In this letter, the low and high frequency limits of the effective density characterizing a limp frame porous medium are investigated. These theoretical limits are compared to the ones found for a classical rigid frame porous medium, and to experimental measurements. While the high frequency asymptotic behaviors of both limp and rigid effective densities are usually only slightly different, their low frequency behaviors are significantly different. Compared to experimental measurements performed on a limp frame fibrous layer, only the limp frame effective density yields good correlations over the whole frequency range.
I. Introduction
Open-cell porous materials used to attenuate sound may be categorized in three frame types: elastic, rigid, and limp. While polymeric foams are common examples of elastic frame porous materials, metal foams and soft fibrous layers are examples of rigid and limp frame materials, respectively. For an elastic porous medium, the Biot theory1,2 is commonly used to describe the propagation of waves in the medium. This theory can be used to model limp and rigid porous media; however, extreme values in the rigidity may create instabilities in a numerical implementation. To prevent instabilities and to reduce the number of degrees of freedom in a numerical poroelastic modeling,3 an equivalent fluid model is often used. In this case, only the acoustic compression wave propagating in the porous medium is considered.4 This propagation is governed by the Helmholtz equation in which the equivalent fluid is characterized by an effective density and an effective bulk modulus accounting for the viscous and thermal dissipations of the acoustic wave. For rigid frame porous media, different models were proposed to evaluate these effective properties.4–10 On the other hand, only a few models were proposed for limp porous media, and few comments on their frequency limits were given.11–14 However, the work by Lai et al.13 has underlined the importance of the limp model in view to prevent erroneous predictions of global acoustic indicators, such as the sound transmission loss, when using a rigid frame model.
The intent of this letter is to develop and experimentally validate the low and high frequency limits of the effective density of the limp model derived as a limit case of the Biot poroelastic mixed model.3 Also, it is intended to show how these limits compare to the classical rigid frame models to explain the aforementioned erroneous predictions.
II. Theory
A. Poroelastic model
For an open-cell porous medium made up from an elastic solid phase (the frame) and a fluid phase, the Biot theory states that three waves propagate simultaneously in the medium: one elastic compression wave, one elastic shear wave, and one acoustic compression wave.1,2 Defining the fluid pore pressure by and the solid phase displacement vector by , the dynamic behavior of the three waves is governed by the following two coupled equations:3
with the volume coupling coefficient given by
and the effective solid phase density
In these equations, is the Laplace operator ( grad), the tilde symbol represents a variable that is complex and frequency dependent, is the open porosity of the porous medium, is the density of the fluid saturating the pores, is the in vacuo bulk density of the medium, is the in vacuo stress tensor, is the bulk modulus of the elastic material from which the frame is made, and is the in vacuo bulk modulus of the frame. The effective density and bulk modulus of the fluid phase may be estimated using different general semiphenomenological models taking into account the viscous and thermal losses of the acoustic compression wave.4–10
B. Rigid frame porous model
When the frame of the porous medium is assumed motionless, the frame does not undergo any displacement (i.e., ) and deformation. This situation occurs under acoustic excitations when the frame is constrained and rigid, heavy, or when the solid-fluid coupling is negligible for an elastic frame.15 Consequently, only the acoustic compression wave propagates in the porous medium, and only the second equation of Eq. (1) remains and simplifies to the following Helmholtz equation:
where and are the effective density and bulk modulus of the so-called rigid frame equivalent fluid medium. These quantities can be used to deduce the more common characteristic impedance and propagation constant of the equivalent fluid medium by and , respectively. Note that the “rigid frame model” appellation is common in the literature to define Eq. (4), and its effective properties. However, it may seem abusive since in dynamics “rigid” only implicates no deformation. As a result, the rigid frame model eliminates de facto the rigid body motion of the solid phase. This points out the potential abusive use of the rigid frame model in the case where the frame is rigid but unconstrained (e.g., highly resistive light foam with a sliding edge condition). This problem was stressed by Lai et al.13
C. Limp frame porous model
When the frame of the porous medium is assumed limp (i.e., flexible), the frame does not resist to external excitations and its stress field vanishes (here, assuming ). This situation occurs for solid particles in suspension in a fluid medium or for porous media with very low shear modulus—in the limit case, the shear modulus is zero. Again, as for the rigid frame model, only the acoustic compression wave exists. Consequently, since the stress field vanishes, taking the divergence of the first equation in Eq. (1), then substituting the result into the second equation of Eq. (1) yields the following limp frame equivalent fluid equation
While the effective bulk modulus stays unchanged (compared to rigid frame media), the effective density of the equivalent fluid is modified as follows:
Since is generally much greater than , the limp effective density may simplify to
where is the total apparent mass of the equivalent fluid limp medium.
Equation (5) is similar to Eq. (4); however, this time the effective density takes into account the inertia added by the limp solid phase. Taking the high bulk density limit of Eqs. (6) or (7), one can show that (as or ). This limit proves that high density limp porous media can be modeled as rigid frame porous media, as mentioned in the previous section. On the other hand, taking the low bulk density limit of Eq. (6) and assuming porosity tends to unity (logical for low density materials), one can show that (as and ). Consequently, as reduces, the viscous losses in the limp frame material reduce, and the thermal loss (introduced by ) stays unchanged.
Finally, it is worth mentioning that the characteristic impedance and propagation constant of the limp frame medium can be computed as for the rigid frame medium; however, this time the limp effective density is used: and . In this case, both properties are modified by the limpness of the frame. In the particular case of the aforementioned low bulk density limit, since the viscous losses vanish and the thermal losses are generally small in porous media, both and approach the characteristic impedance and propagation constant of air. Inversely, as the bulk density increases, the viscous losses take more importance and and may diverge strongly from the air values, and reach the rigid frame behavior when .
III. Results and discussions
Both rigid and limp effective densities are now compared to experimental results, and their high and low frequency limits analyzed. For the simulation results, the semiphenomenological model worked out by Johnson et al.6 is used to compute . This model writes
where the material properties are given in Table I and correspond to a soft fibrous material. For the experimental results, the method worked out by Utsuno et al.16 was used in conjunction with a 100-mm-diam Brüel and Kjaer 4206 impedance tube to deduce the measured effective density.
Material properties of the studied soft fibrous material.
. | Symbol . | Value . | Units . |
---|---|---|---|
Open porosity | 0.98 | ||
Static airflow resistivity | |||
Tortuosity | 1.02 | ||
Viscous characteristic length | 90 | ||
Bulk density | 30 | ||
Density of saturating air | 1.208 | ||
Viscosity of saturating air | |||
Elastic bulk modulus ratio |
. | Symbol . | Value . | Units . |
---|---|---|---|
Open porosity | 0.98 | ||
Static airflow resistivity | |||
Tortuosity | 1.02 | ||
Viscous characteristic length | 90 | ||
Bulk density | 30 | ||
Density of saturating air | 1.208 | ||
Viscosity of saturating air | |||
Elastic bulk modulus ratio |
Figure 1 presents the comparison between the predictions and the measurement of the effective density of the soft fibrous material. As one can note, both rigid and limp models yield good predictions at higher frequencies [see Fig. 1(a)]; however, when zooming at high frequencies [see Fig. 1(b)], a slight discrepancy is observed for the real part of the rigid model. This may be explained by looking at the high frequency limits of the models:
From these limits, it is clear that the asymptotic high frequency behaviors of both models are only slightly different. For a porosity and a tortuosity nearly equal to unity (typical for light fibrous materials), both limits are equal.
Normalized effective density of the soft fibrous material. Comparison between rigid frame predictions, limp frame predictions, and experimental results. (a) Zoom between 0 and . (b) Zoom between 1000 and .
Normalized effective density of the soft fibrous material. Comparison between rigid frame predictions, limp frame predictions, and experimental results. (a) Zoom between 0 and . (b) Zoom between 1000 and .
On the other hand, at low frequencies, the prediction obtained with the rigid model significantly diverges from the measured real and imaginary parts of the effective density. In this case, only the limp model correctly predicts the measured low frequency behavior of the limp material. Again, this may be explained by analyzing the following low frequency limits of the models:
At the low frequency limit, one can observe that the rigid effective density has an asymptotic behavior dominated by a strong imaginary part. On the contrary, the limp effective density yields a purely real static value equal to the global apparent density of the equivalent fluid medium. This static value does not depend on the semiphenomenological model used to compute —in that sense, the limits on may be seen as exact.
The fact the low frequency limit of the limp model is real and equal to the global apparent density of the medium has a major impact on the prediction of global acoustic indicators (absorption and transmission coefficients) of limp porous materials. In fact, the limp model implicitly accounts for the rigid body motion defined at for which all the solid phase particles move in phase (i.e., no deformation). This rigid mode strongly controls the dynamic behavior of the limp porous medium at low frequencies. Also, it explains why the rigid frame model may yield erroneous results at low frequencies.13
Also, beyond the use of the limp model for limp frame porous materials, one can also use the limp model to capture the rigid body motion of an unconstrained rigid frame porous material. In fact, if the sound absorption coefficient or the sound transmission loss of a relatively stiff porous sample (or poroelastic with weak solid-fluid coupling as some polymeric foams15) is measured in an impedance tube, and assuming the sample can move freely along the axis of the tube (i.e., sliding boundary conditions), the measured coefficient will correlate with the limp frame model and not with the rigid frame model at low frequencies.
Finally, if one prefers to directly input the effective properties and obtained from measurements in Eq. (1) (instead of measuring all the physical properties of the medium and using a semiphenomenological model), one needs to ensure that the measured effective density corresponds to . From the previous analysis, if one note that the tested porous sample has a rather limp frame (or unconstrained rigid frame), or that the measured effective density has a limp-like behavior as the one shown in Fig. 1(a), the measured effective density needs to be corrected. Derived from Eq. (7), the correction to apply is
where is assumed to be the measured effective density.
IV. Conclusions
From the previous results and discussions, it was shown that some precautions have to be considered when using an equivalent fluid model for porous media. In many situations, it is important to take into account the static rigid body mode of the porous medium to obtain the proper low frequency behavior in terms of sound absorption and, mainly, in terms of sound transmission. It was shown that the rigid body mode can be captured by the limp frame equivalent fluid model, and not by the rigid frame equivalent fluid model. Consequently, prior to simulating a system containing a porous medium, the practitioner should ensure to select the proper porous model. If some uncertainties remain on the selection of the model, it is suggested to use the full poroelastic model to prevent any erroneous results. However in this case, the elastic properties of the medium are required, and larger numerical systems with potential instabilities may result.3,13
Acknowledgment
The author thanks N.S.E.R.C. Canada and F.Q.N.R.T. Quebec for their financial support.