Figure 1 shows the first advanced viscoelastic model. That model is the Huet model [7] . These components are one spring and one dashpot connected in series. The Huet model is characterized by an infinite number of kelvin Voigt models in series or Maxwell models in parallel. His spectrum is continuous.

The express of complex modulus E^* by the Huet model is done by Equation (1).

with (i) the complex number defined by i² = -1 ; ω = 2π frequency, pulsation; E_∞: limit of the complex modulus when ωτ → ∞; h, k: exponents such that; δ: dimensionless constant; τ: time characteristic whose value varies only with temperature. Huet model is limited at high temperatures and low frequencies because it has not a static modulus E₀.

Huet-Sayegh model solve this problem by added a spring in parallel to Huet model [8] . The spring represents

the static modulus [9] (Figure 2). Equation (2) shows the express of complex modulus by the Huet-Sayegh model.

with (i) the complex number defined by i² = -1; ω = 2π frequency, pulsation; E_∞: limit the complex modulus when ωτ → ∞; E₀ the static modulus of spring; h, k: exponents such that; δ: dimensionless constant; τ: time characteristic whose value varies only with temperature. As Huet model, Huet-Sayegh model is characterized by a continuous spectral.

Di Benedetto and colleagues [10] developed the 2S2P1D model. The model is obtained by added a linear dashpot in series to Huet-Sayegh model (Figure 3). The 2S2P1D model improves the Huet-Sayegh model at high temperatures and low frequencies [9] .

Equation (3) shows the express of complex modulus by the 2S2P1D model.

where (i) is the complex number defined by i² = -1; *ω = 2π frequency pulse; k, h are exponents such that; E₀ (“static module”) when the module ωτ → 0; E_∞ (“glassy modulus”) when the module ωτ → ∞; τ: time characteristic, whose value depends only on the temperature; β: dimensionless constant; η: viscosity Newtonian;. When ωτ → 0, then.

The complex modulus test is used to measure the dynamic modulus |E^*| of asphalt concrete named “Hot Mixture Asphalt” (HMA) at different temperatures and loading frequencies. The test can be conducted in a uniaxial or triaxial condition in either compression or tension. However, the majority of tests during the past years were in compression [12] . When the test is conducted in compression, the specimen experiences creep, in addition to the dynamic response. In the analysis of results, the creep response is typically ignored and the dynamic modulus is taken as the ratio of the amplitude of the dynamic stress to the amplitude of the dynamic strain. The tension/ compression test on cylindrical specimen belongs to the homogeneous tests, i.e. it makes it possible to obtain directly the linear viscoelastic behavior through the complex modulus. The principle of this test consists in soli- citing in traction and compression a cylindrical specimen in a continuous way according to a sinusoidal signal (traction/compression) centered on zero and applied according to the axial direction of the specimen. In the case of a complex modulus test, a low number of cycles are applied to various frequencies [13] . Complex modulus tests were performed using the Direct Traction-Compression (DTC) test on cylindrical samples according to Canadian and European standards (LC 26-700 and NF EN 12697-26, respectively). Six mixtures (Table 1) were studied (BDC, BDD BDF, GDC, GDD and GDF). Each formula undergoes measurements at temperatures of 0˚C, 10˚C, 20˚C, 30˚C, 40˚C and 55˚C and for each temperature the frequencies are 10 Hz, 5 Hz, 1 Hz, 0.3 Hz

and 0.1 Hz. The apparatus used is a MTS Press imposing a strain of 50 μ def. During the test a 4 hours conditioning time is observed for each change in temperature so that it is homogeneous in the sample. The temperature changes are applied from the lowest to the highest. For each temperature, the stress is applied for different target frequencies from the highest to lowest with a limited number of cycles.

Visco-analyse is a LCPC (actual IFSTAR) software developed by Emmanuel Chailleux in 2007 [11] .

It is used to visualize the results of complex modulus test for asphalt concretes and bituminous binders, to build master curve and finally to calibrate the advanced viscoelastic models (Huet, Huet-Sayegh and 2S2P1D). Inputs are the dynamic modulus |E^*| and phase angle measured in several temperatures and frequencies for each temperature. In this part, the complex modulus test results on studied mixtures (BDD, BDC, BDF, GDD, GDC et GDF) are visualized on Visco-analyse. Table 2, present the normalized files input for Visco-analyse Software. In fact two input files are proposed in this software. The first is specific to results of METRAVIB viscoelas- timeter. The second use an ascii normalized file as presented in Table 2.

In the first columns we find the temperatures number, the frequencies number. Data are then rows in temperatures blocks. For each line we find successively, temperature, frequency, dynamic modulus and phase angle. The units are ˚C, Hz, Pa and degree.

First time results are visualized in Cole and Cole plane and in Black Space. After the verification of the uniqueness of Cole and Cole and Black Diagrams, the master curve can be drawn by application of the time- temperature superposition principle. The reference temperature of master curve is 10˚C. Figure 4 shows the uniqueness of Cole and Cole diagram for all mixtures studied.

Figure 5 shows a unicity of Black diagram for all mixtures studied.

The uniqueness of all Cole and Cole and Black diagrams showed above proof that de asphalt concrete studied are thermorheologically simples. And a master curve can be drawn by application of time-temperature superpo- sition principle.

Figure 6 shows the master curves at the reference temperatures (0˚C, 10˚C, 20˚C, 30˚C, 40˚C and 55˚C) for

BDC mixture. Note that the master curve of a reference temperature to the other the master curve obtained at a given reference temperature, is only a translation of the master curve of other reference temperatures. Translation of master curve between temperatures is carried by the “William Landel Ferry” (W.L.F). or Arrhenius law [1] .

After the visualization of complex modulus test results, the proposed viscoelastic models by the software (without modified Huet model) are calibrated. The calibration is a fitting process of dynamic modulus and phase angle. Model parameters are performed by fitting the master curve of complex modulus test results (Cole and Cole and Black Diagrams). First time all parameters are calculated at a reference temperature (10˚C). Models are transposed to the others temperature by using the “Willian Landel and Ferry” law (WLF) on the time parameter. The minimization of prediction error is expressed by the sum of squared deviation (SSD) [11] .

Two algorithms are proposed in the Visco-analyse software, the “gradient” and the “simplex”.

The method “gradient” permits from close game parameters of a solution to quickly reach fine convergence. This algorithm called “trust-region reflective Newton” also gives access to the confidence interval on specified parameters. The second is the “simplex” method that searches a set of very distant from the initial data set parameters, while avoiding getting negative parameters [11] .

In this study the two algorithms are used successively. A first “gradient” optimization is performed. Then the Black curves and Cole and Cole are checked. Then, a second optimization is performed on the “simplex” by ensuring that the curves obtained are not worse than the first and only negative parameters have not been determined. This method significantly reduces the SSD. The final step of the calibration is the calibration of the coefficients A₀, A₁ and A₂. As WLF, these coefficients are used to translate the model to other temperatures by Equation (4).

Où:

= decreasing function of temperature, which reflects the principle of time-temperature equivalence;

A₀, A₁ and A₂ = the regression coefficients;

X = temperature sometime noted.