The geometry of the problem (Figure 1(a)) consists with a stacking of N periods (1 ≤ N ≤ 3), each one made up of two isotropic plates of different acoustic properties. As a case of study (Figure 1(b)), Siryabe et al. investigated the AL/PEp stack, where the polyethylene plate (PE) was perforated (air porosities) [6]. In this study, as an extension, we consider the case of grease inclusions. The aim is to observe the effect of the presence of grease on the reflection and transmission coefficients of the considered multilayer. Here, one will note (PEg) for polyethylene with grease. In both cases, the longitudinal and transverse waves attenuations are taken into account in the polyethylene plate but neglected in the aluminum one [4].

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 [6]. In order to retain the appropriate model to our study, a comparison of the approaches suggested by Voigt [11] [12], Gaunaurd (G-sph) [13] and Hashin-Shtrikman (HS-iso, HS-ort) [14] [15]

is made. For a plate with inclusions, the estimation of effective properties can be done using the following relations [6]:

{ 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 Table 1.

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 [6]:

{ 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)

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 Table 1 according to the different homogenization models for a polyethylene plate containing grease inclusions.

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 Figure 2(a) and Figure 2(b), respectively.

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 Figure 3(a) and Figure 3(b), respectively).

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 Table 2.

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

ρ_eff (kg/m³): 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. [3] [16]. For each period N, one defines a (4 × 4) flexibility matrix:

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_rand 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 previous works, Lenoir et al. [4] for (AL/PE) periodic structures and Siryabe et al. [6] for (AL/PEp) periodic structures showed that reflection coefficients reach their final shape as from the second period (N = 2). It is also the case of our configuration (AL/PEg) (Figure 4(a) and Figure 4(b)). In addition, the reflection coefficients minima tend towards zero and the undulations in both AL/PEp and AL/PEg configurations disappear for N = 1 period. A small shift and interchange is observed on local maxima and minima of reflection coefficients.

In the case of PEg/AL configuration (Figure 4(b)), undulations related to the vibrations of the polyethylene layer are obtained at the period N = 1. We also observed a very great decreasing in the amplitude of the reflection coefficients as the frequency increases. This result was also obtained by Siryabe et al. [6] in the case of PEp/AL. Concerning the minima of reflection; they tend towards zero for the three configurations whatever the frequency and the number of periods.

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

in Figure 4(a) and Figure 4(b) for the configurations (AL/PEg) and (PEg/AL), respectively. From N = 1 period, the reflection minima linked to the propagating Lamb modes in the aluminum plate keep their positions for all configurations. These minima of reflection are observed at the frequencies 1.6, 2.4 and 3.2 MHz, in the bandwidth of the transducer.

The transmission coefficient illustrated by Figure 4(c), shows the forbidden band gaps as from N = 2 periods, similarly as for the previously cited works [4] [6]. More precisely, the transmission coefficients (Figure 4(c)) exhibit a clear reduction of its amplitudes when the frequency and the number of periods increase. Thus, most of the wave energy is attenuated as the frequency and the number of periods increases. For example, if we take the case of the first peak at 1.6 MHz, the amplitude is equal to 0.63, 0.41 and 0.25 for N = 1, 2 and 3 periods, respectively. This leads us to deduce the absorption by the following relationship:

| 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 (Figure 4(d)).

Siryabe et al., [6] observed an increase in the oscillations amplitude of the reflection coefficients when the volume fraction ratio and frequency increase (Figure 5(a)). In the case of our configuration (PEg/AL), for a fixed period N = 3 (Figure 5(b)), a decrease of the oscillations amplitude of reflection coefficients with the volume fraction and the frequency is observed since they are directly linked to the acoustic impedance contrast. This divergence can be explained by the fact that in the case of our configuration, the acoustic impedance contrast is lower than the case of study carried out by Siryabe et al., where air properties were neglected. In addition, we can observe that the minima are periodically spaced out by a shift of frequency Δf = c_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 Table 3, within two adjacent plates (AL and PE), increases the layer number of the period from two to three. The presence of 100 to 500 µm Teflon thickness is analyzed. We observed a shift to low frequencies on the local maxima and minima of reflection. The maximum shift is around 250 - 300 kHz. As illustrated by Figure 6, there are not significant changes depicted on the minima of reflection. By increasing the Teflon thickness, undulations on the transmission spectra increase too. Furthermore, reflection coefficients still have approximately same amplitude, whatever the number of period increases (Figure 6). About the location of the forbidden bands in the periodic structure, as we can see in Figure 6(c), three forbidden bands in the frequency range [0.75; 3.25] MHz are observed: from 1 to 1.5 MHz, 1.7 to 2.3 MHz and 2.5 to MHz. As an illustration, the influence of the variations of Teflon thickness layer on these passbands is depicted in Figure 6(c) where narrow passbands are observed in the frequency range from 1 to 1.5 MHz, reducing the forbidden bands.

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 (Figure 7(a) and Figure 7(b)). This is due to the fact that, the number of layers forming one period has been increased, but also to their arrangements. On the one hand, this result means that the reflection coefficient is related only to the first layers of the multilayer structure, the following ones being dissimulated by the attenuation effect. On the other hand, in the reverse configuration PEg/AL/AL/PEg (Figure 7(b)), the more the frequency increases, the more the amplitude of the reflection coefficients decrease. We also observe a duplication of the spectrum width with the apparition of the news minima of reflection, due to the double thickness of the aluminum layer.

Concerning the transmission coefficients, depending on the reference stack configuration (AL/PEg/PEg/AL) × N (Figure 7(a)) or the symmetric one (PEg/AL/AL/PEg) × N (Figure 7(b)), the insonation side influences significantly the evolution of the transmission coefficients. The forbidden band gap appears at a period N = 1 for the reference configuration (Figure 7(c)), and from N ≥ 2 periods in the symmetric case (Figure 7(d)). This leads us to conclude that, in the reference configuration case (AL/PEg/PEg/AL), the more the layer number

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.