# Three-Dimensional Linear Viscoelastic Properties of Two Bituminous Mixtures Made with the Same Binder

Daniel Perraton, Hervé Di Benedetto, Cédric Sauzeat, Quang Tuan Nguyen, Simon Pouget

► To cite this version:

Daniel Perraton, Hervé Di Benedetto, Cédric Sauzeat, Quang Tuan Nguyen, Simon Pouget. Three-Dimensional Linear Viscoelastic Properties of Two Bituminous Mixtures Made with the Same Binder. Journal of Materials in Civil Engineering, 2018, 30 (11), pp.04018305. 10.1061/(ASCE)MT.1943-5533.0002515 . hal-02458016

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# Three-Dimensional Linear Viscoelastic Properties of Two Bituminous Mixtures Made with the Same Binder

Daniel Perraton<fnref n="1" />; Hervé Di Benedetto<fnref n="2" />; Cédric Sauzéat<fnref n="3" />; Quang Tuan Nguyen, Ph.D.<fnref n="4" />; and Simon Pouget<fnref n="5" />

**Abstract:** Eiffage developed a high-performance bituminous mixture known as GB5. It is based on aggregate optimization method. This paper presents the results of a research project checking whether, in the small strain domain, this type of mixture behaves like the more conventional asphalt mixture GB3 currently used as a base layer in French bituminous pavements. Three-dimensional complex modulus tests are performed on GB5 and GB3, two hot mixtures asphalt (HMA) made with the same bitumen but produced with different aggregate skel- etons. Tension-compression sinusoidal testing is applied to the specimens over a wide frequency and temperature range. Both axial and radial strains are monitored, thereby allowing the computation of the complex Young's modulus and complex Poisson's ratios. From the experimental results, the time-temperature superposition principle (TTSP) is verified for both complex Young's modulus and complex Poisson's ratios in two directions. The obtained shift factors and normalized complex modulus curves for the two mixtures indicate that the bitumen phase drives the viscoelastic behaviour of the asphalt mixture, regardless of the granular skeleton. The effect of air-void content on the static, glassy, and characteristic time values of modulus and Poisson's ratios, which is obtained from simulation using a linear viscoelastic model with two springs, two parabolic elements, one dashpot, is analyzed. The anisotropic properties of the two mixtures and the reproducibility between the two laboratories is also checked.

**Author keywords:** Bituminous mixture; Complex Poisson's ratio; Complex Young's modulus; Modelling; Viscoelastic property; Three-dimensional (3D) behavior.

## Introduction

The complex Young's modulus ($E^*$) and the complex Poisson's ratio ($\nu^*$) are used to define a material's three-dimensional (3D) properties for isotropic linear viscoelastic (LVE) materials. Recently, a detailed review of the literature for the three-dimensional characterization of linear viscoelastic properties of bituminous mixtures was published as state of the art by Task Group 3 (TG3) of RILEM TC 237-SIB (Graziani et al. 2018). Previous studies (Di Benedetto et al. 2001; Airey et al. 2003; Gudmarsson et al. 2014, 2015; Nguyen et al. 2015b, 2017; Perraton et al. 2016; Graziani et al. 2017) showed that classical bituminous mixtures can be considered as LVE if the applied strain amplitude remains small (approximately $10^{-5}$ m/m). Anisotropy can also be observed for bituminous mixtures but its level remains rather limited (Di Benedetto et al. 2016; Perraton et al. 2016). The present

paper investigates the rheological properties of a new type of high-performance asphalt bituminous mixtures known as GB5 and models these properties with a two springs, two parabolic elements, one dashpot (2S2P1D) model developed at the University of Lyon/ Ecole Nationale des Travaux Publics de l'Etat (ENTPE) (Lyon, France) (Olard and Di Benedetto 2003; Di Benedetto et al. 2007b).

Behavior of GB5 is compared with that of a classical GB3 French-type hot mix asphalt (HMA) currently used as a base layer, made with the same bitumen. GB5 HMA is obtained by optimizing the mixture of fine and coarse fractions, resulting in an interactive network of coarse particles which provides a strong resistance (Olard and Perraton 2010; Olard et al. 2010; Perraton et al. 2007).

Tension-compression complex modulus tests were performed on both materials in two laboratories: Laboratoire de Tribologie et Dynamique des Systèmes (LTDS) at the University of Lyon/ ENTPE (TPE) and Laboratoire sur les Chaussées et Matériaux Bitumineux (LCMB) at the University of Québec/École de Technologie Supérieure (ETS) (Québec, Canada). Results were analyzed. Some properties that may be considered as coming from the binder were shown to be common for the two types of mixtures.

## Complex Parameters for Linear Viscoelastic Materials

### Definition

The viscoelastic properties of materials can be obtained from sinusoidal loadings applied during the so-called complex modulus test. This paper reports on tests performed on cylindrical samples of axially loaded [Direction 1 (dir)] HMA. To observe linear behavior, the axial strain amplitude was lower than $10^{-4}$ m/m (Perraton et al. 2016; Di Benedetto et al. 2001; Airey et al. 2003; Nguyen et al. 2015b; Graziani et al. 2014; Godenzoni et al. 2017). In addition to avoid biasing effects such as transient effects

<footnote n="1">¹Professor, École de technologie supérieure, Université du Québec, 1100 Notre-Dame St. West, Montréal, QC, Canada H3C 1K3. Email: daniel.perraton@etsmtl.ca</footnote>

<footnote n="2">²Professor, Ecole Nationale des Travaux Publics de l'Etat-Laboratoire de Tribologie et Dynamique des Systèmes, Université de Lyon, 69518 Vaulx-en-Velin, France. Email: herve.dibenedetto@entpe.fr</footnote>

<footnote n="3">³Professor, Ecole Nationale des Travaux Publics de l'Etat-Laboratoire de Tribologie et Dynamique des Systèmes, Université de Lyon, 69518 Vaulx-en-Velin, France. Email: cedric.sauzeat@entpe.fr</footnote>

<footnote n="4">⁴Lecturer, Dept. of Civil Engineering, Univ. of Transport and Communications, 100000 Hanoi, Vietnam (corresponding author). ORCID: https://orcid.org/0000-0001-5318-0381. Email: quangtuan.nguyen@utc.edu.vn</footnote>

<footnote n="5">⁵Research and Innovation Project Manager, Eiffage Infrastructures, 8 rue du Dauphiné, 69960 Corbas, France. Email: simon.pouget@eiffage.com</footnote>
(Nguyen et al. 2015a; Gayte et al. 2016), self-heating, and thixotropy (Mangiafico et al. 2015; Babadopulos et al. 2017; Riahi et al. 2017) few cycles were applied for each frequency.

For sinusoidal tension/compression loading applied in Direction 1, the axial stress ($\sigma_1$), the axial strain ($\epsilon_1$), and the radial strain values in Directions 2 (dir2) and 3 (dir3) ($\epsilon_2$ and $\epsilon_3$, respectively), can be expressed for linear viscoelastic materials as

$$\epsilon_1(t) = \epsilon_{01} \cdot \sin(\omega t + \phi_{\epsilon1}) \quad (1)$$

$$\sigma_1(t) = \sigma_{01} \cdot \sin(\omega t + \phi_{\sigma1}) \quad (2)$$

$$\epsilon_i(t) = \epsilon_{0i} \cdot \sin(\omega t + \phi_{\epsilon i}) \quad (i = 2, 3) \quad (3)$$

where $\epsilon_{01}$, $\epsilon_{02}$, $\epsilon_{03}$, and $\sigma_{01}$ = axial strain amplitude, radial strain amplitude in Directions 2 and 3, and axial stress amplitude, respectively; $\omega = 2\pi f$ = angular frequency, where $f r$ = frequency; and $\phi$ = phase angles which can be used to characterize the time shift between the different signals.

Considering exponential notation, the complex Young's modulus in Direction 1 is given by

<figure><img src="image_2.png" /><figcaption>Fig. 1. Analogical representation of 2S2P1D model.</figcaption></figure>

$$E^* = \frac{\sigma_{01}}{\epsilon_{01}} e^{j(\phi_{\sigma1} - \phi_{\epsilon1})} = |E^*| e^{j\phi_E} \quad (4)$$

where $j$ = complex number defined by $j^2 = -1$.

The complex Poisson's ratios for Directions 2 and 3 are given by

$$\nu_i^* = -\frac{\epsilon_{0i}}{\epsilon_{01}} e^{j(\phi_{\epsilon i} - \phi_{\epsilon1})} = \frac{\epsilon_{0i}}{\epsilon_{01}} e^{j(\phi_{\epsilon i} - \phi_{\epsilon1} - \pi)} = |\nu_i^*| e^{j\phi_{\nu i}} \quad (i = 2, 3) \quad (5)$$

$$\phi_{\nu i} = \phi_{\epsilon i} - \phi_{\epsilon1} - \pi \quad (6)$$

where $\phi_E$ = phase angle between axial stress and axial strain; $\phi_{\nu2}$ and $\phi_{\nu3}$ = phase angles between the opposite of radial strains in Directions 2 and 3 and the axial strain; and $|E^*|$, $|\nu_2^*|$, and $|\nu_3^*|$ = norms of the complex Young's modulus and of the complex Poisson's ratios in Directions 2 and 3, respectively.

## Modeling

Modeling of LVE properties of tested mixtures was done with the 2S2P1D model developed at University of Lyon/ENTPE (Olard and Di Benedetto 2003; Di Benedetto et al. 2007a). The model is based on a simple combination of physical elements (two springs, two parabolic elements, and one dashpot) (Fig. 1). The 2S2P1D model was successfully considered to simulate the linear viscoelastic unidimensional or tridimensional behaviors of bituminous materials (including binders, mastics, and mixtures) (Di Benedetto et al. 2004; Lachance-Tremblay et al. 2016; Lamothe et al. 2017, 2016; Mangiafico et al. 2014; Md. Yusoff et al. 2013; Perraton et al. 2016; Pham et al. 2015a, b; Pouget et al. 2012; Riccardi et al. 2018; Jiménez del Barco Carrión et al. 2017; Aurangzeb et al. 2017).

The complex modulus and the complex Poisson's ratio of the 2S2P1D model at a fixed temperature are respectively

$$E_{2S2P1D}^*(\omega) = E_{00} + \frac{E_0 - E_{00}}{1 + \delta(j\omega\tau_E)^{-k} + (j\omega\tau_E)^{-h} + (j\omega\beta\tau_E)^{-1}} \quad (7)$$

<figure><img src="image_3.png" /><figcaption>Fig. 2. Diagram of tested specimen production: (a) bituminous mixture specimen obtained from slab; and (b) French LPC compactor used to make bituminous mixture slabs.</figcaption></figure>
$$ \nu_{2S2P1D}^*(\omega) = \nu_{00} + \frac{\nu_0 - \nu_{00}}{1 + \delta(j\omega\tau_\nu)^{-k} + (j\omega\tau_\nu)^{-h} + (j\omega\beta\tau_\nu)^{-1}} \quad (8) $$

where $k$ and $h$ = constants, where $0 < k < h < 1$; $\delta$ = shape constant; $E_{00}$ = static modulus when $\omega \to 0$; $E_0$ = glassy modulus when $\omega \to \infty$; $\nu_{00}$ = static Poisson's ratio when $\omega \to 0$; $\nu_0$ = glassy Poisson's ratio when $\omega \to \infty$; $\eta = (E_0 - E_{00})\beta\tau_E$ = Newtonian viscosity of the dashpot; $\beta$ = parameter linked with $\eta$; and $\tau_E$ and $\tau_\nu$ = characteristic time values, which are the only parameters depending on temperature

$$ \tau_E(T) = a_{TE}(T) \cdot \tau_{0E} \quad \text{and} \quad \tau_\nu(T) = a_T(T) \cdot \tau_{0\nu} \quad (9) $$

where $a_{TE}$ = shift factor at temperature $T$; and $\tau_E = \tau_{0E}$ and $\tau_\nu = \tau_{0\nu}$ at reference temperature $T_{ref}$. Ten constants ($E_{00}$, $E_0$, $\delta$, $k$, $h$, $\beta$, $\nu_{00}$, $\nu_0$, $\tau_{0E}$, $\tau_{0\nu}$) are required to completely characterize the 3D LVE properties (with isotropy hypothesis) at a given temperature. When the temperature effect is considered, the number of constants rises to twelve, including the two Williams-Landel-Ferry (WLF) constants ($C_1$ and $C_2$) [Eq. (10)].

## Experimental Program

Complex Young's modulus and Poisson's ratio measurements were carried out in TPE and LCMB. Bituminous mixture slabs (120 × 400 × 600 mm³) were provided by the laboratory of Eiffage (Vélizy-Villacoublay, France).

## Tested Material and Sample Preparation

Two bituminous mixtures, GB3 and GB5, were tested. These mixtures are used for base-course construction in France. Bituminous mixtures were produced in the laboratory according to the NF EN 12697-33 and 35 standards (CEN 2007a, b).

Cylindrical asphalt samples were cored from slabs compacted in the laboratory with a French LPC compactor (Fig. 2). The plane surface of each specimen was properly marked to situate the location of Directions 2 and 3 [Fig. 2(b)]. The preferential material Directions 1 (wheel rolling displacement), 2 (vertical), and 3 (perpendicular to Directions 1 and 2) are shown in Fig. 2(a).

A nominal aggregate size of 0–14 mm with high-quality, fully-crushed aggregates and a 35/50 Pen grade pure bitumen was used. The same bitumen and aggregate sources were used for both mixtures. The bitumen content was 4.5% by weight of the mixture for both mixtures (GB3 and GB5). Fig. 3 presents

<figure><img src="image_4.png" /><figcaption>Fig. 3. Grading curves of tested GB3 and GB5 bituminous mixtures.</figcaption></figure>

<table><caption>**Table 1.** Geometrical characteristics of tested GB3 and GB5 specimens</caption><thead><tr><th>Mixture</th><th>Name</th><th>Mass (g)</th><th>Diameter (mm)</th><th>Length (mm)</th><th>G<sub>mb</sub><sup>a</sup></th><th>G<sub>mm</sub><sup>b</sup></th><th>Air voids (%)</th></tr></thead><tbody><tr><td rowspan="3">GB3</td><td>GB3-TPE1</td><td>1,572</td><td>73.8</td><td>140.9</td><td>2.610</td><td>2.670</td><td>2.3</td></tr><tr><td>GB3-TPE2</td><td>1,564</td><td>73.7</td><td>140.1</td><td>2.614</td><td></td><td>2.1</td></tr><tr><td>GB3-ETS</td><td>1,375</td><td>74.0</td><td>123.0</td><td>2.615</td><td></td><td>1.5</td></tr><tr><td rowspan="4">GB5</td><td>GB5-TPE1</td><td>1,581</td><td>73.8</td><td>140.5</td><td>2.630</td><td>2.654</td><td>0.8</td></tr><tr><td>GB5-TPE2</td><td>1,585</td><td>73.7</td><td>140.7</td><td>2.641</td><td></td><td>0.5</td></tr><tr><td>GB5-TPE3</td><td>—</td><td>—</td><td>—</td><td>—</td><td></td><td>0.5</td></tr><tr><td>GB5-ETS</td><td>1,425</td><td>74.0</td><td>125.9</td><td>2.644</td><td></td><td>0.7</td></tr></tbody></table>

<sup>a</sup>Bulk specific gravity.

<sup>b</sup>Maximum specific gravity.

the aggregate grading curve of GB3 and GB5 bituminous mixtures. The maximum specific gravity of the mixtures ($G_{mm}$) was 2.670 and 2.654 for the GB3 and GB5 mixtures, respectively. The geometrical characteristics of the tested specimens are listed in Table 1.

## Test Equipment and Measurement Setup

The setups and the instrumentation for measuring the stress-strain values, which were different for the two laboratories (ETS and TPE), are presented in Fig. 4. These systems were also used by Perraton et al. (2016), who described them in more detail.

Tests were conducted in axial strain-control mode using a sinusoidal cyclic signal. The specimens were placed in a thermal chamber to assure temperature regulation.

In both laboratories, the axial strain was measured in the middle part of the specimen using three extensometers [Figs. 4(a and b)] located at 120°, with initial lengths of 75 and 50 mm at TPE and ETS, respectively. The radial strains were measured in dir2 and dir3 using four noncontact sensors (NCS) at TPE. For each directions, two noncontact sensors were placed on a sample diameter in

<figure><img src="image_5.png" /><figcaption>Fig. 4. Setup used by each laboratory: (a) TPE; and (b) ETS.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Experimental values of complex Young's modulus and complex Poisson's ratios of sample GB3-TPE1: (a) Cole-Cole plane of $E^*$; (b) Black diagram of $E^*$; (c) Cole-Cole plane of $\nu_i^*$; and (d) Black diagram of $\nu_i^*$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Experimental results for sample GB3-TPE1: (a) $|E^*|$ isotherms and master curves; (b) $\phi_E$ isotherms and master curves; (c) master curves of $|\nu^*|$; and (d) master curves of $\phi_{vi}$.</figcaption></figure>
opposite directions and aimed at two aluminum targets glued to the sample [Fig. 4(a), top view].

At ETS, the strains were measured in dir2 and dir3 using four strain gauges glued to the lateral surface of the samples. The length of the strain gauges was 50 mm. Two strain gauges were glued face-to-face on the cylinder wall and centered on the transverse axis for each transverse direction (dir2 and dir3) [Fig. 4(b)]. Two strain gauges ($ε_{3a}$ and $ε_{3b}$) were placed face-to-face 45 mm from the top surface of the sample, and the two others were placed at 70 mm ($ε_{2a}$ and $ε_{2b}$). The variations in the strain gauges due to temperature effects were considered in the analysis.

Tests were carried out at six frequencies (0.03, 0.1, 0.3, 1, 3, and 10 Hz), and at nine temperatures (−25°C to 40°C) and three temperatures (0°C, 15°C, and 30°C) at TPE and ETS, respectively. The sinusoidal axial strain (average of the three extensometers) was used to monitor the amplitude of axial strain during cyclic loading. To limit any bias that could exist for this type of test (e.g., heating due to viscous dissipation, thixotropy, and so on) (Di Benedetto et al. 2011; Nguyen et al. 2012; Mangiafico et al. 2015; Babadopulos et al. 2017), fewer than 100 cycles were applied at each testing frequency. The data acquisition and the analysis procedure were detailed by Perraton et al. (2016).

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Results of 2S2P1D model for GB5 samples (continuous lines) and GB3 samples (dashed lines): (a) complex modulus in Black diagram; (b) complex modulus in Cole-Cole plot; (c) complex Poisson's ratio in Black diagram for Direction 2 ($ν_2$); (d) complex Poisson's ratio in Cole-Cole plot for $ν_2$; (e) complex Poisson's ratio in Black diagram for Direction 3 ($ν_3$); and (f) complex Poisson's ratio in Cole-Cole plot for $ν_3$.</figcaption></figure>
# Experimental Results

In addition to analysis of the material behavior, which was the aim of this study, reproducibility of the tests between the two laboratories was checked. Repeatability was also checked at TPE, where one and two replicate tests for the GB3 and GB5 mixtures, respectively, were conducted.

## Results for GB3-TPE1 Test

Figs. 5(a and b) show a unique curve in the Cole-Cole plane and in the Black diagram of the GB3-TPE1 test results. The same trend was found for all specimens of GB3 and GB5 mixtures tested at both laboratories. These unique curves indicate that the materials were thermorheologically simple and respected the time-temperature superposition principle (TTSP). Master curves were plotted at chosen reference temperature ($T_{ref}$) considering the shift factor $a_T$.

Figs. 5(c and d) show the Poisson's ratios obtained from the radial measurements (in dir2 and dir3) in Cole-Cole and Black diagrams. A unique curve can also be considered as a first approximation, which reveals that the Poisson's ratios in Directions 2 and 3 also respected the time-temperature superposition principle. The evolution of the complex Poisson's ratios was opposite that observed for the complex Young's modulus. The values of the norm increased with frequency and decreased with temperature.

The $|E^*|$ and $\phi_E$ master curves obtained for specimen GB3-TPE1 at the reference temperature ($T_{ref}$) 0°C are plotted in Figs. 6(a and b). The norm of complex modulus increased with frequency and decreased with temperature, which was expected; $\phi_E$ increased and then decreased when the equivalent frequency increased.

The $|\nu^*|$ and $\phi_\nu$ master curves for Directions 2 and 3 of GB3-TPE1 sample are plotted at $T_{ref}$ = 0°C in Figs. 6(c and d). As shown by Di Benedetto et al. (2007a), the same shift factor $a_T$ was used for $E^*$ and $\nu^*$.

Perraton et al. (2016) provided a more extensive analysis for GB3 data. They found no significant differences between the complex Poisson's ratios evaluated in the two orthogonal directions. They also reported results obtained for complex Young's modulus and complex Poisson's ratios measured by five laboratories participating in a round-robin test within RILEM Technical Committee SIB activities.

## Black and Cole-Cole Diagrams for GB5 Mixture

To check the repeatability and reproducibility of the complex Poisson's ratios of the GB5 mixture, three samples were tested by TPE and one sample was tested by ETS. Black and Cole-Cole diagrams are plotted in Fig. 7; the complex Young's modulus results for all tested samples were very close both in Cole-Cole plane [Fig. 7(b)] and Black diagram [Fig. 7(a)]. In Figs. 7(a and b), the dashed line indicates the simulation of GB3 results obtained with the 2S2P1D model, and the continuous line represents simulation of GB5 results with the 2S2P1D model.

Figs. 7(c–f) show the results for the complex Poisson's ratios in Directions 2 and 3. Some scattering is observed in these figures, which can be explained by the accuracy limit of the measurement system. The tests were performed at an axial strain amplitude of about 80 µm/m in order to remain within the linear domain. Because the magnitude of the radial strain is often less than half that in the axial direction, measured changes in diameter are quite small. This explains why great accuracy is needed for this measurement

and why some scattering is hardly unavoidable (Graziani et al. 2017).

As shown previously, complex Poisson's ratios are a function of temperature and frequency and respect the time-temperature principle. Taken as a whole, the norms of the complex Poisson's ratio values were in the range 0.20–0.65, increasing with increasing temperature and decreasing with increasing frequency. In the Black and Cole-Cole diagrams, the complex Poisson's ratio data can be considered within the range of scattering for all GB5 samples. The Poisson's ratio values for Direction 2 were somewhat different from those for Direction 3. This difference may indicate anisotropy of the material. This point is treated in the last section of this paper.

The complex Young's modulus data from the two laboratories were very close (Fig. 7). Regarding the complex Poisson's ratios, all the data were within the scatter. These results indicate that the tests on the GB5 material had good reproducibility and good repeatability.

## E* and ν* Shift Factor for GB3 and GB5 Materials

Fig. 8 shows that the experimental shift factor ($a_{TE}$) used to establish the master curves of $|E^*|$, $|\nu^*|$, $\phi_E$, and $\phi_\nu$ at $T_{ref}$ = 0°C was the same for all tested samples of GB3 and GB5 mixtures. This confirms that $a_{TE}$ is fixed by the bitumen and that the granular skeleton has negligible influence on this parameter. The shift factor was modeled with the classical WLF equation (Ferry 1980)

$$ \log(a_T) = \frac{-C_1(T - T_{ref})}{C_2 + (T - T_{ref})} \quad (10) $$

Table 2 reports the $C_1$ and $C_2$ values for the GB3 and GB5 mixtures determined at 0°C ($T_{ref}$).

<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Shift factors ($a_{TE}$) for all tested samples of GB3 and GB5 mixtures. The line represents the WLF fitting curve [Eq. (10)].</figcaption></figure>

<table><caption>**Table 2.** WLF constants at reference temperature 0°C for GB3 and GB5 samples</caption><thead><tr><td>WLF constant</td><td>Value</td></tr></thead><tbody><tr><td>C<sub>1</sub></td><td>29.45</td></tr><tr><td>C<sub>2</sub></td><td>174.4</td></tr><tr><td>T<sub>ref</sub> (°C)</td><td>0</td></tr></tbody></table>
Master Curves of Complex Young's Modulus and Complex Poisson's Ratios for GB5 Samples

In the small strain domain, the behavior of the binder is generally linear, and the TTSP can be considered (Nguyen et al. 2013). Only a single variable, namely the equivalent frequency (fr<sub>eq</sub>), considers the effect of both temperature and frequency because the equivalent frequency is the product of the shift factor a<sub>T,E</sub>, which is a function of the temperature (T) and the chosen reference temperature (T<sub>ref</sub>), by frequency (fr)

$$fr_{eq} = a_{TE}fr \quad (11)$$

Master curves obtained from the results of the tested samples of the GB5 mixture are reported in Fig. 9 at a reference temperature of 0°C. The master curves of the six parameters (|E*|, |ν*<sub>2</sub>|, |ν*<sub>3</sub>|, φ<sub>E</sub>, φ<sub>ν2</sub>, and φ<sub>ν3</sub>) are plotted in Figs. 9(a–f) as a function of the equivalent frequency [Eq. (11)]. The 2S2P1D model simulation of the GB5-TPE2 sample is indicated by continuous line, and the 2S2P1D model simulation of the GB3-TPE1 sample is plotted

<figure><img src="image_10.png" /><figcaption>Fig. 9. Master curves for GB5 tested samples (data points) and 2S2P1D simulation curves for GB5-TPE2 (solid line) and GB3-TPE1 (dashed lines): (a) norm of complex modulus; (b) phase angles of complex modulus; (c) norm of complex Poisson's ratio ν*<sub>2</sub>; (d) phase angle of complex Poisson's ratios ν*<sub>3</sub>; (e) norm of complex Poisson's ratio ν*<sub>3</sub>; and (f) phase angles of complex Poisson's ratios ν*<sub>3</sub>.</figcaption></figure>
<table><caption>**Table 3.** 2S2P1D model *k*, *h*, *δ*, and *β* constants for GB3 and GB5 tested samples</caption><thead><tr><td>Constant</td><td>Value</td></tr></thead><tbody><tr><td>k</td><td>0.15</td></tr><tr><td>h</td><td>0.52</td></tr><tr><td>δ</td><td>1.54</td></tr><tr><td>β</td><td>100</td></tr></tbody></table>

as a dashed line. The 2S2P1D model constants are listed in Tables 2–4. Scattering was also observed for the Poisson’s ratio in Fig. 9. This is explained by the same factors described previously for the Cole-Cole and Black diagrams (Fig. 7).

## Normalized Complex Young’s Modulus of GB3 and GB5 Mixtures

Pouget et al. (2010) proposed normalizing the complex Young’s modulus $E_{\text{norm}}^*$ [Eq. (12)]. Pham et al. (2015a) showed that the curves obtained from normalized values in Cole-Cole and Black diagrams are fixed only by the bitumen and are independent of the granular skeleton. The same bitumen was used for the GB3 and GB5 samples, and Fig. 10 confirms that a unique curve was obtained in the Cole-Cole and Black representations for all tested materials. The only parameters needed to obtain normalized complex modulus [Eq. (12)] are $E_{00}$ (the static modulus obtained at very low and/or high temperature) and $E_0$ (the glassy modulus obtained at very high and/or very low temperature). An implication of the

existence of a unique curve in normalized axes is the consideration of a unique set of *k*, *h*, *δ*, and *β* constants for 2S2P1D modelling of all tested mixtures. These constants are reported in Table 3

$$E_{\text{Norm}}^* = \frac{E^* - E_{00}}{E_0 - E_{00}} \qquad (12)$$

Fig. 10 confirms that the bitumen phase drives the viscoelastic behavior when expressed in normalized axes of the asphalt mixture, regardless of their granular skeleton. From Eqs. (7) and (8), the influence of the granular skeleton can be evaluated considering the evolution of the static modulus ($E_{00}$), the static Poisson’s ratio ($\nu_{00}$) in Directions 2 and 3, the glassy modulus ($E_0$), the glassy Poisson’s ratio ($\nu_0$) in Directions 2 and 3, and the 2S2P1D characteristic time values at a given reference temperature ($\tau_{0E}$ and $\tau_{0\nu}$). These constants are analyzed in next section.

## Static ($E_{00}$ and $\nu_{00}$), Glassy ($E_0$ and $\nu_0$), and Characteristic Time ($\tau_{0E}$ and $\tau_{0\nu}$) Values for GB3 and GB5 Samples

The obtained static ($E_{00}$ and $\nu_{00}$), glassy ($E_0$ and $\nu_0$), and characteristic time ($\tau_{0E}$ and $\tau_{0\nu}$) values [Eqs. (7) and (8)] are presented in Table 4 for the tested GB3 and GB5 samples.

## Static Young’s Modulus

No clear evolution of static modulus $E_{00}$ for both mixtures could be obtained from the data. Previous studies revealed that the degree of uncertainty is significantly high for $E_{00}$ value determination

<table><caption>**Table 4.** Static ($E_{00}$ and $\nu_{00}$), glassy ($E_0$ and $\nu_0$), and 2S2P1D characteristic time ($\tau_{0E}$ and $\tau_{0\nu}$) values determined at 0°C for GB3 and GB5 mixtures in Directions 2 and 3</caption><thead><tr><th rowspan="2">Sample</th><th rowspan="2">Air voids (%)</th><th rowspan="2">E<sub>00</sub> (MPa)</th><th rowspan="2">E<sub>0</sub> (MPa)</th><th rowspan="2">τ<sub>0E</sub> (s)</th><th colspan="3">dir3:ν<sub>3</sub></th><th colspan="3">dir2:ν<sub>2</sub></th></tr><tr><th>ν<sub>00</sub></th><th>ν<sub>0</sub></th><th>τ<sub>0ν</sub> (s)</th><th>ν<sub>00</sub></th><th>ν<sub>0</sub></th><th>τ<sub>0ν</sub> (s)</th></tr></thead><tbody><tr><td>GB5-TPE1</td><td>0.8</td><td>25</td><td>51,000</td><td>6.5</td><td>0.54</td><td>0.21</td><td>194</td><td>0.57</td><td>0.18</td><td>194</td></tr><tr><td>GB5-TPE2</td><td>0.5</td><td>15</td><td>52,000</td><td>5.0</td><td>0.47</td><td>0.22</td><td>150</td><td>0.58</td><td>0.20</td><td>150</td></tr><tr><td>GB5-TPE3</td><td>0.5</td><td>50</td><td>52,000</td><td>4.4</td><td>0.52</td><td>0.20</td><td>133</td><td>0.53</td><td>0.22</td><td>133</td></tr><tr><td>GB5-ETS</td><td>0.7</td><td>65</td><td>50,000</td><td>12.8</td><td>0.56</td><td>0.19</td><td>385</td><td>0.56</td><td>0.24</td><td>385</td></tr><tr><td>GB3-TPE1</td><td>2.3</td><td>65</td><td>46,600</td><td>4.6</td><td>0.45</td><td>0.22</td><td>137</td><td>0.48</td><td>0.22</td><td>137</td></tr><tr><td>GB3-TPE2</td><td>2.1</td><td>50</td><td>47,000</td><td>4.96</td><td>0.47</td><td>0.23</td><td>148</td><td>0.47</td><td>0.23</td><td>148</td></tr><tr><td>GB3-ETS</td><td>1.5</td><td>65</td><td>46,500</td><td>14.0</td><td>0.48</td><td>0.20</td><td>420</td><td>0.45</td><td>0.19</td><td>420</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Normalized complex Young’s modulus data of GB3 and GB5 mixtures: (a) Cole-Cole plane; and (b) Black diagram. The line represents the 2S2P1D simulation [Eqs. (7) and (11)].</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. 11.** Complex Young's modulus and complex Poisson's ratios of tested samples as a function of air-void content: (a) glassy Young's modulus ($E_0$); (b) Young's modulus characteristic time at $T_{ref}$ ($\tau_{0E}$); (c) glassy Poisson's ratios $\nu_0$ and $\nu_{00}$ in Directions 2 and 3, respectively; and (d) Poisson's ratios characteristic time at $T_{ref}$ ($\tau_{0\nu}$).</figcaption></figure>

(high temperatures and/or low frequencies) due to creep and very soft properties of the mixtures at high temperature.

### Glassy Young's Modulus

Fig. 11(a) plots $E_0$ values as a function of void content. Some authors reported a linear decrease of $E_0$ with an increase of void content. For example, Moutier (1991) found a slope of about 2,000 MPa per percentage void content decrease and Pham et al. (2015a) indicated a decrease of about 1,500 MPa per percentage void content. Fig. 11(a) has a slope of 3,054 MPa per percentage void content decrease. This slope can be considered within the range of previously observed values. It is hypothesized that the difference in $E_0$ values observed between the GB3 and GB5 mixtures is independent of aggregates gradation and can be explained only by void content difference. It is worth checking this hypothesis.

### Young's Modulus Characteristic Time Value

Fig. 11(b) shows the characteristic time values ($\tau_{0E}$) of all tested samples determined at a reference temperature $T_{ref} = 0^\circ C$. If the results from each laboratory (TPE and ETS) are considered separately, no significant difference was found between the GB3 and GB5 mixtures. However, there was a clear difference between values from each of the two laboratories. The ETS results from the two laboratories at a reference temperature of $-2.6^\circ C$ [Fig. 11(b), hollow circles] were very similar. This output, previously discussed by Perraton et al. (2016) for GB3 mixture, may be explained by temperature regulation that differed by about 2.6$^\circ C$ between the two laboratories. Taking into account the ETS-corrected values,

the mean characteristic time value for the two tested mixtures (GB3 and GB5) was 4.97 s, with a standard deviation of 0.71 s. Fig. 11(b) indicates an interval of one standard deviation on each side of the average. The calculated temperature interval associated with this interval was $\pm0.40^\circ C$. Based on this result, a unique characteristic time value of 5.0 can be considered for the two tested mixtures (GB3 and GB5).

### Poisson's Ratio Constants in dir2 and dir3

The relationship between the air voids content and the $\nu_{00}$ and $\nu_0$ values is plotted in Fig. 11(c). Similar results were obtained in the two laboratories. Static ($\nu_{00}$) values were higher than glassy ($\nu_0$) values, but no significant difference was observed between Directions 3 and 2. The glassy Poisson's ratios ($\nu_0$) were very close for all samples, whereas the static Poisson's ratios decreased when void content increased.

### Poisson's Ratio Characteristic Time Value

Fig. 11(d) shows the Poisson's ratio characteristic time values ($\tau_{0\nu}$) of the GB3 and GB5 mixtures at the reference temperature ($T_{ref} = 0^\circ C$). The ETS results from both laboratories at a reference temperature of $-2.6^\circ C$ were in the same range. The remark made for the Young's modulus characteristic time value ($\tau_{0E}$) also applies for the Poisson's ratio characteristic time values ($\tau_{0\nu}$), confirming that a 2.6$^\circ C$ difference may exist between the thermal chambers of the two laboratories.

Table 5 gives average values that can be considered for 2S2P1D model simulation of each of the studied mixtures (GB3 and GB5).
<table><caption>**Table 5.** Average static, glassy, and characteristic time values selected for 2S2P1D model simulation of each mixture</caption><thead><tr><th rowspan="2">Mixture</th><th rowspan="2">E<sub>00</sub> (MPa)</th><th rowspan="2">E<sub>0</sub> (MPa)</th><th rowspan="2">τ<sub>0E</sub> (s)</th><th colspan="3">dir2:ν<sub>2</sub></th><th colspan="3">dir3:ν<sub>3</sub></th><th rowspan="2">T<sub>ref</sub> (°C)</th></tr><tr><th>ν<sub>00</sub></th><th>ν<sub>0</sub></th><th>τ<sub>0ν</sub> (s)</th><th>ν<sub>00</sub></th><th>ν<sub>0</sub></th><th>τ<sub>0ν</sub> (s)</th></tr></thead><tbody><tr><td>GB3</td><td>60</td><td>46,700</td><td>5.0</td><td>0.47</td><td>0.21</td><td>150</td><td>0.47</td><td>0.22</td><td>150</td><td>0</td></tr><tr><td>GB5</td><td>39</td><td>51,250</td><td>5.0</td><td>0.56</td><td>0.21</td><td>150</td><td>0.52</td><td>0.21</td><td>150</td><td>0</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Fig. 12.** Norms of complex Poisson's ratio in dir2 and dir3.</figcaption></figure>

## Results on Anisotropic Properties of GB3 and GB5 Samples

Fig. 12 shows the relationship between the norm of the complex Poisson's ratio in dir2 (|ν₂|) and the norm of the complex Poisson's ratio in dir3 (|ν₃|). Most of the data were close to the equality line, which indicates that the anisotropy effect on the Poisson's ratio, if any, was negligible.

## Conclusions

This paper analyzed the results of the complex Young's modulus and complex Poisson's ratios of two different types of hot mixture asphalt (HMA) made with the same bitumen but produced with different aggregate gradations. The two tested bituminous mixtures (GB3 and GB5) are used in France as base-course material in asphalt-structure pavements. Sinusoidal axial strain loading was applied to cylindrical samples. In addition to axial stress and strain values, the transverse strains of the cored samples were measured in two orthogonal directions (dir2 and dir3).

From the obtained results, it can be concluded that

* The GB3 and GB5 mixtures behave very much like a thermomechanical simple material: the time-temperature superposition principle (TTSP) is applicable.

* The same shift factor ($a_{TE}$) was obtained for the two mixtures both for Young's modulus and Poisson's ratio in Directions 2 and 3, which confirms that this parameter is fixed by the binder only.

* The norm of the Poisson's ratio values varied in the range from 0.20 to 0.55, increasing with temperature and decreasing with frequency;

* Based on the normalized complex modulus curves that were superimposed for all GB3 and GB5 tested samples, a unique set of k, h, δ, and β constants can be considered for the 2S2P1D linear viscoelastic model. This result confirms that the bitumen phase drives the viscoelastic behaviour of the asphalt mixture, regardless of the granular skeleton.

* Glassy Young's modulus evolution is explained only with void content;

* Complex Poisson's ratios from dir2 and dir3 were very similar for the GB3 and GB5 samples.

* The reproducibility between the two laboratories was good if a difference of about 2.6°C, which probably exists between the two thermal chambers, is considered.

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