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The linear viscoelastic (LVE) properties of asphalt concrete is investigated in this paper using a controlled-strain triaxial dynamic modulus test over wide frequency, temperature, and confining pressure ranges. The time–temperature-pressure superposition principle (TTPSP) is applied to validate the thermo-piezo-rheological simplicity of the tested materials using triaxial master curves. The LVE response is found highly stress-dependent at intermediate and high temperatures. The Prony series modeling of time-domain properties ascertains that confining pressure strongly correlates with long-term relaxation modulus, the absolute maximum slope of the relaxation modulus, and viscoelastic damage parameter. The stress triaxiality ratio concept is applied, and a new shift model is proposed that takes the triaxiality ratio as an internal state variable in the TTPSP. The model prediction agrees well with the experimental data. Moreover, a relationship between the long-term relaxation modulus and the triaxiality ratio is established. The triaxiality ratio coupled with TTPSP can accurately describe the stress-dependent response of asphalt concrete in the LVE domain.
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Construction and Building Materials 329 (2022) 127106
Available online 16 March 2022
0950-0618/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Thermo-piezo-rheological characterization of asphalt concrete
Mequanent Mulugeta Alamnie
a
,
*
, Ephrem Taddesse
a
, Inge Hoff
b
a
Department of Engineering Science, University of Agder, 4879 Grimstad, Norway
b
Department of Civil and Environmental Engineering, Norwegian University of Science and Technology (NTNU), Høgskoleringen 7a, Trondheim, Norway
ARTICLE INFO
Keywords:
Thermo-piezo-rheology
Linear viscoelastic
Triaxial dynamic modulus
Relaxation modulus
Triaxiality ratio
Asphalt concrete
ABSTRACT
The linear viscoelastic (LVE) properties of asphalt concrete is investigated in this paper using a controlled-strain
triaxial dynamic modulus test over wide frequency, temperature, and conning pressure ranges. The time-
temperature-pressure superposition principle (TTPSP) is applied to validate the thermo-piezo-rheological
simplicity of the tested materials using triaxial master curves. The LVE response is found highly stress-
dependent at intermediate and high temperatures. The Prony series modeling of time-domain properties as-
certains that conning pressure strongly correlates with long-term relaxation modulus, the absolute maximum
slope of the relaxation modulus, and viscoelastic damage parameter. The stress triaxiality ratio concept is
applied, and a new shift model is proposed that takes the triaxiality ratio as an internal state variable in the
TTPSP. The model prediction agrees well with the experimental data. Moreover, a relationship between the long-
term relaxation modulus and the triaxiality ratio is established. The triaxiality ratio coupled with TTPSP can
accurately describe the stress-dependent response of asphalt concrete in the LVE domain.
1. Introduction
Asphalt concrete is a composite, time-dependent material that ex-
hibits elements of elastic, viscous, and viscoelastic properties. The
response of such materials is dependent on loading frequencies and a set
of thermodynamic variables. As a fundamental thermodynamic vari-
able, temperature and pressure signicantly inuence the viscoelastic
and viscoplastic responses of time-dependent materials. The effect of
time (frequency) and temperature is characterized using a joint
parameter called reduced time (or reduced frequency) for a thermo-
rheological simple material. Similarly, a time-pressure shift factor is
used to analyze the joint effect of time and pressure for the piezo-
rheological simple material. Several researchers have validated that
the thermo-rheological simplicity (timetemperature response) of
different asphalt concrete mixtures and the applicability of time-
temperature superposition principle (TTSP) in both undamaged and
damaged states [3,20,29,26,14]. The validity of TTSP in undamaged and
damaged states yields a signicant material saving for the test [5]. The
combined effect of the two fundamental thermodynamic variables
(temperature and pressure) on the viscoelastic response is described
using the Time-Temperature-Pressure superposition principle (TTPSP).
A material that satises the TTPSP principle is called a thermopiezo-
rheological simple material [23,6]. The role of conning pressure on time-
dependent materials has also been studied for several decades, such as
for polymers [8,12]. As a three-phase material, asphalt concrete showed
strong stress-dependent properties. Most studies on the triaxial stress
response of asphalt concrete were focused on the viscoplastic properties
[2,1,19,24]. Some studies such as Yun et al.[26] and Rahmani et al.[18]
have investigated the role of connement on the applicability of TTSP
with growing damage and the effect of conning pressure on Schaperys
nonlinear viscoelastic parameters, respectively. Other studies
[28,29,21] have investigated the effect of conning pressure on linear
viscoelastic (LVE) responses of asphalt concrete mixtures using triaxial
master curves. Previous research focused on proposing ‘verticalshift
models as a function of conning pressure and was mainly involved in
constructing the ‘triaxialmaster curves. The triaxial stress evolution in
the LVE range was not discussed in previous research. Furthermore,
most standards typically use uniaxial dynamic modulus tests for asphalt
concrete LVE properties, including the mechanistic-empirical pavement
analysis methods. However, the triaxial (conned) dynamic modulus
test is more realistic to simulate the in-situ condition and the stress-
dependent LVE properties should be investigated for accurate charac-
terization of asphalt concrete.
In this paper, strain-controlled triaxial dynamic modulus tests were
conducted at wide ranges of conning pressure (0 to 300 kPa), tem-
perature (-10 to 55 C), and frequency (20 to 0.1 Hz). A target on-
* Corresponding author.
E-mail addresses: mequanent.m.alamnie@uia.no (M. Mulugeta Alamnie), ephrem.taddesse@uia.no (E. Taddesse), inge.hoff@ntnu.no (I. Hoff).
Contents lists available at ScienceDirect
Construction and Building Materials
journal homepage: www.elsevier.com/locate/conbuildmat
https://doi.org/10.1016/j.conbuildmat.2022.127106
Received 5 August 2021; Received in revised form 31 January 2022; Accepted 8 March 2022
Construction and Building Materials 329 (2022) 127106
2
specimen strain magnitude of less than or equal to 50 micros is selected
to ensure that the deformation due to sinusoidal stress is within the LVE
domain. A test procedure was proposed in the experimental campaign,
and two different asphalt mixtures were tested. The main objective of
this research is to investigate the stress-dependent LVE properties of
asphalt concrete using the triaxial dynamic modulus test and develop a
simplied triaxial shift model. The objective is achieved; rst by veri-
fying the thermo-piezo-rheological simplicity of the tested materials (con-
structing triaxaial dynamic modulus master curves using existing and
new triaxial shift models), and second by investigating the stress-
dependent time-domain LVE properties using the Prony method.
Furthermore, the role of connement was also explored on the
maximum slope of relaxation modulus and the viscoelastic damage
parameter. Finally, the triaxiality ratio concept is introduced to analyze
the triaxial (3D) stress-state on the LVE responses of asphalt concrete. A
new triaxial shifting model is proposed using the triaxiality ratio as an
internal state variable and validated using experimental data. Moreover,
a simplied model is established between the long-term relaxation
modulus and the triaxiality ratio to explain the stress-dependent visco-
elastic behaviors of asphalt concrete. Unless otherwise stated, the term
pressure in this paper refers to conning pressure (stress).
2. Viscoelasticity
The uniaxial stressstrain constitutive relationship for linear visco-
elastic (LVE) material can be expressed in a Boltzmann superposition
integral form in the time domain.
σ
(t) =
t
0
E(t
τ
)d
ε
d
τ
d
τ
(1)
Where
σ
and
ε
are stress and strain, respectively; t is physical time;
τ
is integral variable; E(t)is relaxation modulus. The one-dimensional
relaxation modulus, E(t)is commonly expressed using a generalized
Maxwell mechanical model (GM in parallel) with the Prony series.
E(t) = E+
M
m=1
Eme(− t/
ρ
m)(2)
Where Eis Long-term (equilibrium) modulus; Em is components of
the relaxation modulus;
ρ
m is components of relaxation time; and M is
the total number of the Maxwell elements (one Maxwell element is
composed of one elastic spring and one viscous dashpot connected in
series). For generalizations into 3D formulations, the deformations
within a material can be decoupled into shear and volumetric compo-
nents. The time-dependent stressstrain response of an isotropic LVE
material in 3D can thus be described in both deviatoric (G(t)) and bulk or
volumetric (K(t)) relaxation moduli, as follow.
G(t) = G+
M
m=1
Gme(t/
ρ
m,G)(3)
K(t) = K+
N
n=1
Kne(t/
ρ
n,K)(4)
Where G and K are Long term (equilibrium) shear and bulk
moduli,Gm,
ρ
m,G and Kn,
ρ
n,K are Prony coefcients of relaxation modulus
and time for shear and bulk, respectively. M and N are the number of
Prony coefcients for shear and bulk relaxation. It is generally assumed
as
ρ
m,G=
ρ
n,K=
ρ
m. For the small stress LVE test, the time-dependent
volumetric deformation of asphalt concrete is negligible. The reasons
include (i) the hydrostatic pressure is usually less than the materials
tensile strength that causes a linear elastic volumetric deformation, (ii) K
(t) is very high and viscous ow is assumed isochronous (linear ow).
Hence, the time-dependent volume change is much smaller than the
corresponding shear distortion on the same material [9] and a constant
Poisson ratio (
υ
) is often assumed for asphalt concrete.
K(t) = E(t)
3(12
υ
),
G(t) = E(t)
2(1+
υ
)
(5)
In theory, the relationships between the three moduli (K(t), G(t), and
E(t)) should be established using a time-dependent Poissons ratio
[27,17].
3. Materials and test method
3.1. Materials
In this study, two different asphalt concrete mixtures (AB11 and
SKA11) collected from asphalt concrete production plants were used.
The AB11 mixture is dense-graded asphalt concrete and SKA11 is a stone
mastic asphalt. Both mixes have an 11 mm nominal maximum aggregate
size (NMAS), where AB11 is a polymer-modied (PMB 65/10560)
while SKA11 is a 70/100 neat binder mixture. The gradation is given in
Table 1. Cylindrical samples were produced by re-heating the loose mix
at 150 C for up to 4 h and compacting using a gyratory compactor
according to the Superpave specication. The nal test specimens (Ø100
mm and 150 mm height) were fabricated by coring and cutting from the
Ø 150 mm and 180 mm height samples.
3.2. Test procedure
A triaxial dynamic modulus test was performed using a servo-
hydraulic universal testing machine (IPC UTM-130). Three sets of
loose core linear variable differential transducers (LVDTs) were moun-
ted on the specimen at 120apart radially with 70 mm gauge length. The
instrumented specimens were conditioned at a target temperature for at
least 2 h. A strain-controlled sinusoidal compressive load was applied
axially with a target on-specimen axial strain of 50 micros or less. The
test was conducted according to AASHTO T378 over a wide range of
temperatures (-10, 5, 21, 40, 55 C), pressures (0, 10, 100, 200, 300 kPa)
and frequencies (20, 10, 5, 2, 1, 0.5, 0.2, 0.1 Hz). Three specimen rep-
licates were tested at each temperature and pressure according to the
following steps.
(1). The instrumented sample is installed in the testing system at the
target temperature. A sinusoidal load is applied from high to low
frequencies without a rest period between sweeps. The average
strain of three LVDTs should not be more than the target 50
micros.
(2). Then, 10 kPa conning pressure is applied for about 15 min to
stabilize the deformation due to connement. Repeat step 1.
(3). Compare dynamic modulus and phase angle values from steps 1
(unconned or uniaxial) and step 2 (10 kPa conned). The dy-
namic modulus results should not vary by a signicant margin.
(4). Apply 100 kPa conning pressure (for 15 min). Repeat step 1.
(5). Apply 200 kPa conning pressure (for 15 min). Repeat step 1.
(6). Apply 300 kPa conning pressure (for 15 min). Repeat step 1.
Table 1
Aggregate gradation.
Sieve size [mm] AB11 [%] SKA11 [%]
16.0 100 100
11.2 95 91.2
8.0 70 53.6
4.0 48 35.7
2.0 36 21.7
0.25 15.5 12.8
0.063 10 8.4
Binder Content [%] 5.6 5.83
M. Mulugeta Alamnie et al.
Construction and Building Materials 329 (2022) 127106
3
4. Dynamic modulus test results
The response of an asphalt concrete material in a dynamic modulus
test under a continuous sinusoidal loading is expressed by a complex
modulus,E*. The three measured parameters (outputs) strain
ε
(t), stress
σ
(t), and dynamic modulus, E* are expressed as follows.
ε
(t) =
ε
osin(
ω
t)(6)
σ
(t) =
σ
osin(
ω
t+φ)(7)
E*=
σ
oei(
ω
t+φ)
ε
oei
ω
t= |E*|(cosφ +isinφ) = |E*|e(8)
|E*|is the norm of dynamic modulus, E= |E*|cosφ and E′′ = |E*|sinφ
are the storage and loss moduli, respectively,
σ
o and
ε
o are stress and
strain amplitudes, respectively. The phase angle, φ [0, 90] is calculated
from time lag (tl) in strain signal and loading period (tp) of the stress
signal,φ=360otl
tp. A material with a phase angle between 0(purely
elastic) and 90(purely viscous) is a viscoelastic material. The dynamic
modulus test results are the average of modulus on three specimens at
each temperature, conning pressure and frequency. The average
specimen-to-specimen variation for dynamic modulus value is generally
less than 10% at high temperatures (40 and 55
o
C). The variation is
overall less than 5% for lower temperatures (21, 5 and -10
o
C).
4.1. Effect of conning pressure on dynamic modulus
The effect of conning stress on the viscoelastic response of asphalt
concrete is investigated by conducting triaxial dynamic modulus tests.
The triaxial dynamic modulus test results of the AB11 mixture are pre-
sented in Fig. 1 (a-e). It is clearly seen that conning pressure has a
signicant role on the viscoelastic response of asphalt concrete at in-
termediate (21 C) and high temperatures (40 C and 55 C) but mar-
ginal or no effect at lower temperatures (such as 5 C). For example, at 1
Hz frequency, the dynamic modulus at 300 kPa is 1.5 times (at 21 C),
2.5 times (at 40 C), and 4.4 times (at 55 C) that of the uniaxial dynamic
modulus. Moreover, the effect of connement at 40 C has some irreg-
ular patterns as compared to the dynamic moduli at 21 and 55 C. The
cause of such variations can be due to the increase in binder viscosity at
high temperatures and low frequencies, resulting in a higher phase
angle. However, the measured phase angles at higher temperatures are
dictated not only by the binder but also by aggregate interactions. The
binder becomes soft at high temperatures and low frequencies and the
elastic aggregate structure dominates the mixture behavior, which is
reected by the reduction of phase angle. The role of connement in
such conditions is retarding the binder ow, and aggregate-to-aggregate
contact is reduced. The other reason can be the transient effects during
the dynamic modulus test [10], in addition to the microstructural
change between 30 and 55 C.
4.2. Isobaric master curves
The dynamic modulus test data are shifted horizontally (along the
logarithmic frequency axis) to construct a master curve at a reference
temperature using a sigmoidal function [16].
0 5 10 15 20
Frequency (Hz)
1
1.5
2
2.5
Dynamic Modulus (MPa)
10
4
(a) -10
o
C
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
0 5 10 15 20
Frequency (Hz)
0.4
0.6
0.8
1
1.2
1.4
1.6
Dynamic Modulus (MPa)
10
4
(b) 5
o
C
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
0 5 10 15 20
Frequency (Hz)
0
1000
2000
3000
4000
5000
6000
7000
Dynamic Modulus (MPa)
(c) 21
o
C
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
0 5 10 15 20
Frequency (Hz)
0
1000
2000
3000
Dynamic Modulus (MPa)
(d) 40
o
C
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
0 5 10 15 20
Frequency (Hz)
0
500
1000
1500
2000
2500
Dynamic Modulus (MPa)
(e) 55
o
C
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
Fig. 1. Dynamic modulus at different temperatures and conning pressures - AB11 mixture.
M. Mulugeta Alamnie et al.
Construction and Building Materials 329 (2022) 127106
4
log(E*) = δ+(
α
δ)
1+exp(
η
γlogfR)(9)
Where E* is the dynamic modulus, δ and
α
are the minimum and
maximum logarithm of the dynamic modulus, respectively,
η
and γ are
shape factors and fR is reduced frequency.
fR=
α
T×f(10)
Where f is the frequency and aT is timetemperature shift factor. The
Williams, Landel, and Ferry (WLF) function [25] is widely used for aT.
log
α
T= − C1(TT0)
C2+TT0
(11)
Where C1 and C2 are WLF constants and T0 is reference temperature.
The isobaric master curves are constructed at 21 C reference temper-
ature using the sigmoid model and WLF shift function for the AB11
mixture, as shown in Fig. 2a. The isobaric master curves did not fall into
a single curve at high temperatures (and low frequencies). The shift
factors in Fig. 2b also showed a slight variation at different conning
pressure levels. Moreover, the effect of conning pressure can be seen
from the energy loss quantity during the dynamic modulus test. The total
energy in cyclic viscoelastic deformation has a dissipated (ΔW=
πσ
o
ε
osinφ) and stored (W=
σ
o
ε
o
2) energy components in J/m
3
per cycle
[22,9]. The ratio of dissipated energy to the maximum stored energy
(ΔW
W) is independent of stress and strain amplitudes, as shown in Eq. (12).
ΔW
W=2
π
sinφ(12)
As shown in Fig. 3, the maximum energy loss ratio is recorded
around 0.1 Hz frequency. At this point, conning pressure contributed
to reducing the dissipated energy (or phase angle) and conning pres-
sure has no signicant effect on energy loss at low temperatures and
high frequencies. Therefore, conning pressure retarded energy loss and
contributed to the elastic energy at high temperatures and low fre-
quencies. These observations verify that the LVE properties of asphalt
concrete are stress-dependent at intermediate and elevated tempera-
tures. Hence, a stress-dependent shift function is necessary to generate a
single, continuous master curve for LVE characterization in a triaxial
stress state.
4.3. Stress-dependent master curve
In the triaxial dynamic modulus test, the conning pressure causes
an increase of dynamic modulus. To construct a stress-dependent master
curve, the modulus at different frequencies, conning pressures and
temperatures are shifted both horizontally and vertically. The time-
temperature shift factor is superposed and modied to couple pressure
in the shifting function. Two models are suggested to construct and
compare stress-dependent or triaxial master curves. The rst model
(Model-1) is the modied WLF function proposed by Fillers, Moonan,
and Tschoegl (FMT) model [8], expressed as follows.
log
α
TP =C1(TT0Γ(p))
C2(P) + TT0Γ(p)(13)
Γ(p) = C3(P)ln 1+C4P
1+C4P0C5(P)ln 1+C6P
1+C6P0
(14)
Where
α
TP is timetemperature-pressure shift factor; P is the pressure
of interest; P0 is reference pressure;C1, C4 and C6 are constants;C2(p),
C3(p)and C5(p)are pressure-dependent parameters. In the FMT model,
the coefcients represent the thermal expansion of the relative free
volume and the pressure-dependent parameter Γ(p)accounts for the
compressibility attributed to the collapse of free volume [7]. The FMT
equation can be reduced to the WLF equation at P=Po=0, and
whenT=To, the FMT function becomes a pressure shift function. A
modied version of Model-1 is proposed in this paper. The pressure-
dependent coefcients C2(P)and C3(P)are approximated as linear
functions and the last component of Eq. (14) can be dropped.
C2(P) = C20 +C21P,
C3(P) = C30 +C31P(15)
Where C20,C21,C30 ,C31 are coefcients. The proposed modied FMT
model (Model-1) time-pressure or vertical shift factor takes the
following form.
Γ(p) = (C30 +C31P)ln 1+C4P
1+C4P0
(16)
The second triaxial shifting model (Model-2) is a sigmoid-type
Zhaos model [30], expressed as,
logλ =− (PPo)
exp[C3+C4log(fR) ] + C5(P+Pa)C6(17)
-4 -2 0 2 4 6
Log Reduced Frequency (Hz)
2
2.5
3
3.5
4
4.5
Log Dynamic Modulus (MPa)
(a) Isobaric Master Curves
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
-20 0 20 40
60
Temperature (
o
C)
-4
-2
0
2
4
6
Shift factor (log(aT))
(b) Shift factor
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
Fig. 2. Isobaric master curves and shift factors at 21
C reference temperature AB11 Mixture.
8
Reduced Frequency (Hz)
4
Energy Loss Ratio
Fig. 3. Energy loss ratio (specic loss) at different conning stress levels
AB11 Mixture.
M. Mulugeta Alamnie et al.
Construction and Building Materials 329 (2022) 127106
5
Where Pa is atmospheric pressure (101.3 kPa); C3C6 are regression
coefcients;P, Po are conning and reference pressures, respectively.
All stress-dependent master curves in the subsequent sections are
constructed at 21 C reference temperature and 100 kPa reference
conning pressure. A total of ten parameters (including four sigmoid
parameters) were optimized to construct the stress-dependent master
curves (see in Table 2 and Table 3).
In Fig. 4 a-b, the triaxial master curves are presented for AB11 and
SKA11 mixtures. Both Model 1 and Model-2 have good prediction ac-
curacy (R
2
=0.98 for model-1 and R
2
=0.95 for Model-2 for AB11) (as
shown in Fig. 5). For the SKA11 mixture, a very close prediction is
observed using both models (Fig. 4 b). It is seen that both models have
the advantage of simplicity while successfully shifting stress-dependent
master curves. Model 1 has a wider reduced frequency range and better
accuracy than Model2. In addition, Model 1 is derived based on the free
volume theory and has a sound physical and theoretical basis. On the
other hand, Model 2 is a mathematical sigmoid function with a char-
acteristic S-shaped. The vertical shift model is independently deter-
mined and added to the sigmoid model (i.e., the model is a two-step
process). Hence, Model-1 is favored and proposed for further analyses in
this paper with simplifaction. As shown in Fig. 6, the vertical shift factor
Γ(p)has an approximate linear function relationship with conning
pressure.
5. Time-domain viscoelastic properties
Although the frequency domain dynamic modulus can give sufcient
information about the viscoelastic properties of asphalt concrete, the
time domain modulus is often used for performance prediction. Inter-
conversion between frequency and time domain is performed using
storage and loss modulus data. Often conversions based on the storage
modulus data provide sufcient accuracy. But it is essential to evaluate
the smoothness of storage modulus data before conversion to time-
domain moduli. As shown in Fig. 3, the dissipated energy due to
phase angle introduces noise and inconsistency to the storage modulus
at high temperatures. Hence, a continuous sigmoidal function [13] in
Eq. (18) is used to smoothen and avoid discreteness, wave or noise in the
data. The error optimization function of min[log10 EfRgfR]is
used.
g(fR) = a1+a2
a2+a4
exp(a5+a6logfR)
(18)
Where a
1, 2,
,
6
are coefcients, fR is the reduced frequency. The
ltered storage modulus data (as shown in Fig. 7) is then utilized to
obtain relaxation modulus using the Prony method.
5.1. Relaxation modulus
The Prony function (Eq. (19)) [15] with error minimization objective
function, OF (Eq. (20)) is used to predict the relaxation modulus from
pre-smoothen storage modulus data.
E(
ω
) = E+
M
m=1
ω
2
ρ
2
mEm
1+
ω
2
ρ
2
m
(19)
OF =1
N
N
i=11|E*(
ω
i)|Predicted
|E*(
ω
i)|Measured2(20)
Where E is long-term relaxation modulus (Mpa), Em and
ρ
m are
Prony coefcients (relaxation modulus [MPa] and relaxation time [sec],
ω
is angular frequency, M is number of Prony coefcients of a general-
ized Maxwell model, and N is number of storage modulus data points.
The data presented in the subsequent sections are only for AB11
asphalt concrete. In Fig. 8, the isobaric and triaxial relaxation modulus
Table 2
Sigmoid and TTPS Shift model coefcients - AB11 Mixture.
Model Coefcients
α
β γ δ C
1
C
20
C
21
C
30
C
31
C
4
1 4.52 0.06 0.46 2.24 10.11 83.42 0.21 1.53 0.02 0.25
2 4.40 0.29 0.68 2.61 8.92 99.11 0.99 0.96 1.50 5.17
Table 3
Sigmoid and TTPS Shift model coefcients - SKA11 Mixture.
Model Coefcients
α
β γ δ C
1
C
2
C
3
C
4
C
5
C
6
1 4.15 0.81 0.83 1.75 7.03 54.44 0.04 0.15 2.5E-03 92.27
2 4.12 0.79 0.85 1.77 6.85 54.5 2.0 2.0 2.2 4.9
10
-6
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
Reduced Frequency (Hz)
2
2.5
3
3.5
4
4.5
Log Dynamic Modulus (MPa)
(a) AB11
Measured (shifted by Model 1)
Model 1
Measured (Shifted by Model 2)
Model 2
10
-6
10
-4
10
-2
10
0
10
2
10
4
10
6
Reduced Frequency (Hz)
1.5
2
2.5
3
3.5
4
4.5
Log Dynamic Modulus (MPa)
(b) SKA11
Measured (shifted by Model 1)
Model 1
Measured (shifted by Model 2)
Model 2
Fig. 4. Stress-dependent Master Curves at 21
C and 100 kPa for AB11 and
SKA11 Mixtures.
M. Mulugeta Alamnie et al.
Construction and Building Materials 329 (2022) 127106
6
master curves using Model-1 are shown. It is seen that the effect of
conning pressure is signicant on the long-term side of the relaxation
modulus. The Prony terms of a generalized Maxwell model presented in
Fig. 9 and Table 4 showed a normal (Gaussian) distribution of relaxation
moduli of the Prony coefcients. A total of twelve Prony coefcients
were used. The coefcients (Ei) at long-time are almost the same and can
be taken as independent of conning stress.
Furthermore, the bulk (volumetric) and shear moduli should be
considered in the triaxial stress analysis for granular materials like
asphalt concrete. From Eq. (5), it can be observed that the bulk relaxa-
tion modulus is more sensitive to change in Poisson ratio than the cor-
responding shear modulus. For incompressible materials (when
υ
=
0.5), the bulk relaxation spectra (Km) are zero or the bulk modulus K(t)
is innite. To illustrate the relationship between the three moduli (E, G
and K), a parametric study at different constant Poisson ratios is shown
in Fig. 10. For example, as Poisson ratio increases from 0.3 to 0.35, the
shear modulus increases by 3.8%, and the bulk modulus is reduced by
2 2.5 3 3.5 4 4.5 5
Measured (Log Dynamic Modulus)
2
2.5
3
3.5
4
4.5
5
Predicted (Log Dynamic Modulus)
(a) AB11
Model 1
Model 2
LOE
1234
5
Measured (Log Dynamic Modulus)
1
2
3
4
5
Predicted (Log Dynamic Modulus)
(b) SKA11
Model 1
Model 2
LOE
Fig. 5. Accuracy of Stress-dependent master curve predicting models for AB11 and SKA11 mixtures.
0 50 100 150 200 250 300
Confining Pressure (kPa)
-5
0
5
10
Vertical Shift factor, Г(p)
(a) AB11
Г(p) = 0.04*P - 3.98
R
2
= 0.999
50 100 150
200
Confining Pressure (kPa)
-0.4
-0.2
0
0.2
0.4
Vertical Shift factor, Г(p)
(b) SKA11
Г(p)= 0.00452*P - 0.449
R
2
= 1
Fig. 6. Vertical shift factor versus conning Pressure (Model 1) for AB11 and SKA11 mixtures.
10
-5
10
-3
10
0
10
3
10
5
Reduced Frequency (Hz)
2
2.5
3
3.5
4
4.5
Log storage Modulus (MPa)
(a) AB11
Measured
Smoothened
10
-4
10
-2
10
0
10
2
10
4
Reduced Frequency (Hz)
1.5
2
2.5
3
3.5
4
4.5
Log storage Modulus (MPa)
(b) SKA11
Measured
Smoothened
Fig. 7. Pre-smoothened Storage Modulus master curves.
10-8 10-6 10-4 10-2 10010210
4
Reduced Time (sec)
102
103
104
105
Relaxation Modulus (MPa)
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
Triaxial (MC)
Fig. 8. Isobaric and triaxial Relaxation Modulus Master Curves
AB11 Mixture.
M. Mulugeta Alamnie et al.
Construction and Building Materials 329 (2022) 127106
7
25%. Similarly, if the Poisson ratio increases from 0.35 to 0.4, the bulk
modulus showed a 33% reduction and the shear component increased by
3.7%. As discussed previously, the applied conning pressure is ex-
pected to be lower than the minimum tensile strength of asphalt con-
crete. Thus, deformation caused by conning stress is assumed an elastic
strain. Moreover, as temperature increases, tensile strength decreases
contrary to the increment of the Poisson ratio. Therefore, the bulk
modulus will ultimately be reduced and vanish due to high viscosity at
υ
=0.5 (incompressible).
5.2. The effect of conning pressure on viscoelastic damage parameter
For a viscoelastic material, the absolute maximum slope of the
relaxation modulus curve is found conning stress-dependent. The
maximum slope of the Log-Log relaxation modulus E(t) time (t) curve
is computed using the following expression.
So=d[logE(t)]
dt =M
m=1Em×et
ρ
m
E+M
m=1Em×et
ρ
m(21)
The maximum slope (So) is a crucial parameter for the viscoelastic
damage prediction of asphalt concrete. The viscoelastic continuum
damage parameter (
α
) is described as
α
=1
So for control-stress and
α
=1
So+1 for control-strain fatigue damage modes [4,11]. As shown in
Fig. 11a, the absolute maximum slope is reduced as conning pressure
increases and due to the increment of relaxation modulus at an innite
time (or when frequency approaches zero,
ω
0) on the high-
temperature side. On the other hand, the damage parameter (
α
) in-
creases as conning pressure increases much faster (5.4 times) than the
slope (So) reduction rate (Fig. 11b). Conventionally, fatigue damage
tests are uniaxial and ignored the role of triaxial (3D) stress conditions. It
is revealed that the triaxial stress-state affects the damage evolution
parameter and subsequently impacts the fatigue life prediction of
asphalt concrete. Further research is underway on this topic by the au-
thors of this paper. The role of connement on the permanent defor-
mation evolution is well understood and incorporated in the strain
hardening phenomenon of asphalt concrete materials.
5.3. Effect of conning Pressure on the Long-term relaxation modulus
The long-term relaxation modulus (E) is a modulus at a very long
time (t→∞) or when the frequency approaches zero (
ω
0). After
removing the applied load, the viscoelastic material gradually recovers
its deformation, and full recovery is possible given sufcient time. The
long-term relaxation modulus is the modulus that governs the stress
relaxation of a material in the long-time limit. The relaxation modulus is
a crucial quantity for performance prediction. From the sigmoid storage
modulus [log(E) = δ+(
α
δ)
1+exp(
η
γlogfR)], the long-term relaxation modulus
is the minimum value as frequency closes to zero (i.e., δ or E10δ).
Different models have been proposed to correlate E with conning
stress [16,28]. As shown in Fig. 9 and Fig. 12, the long-term relaxation
modulus is dependent on conning stress and a linear relation can be
seen between the long-term relaxation modulus and conning pressure.
There is a relatively weak correlation between the long-term relax-
ation modulus and the absolute maximum slope of the relaxation
modulus curve and the damage parameter, as shown in Fig. 13.
5.4. Triaxiality ratio in linear viscoelastic response
In this paper, the stress ratio parameter (triaxiality ratio) is intro-
duced to investigate the linear viscoelastic (LVE) property of asphalt
concrete. The triaxiality ratio (
η
) is dened as the ratio of hydrostatic
pressure (mean stress,
σ
m) and the von Mises equivalent stress (
σ
vm) or
the ratio of the rst stress invariant (I1) to the second deviatoric stress
invariant (J2). It is argued that the stress ratio factor is essential to
describe the stress-dependent LVE response of asphalt concrete.
η
=
σ
m
σ
vm =I1/3

3J2
(22)
Where
η
is triaxiality ratio;
σ
m=
σ
1+
σ
2+
σ
3
3;
σ
vm=
(
σ
1
σ
2)2+(
σ
1
σ
3)2+(
σ
2
σ
3)2
2
;
σ
1,
σ
2,
σ
3 are principal stresses. Simplifying Eq. (22) for triaxial condition,
i.e.,
σ
1=
σ
d+
σ
c and
σ
2=
σ
3=
σ
c (
σ
c is denotes conning pressure and
σ
d is
peak deviatoric stress) takes the following form.
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Relaxation Time (sec)
0
1000
2000
3000
4000
5000
6000
Relaxation Modulus, Ei (MPa)
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
Fig. 9. Prony Coefcient Relaxation Moduli (Gaussian type distribution)
AB11 Mixture.
Table 4
Prony Coefcients of relaxation Spectrum.
Conning Pressure [kPa]
m
ρ
m [sec] Uniaxial 10 100 200 300
1. 1.0E-08 49.93 50.00 148.88 99.91 99.90
2. 1.0E-07 1000.00 128.85 500.00 313.05 1000.00
3. 1.0E-06 3010.97 732.38 1210.12 1368.52 2746.72
4. 1.0E-05 4531.55 2025.15 2820.49 2962.72 4417.43
5. 1.0E-04 5566.01 4315.72 5173.00 5792.28 5628.03
6. 1.0E-03 4439.48 5359.20 5010.70 5268.25 4292.64
7. 1.0E-02 1956.16 2241.81 2027.36 1954.50 1896.97
8. 1.0E-01 708.32 607.34 638.67 621.62 786.14
9. 1.0E +00 224.02 154.84 188.90 183.54 294.38
10. 1.0E +01 83.97 48.50 64.21 60.77 120.83
11. 1.0E +02 33.15 15.41 22.31 19.59 49.20
12. 1.0E +03 18.12 0.01 1.79 9.94 18.09
E[MPa] 149.26 344.35 533.57 921.67 989.65
10-8 10-6 10-4 10-2 100102
Relaxation Time (sec)
0
1000
2000
3000
4000
5000
6000
Prony Coefficients
Em, Gm, Km (MPa)
E(t)
G(t) - υ=0.3
K(t) - υ=0.3
G(t) - υ=0.35
K(t) - υ=0.35
G(t) - υ=0.4
G(t) - υ=0.4
Fig. 10. Parametric study relaxation, shear, and bulk Moduli at different
Poisson ratios.
M. Mulugeta Alamnie et al.
Construction and Building Materials 329 (2022) 127106
8
η
=
σ
c
σ
d+1
3(23)
The triaxiality ratio for the uniaxial or unconned (
σ
c=0)condition
is 1/3. The ratio increases as the connement level dominate the Mises
equivalent stress. Theoretically, the maximum ratio (i.e.,
η
=)is ob-
tained when deviatoric stress is minimum or at very high hydrostatic
stress.
The peak deviatoric stress is obtained from the frequency sweep
dynamic modulus test. Fig. 14 shows the peak deviatoric stress in a
strain-controlled triaxial dynamic modulus test. The stress decreases as
temperature increases and frequency reduces. That means maximum
deviatoric stress is exerted at low temperature and high frequency to
maintain the target strain limit (50 micros). As can be seen in the gure,
the peak deviatoric stress does not vary much with conning pressure at
a particular test temperature. This conrms the consistency of applied
deviatoric stress regardless of the different volumetric stresses.
However, the slight variations observed (e.g., at 21 C) could be due to
transient effects during the sinusoidal test or due to lateral pressures. As
shown in Fig. 15a-b, the triaxiality ratios are computed at each tem-
perature and conning stress. It is clearly seen that
η
is both pressure and
temperature-dependent, and increases with both conning pressure and
temperature in a controlled-strain test. Furthermore, a power relation-
ship (with R
2
over 0.95) is observed between
η
and the two thermody-
namic variables (temperature and pressure). The rate of triaxiality ratio
start decreasing from and after 100 kPa connement (Fig. 15b) and
40 C temperature (Fig. 15a). Although the realistic conning pressure
in the asphalt pavement is not accurately known, the range between 100
and 250 is generally considered as an in-situ conning stress range
(average of 150 to 175 kPa). The surface plot in Fig. 16 also shows that
the triaxiality ratio is critical at the combination of hot temperature and
high connement conditions. This observation indirectly implies that
the linearity limit of asphalt concrete depends on the triaxiality ratio
(Von Mises and mean stresses) and can be determined using different
combinations of conning pressures, temperatures, and deviatoric
stresses. Based on the observations from Figs. 15 and 16, the triaxiality
ratio can be integrated into the time-temperature-pressure shift model to
predict triaxail stress-dependent LVE response of asphalt concrete.
5.5. The proposed model
The triaxiality ratio (
η
) is a fundamental material parameter that can
couple the stress-dependent thermo-piezo-rheology responses of visco-
elastic materials. The Prony method is widely used for viscoelastic
modeling of asphalt concrete. From a mechanistic viewpoint, the
triaxiality ratio is more comprehensive and efcient approach to model
triaxial viscoelastic properties. As discussed in Section 5.4, the triaxi-
ality ratio (
η
) is dependent on both temperature and pressure. This paper
proposes a new model to integrate the triaxiality ratio with the time-
temperature-pressure superposition principle. The proposed model
takes the following form.
50 100 150 200 250 300 350
Confining Pressure (kPa)
0.3
0.35
0.4
0.45
0.5
Maximum Slope (So)
(a)
So = - 0.00056*P + 0.53
R2= 0.94
0 50 100 150 200 250 300
350
Confining Pressure (kPa)
2.8
3
3.2
3.4
3.6
3.8
4
Damage Parameter (α)
(b)
α= 0.003*P + 2.9
R
2
= 0.92
Fig. 11. Conning Pressure versus (a) Maximum slope (S
o) (b) Damage evolution Parameter (a).
Fig. 12. Long-term relaxation Modulus at different conning pressures.
0 200 400 600 800 1000
1200
Long-term Relaxation Modulus (MPa)
0.3
0.35
0.4
0.45
0.5
0.55
Maximum Slope (So)
(a)
So = - 0.00015*E+ 0.54
R
2
= 0.71
0 200 400 600 800 1000
Long-term Relaxation Modulus (MPa)
2.8
3
3.2
3.4
3.6
3.8
Damage Parameter (α)
(b)
α= 0.00082*E+ 2.8
R
2
= 0.67
Fig. 13. Long-term relaxation Modulus versus Maximum slope and viscoelastic damage parameter.
M. Mulugeta Alamnie et al.
Construction and Building Materials 329 (2022) 127106
9
log
α
TP =C1[TT0Γ(p)]
C22 +C23
η
+TT0Γ(p)(24)
Γ(p) = (C7
η
)ln1+C8P
1+C8P0(25)
Where C1,C22,C23,C7,C8 are temperature and pressure shift factor
coefcients. A master curve is constructed using the new proposed
model (Eqs. (24) and (25)) and compared with Model-1 (Eqs. (13), 15,
and 16). As shown in Fig. 17, the proposed triaxiality ratio-based model
tted the measure data well. The main advantage of this model is the
consideration of deviatoric stress in the LVE modeling while being
concise and fewer number of model coefcients. In most dynamic
modulus tests, the load control mechanism is controlled-strain modes.
Limiting the strain output is more controllable than that of the stress
when linear viscoelastic response is concerned. However, there are cir-
cumstances where stress-controlled responses can be more plausible. In
such conditions, the proposed triaxiality ratio shift model is convinient
to predict the linear viscoelastic response.
Furthermore, the relationship between the long-term relaxation
modulus (E) and the triaxiality ratio (
η
) is established.
E=10[κ+λ×ln(
η
)] (26)
Where κ and λ are tting parameters. The exponent in Eq. (26) is
equivalent to the minimum dynamic modulus value δ of the sigmoid
function i.e., min[log(E*)] = δ=κ+λ×ln
η
). As shown in Fig. 18, a good
correlation is observed between long-term relaxation modulus (E) and
the triaxiality ratio (
η
) at high temperatures. This is because the stress
ratio is more critical on the viscous side of asphalt concrete than the
elastic part (as shown in Fig. 16). The tting parameters of the model
(Eq. (26)) are given in Table 5.
6. Conclusion
In this paper, the effect of triaxial stress on the linear viscoelastic
properties of asphalt concrete material was investigated using triaxial
dynamic modulus test over a wide range of temperatures, frequencies,
and conning stresses. Two different asphalt mixtures (neat and
polymer-modied binder) were used and proved as thermo-piezo-
rheologically simple material. The Fillers, Moonan, and Tschoegl
(FMT) model is adopted for timetemperature-Pressure shifting and
compared with another model from literature (Model 2). The stress ratio
concept (triaxiality ratio) is introduced to characterize the stress-
dependent viscoelastic properties of asphalt concrete. The main contri-
butions are summarized as follows.
Fig. 14. Peak deviatoric stress at different Conning Pressures and
temperatures.
-20 -10 0 10 20 30 40 50
60
Temperature (
o
C)
0.2
0.4
0.6
0.8
1
1.2
Triaxiality Ratio
(a)
Uniaxial
10 kPa
100 kPa
200 kPa
300 kPa
0 50 100 150 200 250
300
Pressure (kPa)
0
0.2
0.4
0.6
0.8
1
Triaxiality Ratio
(b)
-10
o
C
5
o
C
21
o
C
40
o
C
55
o
C
Fig. 15. Triaxiality ratio at different Conning Pressures and Temperatures.
Fig. 16. Three-dimensional Surface plot of Triaxiality ratio.
10
-6
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
Reduced Frequency (Hz)
2
2.5
3
3.5
4
4.5
Log Dynamic Modulus (MPa)
Model 1
Proposed Model
Master curves using
Fig. 17. Triaxial Master curves using Model-1 and the New Triaxial Ratio-
based model.
M. Mulugeta Alamnie et al.
Construction and Building Materials 329 (2022) 127106
10
The linear viscoelastic (LVE) property of asphalt concrete in a triaxial
stress state is validated using the timetemperature-pressure super-
position principle (TTPSP). The LVE properties are highly stress-
dependent at intermediate and high temperatures.
A simplied, integral (one-step), and theoretically sound vertical
shift model is proposed by modifying the FMT model to construct
triaxial master curves.
A Prony method time-domain viscoelastic analysis revealed that the
long-term relaxation modulus and maximum slope of a relaxation
modulus curve are strongly stress-dependent. However, the Prony
series coefcients (Ei) are independent of pressure at high relaxation
time.
A slight reduction of the maximum slope of relaxation modulus due
to conning pressure causes more than 5.4 times increment of the
viscoelastic fatigue damage parameter, highlighting the limitations
of uniaxial fatigue life prediction, particularly at intermediate tem-
peratures (from 15 to 25).
The concept of triaxiality ratio is introduced to characterize the 3D
stress effect on the linear viscoelastic responses of asphalt concrete.
For a controlled-strain dynamic modulus test, the triaxiality ratio
increases with temperature and pressure. In addition, the ratio has a
strong correlation with the long-term relaxation modulus.
A new triaxial TTPSP shifting model is proposed and validated. The
triaxiality ratio can indirectly characterize the viscoelastic linearity
limit for the thermo-piezo-rheological simple materials.
CRediT authorship contribution statement
Mequanent Mulugeta Alamnie: Conceptualization, Methodology,
Data curation, Writing original draft, Visualization. Ephrem Tad-
desse: Conceptualization, Supervision. Inge Hoff: Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing nancial
interests or personal relationships that could have appeared to inuence
the work reported in this paper.
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://doi.
org/10.1016/j.conbuildmat.2022.127106.
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0.3 0.4 0.5 0.6
Triaxiality Ratio
2.2
2.4
2.6
2.8
3
3.2
log(E), MPa
(a) -10
o
C
Measured
Fitted
R
2
=0.87
0.4 0.6 0.8
Triaxiality Ratio
2.2
2.4
2.6
2.8
3
3.2
log(E), MPa
(b) 5
o
C
Measured
Fitted
R
2
=0.89
0.4 0.6 0.8 1
Triaxiality Ratio
2.2
2.4
2.6
2.8
3
3.2
log(E), MPa
(c) 21
o
C
Measured
Fitted
R
2
=0.91
0.4 0.6 0.8 1 1.2
Triaxiality Ratio
2.2
2.4
2.6
2.8
3
log(E), MPa
(d) 40
o
C
Measured
Fitted
R
2
=0.95
0.4 0.6 0.8 1 1.2
Triaxiality Ratio
2.2
2.4
2.6
2.8
3
log(E), MPa
(e) 55
o
C
Measured
Fitted
R
2
=0.93
Fig. 18. Triaxiality ratio versus long-term relaxation modulus at different pressure and temperatures.
Table 5
Fitting Parameters for Eq. (26).
Coefcient Temperature [
o
C]
¡10 5 21 40 55
κ 3.87 3.42 3.09 2.93 2.86
λ 1.38 1.00 0.73 0.63 0.60
M. Mulugeta Alamnie et al.
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11
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M. Mulugeta Alamnie et al.
... The calibration of PANDA models used uniaxial creep, uniaxial constant stress creep-recovery, the crosshead strain rate test, and multiple stress creep tests, etc. The influence of confining pressure on the linear viscoelastic as well as nonlinear viscoelastic responses is significant [75,81]. Most studies that tried to calibrate the PANDA models used uniaxial test data, or some used a single confining pressure. ...
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