log ffict
log
Smix
Smax
Smin
ß
γ (increase)
ß (increase)
Constructing the Stiffness Master Curves for Asphaltic Mixes
T. O. Medani, M. Tech, M. Sc. and M. Huurman, ir, Ph. D
Report 7-01-127-3 ISSN 0169-9288
January 2003
2
Constructing the Stiffness Master Curves for Asphaltic Mixes
Delft University of Technology Faculty of Civil Engineering and GeoSciences Road and Railroad Research Laboratory & Steel and Timber Structures
T. O. Medani, M.Tech., M. Sc. and M. Huurman, ir, Ph.D
3
Acknowledgement
The research described herein was supported by the Ministry of Transport,
Public Works and Water Management (Rijkswaterstaat), whose support is
gratefully acknowledged. The study was conducted at the Road and Railroad
Research Laboratory with collaboration with the Steel and Timber Structures of
the Faculty of Civil Engineering and GeoSciences, Delft University of
Technology.
4
CONTENTS
CONTENTS..............................................................................................................4
1. INTRODUCTION...........................................................................................................5
2. TIME-TEMPERATURE SUPERPOSITION PRINCIPLE ..............................................6
2.1. SHIFTING THE EXPERIMENTAL RESULTS .................................................................................6 2.2. ARRHENIUS TYPE EQUATION..................................................................................................8 2.3. WILLIAMS-LANDEL-FERRY (WLF) EQUATION.........................................................................8
3. CONSTRUCTING MASTER CURVE USING SIGMOIDAL MODEL............................9
4. APPLICATION OF THE SIGMOIDAL MODEL TO CONSTRUCT THE STIFFNESS MASTER CURVE FOR MASTIC ASPHALT ..................................................................11
4.1. MIX COMPOSITION...............................................................................................................11 4.2. DETERMINATION OF THE MIX STIFFNESS AT DIFFERENT TEMPERATURES AND FREQUENCIES....11 4.3. APPLICATION OF THE SIGMOIDAL MODEL..............................................................................12
4.3.1. Fitting the experimental data using the Arrhenius equation ...................12 4.3.2. Fitting the experimental data using the Williams-Landel-Ferry equation.............................................................................................................................................13 4.3.3. Comparing the polynomial model and the Sigmoidal model ..................15 4.3.4. Constructing the master curve using the sigmoidal model and data obtained for 3 temperatures ........................................................................................16 4.3.5. Constructing the master curve using the polynomial model and data obtained for 3 temperatures ........................................................................................17
5. CONCLUSIONS.........................................................................................................19
6. REFRENCES ............................................................................................................20
5
1. Introduction
In road engineering, most of the mechanistic design methodologies for asphaltic
pavements are based on estimating the structural response of the pavement, i.e. the
critical stresses/strains due to a certain design load. The critical strains, which are
generally considered, are the horizontal flexural tensile strain at the bottom of the
asphalt layer and the vertical compressive strain at the top of the subgrade. For the
calculation of the stresses/strains, use is made of linear elastic multi-layer program like
BISAR (de Jong et al, 1979) or visco-elastic multi-layer programs e.g. Kenlayer
(Huang, 1993) and VEROAD (Hopman, 1993).
Bituminous mixture stiffness needs to be determined in order to evaluate both the load-
induced and thermal stress and strain distribution in asphalt pavements. Stiffness has
been used as an indicator of mixture quality for pavements and mixture design to
evaluate damage and age-hardening trends of bituminous mixtures both in laboratory
and the field (Epps, et al, 2000).
The mix stiffness is generally estimated from the so-called master curves i.e. the
relationship between the mix stiffness, loading time (or frequency) and temperature. In
practice, the indirect tensile test or the four-point bending test is used to determine this
relationship. This is done by measuring the stiffness of an asphaltic mix at different
temperatures and frequencies.
Typically the stiffness modulus of asphaltic mixes increases with decreasing
temperature and increasing loading frequency. By shifting the stiffness modulus versus
loading time relationship for various temperatures horizontally with respect to the curve
chosen as reference, a complete modulus-time behaviour curve at a constant, arbitrary
chosen, reference temperature Tref can be assembled.
This report describes a methodology to construct the stiffness master curves for
asphaltic mixes. The model described is based on physical observations and it is
believed to give ‘reasonable’ estimates for the mix stiffness at any arbitrary loading
frequency.
6
2. Time-Temperature Superposition Principle
Test data collected at different temperatures can be “shifted“ relative to the time of
loading (or frequency), so that the various curves can be aligned to form a single master
curve.
The master curve can be constructed using an arbitrary selected reference temperature
(Tref) to which all data are shifted. At reference temperature, the shift factor =1.
The technique of the determination of the master curve is based on the principle of
time-temperature correspondence, or thermorheological simplicity, which uses the
equivalence between frequency and temperature for the stiffness modulus of
bituminous mixes.
Tff fict αlogloglog =− (1)
where: ffict :the frequency where the master curve should be read (Hz)
f : loading frequency (Hz)
αt : shifting factor.
The shifting factor αt can be determined in three different ways:
1) by shifting the experimental results,
2) by means of an Arrhenius type equation,
3) by means of the Williams-Landel-Ferry (WLF) equation.
2.1 . Shifting the experimental results
The experimental (stiffness) data are plotted versus log frequency or log loading time.
After choosing a reference temperature the data of the other temperatures are shifted
horizontally until they fit the curve for the reference temperature (the shift can be
obtained by inter- or extrapolation). Then the data obtained at the other temperatures
are shifted until they fix the extended reference curve. This procedure was described by
Germann and Lytton (1977) and is detailed in Figure 1.
7
Figure 1. Setting up of master curve using fitting of experimental data method, after
Germann and Lytton, 1977 Referring to Figure 1:
)]ln()[ln( α−= oldTmaster eT (2)
m
m
ii∑
== 1
)ln()ln(
αα (3)
)ln()ln()ln( ,, inewioldi TT −=α (4)
])[log(,
210 ixTinewT ∆−= (5)
ii
i yyxyx ∆=
∆=∆ .
tan δδ
β (6)
)]log()[log( 21 TTx −=δ (7)
)]log()[log( 21 EEy −=δ (8)
)]log()[log( 2EEy i −=∆ (9)
8
Tnew,i is the individually shifted time-value of data point i, shifted over ln(αi), such that
it exactly matches the reference curve. Tmaster is the shifted time-value of point i, shifted
over the average shift factor for that temperature ln(αi).
2.2 . Arrhenius type equation
A commonly used formula for the shift factor is an Arrhenius type equation (Francken
et al. 1988, Jacobs 1995, Lytton et al. 1993).
−
∆=
−=
refrefT TTR
HeTT
C 11.log11.logα (10)
where: T = the experimental temperature (K)
Tref = the reference temperature (K)
C = a constant (K)
∆H = activation energy (J/mol)
R = ideal gas constant, 8.314 J/(mol.K)
In literature, different values were reported for the constant C.
1) C=10920 K, Francken et al. (1988).
2) C=13060 K, Lytton et al. (1993).
3) C=7680 K, Jacobs (1995).
2.3 . Williams-Landel-Ferry (WLF) equation
Another formula for the calculation of the shift factor is the Williams-Landel-Ferry
(WLF) equation (Williams et al. 1955):
ref
refTfict TTC
TTCff
−+−
−==−2
1 ).(logloglog α (11)
where: ffict = the frequency where the master curve should be read (Hz)
f = loading frequency (Hz)
C1,C2 = empirical constants
and other variables as previously defined.
According to Sayegh [1967] C1= 9.5 and C2=95. It has also been reported by Lytton et
al. [1993] that C1=19 and C2= 92.
9
3. Constructing Master Curve Using Sigmoidal Model
It is quite common to use the generalized power law to describe the frequency
dependant behaviour of bituminous materials at low and moderate temperatures. If
higher temperatures data is included, polynomial fitting functions are also used
(Pellinen and Witczak, 2002). It will be shown later that extrapolation of polynomial
fits can result in some problems. This will mean that if it is desired to include wide
range of frequencies, testing at temperatures higher and lower than the reference
temperature will be needed.
In this report a sigmoidal model similar to the one described by Pellinen and Witczak,
2002 will be presented. It will be shown how the master curve can be constructed
fitting a sigmoidal function using non-linear least square regression techniques.
The shifting will be done using an experimental approach by solving shift factors
simultaneously with the parameters of the model without the need to assume any
functional form for the shift factor equation.
The model is described as follows:
min max minlog( ) log( ) [(log( ) log( )].mixS S S S S= + − (12)
and
10 log1 exp[ ( ) ]fictf
S γ
β+
= − − (13)
where: Smix =Mix stiffness (MPa)
Smin =Minimum mix stiffness (MPa)
Smax =Maximum mix stiffness (MPa)
ffict = Reduced frequency (Hz)
ß, γ =shape parameters
The parameter γ and β are related to the curvature of the S-shaped function and the
horizontal distance from the turning point to the origin, respectively. Smin and Smax are
the minimum and maximum stiffness values (Figure 2).
10
log ffict
log
Smix
Smax
Smin
ß
γ (increase )
ß (increase )
Figure 2. Parameters in the Sigmoidal Model
The justification of using a sigmoidal model for fitting the data is based on physical
observations. The upper part of the sigmoidal model approaches asymptotically to the
maximum stiffness of the mix, which is dependent on the limiting binder stiffness of
the mix. At high temperatures the role of the aggregate skeleton plays a more dominant
role than the viscous binder. The modulus starts to reach a limiting equilibrium value
which depends on the gradation of the aggregates (Pellinen and Witczak, 2002).
11
4. Application of the Sigmoidal Model to Construct the Stiffness Master Curve for Mastic Asphalt
The proposed model has been applied to construct the master curve for the mastic
asphalt mix, which was used for resurfacing the Moerdijk Bridge in the Netherlands in
June 2000. The testing program has been carried out at the Road and Railway Research
Laboratory (RRRL) of Delft University of Technology.
4.1. Mix composition
The mastic asphalt mix consists of stone 2/8 and 2/6 in the ratio 1:1, river sand and fine
sand in the ratio 2:3, weak limestone filler and SBS modified bitumen with a pen of 90.
The mix composition is shown in Table 1.
TABLE 1
MIX COMPOSITION Component Percentage by volume
Aggregate 63
Filler 17.5
Air
Bitumen
1.5
18
4.2. Determination of the mix stiffness at different temperatures and frequencies
The mix stiffness at different temperatures and frequencies has been determined using
the UTM beam fatigue testing machine. The test conditions are:
Type of test : displacement controlled
Frequencies : 0.5, 1, 2, 5 and 10 Hz
Temperatures : 5, 12.5, 20, 27.5, 35 and 42.5 C
Strain amplitudes : 80 µm/m
Loading wave : sine wave
Stiffness measurement: after 100 pulses
12
The mix stiffness at different temperatures and frequencies is shown in Figure 3.
100
1000
10000
0.1 1 10 100
Frequency (Hz)
Mix
stif
fnes
s
T=5T=12.5T=20T=27.5T=35T=42.5
Figure 3: Mix stiffness at different frequencies and temperatures (after Bosch, 2001)
4.3. Application of the sigmoidal model
A reference temperature of 20oC was chosen. By fitting the experimental data to the
sigmoidal model, all the model parameters and the constants of the Arrhenius or the
Williams Landel Ferry equations can be obtained. This can be done by minimising the
sum of the square of the errors using the Solver Function in the Excel spreadsheet.
4.3.1. Fitting the experimental data using the Arrhenius equation
In the Arrhenius equation the shift factor αT is defined as
1 1exp .( )Tref
HR T T
α ∆= −
(14)
The frequency where the master curve should be read ffict is defined as:
.fict Tf fα= (15)
where: f :loading frequency (Hz)
T :the experimental temperature [K]
Tref :the reference temperature [K]
13
∆H :activation energy (J/mol)
R :ideal gas constant=8.314 J/(mol.K)
As explained before, the sigmoidal model parameters and the activation energy ∆H can
be obtained at the same time by minimising the sum of the square of the errors of the
experimental and model values using the Solver Function in the Excel spreadsheet. The
parameters obtained are shown in Table 2.
TABLE 2
THE MODEL PARAMETERS
∆H(J/mol) β γ log Smax log Smin 195.48 10.9131 7.1114 4.0058 2.2634
Figure 4 shows the good fit of the sigmoidal model using the Arrhenius equation for the
shift factor to the experimentally determined mix stiffness.
1
1.5
2
2.5
3
3.5
4
4.5
0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000reduced frequency (Hz)
log
Smix
Figure 4. Stiffness Master Curve at T= 20oC using the sigmoidal model and the Arrhenius equation
4.3.2. Fitting the experimental data using the Williams-Landel-Ferry equation
In the Williams-Landel-Ferry (WLF) equation the shift factor αT is defined as:
ref
refTfict TTC
TTCff
−+−
−==−2
1 ).(logloglog α (16)
14
The frequency where the master curve should be read ffict is defined as:
.fict Tf fα= (17)
where: ffict = the frequency where the master curve should be read (Hz)
f = loading frequency (Hz)
and other variables as explained before.
As explained before, the sigmoidal model parameters and the empirical parameters of
the WLF equation can be obtained at the same time by minimising the sum of the
square of the errors of the experimental and model values using the Solver Function in
the Excel spreadsheet. The parameters obtained are shown in Table 3.
TABLE 3 THE MODEL PARAMETERS
C1 C2 β γ log Smax log Smin
12.0141 101.8896 10.9503 7.0682 3.9992 2.2472
Figure 5 shows the good fit of the sigmoidal model using the WLF equation for the
shift factor to the experimentally determined mix stiffness.
1
1.5
2
2.5
3
3.5
4
4.5
0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000reduced frequency (Hz)
log
Smix
Figure 5: Stiffness Master Curve at T= 20oC using the sigmoidal model and the Williams-
Landel-Ferry equation
A comparison between the Arrhenius equation and the Williams-Landel-Ferry is shown
in Figure 6.
15
100
1000
10000
100000
1.00E-04 1.00E-02 1.00E+00 1.00E+02 1.00E+04 1.00E+06reduced frequency (Hz)
log
Smix
ModelArrhenius
Model WLF
Data
Figure 6: Comparison between the Arrhenius and the WLF equations by fitting the
sigmoidal model for the stiffness master curve at T= 20oC
From the good fit of the sigmoidal model to the experimental data using both the
Arrhenius and the WLF equations for the shift factor (Figure 6), it may be concluded
that either of the two equations can be used to estimate the shift factor.
4.3.3. Comparing the polynomial model and the Sigmoidal model
Using the Williams-Landel-Ferry equation for estimating the shift factor, the
polynomial and the sigmoidal models are compared. For the polynomial model a third
degree polynomial is assumed to describe the relationship between the mix stiffness
and the reduced frequency in the form:
2 30 1 2 3log( ) log( ) (log( )) (log( ))mixS a a f a f a f= + + + (18)
where : ai = regression coefficients
The polynomial model parameters and the WLF constants are shown in Table 4.
TABLE 4 THE MODEL PARAMETERS
C1 C2 a0 a1 a2 a3
27.5134 99.6777 2.9856 0.38469 0.01829 -0.01436 In Figure 7 the master curve for the mix stiffness at 20oC using the polynomial and the
sigmoidal model is shown.
16
1
1.5
2
2.5
3
3.5
4
4.5
0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000frequency (Hz)
log
polynomialdatasigmoidal
Figure 7: Stiffness Master Curve at T= 20oC using the sigmoidal model and the
polynomial model
From Figure 7 it can be seen that the two models fit quite well the experimental data in
the range of the experimental data, but when we move outside that range the
polynomial model may not be correct, as it is known that the stiffness of an asphaltic
mix increases with the increase of frequency till a threshold value (Smax) and it does not
decrease after reaching a maximum value as suggested by the polynomial model.
Furthermore, the stiffness decreases with the decrease of frequency till a threshold
value (Smin) and it does not increase after reaching a minimum value as suggested by
the polynomial model. In other words, the polynomial model does not describe the
observed behaviour of mix stiffness outside the range of the data.
However, if it is desired to increase the frequency range in which the polynomial
relationship is valid more tests at temperatures higher and lower than the reference
temperature will be needed.
4.3.4. Constructing the master curve using the sigmoidal model and data obtained for 3 temperatures The sigmoidal model will again be used to construct the master curve at a reference
temperature of 20oC, but this time using data obtained from only three temperatures
namely 5, 20 and 35oC.
17
In Figure 8 the good agreement between the master curves constructed from data
obtained for 3 and 6 temperatures using the sigmoidal model is shown.
2
2.5
3
3.5
4
4.5
0.0001 0.01 1 100 10000 1000000
reduced frequency [Hz]
logS
mix
model 3 temp.datamodel 6 temp
Figure 8: Stiffness Master Curve at T= 20oC using data obtained for 3 and 6 temperatures using the sigmoidal model
To be able to catch the upper part of the master curve (obtained from a low temperature
data) and the lower part of the master curve (obtained from a high temperature data) at
least two tests are essential: one at a rather high temperature and the other at a rather
low temperature. The third test may be executed at a medium temperature.
4.3.5. Constructing the master curve using the polynomial model and data obtained for 3 temperatures The third degree polynomial model will again be used to construct the master curve at a
reference temperature of 20oC, but this time using data obtained from only three
temperatures namely 5, 20 and 35oC.
In Figure 9 the master curves constructed from data obtained for 3 and 6 temperatures
using the polynomial model are shown.
18
2
2.5
3
3.5
4
4.5
0.0001 0.01 1 100 10000 1000000
reduced frequency [Hz]
log
Smix model 3 temps.
model 6 tempsdata
Figure 9: Stiffness Master Curve at T= 20o C using data obtained for 3 and 6 temperatures using the polynomial model
From Figure 9 it can be noticed that the two models are almost identical in the range of
the data, but outside this range the difference is evident.
19
5. CONCLUSIONS
Based on the material presented in this report, the following conclusions can been
drawn:
• A sigmoidal model can best describe the master curve of mix stiffness of asphaltic
mixes. This model also can explain the physical behaviour of asphaltic mixes
• The parameters for the equations for the shift factor need not be assumed, as they
can be obtained together with the parameters of the sigmoidal model using the
SOLVER function in the spreadsheet program EXCEL.
• The Arrhenius type and the Williams-Landel-Ferry equations for the shift factor
give quite comparable results.
• The polynomial model may give some problems if it is intended to estimate values
of the mix stiffness outside the range of data.
• At least for the mix which has been tested in this program (mastic asphalt), the
sigmoidal model can be described adequately based on the results obtained from
tests executed at only three temperatures. To catch the lower and the upper part of
the curve, at least two tests are essential: one at a rather high temperature and the
other at a rather low temperature; the third test may be executed at a medium
temperature (e.g. room temperature).
20
6. REFRENCES
1 Jong, D.L. de, Peutz, M.G.F, Korswagen, A.R., “Computer Program BISAR,
Layered Systems Under Normal and Tangential Surface Loads,” Koninklijke/Shell
Laboratorium, Amsterdam, Shell Research B.V., 1979.
2 Huang, Y.H., “Pavement Analysis and Design,” Prentice- Hall, Inc. New
Jersey, pp 347-350, 1993.
3 Hopman, P.C., “VEROAD: A Linear Visco-elastic Multilayer Program for the
Calculation of Stresses, Strains and Displacements in Road Constructions. Part I: A
Visco-elastic Halfspace,” Delft University of Technology, December 1993, ISSN-
0169-9288-7-93-500-6, 1993.
4 Epps A., Harvey, J.T., Kim, Y.R., and Roque, R.“ Structural Requirements of
Bituminous Paving Mixtures,” Millennium papers, Transportation Research Record,
2000.
5 Germann, F.P., and Lytton, R.L., "Methodology for Predicting the Reflection
Cracking Life of Asphalt Concrete Overlays," Report No. TTI-2-8-75-207-5, Texas
Transportation Institute of the Texas A&M University, College Station, 1977.
6 Francken, L. and Clauwaert, C., "Characterization and Structural Assessment of
Bound Materials for Flexible Road Structures," Proceedings 6th International
Conference on the Structural Design of Asphalt Pavements, Ann Arbor, 1987;
University of Michigan, pp 130-144, Ann Arbor, MI, USA, 1988.
7 Jacobs, M.M.J. “ Crack Growth in Asphaltic Mixes,” PhD. Thesis, Delft
University of Technology, Netherlands, 1995.
8 Lytton, R.L., Uzan, J., Fernando, E.M., Roque, R., Hiltunen, D. and Stoffels,
S.M., "Development and Validation of Performance Prediction Models and
Specifications for Asphalt Binders and Paving Mixes," SHRP Report A-357,
SHRP/NRC, Washington DC, USA, 1993.
9 Williams, M.L., Landel, R.F. and Ferry, J.D., "The Temperature Dependence of
Relaxation Mechanism in Amorphous Polymers and other Glass Forming Liquids,"
Journal of ACS, Volume 77, pp 3701, 1955.
21
10 Sayegh, G., “ Viscoelastic Properties of Bituminous Mixtures”, Proceedings of
the 2nd International Conference on the Structural Design of Asphalt Pavements, Ann
Arbor, MI, USA, University of Michigan, pp. 743-755, Ann Arbor, MI, USA, (1967).
11 Pellinen, T.K., and Witczak M.W., “Stress Dependent Master Curve
Construction for Dynamic (Complex) Modulus” Annual Meeting Association of
Asphalt Paving Technologists, Colorado Springs, Colorado, USA, March 2002.
12 Bosch, A., “Material Characterisation of Mastic Asphalt Surfacings on
Orthotropic Steel Bridges,” M.Sc. Thesis, Delft University of Technology, the
Netherlands, 2001.