Validation of a Mathematical-Based Model for the Rheological

Characterization of Asphalt Mixtures

FERNANDO MARTINEZ, MARINA CAUHAPE , LUIS ZORZUTTI, SILVIA ANGELONE

Road Laboratory, Institute of Applied Mechanics and Structures,

National University of Rosario,

Riobamba & Berutti, (2000) Rosario,

ARGENTINA

complexity of the integration of the K-K relations, an approximate solution of the K–K relations was used to

develop a Mathematical-Based Model to predict the master curves for the Dynamic Modulus |E*| and the Phase

Angle f. This model was validated using the experimental results of two different asphalt mixtures with

different characteristics. The results indicate that the model is accurate, and could be an effective approach to

mathematically predict the master curves of the asphalt mixture viscoelastic properties in a wide range of

temperatures and frequencies.

Key-Words: - Asphalt Mixtures, Rheology, Viscoelastic Properties, Master Curves, Kramers-Kronig Relations

Received: November 21, 2022. Revised: August 29, 2023. Accepted: October 8, 2023. Published: November 8, 2023.

material resulting from the combination of an

asphalt binder and the aggregate skeleton, linear

viscoelastic (LVE) characterization is usually

considered. For asphalt mixtures, the relationship

between applied stress and resulting strain depends

on loading frequency and temperature and can be

fully described using the complex modulus E*.

the Dynamic Modulus |E*| and the Phase Angle  at

a given temperature and loading frequency.

curve to be used in pavement design procedures.

logistic sigmoidal, non-symmetrical or generalized

functions are equivalent and the model is causal,

[18]. If this requirement is ignored, the parameters

of the models or the time-temperature shift factor aT

for various LVE functions may be different, which

is not consistent with the LVE theory, [19]. Since

physical causality induces relationships between

|E*| and , these two rheological properties are

mathematically interrelated at any frequency, and

K relations were employed in numerical form to

construct the master curves of the dynamic modulus

and the phase angle of asphalt binders and asphalt

mixtures.

wave propagation in anelastic media, [21]. It is

shown in this paper the consistency of the K-K

relations applied to experimental data of rocks.

asphalt mixtures has been evaluated using K-K

constructing the master curves of |E*| and  based

on an approximate solution of the K-K relations that

satisfies the LVE theory. This Unified Model was

validated using the experimental |E*| and  results

of two different asphalt mixtures with different

characteristics as is explained in the next sections.

Mixtures project (VirAM) that is being carried out

at the National University of Rosario, Argentina.

2 Theoretical Background

The Kramers-Kronig relations (K-K) for the

viscoelastic parameters are presented in a couple of

equations as follows, whose detailed derivations can

be found in the literature, [24].

characterization of the asphalt mixtures.

coefficients K1 and K2 result: K1 = 0.256 and K2 =

0.869.

Fig. 3: Applied loads and resulting displacements

the test temperature in the form:

where T = testing temperature in °C, [26].

The phase angle  results:

extreme points,  = 0° corresponds to a purely

elastic material and  = 90° corresponds to a purely

viscous material.

3.4 Obtained Experimental Results

Table 2 shows the obtained experimental results of

the dynamic modulus |E*| in MPa and the phase

angle  in Degrees for the AC30 mixture at the

seven frequencies and the five temperatures for the

two tested samples identified as Sample 1 and 2

The six model parameters considered in Eq. (4) and

(7) (, , , , c1 and c2) and the parameter K of the

time-temperature shift factor aT in Eq. (5) were

simultaneously determined by performing a

nonlinear minimization of the error function defined

as:

Table 1. Volumetric properties of the tested mixtures

Table 2. Experimental |E*| and  results for the AC30 mixture

tightly clustered around the line of equality, which

indicates that these modeled dynamic modulus and

phase angle values are in good match with the

measured values.

model, the correlation of the measured and modeled

values was assessed using goodness-of-fit statistics

and R2 represents the accuracy of the model. Both

parameters have been used to standardize the results

in a subjective goodness classification.

Fig. 12: Measured vs. Modeled  values for the

AC30 mixture

Fig. 13: Measured vs. Modeled |E*| values for the

curves of |E*| and  as shown in Figure 16 and

Figure 17 respectively.

frequencies/low temperatures and stiffer a low

frequencies/high temperatures with a smaller

thermal susceptibility compared to the conventional

AC30 mixture. In addition, the PmB mixture shows

a more elastic behavior compared to the AC30

mixture, with smaller phase angles for the full

frequency range.

asphalt mixtures can be evaluated using the dynamic

compared in arithmetic and logarithmic spaces.

Phase angle values were compared in an arithmetic

space.

data R2 was greater than 0.945 and the Se/Sy ratio

study. Therefore, the model could be evaluated as

Excellent according to the adopted goodness-of-fit

relations.

different rheological behaviors of different asphalt

mixtures was evaluated by the definition of the

Rutting and Fatigue Factors. The mixture

formulated with a polymer-modified asphalt binder

has a higher Rutting Factor than the asphalt mixture

with the conventional asphalt binder, which implies

that the PmB mixture has better resistance to rutting

as compared to the AC30 asphalt mixture. The

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