^{1}

^{*}

^{1}

^{2}

The main purpose of this article is to choose among advanced rheological models used in the French rational design, one that best represents the viscoelastic behavior of asphalt mixtures mixed with aggregates of Senegal. The model chosen will be the basis for the development of computational tools for stress and strain for Senegal. However, the calibration of these models needs complex modulus test results. In opposition to mechanical models the complex modulus can directly characterize the viscoelastic behavior of bituminous materials. Here determination is performed in the laboratory by using several types of tests divided into two groups: homogeneous tests and non-homogeneous tests. The choice of model will be carried out by statistical analysis through the least squares method. To this end, a study was carried out to “Laboratory of Pavement and Bituminous Materials” (LCMB) with asphalt concrete mixed with aggregate from Senegal named basalt of Diack and quartzite of Bakel. In this study, the test used to measure the complex modulus is the Canadian test method LC 26-700 (Determination of the complex modulus by tension-compression). There mainly exist two viewing complex modulus planes for laboratory test results: the Cole and Cole plane and the Black space. The uniqueness of the data points in these two areas means that studied asphalt concretes are thermorheologically simple and that the principle of time-temperature superposition can be applied. This means that the master curve may be drawn and that the same modulus value can be obtained for different pairs (frequency-temperature). These master curves are fitted during the calibration process by the advanced rheological models. One of the most used software in the French rational design for the visualization of complex modulus test results and calibration of rheological models developed tools is named Visco-analysis. In this study, its use in interpreting the complex modulus test results and calibration models shows that, the studied asphalt concretes are thermorheologically simple, because they present good uniqueness of their Black and Cole and Cole and Black diagrams. They allow a good application of the principle of time temperature superposition. The statistical analysis of calibration models by the least squares method has shown that the three studied models are suitable for modeling the linear viscoelastic behavior of asphalt mixtures formulated with the basalt of Diack and the quartzite of Bakel. Indeed their calibration has very similar precision values of “Sum of Squared Deviation” (SSD) about 0.185. However, the lower precision value (0.169) is obtained with the 2S2P1D model.

Asphalt concretes behavior is linear viscoelastic to low deformations and low cycles of loading [^{*} is defined as a complex number that links stress to strain for a linear viscoelastic material subjected to a sinusoidal loading [^{*}| [^{*}| is an approximation of the elastic modulus of a viscoelastic material, which can be used for pavement design when the laws of elasticity are employed. When the laws of viscoelasticity are applied, the mechanical models are used. In pavement design, the asphalt concrete layer can be considered as elastic or viscoelastic materials. The MEPDG [

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

with (i) the complex number defined by i^{2} = -1 ; ω = 2π frequency, pulsation; E_{∞}: limit of the complex modulus when ωτ → ∞; h, k: exponents such that_{0}.

Huet-Sayegh model solve this problem by added a spring in parallel to Huet model [

the static modulus [

with (i) the complex number defined by i^{2} = -1; ω = 2π frequency, pulsation; E_{∞}: limit the complex modulus when ωτ → ∞; E_{0} the static modulus of spring; h, k: exponents such that

Di Benedetto and colleagues [

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

where (i) is the complex number defined by i^{2} = -1; *ω = 2π frequency pulse; k, h are exponents such that_{0} (“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;

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 [

Mix composition | Quartzit dense-dense GDD | Quartzit dense-fine GDF | Quartzit dense-coarse GDC | |||
---|---|---|---|---|---|---|

% in aggregate mixture | % in HMA | % in aggregate mixture | % in HMA | % in aggregate mixture | % in HMA | |

Aggregate 0/3 mm | 58 | 55.05 | 65 | 61.62 | 49.4 | 49.4 |

Aggregate 3/8 mm | 12 | 11.39 | 20 | 18.96 | 23.75 | 23.75 |

Aggregate 8/14 mm | 30 | 28.48 | 15 | 14.22 | 21.85 | 21.85 |

Bitumen 35/50 or PG 70/16 | 5.35 | 5.08 | 5.48 | 5.2 | 5 | 5 |

Mix composition | Basalt dense-dense BDD | Basalt dense-fine BDF | Basalt dense-coarse BDC | |||

% in aggregate mixture | % in HMA | % in aggregate mixture | % in HMA | % in aggregate mixture | % in HMA | |

Aggregate 0/3 mm | 50 | 47.53 | 60 | 56.94 | 32 | 30.53 |

Aggregate 3/8 mm | 15 | 14.26 | 20 | 18.98 | 31 | 29.58 |

Aggregate 8/16 mm | 35 | 33.27 | 20 | 18.98 | 37 | 35.3 |

Bitumen 35/50 or PG 70/16 | 5.2 | 4.94 | 5.38 | 5.11 | 4.81 | 4.59 |

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 [

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.

T (˚C) | F. (Hz) | BDC mix | GDC mix | BDD mix | GDD mix | BDF mix | GDF mix | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

|E^{*}| | δ(˚) | |E^{*}| | δ(˚) | |E^{*}| | δ(˚) | |E^{*}| | δ(˚) | |E^{*}| | δ(˚) | |E^{*}| | δ(˚) | ||

0 | 10 | 8,063,000,000 | 15.3 | 5,851,000,000 | 16.7 | 6,604,000,000 | 14.39 | 10,668,000,000 | 14.9 | 10,028,000,000 | 11.1 | 7,428,000,000 | 15.6 |

0 | 5 | 9,587,000,000 | 12.7 | 7,126,000,000 | 14.1 | 7,670,000,000 | 12.15 | 12,613,000,000 | 12.5 | 11,255,000,000 | 10.4 | 8,760,000,000 | 13.2 |

0 | 1 | 11,146,000,000 | 10.5 | 08,454,000,000 | 11.9 | 8,828,000,000 | 10.20 | 14,674,000,000 | 10.3 | 12,823,000,000 | 9.2 | 10,242,000,000 | 11 |

0 | 0.3 | 12,592,000,000 | 9.1 | 9,636,000,000 | 9.9 | 9,838,000,000 | 9.31 | 16,194,000,000 | 7.7 | 14,231,000,000 | 8.3 | 11,516,000,000 | 9.3 |

0 | 0.1 | 14,105,000,000 | 7.7 | 11,052,000,000 | 8.6 | 11,364,000,000 | 8.02 | 18,107,000,000 | 6.7 | 15,802,000,000 | 7.5 | 13,420,000,000 | 8.3 |

10 | 10 | 3,887,000,000 | 26.3 | 2,331,000,000 | 30.7 | 2,397,000,000 | 27.76 | 4,622,000,000 | 30.5 | 5,259,000,000 | 18.2 | 2,358,000,000 | 31.5 |

10 | 5 | 5,146,000,000 | 22.2 | 3,205,000,000 | 25.7 | 3,254,000,000 | 24.09 | 6,391,000,000 | 24.8 | 6,398,000,000 | 16.6 | 3,355,000,000 | 27.1 |

10 | 1 | 6,724,000,000 | 18.1 | 4,372,000,000 | 21.6 | 4,352,000,000 | 20.26 | 8,540,000,000 | 19.6 | 7,817,000,000 | 14.7 | 4,672,000,000 | 22.6 |

10 | 0.3 | 8,264,000,000 | 14.9 | 5,545,000,000 | 18 | 5,470,000,000 | 17.38 | 10,434,000,000 | 15.8 | 9,286,000,000 | 13.1 | 6,046,000,000 | 19.4 |

10 | 0.1 | 9,872,000,000 | 12.6 | 7,018,000,000 | 15 | 6,744,000,000 | 14.71 | 14,838,000,000 | 12.5 | 10,861,000,000 | 11.7 | 7,648,000,000 | 16.4 |

20 | 10 | 112,200,000 | 40.5 | 597,000,000 | 41.2 | 754,000,000 | 37.68 | 947,000,000 | 51.6 | 2,034,000,000 | 27.9 | 659,000,000 | 41.5 |

20 | 5 | 1,756,000,000 | 36.6 | 973,000,000 | 39.3 | 1,166,000,000 | 34.96 | 1,715,000,000 | 45.4 | 2,799,000,000 | 25.2 | 1,074,000,000 | 39.2 |

20 | 1 | 2,722,000,000 | 31.6 | 1,576,000,000 | 35.4 | 1,811,000,000 | 31.29 | 3,015,000,000 | 37.7 | 3,820,000,000 | 22.6 | 1,775,000,000 | 35.3 |

20 | 0.3 | 3,874,000,000 | 26.9 | 2,337,000,000 | 31 | 2,569,000,000 | 27.86 | 4,593,000,000 | 30.4 | 4,906,000,000 | 20.6 | 2,670,000,000 | 30.8 |

20 | 0.1 | 5,316,000,000 | 22.3 | 3,384,000,000 | 26.2 | 3,579,000,000 | 23.70 | 6,611,000,000 | 24.4 | 6,301,000,000 | 18.4 | 3,894,000,000 | 26.3 |

30 | 10 | 338,000,000 | 40.9 | 186,000,000 | 39.9 | 246,000,000 | 36.46 | 173,000,000 | 60.3 | 696,000,000 | 36.7 | 187,000,000 | 38.3 |

30 | 5 | 550,000,000 | 42.4 | 294,000,000 | 42.1 | 377,000,000 | 38.10 | 349,000,000 | 58.4 | 1,047,000,000 | 34.2 | 293,000,000 | 41.4 |

30 | 1 | 940,000,000 | 41.8 | 511,000,000 | 42.7 | 617,000,000 | 38.38 | 750,000,000 | 54.2 | 1,605,000,000 | 31.7 | 516,000,000 | 42.7 |

30 | 0.3 | 1,515,000,000 | 38.8 | 838,000,000 | 41.5 | 950,000,000 | 37.23 | 1,410,000,000 | 48.5 | 2,281,000,000 | 28.8 | 847,000,000 | 41.9 |

30 | 0.1 | 2,422,000,000 | 34.3 | 1,388,000,000 | 37.7 | 1,511,000,000 | 34.40 | 2,563,000,000 | 41.4 | 3,234,000,000 | 26.0 | 1,461,000,000 | 38.8 |

40 | 10 | 142,000,000 | 29.5 | 87,000,000 | 23.5 | 150,000,000 | 25.43 | 46,000,000 | 45.9 | 218,000,000 | 41.8 | 80,000,000 | 23.8 |

40 | 5 | 194,000,000 | 34.8 | 112,000,000 | 29.6 | 218,000,000 | 29.87 | 80,000,000 | 53.1 | 341,000,000 | 41.1 | 106,000,000 | 30.7 |

40 | 1 | 302,000,000 | 40.2 | 167,000,000 | 36.4 | 326,000,000 | 34.47 | 161,000,000 | 57.5 | 571,000,000 | 39.0 | 160,000,000 | 37.7 |

40 | 0.3 | 484,000,000 | 43.5 | 258,000,000 | 41.5 | 529,000,000 | 38.45 | 325,000,000 | 58.5 | 890,000,000 | 37.0 | 254,000,000 | 42.3 |

40 | 0.1 | 835,000,000 | 44.1 | 438,000,000 | 44.2 | 835,000,000 | 40.24 | 690,000,000 | 56.5 | 1,396,000,000 | 34.5 | 438,000,000 | 45.5 |

55 | 10 | 103,000,000 | 14.7 | 67,000,000 | 10.5 | 87,000,000 | 12.53 | 34,000,000 | 15.8 | 51,000,000 | 38.0 | 60,000,000 | 9.6 |

55 | 5 | 114,000,000 | 17.6 | 70,000,000 | 11.9 | 93,000,000 | 15.17 | 39,000,000 | 20.5 | 76,000,000 | 40.0 | 67,000,000 | 13 |

55 | 1 | 133,000,000 | 22.4 | 78,000,000 | 16.6 | 112,000,000 | 19.52 | 49,000,000 | 28.6 | 121,000,000 | 42.1 | 75,000,000 | 17.9 |

55 | 0.3 | 162,000,000 | 29.1 | 96,000,000 | 22.5 | 134,000,000 | 24.41 | 70,000,000 | 39.4 | 193000000 | 43.3 | 93,000,000 | 24.8 |

55 | 0.1 | 223,000,000 | 37.7 | 124,000,000 | 30.9 | 175,000,000 | 31.81 | 12,000,000 | 51.4 | 322000000 | 43.7 | 126,000,000 | 34.1 |

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.

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.

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 [

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

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 [

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 A_{0}, A_{1} and A_{2}. As WLF, these coefficients are used to translate the model to other temperatures by Equation (4).

Où:

A_{0}, A_{1} and A_{2} = the regression coefficients;

X = temperature sometime noted

The method of least squares is a statistical method analysis, independently developed by Gauss and Legendre.

Method is used to compare the experimental data, usually tainted by measurement errors with a mathematical model meant to describe these data [

The method consists of a prescription (initially empirical) which is the function _{i}) i = 1, N the “optimal” parameters θ within the meaning of the least squares method are those which minimize the quantity [

where r_{i}(θ) are the residues to the model, i.e. the differences between the measurement points (y_{i}) and the model (x; θ).

S(θ) or (SSD) presented in Equation (5) can be considered as a measure of the distance between the experi- mental data and the theoretical model that predicts such data. Prescription least squares command that this dis- tance is minimal [

After the good uniqueness of master curves shows by

Asphalt concrete | Huet model parameters | SSD | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

k | h | T_{ref} (˚C) | A_{0} | A_{1} | A_{2} | ||||||||

BDC | 251540 | 0.16929 | 0.05451 | 22.011 | 0.29782 | 1.07667 | 10 | 0.22928 | −0.367885 | 0.0024178 | 0.187 | ||

GDC | 396393 | 0.17328 | 0.05264 | 43.3680 | 0.15939 | 1.10577 | 10 | 1.69998 | 0.0389322 | 0.0027256 | 0.185 | ||

BDD | 396393 | 0.17328 | 0.05264 | 43.368 | 0.15929 | 1.10577 | 10 | 1.63305 | −0.381401 | 0.0025718 | 0.187 | ||

GDD | 396393 | 0.17328 | 0.05264 | 43.368 | 0.15929 | 1.10577 | 10 | 1.6998 | −0.389322 | 0.0027256 | 0.187 | ||

BDF | 396393 | 0.17328 | 0.05264 | 43.368 | 0.15929 | 1.10577 | 10 | 1.7068 | −0.390838 | 0.0027649 | 0.187 | ||

GDF | 396393 | 0.17328 | 0.05264 | 43.368 | 0.15929 | 1.10577 | 10 | 1.70902 | −0.387427 | 0.0026695 | 0.188 | ||

Asphalt concrete | Huet Sayegh model parameters | SSD | |||||||||||

k | h | T_{ref} (˚C) | A_{0} | A_{1} | A_{2} | ||||||||

BDC | 19.0519 | 255788 | 0.18153 | 0.05636 | 21.657 | 0.27958 | 1.04456 | 10 | 0.95 | −0.367885 | 0.0024178 | 0.187 | |

GDC | 12.3652 | 455823 | 0.18328 | 0.05501 | 48.692 | 0.15213 | 1.12416 | 10 | 1.2232 | −0.389322 | 0.0027256 | 0.185 | |

BDD | 12.3654 | 455796 | 0.18327 | 0.05501 | 48.690 | 0.1520131 | 1.12416 | 10 | 1.26462 | −0.381401 | 0.0025718 | 0.185 | |

GDD | 12.3652 | 455823 | 0.18328 | 0.05501 | 48.692 | 0.15213 | 1.12416 | 10 | 1.2232 | −0.389322 | 0.0027256 | 0.185 | |

BDF | 12.3652 | 455823 | 0.18328 | 0.05501 | 48.692 | 0.15213 | 1.12416 | 10 | 1.2297 | −0.390838 | 0.0027649 | 0.185 | |

GDF | 12.3654 | 455796 | 0.18327 | 0.05501 | 48.690 | 0.1520131 | 1.12416 | 10 | 1.3406 | −0.387427 | 0.0026695 | 0.185 | |

Asphalt concrete | 2S2P1D model parameters | SSD | |||||||||||

k | h | T_{ref} (˚C) | A_{0} | A_{1} | A_{2} | ||||||||

BDC | 21.3872 | 232967 | 21.349 | 0.16688 | 21.349 | 0.56059 | 460.264 | 10 | 0.959055 | −0.367885 | 0.0024178 | 0.169 | |

GDC | 18.2375 | 621995 | 0.24543 | 0.12344 | 65.282 | 0.19585 | 7.75589 | 10 | −0.74466 | −0.389322 | 0.0027256 | 0.185 | |

BDD | 12.3654 | 455796 | 0.18327 | 0.05501 | 48.690 | 0.15213 | 1.12416 | 10 | −2.0076 | −0.381402 | 0.0025718 | 0.186 | |

GDD | 18.1984 | 593230 | 0.24481 | 0.12372 | 62.427 | 0.19749 | 7.8748 | 10 | 1.26702 | −0.389322 | 0.0027256 | 0.185 | |

BDF | 18.1984 | 593230 | 0.24481 | 0.12372 | 62.427 | 0.19749 | 7.8748 | 10 | 1.27305 | −0.390838 | 0.0027649 | 0.185 | |

GDF | 12.3654 | 455796 | 0.18327 | 0.05501 | 48.690 | 0.15213 | 1.12416 | 10 | −1.92479 | −0.387427 | 0.0026695 | 0.186 | |

model linked to the imperfection of the software used (simplex gradient).

François Olard in 2004 [_{0} of Huet Sayegh model was probably linked to aggregates skeleton. The Static modulus presented in _{0} can be impacted by the aggregate skeleton, but this impact is visible for high difference between aggregate mixtures.

In the way of model choice for asphalt concrete mixed with basalt of Diack and quartzite of Bakel, the Huet and the Huet-Sayegh and 2S2P1D models are all representatives for modeling the viscoelastic behavior of asphalt concrete mixed with basalt of Diack and quartzite of Bakel. In fact they all have a low sum of squared deviation (SSD) minimized around 0.185. The best value of 0.169, obtained by the 2S2P1D model with the BDC mixture can suppose that it is the best model representing the viscoelastic behavior of asphalt concrete mixed with aggregate of Senegal. But some irregularities are observed in the fitting of Cole and Cole diagrams by the 2S2P1D model. These irregularities are definitely linked to the software.

The authors would like to acknowledge Professor Meissa FALL (RIP) for his guidance and valuable input in this research project; and the “Mapathé NDIOUCK Enterprise” for supporting the high price shipping of aggregates from Senegal to Canada.

Mouhamed Lamine ChérifAidara,MakhalyBa,AlanCarter, (2015) Choice of an Advanced Rheological Model for Modeling the Viscoelastic Behavior of Hot Mixtures Asphalt (HMA) from Sénégal (West Africa). Open Journal of Civil Engineering,05,289-298. doi: 10.4236/ojce.2015.53029