**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

*Relationships Between the Fatigue Properties of Asphalt Binders and the Fatigue Performance of Asphalt Mixtures*. Washington, DC: The National Academies Press. doi: 10.17226/26302.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

*Relationships Between the Fatigue Properties of Asphalt Binders and the Fatigue Performance of Asphalt Mixtures*. Washington, DC: The National Academies Press. doi: 10.17226/26302.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

*Relationships Between the Fatigue Properties of Asphalt Binders and the Fatigue Performance of Asphalt Mixtures*. Washington, DC: The National Academies Press. doi: 10.17226/26302.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

**Suggested Citation:**"APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA." National Academies of Sciences, Engineering, and Medicine. 2021.

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155 APPENDIX D: ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA INTRODUCTION One of the biggest challenges during NCHRP 9-59 was developing methods to effectively compare mixture fatigue performance to specific binder test data. There are a few problems in making such comparisons, especially when the comparison involves performance of actual pavements. The strains in an asphalt mixture specimen will generally be much different from those in a binder specimen during a specific test. Furthermore, in some tests and in almost all pavements the strains during loading will vary significantly over time. The modulus of the asphalt concrete might also vary. The test temperature and strain used in performing mixture fatigue tests in the laboratory might vary from the tests used to characterize the binder. Candidate binder tests might involve single tests to failure, while fatigue tests generally give results in cycles to failure. These problems are significant and explain why it has been difficult to correlate binder test data to laboratory or field fatigue performance. For similar reasons it has also proven difficult to correlate laboratory fatigue tests to pavement field fatigue performance. The purpose of this appendix is to describe the how the mixture fatigue and binder test data collected as part of NCHRP 9-59 was analyzed, and to discuss the results of this analysis and their implications for the project objectives. In order to effectively analyze these data, the research team developed a new and unique model for the fatigue and fracture of asphalt binders and mixes, called the general failure theory for asphalt binders (GFTAB). The sections below include a short background on fatigue damage analysis, then describe the method of analysis in the form of a handful of equations, followed by a summary of the results of applying this analysis to data collected as part of NCHRP 9-59. Details of the test methods and data used in this analysis are not discussed in detail because this information is presented elsewhere in this report. There are two critical questions that are addressed in this appendix: (1) is the GFTAB model reasonably accurate for the range of data collected during NCHRP 9-59; and (2) are any of the binder tests and/or parameters good predictors of mixture fatigue performance? As discussed below, the evidence seems to clearly suggest that the GFTAB model is accurate, at least for the data collected and analyzed as part of NCHRP 9-59. Concerning the correlation of binder tests to mixture fatigue performance as measured in laboratory tests, both the Christensen- Anderson R-value and the SDENT test are both indicators of binder strain tolerance and so are closely related to asphalt mixture fatigue performance. For non-modified binders, R-value alone can be used as an indicator of fatigue and fracture performance. For polymer-modified binders, both R-value and SDENT must be used for a complete picture of fatigue and fracture performance. MODELING FATIGUE IN ASPHALT CONCRETE There are two general types of fatigue laws. The first and most common expresses fatigue life, typically as cycles to failure, as a function of stress or strain, modulus and one or more volumetric factors such as air void content or voids filled with asphalt. These types of relationships can be called fatigue life equations. An example of this type of fatigue law can be found in the recently developed AASHTO mechanistic-empirical design guide (ARA Inc., 2004)

156 for flexible pavements, which uses the following equation for estimating cycles to failure for bottom-up traffic-associated fatigue cracking, calibrated for general use in the United States: 281.19492.3 1 1100432.0 ï·ï· ï¸ ï¶ ï§ï§ ï¨ ï¦ ï·ï· ï¸ ï¶ ï§ï§ ï¨ ï¦ = HMAt ff E CN Îµ Î² (D-1) Where Î²f1 and C are variables that depend on the asphalt concrete thickness and composition, respectively, Îµt is the maximum tensile strain at the bottom of the asphalt concrete layer, and EHMA is the modulus of the asphalt concrete. This equation is typical for mixture fatigue relationships, which generally contain both strain and mixture modulus as predictors, each raised to some exponent which is ultimately determined through some sort of statistical calibration. More recently, methods have been developed for characterizing the way damage accumulates under fatigue loading in an HMA mixture (Kim and Little, 1990; Little et al., 1997; Kim et al., 2001; Daniel and Kim, 2002; Underwood et al., 2010). Usually this damage is characterized by the ratio of the damaged to undamaged modulus. An example of this type of damage law, developed using continuum damage theory, has been proposed by Underwood and his associates: ( ) ( ) Î±Î± Î± Î¾Îµ + + Îïº ï» ï¹ ïª ï« ï© Îâ= 1 112 2 1 iii R i CdS (D-2) This is a discrete, general form of the damage calculation, where Si is the internal state variable representing damage, ÎµR is the pseudo-strain, C is pseudo-secant modulus, Î¾ is the reduced time under non-constant temperature conditions, and Î± is the damage evolution rate. In actual practice, the calculation of the damage parameter S from real test data is much more complicated than suggested by Equation D-2, as explained in detail by Underwood and his associates (2010). An advantage of this type of fatigue damage law is that it provides a way of estimating the damaged modulus in an asphalt pavement under traffic loading. One disadvantage is that it doesnât really provide a good estimate of when the pavement will actually fail. Zhang and his associates have recently addressed this problem by develop a failure criterion to be used in conjunction with the continuum damage-based damage law (Zhang et al., 2013). However, as with most analysis based on continuum damage theory application of the failure criteria is complicated and not directly based on actual observations of material failure. Christensen and Bonaquist have developed an alternate approach to modeling fatigue damage in asphalt concrete, loosely based on continuum damage principles but in some ways simpler in application. Although simpler than the rigorous approach described above, it also does not directly address when material failure occurs (Christensen and Bonaquist, 2005; Christensen and Bonaquist, 2009; Christensen and Bonaquist, 2012). A NEW APPROACH TO ANALYZING ASPHALT CONCRETE FATIGUE DAMAGE None of the available fatigue models discussed above have proven very useful in correlating binder test data to mixture fatigue performance. Standard laboratory fatigue tests are often not even particularly good indicators of field performance. For these reasons, the NCHRP 9-59 research team developed a new approach for asphalt mixture and binder fatigue analysis. The GFTAB model focuses on material failure rather than the accumulation of damage as manifested

157 in the gradual loss of modulus during loading. This is in part because most fatigue dataâ laboratory test data and field performance dataâis more closely related to material failure rather than damage accumulation. This aspect of fatigue is also emphasized because from a practical point of view this is what pavement engineers are primarily concerned aboutâwhen will a pavement fail? The approach is based on the simple premise that under any given loading condition, the fatigue life of an asphalt concrete specimen is proportional to some ultimate fatigue strain, called here the fatigue strain capacity (FSC), divided by the applied strain, and all raised to a fatigue exponent inversely proportional to the phase angle: ð = Ã( â ) ( â ) (D-3) Where FSC = binder fatigue strain capacity, % VBE = mixture effective binder content, volume % Îµt = mixture maximum tensile strain, % k1 = fatigue exponent coefficient Î´ = binder phase angle, degrees Note that in this equation the mixture tensile strain is divided by the term (VBE/100), which provides a rough estimate of the average binder strain within the mixture. This is done in part because the focus of NCHRP 9-59 is binder testing and not mix testing, but also because this effectively addresses the primary effect of changes in VBE on fatigue resistance in asphalt concrete in a rational way. The fatigue exponent is inversely related to the phase angle, 2 Ã (90/Î´); the fatigue coefficient is determined through statistical analysis of mixture fatigue data, as explained later in this appendix. The value of the coefficient in statistical analysis was found to be 2.08 and was rounded to 2 for simplicity and because in continuum damage theory fatigue exponents are generally rounded to the nearest integer or half integer. An important issue is how to determine the value of the phase angle used in Equation D-3. For many polymer-modified binders, the phase angles are significantly altered by the polymer network. It appears in such cases it is the phase angle of the continuous asphalt binder phase that should be used in Equation D-3. Although determining this value precisely is difficult (or even impossible), a reasonable estimate can be made by determining the Christensen-Anderson R-value at a high modulus value and then estimating the binder modulus using the relationship between phase angle, modulus and R-value. A simple approach to calculating R-valueâand the one used in NCHRP for determining the phase angle value for use in fatigue analysisâinvolves determining the modulus and phase angle at a single point, where the modulus is at least 10 MPa, and using the following equation to estimate R: ð = ððð(2) | â| Ãâ( â ) (D-4) Where R = Christensen-Anderson R (rheologic index) |G*| = dynamic complex modulus, Pa Î´ = binder phase angle, degrees (at same temperature and frequency as |G*|)

158 The phase angle is then calculated from the R-value along with the modulus at the desired temperature and frequency: ð¿ = 90 1 â ðð¥ð ( )Ã | â| Ãâ (D-5) Where the variables are as defined above. In the discussion below, where the phase angle is mentioned or used in an equation, it refers to the phase angle as determined above. For almost all non-polymer-modified binders and some polymer-modified binders there is no significant difference between phase angle values determined this way and from actual measured values at intermediate to high modulus values. For heavily modified binders, this difference can however be substantial and in such cases it is essential to calculate phase angles from the R-value as described above. Another important characteristic of Equation D-3 is that when Îµt /(VMA/100) = FSC, that is when the estimated binder strain is equal to the fatigue strain capacity, the material will fail in a single cycle of loading; that is, Nf = 1. In practical terms this means that FSC should be closely related to measures of binder failure strain. However, the binder within an asphalt concrete mixture is severely constrained and contains numerous flaws and stress concentrations, so the value of FSC will normally be much less than the failure strain in a binder test conducted at similar temperatures and loading rates. An important corollary to this observation is that while standard tensile strength testsâperformed on unconstrained specimens and long aspect ratiosâ generally are very useful in characterizing well defined engineering properties, they probably donât accurately represent what is happening to the binder in an asphalt concrete mixture under loading. The basic relationships for the GFTAB theory, as explained above, can be used to address several problems in the analysis of asphalt binder and mixture fatigue and fracture data. Equation D-3 can be rearranged and solved for FSC: ð¹ðð¶ = ð ( )â ð (D-6) For a given fatigue test where the strain is held constant, Equation D-6 can be used to estimate the value of FSC. However, in many situations, the strain level during fatigue loading will vary significantly. In this case, use can be made of the following damage function D: ð· = â ð ( ) Ã (ððµð¸ 100â ) ( â ) (D-7) Where the subscript i refers to loading segments where the values of Îµt, FSC and FE are approximately equal. If one or more of these vary significantly, the calculation of damage must be done in separate segments. In stress-controlled laboratory testing, for example, the value of FSC and the fatigue exponent will remain constant but the applied strain Îµt will gradually increase. In this case the calculation of damage can be done by breaking up the loading history into numerous small segments with approximately equal strain and calculating and summing the damage according to Equation D-7. Failure will occur when D = 1, which leads to the following equation for calculating FSC for a laboratory fatigue test in which strain is not constant:

159 ð¹ðð¶ = â ð ( )â ( â ) ï¤ ( )â (D-8) Where nf is the total loading cycles to failure. It must be emphasized here that FSC is not a constant but varies significantly with modulus. In fact, it is mostly a function of binder modulus but there are still significant differences in the relationship between FSC and modulus for different binders. These differences are indicative of the inherent fatigue and fracture resistance of different binders, and the primary property that a binder fatigue specification must control. The proposed method can also be used to calculate FSC values for binder fatigue tests, such as the time sweep test and the linear amplitude sweep (LAS) test. In this case, Equations D-6 and D-8 can be applied, but the term (VBE/100) becomes equal to one. This allows a convenient, direct comparison between FSC values calculated from laboratory tests or pavement test sections and those calculated from binder fatigue tests. When FSC is plotted against binder modulus for a wide range of binders under a range of temperatures and loading rates, the data form a well-defined failure envelope, giving FSC as a function of modulus for asphalt binders. Figure D-1 shows two failure envelopes: (1) one proposed by Heukelom for asphalt binders tested in tension using various methods (Heukelom, 1966); and (2) one determined during NCHRP 9-59, from double-edge notched tension (DENT) tests and using direct tension test data from previous research projects. Although FSC is not exactly the same as failure strain, it should be closely related and the FSC failure envelope should be reasonably close to those shown in Figure D-1. However, in reality the FSC values for many asphalts vary from the average or typical failure envelope. For instance, many polymer- modified binders exhibit FSC values substantially higher than typical non-modified binders. As discussed later in this appendix, the rheologic type of the binder also has a significant effect on FSC. The deviation of a particular binder from this typical failure envelope can be characterized by the fracture/fatigue performance ratio (FFPR), which is the ratio of the FSC for the binder to the FSC value for a typical binder: ð¹ð¹ðð = (D-9) FFPR values above one indicate above average fatigue or fracture performance, while values below one suggest poor performance. The concept of a typical FSC value is an important one, since it defines the âstandardâ failure envelope and is also essential to calculating FFPR values and comparing performance among asphalt binders. For this reason, typical FSC valuesâas used to define the standard failure envelopeâare denoted in this appendix as FSC*; when FSC is given without the asterisk, it indicates an FSC value calculated for a specific binder. Using this notation, Equation D-9 becomes: ð¹ð¹ðð = â (D-10) The main advantage of using FFPR to characterize fatigue performance is that it eliminates the effect of modulus on fracture and fatigue properties. Normally strain capacity decreases substantially with increased modulus, which can also result in decreased fatigue life at higher strain levels. This can complicate comparing fatigue and fracture tests among mixtures and binders with different modulus values. The applied strain can also affect the fatigue life of a

160 mixture. Although it is fairly easy to control strain in some mixture fatigue tests, comparing tests results at different strains can be difficult, and binder tests and mixtures tests are rarely performed at comparable strain levels. FFPR provides a way of comparing fatigue and fracture performance largely independent of the effects of modulus and strain. Figure D-1. Binder Failure Envelopes as Proposed by Heukelom (1966) and as Found during NCHRP 9-59, Using Various Binder Tensile Tests. Because of the importance of fatigue failure in the proposed analytical method, some further discussion of what is meant by this term is needed. It is hypothesized here that asphalt concrete contains many voids and micro-cracks, so that crack initiation is not really an essential phase of fatigue damage for this material. Instead, there are two stages of fatigue damage: stable crack growth, and unstable crack growth. Stable crack growth is the initial stage of fatigue damage, where the growth of cracks and flaws is relatively slow. Unstable crack growth is the second stage of damage, and results in rapidly growing crack and flaws. In the case of stress-controlled laboratory testing, unstable crack growth is often almost instantaneously followed by catastrophic failureâspecimen separation. However, in strain-controlled testing catastrophic failure will generally not occur. Instead, unstable crack growth is manifested as an increase in the rate of damage (an increase in the rate of decrease in flexural stiffness). In pavements the definition of failure for purposes of this analysis is a practical oneâthe appearance of surface fatigue cracks which continue to grow with further traffic loading. It is likely that there is a relationship between the onset of unstable crack growth and the appearance of fatigue cracks on the surface of a pavement, but it must be acknowledged that this relationship will depend on the 0.01 0.10 1.00 10.00 100.00 1,000.00 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 Fa ilu re St ra in o r F SC , % Stiffness/3 or G*, Pa From binder tests Heukelom SHRP DENT ALF DENT NCHRP 9-59 DENT Direct tension

161 pavement thickness, among other factors. Therefore, when applying this method of analysis to actual pavements, one or more calibration factors would be needed to address the effect of pavement thickness and the relationship between failure as observed in laboratory fatigue tests and failure as defined in actual pavements. ANALYSIS OF MIXTURE FATIGUE AND BINDER TEST DATA Materials and Test Methods Asphalt bindersâA total of 16 asphalt binders were selected for testing as part of NCHRP 9- 59. These covered a wide range of binder grades and types. Half were polymer-modified, half were not. The binders included two containing recycled engine oil bottoms (REOB), one containing polyphosphoric acid (PPA), and two that were blends of paving grade asphalts and heavily oxidized roofing asphalt. In addition to these binders, the eight core asphalts from the Strategic Highway Research Program (SHRP) were also tested during NCHRP 9-59 (University California, Berkeley, 1994). This was done so that the large database of flexural fatigue tests from SHRP could be included in the NCHRP 9-59 analysis. A few other asphalt binders were subjected to limited validation testing but were not included in the analysis discussed in this appendix. The 16 NCHRP binders were laboratory aged using the rolling thin-film oven test (RTFOT) followed by 40 hours in the pressure aging vessel (PAV). The SHRP binders were aged using the RTFOT only, since the SHRP mixes were only aged using short-term oven conditioning (University of California, Berkeley, 1994). Asphalt mixturesâas part of NCHRP 9-59, a total of 16 mixtures were evaluated using fatigue tests. All 16 of these were tested using uniaxial fatigue tests, following current methodology for simplified visco-elastic continuum damage (SVECD) testing. The mix used a 9.5-mm nominal maximum aggregate size blend of crushed granite, crushed limestone and natural sand. The design gyrations were 80, and the design asphalt content was 6.0 % by weight. All mixtures were aged in the loose condition for 5 days at 95Â°C. This aging protocol was chosen to match the aging occurring in the binder agingâRTFOT followed by 40 hours in the PAV. Binder testsâbinders were tested using three procedures: (1) DSR frequency sweeps at 10, 22, 34 and 46Â°C; linear amplitude sweep (LAS) at a single temperature; and (3) a simplified double-edge notched tension (SDENT) test done at one or two temperatures. From the DSR frequency sweep data, master curves were constructed. The LAS test was only performed at one temperature because it was found that problems occurred outside of this narrow modulus range; the data was inconsistent, probably because of debonding of the asphalt and disc, or because of loss of specimen edge integrity. It was found that the best results were obtained when tests were conducted at a Gâ value of 15 to 20 MPa. The procedure used was as given in AASHTO TP 101- 14, except for the test temperature. Additionally, the data was analyzed using the model and procedures discussed below. Initially, there was some variation in the test temperatures selected for SDENT testing. The final protocol selected was to select the lowest temperature, in 5Â°C increments, for which |G*| did not exceed 6 MPa at 0.33 rad/s. This temperature was selected because it seemed to represent the highest stiffness that could be tested with the SDENT without the likelihood of brittle failureâwhere the specimen failed very quickly, at the very start of loading, providing very little information concerning the properties of the binder. The procedure followed was the FHWA procedure (FHWA-HRT-11-045), except that the loading rate was 50 mm/min, and the test temperature was selected as discussed above.

162 Analysis of Mixture Flexural Fatigue Data The GFTAB model described above was used to analyze the binder test data and mixture fatigue data produced during NCHRP 9-59. Although the final analysis as described in this appendix may not appear overly complicated, developing and fine tuning this analysis was difficult and time consumingâmany different approaches were tried and evaluated before settling on the approach described here. The first step in this analysis was to analyze the mixture flexural fatigue data. This was done first because it was a very robust data set, incorporating eight of the heavily aged NCHRP 9-59 mixtures along with mixtures made from the eight SHRP core asphalt and two different aggregatesâthe SHRP mixes were subjected only to short term oven conditioning. Furthermore, because the flexural fatigue tests were done at constant strain, this greatly simplified the analysis. In the initial trial analyses of these data, the failure envelope was assumed to be the same as that defined by the NCHRP 9-59 test data as shown in Figure D- 1. However, neither this envelope nor Heukelomâs proposed envelope provided ideal resultsâ residuals seemed to show error terms related to the binder modulus, suggesting that the assumed failure envelope was not quite correct. Therefore, a power law function was used to define the failure envelope, with the two power law coefficients determined as part of the analysis. This approach allowed the failure envelope to vary and eliminated the problem with the residuals. An important question in performing this analysis is what test method should be considered the âstandard?â The NCHRP 9-59 fatigue tests were performed using complete strain reversal, while the SHRP fatigue tests were done using Haversine loading (all strains in tension). These two types of loading will probably not produce the same results, so some calibration factor must be included to account for this differenceâand one of the methods must be selected as the standard, with a calibration factor of 1.00. In this case, fully reversed loading (the NCHRP 9-59 fatigue data) was selected as the standard with a calibration factor of 1.00. The calibration factor for haversine loading was determined using a trial-and-error method discussed below. What is meant by an average binder must also be defined. One approach would be to simply define the average binder response as the average of the 17 binders included in the analysis. It was however decided that a better approach would be to establish the average binder as one with a Christensen-Anderson R-value of 2.00, which is typical for lightly to moderately aged binders. As discussed below, the R-value has a strong effect on binder strain capacity, so this approach was useful in ensuring that the resulting failure envelope did in fact represent the response of a typical asphalt binder. This approach to defining a typical binder also had the advantage of being precisely defined. The relationship between mix fatigue FFPR and R-value was also used to determine the calibration factor for the haversine (SHRP) fatigue tests; the calibration factor was varied until a plot of FFPR vs R-value for non-polymer-modified binders for both projects formed a single smooth function. In performing the analysis, Microsoft Excel Solver was used to determine the model coefficients; statistical parameters not provided by solver were calculated using standard analytical methods for non-linear least squares. Calculation of statistical parameters for non- linear least squares analyses is complicated and the interested reader should refer to a standard reference on statistics for details of these calculations (Kutner et al., 2005). The following model, based on Equation D-3 above, was used to predict cycles to failure: ð^ = Ã âÃ( â ) ( â ) (D-11)

163 Where ð^ = predicted cycles to failure FFPRi = fatigue/fracture performance ratio for ith binder FSC* = typical binder fatigue strain capacity, % = ð |ðºâ| where k2 and k3 are power law coefficients and |G*| is the modulus (Pa) for the binder of interest at the test temperature and a frequency of 10 Hz VBE = mixture effective binder content, volume % k1 = fatigue exponent coefficient = 2 Îµt = mixture maximum tensile strain, %, and Î´ = binder phase angle (degrees) for the binder of interest at the test temperature and a frequency of 10 Hz Preliminary analyses showed that the coefficient in the fatigue exponent (k1) had a value very close to one, and in the final analysis it was assumed that the value of this parameter was 2. The tables below summarize the results of the GFTAB calibration using the BBF data. Table D-1 is a summary of the analysis of variance for the model, including the r2 value. Table D-2 gives the coefficients for the two power law parameters for the failure. Also included in this table are the standard deviation, t values (ratio of coefficient to standard deviation) and levels of significance p for the coefficients. Table D-3 lists the FFPR values for the NCHRP 9-59 binders, while Table D-4 lists the FFPR values for the SHRP binders. As in Table D-2, the standard deviation, t- and p-values are also included in these tables. Note that the standard deviation and related statistics for asphalt AAM could not be determined because of restrictions on degrees of freedom in the model. Without fixing the FFPR value of one of the binders, there would be an infinite combination of values for the failure envelope coefficients and binder FFPR values that would explain the data equally well. The FFPR value of AAM was set by trial-and-error to a value (0.81) which resulted in a binder with an R-value of 2.0 having an FFPR of 1.00. Figures D-2 and D-3 show predicted and observed cycles to failure for the flexural fatigue data used in this analysis; Figure D-2 is for the NCHRP 9-59 data, while Figure D-3 is for the SHRP data. The r2 value for the NCHRP 9-59 data is slightly lower than for the SHRP data (88 % versus 90 %), probably because the use of the extended loose-mix aging increased variability in the mixture properties. In both the correlation between predicted and observed cycles to failure is good, in the range typically seen for asphalt mixture fatigue models. Because of the high inherent variability of mixture fatigue tests, it is probably not possible to achieve r2 values much higher than this, so the first of the criteria for evaluating the GFTAB model has been met. Table D-1. Summary of Non-Linear Least Squares Model for Flexural Fatigue. Source Sum of Squares Degrees of Freedom Mean square F-value p-value Model 131.175 17 7.71616 113.4 < 0.001 Error 14.899 219 0.06803 Total 146.074 237 R2 89.8 %

164 Table D-2. Flexural Fatigue Model Parameter Estimates Parameter Coefficient Std. Dev. t-value p-value Log failure envelope coefficient 6.649 0.1669 33.84 0.000 Failure envelope exponent -0.806 0.0203 -39.62 0.000 Table D-3. FFPR Values for NCHRP 9-59 Binders used in the Bending Beam Fatigue Experiment. Binder Code Coefficient Std. Dev. t-value p-value A 0.58 0.0218 -19.46 0.000 B 0.72 0.0218 -12.66 0.000 C 1.02 0.0446 0.55 0.584 I 1.01 0.0352 0.20 0.843 J 0.84 0.0273 -5.91 0.000 K 0.79 0.0229 -9.31 0.000 M 0.92 0.0745 -1.14 0.257 O 1.13 0.0354 3.62 0.000 P 0.64 0.0224 -15.87 0.000 Table D-4. FFPR Values for SHRP Core Asphalts Binder Code Coefficient Std. Dev. t-value p-value AAA 1.11 0.0655 1.65 0.101 AAB 0.95 0.0433 -1.10 0.271 AAC 1.05 0.0520 0.90 0.370 AAD 1.15 0.0622 2.35 0.019 AAF 1.34 0.0557 6.09 0.000 AAG 2.01 0.0785 12.85 0.000 AAK 1.44 0.0626 7.08 0.000 AAM 0.81 N/A N/A N/A

165 Figure D-2. Predicted and Observed Cycles to Failure for NCHRP 9-59 Mixes Tested in Flexural Fatigue. Figure D-3. Predicted and Observed Cycles to Failure for SHRP Mixes Tested in Flexural Fatigue. As mentioned above, the analysis summarized in Tables D-1 and D-2 was set up so that the FFPR value for a binder with an R-value of 2.00 would be 1.00. Because of the way this analysis was done, the t-values were calculated as (FFPR â 1.00)/standard deviation. In other words, an FFPR value of 1.00, indicating average performance, would give a t-value of zero. The standard deviation for the binder FFPR values are all in a similar range, so the t-values and p-values (level RÂ² = 88% 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+02 1.E+04 1.E+06 1.E+08 Pr ed ict ed C yc le s t o Fa ilu re Observed Cycles to Failure RÂ² = 90% 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+02 1.E+04 1.E+06 1.E+08 Pr ed ict ed C yc le s t o Fa ilu re Observed Cycles to Failure

166 of significance) mostly indicate how much the FFPR value for a given binder deviate from the average value of 1.00. Notice that the FFPR values for the NCHRP 9-59 binders are in general significantly lower than those for the SHRP core asphalts, which is attributable to the extended loose mix aging used in NCHRP 9-59. Figure D-4 is a plot showing the +/- 2s confidence limits for all 17 binders included in this analysis. Note that these are not simultaneous confidence intervals (such as Bonferroni) needed for proper comparisons among factor level means but are shown to indicate the general degree of variability in the FFPR values. As part of the analysis, the difference in the overall response in the two data sets (NCHRP 9- 59 and SHRP) was also estimated, although this could not be done in a statistical manner because project and binders are totally confounded. It was possible to estimate the effect of project because of the very good relationship between binder FFPR and the Christensen- Anderson model R-value; this relationship is especially good for binders that are not polymer- modified. As discussed above, the calibration factor for the NCHRP 9-59 data (fully reversed loading) was assumed to be 1.00. The failure envelope for the SHRP binders was multiplied by a calibration factor, found through trial-and-error such that a smooth, continuous relationship between FFPR and R-value was generated, as shown in Figure D-5. This and similar relationships are discussed later in this appendix. In order to match the FFPR vs R-value functions, the resulting calibration factor for the SHRP data/haversine loading was found to be 0.65. The lower FFPR value for the SHRP mixes indicates, as should be expected, that haversine loading in complete tension (at least initially) is somewhat more damaging than fully-reversed sinusoidal loading.

167 Figure D-4. +/- 2s Confidence Limits for Flexural Fatigue FFPR. 0.0 0.5 1.0 1.5 2.0 2.5 A B C I J K M O P AAA AAB AAC AAD AAF AAG AAK AAM Flexural Fatigue FFPR Bi nd er C od e

168 Figure D-5. FFPR from Mixture BBF Testing as a Function of Christensen-Anderson R- value for the Binders. Non-polymer-modified binders only. The good correlation between predicted and observed cycles to failure was the first criteria mentioned for evaluating the GFTAB model, and it has been met. The second criterion was that the failure envelope determined as part of the GFTAB model should be in reasonable agreement with previously published failure envelopes for asphalt binder and should also agree with NCHRP 9-59 binder data. Figure D-6 is a plot showing these comparisons; it shows the same failure envelopes and binder data given in Figure D-1 above, but now also includes the failure envelope determined from the GFTAB analysis. The range of this failure envelope has been limited to modulus values represented in the NCHRP 9-59 data used in the analysis. There is quite good agreement between the GFTAB failure envelope and the other two shown in Figure D-6âthe minor discrepancy is easily explained by differences between the simple state of stress existing in binder tests and the complex, triaxial stress conditions existing in a mixture subject to fatigue. Therefore, the second criteria for evaluating the GFTAB model has been met. RÂ² = 83% RÂ² = 82% 0.00 0.50 1.00 1.50 2.00 2.50 0.00 1.00 2.00 3.00 4.00 FF PR fr om B BF Te st in g Christensen-Anderson R-value NCHRP 9-59 binders SHRP core asphalts

169 Figure D-6. Binder Failure Envelopes as Proposed by Heukelom (1966) and as Found during NCHRP 9-59, Using Various Binder Tensile Tests, along with the Failure Envelope Determined from GFTAB analysis of NCHRP 9-59 and SHRP Flexural Fatigue Data. Analysis of Mixture Uniaxial Fatigue Data Analysis of uniaxial fatigue data is potentially much more complex than for flexural fatigue testing, because the strains are not constant throughout the loading. This requires calculating damage for each specimen in increments. In this analysis, FFPR values were calculated for each mix by using a relationship formed by combining Equations D-6 and D-8: ð¹ð¹ðð = â â ð ( )â ( â ) ï¤ ( )â (D-12) In this equation, the damage is summed up incrementally and then used to calculate the fatigue strain capacity for the individual binder (Equation D-6), and then divided by FSC* (the typical FSC value calculated using the power law model described above) to calculated FFPR (Equation D-8). Because the uniaxial fatigue tests were performed in haversine loading, the strains typically start out completely in tension but gradual revert to fully reversed loading. This makes calculation of damage difficult. To account for this drift in the loading type damage was calculated based only the magnitude of the applied tensile strains. The results of this analysis are summarized in Table D-5. Note that in this case FFPR values are calculated for each specimen, and the average for each binder is simply the average of these individual FFPR values. The statistics in Table D-5 were calculated as for a single factor analysis of variance, using a log 0.01 0.10 1.00 10.00 100.00 1,000.00 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 Fa ilu re St ra in o r F SC , % Stiffness/3 or G*, Pa From binder tests From mix fatigue Heukelom SHRP DENT ALF DENT NCHRP 9-59 DENT Direct tension

170 transformâwhich was applied to keep the variances uniform as a function of FFPR. Figure D-7 is a plot of predicted and observed cycles to failure for the uniaxial fatigue data; note that the same scale has been used for Figures D-2, D-3 and D-7âall comparisons of predicted and observed fatigue lifeâto make comparisons among these plots easier. The r2 for the uniaxial data (87 %) is comparable to the values for the flexural data and is strong verification for the GFTAB analysis. Figure D-8 shows the +/- 2s confidence limits for FFPR values as determined from uniaxial fatigue data; the results are expressed in arithmetic terms rather than using the log transform to make them easier to understand and interpret. Included in this figure are confidence limits from flexural fatigue testing for the nine binders where data is available for comparison. The confidence intervals overlap for eight of the nine binders; the data does not overlap for binder O. It should be noted that these confidence intervals are meant for only rough comparisons, and do not represent simultaneous confidence intervals (such as Bonferonni) which are strictly needed for simultaneous comparison of factor level means. Figure D-9 is a direct comparison of the FFPR values determined from uniaxial fatigue testing and flexural fatigue testing. This is third criteria mentioned above for evaluation of GFTAB. If the theory and analytical approach are correctâor at least reasonably correctâthe two sets of FFPR values should be in reasonable agreement. The agreement appears acceptable, given the large amount of error inherent in asphalt mixture fatigue tests. The error bars for seven of the nine mixtures overlap with the line of equality; the two that do not are O and P. The r2 value for the relationship between the two sets of data is 75 %, which suggests a moderate degree of correlation between the two parameters. As discussed below, another important consideration in evaluating the GFTAB analysis is whether relationships exist between mix FFPR values and binder FFPR values and other binder data. Table D-5. FFPR Values for NCHRP 9-59 Binders used in the Uniaxial Fatigue Experiment. Binder Code LOG FFPR Statistics on log transform of FFPR Std. Dev. t-value p-value A -0.2089 0.0189 -11.0361 0.0000 B -0.1294 0.0189 -6.8357 0.0000 C -0.0183 0.0232 -0.7894 0.4342 D 0.0153 0.0189 0.8086 0.4232 E -0.2094 0.0189 -11.0629 0.0000 F -0.1610 0.0207 -7.7669 0.0000 G -0.1269 0.0175 -7.2419 0.0000 H -0.1249 0.0189 -6.5970 0.0000 I 0.0018 0.0232 0.0769 0.9391 J -0.0791 0.0189 -4.1787 0.0001 K -0.1152 0.0175 -6.5732 0.0000 L -0.4354 0.0207 -20.9996 0.0000 M -0.1814 0.0189 -9.5855 0.0000 N -0.2727 0.0189 -14.4072 0.0000 O -0.0658 0.0155 -4.2580 0.0001 P -0.2446 0.0189 -12.9235 0.0000

171 Figure D-7. Predicted and Observed Cycles to Failure for NCHRP 9-59 Mixes Tested in Uniaxial Fatigue. Dashed line is regression, solid equality. RÂ² = 88% 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 Pr ed ict ed N f Observed Cycles to Failure

172 Figure D-8. +/- 2s Confidence Limits for Binder FFPR from Uniaxial Fatigue. For nine binders where data is available, confidence limits for FFPR from flexural fatigue are included for comparison. 0.0 0.5 1.0 1.5 2.0 A B C D E F G H I J K L M N O P Uniaxial Fatigue FFPR As ph al t B in de r C od e

173 Figure D-9. Comparison of FFPR from Uniaxial Fatigue Testing with FFPR from Flexural Fatigue Testing. Error bars show pooled two standard deviation error limits. Analysis of Binder Test Data The R-values for the binders were calculated from LAS data at 10 Hz using an equation based on the Christensen-Anderson model, and assuming a glassy modulus of 1 GPa (Christensen and Anderson, 1992): ð = ððð(2) | â| Ãâ( â ) (D-13) Where R = Christensen-Anderson R (rheologic index) |G*| = dynamic complex modulus, Pa Î´ = binder phase angle, degrees (at same temperature and frequency as |G*|) This approach was used because it was a relatively high modulus level, ensuring that for polymer-modified binders, the effect of the polymer on the phase angle was minimal. The LAS data also had the advantage of consisting of the average of three separate determinations, so the R-value would represent the average of three measurements. The LAS data was analyzed as a fatigue test, using a variation of Equation D-6: ð¹ð¹ðð = . â â ð (ð¾ ) ( â )â ï¤ ( )â (D-14) RÂ² = 75% 0.40 0.60 0.80 1.00 1.20 0.40 0.60 0.80 1.00 1.20 FF PR fr om U ni ax ia l T es tin g FFPR from Flexural Fatigue

174 Where the variables are as described for Equation D-6 above, except that shear strain Î³ is now used rather than extension strain Îµ. The FFPR in this case represents the ratio of the measured fatigue strain capacity (FSC) of the binder in the LAS test to the expected, or typical value (FSC*). The factor 4.8 in Equation D-12 includes a factor of 3 for converting from shear strain to extensional strain, and a calibration factor of 1.6 for the LAS test geometry. This calibration factor was determined empirically and the fact that it differs from 1.0 is probably due to the fact that the stresses and strains in the LAS test are not uniform but increase from the center of the specimen to the outer edge. Analysis of the SDENT data is complicated by the fact that the heavily notched specimen cannot be analyzed as if it were a tension test, using calculated stresses, strains and modulus. The procedure used instead relied upon comparing measured specimen extension and initial specimen stiffness. Extension in this case was the extension in mm to the post peak point where the load was 20 % of the peak load. This approach was used, rather than trying to determine total extension, in order to avoid the high variability potentially associated with extreme extensions. Specimen initial stiffness was calculated as the specimen load divided by the extension at 3 seconds loading time, expressed in units of N/m. Specimen extension was plotted against specimen stiffness, and Microsoft Excel solver was used to fit the extension data to the following model: ð¸ = ð¹ð¹ðð ð´ð (D-15) Where Ei = extension of ith specimen, mm FFPRi = fatigue/fracture performance ratio of ith specimen (dimensionless) A, B = power law coefficients for relationship between extension and stiffness Si = initial stiffness of ith specimen FFPR values for this model were defined so that a non-polymer-modified binder with an R- value of 2.0 would have an FFPR of 1.0. For the standard SDENT geometryâwith the small, 5 mm ligamentâthe constants A and B were found to be 17.4 and -0.334, respectively, and the fit of the model was excellent with R2 = 99 %. Some early SDENT tests were run with a different geometryâthe large, 15 mm ligament. In this case, the constants A and B were found to be 176 and -0.359, respectively. The fit for this model was also excellent, with an R2 of over 99 %. For calculating FFPR values from SDENT tests using the standard geometry with the 5 mm ligament, the following equation should Be used: ð¹ð¹ðð = . . (D-16) Where the extension is in mm and S is the specimen stiffness at 3 seconds, in N/m. For best results, the initial specimen stiffness should be between about 15 and 35 N/m. It should be emphasized that the FFPR parameter represents the SDENT extension normalized for stiffness and is a representation of the inherent strain tolerance of a binder, independent of stiffness. FFPR values for the SDENT were also calculated based on extension to maximum force and total energy to failure, but these FFPR values showed significantly weaker relationships to mixture fatigue FFPR values and are not considered in this report in the interest of conciseness and clarity. Values of the various binder parameters are summarized for the NCHRP 9-59 and SHRP

175 binders in Table D-6 below. Estimated standard errors are given at the bottom of the table, these are in terms of a coefficient of variation for the reported test result, and accounts for the effect of sample size. For convenience, Table D-6 also includes mixture fatigue FFPR values for the NCHRP 9-59 and SHRP binders. COMPARISON OF MIXTURE FATIGUE FFPR WITH BINDER FFPR AND R-VALUE Figures D-10 through D-13 show the various relationships among the mixture fatigue FFPR values and the three binder parameters. In each of these figures, error bars are shown for both variables, representing +/- two standard errors. In many cases the error bars are so small that they become obscured by the data point. Figures D-10 shows mix FFPR values as a function of R- value. The figures have been coded according to binder type and source of the mixture FFPR value (flexural or uniaxial fatigue). A good correlation (r2 = 83 %) between mixture and binder FFPR is seen in this figure, indicating the R-value is in fact a good indicator of binder performance in both flexural and uniaxial fatigue. Furthermore, it appears to be valid for both polymer-modified and non-modified binders. This finding was initially a bit puzzling to the research team, but it is now believed that the effect of polymer modification on mixture fatigue in highly aged mixtures at intermediate to low temperatures is reduced compared to the effect in binder tests on lightly to moderately aged binders at intermediate temperatures. For instance, as discussed below, polymer modification seems to be significantly more effective in improving SDENT FFPR values than those determined from mixture fatigue tests, probably because of the much lower loading rates used in the SDENT test, which results in significantly lower binder modulus values and enhanced polymer effectiveness.

176 Table D-6. Binder R-Values and FFPR Values. Binder Code Mix FFPR/ Flexural Fatigue Mix FFPR/ Uniaxial Fatigue R-Value Binder FFPR/ SDENT Extension Binder FFPR/ LAS Test A 0.577 0.618 3.140 0.581 0.559 B 0.721 0.742 2.440 0.839 0.956 C 1.024 0.959 2.280 1.075 0.934 D 1.036 2.100 0.934 0.880 E 0.617 2.910 0.759 0.686 F 0.690 2.490 0.773 0.800 G 0.747 2.450 1.313 0.804 H 0.750 2.640 0.978 0.746 I 1.007 1.004 2.100 0.941 0.944 J 0.839 0.834 2.320 1.123 0.900 K 0.787 0.767 2.650 0.739 0.720 L 0.367 3.060 0.582 0.670 M 0.915 0.659 2.560 1.016 0.955 N 0.534 3.210 0.629 0.652 O 1.128 0.859 2.290 1.106 0.973 P 0.645 0.569 2.580 0.812 0.781 AAA 1.108 1.730 1.248 1.142 AAB 0.952 2.080 0.960 0.921 AAC 1.047 2.010 1.048 0.939 AAD 1.146 1.770 1.057 1.200 AAF 1.339 1.930 1.167 1.014 AAG 2.008 1.350 1.545 1.242 AAK 1.443 1.780 1.108 1.170 AAM 0.810 2.440 0.759 0.894 Est. Std. Error % 4.9 4.5 0.63 1.57 4.98

177 Figure D-10. Mixture FFPR as a Function of Christensen-Anderson R-value. Error bars represent +/- 2 Ã standard error. When the very strong correlations between R-value and mixture FFPR were first observed, there was some fear that this was an artifact of the GFTAB model, since in this model R-value effects the binder phase angle, which in turn effects the fatigue exponent, which then might affect the FFPR value. However, a very similar relationship to that shown above is seen between FFPR based on SDENT extension and R-value, as shown in Figure D-11. The relationship here between FFPR from the SDENT test and R-value for non-modified binders is exceptionally good, while the correlation for polymer-modified binder is slightly weaker and shifted with respect to the relationship for non-polymer-modified binders. As discussed above, the shift in the polymer-modified data is probably because the relatively slow loading rate in the SDENT results in enhanced polymer effectiveness compared to the mixture fatigue tests. It is hypothesized that the effectiveness of polymer modifiers in the NCHRP 9-59 mixture tests was attenuated because of (1) the large number of stress concentrations present in asphalt concrete mixtures; (2) the heavy laboratory aging used to condition the mixes prior to testing; and (3) the resulting relatively high modulus values of the mixtures during fatigue testing. It is believed that if the mixes had been only short termed aged and tested at slightly higher temperatures the effect of the polymer modification would probably have been much more pronounced. In any case, what is important in Figure D-11 is that it seems clear that increasing R-value decreases the strain tolerance even for the polymer-modified systems. That is, high R-values will always decrease the strain tolerance of a binder, even when it is polymer-modified. In fact, Figure D-11 suggests that high R-values reduces strain tolerance of modified binders more than non-modified. RÂ² = 83% 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 M ix tu re Fa tig ue FF PR R-value SHRP (non-modified) BBF non-modified BBF polymer modified Uniaxial non-modified Uniaxial polymer modified

178 Figure D-11. FFPR from SDENT Extension as a Function of R-value. Error bars represent +/- 2 Ã standard error. Figure D-12 is a plot of FFPR from the SDENT/extension test as a function of R-value. The overall r2 value here of 63 % is only moderate, but that is in part because of the difference in the performance of the polymer-modified binders as measured by the SDENT test and as indicated in the mixture fatigue tests. If data only from non-polymer-modified binders are included, the r2 value increases to 79 %. It appears that the SDENT test tends to show better performance for polymer-modified binders compared to the mixture fatigue tests. As mentioned above, there are several reasons for this but the most important is perhaps the relatively slow loading rate used in the SDENT test compared to mixture fatigue loading. The resulting relatively low binder stiffness in the SDENT test seems to improve the performance of most of the polymers used in the NCHRP 9-59 binders. This doesnât mean that the SDENT test is wrong, or inaccurateâit simply means that the conditions used in the SDENT test are significantly different from those used in mixture fatigue tests, and as a result, the apparent performance of polymer-modified binders differs in these two tests. It is important to understand that the SDENT testing done as part of NCHRP 9-59 was performed at as low a temperature as possibleâlower test temperatures would produce very quick, brittle fractures providing little information of use on the binder. In many cases, testing at lower temperatures would be impossible using a water bath. One of the reasons for the discrepancy between SDENT tests and mixture fatigue is made clear by Figure D-13, which shows SDENT extension as a function of binder modulus, coded for polymer modification and R-value. As modulus increases, the effectiveness of polymer modification appears to decrease, until at approximately 30 MPa the modification no longer seems to influence the results of the DENT test. The mixture fatigue tests were performed at binder modulus values ranging from about 2 to 85 MPa, so it is likely the effectiveness of the polymer modification was minimal because of the relatively high modulus values of the binder during the tests. It is also possible that the extended loose mix aging used in NCHRP 9-59 had an adverse effect on the performance of polymer-modified binders. It is quite possible that had mixture fatigue tests been done at a significantly lower modulus range and/or using less aggressive RÂ² = 81% RÂ² = 96% 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 SD EN T Ex t. FF PR R-value SHRP (non-modified) NCHRP 9-59 non- modified NCHRP 9-59 polymer- modified

179 laboratory aging methods significant benefit would have been observed from polymer modification. R-value does a good job of characterizing fatigue/fracture performance for non- modified binders and for polymer-modified binders in the intermediate-high modulus range (especially when heavily aged), but probably does not adequately characterize the fatigue/fracture performance of polymer-modified binders in the intermediate-low modulus range. For a completely accurate characterization of fatigue/fracture performance, R-value should be controlled for all binders, and SDENT tests should be performed on polymer-modified binders. Both SDENT and R-value should be controlled for polymer-modified binders because it appears that at lower temperatures, especially for heavily aged mixes, polymer modification becomes ineffective and then the fatigue and fracture performance will be a function of R-value alone. If only SDENT test data is used in a specification, the fatigue and fracture performance of polymer-modified binders at low temperatures could become inadequate. Figure D-12. Mixture FFPR as a Function of FFPR from SDENT Extension. Error bars represent +/- 2 Ã standard error. RÂ² = 63% 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 M ix tu re Fa tig ue FF PR SDENT Extension FFPR SHRP (non-modified) BBF non-modified BBF polymer modified Uniaxial non-modified Uniaxial polymer modified

180 Figure D-13. DENT extension as a function of estimated binder modulus, coded for polymer modification and binder R-value. Figure D-14 shows the relationship between mixture fatigue FFPR values and LAS FFPR. The r2 value for this relationship, like figure D-12 above, is only moderate at 68 %. In this case, the data appears to be similar for polymer-modified and non-modified binders; the reason for the relatively poor correlation appears to be the high variability of FFPR calculated from the LAS test. The relationship between mix FFPR and LAS FFPR also appears to follow a power law, while ideally it should be linearâin fact, the two sets of FFPR values should be approximately equal. The non-linear nature of this relationship is probably a result of the high strains and resulting non-linear behavior occurring during the LAS test. Figure D-14. Mixture FFPR as a Function of FFPR from LAS Test Data. Error bars represent +/- 2 Ã standard error. RÂ² = 98% RÂ² = 73% RÂ² = 56% RÂ² = 91% 5 10 15 20 25 1.E+05 1.E+06 1.E+07 DE NT E xt en sio n, m m |G*|, Pa R < 2.2 2.2 < R < 3.0 R > 3.0 Polymer modified RÂ² = 68% 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 M ix tu re Fa tig ue FF PR LAS FFPR SHRP (non-modified) BBF non-modified BBF polymer modified Uniaxial non-modified Uniaxial polymer modified

181 SUMMARY AND DISCUSSION The results of this analysis show reasonably good correlations among binder and mixture FFPR values, with the results close to equality as predicted by GFTAB theory considering limitations in the various experimental methods. Thus, the results of the analysis suggest that the GFTAB model and theory are reasonably accurate and a useful technique for analyzing fatigue data. FFPR as determined in this analysis is an indicator of overall strain toleranceâthe higher the FFPR, the better the strain tolerance for the binder. FFPR is meant to be independent of modulus, and an indicator of failure strain over a wide range of conditions. Some researchers and engineers might question whether this is a good indicator of fatigue performance, better than simply measuring cycles to failure under a single combination of temperature and loading rate, or perhaps several combinations of conditions. However, the fatigue life of a given mix is a function of applied strain, strain capacity and binder phase angle at the temperature and loading rate of interest. This makes it difficult or impossible to separate the effects of strain tolerance and phase angle on fatigue life without making use of GFTAB theory or some similar model. The NCHRP 9-59 research team believes that FFPR is the best indicator of fatigue performance available to pavement engineers and is a better indicator that simply looking at cycles to failure under a specific set of conditions. Historically, asphalt binders with high R-values, such as heavily oxidized binder, have been prone to premature failure through excessive crackingâboth fatigue cracking and non-load associated cracking. This has been especially true for thin pavements, where strains will be higher and probably closer to critical values. Many researchers have recently linked premature failures with binders containing recycled engine oil bottoms (REOB), which often exhibit very high R-values, especially after aging. That the analysis presented in this appendix shows such a strong correlation between binder R-value and strain tolerance as indicated by FFPR both suggests one reason why these binders tend to be prone to excessive crackingâthey are excessively brittle. As discussed elsewhere in this report, there are at least two other reasons these binders show poor field performance: (1) they tend to exhibit poor healing properties; and (2) they also tend to undergo excessive physical hardening in the BBR test, which results in low temperature grading that is lower than it should be, perhaps by as much as two full grades. Although R-value is probably sufficient for characterizing the fatigue and fracture performance of non-polymer-modified binders, it is probably not adequate for many polymer- modified binders. In such cases the SDENT test provides a more accurate picture of strain tolerance and fatigue/fracture performance. As discussed above, the most effective way of using the SDENT test is to calculate an FFPR value based on extension to 20 % of maximum load. This approach accounts for the effect of stiffness on SDENT extension and so provides an indicator of inherent strain tolerance. CONCLUSIONS Based upon NCHRP 9-59 binder and mixture testing, the binder parameter with the best correlation to mixture fatigue performance as indicated by FFPR values is the Christensen- Anderson R-value. The degree of correlation (r2 = 80 %) is very good for comparison of mixture fatigue data and binder test data and suggests that R-value is in fact suitable for use in an improved binder specification. Determining the R-value of a binder as done in this research is relatively simple, requiring only a measurement of modulus and phase angle where the modulus

182 (|G*|) is at least 10 GPa. Eliminating binders with high R-values from use in asphalt pavements will help ensure that excessively brittle binders are not used in highway construction. Information presented elsewhere in this report also suggests that high R binders also tend to exhibit poor healing and can have BBR grades that are as much as two grades lower than they should be, because of the effects of physical hardening. REFERENCES Christensen, D.W., and Anderson, D.A. (1992). Interpretation of Dynamic Mechanical Test Data for Paving Grade Asphalt Cements. Journal of the Association of Asphalt Paving Technologists, 61, 67-98. Christensen, D.W., and Bonaquist, R.F. (2005). Practical Application of Continuum Damage Theory to Fatigue Phenomena in Asphalt Concrete Mixtures. Journal of the Association of Asphalt Paving Technologists, 74, 963-995. Christensen, D.W., and Bonaquist, R.F. (2009). Analysis of HMA Fatigue Data Using the Concepts of Reduced Loading Cycles and Endurance Limit. The Journal of the Association of Asphalt Paving Technologists, 78, 377-400. Christensen, D. W. and R. Bonaquist (2012) âModeling of Fatigue Damage Functions for Hot Mix Asphalt and Applications to Surface Cracking,â Journal of the Association of Asphalt Paving Technologists, Vol. 81, pp. 241-271. Daniel, J.S., and Kim, Y. Richard. (2002). Development of a Simplified Fatigue Test and Analysis Procedure Using a Viscoelastic, Continuum Damage Model. Journal of the Association of Asphalt Paving Technologists, 71, 619-645. Heukelom, âObservations on the Rheology and Fracture of Bitumens and Asphalt Mixes,â Proceedings of the Association of Asphalt Paving Technologists, Vol. 35, 1966, pp. 358-399. Kim, Y.R., Little, D.N. (1990). One Dimensional Constitutive Modeling of Asphalt Concrete. ASCE Journal of Engineering Mechanics, 116, 751-772. Kim, Y.R., Little, Dallas N., and Lytton, Robert R. (2001). Use of Dynamic Mechanical Analysis (DMA) to Evaluate the Fatigue and Healing Potential of Asphalt Binders in Sand Asphalt Mixtures. Journal of the Association of Asphalt Paving Technologists, 71, 176-199. Kutner, M. H., Nachtsheim, C. J., Neter, J., and Li, W., Applied Linear Statistical Models, 5th Ed., New York: McGraw Hill Irwin, 2005, 1,396 pp. Little, D.N., Lytton, R.L., Williams, D., Chen, C.W. and Kim, Y.R. (1997). Fundamental Properties of Asphalts and Modified AsphaltsâTask K: Microdamage Healing in Asphalt and Asphalt Concrete, FHWA Final Report, Vol. 1, Report No. DTFH61-92-C-00170, Springfield, VA: National Technical Information Service. Underwood, B. S., Kim, Y. R., and Guddati, M. N. (2010). âImproved Calculation of Damage Parameter in Viscoelastic Continuum Damage Model,â International Journal of Pavement Engineering, Vol. 11, No. 6, pp. 459-476. University of California, Berkeley, Asphalt Research Program, Fatigue Response of Asphalt- Aggregate Mixes, Report SHRP-A-404, Washington, D.C.: Strategic Highway Research Program, National Research Council, 1994, 309 pp. Zhang, J., Sabouri, M., Guddati, M. N., and Kim, Y. R. (2013). âDevelopment of a Failure Criterion for Asphalt Mixtures under Fatigue Loading,â Journal of the Association of Asphalt Paving Technologists, Vol. 82, pp. 1-22.