ASPHALT MIXTURE FATIGUE TESTING

ASPHALT MIXTURE FATIGUE TESTING

Influence of Test Type and Specimen Size

Ning LI

ASPHALT MIXTURE FATIGUE TESTING

Influence of Test Type and Specimen Size

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 18 november 2013 om 10:00 uur

door

Ning LI

Master of Science in Material Science and Engineering Wuhan University of Technology, P.R. China

geboren te Hubei Province, P.R. China

Dit proefschrift is goedgekeurd door de promotoren: Prof. dr. ir. A.A.A. Molenaar Prof. S.P. Wu, BSc., MSc., PhD. Copromotor Ir. M.F.C. van de Ven Samenstelling promotiecommissie: Rector Magnificus, Technische Universiteit Delft, voorzitter Prof. dr. ir. A.A.A. Molenaar Technische Universiteit Delft, promotor Prof. S.P. Wu, BSc., MSc., PhD. Wuhan University of Technology, promotor Ir. M.F.C. van de Ven Technische Universiteit Delft, copromotor Prof. dr. ir. S.M.J.G. Erkens Technische Universiteit Delft Prof. dr. A. Scarpas Technische Universiteit Delft Prof. dr. ir. H.E.J.G. Schlangen Technische Universiteit Delft Dr. A. Vanelstraete Belgian Road Research Centre Published and distributed by: Ning Li Road and Railway Engineering Section Faculty of Civil Engineering and Geosciences Delft University of Technology P.O. Box 5048, 2600 GA Delft, the Netherlands E-mail: [email protected], [email protected] ISBN: 978-94-6186-235-8 Key words: Fatigue Test; Endurance Limit; Test Type; Specimen Size; Yield Surface; Safety Factor Printing: Wohrman Print Service, Zutphen (the Netherlands) ©2013 by Ning Li All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior permission of the proprietor.

To my family

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Acknowledgements Looking back to the starting point is helpful for moving forward in a right way. Life is not easy, especial for a PhD student living in a foreign country. During the five years’ study in Delft, many people gave me their guidance, encouragement and support when I faced difficulties. Without their efforts, this research would never have been completed. Therefore, this moment is a good opportunity for me to express my sincere gratitude to all of them. This PhD project was originated from the cooperation between the Delft University of Technology (TUD) and Wuhan University of Technology (WHUT). The research presented in this dissertation was carried out in the Road and Railway Engineering Section of the Faculty of Civil Engineering and Geosciences at the TUD. Firstly, the appreciation goes to those who built up the cooperative link between TUD and WHUT. The author also would like to thank the financial supports from the China Scholarship Council (CSC) and the TUD. I would like to express my deepest appreciation to my promoter, Prof.dr.ir. A.A.A. Molenaar. He always provided me with the valuable guidance and encouragement throughout every stage of my PhD study. I enjoyed the great benefit from not only his academic attainments but also his wisdom of life. His tireless efforts and constructive comments on this dissertation are highly appreciated. At the same time, sincere gratitude goes to my promoter Prof. Shaopeng Wu, who supervised my bachelor, master and PhD study. He advised me to start my PhD study abroad and let me expand my field of vision. His professional knowledge and experience in road industry led me into the road engineering field. I am grateful for my daily supervisors Associate Professor Martin van de Ven. He always gave his patient guidance whenever I need. His careful review on my papers, reports and dissertation are deeply appreciated. I would like to extend my sincere gratitude to Ir. A.C. Pronk for his contribution on the calibration and the modeling work in this research. Under his guidance, I adapted myself to the new study environment quickly and did not feel fear of those complex equations. Even after his retirement, he still came over and discussed with me when I need help. Thanks so much for all your efforts. My sincere appreciation goes to Prof.dr. R.L. Lytton for his arrangement and guidance when I was doing the project in Texas A&M University. Also many thanks to Dr. Rong Luo for her kind help during my stay in Texas. I would like to appreciate Prof. Halil Ceylan from Iowa State University. His valuable suggestions on the data analysis are highly acknowledged. I also would like to thank Dr. Milliyon Woldekidan for the valuable discussion and kind help with the finite element analysis. I would like to thank all the colleges and former colleges of Road and Railway Engineering Section. The extensive laboratory work could not have been successfully finished without the arrangement and support provided by Associate professor Lambert Houben, Abdol Miradi, Marco Poot, Jan-Willem Bientjes, Jan Moraal and Dirk Doedens.

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Particularly, I am grateful to Marco Poot. His rich experience of the test equipments makes the test program smooth and efficient. Many thanks to the secretariats, Jacqueline Barnhoorn and Sonja van de Bos for their kind help with the daily administration affaires. I would like to thank Prof. Tom Scarpas, Prof. Sandra Erkens, Prof. Rolf Dollevoet, Dr. Rien Huurman, Accociate Professor Zili Li and Dr. Xueyan Liu. Their support and guidance are deeply appreciated. Many thanks to all the PhD students, liantong Mo, Jian Qiu, Gang Liu, Xin Zhao, Eyassu, Alemgena Araya, Yue Xiao, Jingang, Sadegh, Mohamad, Oscar, Marija, Maider, Shaoguang, Nico, Dongya, Pengpeng, Chang, Haoyu, Lizuo, Xiangyun and all the new PhD students. I really enjoy the time here with all of them. Special thank goes to my officemates Diederik van Lent, Yuan Zhang and former officemate Dongxing. I feel fortunate that I have sit in the Room 1.29 for 5 years. I will never forget the moments when we were talking about our different cultures and languages, when we were sharing our knowledge and experiences and when we were playing table tennis. I appreciate the wonderful time that I have spent together with all my friends: Associate Prof. Ye, Xu Jiang, Xuhong Qiang, Quantao Liu, Hailing Zhang, Xuming Shan, Ying Wang, Juan Tong, Lili Wu, Huanhuan Mao, Nannan Li, Lin Liu, Huisu Chen, Zhiwei Qian, Yuguang Yang, Qi Zhang, Jinlong Li, Haoliang Huang, Bei Wu, Jiayi Chen, Yun Zhang, Yong Zhang, Yuan Qiu, etc. Thanks all of them for giving me kind help and enjoyable time. Finally, my deepest gratitude goes to my family for their endless love and continues support. My parents have never complained how less time I have spent with them since I left home for my study. I appreciate their encouragement and understanding. My special thanks to my wife Zhuqing Yu for her continuous support, patience and optimistic attitude. No matter what happen, she always stand firm behind me and take good care of my life even when she was also busy with her PhD study. This dissertation is dedicated to my family. Ning Li 李宁

October, 2013, Delft

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Summary Fatigue characterization of an asphalt mixture is commonly estimated by laboratory fatigue tests. Based on the classical fatigue analysis, fatigue lives obtained from different test devices are not comparable even when they are performed at the same test conditions. It is believed that there are two main reasons causing the difference in fatigue results being the difference in stress-strain distribution of the different specimens and the fatigue analysis approach. This research focuses on the harmonization of the fatigue results obtained from the different methods, which are recommended by the European standard EN 12697-24. The main goal is to find a correlation between the different fatigue test methods and to improve the classical fatigue analysis approach to better represent the actual fatigue characteristics of asphalt mixtures. To realize the main objective of the study, an extensive fatigue testing program was carried out on dense asphalt concrete 0/8 (DAC 0/8). In the program, uniaxial tension and compression (UT/C) fatigue tests, four-point bending (4PB) fatigue tests and indirect tensile (IT) fatigue tests were performed. For each fatigue test, specimens with different sizes were tested to explore the size effect on the fatigue results. In order to limit the test program, the tests were performed in two loading modes, at two temperatures and one frequency. For the 4PB fatigue test, the measured displacement highly depends on the properties of the loading frame. Calibration tests on the 4PB test setup were conducted to obtain the pure bending deflection of the beam. Comparison of the fatigue results obtained with the different test methods at the same test condition and loading mode shows that the fatigue life from the 4PB test is the longest and from the IT test is the shortest. Because of the homogeneous tensile strain field, the UT/C and IT fatigue results are not significantly influenced by the specimen size. However, the 4PB test results depend on the dimension of the used specimen, because the stress-strain field of the beam specimen varies along the length and cross section. The partial healing (PH) model was used to determine the relationship between the UT/C and 4PB fatigue results in strain-controlled mode. It is a material model that describes the evolutions of the stiffness and phase angle for a unit volume during testing. This implies that the model can be directly applied to the fatigue results obtained from the “homogenous” tests, the UT/C test, because the strain is uniformly distributed throughout the specimen. When analyzing the 4PB results, the backcalculated stiffness is not the local stiffness of the material but the so-called weighted overall stiffness of the whole specimen. Therefore a weighing procedure is required to calculate the weighted overall stiffness modulus from the local stiffness by taking into account the dimensions and the strain distribution of the specimen. By adjusting the model parameters, the PH model provides a good simulation for the evolutions of the stiffness and phase angle. All the model parameters can be expressed as functions of the applied strain level. The trends of the parameter δγ1 and δγ2 indicate the existence of an endurance limit. The predicted

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endurance limit obtained by the UT/C test is around 68 µm/m for the tested DAC 0/8 mixture, which does not change with specimen size and temperature. For the 4PB fatigue test, the local stiffness at different parts of the beam can be calculated by means of the PH model. The evolution of the calculated stiffness at the surface in the midsection of the beam is comparable with the UT/C fatigue results when the pure bending strain on the beam surface is equal to the tensile strain in the cylinder. Therefore the PH model offers a possibility to compare different fatigue results. In the second part of this research, the yield surface concept was applied to develop a new fatigue analysis approach. As a visco-elastic material, the yield surface of an asphalt mixture highly depends on temperature and strain rate. Therefore, monotonic uniaxial compression (MUC) and monotonic uniaxial tension (MUT) tests were performed at different temperatures and strain rates to derive such yield surfaces. For the three fatigue tests, the yield surface at the critical location of the specimen was determined in the I1-√J2 space. The fatigue results were then interpreted by comparing the actual stress condition with the yield surface. A new parameter R∆ was introduced as an indicator of the “safety against failure”. By comparing the R∆ values at the different locations of the specimen for the IT test, the weakest points are found at the locations with the maximum horizontal tensile strain, which are close to the loading strips, instead of the center of the specimen. A straight line was found by plotting R∆ at the critical location and the fatigue life on a log-log scale. Compared to the traditional fatigue analysis, the size effect on the fatigue results was excluded by using this new fatigue relation. For the stress-controlled mode, the fatigue lines obtained from the UT/C test show a good agreement with the IT fatigue results. Of course the influence of temperature and loading mode still exists in this new fatigue method. In the normal coordinate, when the fatigue life tends to infinity, R∆ becomes a constant value, denoted by Rlimit. The parameter Rlimit represents the endurance limit in a three-dimensional state. This value does not change with specimen size and test type but is influenced by the temperature and loading mode. Based on all the test results and their analysis in this research, it was concluded that the UT/C fatigue results are material properties and not influenced by specimen size. The PH model provides a good simulation of fatigue behavior of the asphalt mixture and is able to find the correlation between the UT/C and 4PB test results. The developed fatigue analysis approach characterizes the fatigue performance of asphalt mixtures in three-dimensional state, which is more close to the field situation. The endurance limit predicted by the new fatigue approach is independent of the specimen size and fatigue test type.

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Samenvatting Het vermoeiingsgedrag van asfaltmengsels wordt in het algemeen bepaald door middel van vermoeiingsproeven in het laboratorium. Wanneer de klassieke vermoeiingsanalyse wordt gebruikt, zijn vermoeiingslevensduren verkregen met verschillende test types niet vergelijkbaar, zelfs niet wanneer zij worden uitgevoerd bij dezelfde proefomstandigheden. Aangenomen wordt dat er twee belangrijke oorzaken zijn voor het verschil in vermoeiingsresultaten, namelijk het verschil in spanning-rek verdeling in de verschillende proef types en de manier waaropn de analyse van de resultaten wordt uitgevoerd. Dit onderzoek richt zich op de harmonisatie van de vermoeiings resultaten van een aantal methoden die worden aanbevolen door de Europese norm EN 12697-24. Het belangrijkste doel is om een correlatie tussen de verschillende vermoeiingsproeven te vinden en de klassieke vermoeiingsanalyse te verbeteren teneinde de werkelijke vermoeiingseigenschappen van asfaltmengsels zo goed mogelijk te kunnen bepalen. Om het hoofddoel van de studie te realiseren is een uitgebreid vermoeiingsonderzoek uitgevoerd op dicht asfaltbeton 0/8 (DAC 0/8). In het onderzoeksprogramma zijn uniaxiale trek/druk vermoeiingsproeven (UT/C), vierpuntsbuigingsproeven (4PB) en indirecte trekproeven (IT) uitgevoerd. Voor elk proeftype zijn proefstukken met verschillende afmetingen getest om het effect van de afmetingen op de vermoeiingsresultaten te onderzoeken. Om het testprogramma te beperken, zijn de proeven uitgevoerd met twee verschillende belastingswijzen (constante spanning en constante rek), bij twee temperaturen en met één frequentie. Voor de 4PB vermoeiingsproef is de gemeten verplaatsing sterk afhankelijk van de eigenschappen van het frame van de proeftopstelling. Om deze reden zijn kalibratieproeven op de 4PB opstelling uitgevoerd om de zuivere doorbuiging van de balk te kunnen bepalen. Bij dezelfde proefomstandigheden en belastingwijze, is de vermoeiingslevensduur in de 4PB proef het langste en die in de IT-test de kortste. Vanwege het homogene trekspanningsveld worden de UT/C en IT vermoeiingsresultaten niet significant beïnvloed door de afmetingen van de proefstukken. Echter, de 4PB resultaten zijn wel afhankelijk van de hoogte van de gebruikte balk, omdat het spanning-rekgebied in de balk varieert over de lengte- en dwarsdoorsnede. Het partial healing (PH) model is gebruikt om de relatie tussen de UT/C en 4PB vermoeiingsresultaten met constante rek te bepalen. Het is een materiaalmodel dat de ontwikkeling van de stijfheid en fasehoek van een volume-eenheid tijdens de vermoeiingsproef beschrijft. Dit impliceert dat het model direct toepasbaar is op de vermoeiingsresultaten van de "homogene" proeven, zoals de UT/C -test, omdat de rek gelijkmatig over het gehele proefstuk verdeeld is. Bij het analyseren van de 4PB resultaten, is de uit de maximale doorbuiging van de balk teruggerekende stijfheid niet de werkelijke stijfheid van het materiaal, maar een gewogen stijfheid van het gehele proefstuk. De werkelijke stijfheid van het materiaal varieert nl over de lengte en de

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hoogte van de balk. Daarom is een wegingsmethode nodig om de gewogen stijfheidsmodulus te berekenen uit de plaatselijke stijfheid door rekening te houden met de afmetingen en de rekdistributie in het proefstuk. Door het aanpassen van de modelparameters, kan met het PH-model de ontwikkeling van de stijfheid en fasehoek worden gesimuleerd. Alle modelparameters kunnen worden uitgedrukt als functie van de aangebrachte rekniveaus. De trends van de parameters δγ1 en δγ2 wijzen op het bestaan van een vermoeiingsgrens. De voorspelde vermoeiingsgrens voor de UT/C proeven is ongeveer 68 µm / m voor het geteste DAC 0/8 mengsel, en deze blijkt onafhankelijk te zijn van de afmetingen van het proefstuk en de temperatuur. Voor de 4PB proeven kan de ontwikkeling van de lokale stijfheid op iedere plaats in het proefstuk worden berekend met behulp van het PH model. De ontwikkeling van de lokaal berekende stijfheid aan het oppervlak in het middelste gedeelte van de balk is vergelijkbaar met de UT/C vermoeiingsresultaten wanneer de zuivere buigingsrek aan het oppervlak van de balk gelijk is aan de trekrek in de cilinder. Hiermeebiedt het PH-model de mogelijkheid om de vermoeiingsresultaten van verschillende types proeven met elkaar te vergelijken. In het tweede deel van dit onderzoek is het yield surface concept toegepast bij de analyse van de vermoeiingsresultaten. Voor een visco-elastisch materiaal zoals een asfaltmengsel, is de yield surface sterk afhankelijk van de temperatuur en de reksnelheid. Om deze reden zijn monotone uniaxiale druk- (MUC) en monotone uniaxiale trek (MUT) proeven uitgevoerd bij verschillende temperaturen en reksnelheden om de contouren van de yield surface te bepalen. Voor de drie types vermoeiingsproeven zijn de kritieke locaties van het proefstuk bepaald in de zogenaamde I1-√J2 ruimte. De vermoeiingsresultaten zijn vervolgens geïnterpreteerd door de werkelijke spanningstoestand te vergelijken met de driedimensionale yield surface voor de diverse proeftypes. Een nieuwe parameter R∆ isgeïntroduceerd als indicator van de "veiligheid tegen bezwijken". Door de R∆ waarden te vergelijken die voor verschillende locaties in het IT proefstukr zijn berekend zijn de locaties gevonden waar falen het eerst op zal treden. Deze bevinden zich dichtbij de belastingstrippen, in plaats van in het midden van het proefstuk. Op log-log schaal is een rechte lijn gevonden bij het plotten van R∆ op de kritieke locatie en de vermoeiingslevensduur. Vergeleken met de traditionele vermoeiingsanalyse, ishet effect van de afmetingen op de vermoeiingsresultaten geëlimineerd bij gebruik van deze nieuwe vermoeiingsrelatie. Voor de spanningsgestuurde proef laten de vermoeiingslijnen verkregen uit de UT/C -proef een goede overeenkomst zien met de IT vermoeiingsresultaten. Maar de invloed van de temperatuur en belastingswijze blijft aanwezig, ook in deze nieuwe analysemethode. Wanneer de vermoeiingslevensduur naar oneindig gaat blijkt R∆ een constante waarde aan te nemen, die aangeduid is met Rlimit. De parameter Rlimit vertegenwoordigt de endurance limit in de driedimensionale toestand. Deze waarde verandert niet met proefstukrgrootte en type proef, maar wordt wel beïnvloed door de temperatuur en belastingswijze. Op basis van alle resultaten en hun analyse in dit onderzoek kan worden geconcludeerd dat de eigenschappen zoals bepaald met de UT/C proef "echte" materiaaleigenschappen zijn die niet beïnvloed worden door de proefstukafmetingen. Het PH model simuleert het

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vermoeiingsgedrag van het asfaltmengsel uitstekend en met dit model kan de correlatie tussen de UT/C en 4PB proefresultaten worden bepaald. De yield surface benadering maakt het mogelijk de vermoeiingsprestaties van asfaltmengsels in een driedimensionale spanningstoestand te bepalen, vergelijkbaar met die welke in de praktijk optreden. De endurance limit die wordt voorspeld met de nieuwe vermoeiingsaanpak is onafhankelijk van de afmetingen van het proefstuk en het type vermoeiingsproef.

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List of Abbreviations HMA Hot Mix Asphalt 2PB Two-Point Bending 3PB Three-Point Bending 4PB Four-Point Bending ITFT Indirect Tensile Fatigue Test NAT Nottingham Asphalt Tester UT/C Uniaxial Tension and Compression IT Indirect Tension S-N Stress amplitude-Life AC Asphalt Concrete TRRL Transport and Road Research Laboratory CEN European Committee for Standardization PI Penetration Index CF Correction Factor ER Energy Ratio RDEC Ratio of Dissipated Energy Change PV Plateau Value GAC Gravel Asphalt Concrete PH Partial Healing 2D Two-Dimensional 3D Three-Dimensional ACRe Asphalt Concrete Response DAC Dense Asphalt Concrete SA Sand Asphalt CDAS Control Data Acquisition System UTM Universal Testing Machine FFT Fast Fourier Transform LVDT Linear Variable Differential Transformer CDAS Control and Data Acquisition System WLF Williams-Landel-Ferry DER Dissipated Energy Ratio MUC Monotonic Uniaxial Compression MUT Monotonic Uniaxial Tension EME Enrobé á Modele Elevé SMA Stone Mastic Asphalt GAC Gravel Asphalt Concrete MPRE Mean Percent Relative Error l.o.c. Level of Confidence HISS Hierarchical Single Surface FEM Finite Element Modelling GUI Graphical User Interface ABAQUS A Finite Element Package RAW Dutch standard Specification for the Civil Engineering Sector

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PReSBOX A Laboratory Asphalt Compactor COD Cracks Opening Displacement ASTM American Society for Testing and Materials AASHTO American Association of State Highway and Transportation Officials RILEM International Union of Laboratories and Experts in Construction Materials

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Table of Contents

Acknowledgements ............................................................................................................ i

Summary........................................................................................................................... iii

Samenvatting..................................................................................................................... v

List of Abbreviations ....................................................................................................... ix

Chapter 1 Introduction.................................................................................................. 1

1.1 Fatigue Damage in Asphalt Pavements ................................................................ 1

1.2 Fatigue Failure Mechanism................................................................................... 2 1.3 Evaluation of Fatigue Properties........................................................................... 3 1.4 Problem Statement and Objectives ....................................................................... 4

1.5 Organization of the Dissertation ........................................................................... 6 References................................................................................................................... 8

Chapter 2 Literature Review ...................................................................................... 11

2.1 Background of Fatigue Research on Asphalt Mixtures ...................................... 11

2.2 Laboratory Fatigue Test Methods....................................................................... 12

2.2.1 Simple Flexure Test ................................................................................. 12

2.2.1.1 Two-Point Bending (2PB) Test .................................................... 12

2.2.1.2 Three-Point Bending Test ............................................................. 13

2.2.1.3 Four-Point Bending Test............................................................... 14

2.2.1.4 Rotating Bending Test .................................................................. 15

2.2.2 Direct Axial Loading Test .......................................................................16

2.2.3 Diametral Loading Test (Indirect Tensile Test)....................................... 17

2.3 Influence of Test Type, Specimen Size and Test Conditions on Fatigue Results................................................................................................................................... 18

2.3.1 Influence of Test Type and Specimen Size.............................................. 19

2.3.1.1 Influence of Test Type .................................................................. 19

2.3.1.2 Influence of Specimen Size .......................................................... 21

2.3.2 Influence of Loading Mode ..................................................................... 22

2.4 Fatigue Analysis Approach................................................................................. 26 2.4.1 Results of Laboratory Fatigue Tests ........................................................ 26

2.4.2 Classical Fatigue Analysis ....................................................................... 29

2.4.3 Dissipated Energy Approach ................................................................... 34

2.4.3.1 Dissipated Energy Theory............................................................. 34

2.4.3.2 Cumulative Dissipated Energy ..................................................... 35

2.4.3.3 Dissipated Energy Ratio ............................................................... 37

2.4.4 Fracture Mechanics Approach ................................................................. 41

2.4.4.1 Theory........................................................................................... 41

2.4.4.2 Determination of Fracture Mechanics Parameters........................ 43

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2.5 Fatigue Prediction Model.................................................................................... 45 2.5.1 Partial Healing (PH) Model ..................................................................... 45

2.5.2 Weibull’s Theory ..................................................................................... 47 2.5.2.1 Survival Probability ...................................................................... 47

2.5.2.2 Calculation of Survival Probability .............................................. 48

2.5.2.3 Application in Fatigue Test........................................................... 50

2.5.3 Mechanical Damage Model ..................................................................... 52

2.5.3.1 Theory........................................................................................... 52

2.5.3.2 Size Effect of the Damage Model................................................. 53

2.5.3.3 Comparison between Model and Experiments ............................. 54 2.6 Summary............................................................................................................. 55 References................................................................................................................. 56

Chapter 3 Research Methodology .............................................................................. 65

3.1 Introduction......................................................................................................... 65 3.2 Research Methodology ....................................................................................... 66 References................................................................................................................. 68

Chapter 4 Mixture Design and Specimen Preparation ............................................ 69

4.1 Selection of the Mixture...................................................................................... 69 4.2 Mixture Design ................................................................................................... 71

4.2.1 Materials .................................................................................................. 71 4.2.2 Mixture Design ........................................................................................ 72

4.3 Specimen Preparation ......................................................................................... 74 4.3.1 Mixture Compaction ................................................................................ 74

4.3.2 Selection of Specimen Size......................................................................76

4.3.3 Specimen Preparation .............................................................................. 77

References................................................................................................................. 82

Chapter 5 Different Laboratory Fatigue Experiments............................................. 83

5.1 Introduction......................................................................................................... 83 5.2 Test Equipment ................................................................................................... 83

5.2.1 Uniaxial Tension and Compression (UT/C) Test .................................... 83

5.2.2 Four-Point Bending (4PB) Test ............................................................... 84

5.2.3 Indirect Tensile (IT) Fatigue Test ............................................................ 90

5.3 Calibration of the 4PB Test Equipment .............................................................. 90

5.3.1 Theory...................................................................................................... 91 5.3.2 Test Program............................................................................................ 93

5.4 Complex Modulus and Fatigue Tests .................................................................96

5.4.1 Complex Modulus Test............................................................................96

5.4.2 Fatigue Test.............................................................................................. 97 5.5 Data Processing................................................................................................... 98

5.5.1 UT/C Test................................................................................................. 98 5.5.2 4PB Test................................................................................................... 99 5.5.3 IT Test.................................................................................................... 101

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5.6 Test Results....................................................................................................... 102 5.6.1 Stiffness Results..................................................................................... 102 5.6.2 Fatigue Test Results............................................................................... 108

5.7 Summary........................................................................................................... 130 5.7.1 Test Setup............................................................................................... 130 5.7.2 Test Results............................................................................................ 131

References............................................................................................................... 132

Chapter 6 Application of Partial Healing Model on Strain Controlled Fatigue

Tests .................................................................................................................... 133

6.1 Introduction....................................................................................................... 133 6.2 Application of PH Model on UT/C Test Results .............................................. 133

6.2.1 PH Model Theory .................................................................................. 133

6.2.2 Determination of PH Model Parameters................................................ 139

6.3 Application of PH Model on 4PB Test Results ................................................ 144

6.3.1 Weighing Procedure for 4PB Test ......................................................... 144

6.3.2 Determination of PH Model Parameters................................................ 148

6.4 Correlation between UT/C and 4PB Fatigue Test Results ............................... 153

6.5 Conclusions....................................................................................................... 155 References............................................................................................................... 157

Chapter 7 Monotonic Uniaxial Tension and Compression Tests .......................... 159

7.1 Introduction....................................................................................................... 159 7.2 Test Equipment ................................................................................................. 159

7.2.1 Monotonic Uniaxial Compression Test ................................................. 159

7.2.2 Monotonic Uniaxial Tension Test ......................................................... 161

7.3 Test Condition................................................................................................... 162 7.3.1 Central Composition Rotatable Design ................................................. 162

7.3.2 Test Conditions for MUC and MUT Tests ............................................ 163

7.4 Test Procedure .................................................................................................. 164 7.4.1 Test Procedure for MUC Tests .............................................................. 164

7.4.2 Test Procedure for MUT Tests .............................................................. 165

7.5 Data Processing................................................................................................. 166 7.5.1 Stress and Strain..................................................................................... 166 7.5.2 Strain Rate, Maximum Stress, Tangent Stiffness and Onset of Dilation168

7.6 Test Results....................................................................................................... 169 7.6.1 MUC Test Results.................................................................................. 169

7.6.2 MUT Test Results .................................................................................. 175

7.7 The Unified Model............................................................................................ 177 7.8 Prediction of the Unified Model Parameters .................................................... 180

7.9 Conclusions....................................................................................................... 195 References............................................................................................................... 196

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Chapter 8 Yield Surface and Fatigue Tests ............................................................. 197

8.1 Introduction....................................................................................................... 197 8.2 Material Model Concept ................................................................................... 197 8.3 Determination of Model Parameters.................................................................202

8.3.1 Model Parameters R and γ ..................................................................... 202 8.3.2 Model Parameter n................................................................................. 205

8.3.3 Model Parameter α................................................................................. 205

8.3.4 Determination of Yield Surface ............................................................. 207

8.4 Critical Location ............................................................................................... 209 8.5 Determination of Yield Surface for Fatigue Test............................................. 212

8.6 Relationship between R∆ and Fatigue Life ....................................................... 215 8.7 Conclusions....................................................................................................... 225 References............................................................................................................... 226

Chapter 9 Conclusions and Recommendations....................................................... 227

9.1 Conclusions....................................................................................................... 227 9.1.1 Conclusions Related to Literature Review ............................................ 227

9.1.2 Conclusions Related to Fatigue Test Equipment................................... 227

9.1.3 Conclusions Related to Stiffness and Fatigue Results........................... 228

9.1.4 Conclusions Related to Partial Healing Model...................................... 228

9.1.5 Conclusions Related to Monotonic Test Results................................... 229

9.1.6 Conclusions Related to Yield Surface Approach................................... 230

9.2 Recommendations............................................................................................. 230 9.2.1 Recommendations Related to Experimental Work................................ 230

9.2.2 Recommendations Related to Partial Healing Model ............................ 231

9.2.3 Recommendations Related to Yield Surface Approach......................... 231

Appendix A Calculations for 4-Point Bending Test................................................... 233

Appendix B Determination for Prony Series Model Parameters............................. 239

Chapter 1 Introduction

1


1.1 Fatigue Damage in Asphalt Pavements Fatigue is defined as the phenomenon of deterioration of a material (reduction in stiffness and strength, ending in fracture) under repeated loading. Similar to other materials, the stiffness and strength of asphalt concrete decrease when it is subjected to repetitive loading [Pell, 1962]. Because of its cost efficiency, reduction in traffic noise, improved safety and comfort and so on, asphalt concrete has been widely used in pavement structures since the beginning of the last century [Hveem and Davis, 1950] [Hindley, 1971]. With the increase of traffic volume and weight, fatigue cracking of the bituminous layer has become one of the major distress modes in flexible road pavements associated with repeated traffic loads. Fatigue cracks decrease the structural capacity of the pavement and increase the maintenance cost. Furthermore, once fatigue cracks propagate through the entire asphalt thickness, water and aggressive agents can infiltrate into the unbound layers, which greatly accelerates the deterioration process. Therefore, understanding the fatigue cracking phenomenon and measuring the fatigue properties of asphalt concrete is essential for the design of flexible pavements. Figure 1-1 shows some examples of fatigue failure in pavements.

Figure 1-1 Fatigue cracks at the surface of asphalt pavement Fatigue cracking is associated with repetitive traffic loading and pavement thickness [Roberts et al., 1996] [McGennis et al., 1994]. Traditionally, it is believed that cracking initiates at the bottom of the hot mix asphalt (HMA) layers where the tensile stress is the highest, then migrates upward toward the surface where it shows up as one or more longitudinal cracks. This is commonly referred to as "bottom-up" or "classical" fatigue cracking. Researchers however found that in thick pavements (≥160 mm), the cracks most likely initiate from the top in areas of high localized tensile stresses resulting from tire-pavement interaction and asphalt binder aging. They then propagate down to a depth of approximately 50 mm; this is called top-down cracking [Molenaar, 1983] [Gerritsen, 1987] [Molenaar, 2004] [Uhlmeyer, 2000]. Up till now, the mechanism of initiation and progression of top-down cracks is not thoroughly understood. It is commonly assumed that high concentration of stresses at the tire-pavement contact surface is the major cause for these cracks.


2

1.2 Fatigue Failure Mechanism In order to have a better understanding of the fatigue cracking mechanism in asphalt pavements, it is necessary to look at how the wheel load is applied to a pavement structure. Figure 1-2 shows the stress states developed in a particular element of the pavement structure when subjected to a moving wheel load.

Figure 1-2 Stresses induced by a moving wheel load on a pavement element [Brown, 1978]

The moving wheel causes vertical, horizontal and shear stresses on an element underneath the wheel. As a result of the passing wheel load, the vertical compressive stress changes following a half sine wave, but in horizontal direction the element is subjected to a tensile or compressive stress alternatively. The bottom-up cracking is mainly caused by the horizontal tensile stress at the bottom of the asphalt layer. Being a visco-elastic material, the properties of asphalt mixtures are time dependent which will have an effect on the magnitude of the tensile strains developed in the structure. The loading time depends on vehicle speed and the depth below the pavement surface [Collop, 1995]. For example, at a velocity of 60 km/h, the loading time will be approximately 0.015 s at a depth of 150 mm. In addition to the loading frequency, environmental conditions, engineering properties of the asphalt concrete, the condition of underlying layers, and the pavement structure are all contributing factors to fatigue cracking. Roads do not crack immediately after traffic starts to use the pavement. It usually takes years or millions of load applications from vehicle tires. The sustained traffic loading results in a decrease in the structural strength of the pavement. If the tensile stress exceeds the local tensile capacity of the material, eventually cracking will occur. In the beginning, cracking manifests itself as a series of parallel longitudinal cracks (cracks along the direction of the flow of traffic) in the top layer of the asphalt pavement. These cracks are initially thin and sparsely distributed. If further deterioration continues, these

Stress

Moving Wheel Load

Pavement Structure

Typical Element

Horizontal Stress

Shear Stress

Vertical Stress

Shear Stress

Horizontal Stress (compressive)

Horizontal Stress (tensile at the bottom of stiff layer)

Vertical Stress (compressive)

Time

(a) (b)


3

longitudinal cracks are connected by transverse cracks forming many sided, sharp-angled pieces. This interlaced cracking pattern resembles chicken wire or the skin of an alligator. Figure 1-3 shows examples of the different levels of severity for fatigue cracking.

Figure 1-3 Alligator crack patterns of differing severity [Miller, 2003]

1.3 Evaluation of Fatigue Properties The fatigue properties of asphalt mixtures are important parameters in pavement design. In order to determine the fatigue resistance of asphalt mixtures, various fatigue tests are carried out in the laboratory at the stress levels, loading times, rest periods and temperature and moisture conditions as realistic as possible. Then the fatigue characteristics of an asphalt mixture derived from the laboratory tests are used as input in the design analysis to predict field performance.

Figure 1-4 Schematic demonstrating the main configurations of fatigue tests [Read, 1996] (a) two point bending; (b) four point bending; (c) three point bending; (d) rotating

bending; (e) direct axial loading; (f) direct axial loading (necked specimen); (g) diametral loading


4

Fatigue tests for asphalt mixtures were developed as early as the fifties of the last century, and different test configurations have emerged since that time. According to the mode of loading the most commonly used tests are classified as simple flexural tests (rotating cantilever, 2-, 3- and 4-point bending), direct axial loading test and diametral loading test [Read, 1996] [Tangella, 1990]. Figure 1-4 gives a schematic representation of the different fatigue tests. The arrows in the figure indicate the direction of the applied loading. A detailed description of each fatigue test will be given in the literature review. During fatigue testing, the specimen is subjected to a repeated load at a certain temperature and frequency. In the classical fatigue analysis, fatigue failure is determined based on the stiffness reduction. For the strain-controlled mode, the failure point is defined as the moment at which the stiffness of the specimen has reduced to 50% of its initial value. For the stress-controlled mode, the failure point is defined as the moment when the specimen has completely fractured. The results of fatigue tests can be interpreted in terms of a relationship between the stress or strain and the number of cycles to failure, which is represented by means of a straight line in a double logarithmic coordinate system [Pell, 1962].

bfN k δ−= ⋅ (1-1)

where: Nf : number of cycles to failure; δ : applied strain level [µm/m] or stress level [MPa]; k and b : coefficients related to the material properties.

1.4 Problem Statement and Objectives As mentioned in Section 1.3, various fatigue test devices are currently used to evaluate the fatigue performance of asphalt concrete. The two-point bending (2PB) test with trapezoidal specimens was adopted by researchers from Shell [van Dijk, 1975] and LCPC [Bonnot, 1986]. The Shell Laboratory at Amsterdam also has used the three-point bending loading equipment to estimate the fatigue life [van Dijk, 1975]. In the USA [SHRP, 1992] and the Netherlands [Pronk, 1996], the four-point bending test (4PBT) is specified. In the UK and Sweden, the standard fatigue test is the indirect tensile fatigue test (ITFT). The Nottingham Asphalt Tester (NAT) was specially designed for this test [Brown, 1995]. In European standard [EN 12697-24], three bending tests and the indirect tensile test are allowed as the standard fatigue test methods. Inevitably, they all give different results for the same material. Di Benedetto et al. reported an inter-laboratory investigation organized by the RILEM 182-PEB Technical Committee [Di Benedetto, 2004]. More than 150 fatigue tests were carried out using eleven different types of test equipment, including uniaxial tension/compression (UT/C), bending and indirect-tension tests. The classic fatigue relations obtained from the different fatigue tests are shown in Figure 1-5.


5

Figure 1-5 Classic fatigue relations obtained from the different fatigue tests [Di Benedetto, 2004]

Fatigue results are very sensitive to load conditions and the used test type. The Indirect Tension Test (ITT) shows the shortest fatigue life due to accumulation of permanent deformation in addition to fatigue damage. For a given strain amplitude, the beam tests generally result in longer fatigue life compared to T/C (Tension/Compression) tests. The conclusion was that fatigue test results obtained from different test equipment are not comparable to each other. It is therefore desirable to find a way to harmonize the fatigue results obtained from the different test methods and improve the classic fatigue analysis approach to better represent the actual fatigue characteristics of an asphalt mixture. In order to achieve these goals, the following objectives were defined in this research: (1) Compare the fatigue results obtained from representative fatigue test methods and

analyze their failure mechanisms. (2) Explore the influence of the specimen size on fatigue results. Three different sizes

(size 0.5, 1.0 and 1.5) are selected in this research, in which size 1.0 corresponds to the standardized size. Size 0.5 is a smaller one, a half of the size 1.0 and size 2 is twice larger than the standardized size.

(3) Develop a material model which is able to describe the fatigue behavior obtained

from the different fatigue tests. (4) Develop a new fatigue analysis approach to decrease if possible exclude the

influence of the test type and specimen size on the fatigue result.

80

1,E+09

1,E+08

1,E+07

1,E+06

1,E+05

1,E+04

1,E+03 100 110 ε0 (µm/m)

Nf50

3PB

4PB

2PB

T/C

ITT

Nf50 = α*ε^(-β)


6

1.5 Organization of the Dissertation This thesis consists of nine chapters. Chapter 1 gives a general introduction of the fatigue failure of asphalt pavements and the assessment of the fatigue characteristics in the laboratory. Problem statements and objectives are described. Chapter 2 provides a literature review on laboratory fatigue test methods and fatigue analysis models. In Chapter 3, the research methodology is presented. Chapter 4 gives a detailed description of the materials used in this study. Ample attention is given to the mixture design and specimen preparation. Chapter 5 describes the experimental work. Three different fatigue configurations, uniaxial tension and compression test, four-point bending test and indirect tension test, were conducted with different specimen sizes. The results of the fatigue tests and the interpretation techniques are also discussed. In Chapter 6, the Partial Healing model is applied to simulate the evolution of the complex modulus and the phase angle for the uniaxial tension and compression and the four-point bending fatigue tests in strain controlled mode. In Chapter 7, the results of the monotonic uniaxial tension and compression test are given at different strain rates and temperatures. In order to determine the yield surface, the unified model is used to calculate the strength and failure strain in both compression and tension. Chapter 8 describes the yield surface concept and its application on the fatigue results. A new parameter R∆ is introduced as safety factor and a new fatigue analysis approach is developed to reduce or exclude the influence of the test type and specimen size on the fatigue life. Chapter 9 presents the conclusions and recommendations.


7

Figure 1-6 Structure of the dissertation

Chapter 3 Research Methodology

Chapter 2 Literature Review


Chapter 4 Mixture Design and Specimen Preparation

Chapter 5 Different Laboratory Fatigue Experiments

Chapter 6 Application of Partial Healing Model on Strain Controlled Fatigue Tests

Chapter 7 Monotonic Uniaxial Tension and Compression Test

Chapter 8 Yield Surface and Fatigue Tests

Chapter 9 Conclusions and Recommendations


8

References Bonnot, J., Asphalt aggregate mixtures, Transportation Research Record 1096, Transportation Research Board, 1986, pp.42-50. Brown, S.F., Practical test procedures for mechanical properties of bituminous materials, Proceeding of ICE Transport 111, 1995, pp. 298-297. Collop, A.C, Cebon, D., A theoretical analysis of fatigue cracking in flexible pavements, J. Mech. Eng. Sci., IMech E, 1995, Vol 209, No C5, pp. 345-361. Di Benedetto H., de la Roche C., Baaj H., Pronk A., Fatigue of bituminous mixtures. Materials and Structures, 2004, vol. 37, n. 3, pp. 202-216. European committee for standardization, Bituminous Mixtures-Test Methods for Hot Mix Asphalt, BS EN 12697: Part 24: Resistance to Fatigue. CEN, Brussels, 2004. Gerritsen, A.H.; van Gurp, C.A.P.M.; van der Heide, J.P.J.; Molenaar, A.A.A. and Pronk, A.C., Prediction and Prevention of Surface Cracking in Asphaltic Pavements. Proceedings, 6th International Conference Structural Design of Asphalt Pavements, The University of Michigan. Ann Arbor, Michigan, July 1987, pp. 378-391. Hindley, G., A History of Roads, the Chausser Press Ltd., Bungay, Suffolk, ISBN 432-06-7361, 1971. Hveem, F. and Davis, H., Some Concepts Concerning Triaxial Compression Testing of Asphalt Paving Mixtures and Subgrade Materials, ASTM Special Technical Publication No. 106: Triaxial Testing of Soils and Bituminous Mixtures, American society for Testing Materials (ASTM), 1916 Race Street, Philadelphia, 1950. McGennis R.B., Anderson R.M., Kennedy T.W., Solaimanian M., Background of Superpave Asphalt Mixture Design and Analysis.Publication, No.FHWA-SA-95-003, 1994. Miller, J.S., Bellinger, W.Y., Distress Identification Manual for the Long-Term Pavement Performance Program, Publication No. FHWA-RD 03-031, June 2003. Molenaar, A.A.A., Bottom-Up Fatigue Cracking: Myth or Reality? in Proc of 5th International RILEM Conference, Limoges, France, 2004. pp. 275-282. Molenaar, A.A.A., Structural Performance and Design of Flexible Road Construction and asphalt Concrete Overlays, Ph.D Thesis, Delft University of Technology, the Netherlands, 1983. Pell, P. S., Fatigue Characteristics of Bitumen and Bituminous Mixes, Proceedings, International Conference on the Structural Design of Asphalt Pavements, 1962.


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Pronk, A.C., The theory of the four point dynamic bending test-Part 1. Report P-DWW-96-008, Rijkswaterstaat, the Netherlands, 1996. Roberts FL, Kandhal PS, Brown ER, Lee DY, Kennedy TW, Hot Mix Asphalt Materials, Mixture Design, and Construction. NAPA Education Foundation, Second Edition. 1996, pp. 603. Strategic Highway Research Program (SHRP), Fatigue response of asphalt-aggregate mixes, Executive summary, National Research Council, 1992. Tangella, S. R., Craus, J., Deacon, J. A. and Monismith, C. L., Summary report on fatigue response of asphalt mixtures, SHRP Report No. TM-UCB-A-003-A, 1990. Uhlmeyer, Jeff S., Willoughby, Kim. Top-down Cracking in Washington State Asphalt Concrete Wearing Courses.". Journal of the Transportation Research Board, 2000, Vol. 1730, pp.110-116. Uhlmeyer, J.S.; Willoughby, K.; Pierce, L.M. and Mahoney, J.P., Top-Down Cracking in Washington State Asphalt Concrete Wearing Courses. Transportation Research Record 1730. Transportation Research Board, National Research Council, Washington, D.C. 2000. pp. 110-116. van Dijk, W., Practical fatigue characterization of bituminous mixes, Proceedings of the Association of Asphalt Paving Technologists, 1975, pp. 38.


10


11

Chapter 2 Literature Review The literature survey on fatigue testing of asphalt mixtures in the laboratory is presented in this chapter. Since in this project, the test type and specimen size effect on the laboratory fatigue results are involved, this Chapter is mainly focused on the following issues. Section 2.1 provides the background regarding the fatigue research on asphalt mixture. In Section 2.2, the widely used laboratory fatigue tests for measuring fatigue behavior of asphalt mixtures are discussed. Some factors affecting fatigue results are summarized in Section 2.3. In Section 2.4, the different analysis approaches to characterize fatigue are reviewed, including the classical analysis, the dissipated energy approach and the fracture mechanics approach. Section 5 discusses some fatigue models which are used to describe the influence of test type and specimen size on the fatigue behavior of asphalt mixtures.

2.1 Background of Fatigue Research on Asphalt Mixtures The first fatigue study was initially done on metal. Richard [1988] stated that Wöhler [1860] conducted systematic investigations of fatigue failure in railroad axles for the German railway industry. His work also led to the characterization of fatigue behavior in terms of stress amplitude-life (S-N) curves. Fatigue considerations in the design of bituminous pavements date back to the early 1940s due to a significant increase in the traffic volume and the magnitude of wheel loads. Many flexible pavement researchers and designers expressed their concern over the fatigue failure of pavements under various loading conditions. Porter noted that flexible pavements failed under deflections as small as 0.02 to 0.03 inch (0.5-0.8 mm) [Porter, 1942]. In 1953, Nijboer and van der Poel showed that cracks which often appeared in the later stages of the life of an asphaltic concrete could be related to the bending stress induced by the moving traffic, exceeding the flexural strength of the material [Nijboer, 1953]. Examinations of the results of the AASHTO Road Test also revealed that cracking and initial failure of the pavement were primarily caused by repeated bending of the bituminous layers. Hveem F.N. [1955] reported fatigue failure caused by repeated loading on asphalt pavements built on highly resilient soils. Hveem concluded that there was a correlation between observations of cracking, fatigue type failures in bituminous pavements, and the measured repeated deflections that the pavement undergoes with each passing wheel. However, it is not fully clear which chemical, physical, and mechanical processes exactly occur in fatigue. It is generally known that, during the fatigue process, successive stages of deterioration occur. These are often named: initiation of microcracks, propagation of microcracks, initiation/formation of macrocracks or coalescence of microcracks into macrocracks, propagation of macrocracks, disintegration or rupture/failure. The use of fundamental, non-empirical mechanical tests to characterize the fatigue behavior of these


12

materials is particularly important when designing and predicting the in-service performance of asphalt mixtures.

2.2 Laboratory Fatigue Test Methods Over the past 60 years, a number of test methods have been developed to simulate the fatigue behavior of bituminous road construction materials. According to the mode of loading the most commonly used tests are classified in three groups, simple flexure, direct uniaxial loading and diametral loading tests, as mentioned in Section 1.3 [Tangella, 1990]. According to the stress-strain distribution in the specimen, the fatigue tests are divided into two types, the homogenous type and inhomogeneous type. If the stress-strain distribution is uniform throughout the specimen, the test is called homogeneous test. An example is the uniaxial loading test. For non-homogeneous tests, such as bending tests and the indirect tensile test, the stress-strain field is not uniform along the specimen and cross section.

2.2.1 Simple Flexure Test

2.2.1.1 Two-Point Bending (2PB) Test

Two-point bending tests on trapezoidal specimens were conducted by researchers of Shell [van Dijk, 1975], the Center of Road Research in Belgium [Verstraeten, 1972], and LCPC researchers [Bonnot, 1986]. Figure 2-1 illustrates the LCPC equipment. The smaller end is subjected to either a sinusoidal displacement [Bonnot, 1986] [van Dijk, 1975] [Verstraeten, 1972] or load [Kunst, 1989]. By properly selecting the dimensions of the trapezoid, the specimen will fail at about mid height where the bending stress is largest and not at the base where the bending moment is largest and the boundary conditions might adversely affect interpretation of test the results. Specimens tested by van Dijk had a base cross section of 55 mm by 20 mm, a top cross section of 20 mm by 20 mm, and a height of 250 mm.

Figure 2-1 2PB fatigue test machine with a trapezoidal specimen [Chkir, 2009] According to EN 12697-24 [EN, 2003], the specimens shall be of an isosceles trapezoidal shape as shown in Figure 2-2, for which the dimensions are given in Table 2-1. The large

Specimen

Displacement sensor


13

base of each specimen should be glued in a groove (about 2mm deep) of a metal base having a minimum thickness of 20mm, as shown in Figure 2-1. The specimen shall be moved sinusoidally at its head at the imposed displacement amplitude until the failure criterion has been reached. The deformation shall be such that at least one third of the element tests provide results with N ≤ 106 and at least one third of the element tests provide results with N ≥ 106.

Table 2-1 Dimensions of the 2PB specimen

Type of mixture D ≤ 14mm 14 < D ≤ 20mm 20 < D ≤ 40mm

B 56 70 70 b 25 25 25 e 25 25 50 h 250 250 250

D: maximum aggregate size of asphalt mixture

2.2.1.2 Three-Point Bending Test

At the Shell Laboratory in Amsterdam, van Dijk [1972] used the center-point loading equipment. The specimen dimensions are 30 mm (1.2 in.) × 40 mm (1.6 in.)× 230 mm (9.2 in.), tests were done in the controlled-deflection (strain) mode. Figure 2-3 shows the test apparatus and the scheme of three point bending test.

(a) (b)

Figure 2-3 Three point bending test apparatus scheme (a) and load characteristics (b)

B

h

e

e

b

Figure 2-2 Geometry of the specimens

F0

F0/2 F0/2 L/2 L/2


14

According to EN 12697-24 [EN, 2003], the dimensions of the test beams shall be (300 ± 10) mm × (50 ± 3) mm × (50 ± 3) mm. The specimen shall be clamped to the support mechanism through the two metallic tubes glued to one of its faces and to the piston rod through the tube glued to the opposite face. The support mechanism shall be capable of moving and tilting its axes. Specimens and extensometer are assembled and brought to the test temperature, 20 ºC. A cyclic sinusoidal displacement of the piston rod shall be applied. The wave frequency shall be 10Hz, and the values of the total amplitude usually range from 80 µm to 350 µm depending on the mixture.

2.2.1.3 Four-Point Bending Test

According to Molenaar [Molenaar, 1987], the specimens for the four-point bending test, can be obtained in the following way. Slabs of about 0.6 × 0.6 m2 are sawn from the AC pavement. From the bottom layer of these slabs, beams are sawn with dimensions of 450 mm × 50 mm × 50 mm, with the length perpendicular to the direction of traffic. For the four-point bending tests, a servo-hydraulic testing rig is used. The setup is schematically shown in Figure 2-4. The distance between the outer supports is 400 mm, between the inner supports 130 mm. The tests are executed with a displacement-controlled fully sinusoidal load signal at 30 Hz and the range of the test temperature is 0~20 ºC. Generally, different preset displacements are chosen, which will result in an expected fatigue life of 105 to 106 load repetitions.

Figure 2-4 Schematic diagram of the four-point bending test According to the ASTM standard D 7460 [ASTM, 1996], the dimensions of the test beam are 380 (length) × 50 (height) × 63 (width) mm. The horizontal spacing of the clamps is 119mm. Before testing, the specimen is placed in an environment which is at 20 ± 0.5 ºC for two hours. The desired initial strain (250 to 750 microstrain) and loading frequency (5 to 10Hz) are selected. The initial stiffness of the specimen is determined at the 50th load cycle, which is used as a reference for determining specimen failure. A deflection level (strain level) is selected such that the specimen will undergo a minimum of 10000 load cycles before its stiffness is reduced to 50 percent or less of the initial stiffness.


15

Figure 2-5 Scheme of the four-point bending test and load characteristics [ASTM, 1996]

2.2.1.4 Rotating Bending Test

(a) Controlled-Stress Rotating Flexure (b) Failure specimen

Figure 2-6 Flexure apparatus used by Pell [Pell, 1965] and specimen before and after testing [Saal, 1960]

At the University of Nottingham, U. K. [Pell et al., 1975 and 1973a] a rotating cantilever machine (Figure 2-6a) was used in which the specimen is mounted vertically on a

bearing Loading head

specimen

seat housing

bearings & electric motor

rev. counter stop switch

weights

tank

wire

pulleys

shaft

chuck


16

rotating cantilever shaft and a single point load is applied through a bearing at the top. This loading system results in a sinusoidally varying bending stress of constant amplitude at any particular cross section of the specimen. Under this system of loading there is no shear stress on the specimen. The specimens were tested in a controlled temperature bath. The majority of the tests were carried out at a temperature of 10°C and a speed of 1,000 rpm. Specimens of the shape have a minimum diameter at the neck of 2.5 in. Figure 2-6b shows a demoulded specimen ready for testing and also a broken specimen which has failed in fatigue after testing.

2.2.2 Direct Axial Loading Test

Raithby [Raithby, 1972a] developed this form of fatigue testing at the Transport and Road Research Laboratory (TRRL) in the United Kingdom. Axial tensile and compressive loading was applied using a servo-controlled electro-hydraulic machine. Specimens were prismoidal, with a 75 mm2 cross section and a length of 225 mm (Figure 2-7). Aluminum caps are glued upon the ends of the specimens, in such a way that these can be mounted in a servo-controlled hydraulic MTS testing machine. Loading frequencies were 16.7 and 25 Hz, and the effects of rest periods, shape of wave form, and the sequence of load application (compression/tension, tension/compression, compression only, and tension only) were evaluated.

Figure 2-7 Schematic representation of a direct axial fatigue test [Raithby, 1972a]

Molenaar [1983] selected the direct tensile test for crack growth experiments. The sketch and picture of the test set-up as used are given in Figure 2-8.

Temperature chamber

LVDT

Command signal

Hydraulic Power supply

Error signal amplifier

Summing junction

C.R.O.

Hydraulic actuator

Servo-valve

Load cell


17

Figure 2-8 Set-up of direct tensile test [Molenaar, 1983] The specimens used in the test were about 0.15 m long and the cross section was approximately 0.05 × 0.05 m. At mid length, an artificial crack was sawn at two opposite sides which had a depth of about 0.005 m. The beam specimens were sawn from slabs. An epoxy resin was used to glue the specimens to the top and bottom loading plate. To ensure a proper alignment of the specimens, hardening of the glue took place while the beam was positioned under the ram of the dynamic loading system. The shape of the load pulse was a haversine and a load to rest period ratio of 1 to 7 was used. During the tests the elastic vertical displacement was continuously recorded.

2.2.3 Diametral Loading Test (Indirect Tensile Test )

Indirect tension testing is done by applying a compressive force to a cylindrical specimen along its vertical diameter to produce tensile stresses perpendicular to the loading axis. Figure 2-9 shows a schematic of the indirect tensile test.

Figure 2-9 Schematic of indirect tension test

Pressure cell

steel plate epoxy resin

specimen

epoxy resin steel plate

LVDT Range ± 1 mm

holder

Loading strip Specimen

Compressive force


18

The test was developed and has been used since the early 1970s to characterize bituminous materials in terms of strength and elastic stiffness [Kennedy, 1968] [Schmidt, 1971]. Since that time the repeated load indirect tensile fatigue test is widely used to evaluate the fatigue properties of asphalt materials by a number of investigators [Kennedy, 1983] [Khosla and Omer, 1985] [Tangella, 1990]. In the UK this test has also been developed as a simple fatigue tool conducted as one of the test modules in the Nottingham Asphalt Tester (NAT) [Brown, 1995]. Normally a haversine load pulse is employed in the test and the diametral specimen is assumed to fail near the load line. Three types of failure patterns were usually observed [Sousa, 1991]: (1) crack initiation at or near the center of the specimen, resulting in complete splitting of the specimen; (2) crack initiation at the top of the specimen, progressively spreading downward in a V-shape, the arms of which originate from the outside edges of the loading platen; and (3) no real cracking occurs, with the specimen being plastically deformed beyond the limiting vertical deformation. Due to the absence of stress reversal, the accumulation of permanent deformation increases. The possibility that under high loads and/or high temperatures either compressive or shear failure occurs in the specimen. Fatigue fracture under indirect tensile loading ideally should occur by splitting of the specimen in two halves with minimum permanent deformation. Read and Collop recommended that the loading time should be 120 ms and test temperatures should be less than 30°C [Read, 1997].

2.3 Influence of Test Type, Specimen Size and Test Conditions on Fatigue Results The European Committee for Standardization (CEN) specifies several tests for characterizing the fatigue of bituminous mixtures, as follows:

1) Two-point bending test on trapezoidal shaped specimens

2) Two-point bending test on prismatic shaped specimens

3) Three-point bending test on prismatic shaped specimens

4) Four-point bending test on prismatic shaped specimens

5) Indirect tensile test on cylindrical shaped specimens

Figure 2-10 shows a schematic illustrating all of the allowed fatigue test methods. Most of these test methods are already discussed in Section 2.2. Because the European Standard does not impose a particular type of testing device, the choice of the test methods and the test conditions depends on the possibilities and the working range of the used device. This might lead to incomparable results obtained from the different test methods. Therefore it is necessary to investigate the influence of the test type, specimen size and some important test conditions on the fatigue results.


19

Figure 2-10 Schematics of the laboratory fatigue tests recommended by the European Committee for Standardization [EN 12697-24, 2003]

2.3.1 Influence of Test Type and Specimen Size

2.3.1.1 Influence of Test Type

A certain bituminous material may show different fatigue results dependent on the various geometries (beam, trapezoidal, cantilever beam, cylinder and other special shapes) and the different types of loading (bending, tension-compression and shear). Many researchers compared the results from different types of fatigue tests. Aguirre et al, found that fatigue lives are shorter in tension-compression than in bending at the same nominal strain, as shown in Figure 2-11 [Aguirre, 1981].

0.00001

0.0001

0.001

1000 10000 100000 1000000 10000000

N cycles

Initi

al s

train

Flexion (labo1)

Flexion (labo2)

Traction Compression(labo3)Traction Compression(labo4)

Traction Compression(labo5)

Figure 2-11 Fatigue behavior at constant stress amplitude (10°C, 10Hz) [Aguirre, 1981]

(1) (2) (3)

(4) (5)


20

Di Benedetto H. et al, reported results from an inter-laboratory investigation organized by the RILEM 182-PEB Technical Committee [Di Benedetto, 2004]. Eleven (11) different test types, including uniaxial tension/compression, 2-, 3- and 4-point bending and indirect-tension tests, were used to evaluate fatigue characteristics of a dense grade asphalt concrete mixture. The loading signal was specified as sinusoidal at a frequency of 10 Hz and a test temperature of 10 °C was chosen. Figure 2-12 shows fatigue results of the used 11 test types. The obtained fatigue lives are significantly influenced by the test method used. The Indirect Tension Test (ITT) shows the shortest fatigue life due to accumulation of permanent deformation in addition to the fatigue damage. For a given strain amplitude, the beam tests generally result in longer life durations compared to T/C (Tension/Compression) tests. The strain value, ε6, which represents the failure at one million cycles, is plotted in Figure 2-13. The ITT has the smallest strain value, ε6, but this value can not be compared with the other strain-controlled tests.

Figure 2-12 Fatigue lives for strain controlled fatigue tests (except ITT)

Figure 2-13 Strain amplitude “ε6” giving failure at 106 cycles from different fatigue tests

80

1,E+09

1,E+08

1,E+07

1,E+06

1,E+05

1,E+04

1,E+03 100 110 ε0 (µm/m)

Nf50

3PB

4PB

2PB

T/C

ITT

Nf50 = α*ε^(-β)

30

5070

90

110

130150

170

190

T/C

2PB 4PB

Average all

ITT


21

According to de la Roche et al., the fatigue life is longer in three-point bending than in two-point bending [de la Roche, 1993]. The relationship of ε6 between these two tests is:

6(3 )

6(2 )

1.4 2PB

PB

εε

= ∼ (2-1)

2.3.1.2 Influence of Specimen Size

Doan carried out two-point bending tests to study the influence of the size effect on fatigue evaluation of trapezoidal specimens with different sizes [Doan, 1973]. He found that the specimen size effect was negligible leading to the same average fatigue life for similar specimen series. Didier Bodin from LCPC, however, showed a significant effect of the trapezoidal beam size in bending fatigue tests [Bodin, 2006]. The fatigue line plots and ε6 of different sizes are presented in Figure 2-14 (a) and (b). Size 1.0 corresponds to the standardized size. Size 0.5, the smaller beam, has a longer fatigue life compared to size 1.0 and size 2.0, the bigger beam. The value of ε6 is also a function of specimen size and decreases with increasing size.

1000

10000

100000

1000000

10000000

100 140 180 220 260Strain anplitude (10 -6)

Fatig

ue li

fe N

F (5

0%

)

size0.5size1.0size2.0

100

120

140

160

180

200

0 0.5 1 1.5 2 2.5Size D/Do

ε 6 [1

0-6]

(a) (b)

Figure 2-14 Fatigue line fitted for each sample size (a) and values of ε6 versus the size of the specimen sets in two-point bending test [Bodin, 2006]

Figure 2-15 shows the influence of the width b of a specimen on the fatigue life, Nf [Jacobs, 1995]. At a chosen strain amplitude of 400 µm/m the ratio in fatigue life between a 50 and a 20 mm wide specimen is around 3.3; for a 250 mm wide specimen the ratio Nf(b=250)/Nf(b=20) is about 19.


22

Figure 2-15 Influence of the specimen width b on the fatigue life Nf [Jacobs, 1995] For the four-point bending tests on beams with a notch subjected to half sine load, Groenendijk developed the function of the number of load repetitions to failure based on the Paris’ law [Groenendijk, 1998]:

( )( )

0

1 /2

0.5 1.5 2.5 3.5 4.5/

/

1.99 2.47 12.97 23.17 24.8

f

nc h

nn n

c hmix

h d c hN

A S c c c c c

h h h h h

ε−

−= ⋅ − + − +

∫ (2-2)

Where: c0 : initial crack length, [mm]; cf : maximum crack length for which stable crack occurs,

[mm]; A,n : parameters depending on the material and on the

experimental conditions; c : crack length, [mm]; Smix : stiffness of asphalt mixture, [MPa]; ε : applied strain, [m/m]; h : height of the specimen, [mm].

2.3.2 Influence of Loading Mode

Normally one of two main modes of loading are applied in a fatigue test: The load (stress) controlled mode – the amplitude of the applied load is held constant

during the test.

b=250mm

b=150mm

b= 50 mm

b= 20 mm


23

The displacement (strain) controlled mode – the amplitude of the applied deformation is held constant.

Strictly speaking it is the load or displacement which is controlled and the stress and strain are calculated from them. However, controlled stress and strain are commonly used terms. Brown indicated that in the controlled stress mode, a repeated stress or load of constant amplitude is applied to a specimen which causes a gradual increase in strain at midspan and a gradual decrease of the stiffness [Brown, 1978]. The increase of strain is rather rapid at the end of the test, till complete fracture occurs. In controlled strain tests, the loading is applied such that the repeated deflection or strain stays constant during the test (see Figure 2-16). During the test the load will gradually decrease because of damage development. Most of the time no real failure will occur and it is therefore assumed that the specimen has failed if the load level has decreased to 50% of the original load level.

(a) (b)

Figure 2-16 Graphical representation of controlled stress (a) and controlled strain (b) modes of loading [Epps and Monismith, 1971]

Tayebali, et al [Tayebali, 1994] showed a typical plot (Figure 2-17) of the stiffness ratio (defined as quotient of stiffness at the ith load repetition to the initial stiffness, Si/S0) versus the number of load repetitions for flexural beam fatigue tests in both controlled-stress and controlled-strain modes of loading. Obviously the stiffness ratio in the controlled-stress mode decreases more rapidly compared to the stiffness ratio results in controlled-strain mode.


24

Figure 2-17 Stiffness ratio versus number of cycles, flexural beam fatigue controlled-stress and controlled-strain tests [Tayebali, 1994]

Figure 2-18 shows the variation of dissipated energy per cycle with the number of load repetitions. The dissipated energy per cycle decreases with an increasing number of load repetitions in the controlled-strain fatigue tests; whereas, for the controlled-stress tests, the dissipated energy per cycle increases as the number of load repetitions increases.

Figure 2-18 Dissipated energy per cycle versus number of cycles, flexural beam fatigue controlled-stress and controlled-strain tests [Tayebali, 1994]

As shown in Figure 2-19, the results from stress and strain controlled tests are different for the same asphalt mixture. Brown [1978] explained this phenomenon as follows: “cracks are initiated at the points of high stress concentration and propagate through the specimen until fracture occurs. The propagation of cracks depends on the stress intensity at the crack tip. In a strain controlled test, the stress gradually decreases with the decrease of the stiffness. Therefore strain controlled tests have a long period of crack propagation. In a stress controlled test, however, the crack propagation is very rapid because the stress is constant in the test”. Each of these modes has been linked to a particular pavement

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,E+02 1,E+03 1,E+04 1,E+05Number of cycles

Stif

fnes

s R

atio

Controlled-Strain Test

Controlled-Stress Test

0

0,02

0,04

1,E+02 1,E+03 1,E+04 1,E+05Num ber of cycles

Dis

sipa

ted

Ene

rgy

(psi

)

Controlled-Strain Test Controlled-Stress Test


25

construction and represents realistic service condition [Pell, P. S., 1973]. Monismith and Salam analyzed pavement structures by using multilayer elastic theory and found that:

(1) The strain controlled mode of loading is characteristic for thin asphalt pavements (≤ 50mm),

(2) The stress controlled mode of loading is more realistic for thick layers (≥ 150mm), (3) For a pavement of average thickness, an intermediate mode is appropriate [Monismith,

1979].

Figure 2-19 Graphical representation of the difference in fatigue response to various modes of loading for one mixture [Read, J. M. 1996]

Many researchers have evaluated the characteristics of these two modes of loading. A brief summary is given in Table 2-2. The results summarized in this table have been obtained from research reported in the following references [Pell, 1972] [Pell, 1973], [Tangella, 1990] [Kim, 1992], [Brennan, 1992], [Tayebali, 1994], [Monismith, 1971], [Francken, 1998].

Table 2-2 Comparison of results obtained with controlled stress and strain loading

Variables Controlled-stress test Controlled-strain test Usual failure criterion Failure of specimen Loss of half of initial force

Magnitude of fatigue life shorter Longer Scatter in fatigue test data Less scatter More scatter

Thickness of asphalt concrete layer

Thicker asphalt-bound layers

Thinner asphalt-bound layers,

< 3 inches Evolution during test Increase of displacement Reduction of force

Rate of crack propagation Faster Slower


26

Attempts have been made to determine the mode of loading that best simulates actual pavement conditions [Monismith and Deacon, 1969] [Monismith et al., 1977]. The type of loading is expressed by means of a mode factor (MF) defined in Equation 2-3:

A B

MFA B

−=+

(2-3)

where MF is the mode factor; A is the percentage change in stress; B is the percentage change in strain due to a fixed reduction stiffness. The MF assumes a value of -1 for controlled-stress conditions and + 1 for controlled-strain conditions.

2.4 Fatigue Analysis Approach

2.4.1 Results of Laboratory Fatigue Tests The measurements obtained during the test are the signals of the applied force and the displacement. The places where they are measured depend on the test device (see Section 2.2). Figure 2-20 shows the force and displacement signals for one loading cycle. The stiffness is calculated based on the amplitudes of the force and defection and the phase angle is obtained from the time lag between the load and displacement.

Figure 2-20 Relationship between force and displacement in one cycle Generally, asphalt materials are considered to be linear visco-elastic materials under relatively low strain applications. For the homogenous fatigue tests, the value of the stiffness is directly obtained from the stress, σ, and strain, ε, and for non-homogenous fatigue tests, the stiffness is calculated from the load and displacement using the specific factors. Table 2-3 shows the calculations of stiffness in different fatigue tests.

Displacement Force

Amplitude

Time

Time lag


27

Table 2-3 Calculation of the stiffness in different fatigue tests

Fatigue test Calculation of stiffness (kPa) Remarks

Two-point bending test [EN 12697-24, 2003]

2

1 22

0

(1 )l l

E K Kd b d

ω νδ

⋅= ⋅ ⋅ + +

⋅ ⋅

ω: horizontal load (kN); l: length of specimen (m); b: the base width (m); δ0: deflection at top of beam (mm); d: top width (m); b: thickness of the specimen (m); ν: assumed Poisson’s ratio; K1, K2: coefficients depending on the geometry of the specimen;

Three- point bending test [Read, J. M. et al, 2003] 34

t

t

LE

bh

σ ωε δ

= =

ω: horizontal load (kN); L: length of specimen (m); δ: maximum deflection at center of beam (m); b: thickness of the specimen (m); h: average specimen height (m);

Four- point bending test [ASTM standard D

7460,1996]

2 2

3

(3 4 )

4t

t

Pa L aE

bh

σε δ

−= =

σt: maximum tensile stress (Pa); εt: maximum tensile strain (m/m); P: load applied by actuator (N); b: average specimen width (m); h: average specimen height (m); δ: maximum deflection at center of beam (m); a: space between inside clamps; L: length of beam between outside clamps;

Uniaxial test t

t

Eσε

= σt: maximum tensile stress (Pa); εt: maximum tensile strain (m/m);

Indirect tensile test [EN 12697-24, 2003]

( 0.27)L

ED t

υ= × +⋅

L: peak value of applied vertical load (N); D: peak horizontal diametral resulting from the applied load (mm); t: mean thickness of the test specimen (mm); ν: assumed Poisson’s ratio.

The stiffness decreases under repeated loading during the fatigue tests. An example of stiffness evolution with the number of cycles is given in Figure 2-21. A typical fatigue process for asphalt mixtures is generally consisting of three distinctive response phases as reported in previous fatigue studies [Di Benedetto H. et al, 1996] [Di Benedetto, 2004].


28

Figure 2-21 Stiffness modulus versus load cycles evolution curve

Phase I -- adaptation phase: During this phase, the stiffness (or stiffness ratio) of the test specimen decreases rapidly due to repetitive loading. The decrease is possibly caused by the combined effects of fatigue damage, heating of the specimen and thixotropy. Microcracks are created in the binder films between the aggregates. Phase II -- quasi-stationary phase: The stiffness is decreasing slowly in this period where the role of fatigue on the stiffness decrease is dominant. Any artifact effects such as thermal heating and thixotropy can be considered to be small compared to the dominant effect of fatigue damage. Microcracks are in a stable growth phase. Phase III -- failure phase: Local crack propagation occurs in this phase. Macrocracks start to develop and failure is obtained at the end of this phase [Jacobs 1995]. The phase angle, φ, is calculated from the time lag between the load and displacement:

360 f tϕ = ⋅ ⋅ ∆ (2-4)

where φ = phase angle, in degrees; f = load frequency, Hz, and t∆ = time [s] lag between the maximum applied load, Pmax, and maximum deflection, δmax, (see Figure 2-20). For a purely elastic material the phase angle is 0°, and 90° for a purely viscous material. Figure 2-22 illustrates the fatigue test results plotted in a phase angle vs. stiffness diagram [Di Benedetto H. et al, 2004]. The gradual increase in phase angle is due to an increased viscous response of the material as damage accumulates in the form of micro-cracks and the test specimen experiences a reduction of its elasticity. However, once the specimen experiences significant structural changes due to macrocracking, it can no longer accumulate damage and consequently the phase angle decreases rapidly [Kim Y. et al. 2006].

Number of cycles

Adaptation Failure Quasi-stationary


29

Figure 2-22 stiffness vs. phase angle during fatigue test [Di Benedetto H. et al, 2004]

2.4.2 Classical Fatigue Analysis In order to evaluate and compare the fatigue properties of a bituminous material, it is necessary to define the failure of the specimen in a consistent way. The stiffness at the beginning of the test (N=50 to 100 cycles) is defined as the initial stiffness. For the controlled stress mode the specimen has a relatively short crack propagation period, so the failure point is close to the moment when the specimen has completely fractured, as shown in Figure 2-23. However, for the controlled strain mode the failure point is not easy to define, due to the long time of crack propagation during the test. Hence an arbitrary specimen fatigue life criterion has been defined. The point of failure is normally defined as the moment at which the stiffness has reduced to 50% of its initial value or when the applied stress is half of the initial stress [van Dijk, W., 1975, Moutier, F. et al, 1988, Bonnaure, F. et al, 1980].

Figure 2-23 Idealized representation of the failure points for the two modes of loading [Read, 1996]

Failure point

Half initial stiffness

Failure poin t in controlled strain mode

Controlled strain mode

Controlled stress mode

Stiffness

Crack propagation time associated with controlled strain mode


30

Rowe, et al, proposed another fatigue failure criterion [Rowe, 2000]. They stated that fatigue failure occurs at the maximum value of the nE/Einitial against n plot, illustrated in Figure 2-24. This definition of fatigue life is considered to be more accurate and reasonable than simply determining failure as an arbitrary condition such a 50% or 90% reduction of initial stiffness. The phase angle increases until a certain maximum value followed by a drop as loading continues (see Figure 2-24). This peak in phase angle has been successfully used as an indication of the fatigue failure point by some researchers [Reese, R. 1997] [Lee, H. J. et al.]. The failure point can also be identified based on the slope of stiffness change with load cycles. There are two rates of change in stiffness modulus against load cycles representing micro-cracking and macro-cracking, respectively, and a transition point between these two rates. This transition point represents the shift from micro- to macro-cracking [Rowe, G.M. et al. 2000] [Kim Y. R. et al. 2002]. This is also shown in Figure 2-24.

Figure 2-24 Comparison of different fatigue life criterions [Cocurullo, A. et al. 2008] Early fatigue research found that the fatigue life could be described using a Wöhler curve described by:

1log logN n kλ= + (2-5)

Where λ is the loading amplitude, either force (F) (or stress) amplitude or displacement (D) (or strain) amplitude. The most important variables from the fatigue test are the intercept and the slope of the fatigue curve, k1 and n respectively. The fatigue coefficients k1 and n are experimentally determined and are used in fatigue based mechanistic design procedures. A large range of values for n is reported. Typically n values are between 3 and 14 based on the database of


31

fatigue testing. Using the Wöhler relationship, as a basis for the description of fatigue results, some researchers found material depending characteristics for the parameters k1 and n. Finn et al. and Monismith et al. proposed a more general formula for the fatigue life as a function of maximum tensile strain and initial mix stiffness [Finn, 1977] [Monismith, 1985]:

1 1

a b

ft mix

N ASε

=

(2-6)

Where: tε = tensile strain, mixS = initial mix stiffness, fN = fatigue life, and A, a and b are

experimentally determined coefficients. This equation was used in the Shell [Shell, 1978] [Shell, 1985] and the Asphalt Institute [AI, 1981] design procedures; the coefficients were determined by considering the amount of cracking, mix type and the thickness of the asphalt-bound layer. The effects of the volumetric bitumen content (Vb) and the air void (Va) content on the fatigue performance of hot mix asphalt (HMA) were introduced by Pell and Cooper [Pell, 1975] as follows:

2 3 4

1

1 1K K K

bf

t mix a b

VN K

S V Vε

= + (2-7)

Where: mixS = stiffness modulus of the HMA mixtures,

bV = volume percentage bitumen content,

aV = air void content,

1K , 2K , 3K and 4K = experimentally determined coefficients.

Different models have been proposed by the Nottingham researchers [Brown et al., 1982] (Equation 2-8), Shell [Shell, 1978] (Equation 2-9), and the Asphalt Institute [AI, 1981] (Equation 2-10) to account for the effects of other factors on fatigue life:

( )&5.13 log 8.63log 15.8

14.39 24.2 40.7& 6

110

10

b R BV T

f b R Bt

N V Tε

+ −−

= ⋅ (2-8)

( ) 0.036 0.20.856 1.08t b mix fV S Nε − −= × + × (2-9)

( )3 3.291 0.85418.4 4.325 10f t mixN C Sε− − −= × (2-10)

Where: &R BT = ring and ball softening point temperature (°C),


32

C = 10M, 4.84 0.69b

b v

VM

V V

= ⋅ − +

,

Vv = volume of air voids. Bonnaure et al. employed a statistical approach using 146 fatigue curves from various fatigue tests and found the following equations in the form of initial strain and fatigue life (Equation 2-11 and 2-12) to determine the fatigue resistance of a mix. [Bonnaure, 1980] For displacement controlled tests:

( )5

1.85 6 1(4.102 0.205 1.094 2.707) 10f b b mix

t

N PI PI V V Sε

− = − ⋅ + − ⋅

(2-11)

For force controlled tests:

( )5

1.45 6 1(0.300 0.015 0.080 0.198) 10f b b mix

t

N PI PI V V Sε

− = − ⋅ + − ⋅

(2-12)

Where: PI = penetration index of the bitumen. Based on the results of 38 fatigue tests including tests on polymer modified mixtures, Medani developed the equation to predict the slope of the fatigue line, n [Medani, 2000].

masnn

CF= (2-13)

( )( )

2 2

log

log

mas

m

nm d S

d t

= = (2-14)

0.541 0.173 0.03524mas aCF n V= + − (2-15)

Where: nmas = n-value determined from the master curve, m = slope of the master curve of stiffness; Sm = mix stiffness, [MPa]; t = loading time, [s]; CF = correction factor. The equation to predict the intercept of the fatigue line k1 (Equation 2-16) was also developed using the results of 108 fatigue tests. The equation has successfully been used to predict the k1 values of 10 asphalt mixtures that were tested in the Strategic Highway Research Program [Medani, 2000].

1 &

3209log 6.589 3.762 2.332log 0.149 0.928 0.0721b

b R Bm a

Vk n V PI T

S V= − + + + + − (2-16)

These fatigue equations, mentioned above, are applicable for a specific value of the


33

applied tensile strain level, εt. For pavements, strains induced in the structure vary widely as a result of variations in the type of axles, their loaded weights, tire pressures, lateral wandering and so on. Accordingly, following the development of these laboratory fatigue relationships, Deacon [1965] demonstrated the applicability of a cumulative fatigue damage hypothesis by using it to consider the development of damage resulting from a range of strains. This relationship is termed the linear summation of cycle ratios, or Miner’s Law [Miner, M.A., 1945]. Miner’s hypothesis for cumulative fatigue damage has been widely used to predict fatigue cracking. The hypothesis is stated as follows,

1 2

1 2

... ...i m

f f if mf

n n n n

N N N N+ + + + + (2-17)

Where i = the i th level of applied strain at a critical point within the pavement structure, in = number of actual traffic load applications at strain leveli ,

ifN = number of allowable traffic load applications at strain leveli .

Fatigue failure is defined when the linear summation of cycle ratios reaches one. In spite of the fact that Miner’s law is difficult to verify for bituminous materials, it is widely used for pavement design purposes because of its simplicity [Peyronne et al, 1984], [Francken, 1979 and 1987].

Figure 2-25 Wöhler curve: Loading amplitude vs. number of loading cycles [Di Benedetto H. et al, 1997]

As shown in Figure 2-25, the material will have infinite fatigue life if the loading amplitude is maintained at levels below the endurance limit. It is also believed by some researchers that an endurance limit exists for an asphalt mixture. Monismith et al. suggested that an endurance limit exists for HMAC and proposed 70 µ-strain as a likely value [Monismith, 1971]. Carpenter et al. concluded that the endurance limit is in the range of 70 to 90 µ-strain at 20°C for a loading frequency of 10 Hz [Carpenter, 2003]. Based on the test results obtained from uniaxial tension-compression testing, Soltani postulated the presence of an endurance limit for two mixtures at 10°C and 10 Hz. The estimated endurance limits are 30 and 80 µ-strain for unmodified binder and modified binder, respectively.

Loading amplitude

Life duration N

Curve sometimes found

Endurance limit


34

2.4.3 Dissipated Energy Approach

2.4.3.1 Dissipated Energy Theory

Energy is dissipated in asphalt mixtures during loading and relaxation because the material behaves substantially visco-elastic at ambient temperatures. The dissipation of energy is demonstrated in Figure 2-26, which shows a comparison between a linear elastic material and a visco-elastic material.

Figure 2-26 Linear elastic versus visco-elastic behavior

For an elastic material, the energy stored in the system is equal to the area under the load-deflection curve and during unloading, all the energy is recovered. By contrast, a visco-elastic material, traces a different path when unloaded to that when loaded. This phenomenon is commonly known as “Hysteresis” and the energy dissipated is equivalent to the area within the loop. When a visco-elastic material, is sinusiodally loaded around a zero position, as in a bending beam fatigue test, a phase lag is observe between the load and measured deflection. If the load is plotted against the deflection, a hysteresis loop is obtained, as shown in Figure 2-27 [Francken L. and Clauwaert C. 1987].

Figure 2-27 Hysteresis loop obtained from plotting load versus deflection [Francken, 1987]


35

The area of the loop can be calculated and if the load-deflection relationship is expressed as stress-strain, the dissipated energy per loading cycle is obtained as follows:

sini i i iw π σ ε φ= (2-18)

Where: iw = dissipated energy in cyclei , iσ = stress amplitude in cyclei , iε = strain

amplitude in cyclei , iφ = phase lag in cyclei .

The dissipated energy is largely associated with viscous flow of the binder which dissipates the energy as heat. The dissipation of energy, more importantly, also relates to the formation of micro-cracks/crack surfaces [Little, 1995].

The amount of energy dissipated per loading cycle changes throughout a fatigue test. In a controlled stress test, the dissipated energy per loading cycle increases whereas in a controlled strain test it decreases, as illustrated in Figure 2-28. This is due to the fact that the dependent variable (strain or stress) and the phase angleiφ in Equation 2-18 do not

remain constant.

Figure 2-28 Variation of dissipated energy per load cycle during controlled stress and strain fatigue tests [Francken, 1987]

2.4.3.2 Cumulative Dissipated Energy

The earliest work using dissipated energy with asphalt materials was reported by Chomton [Chomton, 1972] and van Dijk et al. [van Dijk, 1972]. Chomton and Valayer [1972] presented a relationship in terms of “cumulative dissipated energy” versus number of loading cycles, as follows:

0

sini N

zi i i

i

W ANπ σ ε φ=

== =∑ (2-19)

Where: W = Cumulative dissipated energy to failure, A and z: experimentally determined coefficients.

Fatigue under controlled stress Fatigue under controlled strain


36

Chomton and Valayer presented the results for three materials and suggested that the parameters A and z could be independent of the mixture formulation. With this approach, the fatigue life could be predicted if the dissipated energy was determined for a given mix formulation. Dissipated energy captures both elastic and viscous effects and thus it is possible to predict the relative fatigue behavior of mixes in the laboratory from the results of fatigue tests when strain is the only test variable [Tayebali et al., 1994]. Van Dijk reported further work which demonstrated several important aspects [van Dijk, 1975] [van Dijk 1977]. It was clear that a single relationship could not be used for different materials, as shown in Figure 2-29. For a given mixture the relationship between the dissipated energy and the number of load repetitions to fatigue is valid, independent of the testing method and temperature. In addition, a ratio was developed which was related to the mixture stiffness.

initialW

WΨ = (2-20)

0 0 0sininitialW Nπ σ ε φ= ⋅ ⋅ ⋅ ⋅ (2-21)

Figure 2-29 Accumulative dissipated energy versus fatigue life for a series of mixture [van Dijk 1977]

This ratio was found to be larger than 1 for controlled strain tests and lower than 1 for controlled stress tests (see Figure 2-30). This is a result of decreasing dissipated energy per cycle in a controlled strain test compared to increasing dissipated energy per cycle in a controlled stress test. It can be seen from Figure 2-30 that as the stiffness of the materials increases, the ratio tends to approach unity. This is probably related to the increasing rate of crack growth that occurs with stiffer materials. Clearly, Ψ is a function of the mode of testing and the mixture stiffness.

1: Asphaltic concrete 2: Lean sand asphalt 3: Dense bitumen macdam 4: Gravel sand asphalt 5: Lean bitumen macdam 6: Dense asphaltic concrete 7: Dense bitumen macdam 8: Rolled asphalt base course 9: Grave bitumen 10: asphalt base course 11: Rich sand asphalt 12: Gravel sand asphalt 13: Bitumen sand base course

WFAT, J/m3

NFAT


37

Figure 2-30 Stiffness versus RatioΨ This work has been extended by Himeno et al. [1987] who developed the dissipated energy concept for three-dimensional stress conditions and applied it to the failure of an asphaltic layer in a pavement and by Rowe who has shown that dissipated energy can be used to accurately predict the life to crack initiation [Rowe, 1993].

2.4.3.3 Dissipated Energy Ratio

Figure 2-31 Illustration of energy ratio versus number of cycles in a controlled strain fatigue test

As discussed earlier, the number of cycles to failure is defined differently depending on the mode of loading. Hopman et al. proposed the use of an “Energy Ratio” to define the number of cycles (N1) in a controlled strain test, to a point where cracks are considered to initiate (defined as the merging of micro-cracks to form a sharp crack, which then propagates), as shown in Figure 2-31 [Hopman, 1989]. The “Energy Ratio” is defined as


38

follows:

0

i

n wEnergy Ratio

w= (2-22)

Where: n= number of cycles, 0w = dissipated energy at the start of test, iw = dissipated

energy in cyclei . The “Energy Ratio” plotted against number of cycles reveals a change in behavior at a number of cycles, N1. This point is considered to be the formation of a sharp crack. However, the “Energy Ratio” can be written as Rowe [Rowe, 1993]:

( )( )

0 0 0sin

sini i i

nE R

π σ ε φπ σ ε φ

= (2-23)

Where:

0σ , 0ε and 0φ = stress, strain and phase angle at the start of test,

iσ , iε and iφ = stress strain and phase angle in cyclei .

It is considered thatσ can be replaced by *E ε⋅ , in which *E is the complex stiffness modulus and the strain level is a constant for a controlled strain test. Then Equation 2-23 can be written as follows:

( )( )

* 2 *0 0 0 0 0

** 2

sin sin

sinsin i ii i i

n E n EE R

EE

π ε φ φφπ ε φ

= = (2-24)

The above equation has a constant term, *

0 0sinE φ , which can be removed without

changing the shape of the curve in Figure 2-31. Thus the equivalent ratio, Rε , is reduced

to the following form:

* "sini i i

n nR or

E Eε φ= (2-25)

Where: "iE is the loss modulus at cyclei .

The change insinφ is very small compared to the change in*E and does not significantly alter the shape of the curve in Figure 2-31. Hence, the equation of “Energy Ratio” can be further simplified [Rowe, 1993]:

*i

nR

Eε ≃ (2-26)

For a controlled stress test, the same method is used to simplify the equation, as follows:


39

*

iR n Eσ ≃ (2-27)

For controlled strain test data 1N is defined as the point at which the slope of the energy

ratio versus number of load cycles deviates from a straight line, as shown in Figure 2-32 (a), (b) and (c). The load amplitude remains constant in a controlled stress test and after crack initiation the stress at the crack tip increases rapidly. Fatigue life,1N , can be

determined from the peak of Rσ , as shown in Figure 2-32 (d), (e) and (f). It can be seen

that the 1N condition is very hard to define for controlled strain tests compared to

controlled stress tests. fN is the fatigue life defined by classical analysis. These two

parameters, 1N and fN , correspond to “crack initiation” and “failure”.

(a) (d)

(b) (e)

5000

4000

3000

1000

2000

4,E+05 3,E+05 2,E+05 1,E+05

0

0

Load Cycles

E* (MPa)

5,E+05

CONTROLLED STRAIN

1,E+03

8,E+02

6,E+02

4,E+02

2,E+02

0 0

Load Cycles 1,E+05 2,E+05 4,E+05 3,E+05

Rε CONTROLLED STRAIN

Load Cycles

4.0E+07

6.0E+07

0.0E+00 2,E+04 3,E+04 4,E+04 5,E+04 0 1,E+04

Rσ

1.4E+08

1.0E+08

2.0E+07

1.2E+08

8.0E+07

CONTROLLED STRESS

0

7000

6000

5000

4000

3000

2000

1000

1,E+04 0 2,E+04 3,E+04 4,E+04 5,E+04

E*(MPa)

Load Cycles

CONTROLLED STRESS


40

(c) (f) Figure 2-32 E*, Rσ and wi versus load cycles; trapezoidal controlled strain (left) and

controlled stress (right) fatigue tests [Rowe, 1993] The 1N point occurs generally in a range between 40% and 50% reduction of the initial

extensional complex modulus. In a controlled strain test, the dissipated energy, which is associated with crack propagation, is usually small. In a controlled stress test, the crack growth accelerates rapidly after initiation so life associated with crack propagation is also relatively small. This is illustrated in Figure 2-32 (c) and (f). The shaded area in the figures is the amount of dissipated energy associated with crack propagation. Thus, both fatigue tests give similar relationships for 1N and fN . In addition the dissipated energy

after the 1N stage should only be considered as an apparent energy. Consequently, the 1N

point is considered to be the preferred failure point, since the energy computed to the fN criterion will contain a certain amount of error.

Carpenter and Jansen [1997] showed in their study that the change in dissipated energy is responsible for fatigue damage. They found that the rate of change in dissipated energy versus load cycles to failure produces a unique curve regardless of loading mode and test condition. Therefore, Ghuzlan and Carpenter [Ghuzlan, 2000] defined a ratio of dissipated energy change (RDEC) based on the change in dissipated energy between load cycle i and load cycle i +1 divided by the dissipated energy in load cyclei :

( )1n n

n

DE DERDEC

DE+ −

= (2-28)

Where: RDEC = the ratio of dissipated energy change; DEn = the dissipated energy produced in load cycle n; DEn+1 = the dissipated energy produced in load cycle n+1.

0

800

1000

600

400

200

0 1,E+04 2,E+04 3,E+04 4,E+04 5,E+04

Load Cycles

wi (J/m3) CONTROLLED STRESS wi (J/m3)

1,E+05 2,E+05 3,E+05 4,E+05 5,E+05

200

250

150

100

50

0 0

Load Cycles

CONTROLLED STRAIN


41

The RDEC represents the percentage of dissipated energy causing damage to the material. Using this approach, the percent of input dissipated energy that goes into damage for a cycle can be directly determined during the fatigue test. This ratio is calculated approximately every 100 cycles. As introduced by Ghuzlan and Carpenter [Ghuzlan, 2000], the damage curve represented by RDEC vs. loading cycles can be distinctively divided into three stages. In stage II (plateau stage) the RDEC is almost constant until the dramatic increase in stage III which is the onset of true failure. The schematic is shown in Figure 2-33.

Figure 2-33 Typical RDEC plot with three behavior zones [Carpenter 2003]

The Plateau Value (PV), the nearly constant value of RDEC, characterizes a period where there is a constant percent of input energy being turned into damage. The failure is defined as the number of load cycles at which this ratio begins to increase rapidly. Carpenter and Shen [Carpenter, 2006] developed the relation between the PV value and the number of load cycles to failure Nf:

dfPV cN= (2-29)

Where: c and d are regression constants.

2.4.4 Fracture Mechanics Approach

2.4.4.1 Theory

The basic concepts of fracture mechanics as introduced by Griffith [Griffith, 1921] provide an alternative approach to define the fatigue properties of asphalt mixtures. In this method, fatigue is considered to develop in three phases [Ewalds and Wanhill, 1986]: (1) crack initiation; (2) stable crack growth; and (3) unstable crack propagation, as shown in Figure 2-34. It is assumed that the second phase consumes most of the fatigue life and, consequently, it is for this phase that quantitative models based on fracture mechanics have been proposed. The crack propagation phase was first described by Paris and


42

Erdogan [Paris, 1963]. From experimental data they found by means of regression analysis that the crack propagation rate dc/dN can be described following:

ndcAK

dN= (2-30)

Where: c= crack length (in mm); N = number of load repetitions; K = stress intensity factor describing the stress conditions near the crack tip. ,A n= parameters, depending on the material and on the experimental conditions (waveform, temperature, frequency etc.)

Figure 2-34 Fatigue crack growth

In phase 1, Kth is a threshold value for crack initiation. Above this value the crack growth rate increases rapidly to a constant value, as shown in Figure 2-34. Equation 2-30 is used to describe the stable crack growth process. In phase 3 the crack growth rate becomes infinite. The critical value Kc, called the fracture toughness, is a material property which is not dependent on the type of loading or on the dimensions of the specimen. The stress intensity factor K is dependent on the overall stress conditions and the geometry of the crack [Jacobs 1995]. Based on the experimental data of various Dutch mixtures, Molenaar [Molenaar, 1983] found that the A-value varies between 5×10-3 and 5×10-9 N/mm1.5 and the n-value between 2 and 5 for uniaxial tensile tests, frequencies between 1 and 10 Hz and temperatures between 5 °C and 25 °C. Generally three crack modes are considered, as shown in Figure 2-35 [Broek 1991]. (a) Mode I, the opening mode (tensile or pure bending stress);

(b) Mode II, the sliding or shearing mode (shear stresses);

(c) Mode III, the tearing mode (torsional stresses).


43

Figure 2-35 Three modes of loading to describe the crack growth phenomenon [Broek 1991]

In most cases combinations of crack modes occur in practice, especially for mode I and mode II. For the simple pure tension or bending case the stress intensity factor can be written as [Tseng and Lytton, 1990]:

qc

K c rb

σ = ⋅

(2-31)

Where: σ = nominal tensile or bending stress =E ε⋅ ; E = the elastic stiffness of the beam; ε = maximum strain in the specimen; c = crack length; b = width or height of the specimen; r, q = regression constants. At the moment of failure, K has reached a critical value, Kc. The equation for K will change if the displacement is kept constant during the test. Furthermore K might be influenced if a crack is initiated both at the bottom and the top of the specimen (sinusoidal loading).

2.4.4.2 Determination of Fracture Mechanics Paramet ers

Asphaltic materials show a significant viscous contribution to their behavior. To use the fracture mechanics approach in practice, it is necessary to determine A and n of Paris’ law by means of simple tests or nomographs. Schapery presented a crack growth theory for visco-elastic media for mode I cracking using the so-called generalized J-integral theory [Schapery, 1973] [Schapery, 1975] [Schapery, 1978]. Schapery developed a relationship between A and n and the measurable properties of the material:

( ) ( )

12 1

2 12

2 21 0

1

6 2

m t

m

m

DA w t dt

I

µπσ

∆ +

− = ⋅ ⋅ Γ

∫ (2-32)

11 2 m

KI

πασ

= (2-33)

Mode I Mode II Mode III


44

Where: mσ = maximum tensile stress the material can withstand before failure, MPa;

1I = a factor dependent on the stress conditions at the crack tip, the failure

stress and length of failure zone; α = size of the failure zone in front of the crack tip, mm; 1K = stress intensity factor for mode I loading, N/mm1.5;

2D = intercept of a line drawn tangent to the double-log creep compliance

(D(t)-D0) vs. time plot at t = 1 sec; D(t) = creep compliance, D(t)=D0+D2t

m , MPa-1; D0 = initial creep compliance at t=0 s, MPa-1;

µ = Poisson’s ratio; Γ = fracture energy defined as the work done on a material to produce a unit area of crack surface, N/mm;

( )w t = pulse shape of the stress intensity factor;

m = slope of the compliance curve; t = time, s; t∆ = period of one loading cycle, s. According to Schapery’s theory, the constants A and n from Paris’ law can be determined from simple tests or from nomographs [Heukelom and Klomp, 1964] [Bonnaure et.al 1977]. Also fatigue data at different temperatures and for different geometries can be predicted directly. Furthermore the influence of various material properties can be evaluated explicitly without using extensive series of fatigue tests. Several researchers have verified Schapery’s theory. Molenaar considered that differences exist between the experimentally determined n-values and the theoretical ones [Molenaar, 1983]. These differences are mainly caused by the limitations of the model. Schapery derived his theory for a homogenous material which is not consistent with asphalt mixtures because of the presence of air voids and aggregates. Especially air voids strongly influence the crack growth rate. For mixtures with a void content more than 3% n-values need to be corrected by regression analysis [Molenaar, 1983]. Based on the evaluation of crack growth tests on COD-measurements, similar findings were reported by Jacobs [Jacobs, 1995] who found that correction factors were needed in Schapery relationships to predict values for the parameters A and n adequately.

2n

m CF=

⋅ (2-34)

log 2 log log 2 log2 2m m mix

n nA d a b c Sσ= − − Γ − (2-35)

(log )

(log )mixd S

md t

= − (2-36)

( )( )0 1 2 3exp lnmix bit bit bitCF b b S b S b S S= + ⋅ + ⋅ + ⋅ (2-37)


45

where: a, b, c, d = regression coefficients depending on mix type (a=0.39, b=-0.04, c=0.62, d=-1.29 for dense asphalt concrete and stone matrix asphalt); b0, b1, b2, b3 = regression coefficients depending on mix type (b0=0.34, b1=-3.58×10-4, b2=-6.67×10-3, b3=1.01×10-4 for dense asphalt concrete and stone matrix asphalt); Smix = stiffness modulus of the mix determined from master curves; Sbit = stiffness modulus of bitumen backcalculated from Smix.

Erkens et al. and Sabha et al. did measurements on gravel asphalt concrete (GAC) and renewed the regression analysis. A new correction factor for GAC was expressed as [Erkens, 1995] [Sabha, 1995]:

5 9 20.89981 7.2814 10 7.1487 10mix mixCF S S− −= + × ⋅ + × ⋅ (2-38)

2.5 Fatigue Prediction Model

2.5.1 Partial Healing (PH) Model

The PH model is originally a material model and is based on the viscous elastic dissipated energy per cycle, fatW∆ , which is used to create micro defects and cracks

[Pronk, 2001]. fatW∆ can be regarded as a small portion of the dissipated energy disW∆ .

Pronk stated that the stiffness damage term Q is related to the fatigue consumption

fatW∆ [Pronk, 2005].

disfat dis dis

d d WQ W W Q W

dt dt Tδ δ

∆∴∆ ∴∆ ⇒ = ⋅ ≈ (2-39)

With T= time of one cycle. It is assumed that the stiffness damage Q affects both the loss modulus sinmixS ϕ⋅ and the

storage modulus cosmixS ϕ⋅ in which mixS is the complex stiffness modulus andϕ is the

phase angle. The two expressions of the PH model are given in Equation 2-40 and 2-41:

( ) ( ) ( )( )1 1

0

sin 0 sin 0t

tmix mix

dQ tS t t S e d

dβ τϕ ϕ α γ τ

τ− −= − +∫ (2-40)

( ) ( ) ( )( )2 2

0

cos 0 cos 0t

tmix mix

dQS t t S e d

dβ ττ

ϕ ϕ α γ ττ

− −= − +∫ (2-41)

The terms with the parameters 1,2α in the above equations will be ‘healed’ in time with

time decayβ . The terms with the parameter1,2γ will not heal. Pronk gave the complete

solutions of Equation 2-40 and 2-41 [Pronk, 2000]. From earlier tests [Pronk 1997], it


46

was found that in continuously loaded 4 point bending tests no or negligible healing occurs for the loss modulus. Therefore, the parameter 1α can be taken as equal to zero. An

example of a simulation of a 4PB controlled strain test is presented in Figure 2-36.

Figure 2-36 Data and model fitted for a 4PB controlled strain test at 172 µm/m [Pronk, 2005]

Table 2-4 shows parameter values, obtained from regression analysis. It can be seen thatβ is a function of the applied strain amplitude level and its value increases with increasing strain amplitude, as shown in Figure 2-37.

Table 2-4 Results of the regression from 4PB [Pronk, 2005]

Strain (µm/m) 2α δ⋅ (-) 2γ δ⋅ (-) β (10-5 s-1) 1γ δ⋅ (-)

137 618 36.42 111 6.6 177 618 36.42 146 6.6 217 618 36.42 248 6.6

Figure 2-37 Parameterβ as a function of the strain amplitude (4PB) [Pronk, 2005]


47

The PH model is also applied in strain controlled Tension/Compression (T/C) test on cylindrical specimens. According to Pronk [2005], the model parameters of T/C tests could be deduced from those obtained using 4PB tests. Therefore, the PH model can predict stiffness and phase angle evolution during fatigue tests for both 4PB and T/C tests.

2.5.2 Weibull’s Theory

In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Waloddi Weibull who described it in detail in 1951. The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. Pronk introduced Weibull’s theory into the research on fatigue tests of asphalt concrete [Pronk, 1998].

2.5.2.1 Survival Probability

Pronk [1999] defined the survival probability ( )1 iS x for a unit volume of material in an

uniaxial fatigue test. The parameter ix stands for the loading conditions at a certain

moment during the fatigue test. Survival probability means that the change from the fatigue crack initiation phase into the crack propagation phase has not yet occurred at that moment. This parameter will be between 0 and 1. In case of the Weibull distribution, the survival probability of a unit volume can be expressed as:

( ) ( ),

, .unit volume iC x

unit vol iS x e−= (2-42)

Where: ( ),unit volume iC x is defined as the "risk of change".

In ceramics rupture tests (increasing the stress to a level at which the specimen breaks) the function C, called the "risk of rupture", is defined as [Weibull, 1951]:

( ) 1

0

( )m

ii iC x R

σ σσ

σ−

= =

(2-43)

Where: σ is the applied stress, MPa; σ1 is the stress which there is a zero probability of failure, MPa; σ0 is the mean strength of the material, MPa; m is the Weibull modulus. In case of uniaxial fatigue tests in controlled strain mode, the following expression is supposed to be valid based on Equation 2-43:

( ),

mbi i

unit volume iv

NC x

Q

ε ⋅=

(2-44)


48

Where: iε is the applied strain level; iN is the fatigue life under strain level iε ; vQ is a

reference constant. The ‘risk of change’ for a volume of material is given by:

( ) ( ),volume i unit volume idC x C x dVolume= ⋅ (2-45)

( ) ( ),

mbi i

volume i unit volume iVvolume volume

NC x C x dVolume dVolume

Q

ε ⋅= ⋅ = ⋅

∫ ∫ (2-46)

The survival probability of a certain material volume is given by:

( )mb

i i

Vvolume

NdVolume

Q

volume iS x e

ε ⋅− ⋅

∫

= (2-47)

2.5.2.2 Calculation of Survival Probability

In Equation 2-47, the parameter m is called the Weibull modulus. If in the volume which is subjected to a certain stress, the loading conditions stay constant, such as in a uniaxial push-pull test, the survival probability can be integrated leading to:

( )mb

i i

V

NVolume

Q

volume iS x e

ε ⋅− ⋅ = (2-48)

The value of the parameter m can be determined by taking twice the natural logarithm of the reverse value of the survival probability:

( ) [ ]1log log log log

bi i

volume i V

Nvolume m

S x Q

ε ⋅= + ⋅

(2-49)

The survival probability itself can be calculated from the ranking sequence of the measurements. Two equations are possible:

( ) 11

jS j

T= −

− ;

0.3( ) 1

0.4

jS j

T

−= −

+ (2-50)

Where: T is the total number of measurements; j is the ranking according to the loading

condition b

i ii

v

Nx

Q

ε⋅= .


49

By plotting ( )1

log logS j

againstlog bi iN ε ⋅ , the slope of the linear regression is the

Weibull modulus m. If the survival probabilities of two different uniaxial tests are the same, the following requirement is fulfilled:

1 1 2 2m mVolume x Volume x⋅ = ⋅ (2-51)

In dynamic bending tests, such as a four point bending test, the maximum strain occurs at the surface of the beam and decreases linear to zero at the ‘neutral’ zone in the middle of the beam for tests in constant strain mode. Taking into account that the surface cracks have a significant effect on the initiating of microcracks, the "risk of change" can be calculated by:

, ,area volume unit area unit volume

area Volume

C C C C dArea C dVolume= + = +∫ ∫ (2-52)

( ),

mbi i

unit volume iV

NC x

Q

ε ⋅=

and ( ),

mbi i

unit area iA

NC x

Q

ε ⋅=

In order to calculate the "risk of change" of beams in a four point bending test, two assumptions are made: (1) For a beam with a total length 2L (x axis), a height 2H (z axis), a width 2H (y axis) and the first inner clamp at the distancex A Lα= = , the strain distribution is shown in Figure 2-38.

0

x z

A Hε ε= ⋅ ⋅

for 0 x A≤ ≤ and 0 z H≤ ≤

0

z

Hε ε= ⋅ for A x L≤ ≤ and 0 z H≤ ≤

Figure 2-38 The strain distribution in the beam in four point bending test

A

x

L L

H

y

-H

0

z compressive strain

tensile strain


50

(2) The tensile strains and compression strains equally in creating of fatigue damage. Integration over the total volume of the beam will give the following relationship:

( ) ( )

( )( )

2 2

, ,

0 0 0 0 0

02

4

1 18

1

y B y Bz H x A z H x L

unit volume i unit volume i

y z x y z x A

mbi

v

C x dxdzdy C x dxdzdy

mb NBHL

Qmb

α ε

= == = = =

= = = = = =

+

− + ⋅= ⋅ ⋅

+

∫ ∫ ∫ ∫ ∫ ∫

(2-53)

If taking into account the stressed surface area, the following equation is obtained:

( )( )( ) ( )( ) 0

2

81 1 1

1

mbi

A

L Nmb H mb B

Qmb

εα

⋅ ⋅ ⋅ − + ⋅ + + ⋅ +

(2-54)

Therefore the "risk of change" of beams in a four point bending test can be calculated by combining the two equations above.

( )( )

( )( )( ) ( )( )

02

02

1 18

1

81 1 1

1

bi

v

mbi

A

mb NC BHL

Qmb

L Nmb H mb B

Qmb

α ε

εα

− + ⋅= ⋅ ⋅ +

+

⋅ ⋅ ⋅ − + ⋅ + + ⋅ +

(2-55)

2.5.2.3 Application in Fatigue Test

Pronk carried out the four-point bending fatigue tests in controlled strain mode and compared the test results of beams with different mid span lengths of 90 mm, 130 mm and 190 mm [Pronk, 1999]. The beam dimensions were: a total length of 450 mm (with an effective length between two outer clamps of 400 mm), height of 50 mm and also a width of 50 mm. Figure 2-39 represents the relationships between fatigue life N1 and the applied constant strain amplitudes. Here the fatigue life N1 was determined by the dissipated energy ratio, as discussed in Section 2.4.3. Pronk found that there is little difference between the two Wohler curves for a mid span of 90 and 190 mm and the slope b of the Wohler curve for the standard mid span length of 130 mm is a little bit lower. The slopes of Wohler curves for the mid span of 90 and 190 mm are b1 = 4.471 and b2 = 4.455 respectively. The survival probabilities (calculated by Equation 2-48) are plotted for the measurements with a mid span length of 90 mm and 190 mm, as shown in Figure 2-40. The slope of the curves corresponds to the value for the Weibull modulus m.


51

Figure 2-39 Fatigue lines for different mid span lengths of beams [Pronk, 1999]

(a)

(b) Figure 2-40 Survival probability as a function of the parameter bN ε⋅ Fatigue

measurements with a mid span length of 90 mm (a) and 190 mm (b) [Pronk, 1999]

1000

100

10

200 100

j=1

j=2

j=3

….

j=15

T = 15


52

In order to be able to compare the fatigue lives for the two different mid span lengths, the survival probabilities should be the same. For the Weibull modulus m a mean value of 4.1 is adopted and for the coefficient b of the Wohler curves a mean value of 4.46 is taken. A. For the volume damage the following equation is obtained (Equation 2-53):

( ) ( )4.1 4.14.46 4.46

0 02 2

155 10518.3 1 1 18.3 1 1

200 200

18.3 1 18.3 1i iN Nε ε

⋅ − + ⋅ − + ⋅ ⋅ = ⋅ ⋅ + +

(2-56)

(mid span length of 90 mm) (mid span length of 190 mm)

According to this equation, at the same strain level the ratio in fatigue lives will be 1.15 for the same survival probability. B. For the surface damage the ratio in fatigue lives will be nearly the same: 1.17 (calculated by Equation 2-54).

2.5.3 Mechanical Damage Model

In order to describe the whole damage process during a fatigue test, LCPC has developed an approach based on mechanical damage modeling. The presented modeling and simulation have been performed by Bodin [Bodin, 2002]. This damage approach has been used to describe the local complex modulus decrease induced by microcrack development.

2.5.3.1 Theory

The mathematical model used to describe mechanical damage is an elasticity based damage model for fatigue, which is inspired by the non-local damage model proposed by Mazars and Pijaudier-Cabot [Mazars, 1984] [Pijaudier-Cabot, 1987]. Compared to existing proposals, e.g., by Lee et al. [Lee, 2000], the constitutive relations are simpler since they rely on a scalar, isotropic, damage model which is widely used for concrete and other quasi brittle materials. The influence of microcracking is introduced via a scalar damage parameter, d, that ranges from 0 to 1, which affects the Young’s modulus. The evolution of damage, d, depends on the amount of tensile stress that the material experiences during mechanical loading. The equivalent strain εɶ induced by a principal stress iσ is assumed to lead to damage growth as follows:

( )3

1 1i

i E d

σε +

=

= − ∑ɶ (2-57)

where

+ = Macauley bracket; iσ = principal stress; E = Young’s modulus.


53

For structural computation, the weighted average value of equivalent strain, ε , is used to replace the equivalent strain εɶ .

( ) ( ) ( ) ( )1

r

x x s s dsV x

ε ψ εΩ

= −∫ ɶ , with ( ) ( )rV x x s dsψΩ

= −∫ (2-58)

where Ω = volume of the structure; Vr(x) = representative volume at point x; and Ψ(x-s) = weight function:

( )2

2

4exp

c

x sx s

lψ

− − = −

(2-59)

and cl = internal length of the nonlocal continuum. For concrete, cl is often specified as

equal to three times the size of the largest aggregate fraction. The damage growth rate is defined as a function of the equivalent strain rate [Mazars, 1989]:

( )d f d βε ε+

=ɺ ɺ with ( )3 31

2

1 3 2 2

expd d

f d

α ααα α α α

−

= ⋅ (2-60)

where α1, α2, α3 and β are material parameters, and it can be demonstrated that –(β+1) corresponds to the slope of the Wohler fatigue curve. f(d) is an original law modeling the three phases of the loss of modulus curve during fatigue testing. The non-local damage model has been implemented into a finite element code "Castem 2000", and a specific so-called "jump in cycle" procedure is computed to allow calculations involving large number of cycles. More details on the construction of the model can be found in [Di Benedetto, 1999] and [Bodin, 2002 and 2003].

2.5.3.2 Size Effect of the Damage Model

For the size effect study, Bodin conducted two point bending tests with different specimen sizes [Bodin, 2002]. The specimens are 2D geometrically similar and their thickness is the same as shown in the Figure 2-2 and Table 2-1. Size 1 corresponds to the standardized size. The size effect is illustrated with the stiffness decrease curves. Figure 2-41 shows the results simulated at a strain level of 140×10-6 m/m. Model parameters α1, α2, α3 and β are fixed to take into account the experimental scatter of experimental data. The mean parameter set had been fitted for the average experimental data. The maximum and minimum parameters set had been fitted on experimental data. Compared to the standard size (size = 1), the stiffness decrease is faster for large samples (size = 2) and much slower for small samples (size = 0.5).


54

Figure 2-41 Simulated stiffness decrease calculated with the damage model fitted for the

strain level 140 µε [Bodin, 2002]

2.5.3.3 Comparison between Model and Experiments

Figure 2-42 Comparison of the model prediction with experimental data for the three specimen series for the strain level 140 µε [Bodin, 2006]

Model

Model

Model

Size = 0.5 Size = 1 Size = 2


55

To validate the nonlocal damage model, Bodin repeated the two-point bending tests using similar granular materials but manufactured with a different binder [Bodin, 2006]. Non local fatigue damage simulations are plotted in a gray zone superimposed to experimental data for the three tested sizes, as shown in Figure 2-42. It is observed that the fatigue life is well predicted for the size 2.0 and the difference between model and experiment grows with the decrease of the size. For the smallest size the model predicts failure at a larger number of cycles than it is observed in the lab.

Figure 2-43 illustrates the model fatigue lines with a visualization of the prediction zone given by the error bars for the three loading levels. Model agreement is better for large samples with a fatigue line very close to experimental data. For the other sizes the differences between model and experimental data grows with the decrease of the size.

Figure 2-43 Comparison of model predictions and experiment fatigue line for each size

2.6 Summary

Based on this literature review on fatigue of asphalt mixtures, the following findings can be reported: 1. Fatigue damage, a form of cracking resulting from repeated traffic loading, is recognized as one of the main failure modes of pavement structures. The fatigue characteristics of asphalt concrete mixtures should be known for the thickness design of asphalt pavements.

2. The fatigue response of a asphalt mixture is a complicated phenomenon and has not been understood completely. When asphalt mixtures are subjected to repeated loading, the fatigue cracks propagate gradually, leading to failure at last. However other phenomena are also involved in this process, like self-healing, thermal effects, permanent deformation and so on. It is important to describe these phenomena in a correct way.


56

3. Laboratory fatigue tests are widely used to simulate the situation in the field. However, accurate prediction and evaluation of fatigue is a difficult task not only because of the complex nature of the fatigue phenomena but also because of the fatigue testing itself, which is expensive and time-consuming. In principle, fatigue resistance should be a material property, but fatigue behavior tends to be very sensitive to material type, test characteristics and environmental conditions. Based on the fact that type of test, specimen size and mode of loading all affect the fatigue testing result, laboratory fatigue tests only provide the properties of specimens rather than of materials. Therefore large shift factors have to be applied to ‘adjust’ the laboratory data for practical pavement design purposes.

4. Different countries or institutions now apply different fatigue tests as the standard method for design. The experimental fatigue results obtained for the same asphalt mixture are different and difficult to compare and exchange with each other. For this reason, it is necessary to harmonize the existing test methods so that the evaluation of fatigue performance is more accurate and the test method can be reproduced.

5. The phenomenological approach, widely used to evaluate fatigue performance in the lab, has its limits, because the definition of fatigue life is based on an arbitrary choice. Failure in controlled strain has been defined as a 50% reduction in the initial stiffness modulus. The fatigue life defined by this traditional method is material and test conditions dependent and is not an intrinsic property of the materials. Therefore a new fundamental definition is needed for the fatigue life in order to harmonize the different fatigue tests.

References Aguirre, Morot, De La Taille, Doan Tu Ho, Bargiacchi, Smadja, Udron, Guay et Roncin, Etude comparée des essais de module complexe et de résistance à la fatigue des enrobés bitumineux - Bulletin de Liaison des Laboratoires des Ponts et Chaussées, N° 116, pp.33-43, 1981. ASTM stander D 7460, Standard test method for Determining the fatigue life compacted Hot-Mix Asphalt (HMA) subjected to repeated flexural bending, 1996. Bonnot, J., Asphalt Aggregate Mixtures. Transportation Research Record 1096, Transportation Research Board, Washington, D. C., 1986, pp: 42-50. Bonnaure, F.P., Gest G., Gravoir A. and Ugé P., A new Method of Predicting the stiffness of Asphalt Paving Mixtures, Proceedings AAPT, 1977, Vol.46, pp: 64-105. Bonnaure, F.P., Gravois, A. and Udron, J., A New Method for Predicting the Fatigue Life of Bituminous Mixes, the Association of Asphalt Paving Technologists (AAPT), 1980, Volume 49, pp 499-528. Brennan M.J., Lohan G. and Golden J.M., A laboratory study of the effect of bitumen content, bitumen grade, nominal aggregate grading and temperature on the fatigue


57

performance of dense bitumen macadam, Proceeding of the IVth International Rilem Symposium Budapest, Ed. Chapman & Hall. 1990. Brown, S. F., Pell P.S. and Stock A.F., The Application of Simplified, Fundamental Design Procedures for Flexible Pavements, Proceedings fourth ICSDAP, Ann Arbor, Michigan, 1977, Vol.1, pp: 327-341. Brown, S. F., Material characteristics for analytical pavement design, Developments in Highway Pavement Engineering-1, ed. P.S. Pell, London, 1978 . Brown, S.F., Gibb, J.M., Read, J.M., Scholz, T.V., Cooper, K.E., Design and Testing of Bituminous Material. Volume 2: Research Report, Submitted to DOT/EPSRC LINK Programme on Transport Infrastructure and Operations, 1995. Carpenter, S. H. and Jansen, M., Fatigue behavior under new aircraft loading conditions, In Aircraft/Pavement Technology: In the Midst of Change, Seattle, Washington. Edited by F.V. Hermann, American Society of Civil Engineers, New York, 1997, pp:259-271. Carpenter, S.H. and S. Shen., A Dissipated Energy Approach to Study HMA Healing in Fatigue. In Transportation Research Record: Journal of the Transportation Research Board, No.1970, TRB, National Research Council, Washington D. C., 2006, pp. 178-185. Chkir, R., Bodin, D., Piau, J.M., Pijaudier-Cabot, G., Gauthier, G. and Gallet, T., An inverse analysis approach to determine fatigue performance of bituminous mixes, Mech Time-Depend Mater, 2009, 13: 357–373. Chomton, G. and Valayer, P.J., Applied Rheology of Asphalt Mixes-Practical Applications, Proceedings, Third International Conference on the Structural Design of Asphalt Pavements, London, England, 1972, pp: 214-225. Di Benedetto H., Ashayer Soltani M.A. and Chaverot P., Fatigue damage for bituminous mixtures: a pertinent approach. Journal of the Association of Asphalt Paving Technologists, 1996, 65, 141–176. Di Benedetto H., de la Roche C., Baaj H., Pronk A. and Lundstrom R., Fatigue of bituminous mixtures. Materials and Structures, 2004, 37, No. 3, 202–216. Di Benedetto H., de la Roche C., Francken L., fatigue of bituminous mixtures: Different approaches and RILEM interlaboratory tests, Proceedings of the 5th International RILEM Symposium, Mechanical Tests for Bituminous Materials—Recent Improvements and Future Prospects, Lyon, 1997, 15–18. Di Benedetto H., Ashayer Soltani M.A. and Chaverot P., Étude rationnelle de la fatigue des enrobés: annulation des effets parasites de premier et second ordre, Int. Eurobitume Workshop, Performance related properties for bituminous binders 4p, 1999.


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Dider Bodin, Size effect regarding fatigue evaluation of asphalt mixtures, 2nd European Asphalt Technology Association Meeting, 2006. Dider Bodin, Modèle d’endommagement cyclique: Application à la fatigue des enrobés bitumineux, PhD thesis, EC Nantes, 2002. Dider Bodin, de La Roche, C. and Piau, J.M., A damage approach for asphalt mixture fatigue tests, Paper proposed to the 9th International Conference on Asphalt Pavements-Copenhagen, 2002. Dider Bodin, Chabot, A., de La Roche, C. and Pijaudier-Cabot, G., Endommagement par fatigue des matériaux bitumineux-Recherche de lois d’évolution de l’endommagement, Matériaux, Tours, France, 2002, pp:21-25. Dider Bodin, de La Roche, C., Piau, J.M. and Pijaudier-Cabot, G., Prediction of the intrinsic damage during bituminous mixes fatigue tests, 6th Internal RILEM Symposium: Performance Testing and Evaluation of Bituminous Materials, 2003. Erkens, S.M.J.G., and Moraal, J., A practical method to determine the fatigue characteristics of asphalt concrete, Report 7-95-117-1, DUT, Faculty of Civil engineering, Delft, the Netherlands, 1995. EN 12697-24: 2003 Bituminous mixture – Test methods for hot mix asphalt - Part 24: Resistance to fatigue. European committee for standardization, Brussels, 2003. Ewalds H.L. and Wanhill R.J.H., Fracture Mechanics, Co-publication of the Delftse Uitgevers Maatschappij and Edward Arnold Publishers, London, 1986. Francken L., Fatigue performance of a bituminous road mix under realistic test conditions. Transportation Research Record, 712, Washington, 1979, pp: 30-37. Francken L. and Clauwaert C. Characterization and structural Assessment of Bound Materials for Flexible Road Structures, Proceedings of the Sixth ICSDAP, Ann Arbor, Michigan, 1987 pp: 130-144. Francken L., Bituminous Binders and Mixes, Rilem Report 17, E&FN Spon. 1998. Finn, F.N., Saraf, C., Kulkarni, R., Nair, K., Smith, W. and Abdulah, A., The use of Distress Prediction Subsystems for the Design of Pavement Structures, Proceedings, Fourth International Conference on the structural Design of Asphalt Pavements, University of Michigan, Ann Arbor, 1977, Vol.1, pp: 3-38. Germann F.P. and Lytton R.L., Methodology for Predicting the Reflection Cracking Life of Asphalt Concrete Overlays, Report No. TTI-2-8-75-207-5, Texas Transportation Institue of the Texas A&M University, College Station, 1979.


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Ghuzlan, K, and S. H. Carpenter, Energy-Derived/Damage-Based Failure Criteria for Fatigue Testing, In Transportation Research Record: Journal of the Transportation Research Board, No. 1723, TRB, National Research Council, Washington D.C., 2000, pp. 141-149. Griffith, A.A., The Phenomena of Rupture and Flow in Solids, Philosophical Transactions of the Royal Society, London, Series A, 1921, Vol. 221. Hertzberg, Richard W., Deformation and fracture mechanics of engineering materials, 3rd edition, New York, publisher John Wiley and Son, 1989. Heukelom W. and Klomp A.J.G., Road Design and Dynamic Loading, Proceedings AAPT, 1964, Vol.33, pp: 92-125. Himeno, K., Watanabe, T., and Maruyama, T., Estimation of Fatigue Life of Asphalt Pavements, 6th International Conference on Structural Design of Asphalt Pavements (ISAP), Ann Arbor, 1987, pp 272-288. Hveem, F.N., Pavement Deflections and Fatigue Failures, Highway Research Board, Bulletin 114, Washington D.C., 1955. Hopman P.C., Kunst P.A. and Pronk A.C., A renewed interpretation method for fatigue measurements-Verification of Miner’s rule, Proc. of the 4th Eurobitume Symposium, Madrid, Spain, 1989. Jacobs, M. M. J., Crack Growth in Asphaltic Mixes, Ph.D. dissertation, Delft University of Technology, Delft, the Netherlands, 1995. Kennedy, T. W. and Hudson, W. R., Application of the Indirect Tensile Test to Stabilized Materials, Highway Research Record 235, Highway Research Board, Washington, D.C., 1968. Kennedy, T. W., Characterization of Asphalt Pavement Materials Using the Indirect Tensile Test, Proceedings of the Association of Asphalt Paving Technologists, 1977, Vol. 56. Kennedy, T. W. and Anagnos, J. N., Procedures for the Static and Repeated-Load Indirect Tensile Tests. Research Record 183-14, Center for Transportation Research, University of Texas at Austin, 1983. Khosla, N.P., Omer, M.S., Characterization of Asphaltic Mixtures for Prediction of Pavement Performance, TTR No. 1034, Transportation Research Board. Kim Y., Lee H. J., Little D. N. and Kim Y. R., A simple testing method to evaluate fatigue fracture and damage performance of asphalt mixtures, Journal of the Association of Asphalt Paving Technologists, 2006, Volume 75, 755-788.


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Kim Y. R., Little D. N. and Lytton R. L., Use of dynamic mechanical analysis (DMA) to evaluate the fatigue and healing potential of asphalt binders in sand asphalt mixtures, Journal of the Association of Asphalt Paving Technologists, 2002, Volume 71, 176-206. Kunst, P.A.J.C, Surface Cracking on Asphalt Layers, Working Committee B12, Hoevelaken, Holland, 1989. Lee, H. J., Kim, Y. R. and Lee, S. W., Prediction of Asphalt Mix Fatigue Life with Viscoelastic Material Properties. Transportation Research Record, No. 1832, 139-147. Lee, H. J., Daniel, J.S. and Kim, Y. R., Continuum damage mechanics-based fatigue model of asphalt concrete, Journal of Material in Civil Eng. 105, 2000, pp: 1005-1012. Little, D.N., Investigation of Microdamage Healing in Asphalt and Asphalt Concrete, Task K, Semi-Annual Technical Report Western Research Institute, FHW A Project DTFH61-92-C-00170-Fundamental Properties of Asphalt and Modified Asphalt, 1995. Mazars, J., Application de la mécanique de l’endommagement au comportement non linéaire et à la rupture des bétons de structure, Thèse de Doctorat d’État, Université Pierre et Marie Curie, France, 1984. Medani, T.O., Molenaar, A.A.A., Estimation of fatigue characteristics of asphaltic mixes using simple tests. Heron, 2000, Vol. 45(No. 3): pp. 155-166. Mazars, J. and Pijaudier-Cabot, G., Continuum damage theory – Application to concrete, International Journal of Engineering mechanics, 1989, vol. 115, num.2, pp: 345-365. Molenaar, A.A.A., Fatigue and Reflection Cracking due to Traffic Loads, Proceedings AAPT, 1984, Vol. 53, pp: 440-474. Molenaar, A.A.A., Structural Performance and Design of Flexible Road Construction and asphalt Concrete Overlays, Ph.D Thesis, Delft University of Technology, the Netherlands, 1983. Molenaar, J.M.M., Standard execution of the dynamic four-point bending test; Report MAO-R-87060; RHED; Delft, 1987. Monismith, C. L. and Deacon, J. A., Fatigue of Asphalt Paving Mixtures, ASCE Transportation Engineering Journal, 1969, Vol. 95:2, pp: 317-346. Monismith, C.L., Epps, J.A., Kasianchuk, and McLean, D.B., Asphalt Mixture Behavior on Repeated Flexure. Report No. TE 70-5, University of California, Berkeley, 1971. Monismith, C. L., Inkabi, K., McLean, D. B., and Freeme, C. R., Design Considerations for Asphalt Pavements, Report No. TE 77-1, University of California, Berkeley, March, 1977.


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Monismith, C. L., Fatigue Characteristics of Asphalt Paving Mixtures and Their Use in Pavement Design, Proceedings, 18th Paving Conference, University of New Mexico, Albuquerque, 1981. Monismith, C. L. and Salam, Y.M., Distress Characteristics of Asphalt Concrete mixes, Proceedings of the Association of Asphalt Paving Technologists (AAPT), 1979, Volume 42, pp: 320-349. Monismith, C.L., J.A. Epps and F.N. Finn, Improved Asphalt Mix Design, Proceedings, Association of Asphalt Paving Technologists Technical Sessions, San Antonio, Texas, 1985, pp: 347-406. Moutier, F., Duan, T.H. and Chauvin, J.J., The Effects of Formulation Parameters on the Mechanical Behaviour of Mixes, Proceeding of the Association of Asphalt Paving Technologists (AAPT), 1988, Volume 57, pp 213-242. Nijboer and van der Poel, A study of vibration phenomena in asphaltic road constructions, Proceeding of the Association of Asphalt Paving Technologists, 1953, No. 22, pp:197-231. Paris P.C., and Erdogan K. A Critical Analysis of Crack Propagation Laws, from: Transactions of the ASME, Journal of basic Engineering, Series D, 85, No. 3, 1963. Pijaudier-Cabot, G. and Bazant, Z.P. Non local damage theory, Journal of Engineering Mechanics 113 (10), 1987, pp: 1512-1533. Priest, A., Timm D., Methodology and Calibration of Fatigue Transfer Functions For Mechanistic-Empirical Flexible pavement design, NCAT Report 06-03, Auburn University, Alabama, 2006. Pell, P. S., McCarthy , P.F. and Gardner, R.R., Fatigue of Bitumen and Bituminous Mixes, International Journal of Mechanical Science, Pergamon Press Ltd, 1961, pp: 247-267. Pell, P. S., Fatigue Characteristics of Bitumen and Bituminous Mixes, Proceedings, International Conference on the Structural Design of Asphalt Pavements, 1962. Pell, P.S. Fatigue of Bituminous Materials in Flexible Pavements, Proc. Institution of Civil Engineers, 1965, Vol. 31. Pell, P. S. and Brown, S. F., The Characteristics of Materials for the Design of Flexible Pavement Structures, Proceedings, Third International Conference on the Structural Design of Asphalt Pavements, London, 1972. Pell, P. S. and Hanson, J. M., Behavior of Bituminous Road Base Materials under Repeated Loading, Proceedings Association of Asphalt Paving Technologists, 1973a.


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Pell, P. S., Characterization of Fatigue Behavior, in Structural Design of Asphalt Concrete Pavements to Prevent Fatigue Cracking. Special Report 140, Highway Research Board, 1973b, 49-64. Pell P. S. and Cooper, K. E., The Effect of Testing and Mix Variables on the Fatigue Performance of Bituminous Materials," Proc. The Association of Asphalt Paving Technologists, 1975, Vol. 44. Porter, O.J., Foundations for Flexible Pavements, Proceedings, Highway Research Board, Washington, D.C., 1942. Pronk, A. C., Evaluation of the dissipated energy concept for the interpretations of fatigue measurements in the crack initiation phase, The Road and Hydraulic Engineering Division, Ministry of Transport, Public Work and Water Management, The Netherlands, 1995 Pronk, A. C., Theory of the Four Point Dynamic Bending Test, Research Report, Ministerie van Verkeer en Waterstaat Dienst Weg en Waterbouwkunde, The Netherlands, 1996. Pronk, A. C., Healing during fatigue in 4 point dynamic bending tests, Report: W-DWW-97-095, DWW, Delft, The Netherlands, 1997. Pronk, A. C., Harmonisation of Bending Fatigue Tests: A(n) (Im)possibility? (In Dutch) Proc., Wegbouwkundige Werkdagen 1998. CROW, Ede, Netherlands. Pronk, A. C., Fatigue lives of asphalt beams in 2 and 4 point dynamic bending tests based on a ‘new’ fatigue life definition using the ‘Dissipated Energy Concept’ Phase IV: DAB 0/8 Weibull stressed volume effect, Report: W-DWW-99-070, DWW, Delft, The Netherlands, 1999. Pronk, A. C., Partial healing model-Curve fitting, Report: W-DWW-2000-047, DWW, Delft, The Netherlands, 2000. Pronk, A. C., Partial Healing in Fatigue Tests on Asphalt Specimen, Road Materials and Pavement Design, Vol. 4, Issue 4/2001. Pronk, A. C., Partial Healing, A new approach for the damage process during fatigue testing of asphalt specimen, Symp. Amer. Soc. Civ. Eng., Baton Rouge, USA, 2005. Raithby, K. D. and Sterling, A. B. Some Effects of Loading History on the Performance of Rolled Asphalt, TRRL-LR 496, Crowthorne, England, 1972. Read, J. M. and Collop, A. C., Practical fatigue characterisation of bituminous paving mixtures. Journal of the Association of Asphalt Paving Technologists, 1997, 66, pp: 74-108.


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Read, J. M., Whiteoak D. The Shell Bitumen Handbook, Fifth Edition, Shell Bitumen, U.K., 2003. Reese, R. Properties of aged asphalt binder related to asphalt concrete life, Journal of the Association of Asphalt Paving Technologists, 1997, Volume 66, pp: 604-632. Richard C.Rise, Brian N.Leis, Drew V.Nelson, Henry D.Berns, Dan Lingenfelser, M.R. Mitchell, Fatigue Design Handbook, Society of Automotive Engineers, Inc. 1988. Rowe, G.M., Performance of Asphalt Mixtures in the Trapezoidal Fatigue Test, Proceedings of the Association of Asphalt Paving Technologists (AAPT), 1993, Volume 62, pp: 344-384. Rowe, G.M. and Bouldin, M.G. Improved techniques to evaluate fatigue resistance of asphaltic mixes, Proceedings of the Second Euraphalt and Eurobitume Congress, Barcelona. Foundation Eurasphalt, Breukelen, The Netherlands, 2000, vol. 1, pp. 754–763. Sabha, H., Groenendijk, J. and Molenaar, A.A.A., Estimation of Crack Growth Parameters and Fatigue Characteristics of Asphalt Mixes Using Simple Tests, Delft University of Technology, Road and Railroad Research Laboratory, the Netherlands, 1995. Schapery R.A., A Theory of Crack Growth in Visco-Elastic Media, Report MM 2764-73-1, Mechanics and Materials Research Centre, Texas A&M University, 1973. Schapery R.A., A Theory of Crack Initiation and Growth in Visco-Elastic Media; I: Theoretical Development, II: Approximate methods of analysis, III: Analysis of Continuous Growth, from: International Journal of Fracture, Sijthoff and Noordhoff International Publishers, 1975, Vol.11, No.1, pp: 141-159, Vol.11, No.3, pp: 369-388, and Vol.11, No.4, pp: 549-562. Schapery R.A., A Method for Predicting Crack Growth in Non-homogeneous Visco-Elastic Media, from: International Journal of Fracture, Sijthoff and Noordhoff International Publishers, 1978, Vol.14, No.3, pp: 293-309. Schmidt, R. J., "A Practical Method for Measuring the Resilient Modulus of Asphalt - Treated Mixes," Highway Research Record No. 404, Highway Research Board, pp 22-32. Shell pavement design manual – asphalt pavements and overlays for road traffic, Shell International Petroleum Company Limited, London, 1978. Soltani A. Comportement en Fatigue des Enrobes Bitumineux, Doctoral Dissertation, Ecole Nationale des Travaux Publics de l’Etat-INSA, Lyon, France, 1998.


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Symposium on flexible pavement behavior as related to deflection, Proceedings of the Association of Asphalt Paving Technologists, 1962, Vol. 31, pp: 208-399. Tangella S. R., Craus J., Deacon J. A. and Monismith C. L., Summary report on fatigue response of asphalt mixtures. SHRP Report TM-UCB-A-003A-89-3 for project A-003-A, University of California, Berkeley, 1990. Tayebali A. A., J. A. Deacon, J.S. Coplantz, F.N. Finn, and C.L. Monismith, Fatigue Response of Asphalt-Aggregate Mixes, SHRP-A-404, Strategic Highway Research Program, National Research Council, Washington, D. C., 1994. Tseng K.H. and Lytton R.L., Fatigue Damage Properties of Asphaltic Concrete Pavements, Texas Transportation Institute of the Texas A&M University, submitted to the 69th Annual Meeting of the Transportation Research Board, Washington D.C., 1990. van Dijk, W. Moreaud, H., et al., The Fatigue of Bitumen and Bituminous Mixes, 3th International Conference on Structural Design of Asphalt Pavements (ISAP), London, 1972, Volume 1. van Dijk, W., Practical Fatigue Characterization of Bituminous Mixes, Proceedings The Association of Asphalt Paving Technologists, 1975. van Dijk, W. and Visser, W. The Energy Approach to Fatigue for Pavement Design, Association of Asphalt Paving Technologists (AAPT), 1977, Volume 46, pp: 1-41. Verstraeten, J., Moduli and Critical Strains in Repeated Bending of Bituminous Mixes, Application to Pavement Design, Proceedings 3rd International Conference on the Structural Design of Asphalt Pavements, London, 1972. Weibull, W., A statistical Distribution of Wide Applicability. J. Appl. Mech., 18, 1951, pp. 293-296. Wohler, A., Versuche uber die Festigkeit der Eisenbahnwagenachsen, Zeitschrift fur Bauwesen,10; English summary (1867), Engineering, 1860, Vol.4, pp.160-161.


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3.1 Introduction The literature review on the fatigue of asphalt concrete presented Chapter 2 reveals that the assessment of the fatigue characteristics of asphalt mixtures is a very complicated process and affected by many influence factors. The fatigue characteristics of an asphalt mixture essentially relate to the material properties and volumetric composition of the mixture. In mechanistic pavement design procedures, fatigue tests are normally used to simulate realistic traffic loading and environmental conditions, hence the fatigue characteristics are estimated in the laboratory in a simplified way. To evaluate the fatigue life in the field, more influence factors should be taken into account, such as pavement structure, rest period, lateral wandering, environmental factors, etc [Shell, 1978] [Groenendijk, 1998] [Al-Qadi, 2003]. Shift factors are therefore applied to laboratory results to be able to predict the fatigue life in the field [Lytton, 1993] [Molenaar, 1994]. The process of the fatigue estimation in pavement design is schematically described in Figure 3-1.

Figure 3-1 Assessment of fatigue properties of asphalt mixture and the influence factors

From earlier statements (see Chapter 2), it is clear that laboratory fatigue results depend on the type of fatigue test, specimen size and mode of loading. Until now there is no unified fatigue test method according to testing specifications implemented by many countries or organizations [EN 12697-24:2004] [ASTM D7460-10]. Various types of fatigue tests with their advantages and disadvantages have been used to obtain a

ε

Lab Nf

ε

Field Nf

Asphalt mixture

• aggregate properties • bitumen properties • gradation • volumetric properties: density, void content

• test type • specimen size • test conditions • loading mode

• pavement thickness • rest period • environment • lateral wander

Laboratory fatigue test Field fatigue damage

Influence factors:

Shift factor


66

laboratory fatigue life. With these different fatigue tests, similar conclusions might be found in qualitative analysis for a series of asphalt materials, but there is a large difference in quantitative results for the same mixture and this difference is difficult to be compared with each other. The goal of this research is therefore to determine the influence of the test type and specimen size on laboratory fatigue test results, understanding why these differences occur and developing a procedure to make the results, obtained from different tests, comparable. A major achievement of this study would be if the results help in harmonizing the current European standard on fatigue testing.

3.2 Research Methodology In order to achieve this goal, it was decided to perform uniaxial tension and compression (UT/C) fatigue tests, four-point bending (4PB) fatigue tests and indirect tensile (IT) fatigue tests. Furthermore it was decided that for each fatigue test specimen with different sizes had to be tested to explore the size effect on the fatigue results. It was also decided to conduct the tests in two different loading modes, namely constant load and constant displacement. In order to limit the test program, the tests were performed at only two temperatures and one frequency. By means of the testing program, the influence of test type, testing mode and specimen size can be determined but in itself the test results do not give an explanation why different tests give different test results. Therefore a specific model called the “Partial Healing model” developed by Pronk will be used in an attempt to explain differences between the displacement controlled UT/C and 4PB fatigue test [Pronk, 2001] [Pronk, 2012]. As the name of the model implies, it claims to qualify and quantity healing of a mixture and therefore the PH model will also be used to determine at which stress/strain level damage development is balanced with loading. At such a strain level no damage would occur as a result of repeated loading. This strain level, called “endurance limit”, is a very important parameter in the design of heavy loaded, long life pavements, for which no structural maintenance is needed. Fatigue results are normally obtained from tests in which the specimens are subjected to a uniaxial or at best bi-axial stress condition. Such uni- and bi-axial stress conditions however never occur in pavement where the stress conditions are always tri-axial. It was therefore decided to devote part of this research to apply the yield surface concept to the fatigue results in a similar way as it is done for stone and unbound granular materials. Extensive research at the Road and Railway Engineering Section of the Delft University of Technology on unbound materials has shown that permanent deformation will be limited if the occurring stress condition represented by Mohr’s circles are limited to around 40% of the stress at failure which can be derived from the yield surface. Since it is believed that a similar approach is also applicable to characterize the fatigue behavior of asphalt mixtures, it was decided to perform monotonic compression and tension tests to derive such yield surface as a function of temperature and strain rate. The failure results will then be interpreted by comparing the actual stress condition with the yield surface


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and an endurance limit will be developed. This endurance limit is then also valid for the 3D state of stress at which no fatigue will occur. Based on the selected research approach, the general research layout is shown in Figure 3-2.

Figure 3-2 Schematic of the research program framework

A new fatigue relationship R∆ and fatigue life • Influence of specimen size • Influence of temperature • Influence of loading modes

Mixture design and specimen preparation

Dynamic testing

• Uniaxial tension/compression test • Four-point bending test • Indirect tensile test

The PH model: • Influence of specimen size • Influence of temperature

Stiffness test Fatigue test

Monotonic testing

Monotonic uniaxial tension test

Monotonic uniaxial compression test

Yield surface

Unified model


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References ASTM D7460-10 Standard Test Method for Determining Fatigue Failure of Compacted Asphalt Concrete Subjected to Repeated Flexural Bending, 2010. Al-Qadi, I.L., Nassar, W.N., Fatigue Shift Factors to Predict HMA Performance, International Journal of Pavement Engineering, Volume 4, Number 2, June 2003 , pp. 69-76. EN 12697-24: 2003 Bituminous mixture – Test methods for hot mix asphalt - Part 24: Resistance to fatigue. European committee for standardization, Brussels, 2003. Groenendijk, J., Crack Growth in Asphaltic Mixes, Ph.D. dissertation, Delft University of Technology, 1995. Lytton, R.L., Uzan, J., Fernando, E.M., Roque, R., Hiltunen, D. and Stoffels, S.M., Development and validation of performance prediction models and specifications for asphalt binders and paving mixes; SHRP Report A-357; SHRP/NRC, Washington DC, USA, 1993. Molenaar, A.A.A. Road Material, Part III, Asphalt Materials, lecture notes e52 (CT4850), Delft University of Technology, the Netherlands, 1994. Pronk, A.C., Partial healing in fatigue tests on asphalt specimen, Road Materials and Pavement Design, vol. 4, n. 4, pp 433-445, 2001. Pronk, A.C., Description of a procedure for using the Modified Partial Healing model (MPH) in 4PB test in order to determine material parameters. To be published at the 3rd 4PB conference, Davis, USA, 2012. Shell, Shell pavement design manual: asphalt pavements and overlays for road traffic, London: Shell International Petroleum, 1978.


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4.1 Selection of the Mixture The aim of this research was to investigate the effect of the specimen size and test type on the laboratory fatigue test results of asphalt mixtures. In the study, the test results for the different specimen sizes and fatigue tests were compared to each other. It was considered important to minimize the variability of the specimen as much as possible. This implied that air void content, distribution of particles, etc. had to be kept under strict control. Therefore it was decided to use a relatively homogeneous and isotropic mixture with a small maximum aggregate size. From a previous PhD project it was known that the ACRe mixture, which is a kind of sand asphalt mixture with a maximum grain size of 4 mm, fulfilled these requirements and was specially designed as such [Erkens 2002]. On the other hand, the mixture to be used should be a realistic one implying that it is used in practice. The ACRe mixture is not used in practice. Dense graded asphalt concrete 0/8 and 0/11 (DAC 0/8 and 0/11) mixtures are commonly used as wearing course and this mixture type is durable because of its low voids content and relatively high bitumen content. Furthermore they have a good resistance to deformation, water damage and ageing. Figure 4-1 represents the comparison of the aggregate gradations of the ACRe mixture, DAC 0/8 mixture and DAC 0/11 mixture. More details about the composition of these mixtures are shown in Table 4-1.

0

20

40

60

80

100

0.01 0.1 1 10 100sieve size [mm]

Per

cent

age

pass

ing

[%]

ACReDAC 0/8DAC 0/11

Figure 4-1 Grading curves of dense asphalt concrete, DAC [Data from: Erkens, 2002 and CROW 2005]


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Table 4-1 Compositions of ACRe, DAC 0/8 and DAC 0/11 mixture [Erkens, 2002 and CROW 2005]

Percentage passing by mass [%] Sieve size [mm]

ACRe DAC 0/8 DAC 0/11 11 100 100 97 8 100 97 86

5.6 100 80 68.5 4 96.8 - -

2.8 94.4 - - 2 90.8 45 45 1 65.3 - -

0.5 43.4 - - 0.18 25.2 - - 0.063 14.6 8.23 6.5

Ratio of bitumen and aggregate 9.4 6.6-7 - Furthermore numerous fatigue tests are time consuming and expensive. Due to the high bitumen content and low air void content, fatigue tests on the ACRe mixture would take much more time compared to the other two mixtures. After considering the homogeneity, time needed for testing, the aim to use realistic mixtures and the influence of the maximum grain size on the specimen size (Figure 4-2), it was decided that the tests would be performed on a single mixture, being a DAC 0/8 mixture.

Figure 4-2 Scheme of the selection of mixture

Sand asphalt (SA)

Dense asphalt concrete 0/8 (DAC 0/8)

Dense asphalt concrete 0/11 (DAC 0/11)

Suitable for results comparison

Suitable for application in practice


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4.2 Mixture Design

4.2.1 Materials

The basic components used for the asphalt mixture DAC 0/8 include aggregates, filler and bitumen. The aggregates consisted of crushed Scottish granite (2~8 mm), Norwegian bestone (2~6 mm), and crushed sand (≤ 2 mm). The Scottish granite and Norwegian bestone were obtained from BAM contracting company. To achieve the gradation according to the Dutch RAW specifications [CROW 2005], the Scottish granite and crushed sand were separated into two fractions respectively by using a laboratory sieving machine. For each fraction, the apparent particle density was measured by the saturated surface dry technique [EN 12697-05, 2003].

( )2 1

3 21000 /ap w

m m

V m m

−=

× − −ρ

ρ (4-1)

where: ρa : density of aggregate, [kg/m3]; m1 : mass of pyknometer plus head piece, [g]; m2 : mass of pyknometer plus head piece and aggregate, [g]; m3 : mass of pyknometer plus head piece, aggregate and water,

[g]; Vp : volume of pyknometer, when filled up to the reference mark,

[m3]; ρw : density of water at test temperature, [kg/m3]. The percentage passing a certain sieve size and the apparent density of each fraction are presented in Table 4-2.

Table 4-2 Percentage passing by mass of each fraction of the aggregates

Scottish crushed granite Norwegian

Bestone Crushed sand

Sieve size [mm] 8-5.6 mm 5.6-4 mm 6-2 mm 2-1 mm 1-0 mm

8 87.6 100.0 100.0 100.0 100.0 5.6 2.0 100.0 96.7 100.0 100.0 4 0.0 51.2 71.9 99.9 100.0

2.8 0.0 30.0 30.0 98.0 100.0 2 0.0 13.9 13.8 74.3 100.0 1 0.0 6.9 2.7 2.0 100.0

0.5 0.0 4.6 1.7 0.0 55.5 0.18 0.0 2.6 1.3 0.0 16.1 0.063 0.0 0.3 0.2 0.0 5.8

Apparent density [kg/m3] 2629.9 2700.7 2702.6 2682.7 2647.8 The used filler was Wigras 40K filler, produced by Ankerpoort NV. Table 4-3 shows the properties of this filler.


72

Table 4-3 Properties of Wigras 40K filler

Bitumen number ml/100g

Ca(OH)2 [%] by mass

Mass loss at 110 ºC

[%]

Solvability in H2O [%] by

mass

Density [kg/cm3]

Void [%] by volume

46 8.9 0.5 9.8 2620 43 Cumulative retained [%] by mass

0.063 mm 0.125 mm 2 mm 18 5 0

Bitumen with a 40/60 pen paving grade was provided by Q8 / Kuwait Petroleum B.V. The properties of the virgin 40/60 pen bitumen are given below:

Table 4-4 Properties of bitumen 40/60

Penetration @ 25 ºC [0.1 mm]

Softening point [ºC]

Penetration index Density [kg/cm3]

45 53.6 -0.59 1035

4.2.2 Mixture Design The aim of the DAC mixture design was to obtain a DAC 0/8 that satisfied the mixture specifications for Vehicle Class 4. In order to optimize the binder content and the gradation of the DAC 0/8 mixture, a Marshall mixture design was carried out in accordance with the Dutch RAW specifications [CROW 2005]. Based on the tolerance limits for the DAC 0/8 mixture, several bitumen contents and aggregate gradations were selected. The specimens were compacted with the Marshall hammer with 50 blows on each side. The following properties of the mixture specimen were measured for the optimization: - Marshall stability (Pm), Marshall flow (Fm), Marshall quotient (Qm); - Apparent density and air void content of specimen. Marshall stability and Marshall flow of the mixture were obtained by conducting the Marshall test [J.M.M. Molenaar 2003]. The apparent density of the mixture specimen was measured by the saturated surface dry technique [EN 12697-06, 2003].

1

3 2a w

m

m m= ×

−ρ ρ (4-2)

where: ρa : apparent density of specimen, [kg/m3]; m1 : mass of dry specimen, [g]; m2 : mass of specimen in water, [g]; m3 : mass of saturated surface-dried speicmen, [g]; ρw : density of water at test temperature, [kg/m3]. The air voids content was calculated from Equation 4-3.


73

max

1 100%aaV

= − ×

ρρ

(4-3)

where: Va : air void content of specimen, [%]; ρa : apparent density of specimen, [kg/m3]; ρmax : maximum density of asphalt mixture, [kg/m3]; With the densities of all the fractions in the mixture, the theoretical maximum density was calculated by Equation 4-4 [EN 12697-05, 2003].

max1 2

1 2

100

...a a b

a a b

p p pρ

ρ ρ ρ

=

+ + +

(4-4)

1 2 ... 100%a a bp p p+ + + = (4-5)

where:

ρmax : maximum density of mixture, [kg/m3];

pa1 : the percentage of aggregate 1 in mixture, [%]; ρa1 : the apparent density of aggregate 1, [kg/m3]; pa2 : the percentage of aggregate 2 in mixture, [%]; ρa2 : the apparent density of aggregate 2, [kg/m3]; pb : the percentage of bitumen in mixture, [%]; ρb : the density of bitumen, [kg/m3];

According to the specification [CROW 2005], the optimal material composition for the mixture DAC 0/8 was determined, as shown in Table 4-5. Figure 4-3 shows the aggregate gradation used in the mixture.

0

20

40

60

80

100

0.063 0.18 0.5 1 2 2.8 4 5.6 8sieve size [mm]

Per

cent

age

pass

ing

[%]

DAC 0/8 DesiredMin Max

Figure 4-3 Aggregate gradation of the mixture DAC 0/8


74

Table 4-5 Composition of the DAC 0/8 mixture

Scottish crushed granite

Norwegian Bestone

Crushed sand Wigras

40K Sieve (mm)

8-5.6 5.6-4 6-2 2-1 1-0 0.063-0 Binder Total

Wt. % 11.2 19.6 21.5 16.8 16.4 7.9 6.5 100

Table 4-6 Properties of the Marshall specimen

Specification Measured value

Min Max Marshall stability [N] 9200 7000 - Marshall flow [mm] 3.2 2 4 Marshall quotient 2875 2500 -

maximum density [kg/m3] 2428 - - Air void content [%] 3.5 4

4.3 Specimen Preparation

4.3.1 Mixture Compaction Cylindrical specimens and rectangular beam specimens were fabricated for the uniaxial tension and compression test, the indirect tensile test and the four-point bending test. All specimens were cored or sawn from blocks prepared with the PReSBOX compactor. The rectangular block prepared by the PReSBOX compactor has a fixed length and width of 450 mm × 150 mm. The height can be varied from 145 mm to 185 mm. As one of the input parameters for the block preparation, the height of the block is determined by the maximum density of the mixture and the target air void content. During compaction, asphalt mixtures are compacted with a constant vertical force and a cyclic shear force with a constant maximum shear angle. It is believed that this way of compaction simulates the real compaction process in the field [Qiu, Xuan et al. 2009]. Figure 4-4 illustrates the compaction procedure of the hot asphalt mixture; it consists of the following steps. (1) Heating: The aggregates and filler were placed in an oven at a temperature of 175ºC for at least 4 hours. In addition to the materials, the needed accessories for mixing and compaction (i.e. the mixing bowl, dough hook, trays, steel plates and chute box) were also kept in the oven at the same temperature and for the same duration. Bitumen was preheated for 2 hours in an oven at a temperature of 150ºC. (2) Mixing: Before starting the mixing, the bitumen was poured into the bowl and then the coarse aggregates were poured in the bowl followed by the crushed sand and after that the filler. The total weight of the mixture was around 25 kg. All the materials were mixed using a Hobert H600 mixer for a duration of 3 minutes to make sure that a well mixed material was obtained.


75

Figure 4-4 Illustration of the block production process in the compactor: mixing (top left), laying (top right), compaction (bottom left) and demoulding (bottom right)

(3) Laying: The hot asphalt mixture was transferred from the bowl into two trays. A steel

plate was placed at the bottom of the compaction mold, and the chute box was placed afterwards with the bottom gate closed (Figure 4-4 (top right)). When the temperature of the mixture was around 155ºC, the mixtures were fed carefully into the chute box (Figure 4-4 (bottom left)). Then the bottom gate of the chute box was opened to allow the asphalt mixture to fall into the shear box mold. After that the chute box was removed (Figure 4-4 (bottom right)). The mixture was poked vertically at the four corners of the mold. Then the other steel plate was placed on the surface. Finally, the mold was moved into the frame and ready to be compacted.

(4) Compaction: All the required input data, such as the maximum density, mixture

weight, target height etc. were added to the computer, after that the compaction was started. The compaction process is shown in Figure 4-5 [Qiu, 2012].

.

Figure 4-5 Illustration of the compaction process [Qiu, 2012]

EjectEjectStart

Vertical stressStart

Vertical stress Vertical + Shear stress CycleVertical + Shear stress Cycle CentreCentre


76

Firstly, a vertical force was applied to provide a constant vertical stress of 0.75 MPa. Then cyclic shearing of the framework was applied with a shear angle of 4º. Figure 4-6 shows the development of the vertical and shear stress during the compaction process. Once the target height was reached (in this study the target height was 155 mm), the compaction program was terminated automatically. The framework centered itself and the loads were released. Finally, the asphalt block was ejected by an air pump and removed after it was cooled down.

-2000

-1000

0

1000

2000

0 50 100 150 200

Loading time [s]

Stre

ss [k

Pa]

Vertical stress

Shear stress

Figure 4-6 Loading curves during the compaction process [Qiu, 2012]

4.3.2 Selection of Specimen Size With regard to the determination of the specimen dimensions for each fatigue test, the following principles were taken into account. • From small to large, three specimen sizes were selected, denoted as size 0.5, size 1.0

and size 1.5. The difference between the three specimen sizes had to be large enough to distinguish the size effect on the fatigue results.

• The test equipment to be used could handle all the specimen sizes.

• The minimum length of the specimen should be at least three times larger than the maximum aggregate size of the mixture [EN 12697-24].

• To obtain comparable results from the different fatigue tests, the cross-sectional area in the critical location of specimen size 1.0 should be close to that of the other two specimen types as explained in Figure 4-7.

Figure 4-7 shows the cross-sectional area in the critical location for the three different specimen types. The cross-sectional area of the cylindrical specimen for the UT/C test and the beam specimen for the 4PB can be simply determined. For the IT test, the tensile stress is induced by the compressive stress along the vertical diametrical plane. Fatigue


77

failure is initiated in the region of a relatively uniform tensile stress. According to the stress distribution of the IT specimen [Hondros, 1959] [Kennedy, 1977], this critical area is around 70% of the entire cross section of the cylindrical specimen (see Figure 4-7).

Figure 4-7 Cross-sectional area in the critical location for different specimen types Based on the above considerations, the dimensions for the specimen types are presented in Table 4-7.

Table 4-7 Dimensions of the different specimen types

Test type Specimen size Dimension [mm] Cross area in critical

location [mm2] Size 0.5 ø × h = 25×62.5 491 Size 1.0 ø × h = 50×125 1964 UT/C Size 1.5 ø × h = 75×187.5 4418 Size 0.5 l × w × h = 400×50×25 1250 Size 1.0 l × w × h = 400×50×50 2500 4PB Size 1.5 l × w × h = 400×50×75 3750 Size 1.0 ø × h = 100×30 2100

IT Size 1.5 ø × h = 150×45 4725

Note: size 0.5 = small size; size 1.0 = medium size; size 1.5 = large size ø × h = diameter × height, l × w × h = length × width × height

The UT/C specimens with size 1.0 are also used for monotonic tension and compression tests.

4.3.3 Specimen Preparation One day after the asphalt mixture block was made, it was cut into cylindrical and beam specimens with different sizes. Table 4-8 gives the schematic plan for cutting the different specimens. About 20 mm from the edges of the slab was trimmed-off to avoid inadequately compacted parts of the block; the thickness of the sawing blade was taken into account to get the wanted dimensions.

: Cross-sectional area

UT/C specimen 4PB specimen IT specimen


78

The rectangular beam specimens were cut by using a water-cooled masonry saw and the cylindrical specimens were cored by a water-cooled core drill, as shown in Figure 4-8. After coring, the cylindrical specimens for the UT/C test were polished with a polishing machine to make the upper and lower surfaces smooth and parallel to each other.

Table 4-8 Sawing plan for the different specimens

Specimen name Sawing plan

(left: front view, right: top view) Specimen number per

slab

UT/C-size0.5 160

450

150

160

450

150

450

ΦΦΦΦ=25

h=62.5 450

ΦΦΦΦ=25

h=62.5

30

UT/C-size1.0

160

450

150

160

450

150

450

ΦΦΦΦ=50

h=125

450

ΦΦΦΦ=50

h=125

12

UT/C-size1.5

160

450

150

160

450

150

450

ΦΦΦΦ=74.5

h=186

450

ΦΦΦΦ=74.5

h=186

2

4PB-size0.5

160

450

150

160

450

150

450

h=25

400 450

h=25

400

6

4PB-size1.0

160

450

150

160

450

150

450

h=50

400 450

h=50

400

4


79

4PB-size1.5

160

450

150

160

450

150

450

h=75

400 450

h=75

400

2

IT-size1.0

450

150

170

R=100

450

150

170

450

150

170

R=100

450

30

R=100 450

30

R=100

9

IT-size1.5

450

150

170

R=150

450

150

170

R=150

450

45

R=150

450

45

R=150

4

(a) Masonry saw; (b) Core drill

Figure 4-8 Masonry saw and core drill for preparing the different specimens A numbering system was used to trace from which part of the compacted block the specimens came. The first letter corresponds to the type of sample (C: cylindrical sample for the UT/C test, B: beam, I: cylindrical sample for the IT test), the second number is the number of the slab, and the last number corresponds with the position in the block, which starts from top to bottom, from left to right and from front to back. Figure 4-9 gives an example of the specimen labeling for the cylinders with size 1.0. If the block shown in Figure 4-9 was the first one prepared to obtain cylindrical specimens, then the specimens were named as C-1-1, C-1-2, ······, C-1-12. Specimen C-1-1 means a cylindrical specimen for the UT/C test which is cut from the first block prepared for the UT/C tests and taken from position number 1.


80

21

3 4

21

3 4

1 2

5 6

9 10

1 2

5 6

9 10

(a) Front view (b) Top view

Figure 4-9 Positions of the labeled cylindrical specimens Bulk density and voids content of all the specimens were measured by the saturated surface dry technique and calculated by Equations 4-2 and 4-3. Figure 4-10 compares the air voids of the specimens from the different positions of a block. It shows that the upper part of the block has a high density than the lower part. The same results were also found by Qiu [Qiu, 2012]. Figure 4-11 gives the simulation results of the shear and the vertical stress distribution of the cross-sectional area of the block during the compaction. It can be seen that the material is compacted by both shear and compressive force. The upper part experienced more shear stress than the lower part, which will have consequences on the distribution of the air voids of the specimens taken from different positions. But the difference of the air voids content at the different positions was lower than 1 % in the same slab.

0

1

2

3

4

5

C-8-1 C-8-2 C-8-3 C-8-4 C-8-5 C-8-6 C-8-7 C-8-8 C-8-9 C-8-10 C-8-11 C-8-12

Air

void

s co

nten

t [%

]

Figure 4-10 Air voids content of the cylinders with size 1.0 in one slab

Table 4-9 gives an overview of the measured void contents of totally 190 specimens. The average value of the voids content is very close to the target voids content of the block, 3.5%. The standard deviations show that the variability of the voids content of the specimens is low. Some irregular specimens with extreme air voids of 2.2 % or 4.88 % can also be observed. This might be related to abnormal compacting conditions such as too low or too high compaction temperature, material loss during the laying process and so on.

upper part

Directions in compactor

lower part


81

(a) (b)

Figure 4-11 Shear stress (a) and vertical stress (b) distribution in asphalt mixtures at the maximum shear angle during compaction [Qiu, 2012]

Table 4-9 Statistical analysis of volumetric properties of specimens

Numbers of beams

Target air voids [%]

Average air voids [%]

Max. air voids [%]

Min. air voids [%]

Standard deviation

[%] COV [%]

190 3.5 3.45 4.88 2.20 0.538 15.6 Figure 4-12 gives the distribution of the air voids content of all 190 specimens. The voids content of the specimens should be between 2.75 % and 4.25 % in order to minimize the influence of the volumetric properties on the fatigue test results.

1

2

3

4

5

6

0 50 100 150 200

Specimen number

Air

void

s co

nten

t [%

]

Figure 4-12 Selection of the specimens based on the air voids contents

Before testing, all the selected samples were stored in a climate room at a temperature of 15 ºC. Figure 4-13 shows the beam and cylindrical specimens with different sizes.

4.25%

2.75%


82

(a) Cylinders for UT/C test; (b) Beams for 4PB test; (c) Cylinders for IT test

Figure 4-13 Beam and cylindrical specimens with different sizes

References CROW. In: Standaard RAW Bepalingen, C.R.O.W., Ede (In Dutch). 2005. EN 12697-05: 2003 Bituminous mixture – Test methods for hot mix asphalt - Part 5: Determination of the maximum density. European committee for standardization, Brussels, 2003. EN 12697-06: 2003 Bituminous mixture – Test methods for hot mix asphalt - Part 6: Determination of bulk density of bituminous specimens. European committee for standardization, Brussels, 2003. Erkens, S.M.J.G., Asphalt Concrete Response (ACRe)-Determination, Modelling and Prediction, PhD Thesis, Delft University of Technology, the Netherlands, 2002. Jacobs, M. M. J., Crack Growth in Asphaltic Mixes, Ph.D. dissertation, Delft University of Technology, Delft, the Netherlands, 1995. Molenaar, J.M.M., Performance related characterisation of the mechanical behaviour of asphalt mixtures, PhD Thesis, Delft University of Technology, the Netherlands. 2003. Qiu, J., Xuan D., van de Ven, M.F.C. and Molenaar, A.A.A.,, Evaluation of the shear box compactor as an alternative compactor for asphalt mixture beam specimens. AES - ATEMA'2009, 3rd Int. Conf. on Advances and Trends in Engineering Materials and their Applications. Montreal, Canada, 2009. Qiu, J., Li, N., Pramesti, F.P., van de Ven, M.F.C. and Molenaar, A.A.A.et al., Evaluating Laboratory Compaction of Asphalt Mixtures Using the Shear Box Compactor, Journal of Testing and Evaluation, Vol. 40, No. 5, 2012.


83


5.1 Introduction In this chapter, the uniaxial tension and compression (UT/C), four-point bending (4PB) and indirect tensile (IT) fatigue tests, which were performed at different temperatures and loading modes, will be described. Details of the experimental work are presented in this chapter. The Chapter begins with a description of the used test equipments and the calibration work for the 4PB test setup. Then the stiffness and fatigue test programs are given for the different specimen sizes. Finally the test results from all three test methods are compared and discussed.

5.2 Test Equipment

5.2.1 Uniaxial Tension and Compression (UT/C) Test The uniaxial tension and compression test was performed on a IPC Universal Testing Machine (UTM 25 kN) electro-hydraulic servo system. This system consists of three main components: the testing frame, the environmental chamber, and the control data acquisition system (CDAS); all are shown in Figure 5-1. The temperature range of the environmental chamber is between -15°C and 60°C. All specimens were glued to two steel plates at both ends using a two-component fast curing adhesive glue (X60). Firstly, the specimen was glued to the top platen that is fastened to the actuator. After sufficient glue was placed on the bottom platen, the top platen with the specimen was slowly lowered till the bottom end of the specimen contact with the bottom platen. Due to the stress concentration near the ends of the specimen, enough glue needs to be used to ensure sufficient bond and prevent cracking near the ends. The thickness of the glue was around 3 mm. During the test, the load is applied to the top platen through the actuator. The force is measured with a load cell and the axial deformation is registered by means of three spring-loaded LVDT’s with a range of ±2 mm The LVDT’s are mounted vertically around the side of the specimen at an interval of 120° between two aluminum rings fastened on the top and bottom platens. In terms of data acquisition, the CDAS automatically controls the valve to apply the requested loading during the test. The CDAS with the personal computer collects the data at the same time.


84

Figure 5-1 Uniaxial tension and compression test setup with the temperature chamber (left) and a close-up of a cylindrical specimen for size 0.5 (right)

5.2.2 Four-Point Bending (4PB) Test To allow the use of beam specimens with different dimensions, a four-point bending (4PB) test setup was developed by the Laboratory of the Road and the Railway Engineering Section; it is shown in Figure 5-2a. Like the UT/C test, the test setup was placed in a temperature chamber capable of maintaining a temperature between -10°C to 60°C. Four clamps fix the beam specimen in the bending bed. The distance between the two outer is 400 mm and the distance between the inner clamps is 133.3 mm. The load was applied to the beam specimen through the inner clamps by means of two loading jacks. The two outer clamps prevent vertical movement of the specimen. They consist of two thin steel sheets around the beam with a groove. A small cylindrical spindle lies in the two grooves between clamp and the frame of which the radius is larger than that of the spindle. The clamp allows free rotation and horizontal translation of the specimen at the four supports, as shown in Figure 5-2b.

(a) (b)

Figure 5-2 Set-up of the 4PB fatigue test (a) and local view of inner clamps (b)

LVDT’s

Aluminum ring

Environmental chamber

Testing frame

LVDT

Spring

Bolts

Spindle

Adhesive


85

The clamping force is applied by tightening the bolts with the spring on top of the clamps. A torque wrench was used to apply a specific clamping force, see Figure 5-3a. Figure 5-3b shows the measurement of the clamping force. The force measured by the load cell placed between the clamps was recorded at the applied torque level.

(a) Torque wrench (b) Measurement of the clamping force

Figure 5-3 Measurement of the clamping force

Figure 5-4 gives the relations between the clamping force and the applied torque at the inner and outer clamps. Two different linear relations were found at low torque 0≤ τ ≤2. lb-in and at high torque 2.5 lb-in ≤ τ ≤ 7.5 lb-in (The torque unit “lb-in” is the product of pound and inch, 1 lb-in=0.1129 Nm). The functions representing the relation between torque and clamping force at the inner and outer clamps are listed in Table 5-1.

0

200

400

600

800

0 2 4 6 8 10

Torque [LB-IN]

For

ce [N

]

Inner clamping force

Outer clamping force_rightOuter clamping force_left

Figure 5-4 Relationship between clamping force and torque

Load cell

1

2

τ


86

Table 5-1 Linear equations of the clamping force Linear equation 1

(0≤ τ ≤2.5LB-IN) Linear equation 2

(2.5≤ τ ≤7.5LB-IN) Inner clamping force Fin (N) 48 5 90 2inF . τ .= ⋅ + 82 4 24 3inF . τ .= ⋅ +

Outer clamping force_right For (N) 28 49 2orF τ .= ⋅ + 62 8 34 5orF . τ .= ⋅ −

Outer clamping force_left Fol (N) 21 40 4olF τ .= ⋅ + 58 48 5olF τ .= ⋅ −

The clamping force has a significant influence on the stresses and strains in the beam specimen. In theory the clamping force should be as low as possible to prevent damage near the supports. However, for the test setup shown in Figure 5-2, the sum of the two inner clamping forces should be a little larger than the applied force. If the force supplied by the springs is smaller than the applied force from the actuator, the springs will be deformed and the specimen will be separated from the clamps during the test. The disadvantage of this setup is that the amplitude of the clamping force varies with the applied strain level. At high strain level, implying high load levels and high clamping forces restrain the horizontal movement and rotation of the spindles between grooves and cause high shear stresses on the beam near the supports. In order to determine the stress and strain distributions in the beam specimen, finite element simulations were performed by means of the ABAQUS software. The model geometry of the 4PB specimen is presented in Figure 5-5.

Figure 5-5 Model geometry used in the finite element simulations of the 4PB test

Figure 5-6 shows the horizontal strain and shear stress distribution in the beam during the test. The clamping force on the model is 300N. At the same time, a force of 250 N is applied on top of each inner clamp. The viscoelastic properties of the mixture are described by Prony series. The value of the Prony elements were calculated from the results of the 4PB stiffness tests. Details of this analysis are presented in Appendix B. For the Poisson’s ratio of the mixture, a value of 0.3 was used. The steel clamps on the supports are considered to be linear elastic material with a modulus of 200 GPa and a Poisson’s ratio of 0.27.

133.3 mm

400 mm

Beam specimen

Clamp

20 mm

Loading Loading

450 mm


87

As shown in Figure 5-6, the computational results show that the highest horizontal tensile strain and shear stress occur around the inner clamps. After observing the location of failure in numerous specimens, it was found that most of the specimens failed at the location where the highest stress was computed (see Figure 5-6c). In this case, the cracks on the beam are mainly caused by the combination of horizontal tensile strain and shear stress and not because of pure bending.

(a) Horizontal strain ε11

(b) Shear stress s12 (c) Cracking near the inner clamps

Figure 5-6 Horizontal strain and shear stress distributions in the beam To avoid the unwanted horizontal strain and shear stress concentrations near the inner clamps, the springs on top of the inner clamps were removed from the test setup, as shown in Figure 5-7. After the modification, a certain small clamping force is allowed to be applied for different strain levels. Under a small clamping force, the cylindrical spindle can move and rotate freely between the two grooves.

(a) (b)

Figure 5-7 Modified setup of the 4PB fatigue test (a) and local view of inner clamps (b)


88

Also for this setup, finite element simulations were performed with the ABAQUS software. The same model geometry and loading mode were used. The difference is that the inner clamping forces reduced to 120 N and the clamps rotated with the bended specimen surface. Figure 5-8 shows the horizontal strain and shear stress distribution of the beam at the lower clamping force of 120 N. It can be seen that the maximum horizontal compressive and tensile strains distribute uniformly at the specimen surface between two inner clamps. This implies that in most of the cases, the fatigue cracks will initiate at the surface in the middle section of the beam specimen.

(a) Horizontal strain ε11

(b) Shear stress s12

Figure 5-8 Horizontal strain and shear stress distribution of the beam

0

100

200

300

400

500

600

0 100 200 300 400 500Length of beam [mm]

Hor

izon

tal s

train

[µ

m/m

]

clamping force=120N clamping force=300N

Figure 5-9 Computed horizontal strain at beam surface along the length

Inner clamps


89

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500

Length of beam [mm]

She

ar s

tress

[MP

a]


Figure 5-10 Computed shear stress at beam surface along the length

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 100 200 300 400 500

Length of beam [mm]

She

ar s

tres

s [M

Pa]


Figure 5-11 Computed shear stress in the middle of the beam along the length

Figure 5-9, 5-10 and 5-11 compare the distributions of the horizontal strain at the surface, shear stress at the surface and the shear stress in the middle of the beam. When the applied force is 250 N and the clamping force is 120 N, the horizontal strain increases gradually from the outer supports to the inner supports, in the middle section the horizontal strain takes a constant value (≈160µm/m) and the maximum shear stress at two inner supports is around 0.29 MPa. At the high clamping force of 300 N, the maximum horizontal strain at the inner supports is 500µm/m, while in the middle section of the specimen the strain is almost zero. The maximum shear stress at the inner supports is 0.6 MPa, about 2 times higher than that at the low clamping force. Therefore based on the model analysis, it can be concluded that the fatigue failure of the beam tested in the modified 4PB setup is mainly caused by pure bending when the two clamping forces are as small as possible.

Inner clamps

Inner clamps


90

5.2.3 Indirect Tensile (IT) Fatigue Test Like the UT/C test, the indirect tensile (IT) test is also performed in the IPC Universal Testing Machine (UTM 25 kN). Figure 5-12 shows the loading configuration of the IT test.

Figure 5-12 The set-up of the IT test and loading configuration The cylindrical sample is placed between two arc-shaped steel loading strips. During the test, a repeated compressive load is applied by means of the load actuator along the vertical diametrical plane. This load causes both vertical compressive and horizontal tensile stresses in the specimens. Two LVDT's (range ±2 mm) at either side of the specimen are mounted on the frame to measure the horizontal displacement along the horizontal diameter. The vertical deformations are measured by two LVDT's with a range of ±10 mm glued on the upper loading strip. For specimens with different sizes, different loading strips have been used. The dimensions of specimens and the loading strips are shown in table 5-2.

Table 5-2 Dimensions of the loading strips for IT [EN 12697-26] Nominal specimen

diameter [mm]

Width of loading strip

[mm]

Nominal depth concave segment [mm]

Radius of the load side of the strip [mm]

100 12 ± 1 0.4 ± 0.05 50.5 150 19 ± 1 0.6 ± 0.05 76

5.3 Calibration of the 4PB Test Equipment In the UT/C and IT test, the loading is applied directly to the specimen. So the measured force and displacement are not influenced by the test equipment. However, the 4PB test setup used in this study was specially designed by the Road and Railway Engineering Laboratory and differed from existing commercial test equipment. Furthermore the loading is not directly applied on the beam specimen but through a loading frame. The

Loading strip

Horizontal LVDT

Vertical LVDT

Loading strip

F


91

measured displacement strongly depends on the properties of the loading frame. Three aspects should be taken into account: pure bending deflection of the beam, shear deflection of the beam at the supports and deflection of the frame. However, only the deflection due to pure bending is taken into account in the stiffness calculations. To obtain reliable measurement results, it is necessary to perform calibration tests on the 4PB test setup.

5.3.1 Theory In the whole test setup, the beam and frame from the test setup can be represented by two serial springs, as shown in Figure 5-13.

Figure 5-13 Theoretical model for the deflection of 4PB setup consisting of beam and loading frame

Based on this two serial spring’s model, the following relations can be obtained:

, ,m frame beamm frame beam

F F Fv v v

S S S= = = (5-1)

1 1 1m frame beam

m frame beam

v v vS S S

= + ⇒ = + (5-2)

Where: vm : measured deflection, [m]; vframe : frame deflection, [m]; vbeam : beam deflection between inner clamps, [m]; F : applied force, [N]; Sm : stiffness of whole system, [N/m]; Sframe : frame stiffness, [N/m]; Sbeam : beam stiffness, [N/m]. (1) Correction for shear deflection

The beam deflection includes two parts, bending deflection and shear deflection [Huurman, 2009] [Pronk, 2009]. In the midsection between the two inner clamps, the solution for the bending deflection is given by Equation 5-4 [Pronk, 2009].

beam s b s s bv v v ,v C v= + = ⋅ (5-3)

F F

Sbeam Sframe

vbeam vframe

vm


92

( )2

2

4 1

13 4

beamb s

s

hµ

v Lv ,C

C Aα

L

⋅ + ⋅ ⇒ = =

+ − ⋅

(5-4)

Where: vs : shear deflection, [m]; vb : bending deflection between inner clamps, [m]; Cs : correction coefficient; Μ : Poisson’s ratio; H : beam height, [m]; L : distance between two outer clamps, [m]; A : distance between inner and outer clamps, [m]; Α : shear factor, 0.859. (2) Correction for frame deflection

The beam stiffness in [MPa] is related to the applied force and the bending deflection, as given in Equation 5-5.

b

FE Z

v= ⋅

(5-5)

2 3

3

6 8

8

AL AZ

bh

−= (5-6)

Where: E : theoretical value of the beam stiffness, [MPa]; Z : form factor, [1/m]. The beam stiffness in [N/m] is transformed from the beam stiffness in [MPa] by substituting Equation 5-3 and 5-5 into Equation 5-1.

( ) ( )1 1beambeam s b s

F F ES

v C v Z C= = =

+ ⋅ ⋅ + (5-7)

Therefore the frame stiffness is calculated by the following equation:

( )1 1

1 1 1frames

m beam m

SF Z C

S S v E

= =⋅ +− −

(5-8)

By substituting Equation 5-8 into Equation 5-2, the beam deflection can be calculated from the measured force and deflection if the frame stiffness is known.


93

beam m frame mframe

Fv v v v

S= − = − (5-9)

Finally, the pure bending deflection can be computed by Equation 5-4.

5.3.2 Test Program In order to calculate the frame stiffness of the used 4PB test equipment, a frequency sweep was conducted with an aluminum beam (theoretical stiffness E = 71 GPa). The dimension of the aluminum beam is 450 mm long, 35 mm wide and 34 mm high (see Figure 5-14). During the test, a sinusoidal loading was applied on the beam at different force levels and frequencies (1, 2, 4, 6, 8 and 10 Hz).The inner clamping force was 120 N and kept constant for all force levels. The outer clamping force depended on the applied force level. Table 5-3 presents the applied force level for each outer clamping force.

Figure 5-14 Calibration test for the 4PB setup with aluminum beam

Table 5-3 Applied force levels and used clamping forces Inner clamping force [N] Outer clamping force [N] Applied force level [N]

100 120

200 100 200 180 300 200 300 230 400 300 400

120

270 500


94

Figure 5-15 shows the measured stiffness of the aluminum beam at various applied force and clamping force levels. It can be seen that the measured stiffness of the aluminum beam is independent of the frequency but is strongly influenced by the applied force and clamping force. At a same outer clamping force, a higher applied force leads to a smaller measured stiffness, while at a certain applied force the measured stiffness increases with an increasing clamping force. For each clamping force, the maximum applied force gives in all cases a stiffness of approximately, about 56.4 GPa. This implies that the outer clamping forces should be as low as possible relative to the applied force.

5.0E+04

5.2E+04

5.4E+04

5.6E+04

5.8E+04

6.0E+04

6.2E+04

0 100 200 300 400 500 600Applied force [N]

Mea

sure

d st

iffne

ss [M

Pa]

1Hz-120N 2Hz-120N 4Hz-120N 6Hz-120N 8Hz-120N 10Hz-120N1Hz-180N 2Hz-180N 4Hz-180N 6Hz-180N 8Hz-180N 10Hz-180N1Hz-230N 2Hz-230N 4Hz-230N 6Hz-230N 8Hz-230N 10Hz-230N1Hz-270N 2Hz-270N 4Hz-270N 6Hz-270N 8Hz-270N 10Hz-270N

Figure 5-15 Measured stiffness of aluminum beam at various applied force and outer

clamping force levels

2.0E+07

2.5E+07

3.0E+07

3.5E+07

4.0E+07

4.5E+07

5.0E+07

5.5E+07

0 100 200 300 400 500 600Applied force [N]

Cal

cula

ted

fram

e st

iffne

ss [N

/m]


Figure 5-16 Calculated frame stiffness values at various applied force and outer clamping

force levels

Outer clamping force: 120N 180N 230N 270N


95

As mentioned in the last section, the frame stiffness can be calculated using Equation 5-8. The calculated Sframe values are presented in Figure 5-16; they show a similar trend with respect to the applied force as the measured stiffness. The figure shows that the frame stiffness at the maximum applied force for each clamping force is more or less same. Figure 5-17 gives the measured phase angle at the different applied force levels and clamping forces. Like the stiffness, the measured phase angle does not change with frequency. The phase angle however does increase with increasing applied force and decreases with increasing outer clamping force. For each clamping force, the phase angle measured at the maximum applied force has a linear relationship with the applied force level. Table 5-4 and Figure 5-18 give the average stiffness and phase angle values at the maximum applied force for each outer clamping force. Compared to the stiffness of the aluminum beam 7.47×106, the average stiffness of the 4PB set up is about 4 times higher.

0

0.5

1

1.5

2

2.5

3

0 100 200 300 400 500 600

Applied force [N]

Pha

se a

ngle

[deg

]


Figure 5-17 Calculated phase angle at various applied force and outer clamping force

Table 5-4 Average value of frame stiffness and phase angle

Inner clamping force [N]

Outer clamping force [N]

Applied force level [N]

Calculated Sframe [N/m]

Phase angle [deg]

120 200 3.14E+07 2.0 180 300 3.18E+07 1.5 230 400 3.08E+07 1.6

150

270 500 3.05E+07 1.3


96

y = -0.0021x + 2.3374R2 = 0.8073

y = -3733.6x + 3E+07R2 = 0.697

2.5E+07

3.0E+07

3.5E+07

4.0E+07

4.5E+07

0 100 200 300 400 500 600Applied force [N]

Fra

me

stiff

ness

[N/m

]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Pha

se a

ngle

[deg

]

Sframe

Phase angle

Figure 5-18 Average stiffness and phase angle at the maximum applied force for each

outer clamping force

In the 4PB tests the following regression equations were used to correct the measured stiffness and phase angle. Frame stiffness: ( )03733 6 3 24 07frameS . F . E= − ⋅ + + (5-10)

System phase angle: 00 0021 2 34frameθ . F .= − ⋅ + (5-11)

where F0 = applied force, N.

5.4 Complex Modulus and Fatigue Tests In this study complex modulus tests and fatigue tests are performed. These experiments are conducted according to the test procedures described below.

5.4.1 Complex Modulus Test The complex modulus tests are performed at different temperatures and frequencies. The test programs for the different test equipments are listed in Table 5-5. Due to the capacity of the different test equipments, the used test conditions for UT/C, 4PB and IT tests are slightly different. The testing order always was from low to high temperatures and from high to low frequencies; this is done in order to minimize the damage to the specimens. A five-minute rest period is allowed between each two adjacent different frequencies and at least two and a half hours acclimatization was allowed after changing the testing temperature to achieve thermal equilibrium. All tests are performed according to EN12697-26. The loading levels were carefully adjusted until the sample deformations reach the target value which was considered low enough not to cause damage in the specimen. This implies that the resulting complex modulus represents the modulus of the intact, not damaged specimen. The complex modulus was determined from the average


97

values of load and displacement obtained during the final five cycles of each loading series.

Table 5-5 Complex modulus test programs on different test equipments Test type UT/C 4PB IT

Control mode Strain controlled Strain controlled Displacement control

Strain level [µm/m] 40-70 40-70 Horizontal displacement =3-5µm

Waveform Sine wave Sine wave Haversine wave Frequency [Hz] 0.5, 1, 2, 4, 6, 8, 10 1, 2, 4, 6, 8, 10 0.5, 1, 2, 4, 6, 8,

Temperature [°C] 0, 5, 10, 20, 30, 40 0, 5, 10, 15, 20, 25, 30, 35, 40

0, 5, 10, 15, 20, 25, 30, 35, 40

5.4.2 Fatigue Test The test conditions for the fatigue tests are presented in Table 5-6. Eight strain or stress levels are chosen in each test series. The specimens are conditioned at the target testing temperature for a minimum of 3 hours before conducting the test. The load, vertical and horizontal deformations are monitored and recorded continuously during the test.

Table 5-6 Test conditions for different fatigue test methods Test type Loading mode Specimen size Test condition

size 0.5 size 1.0 strain controlled size 1.5

Temp.: 20 ºC Freq.: 10 Hz Waveform: sinusoidal

UT/C stress and strain

controlled size 1.0


size 0.5 size 1.0 strain controlled size 1.5


4PB stress and strain

controlled size 1.0


size 1.0 IT stress-controlled

size 1.5

Temp.: 5 ºC Freq.: 10 Hz Waveform: haversine

For each set of experiments, 8 strain or stress levels are selected to obtain the relationship between loading level and fatigue life. The loading levels were selected in such a way that the fatigue life would cover a wide range from 5×103 to 2×107 cycles.


98

5.5 Data Processing

5.5.1 UT/C Test For both the stiffness and fatigue tests, the complex stiffness and phase angle are the important test parameters to characterize the material properties. However, there was no test program for the UT/C test data in the software package from the UTM-25 testing machine. In this study, all the UT/C test results were computed from the raw data (time, force and displacement) recorded during the test. The force and displacement in each cycle were described as a sinusoidal function of time t and loading frequency, f, as expressed by Equation 5-12 and 5-13.

0 1sin(2 )F F f t a= ⋅ + +π ϕ (5-12)

0 2sin(2 )L L f t b= ⋅ + +π ϕ (5-13) Where: F : applied force, [N]; F0 : amplitude of the applied force in a cycle, [N]; f : loading frequency, [Hz]; t : testing time[s]; L0 : amplitude of the displacement in a cycle, [mm]; a, b, φ1, φ2 : regression coefficients. Based on the measured results, the regression coefficients in the equations can be simply obtained by the solver function in Excel. Figure 5-19 shows the measured force and the average displacement of the three LVDT’s in a loading cycle. After regression, the stress σ and strain ε were calculated according to Equation 5-14 and 5-15. The phase angle was converted into degrees by using Equation 5-16.

00 2

4F

D=σ

π (5-14)

00

L

H=ε (5-15)

( )2 1 180− ⋅=

ϕ ϕθ

π (5-16)

Where: σ0 : stress amplitude, [MPa]; D : diameter of specimen, [mm]; ε0 : strain amplitude, [mm/mm]; H : height of specimen, [mm]; θ : phase angle in degree, [º].


99

-1600

-1200

-800

-400

0

400

800

1200

1600

0 0.02 0.04 0.06 0.08 0.1 0.12

Time [s]

For

ce [N

]

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Dis

plac

men

t [m

m]

PredictionMeasured forcePredictionMeasured displacement

Figure 5-19 Force and displacement signal in a cycle The absolute value of the complex modulus was then calculated as the ratio of the stress to strain amplitudes.

0

0

E =σε (5-17)

( )0

0

' cosE =σ

θε

(5-18)

( )0

0

'' sinE =σ

θε

(5-19)

Where: E : absolute value of complex stiffness, [MPa]; 'E : storage modulus, [MPa]; ''E : loss modulus, [MPa].

5.5.2 4PB Test The 4PB software was designed and developed in the National Instruments LabView environment to collect and calculate the 4PB test results. The sample frequency is 1000 Hz. Therefore the interval between two data points is 1 ms. In this study the highest test frequency is 10 Hz which means that at least 100 data points are captured per cycle during the test. To minimize noise in the signal, Fast Fourier Transform (FFT) is applied to determine the amplitudes and phase difference between force and deflection. A typical frequency spectrum is displayed in Figure 5-20. Taking into account the specimen shape and the moving mass, the complex stiffness and phase angle are expressed by Equation 5-20 and 5-21. The detailed description of the derivation of these formulas is described in Appendix A.

F0

L 0

θ


100

Figure 5-20 Fast Fourier Transform of the measured force and deflection at 10 Hz.

( )2

2 21 2cos b bs eq eq

b

ZF v vE m m

v F Fϕ ω ω = + ⋅ ⋅ + ⋅

(5-20)

( )( ) 2

sinarctan

cos

smix

bs eq

vm

F

ϕϕ

ϕ ω

= + ⋅

(5-21)

2 3

3

6 8

8

AL AZ

bh

−= (5-22)

Where: E : absolute value of complex stiffness, [MPa]; Z : shape factor, [m-1]; vb : bending deformation, [m]; F : amplitude of force [N]; φs : measured phase lag [º]; meq : equivalent mass [kg] ; ω : circular frequency [rad/s] φmix : phase angle of specimen [º]; A : distance between clamps, next to each other [m]; L : effective length=distance between outer clamps [m]; b : Width of specimen[m] h : height of specimen [m]


101

5.5.3 IT Test

During the test, the control and data acquisition system (CDAS) automatically controls the applied loading and records all the signals, such as vertical load, horizontal and vertical deformation. With the specimen dimension and an estimated value of the Poisson’s ratio for the material, the stiffness modulus can be calculated by the following equations [Hondros, 1959].

Figure 5-21 shows the total horizontal deformation determined from the horizontal displacement transducers. Table 5-7 gives the assumed values for Poisson’s ratio of asphalt mixtures at several temperatures.

0

2

4

6

8

10

12

100.2 100.25 100.3 100.35 100.4 100.45

time [s]

For

ce [k

N]

0.025

0.03

0.035

0.04

Def

orm

atio

n [m

m]

ForceHorizontal LVDT

Figure 5-21 Determination of the horizontal deformation

00

2F

h D=

× ×σ

π (5-23)

0

2 1 3

4rH v

D vε

π π⋅∆ + ⋅ = × + ⋅ −

(5-24)

0

0

E =σε

(5-25)

Where: σ0 : tensile stress at specimen center, [MPa]; F0 : peak value of applied vertical force [N]; h : mean thickness of specimen [mm]; D : specimen diameter [mm]; ε0 : tensile strain at the specimen center [µm/m]; v : Poisson’s ratio; ∆Hr : recovered horizontal deformation, [mm]; E : absolute value of complex stiffness, [MPa];

∆Hr F0


102

Table 5-7 Estimated Poisson’s ratio at different temperatures

Temperature [ºC] 5 10 15 23 35

Poisson’s ratio 0.22 0.24 0.27 0.32 0.4

5.6 Test Results

5.6.1 Stiffness Results Figure 5-22 and 5-23 give typical results from the frequency sweep tests performed at various temperatures for the UT/C test using specimen size 1.5.

UT/C_size1.5

0

5000

10000

15000

20000

25000

30000

0 2 4 6 8 10 12 Frequency [Hz]

Com

plex

mod

ulus

[MP

a]

0ºC 10ºC 20ºC30ºC 40ºC

Figure 5-22 Complex modulus at various temperatures and frequencies

UT/C_size1.5

0

20

40

60

80

100

0 2 4 6 8 10 12Frequency [Hz]

Pha

se a

ngle

[deg

]

0ºC 10ºC 20ºC30ºC 40ºC

Figure 5-23 Phase angle at various temperatures and frequencies


103

@10ºC

0

4000

8000

12000

16000

20000

0 2 4 6 8 10 12

Frequency [Hz]

Com

plex

mod

ulus

[MP

a]

UT/C_Size 0.5 UT/C_Size 1.0 UT/C_Size 1.5

4PB_Size 1.0 4PB_Size 1.5 IT_Size 1.0

IT_Size 1.5

(a)

@20ºC

0

2000

4000

6000

8000

10000

12000

0 2 4 6 8 10 12Frequency [Hz]

Com

plex

mod

ulus

[MP

a]



IT_Size 1.5

(b)

@30ºC

0

1000

2000

3000

4000

5000

6000

0 2 4 6 8 10 12Frequency [Hz]

Com

plex

mod

ulus

[MP

a]



IT_Size 1.5

(c)

Figure 5-24 Comparison of complex modulus at 10, 20 and 30 ºC


104

From these figures it can be seen that, like all viscoelastic materials, the complex modulus and phase angle of the asphalt mixture are significantly influenced by temperature and frequency. An increase in modulus is observed with increasing load frequency and decreasing temperature while the phase angle data show the opposite trend. To make a comparison of the stiffness values obtained with the different test methods, the complex stiffness obtained at temperature of 10, 20 and 30 ºC were plotted vs. the loading frequency; the results are shown in Figure 5-24. In general, the values of complex modulus measured by UT/C and 4PB test are very close at the same test condition and these values are not significantly influenced by specimen size. However the complex stiffness measured by the IT test is always smaller than those measured by the other two tests. The difference might be caused by the loading mode. The UT/C and 4PB test were conducted under a sinusoidal waveform loading (including both tension and compression), while in the IT test the haversine waveform was used. Furthermore, the IT specimen is subjected to a bi-axial stress condition. The rheological properties of the asphalt mixture can be described by a master curve using the time-temperature superposition principle. This principle allows shifting of the data obtained at various temperatures with respect to time or frequency to a selected reference temperature. The curve obtained in this way is plotted as a function of reduced time or frequency. The amount of shifting required at each temperature to the reference temperature is determined using the Williams-Landel-Ferry (WLF) equation [Ferry, 1980].

1 0

2 0

- C ( )

( )TT T

LogaC T T

−=+ − (5-26)

Where: C1 and C2 : model parameters; T : test temperature, ºC; T0 : reference temperature, ºC; aT : shift factor. The resulting master curves for the complex modulus and phase angle are described using a sigmoidal model similar to the one described by Pellinen, [Pellinen and Witczak 2002].

( ) ( ) ( ) ( )min max min 10 log

1log log log log 1 b

r

mix f

a

E E E E

e

+

= + − ⋅ −

(5-27)

( ) ( ) ( ) ( )min max min 10 log

1log log log log 1 b

r

mix f

a

θ θ θ θ

e

+

= + − ⋅ −

(5-28)


105

Where: Emix : mixture complex modulus, MPa; Emin : minimum modulus, MPa; Emax : maximum, MPa; θmix : mixture phase angle, degree; θmin : minimum phase angle, degree; θmax : maximum, degree; fr : reduced frequency, Hz; a, b : shape parameters. To construct the master curves of the complex modulus and phase angle, a reference temperature of 20ºC was chosen. The experimental data were then fitted to the sigmoidal-shape function given by Equation 5-27 and 5-28 in combination with a shift factor as determined by means of Equation5-26. All the model parameters or constants can automatically be obtained by minimizing the mean relative error using the Solver function in the Excel spreadsheet. All the model parameters for complex modulus and phase angle are presented in Table 5-8 and 5-9. The master curves shown in Figure 5-25, 5-26 and 5-27 give a perfect fit for all test data over a wide range of test conditions.

Table 5-8 Master curve parameters for complex modulus

Sigmoidal model WLF factors Model parameters

Emin [MPa] Emax [MPa] a b C1 C2 R2

UT/C-size0.5 0 23058 6.42 2.08 16.30 115.8 0.998 UT/C-size1.0 0 23058 6.39 2.08 15.10 111.2 0.998 UT/C-size1.5 0 18687 7.56 2.79 10.33 76.4 0.991 4PB-size0.5 0 23062 5.87 1.80 15.20 89.7 0.996 4PB-size1.0 0 23058 7.58 2.44 18.77 132.4 0.999 4PB-size1.5 4 23060 8.48 3.11 15.47 111.9 0.998 IT-size1.0 100 16061 9.97 4.86 10.37 72.6 0.998 IT-size1.5 72 22887 9.81 3.93 21.78 150.3 0.998

Table 5-9 Master curve parameters for phase angle

Sigmoidal model WLF factors Model parameters

θmin [º] θmax [º] a b C1 C2 R2

UT/C-size0.5 3.6 76.3 10.45 -4.37 16.30 115.8 0.997 UT/C-size1.0 2.4 77.7 10.97 -3.87 15.10 111.2 0.997 UT/C-size1.5 1.8 73.0 11.57 -3.68 10.33 76.4 0.997 4PB-size0.5 1.9 52.6 11.13 -3.71 11.94 75.8 0.996 4PB-size1.0 2.4 58.1 10.91 -5.10 18.77 132.4 0.999 4PB-size1.5 3.6 51.5 10.63 -5.85 15.47 111.8 0.996


106

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

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

Reduced Frequency [Hz]

Com

plex

mod

ulus

[MP

a]

0

20

40

60

80

100

Pha

se a

ngle

[deg

]UTC-size0.5-EUTC-size1.0-EUTC-size1.5-EUTC-size0.5-θUTC-size1.0-θUTC-size1.5-θ

Figure 5-25 Master curves obtained from UT/C tests at a reference temperature of 20ºC

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06Reduced Frequency [Hz]

Com

plex

mod

ulus

[MP

a]

0

20

40

60

Pha

se a

ngle

[deg

]

4PB-size0.5-E4PB-size1.0-E4PB-size1.5-E4PB-size0.5-θ4PB-size1.0-θ4PB-size1.5-θ

Figure 5-26 Master curves obtained from 4PB tests at a reference temperature of 20ºC

Figure 5-25 shows that the master curves of the different UT/C specimens overlap each other. It means that the complex stiffness and phase angle measured from the UT/C test are not dramatically influenced by the specimen size. The complex modulus measured with the 4PB test is also identical for different specimen sizes (see Figure 5-26). From the comparison of the master curves obtained by the IT test which are shown in Figure 5-27, one can conclude that the complex modulus of the size 1.5specimen is slightly higher than that of the size 1.0 specimen.


107

1.E+02

1.E+03

1.E+04

1.E+05

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06Reduced Frequency [Hz]

Com

plex

mod

ulus

[MP

a]

IT-size1.0 IT-size1.5

Figure 5-27 Master curves obtained from IT test at a reference temperature of 20ºC

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

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

Reduced Frequency [Hz]

Com

plex

mod

ulus

[MP

a]

UTC-size0.5 UTC-size1.0UTC-size1.5 4PB-size0.5

4PB-size1.0 4PB-size1.5IT-size1.0 IT-size1.5

Figure 5-28 Comparison of complex modulus master curves for different test methods at a reference temperature of 20ºC

The comparison of the complex modulus master curves for the different test methods and specimen sizes is shown in Figure 5-28. In most cases, the complex stiffness measured with the IT test is lower than the values measured with the UT/C and 4PB test, but at the lower reduced frequency the IT stiffness becomes larger than the UT/C and 4PB stiffness. To make the difference clear, Table 5-10 presents the predicted modulus from the master curve at several reduced frequencies. As mentioned before, the ABAQUS program was used to simulate the 4PBT. In order to be able to do so prony series parameters were required in the ABAQUS software. The


108

determination of the prony parameters based on the complex modulus master curves is demonstrated in Appendix B. Table 5-10 Comparison of the Predicted modulus values at the reference temperature of

20 ºC

Reduced frequency 0.1 Hz 1 Hz 10 Hz 100 Hz UT/C-size0.5 2054 5270 9840 14494 UT/C-size1.0 1909 5101 9734 14480 UT/C-size1.5 1740 4668 9257 13856 4PB-size0.5 2365 5170 9724 13813 4PB-size1.0 1960 4772 9018 13708 4PB-size1.5 1872 4615 9340 14512 IT-size1.0 1019 2562 5854 10430 IT-size1.5 1365 3240 6916 12181

5.6.2 Fatigue Test Results

5.6.2.1 Fatigue Life Definition

In this study, two different fatigue criteria are used to determine the fatigue life. (1) Classical fatigue definition. For the strain-controlled mode, the point of failure is defined as the moment at which the back calculated stiffness of the specimen has reduced to 50% of its initial value (stiffness at the 100th cycle), denoted as Nf,50. For the stress-controlled mode, the failure point is when the specimen has completely fractured, denoted as Nf. (2) Dissipated energy ratio. In Section 2.4.3, the dissipated energy theory was introduced in detail. The fatigue life is defined as the point at which the slope of the dissipated energy ratio (DER) versus number of load cycles deviates from a straight line [Pronk, 1991]. DER is the ratio of the accumulated dissipated energy up to cycle N and the dissipated energy in cycle N, given as Equation 5-29.

1

n N

in

N

wDER

w

=

==∑

(5-29)

sini i i iw π σ ε θ= (5-30) Where: DER : dissipated energy ratio; wi : dissipated energy in cyclei , J/m3; σi : stress amplitude in cyclei ; εi : strain amplitude in cyclei ; θi : phase lag in cyclei ; wN : dissipated energy in cycle N, J/m3.


109

Figure 5-29, Figure 5-30 and Figure 5-31 show examples of the determination of the classical fatigue life and the fatigue life based on DER for the different fatigue tests.

UT/C_ε=110µε, 20ºC

0

2000

4000

6000

8000

10000

0 100000 200000 300000 400000 500000 600000 700000 800000

Number of cycle

Stif

fnes

s [M

Pa]

0

200000

400000

600000

800000

1000000

1200000

1400000

DE

R

Stiffness DER

50% of initial stiffness

NR Nf,50

Figure 5-29 Fatigue life determination for the UT/C fatigue test in strain-controlled mode

4PB_σ=2.3 MPa, 5ºC

0

5000

10000

15000

20000

25000

30000

0 100000 200000 300000 400000 500000 600000 700000

Number of cycle

Stif

fnes

s [M

Pa]

0

10000

20000

30000

40000

50000

DE

R

StiffnessDER

Figure 5-30 Fatigue life determination for 4PB fatigue test in stress-controlled mode

NR Nf


110

IT-size1.0_σ=1.06 MPa

0

2000

4000

6000

8000

10000

12000

0 20000 40000 60000 80000 100000Number of cycles

Stif

fnes

s [M

Pa]

0

10000

20000

30000

40000

50000

60000

DE

R

Stiffness DER

Figure 5-31 Fatigue life determination for IT fatigue test in stress-controlled mode

5.6.2.2 UT/C Fatigue Test Results

Figure 5-32 shows the stiffness evolutions at different test conditions during the UT/C fatigue test. At 20 ºC, the stiffness curves are rather close for the different specimen sizes at similar strain level. It indicates that the effect of specimen size has no influence on the stiffness reduction for the UT/C fatigue test. The blue dots in the figure represent the stiffness evolution in controlled stress and strain mode at 5 ºC. It can be seen that, under similar initial strain, the initial stiffness in both loading modes is almost same, but the fatigue failure in the controlled stress mode occurred considerably earlier than in the controlled strain mode.

Similar results were also found based on the dissipated energy ratio. The dissipated energy ratio (DER) was plotted against number of cycles, as shown in Figure 5-33. For all the cases, the DER of the specimen increases linearly with the number of cycles following the same slope in the beginning. At 20 ºC, the deviation points of the three specimen sizes are rather close to each other. At the test temperature of 5 ºC, the deviation of the DER curve occurs much earlier at the same initial strain level and the specimen tested in stress controlled mode has the lowest fatigue life determined by DER.

NR Nf


111

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

10 100 1000 10000 100000 1000000Number of cycle

Stif

fnes

s [M

Pa]

size0.5_ε control_116 µm/m size1.0_ε control_118 µm/m


size1.0_σ control_117 µm/m

Figure 5-32 Stiffness evolutions from the UT/C fatigue tests

0

200000

400000

600000

800000

1000000

0 100000 200000 300000 400000 500000 600000Number of cycle

DE

R


size1.5_ε control_116 µm/m size1.0_ε control_118 µm/msize1.0_σ control_117 µm/m

Figure 5-33 DER evolutions from the UT/C fatigue tests

The fatigue lives based on different definitions are presented in Table 5-11. In nearly all cases the fatigue life NR is smaller than the classical fatigue life. At 20 ºC the differences between these two numbers are smaller than at 5 ºC. As discussed in Chapter 2, fatigue relationships for an asphalt mixture are usually represented by means of a straight line in a double logarithmic coordinate system.

T=20 ºC

T=5 ºC

T=20 ºC

T=5 ºC


112

0b

fN k= ⋅ε (5-31)

Where: Nf : fatigue life; ε0 : strain amplitude, [µm/m]; k, b : material coefficients

Table 5-11 Fatigue life of the different specimen sizes in the UT/C fatigue test

Specimen size

Loading mode

Sample code

Strain level [µm/m]

Nf,50 or Nf NR ,50

,50

f R

f

N N

N

−

C-3-18 88 1.6×106 1.5×106 6.3% C-3-25 117 5.4×105 5.2×105 3.7% C-3-26 130 3.0×105 2.7×105 10.0% C-3-9 136 2.8×105 2.6×105 7.1% C-3-24 138 2.3×105 2.2×105 6.9% C-3-4 146 1.3×105 1.0×105 24.2% C-3-11 168 9.0×104 7.5×104 16.7%

Size 0.5 at 20 ºC

C-3-2 186 5.6×104 5.4×104 3.6% C-10-12 68 6.0×106 5.2×105 13.3% C-10-10 84 1.9×106 1.8×105 2.7% C-10-9 110 5.5×105 4.8×105 11.9% C-10-8 131 2.8×105 2.3×105 17.9% C-10-5 155 1.1×105 1.1×105 1.9% C-10-11 172 7.2×104 7.0×104 2.1%

Size 1 at 20 ºC

C-10-7 199 3.2×104 3.4×104 -6.3% C-14-1 68 4.3×106 3.8×106 12.2% C-13-1 89 1.3×106 1.2×106 5.5% C-11-2 115 4.1×105 3.4×105 16.0% C-9-1 133 1.9×105 1.5×105 23.7% C-12-2 161 7.0×104 8.0×104 -14.3%

Size 1.5 at 20 ºC

Strain-controlled

C-11-1 182 4.0×104 4.6×104 -15.0% C-15-4 69 2.7×106 2.3×106 15.1% C-15-11 87 1.5×106 1.2×106 20.7% C-16-5 94 6.6×105 5.6×105 15.2% C-16-10 105 2.8×105 2.3×105 18.1% C-15-6 118 1.6×105 1.4×105 31.7% C-15-9 124 1.6×105 1.2×105 28.6%

Strain-controlled

C-16-10 136 8.3×104 8.2×104 0.6% C-15-5 68 1.5×106 1.2×106 23.6% C-16-9 73 1.2×106 7.0×105 40.0% C-16-2 75 8.4×105 5.2×105 38.3% C-15-7 79 4.7×105 3.6×105 24.1% C-15-8 86 2.9×105 1.9×105 34.5% C-15-3 94 1.6×105 1.1×105 35.4%

Size 1 at 5 ºC

Stress-controlled

C-15-10 118 6.3×104 4.4×104 29.7%


113

Figure 5-34 shows the plot of the classical fatigue life Nf,50 against the initial strain for all the UT/C fatigue tests. The regression coefficients in Equation 5-31 are presented in Table 5-12. It is clear that the fatigue lines of the different specimen sizes are almost identical at the same temperature. With decreasing temperature, the fatigue life becomes shorter at the same initial strain level and the slope of the fatigue lines, b, becomes somewhat larger.

1.E+04

1.E+05

1.E+06

1.E+07

10 100 1000Strain level [ µm/m]

Nf,

50

Size0.5_ε control_20CSize1.0_ε control_20CSize1.5_ε control_20CSize1.0_ε control_5CSize1.0_σ control_5C

Figure 5-34 Fatigue life vs. initial strain level for the UT/C fatigue tests

Table 5-12 Regression coefficients of the fatigue lines

Material coefficients Specimen size Loading mode k b

R2

Size 0.5@20ºC 1.5×1015 -4.60 0.98 Size 1.0@20ºC 5.7×1014 -4.43 0.99 Size 1.5@20ºC

Strain-controlled 1.4×1015 -4.64 1.00

Size 1.0@5ºC Strain-controlled 2.8×1016 -5.40 0.97 Size 1.0@5ºC Stress-controlled 1.6×1017 -6.03 0.94

Compared to the strain controlled mode, the specimens have a shorter fatigue life in the stress controlled mode. This phenomenon can be explained by the dissipated energy theory. In controlled strain tests, a strain with constant amplitude is applied to the specimen and the stress decreases gradually. The dissipated energy per cycle decreases during the testing, as shown in Figure 5-35. In the stress controlled mode, the amplitude of the applied stress stays constant, which causes an increase of the strain amplitude and an increase in the dissipated energy. The strain increases rapidly at the end of the test, till complete fracture occurs.


114

Figure 5-35 Dissipated energy per cycle versus number of cycles in the stress and strain controlled UT/C test

5.6.2.3 4PB Fatigue Test Results As mentioned in Section 5.3.2, the measured displacement was corrected to calculate the complex stiffness. The measured stiffness calculated from the measured deflection vm and the stiffness corrected from the bending deflection, vb, are presented in Table 5-13. Also Figure 5-36 gives information about this.

0

4000

8000

12000

16000

20000

24000

50 100 150 200 250Bending strain [ µm/m]

Initi

al s

tiffn

ess

[MP

a]

Size0.5_measured_20°CSize0.5_corrected_20°CSize1.0_measured_20°CSize1.0_corrected_20°CSize1.5_measured_20°CSize1.5_corrected_20°CSize1.0_measured_5°CSize1.0_corrected_5°C

Figure 5-36 Comparison of measured and corrected initial stiffness in the 4PB test


115

Figure 5-36 shows that the initial stiffness is independent of the strain level. The measured stiffness is always lower than the corrected value. For size 0.5, the bending deflections are close to the measured values, which indicates that during the test the frame and shear deflection are very small. This is because the specimen size 0.5 has a low h/L ratio. With the increase of specimen size, the shear deflection becomes larger and also the deformation of the test frame is playing a more important role because of the higher loads that need to be applied to obtain a certain strain level. So for size 1.5, the differences between corrected and measured stiffness become significant, more than 30 %. At lower temperatures, also a larger force is needed to obtain a certain strain level, so the frame deformation has a significant influence on the measurement. That is why the corrected stiffness is 25 % higher than the measured value. After the deflection correction, the difference of the calculated stiffness between the three specimen sizes is much smaller compared to the measured values.

Table 5-13 Comparison of measured and corrected results in strain controlled mod

Specimen size

Sample code

Stress [MPa]

vm [mm]

vb [mm]

Measured E [MPa]

Corrected E [MPa]

corrected measured

corrected

E E

E

−

B-19-5 0.97 0.135 0.131 9410 9989 5.8% B-14-5 1.17 0.160 0.156 9456 9987 5.3% B-19-2 1.36 0.191 0.187 9514 9908 4.0% B-14-6 1.50 0.218 0.213 9062 9586 5.5% B-19-6 1.69 0.238 0.233 9210 9601 4.1%

Size 0.5 @20ºC

B-19-4 1.81 0.272 0.266 8954 9246 3.2% B-15-2 0.72 0.067 0.058 7282 8857 17.8% B-16-1 0.91 0.082 0.071 7590 8908 14.8% B-16-2 1.06 0.097 0.085 7555 8879 14.9% B-17-4 1.22 0.110 0.096 7600 8976 15.3% B-16-4 1.40 0.124 0.109 7803 9103 14.3%

Size 1.0 @20ºC

B-17-3 1.57 0.140 0.123 7787 9173 15.1% B-20-2 0.61 0.050 0.033 6105 8545 28.5% B-23-1 0.70 0.058 0.041 5658 8430 32.9% B-22-2 0.80 0.063 0.042 5833 9077 35.7% B-22-1 0.80 0.063 0.041 5955 9099 34.6% B-20-1 0.77 0.066 0.046 5134 8225 37.6%

Size 1.5 @20ºC

B-21-2 0.89 0.077 0.053 5365 8146 34.1% B-7-2 1.75 0.079 0.060 14527 19622 26.0% B-18-3 1.80 0.086 0.065 14183 18855 24.8% B-15-3 1.84 0.092 0.070 13987 18517 24.5% B-18-1 2.12 0.099 0.074 14491 19496 25.7% B-15-4 2.25 0.105 0.078 14069 19601 28.2% B-18-4 2.48 0.112 0.085 15025 20467 26.6%

Size 1.0 @5ºC

B-15-1 2.46 0.117 0.090 14298 19639 27.2%


116

The stiffness evolutions of specimens with different sizes at a similar strain are shown in Figure 5-37. It can be seen that the fatigue behavior in the 4PB fatigue tests also depends on temperature and loading mode. In contrast to the UT/C test results, the effect of specimen size becomes distinct. At the same loading level, the stiffness of a larger specimen drops more quickly. These results also can be found on the plot of the dissipated energy ratio, as shown in Figure 5-38.

0

5000

10000

15000

20000

25000

10 100 1000 10000 100000 1000000 10000000Number of cycle

Stif

fnes

s [M

Pa]

size0.5_ε control_100 µm/m size1.0_ε control_105 µm/msize1.5_ε control_116 µm/m size1.0_ε control_109 µm/msize1.0_σ control_108 µm/m

Figure 5-37 Stiffness evolutions from the 4PB fatigue test in strain and stress controlled

mode

0

100000

200000

300000

400000

500000

600000

700000

800000

0 1000000 2000000 3000000 4000000 5000000 6000000Number of cycle

DE

R

size0.5_ε control_100 µm/m size1.0_ε control_105 µm/msize1.5_ε control_116 µm/m size1.0_ε control_109 µm/msize1.0_σ control_108 µm/m

Figure 5-38 DER evolutions from the 4PB fatigue tests in strain and stress controlled

mode

T=20 ºC

T=5 ºC

T=20 ºC

T=5 ºC


117

The classical fatigue life and the fatigue life based on DER, NR, are presented in Table 5-14. As was the case for the UT/C results, the value of NR obtained with the 4PB test is always smaller than the classical fatigue life, but the differences between them are much larger in most of the cases compared to the differences obtained in the UT/C test.

Table 5-14 Fatigue life of the different specimen sizes in the 4PB fatigue test

Specimen size

Loading mode

Sample code


Nf,50 or Nf NR ,50

,50

f R

f

N N

N

−

B-19-5 100 5.7×106 3.1×106 44.8% B-14-5 119 1.3×106 6.8×105 45.9% B-19-2 140 8.5×105 4.3×105 49.5% B-14-6 160 3.8×105 1.6×105 57.5% B-19-6 179 2.2×105 9.5×104 56.4%

Size 0.5 at 20 ºC

B-19-4 201 2.1×105 7.3×104 64.8% B-15-2 87 4.0×106 3.2×106 20.6% B-16-1 105 1.9×106 1.0×106 47.8% B-16-2 124 6.6×105 4.2×105 36.5% B-17-4 142 4.2×105 2.7×105 36.8% B-16-4 161 3.1×105 1.8×105 43.0%

Size 1 at 20 ºC

B-17-3 180 1.9×105 1.1×105 45.6% B-20-2 74 7.9×106 6.2×106 22.1% B-23-1 88 1.7×106 1.5×106 11.7% B-22-2 93 1.6×106 1.0×106 35.9% B-20-1 103 2.5×106 1.7×106 30.2%

Size 1.5 at 20 ºC

Strain-controlled

B-21-2 119 5.8×105 5.0×105 14.0% B-7-2 89 2.0×106 1.7×106 14.5% B-15-3 105 1.6×106 1.0×106 37.8% B-18-1 109 1.3×106 9.1×105 27.2% B-15-4 116 3.4×105 2.8×105 18.0% B-18-4 124 4.7×105 4.2×105 10.7%

Strain-controlled

B-15-1 135 4.1×105 3.4×105 16.4% B-11-4 87 2.4×106 1.6×106 34.2% B-11-1 95 1.7×106 8.9×105 48.8% B-11-2 93 1.9×106 8.3×105 55.5% B-7-3 98 9.1×105 3.2×105 65.1% B-7-4 97 7.3×105 2.5×105 65.4% B-7-1 99 6.0×105 2.1×105 64.6%

Size 1 at 5 ºC

Stress-controlled

B-16-3 108 5.8×105 1.9×105 67.3% The fatigue lines obtained for size 0.5, 1.0 and 1.5 specimens are plotted in Figure 5-39 and all the regression coefficients are gathered in Table 5-15. The effect of specimen size can be observed from the comparison of the fatigue lines. A specimen with a larger height has a shorter fatigue life at the same loading level. One of the reasons is that the shear deflection is related to the height/length (h/l) ratio of the beam specimen, as shown in Equation 5-4. This implies that at the same bending strain level, the shear strain


118

between the inner supports is larger for the specimen with a high h/l ratio, resulting in more damage during the test. The slope and the intercept of the fatigue line increase with a decrease of temperature. In terms of the correlation coefficients, the variation of fatigue results for specimen size 1.5 and low temperature is relatively high. The reason might be that the range of applied loading levels in these cases is not wide enough.

1.E+05

1.E+06

1.E+07

10 100 1000Bending strain [ µm/m]

Nf,5

0

Size0.5_ε control_20CSize1.0_ε control_20CSize1.5_ε control_20CSize1.0_ε control_5CSize1.0_σ control_5C

Figure 5-39 Fatigue life vs. initial strain level for the 4PB fatigue tests

Table 5-15 Regression coefficients of the fatigue lines for the 4PB test


R2

Size 0.5@20ºC 1.1×1016 -4.72 0.95 Size 1.0@20ºC 5.3×1014 -4.20 0.98 Size 1.5@20ºC

Strain-controlled 3.6×1015 -4.69 0.78

Size 1.0@5ºC Strain-controlled 2.0×1015 -4.58 0.74 Size 1.0@5ºC Stress-controlled 1.2×1022 -8.08 0.76

As mentioned in section 5.2.2, the 4PB test setup was modified to avoid stress concentration near the inner clamps. Figure 5-40 gives the comparison of the fatigue lines from the modified and unmodified test setup for each specimen size. It can be seen from the figures, that for size 0.5 the fatigue lines from the two setups are very close. However, with the increase of the specimen size, the fatigue life from the unmodified setup decreases dramatically at lower strain levels compared to that from the modified setup at the low loading range so that the slope of the fatigue line decreases with the increase of the specimen size.


119

y = 5E+15x -4.6029

R2 = 0.9601

y = 1E+16x -4.723

R2 = 0.9532

1.E+05

1.E+06

1.E+07


Nf,

50

Size0.5-unmodified

Size0.5-modified

(a) Specimen size 0.5

y = 3E+11x -2.7383

R2 = 0.9454

y = 5E+14x -4.2037

R2 = 0.9838

1.E+05

1.E+06

1.E+07

1.E+08


Nf,

50

Size1.0-unmodified

Size1.0-modified

(b) Specimen size 1.0

y = 2E+10x -2.1199

R2 = 0.919

y = 4E+15x -4.6946

R2 = 0.7783

1.E+05

1.E+06

1.E+07


Nf,

50

Size1.5-unmodified

Size1.5-modified

(c) Specimen size 1.5

Figure 5-40 Comparison of fatigue results between modified and unmodified 4PB setup


120

5.6.2.4 IT Fatigue Test Results

During the IT fatigue test, a compressive load is applied on the specimen along its vertical diameter. Therefore permanent deformation will develop even at a low temperature of 5ºC. Figure 5-41 shows the load and deformation responses in the 50th and the 60th cycle. In the figure, Dr is defined as the recoverable deformation in the 50th cycle and Dp represents the permanent deformation that developed between the 50th and 60th loading cycle.

0

2

4

6

8

10

4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4

time [s]

For

ce [k

N]

0

0.002

0.004

0.006

0.008

0.01

Rad

ial d

efor

mat

ion

[mm

]

ForceRadial deformation

Figure 5-41 Force and deformation response in the 50th and the 60th cycle

Figure 5-42 shows the horizontal recoverable deformation and vertical permanent deformation for size 1.5. The horizontal recoverable deformation Dr at the various stress levels stays stable in the beginning of a test, because in this period, the material behavior can be considered linear viscoelastic. After a certain number of cycles, Dr increases rapidly and the specimen is damaged in a short time. The vertical permanent deformation, Dp always increases during the test.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

10 100 1000 10000 100000 1000000

Number of cycles

Hor

izon

tal r

ecov

erab

le d

efor

m.

[mm

]

I-3-4_0.56MPa I-2-6_0.61MPaI-4-4_0.64MPa I-3-6_0.69MPaI-2-7_0.83MPa I-4-7_0.88MPaI-4-6_0.93MPa

(a) Horizontal recoverable deformation

Dr

Dp


121

0

1

2

3

4

5

6

7

10 100 1000 10000 100000 1000000Number of cycles

Ver

tical

per

man

ent d

efor

m.

[mm

]

I-3-4_0.56MPa I-2-6_0.61MPaI-4-4_0.64MPa I-3-6_0.69MPa

I-2-7_0.83MPa I-4-7_0.88MPaI-4-6_0.93MPa

(b) Vertical permanent deformation

Figure 5-42 Recoverable and permanent deformation for size 1.5 at 20 ºC

Figure 5-43 gives the evolution of the resilient modulus for the specimen with size 1 and size 1.5 at the various stress levels. Similar to the recoverable deformation results, the material behaves linear elastic in the beginning, as the stiffness stays constant with increasing number of load repetitions. When cracks start to initiate and propagate inside the specimen, the stiffness decreases quickly. Table 5-16 presents the initial stiffness and fatigue life at all the stress levels for both specimen sizes. In general, the initial stiffness of the specimen size 1.5 is a little higher than that of specimen size 1.0, which was also found in the stiffness tests.

0

2000

4000

6000

8000

10000

10 100 1000 10000 100000 1000000 10000000

Number of cycles

Res

ilien

t Stif

fnes

s [M

Pa]

I-4-3_0.46MPaI-3-1_0.49MPaI-1-4_0.58MPaI-1-9_0.71MPaI-1-5_0.89MPaI-1-7_1.06MPaI-1-6_1.28MPa

(a) Size 1.0


122

0

2000

4000

6000

8000

10000

12000

1 10 100 1000 10000 100000 1000000Number of cycles

Res

ilien

t stif

fnes

s [M

Pa]

I-3-4_0.56MPaI-2-6_0.61MPaI-4-4_0.64MPaI-3-6_0.69MPaI-2-7_0.83MPaI-4-7_0.88MPaI-4-6_0.93MPa

(b) Size 1.5

Figure 5-43 Stiffness evolutions for the specimen size 1 (a) and size 1.5 (b) at 5 ºC The fatigue results of the two specimen sizes, based on the initial tensile strain plotted against the number of cycles to failure are shown in Figure 5-44. The regression coefficients of the fatigue functions are presented in Table 5-17. The fatigue lines of size 1.0 and size 1.5 appear to have a similar slope. The fatigue life of the size 1.5 specimen is slightly lower than that of the size 1.0 specimens, but the effect of specimen size is not as significant as for the 4PB fatigue results.

Table 5-16 Initial stiffness and fatigue life for two specimen sizes at 5 ºC

Sample

code σt [MPa]

Initial εt [µm/m]

Initial E [MPa]

Nf NR f R

f

N N

N

−

I-4-3 0.46 51 9361 1.8×106 9.0×105 50.4% I-3-1 0.49 57 9010 7.4×105 4.0×105 45.8% I-1-4 0.58 61 9551 1.0×106 4.5×105 56.8% I-1-9 0.71 78 9381 4.4×105 2.0×105 55.0% I-1-5 0.90 97 9282 1.5×105 1.1×105 28.9% I-1-7 1.06 116 9170 8.4×104 5.5×104 34.3%

Size 1

I-1-6 1.28 143 9005 3.0×104 2.0×104 32.5% I-3-4 0.55 60 9536 7.7×105 4.0×105 48.1% I-2-6 0.61 59 10213 6.9×105 4.3×105 37.7% I-4-4 0.64 70 8474 4.5×105 2.2×105 51.6% I-3-6 0.69 72 9620 3.4×105 1.8×105 47.2% I-2-7 0.83 87 9493 1.6×105 8.0×104 50.2% I-4-7 0.88 89 9820 1.3×105 6.0×104 53.7%

Size 1.5

I-4-6 0.93 95 9893 9.4×104 6.0×104 36.1%


123

1.E+04

1.E+05

1.E+06

1.E+07


Num

ber

of c

ycle

s

Size1

Size1.5

Figure 5-44 Fatigue lines for specimen size 1.0 and size 1.5 in IT fatigue tests

Table 5-17 Regression coefficients of the fatigue lines for the IT test


R2

Size 1.0@5ºC Stress-controlled 6.4×1012 -3.84 0.99 Size 1.5@5ºC Stress-controlled 4.0×1013 -4.33 0.98

5.6.2.5 Comparison of Different Fatigue Results

In the above discussion, the UT/C, 4PB and IT fatigue tests results were presented obtained on specimens with different specimen sizes. To explore the effect of the test type on the fatigue tests, the fatigue lines at the same test conditions obtained with the different fatigue tests are compared. Figure 5-45, Figure 5-46 and Figure 5-47 summarize the fatigue results obtained from the UT/C, 4PB and IT fatigue tests. To make a clear comparison, Table 5-18 presents the fatigue life at an initial strain of 100 µm/m.

1.E+04

1.E+05

1.E+06

1.E+07


Nf,

50

UT/C_Size0.5 UT/C_Size1.0

UT/C_Size1.5 4PB_Size0.5

4PB_Size1.0 4PB_Size1.5

Figure 5-45 Comparison of fatigue lines in strain-controlled mode at 20ºC and 10 Hz


124

1.E+04

1.E+05

1.E+06

1.E+07

0.1 1 10Stress level [MPa]

Nf

UT/C_Size1.0@5C4PB_Size1.0@5CITT_Size1.0@5CITT_Size1.5@5C

Figure 5-46 Comparison of fatigue lines in stress-controlled mode at 5ºC from UT/C and

4PB and IT fatigue tests

1.E+04

1.E+05

1.E+06

1.E+07

10 100 1000Initial strain [ µm/m]

Nf

UTC Size0.5_ε_20°C


UTC Size1.5_ε_20°CUTC Size1.0_ε_5°C

UTC Size1.0_σ_5°C

4PB Size0.5_ε_20°C



4PB Size1.0_ε_5°C4PB Size1.0_σ_5°C

IT Size1.0_σ_5°C

IT Size1.5_σ_5°C

Figure 5-47 Comparison of fatigue lines obtained from all the three fatigue tests

Table 5-18 Comparison of the fatigue life at the initial strain of 100 µm/m

Test type Fatigue life Test type Fatigue life UT/C-size0.5_ε_20 ºC 8.7×105 4PB-size0.5_ε_20 ºC 4.1×106 UT/C-size1.0_ε_20 ºC 7.8×105 4PB-size1.0_ε_20 ºC 2.1×106 UT/C-size1.5_ε_20 ºC 7.3×105 4PB-size1.5_ε_20 ºC 1.5×106 UT/C-size1.0_ε_5 ºC 4.5×105 4PB-size1.0_ε_5 ºC 1.4×106 UT/C-size1.0_σ_5 ºC 1.4×105 4PB-size1.0_σ_5 ºC 8.3×105

IT-size1.0_ε_5 ºC 1.3×105 IT-size1.5_ε_5 ºC 8.6×104


125

For both strain-controlled and stress-controlled mode, the 4PB test gives the longest fatigue life for all the test types and the IT test shows the shortest life duration. Similar results were also found in previous research [Porter, 1975] [Di Benedetto, 2004] [Molenaar, 1983]. The difference in fatigue life obtained from different test types can be explained by the stress-strain distribution in the specimen during the test. In the 4PB test, the stress and strain values vary along the length and height. The maximum strain only occurs at the specimen surface between the two inner supports, while in the UT/C test, the stress-strain field is relatively uniform. The target stress or strain level is distributed throughout most of the cylindrical specimen. In the IT test, a haversine signal was used, while the stress signal in both the 4PB and UT/C test was a full sine, implying alternating tension/compression stresses. Furthermore, the specimen in the IT test is subjected to a biaxial stress state (vertical and horizontal stresses). Therefore, more damage was caused by this complicated loading configuration. In addition to the fatigue damage accumulation of permanent deformation occurs in the IT fatigue test.

5.6.2.6 Dissipated Energy Theory

As discussed in the literature review, the area inside the stress–strain hysteresis loop for a loading–unloading process indicates the amount of dissipated energy for one loading cycle. Figure 4-48 shows the development of the stress–strain hysteresis loop for the UT/C, 4PB and IT fatigue tests in the stress-controlled mode. For the UT/C and 4PB fatigue test, the stress-strain loop rotates around the origin and its area becomes larger with the number of loading cycles. In contrast, during the IT fatigue test, the center of the specimen is only subjected to a tensile stress in the horizontal direction, the loop moves along the strain axis due to the irreversible deformation.

-3

-2

-1

0

1

2

3

-0.0002 -0.0002 -0.0001 -5E-05 0 5E-05 0.0001 0.0002 0.0002

Strain [ µm/m]

Stre

ss [M

Pa]

100 cycle

80000 cycle

160000 cycle

(a) UT/C fatigue test: C-15-3


126

-3

-2

-1

0

1

2

3

-0.0002 -0.0002 -0.0001 -5E-05 0 5E-05 0.0001 0.0002 0.0002

Strain [ µm/m]

Stre

ss [M

Pa]

100 cycle

800000 cycle

1700000 cycle

(b) 4PB fatigue test: B-11-1

0

0.5

1

1.5

0 0.0001 0.0002 0.0003 0.0004

Strain [ µm/m]

Stre

ss [M

Pa]

50 cycle 100 cycle 300 cycle

(c) IT fatigue test: I-1-5

Figure 5-48 Development of strain-stress hysteresis loops for the different fatigue tests It is generally believed that the fatigue life is related to the accumulated dissipated energy. In this section the relationship between fatigue life NR and the total dissipated energy till NR is investigated. NR is determined by the evolution of the dissipated energy and represents the number of cycles to crack initiation. As shown in Figure 5-32, the evolution of the dissipated energy ratio DER in the stress controlled and strain controlled mode follows the same slope till crack initiation. Therefore it is believed that the fatigue life NR for the different loading modes should be comparable. The total dissipated energy is given as follows:

1 1

sinR RN N

fat i i i ii i

W W π σ ε φ= =

= =∑ ∑ (5-32)

Horizontal strain [ µm/m]


127

Where: Wfat : total dissipated energy till NR, J/m3; Wi : dissipated energy at cycle i, J/m3; NR : fatigue life based on DER; σi : stress amplitude at cycle i, MPa; εi : strain amplitude at cycle i, m/m Φi : phase angle at cycle i. Based on previous research [van Dijk, 1972], the following relation was obtained:

( )z

fat RW A N= ⋅ (5-32)

Where: A and z : experimentally determined coefficients Figure 5-49 shows the relationships between total dissipated energy and fatigue life NR obtained from the different fatigue tests. Table 5-19 presents all the regression coefficients. In Figure 5-49a, the NR-Wfat lines from the UT/C test are not influenced by the specimen size like the traditional fatigue analysis, but the specimen size still has influence on the NR-Wfat lines obtained from the 4PB test. From Figure 5-49b, it can be seen that the fatigue relations based on the total dissipated energy obtained from the UT/C and 4PB test are close and both of them are influenced by the loading mode. Compared to the strain controlled mode, the slope of the curves in the stress-controlled mode is a little larger, but the effect of temperature on the fatigue line is negligible. While the fatigue life NR from the IT fatigue test is much shorter at consuming the same amount of total dissipated energy. One of the reasons is that the damage of the IT specimen is not only caused by the dissipated energy, but also by the irreversible deformation (see Figure 5-48c).

(a) Fatigue lines from the UT/C and 4PB test with different specimen sizes


128

(b) Fatigue lines from the different tests at different temperatures

Figure 5-49 Total dissipated energy versus fatigue life NR

Table 5-19 Coefficients of the NR-Wfat lines from the different fatigue tests

Coefficients Specimen size Loading mode A Z

R2

UTC-Size 0.5@20ºC 0.042 0.561 0.99 UTC-Size 1.0@20ºC 0.062 0.523 1.00 UTC-Size 1.5@20ºC 0.030 0.578 1.00 UTC-Size 1.0@5ºC

Strain-controlled

0.015 0.541 0.99 UTC-Size 1.0@5ºC Stress-controlled 0.008 0.658 1.00 4PB-Size 0.5@20ºC 0.025 0.624 0.99 4PB-Size 1.0@20ºC 0.053 0.547 0.98 4PB-Size 1.5@20ºC 0.005 0.691 0.96 4PB-Size 1.0@5ºC

Strain-controlled

0.039 0.558 0.96 4PB-Size 1.0@5ºC Stress-controlled 0.001 0.796 0.99 IT-Size 1.0@5ºC Stress-controlled 0.108 0.304 0.96 IT-Size 1.5@5ºC Stress-controlled 0.030 0.406 0.96

The question now is why the NR-Wfat relation of the IT test is so different from the ones of the other tests. In the UT/C and 4PB test, the outer layer of the specimen is alternately subjected to tension and compression. The dissipated energy calculated for these test consists of a "tensile" and "compressive" part. The dissipated energy calculated from the IT test results is only due to tension. The effect of the dissipated energy due to the high compressive strains in the vertical direction is not taken into account. The influence of this component was investigated and is reported hereafter. Figure 5-50 gives a comparison of the products of the stress and strain in horizontal and vertical direction for the IT test. The values of the horizontal stress σx, horizontal stress εx,


129

vertical stress σy and vertical strain εy are calculated with the ABAQUS software. The details are discussed in Chapter 8. From the figure, it can be seen that the product of the vertical stress and strain is much larger than that of the tension part. Assuming the phase angle in vertical direction is the same as that in horizontal direction, the total dissipated energy in vertical and horizontal direction together is around 7 times larger than that only in horizontal direction. A correction of NR vs. Wfat relationship of the IT test with this factor moves it very close to the other relationships, but the slopes of the curves are still smaller than the other two tests (see Figure 5-51). Furthermore, the low dissipated energy in the IT tests is caused by the measured phase angle. Figure 5-52 gives a comparison of the initial phase angle measured for the different test. At various stress levels, the phase angles measured by the IT test are always smaller than those measured by the UT/C and 4PB test. The lower phase angle causes a lower total dissipated energy as calculated by means of Equation 5-32.

0

0.00005

0.0001

0.00015

0.0002

0.00025

σx*εx σy*εy σx*εx+σy*εy

σ*ε

Figure 5-50 Products of the stress and strain at the center of the specimen I-3-1

Figure 5-51 Total dissipated energy versus fatigue life NR after correcting the total

dissipated energy for the IT test


130

0

5

10

15

20

0 0.5 1 1.5 2 2.5Stress [MPa]

Pha

se a

ngle

[deg

]

UT/C-size1.0@5°C

4PB-size1.0@5°C

IT-size1.0@5°C

Figure 5-52 Comparison of the phase angle from the different test in stress-controlled

mode

5.7 Summary

5.7.1 Test Setup In this Chapter, asphalt specimens with different specimen sizes were tested using the UT/C, modified 4PB and IT fatigue test. With regard to the test setups, the following conclusions were reached: 1. The test configuration of the UT/C test is relatively simple, but the requirements for

specimen preparation and test setup are very strict. To prevent bending failure, the upper and lower steel platens should be parallel to each other and perpendicular to the loading direction. Likewise the polished top and bottom ends of the cylindrical specimen should be exactly parallel to each other and perpendicular to the axis of the cylinder.

2. Compared to the UT/C and IT test, the measurements obtained from the 4PB test are highly influenced by the test equipment. To avoid high shear and tensile stress concentrations near the supports, the clamps should be allowed to rotate freely as well as translate freely at the four supports. Therefore the clamping force should be as low as possible. Another important issue is the calibration of the 4PB test setup. To calculate the stiffness of the specimen, shear deflection of beam and deflection of the frame should be taken into account to get the pure bending deflection of the beam.


131

5.7.2 Test Results Based on the results discussed in this chapter, the following conclusions can be drawn: 1. The complex stiffness measured by means of the UT/C and 4PB test is not

dramatically influenced by the specimen size and the stiffness mastercurves from these two tests are very close to each other. For the IT test, the larger specimens showed have a higher complex stiffness. However the complex stiffness measured by the IT test is always smaller than the complex stiffness measured by the other two tests. The difference is most probably caused by the difference in the mode of loading and the state of stress in the specimen

2. In the UT/C test, the fatigue life NR based on the dissipated energy ratio is smaller than the classical fatigue life Nf,50. Due to the homogeneous stress-strain field, the effect of specimen size on the fatigue life of the cylindrical specimen at the same temperature is not significant. At a lower temperature, the fatigue life becomes shorter and the slope of the fatigue lines becomes steeper.

3. The difference between the NR and Nf,50 fatigue values obtained from the 4PB test is larger compared to the UT/C test. The 4PB fatigue results are significantly influenced by the specimen size. The beam with a higher height/length ratio has a shorter fatigue life at the same loading level. The slope and the intercept of the fatigue line increase at a lower temperature.

4. For the IT test, the effect of specimen size on the fatigue results is not significant.

5. Based on the comparison of the fatigue results between the different test methods, at the same test condition and loading mode, the fatigue life from the 4PB test is the longest and from the IT test is the shortest. The differences can be explained by the difference in stress-strain distributions of the specimens in the different fatigue tests.

6. With regard to the NR-Wfat relation, the effect of specimen size on the 4PB test is obvious, but not for the UT/C and IT test. The NR-Wfat line is independent of temperature but influenced by the loading mode. Compared to the strain controlled mode, the slope of the curves in the stress-controlled mode is a little steeper. The total dissipated energy measured by the IT test is lower compared to the UT/C and 4PB tests at the same fatigue life NR. This difference is caused by the irreversible deformation, biaxial stress state at the center of the specimen and the smaller phase angle measured by the IT test.


132

References Di Benedetto H., de la Roche C., Baaj H., Pronk A. and Lundstrom R., Fatigue of bituminous mixtures. Materials and Structures, 2004, 37, No. 3, 202–216. EN 12697-24: 2003 Bituminous mixture – Test methods for hot mix asphalt - Part 24: Resistance to fatigue. European committee for standardization, Brussels. Ferry, J.D., Viscoelastic properties of Polymers. 3rd edition. New York, 1980. Hondros, G.. 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. Austr. J. Appl. Sci. 1959, 10(3): 243-268. Huurman, M., Pronk, A.C., Theoretical analysis of the 4 point bending test Proceedings of the 7th Int. RILEM Symposium Advanced Testing and Characterization of Bituminous Materials, May 2009, Rhodes, Greece. Li, N., Molenaar, A.A.A., Pronk A.C., van de Ven, M.F.C. and Wu, S., Effect of Specimen Size on Fatigue Behavior of Asphalt Mixture in Laboratory Fatigue Tests, Proceedings of the 7th RILEM International Conference on Cracking in Pavements, Delft, the Netherlands, pp 827-836, 2012. Li, Ning, Molenaar, A.A.A., Pronk A.C., van de Ven, M.F.C.. Investigation into the Size effect on Four Point Bending Fatigue Tests. Proceedings of the 3rd Workshop on 4PB. Davis, California. 2012. Pronk A.C. and Huurman M., Shear deflection in 4PB tests. 2nd Workshop on Four Point Bending, University of Minho, Guimaraes, Portugal, 2009. Pronk, A. C. and Hopman P. C., Energy dissipation: the leading factor of fatigue, Proceedings of the Conference on the United States Strategic Highway Research Program, 1991, pp. 255–267. Porter, B. W., Kennedy, T. W., Comparison of the fatigue test methods for asphalt materials. Research report 183-4, Center for Highway Research, University of Texas at Austin. 1975. Pellinen, T.K., and Witczak, M.W.. “Stress Dependent Master Curve Construction for Dynamic (Complex) Modulus.” Journal of the Association of Asphalt Paving Technologists, Volume 71. Colorado Spring, CO, 2002. van Dijk, W. Moreaud, H., et al. The Fatigue of Bitumen and Bituminous Mixes, 3th International Conference on Structural Design of Asphalt Pavements (ISAP), London, Volume 1. 1972.


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6.1 Introduction In Chapter 5 it was shown that the different fatigue tests give different fatigue results. The main reason for this is the difference in the stress-strain field in the specimen during the test. In this chapter, the partial healing (PH) model is used to simulate the fatigue behavior. With the PH model the differences between the UT/C and 4PB test are explained and the so-called endurance limit is determined. This model was developed and modified by A.C. Pronk and is based on the dissipated energy theory [Pronk, 2001]. It is a material model that describes the evolution of the complex modulus and phase angle for a unit volume during a fatigue test. This implies that the model can be applied directly to the fatigue results obtained from “homogenous” tests [Pronk, 2006]. By taking into account the dimensions and the stress-strain distribution of the specimen, a weighted complex modulus can be obtained for so-called “inhomogeneous” tests, such as bending tests [Pronk, 2010]. Therefore the PH model offers a possibility to compare different fatigue results.

6.2 Application of PH Model on UT/C Test Results

6.2.1 PH Model Theory As discussed in Chapter 2, energy is dissipated in asphalt mixtures during repeated loading, because the material behaves substantially visco-elastic. An example of a stress–strain hysteresis loop during one loading cycle is shown in Figure 6-1. The area inside of the hysteresis loop indicates the amount of dissipated energy in one cycle, which can be calculated with Equation 6-1 [van Dijk, 1972] [Pronk, 1991] [Rowe, 1996].

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-200 -150 -100 -50 0 50 100 150 200

Strain [ µm/m]

Stre

ss [M

Pa]

Figure 6-1 Stress-strain hysteresis loop in one cycle

σi

εi


134

sini i i iw = π σ ε ϕ (6-1) Where: wi : dissipated energy in cyclei , [MPa]; σi : stress amplitude in cyclei , [MPa]; εi : strain amplitude in cyclei , [m/m]; φi : phase lag in cyclei , [°] Normally it is believed that the total energy dissipated into the test device (system losses ∆Wsys) and the specimen (visco-elastic losses ∆Wdis and fatigue damage ∆Wfat), consisting of three components [Pronk, 1996] [Pronk, 2001]: (1) System losses ∆Wsys: The energy is dissipated into the test device. By calibrating the test set-up, the system loss can be diminished. (2) Visco-elastic losses ∆Wdis: Most of the dissipated energy is associated with viscoelastic loss, which is completely transformed into heat and results in an increase of specimen temperature. This amount of energy is approximately equal to the area of the stress-strain loop. So ∆Wdis for a unit volume dV can be calculated by Equation 6-2 for a sinusoidal loading.

( )sindis i i iW∆ = ⋅ ⋅ ⋅π σ ε ϕ (6-2)

Due to the increase of the temperature in the specimen, the visco-elastic losses are responsible for a small part of the decrease in complex stiffness in the beginning of the fatigue test. If a good forced convection is used in the climate chamber, equilibrium in temperature distribution is normally reached within 10000 loading cycles. Therefore the decrease in stiffness modulus due to the increase in temperature is negligible after 10000 loading cycles [Pronk, 1996]. (3) Fatigue consumption ∆Wfat: Only a small part of the total energy relates to the formation of microcracks, which is denoted as fatigue consumption ∆Wfat. This part of the energy is the main reason for the decrease of stiffness and the increase of the phase angle during the fatigue test. In the proposed model, it is assumed that the mathematical formulation of this part is modeled as a very small portion of the visco-elastic losses ∆Wdis. In a strain controlled mode test, ∆Wfat is expressed by Equation 6-3:

( ) ( )20sin sinfat dis i i i i iW W S∆ = ⋅∆ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅δ δ π σ ε ϕ δ π ε ϕ (6-3)

Where: ∆Wfat : dissipated energy related to fatigue damage at cyclei , [MPa]; δ : very small parameter (<<1); Si : stiffness modulus at cycle i, [MPa]; ε0 : strain amplitude, a constant in the strain controlled mode, [m/m].


135

A new parameter, the stiffness damage Q, is introduced, which relates to the fatigue consumption ∆Wfat. The damage factor Q reduces the stiffness modulus, including the loss modulus F and the storage modulus G, expressed by Equation 6-4 and 6-5, respectively.

( ) ( )0 1 1

0

sint

tdQF t S t t F e d

d− − = ⋅ = − + ⋅ ∫

β ττϕ α γ τ

τ (6-4)

( ) ( )0 2 2

0

cost

tdQG t S t t G e d

d− − = ⋅ = − + ⋅ ∫

β ττϕ α γ τ

τ (6-5)

Where: t : test duration, [s]; Ft : loss modulus at testing time t, [MPa]; Gt : storage modulus at testing time t, [MPa]; S t : stiffness modulus at testing time t, [MPa]; φ t : phase angle at testing time t, [°]; F0 : initial loss modulus, [MPa]; G0 : initial storage modulus, [MPa]; Qt : stiffness damage at testing time t, [MPa]; α1, α2, γ1, γ2, β : model parameters. In the damage part (the integral terms in Equation 6-4 and 6-5), the term with α1 and α2 will vanish with time according to the time decay parameter β. So this term represents the reversible damage which will heal in time. It can also be seen as a mathematical formulation for the thixotropic behavior of asphalt mixtures. The terms describe partial healing of the stiffness modulus which is not always the same as partial healing of the strength. The terms with γ1 and γ2 correspond to the irreversible damage which accumulates in time. Because the stiffness damage part Q directly relates to the fatigue damage ∆Wfat. The rate of change of stiffness damage Q is equal to that of ∆Wfat.

( ) ( ) fatfat

Wd dQ t W t

dt dt t

∆= ≈

∆ (6-6)

The change rate of Q in a strain controlled test can be expressed by substituting Equation 6-3 in Equation 6-6.

( ) ( ) ( ) 20 sinS t td

Q tdt t

⋅ ⋅ ⋅= ⋅

∆π ε ϕ

δ (6-7)

Since the loss modulus ( ) ( ) ( ) F t S t sin φ t= ⋅ and the frequency 1

f∆t

= , Equation 6-7

can be rewritten as follows:


136

( ) ( ) ( )20 2

0

sinS t tdQ f F t

dt t

⋅ ⋅ ⋅= ⋅ = ⋅ ⋅ ⋅ ⋅

∆π ε φ

δ δ π ε (6-8)

Where: ∆t : test duration in one cycle, [s]; f : frequency [Hz]. To simplify the equation, the following abbreviations are used:

* 21 0 1fα δ π ε α= ⋅ ⋅ ⋅ ⋅ ; * 2

2 0 2fα δ π ε α= ⋅ ⋅ ⋅ ⋅ (6-9)

* 21 0 1fγ δ π ε γ= ⋅ ⋅ ⋅ ⋅ ; * 2

2 0 2fγ δ π ε γ= ⋅ ⋅ ⋅ ⋅ (6-10)

Then Equation 6-4 and 6-5 can be rewritten as follows:

( ) ( ) ( )* *0 1 1

0

ttF t F F e d− − = − + ⋅ ∫

β ττ α γ τ (6-11)

( ) ( ) ( )* *0 2 2

0

ttG t G F e d− − = − + ⋅ ∫

β ττ α γ τ

(6-12)

For a unit volume, the solutions of the local loss and storage stiffness modulus are given by Equation 6-13 and 6-14, respectively.

( ) ( ) ( )0BtF t F e Cosh Ct DSinh Ct−= + (6-13)

( ) ( ) ( ) ( )( )* *2 2

0 0 *1

1Bt BtG t G F e Sinh Ct e Cosh Ct ESinh CtC

− − = − ⋅ + − ⋅ +

α γγ

(6-14)

where * *1 1

2B

+ +=

α β γ, 2 *

1C B= − βγ , BD

C

−=

β and *1B

EC

−=

γ

In Equation 6-9 and 6-10, π is a mathematical constant and the values of f and ε0 are from the test conditions, but the parameter δ is an unknown constant value. It is not possible to calculate the parameters α1, α2, γ1 and γ2 from the parameters α1

*, α2*, γ1

* and γ2*.

Therefore, in the following analysis, the item of δα1 is considered as one parameter. This is also applied for the items of δα2, δγ1 and δγ2. From previous research, it is noticed that the value of the parameter δα1 is always zero or small enough to be ignored for the UTC and 4PB fatigue test. Figure 6-2, Figure 6-3 and Figure 6-4 show the influence of the model parameters on the evolution curves of the stiffness and phase angle. In each figure, only one parameter is varied and all the other parameters are kept constant. The starting point of the curve is determined by the parameters F0 and G0. The major influence of the parameter δα2 is on the shape and steepness of the evolution curves. With a decrease of the parameter δα2, the curve is more close to a straight line. Figure 6-3 shows that the stiffness is not influenced by the parameter δγ1 and the phase angle grows slower with a larger δγ1. The parameter δγ2 controls both the stiffness and phase angle. A larger δγ2 results in a steeper slope of the evolutions curve.


137

0

2000

4000

6000

8000

10000

0 10000 20000 30000 40000 50000 60000 70000Number of cycles

Stif

fnes

s [M

Pa]

0

20

40

60

80

100

Pha

se a

ngle

[°]

S-δα2=100 S-δα2=1000 S-δα2=2000 S-δα2=3000

φ-δα2=100 φ-δα2=1000 φ-δα2=2000 φ-δα2=3000

Figure 6-2 Influence of δα2 on the evolution of the stiffness and phase angle

0

2000

4000

6000

8000

10000

0 10000 20000 30000 40000 50000 60000 70000

Number of cycles

Stif

fnes

s [M

Pa]

0

20

40

60

80

100

Pha

se a

ngle

[°]

S-δγ1=40 S-δγ1=60 S-δγ1=80 S-δγ1=100φ-δγ1=40 φ-δγ1=60 φ-δγ1=80 φ-δγ1=100

Figure 6-3 Influence of δγ1 on the evolution of the stiffness and phase angle

2000

4000

6000

8000

10000

0 10000 20000 30000 40000 50000 60000 70000

Number of cycles

Stif

fnes

s [M

Pa]

-20

0

20

40

60

80

Pha

se a

ngle

[°]

S-δγ2=50 S-δγ2=100 S-δγ2=150 S-δγ2=200φ-δγ2=50 φ-δγ2=100 φ-δγ2=150 φ-δγ2=200

Figure 6-4 Influence of δγ2 on the evolution of the stiffness and phase angle

δα2 δα2

δγ1

δγ2

δγ2


138

Figure 6-5 gives the side view and the top view of the tensile strain distribution of a cylindrical specimen (size 1.0) as obtained from a finite element simulation. In the model, it is assumed that the cylindrical specimen behaves as a homogeneous material. The viscoelastic properties of the mixture are described with Prony series representation and are determined with the UT/C stiffness test. The boundary conditions are such that the movement of the bottom surface of the cylindrical specimen is fully restricted; the specimen is subjected to a tensile strain of 165 µm/m. (a) (b)

Figure 6-5 Tensile strain distribution of a specimen in the UT/C test: (a) side view; (b) top view

0

25

50

75

100

125

0 50 100 150 200 250Tensile strain [ µm/m]

Hei

ght o

f cyl

inde

r [m

m]

x=0 x=12.5mm

x=20mm x=21mm

x=22mm x=23mm

x=24mm x=25mm

Figure 6-6 Tensile strain along the height at different places from the center of the

cylinder (x=0) to the side (x=25) Figure 6-6 gives the tensile strain distributions along the height at the different positions from the centre to the side of the specimen (from x=0 to x=25). As shown, due to the side effect, the tensile strain differs slightly from the applied strain near the upper and lower

Applied strain, 165 µm/m

Glue area

Glue area

x=0 x=25 x


139

ends. To avoid the unwanted failure near the ends, extra glue was used between the specimen and the steel caps to ensure sufficient bond and confinement. The thickness of the glue is around 5 mm. Except for the ends of the specimen, the tensile strain is uniformly distributed in the vertical and radial directions and the value is very close to the target value. In theory, the stress-strain distribution for the UT/C test is uniform throughout the specimen. It means that the evolution of the stiffness modulus for a small unit volume is equal to that of the whole specimen as measured during the test. So Equation 6-11 and 6-12 can be directly used to fit the measured values for the homogenous fatigue test.

6.2.2 Determination of PH Model Parameters To calculate the parameters F0 and G0, the initial stiffness and phase angle were measured at the 50th loading cycle, as shown in Figure 6-7.

0

5000

10000

15000

20000

25000

30000

0 50 100 150 200 250Strain [ µm/m]

Initi

al s

tiffn

ess

[MP

a]

UT/C_Size0.5@20°C UT/C_Size1.0@20°C


(a) Initial stiffness

0

5

10

15

20

25

30

35

40

0 50 100 150 200 250Strain [ µm/m]

Pha

se a

ngle

[°]



(b) Initial phase angle

Figure 6-7 Initial stiffness and phase angle of the different UT/C specimen sizes


140

At 20 ºC, the initial stiffness does not depend on the strain amplitude and the specimen size. When the temperature reduces to 5 ºC, the initial stiffness of specimen size 1.0 becomes two times higher and the phase angle shows a reverse trend. The parameters F0 and G0 can be calculated from the initial stiffness and phase angle. The model parameters δα1, δα2, δγ1, δγ2 and β are determined by minimizing the differences between the measured and fitted values for S, φ, F (S×sinφ) and G (S×cosφ) in the interval from N=10000 to N=NR (fatigue life determined by DER). This procedure can be simply completed by the Solver function in Excel. In Figure 6-8, the fitted lines of stiffness and phase angle correspond well with the measured values in the first and second phase, because the PH model only describes the crack initiation phase and does not take into account any plastic damage. It is believed that plastic effects are more relevant in the crack propagation phase. In the crack initiation phase the fatigue damage consists of the creation of micro-cracks, dislocations etc, which can be captured by the proposed dWfat term.

0

2000

4000

6000

8000

10000

12000

0 100000 200000 300000 400000 500000 600000 700000 800000Number of cycles

Stif

fnes

s [M

Pa]

24

26

28

30

32

34

36

Pha

se a

ngle

[°]

Stiffness F(S*sinφ) G (S*cosφ)Phase angle Model fit

Figure 6-8 Measured and fitted stiffness and phase angle of cylinder C-10-9 in the UT/C

fatigue test while using a strain level of 110 µm/m

The values of the model parameters are given in Table 6-1. It appears that in most cases the values of the parameter δα1 are nil for the three specimen sizes. The other parameters can be expressed as function of the applied strain. The relationships between the model parameters and the strain levels are illustrated in Figure 6-9. The parameters δα2, δγ1 and δγ2 can be expressed as a linear relationship with the strain level. The parameter β, the time decay, is dependent on the squared value of the strain. The parameters δγ1 and δγ2 represent irreversible damage of the specimen. Its intersection point with the x-axis means that irreversible damage would not occur when the strain level is lower than a certain value. This indicates the existence of an endurance limit. Table 6-2 presents the regression equations of the model parameters and the predicted endurance limit.

Phase I Phase II Phase III


141

Table 6-1 PH model parameters from regressions for the UT/C test

Specimen size

Sample code


δα1 δα2

δγ1 δγ2

β [10-5 s-1]

C-3-18 88 0 964 11.4 25.5 63 C-3-25 117 0 1109 14.9 49.9 138 C-3-26 130 0 1333 19.5 56.8 171 C-3-9 136 0 1263 28.6 64.2 187 C-3-24 138 0 1339 23.4 74.6 191 C-3-4 146 0 1704 32.5 98.1 217 C-3-11 168 0 1773 37.5 114.8 289

Size 0.5 at 20 ºC

C-3-2 186 0 2042 43.3 145.4 397 C-10-12 68 0 755 2.1 8.9 24 C-10-10 84 0 892 9.5 24.1 45 C-10-9 110 0 1134 21.2 48.1 98 C-10-8 131 0 1367 24.6 60.5 175 C-10-5 155 0 1655 39.0 99.9 244 C-10-11 172 0 1980 46.0 127.8 302

Size 1 at 20 ºC

C-10-7 199 0 2142 55.9 171.1 402 C-14-1 68 0 627 3.8 12.1 37 C-13-1 89 0 1060 11.0 23.5 63 C-11-2 115 0 1427 24.4 47.4 106 C-9-1 133 0 1383 36.5 91.7 138 C-12-2 161 0 1719 48.8 110.7 259

Size 1.5 at 20 ºC

C-11-1 182 0 2009 63.2 149.2 318 C-15-4 69 0 606 2.1 18.9 49 C-15-11 87 0 578 7.4 40.8 77 C-16-5 94 0 685 0.9 55.6 61 C-16-1 105 0 617 9.1 90.0 112 C-15-6 118 0 866 8.9 138.9 141 C-15-9 124 0 968 12.2 135.0 156

Size 1 at 5 ºC

C-16-10 136 0 1089 21.9 198.4 226 From Figure 6-9 and Table 6-2, it can be seen that the parameter β does not change with the specimen size and temperature and it only relates to the used strain amplitude. The parameter δα2 is also independent of the specimen size, but the value of δα2 becomes smaller with decreasing temperature at the same strain level. For the parameters δγ1 and δγ2, the slopes of the curves are slightly influenced by the specimen size. The smaller specimen has a steeper slope, but the difference between these three specimen sizes is not significant at the same temperature. When the temperature reduces to 5 ºC, the value of δγ1 becomes smaller and the value of δγ2 becomes larger at the same strain level. It indicates that the more irreversible damage happens at a higher strain level or at a lower testing temperature.


142

0

1000

2000

3000

4000

0 50 100 150 200 250Strain level [ µm/m]

δα

2

Size 0.5@20ºC Size 1.0@20ºC

Size 1.5@20ºC Size1.0@5ºC

(a) Parameter δα2 vs. strain amplitude

0

200

400

600

800

0 50 100 150 200 250Strain level [ µm/m]

β [1

0-5

/s]

Size 0.5@20ºC Size 1.0@20ºC

Size 1.5@20ºC Size1.0@5ºC

(b) Parameter β vs. strain amplitude

0

50

100

150

200

250

0 50 100 150 200 250Strain level [ µm/m]

δγ

1 or

δγ

2

δγ1_Size0.5@20ºC δγ2_Size0.5@20ºC δγ1_Size1.0@20ºCδγ2_Size1.0@20ºC δγ1_Size1.5@20ºC δγ2_Size1.5@20ºCδγ1_Size1.0@5ºC δγ2_Size1.0@5ºC

(c) Parameter δγ1 and δγ2 vs. strain amplitude

Figure 6-9 PH model parameters as functions of strain amplitude for the UT/C test


143

The range of the endurance limit as presented in Table 6-2 is predicted based on the trends of the parameters of δγ1 and δγ2. It can be seen that the predicted endurance limit does not vary with specimen size and temperature for the UT/C fatigue test. The range of the endurance limit is from 66 to 72 µm/m for the different specimen sizes and temperatures.

Table 6-2 Functions of the PH model parameters for the UT/C test

Specimen size Functions of the PH model parameters

Predicted εlimit

[µm/m]

δγ1=0.36×(ε-66) Size0.5@20 ºC δα2=11.0·ε-69.6

δγ2=1.17×(ε-72) 66~72

δγ1=0.41×(ε-63) Size1@20 ºC δα2=11.2·ε-64.0

δγ2=1.32×(ε-69) 63~69

δγ1=0.51×(ε-66) Size1.5@20 ºC δα2=10.8·ε+10.8

δγ2=1.36×(ε-67) 66~67

δγ1=0.24×(ε-68) Size1@5 ºC

δα1=0,

β=(1.01×10-7)·ε2

δα2=7.81·ε-45 δγ2=2.39×(ε-66)

66~68

Table 6-2 shows that all the PH model parameters are related to the applied strain level. The constants in the functions of the parameter δα2 are much lower than the values of the parameter δα2 presented in Table 6-1. For simplification, it is assumed that the fitted line of the parameter δα2 goes through the origin (0, 0). The endurance limits predicted by the parameter δγ1 and δγ2 are much closer to each other and should be a constant value for one mixture type at a certain test condition. Therefore the PH model parameters can be expressed by the following equations.

* 2= ⋅β β ε (6-15) **

2 2= ⋅δα α ε (6-16)

( )**1 1 limit= ⋅ −δγ γ ε ε

(6-17)

( )**2 2 limit= ⋅ −δγ γ ε ε

(6-18)

Where: β*, **

2α , **1γ , **

2γ : regression constants; εlimit : predicted endurance limit, µm/m; ε : applied strain level, µm/m.


144

6.3 Application of PH Model on 4PB Test Results

6.3.1 Weighing Procedure for 4PB Test In an inhomogeneous fatigue test, for example in the 4PB test, the stresses and strains vary along the length and height of the beam specimen. Figure 6-10a gives the horizontal strain distribution of a beam specimen in the 4PB test. This finite element simulation is the same as the one mentioned in Chapter 5 (see Figure 5-8). Figure 6-10b shows that the horizontal strain at the bottom surface increases linearly from the sides to the middle section. In the middle section the horizontal strain is constant. As can be seen from Figure 6-10c, the horizontal strain varies linearly with the height. The maximum strain that occurs at the surface in the middle section is the same as the target value. It indicates that the stiffness varies at the different positions during the 4PB fatigue test. The measured stiffness is not the local stiffness but a weighted overall stiffness for the whole specimen. A weighing procedure is required to calculate the weighted stiffness modulus per cross sectional area (along the height) and the weighted overall stiffness modulus (along the length) from the local stiffness.

(a)

0

40

80

120

160

200

0 100 200 300 400 500Length of beam [mm]

Hor

izon

tal s

trai

n [µ

m/m

]

0

10

20

30

40

50

60

-200 -100 0 100 200Horizontal strain [ µm/m]

Hei

ght o

f bea

m [m

m]

(b) (c)

Figure 6-10 Schematic illustration of the 4PB test modeling (a), variation of the horizontal strain with the length (b) and the linear variation of the horizontal strain with

the height at the centre point (c) In the numerical analysis demonstrated by Pronk [Pronk, 2012], it is assumed that the strain and material properties do not depend on the coordinate y (width of the beam).

Height

Length


145

Because of symmetry, only a quarter part of the beam was considered (0<x<L/2; 0<z<H/2), as shown in Figure 6-11.

Length

Height

x1 x2

x10

x11

z1

z2

z10

……

.

…….0

x=L/2

x=A

z=H/2

Length

Height

x1 x2

x10

x11

z1

z2

z10

……

.

…….0

x=L/2

x=A

z=H/2

Neutral line

Length

Height

x1 x2

x10

x11

z1

z2

z10

……

.

…….0

x=L/2

x=A

z=H/2

Length

Height

x1 x2

x10

x11

z1

z2

z10

……

.

…….0

x=L/2

x=A

z=H/2

Neutral lineNeutral line

Figure 6-11 Schematic diagram of the numerical analysis

It is assumed that the origin of the coordinate system is at the location of the outer support and the inner clamp is at the location x=A. The interval 0<x<A is divided into 10 sections and the interval x=A to x=L/2 is divided into 5 sections, because the weighted stiffness is constant in the middle section (A<x<L/2), which is considered as being only one section. The beam height from z=0 to z=H/2 is also divided into 10 sections with a unit length of H/20. For a unit volume at the point (xj, zi), the local loss and storage stiffness modulus at loading cycle n are given by Equation 6-19 and 6-20.

( ) ( ) ( ) ( ), , 0 Btj iF x z n F e Cosh Ct DSinh Ct−= + (6-19)

( ) ( ) ( ) ( ) ( ) ( )( )* *2 2

*1

, , 0 0 1Bt Btj iG x z n G F e Sinh Ct e Cosh Ct ESinh Ct

C− −

= − ⋅ + − ⋅ +

α γγ

(6-20)

Based on the strain distribution shown in Figure 6-10, the local strain distributions in a beam are given by Equation 6-21 and 6-22.

0

2, 1 j i

j i

x H zx z

A H

−= − ⋅

ε ε when 0<x<A and 0<z<H/2 (6-21)

0

2, 1 i

j i

H zx z

H

− = −

ε ε when A<x<L/2 and 0<z<H/2 (6-22)

Where: ε0 : maximum strain in the midsection of the beam Because of the variation of the bending strain level in the beam specimen, the evolutions of the stiffness and the phase angle are different for each unit volume. This implies that

j=1 j=2 j=3 ………. j=11


146

the PH model parameters in Equation 6-19 and 6-20 are not constant but vary with the local strain level. As discussed in section 6.2.2, all the parameters can be expressed as function of the strain level (Equation 6-15 to 6-18). With the strain distribution and the regression constants β*, **

2α , **1γ and **

2γ , the evolution of the local stiffness with the

loading cycles was simulated. Then the weighted loss and storage stiffness modulus per cross sectional area for a given value of x are calculated by Equation 6-23 and 6-24. This function is based on a weighing function with respect to the distance to the neutral axis. In the case of bending, the

product of stiffness modulus (Smix) times the moment of inertia (3

12

w h⋅) is relevant.

3 310

1

1, , ,

20 10 10i

H i iF x n F x i n

=

− = −

∑ (6-23)

3 310

1

1, , ,

20 10 10i

H i iG x n G x i n

=

− = −

∑ (6-24)

Where: ,F x n ,G x n : weighted loss and storage stiffness modulus per cross

area at location x at the cycle n, [MPa];

The weighted stiffness modulus of each section j on the interval 0<x<A is the mean value of the weighted stiffness at the borders of the section j. The ratios of these weighted stiffness and the initial weighted stiffness are denoted as ,j lossα and ,j storageα (Equation 6-

25 and 6-26).

( ) ( ) ( ) ( )1 ,, , / 2 0j j j j lossF n F x n F x n F− = + = ⋅ α (6-25)

( ) ( ) ( ) ( )1 ,, , / 2 0j j j j storageG n G x n G x n G− = + = ⋅ α (6-26)

Where: ( )jF n ( )jG n : weighted loss and storage stiffness modulus of each section j, [MPa];

,j lossα , ,j storageα : ratio of weighted loss and storage stiffness and the

initial weighted stiffness [Hz].

In the middle section (A<x<H/2), the weighted loss and storage stiffness per section are constant and denoted as:

( ) ( ) ( )11 11, 11, 0 00 sinloss lossF n F S= ⋅ = ⋅α α ϕ (6-27)

( ) ( ) ( )11 11, 11, 0 00 cosstorage storageG n G S= ⋅ = ⋅α α ϕ (6-28)


147

,b lossV and ,b storageV are defined as the deflections corresponding to the loss and storage

stiffness, respectively. By ignoring the influence of any mass on the phase lag, the bending deflections as a result of loss stiffness along the length are determined from the differential equations (Equation 6-29 and 6-30) for a slender beam.

( ) ( )2

,2,

2j b loss

d PF n I V x n x

dx⋅ ⋅ = − ⋅ when 0<x<A (6-29)

( ) ( )2

,2,

2j b loss

d PF n I V x n A

dx⋅ ⋅ = − ⋅ when A<x<H/2 (6-30)

Where: ( ), ,b lossV x n : bending deflection for loss stiffness at location of x at the

number of cycle n, [mm]; P : amplitude of applied force [N]. Because similar equations are also valid for the storage stiffness, in the following analysis the part of the loss stiffness is only mentioned. The general solution for the sections from j=1 to 10 is given by Equation 6-31.

( )3

, , ,,

1,

6b loss loss j loss j lossj loss

x xV x n C D

L L

= − + +

βα

(6-31)

Where: βloss, Cj,loss, Dj,loss : constants, [mm]; The boundary conditions are that both the deflection and its first derivative are

continuous at the cross sections of10j

jx A= . The recursion formulas for the constant

Dj,loss are given by the following equations:

1

3

11

0

1 1 1

3 10

,loss

j ,loss j ,lossj ,loss j ,loss

D

jD D A

L α α+

+

=

= + −

(6-32)

Over the interval of A<x<H/2, the bending deflection has a parabolic shape as shown in equation 6-33.

( )2

, 11,11, 11,

1 1,

2 2b loss loss lossloss loss

A x xV x n D

L L L

= − + +

βα α

(6-33)

At the cross section x=A both the deflection Vb,loss and its first derivative have to be continuous leading to a relation between the constants D10,loss and D11,loss.


148

39

101 1

1 1 1

3 10,lossj j ,loss j ,loss

jD A

L α α= +

= − ∑ (6-34)

3

11, 10,11, 10,

1 1

2 3loss lossloss loss

A AD D

L L

= − − α α

(6-35)

The amplitude of the maximum deflection at x=L/2 is given by Equation 6-36.

( ), 11,11,

1

8b loss loss lossloss

AV n D

L

= +

βα

(6-36)

The expressions of the weighted loss, storage and overall modulus for the whole beam are given by Equation 6-37, 6-38 and 6-39.

( ) ( )

2

0 0

11, 11,

1 1

8 6sin

1

8 loss loss

A

LF n S

D

ϕ

α

− = ⋅

+

(6-37)

( ) ( )

2

0 0

11, 11,

1 1

8 6cos

1

8 storage storage

A

LG n S

D

ϕ

α

− = ⋅

+

(6-38)

( )( ) ( )( )22

S n F n G n= + (6-39)

The model parameters are determined in such a way that the calculated weighted overall modulus stays consistent with the measured stiffness of the beam.

6.3.2 Determination of PH Model Parameters Similar to the procedure for analyzing the UT/C test results, the initial stiffness and phase angle were used to calculate the initial values for the parameter F0 and G0. By means of the weighing procedure, the weighted overall storage modulus, loss modulus, complex modulus and phase angle were fitted with the measured results by varying the regression constants β*, α2

** , γ1** , γ2

** and εlimit. An example of the regression fits is shown in Figure 6-12.


149

0

2000

4000

6000

8000

10000

12000

0 50000 100000 150000 200000 250000

Number of cycles

Stif

fnes

s [M

Pa]

5

10

15

20

25

30

35

Pha

se a

bgle

[º]

Stiffness F(S*sinφ )G(S*cos φ ) Phase angleModel fit

Figure 6-12 Measured and calculated 4PB results for a strain level of 179 µm/m

Table 6-3 PH model parameters for the specimen size 0.5 at 20 ºC

Sample code


**2α [106]

β*

[10-8 s-1] **1γ [106] **

2γ [106] εlimit

[µm/m] B-19-5 100 11.89 7.55 0.38 1.36 73 B-14-5 119 11.89 8.20 0.32 1.38 66 B-19-2 140 11.89 8.58 0.41 1.35 80 B-14-6 160 11.89 8.96 0.39 1.36 72 B-19-6 179 11.89 9.91 0.41 1.35 69 B-19-4 201 11.89 10.17 0.44 1.34 88 Mean 11.89 8.89 0.39 1.36 73.5 St.dev. 0.0003 1.01 0.04 0.01 5.8


Sample code

Strain level

[µm/m]

**2α [106]

β*

[10-8 s-1] **1γ [106] **

2γ [106] εlimit



150


Sample code

Strain level

[µm/m]

**2α [106]

β*

[10-8 s-1] **1γ [106] **

2γ [106] εlimit



Sample code

Strain level

[µm/m]

**2α [106]

β*

[10-8 s-1] **1γ [106] **

2γ [106] εlimit


Table 6-3, Table 6-4, Table 6-5 and Table 6-6 present the regression constants β*, α2

** , γ1

** , γ2** and εlimit at each strain level for all the specimen sizes and their average values

and standard deviations. Similar to the UT/C test, for the 4PB test, the parameter δα2 does not change with the specimen size and becomes lower with a decrease in temperature. The average values of the constant β* for the different specimen sizes and temperatures are very close to each other. It means that the parameter β is independent of the specimen size and temperature and is only related to the used strain amplitude. For the parameters δγ1 and δγ2, the slopes of the curves slightly increase with the specimen size. When the temperature reduces to 5 ºC, the value of γ1

** becomes smaller and the value of γ2** becomes larger. The standard

deviations of these regression constants are relatively larger for the size 1.5 specimens and those tested at the low temperature of 5 ºC. The average values of the regression constants and the predicted endurance limits obtained by means of the UT/C and 4PB test are compared in Figure 6-13, Figure 6-14, Figure 6-15, Figure 6-16 and Figure 6-17, respectively. It can be seen that the values of the constants α2

** , β*, γ1** obtained from the 4PB test are not very much different from

those obtained from the UT/C test. However, the constant γ2** and the predicted


151

endurance limit for a unit volume vary with the 4PB specimen size. The larger specimen has a lower endurance limit. All the data are listed in Table 6-7.

0

3

6

9

12

15

18

Size 0.5@20ºC Size 1.0@20ºC Size 1.5@20ºC Size 1.0@5ºC

α2**

[10

6 ]

4PB UT/C

Figure 6-13 Comparison of the regression constant α2

** for 4PB and UT/C test

0

3

6

9

12

15

18


β*

[10

-8]

4PB UT/C

Figure 6-14 Comparison of the regression constant β* for the 4PB and UT/C test

0

0.2

0.4

0.6

0.8

1


γ1**

[10

6 ]

4PB UT/C

Figure 6-15 Comparison of the regression constant γ1

** for the 4PB and UT/C test


152

0

0.5

1

1.5

2

2.5

3

3.5


γ2**

[106 ]

4PB UT/C

Figure 6-16 Comparison of the regression constant γ2

** for the 4PB and UT/C test

0

20

40

60

80

100


ε lim

it [µ

m/m

]

4PB UT/C

Figure 6-17 Comparison of the predicted endurance limit for the 4PB and UT/C test

Table 6-7 Comparison of the predicted endurance limit for 4PB and UT/C test

Test type

Specimen size **2α [106]

β*

[10-8 s-1] **1γ [106] **

2γ [106] εlimit

[µm/m] Size 0.5@20 ºC 11.0 10.1 0.36 1.17 69 Size 1.0@20 ºC 11.2 10.1 0.41 1.32 66 Size 1.5@20 ºC 10.8 10.1 0.51 1.36 67

UT/C

Size 1.0@5 ºC 7.81 10.1 0.24 2.39 67 Size 0.5@20 ºC 11.89 8.89 0.39 1.36 74 Size 1.0@20 ºC 11.89 10.22 0.49 1.39 63 Size 1.5@20 ºC 11.71 12.21 0.45 2.02 60

4PB

Size 1.0@5 ºC 5.50 9.74 0.18 3.09 63


153

6.4 Correlation between UT/C and 4PB Fatigue Test Results In this section the correlation between the UT/C and 4PB test results for the size 1.0 specimens at 20 ºC is investigated. Due to the inhomogeneous stress-strain distribution in the 4PB test, the stiffness of the beam specimen decreases at different rates along the length and height. With the PH model parameters, the stiffness evolutions from different locations can be calculated. Figure 6-18 shows the comparison of the weighted stiffness evolution in different cross sectional areas.

B-16-1@105µm/m

4000

5000

6000

7000

8000

9000

0 500000 1000000 1500000 2000000 2500000Number of cylces

Wei

ghte

d st

iffne

ss [M

Pa]

A<x<L-A

x=0.9A

x=0.7Ax=0.5Ax=0.3Ax=0.1A

overall

x=Ax=0 x=Lx=L-A

B-16-1@105µm/m

4000

5000

6000

7000

8000

9000

0 500000 1000000 1500000 2000000 2500000Number of cylces

Wei

ghte

d st

iffne

ss [M

Pa]

A<x<L-A

x=0.9A

x=0.7Ax=0.5Ax=0.3Ax=0.1A

overall

A<x<L-A

x=0.9A

x=0.7Ax=0.5Ax=0.3Ax=0.1A

overall

x=Ax=0 x=Lx=L-Ax=Ax=0 x=Lx=L-A

Figure 6-18 Weighted stiffness per cross sectional area and overall weighted stiffness.

It is clear that the stiffness in the midsection (A<x<L-A) decreases dramatically with the number of loading cycles. The fatigue damage becomes less and less from the inner clamps (x=A) to the outer clamps (x=0). Almost no damage develops near the outer clamps. Taking into account the stress-strain distribution, the weighted overall stiffness is calculated, which is higher than the weighted stiffness in the midsection. Along the height direction, the strain reaches the maximum value at the surface and decreases linearly from the surface area to the neutral line at which the strain level becomes zero. During testing, only the surface area in the midsection is always subjected to the target strain. It means that the stiffness evolution at the surface in the midsection of a beam should be comparable to the results from the UT/C test. Three pairs of the cylinders and beams are selected and for each pair the applied strain on the cylinder is similar to the pure bending strain at the surface in the midsection of the beam specimen. Figure 6-19 gives the comparison between the measured stiffness of the cylinder, the weighted overall stiffness of the beam and the calculated stiffness at the surface in the midsection of the beam.


154

0

2000

4000

6000

8000

10000

12000

0 1000000 2000000 3000000 4000000 5000000Number of cylces

Stif

fnes

s [M

Pa]

C-10-10@84µε

B-15-2@87µε_SurfaceB-15-2@87µε_overall

0

2000

4000

6000

8000

10000

12000

0 500000 1000000 1500000 2000000 2500000

Number of cycles

Stif

fnes

s [M

Pa]

C-10-9@110µε

B-16-1@105µε_Surface

B-16-1@105µε_overall

0

2000

4000

6000

8000

10000

12000

0 100000 200000 300000 400000 500000Number of cylces

Stif

fnes

s [M

Pa]

C-10-5@155µε

B-16-4@161µε_Surface

B-16-4@161µε_overall

Figure 6-19 Comparison of stiffness evolutions obtained from UT/C and 4PB test.

From these figures, it can be seen that the local stiffness at the surface of the beam drops more rapidly compared to the weighted overall stiffness, which means that the most serious fatigue damage occurs at the beam surface. It is noted that the stiffness evolution obtained in the UT/C test is consistent with the one valid for the beam surface. For the


155

UT/C fatigue test, the classical fatigue life can be easily determined by the measured stiffness, but for the 4PB test the problem is that the changes in the stiffness are not uniform inside the specimen. The fatigue lives determined at the different locations are also different. Figure 6-20 gives a comparison of the fatigue lines determined from the overall stiffness and the local stiffness on the surface of the beam. As expected, the fatigue line based on the local surface stiffness of the beam is lower than that based on the overall weighted stiffness and it shows a good agreement with the fatigue line obtained from the UT/C fatigue test.

1.E+04

1.E+05

1.E+06

1.E+07

10 100 1000Strain [ µm/m]

Nf,

50

UT/C_Nf,50

4PB_overall

4PB_surface

Figure 6-20 Comparison of classical fatigue life determined by different stiffness curves.

Table 6-8 Regression coefficients of the fatigue lines

Material coefficients k b

R2

UT/C_size1.0 5.7×1014 -4.43 0.995 4PB_ size1.0_overall 5.3×1014 -4.20 0.984 4PB_ size1.0_surface 2.9×1014 -4.25 0.989

6.5 Conclusions In this chapter, the PH model was applied on the strain-controlled UT/C and 4PB fatigue test results. The main findings, remarks and conclusions are summarized below: Application of the PH model on the UTC fatigue test:

1. The PH model describes the evolutions of the complex modulus and phase angle for a unit volume in a strain-controlled fatigue test. The solution can be directly applied on the UT/C test results. By adjusting the model parameters, the PH model provides


156

good simulations for the evolutions of the complex stiffness and phase angle in phase I and phase II (see figure 6-8).

2. All the model parameters can be expressed as function of the applied strain level. The parameter β (time decay) is independent of the specimen size and temperature and is only related to the square of the used strain amplitude. The parameter δα1 (reversible damage in loss modulus) is always nil or close to nil in any case. The parameter δα2 (reversible damage in storage modulus) does not change with the specimen size, but becomes smaller with a decrease in temperature. For the parameters δγ1 and δγ2 (irreversible damage in loss and storage modulus), the difference between the different specimen sizes is not significant. When the temperature reduces to 5 ºC, the value of δγ1 becomes smaller and the value of δγ2 becomes larger at the same strain level. This indicates that in the UT/C test, the healing ability of the asphalt mixture decreases at a lower temperature and more irreversible damage happens at a higher strain level or at a lower testing temperature. The relationships between the PH model parameters and the applied strain level can be expressed by Equations 6-15 to 6-18.

3. The trends of the values of the parameter δγ1 and δγ2 indicate the existence of an endurance limit. The predicted endurance limit obtained by the UT/C test is not influenced by the specimen size and temperature and can therefore be considered as a material property. The range of the endurance limit for the used mixture is from 66 to 69 µm/m at 20 ºC.

Application of the PH model on the 4PB fatigue test:

1. In the 4PB test, the strain is not uniformly distributed throughout the length and height of the beam specimen. The functions of the PH model parameters are used to calculate the local stiffness for each unit volume. By a weighing procedure, the weighted overall stiffness was calculated and by adjusting the regression constants the weighted overall stiffness was fitted to the measured stiffness.

2. The average values of the regression constants α2** , β*, and γ1

** obtained from the 4PB test are close to those obtained from the UT/C test. However, the constant γ2

** and the predicted endurance limit for a unit volume are influenced by the specimen size. The larger specimen has a larger γ2

** value and a lower endurance limit.

3. For the specimen size 1.0, the functions of the PH model parameters obtained from the UT/C and 4PB test are similar to each other. It indicates that the 4PB test results can be predicted by means of the PH model parameters determined from the UT/C test results.

4. By means of the PH model, the local stiffness at different parts of the beam can be calculated. The most serious fatigue damage occurs at the surface in the midsection of the beam. The stiffness evolution of this area is comparable with the UT/C fatigue test results when the pure bending strain on the beam surface is equal to the strain applied to the cylinder. The fatigue line based on the local surface stiffness of the beam is in good agreement with the fatigue line obtained from the UT/C fatigue test.


157

References Li, N., Molenaar, A.A.A., Pronk A.C., van de Ven, M.F.C. and Wu, S., Effect of Specimen Size on Fatigue Behavior of Asphalt Mixture in Laboratory Fatigue Tests, Proceedings of the 7th RILEM International Conference on Cracking in Pavements, Delft, the Netherlands, pp 827-836, 2012a. Li, Ning, Molenaar, A.A.A., Pronk A.C., van de Ven, M.F.C., Investigation into the Size effect on Four Point Bending Fatigue Tests, Proceedings of the 3rd Workshop on 4PB, Davis, California, 2012b. Li, Ning, Molenaar, A.A.A., Pronk A.C., van de Ven, M.F.C. and Wu, S., Comparison of Uniaxial and Four Point Bending Fatigue Tests for asphalt mixtures. The Transportation Research Board (TRB) 92nd Annual Meeting. Washington, D.C. January 13-17, 2013. Pronk, A. C. and Hopman P. C., Energy dissipation: the leading factor of fatigue, Proceedings of the Conference on the United States Strategic Highway Research Program, pp. 255–267, 1991. Pronk, A.C., Analytical description of the heat transfer in an asphalt beam, tested in the 4 point dynamic bending apparatus, DWW report W-DWW-96-006, 1996. Pronk, A. C., Comparison of 2 and 4 point fatigue tests and healing in 4 point dynamic test based on the dissipated energy concept. In: Proceedings of the 8th International Conference on Asphalt Pavements, Seattle, pp. 987–994, 1997. Pronk, A.C., Partial healing in fatigue tests on asphalt specimen, Road Materials and Pavement Design, vol. 4, n. 4, pp 433-445, 2001. Pronk, A.C., Partial Healing, A new approach for the damage process during fatigue testing of asphalt specimen, Proceedings of the Symposium on Mechanics of Flexible Pavements, ASCE, Baton Rouge, 2006. Pronk, A.C. and Molenaar, A.A.A., The Modified Partial Healing Model used as a Prediction Tool for the Complex Stiffness Modulus Evolutions in Four Point Bending Fatigue Tests based on the Evolutions in Uni-Axial Push-Pull Tests, Proceedings of the 11th Int. Conf. on Asphalt Pavements, Nagoya, Japan, 2010. Pronk, A.C., Description of a procedure for using the Modified Partial Healing model (MPH) in 4PB test in order to determine material parameters. Proceedings of the 3rd Workshop on 4PB, Davis, USA, 2012. Rowe, G.M., Application of the dissipated energy concept to fatigue cracking in asphalt pavements, PhD thesis, University of Nottingham, Nottingham, England. 1996.


158

van Dijk, W. Moreaud, H., et al. 1972 The Fatigue of Bitumen and Bituminous Mixes, 3th International Conference on Structural Design of Asphalt Pavements (ISAP), London, Volume 1.

Chapter 7 Monotonic Uniaxial Tension and Compression Tests

159


7.1 Introduction As mentioned in Chapter 3, representing fatigue relations in a 3-Dimensional state of stress rather than a 1-dimensional state of stress was defined as one of the objectives of this research. This was considered necessary since in reality a unit volume of pavement material is always subjected to a 3D state of stress. Therefore it was decided to determine the yield surface for the asphalt mixture, similar to what is done for granular materials. The laboratory tests including triaxial tests, monotonic uniaxial compression (MUC) and monotonic uniaxial tension (MUT) tests are required to determine the yield surface of the asphalt mixture. In this research only the MUC and MUT test are used to develop yield surfaces. The response to the applied load in terms of stress and strain (axial and radial strain) is measured during the tests, when the cylindrical specimens were subjected to uniaxial loading. Because the response of an asphalt mixture is strongly influenced by loading time and temperature, both the MUC and MUT tests are conducted at different temperatures and strain rates. The set-ups of the MUT and MUC test were designed and built at the Delft University of Technology, Road and Railway Engineering Section. A detailed description of these two tests is given in Section 7.2 and 7.4. To reduce the number of tests, the central composite rotatable design method was used for optimization of the experimental program. Section 7.3 briefly describes this experimental design technique and gives the test conditions. The experimental results are shown in Section 7.6

7.2 Test Equipment

7.2.1 Monotonic Uniaxial Compression Test The uniaxial compression test set-up (see Figure 7-1) was designed and built by considering several influence factors on the state of stress and deformation, such as specimen alignment, frame stability, temperature effect, boundary friction, etc [Erkens and Poot, 2000]. The compression test set-up consists of a 3D-space frame in which an MTS 150 kN hydraulic actuator is mounted. The actuator is rigidly connected to the upper loading plate. The frame itself is placed on an elastically supported concrete block. The load is transmitted from the actuator to the specimen through two steel plates placed at the top and bottom of the specimen. The bottom and top plate are kept parallel by using three guidance bars (Φ16 mm) made of Fortal (a strong aluminium alloy), which are connected to the bottom plate and pass through linear bearings in the top plate [Erkens, 2002]. Without any precautions at the contact surface between specimen and the loading plates, the radial deformation would be restrained due to the fact that the plates and the specimen have a different ν/E ratio. The resulting friction would act as a confinement for the top


160

and bottom of the specimen, causing the well-known barrel-shape of specimens in compression. To avoid these stress concentrations, a friction reduction system is applied to the top and bottom ends of the specimen. The friction reduction system consists of two thin steel plates and two pieces of rubber, which are greased with a soft soap at both sides. The specimen is placed between two of these metal-soap-rubber-soap sandwiches.

(A) (B) Figure 7-1 Complete compression test setup (A) and a close-up of a specimen inside of

the temperature cabinet (B)

An insulated cabinet with dimensions 0.6×0.5×0.6 m is placed within the frame, which allows the tests to be performed at temperatures in the range of 0 to 45 °C with an accuracy of ± 0.5 °C. The insulated cabinet is a sandwich construction of wood and foam. The inside of the cabinet is covered with aluminum/plastic insulation foil. A uniform temperature distribution in the specimen was realized by controlling the airflow rate in the chamber. During the tests, the temperature of the plates and the air are monitored. The force data are measured via the MTS load cell, which is positioned between the top plate and the actuator. The axial deformations are recorded by three external displacement transducers (LVDT’s) placed vertically around the specimen. The radial deformations at the middle of the specimen are registered by two circumferential measurement systems (a string and an extensometer). The range of the axial LVDT’s is ± 20 mm. The string and the extensometer have a range of ± 150 mm and ± 3.75 mm, respectively. The purpose of the extensometer is to enable accurate radial deformation measurements to be made in the initial stages of the test. When the extensometer is out of range, the radial deformations are measured by the string. The measurement systems are connected to a PC-based data acquisition system, which produces a single ASCII output file for each test. The measured data are captured at sampling rates ranging from 1 to 1000 Hz.

Hydraulic Actuator

Frame

Load cell Temperature

cabinet

Platen Extension

LVDT’s

String

Extensometer


161

7.2.2 Monotonic Uniaxial Tension Test The tension test set-up consists of a closed temperature cabinet with a 50 kN hydraulic actuator inside. The actuator is connected rigidly to the bottom of the temperature chamber. In the temperature cabinet the specimen is placed in a rigid framework that can resists the high forces that occur during the test without deforming. In the test setup, the specimen is placed between three hinges to ensure that the specimen is subjected to pure uniaxial tension. Two hinges are placed above and one under the specimen to prevent bending moments to occur in the specimen. The specimens are glued to the top and bottom end caps using a 2-component fast curing adhesive, X60. In this case, the radial deformation will cause stress concentrations near the specimen ends. In order to provide confinement at the ends of the specimen, PVC rings were glued around the specimen ends and the caps to prevent specimens from cracking near the ends. Figure 7-2 Tension test set-up (left) and a close-up of a specimen inside the temperature

cabinet (right) The force is measured by means of a load cell, which is positioned between the two hinges above the specimen. An internal 407 MTS controller is used to impose the required controlled deformation rate. The axial deformation is registered by means of three displacement transducers (LVDT’s). These LVDT’s are fixed in an aluminium ring which is placed around the steel cap at the bottom of the specimen. On top of the specimen, the three LVDT’s were positioned such that they touch a second aluminium ring, which is placed around the steel cap on top of the specimen. To obtain an accurate axial displacement curve, the LVDT’s with a range of ± 1 mm are used for measurements at low temperatures (below 20 °C and high strain rates at 20 °C) and the LVDT’s with a range of ± 5 mm are used for high temperatures (above 20 °C and low strain rates at 20 °C). The measurement systems are connected to a PC-based data acquisition system, which produces a single ASCII output file for each test. Moreover, an oscilloscope is used as a backup for the measurements. A close-up of the tension test set-up with an instrumented specimen is shown in Figure 7-2.

Hinge Load cell

Hydraulic Actuator

Hinge

Radial extensometer

LVDT’s

Plastic ring


162

7.3 Test Condition

7.3.1 Central Composition Rotatable Design To investigate the influence of temperature and strain rate, the tests should be conducted at several temperatures and strain rates with a number of repetitions based on a traditional experimental design. If the tests are done at 3 temperatures and 4 strain rates and repeated 4 times for each combination, at least 48 specimens are needed for MUC or MUT test. Therefore, it was decided to use “the central composite rotatable design technique” [Robinson 2000] for the design of the test conditions of the MUC and MUT tests. The procedure is quite simple and fully described by Jansen [2002], Medani [2006] and Muraya [2007], as follows:

1) Determination of the range of the independent variables

In this study, the independent variables are the strain rate denoted by (εɺ ) in the range of

minεɺ to maxεɺ and the temperature denoted by (T) in the range of Tmin to Tmax. The

dependent variables are the compressive and the tensile strength.

Table 7-1 Coded test condition

test codedεɺ Tcoded

1 −1 −1 2 +1 −1 3 −1 +1 4 +1 +1 5 −Ψ 0 6 +Ψ 0 7 0 −Ψ 8 0 +Ψ

9~13 0 0

Figure 7-3 Coded test conditions for the central composite rotatable design

8 4

6

2

7

1

5

3

codedεɺ

+Ψ

9-13 (0, 0)

−Ψ

+1

−1

0

+Ψ −Ψ +1 −1 0

Tcoded


163

2) Each condition of the experiment to be conducted is expressed in coded terms in Table 7-1 and graphically in Figure 7-3. Figure 7-3 shows that 5 tests are repeated in the centre of the range of independent variables (test 9~13) and 8 tests are at the rotatable positions around this centre.

3) Tests 1~8 are called star points and tests 9~13 are called centre points. Any value can

be chosen for Ψ. If the value of Ψ is set at 2 , there may be any number of replicated points in the centre. However, it is recommended by Diamond [2001] to use 5 central points with 2 independent variables.

4) To scale the coded terms of Table 7-1 into values within the range of variables, Ψ is set equal to half the range of the variables and scaling factors for the tests follow from:

( )max min

12

2scalingε ε ε× = × −ɺ ɺ ɺ (7-1)

( )max min

12

2scalingT T T× = × − (7-2)

Where: scalingεɺ : scaling factor for the strain rate;

Tscaling : scaling factor for the temperature;

minεɺ : minimum strain rate, %/s;

maxεɺ : maximum strain rate, %/s; Tmin : minimum test temperature, °C; Tmax : maximum test temperature, °C; Finally the experimental values of the strain rate εɺ and temperature T follow from:

( )max min

1

2scaling coded scalingε ε ε ε ε= × + × −ɺ ɺ ɺ ɺ ɺ (7-3)

( )max min

1

2scaling coded scalingT T T T T= × + × − (7-4)

7.3.2 Test Conditions for MUC and MUT Tests

The MUC and MUT tests are performed under displacement controlled conditions by applying a monotonic increasing axial displacement (constant deformation rate) until complete annihilation of the strength. The chosen range of strain rate and temperature are 0.001 ~ 4 %/s and 5 ~ 35 ºC, respectively. According to the central composite design, the monotonic tension test and compression test are conducted at the target test conditions shown in Figure 7-4 and tabulated in Table 7-2. Two extra tests which are not part of the central composite design are added at 4 and 38ºC to check the accuracy of the experimental design for failure test conditions outside the range of the experimental design.


164

0

1

2

3

4

5

-5 5 15 25 35 45Temperature [ºC]

Str

ain

rat

e [%

/s]

Figure 7-4 Test conditions of the MUC and MUT test

Table 7-2 Test conditions of the MUC and MUT test

MUC test MUT test Temperature

[ºC] strain rate

[%/s] Temperature

[ºC] strain rate

[%/s] 4 1 5 0.01, 0.05, 2 5 2.001 9.4 0.05, 0.587, 3.416

9.4 0.587, 3.414 12.9 0.587, 2 20 0.01, 2 (repeat 5 times), 4 20 0.01, 2 (repeat 5 times), 4

30.6 0.587, 3.414 30.6 0.587, 3.416 35 2.001 35 2 38 3 38 3

7.4 Test Procedure

7.4.1 Test Procedure for MUC Tests

The specimen is placed between the two metal-soap-rubber-soap sandwiches.

The specimen with friction reduction system is placed between the top and bottom loading plates in the temperature cabinet and fitted with the radial measurement systems. One must ensure that all three LVDT’s, string and extensometer are in the measurement range.

Then the temperature cabinet is closed in order to allow the temperature of top and bottom plate to reach the target value. A minimum period of 3 hours is used for temperature stabilization.

Before starting the test, a small pre-load (0.3kN) is applied on the specimen to prevent undesirable moments that can lead to erroneous results [Erkens and Poot, 2000].


165

In the displacement-controlled mode, the load is applied such that a constant displacement rate is achieved, until the specimen breaks. The reported displacement rate is based on the average value of the three external axial LVDT’s and is fed back to the system.

7.4.2 Test Procedure for MUT Tests

Two steel caps are glued to the top and bottom surface of the specimen. The steel caps are 30 mm thick and have a diameter of 80 mm. The caps have to be glued parallel to each other to ensure a state of pure tension during the test. When the caps are not parallel, bending moments are introduced during the test which is unacceptable. The set up for gluing the caps parallel on the specimen is shown in Figure 7-5.

Special PVC rings (thickness 8 mm) are glued around the specimen at the cap to prevent the specimen failure between the cap and the specimen. The glue is a two component fast curing adhesive, X60, consisting of a liquid component (B) and a powder component (A), which are mixed at the weight ratio 1:1. After gluing, the specimen was kept in the set-up for ten minutes.

Figure 7-5 The gluing mould for MUT test

Then the specimen is connected to the upper and lower hinges; three LVDT’s around the steel caps and extensometer in the middle of the specimen are fixed to the specimen.

The temperature cabinet is closed in order to allow the temperature of top and bottom caps to reach the target value. A minimum of 3 hours is used for temperature is stabilization.

In the displacement-controlled mode, a load is applied such that a constant displacement rate is achieved till the specimen breaks. This displacement rate is based on the average value of the three external axial LVDT’s and is fed back to the system.


166

7.5 Data Processing

7.5.1 Stress and Strain

The first step of data processing is to calculate the magnitude of the strains and stresses to be determined during the MUC and MUT tests. They include vertical stress, axial strain, radial strain and volumetric strains and are calculated with Equations (7-5), (7-6), (7-7) and (7-8).

sec

Load sample weightvertical stress

initial cross tion areaσ

+= (7-5)

0a

change in height haxial strain

original height hε

∆= (7-6)

0r

change in radius rradial strain

original radius rε

∆= (7-7)

2v r avolumetric strainε ε ε= + (7-8)

During the MUT test and the beginning of the MUC test, the radial deformation, ∆r, was measured using an extensometer as shown in Figure 7-6 and calculated using Equation (7-9) (MTS manual).

Figure 7-6 Schematic of the measurement of radial deformation using an extensometer

2 sin cos2 2 2

f i

i i i

lr R R

∆∆ = − =

+ −

θ θ θπ

(7-9)

f i∆l l l= − (7-10)

Rf

Ri l i l f

r

θi θf

∆r

End roller of extensometer Chain

l i =initial cord length, mm l f =final cord length, mm r =radius of roller, mm ∆r =radial deformation, mm Ri =initial radius of specimen, mm Rf =final radius of specimen, mm θi =angle subtended by l i, in radians θf =angle subtended by l f, in radians


167

( )2 ci

i

l

R r= −

+θ π (7-11)

Where: ∆l : change in cord length, extensometer output, mm; lc : chain length, remains constant, mm; In the MUC test, the radial deformation is much larger than that in the MUT test and beyond the range of the extensometer. In that case the radial deformation is measured using two potentiometers and a string as illustrated in Figure 7-7 and calculated iteratively from Equation 7-12 (Erkens and Poot 2000).

2 2

2 2 2 21 1 2 22

2 2

2

h hL L R L L R L

Rπ

∆ ∆ ∆ − + + ∆ + − + + ∆ − ∆ = (7-12)

Where: ∆R : radial deformation, mm; ∆L : change in length of the string, mm; ∆h : change in height of the specimen, mm; L1, L2 : distance between potentiometer and the middle of specimen,

mm Figure 7-7 Schematic of the measurement of radial deformation using potentiometers and

string Figure 7-8 gives an example of the radial strains measured by the extensometer and string. It is clear that the signals from extensometer and string are not exactly consistent. A correction was made to combine these two signals. The range of the extensometer is much smaller than that of the string. So the signal from the extensometer is more accurate when the radial deformation is in the small range. Therefore, in the beginning of the test, the radial strain was calculated from the signal of the extensometer.

∆R

∆h

2

∆h Potentiometer Potentiometer

Initial position of the string

Final position of the string

L1 L2


168

With an increasing vertical load, the radial deformation becomes larger as well and will be outside the range of the extensometer at a certain moment (point A). From this moment on, the strain was calculated from the signal of the string. After point A, the signal of the string was shifted to this point.

C-1-5@20ºC, 2.5mm/s

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8 10

Time [sec]

Str

ain

[m/m

]

0.02

0.03

0.04

2.5 2.7 2.9 3.1 3.3

Figure 7-8 Correction of the radial strain measured by extensometer and string

7.5.2 Strain Rate, Maximum Stress, Tangent Stiffnes s and Onset of Dilation Based on the strains and stresses mentioned above, the strain rate, elastic modulus, maximum stress, failure strain and point of onset of dilation can be determined. Figure 7-9 shows an illustration of how these parameters are determined.

C-2-11@30ºC, 4mm/s

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

-0.03-0.025-0.02-0.015-0.01-0.0050

Axial strain [m/m]

Vol

ume

stra

in [m

/m]

-10

-8

-6

-4

-2

0

2

4

Str

ess

[MP

a]

Volume strain

Stress

Figure 7-9 Calculation of strain rate, tangent stiffness, maximum stress, failure strain and

onset of dilation from a MUC test

1

2

3

1- Signal from extensometer 2- Signal from string 3- Corrected signal

2

3

1 A

A

P

D

Etan

Strain rate Calculation

σdil


169

The peak value of the stress curve (point P) is defined as the strength and the corresponding axial and radial strain are the failure strain in vertical and horizontal direction at the maximum stress. The strain rate up to the maximum stress was chosen as the actual strain rate. The slope of the linear part of the stress-strain curve is used to determine the tangent stiffness, Etan. In the beginning of the MUC test, the axial strain is larger than the radial strain, which leads to a reduction in volume strain. The onset of dilation is the point at which the volume starts to increase due to internal crack growth. This point is determined by means of the axial strain versus volume strain plot, in which the lowest point (point D) is defined as the onset of dilation.

7.6 Test Results

7.6.1 MUC Test Results Figure 7-10 shows the failed specimens after the MUC test. At high temperature and low strain rate, the specimen deforms uniformly and does not completely fracture. But at low temperature and high strain rate, the fracture plane lies approximately on one of the diagonals (see Figure 7-10b), which indicates there is still some effect of shear stresses at the loading platens. At the end the cylindrical specimen is completely split into two main parts.

(a) C-2-1@ 30.6 ºC, 0.594 %/s; (b) C-1-10@ 9.4 ºC, 3.416 %/s Figure 7-10 Failed specimens after being tested at different test conditions in the MUC

test

During testing, the force, axial and radial deformation were recorded. The average axial deformation was used to calculate the axial strain. This was done because all three LVDT's did not measure the same displacement rate (see Figure 7-11). This indicates that in spite of all precautions still some bending occurred.


170

C-2-1@30C, 0.594%/s-12

-10

-8

-6

-4

-2

00 5 10 15

Time [s]

Dis

plac

emen

t [m

m]

-10

-8

-6

-4

-2

0

For

ce [k

N]

LVDT1 LVDT2LVDT3 AverageForce

Figure 7-11 Measured axial displacements and force

The vertical displacement measured by the LVDT’s includes two parts, the deformation of the specimen and the deformation of the friction reduction system. In order to remove the influence of the friction reduction system, Pramesti performed the calibration tests to determine the relationship between the compressive stress and the deformation of the friction reduction system [Pramesti, 2013]. During testing, the compressive force was applied to a steel cylinder with the friction reduction system. Three LVDTs with a range of 2 mm were used to measure the deformation. The characteristics of the material and the test conditions are shown in Table 7-3.

Table 7-3 The characteristics of the material and test conditions [Pramesti, 2013]

Specimen Descriptor Test condition

Steel cylinder E = 210,000 MPa H = 130 mm ∅ = 65 mm

1. Force control of 0.02 kN/s 2. T = 20 °C 3. Preload = 0.15 kN 4. Max Force = 60 kN 5. 2 hour precondition before start of the test

Figure 7-12 shows the relation between compressive stress and the measured displacement. In order to determine to what extent the friction reduction deformation influences the overall deformation, a mathematical model relating displacement to stress has been formulated. From Figure 7-12 it can be seen that there is a simple exponential/logarithmic relation between displacement measured by the LVDTs and the stress. It is assumed that there is a maximum displacement when the friction reduction deformation reaches its maximum value. By using the solver function in Microsoft Excel, the constants for the relation between displacement and stress were obtained. The model is expressed in Equation 7-13 and the comparison between measured and predicted values is presented in Figure 7-12.


171

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Stress (MPa)

Dis

plac

emen

t (m

m)

Measuerement Model

Figure 7-12. Measured displacement vs. compressive stress [Pramesti, 2013]

0.9869c0.5998 0.6295 0 15

0.118 15c c

m s f

c

MPaL L L

MPa

σ σ σσ

− + ≤ ≤∆ = ∆ + ∆ = > (7-13)

Where: ∆Lm : measured total deformation, mm; ∆Ls : deformation of the steel cylinder, mm; ∆Lf : deformation of the friction reduction system and frame, mm; σc : compressive stress, MPa

With the height and stiffness of the steel cylinder (see Table 7-3), the deformation of the steel cylinder can be calculated by Equation 7-13.

41306.19 10

210000s

s s s c c cs

HL H

Eε σ σ σ−∆ = = = = × (7-14)

Where: εs : strain of the steel cylinder, m/m; Hs : height of the steel cylinder, mm; Es : stiffness of the steel cylinder, MPa.

By substituting Equation 7-14 into Equation 7-13, the deformation of the friction reduction system can be expressed as follows:

0.9869c

4

0.6004 0.6295 0 15

0.118 6.19 10 15c c

f m s

c c

MPaL L L

MPa

σ σ σσ σ−

− + ≤ ≤∆ = ∆ − ∆ = − × >

(7-15)

Therefore, when testing the asphalt specimen, the deformation of the specimen can be simply calculated by the Equation 7-16.


172

( )( )

0.9869c

4

0.6004 0.6295 0 15

0.118 6.19 10 15

m c c

a m f

m c c

L MPaL L L

L MPa

σ σ σ

σ σ−

∆ − − + ≤ ≤∆ = ∆ − ∆ = ∆ − − × >

(7-16)

Where: ∆La : deformation of the asphalt specimen, mm. Equation 7-16 then can be used to calculate the corrected value of the displacement. Figure 7-13 gives the comparison of the axial strain before and after correction. It can be seen that the influence of the friction reduction system is significant before the compressive stress reaches the peak value.

C-2-8@4°C,1%/s-35

-30

-25

-20

-15

-10

-5

0-0.02-0.015-0.01-0.0050

Axial strain [m/m]

Com

pres

sive

stre

ss [M

Pa]

axial strain

Corrected axial strain

Figure 7-13 Axial strain-stress relation at T = 4°C and εɺ =1 %/s before and after correction

The stress-strain curves at the different test conditions are presented in Figure 7-14 and Figure 7-15. During the test, the stress-strain curve contains two parts, loading and unloading. The peak point and the slope of the loading part represent the strength and the stiffness of the material. It can be seen that the behavior of an asphalt mix depends on temperature and strain rate. Based on the stress-strain curve, the compressive strength fc, tangential stiffness Etan,c, compressive failure strain in axial direction εa,c and in radial direction εr,c were determined. The results are given in Table 7-4.

The friction reduction system influences the tangential stiffness Etan,c, compressive failure strain in axial direction εa,c. Table 7-5 gives the values of Etan,c and εa,c before and after the correction. In general the tangential stiffness increases and the compressive failure strain in axial direction decreases after the correction.


173

-16

-14

-12

-10

-8

-6

-4

-2

0-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

εaxial [m/m] εradial [m/m]

σ [MPa] C-2-4, 20°C, ε=4 %/sC-2-12, 20°C, ε=2.005 %/sC-1-11, 20°C, ε=2.005 %/sC-1-8, 20°C, ε=2.005 %/sC-1-3, 20°C, ε=2.005 %/sC-1-4, 20°C, ε=0.01 %/s

Figure 7-14 Compressive stress versus axial and radial strain at 20 ºC

-40

-35

-30

-25

-20

-15

-10

-5

0-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2


σ [MPa]4°C, ε=1 %/s5°C, ε=2.005 %/s9.4°C, ε=3.416 %/s9.4°C, ε=0.594 %/s

30.6°C, ε=3.416 %/s30.6°C, ε=0.594 %/s35°C, ε=2.005 %/s38°C, ε=3 %/s

Figure 7-15 Compressive stress versus axial and radial strain at other temperatures


174

Table 7-4 Overview of the MUC test results

Sample code

Temp. [ºC]

Target strain rate [%/s]

Real strain rate [%/s]

fc [MPa] Etan,c

[MPa] εa,c [%] εr,c

[%] C-2-8 4 1 1.047 -30.94 9243 -0.666 0.231 C-2-9 5 2.005 2.009 -36.45 8292 -0.598 0.169 C-2-5 9.4 0.594 0.625 -22.27 6239 -0.723 0.288 C-1-10 9.4 3.416 3.364 -35.02 14390 -0.538 0.222 C-1-4 20 0.01 0.0102 -2.22 474 -2.294 1.400 C-2-2 20 2.005 2.002 -14.08 4836 -0.710 0.396 C-2-12 20 2.005 2.091 -11.83 4595 -0.912 0.400 C-1-11 20 2.005 2.037 -11.06 3462 -1.011 0.000 C-1-8 20 2.005 2.011 -11.71 4249 -0.828 0.411 C-1-3 20 2.005 2.02 -12.49 5826 -0.894 0.193 C-2-4 20 4 4.189 -13.86 6549 -0.619 0.265 C-2-1 30.6 0.594 0.620 -3.39 596 -1.985 1.047 C-2-11 30.6 3.414 3.582 -6.45 3079 -1.280 0.705 C-1-9 35 2.005 2.051 -3.63 722 -1.658 1.060 C-1-6 38 3 3.025 -2.75 719 -2.196 1.370

Table 7-5 Influence of the friction reduction system on tangential stiffness Etan,c and compressive failure strain in axial direction εa,c

Sample code

Temp. [ºC]

strain rate

[%/s]

Etan,c

[MPa]

Corrected Etan,c

[MPa] Ratio εa,c

[%]

Corrected εa,c

[%] Ratio

C-2-8 4 1.05 8579 9243 1.08 -0.745 -0.666 0.89 C-2-9 5 2.01 8168 8292 1.02 -0.672 -0.598 0.89 C-2-5 9.4 0.63 5287 6239 1.18 -0.806 -0.723 0.90 C-1-10 9.4 3.36 9475 14390 1.52 -0.612 -0.538 0.88 C-1-4 20 0.01 424 474 1.12 -2.333 -2.294 0.98 C-2-2 20 2.00 4639 4836 1.04 -0.797 -0.710 0.89 C-2-12 20 2.09 3818 4595 1.20 -0.997 -0.912 0.91 C-1-11 20 2.04 3406 3462 1.02 -1.094 -1.011 0.92 C-1-8 20 2.01 4168 4249 1.02 -0.910 -0.828 0.91 C-1-3 20 2.02 3810 5826 1.53 -0.829 -0.894 1.08 C-2-4 20 4.19 4066 6549 1.61 -0.705 -0.619 0.88 C-2-1 30.6 0.62 550 596 1.08 -1.849 -1.985 1.07 C-2-11 30.6 3.58 2411 3079 1.28 -1.351 -1.280 0.95 C-1-9 35 2.05 712 722 1.01 -1.711 -1.658 0.97 C-1-6 38 3.03 654 719 1.10 -2.240 -2.196 0.98 Ratio = (the value after correction)/(the value before correction)


175

7.6.2 MUT Test Results Unlike in the compression tests, failure in a tension test is localized and highly depends on a local defect of the specimen. Therefore the failure plane occurs randomly on the sample; this is shown in Figure 7-16. A compression specimen fails in a kind of overall split cracking pattern, a tension specimen breaks in two and the two halves are mostly undamaged. In a tension test, the end effects do not influence the observed fracturing response if the crack occurs far enough from the end caps.

Figure 7-16 Failed samples after being subjected to the MUT test

0

1

2

3

4

5

6

7

-0.0006-0.0004-0.000200.00020.00040.00060.00080.0010.0012εaxial [m/m] εradial [m/m]

σ [MPa]

5°C, 0.05 %/s5°C, 0.01 %/s9.4°C, 3.414 %/s9.4°C, 0.587 %/s9.4°C, 0.05 %/s12.9°C, 2 %/s12.9°C, 0.587 %/s

Figure 7-17 Tensile stress versus axial and radial strain at low temperature

The average value of the three LVDT’s was also used to determine the strain rate. In Figure 7-17 and Figure 7-18, the tensile stress from the MUT test is plotted versus the axial and radial strain at different test conditions. Similar to the MUC test results, the material behavior of an asphalt mixture in tension also strongly depends on the strain rate and temperature. The difference is that at low temperature or high strain rate, once the cracks initiate, crack propagation happens in a very short time. That is the reason why in Figure 7-17 the stresses suddenly reduce to zero after reaching the peak value and the


176

unloading part of the stress-strain curve is hard to see. Table 7-6 shows the target and actual tests conditions and the results of the MUT tests, namely tensile strength (ft), tangential stiffness modulus in tension (Etan,t), tensile failure strain in axial direction εa,t and in radial direction εr,t.

0

1

2

3

4

5

6

7

8

-0.01-0.00500.0050.010.015


σ [MPa]30.6°C, 0.587 %/s30.6°C, 3.416 %/s35°C, 2.005 %/s38°C, 3 %/s20°C, 0.01 %/s20°C, 2.005 %/s20°C, 4 %/s

Figure 7-18 Tensile stress versus axial and radial strain at high temperature

Table 7-6 Overview of the MUT test results

Sample code

Temp. [ºC]

Target strain rate [%/s]

Real strain rate [%/s] f t [MPa] Etan,t

[MPa] εt,v [%] εt,h [%]

C-2-3 5 0.01 0.010 4.51 9990 0.108 -0.038 C-8-4 5 0.05 0.056 5.75 15316 0.055 -0.015 C-6-5 5 2.005 1.270 6.05 19478 0.045 -0.011 C-6-9 5 2.005 1.274 5.97 16036 0.033 -0.007 C-8-7 9.4 0.05 0.051 5.17 10453 0.100 -0.031 C-8-10 9.4 0.587 0.535 6.29 15361 0.079 -0.015 C-8-5 9.4 3.414 1.148 5.89 17130 0.062 -0.012 C-7-7 9.4 3.414 1.940 6.00 12494 0.071 -0.031 C-6-7 9.4 3.414 1.970 6.36 17865 0.049 -0.023 C-8-6 12.9 0.587 0.620 5.67 12474 0.080 -0.019 C-8-9 12.9 2 1.180 6.02 14793 0.056 -0.015 C-6-1 20 0.01 0.010 1.08 405 0.512 -0.233 C-8-2 20 1 1.560 6.55 8796 0.180 -0.068 C-7-1 20 2.005 1.740 6.20 10213 0.111 -0.064 C-6-2 20 2.005 2.360 6.51 9145 0.104 -0.038 C-7-10 20 2.005 2.127 6.46 7665 0.295 -0.124 C-6-4 20 2.005 2.390 6.76 9392 0.221 -0.108 C-6-3 20 4 3.580 7.43 8739 0.178 -0.069 C-8-1 30.6 0.587 1.289 1.35 2068 0.619 -0.191 C-6-8 30.6 3.416 3.856 3.23 4189 0.378 -0.158 C-6-6 35 2.005 3.350 1.68 2193 0.589 -0.117 C-7-2 38 3 4.370 1.14 956 0.598 -0.238


177

7.7 The Unified Model As a viscoelastic material, material properties of asphalt mixtures such as strength, tangential stiffness modulus and failure strain are related to the temperature and time derivative variables. The unified model, proposed by Medani and Huurman [Medani, 2005] was used to establish the relationships between material properties and the test conditions; the model is shown in Equation 7-17. Based on the Time-Temperature superposition principle, the model allows shifting the data obtained at various temperatures with respect to strain rate to a selected reference temperature. It is believed that when the reduced strain rate converges to 0 or infinite, the material properties tend to increase or decrease to a limit value [Pellinen and Witczak, 2002] [Scarpas et al., 1997] [Erkens, 2002].

( )h l hP P P P S= + − (7-17)

Where: 0

exp ruS

u

γ = −

; r Tu uβ= ; ( )0expT sT T Tβ = − −

P : a material property e.g. compressive strength, tensile strength, MPa; compressive strength, MPa;

Pl : value of P when u→0; Ph : value of P when u→∞; u : time derivative variable e.g. strain rate, 1/s; ur : reduced time derivative variable; u0 : reference value of time derivative variable u; βT : temperature susceptibility function; T : temperature, K; T0 : reference temperature, K; Ts : model parameter, 1/K; γ : model parameter The temperature susceptibility factor, βT, can be determined at a certain reference temperature from the stiffness tests mentioned in Chapter 5. The theoretical minimum value, Pl of the properties derived from the MUC and MUT tests was set to zero since little or no strength and stiffness can be expected at high temperatures without confinement. The other model parameters Ph, u0 and γ can be obtained by minimizing the differences between the measured and fitted values for all material properties. This procedure can simply be completed by means of the Solver function in an Excel spreadsheet. By means of the unified model parameters, the master curves at the reference temperature of 10 ºC (283.15 K) were constructed for the strength, tangential stiffness, axial and radial strain at failure. The results are shown in Figure 7-19, Figure 7-20, Figure 7-21 and Figure 7-22. The model parameters for each material property are presented in Table 7-7.


178

0.1

1

10

100

1E-06 0.00001 0.0001 0.001 0.01 0.1 1Reduced strain rate [1/s]

f c o

r f t

[MP

a]

fc ft

Figure 7-19 Master curves of the compressive and tensile strength at the reference

temperature of 10 ºC

100

1000

10000

100000

1E-06 0.00001 0.0001 0.001 0.01 0.1 1

Reduced strain rate [1/s]

Eta

n,c

or

Eta

n,t [M

Pa]

Etan,cEtan,t

Figure 7-20 Master curves of the compressive and tensile tangent stiffness at the

reference temperature of 10 ºC

0.0001

0.001

0.01

0.1

1E-07 1E-06 1E-05 0.0001 0.001 0.01 0.1 1


ε a,c

or ε a

,t [m

/m]

εa,c εa,t

Figure 7-21 Master curves of the vertical compressive and tensile failure strain at the

reference temperature of 10 ºC


179

0.00001

0.0001

0.001

0.01

0.1

0.0000001 0.00001 0.001 0.1 10


ε r,c

or ε r

,t [m

/m]

εr,c εr,t

Figure 7-22 Master curves of the horizontal compressive and tensile failure strain at

the reference temperature of 10 ºC

Table 7-7 Unified Model parameters for the MUC and MUT tests Model

parameters T0 Ts Pl Ph u0 γ R2

fc 283.15 0.269 0 40.5 2.0×10-2 0.362 0.98 Ec 283.15 0.269 0 9000 2.9×10-3 0.480 0.85 εa,c 283.15 0.269 0 3.8×10-2 9.9×10-7 -0.179 0.94 εr,c 283.15 0.269 0 2.0×10-2 1.2×10-4 -0.354 0.92 ft 283.15 0.269 0 6.1 2.2×10-4 0.694 0.95 Et 283.15 0.269 0 21000 3.9×10-3 0.598 0.83 εa,t 283.15 0.269 0 6.9×10-3 1.6×10-5 -0.357 0.82 εr,t 283.15 0.269 0 2.3×10-3 1.3×10-4 -0.561 0.79

The figures show that the model provides a reasonable description of the test results. However, the graph showing the tension test results exhibits quite some scatter. This might be related to the mechanism of the tensile failure which was discussed in section 7.6.2. By comparing the results from MUC and MUT tests, it can be seen that the compressive strength and failure strain are higher than those obtained in the tension test. The reason for this is that in compression the aggregate skeleton efficiently carries the compression load and the deformation distributes quite uniformly throughout the specimen, while for the tension test, the tensile strength mainly depends on the cohesion and adhesion of the bitumen and the bituminous mortar. Conversely, the tangential stiffness in tension is higher than that in compression at high reduced strain rate. For all material properties, the slopes of the curves in tension are steeper than those in compression. This indicates that the material properties in tension are more sensitive to the temperature and strain rate.


180

7.8 Prediction of the Unified Model Parameters The test procedures described in this chapter, clearly show that in order to obtain reliable results from the MUC and MUT test, many aspects should be carefully taken into account, such as specimen preparation, test setup configuration, data analysis, etc.. Because of their complexity (especially of the tension test) it is not likely that these tests will be done on a routine basis in e.g. contractor’s laboratories. Therefore, it is necessary to find a simple way to predict tensile and compressive strength from the material properties, such as mixture composition without the need to conduct the MUC and MUT test. It should be noted that such procedures are common practice in pavement engineering. They exist for estimating the stiffness and fatigue resistance of asphalt mixtures.

It is reasonable to assume that the shape of the unified model depends on the aggregate gradation, shape of the aggregate, mix composition, binder characteristics and level of compaction. This then should offer possibilities to develop a simplified unified model, which allows the compressive and tensile strength to be directly calculated from the material properties and mixture characteristics. In previous research, MUC and MUT tests have been performed to measure the compressive and tensile strength for a number of road materials [Jansen, 2002] [Muraya, 2007] [Erkens, 2002] [Pungky, 2012]. The test results of 7 different kinds of asphalt mixtures were collected in this Section to establish a relationship between the unified model parameters and the material properties. Hereafter some general information of the involved mixtures is given: • DAC 0/8: Dense asphalt concrete with conventional bitumen (40/60 pen) used in this

research. Detailed information about this mixture was shown in Chapter 4. • EME 0/14: Enrobé á Modele Elevé, a bituminous base course material with “special”

hard binder (10/20 pen). The grain size is between 0 and 14 mm. Details about this mixture can be found in [Jansen, 2002].

• PAC 0/16: Porous asphalt concrete with conventional bitumen (70/100 pen). The grain size is between 0 and 16 mm. Details of this mixture are given in [Muraya, 2007].

• DAC 0/16: Dense asphalt concrete with conventional bitumen (40/60 pen). The grain size is between 0 and 16 mm. Also details of this mixture can be found in [Muraya, 2007].

• SMA 0/11: Stone mastic asphalt mixture with conventional bitumen (70/100 pen). The grain size is between 0 and 16 mm. This mixture was tested by Muraya as well [Muraya, 2007].

• ACRe 0/4: A kind of sand mixture with bitumen (45/60 pen). The grain size is between 0 and 4 mm. This fine grained mixture was used by Erkens for development of the ACRe model [Erkens, 2002].

• GAC 0/32: Gravel asphalt concrete with conventional bitumen (40/60 pen). The grain size is between 0 and 32 mm. This mixture was tested by Pramesti [Pramesti, 2012].


181

Table 7-8 Material properties of various asphalt mixtures

Gradation E [MPa] Va [%] Vb [%] Cc DAC 0/8 5200 3.0 14.9 3.25 EME 0/14 5700 3.4 12.2 0.73 PAC 0/16 2500 20.0 8.2 12.25 DAC 0/16 6900 2.7 12.9 20.48 SMA 0/11 2800 5.2 14.5 0.95 ACRe 0/4 3543 2.6 19.3 0.99

GAC 4700 4.4 8.9 1.60 The strength of an asphalt mixture is not only influenced by temperature and strain rate but also by the properties by the bitumen and the volumetric composition of the mixture. Based on work presented in [Molenaar, 1994], four material properties being stiffness of the mixture E, volume percentage of the air void Va, volume percentage of the bitumen Vb and coefficient of curvature of the gradation curve Cc were taken into account. The material properties of all the asphalt mixtures are collected in Table 7-8. The dynamic stiffness is a very important property, since it reflects the characteristics of the bitumen, the aggregate skeleton and volumetric composition of the mixture. The dynamic stiffness varies with the frequency and the temperature. In previous studies, the stiffness was measured using different tests, different waveforms and loading amplitudes. To obtain comparable results from different mixtures, it is assumed that all stiffness tests were conducted using a triangle waveform. The strain rate was computed as follows from the frequency and strain amplitude.

100t

εε = ⋅ɺ (7-18)

Where: εɺ : approximate strain rate, %; ε : strain amplitude, m/m; t : time duration when the strain level changes from 0 to peak

value in a cycle, s;

In Table 7-8, E represents the dynamic stiffness measured at the strain rate of 0.1 %/s and a temperature of 20 °C. The coefficient of curvature of the gradation curve, Cc is calculated by:

( )30

10 60c

DC

D D=

⋅ (7-19)

Where: D30 : sieve diameter through which 30 % of the aggregates is

passing, mm; D10 : sieve diameter through which 10 % of the aggregates is

passing, mm; D60 : sieve diameter through which 60 % of the aggregates is

passing, mm.


182

With the same procedure as mentioned in Section 7.7, the unified model parameters for the compressive and tensile strength were determined at the reference temperature of 10 °C. The model parameters for compressive and tensile strength are shown in Table 7-9.

Table 7-9 Model parameters for various asphalt mixtures

Mixture type T0 Ts Pl Ph u0 γ R2 DAC 0/8 283.15 0.27 0 48.6 4.4×10-2 0.350 0.84 EME 0/14 283.15 0.27 0 37.1 5.1×10-2 0.279 0.84 PAC 0/16 283.15 0.31 0 18.8 8.6×10-2 0.368 0.90 DAC 0/16 283.15 0.31 0 61.3 9.7×10-2 0.315 0.77 SMA 0/11 283.15 0.33 0 35.1 1.4×10-1 0.320 0.92 ACRe 0/4 283.15 0.30 0 60.6 2.9×10-1 0.304 0.94

Compressive strength

GAC 0/32 283.15 0.25 0 50.6 5.4×10-2 0.316 0.77 DAC 0/8 283.15 0.27 0 6.0 2.3×10-4 0.686 0.93 EME 0/14 283.15 0.27 0 6.0 1.2×10-4 0.400 0.85 PAC 0/16 283.15 0.31 0 2.0 9.8×10-4 0.685 0.89 DAC 0/16 283.15 0.31 0 5.8 2.5×10-4 0.569 0.91 SMA 0/11 283.15 0.33 0 3.9 1.2×10-4 0.538 0.94 ACRe 0/4 283.15 0.30 0 6.1 7.4×10-4 0.529 0.96

Tensile strength

GAC 0/32 283.15 0.25 0 4.9 3.1×10-4 0.551 0.86 Figure 7-23 summarizes the predicted and measured values for compressive strength and tensile strength. It is clear that the unified model as proposed for the different mixtures is capable of providing good estimates for the compressive and tensile strength. Most of the data points lie in the range of ± 20%. The correlation coefficient R2 of each data set listed in Table 7-9 is higher than 0.75.

Unified model

0

10

20

30

40

50

60

0 10 20 30 40 50 60Measured fc [MPa]

Pre

dict

ed fc

[MP

a]

DAC0/8 EME 0/14PAC 0/16 DAC 0/16SMA 0/11 ACRe 0/4GAC 0/32

(a) Compressive strength

Line of equality

-20%

+20%

R2=0.986


183

Unified model

0

2

4

6

8

10

0 2 4 6 8 10Measured ft [MPa]

Pre

dict

ed ft

[MP

a]DAC0/8 EME 0/14PAC 0/16 DAC 0/16SMA 0/11 ACRe 0/4GAC 0/32

(b) Tensile strength

Figure 7-23 Comparison between prediction of the unified model and measurement of compressive strength (a) and tensile strength (b)

As mentioned in Section 7.7, the parameter T0 and Ts relates to the temperature susceptibility and the parameters Pl for various mixtures are always equal to zero. So it is believed that the parameter Ph, u0 and γ are influenced by the material properties. Figure 7-24, Figure 7-25 and Figure 7-26 show the relationships between the selected material properties and the model parameters Ph, u0 and γ. Similar results were also found for the model parameters of the tensile strength. From Figure 7-24, it can be seen that the parameter Ph highly depends on the stiffness of the mixture, the properties of the bitumen and the volumetric composition of the mixture. In general the value of Ph increases with the mixture stiffness E, bitumen content Vb and decreases with void content Va. These observed results are logical. Based on work presented by Molenaar [Molenaar, 1994], also the degree of filling of the voids in the aggregate skeleton by bitumen was considered to be an important parameter. By taking these parameters into account, the following relationship was developed for Ph.

3

21

a

bah

b a

VP a E

V V

= ⋅ +

(7-20)

Where: Ph : value of P when strain rate ε → ∞ɺ ; E : stiffness at the temperature of 20 °C and the strain rate of 0.1

%/s, MPa; Vb : volume content of the bitumen, %; Va : volume content of air void, %; a1, a2 and a3 : model parameters.

Line of equality

-20%

+20% R2=0.949


184

0

10

20

30

40

50

60

70

0 2000 4000 6000 8000

E [MPa]

Ph

0

10

20

30

40

50

60

70

5 10 15 20Vb [%]

Ph

0

10

20

30

40

50

60

70

0 5 10 15 20 25

Va [%]

Ph

0

10

20

30

40

50

60

70

0 5 10 15 20 25Cc

Ph

Figure 7-24 Relations between Ph and the material properties for compressive strength

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2000 4000 6000 8000

E [MPa]

u0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

5 10 15 20Vb [%]

u0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20 25

Va [%]

u0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20 25Cc

u0

Figure 7-25 Relations between u0 and the material properties for compressive strength


185

00.05

0.10.150.2

0.250.3

0.350.4

0 2000 4000 6000 8000

E [MPa]

γ

00.050.1

0.150.2

0.250.3

0.350.4

5 10 15 20Vb [%]

γ

00.050.1

0.150.2

0.250.3

0.350.4

0 5 10 15 20 25

Va [%]

γ

00.05

0.10.15

0.20.25

0.30.35

0.4

0 5 10 15 20 25Cc

γ

Figure 7-26 Relations between γ and the material properties for compressive strength

The parameters in this function can be simply determined by means of the Solver function in Excel. The following equations give the functions of the parameter Ph for compressive and tensile strength.

For compressive strength: 0 623

0 4021 755.

b.h

b a

VP . E

V V

= ⋅ +

(7-21)

For tensile strength: 0 849

0 3080 505.

b.h

b a

VP . E

V V

= ⋅ +

(7-22)

Table 7-10 gives the comparison between the measured and predicted parameter Ph. The mean percent relative error (MPRE) of the prediction value is calculated by means of Equation 7-23. Except for the compressive strength data points of the mixtures ACRe 0/4 and EME 0/14, the relative errors are all less than 20 %.

1 100

npi mi

i mi

P P

PMPRE %

n=

−

= ×∑

(7-23)

Where: Pmi : measured data; Ppi : predicted data; n : number of test data;


186

Table 7-10 Predicted model parameters Ph for various asphalt mixture

Compression tension Mixture type

Measured Predicted MPRE Measured Predicted MPRE DAC 0/8 48.6 48.6 0.0% 6.02 6.02 0.0% EME 0/14 37.1 48.6 31.2% 5.99 5.89 1.6% PAC 0/16 18.8 18.9 0.0% 1.97 1.97 0.0% DAC 0/16 61.3 54.3 11.5% 5.80 6.54 12.7% SMA 0/11 35.1 35.1 0.1% 3.86 4.48 16.3% ACRe 0/4 60.6 43.2 28.7% 6.15 5.62 8.5% GAC 0/32 50.6 40.7 19.6% 4.88 4.84 0.8%

Figure 7-25 and Figure 7-26 show that the change of the parameter u0 and γ with the material properties is not significant. Nevertheless it was assumed that the parameter u0 and γ have a linear relationship with the material properties. By minimizing the MPRE between the predicted and real value, the following equations for u0 and γ were determined. For the compressive strength:

( ) ( ) ( ) ( )5 3 3 30 2 83 10 6 81 10 4 21 10 3 73 10 0 139b a cu . E . V . V . C .− − − −= − × ⋅ + × ⋅ + − × ⋅ + × ⋅ + (7-24)

( ) ( ) ( ) ( )6 4 3 34 36 10 4 24 10 2 62 10 1 03 10 0 311b a cγ . E . V . V . C .− − − −= − × ⋅ + × ⋅ + × ⋅ + × ⋅ + (7-25)

For the tensile strength:

( ) ( ) ( ) ( )7 5 5 50 3 64 10 1 46 10 4 51 10 2 77 10 0 0025b a cu . E . V . V . C .− − − −= − × ⋅ + − × ⋅ + − × ⋅ + × ⋅ + (7-26)

( ) ( ) ( ) ( )5 3 3 31 5 10 4 10 5 86 10 4 85 10 0 503b a cγ . E V . V . C .− − − −= − × ⋅ + × ⋅ + × ⋅ + × ⋅ + (7-27)

On the basis of the calculated model parameters given by Equation 7-21, 7-22, 7-24, 7-25, 7-26 and 7-27, the unified model is simplified and the compressive and tensile strength can be simply predicted from the material properties. All the predictions for the model parameters are listed in Table 7-11. In Figure 7-27, the predicted values by the simplified unified model are compared to the measured results.


187

DAC 0/8

0.1

1

10

100

1E-06 0.0001 0.01 1 100 10000Reduced strain rate [1/s]

fc o

r ft

[MP

a]

fc (Measured)ft (Measured)Prediction

EME 0/14

0.1

1

10

100

0.00001 0.001 0.1 10 1000Reduced strain rate [1/s]

fc o

r ft

[MP

a]


PAC 0/16

0.01

0.1

1

10

100


fc o

r ft

[MP

a]


DAC0/16

0.01

0.1

1

10

100

1E-07 0.00001 0.001 0.1 10 1000


fc o

r ft

[MP

a]fc (Measured)ft (Measured)Prediction

SMA 0/11

0.01

0.1

1

10

100

1E-07 0.00001 0.001 0.1 10 1000Reduced strain rate [1/s]

fc o

r ft

[MP

a]


ACRe 0/4

0.1

1

10

100


fc o

r ft

[MP

a]


GAC 0/32

0.1

1

10

100


fc o

r ft

[MP

a]


Figure 7-27 Prediction of compressive and tensile strength from the material properties


188

Table 7-11 Calculation of the unified model parameters

Model parameters

Prediction

T0, Ts Determined by frequency sweep test Pl 0


0 623

0 4021 755.

b.h

b a

VP . E

V V

= ⋅ +

Ph Tensile strength

0 849

0 3080 505.

b.h

b a

VP . E

V V

= ⋅ +


( ) ( ) ( )( )

5 3 30

3

2 83 10 6 81 10 4 21 10

3 73 10 0 139

b a

c

u . E . V . V

. C .

− − −

−

= − × ⋅ + × ⋅ + − × ⋅

+ × ⋅ +

u0 Tensile strength

( ) ( ) ( )( )

7 5 50

5

3 64 10 1 46 10 4 51 10

2 77 10 0 0025

b a

c

u . E . V . V

. C .

− − −

−

= − × ⋅ + − × ⋅ + − × ⋅

+ × ⋅ +


( ) ( ) ( )( )

6 4 3

3

4 36 10 4 24 10 2 62 10

1 03 10 0 311

b a

c

γ . E . V . V

. C .

− − −

−

= − × ⋅ + × ⋅ + × ⋅

+ × ⋅ +

γ Tensile strength

( ) ( ) ( )( )

5 3 3

3

1 5 10 4 10 5 86 10

4 85 10 0 503

b a

c

γ . E V . V

. C .

− − −

−

= − × ⋅ + × ⋅ + × ⋅

+ × ⋅ +

For the “goodness-of-fit” statistics of the simplified unified model, the statistical parameters, mean percentage relative error (MPRE) and the correlation coefficient (R2) are used. For all mixtures, these parameters are shown in Table 7-12 and Table 7-13, respectively.

( )

( )

2

2 1

2

1

1

n

pi miin

mi mi

P PR

P P

=

=

−= −

−

∑

∑ (7-28)

Where: Ppi : predicted value; Pmi : measured data;

mP : average value of measured data;

From MPRE and R2 values listed in Table 7-12 and Table 7-13, it can be seen that the predictions of the simplified unified model for most of the data sets show a good agreement with the measured results. The prediction for the total compressive strength data has a lower MPRE value. This implies that the model has a better prediction capability for the compressive strength compared to the tensile strength. The same conclusions can also be derived from the plots of predicted versus measured values, (see Figure 7-28).


189

Table 7-12 Mean percentage relative errors for different mixtures

DAC 0/8

EME 0/14

PAC 0/16

DAC 0/16

SMA 0/11

ACRe 0/4

GAC 0/32

Total


11.2% 20.5% 7.5% 10.5% 2.7% 20.5% 20.5% 12.9%

Tensile strength 11.8% 13.2% 14.7% 15.5% 19.5% 13.3% 15.0% 14.6% Overall 13.8%

Table 7-13 Correlation coefficients for different mixtures

DAC 0/8

EME 0/14

PAC 0/16

DAC 0/16

SMA 0/11

ACRe 0/4

GAC 0/32

Total


0.91 0.84 0.98 0.98 1.00 0.82 0.82 0.90

Tensile strength 0.84 0.84 0.90 0.77 0.92 0.94 0.77 0.91 Overall 0.92

Simplified unified model

0

10

20

30

40

50

60

0 10 20 30 40 50 60Measured fc [MPa]

Pre

dict

ed fc

[MP

a]



Simplified unified model

0

2

4

6

8

10


Pre

dict

ed ft

[MP

a]



Line of equality

-20%

+20%

Line of equality

-20%

+20%


190

0

10

20

30

40

50

60

0 10 20 30 40 50 60Measured value [MPa]

Pre

dict

ed v

alue

[MP

a]

(c) Compressive and tensile strength results Figure 7-28 Comparison of predicted and measured values

From the statistical analysis discussed above, it can be seen that in the simplified procedure, the model parameter Ph, u0 and γ can be accurately determined by the proposed equations. It gives a good prediction of compressive and tensile strength directly from the material properties. However, with regard to the regression analysis of the linear equation, the number of independent variables is 4 and the number of measurements is 7. It implies that the degree of freedom is not high enough to provide a reliable regression analysis. Therefore, more test results are required to obtain a reliable linear relationship between model parameter u0 and γ the material properties. From Figure 7-25 and 7-26, it seems that the values of the parameter u0 and γ do not differ too much for the different asphalt mixtures. To make a further simplification, it is assumed that the parameters u0 and γ are independent of the material properties. Then the average values of u0 and γ for all seven mixtures can be used in the unified model. The determinations of the unified model parameters for the compressive and tensile strength are presented in Table 7-14.

Table 7-14 Prediction of the unified model parameters (u0 and γ are constant)

T0 Ts Pl Ph u0 γ


0 0 623

0 4021 755.

b.h

b a

VP . E

V V

= ⋅ +

7.87E-02 0.322

Tensile strength

Determined by stiffness test

0 0 849

0 3080 505.

b.h

b a

VP . E

V V

= ⋅ +

5.44E-04 0.565

By means of the equations and values listed in Table 7-14, the predicted mastercurves of the compressive and tensile strength are compared with the measured values in Figure 7-29.

Line of equality

-20%

+20%


191

DAC 0/8

0.1

1

10

100


fc o

r ft

[MP

a]


EME 0/14

0.1

1

10

100


fc o

r ft

[MP

a]


PAC 0/16

0.01

0.1

1

10

100


fc o

r ft

[MP

a]


DAC0/16

0.01

0.1

1

10

100

1E-07 0.00001 0.001 0.1 10 1000


fc o

r ft

[MP

a]fc (Measured)ft (Measured)Prediction

SMA 0/11

0.01

0.1

1

10

100


fc o

r ft

[MP

a]


ACRe 0/4

0.1

1

10

100


fc o

r ft

[MP

a]


GAC

0.1

1

10

100


fc o

r ft

[MP

a]


Figure 7-29 Prediction of compressive and tensile strength (u0 and γ are constant)


192

Similarly, the mean percent relative error (MPRE) and the correlation coefficients R2 are calculated to estimate the accuracy of the prediction. The values of MPRE and R2 for all the mixtures are presented in Table 7-15 and 7-16, respectively.

Table 7-15 Mean percentage relative errors for each mixture

DAC 0/8

EME 0/14

PAC 0/16

DAC 0/16

SMA 0/11

ACRe 0/4

GAC Total


9.4% 9.9% 26.1% 8.9% 16.3% 11.3% 32.7% 16.2%

Tensile strength 15.9% 29.8% 33.2% 18.9% 50.2% 6.7% 21.3% 25.1% Overall 20.6%

Table 7-16 Correlation coefficients for each mixture

DAC 0/8

EME 0/14

PAC 0/16

DAC 0/16

SMA 0/11

ACRe 0/4

GAC Total


0.92 0.96 0.92 0.99 0.96 0.88 0.64 0.92

Tensile strength 0.73 0.54 0.86 0.77 0.69 0.98 0.72 0.86 Overall 0.94

When it is assumed that the parameters u0 and γ are constant, the predictions have a higher MPRE value and a lower R2 value compared to the linear relations, especially for the tensile strength. The MPRE value for the tensile strength prediction increases from 14.6% to 25.1 %. With regard to the correlation coefficients, in most cases the R2 values are larger than 0.7, except for the mixture EME0/14 and SMA0/11. The same conclusions also can be derived from the comparison of prediction and measurement results (see Figure 7-30). It indicates that when taking the average value for the parameter u0 and γ, the accuracy of the unified model becomes lower, but it still gives acceptable predictions for both compressive and tensile strength.

0

10

20

30

40

50

60

0 10 20 30 40 50 60Measured fc [MPa]

Pre

dict

ed fc

[MP

a]

DAC0/8 EME 0/14 PAC 0/16

DAC 0/16 SMA 0/11 ACRe 0/4GAC


Line of equality

-20%

+20%


193

0

2

4

6

8

10


Pre

dict

ed ft

[MP

a]

DAC0/8 EME 0/14 PAC 0/16

DAC 0/16 SMA 0/11 ACRe 0/4GAC


Figure 7-30 Comparison of predicted and measured values

When it is assumed that all data sets are normally distributed around the line of equality, the probability density function of the MPRE can be expressed as follows.

( )21

22 1

2

x µ

σf x;µ;σ eσ π

− − = (29)

( )2

1

1

n

mi mi

P Pσ

n=

−=

−

∑

(30)

Where: σ : sample standard deviation;

mP : average value of measured data;

Pmi : measured data; n : number of test data; x : a variable of MPRE, %; µ : expected value of MPRE, in this study µ=0; The normal distributions of each data set are plotted in Figure 7-31 and the confidence intervals at 80 % and 95 % confidence levels are presented in Table 7-17.

Line of equality

-20%

+20%


194

0

0.5

1

1.5

2

2.5

3

-1.5 -1 -0.5 0 0.5 1 1.5

Compression_linear relation

Tension_linear relation

Compression_average value

Tension_average value

Figure 7-31 Normal distribution for compressive and tensile strength

Confidence intervals are normally used to indicate the reliability of a prediction. At the same confidence level, the confidence interval of MPRE of the compressive strength is smaller than that of tensile strength. The reason is that, as mentioned before, the specimen in a MUC test is more or less uniformly compressed (if a good friction reduction system and a reliable temperature cabinet are used). This leads to a rather high reproducibility of the experimental results, while for the uniaxial tension test, the failure is localized and the failure plane highly depends on a local defect in the sample. This leads to a lower reproducibility of the experimental results. When using the linear relation to determine the model parameters, with a probability of 80% of the confidence intervals, MPRE of the predicted compressive and tensile strength are within ± 21.1% and ± 24.7%, respectively. In terms of the determination of the parameters u0 and γ, when taking average values, the variation of the unified model becomes larger compared to the case when using the linear relation, especially for the tensile strength.

Table 7-17 80% and 95% level of confidence intervals of MPRE

Equation for u0 and γ

Compressive strength Tensile strength

σ1 80% l.o.c.2 95% l.o.c. σ 80% l.o.c. 95% l.o.c. Linear relation

16.4% ± 21.1% ± 32.2% 19.3% ± 24.7% ± 37.8%

Average value

20.6% ± 26.4% ± 40.4% 36.3% ±46.5% ±71.1%

σ1: sample standard deviation

l.o.c.2: level of confidence

( )2f x;µ;σ


195

7.9 Conclusions Based on the results discussed in this chapter, the following conclusions and recommendations can be made.

1. To make an efficient test plan, “the central composite rotatable design techniques” was used to determine the combination of the temperature and strain rate. For the MUC test, this method significantly reduces the number of tests and the results measured at various test conditions are able to describe a reliable relationship between compressive strength and test conditions. For the MUT test, it is different, because the tensile strength is very sensitive to the strain rate at low temperature. So it is difficult to select an appropriate range of the test conditions. The data points obtained from the central composite rotatable design normally do not distribute uniformly in a wide range, which can not give a full description of the tensile strength evolution. Therefore from the author's viewpoint, before using the central composite rotatable design methods, some preliminary tests are needed to determine an appropriate range of test conditions.

2. The tensile and compressive strength are, like the mixture stiffness highly dependent on the temperature and loading rate. When the loading rate or temperature becomes very high or low, the material properties tend to reach a threshold value. This behavior of asphalt mixtures can successfully be described with the unified model.

3. The compressive strength and strain at failure are higher than those in tension, while the tangential stiffness in tension is higher than that in compression at high reduced strain rate. For all the material properties, the slopes of the curves of tensile strength and failure strain vs. reduced strain rate are normally larger than those in compression. This indicates that the material properties in tension are more sensitive to the temperature and strain rate.

4. The model parameters of the unified model depend on material properties, such as aggregate gradation, mixture composition and binder characteristics. Based on the collected compression and tension test results of 7 different types of asphalt mixtures, the unified model was simplified. Equations to predict the model parameters Ph, u0 and γ were developed for compressive and tensile strength. With the temperature susceptibility function obtained from the stiffness test and the prediction equations of the unified model parameters (see Table 7-11), the compressive and tensile strength at a certain strain rate and temperature can directly be calculated from the stiffness of the mixture, volume content of the bitumen, volume content of air voids and the coefficient of curvature of the aggregate gradation. Based on a statistical analysis, it is concluded that the predictions made by means of the simplified unified model show a good agreement with the measured value; this is especially the case for the compressive strength predictions.

5. For further simplification, the parameter u0 and γ can be considered constant (see Table 7-14). Although the variation of the unified model becomes larger, the predictions of the compressive and tensile strength are still quite acceptable.


196

References Diamond, W.J, “Practical Experiment Design for Engineers and Scientists,” Wiley, New York, U.S.A, 2001. Erkens, S.M.J.G. and Poot, M.R., The Uniaxial Compression Test Asphalt Concrete Response (Acre), Report 7-98-117-4, Delft University of Technology, the Netherlands, 2000. Erkens, S.M.J.G., Asphalt Concrete Response (ACRe)-Determination, Modelling and Prediction, PhD Thesis, Delft University of Technology, The Netherlands, 2002. Jacobs, M.M.J., Crack Growth in Asphalt Mixes, PhD Thesis, Delft University of Technology, the Netherlands, 1995. Jansen, J.P.M. Characterising EME Enrobé à Modele Elevé, Master thesis, Civil Engineering and Geosciences, Delft University of Technology, 2002. Jenkins, K.J., Mix Design Considerations for Cold and Half-Warm Bituminous Mixes with Emphasis on Foamed Bitumen, PhD Thesis, University of Stellenbosch, South Africa, 2000. Molenaar, A.A.A. Road Material, Part III, Asphalt Materials, lecture notes e52 (CT4850), Delft University of Technology, The Netherlands, 1994. Muraya, P.M., Permanent Deformation of Asphalt Mixtures, PhD Thesis, Delft University of Technology, the Netherlands, 2007. Pellinen, T.K. and Witczak M.W., “Stress Dependent Master Curve Construction for Dynamic (Complex) Modulus,” Annual Meeting Association of Asphalt Paving Technologists, Colorado Springs, Colorado, USA, March 2002. Pramesti, F. P., Molenaar, A. A. A., and van de Ven, M. F. C., Fatigue Cracking of Gravel Asphalt Concrete; Cumulative Damage Determination, Proceedings of the 7th International RILEM Conference on Cracking in Pavements, Delft, The Netherlands, June 20–22, 2012. Pramesti, F. P. and Poot M.R., Special report; Correction on compression test result due to friction reduction test set-up (frame test+friction reduction system), Delft University of Technology, 2013. Robinson, G.K. Practical Strategies for Experimenting, Sussex, England, 2000. Scarpas, A., Gurp, C.A.M.P. van, Al-Khoury, R.I.N. and Erkens, S.M.J.G., “Finite Element Simulation of Damage Development in Asphalt Concrete Pavements,” 8th International Conference on Asphalt Concrete Pavements, Seattle, Washington, U.S.A., 1997.


197


8.1 Introduction In this chapter, the yield surface concept is applied to analyze different fatigue test results. By means of the hierarchical single surface (HISS) criterion proposed by Desai [1986], the yield surface at a certain stress state can be determined by using the unified models given in Chapter 7. The distance between the stress state at the critical location and the corresponding yield surface is an indication of the safety margin to failure and it is considered to be related to the number of load repetitions until fatigue damage. A new parameter R∆ is introduced as safety factor. By combining the fatigue results described in Chapter 5 and the unified models given in Chapter 7, a new fatigue analysis approach was developed to reduce or exclude the influence of the test type and specimen size on the fatigue results.

8.2 Material Model Concept The response of a material is normally described by a material model consisting of a yield surface and constitutive relations. The yield surface is the collection of all stress combinations that cause the transition from one type of response to the other. The constitutive relation for each separated response describes the relation between stress and strain. Figure 8-1 shows a simple example of the elastic-plastic material model.

Figure 8-1 Relation between stress and strain for the elastic-plastic model

In this model, the material is subjected to a uniaxial load. When the applied stress is below its yield strength, the material behavior is purely linear elastic and governed by Hooke’s law. When the yield strength is reached, the material behaves ideally plastic associated with the development of irreversible strain. The deformation increases continuously at the same stress level. It means that the constitutive relation has changed. The constitutive equations for these two parts are expressed as follows [Hill, 1950] [Erkens, 2002]:

εe εp ε

σ

fy


198

eσ E ε= ⋅ If σ < fy (Elasticity) (8-1)

( )pσ E ε ε= ⋅ − If σ = fy (Plasticity) (8-2)

e pε ε ε= + (8-3)

( )p

f σε λ

σ

∂=

∂ɺ (8-4)

Where: σ : applied stress, [MPa]; E : elastic modulus, [MPa]; ε : total strain, [m/m]; εp : plastic strain, [m/m]; εe : elastic strain, [m/m]; pεɺ : plastic strain rate, [s-1]; f(σ) : yield function of stress and strain; λ : plastic multiplier, λ=0 for elasticity, λ>0 for plasticity. In the axial case, the yield surface is a point (fy, εe), which separates the responses into an elastic part and a plastic part. In Figure 8-2, the yield surfaces are represented as a ellipse and a surface in a two-dimensional and a three-dimensional principal stress space, respectively [Zienkiewicz, O.C., 1999]. If the yield function f(σ) is negative, the stress state is within the circle or the surface and the response is elastic. If the yield function is equal to zero, the stress state is positioned on the circle or the surface and the response is plastic. States of stress outside the yield surface do not exist.

(a) (b)

Figure 8-2 Yield surface in the two-dimensional (a) and three-dimensional (b) principal stress space

Asphalt concrete is a complicated material exhibiting both visco-elastic and visco-plastic behavior. Under uniaxial monotonic loading to failure in the displacement controlled mode, three phases can be observed in the stress-strain curve being the elastic, hardening and softening phase (see Figure 8-3 [Erkens, 2000]). In the beginning, the material behaves linear-elastic (elastic part). At point 1, the non-linear behavior starts and the slope of the curve becomes smaller, but the load carrying capacity is still increasing

σ1

σ2

σ1

σ2

σ3


199

(hardening part). The nonlinear deformations are basically plastic, in other words upon unloading only the portion of elastic strain εe can be recovered from the total strain. The material is weakened by internal microcracks. After reaching the peak value (point 2), the stress decreases and the softening part starts. Microcracking continues and macrocracks will develop.

Figure 8-3 Material response of asphalt concrete under uniaxial loading [Erkens, 2002] In the three-dimensional situation, the material model describing the response of asphalt concrete was developed in the framework of the ACRe project [Scarpas, 1997] [Erkens, 2002] [Liu, 2003]. The hierarchical single surface (HISS) criterion proposed by Desai [1986] was utilized as the response function of the yield surface. In the stress invariant space it is defined as Equation 8-5:

( )( )

2

1 1

2

20

1 cos 3

n

a a

a

I R I R

P PJf

P

− − − + = − =

−

α γ

β θ

(8-5)

1 xx yy zzI = + +σ σ σ (8-6)

( ) ( ) ( )

( ) ( ) ( )

2 2 2

2 1 2 2 3 1 3

2 2 2 2 2 2

1

61

6 xx yy yy zz zz xx xy yz zx

J σ σ σ σ σ σ

σ σ σ σ σ σ τ τ τ

= − + − + −

= − + − + − + + +

(8-7)

( )( )( )( ) ( ) ( )

3

2 2 2

2xx yy zz xy yz zx

xx yz yy zx zz xy

J p p p

p p p

σ σ σ τ τ τ

σ τ σ τ σ τ

= − − − +

− − − − − − (8-8)

( )( )

33

22

3 3cos 3

2

J

J= ⋅θ

(8-9)

3xx yy zzp

σ σ σ+ +=

(8-10)

Strain

Stre

ss

1

2

Elastic

Hardening

Softening


200

Where: σi : i-th principal deviator stress, [MPa]; I1 : the first stress invariant; J2 : the second deviatoric stress invariant; J3 : the third deviatoric stress invariant; p : the isotropic stress, [MPa]; θ : lode angle; Pa : atmospheric pressure, 0.1 MPa; α, β, γ, n, R : model parameters. Desai's response surface exhibits a spindle shape in the (I1, √J2, lode θ) plane (see Figure 8-4). The size, shape and position of the yield surface are controlled by the model parameters α, β, γ, n and R, which are related to the properties of the material and can be directly determined by means of laboratory tests [Erkens et al., 2002].

Figure 8-4 Schematic of Desai response surface in the (I1, √J2, lode θ) plane

[Erkens, 2002]

Figure 8-5 Influence of model parameter α on the response surface in I1-√J2 space [Erkens, 2002] [Medani, 2006]

By plotting the model in the I1-√J2 space, the yield surface changes from 3D to 2D. Figure 8-5 shows the influence of the model parameter α on the yield surface in the I1-√J2 space. It can be seen that the parameter α determines the size of the response surface. The size of the surface increases with decreasing α value. So the hardening response of the

Strain

Stre

ss

α=C 0<α<C α=0


201

material is controlled by the parameter α. In the elastic part, the parameter α has a constant value. Once the hardening response occurs, the value of α starts to decrease. At the peak stress, the parameter α reduces to zero and the yield surface reduces to a straight line in the I1-√J2 space. The model parameter γ controls the slope of the yield surface in the I1-√J2 space, which increases with increasing γ (see Figure 8-6a). The parameter γ is stress-state independent, but it can vary as a function of temperature and loading rate. The parameter n determines the apex of the surface in the I1-√J2 plane. The apex is defined as the point on the response surface where the tangent is a horizontal line (Figure 8-6b), indicating a fully deviatoric state of stress. It expresses the state of stress after which the material starts to dilate [Scarpas et al., 1997] [Liu, 2003]. The parameter R represents the triaxial strength of the material in tension, which is an indication of the cohesion. For cohesionless materials R=0. For increasing R values, the response surface moves on the positive I1 axis (Figure 8-6c).

(a) (b)

(c) Figure 8-6 Influence of the model parameters on the response surface in the I1-√J2 space

[Medani, 2006] The parameter β determines the shape of the model on the octahedral-plane. For β=0, it is a circle and with increasing β it becomes triangular. To simplify the model, it is assumed that the cross section on the octahedral-plane is a circle (β=0). Then the response surface reduces from Equation 8-5 to:


202

2

2 1 12

0n

a a a

J I R I Rf

P P P

− − = − − + =

α γ (8-11)

8.3 Determination of Model Parameters As described in the previous Section, the parameters α and γ are controlled by the hardening and softening response, respectively. The parameter R is the tensile strength of the material when the deviatoric stress is zero. The parameter n determines the shape of the yield surface. In this Section, the model parameters in Equation 8-11 will be determined using the results obtained from the MUC and MUT test presented in Chapter 7. Thus the model parameters can be expressed as function of the strain rate εɺ and temperature T. The methodology was originally presented by many researchers [Scarpas, 1997] [Erkens, 2002] [Liu, 2003]. In the case of the uniaxial states of stress, the expression of the yield surface can be simplified as follows:

1 2 3 0σ σ ,σ σ= = = ⇒

1 1 2 3I σ σ σ σ= + + = (8-12)

( ) ( ) ( )2 2 2 22 1 2 2 3 1 3

1 1

6 3J = − + − + − =

σ σ σ σ σ σ σ (8-13)

Substitution in Equation 8-11: 22

23

n

a a a

R R

P P P

− − = − +

σ σ σα γ (8-14)

8.3.1 Model Parameters R and γ At the peak stress, the hardening parameter α is zero and the expression for the yield surface is simplified as follows:

( )221

3 c cf f R= − −γ (8-15)

( )221

3 t tf f R= −γ (8-16)

Then the solutions for the parameter R and γ are obtained by substituting Equation 8-16 in Equation 8-15:

2 c t

c t

f fR

f f

⋅=

− (8-17)

( )( )

2

23

c t

c t

f f

f f

−=

+γ (8-18)


203

The parameters R and γ are the intercept and slope of the yield surface in the I1-√J2 plane (see Figure 8-7). Since the uniaxial tensile strength ft and the uniaxial compressive strength fc relate to the strain rate and temperature and can be calculated by the unified models of compressive and tensile strength determined in chapter 7, the parameters R and γ can also be expressed as functions of the strain rate and temperature (see Figure 8-8 and Figure 8-9).

0

5

10

15

20

25

-50 -40 -30 -20 -10 0 10 20 30I1 [MPa]

√J2

[MP

a]

0.001_0°C 0.01_0°C 0.1_0°C 0.001_20°C 0.01_20°C

0.1_20°C 0.001_40°C 0.01_40°C 0.1_40°C

Figure 8-7 Yield surface at the peak stress at different strain rates and temperatures

0

5

10

15

20

25

30

0.0001 0.001 0.01 0.1 1

Strain rate [1/s]

R [M

Pa]

0°C

10°C

20°C

30°C

40°C

Figure 8-8 The parameter R as a function of strain rate and temperature

γ

R


204

0

0.05

0.1

0.15

0.2

0.25

0.3

0.0001 0.001 0.01 0.1 1

Strain rate [1/s]

γ

0°C

10°C

20°C

30°C

40°C

Figure 8-9 The parameter γ as a function of strain rate and temperature The value of the parameter γ also changes during softening. At the stress peak, the parameter γ has a maximum value. After the initiation of the degradation response, the evolution of γ can be expressed as a decaying function of equivalent post fracture strain ξpf, the strain rate and temperature (Equation 8-19). The effective post-fracture plastic strain is defined based on the post-fracture plastic strain increment ξpf. The proposed relationship between γ and the effective post fracture plastic strain ξpf (Figure 8-10) is given by:

Figure 8-10 Evolution of the parameter γ in the softening response [Medani, 2006]

( ) ( )( )2

min max min 1exp pf ηγ γ γ γ η ξ= + − − (8-19)

( ) min 2

min min1 min 3T

γγ γ ε β γ= ⋅ +ɺ (8-20)

0


205

( )( )

2

max 23

c t

c t

f f

f f

−=

+γ (8-21)

( ) 12

1 11 T

ηη η ε β= ⋅ɺ

(8-22)

( )0expT sT T T= − − β

Where: γmin : value of γ at the point of complete annihilation; γmax : value of γ at the point of peak stress; γmin1, γmin2, γmin3 : material constants; η1, η2, η11, η12 : material constants; pfξ : effective post fracture plastic strain; εɺ : strain rate; βT : temperature susceptibility function; T0 : reference temperature; Ts : regression constant.

8.3.2 Model Parameter n The model parameter n is related to the dilation in the specimen. Dilation is the increase in volume that results from the opening of internal cracks. At the beginning of a compression test, the axial strain is larger than the radial strain, which leads to a decrease in volume. This is caused by the closure of initial flaws or the reorientation of grains. Once dilation starts, the volume starts to increase. Therefore, the onset of dilation is related to the point at which the volumetric strain changes from decreasing to increasing [Erkens, 2002] [Liu, 2003]. This point can be determined by means of the axial versus volumetric strain plot and the axial strain versus axial stress plot, as described in Chapter 7. During this dilation phase, whole groups of grains slide past each other [Sitters, 1998].

( )2

2

2

13

dil

dil

n

R

= − −

σ

γ σ

(8-23)

The compressive stress at which dilation starts (σdil) is expressed as a function of strain rate and temperature and is described using the unified model. In addition, the model parameters γ and R are expressed as a function of strain rate and temperature. Thus, the model parameter n can be expressed as a function of strain rate and temperature.

8.3.3 Model Parameter α The model hardening parameters can be determined directly from the expression of the response surface for each state of stress (Equation 8-24).


206

2

2 12

1

a an

a

J I R

P P

I R

P

− − +

=−

γα (8-24)

Based on Equation 8-24, values for α for all states of stress throughout the stress strain curves can be found. As a hardening parameter, the value of parameter α varies between the onset of non-linearity and peak strength. The evolution of the parameter α during the hardening response is shown in Figure 8-11. The value of α evaluated at the stress where the plastic behavior (σ=σplas) starts is given by:

22

2

0

3plas plas

a an

plas

a

R

P P

R

P

− − +

= −

σ σγ

ασ

(8-25)

Like the strength, σplas also depends on the strain rate and temperature and this dependency is described using the unified model. In addition, the model parameter R is expressed as a function of strain rate and temperature too. Thus, the parameter α0 also depends on the strain rate and temperature.

Figure 8-11 Evolution of the parameter α in the hardening response [Medani, 2006] The degradation of α from α0 at the onset of plasticity to zero at the peak stress can be expressed as a decaying function of the effective plastic strain, strain rate and temperature (see Figure 8-11). The proposed relationship is given by Equation 8-26:

0


207

( ) ( )0lim

lim

exp= − − αα

α ξ ξ κ ξξ

(8-26)

( ) lim 2

lim lim1 T= ⋅ ξξ ξ ε βɺ (8-27)

( )( )( )2

1 1 exp T= − − ⋅ ακα ακ κ ε βɺ (8-28)

Where: ξ : value of the effective plastic strain; ξlim : value of the effective plastic strain at peak stress; ξlim1, ξlim2 : material constants; κα, κα1, κα2, : material constants that control the rate of hardening;

8.3.4 Determination of Yield Surface The value of the parameter α is equal to zero when the stress is at the peak value. The yield surface reduces to a straight line in the I1-√J2 space and is expressed as follows:

( )2

22 12 12

a a

J I RJ I R

P P

− = ⇒ = −

γ γ (8-29)

The yield surface can be determined based on the results of the MUC and MUT test. In the uniaxial test, the stress is only applied in one direction. As mentioned above, the compressive and tensile strength can be determined by means of the unified model at a given strain rate and temperature, then the parameter γ and R are calculated by Equation 8-17 and 8-18. The yield surface in the stress invariant space can be directly determined by the two points on the yield surface representing the tensile and compression strength [Ning, 2013a].

In stress invariant: 21 2

1

3c cI f , J f

= =

and 21 2

1

3t tI f , J f

= =

(8-30)

where

fc = compressive strength, MPa; ft = tensile strength, MPa.

The strain-field however is three dimensional and this has to be taken into account when defining the yield surface in the strain invariant space. The parameter γ and R are determined by the strain states at the maximum stress in both compression and tension. For simplicity, the used asphalt mixture was assumed to be laterally isotropic, and therefore the values of the strain in two horizontal directions ε2 and ε3 were considered to be equivalent and both of them are equal to εc,h. The yield surface line also can be determined from the tensile and compressive failure strain, expressed by Equation 8-31 [Ning, 2013b].


208

In strain invariant: ( )2

1 2

12

3c ,v c ,h c ,v c ,hI ε ε , J ε ε

= + = −

and ( )2

1 2

12

3t ,v t ,h t ,v t ,hI ε ε , J ε ε

= + −

(8-31)

where

εc,v = vertical compressive strain when stress is at the peak, µm/m; εc,h = horizontal compressive strain when stress is at the peak, µm/m; εt,v = vertical tensile strain when stress is at the peak, µm/m;

εt,h = horizontal tensile strain when stress is at the peak, µm/m.

Figure 8-12 shows an example of the yield surface in the stress invariant space. At a certain stress state, the stress state point (I1, √J2) is related to the magnitude of the applied load. The corresponding yield surface represents the maximum load carrying capacity of the material. To represent the safety margin to failure of the material at a certain stress state, a new parameter, safety factor R∆ is defined using Equation 8-32 [Ning, 2013b]. This approach is also valid in the strain invariant space. When there is no applied stress, the stress state point is at the original point (0, 0), the value of the safety factor R∆ is equal to zero; when the applied stress reaches the strength of the material, the stress state point is on the yield surface and the value of the safety factor R∆ is equal to 1.

i

tot

R∆

∆=

∆ (8-32)

Where: ∆i : distance between original point and stress state point; ∆tot : distance between a data point and the yield surface along

the line through original point and stress state point.

0

1

2

3

4

5

6

-10 -8 -6 -4 -2 0 2 4 6I1 [MPa]

√J2

[MP

a]

Figure 8-12 Yield surface determined by uniaxial tension and compression test results

(cf , 21

3 cf) (

tf , 21

3 tf)

(1I ,

2J )

∆tot ∆i


209

8.4 Critical Location To obtain the yield surface at the critical location, the stress-strain distributions of the specimen should be taken into account. As discussed in Chapter 5, the stress-strain field is uniform in the UT/C test and the critical stress is reached in the complete cross area. The cracks might occur anywhere inside the specimen. For the inhomogeneous tests, the maximum stress only occurs in the critical location of the specimen. For example, in the 4PB test, the maximum strain occurs at the surface in the midsection of the beam.

(a) Horizontal stress (b) Vertical stress

(c) Horizontal strain (d) Vertical strain

Figure 8-13 Stress and strain distributions of IT specimen For the indirect tensile fatigue test, the stress state of the specimen is more complicated. A repeated compressive loading is applied along the vertical diameter of a cylindrical specimen. The vertical load produces both a vertical compressive stress and a horizontal tensile stress at any point in the tested specimen. The magnitude of the compressive and tensile stresses change along the diameter. The centre of the specimen with a maximum horizontal tensile stress is generally supposed to be the critical location. Figure 8-13 illustrates the horizontal and vertical stress distribution simulated by the finite element (FE) program ABAQUS. It is clear that the horizontal tensile stress reaches the maximum value at the centre and the maximum horizontal strain occurs at the place between the


210

centre and the edge of the specimen, because the horizontal strain is not only caused by the horizontal tensile stress, but also by the vertical compressive stress. Assuming that the specimen is homogeneous, isotropic and behaves linear elastic, the solutions of the horizontal stress, vertical stress and horizontal strain along the vertical diameter are derived by Hondros [1959].

( )

22

22

22 4

22 4

1 2 12

11 2 2x

y ysin θRP Rσ y arctan tanθ

yπad y ycos θ

RR R

− + = − −− +

(8-33)

( )

22

22

22 4

22 4

1 2 12

11 2 2y

y ysin θRP Rσ y arctan tanθ

yπad y ycos θ

RR R

− + = − + −− +

(8-34)

( ) ( ) ( )1x x yε y σ y νσ y

E = −

(8-35)

Where: P : applied load, N; a : loading strip width, mm; d : thickness of specimen, mm; R : specimen radius, mm; θ : half the top angle of the triangle between loading strip

and specimen centre; y : distance to the centre of the specimen, mm; E : Young’s modulus, MPa; v : Poisson’s ratio Figure 8-14 shows the comparison of the stress-strain distribution calculated by the Hondros' equations and the FE model. The vertical and horizontal stresses obtained by these two methods are very close. The maximum horizontal strain occurs at the locations y = ± 36 mm from the centre of the specimen and the value calculated by ABAQUS is a little smaller than the results calculated by Equation 8-35. A relatively large area (from y = 36 to y = -36) is created in which the stress-strain field is more or less uniform. This might be the reason why the size effect on the fatigue life is not significant for IT test.


211

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2

Stress [MPa]

Ver

tical

dia

met

er [m

m] horizontal σx_Eqn.

vertical σy_Eqn. horizontal σx_Abaqus vertical σy_Abaqus

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2

Stress [MPa]

Ver

tical

dia

met

er [m

m] horizontal σx_Eqn.

vertical σy_Eqn. horizontal σx_Abaqus vertical σy_Abaqus

(a)

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

-250 -200 -150 -100 -50 0 50 100

Strain [ µm/m]

vert

ical

dia

met

er [m

m]

horizontal εx_Eqn.

horizontal εx_Abaqus

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

-250 -200 -150 -100 -50 0 50 100

Strain [ µm/m]

vert

ical

dia

met

er [m

m]

horizontal εx_Eqn.

horizontal εx_Abaqus

(b)

Figure 8-14 Stress (a) and strain (b) distributions along vertical diameter of a 100 mm diameter IT specimen

The stress-strain state at the locations of y = 0 (point O), y = 36 (point A) are analysed for the IT specimen. The horizontal and vertical stress and the horizontal strain at each point are presented in Table 8-1. It is clear that the horizontal strain at point A is around 27 % higher than that at the centre point. Therefore in this chapter the point O and point A are considered as the critical locations for the IT test.

IT specimen

Point O (0, 0)

Point A (0, 36)

IT specimen


212

Table 8-1 Stress and strain at different locations for size 1.0 and 1.5

Point O Point A Specimen

size Sample code

Force level [N] σxx

MPa σyy

MPa εxx µm/m

σxx MPa

σyy MPa

εxx µm/m

I-4-3 2218 0.45 -1.38 50 0.28 -3.09 64 I-3-1 2428 0.48 -1.46 54 0.30 -3.27 68 I-1-4 2819 0.57 -1.73 61 0.36 -3.88 78 I-1-9 3320 0.70 -2.13 77 0.44 -4.77 99 I-1-5 4213 0.88 -2.68 98 0.55 -6.01 124 I-1-7 5222 1.04 -3.15 115 0.65 -7.07 146

Size1.0

I-1-6 6108 1.26 -3.82 142 0.79 -8.57 181 I-3-4 6020 0.54 -1.66 60 0.32 -3.68 74 I-2-6 6518 0.60 -1.82 60 0.35 -4.03 74 I-4-4 7011 0.63 -1.93 76 0.37 -4.27 95 I-3-6 7513 0.68 -2.07 72 0.40 -4.58 90 I-2-7 8505 0.82 -2.48 87 0.48 -5.51 108 I-4-7 9491 0.86 -2.63 90 0.51 -5.84 111

Size1.5

I-4-6 9969 0.92 -2.79 95 0.54 -6.19 118

8.5 Determination of Yield Surface for Fatigue Test As described in Section 8.3.4, the yield surface determined by the tension and compression test results is a function of strain rate and temperature. During the UT/C and 4PB fatigue test, the specimen is alternately subjected to a tension and compression load. Since the crack damage is mainly caused by tension, the tension loading is considered as the critical load. During a fatigue test, the specimen is tested under a sine or haversine cyclic loading, so the strain rate is not a constant but follows a cosine. To simplify the calculation of the strain rate, the full sine signal is converted into a triangle signal with the same maximum strain level and time duration (see Figure 8-15). The strain rate of this alternating triangular signal is considered to be an acceptable approximation, computed as follows:

0

t=ɺ

εε (8-36)

Where: ε0 : amplitude of the applied strain, m/m; t : time duration when the strain changes from 0 to peak

value, s;


213

-200

-150

-100

-50

0

50

100

150

200

0 2 4 6 8 10 12

Time [s]

stra

in

Figure 8-15 Conversion from a sine signal to a triangle signal [Erkens, 2002]

Knowing the tensile and compressive strength as a function of temperature and strain rate, the yield surfaces which are applicable for the stress or strain controlled mode fatigue test can be determined by Equation 8-30 or 8-31. Figure 8-16 and 8-17 give the initial stress states and their yield surfaces at different force levels in the IT fatigue test. It can be seen that at the same location the data points determined by the initial stress states are aligned and the straight line also goes through the origin (0, 0). With increasing load, the data points of the initial stress states increase along this path. Based on the plots of the stress state point and its yield surface line, the safety indicator R∆ can be calculated by Equation 8-32.

IT-size1.0@point O

0

2

4

6

8

10

12

14

16

-30 -25 -20 -15 -10 -5 0 5 10I1 [MPa]

ΓJ 2

[MP

a]

2.2kN 2.8kN3.3kN 4.2kN5.2kN 6.1kN

Figure 8-16 Stress state and yield surface for each force level for IT_size 1.0 at point O

εɺ εɺ

∆tot ∆i


214

IT_size1.0@point A

0

2

4

6

8

10

12

14

16

-30 -25 -20 -15 -10 -5 0 5 10I1 [MPa]

ΓJ 2

[MP

a]

2.2kN 2.8kN3.3kN 4.2kN5.2kN 6.1kN

Figure 8-17 Yield surface lines for each force level for IT_size 1.0 at point A

The safety factor R∆ at the different locations of the IT specimen are compared in Figure 8-18 and Figure 8-19. The values of R∆ at point O and point A for the IT specimens with the different sizes are presented in Table 8-2. It is clear that the R∆ value at point A is around 35% larger than that at the centre point O at various force levels. It indicates that although the maximum horizontal stress occurs at the centre of the specimen, point A seems to be the weakest point for the IT fatigue test. In the fatigue analysis, therefore the stress state at the point A should be taken into account instead of the stress state at the centre point.

0.00

0.10

0.20

0.30

0.40

0.50

2.2kN 2.4kN 2.8kN 3.3kN 4.2kN 5.2kN 6.1kN

Force level

R∆

IT-size1@point OIT-size1@point A

Figure 8-18 Comparison of the safety factor R∆ at point O and point A for the IT specimen with size 1.0

∆tot

∆i


215

0

0.1

0.2

0.3

0.4

0.5

6.0kN 6.5kN 7.0kN 7.5kN 8.5kN 9.5kN 10.0kN

Force level

R∆

IT-size1.5@point OIT-size1.5@point A

Figure 8-19 Comparison of the safety factor R∆ at point O and point A for the IT

specimen with size 1.5

Table 8-2 Stress and strain at different locations for size 1.0 and size 1.5

Point O Point A Specimen size

sample code

F [N] εɺ [%/s] R∆ εɺ [%/s] R∆

( ) ( )( )

∆ ∆

∆

R A R O

R O

− Fatigue life Nf

I-4-3 2218 0.10 0.13 0.13 0.20 48.0% 1.8×106

I-3-1 2428 0.11 0.14 0.14 0.21 46.5% 7.4×105 I-1-4 2819 0.12 0.16 0.16 0.24 45.7% 1.0×106 I-1-9 3320 0.16 0.19 0.20 0.27 42.9% 4.4×105 I-1-5 4213 0.20 0.23 0.25 0.33 40.3% 1.4×105 I-1-7 5222 0.23 0.27 0.29 0.37 38.6% 8.4×104

Size 1.0

I-1-6 6108 0.28 0.31 0.36 0.43 36.5% 3.0×104

I-3-4 6020 0.12 0.16 0.15 0.23 45.2% 7.7×105 I-2-6 6518 0.12 0.17 0.15 0.24 41.7% 6.9×105 I-4-4 7011 0.15 0.17 0.19 0.25 45.9% 4.5×105 I-3-6 7513 0.14 0.19 0.18 0.27 41.7% 3.4×105 I-2-7 8505 0.17 0.22 0.22 0.31 39.7% 1.6×105 I-4-7 9491 0.18 0.23 0.22 0.32 38.4% 1.3×105

Size 1.5

I-4-6 9969 0.19 0.24 0.24 0.33 37.7% 9.4×104

8.6 Relationship between R∆ and Fatigue Life Based on the analysis of the stress or strain states, the yield surface at various loading levels in the fatigue test can be determined by Equation 8-30 and 8-31. Table 8-3 shows the stress states and the corresponding yield surfaces for the stress-controlled fatigue tests discussed in Chapter 5. The safety factor R∆ is determined from the plot of the stress states and the yield surfaces in the I1-√J2 space, as shown in Figure 8-20, Figure 8-21, Figure 8-22 and Figure 8-23.


216

Table 8-3 Stress states and corresponding yield surfaces at the critical locations for the stress-controlled fatigue tests at 5 ºC

Stress state Yield surface Test type

Sample code σ1

[MPa] σ2

[MPa] fc

[MPa] ft

[MPa] R γ

R∆ Nf

C-15-5 1.24 0 -23.5 6.23 17.0 0.112 0.204 1.5×106 C-16-9 1.28 0 -24.0 6.23 16.8 0.115 0.216 1.1×106 C-16-2 1.41 0 -24.2 6.23 16.8 0.116 0.234 8.4×105 C-15-7 1.49 0 -24.6 6.23 16.7 0.118 0.247 4.7×105 C-15-8 1.55 0 -25.2 6.23 16.6 0.121 0.260 2.9×105 C-15-3 1.59 0 -25.7 6.23 16.5 0.124 0.281 1.6×105

UT/C-size1.0

C-15-10 1.82 0 -27.3 6.23 16.2 0.132 0.301 8.3×104 B-11-4 1.61 0 -25.3 6.23 16.6 0.122 0.362 2.4×106 B-11-1 1.82 0 -25.8 6.23 16.4 0.125 0.429 1.7×106 B-11-2 1.82 0 -25.7 6.23 16.5 0.124 0.432 1.9×106 B-7-3 1.93 0 -26.1 6.23 16.4 0.126 0.473 9.1×105 B-7-4 2.04 0 -26.0 6.23 16.4 0.125 0.511 7.3×105 B-7-1 2.13 0 -26.1 6.23 16.4 0.126 0.543 6.0×105

4PB-size1.0

B-16-3 2.32 0 -26.7 6.23 16.3 0.129 0.624 5.8×105 I-4-3 0.28 -3.09 -18.5 6.23 18.8 0.082 0.199 1.8×106 I-3-1 0.30 -3.27 -18.9 6.23 18.6 0.085 0.206 1.0×106 I-1-4 0.36 -3.88 -19.7 6.23 18.2 0.090 0.236 7.4×105 I-1-9 0.44 -4.77 -21.3 6.23 17.6 0.100 0.273 4.4×105 I-1-5 0.55 -6.01 -22.9 6.23 17.1 0.109 0.325 1.5×105 I-1-7 0.65 -7.07 -24.0 6.23 16.8 0.115 0.368 8.4×104

IT-size1.0

I-1-6 0.79 -8.57 -25.5 6.23 16.5 0.123 0.426 3.0×104 I-3-4 0.32 -3.68 -19.3 6.23 19.0 0.079 0.226 7.7×105 I-2-6 0.35 -4.03 -19.9 6.23 19.0 0.079 0.242 6.9×105 I-4-4 0.37 -4.27 -20.3 6.23 18.3 0.089 0.253 4.5×105 I-3-6 0.40 -4.58 -20.7 6.23 18.4 0.087 0.266 3.4×105 I-2-7 0.48 -5.51 -22.0 6.23 17.9 0.095 0.305 1.6×105 I-4-7 0.51 -5.84 -22.4 6.23 17.8 0.096 0.319 1.3×105

IT-size1.5

I-4-6 0.54 -6.19 -22.8 6.23 17.7 0.098 0.333 9.4×104 Due to the existence of a vertical compressive stress at the critical location, the value of I1 for the IT fatigue test is negative and the differences of the stress states between the different load levels are relatively large compared to the other two types of fatigue tests.


217

UT/C-size1@5°C_ σ control

0

2

4

6

8

10

12

14

16

18

-30 -20 -10 0 10 20I1 [MPa]

√√ √√J 2

[MP

a]C-15-5 C-16-9C-16-2 C-15-7C-15-8 C-15-3C-15-10

0.6

0.8

1

1.2

1 1.5 2 2.5I1 [MPa]

√√ √√J 2

[MP

a]

Figure 8-20 Stress states and yield surfaces for the UT/C test in stress controlled mode

4PB-size1@5°C_ σ control

0

2

4

6

8

10

12

14

16

18

-30 -20 -10 0 10 20

I1 [MPa]

ΓJ 2

[MP

a]

B-11-4 B-11-1B-11-2 B-7-3B-7-4 B-7-1B-16-3

0.81

1.21.41.6

1 1.5 2 2.5 3I1 [MPa]

ΓJ 2

[MP

a]

Figure 8-21 Stress states and yield surfaces for the 4PB test in stress controlled mode

IT_size1.0@5°C_ σ control

0

2

4

6

8

10

12

14

16

-30 -25 -20 -15 -10 -5 0 5 10I1 [MPa]

ΓJ 2

[MP

a]

I-4-3 I-1-4I-1-9 I-1-5I-1-7 I-1-6I-3-1

Figure 8-22 Stress states and yield surfaces for the IT specimen size 1.0 in stress

controlled mode


218

IT_size1.5@5°C_ σ control

0

2

4

6

8

10

12

14

16

-30 -20 -10 0 10I1 [MPa]

ΓJ 2

[MP

a]I-3-4 I-2-6

I-4-4 I-3-6

I-2-7 I-4-7I-4-6

Figure 8-23 Stress states and yield surfaces for the IT specimen size 1.5 in stress

controlled mode Similarly, the safety factor R∆ at a certain strain level for the strain-controlled fatigue tests can be calculated from the strain state and the corresponding yield surface. Examples of the I1-√J2 plots for the UT/C test with specimen size 1.0 at 20 ºC and 5 ºC are shown in Figure 8-24 and 8-25, respectively. At 20 ºC the yield surface lines at different strain levels show significant differences, because the unified model of the failure strain at 20 ºC is more sensitive to the strain rate than at the relatively low or high temperatures (T ≤ 5 ºC or T ≥ 35 ºC). Due to the Poisson effect, the strain at the critical location for the UT/C and 4PB test changes in three dimensions under the axial loading. So in the determination of the yield surface, the failure strains in three mutually perpendicular directions are taken into account. The values of R∆ and the fatigue lives for some strain controlled fatigue tests are listed in Table 8-4. ε1 means the initial tensile strain along the loading direction, ε2 and ε3 are the strain in the other two directions perpendicular to the strain ε1. For simplicity purposes, ε3 is considered equivalent to ε2 in magnitude.

UT/C-size1@20ºC

0

3000

6000

9000

12000

15000

-2000 -1000 0 1000 2000 3000I1 [µm/m]

√J 2

[µm

/m]

68µε 84µε110µε 131µε155µε 172µε199µε

0

50

100

150

200

0 50 100 150I1 [µm/m]

√J 2

[µm

/m]

Figure 8-24 Strain states and yield surfaces for the UT/C test in strain controlled mode at

20 ºC


219

Table 8-4 Strain states and corresponding yield surfaces at the critical locations for the strain-controlled fatigue tests

Strain state Yield surface Test type

ε1 [µm/m] ε2 [µm/m] εt,v

[µm/m] εt,h

[µm/m] εc,v

[µm/m] εc,h

[µm/m] R∆ Nf,50

88 -26 3105 -1208 -13047 6140 0.043 1.6×106 117 -35 2569 -1071 -12507 5695 0.079 5.4×105 130 -39 2384 -1020 -12304 5531 0.101 3.0×105 136 -41 2310 -999 -12221 5465 0.111 2.8×105 138 -41 2285 -992 -12191 5441 0.116 2.3×105 146 -44 2196 -966 -12085 5357 0.132 1.3×105 168 -50 1996 -906 -11835 5162 0.179 9.0×104

UT/C-size0.5 @20 ºC

186 -56 1862 -863 -11654 5022 0.288 5.6×104 68 -18 3899 -1396 -13785 6772 0.022 6.0×106 84 -25 3195 -1230 -13133 6213 0.040 1.9×106 110 -33 2666 -1097 -12610 5778 0.070 5.5×105 131 -39 2365 -1015 -12284 5515 0.103 2.8×105 155 -47 2109 -940 -11979 5274 0.149 1.1×105 172 -52 1962 -895 -11790 5126 0.187 7.2×104


199 -60 1780 -835 -11537 4933 0.255 3.2×104 68 -21 3642 -1336 -13550 6568 0.027 4.3×106 89 -27 3075 -1200 -13018 6116 0.046 1.3×106 115 -34 2593 -1077 -12533 5716 0.080 4.1×105 133 -40 2351 -1011 -12267 5502 0.110 1.9×105 161 -48 2055 -924 -11911 5220 0.171 7.0×104


182 -55 1888 -871 -11690 5050 0.225 4.0×104 69 -15 637 -183 -7433 2390 0.123 2.7×106 87 -19 624 -160 -7194 2280 0.149 1.5×106 94 -21 620 -153 -7114 2244 0.159 6.6×105 105 -23 615 -143 -7003 2196 0.175 2.8×105 118 -26 611 -134 -6893 2148 0.193 1.6×105 124 -27 609 -130 -6844 2127 0.201 1.6×105

UT/C-size1.0 @5 ºC

136 -30 606 -123 -6754 2090 0.219 8.3×104 100 -30 2843 -1142 -12791 5927 0.045 5.7×106 119 -36 2531 -1061 -12466 5662 0.081 1.3×106 140 -42 2266 -987 -12169 5424 0.116 8.5×105 161 -48 2060 -926 -11918 5226 0.158 3.8×105 179 -54 1909 -878 -11719 5072 0.202 2.2×105

4PB-size0.5 @20 ºC

201 -60 1770 -832 -11522 4922 0.258 2.1×105 87 -26 3134 -1215 -13075 6164 0.041 4.0×106 105 -31 2762 -1122 -12709 5860 0.061 1.9×106 124 -37 2458 -1041 -12387 5598 0.089 6.6×105 142 -43 2240 -979 -12138 5399 0.120 4.2×105 161 -48 2057 -924 -11913 5222 0.159 3.1×105

4PB-size1.0 @20 ºC

180 -54 1908 -878 -11718 5071 0.203 1.9×105 74 -22 3478 -1298 -13398 6438 0.029 7.9×106 88 -26 3100 -1207 -13043 6137 0.042 1.7×106 93 -28 2995 -1181 -12941 6052 0.047 1.6×106 103 -31 2801 -1132 -12748 5892 0.058 2.5×106

4PB-size1.5 @20 ºC

119 -36 2535 -1062 -12471 5666 0.080 5.8×105 89 -20 623 -158 -7170 2269 0.152 2.0×106 105 -23 615 -143 -7009 2198 0.174 1.6×106 109 -24 613 -140 -6965 2179 0.181 1.3×106 116 -26 611 -135 -6908 2155 0.190 3.4×105 124 -27 609 -130 -6843 2127 0.202 4.7×105

4PB-size1.0 @5 ºC

135 -30 606 -123 -6761 2092 0.217 4.1×105


220

UT/C-size1@5ºC

0

1000

2000

3000

4000

5000

6000

-3000 -2000 -1000 0 1000 2000

I1 [µm/m]

√J 2

[µm

/m]

69µε 87µε94µε 105µε118µε 124µε136µε

0

50

100

150

0 50 100I1 [µm/m]

√J2

[µm

/m]

Figure 8-25 Strain states and yield surfaces for the UT/C test in strain controlled mode at

5 ºC For the fatigue test, each stress or strain level corresponds to a R∆ value and also a fatigue life. Assuming that 50% stiffness reduction corresponds to a certain amount of damage, it is possible to find a relationship between R∆ and the fatigue life. 1. Relation in double logarithm coordinates The plots of R∆ versus the fatigue life for the strain-controlled and stress-controlled tests are presented in Figure 8-26 and Figure 8-27. It can be seen that a straight line is found on a double logarithm scale similar to the traditional fatigue line. Equation 8-37 shows the relation between R∆ and fatigue life. All the coefficients for the different fatigue tests are given in Table 8-5.

21

rfN r R∆= ⋅ (8-37)

Where: Nf : fatigue life; R∆ : safety factor, ratio of ∆i and ∆tot; r1 and r2 : regression coefficients.


221

Strain-controlled mode

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

0.01 0.1 1R∆

Nf,

50

UT/C-size0.5@20ºC UT/C-size1.0@20ºCUT/C-size1.5@20ºC UT/C-size1.0@5ºC4PB-size0.5@20ºC 4PB-size1.0@20ºC4PB-size1.5@20ºC 4PB-size1.0@5ºC

Figure 8-26 Relation between R∆ and Nf,50 for the fatigue tests in strain-controlled mode

Stress-controlled mode

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

0.1 1R∆

Nf

UT/C-size1.0@5ºC 4PB-size1.0@5ºC

ITT-size1.0@5ºC ITT-size1.5@5ºC

Figure 8-27 Relation between R∆ and Nf for all the fatigue tests in stress-controlled mode Figure 8-26 shows that in the case of the 4PB fatigue test at 20 ºC, there is no significant difference between the fatigue lines obtained from the specimens with different specimen sizes. The results indicate that the size effect on the fatigue line can be excluded in this new fatigue relation. At 20 ºC, the slopes of the fatigue lines obtained from the UT/C and 4PB tests are very close. With decreasing temperature, the slope of the fatigue line increases. The reason is that the yield surface does not change significantly with the strain rate at low temperature in general. Comparison with the results of the UT/C fatigue test shows that the fatigue lines of the 4PB test are still higher, because the safety factor R∆ represents the safety level at the critical location, being at the surface in the middle part of the beam. The amount of damage in the rest of the beam is not as much as at the critical location. Therefore the fatigue life obtained from the 4PB test is longer than from the UT/C test when the value of R∆ is the same.


222

Table 8-5 Regression coefficients of Nf-R∆ curve determined by the different fatigue tests

Test type r1 r2 R2 UT/C_size0.5@20 oC 3065 -1.93 0.93 UT/C_size1.0@20 oC 2045 -2.11 0.99 UT/C_size1.5@20 oC 2137 -1.99 0.99 UT/C_size1.0@5 oC 5.22 -6.37 0.97 4PB_size0.5@20 oC 3244 -2.17 0.99 4PB_size1.0@20 oC 3866 -2.02 0.97 4PB_size1.5@20 oC 1432 -2.41 0.78

Controlled-strain mode

4PB_size1.0@5 oC 101.5 -5.32 0.74 UT/C_size1.0@5 oC 14.34 -7.37 0.89 4PB_size1.0@5 oC 1.1×106 -3.11 0.89 IT_size1.0@5 oC 523 -4.99 0.99

Controlled-stress mode

IT_size1.5@5 oC 334 -5.12 0.99 With regard to the stress-controlled test, the fatigue life in the 4PB test is also longer than those in the UT/C and IT test. The coefficients r1 (intercept) and r2 (slope) of the fatigue lines are not comparable for the different fatigue tests (see Table 8-5). However, the data points themselves show that UT/C fatigue data are coinciding rather well with the IT fatigue data. The reason is that the FE modeling showed that the distributions of the tensile stress and strain are relatively uniform in the middle part along the diametrical direction. This critical area is around 70% of the whole cross area. Furthermore it is clear that there is no difference between the fatigue lives of the two IT fatigue specimen sizes. 2. Limit value of the safty factor R∆ Figure 8-28 shows the plots of the fatigue life vs. the safety factor R∆ in the normal coordinate system for both strain-controlled and stress-controlled tests. In general, Nf,50 increases with decreasing R∆ value. When the R∆ is lower than a certain value, the fatigue life Nf tends to be inifinite. Taking into account the shape and the trend of the data points, the following regression equation can be used to simulate the R-N (initial stress ratio versus fatigue life) lines:

( )0 1 1 fbaNR R R e−∆ = − − (8-38)

Where: R0, R1, a and b : regression coefficients.


223

Strain-controlled mode

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2000000 4000000 6000000 8000000 10000000 12000000Nf,50

R∆

UT/C-size0.5@20ºC UT/C-size1.0@20ºCUT/C-size1.5@20ºC UT/C-size1.0@5ºC4PB-size0.5@20ºC 4PB-size1.0@20ºC4PB-size1.5@20ºC 4PB-size1.0@5ºC

(a) Strain-controlled mode

Stress-controlled mode

0

0.2

0.4

0.6

0.8

0 500000 1000000 1500000 2000000 2500000 3000000

Nf

R∆

UT/C-size1.0@5°C 4PB-size1.0@5°C

IT-size1.0@5°C IT-size1.5@5°C

(b) Stress-controlled mode

Figure 8-28 Initial stress ratio versus fatigue life Nf,50

The correlation coefficients R2 of the equations for these three test types are all reasonable, as the values of all the specimen types are larger than 0.8. Therefore, it is believed that Equation 8-38 is capable of providing a fair prediction for the R-N lines. When the fatigue life becomes infinite, the safety factor R∆ tends to a limit value Rlimit.

lim 0 1itR R R= − (8-39)

It indicates that the specimen will not have any fatigue damage under the loading condition Rinitial = Rlimit, which relates to the fatigue endurance limit εlimit.


224

Table 8-6 Regression coefficients and equations of three kinds of binder

Regression coefficients Specimen type

R0 R1 a b Rlimit R2

UTC Size0.5_ε_20°C 0.173 0.140 1.60E-06 0.66 0.0332 0.809 UTC Size1.0_ε_20°C 0.319 0.282 3.06E-06 0.460 0.0376 0.904 UTC Size1.5_ε_20°C 0.419 0.393 2.86E-06 0.280 0.0261 0.835 UTC Size1.0_ε_5°C 0.524 0.423 1.33E-07 0.058 0.1011 0.887 UTC Size1.0_σ_5°C 0.412 0.212 2.10E-06 0.428 0.2000 0.901 4PB Size0.5_ε_20°C 0.258 0.213 1.86E-06 0.564 0.0454 0.870 4PB Size1.0_ε_20°C 0.921 0.881 2.42E-06 0.138 0.0404 0.873 4PB Size1.5_ε_20°C 0.895 0.586 2.07E-06 0.135 0.0394 0.900 4PB Size1.0_ε_5°C 0.559 0.416 2.20E-07 0.066 0.1428 0.837 4PB Size1.0_σ_5°C 0.911 0.687 1.65E-07 0.250 0.2238 0.796 IT Size1.0_σ_5°C 0.725 0.521 2.01E-06 0.198 0.2047 0.870 IT Size1.5_σ_5°C 0.727 0.510 3.35E-06 0.197 0.2171 0.920

0 0.05 0.1 0.15 0.2 0.25




UTC Size1.0_ε_5°C

UTC Size1.0_σ_5°C




4PB Size1.0_ε_5°C

4PB Size1.0_σ_5°C

IT Size1.0_σ_5°C

IT Size1.5_σ_5°C

Rlimit

Figure 8-29 The values of Rlimit in all different cases From Figure 8-29, it can be seen that the Rlimit value does not change too much with the specimen size and test type but is strongly influenced by the temperature and loading mode. For all the UT/C, 4PB and IT fatigue tests, the value of Rlimit obtained at the low temperature and stress controlled mode is larger than that obtained from the high temperature and strain controlled mode, respectively.


225

8.7 Conclusions Based on the results presented in this chapter, the following main findings and conclusions are summarized below:

1. By means of the unified models described in Chapter 7, the yield surface based on the Desai yield function can be determined when the strain rate and the temperature are known. At the peak stress, the yield surface reduces to a straight line in the I1-√J2 space.

2. A new parameter, called the safety factor R∆ is introduced as an indicator of the “safety against failure” at a certain stress-strain state. When there is no applied stress, the material is absolutely safe, the value of the safety factor R∆ is equal to zero; when the applied stress reaches the strength of the material, the material starts to yield, the value of the safety factor R∆ is equal to 1.

3. The critical locations of the cylinder and beam specimen can be easily determined for the UT/C and 4PB fatigue tests, respectively. During the IT fatigue test, the specimen is subjected to both horizontal and vertical stresses along the vertical diameter. Based on an FE model, the maximum horizontal tensile stress occurs in the center point (y=0) and the maximum horizontal strain is found at the locations

with 36

50y R= ± (R is the radius). The R∆ value at the points 36

50y R= ± is around

35% larger than at the centre point at various force levels. Therefore these two points seem to be the weakest points instead of the centre point for the IT fatigue test.

4. By converting the sine wave to a triangle wave, the strain rate can be calculated at a given strain amplitude and frequency. For the fatigue test, each load level corresponds to a safety factor R∆ and a fatigue life. In the stress-controlled mode, the value of R∆ is calculated by the stress state and the yield surface determined by the failure stress. In the strain-controlled mode, the value of R∆ is calculated by the strain state and the failure strains.

5. Similar to the traditional fatigue line, a straight line is found by plotting R∆ at the critical location and the fatigue life on a log-log scale. In contrast to the traditional fatigue line, the size effect on the 4PB and IT fatigue results can be ignored by using this new fatigue relation. For the stress-controlled mode, the fatigue lines obtained from the UT/C shows good agreement with the IT fatigue results. However, the 4PB test shows a longer fatigue life when the safety factor at the critical points are the same. The differences of the fatigue relations caused by temperature and loading mode still exist in this new fatigue method.

6. The relationship between R∆ and fatigue life in the normal coordinate can also be

expressed as the function ( )0 1 1 fbaN

R R R e−

∆ = − − . When the safety factor R∆

reduces to a limit value Rlimit, the fatigue life becomes infinite. The Rlimit is independent of specimen size and test type but is strongly influenced by the temperature and loading mode. Rlimit obtained at the low temperature and stress controlled mode is larger than that obtained from the high temperature and strain controlled mode, respectively.


226

References Desai, C.S., Somasundaram, S. and Frantziskonis, G., a Hierarchical approach for Constitutive Modelling of Geologic Materials, International Journal of Numerical and Analytical Methods in Geomechanics, 1986; 10(3): 225-257. Erkens, S.M.J.G., Liu, X., Scarpas, A. and Molenaar, A.A.A., 3D Finite Element Model for Asphalt Concrete Response Simulation, Paper presented at the 2nd International Symposium on 3D Finite Element in Pavement Engineering, Charleston, West-Virginia, USA, 2000. Erkens, S.M.J.G., Asphalt Concrete Response - Determination. Modelling and Prediction. PhD Thesis. Delft University of Technology, the Netherlands, 2002. Hill, R., The Mathematical Theory of Plasticity. Oxford University Press, London, U.K., 1950. Hondros, G., 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. Austr. J. Appl. Sci.. 1959; 10(3): 243-268. Liu, X., Numerical Modelling of Porous Media Response Under Static and Dynamic Load Conditions. PhD Thesis. Delft University of Technology, the Netherlands, 2003. Li, N., Molenaar, A.A.A., van de Ven, M.F.C., Wu S., Application of a New Fatigue Analysis Approach on laboratory fatigue tests. The 5th European Asphalt Technology Association (EATA) conference. Braunschweig, Germany. June 3-5, 2013a. Li, Ning, Molenaar, A.A.A., Pronk A.C., van de Ven, M.F.C., Characterization of Fatigue Performance of Asphalt Mixture Using a New Fatigue Analysis Approach. Construction and Building Materials, 2013b, Vol. 45, pp. 45-52. Medani, T.O., Huurman, M., Superposition Principle to Determine Properties of Bituminous Mixtures in the Time–Temperature Domain. Proc. of the 7th International Conference on the Bearing Capacity of Roads. Railways and Airfields. Norway, 2005. Scarpas, A., Gurp, C.A.P.M., van, Al-Khoury, R.I.N., Erkens, S.M.J.G., Finite Elements Simulation of Damage Development in Asphalt Concrete Pavements. 8th International Conference on Asphalt Pavements. Seattle, U.S.A., 1997. Sitters, C.M.W., Material Models for Soil and Rock, Delft University of Technology, Faculty of Civil Engineering, the Netherlands, 1998. Zienkiewicz, O.C., Chan, A.H.C., Pastor, M., Schreer, B.A., Shiomi, T., Computational Geomechanics, with Special Reference to Earthquake Engineering. Wiley& Sons, 1999.


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Chapter 9 Conclusions and Recommendations The main goal of this research is to investigate the influence of the test type and specimen size on laboratory fatigue test results and to find a way to harmonize the fatigue results obtained from different tests. In Chapter 5, the results of three types of fatigue tests were discussed. For each test, specimens with different dimensions were used. The detailed analysis approaches used in this research were described in Chapter 6 to Chapter 8. In this chapter, the important findings of each Chapter are summarized. Section 9.1 gives the generalized conclusions and Section 9.2 presents the recommendations for future work.

9.1 Conclusions

9.1.1 Conclusions Related to Literature Review

• Since fatigue cracking due to repeated traffic loading is one of the main failure modes of an asphalt pavement, the fatigue characteristics of asphalt concrete are important for the thickness design of asphalt pavements.

• In pavement design methods, normally the fatigue laws are use obtained from laboratory fatigue testing. Various fatigue test devices are currently used to evaluate the fatigue performance of asphalt concrete. Even at the same test conditions, the results from the different tests are difficult to compare to each other. It is therefore important to understand the effect of the test type and specimen size on fatigue results.

• Fatigue testing is normally done in a one-dimensional or two-dimensional stress state. In reality however pavement materials are subjected to a three-dimensional stress state. This makes it necessary to represent the failure condition obtained by means of laboratory testing in a three-dimensional space.

9.1.2 Conclusions Related to Fatigue Test Equipment

• The test configuration of the UT/C test is relatively simple and the results are hardly influenced by the test set up. Due to the uniform stress and strain state, data processing is simple. However, to prevent bending moments, there are high demands for the used specimen and test setup. The upper and lower steel platens should be parallel to each other and perpendicular to the loading direction. The cylindrical specimen should be polished and its top and bottom ends should be exactly parallel to each other and perpendicular to the axis of the cylinder. This asks for very careful preparation of specimens and equipment.

• Compared to the UT/C and IT test, the test results measured by the 4PB test are highly influenced by the test equipment, such as the clamping force, distance between the clamps, structural strength of the whole set up, etc. To avoid high shear


228

and tensile stress concentrations near the supports, the four clamps should be allowed to rotate freely and horizontal translation and the clamping force should be as low as possible. Furthermore, shear deflection of the beam specimen and deformation of the frame should be taken into account in the calculation of the stiffness of the beam specimen.

• The IT test is simple and easy to perform. The specimen can be easily prepared in the laboratory or cored from the field. However the stress-strain distributions in the specimen are complex. For this test, it is impossible to apply a sinusoidal loading to the specimen; a haversine or half sine shaped load pulse need to be applied, which induces permanent deformation during testing even at a low temperature.

9.1.3 Conclusions Related to Stiffness and Fatigue Results

• The complex stiffness values measured by the UT/C and 4PB test are similar and are not influenced by the specimen size. A lower stiffness is obtained from the IT test and the measured IT stiffness is slightly influenced by the specimen size. These differences are most probably caused by the difference in the mode of loading and state of stress.

• The fatigue life NR determined by the dissipated energy ratio is smaller than the traditional fatigue life. The difference between NR and Nf measured by the UT/C test is smaller compared to those from the 4PB and IT test. It indicates that the crack propagation is much shorter during the UT/C fatigue test.

• The effect of specimen size is negligible for the UT/C and IT fatigue results, but for the 4PB fatigue test the influence of the specimen size is significant. The smaller specimen size with a lower height has a longer fatigue life. The reason is that the tensile stress-strain fields inside the UT/C and IT specimen are relatively uniform compared to the 4PB specimen.

• With regard to the influence of the test type, at the same test condition and loading mode, the fatigue life from the 4PB test is the longest and from the IT test it is the shortest. The differences might be explained by the stress-strain distributions of the specimen in the different fatigue tests. Apart from that, failure of the IT is also partly caused by permanent deformation.

• With regard to the NR-Wfat relation, the effect of specimen size on the 4PB test is obvious but not for the UT/C and IT test. The NR-Wfat line is independent of the temperature but influenced by the loading mode. The slope of the curves in the stress-controlled mode is a little steeper than in the strain-controlled mode. The total dissipated energy measured by the IT test is lower compared to the UT/C and 4PB tests at the same fatigue life NR. This difference is caused by the irreversible deformation, biaxial stress state at the center of the specimen and the smaller phase angle measured in the IT test. By taking into account the dissipated energy in vertical direction, the corrected NR-Wfat lines from the IT test is much closed to the results from the other tests.

9.1.4 Conclusions Related to Partial Healing Model

• The PH model provides a good simulation of the evolution of the complex modulus and phase angle for a unit volume in a strain-controlled fatigue test. The results


229

from the UT/C fatigue test can be directly used for the determination of model parameters.

• The model parameters can be expressed as functions of the applied strain level and are not significantly influenced by the specimen size. The parameter δα1 (reversible damage in the loss modulus) is always nil. The parameter δα2 (reversible damage in the storage modulus) becomes smaller with a decrease in temperature. The parameter δγ2 (irreversible damage in storage modulus) shows an opposite tendency. This indicates that in the UT/C test, the healing ability of the asphalt mixture decreases at a lower temperature and the more irreversible damage happens at a higher strain level or at a lower testing temperature.

• For the 4PB test, the strain is not distributed uniformly throughout the beam specimen. The functions of the PH model parameters are used to calculate the local stiffness for different parts of the specimen. In this way the weighted overall stiffness can be calculated by a weighing procedure. For the specimen size 1.0, the functions of the PH model parameters obtained from the UT/C and 4PB test are similar to each other. It indicates that the 4PB test results can be predicted by means of the PH model parameters determined from the UT/C test results.

• By means of the PH model, the local stiffness at the surface in the midsection of the beam is comparable with the UT/C fatigue test results when the pure bending strain on the beam surface is equal to the strain applied to the cylinder. The fatigue line based on the local surface stiffness of the beam shows a good agreement with the results obtained from the UT/C fatigue test.

• The trends of the values of the parameter δγ1 and δγ2 indicate the existence of an endurance limit. The predicted endurance limit obtained by the UT/C test is not influenced by the specimen size and temperature. The range of the endurance limit is between 66 and 69 µm/m. For the 4PB test, the predicted endurance limit for a unit volume is influenced by the specimen size. A larger specimen has a lower endurance limit.

9.1.5 Conclusions Related to Monotonic Test Results

• The tensile and compressive strength are dependent on the temperature and loading rate. When the loading rate or temperature becomes very high or low, the material properties tend to reach a threshold value. This behavior of asphalt mixtures can successfully be described by the unified model.

• The model parameters of the unified model were expressed as functions of the material properties including the stiffness of the mixture, volume content of the bitumen, volume content of air voids and the coefficient of curvature of the aggregate gradation. With the temperature susceptibility function obtained from the stiffness test and the functions of the unified model parameters, the compressive and tensile strength at a certain strain rate and temperature can be predicted directly.

• For further simplification, the parameters u0 and γ can be simply considered to be constant. Although the variation of the unified model becomes larger, the predictions of the compressive and tensile strength are quite acceptable.


230

9.1.6 Conclusions Related to Yield Surface Approach

• By means of the unified models discussed above, the yield surface based on the Desai yield function could be determined at a given strain rate and temperature. At the peak stress, the yield surface reduces to a straight line in the I1-√J2 space.

• The safety factor R∆ is introduced as an indicator of the “safety against failure”. A smaller R∆ value means that the material is safer. The range of R∆ value is from 0 to 1.

• For the IT test, based on the FE model, The R∆ value at the points 36

50y R= ± is

around 35% larger than that at the centre point y=0. Therefore these two points were considered as the higher loaded places instead of the centre point for the IT fatigue test.

• By converting the sine wave to a triangle wave, the strain rate takes a constant value and can be calculated by the given strain amplitude and frequency. For the fatigue test, each loading level corresponds to a safety factor R∆ and a fatigue life. A straight line is found by plotting R∆ at the critical location and the fatigue life on a log-log scale. Compared to the traditional fatigue line, the size effect on the 4PB and IT fatigue results can be ignored by using this new fatigue relation. For the stress-controlled mode, the fatigue lines obtained from the UT/C test show a good agreement with the IT fatigue results. Due to the inhomogeneity of the stress-strain distribution, the fatigue life from the 4PB test is higher than those obtained from the other fatigue tests when the safety factor at the critical point is the same. The differences of the fatigue relations caused by the temperature still exist in this new fatigue method.

• The relationship between R∆ and the fatigue life in the normal coordinate also can

be expressed as the function of ( )0 1 1 fbaN

R R R e−

∆ = − − . When the safety factor R∆

reduces to a limit value Rlimit, the fatigue life becomes infinite. The Rlimit value does not change too much with the specimen size and test type but is strongly influenced by the temperature and loading mode. For all the UT/C, 4PB and IT fatigue tests, the value of Rlimit obtained at the low temperature and stress controlled mode is larger than that obtained from the high temperature and strain controlled mode, respectively.

9.2 Recommendations

9.2.1 Recommendations Related to Experimental Work • To make an efficient test plan, the “central composite rotatable design technique”

was used to determine the combination of the temperature and strain rate. This test design method significantly reduces the number of tests. However, the tensile strength is very sensitive to the strain rate at low temperatures. It is difficult to select an appropriate range of test conditions, which can give a complete description of the tensile strength evolution. Therefore, before using the central composite rotatable design methods, some preliminary tests are needed to determine a appropriate range of the test conditions.


231

• Based on the calibration of the 4PB test set up, it was found that the frame stiffness is only 4 times higher than the stiffness of the aluminum beam. For the large specimen or the test at low temperature, the deformation of the test frame significantly influences the test results because of the relatively high applied force. In future research, the 4PB set up should be strengthened to resist this deformation and to allow applying fairly high load levels when tests need to be performed at low temperatures or when using a large specimen.

• After modification of the 4PB equipment, the two outer clamps are still controlled by the springs so that the outer clamping force varies with the applied force. It is therefore recommended to further improve the outer clamps to keep the outer clamping force constant.

• Since the fatigue test is time-consuming, only one asphalt mixture type was used in this research. It is recommended that different asphalt mixture types should be tested to verify the conclusions based on the results of the DAC 0/8 mixture.

9.2.2 Recommendations Related to Partial Healing Mo del

• At present, the Partial Healing model is only valid for the fatigue test in strain-controlled mode. It is proposed to develop the solutions for the stress-controlled test in the near future.

• By taking into account the stress-strain distribution and the specimen shape, the Partial Healing model is also applicable for the fatigue results from the two-point bending or three-point bending tests.

9.2.3 Recommendations Related to Yield Surface Appr oach

• The model parameters of the unified model depend on material properties. Based on the collected compression and tension test results of 7 different asphalt mixture types, the model parameters were expressed as function of the stiffness of the mixture, volume content of bitumen, volume content of air voids and the coefficient of curvature of the aggregate gradation. In the future, the monotonic test results from more asphalt mixtures are needed to improve those functions of the model parameters.

• The relation between R∆ and fatigue life obtained from the 4PB test shows a big difference compared to the other two tests, because the safety factor R∆ of the 4PB specimen simply represents the stress state at the most severe location and is much larger than those of the UT/C and IT test. It is recommended that for the 4PB test, a new safety factor should be defined to adequately represent the safety condition of the beam specimen.


232

Appendix A Calculations for 4-Point Bending Test

233

Appendix A Calculations for 4-Point Bending Test The four-point bending test is schematically given in Figure A-1. All dimensions are in SI units unless other specified. For the test setup used in this research, the distance between two outer clamps is 400 mm. The distance between inner two clamps A is equal to one third of L.

Figure A-1 Principle four-point bending test

A.1 Shape factor Z [m-1]:

2 3

3

6 8

8

AL AZ

bh

−= (A-1)

where: A : distance between inner clamp and outer clamp, next to each other, [m];

L : effective length=distance between outer clamps, [m]; b : width of specimen, [m]; h : height of specimen, [m].

L

Sample

Clamping Clamping

Actuator


234

For L= 3A (evenly spaced) the formula becomes:

3 3

3 3

460.213

216

L LZ

bh bh= = ⋅ (A-2)

A.2 Weighing function Rx for mass compensation

2 2

2 2

12 1

3 3

12 1

3 3

Lfor x A

A x x A

L L LR x

Lfor x A

x A A x

L L L

> − − =

< − −

(A-3)

Because L= 3A and in this configuration (standard) the sensor is in the middle so x= L/2, Rx becomes:

2 2

36 56.348

2 1 1 13 3

2 2 3

LR = =

− −

(A-4)

And the weighing factor at position x=A becomes (for the moving masses at x=A):

2 2

36 64.8

1 1 13 3

3 3 3

R x = = − −

(A-5)

A.3 Equivalent masses meq[kg]

The equivalent beam mass used in the back calculation depends on the x coordinate where the deflection (used for back calculation) is measured. Here, the sensor is in the middle so x= L/2:

. 4

20.5785eq beam beam beam

LR

m m mπ

= ⋅ = ⋅

(A-6)

The equivalent masses for other moving masses like the clamps and the sensor depend on the x coordinate xback used for the back calculation and the coordinate xplace where the extra mass is located.


235

.

back

eq mass placeplace place

R x R AM M

R x R x=

(A-7)

In general you will use x = L/2 for the back calculation. So, the equivalent mass for the clamps (and all other moving masses at x = A) is given by:

.2

0.8696eq beam clamps beam

LR

m m mR A

= ⋅ = ⋅ (A-8)

The equivalent mass of the sensor used at x=L/2 for the back calculation is given by:

. 1.15

2

eq sensor sensor sensor

R Am m m

LR

= ⋅ = ⋅

(A-9)

So, for the given configuration of the 4-point bending test, the equivalent mass is given by:

0.5785 0.8696 1.15eq sensor clamps sensorm m m m= ⋅ + ⋅ + ⋅ (A-10) where: mbeam : mass of the beam [kg]; mclamps : Moving masses (except beam) between load cell

and outer supports = 1.6 [kg]; msensor : mass sensor =0.14 [kg]. A.4 Stiffness [Pa], (D=0, S=0) Neglecting damping D (non viscous) effects in the system (system losses) and shear, the complex stiffness E*, which consists of a real part (ERe) and an imaginary part (EIm), can be written as:

( ) 2Re cos s eq

b

FE Z m

vϕ ω

= +

(A-11)

( )Im sin sb

FE Z

vϕ

=

(A-12)

where: F : amplitude of the force, [N]; vb : (bending) amplitude of the deformation, [m]; φs : measured phase lag [º] meq : equivalent mass [kg] ; see (A4.4) ω : circular frequency [rad/s]


236

2 2Re ImdynE E E= + (A-13)

Im

Re

arctanmix

E

Eϕ

=

(A-14)

where: F : amplitude of the force, [N]; vb : (bending) amplitude of the deformation, [m]; φs : measured phase lag [º] meq : equivalent mass [kg] ; see (A4.4) ω : circular frequency [rad/s] Substituting Equation A-2 in Equation A-11 and A-12, Equation A-13 and A-14 are expressed as follows [N/m2]:

( )2

2 21 2cos b bdyn s eq eq

b

v vZFE m m

v F Fϕ ω ω = + +

(A-15)

( )( ) 2

sinarctan

cos

smix

bs eq

vm

F

ϕϕ

ϕ ω

= +

(A-16)

In general the equations used for this calculation are given by:

( ) ( )3

21 2cosdyn sb

FLE Q Q

v R x Iϕ= + ⋅ + (A-17)

2beq

vQ m

Fω= (A-18)

31

12I bh= (A-19)

4

eq beam clamps y

R x R x R x R Am m m m

R A R y R yπ

= ⋅ + ⋅ +

∑ (A-20)

A.5 Maximal tensile strain εx [m/m]

34 b

h A R xx v x

Lε

⋅ ⋅= (A-21)

So for L=3A and calculating the strain at the center, at x=L/2, this gives:

2 2

56.34822 12 2 12b b

Lh R

L L hv v

L Lε

⋅ ⋅ = =

(A-22)


237

A.6 Maximum stress σ [Pa]

2 22

132

16

F AM F A FL

W bh bhb hσ

⋅ ⋅= = = =⋅

(A-23)

where: M : bending moment of the beam [Nm]; W : resistance of the cross section against bending [m3]; A.7 Energy dissipation of beam W [J] Total consumed energy for the beam (so including system loss) for one cycle is calculated by:

( ), sinbeam tot s bW v Fπ ϕ= ⋅ ⋅ (A-24)

And the total dissipated (viscous) energy for the beam for one cycle is calculated by:

( ) 2, 3

sinbeam visc mix b

R xW v I

Lπ ϕ= ⋅ ⋅ ⋅ (A-25)

In this set-up Rx=RL/2= 56.348, so Equation A-25 becomes:

( ) ( )2 3

2 3, 3 3

56.348sin 4.7 sin

12b

beam visc s b s

v b hW v b h

L Lπ ϕ π ϕ ⋅ ⋅= ⋅ ⋅ ⋅ = ⋅ (A-26)

A.8 Energy dissipation per unit of volume W [J/m3] And the total dissipated (viscous) energy per unit m3 for one cycle is calculated by:

( )3,sin svisc per m

W π ε σ ϕ= ⋅ ⋅ ⋅ (A-27)

The cumulated dissipated energy per cycle up to cycle n:

3 3, , , ,1

n

sum visc per m visc per m ii

W W=

= ∆∑ (A-28)

If the measurements are taken at intervals Ni we use Simpson for numerical integration:

13 3 3, , , , ,1

1

mii

nn N N

isum visc per m visc per m visc per m iii

W W dn W N== = ∆

==

∑ = ⋅ ≈ ∆ ⋅ ∆ ∑∫ (A-29)


238

Appendix B Determination for Prony Series Model Parameters

239

Appendix B Determination for Prony Series Model Parameters In the Abaqus software, the visco-elasticity of the asphalt mixture is characterized by the Prony series representation. The time domain stiffness relation for this model is given in Equation B-1.

( ) /

1

i

nt

r ii

G t G G e τ−

=

= + ⋅∑ (B-1)

where: Gr : rubbery shear modulus, [MPa]; Gi : shear stiffness, [MPa]; τi : relaxation time; n : number of Prony parameters terms. The model parameters Gr, Gi and τi are determined by conducting the stiffness frequency sweep tests. The parameters determination procedure is incorporated in a user friendly graphical user interface program (GUI) in Matlab environment developed by Woldekidan [Woldekidan, 2011]. By changing the number of Kelvin-voigt elements, optimization can be performed until a satisfactory fit is obtained. The procedure for the parameter determination is summarized as follows:

1. Import the master curve data, frequency (rad/s), complex modulus (MPa) and phase angle (degree) to the Matlab GUI program; in a comma delimited file.

2. In the program interface, fill in the number of Kelvin-voigt elements.

3. Run the optimization to perform Prony series fitting to the imported master curve data.

4. Change the number of Kelvin-voigt terms and re-run the optimization till satisfactory fit is obtained. The weighting factor for complex stiffness, phase angle, loss and storage modulus can also be altered from the program interface helping to obtain better fits.

5. The output data are exported to an excel file in the working direction.

The number of Prony parameter terms is determined based on the quality of the model description. In this study, the number of the Prony terms is 12. Figure B-1 illustrates the obtained regression fit for the specimen UT/C-size 1.0 at a reference temperature of 20°C. Similar qualities of fits were also obtained for the other specimens. The generalized Prony series model parameters for the different specimen types are given in Table B-1 and Table B-2. These data were used as the input data for the material properties in the ABAQUS software.


240

Figure B-1 Matlab GUI for model parameter determination

Table B-1 Prony series model parameters for the UT/C specimen at Tref = 20 oC (12 terms)

n Size0.5 Size1.0 Size1.5 τi Gi τi Gi τi Gi 1 1.2×10-6 1.8×10-1 1.2×10-6 1.7×10-1 1.2×10-6 1.3×10-1 2 7.3×10-6 5.9×10-2 7.3×10-6 5.8E-02 7.3×10-6 1.0×10-1 3 4.3×10-5 1.2×10-1 4.3×10-5 1.2×10-1 4.3×10-5 6.0×10-2 4 2.6×10-4 1.0×10-1 2.6×10-4 1.1×10-1 2.6×10-4 1.1×10-1 5 1.5×10-3 1.2×10-1 1.5×10-3 1.3×10-1 1.5×10-3 9.2×10-2 6 8.9×10-3 1.2×10-1 8.9×10-3 1.3×10-1 8.9×10-3 1.7×10-1 7 5.3×10-2 1.2×10-1 5.3×10-2 1.3×10-1 5.3×10-2 1.4×10-1 8 3.1×10-1 9.8×10-2 3.1×10-1 9.7×10-2 3.1×10-1 1.3×10-1 9 1.8 5.4×10-2 1.8 5.0×10-2 1.8 5.4×10-2 10 11 1.6×10-2 11 1.3×10-2 11 1.5×10-2 11 64 2.1×10-3 64 1.6×10-3 64 2.3×10-3 12 3.8×102 5.0×10-4 3.8×102 3.1×10-4 3.8×102 7.4×10-4

Gr [MPa] 28367 28127 28627


241

Table B-2 Prony series model parameters for the 4PB specimen at Tref = 20 oC (12 terms)

n Size0.5 Size1.0 Size1.5 τi Gi τi Gi τi Gi 1 1.0×10-6 1.5×10-1 1.0×10-6 1.3×10-1 1.0×10-6 1.6×10-1 2 5.0×10-6 3.6×10-2 5.0×10-6 3.5×10-2 5.0×10-6 3.6×10-2 3 2.5×10-5 1.0×10-1 2.5×10-5 9.3×10-2 2.5×10-5 1.0×10-1 4 1.2×10-4 8.0×10-2 1.2×10-4 7.6×10-2 1.2×10-4 7.9×10-2 5 5.9×10-4 1.0×10-1 5.9×10-4 1.0×10-1 5.9×10-4 1.0×10-1 6 2.9×10-3 1.1×10-1 2.9×10-3 1.1×10-1 2.9×10-3 1.1×10-1 7 1.4×10-2 1.1×10-1 1.4×10-2 1.3×10-1 1.4×10-2 1.2×10-1 8 6.9×10-2 1.1×10-1 6.9×10-2 1.3×10-1 6.9×10-2 1.3×10-1 9 3.4×10-1 9.2×10-2 3.4×10-1 1.0×10-1 3.4×10-1 9.2×10-2 10 1.6 6.4×10-2 1.6 6.1×10-2 1.6 4.9×10-2 11 8.0 2.7×10-2 8.0 1.7×10-2 8.0 1.3×10-2 12 39 1.4×10-2 39 1.3×10-2 39 1.1×10-2

Gr [MPa] 25032 22400 24982 Due to the big variation of the measured phase angle, it was decide not to calculate the prony model parameters for the IT test results. In the simulation of the IT specimen, the material was simply defined as an elastic material. References Woldekidan, M.F., Mixture Performance Optimization: Laboratory Investigation and LOT Performance Computations. Report 7-12-186-2, Delft University of Technology, Delft, the Netherlands, 2012. Woldekidan, M.F., Response Modelling of Bitumen, Bitumenous Mastic and Mortar, PhD Thesis, Delft University of Technology, The Netherlands, 2011.


242

243

Curriculum Vitae

Personnel Information

Name: Ning Li (李宁)

Gender: Male

Date of Birth: July 27, 1983

Place of Birth: Tianmen, Hubei Province, P.R. China

Email: [email protected]; [email protected]

Education Experience

Oct. 2008 – Nov. 2013 PhD student in Section of Road and Railway Engineering,

Faculty of Civil Engineering and Geosciences, Delft University of Technology, the Netherlands.

Sep. 2006 – Jun. 2008 Master student in Material Science, School of Materials Science and Engineering, Wuhan University of Technology, Wuhan, P.R.China.

Sep. 2002 – Jun. 2006 Bachelor student in Material Science and Engineering,

School of Materials Science and Engineering, Wuhan University of Technology, Wuhan, P.R.China.

244

Publications

1. Li, N., Molenaar A.A.A., van de Ven, M.F.C., and Wu, S. Estimation of the Fatigue Endurance Limit of HMAC for Perpetual Pavements. Journal of Wuhan University of Technology-Mater. Sci. Ed. Vol.25 No.4, 2010, pp 645-649.

2. Li, N., Molenaar, A.A.A., van de Ven, M.F.C., and Wu S., Characterization of Fatigue Performance of Asphalt Mixture Using a New Fatigue Analysis Approach. Construction and Building Materials. Volume 45, 2013, pp 45–52.

3. Li, Ning, Molenaar, A.A.A., Pronk A.C., van de Ven, M.F.C. and Wu, S., Comparison of Uniaxial and Four Point Bending Fatigue Tests for asphalt mixtures. Journal of Transportation Research Record (TRR), 2013. (Accepted)

4. Qiu J., Li N., Pramesti F.P., Van de Ven M.F.C. and Molenaar A.A.A. Evaluating Laboratory Compaction of Asphalt Mixtures using the Shear Box Compactor. Journal of testing and evaluation. 2012; 40 (5): 844-852.

5. Li, Ning, Molenaar, A.A.A., Pronk A.C. and van de Ven, M.F.C., Effect of Specimen Size on Fatigue Behavior of Asphalt Mixture in Laboratory Fatigue Tests, Proceedings of the 7th RILEM International Conference on Cracking in Pavements, Delft, the Netherlands, pp 827-836, 2012.

6. Li, Ning, Molenaar, A.A.A., Pronk A.C., van de Ven, M.F.C., Investigation into the Size effect on Four Point Bending Fatigue Tests, Proceedings of the 3rd Workshop on 4PB, Davis, California, September 17-18 2012.

7. Li, Ning, Molenaar, A.A.A., Prediction of tensile strength of asphalt concrete, Proceedings of the 2nd International Conference on Sustainable Construction Materials (SusCoM2012). Wuhan, China, October 18-22 2012.

8. Li, N., Molenaar, A.A.A. and van de Ven, M.F.C., Application of a New Fatigue Analysis Approach on laboratory fatigue tests. 5th European Asphalt Technology Association (EATA) conference. Braunschweig, Germany. June 3-5 2013.

9. A.A.A. Molenaar, Ning Li, Pungky Pramesti, Fatigue Characterization of Asphalt Mixtures for Designing Long Life Pavements, 15th AAPA International Flexible Pavements Conference, Brisbane, Australia, September 22-25, 2013.

ASPHALT MIXTURE FATIGUE TESTING

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