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General Introduction
Published in Didier Felbacq, Guy Bouchitté, Metamaterials Modeling and Design, 2017
Didier Felbacq, André Nicolet, Frédéric Zolla
In physics, dispersion most often refers to materials that bring about frequency-dependent effects in wave propagation. To a large extent, all materials are dispersive, except in vacuum. This frequency dependence gives rise to numerous physical phenomena such as light refraction through a prism, rainbows or the spreading of wave packets in optical fibers. As regards to dispersion relations, they specify the link between the angular frequency and some characteristics of materials trough the so-called constitutive relations. In classical electromagnetism, these constitutive relations are encoded in the so-called permittivity ɛ and permeability μ, and link the mesoscopic fields between them (see Chapter 3). For not too large energies, these relations are linear and may be expressed as a convolution operator The use of a Fourier transform in time converts this rather intricate operation into a simple operation of multiplication by a function of ω. It turns out that this function is complex valued and that its imaginary part is related to the phenomenon of leakage: Dispersion and leakage are thus the two sides of the same coin. Once the relative permittivity and permeability are obtained, the norm k of the vector k, which is the associated variable to the position vector x, may be expressed as a function of ω:k = k(ω) . Finally from this last relation, the different notions of velocities such as phase, group, and wave-packet velocities may be derived.
Love wave propagation characteristics in a fluid-saturated cracked double porous layered structure
Published in Mechanics of Advanced Materials and Structures, 2022
Bhanu Pratap Rajak, Santimoy Kundu, Shishir Gupta, Dharmendar Kumar
In this study, Love wave propagation in a heterogeneous fluid-saturated fractured double porous structure has been examined. The heterogeneity in the considered structure are linear, quadratic, and exponential in nature. The purpose of considering different types of heterogeneity is to analyze the impact of various parameters associated with the model on the phase and damped velocity of Love wave in different heterogeneous fluid saturated dual-porosity media. Methodology separation of variables has been used for converting the ordinary differential equation to a partial differential equation. Also, methodology WKB asymptotic approach has been used to calculate the displacement components throughout the structure. With the implementation of ideal boundary conditions, the dispersion equation has been achieved in its compact form, which contains real and imaginary parts of the relation. The real part of this relation is termed as phase velocity, and the imaginary part of this relation is termed as damped velocity. Basically, in physical sciences and engineering fields, the dispersion relation explains the effect of dispersion in an intermediate on the properties of waves traveling within that medium. A dispersion relative relates the wavelength or wavenumber of a wave to its frequency. From this relation, the phase velocity and group velocity of the wave can be calculated. It can be caused by either the boundary conditions or by interface of the waves within the transmitting medium. In addition, geometry-dependent and material-dependent dispersion relations, the overarching Kramers-Kronig relations describe the frequency dependence of wave propagation. So based on that, the phase and damped velocity of the Love wave has been calculated. Furthermore, a special case has been calculated in the absence of heterogeneity, cracked porous, which further reduced to classical Love wave condition [43], which validated the authenticity of the problem. Graphs have been plotted for phase and damped velocity against wave number to analyze the effect of heterogeneity, attenuation and fracture pores associate with layer and half-space for all the considered cases. This study results in decrement of phase and damped velocity with the higher regime of wave number for all the considered cases. Phase and damped velocity increases with increasing values of heterogeneity and attenuation parameters. It can also be noted that fracture pore associated with layer and half-space mitigates and enhances the phase and damped velocity, respectively. This study may be beneficial in the field of petroleum engineering, subsurface hydrology, geophysics, and seismology.
Analysis of shear wave in a FGPE/FGPM structure with imperfect magneto-electro elastic bounding interface
Published in Waves in Random and Complex Media, 2022
Bhanu Pratap Rajak, Santimoy Kundu, Shishir Gupta
In this paper, we have investigated the magneto-electro mechanical coupling, interfacial imperfection, and functionally gradient's effect on the propagation of shear wave in a composite structure consisting of an FGPE stratum and an FGPM semi-infinite substrate. In the section on basic equations, the equation of motion for the composite material has been developed. A detailed discussion of the current problem by using the three-dimensional coordinate system and the implementation of suitable boundary conditions has been drafted in the section ‘formulation of the problem and boundary condition’. The vacuum is assumed as a layer of air [15, 41]. The methodology WKB asymptotic approach has been adopted for finding the mechanical or electrical displacement and potential of the layer and semi-infinite media. With the help of boundary conditions, the solution has been achieved in the form of dispersion relation (both magneto-electrically open and short case). Basically, in physical sciences and engineering fields, the dispersion relation explains the effect of dispersion in an intermediate on the properties of waves traveling within that medium. A dispersion relative relates the wavelength or wavenumber of a wave to its frequency. From this relation, the phase velocity and group velocity of the wave can be calculated. It can be caused by either the boundary conditions or by interface of the waves within the transmitting medium. In addition, geometry-dependent and material-dependent dispersion relations, the overarching Kramers-Kronig relations describe the frequency dependence of wave propagation. So based on that, the phase velocity of the shear wave has been calculated. In the subsection validation of the problem, it has assumed that the FGPE layer and FGPM half-space are free from electric and magnetic effects and become isotropic. Also, it has been considered that the interface between the upper layer and semi-infinite media are perfectly bounded, and there is no gradient effect, then the obtained dispersion relation matches with the stranded Love wave condition [1], which validates the authenticity of the considered problem. Numerical data have been taken to plot the graph (dimensionless phase velocity vs dimensionless wavenumber) for elaborating the behavior of magneto-electro mechanical coupling parameter, mechanical imperfection parameter and functionally gradient parameters have been discussed in the section of numerical results and discussion. The novelty of this article is associated with the parametric study of the different parameters on the phase velocity of shear waves. This study may be helpful in the field of material sciences, geophysics, geology, ocean acoustic, and many other engineering fields.
Actively controllable size-dependent elastic wave band gaps in planar functionally graded micro-lattice
Published in Mechanics of Advanced Materials and Structures, 2022
Notably, tunable band structure is proved to be of great scientific and technological value and hence various approaches have been developed to manipulate the wave propagation and attenuation. On one hand, modifying material characteristics of the periodic lattice is regarded as an effective method to control dispersion relations. Functional graded (FG) materials, a new class of the composite material, has received considerable interests as the composition of which can vary smoothly and continuously in space, alleviating certain problems such as delamination and stress concentrations. The gradual material distribution can further enhance the structural and mechanical properties [17], which promotes the applications using FG materials in various fields, such as the studies focusing on the natural frequencies [18–20], nonlinear vibration characteristics [21, 22] and bulking analysis [23, 24], showing significant influence of the material distribution on adjusting the dynamic behavior. Simultaneously, the improved mechanical performance with FG elements suggests that employing FG structures definitely adds design freedoms to tune wave propagation of periodic structures. Using the SEM and TMM methods, the propagation of longitudinal waves in one-dimensional phononic crystal with FG materials was studied by Wu et al. [25], and the results demonstrate the effects of constituent gradient on adjusting the locations and widths of stop bands. Golub et al. studied time-harmonic elastic SH-waves propagating in periodically laminated composites with FG interlayers. They proposed two different models to deal with the FG interlayers and show the dependence of the power law and the exponent describing the material gradation [26]. Applying the IA mechanisms, the authors present a novel FG metamaterial beam which is able to isolate both longitudinal and transverse wave transmission and obtain multiple stop bands whose attenuation performance can be comparable to that of LR gaps [27]. Moreover, as the planar lattice has shown great potential in engineering applications, Sepehri et al. considered three typical topologies of architected FG structures with hexagonal, rectangular and triangular unit cells to investigate the effects of material distribution on the wave filtering performance by FEM simulations [28]. Motivated by this work, Jafari et al. added FG beam as auxiliary resonators to conventional main structures to constitute the hybrid planar lattice where the band diagram and directionality of wave propagation were analyzed [29]. However, to the best knowledge of authors, the investigations on band gap properties of FG planar lattice are restricted to computing real band diagrams, lacking of attenuation diagrams in two dimensional (2 D) space and exhibiting simplex tuning strategy as only depending on the material gradation. Therefore, the present work aims to develop mathematical models for estimating wave propagation and attenuation in plane and improve the tunability level to explore creative designs of planar FG lattice.