CALIBRATION OF RUTTING MODELS FOR HMA STRUCTURAL AND MIXTURE DESIGN APPENDIX K ADVANCED MATERIALS CHARACTERIZATION AND MODELING

Transcription

1 ALIBRATION OF RUTTING MODELS FOR HMA STRUTURAL AND MIXTURE DESIGN APPENDIX K ADVANED MATERIALS HARATERIZATION AND MODELING Prepared for: National ooperative Highway Researh Program Transportation Researh Board National Researh ounil of National Aademies Washington, D Prepared by: Dr. harles W. Shwartz, P.E., UMd (o-prinipal Investigator); Primary Author Dr. Regis L. arvalho, UMd; Primary Author Mr. Harold L. Von Quintus, P.E., ARA (Prinipal Investigator) Mr. Jagannath Mallela, ARA (Projet Manager) Dr. Ramond Bonaquist, P.E., AAT (o-prinipal Investigator) Submitted by: Applied Researh Assoiates, In North Mays Street, Suite 105 Round Rok, TX (512) May 2012

2 AKNOWLEDGEMENT OF SPONSORSHIP This work was sponsored by the Amerian Assoiation of State Highway and Transportation Offiials, in ooperation with the Federal Highway Administration, and was onduted through the National ooperative Highway Researh Program, whih is administered by the Transportation Researh Board of the National Aademies. DISLAIMER The opinions and onlusions expressed or implied in the report are those of the researh ageny. They are not neessarily those of the Transportation Researh Board, the National Researh ounil, the Federal Highway Administration, the Amerian Assoiation of State Highway and Transportation Offiials, or the individual states partiipating in the National ooperative Highway Researh Program.

3 Table of ontents Table of ontents... i List of Tables... iii List of Figures... iv Aknowledgements... vii Abstrat... viii Exeutive Summary... ix hapter 1 Introdution Problem Statement Objetive Doument Layout hapter 2 Modeling Permanent Deformation of Asphalt onrete Introdution Mehanisti-Empirial Modeling Mehanisti-Empirial Modeling Limitations Asphalt onrete Mehanisti Modeling Linear Visoelasti Behavior Visoelasti ontinuum Damage Behavior Visoplasti Behavior Summary hapter 3 Materials and Test Equipment Mixture harateristis Speimen Preparation Testing Equipment Summary hapter 4 Model alibration Linear Visoelasti omponent omplex Modulus Testing Dynami Modulus Master urve Relaxation Modulus and reep ompliane ontinuum Damage Model onstant Strain Rate Tests to Failure alibration of Damage Funtion alibration of Damage Funtion alibration of Damage Funtion Visoplasti Model Multi-Stress/Load Duration Tests Visoplasti Model alibration Validation with the Flow Number Test alibration of the Visoplasti Model Using the Flow Number Test Validation with the Multi-Stress/Load Duration Test Summary K - i

4 hapter 5 Finite Element Modeling Introdution Finite Element Method Geometri Model and Boundary onditions Meshing and Element Definition Material Property Speifiation Loading onditions Simulation Visoelasti-Visoplasti Model Implementation Numerial Diffiulties and Simplifiations Model Verifiation Summary hapter 6 Numerial Appliations Introdution Simplified Finite Element Modeling Approah Influene of Shear Stress Reversals Pratial Appliations of Finite Element Simulations Predited Rutting omparison between Different Pavement Strutures Analysis of the MEPDG Depth Fator for Rutting Preditions onlusions Field rutting preditions Desription of FHWA ALF Numerial Simulation onlusions hapter 7 onlusions and Reommendations Model alibration Moving Wheel Analyses Effet of Different Pavement Strutures MEPDG Depth Fator for Rutting Preditions Field Rutting Preditions Reommendations Referenes Appendix A. Visoelasti-Visoplasti UMAT Appendix B. Elasto-Visoplasti UMAT K - ii

5 List of Tables Table 1. Laboratory mixture design Table 2. Stresses applied in the dynami modulus test Table 3. Summary of dynami modulus testing Table 4. Dynami master urve and temperature shift fators Table 5. Prony series onstants for relaxation modulus and reep ompliane Table 6. Damage funtion 12 alibration oeffiients Table 7. Deviatori stress and frequeny used on multi-stress/load duration test Table 8. Visoplasti model alibration oeffiients Table 9. omparison between the two visoplasti model alibrations Table 10. Material properties used for studying the effets of the bouning versus the moving wheel Table 11. Struture-based depth funtion oeffiients K - iii

6 List of Figures Figure 1. omponents of measured surfae rutting Figure 2. Effets of rutting onentration in different layers on permanent deformation surfae profile of flexible pavements: (a) asphalt onrete, (b) granular base, and () subbase/subgrade (White et al., 2002) Figure 3. Rutting development versus load appliations Figure 4. Influene of depth funtion β σ3 on alulated permanent vertial strains from the MEPDG model (150 mm HMA layer over rushed stone base, first load yle) Figure 5. omputed total strains beneath tire enter at peak load: (a) vertial; (b) horizontal (Shwartz and arvalho, 2008) Figure 6. Residual total strains from the EPFE analysis after removal of load Figure 7. Residual total vertial strains beneath the tire enter from EPFE analyses for different number of yles Figure 8. Generalized Maxwell model Figure 9. omplex modulus response Figure 10. Example of master urve Figure 11. Example of temperature shift funtion Figure 12. One-dimensional elasto-visoplasti rheologial model Figure 13. Three-dimensional representation of HiSS surfae in prinipal stress spae (Gibson, 2006) Figure 14. Shemati of flow rule and surfae hardening (Gibson, 2006) Figure 15. Asphalt onrete mixture aggregate gradation Figure 16. Axial LVDT instrumentation Figure 17. Radial LVDT setup Figure 18. Dynami modulus master urve Figure 19. Temperature shift funtion Figure 20. omparison between MEPDG and Hirsh master urves Figure 21. Storage modulus master urve Figure 22. Relaxation modulus and reep ompliane Figure 23. Unonfined strain rate tests at 10 : stress versus strain urves for strain rates of and ε/seond Figure 24. onfined strain rate tests at 10 : stress versus strain urves for strain rates of and ε/seond Figure 25. Damage funtion 11 versus S for all repliates at Figure 26. Poisson's ratio variation during the strain rate tests Figure 27. Damage funtion 12 versus S for all repliates at Figure 28. Damage funtion 22 versus S for all repliates at Figure 29. Example of multi-stress/load duration reep and reovery test Figure 30. reep and reovery visoplasti strains versus load yles Figure 31. Visoplasti model alibration using yli reep and reovery tests: (a) unonfined and (b) onfined K - iv

7 Figure 32. Predited versus measured strains in the Flow Number validation test Figure 33. Visoplasti model alibration using the Flow Number test Figure 34. Predited versus measured visoplasti strains from multi-stress/load duration reep and reovery tests: (a) unonfined and (b) onfined. Predited strains omputed using the FN-alibrated model Figure 35. 3D solid model Figure 36. Plan (surfae) view of geometri model Figure 37. Surfae layer mesh in plan view Figure 38. 3D finite element mesh Figure 39. Shemati of moving load Figure 40. Example of HiSS surfae and the limit where normal vetors to the surfae an be alulated Figure 41. Shematis of reep and reovery test at fixed stress level and varying loading time Figure 42. Fixed stress test with 1,500 kpa deviator stress, unonfined, tested at Figure 43. Fixed stress test with 1,500 kpa deviator stress and 250 kpa onfining stress, tested at Figure 44. Influene of temperature on visoplasti behavior of asphalt onrete in a simulated reep and reovery test Figure 45. omparison between visoelasti-visoplasti and elasto-visoplasti simulation of reep and reovery at two different temperatures Figure 46. Permanent deformation omparison between a moving wheel load and repeated vertial load (Brown et al., 1996) Figure 47. Stress distributions over time at the loation of maximum shear for one moving wheel pass Figure 48. Stress distributions over time at the loation of maximum shear for one bouning wheel load Figure 49. omparison between permanent vertial strains indued by loading histories with and without prinipal stress rotations Figure 50. omparison between stress state paths indued by moving and bouning wheel at the loation of highest shear stress Figure 51. omparison between moving and bouning wheel finite element simulations Figure 52. omparison between strain distributions within the asphalt onrete surfae layer when moving and bouning wheel are onsidered. MnRoad rutting distribution from trenh studies is shown in the inset Figure 53. Pavement strutures: (a) onventional flexible, (b) full depth asphalt onrete, and () omposite Figure 54. Asphalt rutting for different pavement strutures Figure 55. omparison of rutting transverse profiles for different pavement strutures Figure 56. omparison of permanent strain distributions with depth for different pavement strutures K - v

8 Figure 57. MEPDG depth funtion analysis for the onventional flexible pavement Figure 58. alibrated depth funtion results for the full depth asphalt onrete pavement Figure 59. alibrated depth funtion results for the omposite pavement Figure 60. Layout of the 12 as-built pavement lanes (Qi et al., 2004) Figure 61. Layer deformation measurement assembly used to measure rut depth (Gibson, 2011) Figure 62. Rutting measurements at Lane 11 of the ALF Figure 63. Load equivaleny equivalent aumulated load yles versus tire pass Figure 64. omputed permanent deformation for 100-yle sequential and 500-yle load equivalent Figure 65. EVP finite element predited rutting for the first 100 yles Figure 66. Ratio a N /a 100 as funtion of number of yles Figure 67. Predited and measured rutting for the ALF lane Figure 68. omparison between preditions using moving and bouning wheel for the ALF lane K - vi

9 Aknowledgements The researh desribed herein was performed under NHRP Projet 9-30A by the Transportation Setor of Applied Researh Assoiates (ARA), In. Mr. Harold L. Von Quintus served as the Prinipal Investigator on the projet. Mr. Von Quintus was assisted by Mr. Jagannath Mallela as the Projet Manager and Engineer on the team. Other management team members and subontrators inluded Dr. harles Shwartz of the University of Maryland, and Dr. Ramon Bonaquist of Advaned Asphalt Tehnologies, L. Both Dr. Shwartz and Bonaquist served as o-prinipal Investigators on the projet. One of the objetives from NHRP projet 9-30A was the ontinuing development of advaned haraterization and modeling of asphalt onrete mixtures. The projet team at the University of Maryland was responsible for this task. All members of the researh team aknowledge and appreiate the support provided by Federal Highway Administration to this task, and speially Dr. Nelson H. Gibson, who provided all laboratory speimens used in this projet task, whih were tested at the University of Maryland laboratory failities, and for sharing laboratory test results onduted at FHWA whih led to the model validation. Drs. harles W. Shwartz and Regis L. arvalho were the primary authors of Appendix K. All members of the researh team also aknowledge and greatly appreiate the support and effort for helping bring this projet to ompletion by Dr. Edward Harrigan with NHRP and the NHRP panel members. K - vii

10 Abstrat Permanent deformation is a major distress in flexible pavements that leads to the development of rutting along the wheel path of heavily traffiked roads. Early detetion of rutting is very important for preventive maintenane programs and design of rehabilitation strategies. Rutting by definition is the aumulated permanent deformation that remains after removal of the load. Rigorous modeling of permanent deformations using nonlinear finite element analysis based on the orret physial mehanism of residual deformations after removal of the load provides important insights into the rutting problem. The doumentation for NHRP Projet 9-30A onsists of a final report and eleven appendies (Appendix A through K). This appendix (Appendix K) douments the study of permanent deformation in asphalt onrete in pavement strutures using a fully mehanisti model based on Shapery s visoelastiity and Perzyna s visoplastiity theories. The model is alibrated and implemented in a 3D finite element ommerial software pakage. Two alibration proedures are performed and disussed. Two immediate pratial appliations are shown and a simulation of full sale aelerated pavement test is performed. K - viii

11 Exeutive Summary This appendix douments the work performed by the University of Maryland at the request of the National ooperative Highway Researh Program (NHRP) Projet 9-30A panel to develop a fully mehanisti model to haraterize permanent deformation in asphalt onrete mixtures. A onstitutive model framework based on Shapery s visoelastiity theory and Perzyna s visoplastiity using a HiSS flow surfae was presented. This model, whih was developed in previous studies (Gibson, 2006; Kim, 2009), has been enhaned in the present work. An improved alibration proess was developed and verified, and the model was implemented in a 3D finite element formulation. Two immediate pratial appliations were shown and a simulation of a full-sale aelerated pavement test was performed. The results and disussion provide important information about the rutting problem in asphalt onrete and how it develops and evolves over traffi loading; this ontributes to the development of better modeling tehniques for pavement performane preditions. This work demonstrates that the Perzyna-HiSS visoplasti model an be suessfully alibrated using either researh-grade reep and reovery tests or the more simple and prodution-oriented Flow Number test. The importane of indued shear stress reversals under a moving wheel load is doumented. The 3D finite element simulation is then used to identify the fundamental differenes on how rutting develops in different pavement strutures in terms of the differenes in the transverse profile and distribution of rutting within the layer. The analysis results are used to develop new pavement-speifi depth funtions for potential future inorporation into the AASHTO Mehanisti-Empirial Pavement Design Guide (MEPDG). Lastly, the 3D finite element model is used to predit rutting at one lane of the FHWA s fullsale Aelerated Load Faility experiment. After orretion for some anomalies during the early loading yles in the experiment, the predited and measured rutting at the enter of the wheel path were in good agreement. K - ix

12 hapter 1 Introdution Permanent deformation ours in most of the pavement layers and auses the development of ruts along the wheel path at the surfae. Early detetion of rutting is very important for preventive maintenane programs and design of rehabilitation strategies. Rutting is defined as a longitudinal surfae depression ourring in the wheel paths of roadways (FHWA, 2003). It is often followed in later stages by an upheaval along the sides of the rut. It an lead to strutural failure and potential danger from hydroplaning. Rutting aumulates inrementally with small permanent deformations from eah load appliation over the pavement s servie life. Rutting is by definition a load-related pavement distress. The total rutting is the ombination of aumulated permanent deformation in all layers in the pavement struture. Some researh suggests that the shape of the surfae profile an indiate whih layer is responsible for the failure of the pavement struture due to rutting. In well-designed flexible pavements, trenh studies show that nearly all of the total rutting ourred within the top 150 mm of surfae asphalt onrete layer. Two mehanisms are the main auses of rutting development. ompation is the primary mehanism at initial stages of loading, in whih the material volume dereases underneath the wheel path with no signifiant upheaval along the sides of the wheel path. After this initial stage is omplete, distortion without volume alteration ours and the material beneath the wheel path migrates to the edge forming the upheaval along the. This deformation mode is essentially aused by shear. Exessive aumulation of shear deformation leads to strutural failure. Three types of models are used to ompute permanent deformations: empirial, mehanisti-empirial and fully mehanisti. Examples of empirial models are regression equations fitted to observed field data. They are often the simplest mathematial form and inlude no material related properties or site parameters and are ommonly used in speifi appliations suh as performane preditions in pavement management systems. The intention in these ases is to estimate future performane based solely on the reorded distress history as a mere extrapolation. Mehanisti-empirial models (M-E) are based on a ombination of simple mehanisti response preditions oupled with empirial distress models, also alled transfer funtions, for prediting field performane (e.g., rutting, raking). During appliation, the mehanisti portion of the model is obtained through a strutural analysis of the pavement subjeted to traffi loading and environmental inputs. Linear elastiity is often used for its relatively simple formulation and fast omputer alulations. K - 10

13 Fully mehanisti models rely on omplex onstitutive models to diretly predit permanent deformations, raking and other distresses. With the vast apabilities of the mehanisti approah, the influene of different loading (e.g., duration, magnitude) and environmental (e.g., temperature) onditions an be evaluated and inorporated diretly into the onstitutive model. Sine mehanisti models are apable of diretly prediting pavement distresses, there is no need for empirial transfer funtions. However, mehanisti models are omplex, expensive to alibrate, and omputationally expensive to exeute. Appliations of advaned onstitutive models to analyze the behavior of asphalt onrete in pavement strutures have been limited. Applied mehanisti modeling is at the leading edge of pavement researh Problem Statement There are signifiant limitations to the use of mehanisti-empirial models to predit permanent deformation in flexible pavements. The most fundamental limitation relies on the simple definition of an empirial relationship i.e., the use of observed data instead of theory to model a phenomenon. The pratial impliation is that purely empirial models are generally not appliable to onditions that are signifiantly different from those used for the model alibration. The inorporation of mehanisti responses redues the limitations by making the model dependent on pavement stress and/or strain response that an be alulated for eah desired ondition. Unfortunately, this improvement has its own limitations and shortfalls. The most basi limitation involves the hoie of stress onditions for alibrating the material response and how these ompare to atual stress states in the field. An example of this is the development of rutting predition models based on unonfined repeated load testing performed in the laboratory. In the field the material is onstantly subjeted to varying onfining stresses that drastially hange its resistane to permanent deformations. There are several issues not yet fully resolved for prediting rutting in the asphalt onrete layers in flexible and omposite pavement systems. There is still no lear onsensus whether rutting is due primarily to axial permanent strains beneath the tire enterline or to shear permanent strains beneath the tire edge. urrent M-E rutting models relate permanent strains (axial or shear) to resilient strains omputed using multilayer elasti theory. However, in the absene of orretion fators, rutting predited from these resilient strains is in sharp disagreement with field observations. Thikness or depth orretion fators are required to bring preditions in line with observations, whih signifiantly weaken the mehanisti linkage between omputed strains and predited rutting. urrent M-E models often assume that the mehanisms and distributions of permanent strains are similar for asphalt onrete layers in any pavement type (e.g., flexible pavements versus overlays on rigid pavements), whih is generally not true. And finally, urrent M-E models are not apable of prediting the entire rutting profile, whih inludes the settlement at the enter of the wheel path and side heaves at the edge of the wheel paths. K - 11

14 Moreover, multidimensional onfinement and plasti flow interations, whih intuitively should strongly influene the permanent deformation response, are treated only in a very approximate way via the empirial thikness/depth orretions in M-E models. There are simpler alternatives based on nonlinear onstitutive models, suh as elasto-plastiity, that are apable of qualitatively orreting these disrepanies, but are not apable of fully addressing the behavior of asphalt onrete mixtures. Advaned mehanisti modeling employs theories of mehanis that are more suitable for desribing the real material behavior. A major shortoming is the omplexity of these theories, and in partiular the onstitutive models. The onstitutive model is the mathematial formulation representing the relationship between stresses, strains, and temperatures (and perhaps other state variables) that governs the material deformation under indued load and environmental onditions. Asphalt onrete is a omplex material in whih reoverable and irreoverable strains are dependent on temperature, stress and strain rates. Therefore, visoelastiity and visoplastiity theories are most appropriate to model the reoverable and irreoverable behavior respetively. The use of advaned mehanisti modeling implemented in a more rigorously nonlinear finite element model an help address some of the issues mentioned and redue the gap between rutting preditions and field measured performane Objetive The objetive of this researh is to implement an advaned onstitutive model in a finite element analysis framework to predit permanent deformation in asphalt onrete pavements. The model, developed in previous researh, is based on Shapery s visoelastiity and Perzyna s visoplastiity theories. To ahieve this objetive, the model formulation is reviewed. Laboratory testing of a modified asphalt onrete mixture used in the Federal Highway Administration s fullsale Aelerated Loading Faility (ALF) is used to alibrate and validate the model. A partiular fous of the alibration is the visoplasti model omponent, whih is the most relevant for the rutting problem. Two alibration proedures are tried, one using a researh-grade reep and reovery test designed speifially for this researh, and one using the Flow Number test intended for routine mix design evaluation. The alibrated model is implemented as a user defined material onstitutive model in the ommerial finite element pakage ABAQUS. The finite element model is used to simulate a moving wheel and to analyze the permanent deformation behavior in the asphalt onrete layer in three dimensional pavement strutures. The results are ompared with the widely used bouning wheel loading, whih is the most typial simplifiation of traffi loading in pavement analyzes. In addition, different pavement strutures are evaluated. The results are used to evaluate the depth orretion funtions used in the AASHTO s Mehanisti- K - 12

15 Empirial Pavement Design Guide (MEPDG). Finally, the model is used to predit rutting in one of the FHWA s ALF tests. This predited rutting is ompared against field measurements Doument Layout The researh is organized as follows. Asphalt onrete permanent deformation modeling tehniques are desribed in hapter 2, inluding a brief overview of past empirial models and desription of more reent mehanisti-empirial models. Advantages and pitfalls observed in urrent models are disussed in greater detail and advaned modeling is introdued as an alternative to overome some of these pitfalls. hapter 2 onludes with a theoretial desription of the visoelasti-visoplasti (VEVP) model used in this researh. hapter 3 desribes the material and the laboratory equipment used to alibrate the VEVP model. The alibration proess is provided in hapter 4, in whih laboratory tests, alibration tehniques and results are disussed. hapter 5 presents the implementation of the VEVP onstitutive model as a user defined material model in ABAQUS finite element software. Numerial appliations of the finite element model are provided in hapter 6. A simplified finite element model is introdued. The importane of onsidering traffi loading as a moving wheel instead of bouning wheel typially used to represent traffi loading is disussed. Two pratial appliations of the finite element model are also presented. hapter 6 onludes with the desription and simulation of an aelerated field pavement testing using the finite element model. The overall summary, onlusions and reommendations from this researh are presented in hapter 7. K - 13

16 hapter 2 Modeling Permanent Deformation of Asphalt onrete 2.1. Introdution Rutting is a major distress in flexible pavements. hanges in traffi onditions, mainly in tire and axle onfigurations, ontributed signifiantly to make rutting a predominant mode of failure of flexible pavements in the 1980s and 90s. Rutting is defined as a longitudinal surfae depression ourring in the wheel paths of roadways (FHWA, 2003). It is often followed in later stages by an upheaval along the sides of the rut. It an lead to strutural failure and potential danger from hydroplaning. Rutting aumulates ontinuously and inrementally with small permanent deformations from eah load appliation. Ruttingg is by definition a load-related pavement distress. There are many tehniques for measuring permanent deformations at the surfae of the pavement layer. One of the simplest approahes is to use a straightedge, as shown in Figure 1. The total measured rutting is a ombination of the settlementt in the enter and the heavee at the edges of the wheel path. Although traffi wander tends to reduee the heave at the edges of the wheel path, it an bee a prominent feature in heavily rutted pavements, espeially in ombination withh hannelized traffi. Figure 1. omponents of measured surfae rutting. The total rutting is the ombination of aumulated permanent deformation in all layers in the pavement struture. Some researh suggests thatt the shape of the surfae profile an indiate whihh layer is responsible forr the failure of the pavement struture due to rutting (Simpson et al., 1995; White et al., 2002). Figure 2 shows three transverse profiles typial of three different t senarios in whih majority of rutting omes from (a) asphalt onrete surfae layer, (b) granular base and () subbase/subgrade. If the majority of rutting originates in the underlying unbound base and subbase layers, little or no heave is observed.. When the asphalt onrete layer is responsible for total rutting, heave is observed. Inn extreme situations of very stiff underlying layers e.g., omposite pavement with a Portland ement (P) slab ating as a base layer the heave may dominate the settlement portion. The majority of the failures from rutting are due to exessive deformation in the asphalt onrete layer (White et al., 2002). K - 14

17 (a) (b) () Figure 2. Effets of rutting onentration inn different layers on permanent deformation surfae profile of flexible pavements: (a) asphalt onrete, (b) granularr base, and () subbase/subgrade (White et al., 2002) Forensi trenhes are the preferred approah for determining the permanent deformation in eah layer. However, very few field pavement test sites have been trenhed. Test traks suh as Westrak and MnRoad are the best soures for trenh data. Researhers found that nearly all of the total rutting ourred within the top 150 mm of HMA at Westrak (Epps et al. 2002). Additional detailed data from the MnRoad test setions, where permanent deformations were measured for every 40 to 50 mm of the asphalt layer, further suggests that most of the rutting ours within the top 100 to 150 mm of the surfae layer (Mulvaney and Worel, 2002). Two mehanisms are the main ausess of rutting development. ompation is the primary mehanism at initial stages of loading. ompation (i.e., densifiation) ours as the material volume dereases underneath the wheel path with no signifiant upheaval along the sides of the wheel path. After this initial stage is omplete, further volume derease of material beneath the wheel path at eah load appliation approximately equals the volume inrease in the upheaval along the sides. This deformation mode is essentially aused by shear (i.e., distortion without volume alteration). When enough distortion has ourred, the asphaltt onrete undergoes shear flow and deformations inreasee rapidly at an inreasing rate termed tertiary flow. Figure 3 depits rutting development versuss load appliation in whih region 1 is mainly aused by material densifiation, regionn 2 is predominately shear deformations, and region 3 is tertiary flow to shear failure. The primary stage, represented by region 1 in Figure 3, happens early on in the pavement s servie life, usually within the first year. The pavement will probably be rehabilitated prior to reahing the tertiary stage (region 3) due to rutting already K - 15

18 reahing the ageny s threshold or another distress triggering the need for maintenane. Therefore, rutting modeling is usually restrited to the seondary stage (region 2). Permanent deformation Load appliations Figure 3. Rutting development versus load appliations. Three types of model are used to ompute permanent deformations: empirial, mehanisti-empirial and fully mehanisti. Examples of empirial models are regression equations fitted to observed field data. They are often the simplest mathematial form for representing relationships in the data. They often inlude no material related properties or site parameters and are ommonly used in speifi appliations suh as performane preditions in pavement management systems. The intention in these ases is to estimate future performane based solely on the reorded distress history as a mere extrapolation. Mehanisti-empirial models (ME) are developed based on a ombination of simple mehanisti response preditions (i.e., often using elastiity theory) with empirial equations alibrated using laboratory testing in whih stress onditions representing field onditions are repliated. The alulated mehanisti response is used as input in the empirial model, also alled transfer funtion, to predit field performane (e.g., rutting, raking). The influene of temperature and loading onditions (e.g., frequeny) an be inorporated. During appliation, the mehanisti portion of the model is obtained through a strutural analysis of the pavement subjeted to traffi loading and environmental inputs. Linear elastiity is often used for its relatively simple formulation and fast omputer alulations. Fully mehanisti models also use a strutural analysis program to evaluate the stresses and strains in the pavement struture due to traffi loading and environmental onditions. omplex onstitutive models are used to represent the different aspets of material behavior and to diretly predit permanent deformations, raking and other K - 16

19 distresses. With the vast apabilities of the mehanisti approah, the influene of different load onditions (e.g., duration, magnitude and temperature) an be easily evaluated and inorporated diretly into the onstitutive model. Sine mehanisti models are apable of diretly prediting pavement distresses, there is no need for empirial transfer funtions. However, mehanisti models are omplex, expensive to alibrate, and omputationally expensive to exeute. Few researhes have been done to implement mehanisti models to predit asphalt onrete behavior. Applied mehanisti modeling is still ahead of the frontline of researh Mehanisti-Empirial Modeling Early attempts to model rutting date to the 1950 s, when Kerkhoven and Dormon (1953) first suggested the use of vertial ompressive strain on the top of the subgrade as a failure riterion to minimize permanent deformation. Dormon and Metalf (1965) inorporated strain-based riteria in a mehanisti-empirial proedure using 1940 s Burmister multi-layer linear elasti solutions. Later, this riterion was used as part of the Asphalt Institute design method (Shook et al., 1982). The lassi pavement design approah onsidered Kerkhoven and Dormon s failure riterion for rutting and a similar one for fatigue raking (i.e., the tensile strain at the bottom of the surfae asphalt onrete layer). The rutting shear resistane of the asphalt onrete was onsidered only in the mix design, through the Marshal method. Following test trak experiments from the 1950 s and 1960 s, inluding the AASHO Road Test, researhers developed rutting models based on regression analysis of observed field data. Finn et al. (1977) developed a rutting regression model relating surfae defletion to the vertial ompressive stress in the asphalt onrete layer. laussen et al. (1977) developed an empirial rutting model based on subgrade vertial strain, whih was inorporated into the Shell pavement design proedure. At this same time there were initial efforts to develop models based on laboratory tests that ould provide a better representation of the loading ondition in the field. One of the first models developed inorporated results from unonfined repeated load permanent deformation (RLPD) tests and was implemented in the VESYS program (Kenis et al., 1977). This followed work done previously with granular base materials and soils done by Barksdale (1972) and Monismith (1975). Moreover, this model formulation was the first to desribe asphalt onrete rutting as power law: p N N (1) in whih p (N) is the inremental permanent vertial strain aused by the N th load yle, is the mehanistially-omputed peak total vertial strain (usually taken as the resilient strain, e ), and and are material parameters determined from laboratory RLPD tests. The VESYS model is onsidered the predeessor of the K - 17

20 urrent power law models. This type of model is partiularly useful beause it fits quite well the seondary stage of rutting behavior (i.e. linear in log-log spae). Repeated load permanent deformation testing beame a typial laboratory proedure for haraterizing rutting behavior of asphalt mixtures. Allen and Deen (1980) developed a regression model based on unonfined RLPD with different deviator stresses and temperatures. The third order polynomial model resulting from the regression analysis had the following general form: p 0 1 N 2 N 3 N 2 3 log log log log (2) Rauhut (1980) presented some quantitative influene of mixture type and deviator stress on and in Eq. (1) based on limited unonfined RLPD data analysis. Leahy (1989) inreased the number of experiments and enhaned these orrelations inluding temperature and some material harateristis (i.e., effetive binder ontent, air voids and binder visosity). Over 250 asphalt onrete speimens enompassing two aggregate types, two binder types, three binder ontents, three stress levels, and three temperatures were tested. The model formulation is as follows: p log logN 2.767logT 0.110log d r 0.118log 0.930logV beff 0.501logV a (3) in whih ε p is the permanent strain, ε r is the resilient strain, N is number of load yles, T is temperature ( F), σ d is deviatori stress (psi), η is binder visosity at 70 o F (x10 6 poise), V beff is effetive binder volume and V a is air voids. The R-squared value for Eq. (3) was Ayres (1997) re-analyzed Leahy s original data plus additional laboratory test data and reommended a model of the form: p log log N logt r (4) in whih the parameters are as defined previously for Eq. (3). The slightly lower R 2 of 0.72 for this model is the onsequene of removing four mix-related parameters from the Leahy model, Eq. (3). This small drop onfirmed the relatively small importane of these parameters as ompared to temperature and number of load repetitions. Kaloush (2001) further improved the rutting model in Eq. (4) by ombining Leahy s original data with the very large number of repeated load permanent deformation test results from NHRP Projet 9-19, yielding a revised model of the form: K - 18

21 p log log N logt r (5) in whih the parameters are as defined previously. The lower R 2 value of 0.64 ompared to Ayres was attributed to the muh broader and more diverse data set analyzed by Kaloush (El-Basyouny, 2004). Kaloush s regression model was the starting point for the rutting model implemented in the newly developed Mehanisti Empirial Pavement Design Guide (MEPDG) (NHRP, 2004). Field alibration of Eq. (5) was performed as part of the MEPDG development in NHRP Projet 1-37A. The Long Term Pavement Performane (LTPP) database was the main soure of data for the alibration (El-Basyouny, 2004). Reent work done under NHRP Projet 1-40D (NHRP 1-40D, 2006) revised the field alibration, produing the rutting model implemented in the most reent version of the MEPDG software (version 1.1): p r 10 T N (6) in whih all variables are defined as previously. The database underlying Eq. (3) through Eq. (5) is based only on unonfined onditions. In reality, substantial horizontal stresses develop in an asphalt layer during wheel loading/unloading, ranging from ompression at the top to tension at the bottom (assuming a linear elasti asphalt onrete response) and varying through the HMA thikness. A depth orretion fator was developed in the MEPDG to adjust the omputed plasti strain due to onfining pressure at different depths: 3 ( 1 2 depth ) depth h A h (7) 1 A h A h in whih depth is depth to the omputational point of strain alulation, and h A is the thikness of the asphalt layer. The final rutting model, inluding the depth fator is: A p r 10 3 T N (8) Another variation of the elasti vertial strain power law model was proposed by Verstraeten (1977, 1982). In this model, the vertial elasti strain is replaed by the ratio between deviatori stress and the dynami modulus of the mixture: K - 19

22 p N A * E 1000 f 1 3 b A (9) in whih, ε p is the permanent shear strain, σ 1 and σ 3 are respetively vertial and radial stresses, E* is the dynami modulus of HMA mixture, N is the number of load yles, f is the frequeny of loads, A and b A are material parameters. Temperature is also impliitly inluded in Eq. (9) through its influene on dynami modulus. For onventional mixtures, the reommended values for the regression oeffiients are A=57.5 and b A =0.25 (D Apuzzo et al., 2004). It is unlear from the referenes how A and b A are determined. However it seems plausible to assume that RLPD tests would suffie in determine these alibration oeffiients. The inlusion of load frequeny as a normalizing fator for the number of load appliation makes this model different from the VESYS power law model. An alternative type of empirial model relates permanent deformation to maximum shear strains observed in the asphalt onrete surfae layer. Researhers at the Westrak developed this type of model, whih exludes densifiation and assumes shear deformations as the solely rutting mehanism (Monismith et al., 2006). Shear plasti strains are omputed using a power law model, whih onsiders the number of load appliations and the elasti shear strains as follows: p b ae en (10) in whih γ p is the permanent shear strain, is mehanistially-determined elasti shear stress, γ e is the mehanistially-determined elasti shear strain, and a, b, and are material parameters. The maximum elasti shear stress and strain are expeted to be loated at the edge of the tire at a depth of 2 inhes below the surfae. This loation was defined based on elasti analysis of two-layer strutures with different load onditions and asphalt onrete stiffness (Sousa et al., 1994). The mehanistially omputed elasti shear strain, γ e,t, varies over time in response to traffi variations and influene of temperature on the asphalt onrete stiffness. A time-hardening priniple similar to that implemented in the MEPDG is used to estimate the aumulation of permanent strains in the asphalt onrete under varying traffi loading onditions: p a N, pt, 1 pt, a t Nt at (11) K - 20

23 b a t ae e, t in whih γ e,t is the elasti shear strain for the t th period of loading, γ p,t is the orresponding permanent shear strain, and ΔN t is the number of load appliations during the t th period. The total rut depth in the asphalt onrete layer is estimated from the permanent shear strain using the following semi-empirial relation: RD K (12) HMA r p, t in whih K r is a oeffiient relating rut depth to permanent strain. K r is a funtion of the asphalt layer thikness; values for K r are determined from finite element analyses of representative pavement strutures range from about 5.5 for a 6 inh layer to 10 for a 12 inh layer (Deaon et al., 2002). The regression oeffiients a, b, and in Eq. (10) and Eq. (11) are determined from repeated load simple shear tests onduted at onstant height (RSST-H). Monismith et al. (2006) desribe the alibration proess in greater detail. The model sheme was developed, alibrated, and validated for the Westrak setions. Mixtures ommonly used by altrans have also been alibrated using RSST-H test data and validated using Heavy Vehile Simulation (HVS) full sale aelerated pavement testing system Mehanisti-Empirial Modeling Limitations There are signifiant limitations to the use of mehanisti-empirial models to predit permanent deformation in flexible pavements. The most fundamental limitation relies on the simple definition of an empirial relationship i.e., the use of observed experimentation instead of theory to model a phenomenon. The pratial impliation is that purely empirial models are generally not appliable to onditions that are signifiantly different from used for the model alibration. The inorporation of mehanisti responses redues the limitations by making the model dependent on pavement stress and or strain response that an be alulated for eah desired ondition. Unfortunately, this improvement also has limitations and shortfalls. The most basi limitation involves the hoie of stress onditions from whih the model has been alibrated and how they ompare to atual stress states in the field, whih brings bak the problem of onditions during alibration versus design/servie. An example of this is the development of rutting predition models based on unonfined repeated load testing performed in the laboratory. In the field the material is onstantly subjeted to varying onfining stresses that drastially hange the resistane to permanent deformations. K - 21

24 The methods for determining the required mehanisti responses are another important soure of limitation in mehanisti-empirial models. Linear elasti multilayer theory is most often used to ompute the mehanisti responses beause it is easy to implement in omputer algorithms and omparatively fast to exeute. However, this simplifiation brings many shortfalls. The mehanisti strains vary with depth and horizontal loation within the HMA layers. The speifi variation will be a funtion of the pavement struture, material properties, load onfiguration, and other fators. M-E rutting models must rationally aount for these strain variations within the HMA and other pavement layers when aumulating the predited permanent deformation at the surfae. Empirial alibration fators attempt to bring the predited surfae deformations into better agreement with measured field performane over a wide range of pavement onditions. There are several issues not yet fully resolved in this framework for prediting rutting in the asphalt onrete layers in flexible and omposite pavement systems. First, there is still no lear onsensus whether rutting is due primarily to axial permanent strains beneath the tire enterline (e.g., NHRP, 2004; D Apuzzo et al., 2004) or to shear permanent strains beneath the tire edge (e.g., Deaon et al., 2002; Monismith et al., 2006). Seond, urrent M-E rutting models relate permanent strains (axial or shear) to resilient strains omputed using multilayer elasti theory. However, in the absene of orretion fators, the influene of layer thikness on the rutting predited from these resilient strains is in sharp disagreement with field observations. Thikness or depth orretion fators are required to bring preditions in line with observations. These orretion fators further weaken the mehanisti linkage between predited rutting and omputed strains in the M-E approah. Third, urrent M-E models often assume that the mehanisms and distributions of permanent strains are similar for HMA layers in flexible pavements vs. HMA overlays on rigid pavements, whih is generally not true. And fourth, urrent M-E models do not expliitly onsider the ontribution of heaving at the edge of the wheel paths (see Figure 1), although this may be impliitly inluded in the field alibration orretions, given how rutting measurements are taken in the field. Both the MEPDG depth funtion β σ3 in Eq. (6) and the Westrak K r fator in Eq. (12) an be viewed as attempts to ompensate for defiienies in using linearly elasti stress and strain distributions to estimate permanent deformations. Figure 4 ompares the variations of the unorreted versus orreted permanent strains from the MEPDG model. (Note: The standard mehanis sign onvention of positive tension applies to this figure.) As ditated by Eq. (6), the unorreted omputed permanent strains are proportional to the mehanistially determined vertial resilient strains, whih are largest at the bottom of the HMA layer due to the ombination of the diret vertial ompression and the ompressive Poisson strains indued by the horizontal tensions. As a onsequene, the permanent deformations are onentrated in the lower depths of the HMA layer ontrary to field experiene. The empirial depth orretion funtion distorts both the magnitude and shape of the permanent strain distribution as K - 22

25 it attempts to fore the majority of the permanent deformations into the upper portions of the HMA layer. The depth orretion funtion dominates the permanent strain values used to ompute the total rutting for the layer and, in the proess, seriously undermines the mehanisti portion of the modeling. Vertial Strain at Peak Load Depth (mm) Elasti Permanent (Before Depth orretion) Permanent (After Depth orretion) Figure 4. Influene of depth funtion β σ3 on alulated permanent vertial strains from the MEPDG model (150 mm HMA layer over rushed stone base, first load yle). The lak on of influene of pavement struture on preditions of mehanistiempirial models an be demonstrated by using a simple finite element exerise (Shwartz and arvalho, 2007 and 2008). One typial flexible pavement was modeled using an elasto-plasti finite element model (EPFE). The HMA layer was modeled as an elasto-plasti material using the Druker-Prager fritional plastiity model with a linear yield surfae and an isotropi pieewise linear hardening law. The other materials were modeled as linear elasti. A fully elasti finite element analysis was also performed for omparison. The impat of plasti yielding on the vertial and horizontal total strains in the HMA layer is shown in Figure 5 for the onventional flexible pavement ase. The vertial ompressive strains beneath the tire enter (Figure 5a) from the elasti analysis monotonially inrease with depth. These are the resilient strains r that are the input for Eq. (6). onsequently, the permanent strains from Eq. (6) will also inrease monotonially with depth. The aumulated rutting will therefore be onentrated in the lower portion of the HMA layer, whih as desribed previously is ontrary to field experiene. The orresponding vertial ompressive strains from the plasti analysis, on the other hand, deviate from the elasti strains in the orret diretion, at least qualitatively, with the peak strain ourring near the enter of the layer at a depth of about 60 mm. However, rutting in this ase is still onentrated at the bottom of the HMA layer. For both analyses, the horizontal strains (Figure 5b) inrease K - 23

26 monotonially from ompression to tension with depth, with the strains from the plasti analysis larger than those from the elasti ase, as expeted. Depth (mm) Total Vertial Strain Depth (mm) Total Horizontal Strain Elasti Analysis Plasti Analysis Elasti Analysis Plasti Analysis (a) (b) Figure 5. omputed total strains beneath tire enter at peak load: (a) vertial; (b) horizontal (Shwartz and arvalho, 2008). Elasti stresses and strains at peak load are the inputs to the rutting models in the M-E predition methodology. However, rutting in physial terms is the permanent deformation remaining after removal of the load. An examination of the residual strains after unloading is therefore instrutive. These residual strains for the EPFE analysis are depited in Figure 6 (the residual stresses and strains for the elasti analysis are zero by definition). The vertial permanent ompressive strains after unloading inrease with depth until about 100 mm, after whih they derease sharply and eventually beome tensile. The yielding at the bottom of the HMA layer results in residual horizontal ompressive stresses after unloading. This residual horizontal ompression indues expansive vertial strains i.e., a derease in the residual vertial ompression--due to the Poisson effet. The residual permanent deformations resulting from this strain distribution will be onentrated in the upper portion of the HMA layer, in better agreement with field experiene. No additional depth orretion funtion is required to bring analysis results into qualitative alignment with physial expetations. K - 24

27 Total Strain Vertial Horizontal Depth (mm) Figure 6. Residual total strains from the EPFE analysis after removal of load. Another noteworthy feature of the EPFE analysis results is the aumulation of inremental permanent deformations over multiple load yles. onventional wisdom often purports that a straightforward strain-hardening plastiity analysis of a onstant amplitude yli tire loading should produe all permanent deformations in just the first load yle; sine all subsequent load yles are to the same load magnitude, no additional plasti yielding and/or deformations should develop in the subsequent yles. As shown in Figure 7, however, this is not the ase. Additional plasti deformations develop in eah load yle, with diminishing magnitude in eah suessive yle. The loation of maximum residual strain also moves upward in the layer, bringing the distribution in even better agreement with field observations. The loked-in residual horizontal stresses that are the onsequene of plasti yielding at the bottom of the HMA layer introdue stress reversals that produe additional plasti yielding and permanent deformation with eah suessive load yle. K - 25

28 Residual Vertial Strain -2.0E-4-1.0E-4 0.0E+0 1.0E Depth (mm) Figure 7. Residual total vertial strains beneath the tire enter from EPFE analyses for different number of yles. learly, the empirial permanent deformation laws desribed earlier have only a very distant relationship to a realisti nonlinear onstitutive response of HMA. Multidimensional onfinement and plasti flow interations, whih intuitively should strongly influene the permanent deformation response, are treated only in a very approximate way via the empirial thikness/depth orretions. Simple nonlinear onstitutive models based on elasto-plastiity are apable of qualitatively orreting these disrepanies. Therefore, these issues and others an be more rigorously addressed via nonlinear finite element analysis inorporating more realisti onstitutive models for the HMA Asphalt onrete Mehanisti Modeling Advaned mehanisti modeling employs theories of mehanis that are more suitable to desribe the real material behavior. The shortoming is the omplexity of these theories, and in partiular the onstitutive models. The onstitutive model is the mathematial formulation representing the relationship between stresses, strains, and temperatures (and perhaps other state variables), and governs the material deformation under indued load and variations of temperature. Asphalt onrete is a omplex material in whih reoverable and irreoverable strains are dependent on temperature, stress and strain rates. Therefore, visoelastiity and visoplastiity theories are most appropriate to model the reoverable and irreoverable behavior respetively. The onstitutive model used in this researh is based on Shapery s nonlinear visoelasti ontinuum damage onstitutive theory (Shapery, 1984). This theory has been used extensively in previous researh to desribe the reoverable portion of the K - 26

29 deformation. Kim and Little (1990) used a one-dimensional formulation to desribe the experimental behavior of asphalt onrete under yli strain loading. Park and Shapery (1997) also used a visoelasti ontinuum damage uniaxial formulation to model reep in solid fuel propellants, while Ha and Shapery (1998) expanded this formulation into a omprehensive multiaxial model. Reently, several other researhers have used Shapery s nonlinear ontinuum damage visoelasti model for prediting the reoverable response of asphalt materials (Lee and Kim, 1998; Daniel and Kim, 2002; Gibson, 2006; Huang et al., 2007; Masad et al., 2008; Huang et al., 2011a, 2011b). There are many approahes for modeling the irreoverable response of asphalt onrete. Often very simple nonlinear modeling an be used with good results, suh as the elasto-plasti analyses desribed in the previous setion. Models based on ratedependent plastiity and reep are found in the literature as well (Perl et al., 1983; Fang et al., 2004). However Perzyna s theory of visoplastiity is the most ommon approah used to model asphalt onrete mixtures. The many instanes of it in the literature differ primarily in the hoie of the yield funtion, type of flow, and anisotropi effets (Lu and Wright, 1998; Gibson, 2006; Masad et al., 2007; Huang, 2008; Huang et al., 2011a, 2011b). In this researh, the Perzyna based visoplasti with the Hierarhial Single Surfae (HiSS) yield funtion model and assoiated plasti flow is used (Gibson, 2006). The key oneptual omponents that govern the omplex behavior of asphalt onrete are: (1) visoelastiity, (2) mirostrutural damage and (3) strain-hardening visoplastiity. The visoelastiity governs the behavior before plasti yielding ours. Mirostrutural damage aounts for hanges in the struture due to the formation of miroraks and is expressed in terms of rate-dependent internal state variables. And finally, the strain-hardening visoplasti model is responsible for determining the plasti deformations (post-yield behavior) and the rate of deformation as the material hardens. Therefore total strains may be separated into visoelasti, visoplasti and damage strains as follows: t ve d vp (13) in whih t is the total strain, ve is the visoelasti strain, d is the damage strain, and vp is the visoplasti strain. The visoelasti strain is assumed to be linear and independent of stress state and damage. The visoelasti strain is solely dependent on rate of loading and temperature, whih for thermorheologially simple materials an be interhanged using time-temperature superposition. Asphalt onrete is ommonly assumed to be thermorheologially simple under small strain (<100με) linear visoelasti onditions. Although these fators are expeted to have diret effet on the magnitudes of the time-dependent omputed internal state variables and on the magnitudes of the K - 27

30 damage strain, it is assumed that the effets of loading rate and temperature an be interhanged using onventional time-temperature superposition as well. It is also assumed that the effets of loading time and temperature on visoplasti strains an be interhanged using a generalized time-temperature superposition. Previous researh has shown that the temperature shift funtion developed for small strain onditions is also valid, at least for engineering purposes, at larger strain levels of interest in pavement analyses (hehab et al., 2002; Gibson, 2006). This immensely simplifies the laboratory testing program for alibrating the model Linear Visoelasti Behavior Linear visoelasti (LVE) materials exhibit elasti and visous linear behavior. The elasti omponent is responsible for the instant response to loading, while the visous omponent is responsible for time- and rate-dependent effets. The linear harateristi means that stresses and strains responses an be superimposed. omplex loading onditions an be broken down into simpler loading onfigurations and superimposed to ahieve the same outome. The onstitutive relationships for LVE materials are ommonly expressed by the following onvolution integrals: t d E t d (14) d t 0 0 d Dt d (15) d in whih E(t) and D(t) are the relaxation modulus and reep ompliane respetively, σ is stress, and is strain. The relaxation modulus is usually obtained through a relaxation test, whih onsists of applying a presribed fixed strain and observing the material relax from the initial indued stress. The relaxation modulus is simply defined as: E t t (16) in whih σ(t) is the indued stress at a given time and 0 is the presribed onstant strain applied to the speimen. onversely, reep ompliane is usually obtained from a reep test in whih the speimen is subjeted to a presribed onstant stress and the strain inrease over time is observed. The reep ompliane is then defined as: 0 K - 28

31 Dt t (17) in whih (t) is the indued strain at a given time and σ 0 is the presribed onstant stress applied to the speimen. Sine relaxation modulus and reep ompliane are just alternate representations of the same underlying visoelasti behavior, they are related as follows: Dt 1 0 Et (18) Visoelasti alulations an be simplified by applying two important fundamental priniples. Time-temperature superposition, as mentioned before, permits the effets of time and temperature to be interhanged through a shift fator. The other important simplifiation is the use of the orrespondene priniple, whih states that the time dependene inherent in visoelasti problems an be removed when physial strains are replaed by pseudo strains defined as follows: d R E t d E (19) d 1 t R 0 (20) E R R in whih R is the pseudo strain, E R is a referene modulus, typially taken as one, and E(t) is the relaxation modulus defined earlier. In addition, reoverable nonlinear behavior an be inorporated through a mirostrutural damage omponent, whih an be onveniently omputed using the orrespondene priniple, as further detailed in the next setion. These simplifiations greatly expedite the laboratory testing and alulations required to alibrate the model. The generalized Maxwell model an be used to represent the relaxation modulus. The generalized Maxwell model onsists of a series of dashpots and springs onneted in parallel, as shown in Figure 8. The material onstants E i and ρ i orrespond respetively to the stiffness of eah Maxwell spring and the relaxation times of eah dashpot. The generalized Maxwell model provides a good fit to the observed behavior of a wide range of visoelasti materials. A Prony series an be onveniently used to represent the relaxation modulus in the generalized Maxwell model (Park and Shapery, 1999): m t i i (21) i1 0 E t E Ee in whih E 0 is the long term equilibrium modulus, E i and ρ i are the elasti springs and relaxation times for the elements in the generalized Maxwell model, and m represents K - 29

32 the number of Maxwell omponents in the generalized model. A great advantage of the Prony series representation of the relaxation modulus is its fairly simple implementation in algorithms. In ombination with reursivee algorithms or evaluating the onvolution integral in Eq. (19), this mathematial modell of the visoelasti behavior is very attrative from the omputation standpoint. In a similar fashion, reep ompliane an also bee onveniently modeled using a Prony series as follows: D t D0 m i De i i1 t (22) in whih the unknown ompliane onstants (D 0, τ j, D j ) an be expressed in terms of the known relaxation onstants of Eq. (21) E 0, ρ i, E i -- by using a tehnique developed by Park and Shapery (1999). Figure 8. Generalized Maxwell model. The prinipal visoelasti properties required for the Shapery model are the Prony series terms for the relaxation modulus and the temperature shift funtionn for the time-temperature superposition. With these two properties, one an model any type of loading (stress or displaement indued) at any rate and any temperature. The omplex modulus test an be used to determine both properties from one single laboratory test. K - 30

33 omplex Modulus The omplex modulus is defined as the ratio of dynami stress to dynami strain under sinusoidal loading. The presribed stress and indued strain an be represented by the following equations: os os t t e 0 0 it t t e 0 0 t i (23) in whih σ 0 is the dynami stress amplitude, ε 0 is the dynami strain amplitude, ω is the loading frequeny, and φ is the phase angle or strain lag, as shown in Figure 9. Based on Eq. (23), the omplex modulus is defined as: (24) * 0 i 0 E e os isin E ie 0 0 in whih the storage modulus, E', and loss modulus, E'', represent the real and imaginary omponents. Stress/Strain σ(t) e(t) σ0 e0 φ Time Figure 9. omplex modulus response. The dynami modulus, E*, defined as the ratio of the dynami stress amplitude to the dynami strain amplitude, is related to the storage and loss moduli as follows: (25) * 0 2 "2 E E E 0 K - 31

34 E E * E E * os sin (26) Dynami modulus is the most ommon way to haraterize the visoelasti behavior of asphalt onrete mixtures in pratie. It is also the main HMA material property in the Mehanisti-Empirial Pavement Design Guide (MEPDG). The generalized Maxwell model an be fit to the storage and loss moduli in the frequeny domain through a Prony series: E E E m 2 2 i i i1 1 i m i1 E 1 E ï i 2 2 i (27) in whih E i are the elasti spring onstants and ρ i are the relaxation times for the m elements in the generalized Maxwell model. These terms are the same as defined for the relaxation modulus Prony series in Eq. (21). Time-Temperature Superposition The effets of temperature and rate of loading on the visoelasti properties of asphalt onrete an be interhanged using the time-temperature superposition priniple. A master stiffness urve is formed by shifting the dynami modulus data measured at different temperatures to a unified referene temperature. The amount of shifting at eah temperature required to align the data along the ommon master urve desribes the temperature dependeny of the material. The master urve is desribed in terms of redued frequeny, whih in turn is omputed based on the temperature shift fators, as follows: r at (28) in whih ω r is the redued frequeny, ω is the test frequeny, a(t) is the temperature shift fator at temperature T. An example of a master urve is provided in Figure 10. The temperature shift fators used to reate the master urve are provided in Figure 11. K - 32

35 100,000 10,000 E* (MPa) 1, Reduedd frequeny y (Hz) Figure 10. Example of master urve Log a(t) Temperature ( ) Figure 11. Example of temperature shift funtion. The master urve in Figure 10 an be desribed using a sigmoidal logisti funtion as follows: log E * 1 e log r e (29) K - 33

36 in whih E * is the dynami modulus, ω r is the redued frequeny, δ is the minimum value of E * (often alled the lower shelf), δ + α is maximum value of E * (often alled the upper shelf), and β and γ are fitting parameters desribing the horizontal loation and slope of the transition region. Researh has shown that δ and α parameters depend primarily on binder ontent, air void ontent, and aggregate gradation while the β and γ parameters depend on the visosity harateristis of the asphalt binder (Bonaquist, 2008) Visoelasti ontinuum Damage Behavior The ontinuum damage behavior is haraterized by marosale stiffness redution due to hanges in the material, mostly due to the development of miroraks. For asphalt onrete, Shapery s work potential theory based on thermodynami priniples has been used suessfully for quantifying damage in HMA (Gibson, 2006; Kim et al, 2009). The material is assumed to be ontinuum and homogeneous. Although asphalt onrete is not homogeneous and no longer ontinuous after miroraks develop, this assumption greatly simplifies the mathematial model and it is reasonably aeptable for pratial purposes of strutural analysis. Uniaxial Formulation Damage is inorporated into the uniaxial visoelasti model by modifying the linear elasti relationship between uniaxial stress and pseudo strains desribed earlier in Eq. (20). Reall that the linear visoelasti problem was onverted into a linear elasti one by using the orrespondene priniple desribed in Eq. (19). Replaing the referene pseudo modulus term, E R, by a damage funtion, Eq. (20) beomes: R S (30) in whih σ and R are defined as previously, (S) is a stiffness funtion that varies with material damage, and S is an internal state variable. Note that if the referene modulus is taken as the unity, (S) is equal to one when there is no damage and zero for a totally damaged material. The variable S quantifies any mirostrutural hanges that result in stiffness redution. The damage evolution law governing the hanges in the damage internal state variable, S, is defined as follows: ds dt S W R (31) in whih α is a material property. For the uniaxial ase, a pseudo work funtion, W R, is defined as: K - 34

37 W R 1 2 S R (32) 2 Assuming that the inrements of time are suffiiently small, ds an be replaed by dt S in Eq. (31) and ombined with Eq. (32) to yield a disrete solution for S as t follows: 1 S S t 2 2 i1 i R i S (33) It is assumed that there is no damage before any loading ours and thus S equals zero and equals one before loading, assuming unity value for the referene modulus. This numerial proess requires the knowledge of the shape of the funtion. An initial funtion an be assumed and through an iterative proess it an be refined until small hanges between iterations is found. A typial funtion for (S) takes the form of: S e (33) After alulating the value of damage, S i, and the inremental damage, S i +ΔS, at a given time step, the orresponding values of are found using Eq. (33). The differene between these values (δ) is then used to alulate damage at the next time step, using Eq. (33). The proess is repeated until all data points are proessed. as b Multiaxial Formulation onsider first the elasti strain energy funtion for a transversely isotropi material (Shapery, 1985): 1 W A e A e 2A e e A A e v 22 d 12 v d s (34) ev in whih ev , es 22 11, ed 33, are the engineering shear ij 3 strains, and A are the five elasti omponents that define the transversely isotropi ij stiffness matrix. Shapery suggested the axis of symmetry be oriented in the urrent maximum prinipal strain diretion beause damage is dominated by miroraks on the planes perpendiular to the maximum prinipal diretion. Therefore, in uniaxial K - 35

38 ompression, the isotropi axis of symmetry is in the axial diretion. This speial ase of multiaxial formulation an be used to determine A omponents of the stiffness ij matrix. Eq. (34) an be simplified as follows: W0 A11ev A22ed 2A12eve d 2 (35) in whih 11 22, es 0, and For a symmetry axis x 3, the following relations are derived from the strain energy funtion: W A 2 A A A A e W A A A e e 0 22 d d 3 v v (36) The stiffness matrix an be onverted into a pseudo stiffness matrix for the visoelasti problem using the orrespondene priniple. Similarly to the uniaxial formulation, the stiffness matrix an be replaed by a pseudo stiffness matrix that depends on damage. For determining the relationship between the omponents of the pseudo stiffness matrix and the damage funtions, Shapery s work potential theory is used. Shapery presented the energy density funtion as a dual energy density funtion of a monotoni uniaxial loading with onfining pressure, p, as follows: (37) Wd S S p S p 2 2 in whih ij are damage funtions, p is the onfining pressure, and is the axial strain. (Note that all the formulation derived here utilizes the onventional mehanis notation of positive tension, with the exeption of the onfining pressure p whih is defined as ompression positive.) The stress-strain relations an be derived from Eq. (37) as follows: Wd p p Wd e v p (37) in whih 33, and p in the multiaxial formulation with symmetry on axis x 3. ombining Eq. (36) and Eq. (37) yields the following damage dependent stiffness omponents, A ij : K - 36

39 A A A E0 A44 A66 G (38) The stiffness omponent A 66 was determined based on the assumption that the undamaged material is isotropi and therefore its value should be equal to the initial shear modulus, G 0. Similar to the uniaxial formulation, the damage funtions, ij, are dependent on one internal state variable, S. It also follows that S is defined by the same damage evolution law presented previously in Eq. (31) Visoplasti Behavior The distintive omponents of a visoplasti onstitutive model are: (a) a yield funtion, whih ontrols the magnitude of visoplasti flow; (b) a hardening law, whih desribes hanges in the strength of the material aompanying aumulated visoplasti straining; and () a potential funtion, whih ontrols the diretion of the visoplasti strain inrements. These omponents an be desribed oneptually in terms of the simply rheologial model for elasti-visoplasti behavior shown in Figure 12. The deformation of the spring (haraterized by the elasti modulus E) represents the instantaneous elasti response while the deformation of the dashpot (ontrolled by the visosity) represents the rate-dependent visoplasti response one the yield strength Y of the parallel slider element has been exeeded. (The linearly elasti spring ould be replaed by a visoelasti spring and dashpot ombination for the more general ase of a visoelasti-visoplasti onstitutive model.) The total strain is then just the sum of the elasti and visoplasti omponents: e vp (39) K - 37

40 σ η σ Y ε vp E ε e Figure 12. One-dimensional elasto-visoplasti rheologial model. For visoplasti strain to our, the applied stress must be larger than the yield stress, Y : f Y 0 (40) in whih f is the yield funtion ontrolling the magnitude of visoplasti flow. Typially, some type of hardening rule (or softening rule) omplements the yield funtion to reflet the tendeny of real materials to beome stronger (or weaker) as plasti strains aumulate over the load duration. For example, a simple linear hardening rule for one dimensional loading an be expressed as: in whih Yo is the initial yield and H is a hardening modulus. Y Yo H vp (41) One flow ommenes (i.e., Eq. (40) is satisfied), the visoplasti strain rate vp is governed by the visous dashpot: Eqs. (40) and (42) an be written in shorthand notation as: vp (42) f g g vp f (43) in whih 1 is a fluidity material parameter and the notation f is interpreted as: K - 38

41 f for f f f 0 0 for f 0 (44) The g term in Eq. (43) is the visoplasti potential funtion ontrolling the is set diretion of plasti flow. In the simplest formulation, the plasti potential g equal to the yield funtion f ; this is termed assoiated flow visoplastiity. For assoiated flow visoplastiity, Eq. (43) beomes: f vp f (45) Equation (45) an be integrated in time to predit the visoplasti strains generated by a given loading history. Multidimensional Perzyna Visoplastiity Equation (43) is the simple uniaxial form of Perzyna s visoplasti onstitutive theory (Perzyna, 1966). Perzyna s general theory is fully multi-dimensional and broad enough to aommodate a wide variety of yield funtions, plasti potentials, and hardening rules. Temperature effets, ritial for asphalt onrete, an be inorporated into Perzyna s theory using time-temperature superposition. Timetemperature superposition implies that loading time and temperature are interhangeable and that their ombined effets an be inorporated via a single redued time variable. Shwartz et al. (2002), hehab et al. (2002), Zhao and Kim (2003), and others have demonstrated that time-temperature superposition for asphalt onrete remains valid at strain levels approahing peak strength and beyond. The multidimensional form of Perzyna s visoplastiity theory an be expressed as: d ij vp f s g s (46) dt R ij vp in whih ij and are speifi omponents of stress and visoplasti strain rate; t ij R is redued time; f (s) is the yield funtion in the multidimensional stress spae s; and g (s) is the visoplasti potential funtion. For assoiated flow visoplastiity, g (s) = f (s). The yield and potential funtions an both be interpreted as surfaes in multidimensional stress spae. Equation (46) states that visoplasti strains develop when the applied stress state lies outside the flow surfae i.e., when f (s) > 0. The magnitude of the visoplasti strain K - 39

42 rate is proportional to the distane between the applied stress state and the flow surfae in multidimensional stress spae. As before, the derivative of the visoplasti potential funtion g s governs the ij diretion of the visoplasti strain inrements. For the ase of assoiated visoplastiity, the diretion of the inremental visoplasti strain vetor is always normal to the yield surfae. As for the uniaxial ase, the key omponents of the multidimensional Perzyna visoplasti onstitutive model in Eq. (46) are the yield funtion, the hardening law, and the visoplasti potential funtion. Most of the differenes among the various visoplasti models in the literature for asphalt onrete enter on speifi hoies for these three omponents. For example, fritional materials like asphalt onrete, aggregates, and soils are best haraterized by yield funtions that inlude the strengthening effets of onfining stress. One example of suh yield funtions is the standard Druker-Prager generalization of Mohr-oulomb fritional yield. Extensions to Druker-Prager theory like the Hierarhial Single Surfae (HiSS) model (Desai and Zang, 1987) add other refinements suh as a nonlinear yield surfae and a ap on visoplasti flow under hydrostati ompression loading. A variety of hardening types (e.g., isotropi vs. kinemati) and various speifi hardening laws have been employed in models in the literature. And although most implementations employ an assoiated plastiity assumption for the visoplasti potential funtion, this is known to overestimate dilatany for many geomaterials and as a onsequene a variety of nonassoiated flow shemes have been proposed. Reent representative examples of visoplasti onstitutive models for asphalt onrete appliations an be found in Shwartz et al. (2004), Gibson et al. (2003), hehab et al. (2003, 2005), Huang et al. (2002, 2004), Masad et al. (2005, 2007), Tashman et al. (2004, 2005), Uzan (1996, 2005), Saadeh et al. (2007), Panneerselvam and Panoskaltsis (2006), ollop et al. (2003), Oeser and Moller (2004), and Lu and Wright (1998). Perzyna-HiSS Model Formulation Fritional materials like asphalt onrete, aggregates, and soils are best haraterized by yield funtions that inlude the strengthening effets of onfining stress. One example of suh yield funtions is the standard Druker-Prager generalization of Mohr-oulomb fritional yield. Extensions to Druker-Prager theory add other refinements suh as a nonlinear yield surfae and a ap on visoplasti flow under hydrostati ompression loading. The present study employs the Hierarhial Single Surfae (HiSS) model, developed by Desai and Zang (1987), with isotropi hardening K - 40

43 and assoiated flow. The HiSS yield funtion has the following mathematial formulation: 2 F 0 J2D I1R I1R n (47) in whih, J 2D and I 1 are the shear and volumetri stress invariants, γ and n are fixed onstants that ontrol the size and shape of the growing flow surfae, ξ is the visoplasti strain trajetory given by the summation of all three prinipal visoplasti strains, and R(ξ) and α(ξ) are parameter funtions governing the size and nature of the apped surfae. Figure 13 shows shematially the HiSS flow surfae in the prinipal stress domain. The funtions R(ξ) and α(ξ) an be formulated as follows: k2 R R0 R A (48) 1 Assoiated flow is assumed,, and: e k (49) 0 f s F A 1 F 0 ' N (50) in whih F is the distane in prinipal stress spae from the applied stress to the hydrostati axis normal to the urrent flow surfae, F 0 is the portion of this distane from the urrent flow surfae to the hydrostati axis, and A is a alibration parameter that depends on the diretion of the plasti flow: A k3 (51) in whih θ is the diretion of the stress vetor in the I 1, J 2D spae and k 3 is a material onstant. Figure 14 desribes shematially the flow rule and surfae hardening. K - 41

44 Figure 13. Three-dimensional representation of HiSS surfae in prinipal stress spae (Gibson, 2006). Figure 14. Shemati of flow rule and surfae hardening (Gibson, 2006) Summary This hapter provided an overview of the rutting problem in pavement strutures. Two mehanisms were identified as the main auses of rutting, ompation at early stages of traffi loading and distortion without volume alteration during the later K - 42

45 stages. These two mehanisms identify the three stages of rutting until failure. Traffi ompation ours fairly quik and it is not often modeled. ommon maintenane praties prevent pavements from reahing the third stage and failure. Therefore the attention to modeling rutting is given to the seondary stage. A few models developed over the past 40 years were briefly disussed. They lay the foundation for the urrent empirial models based on resilient strain. The urrent model used in the AASHTO s mehanisti-empirial design guide, Darwin-ME, is based on vertial resilient strain and is alibrated using axial repeated load permanent deformation test. An alternative model, termed the Westrak model, is based on maximum shear strain and alibrated using the repeated shear test. There are signifiant limitations on the use of empirial models. The most fundamental limitation relies on the appliability of the model. Empirial models are adequate for the onditions at whih they were alibrated. Extrapolations are often hazardous. Mehanisti empirial models rely on preditions of mehanisti response. Linear elasti theory is most used due to simpliity. However most of materials used in pavement onstrution are not linear elasti. In addition, there are several issues not yet fully resolved in the framework for prediting rutting in the asphalt onrete layers using M-E models. There is still no lear onsensus whether rutting is due primarily to axial or shear permanent strains. urrent M-E rutting models relate permanent strains (axial or shear) to resilient strains omputed using multilayer elasti theory. However, in the absene of orretion fators, rutting predited from these resilient strains is in sharp disagreement with field observations. Thikness or depth orretion fators are required to bring preditions in line with observations. These orretion fators further weaken the mehanisti linkage between predited rutting and omputed strains in the M-E approah. urrent M-E models assume that the mehanisms and distributions of permanent strains are similar for HMA layers in flexible pavements vs. HMA overlays on rigid pavements, whih is generally not true. And finally, urrent M-E models do not expliitly onsider the ontribution of heaving at the edge of the wheel paths, although this may be impliitly inluded in the field alibration orretions, given how rutting measurements are taken in the field. The use of advaned mehanisti modeling an help address some of the issues mentioned and redue the gap between rutting preditions and field measured performane. Advaned mehanisti modeling employs theories of mehanis that are more suitable to desribe the real material behavior. The shortoming is the omplexity of these theories, and in partiular the onstitutive models. Asphalt onrete is a omplex material in whih reoverable and irreoverable strains are dependent on temperature, stress and strain rates. Therefore, visoelastiity and visoplastiity theories are most appropriate to model the reoverable and irreoverable behavior respetively. A framework for applying a visoelastivisoplasti model based on Shapery s visoelastiity theory and Perzyna s visoplastiity formulation was presented. As part of this researh, this model will be K - 43

46 reviewed, alibrated and implemented in a finite element model for evaluation of pavement strutures in three dimensions. K - 44

47 hapter 3 Materials and Test Equipment The seletion of the asphalt onrete mixture to be used in this researh was based in three requirements: (1) availability of full-sale performane data, (2) material availability for preparing laboratory test speimens, and (3) full-sale tests onduted with ontrolled environment. Sine the ultimate objetive of this researh was to develop a full 3-D model to predit permanent deformation in full-sale pavement setions, obtaining performane data was ritial. In addition, it was neessary that enough material be available for preparing laboratory speimens for the mixture haraterization. And finally, given the nature of the problem at hand and the omputational effort required to mehanistially predit permanent deformation, the ideal full-sale test should be onduted with ontrolled environmental onditions, more speifially onstant temperature. These onditions were found at the Full-Sale Aelerated Performane Testing for Superpave and Strutural Validation Study onduted at the Federal Highway Administration (FHWA) Turner-Fairbank Highway Researh enter (TFHR), part of the Transportation Pooled Fund Study TPF-5(019). Details of this study are provided elsewhere (FHWA, 2011) This hapter desribes the asphalt onrete mixture used in this projet, mixture volumetris and materials, and testing equipment Mixture harateristis The asphalt onrete mixture hosen was used in lane 11 of the FHWA rutting study. The material is a 12.5 millimeter nominal maximum aggregate size dense graded modified asphalt onrete mixture meeting Virginia Department of Transportation speifiations. The fine and oarse aggregates were all rushed diabase from Loudoun Quarry, VA. One perent hydrated lime was used in the mixture to redue the potential for moisture damage. The aggregate gradation is shown in Figure 15. K - 45

48 Perent Passing Aggregate Gradation ontrol Points Maximum Density Line Restrited Zone mm 2.36 mm 12.5 mm 19 mm Sieve Size (mm), raised to 0.45 power Figure 15. Asphalt onrete mixture aggregate gradation. The binder used in the mixture was a styrene-butadiene-styrene (SBS) elastomeri polymer modified binder with approximately 3 perent by weight linear grafting, denominated SBS-LG. The Superpave performane grade was PG The final mixture design is provided in Table 1. All Superpave mixture design requirements are satisfied. K - 46

49 Table 1. Laboratory mixture design. Gradation (perent passing) Property Design 19 mm mm mm mm mm mm mm mm mm mm 6.7 Gmm 2.7 Gsb PG-grade Binder ontent (% by mass) 5.3 Effetive binder ontent (% by mass) 4.9 Effetive binder ontent (% by volume) 12.7 Design air voids (%) 4.2 VMA at design air voids (%) 16.9 VFA at design air voids (%) Speimen Preparation Twenty-four test speimens were fabriated at FHWA s Turner-Fairbank Highway Researh enter, in MLean, VA. The geometry and instrumentation of the test speimens followed reommendations from the NHRP Projet 9-19 (Witzak, 2005). The reommended dimensions of the ylindrial speimen are 150 mm in height and 100 mm in diameter (height to diameter ratio of 1.5). The laboratory-blended HMA mixtures were short-term aged in the oven for 4 hours at 275 F before ompation (AASHTO PP2). The mixture was then ompated in a Servopa gyratory ompator to a plug of 150 mm in diameter by mm in height. Test speimens were ored from the enter of the gyratory plug, and the speimen ends were sawed parallel to produe the final speimen geometry. The desired target air voids was 5.5%, whih orresponded to the average in-plae air K - 47

50 voids for the same mixture at the TFHR Aelerated Loading Faility (ALF) study 2, lane 11. However, during speimen preparation the atual air voids target was set to a different value orresponding to another lane in the experiment. onsequently, the final average air voids of the speimens was 4.96%. Although this for all pratial purpose is within the +0.5 air void tolerane, the laboratory prepared mixture is nonetheless slightly more dense than the field mixture for the lane being evaluated. Possible impliations of this small disrepany will be offered in a later hapter. Spring-loaded linear variable differential transformers (LVDTs) were used for axial strain measurements. The axial LVDTs were plaed vertially on opposite sides of the speimen as shown in Figure 16. Parallel brass studs used to seure the LVDTs in plae were loated at a distane of 50 mm from the top and bottom of the speimen. The gage length between the studs was 100 mm. The LVDT measurement range was ±5.0 mm. However, as a safety measure, the tests were not allowed to exeed 4 mm deformation (4% strain) to avoid damaging the instrumentation. For radial deformations, four externally mounted LVDTs were aligned horizontally and perpendiular to the enter of the speimen at 90 o intervals. The radial LVDTs set-up is in Figure 17. Surfae frition between the top and bottom of the speimen and the load platens is an important soure of shear stresses at the end of the speimen. Two pairs of rubber membranes lubriated with vauum grease were plaed on the top and bottom of eah speimen during the testing assembly to minimize the possibility of shear stress developing during the tests. K - 48

51 Figure 16. Axial LVDT instrumentation. Figure 17. Radial LVDT setup. All axial strain measurements were the average of two or four axial LVDTs loated at 180 or 90 intervals around the speimen irumferene, depending on the testing being performed. All radial strain measurements were the average of four radial LVDTs loated at 90 intervals around the speimen irumferene. Averaging the LVDTs removed speimen bending effets and redued the overall variability of strain measurements on the speimen Testing Equipment The tests were onduted using a Universal Testing Mahine (UTM) 100, manufatured by IP Global of Vitoria, Australia. The UTM-100 is a servohydrauli feedbak ontrolled testing mahine apable of performing load and displaement ontrolled tests. A photo of the UTM-100 system is provided in Figure 12. The axial load apaity of the mahine is 100 kn. Gain swithes an be used to redue the load range to 50 kn, 20 kn, or 10 kn for more sensitive tests. The mahine is outfitted with an environmental temperature hamber and onfining pressure ell for onfined K - 49

52 tests. Temperature is held onstant within the hamber to the desired temperature ±2 F throughout the test. ontrol and data aquisition is ahieved through a ontrol and Data Aquisition System (DAS) unit interfaed with a P via two serial ables. Two forms of test ontrol software available from the manufaturer were used in this projet. The first software pakage (UTM 3) utilizes pre-programmed test templates for dynami modulus, uniaxial strain rate, and other standard tests. More sophistiated tests an be performed with the seond software pakage (UTM 100), enabling user-defined programs that give the operator muh greater flexibility in speifying loading and data aquisition settings. The UTM-100 is a researh grade mahine in whih almost everything an be adjusted and tuned to ahieve the test needs and objetives. Although this a powerful feature that permits the design of a wide range of tests, it also makes the preparation work for any individual test a little umbersome. The first step is to define setup for the environmental hamber. From previous researh onduted in the same equipment, it was found that variations in room temperature ould have an impat on the final temperature ahieved inside the hamber. Moreover, ports for the wiring required for speimen instrumentation was a soure for heat leakage. Therefore, a sequene of temperature measurements was performed to determine the orret setup for the various testing temperatures. One temperature probe was plaed inside the hamber near the load atuator to verify the temperature reahed at equilibrium for a given temperature setup at the hamber s ontrol panel. One dummy speimen was modified by inserting a temperature probe at its enter of gravity. This probe measured the speimen temperature one equilibrium was reahed inside the hamber. The two equilibrium temperatures (inside the hamber and inside the speimen) were different for all the desired testing temperatures, espeially when the triaxial ell was used. After a sequene of trials, the nominal temperatures to be set at the hamber s ontrol panel were determined so that the speimen would be in equilibrium at the target testing temperature. The time it took to reah equilibrium was also reorded and was used to shedule the beginning of eah test. Given the time required for the speimen temperature reah equilibrium, whih ould be over 7-8 hours for low temperature, an industrial limate hamber available in the laboratory was used to alimatize all speimens prior to testing in order to redue the time spent in the laboratory. Tuning the mahine to respond orretly to the material being tested is also an important step taken to minimize differenes in nominal and target stress/strain levels, as well as load pulse shape. Tuning was arried out at eah of the target test temperatures. K - 50

53 3.4. Summary The asphalt onrete mixture used in this researh was 12.5 millimeter nominal maximum aggregate size dense graded modified asphalt onrete mixture. The binder used in the mixture was a styrene-butadiene-styrene (SBS) elastomeri polymer modified binder, denominated SBS-LG. The Superpave performane grade was PG The target binder ontent was 5.3% by weight and the air voids, 4.2%. One Universal Testing Mahine (UTM), with 100 kn nominal load apaity was used for all laboratory alibration tests. hapter 4 will desribe in greater detail the tests performed and the alibration proedure used to haraterize the mixture behavior. K - 51

54 hapter 4 Model alibration A laboratory testing program was designed to provide the data required to alibrate the linear visoelasti, damage and visoplasti omponents of the onstitutive model. Small-strain frequeny sweep tests were performed to determine the linear visoelasti omplex modulus and the temperature shift fators. Strain rate tests to failure were performed at low temperatures to alibrate the damage omponent of the onstitutive model. And finally, newly designed reep and reovery tests were used to alibrate the visoplasti omponent Linear Visoelasti omponent The prinipal visoelasti properties required for the Shapery model are the relaxation modulus and the time-temperature superposition. With these two properties, one an model any type of loading (stress or displaement indued) at any rate and any temperature. The omplex modulus test is used to determine both properties in a single laboratory test. The visoelasti strain is assumed to be linear and independent of stress state and damage. The visoelasti strain is solely dependent on rate of loading and temperature, whih an be interhanged using time-temperature superposition. The relaxation modulus master urve and the shift fators for time-temperature superposition are determined from the omplex modulus test omplex Modulus Testing omplex modulus tests were performed in unonfined ompression at four temperatures and six frequenies at eah temperature. The dynami modulus obtained in eah frequeny-temperature sweep was used to develop the dynami modulus master urve. Dynami modulus and the measured phase angle an be onverted to the relaxation modulus used in the Shapery linear visoelasti onstitutive model. The dynami modulus test protool was developed in NHRP Projets 9-19 and 1-37A and has been standardized as AASHTO Provisional Standard TP62, Standard Method of Test for Determining Dynami Modulus of Hot-Mix Asphalt onrete Mixtures. The reommended test sequene in AASHTO TP62 for developing the dynami modulus master urve onsists of testing a minimum of two repliate speimens at temperatures of -10, 4.4, 21.1, 37.8, and 54.4 at loading frequenies of 25, 10, 5, 1.0, 0.5, and 0.1 Hz. The 60 dynami modulus measurements are then used to determine the parameters of the master urve by numerial optimization. The dynami modulus test was performed using three repliates in this researh. K - 52

55 The testing protool adopted in this researh was slightly different than what is proposed in the AASHTO provisional standard. The lowest temperature was very diffiult to obtain with the environmental hamber of the UTM. The thermal insulation was insuffiient to keep the temperature at the reommended lowest value. This limitation is not problemati. Reent researh has suggested that only three temperatures and four frequenies are required for developing the dynami modulus master urve, with the temperature values depend on the binder grade used in the mixture (Bonaquist, 2008). The temperatures hosen for the dynami modulus tests were 5, 25, 35 and 50, whih are near the reommended values in the AASHTO provisional standard. The temperature range is also similar to that suggested for the Asphalt Mix Performane Tester (AMPT) dynami modulus test for PG 70-xx binders (i.e., 4, 20 and 40 ). The temperature range is intended to span the operating range for pavements in the field. But more importantly, the range of temperature must be broad enough to provide a full haraterization of the master urve inluding the upper and lower shelves. The frequenies used were 20 Hz, 10 Hz, 3 Hz, 1 Hz, 0.3 Hz, and 0.1 Hz. These frequenies approximate the full range of loading rates pavements are likely to experiene from live traffi ranging from slow ongestion to highway speed. The number of yles applied for eah frequeny varied from the high to low frequeny as follows: 600, 250, 100, 40, 15 and 10. Prior to running the test, preonditioning was performed at 10 Hz using half the stress level defined for this frequeny. This step was suggested as a way to seat any loose aggregates in the speimen and remove any other anomalous strain measurements before the formal frequeny testing (Gibson, 2006). Dynami modulus is the fundamental property that haraterizes the material behavior in the linear visoelasti domain. onsequently the tests must be performed at onditions in whih only visoelasti strains are generated with no damage or residual strains. This is ahieved by limiting the magnitude of the dynami axial strains to values on the order of 100 mirostrain (µ), with a tolerane of ± 25 µ. In equipment designed for routine prodution testing (e.g., the Asphalt Mixture Performane Tester), the ontrol software automatially adjusts the applied stress to produe the strain target range mentioned above. The UTM did not have suh apabilities and a trial and error test was therefore onduted using a sarifiial speimen to determine the target stress levels for eah frequeny and temperature. The final stresses applied are shown in Table 2. The target strain range was ahieved for nearly all temperature and frequenies. The exeptions were observed in tests at the highest temperature (50 ) and the three highest frequenies. The average dynami strain obtained throughout the sweep tests K - 53

56 was about 85 µ, well within the speified limits. A summary of the testing protool used is provided in Table 3. Table 2. Stresses applied in the dynami modulus test. Frequeny (Hz) Test Temperature ( ) Table 3. Summary of dynami modulus testing. Test Temperatures ( ) 5, 25, 35, 50 Frequenies (Hz) (number of yles) 20, 10, 3, 1, 0.3, 0.1 (600, 250, 100, 40, 15, 10) Preonditioning 100 yles at 10 Hz and half the nominal load Dynami axial strain Target range between 75 and 125 µ The tests were onduted starting from the oldest temperature and highest frequeny and marhing on to the lowest frequeny with five minute intervals between frequenies. The frequeny sweep is then repeated at the next warmer temperature until all temperatures have been tested Dynami Modulus Master urve The objetive of the omplex modulus tests is to determine the magnitude of the dynami modulus, E*, and phase angle, φ, between the sinusoidal stress and strain responses at different temperatures and frequenies. These values were omputed for the last six yles in eah frequeny sweep using a built-in algorithm in the IP software ontrolling the test and data aquisition. For a given yle of stress and strain data, the algorithm fits a seond order polynomial over 25% of the period on either side of the peak or valley to determine the peak-to-peak dynami strain, peakto-peak dynami stress, and the phase angle defined as the lag in radians between the stress and strain peaks. After testing was ompleted, a statistial quality hek was performed to assure the results were aeptable for modeling. The oeffiient of variation of the measured K - 54

57 dynami modulus at eah temperature and frequeny averaged 6.8%. The standard deviation of measured phase angle at eah frequeny and temperature averaged 1 with the maximum of 2.6. Suggested values for quality aeptane are 7.5% for the dynami modulus oeffiient of variation and a maximum of 3 for the standard deviation of phase angle (Bonaquist, 2008). Master urves are onstruted using the priniple of time-temperature superposition. First a standard referene temperature is seleted. Next, data at various temperatures are shifted with respet to loading frequeny until the urves merge into a single smooth funtion. The omplete haraterization onsists of the master urve and the shift fators. The assumed shape for the dynami modulus master urve is a sigmoidal funtion of the following form: log E * (52) log e 1 r in whih * E is the dynami modulus; ω r is the redued frequeny (radians/seond), δ is the minimum value of * E (i.e., the lower shelf), δ + α is the maximum value of * E (i.e., the upper shelf), and β and γ are parameters desribing the loation and slope of the transition portion of the sigmoidal funtion. The redued frequeny, ω r, is given by: r a(t ) (53) in whih, ω is the frequeny (radians/seond), a(t) is the shift fator as a funtion of the temperature, T ( ). The approah seleted for developing the master urve was first proposed by Pellinen (2001) and adopted by Gibson (2006). A non-linear optimization algorithm is used to determine the best fit for the master urve equation by adjusting all four parameters in Eq. (52) while simultaneously adjusting the values of a(t) for eah test temperature then fitting a best-fit quadrati urve to the optimized a(t) values at eah temperature. The optimization was ahieved using the Solver tool in Mirosoft Exel. This proedure is similar to that implemented in the MEPDG. The temperature-shifted dynami modulus data and the assoiated fitted master urve are provided in Figure 18. Eah data point in the dynami modulus master urve is the average of three repliates. The temperature shift fators found during the optimization proess and the best-fit quadrati shift funtion are shown in Figure 19. As an be seen in Figure 18, the lower shelf of the master urve, whih orresponds to low frequenies and/or high temperatures, is well haraterized. However, the upper shelf, whih orresponds to high frequenies and/or low temperatures, is less K - 55

58 well haraterized beause of the pratial diffiulties of testing at very low temperatures and/or very high frequenies. 100, ,0000 E* (MPa) 1, Master urve Redued frequeny (Hz) Figure 18. Dynami modulus master urve. 3 2 y = 6E-05x x R² = 1 1 Log a(t) Temperature ( ) Figure 19. Temperature shift funtion. Given the provisional status of the dynami modulus protool used, it was deided to try another proedure for developing the master urve. hristensen et al. (2003) K - 56

59 developed an approah based on binder stiffness and mixture volumetri data using the Hirsh model as part of NHRP Projets 9-25 and Part of this effort was to develop an abbreviated testing protool for development of dynami modulus master urves for routine mixture evaluation and design (Bonaquist, 2008). The need for extreme low temperature testing is avoided by estimating the upper shelf of the master urve using a limiting maximum modulus parameter. The modified master urve equation is given by: log E * Max 1 e log r * in whih, E is the dynami modulus; ω r is the redued frequeny (radians/seond), Max is the limiting maximum modulus and,, and are fitting parameters. The maximum limiting modulus is estimated from mixture volumetri properties using the Hirsh model and a limiting binder modulus of 1 GPa as follows: E* max P 4,200,000 1 VMA VFA xvma 1 P ,000 10,000 1 VMA 100 4,200,000 VMA 435,000(VFA) (55) (54) in whih VMA is the voids in mineral aggregates (%), VFA is the voids filled with asphalt (%), and P is given by: P ,000(VFA) VMA ,000(VFA) VMA The redued frequeny is omputed using the Arrhenius equation: (56) E a 1 1 log log r (57) T Tr in whih, ω is the frequeny (radians/seond), a(t) is the shift fator as a funtion of the temperature, T, T r is the referene temperature, both in K, and E a is the ativation energy, whih is treated as a fitting parameter. The master urve based on the Hirsh model is ompared to the MEPDG master urve in Figure 20. There is a very good agreement between the two fitted master K - 57

60 urves, whihh indiates a good haraterization nluding the upper shelf. The master urve determined using the MEPDG proedure was seleted for the linear visoelasti haraterization of the asphalt onrete mixture. The final parameters are summarized in Table 4. The referene temperaturee seleted was ,000 10,000 E* (MPa) 1, MEPDG Master urve Hirsh Master urve Redued frequeny (Hz) Figure 20. omparison between MEPDG and Hirsh master urves. Table 4. Dynami master urve and temperature shift fators. Dynami Modulus Master urve Temperature Shift Fators δ = α = β = γ = a( (5 ) = a(25 ) = 10-0 a(35 ) = 10-2 a(48 ) = Relaxation Modulus and reep ompliane The relaxation modulus desribes the relationshipp between stress and strain under a onstant strain ondition. onversely, reep ompliane desribes the same relationship under a onstant stress ondition. The omplex modulus testt data desribed in the preeding setion were used to determine the relaxation modulus and reep ompliane properties of the asphalt onrete mixture. K - 58

61 As desribed in hapter 2, the generalized Maxwell model an be used to represent the omplex modulus. Prony series are used to fit the storage modulus, E (ω), and the loss modulus, E (ω). The relaxation modulus an then be desribed using Eq. (21), whih for onveniene is reprodued here: m t i i (58) i1 0 E t E Ee in whih E 0 is the long term equilibrium modulus, E i and ρ i are the elasti springs and relaxation times for the elements in the generalized Maxwell model, and m represents the number of single Maxwell instanes are in the generalized model. reep ompliane an be also represented by the following Prony series: Dt D0 D e t n j j 1 (59) j1 in whih, D 0, D j and τ j are Prony series onstants. Strain reep and stress relaxation are two aspets of the same visoelasti behavior and therefore are related. The relationship is given by Eq. (60). Park and Shapery (1999) developed a proedure to determine the reep ompliane onstants from the relaxation modulus through a system of algebrai linear equations (Gibson, 2006; Kim et al., 2009). t 0 E t D d t (60) For the sope of this researh, the required property is the relaxation modulus, whih is needed for determining the pseudo strains for the ontinuum damage haraterization. reep ompliane, although not neessary, was omputed to omplete the mixture haraterization. The storage modulus was omputed from the dynami modulus measured in the frequeny sweep tests using Eq. (26). A master urve was fit to the storage modulus data following the same proess used for the dynami modulus master urve. Figure 21 illustrates the master urve fit for the storage modulus. One the master urve has been determined, the Prony series desribed in Eq. (58) was determined. The final step was the alulation of the reep ompliane Prony series onstants. The final relaxation modulus and reep ompliane determined from this onversion proedure are illustrated in Figure 22. The final Prony series onstants for Eqs. (58) and (59) are shown in Table 5. K - 59

62 100,000 10,000 E(t) (MPa) 1, Master urve Redued frequeny (Hz) Figure 21. Storage modulus master urve E(t R ) D(t R ) 0.01 E(t), (MPa) D(t), (1/MPa) E E E E E E E+09 wr (rad se-1) Figure 22. Relaxation modulus and reep ompliane. K - 60

63 Table 5. Prony series onstants for relaxation modulus and reep ompliane. i Relaxation Modulus reep ompliane ρ i (se) E i (MPa) τ j (se) D j (MPa -1 ) E E E E E E E E-11 Time-temperature superposition Temperature Log a(t) ontinuum Damage Model The ontinuum damage behavior is haraterized by marosale stiffness redution due to the development of miroraks that eventually oalese into marofratures and rak propagation. The ontinuum damage model was alibrated using onstant strain rate tests to failure at low temperatures. Unonfined and onfined tests at two different strain rates were performed to alibrate the damage funtions as defined by Shapery s work potential theory. The energy density funtion used to relate damage with the stiffness of the material presented previously as Eq. (37) in hapter 2 is reprodued here for onveniene: 2 R 1 R Wd S S p S p (61) 2 2 K - 61

64 in whih, ij are the damage funtions, S is an internal state variable, p is the R onfining pressure and is the pseudo axial strain. From Eq. (61), stress-strain 1 relations an be derived as follows: W R d R 1 p R (62) W p R d e p R v R (63) R in whih the additional variable,, is the pseudo volumetri strain. In the multiaxial v formulation with symmetry on axis x 3, p. The pseudo strains are omputed using the following hereditary onvolution integrals: R d1 1 E t d E (64) 1 t R 0 R R R d R 1 t R dv v E tr R d R ER d 0 R (65) in whih E(t R ) is the relaxation modulus and t R is redued time, E R is the referene R modulus taken here as equal to 1, and v is the pseudo volumetri strain. The internal state variable is governed by a damage evolution law as follows: ds dt S W d (66) in whih α is a material property. alibration of α was found to be very diffiult. The optimum value is the one that makes all damage urves ollapse onto a single master damage urve. Previous attempts to alibrate α were arried out by trial and error using an inremental approah over values varying from 1.25 to 2.25 (Gibson, 2006). It was found that the optimum value ranged between 1.75 and 2.0. Kim et al (2009), summarizing the efforts by other researhers, suggested that α was inversely proportional to the absolute maximum slope of the relaxation modulus master urve, m. This approah suggests a value for α of 2.1. Different α values were tried and ultimately a value of 2.0 was adopted. Following Shapery s work potential theory for a visoelasti media with damage, it is assumed that the material is isotropi in the undamaged state (Ha, 1996). Therefore, Shapery s energy density funtion, Eq. (61), must be equivalent to the strain energy funtion of a typial isotropi material. This assumption yields the following onstraints for the damage funtions when no damage has ourred: K - 62

65 11 Eref When S Eref (67) in whih υ is the initial Poisson s ratio and E ref is the referene modulus in this ase taken as equal to 1. Unonfined onstant strain rate tests to failure were used to determine the damage funtions 11 and 12 by applying both stress-strain relations desribed in Eq. (62). One 11 and 12 were determined, results from the onfined tests were used to determine the remaining damage funtion, 22, using Eq. (63). The referene temperature of 19 was adopted for the damage model alibration. The redued frequeny was alulated using the temperature shift fators provided in Table onstant Strain Rate Tests to Failure Unonfined and onfined onstant strain rate tests to failure were performed at 10. The low temperature minimizes the development of visoplasti strains. Temperatures lower than 10 would have been better, but there were limitations in the environmental hamber enapsulating the speimen, loading and measurement apparatus. After trying to establish equilibrium at 5, it was found that sustained onstant temperature was only possible at 10. Sarifiial dummy speimens were used to define the strain rates for the tests. A trial and error proedure was arried out to find out the fastest loading rate that the mahine would apply and still fail the speimens before reahing its nominal load limit. The triaxial hamber pressure was set at 250 kpa for the onfined tests. For safety reasons and the measurement limits of the LVDTs, the tests were programmed to end when total deformation reahed 4%. Strain rates of and ε/seond were seleted based on this proedure for both unonfined and onfined onditions. Stress versus strain plots from the unonfined and onfined strain rate tests are shown respetively in Figure 23 and Figure 24. Some visoplasti strains were indued along with the visoelasti plus damage strains. To ensure aurate alibration of the damage model, these visoplasti strains were removed from the total measured strains using the alibrated visoplastiity model, whih is desribed in greater detail later in this hapter. K - 63

66 Axial Strains Radial Strains 6000 stress (kpa) RP RP RP RP average average E E E E-02 strain 4.0E E-02 Figure 23. Unonfined strain rate tests at 10 : stress versus strain urves for strain rates of and ε/seond Axial Strains Radial Strains 6000 stress (kpa) RP RP RP RP average average E E E E-02 strain 4.0E E-02 Figure 24. onfined strain rate tests at 10 : stress versus strain urves for strain rates of and ε/seond. K - 64

67 alibration of Damage Funtion 11 Under unonfined onditions (i.e., uniaxial loading), the onfinement term in Eq. (62) disappears and the damage funtion 11 an be defined as: 11 (68) Although 11 an be determined diretly from the test, the internal damage variable, S, annot. As desribed in hapter 2, a disrete solution for S an be defined as: R 1 1 S S t 2 2 i1 i R i S (69) in whih i denotes the time step in the test and i is a simplified notation for 11 at t=t i. The term i is omputed as the differene in the damage funtion aused by a S small variation in S (i.e., 0.1). The internal state variable for the next time step, S i+1, is then omputed using Eq. (69) and the proess repeated until all time steps in the test are alulated. The damage funtion 11 was assumed to have the following form: S e (70) in whih a and b are material onstants. Optimization using the Solver tool in Mirosoft Exel was employed to alibrate the damage funtion by minimizing the sum squared differene between predited and alulated damage. This proess was repeated for every speimen repliate and strain rate tested in unonfined onditions. Figure 25 summarizes the results of the alibration proess for eah repliate and the average alibration onsidering all repliates simultaneously. For this given mixture, it was found that a = and b = as b K - 65

68 11(S) RP01_ RP02_ RP01_ RP02_ alibrated funtion Damage variable, S Figure 25. Damage funtion 11 versus S for all repliates at alibration of Damage Funtion 12 Under unonfined onditions (p=0), 12 an be alulated from Eq. (63) at any time as follows: 12 e (71) The internal state variable, S, has already been determined during the alibration of 11. The damage funtion 12 was plotted against S and the following funtion was fit using a least squares optimization proess: R v R 1 2 S e S (72) S 12 12, ini in whih i are material onstants, and 12, 1 2 to satisfy Eq. (67). ini The strain rate test results showed that the speimens initially exhibited ompressive and later expansive volumetri strains. The variation of the Poisson s ratio with time is shown Figure 26. The variation is attributed to internal damage ourring in the material. Ideally, 12 would be a funtion of a rate-dependent Poisson s ratio, although for simpliity and pratial appliations, the option hosen was to treat 12 as a funtion of the initial Poisson s ratio. The exponential funtions shown in Figure 26 were used to smooth the data points and the interepts were averaged to determine the mixture s initial Poisson s ratio value of This value is within the range reported by other researhers (Kim et al., 2009). It is worth noting that the Poisson s K - 66

69 ratio of 0.5 separates the volume behavior between ontration and dilation. The speimens undergo ontration at the beginning and dilation at the end of the test. Poisson's ratio y = e x R² = y = 0.418e x R² = y = e 0.018x R² = Figure 26. Poisson's ratio variation during the strain rate tests. y = e x R² = RP RP RP RP tr The final 12 funtion is taken as the average of all individually alibrated funtions for eah repliate. The final model is shown in Table 6. Figure 27 presents the measured damage funtion 12 versus the internal damage variable, S, for all repliates and the final alibrated model for all tests. A dereasing 12 is onsistent with an inreasing Poisson s ratio due to growing damage. The point where 12 hanges sign also orresponds to the hange in behavior from ompression to expansion observed in the speimens during testing. Table 6. Damage funtion 12 alibration oeffiients. onstant Value 12,ini x x10-7 K - 67

70 12(S) RP01_ RP02_ RP01_ RP02_ alibrated funtion Damage variable, S Figure 27. Damage funtion 12 versus S for all repliates at alibration of Damage Funtion 22 The last damage funtion to be alibrated was 22. onfined strain rate test results were used to omplete the damage haraterization of the mixture. After the other damage funtions have been established, Eq. (63) was used to solve for 22 as follows: 22 R R ev 121 (73) p in whih the variables are as desribed previously. The solution of Eq. (73) requires a different approah beause the damage variable, S, must be alulated at the same time. The approah onsisted of using Eq. (62) to ompute the deviator stress, Δσ, and minimizing the error by fitting the appropriate urve to the damage funtion 22. The nature of the tests and the approah required to alibrate the damage funtion made it diffiult to alibrate the model for onfined onditions with the same level of auray attained for the unonfined tests. The effets of noise in the data and speimen to speimen variability were magnified beause of the small magnitude of measured strains in the onfined tests, espeially at earlier stages of testing. Moreover, any errors from omputing the damage variable, S, were propagated to the alibration of 22. The funtion hosen to represent 22 was a seond order polynomial funtion: d d S (74) K - 68

71 in whih d 1 and d 2 are material alibration onstants. As desribed previously, the interept was defined by the requirement for transversely isotropi onditions when the material is undamaged. Figure 28 presents the alulated damage funtion 22 versus the internal damage variable, S, for all repliates and the final alibrated model for all tests. The alibrated values of the material onstants are d 1 = and d 2 = x (S) 1.0E E E E E E E E E E RP01_ RP02_ RP01_ RP02_ alibrated funtion Damage variable, S Figure 28. Damage funtion 22 versus S for all repliates at Visoplasti Model The onstitutive model adopted herein is based on the Perzyna-HiSS model developed by Gibson (2006) with some enhanements to the model alibration proess. A new testing proedure was developed to expedite the alibration effort in the laboratory, while maintaining the range of stresses and frequenies defined by Gibson for the reep and reovery tests. The Hierarhial Single Surfae (HiSS) model (Desai and Zang, 1987) employing isotropi hardening and assoiated flow was used. The onstitutive model, detailed previously in hapter 2, is summarized here for onveniene. The HiSS yield surfae is defined as: 2 F 0 J2D I1R I1R n (75) K - 69

72 in whih J 2D and I 1 are the shear and volumetri stress invariants, γ and n are fixed onstants that ontrol the size and shape of the growing flow surfae, ξ is the visoplasti strain trajetory given by the summation of all three prinipal visoplasti strains, and R(ξ) and α(ξ) are hardening funtions governing the size and nature of the apped surfae. The funtions R(ξ) and α(ξ) are formulated as: k2 R R0 R A (76) 1 e k (77) in whih, R 0, R A, k 2, α 0 and k 1 are material onstants. Assoiated flow was assumed, f s g s, and: 0 f s F A 1 F 0 ' N (78) in whih F is the distane in prinipal stress spae from the applied stress to the hydrostati axis normal to the urrent flow surfae, F 0 is the portion of this distane from the urrent flow surfae to the hydrostati axis, A is a alibration parameter that depends on the diretion of the plasti flow: A k3 (79) in whih θ is the diretion of the stress vetor in the I 1, J 2D spae, and k 3 is a material onstant Multi-Stress/Load Duration Tests The visoplasti omponent was alibrated using yli reep and reovery tests in unonfined and onfined onditions. The onfining stress was 250 kpa. A trial was onduted to determine the temperature for the test using sarifiial speimens. Temperatures varying from 60 to 40 were tried. The speimens tested at high temperatures were failing prematurely with few yles. The best results were ahieved at 40. Temperature was measured throughout the test and at the end the final average temperature for all reep and reovery tests was 39. In the original proedure (Gibson, 2006), the model was alibrated using two separate tests, one in whih the deviatori stress was onstant and the duration of the load pulse was inreased with eah yle, and a seond in whih the load duration was onstant and the deviatori stress was inreased with eah yle. This proedure has been updated here to redue the number of speimens and expedite testing. K - 70

73 A ombination of load durations and deviatori stresses were used in a single test to over a wide range of stress onditions, as shown in Table 7. The same ombination of deviatori stresses and frequenies were used in unonfined and onfined tests. An example of the measured total strain history reorded during one test is shown in Figure 29. Eah peak orresponds to one yle. One detailed yle is provided in the figure inset, in whih all omponents of the response (elasti, visoelasti and visoplasti) an be seen. The permanent response at eah yle was reorded at the end of eah rest period. Three repliates were tested in unonfined and three in onfined onditions. The averages of permanent strains measured at the end of eah yle were used in the alibration. Figure 30 shows the average measured axial and radial permanent strains for unonfined and onfined onditions. Table 7. Deviatori stress and frequeny used on multi-stress/load duration test. yle Deviatori Stress (kpa) Duration (s) Rest period (s) K - 71

74 total strain axial radial time (se) Figure 29. Example of multi-stress/load duration reep and reovery test. vp strains axial (unonfined) radial (unonfined) axial (onfined) radial (onfined) yles Figure 30. reep and reovery visoplasti strains versus load yles Visoplasti Model alibration The visoplasti model was written as a Matlab sript and the alibration of all model onstants was performed simultaneously using the minimization funtion in K - 72

75 Matlab. The sum of the squared differene between the alulated and measured permanent strains at all yles in the test was the parameter to be minimized. In order to effetively onsider all stress states and magnitudes, unonfined and onfined test results were used simultaneously in the minimization funtion. The referene temperature of 19 was adopted for the visoplasti model alibration. The redued frequeny was alulated using the temperature shift fators provided in Table 4. The final model alibration oeffiients are presented in Table 8. Predited vs. measured strain plots are shown in Figure 31. Table 8. Visoplasti model alibration oeffiients. Parameter alibration oeffiient Γ 1.659E-09 (kpa.s) -1 γ α k n N R a (kpa) k R (kpa) k K - 73

76 strain strain time (se) (a) time (se) (b) Pred axial VP Pred radial VP Meas axial VP Meas radial VP Pred axial VP Pred radial VP Meas axial VP Meas radial VP Figure 31. Visoplasti model alibration using yli reep and reovery tests: (a) unonfined and (b) onfined. K - 74

77 Validation with the Flow Number Test Researhers at the FHWA Turner-Fairbank Highway Researh enter onduted Flow Number (FN) tests on the same mixture tested in the present projet. The Flow Number test is a pulsed yli load and reovery test with fixed loading/reovery times and a fixed stress level where the permanent strains are measured after eah load yle. The tests an be performed unonfined and onfined on ylindrial speimens. The FN test is used as a performane test for rutting suseptibility of asphalt mixtures. In addition, its results are used for alibration of rutting models suh as the one used in the mehanisti-empirial pavement design guide (MEPDG). The test an be done using the Asphalt Mixture Performane Tester (AMPT). These FN tests provided an exellent opportunity for validation sine they were performed on the same mixture but onduted at a different laboratory and by different tehniians. In addition, the strain measurements were different; high auray digital images of the speimen s deformation were used in the FN test, while onventional LVDTs were used during the multi-stress/load duration tests. The end onditions for applying the load at the two tests were also different. Spherial ball was used at the UTM in the University of Maryland, while flat end platen was used in the FN test. And finally the temperature for the FN test was set at 64, ompared to 40 for the multi-stress/load duration test. The FN tests were performed at unonfined and onfined onditions using three repliates eah. The load was applied as a haversine pulse with a duration 0.1 seonds followed by a rest period of 0.9 seonds for reovery of elasti/visoelasti strains. Visoplasti strains were reorded at the end of eah rest period. The deviatori stress was 207 kpa (30 psi) for the unonfined tests and 827 kpa (120 psi) for onfined. The onfinement was 68.9 kpa (10 psi). The alibrated visoplasti model was used to predit the permanent strains measured in the FN tests. Time-temperature superposition was applied using the temperature shift fators determined from the omplex modulus tests. The omparison between predited and measured strains is shown in Figure 32. The red vertial bars represent the variability observed in the test. The model predited the strains of the onfined test with reasonable auray, but the unonfined strains were underpredited. This is in part due to the small deviatori stress applied in the unonfined test. During alibration, responses to small deviatori stresses were onsistently underpredited as the prie for better preditions at higher stress levels. However, for pratial appliations, the stress onditions used in the onfined FN test are more representative of real pavements and thus more relevant to rutting performane. K - 75

78 Mirostrain yles Measured axial (onfined) Measured axial (unonfined) Measured radial (onfined) Measured radial (unonfined) Predited axial (onfined) Predited radial (onfined) Predited axial (unonfined) Predited radial (unonfined) Figure 32. Predited versus measured strains in the Flow Number validation test alibration of the Visoplasti Model Using the Flow Number Test The FN tests used to validate the researh-grade alibration were also used to realibrate the model as an internal onsisteny hek. The FN test an be done using the AMPT and is being onsidered as the test of hoie for alibrating the empirial rutting model in the MEPDG. Therefore this exerise, if suessful, ould simplify the visoplasti model alibration and minimize the effort for future implementation of a full mehanisti model for rutting predition. Following the opposite path taken for the onventional, researh-grade alibration, the visoplasti model was realibrated using the FN test results. One the model was realibrated, it was validated using the multi-stress/load duration tests. The information about load and duration of eah yle was fed into the Matlab sript and the same optimization algorithm was used to find the material onstants of the visoplasti model. Predited versus measured strains obtained during alibration are shown in Figure 34. The omparison between the two visoplasti model alibrations is provided in Table 9. All parameters in the model are alibrated at the same time, therefore large variations in one or more parameters are expeted. Overall these variations anel eah other out during the minimization of errors. K - 76

79 Mirostrain yles Measured axial (onfined) Measured axial (unonfined) Measured radial (onfined) Measured radial (unonfined) Predited axial (onfined) Predited radial (onfined) Predited axial (unonfined) Predited radial (unonfined) Figure 33. Visoplasti model alibration using the Flow Number test. Table 9. omparison between the two visoplasti model alibrations. Parameter Multi-stress/load duration alibrated model FN alibrated model Γ, (kpa.s) E E-09 γ α k n N R a (kpa) k R 0 (kpa) k K - 77

80 Validation with the Multi-Stress/Load Duration Test The model alibrated using the FN tests was used to predit the results of the multistress/load duration tests. The results, presented in Figure 34, suggest a good agreement between predited and measured permanent strains. The model alibrated using the FN tests predited permanent strains that agree well with the measured data. VP strains Meas axial VP Meas radial VP Pred axial VP Pred radial VP time (se) (a) VP strains Meas axial VP Meas radial VP Pred axial VP Pred radial VP time (se) (b) Figure 34. Predited versus measured visoplasti strains from multi-stress/load duration reep and reovery tests: (a) unonfined and (b) onfined. Predited strains omputed using the FN-alibrated model. K - 78

81 One of the major disadvantages of advaned haraterization of asphalt onrete mixtures is the omplex alibration proess, whih often requires one or more researh-grade tests. This poses as an obstale to pratitioners and limits the use of advaned modeling as tools for pratial appliations. The results shown in Figure 34 suggested that the Perzyna-HiSS visoplasti model an be alibrated using the FN test, a simple test that an be performed easily on a prodution basis using the AMPT Summary This hapter desribed the alibration proess for the visoelasti-visoplasti onstitutive model. The linear visoelasti omponent was alibrated using the omplex modulus test. The dynami modulus master urve, relaxation modulus and reep ompliane were determined. The ontinuum damage omponent was alibrated using unonfined and onfined onstant strain rate tests to failure at low temperatures. The visoplasti omponent was alibrated using unonfined and onfined yli reep and reovery test, termed multi-stress/load duration test, in whih different stress levels and load duration were applied in sequene on the same speimen until failure. This test was designed to expedite the alibration proess and redue the number of speimens required. Flow Number (FN) tests were used to verify the alibrated model. These tests were performed independently by FHWA at the Turner- Fairbank Highway Researh enter. In addition, the same FN tests were used to realibrate the visoplasti model, whih was then verified using the multi-stress/load duration test. The results demonstrated that the Perzyna-HiSS visoplasti model an be suessfully alibrated using the simple FN test. The implementation of the alibrated model in a finite element method is one of the objetives of this researh. This implementation is desribed in detail in hapter 5. A series of appliations follows in hapter 6. K - 79

82 hapter 5 Finite Element Modeling 5.1. Introdution Pavement design requires seleting materials and a struture that will withstand yli loading and limate flutuations over a long period of time. Materials used in pavement onstrution require advaned onstitutive models that are apable of apturing the omplexities observed in their behavior. Finite element (FE) methods are ideal for modeling omplex material behavior. However, simulating large number of load appliations in finite element analyses is a daunting task that requires signifiant omputational effort, whih is often prohibitive for pratial designs. Nevertheless, FE analysis an improve understanding of material behavior and pavement performane, provide insights on ritial loations and behavior phenomena in the pavement struture, and help the design of more effetive strutures and materials. The visoelasti-visoplasti model developed and alibrated in this researh was implemented in ABAQUS (2006), a ommerial finite element pakage widely employed in pavement engineering researh. The objetive was to have a robust but simple to use tool for analyzing permanent deformations in pavements under moving wheel loads. ABAQUS is a good tool for this appliation for several reasons. It is a mature, well-validated, and well-doumented finite element analysis program. It has a user-friendly interfae for pre- and post-proessing, whih failitates reating models and visualizing results after the analysis is omplete. In addition, it has a large variety of onstitutive models in its library that an be used to model other layers in the pavement struture (e.g., elasti, elasto-plasti, et.). Most importantly, ABAQUS provides the option of inorporating user defined material funtions (UMAT) instead of its built-in onstitutive models. The UMAT is written in FORTRAN and it is alled from the analysis module during the simulation proess. The onstitutive model desribed in hapter 4 was implemented in a UMAT. This hapter provides a briefly overview of the key FEM and ABAQUS omponents relevant to the moving wheel analyses and a desription of how the onstitutive model was written in the UMAT. Version of ABAQUS was used in this researh. It was installed in a workstation with 4 proessors Intel ore 2 Extreme, 2.6 GHz, with 4GB of RAM, operated by Windows XP 64-bit. The FORTRAN ompiler was Intel version 9.0. K - 80

83 5.2. Finite Element Method The FEM provides numerial approximations to problems that are diffiult to solve analytially. It is a pieewise formulation in whih the problem is divided in many smaller problems (elements) that are solved simultaneously. The elements are onneted to eah other at nodes, typially the orners but also sometimes at other points (e.g., mid points at element sides). A ontinuous polynomial funtion of the desired response (e.g., displaement) is defined within the element between the nodes, forming an approximate pieewise representation of the response over the entire solution domain. Loads and boundary onditions are applied to the nodes. Equations desribing the behavior of eah element and the interation of elements between nodes are assembled for form a set of linear equations that is solved to find the desired primary response, in this ase the displaement values at the nodes. The proedure for omputational modeling using the FEM onsists of six steps: Geometry modeling, inluding boundary onditions Meshing and element definition Material property speifiation Loading ondition appliation Simulation Visualization Geometri Model and Boundary onditions The pavement geometri model was onstruted by using independent parts in the ABAQUS solid modeler. Eah part represented one layer in the pavement struture. The geometri model was reated in three dimensions (3D) with an axis of symmetry along the longitudinal enter of the tire load. Figure 35 shows one pavement rosssetion; the different shades of gray represent individual layers. The axis of symmetry is taken as the enter of the tire, hene only half of the problem is modeled. Therefore the diretion of traffi was defined along the axis of symmetry. Boundary onditions were applied to all faes of the geometri model to limit displaement in the diretion perpendiular to the fae. The bottom of the last layer modeling the subgrade was limited to no displaement in all diretions (enastre). All preditions of rutting were alulated at the middle ross setion of the model. Several models were tested to ensure that the effets of boundary onditions on the mehanisti responses indued by the tire load were insignifiant. K - 81

84 Figure 35. 3D solid model Meshing and Element Definition The meshing proess divides the problem domain into the set of elements onneted at nodes. The density of elements in a given region of the problem ontrols the auray of the results. In the ase of modeling a pavement subjeted to a tire load, a high element density is desired near the load. Unfortunately, the more elements, the longer the required omputational time. It is therefore neessary to limit the number of elements. The meshing proess typially requires several iterations in order to define on an optimum number of elements that will produe a suffiiently aurate solution at a pratially realisti omputational effort. The element type and respetive number of nodes are defined during the meshing proess. The number of nodes defines the type of funtion that an be used to approximate the solution within an individual element. Simple 4-node quadrilateral elements in 2-D or 8-node brik elements in 3-D only allow linear approximations of the displaements between the orner nodes. Elements with additional nodes (e.g., at the midpoint of eah edge) an aommodate higher order approximating polynomials. However the omputational effort inreases signifiantly. The most ommon approah is to use simple elements and inrease the number of elements in regions of high desired auray. Eight-node brik elements were used in this model. K - 82

85 The tire load ould not be applied instantly in thee nonlinear analyses. Inremental loading was required to bring the tire pressure upp to the desired peak. Sine the ultimate goal was to evaluate responses indued by a moving load, there was no need for additional refinement to the mesh where the tire load wass initially applied and later removed at the end of the yle. In addition,, regions distant from the loading zone ould also be meshed with fewer elements. Therefore, eah layer was divided into several zones and eah zone was meshed differently. This effort greatly expedited omputational time. Figure 36 shows the plan view of the pavement surfae. The ½ tire footprint was modeled as a 0.24 x 0.12 m retangular area. Thee horizontal and vertial lines define the different meshing zones. The enter area was defined as the moving load area, at whih the most refined mesh was defined. The two adjaent areas area where the tire pressure was loaded and unloaded are also shownn in the figure. All layers were modeled using the same zone onfiguration. When zone onfigurationn was ompleted, a meshh study was onduted to determine the optimal mesh density for eah area. Figure 377 shows the final mesh for the surfae layer in plan view. A finer mesh is used in areas lose to the moving wheel load, and a oarser mesh in more remote areas. Figure 36. Plan (surfae) view of geometri model. K - 83

86 Moving Loading Area Figure 37. Surfae layer mesh in plan view. Eah layer in the pavement struture was modeled as one independent part in the ABAQUS solid modeler. The same priniple of meshing by area was applied to all layers in the pavement ross setion, following the zone onfiguration shown in Figure 36. There is no need for oinidental nodee positions between parts when the modeling by parts. This is a great advantage beause one an define the mesh of eah part separately, whih greatly enhane omputational time. The final model mesh is shown in Figure 38. It onsisted of 20,700 8-nodee brik elements with 24,855 nodes. Figure 38. 3D finite element mesh. K - 84

87 Material Property Speifiation ABAQUS has two options for seleting the onstitutive model that governs the material behavior and onsequently the material property inputs. For ommon material behavior, there is a library of onstitutive models enompassing typial linear and non-linear models. For less ommon material behavior, there is an option of reating a user-defined onstitutive model or UMAT. For the analyses in this projet, all pavement layers other than the asphalt onrete were modeled as homogeneous isotropi linearly elasti materials. The asphalt onrete was modeled using the visoelasti-visoplasti model desribed in hapter 2 as implemented in a UMAT. The details about the UMAT are provided in a later setion in this hapter. The remaining material properties used are provided in hapter 6 with the desription of the problems evaluated and results obtained Loading onditions The loading ondition defines the presribed loading of the problem. For strutural analysis it an be in the form of fores, pressures or displaements. Loads are applied to the nodes. Pressure loads are transformed into nodal fores and applied diretly to the nodes within the loaded area. The load annot be applied instantly in the nonlinear analyses; it must be modeled in inrements. To represent instant loading, very small loading time is used. In addition, the type of loading inrement an be defined as well. A linear inrease over the loading time is typial, although many FE software allow different inrement forms, suh as exponential, or even sinusoidal for yli loading. A linear inrease of loading from zero to the defined peak was used in this researh. One the presribed load is fully reahed, a new step is generated whih takes the load and plaes it in an adjaent loation, thus simulating the moving of the wheel. The tire footprint was modeled as a pressure load of 690 kpa (100 psi) applied diretly on the set of elements beneath the wheel. As shown in Figure 39, the moving wheel was simulated by inrementally moving the load footprint from one set of elements to the next by adding a new line of loaded elements in the front end of the tire, while removing one line of loaded elements from the bak end of the tire. This gives the motion aspet of the load. Although time onsuming, this approah is a far better representation of a moving tire than the step-wise proess ommonly found in the literature. In addition, the load when modeled as a pressure area applied diretly on the surfae instead of the atual modeling of the tire as an independent part inreases omputational performane and redues the risk of numerial instabilities that ould happen at the interfae between the two parts (tire and pavement struture). K - 85

88 Figure 39. Shemati of moving load Simulation After the numerial model has been assembled, itt must be solved. The solution to the system of linear equations gives the values for thee response variable at eah node of the mesh. The omputational effort is diretly proportional to the number of equations that are being solved simultaneously, whih in turn is a funtion of the number of elements/nodes. A diret solution algorithm basedd on Gauss elimination is most ommonly used to solve the set of linear equations. However, when one is dealing with nonlinear problems, whih is the ase in thiss researh, the load must usually be applied and repositioned in small inrements in order to trak the nonlinear response. The nonlinear system is approximated as an equivalent linearr system for eah load inrement. An iterative proedure is then applied to solve the equilibrium between the applied loads and the nonlinear stress-strain behavior of the elements. ABAQUS offers a diret linear equation solver based on Gauss elimination and an iterative nonlinear solver based on a modified Newton-Raphson algorithm. Other options available in ABAQUS were explored, but in the end the stability of the Newton-Raph hson method was deisive. The step inrement iss automatially hosen by ABAQUS, but with some onstraints on the maximum step size. Impliit time K - 86

89 integration was employed in the analyses for two main reasons. One was unonditionally stability, whih proved to be ritial even when simple models suh as elasto-plastiity were first explored. The seond was a software limitation; userdefined onstitutive models must be implemented as UMATs using an impliit formulation Visoelasti-Visoplasti Model Implementation The visoelasti-visoplasti onstitutive model was implemented in ABAQUS as a user defined UMAT. The UMAT subroutine is oded in FORTRAN and is ompiled and linked into the ABAQUS exeutable file. The UMAT must provide two outputs to the ABAQUS analysis: (1) updated stress vetors and solution dependent internal variables at the end of eah load step inrement (or iteration), and (2) the material stiffness matrix. The onstitutive model is implemented in a three dimensional formulation but has the apabilities for two dimensional axisymmetri or plane strain problems that are also often used in pavement modeling. Doumentation on how to reate a UMAT is limited. Examples are sare and often poorly doumented. This made the development a diffiult task. The UMAT ode is doumented and presented in Appendix A. In addition to implementing the visoelasti-visoplasti onstitutive model desribed in hapter 2, it an also be used as step-by-step template for reating other UMATs. The key steps an be outlined as follows: 1. Delaration of variables provided by ABAQUS 2. Delaration of loal variables (used within the UMAT during alulations) 3. Definition of material properties and variable initialization 4. alulation of visoelasti responses 5. alulation of visoplasti responses 6. Update of the stress vetor 7. Return of output quantities to ABAQUS All alulation steps are doumented in the UMAT provided in Appendix A. After the displaements are alulated, the stress vetor and the stiffness matrix are updated and returned bak to ABAQUS at the end of the UMAT. At the beginning of eah step inrement alulation, ABAQUS provides ertain variables that may be used in the UMAT alulations. It is mandatory that these variables be delared and imported into the UMAT. Therefore the first step when reating a UMAT is to properly delare the variables that ABAQUS is passing onto the UMAT. The syntax for this delaration is indiated in the doumented UMAT in Appendix A. Loal variables are delared next. These variables are used in the K - 87

90 onstitutive model alulations and are not transferred bak to ABAQUS. All material property values are also defined at this loation for simpliity. After variable delaration is omplete, the onstitutive model alulations are oded, starting with the visoelasti omponent. The objetive is to determine the stress vetor at the end of the step inrement indued by visoelasti strains. The first step is the alulation of pseudo strain. This was ahieved by using the reursive algorithm proposed by Simo and Hughes (1998) based on the strain history provided at the beginning of the step inrement. The prinipal strain tensor is alulated and inversely ordered ( p33 > p22 > p11 ) to aommodate the axis of symmetry at the prinipal axis 3. Aordingly the diretion osines matrix is adjusted to reflet this transformation. This is inorporated to omply with the loally transverse isotropy indued by damage in Shapery s theory. Reall that the UMAT is exeuted at the integration point level. At every step inrement, the load is hanging, either inreasing/dereasing or simply moving from one loation to another when a moving load is simulated. Although the prinipal axes are ontinuously rotating for a given node as the wheel moves on the surfae, it is assumed that there is no rotation within the step inrement. Stresses are omputed using the prinipal pseudo strains and transformed bak into global stresses. This proess was desribed in greater detail in hapter 2 and is doumented step-by-step in the UMAT. In addition to the stress vetor, the stiffness matrix must also be alulated. The stiffness matrix is the derivative of stress with respet to strain inrement at the end of the step. The reursive algorithm by Simo and Hughes (1998) is used. Reall that all alulations are done at the loal oordinate system (i.e., the prinipal pseudo strains were alulated in the loal oordinate system). Therefore the loal stiffness matrix and loal prinipal stress vetor are transformed bak into global stiffness matrix and global stress vetor using the diretion osines matrix defined previously. The final step to omplete the visoelasti alulations is the damage update. A small perturbation in the damage variable is indued and the variation in the pseudo work is omputed. The final value for the damage variable is omputed using the damage evolution law. The proess is also desribed in hapter 2 and fully doumented in the UMAT provided in Appendix A. Time-temperature superposition was onsidered the same way as provided in the model development. Only onstant temperature onditions are onsidered in the implementation. ABAQUS is apable of simulating variations of temperature over time. However, inorporating varying temperatures was beyond the sope of this researh. The visoplasti omponent is initiated by omputing the prinipal stress invariants from the urrent stress vetor. The new stress vetor is alulated from the strain inrement provided by ABAQUS and the stiffness matrix omputed in the visoelasti omponent. Visoplasti strains are alulated following a sequene of K - 88

91 steps to determine if visoplasti flow has ourred based on the position of the prinipal stress vetor in relation to the HiSS flow surfae. If flow has ourred, the point on the HiSS surfae that is normal to the applied stress point must be determined. This was aomplished by using a Newton-Raphson algorithm (Gibson, 2006). The NR algorithm provides the stress vetor at the HiSS surfae, the normal vetor to the surfae in the diretion of the applied stress (strain trajetory) and the relative distane from the applied stress to the hydrostati plane. After the alulation of visoplasti strains is ompleted, the global stress and strain vetors are updated and returned bak to ABAQUS. It is important to note that the final objetive of the UMAT is to provide the stress vetor and the stiffness matrix, both at the global oordinate system, at the end of the step inrement for eah node in the problem. oding and debugging the UMAT was a diffiult task. Any error assoiated with the FORTRAN ode had to be debugged outside ABAQUS. The interfae and interation between the UMAT and ABAQUS is not user friendly and no additional information about the soure of error is the ode is provided. In order to expedite this proess, a standalone ode of the UMAT was reated. The standalone version inluded steps to reate loading senarios that were otherwise provided diretly by ABAQUS. The entire ode was then debugged using the Intel Fortran ompiler. This proved to be vital in the debugging efforts Numerial Diffiulties and Simplifiations Some diffiulties were enountered during the development of the UMAT. The visoelasti omponent was adapted from a previous work done by Hinterhoelzl (1999). His UMAT was written to simulate solid propellant for rokets. The adaptation to asphalt onrete required redoing all the damage funtions and the proedure for updating the damage variables. A slightly different reursive algorithm was implemented to ompute pseudo strains in the hope that it would expedite the alulations. In the end, the visoelasti omponent was the most time onsuming part of the analysis. A full 3-D representation of one moving load yle required about 50 minutes of omputation time, of whih 40 minutes were exlusively dediated to the visoelasti omponent. The main objetive of this researh is the analysis of permanent deformation at high temperatures using the visoplasti omponent of the model. A simplified UMAT was therefore reated in whih the visoelasti omponent was replaed by simple isotropi elastiity. The elasti properties were determined from the dynami modulus master urve of the asphalt onrete for the desired temperature and loading rate in the pavement. The analysis of one moving load yle using the new UMAT required less than 10 minutes of omputation time. This approah was found to have little effet on the K - 89

92 predited permanent deformations, espeially at the intended simulations at high temperatures. Details and examples are provided in hapter 6. Another diffiulty faed during the development of the UMAT was the illonditioned solution of the HiSS funtion at the interept with the volumetri stress invariant axis, desribed in Eq. (75). The normal to the HiSS surfae at the volumetri axis is undefined. The surfae funtion interepts the volumetri axis in a non-normal angle, whih makes the normal undefined. Sine the Perzyna-HiSS model assumes assoiated flow, the diretion of the inremental visoplasti strain vetor is always normal to the yield surfae. Due to the undefined normal vetor of the funtion at the interept, there are stress state regions where the normal vetor annot be omputed. Figure 40 desribes this problem in more detail. It shows the HiSS surfae in the stress invariant spae. The area marked in the plot and identified by a dotted line is the region where normal vetor to the HiSS surfae are undefined. For illustration, 4 ritial stress paths indued by a moving wheel at different loations in the asphalt onrete layer are plotted: (a) point loated at the surfae and enter of the wheel path, (b) below the enter of the wheel path at the bottom of the layer, () surfae and far outside the rutting profile, and (d) below the enter of the wheel path and 50 mm from the surfae. None of the ritial stress paths were loated in the undefined normal vetor region. The laboratory tests for model alibration were all done in ompression, so none had stress paths in the zone where the problem of an undefined yield surfae ours. However, this problem did our when the UMAT was applied to pavement strutures and was the soure for numerial instabilities HiSS (a) () no valid normal (b) (d) J2D^(1/2), kpa I1 (kpa) K - 90

93 Figure 40. Example of HiSS surfae and the limit where normal vetors to the surfae an be alulated. At first, it was thought that only points in pure tension would be suseptible to this problem. A state of pure tension is not usual in pavements, but rather a ombination of tension and ompression. Investigations revealed that the instability problems developed at undisturbed elements near the boundaries of the problem domain when subjeted to small stress levels. It was unlear whether this problem developed during attempts to reah fore equilibrium during or at the end of a given load inrement. During all the heks and tests to identify the problem, the loation remained onfined to regions near the boundaries and at low stress levels. Loations ritial to the analysis (e.g., around the wheel path) were not affeted by this problem, as shown in Figure 40. Therefore the simple solution was to implement a hek during the visoplasti strain alulations to identify if the stress vetor was within the undefined normal zone in the model. When the stress point was found within the undefined normal zone, no visoplasti flow was assumed. Given the remote loations of the problem areas and the infrequeny of the instability, this assumption seemed appropriate to address the issue Model Verifiation The implementation of the UMAT was verified using tests performed previously by Gibson, This provided an independent verifiation without the extra osts and effort of running new tests. Gibson (2006) onduted several tests for alibrating the visoplasti model during its development. One was hosen for this verifiation. It onsisted of a reep and reovery test onduted in uniaxial ompression at fixed stress level and varying loading times. The test shemati is provided in Figure 41. Two test results were used, one unonfined and one onfined. Figure 42 and Figure 43 shows the test results and the preditions obtained during alibration (Gibson, 2006), and the preditions obtained when the tests were simulated using ABAQUS. The omparison between predited during alibration with predited using ABAQUS show very good agreement, indiating that the visoplasti model is suessfully implemented in ABAQUS. K - 91

94 Load Time Figure 41. Shematis of reep and reovery test at fixed stress level and varying loading time ompressive strains Visoplasti strains Radial strains Measured Predited during alibration Predited using ABAQUS Time (se) Figure 42. Fixed stress test with 1,500 kpa deviator stress, unonfined, tested at 35. K - 92

95 ompressive strains Visoplasti strain Radial strains Measured Predited during alibration Predited using ABAQUS Time (se) Figure 43. Fixed stress test with 1,500 kpa deviator stress and 250 kpa onfining stress, tested at Summary This hapter provided a summary of key aspets and steps required to simulate nonlinear strutural response using the finite element method. It also desribed the implementation of the visoelasti-visoplasti model into an ABAQUS user-defined material model subroutine or UMAT. Given the diffiulties in finding good doumentation on how to write and debug UMATs, the UMAT developed in this researh was extensively doumented in order provide some larity on the development proess that an serve as a template for future studies. Diffiulties during the model implementation were also disussed and the approahes for overoming these diffiulties were presented. Finally a simple, minimum ost, independent validation was presented based on previous results using the same model formulation but alibrated for a different mixture. The results suggest that the ode implemented in the UMAT yields results that are in good agreement with the model formulation used during its initial development. K - 93

96 hapter 6 Numerial Appliations 6.1. Introdution The implementation of the visoplasti onstitutive model into the ABAQUS finite element ode was an essential step towards fully mehanisti preditions of permanent deformation in asphalt onrete pavements. The model provides the means for diretly simulating the material behavior that leads to rutting. This hapter desribes a few appliations of the finite element model. The objetive is to use these appliations to provide insights into the rutting problem. The effet of simulating a moving wheel versus the more ommon bouning wheel approah is investigated and the importane of prinipal stress rotations and shear stress reversals indued by the moving wheel is evaluated. The effet of pavement type on the rutting profile and its development over time is also examined; this provides the basis for the development of a mehanisti approah to improve the empirial rutting model used in the urrent Mehanisti-Empirial Pavement Design Guide (MEPDG), a pavement design tool urrently reommended by the Amerian Assoiation of State Highway Transportation Offiials (AASHTO). And finally one field setion from Federal Highway Administration s Aelerated Loading Faility (ALF) is simulated with the finite element model and the results are ompared with field measured permanent deformation Simplified Finite Element Modeling Approah The ommerial finite element pakage ABAQUS was used to simulate pavement permanent deformations in this researh. The UMAT developed for the visoelastivisoplasti onstitutive model desribed in hapter 5 was used for all of the analyses presented here. Most of permanent deformation in pavements ours at high temperature when the binder has low visosity and is thus more fluid. In this ondition, the aggregate skeleton is responsible for arrying most of the traffi load. Plasti deformations are expeted as onsequene of air voids redution (volumetri deformations) and partile reorientation (shear deformations). The influene of the binder visosity is to ause a delay in the material s response to loading. The visoplasti behavior is the dominant ause of permanent deformations. At high temperatures, visoelasti effets are not signifiant and there is far less development of miroraks and damage than at low temperatures. An example of the dominant effet of visoplasti behavior in asphalt mixtures at high temperatures an be seen in Figure 44. This figure shows numerial simulations K - 94

97 of reep and reovery tests on an asphalt onrete mixture at two different temperatures, 19 representing a moderate temperature and 45 a high temperature. (45 was hosen beause it was the temperature during the aelerated load testing analyzed later in this hapter.) It an be seen learly from the figure that the magnitude of visoplasti strains and its ontribution to total strains is signifiantly magnified when the temperature is high. For the same load duration and magnitude, the visoplasti strain is nearly eight times higher at 45 than it is at 19. ve Total Strain ref temp (19 ) 45 vp ve vp Time Figure 44. Influene of temperature on visoplasti behavior of asphalt onrete in a simulated reep and reovery test. The omputational time required to predit the visoelasti-visoplasti response of one moving load yle on a pavement struture was about 50 minutes on a quad ore 2.6 GHz Intel ore 2 Extreme with 4 GB of RAM running 64-bit Windows XP. This may be reasonable for a few yles, but the exeution time beomes prohibitive if one intends to simulate hundreds or even thousands of yles. One way to redue omputational time is to shut off the visoelasti omponent of the model and replae it with a muh simpler onstitutive equation. As illustrated in Figure 44, the visoelasti ontinuum damage ontribution to the total strains is very small at elevated temperatures. The visoelasti ontinuum damage model, whih governs the pre-yielding response, was therefore replaed by a simple isotropi elasti model. This onstitutive model is termed the elasto-visoplasti model (EVP). The linear visoelasti dynami modulus master urve was still used to determine the material s instantaneous elasti modulus based on temperature and load frequeny as related to the tire speed. The EVP UMAT is presented in the Appendix B. K - 95

98 The reep and reovery test simulations using the omplete VEVP model (Figure 44) were repeated using the alternative EVP model. The results are shown in Figure 45. The final total residual strains are the same for both temperatures, whih is expeted sine the visoplasti omponent is the same in both onstitutive models. At the lower temperature, the models predited very different responses to the peak strain and the early portion of the reovery, refleting the influene of the visoelasti response. However, the differene between the two models is negligible at the higher temperature both before and after the peak strain. Based on these results, it was deided to use the alternative, simpler onstitutive model for the numerial simulations of pavement strutures at high temperatures. 45 Total Strain VEVP EVP 19 Time Figure 45. omparison between visoelasti-visoplasti and elasto-visoplasti simulation of reep and reovery at two different temperatures Influene of Shear Stress Reversals Permanent deformation aumulates with load yles over the pavement s life. Simulating thousands or millions of yles is a daunting task even with urrent omputational apabilities. The most realisti approah is a 3D simulation of a moving wheel. Sine suh analysis requires great omputational effort, simplifiations have often been used in the past to ut omputational ost and time. The most typial simplifiation is the assumption of bouning wheel instead of a moving wheel. The bouning wheel applies a yli loading with a period equivalent to the load duration at a ertain travel speed. Bouning wheel analyses an usually be performed assuming 2D axial symmetry, whih greatly streamlines the alulations. However, a bouning wheel does not indue shear stress reversal, whih is a key K - 96

99 mehanism in the development of plasti deformations, espeially for the distribution of permanent strains within the asphalt onrete layer. Limited field studies have found that moving traffi loading on pavement test setions produed higher permanent deformations than did plate loading with similar load magnitude and number of yles. When the wheel ompletes a full pass over a fixed referene point in the pavement struture, the diretion prinipal stresses rotate, ausing shear stress reversals (e.g., ompression to tension or vie-versa). Note that a omplete shear reversal requires a 180 or more rotation of the prinipal stresses. In this projet, the term rotation of prinipal stresses refers to a full rotation that auses shear stress reversals. This is onsistent with the terminology found in the literature where these two terms are used interhangeably to desribe the same phenomenon. This phenomenon has been studied more intensely in the unbound layers, where the effets of shear stress reversals an ause large plasti displaements. Investigations arried out the University of Nottingham s aelerated pavement testing faility suggests that pavements under moving loads develop twie as muh rutting as pavements under yli plate loading, as illustrated in Figure 46 (Brown et al., 1996, 1999). Moreover, bidiretional loading is more harmful than unidiretional loading due to the two-way shear reversals aused by bidiretional traffi (Brown et al., 1999). Similar tests at the Laboratoire entral des Ponts et haussees found that the permanent strains in the granular layers under moving wheel loading were approximately three times as large as those under yli plate loads (Hornyh et al., 2000). Kim and Tutumluer (2005) examined realisti pavement stresses indued on aggregate base layer by moving airraft loads and developed models to predit rutting in the unbound layer that onsidered the shear stress reversal. Equivalent field studies of paved pavement setions have not been found in the literature. Even though the stress rotations are more pronouned in the surfae layer that the underlying unbound granular base in a typial flexible pavement struture, there has been only limited evaluation of the effet of these rotations on the asphalt onrete, none at full sale experiment, nor modeling or numerial simulations. rokford (1993) performed some experimental evaluation of stress rotation effets in asphalt onrete under laboratory onditions. Based on hollow ylinder testing, rokford suggested that stress rotations ause about 2.5 times more plasti strain than speimens tested without the stress rotations. K - 97

100 Figure 46. Permanent deformation omparisonn between a moving wheel load and repeated vertial load (Brown et al., 1996). A simple exerise was performed to evaluate the ability of the Perzyna-Hiss visoplasti model to apture the different material response for stress states with and without shear stress reversal. The stress states indued by a moving wheel were obtained numerially using the 3D model desribed in hapter 5. All materials were modeled as linear elasti, as the inten was just too approximate the indued stress history aused by a moving load. The stress distributions over the loading yle weree omputed at the approximate loation of the maximum shear stress (i.e., about 50 mmm or 2 in below the pavement surfae at the edge off the wheel).. Figure 47 desribes the omputed normal and shear stresses at this loation as a funtion of time. The ase without shear stress reversal was modeled using the stress history for a sinusoidal quasi-stati load applied at the enter of the loading path. The period of the sinusoidal loading was set equal to the load duration simulated in the moving wheel analysis. Stress histories were omputed for the same approximate loation of the maximum shear stress. Figure 48 desribes the omputed normal and shear stresses at this loation as a funtionn of time. K - 98

101 Stress (Kpa) Loading Time (s) transverse longitudinal vertial longitudinal-vertial shear Figure 47. Stress distributions over time at the loation of maximum shear for one moving wheel pass. Stress (Kpa) Loading Time (s) transverse longitidunal vertial longitudinal-vertial shear Figure 48. Stress distributions over time at the loation of maximum shear for one bouning wheel load. The stress distributions shown in Figure 47 and Figure 48 were then applied to a single 3D element using the EVP onstitutive model desribed earlier. The stress histories were applied 1,000 times. K - 99

102 The omparison of permanent strains omputed in the two senarios is shown in Figure 49. The impat of shear stress reversal is lear. The loading with the shear stress reversal indued signifiantly higher permanent strains than the loading without it. It is important to note that the results represent the indued strains at the point of maximum shear stress in the pavement struture. In the ase of the bouning wheel load, the rate of permanent deformation redued and then leveled off after about 300 yles. This is aused by the movement of yield surfae towards the stress point (refer to Figure 14 for theoretial details) and is termed visoplasti saturation. A better representation of the yield surfae and the stress state paths at the loation of maximum shear is provided in Figure 50, in terms of shear and volumetri stress invariants, J 2D and I 1. In the ase of the moving wheel, the stress state path has a larger exursion above the flow surfae during most of the loading yle, as shown in Figure 50. As a result, more plasti strain ours as it takes longer for the yield surfae to reah the stress state path indued by the moving wheel than it does by the bouning wheel. Permanent strain With Stress Rotation "Without Stress Rotation yles Figure 49. omparison between permanent vertial strains indued by loading histories with and without prinipal stress rotations. K - 100

103 Hiss Bouning wheel Moving wheel 200 J2D^(1/2), kpa I1 (kpa) Figure 50. omparison between stress state paths indued by moving and bouning wheel at the loation of highest shear stress. This simple example showed that the visoplasti model is apable of apturing the effets of shear stress reversals on permanent deformations. The large differenes between the plasti strains in the moving and bouning wheel ases in this exerise are possibly magnified as a onsequene of the onstant loading histories over all loading yles. In the field, the state of stress is onstantly hanging as the materials deform, harden/soften, and develop loked-in stresses under the repeated loads. More realisti differenes are expeted when the full pavement setion is simulated. The 3D finite element analyses for both the moving and the bouning wheel loadings were repeated with the surfae HMA modeled using the EVP model. The elasti omponent of the HMA elasto-visoplasti model was omputed from the dynami modulus master urve for the appropriate temperature and loading frequeny. The unbound base layer and subgrade are modeled as linear elasti materials as before. The elasti material properties are summarized in Table 10. The visoplasti material properties for the asphalt onrete were desribed in Table 8 in hapter 4. K - 101

104 Table 10. Material properties used for studying the effets of the bouning versus the moving wheel. Layer Thikness (mm) Elasti Modulus (MPa) Poisson s Ratio HMA (45 o ) Base Subgrade Infinite The numerial simulations were arried out for 500 yles and the predited rutting at the enter of the wheel path is shown in Figure 51. The moving wheel produed 1.6 times more rutting than the bouning wheel. The rate at whih rutting inreases is also different. Visoplasti strain saturation is evident in the bouning wheel analysis, while rutting inreases ontinually throughout the moving wheel analysis, as expeted and illustrated in Figure 50. The rutting values predited in Figure 51 are small, ompared to expeted field rutting. The main reason is that the model doesn t predit as muh densifiation as ours in the field. The problem of densifiation is further detailed later in this hapter where field measurements are ompared with preditions obtained from the finite element analysis. 6.0E E-02 Rutting (mm) 4.0E E E E E+00 Moving wheel Bouning wheel yles Figure 51. omparison between moving and bouning wheel finite element simulations. The strain distribution within the surfae layer is also worth examining. Most of simplified mehanisti-empirial models annot realistially simulate the vertial strain distribution within the asphalt onrete surfae layer. Based on trenh studies at K - 102

105 MnRoad (MN) and Westrak (NV), it is expeted that the majority of permanent deformation will aumulate in the top 100 mm (4 inhes) of the asphalt onrete layer, with the peak permanent strains at about 50 mm (2 inhes). The permanent strain distributions versus depth were ompared for the moving and bouning wheel analyzes. Results from the first yle are plotted in Figure 52. The strain distribution from the moving wheel simulation is qualitatively similar to the distribution observed in field trenhes. Results from the MnRoad test trak are plotted in the inset for omparison. The majority of rutting measured in the field omes from the top two lifts, whih orresponds to about 100 mm (4 inhes). The majority of residual strains predited our at the top 100 mm, with a peak at about 30 mm (1.2 inhes). Although the strain distribution from the bouning wheel also reflets the expeted field distribution, the results are not in the same good agreement. Vertial residual strain -3.5E E E E E E E E % %HMA Rutting 30% 20% 10% 0% Depth (mm) Lift moving wheel bouning wheel Figure 52. omparison between strain distributions within the asphalt onrete surfae layer when moving and bouning wheel are onsidered. MnRoad rutting distribution from trenh studies is shown in the inset. The importane of indued stress reversals in the moving wheel simulation was learly observed in these omparative analyzes. The total amount of rutting observed at the surfae was about 1.6 times greater for the moving vs. bouning wheel loading. There was no visoplasti saturation in the moving wheel simulations as permanent deformation ontinued to aumulate with eah load pass, while a signifiant redution in the rate of permanent deformation with inreasing yles was observed K - 103

106 for the bouning wheel load. In addition, the distribution of permanent strains within the asphalt onrete layer was signifiantly different between the moving and bouning wheel analyzes. For qualitative omparison, results from MnRoad trenh studies were presented to illustrate the good qualitative agreement between distributions measured in the field and the predited distribution obtained from the moving wheel simulations Pratial Appliations of Finite Element Simulations One of the purposes of numerial simulation is to provide insight into phenomena that would otherwise be diffiult or ost-prohibitive to evaluate experimentally. The modeling effort desribed in this researh an be used to evaluate different pavement types, provide understanding of behavior under speial loading onditions (e.g., new tires, new loading gears), and ultimately support the development of improved design tehniques and pavement performane models by extrating simplified relations based on observations drawn from the omplex analyses. As an example, three distintly different pavement types were simulated and the predited rutting results are ompared. The magnitude of rutting omputed at the enter of the wheel path and the transverse permanent deformation profile are evaluated and disussed. The vertial permanent strain distributions are also examined and used to reate pavement type-speifi depth funtions for the MEPDG Predited Rutting omparison between Different Pavement Strutures An example omparative study is provided in this setion. Three pavement types were simulated using the finite element model desribed earlier. The objetive was to ompare quantitatively and qualitatively the predited asphalt rutting. Of partiular interest were the relative magnitudes of the maximum rutting and the shapes of the rutting transverse profiles. The first struture was a onventional flexible pavement onsisting of asphalt onrete as the surfae layer, granular rushed aggregate as the base, and the subgrade. The seond struture was also a flexible pavement, but a full depth asphalt pavement with only one thik layer of asphalt onrete diretly on top of the subgrade. The third struture was a omposite pavement onsisting of an asphalt onrete surfae layer, an underlying stiff Portland ement onrete slab, and the subgrade. The properties of the asphalt onrete, granular base and subgrade were as defined previously in Table 8 (visoplasti properties for the asphalt onrete) and Table 10 (elasti properties for all layers). The elasti properties of the stiff layer in the omposite pavement were defined as 30 GPa for the elasti modulus and 0.25 for the Poisson s ratio. Figure 53 shows the three strutures. Five hundred moving wheel load yles were simulated at an asphalt onrete temperature of 45 for all pavements. K - 104

107 The first differene notieable between the results was the evolution of rutting over number of yles. Figure 54 summarizes the asphalt rutting over load yles for all three pavement strutures. Rutting was omputedd as it is normally measured in the field (i.e., using a straightedge, whihh gives total rutting as the ombination of settlement and heave see Figure 1). The omposite pavement produedd the least rutting, about one-third of that predited for the onventionall flexible pavement at 500 yles. In addition, the harateristi primaryy stage produed signifiantly less rutting and ended sooner than in the flexible pavement ases.. The onventional flexiblee and full depth pavement strutures produed similar rutting preditions. The primary stage is visibly longer than in the omposite ase and the rate of rutting is higher. It would generally be expeted that full depth asphalt pavements exhibit less rutting than a omparable onventional flexible pavement. However, the analyses predited slightly greater rutting for the full depth struture than for the onventional flexible. This is beausee only rutting of the asphalt is onsidered in these analyses. The additional rutting ontributions from the granular base and subgrade layers in real flexible pavements would likely give larger total surfae rutting for the onventional flexible struture. Another explanation for the unexpetedly small differene in predited asphalt rutting between the two flexible pavement strutures is the assumption of onstant temperature throughout the asphalt layers. This does not reflet real senarios in whih daytime high temperatures derease with depth in the layer, whihh results in a stiffer and more permanent deformation resistant material at depth, partiularly for the full depth asphalt layer. (a) (b) () Figure 53. Pavement strutures: (a) onventional flexible,, (b) full depth asphalt onrete, and () omposite. K - 105

108 Asphalt onrete rutting (mm) onventional flexible Full depth omposite yles Figure 54. Asphalt rutting for different pavement strutures. An advantage of advaned finite element modeling over traditional mehanisti empirial models is the ability to predit the entire rutting transverse profile. Figure 55 ompares rutting transverse profiles of all three pavement strutures. The shape of the rutting transverse profile predited for eah of the strutures is different beause of the different distributions of stresses within the asphalt layer. Permanent deformation onsists of settlement underneath the tire load and heave immediately outside the loaded area. The rutting profile observed in the flexible pavement ase is typial of a struture with good quality base. In these ases, the stresses build up in the surfae layer mainly underneath the edge of the tire. This stress build up is the ause of visoplasti flow in the diretion from the enter of the load towards the edge of the tire, thus ausing substantial heave at the edges of the wheel paths. The full depth asphalt onrete pavement produes more rutting underneath the tire load than heave at the edge. It an also be noted that there are permanent deformations beyond one meter from the enter of the load. This is an indiation that the geometri boundaries may have been insuffiiently far away for the full depth analysis. This issue was not observed in any of the other simulations performed. The omposite struture produed the least permanent deformation of all pavements simulated, both in terms of settlement and heave. The P layer supports the majority of the load. The surfae layer remains in ompression at the enter of the load through its thikness, and there are more onfining effets from the horizontal stresses. The onsequene is the predition of less rutting, whih mathes K - 106

109 observations from field strutures. One interesting observation for the omposite pavement is the uneven distribution of rutting underneath the tire. The plot suggests that there is a gradient of deformation inreasing in the diretion towards the edge of the tire. This is likely due to the higher onfinement at the enter and inreased shear stresses at the edge of the tire. Asphalt onrete rutting (mm) Transverse distane from load enter (m) onventional flexible 0.12 Full Depth 0.16 omposite Figure 55. omparison of rutting transverse profiles for different pavement strutures. Different pavement types produe different permanent strain distributions with depth through the asphalt onrete layer. The omparison of the permanent strain distributions for the three different pavements is provided in Figure 56. Only the top 150 mm of the full depth pavement is shown in these omparisons. The peak permanent strain was alulated at a depth of 30 mm for the onventional flexible pavement, 15 mm for the omposite, and 45 mm for the full depth asphalt onrete pavement. The results agreed qualitatively with expetations from observed field data. For example, data olleted at post-mortem trenh studies indiated that the majority of asphalt onrete permanent deformation ourred in the upper 100 mm (4 in) of the surfae layer (MnRoad, 1998; Epps et al., 2002). The results presented here for all three pavement strutures suggest that the loation of the maximum deformation and the zone ontributing most substantially to the total rutting depend on the type of pavement struture. Intuitively it also depends on thikness of the layer, although this fator was not evaluated in this projet. This information an be useful for improving mehanisti-empirial rutting models, suh as those in the MEPDG, that rely on depth adjustment fators for the predited permanent strains. K - 107

110 Depth (mm) Vertial strain 0.0E E E E E Figure 56. omparison of permanent strain distributions with depth for different pavement strutures. Depth (mm) onventional flexible Full Depth omposite Vertial strain 0.0E E E E Analysis of the MEPDG Depth Fator for Rutting Preditions One of the immediate appliations of advaned modeling is to support the improvement of simplified predition models. A prime example is the rutting model urrently used in the MEPDG. A review of the MEPDG rutting model was provided in hapter 2. The main equation and the depth fator adjustment are reprodued here for onveniene. The MEPDG model predits vertial unreoverable strain at the enter of the load using the following empirial funtion: p r T N (80) in whih p is the plasti strain, r is the elasti strain, T is the temperature, N is the number of load appliations. This funtion is based on the reoverable vertial strain omputed at peak load using linear elastiity models. The magnitude of predited permanent deformations is a funtion of number of load appliations and the temperature of the asphalt onrete. Rutting is the permanent deformation after removal of the load. As desribed in hapter 2, previous studies have suggested that the material at the bottom of the asphalt onrete layer yields under triaxial onfined ompression onditions and does not develop horizontal tensions as predited by elasti analyses. Instead, the plasti flow auses the horizontal stresses remain ompressive at all times. When the load is K - 108

111 removed, these ompressive stresses are loked in the struture reating a multidimensional onfinement whih indues residual expansive vertial strains at the bottom of the layer. The permanent deformation resulting from the residual ompressive strain distribution is onentrated in the upper portion of the layer, as onfirmed by field trenh studies. Sine the plasti strains predited in the MEPDG approah using Eq. (80) are proportional to the mehanistially determined elasti vertial strain, the majority of rutting is predited at the bottom of the layer, ontrary to field experiene. Therefore a depth orretion funtion was implemented in the MEPDG to adjust the omputed plasti strain as desribed in Eq. (81). p r T N 3 ( 1 2 depth ) depth h A h (81) 1 A h A h in whih σ3 is the depth orretion funtion, depth is depth to the strain alulation loation, h A is the thikness of the asphalt layer, and the other variables are as defined previously. The depth orretion funtion assumes that the mehanisms and distributions of permanent strains are similar for all asphalt onrete layers, with no differentiation by pavement type. The EVP finite element analyses for the three pavement types (onventional flexible, full depth asphalt, and omposite) provide the atual distribution of residual plasti strains vs. depth through the asphalt layer. The omparison between the residual strain distribution predited by the EVP finite element analysis and the MEPDG omputed residual strain distribution is of interest. For better visualization, the vertial strain distributions were normalized. The intent is to ompare the shape of the distribution not the magnitude of the strains, therefore a unique multiplier was determined as suh that the resulting rutting from the integration of the residual strain distribution was the same. The EVP finite element residual strains were used as referene. The elasti strain distribution was normalized using the same multiplier of the MEPDG omputed residual strain distribution. Figure 57 illustrates the normalized vertial strain distributions in the asphalt onrete layer of the onventional flexible pavement struture. It shows the elasti strain omputed at peak load and the two residual strain distributions under the enter of the tire: (1) one named residual omputed using the EVP finite element model, and the other named MEDPG omputed residual. The MEPDG residual strains were alulated using the urrent depth funtion. It is lear from the figure that the A K - 109

112 MEPDG omputed residual strains do not math the atual residual strains omputed in the finite element analyses. A new depth funtion was developed to better fit the residual response predited by the finite element model. The funtional form was kept onsistent with the MEPDG approah, but the oeffiients were adjusted to provide a best fit to the EVP plasti strain values. The result is shown in Figure 57 and in Eq. (82). Normalized vertial strain -4.0E E E E E E-04 0 Depth (mm) Residual Elasti MEPDG omputed residual New omputed residual Figure 57. MEPDG depth funtion analysis for the onventional flexible pavement. ( depth) depth h h A A (82) h h A A in whih the parameters are as desribed previously. The above approah an be extended to the full depth asphalt onrete and omposite pavements. Figure 58 shows vertial strains distributions in the first 150 mm of the asphalt onrete layer of the full depth struture. The shape of the MEPDGdetermined plasti strain distributions is very different from the residual strain omputed from the EVP analysis. However, when a new depth orretion funtion is alibrated from the EVP strains, the results are in muh better agreement. The new depth funtion for this ase is provided in Eq. (83). K - 110

113 Normalized vertial strain -4.0E E E E E-05 0 Depth (mm) Residual Elasti MEPDG omputed residual New omputed residual Figure 58. alibrated depth funtion results for the full depth asphalt onrete pavement. ( depth) depth h h A A (83) h h A A in whih the parameters are as desribed previously. Figure 59 shows vertial strains distributions in the asphalt onrete layer of the omposite pavement struture. As similar to previous results, the MEPDG adjusted plasti strain does not agree with the residual strain omputed from the finite element analysis. Again, the depth orretion funtion was realibrated to provide better agreement between the two distributions. The new depth funtion for this ase is provided in Eq. (84). K - 111

114 Normalized vertial strain -2.0E E E E E-05 0 Depth (mm) Residual Elasti MEPDG omputed residual New omputed residual Figure 59. alibrated depth funtion results for the omposite pavement ( depth) depth h h A A (84) h h A A in whih the parameters are as desribed previously. The depth funtion equation an be generalized as shown in (85) and the various oeffiients for eah pavement type an be ompared in Table 11. ( depth) a depth a h a h a (85) A 2 A 3 b h b h b A 2 A 3 K - 112

115 Table 11. Struture-based depth funtion oeffiients. Pavement Type a a 1 a 2 a 3 b 1 b 2 b 3 Default MEPDG onventional Flexible Full Depth A omposite onlusions This setion desribed two immediate appliations of the mehanisti model for the permanent deformations in asphalt onrete layers. The first showed omparative analyses between three distint pavement types with asphalt onrete surfae layers. The results suggested that the finite element appliation is apable of identifying fundamental differenes on how rutting develops in different pavement strutures. There were differenes in the evolution of permanent deformation over load yles, different transverse profile patterns, and different distributions of rutting within the layer. The results agree qualitatively with expeted field behavior as determined by trenh data from MnRoad and Westrak. The seond appliation applied the finite element modeling to improve an existing mehanisti-empirial model for prediting asphalt layer rutting. urrent mehanisti-empirial models, suh as the MEPDG, rely on regression models to transform mehanisti responses into distresses. Disrepanies between elasti preditions of response and field expeted permanent strain distribution are resolved by using an empirial depth orretion funtion. Sine the EVP finite element analyses had shown that the shape and magnitude of the permanent strain distribution varies signifiantly with pavement type, pavement-speifi depth funtions are thus neessary (although not inluded in the MEPDG). The finite element model developed was suessfully used to derive pavementspeifi MEPDG depth orretion funtions for three different pavement struture types. The permanent strain distributions after one yle were used to realibrate the MEPDG depth funtion. The new depth funtions have the same mathematial formulation as the urrent MEDPG s depth funtion to failitate implementation; only the oeffiients in the funtion are realibrated. The new depth funtions produe plasti strains preditions that are in muh better agreement with mehanistially omputed residual strains. This approah an be used in future enhanements of the MEPDG. K - 113

116 6.5. Field rutting preditions Desription of FHWA ALF Federal Highway Administration (FHWA) onstruted 12 full-sale lanes of asphalt onrete pavements in 2002 at its Aelerated Loading Faility (ALF) at the Turner- Fairbank Highway Researh enter in MLean, VA. The experiment, the seond of its nature in this faility, was entitled Full-Sale Aelerated Performane Testing for Superpavee and Strutural Validation (Gibson, 2011). The objetive of the study was to validate and refinee hanges being proposed to the Superpave asphalt binder speifiations. The layout of the experiment is presented in Figure 60. Eah lane had a width of 4.0 m (13 ft) and a length of 50 m (165 ft). Eah lanee had four test sites used in various studies. All lanes onsisted of an asphalt onretee surfae layer on top of an unbound, dense-graded, rushed aggregate base (AB) over a uniformly prepared, AASHTO A-4 subgrade soil. The total thiknesss of the HMA and AB is 660 mm (26 in). Lanes 1 through 7 were onstruted with an HMA layer thikness of 1000 mm (4.0 in) and were used to evaluatee raking, while lanes 8 through 12 have a thikness of 150 mm (6.0 in) and were tested for permanent deformation. Eah lane was onstruted with a different binder; these are listed in Figure 60. Figure 60. Layout of the 12 as-built pavement lanes (Qi et al., 2004). K - 114

117 The results provided by FHWA for this researh ame from Lane 11. The binder in this lane was modified with Styrene-Butadiene-Styrene with Linear Grafting (SBSwas 12.5 mmm dense-grade following the Superpavee gradation speifiations. The design binder LG). The Superpave binder grade was PG The mixture gradation ontent was 5.3% at an air voids of 4.2%. The base layer wass rushed aggregate (AB) with a 25 mm (1 in) nominal maximum aggregate size. Additional details of binder, mixture harateristis and base material an be found at Qi et al. (2004). The tire used was a super single wide base tire 425/64R22.5. It is known that this type of tire indues greater damage than onventional dual tires, hene its advantage in aelerated load testing (Gibson, 2011). The tire pressure was 689 kpa (100 psi) under a total load of 44kN (10 kip). There was noo traffi wander for the rutting portion of the experiment. Rutting was measured at the enter of the wheel path using Layer Deformation Measurement Assembliess (LDMA), whih simplyy measure the hange in thikness of the asphalt layer as a onsequene of aumulatedd permanentt deformation. The LDMA setup is shown in Figure 61 the rut depth measurements at the ALF did not inlude the side heaves. Seven LDMAs were installed per test site. Different temperatures were used for speifi groups of lanes and test sites depending on the intended distress development. For the rutting experiment, the temperatures were 45 and 64. On lane 11, the target of this researh, the temperature was kept onstant at 45. The temperature was ontrolledd by measurements obtained from thermoouples embeddedd in the pavement struture. Radiant heaters were mounted along the length of the ALF to provide a soure of heat. Figure 61. Layer deformation measurement assembly used to measuree rut depth (Gibson, 2011) Lane 11 reeived 300,0000 unidiretional load passes. Figure 62 shows the measured rutting versuss number of load passes. It is important to observe that by the time the first measurement was taken at 500 yles, the aumulated permanent deformation was already at 2.5 mm (0.1 in). This aumulatio on representss 40% of the total permanent deformation observed at the end of thee experiment. K - 115