2.1 Geometry description

We consider a periodic permeable material whose fluid network is connected and saturated with a Newtonian fluid. Its frame is made of a very stiff skeleton onto which highly flexible thin flat films are fixed, as illustrated by figure 1. The characteristic length associated with the pores (or the period of the microstructure) is denoted as $l$ . We consider relatively simple pore geometries for which a single characteristic size determines the fluid flow, i.e. no extreme geometrical contrasts in the pore space are considered. The volumes of the fluid network, the stiff solid and the representative elementary volume (REV) are $\unicode[STIX]{x1D6FA}_{f}$ , $\unicode[STIX]{x1D6FA}_{s}$ and $\unicode[STIX]{x1D6FA}$ respectively. The surface of the stiff solid is denoted as $\unicode[STIX]{x1D6E4}_{s}$ .

Figure 1. Diagram of the scales of a permeo-elastic material. The thin films are clamped to the stiff solid frame and the pore fluid network is connected.

On the other hand, it is assumed that the in-plane dimension of the films, $h$ , is of the same order of magnitude as the period of the microstructure, i.e. $h=O(l)$ , while their thickness, $t$ , is much smaller than the in-plane dimension, i.e. $t\ll h$ . Therefore, the volume of the films can be neglected and the porosity of the material is calculated as $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D6FA}_{f}/\unicode[STIX]{x1D6FA}$ . Moreover, the films are made of an isotropic elastic material. These hypotheses allow us to model the films as Love–Kirchhoff plates (Love Reference Love1888) with negligible thickness and surface $\unicode[STIX]{x1D6E4}_{p}$ whose behaviour is characterised by their out-of-plane 2D bending. The edges of the films, $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{p}$ , are clamped to the stiff solid frame. This assumption can be (i) relaxed to consider other boundary conditions such as a combination of a free condition on some parts of $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{p}$ and a clamped condition on other parts of $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{p}$ or (ii) modified by considering membranes in tension instead of plates.

We are interested in long-wavelength sound propagation in permeo-elastic media. The macroscopic characteristic length $L$ is related to the wavelength $\unicode[STIX]{x1D706}$ through $L=|\unicode[STIX]{x1D706}|/2\unicode[STIX]{x1D70B}$ , and the disparity in length scales between the macroscopic and microscopic characteristic sizes provides the small parameter $\unicode[STIX]{x1D700}=l/L\ll 1$ .

2.2 Governing equations at the local scale

The equations that describe the propagation of harmonic sound waves ( $\text{e}^{\text{i}\unicode[STIX]{x1D714}t}$ ) in rigid-frame porous materials are now recalled. These correspond to the linearised equations of conservation of momentum (2.1) and mass (2.2) along with the no-slip (2.3) boundary condition (Biot Reference Biot1956a ,Reference Biot b ; Auriault et al. Reference Auriault, Borne and Chambon1985, Reference Auriault, Boutin and Geindreau2009). It should be noted that, for the sake of clarity in the presentation, adiabatic sound propagation is considered. However, the inclusion of thermal exchanges into the formulation is straightforward, as it will be shown below.

Here, $\unicode[STIX]{x1D748}_{f}$ is the fluid stress tensor, $\unicode[STIX]{x1D702}$ is the dynamic viscosity, $\unicode[STIX]{x1D6FE}$ is the adiabatic exponent, $\unicode[STIX]{x1D63F}(\boldsymbol{v})$ is the strain rate tensor (defined through $2\unicode[STIX]{x1D63F}(\boldsymbol{v})=\text{grad}(\boldsymbol{v})+(\text{grad}(\boldsymbol{v}))^{\text{T}}$ ), $\boldsymbol{v}$ is the fluid velocity, $p$ is the pressure and $\unicode[STIX]{x1D644}$ is the second-rank unitary tensor. The subscript 0 is used to denote equilibrium quantities.

The Love–Kirchhoff (in-vacuum) plate equations (Love Reference Love1888) are now recalled. The scalar out-of-plane displacement of the films, $u$ , is governed by the equations of equilibrium of transverse forces (2.4), moment balance (2.5) and the constitutive law (2.6):

Here, all of the differential operators are acting in the plane $\unicode[STIX]{x1D6E4}_{p}$ of the films. This is represented by the subscript $p$ . For example, $\unicode[STIX]{x1D735}_{p}$ stands for the in-plane gradient operator and $\boldsymbol{e}_{p}(\boldsymbol{a})=(\text{grad}_{p}(\boldsymbol{a})+(\text{grad}_{p}(\boldsymbol{a}))^{\text{T}})/2$ is the in-plane strain tensor with, in the present case, $\boldsymbol{a}=\unicode[STIX]{x1D735}_{p}u$ . The transverse force and the bending moment tensor are respectively denoted as $\boldsymbol{T}$ and $\unicode[STIX]{x1D648}$ . The apparent Young’s modulus of the films is given by $E_{p}=E/(1-\unicode[STIX]{x1D708}^{2})$ , where $E$ is the Young’s modulus and $\unicode[STIX]{x1D708}$ is the Poisson ratio. The inertia moment is given by $I=t^{3}/12$ . It should be noted that the term $E_{p}I$ corresponds to the flexural stiffness, and the films (in vacuum) are loaded by their own inertia $\unicode[STIX]{x1D70C}_{p}t\unicode[STIX]{x1D714}^{2}$ , with $\unicode[STIX]{x1D70C}_{p}t$ being the surface density of the films. We emphasise that under bending, the thickness of the films can be considered as constant so that they are deformed without undergoing volume variation.

As previously mentioned, we consider that the edges of the films, $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{p}$ , are clamped to the solid boundary $\unicode[STIX]{x1D6E4}_{s}$ , i.e.

where $\boldsymbol{n}$ is the outward-pointing normal vector of the edges $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{p}$ .

Sound propagation in permeo-elastic materials is characterised by the interaction between the saturating fluid and the films. Consequently, the equations describing the out-of-plane displacement of the films and the fluid flow through the material are coupled. A two-way coupling is achieved as follows.

First, the boundary condition on the surface $\unicode[STIX]{x1D6E4}_{p}$ expressing the continuity of fluid and film velocities on both faces of the films $\unicode[STIX]{x1D6E4}_{p}^{+}$ and $\unicode[STIX]{x1D6E4}_{p}^{-}$ is added. The uniform out-of-plane displacement across the film thickness allows us to express the boundary condition as

where, by convention, $\boldsymbol{N}$ is the outward-pointing normal vector to $\unicode[STIX]{x1D6E4}_{p}^{+}$ . It should be noted that the film kinematics imposes that the velocity of the fluid is orthogonal to the films.

Second, the transverse force equilibrium equation (2.4) is modified by the stress exerted by the fluid on the two faces of the films. The latter is given by $(\unicode[STIX]{x1D748}_{f}\boldsymbol{\cdot }\boldsymbol{N}^{+}-\unicode[STIX]{x1D748}_{f}\boldsymbol{\cdot }\boldsymbol{N}^{-})$ . Hence, the normal component of the fluid loading on the films can be written as $[\unicode[STIX]{x1D748}_{f}\boldsymbol{\cdot }\boldsymbol{N}]\boldsymbol{\cdot }\boldsymbol{N}$ , where $[\boldsymbol{\cdot }]$ denotes the ‘jump’ across the interface $\unicode[STIX]{x1D6E4}_{p}$ . Then, equation (2.4) becomes

It must be noticed that due to the negligible thickness of the films, the momentum caused by the tangential stress exerted by the fluid does not modify the moment balance equation (2.5).

In summary, the local description is given by (2.1), (2.2), (2.9), (2.5), (2.6) and boundary conditions (2.3), (2.7) and (2.8). This is similar to that in Panasenko & Stavre (Reference Panasenko and Stavre2012) for the case of viscous fluid–thin plate interaction and periodic flow, and that in Lebental & Bourquin (Reference Lebental and Bourquin2012) for water–plate interaction in confined media. The next section deals with the physical analysis of the local description.

2.3 Physical analysis

This section presents the physical analysis of the local description. Assuming scale separation, we focus on describing the most general case for which all of the physical phenomena are present and interact at the local scale. The aim is to derive a general model that (i) captures the contribution of the local physics to the macroscopic behaviour of permeo-elastic media and (ii) reduces to degenerated models in which, for example, one of the mechanisms does not significantly contribute to the macroscopic acoustic behaviour.

As in long-wavelength sound propagation in conventional porous media (Auriault et al. Reference Auriault, Borne and Chambon1985, Reference Auriault, Boutin and Geindreau2009; Lafarge et al. Reference Lafarge, Lemarinier, Allard and Tarnow1997), the pressure fluctuates at the macroscopic scale, and, while the velocity and its rate of deviatoric deformation vary at the microscopic scale, the microscopic divergence itself is of the order of the macroscopic divergence, i.e.

The richest flow regime occurs when the viscous and inertial terms balance the pressure gradient. This means that the three terms in the oscillatory Stokes equation (2.1) are of the same order of magnitude (Auriault et al. Reference Auriault, Borne and Chambon1985, Reference Auriault, Boutin and Geindreau2009), i.e.

In addition, the temporal relative variations of the pressure are balanced out by the rate of volume variations (Auriault et al. Reference Auriault, Borne and Chambon1985, Reference Auriault, Boutin and Geindreau2009), i.e.

For permeo-elastic media, the case of interest occurs when the films are strongly interacting with the fluid. Due to the continuity of fluid and film velocities (2.8), one necessarily has that

This implies that the velocity of the films varies at the local scale. In turn, this leads us to consider that the transverse force and the moment bending tensor also vary at the local scale, i.e.

The most interesting case corresponds to elasto-inertial fluid–film interaction. According to the previous estimates, this leads us to consider that the terms in (2.9) are of the same order of magnitude, i.e.

The last two estimates are justified as follows. Since the deviatoric viscous stress varies at the microscopic scale, the jump of this quantity across the interface also fluctuates at the local scale, i.e. $O([\unicode[STIX]{x1D702}v/l])=O(\unicode[STIX]{x1D702}v/l)$ . However, since the pressure fluctuates at the macroscopic scale, the jump in pressure is assessed as $(p^{+}-p^{-})/l=O([p]/l)=O(p/L)$ .

As a final remark, in accordance with (2.11), (2.15) and (2.13), the co-existence of inertial effects in both the fluid and the films leads to $O(\unicode[STIX]{x1D70C}_{p}t\unicode[STIX]{x1D714}u\unicode[STIX]{x1D714}/l)=O(\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x1D714}v)$ , which results in $O(\unicode[STIX]{x1D70C}_{p}t/h)=O(\unicode[STIX]{x1D70C}_{0})$ .

2.4 Homogenisation methodology

The scale separation between the wavelength and the period size allows us to use the two-scale asymptotic homogenisation method (Sanchez-Palencia Reference Sanchez-Palencia1980; Auriault et al. Reference Auriault, Boutin and Geindreau2009) to derive the equivalent macroscopic description. To represent the evolution at the two spatial scales, one can introduce the following two dimensionless space variables: $x/L=x^{\ast }$ and $x/l=y^{\ast }$ , where $x$ stands for the usual space variable. The space variables $x^{\ast }$ and $y^{\ast }$ are associated with the variations at the wavelength and period scales respectively. One can equivalently use the two dimensional space variables $x=Lx^{\ast }$ and $y=Ly^{\ast }=xL/l=\unicode[STIX]{x1D700}^{-1}x$ by taking $L$ as the reference length. It then follows that any quantity $Q$ is considered to be a function of $(x,y)$ , i.e. $Q=Q(x,y)$ . This brings as a consequence that the usual gradient operator $\unicode[STIX]{x1D735}_{x}$ is changed into $\unicode[STIX]{x1D735}_{(xy)}=\unicode[STIX]{x1D735}_{x}+\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x1D735}_{y}$ (and $\unicode[STIX]{x1D735}_{(xy)}^{2}=\unicode[STIX]{x1D6FB}_{x}^{2}+2\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x1D735}_{xy}+\unicode[STIX]{x1D700}^{-2}\unicode[STIX]{x1D6FB}_{y}^{2}$ ). To reflect the physics of the phenomena, the use of two space variables $(x,y)$ should be combined with a rescaling of the usual equations based upon a single space variable. The reason for the rescaling lies in the fact that when expressed with the two space variables $(x,y)$ , the actual physical gradient of a quantity $Q$ that varies at the large scale, i.e. $\unicode[STIX]{x1D735}_{x}Q$ , becomes $\unicode[STIX]{x1D735}_{(xy)}Q$ . On the other hand, if the quantity varies at the local scale, the actual physical gradient $\unicode[STIX]{x1D735}_{y}Q$ has to be expressed as $\unicode[STIX]{x1D700}\unicode[STIX]{x1D735}_{(xy)}Q$ . Therefore, the gradient of variables oscillating at the local scale should be rescaled. For instance, $\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v}$ should be rewritten as $\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D6FB}_{(xy)}^{2}\boldsymbol{v}(x,y)$ to express that the fluid velocity actually varies at the pore scale.

Using the estimates identified in the physical analysis, equations (2.1), (2.2), (2.9), (2.5), (2.6) and boundary conditions (2.8), (2.3) and (2.7) are rewritten in the following two-space-variable formulation in rescaled form. It should be noted that we adopt the usual homogenisation convention which consists of keeping the same notation as for the single-space-variable formulation for both the variables and the gradient operator. Therefore, in (2.16)–(2.23), $\unicode[STIX]{x1D735}$ stands for $\unicode[STIX]{x1D735}_{(xy)}$ , $\boldsymbol{v}$ represents $\boldsymbol{v}(x,y)$ , etc.

It should be noted that in the rescaled system, the jump in pressure is explicitly of the order of the pressure $O(p)$ . However, it has been shown in the previous section that, physically, the jump in pressure is one order smaller than the pressure. Therefore, to satisfy this physical estimation, the jump in pressure in (2.20) should be rescaled as $\unicode[STIX]{x1D700}^{-1}[p]=O(p)$ .

According to the standard homogenisation procedure, the physical variables are looked for in the form of asymptotic expansions in powers of the small parameter $\unicode[STIX]{x1D700}$ as $Q(x,y)=\sum _{i=0}^{\infty }\unicode[STIX]{x1D700}^{i}Q^{(i)}(x,y)$ , where $Q=\boldsymbol{v},p,u$ . These are then substituted into (2.16)–(2.23) and the terms of the same order are identified. This procedure based upon the physical analysis ensures that the limiting description obtained when $\unicode[STIX]{x1D700}\rightarrow 0$ keeps a physics of the same nature as in the real situation where the scale separation is finite.

2.5 Fluid–structure interaction cell problem

At $\unicode[STIX]{x1D700}^{-1}$ , it follows from the equations of conservation of momentum that $\unicode[STIX]{x1D735}_{y}p^{(0)}=0$ in $\unicode[STIX]{x1D6FA}_{f}$ and $[p^{(0)}]=0$ on $\unicode[STIX]{x1D6E4}_{p}$ . This means that the pressure is a macroscopic variable, i.e. $p^{(0)}=p^{(0)}(x)$ . According to the physical analysis, this result is consistent with the fact that the films are not loaded by the leading-order pressure.

Further identification leads to the following fluid–structure interaction cell problem (with $\unicode[STIX]{x1D748}_{f}^{(1)}=2\unicode[STIX]{x1D702}\unicode[STIX]{x1D63F}_{y}(\boldsymbol{v}^{(0)})-p^{(1)}\unicode[STIX]{x1D644}$ ):

This linear problem, whose unknowns are $\boldsymbol{v}^{(0)}$ , $p^{(1)}$ and $u^{(0)}$ , is forced by the macroscopic pressure gradient $\unicode[STIX]{x1D735}_{x}\,p^{(0)}$ . To further investigate this cell problem, it is convenient to derive its associated variational formulation.

We consider the vectorial space ${\mathcal{W}}$ of complex velocity fields $\boldsymbol{w}$ defined in $\unicode[STIX]{x1D6FA}_{f}\cup \unicode[STIX]{x1D6E4}_{p}$ such that $\boldsymbol{w}$ is $\unicode[STIX]{x1D6FA}$ -periodic, divergence free $\unicode[STIX]{x1D735}_{y}\boldsymbol{\cdot }\boldsymbol{w}=0$ in $\unicode[STIX]{x1D6FA}_{f}$ , with $\boldsymbol{w}=0$ on $\unicode[STIX]{x1D6E4}_{s}$ , $\boldsymbol{w}=w\boldsymbol{N}$ on $\unicode[STIX]{x1D6E4}_{p}$ , and $w=0$ and $\unicode[STIX]{x1D735}_{yp}w\boldsymbol{\cdot }\boldsymbol{n}=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{p}$ . To take advantage of the properties of a Hilbert space, let us introduce the following scalar product on ${\mathcal{W}}$ (it should be noted that $\overline{\boldsymbol{w}}$ is the complex conjugate of $\boldsymbol{w}$ ):

This expression provides a norm since $\Vert \boldsymbol{w}\Vert ^{2}=0$ implies that $\unicode[STIX]{x1D63F}_{y}(\boldsymbol{w})=0$ . Therefore, $\boldsymbol{w}$ is a rigid-body field which reduces to $\boldsymbol{w}=0$ when taking into account the boundary condition on $\unicode[STIX]{x1D6E4}_{s}$ . Equipped with this norm, the space ${\mathcal{W}}$ is a Hilbert space.

The variational formulation of the fluid–structure interaction cell problem is obtained by (i) multiplying the equation of conservation of momentum in the fluid (2.24) by the conjugate of $\boldsymbol{w}\in {\mathcal{W}}$ and integrating it over $\unicode[STIX]{x1D6FA}_{f}$ , (ii) multiplying the momentum balance of the film (2.28) (expressed in terms of the velocity of the films $v^{(0)}=\text{i}\unicode[STIX]{x1D714}u^{(0)}$ ) by $\overline{w}$ and integrating it over $\unicode[STIX]{x1D6E4}_{p}$ and (iii) adding the two integrals. Further use of the divergence theorem, boundary conditions on $\unicode[STIX]{x1D6E4}_{s}$ and $\unicode[STIX]{x1D6E4}_{p}$ , and periodicity leads to the following result (see appendix A, where the mathematical details of the derivation are provided):

where

with

and

From now on, we will refer to ${\mathcal{V}}$ , ${\mathcal{I}}_{f}$ , ${\mathcal{E}}$ and ${\mathcal{I}}_{p}$ as the terms respectively associated with the effects of viscosity and inertia of the fluid, and elasticity and inertia of the films, as indicated in (2.34).

The form $a$ in (2.33) is sesquilinear while $b$ is semi-linear. In addition, by taking $\boldsymbol{w}=\boldsymbol{v}^{(0)}$ in (2.34), one obtains

The term $a$ is a coercive form on ${\mathcal{W}}$ since $|a(\boldsymbol{v}^{(0)},\boldsymbol{v}^{(0)})|\geqslant 2\unicode[STIX]{x1D702}\Vert \boldsymbol{v}^{(0)}\Vert$ . The Lax–Milgram lemma ensures the existence and uniqueness of the solution $\boldsymbol{v}^{(0)}$ in $\unicode[STIX]{x1D6FA}_{f}$ and, by continuity, that of $v^{(0)}$ on $\unicode[STIX]{x1D6E4}_{p}$ .

Since (2.33) is a linear problem forced by the macroscopic pressure gradient, the solution $\boldsymbol{v}^{(0)}$ can be written as a linear combination of the three particular $\unicode[STIX]{x1D6FA}$ -periodic fields $\unicode[STIX]{x1D660}^{J}(y,\unicode[STIX]{x1D714})$ corresponding to $\unicode[STIX]{x1D702}^{-1}\unicode[STIX]{x1D735}_{x}\,p^{(0)}=\boldsymbol{e}_{J}$ , where the $\boldsymbol{e}_{J}$ (with $J=1,2,3$ ) are the unitary basis vectors, i.e.

where the tensor $\widehat{\unicode[STIX]{x1D660}}(y,\unicode[STIX]{x1D714})$ includes visco-elasto-inertial effects. It should be noted that (i) this quantity depends on the geometry of the material and the physical properties of the films and the saturating fluid, and (ii) the motion of the films is linearly related to the macroscopic pressure gradient.

From the fluid velocity field and the equation of conservation of momentum, one can deduce the expression of the pressure field as

where the $\widehat{\unicode[STIX]{x1D70B}}^{J}(y,\unicode[STIX]{x1D714})$ are the particular solutions for $\unicode[STIX]{x1D702}^{-1}\unicode[STIX]{x1D735}_{x}\,p^{(0)}=\boldsymbol{e}_{J}$ and $\widehat{p}^{(1)}(x)$ is an integration constant.

Several remarks regarding the results presented in this section can be made as follows.

(i) The first and second integrals in the term $a(\boldsymbol{v}^{(0)},\boldsymbol{v}^{(0)})$ in (2.37) respectively represent the viscous dissipative power and the inertial power developed in the fluid. The third and fourth integrals account for the elastic stored power and the inertial power in the films respectively. These powers are balanced by the one developed by the pressure gradient, which is accounted for by $b(\boldsymbol{v}^{(0)})$ .

(ii) The terms associated with viscosity and inertial effects in $a(\boldsymbol{v}^{(0)},\boldsymbol{v}^{(0)})$ (i.e. the first and second integrals) along with the term in $b(\boldsymbol{v}^{(0)})$ associated with the pressure gradient correspond to those appearing in the variational formulation of the cell problem found in conventional porous materials. Here, it is of interest to compare a permeo-elastic material with the two following conventional porous materials having the same rigid frame: (1) one in which extremely stiff films are attached and (2) another one where no films are attached. In these three materials, the kinematic conditions imposed on the velocity fields differ only by the condition set on $\unicode[STIX]{x1D6E4}_{p}$ . The latter corresponds to the no-slip condition $\boldsymbol{v}^{(0)}=0$ on $\unicode[STIX]{x1D6E4}_{p}$ for extremely stiff films, $\boldsymbol{v}^{(0)}=v^{(0)}\boldsymbol{N}$ on $\unicode[STIX]{x1D6E4}_{p}$ for elastic films, and continuous fluid velocity in the absence of films. Consequently, there is no direct correspondence between the classical and the present variational formulation as the spaces of solution differ. However, in the three cases, the velocity fields are continuous in $\unicode[STIX]{x1D6FA}_{f}\cup \unicode[STIX]{x1D6E4}_{p}$ . Regarding the fluid stress fields, these are continuous within the fluid space $\unicode[STIX]{x1D6FA}_{f}$ but may differ on the two sides of $\unicode[STIX]{x1D6E4}_{p}$ . However, since the films introduce internal solid boundaries, the condition of continuity across $\unicode[STIX]{x1D6E4}_{p}$ does not have to be considered for the fluid phase. Due to these considerations about the smoothness of the fields in $\unicode[STIX]{x1D6FA}_{f}$ , and taking into account the different kinematic restrictions, one obtains that the spaces of velocity fields, defined strictly in $\unicode[STIX]{x1D6FA}_{f}$ , are structured as $W_{r}\subseteq {\mathcal{W}}\subseteq W$ , where $W_{r}$ and $W$ are the spaces associated with the material with extremely stiff films and without films respectively. As a consequence, the static permeability (respectively high-frequency tortuosity) of the material with extremely stiff films is smaller (respectively larger) than that of the material with no films.

(iii) The presence of the films introduces two effects that are a direct consequence of the fluid–film interaction. These are represented by the elastic term ${\mathcal{E}}$ , which does not appear in the conventional Darcy’s cell problem and introduces an additional mechanism, and the inertial term ${\mathcal{I}}_{p}$ , which is of a similar nature to that in the fluid.

(iv) In contrast to the usual visco-inertial description in conventional porous media which excludes the possibility of inner resonances, the fact that the elastic effects can be balanced out by the inertial ones can lead to inner resonances that result in atypical behaviour.

(v) Finally, we recall that in standard poroelasticity, the elastic effects are of different nature. Specifically, the solid displacement varies at the macroscopic scale so that the fluid flow cell problem is not affected by the deformation of the solid. This brings as a consequence that the visco-inertial permeability does not depend on the solid elasticity. On the other hand, the effective elasticity of the skeleton results from a cell problem defined in the solid domain and forced by the macroscopic strain and fluid pressure. Conversely, in permeo-elastic materials, the fluid flow is affected by the elastic deformation of the films, which is forced by the pressure gradient.

2.6 Macroscopic acoustic description

The leading-order mass balance equation is given by

Integration of this equation over $\unicode[STIX]{x1D6FA}_{f}$ yields

The rightmost term in (2.42) can be written as a surface integral by using the divergence theorem. The resulting integral is null because of periodicity, null velocity $\boldsymbol{v}^{(1)}=0$ on $\unicode[STIX]{x1D6E4}_{s}$ and the continuity of the fluid velocity across the interface $\boldsymbol{v}^{(1)}\boldsymbol{\cdot }\boldsymbol{N}^{+}=\boldsymbol{v}^{(1)}\boldsymbol{\cdot }\boldsymbol{N}^{-}=v^{(1)}$ on $\unicode[STIX]{x1D6E4}_{p}$ .

Thus, the macroscopic description is given by

where the average operator is defined by

and the effective visco-elasto-inertial permeability $\unicode[STIX]{x1D660}(\unicode[STIX]{x1D714})$ is calculated as

It should be noted that the permeability is related to the usual permeability $\text{}\underline{\unicode[STIX]{x1D660}}$ determined by spatially averaging the velocity field with respect to the whole material volume through $\text{}\underline{\unicode[STIX]{x1D660}}=\unicode[STIX]{x1D719}\unicode[STIX]{x1D660}$ . In this paper, we will systematically use the term permeability to refer to that calculated by spatially averaging the velocity field with respect to the pore volume, i.e. that calculated using the operator (2.45).

Equation (2.43) corresponds to the classical case of adiabatic mass balance in conventional rigid-frame porous material. This is because (i) the solid frame and the films in bending experience no volume variation and (ii) the thermal exchanges have been neglected for the sake of clarity in the presentation. It should be noted that accounting for heat exchanges would lead to replacement of the adiabatic bulk modulus $\unicode[STIX]{x1D6FE}P_{0}$ by the dynamic bulk modulus $K(\unicode[STIX]{x1D714})$ . The latter can be calculated, for example, using a semi-phenomenological model, as proposed in Lafarge et al. (Reference Lafarge, Lemarinier, Allard and Tarnow1997).

Conversely, the fluid flow constitutive law given by (2.44) does not correspond to the classical dynamic Darcy’s law since the elasto-inertial effects of the films are accounted for in the associated effective tensor $\unicode[STIX]{x1D660}$ , which is symmetric. This property is inherited from the symmetry of the form (2.34).

Depending on the frequency range and material parameters, it may be convenient to rewrite (2.44) in physically equivalent forms expressed in terms of the effective dynamic flow resistivity $\unicode[STIX]{x1D643}(\unicode[STIX]{x1D714})$ or apparent dynamic density $\unicode[STIX]{x1D746}(\unicode[STIX]{x1D714})$ tensors,

where $\langle \boldsymbol{u}^{(0)}\rangle =\langle \boldsymbol{v}^{(0)}\rangle /\text{i}\unicode[STIX]{x1D714}$ is the mean fluid displacement.

The effective dynamic flow resistivity and apparent dynamic density are related to the visco-elasto-inertial permeability by

Combination of the macroscopic mass balance (2.43) and the constitutive fluid flow law (2.44) leads to the following wave equation:

Considering the case of isotropic permeo-elastic materials, one has that $\unicode[STIX]{x1D660}={\mathcal{K}}\unicode[STIX]{x1D644}$ . Then, the characteristic impedance $Z_{c}$ , wavenumber $k_{c}$ and speed of sound ${\mathcal{C}}$ through the material are given by

To understand the acoustical properties of permeo-elastic materials, it is required to investigate those of the visco-elasto-inertial permeability tensor. This is developed in the next section.

3.1 Characteristic frequencies

In contrast to the classical case of rigid-frame porous media, where only a visco-inertial characteristic frequency can be identified, in permeo-elastic materials, the three mechanisms involved give rise to at least three characteristic frequencies that determine the behaviour of the material. These frequencies can be estimated by considering each one of the three possible combinations of two mechanisms, i.e. visco-inertial, visco-elastic and elasto-inertial. Indeed, when only two of the mechanisms are involved, a specific fluid flow regime arises, which is characterised by a frequency that determines the transition between the two degenerated regimes driven by a single mechanism, i.e. purely viscous, elastic and inertial regimes.

3.2 Quasi-rigid films

The behaviour of the effective properties of permeo-elastic materials is investigated using an asymptotic analysis. First, a dimensionless parameter $\unicode[STIX]{x1D6FC}=\text{i}\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{e}$ is identified from the variational formulation (3.16). The parameter $\unicode[STIX]{x1D6FC}$ tends to zero when $\unicode[STIX]{x1D714}\ll \unicode[STIX]{x1D714}_{e}$ . This is satisfied when either $E_{p}I\rightarrow \infty$ , which corresponds to the case of quasi-rigid films, or $\unicode[STIX]{x1D714}\rightarrow 0$ . This situation leads to a specific fluid–film interaction cell problem, as detailed in appendix B. The analysis is performed by expressing the local fields in the form of expansion series of the type $\boldsymbol{v}^{(0)}=\sum _{i=0}^{\infty }\unicode[STIX]{x1D6FC}^{i}\boldsymbol{v}^{[i]}$ , etc. The solution of the resulting problems at the first two orders of expansion in $\unicode[STIX]{x1D6FC}$ yields for $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{e}\ll 1$ (see appendix B for the details of the resolution)

where $\unicode[STIX]{x1D660}_{r}(\unicode[STIX]{x1D714})$ represents the standard dynamic permeability of a material with perfectly rigid films. The symmetric tensor $\unicode[STIX]{x1D63D}_{r}(\unicode[STIX]{x1D714})$ (in $m^{2}$ ) is related to the fluid flow generated by the elastic motion of the films in response to the excitation by the fluid flow at the leading order (at $\unicode[STIX]{x1D6FC}^{0}$ ).

From (3.20), it is concluded that (i) the elasticity of the films acts as a corrector of the permeability and (ii) the classical dynamic Darcy’s law is recovered for large values of the Young’s modulus and at low frequencies.

Equation (3.20) can be written in the alternative forms (2.47) and (2.48). These will be used in §§ 3.2.1–3.2.3 to investigate the response of the fluid–film system in different fluid flow regimes.

In what follows, the analysis is conducted for an isotropic material (or for a preferential flow direction), i.e. the tensors involved become scalar. For example, the viscous permeability tensor reads as $\unicode[STIX]{x1D660}_{r}(\unicode[STIX]{x1D714})={\mathcal{K}}_{r}(\unicode[STIX]{x1D714})\unicode[STIX]{x1D644}$ . The analysis will be made with reference to figure 2. This figure shows a conceptual diagram representing the asymptotic behaviour of permeo-elastic materials with respect to the visco-inertial frequency $\unicode[STIX]{x1D714}_{vr}$ for the case $\unicode[STIX]{x1D714}\ll \unicode[STIX]{x1D714}_{e}$ . It will become apparent throughout the next subsections under what conditions these regimes are achieved.

Figure 2. Rheological diagram representing the combination of dashpot ( ${\mathcal{V}}$ , 

), spring ( ${\mathcal{E}}$ , 

) and mass ( ${\mathcal{I}}_{f}$ , ○) describing the acoustic behaviour of permeo-elastic materials with respect to the visco-inertial frequency $\unicode[STIX]{x1D714}_{vr}$ for the case $\unicode[STIX]{x1D714}\ll \unicode[STIX]{x1D714}_{e}$ (i.e. quasi-rigid films).

3.3 Very flexible films

The same approach as in § 3.2 is followed in this section to investigate the behaviour of very flexible films. This situation can be studied by considering a small parameter $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D714}_{e}/\text{i}\unicode[STIX]{x1D714}\rightarrow 0$ identified from (3.16). This condition is satisfied when either $E_{p}I\rightarrow 0$ or $\unicode[STIX]{x1D714}\rightarrow \infty$ . As in the previous section, the variables are looked for as expansion series but now in terms of $\unicode[STIX]{x1D6FD}$ . The solution of the resulting problems at the first two orders of expansion in $\unicode[STIX]{x1D6FD}$ yields for $\unicode[STIX]{x1D714}\gg \unicode[STIX]{x1D714}_{e}$ (see appendix C for the details of the resolution)

where $\unicode[STIX]{x1D660}_{s}(\unicode[STIX]{x1D714})$ is the dynamic permeability of a material slightly different from one without films since a condition of normal flow on the films is imposed and its high-frequency behaviour is determined by the inertia of the fluid and that of the films. The term $\unicode[STIX]{x1D63D}_{s}(\unicode[STIX]{x1D714})$ represents the additional flow required to equilibrate the films which are undergoing deformation imposed by the fluid flow at the leading order (i.e. at $\unicode[STIX]{x1D6FD}^{0}$ ).

From (3.37), it is concluded that (i) the elasticity of the films acts as a corrector of the permeability and (ii) a dynamic Darcy’s law slightly different from that of a material without films is obtained when $\unicode[STIX]{x1D714}_{e}/\unicode[STIX]{x1D714}\rightarrow 0$ .

Equation (3.37) can be rewritten in the following alternative forms (see (2.47) and (2.48)) which will be used in §§ 3.3.1–3.3.3:

For simplicity, the analysis is further conducted for an isotropic material (or for a preferential flow direction). Therefore, the tensors involved become scalar. For example, $\unicode[STIX]{x1D660}_{s}(\unicode[STIX]{x1D714})={\mathcal{K}}_{s}(\unicode[STIX]{x1D714})\unicode[STIX]{x1D644}$ . The analysis will be made with reference to figure 3, which shows a conceptual diagram representing the asymptotic behaviour of permeo-elastic materials with respect to the visco-inertial frequency $\unicode[STIX]{x1D714}_{vs}$ for the case $\unicode[STIX]{x1D714}\gg \unicode[STIX]{x1D714}_{e}$ .

Figure 3. Rheological diagram representing the combination of dashpot ( ${\mathcal{V}}$ , 

), spring ( ${\mathcal{E}}$ , 

) and masses ( ${\mathcal{I}}_{f}$ , ○ and ${\mathcal{I}}_{p}$ , ▫) describing the acoustic behaviour of permeo-elastic materials with respect to the visco-inertial frequency $\unicode[STIX]{x1D714}_{vs}$ for the case $\unicode[STIX]{x1D714}\gg \unicode[STIX]{x1D714}_{e}$ (i.e. very flexible films).

4.1 Illustrating examples

This section illustrates the theoretical results obtained in the previous sections for permeo-elastic media. We consider an air-saturated material made of a solid matrix with an embedded array of cylindrical pores into which periodically distributed coaxial short tubes are inserted. The films are fixed on one of the extremes of the short tubes, while the other extremes are open. The geometry of the material is shown in figure 4, together with a 2D axisymmetric geometry of one cylindrical pore with the inserted short tubes with films and the unit cell. We emphasise that the physics of the array is mostly captured by the phenomena occurring in the tubes. The radius of the cylindrical pore is represented by $r_{0}$ , while the short tube inner radius and length are represented by $r_{c}$ and $t_{c}$ respectively. The size of the gap formed in between the tube and the walls of the cylindrical pore is denoted as $g$ , while the thickness of the tube walls is given by $w=r_{0}-r_{c}-g$ . The unit cell of the material has a length $t_{0}$ and the distance between the short tubes is $t_{0}-t_{c}$ .

Figure 4. (a) Array of cylindrical pores with periodically distributed coaxial short tubes with clamped films on one of their extremes (left) and section views (right). (b) Two-dimensional axisymmetric geometry of a cylindrical pore with the inserted short tubes with films (left) and geometrical parameters of the unit cell of the material (right).

The mechanical parameters of a polyetherimide (PEI) film sample are considered as reference. These are $\unicode[STIX]{x1D70C}_{p}=1200~\text{kg}~\text{m}^{-3}$ , $\unicode[STIX]{x1D708}=0.36$ , $t=75~\unicode[STIX]{x03BC}\text{m}$ and $E=E_{0}=6.9~\text{GPa}$ . To illustrate the properties of the materials, we first consider the following geometrical parameters: $r_{g}=23~\text{mm}$ , $r_{c}=17~\text{mm}$ , $t_{c}=20~\text{mm}$ , $w=3~\text{mm}$ and $t_{0}=30~\text{mm}$ , which result in a porosity of $\unicode[STIX]{x1D719}=0.86$ . It should be noted that this porosity corresponds to that of one cylindrical pore with the inserted short tubes.

If one considers two extreme values of the Young’s modulus, $E\rightarrow 0$ and $E\rightarrow \infty$ , corresponding approximately to the cases of materials without films and with perfectly rigid films, the permeabilities and tortuosities are given by $(2.46\times 10^{-5};2.3\times 10^{-7})~~\text{m}^{2}$ and (1.03; 3.16) respectively. One can then conclude that very different acoustical properties can be observed when varying the elastic properties of the films. In addition, it is further noticed that (i) the material with perfectly rigid films will be dominated by inertial effects in the audible frequency range since the Biot frequency is 2.85 Hz and (ii) the in-vacuum fundamental resonance frequency of the clamped films is 313 Hz.

Figure 5. Real part (a) and phase (b) of the normalised apparent dynamic density $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D714})/\unicode[STIX]{x1D70C}_{0}$ of a material with perfectly rigid films (i), and permeo-elastic materials with $E=E_{0}$ (ii) and $E=5E_{0}$ (iii). The vertical dotted lines with downward- and upward-pointing triangles represent $f_{0}=\unicode[STIX]{x1D714}_{0}/2\unicode[STIX]{x03C0}$ and $f_{g}=\unicode[STIX]{x1D714}_{g}/2\unicode[STIX]{x03C0}$ respectively.

In the remaining part of this section, we will present the results for two quantities that are normally used in the field of acoustics of porous media (Allard & Atalla Reference Allard and Atalla2009), namely the apparent dynamic density and the speed of sound.

Figure 5 shows the real part and phase of the apparent dynamic density of a permeo-elastic material compared with those of a material with perfectly rigid films. This effective parameter has been calculated for an infinite array of short tubes with films from the solution of the fluid–structure interaction problem (2.24)–(2.31) in a 2D axisymmetric unit cell (see figure 4). The solution was obtained numerically using the finite element method and the software COMSOL Multiphysics^® (COMSOL 2013). In particular, an arbitrary Lagrangian–Eulerian formulation implemented in this software was used and a fully coupled solver (MUMPS) that involves the simultaneous solution of the fields was considered. Lagrangian linear elements modelled the fluid velocity and pressure (i.e. a P1–P1 formulation with streamline diffusion as stabilisation method), while quadratic elements approximated the film displacement. A triangular mesh in combination with a layered quadrilateral mesh to resolve the boundary layers was used. Furthermore, the mesh coincided on the periodic boundaries of the unit cell and a mesh refining analysis was conducted to ensure the convergence of the solution.

As discussed in § 3.2.3, the films behave as quasi-rigid at low frequencies. Hence, the real part of the apparent dynamic density and its phase approach those of a material with perfectly rigid films. As the frequency increases, this behaviour overlaps with that described in § 3.3.3 and the fluid–film system enters into the resonance zone. The frequency at which the apparent dynamic density tends theoretically to infinity is $f_{0}=241~\text{Hz}$ . The fundamental resonance frequency of the fluid–film system with $E=E_{0}$ is $f_{g}=277~\text{Hz}$ . This is smaller than the in-vacuum fundamental resonance frequency due to the fluid loading on the films (see Filippi Reference Filippi2008, Ch. 3). In between the two elasto-inertial frequencies $f_{0}$ and $f_{g}$ , an atypical band is observed where the real part of the apparent dynamic density becomes negative and its phase tends to $-\unicode[STIX]{x03C0}$ . In the atypical band, the fluid movement is opposite in phase with respect to the carrying wave outside the inner resonating units. As a consequence, the average movement leads to a negative apparent density with an unusual phase value. Such a behaviour is a direct consequence of the fluid–film interaction at the pore scale and does not exist in conventional porous materials. In the latter type of material, the real part of the apparent dynamic density is always positive and its phase varies in between its values for viscous and inertial fluid flow, i.e. from $-\unicode[STIX]{x03C0}/2$ to 0 (Johnson, Koplik & Dashen Reference Johnson, Koplik and Dashen1987; Smeulders, Eggels & van Dongen Reference Smeulders, Eggels and van Dongen1992). As predicted by (3.10), the fundamental resonance frequency for the same material but with films with elastic modulus $E=5E_{0}$ is increased by a factor of approximately $\sqrt{5}$ , i.e. $f_{g}=621~\text{Hz}$ , with respect to that of the material with $E=E_{0}$ . In turn, the frequency $f_{0}$ is also increased by the same factor.

Figure 6. (a) Real part of the normalised speed of sound $\text{Re}({\mathcal{C}}(\unicode[STIX]{x1D714})/c_{0})$ . (b) Normalised phase of the normalised speed of sound $\text{Phase}({\mathcal{C}}(\unicode[STIX]{x1D714})/c_{0})/\unicode[STIX]{x03C0}$ . Material with perfectly rigid films (i), and permeo-elastic material with $E=E_{0}$ (ii) and $E=5E_{0}$ (iii). The vertical dotted lines with downward- and upward-pointing triangles represent $f_{0}$ and $f_{g}$ respectively.

Figure 6 shows the real part and phase of the normalised speed of sound in the material. It should be noted that to calculate this, the bulk modulus (2.51) was calculated using the following semi-phenomenological model (Lafarge et al. Reference Lafarge, Lemarinier, Allard and Tarnow1997): $K(\unicode[STIX]{x1D714})=P_{0}/(1-j\unicode[STIX]{x1D714}F(\unicode[STIX]{x1D714})(\unicode[STIX]{x1D6FE}-1)/\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D714}_{t})$ , with $F(\unicode[STIX]{x1D714})=1/(j\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{t}+\sqrt{1+j\unicode[STIX]{x1D714}M_{t}/2\unicode[STIX]{x1D714}_{t}})$ . Here, $\unicode[STIX]{x1D714}_{t}=\unicode[STIX]{x1D705}/\unicode[STIX]{x1D70C}_{0}C_{p}k_{0}^{\prime }$ is the thermal characteristic frequency, $M_{t}=8k_{0}^{\prime }/{\unicode[STIX]{x1D6EC}^{\prime }}^{2}$ is the thermal shape factor, $\unicode[STIX]{x1D705}/\unicode[STIX]{x1D70C}_{0}C_{p}$ is the thermal diffusivity, $\unicode[STIX]{x1D6EC}^{\prime }=2|\unicode[STIX]{x1D6FA}_{f}|/(|\unicode[STIX]{x1D6E4}_{s}|+|\unicode[STIX]{x1D6E4}_{p}|)$ is the thermal characteristic length and $k_{0}^{\prime }$ is the static thermal permeability. The latter has been numerically calculated by spatially averaging the solution of a static heat conduction problem with respect to the fluid phase, as detailed, for example, in Venegas & Umnova (Reference Venegas and Umnova2011). At low frequencies, the speed of sound of permeo-elastic materials tends to that of a material with perfectly rigid films. A significant decrease in speed of sound is observed in the atypical band. Moreover, the phase of the speed of sound in this band tends to $\unicode[STIX]{x03C0}/2$ , i.e. the waves are overdamped. At the fundamental resonance frequency, and for slightly higher frequencies, supersonic velocities are observed. In conventional porous materials, the phase of the speed of sound varies from $\unicode[STIX]{x03C0}/4$ to 0 and the real part of the speed of sound is bounded by its inertial value, i.e. by the speed of sound in the saturating fluid $c_{0}$ divided by the square root of the tortuosity. It is clear that the acoustic behaviour of permeo-elastic materials is significantly different.

Figure 7 shows the influence of the density of the films on the real part and phase of the normalised apparent dynamic density. Heavier films provide smaller elasto-inertial characteristic frequencies. However, the behaviour is qualitatively similar to that obtained when varying the elastic modulus of the films. As predicted by (3.15), the elasto-inertial frequencies $f_{g}$ and $f_{0}$ become closer to each other as the density of the films is increased. This is shown in the inset plot of figure 7, where $f_{g}$ and $f_{0}$ are plotted as a function of the ratio between the densities of the films and the saturating fluid. The net effect is that the atypical band where the apparent dynamic density becomes negative and its phase shows unusual values becomes much narrower as the density ratio increases. Hence, lightweight films, i.e. $O(\unicode[STIX]{x1D71A})=O(\unicode[STIX]{x1D70C}_{0})$ , are preferred to observe permeo-elastic effects.

Figure 7. Real part (a) and phase (b) of the normalised apparent dynamic density $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D714})/\unicode[STIX]{x1D70C}_{0}$ of a material with film density $\unicode[STIX]{x1D70C}_{p}=0.6\unicode[STIX]{x1D70C}_{p0}$ (i.e. $\unicode[STIX]{x1D71A}=2.785~\text{kg}~\text{m}^{-3}$ ) (i), $\unicode[STIX]{x1D70C}_{p}=\unicode[STIX]{x1D70C}_{p0}$ (i.e. $\unicode[STIX]{x1D71A}=3.839~\text{kg}~\text{m}^{-3}$ ) (ii) and $\unicode[STIX]{x1D70C}_{p}=1.4\unicode[STIX]{x1D70C}_{p0}$ (i.e. $\unicode[STIX]{x1D71A}=4.894~\text{kg}~\text{m}^{-3}$ ) (iii), with $\unicode[STIX]{x1D70C}_{p0}=1200~\text{kg}~\text{m}^{-3}$ . The inset plot shows the elasto-inertial characteristic frequencies $f_{g}$ (grey line) and $f_{0}$ (black line) as a function of the ratio between the film density and the saturating fluid density ( $\unicode[STIX]{x1D70C}_{0}=1.204~\text{kg}~\text{m}^{-3}$ ).

The influence of viscous dissipation on the apparent dynamic density of permeo-elastic materials has been assessed by decreasing the size of the gap $g$ formed in between the short tubes and the cylindrical pore wall (see figure 4). The results are shown in figure 8. The other geometrical and mechanical parameters (with $E=E_{0}$ ) are the same as those in figure 5. The static permeability is decreased when the gap size becomes smaller. For example, this parameter for the material with the narrowest gap ( $g=0.4~\text{mm}$ ) is of the order of 290 times smaller than that with the widest gap ( $g=3~\text{mm}$ ). This results in a higher (respectively lower) visco-inertial (respectively visco-elastic) characteristic frequency. It should be noted, however, that the fundamental resonance frequency $f_{g}$ remains constant. The behaviour of the fluid–film system at low frequencies is determined by the low-frequency inertia of the fluid that is affected by the elasticity of the films, as developed in § 3.2.2. As the frequency increases, the behaviour overlaps with that described in § 3.3.2. This means that the fluid flow regime in the fluid–film system is influenced by viscosity, elasticity and the inertia of the fluid and the films. Then, the fluid–film system enters into the elasto-inertial regime described in § 3.3.3. An increase in the effects of viscosity leads to an increase of the bandwidth of the atypical band. For example, the atypical band covers frequency regions of $B\approx 277-197=80~\text{Hz}$ and $B\approx 277-87=190~\text{Hz}$ when $g=1~\text{mm}$ and $g=0.4~\text{mm}$ respectively. Furthermore, the peak associated with $f_{0}$ decreases in amplitude until it becomes highly damped and no longer appears. The latter occurs when the visco-inertial frequency approaches $f_{g}$ . On the other hand, the wider atypical band leads to a wider frequency region where the sound waves are slowed down, as shown in figure 9, where the normalised speed of sound and its phase are presented. It must be emphasised that in this frequency range, the magnitude of the imaginary part of the apparent dynamic density is much larger than that of its real part, which reflects the fact that viscosity effects dominate in this frequency range. Hence, the first zero value of the real part of the apparent dynamic density in figure 8 does not induce supersonic velocities in the effective fluid–film system.

Figure 8. Influence of viscosity effects on the real part (a) and phase (b) of the normalised apparent dynamic density $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D714})/\unicode[STIX]{x1D70C}_{0}$ : (i) $g=3~\text{mm}$ , (ii) $g=1~\text{mm}$ , (iii) $g=0.6~\text{mm}$ , (iv) $g=0.4~\text{mm}$ . The inset plot shows the values of the phase of $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D714})/\unicode[STIX]{x1D70C}_{0}$ at lower frequencies. The frequency has been normalised to $f_{g}=277$  Hz.

Figure 9. Influence of viscosity effects on the real part of the normalised speed of sound $\text{Re}({\mathcal{C}}(\unicode[STIX]{x1D714})/c_{0})$ (a) and the normalised phase of the normalised speed of sound $\text{Phase}({\mathcal{C}}(\unicode[STIX]{x1D714})/c_{0})/\unicode[STIX]{x03C0}$ (b) for permeo-elastic materials with (i) $g=3~\text{mm}$ , (ii) $g=1~\text{mm}$ , (iii) $g=0.6~\text{mm}$ and (iv) $g=0.4~\text{mm}$ . The inset plot shows the normalised phase of the normalised speed of sound at lower frequencies. The frequency has been normalised to $f_{g}=277$  Hz.

Equation (3.32) predicts that the real part of the apparent dynamic density could be either null or negative in a wide range of frequency depending on the interaction between the low-frequency inertia of the fluid and the elasticity of the films. This is evidenced in figure 10, where the real part of the apparent dynamic density and its phase are shown for three different elastic modulus values, i.e. $E=0.3E_{0}$ , $E=0.65E_{0}$ and $E=0.95E_{0}$ . The geometrical parameters are as in figure 5, except that the gap size is set to $g=0.4~\text{mm}$ . The strongest effect of the elasticity is observed for the material with the smallest Young’s modulus. This shows a wide-band negative real part of the apparent dynamic density. For the material with intermediate Young’s modulus value, the elastic effects compensate those due to the low-frequency inertia of the fluid. This leads to a nearly null real part of the apparent dynamic density. As previously argued, the behaviour in this frequency region is dominated by viscosity effects. Therefore, the nearly null real part of the apparent dynamic density does not lead to supersonic velocities in the fluid–film system, as can be seen in the plot in figure 10(b).

Figure 10. Real part of the normalised apparent dynamic density $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D714})/\unicode[STIX]{x1D70C}_{0}$ (a) and speed of sound ${\mathcal{C}}(\unicode[STIX]{x1D714})/c_{0}$ (b) of permeo-elastic materials with (i) $E=0.3E_{0}=2070~\text{MPa}$ , (ii) $E=0.65E_{0}=4485~\text{MPa}$ and (iii) $E=0.95E_{0}=6555~\text{MPa}$ . The inset plots show the normalised phase values. The frequency has been normalised to (i) $f_{g}=152$  Hz, (ii) $f_{g}=225$  Hz and (iii) $f_{g}=268$  Hz.

4.2 Experimental validation

This section presents results of experiments conducted on a single unit cell of a prototype permeo-elastic material corresponding to that illustrated by figure 4(b) (right). Since the physics of a periodic array is mostly determined by that in a unit cell, our experimental work focuses on measuring the acoustical properties of a single unit cell. This allows us to study the most relevant features of permeo-elastic materials.

For convenience, measurements of surface impedance have been taken in an impedance tube by following the procedure described in ISO 10534-2:2001 (2001). A diagram of the measurement set-up is presented in figure 11. The measured magnitude and phase of the surface impedance $Z_{w}(\unicode[STIX]{x1D714})$ normalised to the characteristic impedance of the saturating fluid, $Z_{0}=\unicode[STIX]{x1D70C}_{0}c_{0}$ , are shown in figure 12. These measurements have been taken on a configuration composed of an air gap backed by a single unit cell of the permeo-elastic material, which is backed by a rigidly backed plenum, as illustrated in figure 11. The film material is PEI, whose parameters are $E=E_{0}=6.9~\text{GPa}$ , $\unicode[STIX]{x1D70C}_{p}=1200~\text{kg}~\text{m}^{-3}$ , $\unicode[STIX]{x1D708}=0.36$ and $t=75~\unicode[STIX]{x03BC}\text{m}$ . The other parameters of the geometry can be inferred from figure 11.

Figure 11. (a) Unit cell of the prototype permeo-elastic material placed in the impedance tube. (b) Diagram of the impedance tube used to measure the surface impedance of a prototype permeo-elastic material. The vertical dashed line represents the front surface of the measured configuration.

Figure 12. Measured and predicted normalised magnitude (a) and phase (b) of the surface impedance of a permeo-elastic material and a material with perfectly rigid films.

The validation of the theory is conducted by comparing the measured experimental data with numerical calculations of surface impedance. The latter was calculated as follows. The dynamic permeability was numerically calculated for an infinite array from the solution of the fluid–film interaction problem (2.24)–(2.31), while the bulk modulus was calculated using the semi-phenomenological model proposed in Lafarge et al. (Reference Lafarge, Lemarinier, Allard and Tarnow1997), whose input parameters were also obtained numerically as detailed above. This parameter and the apparent dynamic density were used to calculate the wavenumber and characteristic impedance using (2.51). Then, a transfer matrix approach (Allard & Atalla Reference Allard and Atalla2009) was used to take into account the air gaps in front of and behind the prototype material (see figure 11).

The predictions of the model shown in figure 12 and the measured values are in reasonable agreement. The developed theory explains and the numerical model captures the local peak of the magnitude of the surface impedance as well as the change in phase around the atypical band where the apparent dynamic density takes negative values. This phenomenon does not exist in the material with perfectly rigid films.

The discrepancies between the numerical and experimental results, i.e. prediction of a narrower band where the phase is shifted and the local peak in the magnitude of the surface impedance being located at a slightly lower frequency, may be due to the following factors. (i) The effective parameters have been determined for an infinite array, while only one period has been measured. This means that, in principle, the theoretical model cannot exactly reproduce the conditions of the prototype material used in the experiment. (ii) The internal damping of the elastic films has not been accounted for. Similarly to viscosity effects, this could lead to a wider atypical band. (iii) The mounting rods used to build the prototype material (see figure 11) are not considered in the 2D axisymmetric numerical model. These rods decrease the effective gap size in between the short tube and the wall of the impedance tube. This smaller gap can lead to a larger influence of viscosity effects which may result in an increase of the bandwidth of the atypical band, as shown in figure 8. (iv) The practical implementation of the perfect clamping condition of the films may not be accurate. This could lead to a variation in the eigenfrequency values.