Theoretical Evidence For Ultrasonic Insulation Using a Fractal-Like Phononic Crystal Membranes

European Journal of Applied Physics
 ISSN: 2684-4451

Theoretical Evidence For Ultrasonic Insulation Using a
 Fractal-Like Phononic Crystal Membranes
 Abdelfattah Elmadani, Abdelmajid Idrissi, Ramdan Braik, Saad Bensallam,
 Abella Bouaaddi, Younes Achaoui, and Hicham Jakjoud

 Abstract — Phononic crystals are artificial engineered materials designed to control and
 manipulate waves. Unusual behaviour of prohibiting the acoustic propagation in some frequency
 bands (Band GAP), is a practical way to produce sound-ultrasound-proof environments with a
 small spatial footprint. In this work, we present a new fractal-like phononic crystal for
 extraordinary ultrasonic insulation. The host material is a silicon plate where the unit cell is
 formed by triangular slice and immersed in water. Our simulation is made between 300 kHz and
 1.2 MHz and show the possibility of obtaining a wideband-gap, inferior to the one described by
 the mass law related to a homogeneous silicon membrane, with an attenuation reaching -70 dB,
 depending on the filling factor.

 Keywords — Acoustic metamaterials, band-gap, Fractal structure, Phononic crystals, Sierpiński
 triangle, Ultrasonic insulation.

 I. INTRODUCTION1
 In the last three decades, researchers were concerned by the development of unnatural phenomena that
acoustic metamaterials exhibit. In fact, the subwavelength restructuration of the materials makes it possible
to perform applications such as acoustic imaging [1], superlensing [2], insulation [3] and sensing [4] etc.
 One of the structures that receive an increasing interest for ultrasonic opacity, the subwavelength
phononic crystal membrane. Pre-existing works on the topic have focused on exploitation of the different
geometric and symmetry constraints, based on Bragg scatterers and serve as the framework for surveying
a variety of acoustic crystals absorbers that can realize previously unattainable absorption spectra [5].
 As research in phononic crystals continued to grow, other configurations were proposed belonging to
localized resonances’ principle [6], for example Quarter-Wave Mode [7], Helmholtz-like resonances [8],
Fabry-Pérot interference [9], or the hybridized coupling effect [10], [11], extraordinary resonated
membranes properties constitute an ambitious and optimized candidate specifically for controlling
ultrasound environments. More subtle symmetries have also been considered, such as quasicrystal [12] and
fractal [13]–[15] phononic crystals − in analogy to similar studies concerned with photonic crystals [16]–
[18].
 In this paper, we show the potential of the put of at least multiple cracks of fractal-like plates in a periodic
array of subwavelength apertures to exhibit different behaviour depending on the incoming ultrasonic
signal.
 Fractal geometry describes complex objects with irregular shapes, ubiquitous in nature, which present
"breaks" at all scales. Fractal objects are distinguished by their self-similarity: they are similar to themselves
at different scales. Very irregular media are difficult to describe by classical Euclidean geometry. They can
however be approximated by statistical analyses and represented by random processes. In search of a
suitable deterministic tool, Mandelbrot introduced the notion of fractal geometry and demonstrated its
universality in 1982 in “The Fractal Geometry of Nature” [19].

 Submitted on June 05, 2022.
 Published on July 01, 2022.
 A. Elmadani, Laboratory of Energy Engineering, Materials and Systems, Physics Department, National School of Applied Sciences
 of Agadir, Ibn Zohr University, Morocco.
 (corresponding e-mail: Abdelfattah.elmadani@gmail.com)
 A. Idrissi, Laboratory of Advanced Materials Studies and Applications, Physics Department, Moulay Ismail University, Morocco.
 R. Braik, Laboratory of Energy Engineering, Materials and Systems, Physics Department, National School of Applied Sciences of
 Agadir, Ibn Zohr University, Morocco.
 S. Bensallam, Hassania School of Public Works, Morocco.
 A. Bouaaddi, Laboratory of Energy Engineering, Materials and Systems, Physics Department, National School of Applied Sciences
 of Agadir, Ibn Zohr University, Morocco.
 Y. Achaoui, Laboratory of Advanced Materials Studies and Applications, Physics Department, Moulay Ismail University, Morocco.
 H. Jakjoud, Laboratory of Energy Engineering, Materials and Systems, Physics Department, National School of Applied Sciences
 of Agadir, Ibn Zohr University, Morocco.

DOI: http://dx.doi.org/10.24018/ejphysics.2022.4.4.182 Vol 4 | Issue 4 | July 2022 1

European Journal of Applied Physics
 ISSN: 2684-4451

 II. DESCRIPTION OF THE PROPOSED FRACTAL STRUCTURE
 We were interested in studying the propagation of acoustic waves in a medium with fractal geometry.
We thus implemented a composite material composed of self-similar patterns and studied them using a
numerical model. The structure considered is silicone plates put into water with triangular patterns in
cascade. The plate thickness is = 1 and have been structured by performing a periodic array, the
periodicity of the 3D sub-wavelength design is = 0.5 , as shown in Fig. 1.
 An acoustic plane wave is generated in one side of the structures. The transmission is evaluated by
comparing the signal that crossed the phononic structures to the source. Due to the periodicity of the system,
we may limit the models to one-unit cell as illustrated in Fig. 1. Perfectly Matched Layers (PMLs) are
considered on both sides, in order to reduce reflections on the system boundaries. In addition, Floquet’s
periodic conditions are considered.

 Fig.1. Schematic of a Fractal-Like phononic unit cell of the proposed structure used as phononic cristal insulator, we distinguish
 incident Po, reflected Pr, and transmitted Pt pressures.

 From a mathematical point of view, the concept of fractal is associated with a geometrical object which:
(1) is self-similar (i.e., the object is exactly or approximately similar to a part of itself) and (2) has a
fractional (or non-integer) dimension. Self-similar structures are obtained by performing a basic operation,
called generator, on a given geometrical object called initiator, and repeating this process on multiple levels,
in each one of them an object composed of sub-units of itself is created that resembles the structure of the
whole object [20]. Our designed structure is a Sierpiński triangle-like, the canonical Sierpinski triangle-like
use an isosceles triangle in a plane with a base parallel to the horizontal axis, the geometrical parameters of
 a2
our fractal of the Cantor triadic set are chosen as (n=√( 2 + ) mm; a mm), the second and the third fractal
 4
order are respectively an subdivide it into smaller congruent isosceles triangles, the pre-fractality remaining
 1 1
segments with two smaller segments of half length ( height and width) (Fig. 2). The physical
 2 2
characteristics of materials (density, longitudinal and transverse wave velocities) used in the simulation are
detailed in Table I.

 a. b. c.
 Fig. 2. 2D Schematic views of the unit cell configurations with different orders fractal Sierpinski gasket structure, a. the Cantor
 equilateral triadic (white triangle), b. the second iterations of the fractal Sierpinski gasket structure (red triangle) and,
 c. (yellow triangle) the third iteration fractal aperture.

 TABLE I: PHYSICAL PROPERTIES OF THE SIMULATED MATERIALS
 Material Density CL (m/s) CT (m/s)
 Silicon 2.329 8433 5843
 Water 1.0 1480 -

DOI: http://dx.doi.org/10.24018/ejphysics.2022.4.4.182 Vol 4 | Issue 4 | July 2022 2

European Journal of Applied Physics
 ISSN: 2684-4451

 III. RESULTS
 The band structure of the thin PC slab, described in detail in Fig. 3. As one can notice, the structure
enables to reach an attenuation down to -70 dB within a relative bandwidth of a 114% and 0.7 MHz as
Mid-gap. This is calculated according to the frequency range, in which the transmittance of the metamaterial
is inferior to the limits described by the conventional mass-density law, characteristic of the homogeneous
plate.

 Fig. 3. The amplitude transmission spectra, related to the Fractal-like membrane in Fig.1 (solid curves). The dashed curve refers
 to the amplitude transmission through a homogeneous silicon membrane (Mass Law).

 The presented arrangement of scatterers based on fractal-like geometries leads to the possibility of
designing devices to control wave propagation, to increase the multiple scattering phenomenon. Thus, the
underlying physical mechanism is destructive interference. Fig. 4 describe the physical origin of those
phenomena by considering the elastic displacement field of the membrane unit cell corresponding to the
modes of vibrations, when observing the displacement distribution, a cross-talking between neighbouring
unit cell components induces high pressure pump motion in both side of the membrane, creating a finite
acoustic impedance ratio between fluid and solid. The resonant behaviour of interest is due to the fluid
spaced each two plates which leads, to the multiple scattering phenomenon, a technique that enables the
creation of a wide bandgap.

 Fig. 4. Pressure and displacement field of the membrane unit cell corresponding to different modes of vibration, at frequencies
 (0.30 MHz, 1.01 MHz, 1.05 MHz, 1.10 MHz, respectively).

 Moreover, the creation of interference phenomena effect is explicit between the different types of waves
traveling through the crystal, the waves propagating in the narrow channels and those re-radiated by
different orders fractal Sierpinski triangle crosses the structure (waves with local interactive mode and
waves resonant).

DOI: http://dx.doi.org/10.24018/ejphysics.2022.4.4.182 Vol 4 | Issue 4 | July 2022 3

European Journal of Applied Physics
 ISSN: 2684-4451

 IV. DISCUSSION
 In this section, we present, the effect of the main factor determining the insulation device performance,
as the iteration and the disposition of the fractal geometry. we present the same acoustic device supporting
Fractal-like resonance structured composite with shunted as illustrated (Fig. 1). We expand our study to the
configuration introduced in (Fig. 5), For the given phononic membrane the periodicity of the 3D sub-
wavelength design is = 0.5 , with a consistent filling factor compared to the one defined in Fig. 1
( = 70% , the amount of matter formed by the scatterers, quantified by the filling fraction (ff)).

 Fig.5. Schematic of a Fractal-Like phononic unit cell of the proposed structure used as phononic cristal insulator, we distinguish
 incident Po, reflected Pr, and transmitted Pt pressures.

 Fig. 6 displays the spectral transmittance of the structure, the numerical curve and the area for a total
transmission. Consequently, it is shown numerically that the variations of the filling factor defining the
ratio of the rigid acoustic material (silicon) over the total volume of the cell, or the disposition of the fractal
component displace drastically the band gap width (with a relative bandwidth of 43% and 0.47 MHz as
Mid-gap), although it improve substantially his efficiency.

 Fig.6. The amplitude transmission spectra, related to the Fractal-like membrane in Fig.5 (solid curves). The dashed curve refers
 to the amplitude transmission through a homogeneous silicon membrane (Mass Law).

 In return, an increase in the bandgap’s width can be obtained only by decreasing the ff. This factor makes
the miniaturization of membrane attributable to the bandwidth limitation or the performance desired (the
pre-fractal stacks subtracted globally transmit less Compared to the traditional Sierpinski fractals described
previously in our literature review).

 V. CONCLUSION
 In conclusion, several acoustic designed resonator cells are reported to date, we have investigated new
class of insulators based on phononic crystals principles, engineered with a kind of fractal-like structures
concept. We have obtained transmission losses up to -70 dB, on a large bandwidth exceeding 114% with
a slight filling factor, depending on the configuration. This new class of structures, increase the flexibility
in design and simplicity in controlling the phononic spectra of materials to better control ultrasonic noise.

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European Journal of Applied Physics
 ISSN: 2684-4451

 We believe that the experimental realization of such a device proves to be difficult to realize ones the
fragility of the membrane (fig. 1–5), but we are convinced that the exploration of the properties inherent to
the fractals-like notion in the phononic crystals field, will open up new perspectives for the extension of
the study feasibility with materials more resistant than silicone.

 CONFLICT OF INTEREST
 Authors declare that they do not have any conflict of interest.

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