An empirical model for modified bituminous binder master curves

Abstract

Modeling the mechanical behavior of asphalt binders and mixtures has been the subject of intensified research in recent decades. Master curves of the norm of the complex modulus |G*| in the linear viscoelastic domain are frequently used for modeling, while phase angle master curves are less frequently considered for this purpose. Therefore in this research, an empirical model is introduced for phase angle master curves of modified and neat bituminous binders. The model is based on a general form of a double-logistic (DL) mathematical function. The |G*| master curve was then modeled using a mutual relationship between the phase angle and |G*|. Master curves of three neat and seven modified binders were generated and used to validate the DL model. The results showed that the model is capable of properly predicting the plateau region of phase angle master curves. In particular, the asymptotic behavior of the master curves at high frequencies can be modeled correctly. The model also describes irregularities in the high temperature range of the phase angle master curve. In general, model outputs such as the phase angle value at the plateau, glassy modulus, rheological index and crossover frequency correctly predict the behavior of the neat and modified binders.

Appendices

Appendix A

The glassy modulus of the binders is calculated using Eq. (6) when simplified as:

$${}\log \left| {G_{g}^{*} } \right| = \lim_{{\log f_{red} \to \infty }} \log \left| {G^{*} } \right| = \log \left| {G_{0}^{*} } \right| + \mathop{\lim}\limits_{{\log f_{red} \to \infty }} \frac{1}{90}\left(- \delta_{P} \cdot {\bf\varvec{H}}\left( {{\bf\varvec{f}}_{red} - {\bf\varvec{f}}_{P} } \right) \left( \log ( {f_{red} } ) - \frac{{\pi^{1/2} }}{{2 \cdot S_{R} }}erf\left( {S_{R} \log \left( {\frac{{f_{red} }}{{f_{P} }}} \right)} \right) - \log \left( {f_{P} } \right) \cdot \frac{{\delta_{P} + \delta_{L} }}{{\delta_{P} }}\right) + \delta_{P} \cdot \log f_{red}\right) = \frac{1}{90}\left( { - \delta_{P} \cdot \left( \mathop {\lim }\limits_{{\log f_{red} \to \infty }} \log \left( {f_{red} } \right) - \frac{{\pi^{1/2} }}{{2 \cdot S_{R} }}\mathop {\lim }\limits_{{\log f_{red} \to \infty }} erf\left( S_{R} \log \left( {\frac{{f_{red} }}{{f_{P} }}} \right)\right) - \log \left( {f_{P} } \right) \cdot \frac{{\delta_{P} + \delta_{L} }}{{\delta_{P} }}\right) + \mathop {\lim }\limits_{{\log f_{red} \to \infty }} \delta_{P} \cdot \log f_{red}}\right)+ \log \left| {G_{0}^{*} } \right| = \frac{1}{90}\left( {\delta_{P} \frac{{\pi^{1/2} }}{{2 \cdot S_{R} }} - \log \left( {f_{P} } \right) \cdot \delta_{P}\frac{{\delta_{P} + \delta_{L} }}{{\delta_{P} }}} \right) + \log \left| {G_{0}^{*} } \right| = \frac{1}{90}\left( {\delta_{P} \frac{{\pi^{1/2} }}{{2 \cdot S_{R} }} + \left( {\delta_{P} + \delta_{L} } \right) \cdot \log \left( {f_{P} } \right)} \right)+ \log\left| {G_{0}^{*} } \right| $$

(A.1)

Rheological index R, as defined by Anderson et al. [3], can be calculated using the glassy modulus. This parameter shows how rapidly the behavior of asphalt cement changes overtime and thus indicates the time-dependency of the asphalt binder [3]. R can be calculated as the difference between the glassy modulus and the modulus of the binder at a point where the loss and storage moduli are equal (or the phase angle is 45°). The crossover frequency f _C can be calculated by setting Eq. (4a) to 45° and extracting the logarithm of the reduced frequency as a function of the model parameters:

$$ {\text{Log}}f_{\text{C}} = {\text{Log}}f_{\text{P}} + {\bf\varvec{H}}\left( {\bf\varvec{f}_{\text{C}} - {\bf\varvec{f}}_{\text{P}} } \right) \cdot \frac{{\sqrt { - {\text{LN}}\left( {\frac{45}{{\delta_{\text{P}}}}} \right)} }}{{S_{\text{R}} }} - {\bf\varvec{H}}\left( {\bf\varvec{f}_{\text{P}} - {\bf\varvec{f}}_{\text{C}} } \right) \cdot \frac{{\sqrt { - {\text{LN}}\left( {\frac{{\delta_{\text{P}} + \delta_{\text{L}} - 45}}{{\delta_{\text{L}} }}} \right)} }}{{S_{\text{L}} }} $$

(A.2)

The above equation denotes that Log(f _C ) is located on either side of the frequency at the plateau (f _P) and can be calculated using either of the Napierian logarithms.

The complex modulus at crossover frequency G ^*_fc can then be calculated by substituting Log(f _C ) from Eq. (A.2) into Eq. (6). Finally, R can be calculated as:

$$ R = {\text{Log}}\frac{{G_{\text{g}}^{*} }}{{G_{{f_{\text{c}} }}^{*} }} $$

(A.3)

Appendix B

The DL model is also capable of modeling asphalt mixture master curves using the same parameters presented above. The high frequency asymptotes of the master curves of asphalt mixtures can be obtained using the same method as for asphalt binders. Here, the phase angle approaches zero and |E*| approaches the glassy modulus. For low frequency ranges (f _red ≤ f _P), the phase angle approaches δ _P + δ _L. The phase angle master curve of an asphalt mixture usually shows an intermediate peak and declines on both sides at higher and lower frequencies. Based on Eq. (6), the |G*| low frequency asymptote will be zero except for δ _L  = −δ _P , where it approaches a fixed value of |G ^*₀ |−10^ (π ^0.5 δ _P /180S _L). It is generally believed that asphalt mixtures show the behavior of a viscoelastic solid. These materials approach an equilibrium modulus of |G ^*_e | at very low frequencies [28], which happens only when δ _L = −δ _P. This is not a limitation, but an advantage, and the number of parameters will reduce to five. The phase angle for extremely low frequencies will also be zero for δ _L = −δ _P. This is a commonly used assumption for asphalt mixtures [28].