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.