Dynamic modulus is defined as the ratio of peak cyclic stress to peak cyclic strain under harmonic loading. It is one of the most important properties of asphalt mixtures, since it determines the strain response characteristics as a function of loading rate and temperature. Different simplified models exist that can predict this variable from mixture composition and binder rheological data, with Witczak and Hirsh models being the most widely accepted. These models have been evaluated in the present study, on the basis of 352 data points from eight asphalt concrete mixtures that were tested between –5 and 60 °C. A new model is also formulated which improves predictions of the previous ones for Spanish mixtures, even though it is a relatively simple equation that requires very limited binder rheological data compared to Witczak and Hirsch models.

Dynamic modulus (∣E*∣) is one of the most important properties of asphalt mixtures, since it determines the strain response characteristics as a function of loading rate and temperature. It is defined as the ratio of peak cyclic stress to peak cyclic strain under harmonic loading. It can be used to predict pavement structural response to traffic loads (

The availability of dynamic modulus testing data corresponding to actual mixtures used in the pavement is not the most common situation. In many occasions, and particularly for secondary roads where the significance of the project does not justify additional testing expenses, only routine quality control test results are available. Even for highly important highway projects, actual mixtures are not available during the pavement design process. This led researchers to develop simplified models to predict dynamic modulus based on the percentage and the mechanical properties of the constituents of the asphalt mixtures. This approach has been introduced in the MEPDG for Levels 2 & 3 designs, when dynamic modulus data are not available but the composition of the actual mixture is known.

There are three widely accepted simplified predictive models. The first one is the Witczak dynamic modulus predictive equation (

The empirical or semi-empirical nature of Witczak and Hirsch models recommends their validation for any database that differs from those used in the calibration process. A number of studies in the open literature present comparisons between these models and actual dynamic modulus data. Results from such studies do not always converge, with some of them reporting sufficiently accurate and robust estimates of ∣E*∣ for use in mechanistic-empirical pavement performance prediction and design, while others report the opposite. Examples of the former are provided by Schwartz (^{2} of 97% in the logarithmic scale, and Robbins (

The above mentioned contradictory results, along with the fact that such a comparison effort has not yet been conducted for Spanish mixtures, requires caution when either of Witczak models or the Hirsch one is used for mixtures and testing protocols typically used in Spain. It should be mentioned that a number of the experimental studies presented above propose alternative simplified predictive equations as a result of Witczak and Hirsch models mispredictions (

The objective of the experimental study presented herein is the evaluation of the applicability of Witczak and Hirsch models for asphalt concrete mixtures and testing protocols typically used in Spain. As a result of this evaluation, it is also considered the validation of an alternative simplified model that can be used to predict dynamic modulus of asphalt concrete from routine mixture design and specification parameters.

Dynamic modulus experimental data for the present research come from eight asphalt concrete mixtures (four different gradations and two asphalt contents) whose formulations are presented in

Asphalt concrete mixtures formulation

G20 | G20+ | S20 | S20+ | S20 mod | S20 mod+ | S20 MAM | S20 MAM+ | |
---|---|---|---|---|---|---|---|---|

binder content (vs aggregate mass) Volumetrics | 3.70% | 4.20% | 3.93% | 4.43% | 4.87% | 5.37% | 4.69% | 5.19% |

VA | 5.1% | 4.5% | 3.9% | 3.3% | 4.2% | 3.6% | 2.5% | 1.8% |

VMA | 12.8% | 13.2% | 12.1% | 12.5% | 14.4% | 14.8% | 12.3% | 12.7% |

VFA | 59.9% | 66.1% | 67.7% | 73.9% | 70.7% | 75.9% | 79.9% | 85.6% |

binder type Gradation | B60-70 | B60-70 | BM-3c | B13-22 | ||||

sieve (mm) | ||||||||

32 | 100% | 100% | ||||||

22 | 100% | 95.0% | 100% | 95.0% | ||||

16 | 79.0% | 71.0% | 82.0% | 76.0% | ||||

8 | 48.0% | 46.0% | 61.0% | 45.0% | ||||

2 | 21.0% | 27.0% | 34.0% | 23.0% | ||||

0.5 | 11.4% | 16.3% | 18.5% | 13.2% | ||||

0.25 | 8.9% | 12.0% | 13.8% | 10.3% | ||||

0.063 | 4.5% | 6.2% | 6.6% | 6.3% | ||||

EN 13108-1 |
AC22 base 50/70 | AC22 bin 50/70 | AC22 bin PMB 45/80-65 | AC22 base 15/25 |

EN 13108-1 “Bituminous mixtures - Material specifications - Part 1: Asphalt Concrete”.

Asphalt mixtures were produced in plant, and reheated in the laboratory in order to prepare gyratory specimens (φ100-mm; 180-mm height). Six specimens were prepared for each of the eight mixtures, and they were tested in stress-controlled cyclic compression in order to determine the complex modulus, E*. E* is a complex number whose modulus, ∣E*∣, is the dynamic modulus and whose argument, ϕ, is referred to as phase angle, both being the so called viscoelastic properties under harmonic loading. Minimum testing temperature was –5 °C for all mixtures, while maximum temperature was adapted to binder rheology (50, 55, and 60 °C, respectively, for binders B60-70, BM-3c, and B13-22). A range of frequencies between 0.1 and 20 Hz were applied at temperature increments of no more than 15 °C. No rest period was introduced between two consecutive frequencies. Load level was adjusted for each frequency to produce ±35 µɛ axial deformation, so that linear range was not exceeded. European standard EN 12697-26:2004 “Test Methods for Hot Mix Asphalt - Part 26: Stiffness”, annex D (cyclic tension-compression), was followed except for the type of loading, which was cyclic compression in the present study. The compressive cyclic loading used in this research is also specified by AASHTO T 342-11, whose methodology is very similar to that used in producing databases for Witczak and Hirsch models calibration. The only remarkable difference between the methodology used in the present experimental study and the AASHTO normative is the rest period between two consecutive frequencies, with the later specifying between 2 and 30 minutes. Coring of φ100-mm testing specimens from φ150-mm gyratory specimens is another prescription of AASHTO T 342 which is not specified by European normative and was not followed in this research.

As a first step, the measured dynamic moduli were compared to Witczak and Hirsch models predictions. This comparison involves 352 data points (44, on average, per mix), where each individual point is the mean of the six replicate specimens. Binder rheological properties to use in the models were determined according to AASHTO TP5-98 “Test Method for Determining the Rheological Properties of Asphalt Binder Using a Dynamic Shear Rheometer (DSR)”. The DSR device applies a torque to a thin film of asphalt binder placed between two parallel plates, while measuring the angular strain used to calculate the complex shear modulus, G*. A range of frequencies between 0.1 and 30 Hz were applied at temperatures between 4 and 82 °C, at 6 °C increments. It would have been desirable that the lower bound of the testing interval had been close to or below –5 °C, which was the lower bound of the corresponding temperature interval for asphalt mixture testing. Unfortunately, measuring binder stiffness at such low temperatures is beyond the loading capacity of the DSR used in this research.

An example of dynamic moduli measured for one of the mixtures is presented in

where, ∣E*∣ is dynamic modulus

_{red} is reduced frequency: log (_{red}) = log(_{T})

a_{T} is time-temperature shift factor; 20 °C is the reference temperature, for which a_{T} =1

δ, α, ϒ, & γ are model parameters

note: log refers to decimal logarithm

Construction of dynamic modulus master curve for G20 mix.

Each of the parameters of the sigmoidal function has a distinct physical meaning. δ represents the minimum dynamic modulus of the mixture (10^{δ}), that is, the lower asymptote of the master curve. δ + α represents the maximum modulus (10^{δ + α}), which is the higher asymptote. ϒ is a location parameter, which determines the horizontal position of the curvature sign change point, while γ represents the maximum slope of the master curve. Sigmoidal function is an analytical curve frequently used in mix/binder viscoelastic characterization within the frame of mechanistic-empirical pavement design. Its robust nature in curve fitting experimental data (both in frequency and time domain) with some noisy makes it also a good choice in pre-smoothing experimental data before fitting a more computational efficient analytical curve (i.e. Prony series) to perform stress-strain analysis of pavement layers using the Finite Element Method (

Binder dynamic shear modulus data were also fitted by a master curve, after the application of the corresponding shift factors. An example is presented in

where, ∣G*∣ is dynamic shear modulus

δ is phase angle

G_{g} is binder glassy modulus (1 GPa)

_{red} is reduced frequency

R, w, & _{c} are model parameters

Dynamic shear modulus master curves for binder B60-70.

Time-temperature shift factors obtained for G20 mixture and B60-70 binder are presented, as an example, in

Time-temperature shift factors obtained for mixture and binder.

Binder DSR minimum testing temperature was 5 °C, so low-temperature shift factors were extrapolated using the well-known Williams-Landel-Ferry equation reflected in

Witczak first predictive equation is based on binder viscosity. This variable was determined by using the approach proposed by Bonaquist et al. (

where, η is binder viscosity in Pa·s

∣G*∣ is dynamic shear modulus in Pa

ω is angular frequency (rad/s)

δ is phase angle

Both Witczak η-based and Hirsch models provided reasonable results when dynamic modulus predictions were compared to actual measured data. R^{2} of Witczak η-based model was 96.2% in the logarithmic scale (88.9% in the arithmetic scale) and mean square error of the prediction (MSE) was 21.1% in the arithmetic scale. In the case of the Hirsch model, R^{2} in the log scale was 95.7% (94.9% in the arithmetic scale) and MSE was 20.6%. These statistics are not very different from those reported by the authors of the two models in the calibration process (_{T} shift factors for the highest temperature, that were explained above. It should be remarked that the high temperature pattern of the prediction error is not in line with a number of references that report both Witczak and Hirsch models to overpredict moduli under this condition (

Evaluation of Witczak η-based and Hirsch models predictions for G20 & G20+ mixes.

Concerning Witczak 1-40D model, it clearly overestimated measured moduli. On average, predictions were 95.2% greater than measured values. R^{2} was 76.3% in the log scale and MSE was 84.3% in the arithmetic scale. The same overprediction is reported by Robbins (

Another issue that can be deduced from

The approach for the formulation of the new model is based on the relationships existing between parameters of the sigmoidal master curve (equation [1]) and mixture volumetric and gradation properties as well as binder rheological variables. There is a general agreement among researchers that δ and α parameters of the sigmoidal function are determined by volumetric and gradation properties of the mixtures rather than by binder rheology (

VMA Influence on maximum modulus of the mixes.

The sum δ + α varied from 4.43 to 4.55 among the different mixtures of this study, as reflected in ^{2} of the master curve calibration was 99.93% in the logarithmic scale, which indicates an almost perfect fit, as can be seen in ^{2} of the new master curve calibration was still 99.82%, indicating that a single set of these two parameters can be used for all mixtures without losing accuracy. ϒ parameter was found to depend on binder stiffness at reference temperature, whether it was formulated in terms of ∣G*∣ or η. This result was expected, as it has been pointed out by Schwartz (^{2} of the model was 98.7% in the log scale (96% in arithmetic scale), and MSE was 12.6% in the arithmetic scale for the model based on G* and 12.5% for the model based on η.

δ + α = 4.466

β= –1.177 + 9.316·VA–0.5345·log(G_{ref}) or +1.609 + 9.316·VA–0.3352log(η_{ref})

ϒ= –0.2094 + 0.09201·β

log(a_{T})= –0.1304·(T–T_{ref})

where, G_{ref} is binder dynamic shear modulus (MPa) at 20 °C (T_{ref}) and 10 rad/s

η_{ref} is binder steady state viscosity (cP) at 20 °C (T_{ref})

VA is voids in asphalt (expressed as ratio)

note: log refers to decimal logarithm

Result from the calibration of the suggested predictive model.

One limitation of the suggested model was related to using only three binder types in the calibration. Specifically, temperature susceptibility was relatively reduced in the three cases, with VTS (viscosity temperature susceptibility) parameter ranging from –2.85 to –2.75, which is a very narrow interval. VTS is the slope of the well-known relationship that exists between double logarithm of viscosity (cP) and logarithm of temperature (in Rankine). In order to overcome this limitation, an additional term was calibrated on the basis of a database generated by Onofre (

Binder rheological data required by the new equation is very limited when compared to Witczak and Hirsch models, since binder stiffness must be only defined for a single temperature. But still, this information will not be available in daily practice of pavement design in Spain, since consistency prescriptions for binders are based on classical penetration and ring and ball tests rather than on DSR or viscosity testing.

A simplified approach is here presented in order to estimate η_{ref} and VTS from available data. First, it is assumed that the ring and ball softening point corresponds to a viscosity of 13000 Poises, which is a hypothesis based on research conducted by Shell Oil and later confirmed by Mirza and Witczak (_{ref}) from binder penetration. This empirical equation was calibrated (R^{2} = 97.5%; MSE = 1.17%) on the basis of ten binders typically used in Spain, that were tested within the frame of a research project focused on binder rheology (

where, η_{ref} is binder steady state viscosity (cP) at 20 °C

P is binder penetration (mm/10) at 25 °C

note: log refers to decimal logarithm

Predictions of the new equation were compared to dynamic modulus data measured for five mixtures coming from the CEDEX accelerated pavement testing facility. Mixtures had been tested between -15 and 40 °C. The simplified approach described above was used to estimate η_{ref} and VTS from binders penetration and ring and ball softening point. The RTFO test residue data were used, since asphalt mixtures were tested after fabrication (no additional aging had taken place). Model's goodness of fit is appreciated in ^{2} of the prediction was 97.9% in the log scale (94.1% in the arithmetic scale) and MSE was 14.4% in the arithmetic scale. No bias is appreciated in the error. These statistics are similar to those obtained in the calibration, which validates the suggested predictive equation.

Result from the validation of the suggested predictive model.

Dynamic modulus has been measured for eight asphalt concrete mixtures that were tested between –5 and 60 °C. They represent typical mixtures used as base and binder courses in Spain. These experimental data have been used to evaluate three widely accepted dynamic modulus predictive models: Witczak η-based equation, which is formulated in terms of binder viscosity, and Witczak 1-40D and Hirsch models, which are based on binder dynamic shear modulus. Witczak 1-40D equation considerably overestimated (95% on average) measured dynamic modulus. On the contrary, both Witczak η-based and Hirsch models provided reasonable results. For both models, R^{2} was 96% in the log scale and the mean square error of the prediction was 21% in the arithmetic scale, which are not very different from the statistics reported by the authors in the calibration process. Nonetheless, a clear pattern was observed in the prediction error, with both models underestimating modulus for low reduced frequencies (high temperatures) while Witczak η-based model overestimating modulus for high reduced frequencies (low temperatures). Besides, the three models presented very low sensitivity versus binder content, which was against experimental results.

A new predictive model was formulated in order to overcome the limitations mentioned above. The suggested model is based on the relationships existing between the four parameters of the sigmoidal master curve (δ, α, β, and Υ) and mixture volumetric and gradation properties as well as binder rheological variables. The model was calibrated on the basis of 352 data points from the eight asphalt concrete mixtures. R^{2} of the calibration process was 98.7% in the log scale (96% in arithmetic scale), and the mean square error of the prediction was 12.5% in the arithmetic scale.

Binder rheological data required by the new model is very limited when compared to Witczak and Hirsch models, since binder stiffness must be only defined for a single temperature, 20 °C, either in terms of the steady state viscosity or in terms the dynamic shear modulus. Binder viscosity temperature susceptibility parameter (VTS) must be also defined. The other predictor variables are voids in asphalt, and voids in the mineral aggregate, which are routine mixture design parameters. Besides, a simplified procedure is proposed for estimating such binder data from penetration and ring and ball test results.

The new model was validated, together with the simplified procedure for binder data estimation, on the basis of the dynamic modulus values measured for five asphalt concrete mixtures that had been tested, within the frame of other research projects, between –15 and 40 °C. R^{2} of the prediction was 97.9% in the log scale and the mean square error was 14.4% in the arithmetic scale. No bias was found in the error.

Authors would like to express their gratitude for the important laboratory effort behind this study. Workers who made it possible are sincerely acknowledged and recognized.