

(kPa) 

Pulse Time (rest period) 

(s, physical time at the testing temperature) 

0 827 0.05 

(200) 

0.1 

(200) 

0.2 

(200) 

0.4 

(200) 

1.6 

(200) 

6.4 

(200) — 

140 552, 827 

0.05 

(200) 

0.1 

(200) 

0.2 

(200) 

0.4 

(200) 

1.6 

(200) 

6.4 

(200) — 

0.82 

*(40) 

1.63 

*(40) 

3.27 

*(40) 

6.53 

*(40) 

26.12 

*(50) 

104.49 

*(60) — 

500 1600, 1800, 

2000 

0.05 

(200) 

0.1 

(200) 

0.25 

(200) 

0.4 

(200) 

1.6 

(200) 

2.0 

(200) 

6.4 

(200) 

* Physical time at 40 °C, which is equivalent in reduced time to physical time at 55 °C. 

— No measurement taken.

Note from table 6 that at a confining pressure of 140 kPa two different pulse times and rest 

period histories are given. Tests at this confining pressure were performed for t-TS verification 

with damage, and the second history was performed at 40 °C for this verification. In RVT testing, 

the loading conditions were the same as for VT testing except for the sequence of loading, which 

was exactly opposite to that in the VT tests. 

3.4.3.3. CLT Test 

CLT testing was conducted to confirm the effects of loading time (or pulse time) on the 

viscoplastic strain development. During CLT testing, the loading time, rest period, and load level 

were kept constant. These tests were performed only at the 500 kPa confining pressure, and only 

a single deviatoric stress level of 1,800 kPa was used. Three different loading times of 0.4, 1.6, 

and 6.4 s were used. 
47

3.4.3.4. VLT Test 

VLT testing incorporated the combined loading histories of the VT and VL testing. The test 

started with a deviatoric stress of 100 kPa and a loading time of 0.05 s, as shown in figure 17, 

figure 18, and table 7. After 200 s of rest following the first loading, another loading followed by 

200 s of rest using the same deviatoric stress but with an increased loading time, similar to the 

VT testing. Following the rest period, the deviatoric stress increased to the next stress level, as 

shown in figure 17 and figure 18. This sequence was repeated until the specimen failed. 

100

300

500

700

900

1100

1300

1500

1800

2100

2400

2700

0

500

1000

1500

2000

2500

3000

3500

1 2 3 4 5 6 7 8 9 10 11 12

Loading Number

Stress (kPa)

Figure 17. Graph. Stress history of VLT testing (140 kPa confinement). 
48

100

300

500

700

900

1200

1500

1800

2100

2400

2800

3200

0

500

1000

1500

2000

2500

3000

3500

1 2 3 4 5 6 7 8 9 10 11 12

Loading Number

Stress (kPa)

Figure 18. Graph. Stress history of VLT testing (500 kPa confinement). 

Table 7. Loading times for VLT test. 

Confining Pressure (kPa) Pulse Time (s) 

140 0.05 0.1 0.2 0.4 1.6 6.4 

500 0.05 0.1 0.25 0.4 1.6 6.4 

3.5. DATA ANALYSIS 

Various data analysis methods were evaluated in this research. The major objective in the data 

analysis was to determine the MVEPCD model coefficients. The time-temperature shift factors 

were determined from the frequency and temperature sweep dynamic modulus test. These factors 

were also used to check if the mixtures exhibited t-TS with growing damage. Dynamic modulus 

mastercurves were used for the determination of MVECD characteristics. 

The viscoelastic model development involved the determination of pseudo strain and the damage 

parameter, S. The constant crosshead rate monotonic tests at 5 °C were used to develop the 

damage characteristic curve for each mixture. The viscoplastic model coefficients were 

determined from either constant crosshead rate tests at high temperatures or creep and recovery 

tests with varying loading times and stress levels. 

The MVEPCD model, developed with the above mentioned testing program, was used to 

calculate the viscoelastic and viscoplastic strains under random cyclic loading. A comparison of 

the calculated and the measured strains revealed the accuracy of the MVEPCD model under 

realistic loading conditions. All of the analysis techniques presented were performed with 

analysis software developed at NCSU. 
49

CHAPTER 4. MVEPCD CHARACTERIZATION AND VERIFICATION IN TENSION 

4.1. INTRODUCTION 

Fatigue cracking of asphalt pavements, along with permanent deformation, is known to be a 

major form of pavement distress in the United States. Such cracking occurs due to extensive 

cyclic applications of loads and/or variations in temperature. The typical understanding of this 

cracking phenomenon suggests that these cycles create areas of tensile strain at the bottom of the 

pavement layer, leading to the initiation of microcracks. Under repeated loadings, these 

microcracks densify, coalesce, propagate, and eventually develop into more visible macrocracks 

on the pavement surface. Further research in recent years also suggests that under certain 

conditions cracking can begin at the top of the pavement and progress downward.(43–45)

Traditional fatigue testing and performance prediction are based on empirical methods that are 

developed from the “bottom-up” understanding of fatigue cracking. For fatigue assessment, these 

methods rely on the simulation of field conditions in the laboratory. As such, typical methods 

apply cyclic bending stresses to asphalt beams of a particular dimension and support 

condition.(46) Tests are performed under either controlled stress or controlled strain conditions. 

Using such methods alone can lead to gross errors in performance prediction when field 

conditions vary considerably from those used in the laboratory. Further complications with these 

methods are evidenced by the combined effects of the elements of both material and structure. 

This shortcoming can be bypassed, however, by separating these two behaviors and focusing on 

the material separately from the structure. 

Results of the Strategic Highway Research Program (SHRP) A-003 project acknowledged the 

need to combine material and structural components in some way.(46,47) Researchers on this 

project concluded that experimental work should be performed to extract fatigue properties and 

that mechanistic analysis should be performed to calculate stresses and strains in the pavement 

structure. Although these researchers made considerable advances in beam fatigue testing—an 

ever-present shortfall of these tests—the inability of the test to truly separate the material from 

the structure could not be overcome. 

A fundamental purpose of this report is to show that material behavior can be accurately 

described using a simplified laboratory program by using constitutive models that make rigorous 

theoretical considerations. The benefits of such an approach lie in the constitutive model’s use of 

the fundamental properties that govern material behavior. That is, changes in external conditions, 

such as temperature and loading level, are easily taken into account without the need for 

additional experimentation. When combined with a structural model, such as finite elements, this 

approach can then account for any combination of pavement structure, traffic, and environmental 

condition. 

In a previous report, a VEPCD model was characterized using the FHWA ALF mixtures.(13)

However, this model was limited because in-service asphalt concrete undergoes a more 

complicated state of stress than that which was simulated in the laboratory. With this in mind, a 

possible drawback to the current VEPCD model formulation is the exclusion of stress state 
50

dependence. In this chapter, the multiaxial extension, first derived by Ha and Schapery, is 

characterized and verified using the Control-2006 mixture.(48) 

As part of the characterization process, results were obtained that allowed further verification of 

the t-TS principle with growing damage, and these results are presented here. The final section of 

this chapter presents findings from a study of the uniaxial VEPCD model as it relates to the 

prediction of cyclic fatigue test results. This section of the report is particularly important 

because it relates directly to potential test protocols that are to be included in the Simple 

Performance Tester (SPT)/Asphalt Mixture Performance Tester (AMPT). 

4.2. MVECD CHARACTERIZATION IN TENSION 

4.2.1. Linear Viscoelastic Characterization 

Characterization of the MVEPCD model began with the development of a stress state dependent 

linear viscoelastic model. Following the test protocols presented in chapter 3, frequencytemperature sweep tests were conducted in both the unconfined and confined state. The results of 

this characterization are shown for the unconfined stress state in table 8, a confining pressure of 

250 kPa in table 9, and a confining pressure of 500 kPa in table 10. Generally, higher variability 

was observed in the confined tests relative to the ones performed under unconfined conditions. 

However, the variability in both was generally small, on the order of 5 percent. It is also 

observed from table 8 through table 10 that the confinement effect became more prominent as 

the temperature increased or the frequency decreased. 
51

Table 8. Linear viscoelastic characterization and variation for Control-2006 in unconfined 

state at selected frequencies and temperatures. 

Temp. 

(°C ) 

Frequency

(Hz) 

Average 

|E*| (MPa)

Average 

Phase Angle 

(°) 

|E*| Coefficient 

of Variation 

(%) 

Phase Angle 

Coefficient of 

Variation (%) 

-10 

25 32,290 4.9 2.8 34.8 

10 31,260 5.4 2.9 31.4 

5 30,416 5.7 3.1 28.4 

1 27,905 6.7 3.1 30.4 

0.5 26,771 7.1 3.0 26.2 

0.1 23,694 8.8 4.0 8.0 

5 

25 23,184 9.7 2.9 17.1 

10 21,243 10.7 3.0 17.2 

5 19,713 11.3 2.8 16.2 

1 16,286 13.7 3.2 12.5 

0.5 14,831 14.8 3.3 13.3 

0.1 11,439 18.3 3.8 13.6 

20 

25 12,305 18.2 2.1 6.6 

10 10,384 20.4 1.7 8.5 

5 8,834 22.8 2.2 5.8 

1 5,732 29.0 3.7 5.6 

0.5 4,701 31.3 3.2 5.1 

0.1 2,736 38.9 2.1 3.4 

40 

25 2,713 38.4 4.0 4.0 

10 1,856 43.2 3.5 2.0 

5 1,358 44.2 3.8 2.6 

1 705 44.5 5.1 2.7 

0.5 527 42.4 5.7 3.5 

0.1 294 34.3 5.2 4.4 

54 

25 764 45.1 8.9 4.1 

10 523 42.8 6.8 2.8 

5 419 38.7 8.7 1.5 

1 229 30.9 9.7 6.6 

0.5 204 25.6 9.5 11.0 

0.1 151 18.2 10.2 34.3 
52

Table 9. Linear viscoelastic characterization and variation for Control-2006 in 250 kPa 

confined state at selected frequencies and temperatures. 

Temp. 

(°C ) 

Frequency

(Hz) 

Average 

|E*| (MPa)

Average 

Phase Angle 

(°) 

|E*| Coefficient 

of Variation 

(%) 

Phase Angle 

Coefficient of 

Variation (%) 

-10 

25 31,769 2.2 — — 

10 30,490 2.5 — — 

5 29,595 2.3 — — 

1 27,011 3.2 — — 

0.5 25,884 3.6 — — 

0.1 22,966 4.6 — — 

5 

25 22,775 6.5 — — 

10 21,065 7.5 — — 

5 19,555 7.5 — — 

1 16,143 9.5 — — 

0.5 14,661 10.9 — — 

0.1 11,312 14.3 — — 

20 

25 12,538 15.0 — — 

10 10,525 16.7 — — 

5 9,173 18.2 — — 

1 6,119 22.8 — — 

0.5 5,069 24.8 — — 

0.1 2,974 29.3 — — 

40 

25 3,801 26.1 — — 

10 2,841 25.8 — — 

5 2,408 24.0 — — 

1 1,554 20.5 — — 

0.5 1,434 18.1 — — 

0.1 1,088 14.0 — — 

54 

25 1,905 22.8 — — 

10 1,555 20.0 — — 

5 1,363 17.4 — — 

1 1,156 13.7 — — 

0.5 1,031 11.2 — — 

0.1 925 8.1 — — 

— Data based on a single test thus variational analysis is not possible. 
53

Table 10. Linear viscoelastic characterization and variation for Control-2006 in 500 kPa 

confined state at selected frequencies and temperatures. 

Temp. 

(°C ) 

Frequency

(Hz) 

Average 

|E*| (MPa)

Average 

Phase Angle 

(°) 

|E*| Coefficient 

of Variation 

(%) 

Phase Angle 

Coefficient of 

Variation (%) 

-10 

25 31,031 4.3 1.9 22.6 

10 30,093 4.7 2.7 24.1 

5 29,207 4.9 2.9 26.3 

1 26,480 5.8 3.0 26.4 

0.5 25,203 6.4 2.9 25.7 

0.1 22,112 7.9 3.1 20.1 

5 

25 22,380 8.3 2.3 18.1 

10 20,573 9.3 3.4 13.9 

5 19,043 9.7 4.0 15.2 

1 15,601 12.1 4.0 13.5 

0.5 14,177 13.4 4.0 10.8 

0.1 10,879 16.8 5.0 9.6 

20 

25 12,628 16.0 5.4 6.4 

10 10,815 17.4 6.3 5.4 

5 9,408 19.1 7.8 2.3 

1 6,561 22.6 10.8 1.7 

0.5 5,623 22.9 12.4 0.5 

0.1 3,994 24.0 6.4 7.0 

40 

25 4,430 24.2 6.5 7.0 

10 3,528 22.9 1.6 6.6 

5 3,049 21.6 2.8 7.0 

1 2,268 18.7 5.8 8.7 

0.5 2,103 17.4 4.6 11.7 

0.1 1,731 14.4 4.6 16.3 

54 

25 2,550 20.1 5.6 9.4 

10 2,154 18.0 4.6 10.7 

5 1,981 16.3 5.5 14.9 

1 1,658 12.8 5.2 20.4 

0.5 1,575 11.6 5.2 24.1 

0.1 1,463 9.5 4.4 29.9 

Table 11. Effect of confining pressure on shift factor function coefficients for 

Control-2006 mixture. 

Parameters 0 kPa 250 kPa 500 kPa 

α 1 0.00069 0.00079 0.00063 

α 2 -0.16245 -0.15411 -0.14816 

α 3 0.79489 0.75076 0.72512 
54

The data presented in table 8 and table 10 have been shifted to form a continuous mastercurve. 

The results are plotted for the three conditions in figure 19, figure 21, and figure 22. The shift 

factor function coefficients are shown in table 11. For all cases, zero-mean deviatoric stress 

conditions (i.e., tension compression) were applied. From figure 19 through figure 21, it is 

observed that at high reduced frequencies, stiffness and phase angle values were insensitive to 

confining pressure, but they diverged as the reduced frequency decreased. Furthermore, it is seen 

from figure 19 and figure 20 that the slope of the confined mastercurve was not as steep as the 

slope of the unconfined mastercurve, thus suggesting reduced time dependence in the material 

under confined stress. In addition, by examining the phase angle mastercurves in figure 21, it is 

seen that at lower reduced frequencies, the material behaved more elastically under confining 

pressure than it did in the uniaxial state. This effect was not significantly dependent on the 

confining stress level. The same observation can be made with regards to the t-TS shift factors 

from figure 22. 

These observations are consistent with those made by other researchers for asphalt concrete.(22,49)

This finding may have implications on the simple performance test protocols since it is known 

that in the pavement structure complicated stress states do exist. Time and resources do not allow 

for an indepth investigation into the effects of this phenomenon, and it is unclear exactly what 

factors influence the phenomenon’s magnitude. Care should be taken in using the findings shown 

in these figures to justify the confined protocol for simple performance testing until the necessary 

analysis has been completed and further testing is performed. Performing confined dynamic 

modulus testing, even in the compression mode, is considerably more complex and timeconsuming than uniaxial tests. In the end, this effort may be wasteful if, through careful study, 

the overall impacts of this behavior are similar across mixtures and would be indirectly 

considered through any mechanistic-empirical analysis calibration process. 

These observations are also the same as those found from unbound paving materials (i.e., higher 

stiffness increases the confining pressure). Therefore, the behavior appears to be related to the 

ability of the asphalt cement to resist dilation of the aggregate skeleton. However the 

deformation that was applied in these tests was generally small, on the order of 60 με , which 

was not large enough to introduce significant aggregate reorientation. Nevertheless, it seems 

reasonable to assume that the mechanisms responsible for the increased stiffness under confining 

pressure were at least similar to those of unbound materials. This hypothesis may also suggest 

that confining pressure effects can be considered without the need to significantly reconsider the 

LVE properties of the material. Support for this hypothesis is given in figure 23 where the 

relaxation spectrum for the confined and unconfined tests is plotted. The relaxation spectrum was 

the crossplot of the Prony coefficients, Ei, and their active or relaxation times, ρi , from 

equation 5. This plot provides an indication of molecular processes occurring in the material and 

the times at which they occurred under fixed displacement.(50,51) Stated more simply, this plot 

provides a snapshot of the LVE processes that occurred in the material. Figure 23 shows that the 

spectrum obtained from collocation for the unconfined test was very similar to that for the 

confined test. The reason for the differences shown in figure 19 can be largely attributed to 

differences in the equilibrium or long-time elastic modulus of the material. In fact, figure 24 

presents the effect of using the uniaxial relaxation spectrum with the confined equilibrium 

modulus, and the overall effect was small. 
55

One may also question whether the differences observed at low reduced frequencies were related 

to a permanent hardening of the material, recoverable hardening, or some other mechanism. To 

examine the possibility of permanent hardening, the frequency sweep test was rerun with no 

confining pressure approximately 1 hour after decompressing from the first 500 kPa test. Results 

of this test are presented in figure 25 along with the average from the original uniaxial 

characterization tests at 54 °C. Similar results were obtained for the two sets of tests, with the 

test performed after the confined test showing slightly higher values. The slightly higher values 

could be related to membrane effects but were more likely due to the relative magnitudes of the 

friction and applied stress levels. The loading rod for the uniaxial tests was not subjected to 

friction, whereas, in the tests run after the confined test (labeled “unconfined” in figure 25), 

friction was present where the rod passes through the triaxial cell lid. Regardless of the influence 

of the friction on the measured modulus, figure 25 shows that the increased stiffness at low 

reduced temperatures was a mostly recoverable phenomenon at the very least. 

0

10000

20000

30000

1.E-08 1.E-05 1.E-02 1.E+01 1.E+04

Reduced Frequency (Hz)

|E*| (MPa)

0 kPa

250 kPa

500 kPa

Figure 19. Graph. Effect of 500 kPa confining pressure on the dynamic modulus 

in semi-log space. 
56

100

1000

10000

100000

1.E-08 1.E-05 1.E-02 1.E+01 1.E+04

Reduced Frequency (Hz)

|E*| (MPa)

Figure 20. Graph. Effect of 500 kPa confining pressure on the dynamic modulus 

in log-log space. 

0

10

20

30

40

50

1.E-08 1.E-05 1.E-02 1.E+01 1.E+04

Reduced Frequency (Hz)

Phase Angle (deg)

0 kPa

250 kPa

500 kPa

Figure 21. Graph. Effect of 500 kPa confining pressure on observed elasticity 

in the Control mixture. 
57

-8

-6

-4

-2

0

2

4

-20 0 20 40 60

Temperature (C)

log a

T

0 kPa

500 kPa

250 kPa

Figure 22. Graph. Effect of 500 kPa confining pressure on the log shift factor function. 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

5.E+06

6.E+06

7.E+06

1E-16 1E-10 1E-04 1E+02 1E+08 1E+14 1E+20

Relaxation Times (s)

Prony Coefficients (kPa)

0 kPa

500 kPa

Figure 23. Graph. Effect of confining pressure on the relaxation spectrum. 
58

1000

10000

100000

1.E-07 1.E-04 1.E-01 1.E+02 1.E+05

Reduced Frequency (Hz)

|E*| (MPa)

Measured

Uniaxial Coefficients

Multiaxial Coefficients

Figure 24. Graph. Use of the uniaxial relaxation spectrum for multiaxial test results. 

100

1000

10000

0.01 0.1 1 10 100

Frequency (Hz)

|E*| (MPa)

Unconfined

500 kPa

Uniaxial-No Cell

Figure 25. Graph. Effect of performing confined temperature/frequency sweep testing 

on the unconfined dynamic modulus. 
59

4.2.2. Modeling Stress State Dependence of the |E*| 

To model the stress state dependence of the |E*| of asphalt concrete, a phenomenological model 

similar to that used for unbound material moduli was utilized. For the purposes of this report, the 

modulus mastercurves determined from 0, 250, and 500 kPa were used along with the 

observations given in figure 19 through figure 25. The relaxation spectrum from the uniaxial test 

was assumed to be independent of confining pressure, and all pressure effects were combined 

into the long-time equilibrium modulus ( E∞ from equation 5). The model used for the 

equilibrium modulus is shown in equation 85. The characterization results for this model are 

shown in figure 26, and it was found that k1 = -1,859, k2 = 0.681, and k3 = 10.01. The degree to 

which these models agreed with the measured responses is shown for the storage modulus in 

figure 27. Note that the MVEPCD model verification efforts in section 4.4 employ the modulus 

predicted from equation 85 instead of the measured modulus. 

2

6

1

3

k

a

a

k E kP

P

θ ∞

⎡ − ⎤ = ⎢ ⎥ ⎣ ⎦ (85) 

Where: 

θ = Bulk stress, σ 11+σ 22+σ 33. 

Pa = Atmospheric pressure. 

k1, k2, k3 = Fitting coefficients. 

0

400000

800000

1200000

1600000

0 100 200 300 400 500 600

Confining Stress (kPa)

Einf (kPa)

Measurement

Model

Figure 26. Graph. Multiaxial equilibrium characterization results. 
60

10

100

1000

10000

100000

1.E-07 1.E-04 1.E-01 1.E+02 1.E+05

Reduced Frequency (Hz)

|E*| (MPa)

0 kPa-Model 0 kPa

250 kPa - Model 250 kPa

500 kPa - Model 500 kPa

Figure 27. Graph. Multiaxial dynamic modulus model strength. 

4.2.3. MVECD Damage Function Characterization 

Characterization of the MVECD model involves developing the functional relationship of C11, 

C12, and C22 with damage, S. It is recalled that the general strain energy density function is given 

as such: 

( )( ) ( ) 2 2

11 12 22

1 1 ( ) 2 2

R RR W C S C S p C Sp d = ++ ε ε (86) 

From this equation, the stress and pseudo dilation can be shown as follows: 

11 12 () () R σ ε = + C S C Sp (87) 

12 22 () () R R υ ε = + C S C Sp (88) 

From equation 87 and equation 88, C11 and C12 may be characterized using uniaxial constant 

crosshead rate tests under conditions where viscoelastic damage mechanisms dominate the 

material behavior. C22, however, must be characterized using multiaxial constant crosshead rate tests. 

4.2.3.1. Characterization of C11(S) 

Under uniaxial conditions in equation 87, the pressure is zero, and C11 may be solved at any time 

as shown in equation 89. 
61

( ) 11

i

i R

i

C σ

ε = (89) 

To find the relationship between C11 and S, the damage is first calculated using the chain rule 

method, equation 90, with C denoting C11 for notational simplicity. 

( ) ( ) ( )

1 2 1

1

1 1 1

1

2

R

i i i i ii S S CC tt

α α

α ε +

+ − − −

⎛ ⎞ = +− − − ⎜ ⎟ ⎝ ⎠ (90) 

These S values are then refined using the method outlined in the NCHRP 9-19 Final Report.(22)

Finally, the functional relationship between C11 and the refined S is determined for each replicate 

test and averaged to obtain the representative C11-S relationship. This relationship is shown for 

the mixture under consideration in figure 28. This figure is constructed at a reference 

temperature of 5 °C and applies only to the material under tensile loading. For this given mixture, 

it is found that a = -0.00157 and b = 0.5320. 

0.00

0.20

0.40

0.60

0.80

1.00

0.0E+00 1.0E+05 2.0E+05 3.0E+05 4.0E+05 5.0E+05

Refined S

C11

Model

5-1-T

5-2-T

5-2-T (2)

5-3-T

11 ( ) b aS CS e =

Figure 28. Graph. C11 versus S for tension for Control-2006 mixture (5 °C reference). 

4.2.3.2. Characterization of C12(S) 

Under uniaxial conditions in equation 88, the pressure is zero, and C12 may be solved at any time, 

as shown in equation 91. 

( ) 12

R

i

i R

i

C υ

ε = (91) 
62

Where: 

The pseudo dilation is calculated using the convolution of the relaxation modulus and the sum of 

the axial and radial strain components. 

Complications arose in C12 characterization due to the differences in time dependence of the 

axial and radial strains (i.e., the time dependence of Poisson’s ratio). The generalized 

correspondence principle did not account for such time dependence in Poisson’s ratio. Instead, 

the correspondence principle placed all of the time dependence with the relaxation modulus or 

creep compliance. 

Through the definitions of pseudo strain and pseudo dilation, it can be observed that C12 = (1-2ν ). 

Thus, C12 defined, in a way, the change in Poisson’s ratio with damage. The initial value of C12

was related to the undamaged Poisson’s ratio of the material, and any change in this value was 

assumed to be related to damage effects alone. When the material of interest did not have a 

constant Poisson’s ratio, the same could be claimed because time effects caused C12 to change as 

well. The data shown in figure 29 demonstrate this effect. Here, a LVE prediction of strains has 

been performed for the axial and radial strains. These predicted strains were then used to 

calculate pseudo strain and pseudo dilation. In turn, these values were used to calculate C12, 

which was plotted against time. Two predictions were performed, one utilizing the timeindependent Poisson’s ratio and one using the time-dependent Poisson’s ratio. Figure 29 shows 

that the prediction utilizing a time-dependent Poisson’s ratio incorporated changes in the values, 

whereas the values from a time-independent Poisson’s ratio did not change with time. 

0

0.2

0.4

0.6

0.8

0 20 40 60 80 100

Time (s)

C12

Time-dependent Poisson's ratio

Time-independent Poisson's ratio

Figure 29. Graph. Effect of time-dependent Poisson’s ratio on C12 calculation. 

To begin, the calculated C12 versus S curves for the asphalt concrete mixture of interest are 

presented in figure 30. This figure shows that C12 decreased drastically at first and then increased, 

suggesting that at small damage levels Poisson’s ratio increased, and then decreased at some 
63

critical damage level. From measurements of the axial and radial strains taken during small strain 

dynamic loading tests and dynamic modulus tests, it was known that the short-time elastic 

Poisson’s ratio for the material of interest was approximately 0.137. When the data in figure 30 

were plotted in semilogarithmic scale, as shown in figure 31, it was observed that at the smallest 

S values, it was not unreasonable to assume an average C12 value of 0.726. This value of C12

corresponded to a Poisson’s ratio of 0.137 and supported the findings from the dynamic modulus 

tests. The reasons for complications in the characterization of C12 became clear. Any change in 

the C12 value was assumed to be due to damage alone. However, because the measurements were 

subject to the time dependence of Poisson’s ratio, a change in C12 could be due either to damage 

growth or time dependence (figure 29). 

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.0E+05 2.0E+05 3.0E+05 4.0E+05 5.0E+05

Refined S

C12

5-1-T

5-2-T

5-2-T (2)

Figure 30. Graph. C12 characteristic curve for Control-2006 mixture. 
64

0.0

0.2

0.4

0.6

0.8

1.0

1.0E-01 1.0E+01 1.0E+03 1.0E+05 1.0E+07

Refined S

C12

5-1-T

5-2-T

5-2-T (2)

Figure 31. Graph. C12 characteristic curve for Control-2006 mixture in 

semi-logarithmic space. 

Because damage affects both axial strains and radial strains, eliminating the time effects from the 

measurements was not possible. Further, it was not possible to separate the time effects from the 

true damage effects seen in figure 30. This situation left only a third possibility, which was to 

assume a constant Poisson’s ratio that was overall representative of the material behavior. With 

this in mind, it was reasonable to assume that the change in curvature of the C12 curves in figure 

30 occurred because the effects of damage overwhelmed those of time. To then characterize the 

material behavior, the value of C12 at the curvature point was averaged for the three replicates 

(approximately 0.23) and was taken as the initial undamaged C12 value. If this operation was 

performed, Poisson’s ratios at points to the left of the curvature point were all overstated. 

To clarify the influence of this error, a series of numerical simulations using LVE analysis were 

performed for tests in the same reduced time range as the characterization tests. In these 

simulations, the value of radial strains calculated using the time-dependent Poisson’s ratio was 

assumed to be correct (the standard). Then, different time-independent Poisson’s ratio values 

were assumed, and comparisons were drawn between these constant Poisson’s ratio predictions 

and the standard. Numerical simulations were performed in lieu of VECD model predictions 

because other errors could be introduced in the prediction process otherwise, and the LVE 

simulation scheme allowed for a more accurate assessment of only the constant Poisson’s ratio 

assumption. Results of the calculations with various assumptions of a constant Poisson’s ratio are 

presented in figure 32. As the figure shows, amongst the values tested, a constant Poisson’s ratio 

of 0.4 performed best. This value was very similar to the point of curvature in the C12 plots 

shown in figure 30. The error in assuming that Poisson’s ratio was equal to the low-bound elastic 

ratio is demonstrated. 
65

-5.E-04

-4.E-04

-3.E-04

-2.E-04

-1.E-04

0.E+00

0 20 40 60 80 100

Reduced Time (s)

Radial Strain

Time-Dependent nu

nu = 0.137

nu = 0.25

nu = 0.40

nu = 0.47

Figure 32. Graph. Effect of different time-independent Poisson’s ratio values on radial 

strain predictions. 

An assumption of 0.25 for Poisson’s ratio did not provide a good prediction. This observation 

was interesting because the generally accepted value for Poisson’s ratio at 5 °C was 0.25. 

However, this acceptance was based on experience with the resilient modulus test, which applied 

a much shorter time of loading than these tests. In brief, this observation further emphasized the 

importance of a reduced time of loading when considering the fundamental behavior of asphalt 

concrete. From the results of the analytical simulations, the assumed lower bound value for C12

appeared to work well for the tests used in characterization. However, each of these tests covered 

a similar range in reduced time. The implication for tests that occurred in a more limited range of 

time, e.g., cyclic tests, was less clear. These tests, when performed at a constant frequency, did 

not display any time dependence in Poisson’s ratio over a given cycle. A cyclic test performed at 

25 Hz and 5 °C, for example, might be represented well by a Poisson’s ratio of 0.25, but the 

model would assume it to be 0.386, thus leading to more erroneous predictions than were 

currently observed in the constant crosshead rate tests. It is unclear exactly how this issue will be 

addressed. It is hoped that the C12 curve can be scaled appropriately depending on the input 

conditions, but more experimental data and theoretical development are needed to determine the 

proper course of action. 

The form taken for C12 as a function of damage is shown with the data in figure 30 and figure 33. 

It was observed that at early damage stages, Poisson’s ratio changed very little, but generally, as 

damage increased, Poisson’s ratio reduced, as indicated by an increase in C12. For this mixture, it 

was found that K1 = 0.228558, K2 = -6.9785 x 10-6, K3 = 0.960065, and K4 = 7.81989 x 10-7. 
66

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.0E+05 2.0E+05 3.0E+05 4.0E+05 5.0E+05

Refined S

C12

Model

5-1-T

5-2-T

5-2-T (2)

3

2

12 1 4 ( )

K K S C S Ke K S = +

Figure 33. Graph. C12 versus S for tensile loading for Control-2006 mixture (5 °C reference). 

4.2.3.3. Characterization of C22(S) 

Characterization of C22 was not as straightforward as the C11 and C12 characterization. For 

multiaxial loading all three relationships, C11, C12, and C22, had to be used to calculate S, but C22

was not yet known. To compensate for this problem, researchers in the NCHRP 9-19 project

developed an iterative technique whereby C22 was substituted, as follows:(22)

12

22

( ) ( )

R R C S C S

p

υ − ε = (92) 

Where: 

υR and εR are calculated from the measured strains based on the relaxation modulus determined 

at the pressure of interest p. 

However, there were some concerns over the NCHRP 9-19 interpretation of the problem.(22) As a 

result of such concern, three methods were examined for the S calculation when the pressure was 

greater than zero, and the C11 and C12 versus S relationships were known. The first method was 

based on the NCHRP 9-19 derivation with only slight differences;(22) the second was based on 

discussions with Schapery; and the third was based on an optimization technique of equation 87. 

In these methodologies, the pseudo strain and pseudo dilation due to pressure were assumed to 

have the following equalities: 

(1 2*0.386) R

pressure ε =− − p (93) 

3 1 2*0.386 ( ) R

pressure υ =− − p (94) 
67

This assumption was necessary to make equation 87 accurate when no load was applied and to 

make equation 88 consistent with the terms in equation 87. In a similar manner, the initial value 

of C22 had to be specified for the consistent transformation from isotropy to transverse isotropy. 

Because C11 equaled 1 when damage was zero, the initial condition of C22 is given as follows: 

C22 = −− + 21 2 1 ( ν )( ν ) (95) 

4.2.3.3.1. Characterization by NCHRP 9-19 Methodology:(22)

By inserting the expression of dual strain energy, equation 86, in the damage evolution law the 

following was found: 

( )( ) ( ) ( ) 2 2

11 12 22

1 1

2 2

R R C S C S p C Sp dS

d S

α

ε ε

ξ

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ∂ ++ ⎜ ⎟ ⎝ ⎠ = − ∂

⎝ ⎠ (96) 

( )2 1 1 11 12 22 2

2 2

dS dC dC dC R R

p p d dS dS dS

α

ε ε

ξ

⎡ ⎤ ⎛ ⎞ = −⎜ ⎟ + + ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ (97) 

From equation 92, the following was found: 

22 12

R dC dC

dS dS p

⎛ ⎞ ε = −⎜ ⎟ ⎝ ⎠ (98) 

Equation 98 assumed that υR

 was either a weak function of S or independent of S. Inserting 

equation 98 into equation 97 yielded the following: 

( )2 11 12 12 1 1

2 2

dS dC dC dC RR R

p p d dS dS dS

α

εε ε

ξ

⎡ ⎤ ⎛ ⎞ = −⎜ ⎟ + − ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ (99) 

For small Δ ξ , the result was as follows: 

( ) ( ) 1 1 11 2 ( ) 12

2 2

i i R R

i i

dC dC

S p dS dS

α

ξε ε

⎡ ⎛ ⎞⎤

Δ =Δ − + ⎢ ⎜ ⎟⎥ ⎢⎣ ⎝ ⎠⎥⎦ (100) 

With change in notation, it appeared as follows: 

( ) ( ) 11 2 ( ) 12

1

1 1

2 2

i i R R

ii i i

dC dC

S S p dS dS

α

ξε ε +

⎡ ⎛ ⎞⎤

= +Δ − + ⎢ ⎜ ⎟⎥ ⎢⎣ ⎝ ⎠⎥⎦ (101) 
68

Where: 

S1 = 0. 

δ S is an increment smaller than ΔS=Si+1 - Si, for example, 0.1. 

δ Ci = C(Si+dS) - C(Si). 

These values of S were then used to calculate C12 at a given time step. Because pressure was 

constant and pseudo dilation and pseudo strain were known as functions of time, equation 88 

may be used to directly calculate C22. These values were then plotted against the values of S

found from the first step, and the C22-S relationship was developed. Due to some numerical 

complications, a condition was set such that if the term inside the parenthesis of equation 101 

was less than zero, the term was replaced by zero. 

4.2.3.3.2. Characterization with Schapery’s Approach: 

Equation 88 was rearranged to solve for C22 (equation 102): 

12

22

R R C C

p p

υ ε = − (102) 

Then, taking the derivative of equation 102 with respect to time and applying the chain rule 

achieved the following: 

22 22 12 12 1 R R R dC dC dS d d dS d dC C

dt dS dt p dt dt dS dt p dt

υ ε ε = =− − (103) 

Solving equation 103 for the damage rate resulted in equation 104: 

12

22 12

1 R R

R

dS C

dt p dC dC

dS p dS

υ ε

ε

⎡ ⎤ ⎢ ⎥ − = ⎢ ⎥ ⎢ ⎥ + ⎢ ⎥ ⎣ ⎦

& &

 (104) 

Similarly, using equation 87 resulted in equation 105: 

11

12 11

1 R

R

dS C

dt p dC dC

dS p dS

σ ε

ε

⎡ ⎤ ⎢ ⎥ − = ⎢ ⎥ ⎢ ⎥ + ⎢ ⎥ ⎣ ⎦

& &

 (105) 

Because equation 104 must equal equation 105, the following can be shown: 

22 12 12 11 12

11

1 RR R R

R

dC C dC dC dC

dS p C dS p dS p dS

υε ε ε

σ ε

⎡ ⎤ − ⎛ ⎞

= + ⎢ ⎥⎜ ⎟ −

⎣ ⎦ − ⎝ ⎠

& &

& & (106) 
69

Substituting equation 106 into equation 97 and simplifying found the following: 

11 12 1

2 2 2

R dS dC RR R dC

p dt dS p dS p

α

λ λ ε

ε ε ε

⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = −⎢ ⎥ ⎜ ⎟ ⎜ ⎟ + + + − ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (107) 

Where: 

λ = A function of C11, C12, and the pseudo dilation, pseudo strain, and stress rates. 

Equation 88 may be used to directly calculate C22. These values were then plotted against the 

values of S found from the first step, and the C22-S relationship was developed. Due to some 

numerical complications, a condition was set such that if the term inside the parenthesis of 

equation 101 was less than zero, the term is replaced by zero. 

4.2.3.3.3. Characterization by Optimization: 

From the triaxial constant crosshead rate tests, it was found that pressure was constant, stress was 

measured, and pseudo strain may be calculated using the measured strains and the relaxation 

modulus. Then, from equation 87, the only unknown was S. In this methodology, S was solved 

for by numeric optimization to minimize the error between measured stress and the stress 

calculated from equation 87. These values of S were then used to calculate C12 at a given time 

step. Because pressure was constant, and pseudo dilation and pseudo strain were known as 

functions of time, equation 88 may be used to directly calculate C22. These values were then 

plotted against the values of S found from the first step, and the C22-S relationship was developed. 

4.2.3.3.4. Comparison of Methodologies: 

Each of the above methods has been used to characterize S. Comparisons were made first with 

the S values as functions of time for a single test. Figure 34 through figure 37 and figure 38 

through figure 41 show the S calculated from each of the methods in both arithmetic and 

logarithmic scales. Figure 42 through figure 45 present the stress calculated using the 

characterized S and equation 87. Method three used optimization; thus, the predicted stress and 

measured stress agree exactly. Figure 46 through figure 49 show the damage characteristic 

relationship by each technique for the four tests. The final representative C22(S) function is 

shown in figure 50. The initial conditions required H1 to be equal to equation 95. Explicitly, the 

coefficients for C22(S) are H1 = -0.633435, H2 = 3.0834 x 10-5, and H3 = 0.83682. 
70

0.E+00

2.E+05

4.E+05

6.E+05

8.E+05

1.E+06

0 20 40 60 80 100 120 140

Reduced Time (s)

Damage Parameter S

Method 1

Method 2

Method 3

 Figure 34. Graph. S as a function of reduced time calculated by three different 

methodologies in arithmetic space for 5-1-T. 

0.E+00

2.E+05

4.E+05

6.E+05

8.E+05

1.E+06

0 50 100 150 200 250 300

Reduced Time (s)

Damage Parameter S

Figure 35. Graph. S as a function of reduced time calculated by three different 

methodologies in arithmetic space for 5-3-T. 
71

0.E+00

2.E+05

4.E+05

6.E+05

8.E+05

1.E+06

0 50 100 150 200 250 300 350

Reduced Time (s)

Damage Parameter S

Figure 36. Graph. S as a function of reduced time calculated by three different 

methodologies in arithmetic space for 5-4-T. 

0.E+00

2.E+05

4.E+05

6.E+05

8.E+05

1.E+06

0 100 200 300 400

Reduced Time (s)

Damage Parameter S

Figure 37. Graph. S as a function of reduced time calculated by three different 

methodologies in arithmetic space for 5-5-T. 
72

1.E+00

1.E+02

1.E+04

1.E+06

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Reduced Time (s)

Damage Parameter S

Method 1

Method 2

Method 3

Figure 38. Graph. S as a function of reduced time calculated by three different 

methodologies in logarithmic space for 5-1-T. 

1.E+00

1.E+02

1.E+04

1.E+06

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Reduced Time (s)

Damage Parameter S

Figure 39. Graph. S as a function of reduced time calculated by three different 

methodologies in logarithmic space for 5-3-T. 
73

1.E+00

1.E+02

1.E+04

1.E+06

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Reduced Time (s)

Damage Parameter S

Figure 40. Graph. S as a function of reduced time calculated by three different 

methodologies in logarithmic space for 5-4-T. 

1.E+00

1.E+02

1.E+04

1.E+06

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Reduced Time (s)

Damage Parameter S

Figure 41. Graph. S as a function of reduced time calculated by three different 

methodologies in logarithmic space for 5-5-T. 
74

0

500

1000

1500

2000

2500

3000

3500

4000

0 20 40 60 80 100 120 140

Reduced Time (s)

Stress (kPa)

Measured

Predicted-Method 1

Predicted-Method 2

Predicted-Method 3

Figure 42. Graph. Predicted and measured stress as a function of reduced time for 

different S calculation methodologies for 5-1-T. 

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300

Reduced Time (s)

Stress (kPa)

Figure 43. Graph. Predicted and measured stress as a function of reduced time for 

different S calculation methodologies for 5-3-T. 
75

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350

Reduced Time (s)

Stress (kPa)

Figure 44. Graph. Predicted and measured stress as a function of reduced time for 

different S calculation methodologies for 5-4-T. 

0

500

1000

1500

2000

2500

3000

0 100 200 300 400

Reduced Time (s)

Stress (kPa)

Figure 45. Graph. Predicted and measured stress as a function of reduced time for 

different S calculation methodologies for 5-5-T. 
76

-12

-9

-6

-3

0

0.E+00 2.E+05 4.E+05 6.E+05 8.E+05 1.E+06

Damage Parameter S

C22

Method 1

Method 2

Method 3

Figure 46. Graph. C22 as a function of S calculated by different methodologies for 5-1-T. 

-12

-9

-6

-3

0

0.E+00 2.E+05 4.E+05 6.E+05 8.E+05 1.E+06

Damage Parameter S

C22

Figure 47. Graph. C22 as a function of S calculated by different methodologies for 5-3-T. 
77

-12

-9

-6

-3

0

0.E+00 2.E+05 4.E+05 6.E+05 8.E+05 1.E+06

Damage Parameter S

C22

Figure 48. Graph. C22 as a function of S calculated by different methodologies for 5-4-T. 

-12

-9

-6

-3

0

0.E+00 2.E+05 4.E+05 6.E+05 8.E+05 1.E+06

Damage Parameter S

C22

Figure 49. Graph. C22 as a function of S calculated by different methodologies for 5-5-T. 
78

-12

-9

-6

-3

0

0.E+00 2.E+05 4.E+05 6.E+05 8.E+05 1.E+06

S

C22

3

2

22 1

H H S C He =

Figure 50. Graph. Representative C22 versus S for tension using optimization methodology 

(5 °C reference). 

4.3. VISCOPLASTIC MODELING OF ASPHALT CONCRETE IN TENSION 

With the damage characteristic functions thus characterized, it was possible to predict the 

material behavior under conditions where viscoelastic damage mechanisms dominate. However, 

it was known that viscoplastic effects may have been significant factors governing the overall 

behavior of asphalt concrete in tension. Currently, viscoplastic effects are considered using an 

empirical strain-hardening model. Because details of this model have been presented previously 

and elsewhere, only the integrated form of the constitutive equation is shown here in 

equation 108.(12,13,25) 

1 1

1 1

0

1 t p p q

vp

p dt

Y

ε σ + ⎛ ⎞ + + ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∫ (108) 

As with the MVECD model, the characterization of equation 108 utilized constant crosshead rate 

tests under confining pressure. The procedure first required prediction of the VECD related 

strains of such tests at high temperatures. Viscoplastic strains were then backcalculated by 

subtracting the predicted MVECD strains from the measured values. This process was repeated 

in multiple tests where viscoplastic strains were considered to be significant. The backcalculated 

values for these tests were then presented to the model, and the coefficients were determined by 

optimization using a genetic algorithm. 

For the prediction of viscoplastic behavior under various confining states, a phenomenological 

observation was utilized. From the characterization, it was found that the coefficients p and q are 
79

not strongly affected by confining states, but the coefficient Y was found to depend heavily on 

confining pressure. This relationship, along with the stress dependent model, is shown in 

figure 51. 

 

y = -161624466.99x + 766528136533.83

6.8E+11

7.0E+11

7.2E+11

7.4E+11

7.6E+11

7.8E+11

0 100 200 300 400 500 600

σ33 (kPa)

Y

Figure 51. Graph. Effect of confining pressure on coefficient Y. 

4.4. MVEPCD MODEL VERIFICATION 

Verification tests for the MVEPCD model were similar to those used for characterization (i.e., 

constant crosshead rate tests under a constant confining pressure). Some of the tests were 

performed at the same confining pressure as characterization tests, but at a different temperature 

(25 °C). A second verification test was performed at the same temperatures as characterization 

but at a different confining pressure (250 kPa). Results of the first verification data set are shown 

in figure 52. Both axial and radial strain predictions are shown in this figure. Although not 

shown here, the agreement observed in figure 52 was approximately the same as that observed in 

the verification of the uniaxial VEPCD model. This agreement was considered a good match 

with the measurements. The results of the second verification data set are shown in figure 53 for 

40 °C and in figure 54 for 5 °C. Although this data set did not show as good an agreement with 

the measured response, the fitting was considered reasonable. 
80

0

500

1000

1500

2000

2500

3000

3500

-4.E-03 -2.E-03 0.E+00 2.E-03 4.E-03 6.E-03 8.E-03 1.E-02

Strain

σ11-σ33 (kPa)

Measured

Predicted Radial Axial

Figure 52. Graph. Results of MVEPCD model simple verification for constant crosshead 

rate tests under 500 kPa confinement at 25 °C. 

0

100

200

300

400

500

600

700

800

900

1000

-1E-02 0E+00 1E-02 2E-02

Strain

σ11-σ33 (kPa)

Measured

Radial Axial Predicted

Figure 53. Graph. Results of MVEPCD model verification for constant crosshead rate tests 

under 250 kPa confinement at 40 °C. 
81

0

500

1000

1500

2000

2500

3000

3500

4000

-4E-03 -2E-03 0E+00 2E-03 4E-03 6E-03 8E-03 1E-02

Strain

σ11-σ33 (kPa)

Measured

Radial Axial Predicted

Figure 54. Graph. Results of MVEPCD model verification for constant crosshead rate tests 

under 250 kPa confinement at 5 °C. 

4.5. t-TS WITH GROWING DAMAGE IN TENSION 

It has been shown repeatedly that dense-graded asphalt concrete, including that with modified 

binders, exhibits t-TS under growing damage in the uniaxial stress state. Less conclusive results 

are available for the behavior of mixtures under confining pressure. Therefore, it was necessary 

to verify the principle of t-TS under confining pressure. Verification was performed following 

the method presented in Chehab.(20) The advantage of utilizing this method was that tests used 

for validation of t-TS may also have utilized in the characterization procedure, thus reducing 

excessive testing. A brief schematic representation of the process is presented in figure 55 

through figure 57. In short, stress as a function of reduced time for constant crosshead rate tests 

performed at different rates and temperatures was plotted for a given strain level. Therefore, each 

test produced a single point in the respective plot. This process was repeated for several strain 

levels covering a broad range in material behavior. If t-TS with growing damage was valid, then 

the resulting plots would appear to be continuous. 
82

σ

ε1

A

B

C

Figure 55. Illustration. Schematic representation of first step for verifying the 

t-TS principle under growing damage, finding stress for different tests at a 

consistent strain level. 

σ

timeR

A

B

C

tRA tRB tRC

Figure 56. Illustration. Schematic representation of the second step for verifying the t-TS 

principle under growing damage, finding reduced time for the stress found in the first step 

for each test. 
83

σ

timeR

tRA tRB tRC

ε = ε1

Figure 57. Illustration. Schematic representation of the third step for verifying the t-TS 

principle under growing damage, plotting the stress from the first step against the reduced 

time from the second step. 

The strain levels examined for t-TS analysis are shown graphically along with the stress-strain 

curves in figure 58. The stress-strain curves in figure 58 represented all the tests noted in table 3, 

and the examined strain levels covered both low damage (low strain levels) and high damage 

(high strain levels) ranges. The resulting stress versus reduced time curves at these strain levels 

are presented in figure 59 through figure 64. At all strain levels examined, the curves appeared 

continuous, and thus, t-TS under deviatoric tension conditions was satisfactorily verified. In 

figure 63 and figure 64, data for 5 °C were not presented due to localization of the strain prior to 

reaching the given strain levels. This observation is also apparent in figure 58, in which the tests 

with the highest strength were performed at 5 °C. 
84

0

1000

2000

3000

4000

0E+00 2E-03 4E-03 6E-03 8E-03 1E-02

Axial Strain

Stress (kPa)

Figure 58. Graph. Strain levels examined for verifying the t-TS principle under growing 

damage and confining pressure for the Control-2006 mixture. 

0

100

200

300

400

500

600

700

800

0.1 1.0 10.0 100.0 1000.0 10000.0

Reduced Time (s)

Stress (kPa)

5 C

25 C

40 C

Figure 59. Graph. t-TS with growing damage under confinement verification 

at a 0.0001ε level. 
85

0

500

1000

1500

2000

0.1 1.0 10.0 100.0 1000.0 10000.0 100000.0

Reduced Time (s)

Stress (kPa)

5 C

25 C

40 C

Figure 60. Graph. t-TS with growing damage under confinement verification at a 

0.0005ε level. 

0

500

1000

1500

2000

2500

3000

0.1 1.0 10.0 100.0 1000.0 10000.0 100000.0

Reduced Time (s)

Stress (kPa)

5 C

25 C

40 C

Figure 61. Graph. t-TS with growing damage under confinement verification at a 

0.001ε level. 
86

0

500

1000

1500

2000

2500

3000

3500

0.1 1.0 10.0 100.0 1000.0 10000.0 100000.0

Reduced Time (s)

Stress (kPa)

5 C

25 C

40 C

Figure 62. Graph. t-TS with growing damage under confinement verification at a 

0.0022ε level. 

0

500

1000

1500

2000

2500

3000

3500

10.0 100.0 1000.0 10000.0 100000.0 1000000.0

Reduced Time (s)

Stress (kPa)

25 C

40 C

Figure 63. Graph. t-TS with growing damage under confinement verification at a 

0.004ε level. 
87

0

500

1000

1500

2000

2500

10.0 100.0 1000.0 10000.0 100000.0 1000000.0

Reduced Time (s)

Stress (kPa)

25 C

40 C

Figure 64. Graph. t-TS with growing damage under confinement verification at a 

0.007ε level. 

4.6. ENGINEERING BEHAVIOR IN CONFINED TENSION 

Verification of t-TS with growing damage under confining stress permitted the development of 

mastercurves for various engineering parameters. Such curves were created in much the same 

way as LVE mastercurves (refer to subsection 2.2.1). In this case, the reduced variable of interest 

was the reduced strain rate. This value was calculated for a given test by multiplying the 

physically observed strain rate by the time-temperature shift factor for that test. The timetemperature shift factor was determined by using the test temperature with the time-temperature 

shift factor function shown in figure 22. 

One such curve of particular interest was the strength mastercurve, which is shown for both the 

confined and unconfined conditions in figure 65. Each point in these curves reflected the result 

from a single test. When examining this figure, the tests performed at higher temperatures 

resulted in lower reduced strain rates, and tests performed at lower temperatures resulted in 

higher reduced strain rates. No tests were performed at intermediate temperatures in the 

unconfined condition, and thus, no data were available in the middle reduced rates for this stress 

condition. From figure 65, results similar to those observed in LVE characterization were seen. 

Specifically, strength did not appear to be a function of confining pressure at high reduced rates 

but did appear to be a strong function of confining pressure at lower reduced rates. Also, less rate 

dependence was observed in the confined test at lower reduced rates. Another similar curve of 

interest was the strain at failure mastercurve. This curve provided a glimpse of the effect of 

temperature and rate on ductility. Plots of the strain at failure mastercurves under both the 

confined and unconfined conditions are shown in figure 66. Given the variability in these 

measurements, it was difficult to find any certain conclusive differences in the ductility with 

pressure. However, the data seemed to indicate an increase in ductility under all conditions, 

particularly at low reduced rates (i.e., high temperatures). The observation regarding asymptotic 
88

behavior of both stiffness and strength at the low reduced frequencies/rates did not seem to 

transfer to the ductility. 

0

1000

2000

3000

4000

1E-09 1E-08 1E-07 1E-06 1E-05 1E-04

Reduced Strain Rate (ε /s)

Peak Stress (kPa)

0 kPa

500 kPa

Figure 65. Graph. Effect of 500 kPa confining pressure on strength mastercurves. 

0.0.E+00

3.0.E-03

6.0.E-03

9.0.E-03

1.2.E-02

1E-09 1E-08 1E-07 1E-06 1E-05 1E-04

Reduced Strain Rate (ε /s)

Axial Strain at Peak

0 kPa

500 kPa

Figure 66. Graph. Effect of 500 kPa confining pressure on ductility in constant crosshead 

rate tests. 
89

4.7. FATIGUE ANALYSIS WITH VECD MODEL 

4.7.1. VECD Integrated Formulation Summary 

Formulas relating to the existing formulation of the VEPCD model are shown in equation 109 

through equation 116. The damage evolution law is given by the following: 

R dS Wd

dt S

α

⎛ ⎞ ∂ = ⎜ ⎟ ∂ ⎝ ⎠ (109) 

Where: 

The dual energy density function for uniaxial loading was as follows: 

( )( ) 1 2

2

R R W CS d = ε (110) 

This leads to the constitutive relationship, which is shown in equation 111. 

( ) R σ = C S ε (111) 

Noting that increments of time are generally small, the following can be assumed. 

dS S

dt t

Δ = Δ , (112) 

Thus, 

1

R

i

i i

W SS t

S

α

δ

δ −

⎛ ⎞ ≅ +Δ −⎜ ⎟ ⎝ ⎠ (113) 

Because only C was a function of damage, equation 113 simplified to the following: 

( )2 ( )

1

1 *

2

R i

i i i

C

SS t

S

α

δ

ε

δ −

⎛ ⎞ ≅ +Δ −⎜ ⎟

⎝ ⎠ (114) 

However, in the case where only stress and time were given, it was convenient to observe that 

the following: 

R

C

σ

ε = (115) 
90

Assuming that pseudo strain was constant over a given increment of time, equation 114 can be 

recast as follows: 

( ) 2

1

1 *

2

i i

i i

i

C

SS t

C S

α

σ δ

δ −

⎛ ⎞ ⎛ ⎞ ≅ +Δ −⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (116) 

For solving equation 116, either an iterative technique or the analytical representation of δ C/δ S

can be used. 

The important aspects of this formulation as they relate to this report are as follows: 

• Damage grew whenever loading was imposed although it did not necessarily grow at an 

almost zero rate. 

• Pseudo strain (true pseudo strain) must also equal zero when stress equaled zero. 

• The time dependence of the material was assumed constant regardless of the damage 

level. 

• Damage was irreversible. 

• Damage was predicted to grow equally during tension and compression. 

• The LVE characteristics of the material were related to the damage characteristics using a 

single variable, C.

• Damage growth was a function of α true pseudo strain (or stress and pseudo stiffness) 

and the current value of damage only. 

4.7.2. Formulation of Lee, Daniel, and Kim 

Formulas relating to the previous formulation as they have been applied in the past are shown in 

equation 117 through equation 123. 

( ) R

e σ ε = ++ I FGH (117) 

Where: 

I = Normalization factor. 

ε e

R

 = Effective pseudo strain = εR - ε s

R

. 

ε s

R

 = Permanent pseudo strain. 

F = Function representing the slope of an individual stress-pseudo strain loop. 

G = Hysteresis function. 

H = Healing function. 
91

It was observed that the damage function, C, is not explicit in equation 117; rather, C enters 

through the function, F, as follows: 

( )

R

m

R R

m s

F CS ε

ε ε

⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ − (118) 

Equation 117 led to the pseudo energy density function shown in equation 119: 

( ) ( ) 1 1 1 2

2 2 2

R

R R m

R R e

m s

W C S G H εε

ε ε

⎡ ⎤ ⎛ ⎞

= + ⎢ ⎥ ⎜ ⎟ +

⎝ ⎠ − ⎣ ⎦ (119) 

The damage evolution law was the same for this approach and the current approach. Because G

and H were not functions of damage, the damage growth equation became the following: 

( ) 1 2

* **

2

R

m R

R R e

m s

dS C

dt S

α

ε

ε

ε ε

⎛ ⎞ ∂ = −⎜ ⎟ − ∂ ⎝ ⎠ (120) 

Previous researchers further simplified this equation by taking only the pseudo strain values at 

the peak and then performing the chain rule operation as follows: 

( )( ) ( )

1 1

1

1 1

1 ** *

2

R R

i i m me i i i i

S S C C t

α α

α ε ε +

+ − −

⎛ ⎞ ≅+ − Δ ⎜ ⎟ ⎝ ⎠ (121) 

Where: 

εR

m,i = Peak pseudo strain of cycle i. 

εR

me,i = Effective pseudo strain at the peak of cycle i. 

Ci = Pseudo stiffness at the εR

m,i. 

Ci-1 = Pseudo stiffness at εR

m,i-1. 

Δt = Cycle pulse time times the number of cycles occurring between observation i and 

observation i - 1. 

If it is assumed that the permanent pseudo strain at the beginning of the cycle was approximately 

equal to the permanent pseudo strain at the peak of the cycle and if it is assumed that the 

permanent pseudo strain was much smaller than the maximum pseudo strain, then equation 121 

can be recast as follows: 

( ) ( )

1 2 1

1

1 1

1 ** *

2

R

ii m i i i

S S C C t

α α

α ε ++ − −

⎛ ⎞ ≅+ − Δ ⎜ ⎟ ⎝ ⎠ (122) 

In terms of the iterative approach currently used, the following was the case: 

( )2

1

1 *

2

R

ii m i

C SS t

S

α

δ

ε

δ −

⎛ ⎞ ≅ +Δ −⎜ ⎟ ⎝ ⎠ (123) 
92

The important aspects of this formulation as they relate to this report are as follows: 

• The effect of damage was only a reduction in stiffness. 

• Other nonlinear aspects were considered but not physically justified using other functions 

(G and H). 

• Damage growth was independent of other nonlinear effects. 

• The pseudo stiffness at the peak value was assumed sufficient for characterization and 

prediction purposes. 

• Damage growth was a function of α , maximum apparent pseudo strain, the current value 

of damage, and permanent pseudo strain levels at the beginning of a cycle. 

• A correction factor was necessary. Rationalization for this factor was provided by the 

assumption that damage only grew when pseudo strain was positive and increasing to 

collapse damage curves for cyclic and monotonic data. 

4.8. SIMPLIFIED ANALYSIS OF CYCLIC FATIGUE DATA 

4.8.1. Correction Factor in Formulation of Lee, Daniel, and Kim 

To make cyclic and monotonic data agree in the previous formulation, it was necessary to 

introduce a pseudo stiffness time change correction factor, M, into the damage calculation 

function. 

( ) ( )

1

1 1 2

1 1

1 * *

2

R

ii m i i i

t SS C C

M

α α α

ε + +

− −

⎛ ⎞ ⎛ Δ ⎞ ≅+ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (124) 

To examine the significance of equation 124, it must be returned to its base form before the 

introduction of factor, M, to equation 121. Undoing the final simplification step for this equation 

is as follows: 

( ) ( ) 1 2 1

1

1 * *

2

R i i

ii m i

C C

S S t

t

α α

ε

+

−

−

⎛ ⎞ − ≅ + ⎜ ⎟ Δ

⎝ ⎠ Δ (125) 

In order to make equation 125 agree with equation 124, it was necessary to introduce the 

correction factor, M, as shown in equation 126. 

( ) ( ) 1 2 1

1

1 * *

2

R i i

ii m i

C C t S S

t M

α α

ε

+

−

−

⎛ ⎞ − Δ

≅ + ⎜ ⎟ ⎝ ⎠ Δ (126) 

From this equation, it was observed that a factor, M (taken as 4 in previous research), introduced 

in the manner shown in equation 124, implied that the pseudo stiffness decreased at a rate of M
93

times the observed cycle averaged rate for a period of 1/M*tp (figure 67). Mathematically, this 

approach can be schematically represented in figure 69. This figure indicates that the problem 

was simplified from that of a cyclic load application with magnitudeεR

m to a square wave form 

of duration 1/M*tp and a magnitude equal toεR

m. Although the square waveform is shown 

centered about the pulse, no real consideration was given to it when the equivalent square pulse 

was located during the cyclic pulse. 

εR

C

Cycle Averaged dC/dt

t

p

t

Not First 

Cycle

Not First 

Cycle

Figure 67. Illustration. A schematic representation of the concept of average dC/dt. 

εR

C

Cycle Averaged dC/dt * M

t

p

t tp/M

Not First 

Cycle

Not First 

Cycle

Figure 68. Illustration. A schematic representation of the effect of the M factor on dC/dt

used in calculation. 
94

t

p

1/M*tp

εR

εm

R

Complete Solution:

( ) ( ) 1 2

0

1 * **

2

pt R

m R

R R e

m s

dC S tdt

dt

α α ε

ε

ε ε

+ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ − ⎝ ⎠ ∫

Complete Solution:

( ) 1 1 2

*

2

R p

m

dC t

S

dt M

α α

ε + ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠

t

p

1/M*tp

εR

εm

R

Complete Solution:

( ) ( ) 1 2

0

1 * **

2

pt R

m R

R R e

m s

dC S tdt

dt

α α ε

ε

ε ε

+ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ − ⎝ ⎠ ∫

Complete Solution:

( ) 1 1 2

*

2

R p

m

dC t

S

dt M

α α

ε + ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠

Figure 69. Illustration. Mathematical equivalence of the formulation used by 

Lee, Daniel, and Kim.(3,4,5) 

4.8.2. Rigorous Analysis of Pseudo Strain History Effects 

The primary complication was that pseudo stiffness values were taken only at the peak. However, 

in order to have a net reduction in pseudo stiffness over a cycle, the damage must evolve during 

loading. Some assumptions had to be made regarding how damage grew between the peak values 

to more rigorously apply equation 120. Ideally, it should be possible to use the measured data to 

determine exactly when damage seems to grow, but complications in the analysis of full-time 

data in cyclic tests prevented such an approach. 

Because the integration was so well defined and repeatable (constant cyclic loading, for 

example), the correction factor could be simplified and applied in a way similar to the previous 

formulation by introducing another derivable factor, Q, as shown in equation 127. This factor 

was known as the pseudo strain shape factor and accounted for the time-varying pseudo strain 

history. 

It was hypothesized that the more rigorously accurate solution to equation 120 is given by 

equation 127 (in contrast to equation 124). 

( ) ( )

( )

1

1 1 2

1 1

1

1 1 2

1

1 * * *

2

1 * * *

2

R

ii m i i i

R

i m betweenpeaks i

t S S QC C M

t S QC M

α α α

α α α

ε

ε

+ +

− −

+ +

−

⎛ ⎞ ⎛ Δ ⎞ ≅ + ⎜ ⎟ − ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ Δ ⎞ =+ Δ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (127) 
95

To derive the factor, Q, the G and H functions in equation 119 had to be ignored and substituted 

with an analytical expression for total pseudo strain as a function of time. This expression was 

denoted by ˆ ( ) R ε t . Then this function was introduced into the damage evolution law, and the 

chain rule to arrive at an equation for the amount of damage occurring within a single cycle 

applied. 

( ) 1 2

1

1 ˆ *

2

N

R

withincycle i

i

C S t

t

α α

ε +

=

⎡ ⎤ ⎛ ⎞ Δ Δ ≅− Δ ⎢ ⎥ ⎜ ⎟ ⎢⎝ ⎠ Δ ⎥ ⎣ ⎦

∑

 (128) 

Equation 128 implies a different form for the constitutive relationship. In particular, the factor, 

ε m

R

/( ε m

R

 - ε s

R ), was lumped in with the effective pseudo strain. This particular factor did not 

affect equation 127; however, it did imply arrival at equation 127 somewhat differently than was 

done previously (equation 120 through equation 122). During the cycle, damage grew over only 

a certain portion of the cycle pulse time, tp. Furthermore, during this time, pseudo stiffness 

reduced linearly at a rate such that the total reduction in pseudo stiffness that occurred over this 

reduced portion was equal to the experimentally observed reduction in pseudo stiffness between 

peaks. Mathematically this phenomenon is as follows: 

* betweenpeaks

p

C C

M

t t

Δ Δ = Δ (129) 

*

p t

t

M N

Δ = (130) 

( ˆ ˆ ) ( ) R R

i i

ε ε = t (131) 

Where: 

M = Factor denoting the inverse of the proportion of total pulse time when damage occurs (i.e., 

if it is assumed that damage occurs during only a quarter of the pulse time, then M = 4). 

tp = Cycle pulse time (1/f). 

N = Number of discrete points used in the numerical calculation. 

Substituting equation 129 and equation 130 into equation 128 and then simplifying found the 

following: 

( )

1 2

1

1

1 ** * ˆ 2 *

N

betweenpeaks R p

withincycle i

p i

C t

S M

t M N

α α α α ε

+

+

=

⎛ ⎞ Δ ⎡ ⎤ Δ ≅−⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ∑ (132) 

( )

1 1 1 2

1

1

1 1 * ** ˆ 2

N

betweenpeaks p R

withincycle i

p i

C t

S

t N M

α α α α

α ε

+ +

+

=

⎛ ⎞ Δ ⎛ ⎞ ⎡ ⎤ Δ ≅−⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ∑ (133) 
96

A comparison of equation 133 and equation 127 illustrated that the factor, Q, could be solved as 

follows: 

( )

( )

1

2

1

1

2

1 ˆ *

N

R

i

i

R

m

N Q

α α α α ε

ε

+

+

=

⎛ ⎞ ⎡ ⎤ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ =

∑

 (134) 

It was then necessary to assume over which range of tp damage would occur. For controlled 

stress testing, it was assumed that damage growth occurred over only half of the loading cycle 

and was centered over the peak stress. It was important to specify the proportion of loading 

where damage occurred and when damage occurred in the loading cycle. The functional form of 

ˆ ( ) R ε t also had to be assumed. For controlled stress loading, this function is represented by 

equation 135: 

( ) ( ) ( ) ( ) 2 2 ˆ sin sin 2 2

R

R RR R m

stress R R ms m

m s

t t t

ε

ε εε ε ω ω

ε ε

⎛ ⎞ =−= ⎡ ⎤ ⎜ ⎟ − ⎣ ⎦ ⎝ ⎠ (135) 

A schematic view of the assumptions made for controlled stress loading conditions is shown in 

figure 70. The shaded region the figure represents the portion of the loading pulse where it was 

assumed that damage was occurring. 

t

p

1/M*tp

εR

1/2*tp

t

p

1/M*tp

εR

1/2*tp

Figure 70. Illustration. Schematic representation of assumptions made for controlled stress 

cyclic loading to develop Q and M factors. 
97

In the case of controlled crosshead loading, the functional form for ˆ ( ) R ε t is given by 

equation 136. 

ˆ ( ) ( ) cos cos () ()

R

R R m R R

Xhd R R m s m

m s

t t t

ε

ε εε ω ε ω

ε ε

⎛ ⎞ = −= ⎡ ⎤ ⎜ ⎟⎣ ⎦ ⎝ ⎠ − (136) 

A schematic view of the assumptions made for controlled crosshead loading is shown in 

figure 71. Again, the shaded region represents the portion of the load pulse where it was assumed 

that damage was occurring. 

tp

εR

1/4*tp

tp

εR

1/4*tp

Figure 71. Illustration. Schematic representation of assumptions made for controlled 

crosshead cyclic loading to develop Q and M factors. 

Finally, combining the pseudo strain shape factor and the pseudo stiffness time factor into a 

single parameter, Z, resulted in the following relationship: 

( ) ( ) ( ) 2 1 1

1

, ,1

1 * *

2

R

cycle m cycle i cycle i p dS C C t Z

α α α ε + +

−

⎛ ⎞ = −⎜ ⎟ − ⎝ ⎠ (137) 

Where: 

Z factor was given by the following equation 138. 
98

1

1

1 1 Z Q *

M

α α

α

+

+ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (138) 

In the work of Lee, Daniel, and Kim, the Z factor was given by equation 139.(3–5)

1

1 1

4

Z +α ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (139) 

The tests used to verify the above derivation are summarized in table 12. The four ALF mixtures 

used in the VEPCD model verification and discussed in detail elsewhere were used here.(12,13) 

Table 12 shows that each of the mixtures had both controlled stress (CS) and controlled 

crosshead (CX) tests and different combinations of levels and temperatures. The Q and Z factors 

for each of these mixtures are summarized in table 13 for both the CS and CX conditions. 
99

Table 12. Cyclic tests performed. 

Material Specimen 

Name 

Test 

Type 

Test 

Designation

Level 

(kPa/ με

Freq. 

(Hz)

Temp. 

( °C ) Nf

Control 

FHWA 101 CS 5-CS-H 1,500/42 4 5 32,000 

FHWA 99 CS 19-CS-H 750/52 4 19 4,700 

FHWA 100 CS 19-CS-L 250/16 4 19 108,720 

FHWA 96 CX 19-CX-L —/262 4 19 14,800 

FHWA 97 CX 19-CX-L(2) —/273 4 19 17,800 

FHWA 98 CX 19-CX-H —/327 4 19 7,900 

FHWA 85 CX 5-CX-H —/89 10 5 51,170 

FHWA 87 CX 5-CX-H(2) —/83 10 5 48,190 

FHWA 88 CX 5-CX-L —/73 10 5 135,000 

CRTB 

CRTB 30 CS 5-CS-H 1,500/90 4 5 4,400 

CRTB 28 CS 19-CS-H 750/90 4 19 2,360 

CRTB 29 CS 19-CS-L 250/25 4 19 106,700 

CRTB 26 CX 19-CX-H —/333 4 19 27,500 

CRTB 27 CX 19-CX-L —/442 4 19 48,762 

SBS 

SBS 28 CS 5-CS-H 1,500/70 4 5 4,100 

SBS 26 CS 19-CS-H 750/110 4 19 670 

SBS 27 CS 19-CS-L 250/30 4 19 54,240 

SBS 25 CX 19-CX-L —/300 4 19 >120,000 

SBS 24 CX 19-CX-H —/523 4 19 > 90,000 

Terpolymer 

EVY 28 CS 5-CS-H 1,500/70 4 5 5,500 

EVY 26 CS 19-CS-H 750/140 4 19 740 

EVY 27 CS 19-CS-L 250/50 4 19 56,230 

EVY 24 CX 19-CX-L —/428 4 19 > 138,000 

EVY 25 CX 19-CX-H —/579 4 19 75,000 

— Initial stress level in CX tests are not reported,

Table 13. Summary of cyclic correction factors. 

Mixture α

Test 

Type Q Z 

PG 70-22 2.12 CS 0.669 0.609

CX 0.814 0.558

CRTB 2.96 CS 0.672 0.624

CX 0.815 0.605

SBS 2.62 CS 0.671 0.619

CX 0.814 0.588

Terpolymer 2.44 CS 0.670 0.622

CX 0.815 0.605
100

The comparisons of the C versus S curves characterized from the monotonic tests and from 

cyclic tests are shown in figure 72 through figure 75. The loading portions of the first cycle in all 

of the tests shown in figure 72 were calculated using the formulation shown in equation 29. 

0.00

0.20

0.40

0.60

0.80

1.00

0.0E+00 1.0E+05 2.0E+05 3.0E+05

S

C

19-CX-L 19-CX-L

19-CX-H 5-CX-H

5-CX-H 5-CX-L

19-CS-H 19-CS-L

5-CS-H Monotonic

Figure 72. Graph. Damage characteristic comparison, cyclic to monotonic using 

equation 127 Control mixture. 

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.0E+05 2.0E+05 3.0E+05

S

C

19-CX-L 19-CX-H

19-CS-H 19-CS-L

5-CS-H Monotonic

Figure 73. Graph. Damage characteristic comparison, cyclic to monotonic using 

equation 127 CRTB mixture. 
101

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.0E+05 2.0E+05 3.0E+05

S

C

19-CX-L 19-CX-H

19-CS-H 19-CS-L

5-CS-H Monotonic

Figure 74. Graph. Damage characteristic comparison, cyclic to monotonic using 

equation 127 SBS mixture. 

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.0E+05 2.0E+05 3.0E+05

S

C

19-CX-L 19-CX-H

19-CS-H 19-CS-L

5-CS-H Monotonic

Figure 75. Graph. Damage characteristic comparison, cyclic to monotonic using 

equation 127 Terpolymer mixture. 

4.8.3. Discussion of Simplified Model Approach 

Figure 72 through figure 75 show that the collapse of the damage characteristic curve between 

cyclic and monotonic data depended on both the mixture and test conditions. The Control and 
102

CRTB mixtures showed the most favorable overall collapse, whereas the SBS and terpolymer 

mixtures showed the least favorable collapse. For each of the mixtures, the degree of collapse 

appeared to reduce as the amount of permanent strain developing in the test increases. In general, 

such permanent strains were significantly higher in the CS tests than they were in the CX tests. 

These strains also tended to increase as a percentage of the total with lower input amplitudes or 

higher temperatures. These results appeared to become significant for only the 19-CS-L test for 

the Control and CRTB mixtures. The SBS mixture showed the effects under both of the 19 °C 

CS test conditions, whereas the terpolymer mixture showed major effects under only the 

19-CS-L and the CX conditions. 

In addition to the effects of permanent strain, the collapse may have been affected by errors in 

some of the aforementioned assumptions made for the calculation of damage. Unfortunately, the 

direct verification of these assumptions was not possible due to uncontrollable data analysis and 

acquisition errors. Additionally, in cyclic loading, it was possible that other mechanisms, such as 

nonlinear viscoelasticity and microcrack healing, may have occurred. If this was the case, these 

mechanisms may not have been triggered by constant rate tension testing. Therefore, it was not 

expected that the curves would collapse until the model was appropriately formulated. Another 

potential source of error lay with the α variable in the VEPCD model. Two different physical 

interpretations of the α variable were used in the literature. However, these interpretations were 

based on the physical consequences for a single macrocrack (i.e., fracture process zone 

characteristics, which may have had no relation to the physical realities of a continuum damage 

problem). In addition, the interpretation of α as a constant value independent of test conditions 

was based on certain LVE characteristics. These particular characteristics were not observed for 

asphalt concrete under every condition. In addition, because α was an exponent in the model, its 

impact was complicated and could have caused an increase or decrease in the damage prediction, 

depending on temperature, mode of loading, and amplitude of loading. 

4.9. REFINED DERIVATION OF SIMPLIFIED FATIGUE MODEL 

The previous formulations had some obvious problems either in the rigor of their derivations or 

in the assumptions made. To address this problem, a new formulation was proposed. In the 

following subsections this refined formulation is given. Then, the refined formulation is further 

simplified into a form which can be easily programmed into a spreadsheet for fatigue life 

predictions. To be clear, the definition used for pseudo stiffness with this formulation is shown in 

equation 140. Notice that pseudo stiffness is given by the variable F to differentiate it from the 

definition used in the previous formulations. Qualitative the meaning of F and C are the same 

even though they differ slightly quantitatively. 

0,

0,

*

ta

R

ta

F

I

σ

ε = (140) 

Where: 

σ 0,ta = The stress tension amplitude. 

ε R

0,ta = The pseudo strain tension amplitude. 

Ι = The normalization factor (0.9–1.1). 
103

4.9.1. Identification of Tensile Loading Time 

To quickly analyze the cyclic data in the simplified mechanics model, it was important to 

identify the actual time that a given cycle was under tensile loading. This time was important 

because it was assumed that fatigue damage occurred only under such conditions. To determine 

this time, it was assumed that regardless of the input condition the following analytical function 

was descriptive of the stress history of any given cycle for the tests used in this study: 

( ) ( ) ( )

0,

1

cos

1 R

ta

σ ξ β ωξ σ β = − + (141) 

Where: 

σ 0,ta = The tension stress amplitude and for generality time was given as reduced time, ξ . 

The β term, defined by equation 142, was a factor that allowed direct quantification of the 

duration that a given stress history was tensile. In equation 142, (σ peak)i was the largest value 

within cycle i, and (σ valley)i was the smallest value within cycle i. 

( peak valley ) ( ) i i

i

peak valley i i

σ σ

β

σ σ

+ = + (142) 

When β = 1, the entire stress (and therefore the pseudo strain minus permanent pseudo strain) 

history for the given cycle was tensile; when β = 0, half of the history was tensile; and when 

β = -1, the entire history was compressive. This last condition was not used for any of the tests 

in this study. 

Noting that the total pulse time was ωR /(2π ) and using the definition of stress shown in 

equation 141, the factor M used in the previous formulation can now be rigorously calculated 

using equation 143. 

( ) 1 1 1 cos

1

M 2

β

π

− ⎛ ⎞ = − ⎜ ⎟

⎝ ⎠ (143) 

With the definition of total stress history and β in equation 141 and equation 142 respectively, it 

can be shown that the tensile amplitude can be calculated from the peak-to-peak values if β is 

known by equation 144. Although equation 144 is shown for stress, it can also be applied to 

pseudo strain. 

0, 0,

1

2 ta pp

β σ σ+ = (144) 

Finally, it can be shown that the times during the load pulse at which tensile loading begins and 

ends for any given cycle are, in terms of the coefficient, β , and the reduced loading frequency, 

ωR , given by equation 145 and equation146. 
104

( ) 1 cos

i

R R

π π β ξ

ω ω

− − = − (145) 

( ) 1 cos

f

R R

π π β ξ

ω ω

− − = + (146) 

4.9.2. Defining Alpha 

The power,α , in the VECD model has been discussed previously. In the reformulated model, the 

power was defined more clearly. Through theoretical arguments that use the macrocracking 

phenomenon, this power was found to relate to linear viscoelastic time dependence.(9) Motivated 

by earlier work on this subject, the maximum absolute value of the log-log slope of the 

relaxation modulus, m, was taken to represent the linear viscoelastic response. (See references 4, 

5, 7, and 8.) According to the theory, if the material’s fracture energy and failure stress are 

constant, then α = 1/m + 1, but if the fracture process zone size and fracture energy are constant, 

then α = 1/m. A review of the literature shows that different researchers have used differing α

values. Some researchers suggest that it is most appropriate to use α = 1/m for the type of CS 

tests that were performed in this current study and α = 1/m + 1 for the CX tests.(3,4) This 

approach is supported by the work of other researchers that uses the constant failure stress and 

energy criteria for the CX tests and by separate studies which have used CS-type tests and 

α = 1/m.

(5,7,12) Therefore, this study used α = 1/m + 1 for the CX tests, and α = 1/m for the CS 

tests. No attempt was made in this work to calibrate these relationships further. A summary of 

the α values used for the mixtures of interest is given in table 14. 

Table 14. Summary of α values for refined model. 

Mixture Test Type α

PG 70-22 CS 2.40539

CX and Monotonic 3.40539

CRTB CS 3.31787

CX and Monotonic 4.31787

SBS CS 2.92686

CX and Monotonic 3.92686

Terpolymer CS 2.72093

CX and Monotonic 3.72093

9.5 Fine CS 3.12658

CX and Monotonic 4.12658

4.9.3. Failure in Cyclic Mode of Loading 

Two criteria were adopted to define the cycle at which data could be used in the VECD 

characterization/verification process. Two different criteria were necessary because the CS and 

CX tests failed in quite different patterns and because each criterion captured one of the two 

possible reasons that it was inappropriate to use a given test result: (1) localization of damage at 

the point of a single macrocrack, or (2) the onset of additional dominating mechanisms that were 

not related to microdamage. To identify the point of macrocrack localization in these tests, the 
105

method suggested in the literature, whereby the cycle at which the phase angle showed a sharp 

decrease, was taken as Nf.

(58) This cycle was identified graphically by plotting the phase angle 

versus cycle number. During the test, the phase angle tended to increase steadily until a point at 

which it dropped rapidly. The cycle where this rapid drop occurred was Nf. 

When processes other than damage mechanisms, such as viscoplasticity, began to have 

significant effect, then a test could no longer be used directly for characterization. Although at 

this time rigorous identification of this occurrence was not possible, it was assumed that the 

onset of other mechanisms was closely related to the total amount of permanent strain 

experienced by the specimen. It was known from experience that the constant crosshead rate 

tests at 5 °C and strain rates above 1 x 10-5 for unmodified mixtures and 1 x 10-4 for polymermodified mixtures were dominated by the viscoelastic damage mechanisms.(7,12) It was also 

known that for any given mixture the tests performed at these rates showed similar strain levels 

at the peak stress. These mixture-dependent strain levels represented a known level below which 

VECD mechanisms dominate. The cycle in the fatigue tests at which the permanent strain 

exceeded this threshold was taken as the point after which data could not be used for VECD 

characterization. To find the permanent strain during the cyclic test, the permanent pseudo strain 

was tracked by cycle. Then these permanent pseudo strain values were compiled as a function of 

reduced time, where reduced time was taken at the end of the beginning of the respective cycle 

where the permanent pseudo strain was known. It was then assumed that the permanent pseudo 

strain varied linearly within the cycle (the final value for cycle i was equivalent to the initial 

value for cycle i + 1). With this assumption, equation 16 could be inverted and used with the 

state variable formulation to solve for permanent strain. Note that while the effect of assuming a 

linear pseudo strain growth is not explored here, the effect of this assumption was small since the 

amount of permanent pseudo strain within a cycle is generally very small. 

When either of these two conditions was met, failure was said to have occurred, and all data 

gathered afterward were ignored. Thus, all of the data shown in any of the subsequent graphs 

represent the data prior to this failure point. It is worth noting that in this study all tests in the CX 

mode failed due to the rapid change in phase angle, while all tests performed in the CS mode 

failed due to exceeding the strain limit. 

4.9.4. Refined Simplified Model Derivation 

The basic concept of the refined model is to separate the damage calculation into two 

components: damage occurring in the first loading path and damage occurring due to the 

repeated loading. These two components were referred to as the transient and cyclic portions 

respectively. The mathematical implications are given in equation 147. 

Transient p

Cyclic p

dS

dS dS

ξ ξ

ξ ξ

⎧ ≤ = ⎨ > ⎩ (147) 

Where: 

ξ p = Reduced cycle pulse time. 
106

This refined method assumed that all of the effects of damage were shown through a reduction of 

the cyclic pseudo stiffness, F. This assumption did not mean that damage affected only the cyclic 

pseudo stiffness but rather that this pseudo stiffness wholly reflected damage effects. It was also 

assumed that damage growth within a given cycle was small. The test data were still normalized 

for specimen-to-specimen variability by the correction factor, I, defined earlier. The transient 

calculation was the same as the rigorous calculation shown in equation 114, but the form of the 

cyclic portion was given by equation 148 where the variable R was a form adjustment factor. The 

subscript k was only a placeholder to represent the fact that the factor R could vary from cycle to 

cycle depending on the particular loading history. During real experiments this might occur due 

to slight deviations in machine control. In general, though, the value of R varied only slightly 

throughout loading. However, the refined model was capable, by including the form adjustment 

factor, to explicitly account for any load form shape (i.e., square, saw-tooth, sinusoidal, etc.), or 

combinations of shapes, as long as that load history did not violate the assumptions made in the 

following paragraphs. 

( ) ( ) ( ) 2 1 1

1

, 0. * *

2

R

cyclic k ta p k

I dS F R

α α α ε ξ + ⎛ ⎞ + =− Δ ⎜ ⎟ ⎝ ⎠ (148) 

This adjustment factor was found by assuming that equation 150 could be used to simplify the 

more rigorous equation 149. 

( )2

* *

2

f

i

R

e

I C S d

S

ξ α

ξ

ε ξ ⎛ ⎞ ∂ Δ= −⎜ ⎟ ⎝ ⎠ ∂ ∫ (149) 

( ) ( ) 2

0, 1

* **

2

R

ta f i

I F S K

S

α

ε ξξ ⎛ ⎞ ∂ Δ=− − ⎜ ⎟ ⎝ ⎠ ∂ (150)

( ) ( ) * * ( ) 0, ( ) R RR R

e ms ta ε ξ ε ε ξε ξ =− = f f (151) 

Setting equation 149 equal to equation 150 and noting that (equation 152): 

C F

S S

∂ ∂

≅

∂ ∂ (152) 

gave the following (equation 153): 

( ) ( ) ( ) 2 2

0, 1

* * * **

2 2

f

i

R R

e ta f i

IF I F d K

S S

ξ α α

ξ

ε ξ ε ξξ ⎛ ⎞⎛ ⎞ ∂ ∂ ⎜ ⎟⎜ ⎟ − = − − ⎝ ⎠⎝ ⎠ ∂ ∂ ∫ (153) 
107

Finally, if it was assumed that within a cycle, ∂F/ ∂ S was nearly constant and if it is recalled that 

the effective pseudo strain history is given by equation 151, then (equation 154): 

( ) ( ) ( ) ( ) ( ) 2 2 2

0, 0, 1

* * **

f

i

R R

ta ta f i f d K

ξ

α α α

ξ

ε ξ ξε ξξ = − ∫ , (154) 

which simplifies to (equation 155): 

( ) ( ) 2

1

1 * *

f

i f i

K f d

ξ

α

ξ

ξ ξ ξ ξ = − ∫ (155) 

The implication of assuming that ∂F/ ∂ S was nearly constant was not explored for this study. 

However, in the context of fatigue where failure often required thousands to millions of cycles, it 

was assumed that such an assumption was valid. It can be shown that the R factor was equal to 

K1 raised to a power that was a function of α as shown in equation 156, where the times for the 

integral in this equation were defined by equation 145 and equation 146. The values for this 

adjustment factor were approximately equal to 0.77 for the CX tests (nearly sinusoidal loading) 

and 0.70 for the CS tests (purely haversine loading) for most mixtures. Should the loading form 

be something other than these two types, the value of R would be different but could be 

calculated using equation 156. 

( ) ( )

1

1

1 2

* *

f

i f i

R fd

ξ α

α

ξ

ξ ξ ξ ξ

+ ⎛ ⎞

= ⎜ ⎟ − ⎝ ⎠ ∫ (156) 

4.9.5. Model Application 

The refined model was applied to the tests given in table 12, and the comparison of the damage 

characteristic curves from the cyclic and monotonic tests are shown in figure 76 through 

figure 79. Comparing these figures to the results shown in figure 72 through figure 75, it was 

seen that the refined method did a much better job in collapsing the damage characteristic curves, 

suggesting that the mathematical rigor and assumptions made in this refined method were more 

appropriate for describing the behavior of asphalt concrete materials. 
108

0.00

0.25

0.50

0.75

1.00

0.E+00 2.E+05 4.E+05 6.E+05 8.E+05 1.E+06

S

F

19-CX-L 19-CX-L (2) 19-CX-H

5-CX-L 5-CX-H (2) 5-CX-H

19-CS-L 19-CS-H 5-CS-H

Monotonic

Figure 76. Graph. Damage characteristic comparison, cyclic to monotonic using 

refined model Control mixture. 

0.00

0.25

0.50

0.75

1.00

0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 5.E+05

S

F

5-CS-H 19-CS-L

19-CS-H 19-CX-H

19-CX-L Monotonic

Figure 77. Graph. Damage characteristic comparison, cyclic to monotonic using 

refined model CRTB mixture. 
109

0.00

0.25

0.50

0.75

1.00

0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 5.E+05 6.E+05 7.E+05

S

F

19-CX-H 19-CX-L

5-CS-H 19-CS-L

Monotonic 19-CS-H

Figure 78. Graph. Damage characteristic comparison, cyclic to monotonic using 

refined model SBS mixture. 

0.00

0.25

0.50

0.75

1.00

0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 5.E+05

S

F

5-CS-H 19-CS-L

19-CS-H 19-CX-H

19-CX-L Monotonic

Figure 79. Graph. Damage characteristic comparison, cyclic to monotonic using 

refined model Terpolymer mixture. 
110

4.9.6. Simplifications to Refined Model 

Compared to the rigorous modeling, the refined model was simple to use. However, the method 

still required the complete calculation of pseudo strain for the entire loading history, e.g., 

evaluation of the integral in equation 16. This process could have been time consuming, so it was 

worthwhile to examine simplifications to the calculation. The easiest simplification was referred 

to as the steady-state assumption because it was rigorously accurate only under steady-state 

loading. This condition was only approximately true for the cyclic portion of loading and is 

mathematically shown in equation 157. 

( ) 0. ( ) 0, (( ) 0, ) 1 1 1 ** * **

2

R

ta cycle i ta i LVE pp i LVE

R R

E E

E E

β ε ε ε+ = = (157) 

Replacing the pseudo strain amplitude in equation 148 with that in equation 157 greatly 

simplified the calculation and made a spreadsheet solution a possibility. The consequences of 

this simplification are shown in figure 80 through figure 84. Note that an additional mixture was 

included in this group of graphs. The test summary for only this mixture is given in table 15; a 

summary of the other mixtures has already been given in table 12. Due to the change in 

definition for pseudo stress, the pseudo stiffness is now referred to as C* to indicate that it was 

computed using the steady-state assumption. Also, recall the definitions of failure in these tests 

have been provided in subsection 4.9.3. 

0.0

0.2

0.4

0.6

0.8

1.0

0.E+00 2.E+05 4.E+05 6.E+05 8.E+05 1.E+06

S

C*

Monotonic 19-CX-L

19-CX-L (2) 19-CX-H

5-CX-L 5-CX-H (2)

5-CX-H 5-CS-H

19-CS-H 19-CS-L

Figure 80. Graph. Damage characteristic comparison, cyclic to monotonic using 

refined simplified model Control mixture. 
111

0.0

0.2

0.4

0.6

0.8

1.0

0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 5.E+05

S

C*

Monotonic

19-CX-L

19-CX-H

5-CS-H

19-CS-H

19-CS-L

Figure 81. Graph. Damage characteristic comparison, cyclic to monotonic using 

refined simplified model CRTB mixture. 

0.0

0.2

0.4

0.6

0.8

1.0

0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 5.E+05 6.E+05 7.E+05

S

C*

Monotonic

19-CX-L

19-CX-H

5-CS-H

19-CS-H

19-CS-L

Figure 82. Graph. Damage characteristic comparison, cyclic to monotonic using 

refined simplified model SBS mixture. 
112

0.0

0.2

0.4

0.6

0.8

1.0

0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 5.E+05

S

C*

Monotonic

19-CX-L

19-CX-H

5-CS-H

19-CS-H

19-CS-L

Figure 83. Graph. Damage characteristic comparison, cyclic to monotonic using 

refined simplified model Terpolymer mixture. 

0.0

0.2

0.4

0.6

0.8

1.0

0.E+00 2.E+05 4.E+05 6.E+05 8.E+05 1.E+06

S

C*

Monotonic 19-CX-VL

19-CX-H 19-CX-VH

5-CX-L 5-CX-H

5-CS-H 19-CS-H

19-CS-L

Figure 84. Graph. Damage characteristic comparison, cyclic to monotonic using 

refined simplified model 9.5-mm Fine mixture. 
113

Table 15. Cyclic tests performed for 9.5-mm Fine mixture 

Material Test 

Type 

Test 

Designation

Level 

(kPa / με

Freq. 

(Hz) 

Temp. 

( °C ) Nf

9.5 Fine 

CS 5-CS-H 1,500/115 10 5 5,900 

CS 5-CS-L 750/40 10 5 29,6450 

CS 19-CS-H 750/108 10 19 2,950 

CS 19-CS-L 250/35 10 19 105,480 

CX 19-CX-H —/240 10 19 46,389 

CX 19-CX-VL —/190 10 19 311,170 

CX 19-CX-VH —/402 10 19 2,670 

CX 5-CX-H —/150 10 5 70,140 

CX 5-CX-L —/126 10 5 145,410 

— Initial stress level in CXH tests is not reported.

4.10. SUMMARY 

Various interpretations of the work potential/pseudo strain class of continuum damage models 

have been explored, and a final refined version is suggested. The two methods shown in this 

report represented a simplification of a more rigorous theory so that it can be easily and 

accurately applied to cyclic fatigue testing and characterized using cyclic fatigue tests that are 

performed in the AMPT. The final suggested version is also further simplified so that it can be 

implemented as a software solution. The earlier versions of this model contained certain 

shortcomings in the rigor of their derivations. The rigorously accurate VECD model has been 

used to gain understanding of, and improve upon, these earlier models. The resulting refined 

formulation unified the results of CS, CX, and monotonic testing and supported earlier findings 

that the damage characteristic curve was a material property independent of temperature and test 

type. In addition, this model allowed rapid characterization and/or prediction of the material 

fatigue response. Coupled with either a structural model or a predictive scheme similar to that 

used in the NCHRP 1-37A Mechanistic Empirical Pavement Design Guide, this model could 

also be used to predict the fatigue response of asphalt concrete pavements.(59) This topic is the 

subject to ongoing efforts by the research team.

115

CHAPTER 5. MVEPCD CHARACTERIZATION AND VERIFICATION IN 

COMPRESSION 

The permanent deformation in HMA is affected by several mixture factors, such as the resistance 

of the binder to flow, aggregate angularity and gradation, the amount of asphalt, the air void 

content, etc. A significant amount of research has been conducted to develop laboratory test 

methods, analysis techniques, and models to study the permanent deformation growth of HMA. 

The nature of the permanent deformation models available in the literature ranges from empirical 

to mechanistic-empirical to completely mechanistic. Further attempts have been made in recent 

years to develop a mechanistic permanent deformation model that involves fundamental material 

characterization.(20,52,53)

The objective of this research was to develop a constitutive model of HMA in compression that 

could predict HMA behavior under loading conditions and temperatures encountered in the field. 

The following sections present theories used in the modeling, the experimental program, the 

characterization process, and predictions for various loading histories. For this effort only, the 

ALF Control mixture has been used, and it should be understood that all figures in this chapter 

are related to this mixture. 

5.1. ENGINEERING BEHAVIOR OF ASPHALT CONCRETE IN COMPRESSION 

5.1.1. Constant Rate Compression 

Figure 85 and figure 86 show the rate-dependent stress-strain curves of HMA at the confining 

pressures of 0 and 500 kPa, respectively. As observed from figure 85 and figure 86, the gross 

trends in stress-strain curves with 500 kPa confinement were quite similar to those with 0 kPa 

confining pressure; the overall strength of the material decreased with increasing temperature 

and with decreasing strain rates. Directly comparing the curves at different temperatures (figure 

87 through figure 90), it was found that at 5 and 25 °C, the response was not sensitive to 

confining pressure. The strengths began to deviate during the 40 °C tests and were significantly 

different at 55 °C, with the confined tests showing higher strengths. This trend was similar to 

that observed in the confined dynamic modulus tests, which was described for tensioncompression loading in subsection 4.2.2 and for compression loading in section 5.2. This finding 

suggests that when the material was very stiff, the effects of confinement were negligible. 
116

0

2000

4000

6000

8000

10000

0% 1% 2% 3% 4%

Axial Strain

Stress (kPa)

5-1 5-2 5-3

5-4 25-1 25-2

25-3 25-4 40-1

40-2 40-3 40-4

55-1 55-2 55-3

55-4

Figure 85. Graph. Stress-strain curves for unconfined constant strain-rate tests. 

0

2000

4000

6000

8000

10000

0% 1% 2% 3% 4%

Axial Strain

Stress (kPa)

5-1 5-2 5-3

5-4 25-1 25-2

25-3 25-4 40-1

40-2 40-3 40-4

55-1 55-2

Figure 86. Graph. Stress-strain curves for 500 kPa confinement constant strain-rate tests. 
117

0

2000

4000

6000

8000

10000

0.0% 1.0% 2.0% 3.0% 4.0%

Strain

Stress (kPa)

5-1-U 5-2-U 5-3-U 5-4-U

5-1-C 5-2-C 5-3-C 5-4-C

Figure 87. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests 

for 5 °C. 

0

2000

4000

6000

8000

10000

0.0% 1.0% 2.0% 3.0% 4.0%

Strain

Stress (kPa)

25-1-U 25-2-U 25-3-U 25-4-U

25-1-C 25-2-C 25-3-C 25-4-C

Figure 88. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests 

for 25 °C. 
118

0

2000

4000

6000

0.0% 1.0% 2.0% 3.0% 4.0%

Strain

Stress (kPa)

40-1-U 40-2-U 40-3-U 40-4-U

40-1-C 40-2-C 40-3-C 40-4-C

Figure 89. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests 

for 40 °C. 

0

1000

2000

3000

4000

5000

0.0% 1.0% 2.0% 3.0% 4.0%

Strain

Stress (kPa)

55-1-U 55-2-U 55-3-U 55-4-U

55-1-C 55-2-C

Figure 90. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests 

for 55 °C. 
119

5.1.2. Repetitive Creep and Recovery Test 

5.1.2.1. Creep and Recovery Test with VL 

Figure 91 through figure 93 present the viscoplastic strains for the given VL stress histories. The 

viscoplastic strains caused by the first loading in each group, except for the first loading group, 

were close to zero or were at least very small because the deviatoric stress that caused the 

viscoplastic strains was quite small compared to that of previous loadings. 

0.0%

1.0%

2.0%

3.0%

4.0%

0 5 10 15 20

Cumulative Loading Time (s)

Viscoplastic Strain

0kPa-variable VL (1)

0kPa-variable VL (2)

Time

...

Stress

0.0%

1.0%

2.0%

3.0%

4.0%

0 5 10 15 20

Cumulative Loading Time (s)

Viscoplastic Strain

0kPa-variable VL (1)

0kPa-variable VL (2)

Time

...

Stress

Figure 91. Graph. Viscoplastic strain versus cumulative loading time (unconfined VL). 
120

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 5 10 15 20

Cumulative Loading Time (s)

Viscoplastic Strain

140kPa-variable VL (1)

140kPa-variable VL (2)

Time

...

Stress

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 5 10 15 20

Cumulative Loading Time (s)

Viscoplastic Strain

140kPa-variable VL (1)

140kPa-variable VL (2)

Time

...

Stress

Figure 92. Graph. Viscoplastic strain versus cumulative loading time 

(140 kPa confinement VL). 

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0 5 10 15 20

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-variable VL (1)

500kPa-variable VL (2)

Time

...

Stress

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0 5 10 15 20

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-variable VL (1)

500kPa-variable VL (2)

Time

...

Stress

Figure 93. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement VL). 
121

5.1.2.2. Creep and Recovery Tests with VT and RVT 

Figure 94 and figure 95 present the viscoplastic strains measured at the end of each rest period in 

unconfined VT and 500 kPa confinement VT, respectively. In both unconfined and confined VT 

testing, the overall reproducibility was sufficiently adequate to identify the effects of load level 

and loading sequence. In RVT testing, less viscoplastic strain was observed than in VT testing 

despite the fact that the test conditions were the same for both tests, with the exception of the 

sequence of loading, as described previously. This difference in viscoplastic strain indicated that 

the sequence of loading plays was an important role in viscoplastic strain development. In 

addition, a change of slope between the two groups of viscoplastic strain, which could not be 

explained by the concepts inherent of the conventional viscoplastic model, was observed. It 

seems the characteristic behavior of HMA was affected by viscoelastic relaxation. 

Although quantifying the variations in the viscoplastic strain rate under a given loading condition 

is necessary for rigorous modeling work, no test protocol was available that could capture only 

the viscoplastic strain rate because HMA showed time-dependent viscoelastic strain, too. 

However, trends for viscoplastic strain rates developed in repetitive creep and recovery testing 

could be estimated by analyzing the VT test results. Figure 96 presents incremental viscoplastic 

strain rates (i.e., incremental viscoplastic strain divided by pulse time). As shown in figure 96, at 

1-percent viscoplastic strain, viscoplastic strain rates from 0.05-s loadings (2.0 x 10-3) were much 

greater than those from 6.4-s loadings (8.0 x 10-5). These results indicated that most of the 

viscoplastic strain developed at the beginning of the loading period and that the viscoplastic 

strain that developed during the remainder of the loading period (i.e., at 6.4 to 0.05 s) was 

relatively small. Because the calculated viscoplastic strain rate was the average of the 

viscoplastic strain rates during loading, the actual viscoplastic strain rates at the end of the 

loading were much smaller than 8.0 x 10-5. This is another important behavior of HMA, along 

with the softening concept presented in the following section. Mathematically, this behavior can 

be formulated in a viscoplastic constitutive model with either increasing viscosity or increasing 

yield stress due to aggregate interlocking. 
122

0.0%

1.0%

2.0%

3.0%

4.0%

0 10 20 30 40 50

Cumulative Loading Time (s)

Viscoplastic Strain

0kPa-827kPa VT (1)

0kPa-827kPa VT (2)

Time

...

Stress

0.0%

1.0%

2.0%

3.0%

4.0%

0 10 20 30 40 50

Cumulative Loading Time (s)

Viscoplastic Strain

0kPa-827kPa VT (1)

0kPa-827kPa VT (2)

Time

...

Stress

Figure 94. Graph. Viscoplastic strain versus cumulative loading time 

(unconfined VT testing). 

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 50 100 150 200

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-1600kPa VT (1) 500kPa-1600kPa VT (2)

500kPa-1800kPa VT (1) 500kPa-1800kPa VT (2)

500kPa-2000kPa VT (1) 500kPa-2000kPa VT (2)

500kPa-1600kPa RVT (1) 500kPa-1600kPa RVT (2)

Time

...

Stress

Time

...

Stress

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 50 100 150 200

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-1600kPa VT (1) 500kPa-1600kPa VT (2)

500kPa-1800kPa VT (1) 500kPa-1800kPa VT (2)

500kPa-2000kPa VT (1) 500kPa-2000kPa VT (2)

500kPa-1600kPa RVT (1) 500kPa-1600kPa RVT (2)

Time

...

Stress

Time

...

Stress

Figure 95. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement for VT and RVT testing). 
123

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

0.0% 0.5% 1.0% 1.5% 2.0% 2.5%

Viscoplastic Strain

Incremental VP Strain Rate

0.05 sec 0.10 sec

0.25 sec 0.40 sec

1.60 sec 2.00 sec

6.40 sec

Figure 96. Graph. Incremental viscoplastic strain rate versus viscoplastic strain 

(500 kPa confinement, 1,600 kPa deviatoric). 

5.1.2.3. Creep and Recovery Tests with Constant Load Level and CLT 

Figure 97 presents the viscoplastic strain history for each loading condition with a confining 

pressure of 500 kPa. A smaller viscoplastic strain was observed as the pulse time increased. Even 

considering the ramp time of 0.005 s, which was not taken into account in the cumulative loading 

time, the difference in the viscoplastic strains was quite significant. 

As shown in figure 97, more viscoplastic strain was observed in the CLT tests that consisted of 

shorter loading times at a given cumulative loading time. The differences were significant. For 

example, at 150 s of cumulative loading time, the viscoplastic strain in 1,800 kPa CLT testing 

with a 0.4-s loading time was over 3 percent, whereas it was around 1.5 percent in 1,800 kPa 

CLT testing with a 6.4-s loading time and was 2.2 percent with a 1.6-s loading time. One reason 

for this behavior could be related to the dynamic effects associated with the ramp to the target 

load because one difference in the tests, as they are presented in figure 97, is the number of load 

applications. Alternatively, because the tests are exposed to different total rest times at given 

cumulative loading times, the differences could be related to material softening. Furthermore, it 

is not beyond reason to suppose that this softening behavior could be rate dependent. For further 

study of this issue, two additional VT tests with 0.1- and 0.05-s rest periods were performed at a 

confining pressure of 140 kPa. These test results were compared with the results from VT testing 

with 200 s of rest. In these tests, the conditions were identical (i.e., the number of loadings and 

rest periods were the same for each), except for the length of the rest period. Figure 98 presents 

the viscoplastic strains measured at the end of the rest periods. For VT tests with 0.05 and 0.1 s 

of rest, 200 s of rest was allowed at the end of the testing to measure the pure viscoplastic strain 

because it was not possible to measure pure viscoplastic strain immediately after 0.05 or 0.1 s of 

rest. The deviatoric stresses were 827 and 552 kPa, and the confining pressure was 140 kPa. 
124

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

0 50 100 150 200

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-1600kPa VT (1) 500kPa-1600kPa VT (2)

500kPa-1800kPa CLT 6.4s (1) 500kPa-1800kPa CLT 6.4s (2)

500kPa-1800kPa CLT 1.6s (1) 500kPa-1800kPa CLT 1.6s (2)

500kPa-1800kPa CLT 0.4s (1) 500kPa-1800kPa CLT 0.4s (2)

Figure 97. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement CLT). 

0.0%

0.5%

1.0%

1.5%

2.0%

0 50 100 150 200 250 300 350

Cumulative Loading Time (s)

Viscoplastic Strain

140kPa-827kPa VT 200s (1) 140kPa-827kPa VT 200s (2)

140kPa-827kPa VT 1.0s (1) 140kPa-827kPa VT 1.0s (2)

140kPa-827kPa VT 0.05s (1) 140kPa-827kPa VT 0.05s (2)

140kPa-552kPa VT 200s (1)

 Figure 98. Graph. Viscoplastic strain versus cumulative loading time 

(140 kPa confinement VT). 
125

As seen in figure 98, even though the loading histories were identical except for the length of the 

rest period, a smaller viscoplastic strain developed as the rest period became shorter. 

Furthermore, the effect of the rest period on viscoplastic development was significant 

considering the amount of viscoplastic strain when using 552 kPa deviatoric stress and 200 s of 

rest. This experimental observation did not demonstrate the effects of the dynamic loading ramp, 

but it did demonstrate the significant effect of rate-dependent softening during unloading. 

Therefore, the modeling effort focused on developing a viscoplastic model that accounted for 

rate-dependent hardening and/or softening. 

5.1.3. t-TS with Growing Damage in Compression 

The principle of t-TS is one of the fundamental and most important concepts for HMA in tension 

modeling because it provides a strong mechanical background and significantly reduces the 

experimental efforts. In order to verify the principle for the compression stress state, stress 

characteristic curves were constructed from constant strain-rate test results by utilizing shift 

factors determined from the dynamic modulus tests. Further verification was also performed 

using repetitive creep and recovery tests. 

 

For the verification of the principle for monotonic loading, a wide range of eight reference-strain 

values were chosen according to the results of both the uniaxial and triaxial compressive constant 

strain-rate tests (figure 85 and figure 86). According to the procedure shown schematically in 

figure 55 through figure 57 and discussed in detail elsewhere, the stress and time values were 

determined for all of these tests at fixed strain levels.(13,20) These plots of stress versus time are 

shown in figure 99 through figure 114. Then, shift factors obtained from small-strain LVE 

testing were applied to determine the reduced time that corresponded to each physical time. If the 

t-TS principle was valid with growing damage, the resulting plots of stress and reduced time 

would appear continuous at all strain levels. This behavior was indeed observed for the 

compression tests under both the confined and unconfined conditions (figure 115 and figure 116). 

These results verified that the t-TS concept held true for mixtures subjected to compressive 

loading as well as to tensile loading, even if there was severe damage and viscoplastic strain. 

However, to verify that the principle held for the physical mechanisms behind the behavior of 

repetitive creep and recovery tests, more rigorous verification was needed—this verification 

compared VT test results at 40 and 55 °C with the same reduced time histories. 

For this verification, VT testing was first performed at 55 °C (the 200 s rest period results are 

used here). Next, the time history was used with the t-TS shift factors to compute the equivalent 

reduced time history at 40 °C. However, because the testing time was estimated to take several 

days (the equivalent time to 200 s at 55 °C is approximately 3,265 s at 40 °C), the following 

analysis was performed to finish the VT testing within a reasonable time. In this analysis, the 

measured strain history during the unloading portion of several load pulses was used to compute 

the strain rate, which was plotted against the rest period time in figure 117. To avoid issues 

related to the initial loading of a test, the 0.05 s data were taken from the second load block, 

whereas the other pulse times were taken from the first loading block of the VT test. As shown in 

figure 117, most of the strain rates became quite small after around 40 s, except for the rest 

periods following the 1.6 and 6.4 s load pulses. For this reason, 40 s was used to compute the 

reduced time for pulse times less than 1.6 s (653 s at 40 °C); 50 s was used for a pulse time of 

1.6 s (816 s at 40 °C); and 60 s was used for a pulse time of 6.4 s (980 s at 40 °C). Note that 
126

strain rates reached an asymptotic value of zero more quickly as the strain level increased, and 

thus, it was conservative to consider the times used in the first loading block as the reference 

times. The results of these two tests are shown figure 118. As the figure shows, viscoplastic 

strains measured at the end of rest periods were well matched to each other. This agreement 

confirmed that the t-TS principle was applicable regardless of loading sequence and the amount 

of damage and viscoplastic strain in asphalt concrete. 

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E+00 1.E+02 1.E+04

Time (s)

Stress (kPa)

5°C 25°C

40°C 55°C

Figure 99. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.0001 strain level under uniaxial conditions. 
127

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E+00 1.E+02 1.E+04

Time (s)

Stress (kPa)

5°C 25°C

40°C 55°C

 Figure 100. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.0005 strain level under uniaxial conditions. 

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E+00 1.E+02 1.E+04

Time (s)

Stress (kPa)

5°C 25°C

40°C 55°C

 Figure 101. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.001 strain level under uniaxial conditions. 
128

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E+00 1.E+02 1.E+04

Time (s)

Stress (kPa)

5°C 25°C

40°C 55°C

 Figure 102. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.003 strain level under uniaxial conditions. 

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E+00 1.E+02 1.E+04

Time (s)

Stress (kPa)

5°C 25°C

40°C 55°C

 Figure 103. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.005 strain level under uniaxial conditions. 
129

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E+00 1.E+02 1.E+04

Time (s)

Stress (kPa)

5°C 25°C

40°C 55°C

 Figure 104. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.01 strain level under uniaxial conditions. 

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E+00 1.E+02 1.E+04

Time (s)

Stress (kPa)

5°C 25°C

40°C 55°C

 Figure 105. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.015 strain level under uniaxial conditions. 
130

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E+00 1.E+02 1.E+04

Time (s)

Stress (kPa)

5°C 25°C

40°C 55°C

 Figure 106. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.02 strain level under uniaxial conditions. 

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Time (s)

Stress (kPa)

5C

25C

40C

55C

 Figure 107. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.0001 strain level under 500 kPa conditions. 
131

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Time (s)

Stress (kPa)

5C

25C

40C

55C

 Figure 108. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.0005 strain level under 500 kPa conditions. 

1.E+02

1.E+03

1.E+04

1.E+05

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Time (s)

Stress (kPa)

5C

25C

40C

55C

Figure 109. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.001 strain level under 500 kPa conditions. 
132

1.E+02

1.E+03

1.E+04

1.E+05

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Time (s)

Stress (kPa)

5C

25C

40C

55C

 Figure 110. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.003 strain level under 500 kPa conditions. 

1.E+02

1.E+03

1.E+04

1.E+05

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Time (s)

Stress (kPa)

5C

25C

40C

55C

 Figure 111. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.005 strain level under 500 kPa conditions. 
133

1.E+02

1.E+03

1.E+04

1.E+05

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Time (s)

Stress (kPa)

5C

25C

40C

55C

 Figure 112. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.01 strain level under 500 kPa conditions. 

1.E+02

1.E+03

1.E+04

1.E+05

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Time (s)

Stress (kPa)

5C

25C

40C

55C

 Figure 113. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.015 strain level under 500 kPa conditions. 
134

1.E+02

1.E+03

1.E+04

1.E+05

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Time (s)

Stress (kPa)

5C

25C

40C

55C

 Figure 114. Graph. Stress-time curves for the Control mixture before the application of 

time-temperature shift factors at a 0.02 strain level under 500 kPa conditions. 

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+00 1.E+02 1.E+04 1.E+06

Reduced Time (s)

Stress (kPa)

0.0001 0.0005 0.001 0.003

0.005 0.01 0.015 0.02

Figure 115. Graph. Stress mastercurves for the Control mixture under uniaxial conditions. 
135

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+00 1.E+02 1.E+04 1.E+06

Reduced Time (s)

Stress (kPa)

0.0001 0.0005 0.001 0.003

0.005 0.01 0.015 0.02

Figure 116. Graph. Stress mastercurves for the Control mixture under triaxial conditions 

(500 kPa). 

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

0 10 20 30 40 50 60

Time (s)

Strain rate

0.1s

0.2s

0.4s

1.6s

6.4s

0.05s

 Figure 117. Graph. Variation of strain rate during unloading. 
136

0.0%

0.5%

1.0%

1.5%

2.0%

0 50 100 150 200 250 300 350

Cumulative Loading Time (s)

Viscoplastic Strain

140kPa-827kPa 55°C VT (1)

140kPa-827kPa 55°C VT (2)

140kPa-827kPa 40°C VT (1)

140kPa-827kPa 40°C VT (2)

 Figure 118. Graph. Viscoplastic strain versus cumulative loading time 

(140 kPa confinement VT at 40 and 55 °C). 

5.2. MVECD CHARACTERIZATION IN COMPRESSION 

5.2.1. Linear Viscoelastic Characterization 

As presented in chapter 2, the first stage in MVECD characterization was the determination of 

the LVE properties of the material. Following the test protocols presented in chapter 3, 

frequency-temperature sweep tests were conducted in both unconfined and confined states. The 

results of this characterization under zero-maximum deviatoric stress conditions are shown for 

the unconfined stress state in table 16, for a confining pressure of 140 kPa in table 17 and for a 

confining pressure of 500 kPa in table 18. The shift factor function coefficients that resulted as 

part of the mastercurve construction process for each of the tests in these tables are shown in 

table 19. 

Table 16 through table 18 show that a trend similar to that observed for the zero-mean deviatoric 

conditions was also observed for the zero-maximum deviatoric tests. Specifically, under 

confinement, the material modulus increased as a function of the confinement level. Additionally, 

under conditions where a given confinement state diverged from the unconfined state, the 

mastercurve moved toward a lower temperature and higher frequency at the higher confined 

stress. Additionally, a higher confining stress generally resulted in a lower phase angle (i.e., 

higher elasticity). Although it may be argued that increased aggregate interlocking was the cause 

of these behaviors, it was assumed that such an interpretation was flawed due to the strain 

magnitudes typically induced during the testing. The net effect of the increased dynamic 

modulus and phase angle was an increase in the relaxation modulus, which is shown for the 

different confining pressures in figure 119. These curves were obtained by the method outlined 

in subsection 2.2.1. Figure 119 shows that the relaxation modulus began to diverge at around 
137

0.01 to 1 s in reduced time. These times generally corresponded to a physical time of between 

1 x 10-5 and 2 x 10-7 s at 54 °C and of between 1 x 10-4 and 1 x 10-7 at 40 °C. Therefore, it was 

important to consider the effect of confining pressure because the time at which the relaxation 

modulus started to diverge was within the range of reduced time that was used in the rutting 

analysis. 

Table 16. Linear viscoelastic characterization and variation for the Control mixture in 

unconfined compression state at selected frequencies and temperatures. 

Temp. 

(°C ) 

Frequency

(Hz) 

Average 

|E*| (MPa)

Average 

Phase Angle 

(°) 

|E*| Coefficient 

of Variation 

(%) 

Phase Angle 

Coefficient of 

Variation (%) 

-10 

25 33,603 4.8 12.6 4.0 

10 32,251 6.1 12.1 10.0 

5 30,865 6.5 11.0 4.6 

1 28,633 7.9 11.7 2.8 

0.5 27,398 8.0 11.4 3.4 

0.1 23,909 9.7 7.9 44.1 

5 

25 23,707 9.7 7.0 45.6 

10 21,323 11.7 5.0 34.3 

5 20,085 12.5 9.1 4.2 

1 16,399 15.1 10.9 4.5 

0.5 14,262 17.2 11.4 3.9 

0.1 11,205 20.7 11.9 5.5 

20 

25 12,444 19.2 11.4 4.3 

10 10,624 21.4 14.7 4.7 

5 8,809 23.2 14.5 5.4 

1 6,229 26.2 7.0 8.4 

0.5 4,941 27.8 1.0 3.0 

0.1 2,818 32.7 5.4 2.4 

40 

25 2,872 32.5 3.2 0.2 

10 1,834 36.2 1.3 1.0 

5 1,386 38.0 2.5 1.7 

1 904 35.7 7.0 3.0 

0.5 640 35.9 7.0 2.6 

0.1 363 32.6 20.4 12.1 

54 

25 973 35.6 4.1 3.3 

10 604 35.8 5.8 2.7 

5 475 34.5 1.0 2.7 

1 321 31.3 20.8 15.4 

0.5 226 29.0 4.7 18.5 

0.1 185 24.3 11.0 38.7 
138

Table 17. Linear viscoelastic characterization and variation for the Control mixture in 

140 kPa confined compression state at selected frequencies and temperatures. 

Temp. 

(°C ) 

Frequency

(Hz) 

Average 

|E*| (MPa)

Average 

Phase Angle 

(°) 

|E*| Coefficient 

of Variation 

(%) 

Phase Angle 

Coefficient of 

Variation (%) 

-10 

25 34,501 9.0 1.6 3.7 

10 33,555 7.8 0.4 4.5 

5 32,396 7.8 0.0 5.7 

1 29,413 8.5 0.0 1.1 

0.5 28,174 9.1 0.5 2.4 

0.1 25,164 10.6 0.9 2.4 

5 

25 24,048 11.1 1.4 0.4 

10 21,655 12.7 1.4 1.6 

5 20,663 13.4 1.2 2.0 

1 17,333 16.3 2.4 2.6 

0.5 15,097 17.9 2.8 1.9 

0.1 11,904 20.6 2.6 1.2 

20 

25 12,773 19.7 3.2 1.3 

10 10,458 22.4 3.2 0.5 

5 9,013 24.3 3.0 0.6 

1 6,278 29.2 4.7 0.2 

0.5 5,176 31.9 5.6 0.3 

0.1 3,376 33.2 1.7 2.2 

40 

25 3,489 33.1 1.2 2.0 

10 2,473 33.5 2.8 4.0 

5 1,978 33.4 3.9 5.1 

1 1,397 30.0 5.0 3.4 

0.5 1,276 27.3 4.8 0.2 

0.1 904 22.4 7.8 16.9 

54 

25 1,459 32.2 5.6 8.4 

10 1,328 28.0 3.9 0.2 

5 1,005 27.4 10.6 18.8 

1 896 19.9 4.5 14.0 

0.5 752 21.9 11.5 27.5 

0.1 740 14.8 5.1 30.4 
139

Table 18. Linear viscoelastic characterization and variation for the Control mixture in 

500 kPa confined compression state at selected frequencies and temperatures. 

Temp. 

(°C ) 

Frequency

(Hz) 

Average 

|E*| (MPa)

Average 

Phase Angle 

(°) 

|E*| Coefficient 

of Variation 

(%) 

Phase Angle 

Coefficient of 

Variation (%) 

-10 

25 32,837 6.3 10.0 0.9 

10 31,947 7.7 11.1 0.9 

5 30,803 8.6 11.6 0.9 

1 27,616 10.3 11.0 0.5 

0.5 26,114 11.1 11.5 0.9 

0.1 22,672 13.5 11.2 0.2 

5 

25 23,317 13.1 8.6 2.9 

10 21,641 13.6 6.7 0.3 

5 19,651 11.9 7.0 0.4 

1 16,013 15.1 7.5 2.1 

0.5 13,834 17.2 7.3 1.6 

0.1 10,903 20.7 8.4 2.2 

20 

25 12,927 18.1 5.1 0.1 

10 11,312 20.1 5.3 0.2 

5 9,540 22.9 5.1 0.3 

1 7,067 27.3 4.0 0.2 

0.5 5,943 23.0 7.0 0.9 

0.1 3,996 24.0 7.5 0.5 

40 

25 4,621 29.2 5.8 6.2 

10 3,631 23.7 0.8 0.3 

5 3,166 23.3 2.6 0.2 

1 2,477 18.7 3.9 1.8 

0.5 2,204 16.7 3.1 1.3 

0.1 1,737 12.5 6.7 9.8 

54 

25 2,664 19.7 3.9 7.2 

10 2,295 17.8 4.0 2.7 

5 2,059 16.3 0.5 3.1 

1 1,674 11.7 2.5 17.6 

0.5 1,528 9.8 1.2 2.5 

0.1 1,398 7.0 3.0 7.9 

Table 19. Effect of confining pressure on shift factor function coefficients for the Control 

mixture in compression state. 

Parameters 0 kPa 140 kPa 500 kPa 

α  0.00071 0.00090 0.00062

α 2 -0.15833 -0.17176 -0.14508

α 3 0.77385 0.83629 0.71001
140

1E+04

1E+05

1E+06

1E+07

1E+08

1.E-17 1.E-11 1.E-05 1.E+01 1.E+07 1.E+13

Time (s)

E(t) (kPa)

500kPa Compression

140kPa Compression

0kPa Tension/Compression

 Figure 119. Graph. Confining stress effect on the relaxation modulus. 

5.2.2. Comparison of Zero-Mean and Zero-Maximum Deviatoric Stress Results 

Figure 120 through figure 123 present the LVE characteristics determined from the zero-mean 

and zero-maximum deviatoric stress states. Only the 0 and 500 kPa results are shown in the 

figures because they were consistent for the two test methods. Overall, the comparisons were 

very favorable, and little difference was seen between the zero-mean and zero-maximum 

deviatoric stress states. However, a noticeable difference was seen in the phase angle results, but 

these results were subject to higher variability. Further, the phase angle calculation was less 

precise due to limitations of the measurement instrumentation. These results indicated that at 

small strains, asphalt concrete was not bimodal, and it was only after some threshold strain had 

been exceeded that such behavior occurred. Such findings are consistent with the work presented 

elsewhere when care is taken in performing the experiments at sufficiently small strains (50 to 

70 με ).(54,55) When such care is not taken, conflicting data are found in the literature.(28,56) For 

modeling then, it is important that the LVE characterization is limited to the 50 to 70 με range; 

in this range, either the zero-mean or zero-maximum method may be used. 

As a final comparison of the two test conditions, the model developed in subsection 4.2.2 was 

applied to the zero-maximum deviatoric stress state tests. The results of this analysis are shown 

in figure 124, and good agreement was observed between the model and measured data. 
141

0

10000

20000

30000

1.E-08 1.E-05 1.E-02 1.E+01 1.E+04

Reduced Frequency (Hz)

|E*| (MPa)

0 kPa-Tension

500 kPa-Tension

0 kPa-Compression

500 kPa-Compression

Figure 120. Graph. Comparison of zero-mean and zero-maximum deviatoric stress 

dynamic modulus mastercurves in semi-logarithmic scale. 

100

1000

10000

100000

1.E-08 1.E-05 1.E-02 1.E+01 1.E+04

Reduced Frequency (Hz)

|E*| (MPa)

Figure 121. Graph. Comparison of zero-mean and zero-maximum deviatoric stress 

dynamic modulus mastercurves in logarithmic scale. 
142

-8

-6

-4

-2

0

2

4

-20 0 20 40 60

Temperature (C)

log a

T

0 kPa-Tension

500 kPa-Tension

0 kPa-Compression

500 kPa-Compression

Figure 122. Graph. Effect of test method on shift factor functions. 

0

10

20

30

40

50

1.E-08 1.E-05 1.E-02 1.E+01 1.E+04

Reduced Frequency (Hz)

Phase Angle (deg)

0 kPa-Tension

500 kPa-Tension

0 kPa-Compression

500 kPa-Compression

Figure 123. Graph. Comparison of zero-mean and zero-maximum deviatoric stress phase 

angle mastercurves. 
143

100

1000

10000

100000

1.E-08 1.E-05 1.E-02 1.E+01 1.E+04

Reduced Frequency (Hz)

|E*| (MPa)

0 kPa-Model 0 kPa

140 kPa - Model 140 kPa

500 kPa - Model 500 kPa

Figure 124. Graph. Application of stress state-dependent model to zero-maximum 

deviatoric stress tests. 

5.2.3. MVECD Damage Function Characterization 

Characterization of the VECD model in compression was identical in analysis and general 

experimental requirements to that in tension. Constant rate compression tests were performed 

under both unconfined and confined conditions, the results of which were then used to calculate 

the damage functions C11, C12, and C22. The process was discussed previously in subsection 4.2.3 

and is shown schematically in the flow diagram shown in figure 125. 

5.2.3.1. Characterization of C11(S) 

Uniaxial constant rate tests were used to characterize the C11 damage function. Test results were 

used with equation 89 and equation 90 to compute the material function and damage parameter, 

respectively. Note that stress-hardening (equation 85) was used for this analysis. As with the 

tension damage function, this relationship was refined following the NCHRP 9-19 

methodology.(22) This relationship is shown for the Control mixture in figure 126. This figure is 

constructed at a reference temperature of 5 °C and is only for the material under compressive 

loading. For this mixture, a1 = 7.679 x 10-6, a2 =1.6971 x 10-6, a3 = 7.6793 x 10-6 , 

a4 = 4.6192 x 10-7, a5 = 1.5128 x 10-7, and a6 = 4.1017 x 10-8. 
144

MVECD Model Characterization

Test Quantity of Interest Material Property

Temperature and Frequency

Sweep |E*| and φ E(t) and aT

Unconfined Monotonic 

at 5°C εR and υR C11(S) and C12(S) 

Confined Monotonic 

at 5°C εR, υR and p C22(S) 

MVECD Model Characterization

Test Quantity of Interest Material Property

Temperature and Frequency

Sweep |E*| and φ E(t) and aT

Unconfined Monotonic 

at 5°C εR and υR C11(S) and C12(S) 

Confined Monotonic 

at 5°C εR, υR and p C22(S) 

Figure 125. Diagram. MVECD model characterization. 

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.0E+07 2.0E+07 3.0E+07

Refined S

C

Model

FHWA 1032

FHWA 1033

FHWA 1034

FHWA 1037

( )

6

11

1

1

3

i a S

i

CS e

i

−

=

= + ∑

Figure 126. Graph. C11 versus S for compression for Control mixture (5 °C reference). 
145

5.2.3.2. Characterization of C12(S) 

In addition to using uniaxial constant rate tests to characterize the C11 damage function, these test 

results were also used to characterize the C12 damage function. As with the tension tests, 

complications arose due to the time-dependent nature of Poisson’s ratio in asphalt concrete. 

Based on the analysis presented in subsection 4.2.3.2, an initial value of approximately 0.179 

was assumed for C12. This assumption was not unreasonable based on the results of the 

characterization tests. A slight difference between the initial values of the compression and 

tension data shown previously was observed due to the use of slightly different mixtures for the 

two sets of tests (Control-2006 for tension and Control for compression). Tension data for the 

Control mixture will be shown in subsequent sections of this report. The C12 material parameter 

was defined from equation 91, and the damage parameter was calculated and refined as before. 

The functional form taken for the compressive C12 damage function is shown along with the data 

in figure 127. It was observed that at early damage stages, Poisson’s ratio changed very little, but, 

in general, as damage increased, Poisson’s ratio reduced, as indicated by an increase in C12. For 

this mixture, it was found that β1 = 3.5598 x 10-8, 1.6982 x 10-8, and 3.2756 x10-10. 

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.0E+00 1.0E+07 2.0E+07 3.0E+07

Refined S

C

Model

FHWA 1032

FHWA 1033

FHWA 1037

() ( ) 1

12 3 2 3 .179 S CS e S β β β β − = − −+

Figure 127. Graph. C12 versus S for compression for Control mixture (5 °C reference). 

5.2.3.3. Characterization of C22(S) 

Confined constant rate tests, along with the other two damage functions, were used to 

characterize the final C22 function. Following the study conducted for the tension 

characterization methodology, a visual basic macro was created in Microsoft Excel® to 

determine the damage function by optimization. The characterized C22 function is shown in 

figure 128 along with the model function. The parameters of the model were H1 = -0.504959 and

H4 = 1.16247 x 10-13.
146

-120

-100

-80

-60

-40

-20

0

0.0.E+00 1.0.E+07 2.0.E+07 3.0.E+07

Refined S

C22

( ) 2 C S h hS 22 = −1 2

Figure 128. Graph. C22 versus S for compression for Control mixture (5 °C reference). 

5.2.4. Comparison of MVECD Behavior in Compression and Tension 

Asphalt concrete, like other composite materials, is known to exhibit bimodal behavior. In 

particular, the strength of the material it is in and its ductility are much higher in the compression 

direction than they are in the tensile direction. Comparisons of the MVECD damage functions 

revealed some interesting behaviors. First, when comparing the primary axial modulus damage 

function, C11, between tension and compression, the material was substantially more resistant to 

damage in compression than it was tension. This behavior is shown in the same scale in figure 

129. This behavior was explained by the notion that the damage parameter, S, represented some 

kind of crack density or volume-averaged cracked area. In tension, these cracks were oriented 

perpendicular to the principal loading direction, whereas, in compression, the cracks ran parallel 

to the principal loading direction. In constant rate loading, cracks only grew when they were 

under a tensile stress. Because tension was only induced (at the microscale level) in compression 

tests and not directly applied, the material was more resistant to a given cracking volume. 

This physical interpretation for damage was also supported by the behavior of the second 

damage function, C12. These damage functions are shown for both tension and compression in 

figure 130. This figure has both the C12 function and Poisson’s ratio between the symmetric axis 

and perpendicular axis (ν 3132) for the convenience of the discussion. From the comparison 

shown in figure 130, it was observed that the primary Poisson’s ratio showed an overall 

decreasing trend in tension, whereas the opposite trend occurred in compression. As with the C11

damage function, the tensile behavior of C12 was observed to be more sensitive to damage. A 

possible explanation of this behavior depended on understanding that Poisson’s ratio indicated 

the degree to which the radius changed relative to a unit change in length. Applying the physical 

interpretation of damage in tension and compression, the physical implications of the patterns in 

both damage curves were identical. The tension curve was increasing, thus indicating that with 
147

higher levels of damage, the direction parallel to the primary cracking direction (radial strain) 

was less sensitive to changes in strain perpendicular to the primary cracking direction (vertical 

strain). For compression, the primary cracking direction was parallel to the loading direction and 

perpendicular to the radial direction, so with the same interpretation (of damage in tension and 

compression), an increase in Poisson’s ratio was observed. In the most general terms, the 

increased opening of a microcrack within a body was not necessarily accompanied by a relative 

increase in the parallel to the crack direction dimension of the body. In fact, the results indicated 

that at higher levels of cracking or damage the material became less likely to change dimension 

as much when loaded perpendicular to the damage orientation. 

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

0E+00 2E+06 4E+06 6E+06 8E+06 1E+07

Refined S

C11

Tension

Compression

Figure 129. Graph. Comparison of tension and compression of C11 damage function. 

The physical interpretation of the C22 damage function was not as straightforward as the other 

two damage functions. As seen in equation 45, the C11 and C12 damage functions can be directly 

translated to elements in the stiffness matrix. However, the C22 damage function entered in the 

denominator of multiple elements of the material stiffness matrix. Nevertheless, this damage 

function can be considered as a sort of volumetric compliance, which has little meaning in a 

transverse isotropic problem. From figure 131, though, it is seen that the material was more 

volumetrically compliant in compression than it was in tension. 
148

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0E+00 2E+06 4E+06 6E+06 8E+06 1E+07

Refined S

C12

0.2

0.4

0.6

Poisson's Ratio

Tension

Compression

Figure 130. Graph. Comparison of tension and compression of C12 damage function. 

-15

-10

-5

0

0E+00 2E+06 4E+06 6E+06 8E+06 1E+07

Refined S

C22

0

5

10

15

Plane Strain Bulk Comp.

Tension

Compression

Figure 131. Graph. Comparison of tension and compression of C22 damage function. 
149

5.3. VISCOPLASTIC MODELING OF ASPHALT CONCRETE IN COMPRESSION 

5.3.1. A Phenomenological Model Considering Pulse Time Effects 

As a first step toward developing a mechanistic material model for the behavior of HMA in 

compression, a series of analyses was performed on VT and VL test data, and a 

phenomenological model was developed. The modeling approach adopted in the 

phenomenological model was based on the existing strain-hardening model presented for 

describing the tensile behavior in section 4.3 and shown in the general form as follows: 

( )

( ) vp

vp

f

g

σ

ε

ε & = (158) 

Where: 

vp ε& = Viscoplastic strain rate. 

σ = Stress. 

ε vp = Viscoplastic strain. 

As shown in equation 158, the viscoplastic strain rate was represented by the combination of two 

functions, f(σ ) and g(ε vp), which allowed the stress rate dependency and strain hardening to be 

taken into consideration in the model. Equation 158 can be generalized as equation 159, which 

accounts for the effect of the pulse time. 

ε εσ &

vp vp p = F t ( , , ) (159) 

Where: 

tp = The loading time. 

The exact form of the function, F, is presented along with experimental data in the following 

sections. 

5.3.1.1. Tests Performed in This Study 

Three types of repetitive creep and recovery tests were performed for the phenomenological 

model development, including the creep and recovery test with VL, the creep and recovery tests 

with VT, and the creep and recovery test with a constant load level and CLT. All the tests were 

conducted at 55 °C under the confining pressure of 500 kPa. Experimental details for these tests 

are given in the following sections because this particular study uses different test conditions 

than those outlined in chapter 3. 

5.3.1.1.1. Creep and Recovery Test with VL: 

The creep and recovery test with a VL test was performed with 200 kPa as the starting load level. 

An incremental factor of 1.0245 was used for the subsequent load levels to increase the load 

level until the complete failure of the specimen. The loading time and rest period were 0.1 and 

10 s, respectively. 
150

5.3.1.1.2. Creep and Recovery Tests with VT: 

The creep and recovery test with a VT test was performed with a loading block consisting of 30 

different loading times. The loading time varied from 0.005 to 2.0 s with an incremental factor of 

1.1356. The rest period for each load cycle was 30 times that of each loading time. The VT tests 

were performed at three different load levels, 1,800, 2,000, and 2,200 kPa. 

5.3.1.1.3. Creep and Recovery Test with CLT: 

In this test, a constant load level and constant loading time were used for each test. Load levels 

and loading times were changed between tests. Three load levels of 1,800, 2,000, and 2,200 kPa 

were used with the loading time of 1.6 s. For the 2,000 kPa load level, the creep and recovery 

tests were conducted with three additional loading times of 0.1, 0.4, and 6.4 s. 

The VL and VT tests were used to understand the effects of load level and loading time on the 

permanent deformation behavior of HMA and to calibrate the phenomenological model. The 

CLT tests were used to verify the calibrated model. 

5.3.1.2. Model Characterization 

By observing the viscoplastic strain rate versus the viscoplastic strain VT in figure 132, a 

constitutive relationship between these viscoplastic media was defined, as shown in equation 160. 

log log , ε&

vp p vp p = + at Dt ( ) ε σ ( ) (160) 

Where: 

a(tp) = Material function of loading time. 

D(tp, σ ) = Intercept of the curve on viscoplastic strain rate axis. 
151

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

0.00 0.01 0.10

Viscoplastic Strain

Incremental Viscoplastic

Strain Rate

First loading block

tp=2.000

Final loading block

tp=0.005

Figure 132. Graph. Incremental viscoplastic strain rate versus viscoplastic strain (500 kPa 

confinement, 2,000 kPa). 

In equation 160, a was a function of the loading time, and D was a function of the load level and 

loading time. Equation 160 can be represented in equation 161, which was a generalized form of 

equation 158. 

( ) ( ) ( ) ( )

( )

( )( )

10 10 p at b d c

vp vp p

vp p

t

f

g kt

σ ε ε

σ

ε

⎡ ⎤ = ⎢ ⎥ ⎣ ⎦

=

&

 (161) 

Where: 

k(tp) = Function of loading time. 

Equation 160 required the determination of a(tp) and D(tp, σ ) to calculate the viscoplastic strain 

rate for a given viscoplastic strain. Values of a(tp) and D(tp, σ ) for given loading times could be 

found by fitting log functions against each viscoplastic strain rate versus viscoplastic strain curve 

corresponding to given loading time. At this time, the values of a(tp) can be represented by the 

second logarithm function, as shown in equation 162. 

( ) 1 2 log p p at a t a = + (162) 

Where: 

a1 and a2 = Material-dependent constants. 
152

In order to determine the form of D(tp, σ ), it was assumed that D(tp, σ ) could be represented by 

the summation of the loading time term and the load level term, as shown in equation 163. 

Dt b t c d ( p p , log σ σ ) = + + ( ) (163) 

Where: 

The function c(σ ) was given by c1log(σ ) when the stress was less than 1,000 kPa and by 

c2σ when the stress was greater than 1,000 kPa. The coefficient b was determined by fitting 

equation 163 against D(tp, σ ) from the VT test. At this time, it was assumed that c(σ ) and d

constitute one constant that accounts for the effect of load level. The function c(σ ) was then 

fitted by the logarithmic function and the linear function. Finally, coefficient d was determined 

by fitting equation 163 for the viscoplastic strain rate versus the viscoplastic strain curve of the 

VL and VT tests. 

Figure 133 and figure 134 show the fitting results and the coefficients that were determined. 

Figure 135 and figure 136 presents predictions for the VT and VL tests, which were then used 

for the characterization process. 

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

1E-02 1E-01 1E+00 1E+01

Loading Time (s)

a(tp)

Measured a(tp)

Fitted a(tp)

Figure 133. Graph. Determined fitting results and coefficients of function a(tp). 
153

-12

-10

-8

-6

-4

-2

0

012345

Stress (MPa)

D(tp,σ)

Measured

Fitted (section 1)

Fitted (section 2)

D(tp,σ) = 1.01300σ - 8.13657

D(tp,σ) = 4.66371 log(σ) - 7.18183

(a) (b)

Figure 134. Graph. Determined fitting results and coefficients of function D(tp σ ). 

0.0%

1.0%

2.0%

3.0%

4.0%

0 40 80 120 160

Cumulative Loading Time (s)

Viscoplastic Strain

2.0MPa VT #1

2.0MPa VT #2

2.0MPa VT #3

Prediction

 Figure 135. Graph. VT predictions. 
154

0.0%

1.0%

2.0%

3.0%

4.0%

0 2 4 6 8 10 12 14

Cumulative Loading Time (s)

Viscoplastic Strain

0.1sec VL #1

0.1sec VL #2

0.1sec VL #3

Prediction

 Figure 136. Graph. VL predictions. 

5.3.1.3. Verification of the Model 

As shown in figure 137 through figure 142, predictions were made for the CLT tests. Although 

the model was able to account for differences in the viscoplastic development for various loading 

times, overall predictions were not as accurate. Several causes for the discrepancy between 

viscoplastic strain predictions and measurements could be suggested. However, the inability of 

the model to consider strain history was a highly probable cause for this discrepancy, given that 

the fitting results for the VT and VL tests were acceptable. 
155

0.0%

1.0%

2.0%

3.0%

4.0%

0 10 20 30 40

Cumulative Loading Time (s)

Viscoplastic Strain

2.0MPa 0.1sec CLT #1

2.0MPa 0.1sec CLT #2

2.0MPa 0.1sec CLT #3

2.0MPa 0.1sec CLT #4

Prediction

 Figure 137. Graph. CLT predictions (2.0 MPa deviatoric stress—0.1-s pulse time). 

0.0%

1.0%

2.0%

3.0%

4.0%

0 10 20 30 40

Cumulative Loading Time (s)

Viscoplastic Strain

2.0MPa 0.4sec CLT #1

2.0MPa 0.4sec CLT #2

2.0MPa 0.4sec CLT #3

2.0MPa 0.4sec CLT #4

Prediction

 Figure 138. Graph. CLT predictions (2.0 MPa deviatoric stress—0.4-s pulse time). 
156

0.0%

1.0%

2.0%

3.0%

4.0%

0 30 60 90 120

Cumulative Loading Time (s)

Viscoplastic Strain

2.0MPa 1.6sec CLT #1

2.0MPa 1.6sec CLT #2

2.0MPa 1.6sec CLT #3

2.0MPa 1.6sec CLT #4

Prediction

 Figure 139. Graph. CLT predictions (2.0 MPa deviatoric stress—1.6-s pulse time). 

0.0%

1.0%

2.0%

3.0%

4.0%

0 50 100 150 200

Cumulative Loading Time (s)

Viscoplastic Strain

2.0MPa 6.4sec CLT #1

2.0MPa 6.4sec CLT #2

2.0MPa 6.4sec CLT #3

Prediction

 Figure 140. Graph. CLT predictions (2.0 MPa deviatoric stress—6.4-s pulse time). 
157

0.0%

1.0%

2.0%

3.0%

4.0%

0 50 100 150 200

Cumulative Loading Time (s)

Viscoplastic Strain

1.8MPa 1.6sec CLT #1

1.8MPa 1.6sec CLT #2

1.8MPa 1.6sec CLT #3

1.8MPa 1.6sec CLT #4

Prediction

 Figure 141. Graph. CLT predictions (1.8 MPa deviatoric stress—1.6-s pulse time). 

0.0%

1.0%

2.0%

3.0%

4.0%

0 10 20 30 40

Cumulative Loading Time (s)

Viscoplastic Strain

2.2MPa 1.6sec CLT #1

2.2MPa 1.6sec CLT #2

Prediction

 Figure 142. Graph. CLT predictions (2.2 MPa deviatoric stress—1.6-s pulse time). 
158

5.3.2. HISS-Perzyna Model 

The HISS-Perzyna model, suggested by the Delft University of Technology and the University 

of Maryland, was investigated with respect to the data set obtained from experimental 

tests.(26,28,36) A prediction using the Delft University of Technology model was not be made 

because of numerical problems. However, the characterization process using the t-TS principle 

and coefficients of the model are described in subsection 5.3.2.1. 

5.3.2.1. Delft University of Technology Model 

The model suggested by researchers at the Delft University of Technology required the 

development of several relationships between the material parameters and the strain rates 

obtained from constant strain-rate tests. In this research, the t-TS principle was utilized to 

simplify the modeling effort and to reduce the number of relationships required. With the 

assumption that the yield stress in deviatoric stress space was presented as circular, ( β =0), 

equation 72 was be reduced to equation 164. 

( )( ) 2

21 1

n

J IR IR = −− − γ α (164) 

Where: 

γ = Softening parameter. 

α = Hardening parameter. 

R = Tensile strength of material when deviatoric stress is 0. 

n = Parameter that determines shape of yield stress. 

 

Figure 143 shows peak stresses for a series of compressive and tensile constant strain-rate tests; 

the strain rates are listed in table 4. These peak stresses were used as fundamental quantities to 

develop relationships between the material parameters and the reduced strain rates. R and γ 0

could be determined as functions of the reduced strain rate by plotting the compressive and 

tensile peak stresses obtained from the constant strain-rate tests under a certain strain rate and 

then taking the slope and x-intercept of the line. In the model, R and γ 0 represented the tensile 

strength for hydrostatic stress and the softening of the material in the post-peak region, 

respectively. The parameter n governs the overall shape and size of the yield function and was 

related to the dilation of the material. The beginning of dilation was defined as the stress at the 

minimum plastic volumetric strain because the plastic deviatoric strain and elastic strain (or 

viscoelastic strain) was assumed to not be associated with the volumetric change of a material. In 

addition, because HMA specimens dilated after a little compression as the compressive stress 

increased, the dilation stress could be determined. Once the dilation stresses were determined for 

several strain rates, the value of n could be determined using equation 165. 

( )

2,

2

1,

2

1 dilation

dilation

n J

γ I R

=

− − (165) 
159

Where: 

I1,diation = I1 at beginning of dilation. 

J2,dilation = J2 at beginning of dilation. 

Once n is determined, α 0 can be readily determined using equation 166. 

2

0 0

n

a

R

P α γ

−

⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (166) 

The sigmoidal function was used to represent relationships between the reduced strain rate and 

the material parameters. The form of the function and the coefficients determined for each 

parameter are listed in equation 167 and table 20. Figure 144 through figure 147 show a 

comparison of measured values versus predicted values at various reduced strain rates. 

( ) ( ) 0 log log or 

1 reduced d

b R a

e ε γ − = +

+ & (167) 

0

1000

2000

3000

4000

5000

6000

-10000 -5000 0 5000

I1

SQRT(J2)

Compression 0kPa

Tension 0kPa

Figure 143. Graph. Compressive and tensile peak stress in SQRT(J2) - I1 space. 

Table 20. Delft material model coefficients functions. 

a b d e 

R 316.86 17,074.55 -9.28 1.75 

γ  0.20 -0.15 -13.91 2.13 

n 2.00 2075.86 0.48 1.06 

α  -0.63 -93.39 -5.01 1.35 
160

Figure 148 shows the strain rate-dependent yield surface that was constructed using the 

characterized parameters when the viscoplastic strain was equal to zero (i.e., the initial yield 

surface). It was observed that the initial yield surface increased as the temperature decreased; the 

reduced strain rate increased. This behavior coincided with observations from constant strain-rate 

tests in which more viscoplastic strains were developed under a small, reduced strain rate (or 

higher temperature). 

As shown in equation 164, the second term in the square root always had to be smaller than the 

first term in order to construct a valid yield surface. However, because of the numerical errors 

involved in the characterization process of α and n, the prediction program was often required to 

calculate the square root of a negative number during analysis. This situation was encountered 

without t-TS, as mentioned by others.(36) 

0.00

0.05

0.10

0.15

0.20

0.25

1E-10 1E-08 1E-06 1E-04 1E-02

LVDT Reduced Strain Rate

γ0

Measured

Fitted

Figure 144. Graph. Determined γ 0 parameter function. 
161

0

3000

6000

9000

12000

15000

18000

1.E-10 1.E-08 1.E-06 1.E-04 1.E-02

LVDT Reduced Strain Rate

R

Measured

Fitted

Figure 145. Graph. Determined R parameter function. 

0

4

8

12

16

20

1.E-10 1.E-08 1.E-06 1.E-04 1.E-02

LVDT Reduced Strain Rate

n

Measured

Fitted

Figure 146. Graph. Determined n parameter function. 
162

1E-29

1E-24

1E-19

1E-14

1E-09

1E-04

1E+01

1.E-10 1.E-08 1.E-06 1.E-04 1.E-02

LVDT Reduced Strain Rate

α

Measured

Fitted

Figure 147. Graph. Determined α 0 parameter function. 

0

500

1000

1500

2000

2500

-20000 -10000 0 10000 20000

I1

SQRT (J2)

5C-1

25C-1

40C-1

55C-1

Figure 148. Graph. Rate-dependent initial yield surface. 
163

5.3.2.2. University of Maryland Model 

As shown in equation 73, a simplified HISS-Perzyna model was suggested by researchers at the 

University of Maryland.(28) Equation 168 represents a general form of the hardening function 

used for the suggested model. 

0

vp e κ ε α α − ⋅ = (168) 

Where: 

α 0 and κ = Material constants. 

However, the observation made in subsection 5.1.2.3 indicates that a single hardening function, 

equation 168, was not sufficient to represent the characteristic behavior of the material, such as 

softening during unloading. Therefore, one more variable, the viscoplastic strain increment 

during loading, was introduced into equation 168. Equation 169 represents the modified 

functionα to incorporate the variation of the viscoplastic strain rate during the pulse time in the 

existing model. α 1 and α 2 in equation 169 describe general variations of α in terms of 

viscoplastic strain and a local variation of α in terms of incremental viscoplastic strain in a pulse, 

respectively. 

α = καα 112 (169) 

Where: 

2

1

vp e κ ε α − = (170) 

( ) 2

2 1 vp e κ ε α

− Δ = − (171) 

Figure 149 presents the variation of α determined by using a modified alpha-viscoplastic 

relationship for five 6.4-s pulses with 1,800 kPa of load level. As shown, α was no longer a 

simple decreasing function of the viscoplastic strain, but had multiple decreasing functions of 

which independent variables were incremental viscoplastic strain during each load pulse and 

overall viscoplastic strain. The incremental viscoplastic strain was reset to zero each time the 

material was unloaded. 

Figure 150 through figure 152 present measured and predicted viscoplastic strains by using a 

modified hardening function. The model was able to describe viscoplastic strain development for 

various loading conditions, such as VT, RVT, and CLT; this capability was not possible in the 

existing HISS-Perzyna model. However, even though incremental viscoplastic strain in a pulse 

described multiple hardening rates at certain viscoplastic strains, it was more reasonable to 

assume that the multiple hardening rates were caused by the viscoelastic property of the material, 

given the rate dependency of the softening. Therefore, a viscoplastic model with a ratedependent hardening-softening function was developed and is presented in subsection 5.3.3. 
164

0.00

0.01

0.02

0.03

0.04

0.05

0.0% 0.4% 0.8% 1.2% 1.6%

VP Strain

Alpha

Pulse_1

Pulse_2

Pulse_3

Pulse_4

Pulse_5

Figure 149. Graph. Variation of α for 1,800 kPa CLT loading (500 kPa confinement). 

0.0%

0.4%

0.8%

1.2%

1.6%

0 1000 2000 3000 4000

Time (s)

Viscoplastic Strain

1.6M VT Measured

1.8M VT Measured

2.0M VT Measured

1.6M VT Predicted

1.8M VT Predicted

2.0M VT Predicted

Figure 150. Graph. Viscoplastic strain predictions for VT tests (500 kPa confinement). 
165

0.0%

0.4%

0.8%

1.2%

1.6%

0 1000 2000 3000 4000 5000

Time (s)

Viscoplastic Strain

1.8M 0.4s Measured

1.8M 1.6s Measured

1.8M 6.4s Measured

1.8M 0.4s Predicted

1.8M 1.6s Predicted

1.8M 6.4s Predicted

Figure 151. Graph. Viscoplastic strain predictions for CLT tests (500 kPa confinement). 

0.0%

0.4%

0.8%

1.2%

1.6%

0 1000 2000 3000 4000 5000

Time (s)

Viscoplastic Strain

1.6M VT Measured

1.6M RVT Measured

1.6M VT Predicted

1.6M RVT Predicted

Figure 152. Graph. Viscoplastic strain predictions for RVT tests (500 kPa confinement). 
166

5.3.3. Development of a Viscoplastic Model Using Rate-Dependent Yield Stress 

The model developed in this research was capable of capturing both additional hardening that 

was due to aggregate interlocking and rate-dependent softening due to viscoelastic relaxation. 

Viscosity in Perzyna’s evolution law is separated into a constant term and a viscoplastic straindependent term that together represent the change of viscosity in viscoplastic flow. The yield 

stress function that takes into account rate-dependent hardening and softening is also described 

in subsection 5.3.3.1. 

5.3.3.1. Flow Rule and Yield Function for Developed Viscoplastic Model 

As an expansion of equation 55, a general flow rule for materials exhibiting kinematic and 

isotropic hardening is represented in equation 172. m amplified or reduced the stress rate 

dependency of the model, and D determined the viscosity in the viscoplastic flow. When D was a 

constant, it was assumed that the effect of the change in viscosity on the response of the material 

was taken into account by the yield stress function. However, when variations of yield stress 

were affected by the viscoelastic property of the material, it seemed reasonable to consider the 

viscosity in the viscoplastic flow as a function that was not subjected to yield stress. 

( ) m

vp

ij

ij

f r f

D

σ α

ε

σ

− − ∂ = ∂ &

 (172) 

Where: 

α = Kinematic hardening function. 

γ = Isotropic hardening function. 

D = Viscosity parameter. 

m = Rate-dependency parameter. 

Therefore, a flow rule that takes into consideration the variations of viscosity in the viscoplastic 

flow is suggested in equation 173 by incorporating Perzyna’s flow rule and Von Mises’ yield 

criterion. In equation 173, D was the viscosity and represented the scalar hardening and softening 

as described above. The anisotropic behavior of the material was also integrated by using Dij. 

Meanwhile, Gij represented the orientation-dependent isotropic hardening function that reflected 

the viscoplastic and viscoelastic property of the material. Because the material was subjected 

only to compressive stress, the kinematic hardening rule was not introduced in this model. The 

viscosity (D) was related to aggregate interlocking and was represented as a function of the 

viscoplastic strain, as shown in equation 174. Because the function could represent both 

increasing and decreasing viscosity according to the viscoplastic strain, it had the potential to 

represent the behavior of HMA mixtures in the tertiary region as well as in the primary and 

secondary regions. 

( ) ( )

( )

( ) ( )

0 ( )

3 1

2

m m

ij ij ij ij ij ij ij ij vp

ij

ij ij ij ij

J G sg J G sg B

D D J G J G

σ σ

ε

η σ σ

−− −− = = − − &

(173) 
167

Where: 

D = Viscosity related to aggregate interlocking, equation 174. 

η 0, m, α , β , γ = Material constants. 

D0 = Initial viscosity. 

sij = Deviatoric stress tensor. 

gij = Deviatoric back stress tensor. 

σ ij = Stress tensor. 

Gij = Yield stress tensor. 

J(σ ij -Gij) = The second invariant of (σ ij - Gij). 

0 sin 1( ) vp D D e γε α β − =+ − ⎡ ⎤ ⎣ ⎦ (174) 

However, as discussed in subsection 5.1.2.3, rate-dependent softening, which implied the 

possibility of a multiple state of the material at certain viscoplastic strains, was observed when 

HMA was subjected to repetitive loading. In order to introduce the characteristic behavior of 

HMA into the viscoplastic constitutive model, equation 175, which was one of the simplest 

forms, was suggested as the hardening-softening function. 

Gg G = (ε ε vp vp , , ) & & (175) 

In equation 175, the yield stress increased as the viscoplastic strain and viscoplastic strain rate 

increased during loading, whereas it decreased during unloading (when the viscoplastic strain 

rate was zero). Figure 153 presents a schematic concept of the variation of yield stress subjected 

to a creep and recovery loading condition. The remaining yield stress, which was yield stress at 

the asymptote, was governed by the viscoplastic strain at the end of the loading condition, ε 0

vp, 

and material constants E1 and E2. The decreasing yield stress during unloading allowed a 

multiple viscoplastic strain rate at a certain viscoplastic strain. 

G

t

1 2

0

1 2

vp E E

E E

ε

⎛ ⎞ + ⎜ ⎟ ⎝ ⎠

E1

η1 E2

G

η0 G

σ

Loading Unloading 

Figure 153. Illustration. Variation of yield stress (Standard Linear Solid model). 
168

Because of their simplicity, equation 176 was suggested as a hardening-softening function to 

confirm the characteristics of the model for an arbitrary stress history, and equation 177 was 

suggested to predict the actual behavior of the HMA mixture. As shown in equation 177, the 

hardening-softening function was represented as the convolution integral, including the 

relaxation modulus. Material constants A and B were introduced into the relaxation modulus to 

develop a relationship between the relaxation modulus, which was the LVE property, and the 

viscoplastic yield stress. Additionally, by utilizing the material constants, the number of 

parameters needed to calibrate the model could be reduced. In calculating the yield stress, the 

state variable approach was used to reduce computational time, as shown in equation 178. 

1 2 1 2

1

2 2

E E vp vp E E GE G ε ε

η η

⎛ ⎞ +

= +−⎜ ⎟ ⎝ ⎠

& &

 (176) 

Where: 

E1, E2, η 1, η 2 = Material constants. 

0

0

i

t t

vp

i

d

G A E B Ee d

d

τ

ρ

ε

τ

τ

− ⎛ ⎞ −

= ⋅ + ⋅Σ ⎜ ⎟

⎝ ⎠ ∫

 (177) 

Where: 

A, B = Material constants. 

E0, Ei = Prony coefficients determined for the relaxation modulus. 

ρ i = Relaxation time. 

Equation 177 was solved using the state variable approach to predict strains and calculate pseudo 

strains. This approach is shown mathematically in equation 178. 

11 1

0

1

m nn n

i

i

G ζ ζ + + +

=

= +∑ (178)

Where: 

1 1

0 0

n n n tt ζ εε A

vp vp

E + + = − ⎡ ⎤ ⎣ ⎦ (179) 

1 1 2 n n i i

t t

n n tt

i iv Bp vp i ee E ρ ρ ζ ζ εε +

Δ Δ − − + ⎡ ⎤

=+− ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ⎢⎣ ⎥⎦ (180)

5.3.3.2. Characteristics of the Developed Model for Arbitrary Stress History 

In order to confirm the characteristics of the developed viscoplastic model, the following 

predictions were made for the arbitrary stress histories by using equation 173 and equation 176. 

In this study, D was considered a constant to simplify the calibration and prediction processes. 
169

Two sets of stress histories were generated, as shown in figure 154 and figure 157. Table 21 

shows the material constants used in this analysis. 

Table 21. Material coefficients used for the developed model analysis. 

m D η  η 2 E1 E2

2 3,000 10 50,000 500 200 

5.3.3.2.1. Effect of Rest Period: 

Figure 154 shows two different stress histories used to check the sensitivity of the viscoplastic 

model to the effects of rest periods. For both stress histories, stress levels and the cumulative 

loading time were fixed to 2,000 units less stress and 160 s, respectively. However, for the first 

stress history, 8.0 s of rest between the loading pulses were allowed, whereas only 1 s of rest was 

allowed for the second loading history. Figure 155 presents the variation of yield stress for each 

stress history, and as expected, the model showed different yield stress developments depending 

upon the rest period. 

Figure 156 presents the viscoplastic strain developed by each stress history. It shows more 

viscoplastic strain for a longer rest period. This result corresponded to the experimental 

observations made from figure 98. 

0

500

1000

1500

2000

2500

0 50 100 150 200 250 300 350

Time (s)

Stress

8 sec rest period 1 sec rest period

Figure 154. Graph. Stress histories for rest period analysis. 
170

0

200

400

600

800

1000

0 50 100 150 200

Cumulative Loading Time (s)

Yield Stress

8 sec rest period

1 sec rest period

Figure 155. Graph. Yield stress versus cumulative loading time (rest period analysis). 

0.0

1.0

2.0

3.0

4.0

5.0

0 50 100 150 200

Cumulative Loading Time (s)

Viscoplatic Strain

8 sec rest period

1 sec rest period

Figure 156. Graph. Viscoplastic strain versus cumulative loading time (rest period analysis). 

5.3.3.2.2. Effect of Loading Time: 

Figure 157 presents another set of stress histories used to check the effects of loading time. For 

these stress histories, the load level, rest periods, and cumulative loading time were fixed to 
171

2,000 units, 4 and 66 s, respectively. However, the first loading history consisted of 6 pulses at 

11 s long, and the second loading history consists of 22 pulses at 3 s long.

The analysis results for the given stress histories are shown in figure 158. The loading history 

with shorter individual loading times showed more viscoplastic strain, which was identical to the 

CLT test results. As shown, the viscoplastic model that incorporated the softening rule appeared 

to account for the pulse time effect. As shown in figure 156 and figure 158, the viscoplastic 

model with the rate-dependent hardening-softening capability could account for the effects of 

loading time. 

0

500

1000

1500

2000

2500

0 50 100 150 200 250

Time (s)

Stress

3 sec x 22 pulse 11 sec x 6 pulse

Figure 157. Graph. Stress history for loading time analysis. 
172

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 10 20 30 40 50 60 70

Cumulative Loading Time (s)

Viscoplastic Strain

3 sec x 22 pulse

11 sec x 6 pulse

Figure 158. Graph. Viscoplastic strain versus cumulative loading time 

(loading time analysis). 

5.4. CHARACTERIZATION AND VERIFICATION OF THE VISCOPLASTIC MODEL 

5.4.1. Calibration 

Prior to the calibration process, data points acquired during the unloading period were filtered to 

reduce the computational time. Strains measured at the end of the rest periods were defined as 

the objective function. The nonlinear optimization function (lsqnonlin) in MatlabTM was utilized 

to minimize errors between the measured and predicted viscoplastic strains. Based on the model 

calibrated for the VT and VL tests, the viscoplastic strains of the other loading conditions, such 

as CLT and VLT, could be predicted. Table 22 shows the coefficients determined from the 

calibration process for 140 and 500 kPa confining pressures. 

Table 22. Compression viscoplastic material model coefficients. 

Confining 

Pressure (kPa) m D0 α β γ A B 

140 3.83 1,951.76 2,758.93 3.07 258.34 0 6.3E-06 

500 9.99 3,460.62 3,569.28 1.87 156.96 0 5.2E-03 

Figure 159 and figure 160 present the calibration results for VT and VL testing at 140 kPa 

confining pressure, and figure 161 and figure 162 present the calibration results for VT and VL 

testing at 500 kPa confining pressure. In general, the predicted and measured viscoplastic strains 

match very well, although there was a slight discrepancy in the VT and VL 500 kPa confining 

pressure results. 
173

0.0%

0.3%

0.6%

0.9%

1.2%

1.5%

1.8%

0 50 100 150 200 250 300 350

Cumulative Loading Time (s)

Viscoplastic Strain

140kPa-827kPa VT 200s (1)*

140kPa-827kPa VT 200s (2)

140kPa-827kPa VT 200s Prediction

Figure 159. Graph. Viscoplastic strain versus cumulative loading time 

(140 kPa confinement VT). 

0.0%

0.3%

0.6%

0.9%

1.2%

1.5%

0 100 200 300 400 500

Cumulative Loading Time (s)

Viscoplastic Strain

140kPa-variable VL (1)*

140kPa-variable VL (2)

140kPa-variable VL Prediction

Figure 160. Graph. Viscoplastic strain versus cumulative loading time 

(140 kPa confinement VL). 
174

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 20 40 60 80 100 120 140

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-1800kPa VT (1)*

500kPa-1800kPa VT (2)

500kPa-1800kPa VT Prediction

Figure 161. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement VT). 

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 5 10 15 20

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-variable VL (1)*

500kPa-variable VL (2)

500kPa-variable VL Prediction

Figure 162. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement VL). 
175

5.4.2. Verification 

Viscoplastic strain predictions for the 140-kPa confining pressure tests are presented in 

figure 163 through figure 166. Figure 163 shows the ability of the developed model to consider 

the effects of rest periods on the viscoplastic strain development even though the viscoplastic 

strains were slightly underpredicted. Figure 164 and figure 165 show the predictions for the VLT 

tests and a low load level VL test, respectively; these predictions were quite good. Figure 166 

presents predictions for complex loading histories, which were a combination of VT test results 

and flow number test results. Up to 0.5-percent strain, the prediction of the viscoplastic strain 

matched well with the measured viscoplastic strain; however, the viscoplastic strain was 

underpredicted for the last strain level. This discrepancy could indicate a need to refine the 

softening function. Figure 167 to figure 172 present the viscoplastic strain predictions made for 

500 kPa confining pressure tests. The overall prediction was good considering the complexity of 

the loading history. 

0.0%

0.3%

0.6%

0.9%

1.2%

1.5%

1.8%

0 50 100 150 200 250 300 350

Cumulative Loading Time (s)

Viscoplastic Strain

140kPa-827kPa VT 200s (1)*

140kPa-827kPa VT 200s (2)

140kPa-827kPa VT 200s Prediction

140kPa-827kPa VT 1s (1)

140kPa-827kPa VT 1s (2)

140kPa-827kPa VT 1s Prediction

140kPa-827kPa VT 0.05s (1)

140kPa-827kPa VT 0.05s (2)

140kPa-827kPa VT 0.05s Prediction

Figure 163. Graph. Viscoplastic strain versus cumulative loading time 

(140 kPa confinement VT). 
176

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 20 40 60 80 100

Cumulative Loading Time (s)

Viscoplastic Strain

140kPa-variable VLT (1)

140kPa-variable VLT (2)

140kPa-variable VLT Prediction

Figure 164. Graph. Viscoplastic strain versus cumulative loading time 

(140 kPa confinement VLT). 

0.0%

0.3%

0.6%

0.9%

1.2%

1.5%

1.8%

0 10000 20000 30000 40000

Time (s)

Viscoplastic Strain

140kPa-827kPa VT (1)*

140kPa-827kPa Prediction

140kPa-552kPa VT (1)

140kPa-552kPa VT Prediction

Figure 165. Graph. Viscoplastic strain versus cumulative loading time 

(140 kPa confinement VT). 
177

0.0%

0.3%

0.6%

0.9%

1.2%

1.5%

1.8%

0 5000 10000 15000 20000 25000 30000 35000

Time (s)

Viscoplastic Strain

552kPa VT+827 kPa Flow Number

552kPa VT+827 kPa Flow Number Prediction

Figure 166. Graph. Viscoplastic strain versus cumulative loading time 

(140 kPa confinement VT + flow). 

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 50 100 150 200 250

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-1600kPa VT (1)

500kPa-1600kPa VT (2)

500kPa-1600kPa VT Prediction

Figure 167. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement 1,600 deviatoric VT). 
178

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 20 40 60 80

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-2000kPa VT (1)

500kPa-2000kPa VT (2)

500kPa-2000kPa VT Prediction

Figure 168. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement 2,000 deviatoric VT). 

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 10 20 30 40 50 60

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-1800kPa CLT 0.4s (1)

500kPa-1800kPa CLT 0.4s (2)

500kPa-1800kPa CLT 0.4s Prediction

Figure 169. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement 0.4 s CLT). 
179

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 10 20 30 40 50 60 70

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-1800kPa CLT 1.6s (1)

500kPa-1800kPa CLT 1.6s (2)

500kPa-1800kPa CLT 1.6s Prediction

Figure 170. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement 1.6 s CLT). 

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 50 100 150 200 250 300 350 400 450

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-1800kPa CLT 6.4s (1)

500kPa-1800kPa CLT 6.4s (2)

500kPa-1800kPa CLT 6.4s Prediction

Figure 171. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement 6.4 s CLT). 
180

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 20 40 60 80 100 120

Cumulative Loading Time (s)

Viscoplastic Strain

500kPa-varaible VLT (1)

500kPa-varaible VLT (2)

500kPa-varaible VLT Prediction

Figure 172. Graph. Viscoplastic strain versus cumulative loading time 

(500 kPa confinement VLT). 
181

CHAPTER 6. ENHANCEMENT OF VEPCD-FEP++ FOR PAVEMENT MODELING 

A complete redesign of FEP++ was one of the major tasks completed in this study. The redesign 

activity resulted in a well designed and modular code base of around 40,000 lines. All of the old 

features of FEP++ have been revamped and tested, and several new features have been added. 

Currently, FEP++ supports the following: 

1. Two-dimensional (2D) and 3D pavement models analysis. 

2. Elastic, viscoelastic, and VECD material models. 

3. Separate analysis for static and dynamic loads. 

4. Linear, nonlinear, and quasilinear analysis. 

5. Special elements for modeling pavements. 

6. Elastic and viscoelastic materials thermal analysis. 

7. Multipoint constraints. 

8. Graphical user interface (preprocessor). 

The following sections discuss the upgraded VECD model, the overall organization of the 

modules in FEP++, and the newly developed preprocessor. 

6.1. DAMAGE IN VISCOELASTIC MATERIALS 

For the sake of completeness, the original VECD model, as detailed elsewhere, is presented first 

in this section.(13) Then, the upgraded VECD model is presented. Last, the implementation of 

temperature dependency for these materials is examined. 

6.1.1. The Original Model 

A one-parameter continuum damage model was incorporated into the viscoelastic material 

module. The new elements account for the evolution of isotropic damage. In this formulation, it 

is assumed that the stress-strain relationship obeys the following form: 

{σ} = ∗ CS E ( )[ ] {ε&} (181) 

Where: 

C(S) = Effect of damage on the stiffness of the material. 

S = Parameter representing damage evolution. 
182

The following evolution law governs the damage parameter: 

S W

t S

α

∂ ∂ ⎛ ⎞ = −⎜ ⎟ ∂ ∂ ⎝ ⎠ (182) 

Where: 

W = Total energy in the material. 

α = Parameter that depends on the type of loading. 

Currently, based on the available experimental data, the following form for the damage function 

is assumed: 

( ) b aS CS e = (183) 

A plot of the above function, assuming a = ―0.001334 and b = 0.5725, is shown in figure 173. 

The formulation presented here is not limited to the form of the damage function shown in 

equation 183. To show the effect of continuum damage, a 10 by 10 patch of 2D plane stress 

elements has been modeled and is shown in figure 174. The specimen is fixed at the bottom and 

loaded (in tension) on five nodes at the top. The loading history is sinusoidal as shown in 

equation 184. 

f = + γ π ⎡1 sin 2( t)⎤ ⎣ ⎦ (184) 

Where: 

γ = The loading amplitude. 

The time history of the vertical displacement for point A () is shown in figure 175. The evolution 

of damage in the specimen and, consequently, a gradual increment in the displacement amplitude 

can be seen from the time history. This figure shows that damage has an effect on the response of 

the material. More quantitative tests on the effect of damage will be performed in the future. 
183

Figure 173. Graph. Damage characteristic relationship used in the finite element 

implementation of one-dimensional VECD model. 

A

Figure 174. Graph. Layout of the numerical experiment specimen. 
184

Figure 175. Graph. Effect of continuum damage evolution on the vertical displacement of 

the test specimen at point A in the test simulation. 

6.1.2. Upgraded Constitutive Model 

The continuum damage formulation was upgraded to a more rigorous model based on the work 

of others.(57) However, a one-parameter damage model was chosen instead of the two-parameter 

model used in the work because the experimental data needed to characterize a two-parameter 

damage model are not yet available. 

The model assumes that a material is isotropic when undamaged and that the growth of damage 

under loading leads to local transverse isotropy (i.e., the material has a local axis of symmetry 

oriented along the maximum principal stress direction). The current framework is formulated for 

the axisymmetric case but can easily be extended to 3D. 

Starting with the work potential theory for an elastic material and making use of viscoelastic 

fracture mechanics and the correspondence principle for viscoelastic materials, a pseudo strain 

energy density function, WR, can be written in terms of the pseudo strains in the local axis as 

follows: (8,9)

(()() ) (( ) ) 22 2 2 2 2

11 22 12 44 13 23 66 12

11 22 33

33

11 22

1 () () 2 ( ) 2

3

R RRR R R R RR

vdd v s

R RR R

v

R

R R v

d

R RR

s

W A e A e Aee A A e

e

e

e

e

γγ γ

εεε

ε

ε ε

⎡ ⎤ = + + + ++ + ⎢ ⎥ ⎣ ⎦

=++

= −

= − (185) 

Where: 

ε 11

R

, ε 22

R

, ε 33

R

, γ 12

R

, γ 13

R

, γ 23

R

 are the pseudo strains along the local axis. 
185

When the local axis is also a principal axis, the shear strains are zero, and equation 185 becomes 

the following: 

2 2 2

11 22 12 66

1 () () 2 () 2

R RRR R R W A e A e Aee A e vdd v s = ++ + ⎡ ⎤ ⎣ ⎦ (186) 

In this case,ε 11

R

, ε 22

R

, and ε 33

R

 are the principal pseudo strains (which lie in the local axis) with 

the axis of isotropy oriented along direction 3. These principal pseudo strains are obtained from 

the pseudo strains along the global axis using standard tensor transformation and knowing the 

angle between the local axis and the global axis. (For the axisymmetric analysis, the hoop 

direction is already a principal axis because there are no shear strains along the rθ -θ z plane.) 

The LVE pseudo strains along the global axis,ε kl

R

(t), are calculated from strains along the global 

axis, ε kl, using the convolution integral. 

0

1 () ( )

t

R kl

kl

R

t Et d

E

ε

ε τ τ

τ

∂ = − ∂ ∫ (187) 

Where: 

ER = Reference modulus that has the same dimensions as the modulus and is usually taken as 

1 E(t). 

1 E(t) = Relaxation modulus for uniaxial loading. 

The calculation of the convolution integral can be very expensive in terms of computation time. 

In practice, therefore, the pseudo strains are calculated using a state variable approach to reduce 

the computational expense. When the relaxation modulus of the material E(t) is represented 

using the Prony series of the form, shown in equation 188, an approximation can be obtained to 

the convolution integral in equation 187. 

1

( ) e i

t M

i

i

Et E E ρ −

∞

=

= +∑ (188) 

Where: 

E∞ = The relaxation modulus at t = ∞. 

Ei = The Prony coefficients corresponding to the relaxation times, ρi . 

The pseudo strains along the global axis can be calculated from the following: 

1 1

1 11

1

/ /(2 )

1 1

1 () () ()

( ) () ( ) n i n i

M

R i

kl n kl n i kl n

R i

i i t t

kl n kl n kl n

t E t Et

E

t e te t ρ ρ

ε εε

εε ε + +

+ ∞+ +

=

−Δ −Δ

+ +

⎛ ⎞

= + ⎜ ⎟ ⎝ ⎠

= +Δ

∑

 (189) 

Where: 

ε kl

i

(tn+1) (i = 1..M) are the internal state variables that record the history of the material up to 

time tn+1, Δ ε kl(tn+1) = ε kl(tn+1) - ε kl(tn), and Δt=tn+1-tn. The pseudo strains at time tn+1 are 
186

calculated based on the strain at times tn and tn+1 which are available at the end of the finite 

element solution step at time tn+1. 

The factors A11, A22, A12, and A66 are stiffness terms that can be related to a damage function, 

C(S) as follows: (48)

[ ]

11

22

12

44 66

1 2(1 ) ( ) 9 (1 2)

(1 2 ) ( ) 2(1 )

1 ( ) 3

2(1 )

R

R

R

R

A CS E

A CS E

A CS E

E A A

ν

ν

ν

ν

ν

⎡ + ⎤

= +⋅ ⎢ ⎥ ⎣ − ⎦

− = +⋅

+

= −

= = + (190) 

Where: 

ν = Poisson’s ratio of the material. 

C(S) = A stiffness function that depends on damage in the material. 

S = A damage parameter used to track the growth of damage in the specimen. 

The principal stresses along the local axis can be found from equation 186 using the following: 

( ) 1, 2, 3

R

ii R

ii

W

σ i

ε

∂ = = ∂ ,

 (191) 

which gives the following: 

11 11 12 12 22 66

22 11 12 12 22 66

33 11 12 12 22

1 1 ( )( ) 3 3

1 1 ( )( ) 3 3

2 2 ( ) ( ). 3 3

R R R

V d S

R R R

V d S

R R

V d

A A e A A e Ae

A A e A A e Ae

A Ae A Ae

σ

σ

σ

=− +− −

=− +− +

=+ ++ (192) 

The stresses along the global axis are then obtained by standard stress transformation and the 

orientation of the local axis with respect to the global axis. 

6.1.3. Damage Model 

The growth of the damage parameter S is modeled by extending the concepts of viscoelastic 

fracture to microcracking.(8) 
187

R dS W

dt S

α

⎛ ⎞ ∂ = −⎜ ⎟ ∂ ⎝ ⎠ (193) 

Where: 

WR = Pseudo strain energy density function shown in equation 185. 

α = Material-dependent parameter. 

From equation 186, the quantity ∂WR

/ ∂ S can be calculated as a function of pseudo strains in the 

local axis as follows: 

( ) ( ) 11 2 2 2

29 3

R

R RR R

V dV d

W C

e ee e

S S

∂ ∂ ⎛ ⎞ = ++ ⎜ ⎟ ∂ ∂ ⎝ ⎠ (194) 

Where: 

C(S) = Damage function that is assumed to be of the form shown in equation 195, based on 

experimental data. 

( ) b aS CS e = (195) 

6.1.4. Finite Element Implementation 

The finite element solution of the problem requires the material tangent stiffness matrix that is 

used to assemble the global tangent stiffness matrix used for the solution of the nonlinear system 

of equations by the Newton-Raphson method. Not to be confused with the damage function, C(S),

the material tangent stiffness matrix, [C], is given by the following: 

[ ] { }

{ }

{ }

{ }

{ }

{ }

R

R

C

σ σ ε

ε ε ε

∂ ∂ ∂ = = ∂ ∂ ∂ (196) 

Where: 

{σ } = {σ rr, σ θθ , σ zz, σ rz} and {ε }={ε rr, ε θθ , ε zz, ε rz}are the stresses and strains for the 

axisymmetric problem. 

[C] = Material tangent stiffness matrix oriented along the global axis. Because the stresses 

along the global axis are obtained by transforming the stresses in the local axis, it is 

easier to construct the material tangent stiffness matrix along the local axis and then 

transform it to the tangent stiffness along the global axis. 
188

The tangent stiffness matrix in the local axis, [CL], is given as follows: 

[ ] { }

{ }

11 12 22 66 11 12 22 66 11 12 22

11 12 22 66 11 12 22 66 11 12 22

11 12 22 11 12 22

21 21 12 000

39 39 39

21 21 12 000

39 39 39

12 12

39 39

L

L R

L

A A AA A A AA A A A

A A AA A A AA A A A

C

AAA AAA

σ

ε

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ −++ −+− +− ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −+− −++ +− ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ∂

= = ⎛ ⎞⎛ ⎞ ∂ ⎜ ⎟⎜ +− +− ⎝ ⎠⎝ ⎠ 11 12 22

66

44

44

4 4 000

3 9

0 0 0 00

0 0 0 00

0 0 0 00

AAA

A

A

A

⎡ ⎤

⎢ ⎥

⎛ ⎞ ⎟⎜ ⎟ + + ⎝ ⎠

⎣ ⎦ (197) 

Where: 

{σ L} = {σ 11, σ 22, σ 33, σ 12, σ 13, σ 23} is the stress vector along the local axis. 

{ε L

R

} = {ε 11

R

, ε 22

R

, ε 33

R

, γ 12

R

, γ 13

R

, γ 23

R

}is the pseudo strain vector along the local axis. 

The pseudo shear strains are zero along the local axis, but the stiffnesses corresponding to them 

are non-zero. The matrix, [C], can be obtained by transforming [CL], as shown in the following: 

[ ] [ ] [ ][ ] 1

C T CT R LR

−

= (198) 

Where: 

[TR] = The rotation and permutation matrix that changes the order of the vector components (the 

axis 3 along the local axis is always oriented along the maximum principal pseudo strain 

direction) and transforms a vector from the local axis to the global axis. 

As seen in equation 193 and 194, the damage growth involves a nonlinear differential equation 

that can be expensive to solve in a large finite element method (FEM) problem. Hence, a semiimplicit method is used to predict the damage parameter in the next time-step, Sn+1, using the 

damage parameter in the current time-step, Sn

, and the pseudo strain vector in the local axis, 

{ε L

R

}n+1, for the next time-step, as follows: 

({ } ) 1 1 , n nn R n

L S SS S t ε + + = + Δ & (199)

This method should provide results similar to those of an exact nonlinear analysis when the timesteps are made small enough. 

6.1.5. Verification 

The continuum damage material model is verified for a monotonic uniaxial test. The test is part 

of the uniaxial testing on cylindrical specimens that is conducted for the damage characterization 

of FHWA mixtures. Verification uses the 5 °C test because there is little or no viscoplastic 

deformation at this temperature. For the FEM simulation, the cylindrical specimen is modeled as 
189

a single quadrilateral (Q4) element that is 150 mm in height and 75 mm in width and is subjected 

to a uniform axisymmetric loading in the vertical direction. The stress history measured from the 

test is used as an input for the problem, with the strains calculated by analysis. Horizontal 

displacements are constrained along the axisymmetric axis of the element, and vertical 

displacements are constrained along the bottom edge of the element, which results in a uniform 

state of stress and strain in the element. Predicted strains are then compared against measured 

strains to verify the FEM implementation of the material model. The damage function is taken as 

C(S) = exp(-0.001S0.5737), based on the monotonic test results from several different specimens. 

The Prony series for the material is obtained from test data for the asphalt concrete mixture used 

in the testing. 

The results of the verification are shown in figure 176 through figure 178. As seen from these 

figures, the prediction of strain is very good, the damage parameter consistently increases with 

time and the pseudo stiffness consistently decreases with time. These combined findings verify 

that the model has been implemented correctly. 

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

0.00E+00 5.00E+01 1.00E+02 1.50E+02

Reduced Time (s)

Strain

Predicted

Measured

Figure 176. Graph. Verification of strain prediction for monotonic uniaxial test using the 

new continuum damage material model. 
190

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

1.20E+05

1.40E+05

1.60E+05

1.80E+05

0.00E+00 5.00E+01 1.00E+02 1.50E+02

Time (s)

Damage Parameter, S

Figure 177. Graph. Evolution of damage parameter, S, for the monotonic test. 

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

0.00E+00 5.00E+01 1.00E+02 1.50E+02

Time (s)

Pseudo-stiffness, C

Figure 178. Graph. Plot of function C(S) with time. 

6.1.6. Implementation of Temperature Dependency 

The variation of temperature in a pavement has two effects: (1) a change in stiffness of the 

asphalt concrete and (2) a change of the thermal stresses due to thermal expansion of the material. 

The thermal stresses are generated in the pavement depending on boundary conditions. These 

two effects of temperature have been implemented in FEP++. 
191

The change in stiffness in the asphalt concrete due to temperature is taken into account using the 

concept of reduced time. The reduced time is calculated as follows: 

( ) ( ) ( ) 0

1 ,

t

t d

a T ξ ττ = ∫ (200) 

Where: 

a(T) = Time-temperature shift factor of the material. The time-temperature shift factor is 

obtained from the characterization of the relaxation modulus of the material using 

dynamic modulus tests at different frequencies and temperatures. The reduced time is 

calculated with respect to a reference time and captures the history of temperature 

variation on the material. The thermal stresses on the material are incorporated by defining 

mechanical strains, ε m(t), as follows: 

{ ( )} { ( )} { }( 0 ). m ε εα t t TT =−− (201) 

Where: 

{ε (t)} = Strain in the material. 

{α } = Coefficient of isotropic thermal expansion in the material (which is assumed constant 

for asphalt concrete) and is given by [α ,α ,α ,0,0,0] in three dimensions. 

T = Current temperature. 

T0 = Reference temperature at which there are no initial stresses. 

The constitutive law for a viscoelastic material undergoing damage is given as follows: 

{ } [ ]{ }, m σ ε = D R (202) 

Where: 

{σ } = Stress vector. 

[D] = Stiffness matrix that is a function of damage. 

{ε m

R

} = Vector of mechanical pseudo strain. 

The i

th component of the pseudo strain vector is given by the following: 

( )' '

, '

0

,

m

m i

R i

d E d

d

ξ ε

ε ξξ ξ ξ = − ∫ (203) 

Where: 

ε i

m = i

th component of the mechanical strain vector. 

E(ξ ) = Relaxation modulus of the material characterized at reduced temperature. ξ is 

computed in accordance with equation 200 and is a function of the temperature history 

of the material. 
192

6.2. REDESIGN OF FEP++ 

The older code base of FEP++ was redesigned during the course of the project, leading to a 

robust, modular, and standards-compliant version. These enhancements greatly improve the 

maintainability of the code base and significantly reduce the training necessary for a new user. 

The FEP++ code base is divided into two major modules, domain and analysis. The following 

sections discuss the features and functionalities of these modules. 

6.2.1. Domain Module 

The domain module represents the core finite element component of FEP++. It encompasses the 

finite element classes that are involved in the modeling of the computational domain. Figure 179 

presents a block diagram and the key components of the module. The following sections briefly 

discuss the main features of the domain module. 

Domain 

Material

NodeLoad

SinglePt

Constraint

MultiPt ElementSet NodeSet

Constraint

Utilities

Element Node

. . .

. .

.

LoadCase

.

. 

.

. .

.

Plane Stress Plane Strain

Elastic

Visco-Elastic

VECD

.

Aggregation

Inheritance

Multiplicity

Aware of

Figure 179. Diagram. Domain module. 

6.2.1.1. Material Model 

FEP++ provides 2D and 3D models of elastic, viscoelastic, and VECD materials for modeling a 

pavement layer. Detailed discussion of these models and their implementation is provided in 

section 6.1. 

6.2.1.2. Elements 

FEP++ provides two main types of elements, the quadrilateral elements for 2D analysis and the 

brick elements for 3D analysis of pavements. Also, special elements developed for modeling 

infinite domains in pavement analysis have been incorporated into FEP++. The following 

subsection describes the special elements and their efficiency in pavement analysis. 
193

6.2.1.2.1. Incorporation of special elements for pavement modeling: 

A finite element mesh generation methodology for incorporating special elements that reduce the 

computational costs of pavement analysis has been developed. An accurate mesh of special 

elements can be difficult to generate because it depends on the characteristics of the load applied. 

Development of such a methodology may also turn out to be somewhat computationally 

expensive if it tries to provide greater accuracy than is necessary for the problem at hand. Hence, 

there is a need to develop a mesh generation methodology that allows the user to easily generate 

a finite element mesh that models the problem at hand with specified accuracy. 

To this end, work is being carried out to generate the best mesh for a specified accuracy through 

optimization techniques. Generating the best mesh is an involved process that could render it 

unfeasible for everyday analysis. Therefore, simplified rules will be developed that allow the 

user to easily generate a mesh that is as close as possible to the best mesh. 

A simple example is given below to demonstrate the computational efficiency achieved. 

Figure 180 shows a layer of elastic material loaded on a rigid base. This layer is infinite in the 

two horizontal directions. For a strip load of width “2B,” the 3D problem can be reduced to a 

2D-plane strain problem. Also, symmetry allows that only one half of the problem needs to be 

modeled. Only the deflection and stresses under the load (the interior region) are of interest here. 

Exterior Interior

Physical Problem

B B

8 8

Exterior

Line of 

symmetry

Figure 180. Illustration. Infinite elastic layer on a rigid base. 

However, despite the emphasis on the interior region, the exterior region also must be modeled 

to a sufficient length to simulate the infinite extent of the layer. As the exterior region is 

extended, the solution in the interior approaches the solution of the physical problem. The 

objective is to model the physical problem to achieve an accuracy of at least 1 percent. A typical 

mesh of finite elements used to simulate this physical problem is Mesh-1, shown in figure 181.

For Mesh-1, the exterior is four times the extent of the interior and hence needs four times as 

many finite elements as the interior to achieve an accuracy of 0.86 percent. 
194

If, on the other hand, special elements are used, then Mesh-2 shown in figure 181 is sufficient 

and provides an accuracy of 0.84 percent. The total number of elements in Mesh-1 is 320, 

whereas the total number in Mesh-2 is 64. In order to achieve almost the same degree of 

accuracy as Mesh-1, far fewer special elements are required for Mesh-2. In short, because the 

computational time is directly proportional to the number of finite elements used to model the 

problem, Mesh-2 with the special elements is far more computationally efficient when compared 

to Mesh-1. 

The development of a finite element mesh generation methodology that incorporates special 

elements is now complete. This methodology is being tested on various finite element models to 

demonstrate its efficacy. Once the results have been verified against theoretical solutions, the 

mesh generation methodology will be coded into FEP++ to make the discretization both 

automatic and efficient. (This task will be performed as part of the ongoing cooperative 

agreement between NCSU and the FHWA.) 

Figure 181. Illustration. Finite element mesh required to model the physical problem with 

1-percent accuracy without special elements. 

Figure 182. Illustration. Finite element mesh required to model the physical problem with 

1-percent accuracy with special elements. 
195

6.2.2. Analysis Module 

The analysis module handles the generation and solution of the linear/nonlinear system of 

equations using the domain module. A block diagram of the classes and relationships is shown in 

figure 183. The following sections discuss the key components of this module. 

6.2.2.1. Solution Algorithm 

The solution algorithm is the component that acts as a liaison between the domain and analysis 

modules. This component uses the information in the domain to generate a set of equations that 

provides the solutions to the problem. Currently, the following static and dynamic analyses of 

linear, quasilinear, and nonlinear problems are available. 

One significant enhancement over the earlier capabilities of FEP++ is the ability to perform 

quasilinear analysis. This improvement is particularly beneficial in the context of pavement 

analysis because the VECD material model is quasilinear. By using a quasilinear solver instead 

of a nonlinear solver, the analysis time is reduced by almost half. 

Analysis

Mapping

SolutionAlgo Solver

Analysis Utilities

Stepper

NonLinear Dynamic

NonLinear Static

QuasiLinear Static

QuasiLinear Dynamic

Linear Static

Linear Dynamic

Iterator

Modified Newton

Linear

Newton Raphson

Newmark(accel)

Static

Newmark(disp)

Aggregation

Inheritance

Is Aware of

Figure 183. Diagram. Analysis module. 

6.2.2.2. Solver 

The solver is the workhorse of the module. It is responsible for solving the set of equations 

generated by the solution algorithm. The solver is also an important component in terms of the 
196

capabilities of FEP++ in solving large computational problems because the solver determines the 

memory requirements and performance output. Any improvements to this module would 

significantly improve the overall capabilities of FEP++. 

Currently, a class of direct methods called Gaussian elimination is used to solve the equations. 

The objective here is to enhance the component by providing an option to interface with external 

linear algebra packages to solve the system of equations. The benefit of such a feature would be 

the ability to use mature, well tested, and highly efficient libraries available in the public domain 

for performing the analysis. This capability would also reduce further interdependence between 

the domain and the analysis modules, thereby enhancing the flexibility and elegance of the code. 

6.3. PREPROCESSOR 

The preprocessor is a graphic user interface (GUI) front end developed for the efficient analysis 

of pavements using FEP++. The motivation behind the tool is to provide the user with a 

simplified and intuitive interface to FEP++ for pavement analysis. The tool can be used either to 

run an analysis directly or to generate the input files for FEP++. It also has visualization 

capabilities to view the mesh discretization, which makes it easy for the user to verify the input 

data for the analysis. Figure 184 shows a screenshot of the main window in the preprocessor. 

Figure 184. Screen capture. Main window of the preprocessor. 
197

As labeled in figure 184, the main window is divided into three functional units, the control 

panel, the canvas panel, and the popular open-source library called Visualization Toolkit (VTK) 

control panel, described below. 

The left-hand unit is called the Control Panel. This panel contains the controls to invoke the dataentry windows and other operations related to the analysis. Each control is either red or green, 

indicating the current state of the analysis. For controls corresponding to data entry operations, 

green indicates that no user action is required. Green, therefore, could either signal that the data 

entry has already been completed or that it is preset with a default set of parameters. Red 

indicates that some action is required from the user. The data input is considered complete only 

when all the controls in the input parameters block (figure 185) are green. 

Figure 185. Screen capture. Control panel. 

The middle unit is called the Canvas Panel. This panel is used for the visualization of the mesh 

discretization and the solid model of the pavement. The underlying graphical engine used to 

render the graphics is VTK. This toolkit was chosen for its stable and mature set of high-level 

graphical libraries. Also, it is popular and widely used in the open-source and scientific 

community. Figure 186 shows a sample mesh discretization in the canvas panel. 

The rightmost unit provides the controls for the visualization operations supported in the 

preprocessor, e.g., zoom, pan, etc., and is called the VTK control panel. The following 

operations are currently supported: 

• Projections along a plane, e.g., along the x-y plane. 

• Image operations, such as zoom, pan, and rotate. 

• Display nodes and cell numbers on the mesh. 
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The results of these operations are displayed in the view window. Figure 187 shows the zoomed 

version of a 2D mesh with node numbers displayed. The visual feedback from these windows is 

useful to verify the data inputs to the analysis. The different stages involved in performing a 

pavement analysis and the ways that the various analytical pieces fit together are outlined in 

figure 188. 

The input stage corresponds to the data-entry windows that receive the input data from a user 

(i.e., it shows the list of data that needs to be entered by the user). The analysis stage shows the 

list of options available to the user once the input stage is complete. The postprocessing stage 

again shows the options after the analysis is completed successfully. The above mentioned stages 

are discussed in detail in the following sections. 

Figure 186. Screen capture. Mesh discretization. 
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Figure 187. Screen capture. Zoom operation on a mesh. 
200

Figure 188. Diagram. Stages involved in an FEP++ analysis. 

6.3.1. Input Stage 

As discussed in the previous section, this stage corresponds to the data entry by the user. Each 

main input data entry for FEP++ corresponds to a data entry window. The user manual, with a 

detailed explanation of each of the data entry windows, is provided in the appendix of this report. 

Figure 189 shows a data entry window for entering the mesh discretization parameters for a 

pavement. 

The data entry windows have been designed in a modular way, and effective error validation 

schemes have been incorporated to ensure consistent input data. Each data entry window has a 

data class associated with it that enforces the validation and consistency checks. There are two 

levels of error validations that are carried out at each data entry window, namely syntactic and 

semantic errors. The syntactic errors refer to invalid data entries, and they are checked by the 

data entry windows themselves, whereas the semantic errors refer to consistency check failures 

imposed by the data classes. 

Once the data entry is complete for a particular block, it is validated against the checks provided 

in the data classes. If the validation is successful, the next data entry window can be accessed. If 

it fails, appropriate error messages are shown, and the user is asked to modify the incorrect data. 

Figure 190 shows a typical error dialog that is displayed when a consistency check fails. 
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Figure 189. Screen capture. Sample data entry window. 

This sequential data entry process may seem to be somewhat limiting, but it greatly simplifies 

the maintenance of consistency in the input data. Also, by strictly enforcing these checks in the 

software, there is less scope for input data corruption. 

6.3.2. Analysis Stage 

Once the data entry has been completed, the analysis section in the control panel turns green, 

signaling that the actions in the analysis stage can now be performed. The following sections 

discuss the actions that can be performed at this stage. 

6.3.2.1. Setting the Output Options 

Output options can be used to control the results that are generated by FEP++. For example, 

FEP++ can be configured to output data that correspond to a particular region in the domain. 

These options are essentially used to control the amount of data output by FEP++. 
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6.3.2.2. Viewing a Summary of the Input Data 

The preprocessor can be used to create a user-readable report of the input data entered for an 

analysis. This report can be used to verify the final state of all the input data before beginning a 

simulation. 

Figure 190. Screen capture. Error dialog for a semantic error. 

6.3.2.3. Generating an FEP++ Input File 

The preprocessor can be used to generate input files for FEP++ for different configurations that 

can then be saved to a remote location. This feature is helpful if the user wishes to perform 

multiple analyses in a script without any user intervention. 

6.3.2.4. Running an Analysis 

Running an analysis is the final step in pavement analysis. The analysis can be started only after 

the run analysis option in the control panel turns green. The preprocessor uses the user input to 

create an input file to FEP++ and invokes it with the generated input file. The preprocessor 

creates a separate instance and displays the progress of the analysis in a window, as shown in 

figure 191. 
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Figure 191. Screen capture. Analysis run-time window. 

6.3.3. Postprocessing Stage 

After an analysis is completed, the run-time window shown in figure 191 exits, and the results

option on the control panel turns green, indicating that result files have been generated that can 

be viewed with a postprocessor. On choosing view results, the default postprocessor is invoked 

to view the results. 

6.4. POSTPROCESSOR 

The postprocessor is an external tool for visualizing the results of the pavement analysis. 

Currently, Tecplot® serves as the default postprocessor, but efforts to provide support for other 

open-source data visualization tools are underway. It has been decided that the VTK format will 

be used for the output file format and that MayaVi, a stable open-source python-based viewer 

that is popular in the finite element community, will be used as a suitable viewer. The support for 

external tools will be completed in the next project.

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CHAPTER 7. A SAMPLE 3D PAVEMENT SIMULATION USING FEP++ 

As mentioned in the previous chapter, FEP++ has full support for 3D finite element analysis of 

elastic and viscoelastic materials. This chapter discusses the details of a 3D analysis performed 

using FEP++ to study the effects of temperature, material, and wheel speed on ALF pavements. 

The following sections discuss the analysis details and the results. 

7.1. THE INPUT DATA 

The pavement was modeled as a 3D cuboid, and the details are shown in table 23. A single 

moving wheel load with the parameters shown in table 24 is used in the analysis. The behavior of 

the pavement was studied for the effects of temperature, material, and wheel speed. The 

following subsections discuss the simulations and the results in detail. 

Table 23. Properties of pavement. 

Depth of AC Layer 46 cm 

Pavement Length 296.5 cm 

AC Material ALF Control/ 

ALF SBS 

Subgrade Type and Stiffness Infinite Subgrade, 

86 MPa 

Ambient Surface Temperature Winter (-5 oC)/ 

Summer (38 oC) 

Table 24. Properties of moving wheel load. 

Contact Pressure 758 kPa

Load Area 19.65 cm by 17.79 cm

Wheel Velocity 13.41 m/s/26.82 m/s

7.2. EFFECT OF TEMPERATURE 

To simulate the effect of temperature, the analysis was performed at representative temperatures 

for winter and summer. The temperature distribution across the depth was found from 

simulations with the Enhanced Integrated Climatic Model (EICM). For these simulations, a 

typical pavement cross section was used with the Raleigh, NC, climatological database. The 

exact temperature variation is shown in figure 192. 

The strains were expected to be larger during the summer as compared to the winter because the 

material stiffness was smaller at higher temperatures leading to larger deformations. Also, the 

material was expected to display a higher degree of viscoelasticity at higher temperatures while 

displaying higher elasticity at lower temperatures. 

Figure 193 through figure 198 show the strain distributions for the above analysis when the 

wheel load reached the center of the pavement. The temperature distribution across the depth is 

also shown in each of the figures. Figure 193 and figure 194 show the vertical strain distribution, 

and, as expected, the strains were smaller for the winter loading than for the summer loading. 
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Also, the strain distribution was more or less symmetrical around the wheel location for winter 

loading as compared to the summer loading which showed some residual strain from the passing 

of the wheel. Thus, the viscoelastic behavior was more prominent at higher temperatures. 

Figure 195 and figure 197 show the strain distributions for longitudinal and transverse strains. 

The strains were consistently higher in the case of summer loading than in winter loading, and 

the elastic nature of the material (symmetry about the wheel) was more prominent during the 

winter loading. Thus, the results of the FEM analysis of the effects of temperature on the 

material properties agreed well with the expected outcome. 

0

0.1

0.2

0.3

0.4

0.5

-5 5 15 25 35

Degrees (C)

Depth (m)

Winter

Summer

Figure 192. Graph. Temperature variations used for simulations. 
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Figure 193. Illustration. Vertical strains in winter. 

Figure 194. Illustration. Vertical strains in summer.
208

Figure 195. Illustration. Longitudinal strains in winter. 

Figure 196. Illustration. Longitudinal strains in summer. 
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Figure 197. Illustration. Transverse strains in winter. 

Figure 198. Illustration. Transverse strains in summer. 
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7.3. EFFECT OF MATERIAL 

The effect of material properties on pavement behavior was analyzed next. Pavements under 

summer conditions, with the ALF Control mixture and the ALF SBS mixture for the asphalt 

concrete layer, were used for the simulations, and a wheel speed of 26.82 m/s was applied. 

Because the SBS pavement was less stiff compared to the ALF Control pavement, the strains 

were expected to be higher for the SBS pavement. 

Figure 199 through figure 204 show the strain distributions for the above analysis when the 

wheel load reached the center of the pavement. Figure 199 and figure 200 show the vertical 

strain distribution; the SBS pavement had higher strains compared to the Control pavement. Also, 

the strains in the SBS pavement located away from the wheel location were higher, indicating 

that the strains in the SBS pavement took longer to recover from the passing of a wheel than 

those of the Control pavement. 

Figure 201 and figure 202 show the longitudinal strains for the two mixtures. Again, the strains 

were larger for the SBS mixture than for the Control mixture, leading to larger regions of tension 

and compression. 

Figure 199. Illustration. Vertical strains for Control mixture. 
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Figure 200. Illustration. Vertical strains for SBS mixture. 

Figure 201. Illustration. Longitudinal strains for Control mixture. 
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Figure 202. Illustration. Longitudinal strains for SBS mixture. 

Figure 203. Illustration. Transverse strains for Control mixture. 
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Figure 204. Illustration. Transverse strains for SBS mixture. 

Figure 203 and figure 204 show the transverse strains. Again, the strains were larger for the SBS 

mixture than for the Control mixture. Also, the region of maximum tension at the bottom of the 

pavement was much larger for the SBS pavement than for the Control pavement. The area of 

influence on the surface of the pavement was higher for the SBS mixture than the Control 

mixture. 

Again, the results were as expected. The SBS pavement deformed more than the Control 

pavement, and the area of influence was larger due to its lower stiffness. 

7.4. EFFECT OF WHEEL SPEED 

Finally, the effect of the wheel speed on pavement behavior was analyzed. The analysis was 

performed on a pavement using the ALF Control mixture for the asphalt concrete layer under 

summer conditions for two cases with wheel speeds of 13.41 and 26.82 m/s, respectively. 

Because the viscoelastic model was quasistatic, dynamic effects (acceleration/velocity) were not 

considered, and the effect of the wheel speed essentially affected only the load duration at each 

point on the pavement. Thus, the effect of a slower wheel speed could be a longer duration of the 

load pulse and, consequently, larger vertical strains compared to a faster wheel speed. Also, the 

slower wheel load could create a larger area of influence because the material recovery time may 

be comparable to the wheel passing time. 

Figure 205 through figure 210 show the strain distribution for the analysis when the wheel load 

reached the center of the pavement. Figure 205 and figure 206 show the vertical strains, which 

were slightly larger for the slower wheel speed. The memory effect was slightly more 
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pronounced for the slower wheel speed, as shown by the larger strains in the pavement region 

passed by the wheel. 

Figure 207 through figure 210 show the longitudinal and transverse strains; these strains 

displayed similar characteristics to the vertical strains, although the effects were much less 

pronounced. The vertical strains seemed to be most sensitive to the wheel speed. 

Figure 205. Illustration. Vertical strains for a wheel speed of 13.41 m/s. 
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Figure 206. Illustration. Vertical strains for a wheel speed of 26.82 m/s. 

Figure 207. Illustration. Longitudinal strains for a wheel speed of 13.41 m/s. 
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Figure 208. Illustration. Longitudinal strains for a wheel speed of 26.82 m/s. 

Figure 209. Illustration. Transverse strains for a wheel speed of 13.41 m/s. 
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Figure 210. Illustration. Transverse strains for a wheel speed of 26.82 m/s. 

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CHAPTER 8. CONCLUSIONS AND FUTURE WORK 

8.1. CONCLUSIONS 

The approach adopted in this research to characterize asphalt concrete over a wide range of 

temperatures and loading rates encountered in the field divided the problem into two 

components: (1) characterizing the viscoelastic response and (2) characterizing the viscoplastic 

response. The VECD model described the time-dependent behavior of asphalt concrete with 

growing microcrack damage. The irrecoverable (whether time-dependent or independent) strain 

was described by the VP model. These two models were integrated based on the strain 

decomposition principle to form the VEPCD model. This model was found to be applicable to 

the tension mode and, in principle, to the compression mode of loading. 

Through the model characterization procedures, it was also found that the dynamic modulus, if it 

was determined according to strict guidelines so that the linear limits of the material were not 

exceeded, was not dependent upon loading direction (tension compression or compression only). 

Further, it was found that this loading direction independence held under different confined 

stress states. From these confined dynamic modulus tests, a state-dependent model similar to the 

one used for unbound paving materials was developed and characterized. 

The viscoelastic damage characteristics were found to differ in the compression and tension 

loading modes, with the compression mode showing the more favorable results (i.e., less 

reduction in the pseudo stiffness, C, for the same amount of increase in the damage parameter, S). 

These results were also consistent with the hypotheses that the damage parameter, S, was related 

to cracking density or crack volume and that the primary direction of this cracking was 

perpendicular to the tensile loading direction or parallel to the compressive loading direction. A 

simple, empirically-based viscoplastic model was found to be sufficient for explaining the tensile 

behavior of asphalt concrete, but it did not explain the compressive behavior. 

For the compressive behavior, a rate-dependent softening mechanism, which operated during 

unloading, was found to be a significant factor that affected the viscoplastic characteristics of 

asphalt concrete. Several existing viscoplastic models that included flow rules and a yield 

criterion had been evaluated, but they were found to be insufficient for describing this softening 

behavior. To account for this characteristic behavior, the rate-dependent, hardening-softening 

function was suggested using Perzyna’s flow rule. The relaxation modulus determined from the 

linear viscoelastic characterization was utilized in this process. It was shown that the developed 

model could account for the effects of rest periods and loading sequence on viscoplastic strain 

development. 

In light of practical concerns related to the use of constant rate tests in the AMPT and due to the 

complexities of performing true time-dependent analysis of cyclic fatigue tests, a simplified 

VECD model was presented. This model utilized results from fatigue tests performed at nominal 

levels that were possible with the AMPT equipment, as well as the VECD model that specialized 

in such loading so as to arrive at a simple formulation to characterize the model. This 

formulation was found to generally agree with the results from the constant rate tests, 

particularly under conditions of minimal viscoplastic strain. 
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Another major finding in this research was the verification of the time-temperature superposition 

principle with growing damage, both in compression and tension, in a confined stress state. This 

principle was proven valid using constant rate tests under various temperatures and strain rates 

with an applied confining stress. Therefore, the response of a mixture with growing damage at 

one temperature could be predicted by shifting its response at another temperature using the 

time-temperature shift factor determined from the LVE complex modulus tests. 

The major contribution of the time-temperature superposition principle and the damage 

characteristic curve was the significant reduction in testing requirements. The model allowed the 

prediction of the material’s behavior at any temperature from a test result obtained from a single 

temperature and the time-temperature shift factors obtained from temperature sweep complex 

modulus tests. The experimental results from the previous FHWA project (DTFH61-03-H00116) and this project verified that the time-temperature superposition principle with growing 

damage was valid in both tension and compression, regardless of the confined state (i.e., 

unconfined or confined). 

To realize the full potential of the VEPCD material model for predicting pavement performance, 

a robust FEP++ was developed to account for the effects of loading and boundary conditions. 

FEP++ was designed using a top-down, object-oriented approach with great care so that further 

enhancements could be made by the research team in an efficient manner. Furthermore, special 

elements and implementation techniques were employed to increase the computational efficiency. 

The 2D version of the resulting software has the ability to predict stresses, strains, and damage of 

the pavement under repeated traffic loading. FEP++ was also extended to 3D stress analysis with 

the ultimate goal of 3D damage modeling of moving traffic loads. All these capabilities were 

made accessible to the user through carefully designed and powerful graphical pre and 

postprocessors. As part of the ongoing HMA-PRS project sponsored by the FHWA (DTFH61-

08-H-00005), FEP++ is now being molded into an integrated software tool that can be used for 

robust pavement performance predictions. 

To illustrate the capabilities of FEP++ and its preprocessor, a full 3D finite-element analysis was 

carried out. The effects of temperature, material, and wheel speed were studied and found to be 

in accordance with the expected results. Namely, the pavement showed an increased viscous 

response with increased temperature and decreased speed. The simulations also showed lower 

strains with higher temperatures, increased wheel speed, and a stiffer asphalt concrete material. 

8.2. FUTURE WORK 

The following items are recommended for future research: 

• Modeling multiaxial VEPCD: Sensitivity analyses considering true in-field conditions 

should be carried out for both tensile and compressive loading to determine the influence 

of simplifications to and the need for the complete multiaxial formulation. If complete 

multiaxial modeling does not substantially change the predicted performance of the 

asphalt concrete, then less testing is needed for accurate pavement performance modeling. 

• Further Simplifying VECD model development: Although constant rate tests are simple to 

perform, the capacity of the recently developed AMPT equipment limits the use of such 
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tests. A simplified model further specialized for more moderate load amplitude cyclic 

tests and derived rigorously will expand the application of the VECD model for further 

routine testing and analysis. 

• Hardening in the MVECD compression model: A possible drawback to the current 

MVECD model for compression is the lack of a function to account for hardening due to 

aggregate interlock. It is known that such a function becomes important, especially at low 

confinement levels and high temperatures, but the implication of aggregate interlocking 

under more moderate conditions is unknown. 

• Yielding criteria in the VP compression model: In this research, the model parameters 

were determined for each specific confining pressure condition. Although these 

parameters are not sufficient to explain the general behavior of HMA pavements, which 

experience an almost uniform confining pressure for certain time periods, they are, 

nonetheless, reasonable for introducing a yield criterion that can account for the entire 

confining pressure. 

• Hardening-softening of the function in the VP compression model: Even though the 

current hardening-softening function reasonably accounts for the rate-dependent 

softening of HMA, some over and underpredictions were observed for certain stress 

histories. These observations may be explained by an inaccuracy in the hardeningsoftening function, and therefore, more research into the hardening-softening function is 

required. 

• Refining analysis program: Current VEPCD modeling is aided by a software package 

developed at NCSU. The program is currently in the development stage and is considered 

to be research-grade software. Usability, therefore, is somewhat limited. Refinement of 

this program into a complete black box package will greatly aid the implementation of 

VEPCD modeling at the industrial level. 

• Advancing material models with FEP++: Currently the 3D FEP++ code includes 

modules for linear elastic and linear viscoelastic materials. The MVECD model should be 

implemented along with other material models and functions. To model unbound paving 

materials accurately, a module should be added to include a nonlinear elastic material 

model. In addition, models that account for the effects of temperature and moisture 

gradients in pavement systems and the effects of aging of the HMA layer on pavement 

performance need to be incorporated into the FEP++. Additionally, anisotropy of the 

HMA and slip elements at the pavement-base interface should be included. The modular 

design of the 3D FEP++ allows for incorporation of new modules that pertain to 

advanced materials, more efficient solution techniques, and fast solvers with little or no 

change to the existing code. 

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APPENDIX. USER MANUAL FOR FEP++ 

A.1. INSTALLATION OVERVIEW 

FEP++ can be installed using the supplied windows installer. The installer file is named 

“Setup.exe.” Executing the binary launches the installation wizard, which then guides through 

the rest of the installation procedure. Once the installation is completed successfully, an entry for 

FEP++ should be created in the Start Menu of Windows®. FEP++ can be launched by choosing 

the program from the Start Menu. 

The following section describes the data entry windows in FEP++ and provides information on 

their functionality and the type of input they accept. 

A.2. LIST OF INPUT DIALOGS 

A.2.1. General Information Dialog 

Figure 211 shows a screenshot of the General Information dialog. This is used to input 

information regarding the model and the type of analysis. This information is used by the 

preprocessor to customize the interface presented. 
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Figure 211. Screen capture. General Information dialog. 

A short explanation of the entry fields is given below. 

A.2.1.1. Data Entry Fields 

Title, Description: This information is used in creating report/summary files and does not affect 

the analysis in anyway. 

Model Type and Analysis Type: These data describe the type of model being solved. Depending 

on the model type, the user interface and analysis parameters are customized. Currently, 3D 

analysis, plane strain, and axisymmetric are supported. 

A.2.2. Material Properties Dialog 

Figure 212 shows a screenshot of the Material Properties dialog, which is used to configure the 

properties of materials used in the analysis. Currently, only elastic and viscoelastic materials are 

supported. 
225

Figure 212. Screen capture. Material Properties dialog. 

A short explanation of the entry fields is given below. 

A.2.2.1. Data Entry Fields 

Material Name: This is a string identification used to uniquely identify a material. This is both 

an entry and a control. On choosing an existing identification, the dialog is populated with the 

properties of the material corresponding to the identification; however, entering a new value 

creates a new material. 

Material Type: This field indicates the type of material. If an existing material is chosen, then 

this field is automatically populated with the material type. On creating a new material, the user 

should choose the material type to continue configuring the specific type of material. Currently 

supported types are elastic and viscoelastic. 

The other fields correspond to parameters depending on the type of material and are discussed in 

subsequent sections. 
226

A.2.3. Elastic Material Properties Dialog 

Figure 213 shows a screenshot of the elastic material properties dialog. 

Figure 213. Screen capture. Elastic material properties dialog. 

A short explanation of the entry fields is given below. 

A.2.3.1. Data Entry Fields 

Material Properties Block: These fields are used to input the material properties like density, 

Poisson’s ratio, and elastic modulus. 
227

Thermal Stresses Block: These fields describe the coefficient of thermal expansion of the 

material. If thermal stresses are to be ignored, then the Include box should be cleared. 

A.2.4. Viscoelastic Material Properties Dialog 

Figure 214 shows a screenshot of the viscoelastic material properties dialog. 

Figure 214. Screen capture. Viscoelastic material properties dialog. 

A short explanation of the entry fields is given below. 

A.2.4.1. Data Entry Fields 

Material Properties Block: These fields are used to input the material properties like density and 

Poisson’s ratio. 
228

Prony Series: This field is used to input the Prony series to be used for characterizing the 

relaxation modulus of the material. Existing Prony series can be chosen by selecting from the 

dropdown list. New Prony series can be added by clicking the View/Edit button to invoke the 

Prony Series dialog. Section A.2.5 describes the configuration of fields on the Prony Series 

dialog. 

Shift Factors Block: These parameters are used along with the Prony coefficients to adjust the 

material properties for the temperature effects. These parameters are used to find the timetemperature shift factor. 

Thermal Stresses Block: These fields describe the coefficient of thermal expansion of the 

material. If thermal stresses are to be ignored, then the Include box should be cleared. 

Damage Parameters Block: These are the parameters for characterizing the damage law as 

discussed in subsection 2.2.4. If damage effects are to be ignored, then the Damage checkbox 

should be cleared. 

A.2.4.2. Control Actions 

View/Edit: This control invokes a Prony series dialog that can be used to add new series or edit 

an existing series. On successful completion, the identification of the selected/configured series 

is displayed in the Prony series dropdown list. 

Delete: This control is active only when an existing material has been chosen. It can be used to 

delete the material from the database. 

Apply: This control performs validations on the modified/input data, and if no errors are found, it 

saves the data to the database; however, if the validation fails, an appropriate error message is 

raised, and the data are not saved to the database. This does not close the dialog on successful 

completion and can be used to make multiple changes without having to open/close the dialog 

multiple times. 

OK: This is similar to the Apply control in functionality; however, on successful completion, this 

control saves the data and closes the dialog. 

 

Cancel: This control discards any changes made to the data and closes the dialog. 

A.2.5. Prony Coefficients Dialog 

Figure 215 shows a screenshot of the Prony Coefficients dialog. This dialog can be used to 

Add/Edit the Prony series. 
229

Figure 215. Screen capture. Prony Coefficients dialog. 

A short explanation of the entry fields is given below. 

A.2.5.1. Data Entry Fields 

Series Name: This field is a string identification used to uniquely identify the Prony series. 

A.2.5.2. Control Actions 

Import Data: This control is used to import the Prony coefficients from a text file. Each record in 

the file is expected to be in the format, “E_i Tai_i”, that is, space separated values of Prony 

coefficients and relaxation times. 

 

Insert Before: This control is used to edit the Prony coefficients table. It adds a new blank row 

above the current selected row. 

Insert After: This control is used to edit the Prony coefficients table. It adds a new blank row 

below the current selected row. 

OK: This control performs validations on the modified data, and if no errors are found, it saves 

the data and closes the dialog. If validation fails, an appropriate error message is raised. 
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Cancel: This control discards any changes made to the data and closes the dialog. 

A.2.6. Layer Properties Dialog 

Figure 216 shows a screenshot of the Layer properties dialog. This dialog is customized 

depending on the analysis type (2D or 3D). This dialog also supports the configuration of the 

subgrade as a half-space. 

Figure 216. Screen capture. Layer properties dialog. 

A short explanation of the entry fields is given below. 

A.2.6.1. Data Entry Fields 

The fields for this dialog are represented in the form of a table. Each row of the table represents a 

layer of the pavement. The column headings specify the property being configured. The columns 

of the table are described as follows: 

Pavement Width (cm): This shows the width of the pavement layer. This can be configured only 

for a 3D analysis. 
231

Pavement Length (cm): This shows length of the pavement section to be simulated. This can be 

configured only for a 3D analysis. 

Layer: This shows a unique string identifying the pavement layer. 

Type: This specifies the type of pavement layer. The valid options are AC, Base, SubBase, 

SubGrade, and InfiniteSubgrade. AC refers to asphalt concrete layer, and InfiniteSubgrade refers 

to a subgrade that is treated as a half-space. This field is not editable. 

Material: This shows the type of material to be associated with the layer. The material can be 

chosen from the material database that was configured using the Material dialog (section A.2.2). 

Thickness: This shows the thickness of the pavement layer. This must be configured for all the 

layers except if the type of the layer is InfiniteSubgrade. In this case, special elements are used to 

treat the subgrade as a half-space. 

Temperature: This shows the mean temperature in the pavement layer. This is used in adjusting 

the viscoelastic material properties for the temperature effects. 

A.2.6.2. Control Actions 

Insert Before: This control adds a pavement layer above the selected pavement layer. 

Insert After: This control adds a pavement layer below the selected pavement layer. 

DeleteRow: This control deletes the selected pavement layer. 

ClearRow: This control clears the selected pavement layer. 

OK: This control performs validations on the modified data. If no errors are found, it saves the 

data and closes the dialog. If validation fails, an appropriate error message is raised. 

Cancel: This control discards any changes made to the data and closes the dialog. 

A.2.7. Mesh Parameters Dialog 

Figure 217 shows a screenshot of the Mesh Parameters dialog. This dialog is used to configure 

the mesh discretization parameters for the finite element analysis. It is customized depending on 

ModelType and AnalysisType that were input in the General Information dialog. 

The model is divided into the following two zones: 

Zone-I: The region corresponding to the actual pavement is called Zone-I. The length and width 

discretization is common for all the layers, but the thickness discretization can be configured 

independently for each layer. A uniform discretization is used for each layer. 
232

Zone-II: Zone-II corresponds to the boundary of the pavements and is modeled using infinite 

elements to simulate the effect of a half-space (semi-infinite medium). A geometric series based 

discretization is used for each region. 

Figure 217. Screen capture. Mesh properties dialog. 

A short explanation of the entry fields is given below. 

A.2.7.1. Data Entry Fields (Mesh Parameters Block) 

Zone-I Length Ratio: This parameter corresponds to the length of the actual pavement. The 

length is calculated as (Zone-I Length Ratio) multiplied by (Load Length). 

Zone-I Length Divisions: This specifies the number of divisions in Zone-I along the length of the 

pavement. 

Zone-I Width Ratio (3D analysis): This parameter corresponds to the width of the actual 

pavement. The width is calculated as (Zone-I Width Ratio) multiplied by (Load Width). 
233

Zone-I Width Divisions (3D analysis): This corresponds to the number of divisions in Zone-I 

along the width of the pavement. 

Zone-II Mesh Ratio: This specifies the geometric ratio used for creating the geometric series 

discretization in Zone-II. 

Zone-II Init Element Ratio: This parameter corresponds to the initial element size in the 

geometric series discretization of Zone-II. The initial element size along direction P is calculated 

as (Zone-II Element Ratio) multiplied by min(element size along perpendicular direction to P). 

Zone-II Num Elements: This parameter specifies the number of elements in Zone-II along any 

direction. 

A.2.7.2. Data Entry Fields (Layer Info Block) 

 The fields Layer, Type, and Thickness are not editable and are taken from the data used to 

configure the Layer properties. 

Divisions: This specifies the number of divisions for the discretization of the pavement layer 

along the thickness. If the layer is Infinite Subgrade, then this field is also not editable, and the 

data input from the Zone-II configuration is used. 

A.2.7.3. Control Actions 

OK: This control performs any validations on the modified data, and if no errors are found, it 

saves the data and closes the dialog. If validation fails, an appropriate error message is raised. 

Cancel: This control discards any changes made to the data and closes the dialog. 

A.2.8. Load Properties Dialog 

Figure 218 shows a screenshot of the Load properties dialog. This dialog is used to configure the 

loading conditions for the model. This interface is customized depending on ModelType and 

AnalysisType fields that were configured using the General Information dialog. In these 

configurations, the loading area is assumed to be rectangular, and the pressure distributions are 

assumed to be uniform in the loading area. 
234

Figure 218. Screen capture. Load properties dialog. 

A short explanation of the entry fields is given below. 

A.2.8.1. Data Entry Fields (Coordinates (in cm) Block) 

X: This shows the x coordinate of the load. 

Y: This shows the y coordinate of the load. 

Z: This shows the z coordinate of the load. 

A.2.8.2. Data Entry Fields (Magnitude and Dimensions Block) 

Load Magnitude: This parameter corresponds to the load magnitude. The user can choose to 

specify either the Total Load or Tire Pressure by choosing the appropriate type in the dropdown 

box. In the screen capture shown in figure 218 the Total Load option is selected. 

Load Length: This parameter specifies the length of the loading area. For the pavement problem, 

this is the length of the tire. 

Load Width: This parameter specifies the width of the loading area. For the pavement problem, 

this is the width of the tire. 
235

A.2.8.3. Data Entry Fields (Moving Load Data Block) 

Velocity: This parameter specifies the velocity at which the loading area moves. For the 

pavement model, this corresponds to the vehicle speed. 

A.2.8.4. Control Actions 

OK: This control performs validations on the modified data, and if no errors are found, it saves 

the data and closes the dialog. If validation fails, an appropriate error message is raised. 

Cancel: This control discards any changes made to the data and closes the dialog. 

A.2.9. Analysis Parameters Dialog 

Figure 219 shows a screenshot of the Analysis Parameters dialog, which is used to configure the 

parameters for the solution algorithm used in the finite element analysis. 
236

Figure 219. Screen capture. Analysis Parameters dialog. 

A short explanation of the entry fields is given below. 

A.2.9.1. Data Entry Fields 

Penalty Number: This parameter corresponds to the Penalty Number used to model stiffness of 

the rigid members in finite element analysis. 

Approximation Scheme Block: The parameters in this block correspond to the parameters used in 

the Newmark Method approximation scheme. 
237

Iteration Scheme Block: The parameters in this block correspond to the standard parameters for 

an iterative solution scheme. The analysis engine currently supports only the Newton Iterations. 

Prony Configuration Block: Maximum Relaxation Time: This specifies an upper bound on the 

Prony coefficients configured for the AC Layers. 

Prony Configuration Block: Minimum Relaxation Time Factor: This specifies a lower bound on 

the Prony coefficients configured for the AC Layers. The lower bound is calculated as 

Min Relaxation Time = (time step size)/(Minimum Relaxation Time Factor). 

Simulation Info Block: Simulation Period: This specifies the total simulation time for the 

analysis. 

Simulation Info Block: Num Time Steps: This specifies the number of timesteps that will be used 

in the simulation and hence fixes the timestep size. 

A.2.9.2. Control Actions 

OK: This control performs validations on the modified data, and if no errors are found, it saves 

the data and closes the dialog. If validation fails, an appropriate error message is raised. 

Cancel: This control discards any changes made to the data and closes the dialog. 

A.2.10. Summary Dialog 

Figure 220 shows a screenshot of the Summary dialog, which is used to view a user readable 

report of the input data. 
238

Figure 220. Screen capture. Summary dialog. 

A short explanation of the entry fields is given below. 

A.2.10.1. Control Actions 

OK: This control closes the dialog. 

A.2.11. Generating an Input File 

The user can generate an input file for FEP++ by choosing the Generate Input File item in the 

Control Panel. The user is prompted to specify a location for saving the generated file, and the 

file is created at that location. This can be used to configure multiple analyses and to store the 

generated input files that can then be run in a batch process. 

A.2.12. Running Analysis 

Figure 221 shows a screenshot of the view when an analysis is being run. The finite element 

analysis can be started by clicking the Run Analysis item in the Control Panel. When a user starts 

a new analysis, it is launched in a separate command window that displays the output from 
239

FEP++. Once the analysis is completed, the window is closed, and the results are gathered and 

stored at the location of the current “db” file. 

Figure 221. Screen capture. FEP++ analysis in progress. 

241

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