Sensitivity of macroscopic properties of a multi-layer panel …past.isma-isaac.be/downloads/isma2014/papers/isma2014_0186.pdf · linked to averaged macroscopic elasto-acoustic properties,

Sensitivity of macroscopic properties of a multi-layer panel including porous material on the micro-level parameters of an open cell porous material

E. Lundberg1, P. Göransson

1, U. Orrenius

2, B. Semeniuk

1

1 KTH Royal Institute of Technology, Department of Aeronautical and Vehicle Engineering,

The Marcus Wallenberg Laboratory for Sound and Vibration Research,

Teknikringen 8, 10044 Stockholm, Sweden,

e-mail: [email protected]

2 Bombardier Transportation, Department CEE/EWA Specialist Engineering Acoustics &Vibration,

Västerås, Sweden

Abstract The micro-structure of an open cell porous material is modelled as an idealized, periodic structure,

allowing anisotropic properties to form on the micro-scale. Using simple analytic descriptions of acoustic

and elastic properties calculated from micro-structure geometry, the microstructural properties can be

linked to averaged macroscopic elasto-acoustic properties, which can be measured and observed. These

macro level properties may be deduced from measurements on a sample of a porous material which is at

least a few centimetres across. The most common of the acoustic properties are the flow resistivity or the

(dynamic) permeability, the porosity and the viscous and thermal characteristic lengths, together with the

tortuosity. For the elasticity, the moduli of the Hooke’s law are the most important. The underlying

motivation for the current work is that the quantities are interdependent since they all depend on the micro

geometry. Thus, to design the macro level acoustic properties for a required performance, physically

relevant models linking these to the micro structure would be necessary. In this paper, a set of such models

are proposed and utilising these, the influence of the anisotropy on acoustic properties of the foam as well

as the transmission loss of a multi-layer plate is investigated.

Typical macro level acoustic properties of open cell poro-elastic materials calculated from micro structure

parameters are found to give results of the same order of magnitude as measured data found in the

literature. Further, the model is applied to evaluate transmission loss of roof constructions with isotropic

and anisotropic foam. It is shown that the sound reduction can be improved without changing the overall

surface weight of the structure by making use of the anisotropy of the foam. The results Underline the

importance of anisotropy and in addition provide guidance for further research.

1 Introduction

Recent studies indicate that transportation is responsible for using a large and increasing part of the

available energy today. With increasing energy prices and strong requirements on reduction of the

emissions of greenhouse gases, a decrease of energy consumption in the transport sector is necessary. One

way to reduce the energy consumption of vehicles is to reduce the vehicle weight. However, in transport

application involving passengers or sensitive goods, the customers still expect the vehicle to meet the

present standards for safety and comfort, in spite of weight reductions.

In vehicle design, multilayer panels are frequently used as a means of fulfilling both structural and

acoustic requirements when designing e.g. floors, walls, and roof of a vehicle. Historically, the most

common way of increasing acoustic performance of a structure, such as for example transmission loss, has

been to increase its weight. With increasing energy cost and demands for sustainable design, other

possibilities must be investigated to achieve the acoustic performance required with limited increase of

weight, or to achieve the same level of acoustic comfort with less weight. For this purpose, porous

2135

materials are often used as a part of the design, adding volume, but very little weight, since as much of the

95-99% of the volume of the porous material is air. The application example used in this study is a roof

design of a passenger train car including porous materials for acoustic and thermal insulation.

Traditionally, the acoustic macro level properties used to characterise a porous material are given as

isotropic quantities. These macro level properties may be deduced from measurements on a sample of a

porous material which is at least a few centimetres across. The most common are the flow resistivity or the

(dynamic) permeability, the porosity and the viscous and thermal characteristic lengths, together with the

tortuosity. These quantities are interdependent since they all depend on the micro geometry. Thus, to

design the macro level acoustic properties for a required performance, physically relevant models linking

these to the micro structure would be necessary. Several such methods exist [14][15][17] [19] many giving

good results, in particular for materials that can be regarded as reasonably isotropic. However, for

materials deviating from this assumption, problems do arise if an isotropic model is to be fitted. As shown

by Van der Kelen [9] certain types of porous materials exhibit directionally dependent macro level

properties, both with regard to elasticity and acoustics. There is however a lack of knowledge as to what

extent the anisotropy on micro level of the porous material affects the elasto-acoustic properties on macro

level and the vibro-acoustic performance of the material. In addition, for those situations where the load

carrying ability as well as the vibration transmission must be taken into account, the elastic properties the

static and anelastic (dynamic and frequency-dependent) Hooke’s matrices are also required. In the present

work, the focus is initially on the acoustic properties.

Simulations using numerical models of the actual, disordered, geometry on micro-scale are still

computationally demanding, in particular if many different configurations are to be evaluated, with the

objective of an optimised design. An alternative approach, which also is the main scope of the present

work, would be if the micro-structure of the open cell porous material is modelled as an idealized, periodic

structure, allowing anisotropic properties on the micro-scale. Following Gibson and Ashby [7], the

microstructural elastic deformations are assumed to be mainly due to bending of the struts forming the

open cell pore. From similar, simplified analytical descriptions of the elastic, anelastic and acoustic

parameters and their dependence on the micro-structure properties, the microstructural properties can be

linked to averaged macroscopic properties. Here, such models taking the anisotropicity of the micro

structure into account are proposed. The basic assumptions for the modelling are that the wavelength is

much longer than the microstructural dimensions, and that the relations both for acoustics and elasticity

are linear.

It is generally known that fibrous materials, such as glass wool, have anisotropic acoustic flow resistivity

[1][9][16]. The flow resistivity normal to the plane of the fibres is in general higher than the flow

resistivity in the plane of the fibres.

Also open cell porous foams can be anisotropic, which has been pointed out by several authors

[5][9][14][15] but measured parameters are normally not determined for more than one direction, nor is

the anisotropy explicitly used in the modelling. The reason for the anisotropy is the method of production,

in which the chemical compounds are mixed, made to form foam, which later is dried. When the foam is

formed, the volume is increased many times. If the compound is confined in a vessel the mixture will rise,

just like dough of bread rises. Depending on the type of foam and particular method of production the

degree of anisotropy will vary. Here the anisotropy is defined by the ratio R of the height of the unit cell

over the width of the unit cell, and with R=1 representing an isotropic structure. In [5] there are examples

of polyurethane foams with anisotropy of 1.2 and 1.4 and in [15] as much as 1.75. In recent work [9] it is

shown that the flow resistivity of a melamine foam could vary 10-20% between the principal directions of

the material and the Young’s modulus for the different directions could vary more than a factor of three

[2]. However, since the most commonly used methods [8] for measuring flow resistivity and elasticity do

not take anisotropy into account, detailed data is scarce and difficult to find.

In this paper a simple anisotropic model on micro level for open cell porous foam is used to investigate the

influence of micro level geometry and anisotropy on the flow resistivity and the viscous and thermal

characteristic lengths in the principal directions of the material. The same foam model is later used in a

multi-layer plate model of a roof -like design to explore the influence on the transmission loss of a

composite structure by the degree of anisotropicity of the porous material. The transmission loss is

calculated with an efficient transfer matrix code for the frequency range 100-10000 Hz.

2136 PROCEEDINGS OF ISMA2014 INCLUDING USD2014

2 Modelling

The proposed modelling approach is intended for open cell materials like melamine foam with at least

90% air. The strut cross section is assumed to be circular, both for the calculation of porosity and flow

resistivity, which seems to be reasonable from photos of the microstructure. The assumed periodic micro-

structure cell is a geometry that completely fill space. It can be viewed as an average or typical size of the

material. The purpose of the proposed micro level model is to link the macro level properties to the micro-

geometry and thereby enable numerical experiments with the objective to explore the possible

performance variations associated with the degree of anisotropic of the micro structure.

The choice of a simple geometry allows for analytical expressions to be derived and used, thus avoiding

the complex and computationally demanding analysis that would be required for more complex

geometries. However, also with a simplified micro structure template, the lack of available experimental

data pertaining to anisotropic porous materials, require further simplifying assumptions to be introduced in

the modelling.

2.1 Geometric model

The basis for the periodic unit cell is the model proposed by Huber and Gibson, according to figure 1. For

the struts in the three principal directions, the variation of the cell structure is here limited to have different

thickness and length, but all struts in the same direction have the same properties. The periodic unit cell is

oriented in the principal directions of the prism in a Cartesian coordinate system x1, x2, x3, where x3

coincides with the rise direction of the foam.

Figure 1. a) The micro structure with connections to neighbouring cells. b) Illustration of the

isotropic and anisotropic unit cell dimensions.

The unit cell consists of one whole strut in x1 direction, one whole strut in the x2 direction and one whole

strut in the x3 direction, with only one corner fully included in the model, see Figure 1a. The grey shaded

structure is the complete micro-structure, and the black is the modelled unit cell with strut lengths l1, l2 and

l3, and strut thicknesses t1, t2, t3 oriented in the x1, x2 and x3 axis directions respectively. This seems like a

reasonable unit, since there are eight corners in the prism, which each is shared by eight cells. Note that the

double counting of the corner is neglected. In this study the only anisotropic variation considered is

that the length directions in the base of the prism (l1 and l2) are reduced by a factor R, see Figure 1b. The

outer cube is the isotropic unit cell with lengths l1,iso , l2,iso and l3,iso. The inner (smaller) shaded prism is the

anisotropic unit cell with lengths l1aniso=l3iso/R, l2aniso=l3iso/R and l3ansio=l3iso.When R increases, the volume

of the unit cell will decrease, illustrated by fitting the anisotropic unit cell inside the isotropic unit cell in

Figure 1.

LIGHTWEIGHT STRUCTURES AND MATERIALS 2137

2 2 21 1 2 2 3 3

1 2 3

11 ( )

4t l t l t l

l l l (1)

2

1

1

4 (1 )

3iso

t

l (2)

2

1

1

4 (1 )

(2 )aniso

t R

l R (3)

The microscopic parameters in the model are the strut thicknesses t1, t2, t3 and the strut lengths l1, l2, l3.

The strut along the x3 direction has length l3 and thickness t3. The measure of the anisotropy is defined the

same way as in Huber &Gibson as a ratio R=l3/l1. In this study the anisotropy is limited to values between

R=1.0 and R=1.5 which is consistent with experimental data [5].

If one full set of (micro) parameters is given, the porosity can be calculated if a cross-section shape of the

struts is assumed. Unfortunately there are few published cases in the open literature with detailed micro-

structural data. Some authors have reported foam data including the porosity together with at least one of

the micro parameters, or information that can be used to deduce a micro-parameter, such as the measured

viscous characteristic length (see equation (11)). To get values for six parameters from only two values

(e.g. porosity and strut thickness), some assumptions have to be made about the micro geometry. It has

been shown [7] that the porosity is proportional to the squared ratio between strut thickness and strut

length. Therefore current scaling is to this ratio, rather than the micro-structural lengths. Together with the

assumptions stated above this proportionality can be used to obtain a full set of micro-parameters which

are of the right order of magnitude as compared to those measured in real open cell porous materials.

Recalling that for an isotropic material the following holds: t1=t2=t3 and l1=l2=l3. In this special case the

porosity and one micro parameter (average strut length or average strut thickness or average cell size) is

sufficient to define the micro parameters. For a transverse isotropic material t1=t2=t3, l1=l2 and l3= R l1,

where is R=l3/l1 is a ratio of anisotropy which has to be assumed unless it can be estimated from e.g.

photos of the micro structure. With R given, together with the porosity and one micro parameter, all the

remaining micro parameters can be calculated.

2.2 Porosity

The porosity is the ratio of volume of air over the total volume of the material, or one minus the volume

fraction of solid material, as stated in equation (1). The current micro structural modelling is restricted to

materials with porosity higher than 90%. With such high porosity the error made in double counting the

solid volume in the strut junction is acceptable. In the overwhelming majority of all published data,

isotropy is assumed, which means that in the best case, one strut thickness and one strut length is given

along with porosity or density. Thus, in order to set the micro level parameters in the present analysis, the

thickness is assumed to be constant for all struts. For materials such as melamine this seems to be a

reasonable assumption, judging from published photos of microstructures [12].

2.3 Static flow resistivity

The flow resistivity is a macro level measure of the pressure drop produced over a material sample when a

static fluid flow is passing. It is measured in [Ns/m4] and can range from 100 to some 1000000 Ns/m

4.

Materials used for reducing sound do in general have a relatively high flow resistivity, of the order of


Figure 2. Lay-out of measurement of flow resistivity over a cylindrical sample in a pipe according

to ISO 9053 [8].

2 1 2 1 2 1 2 1( ) / ( ) ( )

sample

p p F F A F F F F

H H H A Vu u u u (4)

1 1

( / ) ( / ) hdcell dcell

cell

cell cell cell

F F

V au u (5)

10000 Ns/m4

and higher [9][18]. The standardised measurement for isotropic samples of porous material

is defined in the ISO 9053 norm [8].

In the standard measurements, a sample of porous material is placed in a pipe. From the measured data the

flow resistivity is then calculated as,

where (p2-p1) is the measured pressure drop, (F2-F1) is the corresponding drag force, A the sample cross-

section area , H the sample thickness in the flow direction and u denotes the velocity of the air flow . The

volume of the sample is Vsample=AH.

When flow resistivity is calculated on micro-scale the main assumptions are that the dimensions are much

smaller than the acoustic wavelength, and that the fluid inside the pores can be regarded as viscous and

incompressible. The model used here is based on an analytical expression for the drag force around a long

fibre with circular cross section oscillating in a viscous, incompressible medium. The flow generating the

drag force is applied in all principal directions of the unit cell, one direction at a time, resulting in

expressions for each of the diagonal terms in the flow resistivity tensor. Each expression will have two

contributions from transverse flow from the two transverse struts and one contribution from flow

longitudinally along one strut. All the off-diagonal terms are set to zero in the principal direction.

To link the micro level flow in the chosen unit cell to the macro level static flow resistivity, the following

relations based on the standard measurement procedure, ISO 9053 [8] are rewritten for a unit cell.

where σcell is the flow resistivity across the unit cell, Fd,cell is the drag force over the unit cell, a is the cross

section area of the cell perpendicular to the flow direction and h is the thickness of the cell in the flow

direction. Vcell=(ah) is the volume of the periodic cell.

If usample is the flow velocity outside the sample, the velocity inside the sample when represented by unit

cells, will be usample/ϕ due to the continuity requirement at the sample interface, where Φ is the porosity,

and 1- Φ is the volume fraction solid material.

To link the micro model of flow resistivity (including anisotropy) to the macro level flow resistivity, both

expressions have to refer to the same bulk volume of material. When the material is anisotropic, the micro

level cell volume is different compared to the isotropic case, see Figure 1. This has to be accounted for

when the anisotropy changes. If not, the isotropic and the anisotropic model will not refer to the same bulk


3,2 1 2 1 3

1 2 3

, 2

( ) ( ) 1

//

1

/

drag celliso

cell cellaniso

drag call

cell cell

h a FF F f f l

H A a h l l lh a h a

FR

a h

u uu

u

(6)

2 2 21 12 131

1 2 3 1 1 2 3 1 1 2 3 1

2 2 22 21 232

2 1 3 2 2 1 3 2 2 3 1 2

2 2 23 31 323

3 2 1 3 3 1 2 3 3 2 1 3

1 1 1

( ) ( ) ( )

1 1 1

( ) ( ) ( )

1 1 1

( ) ( ) ( )

l t t

l t t

l t t

F F FR R R

u l l l u l l l u l l lF F F

R R Ru l l l u l l l u l l lF F F

R R Ru l l l u l l l u l l l

(7)

2 (1)

1

(1)

0

4 ( ( 2))1

2 ( ( 2)) ( ( 2))

k kktmk m f k

k k k k

H k ttF i u l

k t H k t

(8)

For flow in the longitudinal direction the corresponding expressions are:

(1)

1

(1)

0

( / 2)2

2 ( / 2)k kk

lk k k k f

k k

H k ttF i l u k

H k t for k=1, 2, 3 (9)

volume of material, even if the porosity is kept constant. Therefore, the flow resistivity on micro-scale is

estimated as.

Extending this expression to account for the contributions from all three struts, for each flow direction

leads to the following expressions.

The drag force due to transverse flow, normal to the strut direction is denoted Ftij , where i is flow direction

and j is the axis of strut orientation . For example, Ft13 is the transverse force from flow in the x1 direction

across the x3 oriented strut with length l3. The drag force due to flow longitudinally along the strut is

denoted Fi, where i is the direction of flow as well as the axis of strut orientation.

The drag forces are taken from an analytical expression for the force across a long strut with circular

cross-section [16] derived for fibrous materials.

A time dependence of exp(-iωt) is assumed. Equation (8) is derived assuming that there is no interaction

between struts and no influence of the corners in the cell. This means that the expression is only valid for

materials with a high enough porosity.

The formula has been validated against the results of Tarnow [16] for the extreme case of an anisotropy

ratio of 20, for which the flow resistivity of one single strut is dominating the response. The results agree

very well. An observation is that the flow resistivity from flow normal to the strut is many times greater

than the flow resistivity from flow along the strut.


Figure 3. Static flow resistivity as a function of anisotropy. The static flow resistivity is increasing

with increasing anisotropy. With the kind of orthotropic micro structure used here, the in plane

flow resistivity is higher than the out of plane flow resistivity

2

2

( )2

( )

i wA

iV

v r dA

v r dV (10)

1 1.1 1.2 1.3 1.4 1.50.6

0.8

1

1.2

1.4

1.6x 10

4 Calculated static flow resistivity as function of R

Anisotropy ratio R=l3/l

1

Flo

w r

esis

tivity [N

s/m

4]

3

2

1

With this micro structural model, the flow resistivity in the x1 and x2 directions will be higher than that in

the x3 direction, see Figure 3 which illustrates the static flow resistivity as a function of anisotropy. With

increasing anisotropy R, leading to that l1 and l2 are reduced by a factor R while keeping l3 fixed, more

unit cells can fit within a defined sample volume in the x1 and x2 axis directions. Then the flow will

interact with more struts for flow in the x1 and x2 directions leading to increased flow resistivity. For the

flow in the x3 direction there is no clear explanation. For all of the principal components, the volume

scaling used in the current model will increase the flow resistivity with anisotropy, but to what extent that

is linked to actual physical properties needs to be further examined.

When anisotropy increases while keeping a constant porosity, there will be more struts in the bulk of the

material than for an isotropic material. As a consequence of this the flow through the material will interact

with more struts per volume, leading to an increase of viscous losses related to the flow over the wetted

surface of the struts, and therefore a higher flow resistivity.

2.4 Viscous characteristic length

Another macro level parameter used to characterise porous foam is the viscous characteristic length Λ,

which is a measure of the interaction between the air flow in the volume of the pores and the viscous

boundary layer close to the pore wall. Λ decreases with increasing static flow resistivity in open cell

porous materials with high porosity, according to the relation (14) [3]. In a cylindrical shaped pore Λ is

equivalent to the hydraulic radius divided by two.

Johnson et al [4] gave a general definition of the viscous characteristic length as


2

1 1 4(1 )iso

A

t

t L t (11)

1 2 32

1 2 3

(1 ) 1 _ _

_

4

A

l l l total fibre lengthL

l l l unit volumet

(12)

1 2 31 2

2 3

1 1

( )

l l l

t l lR1 2 3

2 21 3

1 1

( )

l l l

t l lR1 2 3

3 21 2

1 1

( )

l l l

t l lR (13)

81macro static

gc (14)

Where Λ is the viscous characteristic length, v is the flow velocity in the material. The surface integral is

over the wetted surface of the pore and V is the fluid volume of the pore weighted by the squared velocity.

The viscous characteristic length can be regarded as a generalisation of the hydraulic radius for a cylinder.

Allard and Champoux [3] later developed an expression for Λ based on the average length of fibres

(struts) per volume, LA, which is measured in [m/m3]. It can also be expressed as fraction of solid material

divided by strut cross section. The flow is assumed to be normal to the strut orientation,

The anisotropic micro parameter expression for Λ is based on Allard’s expression with the assumption of

normal incidence of flow on a long fibre. Since Allard’s expression includes the total length of the fibres

the flow is encountering in a unit volume, it has to be scaled to the isotropic volume of the unit cell, just

like the flow resistivity previously presented. The LA parameter considers only the fibres normal to the

direction of the flow, since it was developed for fibrous materials. The same assumption is used for the

prism; leading to that only the two struts normal to the flow direction are included. The contribution from

longitudinal flow along the fibre is neglected. For the anisotropic material, the viscous characteristic length

is a tensor. The viscous characteristic length matrix in material principal coordinates has non zero elements

on the diagonal. The equivalent to the LA for the current micro model is the sum of the lengths of the two

struts normal to the flow divided by the volume of the cell. In the same way the viscous characteristic

lengths are defined for the three principal directions as,

The factor (1/R2) is included to normalise the values to the isotropic volume, in the same way as for the

flow resistivity. A third way of obtaining the viscous characteristic length is from relations to the macro

level parameters as shown by Johnson [4]. Approximating the constants cg and α∞ with 1, and data from

[12] gives

where cg is a constant depending on the pore cross section (1 for cylindrical pores), α∞ the tortuosity in the

high frequency limit (explained in a later section), η the dynamic viscosity of air, φ the porosity and σ the


Viscous characteristic length Notation Value [µm]

Measured [12] Melamine 32 Λmeasured 199

Isotropic model with Allard length ΛA 213

Isotropic micro-model, R=1.0 Λmicro,iso 299

Anisotropic micro-model, R=1.3 Λmicro,1 =Λmicro,2 / Λmicro,3 180 /208

From macro parameters Λmacro106

Table 1. Different estimations of viscous characteristic length.

1 2 3

'1 2 3

2 A

V

dAt l l l

l l ldV

(15)

Thermal characteristic length Notation Value [µm]

Measured [12] Melamine 32 Λ´measured 445

Λ´A, is estimated from the Allard

length with rule of thumb Λ´A=2 ΛA 425

Isotropic micro-model, R=1.0 Λ´micro,iso 425

Anisotropic micro-model, R=1.3 Λ´micro,aniso 425

Table 2. Different estimations of Thermal characteristic length

measured static flow resistivity. When applying eq. (14) to a melamine material with properties according

Melamine 32 in [12], Λmacro becomes 106 µm. In Table 1 different estimates of the viscous characteristic

length are compared.

Estimation from macro parameters according to equation (14) gives a value which is almost a factor two

too low. The estimation based on the Allard length gives a better agreement with the measured value. For

such estimation, the strut thickness has to be assessed, just like for the micro model in this work.

2.5 Thermal characteristic length

Yet another parameter characterising the porous material is the thermal characteristic length Λ’, defined as

the ratio between the volume integral over the average pore fluid volume and the surface integral over the

average of the wetted surface of the pore, i.e. the surface area of the struts in contact with the fluid. It only

depends on the geometry, and is independent of the applied flow over the cell. For application to the

present unit cell model the double counting of surface and volume in the corners where the three struts

meet, is neglected. For the type of material with high porosity considered here this assumption is

reasonable and will only introduce a small error. The rule of thumb is that the thermal characteristic length

is twice the viscous characteristic length [3].

The integral over the area is taken as the added surface area of the three struts, and the integral over the

pore volume is simply the porosity times the total volume of the cell. Since it is a purely geometric ratio of

volume to surface, there is no variation with anisotropy.


1 1 1 1 2 1 1 2

2

1 2 1 2

( )a S a S a S a S

a a S S

(16)

(17)

The elements of the compliance matrix based on the approach in [5] may be shown to be:

2.6 Turtuosity

For porous materials with open cells and high porosity, the tortuosity is normally close to one, which is the

lower limit. As the typical variations of tortuosity are small and have minor influence on the final results, a

very simple expression for the tortuosity is used in the present work. Following an approximation from [6]

where the microstructure is assumed to be built up from alternating sequences of cylinders, an expression

depending on cross sectional areas and lengths is obtained.

where a1 is the length of the section of the edge of the pore (the strut thickness) and a2 is the length of the

volume pore inside the cell, i.e. l1 minus the strut thickness. S1 is the fluid cross section in the wall (l2

times l3 minus the cross section for the struts) for flow in the x1 direction and S2 is the fluid cross section in

the pore. The underlying assumption is that the flow velocities are constant within each section. For the

prism model this is probably true in an average sense inside the pore, but less so in the intersections

between the pores. Also, the cross sections in the micro model are square, and not cylindrical. The values

of tortuosity obtained from Eq. (16) are close to unity and also close to what is measured for open cell

foams with porosities of more than 90% [12]. Considering that the tortuosity is not a critical parameter for

the acoustic behaviour, the level of approximation introduced is considered to be appropriate.

2.7 Elasticity

The expressions for the static compliance are given below, and the Hooke´s tensor is obtained from

inversion of the compliance matrix.


4 4

2 1 3 11 1

2 3 3 2s

t l t lE c E

l l l l

4 4

4 3 1 2 2 1 32 2

23 3 23 2

1

s

c l l l l l l

G E t tl l

4 4

1 2 3 22 1

1 3 3 1s

t l t lE c E

l l l l

4 4

4 3 2 1 1 2 32 2

13 3 13 1

1

s

c l l l l l l

G E t tl l

4 4

1 3 2 33 1

1 2 2 1s

t l t lE c E

l l l l

4 4

4 1 3 2 3

12 1 1 2 2

1

s

c l l l l

G E t l t l .(18)

Figure 4. Application example, representing a type of structure used for rail vehicle roof.

3.1 Analysis

The acoustic performance properties of the multi-layer panel is calculated using an efficient method based

on the transfer matrix method [20] to investigate the influence of the anisotropy of the porous material on

e.g. transmission loss of a multilayer panel. To illustrate the possible performance improvement due to the

anisotropic porous foam, the density of the foam is kept constant to study if it is possible to get an acoustic

improvement without adding weight.

The transmission loss is calculated with the porous material oriented with maximum flow resistivity

normal to the plane of the multi-layer panel, as well as an isotropic reference case. All calculations are

carried out with oblique incidence of 45 degrees. A design with isotropic foam is compared to foams with

anisotropy ratios of R=1.2 and R=1.4.

The Es is the Young´s modulus for the frame material. The factors c1 and c4 relates to cross sectional shape

of the struts and to the rigidity of the corners in the micro-structure. Assuming neither relative rotation nor

translation c1 would be of the order of 10 and c4 would be less without scaling or fitting.. For the special

case of all struts having equal thickness and the height l3 = Rl1 = Rl2 and the ratios between E1 and

E3 and G13 and G12 respectively results in the same expression as in Huber& Gibson [5]. With the type of

geometry addressed here, it is not possible to obtain the Poisson´s ratio [5]. It is set to 1/3 in the

calculations.

3 Application example

To illustrate the application of the proposed micro to macro modelling approach in an engineering

application example, the transmission loss of a rail vehicle roof structure is calculated. The lay-up is 1 mm

aluminium, 80 mm porous material, 120 mm air gap and a 1 mm aluminium perforated plate which also

works as ceiling in the interior space, see Figure 4. The perforated plate has ∅=2 mm holes in a square

drilling pattern with a distance between hole centres d=3.5 mm. The acoustic and stiffness properties of the

perforated plate are taken from previous work [13].


Figure 5. Dynamic anisotropic flow resistivity for the principal directions of the anisotropic porous

material with an anisotropy ration of R=1.4.

Figure 6. Transmission Loss for different anisotropy R. Influence of anisotropic material on the

transmission loss of a typical vehicle roof structure. The density and thickness of all the materials

is the same in all cases.

101

102

103

104

1

1.5

2

2.5

3

3.5

4

4.5x 10

4Anisotropic flow resistivity as function of frequency for R=1.4

Frequency [Hz]

Flo

w r

esis

tivity

1

2

3

101

102

103

104

0

10

20

30

40

50

60Transmission Loss for different anisotropy R

Frequency [Hz]

Tra

nsm

issio

n lo

ss [d

B]

R=1.0 isotropic

R=1.2 anisotropic max

out-of-plane

R=1.4 anisotropic max

out-of-plane

The flow resistivity as a function of frequency for the two principal directions of the material is shown in

Figure 5. The flow resistivities in the plane of the multi-layer panel, σ1 and σ2, are higher than the

corresponding out-of-plane flow resistivity σ3. At 500 Hz the difference is 3570 Ns/m4

for the foam with

R=1.4 and the difference increases with frequency.

At frequencies below 800 Hz the influence of the properties of the anisotropic porous material are

negligible for the transmission loss, see Figure 6. Above about 800 Hz for an angle of incidence of 45

degrees, there is a difference of 2 dB at 2000 Hz in favour of the anisotropic material with R=1.2, and 4 dB

for the material with R=1.4. In this case the material has been rotated so that the maximum flow resistivity

is normal to the plane of incidence. At 2000 Hz, the isotropic dynamic flow resistivity is 15 323 Ns/m4,

which shall be compared to the linear average of the flow resistivity in principal components for R=1.2

and R=1.4 which are 21 830 and 25 400 Ns/m4 respectively.


4 Discussion

A model of the micro-structure of an open cell porous material is presented, in which it is possible to

estimate macro-level properties from the geometry of the micro-structure. The model can include effects

of anisotropy, i.e. that the periodic cells are elongated and the flow resistivity, elasticity and viscous

characteristic length have different properties in the different principal directions. The acoustic parameters

obtained with the model, such as flow resistivity and viscous and thermal characteristic lengths, are of the

correct order of magnitude compared to measured (isotropic) data from literature. For a more complete

validation measured material data are needed, including information about the principal directions, e.g.

rise direction of the material, and such data are not available at present.

The main application of the present approach is foam materials with strut cross-sections that can be taken

as being near circular, like melamine. It can be expected to be less applicable for foam materials with a

strut cross-section being far from circular, e.g. polyurethane foams, which often have concave triangular

struts unless in particular the porosity and flow resistivity calculations are adjusted to account for the cross

section shape. Nor is it directly applicable to foams with a high proportion of closed cell membranes.

As input to the micro model, the assumed average thickness of the micro level struts was based on

estimations by Kino [12], in which a large number of measurements have been fitted for a certain type of

melamine. If no microscopic dimensions are available for the material it is possible to estimate a strut

thickness from a measured viscous characteristic length and the porosity according to equation (11) [6].

The thickness reported in [12] is 5.58 µm, and the estimation using equation (11) is 5.57 µm, so for this

particular material it gave a very good agreement.

In addition it is found that, the micro model calculation of the thermal characteristic length gives a

reasonable estimate, see Table 2.

The estimation of the viscous characteristic length gives very different results if it is applied to an

isotropic configuration or an anisotropic one. The isotropic micro model presented in this work strongly

overestimates the viscous characteristic length, while the anisotropic model for R=1.3 gives a value in

good agreement with the measurement of melamine 32 in Kino[12].

The reason why the viscous characteristic length is decreasing with increasing anisotropy is the same as

why the flow resistivity is increasing with anisotropy. As the unit cell is reduced in volume, more struts

can fit in the bulk volume of the sample, and with that LA and similar measures of strut length per volume

increases, which leads to a decrease of the viscous characteristic length according to equation (11). It is

easy to see why σ1 and σ2 increase with increasing R, but the reason why also σ3 is not obvious. It could be

an effect of the modelling principles rather than physical phenomena. The volume scaling introduced

increases all principal components with anisotropy, but should perhaps be modified, in particular for the σ3

component. The linear average of the principal components, as a measure of an isotropic equivalent, will

still increase with anisotropy if two out of three components increase.

Furthermore, the elastic static stiffness predicted without any fitting is found to be much too low. The way

the unit cells are connected in the present model, in the middle of each strut, see Figure 1, is the weakest

possible configuration. A different model for connecting the cells, possibly by introducing a randomized

joint location, would lead to a stiffness increase without influencing the acoustic parameters within the

current hypothesis.

In the application example it is investigated whether the anisotropy of the porous material can have an

effect on the transmission loss. A typical rail vehicle roof structure is analysed with layers of porous

material and air between two metal sheets, one which is perforated. At low frequency the sound

transmission mainly depends on the mass and the properties of the porous material are not significant. At

higher frequencies, from about 500 Hz, the anisotropic material increases the transmission loss compared

to the equivalent isotropic material, where the linear average of the flow resistivity in the principal

components is taken as the isotropic reference.


At 2 kHz the improvement is 2-4 dB with the present modelling approach, depending on the degree of

anisotropy. While 2 dB may be perceived as a rather modest improvement, but it should be kept in mind

that the gain is obtained without neither adding weight nor volume, thus rendering these phenomenon an

interesting aspect worth investigating further. Furthermore, since flow resistivity is even more important

for absorption than for transmission loss, the effect on e.g. the interior absorption in a vehicle depending

on the orientation of an anisotropic porous material has even greater potential of contributing to an

improved acoustic comfort.

5 Conclusion

A model for calculating acoustic and elastic properties of an anisotropic open cell porous material has been

presented. The order of magnitude of the acoustic parameters flow resistivity, viscous characteristic length,

thermal characteristic length and tortuosity corresponds to data found in literature. The model for the

elasticity has to be developed further, and the effect of diffuse incidence needs to be investigated as well.

Calculations on a train roof structure suggest that utilizing the effects of anisotropy of porous materials is

one possibility to improve the acoustic properties of a structure without adding weight.

Acknowledgements

The work has been carried out with funding from ECO2

Centre for Vehicle Design and support from

Bombardier Transportation. The authors thank Juan Pablo Parra Martinez for the use of his code for the

transmission loss calculations.

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