analytical modeling of deformable porous composites · analytical modeling of deformable porous ......

Analytical modeling of deformable porouscomposites

F.-X. Bécot, L. Jaouen, F. ChevillotteMATELYS - Acoustique & Vibrations, 1 rue Baumer, F-69120 Vaulx-en-Velin, France.

SummaryThis paper is concerned with the analytical modeling of the acoustic properties of materials whichcould show interesting combined performances of sound absorption and sound insulation. These ma-terials, made from porous materials having porous inclusions, are called composite porous materialsby analogy with standard composite materials composed by a series of solid (non-porous) fibers em-bedded in a solid (usually non-porous) matrix. The efforts are concentrated here on the analyticalmodeling of the visco-thermal and structural dissipations which occur in the fluid phase and the solidphase respectively. Using dedicated experimental techniques for the characterisation of the porouscomponents, the approach is validated against measured data obtained in impedance tubes. Thiswork finds important applications in the field of the vibro-acoustic properties of sound packageswhich should combine performances both in terms of sound absorption and sound insulation.

PACS no. 43.20.Gp,43.20.Jr,43.55.Ev

1. Introduction

It was demonstrated recently that multi-scale porousmaterials could be used as efficient sound absorbingtreatments. Double porosity or perforated porous ma-terials [1] are examples of such materials as well asporous composites materials obtained by filling theperforations with a porous material of different nature[2] (see Figure 1 for examples of porous composites).Comparatively, whereas sound insulation propertiesof double porosity materials are low [3], materials ofthe second category were proved to offer interestingcombined properties of both sound absorption andsound insulation [4].

Current predictive models for such media are lim-ited by the assumption that the porous frames arerigid and motionless. This restrains the range of ap-plications which could be addressed [2]. In fact, whenporous materials are coupled to elastic structures, de-formations of the porous frame need to be taken intoaccount to predict accurately the response of the sys-tem. The present work aims at overcoming this lack.

The proposed analytical model is based on Biot’stheory [5] and the underlying assumption that theporous frame deformation does not influence the dissi-pation occuring inside the fluid. Accordingly, the pro-posed model handles separately the structural dissi-

(c) European Acoustics Association

Figure 1. Examples of porous composites.

pation in the porous frame and the visco-inertial dis-sipation in the fluid.

The acoustic wave propagation is modeled usingthe derivation of Olny & Boutin theory [6, 7, 8] forrigid framed double porosity materials. The elasticresponse of the composite porous is modeled using aself consistent scheme for solid composites [9, 10, 11].These two dissipation mechanisms are coupled using

Bécot, Jaouen, Chevillotte : Biot Porous CompositesFORUM ACUSTICUM 201127. June - 1. July, Aalborg

a formulation of Biot’s poroelasticity equations usinga (us, p) formalism [12, 13]. The complete model isfinally implemented in Transfer Matrix (TMM ) algo-rithm [14] to associate various layers of different types: impervious films, fabrics, porous or elastic (non-porous) materials.

The paper is organised as follows. The theory ofboth the fluid and structural dissipation is briefly de-scribed in the following two paragraphs. Simulationresults are validated against impedance tube measure-ments of sound absorption and sound transmissioncoefficients for various configurations.

2. Analytical modeling of the poroe-lastic medium

Porous composite materials are above all poroelasticmaterial. Namely they are composed of a solid phase,the porous skeleton, and a fluid phase, the air con-tained inside the pores.

The coupling of the fluid and the solid phase hasbeen proposed by Biot in [5]. This formalism providesthe possibility to model the dissipation inside the fluidseparately from the modeling of the structural dis-sipation in the porous frame. Initially expressed interms of the two displacements of the solid and thefluid phase, works in [12, 13] proposed a formalismin terms of the solid phase displacements us and theinterstitial pressure p. The two coupled poroelasticityequations then write :

∇.σ̃ + ω2ρ̃us = γ̃∇p (1)

∇∇. p+ρ̃22

R̃ω2p =

ρ̃22φ2

γ̃ω2 ∇.us (2)

In this equation, the superscript “ ˜ ” indicates thatthe variable is frequency dependent.σ̃ is the in–vacuo stress tensor of the solid phase. In

addition, ρ̃, ρ̃11, ρ̃22 and ρ̃12 are the modified Biot’sdensities the expressions of which are given below. γ̃ isa coupling factor given by γ̃ = φ(ρ̃12/ρ̃22−Q̃/R̃). Q̃ isa factor which couples the skeleton strain to the fluidstrain, and R̃ can be interpreted as the bulk modulusof a volume of fluid occupying a fraction φmat of theporous media, φmat being the porosity of the medium.

The above parameters relate to the porous materialcharacteristics in the following manner :

ρ̃ = ρ̃11 − ρ̃212/ρ̃22 (3)ρ̃11 = (1− φmat)ρ1 − φmatρ0(α̃mat − 1) (4)ρ̃12 = ρ1 − ρ̃22 (5)ρ̃22 = φ2matρ̃mat (6)

Q̃ = (1− φmat)K̃mat (7)

R̃ = φmatK̃mat (8)

In these equations, ρ1 is the mass density of the porousmaterial and ρ0 is the mass density of the air saturat-ing the porous medium. In addition, α̃mat, ρ̃mat and

K̃mat are the dynamic tortuosity, the dynamic massdensity and dynamic bulk modulus of the porous ma-terial (see also Section 3).

Note that the last two relationships only hold whenthe bulk modulus of the solid phase is very large com-pared to that of the fluid phase and to that of thesolid phase in-vacuo. This condition generally appliesfor the poroelastic materials encountered in most ofsound packages. For interested readers, expressionsvalid for any type of porous materials are availableand can be found in [15].

Once these parameters are determined, the equa-tions of motions are implemented in a TMM algo-rithm [14] and the response in both sound absorptionor sound transmission for any kind of multi-layer sys-tems can be computed.

Therefore, the problematic is to express the effec-tive properties of both the solid phase and of the fluidphase for the porous composite material. For the fluidphase, since the propagation of sound wave inside thepores is dispersive, the effective properties of inter-est here are the dynamic mass density ρ̃mat and thedynamic bulk modulus K̃mat. For the solid phase, ifwe assume that the material is isotropic, the effectiveproperties are Young’s modulus E, Poisson’s ratio ν,structural loss factor η and the mass density ρ1. Thisassumption will be discussed later in the paper.

The following two sections present the separatecomputation of these effective properties.

3. Modeling of the fluid dissipation

The calculation of the dissipation inside the fluid cor-responds to the computation of the effective prop-erties of a rigid and motionless porous material. Itis based on the theory of double porosity materialswhich allows the analytical modeling of a perforatedporous material [8].

In the case of a porous composite, the perforationis filled with another porous material (see Figure 1),which will be called the Client material. The porousmaterial substrate hosting the Client will be calledthe Host material. An analytical model has been pro-posed in [2] for this type of material. The idea is tomodel separately the visco-thermal in the Client andin the Host as if they were infinite media. Then, thetwo viscous permeabilities, respectively the two bulkmoduli, are coupled to represent the visco-inertial ef-fects, respectively the thermal effects, as proposed inthe double porosity theory. By doing so, the couplingis a “volumic” coupling in the sense that the main gov-erning parameter is the rate of inclusion, also calledmeso-porosity and denoted φp. In other words, theinformation about the shape of the inclusion is lost.

To overcome this, the approach used here is slightlydifferent. The coupling of the visco-inertial effects isstill realised via the coupling of the permeabilities, asdone in [2]. The first difference relies on the fact that


the shape of the inclusion is accounted for. In thiswork, only cylindrical inclusions are considered. Thesecond difference is that the coupling of the thermaleffects is realised via the coupling of the thermal per-meabilities instead of the coupling of the bulk moduli.

In the absence of analytical solution for the perme-abilities of a porous cylinder, the viscous and thermalpermeabilities, indexed C for Client, of the cylindri-cal porous inclusion are modeled on the basis of thesound propagation in a cylinder filled with fluid. Wepropose here to use the following expressions :

ΠC = jδ2v

[1− 2

µv√−j

J1(µv√−j)

J0(µv√−j)

](9)

ΘC = jδ2t

[1− 2

µt√−j

J1(µt√−j)

J0(µt√−j)

](10)

In these equations, δv =√η0/ρ0ω and δt =

√η0/Prω

are the dynamic viscous and thermal skin depths,where η0 and ρ0 are the viscosity and mass densityof the air saturating the porous material. µv = Λ/δvand µt = Λ′/δt and J1 and J0 are Bessel function ofthe first kind of zero-th and first order. Lambda andLambda′ are the viscous and thermal characteristiclengths.

The viscous and thermal permeabilities for the Hostmaterial are computed assuming the medium is infi-nite. In a similar manner as done for double poros-ity materials, the coupling between the Host and theClient materials is ensured via the following relation-ships :

Πpc = φpΠC + (1− φp)ΠH (11)Θpc = φpΘC + (1− φp)ΘH × Fd (12)

where the subscript “pc” stands for PorousComposites. The viscous and thermal perme-abilities ΠH and ΘH for the Host material are givenby the model of Johnson-Champoux-Allard-Lafarge(JCAL) model [16, 17, 18].

Note that possible pressure diffusion effects may beaccounted for at this stage of the modeling via thecomplex function Fd the expression of which is givenin [8]. These effects will not be discussed further sincethey occur at frequencies outside the frequency rangeof interest here.

Once these two parameters computed, the dynamicmass density and compressibility of porous compositeare computed using the following standard relation-ships :

ρ̃pc =η0

jΠpc(13)

K̃pc =γP0/φpc

γ − j(γ − 1)Θpc/φpcκ(14)

γ is the ratio of the specific heats, κ is the thermalconductivity of the saturating air, and P0 is the at-mospheric pressure. Here φpc is the total porosity ofthe porous composite, φpc = φpφC + (1− φp)φH .

Client

Host

Figure 2. Geometry used in the computation of the effec-tive elastic properties of the porous composite.

This mass density and bulk modulus correspondsto ρ̃mat and K̃mat and can be inserted into Equation3 to Equation 8 to compute the dissipation inside thefluid phase of the porous composite. The next sec-tion presents the determination of the effective elasticproperties in order to compute the structural dissipa-tion inside the solid phase.

4. Modeling of the effective elasticproperties

The determination of the effective properties of theporous composite is directly inspired from works re-alised for fiber reinforced solids. The geometry is thatdepicted in Figure 2. This situation corresponds toa tranversely isotropic medium for which five elasticpropertes are to be obtained : Young’s moduli for lon-gitudinal and transverse directions, respectively ELand ET , the shear modulus for the longitudinal andtransverse directions, denoted GLT and GTT and thePoisson’s coefficient νLT .

For this geometry, derivation of exact expressionsexist and can be found in [19]. Expressions are quitelengthy and only the expressions of the two Young’smoduli are given below as examples :

EL = ECφp + EH(1− φp) + . . .4(νC − νH)2φp(1− φp)

φp./KH + (1− φp)/KC + 1/GH(15)

ET =2

1/2KL + 1/2G′TT + 2ν2LT /EL(16)

where the subscripts H and C stands for the Host andthe Client materials respectively. KH and KC are thelateral compression moduli and GH is shear modulusfor the Host material. KL is the lateral compressionmodulus of the composite andG′TT is a modified shearmodulus introduced to simplify the expressions.

In addition, the mass density and the structuralloss factor of the porous composite are given using a


Figure 3. Dedicated impedance tube used for the charac-terisation of the acoustic properties of porous materials.

mixing law as :

ρ1,pc = φpρ1,C + (1− φp)ρ1,H (17)ηpc = φpηC + (1− φp)ηH (18)

Knowing the elastic properties of the Host and ofthe Client, the structural effects can be introducedusing the coupled poroelasticity equations as given inEquation 1 and Equation 2. In particular, the bulkmodulus of the solid phase and Biot’s elastic coeffi-cient P̃ are then given by

Kb =2

3GLT

1 + νLT1− 2νLT

(19)

P̃ = (1− φpc)2K̃pc +Kb +4

3GLT (20)

5. Experimental data

To provide with reliable data, a full characterisation ofthe acoustic and elastic material parameters has beenrealised. These procedures, presented in the following,have been applied only on the homogeneous porousmaterials, i.e. taken before arranging them as porouscomposites.

5.1. Characterisation of acoustic parameters

The characterisation of the acoustic parameters con-sists in the determination of the parameters of theJCAL model as described following the method de-scribed in [20, 21, 22]. This method relies on mea-sured data of the dynamic mass density and of thedynamic bulk modulus of the material. These dataare provided using a modified impedance tube usingthree microphone positions, two upstream and one di-rectly placed at the rear side of the material as shownin Figure 3 [23].

The analytical determination of the parameters ofthe JCAL model further requires the prior knowledgeof the static air flow resistivity σ and of the openporosity φ of the material. These two quantities canbe directly measured with a good accuracy using thecorresponding ISO standard [24] for σ and using the

Figure 4. Experimental setup used for the characterisationof the elastic and damping parameters of porous materials.

method described by Beranek in [25] and further mod-ified by Champoux et al. [26] for the open porosity φ.

In total, the procedure allows the determination ofthe following six parameters : the static air flow re-sistivity σ (in N.s.m−4, directly measured), the openporosity φ (dimensionless, directly measured), thehigh frequency limit of the dynamic tortuosity α∞(dimensionless, analytically determined), respectivelythe viscous and thermal characteristic lengths Λ andΛ′ (both in m, analytically determined), and the staticthermal permeability Θ0 (in m2, sometimes also de-noted k′0, analytically determined).

It should be underlined that this method providesthe independent expressions of the acoustic parame-ters, except for the expression of Λ which still dependson α∞. Another consequence is that the visco-inertialeffects from one hand and the thermal effects from theother hand are characterised in a separate manner.

5.2. Characterisation of the elastic anddamping parameters

The elastic and damping parameters have been esti-mated using a method inspired by the works of Lan-glois and co-workers in [27] which assumes an isotropicbehaviour of the material. As for the acoustic pa-rameters, this method has been applied on the non-perforated porous materials.

The basic idea of this method is to reproduce a mass/ spring system where the mass, and thus the stress, isprescribed and the spring is represented by the porousmaterial (see figure 4). Using charts of pre-computedresults of the system response for different values ofstiffness, the apparent Young’s modulus Eapp of thespring, that is of the porous material, is identified.The loss factor ηr is estimated by a -n dB bandwidthmethod.

Moreover, Eapp depends on the shape factor and onthe Poisson’s ratio ν of the tested sample. For brick-like samples, the shape factor is defined as half theradius to thickness ratio (Rtube/2d). Following [27],by testing several samples having different values ofthe shape factor, the “true” Young’s modulus andthe Poisson’s ratio can be determined. In the present


Table I. Acoustic, elastic and damping parameters of the tested materials.

Material σ φ α∞ Λ Λ′ Θ0 E η ν ρ1

Host 150 0.98 1.00 30 35 20 500 0.1 0.1 20

Client 10 0.98 1.00 98 196 100 200 0.1 0.42 11

Screen - - - - - - 1 000 0.01 0.3 1 500

Units k N.s.m−4 - - µ m µ m 10−10 m2 k Pa - - kg.m−3

study, brick-like samples of different thickness havebeen used.

6. Results and discussion

The model has been implemented using the parame-ters characterised and reported in Table I. Sound ab-sorption coefficients and sound transmission loss datahave been measured using the impedance tube shownin Figure 3. For the absorption, the measurement con-ditions correspond to those of the ISO 10534-2 [28].

Due to formatting restrictions, only one arrange-ment is presented here (other material associationswill be presented during the conference). It is com-posed of a 29 mm diameter inclusion of Client ma-terial in a 46 mm diameter sample of Host material(see inserts in the following figures). Results are shownfor three different configurations and are compared tosimulations or measurements when available obtainedfor the Host and for the Client material taken sepa-rately and considered as homogeneous.

In Figure 5, a satisfying correspondence is observedbetween simulation and measurement for the porouscomposite material. The damping predicted by themodel seems to over-estimates that measured. Thismay be due to additional damping due to the actualmounting conditions of the sample inside the tube andat the interface between the inclusion and the Hostmaterial. One could note that the performances forthis latter material compare favourably to those ob-tained for the homogeneous materials.

Figure 6 shows the performance of the same porouscomposite measured in transmission. A fair corre-spondence is observed on these results. These perfor-mances lie in the interval of the performances of Hostand of the Client considered separately.

To prove the efficiency of the approach for multi-layer configurations, the porous composite has beencovered with an impervious screen (see Table I). Forthis system, measurement and simulation comparewell, expect in the frequency range where the quarterwavelength resonance occurs.

7. CONCLUSIONS

Based on Biot’s principle, an analytical approach hasbeen proposed to model the vibro-acoustic response of

0 500 1000 1500 2000 2500 3000 3500 4000 4500

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

Frequency (Hz)

Sound absorption coefficient

PC : MeasurementPC : AnalyticalHost aloneClient alone

Figure 5. Sound absorption coefficient of the Porous Com-posite : simulation Vs. measurements. Comparison withHost (measured) and Client (simulated) materials con-sidered as homogeneous. Total material thickness = 19.5mm.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

2

4

6

8

0

2

4

6

8

0

Frequency (Hz)

Sound transmission lossPC : MeasurementPC : AnalyticalHost aloneClient alone

Figure 6. Sound transmission loss of the Porous Composite: simulation Vs. measurements. Comparison with Host andClient materials considered as homogeneous (simulations).Total material thickness = 19.5 mm.

the porous composite materials. The model has beenimplemented in TMM algorithm to simulate multi-layer configurations.

Measurements and simulations presented aboveproved that these materials could represent an in-


0 500 1000 1500 2000 2500 3000 3500 4000 4500

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

Frequency (Hz)

Sound absorption coefficient

PC+Screen : MeasurementPC+Screen : AnalyticalHost alone+ScreenClient alone+Screen

Figure 7. Sound absorption coefficient of the Porous Com-posite + Impervious Screen : simulation Vs. measure-ments. Comparison with Host and Client materials con-sidered as homogeneous (simulation). Material thickness= 19.5 mm + Screen 7.5 µm thick.

teresting compromise between both sound absorptionand sound transmission performances. With this in-crementation, the proposed model broadens the rangeof applications of porous composites to multi-physicsproblems: acoustic and thermal performances, acous-tic and mechanical properties, . . .

Further developments are in progress to fully ac-count for the transverse isotropic behaviour of thecomposite materials studied.

References

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analytical modeling of deformable porous composites · analytical modeling of deformable porous ......

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