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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 conning 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 conning 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 signicantly inuence 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 (time–temperature 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 signicant 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 satises the TTPSP principle is called a thermo–piezo-

rheological simple material [23,6]. The role of conning 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 connement on the applicability of TTSP

with growing damage and the effect of conning pressure on Schapery’s

nonlinear viscoelastic parameters, respectively. Other studies

[28,29,21] have investigated the effect of conning pressure on linear

viscoelastic (LVE) responses of asphalt concrete mixtures using triaxial

master curves. Previous research focused on proposing ‘vertical’ shift

models as a function of conning pressure and was mainly involved in

constructing the ‘triaxial’ master 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 (conned) 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 conning 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

simplied 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 connement 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 simplied 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 conning pressure (stress).

2. Viscoelasticity

The uniaxial stress–strain 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 E∞ is 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 stress–strain 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 coefcients of relaxation modulus

and time for shear and bulk, respectively. M and N are the number of

Prony coefcients 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 material’s

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(1−2

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 Poisson’s 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-modied (PMB 65/105–60)

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 specication. 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 120◦apart 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 conning pressure is applied for about 15 min to

stabilize the deformation due to connement. Repeat step 1.

(3). Compare dynamic modulus and phase angle values from steps 1

(unconned or uniaxial) and step 2 (10 kPa conned). The dy-

namic modulus results should not vary by a signicant margin.

(4). Apply 100 kPa conning pressure (for 15 min). Repeat step 1.

(5). Apply 200 kPa conning pressure (for 15 min). Repeat step 1.

(6). Apply 300 kPa conning 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*|eiφ (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, conning 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 conning pressure on dynamic modulus

The effect of conning 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 conning pressure has a

signicant 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 connement 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

reected by the reduction of phase angle. The role of connement 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 conning 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 time–temperature shift factor. The

Williams, Landel, and Ferry (WLF) function [25] is widely used for aT.

log

α

T= − C1(T−T0)

C2+T−T0

(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 conning

pressure levels. Moreover, the effect of conning 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, conning pressure contributed

to reducing the dissipated energy (or phase angle) and conning pres-

sure has no signicant effect on energy loss at low temperatures and

high frequencies. Therefore, conning 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 conning pressure causes

an increase of dynamic modulus. To construct a stress-dependent master

curve, the modulus at different frequencies, conning pressures and

temperatures are shifted both horizontally and vertically. The time-

–temperature shift factor is superposed and modied 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 modied WLF function proposed by Fillers, Moonan,

and Tschoegl (FMT) model [8], expressed as follows.

log

α

TP =−C1(T−T0−Γ(p))

C2(P) + T−T0−Γ(p)(13)

Γ(p) = C3(P)ln 1+C4P

1+C4P0−C5(P)ln 1+C6P

1+C6P0

(14)

Where

α

TP is time–temperature-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 coefcients 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

modied version of Model-1 is proposed in this paper. The pressure-

dependent coefcients 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 coefcients. The proposed modied 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

Zhao’s model [30], expressed as,

logλ =− (P−Po)

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.

10

-4

10

-2

10

0

10

2

10

4

10

6

10

8

Reduced Frequency (Hz)

1

1.5

2

2.5

3

3.5

4

Energy Loss Ratio

Uniaxial

10 kPa

100 kPa

200 kPa

300 kPa

Fig. 3. Energy loss ratio (specic loss) at different conning 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); C3–C6 are regression

coefcients;P, Po are conning and reference pressures, respectively.

All stress-dependent master curves in the subsequent sections are

constructed at 21 ◦C reference temperature and 100 kPa reference

conning 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 conning

pressure.

5. Time-domain viscoelastic properties

Although the frequency domain dynamic modulus can give sufcient

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 sufcient 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 EfR−gfR]is

used.

g(fR) = a1+a2

a2+a4

exp(a5+a6logfR)

(18)

Where a

1, 2,

…,

6

are coefcients, 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 coefcients (relaxation modulus [MPa] and relaxation time [sec],

ω

is angular frequency, M is number of Prony coefcients 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 coefcients - AB11 Mixture.

Model Coefcients

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 coefcients - SKA11 Mixture.

Model Coefcients

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

conning pressure is signicant 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 coefcients. A total of twelve Prony coefcients

were used. The coefcients (Ei) at long-time are almost the same and can

be taken as independent of conning 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 innite. 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 conning 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 conning pressure is ex-

pected to be lower than the minimum tensile strength of asphalt con-

crete. Thus, deformation caused by conning 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 conning pressure on viscoelastic damage parameter

For a viscoelastic material, the absolute maximum slope of the

relaxation modulus curve is found conning 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=1−Em×e−t

ρ

m

E∞+M

m=1Em×e−t

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 conning pressure

increases and due to the increment of relaxation modulus at an innite

time (or when frequency approaches zero,

ω

≅0) on the high-

temperature side. On the other hand, the damage parameter (

α

) in-

creases as conning 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 connement on the permanent defor-

mation evolution is well understood and incorporated in the strain

hardening phenomenon of asphalt concrete materials.

5.3. Effect of conning 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 sufcient 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 E∞≅10δ).

Different models have been proposed to correlate E∞ with conning

stress [16,28]. As shown in Fig. 9 and Fig. 12, the long-term relaxation

modulus is dependent on conning stress and a linear relation can be

seen between the long-term relaxation modulus and conning 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 dened 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 conning 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 Coefcient Relaxation Moduli (Gaussian type distribution) –

AB11 Mixture.

Table 4

Prony Coefcients of relaxation Spectrum.

Conning 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 unconned (

σ

c=0)condition

is 1/3. The ratio increases as the connement 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 conning pressure at

a particular test temperature. This conrms 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 conning stress. It is clearly seen that

η

is both pressure and

temperature-dependent, and increases with both conning 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 connement (Fig. 15b) and

40 ◦C temperature (Fig. 15a). Although the realistic conning pressure

in the asphalt pavement is not accurately known, the range between 100

and 250 is generally considered as an in-situ conning 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 connement 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 conning 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 efcient 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. Conning Pressure versus (a) Maximum slope (S

o) (b) Damage evolution Parameter (a).

0 50 100 150 200 250 300

Confining Pressure (kPa)

200

400

600

800

1000

1200

Long-term

Relaxation Modulus (MPa)

E∞= 2.8*P + 2.6e+02

R

2

= 0.94

Fig. 12. Long-term relaxation Modulus at different conning 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[T−T0−Γ(p)]

C22 +C23

η

+T−T0−Γ(p)(24)

Γ(p) = (C7

η

)ln1+C8P

1+C8P0(25)

Where C1,C22,C23,C7,C8 are temperature and pressure shift factor

coefcients. 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 coefcients. 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 conning stresses. Two different asphalt mixtures (neat and

polymer-modied binder) were used and proved as thermo-piezo-

rheologically simple material. The Fillers, Moonan, and Tschoegl

(FMT) model is adopted for time–temperature-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.

-10 0 10 20 30 40 50

Temperature (

o

C)

0

200

400

600

800

1000

Deviatoric Stress (kPa)

Uniaxial

10 kPa

100 kPa

200 kPa

300 kPa

Fig. 14. Peak deviatoric stress at different Conning 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 Conning 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 time–temperature-pressure super-

position principle (TTPSP). The LVE properties are highly stress-

dependent at intermediate and high temperatures.

•A simplied, 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 coefcients (Ei) are independent of pressure at high relaxation

time.

•A slight reduction of the maximum slope of relaxation modulus due

to conning 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 inuence

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.

References

[1] J. Blanc, T. Gabet, P. Hornych, J.-M. Piau, H. di Benedetto, Cyclic triaxial tests on

bituminous mixtures, Road Mater. Pavement Des. 16 (2014) 46–69.

[2] W. Cao, Y.R. Kim, A viscoplastic model for the conned permanent deformation of

asphalt concrete in compression, Mech. Mater. 92 (2016) 235–247.

[3] G. Chehab, Y. Kim, R. Schapery, M. Witczak, R. Bonaquist, Time-temperature

superposition principle for asphalt concrete with growing damage in tension state,

J. Assoc. Asphalt Paving Technol. 71 (2002).

[4] J.S. Daniel, Y.R. Kim, Development of a simplied fatigue test and analysis

procedure using a viscoelastic, continuum damage model (with discussion),

J. Assoc. Asphalt Paving Technol. 71 (2002).

[5] M.K. Darabi, R.K. Abu Al-Rub, E.A. Masad, C.-W. Huang, D.N. Little, A thermo-

viscoelastic–viscoplastic–viscodamage constitutive model for asphaltic materials,

Int. J. Solids Struct. 48 (2011) 191–207.

[6] EMRI, I. 2005. Rheology of solid polymers. Rheology Reviews, 2005, 49.

[7] J.D. Ferry, R.A. Stratton, The free volume interpretation of the dependence of

viscosities and viscoelastic relaxation times on concentration, pressure, and tensile

strain, Kolloid-Zeitschrift 171 (1960) 107–111.

[8] R.W. Fillers, N.W. Tschoegl, The Effect of Pressure on the Mechanical Properties of

Polymers, Trans. Soc. Rheol. 21 (1977) 51–100.

[9] W.N. Findley, J.S. Lai, K. Onaran, Creep and relaxation of nonlinear viscoelastic

materials, North-Holland Publishing, 1976.

[10] P. Gayte, H. di Benedetto, C. Sauz´

eat, Q.T. Nguyen, Inuence of transient effects

for analysis of complex modulus tests on bituminous mixtures, Road Mater.

Pavement Des. 17 (2016) 271–289.

[11] H.-J. Lee, J.S. Daniel, Y.R. Kim, Continuum Damage Mechanics-Based Fatigue

Model of Asphalt Concrete, J. Mater. Civ. Eng. 12 (2000) 105–112.

[12] W.K. Moonan, N.W. Tschoegl, The effect of pressure on the mechanical properties

of polymers. IV. Measurements in torsion, J. Polym. Sci.: Polym. Physics Ed. 23

(1985) 623–651.

[13] S. Mun, G.R. Chehab, Y.R. Kim, Determination of Time-domain Viscoelastic

Functions using Optimized Interconversion Techniques, Road Mater. Pavement

Des. 8 (2007) 351–365.

[14] M.L. Nguyen, C. Sauz´

eat, H. di Benedetto, N. Tapsoba, Validation of the

time–temperature superposition principle for crack propagation in bituminous

mixtures, Mater. Struct. 46 (2013) 1075–1087.

[15] Park, S. W. & Schapery, R. A. 1999. Methods of interconversion between linear

viscoelastic material functions. Part I—a numerical method based on Prony series.

36, 1653-1675.

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).

Coefcient 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.

Construction and Building Materials 329 (2022) 127106

11

[16] T.K. Pellinen, M.W. Witczak, M. Marasteanu, G. Chehab, S. Alavi, R. Dongr´

e, Stress

dependent master curve construction for dynamic (complex) modulus. Asphalt

Paving Technology: Association of Asphalt Paving Technologists-Proceedings of

the Technical Sessions, Association of Asphalt Paving Technologist, 2002,

pp. 281–309.

[17] D. Perraton, H. di Benedetto, C. Sauz´

eat, B. Hofko, A. Graziani, Q.T. Nguyen,

S. Pouget, L.D. Poulikakos, N. Tapsoba, J. Grenfell, 3Dim experimental

investigation of linear viscoelastic properties of bituminous mixtures, Mater.

Struct. 49 (2016) 4813–4829.

[18] E. Rahmani, M.K. Darabi, R.K. Abu Al-Rub, E. Kassem, E.A. Masad, D.N. Little,

Effect of connement pressure on the nonlinear-viscoelastic response of asphalt

concrete at high temperatures, Constr. Build. Mater. 47 (2013) 779–788.

[19] N. Roy, A. Veeraragavan, J.M. Krishnan, Inuence of connement pressure and air

voids on the repeated creep and recovery of asphalt concrete mixtures, Int. J.

Pavement Eng. 17 (2016) 133–147.

[20] C.W. Schwartz, N. Gibson, R.A. Schapery, Time-Temperature Superposition for

Asphalt Concrete at Large Compressive Strains, Transp. Res. Rec.: J. Transp. Res.

Board 1789 (2002) 101–112.

[21] Y. Sun, B. Huang, J. Chen, X. Shu, Y. Li, Characterization of Triaxial Stress State

Linear Viscoelastic Behavior of Asphalt Concrete, J. Mater. Civ. Eng. 29 (2017)

04016259.

[22] N.W. Tschoegl, Energy Storage and Dissipation in a Linear Viscoelastic Material, in:

N.W. Tschoegl (Ed.), The Phenomenological Theory of Linear Viscoelastic

Behavior, Springer Berlin Heidelberg, Berlin, Heidelberg, 1989.

[23] N.W. Tschoegl, Time Dependence in Material Properties: An Overview, Mech.

Time-Dependent Mater. 1 (1997) 3–31.

[24] J. Uzan, Characterization of Asphalt Concrete Materials for Permanent

Deformation, Int. J. Pavement Eng. 4 (2003) 77–86.

[25] M.L. Williams, R.F. Landel, J.D. Ferry, The temperature dependence of relaxation

mechanisms in amorphous polymers and other glass-forming liquids, J. Am. Chem.

Soc. 77 (1955) 3701–3707.

[26] T. Yun, B.S. Underwood, Y.R. Kim, Time-Temperature Superposition for HMA with

Growing Damage and Permanent Strain in Conned Tension and Compression,

J. Mater. Civ. Eng. 22 (2010) 415–422.

[27] Y. Zhang, R. Luo, L. Lytton Robert, Anisotropic Viscoelastic Properties of

Undamaged Asphalt Mixtures, J. Transp. Eng. 138 (2012) 75–89.

[28] Y. Zhao, H. Liu, W. Liu, Characterization of linear viscoelastic properties of asphalt

concrete subjected to conning pressure, Mech. Time-Dependent Mater. 17 (2013)

449–463.

[29] Y. Zhao, Y. Richard Kim, Time–Temperature Superposition for Asphalt Mixtures

with Growing Damage and Permanent Deformation in Compression, Transp. Res.

Rec.: J. Transp. Res. Board 1832 (2003) 161–172.

[30] Y. Zhao, J. Tang, H. Liu, Construction of triaxial dynamic modulus master curve for

asphalt mixtures, Constr. Build. Mater. 37 (2012) 21–26.

M. Mulugeta Alamnie et al.