A generalization of the commonly used pressure jump modeling of thin porous layers is proposed. The starting point is a transfer matrix model of the layer derived using matrix exponentials. First order expansions of the propagating terms lead to a linear approximation of the associated phenomena and the resulting matrix is further simplified based on physical assumptions. As a consequence, the equivalent fluid parameters used in the model may be reduced to simpler expressions and the transfer matrix rendered sparser. The proposed model is validated for different backing conditions, from normal to grazing incidence and for a wide range of thin films. In the paper, the physical hypotheses are discussed, together with the origin of the field jumps.
1. Introduction
Thin porous layers, sometimes referred to as screens, films, veils, etc., are commonly applied as protection, both for aesthetic reasons as well as for prevention of wear and tear. Properly designed they may also contribute to the performance of sound packages in various applications. For example, when used in combination with, e.g., metamaterials (Lagarrigue et al., 2013; Jiménez et al., 2017; Yang et al., 2015) or stacked layers of poroelastic materials (PEMs) (foam, fibre panels, etc.) (Parra Martinez et al., 2016a; Atalla and Sgard, 2007), their properties can be tailored to optimise the system performance, at low weight and without increasing the overall thickness of the installation. They may also be used to improve the absorption at low frequencies and despite their small thickness, usually ranging from 0.1 to 1 mm, these coatings commonly have a noticeable impact and cannot be ignored when simulating the behaviour of the absorbers (Sugie et al., 2006; Allard and Atalla, 2009).
The present work focuses on the numerical modeling of thin permeable films applied as coatings on multilayer poroelastic panels. Depending on which effects need to be accounted for in the simulations, different approaches may be adopted. A minimal one would be the pressure jump model proposed by Pierce (1989) or a slightly more refined rigid-frame model (Allard and Atalla, 2009). This would, however, lead to a rather crude prediction of the response, missing resonances linked to the elasticity of the system but would also have the advantage of requiring only a few parameters to be characterised in order to simulate the acoustic behaviour. A more accurate approach, particularly for woven screens, could be based on models for perforated panels accounting for the vena contracta effect and changes of dynamic tortuosity (Atalla and Sgard, 2007). Another possible strategy is to rely on a porous plate model to account for film bending that may, for instance, be useful in the case of painted films (Etchessahar et al., 2001; Leclaire et al., 2001). Finally, an even more accurate description would consider bi-phase models such as Biot's with the refinements proposed by Johnson, Champoux, and Allard (JCA) for viscous dissipative effects (Biot, 1956; Champoux and Allard, 1991; Johnson et al., 1987) or one of the rewritten sets of equations for PEMs (Bécot and Jaouen, 2013; Dazel et al., 2007). Note that in Bécot and Jaouen (2013) a flexible poroelastic model is presented that may be adapted to any existing equivalent fluid model for the dynamic density and compressibility and , hence allowing for changes in the dissipation model. Even if the thickness is rather small compared to the dimension of the homogenisation volume, the Biot theory has been successfully applied to model screens (Allard and Atalla, 2009; Kanfoud et al., 2009).
The present work aims at proposing a simplified model for acoustic screens accounting for the resonant effects while reducing the number of physical parameters needed in the model. It is intended as an alternative to the more elaborate modeling approaches used in current Transfer Matrix models. The principles used to derive a transfer matrix for a poroelastic layer (Sec. 2) are first presented, followed by an introduction of a number of simplifications in Sec. 3. In these, the propagative terms are linearised and a dimensional analysis is applied to neglect the terms of least importance. The approach is validated using numerical simulations where the results using the proposed model are compared to those obtained from a full Biot-JCA simulation, for several films (Sec. 4). Extensive testing of the proposed model was performed; however, most of these tests are omitted from the paper for conciseness and only two representative films are presented in the results of Sec. 4: one woven and one non-woven. Note that throughout this paper, a positive time convention (ejωt) is used.
2. Transfer matrix as an exponential
To describe the acoustic propagation in a poroelastic layer, a Biot model (Biot, 1956; Biot and Willis, 1957) is used but without taking into account membrane effects that could be significant in films under tension. In the present case, the model uses the two motion equations corresponding to the strain-decoupled formulation (Dazel et al., 2007),
and the associated two constitutive laws,
where and are, respectively, the homogenised solid and total displacements, p is the interstitial pressure, and ε are the in vacuo stress and strain tensors, and δij is the Kronecker symbol. The other symbols are physical parameters of the medium that represent densities (), compressibility (), Lamé coefficients ( and N), and solid/fluid coupling ().
The model is developed assuming a layer of infinite extent in the (x, y) plane and of thickness d excited by a plane wave travelling toward the positive values along the z axis. All the fields have the same dependence and the wave vector, due to the assumed isotropy, is oriented such that ky = 0. The strategy to derive the transfer matrix is then based on the so-called Stroh formalism (Dazel et al., 2013; Parra Martinez et al., 2016a; Serra et al., 2016). This approach leads to a set of first order differential equations in the state vector ,
Note that the physical fields absent from the state vector (such as and ) can be deduced from linear combinations of the ones present.
As the medium is homogeneous, solving Eq. (3) for a layer of thickness d leads to the expression of the transfer matrix for the state vector s written as a matrix exponential,
The derivation of the state matrix is central to the present work. However, it has been addressed multiple times in the literature and thus will be omitted for conciseness. One may refer to Dazel et al. (2013) for this specific case or Parra Martinez et al. (2016b) and Serra et al. (2016) for a more general discussion. The matrix used in the present work is identical to the one proposed in Appendix A.1 of Dazel et al. (2013) (with ):
3. Simplification of the transfer matrix
The key idea of this paper is to use some hypotheses based on the characteristic properties of acoustic screens to simplify the transfer matrix and reduce the number of required parameters. The most important assumption is that the average thickness of these coatings is rather small, especially compared to the wavelengths of the waves in the media (kd 1, with k the wavenumber in air or in the poroelastic layer). This hypothesis allows for an approximation of the matrix exponential of Eq. (4) in terms of its first order Taylor expansion,
with I the identity matrix. This approximation replaces the terms representing the transfer of the corresponding fields through the film, with jumps in the values of the field amplitude values between both faces of the layer. Indeed, Eq. (6) suggests that all the fields are transferred and altered by a combination of the other fields described by the state matrix and proportional to the thickness d.
The next step is to identify the most important contributions in and neglect all others. For this purpose, the second proposed approximation is introduced. Let ks be the largest wavenumber in the layer (i.e., ks ≥ kx,z, noting that ks often corresponds to the solid-borne compressional wave) and using the constitutive Eq. (2), it is easily shown that
From the second and last rows of the transfer matrix (solid displacements), we get
which can be combined with the two inequalities previously introduced to rewrite the jumps for the two components of the solid displacement,
Provided that the frequency or the thickness is sufficiently small such that ksd 1, it is reasonable to assume that (with i being x or z), and thus to neglect all terms on lines 2 and 6 of the matrix in Eq. (6) except the unit diagonal.
Moreover, it is proposed to neglect two other effects: the influence of the saturating fluid on the evolution of in vacuo shear solid stresses ; and the coupling of the tangential solid and normal total displacement. These assumptions, respectively, cancel and , and .
The transfer matrix may then be rewritten as
So far, the proposed modifications mostly impact the structure of the transfer matrix without significantly reducing the number of physical parameters required to describe the system behaviour. Considering the small thickness of the coatings, it is proposed to neglect the effects of tortuosity and set and to use a low-frequency approximation of the equivalent fluid parameters. This last assumption leads to simplified forms for the equivalent fluid quantities and ,
The first of these relations suggests that the compression is isothermal. The imaginary part of the simplified equivalent density represents a phase shift of the fluid-borne wave often referred to as a pressure jump or pressure drop (Pierce, 1989). The real part of the simplified expression of is actually a second order term (whereas the imaginary part is of the first order) that corrects the loss of accuracy at high frequencies due to the simplifications.
This last set of simplifications brings down the number of parameters in the model to six: three mechanical parameters (, N, and the structural loss parameter η), the flow resistivity σ, the porosity ϕ, and the bulk density ρ1. Investigations involving neglected frame mechanical parameters were performed, however, the resulting levels of error were higher than the target. Hence, it was decided to keep the elastic contributions in the proposed model.
4. Validating the simplified model
In this section, the simplified model is tested for different configurations against a reference solution corresponding to a full Biot-JCA Transfer Matrix model. In order to validate the proposed approach, attention must be paid to special cases such as the change of backing condition, grazing incidence, and different materials. A large range of configurations have then been tested in order to assess the robustness of the proposed model. The three configurations shown in Fig. 1 were used in the development process. They present three different types of boundary conditions backing the simplified layer and helps to understand how the errors build up. It has been observed during the refinement of the proposed method that the backing condition featuring an air-gap of Fig. 1(c) leads to results very similar to those obtained with a PEM backing of Fig. 1(b), whereas the rigidly backed configuration of Fig. 1(a) behaves differently. Consequently, only the configurations of Figs. 1(a) and 1(b) are used in the results shown hereafter.
Configurations used during the development of the method and used in the tests.
The numerical values for the parameters used to model the films are taken from a work discussing the characterization procedure for films (Jaouen and Bécot, 2011). These values are gathered in Table 1 along with the parameters of the backing foam. In the simulations, the thicknesses of the films and foam are, respectively, 0.5 and 50 mm and the following properties are used for air: . Note that in this table and for the reference solution, the values for α∞ ≠ 1 for films but they are set to unity for the simulations when evaluating the simplified model.
Physical parameters of the foam and films (woven and non-woven) used in the validation cases of Sec. 4.
Parameters (unit) . | Foam . | Woven . | Non-woven . |
---|---|---|---|
ϕ | 0.994 | 0.72 | 0.04 |
σ (N s m−4) | 9045 | 87 × 103 | 775 × 103 |
1.02 | 1.02 | 1.15 | |
(μm) | 197 | 480 | 230 |
Λ (μm) | 103 | 480 | 230 |
ρ1 (kg m−3) | 8.43 | 171 | 809 |
ν | 0.42 | 0 | 0.3 |
E (Pa) | 194.9 × 103 | 50 × 103 | 260 × 106 |
η | 0.05 | 0.5 | 0.5 |
Parameters (unit) . | Foam . | Woven . | Non-woven . |
---|---|---|---|
ϕ | 0.994 | 0.72 | 0.04 |
σ (N s m−4) | 9045 | 87 × 103 | 775 × 103 |
1.02 | 1.02 | 1.15 | |
(μm) | 197 | 480 | 230 |
Λ (μm) | 103 | 480 | 230 |
ρ1 (kg m−3) | 8.43 | 171 | 809 |
ν | 0.42 | 0 | 0.3 |
E (Pa) | 194.9 × 103 | 50 × 103 | 260 × 106 |
η | 0.05 | 0.5 | 0.5 |
As seen from Fig. 2, the agreement between the proposed simplified model and a complete Biot-JCA model for the film is tested for two materials, two angles (0° and 65°) and two boundary conditions (rigid backing and PEM backing). Clearly, the results obtained by the proposed method stay close to the reference. The agreement is even perfect at low frequencies and some very small discrepancies discussed below are observed at higher frequencies.
(Color online) Absorption coefficient for two different materials (one per sub-figure), two angles of incidence (different colours), and two configurations (different line styles). Left: Non-woven film. Right: Woven film. The graphs showing low absorption (dashed lines) are computed for the configuration of Fig. 1(a) with the film directly laid on the backing, the others correspond to configuration Fig. 1(b) with an inserted PEM layer. The reference results are shown using marker symbols.
(Color online) Absorption coefficient for two different materials (one per sub-figure), two angles of incidence (different colours), and two configurations (different line styles). Left: Non-woven film. Right: Woven film. The graphs showing low absorption (dashed lines) are computed for the configuration of Fig. 1(a) with the film directly laid on the backing, the others correspond to configuration Fig. 1(b) with an inserted PEM layer. The reference results are shown using marker symbols.
In order to systematize the analysis, the evolution of the absolute error
is computed over the (f, θ) plane with αbiot and αscreen being, respectively, the absorption coefficients calculated using the complete Biot-JCA model for the film and the simplified model. The results are shown in Fig. 3 for different films and backing conditions.
(Color online) Evolution of the absolute error in the (f, θ) plane. Left [(a) and (c)]: non-woven film; Right [(b) and (d)]: woven film. Top [(a) and (b)]: configuration as in Fig. 1(a); Bottom [(c) and (d)]: configuration as in Fig. 1(b).
Near grazing incidence, an expected drop of precision is observed, particularly when the film is directly laid on a rigid backing. This difference between the two setups suggests that a compensatory effect is introduced when the backing is a PEM. Indeed, whereas the absorption occurs only in the film when directly laid on a rigid surface, it occurs mainly in the PEM in the other case, smoothing the effect of the film. At grazing incidence, the proposed model (where most of the shear effects have been neglected) obviously breaks down. The path travelled by the waves in the film is also much longer, invalidating the hypothesis of a thin film and leading to excessive absorption. Despite these points, the corresponding relative error is lower than 3% in the observed cases.
5. Conclusion
In the present work, a simplified model for the transfer matrix of acoustic films is proposed. Approximations based on physical reasoning lead to both a simpler propagation model (with expanded matrix exponential) and a reduction in the number of parameters. It is demonstrated that these simplifications hold for thin poroelastic layers and that the resulting model behaves best when used in a transfer matrix model on top of another PEM or air layer. For rigid backing conditions, some discrepancies may be observed, mainly due to some of the parameters being neglected. Despite these aspects, the model still exhibits a reasonable accuracy.