Determination of the dynamic complex modulus of viscoelastic materials using a time domain approach

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Test method

Determination of the dynamic complex modulus of viscoelastic

materials using a time domain approach

Paolo Bon

ﬁglio

*

, Francesco Pompoli

Dipartimento di Ingegneria, Universita degli Studi di Ferrara, Via Saragat 1, 44122 Ferrara, Italy

a r t i c l e i n f o

Article history: Received 19 August 2015 Accepted 25 September 2015 Available online 30 September 2015 Keywords:

Viscoelastic materials Elastic properties Measurement methods

a b s t r a c t

Viscoelastic and poroelastic materials are widely used in multilayer panels for noise control. They are usually used as an inner decoupling layer in double wall systems in order to increase the sound trans-mission loss of a bare plate. In order to correctly simulate the acoustical behaviour of such systems, it is necessary to measure the elastic parameters of these materials (storage and loss moduli, and Poisson's ratio). Physical properties related to pore morphology also need to be determined for open cell struc-tures. Most of the materials used in trimmed panels can show elastic parameters that vary with fre-quency, thus a quasi-static measurement technique is not accurate enough to consider such viscoelasticity effects. This paper focuses on the estimation of complex modulus as a function of the frequency of isotropic viscoelastic materials. In particular, the tested material is positioned between two plates, with one of them being excited by an electromagnetic shaker. Using a sine burst as an excitation signal, the accelerometric response in the time domain is measured at the top and bottom plates. The time ofﬂight between the plates and the envelope function of time domain acceleration at the top plate are then found. A transfer matrix model of the experimental setup is used to inversely estimate the complex modulus of the materials once the remaining mechanical and physical properties have been ﬁxed. The results will be presented and discussed for different materials and compared with well-established quasi-static and dynamic techniques.

1. Introduction

The experimental determination of the mechanical properties of viscoelastic solids as a function of frequency can be performed

using various techniques. These methods may be initially classiﬁed

as quasi-static and dynamic, and they consist generally of mea-surements made in a narrow range of frequencies (possibly carried out at different temperatures). The choice of the appropriate

measurement technique is inﬂuenced by the geometry, damping

factor and frequency range of interest. It has to be considered that, when the tests are not made in vacuum conditions, stiffness de-pends not only on the mechanical parameters of the structure but

also on its interaction with the saturatingﬂuid (that is free air in

practical noise control applications). The inﬂuence of the latter

increases as the airﬂow resistivity and frequency also increase.

A wide review of existing methods for determining mechanical properties of materials used in noise and vibration control was

presented by Jaouen et al. [1]. In that paper, several quasi-static

[2e4] and dynamic [5e12] techniques are discussed and

compared. Apart from the use of the time-temperature

super-position principle[4](presenting several drawbacks in determining

the high loss factors as underlined in Ref.[1]) and the phase velocity

measurement-based methods [10e12] (which require a laser

vibrometer and samples of a large dimension), a common limita-tion of the existing methods is that the maximum frequency is less than 1000 Hz.

Considerable work has been done by Kulik et al.[13] using a

resonant method and a pseudo-spectral approximation of the governing wave equations to determine the complex modulus of cylindrical shaped viscoelastic materials.

The aim of this research is to present a method to determine the

values of the storage and loss moduli (hereafter indicated ad E1[Pa]

and E2[Pa]) in a wide frequency range measuring the time domain

accelerations and using a transfer matrix approach. The transfer

matrix method[14]has been developed for the study of plane wave

propagation within multilayer systems (made of elastic solids, rigid and elastic framed porous materials, membranes, resonators, * Corresponding author.

E-mail address:paolo.bonﬁglio@unife.it(P. Bonﬁglio).

Contents lists available atScienceDirect

Polymer Testing

j o u rn a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / p o l y t e s t

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porous and impervious sheets, etc.). This method can be used for

simulating normal and oblique incidence as well as diffuse ﬁeld

sound absorption and transmission loss.

The paper is organised as follows: the description of the

methodology is given in Section2. Section3reports a description of

the experimental set-up and the results for preliminary numerical simulations and real tested materials compared with quasi-static and dynamic techniques. Finally, concluding remarks are given in the last section.

2. Description of the methodology

The entire measurement procedure can be summarised as fol-lows. The tested material (here assumed to be homogeneous and isotropic) is positioned between two plates (aluminium support plate 10 mm thick and steel top plate 1 mm thick), one of them being excited by an electromagnetic shaker. Using a sine burst as the excitation signal, the accelerometric response in the time

domain is measured at the top and bottom plates.Fig. 1depicts the

measurement layout and the time domain response related to the accelerometers at a given frequency.

From the tests, it is possible to determine the time of ﬂight

between measured signals and the envelope of the top mass

accelerometer, as depicted in Fig 1(b) and 1(c). Afterwards, the

same quantities are used in a minimization procedure for the

determination of complex modulus. The time ofﬂight between the

input and the output signals is measured determining the time delay at which the maximum of their cross correlation function occurs. The envelope of the output signal is calculated by using the following expression:

envacc1¼ ja1ðtÞ þ i$ImðHða1ðtÞÞÞj (1)

a1(t) being the acceleration at the top plate and H the Hilbert

transform of the time domain acceleration signal.

In order to determine the complex modulus of a given material, the proposed methodology requires the measurement layout to be

simulated using a well-established transfer matrix approach[14].

In fact, according to the formalism used in Ref.[14], it is possible to

calculate in the frequency domain the complete set of

vibro-acoustical indicators V2 (pressures, velocities, stresses) from the

transfer matrix model for a given frequency and a semi-inﬁnite

ﬂuid termination, by solving the following expression:

½D2V2¼ F (2)

[D2] being a square matrix obtained from a complete matrix D:

D¼ 2 6 6 6 6 6 6 6 6 6 6 6 4

½

I_{f 1}

½

J_{f 1}

½T1 ½0 ::: ½0 ½0 ½I12 ½ J12½T2 ::: ½0 ½0 ::: ½I23 ½ J23½T3 ½0 ½0 ::: :::

½

I3f

½

J3f

½0 ::: ::: ::: 1 r0c0 3 7 7 7 7 7 7 7 7 7 7 7 5 (3)

when its second column is eliminated, and F represents the vector obtained by multiplying the eliminated second column of matrix D

by1. In Eq.(3)

r

0and c0represent the air density and the sound

speed, respectively.

From Eq.(2), it is possible to demonstrate that, for a given

fre-quency and at normal incidence, the top plate velocity (here

indi-cated as Vout) is given as follows:

Vout¼ V2ðNÞ (4)

N being the dimension of the squared matrix [D2].

In Eq. (3), Iij and Jij are known coupling matrices between

different layers and their exact expressions can be found in Ref.[14]

(pages 257e260), while matrices T1, T2and T3are referred to the

bottom plate, tested material and top plate, respectively. In

particular, T2 depends on density

r

[kg/m3], complex modulus

E¼ E1þ iE2[Pa] and Poisson's ratio

n

[-] of the material to be tested.

In fact, if h indicates the thickness of the material, it is possible to write:

T2¼ ½GðhÞ$½Gð0Þ1 (5)

If we limit the analysis to normal incidence plane wave to the

bottom plate,

G

(x) is equal to:

GðxÞ ¼₂

6 6 4

0 0 iud3sinðd3xÞ ud3cosðd3xÞ

iud1sinðd1xÞ ud1cosðd1xÞ 0 0

u2_{r cosðd} 1xÞ iu2r sinðd1xÞ 0 0 0 0 u2_{r cosðd} 3xÞ iu2r sinðd3xÞ 3 7 7 5 (6)

u

being the angular frequency and:

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d₁¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2_r lþ 2m s and d3¼ ffiffiffiffiffiffiffiffi u2_r m s (7)

where

l

and

m

are theﬁrst and second Lame coefﬁcients,

respec-tively. Finally, Lame coefﬁcients are related to complex modulus and Poisson's ratio as follows:

l¼ n$E

1 2n

ð Þ 1 þ nð Þand m¼

E

2 1ð þ nÞ (8)

When Poisson's ratio is known in advance, it is possible to use the proposed methodology to determine the storage and loss moduli of the material.

In fact, as a_{ﬁrst step, the experimental bottom plate}

accelera-tion signal in the time domain (a0(t) inFig. 1) is converted into a

related frequency domain velocity spectrum (between 0 and Fs/ 2 Hz, Fs being the sampling frequency of the signal) by using a direct Fourier transform and a frequency domain integration operation. The transfer matrix of the three different solid layers (1. support plate, 2. tested material, 3. top mass) is solved in the fre-quency domain in order to calculate the velocity at the top plate

once the velocity on the support plate is known (using Eqs.(2e4)).

Finally, the output velocity spectrum at the top mass layer is con-verted into the time domain acceleration signal using an inverse Fourier transformation and a frequency domain derivative opera-tion. Thus, similarly to the experimental tests, the transfer matrix

model allows the time of _{ﬂight between the input and output}

simulated signals to be determined as well as the envelope of the

top plate accelerometer. The entire procedure is depicted inFig. 2.

Thus, for a given excitation sine burst, by varying the storage and loss moduli in the transfer matrix model and minimising the

dif-ferences in terms of (i) time ofﬂight between the experimental and

numerical input/output sine bursts and (ii) time domain envelope between the experimental and numerical output sine bursts, the

complex modulus of the material isﬁnally obtained. In particular,

the following cost function has been chosen to be minimized:

CF¼XT

t¼0

env*

acc1;expðtÞ env*acc1;modðtÞ (9)

T being the length of the envelope signals and the apex * in-dicates that the envelope of the top plate acceleration signal is

shifted in time domain of a quantity equal to the time ofﬂight for

both experimental and numerical data. The same procedure can be applied at each frequency of interest once the properties of the excitation sine burst are chosen accordingly and, consequently, a curve of complex modulus as a function of frequency can be obtained.

The minimisation procedure is based on a bounded nonlinear

best-ﬁt scheme[15]and has been implemented in Matlab; as an

example,Fig. 3compares the experimental and numerical output

envelopes for a tested material at a frequency of 100 Hz of the sine burst once the above-mentioned parameters have been minimised. In the proposed methodology, the direct and inverse Fourier transforms and differential and integral operators are used since transfer matrix approach allows for the determination of vibro-acoustical indicators (in particular velocities) in frequency domain among different structures of a multi-layer system, while experimental data are determined in time domain using

accelera-tion transducers as described in Secaccelera-tion3.1.

It is worth mentioning that the proposed methodology is considered here for the solid phase of a porous (diphasic) material. However, the method can be easily extended to poroelastic

mate-rials including theﬂuid phase by the use of [6 6] the matrix for

porous materials [Ref. 14 pages 247e251] in T2.

Finally, in order to include experimental errors in determining

the time ofﬂight and envelope of the time signals, among possible

solutions from the minimisation procedure, it has been decided to establish two acceptance criteria:

Dt:o:f ¼t:o:fexp t:o:fmod 1=FS

Dlog ¼ 20$log10 envacc1;exp;rms envacc1;exp;rms ! 0:5 dB (10)

envacc1,exp/mod,rms[m/s2] being the root mean squared values of Fig. 2. Time domain transfer matrix model.

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the top plate acceleration envelope from the experiment and model, respectively. The storage and loss moduli will be presented as the mean value (and related standard deviation) of all possible

solutions satisfying conditions in Eq.(10).

3. Results and discussion

3.1. Measurement set-up and tested materials

The experimental setup for measuring the top and bottom plate accelerations consists of a Data Physics V4 electromagnetic shaker,

a B&K Type 2716C power Ampliﬁer, two PCB 352C22

accelerome-ters (sensitivity 9.65 mV/g and weight 1 gr), a PC equipped with a

NI USB 4431 acquisition device and Labview®software for signal

acquisition and post-processing.

Tests were carried out below theﬁrst natural resonance of the

shaker, in the frequency range between 100 and 1500 Hz (step 100 Hz). At each frequency of interest, a 10 period sine burst (using Hanning time window) was used as the excitation signal.

A procedure was implemented during the tests to calibrate the entire system and minimise any uncertainties from the transfer function between the accelerometers. In particular, both acceler-ometers were located in the same position on the bottom plate and a transmissibility test was carried out in the frequency range of interest. In such conditions, the transmissibility between the ac-celerometers was assumed constant in frequency, and its

magnitude and phase were equal to unity and zero, respectively. Thus, at each frequency of interest, it is possible to identify a correction function in terms of amplitude ratio and time shift be-tween the accelerometers.

A steel top plate with thickness of 1 mm and radius of 22.5 mm was used for the tests (corresponding to a static preload of about 80 Pa), and acceleration of the bottom plate was chosen as about

5 m/s2(rms value) in order to guarantee an adequate signal to noise

ratio at all frequencies of interest.

In order to avoid any lateral sliding of the materials during the

tests, they wereﬁxed to the top and bottom plates by using a thin

adhesive layer. The effect of the adhesive layer on the measure-ments was evaluated by testing a single adhesive layer and veri-fying that the transmissibility is constant in the frequency range of interest with the magnitude and phase equal to unity and zero, respectively.

Experimental tests were carried out on three materials (polyurethane foam, melamine foam, and rubber) and a

description of the tested materials is reported inTable 1. Each

material was also tested by using a quasi-static method[2,16]

(data are given in Table 1) and a well-established dynamic

method based on the transmissibility function in the frequency

domain[17]. It is worth mentioning that the method proposed in

Ref.[17]has a limitation at high frequencies, which depends on

the elastic wavelength to be compared to the thickness of the sample. For all tested materials, the high frequency limit is

0 0.01 0.02 0.03 0.04 0.05 0 200 400 600 800 1000 1200

[m/

s2]

t [samples]

Envelope of top plate acceleraƟon - Test Envelope of top plate acceleraƟon -Model

Fig. 3. Comparison at a frequency of 100 Hz between experimental and numerical output envelopes for a tested material.

Table 1

Description of tested materials.

Material A B C

Description Reconstituted porous rubber (open cells) Polyurethane foam (closed cells) Reticulated foam (open cells)

Density [kg/m3_] ₂₄₂ ₈₇ ₉

Thickness [mm] 26 25 25

Storage modulus [Pa] 806919 121447 54967

Loss modulus [Pa] 363114 27933 4947

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around 500 Hz.

3.2. Numerical validation of the procedure

The transfer matrix approach deals with the inﬁnite lateral size

of the layers while measurements are carried out on cylindrical samples with a reduced radius (22.5 mm in this paper). Moreover, some errors could occur at low frequencies where the longitudinal wavelength of the propagating wave is comparable or even higher than the thickness of the sample. Therefore, in order to investigate the capability of the proposed method to correctly determine the complex modulus, a series of preliminary simulations has been

carried out using results from 2D-axial symmetricﬁnite element

model (FEM) to generate time domain acceleration signals. The values of the FEM simulated combinations are summarised in Table 2.

In particular, the following procedure has been applied: for a given frequency a sine burst with an unit amplitude

(bot-tom plate acceleration) has been built analytically and con-verted into a frequency domain velocity using the Fourier transform and integration operation;

by ﬁxing complex modulus, Poisson's ratio, density and thick-ness of the material, the FEM model has been solved in the frequency domain in order to calculate the top plate velocity, which has been converted into the time domain acceleration using an inverse Fourier transform and derivative operation; the time domain of the top and bottom plate accelerations has

been used in combination with the proposed transfer matrix model and the minimisation based procedure in order to calculate the complex modulus to be compared with the FEM input data. This procedure is identical to that described in

Sec-tion2where experimental tests are used as input data.

Regarding the FEM model, a triangular mesh has been used and the mesh size has been created according to the rule of ten ele-ments per wavelength for both the simulated material and plates. The acceleration at the bottom plate of the model has been imposed as a boundary condition and the top plate acceleration has been evaluated at the corner of the plate close to the axis of symmetry. The linear systems are solved using the Multifrontal Massively Table 2

Description of FEM simulations.

Parameter Values

Frequency [Hz] 100e300e500e1000e1500e2000

Storage modulus [Pa] 1e4e1e5e1e6e1e8

Poisson's ratio [e] 0e0.1e0.2e0.3 0.35e0.4e0.45

Loss factor [e] 0.1e0.2e0.4e0.8

Density [kg/m3_] _{50e100e200e400} Thickness [mm] 10e25e50 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 0.1 0.2 0.3 0.35 0.4 0.45 E1 FEM / E1 TMM Poisson's raƟo 100 Hz 300 Hz 500 Hz 1000 Hz 1500 Hz 2000 Hz

Fig. 4. Ratio between FEM and transfer matrix model storage modulus as a function of Poisson's ratio and frequency.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 0.1 0.2 0.3 0.35 0.4 0.45 E2 FEM / E2 TMM Poisson's raƟo 100 Hz 300 Hz 500 Hz 1000 Hz 1500 Hz 2000 Hz

Fig. 5. Ratio between FEM and transfer matrix model loss modulus as a function of Poisson's ratio and frequency.

1.E+05 1.E+06 1.E+07 0 200 400 600 800 1000 1200 1400 1600 St or ag e Modulus E1 [P a] Frequency [Hz] Quasi-staƟc [ref. 2,15] Dynamic [ref. 16] New method 1.E+05 1.E+06 1.E+07 0 200 400 600 800 1000 1200 1400 1600 Loss Modulus E2 [P a] Frequency [Hz] Quasi-staƟc [ref. 2,15] Dynamic [ref. 16] New method

a

b

Fig. 6. (a) Storage and (b) loss moduli for material A. Comparison between different methods.

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Parallel Sparse (MUMPS) direct.

Fig. 4depicts for simulated materials as described inTable 2the average ratio (and related standard deviation) for each frequency of

interest between the storage modulusﬁxed in the FEM model and

obtained using the new methodology as a function of Poisson's

ratio. The same comparison for the loss modulus is shown inFig. 5.

From the comparisons, it is possible to observe that the ratio be-tween the FEM and proposed transfer matrix model values

in-creases as Poisson's ratio grows. Moreover, for a ﬁxed value of

Poisson's ratio, the ratio between the FEM and proposed transfer matrix model values tends to decrease when the frequency in-creases. Finally, the proposed procedure is able to calculate the storage modulus with higher accuracy if compared with the determination of the loss modulus.

As a general conclusion, for a frequency higher than 100 Hz and Poisson's ratio lower than 0.4, the ratios between FEM and pro-posed transfer matrix model are lower than 1.5 with a maximum standard deviation lower than 0.5.

3.3. Complex modulus: comparison between different measurement techniques

Figs. 6e8depict the comparison between the different mea-surement methods in terms of storage and loss moduli for all the

tested materials summarized inTable 1. It should be recalled that

the storage and loss moduli are presented as the mean value (and

related standard deviation) of all solutions meeting the conditions

in Eq.(10)for each tested material.

From the data shown inFigs. 6e8, is it possible to observe that

the comparison between the different methods is consistent. Moreover, it is possible to observe that, by using the proposed method, the storage modulus is determined with higher accuracy than the loss modulus since the errors are of the same magnitude. As a consequence, the new methodology provides a damping loss factor (calculated as the ratio between the loss and storage moduli)

1.E+04 1.E+05 1.E+06 0 200 400 600 800 1000 1200 1400 1600 St or ag e Modulus E1 [P a] Frequency [Hz] Quasi-staƟc [ref. 2,15] Dynamic [ref. 16] New method 1.E+04 1.E+05 1.E+06 0 200 400 600 800 1000 1200 1400 1600 Loss Modulus E2 [P a] Frequency [Hz] Quasi-staƟc [ref. 2,15] Dynamic [ref. 16] New method

a

b

Fig. 7. (a) Storage and (b) loss moduli for material B. Comparison between different methods. 1.E+03 1.E+04 1.E+05 1.E+06 0 200 400 600 800 1000 1200 1400 1600 St or ag e Modulus E1 [P a] Frequency [Hz] Quasi-staƟc [ref. 2,15] Dynamic [ref. 16] New method 1.E+03 1.E+04 1.E+05 1.E+06 0 200 400 600 800 1000 1200 1400 1600 Loss Modulus E2 [P a] Frequency [Hz] Quasi-staƟc [ref. 2,15] Dynamic [ref. 16] New method

a

b

Fig. 8. (a) Storage and (b) loss moduli for material C. Comparison between different methods. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 200 400 600 800 1000 1200 1400 1600 Loss f actor [-] Frequency [Hz] Material A Material B Material C

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with a higher uncertainty (as depicted inFig. 9and also underlined

in Ref. [1]) or equivalently different values of the loss factor

together with the storage modulus will satisfy Eq.(10).

In all examined cases, the standard deviation is lower than one order of magnitude at each frequency and for all tested materials.

Fig. 10depicts an example of the convergence plot of storage and loss moduli and the cost function for material A at a frequency of 1000 Hz.

In order to validate the results obtained by using the new methodology, a comparison in terms of transmissibility function for acceleration in the frequency domain is used. In particular, the experimental transmissibly (amplitude and phase) is shown

against the transmissibility calculated as follows[17]:

T¼ ½coshðg$lÞ þ ðM=mÞ$g$l$sinhðg$lÞ1 (11)

l being the thickness of the material [m], M the loading mass

[kg], m the mass of the material [kg] and

g

[m1] the propagation

constant depending on frequency [Hz], density of material [kg/

m3], Young's modulus [Pa] and the loss factor [-]. Thus, using

values from quasi-static tests (summarised inTable 1) and

dy-namic values from new methodology (depicted inFigs. 6e8), it is

possible to calculate T to be compared with experimental data. It is

worth of mentioning that Eq. (11) is strictly valid for long rod

samples or for material having a null Poisson's ratio. However,

comparisons are depicted inFigs. 11 and 12for materials A and B

where it is possible to observe good agreement between the tests

and numerical model (Eq.(11)) using mechanical properties from

the proposed methodology. Conversely, when using data from the

quasi-static tests, the comparison cannot be considered

satisfactory.

Finally, as an additional validation of the proposed methodol-ogy, results of loss modulus for tested materials are compared with

an approximated expression suggested by Gross[18]and valid in

linear viscoelastic regime:

E2y

p

2E1

vðlogðE1ÞÞ

vðlogðuÞÞ (12)

u

[rad/s] being the angular frequency. A comparison between

experimental loss moduli and result of Eq. (12) are depicted in

Fig. 13where good agreement can be observed for materials A and

B, while Eq.(12)seems to overestimate value of E2for material C. In

fact, the mean relative errors for material A and B are 9% and 17%, respectively; the same quantity for material C is around 35%. It is worth mentioning the for material B the main discrepancies have been observed at low frequencies where numerical simulation indicate a lower accuracy of the proposed methodology.

4. Conclusions

A novel method to determine the values of the dynamic com-plex modulus of homogeneous and isotropic materials has been proposed and discussed in a wide frequency range using a transfer

5.0E+05 5.0E+06 0 20 40 60 80 100 120 E1 - E2 [P a] IteraƟons

E1

E2

0 1 10 100 0 20 40 60 80 100 120 Cos t func Ɵ on IteraƟons

a

b

Fig. 10. Convergence plot of (a) storage and loss moduli and (b) cost function for material A at a frequency of 1000 Hz. 0 1 2 3 100 1000 Tr ansmissibility - Amplitude [-] Frequency [Hz] Experimental New method QS values 0 1 2 3 4 5 6 100 1000 Tr ansmissibility - Angle [r ad] Frequency [Hz] Experimental New method QS values

a

b

Fig. 11. Comparison in terms of the transmissibility function for acceleration for ma-terials A. (a) Amplitude. (b) Angle.

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matrix approach. Preliminary numerical analyses have shown the capability of the method to predict the complex modulus with good accuracy at frequencies higher than 100 Hz and for values of

Poisson's ratio lower than 0.4. The results of the procedure on real materials have been compared with data from well-established quasi-static and dynamic methods, and the comparison can be considered satisfactory.

The proposed methodology has been applied to the solid phase of a porous (diphasic) material; however, the method can

be easily extended to poroelastic materials including the ﬂuid

phase in the transfer matrix scheme. Moreover, changing the mass of the top plate will allow the mechanical properties to be investigated as a function of the static preload, an aspect that is quite delicate when dealing with materials that exhibit strong viscoelastic behaviour. Finally, the same scheme could also be applied to cubic shaped samples and information about anisot-ropy could also be achieved.

Acknowledgements

Authors would like to thank Dr. Luc Jaouen for providing with very helpful comments on this manuscript.

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Her-mann, Paris, 1953. 0 1 2 3 4 5 6 7 100 1000 Tr ansmissibility - Amplitude [-] Frequency [Hz] Experimental New method QS values 0 1 2 3 4 5 6 7 8 100 1000 Tr ansmissibility - Angle [r ad] Frequency [Hz] Experimental New method QS values

a

b

Fig. 12. Comparison in terms of transmissibility function for acceleration for materials B. (a) Amplitude. (b) Angle.

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 0 200 400 600 800 1000 1200 1400 1600 Loss Modulus E2 [P a] Frequency [Hz]

A-E2 test A-E2 [Eq. 7] B-E2 test B-E2 [Eq. 7] C-E2 test C-E2 [Eq. 7]

Fig. 13. Comparison between experimental loss modulus and values calculated using Eq.(12)for all tested materials.