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41 4.0 ft make use of the viscoelastic thermal stress computation algo- 5.0 ft Lj Lj 5.0 ft rithm (38). The details of the viscoelastic thermal stress compu- tation are given in Appendix I. It makes use of Dirichlet series 1.11 1.11 (Equation 38) to represent the master curve of the relaxation W(t) modulus of the asphalt mixture at any loading time, t. Load Wave n t - r Shape E ( t r ) = E + Ei e Ti (38) i =1 1.11 1.11 where tr is the reduced time of the master curve of the relax- ation modulus of the asphalt mixture. The Dirichlet series (14 + Lj) ft. coefficients, Ei, and relaxation times, Ti, are determined with a (1.11)n (1.11)n collocation process using the Witczak 2006 mixture model (3) and the E is determined from the analysis of the FWD field data. Within the program, the temperature and relaxation [W(t)]n modulus of the overlay at the current tip of the growing reflec- tion crack is calculated for each day, and then used to calcu- late the viscoelastic thermal stress at the tip of the crack, the (1.11)n (1.11)n thermal SIF, and the incremental growth of the crack that occurs that day. The actual thermal loading time is converted t to a reduced loading time using the Williams-Landel-Ferry (WLF) time-temperature shift (39). As with the growth of Figure 36b. Normalized SIF and ak wave patterns for cracks due to traffic stresses, the incremental crack growth tandem shearing loading. each day is accumulated until Position 1 is reached. The num- ber of days required to reach Position 1 is recorded as is the number of days required for the crack to grow up the rest of and thermal stresses is computed. This gives a total of five the way to the overlay surface. These two numbers of days are numbers of days that are computed and recorded in the used in the calibration equations to estimate the value of , the process of growing a reflection crack up through an overlay. All scale parameter of the observed field cracking data. five numbers of days are used in the calibration equations to The viscoelastic thermal stress is calculated using a Boltz- estimate the values of (the scale parameter) and (the shape man Superposition Integral in numerical form (details of this parameter) of the observed field cracking data. calculation are given in Appendix I). The general form of the thermal stress integration is Computational Method for Viscoelastic Thermal Stresses =t r ( - T ) (t r ) = E (t r - ) d (39) =0 In determining the thermal viscoelastic modulus of an in- service mixture for use in calibrating the program, Falling The initial strain is calculated relative to the defining tem- Weight Deflectometer (FWD) data was used to determine the perature of the master curve Td, and the remaining thermal modulus of the overlay and the mixture temperature at the strain value T is the hourly change of thermal strain. It is time of the measurement. The overlay asphalt mixture was the rate of change of the difference between and T that used in the 2006 Witczak modulus model to calculate the accumulates the thermal stress at the tip of the growing ther- modulus of the mixture. Discrepancy between the FWD mal reflection crack. The thermal stress at the tip of the crack modulus and that predicted by the Witczak 2006 model (3) is calculated for every hour of each day and the highest calcu- at the same temperature was ascribed to the rubbery stiffness lated stress is used to calculate the thermal SIF and the incre- of the mixture, E. mental growth of the reflection crack for that day. Paris and Erdogan's Law coefficients were calculated using the modu- E ( t , T )FWD = E + E ( t , T )2006 (37) lus for the critical time and temperature each day to calculate the D1, mmix, and t values. The t was set at one 1 and the ak If the stiffness of the Witczak 2006 model was greater than the was set at 1.0 for the thermal case. FWD back calculated modulus, the value of E was set to zero. The values for the coefficient of thermal expansion (con- Otherwise, the value of E was used in calculating the master traction) can be input at Level 1 and Level 3. The Level 3 curve of the relaxation moduli at 11 different loading times to input is the mean value of the thermal coefficients that were