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This work deals with the study of a plane periodic multilayer structure in which the elementary stack consists of two plates in contact: one in aluminum (AL) and the other one in polyethylene (PE). These isotropic materials, present a high acoustic impedance contrast. The attenuation of the longitudinal and transverse waves is taken into account in the polyethylene but neglected in the aluminum plate. The effect of different defects is analyzed. Firstly, we focus on the effect of the presence of grease inclusion in the polyethylene plate (considering the two plates of the elementary stack in perfect contact). Secondly, the effect of disbond simulated by the insertion of a thin Teflon layer between the interfaces of the two layers constituting the elementary stack of the multilayer structure is investigated. Finally, the effect of the stacking sequences of the multilayer is analyzed. In order to obtain the effective acoustic parameters of polyethylene layer, allowing to evaluate the reflection and transmission coefficients using the stiffness matrix method developed by Rokhlin
*et*
*al*., four homogenization models are analyzed, then the best one to our configuration is chosen. The comparison of the simulation results is carried out.

Periodic multi-layered media are widely studied because of their very interesting properties such as exhibiting absolute or local band gaps in their dispersion curves. The width and position of these bands depend on the physical and geometric characteristics of the materials and their frequency. Indeed, by the modification of the physical or geometric properties of a material constituting one or more periods, the width and the position of the forbidden or passbands can be modified, and very fine bandwidths can appear in the prohibited bands. It is thus possible to perform more or less selective wave filtering depending on the intended application [

The present study is the contribution to the previous work on multilayer periodic structure in which defects were introduced [

The geometry of the problem (

Since the polyethylene layer was considered as containing grease inclusions after manufacturing, thus, it is suitable to determine its new effective properties using homogenization models as was done in [

is made. For a plate with inclusions, the estimation of effective properties can be done using the following relations [

{ E e f f = 9 K e f f G e f f 3 K e f f + G e f f ν e f f = 3 K e f f − 2 G e f f 6 K e f f + 2 G e f f (1)

were {K_{eff}, G_{eff}} are given by homogenization models in

The longitudinal and transverse effective velocities {c_{L,eff}, c_{T,eff}} which constitute the effective acoustic properties of a plate with inclusion, the effective density ρ_{eff} and the effective acoustic impedance Z_{L,eff} are respectively expressed by [

{ c L e f f = E e f f ⋅ ( 1 − ν e f f ) ρ e f f ⋅ ( 1 + ν e f f ) ( 1 − 2 ν e f f ) c T e f f = E e f f 2 ρ e f f ⋅ ( 1 + ν e f f ) (2)

and

{ ρ e f f = ρ + V f ( ρ i − ρ ) Z L , e f f = ρ e f f c L , e f f (3)

In a general way, depending on the homogenization model, the effective mechanical properties X_{eff} of a heterogeneous material can be written as follows:

X e f f = X + a + b V f ( X i − X ) 1 + m + x V f (4)

Model | K_{eff} | G_{eff} |
---|---|---|

Voigt | K + ( K g − K ) V f | G + ( G g − G ) V f |

G-sph | K + 4 G 3 K g + 4 G ( K g − K ) V f 1 − 3 K g − 3 K 3 K g + 4 G V f | G + 14 3 G g + ( G g − G ) V f 1 − 7 8 G g G + 7 8 ( 1 − G g G ) V f |

HS-iso | K + 4 G 3 K g + 4 G ( K g − K ) V f 1 − 3 K g − 3 K 3 K g + 4 G V f | G + 6 G g ( K + 2 G ) 9 K + 8 G + ( G g − G ) V f 1 + 6 G g ( K + 2 G ) G ( 9 K + 8 G ) + 6 ( K + 2 G ) 9 K + 8 G ( 1 − G g G ) V f |

HS-ort | K + G K g + G ( K g − K ) V f 1 − K g − K K g + G V f | G + K G ( G g − G ) K ( G g + G ) + 2 G G g V f 1 − ( K + 2 G ) ( G g − G ) K ( G g + G ) + 2 G G g V f |

ρ: Polyethylene’s density; K: Polyethylene bulk modulus; G: Polyethylene shear modulus; ρ_{g}: Grease’s density; K_{g}: Grease bulk modulus; G_{g}: Grease shear modulus; V_{f}: Volume fraction.

where {x, a, b, m} are either a constant or function depending on the homogenization model, associated to the elastic properties of the matrix X = {K, G, ρ} and V_{f} is the volume fraction of inclusions having the mechanical properties X_{i} = {K_{i}, G_{i}, ρ_{i}}.

Since the effective density ρ_{eff} is considered as that given in Equation (3), the effective properties {K_{eff}, G_{eff}} are summarized in

In this part, the equivalent modulus curves according to the volume fraction V_{f} are plotted for the four models. The normalized {K_{eff}, G_{eff}} coefficients, i.e. {K_{eff}/K, G_{eff}/G} are shown to decrease when V_{f} increases in

This reduction depends on the homogenization model. Moreover, we observed that the Hashin-Shtrikman model for orthotropic inclusions (HS-ort) and isotropic inclusions (HS-iso) gives values of effective mechanical properties lower than those obtained with other models. Consequently, the effective acoustic properties {c_{L,eff}, c_{T,eff}} required also will be lowest possible: what will make it possible as well as possible to describe the behavior of the wave in the structure with porosities (see

For the polyethylene plate containing grease inclusions, the new values of the effective properties for longitudinal and transverse velocities (Equation (2)), density and acoustic impedance (Equation (3) for the four models are calculated. The results are presented in

As we can see, the effective acoustic properties for the HS-ort model are very

Model | ρ_{eff} (kg/m^{3}) | c_{L,eff} (m/s) | c_{T,eff} (m/s) | Z_{L,eff} (MRa) |
---|---|---|---|---|

Voigt | 936 | 2215 | 1074 | 2.07 |

G-sph | 936 | 2176 | 991 | 2.04 |

HS-iso | 936 | 2176 | 991 | 2.04 |

HS-ort | 936 | 2133 | 925 | 1.99 |

ρ_{eff} (kg/m^{3}): Density; {c_{L,eff}, c_{T,eff}} (m/s): Longitudinal and transverse wave velocities; Z_{L,eff} (MRa): Longitudinal acoustic impedance.

lower than effective acoustic properties for the HS-iso model. Thus, in the following, since orthotropic inclusions are considered, effective acoustic properties calculated using Hashin-Shtrikman (HS-ort) model will be retained and will be useful as input data for simulations.

We consider two configurations of insonation: direct insonation (made from the aluminum layer side) and reverse insonation (from polyethylene side). For the direct case, the obtained reflection coefficient is noted R_{d}, and for the reverse one, the reflection coefficient is noted R_{r}. When attenuation is considered in the multilayer structure, those reflection coefficients are different according to the insonation side, either in the direct case R_{d} or in the reverse case R_{r}, whereas the transmission coefficient T is identical whatever the insonation side. These coefficients are obtained from a procedure suggested by Rokhlin et al. [

S n = [ S 11 S 12 S 21 S 22 ] n (5)

where S_{ij} are 2 × 2 submatrices.

The S_{ij} submatrices for N periods are deduced from the ones for (N − 1) periods and those of an additional n layer by the following recursive relationships:

{ S 11 N = S 11 N − 1 + S 12 N − 1 ( S 11 n − S 22 N − 1 ) − 1 S 21 N − 1 S 12 N = − S 12 N − 1 ( S 11 n − S 22 N − 1 ) − 1 S 12 n S 21 N = S 21 n ( S 11 n − S 22 N − 1 ) − 1 S 21 N − 1 S 22 N = S 22 n − S 21 n ( S 11 n − S 22 N − 1 ) − 1 S 21 n (6)

It is shown that the expressions of R_{d}, R_{r}and T coefficients are expressed according to the elements (2, 2) of the S i j N submatrices extracted from global compliance matrix S^{N}, denoted as s i j 22 into the forms:

{ R d = ( s 11 22 − y F ) ( s 22 22 − y F ) + ( s 12 22 ) 2 ( s 11 22 + y F ) ( s 22 22 − y F ) + ( s 12 22 ) 2 R r = ( s 11 22 + y F ) ( s 22 22 + y F ) + ( s 12 22 ) 2 ( s 11 22 + y F ) ( s 22 22 − y F ) + ( s 12 22 ) 2 T = 2 y F s 12 22 ( s 11 22 + y F ) ( s 22 22 − y F ) + ( s 12 22 ) 2 (7)

where y F = cos θ / ( j ω Z F ) .

In this section, the theoretical reflection, transmission and attenuation spectra are plotted according to the insonation side. The multilayer structure is immerged in a coupling medium (water in our case). In

Medium | ρ (kg/m^{3}) | c_{L} (m/s) | c_{T} (m/s) | Z_{L} (MRa) |
---|---|---|---|---|

AL | 2800 | 6380 | 3100 | 17.9 |

PE | 940 | 2370 | 1200 | 2.23 |

Teflon | 2200 | 1350 | - | 2.97 |

Water | 1000 | 1480 | - | 1.48 |

Grease | 920 | 1450 | - | 1.33 |

ρ (kg/m^{3}): Density; {c_{L}, c_{T}} (m/s): Longitudinal and transverse wave velocities; Z_{L} (MRa): Longitudinal acoustic impedance.

calculated using the HS-ort homogenization model presented in _{f} = 20%. The chosen bandwidth frequency is ranging in [0.75; 3.25] MHz.

The constitutive layers of periods have each one 4 mm thickness and are denoted by AL, PE, PEp and PEg for aluminum, polyethylene, perforated polyethylene and polyethylene with grease inclusions, respectively. A computation carried out with plates without losses will give an identical reflection spectrum whatever the insonation side [

In previous works, Lenoir et al. [

In the case of PEg/AL configuration (

The difference of spectrum according to the insonation side is also observed

in

The transmission coefficient illustrated by

| A { d , r } | 2 = 1 − | T | 2 − | R { d , r } | 2 (8)

As a result, the absorption coefficients |A_{{}_{d}_{,r}_{}}|^{2} also depend on the insonation side. The estimated absorption on the reverse side is equal to 0.33, 0.57 and 0.73 at 1.6 MHz (

Siryabe et al., [_{L}/(2d) such as Δf_{PEc} = [267, 242, 221] kHz for PEg/AL configuration, corresponding to the longitudinal velocities c_{L,PEc} = [2133, 1933, 1769] m/s, respectively associated to the volume fraction V_{f} = 20%, 40% and 60%.

The insertion of a thin Teflon layer of which properties are given in

In the structure of type AL/PEg, the reflection coefficients reach their maximum from the second period (N = 2). It is not the case of the structure of type AL/PEg/PEg/AL, where the reflection coefficients do not evolve anymore and this from the period N = 1 (

Concerning the transmission coefficients, depending on the reference stack configuration (AL/PEg/PEg/AL) × N (

of a period increases, the more the forbidden bands are observed in the multilayer. Moreover, amplitudes of the maxima of transmission coefficients decrease considerably when the frequency and the number of period increase. Regarding the number of peaks, it shifted from n_{peaks} to 2n_{peaks} for the reference configuration versus the symmetric one, respectively. In the reference configuration, the position of the maximum of transmission is the same as for the case of AL/Peg, what is to be understood as the resonances of the first AL layer. In the symmetric configuration, additional maxima are observed at 1.2, 2.0 and 2.8 MHz, in between to the three maxima of transmission of the reference stack, respectively. Moreover, in the reference configuration, the positions and widths of the forbidden bands are identical to those of the AL/PEg configuration.

In this study, the effect of grease inclusion, Teflon, volume fraction of grease and the number of layers composing one period were highlighted by the analysis of the reflection and transmission spectra of an immersed multilayered media. As in the porous case of AL/PEp, reflection coefficients reach their maximum as from the second period (N = 2), in the case of AL/PEg, the forbidden bands gap appears in the multilayer stack as from N = 2 periods. In addition, in the case of the studied (PEg/AL) configuration, for a fixed number of periods N = 3, a decrease of the oscillations amplitude of the reflection coefficients with the volume fraction and the frequency is observed, since they are directly linked to the acoustic impedance contrast. The presence of Teflon defects of 100 to 500 µm thick was analyzed. As a result, a shift is observed to low frequencies on the local maxima and minima of reflection. The maximum shift was around 250 - 300 kHz. Finally, concerning the transmission coefficients, unlike to the structures of type (AL/PE or PE/AL) [_{peaks} to 2n_{peaks} for the reference and symmetric configurations, respectively. Further studies are to be led experimentally in order to validate those numerical results. It would also be interesting to include piezoelectric layers as dynamic modulators of the reflection and transmission coefficients. As a perspective, such piezo-active and controllable multilayers have many potential applications as actuators and sensors in the industry. We could plan to evaluate the electromechanical response of the obtained global structure.

The authors declare no conflicts of interest regarding the publication of this paper.

Hatoumva, M., Siryabe, E., Marechal, P., Ntamack, G.E. and Betchewe, G. (2021) On the Simulation of the Influence of Defects on Immersed Plane Periodic Multilayer Viscoelastic Media. Open Journal of Acoustics, 11, 17-30. https://doi.org/10.4236/oja.2021.112002