Morphology influence on the acoustics of permeo-elastic media

Claude Boutin 1

ENTPE Université de Lyon UMR CNRS 5513, Vaulx-en-Velin, France

Rodolfo Venegas 2

University Austral of Chile, Institute of Acoustics Valdivia, Chile

ABSTRACT Permeo-elastic media consist of a rigid porous skeleton to which are attached flexible elastic films. Under acoustic waves the films interact with the gas that saturates the pores and introduces an elastic energy in addition to the viscous dissipation and kinetic energy. The corresponding non- conventional acoustic flow regimes are studied by the two-scale asymptotic homogenization method. The morphology influence on the acoustic features is investigated by considering permeo-elastic media with (i) fully connected pores, (ii) strictly unconnected pores (closed by films) (iii) partially connected pores (iv) mixed morphologies. The analysis shows that in specific frequency ranges, the film-fluid system behaves like an equivalent viscoelastic fluid and that local film-fluid resonances can be observed. These results enables to design materials with either unconventional conductivity or unconventional compressibility, or both, in distinct or common frequency ranges, by playing on the morphological and / or mechanical parameters of the films.

1. INTRODUCTION

This paper studies the acoustic properties of permeo-elastic materials. These fluid-saturated porous materials consist of a rigid skeleton to which thin, highly flexible films are attached. Foams can be an example of such media, the thin membranes (with our without holes) acting as films and the rigid structure being formed by the thick struts. Under acoustic waves unconventional flow regimes appears because the film’s elastic energy of deformation energy combines with the viscous dissipation of the fluid and the inertia of both film and fluid. In particular, over specific frequency ranges, the film-fluid system behaves like an equivalent viscoelastic fluid and local film-fluid resonances can be observed, [1, 2]. For these reasons, the acoustic behavior di ff ers from that of standard porous materials, [3], for which the fluid flow is independent of the solid matrix deformation. The porous network of materials with flexible internal membranes can be fully connected, partially connected or unconnected, see Figure 1. We show that these morphological aspects play an essential

1 claude.boutin@entpe.fr

2 rodolfo.venegas@uach.cl

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

role on the acoustic properties by acting either on the conductivity, or on the compressibility or on both. By assuming acoustics wavelengths much larger than the characteristic pore size, the two-scale asymptotic homogenization method, [4, 5], is used to identify how and under what circumstances the elastic and kinematic energy of the films modifies the e ff ective parameters of conductivity and / or compressibility. This approach provides a guide line for developing new acoustic meta-materials which brings together film e ff ects similar to those observed e.g. [6, 7] and resonator e ff ects as e.g. [8,9]. The paper is organized as follows. The basic assumptions and general framework are first presented, then we successively examine the cases of permeo-elastic materials with :

– Fully connected pores, i.e. no pores are closed by the films,

– Unconnected pores, i.e. all pores are closed by films,

– Partially connected pores, i.e. some pores are closed by the films.

– Mixed morphologies.

For conciseness the theoretical developments are not detailed. The reader may refer to [1], [2], [10]. for further details.

Figure 1: Exemples of possible morphologies of permeo-elastic media

2. GENERAL FRAMEWORK

Consider periodic permeo-elastic media of period Ω and characteristic size ℓ that comprises (i) a fluid network Ω f , (ii) a rigid impervious frame Ω s with surface Γ s = ∂ Ω s , and (iii) highly flexible thin flat films Γ p whose edges ∂ Γ p are clamped on Ω s .The in-plane dimension of the film is h = O ( ℓ ), much larger than the film thickness t , i.e. t ≪ h . The films behave as Kirchho ff plates with apparent Young’s modulus E p = E / (1 − ν 2 ). The bending sti ff ness is E p I , where I = t 3 / 12 is the inertia moment, and ρ p t is the surface density of the films. The saturating gaz is characterized by its dynamic viscosity η , equilibrium density ρ 0 , and adiabatic compressibility β = γ/ P e ; the mean density of the fluid-film system is ϱ = ρ 0 + ρ p t Γ / Ω f . We focus on the propagation of harmonic ( e j ω t ) acoustic waves of long wavelengths for which the ratio between ℓ and the reduced wavelength L = λ/ 2 π is a small parameter ε = ℓ/ L ≪ 1. Furthermore, for the sake of clarity, one assumes that (i) the thermal exchanges can be disregarded (however, their inclusion is easy), (ii) the material is assumed macroscopically isotropic. The local physics is described by the following set of equations

– The fluid in the pore network Ω f is governed by the usual set of linearized equations

3S SB SG

complemented by the continuity of the fluid and film velocities ( v = j ω u ) on Γ p

div( σ ) j ωρ 0 v in Ω f , (1) div( v ) − j ω p β in Ω f , (2) v 0 on Γ s , (3) v j ω u N on Γ p . (4)

In these equations, v and p stand for the oscillatory velocity and pressure, respectively, σ = 2 η D ( v ) − p I , where I is the unitary tensor and D ( v ) = ( ∇ v + ( ∇ v ) T ) / 2 is the strain rate tensor.

– The films undergo out-of-plane 2D bending while their edges ∂ Γ p are clamped on to the solid boundary. The out-of-plane film deflection u is governed by the classical Kirchho ff plates di ff erential set :

f div( T ) − ρ p t ω 2 u − [ σ · N ] · N on Γ p , (5)

T − f div( M ) , (6) M E p I (1 − ν ) e e ( e ∇ u ) + ν e △ u e I
, (7)

u 0 and e ∇ u = 0 on ∂ Γ p . (8)

The tilded operators act in the plane Γ p of the films, e.g. e ∇ is the in-plane gradient vector and the in-plane tensor e e is the symmetric part of the gradient tensor, i.e., 2 e e ( − ) = e ∇ ( − ) + ( e ∇ ( − )) T . T and M denotes the transverse force vector and the bending moment tensor ; e I is the in plane unit tensor. The fluid loading on the films is accounted for by the right-most term in Equation 5 where [ · ] indicates the ’jump’ of the normal component of the fluid stress vector across Γ p .

2.1. Physical analysis and Homogenisation procedure Assuming long-wavelengths induces that, as in conventional porous materials, the pressure and fluid volume variation evolves at the macroscopic scale and balance each other, while the fluid velocity and its rate of deviatoric strain fluctuate locally. Furthermore, the viscous and inertial terms balance the pressure gradient. Thus, in terms of dimensional analysis,

|∇ p | = O (p / L ) ; | div( v ) | = O (v / L ) = O ( ω p β ) ,

| div(2 η D ( v )) | = O ( η v /ℓ 2 ) ; O ( η v /ℓ 2 ) = O ( ρ 0 ω v) = O (p / L ) .

In addition, with regards to the fluid’s stress jump across Γ p , one has O ([ η v /ℓ ]) = O ( η v /ℓ ) but O ([ p ] /ℓ ) = O (p / L ). The continuity of film and fluid velocities leads to O ( v ) = O ( ω u ) so that the film velocity varies at the local scale, as well as the transverse force and bending moment tensor, hence, [1]

! = O  ρ p t ω 2 u  = O  η v

 = O ℓ

L p ! . (9)

O E p I u

ℓ 4

ℓ

The long-wavelength framework ( ε → 0) enable the use of the classical homogenization method [4,5] of periodic media for deriving the macroscopic description. To properly accounts for the above dimensional analysis, the two-scale asymptotic method of homogenization is applied to the following rescaled equations, the others one remains unchanged (for more details see [1,2]).

ε f div( T ) j ωρ p tv − [(2 ηε D ( v ) − ε − 1 p I ) · N ] · N , (10)

T − ε f div( M ) , (11)

M ε 2 E p I

j ω (1 − ν ) e e ( e ∇ v ) + ν e △ v e I
, (12)

ε 2 div(2 η D ( v )) −∇ p j ωρ 0 v . (13)

2.2. Macroscopic description The homogenization approach shows that the more general macroscopic model of permeo-elastic isotropic media presents the same formal structure as for conventional porous media. The physical variables are, P , the uniform pressure at the dominant order in the pores, and, ⟨ v ⟩ , the averaged gaz velocity in the pores. The e ff ective parameters are the conductivity K ( ω, P p , P f ) and the e ff ective compressibility C ( ω, P p ), where P f = { η, ρ 0 , Ω f } , denotes the physical properties of the saturating gaz and pore space geometry while P p = { E p I , ρ e t , Γ p } denotes the mechanical and geometrical properties of the elastic films. Hence, we have

Z

⟨ v ⟩ = 1

Ω f v d Ω − K ( ω, P p , P f ) · ∇ x P (14)

Ω f

∇ x · ⟨ v ⟩ − j ω P C ( ω, P p ) (15)

and the complexe and frequency dependent acoustic wave celerity is

s

j ω K ( ω, P p , P f )

C ( ω ) =

C ( ω, P p ) (16)

However, despite the formal similarity, the acoustic behaviour of permeo-elastic media can deviate significantly from that of conventional porous media because the conductivity and the e ff ective compressibility reflect the e ff ects of elastic films. Moreover, as shown in the next section, these parameters are strongly influenced by the morphology.

3. INFLUENCE OF THE MORPHOLOGY

3.1. Permeo-elastic media with connected pores Let us first consider permeo-elastic medium with connected pores as investigated in [1, 2]. This morphology does not prevent that some films close a pore section but the connection is ensured by another path. This situation could correspond to a foam with open membranes or having with some dispersed closed membranes. In that case the e ff ective description reads :

Z

⟨ v ⟩ = 1

Ω f v d Ω − K O ( ω, P p , P f ) · ∇ x P (17)

Ω f

∇ x · ⟨ v ⟩ − j ω P γ

P e (18)

Note that the compressibility is not modified by the presence of the films (and takes the classical adiabatic value because thermal exchanges are disregarded for simplicity). Conversely the conductivity K O ( ω, P p , P f ) accounts for the presence of the films. Di ff erent flow regimes can be observed :

– Quasi-rigid films . That case corresponds to sti ff films or to low frequency so that the film motion tends to vanish. As a result, the conductivity is assessed as

K O ≈ K r ( ω ) + j ωχ r ( ω )

K . (19)

where K r ( ω ) is the dynamic Darcean pemeability that would be obtain for perfectly rigid films. The elastic corrector K j ωχ r ( ω ) describes the fact that the films are (weakly) deformed in response to the stress induced by the fluid. In this corrector, the dimensionless complex function χ r ( ω ) accounts for the slight deformation of the films while K is related to the elasticity of the films and defined by K Γ p = E p I

Ω f . If furthermore the e ff ect of inertia can be neglected, then one shows

that the fluid flow is characterized by the intrinsic permeability K r = K r (0) /η corresponding to perfectly rigid films and a e ff ective saturating fluid that behaves as a Maxwell fluid, defined by the viscosity of the fluid η connected in series with the e ff ective elasticity of the sti ff films K K r /χ r (0), [1]. – Highly deformable films . That case corresponds to films of weak sti ff ness or to high frequencies so that that whatever the amplitude of the film motion, the elastic energy of deformation is weak. Thus, the conductivity is assessed as :

1 K O ≈ 1 K s ( ω ) + K χ s ( ω )

K s K s

(20)

j ω

where K s ( ω ) is the dynamic Darcean pemeability that would be obtain for infinitely flexible films (i.e. almost as in absence of film except for the ki nem atic restriction, Equation 4, that the films impose on the flow). The elastic corrector K χ s ( ω )

K s K s describes the fact that the film is forced to move by the fluid. In addition, if the e ff ect of inertia can be neglected, the fluid flow is characterized by the intrinsic permeability K s = K s (0) /η corresponding infinitely flexible films and a e ff ective saturating fluid that behaves as a Kelvin fluid, defined by the viscosity of the fluid η connected in parrallel with the e ff ective elasticity of the sti ff films K K s /χ s (0), [1]. To sum up, ignoring for the moment the inertial e ff ects, the e ff ect of viscosity, dominates at both low and high, frequency while the elastic e ff ect are maximized at intermediate frequencies. – Anti-resonant and resonant regimes . Such regimes occurs when (i) the energy dissipated by viscosity is negligible compared to the elastic energy of the films and the kinetic energy of the film and fluid and (ii) when these to latter energy balance each other. The anti-resonance is reached at a frequency ω a such that the internal motions of the fluid and film compensate each other so that the mean velocity vanishes meanwhile the kinetic energy is not zero, while resonance is reached at a frequency ω g such that the fluid-film system moves globally. It can be shown that both ω a and ω g are of the order of K /ϱ. Furthermore we have the following estimates, where ρ ∗ a and ρ ∗ g are dimensionless parameters that account for the structure of the fluid-film flow in these two regimes:

j ω

ω 2 a ) ; j ω K O ( ω )
− 1 ( ω → ω g ) ≈ ϱρ ∗ g (1 − ω 2 g ω 2 ) (21)

j ω K O ( ω )
( ω → ω a ) ≈ 1 ϱρ ∗ a (1 − ω 2

In other words, the presence of the elastic films yields a zero conductivity (hence an infinite apparent density) at anti-resonance, and inversely, an infinite conductivity (hence a zero apparent density) at resonance.

These flow regimes results in acoustic features impossible to achieve with conventional materials e.g. non-classical dispersion, sub-wavelengths band gaps, or enhanced attenuation.

3.2. Permeo-elastic media with closed pores One consider here media in which all the pores are closed by films so that the gas is not connected as in the case of foams made of closed cells. Such material is described by

Z

⟨ v ⟩ = 1

Ω f v d Ω − K F ( ω, P p , P f ) · ∇ x P (22)

Ω f

∇ x · ⟨ v ⟩ − j ω P γ

P e (23)

As for open pores, the compressibility remains classical. However, as expected, the conductivity of closed pores K F ( ω, P p , P f ) di ff ers strongly from that of open pores. Indeed, when the pores are closed, the flow in the pores is governed by the movement of the films. In particular, if the films remain motionless the gas velocity is zero, unlike with open pores where there is always a flow, even with motionless films. Thus, the di ff erent flow regimes must be revisited.

– Quasi-rigid films . This case corresponds to rigid films or to a low frequency so that the film, and therefore the gas, tends to stand still. As a result, the conductivity is assessed as

j ω 1 χ F + η

1 K F = K

κ F + j ωϱζ F , (24)

This shows that to first order, the e ff ective Darcy law appears to be an elastic constitutive law in the sense that the mean displacement depends linearly on the pressure gradient. The dimensionless scalar χ F > 0 accounts for the geometric deformation of the flexed films. This dominant term is corrected by viscous and inertial e ff ects expressed by the parameters κ F = O ( ℓ 2 ) and ζ F (dimensionless). These parameters (i.e. χ F , κ F , ζ F ) are specific and not linked to those of the porous medium constituted by the rigid skeleton. Let us note that, contrary to open pores, the e ff ect of the viscosity is a corrector which appears when the frequency is increased. This regime, dominated by elasticity and low viscosity, induces pressure waves that di ff er greatly from those of standard porous media. In particular, the waves cannot propagate at low frequencies and presents an exponential decay independent of frequency, [10]. – Highly deformable films . Although the pores are closed, the low film sti ff ness or high frequency makes the films behave as if they were extremely flexible. Therefore, the conductivity is assessed in the same way as in the case of open pores, Equation 20, and calls for similar comments

1 K F ≈ 1 K s ( ω ) + K χ s ( ω )

K s K s

(25)

j ω

Thus disregarding for the moment the inertial e ff ects, the e ff ect of elasticity, respectively viscosity, dominates at low, respectively high, frequency and visco-elastic e ff ect arrises at intermediate frequencies. – Anti-resonant and resonant regimes . The theoretical description in these regimes is the same as for open pores, Equation 21, except for the fundamental anti-resonant mode that disappears. Indeed, in closed pores, the film-fluid system moves as a whole, so the movements of the film cannot compensate for those of the fluid. However, higher anti-resonance modes are possible as they allow for di ff erential internal movements, which can lead to a zero average velocity.

In summary, closed-pore permeo-elastic media also exhibit unconventional acoustic characteristics, but they di ff er significantly from those of open-pore media.

3.3. Permeo-elastic media with dead-end pores closed by films The material considered in this section, consists of a porous medium formed by a solid skeleton, a network of connected pores, and dead-end pores (DE-pores, i.e. with rigid walls except the "opening" in contact with the connected pore network that is closed by a flexible film). These situation may be di ffi cult to obtain with foam but can be realized e.g. using 3D printer. For simplicity, the unit cell contains a single DE-pore closed by a film and the fluid domain is then decomposed into Ω f = Ω C f ∪ Ω DE f , where Ω C f = φ Ω f is the volume of the connected pore network and Ω DE f = (1 − φ ) Ω f is that of the dead-end pore. In this case, the corresponding description reads [10]:

Z

⟨ v ⟩ = 1

Ω f v d Ω − K D ( ω, P f ) · ∇ x P (26)

Ω f

∇ x · ⟨ v ⟩ − j ω P C D ( ω, P p ) (27)

Unlike the previous cases, the conductivity is independent of the presence of the film. K D ( ω, P f ) is the classical dynamic Darcean permeability which corresponds to the porous medium with perfectly rigid films. This is due to the fact that at long wavelength the mean flux V = O ( ⟨ v ⟩ ) varies of O ( V ℓ/λ )

between the two extremities of the representative volume. However, by mass conservation, these variations include the flux pulsed by the DE-pores, which is O ( v ) where v is the velocity of the film. Consequently, O ( v ) = O ( V ℓ/λ ) i.e. O ( v ) = ε O ( V ). This means that, at the dominant order, the films appears as immobile as if the dead-end pores were excluded of the pore network. Conversely, the compressibility C D ( ω, P p ) is modified by the presence of films. The reason is that the small flux pulsed by the DE-pores acts as a source term in the mass balance. This flux is derived from the ”drum” behaviour of the DE-pore. The latter results from (i) the dynamic plate behavior of the film loaded by the di ff erence between the pores and DE-pore pressures and (ii) from the gaz compressibility in the DE-pore. The developments, [10], leads to the following expression :

C ( ω, P p ) = φ γ P e + 1 − φ

(28)

γ P e + E p I G ( ω )

where the real function G ( ω ) describes the mean elasto-inertial response of the film, independently of the surrounding gas. Classical plate modal analysis shows at the frequencies ω I of the plate modes having non zero mean values, G is not bounded and changes sign i.e. for 0 < ω < ω 1 , G ( ω ) increases monotonically from G 0 > 0 to ∞ , and for ω I < ω < ω I + 1 , G ( ω ) varies from −∞ to + ∞ . Consequently, in each interval [ ω I − 1 , ω I ] there is specific frequencies ω ∗ I and e ω I such that :

E p I G ( ω ∗ I ) = − γ P e /φ . Then, the e ff ective compressibility of the DE-pore / film system is exactly the opposite of the compressibility of the gas in the connected pores and as a result the compressibility C ( ω → ω ∗ I ) → 0.

–

E p I G ( e ω I ) = − γ P e . Consequently, the e ff ective bulk modulus vanishes and the compressibility C ( ω → e ω I ) →∞ . Note that in each frequency interval [ e ω I , ω ∗ I ], the e ff ective compressibility C is negative.

–

With such properties, even if the possible dissipation not considered in the analysis of C would smooths the behavior, strong e ff ects are expected, particularly around the singular frequencies (at least the first one). Drastic changes in the acoustic behavior occurs (unconventional dispersion and band gaps), as in porous media with embedded Helmholtz resonators [8], [11].

3.4. Permeo-elastic media with with open pores and cells closed by films This section looks at media that contain open pores with films and closed cells. Closed cells are distinguished from DE-pores by the fact that several walls (at least two or more) of the closed cells are made of flexible films. Since at long wavelength the pressure fluctuates at the macroscopic scale, the pressure at the main order is the same in the connected pores and in the closed cells. Therefore, a conductivity similar to that of a permeo-elastic open-pore material is obtained, Equation 17. The compressibility is classical, Equation 18, since the displacement of the whole films closing the cell respects, to first order, the local incompressibility condition. Note that the situation where the pressure in the closed cells would be di ff erent from that in the connected pores (as it happens in dead-end pores) requires wavelengths comparable to the size of the representative elementary volume, which do not respect the scale separation.

3.5. Mixed morphologies The results established for permeo-elastic media with open pores and with DE-pores closed by films provide the description of media that combine both types of morphology. Indeed, since the DE-pores flux is necessarily of a lower order than that of connected pores, the e ff ective conductivity is that of the permo-elastic medium where the DE-pores are excluded form the pores network. Furthermore, the e ff ective compressibility is only a ff ected by the DE pores closed by films. Thus, the description that accounts for the two e ff ects acting independently (at least to the leading order) is the

general one given in Equation 14-Equation 15. Depending on the morphology, the frequency ranges of unconventional conductivity and unconventional compressibility may coincide or be separated, opening up the possibility of a wide variety of acoustic behavior [11].

4. CONCLUSION

Firstly, it should be recalled that the validity of these models requires (i) long wavelengths and (ii) the presence of a rigid skeleton or, at least, a skeleton much sti ff er than the films. This study shows that morphological (and / or mechanical) parameters can be tuned to obtain materials with unconventional conductivity or compressibility or both, and this in distinct or common frequency ranges. This makes it possible to design materials with acoustic properties that are di ffi cult to achieve with standard materials. Examples include foams with controlled morphology, [12], or materials made by 3D printing.

REFERENCES

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ACKNOWLEDGEMENTS

This work was supported by the Chilean National Agency for Research and Development (ANID) through FONDECYT Regular Grant 1211310. Support from CeLyA of Universite ´ de Lyon operated by ANR (ANR 10 Labex 0060 and ANR 11 IDEX - 0007) is also acknowledged.