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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].