Reinaldo Chacón

Lianchao Wang

Qingxiang Ji

Xueyan Chen

Xiaojun Tan

Nicolas Laforge

Julio Andrés

Ugarak, and Thiwanka Arepolage, the doctoral students who became my battle buddies and co-

The classical approach to designing or building any kind of object is thinking in the specific properties of the material, such as the use of glass when transparency is needed, metal when stiffness is sought, or wood for its cost and availability. Such material properties are so important for mankind that we divide History in ages named after the main material in use, i.e. the Stone age, the Bronze age and the Iron age. Nowadays, according to some we are in the age of Silicon or Information, but what is next? Could it be that a complete change of paradigm in the design and fabrication process for materials could occur, such as the one allowed by metamaterials (3D printing)? Rationally designed artificial media are stronger, lighter or even go far off the normal restrictions on effective properties. Many people actively contributing in the meta-'object' area think so.

In the particular field of mechanics, scientists have rather focused on the classical continuum models summarized in [START_REF] Christensen | Vibrant times for mechanical metamaterials[END_REF][START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF][START_REF] Kadic | 3D metamaterials[END_REF]; only a few works have reported the necessity of going beyond this limit [START_REF] Rueger | Strong Cosserat Elasticity in a Transversely Isotropic Polymer Lattice[END_REF][START_REF] Frenzel | Three-dimensional mechanical metamaterials with a twist[END_REF][START_REF] Coulais | A characteristic length scale causes anomalous size effects and boundary programmability in mechanical metamaterials[END_REF]. Indeed, natural materials have a very simple structure which is based on compact arrangement of atoms. Intuitively, the picture of compact spheres connected with springs in any crystalline symmetry tells us a lot about the general properties of solids: mass and stiffness are correlated [START_REF] Ashby | Materials selection in mechanical design[END_REF], shear and bulk stiffness are correlated [START_REF] Milton | The Theory of Composites[END_REF], elasticity tensors are rather symmetric, and structures do not present any chirality. These aspects have been mainly studied in the isotropic cases [START_REF] Milton | On cloaking for elasticity and physical equations with a transformation invariant form[END_REF][START_REF] Lakes | Extreme damping in composite materials with a negative stiffness phase[END_REF][START_REF] Florijn | Programmable Mechanical Metamaterials[END_REF][START_REF] Coulais | Discontinuous Buckling of Wide Beams and Metabeams[END_REF][START_REF] Norris | Acoustic cloaking theory[END_REF]. Controlling deformations (under external or internal stress) and wave 6 CHAPTER 1. INTRODUCTION AND STATE-OF-THE-ART propagation in mechanics is the "holy grail" (see Figure 1.1).

Conversely, one can argue that in many real-life applications, controlling the deformation is not a true requirement and that absorption or insulation are better concepts. Rubber shoe soles or oil-piston shock absorbers in cars exploit viscoelastic effects. Viscoelasticity allows for repeated operation over many cycles, yet the absorbed energy depends crucially on the timescale of loading and unloading [START_REF] Dixon | The shock absorber handbook[END_REF]15,16]. Absorption of large impact energies in car bumpers or lightweight bicycle helmets is based on destructive modifications of foams or metal structures, i.e., the required function can be obtained only once [17]. Clearly, a conceptually ideal shock-absorbing material would exhibit a low weight combined with a large energy-absorption capability in a structurally repeatable manner, allowing multiple use of the material. It has been pointed out that mechanical instability could be exploited to approach that ideal [18,19,16,20,[START_REF] Overvelde | Amplifying the response of soft actuators by harnessing snap-through instabilities[END_REF]. Proof-of-principle experiments on 3D polymer microstructures made by laser lithography are presented in Ref. [START_REF] Frenzel | Tailored buckling microlattices as reusable light-weight shock absorbers[END_REF]. However this approach lacks scalability toward low densities as the buckling process disappears. Assisted bucking via magnetic inclusion will be used as an alternative and new direction to all these presented works. The insertion of discrete continua with frictional contacts (dry or wet) and fluid structure inclusion will be also a great opportunity for re-usability and avoiding plastification and destruction.

In this thesis, I will show some extremal properties that can be obtained for acoustic and elastic metamaterials for their dynamical and static use. I will start (Chapter 1) by defining all necessary properties of acoustics and continuum mechanics and their recent progress, before entering the core work of Chapters 2 to 5, where we will design acoustical metamaterials with huge band gaps, tailor to will their dispersion relations and control their propagation directions before focusing on programmable and extremal elastic structures.

GENERAL INTRODUCTION

.1: Gallery of metamaterial unit cells and corresponding experimental realizations. Taken from [START_REF] Kadic | 3D metamaterials[END_REF] a) Unit cell with internal mass-spring resonance that leads to negative effective mass density. b) 3D labyrinthine metamaterials. c) Pentamodes. d) 3D chiral mechanical metamaterial. E) Buckling, multistable and programmable metamaterial. F) Truss lattices with a large coordination number. G) Programmable mechanical metamaterial. H) Two-component cell supporting sign reversal of thermal expansion it is a stimuli-responsive metamaterial. (See [START_REF] Kadic | 3D metamaterials[END_REF] for more information regarding the functionalities of each unit cell).

1.2/ METAMATERIALS AND PHONONIC CRYSTALS

For the last 50 years, a considerable effort has been made to design novel materials by structuration (composites, fibrous materials, multilayers) [START_REF] Milton | The Theory of Composites[END_REF], or by topology optimization in quasi-static conditions [START_REF] Yu | Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review[END_REF]. In the field of acoustics and mechanics, the holy grail has been to achieve the best stiffness or toughness, decrease the mass density, or to absorb/reflect waves or energy [START_REF] Yu | Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review[END_REF][START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF][START_REF] Lee | Micro-/Nanostructured Mechanical Metamaterials[END_REF]. Probably, one of the pioneer works on the idea of mechanical metamaterials was formulated in 1995 by Milton and Cherkaev [START_REF] Milton | Which Elasticity Tensors are Realizable?[END_REF] where the key point of achieving any elasticity tensor was made.

Later on, transformational elastodynamics [START_REF] Milton | On cloaking for elasticity and physical equations with a transformation invariant form[END_REF][START_REF] Norris | Acoustic cloaking theory[END_REF] and the generalized continua such as micropolar, micromorphic or Cosserat models [26]) led to the higher order gradient theories of elasticity [START_REF] Achenbach | Waves in elastic solids[END_REF] and to the modification of Newton's second law by Willis and Milton [START_REF] Milton | On modifications of Newton's second law and linear continuum elastodynamics[END_REF].

A complete review on mechanical materials can be found, for instance, in Ref. [START_REF] Kadic | 3D metamaterials[END_REF].

Here, we will only emphasize a few notable works, on mechanical metamaterials. The first class can be called lattice metamaterials which are often treated by Timoshenko's and Euler-Bernouilli beam theories. An extensive literature has been produced on the design of lattice metamaterials [START_REF] Martinsson | Vibrations of Lattice Structures and Phononic Band Gaps[END_REF].

The second class can be called extremal metamaterials for which some elastic properties are rather extremal compared to the others. In the isotropic case, the ideal pentamode metamaterials were introduced by Milton and Cherkaev [START_REF] Milton | Which Elasticity Tensors are Realizable?[END_REF]. Pentamodes are expected to have a low shear modulus, G, compared to a large bulk modulus, B, [START_REF] Kadic | On the practicability of pentamode mechanical metamaterials[END_REF]. Conversely, auxetics or dilational metamaterials have a large shear compared to bulk modulus [START_REF] Bückmann | On three-dimensional dilational elastic metamaterials[END_REF].

Shape morphing [START_REF] Neville | Shape morphing Kirigami mechanical metamaterials[END_REF][START_REF] Florijn | Programmable Mechanical Metamaterials[END_REF][START_REF] Coulais | Combinatorial design of textured mechanical metamaterials[END_REF][START_REF] Coulais | Multi-step self-guided pathways for shape-changing metamaterials[END_REF][START_REF] Wenz | Designing Shape Morphing Behavior through Local Programming of Mechanical Metamaterials[END_REF][START_REF] Zhu | Elastically and Plastically Foldable Electrothermal Micro-Origami for Controllable and Rapid Shape Morphing[END_REF][START_REF] Gladman | Biomimetic 4D printing[END_REF][START_REF] Mirzaali | Shape-matching soft mechanical metamaterials[END_REF], utilizing topological or space time properties [START_REF] Fleury | An invisible acoustic sensor based on parity-time symmetry[END_REF][START_REF] Liu | Designing 3D Digital Metamaterial for Elastic Waves: From Elastic Wave Polarizer to Vibration Control[END_REF] and the ability to control mechanical properties [START_REF] Milton | The Theory of Composites[END_REF] of functional materials [START_REF] Milton | Complete characterization of the macroscopic deformations of periodic unimode metamaterials of rigid bars and pivots[END_REF][START_REF] Jackson | Field responsive mechanical metamaterials[END_REF][START_REF] Qi | Recent Progress in Active Mechanical Metamaterials and Construction Principles[END_REF][START_REF] Galea | Reconfigurable magneto-mechanical metamaterials guided by magnetic fields[END_REF][START_REF] Korpas | Temperature-Responsive Multistable Metamaterials[END_REF][START_REF] Xin | 4D Printing Auxetic Metamaterials with Tunable, Programmable, and Reconfigurable Mechanical Properties[END_REF][START_REF] Rafsanjani | Programming soft robots with flexible mechanical metamaterials[END_REF][START_REF] Cai | Mechanomaterials: A Rational Deployment of Forces and Geometries in Programming Functional Materials[END_REF][START_REF] Overvelde | Rational design of reconfigurable prismatic architected materials[END_REF][START_REF] Song | Thermomechanically Triggered Reversible Multi-Transformability of a Single Material System by Energy Swapping and Shape Memory Effects[END_REF][START_REF] Li | Controlling Liquid Crystal Orientations for Programmable Anisotropic Transformations in Cellular Microstructures[END_REF] remain some of the main challenges in materials science. At the same time, the possibility of constructing materials possessing such properties is in high demand as it may lead to the design of structures that are superior to currently known biomedical and other devices used in various industries. In addition, the task of designing and manufacturing such materials becomes even more difficult at the small scale, such as the microscale, that is crucial from the point of view of many new types of applications, e.g. flexible electronics [START_REF] Russo | Pen-on-Paper Flexible Electronics[END_REF][START_REF] Xu | Flexible Visible-Infrared Metamaterials and Their Applications in Highly Sensitive Chemical and Biological Sensing[END_REF] and specialized medical equipment 1.2. METAMATERIALS AND PHONONIC CRYSTALS Figure 1.2: a) An ideal periodic unit cell of a pentamode metamaterial. Taken from [START_REF] Kadic | On the practicability of pentamode mechanical metamaterials[END_REF] with permission of the authors. [START_REF] Kolken | Rationally designed meta-implants: A combination of auxetic and conventional meta-biomaterials[END_REF][START_REF] Manen | 4D printing of reconfigurable metamaterials anddevices[END_REF]. However, despite the level of difficulty, over the years, it has been possible to note an emergence of promising directions of studies that address the possibility of observing these effects. In fact, one of the most interesting directions of studies related to this topic are mechanical metamaterials [START_REF] Evans | Molecular network design[END_REF][START_REF] Grima | Materials that push back[END_REF][START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF][START_REF] Coulais | Combinatorial design of textured mechanical metamaterials[END_REF][START_REF] Kadic | 3D metamaterials[END_REF][START_REF] Kern | Theory of the Hall effect in three-dimensional metamaterials[END_REF], i.e. structures that can exhibit counterintuitive mechanical behavior based primarily on their design. Auxetic behavior [START_REF] Wojciechowski | Two-dimensional isotropic system with a negative poisson ratio[END_REF][START_REF] Lakes | Foam structures with a negative Poisson's ratio[END_REF][START_REF] Evans | Auxetic Materials: Functional Materials and Structures from Lateral Thinking![END_REF][START_REF] Novak | Experimental and computational evaluation of tensile properties of additively manufactured hexa-and tetrachiral auxetic cellular structures[END_REF][START_REF] Lim | Composite microstructures with Poisson's ratio sign switching upon stress reversal[END_REF][START_REF] Mizzi | Auxetic metamaterials exhibiting giant negative Poisson's ratios[END_REF][START_REF] Grima-Cornish | Smart Honeycomb "Mechanical Metamaterials" with Tunable Poisson's Ratios[END_REF][START_REF] Farrugia | Tuning the Mechanical Properties of the Anti-Tetrachiral System Using Nonuniform Ligament Thickness[END_REF], negative stiffness [START_REF] Florijn | Programmable Mechanical Metamaterials[END_REF][START_REF] Hewage | Double-Negative Mechanical Metamaterials Displaying Simultaneous Negative Stiffness and Negative Poisson's Ratio Properties[END_REF] and negative compressibility [START_REF] Nicolaou | Mechanical metamaterials with negative compressibility transitions[END_REF][START_REF] Baughman | Materials with negative compressibilities in one or more dimensions[END_REF] are essential in many industries. However, in a vast majority of cases, standard mechanical metamaterial have proved insufficient during the search for structures capable of exhibiting versatile deformation patterns. This, in turn, has led to intensive studies devoted to hierarchical mechanical metamaterials [START_REF] Lakes | Materials with structural hierarchy[END_REF][START_REF] Gatt | Hierarchical Auxetic Mechanical Metamaterials[END_REF][START_REF] Cho | Engineering the shape and structure of materials by fractal cut[END_REF], i.e. structures composed of elements having their own geometry that normally can deform irrespective of the rest of the system. Despite the wide popularity of hierarchical mechanical metamaterials, it is a relatively new thread in the field of mechanical metamaterials. It seems that some of the first studies devoted to this topic were published by Gatt et al. [START_REF] Gatt | Hierarchical Auxetic Mechanical Metamaterials[END_REF] and Cho et al. [START_REF] Cho | Engineering the shape and structure of materials by fractal cut[END_REF] where the famous hierarchical rotating square system was proposed. In the following years, it was demonstrated [START_REF] Gatt | Hierarchical Auxetic Mechanical Metamaterials[END_REF][START_REF] Cho | Engineering the shape and structure of materials by fractal cut[END_REF][START_REF] Tang | Design of Hierarchically Cut Hinges for Highly Stretchable and Reconfigurable Metamaterials with Enhanced Strength[END_REF][START_REF] Dudek | Controllable Hierarchical Mechanical Metamaterials Guided by the Hinge Design[END_REF][START_REF] Cai | Hierarchical kirigami-inspired graphene and carbon nanotube metamaterials: Tunability of thermo-mechanic properties[END_REF] that this structure can exhibit tunable auxetic behavior where the extent of auxeticity depends on the relative rate of the deformation of hierarchical levels constituting the system. Recently, it was also reported that the design of the hierarchical square system can be modified in order to change its characteristics [START_REF] Li | Programmable mechanical metamaterials based on hierarchical rotating structures[END_REF][START_REF] Mizzi | Lightweight mechanical metamaterials designed using hierarchical truss elements[END_REF][START_REF] Coulais | Multi-step self-guided pathways for shape-changing metamaterials[END_REF][START_REF] An | Programmable Hierarchical Kirigami[END_REF][START_REF] Billon | Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials[END_REF]. Similar studies based primarily on the extent of the observed aux-eticity were also conducted for structures based on rotating rectangles [START_REF] Tang | Design of cut unit geometry in hierarchical kirigamibased auxetic metamaterials for high stretchability and compressibility[END_REF] as well as re-entrant and other honeycombs [START_REF] Tang | Design of cut unit geometry in hierarchical kirigamibased auxetic metamaterials for high stretchability and compressibility[END_REF][START_REF] Oftadeh | Optimal Fractal-Like Hierarchical Honeycombs[END_REF][START_REF] Mousanezhad | Hierarchical honeycomb auxetic metamaterials[END_REF][START_REF] Chen | 3D printed hierarchical honeycombs with shape integrity under large compressive deformations[END_REF][START_REF] Zhang | Crushing of vertex-based hierarchical honeycombs with triangular substructures[END_REF].

If we come to the field of dynamic metamaterials, we can distinguish two main domains.

The first one is periodic structures in mechanics, that also called composites by some people from the community. There has been a noticeable amount of work on such phononic crystals over the last 40 years. We can classify those crystals by the relevant dimension. For example, a large literature exists about plate-based crystals where authors study flexural or Love waves [START_REF] Korotyaeva | Love waves in two-dimensional phononic crystals with depth-dependent properties[END_REF]. Another example are the so-called surface elastic waves such as Rayleigh waves [START_REF] Benchabane | Surface-Wave Coupling to Single Phononic Subwavelength Resonators[END_REF]. When we consider fully 3D periodic structures, a considerable amount of crystals have been proposed for gap control, or for negative effective mass or negative bulk moduli [START_REF] Kadic | 3D metamaterials[END_REF].

In my thesis, as I will compete only in the fields of auxetic metamaterials and phononic crystals with a large gap, I will later cite the specific references needed for comparison.

Finally, for the scalar problem, i.e. acoustics, we can also find a large literature from bandgap creation [START_REF] Craster | Acoustic metamaterials: Negative refraction, imaging, lensing and cloaking[END_REF][START_REF] Christensen | Vibrant times for mechanical metamaterials[END_REF] at audible frequencies, sound absorption [START_REF] Khelif | Coupling characteristics of localized phonons in photonic crystal fibers[END_REF][START_REF] Khelif | Octave Omnidirectional Band Gap in a Three-Dimensional Phononic Crystal[END_REF], topological acoustics [START_REF] Ni | Observation of higher-order topological acoustic states protected by generalized chiral symmetry[END_REF][START_REF] Yang | Topological acoustics[END_REF][START_REF] Ma | Topological phases in acoustic and mechanical systems[END_REF][START_REF] Fleury | Sound isolation and giant linear nonreciprocity in a compact acoustic circulator[END_REF][START_REF] Laforge | Acoustic topological circuitry in square and rectangular phononic crystals[END_REF], low sound velocities [START_REF] Martin | Phonon band structures of three-dimensional pentamode metamaterials[END_REF] or ultrasound imaging [START_REF] Fenster | Three-dimensional ultrasound imaging[END_REF][START_REF] Wells | Ultrasound imaging[END_REF].

I will specifically present in Chapter 4 a contribution to topological phononics based on periodic acoustic waveguides with a glide-reflection symmetry and discuss the relevant literature there.

1.3/ WAVES IN PERIODIC MEDIA 1.3.1/ BLOCH THEOREM

Obtaining the effective properties of composites and heterogeneous structures is not an easy task. Along the general methods, one can find low and high-frequency homogenization [START_REF] Craster | Acoustic metamaterials: Negative refraction, imaging, lensing and cloaking[END_REF]. However, in this thesis, I will not use this method and will rely on the fact that we will only treat periodic media such as macroscopic crystals. The Bloch theorem is then very natural and can be easily applied to obtain the dispersion relations for different physical problems. For example, each Bloch state of a crystal can be written in an explicit form Ψ n along the following lines. For a periodic potential, the Floquet-Bloch theorem establishes that the eigensolutions of the system are in the form of a periodic function, with the same periodicity as the potential, times a plane wave or Bloch wave,

Ψ n = exp(ik B • r)u (1.1)
with k b the Bloch vector or the Floquet exponent, and u a function with the same periodicity as the potential of the problem. For example, in this thesis, it will be used for the displacement field in mechanics and for the pressure variation in acoustics. It could be used for the electric field in electrodynamics.

The Bloch theorem only applies to periodic media, often referred to as crystals. To restrict this introduction, we will only consider the case of periodicity and define the relevant quantities.

Figure 1.3 shows a crystal with a square lattice symmetry and the corresponding first Brillouin zone. The solution of the problem in reciprocal space also shares some symmetries as for the unit cell. Because of this, it is possible to establish a cell that fills up the reciprocal space, called the first Brillouin zone. Furthermore, the lattice symmetries of the unit cell translate into symmetries in reciprocal space. Some points are the most symmetrical and are called high-symmetry points. In this example, we have 9 high-symmetry points but only 3 of them are really different, Γ at the origin, X at (0, π/a) and M at the corner, with coordinates (π/a, π, a). We also depict the acoustic dispersion relation for the model treating the inclusion as rigid and the surrounding medium as air.

In order to illustrate the Bloch theorem, we depict in Figure 1.4, the pressure field for

1.3.2/ SYMMETRIES AND CRYSTALS

As stated previously, the Bloch theorem can only be of use in periodic media and is very convenient when it comes to crystals. The properties that a crystal fills up all space only using translation symmetry, with an underlying Bravais lattice and point group, and that the whole symmetries including those of the unit cell are described by a space group, play a key role for the generation of certain properties, i.e. elastic anisotropy and piezoelectricity.

In crystallography, a space group is defined as the symmetry group of an object in space [START_REF] Authier | International Tables for Crystallography[END_REF], i.e. in three dimensions. The elements of a space group are the rigid transformations of an object that leave it unchanged, or isometries. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct.

In 2D there are 17 space groups, also called wallpaper groups. Bravais lattices only account for the periodic translations of the unit cell. All together, there are 14 different Bravais lattices in 3D and 5 in 2D. We depict them in Figure 1.5.

In addition, when it comes to 2D problems but with only one direction of periodicity, we have only 7 1D space groups that are also called frieze group. These patterns describe the symmetry of pseudo 2D crystal waveguides, that we will study in later chapters. In

1.4/ CONTINUUM MECHANICS AND ELASTIC WAVES

In this section I will introduce the basic laws of continuum mechanics, first in the static case and then in the governing dynamical case.

1.4.1/ ELASTOSTATICS

The fundamental relation for any elastic interaction can be said to be Hooke's law, which for a spring is a linear relationship between force and displacement (elongation), the proportionality constant being the so-called elastic constant k. This law generalizes for a three dimensional continuum to a linear relationship between stress and strain, with a rank-4 stiffness tensor C describing the fundamental elastic properties of a material,

σ (r, t) = C (r, t) : ϵ (r, t) (1.2)
with σ (r, t) the stress tensor and ϵ (r, t) the strain tensor. Using thermodynamical principles applied to the intrinsic relationship between state parameters, it is possible to define and obtain a relation for the strain energy density W. The only condition for 1.3 to hold true, using Einstein's notation, is for a differentiable energy function to exist and to describe a stable system,

C i jkl = ∂ 2 W ∂ϵ i j ∂ϵ kl . (1.3)
The elastic tensor has the major symmetry C (i j)(kl) = C (kl)(i j) . Furthermore, the stress tensor σ i j = σ ji is also symmetrical. This results from the fact that for a force equilibrium neither rotations nor body torques are taken into account. This implies the minor sym-

metries C i jkl = C i jlk = C jikl = C jilk .
It is important to notice that to take into account the rotational parts of local deformations, one has to acknowledge that the classical continuum theory is not sufficient and that other theories, such as micropolar mechanics [START_REF] Eremeyev | Foundations of Micropolar Mechanics[END_REF],

have to be considered.

The major symmetry reduces the number of independent components from 81 to 45, and to 21 taking minor symmetries into consideration. Then, the full elastic tensor can be represented by a matrix of size 6 × 6 using Voigt's notation [START_REF] Hahn | International tables for crystallography. Volume A, Space-group symmetry[END_REF]:

i j = ⇓ α = 11
⇓ ⇓ ⇓ ⇓ ⇓ ⇓ 1 2 3 4 5 6
.

In Voigt notation, the elasticity tensor reads

C i jkl ⇒ C αβ =                                           C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66                                           .
This tensor further simplifies according for the point group symmetry of the material.

For the isotropic and cubic cases, it reduces to 9 non zero parameters which are not independent. In the isotropic case, there are only two independent elements 1.4, and three independent elements in the cubic case.

In the isotropic case, the two independent elements can be arbitrarily picked depending on the usefulness of the elastic relation and on the particular application. For example, in continuous mechanics the two most commonly used parameters are Young's modulus E,

C = E (1 + ν) (1 -2ν)      . (1.4)
But there also exist Lamé parameters λ and µ, which are used for ultrasonics and seismology, since these parameters directly relate to pressure and shear waves. Young's modulus directly relates the stress and the strain of an object subject to compression or tension. Poisson's ratio is the negative ratio of transverse strain to axial strain, Eq. (1.5)

ν = - ϵ load ϵ axial . (1.5)
For almost all materials found in nature this ratio is positive, but it has been shown that thanks to the metamaterial principles it is possible to obtain negative Poisson's ratios, at least in an effective sense. This type of materials are called auxetic metamaterials, the schematic of which is depicted in figure 1.8.

1.4.2/ ELASTODYNAMIC EQUATION

The elastodynamic equation can be derived from the momentum conservation principle and is

∇ • σ + b (r, t) = ρ (r, t) ∂ 2 u ∂t 2 . (1.6)
b is the body force per unit volume, or force density, and u is the displacement vector.

Using Hooke's law (1.2) we write the classical elastodynamic governing equation as

∇ • C (r, t) : ∇u = ρ (r, t) ∂ 2 u ∂t 2 . (1.7)
Imposing the solution in a homogeneous medium to be a harmonic plane wave at angular frequency ω, with arbitrary linear polarization u, and wavevector kn, the Christoffel equation is obtained

Γ -ρv 2 I u = 0 (1.8)
with Γ the Christoffel tensor of the form

Γ il = C i jkl n j n k (1.9)
and v = ω/k the phase velocity.

Since Γ is a 3×3 matrix, the eigenvalue problem has 3 solutions defining 3 normal modes.

For a given direction n, there are 3 bulk elastic waves. In addition, if the material is isotropic then there are 1 purely longitudinal/pressure wave, or P-wave, and two purely shear waves, or S-waves. In figure 1.9, the displacements associated with transverse and longitudinal waves in an elastic homogeneous medium are depicted.

1.5/ ACOUSTIC WAVES

Let's now focus on the scalar wave problem. If instead of solids, we want to describe the propagation of waves in fluids, then we enter the world of acoustics. It is important to note that the same governing equations are used for gazes and for fluids.

To derive the acoustic equation there are several paths, which involve in various ways mass conservation or the Navier-Stokes equations. Following the mass conservation and the momentum conservation principles, and making a linear approximation, we find the following equations in the absence of sources

∂ρ ∂t + ρ 0 c 2 0 ∇ • v = 0, (1.10)
ρ 0 ∂v ∂t + ∇p = 0, (1.11)
where p is the pressure variation, v is the fluid velocity, ρ 0 is the initial density of the fluid, and c 0 is the speed of sound in the fluid; The relationships p = c 2 ρ and c 2 = ∂p ∂ρ hold. These constitute the linear acoustic equations. Furthermore, combining equations 1.10 and 1.11 we reach the linear acoustic pressure wave equation

1 ρ (r) c 2 0 (r) ∂ 2 p ∂t 2 -∇ • 1 ρ (r) ∇p = 0. (1.12)
To describe experiments involving pressures waves interacting with static solid objets or walls, we will need boundary conditions. The two most common boundary conditions prescribe pressure or its normal derivative. The clamped or Dirichlet boundary condition is p = p 0 , and the free or Neumann boundary condition is ∂p ∂n = γ. The latter with γ = 0 describes a perfect wall cancelling the normal acceleration of the fluid; it is typically used in chapter 3. The former with p 0 = 0 describes a perfectly soft wall or interface with a low pressure gas, neglecting surface tension effects. Another particular boundary condition we will need to describe the interaction of pressure waves in water with elastic waves in solid inclusions is introduced in the following section. As a note, we will neglect in a first approximation the effects of losses, described by viscosity or thermo-viscosity, as they are not essential for our experiments.

1.6/ ACOUSTO-ELASTIC COUPLING

In some cases, an elastic body is immersed in a fluid like water. As seen previously, acoustic wave propagation in water is described by the scalar wave equation

∇ • (ρ -1 1 ∇p) = 1 κ ∂ 2 p ∂t 2 (1.13) where ∇ = (∂/∂x 1 , ∂/∂x 2 , ∂/∂x 3 ) T , T denotes the transpose, p(x, t) = p(x 1 , x 2 , x 3 , t)
is the deviation of the acoustic pressure from the ambient pressure, ρ 1 is the homogeneous mass density (in units of kg.m -3 ) of the fluid and κ = ρ 1 c 2 0 (in units of Pa) is the corresponding homogeneous bulk modulus. It is tempting to assume an almost infinite contrast as the bulk modulus of steel is almost two orders of magnitude larger than the bulk modulus of water. However, it has been shown that the elastic waves excited in the solid rods affect the dispersion of the crystal, so that their coupling with acoustic waves in water must be taken into account [START_REF] Laude | Phononic Crystals: Artificial Crystals for Sonic, Acoustic, and Elastic Waves[END_REF]. Thus, we model the metal as an elastic body by using the elastodynamic equation 1.7:

∇ • [C : ∇u] = ρ 2 ∂ 2 u ∂t 2 , (1.14)
where the displacement field u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) T , and C is the rank-4 (symmetric) elasticity tensor and ρ 2 (in units of kg.m -3 ) is the mass density of the solid. Equations (1.13) and (1.14) are related to each other via two coupling conditions at the interface between the fluid and the solid. First, the pressure in the fluid is continuous with the normal traction in the solid, or T i j n j = -pn i where T = C : ∇u is the stress tensor and n is the normal to the interface entering the fluid. Second, the normal acceleration is continuous and proportional to the gradient of pressure in the fluid according to

n • ρ 1 -1 ∇p = -n • ∂ 2 u ∂t 2 (1.15)
where the normal n now enters the solid. Therefore, by using this coupled acousto-elastic model we take into account the conversion of pressure waves, in the fluid, into coupled shear and longitudinal elastic waves, in the solid. This allows us to include subtle physical effects that would otherwise be missed by simpler models [START_REF] Wang | Hybridization of resonant modes and bloch waves in acoustoelastic phononic crystals[END_REF] for example, a model that comprises of the scalar wave equation (1.13) alongside rigid Neumann boundary conditions. We solve the coupled acousto-elastic equations using the finite element method (FEM) following the techniques detailed in [START_REF] Laude | Phononic Crystals: Artificial Crystals for Sonic, Acoustic, and Elastic Waves[END_REF][START_REF] Wang | Hybridization of resonant modes and bloch waves in acoustoelastic phononic crystals[END_REF].

1.7/ TOPOLOGICAL PHONONICS

In this section we first introduce briefly concepts related to Berry phase and different topological invariants, before discussing one of the simplest examples consisting of a diatomic 1D lattice that is often called the Su-Schrieffer-Heeger (SHH) model.

1.7.1/ BERRY PHASE

The general concept of topological states is best seen via some invariants of the system.

When the parameters R describing the evolution of the considered system change, the state |Ψ n ⟩ is expected to change along with it. If the parameters change slow enough (compared to the relevant characteristic times), it is said that there has been an adiabatic transformation. When this adiabatic transformation induces a difference between the state of the initial system and the system after the transformation, it can be characterised by an arbitrary complex phase, β(R) that is dependent on the parameter changes

|Ψ n ⟩ → exp(-iβ(R))|Ψ n ⟩ (1.16)
Since the physical content does not change by the multiplication with a complex phase factor, it is said to be a gauge transformation. This phase can be, then, separated between what is called the geometrical phase γ(R) (phase change of the eigenvector) and the dynamic phase (phase change of the eigenvalue). When the path of the transformation in time is a closed path, then continuity requires that the geometrical phase can only change by a factor of 2π, ensuring the state is the same at the end of a cycle as the initial state. When the evolution of the parameters is cyclic, then this geometric phase is called the Berry phase [START_REF] Shen | Topological insulators[END_REF]

γ n = C dR • A n (R) (1.17)
where

A n (R) = i⟨u n (R)|∇ R |u n (R)⟩ (1.18)
is called the Berry connection.

In the 1D periodic case, this phase is called the Zak phase [104]

Z = i π/a -π/a dk⟨u n |∂ k |u n ⟩. (1.19)
For other systems, other invariants can be selected. For example the Chern number is often used in 2D systems [START_REF] Shen | Topological insulators[END_REF] and is defined as

C = 1 2π S dk • Ω n (k) , (1.20)
with Ω n the Berry curvature of Bloch band n defined by

Ω n = i ⟨∂ k x u n |∂ k y u n ⟩ -⟨∂ k y u n |∂ k x u n ⟩ (1.21)
where integration is over the first Brillouin zone. Note that the Berry curvature appears after application of a Stokes-like theorem on the integral of the Berry connection along a closed path around the first Brillouin zone. The main representation for the SSH model is quantum and uses the Hamiltonian formalism. Here, I will use the dynamic matrix alternative since the method is preferable in elasticity and acoustics [some reference for SSH using dynamic formalism [START_REF] Deymier | Sound Topology, Duality, Coherence and Wave-Mixing[END_REF][START_REF] Laude | Phononic crystals: artificial crystals for sonic, acoustic, and elastic waves[END_REF]. This dynamic matrix comes from the general dynamic equation with no losses, Ku n = ω 2 n Mu n . Then, if the mass matrix is invertible, a dynamic matrix can be defined as

D = M -1K D (q) γ (q) = ω 2 γ (q) . (1.22)
As is shown in Figure 1.10, the model consists in two sites, in red an blue, that are coupled to each other with different hopping amplitudes, v and w, with a lattice constant a. Taking into account periodicity and after taking a Fourier transform, we get the dynamic matrix in reciprocal space

D (q) =           0 v + w exp(ıq) v + w exp(-ıq) 0           (1.23)
with q = ka the normalized wavenumber. Solving the eigenvalue problem, we can obtain the dispersion relation ω(q), very similar to the classical bilinear chain example (consisting of an infinite periodic structure an each unit cell made up of two different masses or springs)

(ω(q) 2 -ω 2 0 ) 2 = v 2 + w 2 + 2vw cos(q). (1.24)
It is not hard to see that in the two extreme cases when the hopping factor is 0, called the dimerized chain, and with equal coefficients, we go back to the linear chain with no gap, but for any other case a gap is formed. But, if this chain is finite, then the dimerized model have two cases, show in Figure 1.10 b) and c). When all the sites are connected with another site (w = 0) and when there are isolated sites at the edges (v = 0).

Since D(q) is a complex matrix, it can be separated into real and imaginary parts. In this case, we introduce the customary relation d(q) = v + w cos(q)iw sin(q) = d x + id y . We can then get D(q) and show if it is Hermitian, figure 1.11. The eigenvectors are as follows

|u⟩ = 1 √ 2           exp(-iϕ(q)) ±1           (1.25)
with ϕ(q) the phase of d(q). Using the equation 1.19 we can calculate the Zack phase Z = 1 2 dq dϕ(q) dq = ±π or 0.

(1.26)

This phase will have a value of either π or 0, depending on the parameters. For v > w it is always 0, hence we are in the topological trivial case by definition, and for w > v the phase is ±π and we are in the topological nontrivial case. An important remark is that the selection of the initial order of the cells produces an inversion of the classification of topological and trivial, meaning that the actual value of the Zak phase or any other topological invariant is not that significant. The crucial information is the change in the value, at this process refers to a topological transition. But since the void has 0 as topological invariant value, or Zak phase, if the phase changes from π to 0 there will be a transition at the edges, generating topological edges effects such as the appearance of an edge mode [START_REF] Laforge | Acoustic topological circuitry in square and rectangular phononic crystals[END_REF]. An example is depicted in Figure 1.12, where a finite chain of 10 units shows the opening of the gap and the appearance of edges modes when the condition w > v is fulfilled. most applications require to work with all spatial directions. In the past [START_REF] Khelif | Octave Omnidirectional Band Gap in a Three-Dimensional Phononic Crystal[END_REF], the process of fabrication of 3D phononic crystals has been an intensive manual labor and resulted in gaps opening as high as 500 kHz. Nowadays, it has been shown that additive manufacturing enables the realization of complex unit cell designs and thus the optimization of the bandgap width [START_REF] Belloni | Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal[END_REF][START_REF] Warmuth | Single phase 3D phononic band gap material[END_REF][START_REF] Lucklum | Bandgap engineering of threedimensional phononic crystals in a simple cubic lattice[END_REF][START_REF] Mcgee | 3D printed architected hollow sphere foams with lowfrequency phononic band gaps[END_REF]. However, mainly because of the limited printing resolution and available print materials, available frequencies remain well below 1 MHz, table 2.1.

In this chapter we design, optimize, manufacture and characterize a 3D phononic crystal at the micro scale operating in the megahertz range and exploiting the exceptional capabilities of direct printing technology.

Symmetry

Technique ω g ∆ω/ω g pc [START_REF] Iglesias Martínez | Three-dimensional phononic crystal with ultra-wide bandgap at megahertz frequencies[END_REF] 3D printing (TPL) 4 MHz 170% fcc [START_REF] Khelif | Octave Omnidirectional Band Gap in a Three-Dimensional Phononic Crystal[END_REF] steel balls and epoxy 500 kHz 60% hcp [START_REF] Tragazikis | Elastodynamic response of three-dimensional phononic crystals using laser Doppler vibrometry[END_REF] steel balls and paraffin 635 kHz 72% pc [START_REF] Belloni | Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal[END_REF] 3D printing (SLS) 11.36 kHz 132% sc [START_REF] Lucklum | Bandgap engineering of threedimensional phononic crystals in a simple cubic lattice[END_REF] 3D printing (SLA) 55 kHz 166% bcc [START_REF] Mcgee | 3D printed architected hollow sphere foams with lowfrequency phononic band gaps[END_REF] 3D printing (SLS) 11.3 kHz 48% sc [START_REF] Warmuth | Single phase 3D phononic band gap material[END_REF] 3D printing (SBEM) 90 kHz 22%

Figure 2.1: Principle of Direct Laser Writing: a laser is focused to a photoresist where passing an energy threshold two-photon abortion will polymerize a voxel. By moving the laser focal point or the substrate the voxel will create and arbitrary 3D structure.

2.1/ DIRECT LASER 3D PRINTING

Additive manufacturing is the process in which a structure is constructed layer by layer to make complex 3D designs varying from the meter scale all the way own to nanometer scale. One of the most well understood process is stereolithography (SLA), that consists in the photo polarization of each of the layers with the help of normally UV-light sources. However, the resolution of commonly used processes is not very high, normally it is around 100 µm. To increase the resolution, two-photon lithography (TPL) is a 3D direct laser writing technique inside polymers, developed during the last two decades, that has enabled researchers to manufacture objects with an overall size of a few millimeters and features as small as 100 nm.

This is thanks to the non linearity of the TPL process, which has a square dependency with respect to intensity, as figure 2.1 depicts. Taking this fact into account when we focus a laser beam, the writing light field for a Gaussian beam profile has twice the exponential decrement absorption as compared to a linear process such as one-photon absorption, as commonly for SLA. This property in turn drastically increases the resolution, but requires much more laser intensity or fluence. To meet such conditions, infrared femtosecond pulsed lasers at 780 nm are commonly used. The pulsed nature of the light field does not create a continuous structure but rather a discrete egg-shaped region of space called a voxel. Voxels are stacked with some overlap that can be controlled by the different parameters of the technique, such as the writing speed and the laser power, as figure 2.2 depicts.

As with any polymer, the final mechanical properties will be highly dependent on the fabrication process. In fact, we have used the distance between the voxels in the xy direction, called the hatching distance, and the direction in the z direction, called the slicing distance, as well as the laser power, in order to adjust the dose given to the monomer.

TPL has a flexibility that efficiently relieves fabrication and geometrical constraints, because, it enables the fabrication of overhanging parts that would appear to violate the law of gravity if one forgets about the viscosity of monomers and the extreme writing speeds [START_REF] Frenzel | Three-dimensional mechanical metamaterials with a twist[END_REF] that leave almost no time for the parts to fall down before getting fully connected.

A 3D phononic crystal can be indeed extremely complex and the extraordinary flexibility of 3D direct laser writing [START_REF] Yang | On the Schwarzschild Effect in 3D Two-Photon Laser Lithography[END_REF] makes the fabrication possible. Not only that, but thanks to the high resolution, TPL enables the possibility to work at higher frequencies. We used this technique to create a 3D phononic crystal with a complete bandgap working in the MHz regime.

2.2/ DESIGN OF THE UNIT-CELL

The space group is chosen to conform to symmetries of the perovskite structure, in the class of cubic crystals. Previous studies have indicated that very wide phononic bandgaps can be achieved based on this structure, benefiting from both Bragg scattering and local resonance [START_REF] Belloni | Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal[END_REF][START_REF] Lucklum | Bandgap engineering of threedimensional phononic crystals in a simple cubic lattice[END_REF]. Each cubic unit cell contains three balls or 6 half-spheres connected by thin bars. Figure 2.3 depicts the unit cell of the phononic crystal that we consider. We set the diameter of the bars to a small value ensuring mechanical stability during TPL, d c = 6 µm. Then the lattice constant is fixed as a = 300 µm, because this is the limit of the working area for the TPL without the necessity to move the stage, hence making fabrication more reliable. Only one free geometrical parameter remains, the radius of the spheres which is linked to their separation distance h. Optimization of the bandgap width then yields h = 40.8 µm and d s = 171.3 µm. The filling fraction is 39%.

The geometrical dimensions are chosen to respect fabrication constraints. Taken from [START_REF] Iglesias Martínez | Three-dimensional phononic crystal with ultra-wide bandgap at megahertz frequencies[END_REF] The phononic crystal is coated with a thin chromium coating to enhance vibration metrology. The top plate is here removed to show the internals of the crystal but was present during vibration experiments. (b) Phononic band structure obtained with the finite element method, with complete bandgaps outlined in gray. 5 × 7 × 0.5 mm. The femtosecond laser beam, centered at 780 nm, was focused by a 25× (0.8 NA) objective lens. After the writing, the sample was immersed in a PGMEA The phononic properties shown in Figure 2.4(b) are evaluated numerically with the finite element method [START_REF] Laude | Phononic crystals: artificial crystals for sonic, acoustic, and elastic waves[END_REF]. The phononic band structure is obtained by solving the eigenvalue problem of the elastodynamic equation

-ρω 2 u = E 2(ν + 1) ∇(∇ • u) + E ν + 1 ∇ 2 u (2.1)
where ρ is the mass density, under Bloch periodic boundary conditions. Examples of the modes at the lower band at the Γ point at the frequency around 0.5 MHz and at the upper band at around 8 MHz are shown in figure 2.5, a and b, respectively. Furthermore, transmissions are calculated using a frequency sweep under a time-harmonic excitation of the bottom surface of the crystal. The key point for the optimization of the bandgap is that the modes of the bars of the structure should match the higher bands of the structure such as the local resonances of the spheres, while keeping bars as thin as possible. The thinnest value for the bar is given by the geometry and the manufacturing process.

One of the biggest disadvantages of using polymers is that the material properties can change significantly depending on the manufacturing process, as well as the dependency with respect to the frequency. For that reason, it is needed to evaluate the mechanical properties of the bulk material. Special attention is paid to Young's modulus which is the fundamental property that has the biggest impact on the frequency response and on the band structure. There are a lot of methods for the characterization of Young's modulus, but in most cases samples with centimetre length are considered. For small samples, one of the most common method is micro-indentation. This test gives information on the formation of an indent as a function of the applied load. However, this test has two main drawbacks. The first is that the test does not give a direct measurement of Young's modulus but rather of the reduced Young's modulus which is related to Poisson's ratio. It is furthermore not a dynamical test so that only the static properties are obtained. For this reason, a different method has to be used for a correct characterization. The method selected was the mode localization of a clamped cantilever, with a theoretical value given by Euler-Bernoulli's beam theory [START_REF] Hagedorn | Vibrations and waves in continuous mechanical systems[END_REF]. The first longitudinal mode, then, depends only on the geometrical properties, and on Young's modulus, E, and the density, ρ. In equation (2.2) , we can see the formula for the resonant frequency, f b , of the first bending mode for a cantilever with a rectangular cross section with a length l and a thickness t. Hence, varying a geometrical parameter such as the length of the cantilever is sufficient.

f b = (1.875) 2 t 2πl 2 E 12ρ (2.2)
We use a laser Doppler vibrometer (Polytec, model MSA 500) and a piezoelectric patch that is directly connected to a power source exciting the sample. The out of plane displacement is measured, figure 2.6, to find the fundamental mode of the cantilevers. We have fabricated cantilevers with various lengths l using the same writing parameters as used for crystals. A SEM image of the fabricated cantilevers is shown in As an important difference between 2D and 3D crystals, in the 2D case there is a particular lattice, the triangular lattice, which is indistinguishable from an isotropic material in the long wavelength limit. This is not the case for 3D where there does not exist any lattice type implying a general isotropic behavior. This is important since in many applications we want to have an isotropic material. For this kind of phononic crystal with huge band gap, however, the anisotropic properties are also huge. To compare our structure with others we use the anisotropic universal anisotropy index [START_REF] Ranganathan | Universal Elastic Anisotropy Index[END_REF], A U . This index presents some advantages compared to other indices such as the Zenner ratio that only works for cubic symmetries, or the Ledbetter-Migliori index that is the square ratio between the maximum and the minimum of the shear velocities but cannot work for materials with Zenner ratios less than 1. The Universal index varies from 0 to ∞ and works for all symmetries and lattices, and is defined as in equation (2.3) , with C V the stiffness Voigt estimate tensor, and S R the compliance Reuss estimate:

A U = C V : S R -6.
(2.3)

For the chosen structure with a cubic symmetry we have only 3 different and non zero coefficients of the elastic tensor, C 11 , C 12 and C 44 , using the Voigt notation. These effective properties can be calculated using the Christoffel equation, as shown in [START_REF] Tsang | Sound velocity anisotropy in cubic crystals[END_REF][START_REF] Bückmann | On three-dimensional dilational elastic metamaterials[END_REF],

which results in the relations shown below. The phase velocities for different polarizations

EXPERIMENT AND TRANSMISSION SPECTRA

and the elastic elements of the elastic tensor C are related. It is convenient to select the velocities for the ΓM or [START_REF] Belloni | Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal[END_REF] direction, since in this direction it is possible to calculate all the coefficient extracting the velocities for the three polarizations, for one pure longitudinal (L) elastic wave and two shear waves polarized along z, and xy.

C 44 = ρ v T,z 110 2 , (2.4
)

C 12 = ρ v L 110 2 -C 44 -ρ ν T,xy 110 2 , (2.5)
C 11 = 2ρ v T,xy 110
2 + C 12 .

(2.6)

For the structure our equals A U = 0.52, which is much closer to the isotropic value of 0 than for the BCC symmetry [START_REF] Jia | Designing Phononic Crystals with Wide and Robust Band Gaps[END_REF] and the SC symmetry [START_REF] Lucklum | Bandgap engineering of threedimensional phononic crystals in a simple cubic lattice[END_REF], for which it can be calculated to be around 31 and 28, respectively.

2.3/ EXPERIMENT AND TRANSMISSION SPECTRA

In the experimental validation, we test only direction [START_REF] Hahn | International tables for crystallography. Volume A, Space-group symmetry[END_REF], or ΓX. Though other symmetry directions should in theory be included as well, especially ΓR, band foldings along the ΓX direction implies that the gap width is almost the same in all principal directions.

Furthermore, in the experiments we used a source of longitudinal elastic waves and we detected out-of-plane vibrations only. It could then be questioned if shear waves may propagate through the structure when longitudinal waves cannot. In the case of 3D phononic crystals, however, there is generally no decoupling of shear and longitudinal waves and we thus do not expect the presence of shear vibrations in the absence of out-of-plane vibrations. transmittance parameter S 12 , with port 1 connected to the transducer and port 2 receiving the output displacement signal from the vibrometer. The input power signal was 20 dBm.

The bandwidth of the measurements was set to 100 Hz, with an average factor of 10 to reduce measurement noise. An example of a single-frequency scan of the top surface is shown in Fig. 2.8(a). Measurements were performed in several positions at the top surface and were then averaged to improve the signal to noise ratio.

Measurements are reported in Fig. 2.9(a). The harmonic velocity ω|u z | is plotted as a function of frequency. Vibrations at the bottom surface were measured to check the frequency response of the PZT transducer. The wide bandwidth emission of elastic waves covers from almost a few KHz up to 10 MHz. The frequency response of the phononic crystal is compatible with a wide bandgap extending approximately from 0.6 MHz to 7.5 MHz.

Significantly, there is a sharp drop at the bandgap entrance and a transmission recovery above the bandgap, peaking around 8.5 MHz. The numerical displacement transmission plotted in Fig. 2.9(b) confirms those observations.

2.4/ CONCLUSIONS

In this chapter, we have shown the design of an ultra wide bandbap 3D phononic crystal for ultrasonic waves in the MHz range. Samples with complex unit cells were manufactured using two-photon lithography with lattice constants in the range of hundreds of micrometers, 300 µm. The experimental characterization unambiguously shows a bandgap extending from 0.6 MHz to 7.5 MHz. This study paves the way for the use of three dimensional two-photon lithography for ultrasonic applications.

ROTON-LIKE METAMATERIAL

For the past two decades, most of the work has been focusing on shaping wave propagation in phononic crystals and metamaterials for higher bands, with key attention for the bandgaps, spatial dispersion, local resonances and, more recently topological effects such as the one found in topological insulators. Strangely, engineering of the lowest band has fallen behind. Interaction beyond the nearest neighbors pave the way for a systematic route towards obtaining a large variety of behaviors for the lowest band.

Using the power of metamaterials [START_REF] Chen | Mapping acoustical activity in 3D chiral mechanical metamaterials onto micropolar continuum elasticity[END_REF][START_REF] Fleury | Non-local oddities[END_REF], possibility has been shown of the control of the coupling factor for interactions beyond the nearest neighbors, or non-local interactions. Using this mechanism it is possible to shape bands to have a "roton" minimum in the first band that has deep similarities with the elementary excitation in superfluid 4 He at low temperatures. Detailed theoretical [START_REF] Bogoliubov | On the theory of superfluidity[END_REF][START_REF] Feynman | Energy Spectrum of the Excitations in Liquid Helium[END_REF][START_REF] Schneider | Theory of the Superfluid-Solid Transition of $^4\mathrmHe[END_REF][START_REF] Glyde | Zero sound and atomiclike excitations: The nature of phonons and rotons in liquid $^4\mathrmHe[END_REF][START_REF] Kalman | Correlational origin of the roton minimum[END_REF] and experimental [START_REF] Woods | Neutron Inelastic Scattering from Liquid Helium at Small Momentum Transfers[END_REF][START_REF] Beauvois | Microscopic dynamics of superfluid $^4\math-rmHe$: A comprehensive study by inelastic neutron scattering[END_REF][START_REF] Maris | Phonon-phonon interactions in liquid helium[END_REF] investigations of this highly unusual dispersion relation in liquid 4 He remain subject of research until today [START_REF] Godfrin | Dispersion relation of Landau elementary excitations and thermodynamic properties of superfluid $^4\mathrmHe[END_REF].

In 2020 [START_REF] Kishine | Chirality-Induced Phonon Dispersion in a Noncentrosymmetric Micropolar Crystal[END_REF] and 2021 [START_REF] Chen | Mapping acoustical activity in 3D chiral mechanical metamaterials onto micropolar continuum elasticity[END_REF][START_REF] Fleury | Non-local oddities[END_REF] respectively, a few papers showed by theoretical calculations that roton-like dispersion relations may also occur in designed crystals or periodic metamaterials [START_REF] Cummer | Controlling sound with acoustic metamaterials[END_REF][START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF]. Here, quantum effects and correlations would play no role and low temperatures, which often hinder applications, would not be needed. The first paper [START_REF] Kishine | Chirality-Induced Phonon Dispersion in a Noncentrosymmetric Micropolar Crystal[END_REF] builds on micropolar continuum elasticity theory [START_REF] Eringen | Linear theory of micropolar elasticity[END_REF][START_REF] Eringen | Theory of Micropolar Elasticity[END_REF][START_REF] Chen | Mapping acoustical activity in 3D chiral mechanical metamaterials onto micropolar continuum elasticity[END_REF]. In this context, chirality based on broken centrosymmetry is a necessary condition for coupling to micro-rotations and hence for obtaining rotons [START_REF] Kishine | Chirality-Induced Phonon Dispersion in a Noncentrosymmetric Micropolar Crystal[END_REF]. The second paper [START_REF] Chen | Roton-like acoustical dispersion relations in 3D metamaterials[END_REF] suggests achiral and chiral three-dimensional periodic micro-and macrostructures. Herein, the mechanism for rotons is based on tailored third-nearest-neighbor interactions in addition to the usual nearest-neighbor interactions. For pronounced roton behavior, the effective strengths of the two interactions must be comparable.

In this chapter, we follow the specific achiral structure blueprints of Reference [START_REF] Chen | Roton-like acoustical dispersion relations in 3D metamaterials[END_REF] and manufacture the corresponding three-dimensional metamaterial by additive manufacturing. For each unit cell, we measure the acoustic pressure at hundreds of Hertz. The unusual acoustical-phonon dispersion relations obtained by Fourier transformation agree well with calculated roton band structures as well as with numerical finite-element calculations for the finite-size samples.

3.1/ EFFECTIVE-MEDIUM DESCRIPTION

We start with the simplest mathematical model for metamaterials which consist in point masses and perfect linear springs in 1D. The masses are identical and equally separated by the lattice constant a. The masses are connected with each other to their nearest neighbor with a linear spring with a constant K 1 , and with linear spring with constant K N connecting to the N nearest neighbor. Examples for the cases of N = 1, N = 2, and N = 3 are shown in figure 3.1, on panel a, b, and c respectively. It is important to mention that the springs are not interacting with each other, if this happened, then, the coupling of the masses would not be independent from their neighbors. A dynamical model for the position of the n mass, u n , is given by Hooke's and Newton's laws.

d 2 u n dt 2 = K 1 (u n+1 -2u n + u n-1 ) + K N (u n+N -2u n + u n-N ) , (3.1)
The solution of this dynamic equation are Bloch waves, u n (t) = u 0 exp (i (ωtkna)). Inserting this into equation (3.1) , yields the dispersion relation, ω (k), in the form

ω (k) = 2 K 1 m sin 2 ka 2 + K N m sin 2 Nka 2 . (3.2)
It is possible to argue that the interplay of N = 1 and N = 3 contributions leads to a phonon mode hybridization and thereby to extraordinary Bragg reflections that give rise to the occurrence of the roton minimum within the first Brillouin zone of the metamaterial crystal. Interestingly, while this statement is valid, it is not necessary to invoke Bragg reflections to understand or reproduce the roton minimum. To appreciate this point, we make the transition to an effective-medium description. As usual, the effects of Bragg reflections do not occur in effective-medium continuum theory, for which one considers the limit a z → 0. For simplicity and corresponding to our experiments, we consider only waves propagating along the z-direction with wavenumber k z for both, airborne pressure waves with the air-pressure modulation P, and elastic waves with the transverse displacement u. The wave amplitude, A, shall stand for either P or u. For an infinitely extended periodic lattice, we have the equation of motion for the amplitude A n = A n (t) at the integer lattice sites n = z/a z along the z-direction

d 2 A n dt 2 = C 1 (A n+1 -2A n + A n-1 ) + C 3 (A n+3 -2A n + A n-3 ) . (3.3)
For the case of elastic waves, where A n = u n stands for the transverse displacement u n of a mass m, we start from the mass-and-spring toy model discussed earlier. It is immediately obvious, looking at (3.1) that the coupling coefficients are given by

C 1 = K 1 /m
and C 3 = K 3 /m, with the Hooke's spring constants K 1 and K 3 .

3.2/ METAMATERIAL DESIGN

Since the interaction between the springs is unwanted, it is necessary to avoid any crossing in the design of the metamaterial that would act as an effective spring. However, it is important to point out that it is impossible to make it in 1D for N = 2 and above cases, and it is needed to pass at least to 2D, and for achieving N = 3 and above it is necessary to add yet another dimension passing to 3D, hence avoiding the interaction between springs, and making possible the consideration of high order interactions that are even stronger that the first neighbor interaction. Taking this into consideration, a blueprint was first depicted theoretically in Ref. [START_REF] Chen | Roton-like acoustical dispersion relations in 3D metamaterials[END_REF], for the acoustic case. The suggested metamaterial structures are illustrated in 3.2, with tetragonal symmetry and achirality. For the structure discussed in figure 3.2, since it is purely acoustic, only longitudinal airborne pressure waves propagate in the voids within the material, and the material itself ideally merely provides rigid (Neumann) boundaries to the air flow in the channel system. The waves in the structure in figure 3.2 is closer to rotons in superfluid helium than the ones in the elastic case in the sense that both are longitudinal waves.

The lattice constant is not fixed and therefore is free to choose, and the ratio a xy = 2a z . The metamaterial in figure 3.2 was selected to have a lattice constant of a z = 5 cm, so as to operate at audible frequencies and not to worry about imperfections in for the roughness of the walls. The metamaterial-beam sample considered has N z = 50 unit cells along the propagation direction (z-axis) and a finite cross section of N x × N y = 1 × 1 unit cells in the perpendicular xy-plane.

The mechanism leading to roton dispersion relations is the same for the metamaterial, as

show in the toy model of figure 3.1 (c). The small cubic voids, in light yellow, act as point masses or "atoms". The short (blue) channels mediate the nearest-neighbor interaction (N = 1) between the atoms. All other beams and channels (red), respectively, serve to mediate the third-nearest-neighbor interaction (N = 3). Intuitively, the length and cross section of the cylindrical elements effectively determine the strength of the corresponding interaction. Before the Fourier transformation, we multiply a Hann window [START_REF] Smith | Spectral audio signal processing[END_REF] onto the data to suppress possible artifacts from the two sample ends at n = 1 and n = 50, respectively. In the graphical representations below, we omit the negative wavenumbers due to the symmetry

ω (k z ) = ω(-k z ).
The other end of the beam is closed for the channel-based pressure-wave system, such that 100% of the wave is reflected at that end. In the presence of finite wave damping, the resulting waves are therefore a mixture of standing waves and propagating waves along the metamaterial-beam axis. At the launching end, the surface termination decides how well the excitation couples to the different modes at a given angular frequency, thus it determines the weight of the mode in the band structures to be shown below, figure 3.5. We have optimized the surface terminations of the two three-dimensional structures for roughly equal weights based on numerical finite-element calculations in order to avoid a tedious experimental trial-and-error procedure. In the higher-order-gradient effective-medium description outlined in the previous section, the effect of the surface termination, which is not expected for ordinary acoustical phonons, enters via the fourthand sixth-order spatial derivatives, which must be matched at an interface in addition to the first and second-order spatial derivative. The weight of the peaks in the false-color representation of the measured roton band structures in 3.5 should be taken with caution, especially in the frequency region where three peaks versus k z occur at a given frequency. First, at each frequency, we have normalized the Fourier transforms to unity power density to obtain a unique and unambiguous normalization. As a result, a single peak with fixed shape versus k z will always This means that one can choose between three different wavelengths and behaviors.

This aspect brings an entirely new design quality to the interface or transition region.

3.3.1/ ACOUSTIC SINGLE-FREQUENCY EXCITATION MEASUREMENT

A sinusoidal signal at a given frequency is produced by a computer, subsequently am-

3.4/ CONCLUSIONS

In this chapter, third neighbor interaction has been used to shape the first band of the dispersion relation. As a result, a metamaterial-beam dispersion relation shows a minimum, like in the case of superfluid 4 He at low temperature. The metamaterial was fabricated using additive manufacturing and was experimentally measured. The dispersion relation is obtained by recording the acoustic signal of the longitudinal airborne waves at each of the 50 unit cells. The metamaterial operates in the hundreds of Hertz, with a bandgap starting at around 450 Hz. Higher bands have also been measured and show a similar behavior as well as a flat band in the second band.

Finally, our experimental results for longitudinal airborne pressure waves in metamaterials can likely also be transferred to other areas of physics, for example to elastic and optical waves. Experiments for the elastic case has also been done at the KIT institute at the micro scale with equally convincing results.

As with any inflexion point, the roton presents an zero group velocity for a given wavenumber that could be used for energy harvesting. [START_REF] Wu | Scheme for Achieving a Topological Photonic Crystal by Using Dielectric Material[END_REF][START_REF] He | Acoustic topological insulator and robust one-way transport[END_REF][START_REF] Pal | Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect[END_REF]. In both cases, however, the available bandwidth for the dispersion of the guided wave is limited by the effective opening of the band gap that the control parameter allows [START_REF] Laude | Principles and properties of phononic crystal waveguides[END_REF]. In contrast, artificial crystals have long been designed to present very wide complete band gaps [START_REF] Vasseur | Experimental and Theoretical Evidence for the Existence of Absolute Acoustic Band Gaps in Two-Dimensional Solid Phononic Crystals[END_REF][START_REF] Laude | Phononic crystals: artificial crystals for sonic, acoustic, and elastic waves[END_REF][START_REF] Iglesias Martínez | Three-dimensional phononic crystal with ultra-wide bandgap at megahertz frequencies[END_REF] that the guided bands could in principle cover. Phononic crystal waveguides formed by coupling a sequence of crystal defects, however, lack topological protection and are generally multimodal, leading to a competition of the guided bands inside the complete band gap that can severely flatten the guided bands [START_REF] Laude | Principles and properties of phononic crystal waveguides[END_REF].

The Chapter is organized as follows. We first discuss the topology of the band structure of a square-lattice crystal and its transformation under a glide dislocation. We show how the dispersion of waves guided along the glide dislocation closes the complete band gap exactly for a half-lattice glide dislocation. Tuning the glide parameter, the spectral transmission can be changed continuously from no transmission at all to full transmission through the phononic band gap. The glide-reflection symmetric crystal waveguide offers wide bandwidth, single mode operation, and symmetry-protected backscattering immunity. An experiment performed with ultrasonic acoustic waves around 0.5 MHz and a crystal of steel rods in water demonstrates the operation of the glide-reflection symmetric phononic crystal waveguide.

4.1/ TOPOLOGICAL WAVES GUIDED BY A GLIDE-REFLECTION

Can we obtain topological crystal waveguides that make full use of a wide complete The guided waves for g 0 appearing inside the bandgap originate from the N-th and (N + 1)-th bands. Actually, as Fig. 4.2(b) shows for glide parameter g = a/2, all bands are degenerate by pairs at the X point of the first Brillouin zone. This essential property is obtained only for a half-lattice glide; it is shown later to signal a topological transition of the band structure occurring at g = a/2. Since the N-th and the (N + 1)-th bands were repelling and thus sitting on opposite sides on the bandgap for g = 0 and they are degenerate at the X point for g = a/2, they have to move inside the bandgap as g is tuned continuously between those two values. When g > a/2 and is tuned toward g = a, the gap closes continuously and symmetrically from the case g < a/2.

Why the N-th and (N + 1)-th bands hold a pair of guided Bloch waves can be understood based upon the transformation of the band structure under the continuous change of glide parameter g from 0 to a/2. The dispersion of the guided wave extends inside the complete bandgap, with a real wavevector k x ; along the y direction the guided wave is evanescent, i.e. its amplitude is decreasing exponentially.

4.2/ ZAK PHASE

Zak phase was originally introduced [START_REF] Zak | Berry's phase for energy bands in solids[END_REF] for 1D crystals as the integral of the Berry connection along the 1D Brillouin zone. The 2D Zak phase for 2D crystals [START_REF] Liu | Novel Topological Phase with a Zero Berry Curvature[END_REF][START_REF] Liu | Topological photonic crystals with zero Berry curvature[END_REF] is a natural generalization where the integral of the Berry connection is taken along a 1D contour, chosen as the interface direction in reciprocal space [START_REF] Ma | Topological phases in acoustic and mechanical systems[END_REF]. Namely, Zak phase for band n is

γ n = C dR • A n (R), (4.1)
with R = k x at fixed k y and C = [-π/a; π/a]. The Berry connection is

A n (R) = i⟨u n (R)|∇ R |u n (R)⟩, (4.2)
with u n (R) a Bloch wave defined over the 2D unit cell and ⟨.⟩ denoting the scalar product in real space defined on this unit cell. Note that the integration contour chosen, C, is different from the one used to define Chern numbers, that is the boundary enclosing the first 2D Brillouin zone. By construction, the bottom crystal B is the glide-reflection (GR) image of crystal A. The glide operation implies a phase change for every Bloch wave ϕ(k x ) = -gk x (a translation of the origin by g). Since the Berry connection changes as

A n (R) → A n (R) -∂ϕ ∂k
x under any phase change, we have

γ n (B) = γ n (A) + 2πg/a. (4.3)
Hence there is a π change of the 2D Zak phase across the interface for every band, for g = a/2 exactly. Since the Zak phase is 2π-periodic, its value alternates by π between both crystal images.

Why degeneracy occurs at the X point of the Brillouin zone can be explained considering a compact demonstration based on operators of the 1D crystal interface. For any glide parameter g, the composition G a-g •G g is the translation by one lattice constant T a in direct space. In reciprocal space, this implies G a-g (k)G g (k) = exp(ika). For g = a/2, we then have Summarizing, each complex conjugate eigenvector pair shares a degenerate eigenvalue at the X point. There is in addition a vertical mirror symmetry, resulting in a total of four combinations. (b) At the X point of the first Brillouin zone, the symmetry of the complete Bloch waves is the combination of the symmetry of the periodic part of an alternating signum function with the periodicity of the lattice. The dispersion of the S and AS GR Bloch waves are degenerate at the X point, since those waves are images in a vertical reflection. Note that the periodicity of Bloch waves is 2a. (c) At the Γ point of the first Brillouin zone, the symmetry of the Bloch wave is simply the symmetry of its periodic part. The S and AS GR modes are never degenerate. Note that the periodicity of Bloch waves is a.

G 2 g (k) = -1

4.3/ MODAL SYMMETRY

In this section, we examine the degenerescence of Bloch waves at the X point of the Brillouin zone for g = a/2 in direct space. For simplicity, we consider the case of scalar pressure waves.

The 2-periodic crystal initially has symmetry space group p4m and the supercell has frieze group p2mm for g = 0. When the glide operation is applied, the structure looses one periodicity and the horizontal reflection is replaced by a glide-reflection; the space group or frieze group of the waveguide structure becomes p2mg. The glide-reflection is the essential symmetry operation of the p2mg space group. This implies that periodic eigenmodes separate into two orthogonal families: they are either GR symmetric or GR antisymmetric. At the X point of the first Brillouin zone, ka = π and the plane wave term exp(ıka) = -1 results in an alternation of the sign of the amplitude of Bloch waves between opposite sides of the unit cell. As a whole, the combination of GR symmetry and sign alternation leads to degeneracy of pairs of Bloch waves, as we explain next.

We first observe that Bloch waves combine the symmetry of the crystal, in direct space, and of the Bloch wavevector, in reciprocal space. Indeed, the Bloch wave is the product of a periodic function P(r), having the periodicity of the crystal, and of a plane-wave like term depending on the Bloch wavevector k:

P(r, t) = P(r) exp(ı(ωt -k • r)). (4.4)
P(r) inherits its symmetry from the space group of the crystal. In the case of the squarelattice crystal of circular rods subjected to an horizontal glide operation, there are a vertical mirror and a horizontal glide-reflection. P(r) can be symmetric (S) or anti-symmetric (AS) with respect to these two basic symmetries. In Fig with period 2a and alternates its sign between successive unit cells. The product of P(r)

and exp(-k • r) hence recomposes the distribution of the Bloch wave P(r, t) at a fixed time t, as depicted in Fig. 4.3(c). All possible spatial distributions are identical up to a vertical mirror symmetry, so that the eigenvectors are orthogonal but lead to the same degenerate eigenvalue. This reasoning is only valid exactly at the X point and for g = a/2.

4.4/ OBLIQUE LATTICE

The glide-reflection symmetric interface created from a square lattice crystal, as considered in the main text, possesses the additional symmetries of the p2mg frieze group: a vertical reflection and an inversion. In order to test the influence of those additional symmetries, we consider the oblique lattice in In Fig. 4.4(a-b) we consider the case of a glided inversion-symmetric interface. Whatever the value of the glide parameter, the frieze group of the interface remains p2. As is seen immediately, the two guided waves obtained for g = a/2 repulse each other at the X point of the first Brillouin zone. Hence, it is readily concluded that inversion symmetry does not account for the phenomenon described in the main text.

Next, we consider in Fig. 4.4(c-d) the case of the GRS interface. The frieze group of the interface is p11m for g = 0, p11g for g = a/2, and p1 for any intermediate value.

Clearly, the two guided waves for g = a/2 cross without interacting at the X point of the first Brillouin zone and degeneracy of the eigenvalues applies. Hence, we conclude that glide-reflection symmetry of the interface is the required symmetry for observing the phenomenon described in the main text. Note that 6 of the 7 frieze groups have been considered so far (p2mm, p2 and p2mg for the square lattice; p11m, p1 and p11g for the oblique lattice), and that the remaining p1m1 frieze group only combines a vertical reflection and lattice translations, so that all frieze groups have been exhausted in our discussion.

4.5/ RESILIENCE OF GUIDED WAVES TO DISORDER

Resilience to disorder is often advocated as a property seperating topological structures from trivial ones, such as defect crystal waveguides. In order to test the resilience of the guided waves of the GRS crystal interface, we add random fluctuations on the position and the diameter of the inclusions. The random noise is uniformly distributed in a given interval around the mean value. Because d/a = 0.9 in our crystal structures, we limit the interval to 5% of the lattice constant for position disorder and to 10% for diameter disorder.

Beyond those values it is likely that the inclusions could overlap.

The disordered crystal interface is no more periodic and only approximately glidereflection symmetric. Hence instead of a band structure we consider the frequency response function (FRF) of a finite crystal structure, as shown in Fig. 4.5. Incoming waves are numerically excited from an internal boundary in the shape of an arc of a circle, focused on the left entrance of the waveguide. The normal acceleration on this boundary is enforced weakly in the finite element model, in order to represent a source of acoustic waves in water. A similar internal boundary is placed at the right exit of the waveguide and collects the mostly cylindrical acoustic waves radiated from the exit. The FRF is defined as the pressure integrated over that boundary.

As Fig. 4.5 shows, the FRF is not significantly affected by position disorder compared to the reference value (no disorder). As position disorder increases from 1% to 2% and then to 5 %, transmission of waves is always observed over the passband of the waveguide.

Diameter disorder affects the FRF even less than position disorder. Taken from [START_REF] Iglesias Martínez | Glide-Reflection Symmetric Topological Phononic Crystal Waveguide[END_REF]. For a half-lattice glide parameter (g = a/2), the (b) numerical and (c) experimental acoustic wave transmission covers most of the complete phononic band gap (highlighted with the grey color). The reference for transmission is the measurement in the absence of the phononic crystal waveguide.

We now turn to the experimental results for the glide-reflection symmetric phononic crystal waveguide. [START_REF] Iglesias Martínez | Glide-Reflection Symmetric Topological Phononic Crystal Waveguide[END_REF]. (a) For g a/2, a mini-gap for guided waves opens in the phononic band structure at the X point of the Brillouin zone. (b) Experiment confirms the opening of the mini-gap, for g = a/4 and g = 3a/8. (c) The eigenfrequencies of the two guided waves at the X point vary with the glide parameter (blue line: S waves; red line: AS waves). For exactly g = a/2, the waveguide is glide-reflection symmetric and the guided wave gap closes. This gap opens symmetrically on either side of that value.

parameter g = a/2.

4.8/ OTHER MATERIAL SYSTEMS

In this section, we review a number of material systems for phononic crystals and show that the main result of the main text applies to all of them: the glide-reflection symmetry creates a crossing at the Brillouin zone boundary of the waveguide that implies the existence of pairs of symmetry-protected guided waves.

4.8.1/ SONIC CRYSTAL OF RIGID INCLUSIONS IN A FLUID

Previously, the full coupling of acoustic waves in water and elastic waves in steel was taken into account. Here, we consider a simplified model of perfectly rigid rods in water whereby no elastic waves are excited in the solid parts and the acoustic waves are perfectly reflected at the water/rod boundary. The corresponding boundary condition is the Neumann or free boundary condition (the normal derivative of pressure vanishes along the water/rod boundary). Such a simplified model is generally used for acoustic waves in air, since the elastic contrast with steel or most solids is huge. The phononic band structures in Fig. 4.9 show the opening of the complete phononic band gap for g = 0 and the appearance of the pair of non-interacting guided waves for g = a/2. The result is essentially similar to the discussion in the past sections, with a slight frequency shift of all bands since acousto-elastic coupling is not taken into account (note that taking this coupling into account is essential is order to match with experimental results).

Furthermore, the result applies to all single-phase fluids. Indeed, the acoustic wave equation for pressure is

-∇ • 1 ρ(r) ∇p(r) = ω 2 1 B(r) p(r), (4.5)
with B and ρ the elastic modulus and the mass density of the fluid. For a single-phase fluid with an arbitrary spatial distribution, ρ -1 (r) = ρ -1 0 w(r) and B -1 (r) = B -1 0 w(r), where w(r) is a window function that equals 1 in the fluid and 0 in the rod. For water, we take B 0 = 2.2 GPa and ρ = 1000 kg/m 3 . As a result, the acoustic wave equation becomes

-∇ • (w(r)∇p(r)) = ω 2 c 2 w(r)p(r), (4.6

4.8.3/ PHONONIC CRYSTALS OF STEEL RODS IN EPOXY

Consider the case of solid inclusions embedded in a solid matrix. For definiteness, we consider the case of steel rods in epoxy, for which a very wide phononic band gap is known to exist [START_REF] Vasseur | Experimental and Theoretical Evidence for the Existence of Absolute Acoustic Band Gaps in Two-Dimensional Solid Phononic Crystals[END_REF]. Independent material constants for isotropic steel are c 11 = 264 GPa,

4.8.4/ CONCLUSION

Summarizing the results above, the interface waves are protected by a class of topology relying on spatial symmetries and thus belong to crystalline topological insulators [START_REF] Prodan | Bulk and Boundary Invariants for Complex Topological Insulators[END_REF]. Whereas crystalline topological phases generally induce interface waves that have a gapped spectrum, because the interface breaks the corresponding spatial symmetries, the glide symmetry of the interface ensures a gapless Dirac point at the X point of the Brillouin zone.

The importance of glide-reflection symmetry is further verified in by considering the oblique lattice instead of the square lattice. It is specifically found that inversion symmetry [START_REF] Li | Topological interface states in the low-frequency band gap of one-dimensional phononic crystals[END_REF] combined with the glide operation leads to a gapped spectrum, unlike GRS.

We note that glide-reflection symmetric waveguides have been considered before, e.g. for microwaves [START_REF] Quevedo-Teruel | On the Benefits of Glide Symmetries for Microwave Devices[END_REF] or acoustic waves [START_REF] Jankovi Ć | Glide-Symmetric Acoustic Waveguides for Extreme Sensing and Isolation[END_REF], but that waveguiding is in this case ensured by structural boundaries rather than by a phononic band gap. The existence of a complete band gap without glide (g = 0) is indeed essential to our result.

The resilience of the interface waves to disorder, a direct check of symmetry protection.

It is observed numerically that they survive a position disorder of at least 5% of the lattice constant and an inclusion diameter disorder of 10%.

The discussion so far has been limited to the scalar case of acoustic waves and to a square lattice crystal of steel rods in water. It is obvious, however, that the symmetry principles involved extend the existence of glide-reflection symmetric crystal waveguides to other material systems and lattices. As illustrated for the cases of acoustic waves in a fluid with rigid inclusions and of vector elastic waves in a solid perforates with cylindrical holes or containing solid inclusions. By virtue of the well-established analogies between acoustic/elastic waves and optical/electromagnetic waves [START_REF] Kushwaha | Acoustic band structure of periodic elastic composites[END_REF], the transposition to photonic crystals is straightforward. Other wave systems such as plasmonic crystals, gravitycapillary waves at the surface of water [START_REF] Laforge | Observation of topological gravity-capillary waves in a water wave crystal[END_REF], or solutions of the Schrödinger equation are likely to present similar properties too.

On the practical side, the glide operation offers the opportunity to design a continuously varying transmission that can be changed from no transmission (for g = 0) to full transmission through the phononic band gap (for g = a/2). As a waveguide for transmission of information, the glide-reflection symmetric crystal waveguide offers wide bandwidth, single mode operation, and symmetry-protected backscattering immunity.

AUXETIC HIERARCHICAL MECHANICAL

METAMATERIALS

In this chapter, I show step by step the design of a novel auxetic hierarchical mechanical metamaterial and the way it can be used for shape morphing.

5.1/ DESIGN AND FABRICATED SAMPLES

The starting point is the design of a unit cell which should have the ability to exhibit a wide range of auxetic behavior as well as shape morphing. As shown in Figure 5.1(a), the 2D version of the considered system is a two-level hierarchical structure. However, unlike a vast majority of already known hierarchical mechanical metamaterials, the deformation mechanisms corresponding to each of the hierarchical levels of the system are very different. More specifically, as shown in Figure 5.1(b), the structure selected to form level 0 of the system corresponds to the non-auxetic rotating triangles structure with its unit-cell having a shape of a rectangle. This geometry was first proposed by Milton et al.

[41] where it was used in order to construct mechanical expanders capable of significantly changing their dimensions during the deformation process. A few years later, Dudek et al.

[171] demonstrated that this specific mechanical metamaterial never exhibits auxetic behavior in the axial directions, but instead that it can be characterized by the negative linear compressibility irrespective of the geometric parameters used to define it. On the other hand, the deformation mechanism associated with level 1 of the structure corresponds to the rotating square [START_REF] Grima | Auxetic behavior from rotating squares[END_REF] / rectangle [START_REF] Grima | On the Auxetic Properties of 'Rotating Rectangles' with Different Connectivity[END_REF] system proposed initially by Grima et al. with the rotating square-like geometries being famous for their ability to exhibit auxetic behav- ior. In view of this, the deformation of the hierarchical system considered in this work corresponds to the interplay between the two hierarchical levels associated with a positive (level 0) and negative (level 1) Poisson's ratio respectively (this interplay is referred to as mutually-competing substructures). Thus, depending on the extent of the mechanical deformation of the two hierarchical levels, it is expected that the Poisson's ratio of the entire structure can be significantly modified. At this point, it should be mentioned that even though in this work a specific configuration of rotating triangles is considered, in the literature, there are also other types of triangle-based geometries [START_REF] Grima | Auxetic behaviour in non-crystalline materials having star or triangular shaped perforations[END_REF][START_REF] Grima | Auxetic behavior from rotating triangles[END_REF][START_REF] Zhou | Negative linear compressibility of generic rotating rigid triangles[END_REF][START_REF] Grima | On the auxetic properties of generic rotating rigid triangles[END_REF] that in general may exhibit other forms of unusual mechanical behavior.

In terms of the geometry, the two-dimensional analog of the analyzed structure corresponds to the two-level hierarchical structure where level 0 is composed of structural elements closely resembling isosceles triangles (see Figure 5.1(a)). On the other hand, level 1 structural elements correspond to rectangle-like motifs that for the specific case of θ 0 = 90 • assume a shape of a square. In the case of the provided example (see Fig- Furthermore, one should note that even though the structure portrayed in Figure 5. 1(d) is being referred to as being two-dimensional, in reality, it has a non-zero out-of-plane thickness that corresponds to the variable l s , or quasi-2D . In addition, it should be emphasized that the unit-cell of the 3D version of the considered hierarchical mechanical metamaterial is very similar to its 2D equivalent. As shown in Figure 5.1(c), walls of the unit-cell of the 3D system are almost identical to the 2D unit-cell with the addition of solid blocks having dimensions l × l × h that help to connect adjacent walls.

5.2/ FABRICATION

The micro-scale samples investigated in this work were fabricated by polymerizing individual unit-cells assuming a small overlap between the adjacent cells. Such overlap was introduced to ensure that the adjacent unit-cells would be connected to each other to form larger composites analyzed in this study. To this aim, the commercial 3D printer (Photonic Professional GT+, Nanoscribe GmbH) operating based on the two-photon lithography method was used. The photoresin selected to produce the analyzed mechanical metamaterials was the commercial negative tone IP-S resin (Nanoscribe GmbH) that is customized to work well with the Nanoscribe 3D printer. Furthermore, the slicing and hatching distances associated with the samples were set to be equal to 1 µm and 0.5 As a result, the bottom part of the samples was constrained from the movement during the deformation process.

µm

5.3/ MECHANICAL TESTING AND NUMERICAL MODELS

As stipulated in the Appendix A, I have used the full advantage of my developed setup to get the stress strain behaviour of the samples and at the same time to monitor the local strains in order to obtain Poisson's ratio for each sample.

To test the mechanical properties of the analyzed samples, they were uniaxially compressed at the constant rate of 1 µm/s. The indenter used to induce the compression had a circular polished surface corresponding to the diameter approximately equal to 3 mm. Each sample was placed on top of the leveling table in order to ensure its correct orientation during the experiment. In addition, to ensure the better stabilization, the entire set-up was also placed on top of the optical table. To record the deformation process, an optical camera equipped with a lense corresponding to the 20X and 10X magnification factor was used. It is also worth mentioning that prior to the testing, in order to have high-resolution pictures showing the structural details of the printed samples, the SEM microscope was used (Apreo S -Thermofisher). Finally, it should be emphasized that the experiments were continued until the point when hinges connecting adjacent structural elements were breaking at large strains. In addition, to assess whether a given breaking hinge was significantly affecting the deformation pattern, the force vs displacement relationship was analyzed throughout the experiment. It should be also emphasized that if one was to construct the system with more durable hinges, then considered structures could be deformed to an even larger extent than is the case in this study. Furthermore, it should be mentioned that the experimental data presented in Figure 5.2 were obtained based on pictures taken during the experiment by means of the optical camera. To this aim, positions of characteristic vertices (e.g. edges of hinges connecting level 0 and level 1 blocks) were extracted from respective pictures and utilized to calculate the desired geometric parameters by means of standard mathematical operations.

To ensure the validity of the experimental results, the mechanical behavior of the analyzed hierarchical metamaterials was analyzed by means of the FEA simulations. To do this, COMSOL Multiphysics software was used. Since the behavior of the structure was investigated solely from the point of view of its deformation, only the Structural Mechanics module was implemented. In addition, to minimize the computation time, the geometries of the quasi-2D experimental prototypes analyzed in this study were converted into regu-lar 2D structures. Furthermore, to replicate the behavior of the experimental prototypes, the surfaces at the bottom of the structure were fixed in space. Thus, as the samples were compressed, the bottom part of the system could not move. Such deformation process associated with the uniaxial compression of the system was induced through the use of the external indenter pushing the topmost part of the structure downwards. Finally, it should be mentioned that the material corresponding to the analyzed samples was assumed to be isotropic and its properties were set to be the following: ν = 0.4, E = 4 GPa.

It should be noted that for the considered structures, the selection of the model of elasticity does not significantly affect the mechanical behavior of the system since it depends primarily on its design. Furthermore, it should be emphasized that while the considered geometry is nonlinear, the conducted FEM simulations were linear. This stems from the fact that the considered structures are deforming primarily through hinging of structural elements that are retaining their shape throughout the deformation. Thus, the plasticity of a material is not significant from the point of view of the deformation of considered structures and internal stresses are relatively small in comparison to the Young's modulus of the polymer. At this point, even though the following does not belong to the scope of this work, it should be also mentioned that deriving dynamics-related properties would be very difficult in the general case for this work since they would always depend on the material's non-linear properties [START_REF] Parnell | Employing pre-stress to generate finite cloaks for antiplane elastic waves[END_REF].

5.4/ 2D HIERARCHICAL METAMATERIALS -CONTROL OVER ME-CHANICAL PROPERTIES

In order to assess the ability of the considered model to exhibit a tunable Poisson's ratio, three different quasi-2D hierarchical structures (see Figure 5.2) are considered with the main difference between them being the thickness of hinges connecting level 0 and level 1 elements.

As shown in Figure 5.2(a), the first of the analyzed structures (case 1) corresponds to the hierarchical system where level 1 building blocks assume the shape of a square, since θ 0 = 90 • . Based on the pictures taken during the experiment, one can note that during the deformation process corresponding to compression along the z-axis, the considered part of the structure, i.e. the central topmost unit-cell, acts as a typical rotating square system. For all mechanical metamaterials portrayed on panels a-c), the red and blue dashed lines correspond to the outline of the initial and deformed unit-cell respectively. All of the experimental pictures shown on panels a-c) were taken by means of the optical camera. Finally, the surface stress presented graphically for the FEA results on panels a-c) corresponds to the von Mises stress model implemented in the COMSOL Multiphysics software.

In other words, level 1 structural blocks retain their shape throughout the deformation and rotate with respect to each other which leads to a change in the angle θ 1 . As presented in Figure 5.2(d), this deformation mechanism results in the highly negative and almost isotropic Poisson's ratio close to -1, i.e. the extent of auxeticity characteristic for the rotating square system. The specific behavior of the structure referred to in this work as case 1 stems from the fact that hinges connecting level 0 elements are thicker than hinges connecting level 1 blocks (d 1 ≈ 9.3 µm, d 2 ≈ 6.1 µm). At the same time, all hinges are thin enough so that all of the unit-cells deform primarily through hinging between their structural elements. As a result, it is much easier for the level 1 blocks to rotate with respect to each other than is the case for level 0 elements where the resistance to the rotational motion is significantly larger. Hence, during the deformation process, θ 1 changes to a large extent while θ 0 approximately retains its value (see Figure 5.2(e)). At this point, it should be also noted that even though the initial configuration of the system allows for a very large extent of the mechanical deformation, it does not correspond to

the fully-open configuration, where θ 0 = 90 • and θ 1 = 90 • . This stems from the fact that for such system, it would be difficult to predict the direction of rotation of structural elements since it would depend on the polymer's composition and printing imperfections.

According to Figure 5.2(b), the hierarchical structure referred to as case 2 exhibits a very different behavior than was observed for case 1. More specifically, during the deformation process, level 1 blocks having a rectangular shape become significantly flattened in one dimension with the orthogonal dimension being slightly increased (i.e. level 1 blocks exhibit the negative linear compressibility behavior characteristic for the non-auxetic rotating triangle system). Furthermore, the change in the shape of the level 1 blocks is accompanied by the mutual rotation of level 0 elements to a significant extent (see Figure5.2(e)).

As shown in Figure 5.2(d), this change in the behavior of the hierarchical structure significantly changes the Poisson's ratio that could be observed for the system referred to as case 1. In fact, the Poisson's ratio assumes much less negative values throughout the entire deformation process that reach the level of -0.38 for the experiment. This drastic change in the behavior of the analyzed system originates from the fact that hinges connecting level 0 elements are significantly thinner than hinges corresponding to level 1 of the system (d 1 ≈ 2.7 µm, d 2 ≈ 10.0 µm). As a result, despite a large number of hinges associated with level 0 of the system (16 within the unit-cell), the overall resistance of level 0 elements to the rotational motion is not much different than in the case of level 1. Thus, both hierarchical levels could deform to a significant extent which process is reflected by a large concurrent change in angles θ 0 and θ 1 (see Figure5.2(e)). At this point, it should also be noted that for cases 1 and 2, a part of the bottommost row of unit-cells is not captured on the provided pictures. This stems from the fact that the lenses used in the experiment (see Methods section) were selected in a way ensuring a high quality of this part of the picture that was taken into account while assessing mechanical properties of the system.

The last of the analyzed 2D structures is referred to as case 3 (see Figure5.2(c)) and corresponds to a very similar system to the one referred to as case 2. In fact, all of the initial angles, as well as thicknesses of hinges, are almost the same in terms of the aspect ratio for both structures. The only significant difference between them corresponds to the number of level 0 elements constituting level 1 blocks, i.e. for case 3, level 1 blocks consist of 16 level 0 elements instead of 4. As shown in Figure5.2(c), this small difference in the design of the structure results in a very different behavior of the hierarchical system during the deformation process. More specifically, level 1 blocks do not get visibly flattened with respect to the initial configuration as was the case for case 2. Instead, they approximately retain their shape throughout the entire deformation process. According to Figure 5.2(d),

the change in the number of level 0 units also significantly affects mechanical properties of the system resulting in the greater extent of the auxeticity. This change in the behavior between case 2 and case 3 can be explained upon having a closer look at the number of hinges present within the system. More specifically, upon increasing the number of level 0 units constituting level 1 blocks, the number of hinges connecting level 0 elements is also significantly increased. At the same time, the number of hinges connecting level 1 blocks remains the same. Thus, upon comparing cases 2 and 3, it is much more difficult to rotate level 0 elements corresponding to case 3. As a result, the deformation of this structure corresponds primarily to the change in the value of θ 1 as opposed to the variation in θ 0 (see Figure5.2(e)).

At this point, it is also worth noting that all of the structures considered in this study deform primarily through the rotation of their structural elements and not their flexing. More specifically, as shown in Figure 5.2, according to the von Mises stress distribution corresponding to the FEA simulations, the main accumulation of the stress can be observed in the vicinity of hinges connecting level 1 blocks. On the other hand, in the remaining parts of the structure, it assumes much lower values not exceeding 50 MPa. Since Young's modulus of the polymer used to manufacture the samples is equal to 4GPa (see Methods section), this means that the local stresses do not result in the change of the shape of structural elements. Instead, as shown on the provided pictures, structural elements approximately retain their shape throughout the entire deformation process. Furthermore, it should be noted that even though in this work we focus on the longitudinal compression of the analyzed structures, in general, one can also compress them from other directions including the transverse direction. As shown the compression of the considered systems in the transverse direction may affect their mechanical properties. Nevertheless, not all of them are affected by a change in the direction of compression to the same extent. More specifically, the behavior of the case 1 system seems to be only marginally affected by the direction of compression. On the other hand, especially at low strains, the Poisson's ratio corresponding to the case 2 structure is visibly altered.

5.5/ SHAPE MORPHING

In addition to the possibility of controlling the magnitude of the Poisson's ratio exhibited by the analyzed system, another very interesting direction of studies corresponds to the ability of the hierarchical structure to change its shape during the deformation process.

In fact, in Figure 5.2, it was shown that depending on the thicknesses of hinges, the entire unit-cell may assume different shapes. Thus, in theory, if one would construct the composite mechanical metamaterial consisting of unit-cells associated with multiple different sets of geometric parameters, then it would be possible to obtain the hierarchical structure with the local change in shape as well as mechanical properties. To demonstrate it, one such example is considered in this study.

As shown in Figure 5.3, the hierarchical mechanical metamaterial composed of 4 × 3 × 4 case 1 unit-cells and 4 × 3 × 2 case 2 unit-cells (see Figure 5.3(a-b)) is considered in order to analyse its ability to change shape in a controllable manner. More specifically, the objective is to change the shape of the initial cuboid-like structure to match the shape of the reference object in the form of the mug presented in Figure 5.3(d). According to Figure 5.3(c), upon compressing the system, its shape changes drastically with the bottom part composed of case 1 cells shrinking significantly in the lateral dimension. This effect originates from the strong auxetic behavior that is characteristic for case 1 unit-cells (see Figure 5.2). On the other hand, even though the top part of the hierarchical system consisting of case 2 unit-cells also shrinks in the lateral dimension, the extent of such shrinkage is much smaller than is observed for the bottom part of the structure. This change in the behavior can be explained by the much smaller extent of the auxeticity corresponding to the unit-cell referred to as case 2. Thus, overall, Figure 5.3(c-d) shows that by designing the hierarchical structure considered in this work in the specific compositelike manner, the entire system may undergo a shape morphing into a much more complex Taken from [START_REF] Dudek | Micro-Scale Auxetic Hierarchical Mechanical Metamaterials for Shape Morphing[END_REF]. a) A graphical representation of unit-cells corresponding to case 1 and case 2 that were used in the design of the structure that is analyzed from the point of view of its ability to change shape in a predefined manner. b) Picture of the prototype used in the experiment together with the zoomed-in images of types of structures used in the design of the resultant composite material. c) The experiment corresponding to the compression of the considered structure that results in the change in its shape. d) The real-life mug used as an inspiration for the shape that the analyzed hierarchical system is supposed to assume upon being deformed.

configuration. Of course, the considered example is relatively simple but it proves that in general, one can use the concept presented in this work in order to observe very complex shape morphing at the micro-scale. At this point, it is also worth to mention that in the case of the specific design considered in this study, the two types of materials were connected by means of an additional thin plate that was constraining topmost case 1 cells and bottommost case 2 cells. As a result, at the end of the deformation process, one can observe a big difference in the horizontal dimension of the middle and the top part of the case 1 structure. In fact, the same effect could be achieved if one was to replace the plate with one additional row of very stiff unit-cells. Of course, without the additional plate, a similar behavior is still expected to be observed due to a large difference in the Poisson's ratio between the two types of structures. Nevertheless, such system would be more difficult to manufacture and the change in the horizontal dimension of the middle and top part of the case 1 structure would be expected to be smaller.

The results reported in this study clearly show that the considered hierarchical structure can be constructed at the microscale and that its mechanical properties with the emphasis on the exhibited Poisson's ratio may be controlled and significantly modified depending on the small change in the design parameters. It should be also mentioned that in theory, such control over the extent of the exhibited auxeticity could be achieved in an active manner, i.e. without the need of reconstructing the system. In this regard, it should be noted that in the case of this work, the behavior of the hierarchical structures is considered for the quasi-static compression where the moment of inertia of structural elements does not affect the results. However, for fast deformations, the rate at which the structure is being compressed could become another factor that allows to determine whether both hierarchical levels are deforming to a large extent. Of course, for this effect to be observed, hinges would have to be appropriately thin with respect to the considered deformation rates. In addition, the active control over the behavior of the structure could be achieved through the use of external stimulus such as a change in temperature assuming the use of materials corresponding to the high thermal expansion coefficient while designing the hinges.

5.6/ ADDITIONAL STUDIES ON THE DIRECTION OF COMPRES-SION, DESIGN AND MECHANICAL PROPERTIES

In order to assess the mechanical properties of the considered quasi-2D structures, they were subjected to compression in the longitudinal direction. However, it is also interesting to check how would these properties change if one was to compress the samples in the other direction. Particularly important in this regard is the transverse direction.

To assess the effect of the change in the direction of compression on the Poisson's ratio, we selected case 1 and case 2 structures that, as discussed in the main text, correspond to very different deformation profiles. As shown in Figure 5.4, the Poisson's ratio corresponding to case 1 of the system is not significantly affected by the change in the direction of compression. In fact, this should be expected since it acts similarly to the rotating square system. On the other hand, it is possible to note a large change in the exhibited Poisson's ratio for the case 2 system. This is particularly visible at low strains.

To gain a better understanding of the properties of structures analyzed in this work, in addition to studies related to the Poisson's ratio, also other properties were characterized for 3 types of the system defined in the main text.

As shown in Figure 5.5, the force vs displacement results recorded during the experiment allow us to estimate the energy absorbed by analyzed structures during the experiment.

These results can also be used to analyze the stiffness exhibited by the system during the deformation. According to Figure 5.5(d), all of the structures exhibit the positive stiffness during the initial part of the deformation process in order to later exhibit negative stiffness.

In addition, it should be emphasized that depending on the type of the structure, the stiffness of the system can be significantly adjusted. This in turn may prove to be of great significance in the design of vibration damping devices where the ability to control the extent of the exhibited stiffness and the possibility of exhibiting negative stiffness is of great significance.

In Figure 5.5(e), it is shown that based on the recorded experimental results and information about the mass of considered structures, it is possible to estimate the Specific Energy Absorption (SEA) rates corresponding to each of the systems. More specifically, it is shown that the largest amount of energy per unit of mass was absorbed by case 1 structure (around 386.6 J kg -1 ), i.e. the structure that corresponds to the most negative Poisson's ratio. These results also indicate that depending on the type of the analyzed structure, the SEA can be significantly adjusted which is very promising from the point of view of potential applications related to tunable shock-absorbing devices.

Finally, based on Figure 5.5(f), it can be noted that Young's moduli corresponding to analyzed structures are significantly different. The largest value of this parameter is associated with the structure referred to as case 1 where it assumes the value of E1 = 19.88

MPa. In fact, this result should be expected since, for case 1, the relative thickness of hinges is significantly larger than for other cases. Furthermore, E2 = 11.73 MPa and E3 = 6.19 MPa. The large difference between values of E2 and E3 stems from the fact that their thickness of hinges was very similar while level 1 building blocks corresponding to case 3 were much larger. Hence, it is easier to deform the case 3 system.

In this work, the initial geometry does not correspond to the fully-open configuration of the system but is relatively similar to it. More specifically, case 1 structure is considered for initial angles θ 0 = 90 are not geometrically-inclined to start rotating and the deformation process corresponds primarily to the decrease in the vertical direction while the transverse dimension of the system is not decreasing. However, eventually, the internal stress and material imperfections force structural elements to commence the rotation. As a result, after exceeding a relatively small strain of approximately 0.03, the structure starts acting similarly to the typical rotating square system with its Poisson's ratio being in the vicinity of -1.

In order to analyze the effect of the strain rate on the behavior of the system, we decided to compress the case 1 structure at two very different rates. Based on Figure 5.7(a), one can note that the deformation of this structure compressed at the very fast rate of 2 mm/s appears to be similar to what can be observed upon subjecting it to the much slower deformation. In fact, this observation is also confirmed in Figure 5.7(b) where it is shown that the Poisson's ratio is only marginally affected by a change in the deformation rate. This stems from the fact that hinges connecting level 0 elements are relatively thick in comparison to hinges connecting level 1 blocks. As a result, this structure deforms similarly to the rotating square system irrespective of the deformation rate. Of course, a non-negligible dependence of the behavior of the system on the deformation rate could potentially be observed for structures where hinges are very thin. The reason is the fact that in such a case, the moment of inertia of rotating structural elements would become a significant factor affecting the behavior of the deforming system. On the other hand, in the case of the relative hinge thickness selected for the system presented in Figure 5.7, the effect of the moment of inertia on the behavior of the system remains relatively insignificant in comparison to the resistance offered by hinges.

In view of all of the above, it is important to emphasize the fact that the results of this work may prove to be of great significance in the case of numerous applications. First of all, the ability of the considered hierarchical mechanical metamaterial to exhibit a very broad range of auxetic behavior may lead to the design of highly efficient protective equipment (especially for the macroscopic version of the considered concept). In fact, depending on the type of the analyzed structure, the specific energy absorption of the system may be significantly adjusted with its estimated values ranging between 133.2 J kg -1 and 386.6

J kg -1 . Hence, the considered tunable properties are also interesting from the perspective of impact absorption devices. In addition, the concept proposed in this work may be used in the design of tunable vibration dampers utilizing local vibration isolation. This stems from the fact that as shown the stiffness of the system can be adjusted depending amongst others on the thickness of its hinges. This in turn is a commonly desired feature related to vibration damping devices. In addition, the considered concept is particularly interesting since the model proposed in this work is not solely limited to two-dimensional structures where the devices making use of the provided results could only exhibit their properties in a single plane. Furthermore, the ability of the analyzed system to exhibit shape morphing may significantly increase the efficiency of multiple medical devices including stents by providing localized support for specific parts of the blood vessel that need additional reinforcement. Finally, the shape morphing ability could also be utilized in the design of flexible electronics and equipment used on spacecraft that could adjust its shape upon being deployed from the initially confined configuration that does not waste space onboard prior to being used.

5.7/ 3D HIERARCHICAL SYSTEM

As mentioned earlier, almost all of the known hierarchical mechanical metamaterials are two-dimensional structures which limits their applicability as they can only exhibit their properties in one plane. Thus, it is essential to identify structures that could also exhibit a controllable mechanical response in the case of 3D systems. In fact, one of the first such structures is the hierarchical metamaterial proposed in this work as its ability to exhibit a very broad range of the auxetic behavior is not limited to 2D structures and can be conveniently extended to 3D models. To demonstrate it, we selected two very similar structures referred to as type A and type B. These structures have unit-cells defined as shown in Figure 5.1(c). In addition, the aspect ratios of hinges and other geometric dimensions corresponding to type A and type B mechanical metamaterials are the same as in the case of the quasi-2D structures named case 1 and case 2 respectively. Hence, based on the already generated results, one should expect the type A structure to be able to exhibit a more auxetic behavior than the type B system. It should be also noted that type A and type B structures look almost identical with the main difference between them being the thickness of hinges. Thus, as an example, a picture of the type A system is shown in Figure 5.8(b).

As shown in Figure 5.8(a-b), it is indeed possible to construct the microscopic 3D equivalent of the quasi-2D models considered in this work. In addition, upon designing the system composed of unit-cells having the same thickness of hinges as the 2D structure As shown in Figure 5.9, the evolution of the case 1 system corresponding to the or-

thotropic elasticity model (E y = E z = 2 GPa, E x = 4 GPa, G xy = G xz = 0.71 GPa, G yz = 4/(2(1 + ν yz )) GPa, ν yz = 0.4, ν xz = ν xy = 0.
2) is very similar to what could be observed in the main text. More specifically, the evolution of the system corresponds primarily to the rotation of level 1 structural blocks that approximately retain their shape throughout the deformation process. Furthermore, based on Figure 5.9(b), it is possible to note that the use of the aforementioned anisotropic model only marginally affects the exhibited Poisson's ratio. Thus, the behavior of the considered system depends mainly on its design rather than the selection of the model of elasticity.

5.9/ CONCLUSION

In this chapter, it is shown that the novel microscopic hierarchical mechanical metamaterial can exhibit a wide range of the auxetic behavior depending solely on the change in the thickness of hinges connecting its structural elements. It has also been demonstrated that the considered concept can be observed both for the quasi-2D and 3D version of the structure which enables the system to exhibit a desired mechanical response upon being subjected to a deformation induced from different directions. Furthermore, it is presented that the considered structure can exhibit shape morphing allowing it to match the shape of the reference object upon being deformed. It should be noted that different aspects of these results offer a new perspective with regards to the capabilities of hierarchical mechanical metamaterials and are likely to pave the way for other researchers to design similar microscopic functional materials that will further enhance their properties. Finally, it should be emphasized that the presented results may be utilized in numerous applications. More specifically, the ability of considered hierarchical metamaterials to exhibit a broad range of the auxetic behavior could be important in the design of the protective equipment and vibration dampers. On the other hand, the ability of the considered structure to exhibit shape morphing is very promising for example from the point of view of medical devices capable of locally adjusting their properties as well as flexible electronics.

GENERAL CONCLUSIONS AND PERSPECTIVES 6.1/ GENERAL CONCLUSIONS In this thesis, I have studied the topics of extremal acoustics (large band gaps and nonmonotonic dispersion relation) and elastic metamaterials (extreme and tunable auxeticity) from both the numerical and the experimental sides.

First, I have shown the design of an ultra wide band gap 3D phononic crystal for ultrasonic waves operating in the MHz range. Samples with complex unit cells were manufactured using direct laser writing with lattice constants of about 300 µm. Experimental characterization was used to confirm the existence of the ultra wide gap extending from 0.6 MHz to 7.5 MHz.

Second, I have designed, fabricated and experimentally tested a novel acoustic metamaterial having third neighbor interactions to shape the first band of the dispersion relation.

FEM calculations and experiments confirm that the dispersion relation shows a minimum, like the roton in some superfluid in the low dispersion band. The metamaterial was fabricated using additive manufacturing and was experimentally measured. Most interestingly, at the inflexion point the roton presents a zero group velocity for a given wavenumber which could be used for energy harvesting.

Third, I have shown that using a glide-symmetric dislocation in a 2D crystal, a pair of wide-bandwidth, single-mode, and symmetry-protected guided waves appear in the bulk band gap. The 2D Zak phase changes by π from one side of the interface to the other, CHAPTER 6. GENERAL CONCLUSIONS AND PERSPECTIVES providing a topological invariant protected by glide-reflection symmetry at the X point of the Brillouin zone. An experiment was performed with acoustic waves in water, at ultrasonic frequencies, demonstrating the continuous tuning of transmission as a function of the glide parameter.

Fourth, as a starting step, I have participated in the design of a novel microscopic hierarchical mechanical metamaterial showing a wide range of auxetic behavior (for both the quasi-2D and the 3D cases). Furthermore, we have shown that the structure can exhibit shape morphing, which paves the way for other researchers willing to design similar microscopic functional materials. This demonstration is a crucial step to make our previous crystals and metamaterials in general stimuli-responsive, with the existence of continuous phase transitions and the property of tunability.

Finally, I would like to remark that I have also done a lot of instrumentation work, necessary for my thesis but also for the other 6 PhD students who have been present in our group during my own doctoral work.

Last but not least, some of our works were advertised and promoted, either on the cover of journals or for some as highlights of CNRS. Altogether, these show the timeliness of the results published in this thesis.

6.2/ PERSPECTIVES: FROM PASSIVE EXTREMAL METAMATERIALS TO DYNAMIC AND STIMULI-RESPONSIVE METAMATERIALS

In the future, I would like to investigate the effects of external stimuli in order to promote active properties in metamaterials and phononic crystals.

A first possible extension would be to investigate the effect of a change on the gap when the geometry of a crystal is changed. This change can be made active using an external magnetic field and it has been proven to be possible to change drastically the elastic effective properties of metamaterials. In Another perspective that would be desirable to explore is the addition of a flow in an acoustic crystal, thus breaking the PT symmetry and potentially opening a gap at will by controlling this flow. To control the flow, Figure 6.2, a 2D array of fans may be constructed so the rotational flow can open an gap. This process can be further extended to elasticity using piezoelectric patches (creating small perturbations) or magnetostrictive media. In this work, my contribution was the realisation of the compression experimental test and help with the image processing. Experiments and simulations demonstrate that both the elastic modulus and the yield strength of the simple-cubic closed tubular lattice are significantly larger than those of the simple-cubic truss lattice, regardless of the loading direction. At a relative density of 0.1 and compared to the truss lattice, the closed tubular lattice can absorb respectively 4.45 times and 6.14 times as much energy along directions [START_REF] Hahn | International tables for crystallography. Volume A, Space-group symmetry[END_REF] and [START_REF] Belloni | Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal[END_REF]. The average normalized Young's modulus and yield strength are respectively 28% and 53% larger than those of the most outstanding shellular metamaterial with the same mass. Such excellent mechanical properties make it a potential candidate for applications to load-bearing and energy absorption. In this work, a novel hierarchical mechanical metamaterial is proposed that is composed of re-entrant truss-lattice elements. It is shown that this system can deform very differently and can exhibit a versatile extent of the auxetic behaviour depending on a small change in the thickness of its hinges. In addition, depending on which hierarchical level is deforming, the whole structure can exhibit a different type of auxetic behaviour that corresponds to a unique deformation mechanism. This results in a dual auxetic structure, shown in Figure

A.9, where the interplay between the two auxetic mechanisms determines the evolution of the system. It is also shown that depending on the specific deformation pattern, it is possible to observe a very different behaviour of the structure in terms of frequencies of waves that can be transmitted through the system. In this particular study, my contribution was the band diagram analysis and simulation of the proposed hierarchical metamaterial.

In fact, it is demonstrated that even a very small change in the parametric design of the system may result in a significantly different band gap formation that can be useful in the design of tunable vibration dampers or sensors. The possibility of controlling the extent of the auxeticity also makes the proposed metamaterial to be very appealing from the point of view of protective and biomedical devices. The construction of extreme acoustic and elastic metamaterials is considered from both the numerical and the experimental point of views. Large phononic bandgaps, non-monotonic dispersion relations, and extreme and tunable auxeticity are the specific goals discussed in the manuscript. In the first chapter, we introduce the concepts behind waves and metamaterials, and the constitutive equations in elasticity and acoustics. In the second chapter, we describe the design of an ultra wide bandgap 3D phononic crystal for ultrasonic waves operating in the MHz range. In the third chapter, we design, fabricate and test experimentally a novel acoustic metamaterial having third neighbor interactions to shape the first band of the dispersion relation like the roton in superfluids in the low dispersion band.

In the fourth chapter, we show that using a glidesymmetric dislocation in a 2D crystal, a pair of widebandwidth, single-mode, and symmetry-protected guided waves can be created in the bulk band gap. The 2D Zak phase changes by π from one side of the interface to the other, providing a topological invariant protected by glide-reflection symmetry at the X point of the Brillouin zone. The corresponding experiment was performed with acoustic waves in water, at ultrasonic frequencies, and demonstrates the continuous tuning of transmission as a function of the glide parameter. In the last chapter, we show the design of a novel microscopic hierarchical mechanical metamaterial exhibiting a wide range of auxetic behavior (for both the quasi-2D and the 3D cases). Furthermore, we show that the structure can exhibit shape morphing. Dans le troisième chapitre, nous concevons, fabriquons et testons expérimentalement un nouveau métamatériau acoustique présentant des interactions au troisième voisin pour façonner la première bande de la relation de dispersion suivant celle du roton dans les superfluides. Dans le quatrième chapitre, nous montrons qu'en utilisant une dislocation symétrique par réflection glissée dans un cristal 2D, une paire d'ondes guidées à large bande passante, monomodes et protégées par symétrie peut être créée dans la bande interdite. La phase de Zak 2D change par π d'un côté de l'interface à l'autre, fournissant un invariant topologique protégé par la symétrie de réflexion glissée au point X de la zone de Brillouin. L'expérience correspondante a été réalisée avec des ondes acoustiques dans l'eau, à des fréquences ultrasonores, et démontre l'accord continu de la transmission en fonction du paramètre de glissement. Dans le dernier chapitre, nous montrons la conception d'un nouveau métamatériau mécanique hiérarchique microscopique présentant une large gamme de comportement auxétique (à la fois pour les cas quasi-2D et 3D). De plus, nous montrons que la structure peut présenter un morphing de forme.

1

  Introduction and state-of-the-art 1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Metamaterials and Phononic Crystals . . . . . . . . . . . . . . . . . . . . . 1.3 Waves in Periodic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Symmetries and Crystals . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Continuum Mechanics and Elastic Waves . . . . . . . . . . . . . . . . . . . 1.4.1 Elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Elastodynamic Equation . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Acoustic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Acousto-Elastic Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Topological Phononics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Berry Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Su-Schrieffer-Heeger (SSH) Model . . . . . . . . . . . . . . . . . . . 2 Ultra-Wide Bandgap Metamaterials 2.1 Direct laser 3D printing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Design of the Unit-Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Experiment and Transmission Spectra . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 3 :

 13 Figure 1.3: (a) A 2D square-lattice periodic crystal, with lattice constant a, composed of rigid cylinders in air. In (b) we depict the corresponding Brillouin zone, the path ΓXM and we plot ω(k) for different directions.

Figure 1 . 4 :

 14 Figure 1.4: Bloch solutions for the crystal depicted in Figure 1.3 for the first band and for different values of Bloch vectors. (a) k B = Γ, (b) k B = X, (c) k B = M.

Figure 1 .

 1 Figure 1.6 we depict all possible frieze groups.

Figure 1 . 5 :

 15 Figure 1.5: All different Bravais lattices in (a) 2D and (b) in 3D.

Figure 1 . 6 :

 16 Figure 1.6: All 7 possible frieze groups. p1: translation only. p11g: glide-reflections and translations. p1m1: vertical reflection lines and translations. p2: translations and 180°rotations. p2mg: vertical reflection lines, glide reflections, translations and 180°rotations. p11m: translations, horizontal reflections, glide reflections. p2mm: horizontal and vertical reflection lines, translations and 180°rotations.

Figure 1 . 7 :

 17 Figure 1.7: Schematic definitions for bulk modulus B a), Young's modulus E b) and shear modulus G c).

  22 33 23, 32 13, 31 12, 21

Figure 1 . 8 :

 18 Figure 1.8: Poisson's ratio schematic representation. Constant stress, σ, is applied to the material. The common positive ratio case on the left, and the auxetic case with a negative value, on the right, show in both cases the initial state of the object with a transparent color and the final state with a solid color.

Figure 1 . 9 :

 19 Figure 1.9: General illustration of two different polarised waves in elastodynamics. (a) A longitudinal wave. (b) a transverse wave.

Figure 1 . 10 :

 110 Figure 1.10: SSH model. a) Each unit cell consists of two sites (red and blue). The intercell hopping v, single line, is different from the intracell hopping w, double line. The full unitcell is marked with a dashed rectangle containing the two sites in blue and red. b)and c) Dimerized limits of the model, when the chain is disconnected. b) v = 1, w = 0, trivial case. c) v = 0, w = 1 the topological case where the sites are disconnected but there is a site in blue and red isolated at the edges. At these sites some topological modes such as edges modes can appear.

Figure 1 .

 1 Figure 1.11: a) Band diagram for infinite chain with different hopping values permutations. The gap is always present when w v. b) Complex d(q), bulk function, to show a trivial case, w < v when curve does not wind around the origin and c) a topological case, w > v that wind around it making a to a non trivial Zak phase.

Figure 1 . 12 :

 112 Figure 1.12: Opening of the gap of the topological modes for a finite chain of 10 units, a) w = 1 as constant changing w and b) v = 1 as constant changing w.

Figure 2 . 2 :

 22 Figure 2.2: Principle of direct laser writing of voxels with different laser power. Taken from [116] (a) Definition of a voxel as used in our printing setup, and of the hatching distance l x and the slicing distance l z . (b) Schematic representation of voxels for a laser power larger than in (a). A larger voxel is obtained, which for the same slicing results in a larger overlap of adjacent voxels (overlaps are depicted with blue ellipses).

Figure 2 . 3 :

 23 Figure 2.3: Schematic view of the unit cell of the perovskite cubic three-dimensional phononic crystal. Taken from [114] Designed dimensions are a = 300 µm, h = 40.8 µm, d s = 171.3 µm, and d c = 6 µm.

Figure 2 .

 2 Figure 2.4(a) shows a scanning electron micrograph (SEM) view of the fabricated phononic crystal. Two-photon polymerization (Nanoscribe, model GT) was performed with a negative tone photoresist "Ip-S" [5] dropped on a PZT-5A piezoelectric patch of

Figure 2 . 4 :

 24 Figure 2.4: (a) SEM view of the phononic crystal sample fabricated by TPL.Taken from[START_REF] Iglesias Martínez | Three-dimensional phononic crystal with ultra-wide bandgap at megahertz frequencies[END_REF] The phononic crystal is coated with a thin chromium coating to enhance vibration metrology. The top plate is here removed to show the internals of the crystal but was present during vibration experiments. (b) Phononic band structure obtained with the finite element method, with complete bandgaps outlined in gray.

( 1 -

 1 methoxy-2-propanol acetate) solution for 20 minutes to remove the monomer photoresist. The scanning mode (at constant speed 100 mm/s) with a goniometric mirror was used. Post-fabrication measurements yield a = 299.6 µm, h = 37.1 µm, d s = 170.8 µm, and d c = 6.6 µm for the crystal sample.

Figure 2 . 5 :

 25 Figure 2.5: Taken from [114]. (a) Lower band mode at 0.5 MHzat the Γ point and (b) Local resonance of the bars at higher bands at 8 MHzat Γ.

Figure 2 . 6 :

 26 Figure 2.6: Displacement of the cantilever tip for different lengths, which are show with different colors for clarity. the increase in the resonant frequency with length is clearly apparent.

Figure 2 .Figure 2 . 7 :

 227 Figure 2.7: Taken from [114] (a) SEM view of fabricated cantilevers with various length. (b) Measured resonance frequency f b of the fundamental bending mode of the cantilever versus length, l. Formula(2.2) , valid under Euler-Bernoulli theory for beams with a rectangular cross-section, is used for the fit.

Figure 2 .

 2 Figure 2.8 illustrates the experimental set-up. Out-of-plane displacements were measured with a laser Doppler vibrometer (Polytec, model MSA 500). For the source of elastic waves, we used a 0.25 mm thick and 6 mm wide PZT patch (PI Ceramic PIC255) in longitudinal mode. A network analyzer (Rhode & Schwartz, model ZV) measured the

Figure 2 . 8 :Figure 2 . 9 :

 2829 Figure 2.8: Taken from [114] (a) Principle of vibration metrology. The sample is placed under the scanning laser beam and its reflection is recorded in time. An example of a scan of the out-of-plane displacement field at 100 kHz is overlaid on the top surface. (b) Schematic of the experimental setup.

Figure 3 . 1 :

 31 Figure 3.1: Toy model for the higher neighbor interaction. Masses shown in orange are connected by a spring with a K n spring constant, from the n th nearest neighbor; first, second, and third nearest neighbor interaction in the panels a, b, and c respectively, with a lattice constant a. On panel b and c are also depicted the first and the n th neighbor interactions.

  m

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: Blueprint for roton metamaterials.Taken from[START_REF] Iglesias Martínez | Experimental observation of roton-like dispersion relations in metamaterials[END_REF] The unit cell of the channel-based metamaterial beam supporting rotons for airborne longitudinal pressure waves along the z-direction in the channel system is shown. It is composed of a bottom piece and an upper piece, whose front half is intentionally removed to show the inner compartment (yellow). The cylindrical channels for air pressure propagation are rendered semi-transparent in red and blue, respectively. Here, the masses in the toy model of figure3.1 correspond to cylindrical compartments. A microphone is installed in the through-hole with diameter d on the front wall of the lower piece. The other holes are for alignment and assembly.

Figure 3 .

 3 Figure 3.4 describes and illustrates the sample. Each unit cell of one acoustic metamaterial sample is divided into two identical parts, for each of the N z = 50 unit cells, thanks to the symmetry with respect to the horizontal plane parallel to the xy-plane. A set of 100 identical pieces is fabricated by 3D printing fused deposition modeling with PLA (3DHubs). The two parts forming each of the 50 unit cells are joined together with the help of screws and an acrylic plastic filler (Mastic, Axton) to ensure sealing with respect

Figure 3 . 4 :

 34 Figure 3.4: Roton metamaterial for airborne sound. Taken from [138]. (A) The 3D printed polymer sample follows the blueprint shown in 3.2(B). It has been assembled from 100 individual pieces, 2 for each of the N z = 50 unit cells. The metamaterial sample has a length of 2 malong the z-direction. Therefore, only the bottom part and the top part are shown here. (B) One of the two pieces for one unit cell. (C) Zoom-in view of the highlighted rectangle region in (B) showing an installed microphone on the side wall.

Figure 3 .

 3 Figure 3.5 summarizes resulting exemplary measured raw data for the two experiments

Figure 3 . 5 :

 35 Figure 3.5: Measured and calculated roton dispersions. Taken from [138]. (a) Measured raw data (left) versus position and frequency and derived roton band structure (right). (b) Corresponding numerically calculated behavior for the same finite sample length and including damping. The white solid curves are the calculated roton band structures for a lossless metamaterial beam that is infinitely extended along the z-direction. We use the geometrical parameters: a xy = 100 mm, a z = 50 mm, 2r 1 = 10 mm, 2r 2 = 16 mm, 2r 3 = 30 mm, t 2 = 30 mm, d = 9.8 mm, r 4 = 7.5 mm, and L = 120 mm. The gray curves in panels (b) correspond to the approximate analytical dispersion relations of the higher-order-gradient effective-medium model with parameters c 2 , c 4 , and c 6 fitted to the interval k z ∈ [0, 0.6 × π/a z ].

Figure 3 . 6 :

 36 Figure 3.6: Taken from [138]. Full view of the air-borne sound roton metamaterial sample.

Figure 3 . 7 :

 37 Figure 3.7: Taken from [138]. Extra unit cell with a circular face to connect the loudspeaker to the airborne sound roton metamaterial. (A) Front-side view and (B) back-side view.

Figure 3 . 8 :

 38 Figure 3.8: Example that for the 25 th cell of the normalized acoustical pressure wave signal take it using the discrete stepping in single frequency, P, (a) the amplitude, and (b) the phase.

Figure 3 . 9 :

 39 Figure 3.9: Example that for the 25 th cell of the normalized cross power spectral density, P xy (a) the amplitude, and (b) the phase.

Figure 3 . 10 :

 310 Figure 3.10: Taken from [138]. Measured and calculated acoustic roton dispersions using white-noise excitation. (A) Measured raw data for the sample in Fig. 3 versus position and frequency. (B) Derived roton band structure. The solid curves are the calculated roton band structure for a lossless metamaterial beam that is infinitely extended along the z-direction.

4 TOPOLOGICAL

 4 CRYSTAL WAVEGUIDETopological phononics promises unprecedented wave properties inspired by the concepts of topological insulators[START_REF] Hasan | Colloquium: Topological insulators[END_REF][START_REF] Fleury | Sound isolation and giant linear nonreciprocity in a compact acoustic circulator[END_REF][START_REF] Yang | Subwavelength perfect acoustic absorption in membrane-type metamaterials: a geometric perspective[END_REF][START_REF] Zhang | Formation of highquality vortex laser beams with different orbital angular momenta in the laser resonator[END_REF][START_REF] Ma | Topological phases in acoustic and mechanical systems[END_REF][START_REF] Gao | Topological vortices for sound and light[END_REF]. One promising direction is the achievement of uni-directional and backscattering-free guided wave propagation along a boundary of a crystal or a domain wall between two crystal phases. Passive topological waveguides, for instance of the valley-Hall[START_REF] Lu | Observation of topological valley transport of sound in sonic crystals[END_REF][START_REF] Yan | On-chip valley topological materials for elastic wave manipulation[END_REF] and the quantum-Hall type[START_REF] Wu | Scheme for Achieving a Topological Photonic Crystal by Using Dielectric Material[END_REF][START_REF] He | Acoustic topological insulator and robust one-way transport[END_REF][START_REF] Miniaci | Experimental observation of topologically protected helical edge modes in patterned elastic plates[END_REF], lead to symmetry-protected, single-mode guided waves along a domain wall separating two phases of the same crystal with different topological invariants. The topological properties of the waveguide are inherited from those of the two-dimensional (2D) bulk crystal according to the bulk-boundary correspondence principle[START_REF] Mong | Edge states and the bulk-boundary correspondence in Dirac Hamiltonians[END_REF]. The one-dimensional (1D) domain wall hetero-structure is formed without breaking the periodicity of the 2D lattice, by tuning a continuous geometrical parameter that controls a topological transition[START_REF] Lu | Dirac cones in twodimensional artificial crystals for classical waves[END_REF][START_REF] Chen | Topological phase transition in mechanical honeycomb lattice[END_REF]. For instance, in valley-Hall crystals, a triangular inclusion is rotated continuously to reduce the symmetry of a 2D crystal possessing a band structure with a Dirac point at the K point of the first Brillouin zone, causing a gap to open there[START_REF] Lu | Valley vortex states in sonic crystals[END_REF][START_REF] Lu | Observation of topological valley transport of sound in sonic crystals[END_REF][START_REF] Zhu | Design and Experimental Observation of Valley-Hall Edge States in Diatomic-Graphene-like Elastic Waveguides[END_REF]. In quantum-Hall crystals with C6v symmetry, a double Dirac at the Γ point undergoes a topological transition under the continuous tuning of the internal structure of the unit cell of the crystal

Figure 4 . 1 :

 41 Figure 4.1: Principle of the glide-reflection symmetric topological phononic crystal waveguide. Taken from[START_REF] Iglesias Martínez | Glide-Reflection Symmetric Topological Phononic Crystal Waveguide[END_REF]. (a) A 2-periodic square-lattice phononic crystal is composed of steel rods in water (lattice constant a, diameter d = 0.9a). For every frequency within the complete bandgap, transmission of incident acoustic waves is forbidden. (b) Two pieces of the same square-lattice crystal are now glided along the x-axis. The glide parameter g is periodic with period a. For a half-lattice glide parameter (g = a/2), waves are guided along the glide dislocation, for frequencies within the complete bandgap. In numerical simulations, P is the normalized pressure field, frequency is taken at the center of the bandgap, and waves are incident from the left.

Figure 4 .

 4 2(a) shows a super-cell and the phononic band structure for the 2-periodic crystal for g = 0; Fig. 4.2(b) shows similar information for the glidereflection symmetric waveguide for g = a/2. The super-cell is the numerical device used to obtain the dispersion relation of crystal waveguides. Periodic boundary conditions are applied on the vertical boundaries of the supercell while the horizontal boundaries are left free (Neumann boundary condition). In the figure, we consider N = 10 unit cells in the vertical direction and 1 unit cell in the horizontal direction. The band structures account for acousto-elastic coupling between pressure waves in water and elastic waves in steel [94]. The band structure for g = 0 in Fig. 4.2(a) shows the complete bandgap separated by groups of bands. Counting the bands, there are exactly N + 1 bands below the bandgap. Those bands are actually sampled from the original Brillouin zone as (k y a/π = n/N, k x ) with n = 0, • • • , N [107]. Hence, when n = N, the Bloch wavevector varies along the YM edge of the first Brillouin zone (k y a/π = 1).Frequency, ω/2π (MHz)

Figure 4 . 2 :

 42 Figure 4.2: Band structure topology of the glide-reflection waveguide, computed for a supercell made of N = 10 unit cells of the crystal. Taken from [160]. (a) For g = 0, the supercell simply repeats N times vertically the primitive cell of the 2D square-lattice crystal (d = 2 mm, a = 2.22 mm). The band structure of the waveguide is obtained from the projected band structure of the 2D crystal. (b) For g = a/2, the bands group by pairs of symmetric (S) / anti-symmetric (AS) Bloch waves, with respect to the glide reflection symmetry. They are degenerate at the Brillouin zone edge (the X point), causing a pair of guided waves to appear inside the complete bandgap. The modal distributions of the S (red color band) and AS (yellow color band) guided waves are shown on the left for k x a/π = 0.8.

  Fig. 4.2(b) illustrates that property for one particular value of k x . In contrast to the case g = 0, the Bloch waves of the 2-periodic phononic crystal do not translate directly into Bloch waves of the waveguide structure for g 0. However, they can still be used as a functional basis to express the 1-periodic guided mode. Bloch waves of the 2-periodic crystal are all evanescent for frequencies inside the complete bandgap. Hence their wavevector satisfies ℜ(k y )a/π = 1: the real part of the Bloch wavevector is restricted to the top edge of the 2D first Brillouin zone. Furthermore, the imaginary part of the Bloch wavevector can only be directed along the k y direction in reciprocal space, since propagation is lossless along the x-axis. ℑ(k y ) 0 hence provides the necessary exponential decrease away from the glide interface such that the guided wave is confined. It can be checked visually in the modal shapes of Fig. 4.2(b) that the exponential decrease in amplitude along the y-axis is accompanied by an alternation of the sign from one unit cell to the next, in accordance with the condition ℜ(k y )a/π = 1.

  at the X point of the 1D Brillouin zone (ka = π). Hence the eigenvalues of G g (π/a) are ±i. Its eigenvectors form complex conjugate pairs, since G g (π/a)u = iu implies G g (π/a)u * = -iu * . Since the glide operator commutes with the dynamical operator for the wave equation, they share common eigenvectors. Hence Du = ω 2 u implies Du * = ω 2 u * since the wave equation has real coefficients because of its time reversal invariance (TRI).

Figure 4 . 3 :

 43 Figure 4.3: Glide reflection symmetry of Bloch waves. Taken from[START_REF] Iglesias Martínez | Glide-Reflection Symmetric Topological Phononic Crystal Waveguide[END_REF]. (a) The periodic part of the Bloch wave can be either symmetric (S) or antisymmetric (AS) with respect to the horizontal glide-reflection. There is in addition a vertical mirror symmetry, resulting in a total of four combinations. (b) At the X point of the first Brillouin zone, the symmetry of the complete Bloch waves is the combination of the symmetry of the periodic part of an alternating signum function with the periodicity of the lattice. The dispersion of the S and AS GR Bloch waves are degenerate at the X point, since those waves are images in a vertical reflection. Note that the periodicity of Bloch waves is 2a. (c) At the Γ point of the first Brillouin zone, the symmetry of the Bloch wave is simply the symmetry of its periodic part. The S and AS GR modes are never degenerate. Note that the periodicity of Bloch waves is a.

  . 4.3(a), the sign of the real and imaginary parts of the complex field P(r) are represented schematically for both A and AS Bloch waves with respect to the glide-reflection, with a total of four possibilities. Next, we introduce the influence of the value of the Bloch wavenumber. At the Γ point (k x = 0 and hence exp(-k • r) = 1), in Fig. 4.3(b), the symmetry of the Bloch wave essentially reproduces the symmetry of the periodic function P(r). The four possibilities depicted in the figure can never be degenerate and Bloch waves crossings at the Γ point are accompanied with Bragg interference. Note that P(r, t) is spatially continuous between unit cells, so when the sign changes across adjacent cells the amplitude goes to zero at the boundary. At the X point of the Brillouin zone, k x = π/a and exp(-k • r) = exp(-πx/a) is periodic in x

Figure 4 . 4 .

 44 The orthogonality of the primitive lattice translations is removed. Compared to the horizontal axis, the vertical boundaries make an angle of 85 • compared to the initial 90 • and hence the vertical reflection is removed. The choice of changing the angle by only 5 • is made so that the complete band gap is not significantly affected. It would actually close for an angle of 60 • , or the hexagonal lattice.

Figure 4 . 4 :

 44 Figure 4.4: Band structure of different crystalline topological insulators, computed for a supercell made of N = 10 unit cells of the oblique lattice crystal. The two lattice directions make an angle θ = 85 • and the two lattice constants are equal.In the case of inversion symmetry with (a) g = 0 there is a full band gap, whereas with (b) g = a/2 the pair of interface waves is present but the spectrum is gapped. Taken from[START_REF] Iglesias Martínez | Glide-Reflection Symmetric Topological Phononic Crystal Waveguide[END_REF]. In the case of glide-reflection symmetry with (c) g = 0 there is a full band gap, whereas with (d) g = a/2 a gapless Dirac point exists at the X point of the Brillouin zone.

Fig. 4

 4 .5 presents three example field maps at frequencies 300 kHz, 350 kHz, and 400 kHz, in the worst case scenario (5% position disorder plus 10% diameter disorder). For all frequencies the excitation of waves guided along the interface is clearly observed.

Figure 4 . 5 :

 45 Figure 4.5: Numerical simulation of the effect of disorder on the frequency response function (FRF) of the glide-reflection symmetric crystal waveguide, for steel rods in water. Taken from [160]. (a) The FRF as a function of frequency shows that interface waves survive a uniformly distributed disorder acting on both the center position of inclusions and their diameter. The grey rectangular region indicates the reference waveguiding bandwidth. Field pressure maps are shown at the selected frequencies (a) 300 kHz, (b) 350 kHz, and (c) 400 kHz.

Figure 4 . 6 :Figure 4 . 7 :

 4647 Figure 4.6: Two pieces of the same square-lattice crystal of steel rods in water (diameter d = 2 mm; lattice constant a = 2.22 mm) are glided. Taken from [160]. (a), Schematic of the experiment for transmission measurement of the water pressure waves experiment where two ultrasonic transducers (Olympus, A301S-SU) of short pulses where used, one as the emitter (E) and as the receiver (R). (b), Photograph of the experimental sample for glide parameter g = a/2, with a the lattice constant of the square-lattice phononic crystal of steel rods in water. The two glided halves of the crystal have been colored numerically to separate them visually.

Fig. 4 .

 4 7(a) shows the numerical transmission as a function of frequency. The experimental transmission of Fig. 4.7(b) is obtained based on the ultrasonic pulse-echo technique described for instance in Ref. [163]. The complete band gap extends from 0.28 MHz to 0.46 MHz whereas the guided mode transmission covers the range from 0.28 MHz to 0.43 MHz, in accordance with theory. For other glide parameter values, transmission is observed as well inside the complete band gap but within a reduced frequency range. Actually, for g a/2 the left and right propagating guided Bloch waves interfere and form an anti-crossing and thus a mini band gap at the X point, as Fig. 4.8(a) shows. The experiments reported in Fig. 4.8(b) clearlyshow the mini band gap opening when g = a/4 and g = 3a/8 and the corresponding reduction of the transmission range. Furthermore, Fig.4.8(c) shows the variation with g of the N-th and the (N + 1)-th band intersection with the X point. The phononic band gap is completely opened for g = 0 and g = a, closes midway for g = a/2, and varies continuously and symmetrically between these points. Since the gap is fully opened for g = 0 and closed for g = a/2, and the glide parameter can be continuously tuned with periodicity a, a continously-tunable transmission filter is obtained. Note that symmetry protection against backscattering of the guided waves is only achieved when the glide CHAPTER 4. TOPOLOGICAL CRYSTAL WAVEGUIDE g = a/4 g =

Figure 4 . 8 :

 48 Figure 4.8: Gap opening as a function of the glide parameter. Taken from[START_REF] Iglesias Martínez | Glide-Reflection Symmetric Topological Phononic Crystal Waveguide[END_REF]. (a) For g a/2, a mini-gap for guided waves opens in the phononic band structure at the X point of the Brillouin zone. (b) Experiment confirms the opening of the mini-gap, for g = a/4 and g = 3a/8. (c) The eigenfrequencies of the two guided waves at the X point vary with the glide parameter (blue line: S waves; red line: AS waves). For exactly g = a/2, the waveguide is glide-reflection symmetric and the guided wave gap closes. This gap opens symmetrically on either side of that value.

Figure 4 . 9 :

 49 Figure 4.9: Taken from [160]. (a) The finite element mesh (FEM) for a supercell with N = 10 unit-cells of a square-lattice sonic crystal of rigid cylinders in water is shown. The rod diameter to lattice constant is d/a = 0.9. (b) The FEM mesh of the glide-symmetric waveguide structure formed in the same crystal by shifting horizontally the 5 bottom rows by half a lattice constant (g = a/2). (c) The phononic band structure for g = 0 plotted with yellow dots shows a complete band gap (outlined gray region). The phononic band structure for g = a/2 plotted with blue dots shows a pair of guided waves with smooth dispersion (Bloch waves number N and N + 1). (d) The real part of the pressure distribution of the guided Bloch wave number n = N + 1 = 11 is displayed at ka/(2π) = 0.1, ωa/(2π) = 959 m/s.

Figure 4 . 10 :

 410 Figure 4.10: Taken from[START_REF] Iglesias Martínez | Glide-Reflection Symmetric Topological Phononic Crystal Waveguide[END_REF]. Phononic band structure for the glide-symmetric waveguide, for a squarelattice phononic crystal of circular holes in silicon, supporting vector elastic waves. In-plane elastic waves described by two displacements decouple from scalar out-of-plane elastic waves. The FEM mesh is the same as in Figure4.9(b). (a) For in-plane waves, The color bar in the phononic band structure indicates the relative longitudinal polarization contents, from 0 (pure shear) to 1 (pure longitudinal). (b) The phononic band structure for scalar out-of-plane waves essentially looks similar to the scalar fluid case.

c 44 =

 44 [START_REF] Zhang | Crushing of vertex-based hierarchical honeycombs with triangular substructures[END_REF] GPa, and ρ = 7780 kg/m 3 . Independent material constants for isotropic epoxy are c 11 = 7.54 GPa, c 44 = 1.48 GPa, and ρ = 1142 kg/m 3 . The elastic contrast between steel and epoxy is very high; the shear velocity of epoxy is especially very slow compared to the shear velocity of steel. For this reason, the Bragg interaction at the Γ point of the Brillouin zone is very strong and acts against the frequency width of the guiding bandwidth for the glide-reflection symmetric waveguide. In Fig.4.11, an additional a/4 vertical spacing has been added between the two glided crystal phases. This addition slightly reduces the Bragg interaction at the Γ point without affecting glide symmetry. Overall, the observations are similar to the case of the single-phase solid phononic crystal. In the case of scalar out-of-plane elastic waves in Fig.4.11(b), a pair of symmetry-protected guided waves appears for g = a/2. This occurs for two successive band gaps separated by a group of flat propagating bands. In the case of in-plane elastic waves in Fig.4.11(a) we again observe both the pairs of symmetry-protected guided waves and surface waves propagating along the bottom and top surface of the phononic crystal strip.

Figure 4 . 11 :

 411 Figure 4.11: Taken from[START_REF] Iglesias Martínez | Glide-Reflection Symmetric Topological Phononic Crystal Waveguide[END_REF]. Phononic band structure for the glide-symmetric waveguide, for a squarelattice phononic crystal of circular steel rods in epoxy, supporting vector elastic waves. In-plane elastic waves described by two displacements decouple from scalar out-of-plane elastic waves. The FEM mesh is the same as in Figure4.2(b). Phononic band structures are shown for (a) in-plane waves and (b) scalar out-of-plane waves. For in-plane waves, The color bar in the phononic band structure indicates the relative longitudinal polarization contents, from 0 (pure shear) to 1 (pure longitudinal). Note that the waveguide width is a/4 (an additional vertical spacing is added between the two crystal phases).

Figure 5 . 1 :

 51 Figure 5.1: The design of 2D and 3D hierarchical structure considered in this work. Taken from [170]. a) A cross-section of the unit-cell of the considered model having level 1 structural elements composed of 4 (2 × 2) level 0 units. b) Conceptual deformation of substructures corresponding to level 0 and level 1 of the analyzed system. c) The unit-cell of the 3D model considered in this study. d) SEM (Scanning Electron Microscopy) picture of the quasi-2D experimental prototype composed of 3×3×3 unit-cells. e) SEM picture of the 3D experimental prototype consisting of 3×3×4 unit-cells.

ure 5 .

 5 1(a)), level 1 structural elements are composed of 2 × 2 level 0 units. However, in general, within the frame of the considered model, level 1 blocks can be constructed by means of a different number of 2N x × 2N y level 0 elements (e.g. 4 × 4) assuming that they form a rectangle-like lattice and are connected to each other as shown in Figure5.1.

  respectively. A drop of IP-S resin was deposited on a ITO-coated soda lime glass substrate with dimensions 25 × 25 × 0.7 mm 3 and photopolymerized with a femtosecond laser operating at λ = 780 nm and a 25X-objective. After printing, the sample was developed for 25 min in Propylene glycol methyl ether acetate (PGMEA) solution to remove the unexposed photoresist and rinsed for 3 min in Isopropyl alcohol (IPA) to clear the developer. A Laser Power of 90 and a Galvanometric scan speed of 100 mm s -1 were used for the whole fabrication process. Finally, to improve the adhesion properties of IP-S resin and so to ensure a better connection of the printed sample with the substrate, ITO-coated substrates are pre-treated with oxygen plasma (Corial 200R, RIE etch system) for 5 min.

Figure 5 . 2 :

 52 Figure 5.2: Deformation of the three examples of 2D systems considered in this work that correspond to different thicknesses of hinges. Taken from [170]. a) The first of the analyzed structures (case 1) where for the predeformed structure θ 0 = 90 • , θ 1 = 70 • , d 1 ≈ 9.3 µm, d 2 ≈ 6.1 µm. b) Case 2 with the initial parameters defined as θ 0 = 100 • , θ 1 = 70 • , d 1 ≈ 2.7 µm, d 2 ≈ 10.0 µm. c) Case 3 where for the predeformed system θ 0 = 100 • , θ 1 = 70 • , d 1 ≈ 2.6 µm, d 2 = 9.6 µm. d) Poisson's ratio measured for the compression along the z-axis calculated for the central topmost unit-cell. e) FEA results corresponding to the variation in the θ 0 and θ 1 angles throughout the deformation process. For all structures, l s ≈ 1200 µm. Mechanical metamaterials presented on panels a-b) and c) are composed of 3 × 3 × 3 and 3 × 3 × 2 unit-cells respectively. The remaining geometric parameters used for structures shown on panels a-c) were set to be the following: a) a 1 ≈ 77 µm, a 2 ≈ 78 µm, b ≈ 48 µm, d 3 ≈ 6.3 µm, d 4 ≈ 6.4 µm, b) a 1 ≈ 79 µm, a 2 ≈ 81 µm, b ≈ 48 µm, d 3 ≈ 12.7 µm, d 4 ≈ 10.4 µm and c) a 1 ≈ 77 µm, a 2 ≈ 79 µm, b ≈ 47 µm, d 3 ≈ 14.3 µm, d 4 ≈ 10.2 µm.For all mechanical metamaterials portrayed on panels a-c), the red and blue dashed lines correspond to the outline of the initial and deformed unit-cell respectively. All of the experimental pictures shown on panels a-c) were taken by means of the optical camera. Finally, the surface stress presented graphically for the FEA results on panels a-c) corresponds to the von Mises stress model implemented in the COMSOL Multiphysics software.

Figure 5 . 3 :

 53 Figure 5.3: Shape morphing of the considered hierarchical mechanical metamaterials aimed towards the possibility of replicating the shape of the reference object. Taken from[START_REF] Dudek | Micro-Scale Auxetic Hierarchical Mechanical Metamaterials for Shape Morphing[END_REF]. a) A graphical representation of unit-cells corresponding to case 1 and case 2 that were used in the design of the structure that is analyzed from the point of view of its ability to change shape in a predefined manner. b) Picture of the prototype used in the experiment together with the zoomed-in images of types of structures used in the design of the resultant composite material. c) The experiment corresponding to the compression of the considered structure that results in the change in its shape. d) The real-life mug used as an inspiration for the shape that the analyzed hierarchical system is supposed to assume upon being deformed.

Figure 5 . 4 :

 54 Figure 5.4: Comparison of the Poisson's ratio plotted against strain for case 1 and case 2 structures compressed in the longitudinal and transverse directions respectively. Taken from [170].

Figure 5 . 5 :

 55 Figure 5.5: Properties of the experimental prototypes analyzed in this study. Taken from [170]. a) Dependence of the reaction force on the displacement of the topmost part of the sample. b) Force plotted against the strain measured in the direction of compression. c) An estimate of the total absorbed energy for each of the structures (area under the graph). d) The stiffness exhibited by each of the analyzed systems plotted against strain. e) Force plotted against the displacement and the corresponding Specific Energy Absorption for each of the structures. f ) Stress plotted against strain as well as auxiliary straight black dashed lines used in order to determine Young's modulus corresponding to each of the systems.

  • and θ 1 = 70 • . However, it is also interesting to check what would happen if the initial configuration of the system was indeed the same as for the fully-open structure (θ 0 = 90 • and θ 1 = 90 • ).

Figure 5 . 6 :

 56 Figure 5.6: Behavior of the case 1 structure corresponding to different initial angles θ 0 and θ 1 . Taken from [170]. a) Diagrams extracted from the simulation showing the topmost central unit-cell of the initially fullyopen structure subjected to the uniaxial compression. b) The change in the Poisson's ratio for the structures corresponding to the fully-open configuration and the configuration discussed in the main text.

Figure 5 . 7 :

 57 Figure 5.7: The behavior of the case 1 structure deformed at different rates. Taken from [170]. a) Diagrams extracted from the simulation showing the topmost central unit-cell of a considered system throughout the deformation process. b) The change in the Poisson's ratio for the case 1 system compressed at different rates.

Figure 5 . 8 :Figure 5 . 9 :

 5859 Figure 5.8: The behavior of the type A and type B 3D hierarchical structures composed of 3 × 3 × 4 and 3 × 3 × 3 unit-cells respectively. Taken from [170]. Both hierarchical models are subjected to the compression along the z-axis. a) Three different stages of the deformation process corresponding to the type A system. Provided pictures were captured by means of the optical camera and portrayed from the perspective of the xz plane. b) SEM picture of the type A prototype. c) The Poisson's ratio for compression along the z-axis calculated for the topmost row of unit-cells for both types of the considered 3D systems. The geometric parameters corresponding to the predeformed type A structure were the following: θ 0 = 90 • , θ 1 = 70 • , a 1 ≈ 33.2 µm, a 2 ≈ 33.8 µm, b ≈ 22.4 µm, d 1 ≈ 7.0 µm, d 2 ≈ 4.0 µm, d 3 ≈ 6.6 µm, d 4 ≈ 4.1 µm, l ≈ 88.7 µm and h ≈ 16.4 µm. On the other hand, for the type B system, these parameters were set to be as follows: θ 0 = 100 • , θ 1 = 70 • , a 1 ≈ 73.5 µm, a 2 ≈ 74.5 µm, b ≈ 44.4 µm, d 1 ≈ 2.6 µm, d 2 ≈ 9.2 µm, d 3 ≈ 14.5 µm, d 4 ≈ 12.5 µm, l ≈ 180 µm and h ≈ 33 µm . For pictures on panels (a-b), the blue dashed line indicates the dimensions of the undeformed structure in the xz plane.

Figure 6 . 1 ,

 61 it is shown that a simple angle change in a given crystal can drastically change the dispersion relation as depicted in the second panel. The gap can be seen as a tunable function of the angle α. We can, in principle, place magnets on such structure and use an external magnetic field to change the angle, and thus tune or select the required band gap.

Figure 6 . 1 :

 61 Figure 6.1: Active reconfiguration and transition in the properties of the system induced by the application of an external magnetic field. a) Diagram showing the concept of the reconfiguration of the magnetomechanical structure. b) Effect of the elongation of the isosceles triangles, corresponding a metamaterial composed of simple triangles, on the band gap formation. Phonon band structures corresponding to the analyzed structures. c) The first three band gaps associated with structures corresponding to different values of the b/a ratio. The geometric parameters used for all of the structures were set to be the following: a = 200 µm, l 1 = 8 µm, l 2 = 4 µm, l 3 = 4 µm, l 4 = 4 µm, l 5 = 9 µm.

Figure 6 . 2 :

 62 Figure 6.2: (a) Unit cell of considered crystal, with r 1 = 0.315a. (b) Calculated dispersion relation along the ΓX direction for two cases: rotating fluid or non-rotating fluid.

  Overall, I would like to extend the work of my thesis from simple passive media to stimuliresponsive and time-dependent media. , I have been the first one to build a characterisation setup that we have used for mechanical testing of micro-scale samples. It should be noted that sample lengths range from few tens of microns to about 1 mm. A serious attention was given to the optical visualisation, and to displacements and force measurements. This was not my main PhD topic but I have been in charge of making all instrumentation of this device, calibrations and all data treatments such as static cross-correlation analysis for strain measurements, displacement under dynamic excitation, and stress-strain curves calibration. Overall, this part took a lot of time and my contribution has always been made as co-author of these studies. Here, I will not go through all details but I will rather emphasize my contribution to a series of published studies, along with an illustration. A.1/ CLOSED TUBULAR MECHANICAL METAMATERIAL AS LIGHTWEIGHT LOAD-BEARING STRUCTURE AND ENERGY ABSORBER Periodic truss-lattice materials, especially when combined with current additive manufacturing techniques, are attracting attention in lightweight material engineering. As a member of the elementary cubic truss family, the simple-cubic truss lattice possesses the highest stiffness and strength along the principal directions and plays an important role in load-bearing mechanical metamaterials. High anisotropic mechanical properties and low resistance to buckling loading and shearing loading, however, limit its use in energy absorption. Here, we present a class of simple-cubic closed tubular lattice, Figure A.1, with limited loading direction dependence along with high mechanical properties and irregular stable post-yield response. The fabrication of its complex structure was made possible by direct laser writing at the microscale, depected of the SEM image in the Figure A.1.

Figure A. 1 :

 1 Figure A.1: SEM images of simple-cubic polymeric samples fabricated via 3D printing technology. Taken from [179]. Isometric views and zoom-in views are shown for truss and closed tubular lattices with relative density 0.1. (A) [100] truss lattice and (B) [100] closed tubular lattice. (C) [110] truss lattice and (D) [110] closed tubular lattice.

Figure A. 2 :

 2 Figure A.2: Principle of the micro-stepper. Taken from [180]. a) An oblique view shows that the stepper contains a chiral elastic column, based on a mechanical metamate-rial, that is twisting under a unidirectional compression; the chiral metamaterial part is depicted as a subpanel. The bottom subpanel is a zoom on the central part that shows the ratchet moving under frictional contact with the motionless central plate. A pawl is placed as a blocking mechanism and a keyway is included for rotation monitoring. A full CAD representation of the structure is presented in Figure S1 (Supporting Information). b) Under uniaxial compression, an oblique view and top views are shown of the deformed structure for 3 different pre-deformed states: initial state, compression to the nominal stepping rotation angle, and compression-released case leaving the ratchet in its first stepping angle. Geometrical parameters are the height of the chiral metamaterial part h = 57.57µ m, the radius of the chiral metamaterial part R = 36µm, and the angle of the chiral elastic column θ = 45deg. The colorbar for total displacement shows which parts are actually moving during operation.

Figure A. 3 :

 3 Figure A.3: SEM imaging of the fabricated micro-stepper. Taken from [180]. a) The full micro-stepper is fabricated using the DLW microprinting technique. b) Zoom on the locking mechanism (pawl) that prevents the central plate from moving backward during the unloading phase. c) Top view of the stepper.

Figure A. 4 :

 4 Figure A.4: Experimental tests. Taken from [180]. a) Side optical images of the deformation process from initial via compression to released state. b) Top optical images show the rotation process during multiple actuation of the micro-stepper. A red arrow has been added to indicate the rotation of the ratchet. Images are shown for the n = 0, 7, 14, 21, 30 loading cycles for a sample without surface processing. c) Same as (b), but for a chromium sputtered (layer thickness about 5 nm) sample, for cycles n = 0, 15, 30, 45, 53. In both panels (b) and (c) the 0th cycle is used as the initial state. Force-displacement (Dis.) curves obtained during cycling are shown for the d) clean polymer sample and for the e) chromium coated sample.

Figure A. 5 :

 5 Figure A.5: Schematic view of the unit cell, its effective isotropic behaviour and the optical imaging during the compressional tests. Obtained results are plotted versus loop loads. Taken from [83].

Figure A. 6 :

 6 Figure A.6: Design and methodologies. Five different bio-inspired unit cells are designed from the inspiration of the parrot's beaks. We depict the different meta-structures constructed from primitive motifs and embedded in a periodic manner to produce later the cylindrical-like shapes.

Figure A. 7 :

 7 Figure A.7: Micro-structures: (a) SEM Images of Fabricated Micro-structures. Samples corresponding to "A", "B", and "E" designs are shown and a corresponding zoom onto the critical regions of the structures is depicted under each column. Two scale bars are used: the white corresponds to 100µm and the red to 50µm. (b) The corresponding measured Stress-strain curves. (c) deformation patterns at strain level of 0.3.

Figure A. 8 :

 8 Figure A.8: Compression experiments with BCC-SC micro-lattice samples fabricated by two-photon polymerization along directions [100] and [110]. (a) Nonlinear mechanical response and corresponding frames acquired during compression tests on samples with a relative density of 1.4%. Frames I to IV show the initial, collapse deformation, global unstably or steady deformation, and large compression up to maximum applied strain. (b) Nonlinear mechanical response and corresponding frames acquired during compression tests on samples with a relative density of 5.9%. Frames I to IV show the initial, collapse deformation, inelastic buckling, and large compression up to maximum applied strain. All scale bars are 200 m long.The dotted line at 20% strain marks the upper limit of available numerical simulation data.(c-e) Stress-strain curves are shown for samples with higher relative densities. The dotted line at 20% strain marks the upper limit of available numerical simulation data. Taken from[START_REF] Chen | 3D lightweight mechanical metamaterial with nearly isotropic inelastic large deformation response[END_REF]

Figure A. 9 :

 9 Figure A.9: The unit-cell of the structure analysed in this work from the point of view of its mechanical properties. a) A diagram of the unit-cell of the system with the definition of all of the dimensions and geometric parameters of the structure. b) Different deformation patterns exhibited by the considered system in the case of the isolated deformation of the level 0 or level 1 elements of the hierarchical structure.

Table 2 .

 2

	1: 3D phononic crystals. pc: perovskite-cubic; fcc: face-centered cubic; hcp: hexagonal com-
	pact; sc: simple cubic; bcc: body-centered cubic; SLS:selective laser sintering; SLA: stereolithography
	apparatus; SBEM: Selective Electron Beam Melting.

  La construction de métamatériaux acoustiques et élastiques extrêmes est considérée à la fois d'un point de vue numérique et expérimental. De larges bandes interdites phononiques, des relations de dispersion non monotones et une auxéticité extrême et accordable sont les objectifs spécifiques discutés dans le manuscrit. Dans le premier chapitre, nous introduisons les concepts des ondes et des métamatériaux, ainsi que les équations constitutives en élasticité et en acoustique. Dans le deuxième chapitre, nous décrivons la conception d'un cristal phononique 3D à bande interdite ultra large pour les ondes ultrasonores fonctionnant dans la gamme des MHz.

Titre : Métamateriaux acoustiques et élastiques présentant des propriétés extrêmes

Mots-clés : Cristaux Phononiques, Metamatériaux, Relations de dispersion, Guides d'onde Topologiques, Matériaux Auxétiques Résumé :

ACKNOWLEDGEMENTS

I am deeply grateful to my supervisors Vincent Laude and Muamer Kadic who not only gave me the opportunity to study this fascinating subject and helped to enter to the Metamaterials and Phononic crystal world, but also gave me great discussions that allowed us