MECHANICS OF ASPHALT CONCRETE: ANALYTICAL AND COMPUTATIONAL STUDIES

Transcription

1 MECHANICS OF ASPHALT CONCRETE: ANALYTICAL AND COMPUTATIONAL STUDIES DINESH PANNEERSELVAM Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Dissertation Advisor: Prof. Vassilis P. Panoskaltsis Department of Civil Engineering CASE WESTERN RESERVE UNIVERSITY May, 2005

2 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of candidate for the Ph.D. degree *. (signed) (chair of the committee) (date) *We also certify that written approval has been obtained for any proprietary material contained therein.

3 Dedicated to my grandparents and my parents for their love and support iii

4 TABLE OF CONTENTS TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES ACKNOWLEDGEMENTS ABSTRACT iv vii xi xii xiii CHAPTER ONE: INTRODUCTION. Pavements.2 Asphalt as a pavement material 2.3 Design Factors 4.4 Failure Criteria 4.4. Rutting Fatigue Cracking Thermal Cracking 5.5 Laboratory Experiments 6.5. Repetitive Simple Shear Test at Constant Height Triaxial Load Test 9.6 Modeling of Asphalt Concrete.7 Overview 6 CHAPTER TWO: ASPHALT CONCRETE CHARACTERIZATION Introduction Characteristics of Asphalt Concrete Evidence of Temperature Dependence in the Behavior of the Mix Evidence of Plasticity in Asphalt Concrete Evidence of Rate Dependent Permanent Deformations Evidence of Dilation in the Mix 30 CHAPTER THREE: HYPERLEASTICITY Introduction Hyperelastic Models Hyperelastic Model for Asphalt Concrete Determination of Material Parameters Shear Test Volumetric Test Uniaxial Test 50 iv

5 3.5 Parameter Estimation Stability Analysis: Positive Definiteness of Tangent Stiffness Stability Analysis for a Uniaxial Test Stability Analysis for a Volumetric Test Stability Analysis for a Shear Test Stability Analysis for the Second set of Experiments Closure 6 CHAPTER FOUR: VISCOELASTICITY 7 4. Introduction The Modified Kuhn Model Generalized Kuhn Model Derivation of Loss Tangent Need for a Temperature Dependent Model Time Temperature Superposition Principle The Generalized Kuhn Model with Temperature 86 CHAPTER FIVE: VISCOPLASTICITY Introduction ` Viscoplastic Models Viscoplastic Models Based on Loading Surface Viscoplastic Model for Asphalt Concrete Loading Surface Flow Rule Hardening Law Summary of Equations Anisotropy Damage 27 CHAPTER SIX: NUMERICAL IMPLEMENTATION Introduction Numerical Implementation of the Complete Model Basic Algorithmic Setup: Strain Driven Problem Predictor Phase Corrector Phase Global Solution Triaxial Experiments Parameter Estimation Numerical Implementation of the combined hyperelastic viscoelastic viscoplastic model 58 v

6 CHAPTER SEVEN: RESULTS FROM FINITE ELEMENT ANALYSIS Introduction Hyperelastic Model Implementation in ABAQUS Finite Element Model of RSST-CH in ABAQUS Hyperelastic-viscoplastic-damage model implementation in ABAQUS Finite Element Modeling of Triaxial Experiments Finite Element Modeling of Repeated Simple Shear Test at Constant Height Finite Element Modeling of Pavement 8 CHAPTER EIGHT: CONCLUSIONS AND FUTURE WORK : Summary : Future Work 206 BIBLIOGRAPHY 208 vi

7 LIST OF FIGURES Figure.a: Severe Rutting accompanied by upheavals to the side 8 Figure.b: Rut Depth Measurement 8 Figure.2: Correlation between laboratory test results and field test data 9 Figure.3: Permanent Shear Strain versus Number of cycles on a log-log scale 20 Figure.4: Evolution of axial permanent shear strain with number of cycles. 2 Figure.5: Regression constants a and b when plotted on a log-log scale. 22 Figure.6: Axial permanent strain versus number of cycles on a log-log scale. 23 Figure 2.: Rut Depth vs. Repetitions of loading at 40 o C and 50 o C 3 Figure 2.2: Permanent Shear Strain vs. Number of cycles of Shear loading on asphalt pavement specimens. 32 Figure 2.3: Evolution of permanent shear strain in a sequence of RSST-CH test at different temperatures. 33 Figure 2.4: Evolution of permanent shear strain in a sequence of RSST-CH tests at different stress levels. 34 Figure 2.5: Evolution of permanent shear strain in RSST-CH experiment at different loading rates. 35 Figure 2.6: Development of normal strain in RSST-CH experiment. 36 Figure 3.: Shear Stress vs. Shear Strain for RSSTCH: Experimental data and Model fit. 63 Figure 3.2: Volumetric Stress vs. Volumetric Strain for RSSTCH: Experimental data and Model fit. 64 Figure 3.3: Axial Stress vs. Axial Strain for RSSTCH: Experimental data and Model fit. 65 Figure 3.4: Axial Stress vs. Shear Strain for RSSTCH: Experimental data and Model fit. 66 vii

8 Figure 3.5: Strain Energy vs. Axial Strain. 67 Figure 3.6: Shear Stress vs. Shear Strain for RSSTCH: Experimental data and Model fit. 68 Figure 3.7: Volumetric Stress vs. Volumetric Strain for RSSTCH: Experimental data and Model fit. 69 Figure 3.8: Axial Stress vs. Axial Strain for RSSTCH: Experimental data and Model fit. 70 Figure 4.: Rheological Representation of Modified Kuhn Model 95 Figure 4.2: Rheological Representation of Generalized Kuhn Model. 96 Figure 4.3: Experimental results and Generalized Kuhn model prediction for Asphalt Concrete. 97 Figure 4.4: Generalized Kuhn model fit for * G vs. frequency. 98 Figure 4.5: Rut Depth vs. Repetitions of loading at 40 o C and 50 o C 99 Figure 4.6: Permanent Shear Strain vs. Number of cycles of Shear loading on asphalt pavement specimens. 00 Figure 4.7: Creep compliance at different temperatures. 0 Figure 4.8: Shift factor versus temperature. 02 Figure 4.9: Loss tangent vs. Frequency: Model fit to Experimental Data 03 Figure 4.0: Model fit for Phase Angle vs. Temperature at 7.8Hz 00 Figure 5.: The quasistatic yield surface and the dynamic loading surface 32 Figure 5.2: Representation of Overstress in deviatoric stress space. 33 Figure 5.3: Loading Unloading cycle for a ratcheting behavior 34 Figure 5.4: Loading Unloading in stress space. 35 Figure 5.5: Evolution of permanent shear strain in RSST-CH for different asphalt concrete mixes. 36 Figure 5.6: Ratcheting behavior in granular materials 37 viii

9 Figure 5.7: Particle Orientation on Vertical Section, vector magnitude 38 Figure 6.: Axial Stress vs. Axial Viscoplastic Strain. Model fit to triaxial experimental data at 0-psi confinement pressure 68 Figure 6.2: Axial Stress vs. Axial Viscoplastic Strain. Model fit to triaxial experimental data at 5-psi confinement pressure 69 Figure 6.3: Axial Stress vs. Axial Viscoplastic Strain. Model fit to triaxial experimental data at 30-psi confinement pressure 70 Figure 6.4: Axial Stress vs. Axial Viscoplastic Strain. Model fit to triaxial experimental data at 30-psi confinement pressure 7 Figure 7.: Axial Stress vs. Axial Strain for RSST-CH experiments. Single element test in ABAQUS. 83 Figure 7.2: Shear Stress vs. Shear Strain for RSST-CH experiments. Single element test in ABAQUS. 84 Figure 7.3: Finite Element Model of the cylindrical specimen used for RSST-CH experiments. 85 Figure 7.4: Reaction force vs. Displacement in the axial direction. 86 Figure 7.5: Axial Stress vs. Axial Strain. Comparison of Finite Element model with Experimental data. 87 Figure 7.6: Reaction Force vs. Displacement in -3 direction. 88 Figure 7.7: Shear Stress vs. Shear Strain. Comparison of Finite Element model with Experimental data. 89 Figure 7.8: Deformation of cylindrical specimen. 90 Figure 7.9: Shear stress contour on the deformed cylindrical specimen. 9 Figure 7.0: Evolution of normal stress with shear strain. 92 Figure 7.: Finite element mesh of triaxial specimen 93 Figure 7.2: Deformed shape of the cylinder 94 Figure 7.3: Axial Stress distribution in the cylinder 95 Figure 7.4: Axial stress vs. axial viscoplastic strain. Comparison of finite element ix

10 analysis results to triaxial experiments at 0psi confining pressure. 96 Figure 7.5: Axial stress vs. axial viscoplastic strain. Comparison of finite element analysis results to triaxial experiments at 5psi confining pressure. 97 Figure 7.6: Finite element mesh of RSST-CH specimen 98 Figure 7.7: Evolution of permanent shear strain with no. of cycles. 99 Figure 7.8: Schematic of the pavement model 200 Figure 7.9: Finite Element model of the pavement 20 Figure 7.20: Deformed shape of the pavement 202 Figure 7.2: Stress distribution in the deformed pavement 203 x

11 LIST OF TABLES Table 2.: Rut Depth measured at various slopes along a hilly highway in Portugal. 29 Table 2.2: Rut Depth measured at various distances from the intersection 29 Table 3.: List of Parameters. 52 Table 3.2: List of parameters for second order hyperelastic model fit. 59 Table 4.: List of parameters for * G vs. frequency. 82 Table 4.2. List of parameters for various temperatures fit. 89 Table 4.3: List of parameters for Phase Angle vs. Temperature fit. 93 Table 6.: Elastic-Viscoplastic operator split 44 Table 6.2: Predictor Corrector Algorithm for Hyperelastic-Viscoplastic Model 54 Table 6.3: Values of parameters for 0-psi confinement pressure data. 65 Table 6.4: Values of parameters for 5-psi confinement pressure data. 66 Table 6.5: Values of parameters for 30-psi confinement pressure data. 64 Table 7.: Parameter values for the hyperelastic model 78 xi

12 ACKNOWLEDGEMENTS I am very grateful for having an exceptional advisor, Prof. Vassilis Panoskaltsis and wish to thank him for his guidance, generous time and commitment. His encouragement and continuous support through out my doctoral studies is greatly appreciated. He continually stimulated my analytical thinking and greatly assisted me with scientific writing. I would also like to thank Prof. Mullen, Prof. Huckelbridge and Prof. Calvetti for serving on my doctoral committee. I also wish to thank the department and Chairman Prof. Mullen for the Case Prime Fellowship and their continuous support for my education and also for the opportunities extended to me to develop my professional career. I am extremely grateful for the assistance I received from Ms. Bernie Strong, Ms. Kathleen Ballou and Ms. Sheila Campbell during my graduate studies. I extend many thanks to all my colleagues and friends, especially Dr. Saurabh Bahuguna and Guruprasad Ramanathan for their help and support. Finally, I would like to thank my family and relatives for their constant encouragement; my father is my inspiration, while my mother is a constant source of support through the years. I am extremely grateful to my brothers for their encouragement and enthusiasm. xii

13 Mechanics of Asphalt Concrete: Analytical and Computational Studies Abstract by DINESH PANNEERSELVAM Permanent deformation under repeated loading (rutting) is the most prominent distress mechanism in pavements. In properly compacted pavements shear flow is considered to be the primary rutting mechanism. Also the mix exhibits volumetric/deviatoric coupling behavior which is observed when the mix dilates under shear loading. A new multi-dimensional hyperelastic-viscoelastic-viscoplastic-damage model is developed to describe the permanent deformations and coupling behavior of asphalt concrete. The elastic component of the asphalt concrete response is modeled by a second order hyperelastic model. Asphalt concrete mix exhibits volumetric/deviatoric coupling behavior even at very small strain values and hence an elastic model is used to capture this phenomenon. The viscoelastic model developed in this study is based on a new viscoelastic model which has been recently developed by Panoskaltsis and co-workers (Panoskaltsis, V.P., et al. The Generalized Kuhn model of Viscoelasticity, submitted for Publication). In the work presented here, the Generalized Kuhn model is applied for the description and prediction of asphalt behavior. The response of the material for frequency sweep test is captured and the model predicts the loss tangent for asphalt concrete very well. xiii

14 Furthermore, the model has been appropriately modified in order that the important asphalt temperature effects are taken into account through time temperature superposition principle. The viscoplastic component captures the rate dependent behavior and is based on Perzyna s theory of viscoplasticity. This theory is used to model the ratcheting behavior exhibited by the mix and also the evolution of the permanent strain with number of loading cycles is captured. The loading surface used in this model is based on Vermeer loading surface, which was used successfully for soils. A nonassociative flow rule for the plastic strains as well as an evolution equation for the hardening parameter is given. This model takes into account the anisotropy in the material and also isotropic damage based on effective stress theories is included in the model. Numerical implementation and algorithmic aspects of the multi- dimensional hyperelastic-viscoplastic-damage model are presented. A robust integration algorithm for the nonlinear differential equations is described; also the algorithmic (consistent) tangent moduli are derived. The model is implemented into the finite element environment ABAQUS to study boundary value problems. Triaxial and Repeated simple shear tests at constant height (RSST-CH) are studied as boundary value problems and results are compared to experiments. Finally, a section of the pavement is studied as a boundary value problem to asses the capability of the model to predict rutting in the pavement. xiv

15 CHAPTER ONE INTRODUCTION. Pavements Pavements are broadly divided into two categories: Flexible pavements Rigid pavements. Flexible pavements are pavement systems in which total pavement structure deflects, or flexes, under loading. A flexible pavement structure is typically composed of several layers of material. Each layer receives the loads from the above layer, spreads them out, and then passes on these loads to the next layer below. Asphalt concrete is predominantly used as a construction material for flexible pavements. Rigid pavements are pavement systems in which the pavement structure deflects very little under loading due to the high modulus of elasticity of their surface course. A rigid pavement structure is typically composed of a portland cement concrete (PCC) surface course built on top of either () the subgrade or (2) an underlying base course. Because of its relative rigidity, the pavement structure distributes loads over a wide area with only one, or at most two, structural layers. Rigid Pavements are constructed using concrete. This study focuses on flexible pavements and asphalt concrete as a construction material for flexible pavements.

16 .2 Asphalt as a pavement material Asphalt has a long history of usage as a pavement material. Ever since its discovery, it has been one of the most important materials for civilizations. It has been traditionally used as a pavement material and also as a waterproofing material in ship building industries. The first recorded use of asphalt as a road building material was in Babylon around 625 B.C., in the reign of King Naboppolassar. Hugh Gillespie, in his report, In A Century of Progress: The History of Hot Mix Asphalt, published by National Asphalt Pavement Association in 992, quotes the following an inscription on a brick records the paving of Procession Street in Babylon, which led from his palace to the north wall of the city, with asphalt and burned brick.. Through the centuries, the use of asphalt as a pavement material has been on the rise and it is the popular choice as a pavement material worldwide. Today, asphalt covers more than 94% of the paved roads in the United States. In addition to roads, it is used as a surface material for driveways, parking lots, racetracks, airport runways among other uses. Asphalt is a tar like substance which is obtained as a residual during fractional distillation process of crude natural oil. It is a highly viscous material and acts as a fluid at high temperatures. This characteristic property is used to produce asphalt concrete, which is used as a pavement material and is frequently referred to as Hot Mix Asphalt in pavement literature and industry. Asphalt concrete is a mixture of gravel and sand bonded together by bitumen. The word asphalt comes from the Greek word asphaltos, meaning secure. In asphalt mixture, the aggregates constitute a total ninety-three (93) to ninetyseven (97) percent by weight of the total mixture and are mixed with three (3) to seven 2

17 (7) percent asphalt. It is manufactured in a central mixing plant where the asphalt and aggregates are heated to a temperature of approximately 300 degrees Fahrenheit, properly proportioned and mixed. At this temperature, bitumen acts as a fluid and aids in preparing a good mixture. An important aspect of this viscous property of asphalt concrete is that it is relatively very flexible. It deforms without forming cracks and is compatible with subgrade deformations. This makes it a highly preferred pavement material. Developing mix designs which meet the pavement design requirements has been an area of active research for the last few decades. To put it in one line, the objective of the research was to develop an economic blend of aggregates and bitumen to meet the design requirements. Historical mix design, methods still traditionally used by many pavement organizations are Marshal Test Hveem Test New mix design methods like superpave mix are replacing these traditional performance test methods. One of the drawbacks of these mix design methods is that they were empirical and lacked mechanics basis. Hence, they do not provide a real time assessment of the pavement performance. To overcome this, design mixes based on mechanics were preferred. Over the last few years, the focus of the research in this area has shifted to finding more mechanistic-based design mixes. A mechanistic design method is based on the mechanics of materials that relates an input, such as wheel load, to an output or pavement response, such as stress or strain. The response values are used to predict distress based on laboratory test and field performance data. 3

18 This thesis deals with mechanistic based mix design for Hot Mix Asphalt..3 Design Factors The design factors can be broadly divided into four categories: Traffic and Loading Environment Materials Failure Criteria. In the traffic and loading category, axle loads, number of repetitions, tire contact areas and vehicle speeds are included. Temperature and precipitation effects are classified into environmental factor. One of the important design factors to be considered is the failure criteria. This includes rutting, fatigue cracking and thermal cracking..4 Failure Criteria.4. Rutting Rutting is a failure, characteristic of flexible pavements. Rutting is defined in ASTM Standard E 867 as a contiguous longitudinal depression deviating from a surface plane defined by transverse cross slope and longitudinal profile. The longitudinal depressions (sometimes referred to as ruts ) are accompanied by upheavals to the side. The longitudinal depressions reduce drainage capacity of pavements and result in accumulation of water in the wheel paths. Accumulation of water accelerates damage of pavements due to moisture and also results in hydroplaning causing traffic accidents. In warmer climates, rutting causes bleeding, a phenomenon in which the asphalt binder 4

19 (bitumen) rises to the surface. As a result, the surface becomes very smooth resulting in loss of traction for vehicle tires. Rutting also increases chances of fatigue failure of pavements. The thickness of the pavement under the wheel path reduces considerably and the pavement gets susceptible to fatigue cracking. Fig.. shows the failure in pavements due to rutting..4.2 Fatigue Cracking The fatigue cracking of flexible pavements is based on the horizontal tensile strain at the bottom of hot mix asphalt (HMA). Cracks vent moisture movement into the pavement and result in damage of pavements. The moisture trapped in the cracks expand due to ice formation during cold temperatures and this results in further damage to the pavements. This failure criterion relates the allowable number of load repetitions to the tensile strain of HMA..4.3 Thermal Cracking This type of distress includes both low-temperature cracking and thermal fatigue cracking. Low-temperature cracking is usually associated with flexible pavements in colder regions. Thermal fatigue cracking occurs in much milder regions if excessively hard asphalt is used or the asphalt becomes hardened due to aging. In an industry wide survey conducted by researchers at the University of Maryland, College Park, rutting was rated the most significant distress type regarding damage in pavements (Witczak, 998, Christensen et. al. 2000, Witczak et al. 2002). A 5

20 thorough understanding of the rutting phenomenon is required in order to improve pavement design and performance. Rutting in pavements accumulates as the number of traffic loads on the pavements increases. In flexible pavements, both the top asphalt concrete layer and the subgrade contribute to rutting. In the asphalt concrete layer, the rutting is caused by a combination of densification (compaction) and shear flow. The initial rut is caused by densification of the pavement under the path of the wheel. However, the subsequent rut is a result of shear flow of the mix. This can be seen in the upheaval of the pavement in between the path of the wheels; see Fig... In properly compacted pavements, however, it has been found that shear flow in asphalt concrete layer is the primary rutting mechanism. See e.g. Eisenmann and Hilmer (987). A great need for a constitutive model for asphalt concrete exists in order to predict the rutting caused by this layer of the pavement. In this study, the focus is on the rutting in the asphalt concrete layer..5 Laboratory Experiments used in evaluating Asphalt Concrete mixes with respect to Rutting The evaluation of asphalt concrete mixes for their tendency to cause rutting has been an active area of (experimental mainly) research for several years and a number of experimental tests have been proposed for this purpose; all these tests have their advantages and disadvantages. (Two major programs dealt with this: a Strategic Highway Research Program (SHRP) - end of 80s - beginning of 90s and an ongoing National Cooperative Highway Research Program (NCHRP) Project 9-9). The focus of the NCHRP project Simple Performance Test for Super Pave Mix Design, Report 465 was 6

21 to develop a Simple Performance Test (SPT) of Hot Mix Asphalt (HMA) that would aid in design of Superpave Mix and would enhance the performance of the pavements. The Simple Performance Test (SPT) would measure a fundamental engineering property that can be linked back to the advanced material characterization methods needed for detailed distress prediction models. This measurement would enable the use of a simple performance model in the development of criteria for HMA mixture design. The definition for the Simple Performance Test (SPT) as given in the report is A test method(s) that accurately and reliably measures a mixture response characteristic or performance that is highly correlated to the occurrence of pavement distress (e.g, cracking and rutting) over a diverse range of traffic and climatic conditions. A utility analysis was initially used to rank each of the test methods that were considered candidates for use as a Simple Performance Test. Four testing methods listed below Static Creep Parameters versus rutting Triaxial Repeated Load Permanent Deformation Simple Shear Repeated Load Permanent Deformation Triaxial Shear Strength were aimed at evaluating their effectiveness with respect to rutting. The laboratory results of the test candidates selected were actually evaluated against field test data. The sites selected for the field tests are MnRoad, Accelerated Loading Facility-Turner Fairbanks, and Wes Track. The correlation between the laboratory test results and the field test data 7

22 for the various experiments is given in Fig..2. This correlation study reports that the socalled Repetitive Simple Shear Test at Constant Height (RSST-CH) and the Triaxial Repeated Load Test showed the best correlation to the measured (in the field) rut depth. See NCHRP Report No. 465, pages 43 and Repetitive Simple Shear Test at Constant Height (RSST-CH). RSST-CH was developed by Monismith and his co-workers at U.C. Berkeley (see e.g. Monismith 994, Sousa et al. 993) and has been very effective in grading mixes with respect to their propensity to rut. This test is widely accepted in the asphalt community and has become recognized as perhaps the most reliable laboratory test for characterizing rut resistance (quote from the report by Christensen et al. 2000, pg. 59). This recently released report, by the Penn State University and The Pennsylvania Transportation Institute, deals extensively with the analysis of many experimental data in order to evaluate the triaxial test for predicting the rut resistance of asphalt concrete and states the wide use and acceptance of RSST-CH. (See in particular the pages 3, 59, 60 and 6.) The RSST-CH test is performed on cylindrical specimens with a height to diameter ratio of about :3. The height is kept constant and a haversine shear stress is applied and the evolution of permanent shear strain with increasing number of cycles is observed. Since the asphalt concrete mixes have a tendency to dilate under shear loading and as the height of the specimen is kept constant, compressive normal stresses evolve during the test. These stresses form one of the stabilizing mechanisms against rutting in the mix. This mechanism along with the stiffness of the binder restricts the development of permanent deformation in the shearing direction. The RSST-CH test 8

23 hypothesizes two mechanisms, one related to the asphalt binder stiffness and the other being the stability due to the aggregate structure. Stiffer binders resist permanent deformation in asphalt because the magnitude of shear strains is reduced with each load application. The rate of accumulation of permanent deformations is strongly related to the magnitude of shear strains. A stiffer asphalt binder, therefore, will have increased rutting resistance because it minimizes shear strains in the aggregate skeleton. In the second mechanism, the axial stresses developed in the specimen act as a confining pressure and tend to stabilize the mixture. A well compacted mixture with a strong aggregate structure will develop high axial forces at very small shear strain levels. Poorly compacted mixtures can also generate similar levels of axial stresses but it requires much higher shear strains. In the RSST-CH test, these two mechanisms are free to fully develop their relative contribution to the resistance of permanent deformation. It should be emphasized that the so called simple shear test has no relation to the simple shear of mechanics. The permanent shear strain versus the number of load repetitions plotted on a log-log scale is linear. Fig..3 shows the evolution of permanent shear strain with number of cycles for various specimens..5.2 Triaxial Repeated Load Permanent Deformation Tests In the triaxial repeated load test, several thousand loading cycles are performed on the sample and the cumulative permanent strain as a function of the number of loading cycles (i.e. repetitions) is recorded. The plot of axial permanent shear strain versus the number of cycles on a log-log scale is a straight line. This is very similar to that observed 9

24 in the RSST-CH test. Fig..4 shows the evolution of axial permanent strain with number of cycles. This linear relation in the log-log scale is very significant in characterizing the asphalt concrete mixes. The slope and intercept are used to characterize the permanent deformations of the Hot Mix Asphalt mixture. Fig..5 shows a qualitative plot of plastic shear strain versus number of cycles on a log-log plot. The intercept a represents the permanent strain at N= whereas the slope b represents the rate of change of the permanent strain as a function of the change in loading cycles. Fig..6 shows the plot of axial permanent shear strain versus number of cycles on a log-log scale. The classic power law model, mathematically expressed by Eq. (.) is typically used to analyze the test results. b ε p = an (.) where ε p is the plastic strain, N is number of cycles of loading, a and b are parameters interpreted as intercept and slope respectively. This model was developed as part of phase of NCHRP -26. See the report submitted to TRB, National Research Council, Washington, D.C., 990. A similar model was also proposed by researchers at Ohio State University for use in a pavement design system developed for the Ohio Department of Transportation. The OSU permanent strain accumulation model was proposed by Majidzadeh, et al. (98). It has been reported in NCHRP Report No. 465 that the permanent shear strain measured at 3000 and higher loading cycles, using RSST-CH gave excellent correlations with the actual field test data of rut depths. The cumulative permanent strain measured at 0

25 000 and fewer loading cycles using triaxial repeated load test was reported as having the best relation with the rut depths measured on the field. As mentioned earlier, rutting in pavements is initially caused by densification of pavements and by shear flow later. In a specimen loaded axially, the permanent deformation occurring in the first few cycles is predominantly due to the densification of the specimen. Hence, triaxial repeated load test correlates best with field test data of rut depths measured at 000 and fewer loading cycles. In pavements, after initial densification, rutting is due to shear flow and this is captured well by RSST-CH experiments and hence RSST-CH tests correlates best with field test data measured at 3000 and higher loading cycles. The aforementioned tests reveal a great need for duplicating their results through a constitutive model. This model can then be implemented into a Finite Element environment, and a pavement section can be analyzed. If the constitutive model is able to capture the primary mechanisms causing rutting, then the Finite Element results can be used to predict rutting in pavements for particular asphalt concrete mix. This would eliminate the need to have multi million dollar testing facilities to evaluate pavements..6 Modeling of Asphalt Concrete Until early 920 s, a fixed thickness (conservative thickness) of asphalt concrete course was laid on pavements to negate the effects of wear and damage regardless of the conditions. Modernization and industrial revolution led to the development of more sophisticated automobiles, contributing to higher traffic on pavements. This called for a more improved pavement technology to cope with the increased severity in loading conditions on roads, leading to new design methodologies. Early design procedures were

26 primarily empirical and were based on practical experience. In the 930 s, Francis Hveem developed a mixture design procedure in which a compacted mixture was tested to establish the strength of the bond between the aggregates and also the strength of the aggregate skeleton. See e.g. Hveem and Davis (950). Bruce Marshall (965) devised a design methodology, now known as the Marshall test, measuring the stability, which is the failure load of the material and the flow index of asphalt concrete mixes. The stability is a measure of cohesion, while flow index measures the internal friction of asphalt concrete mixes. Early mechanics based research studies mainly focused on the viscous properties of asphalt concrete. In his series of papers, Van der Poel (954a, 954b and 954c) advocated a linear viscoelastic model for asphalt. He recognized stiffness as an important parameter of asphalt and employed the ring and ball softening point and the penetration index methods to measure stiffness. The justification for the assumption of a linear behavior was based on the observation that at low temperatures and short times of loading, asphalt behaves linearly, while for long loading times and high temperatures, asphalt seems to behave like a Newtonian fluid. Using these ideas, and measuring the penetration index and ring and ball softening point for 47 samples from different parts of the world, a nomograph was proposed for finding the stiffness of bitumen. Similar nomographs for finding the stiffness of asphalt concrete knowing the stiffness of asphalt were proposed later for Shell Pavement Design Manual (978) and for Asphalt Institute Pavement Design Manual (982). However, the assumption of linear behavior could not predict the complex behavior of asphalt concrete and further research studies focused on overcoming this set back. 2

27 Wood and Goetz (958) studied the relationship between the unconfined compressive strength of a bituminous mixture and the viscosity of the binder. Their study revealed that there are other factors apart from viscosity of binder which affect the properties of asphalt mix. In a series of papers, Brodnyan (958), Gaskins et al. (960) and Brodnyan et al. (960) showed the validity of time-temperature principle for asphalt materials in the practical temperature range, 25 o C-00 o C. Subsequent research efforts concentrated on developing various viscoelastic models for asphalt concrete, e.g. Papazian (962), Monismith et al. (966), Kim, Drescher and Newcomb (99), Judycki (992), Park et al. (996), Lee and Kim (998) among others. However, for better pavement performance, the focus of the research shifted from developing models for the mixture to developing pavement design procedures. One of the early road design manual developed was the Shell Pavement Design Manual (Shell 978). The manual consisted of a series of charts based on research and accumulated results from previous studies. One of the main aspects of the design procedures was to minimize distress in pavements and this resulted in development of distress prediction models based on continuum mechanics principles. In these models, the measure of distress (number of loading cycles to failure, amount of permanent deformation) is related to material properties through constitutive theory. The motivation for a more mechanistic approach instead of traditional empirical approach was provided by the Asphalt Aggregate Mixture Analysis System (AAMAS) developed as part of the National Cooperative Highway Research Program between 986 and 99. See Von Quintus et al. (99). The shift from empirical to an mechanistic approach resulted in the 3

28 development of SUPERPAVE mix design methods comprised of performance based material specifications, mechanistic test methods, design and analysis procedures and mix design for asphalt concrete. See NRC (994). In parallel, a Strategic Highway Research Program (SHRP) project entitled Performance related testing and measuring of asphalt-aggregate interactions and mixtures, was carried out by Monismith et al. to investigate relationship between asphalt concrete properties and pavement performance. See e.g. Monismith et al. (990), Monismith et al. (99). A number of test methods for e.g uniaxial, triaxial and simple shear tests were conducted under repeated, creep and dynamic loading conditions. These tests were conducted in order to differentiate between various asphalt mixes with respect to their rutting propensities. The repetitive simple shear test under constant height (RSST-CH) performed on cylindrical specimens showed promising results with respect to field observations. Although a considerable amount of work has been performed on several aspects of asphalt behavior, such as viscoelastic (rheological) modeling and characterization and fatigue cracking analysis and modeling, not much effort has been devoted into modeling the accumulation of permanent deformation in the asphalt concrete mixes. Early work on developing distress prediction models was mainly based on viscoelasticity as asphalt was presupposed to be a viscoelastic material. See e.g. Kim et al. (99), Judycki (992), Callop et al. (995), Park et al. (996), Lee and Kim (998) among others. However, viscoelasticity is not sufficient to capture permanent deformation evaluation, i.e. rutting in pavements and efforts to do so by viscoelasticity have been unsuccessful, see e.g. 4

29 Sousa, Weissman, Sackman and Monismith (993). Others like Liu (993) have utilized classical plasticity theory, but they have been limited by the nature of the theory in modeling repetitive loading on the mix. In this case a plasticity model is used to calculate the permanent strain at the end of the first loading cycle and then a linear relation between permanent strains and number of cycles on a log-log scale (in order to simulate the observed experimental results) is assumed to calculate the permanent strain at any loading cycle. In other words, in this model a phenomenological theory (plasticity) is combined with an empirical relation resembling the experiments. Recently, constitutive models to capture rate effects and permanent deformations have been developed by Panoskaltsis et al. (2003), Bahuguna (2003) and Tashman et al. (2004). The model developed by Tashman et al. is a viscoplastic model aimed at characterizing the behavior of asphalt concrete under triaxial experiments. The model also includes anisotropy and isotropic damage. However, the equation for the hardening parameter is an explicit relationship - of a power law type- of an effective viscoplastic strain. This explicit relationship is decoupled from the evolution of the viscoplastic strain and therefore is independent of its history. The constitutive model proposed by Bahuguna is based on hyperelasticity-viscoplasticity theories. The model captures the ratcheting behavior exhibited by asphalt concrete very well. This model has been further enhanced in this present work. In this present research work, this model has been enhanced in several ways: it accounts for anisotropy and damage has been included to describe the softening phenomenon in asphalt concrete. Furthermore, coupling is provided through out the whole history of the material s deformation. The viscoelastic part has been enhanced to 5

30 account for temperature and this is done through time-temperature superposition principle using the classical approach as well as a novel approach within the framework of the internal variable theory. The model will be verified with the results of the aforementioned RSST CH and triaxial experiments. The resulting constitutive model will be implemented in Finite Element Program to study boundary value problems. The pavement section is then modeled as a plane strain problem and the amount of permanent deformation that is developed for a given mix is tested against data from studied pavements..7 Overview Chapter 2 discusses mechanistic behavior of asphalt concrete. Presence of elastic, viscoelastic, plastic, and viscoplastic components in the permanent deformation of asphalt concrete is established and accordingly a constitutive model in series is suggested. A second order hyperelastic material model is proposed to model the elastic component of the permanent deformation of asphalt concrete in Chapter 3. This model also captures the dilation in the material. The model is validated against uniaxial, volumetric and simple shear tests. Chapter 4 presents a viscoelastic model to capture the time dependent recoverable component of the material. Temperature-dependence is captured with this model through the Time Temperature Superposition principle. Chapter 5 deals with the viscoplastic component of the material deformation. A non-associative viscoplastic constitutive model is proposed to model the rate effects in the material. Viscoplastic model is developed within the general framework of Perzyna s theory of viscoplasticity. Anisotropy and damage are also incorporated into this model. The model 6

31 is validated against RSST-CH and triaxial experiments. The complete constitutive model and its numerical implementation is presented in chapter 6. An effective return mapping algorithm is provided for time integration of the model. The model is then implemented into a finite element environment to study boundary value problems, in particular the pavement problem. Chapter 7 presents the results and model prediction against experimental results. Implementation of this model in ABAQUS through UMAT is discussed in detail in this chapter. Chapter 8 concludes the work and various suggestions for future work to enhance the model for further applications have been given. 7

32 Figure.a. Severe Rutting accompanied by upheavals to the side Figure.b. Rut Depth Measurement (Courtesy: Montana Department of Transportation) 8

33 Figure.2. Correlation between laboratory test results and field test data (NCHRP Report No. 465) 9

34 Figure.3. Permanent Shear Strain versus Number of cycles on a log-log scale (Sousa) 20

35 Figure.4. Evolution of axial permanent shear strain with number of cycles. (NCHRP Report No. 465) 2

36 Figure.5. Regression constants a and b when plotted on a log-log scale. (NCHRP Report No. 465) 22

37 Figure.6. Axial permanent strain versus number of cycles on a log-log scale. (NCHRP Report No. 465) 23

38 CHAPTER TWO ASPHALT CONCRETE CHARACTERIZATION 2. Introduction Constitutive, or stress-strain laws, represent mathematical models that describe the macroscopic material behavior that results from the internal constitution of a material. The importance of constitutive laws has been enhanced significantly with the increase in development and application of many modern computer-based techniques such as finite element, finite difference, and boundary integral equation methods. In any engineering analysis, these constitutive laws play a crucial role providing reliability to the results obtained from the numerical procedures. The focus of this study is to develop a constitutive model to characterize asphalt concrete mix with respect to permanent deformations in pavements. A complete 3D constitutive model will aid in studying the performance of a pavement as a boundary value problem. 2.2 Characteristics of asphalt concrete mix In order to develop a constitutive model, it is essential to know the scope and limitations of the constitutive equations and hence it is necessary to review the main attributes of asphalt-aggregate mixes that can be directly associated with permanent deformation. The main attributes of asphalt concrete mix which can be associated with permanent deformations are as follows (See Sousa (994), Huang (993)): Mix behavior is rate and temperature-dependent. 24

39 Asphalt concrete mix exhibits dilatancy. There is volumetric deviatoric coupling in the sense that deviatoric stresses lead to volume change. Exhibits different properties in tension and compression. Cyclic loading (both mechanical as well as thermal) leads to crack development. Residual deformations are observed at the end of loading cycles. Mix behavior is dependent on air void contents. Aging plays an important role in development of permanent shear deformations. Moisture damage also plays an important role in development of permanent deformations. A complete constitutive law must accommodate all of the above characteristics. However, the model is developed in increments i.e., prioritizing the requirements and accordingly accommodating each attribute. The most significant characteristics of asphalt concrete with respect to rutting are rate and temperature dependence, dilatancy and damage and these attributes are considered for the development of the constitutive model. The permanent deformation in asphalt concrete mix is divided into the following four components. Reversible rate-independent deformation : Elasticity Reversible rate-dependent deformation : Viscoelasticity Irreversible rate-independent deformation: Plasticity Irreversible rate-dependent deformation. : Viscoplasticity 25

40 Each of these forms of deformation can be modeled using the various theories of mechanics as indicated above. In the reminder of this chapter, experimental evidence for various relevant attributes of asphalt concrete is discussed. 2.3 Evidence of temperature dependence in the behavior of the mix Bitumen is one of the main components of asphalt concrete mixes. It is a highly temperature sensitive material; in other words, its rheological properties are temperaturedependent. Accordingly, the asphalt concrete mix behavior is also highly temperature dependent. This is also evident from the fact that rutting is severe in hotter climates. Research conducted by Harvey et al. at the Institute of Transportation Studies, U.C.Berkeley shows that the number of traverses of a single-axle load, before a.3-cm rut is created, is much larger when the pavement temperature is lowered from 50 C (22 F) to 40 C (04 F) (measured 5 cm below the surface). This suggests that failure due to rutting can be delayed if the pavement is kept cooler. Fig. 2. shows the rut depth variation with number of cycles at various temperatures. An asphalt pavement that is subjected to repeated simple shear permanently deforms over time. This permanent deformation of pavement severely decreased as the temperature of the pavement is decreased from 60 C (40 F) to 40 C (04 F). Fig. 2.2 shows the variation in permanent deformation with number of repetitions of loading cycles at various temperatures. Sousa et al. conducted RSST-CH experiments at different temperature levels to see the effect of temperature on the evolution of permanent deformations. RSST-CH experiment was conducted at two different temperature levels: 35 o C and 45 o C. 400 cycles of a 0. sec 26

41 haversine pulse shear stress with 0 psi amplitude was applied with 0.6 sec rest periods. The height of the specimen was kept constant within +/ inches. A series of 5 identical test sequences was executed on the same specimen in the following order:. 400 cycles at 35 o C cycles at 45 o C cycles at 35 o C cycles at 45 o C cycles at 35 o C. The results for the two temperature levels are shown in Fig The rate of accumulation of permanent deformations at 35 o C significantly reduces after performing tests at 45 o C on the specimen, while the rate of accumulation of permanent strains at 45 o C does not change after the application of loads at 35 o C. This experimental observation leads to a conclusion that hardening of the mix is dependent on the temperature. The mix hardens, i.e. the elastic region expands much more on application of loads at 45 o C. As a result, when load is applied at 35 o C the second time, the accumulation of permanent strains is much less. 2.4 Evidence of Plasticity in Asphalt Concrete The presence of aggregates in the mix is responsible for the plastic behavior of the mix. The aggregates tend to roll over and tend to form new stable aggregate skeleton thus inducing permanent deformations in the material. Experimentally, presence of plasticity was established by performing RSST-CH test on the specimen using a sequence of different load levels. The height of the specimen was kept constant to within +/

42 inches while 800, 900, 000 and 200 cycles of 0. sec haversine pulse shear stress with stress amplitude 4, 6, 8, and 0 psi respectively was applied with a 0.6 sec rest period. This sequence of tests was repeated three times. Fig 2.4 shows the evolution of permanent strains with number of cycles on a log-log scale. The figure shows that the rate of accumulation of permanent deformations at a particular stress level after a higher stress level has been applied is significantly lower than the rate of accumulation if the higher stress level was not applied. At higher stress levels such as 8 psi, the rate of accumulation of permanent deformations after the specimen was subjected to 0 psi is lower than the first time the 8 psi stress was applied. At the 4 psi and 6 psi levels, the rate of accumulation is almost negligible after the specimen has been exposed to the 8 and 0 psi stress levels. This indicates that higher stress levels permit the creation of stronger aggregate structures that the lower stress levels cannot unlock. In mechanics, this is explained by hardening, a characteristic of plastic materials. 2.5 Evidence of rate dependent permanent deformation Rutting in pavements is prominent at traffic intersections, ramp sections in hilly areas and other pavement sections where traffic speed is low. This is confirmed by studies by Pereira et al., (999) and Kandhal et al., (998). A hilly section of a highway in Portugal was chosen for this study by Pereira et al. As vehicles climb the hill, the speed reduces from around 85Km/h to about 20Km/h. Rut depth was measured at four different cross sections. The measured rut depths with corresponding slopes is tabulated in Table 2. 28

43 Slope Rut Depth % 5mm 4% 5mm 6% 8mm Table 2.. Rut Depth measured at various slopes along a hilly highway in Portugal. As the slope increases, the rut depth increases. From this observation, it can be inferred that rut depth increases as the rate of loading decreases. Rate of loading is directly proportional to traffic speed. Kandhal et al. measured rut depths at traffic intersections and observed that rut depths decreases as the distances from the intersection increases. This is tabulated in Table 2.2. Highway Rut Depth at Intersection (mm) Distance from the Intersection. (mm) Table 2.2. Rut Depth measured at various distances from the intersection Also, rate dependence is exhibited in RSST-CH and triaxial experiments. An RSST-CH experiment with varying loading rates was performed on specimens obtained from highway pavements. The test was performed with 0., 0.2, 0.3 and 0.4 sec haversine 29

44 loading and rest period of 0.6 second. The tests were performed at 50 o C at 70kPa load level. Fig. 2.5 shows the response of the mix to different loading times. As is evident in the figure, the accumulation of permanent deformation increases as the loading rate decreases. Triaxial results presented my Tashman et al. (2004) show the response of asphalt concrete for different rates of loading. Also, triaxial experiment results show softening in asphalt concrete which is a strong evidence for damage in the material. 2.6 Evidence of Dilation in the mix Dilation is an inherent property exhibited by granular materials. This can be attributed to the tendency of aggregates to role over leading to volume change in the material. The same phenomenon is also exhibited by asphalt mixes. This is best exhibited in a RSST-CH experiment. Asphalt concrete specimens tend to dilate when subjected to shear stresses and to keep the height constant, stresses are developed in the normal direction. This is shown in Fig. 2.6; the figure shows the development of normal stresses with shear strain in an RSST-CH experiment. 30

45 Figure 2.. Rut Depth vs. Repetitions of loading at 40 o C and 50 o C (The data is from Dr. John Harvey, Institute of Transportation Studies, U.C. Berkeley) 3

46 Figure 2.2. Permanent Shear Strain vs. Number of cycles of Shear loading on asphalt pavement specimens. (The data is from Dr. John Harvey, Institute of Transportation Studies, U.C. Berkeley) 32

47 Figure 2.3. Evolution of permanent shear strain in a sequence of RSST-CH tests at different temperatures. 33

48 Figure 2.4. Evolution of permanent shear strain in a sequence of RSST-CH tests at different stress levels. 34

49 Figure 2.5. Evolution of permanent shear strain in RSST-CH experiment at different loading rates. 35

50 Figure 2.6. Development of normal strain in RSST-CH experiment. 36

51 CHAPTER THREE HYPERELASTIC MODEL 3. Introduction In chapter two, the various components of the deformation in asphalt concrete were discussed and also, it was proposed that the rate-independent recoverable component would be modeled by an elastic model. It is known that asphalt concrete exhibits volumetric/deviatoric coupling within its elastic region. This volumetric/deviatoric coupling behavior is clearly evident in a RSST-CH experiment during which, stresses are developed in the normal direction when the specimen is subjected to shear loading in the transverse direction. The tendency of the aggregates to roll over each other when the specimen is subjected to shear type of loading is attributed to this behavior of asphalt concrete mix. The bituminous medium surrounding the aggregates, constraints the expansion (expansion in the normal direction) when the aggregates tend to roll over and thus normal stresses are developed. This coupling behavior exhibited by asphalt mixes is very similar to that observed in shear tests on granular materials like soils. Taylor (948) proposed a model, called the shear box model, to describe the coupling behavior of dry sand under shear deformations. In his model, he proposed that two factors contributed to the strength of the soils; the frictional resistance between particles as they slip during shear distortion of the soil and a second factor called interlocking which causes volume increase during shear distortion. Based on these assumptions, Deshpande et al. (999) proposed a shear box model for asphalt mixes. This 37

52 model is analogous to the shear box model of Taylor (948) and the Cam-Clay model by Roscoe et al. (958). In the past, the coupling behavior in granular soils has been successfully characterized by models based on plasticity theory; see e.g. Vermeer (982). However, as discussed earlier, in the case of asphalt concrete mix, the normal stresses developed during shear loading return to zero upon shear unloading. This is a strong evidence that asphalt concrete exhibits coupling behavior in elastic region as well, and this material nonlinearity must be modeled by elasticity. A second order hyperelastic model is proposed to capture the characteristic volumetric/deviatoric coupling in asphalt concrete, as a linear elastic model, which can be considered as a first order elastic model, would not capture this behavior. 3.2 Hyperelastic Materials For an elastic body, the current state of stress depends only on the current state of deformation; that is, the stress is a function of current strain and not a function of history of strain. Materials which satisfy this fundamental requirement of an elastic body are called Cauchy elastic materials. A Cauchy elastic material is characterized as a material which does not depend on the history of the deformation process and general constitutive relation for such materials is given by Eq. (3.) ( (,t)) σ = G F X, (3.) 38

53 where the function G is the elastic material response function, F is the deformation gradient, X is the spatial coordinate in the reference configuration and t is the time. The response function depends only on the current value of the deformation gradient and not on its history. The stress computed for a Cauchy elastic material using Eq. (3.) is independent of the deformation history, however the work done by the body and traction forces, may depend on the deformation history or load path. In other words, a Cauchy elastic material exhibits energy dissipation violating the energy dissipation properties of elastic materials. For elastic bodies, there exists an unstressed state called the natural state and when external forces are applied, the body deforms and reaches a different energy state. When the external forces are removed, the body regains its natural state and there is no dissipation of energy during the deformation process. Materials for which the work which is performed is independent of the load path are said to be hyperelastic materials (or in the literature often called Green-elastic materials). As will be shown later in this section, materials for which work is independent of the load path, are characterized by the existence of a potential for stress or strain. Hence, hyperelastic materials are elastic materials, which are characterized by the existence of a strain energy function which is the potential for the stress or the complementary energy function which is the potential for the strain. In other words, the elastic response function defined in Eq. (3.) is further restricted by the existence of such potentials. Green (839, 84) first proposed the existence of such a potential for an elastic material and hence hyperelastic materials are often called Green-elastic materials in literature. See e.g. Truesdell and Toupin (960) and Truesdell and Noll (965). Eq. (3.2) describes a general hyperelastic constitutive law. 39

54 S = W ( E ) E, (3.2) 6 where ( ): W E R R is a scalar-valued function, called the elastic strain energy function and is a function of the Green-St Venant strain E. W ( E ) is the strain energy per unit reference volume. In the above constitutive law given by Eq.(3.2), S is the second Piola-Kirchhoff stress tensor. As mentioned earlier, a consequence of the existence of a stored energy function is that the work done on a hyperelastic material is independent of the deformation path. This is illustrated as follows; see e.g. Holzapfel (2000). Consider a deformation process of a continuum body within some closed time interval denoted by[ t, t ]. During the process the body deforms according to the 2 deformation gradient F = F ( t) with t [ t, t ] 2. A deformation process is said to be closed if the deformation gradient F is identical for the initial and final configuration, i.e. ( t ) = ( t ) F F. (3.3) 2 2 Hence, the strains at the beginning and end of the process will be T T E( t) = ( F F I) = ( F2 F2 I) = E 2( t2). (3.4) 2 2 Eq. (3.4) because of W W( ) = E implies the following relation 40

55 W ( ) = W ( ) E E. (3.5) 2 The work done by the stress field on the continuum body Ω during the time interval [ t t ], is given by Eq.(3.6) 2 t2 t2 ( E ) W S: E dt dω= : E dt dω, (3.6) E Ω t Ω t where the superimposed dot indicates the material time derivative. Eq. (3.6) because of Eq. (3.2) can be rewritten as t2 Ω t dw ( E ) dt ( ( ) ( )) dω dt = W E W E dω= (3.7) Ω 2 0 Hence, unlike a Cauchy elastic material, the actual work done by the stress field on a hyperelastic material during a certain time interval is path independent and depends only on the initial and final configurations and therefore the work done is zero in closed deformation processes. Thus far, the energy function has been defined in terms of strain and the same energy function can also be defined in terms of stress tensor. The counterpart of Eq. (3.2) can be postulated, assuming the existence of a complementary energy function 4

56 Ω 6 ( S ) : R R, which is a function of the second Piola-Kichhoff stress. The constitutive law is given by Eq. (3.8) Ω( S ) E =. (3.8) S For a detail discussion on hyperelasticity, refer to Malvern(967), Ciarlet (988), Ogden (997), Holzapfel (2000), and Chen and Mizuno (990) among others. One of the focuses of research on hyperelasticity has been on proposing various appropriate energy functions and in this regard, strain energy functions corresponding to different material behavior have been proposed in the literature. The hyperelasticity theory discussed so far holds for both small and large strain theories. The hyperelastic model proposed for asphalt concrete in the next section is under the realm of small strain theory. The proposed model captures only material nonlinearity and not geometric nonlinearity as opposed to hyperelastic models for large strain theories which capture both material and geometric nonlinearity. 3.3 Hyperelastic model for asphalt concrete: A second order hyperelastic model is proposed to capture the characteristic volumetric/deviatoric coupling in asphalt concrete as a simple first order elastic model would not capture this behavior. An extension of Eq. (3.2) into the small strain theory yields the following relation. 42

57 σ ij ( ) W ε =, (3.9) ε ij 6 where ( ): W ε R R is the strain energy function in terms of the infinitesimal strain tensor ε. For an isotropic material, such as asphalt concrete in its elastic region, the scalar valued function W ( ε ) is an isotropic function and therefore it can be expressed in terms of the invariants of strain. Thus, one can derive hyperelastic models of various orders according to the degree of the polynomial in terms of the strain invariants. (,, ) W = W I I I, (3.0) 2 3 where I, I2, I 3 are the strain invariants and are given by I = tr( ε ) = εii 2 I2 = tr( ε ) = εε ij ij I3 = tr( ε ) = εikεkmεmi. 3 3 From Eq. (3.9), the stress tensor is given by σ ij W I W I W I3, (3.) I ε I ε I ε 2 = + + ij 2 ij 3 ij 43

58 where I ε ij = δ, (3.2) ij I2 ε ij = ε, (3.3) ij I3 ε ij = ε ε. (3.4) im mj Introducing Eq. (3.2), Eq. (3.3) and Eq. (3.4) into Eq. (3.), the following expression for stress is obtained W W W σ = δ + ε + ε ε ij I ij ij im mj I2 I3. (3.5) In order to model the nonlinear elastic behavior of asphalt concrete materials, a third order hyperelastic model has been proposed by Sousa et al. (993) and Bahuguna (2003). However, to reduce the number of parameters, a second order hyperelastic model is proposed here. In contrast to a third order model, which has 9 parameters, the second order Green elastic model proposed here has only 5 parameters. This saves computational time considerably as well as it aids in easier identification of the parameters with material properties. For the second-order model, terms up to the second power of strain in the stressstrain law are retained. Hence, the strain energy density must be expressed as a cubic 44

59 function in strains or invariants of strain. The third order strain energy function considered in the model development is given by Eq. (3.6) W= b + bi + bi + bi + bii + bi + bi, (3.6) where b 0, b, b 2, b 3, b 4, b 5, and b 6 are material constants. The model should be able to represent the material at its natural state. In other words, at zero strain state, the value of the strain energy function and the stresses in the material are equal to zero. This condition requires the parameters b 0 and b to be equal to zero. Then, the strain energy function is given by W = b I + b I + b I + b I I + b I, (3.7) Linear elastic behavior can also be modeled as a special case with this model. If the third order terms are ignored, Eq. (3.7) reduces to W = b I + b I, (3.8) which represents the first order or linear elastic model which is given in Eq. (3.9) below. σ = λi δ + µε, (3.9) ij ij 2 ij 45

60 where λ and µ are the Lamé s constants. λ = µ = ν E ( + ν )( 2ν ) E 2 ( + ν ), (3.20). (3.2) From Eq. (3.5), Eq. (3.8), and Eq. (3.9) parameters b 2 and b 5 are given by b 2 λ =, (3.22) 2 b 5 = 2µ. (3.23) Differentiating the strain energy function given by Eq. (3.7) with respect to strain invariants, the following relations are obtained. W I = 2bI + 3bI + bi, (3.24) W = b 5 + b 4 I, (3.25) I 2 W I 3 = b. (3.26) 6 From Eq. (3.9), Eq. (3.24), Eq. (3.25) and Eq. (3.26), the stress tensor is given by 46

61 2 ( 2 3 ) ( ) σ = bi + bi + bi δ + b + bi ε + bε ε. (3.27) ij ij 5 4 ij 6 im mj The stress tensor given by Eq. (3.27), represents a second order hyperelastic stress strain relation. 3.4 Determination of Material Parameters To determine the material parameters in the second order hyperelastic model, the tests conducted by Sousa et al. (993) at the University of California, Berkeley are used. The following three experiments: simple shear test, volumetric test and uniaxial compression test were conducted in the elastic range. In other words, the experiments were conducted at low strain levels to minimize the plastic effects and at low temperature (4 o C) to minimize the viscous effects. The second order hyperelastic stress strain relation contains 5 material parameters. The model was made to fit simultaneously to all the three experimental data obtained from the above mentioned experiments and parameters were evaluated based on a nonlinear optimization scheme. The objective function for the optimization procedure is the least square function constructed using the model predictions and experimental results. The optimization procedure is carried out using the optimization toolbox of the mathematical software MATLAB, Mathworks Inc. Using the symmetry of the stress and strain tensors, the stress and strain tensors are written in the following vector forms. [ ] σ = σ σ σ σ σ σ, [ ] ε = ε ε ε ε ε ε, T T 47

62 where σ = σ, σ = σ, σ = σ, σ = σ, σ = σ, σ = σ, and ε = ε, ε = ε, ε = ε, ε = ε, ε = ε, ε = ε Shear Test: when The analysis is based on the assumption that a pure shear state of shear is obtained ε2 = ε2 = ε0, and all other components are equal to zero. The state of strain can be given as ε 0 ε 0 0 ε 0 0. = From Eq. (3.2), Eq. (3.3) and Eq. (3.4) I = 0, I 2 2 ε 0 =, 2 I 3 = 0. 48

63 Using Eq. (3.27), the shear stress is given by σ = 2b ε (3.28) and the normal stress developed due to the coupling of deviatoric and volumetric components is calculated as 2 ( b b ) ε σ = +. (3.29) Volumetric Compression Test: The state of strain in this case can be written as ε ε ε 0, = ε 0 From Eq. (3.2), Eq. (3.3) and Eq. (3.4) I = 3ε, 0 I I 2 2 3ε 0 =, 2 = ε

64 Using Eq.(3.27), the compressive stress is given by 2 σ ( 2b2 b5) ε 9 = b3+ b4 + b6 ε0. (3.30) Uniaxial Test: In this test the axial stress is applied while the confining stress keeps the perimeter of the cylindrical specimen constant. Confining stresses develop in the specimen as a result of this arrangement. The strain tensor for this test is given by ε ε = 0 0 0, From Eq. (3.2), Eq. (3.3) and Eq. (3.4) I = ε, 0 I I ε 0 =, 2 3 ε 0 =. 3 Using Eq. (3.27), the axial stress is given by 50

65 2 σ 3 = 3b3+ b4 + b6 ε0 + ( 2b2 + b5) ε0 2 (3.3) and the confining stress is given by ( ) 2 σ = 2bε + 3b + b ε. (3.32) Parameter Estimation: The second order hyperelastic stress strain relation has 5 material parameters. The model was made to fit simultaneously to all the three experimental data and parameters were evaluated based on a nonlinear optimization scheme. The objective function for the optimization procedure is a least square function given by N N N f = σ σ + σ σ + σ σ ( i exp, i) ( i exp, i) ( i4 exp, i) (3.33) i= uniaxial i= Volumetric i= Shear where N, N 2, and N 3 are the number of data points taken from the uniaxial, volumetric and shear experiments, σ i is the i th point obtained from the hyperelastic model for σ component and similarly σ i4 is the i th point obtained from the model for the σ 4 stress component. The minimization of the objective function was performed using the optimization toolbox of MATLAB. The values obtained for the parameters are tabulated in Table 3. 5

66 Parameter b 2 b 3 b 4 b 5 b 6 Value x x x x x 0 8 Table 3.. List of Parameters. As discussed earlier, the parameter b 2 and b 5 are elastic constants and have to be positive for a hardening material. For the model, the values obtained for these parameters are positive and is in agreement with the requirement. Fig 3. shows the results for a uniaxial test. Fig 3.2 and Fig 3.3 show the model fit for volumetric and shear tests respectively. All the figures show excellent fit between the model results and experimental data. With the values of the parameters obtained from the parameter estimation procedure, the model s predictions of dilatancy behavior are compared to experimental results. Fig. 3.4 shows model s predictions of the normal force developed during the repeated simple shear test and the experimental data for the same. The plot shows the development of normal stress with shear strain. 3.6 Stability Analysis: Positive Definiteness of tangent stiffness. It is well known that the work done on an elastic body by any external agency on the changes in displacements it produces is positive. This is manifest in terms of stresses and elastic strains as follows: σ ε > 0 (3.34) ij ij 52

67 and materials which satisfy the above condition are called stable materials. It is necessary that constitutive laws proposed to characterize elastic material behavior must satisfy the stability criterion given by Eq. (3.34). This stability criterion when applied to the hyperelastic model developed in this chapter, requires that the Hessian matrix be positive definite. This can be shown as follows. From Eq. (3.34) and Eq. (3.9) σ W σ ε ε, (3.35) 2 ij e e ij = e kl = kl εkl εij εkl Substituting Eq. (3.35) into the stability condition Eq. (3.34), the following inequality is obtained. 2 W ε ε ij kl εε > 0, (3.36) ij kl that is, the quadratic form given in Eq. (3.36) must be positive definite for arbitrary values of the components of strain rate. The inequality may be rewritten as H ijklε ijε kl > 0, (3.37) where H ijkl is the tangent material stiffness and is a fourth order tensor given by 53

68 H ijkl 2 W =. (3.38) ε ε ij kl Rewritting Eq. (3.37) in matrix form, T ε H ε > 0, (3.39) where ε is the [6x] vector containing strain rates and H is the [6x6] tangent stiffness matrix. The Hessian matrix H is positive definite since Eq. (3.37) holds for all ε 0. Mathematically, the matrix of the components of H ijkl is known as the Hessian matrix of the function W. When strain is expressed in a vector form, the elements of the Hessian matrix for W are written as [ H ] W W W W W W 2 ε ε ε2 ε ε3 ε ε4 ε ε5 ε ε W W W W W 2 ε ε 2 2 ε3 ε2 ε4 ε2 ε5 ε2 ε W W W W 2 ε ε 3 3 ε4 ε3 ε5 ε3 ε 6 = W W W 2 ε ε 4 4 ε5 ε4 ε W W 2 ε ε 5 5 ε 6 2 W 2 ε6. (3.40) 54

69 Positive definiteness of the Hessian matrix ensures that the surfaces corresponding to constant W and Ω in strain and stress space, respectively, are convex. This can be proved mathematically as follows. Consider two different strain tensors and b a ε ij in the strain space and the corresponding strain energies are given by W ( εij ) b ( ij ) W ε respectively. Using Taylor series expansion, the difference in the strain energies can be approximated as a ε ij and a b W W ( εij ) W ( εij ) εij + H ijkl ε b ij εkl ε 2, (3.4) ε ij ij b ε ij where a b εij = εij εij and ijkl ε b ij H is the Hessian matrix of W calculated at ε b ij. Note that the higher order terms are neglected in the above given Taylor series expansion. As discussed earlier, the second term in Eq. (3.4) is positive definite and thus the following inequality holds W W ε ij a b a b ( εij ) W ( εij ) > ( εij εij ) a ij ε. (3.42) The inequality given in Eq. (3.42) is the condition for strict convexity of W. The positive definite nature of the Hessian ensures convexity of the strain energy function. 55

70 The stability check is performed on the model for the different forms of loading specified in the earlier section. The strain range in which the proposed second order hyperelastic model is stable is detailed for the various forms of loading Stability analysis for a Uniaxial Test: The strain tensor is given by ε ε = 0 0 0, For a uniaxial case, the Hessian matrix contains only one component. The only nonzero component of the Hessian matrix is given in Eq. (3.43) 2 W H = = b + b + b + b + b ( ) ε ( 2 ) ε 0. (3.43) Substituting the values of the parameter from Table 3., ( ) H = 2.785x0 ε + 5.8x0 (3.44) H 4 > 0 ε 0 < 2.08x0. (3.45) 56

71 The above condition in Eq. (3.45) requires that for Hessian to be positive definite, the magnitudes of compressive strains have to be less than 2.08 x 0-4. This condition is satisfied as observed from the experiments and hence the Hessian is positive definite and the strain energy function is convex Stability analysis for a Volumetric Test: The strain state is given by ε ε 0 0, 0 = 0 ε ε 0 For a volumetric compression test, the Hessian matrix has all elements equal to zero except the first three main diagonal elements. 9 H = H2222 = H3333 = 2 27b3 + b4 + b6 ε 0 + 2b2 + b5, (3.46) 2 substituting the values of the parameters from the Table 3. ( ) H = H = H = 38.92x0 ε + 5.8x0 (3.47) H > 0 for ε 0 <.49x0. (3.48) 57

72 4 The Hessian matrix is positive definite for ε 0 <.49x0 and the strain energy function is convex Stability analysis for a Shear Test: The strain state is given by ε 0 ε 0, 0 = ε The Hessian matrix has only one non zero element which is = 2 5 = > 0 H b x and the eigenvalues of the Hessian matrix are positive. The positive definiteness of the Hessian matrix ensures that the strain energy function is convex. In Fig. 3.5, the strain energy function corresponding to an uniaxial test is plotted against axial strain. From the graph it can be seen that W ( ε ) is Convex for ε 0 < 2.0x0 Concave for ε 0 > 2.0x0 4 4 This is in complete agreement with the limiting value found earlier for the uniaxial case. Similar plots could also be shown for shear and volumetric tests. The second order hyperelastic model is validated against a second set of experiments reported by Sousa et al. (994). The model fit for uniaxial, volumetric and 58

73 shear test data are shown in Fig. 3.6, Fig. 3.7 and Fig. 3.8 respectively. The model parameters obtained by the optimization technique are listed in Table 3.2. Parameter b 2 b 3 b 4 b 5 b 6 Value 8.9x x x x x 0 6 Table 3.2: List of parameters for second order hyperelastic model fit. 3.7 Stability Analysis for the second set of experiments: A detailed stability analysis was performed for the fist set of experiments. Similarly, stability analysis is also performed for the second set of experiments and details are given below. Uniaxial Test: The strain tensor is given by ε ε = 0 0 0, For a uniaxial case, the Hessian matrix contains only one component. The only nonzero component of the Hessian matrix is given in Eq. (3.49) W H b b b b b 2 3 = = ε ε 2 ( ). (3.49) 59

74 Substituting the values of the parameter from Table 3.2, ( ) H = 4.27x0 ε x0 (3.50) H 3 > 0 ε 0 >.47x0. (3.5) The above inequality requires that for Hessian to be positive definite, the magnitude of compressive strains has to be less than.47 x 0-3. This condition is satisfied as observed from the experiments and hence the Hessian is positive definite and the strain energy function is convex. Volumetric Test: The strain state is given by ε ε 0 0, 0 = 0 ε ε 0 For a volumetric compression test, the Hessian matrix has all elements equal to zero except the first three main diagonal elements. 9 H = H2222 = H3333 = 2 27b3 + b4 + b6 ε 0 + 2b2 + b5, (3.52) 2 Substituting the values of the parameters from the Table

75 ( ) H = H = H = 6.40x0 ε x0 (3.53) H > 0 for ε 0 < 3.28x0. (3.54) The Hessian matrix is positive definite for all strain values satisfying the condition given in Eq. (3.54) and accordingly the hyperelastic model is stable. Shear Test: The strain state is given by ε 0 ε 0, 0 = ε Hessian matrix has only one non zero element which is H = b = x > and the eigen values of the Hessian matrix are positive. This ensures that the strain energy function is convex for all values of shear strain and hyperelastic model is stable. 3.8 Closure A second order hyperelastic model is proposed to capture the nonlinear recoverable deformations of asphalt concrete. The model is validated against two sets of experimental data and the model s predictions are very good. These results show that the second order hyperelastic model can be used to model the elastic behavior of asphalt 6

76 concrete materials. The model is further implemented into a finite element environment to study boundary value problems. This is illustrated in chapter 7 in this dissertation. 62

77 Figure 3.. Shear Stress vs. Shear Strain for RSSTCH: Experimental data and Model fit. 63

78 Figure 3.2. Confining Stress vs. Radial Strain for RSSTCH: Experimental data and Model fit. 64

79 Figure 3.3. Axial Stress vs. Axial Strain for RSSTCH: Experimental data and Model fit. 65

80 Figure 3.4. Axial Stress vs. Shear Strain for RSSTCH: Experimental data and Model fit. 66

81 Figure 3.5. Strain Energy vs. Axial Strain. 67

82 Figure 3.6. Shear Stress vs. Shear Strain for RSSTCH: Experimental data and Model fit. (Experimental results from Sousa et al., 994) 68

83 Figure 3.7. Confining Stress vs. Radial Strain for RSSTCH: Experimental data and Model fit. (Experimental results from Sousa et al., 994) 69

84 Figure 3.8. Axial Stress vs. Axial Strain for RSSTCH: Experimental data and Model fit. (Experimental results from Sousa et al., 994) 70

85 CHAPTER FOUR VISCOELASTIC MODEL 4. Introduction In this chapter a viscoelastic model to predict the time dependent behavior of asphalt concrete will be presented. The inclusion of viscoelastic component is not very crucial for predicting the permanent deformations of asphalt concrete; however, it is essential for modeling the fatigue behavior of the material. The model developed in this work is a temperature dependent viscoelastic model, which is based on a new viscoelastic model developed by Panoskaltsis and his co-workers (Panoskaltsis et al. The Generalized Kuhn Model of Viscoelasticity, submitted for publication). The generalized Kuhn model is based on a variation of the modified Kuhn model developed by Panoskaltsis (992), which was used successfully for modeling viscoelastic behavior of concrete. Also see Lubliner and Panoskaltsis (992). The modified Kuhn model and the generalized Kuhn model are discussed in detail in this chapter. The viscoelastic model presented in this chapter captures time as well as temperature dependent deformations of asphalt concrete. The model s response to frequency response experiments on asphalt concrete is presented in this chapter. 4.2 The modified Kuhn model The modified Kuhn model is a linear viscoelastic model and describes the creep behavior of frictional materials. This model is a modification of the model proposed by 7

86 Kuhn and his co-workers (947) to describe rubber. The creep function described by the modified Kuhn model is given by t dτ Jt () = A+ B ( e τ ), (4.) τ / C where A,B and C are parameters and τ is the time dummy variable. The model accounts for the instantaneous elastic response, logarithmic creep in long time limit and elastic response in the high frequency limit. The creep function given by Eq. (4.) represents the continuous modified Kuhn model, as it is based on a continuous retardation-time spectrum. The modified Kuhn model has furthermore, both a rheological description, as a series of Kelvin type elements, as well as an equivalent internal variable description; this representation has the unique property that no matter how many Kuhn (Kelvin) elements are used, the number of parameters remains the same, a factor that keeps the computational time unaffected. In the model, the Kuhn elements (Kelvin elements with special stiffness and relaxation time properties as it will be explained next) account for the viscous strain and increasing the number of such elements increases the predicting capability of the model. The rheological representation of this model, called the discrete modified Kuhn model, has a creep function given by N m Ct r () = + ln ( ) JN t A B r e (4.2) m= 0 72

87 where A, B, C are the same parameters as in Eq. (4.) and N+ is the number of Kuhn elements. The above equation denotes a truncated Dirichlet series. This series represents the creep function of a generalized Kelvin model, as shown in Fig. 4., with an external + compliance A, compliance of the spring in each element Blnr, where r, distributed relaxation times r m m m C and viscosity coefficients η = r BCln r, where m refers to the m th element. It has been proved by Panoskaltsis (992) (See also Lubliner and Panoskaltsis (992)) that J ( ) N t has a finite limit and tends to J(t) for large N; in other words the discrete modified Kuhn model has as a limit the continuous modified Kuhn model for large number of elements. The modified Kuhn model describes satisfactorily the creep of concrete for short periods of time (since aging is not included), as well as the frequency dependence of the loss tangent. 4.3 Generalized Kuhn model Although the modified Kuhn model predicts correctly the frequency dependence of the loss-tangent of concrete materials, it cannot predict the loss tangent for highly dissipative materials like asphalt. A new viscoelastic model capable of predicting loss tangent for highly dissipative materials has been recently developed by Panoskaltsis and co-workers (Panoskaltsis, et al. The Generalized Kuhn Model of Viscoelasticity, submitted for publication). The model is given both a rheological as well as an internal variable representation and has several advantages over the Prony series type of models. Moreover, the generalized Kuhn model has been developed within the linear and nonlinear viscoelastic domain. In this chapter, the generalized Kuhn model has been 73

88 applied for the description and prediction of asphalt behavior. A brief overview of the generalized Kuhn model is given below. A power law kernel Kt ( ) given by Eq. (4.3) Kt () =, (4.3) α t where 0 α, is introduced instead of a traditional singular kernel given by Eq. (4.4), which was used in the modified Kuhn model, Kt () =. (4.4) t Accordingly, the creep compliance is modified and is given by Eq. (4.5) C t d ( ) B τ Jt () = A+ B e τ α, (4.5) τ and the creep compliance of the discrete generalized Kuhn model is N Bln( r) Ct αm m r Jt () = A+ r e α C. (4.6) m= 0 In the generalized Kuhn model the stiffnesses of the springs are not constant for all connected Kelvin elements, unlike the modified Kuhn model. The stiffness of the spring 74

89 and viscosity coefficient of the dashpot in the m th element are denoted by E m and η m respectively, are E m αm α r C = (4.7) B ln( r) and η m ( α ) m r =. (4.8) ( α ) C Bln( r) m m η r The retardation time in each element isτ m = =. A rheological representation of the m E C generalized Kuhn model is given in Fig The generalized Kuhn model can be also given an internal variable representation. Eq. (4.9) and Eq. (4.0) represent the evolution equations of the internal variables, describing the discrete generalized Kuhn model. ( α ) ( ) d d q ln m s C B r q m + = = s, (4.9) m( α ) τ η r m m θm C θm + = trσ, (4.0) m τ r A m where d q m and θ m are the deviatoric and volumetric components of the viscous strain tensors in the m th element, i.e. 75

90 d qm = qm + ( trq m), (4.) 3 θ = trq, (4.2) m m where q m is the total strain tensor in the m th Kuhn element and is the identity tensor of rank 2, s is the deviatoric stress and tr stands for the trace operator. The total viscous v strain tensor ε is N v ε = qm. (4.3) m= 0 The total strain tensor is the sum of elastic and viscous strain tensors, i.e. el v ε = ε + ε. (4.4) 4.3. Derivation for loss tangent In this section, the loss tangent for the generalized Kuhn model is derived. The stress strain relation for a viscoelastic material is given by t σ( t) = G( t, ) ε( ) + G( t, t ) ε( t ) dt, (4.5) 76

91 where Gtt (, ) is the relaxation function, t ' the dummy time, t the current time and the superimposed dot indicates the derivative with respect to t '. The integral over the history of strain in the equation is referred to as hereditary integral. Materials, whose constitutive equations contain such hereditary integrals, are referred to as materials having memory. If the strain is zero before a certain time t, then Eq. (4.5) results to the following equation, t σ() t = Gtt (, ) ε( t) + Gtt (, ) ε( t ) dt. (4.6) t As a consequence of the existence of an unique inverse, the above constitutive relation can be written as t ε() t = J(, t t ) σ( t ) + J(, t t ) σ( t ) dt, (4.7) t where Jtt (, ) is the creep or compliance function. For materials whose properties are time-homogenous (non-aging materials), Eq. (4.7) becomes t ε() t = J( t t) σ( t) + J( t t ) σ( t ) dt. (4.8) t 77

92 Performing integration by parts, the above equation yields t ε() t = J(0) σ() t + J ( t t ) σ( t ) dt, (4.9) t In compact notation involving singular kernel, Eq. (4.9) can be written as t ε() t = γ( t t ) σ( t ) dt, (4.20) t where γ() t = J(0) δ() t + J () t H() t, (4.2) where δ is the Dirac delta function and H is the Heaviside step function. Frequency representation is accomplished with the aid of Fourier transforms. See e.g Graham and Golden (988). Performing Fourier transform of Eq. (4.2), the following equation is obtained, iωt γω ( ) F γ() t γ() t e dt, (4.22) { } = = 0 78

93 where F{}. is the Fourier transform and γ ( ω) is the complex creep compliance. For the generalized Kuhn model, the creep compliance is given by Eq. (4.6), repeated here for conveninence, i.e. N Bln( r) Ct αm m r Jt () = A+ r e α C. (4.23) m= 0 From Eq. (4.23)and Eq. (4.2), the following equation is obtained, γ Bln r. (4.24) N Ct ( α ) m m r () t = Aδ() t + r e H() t α C m= 0 Fourier transform of Eq. (4.24) gives the complex compliance, N 2 αm ( α+ ) m Bln r C r Cωr γω ( ) = F { γ( t) } = A+ i α 2 2 2m 2 2 2m. (4.25) C m= 0 C + ω r C + ω r If i 2 γ ( ω) = γ ( ω) γ ( ω), then the loss tangent is defined as tanδ γ ( ω) γ ( ω) 2 =. (4.26) Frequently, γ ( ω ) and γ ( ) 2 ω are referred to as storage and loss modulus respectively. 79

94 Hence, from Eq. (4.25) and Eq. (4.26), the loss tangent for the generalized Kuhn model is given by tanδ = N ( + α ) m ω r 2 2m C m= 0 ω r + 2 C α N αm AC r + Bln r m= 0 ω r + C 2 2m 2. (4.27) Note that when α = 0, the modified Kuhn model is completely recovered. The complex modulus µ ( ω ) (often referred to as * G in the asphalt concrete and polymer literatures) for the generalized Kuhn model can be derived as follows. Using the following relation, between complex modulus and complex compliance functions, γ ( ω) =, (4.28) µ ω ( ) the complex modulus µ ( ω ) can be written as µω ( ) = = = γ ω A+ i α 2 2 2m 2 2 2m C m= 0 C + ω r C + ω r P+ iq N 2 αm ( α+ ) m 2 2 ( ) Bln r C r Cωr P + Q, (4.29) where P and Q are given by 80

95 P = A+ N 2 αm Bln r C r α m C m= 0 C + ω r Q = ( α + ) Bln r Cωr α C C r N m m m= 0 + ω The modulus of the complex modulus is given by * ( ) G µ ω = = P + Q 2 2. The loss tangent predicted by the generalized Kuhn model fits the experimental results of Dietrich et al. (998), see Bahuguna, (2003). In these tests, the asphalt specimens are subjected to frequency sweeps and the variation of phase angle with frequency is shown in Fig The model s fit is compared with the experimental results and it is shown in Fig As can be seen from the figure, the model s prediction is in good agreement with the experimental results. Sousa et al. (993) studied the influence of asphalt and aggregate type on the mix stiffness and the variation of * G with frequency for four mixes. The model fits of the generalized Kuhn model to these experimental results are shown in Fig. 4.4 and the corresponding optimized parameter values are tabulated in Table 4. below. The model fit is very good and it reflects the capability of the generalized Kuhn model in predicting the time-dependent recoverable deformations of asphalt concrete as well as its capacity to describe the energy dissipation of the material in the viscoelastic range. 8

96 V0W0 VT0 B0W0 BW A 0.67e e e e3 B 3.224e e3.244e e3 C e4.926e e e3 r α Table 4.. List of parameters for fitting of * G vs. frequency. The value of the parameter N, which is the number of Kuhn elements in the model, is 5. The symbols V0W0, VT0, B0W0 and BW listed in the above table refer to nomenclature adopted by the experimentalists to distinguish between various asphalt concrete mixes. The asphalt concrete mixes contained aggregates RB and RL and asphalts AAK- and AAG- (materials from the SHRP material reference library). The asphalt concrete mixes used for the experiments contain V0W0 - Aggregate RB, Asphalt AAG-, low asphalt content, low void content. B0W0 - Aggregate RB, Asphalt AAK-, low asphalt content, low void content. BT - Aggregate RL, Asphalt AAK-, high asphalt content, high void content. V0T - Aggregate RL, Asphalt AAG-, low asphalt content, high void content. In the following section, the generalized Kuhn model is further enhanced by including temperature effects, in order to model the time and temperature response of asphalt mixes. 82

97 4.4 Need for a temperature dependent model Bitumen is one of the main components of asphalt concrete mixes. It is a highly temperature-sensitive material, in other words, its rheological properties are temperature dependent. Bitumen exhibits three different characteristic behaviors within three temperature regions and these characteristic behaviors are taken as reference for the study of properties of asphalt concrete. At temperatures higher than 00 o C, bitumen behaves as a Newtonian fluid and is characterized by its viscosity parameters. For workability reasons of asphalt concrete, this viscosity parameter is kept within some acceptable limits. At intermediate temperatures (0 o C-85 o C), the range of interest for this study, bitumen exhibits viscoelasticity. The rheological properties are characterized by the complex shear modulus G * and the phase angle δ. In pavement design, minimum value of * G sinδ measured at 0 rad/sec frequency is desired in order to avoid early rutting, see e.g. Cerni, (200). At lower temperatures (below 0 o C), asphalt concrete is elastic and fragile. In these conditions, the material is more vulnerable to thermal cracks. As already mentioned, the intermediate range is the range of interest as far as viscoelastic properties of asphalt concrete are concerned. Any model developed to predict the long-term timedependent behavior of asphalt material, should also be temperature-dependent as the modulus is both time and temperature-dependent. Bituminous materials are thermorheologically simple materials and hence time and temperature can be interchanged, see e.g. Pagen (965), Monismith (966). This is explained in detail by the so-called time temperature superposition principle. The proposed temperature-dependent viscoelastic model is developed along the same lines. In addition, a novel approach based on the 83

98 internal variable formulation is proposed. A short note on time temperature superposition principle is given in the next section. Experimental studies on temperature dependence of rutting of pavements were performed by Harvey, at the Institute of Transportation Studies, U.C. Berkeley. It has been reported that the number of traverses of a single-axle load before a.3 cm rut is created, is much larger when the pavement temperature is lowered from 50 C (22 F) to 40 C (04 F) (temperatures are measured 5 cm below the surface). This suggests that failure due to rutting is delayed if the pavement temperature is low. Fig. 4.5 shows the variation of rut depth with number of cycles at two different pavement temperatures. A cylindrical asphalt pavement specimen when subjected to shear tests at constant height develops permanent shear strains after each loading cycle. A similar test was performed by Harvey, et al., at the Institute of Transportation Studies, at U.C. Berkeley. A plot of permanent shear strain versus number of shear load applications at various temperatures is shown in Fig It can be observed from the figure that the permanent deformation of the pavement specimen severely decreased as the temperature of the pavement is decreased from 60 C (40 F) to 40 C (04 F). These experiments strongly suggest that the response of asphalt pavement is highly temperature-dependent. 4.5 Time Temperature Superposition Principle When asphalt concrete is subjected to stress and higher temperatures, it exhibits accelerated deformations as if the time scale has been compressed. A qualitative plot of creep compliance versus time on log scale is given in Fig. 4.7, (from Huang (993)). From the plot it can be observed that at a given time, the creep compliance at a lower 84

99 temperature is lesser than that at a higher temperature. If a creep compliance is known at a reference temperature T o, then the creep compliance at any other temperature T is obtained by using a time temperature shift factor a T, which was first introduced for polymers, by Leaderman (943), See also, Ferry (950), Williams, Landel and Ferry (955). For asphalt concrete, based on numerous experiments, log ( ) linearly varying with temperature. The plot of log ( ) T a is observed to be T a versus temperature is a straight line as shown in Fig. 4.8, (FHWA 978). The shift factor for asphalt concrete is defined as a T t t T =, (4.30) To where t T is the time to obtain the creep compliance at temperature T, and tt o is the time to obtain the same creep compliance at temperature T o, see Pagen (965). The slope of the curve is given by t T log log ( at ) t To β = = T T T T o o (4.3) In other words, the time to obtain the creep compliance at temperature T is given by Eq. (4.32) 85

100 ( β ( )) t = t exp T T. (4.32) T To o where = loge 0. Note that in Eq. (4.3), logarithm is taken to the base 0 and in Eq. (4.32), exponential function exp is with respect to base e. From the above principle it can be easily seen that the value of creep compliance at any other temperature is obtained by scaling the time with respect to temperature. This is given in Eq. (4.32). Such temperature dependent behavior can be incorporated into viscoelastic models and the scaling of time is reflected by change in relaxation time. 4.6 The generalized Kuhn model with temperature The creep compliance as a function of t T o is given by N CtT Bln( r) o m αm r Jt ( T ) = A+ r e o α C, (4.33) m= 0 and similarly as a function of t T by N CtT Bln( r) αm m r Jt ( T ) = A+ r e α C m= 0. (4.34) Substituting Eq. (4.32) into Eq. (4.34), the following equation is obtained, 86

101 N Ct B ln( r) To m αm r Jt ( T ) = A+ r e α C, (4.35) m= 0 where ( β ( )) C = Cexp T To, (4.36) and ( β ( ) α ) B = Bexp T To. (4.37) The above equations show the temperature dependence of the parameters B and C. Using Eq. (4.27), the loss-tangent at temperature T is accordingly given by tanδ = N ( + α ) m ω r 2 2m C m= 0 ω r + 2 C α N αm AC r + B ln r m= 0 ω r + C 2 2m 2. (4.38) Thus, the temperature dependence of the viscosity coefficient can be shown as follows. * Let η m and η m be the viscosity parameter for the m th Kuhn element at temperature T and reference temperature T o. Recalling Eq. (4.8), and using Eq. (4.36) and Eq. (4.37), the following relation between the two viscosity parameters is established, ( α) ( α) * m m ηm r r exp ( ) ( ) ( β ( T T α α o )) η = C B ln( r) C Bln( r) = (4.39) m 87

102 Note that the stiffness of the spring remains independent of temperature but the relaxation time changes with temperature. * * * τm ηm Em ηm = = = exp( β ( T To )) (4.40) τ η E η m m m m So far, the temperature dependence of the parameters of the Kuhn element has been established for one-dimensional case. The same relations between the parameters could be established for three-dimensional cases using a novel approach to incorporate the temperature into the model through an internal variable formulation, which is presented below. This approach provides a powerful way of extending the time temperature superposition principle into multi-dimensional formulations. Differentiating Eq.(4.32) with respect to t T o, the following equation is obtained. dt dt T To ( β ( T To )) = exp (4.4) The evolution equation of the internal variables given by Eq. (4.9) at temperature T o can be written as dq η + q = s, (4.42) d m d m Em m dtt 0 88

103 dθm τm dt T0 + θm = trσ, (4.43) A and at temperature T by dq η + q = s, (4.44) d m d m Em m dtt dθm τm dt T + θm = trσ. (4.45) A Eq. (4.44) and Eq. (4.45) can be rewritten as η dq dt + q = s, (4.46) d m T0 d m Em m dtt dt 0 T dθ dt m T 0 τm + θm = trσ. (4.47) dt dt A T0 T From Eq. (4.4), Eq. (4.46) and Eq. (4.47) the following equations are obtained, dq η + q = s, (4.48) d * m d m Em m dtt 0 89

104 dθ τ + = trσ, (4.49) * m m θm dtt A 0 where * η m and * τ m represents the modified viscous parameter and modified relaxation time at temperature T respectively, where η η * m m dt T = = exp dt To ( β ( T To )). (4.50) τ τ * m m dt T = = exp dt To ( β ( T To )). (4.5) Eq. (4.50) shows the relation between the viscosity coefficient at temperature T and its value at the reference temperature T 0. This is identical to the relationship obtained in Eq. (4.39). Similarly, Eq. (4.5) shows the relation between the relaxation times at temperature T and its value at the reference temperature T 0 and this is identical to the relationship obtained in Eq. (4.40). The loss-tangent given by Eq. (4.38) is used to fit the experimental results reported by Cerni, (200). Bitumen specimens are subjected to a frequency sweep experiment at various temperatures. The plots of phase angle δ versus frequency in log scale at temperatures 6 o C, 5 o C and 25 o C are obtained. The parameters are estimated from model fit of the plots at 6 o C and 5 o C and using the values of these parameters, the temperature dependent parameter values (B and C) corresponding to 25 o C are calculated. 90

105 These calculated values are compared with the parameter values obtained by fitting the model to the experimental results at 25 o C. The parameter estimation is done by nonlinear optimization techniques and programming is done in MATLAB. The objective function used for the parameter estimation is given by Eq. (4.52) f 2 M ltmi, lt exp ti, = i= 0 lt exp ti, (4.52) where lt mi, and lt exp ti, are i th model and experimental loss tangent values and M is the total number of experimental data points. Fig. 4.9 shows the model fit at 6 o C, 5 o C, and 25 o C. The values of the parameters are listed in Table 4.2. Temperature A B C α r T =6 0 C B = C = T 2 =5 0 C B 2 =0.228 C 2 = T 3 =25 0 c B 3 = C 3 = Table 4.2. List of parameters for various temperatures fit. Using the parameters estimated from model fit of plots 6 o C and 5 o C, the slope β is calculated as shown. From Eq. (4.36) ( β ( )) C = C exp T T, (4.53) 2 2 9

106 where T, T 2, C and C 2 are given in Table 4.. From the Eq. (4.53), the value of the slope β is calculated as β = The value of the parameter B and C for 25 0 C are calculated from the following equation ( β ( )) C = C exp T T = (4.54) 3 3 ( β( ) α) B = B exp T T = (4.55) 3 3 These calculated values are almost equal to the values B 3 and C 3 predicted by the model fit which are given in Table 4.2. This provides an excellent validation of the model. The model is validated against a second set of experiments reported by Corte, (200). In this case, dynamic shear tests were conducted at various temperatures on four different Asphalt Concrete mixes with penetration grades (PG) 0/20, 25/35, 35/50, and 50/70. Phase angle variation with temperature at 7.8Hz has been recorded. Fig. 4.0 shows the experimental results and the model s fit to the experimental data. The parameters are tabulated in Table 4.3 and the number of Kuhn elements used for the fitting is 5. It is evident from the Fig. 4.0 that generalized Kuhn model predicts phase angle variation with respect to temperature with good accuracy. The reference temperature T 0 is taken to be 0 o C and hence the temperature dependent parameters B and C listed in the table below correspond to 0 o C. The values of B and C at any other 92

107 temperature can be obtained at by using Eq. (4.37) and Eq. (4.36) and thus the phase angle value at any temperature can be predicted by the model. Parameters PG 0/20 PG25/35 PG35/50 PG50/70 A B C α r β Table 4.3: List of parameters for Phase Angle vs. Temperature fit. 93

108 m=n m= E m = Bln r ( ) m r τ m = ;m = 0,,...N C m r η m = CB ln( r) η m A E m m=0 Figure 4.. Rheological Representation of Modified Kuhn Model 94

109 m=n m= E m αm r C = Bln r ( ) m r τ m = ;m = 0,,...N C m( α ) r ηm = ( α ) C Bln( r) α η m A E m m=0 Figure 4.2. Rheological Representation of generalized modified Kuhn model. 95

110 Figure 4.3. Experimental results and Generalized Kuhn model prediction for Asphalt Concrete. (Experimental data from Dietrich et al.,998 and model fit by Bahuguna, 2003) 96

111 Figure 4.4. Generalized Kuhn model fit for complex shear modulus versus frequency. 97

112 Figure 4.5. Rut Depth vs. Repetitions of loading at 40 o C and 50 o C (The data is from Harvey, Institute of Transportation Studies, U.C. Berkeley) 98

113 Figure 4.6. Permanent Shear Strain vs. Number of cycles of shear loading on asphalt pavement specimens. (The data is from Harvey, Institute of Transportation Studies, U.C. Berkeley) 99

114 Figure 4.7. Creep compliance at different temperatures. 00

115 Figure 4.8. Shift factor versus temperature. 0

116 Figure 4.9. Loss tangent vs. Frequency: Model fit to Experimental Data (Experimental data is from Cerni, 200) 02

117 Figure 4.0. Model fit for Phase Angle vs. Temperature at 7.8Hz (Experimental data from Corte, 200) 03

118 CHAPTER FIVE VISCOPLASTICITY 5. Introduction In the second chapter, the rate dependent irrecoverable deformation component of the asphalt concrete has been discussed in detail and an inclusion of a viscoplastic component in the constitutive model has also been stressed. The plot of permanent strain versus number of loading cycles on a log-log scale is a straight line for asphalt concrete. This behavior is very similar to ratcheting in metals and it is the primary consideration in choosing theory of viscoplasticity to model irrecoverable permanent deformations of asphalt. Theory of viscoplasticity has been successfully used to model ratcheting behavior in metals, see Krempl and Yao (987). 5.2 Viscoplastic Models The theory of viscoplasticity models rate dependent material behavior. Traditionally, three different approaches are followed in modeling viscoplastic materials; accordingly viscoplastic models reported in the literature belong to one of the three approaches, which will be discussed briefly next. First type of viscoplastic models are based on the existence of a yield surface. In models based on yield surface, the elastic domain is distinctly defined with the help of a yield surface and states of stress outside the yield surface lead to the evolution of inelastic deformations. This type of viscoplastic models was first introduced by Perzyna, (Perzyna, 963). This theory is a generalization of the classical theory of plasticity, and classical plasticity is included in as a special case. Perzyna s theory is based on the existence of a 04

119 yield function F: R 6 R, where R 6 denotes the six-dimensional Euclidean stress space, such that the viscoplastic strain rate is zero when F 0 and is finite when F > 0. Complying with this condition, the viscoplastic strain rate is given by ( F ) F vp o ij = γ Φ, σ ij ε o where γ is a material constant and 0 for F 0 Φ ( F ) =. Φ ( F) for F > 0 The function Φ ( F ) is chosen differently to describe different experimental results. The yield function can be expanded to include hardening phenomenon as well and the resulting function is called loading function, this is illustrated in Fig. 5.. The function F = 0 is the yield surface and within this surface the viscoplastic strain rate is zero. At stress points outside the yield surface, the viscoplastic strain rate is different than zero and is normal to the surface F coordinates outside the yield surface. F = C where C is the value of F corresponding to the stress = C belongs to the same family of surfaces as the yield surface and is called the dynamic loading surface. The surface F = 0 is called the quasistatic yield surface since it separates the region of stress space based on the value of the viscoplastic strain rates. This theory is further discussed in detail later in the chapter. 05

120 In the second type of models, there is no need of defining yield surfaces. According to Bodner (968), yielding is not a separate and independent criterion but is a consequence of a general constitutive law of the material behavior. A system of nonlinear equations describes the material behavior. In addition to state variables like stresses and strains, a set of internal variables are also involved in characterizing the material. These internal variables are process dependent and account for the history of loading. In literature, this type of models are referred to as unified viscoplasticity models. See e.g. Bodner and Partom (975), Valanis (97), and Lubliner (973). By a development in irreversible thermodynamics based on the concept of internal variables, Valanis (97) proposed a plasticity theory with out yield surface which he called the endochronic theory. The word endochronic comes from Greek, endo meaning inner and chronic meaning time. The constitutive relations are derived based on the principles of irreversible thermodynamics using internal variables. The derived relations are analogous to those of equations based on viscoelasticity. The difference is that the real time t is replaced by a intrinsic time measure z. This theory was developed and modified further by Valanis in a series papers published in the 70 s and 80 s. See. e.g. Valanis (97), Valanis (980), Valanis (984) among others. Valanis (980) modified his original version by introducing a new measure of intrinsic time which has made the theory more versatile and powerful in the analysis of plastic deformation. The details of endochronic theory and unified viscoplasticity models are not discussed any further in this chapter as it is not the primary focus of this study. The third approach is based on an additive decomposition of stress into equilibrium stress and overstress. Overstress is defined as the difference between the 06

121 current stress and the corresponding stress point on the quasi static surface. Quasi static surface is a multi dimensional generalization of the one dimensional equilibrium curve, which is defined as a limit where the rate of strain becomes zero. The reader is referred to Phillips (986) who invented the notion of equilibrium curve for more details. Also, a detailed discussion on equilibrium curve can be found in Panneerselvam (2002). The additive decomposition of stress state into equilibrium stress and overstress renders the relation between the rate independent equilibrium stress and rate dependent overstress to be uncoupled. In this theory it is assumed that the hydrostatic pressure does not cause yielding and does not influence the viscoplastic properties of the material. Taking this assumption into consideration, the overstress, also called as excess stress, is defined as the perpendicular distance from the current stress point to the quasistatic yield surface in the deviatoric stress space. See Fig In this theory unlike Perzyna s theory, the viscoplastic strain rate is not normal to the dynamic loading surface but it is normal to the quasistatic yield surface. In Perzyna s theory the viscoplastic strain rate is always normal to the dynamic loading surface. See Phillips et al. (973). In this respect, it is true to say that in Perzyna s theory, the current dynamic loading surface is always an isotropic expansion of corresponding subsequent static yield surface but this is not the case in the viscoplasticity theory based on overstress. For a detailed discussion on viscoplasticity theory based on overstress the reader is referred to Phillips et al. (973), Krempl (979, 987, and 998) among others. 07

122 5.3 Viscoplastic Model based on loading surface Phillips and Sierakowski (964) introduced in the theory of plasticity, the concept of a family of loading surfaces that are distinct from the yield surface. In their classic paper, difference between yield surface and a loading surface is very well discussed and it is evident that ratcheting behavior can be modeled using theories based on loading surfaces. Consider ratcheting phenomenon in one dimension as shown in Fig This figure represents stress strain curve of a material point in continuum when subjected to the following loading cycle. The material point is loaded upto point A, unloaded to point B and then is reloaded to point C. As seen in the figure, unloading process AB produces only elastic strain but reloading process BDC produces both plastic as well as elastic strains. The material point develops plastic strain when reloaded beyond the level represented by point D. This one dimensional process when generalized into multi dimension, shown in Fig. 5.4, is represented by two surfaces, one corresponding to point A in Fig. 5.3 and the other corresponding to the point D in Fig The first surface is called loading surface and the second one is called the yield surface. Note that both these surfaces depend on the amount and history of plastic strain. In Fig. 5.4, motion from O to a by means of loading paths corresponding to different total plastic strains, or to the same total plastic strain but through different histories of plastic strain, will produce different yield surfaces and different loading surfaces. The yield surface encloses purely elastic region and any path within the region enclosed by the yield surface produces only elastic strains. The loading surface can be defined as follows. From the point a on the loading surface, the motion can be either in a direction of loading producing additional plastic strains or in a direction of unloading without additional plastic strain. In addition 08

123 to these directions, there exist directions such that motions along these directions will not produce any plastic strain and also there is no change in yield or loading surface. Such directions are called neutral loading directions. As seen from Fig. 5.4, the loading surface encloses yield surface. Partial unloading from point a to point f produces intermediate loading surface passing through point f. Such intermediate loading surfaces can be defined for each and every point in the region between yield and loading surface. Thus, it can be seen that yield surface is a limit to such intermediate loading surfaces. Viscoplastic models based on loading surface are very similar to viscoplastic models based on yield surface but the concept of yield surface is replaced by loading surface. In this theory the main assumption is the existence of a loading surface f ( σ, T, ) q where q is the vector of internal variables and T is the temperature. The loading surface given by ( σ ) n f, T, q : R R is a continuous function and is dependent on stress state, temperature and an array of internal variables denoted by q. For f ( T ) σ,, q < 0 at a given set of internal variables and temperature, the loading surface represents a region in stress space such that the inelastic strain rate tensor i ε vanishes in that region but not outside it. The concept of the elastic domain follows as the region in stress space where the following inequality holds f ( σ T ),, q 0. (5.) In region of stress space occupied by loading surface, reloading produces plastic strains, while unloading produces only elastic strain. This is in contrast to regions 09

124 bounded by yield surface, where both reloading and unloading produces only elastic strains. From this statement, it can be deduced that yield stress is a limit to loading surfaces. The boundary of the elastic domain given by the surface f ( T ) σ,, q = 0 can be viewed as a yield surface. The accumulation of plastic strain upon reloading from loading surface facilitates the model to capture ratcheting, see Bahuguna (2003). This theory differs from classical plasticity theories in the fact that stress states outside the yield surface are permissible. In the elastic domain defined by a loading surface, the vanishing of the viscoplastic strain rate does not imply vanishing of all the internal variable rates. If this was the case, the description of the phenomenon of strain aging would not be possible. However, for materials where strain aging is not a significant phenomenon, the rates of internal variables could be assumed to be zero in an elastic domain. The set of internal variables are associated with the microstructure of the material and any change in the microstructure is reflected by changes in the internal variables. In case of aging materials, the microstructure changes even in the elastic region and such changes are captured by the internal variables. Thus, in the elastic domain, the rates in internal variables do not essentially vanish. In models based on Perzyna type, the rates of internal variables depend on the loading surface. Thus, the internal variables evolve even when viscoplastic strain rates vanish. Based on these arguments, the evolution equations of internal variables are formulated in terms of scalar function ( ) conditions. n g σ, T, q : R R which satisfies the following 0

125 > 0 if f ( σ, q, T ) > 0 g = 0 if f ( σ, q, T ) 0 (5.2) Perzyna(963) redefined the scalar function g as given below g = γ ( T) g( f), (5.3) where γ ( T ) is a temperature dependent viscosity coefficient and g ( f ) 0if f 0 = g( f ) for f. (5.4) > 0 The evolution equations for the viscoplastic models based on loading surface can now be expressed as: γ = η ( T ) ( σ, q) vp, (5.5) ε ( T ) Ω σ ( σ, q) γ Ω q = D, (5.6) η σ ( ( σ )) γ = g f, q, (5.7) where η ( T ) is the temperature dependent viscous parameter. In the above mentioned evolution equations, the viscoplastic model follows associative flow rule when the

126 viscoplastic potential ( σ ) n Ω, q : R R is equal to the loading surface and is non associative when viscoplastic potential is different from loading surface. 5.4 Viscoplastic model for Asphalt concrete The permanent shear strain accumulation with increasing number of loading cycles in a repetitive shear test is shown in Figure 5.5. If loading and unloading cycle is performed on asphalt concrete specimen up to a certain stress limit, accumulation of plastic strain is observed with increasing number of cycles. This behavior is similar to plastic strain accumulation in metals and this phenomenon is commonly called ratcheting. A general representation of ratcheting phenomenon in granular materials is shown in Fig In a cyclic loading within a fixed stress limit, classical plasticity theory predicts evolution of plastic strain in the first cycle only and cannot capture plastic strain developed in subsequent cycles. Hence, ratcheting behavior cannot be modeled using classical plasticity theory based models. The ratcheting phenomenon in rate dependent materials like asphalt concrete can be captured by viscoplastic models based on loading surfaces. In the past, viscoplasticity models were successfully used to model ratcheting behavior in metals. See Krempl and Yao (987). In the present section a viscoplastic model based on loading surface is proposed to capture the time dependent permanent deformations in asphalt concrete. The model is then appropriately modified to account for anisotropy in the material. Damage is also included in the model to characterize the softening behavior of asphalt concrete. The model presented in this section is an extension of the successful Perzyna type viscoplastic model proposed by Bahuguna (2003). The viscoplastic model proposed by Bahuguna 2

127 captures the evolution of viscoplastic strains in asphalt concrete materials. The details of this model are given below Loading Surface: In viscoplastic models based on loading surfaces, the focus is to define an appropriate loading surface. The loading surface equations in viscoplastic models are very similar to the yield surface equations of classical plasticity theory. Asphalt concrete consists of aggregates bound together by the viscous bituminous material. Due to the presence of these aggregates the plastic behavior of asphalt concrete is considered similar to the behavior of granular materials. The shear dilation developed during repeated simple shear tests is a strong evidence for this analogy. Vermeer (982) proposed a very successful plasticity model to capture the behavior of sand. The yield surface for the plasticity model proposed by Vermeer has been adopted as the loading surface for the viscoplastic model presented in this section. The form of the loading surface considered is f = II + α I Hκ (5.8) 2 3 where H is the hardening parameter, κ is the isotropic hardening variable and I, I 2, I3 are the invariants of the stress tensor. In pavements, the traffic passes in only one direction and hence consideration of only isotropic hardening is adequate. I = σ (5.9) ii 3

128 I I 2 = ( σσ ij ji I ) (5.0) det ( σ ) =. (5.) In Vermeer model the parameter α is defined as 2 9 sin φm α =, (5.2) 2 cos φ m where φ m is the friction angle. The internal friction angle indicates the degree of interaction among aggregate particles and is an important parameter for granular materials. Materials consisting of strong, cubicle aggregates will have high value of friction angle and the strength of such materials is strongly dependent on confining stress. Materials consisting of smooth, spherical particles will have smaller values for friction angles and the strength is not dependent on confining stress. According to a study conducted by Christensen et al (2000), the rutting behavior in asphalt concrete pavements was observed to be dependent on the internal friction angle along with various other parameters Flow Rule: The evolution of plastic strain in a deformation process is given by the flow rule. The general form of flow rule is similar to constitutive equation of a viscous fluid. Eq. (5.3) represents a flow rule in general form 4

129 i (,, q) i ν, (5.3) ε = hλ σ ε where λ represents the direction of the plastic flow and is given by Eq.(5.4) f λ =, (5.4) σ f is the loading surface and h is a scalar function of loading surface and enforces the defining property of the loading surface. The following form of h is chosen after observing evolution of permanent shear strains in RSST-CH, Bahuguna (2003). f h = η m, l x + κ (5.5) where η is the viscosity parameter and κ is the hardening parameter. x, m and l are scalar constants. The above form of h was taken specifically to model the straight line evolution of plastic strains on a log-log plot in the RSST-CH experiment. The parameter l is used to model the slope and m is used to model the variation of permanent shear strain between various stress levels. Following the form of h, the evolution equation for plastic strains can be written as 5

130 m vp f ε = ν. (5.6) η l ( x + κ ) where ν is the normal to the viscoplastic potential surface denoted by Ω and is given by Ω ν = σ. (5.7) If the viscoplastic potential Ω is same as the loading function, then the plastic flow is normal to the loading surface and the flow rule represents an associative flow rule. However, if the viscoplastic potential is different from the loading surface, the plastic flow is not normal to the loading surface and the flow rule represents a nonassociative flow rule. Associative flow rule predicts excessive plastic dilatancy and to avoid excessive plastic dilatancy in the model, a nonassociative flow rule is assumed. In non associative flow rule, the direction of the plastic flow is not normal to the viscoplastic loading surface and hence the evolution of plastic dilatancy is very less. As mentioned earlier, the viscoplastic potential is chosen different from the loading surface. In this case, the viscoplastic potential chosen is of Drucker-Prager form. It is expressed as Ω= J + γ I (5.8) 2 where J 2 is the second invariant of the deviatoric stress tensor given by 6

131 ij ij J 2 = ss. (5.9) 2 and S is the deviatoric stress tensor. Hence, the evolution equation for the viscoplastic strain can then be expressed as: m vp f ε = ν, (5.20) η l ( x + κ ) where ν is the normal to the viscoplastic potential, η is the temperature dependent viscosity parameter, κ is the hardening variable. x, l, and m are parameters. The normal to the viscoplastic potential is given by Ω s ν = = + γ I. (5.2) σ 2 J Hardening Law: The loading function described in Eq. (5.8) is a function of hardening variable κ which is an internal variable and this accounts for the hardening properties of the material. A discussion on hardening can be found in Lubliner (990), pg. 03. In a deformation process for the state of stress outside the loading surface, if a creep curve approaches the static equilibrium curve with the evolution of hardening parameters, the material is said to be hardening. If the creep curve tends to move away from the 7

132 equilibrium curve, the material is said to be softening. This argument can be described mathematically as given below. For a constant stress state and at a given temperature, the rate of change of loading surface is given by Y f = ( σ, q) q : q (5.22) If Y>0, the material is said to be hardening and is said to be softening otherwise. The limiting case Y=0, describes a perfectly plastic material. For the Vermeer loading surface given by Eq. (5.8), ( σ, q) f Y = : q = H κ (5.23) q The evolution equation of the hardening variable is given by m f κ =, (5.24) η l ( x + κ ) where f is the loading surface and x, l, and m are parameters as mentioned earlier. The form chosen for the hardening variable ensures that its rate is always positive. Hence, the hardening parameter H has to be positive to satisfy the condition for hardening. 8

133 5.5 Summary of Equations Loading Surface: f = II + α I Hκ (5.25) 2 3 Flow Rule: m vp f ε = ν (5.26) η l ( x + κ ) Ω s ν = = + γ I σ 2 J 2 (5.27) Hardening variable: Elastic Relation: m f κ = (5.28) η i ( ) l ( x + κ ) σ = D ε ε (5.29) where D is the elastic stiffness matrix. 5.6 Anisotropy The aggregates in asphalt concrete mix are seldom spherical in shape. Hence it is essential to include anisotropy in the constitutive model. Anisotropy in granular materials like soils and asphalt concrete is broadly classified into two types based on the cause of anisotropy inherent anisotropy and stress induced anisotropy. In case of soils, Casagrande and Carillo (944) first discriminated between the inherent anisotropy and stress induced anisotropy. The former is produced through the sedimentation of particles, while the latter is induced during the process of inelastic deformation. In case of granular 9

134 materials like soils the inherent anisotropy is observed to be caused by the following three factors.. The anisotropic distribution of contact normal which is indicative of the mutual relation among particles. 2. The preferred orientation of non-spherical voids. 3. The preferred orientation of non-spherical particles. In biaxial compression tests on two dimensional assemblies of rods conducted by Oda et al. (985), it is observed that the inherent anisotropy by () and (2) tends to be completely altered during the early stage of inelastic deformation. The inherent anisotropy by (3) still remains at later phases of deformation. In this study inherent anisotropy caused by the preferred orientation of nonspherical particles is considered. Accordingly, anisotropy is introduced in the material by the preferred orientation of non-spherical aggregate particles. Anisotropy is included in the model through a tensor F ij called the fabric tensor, first introduced by Oda and Nakayama (989). A tensor, called the fabric tensor is introduced as an index showing the anisotropy due to the preferred orientation of aggregate particles and this is based on the spatial arrangement of the constituent particles in the matrix. A typical direction of each particle is labeled by a unit vector n. This unit vector is parallel to the longest axis for particles which are ellipsoid in shape and for flaky particles like clay, n is taken perpendicular to the major plane. Then, a fabric tensor is introduced as F ij i j Ω ( ) = nn E n dω, (5.30) 20

135 in which (, 2, 3) n i = are the components of the unit vector n projected on the i orthogonal reference axes; and Ω is a solid angle. E ( n) is a density function such that E( n) dω corresponds to the rate of unit vectors oriented within a small solid angle dω. The density function satisfies the following condition. ( ) Ω= E n d. (5.3) Ω Granular materials exhibit symmetry about vertical axis making it easy to introduce anisotropy though a equivalent two dimensional fabric tensor formulation. To this end, the specimen is cut in the vertical and horizontal directions an the particles on the resulting surface are assigned unit vectors m similar to the unit vector n introduced in three dimensions. m is the unit vector pointing in the direction of the longest axis of a particle visible on the section. See Fig.(5.7). A fabric tensor F ij, which is a two dimensional equivalent of Eq. (5.30) is defined as follows: F ij i j Ω ( ) = mm E m dω (5.32) where Ω= 2π since all the unit vectors are placed on a plane. Consider a vertical section k including a horizontal axis x 3 and a vertical axis x. Let θ be an inclination angle of a unit vector m k (corresponding to a k th particle) to the horizontal axis as shown in 2

136 Fig.(5.7). Accordingly, the components of the fabric tensor F ij are calculated using Eq. (5.32) as follows. F M 2 k = Sin θ (5.33) M k = M k k F3 = Sinθ Cosθ (5.34) M k = F M 2 k = Cos θ (5.35) M k = where M is total number of measurements. Since F ij is a symmetric, second rank tensor. The tensor can be written in terms of principal values with corresponding principal directions as follws: F 2 = + ± F 2 + = ± ( F F33 ) ( F F33 ) F3 ( ) (5.36) θ 2F3 = arc tan θ 2 F F 33 (5.37) Where is an index measure to show the intensity of the preferred orientation of particles, which was first introduced by Curray (956) called as the vector magnitude. 22

137 = + M M M 2 k 2 k 2 ( cos 2θ ) ( sin 2θ ) (5.38) k= k= where k θ is the inclination angle of a unit vector k m corresponding to a th k particle on a π π two dimensional section of the material and ranges between and and M is total 2 2 number of measurements. This is illustrated in figure 5.7. The vector magnitude ranges from zero to one, zero is analogous to isotropic materials and one indicates that all the aggregates are oriented in one direction. This two dimensional fabric tensor is extended into three dimension based on the observation that the material exhibits symmetry about the vertical axis. Hence, the three dimensional fabric tensor can be written in terms of the vector magnitude as follows. ( ) ( + ) F = 0 ( + ) ( 3+ ) ( + ) ( + ) (5.39) Anisotropy is incorporated into the model by appropriately modifying the loading function by incorporating the fabric tensor into the loading function. The fabric tensor is incorporated into the loading surface by modifying the stress invariants as follows ( δ ) I = a + a F σ, (5.40) ij 2 ij ij 23

138 ii 2 ij ij 2 T ( ) ( ) I = aσ + a Fσ = atr σ + a tr F σ (5.4) ( 2 4 ) I = bδ δ + b F δ σ σ, (5.42) 2 6 ik jl 7 ik lj ij kl T ( ) T T ( ) ( ) I = 2bσ σ + 4b F σ σ = 2btr σ σ + 4b tr F σ σ (5.43) 2 6 ik jk 7 ik kj ij 6 7 ( ) I = cδ δ δ + c F δ δ σ σ σ. (5.44) 3 il kn jm 2 il kn jm ij kl mn ( σσ lj nlσ jn ilσσ ij nlσ jn ) 3 T T T ( σ ) ( σ) ( σ σ) I = c + c F 3 2 = ctr + ctr F 2 ( ) (5.45) where ( D ) 2 a = λ 2, (5.46) a b b c c 2 ( D ) 2 = 3λ 2, (5.47) 2 2 µ = ( 2D ) 2, (5.48) = µ ( 2D ) 2, (5.49) ( 2D ) 2 = ς, (5.50) 2 ( D ) 2 = 3ς 2. (5.5) 3 2 λ, µ and ς are anisotropy material constants which are experimentally determined. D 2 is the second invariant of the deviatoric fabric tensor and is given by D 2 4 = 3(3 + ). (5.52)

139 The value of the vector magnitude as mentioned earlier is a measure of the aggregate orientation. It depends on the aggregate shape, size and compaction method used in preparation of the asphalt concrete specimen. The value of the vector magnitude is measured by the image analysis of specimen cut sections. See Tashman et al. (200). Introduction of anisotropy into the viscoplastic model modifies the associated equations as follows. The loading surface is given by f = II + αi Hk, (5.53) 2 3 Flow rule is given by m vp f ε = ν (5.54) η l ( x + κ ) where ν is the normal to the viscoplastic potential. Viscoplastic potential is modified as follows: Ω= J + γ I (5.55) 2 where J 2 is the second invariant of the deviatoric stress tensor. 25

140 ( 2 δ δ 4 δ ) J = b + b F s s, (5.56) 2 6 ik ji 7 ik lj ij kl J = 2b s s + 4b F s s 2 6 ij ij 7 ik kj ij T ( ) T T ( ) ( ) = 2btr s s + 4btr F s s 6 7 (5.57) And J 2 = ( 4b + 8b F ) s+ 4b ( F F ) DJJ σ (5.58) where 4 3 s 2s2 2s3 2 DJJ 2 = 2s2 3 s 0 2 2s3 0 3 s Ω ν = = σ 2 J 2 ( ( )) ( 4b 8b F ) s 4b ( F F ) DJJ γ a I a tr( F) (5.59) For numerical implementation, the following matrices are introduced P = ; P2 = ; P3 = (5.60) 26

141 [ φpσ + φ2p2σ] (( Pσ) ( Pσ) ) + φ2 ( P3σ) ( P3σ) ν = + γ + φ ( ) T T 2 2 ( ai atr 2 ( F) ) (5.6) Where φ = 4b + 8b F (5.62) ( ) φ = 4b F F (5.63) Hardening variable is given by m f κ = (5.64) η l ( x + κ ) 5.7 Damage In this section, a damage model based on continuum damage mechanics (abbreviated as CDM) principles is developed to model the softening behavior in asphalt. The approach followed is purely phenomenological and not micromechanical as micromechanical approach requires a strong knowledge of the micromechanical behavior of the material. However, damage models based on phenomenological approach describe only the macroscopic constitutive behavior of the material but developed models are easy to implement. The micromechanical models are mathematically complex as compared to phenomenological models and often numerical implementation is difficult. The damage model developed in this section is incorporated in the viscoplastic model to capture the softening behavior of asphalt concrete. In the phenomenological 27

142 approach followed here, the material is treated as a continuum and the damage initiation at the microstructure level is assumed to be isotropic. Hence, the damage model included in this model is isotropic in which case, it is assumed that microcracks and voids are equally distributed in all directions. Effective stress theory proposed by Kachanov forms the basis for the damage model included in the study. The damage variable is a scalar unlike in anisotropic damage theories where damage variable is a tensor. The damage variable ω is defined as A ω = ;0 ω. (5.65) A 0 where A 0 is the initial area of the undamaged section and A is the area lost due to damage. It is a positive monotonic increasing function, i.e. ω > 0. In other words, the damage is irreversible. The function ψ defined as ψ = ω, is called continuity. It follows that A A ψ ω ψ A 0 = = ; 0 (5.66) 0 The effective stress denoted by σ a is defined as σ σ a =, (5.67) ψ 28

143 where σ is the nominal stress. The damage is incorporated in the viscoplastic model, presented earlier in this chapter. Utilizing Eq.(5.67) the stress tensor is updated with damage and the resulting stress tensor is called effective stress tensor. Invariants of this effective stress tensor are given as follows. c I I I = =, (5.68) ω ψ c I I I = = ψ ( ω) c I I I = = ψ ( ω), (5.69), (5.70) J = J c ( ω) ( ω), (5.7) T ( ) c T T J2 = 2btr 2 6 ( s s) + 4b7tr ( F s) s, (5.72) where I, I2, I 3 are the stress invariants defined in Eq.(5.4), Eq.(5.43) and Eq.(5.45). J 2 is the second invariant of the deviatoric strain and is given in Eq.(5.57). Eq.(5.68) to Eq.(5.72) represent stress invariants which are modified to incorporate both anisotropy and damage effects. Accordingly, the modified loading surface is given by f = II + α I Hk. (5.73) c c c c

144 The damage variable ω is assumed to be a function of the total plastic strain and is given by the following form, p d ( ε ) 2 ω = d, (5.74) where d and d 2 are parameters. Finally, a summary of equations for the viscoplastic model with both anisotropy and damage incorporated is given below. The numerical implementation of this model is shown in detail in the next chapter. Laoding Surface: f = II + α I Hk (5.75) c c c c 2 3 Flow Rule: Normal to the plastic potential: m c f vp ε = ν (5.76) l η ( x + κ ) 30

145 [ φpσ + φ2p2σ] ν = + γ ( ai + atr 2 ( F) ) (5.77) ( w) φ T T 2 (( Pσ) ( Pσ) ) + φ2( ( P3σ) ( P3σ) 2 ) Hardening variable: m c f κ = (5.78) l η ( x + κ ) These equations are supplemented with the following elastic relation. Elastic Relation: i ( ) σ = D ε ε where D is the elastic stiffness matrix for the second order hyperelastic model presented in chapter three. 3

146 Figure 5.. The quasistatic yield surface and the dynamic loading surface (Perzyna s theory) 32

147 Figure 5.2. Representation of Overstress in deviatoric stress space. 33

148 Figure 5.3. Loading Unloading cycle for a ratcheting behavior 34

149 Figure 5.4. Loading Unloading in stress space. 35

150 Figure 5.5. Evolution of permanent shear strain in RSST-CH for different asphalt concrete mixes. 36

151 Figure 5.6. Ratcheting behavior in granular materials (Marroquin and Herrmann, 2004) 37

152 X m θ X 3 Figure 5.7. Particle Orientation on Vertical Section, vector magnitude 38

153 CHAPTER SIX NUMERICAL IMPLEMENTATION 6. Introduction This chapter deals with the numerical implementation aspects of the complete constitutive model developed in the previous chapters. The constitutive model developed to describe the permanent deformations of asphalt concrete comprises of two parts: Hyperelastic model developed in chapter 3 and viscoplastic model developed in chapter 5. The complete model can be thought of as a model in series and accordingly, the total strain is additively decomposed into elastic and viscoplastic parts. e vp ε = ε + ε. (6.) As discussed earlier, the viscoelastic model developed in chapter 3 is not very crucial in predicting the rutting behavior of asphalt concrete and hence it is not included in the complete model. This component would be essential to model fatigue behavior and energy dissipation in asphalt concrete. Summary of equations for the complete hyperelastic-viscoplastic model, with anisotropy and damage incorporated, are given below. The details of these equations as well as the definitions of the symbols are given in chapter 5. Laoding Surface: f = II + α I Hκ. (6.2) c c c

154 Flow Rule: m vp f ε = ν. (6.3) η l ( x + κ ) Normal to the plastic potential: [ φpσ + φ2p2σ] ν = + γ ( ai + atr 2 ( F) ) (6.4) ( ω ) φ T T 2 (( Pσ ) ( Pσ )) + φ2( ( P3σ ) ( P3σ ) 2 ) Hardening variable: m f κ =. (6.5) η l ( x + κ ) where η,x, l and m are parameters. For the definition of all the symbols involved in the equations, see chapter 5. The above equations are supplemented by the elastic stress strain equations given by the second order hyperelastic model which was developed in chapter 3. The Elastic Relations are: W σ = e ε e ( ε ). (6.6) 40

155 6.2 Numerical implementation of the complete model For implementation, the stress and strain tensors are written in vector form as follows: [ ] σ = σ σ σ σ σ σ. (6.7) [ ] ε = ε ε ε ε ε ε. (6.8) Basic Algorithmic Setup: Strain-Driven Problem Let [ 0,T] Rbe the time interval of interest. At time t [ 0, T], it is assumed that the total strain and viscoplastic strain fields, as well as internal variables are known, that is ε : total strain tensor ε n vp n : viscoplastic strain tensor κ n : isotropic hardening variable are given at time t n. The elastic strain is trivially obtained by Eq. (6.9) ε = ε ε, (6.9) e p n n n and the stress tensor is obtained using the elastic stress strain relationships. W σ = e ε e ( ε ), (6.0) 4

156 where W is the strain energy function and which is given by W = ( b I + b I ) + b I + b I I + b I, (6.) where I, I 2, and I 3 are the first, second and third invariants of strain respectively and b 2, b 3,..b 6 are parameters associated with the hyperelastic model, see chapter 3. The stress components can be calculated using Eq. (6.0) 2 ( 2 3 ) ( ) σ = bi + bi + bi δ + b + bi ε + bε ε. (6.2) ij ij 5 4 ij 6 im mj i Let un+ be the incremental displacement field, which is assumed to be resulting from the i th iteration of the global equilibrium equations at time step t n +. The basic problem is to update the field variables to t [ T] n+ 0, in a manner consistent with elastoviscoplastic constitutive equations developed in the previous chapters. The problem is strain driven in the sense that the total strain tensor is updated according to i ( ) ε = ε + u. (6.3) s n+ n n+ s where ( ) denotes the symmetric gradient operator, the subscript n + refers to the values at time step t n + and subscript n refers to the values at time step t n. The evolution equations for the continuum problem are transformed into discrete, algebraic 42

157 equations by applying an implicit backward Euler difference scheme, which is first order accurate and unconditionally stable. Applying the backward-euler difference scheme and using initial conditions, a system of coupled nonlinear equations is obtained, which is given below. m vp vp fn+ ν n+ εn+ = ε n + t, (6.4) l η x + κn+ κ = κ + f t, (6.5) n+ n+ n l η x + κn+ m where the subscript n + refers to the values at time step t n + and subscript n refers to the values at time step t n, t is the time step increment, i.e. t = tn+ tn. These equations are supplemented by Eq. (6.0) and Eq. (6.3). This system of nonlinear coupled equations is solved under the general framework of predictor-corrector method. The predictor phase consists of an elastic problem and the internal variables are assumed to be frozen. In the corrector phase, the internal variables evolve, and the total plastic strain and hardening parameter are updated to time step t n +. This is obtained as a solution of the evolution equations Eq. (6.4) and Eq. (6.5) by an iterative Newton scheme (in multi dimension) as shown later in the chapter. The predictor-corrector method is manifest as an elastic viscoplastic operator split as shown in Table 6.. See Simo and Hughes (998), pg

158 ε vp Total Elastic Predictor Viscoplastic Corrector s s ε = u ε = u ε = 0 m ( ) f f = η σ f κ = η m l ( x + κ ) l ( x + κ ) ( ) ε vp = 0 κ = 0 ε vp m f f = η σ f κ = η l ( x + κ ) m l ( x + κ ) Table 6.. Elastic-Viscoplastic operator split Predictor Phase: In the predictor phase the trial state is assumed to be elastic and therefore the internal variables do not evolve in this phase. From a physical standpoint the trial elastic state is obtained by freezing plastic flow during the predictor phase. In an elastic state where f + <, the rate of change of internal variables is zero and also the rate of change of trial n 0 plastic strain is zero. Internal variables and plastic strain are updated at time step t n + as follows: ε ε, (6.6) vp = n+ vp n κ = + κ. (6.7) n n 44

159 trial However in case of plastic step, where, f + > 0, the values of the internal variables and plastic strain have to be updated based on evolution equations given by Eq.(6.3) and Eq.(6.5). This is done in the corrector phase. The trial elastic state is given by the following three equations. n ( u ) ε = ε ε = ε + ε, (6.8) etrial vp s i vp n+ n+ n n n+ n ( ) ( ) W ( u ) σ = W ε = ε + ε, (6.9) trial etrial s i vp n+ n+ n n+ n κ + = κ. (6.20) trial n n etrial where ε denotes the elastic trial state. The stress state and therefore the stress n+ invariants are then updated with anisotropy and damage as discussed in section 5.6 and section 5.7 in chapter 5. The damage variable ω is calculated from Eq. (5.63) using the value of the total plastic strain at time t n. From Eq. (6.2), the loading function is calculated as (, ) f = f σ κ = I I + αi Hκ, (6.2) trial trial trial c c c trial n+ n+ n+, n+ 2, n+ 3, n+ n+ c c where I, n, I and c + 2, n+ I 3, n+ are the three equivalent stress invariants which are computed using Eq. (5.57), Eq. (5.58) and Eq. (5.59) at t n +, by using the trial stress values. The superscript trial on the invariants are omitted for simplicity. 45

160 In Eq.(6.4), the symbol ν n + denotes the direction of plastic flow which is normal to the plastic potential given at t n + time step by Eq. (6.4). The trial normal to the plastic potential is obtained from Eq. (6.4) and is given by ν φ Pσ + φ Pσ ( ai atr( F) ) φ trial T trial trial T trial 2 (( Pσn+ ) ( Pσn+ )) + φ2( ( P3σn+ ) ( P3σn+ 2 )) trial trial trial n+ 2 2 n+ n+ = + γ + 2 ( ω ) where P, P 2 and P 3 are the three matrices introduced in Eq. (5.49). These matrices are mainly introduced for computational implementation purposes. In plasticity models, the loading unloading criteria are imposed using the Kuhn- Tucker conditions however, in viscoplastic models, loading-unloading is decided solely trial from f n + according to the following conditions. trial vp f + < 0 elastic step κ = 0, ε = 0 (6.22) n trial vp f + > 0 plastic step κ > 0, ε > 0 (6.23) n Here, it should be emphasized that in our case, a theory of viscoplasticity has been employed; with one major difference with the theory of plasticity being that the concept of the loading surface has been introduced instead of the concept of yield surface of plasticity. In case of classical plasticity, the Kuhn Tucker conditions impose an additional constraint on the stress state i.e. the stress state always lies on the yield surface. This 46

161 condition renders a classical plasticity problem to be a constrained optimization problem. A detailed description of a robust numerical scheme for solving the constrained system of equations can be found in Bahuguna (997). In case of Perzyna type of viscoplasticity where the concept of loading surface is accepted, the Kuhn Tucker conditions do not impose any additional constraints. The absence of such conditions has significant computational implications. In this case, the loading-unloading conditions are given by Eq. (6.22) and Eq. (6.23) Corrector Phase: trial In the corrector phase, where f + > 0, the values of the internal variables and n plastic strain have to be updated based on the evolution equations given by Eq. (6.3) and Eq. (6.5). The initial values of the variables in the corrector phase will be those of the trial solution. ( 0) ε ε, (6.24) vp n+ = ( 0) n n vp n κ = + κ, (6.25) ( 0) σ = σ, (6.26) trial n+ n+ where the terms in the parenthesis in the superscript refers to the iterate number. In this phase, value of the isotropic hardening variable and the value of the viscoplastic strain are updated at time tot n +. Eq. (6.4) and Eq. (6.5) can be rewritten in the form F(x)=0 by defining the residuals ( k ) k R + and S + given by n ( ) n 47

162 R S ( ) ( ) ( k) ( k) k vp k vp n+ n+ n+ n+ n ( k ) η x + κn+ m = ε ε f ν t, (6.27) ( ) ( ) ( k ) = κ f κ t. (6.28) k k n+ n+ n+ n ( k ) η x + κn+ m where ( k) c( k) c( k) c( k) ( k) f = I I + α I Hκ. (6.29) n+, n+ 2, n+ 3, n+ n+ Linearization of the residual equations yields the following Newton scheme: m ( k) ( k) ( k) k m k ( k) vp( k) f n+ fn ν n f + + n+ νn+ n+ ε n+ lk ( ) lk ( ) η η x+ κ η n+ x+ κn+ R + m t t m ( k) ( k) f ν η + lk ( ) ( x κn+ ) lκ ( kl ) n+ n+ + 2 n+ = t 0 (6.30) ( k) m ( k) ( k) fn t fn t ( kl ) ( k) S κ + + n n m fn l κ κ n+ n+ 0 η + = η κ η, (6.3) + κ ( kl ) ( x + n ) ( k) ( kl ) ( x n ) + + m where ( ) ( k) ( ) ( k+ ) ( ) ( k) = at time step t + and (k) denotes the current iterate. ν n+ is obtained from Eq. (6.4). n 48

163 ( k) ( k ) φp n φ2p σ σn+ T ( ) (( ) ( )) T ( ) ( ) ( ) ( ω) φp n φ2p σ + + 2σn+ φ( ( Pσn+ ) ( P σn+ ) ) + 2φ2( ( P3σn+ ) ( P3 σn+ ) ) 3 φ T T 2 ( k) ( k) ( k) ( k) 2 (( Pσn+ ) ( Pσn+ ) ) + φ2( ( P3σn+ ) ( P3σn+ 2 )) φ T ( k) ( k) ( k) k 2 (( Pσn+ ) ( Pσn+ ) ) + φ2 P3σn+ P3σn+ 2 ( k ) ν n+ = T ( k) ( k) ( k) k k k The variation in loading surface is given by Eq. (6.32) T f = Z σ H k, (6.32) where ( σ 2 σ 2 σ 2 ) 2σσ 3σσ 2σσ σ 2 σ 2 α ( σσ σ 2 ) ( σ 2 σ 2 σ 2 ) 2σ σ 3σ σ 2σ σ σ 2 σ 2 α ( σ σ σ 2 ) ( σ 2 σ 2 σ 2 ) 2σσ 3σσ 2σσ σ 2 σ 2 α ( σσ σ 2 ) Z = AB C Z = AB C Z = AB C Z = AB 2σ σ + 2σ σ + 2σ σ + 2αC σ σ σ σ, ( ) ( ) ( 2σ σ 2σ σ 2σ σ ) 2α ( σ σ σ σ ) ( 2σ σ 2σ σ 2σ σ ) 2α ( σ σ σ σ ) Z = AB C, Z = AB C, ,,, a a = A; 2 6 4b c2 b + = B; c + = C. 3 3 a, a2, b6, b7, c, and c 2 are the anisotropy parameters. From Eq. (6.32), written as follows ( k ) can be f n + ( k) ( k) T ( k) ( k) f = Z σ H κ. (6.33) n+ n+ n+ n+ 49

164 In classical plasticity, the corrector phase of the algorithm is based on return mapping algorithm. Return mapping algorithm consists of a relaxation procedure towards the yield surface, which in turn is continuously evolving. In case of viscoplasticity with a loading surface, solution of the algebraic system of nonlinear time-discretized equations is obtained by an iterative Newton scheme in multi-dimensions as shown below. In this case, a yield surface on which the trial solution returns does not exist. For simplicity, the linearized residual equations given by Eq. (6.30) and Eq. (6.3) can be rewritten as two equations in two variables, i.e. ( k) vp( k) ( k) ( k) ( k) ( k) R + ε + X2 σ + X κ = 0, (6.34) n+ n+ n+ n+ n+ n+ ( k) ( k) ( k) ( k) ( k) S + Y2 σ + Y κ = 0, (6.35) n+ n+ n+ n+ n+ where m ( k) ( k) ( k) ( k) ( k ) f Z ν f X2 = m t ( ) * XX2 k η + η x κ η + x + κ ( ω) XX 2 n+ n+ n+ n+ n+ l l n+ ( k ) n+ [ φ P + φ P ] 2 2 T φ ( k) ( k) ( k) k 2 (( Pσn+ ) ( Pσn+ ) ) + φ2 P3σn+ P3σn+ 2 φ Pσ + φ Pσ φ Pσ P + φ Pσ P φ 2 2 m ( k ) T ( ) (( ) ( )) T T ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) 3 T T ( k) ( k) ( k) ( k) 2 (( Pσn+ ) ( Pσn+ ) ) + φ2( ( P3σn+ ) ( P3σn+ ) ) = ( k) ( k) ( k) k X n+ 2 2 n+ n+ 2 3 n+ 3 m m ( k) ( k) ( k) ( k) ( k) l ( k ) n+ νn+ n+ νn+ κn+ n+ = m ( kl ) 2 η + kl η( x κ n ) η κ + ( ) ( x n ), (6.36), f H t f l t, (6.37) 50

165 Y2 ( ) ( k) ( k) T + + ( kl ) ( x + n+ ) f Z t = m η η κ k n n n+, (6.38) m m ( k) ( k) l ( k ) f n+ H t f n+ lκ n+ t n+ = + m ( kl ) 2 η + kl η( x κ n ) η κ + Y ( ) ( x n ). (6.39) Note that total strain is constant in the corrector phase, therefore ( k) e( k) vp( k) ε = ε + ε =, (6.40) n+ n+ n+ 0 and thus stress increment can be written as ( ) ( ) ( ) ( ) ( ) ( ) D( ) σ = D ε ε = ε ε, (6.4) k e k e k e k vp k n+ n+ n+ n+ n+ Solving the Eq. (6.34) and Eq. (6.45), and utilizing Eq. (6.4), the plastic strain increments and increment in hardening parameter can be computed, ( k) ( k) ( k) ( k) ( k) vp( k ) ( k ) ( k ) Xn+ Y2n+ D n ( k ) Xn S + + n+ n+ X2n+ Dn+ R ( k) n+ ( k) Yn+ Yn+ ε = + +, (6.42) ( k) ( k) ( k) ( k) vp( k) κ n + = S ( ) n 2 k + + Y n + n + n + Y D ε. (6.43) n+ 5

166 Substituting Eq. (6.42) into Eq. (6.4), the stress increment can be obtained. In the above equation, D ( k ) n+ is the Hessian matrix associated with the second order hyperelastic model. Having obtained ( ) ( ) ( ) ε, κ, σ, the state variables are updated for the vp k k k n+ n+ n+ (k+) iterate as shown below ( k+ ) ( k) ( k) σ = σ + σ, (6.44) n+ n+ n+ ( + ) ( ) ( ) ε = ε + ε, (6.45) vp k vp k vp k n+ n+ n+ ( k+ ) ( k) ( k) κ = κ + κ, (6.46) n+ n+ n+ Using these updated values, new residuals ( k R + ) + and ( k S + ) + are computed from Eq. (6.27) and Eq. (6.28). If these residuals and the loading surface are within some specified tolerance limits, iteration moves to next step, else, the whole process is repeated. The complete 3 dimensional implementation procedure is summarized in Table 6.2. n n. Predictor Phase Compute elastic predictor ( 0) s σ = W u ε vp ( ( ) ) n+ n+ n Update with damage ( 0) ( 0) = σ n+ σ n+ ω Compute loading surface IF ( 0) ( n ) 0 f ( n ) ( 0) ( 0) = f σ n+ + f σ + THEN Elastic phase vp vp Set ε = n+ ε n and EXIT 52

167 2. Corrector Phase ( 0) ( n ) ELSE f σ + 0 Plastic phase GO TO 2. ENDIF ( k) s vp( k) ( ( u ) ) σ = W ε Update with damage n+ n+ n ( k ) ( k ) = σ n+ σ n+ ω 2a. Compute Residuals f R ( n ) ( k) ( k) = f σ n+ + ( ) ( ) ( k) ( k) = ε ε f ν t k vp k vp n+ n+ n+ n+ n ( k ) η x + κn+ ( k ) ( k) ( k) f n+ Sn+ = κn+ κn t ( k ) η x + κn+ 2b. Check Convergence IF ( n ) ( k) ( k) R > TOL or f σ > TOL THEN n+ + 2 Compute k th increments ( k ), ε κ n + ( ) vp k n+ m m Update Stress Tensor, Plastic Strains and Hardening Parameter ( k+ ) ( k) ( k) σ = σ + σ n+ n+ n+ ( + ) ( ) ( ) ε = ε + ε vp k vp k vp k n+ n+ n+ ( k+ ) ( k) ( k) κ = κ + κ n+ n+ n+ ω = d vp( k + ) ( εn+ ) d2 Set k k+ and GO TO 2a. 53

168 ELSE ENDIF EXIT vp Set ε = ε ( ) vp k n+ n+ Table 6.2: Predictor Corrector Algorithm for Hyperelastic-Viscoplastic Model Global Solution The algorithm described so far is for solving the constitutive equations at a Gauss point. The process summarized in Table 6.2 determines the values of the plastic strain ε + and hardening parameter n p n κ + for given strains s i n+ = n + ( un+ ) ε ε at a Gauss point. The algorithm developed in the preceding section needs to be integrated with the global governing equations to solve boundary value problems within the context of finite element method. In other words, the problem can be stated as follows. Find the admissible displacement field u(., ) t such that the weak form of the equilibrium equation is satisfied. The stresses in the equilibrium equations satisfy the local constitutive equations, which is detailed in Table 6.2. The complete problem statement for solving a boundary value problem can be stated as follows. At time t n + find admissible u n+, the updated displacement field vp un+ = un + u n+, the updated internal variables { n+, κn+ } ε and the stress field σ n+ such that the following conditions are satisfied. int ext int. The equilibrium condition F ( σ ) F = where ( σ ) n+ n+ 0 F n+ is the global internal ext force vector obtained through the element internal force vector and F n + is the global 54

169 external force vectors obtained through the element external force vector. See Simo and Hughes (997), pg. no The constitutive equations given in Table 6.2 The solution to the problem is obtained by an iterative solution procedure both at the local and the global level. The solution procedure at the local level has been described in detail in the previous section. The solution at the global level is given by the standard finite element iterative solution. See for e.g Simo and Hughes (997), pg. 50, 5 for a more detailed description. To complete the global procedure, the algorithmic tangent moduli, which is defined as the derivative of stress with respect to strain at time step t n + after convergence at the local level, is calculated. See e.g. Nagtegaal (982), Simo and Taylor (985), Simo and Huges (988) and Lubliner (990). The algorithmic tangent moduli for the complete model is derived as follows. C ε ε ε ε e vp vp n+ n+ n+ n+ alg = = + = D + σ n+ σn+ σn+ σn+, (6.47) where D is the Hessian matrix which comes from the hyperelastic model and m m f ν t f ε σ n+ n+ n+ m vp f ( ) η x κ n n+ + ν + σ n+ n+ n+ = t + m l n η ( x κn ) σ n+ fn+ tlκ n+ + t 2 η ( x + κn+ ), (6.48) 55

170 Hence from Eq. (6.47)and Eq. (6.48) algorithmic tangent moduli can be written as C alg ε = D + σ vp n+ n+. (6.49) The solutions to boundary value problems are shown in the next chapter. However, the model predictions to experimental results, when they are interpreted as a material point, are presented here. The algorithm described in Table 6.2 is implemented in MATLAB, and the model is validated against triaxial experimental results. 6.3 Triaxial Experiments The triaxial experiments considered in this study were performed at the Texas A&M University by Tashman et al. See Tashman et al. (2004). The experiments were strain controlled triaxial compressive strength tests, tested at five displacement rates and three confining pressures. The displacement rates were 0.mm/min, 0.5mm/min, 2.5mm/min, 2.5mm/min and 62mm/min and confining pressures were 0-psi, 5-psi and 30-psi. All the tests were conducted at 30 o F. The cylindrical test specimens were compacted using a ServoPac gyratory compactor to a size of approximately 4 inch diameter and 6.2 inch height. The applied load deformed the specimens at strain rates of 0.066%/min, 0.38%/min,.6%/min, 8.03%/min and 46.4%/min. At an average axial strain of approximately 0.5%, the material started to dilate and the stress-strain curve deviated from the straight line. This level was considered by the experimentalists to mark the initiation of plastic deformation. 56

171 6.4 Parameter Estimation The hyperelastic viscoplastic model s response was compared to experimental data for strain rates 46.4%/min, 8.03%/min and.6%/min and at confining pressures 0- psi, 5 psi and 30 psi. The model was fitted to axial stress vs. viscoplastic axial strain curves and parameters were evaluated based on a nonlinear optimization scheme. The objective function for the optimization procedure is a least square function defined as N i= ( ) 2 i expt f = σ σ. (6.50) where N is the number of data points. σ i is the value of the stress predicted by the model for the i th data point and σ exp ti, is the i th experimental data point. The stress σ i is computed using the algorithm described in Table 6.2. The minimization of the objective function was performed using the optimization toolbox in MATLAB. The values of the parameters are tabulated in Table 6.3 for 0-psi and Table 6.4 for 5-psi confining pressure. The model s fit to the various experimental data is shown in Fig. 6. and Fig Fig.6.3 and Fig. 6.4 show the model fit for experimental data for 30-psi confining pressure and corresponding parameter values for the 30-psi confining pressure experiment is tabulated in Table 6.5. The values of the friction angle decreases with increase in rate of loading while the viscosity parameter increased with increase in rate. This is in complete agreement with the observations reported by Nijboer (948) based on his experimental studies on 57

172 asphalt concrete. Nijboer describes the basic principle of triaxial strength test and develops a comprehensive theory for explaining the behavior of bituminous materials during triaxial test. The same observations were also reported by Hewitt (964) based on triaxial experiments on granular materials. For granular materials, Hewitt found that cohesion increased with increase in loading rate while friction angle decreased with increase in loading rate. His observations were primarily relying on triaxial strength testing of granular materials. The effect of friction angle and cohesion on the triaxial strength tests were also studied by Christensen et al. (2000). Their experimental observation are in agreement with the observation reported by Nijboer. 6.5 Numerical Implementation of the combined hyperelastic viscoelastic viscoplastic model In this section, the numerical implementation of the combined hyperelastic viscoelastic viscoplastic model is presented. Unlike the elastic viscoplastic case, in which the predictor phase consists of purely elastic relations, the trial stress state is found from a solution of a system of nonlinear equations, resulting from the fact that during the predictor phase the viscoelastic strains associated as internal variables with the viscoelastic model change. The basic algorithmic setup is strain driven problem and the problem statement can be stated as follows. Let [ 0,T] Rbe the time interval of interest. At time t [ 0, T] the total and plastic strain fields and internal variables are known, that is ε : total strain tensor n p ε : viscoplastic strain tensor n n, it is assumed that 58

173 κ n : isotropic hardening variable v ε n : viscoelastic strain tensor are given at time t n. Given the incremental displacement field u, which results form the iteration of the solution of the global equilibrium equations at time step t n +, the basic problem is to update the field variables to t n + in a manner consistent with elasto-viscoplastic constitutive equations described in the previous chapters. The evolution equations of the internal variables associated with viscoplastic model are given earlier in the chapter. The evolution equation for the viscoelastic strain component is given by the following equation. ( α ) ( ) d d q ln m s C B r q m + = = s (6.5) m( α ) τ η r m m θm C θm + = trσ (6.52) m τ r A m where d q m and θ m are the deviatoric and volumetric components of the viscous strain tensors in the m th element, i.e. d qm = qm + ( trq m), (6.53) 3 θ = trq, (6.54) m m 59

174 where q m is the total viscoelastic strain tensor in the m th Kuhn element and the total viscous strain tensor ε v and the deviatoric stress tensor s are given by the following equation, v ε = + θ I, (6.55) N N d qm m= 0 3 m= 0 m s = σ ( ). 3 trσ (6.56) where is the identity tensor of rank 2 and σ the stress tensor. The evolution equation is time discretized as follows. d d ( ) ( ) ( α ) q C t C t C t m = ( ) ( ln ) n m m B r P n m α + m n+ r q r r σ (6.57) C t C t C t θm n+ m θm n m m σ n+ I (6.58) r r r A ( ) = + ( ) + + tr ( ) where P is the projection matrix by means of which the deviatoric stress is expressed in terms of stress tensor. For convenience, let X(m), Y(m), and Z(m) be three constants defined as follows. ( ) X m C t = + m r (6.59) 60

175 ( α ) C t Y( m) = Bln r m( α ) r (6.60) ( ) Z m C t = (6.6) m r A Using these constants, Eq. (6.57) and Eq. (6.58) can be rewritten in the following form d d ( m) X ( m)( m) X ( m) Y( m) P n + q q σ (6.62) = + n+ n ( ) ( )( ) ( ) ( ) ( ) θ = m X m θ + n m Y m Z m tr σ I (6.63) + n n+ The total viscoelastic strain at ( ) th v n + step given by ε n+ is obtained by summing the viscous strains associated with each of the Kelvin element, ε = + θ I (6.64) N N d ( q ) ( ) ν n+ m n+ m n+ m= 0 3 m= 0 Adding all the deviatoric and volumetric parts of strain for each of the Kelvin element, we obtain N N N d d ( m) = X ( m )( m) + X ( m ) Y ( m ) P n + q q σ (6.65) n+ n m= 0 m= 0 m= 0 N N N ( m) = X ( m )( ) Y ( m ) Z ( m ) tr ( ) n+ m + n n I + θ θ σ (6.66) m= 0 m= 0 m= 0 6

176 The stress tensor can be decomposed into deviatoric and volumetric parts as follows. σn+ = tr ( σ n+ ) I +s (6.67) 3 In the predictor phase the trial stresses are computed as follows: ( ) σ = D ε ε ε (6.68) trial p v n+ n+ n+ n+ Using Eq. (6.67)and Eq. (6.68), Eq. (6.65)and Eq. (6.66) can be rewritten as linear equations in two variables; the total deviatoric strain and total volumetric strain. I + X m Y m PD q + Y m X m PD N N N N d ( ) ( ) ( ) ( ) ( ) ( θ ) m n+ m n+ m= 0 m= 0 3 m= 0 m= 0 N = X m q + X m Y m PD n m= 0 m= 0 N d p ( )( m) ( ) ( ) ( εn+ εn+ ) (6.69) I + X m Z m I P D + Z m X m I P D q N N N N d ( ) ( ) ( ) ( θm) 3 ( ) ( ) ( ) n ( m) + n+ m= 0 m= 0 m= 0 m= 0 N = X m + X m Z m I P D m= 0 m= 0 N p ( )( θm) ( ) ( ) 3( ) n ( εn+ εn+ ) (6.70) Solving Eq. (6.69) and Eq. (6.70) for the total deviatoric viscoelastic strain and total volumetric viscoelastic strain, we obtain 62

177 m d ( q ) ( ) ( m a 2 a b b 2 a 2 c b c2) =, (6.7) n + n+ = (6.72) m ( θ ) ( ) ( m I a a2 b 2 b a c b 2 c2) where a, a2, b, b2, c, and c 2 are parameters defined as follows. N N a = I + X m Y m PD a2 = Y m X m PD m= 0 3 m= 0 ( ) ( ) ; ( ) ( ), (6.73) N N b = I + X m Z m I P D b2 = Z m X m I P D m= 0 m= 0 ( ) ( ) ( ) ; 3 ( ) ( ) ( ),(6.74) c = X m q + X m Y m PD ε ε N N d ( )( ) p m ( ) ( ) ( n+ n+ ), (6.75) n m= 0 m= 0 c = X m + X m Z m I P D N N p ( )( θ ) m ( ) ( ) 3( ) n ( εn+ εn+ ), (6.76) 2 m= 0 m= 0 The trial stress tensor can then be calculated using Eq. (6.68) p ν ( ) σ = D ε ε ε (6.77) trial n+ n+ n n+ trial p d σn+ = D εn+ εn D qm + θ n m I + n+ m= 0 3 m= 0 N N ( ) ( ) ( ) (6.78) If the computed trial stress satisfies the loading condition, 63

178 trial ( n ) F σ + <, 0 then a valid solution has been obtained and the solution step moves to the next Gauss point or to the next global iterate. However, if the computed trial stress does not satisfy the loading condition, then the solution step moves into corrector phase where the internal variables associated with viscoplastic model evolve. This corrector step has been discussed in detail earlier in the chapter. 64

179 46.42%/min 8.030%/min.6%/min b e2 70.3e2 70.3e2 b 3 -.8e5 -.8e5 -.8e5 b e e e5 b e e e5 b e e e5 λ µ ξ H α η 8.23e4 4.7e4.34e4 m x.03e-3.03e-3.03e-3 l a a γ Table 6.3. Values of parameters for 0-psi confinement pressure data. 65

180 46.42%/min 8.030%/min.6%/min b e2 70.3e2 70.3e2 b 3 -.8e5 -.8e5 -.8e5 b e e e5 b e e e5 b e e e5 λ µ ξ H α η 6.34e4 5.7e4 4.2e4 m x.03e-3.03e-3.03e-3 l a a γ Table 6.4. Values of parameters for 5-psi confinement pressure data. 66

181 8.030%/min.6%/min b e2 70.3e2 b 3 -.8e5 -.8e5 b e e5 b e e5 b e e5 λ µ ξ H α η 5.47e4 4.59e4 m.0.0 x.03e-3.03e-3 l a 5 5 a γ -5-5 Table 6.5. Values of parameters for 30-psi confinement pressure data. 67

182 Figure 6.. Axial Stress vs. Axial Viscoplastic Strain. Model fit to triaxial experimental data at 0-psi confinement pressure (Experimental data from Tashman et al, 2004) 68

183 Figure 6.2. Axial Stress vs. Axial Viscoplastic Strain. Model fit to triaxial experimental data at 5-psi confinement pressure (Experimental data from Tashman et al, 2004) 69

184 Figure 6.3. Axial Stress vs. Axial Viscoplastic Strain. Model fit to triaxial experimental data at 30-psi confinement pressure (Experimental data from Tashman et al, 2004) 70

185 Figure 6.4. Axial Stress vs. Axial Viscoplastic Strain. Model fit to triaxial experimental data at 30-psi confinement pressure (Experimental data from Tashman et al, 2004) 7

186 CHAPTER SEVEN RESULTS FROM FINITE ELEMENT ANALYSIS 7. Introduction A multidimensional hyperelastic-viscoplastic-damage model for asphalt concrete has been developed and presented in the previous chapters of this dissertation. The complete three dimensional numerical implementation of the model is described in chapter 6. The results, presented in chapter 3 and chapter 6 are the model s response at a material point in a continuum. In other words, the experiments are simulated on the assumption that the specimen in the experiment represents a material point. In order to study the experiment as a boundary value problem, the complete model needs to be implemented within the context of the finite element program. In this chapter, the second order hyperelastic model and the combined hyperelastic-viscoplastic-damage model is implemented within the context of the finite element program ABAQUS, which is a commercial finite element code developed and marketed by ABAQUS, Inc., RI, USA. The implementation is done through user subroutine modules UHYPER and UMAT. The RSST-CH and triaxial experiments are modeled as boundary value problems and the results obtained from finite element analysis are compared to experimental results. Also, a section of a pavement is modeled as a plane strain problem and the model s response to repeated loading is studied. 7.2 Hyperelastic model implementation As discussed in the introduction, the results presented in chapter 3 are the hyperelastic model s response at a material point in a continuum. In order to study the 72

187 experiments as a boundary value problem, the second order hyperelastic model developed in chapter 3 is implemented within the context of a finite element program. The model is implemented in thefinite element program ABAQUS through a user material subroutine UHYPER. At each Gauss integration point, the user material subroutine UHYPER is called for externally by the main program and stresses corresponding to strain states are obtained. This provides the user material subroutine as a module to the main program unlike other codes like CASTEM2000 where the material model is implemented in the main finite element code itself. The subroutine is written in FORTRAN 90 and is compiled by ABAQUS processors whenever a call is made to this UHYPER subroutine. In the subroutine the strain energy function is supplied in terms of deviatoric strain invariants and also the derivatives of strain energy function with respect to deviatoric strain invariants are provided. The second order strain energy function proposed in chapter 3 is rewritten in terms of deviatoric strain invariants as follows: Strain invariants in terms of principal stretches are given by I = λ + λ + λ, (7.) I = λ λ + λ λ + λ λ, (7.2) I = λ λλ, (7.3) where λ, λ 2, λ3 are called the principal stretches and are the eigenvalues of the right stretch tensor. I, I 2, I 3 are first, second and third invariants of strain, respectively. The stretch λ is defined as the ratio of final length to initial length of a line element in a 73

188 deformation process. A line element is said to be extended, unextended or compressed according to λ >, λ = or λ <, respectively. The total volume change, denoted by J, at a point in the material, is given by Eq. (7.5) J = det ( F ), (7.4) J = λ λλ, (7.5) 2 3 where F is the deformation gradient. The deviatoric part of the deformation gradient can be written as follows 3 F = J F, (7.6) and the deviatoric stretch matrix (the left Cauchy-Green tensor) of F is defined as T b = FF (7.7) The first and second strain invariant of b can be defined as I ( ) = tr b, (7.8) ( 2 ( 2 )) I2 = I tr b. (7.9) 2 74

189 The deviatoric strain invariants I and I 2 can also be written in terms of strain invariants as I = J I = I I, (7.0) I = J I = I I. (7.) Using Eq. (7.0) and Eq. (7.) the following relations are obtained. I 2 3 = IJ (7.2) I = IJ 2 (7.3) Strain energy function introduced in chapter 3 in Eq. (3.7) is rewritten in terms of the deviatoric strain invariants as follows. (,, ) U I I J = b I J + b I J + b I J + b I I J + b J. (7.4) The strain energy function in the form specified in Eq. (7.4) is supplied to ABAQUS through the user subroutine UHYPER Finite Element Model of RSST-CH in ABAQUS In order to validate the UHYPER code against solutions at a material point presented in chapter 3, a single element testing was performed for both uniaxial as well as 75

190 shear tests. In other words, a single cubic reduced integration element was tested with this material model. The single element test can be regarded as a test very close to testing the model at a material point in a continuum. The values of the parameters used were the same parameters reported in chapter 3, Table 3.. The response of the model for a single element test is shown in Fig. 7. and Fig. 7.2 for uniaxial test and shear test respectively. In the following section, the RSST-CH experiments performed by Sousa et al. are modeled as boundary value problems and solved for in ABAQUS. This is done in order to study the model s response to physically more realistic situations like modeling a pavement section within the context of finite elements. As already mentioned in chapter, cylindrical specimens of 6 inch diameter and 2 inch height are subjected to uniaxial and shear loading. The model of the cylindrical specimen developed using ABAQUS CAE is shown in Fig The elements used are 8 noded linear fully integrated brick elements. The computational time is more when fully integrated elements are used as compared to reduced integration elements but the use of reduced integration element results in mesh instability, commonly referred to as hourglassing. The hourglass mode does not cause any strain and hence, does not contribute to the energy integral in the Gauss integration. It behaves like a rigid body mode. Hourglassing can occur both in geometrically linear as well as nonlinear problems. In geometrically linear problems, hourglassing usually does not affect the stresses but in case of geometrically nonlinear analysis, the hourglass modes tend to interact with the strains at integration points, leading to inaccuracy and/or instability. As mentioned earlier in chapter three, the second order hyperelastic model is developed under the realm of small strain theory; however the implementation in ABAQUS is under finite strain theory. 76

191 A reference point is created at the center of the top surface and all the nodes on the top surface are linked to the reference point through kinematic coupling. This allows for a convenient way of prescribing loads, displacements and boundary conditions. A uniaxial compression test is performed using displacement control test in the axial direction that is, a prescribed displacement is applied at the reference point in the axial direction. The reaction force vs. displacement at the reference point is shown in Fig. 7.4 and also stress vs. strain curve is shown in Fig The finite element results are compared to experimental results and this is shown in Fig Note that the axial stress and axial strain shown in Fig. 7.5 is the average axial stress and average strain values. The average axial stress is calculated by dividing the reaction force at the reference point by the cross section area of the cylinder. Similarly, average strain is calculated by using the height of the sample as a gauge length. In order to study the behavior of the cylinder when subjected to shear loading, a shear test is also performed. A prescribed amount of displacement in the -3 direction is applied at the reference point and this reference point is constrained so that it cannot move in the vertical direction. This boundary condition keeps the height of the specimen constant throughout the test. Reaction force vs. displacement of the reference point in the -3 direction is shown in Fig. 7.6 and corresponding stress vs. strain plot is shown in Fig Here again, the shear stress and shear strain in Fig. 7.7 represent the average values. This assumption is valid as the strain values involved in the test are small and also as mentioned in chapter 3, section 3.2, the hyperelastic model proposed in this work captures material nonlinearity and not geometric nonlinearity. The response of the shear test is compared to the experimental data and is shown in Fig

192 b 2 b 3 b 4 b 5 b 6 Solution at a x x x x x0 8 material point FEM(Single x x x x x0 8 Element Test) FEM (RSST-CH) x x x x x0 8 Table 7.. Parameter values for the Hyperelastic model The deformed cylindrical specimen after the application of the load is shown in Fig. 7.8 and Fig. 7.9 shows the shear stress contour on the deformed configuration. The shear test is performed under constant height, in other words, the height of the specimen is kept constant throughout the experiment. As the height of the specimen is kept constant, the specimen develops stresses in the normal direction due to the volumetric deviatoric coupling behavior. The evolution of normal stresses with shear strain is shown in Fig Hyperelastic-viscoplastic-damage model implementation in ABAQUS 7.3. Finite Element Modeling of Triaxial Experiments To study the response of the combined model for boundary value problems, a multidimensional implementation of the combined model within a framework of finite element environment is performed. The algorithmic aspects of this implementation procedure are described in chapter 6. Also, the consistent tangent moduli of the model are derived. The model s response to triaxial experiments given in chapter 6 is based on the assumption that the specimen represents a material point in the continuum. In this 78

193 section, the triaxial experiments used to validate the model in chapter 6 will be studied as boundary value problems in a finite element environment. The predictor corrector algorithm for the combined model presented in Table 6.2 is implemented in ABAQUS through a user subroutine module UMAT. At each Gauss integration point, the user material subroutine UMAT is called for externally by the main program and stresses corresponding to strain states are obtained. The internal variables are also updated at each time step through the user subroutine SDVINI. These subroutines are written in FORTRAN 90 and are compiled by ABAQUS processors, whenever a call is made to these subroutines. In the following section, the triaxial experiments performed by Tashman et al., at Texas A&M University are modeled as boundary value problems and solved for in ABAQUS. Cylindrical specimens of 4 inch diameter and 6 inch height are subjected to uniaxial stresses at different rates and different confining pressures. The specimens were loaded at three rates 46.42%/min, 8.03%/min, and.6%/min and at two confining pressures 0 psi and 5 psi. The model of the cylindrical specimen developed using ABAQUS CAE is shown in Fig. 7.. Fig. 7.2 shows the deformed shape of the cylinder after the application of stress in the vertical direction and the compression stress contour is shown in Fig The elements used are 8 noded linear fully integrated brick elements. The reason for using fully integrated elements has been discussed earlier in this chapter. The base of the cylinder is fixed and top of the cylinder has displacement boundary conditions in the axial direction. The uniaxial compression test is performed using displacement control in the axial direction taking into account the three different rates and at two different confining pressures 0 psi and 5 psi. The reaction force and 79

194 displacement at the reference point is recorded and stress vs. strain curve is obtained for the various experiments. Note that, the axial stress and axial strain are the average values. The results from the finite element analysis are compared to the experimental data and this is shown in Fig. 7.4 and Fig. 7.5 for 0 and 5 psi confining pressure respectively Finite Element Modeling of Repeated Simple Shear Test at Constant Height In this section, the repeated simple shear test at constant height is modeled as a boundary value problem. The RSST-CH experiment is conducted on a cylindrical specimen of asphalt concrete, where the specimen is subjected to repeated shear loading with the height of the specimen kept constant. A haversine load of 0.05 sec loading and 0.05 sec unloading time is applied on a 0.2 inch thick steel plate which is glued to the cylindrical specimen of asphalt concrete of dimensions 6 inches diameter and 2 inch height. The evolution of permanent shear strain with the number of cycles is a straight line on a log-log scale and also the accumulation of permanent strain with increasing number of cycles is observed. A model of the cylindrical specimen with the steel plate glued to it, is developed using ABAQUS CAE and is shown in Fig The elements used for analysis are 8 noded linear brick elements. The asphalt specimen is subjected to cyclic shear loading at two different loading amplitudes, namely 8psi and 0psi. The evolution of permanent shear strain with number of cycles predicted by the model is compared with the experimental data for the two loading amplitudes and this is shown in Fig With the same set of parameters, which are listed below, the model gives good predictions for the two different loading amplitudes. 80

195 b = 4e4, b = 4.75e7, b = 5.33e7, b = 5e5, b = 4.85e α = 2, η = 2.82e6, H = 5, m=, x=.3e 3, γ = 2.3, a = 3.7, a = This shows that the model is capable of predicting the evolution of permanent shear strain in asphalt concrete at different stress levels. It is to be noted that anisotropy has not been included in the model as there is no experimental data corresponding to the anisotropic properties of the material. 7.4 Finite Element Modeling of Pavement The complete constitutive model developed in this work has been validated against laboratory experiments and the same has been presented in this chapter earlier. In this section, a section of a pavement is studied for rutting, as a boundary value problem under plane strain conditions. The pavement section is subjected to repeated tire pressure loading and the model s response to the evolution of ruts in the asphalt pavement section is studied. The pavement section, shown in Fig. 7.8 represents half of a full depth, one lane pavement section. The finite element model of this section is shown in Fig This is a two dimensional model, the two wheel pressure loads have been simulated as continuous loading strips consistent with the plane strain assumption. The boundary conditions applied to the model are consistent with physical conditions of the pavement and also, the symmetry in the road section has been used effectively. This results in more efficient computational time. The asphalt concrete layer is taken to be 5 inches and rests on a 40 inch deep subgrade. The bottom edge and also the sides of the subgrade are fixed. The width of the half pavement is 80 inches and the width of the subgrade is 60 inches. The outer edge of asphalt pavement is assumed to be unsupported and is free to deform. 8

196 Traffic load is applied as two tire pressure loads of magnitude 00 psi, the wheel loads are 0 inches in width and are separated by 2 inches. This complete model of the pavement section is studied under plane strain conditions and is a good assumption for a section of a highway where the traffic does not stop or start. A more complicated three dimensional pavement model could be used to represent pavement sections such as intersections, where traffic accelerates, decelerates or stands still. The asphalt concrete pavement is modeled using the hyperelastic-viscoplasticdamage model developed in this work. The subgrade in reality is a granular material and a nonlinear material model would best represent its behavior. However, for simplicity and to keep the computational time down, the subgrade is modeled as a linear elastic material with Young s modulus of psi and a Poisson s ratio of 0.3. See Sousa et al. (993). Fig shows the deformed shape at the end of 5 cycles magnified 00 times. The figure shows permanently deformed shape and also shows the upheavals to the sides due to shear flow and dilation and is similar to those observed in rutted pavement sections. The stress distribution in the pavement section model is shown in Fig This simulation of rutting pattern illustrates the capability of the model to study boundary value problems representing pavement sections and also to simulate rutting patterns. 82

197 Figure 7.. Axial Stress vs. Axial Strain for RSST-CH experiments. Single element test in ABAQUS. 83

198 Figure 7.2. Shear Stress vs. Shear Strain for RSST-CH experiments. Single element test in ABAQUS. 84

199 Figure 7.3. Finite Element Model of the cylindrical specimen used for RSST-CH experiments. 85

200 Figure 7.4. Reaction force vs. Displacement in the axial direction. (RSST-CH experiment) 86

201 Figure 7.5. Axial Stress vs. Axial Strain. Comparison of Finite Element model with Experimental data. 87

202 Figure 7.6. Reaction Force vs. Displacement in -3 direction. 88

203 Figure 7.7. Shear Stress vs. Shear Strain. Comparison of Finite Element model with Experimental data. 89

204 Figure 7.8. Deformation of cylindrical specimen, magnified. 90

205 Figure 7.9. Shear stress contour on the deformed cylindrical specimen. 9

206 Figure 7.0. Evolution of normal stress with shear strain. 92

207 Figure 7.. Finite element mesh of triaxial specimen 93

208 Figure 7.2. Deformed shape of the cylinder 94

209 Figure 7.3. Axial Stress distribution in the cylinder 95

210 Figure 7.4. Axial stress vs. axial viscoplastic strain. Comparison of finite element analysis results to triaxial experiments at 0psi confining pressure. 96

211 Figure 7.5. Axial stress vs. axial viscoplastic strain. Comparison of finite element analysis results to triaxial experiments at 5psi confining pressure. 97

212 Figure 7.6. Finite element mesh of RSST-CH specimen. 98

213 Figure 7.7. Evolution of permanent shear strain with no. of cycles. 99

214 Figure 7.8. Schematic of the pavement model 200

215 Figure 7.9. Finite element model of the pavement 20

216 Figure Deformed shape of the pavement 202

217 Figure 7.2. Stress distribution in the deformed pavement 203