A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers (2018)

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D-1 APPENDIX D. DETERMINATION OF FRACTURE COEFFICIENTS IN PSEUDO J- INTEGRAL BASED PARIS’ LAW The modified Paris’ law was early proposed as (1):  nR d A J dN   (D-1) where  is the damage density; RJ is the pseudo J-integral; and A and n are modified Paris’ law parameters associated with the evolution of the damage density. The purpose of this appendix is to determine A and n from different test sources and identify the relationship between them. Three data sources are identified for this purpose:  Jacobs (2)’s dynamic uniaxial tensile test data;  Luo et al. (1)’s controlled-strain RDT test data; and  Gu et al. (3)’s overlay test data. Jacobs (2) used dynamic uniaxial tensile tests to determine fracture characteristics of different asphalt mixtures. During the tests, the crack opening displacement (COD) measurements were carried out to provide information about the complete fracture process from microcracks to macrocracks. Then, this information was utilized to determine A and n in Equation D-1 with the stress intensity factor IK , and the results are documented in Appendix 7A of Jacobs (2). The values of A and n in Appendix 7A must be converted to that of A and n first. The conversion needs to consider the unit of the variables as follows: 11 MPa mm nn I Rdc K JA A dN N m              (D-2) There is the following relationship:   2 2 2 21 1 R I II III R R J K K K E E       (D-3) Substituting Equation D-3 into D-2 gives:     2 21 1 1 MPa mm n nn n n I I R A K A K E N m                     (D-4) Compare the left and right sides of Equation D-4, which yields: 2 nn  (D-5) 2 2 3 1 1 MPa 10 n REA A         (D-6) The reference modulus in Equation D-6 is computed as reE by the following equation: * 1 1 2 2 p p re f t t E E E t              (D-7) where reE is the representative elastic modulus; *E is the dynamic modulus; and  E t is the relaxation modulus; f is the frequency of a load pulse; and pt is the pulse time of a load.

D-2 The Poisson’s ratio is assigned the same value of 0.35 as in Jacobs (2). The dynamic modulus test results at different frequencies and temperatures in Appendix 6A of Jacobs (2) are used to determine reE . The procedure is similar to that of utilizing the tensile creep and recovery test to compute reE presented above, which is: 1) Construct the dynamic modulus master curve at the reference temperature of 25°C using the sigmoidal model:   3 4 2 1 loglog 1 rr c c t cE t c e     (D-8) where rt is the reduced time of loading at the reference temperature; and 1c , 2c , 3c , and 4c are fitting parameters. 2) Convert the dynamic modulus master curve to the relaxation modulus using the following forms: a. Compute the dynamic modulus by (Findley et al. 1989):   2 22 2 * 2 2 2 2 1 11 1 M M j j j j j jj j E E E E                              (D-9) a. Assume a Prony series form of the relaxation modulus as below and determine the Prony series’ coefficients by fitting the Equation D-10 to the master curve of the relaxation modulus constructed in step 2:   1 j tM j j E t E E e       (D-10) where E is the long term relaxation modulus; jE are the relaxation modulus coefficients; and j are the relaxation times. 3) Figure D-1 shows an example of the dynamic modulus master curve produced by the data in Appendix 6A of Jacobs (2). The materials used in Jacobs (2) to obtain A and n include five types of asphalt mixtures: two dense asphalt mixtures, denoted as “DAC8” and “DAC16”; one dense asphalt mixture with a modified binder, “DACmod”; one stone mastic asphalt mixture, “SMA”; and one sand asphalt mixture, “SA”, which are shown in Table D-1. For each mixture type, several replicate specimens are fabricated and subjected to the dynamic tests under the frequencies and temperatures shown in Table 2. For example, under Test 1a of DACmod, three specimens are tested at a frequency of 8 Hz at 15°C. Such details can be found in Appendix 7A of Jacobs (2). There are a total of 181 data points in Appendix 7A. Under some circumstances, a relatively high variability exists among the replicates that are tested under the same condition. To reduce the data variance, the average of the replicates is used for the same test condition. Thus Jacobs (2)’s data reduce from 181 to 57 different pairs. In addition, the representative elastic modulus for a specific pair of frequency and temperature is computed following the procedure above and given in Table 2. With known RE , n and A of Jacobs (2)’s materials are determined by Equations D- 5 and D-6.

D-3 Figure D-1. Example of Dynamic Modulus Master Curve at Reference Temperature 25˚C Luo et al. (1) used controlled-strain RDT test to determine A and n of a variety of asphalt mixtures. The load and displacement of the tested specimen are measured, and then an Energy-Based Mechanistic (EBM) approach is utilized to analyze the stress and strain data. A primary outcome of the EBM approach is the evolution curve of the damage density. This curve is used to determine A and n of the modified Paris’ law in Equation D-5 and D-6. Detailed procedures are not repeated; only the results of A and n are presented herein. Accordingly the values of A and n are recalculated in this study using the same test data in Luo et al. (1). The materials in Luo et al. (1) include the 12 types of asphalt mixtures illustrated in Table 1 and another 8 types, totally 20 mixture types. These additional eight types of mixtures are made of two asphalt binders: AAD and AAM from the Strategic Highway Research Program Materials Reference Library (Jones, 1993) and one aggregate type: Texas limestone from San Marcos, Texas. In the mixture design, two air void contents (4% and 7%) and two aging periods (0 and 6 months) are chosen. As a result, there are 20 data points collected from Luo et al. (1). Gu et al. (3) utilized the overlay test to study fracture properties of both hot mix asphalt (HMA) and warm mix asphalt (WMA). The stress and strain data are obtained from the measured load and displacement of the tested specimen. Then a combined analytical and numerical approach is employed to calculate A and n in Equations D-5 and D-6. More details can be referred to Gu et al. (3). The materials are laboratory–mixed-laboratory-compacted asphalt mixtures, including one type of control HMA and two types of WMA mixtures. The two WMA mixtures are produced respectively by the Evotherm DAC and water-based foaming technologies. The target air void content is 7%. For each mixture type, four replicate specimens are prepared and tested. Thus, there are 12 data points collected from Gu et al. (3).

D-4 Table D-1. Material Information and Calculated Representative Elastic Modulus of Jacob (2)’s Test Data for Determining Fracture Coefficients Mixture Type Test Temperature (˚C) Frequency (Hz) reE (MPa) Air Void Content (%) Binder Content (%) DACmod 1a, 3a, 3b, 3c, 4a, 4b 15 8 3098 2.0 6.0 1b 25 8 1714 1c 5 8 5190 2a 15 2 2178 2b 15 4 2616 2c 15 16 3609 SMA 1a, 3a, 3b, 3c, 4a, 4b 15 8 3661 3.0 7.0 1b 25 8 1708 1c 5 8 6286 2a 15 2 2446 2b 15 4 3027 2c 15 16 4334 SA 1a, 3a, 3b, 3c, 4a 15 8 3414 8.3 10.0 1b 25 8 2116 1c 5 8 4873 2a 15 2 2167 2b 15 4 2766 2c 15 16 4082 DAC8 1a, 3a, 3b, 3c, 4a, 4b, 4c, 5a, 5b, 5c 15 8 3344 7.9 6.8 1b 25 8 1589 1c 5 8 5920 2a 15 2 2022 DAC16 1a 15 8 3177 1.1 6.2 1b 25 8 1444 1c 5 8 5522 2a 15 8 3177 5.1 2b 1.6 3a 1.5 5.7 3b 0.8 5.95 3c 1.3 6.7 4a 2.0 6.2 4b 2.35a 0.3 5b 0.8

D-5 From the three studies above, there are altogether 89 different pairs of A and n. The plot of  log A versus n is shown in Figure D-2. When a straight line is fitted to the data in Figure D-2, the R-squared of the linear function is 0.892. This suggests a strong correlation between the modified Paris’ law coefficients. Once one of them is known, the other coefficient can be estimated by:  1.246 3.61510 nA    (D-11) Figure D-2. Relationship Between Fracture Coefficients A’ and n’ References: 1. Luo, X., Luo, R., and Lytton, R.L. (2013c). “Modified Paris’ Law to Predict Entire Crack Growth in Asphalt Mixtures.” Transportation Research Record: Journal of the Transportation Research Board, 2373, 54–62 2. Jacobs, M.M.J. (1995). “Crack Growth in Asphaltic Mixes.” Ph.D. dissertation, Delft University of Technology, The Netherlands. 3. Gu, F., Y. Zhang, X. Luo, R. Luo, and R.L. Lytton. (2015). “Improved Methodology to Evaluate Fracture Properties of Warm Mix Asphalt Using Overlay Test”, Transportation Research Record: Journal of the Transportation Research Board, No. 2506, pp. 8-18.