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Modeling of Asphalt Concrete

About the Editor Y. Richard Kim, Ph.D., P.E., is a professor in the Department of Civil, Construction, and Environmental Engineering at North Carolina State University in Raleigh, North Carolina.

Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.

Modeling of Asphalt Concrete Y. Richard Kim, Ph.D., P.E.

Editor

Professor, North Carolina State University

New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

Copyright © 2009 by the American Society of Civil Engineers. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-159651-8 The material in this eBook also appears in the print version of this title: 0-07-146462-X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at george_hoare@ mcgraw-hill.com or (212) 904-4069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. DOI: 10.1036/007146462X

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To the glory of God For my better half, Jee Hye, and my children, Frances, Daniel, and Christie

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Contents Contributors 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Modeling of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future of Asphalt Concrete Modeling .......................... Pavement Response Model versus Performance Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiscale Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Virtual Testing of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . Organization Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 1—Asphalt Binder Rheology . . . . . . . . . . . . . . . . . . . . . . . . . Part 2—Stiffness Characterization . . . . . . . . . . . . . . . . . . . . . . . . . Part 3—Constitutive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 4—Models for Rutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 5—Models for Fatigue Cracking and Moisture Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 6—Models for Low-Temperature Cracking . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 3 4 5 5 5 6 6 6 7 7 7 7

Part 1 Asphalt Rheology 2

Modeling of Asphalt Binder Rheology and Its Application to Modiﬁed Binders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Critical Properties of Asphalt Binders . . . . . . . . . . . . . . . . . . Traditional Rheological Properties . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Susceptibility Parameters . . . . . . . . . . . . . . . . . . . . . . . . . Linear Viscoelastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Viscoelastic Nature of Asphalt Binders . . . . . . . . . . . . . . . . . . . . . . Asphalt Viscoelastic Properties and Pavement Performance . . . . . . . Modeling of the Viscoelastic Properties of Asphalts . . . . . . . . . . . . . . . The Asphalt Viscoelastic and Failure Properties Selected in the SHRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Need for Asphalt Modiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Modiﬁcation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Modiﬁers Currently Used . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 12 16 19 20 22 23 25 29 29 30

vii

viii

Contents Critical Properties of Modiﬁed Asphalts . . . . . . . . . . . . . . . . . . . . . . . . Effects of Modiﬁcation on Viscoelastic Properties . . . . . . . . . . . . Effects of Modiﬁcation on Failure Properties . . . . . . . . . . . . . . . . Effects of Modiﬁcation on the Superpave Grading Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complexity of Modiﬁed Binders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Assumptions in the Superpave Binder System . . . . . . . . . . . New Classiﬁcation of Asphalt Binders . . . . . . . . . . . . . . . . . . . . . Damage Resistance Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of New Tests for Binder Damage Behaviors . . . . . . . . . Rutting Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binder Fatigue Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of New Damage Behavior Parameters . . . . . . . . . . . . . . . . . . Binder Rutting Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binder Fatigue Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Findings for the Damage Resistance Parameters . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 34 36 38 38 41 42 45 45 48 50 50 52 55 55 56 56

Part 2 Stiffness Characterization 3

Comprehensive Overview of the Stiffness Characterization of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Concrete Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Measuring Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Location of Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of the Test Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate, Temperature, and Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Importance of Asphalt Concrete Stiffness . . . . . . . . . . . . . . . . . . . . . . . . Use of Stiffness in Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tests to Determine Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors on Which Asphalt Concrete Stiffness Depends . . . . . . . . . . . . Temperature and Rate of Loading . . . . . . . . . . . . . . . . . . . . . . . . . Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregate Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Binder Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air Voids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 65 66 66 66 66 66 67 67 67 68 68 68 69 70 70 70 74 74 77 77 77

Contents Characterization of Asphalt Concrete Stiffness . . . . . . . . . . . . . . . . . . . Undamaged Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Microcracks on Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78 78 80 87 87

4

Complex Modulus Characterization of Asphalt Concrete . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressive Axial versus SST-Shear Modulus . . . . . . . . . . . . . . Stiffness as the Asphalt Mix Performance Indicator . . . . . . . . . . Analysis of Cyclic Sinusoidal Test Data . . . . . . . . . . . . . . . . . . . . Mastercurve Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 90 90 92 96 98 101 111 116 116 117 117

5

Complex Modulus from the Indirect Tension Test . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Viscoelastic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Modulus Testing of HMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specimen Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Dynamic Moduli Values . . . . . . . . . . . . . . . . . . . . . . . . . Graphical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Phase Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 121 122 122 126 126 128 128 131 131 131 134 135 135 137 137

6

Interrelationships among Asphalt Concrete Stiffnesses . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of LVE Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creep Compliance and Relaxation Modulus . . . . . . . . . . . . . . . . Complex Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of LVE Response Functions . . . . . . . . . . . . . . . . . . . . . . Analytical Representation of LVE Response Functions . . . . . . . . . . . . Power-Law Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pure Power Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 139 140 140 141 142 142 143 143

ix

x

Contents Generalized Power Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modiﬁed Power Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prony Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting of Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Presmoothing Experimental Data Prior to Prony Series Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interconversion between LVE Response Functions . . . . . . . . . . . . . . . . Approximate Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . Approximate Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 144 144 145 145 146 148 149 155 157

Part 3 Constitutive Models 7

VEPCD Modeling of Asphalt Concrete with Growing Damage . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TTS with Growing Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The VECD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain-Hardening Viscoplastic Model . . . . . . . . . . . . . . . . . . . . . . VEPCD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration of the VEPCD Model in Tension . . . . . . . . . . . . . . . . Validation of the VEPCD Model in Tension . . . . . . . . . . . . . . . . . . . . . . TSRST Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VEPCD Model in Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Veriﬁcation of the VEPCD Model in Compression . . . . . . . . . . . . . . . . Finite Element Implementation of the VEPCD Model . . . . . . . . . . . . . Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 163 164 165 167 179 180 180 186 187 191 194 198 200 200 200

8

Uniﬁed Disturbed State Constitutive Modeling of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approaches for Pavement Analysis and Design . . . . . . . . . . . . . Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review of RM Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniﬁed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors in Mechanistic Uniﬁed Model . . . . . . . . . . . . . . . . . . . . . . Disturbed State Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbed State Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 206 206 206 209 209 210 212 213 213 213 214

Contents Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capabilities and Hierarchical Options . . . . . . . . . . . . . . . . . . . . . . . . . . Plasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creep Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Healing or Stiffening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interfaces and Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation for Laboratory Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation for Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . Computer Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validations and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reﬂection Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniﬁed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

DBN Law for the Thermo-Visco-Elasto-Plastic Behavior of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solicitations in Bituminous Roadways . . . . . . . . . . . . . . . . . . . . . . . . . . General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trafﬁc Inﬂuence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Inﬂuence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavior of Bituminous Materials in Pavement Structures . . . . Presentation of the DBN Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of the EPi Body Behavior . . . . . . . . . . . . . . . . . . . . . . Description of the Vi Body Behavior . . . . . . . . . . . . . . . . . . . . . . . Brittle Behavior of the Spring of Rigidity, E0 . . . . . . . . . . . . . . . . . How the Same Formalism Can Be Adapted to Describe the Different Typical Kinds of Mix Behavior . . . . . . . . . . . . . . . . . . . . Current Developments for the DBN Law . . . . . . . . . . . . . . . . . . . . . . . . Small Strains and Small Number of Cycles: Linear Viscoelastic (LVE) Behavior . . . . . . . . . . . . . . . . . . . . . . . Large Strains and Small Number of Cycles: Nonlinearity and Viscoplastic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Strains at Low Temperatures: Brittle Failure . . . . . . . . . . .

215 215 218 220 220 221 222 222 222 223 223 224 225 225 226 227 229 229 231 236 238 241 241 241 245 245 245 246 246 246 247 248 249 250 251 251 252 252 252 254 257

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Contents Developments in Progress for the DBN Law . . . . . . . . . . . . . . . . . . . . . Small Strains and Great Number of Cycles: Fatigue Damage Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic Loading from Stress Control Tests: Accumulation of Permanent Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thixotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Healing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Dimensional Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

258 258 259 260 261 261 262 263

Part 4 Models for Rutting 10

11

Rutting Characterization of Asphalt Concrete Using Simple Shear Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanics of Permanent Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . Volume Change versus Shape Distortion . . . . . . . . . . . . . . . . . . . Representative Volume Element and Laboratory Test Specimen Size Considerations . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Response Characteristics of Asphalt Aggregate Mixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Laboratory Simple Shear Test to Characterize Permanent Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Equipment and Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Variability and Reliability Considerations . . . . . . . . . . . . . . Test Specimen Size and Preparation Considerations . . . . . . . . . . Mix Design and Analysis, Performance Evaluation . . . . . . . . . . . . . . . Mix Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Warrantied Pavement, I-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Long Life Pavement Rehabilitation, I-710 . . . . . . . . . . . . . . . . . . . Mix Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of Mix Design Criteria for Taxiway Subjected to Heavy Aircraft Loading . . . . . . . . . . . . . . . . . . . . . Creep versus Repeated Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recursive Rut Depth Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . Constitutive Relationship Formulation for HMA Behavior at Elevated Temperatures . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permanent Deformation Assessment for Asphalt Concrete Pavement and Mixture Design . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269 269 269 270 271 275 279 280 280 285 286 288 288 292 293 295 296 299 301 308 311 313 317 317 317

Contents Mechanistic-Empirical Rutting Models . . . . . . . . . . . . . . . . . . . . . . . . . . Subgrade Rutting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permanent Strain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permanent to Resilient Strain Ratio Models . . . . . . . . . . . . . . . . . Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Constitutive Models for Rutting . . . . . . . . . . . . . . . . . . . . . . Linear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuum Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Performance Test for Mixture Rutting . . . . . . . . . . . . . . . . . . . . Static Creep Permanent Deformation Test . . . . . . . . . . . . . . . . . . Repeated Load Permanent Deformation Test . . . . . . . . . . . . . . . . Testing Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mix Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

318 318 319 322 325 326 326 327 328 332 336 338 339 340 342 342 344 346 347 347

Part 5 Models for Fatigue Cracking and Moisture Damage 12

Micromechanics Modeling of Performance of Asphalt Concrete Based on Surface Energy . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schapery’s Fundamental Law of Viscoelastic Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adhesion and Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Micromechanics Modeling of Fatigue and Healing of Asphalt Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture Damage Prediction Model . . . . . . . . . . . . . . . . . . . . . . . Measurement of Surface Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Energy of Asphalt Binder . . . . . . . . . . . . . . . . . . . . . . . . . Surface Energy of Aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation of the Fatigue and Healing Models . . . . . . . . . . . . . . . . . . . Cohesive and Adhesive Bond Energies . . . . . . . . . . . . . . . . . . . . . Cohesive Fatigue and Healing . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adhesive Fatigue and Healing . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture Damage Modeling and Its Mechanistic Validation . . . . . . . Stripping Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accelerated Moisture Damage Testing on Asphalt-Aggregate Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . .

355 355 355 357 357 357 358 359 363 366 366 372 377 377 378 380 382 384 385

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Contents

13

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387 388 389

Field Evaluation of Moisture Damage in Asphalt Concrete . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanisms of Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Reaction and Molecular Orientation Theory . . . . . . . Surface Energy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Field Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visual Inspection of Split Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . Image Analysis of Split Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strength Measurements of Cores . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

391 391 391 392 392 392 393 393 394 394 395 395 395 396 401

Part 6 Models for Low-Temperature Cracking 14

Prediction of Thermal Cracking with TCMODEL . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Cracking Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . Previous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pavement Response Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Governing Constitutive Equation for Stress Prediction and Relaxation Modulus . . . . . . . . . . . . . . . . . . . . . . Viscoelastic Interconversion: Creep Compliance to Relaxation Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Methods for Applying Hereditary Integral: Hourly Stress Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pavement Distress Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Intensity Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack Growth Prediction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . Probabilistic Crack Amount Model . . . . . . . . . . . . . . . . . . . . . . . . Model Calibration and Sample Output . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration Methods and Results . . . . . . . . . . . . . . . . . . . . . . . . . . Model Changes Implemented under NCHRP 1-37A . . . . . . . . . . . . . . Use of Three Input Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modiﬁed Calibration Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

405 405 405 406 407 408 408 411 411 413 413 414 414 415 416 419 419 419 419 421 422

Contents Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

424 425

Low-Temperature Fracture in Asphalt Binders, Mastics, and Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Stress Buildup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymers and the Thermal Stress Restrained Specimen Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interface Studies in Asphalt Systems . . . . . . . . . . . . . . . . . . . . . . . Low-Temperature Failure in Asphalt Binders and Mastics . . . . . . . . . Toughness, Brittleness, and Ductility . . . . . . . . . . . . . . . . . . . . . . . Brittle-to-Ductile Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toughening Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enhanced Yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toughness in the Brittle State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Temperature Fracture in Asphalt Mixtures . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

430 431 431 432 433 435 436 443 446 448 449 449

Index

453

......................................................

429 429 429 430

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Contributors Professor, Department of Civil and Environmental Engineering, University of Wisconsin, Madison, Wisconsin (Chap. 2)

Hussain U. Bahia

Amit Bhasin Associate Research Scientist, Texas Transportation Institute, Texas A&M University, College Station, Texas (Chap. 12)

William G. Buttlar Associate Professor, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois (Chap. 14) Assistant Professor, Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, Pennsylvania (Chaps. 6, 7)

Ghassan R. Chehab

Assistant Professor, Department of Civil Engineering, University of New Hampshire, Durham, New Hampshire (Chap. 7)

Jo S. Daniel

Regents’ Professor, Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, Arizona (Chap. 8)

Chandrakant S. Desai

Hervé Di Benedetto Professor, Département Génie Civil et Bâtiment, URA CNRS 1652, Ecole Nationale des TPE, Vaulx-en-Velin, France (Chap. 9) John T. Harvey Professor, Department of Civil and Environmental Engineering, University of California, Davis, California (Chap. 10) Associate Professor, Department of Chemistry, Queens University, Kingston, Ontario, Canada (Chap. 15)

Simon A. M. Hesp

Associate Professor, Department of Civil and Coastal Engineering, University of Florida, Gainesville, Florida (Chap. 14)

Dennis R. Hiltunen

Kamil E. Kaloush Assistant Professor, Department of Civil and Environmental Engineering, Arizona State University, Tempe, Arizona (Chap. 11) Y. Richard Kim Professor, Transportation Systems and Materials, Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, North Carolina (Chaps. 1, 5, 6, 7) Department of Civil & Environmental Engineering, Sejong University, Kwangjin Ku, Seoul, Republic of Korea (Chap. 7)

H. J. Lee

Dallas N. Little E. B. Snead Chair Professor, Department of Civil Engineering, Texas A&M University, College Station, Texas (Chap. 12)

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Contributors Robert L. Lytton Benson Chair Professor, Department of Civil Engineering, Texas A&M University, College Station, Texas (Chaps. 3, 12)

G. W. Maupin, Jr. Principal Research Scientist, Virginia Transportation Research Council, Charlottesville, Virginia (Chap. 13)

Mostafa Momen North Carolina Department of Transportation, North Carolina State University, Raleigh, North Carolina (Chap. 5)

Carl L. Monismith Robert Horonjeff Professor of Civil Engineering, Institute for Transportation Studies, University of California, Berkeley, California (Chap. 10)

François Olard Research and Development Direction, EIFFAGE Travaux Publics, Corbas, France (Chap. 9) Terhi K. Pellinen Professor, Highway Engineering, Department of Civil and Environmental Engineering, Helsinki University of Technology, Espoo, Finland (Chap. 4)

Reynaldo Roque Professor, Department of Civil and Coastal Engineering, University of Florida, Gainesville, Florida (Chap. 14)

Charles W. Schwartz Associate Professor, Department of Civil and Environmental Engineering, University of Maryland, College Park, Maryland (Chap. 11) Senior Researcher, Highway & Transportation Technology Institute, Hwaseong, Gyeonggi, South Korea (Chap. 5)

Youngguk Seo

Shane Underwood Research Scientist, Department of Civil, Construction, and Environmental Engineering , North Carolina State University, Raleigh, North Carolina (Chap. 7) Shmuel L. Weissman

President and CEO, Symplectic Engineering Corporation, Berkeley,

California (Chap. 10)

T. Y. Yun Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, North Carolina (Chap. 7)

Modeling of Asphalt Concrete

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CHAPTER

1

Modeling of Asphalt Concrete Y. Richard Kim

Introduction 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 varying 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. It is performance models of asphalt concrete that provide the links among various processes involved in asphalt mixture design, pavement design, construction, and rehabilitation. Various factors affect the deformation behavior and performance of asphalt concrete, including time (i.e., rate of loading, loading time, rest period), temperature, stress state, mode of loading, aging, and moisture. Models have been developed to capture the effects of these factors on asphalt concrete performance. Most of these models, developed prior to the Strategic Highway Research Program (SHRP), are empirical in nature. The primary reason for the empirical nature of these models is the lack of computing power necessary to calculate the long-term performance of asphalt concrete and, therefore, asphalt pavements. The SHRP recognized the importance of mechanistic models for material specifications, mixture design, and pavement design and developed a range of research products based on the principles of mechanics. The paradigm shift from empiricism to mechanics during the SHRP made a significant impact on the role of models in asphalt pavement engineering. Development of a fundamentally sound performance model serves two important purposes. For pavement engineers, such a model can provide accurate information about the performance of asphalt concrete under realistic loading conditions, thus leading to a better assessment of the service life of a new pavement or the remaining life of an existing pavement. For materials engineers, the performance model founded on

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Chapter One basic principles of mechanics provides relationships between material properties (chemical or mechanical) and model parameters, which can be used for the selection or design of better performing binders or mixtures.

Performance Characteristics Performance of asphalt concrete can be categorized into two major types of distress: cracking and permanent deformation. Cracking of asphalt concrete can be caused by mechanical loading from repetitive traffic and/or thermal loading from changes in temperature. When the asphalt concrete is subjected to repeated loading, whether it is mechanical or thermal, distributed microstructural damage occurs primarily in the form of microcracks. A microscopic video image of the cracking area in asphalt concrete is presented in Fig. 1-1 to display the formation of micro- and macrocracks under tensile stress. As shown in this figure, microcracks exist ahead of the macrocrack tip, forming a so-called damage zone. Propagation, coalescence, and rebonding of these microcracks in the damage zone affect the macrocrack growth and healing and, thus, the fatigue behavior of asphalt concrete. That is to say, the modeling of the fatigue behavior of asphalt concrete requires an evaluation of the effects of both micro- and macrocracks and their interaction on the global behavior of the mixture. At high temperatures and/or slow loading rates, 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 caused by shear flow. The degree of aggregate interlocking and anisotropy in asphalt concrete caused by aggregate orientation under compaction become important factors in the accurate prediction of the permanent deformation behavior of asphalt concrete.

FIGURE 1-1 Microscopic surface image of cracking area in asphalt concrete. (Kim et al. 1997, with permission from International Society for Asphalt Pavements.)

Modeling of Asphalt Concrete

Future of Asphalt Concrete Modeling The modeling of asphalt concrete is an evolving subject. Continuing developments and improvements in computational power and test techniques will allow asphalt materials and pavement engineers to use more realistic, powerful models to predict the performance of asphalt materials and pavements. The following subsections attempt to shed some light on the possibilities for future models of asphalt concrete.

Pavement Response Model versus Performance Model A traditional approach to asphalt pavement performance prediction is divided into two steps: pavement response prediction and pavement performance prediction. In this approach, responses of an undamaged pavement (e.g., tensile strain at the bottom of the asphalt layer) are estimated from a structural model (e.g., the multilayered elastic theory) using initial, undamaged properties of the layer materials. Asphalt concrete performance models are developed using laboratory test results and relate the initial response of the asphalt concrete specimens to the life of those specimens. The responses estimated from the structural model are then input to the performance model to determine the life of the pavement. This approach is the state-of-the-practice method that is adopted in most recent mechanistic-empirical pavement design methods, including the Mechanistic-Empirical Pavement Design Guide (MEPDG) developed under the NCHRP project 1-37A (2004). These models are simple to use because the only measured response of the mixture is at the initial stage of fatigue testing. Such models deserve credit for the basic foundation of current mechanistic-empirical pavement designs. However, there are several weaknesses in this traditional approach. First, damage evolution in complex structures and modified materials may not be captured accurately. For example, complex combinations of layer material types and thicknesses in perpetual pavements make it more difficult to accurately predict the failure mechanisms using conventional hot mix asphalt (HMA) performance prediction models and pavement response models. Secondly, most performance models used in the two-step approach are mode-ofloading dependent. These models are developed using results obtained from laboratory tests, which are conducted either in controlled stress mode or in controlled strain mode. Currently available two-step approaches do not have the ability to discern the mode of loading in a mechanistic manner and, therefore, could result in an unreliable performance prediction. Thirdly, the laboratory test methods used in the traditional two-step approach are designed to simulate the boundary conditions of pavement structures rather than to define the material’s constitutive behavior in the representative volume element (RVE). Often these laboratory test methods predict the performance under only some selected pavement conditions. So, because the test methods simulate the pavement boundary conditions rather than capture the behavior of the RVE, the number of tests needed to cover the wide range of pavement conditions that are expected in the field is undesirably large. The weaknesses of the two-step approach can be overcome using a mechanistic approach that combines HMA material models and the pavement response model. In this approach, the material model describes the stress-strain behavior of the material in the RVE. The material model is then implemented into the pavement response model where boundary conditions of the pavement structure in question are applied. This

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Chapter One approach allows the accurate evaluation of the effects of changes in layer stiffnesses due to damage growth on pavement performance. Prediction of multiple performance characteristics and their interactions is possible in a realistic manner, although the material models in both tension and compression are needed. The lack of computing power needed to calculate damage evolution for the entire life of the pavement forced earlier researchers to develop the two-step approach to pavement performance prediction, as opposed to the more realistic one-step integrated approach. However, improvements in computing power and numerical techniques now allow modelers to implement more powerful material models into the pavement response model and to predict the pavement performance directly from the integrated model.

Multiscale Model Two general approaches in mechanics can be used for modeling the changes in the stressstrain behavior of asphalt concrete: a micromechanical approach and a continuum approach. In the micromechanical approach, defects that constitute the damage are described by microscopic geometrical parameters, such as microcrack size, orientation, and density. These parameters are evaluated through an appropriate microstructural evolution law, such as the microcrack growth law. Mechanics is then applied typically on an idealized RVE to determine the effects of the distribution of microdefects on the macroscopic constitutive parameters, such as the effective stiffness of the damaged body. Such analyses are, in general, difficult to perform because of the intrinsic complexity of the microstructure and the micromechanisms and also due to the interactions among the defects. Therefore, without proper simplifications and assumptions both in modeling and analysis, the micromechanical approach may fail to provide realistic information about the macroscopic constitutive framework for modeling the progressive degradation of the mechanical properties of solids (Park et al. 1996). On the other hand, in the continuum approach, or so-called continuum damage mechanics, the damaged body is represented as a homogeneous continuum on a scale that is much larger than the flaw sizes. The state of damage is quantified by internal state variables (ISVs) within the context of the thermodynamics of irreversible processes. That is, the growth of damage is governed by an appropriate damage evolution law. The choice and interpretation of the ISVs are somewhat arbitrary, and the functional form of the thermodynamic potential (typically Helmholtz or Gibbs free energy) and the resulting stress-strain relations are postulated usually on a phenomenological basis. The stiffness of the material, which varies with the extent of damage, is determined as a function of the ISVs by fitting the theoretical model to the available experimental data. The phenomenological continuum damage models thus provide a viable constitutive framework for the efficient modeling of macroscopic mechanical behavior of materials with distributed damage without requiring explicit descriptions of microstructural evolution kinetics (Park et al. 1996). Recently, significant advancements in the modeling of asphalt concrete have been made in both micromechanics and continuum damage mechanics. In future models of asphalt concrete, micromechanical and continuum damage models will be coupled to describe the behavior and performance of asphalt pavements using the properties of their component materials (i.e., binder and aggregate). This multiscale model will take advantage of the strengths of both micromechanics and continuum damage mechanics, that is, the ability of the micromechanical model to describe mixture behavior using component material properties and that of the continuum damage model to describe

Modeling of Asphalt Concrete the global stress-strain behavior of asphalt concrete in predicting the pavement performance. The challenge in this combined approach is to determine the material properties at the proper scales, as it is expected that some material properties are scale dependent.

Virtual Testing of Asphalt Concrete One of the fastest growing techniques that can aid asphalt concrete modeling is the imaging technique, including digital imaging, laser, and x-ray tomography, to name a few. These techniques allow engineers to view and construct two- and three-dimensional microstructures of the mixture. The imaging techniques can be combined with advanced models of asphalt concrete and provide the tools to perform virtual testing of asphalt concrete. In this approach, virtual microstructures of asphalt concrete are generated from the imaging technique, and virtual testing is conducted on the virtual microstructure using the advanced numerical models. These virtual testing techniques will help asphalt material and pavement engineers to evaluate the effects of any change in the component material properties on the mixture behavior and performance without any laboratory testing. The virtual testing will also be an efficient tool in undergraduate and graduate asphalt materials courses to demonstrate the effects of changing testing conditions and mixture design parameters on the behavior and performance of asphalt concrete.

Organization Summary Part 1 (Chap. 2) is dedicated to asphalt binder modeling. Various aspects of the stiffness characterization of asphalt concrete are described in Part 2 (Chaps. 3 through 6). Part 3 (Chaps. 7 to 9) presents different constitutive modeling approaches for asphalt concrete. Part 4 (Chaps. 10 and 11) examines models for rutting. Part 5 (Chaps. 12 and 13) addresses models for fatigue cracking and moisture damage. Last, Part 6 (Chaps. 14 and 15) addresses models for low-temperature cracking. Note that this book does not necessarily describe all the models that are currently available within each aspect of asphalt concrete modeling (as outlined in the six parts, accordingly). However, it should provide sufficient information about a wide range of models that are available. The following descriptions provide a summary of the contents of each part.

Part 1—Asphalt Binder Rheology Part 1 of this book deals with issues pertaining to asphalt binder rheology. The historical use of rheological indices in the asphalt industry is discussed to provide perspectives on the development and rationale for the findings from the SHRP project. The influence of binder properties on mixture performance is discussed. Background on polymer modification of asphalt binders is presented and, subsequently, an argument for the enhancement of asphalt binder performance by adding polymer-modifying agents is given. A rheological modeling approach capable of capturing the beneficial aspects of polymer-modified binders is presented along with results from an accompanying experimental study. Since the focus of this book is the modeling of asphalt concrete, only one chapter is allotted to a discussion of asphalt binders.

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Chapter One

Part 2—Stiffness Characterization Part 2 of this book focuses on asphalt concrete stiffness. Stiffness is critically important for mechanistic modeling of both the pavement response and the pavement performance. Chapter 3 discusses explicitly the importance of this factor for such analysis and also details the major factors affecting the material stiffness. Particular attention is paid in Chaps. 4 and 5 to the stiffness characterization of asphalt concrete via the complex modulus. Two different test methods are demonstrated. The first is part of the proposed simple performance test protocol and involves testing cylindrical asphalt concrete specimens in the axial direction. The second method strives to overcome shortcomings associated with using the geometry of the first method to evaluate the stiffness of field cores using the indirect tension test. There are numerous advantages to assessing material stiffness via the dynamic modulus in the frequency domain; however, many of the mechanistic models presented in this book require stiffnesses in the time domain. Linear viscoelastic theory and mathematical manipulation are used in Chap. 6 to demonstrate different methods of converting the dynamic modulus into time domain functions such as creep compliance and the relaxation modulus.

Part 3—Constitutive Models Part 3 of this book focuses on the constitutive modeling of asphalt concrete. Three approaches are presented in detail in this part. These approaches utilize different principles to describe the deformation behavior and performance of asphalt concrete, but are similar in that they attempt to form a unified model encompassing different performance characteristics by accounting for various constitutive factors. Chapter 7 in this part incorporates the theory of viscoelasticity, continuum damage mechanics, and the theory of viscoplasticity to arrive at a so-called viscoelastoplastic continuum damage (VEPCD) model as a constitutive relationship for the behavior of asphalt concrete. Implementation of the VEPCD model into the finite element program is discussed. Chapter 8 presents a constitutive model based on the hierarchical disturbed state concept (DSC). The chapter describes the capabilities of the DSC for various pavement distresses such as permanent deformation and different types of cracking. Analysis of both two- and three-dimensional pavement problems is given using the DSC model, and a unified methodology with DSC for design, maintenance, and rehabilitation of pavement structures is proposed. Chapter 9 uses the DBN (Di Benedetto and Neifar) law to describe the behavior of asphalt concrete under a broad range of conditions. It explains how the different types of behavior can be modeled using the same formulation.

Part 4—Models for Rutting In this part, the mechanisms of permanent deformation are described and modeled in two chapters. Information documented in Chap. 10 is the result of the SHRP A-003 study and illustrates that shear deformation contributes a significantly greater portion of total permanent deformation (rutting) in asphalt concrete than volume change. Based on these findings, the shear test was proposed to measure the propensity of a mix for rutting. The issue of sample size is discussed in the light of RVEs. The data presented illustrate the efficacy of the simple shear test, performed in the repeated load, constant height mode, for mix design and performance evaluation. Chapter 11 summarizes the findings from the more recent NCHRP 9-19 project. This chapter is composed of three main sections: (a) a review of mechanistic-empirical modeling approaches, and in particular the permanentto-resilient strain ratio model adopted for the NCHRP 1-37A MEPDG; (b) the VEPCD

Modeling of Asphalt Concrete model for the compression behavior of asphalt concrete; and (c) a simple performance test to identify the rutting potential of mixtures during the design process, based on the measurement of fundamental engineering responses and properties. It is noted that the VEPCD model adopted in this chapter employs the same principles as found in Chap. 7 in Part 3, except that the HiSS-Perzyna model is used to describe the viscoplastic strain of asphalt concrete instead of the strain-hardening model used in Chap. 7.

Part 5—Models for Fatigue Cracking and Moisture Damage The detrimental effects of moisture and fatigue damage are discussed in Part 5 of this book. Chapter 12 focuses primarily on the fatigue damage mechanisms with a particular interest in the acceleration of such damage growth with additional moisture damage. Surface energy principles, fracture mechanics, and continuum damage mechanics are utilized for this argument. In Chap. 13 more attention is given to the moisture damage phenomenon. A review of current procedures for moisture damage assessment is given as a precursor to more advanced, objective techniques for the assessment of moisture damage.

Part 6—Models for Low-Temperature Cracking Part 6 of this book discusses thermal cracking of asphalt concrete pavements. Mechanisms and events leading to thermal cracking are discussed in detail in Chaps. 14 and 15. Chapter 14 presents the TCMODEL, which has been implemented into the NCHRP 1-37A MEPDG to predict thermal cracking performance. The second chapter, Chap. 15, casts the phenomenon in the light of fracture mechanics and presents experimental results of multiscale modeling efforts encompassing binder, mastic, and mixture modeling.

Concluding Remarks This book attempts to document models of asphalt concrete and should be regarded as an evolving document, as some of the models are still being refined and improved. It also may be noted that this book focuses mostly on continuum models in order to maintain a reasonable length. Significant advancements in micromechanical modeling of asphalt concrete have also been made. Nonetheless, this book should provide a fair presentation and sufficient review of mechanistic models that are currently available at the time of publication.

References Kim, Y. R., H. J. Lee, Y. Kim, and D. N. Little, Mechanistic Evaluation of Fatigue Damage Growth and Healing of Asphalt Concrete: Laboratory and Field Experiments, Proceedings of the Eighth International Conference on Asphalt Pavements, International Society for Asphalt Pavements, University of Washington, Seattle, Washington, 1997, pp. 1089–1107. NCHRP 1-37A Research Team, “Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures,” Final Report, NCHRP 1-37A, ARA, Inc. and ERES Consultants Division, 2004. Park, S. W., Y. R. Kim, and R. A. Schapery, “A Viscoelastic Continuum Damage Model and Its Application to Uniaxial Behavior of Asphalt Concrete,” Mechanics and Materials, Vol. 24, No. 4, December 1996, pp. 241–255.

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PART

Asphalt Rheology CHAPTER 2 Modeling of Asphalt Binder Rheology and Its Application to Modiﬁed Binders

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CHAPTER

2

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Hussain U. Bahia

Introduction Rheology is the science of flow and deformation. It is a science for the study of materials whose deformation characteristics vary not only with load but also with time rate of load application. Asphalt binders are rheological materials; that is, their behavior depends on temperature and rate (or time) of loading. At any combination of time and temperature, the time-dependent (i.e., viscoelastic) behavior of asphalt binders, within the linear range, is best characterized by two properties: the total resistance to deformation under load and the relative distribution of that deformation between elastic and viscous parts (Bahia and Anderson 1995). Although there are many methods of characterizing viscoelastic properties, cyclic (oscillatory) testing and creep testing are two of the best techniques to represent the uniqueness of the behavior of this class of materials. What is unique about asphalt binders is their high sensitivity to temperatures within the range of applications. The stiffness of asphalt can vary by as much as eight orders of magnitude, and their phase angle (relative distribution of response between elastic and viscous) by as much as 85°, between peak summer and peak winter conditions. It can also vary by similar amounts in response to standing traffic and high-speed traffic (Anderson et al. 1994). Similar to viscoelastic properties, failure properties and damage resistance properties of asphalt binders are also very sensitive to temperature and loading rate. Stress and strain at failure can change by an order of magnitude by a change of only 10°C (Dongre et al. 1995). Fatigue life of binders can change by orders of magnitude in response to a change of a few degrees of temperature or by changing loading frequency within the application range (fast versus slow traffic) (Pell and Cooper 1975; Bahia et al. 1999; Bonnetti et al. 2002). Conventional methods of asphalt refining have their limitations which prompted the introduction of modified binders. In 2005 it is estimated that 20 to 25 percent of the

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Chapter Two asphalts used in the United States are modified asphalts. Modification of asphalts is, therefore, an integral part of this field and understanding its variables is necessary. Modification of asphalt binders is usually performed to improve one or more of the basic asphalt properties that are related to one or more of the pavement distress modes (Bahia 1995; Terrel and Epps 1989). The basic properties that have been targeted include Rigidity: Total resistance to deformation which can be measured by complex moduli like G∗ under dynamic loading or by creep stiffness, S(t), under quasistatic loading. Higher rigidity is favorable at high temperatures or low loading rates to resist rutting while lower rigidity is favorable at intermediate and low temperatures to resist fatigue and thermal cracking, respectively. Elasticity: Recovery of deformation using stored energy applied. It can be measured either by the phase angle (d) or by the logarithmic creep rate (m). To resist rutting and fatigue damage more elasticity is favorable. To resist thermal cracking, less elasticity and more ability of relaxing stress by flow is favorable. Brittleness: Failure at low strains is the best definition of brittleness. To improve resistance for fatigue and thermal cracking, brittleness should be reduced by enhancing strain tolerance or ductility. Storage stability and durability: Oxidative aging, physical hardening, and volatilization are key durability properties. Resistance to all of these changes is favorable. Resistance to accumulated damage: Rutting and fatigue damage are recognized as the two most important load induced types of distress. They represent progressive failure mode which is not necessarily measured using small stress or small strain testing.

Modeling Critical Properties of Asphalt Binders Critical properties of asphalt binders can be classified into two main groups: mechanical properties and durability properties. Testing of mechanical properties can be further sorted, based on progress in testing technologies into three groups: traditional rheological, linear viscoelastic, and damage resistance characterization. Durability properties include aging during production and construction and for modified binders, compatibility or storage stability are important properties. The following sections describe these different properties.

Traditional Rheological Properties Traditional or index rheological properties include many standardized tests that were used mostly before the completion of the Strategic Highway Research Program (SHRP) in the early 1990s. Although in North America and some European countries a new set of rheological test methods are rapidly being introduced, the traditional tests continue to be used widely in other parts of the world. Table 2-1 includes a list of the main traditional tests (Isacsson and Lu 1995). The most common independent traditional measures used for asphalts include penetration, ductility, softening point, and viscosity. The problem with the first three measures, in addition to being one point measurements, is that the empirical nature complicates the derivation of any meaningful engineering property. The problem with

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders

Standard Protocol

Test

Purpose of Test and Applicability for Modified Asphalt Binders (MABs)

Index (Empirical) Tests Softening Point (R&B)

ASTM D36

An index of consistency at very high temperatures. Used widely around the world for compliance but not considered reliable for assessing the high temperature performance in paving applications. Test can be used to measure instability of modified asphalts.

Penetration @ 25°C (dmm)

ASTM D5 NF T 66-004

Considered an index for consistency of binder at intermediate temperature. Although test is questionable and of little value, it is used in many MAB specifications in Europe, Australia, and Japan. The effect of modifiers can be an increase or a decrease in penetration.

Frass Breaking Point

IP 80

An index of resistance to cracking at low temperatures under repeated flexing and reducing temperature. Test is mostly used in Europe as an indicator of brittleness. Modifiers are known to result in decreasing Frass point, particularly for elastomers.

Viscosity (Resistance to Flow) Tests Rotational Viscosity @ 135– 165°C

ASTM D402

A measure of resistance to flow at production and construction temperatures of HMA. Used widely to measure workability of MABs.

Absolute Viscosity @ 60°C

ASTM D2170

Measured using the capillary tube under vacuum. Because many modified asphalts are non-Newtonian, it has been used with caution and is not used in specification widely. Modifiers are known to increase the value of this viscosity and also increase shear rate dependency.

Cone and Plate Apparent Viscosity @ 25–60°C

ASTM D3205

Developed to measure creep response under increasing creep loads to estimate apparent viscosity at intermediate to high temperature. Not used widely and standard was discontinued in 2000. Most polymer modifiers are expected to increase viscosity.

Tensile (Extensional) Properties Ductility @ 4°C and @ 25°C

TABLE 2-1

ASTM D113

An index of flexibility at low temperatures when used at 4°C. Also considered an index of compatibility particularly when used at 25°C. Effect of modifiers on ductility varies significantly depending on nature of modifier. Elastomeric modifier tends to increase ductility at these temperatures, plastomers have minimal or negative effect, and chemical or oxidation tends to decrease ductility particularly at low temperatures. Used in Europe at various temperatures. Also used in North America in some states.

List of Asphalt Traditional Tests

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Chapter Two

Standard Protocol

Test

Purpose of Test and Applicability for Modified Asphalt Binders (MABs)

Tensile (Extensional) Properties Forced Ductility @ 4°C

Not a standard

An index of tensile strength and energy required for complete failure. Specifically developed for polymer modified asphalts and used widely in North America. Response usually includes an initial and a secondary peak of the stress which is used to calculate a ratio showing effect of the modifier. Test taken from joint sealant testing field.

Elastic Recovery @ 25°C

D8084

An index of the capability of modified asphalt for elastic recovery. Measured using the conventional ductility set up but sample is stretched and then cut to measure recovery of cut ends. One of the most widely used to determine if modified binder includes elastomers. Used in North America, Australia and Europe. Method has been modified several times and is run using sliding plate rheometer, ARRB Elastometer, Consistometer, and torsional loading setup.

Toughness and Tenacity @ 25°C

Not a standard test

An index of energy to failure used to detect modifiers and assess their contribution to toughness. A hemispherical head is inserted in an asphalt container and then pulled out. The area under the load deformation curve is divided into an initial peak area and a terminal tenacity area. The sum is the toughness. Elastomeric modifiers could have a significant effect on tenacity and on toughness particularly if they are cross-linked.

TABLE 2-1

List of Asphalt Traditional Tests (Continued)

these measures has been recognized by several researchers, and many attempts have been made to correlate them with more fundamental rheological properties. For example, formulas were proposed to calculate the coefficient of viscosity from penetration (Saal and Labout 1958; Van der Poel 1954; Heukelom 1973; Davies 1981). Ductility was considered by some researchers as an indicator of internal structure (Halstead and Zenewitz 1961; Barth 1962), while others showed that it may be correlated with certain shear susceptibility parameters (Traxler 1961; Traxler et al. 1944; Kandhal and Wenger 1975). Other researchers tried to combine the two measures (penetration and ductility) and relate them directly to pavement performance (Halstead 1961; Serafin et al. 1967). The softening point was also correlated with more fundamental measures; Van der Poel (1954) indicated that it can be considered as equipenetration temperature, while Jongepier and Kuilman (1969) indicated that softening point is an equimodulus temperature rather than an equiviscous temperature. These correlation studies have relied on some general correlations that suffer in many instances from exceptions, low statistical significance, and limited sampling of asphalts. Viscosity, also a single point measure, is a fundamental material property expressed in absolute units. However, the coefficient of viscosity is only a fundamental absolute measure for Newtonian fluids. Newtonian fluids are those whose properties are

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders independent of rate of loading or stress level. Asphalts exhibit Newtonian behavior only at very high temperatures, (above softening point), or at very low shear rates, which are seldom seen by asphalts in their applications in road pavements. At low temperatures, or short loading times asphalts are not Newtonian and cannot be described by an absolute value of coefficient of viscosity. To solve this problem and make it more relevant to the domain of application conditions, the “apparent viscosity,” a measure dependent on shear rate, was introduced. Questions then arise regarding where, in the time, stress and temperature senses, that the measurement should be taken. The selection of appropriate ranges varied amongst researchers and became a matter of experimental convenience. Traxler and his coworkers (Traxler and Schweyer 1936; Romberg and Traxler 1947; Traxler 1947) selected a temperature level of 77°F and a constant power input (constant value of the product of stress times strain rate) was proposed. For these experimental studies a power input of 1000 ergs was used because aged and unaged asphalts could be measured at that value of power input, with the available viscometer without any extrapolation (Romberg and Traxler 1947). Several researchers used this approach in asphalt aging studies, good examples are given in Moavenzadeh and Stander (1967), Majidzadeh (1969), and Page and coworkers (1985). With the introduction of the Shell sliding plate viscometer (Griffin et al. 1955), apparent viscosity at 77°F and a constant shear rate of 0.05 s−1 was introduced for experimental convenience and suitability of the device for this measurement. A great number of research groups followed this approach; apparent viscosity at 77°F and 0.05s −1 became the most common measure for evaluating asphalt rheology of aged and unaged asphalts. Early as well as recent studies have used this measure (Heithus and Johnson 1958; Gallaway 1957; Kemp and Predoehl 1981; Button and Epps 1985). Despite the wide acceptance of the constant strain rate viscosity measure Mack (1965) indicated that stress level is equally important as the shear rate, and that apparent viscosity should not only be compared at a constant temperature and strain rate but also at a constant stress level. Chipperfield and Welch (1967) suggested that using a constant stress level is more accurate than constant strain rate. Based on an extensive field study, the authors indicated that a constant stress apparent viscosity, although not the ultimate choice, is a much better indicator of asphalt hardening due to aging than the constant strain rate apparent viscosity. This other approach also had its followers (Schmidt 1972). The apparent viscosity was one of the measures adopted into the American Society for Testing and Materials (ASTM ) standards. It is, however, subject to many questions. The common method of determining apparent viscosity is by incremental creep tests, where a series of loads are added in sequence and the strain is measured with time. At each load level the strain is monitored until it shows a constant rate which is selected to calculate viscosity from the corresponding stress application. Then, the next load is added and the procedure is repeated to calculate the viscosity at the new shear rate. With several measured viscosities at different strain rates, the viscosity is calculated by interpolation (ASTM D 3205). For a non-Newtonian material, such as asphalts at temperatures below 140°F, the strain rate is a strong function of loading time and becomes more so as the temperature is reduced. It may take several hours or even days of loading time so that the strain rate reaches a constant and an asphalt starts behaving like a truly viscous material. Moreover, the proximity to the steady-state viscosity is highly asphalt specific. The other fundamental problem with apparent viscosity is the possibility of reaching the nonlinear region because of the geometry of the specimen or the stress level being

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Chapter Two used. Keeping the load for long times so as to reach an approximately constant strain rate, may easily put the material in the material nonlinearity state (high stains), or in the geometrically nonlinear region which complicates the interpretation of the measurement. Increasing the load level to very high values, in order to reach a constant strain rate, may also result in stress nonlinearity. These fundamental problems with apparent viscosity are not new subjects; the same researchers who promoted the measure have warned continuously of these problems. Wood and Miller (1960) indicated that testing with the sliding plate microviscometer may require several hours before a constant strain rate may be reached. Labout and van Ort (1956), Griffin and coworkers (1955), Fink and coworkers (1961), and Evans and Griffin (1963), all suggested using certain geometries for the sliding plates and used different film thicknesses to limit the strain rates and to show that the measure may be significantly affected by the geometry or the total amount of deflections are exceeded. Others have also reported on the effect of stress level and more specifically the stress history on the apparent viscosity; Puzinauskas (1967, 1979), and Majidzadeh and Schweyer (1965) are among those who thoroughly evaluated the effect of stress history on apparent viscosity and reported sufficient evidence of the stress dependence. Shortfalls of the traditional tests have been recognized for some time, and various attempts were made to use these tests to estimate fundamental rheological properties. These attempts started more than 50 years ago and have demonstrated that it is best that asphalt cements be characterized as linear viscoelastic materials that are thermorheologically simple. However, the difficulty of conducting fundamental rheological testing has led many to simplify the rheology of asphalts, and depend on what is available in the traditional labs, despite the empiricism involved and the insufficient characterization that followed. Van der Poel introduced his widely used nomograph in 1954 and indicated that asphalts can be successfully characterized using simple rheological indices, derived from empirical measures (Van der Poel 1954). The concepts of using the so-called “temperature susceptibility parameters, and shear susceptibility parameters” was introduced to model behavior of asphalts.

Asphalt Susceptibility Parameters To advance modeling of asphalt, rheological derivatives of the traditional tests were introduced. The motivation was to derive a better relationship with performance that the traditional tests fail to predict. These estimated, pseudorheological properties, can be divided into two sets: the temperature susceptibility parameters and the shear susceptibility parameters.

The Temperature Susceptibility Parameters The change in consistency with temperature is the general definition of temperature susceptibility. Several types of such parameters have been proposed and used. They vary basically in two aspects: the type of consistency measurement used and the range of temperature covered. The early works used penetration as the consistency measure. The penetration at different temperatures was measured and the ratio of penetration, the difference in penetration, the temperature required to increase the penetration by a certain number, or the slope of log penetration versus temperature have been used to characterize temperature susceptibility (Pfieffer and van Doormaal 1936; Van der Poel 1954; Neppe 1952; Barth 1962). More fundamental approaches used viscosity as the consistency

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders measure and to define a temperature susceptibility parameter. Again several types of relations have been used: slope of log viscosity versus temperature (Traxler and Schweyer 1936), log viscosity versus reciprocal of temperature raised to a variable power (Traxler and Schweyer 1936; Cornelissen and Waterman 1956), and log-log viscosity versus log temperature (Fair and Volkmann 1943). The latter measure is probably the most widely accepted and is referred to as viscosity-temperature susceptibility or VTS. A third type of approach used a combination of consistency measures. The penetration index (PI) for example combines penetration and softening point (Van der Poel 1954), and the penetration viscosity number (PVN) combines penetration and viscosity (McLeod 1972). The best indication of how well these different susceptibility measures represent the rheological properties of asphalts is probably found in durability studies that used these measures to define asphalt performance with age hardening. The conclusions about the effect of aging on temperature susceptibility are controversial: Pfieffer (1950) and Blokker and Hoorn (1959), using the penetration index, reported a decrease in temperature susceptibility upon oxidative aging. Halstead and Zenewitz (1961) using plots of log-log viscosity versus temperature, in the range of 60 to 95°F, indicated that some asphalts show a decrease in temperature susceptibility while others show an increase upon aging. McLeod (1972) used the penetration index (PI) and his penetration viscosity number (PVN) and observed that after 9 years of service asphalts showed either no change or a significant increase in temperature susceptibility. However, the PI did not show the same change in temperature susceptibility. Puzinauskas (1979) used the VTS and concluded that the majority of asphalts show an increase in temperature susceptibility. Anderson and his coworkers (1983) used three different parameters (PI, PVN, and VTS) and concluded that PI and VTS show a general increase in temperature susceptibility, while PVN does not show significant changes with aging. The authors, experimenting on a very large number of asphalts, indicated that these measures do not measure the same property, and no simple explanation can justify the difference in the effect of oxidative aging on the values of these parameters. Button and his coworkers (1983) confirmed Anderson’s findings. The authors found that, using the PVN (estimated from pen77 and vis140) as the temperature susceptibility measure, asphalts that were originally highly temperature susceptible became more susceptible with oven aging, while those that were less temperature susceptible became even less susceptible with aging. Others have also used apparent viscosity as the measure of consistency. Moavenzadeh and Stander (1967) measured apparent viscosity at a constant power input (1000 ergs) over a range of temperatures between 10 and 160°C. The authors indicated that aged asphalts are less temperature susceptible in the lower temperature range while more susceptible in the higher temperature range. In other words, the temperature susceptibility shows different changes at different ranges of temperature. Kandhal and his coworkers (Kandhal et al. 1973; Kandhal and Wenger 1975) confirmed Moavenzadeh and Stander findings. Here researchers measured apparent viscosity at 39, 77, 140, and 275°F and observed that temperature susceptibility measured in terms of log-log viscosity versus log temperature, decreased in the lower temperature range with age hardening while it increased in the higher temperature range. No doubt the above review is confusing. Two fundamental reasons can be given to explain the problems with temperature susceptibility parameters: 1. The temperature susceptibility, as shown in linear viscoelastic characterization studies, is both temperature range and time of loading dependent. The relations are not linear nor can be approximated by a linear relation such that a few

17

18

Chapter Two measurements, even if done properly, can be used to represent them. Therefore, any two measures should consider both time of loading and temperature range. In addition oxidative hardening may change the shape of the relation differently at different times of loading. This observation dictates then that unless the parameters of susceptibility are referring to the same narrow temperature range and times of loading they will naturally give different indications of the effect of aging. Except for the VTS, none of these requirements are satisfied in the used parameters. 2. The penetration, softening point, and apparent viscosity have been shown in the previous section to involve serious problems. Using them for temperature susceptibility measurements may only lead to more difficulties. The simple reasoning is that as the temperature decreases, or as the asphalt is aged, the time required to reach the specified shear rate will be different which means we are using measures at the same temperature but two different times of loading are being used. In summary, it is rather difficult to consider the temperature susceptibility parameters discussed here as reliable measures to characterize asphalts or to even measure the effect of oxidative hardening.

Shear Susceptibility Parameters Two types of parameters have been used to represent the shear susceptibility of asphalt cements: The degree of complex flow “C” and the shear index. Traxler and coworkers (1944) were the first to introduce the use of this parameter for paving asphalts. They indicated that measurements on different asphalts show that the slope of log shear stress versus log strain rate may be considered constant and, therefore, concluded that asphalt rheological properties may be approximated by a complex flow equation that is usually used for power low fluids:

Degree of Complex Flow

M=

T SC

(2-1)

where M = constant T = shear stress S = shear rate C = degree of complex flow For C = 1, the asphalt is a Newtonian fluid and M is the steady-state coefficient of viscosity. Therefore, C was considered a good measure of non-Newtonian behavior. A large number of researchers accepted the approximation and the change in the C value was used as an indicator of the asphalt rheology and an indicator of effect of oxidative aging on the properties of asphalts (Gallaway 1957; Moavenzadeh and Stander 1967; Jimenez and Gallaway 1961). The degree of complex flow C, in fact, looks very attractive for the study of asphalt rheology. However, the measure also has its own problems that have been shown in several studies: 1. It assumes the relation between shear stress and shear rate to be linear when C = 1. This is true only over a small range of stress or strain rate. At very low shear rates or at very small stress levels almost all asphalts will show Newtonian behavior (C = 1.0). This behavior, depending on the type of asphalt, will start

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders changing to non-Newtonian gradually as the shear rate or stress level increases (Puzinauskas 1967). Therefore, the relation is not linear and the value of C will depend on which range is referred to. 2. The parameter, being a measure of non-Newtonian behavior, will naturally depend on the stress history. Majidzadeh and Schweyer (1965) conducted an experiment where the C value was measured for the same asphalt under the same conditions, but the sequence of loading applications was changed. They compared Ci measured when the load increments where added in a decreasing sequence of load values with “Cd “ measured with an increasing sequence of load step values. 3. In addition to the above problems with the C measure another point may be added here. Constructing the stress-shear rate relation is subjected to the same controversy associated with apparent viscosity. From a standard creep curve measured for a non-Newtonian material, the shear rate may never come to an equilibrium within reasonable time or strain level, therefore, the value of the shear rate used to construct the stress-shear rate curve will depend on the time of loading at which this rate is determined. In other words the C value is time of loading dependent. Apparent viscosity is plotted versus shear rate on a log-log scale. The slope of the relation between two different shear rates is determined and considered as the shear susceptibility parameter. Although early works tried to approximate the relation as linear (Jimenez and Gallaway 1961; Gallaway 1959), a large number of more recent studies showed clearly that this is not true. Apparent viscosity is constant up to the Newtonian behavior limit after which the viscosity starts decreasing continuously with increasing shear rate. For asphalt aging studies, the shear index has been used by many researchers. Zube and Skog (1969), Culley (1969), and Kandhal et al. (1973) all compared viscosity-shear rate plots of unaged with lab or field aged asphalts. However, since the plots are nonlinear and temperature dependent researchers selected different ranges of shear rates and different temperatures. All agreed that an increase in the shear susceptibility is observed upon aging. As with other susceptibility parameters, the shear index may be considered as an arbitrary value. The exact value depends, to a great extent, on the range of shear rate considered and the temperature at which it is measured. A single shear index may only indicate the type of behavior expected for an asphalt at the temperature and shear rate range it is measured at; no simple extrapolation or even interpolation can be made. Moreover, it still uses the apparent viscosity which is time of loading dependent and if this is not constant then the plot is very difficult to interpret. In summary, shear susceptibility parameters are not much better than temperature susceptibility parameters, or even the single measurements. They are also arbitrary in nature and do not show high promise for use in asphalt rheological characterization.

The Shear Index

Linear Viscoelastic Properties Although using rheological concepts to characterize asphalts dates back more than 50 years, the cost and availability of equipment hampered the spread of using viscoelasticity to qualify asphalts and study effect of modifiers. This shortcoming was overcome

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20

Chapter Two through the SHRP project where major emphasis was placed on the development of testing equipment and/or test protocols to standardize the use of asphalt tests that were relatively blind to the binder composition. The test protocols, although developed with the intention of being suitable to unmodified and modified binders, focused on linear viscoelasticity as a compromise between the difficulty of rheological testing and the need to produce practical and realistic test protocols for an industry that has historically used mainly simplistic index testing to qualify asphalts (Anderson et al. 1991). In 1992–1993 a new set of testing techniques and a new grading system was introduced. The testing and grading systems are based on fundamental properties that are related in a more rational way to pavement performance. The viscoelastic properties of asphalt binders are measured at application temperatures and grading criteria are based on sound understanding of pavement failure mechanisms.

The Viscoelastic Nature of Asphalt Binders At any combination of time and temperature, viscoelastic behavior, within the linear range, must be characterized by at least two properties: the total resistance to deformation and the relative distribution of that resistance between an elastic part and a viscous part. Although there are many methods of characterizing viscoelastic properties, dynamic (oscillatory) testing is one of the best techniques to represent the behavior of this class of materials. In the shear mode, the dynamic modulus (|G∗|, for simplicity, denoted as G∗ hereinafter) and phase angle (d) are measured. G∗ represents the total resistance to deformation under load, while d represents the relative distribution of this total response between an in-phase component and an out-of-phase component. The in-phase component is an elastic component and can be related to energy stored in a sample for every loading cycle, while the out-of-phase component represents the viscous component and can be related to energy lost per cycle in permanent flow. The relative distribution of these components is a function of the composition of the material, loading time, and temperature. Rheological properties can be represented either by the variation of G∗ as a function of frequency at a reference temperature (commonly referred to as a mastercurve) or by the variation of G∗ and d with temperature at a selected frequency or loading time, commonly called isochronal curve. Although time and temperature dependency can be related using a temperature-frequency shift function (Ferry 1980), for practical purposes it is much easier to present data with respect to one of the variables. Figure 2-1 depicts typical rheological properties of an AC-40 and an AC-5 asphalt binder at a wide range of temperatures and frequencies. Figure 2-1a is a mastercurve at 25°C and Fig. 2-1b is an isochronal curve at 10 rad/s. Some common unique characteristics of the rheological behavior of asphalt can be seen in the typical plots of Fig. 2-1: • At low temperatures or high frequencies, both asphalts tend to approach a limiting G∗ value of approximately 1.0 GPa and a limiting d value of 0°. The 1.0 GPa reflects the rigidity of the carbon hydrogen bonds as the asphalts reach their minimum thermodynamic equilibrium volume. The 0° value d represents the completely elastic nature of the asphalts at these temperatures. • As the temperature increases or as the frequency decreases, G∗ decreases continuously while d increases continuously. The first reflects a decrease in resistance to deformation (softening) while the second reflects a decrease in elasticity or ability to store energy. The rate of change is, however, dependent

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders 11

100 AC-40 (G∗) AC-5 (G∗)

AC-5 (Delta) 80

9 8 7

60

6 40

5 4 3

Phase Angle at 25°C

Complex Modulus at 25°C, log Pa

10

AC-40 (Delta)

20

2 1 −5 −3

−1 1 3 5 7 9 11 13 Loading frequency, log rod/s

0 15

(a) Frequency master curves 11 AC-40 (Delta) AC-5 (Delta)

9

100

80

8 7

60

6 40

5

Phase Angle, °

Complex Modulus at 10 rad/s, log Pa

10

AC-40 (G∗) AC-5 (G∗)

4 3

20

2 1 −50

−30

−10

10 30 50 Temperature, °C

70

0 90

(b) Isochronal curves

FIGURE 2-1

Typical rheological spectra for two asphalt binders.

on the composition of the asphalt. Some will show a rapid decline with temperature and frequency; others will show a gradual change. Asphalts within this range may show significantly different combinations of G∗ and d. • At high temperatures, the d values approach 90° for all asphalts, which reflects the approach to complete viscous behavior or complete dissipation of energy in viscous flow. The G∗ values, however, vary significantly, reflecting the different consistency properties (viscosity) of the asphalts.

21

Chapter Two

Asphalt Viscoelastic Properties and Pavement Performance Figure 2-2 is an isochronal plot that depicts rheological properties of asphalt in its unaged condition and after aging in the field under a moderate climate for approximately 16 years. To relate asphalt properties to pavement performance, reference can be made to three temperature zones. At temperatures in the range of 45 to 85°C, typical of highest pavement in-service temperatures, the main distress mechanism is rutting (Chaps. 10 and 11), and, therefore, the G∗ and d need to be measured. A measure of viscosity alone cannot be sufficient, since viscosity measurements are done on the assumption that asphalt response has only a viscous component. For rutting resistance, a high G∗ value is favorable because it represents a higher total resistance to deformation. A lower d is favorable because it reflects a more elastic (recoverable) component of the total deformation. Within the intermediate temperature zone, asphalts are generally harder and more elastic than at higher temperatures. The prevailing failure mode at these temperatures is fatigue damage (Chaps. 12 and 13). For viscoelastic materials, like asphalt binders, both G∗ and d play a role in damage caused by fatigue. They are both important because during every cycle of loading the damage is dependent on how much strain or stress is developed by the cyclic load and how much of that deformation can be recovered or dissipated. Under strain-controlled conditions, a softer material and a more elastic material will be more favorable to resist fatigue damage because the stress developed for a given deformation is lower and the asphalt will be more capable of recovering to its preloading condition. Similar to rutting, a single measure of hardness or viscosity cannot be sufficient to select better-performing asphalts with respect to fatigue resistance. Rutting and fatigue damage are both functions of frequency of loading, and, therefore, the rate Unaged (G∗) Field aged (G∗)

Unaged (delta) Field aged (delta)

10

100

9

90

8

80

7

70

6

60

5

50

4

40

3

30

2

20

1

10

0 −1 −50

Thermal cracking

−30

−10

Fatigue cracking

30 50 10 Temperature, °C

Rutting

70

Delta at 10 rad/s, °

G∗ at 10 rad/s, log Pa

22

0 −10 90

FIGURE 2-2 Typical rheological behavior of asphalt binders before and after aging in the ﬁeld in relation to pavement main distress modes.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders of loading of the pavement under traffic needs to be simulated in measurement to obtain a reliable estimate of binder contribution to pavement performance. The third temperature zone is the low-temperature zone at which thermal cracking (Chaps. 14 and 15) is the prevailing failure mode. During thermal cooling, asphalt stiffness increases continuously and thus results in higher stresses for a given shrinkage strain. Simultaneously, thermal stresses relax due to viscoelastic flow of the binder. To reliably predict binder contribution to cracking, both the stiffness of a binder and its rate of relaxation need to be evaluated. The stiffness of the binder is directly proportional to G∗ and the rate of relaxation is directly related to d. A lower stiffness and higher rate of relaxation are favorable for resistance to thermal cracking. As with other temperature zones, a single measure of the stiffness or viscosity of the binder is not sufficient to select better binders that will resist cracking at the lowest pavement temperatures. The above discussion of the relation between asphalt binder properties and pavement performance is further complicated by the aging phenomenon. Asphalts are hydrocarbon materials that oxidize in the presence of oxygen from the environment. This oxidation process changes the rheological and failure properties of the asphalt. As shown in Fig. 2-2, the rheological mastercurve slope decreases with aging, which indicates higher G∗ and smaller d values for the unaged binder at all temperatures. These changes translate into less sensitivity of G∗ and d to temperatures or loading frequency and into more elastic component (lower d). Significant oxidation effects usually appear after considerable service life. Increased G∗ values and lower d values are favorable changes with respect to rutting performance, but they are unfavorable for thermal cracking performance. For fatigue cracking, the increase in G∗ is not favorable while the decreased d is generally favorable, depending on the type of pavement and mode of fatigue damage.

Modeling of the Viscoelastic Properties of Asphalts Many attempts have been made to use simple mechanical analog, such as the generalized Burgers model and the Prony series, and phenomenological models, defined by curve fitting of experimental data, to describe the viscoelastic properties. The latter approach has seen more acceptance particularly with the advancement of computers and the flexibility of these models. Some of the most notable models that followed the pioneering work by Van der Poel in the early 1950s include work by Jongepier and Kuilman (1969) who proposed that asphalts can be considered as simple liquids whose rheological behavior can be approximated by log Gaussian distribution of relaxation times. These authors used a width parameter to represent the dependency of rheological behavior on loading time and an equiviscous temperature to represent the temperature dependence. Dickinson and Witt (1974) reported a study related to the work done by Dobson (1969). The authors proposed a new mathematical function for representing the loading-time dependency of the rheological parameters and adopted the same mathematical functions developed by Dobson for the temperature dependency (Dobson 1969). These early models were evaluated in several following works, and the accuracy of the models have been tested using many types of unaged as well as aged asphalts (Pink et al. 1980; de Bats and Gooswilligen 1995; Maccarrone 1987). These researchers, although sometimes not consistent, in their observation all agree that asphalts can indeed be represented as linear viscoelastic materials that are thermorheologically simple. They also agree that to characterize such materials two behaviors need to be defined: the dependency of rheology on loading time, and the dependency of rheology on temperature.

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24

Chapter Two More recently, during the SHRP, Christensen and Anderson (1992) proposed using a function derived from the Weibel distribution to represent asphalt rheology. Following that work, Marasteanu, working with Anderson (Marasteanu and Anderson 1999), offered a modification of the original Christensen–Anderson model and introduced what is called the CAM model. The CAM model was used in many studies and is considered an effective phenomenological model for unmodified asphalt binders whose properties are within the linear viscoelastic range. As a result of studying the rheology of modified binders Zeng et al. 2001 introduced a generalized model to represent the complex behavior of modified binders and mixtures, which allows for a shift for the nonlinearity (strain dependency) and the plateau region at high temperatures or very long loading times (Zeng et al. 2001). The model reduces dynamic test data, measured at multiple temperatures and strains, by constructing single complex modulus and phase angle curves. The model was considered universal because it is used to reduce the test data for the binders and mixtures with four formulations for complex modulus mastercurve, phase angle mastercurve, temperature shift factor, and strain shift factor. Details of the development of the model are found elsewhere (Zeng et al. 2001). The following is a brief review: The complex modulus mastercurve of both asphalt binder and mixture may be expressed by the following equation:

G* = Ge* +

where

Gg* − Ge*

(2-2)

[1 + ( fc f ′)k ]m /k e

G∗e = G∗(f → 0), equilibrium complex modulus G∗g = G∗(f → ∞), glass complex modulus fc = location parameter with dimensions of frequency f´ = reduced frequency k, me = dimensionless shape parameters

The phase angle mastercurve is represented by the following equation: ⎧⎪ ⎡ log( fd f ′) ⎤ 2 ⎫⎪ δ = 90I − (90I − δ m ) ⎨ 1 + ⎢ ⎥⎦ ⎪⎬ Rd ⎪⎩ ⎣ ⎭ where

−md

2

(2-3)

dm = phase angle constant f´ = reduced frequency fd = location parameter with dimensions of frequency Rd and md = dimensionless shape parameters I = 0 if f > fd I = 1 if f < fd for binders I = always 0 for mixtures

Equation (2-3) satisfies the requirement that the phase angle varies from 90 to 0° when the frequency is increased from zero to infinity for asphalt binders. For asphalt mixtures, this equation satisfies the requirement that the phase angle increases from 0° to a peak value and returns to 0° when the frequency is increased from zero to infinity.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Formulatifon of temperature and strain dependencies followed the WLF (WilliamsLandel-Ferry) equation (Williams et al. 1955) to express the temperature shift factor log

c (T − T0 ) aT (T ) =− 1 aT (T0 ) c2 + (T − T0 )

(2-4)

where T0 = reference temperature c1 = constant c2 = temperature constant In view of the similarity of strain and temperature dependencies for constructing single curve purpose, the WLF equation is utilized to express the strain shift factor: log

aγ (γ ) d (γ − γ 0 ) =− 1 aγ (γ 0 ) d2 + (γ − γ 0 )

(2-5)

where g0 = reference strain d1 = constant d2 = strain constant The reduced frequency in Eqs. (2-2) and (2-3) is defined as follows: log

f = log a = log aT + log aγ f′

(2-6)

where a = overall shift factor aT = strain shift factor ag = strain shift factor The reference temperature, T0, and reference strain, g0, in Eqs. (2-4) and (2-5) can be arbitrarily chosen at convenience. At the reference values, the respective shift factors are unity, or their logarithms are zero. The logarithms of both shift factors are the amounts of shift to form a single curve; one unit represents a shift of one logarithmic decade, positive to the direction of high frequency or low loading time. Based on testing a relatively large number of binders and mixtures Zeng and Bahia characterized the models in Eqs. (2-2) through (2-5). The reference temperature and reference strain were chosen to be 52°C and 0 percent, respectively. The reference strain is chosen assuming that material properties at zero strain represent linear properties, which cannot be directly measured but can be projected from measurements at strains greater than zero. For binders, constants G∗g = 1.0 GPa, G∗e = 0, and me = 1 were assigned as it has been shown that Gg ≈ Eg/3 ≈ 3.0/3, GPa = 1.0 GPa and that asphalt binders are nearly a linear viscoelastic fluid; parameters k and fc were estimated using a minimum square of error fitting procedure. For mixtures, all five parameters were estimated to give the best fit by minimizing the square error routine. An example of the fitting results for binder and mixture is presented in Figs. 2-3 and 2-4. The binder in the example is the PG 76-22 modified by ethylene terpoly; the aggregate in the mixture is crushed limestone with fine gradation.

The Asphalt Viscoelastic and Failure Properties Selected in the SHRP Although there are many methods of characterizing viscoelastic properties, SHRP researchers selected the dynamic (oscillatory) testing as the best technique to represent

25

26

Chapter Two

FIGURE 2-3 Example of binder and mixture complex modulus and phase angle master curves (PG 76-22 ethylene terpoly modiﬁed binder and ﬁne limestone aggregate).

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders

FIGURE 2-4

Temperature and strain shift factors associated with Fig. 2-3.

the behavior of this class of materials. As previously discussed, in the shear loading mode, the dynamic modulus (G∗) and the phase angle (d) are measured. G∗ represents the total resistance to deformation under load, while d represents the relative distribution of this total response between an in-phase component and an out-of-phase component. In addition to the prefailure properties of asphalts as measured by rheology, their failure properties need to be characterized. Asphalt’s failure behavior is also highly dependent on temperature and time of loading (Dongre 1994). At low temperatures,

27

28

Chapter Two failure occurs in a brittle manner with a plateau zone showing a strain at failure that is relatively small (limiting value of approximately 1.0 percent strain). As temperature increases, a transition from brittle to ductile failure can be observed which, at high temperatures, converts into a flow zone. The most critical part of this behavior for pavement applications is the temperature and loading rate at which the transition from the brittle to the ductile behavior occurs. For many unmodified asphalts, there is some correlation between stuffiness measured at small strains (rheological prefailure properties) and this transition. The correlation, however, does not hold for modified asphalts or specially produced asphalts (Bahia 1995). Failure properties can be measured using the direct tension test (DTT) to measure the strength and strain tolerance of asphalts. In the original version of the SHRP binder specification it was determined that most unmodified asphalts show very similar strength value and thus strength is not needed as a specification parameter. Strain tolerance, however, was found to vary significantly depending on asphalt source (chemistry) and aging. It was also observed that asphalts vary significantly in the temperature zone at which their failure behavior transitions from brittle to ductile behavior. To ensure that asphalts at minimum pavement temperatures are within the ductile region, a minimum value for the strain at failure was included in the specifications. A minimum value of 1 percent strain at failure measured at a specified strain rate was selected as a suitable criterion to ensure ductility at minimum pavement temperature. Figure 2-5 shows a schematic explaining the concept of brittle to ductile transition of asphalt binders. From the earlier discussion of asphalt properties, it is expected that without measuring the rheological and failure properties at the temperature and loading frequency ranges that correspond to pavement climatic and loading conditions, selection of asphalt binders for better-performing pavements and selection of modifiers that can improve the properties of these binders is very difficult. In the NCHRP 9-10 project the use of rheological and failure properties to differentiate between modified asphalts played a different role (Bahia et al. 2001). It was observed that although binders can show similar linear viscoelastic behavior, their nonlinear behavior and resistance to damage can vary significantly. It was clearly observed that the assumption that energy dissipated during viscoelastic testing (particularly within the linear range) could be dissipated in more

FIGURE 2-5

Schematic showing the concept of brittle—ductile transition for asphalt binders.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders than one form. A significant part could be dissipated in damping as opposed to damage and thus testing for accumulated damage is necessary to differentiate between the two mechanisms. This issue was found to be particularly important for modified asphalts since most are manufactured to enhance elasticity thus causing an increase in delayed elasticity (cause for damping). New testing protocols were, therefore, introduced to capture the rate of damage accumulation in the linear and nonlinear viscoelastic range. Depending on G∗sind and G∗/sind parameters has proven to be insufficient because these parameters are derived based on the concept that all energy dissipated is lost in damage. The subject of damage resistance is covered in a subsequent section.

The Need for Asphalt Modification Modification of asphalts is prompted mainly by the limitation of the conventional refining practices used today in producing asphalts from crude petroleum that can resist distresses. The chemical composition of asphalt and, in consequence its properties, are largely dependent on the crude source and the refining process. Asphalt production in most refineries is a secondary process that cannot compete with fuel and other products in revenue generation. Therefore, production of better-performing asphalts is not one of the common strategies in petroleum refining. When the produced asphalt does not meet climate, traffic, and pavement structure requirements, modification has been used as one of the attractive alternatives to improve asphalt properties. In effect, what is called conventional asphalts or straight run asphalts have a range of rheological and durability properties that are not sufficient for resistance on distresses caused by the increase in traffic and total loading on current highways. Modification by specialized refining practices, chemical reaction, and/or additives has been found to improve contribution of asphalt binders to resistance of asphalt mixtures to various modes or pavement distress. This improvement is recognized to result in life-cycle cost savings and thus use of modified asphalts has been steadily increasing for the last 20 years or so. In a recent survey of the state highway agencies in the United States, 35 out of 47 agencies that responded indicated that they plan on increasing the use of modified binders in road construction, 12 were expecting to use the same amount of modified asphalt, and none indicated that they plan on reducing amount of modified asphalts. The majority of the agencies have cited premature distress such as rutting and fatigue cracking as the main reasons for justifying the use of modified binders, which on average increase the initial cost of construction (Bahia et al. 1998). Asphalt modification using additives dates back to the last century (King et al. 1999). Patents for using polymers to modify asphalts date back to 1823 (Isacsson and Lu 1995). Test projects were placed in Europe in the 1930s and in North America in the 1950s. In the early and mid-1980s the introduction of newer polymers and European technologies resulted in proliferation of asphalt modification in the United States. By 1982 more than 1000 technical articles had been published on polymer modified asphalts or mixtures, and more recently there is a continuing emphasis on this subject (Bahia et al. 2001).

Asphalt Modification Strategies A modifier can be selected to improve one or more of the main performance related properties of asphalts. Also, different modifiers that affect different properties can be combined to improve several properties. There have been numerous tests used to quantify each of the properties and measure the effectiveness of certain additives to

29

30

Chapter Two

FIGURE 2-6 Schematics shown the target change in rheological and failure properties expected from modiﬁcation.

improve asphalt properties. This chapter covers some of the key techniques to measure properties of modified asphalts and gives the background of these techniques. Ideally a modifier will change rheological properties to match requirements as defined by resistance to pavement distresses as shown in the first part of Fig. 2-6. It would also change failure properties such that binders would tolerate higher stresses and strains before failure due to static or repeated loading as shown in the second part of Fig. 2-6.

Asphalt Modifiers Currently Used There are currently a large number of modifiers used for paving grade asphalts. Table 2-2 lists several published surveys of asphalt modifiers and identifies the general types of modifiers that each study had identified. Asphalt modifiers can be classified based on the mechanism by which the modifier alters the asphalt properties, based on composition and physical nature of the modifier, or based on the target asphalt property that needs to be modified. In the NCHRP 9-10 project (Bahia et al. 2001) a list of modifiers classified based on the nature of the modifier and the mechanism by which it alters asphalt properties was developed. A total of 55 modifiers classified in 17 generic classes were identified. The information gathered about these modifiers indicates that they vary in many respects. Some modifiers are particulate matters while others disperse completely or

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders

Terrel and Epps 1989

Peterson 1993

Romine et al. 1991

Moratzai and Moulthrop 1993

McGennis 1995

Isacsson and Lu 1995

Banasiak and Geistlinger 1996

Reference

Thermoplastic Polymers

X

X

X

X

X

X

X

Thermoset Polymers

X

X

X

X

X

X

X

Fillers/Reinf. Agents/ Extenders

X

X

X

X

X

X

X

Adhesion Promoters

X

X

X

X

X

X

Catalysts or Chemical Reaction Modifiers

X

X

X

X

X

X

X

Aging Inhibitors

X

X

X

X

X

X

X

Others

X

X

X

X

Total number of existing brands or types

-

-

82 (27 ASA)

48

31

62

Modifier

46

ASA: Antistripping additives

TABLE 2-2

Recent Surveys of Asphalt Modifiers

dissolve in the asphalt. The modifiers range from organic to inorganic materials, some of which react with the asphalt while others are added as inert fillers. The modifiers generically vary in their specific gravity as well as other physical characteristics. They are expected to react differently to environmental conditions, such as oxidation and moisture effects. Not all these types of modifiers are being currently used. The frequency of use varies significantly depending on marketing of the modifiers, experience of contractors and agencies, and cost. In a survey conducted in 1997, state highway agencies indicated that polymer modifiers are among the most widely known and used in practice. Table 2-3 depicts the results of the survey (Bahia 1998b). Table 2-3 shows that the elastomeric polymer styrene-butadiene-styrene (SBS) is the most frequently used to target a variety of distresses among which permanent deformation is the most common. As indicated in Table 2-3, many agencies did not indicate the specific target distress for which they are using the modifier.

Critical Properties of Modified Asphalts Effects of Modification on Viscoelastic Properties Measuring properties of modified binders has in general followed the testing technology used for unmodified asphalts but in many specifications expanded to capture some

31

32

Chapter Two

Target Distress/Property (No. of Agencies) Type

No. of Agencies

PD∗

FC†

LTC‡

MD§

AR¶

Styrene Butadiene Styrene (SBS)

28

18

8

10

3

6

Styrene Butadiene (SB)

16

13

5

5

0

2

Styrene Butadiene Rubber Latex (SBR)

17

10

4

4

1

2

Tire Rubber

3

1

Ethyl Vinyl Acetate (EVA)

6

3

Fatty Amidoamines

8

4

Polyamines

6

4

Hydrated Lime

4

3

Others

7

3

Natural Asphalts

6

5

12

3

1

7

4

1

Class

PolymerElastomer

PolymerPlastomer

Anti-Stripping Agents

Hydrocarbons

Cellulose Polypropylene

Fibers

1 1

1

Polyester

6

4

Mineral

3

1

ProcessedBased

Air Blowing

4

2

Mineral Fillers

Lime

4

1

Anti-Oxidants

Hydrated Lime

7

4

Extenders

Sulfur

4

∗

Permanent Deformation. Aging Existence.

†

Fatigue Cracking.

‡

1 1

Low Temperature Cracking.

1

1

§

Moisture Damage.

¶

TABLE 2-3

Modifiers Most Commonly Used by State Highway Agencies

unique properties that modifiers were used to induce or enhance in the base asphalt (Bahia 1995; King et al. 1992; Brule and Maze 1995). As discussed at the beginning of this chapter modification targets included rigidity, elasticity, strength, brittleness, and damage resistance. To show the general trends in effect of modifiers on asphalt viscoelastic properties, typical examples are shown in Fig. 2-7. The figure includes two plots, plot (a) depicts

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders 10

100 c%

2c % 80

8

60 6 40 4

Delta at 10 r/s, °

G* at 10 r/s, log Pa

0%

20

2

−30

0 30 Temperature, °C

60

0

(a) Modified with an SB-polymer

9

Phase angle, °

80 G* CRM 60 40

Delta CRM G* AC

7 5

Delta AC 3

20 0 −5

0 5 10 Reduced frequency, log(s)

G* at 25°C, log Pa

100

1 15

(b) Modified with crumb rubber

FIGURE 2-7 Isochronal rheological curves for an asphalt before and after modiﬁcation with two types of additives. (a) Modiﬁed with an SB polymer; (b) modiﬁed with crumb rubber.

rheological mastercurves measured using a dynamic shear rheometer for an asphalt before and after modification with an styrene-butadiene (SB) polymer at two different concentrations (c = 3 and 2c = 6 percent). Changes in both G∗ and d as a function of temperature are shown. The effects of this modifier show favorable trends of change: at high temperatures, G∗ is higher while d is lower. This indicates an increase in rigidity and in elasticity, which results in better resistance to permanent deformation. At intermediate temperatures (0 to 30°C) lower values of G∗ can be observed while d values remain indifferent. The reduction in G∗ values is favorable for fatigue cracking under straincontrolled conditions, typical of conditions for thin pavements. At low temperatures (−20 to 0°C), a similar, or more pronounced, reduction is observed for G∗ and a minor increase in d is seen. Both these effects are favorable since they make the binder less rigid and less elastic or more prone to stress relaxation under load. The changes shown appear to improve the properties with respect to pavement performance at all temperatures. Considering the relative changes in the G∗ and d, it is evident that the main effect is the change in the rigidity of the binder as measured by G∗. The data shown in Fig. 2-7(a) indicate that while the G∗ value is increased at 60°C by 100 to 200 percent, the d value is reduced by approximately 16 to 30 percent. At low temperatures the same trend can be

33

34

Chapter Two observed; the G∗ value is reduced by 40 to 50 percent while the d value is increased by only few degrees. Similar trends in changes were observed for the other types of polymers that were used in the study. Considering the fact that energy dissipation and rate of relaxation of binders are functions of sind or tand, it appears that effects of these commonly used polymeric additives on binders at small strains or stresses are mainly caused by change in rigidity while only secondary effects are caused by changes in elasticity. It is, however, recognized that type of polymer and concentration can have different effects (Bouldin et al. 1991; Brule et al. 1986 and 1988; King et al. 1992; Anderson and Lewandowski 1993; Collins and Bouldin 1991; Lesueur et al. 1997; Masson and Lauzier 1993). Figure 2-7(b) depicts master rheological curves for an asphalt before and after modification with a crumb rubber modifier (CRM), at 15 percent weight concentration (Bahia and Davies 1994). The figure is in terms of loading frequency rather than temperature. As discussed earlier, frequency and temperature are interchangeable; the effect of high temperature corresponds to that of low frequencies, and vice versa. Changes in mastercurves are similar to the changes observed for polymer modification shown in Fig. 2-7(a). G∗ values increase at low frequencies (high temperatures) while they decrease at intermediate and high frequencies (intermediate and low temperatures). The d values are lower at low frequencies but higher at high frequencies. The relative changes of either parameter are of the same order of magnitude as for the polymer modification. The effects of CRM can, therefore, be also described as mainly changes in rigidity of the asphalt. The mechanism by which CRM changes properties is, however, different; while for most polymer modifiers the polymer is completely dispersed in the asphalt and causes changes in the molecular structure of the asphalt, the CRM is observed to keep its physical identity and behave as a flexible particulate filler in the asphalt. The overall effect of CRM on rheological mastercurve is reduction of dependency of G∗ and d on frequency. This effect is similar in nature to the effect of polymer modification despite the difference in the nature of material. Polymer modification usually results in a more homogeneous binder which is more favorable than the nonhomogeneous CRM modification. The trade-off, however, is the relatively higher cost of the polymer modifiers compared to the CRM. Various researchers have conducted extensive evaluation of CRM modifications (Oliver 1982; Chehoveits et al. 1982; Bahia and Davies 1994).

Effects of Modification on Failure Properties Using the direct tension test developed by SHRP, the binders modified with the different additives were tested at temperatures ranging between −30 and 0°C. The tests were conducted at a deformation rate of 1.0 mm/min in three replicates and the stress and strain at failure were calculated. To evaluate the effect of the modifiers, the stress and strain values at failure of the base and the modified binders are compared. Figure 2-8 depicts the results for the same additives discussed in the pervious section. Figure 2-8(a) shows strain at failure and stress at failure plots as a function of temperature for an asphalt before and after modification with 3 and 6 percent of the SBbased polymer. The strain curves show that the polymer increases the strain at failure within the brittle and the brittle-ductile zones but converge to the same values as the flow zone is approached. The effect can be considered as shifting the strain at failure curve horizontally to lower temperatures without significant changes in shape of curve. The effect of polymer addition is favorable as it tends to increase the strain at failure within the critical region. The results shown also indicate that the effect is more favorable with higher concentration of the polymer. The stresses at failure are similar for all binders which indicate that the polymer does not result in significant changes in strength of binders.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders 100 Unmodified 3 % SB 6 % SB

10

10

Stress

1

1

Stress at failure, MPa

Starin at failure, %

100

Strain

Test Temperature, °C −30

−25 −20 −15 −10 −5 (a) Modified with SB-Polymer

100

100

Starin at failure, %

.1 0

Unodified CRM1 CRM2

10

10

Stress

1

1

Stress at failure, MPa

.1 −35

Strain

Test Temperature, °C .1 −35

−30 −25 −20 −15 −10 −5 (b) Modified with crumb rubber

.1 0

FIGURE 2-8 Failure strain isochronal curves for an asphalt before and after modiﬁcation. (a) Modiﬁed with SB-polymer; (b) modiﬁed with crumb rubber.

The results shown may not apply to all types of polymer modifiers. The effect of different polymers on failure properties is expected to depend largely on the type of interaction between the asphalt and polymer, the molecular nature of the polymer additives, and the way it is dispersed in the asphalt. The effect of polymers on failure properties can be hypothesized in different ways. One hypothesis is that polymers form some kind of molecular network inside the asphalts resulting in more strain-tolerant

35

36

Chapter Two material. Another hypothesis is that the dispersed polymer particulates may serve as reinforcements, arresting microcrack propagation and increasing toughness of binders. The typical trend that can be observed from review of polymer modification works, however, is that not many polymers used currently improve low-temperature failure properties. This may be attributed to the fact that until recently there has been no simple technique to measure the brittle failure of asphalt, and also to the fact that none of the used binder specifications address the brittleness of asphalt in a rational and fundamental form. These issues did not encourage many polymer modifier producers to concentrate on designing a modifier to mainly enhance low-temperature failure properties. Figure 2-8(b) depicts failure plots for an asphalt before and after modification with crumb rubber at 10 percent (CRM1) and 20 percent (CRM2) concentration by weight of total binder. The effect of the CRM is similar to the polymer modification with respect to the strain at failure values; higher strains are observed at low temperatures but similar strains are observed as the flow region is reached by the binders. The effect also represents a shift of the failure curve along the temperature scale toward lower temperatures. The shift is larger for the higher CRM content. The stresses at failure, however, show a trend different from the polymer modification. The CRM results in stress values that are significantly higher than the unmodified asphalt at all temperatures. This behavior can be attributed to the reinforcing effect of the rubber particles. The crumb rubber particles do not dissolve in asphalt; the particles maintain their integrity and tend to swell in asphalts resulting in effective volumes that are larger than their initial volume (Bahia and Davies 1994; Chehoveits 1982). It is speculated that the swelling results in selective absorption and/or adsorption of certain components of the asphalt. Such interactions are expected to reinforce the matrix of the binder and result in higher strength, as observed in the figure. The increase in strain and stress at failure is favorable for paving grade asphalts, particularly when it is not accompanied by an increase in stiffness.

Effects of Modification on the Superpave Grading Properties The Superpave performance grading (PG) system was developed to evaluate binder properties at specific temperatures with specific testing systems (Anderson and Kennedy 1993; Anderson et al. 1994; McGennis 1995). The AASHTO MP1 procedure includes details of testing and grading of binders. Figure 2-9 is a schematic describing

FIGURE 2-9

The Superpave grading system.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders the different test systems and pavement distresses covered in the PG system. Workability is measured at 135°C using a rotational viscometer; resistance to permanent deformation and fatigue cracking is measured using a dynamic shear rheometer (DSR) at maximum pavement temperature and intermediate pavement temperatures. For evaluating the resistance to low-temperature cracking, the bending beam rheometer (BBR) and the direct tension testing (DTT) device are used. Numerous studies have been conducted to evaluate how different modifiers affect the grades of conventional asphalts. Figure 2-10 depicts an example of effect of three different types of crumb rubber based additives on performance measures. The effect is calculated in terms of ratios of the value of the parameter after modification to the parameter before modification. It is apparent that for the rutting parameter (G∗/sind) the ratios are higher than one while for the fatigue parameter, the ratios of G∗sind is much lower than one. S(60) ratios are also lower than one while the ratios of m(60) and strain at failure are very close to the value of 1.0 (no change). The data in Fig. 2-10 show that this type of modification has its main effects at high temperatures. This is expected when the nature of the CRM modifier is considered. Crumb rubber acts mainly as a flexible filler; at high temperatures it is stiffer than the asphalt and thus contributes significantly to the increased moduli. With decreasing temperatures, the asphalt becomes stiffer while the crumb rubber properties do not change significantly. At a certain temperature, the asphalt may become stiffer than the crumb rubber and thus a reduction in stiffness can be observed for the modified binder. Crumb rubber, at moderate concentrations that are used in practice (10 to 20 percent), however, cannot reduce stiffness by large margins because of its own relatively high stiffness at low temperatures. It is, therefore, expected that the main effects of crumb rubber remain to be seen at higher temperatures and to affect mainly the rutting parameter. The use of the Superpave technology to evaluate modification is rather widespread today and it became the standard for estimating the relative value of additives and to justify some of the added initial cost to buy and use selected modifiers. Concerns, however, were raised about using this technology without verification that it applies to modified binders. The concerns were the results of a few field problems and because the original SHRP did not include a variety of modified asphalts used in practice after the program was completed. The following sections address these concerns (Bahia et al. 1998b; Bahia et al. 2001).

Relative change (CRM/neat)

10 RB1

RB2

RB3

8 6 4 2 0 G*/sind

FIGURE 2-10

G*/sind

S(60)

m(60)

F.strain

Typical effects of CRM-modiﬁer on the Superpave grading parameters.

37

38

Chapter Two

Complexity of Modified Binders As discussed previously, one of the main objectives of the SHRP was to develop test methods for characterization of asphalts that are equally applicable to unmodified or modified asphalt cements, collectively called asphalt binders (Anderson et al. 1994). There were, however, two problems that raised concerns about applicability of PG specification to all asphalt binders. The first was that the majority of the testing during SHRP was done on unmodified asphalts of certain PG grades that did not cover the extreme grades required by the new specifications. A review of the asphalts included in the SHRP Materials Reference Library (MRL) indicates that they range between a PG 64-28 and a PG 46-34 with one PG 70-22. This range of grades does not cover extreme grades that are being specified for high-volume traffic in warm regions and grades being considered in many cold regions. The second problem was the fact that these extreme grades, such as 76-22, 82-22, 64-34, and 58-40 did not exist at the time SHRP research was active. The concerns about the Superpave binder specification applicability to all asphalt binders resulted in the initiation of the NCHRP 9-10 project, “Superpave Protocols for Modified Asphalt Binders” (Bahia et al. 2001). The first phase of the project included a survey of users and producers of modified binders to identify the types of asphalt additives most commonly used in practice, to summarize concerns about the use of Superpave protocols for modified asphalts, and to define the current and future needs for modified asphalts. It also included a comprehensive literature review to evaluate the research done to evaluate modified binders using the Superpave protocols. The first phase resulted in recommendation for classifying asphalt binders into simple and complex binders. Based on this classification, it is recommended that the Superpave binder specification be used for asphalts that exhibit simple rheological behavior. The first phase has also resulted in the recommendation for the addition of new or revised testing procedures to characterize specific properties that are important for asphalts modified with additives. These procedures include modification of the rolling thin film oven test (RTFOT) procedure, development of the particulate additive test (PAT) and the laboratory asphalt stability test (LAST) (Bahia et al. 1998). The following sections cover the details of the deficiencies in the existing SHRP PG grading protocols and the recommended modifications.

The Assumptions in the Superpave Binder System The Superpave binder specification contains criteria based on assumptions that were made to simplify the testing required and evaluate characteristics that are most critical to pavement performance. These assumptions although were validated for neat asphalts, may not be valid for asphalts modified with different additives. Based on detailed review of the SHRP Project A-002A report (Anderson and Kennedy 1993; Anderson et al. 1994) and other recent published literature (Bahia et al. 1998; Bahia et al. 1999), the following assumptions are found to be the most important that are related to the behavior of modified binders. 1. No strain/stress dependency of rheological response (wide linear viscoelastic range). 2. No shear rate dependency of viscosity (wide Newtonian range). 3. Testing at one loading rate is sufficient (similar loading rate dependency).

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders 4. Binders are homogeneous and isotropic (no sample geometry or particulate additives effects). 5. Similar time-temperature equivalency for all binders (one shift is used). 6. Binders are not thixotropic (no effect of mechanical working). 7. Stability of asphalts is affected mainly by oxidation. The essence of the above assumptions is that asphalt binders are simple systems that can be characterized using linear viscoelasticity and simple geometry within which stress and strain fields are simple to calculate. Except for number 7, the assumptions in the specification are related to rheological and thermorheological behavior. Assumption number 7 may not be very important for neat asphalt, but it is critical for asphalts modified with additives. In a recent study conducted as part of the NCHRP 9-10 project, each one of the listed assumption was challenged and data were collected to prove that many modified binders used in practice do not satisfy these assumptions. There were, however, important reasons why researchers in SHRP had considered these assumptions to be reasonable among which the two main are: (1) Most asphalts tested during the SHRP showed a relatively wide linear range. (2) Asphalt pavement structural design should be selected such that materials will be exposed to small stresses and strains. Therefore, the specification testing was designed to measure behavior within the linear range expecting that asphalts in the field will be mainly performing within this region. Some of the important behaviors that will need to be characterized were, however, not addressed due to need for simplification. Of particular importance are the possible thixotropic effect and the dissimilarity of the effect of repeated loading. Figure 2-11 is an example of possible effects of repeated loading at two different strain levels. It is clear that some modified asphalts can show significant reductions in G∗

FIGURE 2-11 Thixotropic behavior of modiﬁed binder at 10 percent strain.

39

40

Chapter Two due to mechanical working (repeated loading). In the field of asphalt emulsions, technology is available to produce such thixotropic materials, such as high-float emulsions. It is reasonable to assume that this same technology will find its way in the production of paving grade asphalts. The dependence of rheological response on loading rates and temperature is known in many fields to be material specific. In the asphalt field, behavior of modified and unmodified asphalts varies significantly in responding to change of traffic speed in the field, or testing rate in the lab. Because of this variation, using common time-temperature equivalency factors could result in significant errors in estimating the effect of traffic speed on response of various asphalts. Figure 2-12 depicts that the relationship between effect of changing the loading rate and effect of changing temperature on G∗ for typical paving grade asphalts. The R2 values are 0.17 for the shift from 5 to 10 rad/s and 0.30 for the shift from 1 to 10 rad/s. This clearly indicates that there is hardly a correlation between the effects of loading time and temperature changes on changes in G∗ values. In other words, the practice of changing the temperature grade to compensate for traffic speed cannot be justified. The polymer-modified asphalts appear to be significantly less affected by a change in traffic speed (frequency) than the unmodified asphalts. Also, the effect on G∗ of changing temperature by 6°C is highly variable, even for the unmodified asphalts. For the asphalts examined, which include 32 sources tested during the SHRP, the change ranged from 1.7 to 2.6 folds. This result is expected, since temperature susceptibility is highly asphalt source specific. The simplification used in the current specification is, therefore, not acceptable. Regarding the assumption of homogeneity, the Superpave binder testing is done using selected geometries for different temperatures and response types. These geometries were selected to give stress fields that can be easily estimated and thus used in calculating the material response. The concept is based on the assumption that binders are homogeneous materials that exhibit isotropic behavior. Therefore, testing in one mode of loading, using a single geometry, can give a comprehensive evaluation of the material behavior under different geometric and loading conditions. For some modified asphalts, however, the additive used results in an anisotropic binder and thus

FIGURE 2-12

Relationship between 6°C temperature change and loading rate.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders the response under loading can be highly dependent on sample geometry. Additives such as fibrous materials with high aspect ratio, and other particulate additives of relatively large random shapes are examples that can result in anisotropic behavior of the modified binders. It is known that as the ratio of particle size to sample size increases, and/or as the volume concentration of the additive increases, the geometry of the sample can interfere with the measurement. The size and the volume concentration of particulate additives are not fully controlled in the current specification. The only existing limitation is particulates should not exceed 250 μm, which was selected arbitrarily (AASHTO TP5). Size and volume concentrations are both required in limiting the amount of additive that can result in interference with measurements. It is, therefore, necessary that the presence of solid additives be detected and the nature of the additive be checked. A new test, called the PAT is described in this article to address this need. Stability of binders in the current Superpave binder specification has also been oversimplified by considering oxidation as the only mechanism by which asphalt binder properties can change in service. Modified asphalt binders, however, can undergo changes due to factors other than oxidation. It is recognized that modified binders are multiphase systems in which the modifiers are dispersed into the asphalt cement phase. This dispersion is generally accompanied by a degree of incompatibility that is affected by various physical, thermal, and chemical factors. Excessive incompatibilities can negatively affect the performance of these binders as gross separation occurs during storage and handling of the binder, production and transportation of asphalt mixtures, and during the construction of pavement layers. There are four separation mechanisms (physical, thermal, chemical, and oxidation) that need to be considered. In the current Superpave system, there are protocols for characterizing the oxidative stability [Pressure Aging Vessel PAV and RTFO]. There are also proposed protocols for measuring physical separation (cigar tube test). There are, however, no provisions to separate these effects and to take into account the different physical, thermal, and chemical treatments expected in the field.

New Classification of Asphalt Binders Two main conclusions can be drawn from the review of the Superpave binder test protocols and specification and the existing knowledge of modified binders: 1. The existing protocols cannot be used to fully characterize all asphalt binders modified with different additives. The main reason is that they are based on simplifying assumptions that cannot be reliably extended to modified binders. 2. Some additives can result in binders that are too complex to be evaluated by any binder-only protocols. Such additives will result in anisotropy or interference with testing geometry such that only actual replication of films that will exist in mixtures will allow reliable estimation of their role in pavement performance. To apply the current Superpave binder protocols for modified binders these two conditions have to be satisfied. In other words, the modified binders have to be “simple” rheological systems. Based on this concept, asphalt binders should be classified into simple binders and complex binders, as follows. Simple Binders: Asphalt binders with simple behavior that do not violate the assumptions, which the PG-grading system is based upon; these assumptions include

41

42

Chapter Two 1. Wide linear range (independence of strain) 2. Nonthixotropy (independence of mechanical working effects) 3. Isotropy and independence of sample geometry (no additives that result in geometric effects) Complex Binders: Asphalt binders that cannot be classified as simple binders because their behavior violates one or more of the PG-grading system assumptions. This new classification is based on the hypothesis that the role of simple binders in mixture and pavement performance can be estimated using the existing (or slightly revised) Superpave binder protocols, regardless of their constituents or the method of production. The role of complex binders in mixture and pavement performance, on the other hand, cannot be estimated using binder testing. Mixture testing will have to be used. An asphalt binder can be classified as a complex binder because of the physical characteristics of the modifier or because of the nature of the effect of the modifier. Binders modified with particulate matter can be complex because of their dependency on sample geometry. Other binders can be complex because they are thixotropic or strain dependent. To qualify asphalt as a simple binder, two initial tests were recommended: (a) strain sweep, and (b) time sweep. Each should be done at high and intermediate temperatures. Both these tests are easily performed using the current version of the Dynamic Shear Rheometer (DSR). A change by more than 10 percent for either test will indicate a complex behavior. If the binder passes both tests, an evaluation of the nature of additive was recommended. A new test called the PAT is proposed to separate the additive and evaluate its nature. If the binder does not contain more than 2 percent by volume of a particulate additive, and passes the strain and time sweep tests, it can be graded as a simple binder and can be graded according to the PG system. The current Superpave-testing protocols do not cover stability under nonoxidative conditions. The stability under nonoxidative conditions requires a new test method. The LAST was proposed to measure potential for phase separation and thermal degradation of asphalts. The LAST can be used to simulate the conditions of hot storage with and without agitation under minimal exposure to oxidation. It is proposed that potential rate factors for degradation (Kd) and for separation (Ks) be calculated from the results of this test used to evaluate the stability of asphalts. In addition to these new tests, the RTFO test needs modification to make it more effective in handling the highly viscous polymer modified asphalts. The following sections give a brief overview of the PAT and the LAST. A detailed description of the background and the initial data collected is available in another publication by the author (Bahia 1999). The finding of the study on modified asphalts in the NCHRP 9-10 project (Bahia et al. 2001) resulted in considering a different approach to testing of asphalts in general. The approach is to evaluate damage resistance as apposed to linear viscoelastic properties. It is becoming clear that the critical role of modification is not reflected in the linear viscoelastic range but is more profoundly shown in the damage resistance range. The following section describes this approach.

Damage Resistance Characterization In the previous section the limitations of applying the Superpave binder specification to modified binders was described. It was shown that modified asphalts are generally nonlinear viscoelastic materials that exhibit complex rheology. They also exhibit unique

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders stability and composition characteristics that are not typical of unmodified asphalts and thus require special testing. In this section a new approach to test performance related properties in the linear and nonlinear domains is described. The approach is based on the concept of characterizing resistance to damage as a better alternative to link lab testing results to pavement performance. To understand the causes of pavement distresses and study their effects on the pavement performance has been the goal of asphalt researchers attentions for several decades (Kim et al. 1997; Majidzadeh et al. 1972; Dijk 1975; Dijk and Visser 1977; Moavenzadeh et al. 1974; Monismith and Deacon 1969). Among these distresses, rutting and fatigue, which are recognized as associated with the increasing traffic volume, have led to an apparent reduction in the long-term performance of flexible pavement. To predict these damages efficiently, many research projects have been designed to evaluate the effects of different factors on the performance of the asphalt pavement. Although it is recognized that these distresses are mainly caused by the deformation and/or damage within the asphalt binders, very few studies have used binder testing to evaluate damage behaviors of binders under simulated testing conditions (Bahia et al. 2001). While it is recognized that mixture factors and pavement structure factors can have important effects, efforts to understand damage behaviors are very limited. In the existing Superpave specifications, the parameter G∗sind is used to rate the binder contribution to fatigue damage resistance, while G∗/sind is used to evaluate the rutting damage resistance (Bahia and Anderson 1995). Both parameters were selected based on the dissipated energy concept as applied to linear viscoelastic range. However, there is a significant lack of information about the role of binder composition or rheological properties of binders in damage progression under cyclic loading. As part of the NCHRP 9-10 project (Bahia et al. 2001) two sources (gravel and crushed limestone) and two gradations (12.5 mm coarse and 12.5 mm fine) of aggregates were used to study the relationship between mixture damage behavior and the Superpave binder parameters. One asphalt binder content was used throughout the testing. Nine modified asphalts were included in the study. All of these asphalts were modified from one base asphalt. Five asphalts were modified with elastomers: ethylene-propylene diene terpolymer (Ethylene terpoly), styrene-butadiene-styrene (SBS) radial, styrene-butadiene diblock (SB), styrene-butadiene-styrene (SBS) linear, and styrene-butadiene rubber (SBR). One asphalt was modified using stabilized polyethylene (PE) and three oxidized asphalts were also included in this study. The oxidized asphalts were produced by steam distilled, oxidized by back blending (BB) and oxidized by straight run (SR). All of the asphalts were aged using an RTFO prior to conducting the binder tests. To study the effect of modified asphalts on the rutting behavior of asphalt mixtures the Repeated Shear Constant Height (RSST-CH) test (Chap. 10) was used. In order to evaluate the effect of binder modification on the results of the RSCH test results, certain mixture behavior indicators are needed. These indicators are commonly derived from the typical power law model recommended for representing rutting by the SHRP. The model, defined in Eq. (2-7), includes an initial strain factor (ep(1)) and a slope (S) factor. log ε p = log ε p(1) + S log N

(2-7)

where ep(1) is the total accumulated permanent strain and N is the number of cycles.

43

Chapter Two

FIGURE 2-13 Correlation between average rate of accumulation of strain (S) and G ∗/sind.

It is believed that the initial permanent strain can be affected by many mixture factors that are not related to the binders and this effect is carried to the total strain that is commonly used in comparing mixtures. Based on this, the value of the logarithmic slope (S) is selected as the more representative parameter that needs to be considered for studying the role of the binders in permanent deformation of the mixtures. Figure 2-13 shows the correlation between the average mixture rate of accumulated strain (S) and the parameter G∗/sind measured at 10 rad/s. As can be seen, there is hardly any reasonable correlation. Correlations with mixture rutting of individual aggregate blends showed similar lack of correlations. These results made it necessary to explain the reason of this lack of correlations and search for a better indicator of binders’ contribution to mixtures’ rutting performance. To study effect on the fatigue behavior of mixtures, the flexural beam fatigue test was used in the strain-controlled mode and the test was terminated at 50 percent reduction in modulus of mixture (N50). To look at the overall relations, the fatigue lives of mixtures (N50) were averaged across the aggregate types and correlated with the binder G∗sind values as shown in Fig. 2-14. Average for all Aggregates 1.1E+07 1.0E+07 Binder G∗ sin delta

44

y = −27.925x + 8E + 06 R2 = 0.1874

9.0E+06 8.0E+06 7.0E+06 6.0E+06 5.0E+06 4.0E+06 0

40000 60000 80000 20000 Average mixture fatigue life (N50)

100000

FIGURE 2-14 Correlation between G ∗sind at 10 rad/sec and mixture fatigue life measured at 10 Hz.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders The same lack of correlation is observed, as the R-squared value is lower than 20 percent. Similar values of correlation were observed for the individual aggregates. These results show clearly that the G∗sind of the binders is a poor indictor of the fatigue life of the mixtures (N50), as defined by the number of cycles at which the initial mixture modulus reduces by 50 percent. These findings started the process of looking for a better indicator of the role of binder in fatigue damage and resulted in putting more emphasis on the development of a better binder test for the evaluation of binder fatigue behavior, as will be discussed next.

Development of New Tests for Binder Damage Behaviors Rutting Test Several protocols were investigated to select a test procedure and a rheological parameter that could be used as a more effective indicator of the role of binders in mixture rutting than the parameter G∗/sind. The selection process was based on two main hypotheses: 1. The strains in the binder domains within the mix are significantly larger than those at which the binders are subjected to in the DSR. 2. The cyclic loading with complete reversal in strain or stress is not appropriate for rating the binder’s contribution to rutting resistance caused by cyclic irreversible loading (also called non-steady-state cyclic deformation or more simply repeated creep). The first hypothesis is based on the data collected in a previous study by the author with coworkers (Bahia et al. 1999), which indicated that modified binders vary significantly in their strain dependency. It is also based on the finding that mixture rheological behavior was found highly sensitive to strain level (Monismith 1994). The second hypothesis was based on the concept of the RSCH and the review of literature related to the concept of energy dissipation. To test these hypotheses different testing protocols were used to find a relationship with mixture rutting performance. The protocols included strain sweeps, stress sweeps, time sweeps at constant strain and time sweeps at constant stress. In addition, a repeated creep test was developed and conducted to measure permanent strain behavior of binders. The analysis of the results led to believe that these strain sweeps and time sweeps are not promising. None of the cyclic-reversible tests showed a clear potential to differentiate between binders and to relate strongly to mixture performance. As a result of this finding a detailed review of the dissipated energy concept and the derivation of the binder parameters were initiated. This review indicated that although the cyclic reversible loading could be used to estimate the total energy dissipated during a loading cycle, for viscoelastic materials that combines permanent deformation and delayed elasticity, this type of test does not allow the separation between these two different types of dissipated energy. As shown in Fig. 2-15, during the cyclic reversible loading only the total energy dissipated is possible to estimate. The rutting mechanism, as described by many research efforts and measured in the field, does not include reversible loading required to bring the pavement material to zero deformation. As shown in Fig. 2-16, rutting is in fact a repeated creep mechanism with sinusoidal loading pulses. In this case the pavement layer is not forced back to zero deflection but would

45

46

Chapter Two

FIGURE 2-15

Current concept used in deriving the G ∗/sind for the binder speciﬁcation (MP1).

FIGURE 2-16 Improved application of the dissipated energy concept to derive a fundamental rutting parameter for binders.

recover some deformation due to elastic stored energy in the material of the layers. Under this type of loading, the energy is dissipated in damping (also called viscoelasticity or hysteresis) and in permanent flow. The damping energy is mostly recoverable but requires time to be effectively utilized. The energy related to the permanent flow, however, is lost and thus called permanent. The permanent part of the dissipated energy is believed to be the main contributor to the rutting behavior of asphalt mixtures and pavements. The main problem with the reversible cyclic loading used today is the inability to distinguish between these two mechanisms that result in energy dissipation. Based on this analysis the focus of this study shifted toward developing a creep and recovery testing procedure to better simulate the contribution of binders to rutting resistance based on more fundamental understanding of rutting behavior. Realizing that the cyclic reversible tests that have been tried in the project do not offer a good indicator, and recognizing that there is a fundamental problem with estimating energy dissipation during repeated creep from the cyclic reversible loading, it was decided that a new approach that utilizes repeated creep testing is required. Evaluation of the DSR software and capabilities indicated that a repeated creep test could be conducted on binders using same geometry and temperature range. The testing was, therefore, started to develop a repeated creep test using the DSR.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Creep Tests at 70°C, 300 Pa Shear Stress (Loading is Recovery 9s) 100 cycles 14.0

Accumulated strain

12.0 10.0

PG 82-Oxidized

8.0

PG 82-PEs

6.0 4.0 PG-82-SBSr

2.0 0.0

0

200

400

600 Time (s)

800

1000

FIGURE 2-17 Results of the accumulated strain under repeated creep testing for 3 PG 82 binders at selected conditions of 1-second loading and 9 seconds recovery.

Figure 2-17 depicts the results of the repeated creep testing of 3 binders of the PG 82 grade at selected loading conditions of 1 second loading and 9 seconds unloading. The results show clear distinction between the accumulated permanent strains of the binders which could not be detected using the G∗/sind. It can be observed that the elastomeric binder (SBS) is offering significantly higher recovery during the creep testing and thus results in less accumulated deformation compared to the other binders. The results of the creep and recovery and the G∗ and phase angle are listed in Table 2-4. As shown, the accumulated strain after 100 cycles at 70°C is only 20 percent SBS Modified PG 82-22 (R1B02)

PEs Modified PG 82-22 (R1B09A)

Oxidized PG 82-22 (R1B15)

2.389

6.948

11.599

G at 300 Pa

10989

11379

15272

d at 300 Pa

56.2

60.3

73.9

sind

0.831

0.869

0.961

G∗/sind

13224

13100

15895

Ratio of strain at end of loading cycle (eL) to permanent strain at end of recover (ep) at 1 cycle

2.787

1.656

1.169

eL/ep at 100 cycle

5.571

1.764

1.179

Total strain at 100 cycles, etotal ∗

TABLE 2-4 Comparison of the Creep and Recovery Indicators with the Values of G ∗ and sind Measured with the DSR at Small Strains

47

48

Chapter Two

FIGURE 2-18 Correlation between binder rutting parameter and the average mixture rutting parameter. Slope measured in strain per stress (1/kPa).

of the accumulated strain of the oxidized binder of the same grade. The G∗/sind of the oxidized binder at the testing temperature is 15,900 Pa which is higher than the elastomeric binder that has a G∗/sind value of 13,000 Pa. This inversion of ranking is very critical and can be well explained by the ability of the elastomeric binder to recover under the testing conditions. The recovery, however, is not being captured by the G∗/sind due to the fact that the parameter cannot distinguish between total energy dissipated and the energy dissipated in permanent flow. The results from the creep and recovery test can also be explained by what is known about the molecular nature for these materials. To evaluate the effectiveness of using the creep and recovery binder test, 9 binders of various grades were aged in the RTFO and tested at condition that matches the temperatures and loading-time conditions at which the RSCH testing was conducted. The rates of accumulation of permanent strain of the mixtures were plotted versus the rate from the new binder test, as shown in Fig. 2-18. The correlation between the mixture and binder properties has improved significantly from 23 percent to approximately 68 percent. It appears that the new approach based on the creep and recovery testing is very promising. The correlations with the individual aggregate blends varied between higher than the average value for the crushed limestone to very low correlations for the gravel aggregates. The poor correlations for certain binders were expected since aggregates play a main role in the rutting performance. An analytical approach was used to isolate the effect of binders from the effect of aggregates based on modeling the relationship between mixture and binder accumulation of permanent strain. This type of analysis enabled the separation of the main effect of binders which showed correlations of the mixture and binder rutting behavior in the range of 80 to 90 percent (Bahia et al. 2001).

Binder Fatigue Test As discussed earlier, it is found that the parameter G∗sind is not well related to the accumulation of fatigue damage of mixtures as measured in a beam fatigue test, under strain-controlled conditions. It is believed that the main reason is that the parameter G∗sind is measured in the linear viscoelastic range using small strains. There is a

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders fundamental problem with this approach because it is unlikely to be useful in representing the effect of repeated cyclic loading and the changes in binder properties with accumulation of damage. The fatigue damage behavior was the subject of a previous publication that showed the effect of modification on the nonlinear behavior and specifically the damage behavior (Bonnetti et al. 2002). The effort to develop a new test was focused on simulating the fatigue phenomenon in a binder-only fatigue test. The DSR was used to conduct what is called a time sweep test. The test provides a simple method of applying repeated cycling of stress or strain loading at selected temperatures and loading frequency. The initial data collected were very promising and showed that the time sweeps are effective in measuring binder damage behavior under repeated loading in shear (Bahia et al. 1999). To understand the test results and to establish the best testing conditions that could lead to effective characterization of binder fatigue behavior, all nine binders used in the production of mixtures were tested at conditions that match the mixture beam fatigue conditions. The binders were aged in the RTFO to simulate the effect of mixing and compaction and the testing was conducted at 10 Hz at temperatures as close as possible to the mixture beam fatigue temperatures. The testing of binders was conducted in strain-controlled mode, and to match the mixture strain level, an estimated strain of 3 percent was used for all binders. Figure 2-19 shows the results of the binder testing and indicates that although the initial G∗ values are similar; these binders show significantly different fatigue behavior. Some of these binders did not reach 50 percent of the initial G∗ even after applying close to 1,000,000 cycles while others reached this level of G∗ after only 10,000 cycles. In all these tests the maximum strain at edge of plate was kept constant at 3 percent. To see if the binder fatigue life measured in the strain-controlled binder test has any relationship to the mixture strain-controlled fatigue life, Fig. 2-20 was prepared to show

FIGURE 2-19 Binder fatigue results at 10 Hz, 3 percent strain, and the temperatures selected for the mixture beam fatigue testing (binders were RTFO aged).

49

50

Chapter Two

FIGURE 2-20 Correlation between binder fatigue life and average mixtures fatigue life measured at the same temperature and frequency.

the relationship between the average mixture performance and the binder fatigue life as determined by the number of cycles to 50 percent of the initial G∗ value. As shown in the figure, there is a high correlation (R-squared = 84 percent) for the nine binders that showed binder fatigue failure. This result is very encouraging and indicates that the newly developed binder fatigue test is promising and could be a better indicator of fatigue damage of mixtures.

Selection of New Damage Behavior Parameters Binder Rutting Parameter Based on the above analyses of the repeated creep tests, it was determined that a creep and recovery test would significantly improve the estimation of resistance to accumulation of permanent strain of binders and their contribution to resistance of mixture rutting. To derive a new parameter for rutting resistance, a basic approach based on viscoelasticity was followed. The approach is based on the well-known power law model that represents the relationship between secondary creep rate as a function of number of cycles of loading [Eq. (2-7)]. Although this concept has been used extensively for asphalt mixtures, it has not been used for binders before. The data collected in this study indicate that modified asphalt binders can be evaluated using the same concept. The binder data, however, indicated that the rate of secondary creep of binders is a simple direct function of the number of cycles and thus the logarithmic transformation is not needed. The following model proved to be reliable:

εa = I + S × N

(2-8)

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders where ea = accumulated permanent strain I = intercept with permanent strain axis (arithmetic strain value, not log value) N = number of load applications S = slope of the linear portion of the logarithmic relation It is rather difficult to use the parameter of creep rate S as a specification parameter because it is an experimental parameter that is affected by a few testing attributes such as stress, loading time, and number of cycles. A better choice for a specification parameter is to use rheological models that combine fundamental behaviors to understand the performance of the material. Although several models have been used to describe the behavior of asphalt binders, the “four-parameter” (Burgers) model, shown in Fig. 2-21, was shown to offer a good representation of the binder behavior (Bahia et al. 2001). This model is a combination of a Maxwell model and a Voigt model. The total shear strain versus time is expressed as follows:

γ (t) = γ 1 + γ 2 + γ 3 =

τ0 τ0 τ + (1 − e − t/τ ) + 0 t G0 G1 η0

(2-9)

By normalizing the strain to the stress applied, the following equation representing the creep compliance, J(t), in terms of its elastic component (Je), the delayed-elastic (Jde), and the viscous component (Jv) could be defined: J(t) = J e + J de + J v

(2-10)

The viscous component is inversely proportional to the viscosity (h) and directly proportional to stress and time of loading. Based on this separation of the creep response, the compliance could be used as an indicator of the contribution of binders to rutting resistance. Instead of using the compliance (Jv), which has a strange unit of 1/Pa, and to be compatible with the concept of stiffness introduced during SHRP, the inverse of the compliance (Gv) could be used. This term is defined as the viscous component of the

FIGURE 2-21

Four-element (burgers) model and its response.

51

52

Chapter Two creep stiffness. The creep and recovery response measured with the DSR could be used to estimate the Gv value and the accumulated permanent strain for any selected combination of loading and unloading times. This finding implies that the accumulated permanent deformation is a function of viscosity, load, and loading time.

γ 1 = f (η , τ , t)

(2-11)

S = f (η , τ , t)

(2-12)

By selecting the appropriate testing stress (t) and the appropriate time of loading (t) the viscous component of the stiffness Gv could be directly related to the rate of accumulation of permanent deformation S and thus used as a fundamental indicator of rutting resistance of asphalt binders.

Binder Fatigue Parameter Although there are different loading modes that could be used in fatigue testing, a reliable indicator of fatigue failure should be independent of the loading mode. It should provide a consistent indication of the level of damage and progression of damage in the material in terms of changes in mechanical behavior under any loading conditions. The most commonly used definition of fatigue failure in asphalt mixtures is a decrease in the initial stiffness by 50 percent, as was indicated in the previous sections. This arbitrary definition, however, does not allow evaluation of the distinctly different mechanism by which a material would respond to the energy input during a loading history for the different loading modes. Researchers have, therefore, focused on using the concept of dissipated energy to explain fatigue behavior of asphalt mixtures. For many decades researchers have used the loss modulus as an indicator of fatigue resistance because of the relationship between this modulus (G∗sind) and the energy dissipated per cycle. The success of this approach has been questioned, however, in many studies because this parameter tends to give different results at different loading conditions. Recent advancements in fatigue research have indicated that a better indicator of fatigue is the rate of change of dissipated (distortion) energy per load cycle. There are several approaches to present the criterion of fatigue based on the rate of change in the dissipated energy. The most promising approaches are presented by Carpenter and coworkers (Carpenter and Jansen 1996; Ghuzlan and Carpenter 2000), and by Pronk and coworkers (Pronk 1995; Pronk and Hopman 1990).

Rate of Change of Dissipated Energy Ghuzlan and Carpenter (2000) defined the ratio of dissipated energy as ΔDE Wi − Wi+1 = DE Wi

(2-13)

where Wi is the total dissipated energy at cycle i calculated by area within hysteresis loop and Wi+1 is the total dissipated energy at cycle i+1. Plotting the values of this ratio versus loading cycles gives a curve that can be used to determine the fatigue life (Np) by identifying the sudden change in the rate. The problem with this approach is that the data points, especially for the constant stress tests, are scattered widely, which makes it difficult to determine an accurate Np value.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders

FIGURE 2-22 Application of rate of change of energy dissipation concept to binder straincontrolled testing data.

Applying this concept to binder data, it is observed that the approach is useful for the stress-controlled testing, but is not useful for the strain-controlled testing. As shown in Fig. 2-22, the strain-controlled measurements result in scattered points, which makes it almost impossible to clearly define the Np value. Conceptually, it is difficult to apply this approach for most binders because of the nature of the strain-controlled test. Since the test is strain-controlled, the rate of energy dissipation will stabilize when the damage starts accumulating because the material will soften due to damage resulting in reduction in stress required to cause same strain. The test can, therefore, take a long time to result in transition of material from the crack initiation stage to the propagation stage.

Cumulative Dissipated Energy Ratio Pronk (1995) defined the dissipated energy ratio as follows: n

Dissipated energy ratio =

∑W i=1

Wn

i

(2-14)

where Σ nn=1Wi is the total sum of the dissipated energy up to cycle n and Wn is the dissipated energy at cycle n. Figure 2-23 shows examples of applying this concept to binder data. The results shown indicate that binders can be evaluated effectively using this method. The curves of the binder data are similar to the mixture data published earlier by other researchers, which is very promising. This also implies that the main factor in the fatigue behavior of mixtures could be well related to the fatigue damage in the binder. It is also observed that the slope of the relationship between the energy ratio and the number of cycles to failure is equal to 1.0 when the material is not undergoing

53

54

Chapter Two

FIGURE 2-23

Using the dissipated energy ratio to analyze fatigue data.

fatigue damage as indicted by the constancy of the dissipated energy per cycle. This actually can be derived from Eq. (2-15) by assuming that Wi is constant and is equal to Wn: n

∑W = N ⋅ W i=1

i

n

(2-15)

Based on this derivation, an understanding of the fatigue curve could be introduced. It is assumed that the first portion represents the stage during which the energy per cycle is dissipated in viscoelastic damping with negligible damage. In the next stage, cracking initiation consumes an additional amount of energy beyond the viscoelastic damping. In the third stage, crack propagation starts and a noticeable increase in dissipated energy per cycle is observed. This is assumed to be the most critical stage during which damage per cycle is so high that healing and recovery from damage are difficult to occur.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders

Summary of Findings for the Damage Resistance Parameters Based on the results and analysis of the binder rutting and fatigue studies, the following points provide a summary of the findings: 1. There are critical questions about the validity of the binder parameter G∗/sind. The correlation between the mixture rutting indicators and G∗/sind is very poor. The parameter is derived from testing that does not provide a good representation of traffic loading in the field. The parameter could not be found useful in describing the accumulation of permanent flow, which is important in rutting evaluation. 2. Repeated creep testing of binders is introduced as a better method for estimating binder resistance to permanent strain accumulation. The viscous component of the creep stiffness (Gv) is found to be a good indicator of the rate of permanent strain accumulation for binders. It is proposed as a better specification parameter. 3. Compared to the current binder protocol, the repeated creep test protocol for measuring binder accumulated permanent strain represents improvements in the theoretical and practical concepts for better rating of binder properties as related to rutting of pavements. 4. The lack of correlation between the mixture fatigue life and current binder fatigue resistance indicator, G∗sind, at the intermediate temperatures, indicates that a new test for the binder fatigue resistance is needed to determine the relation between mixture fatigue life and binder rheological properties. 5. Time sweep test is introduced as a promising binder-only fatigue test to evaluate the fatigue resistance of the binder. The test can be conducted using the current DSR within a relatively short testing time. 6. The dissipated energy ratio approach is employed as the method to determine the fatigue life of binders, because of the independent nature of this approach to the loading modes. Although the geometry has certain effects on the results, by selecting the proper test conditions, this approach can give reliable results that are found to correlate well with mixture performance.

Acknowledgments A major part of the work reported in this chapter in regard to modified asphalts is based on the SHRP and the NCHRP 9-10 project, which is sponsored by the American Association of State Highway and Transportation officials, in cooperation with the Federal Highway Administration. It was conducted as part of the National Cooperative Highway Research Program, which is administrated by the Transportation Research Board of the National Research Council. The author gratefully acknowledges this support and also acknowledges the support of the NCHRP Project Officer Dr. E. Harrigan, and the members of the project panel for their continuous encouragement and feedback. Also the author gratefully acknowledges the technical support of the engineers and scientist that contributed directly and indirectly to the results presented.

55

56

Chapter Two

Disclaimer The opinions and conclusions expressed or implied in the report are those of the author. They are not necessarily those of the Transportation Research Board, the National Research Council, the Federal Highway Administration, the American Association of State Highway and Transportation Officials, or of the individual states participating in the National Cooperative Highway Research Program.

References Anderson, D. A., D. W. Christensen, H. U. Bahia, R. Dongre, M. G. Sharma, and J. Button, “Binder Characterization and Evaluation,” Volume 3: “Physical Characterization,” Report No. SHRP-A- 369, The Strategic Highway Research Program, National Research Council, Washington, D.C., 1994. Anderson, D. A., and T. W. Kennedy, “Development of SHRP Binder Specification,” Journal of the Association of Asphalt Paving Technologists, Vol. 62, 1993, p. 481. Anderson, D. A., D. W. Christensien, and H. U. Bahia, “Physical Properties of Asphalt Cement and the Development of Performance-Related Specifications,” Journal of the Association of Asphalt Paving Technology, Vol. 60, 1991, p. 437. Anderson, D. A., E. L. Dukatz, and J. L. Rosenberger, “Properties of Asphalt Cement and Asphaltic Concrete,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 52, 1983, pp. 291–324. Anderson, G. L., and L. H. Lewandowski, “Additives in Bituminous Materials and FuelResistant Sealers,” National Technical Information Service, A847582, DOT/FAA/RD93/30, 1993. Bahia, H. U., and R. Davies, “Effect of Crumb Rubber Modifiers (CRM) on Performance Related Properties of Asphalt Binders,” Journal of the Association of Asphalt Paving Technologists, Vol. 63, 1994, pp. 414–449. Bahia, H. U., and D. A. Anderson, “The New Proposed Rheological Properties: Why are They Required and How Do They Compare to Conventional Properties,” ASTM STP 1241, John C. Harden, Ed. American Society for Testing and Materials, 1995, pp. 1–27. Bahia, H. U., “Critical Evaluation of Asphalt Modification Using the SHRP Concepts,” TRR 1488, Washington, D.C., 1995, pp. 82–88. Bahia, H. U., H. Zhai, and A. Rangel, “Evaluation of Stability, Nature of Modifier, and Short Term Aging of Modified Binders Using New Tests: The LAST, The PAT, and the MRTFO,” TRR 38, Washington, D.C., 1998a, pp. 64–71, Bahia, H. U., W. Hislop, H. Zhai, and A. Rangel, “The Classification of Asphalts into Simple and Complex Binders,” Journal of the Association of Asphalt Paving Technologists, Vol. 67, 1998b, pp. 1–41. Bahia, H. U., H. Zhai, S. Kose, and K. Bonnetti, “Non-linear Viscoelastic and Fatigue Properties of Asphalt Binders,” Journal of the Association of Asphalt Paving Technologists, Vol. 68, 1999, pp. 1–34. Bahia, H. U., D. Hanson, M. Zeng, H. Zhai, and A. Khatri, NCHRP Report 459, “Characterization of Modified Asphalt Binders in Superpave Mix Design,” NCHRP Report 459, National Academy Press, Washington, D.C., 2001. Banasiak, D., and L. Geistlinger, “Specific’s Guide to Asphalt Modifiers,” Roads and Bridges Magazine, May 1996, pp. 40–48. Barth, E. J., Asphalt Science and Technology, New York, Gorden and Breach, 1962.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Blokker P. C., and Van Hoorn, Proceedings of the 5th World Petroleum Congress, June 1959. Bonnetti, K., K. Nam , and H. U. Bahia, “Fatigue Behavior of Modified Asphalt Binders,” Journal of the Transportation Research Board, TRR No. 1810,Washington, D.C., 2002, pp. 33–43. Bouldin, M. G., J. H. Collins, and A. Berker, “Improved Performance of Paving Asphalt by Polymer Modification,” Rubber Chemistry and Technology, Vol. 64, 1991, p. 577. Brule, B., and M. Maze, “Application of SHRP Binder Tests to the Characterization of Polymer Modified Bitumens,” Asphalt Paving Technology, Vol. 64, 1995. Brule, B., G. Ramond, and C. Such, “Relationship between Composition, Structure, and Properties of Road Asphalts: State of Research at the French Public Works Central Laboratory,” Transportation Research Record, 1096, National Academy Press, Washington, D.C., 27 1986. Brule, B., Y. Brion, and A. Tanguy, “Paving Asphalt Polymer Blends: Relationship between Composition and Properties,” Journal of the Association of Asphalt Paving Technologists, Vol. 57, 1988, pp. 41–65 Button, J. W., D. N. Little, B. M. Gallaway, and J. A. Epps, “Influence of Asphalt Temperature Susceptibility on Pavement Construction and Performance,” NCHRP Report No. 268, 1983 Transportation Research Board, Washington, D.C., 2001. Button, J. W., and J. A. Epps, “Identifying Tender Asphalt Mixtures in the Laboratory,” Transportation Research Record, 103.4, 1985, pp. 20–26. Button, J. W., D. N Little, Y. Kim, and J. Ahmed, “Mechanistic Evaluation of Selected Asphalt Additives,” Journal of the Association of Asphalt Paving Technologists, Vol. 56, 1987, p. 62. Carpenter, S. H., and M. Jansen, “Fatigue Behavior under New Aircraft Loading Conditions,” Proceedings of Aircraft/Pavement Technology, ASCE, Washington, D.C., 1996. Chehoveits, J. G., R. L. Dunning, and G. R. Morris, “Characteristics of Asphalt-Rubber by the Sliding Plate Microviscometer,” Proceedings of the Association of Asphalt Paving Technologies, Vol. 51, 1982, p. 240. Chipperfield, E. H., and T. R. Welch, “Studies on the Relationships between The Properties of Road Bitumens and Their Service Performance,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 36, 1967, 421–488. Christensen, D. W., and D. A. Anderson, “Interpretation of Dynamic Mechanical Test Data for Paving Grade Asphalt,” Proceedings of the Association of Asphalt Pavement Technology, Vol. 61, 1992, pp. 67–116. Collins, J. H., and M. G. Bouldin, Long and Short-Term Stability of Straight and Polymer Modified Asphalts, Rubber Division, American Chemical Society, Detroit, Mich., 1991. Cornelissen, J., and Waterman H. I., Anal. Chim. Acta., 15, 401 (1956). Culley, R. W., “Relationship between Hardening of Asphalt Cements and Transverse Cracking of Pavements in Saskatchewan,” Proceedings, Association of Asphalt Paving Technologists, Vol. 38, 1969, pp. 1–15. Davies, R. L., “The ASTM Penetration Method Measures Viscosity,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 50, 1981, pp. 116–149. de Bats and G van Gooswilligen, Practical Rheological Characterization of Paving Grade Bitumens, Shell Research, Amsterdam, 1995. Dickinson, E. J., and H. P. Witt, “The Dynamic Shear Modulus of Paving Asphalts as a Function of Frequency,” Transaction of the Society of Rheology, Vol. 18, No. 4, 1974, p. 591. Dijk, W., “Practical Fatigue Characterization of Bituminous Mixes,” Journal of the Association of Asphalt Paving Technologists, Vol. 44, 1975, pp. 38–72.

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Chapter Two Dijk, W., and W. Visser, “The Energy Approach to Fatigue for Pavement Design,” Journal of the Association of Asphalt Paving Technologists, Vol. 46, 1977, pp. 1–37. Dobson, G. R., “The Dynamic Mechanical Properties of Bitumen,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 38, 1969, p. 123. Dongre, R., “Development of Direct Tension Test Method to Characterize Failure properties of Asphalt Cements,” Ph.D. Thesis, the Pennsylvania State University, 1994. Dongre, R., M. G. Sharma, and A. A. Anderson, “Chracterization of Failure Properties of Asphalt Binders,” Physical Properties of Asphalt Cement Binders, ASTM STP 1241, 1995, pp. 117–136. Evans, C. G., and R. L. Griffin, “Modified Sample Plates for Tests of High Consistency Materials with the Microviscometer,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 32, 1963, pp. 64–81. Fair, W. F., Jr., and Volkmann, W., Ind. Eng. Chem., Anal. Ed., Vol. 15, 1943, pp. 240–242. Ferry, J. D., Viscoelastic Properties of Polymers, chapter 11, New York: John Wiley and Sons, 1980. Gallaway, B. M., “Durability of Asphalt Cements Used in Surface Treatments,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 26, 1957, p. 151. Gallaway, B. M., “Factors Relating Chemical Composition and Rheological Properties of Paving Asphalts with Durability,” Proceedings, The Association of Asphalt Pavino Technologists, Vol. 28, 1959, pp. 280–293. Ghuzlan, K. A., and S. H. Carpenter, “An Energy-Derived/Damage-Based Failure Criteria for Fatigue Testing,” Preprints of the 79th TRB Annual Report, 2000. Griffin, R. L., T. K. Miles, and C. J. Penther, “Microfilm Durability Test for Asphalt,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 24, 1955, pp. 31–62. Halstead, W. J., and J. A. Zenewitz, “Changes in Asphalt Viscosities During the Thin Film Oven and Microfilm Durability Tests,” American Society for Testing and Materials, ASTM Special Technical Publication, No. 309, 1961, p. 133. Heithaus, J. J., and R. W. Johnson, “A Microviscometer Study of Road Asphalt Hardening in Field and Laboratory, “ Proceedings of the Association of Asphalt Paving Technologists, Vol. 27, 1958, pp. 17–34. Heukelom, W., ”An Improved Method of Characterizing Asphaltic Bitumens with the Aid of their Mechanical Responses,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 42, 1973, pp. 67–98. Isacsson, U., and X. Lu, “Testing and Appraisal of Polymer Modified Road BitumensState of the Art,” Material and Structures, Vol. 28, 1995, pp. 139–159. Jimenez, R. A., and B. M. Gallaway, “Laboratory Measurements of Service Connected Changes in Asphaltic Cement,” Proceedingsof the Association of Asphalt Paving Technologists, Vol. 30, 1961, p. 328. Jongepier, R., and B. Kuilman,”Characterization of the Rheology of Bitumens,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 38, 1969, pp. 98–122. Kandhal, P. S., L. D. Sandvig, and M. E. Wenger, “Shear Susceptibility of Asphalts in Relation to Pavement Performance,” Proceedings, Association of Asphalt Paving Technologists, Vol. 42, 1973, pp. 99–111. Kandhal, P. S., and M. E. Wenger, “Asphalt Properties in Relation to Pavement Performance,” Transportation Research Record_544, 1975, pp. 1–13. Kemp, G. R., and N. H. Predoehl, “A Comparison of Field and Laboratory Environments on Asphalt Durability,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 50, 1981, pp. 492–533.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Kim, Y. R., H. J. Lee, and D. N. Little, “Fatigue Characterization of Asphalt Concrete Using Viscoelasticity and Continuum Damage Theory,” Journal of the Association of Asphalt Paving Technologists, Vol. 66, 1997, pp. 633–685. King, G. N., H. W. King, O. Harders, P. Chavenot, J. P. Planche, “Influence of Asphalt Grade and Polymer Concentration on the High Temperature Performance of Polymer Modified Asphalt,” Asphalt Paving Technology, Vol. 60, 1992. King, G. N., H. W. King, O. Harders, A. Wolfgang, J. P. Planche, and P. Pascal., “Influence of Asphalt Grade and Polymer Concentration on the Low Temperature Performance of Polymer Modified Asphalt,” Journal of the Association of Asphalt Paving Technologists, Vol. 62, 1993, pp. 1–22. King, G. N., H. King, R. D. Pavlovich, A. L. Epps, and P. Kandhal, “Additives in Asphalt,” Journal of the Association of Asphalt Paving Technologists, Vol. 68A, 1999, pp. 32–69. Labout, J. W. A., and W.P. van Ort, “Micromethod for Determining Viscosity of High Molecular Weight Materials,” Analytical Chemistry, Vol. 28, 1956, p. 1147. Lesueur, D., J. F. Gerard, P. Claudy, et al., “Relationships between the Structure and the Mechanical Properties of Paving Grade Asphalt Cements,” Preprint of the AAPT Annual Meeting, Salt Lake City, Utah, 1997. Mack, C., “An Appraisal of Failure in Bituminous Pavement, “Proceedings of the Association of Asphalt Paving Technologists. Vol. 34, 1965, pp. 234–247. Maccarrone, S., “Rheological Properties of Weathered Asphalts Extracted from Sprayed Seals Nearing Distress Conditions,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 56, 1987, pp. 654–687. Majidzadeh, K., and H. E. Schweyer, “Non-Newtonian Behavior of Asphalt Cements,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 34, 1965, p. 20. Majidzadeh, K., “Rheological Aspects of Aging: Part II,” Highway Research Record, 273, 1969, pp. 28–41. Majidzadeh, K., E. M. Kaufmann, and C.L. Saraf, “Analysis of Fatigue of Paving Mixtures from the Fracture Mechanics Viewpoint,” Fatigue of Compacted Bituminous Aggregate Mixtures, ASTM, STP 508, 1972, pp. 67–83. Marasteanu, M. O., and Anderson, D. A., “Improved Model for Bitumen Rheological Characterization,” Eurobitume Workshop on Performance Related Properties for Bituminous Binders, Luxembourg, May 1999. Masson, J-F, and C. Lauzier, Methods for the Analysis of Polymers in Polymer Modified Asphalts, National Research Council Canada, Institute for research in Construction, Ottawa, A-2053.2, 1993. McGennis, R. B., “Asphalt Modifiers are Here to Stay,” Asphalt Contractor Magazine, April 1995, pp. 38–41. McLeod, N. W., “A 4-Year Survey of Low Temperature Transverse Pavement Cracking on the Three Ontario Roads,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 41, 1972, p. 424. Moavenzadeh, F., and R. R. Slander, “Durability Characteristics of Asphaltic Materials.” Research Report EES-259, 1966, p. 236. Moavenzadeh, J., and R. R. Stander, Jr., “Effect of Aging on Flow Properties of Asphalts,” Hwy. Res. Brd. Rec., Vol. 178, 1967, pp. 1–29. Moavenzadeh, F., J. E. Soussou, and H. K. Findakly, “Synthesis for Rational Design,” Final Report for FHWA, Contract 7776, Vol. 2, 1974. Monismith, C. L., and J. A. Deacon, “Fatigue of Asphalt Paving Mixtures,” Transportation Engineering Journal, Proceedings of the ASCE, Vol. 95, TE2, May, 1969, pp. 317–346.

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Chapter Two Monismith, C. L., “Fatigue Response of Asphalt-Aggregate Mixes,” Strategic Highway Research Program, SHRP-A-404, National Research Council, 1994. Mortazavi, M., and J. S. Moulthrop, The SHRP Materials Reference Library, SHRP-A-646 Report, The Strategic Highway Research Program, National Research Council, Washington, D.C., 1993. Neppe, S. L., “Durability of Asphaltic Bitumen as Related to Rheological Characteristics,” Transaction, South African Institute of Civil Engineers, Vol. 2, 1952, p. 103. Oliver, J. W. H., “Optimizing the Improvements Obtained by the Digestion of Comminuted Scrap Rubbers in Paving Asphalts,” Proceedings of the Association of Asphalt Paving Technologies, Vol. 51, 1982, p. 169. Page, G. C., K. H. Murrphy, B. E. Ruth, and R. Roque, “Asphalt Binder Hardening— Causes and Effects,”Proceedings of the Association of Asphalt Paving Technologists, Vol. 54, 1985, pp. 140–167. Pell, P. S., and K. E. Cooper, “The Effect of Testing and Mix Variables on the Fatigue Performance of Bituminous Materials,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 44 (1975), pp. 1–37. Peterson, K., “Specific’s Guide to Asphalt Modifiers,” Roads and Bridges Magazine, May 1993, pp. 42–46. Pfeiffer, J. Ph., and P. M. van Doormaal, “The Rheological Properties of Asphaltic Bitumen,” Journal of Institute of Petroleum Technologists, Vol. 22, 1936, p. 414. Pfeiffer, J. Ph., The Properties of Asphaltic Bitumen, Section II, Elsevier Publishing Co., Amsterdam, 1950. Pink, H. S., R. E. Merz, and D. S. Bosniak, “Asphalt Rheology: Experimental Determination of Dynamic Moduli at Low Temperature,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 49, 1980, p. 64. Pronk, A. C., and P. C. Hopman, “Energy Dissipation: The Leading Factor of Fatigue,” Proceedings of Strategic Highway Research Program: Sharing the Benefits, London, 1990. Pronk, A. C., “Evaluation of the Dissipated Energy concept for the Interpretation of Fatigue Measurements in the Crack Initiation Phase,” The Road and Hydraulic Engineering Division (DWW), Netherlands, P-DWW-95-001, 1995. Puzinauskas, V. P., ”Evaluation of Properties of Asphalt Cements with Emphasis on Consistencies at Low Temperature,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 36, 1967, p. 489. Puzinauskas, V. P., “Properties of Asphalt Cements,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 48, 1979, pp. 646–710. Romberg, J. W., and R. N. Traxler, “Rheology of Asphalt,” Journal of Colloid Science, Vol. 2, 1947, pp. 33–47. Romine, R. A., M. Tahmoressi, R. D. Rowlett, and D. F. Martinez, “Survey of State Highway Authorities and Asphalt Modifier Manufacturers on Performance of Asphalt Modifiers,” Transportation Research Record no. 1323, 1991, p. 61. Saal, R. N. J., and J. W. A. Labout, “Rheological Properties of Asphalts,” in Rheology: Theory and Applications, Vol. 2, 1958, F.R. Eirich, Ed., New York, Academic Press. Schmidt, R. J., “Laboratory Measurement of the Durability of Paving Asphalts,” ASTM SPT 532, 1973, pp. 79–99. Serafin, P. J., L. L. Kole, and A.P. Chirtz, “Michigan Bituminous Experimental Road— Final Report,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 36, 1967, pp. 582–614. Terrel, R. L., and I. A. Epps, Using Additives and Modifiers in Hot Mix Asphalt, QI Series 114, National Asphalt Pavement Association (NAPA), Riverdale, Md., 1989.

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Traxler, R. N., Asphalt: Its Composition, Properties and Uses, Reinhold, New York, 1961. Traxler, R. N., H. E. Schweyer, and H. W. Romberg, “Rheological Properties of Asphalt,” Industrial and Engineering Chemistry, Vol. 36, No. 9, 1944, p. 823. Traxler, R. N., and H. E. Schweyer, “Increase in Viscosity of Asphalts with Time,” Proceedings of the American Society for Testing and Materials, Vol. 36, Part II, 1936, pp. 544–551. Traxler, R. N., “Review of the Rheology of Bituminous Materials,” Journal of Colloidal Science, Vol. 2, 1947, p. 49. Van der Poel, C., “A General System Describing the Visco-Elastic Properties of Bitumens and Its Relation to Routine Test Data,” Journal of Applied Chemistry, Vol. 4, 1954, p. 221. Williams, M. L., R. F. Landel, and J. D. Ferry, “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-Forming Liquids,” The Journal of the American Chemical Society, Vol. 77, 1955, pp. 3701–3707. Wood, P. R., and H. C. Miller, “Rheology of Bitumens and the Parallel Plate Microviscometer,” Highway Research Board Bulletin, National Research Council, D.C., Vol. 270, 1960, pp. 38–46. Zeng, M., H. U. Bahia, H. Zhai, M. Anderson, and P.Turner, “Rheological Modeling of Modified Asphalt Binders and Mixtures,” Journal of the Association of Asphalt Paving Technologists, Vol. 70, 2001, pp. 403–441. Zube, E., and J. Skog, “Final Report on Zaca-Wigmore Asphalt Test Road,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 38, 1969, pp. 1–39.

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PART

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Stiffness Characterization CHAPTER 3 Comprehensive Overview of the Stiffness Characterization of Asphalt Concrete

CHAPTER 5 Complex Modulus from the Indirect Tension Test

CHAPTER 4 Complex Modulus Characterization of Asphalt Concrete

CHAPTER 6 Interrelationships among Asphalt Concrete Stiffnesses

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CHAPTER

3

Comprehensive Overview of the Stiffness Characterization of Asphalt Concrete Robert L. Lytton

Abstract Stiffness of asphalt concrete is a material property that is central to the performance of asphalt pavements. This chapter reviews various factors affecting the stiffness of asphalt concrete, methods of measuring stiffness, and importance of stiffness in pavement response and performance prediction. Special attentions are given to describe how asphalt concrete stiffness is affected by different factors in a mechanics sense.

Introduction The first part of this chapter is conceptual and only the latter part becomes analytical. The chapter starts with five basic questions concerning asphalt stiffness and some conceptual answers to those questions. The questions are What is asphalt concrete? What is stiffness? How is stiffness measured? Why is stiffness important? How is stiffness used in computations? Although the first of these questions has been answered in the previous chapters, it is answered again here as a preparation for the conceptual answers concerning stiffness. The remaining parts of the chapter present the tests that can determine asphalt concrete

65 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.

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Chapter Three stiffness in the laboratory and in the field, the composition and condition factors that stiffness depends upon, and finally the methods of mathematical characterization of asphalt concrete stiffness in both the damaged and undamaged states.

Asphalt Concrete Asphalt concrete is a composite material made up of aggregate particles, bitumen, air, and other components such as additives, modifiers, fines, and water in either liquid or vapor forms. The emphasis in this chapter is that the material is a composite and its composition is important in determining its useful properties for engineering and construction applications.

Asphalt Concrete Stiffness Asphalt concrete stiffness is a material property. Precisely, it is a slope of a stress-strain curve of the asphalt concrete. What is unique about a material property is that it is independent of the test apparatus, or the sample size or geometry that is used to measure it. The converse of this is that if a measured result of a test does depend upon the test apparatus or the sample size or geometry, then that measured result is not and cannot be a material property. There are several types of stress-strain curves from which asphalt concrete stiffness can be measured as a material property.

Methods of Measuring Stiffness Asphalt concrete stiffness can be measured either in the laboratory, as will be discussed in Chaps. 4 and 5, or in the field. The geometry of the test specimen is important because it determines whether the material property can be measured directly with instruments mounted on the sample or must be inferred by analyzing the response of the test sample to the imposed load. The loading pattern that is applied to the test sample also determines what the measured stiffness will be. The rate of loading and the temperature and moisture at which the loading is applied will also determine the measured stiffness. Finally, the age of the test sample will also affect the measured stiffness.

Location of Test In the field, asphalt concrete stiffness may be measured either destructively or nondestructively. In the laboratory, tests can be conducted on cores taken from the field or from samples that were compacted in the laboratory. Destructive tests in the field include “small aperture” testing instruments such as cone penetration or pressure meter devices or “large aperture” methods such as test pits, or accelerated loading tests using full-scale or scale-model vehicle simulation devices. Nondestructive tests include static, cyclic, impulse, and surface wave instruments. In general, the test data from all of these devices must be analyzed by some form of inverse analysis in order to produce a material property of the asphalt concrete, and specifically, its stiffness.

Geometry of the Test Specimen The geometry of a test specimen, either in the field or in the laboratory, is an important consideration because if some geometries are selected, material properties can be inferred directly from the test measurements without the need for the inverse analysis

Overview of the Stiffness Characterization of Asphalt Concrete of the test data. Test specimen geometry in situ is usually dictated by the pavement structure. In the laboratory, there are many more options such as uniaxial tension and compression, triaxial tests in compression and extension (Chaps. 4, 7, and 11), various forms of shear tests (Chap. 10), bending and torsion tests, and indirect tension tests (Chap. 5) of a test sample. Some test specimens are constructed and tested in the laboratory with load, layer, and test conditions that are intended to be severe simulations of field conditions such as moisture or temperature exposure. These are commonly referred to as torture tests. It is uncommon to be able to extract material properties, and asphalt concrete stiffness specifically, from torture tests. Torture tests are commonly used as screening tests for the suitability of the composition of the asphalt concrete. Other test specimens are constructed with such simple geometries that the direct measurement of both stresses and strains make the direct determination of asphalt stiffness possible. Examples of these are the uniaxial and triaxial tension and compression tests and torsional tests on cylindrical samples. These tests are characterized by a high degree of accuracy, precision, and repeatability, as will be shown in subsequent chapters.

Loading Pattern A variety of loading patterns are used to measure asphalt concrete stiffness under a variety of loading conditions. These include monotonic loading of both stress-controlled or strain-controlled tests, frequency sweep tests, impulse and wave propagation tests, repeated load tests, creep, relaxation, and creep and recovery tests. A creep test is one in which the applied stress is held constant and the strain is measured as it grows with time. The ratio of the strain divided by the constant stress is termed the creep compliance. A relaxation test is one in which the applied strain is held constant and the diminishing stress is measured with time. The ratio of the stress divided by the constant strain is the relaxation modulus.

Rate, Temperature, and Age The stiffness of asphalt concrete rises as the rate of loading increases and it decreases as the temperature increases. The age of the asphalt concrete is determined less by its chronological age than by its exposure to air, heat, and solar radiation conditions which will increase its rate of reaction with oxygen and make its stiffness increase along with its susceptibility to brittle fracture.

Moisture The stiffness of asphalt concrete is affected by the amount of moisture that is held within the asphalt binder; the solubility of various components of the asphalt; the amount of asphalt that becomes emulsified; the strength of the adhesive bond between the asphalt and aggregates, with and without water present on the interface; and the strength of the cohesive bond within the asphalt, with and without water present on the surface of microcracks within the asphalt. The fact that asphalt can absorb water within the thin films of mastic that are present within a mixture, together with the fact that different asphalts can absorb widely differing amounts of water at a given level of water vapor pressure, makes the effect of water on the stiffness of asphalt concrete very highly dependent on the composition of the bitumen. It is also a fact that the rate at which water can diffuse through asphalt films differs greatly with the composition of the bitumen.

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Chapter Three

Summary With the stiffness of asphalt concrete varying with so many different factors and conditions, there is the possibility of an endless round of confusion about how each relates to the other and how to combine the effects of separate investigations of these different effects on stiffness into a practical, usable predictable result. The secret to combining all of these effects in a realistic synthesis is by the use of mechanics, as will be discussed later in this chapter.

Importance of Asphalt Concrete Stiffness Subsequent chapters present constitutive, mechanistically based models for describing the behavior of asphalt concrete. An important theme in each of these models is the need for accurate characterization of the fundamental material stiffness. Just as Young’s modulus is paramount for predicting the deflection of steel beams in a structure, the stiffness of asphalt concrete is critical for predicting the behavior of the material in pavement structures. The materials that pavements are built with are very complex despite the fact that they are so commonplace. Such complexity necessitates the use of numerical procedures, such as finite elements, which require significant computation effort. Technology advances in recent years have resulted in great improvements in computation speed, and in principle it is now possible to predict the time of appearance and the rate of deterioration of asphalt concrete distresses. That is not to say that all distresses can be predicted reliably at the present time with existing models but it is now possible to construct a model that will provide reliable predictions in a rapid manner. The only impediment to developing such a model is the willingness to understand the mechanics of the selected type of distress, identify the relevant materials properties, devise test methods that will provide these materials properties, select the most appropriate numerical method to use, and assemble the computer model. Having such a model available makes it possible to interpret correctly and quantitatively the results of in-service pavement, test track, and accelerated pavement tests and to extrapolate these results to other pavements. It also makes possible coordinated laboratory tests that are simple, accurate, precise, and inexpensive with which to obtain materials properties. It also makes possible the coordinated use of nondestructive testing in the field that will provide on site measurements of the same materials properties that are the central focus of all of these mechanics-related activities. Computer-predicted distresses make possible and practical the use of performancebased specifications, warranties, and construction quality assurance and quality control. They also make possible the prediction of remaining life to allow the planning of pavement maintenance and rehabilitation activities as well as the overall asset management of the pavement network, including the effects of safety and cost profiles with time. All of this is possible if stiffness, as a material property, is measured and used properly in numerical prediction methods employing the computers that are now capable of handling the computational tasks that pavements require.

Use of Stiffness in Computations Asphalt concrete stiffness is used in numerical computations to calculate both the primary responses and the mechanisms of distress. The primary responses are the deflections, stresses, and strains in a pavement structure under the variety of

Overview of the Stiffness Characterization of Asphalt Concrete loading, thermal, and moisture conditions that are applied to pavements. The principal mechanisms of distresses are fracture and flow. Fracture is predicted by the use of fracture mechanics while flow is predicted by the use of the various forms of plasticity theory. Both processes use energy in causing the asphalt to deteriorate and this fact is used in damage theory, which describes in mathematical form how the various components of energy are used in predicting the rate and magnitude of the damage that is done by the loads and environmental stress that are applied to the asphalt concrete. Because asphalt concrete is a viscoelastic material, the components of energy that affect its response to loading are that which is stored and can be recovered and that which is dissipated in a loading and unloading process. Some of that energy is used in overcoming the viscous resistance of the material while the rest of the dissipated energy is available to damage the material. The correct representation, measurement, and use of asphalt concrete stiffness as a material property are essential to being able to predict correctly this partitioning of the energy.

Tests to Determine Stiffness In order to be useful in mechanics- and computer-based numerical predictions of both primary responses and distress of pavements, asphalt concrete stiffness must be a material property and not an index property. In the laboratory, the material property must be measured accurately, precisely, and repeatably by testing machines and transducers that are operating within the response range of which they are capable including response time or frequency, stroke, and magnitude. The test measurements must be made in that part of the test specimen where there is a uniform stress field, a uniform strain field, and the sample is held in a condition of uniform temperature and moisture. Examples of this are measurements in the middle third of triaxial compression and tensile tests and measurements in the center of the indirect tension test. It is only in this way that the measured material property can be assured to be independent of test apparatus, sample size, and geometry. Failure to do this produces large variances of the test results, low repeatability, large variances in the predicted results, and higher levels of risk to those who must rely upon them. Material properties such as asphalt concrete stiffness can be back-calculated from tests where there is no uniformity of stress, strain, temperature, or moisture but this requires the use of a computer program that is based upon the mechanics of the sample loading and geometry and may need to include the buildup of damage in all three dimensions. The material properties that are determined by this indirect method cannot be determined directly from the test measurements and this is a major source of their higher variability. The best tests to use for determining the asphalt concrete stiffness and, in fact, all of the material properties of asphalt concrete are those tests in which both the load and the displacement of the test sample are both known precisely by actual measurement on the sample and by the use of feedback control of the test with such rapid response that it is much faster than the loading or frequency rate being used in the test. These are general principles. It is understood that nondestructive tests in the field will always require the use of back calculation in order to arrive at material properties. This is inherently the reason for the larger variances of the results.

69

Chapter Three

Factors on Which Asphalt Concrete Stiffness Depends Asphalt concrete depends upon the strain rate of loading, the temperature, the moisture content of the bitumen, the stress state, the aggregate particles, the bitumen itself, the fines in the mastic, the water in liquid and vapor form and its location with the mixture, the air in the mixture, its age, and its reactivity with oxygen, and any additives or modifiers that have been added to the mixture. Each of these is discussed subsequently.

Temperature and Rate of Loading The stiffness of asphalt concrete depends on the temperature and strain rate of loading. At any given temperature, asphalt concrete will deform slowly and permanently if it is loaded slowly, while if it is loaded at a higher rate, it will be much stiffer and will be subject to fracture. At any given strain rate of loading, there is a temperature above which the material will relax quickly enough that no stress will accumulate in the test sample. These two known facts about materials in general can be illustrated in a qualitative graph of strain rate versus temperature that shows the stress-free temperature and strain rate above which the material will experience microcracking and healing and below which the material will undergo plastic flow and the properties of the aggregates in the mixture will be important in limiting the size and shape of the flow patterns. This conceptual graph is shown in Fig. 3-1. Moisture has a similar effect on the stiffness and type of damage that occurs in asphalt concrete. Although the mechanisms are distinctively different, high moisture and high temperature both result in plastic flow. At any given strain rate, there is a moisture content in the asphalt, above which the material relaxes faster than the stress can build up in the material. On a graph of strain rate versus moisture content there will be a similar stress-free boundary between the brittle, fracture-prone behavior and the soft, plastic flow-prone behavior.

Stress State The state of stress of asphalt concrete modifies its stiffness. In an isotropic mixture, the stiffness depends upon the level of the first and the second deviatoric stress invariants I1 and J2′. In an anisotropic mixture, which is the usual case, the stiffness is directional and depends upon both the two stress state invariants just mentioned and the Asphalt Concrete Damage

Microcracking healing

te m p ra era te tur e

Fracture (Binder in tension)

in fre stra s s d re n St a

e

Strain rate

70

Plasticity flow (Aggregate properties important)

Temperature (Moisture)

FIGURE 3-1

Asphalt concrete damage dependence on temperature and loading rate.

Overview of the Stiffness Characterization of Asphalt Concrete

B C A

I1

D

J ′2

FIGURE 3-2

Loading and unloading stress path for an isotropic stress sensitive material.

components of the stress tensor. Because of the shape of the particles and the way that an asphalt concrete mixture is compacted, there is a vertical modulus, a horizontal modulus, and a shear modulus. There are also two Poisson’s ratios, one in the vertical plane and the other in the horizontal plane. Testing and analysis of triaxial test data have been able to determine all five of the moduli of this cross-anisotropic case. The fact of cross-anisotropy has several important implications for the way that pavements resist both fracture and plastic deformation. In a stress-sensitive material like asphalt concrete, an elastic material is one in which loading and unloading follow a different stress path but one which closes upon itself as is illustrated schematically in Fig. 3-2. The figure is a graph of the first stress invariant plotted against the second deviatoric stress invariant and illustrates how the invariants during loading and unloading follow the path A-B-C-D-A. The elastic work potential for this kind of material is given by (Lade and Nelson 1989) W=

∫

ABCDA

dJ 2′ ⎞ ⎛ I l dI1 ⎜⎝ 9K + 2G ⎟⎠

(3-1)

where W = elastic work potential I1 = first invariant of the stress tensor J 2′ = second invariant of the deviatoric stress tensor K,G = bulk and shear modulus of the material If the elastic modulus of the material depends upon the same two stress invariants according to the equation, E = K1 ( I1 )K ( J 2′ )K 2

3

(3-2)

The requirement that the elastic work potential results in no net work produces a partial differential equation that Poisson’s ratio must satisfy (Lytton et al. 1993). −

k ⎞ ⎛ 1 k3 k ⎞ 2 ∂ν 1 ∂ν ⎛ 2 k3 + 22 ⎟ + =ν ⎜ + 22 ⎟ + ⎜ − I1 ∂I1 3 ∂J 2′ I1 ⎠ ⎝ 3 J 2′ I1 ⎠ ⎝ 3 J 2′

(3-3)

71

Chapter Three

Computed Poisson’s ratio

1.2 1.0 0.8 0.6 0.4 0.4

FIGURE 3-3

Where

0.8 0.6 Measured Poisson’s ratio

1.0

1.2

Measured versus predicted stress dependent Poisson’s ratio.

n = Poisson’s ratio k2, k3 = coefficients which satisfy the boundary conditions of the partial differential equation

One of the implications of the solution of this differential equation is that if the modulus is stress dependent, then the Poisson’s ratio must be also. A graph of the Poisson’s ratio measured by Allen (1973) and predicted by the solution of Eq. (3-3) is shown in Fig. 3-3. The figure shows Poisson’s ratios that rise well above 0.5 which is the maximum that it can be in a material that has a constant elastic modulus. The measurement of Poisson’s ratios that are above 0.5 is a common observation in stress-dependent materials such as asphalt concrete and unbound aggregate base course materials. It also varies with the frequency and direction of loading. A typical pattern is shown in Fig. 3-4. The tensile Poisson’s ratios remain below 0.5 while the compressive Poisson’s ratios rise above 0.5 once the loading frequency rises above about 1 Hz. Highway traffic loading is typically above 8 Hz and this means that an asphalt concrete layer, when loaded by traffic traveling at normal highway speeds, tries to expand laterally. When prevented from doing so, the asphalt concrete layer builds up a confining pressure that

1.0

Poisso’s ratio

72

0.1 0.5

0.2

2

res

si o

n

10

Tensio n

Intersection 0.0

FIGURE 3-4

1

p Com

Street

Frequency of loading, Hz

Poisson’s ratios of asphalt concrete.

Highway

20

Overview of the Stiffness Characterization of Asphalt Concrete Tire

Confinement pressure

Expansion

FIGURE 3-5

Effect of large compressive Poisson’s ratios.

stiffens the asphalt concrete, resists lateral plastic deformation, and presses closed any microcracks that may be growing in the asphalt. This is illustrated in Fig. 3-5. More recent work on cross-anisotropic pavement materials has shown that the same phenomenon of large radial strains can be predicted using the elastic work potential shown in Eq. (3-4) without requiring either of the Poisson’s ratios to rise above 0.5 (Lytton 2000).

W=

where

∫

ABCDA

α

I1 dI + βτ zx dτ zx dJ 2′ 9 1 + Exx 2Gxy

(3-4)

1 − 4n − 2r m s β = 2 + 2r − m Eyy m= Exx n = nxy r = nxz E s = xx Gxy

α =2+

Exx = horizontal modulus Eyy = vertical modulus W = cross-anisotropic elastic work potential It is still useful to use an “effective” Poisson’s ratio to describe the formation of large radial strains in a cross-anisotropic elastic asphalt concrete. The determination of all five of the material properties of a cross-anisotropic material cannot be determined from the measurements of the axial and radial stresses and strains of a triaxial test using the stress-strain relations of the material alone. Instead, it requires the use of an extra relation which solves for the shear modulus using the deviatoric strain energy that is measured during a test in which only the second invariant of the deviatoric stress tensor is applied (Adu-Osei 2000). Other methods using a constrained optimization approach to get a realistic estimate of the shear modulus have been suggested (Tutumluer and Seyhan 2002).

73

Chapter Three 100 e

Percent passing

74

lin

n

um xim

a

M

0

FIGURE 3-6

y sit

de

Large departure Moderate departure

Aggregate size

Effect of particle size distribution on “effective” poisson’s ratio.

Aggregate Particles The principal reason for the emergence of the cross-anisotropic formulation of the stiffness of asphalt concrete is because of the shape of the aggregate particles. When they are compacted, the oblong particles tend to lie flat resulting in a modulus that is greater in the vertical direction than in the horizontal direction. In addition to the shape, the size and size distribution of the particles and the texture of the particles also have an effect upon the directional stiffness and the effective Poisson’s ratio of asphalt concrete. The graph in Fig. 3-6 shows qualitatively the effect of particle size distribution on the “effective” Poisson’s ratio of the mixture. The closer that the particle size distribution approaches the maximum density distribution line, the higher will be the “effective” Poisson’s ratio.

Asphalt Binder Properties The material properties of the asphalt binder which affect the stiffness of asphalt concrete are the compliance, the mastic film thickness, the aging, and the wetting and dewetting components of the surface energy. The compliance is a material response to a constant applied stress. When a stress is applied and held constant in a uniaxial test, the strain increases with the time after the load was applied. This time-dependent strain divided by the constant stress is the creep compliance. A commonly used relation of the creep compliance to the loading time is given in the modified power law, which is given in Eq. (3-5) (Daniel and Kim 1998): D(t) = D0 +

D∞ − D0

τ0 ⎞ ⎛ ⎝1+ t ⎠

n

(3-5)

where D0 , D∞ are minimum and maximum compliance and t0, n are modified power law coefficient and exponent. A log-log graph of the modified power law is shown in Fig. 3-7. The curve begins with a nearly horizontal slope at short loading times and rises to approach a straight line in its midrange and finally can return to horizontal at long loading times as it

Overview of the Stiffness Characterization of Asphalt Concrete

Creep compliance Iog D(t)

D∞

n 1 Do

Do, D∞, to, n

FIGURE 3-7

Log (time)

Creep compliance.

approaches the asymptote of D∞. The slope n can never be greater than 1.0. The creep compliance is related to the relaxation modulus. This is determined in a uniaxial test in which a strain is imposed on the material and held constant. The stress in the material then relaxes as the time after the strain was first applied increases. The relaxation modulus is the time-dependent stress divided by the constant strain. Chapter 6 deals exclusively with the relationship between this property and the creep compliance. At this point it simply needs to be known that the theory of linear viscoelasticity may be used to convert between any of the linear viscoelastic response functions as long as the measurements actually represent linear viscoelastic behavior.

Surface Energies The surface energies of the asphalt can be measured in a number of different test devices. The instrument which was used in measuring the surface energies that are tabulated in Table 3-1 were measured with the Wilhelmy plate apparatus. The apparatus is pictured in Fig. 12-3, and a detailed explanation of the testing procedure is given there. There are three components of surface energy: the nonpolar or Lifshitz-van der Waals, and the Lewis acid and Lewis base polar components. There is a hysteresis effect between the wetting and the dewetting surface energies. The wetting surface energies are associated with the healing of the microfractures in the asphalt while the dewetting surface energies are associated with the fracture of the asphalt. The total surface energy is made Surface Energy Components, ergs/cm2

Water

Wetting

Dewetting

Wetting

ΓLW

21.6

8.80

13.62

14.91

6.76

Γ–

25.5

1.50

18.87

1.74

15.28

Γ

25.5

2.81

10.52

1.07

15.03

Γ

51.0

4.13

28.18

2.64

30.30

Γ

72.6

12.93

41.80

17.55

37.06

+ AB TOTAL

TABLE 3-1

New Asphalt

Aged Asphalt

Wetting and Dewetting Surface Energies of Asphalt and Water

Dewetting

75

76

Chapter Three up of a sum of the nonpolar and the combined effect of the polar components. The relation is shown in Eq. (3-6) (Good and van Oss 1991): Γ = Γ LW + 2 Γ + Γ − where

(3-6)

Γ = total surface energy ΓLW = nonpolar Lifshitz-van der Waals surface energy Γ+ = acid component of the polar surface energy Γ− = basic component of the polar surface energy

In Table 3-1, the wetting and dewetting components and the total surface energies of new and aged asphalts are given along with the same components of water. Table 3-1 shows that as an asphalt ages, the surface energies change so as to reduce healing and to make fracture easier. Thus, the nonpolar portion of the wetting surface energies grows larger and the polar portions grow smaller as the asphalt ages. At the same time, both of the portions of the dewetting surface energies decrease so as to decrease the work of fracture with age. More detailed tables and figures illustrating these effects on the asphalt binder are presented in Chap. 12. Once a microcrack, or a crack, forms in the asphalt binder, the surface energies on each face of the crack interact to provide the cohesive bond strength against fracture and to provide the surface energy to promote healing. The computation of the cohesive bond strength on this interface, both dry and in the presence of water, is presented in detail in Chap. 12. The aggregate particles also have surface energy components which interact with the surface energies of the asphalt binder to produce the adhesive bonding strength at the interface between the two. The method used to measure the surface energies of the aggregates is the universal sorption device (USD) which is illustrated in Figs. 12-5 and 12-6. The USD is used to deposit vapor molecules on the surface of the aggregate particles in a vacuum. The accumulation of the mass of the vapor molecules on the particle surfaces at different levels of vapor pressure is used to calculate the specific surface area of the aggregate particles and to determine the wetting and dewetting components of the aggregate surface energies. The method of calculating the adhesive bond strength of an asphalt binder with an aggregate particle, both when the two surfaces are dry and when there is water present on the interface, is presented in Chap. 12. It is demonstrated there, and noted here for emphasis, that when water is on the interface between asphalt and an aggregate surface, the water acts to destroy the adhesive bond. The intensity of the action of the water varies greatly between various combinations of asphalt and aggregates as is demonstrated in Table 12-16. This is the scientific basis for determining which combinations of asphalt and aggregate will strip and which will not. It explains why some aggregates will strip with some asphalt but not with others. There are two components of moisture damage to asphalt concrete stiffness: one due to soaking and the other due to repeated loading progressively opening adhesive debonding interface zones along the surface of the aggregates in the asphalt concrete mixture. The soaking damage depends upon the rate of moisture diffusion and the amount of water that the asphalt film can hold. The rate of moisture diffusion depends upon the relative vapor pressure in the immediate vicinity of each aggregate particle and the thickness of the mastic film surrounding the aggregate. Each asphalt has a unique water versus relative vapor pressure characteristic curve. Some asphalts hold more water at the same level of relative vapor pressure than others. Based upon measurements made to the present, this vapor pressure characteristic curve is the crucial

Overview of the Stiffness Characterization of Asphalt Concrete element in determining how much damage will be done to the asphalt concrete by moisture diffusion. Those asphalts which hold more moisture will experience more damage due to soaking. Also, based upon the results of recent measurements, those asphalt-aggregate mixtures in which the work of the water in assisting the fracture of the interface is the greatest will be the ones that are most damaged by repeated loading in the presence of moisture.

Fines The fines in an asphalt concrete mix are all of the particles that are smaller than 0.075 mm. They make up around half of the volume of the mastic. The stiffness of the mix will be affected significantly by how well the fines bond with the asphalt binder, the size and size distribution of the fines, how well they are dispersed in the mastic, and their surface energy compatibility with the asphalt both with and without water. The size, size distribution, and dispersion all work together to arrest microcracks when they are small and easier to stop. Microcracks start out as a cloud of many small, dispersed flaws in the mastic. They grow as strain energy from repeated loading is made available to the mixture to extend the cracks. If any of these microcracks encounters a fine particle blocking its path, the particle will act as a crack arrester, and that microcrack will stop growing. If there are many well-dispersed fine particles in the mastic, many of these microcracks will be arrested. One of the major effects of microcracks growing in an asphalt-aggregate mixture is a progressive reduction of stiffness and one of the major effects of a well-dispersed distribution of fines is to retain the stiffness, precisely because of its actions in arresting cracks. This also means that the fine particles must bond well with the asphalt, especially with water present. As discussed before, the adhesive bond strength between the two is determined by the surface energy characteristics of both the asphalt and the fine particles. The same can be said for additives that are expected to improve the stiffness, strength, and ductility of asphalt-aggregate mix. Regardless of their composition, the particles of the additive must be small and well dispersed and must bond well with the asphalt in order to improve the mechanical properties of the mix.

Air Voids Air voids may be viewed with some accuracy as small particles with zero stiffness. Small, well-dispersed air voids in the asphalt concrete mixture will provide several benefits to the mix, including acting as microcrack arresters and providing well-dispersed volumes for the asphalt to expand into at high temperatures. Too much air will accelerate the growth of microcracks and too little air will cause bleeding (or flushing) and promote large plastic deformations. Too much air will also provide ready access of both air and water into the interior of an asphalt concrete layer and will accelerate aging and moisture damage. As with the fine particles, the air voids must be small and well dispersed in order to have its desired effect on the stiffness of the mixture.

Summary The stiffness of asphalt concrete depends upon numerous factors such as stress state, temperature, rate of loading, composition, and the mechanical and surface energy properties of the components of the mixture. It would be impossible to anticipate empirically how all of these factors will interact to provide an instantaneous value of

77

78

Chapter Three the asphalt concrete stiffness. It is fortunate that the disciplines of micromechanics and viscoelastic fracture mechanics have provided principles that may be used to determine how these factors will affect the mixture stiffness. An overview of these principles as applied to an asphalt concrete mixture is presented in the next section.

Characterization of Asphalt Concrete Stiffness Characterization of asphalt concrete stiffness refers to the mathematical relations between stress, strain, temperature, loading rates, moisture, and composition that are found in the disciplines of mechanics, including micromechanics and fracture mechanics. The stiffness of asphalt in an undamaged state is different from that in a damaged state and this difference is directly a result of the surface energy, adhesive and cohesive fracture and healing, plasticity and viscoplasticity, and moisture damage characteristics of the material. There is a large body of technical literature relating to the micromechanics of representing the properties of a composite material when the properties of the component materials are known. The discussion that follows will provide some of the high points that are relevant to the characterization of both damaged and undamaged asphalt concrete stiffness.

Undamaged Stiffness There are several approaches in micromechanics, all of which require that the strain energy that is put in to a composite material when it is being loaded is completely accounted for in being stored in the component materials of the composite. The same is true of energy that is released when the composite is unloaded. The objective is to arrive at a single stiffness that treats the composite as a homogeneous material but one that absorbs and releases strain energy in the same way that the composite does. Several of the principal micromechanics approaches give an upper and lower bound to this stiffness. One result of this mathematical process is the ratio of the shear modulus of the composite G* to the shear modulus of the matrix in which a solid inclusion is imbedded Gm. The relation is in Eq. (3-7) (Christensen 1991; Aboudi 1991):

G = 1− Gm

⎡ ⎛ G 15(1 − ν m ) ⎢ 1 − ⎜ i ⎝ Gm ⎣

⎞⎤ ⎟⎠ ⎥ ci ⎦ ⎛ Gi ⎞ 7 − 5ν m + 2(4 − 5ν m ) ⎜ ⎝ Gm ⎟⎠

(3-7)

where Gm , Gi = shear moduli of the matrix and the inclusion, respectively , nm = Poisson s ratio of the matrix ci = fraction of the total volume occupied by the inclusion Two well-known results are when the inclusion is rigid and when it is a void, corresponding roughly to aggregate particles in an asphalt matrix and air voids in an asphalt medium. The rigid particle relation is in Eq. (3-8): G 5 = 1 + ci Gm 2

(3-8)

This relation was first published by Einstein (Einstein 1956). It has been used in a somewhat modified form in the Shell nomograph to estimate the stiffness of a mixture

Overview of the Stiffness Characterization of Asphalt Concrete from the stiffness of the bitumen (Heukelom and Klomp 1964, Van der Poel 1954). The air void relation is in Eq. (3-9): G 5 = 1 − ci Gm 2

(3-9)

It is possible to use these equations to get a rough estimate of how the shear stiffness of the asphalt concrete is related to the shear stiffness of the asphalt [Eq. (3-8)] or how air bubbles in the asphalt will alter the stiffness of the asphalt [Eq. (3-9)]. A corresponding relation for the bulk modulus of the composite is in Eq. (3-10) (Christensen 1991): K − Km (4Gm + 3K m ) = ci Ki − K m [4Gm + 3Ki + 3(K m − Ki )ci ] where

(3-10)

K = bulk modulus of the composite Km, Ki = bulk modulus of the matrix and inclusion, respectively Gm = shear modulus of the matrix material ci = volume fraction occupied by the inclusion

There are other, different relations predicting the bulk modulus of the composite that have been developed by using slightly different mathematical approaches, all of which have their limitations. Using the bulk and shear moduli of the composite as estimated from one of these formulas, it is possible to make further estimates of the Young’s modulus and Poisson’s ratio. Recognizing these limitations, the Method of Cells was developed to provide a numerical method for estimating these and other material properties, taking into account the one-, two-, or three-dimensional geometry of the inclusion (Aboudi 1991). The numerical results are able to match measured results very closely. Other properties that are estimated by use of these micromechanics methods include the coefficient of thermal expansion, thermal conductivity, creep compliance and relaxation modulus, the timetemperature shift function, electrical conductivity and dielectric constant, and anisotropic yield strength of the composite among others (Christensen 1991). Because asphalt concrete stiffness is a viscoelastic, rather than an elastic property, it is frequently necessary to convert the viscoelastic properties of the components of the composite into the effective viscoelastic property of the composite. This can be done using the same micromechanics formulas and the correspondence principle which is illustrated in Eqs. (3-11), (3-12), and (3-13). The effective elastic bulk modulus formula in Eq. (3-10) is rewritten with bars over the modulus terms as in Eq. (3-11) (Christensen 1991): K − Km 4G m + 3K m = ci Ki − Km [4G m +3K i +3(K m − K i )ci ]

(3-11)

The meaning of the bar over either K or G is the Laplace transform of the relaxation modulus multiplied by s (Carson transform), the Laplace transform parameter in Eq. (3-12). ∞

K (s) = s ∫ K (t) e − st dt o

(3-12)

Equation (3-11) is solved for the Laplace transform of the composite K (s), and the entire expression is inverted to produce the effective bulk relaxation function of the composite K(t). A similar exercise will produce the effective shear relaxation function of the composite G(t). The inversion of this expression is usually done numerically although closed forms are possible using Schapery’s approximate inverse Laplace transform

79

80

Chapter Three method (Schapery 1962, 1965). Because there are different methods of estimating the bulk and shear moduli of elastic composite, it is usually necessary to verify that the conversion produces reliable results by comparison with actual measurements made on the composite. A similar transformation can be used to convert the complex moduli of the components into the effective complex moduli of the composite. In this way, the formula for the elastic bulk modulus of the composite is converted into the formula for the complex bulk modulus of the composite as in Eq. (3-13) (Christensen 1991). 4Gm* + 3K m* K * − K m* * * = ci * Ki − K m [4Gm + 3Ki* + 3(K m* − Ki* )ci ]

(3-13)

The K and G terms with the asterisks in this equation are the complex bulk and shear modulus of the matrix and the inclusion, all of which have a real and an imaginary component as in Eqs. (3-14a), (3-14b), and (3-14c) (Christensen 1991): K m* (ω ) = K m′ (ω ) + i K m′′ (ω )

(3-14a)

Ki* (ω ) = Ki′(ω ) + i Ki′′(ω )

(3-14b)

Gm* (ω ) = Gm′ (ω ) + i Gm′′ (ω )

(3-14c)

If the material is nonlinear viscoelastic, the equations given above must be treated as approximations, but they provide correct forms of equations that take into account the strain-energy storage of each of the components of the composite material.

Effects of Microcracks on Stiffness Because of the ability to use the correspondence principle to convert elastic solutions into viscoelastic equations for asphalt concrete stiffness, it is possible to derive relations using elastic theory with the confidence that they can be converted into the appropriate viscoelastic form, either the creep compliance, the relaxation modulus, the complex compliance, or the complex modulus. When a repeated load test is made on an asphalt concrete, its stiffness appears to decrease with increasing numbers of load applications. However, what is really happening is that small microcracks are forming in the material, producing an apparently smaller modulus. This is illustrated in Fig. 3-8 with two straps being subjected to the same tensile stress.

Microcrack size Actual E

Apparent E′

2c

Same stretch E′ < E

FIGURE 3-8

Effect of microcrack size on apparent modulus.

Overview of the Stiffness Characterization of Asphalt Concrete On the left is the actual strap with a crack in it of length 2c. On the right is an intact strap with an apparent elastic modulus E′. If it is required to find the modulus E′ that will store as much strain energy as the real strap on the left where the actual modulus E remains unchanged, the ratio between the apparent and the real modulus of the material is given by Eq. (3-15): E′ = E where

1 c2 ⎡ 8ΓE ⎤ 1 + 2π 1− 2 ⎥ bh ⎢⎣ σ c⎦

(3-15)

c = b, h = Γ= E, E′ =

half crack length width and length of the strap, respectively fracture surface energy of the material undamaged and apparently damaged mod dulus of the material, respectively σ = tensile stress applied to the strap

If the same strap is loaded repeatedly and the crack grows with each load repetition, the relation between the real and the apparent modulus is in Eq. (3-16): E′ = E

where

1 ⎡ c2 1 + 2π ⎢ 1 − 2Γ bh ⎢ ⎢ ⎣

1 ⎤ ⎛ 4At ⎞ 1+ n ⎥ ⎜ dW ⎟ ⎥ ⎜⎝ ⎟ dN ⎠ ⎥⎦

(3-16)

t = thickness of the strap A, n = Paris-Erdogan fracture law parameters dW = rate of change of dissipated strain eneergy per load cycle dN N N = number of load repetitions

If, instead of there being a single large crack of length 2c, there is a distribution of microcracks of various sizes with a crack density of (m/bh), the ratio between the apparent and the real modulus is given in Eq. (3-17): E′ = E

where

1 ⎡ 2 ⎛ m ⎞ 1 + 2π c ⎜ ⎟ ⎢ 1 − 2Γ ⎝ bh ⎠ ⎢ ⎢ ⎣

1 ⎤ ⎛ 4At ⎞ 1+ n ⎥ ⎜ dW ⎟ ⎥ ⎟ ⎥ ⎜⎝ dN ⎠ ⎦

(3-17)

m = number of microcracks in the strap (m/bh) = microcrack density c = mean crack size

Equations (3-16) and (3-17) both show that the apparent loss of stiffness depends upon the fracture properties of the material, A and n, the microcrack density (m/bh), the instantaneous rate of change of the dissipated strain energy per load cycle (dW/dN), and the average microcrack size c.

81

Chapter Three

FIGURE 3-9

Dissipated pseudo-strain energy in a beam fatigue test.

If the material is viscoelastic instead of being elastic as in the previous examples, a certain amount of the energy that is expended with each load cycle is used up in overcoming the viscous resistance of the material and not in contributing to the damage of the material. In order to have the correct relation between the apparent and real relaxation moduli of the material, it is necessary to correct for the amount of dissipated energy that is not used directly in damaging the material. In order to make this correction, the concept of pseudostrain energy has been introduced. Although the pseudostrain concept will be presented in detail in Chap. 7, it is repeated here to compliment slight differences in its application to the work presented in this chapter. Pseudostrain energy is the amount of dissipated energy that is available to damage the material. A typical fatigue test on a beam fatigue sample is illustrated in Fig. 3-9. As the load is applied and then reduced it is necessary to apply compressive force in the opposite direction in order to return the beam to its original unstrained location. The graph of applied load versus deflection must be corrected to subtract the amount of energy that has been used to overcome the viscous resistance of the beam to upward and downward movement. This can be done by first finding out what the relaxation modulus of the beam material is by running a relaxation test on the material at a low stress level. Then the beam strain rate history is combined with the convolution integral to predict the linear viscoelastic stress history of the beam sLVE(t). If this calculated linear viscoelastic stress is plotted against the measured stress and a straight line such as shown in Fig. 3-10

Measured stress s

82

Linear x x x

Nonlinear

x x Calculated viscoelastic stress s LVE

FIGURE 3-10

Measured viscoelastic stress versus calculated linear viscoelastic stress.

Overview of the Stiffness Characterization of Asphalt Concrete

Measured stress s

Dissipated pseudostrain energy x x

x

x x

x x

x x

x x x

e R = s LVE/ER = “Pseudo’’ or reference strain

FIGURE 3-11

Illustration of dissipated pseudostrain energy.

is the result, it means that the material is behaving as a linear viscoelastic material. On the other hand, if a closed, curved figure such as is also shown in Fig. 3-10 is the result, then the material is behaving as a nonlinear viscoelastic material. If the calculated linear viscoelastic stress is divided by a reference modulus, the result has the dimensions of strain and is called pseudostrain. By this means, the graph of measured against calculated linear viscoelastic stress has been converted into a stress versus pseudostrain graph and the area within the closed, curved loop has been converted into dissipated pseudostrain energy, as illustrated in Fig. 3-11. This dissipated pseudostrain energy is the energy that has been lost in loading and unloading the material minus the energy that has been lost in overcoming the viscous resistance of the material, and therefore it represents the energy that is available to do damage to the material. However, if the material is loaded and unloaded repeatedly and the closed loop does not change its size or area, then that is an indication that the material is not changing and is not being damaged. Damage to the material is indicated by a change in the shape and area of the dissipated pseudostrain energy loop. Figure 3-12 illustrates a slight change in the dissipated pseudostrain energy loop.

Actual stress

Dissipated Pseudostrain Energy Change

Pseudostrain =

Dissipated pseudostrain energy

Linear viscoelastic stress (from strain history) Reference modulus

FIGURE 3-12

Change in dissipated pseudostrain energy indicating damage.

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Chapter Three Changes this slight with one loading and unloading cycle can accumulate with repeated load cycles and the damage can grow accordingly. In order for the pseudostrain energy to match actual measured energy expenditures, the reference modulus ER must be a real modulus of the material which is found by dividing the maximum measured stress by the maximum measured strain. The rate of change of this dissipated pseudostrain area with respect to the change of microcrack surface area is defined as the pseudo-J-Integral and is used in the fundamental law of fracture mechanics from which all fracture predictions are derived. This fundamental law was stated by Schapery (Schapery 1984) to be 2Γ = ER D(tα ) J R where

(3-18)

Γ = cohesive fracture surface energy of the material JR = pseudo-J-integral D(ta) = compliance of the material ta = time required for the crack to grow through the distance of the fracture process zone which is of length, a ER = reference modulus of the material

Several examples of the application of this fundamental law of fracture are given in Chap. 12 and will not be repeated here. The law given in Eq. (3-18) is for cohesive fracture. There is a slightly different formulation for adhesive fracture which takes into account the interaction of the materials that are bonded at an interface, and even include the effect of a third material, such as water, which may be present on the interface. The calculation of the adhesive and cohesive bond strength from the individual surface energies of the component materials is also illustrated in Chap. 12. Thus the expression of the ratio between the damaged and undamaged modulus of a nonlinear viscoelastic material, taking into account the rate of change of dissipated pseudostrain energy in repeated loading is given in Eq. (3-19): ⎧ ⎡ E ⎛ m ⎞ ⎪ 2 ⎢ 1 − 2 Γ ⎛ 4 At = 1 + 2π ⎜ ⎟ ⎨ ∫ c ⎢ ⎜ dW ⎝ bh ⎠ ⎪ E′ R ⎜⎝ ⎢ dN ⎣ ⎩

1 ⎤ ⎫ ⎞ 1+ n ⎥ ⎪ ⎟ ⎥ p(c)dc ⎬ ⎪ ⎟⎠ ⎥ ⎦ ⎭

(3-19)

If a material that has been damaged by cracking is allowed time between repeated loads for the cracks to close and to reform the broken bonds, it will heal. Healing is a complementary process to fracture except that the polar and nonpolar components of wetting or healing surface energy play differing roles than they do in fracture. In fracture, they both resist the fracture. In healing, the stronger the polar wetting surface energies, the more they assist in forming healing bonds whereas the stronger the nonpolar surface energies of wetting or healing, the more they resist the reforming of the broken bonds. The nonpolar surface energies, which are primarily Lifshitz-van der Waals forces, affect the rate of short term healing, in the range of seconds. The polar surface energies, which are primarily hydrogen bonds, affect the long-term rate of healing in the range of minutes and hours. Thus, it is possible for a pavement to have microcracks and even large propagating shear cracks develop under repeated traffic loading, as illustrated in Fig. 3-13. But if the pavement is built of a good asphalt concrete which both resists fracture and heals well, it is possible to have the microcracks and even some of the propagating cracks to heal substantially and to recover much of the original strength and stiffness of the material during low traffic periods.

Overview of the Stiffness Characterization of Asphalt Concrete

Propagating

Microcracks

Shear cracks

FIGURE 3-13

Microcracks and propagating cracks under trafﬁc loading. Microcracking “Tertiary Creep”

Plastic pseudostrain e R

e 0R

N No. of load cycles

FIGURE 3-14 Plastic pseudostrain accumulation with repeated loading.

Another concern in the performance of asphalt concrete pavements is its plastic stiffness. A material which ruts substantially or at a high rate is undesirable. A typical graph of the plastic strain that an asphalt concrete undergoes under repeated loading is shown in Fig. 3-14. The rate of rutting is rapid initially and slows down as the asphalt work hardens and acquires lower plastic compliance, eventually approaching a horizontal asymptote at a plastic strain level of ε 0P . Occasionally, the asphalt concrete becomes too stiff and brittle and becomes susceptible to microcracking. At that point, the plastic strain curve begins to climb steeply as the asphalt concrete softens in accordance with the microcracking process described previously. The equation that describes this process of work—hardening and slowing the rate of increase of the plastic compliance of the material is in Eq. (3-20): ρ ⎛ ε P ⎞ −⎛⎜ ⎞⎟ DP (N ) = ⎜ 0 ⎟ e ⎝ N ⎠ ⎝ Δσ ⎠

(3-20)

where DP(N) = plastic compliance of the asphalt concrete which increases at a decreasing rate with repeated loading ε 0P = maximum or asymptote value of the plastic strain r, b = scale and shape factor of the plastic compliance curve Δs = repeated stress

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Chapter Three The coefficient of this equation r is a scale factor and the exponent b is a shape factor which has a considerable physical significance. It measures the logarithmic rate of decrease of the plastic compliance of the asphalt concrete. The coefficient ε 0p is the asymptote that is approached by the accumulating plastic strain. If this asymptote plastic strain is divided by the repeated stress that causes it, it is the maximum plastic compliance of that material. The maximum plastic compliance is given in Eq. (3-21): D∞P =

ε 0P Δσ

(3-21)

The logarithmic rate of change of the plastic compliance D of the material is a constant b, as shown in Fig. 3-15. The figure also shows that as the plastic compliance gets smaller, it becomes stiff enough to promote the formation of microcracks. The curve departs from the straight line curving upward as microcracks grow and multiply, softening the material and allowing plastic deformation to accelerate. It can be shown mathematically that the exponent b is equal to the creep compliance exponent n if the rate of change of the plastic compliance DP(N) with respect to log(N) is a constant. Even if this rate of change is not constant, the slope b is closely approximated by the creep compliance exponent n. This exponent plays an important role in the fracture and healing of asphalt concrete, and it is also a close approximation of the logarithmic rate of accumulation of plastic strain in the same material. It also governs the rate at which the microcracks soften the asphalt concrete and accelerate the permanent deformation of the material in a process that has been called tertiary creep. This means that these two exponents are very important measures of the rate at which a pavement will deteriorate. It will be wise to devise ways of measuring these properties in the field in in-service pavements. Microcracking and plasticity are closely related phenomena. The first reduces the stiffness of asphalt concrete while the second increases it. Other types of damage, which reduce this stiffness, include moisture damage and aging, the latter of which makes the asphalt concrete more brittle and more susceptible to microcracking. The causes, measurement, and prediction of moisture damage are discussed in detail in Chap. 12.

Plastic deformation 1 log

∂ (log D p)

b

Microcracking

∂ (log N )

Iog N

FIGURE 3-15 Microcracking and plastic deformation.

Overview of the Stiffness Characterization of Asphalt Concrete

Conclusions The stiffness of asphalt concrete is a material property that is central to the performance of asphalt pavements. It depends upon many factors including stress state, temperature, moisture, strain rate, and damage condition. Being able to measure it precisely and accurately in the laboratory and the field is essential to making the design, construction, and management of pavements possible in the present and in the future. Subsequent chapters present methodologies and considerations for measuring the stiffness of asphalt concrete. A large part of the need for making these measurements is the greatly increased role that numerical predictions using mechanics models on computers will have on all engineering and construction operations related to pavements. Mechanics models require material properties of which asphalt concrete stiffness is one of the more important. The composition of the asphalt concrete determines its stiffness in both its damaged and undamaged conditions and also determines its response to the various stresses that are imposed upon it under traffic. Tiny variations of critical components of the mixture can have large scale effects on how the mix behaves under load and this is the reason that construction quality control will become an increasingly critical factor in the future. Defining in a mechanics sense how asphalt concrete stiffness is affected by all of these factors is what makes it possible to identify the critical components and to incorporate them in specifying the performance that those pavements will need to provide in order to meet taxpayer, safety, and public policy expectations.

References Aboudi, J. (1991), Mechanics of Composite Materials: A Unified Micromechanical Approach, Elsevier, New York. Adu-Osei, A. (2000), “Characterization of Unbound Granular Layers in Flexible Pavements,” Ph.D. dissertation, Texas A&M University, College Station, Tex., December. Allen, J. J. (1973), “The Effects of Non-Constant Lateral Pressures on Resilient Response of Granular Materials,” Ph.D. dissertation, University of Illinois at Urbana-Champaign. Christensen, R. M. (1991), Mechanics of Composite Materials, Krieger Publishing Company, Malabar, Fla. Daniel, J. S., and Kim, Y. R. (1998), “Relationships among Rate-Dependent Stiffnesses of Asphalt Concrete Using Laboratory and Field Test Methods,” Transportation Research Record No. 1630, Transportation Research Board, National Research Council, Washington, D.C., pp. 3–9. Einstein, A. (1956), Investigations of the Theory of Brownian Movement, Dover, New York. Findley, W. N., Lai, J. S., and Onaran, K. (1989), Creep and Relaxation of Nonlinear Viscoelastic Materials, Dover, New York. Good, R. J., and van Oss, C. J. (1991), “The Modern Theory of Contact Angles and the Hydrogen Bond Components of Surface Energies,” in Modern Approaches to Wettability (M. E. Schrader and G. Loeb, eds.), Plenum Press, New York. Heukelom, W., and Klomp, A. J. G. (1964), “Road Design and Dynamic Loading,” Proceedings, Association of Asphalt Paving Technologists, Ann Arbor, Mich. Lade, P. V., and Nelson, R. D. (1987), “Modeling the Elastic Behavior of Granular Materials,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 11, No. 5, pp. 521–542.

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Chapter Three Lytton, R. L., Uzan, J., Fernando, E. M., Roque, R., Hiltunen, D., and Stoffels, S. M. (1993), “Development and Validation of Performance Prediction Models and Specifications for Asphalt Binders and Paving Mixes,” SHRP A-357 Report, National Research Council, Washington, D.C. Lytton, R. L. (2000), “Characterizing Asphalt Pavements for Performance,” Transportation Research Record No. 1723, Transportation Research Board, National Research Council, Washington, D.C., pp. 5–16. Schapery, R. A. (1962), “Approximate Methods of Transform Inversion for Viscoelastic Stress Analysis,” Proceedings, 4th U.S. National Congress of Applied Mechanics, p. 1075. Schapery, R. A. (1965), “A Method of Viscoelastic Stress Analysis Using Elastic Solutions,” Journal of the Franklin Institute, Vol. 279, No. 4, pp. 268–289. Schapery, R. A. (1984), “Correspondence Principles and a Generalized J-Integral for Large Deformation and Fracture Analysis of Viscoelastic Media,” International Journal of Fracture, Vol. 25, pp. 195–223. Tseng, K.-H., and Lytton, R. L. (1989), “Prediction of Permanent Deformation in Flexible Pavement Materials,” in Implications of Aggregates in the Design, Construction, and Performance of Flexible Pavements, STP 106, ASTM, Philadelphia, Pa., pp. 154–172. Tutumluer, E., and Seyhan, U. (2002),“Characterization of Cross-Anisotropic Aggregate Base Behavior from Stress Path Tests,” Proceedings, 15th ASCE Engineering Mechanics Division Conference, Columbia University, New York. Van der Poel, C. (1954), “A General System Describing the Visco-Elastic Properties of Bitumens and Its Relation to Routine Test Data,” Shell Bitumen Reprint No. 9, Shell Laboratorium-Koninklijke, Amsterdam, Netherlands.

CHAPTER

4

Complex Modulus Characterization of Asphalt Concrete Terhi K. Pellinen

Abstract There are two complex modulus tests that have been used for characterization of asphalt mixtures in the United States: The dynamic modulus |E∗| test and the dynamic shear modulus |G∗| test, better known as the simple shear tester (SST) shear frequency sweep test. The dynamic modulus test has been selected to be used in the new AASHTO pavement design guide and it will replace the resilient modulus test currently used in pavement design. A new test protocol development for the dynamic modulus test has involved improving specimen instrumentation techniques and analysis methods for the test data. Research has shown that the modulus is less sensitive for test data imperfections than the phase angle when obtained using different signal analysis methods. However, a limit for feedback waveform for deviations of a perfect sine wave would improve data quality for both the modulus and phase angle. Comparison between uniaxial |E∗| and SST shear modulus |G∗| has shown that measured shear modulus was 2 to 30 times lower than theoretically would be expected, and deviations increased as the test temperature increased. Also, the phase angle values obtained from the SST testing were substantially higher than that of the uniaxial testing. It is assumed that instrumentation and specimen size problems in the SST testing are contributing to the differences. Also, differences in the specimen loading modes may have affected obtained parameter values. A new method of constructing mastercurves for cyclic modulus data has been developed that uses a sigmoidal fitting function and experimental shifting. The sigmoidal function approaches asymptotically to the limiting mix stiffness at cold and warm temperatures. At cold temperatures, the limiting stiffness is dependent of the glassy modulus of the binder and at warm temperatures it is dependent of the modulus of aggregate skeleton. Research has also shown that the stiffness correlates well with rutting and fatigue cracking and could be used as a simple performance test to complement the Superpave volumetric mix design method.

89 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.

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Chapter Four

Introduction Pavement design using the elastic layer theory needs two elastic parameters for each material layer used: Young’s modulus (stiffness) and Poisson’s ratio. One of the more widely used stiffness parameters for asphalt mixtures employed in mechanisticempirical structural pavement design procedures has been the dynamic modulus |E∗|. The dynamic modulus has also been selected to characterize the asphalt mixtures in the new AASHTO 2002 Guide for the Design of Pavement Structures, which has been in development in the NCHRP 1-37A project at Arizona State University (ASU). Additionally, the importance of dynamic modulus for mechanistic modeling will be discussed in other chapters in this book. Dynamic modulus will replace the resilient modulus test currently used for pavement design. This chapter discusses a new test protocol development for the dynamic modulus test and presents some considerations of the analysis of imperfect sinusoidal cyclic test data. Also, stiffness as a performance indicator for hot mix asphalt (HMA) will be discussed. A key feature in the material characterization is to construct a mastercurve of the mix. Through the mastercurve it is possible to integrate traffic speed, climatic effects, and aging for the pavement response and distress models. A new method to construct an asphalt mix mastercurve by using a sigmoidal fitting function and experimental shifting is discussed and a stress-dependent master-curve construction method is introduced. There are two complex modulus tests that have been used for characterization of asphalt mixtures in the United States: The dynamic modulus |E∗| test and the shear modulus |G∗| test. Differences of these two tests are discussed related to the mix design and pavement design applications.

Complex Modulus Complex mathematics gives a convenient tool to solve the viscoelastic behavior of the asphalt mixtures and binders in cyclic loading. The sinusoidal one-dimensional loading can be represented by a complex form:

σ ∗ = σ 0 e iω t

(4-1)

ε ∗ = ε 0 e i(ω t−ϕ )

(4-2)

and the resulting strain

The axial complex modulus E∗(iw) is defined as the complex quantity

σ∗ ⎛σ ⎞ = E∗ (iω ) = ⎜ 0 ⎟ e iϕ = E1 + iE2 ε∗ ⎝ ε0 ⎠

(4-3)

in which σ0 is the stress amplitude, e0 is strain amplitude, and w is angular velocity, which is related to the frequency by

ω = 2π f

(4-4)

In the complex plane, the real part of the complex modulus E∗(iw) is called the storage or elastic modulus E1 while the imaginary part is the loss or viscous modulus E2, shown

Complex Modulus Characterization of Asphalt Concrete

FIGURE 4-1

Complex plane.

in Fig. 4-1. For elastic materials j = 0, and for viscous materials j = 90°. The alternative nomenclature is to call the storage modulus as E′ and loss modulus as E″. If a linearly viscoelastic material is subjected to a uniaxial compressive, tensile, or shear loading, the resulting steady-state strain ε = ε 0 sin(ω t − ϕ ) will be out of phase with the stress by the lag angle j, as shown in Fig. 4-2. The ratios of stress and strain amplitudes s0/e0 define the dynamic (or cyclic1) modulus |E∗(w)|, shown in Eq. (4-5): E* (ω ) = E12 + E2 2 =

σ0 ε0

(4-5)

where E1 and E2 can be expressed as a function of phase lag or lag angle E1 =

σ 0 cos ϕ ε0

E2 =

and

σ 0 sin ϕ ε0

(4-6)

The loss tangent defines the ratio of lost and stored energy in a cyclic deformation: tan ϕ =

E2 E1

(4-7)

Figure 4-1 shows that the quantity dynamic modulus presents a magnitude, that is, length of the complex modulus vector E∗ in a complex plane. It should be noted that since the test can be done using either normal or shear stress, the norm of the complex modulus can be defined either |E∗| or |G∗|. In applied viscoelasticity for the asphaltic

FIGURE 4-2

Sinusoidal stress and strain in cyclic loading.

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Chapter Four

FIGURE 4-3 |E∗| and j shown (a) in complex plane and (b) in the Black space. (Pellinen et al. 2002, ASCE.)

materials, the phase angle is usually denoted by j or f for the mix and d for the binder. The complex plane also called Cole and Cole plane (Di Benedetto and de la Roche 1998), or the Black space, can be used to check the quality of test data. In complex plane, the storage modulus E1 is plotted to the real axis (x-axis), and the loss modulus E2 is plotted to the imaginary axis (y-axis). Figure 4-3(a) presents an example of the dynamic modulus test results in the complex plane that allows assessment of data at intermediate and low temperatures. The plotted complex modulus E∗ points on a complex plane should form one unique curve, which is independent of frequency or temperature. In Black space, the modulus values are plotted in log space and phase angle values in arithmetic space, which gives better assessment of the data at high temperatures, as shown in Fig. 4-3(b). Similar to complex plane, the Black space shows the frequency and temperature independent relation of the complex modulus and phase angle. It also allows one to estimate the pure elastic component E(j = 0) of the complex modulus at very low temperatures. The Black space diagram has been shown in the literature with either parameter |E∗| or j in the x-axis.

Test Protocols There are two complex modulus tests that have been used for material characterization of asphalt mixtures in the United States; the dynamic modulus |E∗| test, and the shear modulus |G∗| test, better known as the simple shear tester (SST) shear frequency sweep test. There are other test configurations and specimen geometry that can be used to obtain the complex modulus of the mix, such as bending tests using beams and various shearing tests (Di Benedetto et al. 2001). Some of the tests are homogeneous, that is, tests have direct access to stress and strain and therefore to the constitutive law. Some of the tests are nonhomogeneous, that is, they call for postulating the constitutive law first (such as linear elasticity), and specimen geometry needs to be taken into account to get parameters to the constitutive law. This chapter concentrates on the axial dynamic modulus and SST shear modulus tests although some of the data analysis and instrumentation issues can be applied to any cyclic testing. Both of these tests are homogeneous tests.

Complex Modulus Characterization of Asphalt Concrete

FIGURE 4-4

Schematic of shear frequency sweep test.

SST-Shear Frequency Sweep Test The shear frequency sweep test conducted with the SST was developed in the SHRP research program. The test protocol was first introduced as SHRP Designation M-003: “Standard Method of Test for Determining the Shear Stiffness Behavior of Modified and Unmodified Hot Mix Asphalt with Superpave Shear Test Device” (Harrigan et al. 1994). Later the test protocol was adopted by the American Association of State Highway and Transportation Officials (AASHTO) as a Provisional Standard: AASHTO Designation: TP7-94 (AASHTO 1994). The shear frequency sweep at constant height is a strain-controlled test; the maximum shear strain is limited to 100 microstrains. During the test, a horizontal shear strain is applied at a frequency of 10 to 0.01 Hz using a sinusoidal straining pattern. At the same time the specimen height is kept constant by compressing or pulling the specimen axially based on the closed loop feedback given by the vertical LVDTs (linear variable differential transformer) attached to the sides of the specimen. The specimen is sheared from the bottom as Fig. 4-4 schematically presents. The cylindrical test specimen diameter is 150 mm, height 50 mm and it is glued between two aluminum platens. Based on the TP7 protocol, testing is conducted at 4, 20, and 40°C, but higher temperatures have been used. The strain control mode in both actuators makes the test difficult to run and very soft mixtures can cause severe control problems at high temperatures. Also, test temperatures are limited above 4°C because at colder temperatures mix stiffness may exceed the stiffness of the glue and specimens may shear off the platens.

Compressive Dynamic Modulus Test Background The complex dynamic modulus test was originally adopted in 1979 by the American Society of Testing and Materials (ASTM) as a standard method: “Test Method for Dynamic Modulus of Asphalt Concrete Mixtures” ASTM D 3497-79 (ASTM 1979). The test was conducted applying a haversine load between 0 and 241 kPa using temperatures of 5, 25, and 40°C and frequencies of 1, 4, and 16 Hz. Additional temperatures and frequencies have been added to the protocol by Witczak et al. at University of Maryland (UMd) (Witczak et al. 1996; Witczak and Kaloush 1998; Pellinen and Witczak 1998) to enable construction of a full mastercurve

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Chapter Four

FIGURE 4-5

Loading pattern for compressive dynamic modulus testing.

of the asphalt mix. They used test temperatures ranging from 15 to 55°C and frequencies from 25 to 0.1 Hz. In the original ASTM protocol, capped cylindrical specimens, having a height to diameter ratio of ≥2, are loaded with a uniaxial compressive haversine stress pattern shown in Fig. 4-5. Testing is performed in load control keeping the load constant throughout the frequency sweep. However, load levels are decreased as a function of test temperature to keep measured strain levels small and to achieve steady-state straining pattern after the initial creep, that is, recoverable and nonrecoverable (permanent) strain accumulation, of the specimen. This same loading approach was used by Witczak et al. in their testing program at UMd, where they created a large database that was used to develop the predictive dynamic modulus equation (latest form by Andrei et al. 1999), which will be used in the new pavement design guide as an initial approximation of the stiffness of asphalt mixtures. In recent years, the simple performance test and the new pavement design guide work has led to a new research and protocol development for the dynamic modulus test. The next sections discuss protocol issues that should be addressed when conducting cyclic sinusoidal testing.

Specimen Fabrication and Instrumentation Different instrumentation techniques have evolved in the testing over the years. In the early development of the test, strain gauges were used to measure displacements, whereas today, the deformations are measured using spring-loaded LVDTs. In the early approach used by Witczak et al. at UMd, these LVDTs were clamped vertically on diametrically opposite specimen sides. A new method of specimen instrumentation has been developed by the Superpave Models Management research team (Witczak et al. 2000). Axial deformations are measured with LVDTs mounted between gauge points glued to the specimen, as shown schematically in Fig. 4-6. LVDTs are secured in place using brackets screwed to the studs. In addition, guiding rods, which prevent the LVDTs from bulging out at high temperatures, were added to the instrumentation. This instrumentation allows the use of membrane around the specimen for confined testing. It also prevents any constraining that the clamp around the specimen may have at higher temperatures. The deformations are measured at a minimum of two locations 180° apart; however, three locations located 120° apart are recommended to minimize the number of replicate specimens required for testing. This new approach has been found to be mandatory for asphalt specimens that exhibit large nonrecoverable deformation during testing. Compaction methods for fabricating test specimens in the laboratory have varied over the years. The early compaction methods included kneading compactor and rolling

Complex Modulus Characterization of Asphalt Concrete

FIGURE 4-6 General schematic of gauge points (not to scale). (Witczak et al. 2000, with permission from Association of Asphalt Paving Technologists.)

wheel compaction method. In the Superpave volumetric mix design method, the specimens are compacted using Superpave Gyratory Compactor (SGC), which is also used for compacting test specimens for the simple performance testing. In addition, the SGC will be used to compact test specimens for the new pavement design guide material characterization testing. The standard protocol for laboratory determination of asphalt concrete stiffness dictates that the specimen size and boundary conditions produce a homogeneous stress distribution in the test specimen. A study by Witczak et al. (2000) determined the minimum test specimen dimensions that provide measured responses and material properties that are independent of the test specimen size (end effects) and aggregate size using gyratory compacted laboratory test specimens. The experiment included four height-to-diameter ratios being 1, 1.5, 2, and 3, and three specimen diameters being 70, 100, and 150 mm. The specimen diameter was also used as gauge length for the LVDTs. Testing was conducted with three mixtures with nominal aggregate sizes of 12.5, 19.0, and 37.5 mm. For the dynamic modulus test, the 70-mm-diameter specimens with height-to-diameter ratio of 1.5 or greater gave acceptable test results when tested at 4 and 40°C. The specimen size of 100 mm in diameter and 150 mm in height has been selected for the new dynamic modulus test protocol development at ASU. The gauge length used is 100 mm. After compaction, the 100-mm-diameter cylindrical test specimens are cored from the center of the 150-mm-diameter gyratory specimens. If the desired specimen height is 150 mm, the gyratory specimen has to be compacted to a height of at least 170 mm in the gyratory mold. This gives 10 mm of each specimen end that can be sawed to obtain smooth ends perpendicular to the axis of the specimen. A recommended tolerance for perpendicular ends is 0.05 mm across any diameter, which can be checked using a straight edge and feeler gauges. The specimen end should not depart from perpendicular to the axis of the specimen by more than 0.5°. The ASTM protocol required capping specimens with sulfur mortar to ensure parallel specimen ends to prevent any misalignment and rocking during testing. Witczak et al. (2000) have recommended that specimens should not be capped and friction-reducing treatments should be placed between the specimen ends and the platens (hardened steel disks). This treatment is needed to have homogenous stress distribution in the specimens by avoiding any restraints capping may introduce. The end treatment consists of two 0.5-mm-thick latex sheets separated with silicone or vacuum grease.

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Chapter Four Control Mode In the original ASTM protocol, the dynamic modulus test was conducted in load control by applying constant compressive cyclic loading throughout the frequency sweep. This same approach was planned to be adapted to the new test protocol. However, the applied constant stress causes the subsequent resilient strain to increase as frequencies decrease in the frequency sweep due to the viscoelastic nature of the mix. The proposed new test protocol considers 150 microstrains as the limit for the linear viscoelastic region. However, at warm temperatures the low-frequency strain can be up to three times larger than strain at high frequencies, when constant stress is used. Then, at high temperatures the magnitude of the strain may exceed 150 microstrains at low frequencies although the strain at high frequencies stays under 50 microstrains. This may cause some damage accumulation in the specimen, in addition to a creep imposed to the cyclic straining pattern. This needs to be accounted for in the data analysis when obtaining modulus and phase angle values. Therefore, some European researchers (Di Benedetto and de la Roche 1998; Doubbaneh 1995) prefer to do the cyclic testing in strain control mode by applying tension-compression loading. However, specimens need to be glued to the platens to apply tension. This loading model gives zero average stress, which enables the steadystate straining pattern and, thus, eliminates the creep. The strain amplitude can also be controlled to stay below desired limit during testing. In pure tension testing, strain keeps creeping as a function of time without achieving steady-state pattern. A rest period between frequencies in the frequency sweep is not discussed in the proposed new test protocol. However, some of the controllers used in the past and perhaps some current ones cannot produce continuous frequency sweeps; each frequency must be programmed separately in such a manner that there will be some lag time or rest period between each frequency. The rest period helps to prevent specimen from heating during cyclic testing, although for modulus tests the number of cycles is usually limited to fewer than 200 and heat increase may not be a problem. However, the rest period allows some of the transient strains to recover during testing which may have some effect on the measured modulus values and selection of suitable data analysis methods.

Rest Period in the Frequency Sweep

Compressive Axial versus SST-Shear Modulus Correlation of Modulus and Phase Angle Values A study conducted by Witczak et al. (2000) compared SST frequency sweep and uniaxial dynamic modulus tests and concluded that the shear modulus value |G∗| obtained from the SST test was not a true fundamental property such as |E∗| due to the instrumentation and specimen size problems. Thus, according to Witczak et al., test results from the SST device can be categorized as representing an index value of shear modulus of the asphalt mixture, not a fundamental material property. This means that the |G∗| cannot be used directly to replace the |E∗| in the pavement design applications where mixture stiffness is needed. Equation (4-8) gives a conversion model developed by Christensen, Pellinen, and Bonaquist (2003), which can be used to convert |E∗| to |G∗|. Both moduli values have units of psi in the equation: |G∗ |= 0.0603|E∗ |1.0887

R 2 = 0.93

(4-8)

Complex Modulus Characterization of Asphalt Concrete A study by Pellinen and Witczak (2002a) suggests similar results. The overall linear correlation between measured dynamic modulus |E∗| and SST-shear modulus |G∗| values within the linear viscoelastic region for the various dense graded mixtures was R2 = 0.87. Figure 4-7(a) shows that the deviations in test results are greater at low- and high-temperature extremes, greatly exceeding the stiffness ratio of |E∗| ≈ 2.5 to 3|G∗|, which is based on theoretical equivalency using Poisson’s ratio values from 0.2 to 0.5. The overall linear correlation of the phase angle values from dynamic modulus and SST-shear modulus measurements was R2 = 0.61 and, at best, is indicative of a fair correlation. More importantly, it can be observed from Fig. 4-7(b) that there is no direct

FIGURE 4-7 Correlation of axial and shear (a) modulus and (b) phase angle. (Pellinen and Witczak 2002a, with permission from National Academies Press.)

97

98

Chapter Four agreement between the phase angles measured in axial and shear conditions. The SST device measured phase angle values ranging from 12 to 70°, while the axial testing gave values ranging between 12 and 42°. It can be hypothesized that two possibilities contribute to these findings: the SST device may have control problems at very high and low test temperatures, or the difference in the specimen loading mode may contribute to the different material responses. In compressive axial loading, the average stress amplitude is always larger than zero, but in the SST device, the average shear strain and stress amplitude are zero due to the shear straining through zero, as Fig. 4-4 shows. This may cause compressive testing to be more influenced by the aggregate skeleton effect (which is elastic) in the higher test temperatures, causing lower phase angle values and higher modulus values. Also material inhomogeneity and anisotropy may cause some differences.

Stiffness as the Asphalt Mix Performance Indicator Introduction A new mix design procedure, the Superpave volumetric mix design, was developed in the Strategic Highway Research Program (SHRP) in the mid-1990s. However, unlike the Marshall mix design method, the new Superpave volumetric mix design procedure did not include any mechanical test to check the mixture performance after the volumetric part of the design procedure had been completed. Experience from the implementation process over recent years has shown that the volumetric mix design procedure without a performance test is inadequate for ensuring acceptable mix performance. Work leading to the development of a simple performance test has been in progress through the NCHRP 9-19 project: “Superpave Support and Performance Models Management,” Task-C at ASU (Pellinen and Witczak, 2002a). The focus of the proposed simple performance test (SPT) has been to measure a fundamental engineering material property that can be linked back to the advanced material characterization measurements that are needed for a detailed distress analysis. The three main asphalt mixture distresses considered in the design process are permanent deformation, fatigue cracking, and thermal cracking. The main objective of the overall NCHRP 9-19 Task-C research effort was to recommend from several candidate tests the most promising fundamental SPT for use with the Superpave volumetric mix design procedure. The potential simple performance tests that have been studied can be categorized as stiffness-related tests, deformability tests, and cracking tests.

Stiffness-related Tests: Recommendations for SPT The laboratory test program for stiffness-related test by Pellinen (2001) and Pellinen and Witczak (2002a) included duplicating mixtures from three different experimental test sites in the United States. These sites were the MnRoad, FHWA-ALF, and WesTrack sites. All mixtures from these sites were dense graded mixtures. However, two stone mastic asphalt (SMA) mixtures and two dense graded mixtures from the Finnish Asphalt Pavements Research Project (ASTO) were also studied. ASTO project was conducted from 1987 to 1992 including comprehensive laboratory and field research effort on Finnish asphalt mixtures (Saarela, 1993). The ASTO test specimens were fabricated by the Technical Research Center of Finland and tested at ASU. Testing was conducted using two replicate specimens. Specimen instrumentation was as shown in Fig. 4-6. Each specimen was tested in an increasing order of temperature

Complex Modulus Characterization of Asphalt Concrete using dynamic stress levels of 138 to 965 kPa for colder temperatures of −9, 4.4, and 21.1°C. For warmer temperatures of 37.8 and 54.4°C, stress levels of 46–68 kPa and about 21 kPa were used, respectively. For each temperature level, specimens were tested in a decreasing order of frequency; frequencies used were 25, 10, 5, 1, 0.5, and 0.1 Hz. A 60-second rest period was used between each frequency to allow some specimen recovery before applying new loading at lower frequency. This was done to reduce the possible damage and heat accumulation during testing. This testing was conducted trying to stay under 150 microstrains at all temperature and frequency levels. For the confined testing program, the testing frame was equipped with a triaxial cell capable of applying cell pressure up to 690 kPa. High-temperature stress levels were determined based upon the stress to strength ratios determined using cohesion and friction parameters from the triaxial strength test at 54.4°C. At colder temperatures, the deviatoric stress was changed depending upon the stiffness of the mix. A 60-second rest period was used between each frequency similar to the unconfined testing. The instrumentation and testing setup is illustrated in two photos for confined dynamic modulus testing. Figure 4-8(a) demonstrates the loading of a specimen to the pressure cell, and Fig. 4-8(b) shows the pressure cell sealed and ready for testing. Based on the research, a provisional recommendation for the stiffness parameter for rutting was the dynamic modulus |E∗| of the mix. For dense graded mixtures, the |E∗| parameter could be obtained at an unconfined stress state using small stress levels to stay in the linear viscoelastic region. The provisional recommended test temperature was 54.4°C and the frequency was 5 Hz. These were the conditions under which all mixtures exhibited the best correlation to rutting. Time-temperature superposition principle can be used to translate test results to the desired climatic conditions and traffic speed for the performance criteria. The analysis of the Finnish SMA mixtures indicated that confinement is needed to capture performance of SMA mixtures compared to dense graded mixtures. Therefore, confinement was recommended for testing of SMA and open-graded mixtures. However, the level of confinement needs to be determined for mixtures constructed in the United States. In addition to the axial stiffness, the SST shear modulus |G∗| was also included in the research program. Testing was conducted by Advanced Asphalt Technologies. The overall correlation to rutting was the same for both |E∗| and |G∗| values, the correlation coefficient being 0.79. However, the order or ranking of the mixtures was different. This indicates that these two modulus values are not interchangeable, as discussed above.

(a)

FIGURE 4-8

Conﬁned |E∗| test set up.

(b)

99

100

Chapter Four Most likely mixtures that have binder as a dominating stiffness contributor will be ranked similarly, while mixtures that have different gradations, such as dense graded versus SMA, may be ranked differently. The recommended provisional stiffness parameter for fatigue cracking was also found to be the unconfined compressive dynamic modulus of the mix. However, this conclusion is based on limited data. The dynamic modulus of asphalt mixtures was not an adequate performance indicator for thermal cracking.

|E∗| versus |E∗|/sinj The Superpave binder specification defines and places requirements on a rutting factor, |G∗|/sind, which presents a measure of the high-temperature stiffness or rutting resistance of asphalt binder. |G∗| is a shear modulus of binder and d is the phase lag between stress and strain. According to Bahia and Anderson (1995), the work W dissipated per loading cycle is inversely proportional to the parameter |G∗|/sind: 1 ⎡ ⎤ W = πσ2 ⎢ * ⎥⎦ | |/sin G δ ⎣

(4-9)

To minimize permanent deformation, the work dissipated during each load cycle should be minimized. In a similar manner, a rutting factor, |E∗|/sinj, can be defined for asphalt mixtures, where j is the phase angle of the mix. For fatigue cracking, a performance factor in the Superpave binder specification is |G∗|sind. Thus, the equivalent performance factor for the mix is |E∗|sinj. The Superpave binder specification has a minimum value for the |G∗|/sind parameter against rutting, and a maximum value for the |G∗|sind for fatigue cracking. The correlation between rutting and rutting factor |E∗|/sinj was R2 = 0.91, and between rutting and |G∗|/sinj only R2 = 0.74, although the modulus itself gave similar correlations. As discussed earlier, Fig. 4-7(b), the correlation between phase angles of the axial and shear modulus was poor to fair, which may explain the poorer correlation of |G∗|/sinj and rutting. However, it should be noted that these results apply only to the dense graded mixtures used in the testing program. The reason that the |E∗|/sinj was not recommended as the SPT, regardless that it gave better correlation to rutting than modulus itself, is that the phase angle of the asphalt mix is dependent of frequency and temperature differently than the phase angle of the conventional binder. For conventional binders, the phase angle is an increasing function of temperature, while the asphalt mix phase angle first increases as temperature increases and then starts to decrease. This is illustrated in Fig. 4-9 that shows binder and mix data in the Black space. The binder phase angle approaches 90° at high temperatures when the mix phase angle approaches some limiting value initiated by the damping of aggregate skeleton. This occurs because the elastic effect of aggregate skeleton pushes through the viscous effect of binder at high temperatures. Therefore, if the phase angle decreases it may be due to more elastic binder or more viscous binder that allows the elastic effect of aggregate skeleton to influence the phase angle value. Therefore |E∗|/sinj is not a stable performance parameter for asphalt mixtures. This same phenomenon can also be seen for some of the modified binders for which the phase angle is not an increasing function of temperature.

Effect of Confinement and High Stress Levels One potential limitation of using stiffness measurements as a performance indicator at low stress/strain test applications is that the effect of aggregate shape and hence internal

Complex Modulus Characterization of Asphalt Concrete

FIGURE 4-9

Binder and mix behavior in the Black space.

friction may not be accounted for adequately enough. To assess this possibility, axial testing was conducted employing both low and high levels of deviatoric stress with different levels of confinement. It was hypothesized that high stress levels with confinement would mobilize the internal friction in the mixtures. However, confinement (138 to 206 kPa) or high stress levels (up to 552 kPa) did not improve the correlation to rutting.

Analysis of Cyclic Sinusoidal Test Data It is often difficult to obtain a perfectly sinusoidal feedback signal from the highfrequency testing due to the test equipment limitations and operator errors. If the feedback signal is not a perfect sine wave, it is noisy, or if there is transient recoverable and permanent deformation imposed over the sinusoidal signal, the computed modulus and phase angle values may differ depending on the method used for filtering and phase referencing the signal. Fast Fourier Transform is one of the filtering methods that can be used to process the stress and strain signals. Also, different regression techniques have been used to smooth the data.

Imperfections in the Cyclic Test Data Figure 4-10 shows some examples of various dynamic modulus test data imperfections in cyclic testing by Pellinen and Crockford (2003). Figure 4-10(a) shows data where the applied load signal (stress) is slightly skewed to the left of the strain signals denoted as Axial1 and Axial2 in the legend. This imperfection causes a large deviation (linear regression standard error of 8.9% from a perfect sine wave) as Table 4-1 later on shows. Figures 4-10(b) and 4-10(d) show fairly good data, which is creeping due to the transient recoverable and nonrecoverable deformation. Linear regression standard error for load from perfect sine wave is between 1.5% and 2.1%, and between 2.8% and 6.5% for displacement. Figure 4-10(c) shows noisy displacement and load data in which standard error for load is 11.6% and for displacement 14.5%. Figure 4-10(e) shows load data that is skewed left and displacement transducers that are deviating from each other in large amounts. Standard error is 8.3% for load and 9% for displacement. Figure 4-10(f) shows fairly good load data signal and somewhat noisy displacement signals, standard error for load is 3.7% and for average displacement 6.2%.

101

102

Chapter Four

FIGURE 4-10 Examples of different imperfections of the sinusoidal test data. (Pellinen and Crockford 2003, with permission from RILEM Publications.)

Fast Fourier Transform The following short introduction of signal processing and Fast Fourier Transform (FFT) has been summarized from the Fundamentals of Signal Analysis by Hewlett Packard (1989) and from a paper by Ramsey (1975). The FFT is an algorithm for transforming data from time domain to frequency domain. A finite number of data points need to be sampled using discrete intervals of time separated by Δt. Then, time record is N equally spaced samples of input signal and the requirement for the Fast Fourier analysis is that the dataset have exactly N = 2m data

Complex Modulus Characterization of Asphalt Concrete

Standard Error for Load, %

Avg. Standard Error for Displ., %

Hz Temp.

25

Hz 10

5

1

0.5

0.1

25

10

5

1

0.5

0.1

9.6

14.4

5.8

3.8

10.4

3.3

−9°C

8.9

6.0

3.4

1.2

0.8

0.6

4.4°C

8.6

3.7

2.3

1.2

1.0

1.5

8.7

4.9

4.0

3.1

3.0

2.8

21.1°C

9.5

5.6

3.6

2.4

2.0

2.2

12.0

9.5

3.9

3.1

6.9

6.5

37.8°C

11.6

7.3

5.5

3.2

3.1

2.1

14.5

7.6

6.4

6.8

7.4

8.4

54.4°C

8.3

9.1

6.8

5.3

4.4

3.7

10.0

9.0

7.2

6.8

6.1

6.2

Average

9.4

6.3

4.3

2.7

2.3

2.0

11.0

9.1

5.5

4.7

6.8

5.4

St.Dev.

1.3

2.0

1.8

1.7

1.5

1.1

2.3

3.5

1.5

1.9

2.7

2.3

Source: Pellinen and Crockford 2003, with permission from RILEM Publications.

TABLE 4-1

Standard Error of Estimate for Deviation of Sine Wave

points (m > 2). To completely describe a given frequency, two values are required: the magnitude and phase, or real part and imaginary part. Consequently, N points in timedomain can yield N/2 complex quantities in the frequency domain. The linear Fourier spectrum is a complex valued function that results from the Fourier transform of a time waveform. A System Transfer Function H(f ) is used to transfer the input data x(t) and output data y(t) to frequency domain Sx(f ) and Sy(f ), where Sx and Sy are linear Fourier spectrums of x(t) and y(t). Thus, Sx and Sy have real (in phase or coincident) and imaginary (quadrature) parts, respectively. In general, the result of any continuous linear system on any time domain input signal x(t) may be determined from the convolution of the system impulse response h(t), with the input signal x(t), to give the output y(t). y(t) =

∫

∞

−∞

h(τ )(t − τ )dτ

(4-10)

Applying Fourier transform to the convolution integral Sy ( f ) = Sx ( f )H ( f )

(4-11)

the transfer function H can be defined H=

OUTPUT Sy = INPUT Sx

(4-12)

Power spectrum of the input x(t) is defined as Gxx = SxSx∗, where Sx∗ is complex conjugate of Sx. Power spectrum of output y(t) is defined as Gyy = SySy∗, where Sy∗ is a complex conjugate of Sy. Gross power spectrum is Gyx = SySx∗ and it contains the phase information. Then, the transfer function H, which can be applied to any waveform, can be defined as: H=

Sy Sx* Gyx ⋅ = Sx Sx* Gxx

(4-13)

103

104

Chapter Four The quality of transfer function determines if the system output is totally caused by the system input. Noise and/or nonlinear effects can cause large errors at various frequencies, thus including errors when estimating the transfer function. Coherence function g 2 can be used to estimate the quality of system, where

γ2 =

Response power caused by applied input Measuredd response power

γ = 2

Gyx

(4-14)

2

GxxGyy

where 0 ≤ g 2 ≤ 1

(4-15)

If g 2 = 1 at any specific frequency, the system is said to have perfect causality at that frequency. If g 2 < 1, then extraneous noise is also contributing to the output power.

Differences in Employed Analysis Methods: Time Domain Methods A study by Pellinen and Crockford (2003) compared three different filtering methods and two different phase referencing methods of computing modulus and phase angle from compressive dynamic modulus test data. The methods discussed were limited to time domain techniques (other than FFT method) applied to cyclic loading in compression. Study showed that the computed modulus values were less sensitive to different analysis techniques than the phase angles. Dynamic modulus test data of a dense graded asphalt mix obtained at five different temperatures and six different frequencies were analyzed in this study. This type of a data set is needed to construct a mastercurve of the mix required in the new pavement design guide. Testing was conducted as a stress controlled test applying compressive stress with a frequency sweep of 25, 10, 5, 1, 0.5, and 0.1 Hz. The testing system capable of maximum sampling rates of 1 kHz and sampling rates adequate to eliminate aliasing at the frequency of loading were used. Additional firmware and sampling techniques were employed to minimize skewing of sequential samples and noise in the incoming signal. In the presence of the noise remaining after firmware filtering, the peaks of the waveforms generally exhibit the largest noise amplitude in a single cycle and are therefore the worst locations to perform analyses that determine phase angles. Greater noise at the peaks is due in part to the test machine actuator and the transducers reversing their direction of movement at those times. It is assumed that it is much better to determine phase angle from the “middle” of the waveform where the machine and transducers are all moving in a relatively “steady state.” Using unfiltered peaks can easily create phase shifts that are due to noise instead of fundamental signal peaks. The modulus must be computed from the peaks, so additional software filtering is usually performed to improve the peak measurement. This filtering must be carefully done so that time skewing and alteration of the fundamental signal magnitude are minimized. When any type of cyclic forcing function is applied to a material such as asphalt concrete under load control, a strain response that mirrors the forcing function but with different amplitude and a phase shift is expected. This is an oversimplification even in the case of strictly compressive loading: • Even if the forcing function’s wave shape is perfect, there is a nonzero average stress level during the cyclic loading which causes the cyclic strain response curve to be superimposed on a creep curve. For asphalt and polymers, this is more apparent at high temperatures. Reducing the load may minimize the creep, but typical load amplitude requirements do not eliminate the creep.

Complex Modulus Characterization of Asphalt Concrete • Engineering behavioral characteristics of the material can cause the response curve to deviate from one which mirrors the forcing function. Characteristics that are important in causing these deviations include anisotropy (transversely isotropic and orthotropic degrees of anisotropy are particularly relevant to asphalt which has been compacted in the field or in a gyratory compactor), and what are sometimes referred to as bimodular properties. An additional response occurs if there are phenomena such as damage or strain softening embedded in the creep response. If, in each cycle of loading, the stress changes from compression to either extension or tension, an additional response called the Bauschinger effect (Chen and Han 1988) arises from plasticity theory. Since it is relatively unimportant to differentiate between strain softening and the Bauschinger effect, a single plasticity/damage response is the final component of a generalized strain response curve in which plasticity/damage effects may cause changes in response amplitude over time.

Studied Analysis Techniques The three studied data filtering methods were (1) no filtering, (2) Spencer’s 15 point, and (3) regression; the two-phase referencing methods were (1) peak picking and (2) central waveform bracketing. These produced seven combinations of analyzed methods. Methods A and B deviate only in how the phase angle is obtained, as do methods E and F: • Method A: Spencer’s 15-point data filtering and central waveform bracketing • Method B: Spencer’s 15-point data filtering and peak picking • Method C: Second-order polynomial over 25% of data and peak picking • Method D: Second-order polynomial over 10% of data and peak picking • Method E: No filtering and central waveform bracketing • Method F: No filtering and peak picking • Method G: Sinusoidal over 100% of data and regression coefficients Figure 4-11(a) shows an example of the filtering and phase referencing for methods C and D. These methods were used in the UMd testing program (Mirza and Witczak

FIGURE 4-11 Example of (a) method C and (b) method A.

105

106

Chapter Four 1994). Signals were filtered using second order polynomial over 25% of data, and + marks in the signal peaks show the data points used in the modulus and phase angle computations. The number of points included in the regression analysis was based on the total number of points available in a cycle. The second order polynomial seems to be an adequate representation of the waveform in the local region of the peak even though a perfect waveform would be sinusoidal. The peak picking method has embedded the filtering method within it. That is, the peak with no filtering (method F) is the peak of the noise and the peak with regression filtering over less than 100% of the data coincident with a zero in the first derivative of the regression equation (methods C and D). Figure 4-11(b) shows an example of the filtering and phase referencing for method A. Signals were filtered using Spencer’s 15-point filter and + marks in the stress signal peaks show the data points used in the modulus computations. Phase angle computations were done using data points from the central part of the wave signals averaging the lag time between stress and strain. Spencer’s 15-point filter (Kendall 1951) is similar to a running cubic spline, and it seems to be quite effective at reducing the tendency to skew that is normally seen with running mean techniques. Cubic spline is an interpolation function that passes a curve through a set of points in such a way that the first and second derivatives of the curve are continuous across each point. Central waveform bracketing combines regression across peaks with numerical search methods to find a line (or curve) that defines the waveform period and location of central reference points to compute the phase (Crockford 2001). An assumption is made that the underlying creep in the signal is linear. This is known to be an approximation, but the number of cycles used for the analyses is relatively small justifying the assumption. The sinusoidal regression over 100% of the data (method G) compares the data to the perfect sine wave (AAT 2001). The phase reference comes from the regression parameter estimates, and again the underlying creep is assumed to be linear.

Variability of Modulus and Phase Angle Values An analysis of variability of modulus and phase angle values was performed for methods A, B, C, D, E, and F. At each loading frequency, analysis was done using the last five cycles. The modulus and phase angle values were determined for each five cycles separately and then values were averaged from the test data obtained at temperatures ranging from –9 to 54.4°C and frequencies between 25 to 0.1 Hz. Standard error of estimate Se% was computed as percent of average value. This analysis could not be performed for the method G because the analysis calculation scheme used produced only a single estimate of amplitude and phase over a group of cycles. Table 4-2 shows results of the analysis for all of the studied methods for the modulus of the mix using 25, 5, and 0.1 Hz test data at all test temperatures. For the modulus computations, the method A and B were the same and produced the same modulus values, as did the methods E and F. The Se% varied between 0.3 and 1.2 with the lower frequency values and higher temperature data giving slightly higher variation. Table 4-3 shows analysis results for the phase angle values. Methods B and F turned out to be very unstable, Se% were 12.8 and 19.7, respectively. Both methods used the peak picking technique to obtain data points, but signals for method B were filtered while method F had unfiltered data. For the other methods Se% was between 1.5 and 4.1.

Complex Modulus Characterization of Asphalt Concrete

−9°C ∗

4.4°C

21.1°C

|E | (MPa)

Se (%)

|E | (MPa)

Se (%)

|E | (MPa)

Se (%)

|E | (MPa)

Se (%)

|E∗| (MPa)

Se (%)

25

A,B

16825

0.3

9762

0.2

4229

0.8

814

0.7

280

0.5

TABLE 4-2

∗

54.4°C

Case

0.1

∗

37.8°C

Hz

5

∗

C

16759

0.3

9686

0.3

4234

0.7

810

0.6

276

0.4

D

16876

0.4

9748

0.4

4243

0.7

815

0.9

282

0.4

E,F

16576

0.5

9563

0.3

4074

1.5

777

0.6

282

0.5

G

16734

–

9673

–

4233

–

810

–

272

–

A,B

14129

0.3

7553

0.4

2544

0.6

451

1.1

199

0.8

C

14401

0.2

7618

0.1

2606

0.2

452

0.5

195

0.6

D

14186

0.2

7581

0.1

2558

0.4

444

0.8

194

1.0

E,F

13682

0.4

7464

0.5

2533

0.7

452

1.2

200

0.8

G

14324

2567

–

7562

–

–

450

–

194

–

A,B

8928

0.3

3185

0.2

831

0.4

173

0.6

128

0.4

C

8881

0.2

3181

0.1

828

0.2

173

0.4

124

0.3

D

8923

0.3

3185

0.2

827

0.1

173

0.5

128

0.4

E,F

8795

0.3

3160

0.2

817

1.0

174

0.9

131

0.9

G

8720

–

2973

–

750

–

166

–

118

–

Variation of Modulus Values for Different Analysis Methods

Based on this analysis, it is clear that methods B and F did not produce stable phase angle parameter values. An analysis of variance and a Tukey test were conducted for methods A, C, D, and G to assess statistical differences of the average modulus and phase angle values. The Tukey test showed that, generally, there were only slight statistical differences in the modulus values between the methods A, C, D, and G at 25- or 5-Hz test data. However, method G was computing systematically lower modulus values up to 11% for the 0.1-Hz data (a = 5%). For the phase angle, the most deviations occurred at 54.4°C and 0.1-Hz frequency range being up to 22% between method C and G.

Deviations from Perfect Sine Wave Test data was analyzed using method G provided by AAT (2001) to estimate the deviations from a perfect sine wave. Table 4-3 summarizes analysis results. The load feedback data, obtained at 25 and 10 Hz frequency, systematically exceeded a 5% standard error value from a perfect sine wave, which has been considered to be a cutoff value for rejecting the data in the proposed new test protocol (AAT 2001). The closest match for the perfect sine wave was with 0.1-Hz load data at all test temperatures. This trend may be explained by incorrect PID parameters that adjust the waveshape in the feedback loop, failure to include an adaptive level control, or incapacity of the hardware (servovalve, actuator, and associated hydraulic flow controls) to deliver the desired waveshape.

107

108

Chapter Four

−9°C

4.4°C Se (%)

Se (%)

Se (%)

Se (%)

j (°)

Se (%)

j (°)

25

A

10.2

1.7

17.4

2.4

31.0

1.4

34.1

4.2

32.7

1.1

B

7.2

25.0

19.8

9.1

41.4

5.3

41.4

8.7

43.2

7.8

C

9.0

5.0

18.2

1.6

32.0

1.6

34.5

1.2

33.2

0.6

D

9.9

6.8

20.0

4.5

42.8

7.8

39.0

4.3

40.7

3.7

E

9.9

3.2

17.3

1.8

31.8

2.4

34.4

7.0

32.9

1.8

F

7.2

61.2

14.4

15.3

43.2

12.1

48.6

12.6

50.4

4.4

G

10.8

–

19.5

–

33.1

–

36.8

–

34.7

–

A

12.1

1.7

21.1

0.4

33.3

1.1

31.9

1.3

28.7

2.9

B

12.2

37.9

22.3

18.0

38.2

14.2

40.3

18.4

31.0

7.9

C

12.0

1.8

22.7

1.5

34.0

0.9

31.3

1.5

29.3

1.5

D

12.6

6.2

23.9

2.8

34.0

2.7

34.7

8.2

29.3

4.0

E

11.9

3.4

20.9

0.7

34.0

1.2

32.4

1.3

28.7

3.7

F

7.9

73.9

25.2

20.7

41.0

17.7

40.3

24.0

38.2

13.5

G

12.1

–

21.8

–

32.3

–

30.7

–

27.3

–

A

16.1

1.0

29.9

0.5

31.5

1.3

23.6

1.7

21.7

1.2

B

17.3

7.8

32.4

3.5

33.1

7.2

27.4

6.7

21.6

13.9

C

17.2

1.1

33.3

0.2

36.5

0.7

27.4

1.2

25.1

1.4

D

17.8

3.2

33.4

0.5

36.2

2.1

26.9

1.6

22.3

3.6

E

15.6

1.1

30.0

0.5

31.4

1.5

23.5

1.7

21.2

1.8

F

14.4

11.2

31.7

6.6

36.0

7.7

35.3

8.2

27.4

6.7

G

16.3

–

29.8

–

31.3

–

23.0

–

20.6

–

TABLE 4-3

j (°)

54.4°C

Case

0.1

j (°)

37.8°C

Hz

5

j (°)

21.1°C

Variation of Phase Angle Values for Different Analysis Methods

In Fig. 4-12, the same test data is plotted in the Black space, where the data should form a single temperature and frequency independent curve. Figure 4-12(a) shows method F and the combination of methods C and D, which was obtained by using either method based on the accuracy of the peaks at a given temperature and frequency. Method F produced very noisy data and should not be used for data analysis. Figure 4-12(a) also shows methods A and G. Method C and D gives higher phase angles compared to methods A and G, which seem to give very similar results, although method A has a slightly better correlation. Figure 4-12(b) presents a subset of the original data set that excludes the 25- and 10-Hz data in which the load waveform standard error exceeded 5%. The selection of this subset noticeably cleaned the A and G data sets. This analysis suggests that the most robust method to obtain the parameters from the dynamic modulus test data is the combination of methods C and D, because data reduction increased the R2 value only 0.5%. However, this method seems to overpredict

Complex Modulus Characterization of Asphalt Concrete

FIGURE 4-12 Test data plotted in the Black space. (Pellinen and Crockford 2003, with permission from RILEM Publications.)

the phase angle values at intermediate temperatures compared to the methods A and G. This analysis also suggests that method A is slightly more robust than method G because the data reduction increased the R2 value only 2.4% compared to 5.7% increase for the method G. Overall, based on this analysis it seems reasonable to have some limit for deviations of the controlling load waveform from a perfect sine wave to obtain good quality data, but further study would be necessary to determine whether that limit should be 5% or some other value.

Differences in Employed Analysis Methods: FFT versus Time Domain Methods The same data set described above was also analyzed using Fast Fourier Transform in the Mathcad software. The original dataset did not have the required amount of data points, and interpolation functions cspline(x,y) and interp(S,X,Y,x) were needed to create the required amount of data points. After manipulation, the stress and strain dataset were analyzed using FFT function fft(v) in Mathcad. If the dataset would have exactly N = 2m data points, Excel spreadsheet and a Fourier analysis toolset could also be used to analyze the data. Before applying FFT analysis, the measured strain signals were modified to remove the drift caused by the creep to obtain steady-state sine wave. The drift was removed by assuming linear creep upon the cyclic complex modulus signal. Also, stress and strain datasets were normalized through zero to transfer compressive haversine waveform to be sinusoidal waveform. This is illustrated in Fig. 4-13. For more complex data manipulation, the following model presented by Neifar, Di Benedetto, and Dogmo (2003) can be used for modeling axial cyclic strain:

ε ax (N , t) = α ax (N )t + ε ax av (N ) + ε 0 (N )* sin[ω t + ϕ ε (N )] ax

where 0 ≤ t < 2T and ω = 2π/T.

ax

(4-16)

109

110

Chapter Four

FIGURE 4-13 (a) Raw cyclic data and (b) manipulated data for FFT analysis.

T is the period of the cyclic loading and j is the phase angle of the mix. The formula ε ax av (N ) represents the axial permanent deformation and ε 0 (N ) is the amplitude of the axial sinusoidal strain component at cycle N, which can be considered as linear viscoelastic response (complex modulus) when creep is eliminated. α ax (N ) is the slope of the average deformation at cycles N (and N+1). A similar approach for modeling the cyclic dynamic modulus signals is used by AAT (2001) in the method G discussed earlier. Figure 4-14 compares FFT and time domain techniques discussed above. Two different ways of obtaining modulus and phase angle values using FFT analysis are shown in the figure. A method designated as FFT was conducted by manipulating stress and strain signals as discussed above, and a method designated as FFT-haversine was conducted by applying fft(v) function to the stress and strain data, which was not normalized through zero (rectified sinusoidal data). As an example, Fig. 4-13 shows rectified data on the left and normalized data on the right. A quadratic polynomial function was fitted through each data set to investigate relative variation of data points among them. ax

FIGURE 4-14 FFT versus time domain techniques.

Complex Modulus Characterization of Asphalt Concrete The FFT analysis seems to be very close to the method G. However, if the data is analyzed without normalizing the data through zero (FFT-haversine), the analysis results were closer to the methods C and D. This indicates that the increased phase angle values are related to the phase referencing method. The peak picking produces larger phase angle values than central waveform bracketing due to the skewness of the stress and strain signals, as was hypothesized in the beginning of the research. However, the moduli values are not affected in the same extent as Fig. 4-14 shows. This also explains why the moduli values are not so sensitive for the analysis method compared to the phase angle values. Therefore, if other than FFT or method G is used for the data analysis, the phase referencing should be done from the center of the signal to obtain the phase angle. Based on this analysis, methods A, G, and FFT should be used for analyzing less perfect sinusoidal cyclic test data. Although this research was done using dynamic modulus test data, these results can be applied also for shear modulus data or any sinusoidal cyclic test data.

Mastercurve Development A full characterization of asphalt mixtures requires one to construct a mastercurve, which defines the viscoelastic material behavior as a function of both temperature and loading time.

Time Temperature Superposition Principle Test data collected at different temperatures can be “shifted” relative to the time of loading or frequency, so that the various curves can be aligned to form a single mastercurve. The shift factor a(T) defines the required shift at a given temperature, that is, a constant by which the time must be divided to get a reduced time for the mastercurve. In frequency domain, the frequency must be multiplied by shift factor a(T) to obtain reduced frequency x:

ξ = f ⋅ a(T )

or

log(ξ ) = log( f ) + log[a(T )]

(4-17)

Mastercurve can be constructed using an arbitrarily selected reference temperature T0 to which all data are shifted. At the reference temperature, the shift factor a(T0) = 1 or log a(T0) = 0. The advantage of this procedure is that once the mastercurve is established, it is possible to derive interpolated values of asphalt mixture stiffness for any combination of temperature T or time of loading (frequency) inside the range covered by the measurements. In addition, this gives the possibility of comparing the results obtained by two laboratories with different sets of test conditions such as frequencies and temperatures.

Shifting Techniques The three different functions that commonly have been used to model the timetemperature superposition relationship in the bituminous viscoelastic materials are loglinear, Arrhenius, and Williams, Landel, and Ferry (WLF) equations. The Arrhenius and WLF equations have been used to characterize the relationship with a(T) and temperature for asphalt binders, and the Arrhenius and log-linear equations have been used for asphalt mixtures as well (Partl and Francken 1998; Huang 1993). There are two different functional forms that mainly have been used by different researchers to mathematically model the response of bituminous mixtures to construct

111

112

Chapter Four the mastercurve. For time or frequency dependency, the generalized power law has been used at low to intermediate temperatures when shifting creep or relaxation test data for asphalt mixtures (Rogue and Buttlar 1992; Christensen 1998). As the higher temperature data is included, polynomial fitting functions have been used to capture the form of mastercurve initiated by the material behavior (Francken and Verstraeten 1998). Gordon and Shaw (1994) have used piecewise fitting of polynomial functions through the test data to construct a mastercurve. Rowe and Sharrock (2000) have modified this approach by adding the cubic spline method to shift the data to the reference temperature.

Experimental Shifting and Sigmoidal Fitting Function A study by Pellinen, Witczak, and Bonaquist (2002) and Pellinen (2001) developed a method of constructing the full mastercurve using an “experimental” shifting technique using a sigmoidal fitting function. The experimental shift solves shift factors simultaneously with the coefficients of the fitting function. In this way the form of the shift function is not forced to the mastercurve. However, shift factors may absorb some of the experimental error from the test data. As mentioned earlier, polynomial fitting functions have been used to shift the asphalt mix test data using piecewise fitting approach. However, a single polynomial model cannot be used for fitting the whole mastercurve because the polynomial swing at low and high temperatures causes irrational modulus value predictions when extrapolating outside the range of data. To avoid this problem a new functional form, sigmoidal function Eq. (4-18), was selected to fit the dynamic modulus test data obtained from temperatures ranging from −18°C to 55°C. log(|E∗ |) = δ +

α 1 + e β −γ log(ξ )

(4-18)

where |E∗| = dynamic modulus x = reduced frequency d = minimum modulus value a = span of modulus values b, g = shape parameters Parameter g influences the steepness of the function (rate of change between minimum and maximum) and b the horizontal position of the turning point, shown in Fig. 4-15.

FIGURE 4-15 Sigmoidal function (Pellinen et al. 2002, ASCE).

Complex Modulus Characterization of Asphalt Concrete The justification of using a sigmoidal function for fitting the compressive dynamic modulus data is based on the physical observations of the mix behavior. The upper part of the sigmoidal function approaches asymptotically to the maximum stiffness of the mix, which is dependent on the limiting binder stiffness (glassy modulus) at cold temperatures. At high temperatures, the compressive loading causes aggregate influence to be more dominant than the viscous binder influence causing mix stiffness to approach a limiting equilibrium value, which is dependent of the aggregate gradation. Thus, the sigmoidal function captures the physical behavior of the asphalt mixtures observed in the mechanical testing using compressive cyclic loading through the entire temperature range. The advantage of using the sigmoidal fitting function is that a mastercurve can be constructed using Excel spreadsheets and Solver Function. The Solver Function is a tool for performing nonlinear least squares regression in the Excel spreadsheet. However, it should be noted that if the dataset does not include modulus values for full temperature range, caution should be used if the sigmoidal function is employed in the mastercurve construction. One way is to confine the asymptotic high and low modulus values to some assumed default values. Then, the asymptotic parameter values d and d +a need to be constrained to proper modulus values to obtain adequate mastercurve. Witczak et al. (Fonseca and Witczak 1996; Andrei et al. 1999) introduced the sigmoidal function to model the behavior of asphalt mixtures in conjunction with the dynamic modulus predictive equation, which predicts mixture stiffness from volumetric and raw material information.

Stress-Dependent Mastercurve for HMA The stiffness of the hot mix asphalt (HMA) varies as a function of test temperature and loading frequency as discussed above, however, the applied stress levels also affect the measured modulus values. Figure 4-16 presents three separate mastercurves constructed using test data obtained by applying four different combinations of dynamic deviatoric stress sd and confinement sc. The mastercurves were constructed using a sigmoidal fitting function and experimental shifting. All mastercurves approached the same

FIGURE 4-16 Master curves for varying conﬁnement levels.

113

114

Chapter Four asymptotic mix stiffness values at cold temperatures. At warm temperatures mastercurves deviated indicating that the mixture stiffness was affected by the applied stress state, that is, stress to strength ratio and bulk stress q. At cold and intermediate temperatures the measured strain levels stayed under 100 microsstrains for all applied stress levels, but at high temperatures the high stress levels produced resilient strains up to 1000 microstrains.

Model Development Equation (4-19) shows the universal material model, also called the k1-k3 model, proposed by Witczak and Uzan (1988) for unbound materials k2

⎛ θ ⎞ ⎛τ ⎞ E = (k1 pa ) ⎜ ⎟ ⎜ oct ⎟ ⎝ pa ⎠ ⎝ pa ⎠

k3

(4-19)

where E = resilient elastic modulus pa = atmospheric pressure q = bulk stress = I1 (first stress invariant) (= s1+ s2 + s3 with si = principal stress) toct = octahedral shear stress = √(2/3)√J2 (second stress invariant) {= [(√2)/3]sd with sd = deviatoric stress} ki = model coefficients In a study conducted by Pellinen (2001) and Pellinen and Witczak (2002b) it was hypothesized that the three separate mastercurves shown in Fig. 4-16 can be combined to form a stress-dependent mastercurve of the mix using the approach suggested by Witczak and Uzan for unbound materials. A proposed model form to fit a stressdependent mastercurve of a mix is described in Eqs. (4-20) and (4-21): 6 log|E∗ |= δ +

α −δ 1 + exp( β −γ log(ξ ) )

k k ⎡ θ ⎛τ ⎞ ⎤ δ = ⎢ (k1 pa ) ⎛⎜ ⎞⎟ ⎜ oct ⎟ ⎥ ⎝ pa ⎠ ⎝ pa ⎠ ⎦ ⎣ 2

(4-20)

3

(4-21)

where log|E∗| = log of stress-dependent dynamic modulus, pa = atmospheric pressure q = bulk stress toct = octahedral shear stress ki = regression coefficients a, b, g = regression coefficients for sigmoidal function X = reduced frequency The stress dependency has been incorporated into the equilibrium modulus value, that is, parameter d in the mastercurve, instead of incorporating it into the shift factors. This approach differs from the approach that Schapery (1969) has proposed for constructing a mastercurve for nonlinear viscoelastic materials. He incorporates the strain dependency to the reduced time by combining vertical strain-dependent ae (e) and horizontal time-dependent aT(T) shift to a combined shift factor aeT(eT).

Complex Modulus Characterization of Asphalt Concrete The studied mixtures were from the FHWA-ALF, MnRoad and WesTrack experimental test sites. The calibrated model coefficients for the studied mixtures were k1 = −0.0124, k2 = −0.59063, k3 = 0.54011, and a = 1.395045, b = 0.464119, and g = −0.04893. Both the bulk stress and octahedral shear stress had an effect on the modulus values as the coefficients k2 and k3 indicate. A separate verification of the model was conducted by testing a single specimen of asphalt mixture with randomly varying confinement and deviatoric stress levels. The difference between measured and predicted modulus values is less than a factor of 1.5 indicating relatively good prediction accuracy. It can be speculated that deviations between predicted and measured values were caused by the damage accumulation because applied stress states were in the dilative side of the phase change line most of the time.

Stress-Dependent Stiffness Predictive Equation Stiffness of asphalt mixture can be predicted using models such as the dynamic modulus predictive equation (Andrei et al. 1999) and the Hirsch model (Christensen et al. 2003). The advantage of these models is that they provide an approximate but useful way of estimating mixture stiffness (modulus) for various design purposes. These models, however, only model the modulus obtained in the linear viscoelastic region. Furthermore, there are no simple models available to estimate the effect of nonlinearity and confinement on HMA modulus values. Pellinen and Witczak (2002b) also developed the stress-dependent stiffness predictive equation which is based on Eqs. (4-20) and (4-21). Equations (4-22), (4-23), and (4-24) show the model form, and Eq. (4-25) describes mix gradation (Ga) as an average of the percent passing four sieve sizes of 0.074 mm, 4 mm, 9.5 mm, and 19 mm (No. 200, No. 4, 3/8 in, and 3/4 in). Table 4-4 gives the model coefficients. The model expects a minimum bulk stress value of 21 kPa and octahedral shear stress value of 9.9 kPa for unconfined linear viscoelastic stress case predictions. log(|E∗ |) = δ + A +

α − (δ + A) + a4Ga + a5 VFA 1 + exp β +γ log( f )− c log(η )

A = a0 + a1Ga + a2 VFA + a3 log(η)

(4-22) (4-23)

Material Coefficients

k1–k3 Model Coefficients

Sigmoidal Model Coefficients

a0 = −10.429150 a1 = 0.004106 a2 = −0.015376 a3 = 0.013351 a4 = −0.000808 a5 = 0.001594 c = 0.625379

k1 = 0.099088 k2 = 0.0217941 k3 = −0.011816

a = 1.324132 b = 0.615775 g = −0.584201

Source: Pellinen and Witczak 2002b, with permission from Association of Asphalt Paving Technologists.

TABLE 4-4

Predictive Model Coefficients

115

116

Chapter Four where

log|E∗| = log of stress-dependent dynamic modulus (106 kPa) d = equilibrium modulus Ga = average gradation (passing %) VFA = Voids filled with asphalt (volume %) h = binder viscosity (106 P) F = frequency (Hz) a0, a1, a2, a3, a4, a5 = regression coefficients k k ⎡ θ ⎛τ ⎞ ⎤ δ = ⎢ (k1 pa ) ⎛⎜ ⎞⎟ ⎜ oct ⎟ ⎥ ⎝ pa ⎠ ⎝ pa ⎠ ⎦ ⎣ 2

where

3

(4-24)

d = equilibrium modulus q = bulk stress (kPa) toct = octahedral shear stress (kPa) pa = atmospheric pressure, 103.3 kPa k1, k2, k3 = regression coefficients Ga =

p200 + p4 + p3/8 + p3/4 4

(4-25)

where Ga = average gradation (%) p200 = passing 0.074 mm (%) p4 = passing 4.36 mm (%) p3/8 = passing 9.5 mm (%) p3/4 = passing 19 mm (%)

Summary This chapter discusses the use of the axial dynamic modulus |E∗| test and the SST shear modulus |G∗| from the shear frequency sweep test to obtain complex modulus of asphalt mixture. More specifically, specimen fabrication, instrumentation, test control modes, and analysis of test data to obtain modulus and phase angle are discussed. In addition, a new method of constructing a mixture mastercurve is presented. Research has shown that these two test methods are not interchangeable for pavement design applications because the theoretical linear elastic law of |E∗| being approximately three times larger than the |G∗| when the Poisson’s ratio is 0.5, is not fulfilled. A conversion equation needs to be used to estimate the |E∗| from the SST |G∗| test results. Research has also shown that both of these test methods can be used as performance indicators for asphalt mixtures. However, they do not rank the mixtures necessarily in the same order for rutting performance. Most likely mixtures that have binder as a dominating stiffness contributor will be ranked similarly, while mixtures that have different gradations, such as dense graded versus SMA, may be ranked differently.

Acknowledgments The author wishes to acknowledge Dr. Donald W. Christensen from Advanced Asphalt Technologies, LLC, for his help in analyzing data using the method G he developed for National Cooperative Highway Research Project contract, NCHRP 9-29, for testing

Complex Modulus Characterization of Asphalt Concrete equipment development. The author also wishes to acknowledge Mr. Bill Crockford from ShedWorks for his help in writing the section of Analysis of Cyclic Sinusoidal Test Data. In addition, the author wishes to acknowledge professor Matthew W. Witczak from Arizona State University for his help and advice in the course of dynamic modulus testing development.

Endnote 1. According to Di Benedetto and de la Roche (1998), the word dynamic should be used in the tests with nonnegligible inertia effects inside the sample (i.e., when wave propagation is observed). Then, a complex modulus test is not a dynamic test but a cyclic test with repeated loading and is therefore interpreted as a static test. However, the author is using the word dynamic as it is traditionally used in literature in the United States.

References Advanced Asphalt Technologies, LLC (AAT). (2001). First Article Equipment Specifications for the Simple Performance Test System. NCHRP Project 9-29: Simple Performance Tester for Superpave Mix Design, Sterling, Va.: Advanced Asphalt Technologies, LLC, November 2001, p. 89. American Association of State Highway and Transportation Officials (AASHTO). (1994). “Shear Device, AASHTO Designation: TP-7-94.” AASHTO Provisional Standards. American Society for Testing and Materials (ASTM). (1979). “Test Method for Dynamic Modulus of Asphalt Concrete Mixture.” Annual Book of ASTM Standards, ASTM D Vol. 04.03, 3497-79. Andrei, D., Witczak, M. W., and Mirza, M. W. (1999). Development of Revised Predictive Model for the Dynamic (Complex) Modulus of Asphalt Mixtures. Development of the 2002 Guide for the Design of New and Rehabilitated Pavement Structures, NCHRP 1-37A. Interim Team Technical Report. Department of Civil Engineering, University of Maryland of College Park, Md. Bahia, H., and Anderson, D. (1995). “The SHRP Binder Rheological Parameters: Why they are Required and How They Compare to Conventional Properties.” Transportation Research Board, 75th Annual Meeting, Washington, D.C. Chen, W. F., and Han D. J. (1988). Plasticity for Structural Engineers. Springer-Verlag, New York, p. 10. Christensen, D. W., Pellinen, T., and Bonaquist, R. F. (2003). “Hirsch Model for Estimating the Modulus of Asphalt Concrete.” Proceedings of the Association of Asphalt Paving Technologists. March 10–12, 2003, Lexington, Ky. Christensen, D. W. (1998). “Analysis of Creep Data from Indirect Tension Test on Asphalt Concrete.” Journal of the Association of Asphalt Paving Technologists, Vol. 67, 458–492. Crockford, W. W. (2001). Data Analysis—Load Controlled Dynamic Tests. ShedWorks Inc. Technical Note, 2nd ed. Di Benedetto, H., and de la Roche, C. (1998). “State of the Art of Stiffness Modulus and Fatigue of Bituminous Mixtures.” RILEM Report 17, Bituminous Binders and Mixes, Edited by L. Francken, 137–180, London. Doubbaneh, E. (1995). “Comportement Mécanique Des Enrobes Bitumineux Des “Petites” Aux ‘Grandes‘ Déformations.” Grade de Docteur. L’institut Nations Des Sciences

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Chapter Four Appliquées De Lyon, Laboratoire Géomatériaux du Département de Génie Civil, Lyon. Di Benedetto, H., Partl, M., and de la Roche, C. (2001). “Stiffness Testing for Bituminous Mixtures.” RILEM TC 182-PEB Performance Testing and Evaluation of Bituminous Materials. Materials and Structure, Vol. 34 (236), March, 2001, pp. 65–70. Francken, L., and Verstraeten, J. (1998). “Interlaboratory Test Program on Complex Modulus and Fatigue.” RILEM Report 17, Bituminous Binders and Mixes, Edited by L. Francken, 182–215 London. Fonseca, O. A., and Witczak, M. W. (1996). “A Prediction Methodology for the Dynamic Modulus of In-Placed Aged Asphalt Mixtures.” Journal of the Association of Asphalt Paving Technologists, Vol. 65, 532–572. Gordon, G. V., and Shaw, M. T. (1994). Computer Programs for Rheologists. Hanser/Gardner Publishers, New York. Harrigan, E. T., Leahy, R. B., and Youtcheff, J. S. (1994). The Superpave Mix Design Manual Specifications, Test Methods, and Practices, SHRP-A-379. Strategic Highway Research Program. National Research Council. Washington, D.C.: National Academy of Science. Hewlett-Packard Co. (1989). The Fundamentals of Signal Analysis. Application Note 243. Hewlett-Packard Co. Printed in USA, 5952-8898. Huang, Y. (1993). Pavement Analysis and Design. Appendix A, Theory of Visco-elasticity. Prentice Hall, Englewood Cliffs, N.J. Kendall, M. G. (1951). The Advanced Theory of Statistics. Vol. II, Hafner, New York, p. 377. NCHRP 1-37A Draft Document. (2002). 2002 Guide for the Design of New and Rehabilitated Pavement Structures. ERES Division of ARA Inc., Champaign, Ill. Mirza, M. W., and Witczak, M. W. (1994). Bituminous Mix Dynamic Material Characterization Data Acquisition and Analysis Programs Using 458.20 MTS Controller. University of Maryland, Department of Civil and Environmental Engineering, College Park, Md. Neifar, M., Di Benedetto, H., and Dogmo, B. (2003). Permanent Deformation and Complex Modulus: Two Different Characteristics from a Unique Test. Proceedings in the 6th International RILEM Symposium, April 14–17, 2003, Zurich, Switzerland, 316–323. Partl, M. N., and Francken, L. (1998). “Introduction.” RILEM Report 17, Bituminous Binders and Mixes. edited by L. Francken, 2–10 London. Pellinen, T. (2001). Investigation of the Use of Dynamic Modulus as an Indicator of HotMix Asphalt Performance. A Dissertation Presented in Partial Fulfillment of the Requirements of the Degree of Doctor of Philosophy. Submitted to the Faculty of Graduate School of the Arizona Sate University, Tempe, Ariz. Pellinen, T., and Crockford, B. (2003). “Comparison of Analysis Techniques to Obtain Modulus and Phase Angle from Sinusoidal Test Data.” Proceedings in the 6th International RILEM Symposium, April 14–17, Zurich, 2003, Switzerland, 307–309. Pellinen, T., and Witczak, M. W. (1998). Assessment of the Relationship Between Advanced Material Characterization Parameters and the Volumetric Properties of A MDOT Superpave Mix. Submitted to Maryland Department of Transportation State Highway Administration. Department of Civil Engineering, University of Maryland of College Park, Md. Pellinen, T, K., and Witczak, M. W. (2002a). “Use of Stiffness of Hot-Mix Asphalt as a Simple Performance Test.” Journal of Transportation Research Records, No 1789, 80–90.

Complex Modulus Characterization of Asphalt Concrete Pellinen, T. K., and Witczak, M. W. (2002b). “Stress Dependent Mastercurve Construction for Dynamic (Complex) Modulus.” Journal of the Association of Asphalt Paving Technologists, Vol. 71, 281–309. Pellinen, T., Witczak M. W., and Bonaqusit, R. (2002). “Master Curve Construction Using Sigmoidal Fitting Function with Non-linear Least Squares Optimization Technique.” Proceedings of the 15th ASCE Engineering Mechanics Division Conference, Columbia University, June 2–5, 2002, New York. Ramsey, K. A. (1975). “Effective Measurements for Structural Dynamics Testing.” Sound and Vibrations, November 1975, 24–35. Rogue, R., and Buttlar, W. G. (1992). “The Measurement and Analysis System to Accurately Determine Asphalt Concrete Properties Using the Indirect Tensile Mode.” Journal of the Association of Asphalt Paving Technologists, Vol. 61, 304–328. Rowe G. M., and Sharrock, M. J. (2000). “Development of Standard Techniques for the Calculation of Master Curves for Linear-Visco Elastic Materials.” The 1st International Symposium on Binder Rheology and Pavement Performance. The University of Calgary, August 14–15, 2000, Alberta, Canada. Saarela, A., (1993). Asfalttipäällysteiden tutkimusohjelma ASTO 1987-1992 (Asphalt Pavements Research Program ASTO 1987-1992), Loppuraportti. Technical Research Centre of Finland. Road, Traffic and Geotechnical Laboratory. Espoo. Schapery, R. (1969). “On the Characterization of Non-linear Visco-elastic Materials.” Polymer Engineering and Science, Vol. 9, 295–310. Witczak, M. W., and Kaloush, K. (1998). Performance evaluation of Asphalt Modified Mixtures Using Superpave and P-401 Mix Grading. Submitted to: Maryland Department of Transportation, Maryland Port Administration. Department of Civil Engineering, University of Maryland of College Park, Md. Witczak, M. W., and Uzan, J. (1988). The Universal Airport Pavement Design System— Report I of IV: Granular Material Characterization. Department of Civil Engineering, University of Maryland of College Park, Md. Witczak, M. W., Bonaquist, R., Von Quintus, H., and Kaloush, K. (2000). “Specimen Geometry and Aggregate Size Effects in Uniaxial Compression and Constant Height Shear Test.” Journal of the Association of Asphalt Paving Technologists, Vol. 69, 733–793. Witczak, M. W., Hafez, I., Ayres, H., and Kaloush, K. (1996). Comparative Study of MSHA Asphalt Mixtures Using Advanced Dynamic Material Characterization Tests. Submitted to: Maryland Department of Transportation, Maryland Port Administration, Department of Civil Engineering, University of Maryland of College Park, Md.

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CHAPTER

5

Complex Modulus from the Indirect Tension Test Y. Richard Kim, Youngguk Seo, and Mostafa Momen

Abstract This chapter presents the results from an analytical/experimental study on the dynamic modulus testing of hot mix asphalt (HMA) using the indirect tension (IDT) mode. An analytical solution for the dynamic modulus in the IDT mode is developed using the theory of linear viscoelasticity. To verify the analytical solution, temperature and frequency sweep tests were conducted on 24 asphalt mixtures commonly used in North Carolina, using both axial compression and IDT test methods. Graphical and statistical comparisons of results from the axial compression and IDT test methods show that the dynamic modulus mastercurves, phase angles, and shift factors derived from the two methods are in good agreement.

Introduction One of the issues related to the role of dynamic modulus in pavement management is its use in forensic studies and pavement rehabilitation design. The current dynamic modulus protocol, AASHTO TP-62, calls for the uniaxial compression testing of 100-mm-diameter and 150-mm-tall asphalt concrete specimens. It is often impossible to obtain this size specimen from actual pavements. Given that a typical asphalt layer thickness is less than a few inches and that coring is the most effective method of obtaining specimens from actual pavements, the indirect tension (IDT) test of cores seems to be more appropriate for the evaluation of existing pavements. In this chapter, linear viscoelastic solutions are presented for the determination of dynamic modulus (|E∗|) of asphalt concrete from the IDT test data. The IDT |E∗| values are then compared with the values determined from the axial compression complex modulus testing for 24 different asphalt mixtures typically used in North Carolina.

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122

Chapter Five

Theoretical Background Historically, the pavement community utilized the elastic solutions for IDT testing that Hondros (1959) derived using the plane stress assumption until Roque and Buttlar (1992) introduced correction factors that accounted for the bulging effect of the specimen. Later, Kim et al. (2000) introduced viscoelastic solutions for the IDT creep test using the theory of linear viscoelasticity. Unlike those of the uniaxial test specimens, the stress and strain distributions in IDT specimens are biaxial. This biaxial state of stress and strain can cause errors in determining the material properties obtained from the IDT test unless the derivation of the properties is carefully handled. To illustrate this point more clearly, Hooke’s law, the governing equation for elastic materials, is presented below for both uniaxial and biaxial cases: Uniaxial case:

σ y = E × εy

or

εy =

σy E

(5-1)

Biaxial case:

εx =

1 (σ − νσ y ) E x

(5-2)

where x and y denote the loading direction (i.e., the vertical direction) and the direction perpendicular to the loading direction (i.e., the horizontal direction), respectively. In the uniaxial case (i.e., the axial compression dynamic modulus test) in Eq. (5-1), one can divide the axial stress (sy ) by the axial strain (ey ) to obtain the modulus. However, in the biaxial case (i.e., the IDT dynamic modulus test) in Eq. (5-2), one cannot obtain the modulus by dividing the horizontal stress (sx) by the horizontal strain (ex). Rather, the correct way to determine the modulus of the material is to divide the biaxial stress (i.e., sx–nsy) by the horizontal strain (ex). If the incorrect solution (i.e., sx /ex) is used to represent the modulus of the material, then that modulus should not be considered the same as the modulus determined from the axial test.

Linear Viscoelastic Solution The linear viscoelastic solution for the complex modulus of HMA under the IDT mode has been developed by Kim et al. (2004), and is presented in this section. Assuming the plane stress state, Hondros (1959) developed the following expressions for stresses and strains along the horizontal diameter of the IDT specimen subjected to a strip load, shown in Fig. 5-1:

εx =

1 (σ − νσ y ) E x

(5-3)

with

σ x ( x) = =

2 2 2P ⎡ (1 − x 2 / R 2 )sin 2α ⎫⎤ −1 ⎧ 1 − x / R ⎨ ⎢ 4 − tan 2 2 tan α ⎬ ⎥ 2 2 4 π ad ⎣ 1 + 2 x / R cos 2α + x / R 1 + x R / ⎩ ⎭⎦

2P [ f (x) − g(x)] π ad

(5-4)

C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t

FIGURE 5-1 Schematic of the IDT specimen subjected to a strip load. (Kim et al. 2004, with permission from Transportation Research Board.)

σ y ( x) = − =−

1 − x2 / R2 2P ⎡ (1 − x 2 / R 2 ) sin 2α ⎫⎤ −1 ⎧ + tan tan α ⎬ ⎥ ⎨ ⎢ 2 2 π ad ⎣ 1 + 2 x 2 / R 2 cos 2α + x 4 / R 4 + 1 x R / ⎩ ⎭⎦ 2P [ f (x) + g(x)] π ad

(5-5)

where x = horizontal distance from the center of the specimen face P = applied load a = loading strip width, m d = thickness of specimen, m R = radius of specimen, m a = radial angle E = Young’s modulus n = Poisson’s ratio For the viscoelastic materials subjected to the sinusoidal load in a steady state, Eq. (5-3) can be rewritten as

εx =

1 (σ − νσ y ) E∗ x

(5-6)

where E∗ is the complex modulus. It is often helpful to have E∗ in polar form, E ∗ = E∗ ⋅ e iφ

(5-7)

123

124

Chapter Five where |E∗| is the dynamic modulus and f is the phase angle calculated from the time lag between the load and the displacement. The response to the sinusoidal load applied in the complex modulus test is the imaginary component of the response due to the complex load P shown below: P = P0 e iwt = P0 (cos wt + i sin wt)

(5-8)

where P0 and w are the amplitude and the angular frequency of the sinusoidal load used in the complex modulus test, respectively. Substituting Eqs. (5-4), (5-5), (5-7), and (5-8) into Eq. (5-6) results in 2 P0 e i( wt−φ ) [(1 + ν ) f (x) + (ν − 1) g(x)] E∗ π ad

ε x ( x , t) =

(5-9)

Integrate Eq. (5-9) over the gauge length to determine the horizontal displacement U(t) and obtain l

U (t) = ∫ ε x (x , t) dx = −l

l l ⎡ ⎤ 2 P0 i ( wt −φ ) e ( + ν ) f ( x ) dx + ( ν − 1 ) g(x) dx ⎥ 1 ⎢ ∫ ∫ E∗ π ad ⎢⎣ ⎥⎦ −l −l

(5-10)

where l is half of the gauge length. One may extract the response that only occurs due to the sinusoidal input by taking an imaginary part of the total response. Therefore, the dynamic modulus derived can be expressed using the horizontal displacement U(t) as follows: E∗ =

2 P0 sin(wt − φ ) A π ad ⋅ U (t)

(5-11)

where l l ⎡ ⎤ A = ⎢ (1 + ν ) ∫ f (x)dx + (ν − 1) ∫ g(x)dx ⎥ ⎢⎣ ⎥⎦ −l −l

(5-12)

with f ( x) =

and

(1 − x 2 / R 2 )sin 2α 1 + 2 x 2 / R 2 cos 2α + x 4 / R 4

⎫ ⎧ 1 − x2 / R2 g(x) = tan −1 ⎨ 2 2 tan α ⎬ ⎭ ⎩1+ x /R

(5-13)

(5-14)

Similarly, the analogous expression for the dynamic modulus using the vertical displacement V(t) is E∗ =

2 P0 sin(wt − φ ) B π ad ⋅ V (t)

(5-15)

C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t where l l ⎡ ⎤ B = ⎢ (ν − 1) ∫ n( y )dy − (1 + ν ) ∫ m ( y ) dy ⎥ ⎢⎣ ⎥⎦ −l −l

(5-16)

with m( y ) =

(1 − y

/ R 2 ) sin 2α 1 − 2 y 2 / R 2 cos 2α + y 4 / R 4 2

⎫ ⎧ 1 + y 2 / R2 n( y ) = tan −1 ⎨ 2 2 tan α ⎬ − / y R 1 ⎩ ⎭

and

(5-17)

(5-18)

By equating Eqs. (5-11) and (5-15), one can obtain A ⋅ V (t) = B ⋅ U (t)

(5-19)

Then, one may derive the expression for Poisson’s ratio as follows:

β 1U (t) − γ 1V (t) − β 2U (t) + γ 2V (t)

ν=

(5-20)

where l

l

β1 = − ∫ n( y )dy − ∫ m( y )dy −l

−l

l

l

−l

−l

l

l

−l

−l

β2 = ∫ n( y)dy − ∫ m( y )ddy γ1 =

∫ f (x)dx − ∫ g(x)dx l

γ2 =

∫

−l

−l

f (x) dx + ∫ g(x) dx

(5-21)

−l

Combining Eqs. (5-11) and (5-15) yields a single form of the dynamic modulus, as shown below: E∗ =

P0 sin(wt − φ ) AV (t) + P0 sin(wt − φ ) BU (t) π ad ⋅ V (t) ⋅U (t)

(5-22)

After substituting Eqs. (5-12) and (5-16) into Eq. (5-22), one can obtain E∗ = 2

P0 sin(wt − φ ) β1γ 2 − β 2γ 1 π ad γ 2V (t) − β 2U (t)

(5-23)

125

126

Chapter Five

Specimen Diameter (mm)

Gauge Length (mm)

b1

b2

g1

g2

101.6

25.4

−0.0098

−0.0031

0.0029

0.0091

101.6

38.1

−0.0153

−0.0047

0.0040

0.0128

101.6

50.8

−0.0215

−0.0062

0.0047

0.0157

152.4

25.4

−0.0065

−0.0021

0.0020

0.0062

152.4

38.1

−0.0099

−0.0032

0.0029

0.0091

152.4

50.8

−0.0134

−0.0042

0.0037

0.0116

Source: Kim et al. 2004, with permission from Transportation Research Board.

TABLE 5-1

Coefficients for Poisson’s Ratio and Dynamic Modulus

The vertical and horizontal displacements can be expressed in sine functions as follows: V (t) = V0 sin(wt − φ )

(5-24)

U (t) = U 0 sin(wt − φ )

(5-25)

where V0 and U0 are the constant amplitudes of vertical and horizontal displacements, respectively. Therefore, the final form of the dynamic modulus is E∗ = 2

P0 β1γ 2 − β 2γ 1 π ad γ 2V0 − β 2U 0

(5-26)

Likewise, the expression for Poisson’s ratio can be simplified as

ν=

β1U 0 − γ 1V0 − β 2U 0 + γ 2V0

(5-27)

The coefficients, b1, b2, g1, and g2, in Eqs. (5-26) and (5-27), are calculated for different specimen diameters and gauge lengths, and are presented in Table 5-1. Equations (5-26) and (5-27) are based on the plane stress assumption. Kim et al. (2000) used the threedimensional finite element analysis to calculate the center strain in the IDT specimen and concluded that the error due to the plane stress assumption is negligible.

Dynamic Modulus Testing of HMA Included in this section are the HMAs selected for testing, the procedure for specimen fabrication, and axial compression and IDT dynamic modulus test methods.

Materials A total of 24 asphalt mixtures with varying aggregate and binder characteristics were tested in the axial compression and IDT modes. These mixtures are typical HMAs used in paving construction in North Carolina. Table 5-2 summarizes the mixture variables for all the mixtures. Granite aggregates from six different sources from the mountains

C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t

Agg. Source

Binder Source

Binder Grade

Binder Content

Gmm

S 9.5 A -C

Morganton

Inman, SC

PG 64-22

5.8%

2.615

S9.5A-F

Charlotte-Pineville

Citgo-Wilmington

PG 64-22

6.4%

2.668

S9.5B-C

Haw River

Citgo-Wilmington

PG 64-22

5.9%

2.579

S9.5B-F

Morganton

Inman, SC

PG 64-22

6.3%

2.579

S9.5B-F

Charlotte-Pineville

Citgo-Wilmington

PG 64-22

5.8%

2.689

S9.5C-C

Holly Springs

Citgo-Wilmington

PG 70-22

5.3%

2.486

S9.5C-F

Garner

Citgo-Wilmington

PG 70-22

5.0%

2.456

S12.5B-C

Haw River

Citgo-Wilmington

PG 64-22

5.5%

2.595

S12.5B-F

Holly Springs

Citgo-Wilmington

PG 64-22

5.3%

2.480

S12.5C-C

Morganton

Inman, SC

PG 70-22

4.6%

2.663

S12.5C-F

Concord-Cabarrus

Citgo-Wilmington

PG 70-22

5.0%

2.570

S12.5D-C

Concord-Cabarrus

Citgo-Wilmington

PG 70-22

5.0%

2.582

S12.5D-F

Concord-Cabarrus

AA-Salisbury

PG 76-22

4.7%

2.571

I19.0B-C

Haw River

Alpaso-Apex

PG 64-22

5.0%

2.633

I19.0B-F

Garner

Citgo-Wilmington

PG 64-22

5.4%

2.441

I19.0B-F

Charlotte-Pineville

Citgo-Wilmington

PG 64-22

4.3%

2.773

I19.0C-C

Garner

Citgo-Wilmington

PG 64-22

4.7%

2.472

I19.0C-F

Concord-Cabarrus

Alpaso-Charlotte

PG 64-22

4.8%

2.582

I19.0D-C

Charlotte-Pineville

Citgo-Wilmington

PG 70-22

4.3%

2.770

I19.0D-F

Concord-Cabarrus

AA-Salisbury

PG 70-22

4.1%

2.597

B25.0B-C

Holly Springs

Citgo-Wilmington

PG 64-22

4.5%

2.503

B25.0B-F

Garner

Citgo-Wilmington

PG 64-22

4.2%

2.485

B25.0C-C

Haw River

Citgo-Wilmington

PG 64-22

4.0%

2.678

B25.0C-F

Concord-Cabarrus

Alpaso-Charlotte

PG 64-22

4.4%

2.599

Mixture ID *

† ‡

§

∗

S for surface mix, I for intermediate mix, and B for base mix. Nominal maximum aggregate size (NMAS) in mm. ‡ Traffic volume indicator. § Aggregate gradation type (C for coarse and F for fine). Source: Kim et al. 2004, with permission from Transportation Research Board. †

TABLE 5-2

Summary of Mixture Characteristics

to the coast of North Carolina were utilized with four different nominal maximum aggregate sizes (NMASs) (9.5, 12.5, 19.0, and 25.0 mm). Also, binders from six different sources with performance grades (PG) of 64-22, 70-22, and 76-22 were used in making these mixtures. Superpave volumetric mix designs were performed on these mixtures and the resulting optimum asphalt content and the maximum specific gravity (Gmm) for each of the mixtures are shown in Table 5-2.

127

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Chapter Five

Specimen Fabrication Asphalt mixtures were mixed and compacted at temperatures that were in accordance with the requirements for each binder. All mixtures were aged at 135°C for 4 hours (i.e., short-term oven aging) before compaction. For mixtures tested in the axial compression mode, samples were compacted into gyratory plugs of 150 mm in diameter by 178 mm in height. Later, they were cut and cored to cylindrical specimens with dimensions of 100 mm in diameter and 150 mm in height. For mixtures tested in the IDT mode, samples were compacted into gyratory plugs of 150 mm in diameter by 60 mm in height, and they were cut to 38 mm height. Both ends were cut to ensure a more consistent air void distribution along the height of the test specimens. The target air voids for the final cut and cored specimen were 4% ± 0.5%. In order to achieve this density, the target air voids for the gyratory plug had to be higher than that of the cut and cored specimen. The difference between the target air voids for the gyratory plug and for the cut and cored specimen typically increased as the NMAS increased with an average being about 1.5% to 2%. For IDT specimens, a small study was conducted using four mixtures with different NMASs to estimate the reduction in air voids when the specimen is cut from a 60 mm height to 38 mm. The study showed that, in general, the reduction in air voids is around 1% for 9.5-mm NMAS and the reduction increases by approximately 0.5% increments as the NMAS increases to 12.5, 19, and 25 mm. Air voids were measured using the Corelok vacuum sealing device. Specifications provided in ASTM D6752-03 were followed in taking these measurements and making the calculations. Appropriate adjustments were made to account for the density of water when measurements were taken at temperatures other than 25°C. For each of the 24 mixtures there were 3 replicates of each tested. During fabrication or testing, if errors were made or densities were not met, then the specimen was discarded and an additional specimen was manufactured and tested.

Testing and Analysis Test Setup Testing was performed using a closed-loop servo-hydraulic machine, manufactured by Material Testing System (MTS). A temperature chamber, cooled by liquid nitrogen, was used to control the test temperature. Dummy specimens with thermocouples embedded in the middle of the specimen were used to monitor the temperatures to which the specimens were subjected. For axial compression testing, 100-mm-diameter metal end plates attached to the top and bottom rams were used to apply the compressive load to the specimen. Frictionreducing end treatments were used to lessen the confinement that occurs as a result of the friction against the specimen and the metal end plates. The end treatments were made of two latex membranes, each 0.0125 in. thick. A very small amount of silicon grease was smeared between the latex membranes to allow the specimen to move (dilate) with respect to the metal end plates. Caution was used to apply only the minimum amount of grease necessary because excess grease creates a slick interface that causes the specimen to slip out of the test setup under high load and high frequency combinations. Also, an approximate 25-mm-diameter hole was introduced in the center of the membrane to allow a small amount of contact between the metal plate and the specimen, thereby increasing the friction and reducing the movement of the specimen during testing.

C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t For the IDT test, the Load Guide Device (LGD), developed from the Strategic Highway Research Program (SHRP), was used as the loading apparatus. This device is shown in Fig. 5-2(a). From the NCHRP 1-28 study, it was found that when compared against other loading devices with no column or four columns, the SHRP LGD with

(a)

(b)

FIGURE 5-2 (a) IDT test setup with SHRP LGD and (b) surface-mounted LVDTs. (Kim et al. 2004, with permission from Transportation Research Board.)

129

130

Chapter Five

FIGURE 5-3 LVDT mounting and spacing in axial compression testing.

two guide columns resulted in the least amount of “rocking” of the IDT specimen without causing significant friction between the upper loading plate and guide columns under repetitive loading (Barksdale et al., 1997). For axial compression testing, vertical deformations were measured using four loose-core, CD type LVDTs (linear variable differential transducers) at 90° radial intervals. Targets were glued to the specimen face in the middle two-thirds of the specimen (100 mm), and the LVDTs were mounted to the targets. A gluing device was used to maintain consistent spacing between the LVDT targets. The LVDT setup for axial compression testing is shown in Fig. 5-3. For IDT specimens, the vertical and horizontal deformations were measured using loose-core type miniature XSB LVDTs. These were mounted on each of the specimen faces using a 50.8-mm gauge length, as shown in Fig. 5-2(b).

Data Acquisition System The data acquisition system used in this project is composed of LabView software and a 16-bit data acquisition board by National Instruments. One channel was dedicated to the load cell on the machine, one to the actuator LVDT, and four to the on-specimen LVDTs. The data acquisition rate was 100 points per cycle.

Test and Analysis Methods In principle, the AASHTO TP-62 protocol was followed. To ensure that the testing captures the linear viscoelastic behavior of the material, 75 microstrain was used as the maximum allowable axial and horizontal strains for the axial compression and IDT testing, respectively. Testing was performed by applying sinusoidal loadings at different frequencies and temperatures. Prior to applying the first frequency at each temperature, the preconditioning cycles were applied at 25 Hz and one-half the load used in actual

C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t testing for 25 Hz. Following each loading frequency, a 5-minute rest period was allowed before the next frequency was applied. Averaged deformations were used to calculate the dynamic modulus and phase angle. In the uniaxial case, readings from four LVDTs spaced at 90° intervals were averaged. In IDT, the vertical deformations and horizontal deformations from two surfaces were averaged to determine the deformation in each axis.

Comparison of Dynamic Moduli Values Graphical Comparison The IDT test data of the 24 mixtures were analyzed using the viscoelastic solutions in Eq. (5-26). The resulting dynamic modulus mastercurves from these analyses are plotted in Figs. 5-4 to 5-6 for the 12 representative mixtures. The reference temperature of 10°C was used as the basis of shifting the data. The data presented in these figures are the averages of the three replicates. The rest of the data can be found in Kim et al. (2005). It can be observed from these figures that the dynamic modulus mastercurves developed from the IDT test using the biaxial linear viscoelastic solution are generally in good agreement with those determined from the axial compression test. It was also found that the time-temperature shift factors obtained during the construction of the mastercurves are essentially identical for the axial compression and IDT tests.

Statistical Analysis Using P-Value Recognizing that a sample-to-sample variation exists, a statistical analysis was conducted using the unequal variance t-test for each mixture at two frequencies for each testing temperature. In this analysis, all the individual replicates (three from the axial compression and three from the IDT tests) were used. The null hypothesis is that the dynamic modulus from the IDT test is the same as that from the axial compression test. The P-value was calculated and compared with the critical value of 0.05 to reject or accept the null hypothesis. The P-value indicates the extent to which a computed test statistic is unusual in comparison with what would be expected under the null hypothesis. Therefore, in this study a P-value greater than 0.05 indicates that the dynamic modulus from the IDT test is statistically the same as that from the axial compression test. A summary of the P-values for 144 tests (2 frequencies × 3 temperatures × 24 mixtures) is given in Table 5-3. About 19% of the tests indicate that the dynamic modulus from the IDT test is statistically different from the dynamic modulus from the axial compression test.

Using Percent Difference In addition to the statistical analysis, the percent difference was calculated for the dynamic moduli determined from the axial compression and IDT tests for 288 combinations of temperature and frequency (8 frequencies × 3 temperatures × 12 mixes). These values are also summarized in Table 5-3. A comparison of the data in this table and further investigation of individual test data resulted in several important observations.

131

132

Chapter Five

FIGURE 5-4 Dynamic modulus mastercurves for (a) S9.5A-Fine; (b) S9.5B-Coarse; (c) S9.5C-Fine; and (d) S9.5C-Coarse. (Kim et al. 2004, with permission from Transportation Research Board.)

FIGURE 5-5 Dynamic modulus mastercurves for (a) S12.5C-Fine; (b) S12.5D-Coarse; (c) I19.0B-Fine; and (d) I19.0C-Coarse. (Kim et al. 2004, with permission from Transportation Research Board.)

C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t

FIGURE 5-6 Dynamic modulus mastercurves for (a) I19.0D-Fine, (b) I19.0D-Coarse, (c) B25.0B-Fine, and (d) B25.0B-Coarse. (Kim et al. 2004, with permission from Transportation Research Board.)

NMAS

P-Value

% Difference

9.5 mm

12.5 mm

19.0 mm

25.0 mm

All

<0.05

15%

14%

19%

33%

19%

>0.05

85%

86%

81%

67%

81%

<5%

48%

46%

28%

25%

39%

5−10%

21%

33%

42%

19%

31%

10−20%

21%

21%

26%

19%

22%

>20%

10%

0%

4%

38%

8%

TABLE 5-3 Summary of P-Values and Percent Difference in Dynamic Moduli from IDT and Axial Compression Tests

First, about 70% of the tests had a percent difference below 10%. Secondly, although it is not shown in Table 5-3, further investigation of the testing conditions that have high percent differences reveals that most of the high percent difference come from 35°C data due to very small dynamic modulus values in the denominator of the percent difference calculation. Finally, it can be observed in Table 5-3 that the percent difference becomes greater as the NMAS increases. An investigation of the individual test data reveals the same trend; that is, the variability among the replicates increases as the NMAS increases. These observations may be related to the ratio of gauge length to NMAS. Typically, a factor of 3 is recommended to maintain a representative volume element (RVE). Meeting

133

134

Chapter Five this requirement is not a problem in the axial compression test geometry. However, in the IDT tests with a 50.8-mm gauge length, this requirement is satisfied for the 9.5- and 12.5-mm mixes, but not for the 19.0- and 25.0-mm mixes, resulting in a higher variability among replicates and a higher percent difference in the 19.0- and 25.0-mm mixes. Another observation made from a detailed data analysis is that, in some replicates of the 25.0-mm mix, a significant difference was found between displacements from the front and back surfaces of the IDT specimens. These observations suggest that the positions of large aggregate particles within the gauge length affect the data, and that a larger gauge length is required for 25.0-mm mixes. The visual observation of the average mastercurves in Figs. 5-4 to 5-6 and further statistical analysis suggest that the dynamic modulus determined from the IDT test using the linear viscoelastic solution in Eq. (5-26) is statistically the same as the one measured from the axial compression test. A question may arise regarding why the effect of different relationships between the compaction direction and the direction in which the stress-strain analysis is performed in the axial compression and the IDT tests seems to be insignificant. This difference and possibly anisotropy may exist when the axial compression cylinders and the IDT specimens are compared. However, due to very small strain levels used in these tests (50 to 80 microstrains), the dynamic modulus test more or less “tickles” the mastic and does not fully capture the effect of these differences that are mostly related to the large aggregate orientation.

Comparison of Phase Angles It was observed that the phase angle obtained from axial compression testing is normally between the phase angles calculated from the horizontal and vertical strains in IDT testing. Based on this observation, the phase angles calculated from the horizontal and vertical strains were averaged and plotted in Figs. 5-7 and 5-8. The

FIGURE 5-7 Phase angle mastercurves for S9.5A-Fine mixture. (Kim et al. 2004, with permission from Transportation Research Board.)

C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t

FIGURE 5-8 Phase angle mastercurves for I19.0D-Fine mixture. (Kim et al. 2004, with permission from Transportation Research Board.)

averaged phase angles are close to the values from the axial compression testing. This finding needs to be refined further using a more rigorous approach. Kim et al. (2005) contains more phase angle comparisons between IDT and axial compression tests.

Poisson’s Ratio In order to investigate the cause(s) for the difference in the phase angles determined from the horizontal and vertical strains, Poisson’s ratios of the four representative mixtures were calculated from the IDT test results and plotted in Fig. 5-9. First to note from this figure is that the average values of Poisson’s ratio at different temperatures seem to be reasonable, that is, about 0.18 at −10°C, about 0.25 at 10°C, and around 0.45 at 35°C. Some of Poisson’s ratio values at lower frequencies and 35°C exceeded the linear elastic limit of 0.5, indicating that at these conditions specimens were damaged during the dynamic modulus test. Another important observation from Fig. 5-9 is that there is a slight dependence of Poisson’s ratio on the loading frequency. This frequency dependence of Poisson’s ratio results in the phase lag between the vertical and horizontal strains, which in turn produces different phase angles calculated from the vertical and horizontal strains in Figs. 5-7 and 5-8.

Conclusions In this chapter, an analytical solution for the dynamic modulus of asphalt mixtures tested in the IDT mode is developed using the theory of linear viscoelasticity. The accuracy of this solution was successfully validated using the experimental data

135

136

Chapter Five

FIGURE 5-9 Poisson’s ratios at (a) −10°C; (b) 10°C; and (c) 35°C. (Kim et al. 2004, with permission from Transportation Research Board.)

obtained from the axial compression and IDT testing of 24 commonly used North Carolina mixtures. It was also found that Poisson’s ratio is a weak function of loading frequency. This frequency dependence of Poisson’s ratio results in different phase angles, depending on whether the horizontal strain or the vertical strain is used. An average of these two phase angles yields similar values to the axial compression phase angle. The findings presented in this chapter provide the analytical solutions that may be used in a standard test and analysis protocol for the determination of dynamic modulus and phase angle mastercurves of asphalt mixtures using the IDT test. This protocol will be an essential means of using the dynamic modulus in forensic analysis that utilizes the mechanistic principles in the NCHRP 1-37A Guide for Design of New and Rehabilitated Pavement Structures developed under the NCHRP project 1-37A (2004).

C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t

Acknowledgment This material is based on work supported by the North Carolina Department of Transportation under Project No. HWY-2003-09. The authors gratefully acknowledge this support.

References ASTM D6752-03, 2003, Standard Test Method for Bulk Specific Gravity and Density of Compacted Bituminous Mixtures Using Automatic Vacuum Sealing Method. American Society for Testing and Materials. Barksdale, R. D., J. Alba, N. P. Khosla, Y. R. Kim, P. C. Lambe, and M. S. Rahman, 1997, Laboratory Determination of Resilient Modulus for Flexible Pavement Design. Final Report, National Cooperative Highway Research Program 1-28 Project, National Research Council, Washington, D.C. Hondros, G, 1959, Evaluation of Poisson’s Ratio and the Modulus of Materials of a Low Tensile Resistance by the Brazilian (Indirect Tensile) Test with Particular Reference to Concrete. Australian Journal of Applied Science, Vol. 10, No. 3, pp. 243–268. Kim, Y. R., J. Daniel, and H. Wen, 2000, Fatigue Performance Evaluation of WesTrack and Arizona SPS-9 Asphalt Mixtures Using Viscoelastic Continuum Damage Approach. Final report to Federal Highway Administration/North Carolina Department of Transportation. Kim, Y. R., Y. Seo, M. King, and M. Momen, 2004, Dynamic Modulus Testing of Asphalt Concrete in Indirect Tension Mode. Journal of Transportation Research Board, No. 1891, National Research Council, Washington, D.C., pp. 163–173. Kim, Y. R., M. Momen, and M. King, 2005, Typical Dynamic Moduli for North Carolina Asphalt Concrete Mixtures. Final report to the North Carolina Department of Transportation, Report No. FHWA/NC/2005-03. NCHRP 1-37A Research Team, 2004, Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures, Final Report. NCHRP 1-37A, ARA, Inc. and ERES Consultants Division. Roque, R., and W. G. Buttlar, 1992, The Development of a Measurement and Analysis System to Accurately Determine Asphalt Concrete Properties Using the Indirect Tensile Mode. Proceedings, The Association of Asphalt Paving Technologist, pp. 304–333.

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CHAPTER

6

Interrelationships among Asphalt Concrete Stiffnesses Ghassan R. Chehab and Y. Richard Kim

Abstract The chapter discusses the three major response functions used for characterizing the linear viscoelastic behavior of asphalt concrete mixtures. Definitions and analytical representations of those functions: creep compliance, relaxation modulus, and complex modulus, are presented. Methods for the determination of the functions’ analytical parameters through experimental tests are introduced. Additionally, numerical and analytical interconversion techniques to determine one LVE response function from the other are also presented and compared. Numerical examples and plots are included to supplement the methodologies presented.

Introduction Asphalt concrete exhibits time/rate dependence, where the material response is not only a function of the current input, but also of the current and past input history. When the loading conditions do not cause damage to the asphalt mixture, the response could be defined as linear viscoelastic and expressed through the convolution (hereditary) integral. While viscoelasticity is typically associated with a system’s time-dependent response, linearity is associated with systems where the conditions of homogeneity and superposition are satisfied: Homogeneity: R {AI} = A R {I} Superposition: R {I1 + I2} = R {I1} + R {I2}

(6-1) (6-2)

where I, I1, I2 = input histories R = response A = arbitrary constant

139 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.

140

Chapter Six The brackets { } indicate that the response is a function of the input history. The homogeneity condition, also referred to as proportionality, essentially states that the output is directionally proportional to the input, for example, if the input is doubled, the response doubles as well. The superposition condition states that the response to the sum of two inputs is equivalent to the sum of the responses from the individual inputs. For linear viscoelastic (LVE) materials, the input-response relationship is expressed through the hereditary integral which allows the response to any input history to be calculated as follows: t

R=

∫R

−∞

H

(t , τ )

dI dτ dτ

(6-3)

where RH = unit response function I = input t = time of interest t = integration variable With a known unit response function, the lower limit of the integration can be reduced to 0− (just before time zero) if the input starts at time t 0 and both the input and response are equal to zero at t < 0. The value of 0− is used instead of 0 to allow for the possibility of a discontinuous change in the input at t 0. For notational simplicity, 0 is used as the lower limit in all successive equations and should be interpreted as 0− unless specified otherwise. Equation (6-3) is applicable to an aging system in which the time zero is the time of production of the material rather than the time of load application. In a nonaging system, time zero corresponds to the onset of load application, irrespective of when the material was produced. In this text, asphalt concrete will be treated as a nonaging system; thus, Eq. (6-3) reduces to t

R = ∫ RH (t − τ ) 0

dI dτ dτ

(6-4)

Types of LVE Response Functions Several viscoelastic response functions can be used to characterize the LVE behavior of asphalt concrete (AC), the most fundamental ones being the relaxation modulus E(t), creep compliance D(t), and complex modulus E*. The need for different types of response functions is attributed to a number of factors including type of loading application, conditions under which the material is characterized, and the experimental testing difficulties that constrain the determination of those functions. While those response functions, or linear viscoelastic properties, are fundamental for characterizing the AC in the linear viscoelastic range, they also serve as key components in constitutive models that describe the nonlinear behavior of AC under damage. Additionally, when AC specimens are used in experimental testing, response functions can be used as “viscoelastic fingerprints” to evaluate the specimen-to-specimen variation and/or determine if the material has been damaged or not.

Creep Compliance and Relaxation Modulus The creep compliance D(t) is the ratio of strain response to a constant stress input; while the relaxation modulus E(t) is the ratio of stress response to a constant strain input. If

Interrelationships among Asphalt Concrete Stiffnesses asphalt concrete were purely elastic, then D(t) and E(t) would be the reciprocal of each other. However, due to the viscoelastic nature of asphalt concrete this is only true in the Laplace transform domain. In equation form D(t) =

ε (t) , σ0

E(t) =

σ (t) ε0

and

(6-5)

(6-6)

where D(t) and E(t) = creep compliance and relaxation modulus, respectively s0 and e0 = constant input stress and strain, respectively s (t) and e(t) = stress and strain response, respectively For uniaxial loading, and nonaging, isothermal conditions, the linear viscoelastic stress-strain relationships are represented by the Boltzmann convolution integral as follows: t

σ = ∫ E(t − τ ) 0

t

ε = ∫ D(t − τ ) 0

dε dτ dτ

(6-7)

dσ dτ dτ

(6-8)

where t is an integration variable. Substituting e in the right-hand side of Eq. (6-7) by its equivalent from Eq. (6-8) results in the constitutive equation relating D(t) and E(t) as follows: t

1 = ∫ E(t − τ ) 0

dD(t) dτ dτ

(6-9)

Complex Modulus The complex modulus E∗ is a response function that relates stresses and strains for a linear viscoelastic material subjected to sinusoidal loading. It is composed of two components: dynamic modulus |E∗| and phase angle f, defined as follows:

E* =

σ amp ε amp

,

and

φ = 2π fΔt where samp = stress amplitude eamp = strain amplitude f = loading frequency Δt = time lag between stress and strain response

(6-10)

(6-11)

141

142

Chapter Six Being a complex function E* is composed of real and imaginary components, referred to as the storage and loss moduli, respectively, and mathematically expressed as follows: E * = E′ + iE′′ where

(6-12)

E′ = E * cos φ storage modulus E″ = E * sin φ loss modulus E * = (E′)2 + (E″ )2 i = −1

Determination of LVE Response Functions Viscoelastic response functions can be determined either through experimental testing conducted in the LVE range or through interconversion from other known response functions. From the theory of viscoelasticity, it can be shown that all LVE response functions are interrelated; thus, any function can be obtained if another is known. Both the creep compliance and complex modulus tests are simple mechanical tests that allow for the accurate characterization of AC in the LVE range. From the creep test D(t) is determined as a function of time; whereas, from the complex modulus test |E*| and f are determined as a function of frequency. The simplicity of obtaining D(t) and E* from mechanical tests is countered with a difficulty in obtaining E(t) from the relaxation test which is more difficult to conduct and requires a high capacity and robust testing machine. Therefore, it is often the case where E(t) is obtained through interconversion from D(t) or E*. Interconversion can also be necessary where one material function cannot be determined from a single test type over the entire range of the domain needed. For example, E(t) and D(t) cannot be determined at very short times; in this case, E* is determined by conducting a complex modulus test for the corresponding range of interest in frequency domain and then converted to E(t) and D(t). Prior to elaborating on the different interconversion techniques, it is necessary to present and discuss the analytical representation of the response functions since that will impact the choice and accuracy of the interconversion method used.

Analytical Representation of LVE Response Functions For accurate material characterization to be achieved, it is essential that representative analytical expressions of LVE response functions be established regardless of how those functions are obtained. For example, if an analytical expression is to be established for E*, complex modulus tests are first conducted at several temperatures and frequencies. The time-temperature superposition principle is then applied to obtain a single mastercurve for |E*| and f as a function of reduced frequency at a reference temperature of choice. This has been covered in more depth in Chap. 4. A mathematical function is then fit to the mastercurve to arrive to a representative analytical expression of that response for a broad frequency (time) range.

Interrelationships among Asphalt Concrete Stiffnesses

Power-Law Representations Various forms of power-law expressions have been used to represent the viscoelastic response. Usually, those representations, although rough and simplistic, yield fits that are globally smooth and stable (Park et al. 1996). Presented below are some of the common power-law expressions commonly used in fitting viscoelastic response functions. Although the expressions are presented for D(t), they can similarly apply to E(t).

Pure Power Law The pure power law (PPL) is the simplest among all power-law representations; it is given by the following form: D(t) = D1 t n

(6-13)

where the constant D1 is the value of the creep compliance at t = 1 and the exponent n is obtained by identifying the representative slope of the experimental data over the transient region plotted on a log-log scale. The limitation of the PPL is its inability of representing the data in regions other than the transient zone (Park and Kim 1996).

Generalized Power Law The generalized power law (GPL) can also be used to represent the response functions. It is expressed as D(t) = Dg + D1 t n

(6-14)

where Dg is the glassy compliance: Dg = lim D(t) . A sample graphical representation is t→ 0 given in Fig. 6-1 with Dg = 7.0E − 5 MPa−1, D2 = 2.3E − 3 MPa−1, and n = 0.45. The GPL fits the response data more accurately than the PPL in that it simulates the short-time

FIGURE 6-1 Different analytical representations of the experimental creep data. (Park et al. 1996.)

143

144

Chapter Six behavior more realistically due to the presence of the factor Dg ; however, it fails to simulate the data at long times.

Modified Power Law The representation of the modified power law (MPL) (Williams 1964) is of the following form: D(t) = Dg +

De − Dg

τ⎞ ⎛ ⎜⎝ 1 + t ⎟⎠

n

(6-15)

and is plotted in Fig. 6-1 with Dg = 7.0E − 5 MPa−1, De = 6.6E − 2 MPa−1, t = 1.5E + 3 seconds, and n = 0.45. The constant De is the long-time equilibrium or rubbery compliance which is defined by De = lim D(t). Constants Dg and De can be determined through inspection of the t→∞ experimental data. As for t and n, their determination typically requires nonlinear analysis; however, both can be reasonably estimated by an appropriate simplified procedure. The MPL fits the data much better than the two aforementioned power laws by generating a characteristic, broad-band, S-shaped curve. In particular, the MPL is capable of describing the glassy and rubbery behavior at short and long times, respectively. The exponent n gives the slope of the creep curve through the transition region between the glassy and rubbery behavior, and t fixes a characteristic retardation time (Park et al. 1996). As observed, the MPL representation ceases to fit the data satisfactorily at the top and bottom asymptotes (where the curvatures are maximum), a shortcoming attributed to the limited degrees of freedom associated with the expression. Enhancing the fit requires expanding the degrees of freedom such as in the case of a power-law series representation (Park et al. 1996). To enhance the fit of the experimental data, Park et al. (1996) investigated the power-law series representation in the following form: M

D(t) = Dg + ∑ i =1

ˆ D i

τˆi ⎞ ⎛ ⎜⎝ 1 + t ⎟⎠

n

(6-16)

ˆ and τˆ (i = 1, …, M), n, M are all constants. A fixed value for n is chosen a priori where D i i in order to allow only two degrees of freedom per term in series [Eq. (6-16)] just as in the PPL. Although a single n yields a satisfactory fitting, one may consider a discrete set of exponents, ni ( i = 1,…, M) instead of using a single, fixed n. Four cases with different Ms and ti ’ s were considered in fitting creep mastercurves. In general, the fit improves as the number of terms increases; however, the five-term (M = 5) representation was found to be sufficient in accurately fitting the experimental data as shown in Fig. 6-1.

Prony Series A Prony (Dirichlet) series consisting of a sequence of decaying exponentials has been widely used to represent viscoelastic response (Schapery 1961; Tschoegl 1989). The popularity of this series representation is attributed mainly to its ability of describing a wide range of viscoelastic response and to the relatively simple and rugged computational efficiency associated with its exponential basis functions. In addition, the Prony series representation of an LVE response function has a physical basis in the theory of mechanical models dealing with linear springs and dashpots (Park et al. 1996).

Interrelationships among Asphalt Concrete Stiffnesses

Analytical Representation For creep compliance, the Prony series representation is of the following form: M

D(t) = D0 + ∑ Dm [1 − e − t/τm ]

(6-17)

m

where tm = retardation time Dm = regression coefficient D0 = glassy compliance D(t) . The glassy compliance denotes the short-time creep behavior, that is, D0 = lim t→ 0 The right-hand side of Eq. (6-17) represents a mechanical model often referred to as the generalized Voigt model. It implies that the viscoelastic retardation process of the material can be regarded as a superposition of elementary processes in which the strain retards exponentially. For relaxation modulus, the Prony series expression takes a similar form: M

E(t) = E∞ + ∑ Em e − t/ρ

m

(6-18)

m= 1

where E∞, rm, and Em are long-time equilibrium modulus, relaxation time, and Prony regression coefficients, respectively. Physically, this representation is related to the Wiechert (or generalized Maxwell) model. Figure 6-2 provides a graphical illustration of the assembly of springs and dashpots associated with the Voigt and Wiechert models, respectively.

Fitting of Experimental Data There are a number of ways of fitting the Prony series expression to the given experimental data. Cost and Becker (1970) developed the so-called multidata method based on the least squares scheme and applied it to fit the Laplace-transformed Prony series to the Laplace-transformed data. Schapery (1961) used the collocation method to fit a Prony series model to experimental data obtained from tensile relaxation of polymethyl methacrylate and dynamic shear creep tests of polyisobutylene, while Park and Kim (1996) used it on data from relaxation and creep tests of sand-asphalt and asphalt concrete mixtures. An illustration of the collocation method follows. In fitting a creep mastercurve using a Prony series expression, there are 2N unknowns including Di and ti (i = 1,…, N) and a corresponding system of 2N nonlinear equations. However, in order to avoid the complexity that often arises in solving for the unknowns, the relaxation times ti are usually specified a priori from experience, thus leaving Di the only unknowns to be determined by solving the resulting linear system of equations. Usually, one-decade intervals of ti are adequate (Schapery 1974), and the glassy compliance D0 is obtained by extending the experimental curve asymptotically to t = 0. Although the aforementioned methods are simple and straightforward; however, they often yield negative values for the Prony coefficients, a case that is physically unrealistic and often results in oscillations in the reconstructed curve. In an effort to overcome the problem of negative coefficients in a Prony series representation, a number of improved fitting methods have been proposed. Emri and Tschoegl (1993, 1994, 1995) and Tschoegl and Emri (1992, 1993) designed a recursive algorithm to avoid negative coefficients by using only well-defined subsets of the complete set of experimental data. Kashhta and Schwarzl (1994a, 1994b) developed a method that ensures positive coefficients through an interactive adjustment of relaxation or retardation times. Others

145

146

Chapter Six

FIGURE 6-2 (a) Generalized Voigt model: where h m is the coefﬁcient of viscosity and Dm is the compliance for the mth term and (b) Wiechert model: where h m is the coefﬁcient of viscosity and Em is the stiffness for the mth term.

applied the so-called Tikhonov regularization techniques (Honerkamp and Weese 1989, Elster et al. 1991) and the maximum entropy method (Elster and Honerkamp 1991) to overcome the difficulties in finding the coefficients due to the ill-posed nature (or nonuniqueness) of the problem. Baumgaertel and Winter (1989) found a series representation through a nonlinear regression in which the spectra, time constants, and the number of terms in the series were variable; they started with a large number of relaxation modes (usually 0.5 to 1 decade apart) and merged or eliminated unnecessary ones whenever negative coefficients occurred. Mead (1994) presented a numerical method of determining discrete line spectra based on a constrained linear regression.

Presmoothing Experimental Data Prior to Prony Series Fit It is very desirable to use the Prony series representation of linear viscoelastic response due to its amenability to mathematical operations as long as the local waviness of the fit and negative values of some coefficients that might result are resolved. Since those problems are mainly caused by the wide scatter of unsmoothed data, it would be advantageous to presmooth the experimental data to eliminate the scatter prior to fitting it using the Prony series.

Interrelationships among Asphalt Concrete Stiffnesses As concluded previously, the power-law series provides a globally smooth and broadband representation of the LVE response data. Its main drawback is the associated analytical difficulty when conducting mathematical operations. However, the latter fact does not eliminate the importance of power-law series as an ideal candidate to presmooth the data to which a Prony series is subsequently fit. The resulting Prony series fit to the presmoothed experimental data yields a smoother reconstructed curve that is free of local waviness and that is analytically simple for mathematical operations. In an exercise conducted by (Park and Schapery (1999). on AC relaxation data, a fiveterm power-law series representation (M = 5) was used to presmooth the experimental data that was later fit to a Prony series. The reconstructed curve from the resulting fit was graphically indistinguishable from the five-term power-law series representation. Figure 6-3 illustrates the effect of presmoothing on the quality of the Prony series fit to relaxation data. The above mentioned fitting techniques also hold true for other response functions such as relaxation modulus. After fitting the relaxation data using an MPL series, a Prony series is fit with the coefficients determined using the collocation method (Chehab 2002), as follows: Formulating Eq. (6-18) in column vectors ({A} and {C}) and matrix [B], the regression coefficients are determined using the following equation (Mun et al. 2007): M

E(tn ) − E∞ = ∑ exp(−tn / ρm ) E m , n 1,…, N m= 1 {C} {A}

(6-19)

[B]

The nonnegative coefficients {C} are solved for using the embedded linear programming function provided by MATLAB. The following rearranged form forces the coefficients to be positive while still satisfying Eq. (6-19): minimize |[B]{C}−{A}|

such that {C} ≥ 0

FIGURE 6-3 Prony series ﬁts to unsmoothed and smoothed E(t).

(6-20)

147

148

Chapter Six

FIGURE 6-4

MPL and Prony series ﬁts to E(t).

Figure 6-4 shows the MPL series fit of the experimental data and the Prony series fit. As observed, both fits match closely, with the Prony fit being more advantageous due to the analytical simplicity in mathematical applications.

Interconversion between LVE Response Functions Being mathematically equivalent, LVE response functions can be obtained from one another using several mathematical interconversion techniques (Schapery et al. 1999). This holds true for both shear and uniaxial modes of loading. As presented earlier, interconversion may be required when conditions do not allow a response function to be determined through direct experimental testing. A common example is the difficulty in obtaining E(t) from the relaxation test which requires a robust testing machine which may or may not be available. It is therefore more common to obtain E(t) through interconversion of D(t) or E*, both of which are generally easier to obtain from experimental testing. In other instances, a response function cannot be determined over the complete range of its domain from a single test type; in these cases, the response can be obtained for the desired broad range through the superposition of responses obtained through different types of tests. Interconversion is also used when an accurate shorttime response which is difficult to obtain from a test with a transient excitation is alternatively obtained from a test with steady-state sinusoidal excitation. This normally requires an interconversion between responses in time and frequency domains (Park and Schapery 1999). Hopkins and Hamming (1957) were among the early researchers who dealt with the subject of interconversion between linear viscoelastic functions by developing a numerical technique for relating E(t) and D(t). The approach was later improved by Knoff and Hopkins (1972) and Baumgaertel and Winter (1989) who established analytical conversion techniques using interrelationships in the Laplace transform domain and

Interrelationships among Asphalt Concrete Stiffnesses the Prony series representation of both the source and target response functions. Mead (1994) presented a numerical interconversion method based on constrained linear regression with regularization, while Ramkumar et al. (1997) proposed a regularization technique that used a quadratic programming technique. The aforementioned methods as well as others not mentioned have been covered extensively by Schwarzl and Struik (1967), Ferry (1980), and Tschoegl (1989). This chapter sheds light on some of the mostly used techniques in converting between LVE response functions for asphalt materials. Both the E(t) and D(t) are essential elements for characterizing viscoelastic responses. They have been incorporated in several constitutive models that have been developed and used over the years including some of those presented in this book (Schapery 1961 and 1974; Ferry 1980; Christensen 1982; Tschoegl 1989; Kim et al. 1990 and 1997; Lee and Kim 1998; Uzan 1996; Bahia et al. 2000; Daniel and Kim 2002; Roque et al. 2002; Chehab et al. 2003). As for E*, its acceptance as an indicator of the AC mix behavior has been on the rise especially after it was chosen as a Superpave simple performance test for mix design, a QC/QA tool, in addition to an indicator of stiffness in the M-E design guide. It is also used for depicting the material’s response under sinusoidal loading. Thus, with the simplicity and wide use of the complex modulus test, using interconversion methods to obtain E(t) and D(t) from E* becomes more favorable than conducting additional experimental tests to obtain them. Interconversion is not always simple and straightforward. Using exact solutions may require integration over an infinite range, a task that is often complicated whether done analytically or numerically. In addition, experimental data required for the interconversion may only be available for a limited range of the required time or frequency domain. To overcome such difficulties, adopting approximate analytical and numerical techniques becomes necessary. In what follows, both approximate and analytical methods of interconversion are presented.

Approximate Numerical Methods Several numerical methods have been used in converting between LVE response functions. According to Taylor et al. (1970) and Park and Schapery (1999), those methods are especially useful when the response functions are expressed in the Prony series form. In what follows, t and w are used as symbols for time and radial frequency, respectively, or as their corresponding temperature-reduced quantities.

Interconversion between E(t) and D(t) The exact relationship between relaxation modulus E(t) and creep compliance D(t) in time domain is given by the following integral: t

∫ E(t − τ ) 0

dD(τ ) dτ = 1 dτ

(6-21)

Taking the Laplace transform of Eq. (6-21) yields E(s) D(s) = ∞

1 (for t > 0) s2

(6-22)

where f (s) ≡ ∫ f (t)e − st dt denotes the Laplace transform of f(t) and s is the transform 0 parameter. In solving the integral, a typical numerical approach requires that the integral be divided into a large number of time segments. This can be easily achieved when the functions are presented in their Prony series form. This being the case, when

149

150

Chapter Six {ri, Ei(i = 1,…, m) and Ee} or {tj, Dj (j = 1,…, n), Dj and h0} are known and the target time constants specified, the unknown set of constants can be determined through a system of linear algebraic equations. For example, creep compliance in its Prony series form, Dj (j 1,…, n), can be determined from relaxation modulus E(t) as follows: [ A]{D} = {B}

(6-23)

or AkjDj = Bk (summed on j; j = 1,…, n; k = 1,…, p) where

ρi Ei −(t /ρ ) (e − e − ( t /ρ ) ) ρi − τ j

m

{

Akj = Ee (1 − e −(t /τ ) ) + ∑ k

j

k

i=1

i

k

i

ρi ≠ τ j

(6-24)

or Ee (1 − e −(t

k

/τ j )

m

tk Ei −(t (e τj i=1

)+∑

k

/ ρi )

)

ρi = τ j

and m ⎛ ⎞ Bk = 1 − ⎜ Ee + ∑ Ei e −(t /ρ ) ⎟ ⎝ ⎠ i =1 k

i

m ⎛ ⎞ + E Ei ⎟ ∑ ⎜⎝ e ⎠

(6-25)

i =1

The symbol tk (k 1,…, p) denotes a discrete time corresponding to the upper limit of integration in Eq. (6-21). Once the model constants Dg, Dj, and tj are determined, then function D(t) can be obtained in its Prony series form. Similarly, E(t) can be determined from D(t) by determining the unknown constants of the Prony series representation of E(t): Ei (i 1,…, m). As for Dg and Ee, they are related as follows: Dg ≡

1 m

Ee + ∑ Ei i =1

Ee ≡

1

(6-26)

n

Dg + ∑ D j j =1

Equations (6-23) through (6-26) were used in converting E(t) data of an AC mixture to D(t). The source and target Prony series coefficients are presented in Table 6-1, with corresponding fitted data plotted in Fig. 6-5. For additional details and derivations of the above equations, the reader is referred to Park and Schapery (1999) where other approximate interconversion techniques available in literature are documented and compared. Only a limited number of common techniques are presented in the next section.

Quasi-Elastic Interrelationship The most intuitive and crude interrelationship between the E(t) and D(t) is the one based on quasi-elastic approximation: E(t)D(t) ≅ 1 (for t > 0)

(6-27)

Interrelationships among Asphalt Concrete Stiffnesses

Relaxation Modulus (MPa) 34.5

E•

Creep Compliance (1/MPa) D0

4.00E–05

ti (sec)

Di

Prony Coefficients ri (s)

Ei

1.0E−06

9.34E+02

1.0E−06

1.26E−06

1.0E−05

4.09E+03

1.0E−05

6.53E−06

1.0E−04

5.39E+03

1.0E−04

1.54E−05

1.0E−03

4.84E+03

1.0E−03

2.48E−05

1.0E−02

5.19E+03

1.0E−02

5.27E−05

1.0E−01

2.96E+03

1.0E−01

1.64E−04

1.0E+00

1.31E+03

1.0E+00

3.33E−04

1.0E+01

1.33E+02

1.0E+01

2.93E−03

1.0E+02

1.14E+02

1.0E+02

3.21E−03

1.0E+03

3.57E+01

1.0E+03

1.63E−02

1.0E+04

1.44E+00

1.0E+04

1.0E+05

−3.00E–01

1.0E+05

TABLE 6-1

Prony Series Coefficients for E(t) and D(t) for an AC Mix

FIGURE 6-5 E(t) and D(t) from interconversion for an AC mixture.

7.48E–03 –1.52E–0

151

152

Chapter Six The interconversion in Eq. (6-27) provides a good relation between the response functions when the material exhibits mostly elastic behavior with minimal viscoelasticity.

Power-Law-Based Interrelationship An LVE material can be approximately represented by simple power law for small ranges in their transition zone. Representing E(t) and D(t) in pure power-law form [Eqs. (6-28) and (6-29), respectively], interrelationship Eq. (6-30) is obtained. E(t) = E1t − n

(6-28)

D(t) = D1t n

(6-29)

E(t)D(t) =

sin nπ nπ

(6-30)

where E1, D1, and n are all positive constants. Equation (6-30) was first given by Leaderman (1958). It is accurate in regions in which E(t) and D(t) are represented approximately by straight lines on log-log scales, with the exponent n being the absolute value of the slope of these lines. When n approaches zero, that is, for an elastic material, the right-hand side of Eq. (6-30) becomes unity and Eq. (6-27) becomes exact. In equation form, n is represented as n=

d log RH (t) d log t

(6-31)

where RH(t) is the unit response function of interest.

Interrelationship by Christensen Christensen (1982) developed an approximate interconversion method between E(t) and D(t) using approximate relationships between the real and imaginary parts of a complex material function, and between the transient function and the real part of the complex material function: If E(t) is known then D(t) can be determined according to Eq. (6-32): E(t)

D(t) ≅ E 2 (t) +

π 2 t 2 ⎧ dE(t) ⎫ ⎨ ⎬ 4 ⎩ dt ⎭

2

(6-32)

Equation (6-32) is also applicable when D(t) and E(t) are interchanged. In terms of n, Eq. (6-32) can be expressed as follows: E(t)D(t) ≅

1 n2 π 2 1+ 4

(6-33)

Interrelationship by Denby An approximation similar to that presented above was proposed by Denby (1975), yielding the following interrelationship: E(t)D(t) ≅

1 n2 π 2 1+ 6

(6-34)

Interrelationships among Asphalt Concrete Stiffnesses Park et al. (1996) found that Eqs. (6-30), (6-33), and (6-34) yield comparable and accurate relationships for interconversion. This holds true as long as both functions E(t) and D(t) exhibit broadband and smooth variations on log-log scales. On that same premise, the following interconversion scheme has also been proposed: D(t) =

1 , E(α t)

E(t) =

1 ⎛ t⎞ D⎜ ⎟ ⎝α⎠

and

(6-35)

(6-36)

where ⎛ sin nπ ⎞ α =⎜ ⎝ nπ ⎟⎠

1

n

(6-37)

Interconversion between E* and D* The significance of using response functions in their complex domain in characterizing the LVE behavior of asphalt materials has been highlighted here as well as in other chapters. On that premise, it thus becomes important to establish a relationship between the material functions in the complex domain, specifically between E* and D*. Deriving such a relationship is more convenient when E(t) and D(t) are represented in the Laplace domain as follows (Tschoegl 1989): ∞

E (s) ≡ s ∫ E(t)e − st dt 0

(6-38)

∞

(s) ≡ s D(t)e − st dt D ∫ 0

(6-39)

where the integrals in Eqs. (6-38) and (6-39) are the Laplace transforms of E(t) and D(t), respectively. Incorporating the above equations with Eq. (6-21) yields the following relationship between the relaxation and creep compliance in the Laplace domain: (s) = 1 E (s)D

(6-40)

Complex material functions arise from the response to steady-state sinusoidal loading with angular frequency w, and are related to the Laplace-transformed functions according to the following relationship proposed by Tschoegl (1989): E * (w) = E (s) (s) D * (w ) = D

s→iw

s→iw

(6-41) (6-42)

Hence, the relationship between E* and D* can be deduced from Eqs. (6-40) to (6-42): E * (w)D* (w) = 1

(6-43)

153

154

Chapter Six The real and imaginary parts, denoted with primes and double primes, respectively, are expressed as follows: E* (w) ≡ E ′(w) + iE ′′(w)

(6-44)

D* (w) ≡ D′(w) − iD′′(w)

(6-45)

where D″ is positive. The real component of the response functions is also referred to as the storage component, while the imaginary is referred to as the loss component. The Prony series expression of the storage and loss components of modulus and creep are presented below (Park and Schapery 1999): m

E′(w) = Ee + ∑ i=1

m

E′′(w) = ∑ i=1

w 2 ρi2Ei w 2 ρi2 + 1

(6-46)

wρiEi w 2 ρi2 + 1

(6-47)

Dj w τ 2j + 1 j=1 n

D′(w) = Dg + ∑ D′′(w) =

(6-48)

2

n wτ j Dj 1 +∑ 2 2 η0 w j=1 w τ j + 1

(6-49)

It is observed from Eqs. (6-46) to (6-49) that if the Prony series of either the real or imaginary components of a complex function are known, then the series representation of the other component can be determined. The components of E* and D* can be interrelated using Eqs. (6-43), (6-44), and (6-45): D′ =

E′ (E ′)2 + (E ′′)2

(6-50)

D″ can then be obtained in terms of the same set of constants. E* and D* can be related by expressing E′ E″, and D′ in Eq. (6-50) by their Prony series representations in Eqs. (6-46) to (6-48), respectively. The interconversion can be achieved using the same form of Eq. (6-23) with the following Akj and Bk: Akj =

1 , wk2τ 2j + 1

and m

Bkj =

Ee + ∑ i=1

(6-51)

w 2 ρi2Ei w 2 ρi2 + 1 2

⎛ w ρ Ei ⎞ ⎛ wρiEi ⎞ ⎜⎝ Ee + ∑ w 2 ρ + 1 ⎟⎠ + ⎜⎝ ∑ w 2 ρ 2 + 1 ⎟⎠ i i=1 i=1 m

2

2 i 2 i

m

2

−

1 m

Ee + ∑ Ei

(6-52)

i=1

where wk (k = 1,…, p) is the angular frequency. It is selected using the same procedure used in selecting tk in Eqs. (6-24) and (6-25). Similarly, Dg can be computed from Ee and Ei according to Eq. (6-26).

Interrelationships among Asphalt Concrete Stiffnesses

Approximate Analytical Methods Numerous approximate analytical interconversion methods with different bases, simplifications, assumptions, and accuracies have been proposed by Tschoegl (1989), Schapery (1962), and Park et al. (1996), and others. Methods of primary interest are those that have been widely used by asphalt researchers to relate between the time, frequency, and Laplace domain representations of modulus and compliance. Schapery (1962) presented two approximate analytical methods for interconversion between the uniaxial relaxation modulus E(t) and the operational modulus E (s), defined as the Carson transform or the s-multiplied Laplace transform of E(t); refer to Eq. (6-53). ∞

∞

0

−∞

E (s) ≡ s ∫ E(t)e − st d(ln t) = s ∫ E(t)te − st d(ln t)

(6-53)

The first relationship is given by E(t) ≅ E (s)

or

s =α t

E (s) ≅ E(t) t=α

s

(6-54)

where E (s) ≡ sE(s) , E(s) is the Laplace transform of E(t); and a = e−C, where C is Euler’s constant, thus yielding a ≅ 0.56. The second relationship is given as follows: E (s) ≅ E(t) t= β

(6-55)

s

−1

where β = {Γ(1 − n)} n , Γ(.) is the gamma function, and n is the local log-log slope of the E ( s ) E( t ) source function given either as n ≡ d log or n ≡ d log . If the moduli in Eqs. (6-54) d log s d log t ∼ and (6-55) are replaced by compliances, analogous relationships between D(t) and D (s) are obtained. Christensen (1982) proposed an approximate interconversion between E(t) and the storage modulus E′(w) as follows: E(t) ≅ E ′(w) w =2 π t

or

E ′(w) ≅ E(t) t= 2 π w

(6-56)

Analogous relationships hold for compliance functions when E′s in Eq. (6-56) are replaced by D′s. Staverman and Schwarzl (1955) gave the following approximate conversion from storage modulus E′(w) to the loss modulus E″(w): E′′(w) ≅

π dE ′(w) 2 d ln(w)

(6-57)

Booij and Thoone (1982) proposed the following conversion from E″(w) to E′(w): E ′(w) ≅ Ee −

π w d[E ′′(w)/w] , d ln w 2

E ′(w) ≅ Ee +

d ln E ′′ ⎞ π⎛ E ′′(w) 1− d ln w ⎟⎠ 2 ⎜⎝

or

(6-58) (6-59)

where Ee is the equilibrium modulus. Equations (6-57) and (6-59) also apply to compliance components when E′(w) and E″(w) are replaced by D′(w) and −D″(w), respectively.

155

156

Chapter Six A more recent interconversion technique has been developed by Schapery and Park (1999). The method employs variable adjustment factors dictated by the slope of the source function on a log-log scale. Presented below is a set of relationships for interconversion of relaxation modulus and its components. Similar relationships apply for creep compliance when the appropriate changes in parameters are made (Schapery ∼ ∼ and Park 1999), as follows: E( ) → D( ), n → −n, E → D, E′ → D′, and E″ → −D″. The sign change in E″ → −D″ necessitates a change in the sign of n so that the arguments of the trigonometric functions appearing in the set of l parameters used in the interconversion remain the same when compliance and modulus are interchanged. 1 E(t) ≅ E (s) s=(1/t ) λ E(s) ≅ λ E (t)

λ = Γ(1 − n)

(6-60)

t = (1 s)

E(t) ≅

1 E ′ (w ) w = ( 1 t ) λ′

E ′(w) ≅ λ ′E (t) t = (1 w )

E(t) ≅

1 E ′′ (w) w =(1 t ) λ ′′

E ′′(w) ≅ λ ′′E (t) t=(1/w ) 1 E (s) ≅ E ′ (w) w = s λ E ′(w) ≅ λ E (s) s= w 1 E (s) ≅ E ′′ (w) w = s λ E ′′(w) ≅ λ E (s) s= w 1 E ′(w) ≅ E ′′ (w ) w = w λ E ′′(w) ≅ λ E ′ (w ) w = w

λ ′ = Γ(1 − n)cos(nπ 2)

(6-61)

λ ′′ = Γ(1 − n)sin(nπ/2)

(6-62)

λ = cos(nπ /2)

(6-63)

λ = sin(nπ 2)

(6-64)

λ = tan(nπ 2)

(6-65)

In obtaining a Prony series representation for E(t) of an AC mix, the approximate interconversion in Eq. (6-61) was applied to experimental data obtained from E* testing. The E(t) obtained from interconversion were then fit to a Prony series as shown in Fig. 6-6.

Interrelationships among Asphalt Concrete Stiffnesses

FIGURE 6-6

Conversion from E′ to E(t) for an AC mix.

References Bahia, H., Zeng, M., and Nam, K. (2000), “Consideration of Strain at Failure and Strength in Prediction of Pavement Thermal Cracking,” Journal of Asphalt Paving Technology, AAPT, Vol. 69, pp. 497–539. Baumgaertel, M., and Winter, H. H. (1989), “Determination of Discrete Relaxation and Retardation time Spectra from Dynamic Mechanical Data,” Rheologica Acta, Vol. 28, pp. 511–519. Booij, H. C., and Thoone, G. P. (1982), “Generalization of Kramers–Kronig Transforms and Some Approximations of Relations between Viscoelastic Quantities,” Rheologica Acta, Vol. 21, pp. 15–24. Chehab, G. R. (2002), “Characterization of Asphalt Concrete in Tension Using a Viscoelastoplastic Model,” Ph.D. dissertation, North Carolina State University, Raleigh, N.C. Chehab, G. R., Kim, Y. R., Schapery, Y. R., Witczack, M., and Bonaquist R. (2002), “TimeTemperature Superposition Principle for Asphalt Concrete Mixtures with Growing Damage in Tension State,” Journal of Asphalt Paving Technology, AAPT, Vol. 71, pp. 559–593. Chehab, G. R., Kim, Y. R., Schapery, R. A., Witczack, M., and Bonaquist, R. (2003), “Characterization of Asphalt Concrete in Uniaxial Tension Using a Viscoelastoplastic Model,” Journal of Asphalt Paving Technology, AAPT, Vol. 72, pp. 315–355. Christensen, R. M. (1982), Theory of Viscoelasticity, 2d ed., Academic Press, New York, 1982, Section 4.6. Cost, T. L., and Becker, E. B. (1970), “A Multi-Data Method of Approximate Laplace Transform Inversion,” International Journal for Numerical Methods in Engineering, Vol. 2, pp. 207–219.

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158

Chapter Six Daniel, J. S., and Kim, Y. R. (2002), “Development of a Simplified Fatigue Test and Analysis Procedure Using a Viscoelastic Continuum Damage Model,” Journal of Asphalt Paving Technology, Vol. 71, pp. 619–650. Elster, C., and Honerkamp, J. (1991), “Modified Maximum Entropy Method and Its Application to Creep Data,” Macromolecules, Vol. 24, pp. 310–314. Elster, C., Honerkamp, J., and Weese, J. (1991), “Using Regularization Methods for the Determination of Relaxation and Retardation Spectra of Polymeric Liquids,” Rheologica Acta, Vol. 30, pp. 161–174. Emri, I., and Tschoegl, N. W. (1993), “Generating Line Spectra from Experimental Responses, Part I. Relaxation Modulus and Creep Compliance,” Rheoogica Acta, Vol. 32, pp. 311–312. Emri, I., and Tschoegl, N. W. (1994), “Generating Line Spectra from Experimental Responses, Part IV. Application to Experimental Data,” Rheologica Acta, Vol. 33, p. 6070. Emri, I., and Tschoegl, N. W. (1995), “Determination of Mechanical Spectra from Experimental Responses,” International Journal of Solid Structures, Vol. 32, pp. 817–826. Ferry, J. D. (1980), Viscoelastic Properties of Polymers, 3d ed., John Wiley & Sons, Inc., New York. Honerkamp, J., and Weese, J. (1989), “Determination of the Relaxation Spectrum by a Regularization Method,” Macromolecules, Vol. 22, pp. 4372–4377. Hopkins, I. L., and Hamming, R. W. (1957), “On Creep and Relaxation,” Journal of Applied Physics, Vol. 28, pp. 906–909. Kashhta, J., and Schwarzl, F. R. (1994a), “Calculation of Discrete Retardation Spectra from Creep Data: I. Method,” Rheologica Acta, Vol. 33, pp. 517–529. Kashhta, J., and Schwarzl, F. R. (1994b), “Calculation of Discrete Retardation Spectra from Creep Data: II. Analysis of Measured Creep Curves,” Rheologica Acta, Vol. 33, pp. 530–541. Kim, Y. R., and Little, D. L. (1990), “One-Dimensional Constitutive Modeling of Asphalt Concrete,” ASCE Journal of Engineering Mechanics, Vol. 116, No. 4, pp. 751–772. Kim, Y. R., Lee, H. J., and Little, D. N. (1997), “Fatigue Characterization of Asphalt Concrete Using Viscoelasticity and Continuum Damage Theory,” Journal of Asphalt Paving Technology, Vol. 66, pp. 520–569. Knoff, W. L., and Hopkins, I. L. (1972), “An Improved Numerical Interconversion for Creep Compliance and Relaxation Modulus,” Journal of Applied Polymer Science, Vol. 16, pp. 2963–2972. Leaderman, H. (1958), “Viscoelasticity Phenomena in Amorphous High Polymeric Systems,” Rheology, Vol. II, F. R. Eirich, Academic, New York. Lee, H. J., and Kim, Y. R. (1998), “A Uniaxial Viscoelastic Constitutive Model for Asphalt Concrete under Cyclic Loading,” ASCE Journal of Engineering Mechanics, Vol. 124, No. 11, pp. 1224–1232. Mead, D. W. (1994), “Numerical Interconversion of Linear Viscoelastic Material Functions,” Journal of Rheology, Vol. 38, pp. 1769–1795. Mun, S., Chehab, G. R., and Kim, Y. R. (2007), “Determination of Time-Domain Viscoelastic Functions Using Optimized Interconversion Techniques,” Road Materials and Pavement Design, Lavoisier, Vol. 8, No. 2, pp. 351–365. Park, S. W., and Schapery, R. A. (1999), “Methods of Interconversion between Linear Viscoelastic Material Functions, Part I—A Numerical Method Based on Prony Series,” International Journal of Solids and Structures, Vol. 36, pp. 1653–1675.

Interrelationships among Asphalt Concrete Stiffnesses Park, S. W., Kim, Y. R., and Schapery, R. A. (1996), “A Viscoelastic Continuum Damage Model and Its Application to Uniaxial Behavior of Asphalt Concrete,” Mechanics of Materials, Vol. 24(4), pp. 241–255. Ramkumar, D. H. S., Caruthers, J. M., Mavridis, H., and Shroff, R. (1997), “Computation of the Linear Viscoelastic Relaxation Spectrum from Experimental Data,” Journal of Applied Polymer Science, Vol. 64, pp. 2177–2189. Roque, R., Birgisson, B., Sangpetngam, B., and Zhang, Z. (2002), “Hot Mix Asphalt Fracture Mechanics: A Fundamental Crack Growth Law for Asphalt Mixtures,” Journal of Asphalt Paving Technology, AAPT, Vol. 71. Schapery, R. A. (1961), “A Simple Collocation Method for Fitting Viscoelastic Models to Experimental Data,” Report GALCIT SM 61-23A, California Institute of Technology, Pasadena, California. Schapery, R. A. (1962), “Approximate Methods of Transform Inversion for Viscoelastic Stress Analysis,” Proc. 4th U.S. Nat. Cong. Appl. Mech., pp. 1075–1085. Schapery, R. A. (1974), “Viscoelastic Behavior and Analysis of Composite Materials,” Composite Materials, Chap. 4, Vol. 2, G. P. Sendeckyj Ed., Academic Press, pp. 85–168. Schapery, R. A., and Park, S. W. (1999), “Methods of Interconversion between Linear Viscoelastic Material Functions. Part II—An Approximate Analytical Method,” International Journal of Solids and Structures, Vol. 36, pp. 1677–1699. Schwarzl, F. R., and Struik, L. C. E. (1967), “Analysis of Relaxation Measurements,” Advances in Molecular Relaxation Processes; Vol. 1, pp. 201–255. Taylor, R. L., Pister, K. S., and Goudreau, G. L. (1970), “Thermomechanical Analysis of Viscoelastic Solids,” International Journal for Numerical Methods in Engineering, Vol. 2, pp. 45–49. Tschoegl, N. W. (1989), The Phenomenological Theory of Linear Viscoelastic Behavior, SpringerVerlag, Berlin. Tschoegl, N. W., and Emri, I. (1992), “Generating Line Spectra from Experimental Responses. Part III: Interconversion between Relaxation and Retardation Behavior,” International Journal of Polymeric Materials, Vol. 18, pp. 117–127. Tschoegl, N. W., and Emri, I. (1993), “Generating Line Spectra from Experimental Responses. Part II: Storage and Loss Functions,” Rheologica Acta, Vol. 32, pp. 322–327. Uzan, J. (1996), “Asphalt Concrete Characterization for Pavement Performance Prediction,” Journal of Asphalt Paving Technology, AAPT, Vol. 65, pp. 573–607. Williams, M. L. (1964), “Structural Analysis of Viscoelastic Materials,” AIAA Journal, Vol. 2(5), pp. 785–808.

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PART

Constitutive Models CHAPTER 7 VEPCD Modeling of Asphalt Concrete with Growing Damage

CHAPTER 9 DBN Law for the Thermo-Visco-ElastoPlastic Behavior of Asphalt Concrete

CHAPTER 8 Uniﬁed Disturbed State Constitutive Modeling of Asphalt Concrete

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CHAPTER

7

VEPCD Modeling of Asphalt Concrete with Growing Damage Y. Richard Kim, Shane Underwood, Ghassan R. Chehab, Jo S. Daniel, H. J. Lee, and T. Y. Yun

Abstract This chapter presents the development of a viscoelastoplastic continuum damage (VEPCD) model for the behavior of asphalt concrete in tension and compression. The modeling strategy adopted is based on (1) the elastic-viscoelastic correspondence principle, (2) continuum damage mechanics to account for the effect of microcracking on the constitutive behavior, (3) a time- and stress-dependent viscoplastic model to account for the plastic and viscoplastic behavior, and (4) the time-temperature superposition (TTS) principle with growing damage to describe the effect of temperature on the constitutive behavior. The resulting models are integrated by the strain decomposition approach to form the VEPCD model. The VEPCD model in tension is developed for four asphalt-aggregate mixtures with three of the four mixtures modified by polymers. The model is shown to accurately predict the material behavior in tension over a range of conditions different from those used to characterize the model, including the results from the thermal strain restrained specimen tensile (TSRST) tests at several different cooling rates. Finally, a brief discussion on the VEPCD model in compression and the finite element implementation of the model is given.

Introduction Developing a realistic mathematical model of the mechanical behavior of asphalt concrete with growing damage is a complicated problem. The complexity is attributed to the viscoelastic hereditary effects of the binder, the complex n