Stiffness of Bituminous Mixtures Using Ultrasonic Wave Propagation

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Stiffness of Bituminous Mixtures Using UltrasonicWave PropagationHervé Di Benedetto a , Cédric Sauzéat a & Juliette Sohm aa Université de Lyon, Ecole Nationale des TPE, Département Génie Civil et Bâtiment(URA CNRS) , Rue Maurice Audin, F-69518, Vaulx-en-Velin Cedex E-mail:Published online: 19 Sep 2011.

To cite this article: Hervé Di Benedetto , Cédric Sauzéat & Juliette Sohm (2009) Stiffness of Bituminous Mixtures UsingUltrasonic Wave Propagation, Road Materials and Pavement Design, 10:4, 789-814

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Road Materials and Pavement Design. Volume 10 – No. 4/2009, pages 789 to 814

Stiffness of Bituminous Mixtures UsingUltrasonic Wave Propagation

Hervé Di Benedetto — Cédric Sauzéat — Juliette Sohm

Université de Lyon, Ecole Nationale des TPEDépartement Génie Civil et Bâtiment (URA CNRS)Rue Maurice AudinF-69518 Vaulx-en-Velin Cedex{herve.dibenedetto; cedric.sauzeat}@[email protected]

ABSTRACT. Wave propagation tests were performed on several specimens of bituminous mixes.Back analysis is made within the framework of linear viscoelastic materials to obtain dynamicmoduli. Comparison with moduli obtained from traditional cyclic tension-compressioncomplex modulus test, shows that dynamic tests and cyclic tests results fit very well. A uniquemaster curve is obtained and the time-temperature superposition is validated for very highfrequencies (several 10 kHz). Simulation with the model developed at University of Lyon,ENTPE laboratory (2S2P1D) confirms the ability of this model to cover the wide frequencyand temperature range. Furthermore, moduli obtained in different directions from dynamic(wave propagation) tests, reveal that different compaction techniques create different types ofanisotropy.KEYWORDS: Bituminous Materials, Wave Propagation Test, Complex Modulus,Tension/Compression Test, Time-temperature Superposition.

DOI:10.3166/RMPD.10.789-814 © 2009 Lavoisier, Paris

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790 Road Materials and Pavement Design. Volume 10 – No. 4/2009

1. Introduction

Properties measurement with ultrasonic (US) wave propagating in civilengineering material is an interesting investigation technique. It is rather simple toperform and does not damage the specimen. The two main possibilities are 1)emission technique (Miller, 1987; Li et al., 2006; Cordel et al., 2003; Sauzéat et al.,2007), and 2) propagation techniques (Arnaud et al., 2007; Di Benedetto et al.,2005; Duttine et al., 2007; Hochuli et al., 2001; Kweon et al., 2006; Lacroix et al.,2009; Stephenson et al., 1972 etc.). The first technique consists of hearing thematerials, while the second technique analyses the waves created by a shock or animpulse at the surface of the sample. These kinds of tests are clearly dynamic testsas inertia (or acceleration) effects are of main importance.

In this paper, measurement of wave velocities propagating in bituminousmixtures, at different temperatures, are presented. The proposed dynamicexperimental procedure is presented and validated. The data are interpreted in theframework of linear viscoelastic theory (VEL) for isotropic media. This backanalysis method gives the values of the complex modulus at very high frequencies(several 10 kHz).

Classical cyclic tension/compression complex modulus tests, including Poisson’sratio measurements, are also presented for the same materials. Temperaturesbetween -20°C and 70°C are considered. Frequencies which range from 0.03 Hz to10 Hz are far below the frequencies involved during dynamic tests.

Analysis of the data reveals that a unique modulus master curve is obtained fromdynamic and cyclic tests. This confirms theoretical results that dynamic and cyclictests give complementary rational information. It also shows a validation of thetime-temperature superposition (TTSP) principle for high frequencies. This newresult is of main importance, as it confirms the feasibility of wide application of theTTSP.

A complementary study with the developed dynamic test, on specimenscompacted using 2 different techniques (French LPC wheel compactor and gyratorycompactor), is also presented. This study aims at characterizing the anisotropy of thespecimens. The measured wave velocity is different following the direction ofpropagation and, surprisingly, the vertical direction appears as either softer (forwheel compactor specimen) or stiffer (for gyratory compactor specimen) than thetwo other directions. The influence of the compaction process is then clearlyoutlined.

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Stiffness of Bituminous Mixtures 791

2. Theoretical analysis

2.1. Wave propagation

A wave is generated by a shock on a piezoelectric sensor placed on one end ofthe specimen and is received at the other end of the specimen by anotherpiezoelectric sensor. The distance between the two sensors, L, and the time requiredby the wave to cover this distance, t, are used to calculate the propagation velocity,C, defined by: C = L / t

After determining the propagation velocity CP of the fastest wave generallycalled “compression wave” (P wave), the norm of complex Young modulus Ε∗can be calculated, using Equation [1], which gives the velocity of this wave in alinear isotropic viscoelastic material (Mandel, 1966) having a constant Poisson’sratio:

*(1 )1(1 )(1 2 )cos

2

P

EC

ν

φ ν ν ρ

−=

+ −

[1]

with:ρ : densityν : Poisson’s ratio (constant in the considered frequency range)Ε∗ : absolute value (norm) of complex Young’s modulusφ : phase angle or phase lag of the complex Young’s modulus.

2.2. The 2S2P1D model

Extensive works performed in our laboratory (at University ofLyon/ENTPE/DGCB) on the linear viscoelastic properties of bituminous bindersand mixes (Olard et al., 2003; Di Benedetto et al., 2004) allowed formulating ageneral unidirectional (1D) linear viscoelastic model with a continuum spectrumcalled 2S2P1D (two Springs, two Parabolic elements, one Dashpot). This modelbased on combination of simple physical elements is presented Figure 1. A tri-directional generalization of the 2S2P1D model (2S2P1D in 3D) is presented furtherin the paper.

The 2S2P1D model consists of a generalization of the Huet-Sayegh model(Sayegh, 1965). It has been shown in previous publications (Olard et al., 2003; DiBenedetto et al., 2004) that the 2S2P1D model is powerful to determine the linearviscoelastic behaviour (in the small strain domain) of bituminous binders, masticsand mixes, over a very wide range of frequencies and temperatures.

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Figure 1. Analogical form of the 2S2P1D model, Pk and Ph represent parabolicelements

The complex modulus expression of the 2S2P1D model is given in Equation [2].It requires only seven constants ( E00, E0, δ, k, h, η, τ ) at a given temperature.

( )( ) ( ) ( )

* 0 002 2 1 00 11

S P D k hE E

E Ej j j

ωτδ ωτ ωτ ωβτ− − −

−= +

+ + + [2]

where:j is the complex number defined by j2 = - 1ω is the pulsation, ω = 2 π f, (f is the frequency )k, h: constants such as 10 ≤≤≤ hkδ: constantE00: the static modulus, which is the value of E* when ω tends toward 0

( 0→ω )E0: the glassy modulus, which is the value of E* when ω tends toward infinity

( ∞→ω )η : Newtonian viscosity 0 00( )E Eη βτ= −

τ : characteristic time, whose value varies only with temperature. τ evolutionmay be approximated by a WLF type law (Ferry, 1980) in the range of consideredtemperatures. It accounts for the Time-Temperature Superposition Principle:

0).()( ττ TaT T=

where )(TaT is the shift factor at temperature T. )(0 refTττ = is determined at Tref.

The shift factor at temperature T, )(TaT , can be determined by means of theWLF equation (Ferry, 1980) for bituminous materials (Equation [3]).

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1

2

( )log( ) ref

Tref

C T Ta

C T T−

= −+ −

[3]

It has to be emphasized that only seven constants ( E00, E0, δ, k, h, η, τ ) areneeded to entirely determine the linear viscoelastic behaviour of the material, at agiven temperature. If the hypothesis of a linear viscoelastic thermorheologicallysimple behaviour can be applied to the materials (which means the Time-Temperature Superposition Principle holds (TTSP)), only the parameter τ dependson temperature. If the TTSP holds, the two additional constants C1 and C2

(calculated at the reference temperature Tref of the WLF equation) are needed. Thenthe total number of constants of the unidirectional expression of the model amountsto nine.

A generalization of the 2S2P1D model for the three dimensional case (2S2P1Din 3D) is proposed in (Di Benedetto et al., 2007a and 2007b). For isotropic linearviscoelatic materials, only the expression of the Poisson’s ratio ν* is needed.Experiments show that Poisson’s ratio ν* is a complex number. It is modelled by theexpression given in Equation [4]

*2 2 1 0

100 0

11 ( ) ( ) ( )

S P Dk hj j j

ν νν ν δ ωτ ωτ ωβτ− − −

−=

− + + + [4]

where only two new constants are added:

00ν which is the value of ν* when ω tends toward 0 ( 0→ω )

0ν which is the value of ν* when ω tends toward infinity ( ∞→ω )

The other constants are the same than in Equation [2].

From Equations [2] to [4], it can be checked that the global expression of the2S2P1D model in the three directional case for isotropic linear viscoelastic materialrequires 11 constants. This model allows simulating any loading conditionsincluding temperature changes.

3. Calibration of the 2S2P1D model from tension/compression (push/pull)complex modulus tests

In order to use the 2S2P1D model to interpret the dynamic wave propagationtests a calibration of the model in the quasi static cyclic regime is first realized. Let’sunderline that the quasi static hypothesis, which consists in neglecting the inertiaeffects for the interpretation of the test, are justified for traditional tests onbituminous materials up to some tens hertz. As a general remark, the name DSR(Dynamic Shear Rheometer) is then a very bad choice as no dynamic effect existsduring this test.

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Tension/compression tests were performed on the same materials than the onetested with our ultrasonic wave propagation system.

The test (Di Benedetto et al., 2007a; Delaporte et al., 2008) consists ofmeasuring the sinusoidal axial 1( )ε and radial 2( )ε strains when a sinusoidalloading 1( )σ is applied on cylindrical specimens of asphalt mixture.

Small axial strain amplitudes (about 6.10-5m/m) were applied ensuring thebehaviour to remain inside the linear domain (Doubbaneh, 1995; Di Benedetto et al.,2004, 2007a; Delaporte et al., 2007). The sinusoidal evolution with time of the threemeasured values is defined by the following equations:

)sin()( 011 tt ωεε = ,

1 01( ) sin( )t tσ σ ω φ= +

2 02 02( ) sin( ) sin( )t t tν νε ε ω π φ ε ω φ= + + = − + .

φ is the classical complex modulus (E*) phase angle between the axial strainand the axial stress. νφ is the phase angle of the complex Poisson’s ratio (ν*)between the axial strain and the opposite of radial strain.

Considering complex notations where j is the complex number defined by12 −=j , the measured values are written *

1 01( ) j tt e ωε ε= , * ( )1 01( ) j tt e ω φσ σ += and

* ( )2 02( ) j tt e ω φνε ε += − .

Complex modulus *E and complex Poisson’s ratio *ν of materials can then beobtained from these homogeneous tension/compression tests with Equations [5] and[6]:

** *1

*1

( ) jE E e φεω

σ= = [5]

** *2

*1

( ) je φνεν ω ν

ε= − = [6]

Results from these tests are presented further in the paper

4. Ultrasonic wave propagation device and experimental procedure

Two piezoelectric sensors from Euro Physical Acoustic Company are used. Thecontact surface is circular (15mm diameter). These sensors, whose frequency rangeis between some 10 kHertz up to 100 kHertz, were used differently (acousticemission) in previous studies to detect the damage during TSRS Tests (ThermalStress Restrained Specimen Test) (Cordel et al., 2003, Sauzéat et al., 2007). This

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section presents the ultrasonic (US) wave propagation test and a validation of thechosen procedure.

4.1. Presentation of the test

The two sensors are fixed on each side of the specimen with lubricate silicone.The specimen is placed on a very soft polystyrene base in order to avoid effect ofwave propagation in the support. An impact is imparted on sensor 1, placed on oneend of the specimen, with a bolt hanged from a wire (Figure 2).

Figure 2. Disposition of the two piezoelectric sensors on a cylindrical bituminousmix specimen and system used to produce the impact

Waves generated through sensor 1 propagate inside the specimen before beingdetected by sensor 2, placed at the oposite end of the specimen. Signals of bothsensors are recorded and analysed using an oscilloscope and a computer.

As compression wave (P-wave) is the fastest, it is therefore the first one to reachthe receiver (Brignoli et al., 1996; Duttine et al., 2007; Di Benedetto et al., 2005).No reflected waves could arrive before P waves.

An example of waves recorded by the two sensors for bituminous mixture test at26°C is given in Figure 3. The arrival time of the P-wave can be easily identifiedfrom the two piezoelectric sensors’records.

Propagation time of the P-wave, tp, is the difference between the two arrivaltimes. The P-wave velocity Cp is obtained by dividing the height of the sampleby tP.

The next section presents a rapid validation of the system.

Sensor 1

Sensor 2Bolt usedfor impact

14 cm

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0 20 40 60 80 100 120 140

-10-8-6-4-202468

10Arrival Time

Am

plitu

de (V

)

Time (µs)

Sensor 1 Sensor 2

Propagation Time

Arrival Time

Figure 3. Example of recorded signals and arrival times of P-wave in bituminousmixture specimen at 26°C

4.2. Validation of the experimental procedure

4.2.1. Check of the calibration of the sensors

The two sensors were placed on the same side of a bituminous mix specimen atthe same distance from the cylindrical axis of symmetry. An impact was made onthe other end, in the centre of the specimen. Figure 4 shows the disposition of thesensors and the impact position.

Sensors Impact

Figure 4. Position of the two sensors and impact point

The two recorded signals give the same arrival time and similar evolution withtime, which means that the two sensors have similar calibration.

4.2.2. Check of the time delay in the acquisition system

Time delay is measured by placing the two sensors directly in contact. Themeasurements show that the time delay is very small and can be neglected whencomparing to the flying time of the wave in the sample.

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4.2.3. Repeatability of impacts and wave velocities

Wave propagation tests using configuration presented Figure 2 are realisedseveral times on the same specimen. The P-wave velocities (Cp) obtained from thedifferent impacts created by the bolt having initially close positions are measured.Even though the amplitudes of the signals are not identical because the impacts werenot realized with the same intensity, the P-wave velocities are very close. Forexample, Table 1 presents the results of five replicates on a bituminous mixturespecimen at three different temperatures (-19°C, 26°C and 42°C). The maximumcoefficient of variation (COV) observed at 42°C, is less than 4%. For the two othertemperatures the COV is lower than 1%.

Even though the repeatability is very good, all presented data from ourexperimental work is an average of five wave propagation tests made successively.

Table 1. Values of P-wave velocities for 5 replicates at three temperatures(sample 2.2)

Cp (T=-19°C) (m/s) Cp (T=26°C) (m/s) Cp (T=42°C) (m/s)

Test 1 3922 3448 2734

Test 2 3933 3491 2740

Test 3 3966 3457 2546

Test 4 3955 3491 2778

Test 5 3922 3483 2602

Average 3939 3474 2680

Standard Deviation 20.2 20.2 100.3

Stand. dev./average(COV) (%) 0.5 0.6 3.7

4.2.4. Frequency of the wave

The Fourier transform of the recorded signal considering disposition of Figure 2,is presented in Figure 5. The frequencies range between about 10 kHz to 70 kHzcontain some energy. The value of the main frequency, which does not change withthe considered range of temperature (-19°C to 42°C), is about 30 kHz. This mainfrequency of 30 kHz is also obtained when applying an impact on a sensor lying ona polystyrene plate (Figure 6). Then, it can be concluded that 30 kHz is the mainoscillatory frequency for the transducer. In the back calculation of the test, thisfrequency is chosen as the main loading frequency of the impact.

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0 20 40 60 80 1000

4000

8000

12000

16000A

mpl

itude

Frequency (kHz)

Fourier transform sensor 1 Fourier transform sensor 2

Figure 5. Fourier transform of the signals obtained from the 2 sensors fixed on abituminous mixture specimen, as shown in Figure 2, at 26°C

0 20 40 60 80 1000

20000

40000

60000

Am

plitu

de

Frequency (kHz)

Figure 6. Fourier transform of the signal obtained in realizing an impact on asensor which is freely placed on a polystyrene plate at 26°C

4.2.5. Tests on stainless steel specimen

The same disposition as presented in Figure 2 is considered for a stainless steelcylindrical specimen having 15 cm height and 10 cm in diameter. The Youngmodulus, density and Poisson’s ratio of this elastic material are about 200 GPa,7800 kg/m3 and 0.24, respectively.

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Equation [1] between the velocity of the P-waves and the Young modulus Etakes the following form in the case of elastic material :

21 ( 1)(

(2 1)P

EC νρ ν ν

−=

+ −[7]

Considering a Poisson’s ratio of 0.24, the Young’s modulus obtained for thespecimen is 213 GPa. Wave propagation tests were replicated 5 times. The value ofthe obtained standard deviation is 8 GPa. The value obtained using dynamic tests isclose to the classical given value of 200 GPa (7% difference). Considering that theYoung’s modulus and the Poisson’s ratio of the stainless steel specimen are not veryprecisely known, this small difference makes us confident in our US wavepropagation test interpretation.

5. Tests on cylindrical specimens: validation of time-temperature SuperpositionPrinciple at high frequencies for mixes

Analyses of tests on two different bituminous mixtures (ENTPE 2 and ENTPE3)are presented in this section.

Bituminous mixture specimens were cored from samples obtained with agyratory compactor (standard EN 12697-31).

Table 2. Characteristics of the two mixtures

Mixture name ENTPE-2 ENTPE-3

Washed 0/2* aggregates (% weight) 28.8 29.9

2/6* aggregates (% weight) 21.7 22.6

6/10* aggregates (% weight) 42 43.6

Silica filler (% weight) 3.75 3.9

limestone filler (% weight) 3.75 0.00

binder content (% weight of aggregates) 6 6

Compacity 0.962 0.960

* x/y means that the aggregates size is between x mm and y mm.

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Complex modulus measurements using tension compression cyclic tests andwave propagation tests were performed on both mixtures. The two mixtureformulations are based on a continuous 0/10 mm diorite grading prepared with a50/70 pure bitumen proportioned at 6% (dry weight of the aggregate). The onlydifference between the two mixtures is the type and content of fillers as indicated inTable 2.

Before mixing the materials, aggregates have been washed to remove fillermainly contained in the sand. Then, the filler content in the asphalt concrete iscompletely controlled.

5.1. Wave propagation tests

Dynamic tests were performed at three different temperatures: -19°C, 26°C and42°C (Sohm, 2007). A sensor was fixed on each side of the cylindrical specimen andan impact was made on the sensor 1, as presented in Figure 2.

The cylindrical specimen 2.2 of mixture ENTPE-2 was 14 cm high and 7.5 cmdiameter. Its density is 2440 kg/m3. The cylindrical specimen 3.1 of mixtureENTPE-3 was 11.4 cm high and 7.5 cm diameter. Its density is 2432 kg/m3.

The measured P-wave velocities (Cp) are given in Table 3 for the 2 specimens.Each velocity is an average of five tests. As indicated in Section 4, the process iswell repeatable.

Table 3. Measured values of P wave velocities (Cp) for materials ENTPE-2 andENTPE-3 (5 replicates)

Mixture ENTPE-2 ENTPE-2 ENTPE-2 ENTPE-3 ENTPE-3 ENTPE-3

Specimen 2.2 2.2 2.2 3.1 3.1 3.1

Temperature (°C) -19 26 42 -19 26 42

P wave velocity (m/s) 3939 3474 2680 3991 3190 3095

Coefficient of variation (%) 0.5 0.6 3.7 0.8 0.6 0.3

5.2. Tension-compression cyclic tests and calibration of the 2S2P1D model

In order to obtain the complex modulus E* and the Poisson’s ratio ν* asspecified in Section 3, an hydraulic press was used at the ENTPE/DGCB laboratory.The applied frequencies range from 0.03 Hz to 10 Hz, over a wide range of

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temperatures from -20°C to 70°C (one isotherm every 10°C). 120 mm highcylindrical (diameter = 80 mm) specimens were cored from cylindrical PCG(Gyratory Compactor) samples of mixtures ENTPE-2 and ENTPE-3 (cf. Table 2).The samples were tested in tension/compression. At each cycle, the strain wasconsidered as the mean value of the measurements given by three displacementtransducers placed at 120° around the sample (Di Benedetto et al., 1996; Baaj et al.,2005). The hydraulic press was controlled in strain mode on the mean value of thetransducers, with amplitude fixed at 60.10-6 m/m. This small strain amplitudeassures to remain inside the domain of linear viscoelastic behaviour. As explained inSection 3, the isotropic linear viscoelastic theory for thermorheologically simplebodies (i.e. the time-temperature superposition principle is verified) is verified forthe mixtures and considered in the analysis of the data.

Then, norm of the complex modulus (E*) master curves and norm of thecomplex Poisson’s ratio (ν*) master curve can be plotted using the shiftingprocedure. As an example, Figure 7 represents the values of ν* obtained atdifferent frequencies and temperatures and the master curve obtained at a referencetemperature of 0°C, for sample 2.1 made with mixture ENTPE-2. It should beemphasized that the values of the shift factor aT are very close for the complexmodulus and the complex Poisson’s ratio, which confirms the previousinvestigations presented in Di Benedetto et al. (2007a, 2007b, 2008). The mastercurve of E* for sample 2.1 (mixture ENTPE-2) and sample 3.2 (mixture ENTPE-3) are plotted at the same reference temperature of 0°C in Figures 10 and 11,respectively.

The values obtained for the shift factor aT using the shift procedure and fittedcurves using WLF equation (Equation [3] and constants given in Table 4), areplotted Figure 8 for the 2 mixtures ENTPE-2 and ENTPE-3, at a referencetemperature of 0°C.

From Figure 7, the Poisson’s ratio is not constant and varies with frequency andtemperature. It decreases when decreasing temperature or increasing frequency. Thischange should be taken into account to back analyse the wave propagation tests, inparticular when using Equation [1]. Meanwhile at a given temperature and forfrequencies values having few difference (up to a factor 5), it can be seen that thePoisson’s ratio can be considered as constant. Values of the Poisson’s ratio phaseangle (φν), not presented in this paper, vary between 0° and 5°. These small valuesconfirm that the Poisson’s ratio can be considered as real in our analysis. Meanwhileviscous lag could be taken into account for other specific applications needing moreaccuracy.

The tension compression test data are used to calibrate the 2S2P1D in 3D model(Section 2). The values of the 11 constants of the model obtained for materialsENTPE-2 and ENTPE-3 are given in Table 4. Simulation with the three dimensionalformulation of the 2S2P1D model using these constant are plotted together with theexperimental data of the complex modulus in Figures 10 and 11. It can be seen on

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these figures that the predicted values are in very good agreement with experimentaldata over the wide range of temperatures and frequencies used during cyclic tests.

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

1,E-06 1,E-04 1,E-02 1,E+00 1,E+02 1,E+04 1,E+06

a T . frequency (Hz)

|ν*|

30°C

19.8°C

8.5°C

-2.1°C

-12.5°C

-22.3°C

aT(ν) 19,8°C

Master curve at 0°C

Figure 7. Poisson’s ratio values at temperatures ranging from -22,3°C to 30°Cobtained from compression/tension tests and master curve at 0°C. (MixtureENTPE-2)

1,E-08

1,E-06

1,E-04

1,E-02

1,E+00

1,E+02

1,E+04

1,E+06

-40 -20 0 20 40 60 80

Temperature (°C)

shif

t fa

cto

r a

T

aT 2-1aT WLF 2-1aT 3-2aT WLF 3-2

Tref = 0°C

Figure 8. Shift factor aT data using the shift procedure and fitted curves using WLFequation (Equation [3] and constants given in Table 4), at a reference temperatureof 0°C for mixtures ENTPE-2 and ENTPE-3

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Table 4. Constants of the 2S2P1D in 3D model for the two mixtures (ENTPE-2 andENTPE-3) at a reference temperature of 0°C

No.E00

(MPa)E0

(MPa)k h δ

τ0

(s)β ν00 ν0 C1

C2

(°C)

2 3,0E+01 4,0E+04 0,21 0,55 2,3 9,0 1,0E+20 0,40 0,15 24,3 147,4

3 1,0E+01 3,5E+04 0,21 0,55 2,3 1,5 1,0E+20 0,43 0,22 22,9 141,1

The previous calibration of the 2S2P1D model allows calculating phase angle ofE* and Poisson’s ratio at 30 kHz, which is the main frequency of the wavepropagation tests. These values are reported in Table 5, for the three consideredtemperatures and for the two mixtures ENTPE-2 and ENTPE-3. Theses obtainedvalues are used to interpret wave propagation tests as explained in next section.

Table 5. Values of the phase lag φ of E* and norm of the complex Poisson’sratioν=ν∗ obtained at 30 kHz with the 2S2P1D model, for the three temperaturesand for the two mixtures ENTPE-2 and ENTPE-3

ENTPE-2 ENTPE-3

φ (°) ν φ (°) ν

T = -19°C 0.3 0.15 0.3 0.22

T = 26°C 7.5 0.24 7.2 0.30

T = 42°C 15.5 0.31 14.3 0.35

5.3. Back analysis of wave propagation tests (dynamic tests)

As the Poisson’s ratio and the phase angle of bituminous mixture specimens areknown from the 2S2P1D model (calibrated with cyclic tension/compression testsdata) (see Section 5.2), Equation [1] can be used to obtain the E* values from thevelocities of the P-waves. Then, the complex Young modulus can be determined forthe three temperatures for the 2 mixtures.

The values of the norm of the complex modulus E* obtained from wavepropagation velocities Cp given in Table 3 using φ and ν given in Table 5, areplotted Figure 9 for mixture ENTPE-2. The complex modulus values E* obtainedat different temperatures and frequencies during cyclic tension-compression tests arealso plotted in this Figure 9.

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These values of the norm of the complex modulus E* obtained from wavepropagation velocities Cp are also reported in Table 6 for the 2 mixtures at the 3considered temperatures.

Table 6. Moduli obtained from wave propagation tests and moduli obtained from2S2P1D in 3D model calibrated from cyclic tension/compression tests, for the twospecimens ENTPE-2 and ENTPE-3, for the three considered temperatures. 2S2P1Dsimulations are made at 30kHz

ENTPE-2 ENTPE-3

E* (wavetests) (MPa)

E* (2S2P1Dfrom cyclictests) (MPa)

difference(%)

E* (wavetests) (MPa)

E* (2S2P1Dfrom cyclictests) (MPa)

difference(%)

T = -19°C 35900 39300 8.7 34000 34400 1.2

T = 26°C 25400 24800 2.5 20700 22700 8.8

T = 42°C 11700 14400 18.3 13500 14000 3.5

The moduli simulated by the 2S2P1D model are also given in Table 6. It can beseen that the difference between the values of the complex modulus simulated fromcyclic tests and the one obtained with dynamic tests is rather small (less that 10%except for one test at 42°C where it reaches 18.3 %).

f = 30kHz

Figure 9. Complex modulus data obtained during tension/compression tests andvalues given by wave propagation tests for mixture ENTPE-2

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These results confirm that, i) the dynamic back calculation seems pertinent and,ii) the 2S2P1D model is able to give good simulation for high frequencies. This lastresults as never been obtained previously and is of great importance for pavementloading analysis.

Considering the Time-Temperature Superposition Principle it is possible to plotthe obtained dynamic moduli on the master curve of the complex modulus. Thechosen reference temperature is 0°C.

The shift factor aT for the 3 temperatures (42°C, 26°C and -19°C) is determinedfrom cyclic tension/compression tests using shift procedure and WLF equation, asexplained before (Equation [3] and Figure 8). Values of the equivalent frequenciesfor the 2 mixtures are reported in Table 7.

Table 7. Values of equivalent frequencies (aT *30 kHz) at a reference temperature of0°C, for the three temperatures and the two specimens from mixtures ENTPE-2 andENTPE-3

aT*30 kHz (Hz)

ENTPE-2 ENTPE-2

T = -19°C 1.18E+8 1.10E+8

T = 26°C 6.83E+0 8.22+0

T = 42°C 1.23E-1 1.68E-1

Values of complex Young modulus obtained from wave propagation tests(Table 6) using equivalent frequencies given in Table 7, are reported on the mastercurve plotted with tension/compression tests in Figures 10 and 11 for mixturesENTPE-2 and ENTPE-3, respectively. The simulated master curve with the 2S2P1Dmodel (constants given in Table 4) is also plotted in these 2 figures.

From Figures 10 and 11, the dynamic data fit very well with the cyclic data. Toconfirm this qualitative affirmation, an analysis of the error in back analysing thedynamic tests is presented in Section 5.4.

A first and main output from the presented analysis is that the time-temperatureprinciple (TTSP) is validated for high frequencies (up to several 10 kHz). Thisresult, which is new, is quite important and provides a generalisation of the TTSP. Itis of great importance for general pavement loading analysis.

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1,E+01

1,E+02

1,E+03

1,E+04

1,E 05

1,E-11 1,E-08 1,E-05 1,E-02 1,E+01 1,E+04 1,E+07 1,E+10aT . frequency (Hz)

|E*|

(M

Pa) 2S2P1D

ENTPE2-1

test 26°C

test -19°C

test 42°C

Wave propagation tests

Figure 10. Master Curve of the complex modulus at the reference temperature of0°C, data (plain black label) obtained experimentally from tension/compressiontests, simulation with the 2S2P1D model and values obtained from wavepropagation tests, for mixture ENTPE-2 (Table 2)

1,E+00

1,E+01

1,E+02

1,E+03

1,E+04

1,E+05

1,E-11 1,E-08 1,E-05 1,E-02 1,E+01 1,E+04 1,E+07 1,E+10

aT . frequency (Hz)

|E*|

(M

Pa) 2S2P1D

ENTPE3-2

test 26°C

test -19°C

test 42°C

Wave propagation tests

Figure 11. Master Curve of the complex modulus at the reference temperature of0°C, data (plain black label) obtained experimentally from tension/compressiontests, simulation with the 2S2P1D model and values obtained from wavepropagation tests, for mixture ENTPE-3 (Table 2)

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5.4. Evaluation of the error in back analysis of dynamic tests

Equation [1] can be rewritten as:

* 2 2 (2 1)( 1)cos ( )2 ( 1)PE C φ ν νρ

ν− +

=−

[8]

As Cp=L/t, the relative error of measurement of complex modulus ( * *E E∆ ) is

given by Equation [9] obtained from derivation of Equation [8].

*

*2 (2 )2 tan( )

2 (1 )(1 )(1 2 )

E ttE

φ ν νφ νν ν ν

∆ ∆ −= + ∆ + ∆

+ − −[9]

Where:

tt∆2 represents the error of measurement associated to the determination of

the propagation time. From experimental curves, t∆ is estimated equal to 1 µs.

φφ ∆)2tan( represents the error of measurement associated to the complex

modulus phase angle. It is linked with temperature and frequency variations.2 (2 )

(1 )(1 )(1 2 )ν ν ν

ν ν ν−

∆+ − −

represents the error of measurement linked to the

Poisson’s ratio. It is linked with temperature and frequency variations.

φ∆ and ν∆ are determined with the 2S2P1D in 3D model considering that thetemperature variation (∆T) is estimated to 1°C and that the value of the frequencyvariation (∆fr) is 20 kHz ( between 10 kHz and 50 kHz).

The calculated relative error of measurement on *E ( * *E E∆ ) for the three

different temperatures for the specimens from mixtures ENTPE-2 and ENTPE-3 areindicated in Tables 8 and 9, respectively.

From this analysis of error (Tables 8 and 9) for dynamic tests it can be concludedthat:

– the error introduced by the determination of the phase lag φ is negligible. Itrepresents less that 1.5 percent whatever the temperature,

– the error produced by the estimation of the Poisson’s ratio ν is more importantbut remains rather small. It is negligible at low temperature and reaches about 10percent at 42°C.

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– the total error on *E increases from about 7% at low temperature up to about

17% at 42°C. This error is small when comparing with the huge variation ofmodulus observed when changing temperatures and frequencies in the consideredrange. It is also quite compatible with the differences obtained between dynamicmoduli and calculated moduli with the 2S2P1D model, as presented in Table 6.

Table 8. Values of relative errors for dynamic tests for mixture ENTPE-2, values ofE* are reported in table 6

Error linked to T and frError* *E E∆

Error linked with themeasurement of t (%)

Error from φ (%) Error from υ (%)

Total error(%)

T = -19°C 5.6 0.0 0.8 6.4

T = 26°C 5.0 0.4 5.6 11.0

T = 42°C 3.8 1.3 12.2 17.3

Table 9. Values of relative errors for dynamic tests for mixture ENTPE-3, values ofE* are reported in Table 6

Error linked to T and frError* *E E∆

Error linked with themeasurement of t (%)

Error from φ (%) Error from υ (%)

Total error(%)

T = -19°C 7.1 0.0 0.0 7.1

T = 26°C 5.7 0.3 8.4 14.4

T = 42°C 5.5 1.0 8.8 15.3

6. Application of dynamic tests: anisotropy characterisation and influenceof the specimen preparation method

In order to investigate the anisotropy of bituminous mixtures, wave propagationtests were performed in different directions on samples prepared using two methods.

The first parallelepipedic tested specimens were sawed from plate (60cm * 40cm* 15 cm) made using the French LPC wheel compactor (EN12697-33). Orientationof the sample relatively to the compactor wheel displacement is indicated in Figure12. X direction corresponds to the displacement of the wheel, Z is the verticaldirection and Y is perpendicular to X and Z. The dimensions of the specimen areindicated in Table 10.

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Figure 12. Parallelepipedic specimen sawed in plate made with the French LPCcompactor. X, Y and Z orientation are given from vertical direction and wheeldisplacement

The second type of tested specimens was cored from samples made with theFrench gyratory compactor (EN12693-31). Orientation of the sample relatively tothe compaction direction is indicated in Figure 13. Z direction corresponds to thedirection of compaction, which is also vertical during the confection process. Asindicated in Table 11, the heights of the specimens were nominally 14cm and itsdiameter is 7.5cm.

Wave propagation tests and interpretation using the back analysis presented inprevious sections were done for both types of specimens in directions X, Y and Z ata temperature of 26°C. The disposition of the 2 piezoelectric sensors (cf. Figure 2)was adapted so that the wave velocity measurements could be done successively indirection X, Y and Z. As the obtained anisotropy is less than 15% (see resultsbelow), the hypothesis of isotropy could still be considered with a goodapproximation for the dynamic back analysis.

Table 10. Values of propagation time, velocity of P waves and obtained complexmodulus for the parallelepipedic specimen (T=26°C)

Direction of wave propagation Tp(µs) L (cm) CP (m/s) E* (Mpa)

X 20.8 7.5 3625.3 23750

Y 18.3 6.5 3581.6 23200

Z 21.5 7.1 3284.8 19500

The Poisson’s ratio and the phase angle of parallelepipedic samples, that areunknown, were evaluated to: φ=10° and υ=0.3. From these values and the dynamictests, complex modulus can be determined using previously presented methods.

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Values of propagation times, velocities of the P waves and obtained complexmodulus are given in Table 10.

From Table 10, the complex moduli in direction X and Y are close; whilemodulus in direction Z is about 10% smaller. It comes:

* * *( ) ( ) ( )E X E Y E Z≈ ≥ [10]

Direction of compaction

Figure 13. Cylindrical specimen cored from sample made with the gyratorycompactor

The cylindrical type specimen (cf. Figure 13) was also tested using complexmodulus tension/compression test in Z direction so that the Poisson’s ratio and thephase angle could be determined with experimental data and 2S2P1D in 3D model(as explained in Section 5.2). The obtained Poisson’s ratio is 0.31 and the phaseangle is 6.9°.

The results of wave propagation tests on cylindrical specimen, in the 3directions, are reported in Table 11.

Table 11. Values of propagation time, velocity of P waves and obtained complexmodulus for the cylindrical specimen (T=26°C)

Direction of wave propagation Tp(µs) L (cm) CP (m/s) E* (Mpa)

Z 45.9 14 3030.3 16100

X or Y 25.9 7.5 2899.7 14700

For this type of specimen, the complex modulus in X and Y direction areidentical, which is an expected result when considering the material cylindricalsymmetry. The great difference with the parallelepipedic specimen is that the

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modulus in the Z direction is about 10% higher than complex modulus in directionX or Y. Then it comes:

* *( ) ( , )E Z E X Y≥ [11]

This last observation reveals that the specimens made with the two methods(French wheel compactor and French Gyratory compactor) are both slightlyanisotropic but don’t have the same type of behaviour. For plate compactorspecimen, compacting direction (Z) is the softer while it is the stiffer for specimenfrom gyratory compactor. This is due to a different geometrical organisation of thegranular skeleton, whose effects are far to be negligible. Similar conclusion ispresented in Ezaoui et al., 2007, from results on dry sand prepared using differenttechniques.

An important output from these data concerns the representativity of specimenstested in laboratory. The compositions of the two tested mixtures are not exactly thesame but clear qualitative information can be outlined. As two different preparationtechniques give specimens having different mechanical properties, the specimenshaving the closest properties that the one present in the pavement should be chosen.It is the opinion of the authors that the wheel compactor provides specimens thatreconstitute better materials present in the road. Then, our results confirm that usinggyratory compaction preparation method does not appear as a good choice, to obtainmechanical properties of pavement mixtures.

7. Conclusion

A series of wave propagation tests were performed on different types ofbituminous mixtures specimens. The complex modulus is obtained at differenttemperatures from P wave velocities measurements and rational approachconsidering isotropic linear viscoelastic behaviour. Complex modulus values arealso obtained, using traditional cyclic tension-compression tests, on the samematerials. These values are considered in the comparison of the 2 kinds of tests(dynamic and cyclic).

The developed method is validated and reveals to be quite efficient. A first andmain output from the data is that the time-temperature superposition principle(TTSP) is validated for high frequencies (up to some 10 kHz). This new result isquite important. It provides a generalisation of the TTSP and is of great importancefor general pavement loading analysis.

The dynamic back calculation is consolidated with the 2S2P1D in 3 dimensionsmodel, which gives good simulations for high frequencies.

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The proposed dynamic test method is also applied to measure values of themoduli in different directions on specimens made using 2 different preparationtechniques: French wheel compactor and gyratory compactor.

A slight anisotropy of about 10% is obtained for the 2 types of specimens.Surprisingly, the compaction direction (Z) is softer for wheel compactor specimenand stiffer for “gyratory” specimen than the 2 other directions (X and Y).

These results confirm that gyratory compaction preparation method providessamples that have different mechanical properties than wheel compactor one. Aswheel compactor process is most probably rather close to the in situ compactionprocess, it is underlined that gyratory technique does not appear as a good choice toprovide specimens having mechanical properties of road mixtures.

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Di Benedetto H., Geoffroy, A. Duttine, C. Sauzéat, B.Chau, « Comportement anisotrope dessols et caractérisation d’un site à partir d’essais de propagation d’ondes », Congrès

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Received: 1 December 2008Accepted: 30 October 2009

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