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Love waves are antiplane elastic waves which propagate along the surface of a heterogeneous medium. Under time-harmonic regime, they are governed by a scalar equation of the Helmholtz type. We exploit the invariance of this governing equation under an in-plane arbitrary coordinate transformation to design broadband cloaks for surface defects. In particular, we apply transformation elastodynamics to determine the anisotropic, position dependent, mechanical properties of ideal cloaks able to hide triangular and parabolic-shaped defects. Dispersion analysis and time-harmonic numerical simulations are employed to validate the proposed strategy. Next, we utilize layered monoclinic materials, with homogenized properties matching those of the ideal cloaks, to design feasible triangular-shaped cloaks. The performance of the layered cloaks is validated via parametric analysis of the dispersion curves, which converge to those of the ideal cloak when the unit cell-wavelength ratio vanishes. Finally, time-harmonic numerical simulations confirm a significant reduction of the defect-generated scattered fields by the layered cloaks.

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... In this study we recalled a brief theoretical background and developed a strategy for 3-D elastic cloaking from surface waves, assuming cylindrical symmetry. This idea is an expansion of [4] where only the case of Love waves was considered. We believe that this work can support future extensions, including the realization of a fully feasible and functional elastic cloaking device. ...

... In the context of elastodynamics, materials dressed with space, time, and space-time modulations are conveniently molded to manipulate wave motion at will. Examples have shown different forms of waveguiding that rely on defect mechanisms [1], topological insulators [2][3][4][5][6][7][8][9][10], metagradings [11][12][13][14][15][16][17], transformation optics and acoustics [18][19][20][21][22][23][24][25][26][27][28], which are passive strategies effectively employed to accomplish relevant technological behaviors, such as communication, focusing, and invisibility. ...

We provide a theoretical framework to mold time-modulated lattices with frequency conversion and wave-steering capabilities. We initially focus on 1D lattices, whereby a sufficiently slow time-modulation of the stiffness is employed to convert the frequency content of impinging waves. Based on the adiabatic theorem, we demonstrate that undesired reflections, which emerge in time-discontinuous materials, can be eliminated by a careful choice of the modulation velocity. The concept is later explored in the context of 2D lattices, whereby a slow time modulation of the stiffness not only induces frequency conversion without back-scattering, but also serves as a mechanism to steer waves. Our paper explores a new and exciting way to control wave propagation in elastodynamics with scattering-free guiding capabilities, and may open new avenues for the manipulation and transport of information through elastic waves.

This paper aims to investigate the propagation of low-frequency seismic surface waves in periodic in-filled barriers from the viewpoint of bandgaps. Based on Bloch-Flouquet theory, four configurations of periodic pile barriers with the same filling rate but different cross sections are studied. The periodic boundary conditions are imposed to derive the characteristics equation. Simultaneously, the finite element method is used to calculate the numerical dispersion relation and vibration modes of four types of periodic pile barriers with various inclusions. The sound cone and the energy parameter method are introduced to identify the surface wave modes further. Additionally, the mitigation effectiveness of a finite number of periodic barriers is evaluated using a three-dimensional transmission model. The results show that the initial frequency and bandwidth of the surface bandgaps are sensitive to the shape of the cross-section and mechanical parameters of the inclusions. The surface wave bandgaps within 9.64 ∼ 13.92 Hz are observed for the periodic barrier composed of cross-like polyfoam sections. Moreover, the radiation modes formed by the leakage of some surface waves into the sound cone are expected to widen the attenuation zones of the periodic barriers. The numerical results in frequency and time domains further validate the attenuation effect of finite barrier in the bandgap and the attenuation widening phenomenon caused by the dissipation of the leak modes into the bulk. This study provides new insights into the design of periodic pile barriers. In the future, it is expected to mitigate the low-frequency ground motion by fully designing the section shape and filling materials of pile barriers.

Coordinate‐transformation‐inspired optical devices have been mostly examined in the continuous‐wave regime: the performance of an invisibility cloak, which has been demonstrated for monochromatic excitation, is likely to deteriorate for short pulses. Here, pulse dynamics of flexural waves propagating in transformed plates is investigated. A practical realization of a waveshifter and a rotator for flexural waves based on the coordinate transformation method is proposed. Time‐resolved measurements reveal how the waveshifter deviates a short pulse from its initial trajectory, with no reflection at the bend and no spatial and temporal distortion of the pulse. Extending the strategy to cylindrical coordinates, a wave rotator is designed. It is demonstrated experimentally how a pulsed plane wave is twisted inside the rotator, while its wavefront is recovered behind the rotator and the pulse shape is preserved, with no extra time delay. The realization of the dynamical mirage effect is proposed, where an obstacle appears oriented in a deceptive direction. The pulse dynamics of flexural waves propagating inside a 3D‐printed transformed plate is investigated. Time‐resolved measurements reveal how a pulsed plane wave is twisted inside the rotator, while its wavefront is recovered behind the rotator and the pulse shape is preserved, with no extra time delay.

We propose a design of cylindrical cloak for coupled in-plane shear waves consisting of concentric layers of sub-wavelength resonant stress-free inclusions shaped as Swiss rolls. The scaling factor between inclusions’ sizes is according to Pendry’s transform. Unlike the hitherto known situations, the present geometric transform starts from a Willis medium and further assumes that displacement fields u in original medium and u ′ in transformed medium remain unaffected ( u ′ = u ). This breaks the minor symmetries of the rank-4 and rank-3 tensors in the Willis equation that describe the transformed effective medium. We achieve some cloaking for a shear polarized source at specific, resonant sub-wavelength, frequencies, when it is located in close proximity to a clamped obstacle surrounded by the structured cloak. The structured medium approximating the effective medium allows for strong Willis coupling, notwithstanding potential chiral elastic effects, and thus mitigates roles of Willis and Cosserat media in the achieved elastodynamic cloaking.

We inspect the propagation of shear polarized surface waves akin to Love waves through a forest of trees of the same height atop a guiding layer on a soil substrate. An asymptotic analysis shows that the forest behaves like an infinitely anisotropic wedge with effective boundary conditions. We discover that the foliage of trees brings a radical change in the nature of the dispersion relation of these surface waves, which behave like spoof plasmons in the limit of a vanishing guiding layer, and like Love waves in the limit of trees with a vanishing height. When we consider a forest with trees of increasing or decreasing height, this hybrid "spoof Love wave" is either trapped within the trees or converted into a downward propagating bulk (shear) wave. These mechanisms of wave trapping and wave conversion appear to be robust with respect to perturbations of height or position of trees in the metawedge and with respect to three-dimensional effects such as regarding a potential change of elastic wave polarization.

We inspect the unusual scattering properties reported recently for structures alternating dielectric layers of subwavelength thicknesses near the critical angle for total reflection. In TE polarization, the unusual scattering properties are captured by an effective model with an accuracy less than 1% up to kd∼0.1. It is shown that the propagation is simply dispersive with local dispersion while the boundary layer effects are captured through a nonintuitive transmission condition. The resulting model involves two parameters depending only on the characteristics of the multilayer and which are given in closed forms. Besides, we show that a discrete description of the spectrum using the layer thickness d as unit of measure misses the complexity of the continuous spectrum exhibiting strong variations within the scale d. This ultrasensitivity to variations below d is attributable to strong boundary layer effects and, for large structures, to a cooperation between the boundaries and the phase accumulation within the structure.

Metasurfaces of mechanical resonators have been successfully used to control in-plane polarized surface waves for filtering, waveguiding and lensing applications across different length scales. In this work, we extend the concept of metasurfaces to anti-plane surface waves existing in semi-infinite layered media, generally known as Love waves. By means of an effective medium approach, we derive an original closed-form dispersion relation for the metasurface. This relation reveals the possibility to control the Love waves dispersive properties by varying the resonators mechanical parameters. We exploit this capability to manipulate the metasurface refractive index and design two gradient index (GRIN) metalenses, i.e. a Luneburg lens and a Maxwell lens. We confirm the performance of the designed lenses using full 3D finite element simulations. Our work demonstrates the possibility of realizing wave control devices for anti-plane waves.

Hyperelastic materials possess the appealing property that they may be employed as elastic wave manipulation devices and cloaks by imposing pre-deformation. They provide an alternative to microstructured metamaterials and can be used in a reconfigurable manner. Previous studies indicate that exact elastodynamic invariance to pre-deformation holds only for neo-Hookean solids in the antiplane wave scenario and the semi-linear material in the in-plane compressional/shear wave context. Furthermore, although ground cloaks have been considered in the acoustic context they have not yet been discussed for elastodynamics, either by employing microstructured cloaks or hyperelastic cloaks. This work therefore aims at exploring the possibility of employing a range of hyperelastic materials for use as antiplane ground cloaks (AGCs). The use of the popular incompressible Arruda-Boyce and Mooney-Rivlin nonlinear materials is explored. The scattering problem associated with the AGC is simulated via finite element analysis where the cloaked region is formed by an indentation of the surface. Results demonstrate that the neo-Hookean medium can be used to generate a perfect hyperelastic AGC as should be expected. Furthermore, although the AGC performance of the Mooney-Rivlin material is not particularly satisfactory, it is shown that the Arruda-Boyce medium is an excellent candidate material for this purpose.

Surface wave methods gained in the past decades a primary role in many seismic projects. Specifically, they are often used to retrieve a 1D shear wave velocity model or to estimate the VS,30 at a site. The complexity of the interpretation process and the variety of possible approaches to surface wave analysis make it very hard to set a fixed standard to assure quality and reliability of the results. The present guidelines provide practical information on the acquisition and analysis of surface wave data by giving some basic principles and specific suggestions related to the most common situations. They are primarily targeted to non-expert users approaching surface wave testing, but can be useful to specialists in the field as a general reference. The guidelines are based on the experience gained within the InterPACIFIC project and on the expertise of the participants in acquisition and analysis of surface wave data.

Transformational elastodynamics can be used to protect sensitive structures from harmful waves and vibrations. By designing the material properties in a region around the sensitive structure, a cloak, the incident waves can be redirected as to cause minimal or no harmful response on the pertinent structure. In this paper, we consider such transformational cloaking built up by a suitably designed metamaterial exhibiting micropolar properties. First, a theoretically perfect cloak is obtained by designing the properties of an (unphysical) restricted micropolar material within the surrounding medium. Secondly, we investigate the performance of the cloak under more feasible design criteria, relating to finite elastic parameters. In particular, the behavior of a physically realizable cloak built up by unrestricted micropolar elastic media is investigated. Numerical studies are conducted for the case of buried as well as surface breaking structures in 2D subjected to incident Rayleigh waves pertinent to seismic loading. The studies show how the developed cloaking procedure can be utilized to substantially reduce the response of the structure. In particular, the results indicate the performance of the cloak in relation to constraints on the elastic parameters.

We present a proof of concept design for a structured square cloak enclosing
a void in an elastic lattice, subjected to flexural vibrations. In addition to
the theoretical design, we fabricate and experimentally test an elastic
invisibility cloak for flexural waves in a mechanical lattice. Further
verification and modelling is performed numerically through finite element
simulations. The primary advantage of our square lattice cloak, over other
designs, is the straightforward implementation and the ease of construction.
The elastic lattice cloak shows an impressive efficiency within the predicted
frequency range of 100-250Hz.

The dynamic response of a coated inclusion is considered in the context of active cloaking. The active cloak is achieved for
a coated inclusion in the presence of flexural and membrane waves. In this article, we investigate the design of an active
cloak for a coated inclusion in three frequency regimes: the low-frequency range, the intermediate range and the higher frequency
range in which scattering resonances occur. In the first of these ranges, we extend previous work, which resulted in a simple
mass-compensation design for the monopole scatterer, while in the second and third ranges, a combination of the use of an
appropriate coating and the appropriate choice of the amplitudes of the active cloaking sources is necessary. We show that
such cloaking can indeed be effective in the region of strong scattering resonances. We give closed form analytic expressions
for the required amplitudes of the active cloaking sources in the three frequency regions and provide asymptotic estimates
and numerical illustrations.

We analyse basic thermal cloaks designed via different geometric transforms
applied to thermal cloaking. We evaluate quantitatively the efficiency of these
heterogeneous anisotropic thermal cloaks through the calculation of the
standard deviation of the isotherms. The study addresses the frequency regime
and we point out the cloak's spectral efficiencies. We find that all these
cloaks have comparable efficiency irrespective of whether or not they have
singular conductivity at their inner boundary. However, approximate cloaking
with multi-layered cloak critically depends upon the homogenization algorithm
and a large number of thin layers (at least fifty) is required to reduce
substantially the standard deviation of the isotherms.

Metamaterials are rationally designed man-made structures composed of functional building blocks that are densely packed into an effective (crystalline) material. While metamaterials are mostly associated with negative refractive indices and invisibility cloaking in electromagnetism or optics, the deceptively simple metamaterial concept also applies to rather different areas such as thermodynamics, classical mechanics (including elastostatics, acoustics, fluid dynamics and elastodynamics), and, in principle, also to quantum mechanics. We review the basic concepts, analogies and differences to electromagnetism, and give an overview on the current state of the art regarding theory and experiment-all from the viewpoint of an experimentalist. This review includes homogeneous metamaterials as well as intentionally inhomogeneous metamaterial architectures designed by coordinate-transformation-based approaches analogous to transformation optics. Examples are laminates, transient thermal cloaks, thermal concentrators and inverters, 'space-coiling' metamaterials, anisotropic acoustic metamaterials, acoustic free-space and carpet cloaks, cloaks for gravitational surface waves, auxetic mechanical metamaterials, pentamode metamaterials ('meta-liquids'), mechanical metamaterials with negative dynamic mass density, negative dynamic bulk modulus, or negative phase velocity, seismic metamaterials, cloaks for flexural waves in thin plates and three-dimensional elastostatic cloaks.

In this paper, we investigate how the form of the conventional elastodynamic equations changes under curvilinear transformations. The equations get mapped to a more general form in which the density is anisotropic and additional terms appear which couple the stress not only with the strain but also with the velocity, and the momentum gets coupled not only with the velocity but also with the strain. These are a special case of equations which describe the elastodynamic response of composite materials, and which it has been argued should apply to any material which has microstructure below the scale of continuum modelling. If composites could be designed with the required moduli then it could be possible to design elastic cloaking devices where an object is cloaked from elastic waves of a given frequency. To an outside observer it would appear as though the waves were propagating in a homogeneous medium, with the object and surrounding cloaking shell invisible. Other new elastodynamic equations also retain their form under curvilinear transformations. The question is raised as to whether all equations of microstructured continua have a form which is invariant under curvilinear space or space-time coordinate transformations. We show that the non-local bianisotropic electrodynamic equations have this invariance under space-time transformations and that the standard non-local, time-harmonic, electromagnetic equations are invariant under space transformations.

We derive the elastic properties of a cylindrical cloak for in-plane coupled shear and pressure waves. The cloak is characterized by a rank 4 elasticity tensor with spatially varying entries, which are deduced from a geometric transform. Remarkably, the Navier equations retain their form under this transform, which is generally untrue [G. W. Milton etal, N. J. Phys. 8, 248 (2006)]. The validity of our approach is confirmed by comparison of the analytic Green’s function in homogeneous isotropic elastic space against full-wave finite element computations in a heterogeneous anisotropic elastic region surrounded by perfectly matched layers.

A theory is presented showing that cloaking of objects from antiplane elastic
waves can be achieved by elastic pre-stress of a neo-Hookean nonlinear elastic
material. This approach would appear to eliminate the requirement of
metamaterials with inhomogeneous anisotropic shear moduli and density. Waves in
the pre-stressed medium are bent around the cloaked region by inducing
inhomogeneous stress fields via pre-stress. The equation governing antiplane
waves in the pre-stressed medium is equivalent to the antiplane equation in an
unstressed medium with inhomogeneous and anisotropic shear modulus and
isotropic scalar mass density. Note however that these properties are induced
naturally by the pre-stress. Since the magnitude of pre-stress can be altered
at will, this enables objects of varying size and shape to be cloaked by
placing them inside the fluid-filled deformed cavity region.

Conceptually, all conceivable three-dimensional mechanical materials can be
built from pentamode materials. Pentamodes also enable to implement
three-dimensional transformation acoustics - the analogue of transformation
optics. However, pentamodes have not been realized experimentally to the best
of our knowledge. Here, we investigate inasmuch the pentamode theoretical ideal
suggested by Milton and Cherkaev in 1995 can be approximated by a metamaterial
with current state-of-the-art lithography. Using numerical calculations
calibrated by our fabricated three-dimensional microstructures, we find that
the figure of merit, i.e., the ratio of bulk modulus to shear modulus, can
realistically be made as large as about 1,000.

This chapter consists of three parts. In the first part we recall the
elastodynamic equations under coordinate transformations. The idea is to use
coordinate transformations to manipulate waves propagating in an elastic
material. Then we study the effect of transformations on a mass-spring network
model. The transformed networks can be realized with "torque springs", which
are introduced here and are springs with a force proportional to the
displacement in a direction other than the direction of the spring terminals.
Possible homogenizations of the transformed networks are presented, with
potential applications to cloaking. In the second and third parts we present
cloaking methods that are based on cancelling an incident field using active
devices which are exterior to the cloaked region and that do not generate
significant fields far away from the devices. In the second part, the exterior
cloaking problem for the Laplace equation is reformulated as the problem of
polynomial approximation of analytic functions. An explicit solution is given
that allows to cloak larger objects at a fixed distance from the cloaking
device, compared to previous explicit solutions. In the third part we consider
the active exterior cloaking problem for the Helmholtz equation in 3D. Our
method uses the Green's formula and an addition theorem for spherical outgoing
waves to design devices that mimic the effect of the single and double layer
potentials in Green's formula.

A new cloaking method is presented for 2D quasistatics and the 2D Helmholtz equation that we speculate extends to other linear wave equations. For 2D quasistatics it is proven how a single active exterior cloaking device can be used to shield an object from surrounding fields, yet produce very small scattered fields. The problem is reduced to finding a polynomial which is close to 1 in a disk and close to 0 in another disk, and such a polynomial is constructed. For the 2D Helmholtz equation it is numerically shown that three exterior cloaking devices placed around the object suffice to hide it.

A new type of cloak is discussed: one that gives all cloaked objects the appearance of a flat conducting sheet. It has the advantage that none of the parameters of the cloak is singular and can in fact be made isotropic. It makes broadband cloaking in the optical frequencies one step closer.

The transformation method is a powerful tool providing the constitutive parameters necessary for arbitrary geometric transformations of solution fields. These constitutive parameters, in elasticity, describe a constitutive law that, unlike conventional Hooke's law, is polar and chiral and that no known solids exhibit. This raises the question of whether polar and chiral elastic solids can be designed from the bottom up as architected lattice-based materials; this design task is a major challenge in the field of transformation elasticity. The present study aims to provide a theoretically justified design methodology based on a discrete transformation method. The key idea is to let the gradient of the geometric transformation operate not only on the elastic properties but on the underlying lattice-based architecture of the solids. As an outstanding application, we leverage the proposed design paradigm to construct a polar lattice metamaterial subsequently used for elastic carpet cloaking purposes. Numerical simulations are carried to show excellent cloaking performance under different static and dynamic mechanical loads and thus demonstrate the validity of the proposed designs. The approach presented herein could promote and accelerate new designs of lattice topologies for transformation elasticity in particular and can be extended to realize other emergent elastic properties and to unlock peculiar field-warping functions other than cloaking in static and dynamic contexts.

Cloaking elastic waves has, in contrast to the cloaking of electromagnetic waves, remained a fundamental challenge: the latter successfully uses the invariance of Maxwell’s equations, from which the field of transformational optics has emerged, whereas the elastic Navier equations are not invariant under coordinate transformations. Our aim is to overcome this challenge, at least in practical terms, and thereby unlock applications in mechanics, ultrasound, vibration mitigation, non-destructive evaluation and elastic wave control. We achieve near-cloaking by recognising that, despite the lack of invariance, a decoupling into a system of form invariant potential equations together with a quantifiable approximation, can be used effectively in many cases to control the flow of elastodynamic waves. Here, in particular we focus on the efficiency and practicability of the proposed near-cloaking which is illustrated using carpet cloaks to hide surface defects from incoming compressional and shear in-plane waves and from surface elastic Rayleigh waves.

Metamaterial thermal energy devices obtained from transformation optics have recently attracted wide attention due to their vast potential in energy storage, thermal harvesting or heat manipulation. However, these devices usually require inhomogeneous and extreme material parameters which are difficult to realize in large-scale applications. Here, we demonstrate a general process to design thermal harvesting devices with available natural materials through optimized composite microstructures. We first design a cross-shaped microstructure and apply two-scale homogenization theory to obtain its effective properties. Optimal Latin hypercube technique, combined with a genetic algorithm, is then implemented on the microstructure to achieve optimized geometrical parameters. The optimized microstructure can accurately approximate the behavior of transformed materials. We design such devices and numerically characterize good thermal energy harvesting performances. To validate the wide application range of our approach, we illustrate other types of microstructures like split rings and rectangles, and show that they mimic well the required constitutive parameters. The approach we propose can be used to design novel thermal harvesting devices available with existing technology, and can also act as a beneficial vehicle to explore other transformation opticcs enabled designs.

Cloaks make objects invisible to their surroundings. Cloaking against elastic disturbances, in particular, has been demonstrated using several designs. None, however, tolerate the coexistence of normal and shear stresses due to a shortage of transformation-invariant elastic materials. Here, we overcome this limitation thanks to a new class of polar materials that exhibit asymmetric stresses. A static cloak for full two-dimensional elasticity is thus constructed based on the transformation method. The proposed cloak is made of a functionally graded multi-layered lattice embedded in an isotropic continuum background. While one layer is tailored to produce a target elastic behavior, the other layers impose a set of kinematic constraints equivalent to a distribution of body torques that breaks stress’ minor symmetry. Experimental testing under static compressive and shear loads demonstrates encouraging cloaking performance as “neutral inclusions”, which is also in good agreement with our theoretical prediction. The work sets a precedent in the field of transformation elasticity and should find applications in mechanical stress shielding and stealth technologies.

Rotationally resonant metamaterials are leveraged to answer a longstanding question regarding the existence of transformation-invariant elastic materials and the ad hoc possibility of transformation-based passive cloaking in full plane elastodynamics. Combined with tailored lattice geometries, rotational resonance is found to induce a polar and chiral behavior, that is, a behavior lacking stress and mirror symmetries, respectively. The central, and simple, idea is that a population of rotating resonators can exert a density of body torques strong enough to modify the balance of angular momentum on which hang these symmetries. The obtained polar metamaterials are used as building blocks of a cloaking device. Numerical tests show satisfactory cloaking performance under pressure and shear probing waves, further coupled through a free boundary. The work sheds new light on the phenomenon of resonance in metamaterials and should help put transformation elastodynamics on equal footing with transformation acoustics and optics.

Cloaking of a circular cylindrical elastic inclusion embedded in a homogeneous linear isotropic elastic medium from antiplane elastic waves is studied. The transformation or change-of-variables method is used to determine the material properties of the cloak and the homogenization theory of composites is used to construct a multilayered cloak consisting of many bi-material cells. The large system of algebraic equations associated with this problem is solved by using the concept of multiple scattering with wave expansion coefficient matrices. Numerical results for cloaking of an elastic inclusion and a rigid inclusion are compared with the case of a cavity. It is found that while the cloaking patterns for the three cases are similar, the major difference is that standing waves are generated in the elastic inclusion and the multilayered cloak cannot prevent the motion inside the elastic inclusion, even though the cloak seems nearly perfect. Waves can penetrate into and cause vibrations inside the elastic inclusion, where the amplitude of standing waves depend on the material properties of the inclusion but are very much reduced when compared to the case when there is no cloak. For a prescribed mass density, the displacements inside the elastic cylinder decrease as the shear modulus increases. Moreover, the cloaking of the elastic inclusion over a range of wavenumbers is also investigated. There is significant low frequency scattering even if the cloak consists of a large number of layers. When the wavenumber increases, the multilayered cloak is not effective if the cloak consists of an insufficient number of layers. Resonance effects that occur in cloaking of elastic inclusions are also discussed.

A doubly periodic geometric transformation is applied to the problem of out of plane shear wave propagation in a doubly periodic perforated elastic medium. The technique leads to the design of a system of radially anisotropic and inhomogeneous shells surrounding the void inclusions, that can be tuned to give the desired filtering properties.
For a regular transformation, the transformed elastic system displays the same dispersion properties than the original homogeneous one, but for overlapping and unfolding transformations new filtering properties can be obtained, which include anomalous resonances at zero and finite frequencies. Low-frequency homogenisation reveals how it is possible to tune the phase and group velocity in the long-wave limit at any value, or to obtain a zero frequency band gap for Neumann boundary conditions. The dispersion properties of the medium are studied both semi-analytically by the multipole expansion and numerically by the finite element methods.
Several applications are shown, including the transmission problem throughout a grating of void inclusions and an interface in a waveguide, where the capability of the proposed model is quantitatively demonstrated by computing the transmitted power flow. Finally, we gave a demonstration of defect modes in a waveguide and of tuning the transformation for the research of Dirac points.

Elastic material with its elastic tensor losing minor symmetry is considered impossible without introducing artificially body torque. Here we demonstrate the feasibility of such material by introducing rotational resonance, the amplified rotational inertia of the microstructure during dynamical loading breaks naturally the shear stress symmetry, without resorting to external body torque or any other active means. This concept is illustrated through a realistic mass-spring model together with analytical homogenization technique and band structure analysis. It is also proven that this metamaterial model can be deliberately tuned to meet the material requirement defined by transformation method for full control of elastic wave, and the relation bridging the microstructure and the desired wave functionality is explicitly given. Application of this asymmetric metamaterial to design elastic wave cloak is demonstrated and validated by numerical simulation. The study paves the way for material design used to construct the transformation media for controlling elastic wave and related devices.

In elasticity, the design of a cloaking for an inclusion or a void to leave a vibrational field unperturbed by its presence, so to achieve its invisibility, is a thoroughly analysed, but still unchallenged, mechanical problem. The 'cloaking transformation' concept, originally developed in electromagnetism and optics, is not directly applicable to elastic waves, displaying a complex vectorial nature. Consequently, all examples of elastic cloaking presented so far involve complex design and thick coating skins. These cloakings often work only for problems of unidirectional propagation, within narrow ranges of frequency, and considering only one cloaked object. Here, a new method based on the concept of reinforcement, achieved via elastic stiffening and mass redistribution, is introduced to cloak multiple voids in an elastic plate. This simple technique produces invisibility of the voids to flexural waves within an extremely broad range of frequencies and thus surpassing in many aspects all existing cloaking techniques. The proposed design principle is applicable in mechanical problems ranging from the micro-scale to the scale of civil engineering. For instance, our results show how to design a perforated load-bearing building wall, vibrating during an earthquake exactly as the same wall, but unperforated, a new finding for seismic protection.

A general process is proposed to experimentally design anisotropic inhomogeneous metamaterials obtained through a change of coordinates in the Helmholtz equation. The method is applied to the case of a cylindrical transformation that allows cloaking to be performed. To approximate such complex metamaterials we apply results of the theory of homogenization and combine them with a genetic algorithm. To illustrate the power of our approach, we design three types of cloaks composed of isotropic concentric layers structured with three types of perforations: curved rectangles, split rings and crosses. These cloaks have parameters compatible with existing technology and they mimic the behavior of the transformed material. Numerical simulations have been performed to qualitatively and quantitatively study the cloaking efficiency of these metamaterials.

The paper initiates a constitutive theory of isotropic linear elastic solids where Cauchy’s stress tensor is not necessarily symmetric thus earning them the attribute “polar”. Mainly, expressions for the fourth-order elasticity tensors of isotropic polar solids in two and three space dimensions are derived. It is found that the constitutive tensors feature, in addition to the usual bulk and shear moduli, one extra independent elastic constant in 3D, and two extra constants in 2D. In the latter case, the new constants quantify to which degree stress and mirror symmetries are broken. Indeed, it turns out that in 2D, stress asymmetry enables isotropic yet chiral behaviors and the interplay between chirality and polarity is subsequently investigated.
To motivate the theory, it is shown that 2D isotropic polar solids naturally arise in the application of the transformation method when the background medium is isotropic and when the underlying spatial transformation is conformal (i.e., angle-preserving). Accordingly, isotropic polar solids, in conjunction with conformal transformations, can simplify the design of invisibility cloaks and other wave-steering devices by circumventing the need for anisotropic behaviors. As a demonstration, a conformal carpet cloak is designed out of graded hexachiral lattices and numerically tested. The cloaking performance is shown to be satisfactory over a range of pressure-shear coupled dynamic loadings.

We present a practical design of underwater acoustic carpet cloak with 2-dimensional
version of pentamode lattice. The quasi-conformal transformation (QCT), which is
achieved by inverse Laplace's equations with Neuman and Dirichelet boundaries, is
used to obtain the required parameters of the impedance matching carpet cloak. The
theoretical carpet cloak is pre-divided into 300 cells and then �lled by the correspond-
ing PM unit cells to achieve the latticed pentamode carpet cloak. The simulation
results indicate that the proposed carpet cloak has a good and broadband cloaking
effect. Moreover, the technique in this work can also be used to design arbitrary
shaped devices.

We inspect the propagation of shear polarized surface waves akin to Love waves through a forest of trees of same height atop a guiding layer on a soil substrate. We discover that the foliage of trees { brings a radical change in} the nature of the dispersion relation of these surface waves, which behave like spoof plasmons in the limit of a vanishing guiding layer, and like Love waves in the limit of trees with a vanishing height. When we consider a forest with trees of increasing or decreasing height, this hybrid "Spoof Love" wave is either reflected backwards or converted into a downward propagating bulk wave. An asymptotic analysis shows the forest behaves like an anisotropic wedge with effective boundary conditions.

We present a two-dimensional carpet cloak for static magnetic field, a design that renders the magnetic response of a given volume invisible from its exterior, without altering the external magnetic fields. The device is designed using transformation optics method and can be implemented with alternating superconducting and magnetic material layers. Through the proper design of the constitutive tensors and relative thicknesses of each slab, we achieve the perfect performance of invisibility. Full wave numerical simulations confirm our design.

Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.

Following a theoretical proposal [M. Farhat et al., Phys. Rev. Lett. 103, 024301 (2009)], we design, fabricate, and characterize a cloaking structure for elastic waves in 1 mm thin structured polymer plates. The cloak consists of 20 concentric rings of 16 different metamaterials, each being a tailored composite of polyvinyl chloride and polydimethylsiloxane. By using stroboscopic imaging with a camera from the direction normal to the plate, we record movies of the elastic waves for monochromatic plane-wave excitation. We observe good cloaking behavior for carrier frequencies in the range from 200 to 400 Hz (one octave), in good agreement with a complete continuum-mechanics numerical treatment. This system is thus ideally suited for demonstration experiments conveying the ideas of transformation optics.

The transformation theory of optics and acoustics is developed for the
equations of linear anisotropic elasticity. The transformed equations
correspond to non-unique material properties that can be varied for a given
transformation by selection of the matrix relating displacements in the two
descriptions. This gauge matrix can be chosen to make the transformed density
isotropic for any transformation although the stress in the transformed
material is not generally symmetric. Symmetric stress is obtained only if the
gauge matrix is identical to the transformation matrix, in agreement with
Milton et al. (2006). The elastic transformation theory is applied to the case
of cylindrical anisotropy. The equations of motion for the transformed material
with isotropic density are expressed in Stroh format, suitable for modeling
cylindrical elastic cloaking. It is shown that there is a preferred approximate
material with symmetric stress that could be a useful candidate for making
cylindrical elastic cloaking devices.

Control of waves with metamaterials is of great topical interest, and is fueled by rapid progress in broadband acoustic and electromagnetic cloaks. We propose a design for a cloak to control bending waves propagating in isotropic heterogeneous thin plates. This is achieved through homogenization of a multilayered concentric coating filled with piecewise constant isotropic elastic material. Significantly, our cloak displays no phase shift for both backward and forward scattering. To foster experimental efforts, we provide a simplified design of the cloak which is shown to work in a more than two-octave frequency range (30 Hz to 150 Hz) when it consists of 10 layers using only 6 different materials overall. This metamaterial should be easy to manufacture, with potential applications ranging from car industry to anti-earthquake passive systems for smart buildings, depending upon the plate dimensions and wavelengths.

We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.

An acoustic cloak envelopes an object so that sound incident from all directions passes through and around the cloak as though the object were not present. A theory of acoustic cloaking is developed using the transformation or change-of-variables method for mapping the cloaked region to a point with vanishing scattering strength. We show that the acoustical parameters in the cloak must be anisotropic: either the mass density or the mechanical stiffness or both. If the stiffness is isotropic, corresponding to a fluid with a single bulk modulus, then the inertial density must be infinite at the inner surface of the cloak. This requires an infinitely massive cloak. We show that perfect cloaking can be achieved with finite mass through the use of anisotropic stiffness. The generic class of anisotropic material required is known as a pentamode material. If the transformation deformation gradient is symmetric then the pentamode material parameters are explicit, otherwise its properties depend on a stress like tensor which satisfies a static equilibrium equation. For a given transformation mapping the material composition of the cloak is not uniquely defined, but the phase and wave speeds of the pseudo-acoustic waves in the cloak are unique. Examples are given from 2D and 3D.

On form invariance of the kirchhoff-love plate equation

- L Pomot
- S Bourgeois
- C Payan
- M Remillieux
- S Guenneau

L. Pomot, S. Bourgeois, C. Payan, M. Remillieux, S. Guenneau, On form
invariance of the kirchhoff-love plate equation, 2019, arXiv preprint arXiv:
1901.00067.

The Theory of Composites

- G W Milton

G.W. Milton, The Theory of Composites, 2002.