Investigation of the effects of the microstructure on the sound
absorption performance of polymer foams using a
computational homogenization approach
Citation for published version (APA):Gao, K., van Dommelen, J. A. W., & Geers, M. G. D. (2017). Investigation of the effects of the microstructure on the sound absorption performance of polymer foams using a computational homogenization approach.
European Journal of Mechanics. A, Solids, 61, 330-344. https://doi.org/10.1016/j.euromechsol.2016.10.011
DOI:
10.1016/j.euromechsol.2016.10.011
Document status and date: Published: 01/01/2017 Document Version:
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Investigation of the e
ffects of the microstructure on the
sound absorption performance of polymer foams using a
computational homogenization approach
K. Gaoa, J.A.W. van Dommelena,∗, M.G.D. Geersa
aDepartment of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The
Netherlands
Abstract
In this paper, a computational homogenization approach is exploited to study the effects of the microstructure of polymer foams on their acoustic properties. A Kelvin cell with partially-open thin membranes is adopted to represent the mi-crostructure of the foam. By applying the homogenization approach, the effective material parameters are obtained based on a microscopic representative volume element (RVE) subjected to different loading conditions. Geometrical proper-ties, including the opening and the thickness of the thin membranes and the cell size, are investigated. It is shown that when the opening of the membranes or the cell size are smaller, the sound absorption performance at low frequencies can be improved, at the expense of the mid-high frequency performance. Moreover, the optimal opening and optimal cell size for best sound absorption performance depend on the target frequency range. The effect of solid properties, including the stiffness and the loss factor, are also discussed. For low-stiffness materials, local resonance of the solid frame greatly affects the effective fluid density and
∗Corresponding author
the frame-borne wave, whereas global resonances can be utilized to improve the absorption performance in a specific frequency band.
Keywords: microstructure, Kelvin cell, homogenization, acoustic porous materials
1. Introduction
Acoustic porous materials, such as acoustic foams, are widely used as sound absorbers. Their sound absorption performance is strongly related to the underly-ing microstructure. For open-cell foams, an important factor of the performance is the cell size triggering the viscous effects. For example, by investigating 12 fully reticulated polyurethane (PU) foams, Cummings and Beadle discussed the rela-tionship between the flow resistivity and the cell size [1]. Han et al. tested several open-cell Al foams under different conditions and found that the foam samples with the smallest pore size of 0.5 mm exhibit the best absorption capacities when there is no air gap behind the sample [2]. For partially-open and fully-closed foams, thin membranes or thick walls exist in the faces of cells [3]. In this case, the opening of the membrane or the wall is important for the absorption. Lu et al. discovered that the sound absorption performance of Al foams can be enhanced by drilling holes in the cell wall [4]. Moreover, based on the microstructure char-acterization of 15 PU foams, Doutres et al. discussed the effects of membranes in the cells and they pointed out that the presence of membranes enhances the viscous effects in the foams [5]. Furthermore, compared with fully-closed foams, Zhang et al. observed that the sound absorption performance of PU foams was improved for the low-mid frequency range by using an open cell structure [6].
depending on the fabrication processes, such as foaming with inert gases, cen-trifugal casting, powder metallurgy, 3D printing, etc. For example, Zhou et al. fabricated a polymer foam, in which the cell size gradually changed from 20 to 100 µm, via supercritical carbon dioxide foaming [7]. Ghaffari Mosanenzadeh et al. applied a combination of a particulate leaching technique and a compression molding method to closely control the cell structure of open-cell foams with a graded cell size changing from 200 µm to 600 µm [8]. It is also reported that ultrasound can be applied for a better control and monitoring of the production process [9, 10, 11, 12]. Moreover, it was shown that the opening ratio of the membranes is influenced by the concentration of the reactants [6].
To understand effects of the microstructure, many methods or models have been developed. When the solid deformation can be ignored, effective fluid mod-els and their derivatives can be applied [13, 14, 15, 16, 17]. Among them, the Johnson-Champoux-Allard-Pride-Larfarge (JCAPL) model [18, 19, 20, 21] is note-worthy, providing an adequate description of the effective density and the effective bulk modulus of the fluid. By using a numerical microstructure-based approach to obtain the non-acoustical parameters used in the JCAPL model, Perrot et al. discussed the effects of throat size, pore size and cross-section shape on the sound absorption performance of porous fibrous materials [22] and also evaluated the approach on foams with experimental measurements [23]. By following the same approach, Hoang and Perrot studied the effects of the thin membranes [24] and Zieli´nski discussed the influences of a microstructure with randomly-distributed spherical pores [25]. On the other hand, the non-acoustical parameters can also be obtained by using semi-phenomenological models. For instance, Doutres et al. proposed a 2-parameter model and a 3-parameter model for fully-open and
partially-open foams respectively [26] and used their models in a corresponding sensitivity study [27]. Moreover, Yang et al. developed an empirical model based on a simple cubic unit cell [28].
The solid deformation cannot be ignored in vibro-acoustic problems or when the material has a relatively low stiffness, such as for polymers. Considerable work has been done by focusing on mechanical properties of polymer foams [29]. By coupling the fluid and the solid, Biot’s poroelastic theory [30] is nowadays widely adopted in vibro-acoustic problems [31, 32, 33, 34, 35, 36]. The effective parameters required in Biot’s theory can be linked to the microstructure by using a direct volume average [37] and semi-phenomenological scaling laws [34, 38]. Nevertheless, the total microscopic mechanical energy is not consistent with the macroscopic mechanical energy in the direct volume average method [39]. The semi-phenomenological scaling laws can only be applied for a specific type of microstructure since they are usually derived for simplified geometries only [40]. To overcome these shortcomings, homogenization methods based on the scale separation principle can be applied without using semi-phenomenological or em-pirical relations. For example, the asymptotic homogenization method has been applied to porous materials [41, 42, 43, 44]. As an alternative, the authors pro-posed a computational homogenization approach for acoustic porous materials in earlier work [39, 45]. It straightforwardly assesses the macroscopic influence of the microstructure by considering the viscous-thermal gaseous fluid flowing in an elastic solid frame. By using Kelvin cells with thin membranes, a comparison with experimental measurements showed the versatility of this approach [46].
Particularly, polymer foams of a solid material with a low stiffness cannot be considered as rigid. The influence of the microstructure of these foams should be
investigated by considering both the flowing fluid and a deformable solid. The purpose of the present paper is to employ the recently presented computational homogenization approach to obtain insights in the role of various morphological characteristics and solid properties for sound absorption of polymer foams based on isotropic Kelvin cells. The influences of the geometrical properties and the solid properties are investigated by focusing on the effective material parameters involving both mechanical and acoustical properties of polymer foams. For the ge-ometrical properties, earlier investigations on these factors [22, 23, 24, 25, 26, 27] were based on the JCAPL model, in which the solid deformation is not considered. In this paper, the effects of the thin membrane and the cell size are analysed sys-tematically by applying the homogenization approach to the corresponding Kelvin cells. Thereafter, influences of solid properties, including the Young’s modulus and the loss factor are discussed and particular attention is paid to the case of a low-stiffness solid with strong fluid-solid coupling effects. Significant phenom-ena, such as absorption peaks triggered through global resonance and anomalous behaviour induced by local resonance, are observed in the investigation.
2. Homogenization approach
When an acoustic wave is propagating in a porous material, two coupled prob-lems at different scales can be considered: at the macroscopic scale, the porous material is replaced by an equivalent homogeneous medium, whereas at the mi-croscopic pore scale, the material is intrinsically inhomogeneous. In the homog-enization approach, the macroscopic characteristic length related to the external excitation is assumed to be much larger than the microscopic characteristic length, i.e. the two problems at the macro- and micro-scales can be separated. The two
problems are studied in the frequency domain and the time derivative is replaced by jω where j is the imaginary unit and ω is the angular frequency. At the macro-scopic scale, the solid displacement usM and the fluid pressure pMf are chosen as the macroscopic field variables. Momentum conservation of the solid and mass conservation of the fluid govern the macroscopic behaviour:
fMs = ∇M· σsM
T ,
and Mf − ∇M ·u f
M = 0 . (1)
Here, the operator ∇Mrepresents the spatial gradient at the macroscopic scale. In
the first equation, σsM is the macroscopic Cauchy stress of the solid and fMs is the inertial force exerted on the solid. In the second equation, Mf is the macroscopic volumetric change of the fluid and uMf is the fluid displacement.
In the microscopic representative volume element (RVE), the solid and the fluid are coupled through a continuous interface condition. In the solid, conserva-tion of linear momentum is applied for the mechanical problem and Fourier’s law is adopted for the thermal diffusion:
−ω2ρ0sums = ∇m·σsm ρs 0C s pjωθ s m= − ∇m· −ks∇mθms . (2)
Here, ρ0s is the static density of the solid; σms is the microscopic Cauchy stress
of the solid; Cs
p is the thermal capacity at constant pressure; θ s
m is the
tempera-ture difference of the solid; and ks is the thermal conductivity of the solid. The isotropic linear elastic constitutive law is applied, and thermal expansion effects are ignored: σs m = K s− 2 3G s ! tr(εms)I+ 2Gsεms , (3) with Ks and Gs the bulk and shear moduli of the solid. The infinitesimal linear strain of the solid is given by εs
m= 1 2 h ∇mums + (∇mums)T i
the linearized Navier-Stokes-Fourier equations: −ω2ρf 0u f m= − ∇mpmf + ∇m· " jωµf ∇mumf + (∇mumf) T− 2 3(∇m·u f m)I !# ρf 0C f pjωθ f m= jωp f m− ∇m· −kf∇mθmf pmf P0 = θ f m T0 − ∇m·u f m, (4)
where ρ0f is the static density of the fluid; umf is the microscopic fluid displacement;
pmf is the difference of the fluid pressure; µf is the viscosity of the fluid; C f p is the
thermal capacity of the fluid at constant pressure; θmf is the temperature difference
of the fluid; kf is the thermal conductivity of the fluid; and P0 = 0.101 MPa and
T0 = 293 K are the ambient pressure and temperature.
A periodic boundary condition is adopted on the solid external surface based on the macroscopic solid deformation:
ums+−ums− = (∇MusM) T· xs+ m − x s− m , (5) where+and−denote opposite boundary points. For the fluid, the pressure on the boundary is assumed to depend on the variation of the macroscopic fluid pressure and its gradient
pmf = p f M+ ∇Mp f M· x f m (6)
whereby the viscous boundary traction is ignored. Furthermore, a periodic bound-ary condition for the thermal flux is applied, so that the total thermal flux leaving the RVE is zero.
The microscopic RVE problem is solved using the finite element method. Based on the microscopic response, the condition of energy consistency is en-forced to retrieve the work-conjugate macroscopic solid stress and macroscopic
fluid displacement: (1 − φ) fMs = 1 V Z Ses (n · σms)dA , (1 − φ)σ s M = 1 V Z Ses (n · σsm)xmdA φf M = 1 V Z Sef n · umfdA , φuMf = 1 V Z Sef (n · umf)xmfdA , (7) where Sesand S f
e are the external solid surface and the external fluid surface,
re-spectively. Since the microscopic RVE problem is linear, only effective material parameters are required and the macroscopic balance equations, i.e. Eq. (1), can be rewritten as: (1 − φ) fMs = −ω2ρs0ηs·usM+ ω2ρc·uMf −usM −φ∇Mp f M = −ω 2ρc· uMf −usM−ω2ρ0fηf ·uMf , (8)
with a linear stress-strain relation:
(1 − φ)σsM =4D : εsM+ QMf −φpf M = Q : ∇Mu s M+ R f M . (9)
Here, the effective parameters ηs, ηf, ρc, 4D, Q and R are obtained numerically
from the microscale response.
3. Effects of microscopic geometry
In this section, the effects of geometrical characteristics of a foam are investi-gated by studying an isotropic Kelvin cell with partially-open faces. As illustrated in Fig. 1, the skeleton of the cell is a 14-sided truncated octahedron and the length of a strut can be calculated as L =hV/(8
√
2)i1/3, with V the cell size. The faces of the cell are covered by thin membranes with a circular opening in the centre. The opening ratio of a face r is defined as the area of the opening relative to the area of the face. Here, the opening ratios of all faces in the Kelvin cell are identical.
(a) Solid domain (b) Fluid domain
Figure 1: Illustration of an isotropic Kelvin cell with membranes (discretized, r= 0.2). In the solid domain, the blue regions represent the skeleton and the grey regions reveal the thin membranes with circular holes. In the fluid domain, the blue regions are the interfaces between the fluid and the skeleton and the grey regions show the fluid.
The boundary value problem of the microscopic RVE is solved by COMSOL 4.3b with quadratic elements. The weak form PDE module is used to simulate the fluid part and tetrahedron elements are used. Tetrahedron solid elements are used for the skeleton and triangular shell elements are adopted in the region of the thin membranes. Continuity of the displacements is used to couple the shell elements, the fluid elements and the solid elements. Furthermore, the rotation between the shell elements and the solid elements is suppressed. The frequencies considered range from 100 Hz to 5000 Hz with a step size of 0.02 on a logarithmic scale. In the discussion, the low frequencies refer to 100-300 Hz and the mid frequencies range from 300 Hz to 1000 Hz. The 1000-5000 Hz range is the high-frequency
regime.
In the microscopic RVE, air properties are used for the fluid. The static solid density ρ0sis taken as 1000 kg m−3and Esis 100 MPa. The Poisson’s ratio ν is 0.4 and the thermal properties Cs
p and k
s are 1800 J kg−1
K−1 and 0.022 W m−1K−1. The effective material parameters of the foams, which are represented by the Kelvin cells with various geometrical parameters, are numerically calculated us-ing the computational homogenization approach. Due to the inertial effects, the stiffness tensor4D is asymmetric and the asymmetry becomes more pronounced
for higher frequencies. However, for all microstructures studied in this paper, the relative differences of the components Di jkland Dkli j at 100 Hz are of the order of
10−6and of 10−3at 5000 Hz. Therefore, the asymmetry of4D is further ignored. Considering the cubic symmetry of the Kelvin cell, five loading conditions (listed in Table 1) are used to extract the effective material parameters.
Set usM [m] ∇Mp f M [Pa/m] ∇MusM[-] p f M [Pa] 1 10−3e1 0 0 0 2 0 103e 1 0 0 3 0 0 0 1 4 0 0 10−3e 1e1 0 5 0 0 10−3e1e3 0
Table 1: Loading conditions used to calculate the parameters. The vectors eiindicate the principal
directions of the Cartesian coordinate system.
Besides the elastic and acoustic properties, foams with various microstructures are evaluated by macroscopic sound absorption simulations as shown in Fig. 2, using Eq. (8) and (9). In these simulations, the foam is surrounded by a rigid wall
and backed by an impermeable wall. The boundary condition of a rigid wall is expressed as
usM ·n= 0 and uMf ·n= 0 , (10)
and the boundary condition of an impermeable wall reads
uMf ·n= 0 . (11)
Moreover, only a quarter of the cylinder is modeled by considering the symmetry.
x 5 cm 5 cm foam impermeable wall rigid wall rigid wall
Figure 2: Macroscopic sound absorption simulation of a cylinder of foam.
3.1. Opening of membranes
Yasunaga et al. classified the membranes in partially-open foams into four types: fully-open, partially-open, closed and pin holes [3]. In this paper, only the fully-open and partially-open faces are discussed since the single Kelvin cell model combined with the homogenization approach is not applicable to foams with many fully-closed faces [46]. Effects of the opening of membranes are dis-cussed by comparing the homogenization results of 6 Kelvin cells with various
opening ratios for a fixed cell size of 0.05 mm3. The corresponding effective pa-rameters in the cubic-symmetric Kelvin cell have the following symmetry proper-ties: ηs i j = η sδ i j, η f i j = η fδ i j , ρci j = ρ cδ i j , Qi j = Qδi j , D1111= D2222 = D3333, D1212= D1313 = D2323 , D1122= D1133 = D2233= D2211 = D3311 = D3322. (12)
Here, δi jis the Kronecker delta. Other components of4D are all zero. In this case,
the effective Young’s modulus Eeff is defined as Eeff =
D21111+ D1111D1122− 2D21122
D1111+ D1122
. (13)
Because the solid is much stiffer than the air, the inertial effects and the fluid-solid coupling effects have negligible influences on the effective elastic properties in the studied frequency range. The real parts of Eeffof the different foams at 100 Hz are shown with respect to the opening ratio in Fig. 3, demonstrating that the effective Young’s modulus of the foam increases with a decreasing opening ratio.
The effective density and the effective bulk modulus of the fluid are depicted in Fig. 4. The effective fluid density ρefff is calculated by
ρf eff = η fρf 0 + ρ c . (14) In Fig. 4a, a smaller opening ratio, i.e. a smaller hole in the membrane, increases both the real part and the imaginary part over the analysed frequency range. The real part of the effective fluid density stands for the inertial coupling effects and the imaginary part represents the viscous coupling effects [13]. As indicated in Fig. 4a, the inertial-viscous coupling effects are both enhanced by closing the membrane. On the other hand, R is related to the effective fluid bulk modulus and
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 1 2 3 4 5 0.01 0.1 0.3 R e ( E e f f / E s ) [ 1 0 -3 ] Opening Ratio [-]
Figure 3: Real parts of Eeff(normalized by Es) of the foams with various opening ratios at 100 Hz.
The Kelvin cells with opening ratios of 0.01, 0.1 and 0.3 are shown from left to right.
the imaginary part stands for the thermal dissipation. When the membrane open-ing becomes smaller, both the real and imaginary part of R decrease in the 100-1000 Hz range. When the frequency is around 100-1000-1500 Hz, the imaginary part reaches its maximum. This maximum decreases and occurs at a higher frequency for a smaller opening ratio. The trends agree with Hoang and Perrot’s work [24], which shows that for larger membrane closures, the viscous characteristic length is decreasing (i.e. the viscous effects are increasing), but the tortuosity and the static thermal permeability are increasing (i.e. the inertial effects are increasing and the thermal dissipation is less). For higher frequencies, a smaller opening ra-tio increases the thermal dissipara-tion because the thermal boundary layer is thicker.
Complex wave numbers k1and k2of the two compression waves are displayed
in Fig. 5. The wave numbers are calculated as shown in Appendix A. Here, k1
2 3 4 0.01 0.3 0.3 R e ( f e f f ) [ kg/ m 3 ] r = 0.01 r = 0.15 r = 0.05 r = 0.2 r = 0.1 r = 0.25 r = 0.3 0.01 100 1000 5000 1 10 -I m ( f e f f ) [ kg/ m 3 ] Frequency [Hz] (a) ρefff 1.0 1.1 1.2 1.3 0.3 0.01 R e ( R / P 0 ) [ -] r = 0.3 r = 0.1 r = 0.25 r = 0.05 r = 0.2 r = 0.01 r = 0.15 100 1000 5000 0.00 0.03 0.06 0.09 0.12 0.3 0.01 I m ( R / P 0 ) [ -] Frequency [Hz] (b) R
Figure 4: Effective densities and effective bulk modulus of the fluid in foams with different mem-brane opening ratios, indicated with different colours.
frame-borne wave. The real parts Re(ki) are normalized by the corresponding
wave number in the air (= ω/cair with cair = 343 m/s). As illustrated in Fig. 5a,
Re(k1)/(ω/cair) becomes larger when the opening ratio is reduced. The reciprocal
of Re(k1)/(ω/cair) represents the normalized phase velocity of the airborne wave
and it shows that a smaller opening ratio reduces the phase velocity of the air-borne wave. Decreasing the opening ratio also leads to a higher attenuation of the airborne wave (i.e. Im(k1)). On the other hand, a smaller opening ratio speeds up
the frame-borne wave, with a smaller attenuation for low-mid frequencies. When the opening ratio becomes smaller, the airborne wave is more likely to propagate in the air (as also indicated in Fig. A.21a in Appendix A), entailing a lower phase velocity and a higher attenuation. As the frame-borne wave propagates in both the solid frame and the air, the decreasing of the opening ratio, which makes the foam stiffer, leads to a faster phase velocity and a lower attenuation.
1 2 3 4 5 0.3 0.3 R e ( k 1 ) / ( / c a i r ) [ -] r = 0.01 r = 0.2 r = 0.05 r = 0.25 r = 0.1 r = 0.3 r = 0.15 100 1000 5000 5 10 20 30 40 50 0.01 0.01 -I m ( k 1 ) [ m -1 ] Frequency [Hz]
(a) Airborne wave
1.2 1.4 1.6 0.3 R e ( k 2 ) / ( / c a i r ) [ -] r = 0.01 r = 0.2 r = 0.05 r = 0.25 r = 0.1 r = 0.3 r = 0.15 100 1000 5000 10 -2 10 -1 10 0 10 1 0.3 0.01 0.01 -I m ( k 2 ) [ m -1 ] Frequency [Hz] (b) Frame-borne wave
Figure 5: Complex wave numbers of the two compression waves in the analysed foams with different membrane opening ratios.
frequency is smaller than ±800 Hz, the foam with a smaller opening ratio has a higher sound absorption coefficient. Because the outer dimensions of the foams and the macroscopic boundary conditions are the same, the first absorption peak depends only on the openings of the membranes. The foam with the smallest opening ratio corresponds to the lowest peak (occurring at the lowest frequency). After the first peak, the foam with the weakest sound absorption performance is the one with the smallest opening ratio r = 0.01. Therefore, although a small opening ratio improves the sound absorption performance at low frequencies, it may reduce the performance for mid-high frequencies.
The average sound absorption coefficients in three frequency bands: 100-300 Hz, 100-300-1000 Hz and 1000-5000 Hz are shown in Fig. 7a. A smaller opening ratio is better at low-mid frequencies, whereas r = 0.25 has the best absorption performance in the high-frequency regime. The sound absorption coefficients at three target frequencies are shown in Fig. 7b. At a target frequency of 500 Hz, the foam with r = 0.01 has the largest sound absorption coefficient; at 1000 Hz, the
100 1000 5000 0.0 0.2 0.4 0.6 0.8 1.0 0.3 0.01 r = 0.01 r = 0.05 r = 0.1 r = 0.15 r = 0.2 r = 0.25 r = 0.3 S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Frequency [Hz]
Figure 6: Sound absorption coefficients of the analysed foams with different opening ratios.
foam with r= 0.15 performs best and at 2000 Hz, r = 0.30 is optimal.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.2 0.4 0.6 0.8 1.0 100-300 Hz 300-1000 Hz 1000-5000 Hz A v e r a g e S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Opening Ratio [-]
(a) In frequency bands
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.2 0.4 0.6 0.8 1.0 500 Hz 1000 Hz 2000 Hz S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Opening Ratio [-] (b) At target frequencies
Figure 7: Sound absorption coefficients of the analysed foams with different opening ratios in different frequency bands. The maximum values are highlighted with circles.
3.2. Thickness of membranes
As shown in Fig. 8, the membrane thickness has a significant influence on the effective elastic stiffness, which is in agreement with the work of Hoang et al. [47]. Here, the influence of the thickness of the membranes is further
stud-ied for thicknesses of the order of 1 µm, which is representative for flexible PU foams [3, 48]. Three different thicknesses of 0.5 µm, 1 µm and 2 µm are explored. The corresponding effective fluid properties and the complex wave numbers of the airborne waves are shown in Fig. 9. The results are close to each other and exhibit small differences at high frequencies. Furthermore, as shown in Fig. 9d, when the frequency is smaller than 1000 Hz, the sound absorption coefficients are nearly identical and differences only appear at higher frequencies due to the global resonance behaviour. Therefore, for very thin membranes, the thickness has an ignorable influence on the absorption performance when resonance does not occur. 0 1 2 3 4 5 0 2 4 6 8 R e ( E e f f / E s ) [ 1 0 -3 ] Membrane Thickness [ m]
Figure 8: Effective elastic stiffness of the foams for different membrane thicknesses.
3.3. Cell size
The effects of the cell size are investigated through Kelvin cells with two dif-ferent geometries: one with an opening ratio of 0.2 (referred to as the membrane cell) and another without membranes (referred to as the skeleton cell). The strut cross-sectional area A is adjusted to preserve a constant value of LA/V (the thick-ness of the thin membrane remains unchanged). Resulting variations in porosities
1.5 2.0 2.5 3.0 R e ( f e f f ) [ kg/ m 3 ] h = 0.5 m h = 1.0 m h = 2.0 m 100 1000 5000 1 10 -I m ( f e f f ) [ kg/ m 3 ] Frequency [Hz] (a) ρefff 1.0 1.1 1.2 1.3 R e ( R / P 0 ) [ -] h = 0.5 m h = 1.0 m h = 2.0 m 100 1000 5000 0.00 0.03 0.06 0.09 0.12 I m ( R / P 0 ) [ -] Frequency [Hz] (b) R 2 3 4 R e ( k 1 ) / ( / c a i r ) [ -] h = 0.5 m h = 1.0 m h = 2.0 m 100 1000 5000 6 8 12 16 24 32 -I m ( k 1 ) [ m -1 ] Frequency [Hz] (c) k1 100 1000 5000 0.0 0.2 0.4 0.6 0.8 1.0 S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Frequency [Hz] h = 0.5 m h = 1.0 m h = 2.0 m
(d) Sound absorption coefficients Figure 9: Effective fluid density, effective fluid bulk modulus, wave numbers of the airborne waves and sound absorption coefficients of the foams for different membrane thicknesses.
are smaller than 0.0001 for the skeleton cells and 0.001 for the membrane cells. The real parts of the effective Young’s modulus at 100 Hz are displayed in Fig. 10 with respect to the cell size. When the cell size is reduced, the stiffness of the foam with membranes increases. This effect is due to the relative contribution of the membranes. The increase of the stiffness of the skeleton cell is much smaller in Fig. 10 because of the constant porosity.
0.001 0.010 0.100 0 2 4 6 8 Skeleton r = 0.2 R e ( E e f f / E s 0 ) [ 1 0 -3 ] Cell Size [mm 3 ]
Figure 10: Effective Young’s modulus of the foams with different cell sizes at 100 Hz. Fig. 11. Both the real part and the imaginary part of the effective fluid density increase for a smaller cell size, while the change of the imaginary part is more significant. In contrast, decreasing the cell size results in a shift of the curve of R towards higher frequencies. Therefore, reducing the cell size can significantly amplify the viscous dissipation, while the thermal dissipation is reduced at low frequencies and enhanced at high frequencies, which is in agreement with Lu et al.’s research on Al foam [4].
The wave numbers of the airborne wave in the foams with various cell size are shown in Fig. 12a and Fig. 12b. Decreasing the cell size results in a smaller phase velocity and a higher attenuation of the airborne wave. For the frame-borne wave, in Fig. 12c, the attenuation becomes insensitive to the frequency when the frequency is sufficiently high. At low frequencies, both Re(k2) and the
attenuation decrease for a smaller cell size. However, a smaller cell size results in a higher attenuation at high frequencies. On the other hand, in the membrane cells, Re(k2) decreases with the cell size (except for the case of 0.075 mm3 at
1.2 1.6 2.0 2.4 0.125 0.001 0.001 0.001 mm 3 0.05 mm 3 0.0025 mm 3 0.075 mm 3 0.01 mm 3 0.1 mm 3 0.025 mm 3 0.125 mm 3 R e ( f e f f ) [ kg/ m 3 ] 100 1000 5000 0.1 1 10 100 0.125 -I m ( f e f f ) [ kg/ m 3 ] Frequency [Hz]
(a) Skeleton cells, ρefff
1.2 1.6 2.0 2.4 R e ( f e f f ) [ kg/ m 3 ] 0.001 mm 3 0.05 mm 3 0.0025 mm 3 0.075 mm 3 0.01 mm 3 0.1 mm 3 0.025 mm 3 0.125 mm 3 100 1000 5000 0.1 1 10 100 0.125 0.125 0.001 0.001 -I m ( f e f f ) [ kg/ m 3 ] Frequency [Hz] (b) Membrane cells, ρefff 0.9 1.0 1.1 1.2 1.3 1.4 0.125 R e ( R / P 0 ) [ -] 0.001 mm 3 0.05 mm 3 0.1 mm 3 0.0025 mm 3 0.075 mm 3 0.125 mm 3 0.01 mm 3 0.025 mm 3 100 1000 5000 0.00 0.03 0.06 0.09 0.12 0.15 0.001 0.125 0.001 I m ( R / P 0 ) [ -] Frequency [Hz] (c) Skeleton cells, R 0.9 1.0 1.1 1.2 1.3 1.4 0.125 R e ( R / P 0 ) [ -] 0.001 mm 3 0.05 mm 3 0.0025 mm 3 0.075 mm 3 0.01 mm 3 0.1 mm 3 0.025 mm 3 0.125 mm 3 100 1000 5000 0.00 0.03 0.06 0.09 0.12 0.15 0.001 0.125 I m ( R / P 0 ) [ -] Frequency [Hz] 0.001 (d) Membrane cells, R
Figure 11: Effective density and effective bulk modulus of the fluid in the foams with different cell sizes.
attenuation is enhanced for a larger cell size because it is determined by the elastic properties. For cells smaller than 0.025 mm3, the attenuation is controlled by the inertial-viscous coupling effects between the fluid and the solid, which becomes higher for a smaller cell size.
The corresponding sound absorption coefficients are shown in Fig. 13. For the skeleton cells, when the cell size is larger than 0.01 mm3, decreasing the cell size improves the absorption performance from around 300 Hz to 5000 Hz and
4 8 12 R e ( k 1 ) / ( / c a i r ) [ -] 0.001 mm 3 0.05 mm 3 0.0025 mm 3 0.075 mm 3 0.01 mm 3 0.1 mm 3 0.025 mm 3 0.125 mm 3 100 1000 5000 1 10 100 0.125 0.125 0.001 0.001 -I m ( k 1 ) [ m -1 ] Frequency [Hz]
(a) Airborne wave, skeleton cells
4 8 12 R e ( k 1 ) / ( / c a i r ) [ -] 0.001 mm 3 0.05 mm 3 0.0025 mm 3 0.075 mm 3 0.01 mm 3 0.1 mm 3 0.025 mm 3 0.125 mm 3 100 1000 5000 1 10 100 0.125 0.125 0.001 0.001 -I m ( k 1 ) [ m -1 ] Frequency [Hz]
(b) Airborne wave, membrane cells
1.0 1.2 1.4 1.6 1.8 0.001 mm 3 0.05 mm 3 0.0025 mm 3 0.075 mm 3 0.01 mm 3 0.1 mm 3 0.025 mm 3 0.125 mm 3 R e ( k 2 ) / ( / c a i r ) [ -] 100 1000 5000 10 -3 10 -2 10 -1 10 0 10 1 0.125 0.125 0.001 0.001 -I m ( k 2 ) [ m -1 ] Frequency [Hz]
(c) Frame-borne wave, skeleton cells
1.0 1.2 1.4 1.6 1.8 0.125 R e ( k 2 ) / ( / c a i r ) [ -] 0.001 mm 3 0.05 mm 3 0.0025 mm 3 0.075 mm 3 0.01 mm 3 0.1 mm 3 0.025 mm 3 0.125 mm 3 100 1000 5000 10 -3 10 -2 10 -1 10 0 10 1 0.125 0.001 0.001 -I m ( k 2 ) [ m -1 ] Frequency [Hz]
(d) Frame-borne wave, membrane cells Figure 12: Complex wave number of the two compression waves in the foams with different cell sizes.
the absorption coefficients in the 100-300 Hz regime are decreased slightly (note that this is not well visible in the figure because the curves are close on a scale from 0 to 1). When the cell size is smaller than 0.01 mm3, decreasing the cell size improves the low-frequency absorption performance but reduces the performance at mid-high frequencies. This effect is more significant in the membrane cells. A similar phenomenon was also observed in Al foams [49].
100 1000 5000 0.0 0.2 0.4 0.6 0.8 1.0 0.125 S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Frequency [Hz] 0.001 mm 3 0.0025 mm 3 0.01 mm 3 0.025 mm 3 0.05 mm 3 0.075 mm 3 0.1 mm 3 0.125 mm 3 0.001
(a) Skeleton cells
100 1000 5000 0.0 0.2 0.4 0.6 0.8 1.0 0.125 0.001 0.001 mm 3 0.0025 mm 3 0.01 mm 3 0.025 mm 3 0.05 mm 3 0.075 mm 3 0.1 mm 3 0.125 mm 3 S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Frequency [Hz] (b) Membrane cells Figure 13: Sound absorption coefficients of the foams with different cell sizes.
shown in Fig. 14a and those at specific frequencies are presented in Fig. 14b. For the skeleton cells, 0.001 mm3 behaves best in the low-mid frequency regime. In the 1000-5000 Hz regime, 0.01 mm3has the maximum average absorption
coef-ficient. For the membrane cells, 0.001 mm3 and 0.01 mm3 are the best size for a foam in the 100-300 Hz and 300-1000 Hz bands, respectively. For 1000-5000 Hz, cells larger than 0.025 mm3 all show good performances. The optimal cell sizes for the specific target frequencies are the medium cell sizes (0.01-0.025 mm3)
except for the cases of skeleton cells at 500 Hz (0.001 mm3is the best) and mem-brane cells at 1000 Hz (0.05 mm3is the best). It suggests that a medium cell size
ranging from 0.01 mm3 to 0.025 mm3 may be an adequate choice to maximally absorb the mid-high frequency noises. Reducing the cell size can increase the flow resistivity and enhance the sound absorption performance. However, when the cell size becomes too small, the high flow resistivity actually deteriorates the absorption ability because it obstructs the fluid flow through the material.
0.001 0.010 0.100 0.0 0.2 0.4 0.6 0.8 1.0 Skeleton: 100-300 Hz 300-1000 Hz 1000-5000 Hz Membrane: 100-300 Hz 300-1000 Hz 1000-5000 Hz A v e r a g e S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Cell Size [mm 3 ]
(a) In frequency bands
0.001 0.010 0.100 0.0 0.2 0.4 0.6 0.8 1.0 S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Cell Size [mm 3 ] Skeleton: 500 Hz 1000 Hz 2000 Hz Membrane: 500 Hz 1000 Hz 2000 Hz (b) At target frequencies
Figure 14: Sound absorption coefficients of the foams with different cell sizes in different fre-quency bands. The maximum values are indicated by circles.
4. Influence of solid properties
In this section, the influence of the solid properties is discussed. First, the Young’s modulus is real and is varied from 1 MPa to 1000 MPa. Thereafter, the influence of the viscoelasticity of the solid is studied by using a complex Young’s modulus.
4.1. Stiffness
Four stiffnesses are chosen for the study and the normalized effective stiffness is shown in Fig. 15. It shows that the normalized Eeff at both 100 Hz and 5000 Hz
are almost constant for an Es of 100 MPa and 1000 MPa. When Esis reduced to
10 MPa and 1 MPa, the normalized Eeff at 100 Hz increases slightly, whereas it
decreases significantly at 5000 Hz (relative to the high-stiffness response). This suggests that the fluid-solid coupling effects and the inertial effects become im-portant in cases of a low stiffness solid.
1 10 100 1000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 100 Hz 5000 Hz R E ( E e f f / E s ) [ 1 0 -3 ] E s [MPa]
Figure 15: The normalized effective Young’s modulus of the foam at 100 Hz and 5000 Hz for different solid stiffnesses.
As shown in Fig. 16, both ρefff and R for the foams with different solid stiff-nesses are nearly identical at low-mid frequencies. Attention is focused on the result of the lowest stiffness of 1 MPa. When the frequency becomes higher, there are peaks around 2700 Hz for both the real part and the imaginary part of the e ffec-tive fluid density. This is due to local resonance occurring in the microscopic RVE around this frequency. The second resonance frequency occurs around 4000 Hz with smaller peaks in Fig. 16a. Note that some details of the resonance behaviour may be missed due to the step size in the simulations. Moreover, when the fre-quency is higher than about 3000 Hz, because the stiffness of the solid (1 MPa) is quite low, the inertial coupling effects are more significant than the viscous dissipation, resulting in positive imaginary parts in this regime. Besides, the dif-ferences of the foam with Es = 10 MPa at 5000 Hz reveal a shift of the local resonance to a higher frequency.
The complex wave numbers are shown in Fig. 17. The airborne wave of 100 MPa and 1000 MPa are quite close. The phase velocity of the airborne wave
-4 -2 0 2 1 MPa 10 MPa 100 MPa 1000 MPa R e ( f e f f ) [ kg/ m 3 ] 100 1000 5000 -5 0 5 10 15 20 -I m ( f e f f ) [ kg/ m 3 ] Frequency [Hz] (a) ρefff 0.9 1.0 1.1 1.2 R e ( R / P 0 ) [ -] 1 MPa 10 MPa 100 MPa 1000 MPa 100 1000 5000 0.00 0.04 0.08 0.12 I m ( R / P 0 ) [ -] Frequency [Hz] (b) R
Figure 16: Effective density and effective bulk modulus of the fluid in the foams with different solid stiffnesses.
in the 10 MPa foam clearly differs from the other three foams at low frequencies. The attenuations in the 10 MPa and 1 MPa foams are generally less pronounced than in the high-stiffness foams. These differences are due to a stronger fluid-solid coupling effect in the low-stiffness foams. By checking the fluid-solid ratio of the eigen-velocity, Fig. 17c shows that the airborne wave before 200 Hz be-comes the frame-borne wave after 200 Hz, leading to discontinuities of the wave number. For the frame-borne wave shown in Fig. 17b, at low-mid frequencies, when the stiffness is increasing, the phase velocity is increasing and the attenua-tion tends to zero. At high frequencies, the foam with a solid stiffness of 1 MPa shows high attenuations and significant oscillations in the phase velocity, where a negative phase velocity can occur (not shown in the figure). This is due to the local resonance of the thin membrane as observed in Fig. 18, where the amplitude of the solid displacement in the thin membrane becomes much higher than the displacement in the struts.
2 3 4 5 6 R e ( k 1 ) / ( / c a i r ) [ -] 1 MPa 10 MPa 100 MPa 1000 MPa 100 1000 5000 0 10 20 30 40 -I m ( k 1 ) [ m -1 ] Frequency [Hz]
(a) Airborne wave
10 -2 10 -1 10 0 10 1 10 2 R e ( k 2 ) / ( / c a i r ) [ -] 1 MPa 10 MPa 100 MPa 1000 MPa 100 1000 5000 10 -2 10 0 10 2 10 4 -I m ( k 2 ) [ m -1 ] Frequency [Hz] (b) Frame-borne wave 100 1000 5000 0.1 1 10 100 | air | | frame | 1 MPa 10 MPa 100 MPa 1000 MPa A b s ( ) [ -] Frequency [Hz]
(c) Ratio of the eigen-velocity
Figure 17: Complex wave numbers and ratio of the eigen-velocity in the fluid over the one in the solid for the foams with different solid stiffnesses.
stiffness of 100 MPa and 1000 MPa are nearly overlapping at low-mid frequen-cies and their differences at high frequencies result from the global resonance of the foams. Except for the global resonance around 400 Hz, the response of the foam with Es = 10 MPa agrees qualitatively with the one for a solid stiffness of
1000 MPa with only minor quantitative differences. The global resonance of the foam with 1 MPa at low frequencies significantly increases the sound absorption coefficient. Moreover, the foam with a solid stiffness of 1 MPa has a lower and
Figure 18: The amplitude of the microscopic solid displacement (mm) at ∼2630 Hz (close to the first resonance frequency) when the boundary condition Set 4 in Table 1 is adopted.
flatter absorption peak around 1000 Hz.
100 1000 5000 0.0 0.2 0.4 0.6 0.8 1.0 1 MPa 10 MPa 100 MPa 1000 MPa S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Frequency [Hz]
4.2. Viscoelastic solid properties
The influence of viscoelasticity of the solid is studied by using a complex solid stiffness Es. Four constant loss factors ηs = Im(Es)/Re(Es) are considered here.
Because the fluid-solid coupling effects are not significant at high stiffnesses, only the case of a solid stiffness of Re(Es)= 1 MPa is analysed. Fig. 20a shows that the
amplitude of the oscillation of ρefffat high frequencies is reduced by increasing the loss factor. The airborne wave hardly changes when the loss factor is increased. For the frame-borne wave, the oscillations of the phase velocity and the attenua-tion are smoother. The sound absorpattenua-tion coefficients in Fig. 20e reveals that the loss factor only reduces the oscillation amplitude.
5. Conclusions
In this paper, a computational homogenization approach has been applied to study the multiscale effects of the microstructure of porous materials. The theory was presented before [39, 45], but there the application was limited to very simple and unrealistic structures. Here the new homogenization method is employed for the first time to obtain insight in the role of various morphological characteristics for sound absorption based on a more realistic geometry. The microstructure of the material was described by a Kelvin cell with partially-open faces. By applying the homogenization approach, the effective material parameters were obtained. The influence of the geometrical properties and the solid properties, including the opening and the thickness of the membrane in the face, the cell size, the stiffness and the loss factor of the solid were analysed. The effective fluid properties, the complex wave numbers of the two compression waves and the sound absorption coefficients were compared.
When the opening of the membrane or the cell size is smaller (all other geo-metrical properties unchanged), the inertial-viscous coupling effects are both en-hanced (decreasing the cell size mainly strengthens the viscous effects) and the thermal dissipation is reduced at low frequencies. The propagation of the airborne wave becomes slower and the attenuation is higher. Decreasing the cell size or the opening of the membrane has opposite effects on the frame-borne wave. The sound absorption performance at low frequencies may be improved by amplifying the viscous effects. However, this may negatively affect the performance at mid-high frequencies since it may prohibit the fluid flow through the material. More-over, an optimal opening of the membrane or an optimal cell size is required to optimize the sound absorption performance in a specific frequency range. When the membrane thickness is of the order of 1 µm, the variation of the thickness has negligible effects on the effective fluid properties and the sound absorption performance.
On the other hand, it is found that the solid stiffness has a limited influence on the sound absorption performance at least, if the stiffness is sufficiently high. In low-stiffness cases, the fluid-solid coupling effects become significant. Notable difference between the displacements in the thin membranes and the ones in the solid struts may be observed at some frequencies. This local resonance can lead to anomalous behaviour, such as a negative real part of the effective fluid density and a negative phase velocity of the frame-borne wave. Furthermore, the global resonances in the low-stiffness cases can be exploited to improve the absorption performance in a specific frequency band. The loss factor is studied by using a complex Young’s modulus. As expected, a higher loss factor can be used to remove sharp oscillations.
In practice, increasing the thickness of the sample is usually a simple and di-rect way to improve the sound absorption performance for foams with low- and mid-resistivity. However, in most cases, the maximum sample thickness is lim-ited due to volumetric constraints. Therefore, finding an optimal microstructure is essential to maximize the sound absorption performance in a frequency band or at a specific target frequency. It has been shown that the opening of the membrane and the cell size both need to be fine-tuned to achieve the best performance. Fur-thermore, exploiting resonance effects for low-stiffness foams can be helpful for this purpose.
Appendix A. Two compression waves in a transversely-isotropic porous medium In this section, the two compression waves in a porous material are discussed based on Eq. (8) and (9). Since only the macroscopic scale is involved in this section, the subscript M is omitted and the Einstein notation is used for
simplic-ity. By substituting Eq. (9) into Eq. (8), the macroscopic governing equations are obtained: −ω2(ρs 0η s i j+ ρ c i j)u s j+ ω 2ρc i ju f j = 1 2Di jkl ∂2us l ∂xj∂xk + ∂2u s k ∂xj∂xl ! + Qi j ∂2uf k ∂xj∂xk ω2ρc i ju s j−ω 2 (ρ0fηi jf + ρci j)u f j = Qkl ∂2us k ∂xi∂xl + R ∂ 2uf k ∂xi∂xk . (A.1)
By following Biot’s poroelastic theory [50, 51, 52], for studying the compression waves, two scalar potentials ϕsand ϕf are introduced for the solid frame and the air with uis= ∂ϕ s ∂xi and uif = ∂ϕ f ∂xi . (A.2)
By substituting Eq. (A.2) into Eq. (A.1) and assuming that the third derivatives of ϕsand ϕf are still differentiable, it can be shown that
−ω2(ρ0sηi js + ρ c i j)ϕ s+ ω2ρc i jϕ f = D i jkl ∂2ϕs ∂xl∂xk + Qi j ∂2ϕf ∂xk∂xk ω2ρc i jϕ s−ω2(ρf 0η f i j+ ρ c i j)ϕ f = Q jl ∂2ϕs ∂xl∂xi + R∂x∂2ϕf i∂xj . (A.3)
Considering the cubic symmetry properties in Eq. (12), and summing up the equations for i= j, Eq. (A.3) is rewritten as
−ω2(ρ0sηs+ ρc)ϕs+ ω2ρcϕf = (D1111+ 2D1122) ∂2ϕs ∂xk∂xk + Q∂x∂2ϕf k∂xk ω2ρcϕs−ω2(ρf 0η f + ρc )ϕf = Q ∂ 2ϕs ∂xk∂xk + R∂x∂2ϕf k∂xk . (A.4) By denoting ϕ ˜ = ϕs ϕf , ρ= ρs 0η s+ ρc −ρc −ρc ρf 0η f + ρc , and M= D1111+ 2D1122 Q Q R , (A.5) Eq. (A.4) is rewritten as
−ω2ρ ϕ
˜ = M∇
2ϕ
˜
. (A.6)
The above equations can be further rewritten as −ω2 M−1ρ ϕ ˜ = ∇ 2ϕ ˜ . (A.7)
Let k21and k22be the eigenvalues of the part ω2M−1ρ and the corresponding eigen-vectors are ϕ ˜1 and ϕ ˜2 . Then, −k2iϕ ˜i = ∇ 2ϕ ˜i for i= 1, 2 . (A.8)
The eigenvalues k1and k2 are the complex wave number of the two compression
the imaginary part represents the attenuation. Based on the wave speed, the two compression waves can be distinguished as a fast wave and a slow wave. For the corresponding eigenvectors ϕ ˜i = [ϕ s i, ϕ f i]
T, the following two ratios are defined
∆i = ϕf i ϕs i for i= 1, 2 , (A.9)
representing the ratio of the eigen-velocity of the fluid over that of the solid frame. They indicate the preferred medium for the two compression waves. Usually, the slow wave mainly propagates in the fluid and the fast wave propagates in both the air and the solid frame. When the fluid is air, the compression wave mostly propagating in the air is referred to as the airborne wave and the other compression wave is the frame-borne wave. Fig. A.21 demonstrates the ratios of both waves in the foams for different membranes and cell sizes.
Acknowledgement
This research was supported by the Dutch Technology Foundation STW, ap-plied science division of NWO, and the Technology Program of the Ministry of Economic Affairs (under grant number 10811).
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-4 -2 0 2 R e ( f e f f ) [ kg/ m 3 ] s = 0 s = 0.01 s = 0.04 s = 0.16 100 1000 5000 -5 0 5 10 15 20 -I m ( f e f f ) [ kg/ m 3 ] Frequency [Hz] (a) ρefff 0.9 1.0 1.1 1.2 s = 0 s = 0.01 s = 0.04 s = 0.16 R e ( R / P 0 ) [ -] 100 1000 5000 0.00 0.04 0.08 0.12 I m ( R / P 0 ) [ -] Frequency [Hz] (b) R 1 2 3 4 R e ( k 1 ) / ( / c a i r ) [ -] s = 0 s = 0.01 s = 0.04 s = 0.16 100 1000 5000 2 4 8 16 32 -I m ( k 1 ) [ m -1 ] Frequency [Hz] (c) Airborne wave 0.01 0.1 1 10 100 R e ( k 2 ) / ( / c a i r ) [ -] s = 0 s = 0.01 s = 0.04 s = 0.16 100 1000 5000 10 100 1000 -I m ( k 2 ) [ m -1 ] Frequency [Hz] (d) Frame-borne wave 100 1000 5000 0.0 0.2 0.4 0.6 0.8 1.0 S o u n d A b s o r p t i o n C o e f f i c i e n t [ -] Frequency [Hz] s = 0 s = 0.01 s = 0.04 s = 0.16
(e) Sound absorption coefficients
Figure 20: Effective fluid density, effective fluid bulk modulus, complex wave numbers and sound absorption coefficients in the foams with different solid loss factors ηs.
8 10 12 14 16 100 1000 5000 0.8 1.0 1.2 1.4 A bs ( a i r ) [ -] r = 0.01 r = 0.2 r = 0.05 r = 0.25 r = 0.1 r = 0.3 r = 0.15 A bs ( f r a m e ) [ -] Frequency [Hz]
(a) Opening ratios
6 9 12 15 100 1000 5000 0.9 1.0 1.1 1.2 1.3 A bs ( a i r ) [ -] h = 2 m h = 1 m h = 0.5 m A bs ( f r a m e ) [ -] Frequency [Hz] (b) Thicknesses 10 20 30 40 50 60 100 1000 5000 0.0 0.2 0.4 0.6 0.8 1.0 0.001 mm 3 0.05 mm 3 0.0025 mm 3 0.075 mm 3 0.01 mm 3 0.1 mm 3 0.025 mm 3 0.125 mm 3 A bs ( a i r ) [ -] A bs ( f r a m e ) [ -] Frequency [Hz]
(c) Cell size, skeleton cells
8 12 16 20 24 100 1000 5000 0.8 1.0 1.2 1.4 A bs ( a i r ) [ -] 0.001 mm 3 0.05 mm 3 0.0025 mm 3 0.075 mm 3 0.01 mm 3 0.1 mm 3 0.025 mm 3 0.125 mm 3 A bs ( f r a m e ) [ -] Frequency [Hz]
(d) Cell size, membrane cells Figure A.21:∆iof the foams with different geometrical properties.
