## Abstract

A porous solid may be characterized as an elastic–viscoelastic and acoustic–viscoacoustic medium. For a flexible, open cell porous foam, the transport of energy is carried both through the sound pressure waves propagating through the fluid in the pores, and through the elastic stress waves carried through the solid frame of the material. For a given situation, the balance between energy dissipated through vibration of the solid frame, changes in the acoustic pressure and the coupling between the waves varies with the topological arrangement, choice of material properties, interfacial conditions, etc.

Engineering of foams, i.e. designs built on systematic and continuous relationships between polymer chemistry, processing, micro-structure, is still a vision for the future. However, using state-of-the-art simulation techniques, multiple layer arrangements of foams may be tuned to provide acoustic and vibrational damping at a low-weight penalty.

In this paper, Biot's modelling of porous foams is briefly reviewed from an acoustics and vibrations perspective with a focus on the energy dissipation mechanisms. Engineered foams will be discussed in terms of results from simulations performed using finite element solutions. A layered vehicle-type structure is used as an example.

## 1. Introduction

Engineering of foams, with the specific aim to reduce unwanted sound (noise) and vibrations, is a multi-dimensional design task involving proper choices of materials, topology, geometry, interfaces, etc. The foam behaviour, under dynamic, elastoacoustic loading conditions, may be understood in terms of its elastic properties (i.e. stiffness controlled by material, topology, geometry, interfaces), its viscoelastic properties (i.e. solid damping controlled by material, geometry), its acoustic properties (which are governed by the fluid medium) and its viscoacoustic properties (i.e. fluid damping controlled by geometry, topology, interfaces). Given this spectrum of design variables, it is clear that, within certain constraints related to e.g. cost, production, environment, etc., multi-functional foams may be tailored to meet simultaneous requirements on stiffness, mass and damping.

The interest in engineering of foams has been fuelled by the challenge facing almost all branches of transport vehicle engineering to reduce overall weight while avoiding negative effects related to increases in noise and vibration levels. Unfortunately, engineering of foams on the individual material level, i.e. designs built on systematic and continuous relationships between polymer foam chemistry, foam processing, foam micro-structure and the resulting elastic–viscoelastic and acoustic–viscoacoustic properties, is still a vision for the future. However, the design parameter space available for a multi-functional, foamed material (or combinations thereof in a multi-layer arrangement), utilized within the frame of state-of-the-art numerical simulation techniques, allows for multiple layer arrangements of foams with different properties to be properly tuned to provide acoustic and vibrational damping performance at a low-weight penalty.

For control of noise and vibration, flexible foams with open cells are most commonly used. For such porous foams, the vibroacoustic energy is carried both through the airborne path, i.e. the sound pressure waves propagating through the fluid in the pores, and through the structure borne path, i.e. the elastic stress waves carried through the solid frame of the material. These waves are strongly coupled, i.e. they simultaneously propagate in both the fluid and the solid frame but with different strengths and relative phase. A characteristic of this coupled wave propagation, is that the vibroacoustic energy is dissipated and converted into heat as the wave travels through the material. This loss of mechanical–acoustical energy is mainly related to viscoelastic and viscoacoustic phenomena in the solid frame and at the interface between the solid frame and the fluid in the pores. All these dissipation mechanisms are in general functions of frequency and furthermore, vary with frequency in strength and character. Clearly for a given situation, the balance between energy dissipated through vibration of the solid frame and changes in the acoustic pressure varies with the topological arrangement, choice of material properties, geometrical dimensions, interfacial conditions, etc.

Traditionally, and also in the present paper, the dynamic behaviour of porous materials is described in terms of macroscopic, space averaged quantities, such as acoustic pressure, elastic stress, solid and fluid displacements. Here parts of the modelling paradigm of porous foams, known in the literature as Biot's theory, will be briefly reviewed from an acoustics and vibrations perspective. A full review is beyond the scope of the present paper and the reader is referred to Biot's original publications for an in-depth coverage.

In the theory attributed to Biot, the spatially averaged quantities are characterized by a number of bulk material parameters which either may be computed for idealized geometrical arrangements or from dedicated measurements performed on samples of porous materials. Such parameters are required for the conservative parts of the fluid and the solid frame energies as well as for the dissipation mechanisms described above. To be useful for design purposes, these parameters should preferably be material parameters, in the sense that they, for a linear system, should be independent of the deformation, the direction of the propagating waves, the boundary conditions and the frequency.

In the present paper, the balance between the energy dissipation mechanisms mentioned above is first discussed. A simple scaling law, relating micro-structural dimensions to the macro-scale, viscoacoustic bulk parameters is then proposed. Together with existing scaling laws for elasticity, the flow resistivity scaling law is used to discuss the foam engineering process in terms of results from simulations performed using convergent, finite element solutions to Biot's equations. As a final demonstration, a layered structure typical for most vehicle applications, metal sheet, foams, barrier and felt materials, is presented. The performance rating in terms of some common acoustical and vibrational measures (e.g. average vibration velocity, transmission loss) is evaluated against the different dissipation mechanisms studied.

## 2. Modelling aspects

The first work on modelling acoustic wave propagation through a porous medium was published by Lord Rayleigh (Rayleigh 1877), for rigid, parallel fibres. Most of the fundamental work for the present theoretical treatment of elastoacoustic waves were developed during the first half of the twentieth century, by Zwikker and Kosten (e.g. Zwikker & Kosten 1949) and by Biot (e.g. Biot 1956*a*,*b*).

A useful model of a porous material for damping of acoustic, i.e. noise, and elastic, i.e. vibrations, requires theoretical rate of change relations for all the relevant mechanisms introduced above. To introduce the concept of elasto–viscoelastic–acoustic–viscoacoustic modelling of porous materials, a brief overview of the different models which together constitute the theoretical modelling basis is given in this chapter. The starting point is the homogenized, macro-scale Biot's equations.

### (a) Biot's equations

The most widely used theory for porous materials is due to Biot (see e.g. 1956*a*,*b*), who formulated a general three-dimensional continuum theory for elastic porous media. In a sense, and for specific parameter choices, Biot's early theory is similar in nature to the works by Zwikker & Kosten (1949), but with the important difference that Biot also included effects of shear in the elastic frame of the porous medium. The resulting equations may be written in several different forms, here they will be given in a slightly different form than the original proposed by Biot, treating the solid phase as separated from the fluid phase, and extended to include also the internal losses in the solid frame.

Having as dependent variables the macro-scale, solid frame structure and pore fluid displacement potential, assuming constant material properties within a volume, the equations may be written in Cartesian tensor notation, for a harmonic time-dependence , as (notation and explanation of parameters may be found in appendix A),(2.1)(2.2)The elastic–viscoelastic effects are represented by the complex, frequency-dependent correspondents to Lamé's constants and , see §2*b*(ii), and are homogenized, material properties of the solid, frame material.

The acoustic–viscoacoustic effects are described through three homogenized, frequency-dependent constants, *b*, *Q* and *R*, the first being the viscous drag constant, see §2*b*(i), the second being the dilatational coupling constant and the third being the pore fluid bulk modulus.

The thermal interaction between the fluid and the frame is introduced via *R*(*ω*) and *Q*(*ω*). This mechanism is mainly active through the compressibility of the fluid, essentially going from being isothermal at low frequencies (the compression/expansion of the fluid is slow enough for the surrounding frame to regulate the produced heat) to become adiabatic at high frequencies (the compression/expansion locally affects the temperature of the fluid). In between, a relaxation of the vibroacoustic energy occurs. For a detailed discussion on the thermal interaction mechanisms the reader is referred to Zwikker & Kosten (1949), Allard (1993) and Lafarge *et al*. (1997).

Finally, there is another coupling effect, related to the pore geometry and the openness of the material, mainly introducing the added inertia due to curvature of the fluid paths through the pores, *ρ*_{a}, see §2*b*(iii).

### (b) Homogenized model

The origin of the coupled wave propagation described by Biot's equations is complex. Along the solid frame walls of the porous material, the fluid is interacting with the solid both through inviscid, thermal as well as through viscid mechanisms. All these phenomena are related to conditions along the boundaries, i.e. *surface* fluid–structure interaction. Surface fluid–structure interaction typically enters the problem to be solved through boundary integrals, coupling quantities of the different media to each other. An example of this is fluid and solid displacement continuity together with pressure and normal stress continuity along an elastic, moving boundary. However, the fluid–structure interaction taking place in the interior of the porous material, the *volume* fluid–structure interaction, will not appear as boundary integrals but as volume coupling terms as discussed below. This change from surface to volume fluid–structure interaction is a consequence of the spatial averaging required to obtain Biot's equations.

In the literature several different derivations of equivalents of Biot's equations may be found, here the approach taken by Pride *et al*. (1992) will be used to illustrate the correspondence between the homogenized, macroscopic, volume fluid–structure couplings and the micro-level field variables.

The starting point is Navier's equations, for the fluid displacement field in the pores and for the elastic displacement of the solid frame, equations (2.3), (2.4) and (2.5),(2.3)(2.4)(2.5)Averaging these over a sufficiently large volume, using Slattery's averaging theorem (Slattery 1967), to relate the average of a gradient to the gradient of an average,(2.6)equation (2.3), with *ξ*=*f*, and equation (2.5) become,(2.7)(2.8)where the two boundary integrals account for the coupling forces along the wetted, pore solid surfaces and the volume averaged viscous terms are neglected. Rewriting equation (2.7), neglecting fluid compressibility locally on the micro-scale,(2.9)where(2.10)is the coupling between the solid frame and the pore fluid due to the drag forces induced by the viscous fluid, the first term on the right hand side being related to shape and the second being due to friction.

The volume averaged drag force, expressed by equation (2.10), is a relation between the fluid motion and the motion of the solid frame. Boundary layers form along *S*_{w} with a vanishing relative velocity between the fluid and the solid at the surface due to the viscosity of the fluid. At the same time the macroscopic drag force amplitude increases with an increasing relative displacement, since the velocity gradient at the surface must increase correspondingly. Outside of the boundary layers, the viscous effects may be neglected and the fluid response considered to be inviscid. Traditionally, this latter assumption has been used to find a principal, yet paradoxical, choice for a reformulation of equation (2.10) as(2.11)which is similar to Biot's original model, and where the averaged fluid displacement relates to an inviscid fluid. The determination of *D*(*ω*) will be discussed in §2*b*(i).

As a final remark and in analogy to the discussion above, the last term at the right hand side of equation (2.8), representing the change in pore pressure due to deformation of the solid frame, may be shown (Pride *et al*. 1992) to be equivalent to(2.12)

#### (i) Viscous dissipation

Based on equations (2.9), (2.10) and (2.11), expressions for *D*(*ω*) may be derived. Zwikker & Kosten (1949) and Biot (1956*a*,*b*) assumed idealized pore geometries; circular, rectangular tubes, etc. and arrived at expressions valid for low frequencies, based on Poiseuille flow, and high frequencies, based on Helmholz flow. Even though far away from real pore geometries, these models intrinsically describe the main features of the frequency dependence of the viscous drag effects. Pride *et al*. (1992) extended these works to pore channels having varying widths. However, for the purpose of modelling the type of porous materials of interest here, the work by Johnson *et al*. (1987) merits a more thorough discussion.

Introducing a viscous response function, relating the relative, homogenized displacement to a macroscopic pressure gradient,(2.13)where is given in Johnson *et al*. (1987) and here in a slightly modified form as(2.14)The derivation of equation (2.14) distinguishes between the response at low and high frequencies, the difference being related to the relation between the viscous skin depth,(2.15)and the typical pore dimensions. At high frequencies, *δ* is small compared to the geometrical properties of the pores and the fluid motion is of a potential flow type. At low frequencies, *δ* is large and approaches the static flow resistivity value obtained from a measurement of a sample material subjected to a static pressure drop (Allard 1993). At intermediate frequencies, the balance between the energy contained in the viscous boundary layer and the total energy in the fluid flow has to be established. Johnson *et al*. (1987) divides the microscopic fluid field in the pores into two contributions, the bulk of the pore space where potential flow may be assumed and the boundary layer where the fluid velocity varies exponentially with the distance from the surface up to the potential flow field value. Relating the losses in the boundary layer, which is our primary interest, to the velocity at the surface it may be shown that the previously mentioned energy balance leads to an expression for the characteristic viscous length, *Λ*, written as(2.16)where *u*_{p} is the fluid potential flow field, i.e. evaluated as if the fluid was inviscid.

Having established equations (2.13), (2.14) and (2.16), it is now possible to link these results to the previously derived equation (2.9), rewritten in the form(2.17)thus identifying(2.18)which gives for the viscous drag term in equations (2.1) and (2.2)(2.19)and for the inertial coupling term(2.20)Through this mechanism, energy is dissipated from both the fluid as well as from the solid frame. As seen from equation (2.18), for a given fluid, the viscous drag depends on the frequency, the geometrical dimensions of the pores and the pore fluid properties.

At low frequencies, the viscous dissipation is characterized by the static flow resistivity (or equivalently the viscous permeability), while at higher frequencies viscous boundary layers formed along the frame walls tend to decrease the resistance against the fluid movement through the voids of the material, leading to decreasing viscous drag and viscous relaxation.

#### (ii) Solid frame dissipation

The internal losses in the solid frame of the porous material occur due to stress–strain relaxation as the frame is deformed. In the model discussed here, it enters the equations above through the generalized Lamé constants in the Augmented Hooke's Law (AHL), introduced by Dovstam (1995). AHL, which is similar to Biot's model of a viscoelastic porous material (Biot 1956*c*), is formulated for isothermal conditions. Thus, for the solid frame the heating due to the expansion/compression of the fluid is neglected. For low frequencies (with respect to the lowest relaxation frequency) the solid frame is characterized by the static Young modulus, while for high frequencies the real valued, unrelaxed material modulus governs the behaviour.(2.21)where(2.22)(2.23)and(2.24)(2.25)For a commercially available foam, Tramico Polyurethane 22.1 kg m^{−3}, the parameters found (Göransson & Lemarinier 1998) suitable for representing the relaxation behaviour at low frequencies are given in appendix B, table 1.

#### (iii) Inertial coupling

In addition to the dissipative coupling mechanisms and losses discussed above, the geometry of the pore space must be accounted for in the spatially averaged representation of the porous material. One parameter, the structure factor *K*_{s}, was originally introduced by Zwikker & Kosten (1949). According to their model of the porous medium, the fluid might be partially trapped in voids of the material, which are inclined to the pressure gradient leading to an apparent increase in the static flow resistivity. Another geometrically related effect is attributed to the variation of the pore shapes/dimensions throughout the material (Allard 1993), which also might be modelled through a structure factor now increasing the fluid density. The effects related to pore geometry are commonly grouped into a concept known as tortuosity, denoted by *α*_{∞}. Biot (1956*a*) showed that the effects of the tortuosity also appear as an inertial interaction mechanism. An acceleration of the fluid results in an inertial force in the solid frame and vice versa. This is reflected by the coupling parameter *ρ*_{12} in Biot's equations.

### (c) Mechanics

The focus of the present paper is engineering of foams for acoustics and vibration damping purposes. This requires a set of continuous, systematic relations between the foam micro-structural mechanical properties and its equivalent macro-properties. Such models, applicable for mechanics, have been the focus of much research during the last decades, see e.g. Gibson & Ashby (1988) and Warren & Kraynik (1988).

Here, the scaling laws introduced by Gibson & Ashby (1988) are used. For simplicity they will be given in their isotropic forms, i.e. assuming cubic symmetry of the cellular structure.

Assuming a micro-structure as shown in figure 1, and furthermore assuming strut bending as the primary mode of deformation, the bulk modulus of the foam might be shown to be proportional to the material bulk modulus and the relative density and hence also to the thickness and length of the struts forming the cell geometry.(2.26)Similarly the bulk density of the porous foam may be shown to be proportional to the frame material density, the length and the thickness of the struts, see equations (2.26) and (2.27).(2.27)

### (d) Acoustics

#### (i) Viscous interaction

To derive scaling laws applicable to the viscoacoustic macroscopic parameters appearing in Biot's equations, relations between the micro-structure dimensions and the macroscopic properties are needed. Allard & Champoux (1992) showed that for a circular cylinder with radius *d*_{s} the characteristic length, *Λ*, may be related to the porosity as,(2.28)provided thatand may be rewritten in terms of the solid frame bulk and material densities as(2.29)Furthermore, Allard (1993) has shown that the viscous characteristic dimension may be expressed in terms of the static flow resistivity as,(2.30)where *c*=1 for a cylindrical geometry.

Now, combining equations (2.29) and (2.30), we may write(2.31)which may to first order be simplified as(2.32)which then serves as the desired scaling law for the viscoacoustic properties.

Using equation (2.32), a parameter map may be created to illustrate the use of the scaling law for the viscoacoustic interaction. Figure 2 shows the functional dependence between the bulk density and the static flow resistivity for varying strut dimensions. The base material used to obtain the proportionality constants is given in appendix B and has an average strut thickness of 4.4×10^{−6} m.

Note that keeping the bulk density constant, changing the strut thickness implies a growing pore volume as well as a constant modulus of elasticity. On the other hand keeping the strut thickness constant while increasing the bulk density, implies a shrinking pore size and an increasing stiffness.

### (e) Finite element modelling

To predict the propagation of elastoacoustic waves through a porous medium, for an arbitrarily complex arrangement of different materials, geometries, boundary conditions and multi-layer structures, it is necessary to resort to some kind of numerical solution procedure. It is clear that such a suitable solution procedure has to take an extended set of physical quantities into account, i.e. describing the state of both the fluid in the pores and the solid frame structure and the coupling between these. At this point it should also be mentioned that the couplings, dealt with above, all relate to the fluid–structure interaction occurring throughout the volume of the porous medium. In addition, along the boundaries to other media, both fluid, porous and solid, special treatment and careful attention have to be paid to the kinematic conditions, the mass flow continuity requirements as well as to the relevant stress balances.

The results discussed in the present paper are obtained from finite element solutions of Biot's equations based on higher order polynomial, hierarchical functions. As has been shown by Hörlin *et al*. (2001), convergent solutions to Biot's equations require special care in the selection of the trial functions. This is even more pronounced in the case of multiple layers of porous materials, where a combination of higher order polynomials for the element base functions and mesh refinement (hp-FEM) is adopted (Hörlin 2005). The results discussed below are all computed with an accuracy better than 10% in the displacements and the acoustic pressures.

## 3. Wave propagation

In a fluid filled porous medium, there exist two compressional and one rotational waves. All the three wave types represent coupled wave motion, i.e. they propagate simultaneously in both the fluid and the solid frame. However, the relative amplitude and the phase between the fluid and the solid frame will be different resulting in a varying degree of spatial attenuation for each wave. Typically they also propagate with different phase velocity. The slower waves have close to out of phase motion between the fluid and the solid frame, while for the fast waves the amplitudes are almost equal and the fluid and the solid frame are moving in phase. The spatial decay of the slow waves is mostly much larger, being controlled by the viscous dissipation, than for the fast wave, being controlled by inertial interaction.

Following Biot (1956*a*) and Allard (1993), the solid and fluid displacements in the porous medium may be decomposed into a scalar potential dilatational field and a vectorial potential rotational field,(3.1)(3.2)The potentials may then be written as(3.3)(3.4)with *ζ*=1, 2 and *κ*=s, f.

The relative amplitude of the compressional waves may be calculated from equation (3.5) and are shown for the foam given in appendix B in figure 3.(3.5)At low frequencies the fluid and solid displacement amplitudes are the same in one compressional wave while the fluid has a higher amplitude in the other. The first wave is almost in phase while the other is close to out of phase. The latter is also more damped, see figure 3*b*.(3.6)In the rotational wave, displacements are similar in amplitude over the studied frequency range, the motion is in phase and the damping is relatively low.

To illustrate the change in the wave properties for a modified material, derived through the scaling laws, i.e. equations (2.1), (2.2) and (2.7), examples are given in figure 4 for a material with bulk density 250 kg m^{−3} and strut thickness 5×10^{−7} m.

The high-flow resistivity of the finer and heavier foam locks the displacement fields together, see figure 4*a*, with a corresponding heavy damping effect for the out of phase wave and only low damping to the in-phase d : o, see figure 4*b*. In fact the damping effect is in the latter case related to the viscoelastic losses in the solid, implying a potential benefit in using a foam with a highly viscoelastic frame.

## 4. Engineering porous foams for acoustics and vibrations

To illustrate the foam engineering process in relation to acoustics and vibrations, a series of multi-layered arrangements are discussed below. All have the same base steel plate of 0.000 85 m thickness and the same top viscoelastic layer of 0.001 m thickness. Sandwiched in between these is then the foam treatment which should be engineered to a certain performance level. The performance is evaluated through two measures, RMS of the top surface displacement and the transmission of acoustic energy. Of primary interest is to achieve low displacement and low transmission at the lowest total added weight.

The sequence of examples are: the foam 1 given in table 1 in appendix B as a single layer, a viscoelastic foam, i.e. foam 2 in table 1 in appendix B, with the same strut thickness as a single layer and finally three layers of foam tailored to give high performance with low total weight. Alternative arrangements have been discussed in a recent paper by Göransson *et al*. (2005).

### (a) Performance measures

The performance measures used in the discussion below are common indicators for automotive applications. The square root of the surface averaged squared displacements of the viscoelastic layer, equation (4.1),(4.1)the sound transmission loss, the ratio between incident power to transmitted power, here assuming plane acoustic waves, equation (4.2).(4.2)For reference the sound transmission calculated for an infinite wall having the same surface mass and no stiffness, i.e. the mass law (Fahy 1985), equation (4.3),(4.3)

### (b) Multi-layer trim component

#### (i) Geometrical dimensions and material parameters

The geometrical arrangement used for the evaluation of the performance of the multi-layered foam arrangements is shown in figure 5. The dimensions of the set-up are 0.4 m by 0.5 m. The acoustic ducts are echoically terminated, the excitation being in the form of an acoustic incident wave in one of the ducts. The material parameters used for the steel and the viscoelastic material are shown in table 2 in appendix B.

### (c) Results

#### (i) Steel, foam 1 and viscoelastic damping layer

With the comparably light foam, having a low-flow resistivity, all three waves propagate through the material. Indeed, looking at the RMS displacement of the top surface.

The area weight of the package is for this case 2.1 kg m^{−2}. It is obvious from the results shown in figure 6, that this foam does not provide any important damping effects, in fact the displacement of the viscoelastic layer is amplified relative to the steel plate at high frequencies.

#### (ii) Steel, viscoelastic foam 2, viscoelastic layer

This case, still involving a single foam layer, is different in the bulk density, the flow resistivity of the foam, as well as the introduction of viscoelastic losses in the solid frame of the foam.

The area weight of the package is for this case 5.3 kg m^{−2}. With this foam, foam 2 in table 1 in appendix B, a decoupling effect is visible in figure 7*a*. Above about 280 Hz, the top layer displacement is lower than the steel plate. This is also immediately reflected in the sound transmission curve, figure 7*b*, above this frequency the isolation effect increases rapidly.

#### (iii) Steel, 3 layers of foam, viscoelastic layer

Here, engineering the foam properties has resulted in a combination of foams each with different functions tailored to achieve a high performance at a low weight.

The area weight of the package is for this case 2.7 kg m^{−2}. Also here we see the decoupling effect is visible in figure 8*a*. Above about 240 Hz, the top layer displacement is lower than the steel plate and the isolation effect increases rapidly.

### (d) Discussion

Comparing the three cases, it is interesting to notice that with a combination of tailored foams, having a comparably low total area weight, it is possible to achieve the same performance as a single layer, having almost twice the area weight. This is achieved through a careful tailoring based on equations (2.1), (2.2) and (2.7), and the use of convergent finite element solutions.

A closer look at the actual deformation and pressure fields, for a given frequency, reveal some more insight into the phenomena dealt with. Figure 9 shows the deformation of the multi-layer structure at different frequencies. The light foam, figure 9*a*, gives low damping and the motion of the viscoelastic surface is amplified relative to the steel plate at the bottom of the figure. The heavy, viscoelastically damped foam seems to dissipate most of the energy, both the acoustic and the elastic, and the upper surface has a considerably lower vibration level, figure 9*b*. The same holds for the three layer combination, figure 9*c*, which only has half the area weight compared to the previous case.

## Footnotes

One contribution of 18 to a Discussion Meeting Issue ‘Engineered foams and porous materials’.

- © 2005 The Royal Society