On the role of geometric phase in the dynamics of elastic waveguides.

The geometric phase provides important mathematical insights to understand the fundamental nature and evolution of the dynamic response in a wide spectrum of systems ranging from quantum to classical mechanics. While the concept of geometric phase, which is an additional phase factor occurring in dynamical systems, holds the same meaning across different fields of application, its use and interpretation can acquire important nuances specific to the system of interest. In recent years, the development of quantum topological materials and its extension to classical mechanical systems have renewed the interest in the concept of geometric phase. This review revisits the concept of geometric phase and discusses, by means of either established or original results, its critical role in the design and dynamic behaviour of elastic waveguides. Concepts of differential geometry and topology are put forward to provide a theoretical understanding of the geometric phase and its connection to the physical properties of the system. Then, the concept of geometric phase is applied to different types of elastic waveguides to explain how either topologically trivial or non-trivial behaviour can emerge based on the geometric features of the waveguide. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Nonlocal elastic metasurfaces: Enabling broadband wave control via intentional nonlocality.

While elastic metasurfaces offer a remarkable and very effective approach to the subwavelength control of stress waves, their use in practical applications is severely hindered by intrinsically narrow band performance. In applications to electromagnetic and photonic metamaterials, some success in extending the operating dynamic range was obtained by using nonlocality. However, while electronic properties in natural materials can show significant nonlocal effects, even at the macroscales, in mechanics, nonlocality is a higher-order effect that becomes appreciable only at the microscales. This study introduces the concept of intentional nonlocality as a fundamental mechanism to design passive elastic metasurfaces capable of an exceptionally broadband operating range. The nonlocal behavior is achieved by exploiting nonlocal forces, conceptually akin to long-range interactions in nonlocal material microstructures, between subsets of resonant unit cells forming the metasurface. These long-range forces are obtained via carefully crafted flexible elements, whose specific geometry and local dynamics are designed to create remarkably complex transfer functions between multiple units. The resulting nonlocal coupling forces enable achieving phase-gradient profiles that are functions of the wavenumber of the incident wave. The identification of relevant design parameters and the assessment of their impact on performance are explored via a combination of semianalytical and numerical models. The nonlocal metasurface concept is tested, both numerically and experimentally, by embedding a total-internal-reflection design in a thin-plate waveguide. Results confirm the feasibility of the intentionally nonlocal design concept and its ability to achieve a fully passive and broadband wave control.

Applications of Distributed-Order Fractional Operators: A Review.

Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

Variable-order particle dynamics: formulation and application to the simulation of edge dislocations.

This study presents the application of variable-order (VO) fractional operators to modelling the dynamics of edge dislocations under the effect of a static state of shear stress. More specifically, a particle dynamic approach is used to simulate the microscopic structure of a material where the constitutive atoms or molecules are modelled via discrete masses and their interaction via inter-particle forces. VO operators are introduced in the formulation in order to capture the complex linear-to-nonlinear dynamic transitions following the translation of dislocations as well as the creation and annihilation of bonds between particles. Remarkably, the motion of the dislocation does not require any a priori assumption in terms of either possible trajectory or sections of the model that could undergo the nonlinear transition associated with the creation and annihilation of bonds. The model only requires the definition of the initial location of the dislocations. Results will show that the VO formulation is fully evolutionary and capable of capturing both the sliding and the coalescence of edge dislocations by simply exploiting the instantaneous response of the system and the state of stress. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Anomalous diffusion of acoustic waves in 2D periodic media: Radiative transport and renormalization analysis.

This work investigates the occurrence of anomalous transport of acoustic waves propagating in two-dimensional (2D) perfectly periodic media and presents dedicated analysis tools to explore and understand the properties of the medium controlling the transitions between different transport regimes. By leveraging a two-fold approach that combines both radiative transport and renormalization theory, the propagation properties of the inhomogeneous medium can be characterized both near and at the transition from normal to anomalous diffusion. The proposed approach builds upon the classical radiative transfer theory of bulk materials, and it is specifically designed to study 2D systems. The ability to simulate and interpret the field quantities that describe such transport mechanisms can play a significant role in the development of wave-based imaging technology for highly inhomogeneous and scattering media.

Anomalous Refraction of Acoustic Guided Waves in Solids with Geometrically Tapered Metasurfaces.

The concept of a metasurface opens new exciting directions to engineer the refraction properties in both optical and acoustic media. Metasurfaces are typically designed by assembling arrays of subwavelength anisotropic scatterers able to mold incoming wave fronts in rather unconventional ways. The concept of a metasurface was pioneered in photonics and later extended to acoustics while its application to the propagation of elastic waves in solids is still relatively unexplored. We investigate the design of acoustic metasurfaces to control elastic guided waves in thin-walled structural elements. These engineered discontinuities enable the anomalous refraction of guided wave modes according to the generalized Snell's law. The metasurfaces are made out of locally resonant toruslike tapers enabling an accurate phase shift of the incoming wave, which ultimately affects the refraction properties. We show that anomalous refraction can be achieved on transmitted antisymmetric modes (A_{0}) either when using a symmetric (S_{0}) or antisymmetric (A_{0}) incident wave, the former clearly involving mode conversion. The same metasurface design also allows achieving structure embedded planar focal lenses and phase masks for nonparaxial propagation.

Numerical analysis of the vibroacoustic properties of plates with embedded grids of acoustic black holes.

The concept of an Acoustic Black Hole (ABH) has been developed and exploited as an approach for passively attenuating structural vibration. The basic principle of the ABH relies on proper tailoring of the structure geometrical properties in order to produce a gradual reduction of the flexural wave speed, theoretically approaching zero. For practical systems the idealized "zero" wave speed condition cannot be achieved so the structural areas of low wave speed are treated with surface damping layers to allow the ABH to approach the idealized dissipation level. In this work, an investigation was conducted to assess the effects that distributions of ABHs embedded in plate-like structures have on both vibration and structure radiated sound, focusing on characterizing and improving low frequency performance. Finite Element and Boundary Element models were used to assess the vibration response and radiated sound power performance of several plate configurations, comparing baseline uniform plates with embedded periodic ABH designs. The computed modal loss factors showed the importance of the ABH unit cell low order modes in the overall vibration reduction effectiveness of the embedded ABH plates at low frequencies where the free plate bending wavelengths are longer than the scale of the ABH.

A normalized wave number variation parameter for acoustic black hole design.

In recent years, the concept of the Acoustic Black Hole has been developed as an efficient passive, lightweight absorber of bending waves in plates and beams. Theory predicts greater absorption for a higher thickness taper power. However, a higher taper power also increases the violation of an underlying theory smoothness assumption. This paper explores the effects of high taper power on the reflection coefficient and spatial change in wave number and discusses the normalized wave number variation as a spatial design parameter for performance, assessment, and optimization.

Airframe structural damage detection: a non-linear structural surface intensity based technique.

The non-linear structural surface intensity (NSSI) based damage detection technique is extended to airframe applications. The selected test structure is an upper cabin airframe section from a UH-60 Blackhawk helicopter (Sikorsky Aircraft, Stratford, CT). Structural damage is simulated through an impact resonator device, designed to simulate the induced vibration effects typical of non-linear behaving damage. An experimental study is conducted to prove the applicability of NSSI on complex mechanical systems as well as to evaluate the minimum sensor and actuator requirements. The NSSI technique is shown to have high damage detection sensitivity, covering an extended substructure with a single sensing location.

Synthetic Kramers Pair in Phononic Elastic Plates and Helical Edge States on a Dislocation Interface.

In conventional theories, topological band properties are intrinsic characteristics of the bulk material and do not depend on the choice of the reference frame. In this scenario, the principle of bulk-edge correspondence can be used to predict the existence of edge states between topologically distinct materials. In this study, a 2D elastic phononic plate is proposed with a Kekulé-distorted honeycomb pattern engraved on it. It is found that the pseudospin and the pseudospin-dependent Chern numbers are not invariant properties, and the ℤ 2 number is no longer a sufficient indicator to examine the existence of the edge state. The distinctive pseudospin texture and the pseudomagnetic field are also revealed. Finally, the synthetic helical edge states are successfully devised and experimentally implemented on a dislocation interface connecting two subdomains with bulk pattern identical up to a relative translation. The edge state is also imaged via laser vibrometry.

Structural damage identification in plates via nonlinear structural intensity maps.

A nonlinear structural intensity concept is presented as an approach for the identification of defects displaying nonlinear vibration behavior. The nonlinear structural dynamic response exhibited by a riveted joint with loosened fasteners connecting a stiffener with a flat panel is investigated. The excitation, generating elastic waves with dominant bending components, triggers the nonlinear contact between the plate and the stiffener inducing a dynamic response rich with nonlinear harmonics. Experimental structural intensity maps are evaluated at the super-harmonic frequencies. This technique provides an experimental approach for the characterization and two dimensional visualization of nonlinear types of defects.

A generalized fractional-order elastodynamic theory for non-local attenuating media.

This study presents a generalized elastodynamic theory, based on fractional-order operators, capable of modelling the propagation of elastic waves in non-local attenuating solids and across complex non-local interfaces. Classical elastodynamics cannot capture hybrid field transport processes that are characterized by simultaneous propagation and diffusion. The proposed continuum mechanics formulation, which combines fractional operators in both time and space, offers unparalleled capabilities to predict the most diverse combinations of multiscale, non-local, dissipative and attenuating elastic energy transport mechanisms. Despite the many features of this theory and the broad range of applications, this work focuses on the behaviour and modelling capabilities of the space-fractional term and on its effect on the elastodynamics of solids. We also derive a generalized fractional-order version of Snell's Law of refraction and of the corresponding Fresnel's coefficients. This formulation allows predicting the behaviour of fully coupled elastic waves interacting with non-local interfaces. The theoretical results are validated via direct numerical simulations.

Applications of variable-order fractional operators: a review.

Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.

Stochastic scattering model of anomalous diffusion in arrays of steady vortices.

We investigate the occurrence of anomalous transport phenomena associated with tracer particles propagating through arrays of steady vortices. The mechanism responsible for the occurrence of anomalous transport is identified in the particle dynamic, which is characterized by long collision-less trajectories (Lévy flights) interrupted by chaotic interactions with vortices. The process is studied via stochastic molecular models that are able to capture the underlying non-local nature of the transport mechanism. These models, however, are not well suited for problems where computational efficiency is an enabling factor. We show that fractional-order continuum models provide an excellent alternative that is able to capture the non-local nature of anomalous transport processes in turbulent environments. The equivalence between stochastic molecular and fractional continuum models is demonstrated both theoretically and numerically. In particular, the onset and the temporal evolution of heavy-tailed diffused fields are shown to be accurately captured, from a macroscopic perspective, by a fractional diffusion equation. The resulting anomalous transport mechanism, for the selected ranges of density of the vortices, shows a superdiffusive nature.