REVIVAL OF THE HUET-SAYEGH RESPONSE MODEL

Notes on the Huet - Sayegh Rheological Model

REVIVAL OF THE HUET-SAYEGH RESPONSE MODEL

Notes on the Huet - Sayegh Rheological Model

REVIVAL OF THE HUET-SAYEGH RESPONSE MODEL

Notes on the Huet - Sayegh Rheological Model

Abstract A short introduction is given of the Huet & Sayegh rheological model and the

application on (the results for) three very different Dutch asphalt mixes. In contrast with the

more known Burger’s model the H&S model fits the master curve for an asphalt mix over a

large range of frequencies and temperatures perfectly. The only disadvantage is the lack of an

element representing the permanent deformation characteristics of asphalt.

Contents

1. Introduction

2. Huet & Sayegh (H&S) Model

2.1 The variable dashpot

2.2 The general H&S model

3. Application of the H&S Model (Examples)

3.1 Porous Asphalt (ZOAB; very open graded mix)

3.2 “Guss Asphalt” (penetration asphalt mix for dykes)

3.3 Gravel Asphalt Concrete (GAB)

4 Short Comparison between the H&S, Burger’s and Zener model

5. Conclusions and Recommendations

6. References

Annex

DWW-2003-29 Author: A.C. Pronk

Date: March 2003

Disclaimer

This working paper is issued to give those interested an opportunity to acquaint themselves with progress in this particular field

of research. It must be stressed that the opinions expressed in this working paper do not necessarily reflect the official point of

view or the policy of the director-general of the Rijkswaterstaat. The information given in this working paper should therefore be

treated with caution in case the conclusions are revised in the course of further research or in some other way. The Kingdom of

the Netherlands takes no responsibility for any losses incurred as a result of using the information contained in this working

paper.

REVIVAL OF THE HUET-SAYEGH RESPONSE MODEL

NOTES ON THE HUET & SAYEGH RHEOLOGICAL MODEL

1. Introduction

Today the Burger's model is one of the traditional models for the characterization of the

rheological behaviour of bituminous mixes. However, at a chosen temperature this model

describes the response on a loading quite well but only for a limited range of frequencies. If the

frequency of a sinusoidal load is changed too much the figures for the elements in the Burger's

model have to be changed too. In the past another model has been developed by C. Huet (1),

which is valid over a wide range of frequencies. However the formulation of this response

model (Huet & Sayegh) is rather complex.

2. Huet & Sayegh (H&S) Model

2.1 The Variable Dashpot

The H&S model (see cover) looks like a Zener model but instead of one linear dashpot (Zener

model) it has two variable dashpots. The mathematical operation for the variable dashpot is only

defined for sinusoidal signals as:

t iat ia1at iei.ee

[1]

Regarding equation [1] the variable dashpot can be seen as a rheological element between the

linear spring (a=0) and the linear dashpot (a=1). See also figure 1.

Figure 1. The response a linear spring (purple), a linear dashpot (red), and a variable

dashpot (blue) on a sinusoidal load.

S.sin()

Variable Dashpot

Linear Dashpot

Linear Spring S.cos()

= /2

= a./2

2.2 The general H&S model

For sinusoidal signals (ei ) the response equation of a general “H&S” model will be

(Annex):

) i ( .+ ) i ( . + 1

E -E + E= ) ( E

h -

22

k -

11

oo

[2]

with the following explanations:

i+ 0 2

i.sin2

cos= e= i 0 ;> k> h> 1

[Pa]; }S{= E ;[Pa] 0}S{= E

][parameters Model EE.

;[s] constantsTime = ;]s/rad[frequency=

2

.i +

o

2,1

02,12,1

2,1

[3]

Because this general model has already 6 parameters C. Huet [1] decreased the number by

taking only one time constant and one model constant [4]

The response S{} can be rewritten as: i.BA.BA

EEE}{S

22

00

[5]

with:

) . (

) 2

. h ( sin

+ ) . (

) 2

. k ( sin

. + 0 = B; ) . (

) 2

. h ( cos

+ ) . (

) 2

. k ( cos

. + 1= Ahkhk

[6]

Based on these formulas an Excel file has been made for a regression analysis. For this analysis

measurements (4PB tests) are needed at different frequencies and different temperatures. In

contrast with the Burger’s model the temperature influence can be included quite easily by

adopting only an influence on the time decay constant . Moreover the relationship between the

time decay constant and the temperature is very simple and given in equation [7].

2111 T.CT.BA

e

[7]

In most cases the coefficient C1 can be taken equal to zero. Two approaches are developed. In

the first one an integrated regression is made by using equation [7] directly for all measurements.

In the second approach a time decay constant is calculated for each temperature separately.

Afterwards these constants are fitted with the aid of equation [7]. In most cases it don’t make

much difference. In the regression analysis some restrictions are used. The most important one is

the restriction that the static modulus E0 should be larger or equal than 1 MPa. It turns out that if

the measured curve doesn’t have data at low frequencies and/or high temperatures the regression

analysis (the solver option in Excel) might lead to zero or negative E0 values.

3. Application of the H&S model (Examples)

The complex stiffness modulus of the original H&S model is given in the equation below:

) i (+ ) i ( + 1

E -E+ E= ) ( S

h -k -

oo

[8]

There are six independent variables. In contrast with the Burger’s model, only the time constant

has to be a function of the temperature for a master curve. In the Burger’s model all four

parameters change if the temperature changes. In the next sub paragraphs some results will be

shown for actual Dutch asphalt mixes. The relationship, which is adopted for the dependency of

the time decay t with the temperature T, has the following form:

0) to equal takenbe can C Often C; in T (with e0T.CT.BA

2 [9]

3.1 Porous Asphalt (ZOAB; very open graded mix)

The stiffness measurements are carried out with the aid of four point bending tests in controlled

strain mode (with the “old” clamping device). In the following figures the fitting is shown with

the H&S model

Figure 2. A measured master curve for a Porous Asphalt mix (ZOAB) and the fitted

H&S model

0

500

1000

1500

2000

2500

0 5000 10000 15000

|S|.Sin() [MPa]

|S|.

Co

s(

) [

MP

a]

Measurements Model

Figure 3. Measured stiffness moduli at 7 temperatures as a function of the applied

frequency and the regression fit by the H&S model

Figure 4. The by regression calculated time decay constants for 7 temperatures and

the log-linear regression fit.

0

2000

4000

6000

8000

10000

12000

14000

0 10 20 30 40 50 60Frequency [Hz]

Sm

ix [

Mp

a]

= 0.0066e-0.2539.T

R2 = 0.9911

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

-20 -10 0 10 20 30Temperature T [

oC]

Tim

e d

ecay

[s]

Figure 5. Calculated stiffness values by the H&S model as a function of the measured

stiffness values for the porous asphalt mix.

Figure 6. Calculated phase lags by the H&S model as a function of the measured

phase lags for the porous asphalt mix.

y = 1.00546x

R2 = 0.997

0

2000

4000

6000

8000

10000

12000

14000

0 5000 10000 15000

Smix Measured [Mpa]

Sm

ix C

alc

ula

ted

[M

pa]

y = 1.0114x

R2 = 0.991

0

10

20

30

40

0 10 20 30 40 50

Measured [o]

C

alc

ula

ted

[o

]

Figure 7. Calculated real parts of the stiffness values by the H&S model as a function

of the measured real parts of the stiffness values for the porous asphalt mix.

Figure 8. Calculated imaginary parts of the stiffness values by the H&S model as a

function of the measured imaginary parts of the stiffness values for the

porous asphalt mix.

y = 1.0039x

R2 = 0.888

800

1000

1200

1400

1600

1800

2000

2200

800 1000 1200 1400 1600 1800 2000 2200

Simag Measured [Mpa]

Sim

ag*

Calc

ula

ted

[o

]

y = 1.00550x

R2 = 0.997

0

2000

4000

6000

8000

10000

12000

14000

0 2000 4000 6000 8000 10000 12000 14000

Sreal [Mpa]

Sre

al*

Calc

ula

ted

[M

pa]

3.2 “Güss Asphalt” (penetration asphalt mix for dykes)

“Güss Asphalt” or in Dutch “Gietasfalt” is an asphalt mix used for (penetrating) sea dykes.

In figures 9 to 11 the results are given of the regression analysis. As can be seen in these

figures a perfect comparison is obtained between model and measurements.

Figure 9. Calculated master curve by the H&S model and the measured master

curve for a “Gietasfalt” mix.

Figure 10. Calculated stiffness values by the H&S model as a function of the

measured stiffness values for a “Gietasphalt” mix.

Huet & Sayegh model

0

500

1000

1500

2000

2500

3000

3500

4000

0 5000 10000 15000 20000 25000 30000

Smix real [M Pa]

Sm

ix im

ag

ina

ir [

M P

a]

y = 0.996x

R2 = 0.997

0

5000

10000

15000

20000

25000

0 5000 10000 15000 20000 25000 30000

Smix measured [M Pa]

Sm

ix c

alc

ula

ted

[M

Pa

]

Figure 11. Calculated phase lags by the H&S model as a function of the measured

phase lags for a “Gietasphalt” mix.

3.3 Gravel Asphalt Concrete (GAB)

This asphalt mix was used in trial sections of the LINTRACK (2). The stiffness modulus is

measured in a four point bending test (4PB) in controlled strain mode. The values are taken at

cycle 22.

Figure 12. Calculated master curve by the H&S model and the measured master

curve for Dutch Gravel Asphalt Concrete (GAB).

y = 0.998x

R2 = 0.990

0

10

20

30

40

50

0 5 10 15 20 25 30 35

measured [o]

c

alc

ula

ted

[o

]

0

500

1000

1500

2000

2500

3000

3500

0 5000 10000 15000 20000 25000 30000

Sreal [MPa]

Sim

ag

[M

Pa

]

Figure 13. Measured stiffness moduli at 5 temperatures as a function of the applied

frequency and the regression fit by the H&S model

Figure 14. The by regression calculated time decay constants for 5 temperatures and

the log-linear regression fit.

0

5000

10000

15000

20000

25000

30000

0 5 10 15 20 25 30Frequency [Hz]

Sm

ix [

Mp

a]

= 0.0066e-0.2539.T

R2 = 0.991

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

-20 -10 0 10 20 30 40Temperature T [

oC]

Tim

e d

ec

ay

[

s]

Figure 15. Calculated stiffness values by the H&S model as a function of the measured

stiffness values for the gravel asphalt concrete.

Figure 16. Calculated phase lags by the H&S model as a function of the measured

phase lags for the gravel asphalt concrete.

y = 1.00546x

R2 = 0.997

-1000

1000

3000

5000

7000

9000

11000

13000

15000

0 5000 10000 15000

Smix Measured [Mpa]

Sm

ix C

alc

ula

ted

[M

pa

]

y = 1.0114x

R2 = 0.991

0

10

20

30

40

50

0 10 20 30 40 50

Measured [o]

C

alc

ula

ted

[o

]

Figure 17. Calculated real parts of the stiffness values by the H&S model as a function

of the measured real parts of the stiffness values for the gravel asphalt

concrete.

Figure 18. Calculated imaginary parts of the stiffness values by the H&S model as a

function of the measured imaginary parts of the stiffness values for the

gravel asphalt concrete.

y = 1.00550x

R2 = 0.997

0

5000

10000

15000

0 2000 4000 6000 8000 10000 12000 14000

Sreal [Mpa]

Sre

al*

Ca

lcu

late

d [

Mp

a]

y = 1.0039x

R2 = 0.888

800

1000

1200

1400

1600

1800

2000

2200

800 1000 1200 1400 1600 1800 2000 2200

Simag Measured [Mpa]

Sim

ag*

Ca

lcu

late

d [

o]

4. Short Comparison between the H&S, Burger’s and Zener model

First of all a short description of the Zener model will be given. The Zener model consists out

of a linear spring E1 in series with a circuit of a parallel linear spring E2 and a linear dashpot

2. In fact it is a Burger’s model without the serial linear dashpot 1. By a so called Z

transform the Zener model is equivalent to a circuit of a linear spring E1* parallel to a series of

a linear dashpot 2*and a linear spring E2

*. In this form it resembles a H&S model with one

linear dashpot. Both the Zener model and H&S model do not have a permanent deformation

element. The permanent deformation in the Burger’s model is represented by the serial

dashpot 1. However, the adoption of a variable dashpot in series with a linear dashpot can

simulate the observed evolution of permanent deformation in a creep test as shown in figure 19

(3).

It is just a simulation because in the calculation of the curve, the contribution of the first and

following loads diminish in time. This implicates that for t2>t1>t0 at time t2 the contribution

to the deformation by a load at time t1 is more than the contribution by a load at time t0.

If in figure 19 after load n=10000 no loads are applied the calculated response will eventual

drop to the real permanent deformation which is caused by the linear dashpot.

Figure 19. Simulation of creep behavior in a (cyclic) creep test (3)

Another big difference between the Zener and Burger’s model at one hand and the H&S model

at the other hand is the difference in the behavior of the master curves for and for .

In the Zener and Burger’s model the angle is 90o while in the H&S model these angles can be

different from each other and are usually not equal to 90o (1; figure 26)

"Permanent Deformation in cyclic creep test

0

50

100

150

200

250

300

350

400

450

500

0 2000 4000 6000 8000 10000

Number of pulses

Measu

red

defo

rmati

on

5. Conclusions and Recommendations

For the three Dutch asphalt mixes the H&S model is a perfect rheological model.

The H&S model calculates both the stiffness modulus and the phase lag very accurately.

The H&S model has five parameters per temperature in contrast with the four-parameter

model of Burger. However the far larger frequency range and better fit are of much

greater weight.

It is recommended to investigate if the H&S model can be used as a better prediction

model than the traditional Burger’s model. Because four of the five parameters do not

depend on the temperature, it should be possible to relate in an easy way these

parameters to physical quantities of the mix (e.g. air voids, pen., TR&B etc.). In the

Burger’s model all four parameters depend on the temperature, which increases also the

number of ‘constants’ (around 12) if the model should be applicable for different

temperatures.

The only disadvantage of the H&S model is the lack of a permanent deformation

element like the serial dashpot in the Burger’s model. However, by adopting an extra

serial variable dashpot a better simulation of a creep curve can be obtained than with the

viscous response of the Burger’s model (3).

6. References

1. Huet, C.,“Étude, par une méthode d’impédance, du comportement viscoélastique des

matériaux hydrocarbonés (Study of the viscoelastic behaviour of bituminous mixes by

method of impedance)’’ , Thesis, Faculté des sciences de l’université de Paris, Paris,

1965

2. Montauban, C.C., “Construction asphalt pavement GAB trial section (MEVA-3)”,

MAO-R-89005, 1988, RHED, Delft (in Dutch)

3. Pronk, A.C., “The Variable Dashpot”, DWW-2003-030, RHED, Delft, 2003.

ANNEX

In this annex a popular description will be given of the operator a when this operator is used

directly in the time domain for sinusoidal signals from t = - to t = + Also a short

description will be given of the H-S model response on sinusoidal signals. The operator a can

be regarded as a kind of differential operator of the order a with 0 < a < 1. In fact a

differentiation of an order a, in which a has a (positive) non-integer value, has no physical and

mathematical meaning. Nevertheless it is allowed to define an arbitrarily operator for certain

signals.

The “differential” equation for the variable dashpot (mark that the parameter has the viscosity

dimension [Pa.s]) is:

= . .d

d t

a-1

a

a

[A1]

The mathematical manipulation in equation [A1] is only allowed and defined for sinusoidal

signals. This implicate that the load/stress signal has to be a sinusoidal signal too:

= .eoi. .t

t to t from [A2]

The response for a linear dashpot (a = 1) on this stress signal is:

e. )+.ti.(0

2.i

000 ei ;

2 and

...

. with

[A3]

In case of a variable dashpot:

.)..(i.1

.= .).(i..= aa1-a

.e... ora.

2.i

a

[A4]

The beauty of equation [A4] is that it describes the response of a rheological element

changing from a linear spring (a=0) with modulus E = / and an argument = 0 to a linear

dashpot (a=1) with a modulus of E = (.)./ = . and an argument = + /2 .

In the intermediate range the modulus E = (.)a./ and the argument = + (/2).a

H&S Model

The response (transfer function) for the H&S model is given in equation [A5] in which the

spring in line with the two variable dashpots is denoted by EL.

= . E + E .1

1 + E .

.(i. . ) +

E .

.(i. . )

o LL

1

k

L

2

h

[A5]

For the response equals E0 and for Huet denoted the response by E .

To obtain this answer the spring EL has to be equal to EL = E - E0. For the response of the

original H&S model C. Huet also adopted a relationship between 2, , 0, and , which

reduces the amount of independent variables. Equation [A5] can be written as:

= . E + ( E - E ).1

1 + .(i. . ) + (i. . )o o -k -h

[A6]

The relationship between 2, , 0, and and the definition of the parameter are given by

equation [A7]

2 o = ( E - E ). ; 1

o =

( E - E ).

[A7]

In a more general model one can have two and two parameters (equation [A8]).

1

o 1

1

= ( E - E ).

; 2

o 2

2

= ( E - E ).

[A8]

The H&S model is thus a simplification of a more general model. However, it seems logical

to adopt only one parameter (“time decay constant”). The differential equation for this

model with two linear dashpots is given in equation [A9].

1

E.d

dt+(

1+

1). =

E

E+1 .

d

dt+ E .

1+

1.

L 1 2

o

L

o

1 2

[A9]

The solution of the homogenous differential equation ( = 0) gives only one time decay

constant (equation [A10]):

=

E + E

E .E.

.

+ ; E + E = E

o L

L o

1 2

1 2

o L

[A10]

If the operator a is used as a “normal” differential operator equation [A11] will be obtained

for the homogenous “differential” equation of the general H&S model:

1+E

E.d

dt +

E

..

d

d t +

E

..

d

d t = 0

o

L

o

1 1k-1

1-k

1-k

o

2 2h-1

1-h

1-h

[A11]

Notice that for a “positive” exponent of the differentiation both parameters h and k have to be

less than 1. The solution of this type of differential equations can be obtained by filling in

equation [A12]. Mark the minus sign in the exponent.

= .eo

t-

[A12]

0

1.

.

E1.

.

E1.

E

E1

h1

h1

1h22

0

k1

k1

1k11

0

L

0

[A13]

It seems logical to take equal to 1 and 2. In that case equation A[13] becomes similar to

equation [A9]. In order to reduce the number of independent variables C. Huet adopted a

relationship for the parameters of the second dashpot [A7].