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38 Table 22. Mean CAM model parameters for the four climatic regions. Climatic Region rm, (rad/sec) R Td (°C) C1 C2 Gg (Gpa) Wet-Freeze 0.01516 1.935 -5.8 31.57 199.2 0.861 Wet-No Freeze 7.06E-05 2.261 -6.41 42.49 259.3 0.906 Dry-Freeze 0.001397 2.286 -6.22 38.77 239.0 1.571 Dry-No Freeze 0.000845 2.032 -6.07 41.55 266.9 0.532 user may input these six properties with Level 1 input. In Models of Paris and Erdogan's Law Fracture Level 2 input, the user may specify the Performance Grade Coefficients A and n (PG) of the binder and the climatic region in which the over- lay is to be placed and the program will internally calculate Earlier studies (4) provided formulas for the Paris and Erdo- the six CAM parameters that correspond to the PG specified. gan's Law fracture coefficients A and n which were found to In Level 3 input, the user only needs to specify the climatic work well for predicting reflection cracking without being region where the overlay will be built. These simplifications altered. The formulas presented in these studies had been cal- can be made because the mean values of the six CAM param- ibrated to field fatigue cracking data in each of the four climatic eters for each of the four climatic regions in North America zones. The form of the equations for both A and n were taken are stored. A total of 48 sets of CAM parameters were mea- from viscoelastic crack growth theory by Schapery (36, 37) sured on binders extracted from cores (4); the mean values (some details on developing these formulas using a Systems for each of the climatic regions are listed in Table 22. Identification method are presented in Appendix J). The use of the ANN algorithms permits the calculation of g1 the magnitude and phase angle of a mixture modulus at any n = g0 + (25) m mix temperature and loading rate. The input of mixture moduli at different loading times to the viscoelastic thermal stress g3 log A = g 2 + log D1 + g 4 log t (26) subprogram is also generated by use of the ANN algorithm. mmix sin (mmix ) Models of Tensile Strength of Mixtures D1 = (27) E1mmix The tensile strength of a mixture depends upon the tem- perature and loading rate application. In order to have a real- log ( E ( t ,T )) = log ( E1 ) - mmix log ( t ) (28) istic estimate of the tensile strength of a mixture in the field, the relationships between the mixture modulus and tensile where strength that were measured in earlier research (4) were used mmix = the log-log slope of the mixture modulus versus and converted from the U.S. Customary units to the Interna- loading time graph for the current temperature tional System (SI) of units. The relationship developed for the and loading rate; slowest rate (0.005 in./min) was used for the thermal fracture E (t, T) = the mixture relaxation modulus (in MPa) at load- properties (Equation 23) and that developed for the most ing time, t (in sec.), and temperature, T (in °C); rapid rate (0.5 in./min) was used for the traffic fracture prop- D1 = the coefficient of the mixture creep compliance erties (Equation 24). expressed in a power law form (in kPa-1); and 1 t = tensile strength (in kPa). E ( t ,T ) × 1000 1.95 t = 6.895 (23) The magnitudes of the Complex Modulus at three differ- 6.895 × 21.3 ent loading times, t, which are set at one log cycle apart and 1 at the current temperature, were determined by the selected E ( t ,T ) × 1000 1.56 ANN Witczak model and fit by linear regression to produce t = 6.895 (24) the values of E1 and mmix that are used in defining the current 6.895 × 45.5 value of the Creep Compliance coefficient, D1. The values of Where E(t,T) designates the mixture modulus in MPa and D1 and mmix and the tensile strength, t, which is computed the t designates tensile strength in kPa. from the current value of E(t, T) are then used in the Schapery