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Publication Number: FHWA-HRT-10-035
Date: September 2011

 

LTPP Computed Parameter: Dynamic Modulus

2.0 PREDICTIVE MODELS

Several alternative predictive relationships have been developed to estimate |E*| from simpler material properties and volumetrics. These predictive relationships can be used to populate the LTPP database with estimated |E*| values. Table 1 lists the predictive relationships identified by the research team. These relationships are described briefly in the following subsections.

Table 1. Predictive relationships for |E*|.
Model Number Model
1 Original Witczak equation (NCHRP 1-37A)(2,11)
2 Modified Witczak |G*| equation (NCHRP 1-40D)(5)
3 Hirsch model(6)
4 Law of mixtures parallel model(7)
5 ANN model
6 MR-|E*| model

2.1 ORIGINAL WITCZAK EQUATION (NCHRP 1-37A)

Andrei et al. revised the original Witczak |E*| predictive equation based on data from 205 mixtures with 2,750 data points.(11) The revised equation is as follows:

dfs (1)

Where:

p200 = Percentage of aggregate passing #200 sieve.
p4 = Percentage of aggregate retained in #4 sieve.
p3/8 = Percentage of aggregate retained in 3/8-inch (9.56-mm) sieve.
p¾ = Percentage of aggregate retained in ¾-inch (19.01-mm) sieve.
Va = Percentage of air voids (by volume of mix).
Vbeff = Percentage of effective asphalt content (by volume of mix).
f = Loading frequency (hertz).
η = Binder viscosity at temperature of interest (106 P (105 Pas).

Witczak's equation is based on a nonlinear regression analysis using the generalized reduced gradient optimization approach in Microsoft® Excel's Solver. This model incorporates mixture volumetrics and aggregate gradation and is currently one of two options for level 3 analysis using the NCHRP 1-37A MEPDG program.(2) For the viscosity term in equation 1, the program converts all level 2 and 3 inputs into regression intercept-regression slope of viscosity temperature susceptibility (A-VTS) values for the formulation of the |E*| mastercurve. Furthermore, Witczak's model has an equation (not listed) to convert A-VTS coefficients from virgin or tank binders to RTFO- and PAV-aged binder values.

The limitations of Witczak's equation, acknowledged by Bari, include relying on other models to translate the currently used |G*| measurement into binder viscosity.(5) Because the original Witzcak predictive equation is based on regression analysis, extrapolation beyond the calibration database should be restricted. Bari also mentions that there is limited volumetric influence (precision) when the model is compared to the Shell Oil model.(5) Other researchers have also noted the need for improved sensitivity to volumetrics, such as the percentage of voids in mineral aggregate (VMA), the percentage of voids filled with asphalt (VFA), asphalt concrete percentage (AC), and Va.(12)

2.2 MODIFIED WITCZAK EQUATION BASED ON |G*| (NCHRP 1-40D)(5)

To include binder |G*| in the predictive model, Witczak reformulated the model to include the binder variable directly. The updated model is as follows:

Equation 2. Modified Witczak model for predicting dynamic modulus of hot mix asphalt concrete. The logarithmic base 10 of vertical line E superscript star vertical line equals −0.349 plus 0.754 times parenthesis vertical line G superscript star vertical line subscript b to the power of −0.0052 end parenthesis times parenthesis 6.65 minus 0.032 times rho subscript 200 plus 0.0027 times parenthesis rho subscript 200 end parenthesis squared plus 0.011 times rho subscript 4 minus 0.0001 times parenthesis rho subscript 4 end parenthesis squared plus 0.006 times rho subscript three-eighths minus 0.00014 times parenthesis rho subscript three-eighths end parenthesis squared minus 0.08 times V subscript a minus 1.06 times parenthesis V subscript beff divided by the sum of V subscript beff plus V subscript a end parenthesis plus the quotient of 2.558 plus 0.032 times V subscript a plus 0.713 times parenthesis V subscript beff divided by V subscript beff plus V subscript a end parenthesis plus 0.0124 times rho subscript three-eighths minus 0.0001 times parenthesis rho subscript three-eighths end parenthesis squared minus 0.0098 times rho subscript three-fourths divided by the sum of 1 plus exponential of parenthesis  0.7814 minus 0.5785 times logarithmic base 10 vertical line G superscript star vertical line subscript b plus 0.8834 times logarithmic base 10 delta subscript b end parenthesis. (2)

Where:

G*|b = Dynamic shear modulus of asphalt binder (pounds per square inch).
δb = Binder phase angle associated with |G*|b (degrees).

As with the NCHRP 1-37A model, equation 2 is based on a nonlinear regression analysis using 346 mixtures with 7,400 data points. The measured results of the unmodified binders have a better correlation with the model (R2 = 0.87) compared to those of the modified binders (R2 = 0.79) in arithmetic scale. In logarithmic scale, both binder types have R2 = 0.99. The binder phase angle is predicted using an empirical equation (R2 = 0.83). This equation is one of two options for level 3 analysis in the most current MEPDG program.

Because some of the mixtures in this database do not contain |G*|b data, the Cox-Mertz rule, using correction factors for the non-Newtonian behaviors (see equations 3–5), is used to calculate |G*|b from A-VTS values as follows:

Equation 3. Modified Witczak model for predicting dynamic shear modulus of asphalt binder. Vertical line G superscript star vertical line subscript b equals 0.0051 times f subscript s times eta subscript f subscript s and apostrophe T times parenthesis sine of delta subscript b end parenthesis raised to the power of 7.1542 minus 0.4929 times f subscript s plus 0.0211 times f subscript s squared. (3)
Equation 4. Witczak model for predicting binder phase angle. Delta subscript b equals 90 plus the sum of parenthesis −7.3146 minus the product of 2.6162 times VTS apostrophe end parenthesis times logarithmic base 10 parenthesis f subscript s times eta subscript f subscript s and apostrophe T end parenthesis plus the sum of parenthesis 0.1124 plus the product of 0.2029 times VTS apostrophe end parenthesis times logarithmic base 10 parenthesis f subscript s times eta subscript f subscript s and apostrophe T end parenthesis squared. (4)
Equation 5. Witczak model for predicting viscosity of asphalt binder as a function of temperature and loading frequency. The logarithmic base 10 of the logarithmic base 10 of eta subscript f subscript s and apostrophe T equals 0.9699 times f subscript s raised to the power of −0.0527 times A plus 0.9668 times f subscript s raised to the power of −0.0575 times VTS times logarithmic base 10 T subscript R. (5)

Where:

fs = Dynamic shear frequency.
δb = Binder phase angle predicted from equation 4 (degrees).
ηfs,T = Viscosity of asphalt binder at a particular loading frequency (fs) and temperature (T) determined from equation 5 (centipoise).
TR = Temperature in Rankine scale.

2.3 HIRSCH MODEL

Christensen et al. examined four different models based on the law of mixtures parallel model and chose the model that incorporates the binder modulus, VMA, and VFA because it provides accurate results in the simplest form.(6,7) The other more complicated forms attempt to incorporate the modulus of the mastic or the film thickness, which are difficult parameters to measure. The suggested model for |E*| estimation is provided in equations 6–8 as follows:

Equation 6. Hirsch model for predicting dynamic modulus of hot mix asphalt concrete. Vertical line E superscript star vertical line subscript m equals the product of P subscript c bracket 4,200,000 times the difference of parenthesis 1 minus VMA divided by 100 end parenthesis plus 3 times vertical line G superscript star vertical line subscript b times the product of parenthesis VFA times VMA divided by 10,000 end parenthesis end bracket plus the quotient of parenthesis 1 minus P subscript c end parenthesis divided by the quotient of parenthesis the difference between 1 minus VMA divided by 100 end parenthesis divided by 4,200,000 plus the quotient of VMA divided by 3 times vertical line G superscript star vertical line subscript b times parenthesis VFA end parenthesis. (6)
Equation 7. Hirsch model for predicting phase angle of hot mix asphalt. Phi equals the product of -21 time parenthesis logarithmic base 10 times Pc end parenthesis squared minus 55 times logarithmic base 10 times Pc. (7)
Equation 8. Hirsch model for predicting the aggregate contact volume. P subscript c equals the quotient of parenthesis 20 plus 3 times vertical line G superscript star vertical line subscript b times parenthesis VFA end parenthesis divided by parenthesis VMA end parenthesis, end parenthesis raised to the power of 0.58 all divided by 650 plus parenthesis 3 times vertical line G superscript star vertical line subscript b times parenthesis VFA end parenthesis divided by parenthesis VMA end parenthesis, end parenthesis raised to the power of 0.58 (8)

Where:

|E*|m = Dynamic modulus of HMA (pounds per square inch).
Pc = Aggregate contact volume.
ø = Phase angle of HMA.

A strength of this model is the empirical phase angle equation, which is important for the interconversion of |E*| to the relaxation modulus or creep compliance. Weaknesses of the model include a lack of a strong dependence on volumetric parameters, particularly at low Va and VFA conditions. Also, questions arise regarding the ability of the |G*|b parameter to account for the possible beneficial effects of modifiers.(7) It must be noted that only 206 data points were used to determine the coefficients in the Hirsch model compared to 2,750 data points for the original Witczak model and 7,400 data points for the modified Witczak model.

2.4 LAW OF MIXTURES PARALLEL MODEL (AL-KHATEEB MODEL)

Based on their findings from the Hirsch model, Al-Khateeb et al. suggest the following model:(7)

Equation 9. Law of mixtures parallel model for predicting dynamic modulus of hot mix asphalt concrete. Vertical line E superscript star vertical line subscript m equals 3 times parenthesis the difference of 100 minus VMA divided by 100 end parenthesis times the quotient of bracket parenthesis 90 plus the product of 10,000 times parenthesis vertical line G superscript star vertical line subscript b divided by VMA end parenthesis, end parenthesis all raised to the power of 0.66 divided by 1,100 plus parenthesis the product of 900 times parenthesis vertical line G superscript star vertical line subscript b divided by VMA end parenthesis, end parenthesis, all raised to the power of 0.66 end bracket times vertical line G superscript star vertical line subscript g. (9)

Where:

|G*|g = Dynamic shear modulus of asphalt binder at the glassy state (assumed to be 145,000 psi (999,050 kPa).

Like the Hirsch model, this formulation is based on the law of mixtures for composite materials. In this model, the different material phases (aggregate, asphalt binder, and air) are considered to exist in parallel. Therefore, this model is a simpler interpretation of the Hirsch model. The researchers note that their model addresses one of the primary shortcomings of the Hirsch model (i.e., the Hirsch model's inability to accurately predict |E*| of the mixture at low frequencies and high temperatures).

Strengths of this model include the improved prediction of high-temperature and low-frequency |E*| data for mixtures used in the FHWA accelerated loading facility (ALF) test strips. Weaknesses include a lack of model verification and the fact that the researchers who developed this model did so based on |E*| values obtained from tests at higher than recommended strain amplitudes (200 µε versus the recommended maximum of 75–150 µε).

2.5 ANN MODELS

The NCSU research team employed the ANN technique to develop new |E*| predictive models. The primary advantage of this approach over statistical regression is that the functional form of the relationship is not needed a priori. Considering that so many variables affect |E*| values and their interactions, the ANN technique captures complicated nonlinear relationships between |E*| and other mixture variables better than regression analysis.

The ANN technique was used in this research to develop several different models. The first model is the ANN model that predicts |E*| values using the input variables employed in the modified Witczak equation (i.e., binder dynamic modulus and phase angle, aggregate gradation, and volumetrics of the HMA mixture). The effort to develop this ANN model is described later in this report. The ANN technique was also applied to backcalculate |E*| values from MR. During the FHWA DTFH61-05-RA-00108 project, the NCSU research team developed a mechanistic approach to compute MR from |E*| of HMA.(13,14) This approach was verified successfully using measured data from mixtures with varying gradations and binder characteristics. The verified solutions were then applied to an available |E*| database to estimate the MR values corresponding to the |E*| values. This database was used to develop an inverse algorithm based on the ANN technique that can predict |E*| from MR. The development and verification of MR-|E*| ANN are presented later in this report. Finally, the ANN technique was used to develop an |E*| predictive model based on binder viscosity information. This ANN model is also described in this report.

2.6 SUMMARY OF INPUT VARIABLES

Table 2 presents the necessary input variables for each predictive relationship discussed previously. For models that utilize |G*|b, predictions are only possible at the temperatures and frequencies where |G*|b values are available. Having |G*|b only at the conditions used in Superpave™ testing is not sufficient for generating |E*| values over the range of conditions typically needed for mechanistic analysis (14–129.2 °F (-10–54 °C)). For example, if the user has |G*|b at only 147.2 °F (64 °C) and 10 radians per second (rad/s), then it is possible to predict the |E*| value only at 147.2 °F (64 °C) and 10 rad/s. To predict |E*| at 77 °F (25 °C) and 25 rad/s, the user must measure |G*|b at 77 °F (25 °C) and 25 rad/s.

Table 2. Model variables.
Variable Description Model Number LTPP Data Availability
1 2 3 4 5 6 Specific Pavement Study (SPS)1 General Pavement Study (GPS)1
Mixture MR           X Yes Yes
|G*|b   X X X X   Yes2 No
σb   X     X   Yes No
VMA (percent)     X X     Yes Yes4
VFA (percent)     X       Yes Yes4
Aggregate passing #200 sieve (percent) X X     X   Yes Yes
Aggregate passing #4 sieve (percent) X X     X   Yes Yes
Aggregate passing 3/8-inch sieve (percent) X X     X   Yes Yes
Aggregate passing ¾-inch sieve (percent) X X     X   Yes Yes
Va (by volume of the mix (Vmix)) (percent) X X     X   Yes Yes
Effective asphalt content (by total volume of the mixture) (percent) X X     X   Yes Yes4
Loading frequency (Hz) X           Yes No
A-VTS X           Yes3 Yes5

1 inch = 25.4 mm
1The in-service pavement sections are classified in the LTPP program as GPS and SPS.
2|G*|b was tested for SPS-9 sections only.
3Data for penetration at 77 and 115 °F (25 and 46 °C), cone and plate viscometer at 77 °F (25 °C), absolute viscosity at 140 °F (46 °C), and kinematic viscosity at 275 °F (135 °C) are available.
4Indicates information was reported by an agency.
5Data for ring and ball softening point, penetration at 39.2 and 77 °F (4 and 25 °C), absolute viscosity at 140 °F (46 °C), and kinematic viscosity at 275 °F (135 °C) are available (reported by agency).
Note: Blank cells indicate that the input parameter is not required in the model.

A preliminary review of available information in the LTPP materials database revealed that measured |G*|b data are only available for most of the SPS-9 projects and only at 10 rad/s at multiple temperatures. Even the available |G*|b data are measured from binders aged at different levels (i.e., RTFO-aged versus PAV-aged binders). The lack of complete |G*|b data at multiple temperatures and multiple frequencies is a serious problem because all of the models except the original Witczak equation require this information. (Note that the MR-|E*| ANN model needs the binder t-T shift factor as well.) During the course of this project, the NCSU research team developed empirical models that allow the estimation of the RTFO-aged |G*|b values at multiple temperatures and frequencies from the |G*|b values obtained at a single frequency and multiple temperatures and aging levels.

 


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