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7 log [ a(T )] = c [10 A+VTS log TR - log(70RTFOT )] (5) where E * = dynamic modulus; where = loading frequency, Hz; a(T) = shift factor as a function of temperature; TR = temperature, Rankine; TR = temperature, Rankine; 70RTFOT = viscosity at the reference temperature of 70 F, 70RTFOT = viscosity at the reference temperature of 70 F, AASHTO T240 residue; AASHTO T240 residue; A, VTS = viscosity-temperature parameters for AASHTO A, VTS = viscosity-temperature parameters for AASHTO T240 residue; T240 residue; and Max = specified limiting maximum modulus; and c = fitting parameter. , , and c-fitting parameters. Substituting Equation 5 into Equation 2b and the result The four unknown fitting parameters are still estimated into Equation 1 yields the form of the dynamic modulus using numerical optimization of the test data, but since the master curve relationship used in the MEPDG for the devel- limiting maximum modulus is specified, data at low test opment of master curves from laboratory test data. temperatures are no longer needed. Equations 8 and 9 present the Hirsch model, which allows estimation of the modulus of the mixture from binder stiff- log E * = + 1+ e { } + log( )+c 10 A+VTS log TR - log( 70RTFOT ) (6) ness data and volumetric properties of the mixture. where | E *|mix = E * = dynamic modulus; VMA VFA x VMA Pc 4, 200, 000 1 - + 3 | G *|binder = loading frequency, Hz; 100 10, 000 TR = temperature, Rankine; 1 - Pc + 70RTFOT = viscosity at the reference temperature of 70 F, VMA 1- AASHTO T240 residue; 100 VMA (8) + A, VTS = viscosity-temperature parameters for AASHTO 4, 200, 000 3 VFA | G *|binder T240 residue; c = fitting parameter; = limiting minimum value of E * ; where + = limiting maximum value of E * ; and 20 + VFA 3 | G * |binder 0.58 , = parameters describing the shape of the sigmoidal Pc = VMA function. (9) VFA 3 | G * |binder 0.58 650 + The fitting parameters (, , , , and c) are determined VMA through numerical optimization of Equation 6 using mixture E * = dynamic modulus of the mixture, psi; test data collected in accordance with AASHTO TP62. Due to equipment limitations, neither the limiting maximum nor VMA = Voids in mineral aggregates, %; VFA = Voids filled with asphalt, %; and limiting minimum modulus can be measured directly; there- | G * |binder = dynamic shear modulus of binder, psi. fore, these parameters are estimated through the curve fitting process. Based on research conducted during the Strategic Highway Research Program (SHRP), all binders reach a maximum shear modulus of approximately 1 GPa or 145,000 psi (6). 2.3 Proposed Dynamic Modulus Substituting this value into Equations 8 and 9 yields the rec- Master Curve Modification ommended equation for estimating the limiting maximum modulus of asphalt concrete mixtures from volumetric data. The modification proposed in this project is to estimate the limiting maximum modulus based on binder stiffness and VMA | E *|max = Pc 4, 200, 000 1 - mixture volumetric data using the Hirsch model developed 100 (10) in NCHRP Projects 9-25 and 9-31 (7). For a known limiting A VMA VFA 1 - Pc maximum modulus, the MEPDG master curve relationship + 435, 000 + 10, 000 VMA given in Equation 6 reduces to: 1- 100 VMA + ( Max - ) 4, 200, 000 435,000(VFA) log E * = + (7) 1+ e { } + log( )+c 10 A+VTS log TR - log( 70RTFOT )