19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007

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1 19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 FREQUENCY DEPENDENCY AND ANISOTROPY OF THE ELASTIC CONSTANTS OF (NON-)POROUS MATERIALS AND THEIR INFLUENCE ON THE USAGE IN BUILDING ACOUSTICAL APPLICATIONS PACS: Hq, Jr, Ks, Mv, Mr Boeckx,Laurens 1 ; Descheemaeker, Jan 1 ; Nathalie, Geebelen 1 ; Khurana, Poonam 1 ; Vermeir, Gerrit 1 ; Desmet, Wim 2 ; Lauriks, Walter 1 1 Laboratory of Acoustics and Thermal Physics, Katholieke Universiteit Leuven, Celestijnenlaan 200D, BE-3001 Leuven, Belgium. Laurens.boeckx@fys.kuleuven.be 2 Division PMA, Katholieke Universiteit Leuven, Celestijnenlaan 300B, BE-3001,Leuven, Belgium ABSTRACT The elasticity and other material parameters are crucial in modeling wave propagation in building acoustical applications. Measurement data and techniques regarding the frequency dependency and anisotropy of the elastic constants of these often soft and highly damped materials are however scarce. Recently a new technique that makes use of guided wave and/or surface waves to characterize the rigidity of (non)porous materials has been presented. This contribution will present this technique and data regarding dynamically measured elastic constants of (an) isotropic (non-) porous materials. The frequency dependent measurement data is compared to data obtained using the temperature-frequency superposition. Master curves of materials commonly used in building acoustical applications are presented. The use of master curves is validated by comparing the elasticity -measured in lab conditions- to data obtained from in situ testing. The data is used to model the measured acoustical behavior of the materials (transmission loss, absorption, etc ). INTRODUCTION Noise control materials are characterized by a wide range of material properties and more specifically the elasticity. The stiffness of these materials can vary from 10 4 Pa up 10 6 Pa. These materials can also be anisotropic, porous and or the elasticity can exhibit a frequency dependency. In the past years considerable research effort has gone to the determination of the parameters of the Biot-Allard [1] model which determine the airborne wave propagation throughout porous media. These parameters solely determine the acoustical behavior in many applications where a sound wave is impinging from the air directly onto the surface of an opencelled porous material; elastic coupling can in these cases often be neglected. Classical ultrasound methods [2, 3] provide an elegant tool for estimating the Biot-Allard parameters. They are fast, non-destructive and user-friendly. Furthermore ultrasonic characterization has the potential of being used as an inline technique for quality control and for the validation of the acoustical properties. In many cases however the porous material cannot be treated as an equivalent fluid and accurate knowledge of the elastic constants of the porous material is needed for modeling purposes. For many applications it is also not clear in which geometrical configuration the porous material is combined with the other components of the multilayered system since the manufacturer differs from the end-user of the porous material. The range of the elastic constants of these materials varies between 10 4 Pa up 10 6 Pa. Furthermore, the elastic response of these materials is influenced by visco-elasticity, anisotropy and high attenuation. Elastic constants are therefore very difficult to measure accurately. Accurate material data are however of crucial importance for numerical modelling, optimizing or fine tuning multilayered

2 systems and an appropriate choice of materials depending upon the application (vibro- acoustical problem, noise control, airborne sound insulation, ). Classical techniques [4,5] are limited to the lower frequency range and accuracy is often dependent on the size and shape of the sample under investigation. Complicated numerical inversion schemes (e.g. 3D finite element modeling) are often needed to obtain the elastic constants. A method which provides accurate material dataa at a higher frequency range with non- source and the accuracy is not dependent upon sample shape. Recently surface waves [7] and guided waves [8,9] have been proposed as a method for contact excitation has been given by Allard [6]. A frame resonance is induced by a monopole investigating the elastic response of porous materials over a wider frequency range. The use of surface waves as a tool for material research has some convenient advantages in that their application does not depend on the size and shape of the sample, their use is not restricted to the lower frequency range and the measurement techniques are non-destructive. Indeed surface/guided wave propagation depends only on the bulk wave propagation throughout the porous material and on the boundary conditions at the free surface. This paper reviews briefly the method for measuring the elastic response using guided waves and surface waves. Phase velocity dispersion curves are presented for three materials and the effect on the acoustical behavior is presented. MEASURING SURFACE/GUIDED WAVES IN PORO-ELASTIC MEDIA The experimental setup for measuring the phase velocity of surface waves and guided waves in (poro)-elastic media is given by Figure 1. A sample is glued upon two rigid endings which and a thin metal plate which acts as a line source is attached to the shaker and to the sample under investigation. A stepwise harmonical excitation is given to the shaker. The detection occurs by means of a Laser Doppler vibrometer and a mirror lens arrangement which allows measuring the displacement pattern over the whole length of the sample. The measured spatial interference pattern is governed by wavelength of guided waves which are allowed to propagate in the system. A more detailed description of the measurement principle and the model used to determine the allowed guidedd waves in the system is given by [8,9]. Figure 1.-Shematical representation of the Guided/ /Surface wave setup in Lamb conditions The measured interference pattern allows us to determine the phase velocities of the guided waves as a function of frequency. The use of the exact guided wave model provides information regarding the frequency (in)dependent behaviour of the elastic constants. In general the frequency range of the guided wave experiment is limited to 2 or 3 khz depending on the material. For higher frequencies it is harder to setup a spatial interference pattern. However at higher frequencies one can excite a couple of periods of sinusoidal and measure the time of flight. For this measurement the source has to mounted on the same side at which the detection of the displacement pattern occurs. 2

3 PHASE VELOCITY DISPERSION CURVES: CASE STUDY OF THREE MATERIAL TYPES Elastic porous material Figure 2 provides the measured phase velocities of guided waves of a sound absorbing material whichh is used in building applications. The phase velocities of several guided waves were measured) by means of the guided wave experiment (full circles) and by measuring the phase velocity of the Rayleigh wave (full squares). The phase velocity of the a 0 mode was measured up to 3kHz. This mode corresponds to the bending mode and at low frequencies the phase velocity of this mode varies as. The s 0 mode has a very small displacement component along the normal direction at low frequencies which causes the lack of data points at this frequency range for the s 0 mode. The start frequency for which the a 1 mode has a measurable displacement component corresponds to the shear resonance of the material which is equal to ( is the transversal bulk phase velocity and is the thickness of the material). Figure 2.- Phase velocity dispersion curves of a sound absorbing material. Full circles and full squares represent measured phase velocities. Green solid lines are the calculated phase velocities. The phase velocity dispersion curves (Figure 2) provide a estimation of the frequency (in)dependency of the elastic constants by minimizing the difference between the measured phase velocities and calculated phase velocities, with respect to the elastic constants. For the material presented in figure 3 no frequency dependency could be observed up to 2kHz. The full squares deviate from the green solid lines which present the phase velocities calculated with a fixed shear modulus N= 110 kpa. The Poisson ratio needed to fit the measured dataa was estimated at The bending mode phase velocity is sensitive to the value of the shear modulus N, while the phase velocity of the s 0 is sensitive to the Poisson ratio. Visco-elastic material The phase velocity dispersion curves of rubber-like material (ρ= 1000kg/m³) used for vibration insulation are given in Figure 3. The phase velocity of the a 0 mode or bending mode was measured up to 3kHz using the guided wave experiment (indicated by full circles in Figure 3). Additional data points (full squares) were obtained by measuring the phase velocity of a surface wave which was excited by means of a sinusoidal burst. The dashed lines (green and black) provide two alternative ways of modelling the material. The dashed green lines are obtained by modelling the material with a frequency independent shear modulus. The dashed black lines represent the phase velocities of a material with a frequency dependent shear modulus. The 3

4 difference in behaviour is suggested by (>3kHz). the phase velocities measured at higher frequencies Figure 3.- Phase velocity dispersion curves of a rubber-like material. Full circles and full squares represent measured phase velocities. Green dashed line is the calculated phase velocities for a frequency independent elastic constants. Black dashed line is calculated phase velocity for a frequency dependent phase velocity. Visco-elastic porous material A clear frequency dependency can be found in Figure 4. The phase velocity of the bending mode of a highly porous very soft poro-elastic material was measured between 100Hz and 2.5 khz. The material is used as vibration insulation in floating floor constructions. Clear deviations from normal frequency independent behaviour of the phase velocity of the bending mode are found. The measurement was performed at 10 C. The measured phase velocities are indicated by the red circles and the black dashed lines are phase velocities calculated with a fixed value of the shear modulus N, namely N=1.01 MPa and N=500 kpa. Figure 4.- Phase velocity of the bending mode for a visco-elastic porous material. Circles represent measured phase velocities. Black dashed lines are calculated phase velocities using fixed values of the shear modulus, N= =1.01MPa and N=500kPa. 4

5 The airborne sound insulation was calculated for a sandwich construction in which the measured material properties of the material (figure 4) were used. The other material properties (porosity, flow resistivity, tortuosity, viscous and thermal length) were measured using the normal standardized techniques [e.g. 2,3]. The material was incorporated in sandwich structure with a two gypsum board walls with a thickness of 1.8 cm and an air gap of 2cm. Although the effect of the static loading on the material is not included, a clear difference in acoustical behaviour is noticeable in the upper frequency range (500Hz-3kHz) for which the material depicts a clear visco-elastic behaviour. Figure 5.-Calculated airborne sound insulation for a visco-elastic porous material (figure 4) and for the same material with a frequency independent elastic constant, N= CONCLUSIONS A method for measuring the frequency dependency of the elastic constants in the audible frequency range of noise control materials is presented. The characteristics of three different types of materials were presented: a porous elastic material used for sound absorption, a rubber-like material used for vibration insulation purposes and a porous visco-elastic material used for vibration insulation. The airborne sound insulation of the latter material was calculated at the measured frequencies and a clear influence of the visco-elasticity on the sound insulation was shown. The influence of the static loading was not taken into account the measurement method. ACKNOWLEDGEMENTS Financial support was obtained by the Institute for the promotion of Innovation through Science and Technology in Flanders (IWT, Flanders, Belgium). References: [1] J.F. Allard, Propagation of sound in porous media modelling sound absorbing materials, Elsevier, London, [2] J.F. Allard, B. Castagnede, M. Henry, W. Lauriks, Evaluation of tortuosity in acoustic materials saturated by air, Rev. Sci. Instr., 65, , [3] Ph. Leclaire, L. Kelders, W. Lauriks, C. Glorieux, J. Thoen, Determination of the viscous characteristic lengths in air-filled porous materials by ultrasonic attenuation measurements, J.Acoust. Soc. Am., 99, , [4] T.Pritz, Transfer function method for investigating the complex modulus of acoustic materials: spring-like specimen, J. Sound and Vibr., 72, , 72. 5

6 [5] [AST-98]: ASTM. E756-98, Standard test method for measuring vibration-damping properties of materials, American Society for Testing and Materials, [6] N.Geebelen, L. Boeckx, G. Vermeir, W. Lauriks, J.F.Allard, O. Dazel, Measurement of the rigidity coefficients of a melamine foam, Accepted for publication in Applied Acoustics. [7] J. F. Allard, G. Jansens, G. Vermeir and W. Lauriks, Frame-borne surface waves in airsaturated porous media, J. Acoust. Soc. Am. 111, , [8] L.Boeckx, P.Leclaire,P.Khurana,C.Glorieux, W.Lauriks and J.F.Allard, Investigation of the phase velocities of guided waves in soft porous materials, J. Acoust. Soc. Am., 117, , [9] L. Boeckx, P. Leclaire, P. Khurana, C.Glorieux, W. Lauriks and J.F. Allard, Guided elastic waves in porous materials saturated by air in Lamb conditions, Jour. Appl. Phys., 97(1) ,