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36 8 6 Daily Average Temperature (Degree C) Temperature Flutuation Pattern Around 4 2 0 -2 -4 Texas (48-1000) Texas (48-0800) Texas (48-A800) -6 South Dakota (46-0800) Utah (49-0800) -8 Nevada (32-0100) 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Hours Figure 32. Predicted daily air temperatures at six different LTPP test sites. models is described. The tensile strength was determined by 0.68 and 0.98 for the 1999 Witczak model and the ANN algo- Schapery (36, 37) to be an important variable in making real- rithm, respectively, when using the source input data istic estimates of the Paris and Erdogan's Law fracture coeffi- The binder input data for the 2006 Witczak model is the cient, A. Earlier studies reported tensile strengths obtained of magnitude of the shear modulus, G*, of the binder and its phase field cores taken from pavement sections well distributed angle, c, in degrees, as shown in Table 21. Also shown in that around the United States and Canada (4, 38) and were con- table are the other required input variables for the model, which sidered to be representative of HMA mixtures. The calibra- include the same gradation and volumetric composition as in tion coefficients from these studies (4) could be used to the 1999 model. The output is the magnitude of the Complex predict both thermal and traffic related reflection cracking Modulus. A comparison of the behavior of the 2006 Witczak and healing between traffic loads. model (3) and the corresponding ANN algorithm is shown in Figure 34. The R2 values are 0.77 and 0.96 for the 2006 Witczak model and the corresponding ANN algorithm, respectively. Artificial Neural Network Algorithms The user input to the ANN mixture modulus model is the for Witczak's Complex Modulus Models gradation and volumetric parts of the mix design and the The binder input data required for the 1999 Witczak model binder data. In keeping with the MEPDG format, the binder (2) are the viscosity, , and frequency of loading in Hertz, as data can be input at any of three levels. The binder data are shown in Table 20. Figure 33 shows a graphic comparison the six properties of the CAM model (i.e., Gg, the glassy mod- between the data obtained using Witczak's 1999 model and ulus in Gpa; R, the Rheological Index; rm, the cross-over fre- those developed using the ANN algorithm (35) (statistical quency in rad/sec; Td, the defining temperature in °C; and the comparisons are provided in Appendix G). The R2 values are two time-temperature shift parameters, C1 and C2) (39). The Table 20. Data input for 1999 Witczak model (viscosity, , and frequency of loading in Hertz). Input R2 Output Gradation Volumetric Binder Data Witczak ANN 3/4 3/8 #4 #200 Vbeff Va (%) fc Hz E* psi 0.68 0.98 (%) (%) (%) (%) (%) 106 poise
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37 Figure 33. Comparison of Witczak 1999 model with ANN algorithm. Table 21. Input data for the 2006 Witczak model (magnitude of the shear modulus of the binder and its phase angle in degrees). Input R2 Output Gradation Volumetric Binder Data Witczak ANN 3/4 3/8 #4 #200 Va Vbeff Log|G*| 106 c E* psi 0.77 0.96 (%) (%) (%) (%) (%) (%) psi deg Figure 34. Comparison of Witczak 2006 model with ANN algorithm.