A Nonlinear Elastic Resonator 

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Phononic Crystals and Their Applications

Abstract

Phononic crystals (PnCs) are the analogue of photonic crystals for the acoustic waves. By adding scattering periodic inclusions, following a crystalline lattice, inside an homogeneous host material, some ranges of acoustic frequencies are totally reflected and thus forbidden in the resulting material. This creates acoustic band-gaps.
Initially, the constructed crystals were limited to frequencies below 1 MHz, but the micro-crystals made by micro-fabrication can reach very high and even ultra high frequencies, with an example above 1 GHz [2].
The key issue in the field of PnCs is tuning the acoustic band-gaps in order to tailor the behavior of acoustic waves and create new functions, such as selective frequency filtering and wave guiding, and new devices, such as wavelength demultiplexers. This can be done by using the nonlinear properties of the materials that constitute the crystals as well as using the geometrical nonlinearities of the structures [1].
The initial purpose of this thesis was to demonstrate, numerically, that one can dynamically tune the band gaps and localized modes in PnCs by using magnetoelastic materials. It then evolved in studying the nonlinear effects in various phononic structures. This thesis does not only concern the analytical study of nonlinearities in PnCs but also the methods used to simulate numerically and efficiently the acoustic wave behavior in such structures. So, some facts and recent breakthroughs about the resolution of the acoustic wave equation will be explained: a state-of-the-art highly parallelizable finite element method called the Discontinuous Galerkin method will be presented. This introduction will first provide a general overview about PnCs. Then, it will explain the different methods that can be used for tuning PnCs and detail what has already been done in some specific areas of this field, namely granular PnCs and nanoscale PnCs, and what is still to be done. It will present both the history and state of the art and provide a review of the literature on this subject.
Figure 14 – Example of a phononic crystal. This example shows the lattice parameters for the Bragg resonance on the left and the resulting band diagram on the right, as well as the directions of propagation in the Brillouin zone.

General Overview

Principle

PnCs are periodic arrangements of inclusions inside an elastic or viscoelastic material or a fluid, for example metal in air, polymer in water, air in epoxy, etc. With such a structure, band-gaps may appear and are independent of the direction of propagation of the incident elastic wave. In this case, the PnC behaves like a perfect non absorbing acoustic mirror for the rejected frequencies [1].
The central frequency of the band-gap is determined by the size, periodicity, filling and arrangement of the inclusions. Physically, the phononic band-gap is due to the diffraction of the elastic wave at the interface between the matrix and the inclusions [2].
Creating defects by replacing or removing some of the inclusions in a PnC, by making the arrangement irregular, allows certain frequencies to exist within the band-gap. Those defects behave like impurities in doped semiconductors. Thus, it is possible to tailor the acoustic properties of the crystal [1].
Typical PnCs obey the Bragg and Mie resonance conditions. Figure 14 represents a square lattice crystal. On this figure, r is the inclusion radius and a is the distance between the centers of the inclusions. The irreducible Brillouin zone has three directions, X√, XM and M . The fundamental Bragg resonance frequencies are V /(2a) for the X direction and V /(2 2a) for the M direction, where V is the average acoustic velocity in the crystal. This velocity depends on the filling ratio, r/a [2]. PnCs are elastic or acoustic depending on whether the host material can (solid) or cannot (gas or liquid) support transversely polarized waves [8].

History

The very first known experimental observation of PnCs was in 1979, when Narayanamurti et al. investigated the propagation of high-frequency phonons through a GaAs/AlGaAs superlattice. Their superlattice can be considered as a 1D PnC [3].
As for theoretical work, Sigalas and Economou demonstrated for the first time in 1992 that frequency gaps appear for elastic waves in periodic arrangements of spheres with high density compared to that of the host material [4]. Then, Kushwaha et al. calculated full band-structure for periodic, elastic composites in 1993 [5].
However, the first complete phononic band-gap was only observed in 1998 by Montero de Espinosa et al., using an aluminum alloy plate with a square periodic arrangement of cylindrical holes filled with mercury. A band-gap appeared in the frequency range between 1000 and 1120 kHz [6]. Recently, the variety of materials used in the fabrication of phononic devices allowed great improvements on the reachable frequencies for the band-gap, making it increase from about 1 MHz to very high frequencies (VHF: 30–300 MHz) and even ultra high frequencies (UHF: 300–3000 MHz) [2].
Gorishnyy demonstrated an hypersonic (above 1 GHz) PnC using air scattering inclusions in epoxy in 2007. The band-gap was measured by Brillouin light scattering [7].

Applications

The domains in which PnCs have potential applications are radio-frequency communications and ultrasound imaging for medicine and nondestructive testing. Focusing devices made with PnCs could miniaturize acoustic lenses, adapt impedance and decouple the transducer size from the aperture [2].
Moreover, sub-diffraction-limited resolution by transmitting the evanescent components of the wave and acoustic shielding could be reached by using them as wave-guides [8]. Using the photon-phonon interaction would allow modulation and optical cooling [2]. They could also improve direct energy conversion by thermoelectric and thermophotovoltaic effects [9].
At the micro-scale, PnCs are used to isolate resonating structures, such as Coriolis force gyroscopes, mechanical resonators, filters and oscillators, from external vibrations and noise [2]. Therefore, they allow the rigid attachment of these devices to the substrate in a vibration-free environment, which allows to make high-precision mechanical systems [1].

Tunable Phononic Crystals

As discussed above, adding defects in the crystalline structure is an easy way to tailor the acoustic band-gaps of the PnC. Tunability of PnCs can also be achieved by changing the geometry of the inclusions through stress or thermal effects [11].
Another way is to vary the elastic characteristics of the constitutive materials by applying external stimuli such as electric field, temperature or stress. However those techniques either require physical contact or the application of very large stimuli for a very small result [10].
In 2009, Robillard et al. demonstrated the possibility of controlling the band-gaps using an external magnetic field. This field would use the giant magnetostriction effect to modify the nonlinear elastic constants of Terfenol-D, a giant magnetostrictive material, by more than 50 % which allows contact-less tunability of the PnC [10]. Bou Matar et al. extended this work by explaining how it would be possible to tailor the band structure of the PnC by using giant magnetostriction and spin reorientation transition effects. They detail the modeling equations of a piezomagnetic PnC [11].
In optics, Soljačić et al. published in 2002 a description of a nonlinear photonic crystal capable of performing optimal bistable switching. Their analytical model accompanied with numerical simulations describe a resonator consisting of a cavity with a nonlinear optical index. When increasing the wave amplitude, the nonlinear frequency shift brings the wave frequency to the resonance frequency of the cavity. This leads to a higher transmission of the wave through the crystal. When decreasing the wave amplitude, the transmitted power delineates an hysteresic curve and the system reaches full transmission [20].
This kind of behavior has not yet been observed in acoustics even if the approach proposed by Robillard and Bou Matar seems to prove the feasibility of an acoustic bistable switch using piezomagnetic PnCs.

Granular Phononic Crystals

The discussion above presented general facts and breakthroughs about PnCs. Despite progresses have been made to discover their properties and try to tailor their band structures, very few attempts have been done to understand their nonlinear behavior, except in the specific case of granular PnCs which will be presented now.
Nonlinear granular PnCs are composed of statically compressed chains of particles, confined in a guide, that interact nonlinearly through Hertzian contacts [21]. Uncompressed granular crystals are called “sonic vacuum type crystals” and are incapable of transmitting linear elastic waves [22].
How waves propagate in statically and dynamically loaded granular media have lead to an extensive research work since the 1980s, for example with Nesterenko, who studied the existence and interaction of solitons in packed spherical granules, using numerical methods and Shukla et al. who modeled the wave propagation in these media and computed it using experimental data [23, 24].
In 2004, Daraio et al. designed 1D and 3D crystals made of 0.03 to 0.5 g steel spheres in a silicon or PTFE matrix and observed trains of strongly nonlinear solitary waves with small amplitudes (corresponding to forces of about 0.3 N) which propagate at 317 m/s which is below the speed of sound in air [25].
Then, in 2006, the same team showed that the nonclassical, strongly nonlinear wave behavior appears in granular materials if the system is weakly compressed, which means that the precompression force is very small with respect to the wave amplitude. Oppositely, strongly compressed chain behavior approaches linear wave behavior [26].
The derivation of the dispersion relation of 3D granular crystals made of hexagonal arrangements of spheres was performed by Merkel et al. in 2010. They predict the existence of translational, rotational and coupled translational/rotational modes and show that the longitudinal modes are not changed when the rotational degree of freedom is taken into account [27].
In 2011, Boechler et al. used granular crystals similar to the one shown on Figure 15 with a defect to design a rectifier (device that allows the propagation of some frequencies only in one direction) and an acoustic switch with sharp transitions between states [21]. Nonlinear resonances in diatomic granular chains have also been observed by Cabaret et al. who use the amplitude-dependent behavior that result from the geometry of the structures, specially the Hertzian contacts between the particles [28].
Finally, in 2012, Yang and Daraio studied the propagation of stress waves in granular crystals composed of diatomic unit cells in bent elastic guides. The cells were made of centimeter-long alternating spherical and cylindrical particles. The waves were generated by striking the particle on the top with a force of 50 to 1000 N. Those structures possess band gaps around 5 and 10 kHz which can be tuned by modifying the precompression of the chain. They show highly nonlinear behaviors and propagate solitary waves [22].

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Tunability

One of the interests of granular crystals is the possibility to tune their behavior from near-linear to highly nonlinear by varying the precompression which changes the ratio of static to dynamic particle displacement [21].
A simulation by Göncü et al. proved that the band structure of a 2D granular crystal composed of silicon rubber and polytetrafluoroethylene (PTFE) cylinders could be tuned more efficiently using pattern transformation rather than changing the particles’ mechanical properties, creating new gaps around 5 kHz [29].
A way of tuning the solitary waves speed by a factor of two by applying a precompression of 2.38 N with a magnetically-induced interaction or only the gravitational preload of 0.017 N was demonstrated by Daraio et al. [26].
A numerical model by Spadoni and Daraio shows an acoustic lens capable of generating acoustic pulses with a pressure amplitude of 675 Pa, corresponding to 57 dB, which is two orders of magnitude larger than what is reachable with linear lenses. The focal point position is tuned by varying the time and space distribution of the static precompression of the chains that compose the lens. Experiments using an array of steel spheres in Teflon sheets produced focused waves in a polycarbonate plate in accordance with the simulations. The wavelength is determined by the size of the particles [30].

Applications

Nonlinear acoustic lenses created by Spadoni and Daraio with granular PnCs can achieve better focusing and allow higher focal power than linear ones. So, they are expected to improve the performances of current devices for biomedical imaging, nondestructive testing and sonars. They could also be used to constitute nonintrusive scalpels, for cancer treatment, for example [30].
The ability of nonlinear granular PnCs to behave as rectifiers or switches suggest that they could be used to control the flow of energy in several applications, such as energy-harvesting materials with frequency-dependent absorption and emission and thermal computers. Their nonlinear response allows them to change their state when small perturbations are applied, which makes them suited for sensing applications [21].
The control of nonlinear resonances in diatomic granular chains are expected to lead to the creation of new devices, such as passive amplitude-dependent amplifiers and attenuators [28].
One of the possible future direction in research about those crystal could be the ability to engineer the dispersion relation and create gaps to create tunable vibration filtering devices and systems that could be insulated from noise vibrations [31].

Nanoscale Phononic Crystals

The first part provided a general description of PnCs. The second part explained the specific case of granular PnCs, whose nonlinear properties have been investigated. Another particular domain, for which a brief review is now provided, consists of nanoscale PnCs. Those crystals deal with very high frequencies (THz) and thermal effects. In this field, recent efforts have been made to engineer the band gaps. The most considered materials for those crystals are graphene and nanoporous silicon.

History

In 2009, Gillet et al. demonstrated that 3D arrays of germanium quantum dots in silicon reduce the thermal conductivity by several orders of magnitude compared to bulk silicon. This effect is produced by THz phonon confinement and reflection on layer interfaces which decrease the phonon group velocity [32].
A similar effect was observed by Marconnet et al. in silicon nanobridges where the thermal conductivity can be reduced to 3 % of its value in bulk material, without affecting the electrical conductivity. They measured a thermal conductivity one order of magnitude lower than predicted by the model [33].
Then, this effect was explained in 2012 by Dechaumphai and Chen who modeled the coherent (wave-like) and incoherent (particle-like) behavior of the phonons and computed the dispersion relation with a FDTD method. They found that the zone folding effect (formation of mini-bands in the superlattice because the Brillouin zone edge for two materials is smaller than the Brillouin zones of each material) has a major impact on the thermal conductivity [34].
A simulation by Sgouros et al. models PnCs made of graphene whose defects are constituted by adding carbon atoms, removing some or replacing some by silicon atoms. This study shows that only the substitution by silicon atoms creates band gaps [35].
As explained by Maldovan in 2013, the control of thermal conduction can be achieved by transforming the heat flow to wave phonon transport by applying a 2D holes pattern to nanostructured alloys of silicon. By blocking high-frequency phonons (above 1 THz), preventing them from existing, the heat transfer can only be done by low-frequency phonons (below 1 THz) which can be guided by nanoscale crystals [36].

Table of contents :

Abstract 
Acknowledgements
Contents
Résumé en français 
Introduction
Étude analytique et numérique des super-réseaux 1D
Dispersion des ondes élastiques dans une structure osseuse
Étude d’un résonateur élastique non-linéaire
Lois de mélange pour les paramètres élastiques quadratique et cubique
Étude numérique des cristaux phononiques non-linéaires 2D
Conclusion et perspectives
Introduction 
1 Analytical and Numerical Study of 1D Superlattices 
1.1 Introduction
1.2 One-Dimensional Superlattices
1.3 Analytical Method: Transfer Matrix Method
1.3.1 Propagation in a Layer
1.3.2 Propagation in a Bilayer
1.3.3 Band Structure in a 1D Phononic Crystal
1.3.4 Propagation of Amplitudes
1.3.5 Transmission Through a Bilayer
1.3.6 Transmission Through N Bilayers
1.3.7 Transmission Through N Bilayers with a Defect
1.3.8 Reflected Impedance Through a Multilayer
1.3.9 Transmission Through N Bilayers with Quarter-Wavelength Layers
1.4 Numerical Methods
1.4.1 Pseudospectral Method
1.4.2 Finite Difference Time Domain
1.4.3 Spectral Energy Density
1.5 Conclusion
2 Dispersion of ElasticWaves in a Bone Structure 
2.1 Abstract
2.2 Introduction
2.3 Models
2.3.1 Thermodynamics of a Stressed Solid Solution
2.3.2 Nonlinear Young’s Modulus of Collagen
2.4 Methods
2.4.1 Matrix Transfer Method
2.4.2 SED-FDTD Method
2.5 Results
2.5.1 Matrix Transfer Method
2.5.2 SED-FDTD Method
2.6 Conclusion
3 A Nonlinear Elastic Resonator 
3.1 Abstract
3.2 Introduction
3.3 Nonlinear Oscillators
3.3.1 Base Equation
3.3.2 Study of the Resonance Curves
3.4 Analytical Model
3.4.1 Derivation of the Resonator Equation
3.4.2 Complete Model of the Transmission
3.4.3 Study of the “S” Curve
3.5 Models and Methods
3.6 Results
3.6.1 Linear Resonator
3.6.2 Nonlinear Resonator
3.7 Discussion
3.7.1 Parametric Study
3.8 Conclusion
4 Mixing Laws for the Quadratic and Cubic Elastic Constants 
4.1 Introduction
4.2 1D Mass-Spring System Model
4.2.1 Low-concentration Case
4.2.2 High-concentration Case
4.3 Mixing Law for a Fluid in 3D
4.4 Relations Between Constants for Isotropic Solids
4.4.1 Compression Modulus
4.4.2 Tensor for the Linear Constants
4.4.3 Tensor for the Nonlinear Quadratic Constants
4.5 Landau Coefficients in a Heterogeneous Medium
4.5.1 Derivation to the Second Order
4.5.2 Expressions for the Quadratic Nonlinearities
4.5.3 Derivation to the Third Order
4.5.4 Expressions for the Quadratic and Cubic Nonlinearities
4.5.5 Amplification of the Nonlinear Effective Parameters
4.6 Interpretation and Exploitation
4.6.1 Interpretation
4.6.2 Values of the Constants
4.6.3 Exploitation
4.7 Conclusion
5 Numerical Study of 2D Nonlinear Phononic Crystals
5.1 Introduction
5.2 A tool for 2D Nonlinear Elastodynamics: Hedge
5.2.1 The Discontinuous Galerkin Finite Elements Method
5.2.2 Implementation of an Elastodynamics Operator
5.2.3 The Nearly Perfectly Matched Layers
5.2.4 Validation of the Elastodynamics Operator
5.3 Numerical Studies on 2D Structures
5.3.1 Validation of the Nonlinear Parameters
5.3.2 Validation of the Nonlinear Mixing Law in a Propagative System
5.4 Conclusion
General Conclusion 
Conclusion
Prospect
Bibliography 

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