A review of experimental methods for the elastic and damping characterizations of acoustical porous materials M. Devergea and L. Jaouenb a,b

LAUM, UMR CNRS 6613, Universit´e du Maine, 72 085 Le Mans Cedex, France a,b

[mickeal.deverge;luc.jaouen]@univ-lemans.com

Abstract [165] Acoustical porous materials like polymer foams or mineral wools are widely used in noise and vibration control. Their efficiencies, in this domain, can be influenced by their elastic and damping properties. The objective of this communication is to present a review of measurement methods and difficulties faced in the estimations of the Young’s or shear moduli, Poisson’s ratios and structural damping coefficients of these materials. Two quasistatic well known compression and torsion tests are performed on a couple of polymeric foams (a melamine and a polyurethane ones) to illustrate the difficulties: non linear deformation, anisotropy, coupling between the material phases, etc. Three dynamic, resonant, methods based on longitudinal or bending vibrations of beam-like or plate-like samples are also used to investigate the frequency and temperature dependences of the foams properties. The measurement results obtained from these methods are discussed.

1

INTRODUCTION

In the past few years, new experimental methods have been proposed to characterize the elastic and damping properties of polymer foams or fibrous materials used in the field of acoustics. This denotes the actual effort in the understanding of mechanical and particularly elastic and damping behaviors of such porous materials which play an important role in sound insulation. This paper presents a review of common and recent experimental methods for the determination of Young’s or shear moduli, Poisson’s ratios and structural damping coefficients of acoustical open-cell polymer foams. Polymer foams are composed of two phases: a solid one, the polymeric skeleton, and a saturating fluid one, the air. The viscoelastic behavior of polymers, and thus of their foams, is intermediate between the pure elastic solid and the ideal viscous liquid ones. The usual framework of the studies of polymer foams behavior is often limited to the field of linear viscoelasticity [11, 6]. Measurements presented here are carried out under this small deformations assumption and complex moduli are used to account for this viscoelastic behavior of acoustical porous materials subjected to periodic loadings: E = E 0 (ω) + jE 00 (ω) = E 0 (1 + jη(ω)), E 00 (ω) with η(ω) = , E 0 (ω)

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(1) (2)

where E 0 (ω) is a storage modulus, E 00 (ω) a loss √ modulus and η(ω) the corresponding structural damping coefficient. j is the square root of −1. Another remarkable point is that, due to the foaming process [13, 2], foams are usually anisotropic. Especially they often exhibit an orthotropic or transverse isotropic behavior [19], the foam cells being longer in the growth direction. However, despite this statement, few 2-Dimensional or 3-D methods take into account this anisotropy. A spatial inhomogeneity can also be noted for some materials. For this reason, the foam samples used in the following experiments are cut from a same small volume of material. The experimental methods presented in the next sections, which almost derived from polymers or metal ones [2, 6, 14], are classified in two groups: quasistatic methods, where inertia effects are neglected, and dynamic, resonant, ones.

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Load:

A

Compression

B C D

Torsion

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2

LASER

B

2

Traction−compression

1

1

Bending

4

3

A C

LASER

1

1

Shaker or rotor

2

Sample

3 4

Accelerometer or force/torque transducer

3

D 1

2

4 2

3

Figure 1: Schematic representations of experimental set-ups for the elastic and damping characterization of porous materials under various types of loading conditions and for different sample shapes.

2

QUASISTATIC EVALUATIONS

Non resonant methods in the quasistatic regime take advantage of the low inertia effects under the first resonance of the system. A low coupling assumption between the material phases is also assumed in this regime. Porous materials are thus modeled as solid media. 2.1 Compression Figure 1A presents the compression set-up described by Mariez et al. [18, 19]: a cubic foam sample (40×40×40 mm3 ) is placed between two parallel rigid planes. The lower plane is transversally excited by an electrodynamic shaker while the higher plane is fixed. The planes are

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covered with sandpaper to avoid sliding of the sample and the imposed deformation is of the form: ε = εs + εd sin(ωt). εs is the static strain fixed to circumvent the surface inhomogeneity of the porous sample and εd is the dynamic amplitude of deformation. The elastic characterization is realized in two steps. First, a measurement of the ratio between the imposed transverse displacement and the induced displacement by Poisson’s effect in a perpendicular direction, by means of a laser vibrometer, allows an estimation of this Poisson’s ratio. Second, a measurement of the stiffness of the sample is done from the measured compression force and the imposed displacement. The complex Young’s modulus E is estimated using an inverse method based on a 3-D solid finite elements model. These operations can be repeated changing the observation side for the measurement of Poisson’s ratios and the compressed side to estimate three complex Young’s moduli Ei and six Poisson’s ratios νij , (i, j) = {1, 2, 3} with i 6= j. 6

10

i

Evolutions of E (N.m−2)

Real part of E i: Ei’

5

10

Imaginary part of E i: Ei’’

4

10

1

10

Frequency (Hz)

2

10

Figure 2: Estimations of a melamine foam complex Young’s moduli at 18o C. Compression test [18, 19]. ◦: side 1, 4: side 2, : side 3.

Measurements of the Poisson’s ratio ν13 in the frequency range [70 100] Hz show constant and quite real ratios of values 0.44 (which indicate an elastic anisotropy). These values are close to the one obtained from a embedded version of this experimental setup for cylindrical shaped samples [16]: 0.45. Results for the estimations of Young’s moduli Ei and structural coefficients ηi for the three sides of the cubic sample at 18o C are presented on figure 2. These figures confirm the elastic anisotropic behavior of the melamine foam and, more precisely, that this material has a symmetry close to an orthotropic one (assuming the principal axis are parallel to the cube sides). Another remarkable point is a slow increase of the moduli for frequencies below 70 Hz which confirm the viscoelastic behavior of the foam. Values above 70 Hz are reported although the low coupling effects assumption between the material phases for this foam and this configuration is no more valid above this frequency [9, 8]. The Determination of the elastic and damping behaviors of materials can be improved with measurements at various temperatures. An example for a torsion test follows. 2.2 Torsion Figure 1B is a schema of this non resonant experiment for which commercial devices are available. One of the main advantages of a torsion test compared to the previous compression one

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is that it ensures a constant volume of the material during the loading. Consequently, the fluid-structure coupling can be considered ever lower compared to others methods. The experimental set-up for this test is quite similar to the compression one except that one of the plane is harmonically excited in torsion and that the sample is of cylindrical shape. The shear stress and strain, obtained from measurements of a torque and an angular displacement transducers are used to calculate the complex shear modulus G. Measurements for the melamine foam, at temperatures from 0o C to 40o C, with a commercial Rheometrics RDA II apparatus are reported on the two graphs of figure 3. Radius and height of the samples are 31 mm and 10 mm respectively. 2e+04

Real part of shear modulus G 13 (N.m−2)

Imaginary part of shear modulus G 13 (N.m−2)

2e+05

1e+04 9e+03 8e+03 7e+03

1e+05 9e+04

6e+03

8e+04

5e+03

7e+04

4e+03

6e+04 5e+04 −2 10

−1

10

0

10 Frequency (Hz)

3e+03 −2 10

1

10

−1

10

0

10 Frequency (Hz)

1

10

Figure 3: Variations of the shear modulus G13 real and imaginary parts (G013 and G0013 ) with temperature and frequency for the melamine foam. +: 0o C, o: 10o C, ?: 24o C, : 40o C

A state transition is clearly observed at 24o C, on the right graph of Fig. 3, in the [0.01 10] Hz frequency range. However, without any additional information, no conclusion can be made on the nature of this transition. The complex shear moduli of a material can also be estimated with a pure shear test. An application to an open-cell polyurethane foam is presented in [10]. 2.3 Discussion on quasistatic evaluation methods The above illustrated methods and in general all methods for the quasistatic evaluation of the elastic and damping properties of acoustical foams, present the interest to allow a sharp frequency resolution. However, one can note that these methods present some disadvantages: these methods do not really characterize the material the way it is used and do not take into account the diphasic nature of the porous medium. Another point to mention is that the usable frequency ranges remain low compared to acoustical frequencies. Estimations of the material properties at higher frequencies are possible from measurements at various temperatures by using the Time-Temperature Superposition principle (TTS) [11, 6]. Nevertheless, it is perilous to use the TTS principle from measurements that consider only the solid phase of a diphasic foam. As an alternative to the use of the TTS, the next section presents experimental methods for the dynamic evaluation of these material properties.

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3

DYNAMIC EVALUATION

Three experimental dynamic methods, based on mechanical loadings, are presented below. The classical mass-spring experiment and a method based on an acoustical loading of the porous material are reported in [10, 12]. 3.1 Beam under longitudinal vibrations Estimations of the complex Young’s moduli Ei of a porous medium can be calculated from the modal analysis of a beam excited into longitudinal vibrations as described by Pritz [21] for close cells materials. This method has been used on a 185 mm wide and 10×10 mm2 cross-section melamine foam beam at a temperature of 25o C. E10 and η1 are calculated from the resonance frequencies and resonance bandwidths of the frequency response function defined as the ratio between the imposed displacement at one end of the beam over the displacement at the other tip. This latter displacement is measured by the means of a laser vibrometer or a light accelerometer. Although the melamine foam is an open-cell material, this method gives coherent results for the first two modes: (165 000 N.m−2 , 0.048) at 188 Hz and (169 200 N.m−2 ) at 576 Hz. The modeling used in [21], which does not take into account the material fluid phase, shows nevertheless its limitations with increasing frequency when the fluid phase and its interactions with the solid phase become important [9, 8]. A more complex modeling of open-cell porous media, as the Biot-Johnson-Champoux-Allard’s [5, 1] model, is thus needed to account for the inertial, viscous and thermal interactions between the material two phases and their effects on the vibration response of the sample. Despite a simple modeling, this method presents the advantage to offer a simple 1-D approach and can give rough estimations of Ei and ηi for very low flow resistivity (cf. [5, 1]) and high density materials. Another important point on which we will focus in the next sections is the fact that no adhesive layer is needed in this longitudinal vibrations based method. 3.2 Beam under bending vibrations A beam sample can also be tested under bending vibrations as described in the set-up D of Fig. 1 [26]. In this method, derived from the Oberst’s beam method [3], the shaker is now glued to the center of a base metal beam supporting the foam layer and imposes a transverse displacement. A laser vibrometer measures the transverse velocity of the base beam at one free tip. A constrained metal beam can be added on the foam layer when the porous Young’s modulus to measure is low, typically less than 106 N.m−2 . This last configuration is called the “three layers configuration”. Like in the Oberst’s beam method, the determination of the material Young’s modulus and the structural damping coefficient in the direction of the beam axis is deduced from an inverse calculation. However, to observe a significant modification in the vibration behavior of the metal base beam, the material thickness should overcome the RKU’s model assumptions [24]. Numerical calculations are then required. In spite of this numerical constraint, one advantage of this method is to allow a diphasic modeling of the porous medium. Graphs at Fig. 4 show results of this method applied to the melamine foam. The method presented in the next section attempts to save computational resources by using a simplified calculus to model a plate-porous configuration [7, 26].

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3.3 Plate under bending vibrations In practice, a 2-D multilayer configuration is more widely used than a 1-D configuration. On the basis of this statement, Jaouen et al. [15] present an extension of the previous method to a 2-D space: a layer of the porous material is bonded onto a metal plate. A shaker imposes a transverse point load on this base metal plate. The input force on the plate is measured by means of a force transducer and its transverse velocity is measured by a laser vibrometer or by the integration of a light accelerometer signal. A simplified model, based on the mixed displacement-pressure formulation of the BiotAllard theory [4] is used to predict this two layers plate vibration behavior. The base plate and the porous solid phase are described as an equivalent viscoelastic plate and assumed admissible functions are used to describe the fluid phase pressure. The poroelastic layer is assumed to be isotropic once again. A perfect bond is supposed at the base plate-porous boundary interface. A non-linear inversion algorithm is then used to estimate the porous material Young’s modulus and structural damping coefficient, at the system resonances, from the measurements and the numerical simulation. Graphs at Fig. 4 and 5 show results of this last method applied to the previous melamine foam and to a polyurethane foam. The foams have been considered isotropic (even if obviously the melamine one is not). The Poisson’s ratios are supposed to be real and equal to their quasistatic values: ν13 = 0.44 for the melamine foam and ν13 = 0.30 for the polyurethane foam. In addition to a simplified modelization, this method allows a spatial averaging of the porous material properties. However, a new difficulty is encountered: the acoustic radiation can be no more negligible, in this configuration, compared to previous experiments. Yet, there is no model to quantify it to the authors knowledge. −1

10

6

Estimations of η1

−2

Estimations of E1’ (N.m )

10

−2

5

10 2 10

10

3

Frequency (Hz)

10

2

10

3

Frequency (Hz)

10

Figure 4: Comparisons of estimation results for the Young’s modulus E10 and the structural damping coefficient η1 of the melamine foam. : beam bending - three layers configuration (25o C), o: plate bending (23o C).

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6

0

10

Estimations of η1

Estimations of E1’ (N.m−2)

10

−1

10

5

10 0 10

1

10

2

3

10 Frequency (Hz)

0

10

10

1

10

2

10 Frequency (Hz)

3

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Figure 5: Comparisons of estimation results for the Young’s modulus E10 and the structural damping coefficient η1 of a polyurethane foam. o: plate bending (23o C), 4: quasistatic compression test (22o C).

3.4 Discussion on dynamic evaluation methods Considering the experimentally evaluated 10 to 20% of uncertainties in the moduli evaluations, the previous dynamic evaluation methods give coherent results with each other and with quasistatic evaluation methods. An important point to mention is the major assumption assuming that Poisson’s ratios νi are real and constant in each frequency range of measurement although this approximation is valid only for small frequency ranges [23]. 4

CONCLUSION

A certain number of methods for the characterization of elastic and damping properties of porous acoustical materials, and particularly acoustical foams, exist. Some of them have been illustrated in this communication by measurements for a melamine foam and a polyurethane foam. From these results, three concluding points can be made. • Among all these methods, which present their own advantages and disadvantages, none can be considered as a unique solution to all characterization problems (viscoelasticity of the skeleton, anisotropic behavior, spatial inhomogeneity in the properties...). The complete elastic and damping characterization of a material requires a combination of these quasistatic and dynamic methods. • It has been shown that the fluid phase has an influence on the estimations for quasistatic methods [9, 8], the influence of acoustical properties uncertainties for the dynamic evaluations of elastic and damping properties of porous materials is still an open question. • An effort is needed to consider the anisotropy of acoustical porous materials, which is actually ignored in too much work. ACKNOWLEDGEMENT The authors would like to thank L. Benyahia for providing the torsion results of the melamine foam.

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