towards a viscoelastic mechanical characterization of asphalt materials by ultrasonic measurements

ORIGINAL ARTICLE

Towards a viscoelastic mechanical characterizationof asphalt materials by ultrasonic measurements

Nicolas Larcher • Mokhfi Takarli •

Nicolas Angellier • Christophe Petit •

Hamidou Sebbah

Received: 11 April 2013 / Accepted: 12 December 2013

� RILEM 2014

Abstract This study focuses on the use of P-wave

propagation measurements in order to evaluate the

complex modulus, more specifically for the reversible

and dissipated parts of asphalt materials. Both the

wave velocity and attenuation factor have been

measured by means of an ultrasonic transmission test,

at frequencies between 200 and 300 kHz and temper-

atures ranging from -20 to 40 �C. Based on this wave

velocity and attenuation factor, the high-frequency

complex modulus and its components are computed

by considering a 2D propagation of waves in an

isotropic viscoelastic medium. Results are plotted with

respect to the master curve, Cole–Cole and Black

spaces. The ultrasonic test results agree with results

obtained by complex modulus test and then fitted by

the 2S2P1D rheological model. This paper shows in

Cole–Cole space that ultrasonic data can facilitate the

determination of important rheological parameters as

one of the two parabolic dashpots (k) and the glassy

modulus E0. The phase angle, which is also a key

viscoelastic identification parameter, can be deter-

mined at high frequency in a Black space

representation.

Keywords Asphalt concrete � Complex

modulus � Ultrasonic wave propagation �Velocity � Attenuation

1 Introduction

In the field of civil engineering, non-destructive

testing (NDT) and in particular wave propagation

methods provide a valuable tool for structural diag-

nostics (whether in-service, in situ or field NDT) as

well as for materials characterization (laboratory

NDT).

When applied in the laboratory, these techniques

allow studying several aspects of asphalt concrete

characterization, namely: microstructural characteriza-

tion (e.g. density, air voids, bitumen content, asphalt

viscosity, gradation and filler content, aggregate segre-

gation, anisotropy) [1–4]; analysis of fatigue behavior

[5, 6]; moisture damage investigation [7]; acoustic

properties (impedance, absorption and damping) for

assessing noise reduction performance [8–10]; and

determination of mechanical properties (complex mod-

ulus, high-frequency or dynamic modulus) [11–19];

N. Larcher � M. Takarli � N. Angellier �C. Petit (&) � H. Sebbah

Laboratoire GEMH-GCD, Universite de Limoges, 17

Boulevard Jacques Derche, 19300 Egletons, France

e-mail: [email protected]

N. Larcher


M. Takarli


N. Angellier


H. Sebbah


Materials and Structures

DOI 10.1617/s11527-013-0240-0

This paper focuses on determining complex mod-

ulus, more specifically in the reversible and dissipated

parts, since this is a key parameter in pavement design.

In order to measure this mechanical property for

asphalt concrete, the viscoelastic approach for wave

propagation is considered. An ultrasonic P-wave

propagation technique will be used to determine the

wave propagation parameters (i.e. wave velocity and

attenuation factor). Asphalt concrete is a heteroge-

neous material with viscoelastic behavior, thus mak-

ing ultrasonic parameters a source of frequency and

temperature dependence. To highlight this depen-

dence, measurements were performed for tempera-

tures of between -20 to 40 �C, with the frequency

bandwidth of analysis ranging from 200 to 300 kHz.

The complex modulus, its real and imaginary parts and

the phase angle are all computed using experimental

ultrasonic data. It has been demonstrated that atten-

uation is the most difficult parameter to analyze and

identify given that the many complex mechanisms

created by various sources are responsible for a wave

amplitude decrease. Moreover, the major interest of

this study lies in its use of a viscoelastic approach for

wave propagation analysis. Based on this approach,

both real and imaginary parts of the complex modulus

in addition to the phase angle can be determined with

ultrasonic parameters (velocity and attenuation fac-

tor). These results are plotted respectively in Cole–

Cole and Black spaces, and a good fit with mechanical

and rheological data can be observed. This comparison

shows the potential of an ultrasonic test to achieve

separation of the reversible modulus from the dissi-

pated modulus (E1 and E2 or E* and phase angle). In

focusing exclusively on the Cole–Cole space, it thus

becomes obvious that the ultrasonic results allow

determining rheological parameters (k and E0). This

constitutes the first step towards identifying the

rheological parameters of a viscoelastic model, such

as the well-known 2S2P1D model, which proves very

effective for asphalt materials [20].

2 Background

The potential of an ultrasonic laboratory test to

measure complex modulus at high frequencies

(10–200 kHz) has been clearly demonstrated in the

literature review. An impulse echo method, developed

by Dos Reis et al. [11], allows estimating the complex

modulus according to an energy approach. Hochuli

et al. [12] generated a flexural wave in asphalt concrete

and measured the complex modulus at various

frequencies. Another method introduced in [13–15]

is the free resonant technique; these authors investi-

gated the resonant frequency and damping ratio in

order to derive the dynamic modulus value. The high-

frequency modulus has been accurately determined by

Barnes et al. [16] using both a spectral analysis of

surface waves and a multichannel analysis of surface

waves. Compression wave propagation has also been

employed in considering different hypotheses. Nor-

ambuena-Contreras et al. [17] applied an elastic

hypothesis and computed dynamic modulus at high

frequency simply with wave velocity at the test

temperature of 22 �C. More recently, several authors

have considered the viscoelastic behavior of asphalt

concrete by incorporating into the computation of

complex modulus either the phase angle or attenuation

factor. Mounier et al. [15] and Di Benedetto et al. [18]

associated the P-wave velocity derived from ultra-

sonic testing with the phase angle output by the

2S2P1D rheological model. Van Velsor et al. [19]

determined two wave propagation parameters for

compression, i.e. wave velocity and attenuation factor,

for introducing viscoelastic behavior. In these last

studies, test temperatures ranged roughly from -20 to

40 �C. The authors observed a significant difference

between the ultrasonic prediction and the complex

modulus test data. They explain this difference by the

wave scattering from air voids and aggregates.

Through this literature review, it was shown that

different types of waves can be used in the evaluation

of the complex modulus (Structural waves, surface

waves, compression and shear waves). However it

should be noted that group velocity and elastic wave

propagation hypotheses are most often adopted. The

viscoelastic approach is employed less frequently and

requires either combination with a rheological model

to identify viscous parameters, such as phase angle, or

an experimentally measured ultrasonic attenuation.

3 Materials and methods

3.1 Materials

The asphalt concrete specimen is prepared in accor-

dance with requirement of French Standard NF P 98-


250-(1&2), with a size of 70 9 395 9 595 mm3

(depth 9 width 9 length). The design gradation is

presented in Table 1. The binder selected is a 35–50

penetration degree asphalt binder and the mixture

density is 2.533 kg/m3.

3.2 Viscoelastic behavior

3.2.1 Principle of viscoelastic behavior

The complex modulus E* is a complex number

defined as the ratio between the complex amplitude

of the sinusoidal stress of pulsation x applied to the

materials, r tð Þ ¼ r0 sin xtð Þ, and the complex ampli-

tude of the sinusoidal strain resulting in a steady state

[21]. Due to the viscoelastic nature of the material, the

strain lags behind stress, as reflected by a phase angle

u between the two signals, e tð Þ ¼ e0 sin xt � uð Þ.Given this definition, the complex modulus is not a

function of time but instead depends on the pulsation

x for a fixed temperature (Eq. 3).

r tð Þ ¼ Im r� tð Þ½ � with r� tð Þ ¼ r0 � expixt ð1Þ

e tð Þ ¼ Im e� tð Þ½ � with e� tð Þ ¼ e0 � expiðxt�uÞ ð2Þ

E� xð Þ ¼ r0 � expixt

e0 � expiðxt�uÞ ¼ E�j jexpiu ð3Þ

In these expressions, i is the imaginary number, |E*|

is the norm of the complex modulus, x the pulsation

and u is the phase angle. The complex modulus can

also be written, E�j j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

E21 þ E2

2

p

, where E1 is the

instantaneous or elastic response and E2 is the deferred

or viscous response. The complex modulus depends

on both temperature and frequency.

3.2.2 Complex modulus test

The complex modulus is obtained through mechanical

cyclic tests, which can be divided into two categories

[22]: homogeneous tests (tension/compression, shear

test, constant-height shear test, coaxial shear test); and

non-homogeneous tests (2, 3 and 4-point bending

tests, indirect tensile test). To thoroughly evaluate the

temperature and frequency dependence of the stiffness

modulus, a minimum temperature and frequency

range must be applied. Such cyclic tests allow

computing the real and imaginary parts of the complex

modulus (E1, E2) or else the complex modulus |E*| and

phase angle u. Several representations of these results

are possible as isochronous and isotherm curves,

Cole–Cole and Black spaces. In this study, the

complex mechanical modulus tests have been per-

formed by introducing sinusoidal bending on trape-

zoidal samples (250 mm high, 56 mm large base,

25 mm small base, and 25 mm thick) at the LRPC

(Laboratoire Regional des Ponts et Chaussees,

France).Various frequencies were applied (1, 3, 10,

25, 30 and 40 Hz) for temperatures ranging from -20

to 40 �C. The isotherm curves of the complex modulus

are plotted in Fig 1a.

3.2.3 Master curve

The master curve, for the reference temperature of

15 �C used in French pavement design, has been

plotted in Fig. 1a by applying a mathematical-based

procedure [23]. The method is based on the Kramers-

Kronig relations linking modulus and phase angle of a

complex function. It seems to be suitable for binders

and mixtures as soon as their behavior is in agreement

with Time–Temperature Superposition Principle

(TTSP). This principle is based on the premise that

any given modulus value on the isotherm curve can be

obtained by different pairings of frequency and

temperature. TTSP yields the expression of E*(x,T)

as E*(x,f(T)). The master curve was derived by a

translation operation using a shift factor called aT,

which is computed by either a type of Arrhenius

equation or the WLF formula (which has been chosen

herein [23]). The WLF formula requires determining

two coefficients (C1 and C2) for the reference

temperature. In this case, the C1 and C2 values for a

reference temperature of 15 �C are respectively equal

to 19 and 130.

Table 1 Aggregate gradation and % of binder by mass

Sieve size (mm) Pass. aggregate (%)

0.08 10.4

0.2 17.1

0.4 21.8

1 42.1

2 62.3

4 70.8

6.3 98.4

8 100

Binder (by mass of aggregate) 6.85

Air voids 4.4


3.2.4 2S2P1D model

Once the master curve has been determined by the

mathematical-based procedure, a rheological model

can be fitted and then used to predict the complex

modulus at frequencies not reached by the experi-

mental master curve. Various rheological models are

available in the literature, yet those with a continuous

spectrum are more interesting in viscoelastic analyses

of asphalt concretes. These models allow the simula-

tion of their mechanical behavior for a great frequen-

cies range. Let’s cite, for example, Huet, Huet-Sayegh

and 2S2P1D (two springs, two parabolic dashpots, and

one linear dashpot, respectively) (Fig. 1b). The Huet-

Sayegh set-up offers an improvement over the Huet

model reported by Sayegh in that it has added in

parallel a spring with low stiffness. This spring

represents the static modulus of asphalt concrete and

moreover predicts the complex modulus at low

frequency or high temperature. More recently, Olard

et al. [20] added a linear dashpot in tandem with the

two springs (h and k) as well as the stiffening spring

(E0 - E00). This new rheological model, called

2S2P1D, is able to more accurately predict the

complex modulus than any other rheological model

at low frequency or high temperature. In this study, we

have introduced the 2S2P1D model, which in turn has

yielded the complex modulus via Eq. 4. The fitting

parameters are given in Fig. 1b.

E�j j ¼ E00 þE0 � E00

1þ d ixsð Þ�kþ ixsð Þ�hþ ixbsð Þ�1ð4Þ

In this equation, i is the complex number defined by

i2 ¼ �1, x is the pulsation with x ¼ 2pf where f is the

frequency, k and h are exponents such that 0 \ k \h \ 1, d is a dimensionless constant, E00 is the static

modulus when xs! 0, E0 is the glassy modulus when

xs!1, s is a characteristic time dependent only on

temperature, b is a dimensionless constant, g is the

Newtonian viscosity, and g ¼ ðE0 � E00Þbs when

Real part E1

Imag

inar

ypa

rt E

2

h /2 k /2

h k

E00 E0

E00

(b)

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E-04 1.E-01 1.E+02 1.E+05 1.E+08 1.E+11

Nor

m o

f th

e co

mpl

ex m

odul

us I

E*I

(G

Pa)

Equivalent frequency f.aT (Hz)

Experimental isotherms

Experimental master curve

2S2P1D model

-20°C-10°C

0°C10°C15°C

20°C

30°C

40°C

(a)

Fig. 1 a Standard dynamic tests plotted in terms of isotherms and master curves b Schematic diagram of the 2S2P1D rheological

model used to construct the theoretical master curve


xs! 0 then E � ixsð Þ ! E0 þ ix E0 � E00ð Þbs. It

must be notified that the fitting of this rheological

model required extrapolating the mechanical results in

order to determine a number of calibration parameters,

such as E0, E00, h and k.

3.3 Ultrasonic test

The aim of our ultrasonic test is to measure the wave

propagation parameters (velocity and attenuation),

thus making it possible to determine the complex

modulus by considering 2D wave propagation in an

isotropic viscoelastic medium.

3.3.1 Theoretical background

By adopting the hypothesis of a 2D P-wave propaga-

tion in isotropic viscoelastic material [24] (Eq. 5), the

phase velocity V/;p and attenuation factor ap of P-

wave can be given by using the Poisson’s ratio m, the

pulsation x, the complex modulus E�j j and the phase

angle u (Eqs. 6 and 7).

u x; tð Þ ¼ u0 exp �apxð Þ exp i xt�kxð Þð Þ ð5Þ

V/;p ¼1

cos u2

� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

E�j j 1� mð Þq 1þ mð Þ 1� 2mð Þ

s

ð6Þ

ap ¼ x sinu2

� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q 1þ mð Þ 1� 2mð ÞE�j j 1� mð Þ

s

ð7Þ

In these equations, k is the wave number and q is the

density (kg/m3). By combining Eq. 6 and 7, both the

complex modulus (Eq. 8) and phase angle can indeed

be determined (Eq. 9).

E�j j ¼qV2

/;px2 1þ mð Þ 1� 2mð Þ

x2 þ a2pV2

/;p

� �

1� mð Þð8Þ

u ¼ tan�1 2apV/;px

x2 � a2pV2

/;p

� �

0

@

1

A ð9Þ

3.3.2 Principle of the ultrasonic test

The ultrasonic test device is composed of: an arbitrary

waveform generation card (e.g. sinus, dirac, sweep)

with frequency modulation; piezoelectric compression

transducers with resonant frequencies of 100 to

500 kHz; a 40 dB analog pre-amplifier; a 40 Ms/s

acquisition and sampling card; and an appropriate

software for waveform processing and analysis. For

this study, the sinus waveform was chosen with a

300 kHz frequency. The sampling rate was set at

10 Ms/s.

In bituminous materials, transmission is a more

appropriate mode than reflection since attenuation is

very high and increases with temperature and fre-

quency. In this study, velocity and wave attenuation

have been computed with signals obtained for two

distinct propagation distances. Two samples with two

different lengths (L1 and L2) were subjected to the

same temperature variation with a simultaneous

occurrence (Fig. 2). Length L2 is considered less

advantageous, as regards attenuation measurement,

for signal acquisition at the highest temperature

(40 �C). Length L1, which is less than L2, is assumed

to satisfy the far-field criterion or Fraunhofer field

[25]. Here, lengths L1 and L2 are equal to 44 and

93 mm, respectively, and the cross-section of these

specimens is 70 9 70 mm2. The test temperature

ranges from -20 to 40 �C, with a 10 �C incremental

step from -20 to 0 �C and a 5 �C step from 0 to 40 �C.

For the wave velocity measurement, many calcula-

tion methods are available in the literature. Using

temporal analysis (Fig. 3a, b), the group velocity can

be determined according to several methods: (i) the

temporal difference into the first zero crossing for two

different signals can potentially be used, though this

method is highly dependent on the signal-to-noise ratio

[26]; (ii) the temporal difference between the maxi-

mum amplitudes of signal envelopes obtained by

applying the Hilbert Transform; and iii) the temporal

difference between the peaks (positive or negative). In

frequency analysis (Fig. 3c, d), the phase velocity can

be determined [27] by the following equation (Eq. 10):

V/;p ¼xDL

D/¼ 2pf DL

D/ð10Þ

In this expression, f is the frequency, DL the travel

difference between two signals, and D/ the phase shift

between the two signals (Fig. 3d). To identify the

frequency value, a cross-spectrum (Fig. 3c) is pro-

duced from the Fast Fourier Transform (FFT) of each

signal.

Like for wave velocity, wave attenuation factor

aUS;p (which is experimentally determined by consid-

ering ultrasonic P-wave propagation and expressed in

Np/m), can be measured in terms of several


parameters. In a temporal analysis (Fig. 3a and 3b),

these parameters are: the peak-to-peak amplitude ratio

[28]; and the ratio of maximum Hilbert envelope

amplitudes. In the frequency domain (Fig. 3c), mea-

surements rely on: the magnitude of the FFT ratio

(Eq. 11); and the spectrum area ratio. Other methods

are also available in the literature review.

aUS;p ¼ln

Magnitude FFTof signal L1

Magnitude FFT of signal L2

� �

DLð11Þ

Fig. 2 Schematic principle

of the ultrasonic test

0

1

2

3

4

5

6

7

0 50 100 150 200 250 300 350 400 450 500

Am

plitu

de F

FT

Frequency (kHz)

FFT Signal 1

FFT Signal 2

Cross Spectrum

0

10

20

30

40

50

60

0 50 100 150 200 250 300 350 400 450 500

Sig

nal p

hase

(ra

dian

)

Frequency (kHz)

Phase Signal 1

Phase signal 2

-3

-2

-1

0

1

2

3

118 122 126 130 134 138 142 146 150 154

Am

plitu

de (

V)

Time (µs)

Signal L2

Hilbert transform of signal 2

(b)

(c)

(d)

-10

-8

-6

-4

-2

0

2

4

6

8

10

118 122 126 130 134 138 142 146 150 154

Am

plitu

de (

V)

Time (µs)

Signal L1

Hilbert transform of signal 1

(a)

Fig. 3 a, b Time ultrasonic waveforms obtained for samples L1 and L2 at 10 �C, and the Hilbert Transform c Fast Fourier Transform

(FFT) amplitudes d Phase shift between L1 and L2


4 Results and discussion

Laboratory measurements of P-wave propagation

parameters are performed at temperatures ranging

from -20 to ?40 �C. The results output of velocity

and attenuation are the averages of ten measurements

at each test temperature. These wave propagation

parameters are used to calculate the high-frequency

modulus by using Eq. 8. Literature review shows that

Poisson’s ratio of asphalt concrete can ranges between

0.2 to 0.5 according the temperature and frequency

[21, 29]. In this study, we consider that the Poisson’s

ratio is constant (0.36) by taking into account mean

value of the variation. This value was also used in

several studies based on ultrasonic wave propagation

methods for mechanical characterization of asphalt

concrete [16, 17]. The real and the imaginary parts of

the high-frequency modulus are computed by the

combination of Eqs. 8 and 9.

The 2S2P1D model allows fitting the master curve

and predicting the complex modulus at high frequen-

cies. Thus, for each experimental ultrasonic fre-

quency, theoretical values of phase velocity and

attenuation factor can be expressed by using Eqs. 6

and 7.

4.1 Comparison between theoretical

and experimental wave propagation

parameters

The velocities obtained by the previously cited

methods (phase velocity, peak to peak velocity and

Hibert envelope velocity, see Sect. 3.3.2) are com-

pared in Fig. 4. It is apparent that similar values are

found for the methods being compared at temperature

above 0 �C. These values match well with the

theoretical values of velocities expressed in Eq. 6.

For the method based on the Hilbert transform,

velocities are overestimated at low temperature (less

than 0 �C). To evaluate the complex modulus, phase

velocity with a known frequency seems to be more

appropriated.

Figure 5a illustrates the comparison of P-wave

attenuation factor values computed with the methods

cited in Sect. 3.3.2. It is observed that these values are

relatively similar: the mean standard deviation is

approx. 1.70 Np/m. The FFT ratio magnitude at the

frequency determined by the cross-spectrum has been

selected for this complex modulus evaluation.

Next, by comparing experimental results with

theoretical findings with respect to attenuation factor

0

50

100

150

200

250

300

3000

3500

4000

4500

5000

5500

-30 -20 -10 0 10 20 30 40 50

Cro

ss s

pect

rum

freq

uenc

y (k

Hz)

P-w

ave

velo

city

(m

/s)

Temperature (°C)

Hilbert envelope velocity

Peak to peak velocity

Phase velocity

Theoretical velocity

Cross spectrum frequency

Fig. 4 Comparison

between experimental and

theoretical velocities and

frequencies determined

from a cross-spectrum


parameters (Fig. 5b), a number of observations can be

drawn. For starters, a similar overall shape is noted in

the attenuation factor increase with temperature gains.

This attenuation factor increase becomes more signif-

icant above 0 �C. Moreover, a constant difference

between experimental and theoretical values of about

21.55 Np/m (with a standard deviation of 1.57 Np/m)

is systematically displayed. The attenuation overesti-

mation can be explained by several factors: (i) first, the

additional attenuation sources (geometric spreading

and scattering) are not taken into account by the

rheological model; (ii) secondly, Poisson’s ratio

required for computing the attenuation factor is

supposed constant despite it possible variation

between 0.2 and 0.5 as shown in the literature review

[21, 29]. Usually, the minimum value of m (0.2) is

obtained at low temperature of –20 �C. Thus, the

relative error which can be caused by assuming

m = 0.36 at this temperature is negligible (about

–0.46 Np/m) compared to the observed constant

difference between experimental and theoretical val-

ues (about ?21.55 Np/m). However, when m value is

0

10

20

30

40

50

60

0 10 20 30 40 50

Atte

nuat

ion

(Np/

m)

Temperature (°C)

Maximum amplitude of Hilbert envelopes ratio

Magnitude FFT ratio

Peak to peak amplitude ratio

Spectrum area ratio

0

5

10

15

20

25

30

0

10

20

30

40

50

60

-30 -20 -10

-30 -20 -10

0 10 20 30 40 50W

avel

engh

t (m

m)

Atte

nuat

ion

(Np/

m)

Temperature (°C)

Magnitude FFT ratio

Theoretical attenuation

Wavelenght

(a)

(b)

Fig. 5 a Experimental

attenuation factor according

to two different temporal

and frequency analyses

b Comparison between the

theoretical attenuation

factor and the experimental

attenuation factor; the

experimental wavelength


increased above 0.36 at temperature of 40 �C the

difference between theory and experimentation

becomes more larger. In conclusion, the systematic

overestimation of attenuation factor can’t be totally

explained by the value of Poisson’s ratio which is

adopted as constant (0.36) for all temperatures and

frequencies; (iii) thirdly, hypothesis of isotropy in

asphalt concrete needs to be inspected by ultrasonic

measurements. For all computations therefore, each

experimental attenuation value is corrected by this

constant value. Moreover, without this correction,

results will not be in agreement with mechanical data,

and the complex modulus will be underestimated

while the phase angle will be overestimated (as shown

in [20]).

4.2 Determination of high-frequency modulus

The results reported in the previous sections have

indicated that wave propagation parameters (velocity

and attenuation factor) are consistent with the

mechanical results fitted by the 2S2P1D rheological

model. A viscoelastic hypothesis for wave propaga-

tion was necessary to determine both the theoretical

velocity and attenuation factor from the rheological

model.

The values for the norm of the complex modulus, as

obtained from wave propagation parameters (P-wave

velocity and attenuation factor, Eq. 8), are plotted in

Fig. 6. It can be observed that the ultrasonic

frequencies range from 200 to 300 kHz and the

complex modulus from 15 to 30 GPa. The ultrasonic

moduli were translated by applying the TTSP and by

introducing the shift factor aT derived from the

mechanical tests. Thus, the ultrasonic master curve

was also plotted, and Fig. 6 illustrates its agreement

with both the experimental master curve and 2S2P1D

model, as previously shown in several studies [11–19].

At the reference temperature of 15 �C, this ultrasonic

modulus translation yields frequencies of approxi-

mately 200 Hz.

The modulus at this frequency cannot be used in

pavement design because the required frequency has

been set at 10 Hz. However, with new measurements

at temperatures above 40 �C, the frequency needed for

pavement design can ultimately be reached. However,

the ultrasonic modulus at a given temperature can

serve as an indicator of material degradation under

mechanical and/or physicochemical loading.

4.3 Contribution to the viscoelastic approach

The characterization and modeling of asphalt concrete

behavior requires a complex representation called the

Cole–Cole space (Fig. 1b). According to this set-up,

the imaginary part is plotted vs. the real part, which

respectively represent the viscous and elastic parts

of the complex modulus. Such a representation

allows determining several parameters, e.g. h, k, E0

and E00, used in rheological modeling by means of

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E-04 1.E-01 1.E+02 1.E+05 1.E+08 1.E+11

Nor

m o

f the

com

plex

mod

ulus

IE*I

(G

Pa)

Equivalent frequency f.aT (Hz)

Experimental master curve

2S2P1D model

US modulus

US equivalent modulus

TTSP

TTSPFig. 6 Mechanical master

curve fitted by the 2S2P1D

rheological model, the

ultrasonic modulus, and the

ultrasonic equivalent

modulus obtained by the

TTSP (feq = f.aT)


extrapolating mechanical data. The extrapolation

needed to determine the k and E0 parameters may be

simplified by an ultrasonic test that employs the

viscoelastic approach. With this approach, both the

real and imaginary parts of the complex modulus can

in fact be expressed by wave propagation parameters,

and especially the attenuation factor, representing the

viscous material behavior. In Fig. 7, the real and

imaginary parts of the complex modulus obtained by

the ultrasonic test are shown in a Cole–Cole space.

First of all, it can be observed that the ultrasonic test

gives some values outside the range of the mechanical

test results. Moreover, these results are in good

agreement with the mechanical test as well as the

rheological model.

Secondly, the tangent to the mechanical data is

plotted to enable determining the k parameter used to

fit the 2S2P1D rheological model for this material

(k = 0.145). It is important to underscore that this

tangent passes through the ultrasonic data, which

0

1

2

3

0 5 10 15 20 25 30 35

Imag

inar

y pa

rt o

f com

plex

mod

ulus

E2(G

Pa)

Real part of complex modulus E1 (GPa)

Mechanical data2S2P1D modeUS dataTangent

kπ/2

E0

Fig. 7 Representation of

ultrasonic results in a Cole–

Cole space: determination of

k and the glassy modulus E0

0

5

10

15

20

25

30

2 7 12 17 22 27 32

Pha

se a

ngle

(

)

Norm of the complex modulus IE*I (GPa)

Mechanical data

2S2P1D model

US data

Fig. 8 Representation of

ultrasonic results in a Black

space


complete the dataset generated by mechanical tests.

Otherwise, the glassy modulus, which is a key

parameter in identifying the compaction performance

of the aggregate skeleton, could only be estimated

with ultrasonic data.

The representation of these results can be also

plotted using Eqs. 8 and 9 in a Black space, which

presents the viscous behavior of asphalt concrete in

terms of phase angle and complex modulus (Fig. 8).

This representation also yields good agreement

between the mechanical, rheological and ultrasonic

results. It must be highlighted that the phase angle is

key parameter to determine the shift factor introduced

in the WLF formula [23].

5 Conclusion

In this study, three approaches to characterizing the

complex modulus of asphalt concrete have been

compared: complex modulus test, 2S2P1D rheological

model, and ultrasonic measurements. The test tem-

perature has ranged from -20 to ?40 �C. The results

of these various approaches have been presented on a

master curve (obtained using TTSP), as well as on

Cole–Cole and Black spaces and then compared.

The measured wave propagation parameters (phase

velocity and attenuation factor) associated with 2D

wave propagation in an isotropic viscoelastic medium

have allowed computing the complex modulus values

at high frequencies in the 200–300 kHz range. As

expected, these values reported on the master curve by

the use of equivalent frequencies show good agree-

ment with the standard dynamic test and the modeling

data. This agreement has been verified for a wide

range of equivalent frequencies (200 Hz to 50 GHz).

A comparison of results in the Cole–Cole space

indicates that ultrasonic results yield values in terms of

both the real E1 and imaginary E2 parts of the complex

modulus not reached by mechanical tests (Fig. 7), and

which usually require an extrapolation with rheolog-

ical model. The potential of an ultrasonic test to

estimate rheological parameters has been clearly

demonstrated. As a matter of fact, k and the glassy

modulus E0, which respectively represent one of the

two parabolic creep elements in the rheological model

and the aggregate skeleton mixes, can only be

estimated using the ultrasonic test. Moreover, the

phase angle, a key parameter in viscoelastic behavior,

is also determined with the ultrasonic test.

In conclusion, the proposed ultrasonic test offers an

attractive method for the laboratory characterization

of bituminous materials.

The ability of ultrasonic testing to evaluate the

complex modulus is however primarily based on a

combination of the wave velocity and the wave

attenuation factor. A better determination of this

experimental attenuation factor would require quan-

tifying the attenuation caused by other sources. In

addition, the literature review has shown that Pois-

son’s ratio for asphalt concrete can range between 0.2

and 0.5 depending on temperature and frequency [21],

which demonstrates the necessity of shear wave

propagation measurements.

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towards a viscoelastic mechanical characterization of asphalt materials by ultrasonic measurements

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