Dispersion Relations Of Elastic Waves Research Articles

Propagation of Elastic Waves along a Cylindrical Rod of Micropolar Elastic Solid with Stretch under Impedance Boundary Conditions

Objective: The objective of this article is to derive the dispersion relations of elastic waves along a cylindrical rod of homogeneous, isotropic micropolar elastic solid with stretch under impedance boundary conditions. Method: The basic equations and constitutive relations are converted into cylindrical coordinates and solved by adopting the method of plane harmonic solution. Findings: Displacements, micro rotations, and micro stretch relations are derived analytically. The dispersion relations pertaining to elastic waves with using impedance boundary conditions also derived analytically and these relations are compared with the results of Tomar. Novelty: With the use of the MATLAB program, the effects of impedance parameters, micropolarity and stretch are discussed for a particular model. The speed of the elastic wave is proportional to the ratio of the impedance parameter in microstretch elastic solids, but there is no significant effect of the impedance ratio in micropolar elastic solids. Keywords: Elastic Waves, Cylindrical rod, Micropolarity, Micro stretch, Impedance Boundary Conditions

Graphene-based phononic crystal lenses: Machine learning-assisted analysis and design

The advance of artificial intelligence and graphene-based composites brings new vitality into the conventional design of acoustic lenses which suffers from high computation cost and difficulties in achieving precise desired refractive indices. This paper presents an efficient and accurate design methodology for graphene-based gradient-index phononic crystal (GGPC) lenses by combing theoretical formulations and machine learning methods. The GGPC lenses consist of two-dimensional phononic crystals possessing square unit cells with graphene-based composite inclusions. The plane wave expansion method is exploited to obtain the dispersion relations of elastic waves in the structures and then establish the data sets of the effective refractive indices in structures with different volume fractions of graphene fillers in composite materials and filling fractions of inclusions. Based on the database established by the theoretical formulation, genetic programming, a superior machine learning algorithm, is introduced to generate explicit mathematical expressions to predict the effective refractive indices under different structural information. The design of GGPC lenses is conducted with the assistance of the machine learning prediction model, and it will be illustrated by several typical design examples. The proposed design method offers high efficiency, accuracy as well as the ability to achieve inverse design of GGPC lenses, thus significantly facilitating the development of novel phononic crystal lenses and acoustic energy focusing.

Bandgaps in functionally graded phononic crystals containing graphene origami-enabled metamaterials

This paper investigates the dispersion characteristics of elastic waves propagating along the thickness direction in functionally graded laminated phononic crystals (FGLPCs) containing novel auxetic metamaterials enabled by graphene origami that is created with the aid of hydrogenation. Both graphene weight fraction and hydrogen coverage which are the key parameters governing the auxetic property are nonuniformly distributed in unit cells of FGLPCs whose material properties are determined by genetic programming-assisted micromechanical models. The dispersion relations of elastic waves in the structure are obtained based on the state space approach and the method of reverberation-ray matrix. A comprehensive parametric study is conducted to discuss the effects of graphene origami weight fraction and hydrogen coverage on bulk waves in elastic solids made of the metamaterial and elastic waves in FGLPCs. It is found that introducing auxetic metamaterials into FGLPCs can effectively manipulate elastic waves. The graded distribution of weight fraction in FGLPCs can lead to bandgaps for both transverse and longitudinal waves, while a through-thickness graded pattern in hydrogen coverage can trigger broad bandgaps for longitudinal waves only with transverse waves nearly unchanged.

The impact of the fiber orientation angle on the dispersion characteristic of guided waves for multilayered GLARE composites

The present paper is devoted to the parametric analysis of the impact of the fiber orientation angle on the dispersion relations of elastic waves in the case of GLARE composite material. The studied material consists of 13 layers. The layers are made of aluminum alloy AW-6060 and glass fiber/epoxy resin TVR380M12/R-glass. It is assumed that general configuration of the GLARE composite is as follows: [Al, +θ, −θ, +θ, −θ, +θ, Al, +θ, −θ, +θ, −θ, +θ, Al] and the angle θ is the parameter, which values are taken from the following range [0°, 90°]. In order to determine the dispersion curves the stiffness matrix method is applied. Moreover, the special computer application is created with the use of C++, where the multithreading mechanism is adopted to significantly reduce the time of calculations. As a result of the variations of the fiber orientation angle θ the change of the phase velocity of the fundamental symmetric and shear horizontal mode is mainly observed. The sensitivity of the other modes on the angle θ is not significant.

Dispersion relations of elastic waves in three-dimensional cubical piezoelectric phononic crystal with initial stresses and mechanically and dielectrically imperfect interfaces

• The theoretical model of a 3D cubical piezoelectric phononic crystal has been established. • Contrastive influences of non-dimensional structural and modal parameters have been considered. • Contrastive influences of initial stresses and imperfect interfaces have been considered. • Properties of partial and global dispersion relations have been discussed. A three-dimensional cubical piezoelectric phononic crystal is theoretically studied in this paper, formed by pre-stressed piezoelectric rectangular blocks and imperfect interfaces. Firstly, transfer matrices of pre-stressed piezoelectric rectangular blocks and imperfect interfaces are obtained based on the structural characteristics. The Bloch waves consist of three groups of sub-coupled elastic waves corresponding with three orthogonal periodical directions in the three-dimensional periodical structure. Furthermore, based on the transfer matrices of typical single cell and the Bloch theorem, it is established that the theoretical model of above-mentioned three-dimensional cubical piezoelectric phononic crystal to obtain the dispersion relations of Bloch waves. Finally, the influences of non-dimensional geometrical parameters (structural and modal parameters) and physical parameters (initial stress and mechanically and dielectrically imperfect interface parameters) on the dispersion relations are discussed based on the graphically numerical results. Numerical calculation results show that the existences of initial stresses and mechanically and dielectrically imperfect interfaces equivalently lead to the alterations of structural flexibilities or rigidities. The theoretical model and numerical discussions will provide a direct guidance of multi-material additive manufacturing for pre-stressed and inhomogeneous periodic structures with partial and global dispersion properties.

Dispersion relations of elastic waves in two-dimensional tessellated piezoelectric phononic crystals

A type of two-dimensional tessellated piezoelectric phononic crystal is theoretically studied in this paper, formed by homogeneous piezoelectric and inhomogeneous functionally graded rectangular columns. Firstly, the propagation properties of in-plane and anti-plane Bloch waves in each piezoelectric rectangular column are systematically investigated. Furthermore, the total transfer matrices of Bloch waves are obtained based on transfer matrices of homogeneous piezoelectric and inhomogeneous functionally graded rectangular columns. Finally, the Bloch theorem is used to obtain the dispersion relations of in-plane and anti-plane Bloch waves. The influences of non-dimensional geometrical parameters and gradient profile functions on the dispersion relations are discussed based on the graphically numerical results. Under a special value of modal parameter, the dispersion surfaces and curves of in-plane Bloch waves approximatively have plane and axis symmetries respectively, and an approximate total band-gap of anti-plane Bloch waves arise. The theoretical models and numerical discussions will provide a direct guidance of multi-material 3D printing for inhomogeneous periodic structures with dispersion and band-gaps properties.

Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with functionally graded interlayers

The effects of functionally graded interlayers on dispersion relations of elastic waves in a one-dimensional piezoelectric/piezomagnetic phononic crystal are studied in this paper. First, the state transfer equation of the functionally graded interlayer is derived from the motion equation by the reduction of order (from second order to first order). The transfer matrix of the functionally graded interlayer is obtained by solving the state transfer equation with the spatial-varying coefficient. Based on the transfer matrixes of the piezoelectric slab, the piezomagnetic slab and the functionally graded interlayers, the total transfer matrix of a single cell is obtained. Further, the Bloch theorem is used to obtain the resultant dispersion equations of in-plane and anti-plane Bloch waves. The dispersion equations are solved numerically and the numerical results are shown graphically. Five kinds of profiles of functionally graded interlayers between a piezoelectric slab and a piezomagnetic slab are considered. It is shown that the functionally graded interlayers have evident influences on the dispersion curves and the band gaps.

Dispersion relations of elastic waves in one-dimensional piezoelectric phononic crystal with initial stresses

The effects of initial stresses on dispersion relations of elastic waves in a one-dimensional piezoelectric phononic crystal are studied in this paper. It is taken into account that initial normal and shear stresses acted on piezoelectric slabs and their influences on constitutive equations, governing equations and boundary conditions under in-plane and anti-plane strain cases. Based on the transfer matrices of ingredient piezoelectric slabs, the total transfer matrix of a single cell is derived. Furthermore, the Bloch theorem is used to obtain the dispersive equation of in-plane and anti-plane Bloch waves. The dispersive equations are solved numerically and the numerical results are shown graphically. In the case of normal propagation of elastic waves within piezoelectric slabs, the analytical expressions of the dispersion equation are derived and compared with other literatures. The influences of initial stresses on the dispersive relations are discussed based on the numerical results. It is found that the initial normal stresses have more evident influences than that of initial shear stresses, no matter that in-plane Bloch waves or anti-plane Bloch wave are concerned. Moreover, the influences of initial stress are more evident on the dispersion curves at high frequency than that at the low frequency.

Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses

The dispersion relations of elastic waves in a one-dimensional phononic crystal formed by periodically repeating of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are studied in this paper. The influences of initial stress on the dispersive relation are considered based on the incremental stress theory. First, the incremental stress theory of elastic solid is extended to the magneto-electro-elasto solid. The governing equations, constitutive equations, and boundary conditions of the incremental stresses in a magneto-electro-elasto solid are derived with consideration of the existence of initial stresses. Then, the transfer matrices of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are formulated, respectively. The total transfer matrix of a single cell in the phononic crystal is obtained by the multiplication of two transfer matrixes related with two adjacent slabs. Furthermore, the Bloch theorem is used to obtain the dispersive equations of in-plane and anti-plane Bloch waves. The dispersive equations are solved numerically and the numerical results are shown graphically. The oblique propagation and the normal propagation situations are both considered. In the case of normal propagation of elastic waves, the analytical expressions of the dispersion equation are derived and compared with other literatures. The influences of initial stresses, including the normal initial stresses and shear initial stresses, on the dispersive relations are both discussed based on the numerical results.

Dispersion relations of elastic waves in one-dimensional piezoelectric phononic crystal with mechanically and dielectrically imperfect interfaces

The effects of mechanically and dielectrically imperfect interfaces on dispersion relations of elastic waves in a one-dimensional piezoelectric phononic crystal are studied in this paper. Six kinds of imperfect interfaces between two different piezoelectric materials constituting the phononic crystal are considered. These imperfect interfaces include: the mechanically compliant dielectrically weakly conducting interface, the mechanically compliant dielectrically highly conducting interface, the grounded metallized interface, the low dielectric interface, the tangent fixed interface and the tangent slippery interface. Based on transfer matrices of piezoelectric slabs and imperfect interfaces, the total transfer matrix of a typical single cell in the periodical structure is obtained. Furthermore, the Bloch theorem is used to obtain the dispersive equations of in-plane and anti-plane Bloch waves. The dispersive equations are solved numerically and the numerical results are shown graphically. In the case of normal propagation of elastic waves within piezoelectric slabs, the analytical expressions of the dispersion equations are derived and compared with other literatures. The influences of mechanically and dielectrically imperfect interfaces on the dispersive relations are discussed based on the numerical results.

Effects of surface piezoelectricity and nonlocal scale on wave propagation in piezoelectric nanoplates

Abstract In this paper, the dispersion characteristics of elastic waves propagating in a monolayer piezoelectric nanoplate is investigated with consideration of the surface piezoelectricity as well as the nonlocal small-scale effect. Nonlocal electroelasticity theory is used to derive the general governing equations by introducing an intrinsic length, and the surface effects exerting on the boundary conditions of the piezoelectric nanoplate are taken into account through incorporation of the surface piezoelectricity model and the generalized Young–Laplace equations. The dispersion relations of elastic waves based on the current formulation are obtained in an explicit closed form. Numerical results show that both the nonlocal scale parameter and surface piezoelectricity have significant influence on the size-dependent properties of dispersion behaviors. It is also found that there exists an escape frequency above which the waves may not propagate in the piezoelectric plate with nanoscale thickness.

Harmonic wave propagation in materials with periodic beam-structure

The characteristics of harmonic waves propagating in periodic beam structures are investigated. For this purpose, a very effective method for the calculation of dispersion relations of elastic waves in these materials is developed by applying dynamic theory of crystal lattices to discrete models of periodic beam structures. This method is applicable to general three-dimensional periodic beam structures. Results presented show that the solutions converge to the exact solution as the number of atoms in the discrete models increases. The dispersion relations of plane harmonic waves in the micropolar continuum model, developed in a previous study, are also calculated. They are compared with the exact solutions to examine the applicability of the continuum models to dynamic problems.

Dispersion Relation of Elastic Waves in Bars of Rectangular Cross Section

A formal solution for travelling waves in isotropic linear elastic bars of rectangular cross section which have traction-free lateral surfaces is presented. The solution is obtained by crosswise superposition of two single series, each term of which gives zero shearing stresses on the lateral surfaces. Frequency equations, which define the dispersion relations of elastic waves, are given by infinite determinants. Based on the approximate frequency equations, dispersion relations for a longitudinal mode, a torsional mode, and bending modes were calculated with the aid of a digital computer. Numerical results were compared with those given by other authers.