Propagation Of Longitudinal Elastic Waves Research Articles

TRANSFORMATION OF A LONGITUDINAL ELASTIC WAVE INTO A POCHHAMMER WAVE

The finite element method in a dynamic formulation was used to characterize the propagation of longitudinal elastic waves in cylindrical plates and Pochhammer waves in rods. It was shown that in compact bodies first only one longitudinal wave propagates until unloading waves from the side surfaces begin to interact, and then two types of longitudinal elastic waves propagate simultaneously. In rods, a longitudinal elastic wave propagates at the initial stage. Over time, the influence of unloading from the side surfaces of the rod leads to the simultaneous propagation of two types of elastic waves: a longitudinal elastic wave and a Pochhammer wave. Using aluminum as an example, the range of geometric parameters was determined for nearly compact bodies, whose elastic properties cannot be evaluated by ultrasonic methods.

A Mechanical Model for Elastic Wave Propagation in Nacre-Like Materials With Brick-and-Mortar Microstructures

Abstract Many biological materials have outstanding properties unparalleled by artificial materials because of their periodic and/or hierarchical structures. In this paper, the longitudinal elastic wave propagation in nacre-like materials with brick-and-mortar microstructures has been studied theoretically by taking the different deformation mechanisms into account. A viscoelastic model in relation to the sacrificial bond behavior for the organic matrix has been developed. The shear-lag model is adopted to simulate the deformation of the brick-mortar structure. Complex band structures have been analyzed and the mechanical model has been validated by finite element method. Results have shown that bandgaps with lower edge close to vanishing frequency may arise in such bio-inspired periodic structures and attenuated propagation of elastic waves may be dominant in the range of higher frequencies. It may cast light on the dynamic strengthening mechanisms in nacre-like materials and provide a scientific basis for the design of bio-inspired metamaterials with low-frequency bandgaps.

Elastic wave propagation in metamaterial rods with periodic shunted piezo-patches

The wave propagation attenuation in low frequencies by using piezoelectric elastic metamaterials has been developed in recent years. These piezoelectric structures exhibit abnormal properties, different from those found in nature, through the artificial design of the topology or exploring the shunt circuit parameters. In this study, the wave propagation in a 1-D elastic metamaterial rod with periodic arrays of shunted piezo-patches is investigated. This piezoelectric metamaterial rod is capable of filtering the propagation of longitudinal elastic waves over a specified range of frequency, called band gaps. The complex dispersion diagrams are obtained by the extended plane wave expansion (EPWE) and wave finite element (WFE) approaches. The comparison between these methods shows good agreement. The Bragg-type and locally resonant band gaps are opened up. The shunt circuits influence significantly the propagating and the evanescent modes. The results can be used for elastic wave attenuation using piezoelectric periodic structures.

Stationary and Self-Similar Waves in a Rod with Bimodular Nonlinearity, Dissipation, and Dispersion

Propagation of longitudinal elastic waves in a rod with bimodular nonlinearity, linear dissipation, and geometrical dispersion of phase velocity is analyzed. Analytical and numerical solutions are obtained for stationary and self-similar waveforms and pulsed perturbations that propagate in the rod without shape changes, and a corresponding graphical analysis is presented.

Стационарные и самоподобные волны в стержне с разномодульной нелинейностью, диссипацией и дисперсией

STATIONARY AND SELF-SIMILAR WAVES IN A ROD WITH BIMODULAR NONLINEARITY, DISSIPATION AND DISPERSION V.E.Nazarov, S.B.Kiyasko The propagation of longitudinal elastic waves in a rod with bimodular nonlinearity, linear dissipation and geometric dispersion of the phase velocity is investigated. Analytical and numerical solutions for the profiles of stationary waves and self-similar periodic waves and impulsive perturbations propagating in such rod without changing the shape are obtained; their graphical analysis was carried out.

Longitudinal wave propagation in one-dimensional waveguides with sinusoidally varying depth

In this paper, longitudinal elastic wave propagation in one-dimensional waveguides with sinusoidally varying depth is investigated. Furthermore, different types of such waveguide designs are explored to understand their capability to attenuate the group speeds as well as the velocity amplitudes. The plane waveguide configurations with sinusoidally varying depth, of three types, namely Convex, Concave and Full, along with their combinations and variants, whose segmental depth along the length of the waveguide is modeled by a sine function having an amplitude parameter α and the half-period p, which leads to a governing differential equation with variable coefficients, are considered in this study. To study the longitudinal wave propagation in these inhomogenous waveguides, a novel superconvergent finite element formulation is developed, which gives an exact stiffness matrix. In addition to the wave propagation analysis, static and free vibration behaviors in these waveguides are also studied. The implemented superconvergent finite element formulation for these studies is validated with the commercial finite element software Abaqus. In the first part of the wave propagation analysis, abilities of the three plane waveguide configurations with sinusoidally varying depth to attenuate high amplitude and frequency waves are investigated for different values of α. Next, four different waveguides composed of Convex-Concave combinations are studied with an aim to get better attenuation. Following this, the waveguides' segmental parameters, α and p, are varied across the segments along the length of the waveguide, using a sine function and polynomial power law to see if such graded variations give better energy absorption properties compared to the plane waveguides whose parameters are unvarying. In the last part, a developed inhomogenous rod of a certain length is inserted in the middle of a uniform waveguide to study the possibility of changing the longitudinal wave propagation characteristics of the uniform waveguide. The results from the analyses show that the group speed and amplitude of the longitudinal wave in these configurations change significantly with space as well as with frequencies, especially for high values of α. The Concave and Full waveguides delay the propagation of longitudinal waves significantly, which translates into the reflected waves appearing later in the chosen large time window. The Convex waveguide and its variants reduce the wave amplitudes significantly. Building on these results, some waveguides with notable attenuation characteristics are proposed.

Realizing passive direction-bias for mechanical wave propagation using a nonlinear metamaterial

The possibility of realizing amplitude-activated passive direction-bias in longitudinal elastic wave propagation using a nonlinear acoustic metamaterial is demonstrated. Applying the method of multiple scales, approximate analytical solutions are derived for the dispersion curve and bandgap shifts due to the presence of nonlinear hardening local oscillators within the metamaterial using its lumped parameter effective-mass model. The configuration for a structural waveguide consisting of a tuned combination of a nonlinear and a linear acoustic metamaterial that takes advantage of these shifts to produce amplitude-activated direction-bias in propagation was postulated and verified using discrete element simulations. A numerical routine to generate root profile geometries that enable contact-based hardening response in tip-loaded cantilever beam resonators was developed and implemented. Utilizing a hybrid fabrication process involving additively manufactured and machined components, a prototype test article was constructed. Experiments using a structural waveguide test rig verify the existence and extent of bandgaps and provide evidence for the passive direction-bias phenomenon. Follow-on investigations on configurations with different types of nonlinearities could pave the way for development of additional passive acoustic wave manipulation devices with enriched dynamics.

One-dimensional propagation of longitudinal elastic waves through functionally graded materials

The one-dimensional propagation of longitudinal elastic waves along the thickness of a plate made of functionally graded materials excited by a harmonic force is reported in this article. The material properties of the plate are assumed to be graded along the thickness direction according to a trigonometric law distribution. This distribution smoothly connects the material properties of the upper and lower homogeneous materials that bounds the plate. The corresponding propagation equation is Ince-type equation that can be transformed to Heun’s equation a local exact solution of which is expressed in terms of local Heun functions. The general nature of these functions is demonstrated based on four degenerate cases of Heun’s equation. The transfer matrix method is used to study the elastic waves propagating in the inhomogeneous domain. The calculation of the transfer matrices requires the evaluation of the general solution in the interval containing two regular singular points. For this purpose, the modified Heun function is introduced. Based on the transfer matrices, the influence of both the asymmetry of the unit cell and various constituent materials on the transmission coefficient spectrum is studied. The transmission coefficient is also calculated for the locally periodic structures with the help of the Chebyshev polynomials.

Stationary waves in a bimodular rod of finite radius

Theoretical study of the propagation of longitudinal elastic waves in a bimodular rod of finite radius with dispersion is carried out. The exact analytical solutions are derived for the stationary waves propagating in such a rod without changing in their shape.

Longitudinal elastic wave propagation characteristics of inertant acoustic metamaterials

Longitudinal elastic wave propagation characteristics of acoustic metamaterials with various inerter configurations are investigated using their representative one-dimensional discrete element lattice models. Inerters are dynamic mass-amplifying mechanical elements that are activated by a difference in acceleration across them. They have a small device mass but can provide a relatively large dynamic mass presence depending on accelerations in systems that employ them. The effect of introducing inerters both in local attachments and in the lattice was examined vis-à-vis the propagation characteristics of locally resonant acoustic metamaterials. A simple effective model based on mass, stiffness, or their combined equivalent was used to establish dispersion behavior and quantify attenuation within bandgaps. Depending on inerter configurations in local attachments or in the lattice, both up-shift and down-shift in the bandgap frequency range and their extent are shown to be possible while retaining static mass addition to the host structure to a minimum. Further, frequency-dependent negative and even extreme effective-stiffness regimes are encountered. The feasibility of employing tuned combinations of such mass-delimited inertant configurations to engineer acoustic metamaterials that act as high-pass filters without the use of grounded elements or even as complete longitudinal wave inhibitors is shown. Potential device implications and strategies for practical applications are also discussed.

Control of elastic wave propagation in one-dimensional piezomagnetic phononic crystals.

Two ways of controlling the acoustic waves propagation by external inductance or capacitance in a one-dimensional (1-D) piezomagnetic phononic crystal are investigated. The structure is made of identical bars, constituted of a piezomagnetic material, surrounded by a coil and connected to an external impedance. A model of propagation of longitudinal elastic waves through the periodic structure is developed and the dispersion equation is obtained. Reflection and transmission coefficients are derived from a 2 × 2 transfer matrix formalism that also allows for the calculation of elastic effective parameters (density, Young modulus, speed of sound, impedance). The effect of shunting impedances is numerically investigated. The results reveal that a connected external inductance tunes the Bragg band gaps of the 1-D phononic crystal. When the elements are connected via a capacitance, a hybridization gap, due to the resonance of the LC circuit made of the piezomagnetic element and the capacitance, coexists with the Bragg band gap. The value of the external capacitance modifies the boundaries of both gaps. Calculation of the effective characteristics of the phononic crystal leads to an analysis of the physical mechanisms involved in the wave propagation. When periodically connected to external capacitances, a homogeneous piezomagnetic stack behaves as a dispersive tunable metamaterial.

Intrinsic transfer matrix method and split quaternion formalism for multilayer media

The intrinsic transfer matrix method for 1D longitudinal elastic wave propagation through multilayer media is used to obtain an equivalent split quaternion formalism. Periodic media are analysed in this framework and the presence of a defect is considered. A simple one layer defect and an inversion defect are analysed. A commutative type of split quaternion is identified, which corresponds to defects of the periodic structure that can be placed in any position, the overall acoustic properties of the medium being conserved. Also, a nonperiodic medium composed of such commutative elements has a behaviour independent of the order of elements. Several possible applications in sound wave measurement and processing are outlined. The proposed split quaternion formalism is compact and can make analytical and numerical computations easier.

Quaternion Formalism for the Intrinsic Transfer Matrix

A quaternion formulation is applied to the intrinsic transfer matrix for longitudinal elastic wave propagation through a multilayer medium in order to find the spectral response of a sonic crystal. Resonance conditions and the band structure of the crystal are obtained. The presence of a defect is also analysed. The analysis is carried out theoretically and through simulations. A coupled oscillators model is used to validate the obtained results from a phenomenological point of view. Experimental measurements are carried out for some periodic multilayer arrangements and they are correlated with theory. The obtained spectral response and band structure are essential in characterising the sonic crystal and also in optimising its structure in order to obtain specific passbands and stopbands. The adaptedness of the quaternion formulation to periodic structures and to the inclusion of defects is considered.

Numerical analysis of elastic wave propagation in unbounded structures

The main objective of this paper is to show the effectiveness and usefulness of the concept of an absorbing layer with increasing damping (ALID) in numerical investigations of elastic wave propagation in unbounded engineering structures. This has been achieved by the authors by a careful investigation of three different types of structures characterised by gradually increasing geometrical and mathematical description complexities. The analysis includes propagation of longitudinal elastic waves in a 1-D semi-infinite isotropic rod, modelled according to the classical 1-mode theory of rods, propagation of coupled shear and flexural elastic waves in a 1-D semi-infinite isotropic beam modelled according to the Timoshenko beam theory, as well as propagation of elastic waves in a 3-D semi-infinite isotropic half-pipe shell modelled by a 6-mode theory of shells. The concept of the ALID has been not only presented by the authors, but certain relations between the ALID properties and the characteristics of propagating elastic waves have been given that can help to maximise the ALID performance in terms of its damping capability. All results of numerical calculations presented by the authors in this work have been obtained by the use of the Time-domain Spectral Finite Element Method (TD-SFEM).

Dynamic behaviour of a metamaterial system with negative mass and modulus

Different models of metamaterials have been developed to generate negative mass and/or negative modulus. The resulting mass and modulus in existing works, however, cannot be independently controlled. The current study presents a new representative cell of elastic metamaterials in an effort to provide a comprehensive model for generating negative mass and/or negative modulus. The current model consists of a series of properly arranged rigid bodies and linear springs. By introducing both translational and rotational motions in the representative cell, negative mass and negative modulus can be obtained in a controlled manner. The mechanisms and conditions under which negative mass and/or negative modulus can be achieved are studied in detail. Numerical examples indicate that by varying the design of the representative cell, different properties of the material system can be reliably generated, i.e., double negative (mass and modulus) or single negative (mass or modulus). The dynamic behaviour of the developed material system under different loading frequencies is evaluated and the longitudinal elastic wave propagation in such metamaterials is studied.

Investigation of Longitudinal Elastic Wave Propagation Through Intersecting Welded bars

The main purpose of this paper is to express a general formulation for the theoretical analysis of longitudinal elastic wave propagation through a general and in particular the T and十 junctions. The bar joint is modeled as a rigid block. The bars are assumed to be all of the same thickness but may differ in width and may be of different materials. Elementary theory is applied to investigate the propagation of longitudinal wave. The present study indicates that for a T junction there would be no transmission of the longitudinal elastic wave. For a 十junction there would be no transmission of the longitudinal elastic wave into the perpendicular branches but the transmission into the horizontal bar was happened.

Acoustical and optical branches of wave propagation in random particulate composites

The work is dedicated to the analysis of acoustical and optical branches of longitudinal elastic wave propagation in the medium with a random set of spherical inclusions. The effective field method and quasicrystalline approximation are used for the construction of the dispersion equation for the wave number of the mean (coherent) wave field propagating in the composite. This dispersion equation serves for all frequencies of the incident field, properties and volume concentrations of inclusions. Different branches of the solutions of this equation are obtained and analyzed. Each of these branches may be interpreted as a specific mode of wave propagation, and its input in the mean (coherent) wave field is essential only in a certain frequency region. The predictions of the method are compared with some experimental data existing in the literature.

Propagation of longitudinal elastic waves in composites with a random set of spherical inclusions (effective field approach)

The work is devoted to the problem of plane monochromatic longitudinal wave propagation through a homogeneous elastic medium with a random set of spherical inclusions. The effective field method and quasicrystalline approximation are used for the calculation of the phase velocity and attenuation factor of the mean (coherent) wave field in the composite. The hypotheses of the method reduce the diffraction problem for many inclusions to a diffraction problem for one inclusion and, finally, allow for the derivation of the dispersion equation for the wave vector of the mean wave field in the composite. This dispersion equation serves for all frequencies of the incident field, properties and volume concentrations of inclusions. The long and short wave asymptotics of the solution of the dispersion equation are found in closed analytical forms. Numerical solutions of this equation are constructed in a wide region of frequencies of the incident field that covers long, middle, and short wave regions of propagating waves. The phase velocities and attenuation factors of the mean wave field are calculated for various elastic properties, density, and volume concentrations of the inclusions. Comparisons of the predictions of the method with some experimental data are presented; possible errors of the method are indicated and discussed.

Propagation of longitudinal elastic waves in a fluid-saturated porous medium with spherical inclusions

The Frenkel-Biot theory is used to study the propagation of a longitudinal harmonic wave of the first kind in an isotropic porous matrix with inclusions contrasting in elastic properties and hydrodynamic permeability. The generation of elastic waves of the second kind at the boundaries of inclusions is taken into account. The effective wave number of the longitudinal wave is calculated using the equations of multiple scattering theory. The characteristic size of inhomogeneities is assumed to be much greater than the size of pores. The parameters of the model used for calculations correspond to sandstone with centimeter-scale inhomogeneities. The presence of such inhomogeneities is typical of sedimentary rocks. Calculations show that, in the frequency range of acoustic logging, the effective attenuation factor of the longitudinal wave may noticeably exceed the attenuation factors of longitudinal waves of the first kind in both matrix and inclusions. From the results obtained, it follows that, when studying the propagation of elastic waves in fluid-saturated porous media, it is necessary to take into account the hydrodynamic effects associated with the filtration overflows that arise at the boundaries of inhomogeneities.

Quasi-One-Dimensional Periodic Structure with Locally Resonant Band Gap

The propagation of longitudinal elastic waves in quasi one-dimensional structure consisting of harmonic oscillators periodically jointed on a slender beam is studied. Sub-frequency locally resonant band gap with highly asymmetric attenuation is observed in both theoretical and experimental results, and both results match well. The stiffness and mass ratios are found analytically as two factors that influence the actual attenuation in the band gap of the locally resonant phononic crystals. The study on the weights of the two factors shows that the stiffness ratio is the key one. Thus, the reason for the mismatch between the regions of the sharp attenuation and the theoretical band gap in the locally resonant phononic crystals is discovered.