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Exceptional points (EPs) are degeneracies of two or more natural modes of open systems, which lead to unusual wave phenomena. Despite the robustness against imperfections of spatial EPs, they are less studied relative to temporal EPs, particularly in elastodynamics. However, elastic waves exhibit features not found in sound and light, which have proven useful for forming spatial EPs. Here, we harness these features to tune the coalescence of three eigenmodes in the Bloch spectrum of planar elastic laminates. We show that these third-order EPs give rise to axially frozen modes : anomalous transmitted waves with zero axial group velocity and finite transmittance. These modes, which were first reported in optics and required three-dimensional laminates, are achieved here in a planar setting thanks to elastodynamics tensorial structure, and expand the toolbox for elastic wave shaping. [ABSTRACT FROM AUTHOR]

We present a generalized Bloch wave framework for the dynamic analysis of structures with nonlocal interactions and apply it to the design of origami acoustic metamaterials. Specifically, we first discretize the origami structures using a customized structural bar-and-hinge model that minimizes the degrees of freedom in the associated unit cell, while being sufficiently accurate to capture the behavior of interest. Next, observing that this discretization results in nonlocal structural interactions—the stiffness matrix has nonzeros between nodes that are not nearest neighbors due to the coupled deformations arising during folding or bending—we generalize the standard Bloch wave approach used in structural analysis to enable the study of such systems. Utilizing this framework, choosing the geometry of the unit cell as well as the folded state of the structure as design variables, we design tunable and programmable Miura-ori and eggbox strips, sheets , and composites that are large band, low frequency acoustic switches. In doing so, we find that the number of bandgaps in the sheets is significantly smaller than their strip counterparts and also occur at relatively higher frequencies, a limitation which is overcome by considering composite structures that have individual panels made of different materials. Overall, we have found origami structures to be ideal candidates as acoustic metamaterials for noise control, vibration isolation, impact absorption, and wave guides. [ABSTRACT FROM AUTHOR]

This paper addresses the modeling of large scale modulations of high-frequency mechanical waves propagating in discrete and continuum periodic reticulated structures. By means of an asymptotic approach implemented using a multi-cells approach, we derive (i) the equations governing the large scale modulations, (ii) the specific features of the modulation propagation, and (iii) the domain of validity of the description. This approach provides specific information on the physics of large scale modulations that is not directly accessible from the Bloch wave decomposition nor from a multi-scattering approach. The theoretical formulation enables (i) the identification of the frequency range for which evolutions of the rapidly oscillating wave motions arise over long distances, and (ii) simple calculation of the high-frequency wave field based on a two-step procedure that separate the (multi-) cell and the large modulation scales. The asymptotic method is introduced by studying the academic case of a 1D spring-mass chain. This allows us to establish the correspondence between the modulation phenomena and the classical approaches of homogenization and Floquet–Bloch waves. High-frequency modulations in reticulated beams are then investigated. The dispersion equation for longitudinal and transversal vibrations are first established. Then, from the asymptotic analysis, the equations governing the structure behavior in the standard homogenization regime and in the modulation regimes are derived. These results are illustrated and supported by numerical simulations. General comments on modulation and potential application are developed in the conclusions. [ABSTRACT FROM AUTHOR]

We consider a vibrating triangular mass-truss lattice whose unit cell contains a resonator of triangular shape. The resonators are connected to the ambient lattice by trusses. Each resonator is tilted, i.e. it is rotated with respect to the ambient lattice’s unit cell through an angle ϑ 0 . This geometrical parameter is responsible for the emergence of a resonant mode in the Bloch spectrum for elastic waves and strongly affects the dispersive properties of the lattice. Additionally, the tilting angle ϑ 0 triggers the opening of a band gap at a Dirac-like point. We provide a physical interpretation of these phenomena and discuss the dynamical implications on elastic Bloch waves. The dispersive properties are used to design a structured interface containing tilted resonators which exhibit negative refraction and focussing, as in a “flat elastic lens”. [ABSTRACT FROM AUTHOR]

This work is concerned with the reflection and transmission of waves at a plane interface between a homogeneous elastic half-space and a half-space of elastic material that is periodically laminated. The lamination is always in the direction of the x 1 -coordinate axis and the displacement is always longitudinal shear, so that the only non-zero displacement component is u 3 ( x 1 , x 2 , t ) . After an initial discussion of Floquet–Bloch waves in the laminated material, brief consideration is given to the reflection–transmission problem, when the interface between the two media is the plane x 1 = 0 . Nothing unusual emerges: there are just a single reflected wave and a single transmitted wave, undergoing positive group-velocity refraction. Then, the problem is considered when the interface between the two media is the plane x 2 = 0 . The periodic structure of the interface induces an infinite set of reflected waves and an infinite set of transmitted waves. All need to be taken into account, but most decay exponentially away from the interface. It had previously been recognized that, if the incident wave had appropriate frequency and angle of incidence, a propagating transmitted wave would be generated that would undergo negative group-velocity refraction – behaviour usually associated with a metamaterial. It is established by an example in this work that there is, in addition, a propagating transmitted wave with smaller wavelength but larger group velocity that undergoes positive group-velocity refraction. The work concludes with a brief discussion of this finding, including its implications for the utility (or not) of “effective medium” theory. [ABSTRACT FROM AUTHOR]

The theory of elastodynamic homogenization initiated by J.R. Willis is revisited for periodically inhomogeneous media through a careful scrutiny of the main aspects of that theory in the 3D continuum context and by applying it to the thorough treatment of a simple 1D discrete periodic system. The Bloch theorem appears to be central to appropriately defining and interpreting effective fields. Based on some physical arguments, three necessary conditions are derived for the transition from the microscopic description to the macroscopic description of periodic media. The parameters involved in the Willis effective constitutive relation are expressed in terms of two localization tensors and specified with the help of the corresponding Green function in the spirit of micromechanics. These results are illustrated and discussed for the 1D discrete periodic system considered. In particular, inspired by Brillouin's study, the dependency of the effective constitutive parameters on the frequency is physically interpreted in terms of oscillation modes of the underlying microstructure. [ABSTRACT FROM AUTHOR]

Cauchy-elastic solids include hyper-elasticity and a subset of elastic materials for which the stress does not follow from a scalar strain potential. More in general, hypo-elastic materials are only defined incrementally and comprise Cauchy-elasticity. Infringement of the hyper-elastic 'dogma' is so far unattempted and normally believed to be impossible, as it apparently violates thermodynamics, because energy may be produced in closed strain cycles. Contrary to this belief, we show that non-hyper-elastic behavior is possible and we indicate the way to a practical realization of this new concept. In particular, a design paradigm is established for artificial materials where follower forces, so far ignored in homogenization schemes, are introduced as loads prestressing an elastic two-dimensional grid made up of linear elastic rods (reacting to elongation, flexure and shear). A rigorous application of Floquet–Bloch wave asymptotics yields an unsymmetric acoustic tensor governing the incremental dynamics of the effective material. The latter is therefore the incremental response of a hypo-elastic solid, which does not follow from a strain potential and thus does not belong to hyper-elasticity. Through the externally applied follower forces (which could be originated via interaction with a fluid, or a gas, or by application of Coulomb friction, or non-holonomic constraints), the artificial material may adsorb/release energy from/to the environment, and therefore produce energy in a closed strain loop, without violating any rule of thermodynamics. The solid is also shown to display flutter, a material instability corresponding to a Hopf bifurcation, which was advocated as possible in plastic solids, but never experimentally found and so far believed to be impossible in elasticity. The discovery of elastic materials capable of sucking up or delivering energy in closed strain cycles through interaction with the environment paves the way to realizations involving micro and nano technologies and finds definite applications in the field of energy harvesting. [ABSTRACT FROM AUTHOR]

Abstract: A method for the homogenization of a layered elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations without the need for a point-wise solution of the field equations. When the spatial variation of the field variables is restricted by Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress mode-shapes and, necessarily, the dispersion relations. The method can also give the point-wise solution of the elastodynamic field equations (to any desired degree of accuracy), which, however, is not required for the calculation of the average overall properties. The resulting overall dynamic constitutive relations are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micromechanical modeling of a representative unit cell of the composite. For waves in periodic layered composites, the overall effective mass-density and compliance (stiffness) are always real-valued whether or not the corresponding unit cell (representative volume element used as a unit cell) is geometrically and/or materially symmetric. The average strain and linear momentum are coupled and the coupling constitutive parameters are always each others'' complex conjugates. We separate the overall constitutive relations, which depend only on the composition and structure of the unit cell, from the overall field equations which hold for any elastic composite; i.e., we use only the local field equations and material properties to deduce the overall constitutive relations. Finally, we present solved numerical examples to further clarify the structure of the averaged constitutive relations and to bring out the correspondence of the current method with recently published results. [Copyright &y& Elsevier]

Abstract: We consider a net created from elastic strings as a model structure to investigate the propagation of waves through semi-discrete media. We are particularly interested in the development of continuum models, valid at high frequencies, when the wavelength and each cell of the net are of similar order. Net structures are chosen as these form a general two-dimensional example, encapsulating the essential physics involved in the two-dimensional excitation of a lattice structure whilst retaining the simplicity of dealing with elastic strings. Homogenization techniques are developed here for wavelengths commensurate with the cellular scale. Unlike previous theories, these techniques are not limited to low frequency or static regimes, and lead to effective continuum equations valid on a macroscale with the details of the cellular structure encapsulated only through integrated quantities. The asymptotic procedure is based upon a two-scale approach and the physical observation that there are frequencies that give standing waves, periodic with the period or double-period of the cell. A specific example of a net created by a lattice of elastic strings is constructed, the theory is general and not reliant upon the net being infinite, none the less the infinite net is a useful special case for which Bloch theory can be applied. This special case is explored in detail allowing for verification of the theory, and highlights the importance of degenerate cases; the specific example of a square net is treated in detail. An additional illustration of the versatility of the method is the response to point forcing which provides a stringent test of the homogenized equations; an exact Green''s function for the net is deduced and compared to the asymptotics. [Copyright &y& Elsevier]

We formulate and implement Helical Density Functional Theory (Helical DFT) — a self-consistent first principles simulation method for nanostructures with helical symmetries. Such materials are well represented in all of nanotechnology, chemistry and biology, and prominent examples include nanotubes, nanosprings, nanowires, miscellaneous chiral structures and important proteins. The overwhelming preponderance of such helical structures in all of science and engineering and the likelihood of these systems being associated with exotic materials properties, provides the motivation to develop systematic and predictive tools for their study. Following this line of thought, we develop a mathematical and computational framework in this contribution, that allows helical structures to be studied ab initio , using Kohn–Sham theory. We first show that the electronic states in helical structures can be characterized by means of special solutions to the single electron problem called helical Bloch waves. We rigorously demonstrate the existence and completeness of such solutions, and then describe how they can be used to reduce the Kohn–Sham Density Functional Theory (KS-DFT) equations for helical structures to a suitable fundamental domain. Next, we develop a symmetry-adapted finite-difference strategy in helical coordinates to discretize the governing equations, and obtain a working realization of our proposed approach. We verify the accuracy and convergence properties of our numerical implementation through examples. Finally, we employ Helical DFT to study the properties of zigzag and chiral single wall black phosphorus (i.e., phosphorene) nanotubes. Specifically, we use our simulations to evaluate the torsional stiffness of a zigzag nanotube ab initio. Additionally, we observe an insulator-to-metal-like transition in the electronic properties of this nanotube as it is subjected to twisting. We also find that a similar transition can be effected in chiral phosphorene nanotubes by means of axial strains. The strong dependence of the band gap of these materials on various modes of strain suggests their possible use as nanomaterials with tunable electronic and transport properties. Notably, self-consistent ab initio simulations of this nature are unprecedented and well outside the scope of any other systematic first principles method in existence. We end with a discussion on various future avenues and applications. [ABSTRACT FROM AUTHOR]

A lattice (or 'grillage') of elastic Rayleigh rods (possessing a distributed mass density, together with rotational inertia) organized in a parallelepiped geometry can be axially loaded up to an arbitrary amount without distortion and then be subject to incremental time-harmonic dynamic motion. At certain threshold levels of axial load, the grillage manifests instabilities and displays non-trivial axial and flexural incremental vibrations. Including every possible structural geometry and for an arbitrary amount of axial stretching, Floquet–Bloch wave asymptotics is used to homogenize the in-plane mechanical response, so to obtain an equivalent prestressed elastic solid subject to incremental time-harmonic vibration, which includes, as a particular case, the incremental quasi-static response. The equivalent elastic solid is obtained from its acoustic tensor, directly derived from homogenization and shown to be independent of the rods' rotational inertia. Loss of strong ellipticity in the equivalent continuum coincides with macro-bifurcation in the lattice, while micro-bifurcation remains undetected in the continuum and corresponds to a vibration of vanishing frequency of the lowest dispersion branch of the lattice, occurring at finite wavelength. Dynamic homogenization reveals the structure of the acoustic branches close to ellipticity loss and the analysis of forced vibrations (both in physical space and Fourier space) shows low-frequency wave localizations. A perturbative approach based on dynamic Green's function is applied to both the lattice and its equivalent continuum. This shows that only macro-instability corresponds to localization of incremental strain, while micro-instabilities occur in modes which spread throughout the whole lattice with an 'explosive' character. In particular, extremely localized mechanical responses are found both in the lattice and in the solid, with the advantage that the former can be easily realized, for instance via 3D printing. In this way, features such as shear band inclination, or the emergence of a single shear band, or competition between micro and macro instabilities become all designable features. The comparison between the mechanics of the lattice and its equivalent solid shows that the homogenization technique allows an almost perfect representation, except when micro-bifurcation is the first manifestation of instability. Therefore, the presented results pave the way for the design of architected cellular materials to be used in applications where extreme deformations are involved. [ABSTRACT FROM AUTHOR]

We consider a body , homogeneous or periodic, equipped with a structure composed of dynamic inhomogeneities uniformly distributed along a line, and study free and forced sinusoidal waves (Floquet - Bloch waves for the discrete system) in such a system. With no assumption concerning the wave nature, we show that if the structure reduces the phase velocity, the wave localizes exponentially at the structure line, and the latter can expand the transmission range in the region of long waves. Based on a general solution presented in terms of non-specified Green's functions, we consider the wave localization in some continuous elastic bodies and a regular lattice. We determine the localization-related frequency ranges and the localization degree in dependence on the frequency. While 2D-models are considered throughout the text, the axisymmetric localization phenomenon in the 3D-space is also mentioned. The dynamic field created in such a structured system by an external harmonic force is obtained consisting of three different parts: the localized wave, a diverging wave, and non-spreading oscillations. Expressions for the wave amplitudes and the energy fluxes in the waves are presented. [ABSTRACT FROM AUTHOR]

The momentum of deformable materials is coupled to their velocity. Here, we show that in piezoelectric composites which deform under electric fields, the momentum can also be coupled to the electric stimulus by a designed macroscopic property. To this end, we assemble these materials in a pattern with asymmetric microstructure, develop a theory to calculate the relations between the macroscopic fields, and propose a realizable system that exhibits this coupling. In addition to its fundamental importance, our design thus forms a metamaterial for mechanical wave control, as traversing waves are governed by the balance of momentum, and, in turn, the engineered electro-momentum coupling. While introduced for piezoelectric materials, our analysis immediately applies to piezomagnetic materials, owing to the mathematical equivalence between their governing equations, and we expect our framework to benefit other types of elastic media that respond to non-mechanical stimuli. [ABSTRACT FROM AUTHOR]