New Predictive Equations for Dynamic Modulus and Phase Angle Using a Nonlinear Least-Squares Regression Model

The sigmoidal equation is usually used to make predictions for dynamic modulus ($|E^{*}|$), but equations to predict phase angle are still unavailable. In this study, equations to predict both the $|E^{*}|$ and phase angle for asphalt mixtures were proposed. The expression for the $|E^{*}|$ prediction in the proposed model is the same as the sigmoidal equation, while the predictive equation for the phase angle was derived based on the Kramers-Kronig equation, which reveals the relation between the $|E^{*}|$ and phase angle. The new predictive equation set is called the initial proposed (IP) model, as differentiated from the modified proposed (MP) model, in which an additional coefficient was given to the equation. To validate the IP and MP models, the equations were used to fit the $|E^{*}|$ and phase angle data at a wide range of temperatures and frequencies from 14 different asphalt mixtures. The coefficients and shift factors in the equations were obtained from a nonlinear least-squares optimization using optimization software. The predicted results using the sigmoidal equation and the IP and MP models were compared through the measured $|E^{*}|$ and phase angle values, $|E^{*}|$ and phase angle master curves, shift factors, Cole-Cole plots, and black space diagrams. The results showed that both the IP and MP models predicted the $|E^{*}|$ and phase angle well, and the MP model made more precise predictions for both the $|E^{*}|$ and phase angle as compared to the IP model. The MP model and the sigmoidal equation obtained close $|E^{*}|$ master curves and shift factors but slightly differed from the IP model. In addition, the IP and MP models successfully predicted the existence of the maximum and minimum values of the $|E^{*}|$ and phase angle through the construction of master curves. The proposed models developed in this study can be an effective approach to predict the $|E^{*}|$ and phase angle in a wide range of temperatures and frequencies for asphalt mixtures.