Top-Down Cracking of Hot-Mix Asphalt Layers: Models for Initiation and Propagation (2010)

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A-i APPENDIX A. DEVELOPMENT OF THE VECD-BASED MODEL

A-ii TABLE OF CONTENTS page TABLE OF CONTENTS ............................................................................................................ A-ii APPENDIX A DEVELOPMENT OF THE VECD-BASED MODEL ....................................................... A-1  A.1 Development of Model Components ............................................................................ A-3  A.1.1 Laboratory Specimen Fabrication and Test Method .......................................... A-3  A.1.2 Material Property Sub-models ............................................................................ A-5  A.1.2.1 Linear viscoelastic (LVE) model ............................................................. A-5  A.1.2.2 Viscoelastic continuum damage model (VECD) ..................................... A-9  A.1.2.3 Fatigue life prediction ............................................................................ A-14  A.1.2.4 Statistical analysis for dynamic modulus test results of the AL mix ..... A-25  A.1.2.5 Viscoplasticity ........................................................................................ A-27  A.1.2.6 Thermal stress ......................................................................................... A-30  A.2 Reasonableness and Sensitivity Checks ..................................................................... A-32  A.2.1 Reasonableness ................................................................................................. A-32  A.2.2 Sensitivity ......................................................................................................... A-39  A.2.2.1 Support condition effect ......................................................................... A-39  A.2.2.2 Aging effect ............................................................................................ A-41  A.2.2.3 Thermal stress effect .............................................................................. A-42  A.2.2.4 Viscoplastic effect .................................................................................. A-43  A.3 Example Simulation and Parametric Study Contour Plots ......................................... A-45  A.3.1 Example Simulation Contour Plots .................................................................. A-45  A.3.2 Parametric Study Contour Plots ....................................................................... A-46  A.4 List of References ....................................................................................................... A-55 

A-1 CHAPTER 1 DEVELOPMENT OF THE VECD-BASED MODEL Top-down cracking has been proven to be one of the major distresses in hot-mix asphalt (HMA) pavements in the United States. Many field studies have shown that different patterns of top-down cracking exist, including cracks that start from the pavement surface and propagate downward, cracks that start as a top-down crack and propagate horizontally at the layer interface, and cracks that start from both the top and bottom simultaneously, forming a conjoined cracking pattern. The complex state of stress that exists in HMA pavements with various layer materials and thicknesses makes it difficult to determine the location of crack initiation and to predict crack propagation. For example, a crack may initiate at the bottom of the HMA layer, but changes in the state of stress during the propagation of that crack may result in the initiation and propagation of another crack at the top of the HMA layer. These complex mechanisms involved in crack initiation and propagation can make it difficult to reliably predict the service life of HMA pavements using conventional HMA performance prediction models and pavement response models. With the goal of accurate pavement performance evaluation, researchers at North Carolina State University (NCSU) have been developing advanced models for HMA under complex loading conditions. Over the past decade, they have successfully developed material models that can accurately capture various critical phenomena, such as: microcrack-induced damage, which is critical in fatigue modeling; strain rate-temperature interdependence; and viscoplastic flow, which is critical in high temperature modeling. The resulting single model is termed the viscoelastoplastic continuum damage (VEPCD) model. To extend the strengths of the VEPCD model to the fatigue cracking evaluation of pavement systems, the viscoelastic continuum damage (VECD) model has been incorporated

A-2 into the public domain finite element code, FEP++. The resulting product, the so-called VECD- FEP++, allows the accurate evaluation of boundary condition effects (e.g., layer thickness) on the material behavior. In the VECD-FEP++, the damage is calculated for each element based on its state of stress, temperature, loading rate, and boundary conditions. Therefore, it is not necessary to assume a priori the location of distress initiation, nor the path of distress evolution. Not having to make such assumptions is a feature of the VECD-FEP++ that is essential in modeling top-down cracking initiation in various HMA pavements. The flexible nature of the VECD-FEP++ modeling technique allows cracks to initiate wherever the fundamental material law suggests. As a result, much more realistic and accurate cracking simulations can be accomplished using the VECD-FEP++. The capability of the VECD-FEP++ to predict the performance of the pavement structure has been validated using different pavement structures. One example uses the Federal Highway Administration Accelerated Load Facility (FHWA ALF) pavements and another example is the Korea Express Highway (KEC) test road in South Korea (13). In the FHWA ALF research, twelve asphalt pavement lanes (that use various binders) were constructed at the FHWA Turner- Fairbanks Highway Research Center to rapidly collect data on pavement performance under conditions in which axle loading and climatic conditions are controlled. The KEC test road was constructed in December 2002 to serve as the basis for Korea’s new pavement design guide, giving consideration to actual traffic loads and environmental changes. This test road is a 7.7 km two-lane highway consisting of 25 Portland cement concrete (PCC) pavement sections and 15 asphalt concrete (AC) pavement sections as well as three bridges and two geotechnical structures. The variables considered in the Korean study for AC pavements are surface layer type, base layer type, base layer thickness, and sublayer properties. Field performance surveys

A-3 have been performed since the test road was constructed and will be continued. Predictions made using the VECD-FEP++ for the KEC test road sections demonstrate field performance trends and also reveal a generally positive relationship between model predictions and field observations for all the sections. In an attempt to demonstrate the ability of the VECD-FEP++ to integrate the effects of variables that are important in top-down cracking behavior, several sub-models are refined from existing models and incorporated into the VECD-FEP++. These models account for aging, healing, thermal stress, viscoplasticity and mode-of-loading. The Enhanced Integrated Climatic Model (EICM) is also integrated into the framework. It must be noted that these sub-models are incorporated into the VECD-FEP++ framework as place-holders only. Their accuracy needs to be verified and calibrated with further testing using various mixtures and in-depth analysis of the results. However, the reasonableness of the VECD-FEP++ with these sub-models is demonstrated using sensitivity analysis and a parametric study. A.1 Development of Model Components A.1.1 Laboratory Specimen Fabrication and Test Method All specimens were compacted by the Servopac Superpave gyratory compactor, manufactured by IPC Global of Australia, to dimensions of 178 mm in height and 150 mm in diameter. To obtain specimens of uniform quality for testing, these samples were cored and cut to a height of 150 mm and a diameter of 75 mm. After obtaining specimens of the appropriate dimensions, air void measurements were taken via the Core-Lok method, and specimens were stored until testing. It is noted that the air voids for all tests in this research are between 3.5% and 4.5%. During storage, specimens were sealed in bags and placed in an unlit cabinet to reduce the aging effects. Further, no test specimens were stored for longer than two weeks before testing.

A-4 Prior to all testing, the steel end plates were glued to the specimen using DEVCON steel putty. Extreme care was taken to completely clean both the end plates and the specimen ends before each application to prevent failure at the glued area. To ensure the specimens were properly aligned, a special gluing jig was employed so that the end plates were parallel, thus minimizing any eccentricity that might occur during the test. Measurements of axial deformations were taken during testing at 90° intervals over the middle 100 mm of the specimen using loose-core linear variable displacement transducers (LVDTs). Load, crosshead movement and specimen deformation data were acquired using a 16- bit National Instruments data acquisition board and collected using LabVIEW software. The data acquisition rate varied depending on the nature of the test so that the appropriate amount of data could be acquired for analysis. A MTS closed-loop servo-hydraulic loading frame was used for all the tests. Depending on the nature of the test, either an 8.9 kN or 25 kN load cell was used. An environmental chamber, equipped with liquid nitrogen coolant and a feedback system, was used to control and maintain the test temperature. Three different types of laboratory testing were performed for the aging study: the complex modulus test, constant crosshead rate test, and cyclic fatigue test. The complex modulus test and constant crosshead rate test were carried out with the aim of capturing the aging effects in the current VECD model. For a more direct investigation of aging effects, cyclic fatigue tests were performed in controlled crosshead (CX) and controlled stress (CS) modes. The complex modulus test was performed in stress-controlled uniaxial tension- compression mode. The test was performed at frequencies of 25, 10, 5, 1, 0.5, and 0.1 Hz and temperatures of -10°, 5°, 20°, 40°, and 54°C. The load level was adjusted for each condition to produce total strain amplitudes of about 50 to 70 microstrains, which is in the linear viscoelastic

A-5 (LVE) range. The constant crosshead rate test was conducted in uniaxial tension mode at different on-specimen LVDT strain rates at both 5°C and 40°C until failure. Instead of testing several replicates at a limited set of rates and temperatures, tests were conducted at three different rates with one replicate per rate. Both types of cyclic fatigue tests, CX and CS, were conducted only in tension mode with a haversine loading at a fixed level. Because a true controlled strain test using cylindrical specimens is difficult to run and can damage equipment if improperly performed, the CX test was utilized. Such a test results in a mixed mode of loading that is neither controlled stress nor controlled strain. A.1.2 Material Property Sub-models A.1.2.1 Linear viscoelastic (LVE) model A.1.2.1.1 Linear viscoelasticity Viscoelastic materials exhibit time and temperature dependence, meaning that the material response is not only a function of the current input, but the entire input history. By contrast, the response of an elastic material is dependent only on the current input. For the uniaxial loading considered in this research, the non-aging, LVE constitutive relationships are expressed in the convolution integral form, as shown in Equations (A-33) and (A-34): ( ) 0 t dE t d d εσ τ τ τ = −∫ and (A-33) ( ) 0 t dD t d d σε τ τ τ = −∫ , (A-34) where ( )E t and ( )D t are the relaxation modulus and creep compliance, respectively (the τ term is the integration variable). The relaxation modulus and creep compliance are important material

A-6 properties, along with the complex modulus, in LVE theory. Because these two properties are the responses for respective unit inputs, they are called unit response functions. These unit response functions can be obtained either by experimental tests performed in the LVE range or by converting another unit response function, as suggested by Park and Schapery (2). A.1.2.1.2 Unit response functions and their interrelationships The unit response functions presented in Equations (A-33) and (A-34) are often measured in the frequency domain via the complex modulus test, because it is often difficult to obtain unit response functions in the time domain due to the limitations of the machine capacity or testing time. The complex modulus provides the constitutive relationship between the stress and strain of a material loaded in a steady-state sinusoidal manner, e.g., in the frequency domain. The storage modulus can be determined from the complex modulus and it can be converted to a time- dependent property, such as ( )E t and ( )D t , through LVE theory. When the storage modulus is expressed in terms of reduced angular frequency, Rω , as shown in Equation (A-35), it can be expressed using the Prony series representation given in Equation (A-36) (2, 3). ( ) ( ) ( )( )*' *sinR R RE Eω ω φ ω= and (A-35) ( ) 2 2 2 1 ' 1 m R i i R i R i EE E ω ρω ω ρ∞ = = + +∑ , (A-36) where E∞ = the elastic modulus, Rω = the angular frequency (= 2 Rf tπ ∆ ), t∆ = the time lag between the stress and strain, iE = the modulus of the i th Maxwell element (fitting coefficient), and iρ = the relaxation time (fitting coefficient).

A-7 The coefficients determined from this process are then used with Equation (A-37) to find the relaxation modulus. ( ) 1 i tm i i E t E E e ρ − ∞ = = + ∑ (A-37) Using the theory of viscoelasticity, the exact relationship between the creep compliance and the relaxation modulus is given by Equation (A-38). ( ) ( ) 0 1 t dD E t d d τ τ τ τ − =∫ (A-38) If the creep compliance can be written in terms of the Prony representation (Equation (A-39)), substituted into Equation (A-38) along with Equation (A-37) and simplified, the result can be expressed as a linear algebraic equation, Equation (A-40). The coefficients, {D} in this equation, are solved by any proper numerical method. ( ) 1 1 j n t g j j D t D D e τ − =   = + −    ∑ and (A-39) [ ]{ } [ ]A D B= , (A-40) where [ ] 1 1 1j jm t ttM N m m j m m j EA e e E eτ τρρ ρ τ −− − ∞ = =          = − + −    −      ∑ ∑ ; { } jD D= ; and [ ] 1 1 1 1 m tN mN m m m B E E e E E ρ − ∞ = ∞ =   = − +    + ∑ ∑ .

A-8 Once the coefficients, jD , are determined, they are substituted into Equation (A-39) to find the creep compliance. A.1.2.1.3 Time-temperature superposition principle for linear viscoelastic material The effects of time and temperature on viscoelastic material behavior can be combined into a single parameter, called reduced time, through the time-temperature superposition principle. Viscoelastic properties obtained in the LVE range at different temperatures can be superposed to develop a single mastercurve by shifting them horizontally to a certain reference temperature. The horizontal distance required to superpose a curve to a reference curve in logarithmic space is the log of the time-temperature shift factor ( Ta ). A material that exhibits a single mastercurve by this method is called thermorheologically simple. Equation (A-41) represents the general mathematical definition of reduced time. ( ) ( )0 1t T t d a ξ τ τ = ∫ , (A-41) where Ta = the time-temperature shift factor. When the temperature is constant, for example in a relaxation modulus experiment, Equation (A-41) simplifies to the more common form shown in Equation (A-42). For frequency domain conditions, such as when measurements are taken in a dynamic modulus test, the reduced frequency is similarly computed using Equation (A-43). T t a ξ = (A-42) *red Tf f a= (A-43)

A-9 A.1.2.2 Viscoelastic continuum damage model (VECD) A.1.2.2.1 Continuum damage On the simplest level, continuum damage mechanics considers a damaged body with some stiffness as an undamaged body with a reduced stiffness. Continuum damage theories thus attempt to quantify two values: damage and effective stiffness. Further, these theories ignore specific microscale behaviors and, instead, characterize a material using macroscale observations, i.e., the net effect of microstructural changes on observable properties. In the macroscale, the most convenient method to assess the effective stiffness is to use the instantaneous secant modulus. As discussed in the subsequent sections, direct use of the stress- strain secant modulus in asphalt concrete (AC) is complicated by time dependence. Damage is oftentimes more difficult to quantify and generally relies on macroscale measurements combined with rigorous theoretical considerations. For the VECD model, Schapery’s work potential theory, which is based on thermodynamic principles, is appropriate for the purpose of quantifying damage. Within Schapery’s theory, damage is quantified by an internal state variable ( S ) that accounts for microstructural changes in the material. A.1.2.2.2 Correspondence principle The correspondence principle states that viscoelastic problems can be solved with elastic solutions when physical strain is replaced by pseudo strain. ( ) 0 1 tR R dE t d E d εε τ τ τ = −∫ , (A-44)

A-10 where RE is a particular reference modulus included for dimensional compatibility and typically taken as one. Using pseudo strain in place of physical strain, the constitutive relationship presented in Equation (A-33) can be rewritten as R REσ ε= . (A-45) It is seen from Equation (A-45) that a form corresponding to that of a linear elastic material (Hooke’s Law) is used when strain is replaced by pseudo strain. In a practical sense, pseudo strain is simply the LVE stress response to a particular strain input. The most important effect of pseudo strain is seen when plotting stress, because any time effects are removed from the resulting graph. This property of the stress-pseudo strain relationship is used in the modeling approach presented here. The basic consideration of the continuum damage theory is that any reduction in stiffness is related to damage. Graphically, this phenomenon is seen in a reduction of the stress-strain modulus; recall that continuum damage theories typically use a secant modulus to quantify the effect of damage. For viscoelastic materials, a reduction in the secant modulus is also related to the time effects. However, in stress-pseudo strain space the time effects are removed, and any reduction in the pseudo secant modulus (the secant modulus in the stress-pseudo strain space) is a direct consequence of damage. A.1.2.2.3 Viscoelastic continuum damage theory Continuum damage theory states that the stiffness reduction is defined by the pseudo secant modulus (pseudo stiffness). This quantity is typically normalized for specimen-to- specimen variability by the factor, I , and denoted as C (Equation (A-46)). RC I σ ε = × (A-46)

A-11 The LVE relationship represented by the pseudo strain in Equation (A-45) can be modified to Equation (A-47) when the microcracking damage grows (4). Then, by substituting Equation (A-47) into Equation (A-34), the nonlinear constitutive relationship for strain is given in Equation (A-48). ( ) RC Sσ ε= (A-47) ( ) ( ) 0 t ve R d C S E D t d d σ ε τ τ τ       = −∫ (A-48) Equations (A-47) and (A-48) require the determination of an internal state variable, S . This internal state variable quantifies any microstructural changes that result in observed stiffness reduction. The relationship between damage ( S ) and the normalized pseudo secant modulus (C ) is known as the damage characteristic relationship and is a material property that is independent of loading conditions. A.1.2.2.4: Simplified VECD model Recently, a new simplified model for the analysis and prediction of cyclic data has been developed using the current VECD model constitutive functions. The major difference between this simplified VECD (S-VECD) model and the more rigorous VECD model is the use of a simplified pseudo strain calculation methodology for the cyclic portion, as shown in Equation (A-49). The overall effect of such a simplification on the value of pseudo strain is found to be small, but the simplification is found to save a great deal of computational time without causing a large error in the calculations. In addition, the resulting formulation unifies the results of the CS, CX, and monotonic testing and supports earlier findings that the damage characteristic curve

A-12 is a material property independent of temperature and test type. The simplified model formulations are shown in the following equations with descriptions of the variables (20). ( ) ( ) ( )( ) 0 0. 0, 1 ( ) 1 * * 2 R p R R R ta pp pLVEii d E d E d E ξ ε ε ξ τ τ ξ ξ τε βε ε ξ ξ  = − ≤=  + = > ∫ (A-49) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 12 1 1 1 12 1 1 0. 1 2 * 2 R Transient j ptimestep j jj R cyclic ta i p pcycle i ii dS C dS IdS C R α α α α α α α ε ξ ξ ξ ε ξ ξ ξ + + + + +    = − ∆ × ∆ ≤    =    = − ∆ × ∆ × >     (A-50) 0, 0, * pR ta pR ta C I C C I σ ξ ξ ε σ ξ ξ ε  = ≤ =   = >  (A-51) ( ) ( )peak valleyi i i peak valleyi i σ σ β σ σ + = + (A-52) ( )( )21 * * f if i R f d ξ α ξ ξ ξ ξ ξ = − ∫ (A-53) where ξp = reduced pulse time of the loading pulse, ε0R = cyclic pseudo strain amplitude, |E*|LVE = linear viscoelastic dynamic modulus of the material at the particular temperature and frequency of the test, C* = cyclic pseudo stiffness, ε0 = cyclic strain amplitude, β = a factor that allows direct quantification of the duration that a given stress history is tensile, R = form adjustment factor, ξi = reduced time within loading cycle when tensile loading begins, ξf = reduced time within loading cycle when tensile loading ends, f(ξ) = normalized time function found from loading type assumption,

A-13 pp = peak-to-peak amplitude, and ta = tension amplitude. The transient portion of the loading history, i.e., the first half of the first cycle, is important because it is used to define the specimen-to-specimen correction factor, I , and because damage growth in this first loading path can be significant. The α in the simplified model formulations is defined as (1/u+1) for the constant rate and CX tests, and as (1/u)for the CS tests. This definition of α is based on theoretical arguments and has been verified using short-term aged (STA) materials only. Its universality with regard to materials tested at other aging conditions was not verified in the original work (20). Because the CS and CX tests fail in different patterns, two different criteria are necessary. From the study by Underwood et al. (20), two criteria have been adopted to define the cycle in which data can be used in the VECD characterization process: 1) the phase angle criterion, and 2) the threshold criterion. The phase angle criterion is the same as that used in defining the number of cycles to failure ( fN ) suggested by Reese (19). The concept of the threshold criterion is that when processes other than damage mechanisms, such as viscoplasticity, begin to have a significant effect, then a test no longer can be used directly for characterization. It is believed that the onset of other mechanisms is closely related to the total amount of permanent strain experienced by the specimen. From experience with the constant crosshead rate tests at 5°C, it is known that for any given mixture, tests performed at certain rates show similar strain levels at the peak stress, as shown in Figure 1-1 (a). In this figure, the straight lines represent the averaged strain levels for each aged mixture, and these strain levels are clearly ranked according to aging level. These mixture-dependent strain levels represent a known level below which VECD mechanisms dominate. The cycle in the fatigue tests at which the permanent strain (backcalculated from the measured permanent pseudo strain) exceeds this threshold is taken as

A-14 the point after which the data cannot be used for VECD characterization. Figure 1-1 (b) shows the cut-off point schematic for the AL-STA mix. As can be seen, the cut-off criterion is applied only to the CS tests. 0.E+00 1.E-03 2.E-03 3.E-03 1E-06 1E-05 1E-04 Red. Strain Rate St ra in a t P ea k St re ss STA LTA1 LTA2 LTA3 (a) 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E-01 1.E+01 1.E+03 1.E+05 1.E+07 Reduced Time (s) B ac kc al cu la te d VP S tra in 19-CX-H 19-CS-H 19-CS-L 5-CX-H 5-CS-H 5-CS-L Cutoff VP Strain (b) Figure 1-1. (a) Strain at peak stress and (b) Cut-off viscoplastic strain level for AL-STA mix A.1.2.3 Fatigue life prediction The most recent study of material model simulations using the simplified VECD model is presented in this subsection for a better understanding of the verification and/or the application of the simplified VECD model, including the general failure criterion (7). A.1.2.3.1 Simplified VECD model Controlled strain test simulation All the cyclic fatigue test data were fitted to analytical forms to obtain the damage characteristic curves for each mixture. The power law function (Equation (A-54)) was found to fit the experimental results better than the exponential function (Equation (A-55)). Once the simplified VECD model was calibrated, i.e., the 11C and 12C coefficients in Equation (A-54) were found for each mixture, the analytical function of the damage characteristic curve could be substituted into Equation (A-50) to derive the function needed to simulate the cyclic tests, as shown in Equation (A-56). 12* 111 CC C S= − (A-54)

A-15 nmSC e= (A-55) ( ) ( )122 11 0. 11 1212 CR i i taS S C C S R d α ε ξ−+  = +     (A-56) In these simulations the pseudo strain history was taken directly from the measured pseudo strain and the pseudo stiffness, and hence, the stress response was predicted. The predicted and measured pseudo stiffness values for a typical good prediction are shown in Figure 1-2, and the results from a typical bad prediction are shown in Figure 1-3. 0.0 0.2 0.4 0.6 0.8 1.0 0.0E+00 1.0E+05 2.0E+05 3.0E+05 4.0E+05 5.0E+05 Reduced Time (s) C * Predicted Measured Figure 1-2. Typical good pseudo stiffness prediction (RI19B-5) 0.0 0.2 0.4 0.6 0.8 1.0 0.0E+00 5.0E+02 1.0E+03 1.5E+03 2.0E+03 2.5E+03 3.0E+03 Reduced Time (s) C * Predicted Measured

A-16 Figure 1-3. Typical bad pseudo stiffness prediction (I19C-10) Simulation failure criterion The simplified fatigue model does not account for changing time dependency and, therefore, it was not possible to observe a sudden decrease of the phase angle in the simulation. For this reason, an empirical observation of all the tested mixtures was made to determine the failure criterion. The observation is shown in Figure 1-4 where the pseudo stiffness at failure ( *fC ) is plotted against reduced frequency for multiple mixtures. Reduced frequency, Rf , is computed by multiplying the actual test frequency, f , by the shift factor for the temperature of the test, Equation (A-57), where the coefficients 1α , 2α , and 3α are all characterized as part of the LVE characterization process. 2 1 2 3* *10 T TR Tf f a f α α α+ += = (A-57) Note that only mid-failure test results are used here, because the goal is to predict localization failure, not pre-localization failure. 0 0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1 10 100 Reduced Frequency C * f S9.5C S9.5B I19C I19B RI19B B25B RI19C RB25B RS12.5C

A-17 Figure 1-4. Experimental observation of pseudo stiffness values at failure against reduced frequency It is found from Figure 1-4 that, in general, the pseudo stiffness at failure ( * fC ) value increases with reduced frequency. It is also found that for the non-RAP mixtures at reduced frequencies at or below 0.01 Hz (a condition that corresponds physically to a temperature of approximately 27°C and frequency of 10 Hz), failure occurs at a pseudo stiffness value of approximately 0.28. This value is similar to that observed by Daniel and Kim (8) for their tests, which were performed at 25°C. As the reduced frequency increases, failure tends to occur at a higher level of pseudo stiffness. Although the data scatter is significant, it is also observed that the rate of this increment is aggregate size-dependent. From the data obtained at a reduced frequency of around 0.1 Hz, it is found that RAP mixtures have a higher failure pseudo stiffness value than non-RAP mixtures. These observations lead to the piecewise failure function, given in Equation (A-58). * 0.01 (log( ) log(0.01)) 0.01 10 R f R R b f C a f b f < =  ⋅ − + ≤ < (A-58) Note that: the failure criterion is a linear function, dependent on nominal maximum aggregate size (NMAS), in semi-log space between the reduced frequencies of 0.01 to 10 Hz; the failure criterion is constant, as a function of RAP or non-RAP mixtures, at reduced frequencies lower than this range; and at temperatures above this range no interpretation is made. To characterize the coefficients of this function an optimization approach is taken. In this approach, the error between the measured and predicted fatigue life is minimized by systematically changing the coefficients of Equation (A-58) as a function of NMAS and for RAP versus non-RAP mixtures. Due to the inherent complexity involved in this optimization program,

A-18 a genetic algorithm technique is used. A commercial Excel-based macro language add-in, Evolver, is used for this purpose. 0 0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1 10 100 Reduced Frequency C * f S9.5C S9.5B I19C I19B RI19B B25B RI19C RB25B RS12.5C NR 9.5 NR12.5 NR 19 NR 25 R 9.5 R 12.5 R 19 R 25 Figure 1-5. Optimized failure criterion The results in Figure 1-5 show that the optimized failure criterion matches with the experimental data points. The value of the intercept coefficient, b, for the RAP mixtures is greater than that for non-RAP mixtures, and the value of the slope coefficient, a, decreases with the increase in nominal maximum aggregate size (NMAS). The final functional forms for these coefficients are shown in Equations (A-59) and (A-60). ( )0.0045 0.12a NMAS= − + (A-59) 0.26 0.30 non RAP b RAP − =   (A-60) Fatigue life prediction results The simulation failure criterion developed in the previous section is applied to predict the fatigue life for the mid-failure cyclic tests. The results of this prediction process are shown in the form of line-of-equality (LOE) plots in both arithmetic and logarithmic space in Figure 1-6.

A-19 0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 5.E+05 0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 5.E+05 Measured Nf Pr ed ic te d N f S9.5C S9.5B I19C B25B I19B RS12.5C RI19B RI19C RB25B LOE (a) R2=0.932805 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 Measured Nf Pr ed ic te d N f S9.5C S9.5B I19C B25B I19B RS12.5C RI19B RI19C RB25B LOE (b) R2=0.989562 Figure 1-6. Comparison of measured and predicted fatigue life for mid-failure cyclic tests in (a) arithmetic and (b) logarithmic space The findings from Figure 1-6 are encouraging, as the relationship shows a high degree of statistical significance as evidenced by the high correlation coefficients in both arithmetic and logarithmic scales. Although it cannot be assessed directly from this figure, the arithmetic fatigue life prediction error for all the available mid-failure cyclic tests is 17 ± 12%. A slight tendency to over-predict is evident, which may be caused by the viscoplasticity in some of the high temperature and low strain level tests.

A-20 To validate the failure criterion, fatigue test results from mixtures not used in the calibration of Equations (A-58) through (A-60) are utilized. In total, three mixtures, each from the FHWA ALF experiment, are included in this effort: CRTB, Terpolymer, and Control mixtures. Details of these mixtures, including the experimental results used to generate the data in Figure 1-7, are given elsewhere (26, 10). The measured and predicted cycles to failure for these mixtures are shown in both arithmetic and logarithmic space in Figure 1-7. 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 Measured Nf Pr ed ic te d N f Control CRTB Terpolymer LOE (a) 0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 Measured Nf Pr ed ic te d N f Control CRTB Terpolymer LOE (b) Figure 1-7. Validation of calibrated failure criterion with non-calibration mixture cyclic tests in (a) arithmetic and (b) logarithmic space Overall, the agreement between the measured and predicted number of cycles to failure ( fN ) is very good. One major exception occurred for one of the CRTB tests, which shows a significant under-prediction. After examining the measured and predicted behaviors more closely

A-21 for the CRTB experiments, it was found that the number of cycles to failure values for the two tests do not follow the expected trends. Specifically, the test with the higher input strain values produced a larger number of cycles to failure than the experiment with the lower input strain values. In light of this anomaly and in light of the fact that, besides the single CRTB test, all other predictions agree favorably with the measured data, the failure criterion is deemed acceptable. Further validation and improvement of this failure criterion is part of further research work. A.1.2.3.2: Direct tension fatigue simulation using simplified VECD model Having verified the simplified VECD model for the case of mixed loading mode, attention now turns to its use for simulating pure controlled strain and controlled stress direct tension fatigue tests. Controlled Strain Test Simulation Starting with the simplified VECD model formulation (Equation (A-50)), assuming the power law damage model (Equation (A-54)), and after rearranging, integrating and simplifying, the following relationship can be obtained to find the fatigue life for a pure controlled strain direct tension cyclic test: ( )( ) ( )( ) ( )( )( ) 12 13 2 12 11 12 0, 2 1 1 * C R f f pp LVE f S N C C C E R α αα ααα α β ε − + =  − + +  , (A-61) where fS is the value of the damage parameter at failure. This equation is derived based on the assumption that the damage at failure is much greater than the initial damage at the first cycle of loading, i.e., fS >> iS , and the assumption that the fatigue life value is much greater than one. Both of these assumptions are true for most real experiments.

A-22 Equation (A-61) can be used directly to find the effect of strain amplitude, loading frequency and testing temperature on fatigue life. Analysis was performed to simulate fatigue tests at 5°, 19° and 27°C. Figure 1-8 presents the simulation results for the S9.5C mixture in the plot of the standard logarithmic strain level versus the fatigue life. Note that similar plots are generated when empirically characterizing the fatigue resistance of AC mixtures. Also note that all simulated tests are in a zero mean strain condition, i.e., β = 0, and the loading frequency is 10 Hz. The value of the damage parameter at failure, fS , is determined based on the same mixture type-dependent failure criterion proposed in Equation (A-58). Figure 1-8 also shows that Equation (A-61) effectively captures the same major trends suggested by other empirical research (11-16). That is, the fatigue life decreases as the strain level increases, and stiffer materials (i.e., materials at a low temperature) fail earlier than softer materials for the same strain level. The model also suggests that the effect of temperature (when the frequency and material type are fixed) is to shift the fatigue envelopes such that the curves become parallel. Anecdotal evidence of this behavior is supported in the literature by the form taken for various empirical predictive models (14, 16, 17-22). 1.E-04 1.E-03 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 Nf St ra in L ev el (5 0t h cy cl e) 5C Simulation 19C Simulation 27C Simulation Figure 1-8. Controlled strain direct tension fatigue simulation results for S9.5C mixture

A-23 Controlled Stress Test Simulation The simplified VECD model is also applied to simulate the controlled stress direct tension fatigue test. The formulation of the stress-based model is not as straightforward as that of the strain-based model due to the complexity of the integration, as evident in Equation (A-62). ( )( ) ( )( ) ( ) 12 12 2 ˆ23 11 2 1 ˆ 11 120, 1 ˆˆ12 * ˆ ˆˆ1 f i C S f C Spp C Sf E N dS C C SK α αα α σ β −   − ⋅ ⋅  =   +      ∫ , (A-62) where Sˆ is pseudo stress-based damage as compared to the pseudo strain-based damage, S. The two damage parameters are simply related by a factor, 2K . ( ) ( ) ( )2ˆ *cycle i cycle iS S K∆ = ∆ (A-63) The 2K factor and 11Cˆ term can be obtained using Equations (A-32) and (A-33), respectively. ( ) 2 1 2 * LVEK E α α += (A-64) ( ) 1211 11 2ˆ CC C K= (A-65) Similar to the controlled strain test simulation, different fatigue life values are found at different stress levels and testing temperatures using Equation (A-62). This analysis was performed at 5°, 19° and 27°C with a constant loading frequency of 10 Hz. For these simulations the load levels were assumed to be constant and tensile, i.e., β = 1. Typical results from this type of simulation are shown in Figure 1-9.

A-24 1.E+02 1.E+03 1.E+04 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 Nf St re ss L ev el (k Pa ) 5C Simulation 19C Simulation 27C Simulation (a) 1.E-04 1.E-03 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 Nf In iti al S tra in L ev el 5C Simulation 19C Simulation 27C Simulation (b) Figure 1-9. Controlled stress direct tension fatigue simulation results for S9.5C mixture in plots for: (a) stress vs. fatigue life, and (b) strain vs. fatigue life Figure 1-9 (a) shows the simulated fatigue envelope for the S9.5C mixture in the stress level versus fatigue life plot. At the same testing temperature, the fatigue life decreases as the input stress level increases. Also, it is interesting to see that the position of the fatigue envelopes at different temperatures flips, as compared to the controlled strain test simulation results shown in Figure 1-8. That is, under the same stress level input, the fatigue life increases as the temperature decreases. This behavior is also consistent with the controlled stress flexural

A-25 bending test results from the SHRP project A-003A (13) and also consistent with findings from material level testing on the ALF mixtures (26). Figure 1-9 (b) plots the same simulation results, but in a different way. Here, the initial strain level is shown instead of the initial stress level. This initial strain level is obtained by dividing the stress level by the material’s LVE modulus value. For comparative purposes, Figure 1-9 (b) is plotted in the same scale as Figure 1-8. It is found that now the positions of the fatigue envelopes at different temperatures are in the same order that appeared in the controlled strain simulation. However, the fatigue life obtained from the controlled stress test is much shorter than that from the controlled strain test. This observation is reasonable, because during a pure controlled stress test, the material’s stiffness decreases with the number of loading cycles, and the actual on-specimen strain amplitude increases. This situation eventually causes earlier failure than the controlled strain test with the same initial on-specimen strain level. This phenomenon has also been identified with beam fatigue experiments (12, 21). A.1.2.4 Statistical analysis for dynamic modulus test results of the AL mix A statistical analysis of the dynamic modulus (|E*|) data at different aging levels for the AL mixture has been performed to determine the significance of the graphical observations. Because all the tests were not conducted under exactly the same conditions, each replicate was first processed using interpolation to build a data set for precisely the same temperature and frequency conditions. The specific temperatures and frequency combinations that were compiled are: -10°C, 5°C, 20°C, 40°C, and 54°C and 25, 10, 5, 1, 0.5, and 0.1 Hz, respectively. As a first cut analysis, the average dynamic modulus value from each long-term aging level (LTA1, LTA2, and LTA3) is plotted against STA dynamic modulus values at a consistent temperature and frequency condition, as shown in Figure 1-10 (a) and (b). Error bars are shown in these figures and represent a single standard deviation from the mean. The analysis of these

A-26 figures clearly shows that the test results, i.e., the means plus a single standard deviation, of the LTA specimens are higher overall than those from the STA specimens. Although this graphical technique led the research team to conclude that the differences between the STA and LTA samples are significant, a more comprehensive statistical analysis of these values using the step-down bootstrap method has also been performed. This method is used in lieu of multiple paired t-tests due to the effect of experiment-wise error rates, which can result in incorrect conclusions when making multiple comparisons. Failing to account for this error rate increases the probability of finding significance when none is present. The statistical analysis results are shown by temperature and frequency in Table 1-14. Note that in this table, the conditions under which the means are statistically similar (based on a 95% significance level) are highlighted. From Table 1-14 the first comparisons to review are those between adjacent aging level test results, i.e., STA versus LTA1, LTA1 versus LTA2, and LTA2 versus LTA3. Overall, statistically different values are found between each adjacent aging level except at the extreme conditions, i.e., a fast frequency at a low temperature and a slow frequency at a high temperature. Based on the standard deviation values shown in Figure 1-10, this finding may be explained by the higher degree of replicate variation under extreme conditions as compared to that under less extreme conditions. A comparison of the two extreme aging conditions, STA versus LTA3, shows that significant differences exist for almost all the conditions except 0.1 Hz at 54°C. Because this condition has the highest amount of variability and is also the most likely to contain experimental errors due to accumulated permanent strain, it is reasonable to conclude that overall a statistically significant effect on the dynamic modulus values exists due to the laboratory aging procedures.

A-27 0.0.E+0 5.0.E+3 1.0.E+4 1.5.E+4 2.0.E+4 2.5.E+4 0.0.E+0 5.0.E+3 1.0.E+4 1.5.E+4 2.0.E+4 2.5.E+4 |E*| of STA (MPa) |E *| of L TA s (M Pa ) STA vs. LTA1 STA vs. LTA2 STA vs. LTA3 (a) |E *| of L TA s (M Pa ) 1.0.E+2 1.0.E+3 1.0.E+4 1.0.E+5 1.0.E+2 1.0.E+3 1.0.E+4 1.0.E+5 |E*| of STA (MPa) |E *| of L TA s (M Pa ) STA vs. LTA1 STA vs. LTA2 STA vs. LTA3 (b) |E *| of L TA s (M Pa ) Figure 1-10. Comparison of dynamic modulus of long-term aging levels against the short-term aging level in (a) arithmetic scale and (b) logarithmic scale A.1.2.5 Viscoplasticity Viscoplasticity is commonly considered to be a material level factor related to rutting distress. However, as results from the NCHRP 9-30A project suggest, viscoplasticity may interact with the pavement structure to change the distribution of stress and strain within the

A-28 pavement. Modifications to these distributions may drastically alter the potential for bottom-up cracking because these structural viscoplastic effects have been shown to be the most drastic near the pavement base. Table 1-14. Statistical Analysis Summary of Dynamic Modulus of AL Mix Temperature (°C) Frequency (Hz) p-value STA vs. LTA1 LTA1 vs. LTA2 LTA2 vs. LTA3 STA vs. LTA2 LTA1 vs. LTA3 STA vs. LTA3 -10 25 0.486 0.070 0.100 0.045 0.019 0.019 -10 10 0.589 0.071 0.105 0.044 0.016 0.005 -10 5 0.694 0.047 0.087 0.040 0.014 0.004 -10 1 0.855 0.038 0.034 0.023 0.005 0.005 -10 0.5 0.730 0.035 0.033 0.022 0.006 0.003 -10 0.1 0.460 0.005 0.015 0.001 0.003 0.001 5 25 0.241 0.008 0.031 0.004 0.009 0.009 5 10 0.194 0.012 0.026 0.005 0.010 0.006 5 5 0.142 0.010 0.025 0.004 0.003 0.005 5 1 0.029 0.002 0.040 0.002 0.003 0.006 5 0.5 0.021 0.001 0.033 0.003 0.002 0.008 5 0.1 0.012 0.001 0.024 0.003 0.006 0.011 20 25 0.031 0.004 0.011 0.004 0.006 0.006 20 10 0.047 0.003 0.008 0.006 0.003 0.008 20 5 0.009 0.002 0.012 0.001 0.005 0.005 20 1 0.019 0.002 0.013 0.001 0.003 0.003 20 0.5 0.014 0.003 0.013 0.001 0.003 0.004 20 0.1 0.002 0.000 0.029 0.001 0.008 0.006 40 25 0.429 0.003 0.026 0.006 0.007 0.020 40 10 0.015 0.002 0.011 0.003 0.003 0.001 40 5 0.020 0.001 0.008 0.001 0.003 0.001 40 1 0.023 0.006 0.011 0.002 0.010 0.025 40 0.5 0.025 0.006 0.008 0.001 0.004 0.001 40 0.1 0.054 0.066 0.048 0.017 0.022 0.006 54 25 0.022 0.013 0.021 0.002 0.013 0.002 54 10 0.009 0.001 0.002 0.001 0.003 0.001 54 5 0.022 0.007 0.029 0.002 0.006 0.005 54 1 0.074 0.022 0.140 0.016 0.009 0.016 54 0.5 0.151 0.038 0.132 0.032 0.014 0.023 54 0.1 0.419 0.265 0.507 0.199 0.147 0.132 To develop and implement a rigorous viscoplastic model into the FEP++ code would require substantial effort beyond the scope of the current project. Therefore, viscoplasticity is considered in terms of the sensitivity ratio to viscoplasticity of the given material and structure, the so-called damage correction factor (DCF). The DCF is calculated by using a simple strain-

A-29 hardening viscoplastic material model and pavement responses predicted using the FEP++ without damage for the pavement structure under evaluation. To model the viscoplastic behavior of AC under tensile loading Uzan (23) and Schapery (24) suggest a simple relationship, evidenced by Equation (A-66), which assumes that viscosity obeys a power law in viscoplasticity. Several researchers (24, 25) have shown that the model is applicable to monotonic behavior in tension. ( ) ( )vp vp g σ ε η ε = , (A-66) where ( )g σ = stress function and η = viscosity. Assuming that η is a power law in the viscoplastic strain, Equation (A-67) becomes ( ) vp p vp g A σ ε ε = , (A-67) where A and p are model coefficients. Rearranging and integrating yield ( )p vp vp g dt d A σ ε ε = and (A-68) ( )1 0 1 tp vp p g dt A ε σ+ += ∫ . (A-69) Raising both sides of Equation (A-69) to the 1 ( 1)p + power yields ( ) 11 11 0 1 t pp vp p g dt A ε σ ++  + =        ∫ . (A-70) Letting ( ) qg Bσ σ= , and coupling coefficients A and B into coefficient Y , Equation (A-70) becomes

A-30 11 11 0 1 t pp q vp p dt Y ε σ ++  + =        ∫ . (A-71) In the current work, the coefficients, p , q and Y , are pressure-dependent quantities. Viscoelastic Plastic Elastic Elastic Viscoelastic + Viscoplastic Time Axial Strain Figure 1-11. Strain decomposition from creep and recovery testing Typically, viscoplastic models are characterized using creep and recovery tests. These tests allow relatively easy separation of the viscoplastic and viscoelastic components, as shown in Figure 1-11. However, it is difficult (if not impossible using some machines) to maintain zero load during the recovery period of the creep and recovery test in tension. Therefore, in tension, viscoplastic characterization uses constant rate tests in which the VECD model is used first to predict the viscoelastic strain. This viscoelastic strain is then subtracted from the measured strain to provide the viscoplastic strain that is needed for curve fitting to Equation (A-71). A.1.2.6 Thermal stress Another main source of top-down cracking in asphalt pavements is thermal stress. Thermal stress can contribute to top-down cracking in two ways: (1) through thermal fatigue, expressed

A-31 by repeated, thermally-induced tensile stress at the pavement surface, which can gradually damage the pavement and contribute to surface-induced cracking; and (2) through acute thermal cracking, where very low temperatures cause sudden fracture of the pavement surface. In evaluating thermal cracking and traditional bottom-up cracking, failure is determined when the maximum tensile stress exceeds the tensile strength of the asphalt concrete. However, the situation is more complicated in top-down cracking analysis, because the stress induced at the surface of the pavement involves both normal and shear components. The ability of the viscoelastoplastic continuum damage (VEPCD) model to accurately characterize the tensile behavior of AC under thermally induced loading has been confirmed by Chehab and Kim (26). Measured responses and fracture parameters from thermal strain- restrained specimen tensile (TSRST) strength tests were compared with those predicted using the VEPCD, VECD, and LVE models. Excellent agreement between the measured and predicted responses was found, especially in the VEPCD and VECD models, as explained in the next paragraph. The stress histories predicted for the three cooling rates via the three models are plotted in Figure 1-12 as a function of time. Also plotted are the average measured stress values from all replicates tested at each rate. As is apparent from visual inspection, the stress predicted using the LVE model is greater than the measured stress, with the difference increasing as time increases and the cooling rate decreases. This discrepancy is due to the fact that the LVE model does not account for stress relaxation due to microcracking. The error between the VECD-predicted stress and the measured stress is much smaller than that for the LVE case for all cooling rates. Moreover, the error lessens with an increase in time and a decrease in cooling rate. The VEPCD- predicted stress matches the measured stress very well, with discrepancies being greatest at the

A-32 slowest cooling rate. From a comparison of the predicted stresses among each other, it is evident that the VEPCD model yields the most accurate predictions, slightly better than the VECD model. Another important observation is that the rate of increase in VECD-predicted stress with time deviates from stress that corresponds to the measured and other predictions. Figure 1-12. Average measured and predicted stress histories for different material models and cooling rates A.2 Reasonableness and Sensitivity Checks A.2.1 Reasonableness To verify the modeling framework, first a reasonableness check is performed. Because the purpose here is to verify that the VECD model is implemented correctly, the analytical sub- models (healing, aging, thermal damage, etc.) are not included. Table 1-15 summarizes the simulation details for the analysis to verify reasonableness. Three factors were chosen based on the general effects that are expected of such factors from experience. Four temperature profiles were selected to represent the critical conditions: Tallahassee, Florida in the summer (FL-Jun) at

A-33 5:00 AM (5:00) and at 2:00 PM (14:00), and Laramie, Wyoming in the winter (WY-Dec) at 5:00 AM and at 2:00 PM. The temperature profiles for these four conditions are shown in Figure 1-13. For both geographical locations, the 5:00 AM temperature profile shows a low temperature at the top of the pavement and a high temperature at the bottom. Due to heat from the sun, the opposite trend occurs for both locations at 2:00 PM. Differences between the average temperatures for the two locations are also evident from Figure 1-13. The two pavement structures shown in Figure 3- 34 and the two asphalt mixtures from the ALF study (Control and SBS) were utilized in these simulations. Table 1-15. Simulation Details for Reasonableness Check Item Number of Cases Detail Region 4 FL-Jun-5:00, FL-Jun-14:00, WY-Dec-5:00, WY-Dec-14:00 Structure 2 Thin (127 mm or 5 in.), Thick (304.8 mm or 12 in.) AC Material 2 ALF Control, ALF SBS Support Condition 1 Moderate Aging 1 No Aging Healing 1 None Thermal Stress 1 None DCF 1 None Total Number of Cases 16 Simulation results are presented in Table 1-16 to Table 1-18 in terms of the tensile stress and radial strain at the peak loading time (0.05 sec) for all 16 test cases. For easier comparisons, all contours for the same variables (stress or strain) are plotted using the same scale. From the contours shown in Table 1-16 to Table 1-18, the following comparisons can be made: • Highest versus lowest temperature profile by region (Florida and Wyoming) • Florida versus Wyoming by time of day (5:00 AM or 2:00 PM) • Control versus SBS mixtures under the same condition(s) • Thin versus thick structure under the same condition(s)

A-34 0 2 4 6 8 10 12 14 -10 0 10 20 30 40 50 Temperature (°C) D ep th (c m ) FL-Jun-5:00 FL-Jun-14:00 WY-Dec-5:00 WY-Dec-14:00 Figure 1-13. Temperature profile for reasonableness check A key observation from the plots shown in Table 1-16 to Table 1-18 is that the tensile strain at the pavement bottom is greater at higher temperatures. The magnitude of stress also shows a change, a decrease, with increasing temperatures. This effect is expected because the support conditions are constant for all cases, and the modulus value of the asphalt layer increases with reductions in temperature. However, it is interesting that daily temperature variations have little effect on the stress distribution. Instead, a large difference in the mean temperature is needed in order to change the stress distribution noticeably. This effect is more obvious in the thin pavement section than in the thick pavement section, and results from increased bending- related stress in the thin sections. Of course, because the stress distribution changes very little due to temperature variations, the strain magnitude increases with the temperature because the material is softer at higher temperatures. Unlike the stress distributions, though, these changes in strain magnitude may be noticeably different during daily temperature fluctuations.

A-35 Comparisons of material type reveal that when the SBS materials are used, higher strain magnitudes result. If the temperatures shown in Figure 1-13 are combined with the data shown in Figure 3-35, i.e., the mastercurve, appropriate time temperature shift factors, and a quasi- approximate time-frequency conversion (27), it can be seen that the most extreme temperature in the simulations, 41°C, and the loading time of 0.1 second, correspond approximately to a reduced frequency of 6.0 x 10-5 Hz. At this reduced frequency, the SBS material shows a softer LVE response than the Control mixture, as seen in Figure 3-35. If, on the other hand, the simulation is performed at the high temperature PG grade (continuous grade-based) of the two mixtures, 72°C, then the approximate equivalent reduced frequency would be 1.5 x 10-7 Hz. At this reduced frequency, the SBS mixture is stiffer (in terms of linear viscoelasticity) than the Control mixture, and it should be expected that the radial strain would be less for the SBS mixture than for the Control mixture. Table 1-19 shows the contours for a thin pavement at 0.05 second and at 72°C for both the SBS and the Control mixtures. From this table it is seen that the SBS mixture shows less total strain than the Control mixture, which, along with the data in Table 1-16 to Table 1-18 and Figure 3-35, confirms the reasonableness of the response predictions with regard to material type. Finally, comparisons between the thin and thick pavements show that the magnitude of the tensile strain at the bottom of the thin pavement is always greater than the strain in the equivalent thick pavement case. The difference is more noticeable in the Wyoming simulations because of the scaling used in the contours and because the Wyoming conditions result in a mixture with a high modulus. The Florida simulations show similar gray scale patterns of the thick and thin pavements, which is a result of setting a scale capable of delineating strain under all of the conditions shown. These cases are thus misleading; in reality, the strain in the thin pavement

A-36 case is greater than in the simulations. It is also observed that the approximate pavement shear center, as indicated by a bulb of radial strain extending near the loading edge, is relatively higher in the thick pavement (at 20% of total depth) than it is in the thin pavement case (at 40% of total depth). This behavior is to be expected in layered analysis. Table 1-16. Tensile Stress and Strain Contours for Reasonableness Check: Thin Pavements – Florida and Wyoming Case Tensile Stress at t = 0.05 sec (kPa) Tensile Strain at t = 0.05 sec FL-Jun-5:00 Thin-Control 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 FL-Jun-5:00 Thin-SBS 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 FL-Jun-14:00 Thin-Control 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 FL-Jun-14:00 Thin-SBS 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 WY-Dec-5:00 Thin-Control 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 WY-Dec-5:00 Thin-SBS 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 WY-Dec-14:00 Thin-Control 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 WY-Dec-14:00 Thin-SBS 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 Contour Legend Srr: -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 Err: -1.6E-04 -1.3E-04 -1.0E-04 -7.0E-05 -4.0E-05 -1.0E-05 2.0E-05 5.0E-05 8.0E-05 1.1E-04 1.4E-04

A-37 Table 1-17. Tensile Stress and Strain Contours for Reasonableness Check: Thick Pavements - Florida Case Tensile Stress at t = 0.05 sec (kPa) Tensile Strain at t = 0.05 sec FL-Jun-5:00 Thick-Control 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 FL-Jun-5:00 Thick-SBS 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 FL-Jun-14:00 Thick-Control 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 FL-Jun-14:00 Thick-SBS 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 Contour Legend Srr: -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 Err: -1.6E-04 -1.3E-04 -1.0E-04 -7.0E-05 -4.0E-05 -1.0E-05 2.0E-05 5.0E-05 8.0E-05 1.1E-04 1.4E-04

A-38 Table 1-18. Tensile Stress and Strain Contours for Reasonableness Check: Thick Pavements - Wyoming Case Tensile Stress at t = 0.05 sec (kPa) Tensile Strain at t = 0.05 sec WY-Dec-5:00 Thick-Control 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 WY-Dec-5:00 Thick-SBS 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 WY-Dec-14:00 Thick-Control 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 WY-Dec-14:00 Thick-SBS 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 Contour Legend Srr: -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 Err: -1.6E-04 -1.3E-04 -1.0E-04 -7.0E-05 -4.0E-05 -1.0E-05 2.0E-05 5.0E-05 8.0E-05 1.1E-04 1.4E-04

A-39 Table 1-19. Tensile Stress and Strain Contours for Reasonableness Check: Thin Pavements – Full Depth 72°C Case Tensile Stress at t = 0.05 sec (kPa) Tensile Strain at t = 0.05 sec HighTemp. 72ºC Thin-Control 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 HighTemp. 72ºC Thin-SBS 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 Contour Legend Srr: -400 -350 -300 -250 -200 -150 -100 -50 0 50 Err: -1.0E-03 -8.0E-04 -6.0E-04 -4.0E-04 -2.0E-04 0.0E+00 2.0E-04 4.0E-04 6.0E-04 A.2.2 Sensitivity This section discusses in detail the effect of each analytical sub-model on pavement performance. Simulation details are shown in Table 1-20. Here, the pavement structure and asphalt material are fixed as the thick pavement and Control mix, respectively. Temperature data for two regions (Florida and Wyoming) are utilized for 1 year or 20 years, depending on the purpose. It was decided that the DCF should be included in all cases, because this factor has a significant effect, especially in Florida, which experiences relatively higher temperatures than Wyoming. For the support condition, sensitivity is evaluated independently of the other factors, but a specific condition was chosen for the sensitivity analysis of each variable (i.e., a weak support condition was used to assess the DCF, and a moderate support condition was used to assess aging). The rationale behind selecting which support condition to use for which variable is to amplify the effect of the variable under investigation; i.e., the support condition is chosen to represent the worst case scenario. A.2.2.1 Support condition effect Table 1-21 displays a comparison of the damage contours for Florida and Wyoming to show the effects of varying the support condition. From this table it is evident that a strong

A-40 support condition leads to an overall reduction in damage distribution. The effect is most significant for the top-down damage, but also affects the damage at the bottom of the pavement. It is also seen that the effect of the support condition is most noticeable in the case of the Wyoming climatic condition. Table 1-20. Simulation Details for Sensitivity Check Item Number of Cases Details Region 2 FL, WY Structure 1 Thick (304.8 mm or 12 in.) AC Material 1 ALF Control Support Condition 3 Weak, Moderate, Strong Aging 2 No Aging, Aging Healing 0 --* Thermal Stress 2 High vs. Low Thermal Coefficient DCF 2 1-yr EICM or 10-yr EICM Total Number of Cases 32 * Not assessed due to time and resource limitations Table 1-21. Support Condition Effect in Extreme Regions: Florida and Wyoming Conditions FL WY Control Mix SC – Weak DCF No Aging No Healing No Thermal 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 Control Mix SC – Medium DCF No Aging No Healing No Thermal 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 Control Mix SC – Strong DCF No Aging No Healing No Thermal 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3

A-41 A.2.2.2 Aging effect To investigate the effect of aging, FEP++ simulations were conducted, and the results are shown in Table 1-22. In the simulations a single year run was performed, but the input material properties correspond to either the un-aged properties (denoted as No Aging in Table 1-22) or the 20 year-aged properties (denoted as Aging in Table 1-22). Because the amount of field aging is related directly to environmental conditions, two extreme cases, Laramie, Wyoming and Tallahassee, Florida, were used in this sensitivity analysis. The ALF Control mixture was used for both cases. Note that, because the purpose of the first simulation runs was to assess only the sensitivity of the aging model, the healing and thermal damage models were not active. The following observations can be made from the contours shown in Table 1-22: • Aging can have a noticeable effect on the damage growth in a pavement, particularly at the pavement surface; • The effects of aging are more significant in climates with higher annual pavement temperatures (Florida) than in cooler climates (Wyoming); • Although the effects of aging are most noticeable at the pavement surface, aging affects the way stress distributes throughout the pavement and, thus, the way that damage accumulates throughout the whole pavement structure. Table 1-22. Aging Effect in Two Extreme Regions: Florida and Wyoming Conditions FL WY Control Mix SC – Medium DCF No Aging No Healing No Thermal 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 Control Mix SC – Medium DCF Aging No Healing No Thermal 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3

A-42 A.2.2.3 Thermal stress effect Thermal stress is the direct result of structural constraints on material expansion. When a material is heated or cooled it tends to expand or contract by an amount directly proportional to the change in temperature. The constant of proportionality is known as the thermal coefficient, and for the purposes here is assumed to be a material constant. This parameter is not typically measured for AC, but it can be estimated from Equation (A-72) (29). A parametric study using this equation with typical in-service input values shows that this property may range from 1.3 x 10-5 to 2.5 x 10-5 Cε  . This range was further truncated for this study, and two mixα values, 1.2 x 10-5 and 2.1 x 10-5, were used in the sensitivity analysis. * * 3* AC AGG agg mix Total VMA V V α α α + = , (A-72) where mixα = thermal coefficient of the mixture, ACα = thermal coefficient of the asphalt cement (3.45 x 10 -4 Cε  ), aggα = thermal coefficient of the aggregate particles (6.5 x 10 -6 Cε  ), VMA = percentage of voids in the mineral aggregate, AGGV = percentage of aggregate in the mixture, by volume, and TotalV = percentage of total volume, 100. Laramie, Wyoming and Tallahassee Florida are the regions selected for these simulations. Because thermal damage is most severe during the winter months when the material is stiff and cools relatively rapidly, only the damage that occurs during December was simulated. The results are shown in Figure 1-14 for the four different conditions. From this figure it is clearly observed that the damage is greater with the larger thermal coefficient. The effect of the thermal coefficient varies by region because the modulus is affected by the mean temperature. In the case of the Florida simulation, the material is much more viscous than it is in the Wyoming case. As a

A-43 result, the material is more likely to relax and absorb thermally-induced dimensional changes and, therefore, experience less damage. Because the overall effect of thermal damage is reduced, the apparent effect of the thermal coefficient is also reduced for Florida. In Wyoming, most of the damage in December occurs due to thermal damage and, as a result, it is concluded that the total damage growth for Wyoming is quite sensitive to the thermal coefficient. Similar ratios of damage growth for the two coefficients would be observed for other months, but the total damage growth for those months might not be as large as is seen for December, so the thermal stress effect may be smaller. 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 WY-Low WY-High FL-Low FL-High R ed uc tio n of C Thermal Load Mechanical Load 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 WY-Low WY-High FL-Low FL-High R ed uc tio n of C Thermal Load Figure 1-14. Sensitivity of thermal damage growth to selected thermal coefficient: (a) thermal and mechanical, and (b) thermal damage only A.2.2.4 Viscoplastic effect The effect of viscoplasticity on pavement performance, especially cracking, could be significant, depending on the material, structure, and/or temperature. As described in Section 1.2.3.2, the effect of viscoplasticity on the pavement performance can be handled through the DCF. To evaluate the viscoplastic effect, simulations were conducted by making only the DCF analytical model active. The other sub-models – healing, aging and thermal – were inactive. Some typical results are shown in Table 1-23. To amplify the effects of the DCF, the support conditions for the contours are all weak, as shown in Table 1-23. These simulations were run for

A-44 one year with and without the DCF. As is clearly seen in Table 1-23, the DCF noticeably reduces the total damage growth for both the Wyoming and Florida regions. The effects of the DCF are somewhat more pronounced in the Florida case and much more pronounced near the pavement base. Table 1-23. Viscoplastic Effect in Two Extreme Regions: Florida and Wyoming Conditions FL – 1 year WY – 1 year Control Mix SC – Weak No DCF No Aging No Healing No Thermal 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 Control Mix SC – Weak DCF No Aging No Healing No Thermal 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 Another important aspect of the DCF that should be examined is the temperature used in the computations. This issue is important because most of the temperature data compiled in the EICM are less than 10-year data, and to simulate a pavement condition requires repeating the data. The effect of sequential year temperature data on the DCF calculation was evaluated. Two methods to calculate the DCF were chosen: • Calculate the DCF using 1-year EICM data and repeat both the temperature profiles and the DCF 20 times. • Calculate the DCF year by year and use annual specific temperature profiles by repeating the 10-year EICM data. As with the other DCF sensitivity analysis, no other analytical sub-models were active in these simulations. For simulation purposes, the D.C. temperature data were selected as representative for all regions. The ALF Control mix and a moderate modulus were utilized for

A-45 the asphalt layer and subgrade, respectively. As shown in Table 1-24, the two methods generate similar results. As a result of the analysis shown in Table 1-24, it is concluded that the DCF calculation is not sensitive to yearly temperature fluctuations and, as a result, only a single annual temperature profile needs to be used in computing its value. Because thermal effects were considered by sequentially repeating five years of EICM data, the DCF was computed based on the average of five years of EICM data. Table 1-24. Sensitivity of the Damage Factor Calculation Conditions Method DC -20 years Control Mix SC – Medium DCF No Aging No Healing No Thermal EICM-1 yr DCF-1 yr 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 EICM-10 yr DCF-20 yr 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 C 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 A.3 Example Simulation and Parametric Study Contour Plots A.3.1 Example Simulation Contour Plots Results from the simulations are shown as a series of contour plots. The variable of interest in these contours is the condition index (CI). The resulting contour plots for this simulation condition are summarized in Table 1-25. In this table, the month column contains two items of information: the simulation month number (all simulations began in July) and the abbreviated month that corresponds to the simulation. Simulations for all cases were performed until one element reached a pseudo stiffness value of 0.25 or until ten years had been simulated. In the

A-46 case of the thin pavement, the length of time was somewhat longer than a full year, but the thick pavement was simulated for the full ten years. Table 1-25 shows the contours for the months of October through July of the first simulation year for both the thick pavement and the thin pavement. In addition, the final nine months of the thick pavement simulation are shown in the far right column of the table. All contours are interpreted in terms of the extent and severity of the damage. A.3.2 Parametric Study Contour Plots The simulated CI contours are shown for the thin pavements in Table 1-26 and for the thick pavements in Table 1-27 through Table 1-29. Note that the contours in Table 1-26 and Table 1-27 are all at the end of the ninth month (March), whereas the contours in Table 1-28 and Table 1-29 are also for March, but at year 5 and year 10, respectively. Also note the naming convention used for the different simulations. For the evaluation of the effect of asphalt mixture properties, the damage progression contours from October through June are shown in Table 1-30. The letters stand for, in order: pavement type, thick (T) or thin (t); material type, ALF control (C) or ALF SBS (S); and support condition, weak (W) or strong (S).

A-47 Table 1-25. Damage Contours for Example Simulations Mon. t-C-M-DC T-C-M-DC Mon. T-C-M-DC 4 (Oct) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 112 (Oct) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 (Nov) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 113 (Nov) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 6 (Dec) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 114 (Dec) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

A-48 7 (Jan) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 115 (Jan) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 8 (Feb) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 116 (Feb) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 9 (Mar) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 117 (Mar) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 (Apr) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 118 (Apr) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

A-49 11 (May) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 119 (May) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 12 (Jun) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 120 (Jun) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 13 (Jul) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

A-50 Table 1-26. Parametric Study Simulations for Thin (t) Pavements Condition∗ FL DC WY t-C-W 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 t-C-S 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 t-S-W 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 t-S-S 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ∗ Four conditions were considered for thin pavements: 1) t-C-W stands for a thin pavement with an ALF Control material layer over a weak supporting layer; 2) t-C-S stands for a thin pavement with an ALF Control material layer on a strong supporting layer; 3) t-S-W stands for a thin pavement with an ALF SBS material layer over a weak supporting layer; and 4) t-S-S stands for a thin pavement with an ALF SBS material layer on a strong supporting layer. Similar conditions were considered for thick pavements (see Tables 1-27 to 1-29), except that t (thin) was replaced by T (thick).

A-51 Table 1-27. Parametric Study Simulations for Thick (T) Pavements: Year 1 Condition FL DC WY T-C-W 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T-C-S 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T-S-W 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T-S-S 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

A-52 Table 1-28. Parametric Study Simulations for Thick (T) Pavements: Year 5 Condition FL DC WY T-C-W 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T-C-S 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T-S-W 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T-S-S 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

A-53 Table 1-29. Parametric Study Simulations for Thick (T) Pavements: Year 10 Condition FL DC WY T-C-W 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T-C-S 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T-S-W 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T-S-S 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

A-54 Table 1-30. Damage Progression and Healing in SBS and Control Pavements Mon. t-C-W-DC t-S-W-DC 4 (Oct) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 6 (Dec) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 8 (Feb) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 (Apr) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 12 (Jun) 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 CI 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

A-55 A.4 List of References 1. Kim, Y. R., C. Baek, B. S. Underwood, V. Subramanian, M. N. Guddati, and K. Lee. Application of Viscoelastic Continuum Damage Model Based Finite Element Analysis to Predict the Fatigue Performance of Asphalt Pavements. KSCE, Journal of Civil Engineering, Vol. 12, No. 2, 2008, pp. 109-120. 2. Park, S. W. and R. A. Schapery. Methods of Interconversion between Linear Viscoelastic Material Functions. Part I – A Numerical Method based on Prony Series. International Journal of Solids and Structures, Vol. 36, 1999, pp. 1653-1657. 3. Schapery, R. A. A Simple Collocation Method for Fitting Viscoelastic Models to Experimental Data. GALCIT SM 61-23A. California Institute of Technology, Pasadena, CA. 1961. 4. Schapery, R. A. Correspondence Principles and a Generalized J-integral for Large Deformation and Fracture Analysis of Viscoelastic Media. International Journal of Fracture, Vol. 25, 1984, pp. 195-223. 5. Underwood, B. S., Y. R. Kim, S. Savadatti, S. Thirunavukkarasu, M. N. Guddati. Response and Fatigue Performance Modeling of ALF Pavements Using 3-D Finite Element Analysis and a Simplified Viscoelastic Continuum Damage Model. Journal of the Association of Asphalt Paving Technologists, 2009. In Press. 6. Reese, R. Properties of Aged Asphalt Binder Related to Asphalt Concrete Fatigue Life. Journal of the Association of Asphalt Paving Technologists, Vol. 66, 1997, pp. 604-632. 7. Hou, T., B. S. Underwood, and Y. R. Kim. Fatigue Performance Prediction of North Carolina Mixtures Using the Simplified Viscoelastic Continuum Damage Model. Journal of the Association of Asphalt Paving Technologists, Vol. 79, 2010. In press. 8. Daniel, J. S. and Y. R. Kim. Development of a Simplified Fatigue Test and Analysis Procedure Using a Viscoelastic Continuum Damage Model. Journal of the Association of Asphalt Paving Technologists, Vol. 71, 2002, pp. 619-650. 9. Underwood, B. S., Y. R. Kim and M. N. Guddati. Characterization and Performance Prediction of ALF Mixtures Using a Viscoelastoplastic Continuum Damage Model. Journal of the Association of Asphalt Paving Technologists, Vol. 75, 2006, pp. 577-636. 10. Underwood, B. S., Y. R. Kim, and M. N. Guddati. Improved Calculation Method of Damage Parameter in Viscoelastic Continuum Damage Model. International Journal of Pavement Engineering, 2009. In Press. 11. Tangella, S. C. S. R., J. Craus, J. A. Deacon, and C. L. Monismith. Summary Report on Fatigue Response of Asphalt Mixtures. Prepared for SHRP Project A-003-A. Strategic Highway Research Program, National Research Council, Washington, D.C., 1990.

A-56 12. Strategic Highway Research Program (SHRP). Fatigue Response of Asphalt-Aggregate Mixes. SHRP-A-404, National Research Council, Washington, D.C., 1994. 13. Tayebali, A. A., J. A. Deacon, and C. L. Monismith. Development and Evaluation of Surrogate Fatigue Models for SHRP A-003A: Abridged Mix Design Procedure. Journal of the Association of Asphalt Paving Technologists, Vol. 64, 1995, pp. 340-364. 14. Asphalt Institute. Research and Development of the Asphalt Institute’s Thickness Design Manual (MS-1), 9th edition. Research Report 82-2. 1982. 15. Tayebali, A. A., J. A. Deacon, J. S. Coplantz, and C. L. Monismith. Modeling Fatigue Response for Asphalt-Aggregate Mixtures. Journal of the Association of Asphalt Paving Technologists, Vol. 62, 1993, pp. 385-421. 16. Huang , Y. Material Characterization and Performance Properties of Superpave Mixtures. Ph.D. Dissertation, North Carolina State University, Raleigh, NC. 2004. 17. Finn, F., C. L. Saraf, K. Kulkarni, K. Nair, W. Smith, and A. Abdullah. Development of Pavement Structural Subsystems. Final Report NCHRP 1-10B, National Cooperative Highway Research Program, National Research Council, Washington, D.C., 1977. 18. Shell International Petroleum. Shell Pavement Design Manual: Asphalt Pavements and Overlays for Road Traffic. London. 1978. 19. Craus, J., R. Yuce, and C. L. Monismith. Fatigue Behavior of Thin Asphalt Concrete Layers in Flexible Pavement Structures. Journal of the Association of Asphalt Paving Technologists, Vol. 53, 1984, pp. 559-582. 20. Advanced Research Associates. 2002 Design Guide: Design of New and Rehabilitated Pavement Structures. NCHRP 1-37A Project, National Cooperative Highway Research Program. National Research Council. Washington, D.C., 2004. 21. Epps, J. A. and C. L. Monismith. Influence of Mixture Variables on the Flexural Fatigue. Journal of the Association of Asphalt Paving Technologists, Vol. 38, 1969, pp. 423-464. 22. Pell, P. S. and K. E. Cooper. The Effect of Testing and Mix Variables on the Fatigue Performance of Bituminous Materials. Journal of the Association of Asphalt Paving Technologists, Vol. 44, 1975, pp. 1-37. 23. Uzan, J., M. Perl, and A. Sides. Viscoelastoplastic Model for Predicting Performance of Asphalt Mixtures. In Transportation Research Record: Journal of the Transportation Research Board, No. 1043, Transportation Research Board of the National Academies, Washington, D.C., 1985, pp. 78-79.

A-57 24. Schapery, R. A. Nonlinear Viscoelastic and Viscoplastic Constitutive Equations with Growing Damage. International Journal of Fracture, Vol. 97, 1999, pp. 33-66. 25. Chehab, G. R., Y. R. Kim, R. A. Schapery, M. Witczack, and R. Bonaquist. Characterization of Asphalt Concrete in Uniaxial Tension Using a Viscoelastoplastic Model. Journal of the Association of Asphalt Paving Technologists, Vol. 72, 2003, pp. 315-355. 26. Chehab, G. R. and Y. R. Kim. Viscoelastoplastic Continuum Damage Model Application to Thermal Cracking of Asphalt Concrete. ASCE Journal of Materials in Civil Engineering, Vol. 17, No. 4, 2003, pp. 384-392. 27. Underwood, B. S. and Y. R. Kim. Determination of the Appropriate Representative Elastic Modulus for Asphalt Concrete. International Journal of Pavement Engineering, Vol. 10, No. 2, 2009, pp. 77-86. 28. Lytton, R. L., J. Uzan, E. G. Fernando, R. Roque, D. Hiltunen, and S. M. Stoffels. Development and Validation of Performance Prediction Models and Specifications for Asphalt Binders and Paving Mixes. SHRP-A-357, Strategic Highway Research Program, National Research Council, Washington, D.C., 1993.