Flexural Wave Dispersion Research Articles

Mechanical metastructure with embedded phononic crystal for flexural wave attenuation

Destructive interference-based metamaterials have shown excellent characteristics in elastic wave manipulation and vibration attenuation. Nevertheless, challenges persist in the application due to limited space and lightweight design, as current metastructures require additional beam structure. To simplify the design of metamaterials for flexural wave manipulation, this paper presents a new class of embedded phononic crystal for manipulating flexural wave propagation in both one and two-dimensional space by taking advantage of destructive interference, which can effectively suppress the mechanical vibration of a beam structure with a broad band gap. The flexural wave dispersion characteristic in a non-uniform beam structure is derived based on the Euler–Bernoulli beam theory, and an embedded phononic structure with the mechanism of destructive interference is presented to demonstrate its effectiveness in mitigating mechanical vibration. Subsequently, four typical units of embedded phononic structures are designed for attenuating flexural wave propagation in a beam structure. Finally, both numerical simulations, including one and two-dimensional phononic crystals, and physical experiments are implemented to evaluate the performance of the presented metastructure for flexural wave manipulation, which indicates that the proposed embedded phononic structures can effectively mitigate structural vibration in the low-frequency domain. To the best of our knowledge, it is the first attempt to design the metabeam with embedded phononic structures by taking advantage of destructive interference.

On Flexural Wave Dispersion of a Higher-Order Metamaterial Sandwich Composite Plate Based on a Visco-Pasternak Foundation

This paper elaborates on the wave propagation characteristics of a smart sandwich plate consisting of smart piezo composite layers on a plate’s upper and lower borders via an auxetic core. This study’s equation of motion was derived using a refined version of the Shear-deformation theory at a higher order, or HSDT. Additionally, detecting the properties of the auxetic core and piezoelectric composite layers employs a micromechanical model. The fundamental equations of the smart sandwich plates were formulated utilizing the Hamiltonian principle and Maxwell’s law. Then, the governing equations of a sandwich plate are solved using an exponential form and an analytical method. Furthermore, the phase velocity of the intelligent sandwich plate is provided based on the layer thicknesses of the auxetic core and thicknesses of intelligent piezoelectric composite layers. In addition, each figure calculates and presents the weight of the Winkler–Pasternak coefficient, the viscoelastic substances factor, the source of the outside electricity, and the volume percent of reinforcement in the matrix on the phase velocity.

Realization of low-frequency stop band in Timoshenko beam with periodic IDVAs

The most prominent dynamic characteristic of metamaterials is that they possess the frequency pass bands and band gaps. In the bandgap, the elastic waves will attenuate significantly, and this property is potentially used to control low-frequency vibrations in civil engineering. To open the low-frequency bandgap of the Timoshenko beam, inerter-based dynamic vibration absorbers (IDVAs) are arranged periodically in the beam. An analytical methodology based on transfer matrix method was developed to predict the flexural wave dispersion characteristics, and was validated by test and numerical experiments. The band gaps can be simultaneously tuned by the parameters of the IDVAs, including spring stiffness, inertance and attached mass. Moreover, the band gaps generated by single IDVAs are independent of each other, and a super-wide bandgap can be achieved by splicing the band gaps for metamaterial beams with graded IDVAs. Explicit formulas with respect to the parameters of IDVAs have been developed to determine the condition of bandgap merging, which can broaden the bandwidth effectively. Finally, a computer-based program was presented to determine the reasonable design parameters.

Symmetrically partitioned isotropic model for quantitative equivalent simulation of circumferential anisotropy

Abstract Simulation wells with known parameters are vital in the study of anisotropic methods and logging equipment development. We propose a symmetrically partitioned isotropic (SPI) model, which can quantitatively control the parameters of equivalent circumferential anisotropy. First, the circumferential anisotropy of the flexural wave dispersion in the SPI model is analysed using a perturbation method, and the azimuth characteristics of the flexural wave and the wave velocity in the model are evaluated using a numerical simulation algorithm. Next, the circumferential anisotropy characteristics of the model are verified. Finally, the linear relationship between the fast and slow shear-wave velocities of the model and the model parameters is obtained through multiple linear regression analyses, and then, the equivalent circumferential anisotropy is predicted. The feasibility of the quantitative equivalent simulation of circumferential anisotropy media using the SPI model is verified.

Flexural wave dispersion characteristics of imperfect Ti-6Al-4V foam circular cylindrical shells in a thermal environment

The present paper is mainly focused on analyzing the flexural wave dispersion of imperfect Ti-6Al-4V foam circular cylindrical shells in a thermal environment. The pores were supposed to be distributed across the thickness (z-direction) in the form of three different patterns as follows: symmetric porosity distribution (SPD), asymmetric porosity distribution (ASPD) and uniform porosity distribution (UPD). Besides, various kinds of temperature variations, including sinusoidal, linear and uniform temperature variations, were studied. The strain-displacement relationship of the shell was derived based on the first-order shear deformable theory (FSDT) of shells. Hamilton’s principle was also applied to obtain the governing equations of metal foam shells which were then solved using an analytical method. Finally, influences of different parameters including circumferential wave number, different kinds of temperature variation, temperature change, radius to thickness ratio , types of porosity distribution across the thickness, porosity coefficient and mode number on the variations of phase velocity and wave frequency were investigated and the results were illustrated and discussed.

On the wave dispersion in functionally graded porous Timoshenko-Ehrenfest nanobeams based on the higher-order nonlocal gradient elasticity

Dispersion characteristics of flexural waves in a functionally graded (FG) porous nanobeam are analytically examined. Kinematics of the nano-sized beam is assumed to be consistent with the Timoshenko-Ehrenfest beam model. Material behavior of the FG porous beam is considered to vary symmetrically along the beam thickness while appropriate symmetric porosity models are applied to account for two diverse distributions of porosity with variable volume of voids. For the first time, size-dependent response of the symmetric FG porous nanobeam is realized within the framework of the higher-order nonlocal gradient elasticity theory. The integro-differential constitutive laws of the stress resultant fields are established and reinstated with the equivalent differential relations equipped with non-standard boundary conditions. The closed-form solution of the phase velocity of flexural waves is analytically determined. The ensuing numerical results of the flexural wave dispersion detect new benchmarks for numerical analysis and can be effectively exploited in design and optimization of composite nano-structural elements of advanced nano-electro-mechanical systems.

Advanced monopole and dipole sonic log data processing — Part 1: Real time

Real-time open-hole wireline sonic logging data processing becomes a nontrivial task to accurately, automatically, and efficiently evaluate the compressional and shear slowness of a borehole rock formation when human interaction is not possible and the signal processing time is limited to the elapsed time between different transmitter excitations. To address real-time sonic data processing challenges, we have developed self-adaptive, data-driven methods to accurately measure the formation P- and S-wave slowness from monopole and dipole waveforms in all types of formations. These new real-time processing techniques take advantage of the fact that advanced wireline sonic logging tools have wide frequency responses and little to no detectable tool body arrivals. These technology improvements provide an opportunity to implement a first-motion-detection technique that detects the onset of P-waves in the monopole array waveforms. The knowledge of compressional arrival time and corresponding slowness are then used to project an appropriate slowness-time window to identify the monopole refracted S-wave and its slowness based on the range of possible compressional to shear velocity ratio ([Formula: see text]/[Formula: see text]) for earth rock formation. To process the borehole dipole flexural waves, we develop a new, data-driven frequency-domain method that enables the evaluation of the full flexural-wave dispersion response and its corresponding low-frequency shear slowness asymptote. Field data processing results indicate that our methods provide high-quality compressional slowness (delta-T compressional) and shear slowness measurements that are not affected by other borehole modes or dispersion complications in most formation types.

Band Gaps Characteristics Analysis of Periodic Oscillator Coupled Damping Beam.

The vibration of the periodic oscillator coupled damping beam model is reduced through the band gaps designing method, which can be applied in equivalent engineering structures. In this paper, the flexural wave dispersion relations of the infinite long periodic oscillator coupled damping beam were calculated using the reverberation-ray matrix method combined with the Bloch theorem. The flexural wave vibration frequency response function of the finite long periodic oscillator coupled damping beam was carried out using the finite element method. The flexural wave vibration band gaps occur in the infinite long periodic oscillator coupled damping beam model in both the analytical and numerical results. In these band gaps, flexural waves’ propagation is prohibited, and flexural vibration is significantly suppressed. Furthermore, the effects of structure and material parameters on the flexural wave vibration band gaps characteristics are studied. Thus, the structural vibration reduction design can be realized by adjusting the related parameters of the periodic coupled damping beam structures and the equivalent 2D periodic stiffened plate structures.

Localization of unbounded contacts on vibrating elastic plates

Detection and localization of unbounded contacts in industrial structures are crucial for user safety. However, most structural health monitoring techniques are either invasive, power-consuming, or rely on time-varying baseline comparison. A passive acoustic method is proposed to localize unbounded contacts in plate-like structures, using the acoustic emissions by the contacts when they are excited by ambient noise. The technique consists of computing the correlation matrix of the signals measured by a set of receivers and applying to this matrix a beamforming algorithm accounting for flexural wave dispersion. To validate the technique, an experimental setup is developed in which three idealized unbounded contacts are created on a thin plate excited by a shaker. How the quality of the defect localization depends on the defect type, receiver number, and the characteristics of the noise is investigated. Finally, it is shown that the localization of unbounded contacts is possible using either an acoustic ambient noise source or a more realistic jet engine noise.

RETRACTED: Flexural wave dispersion of nonlocal bi-Helmholtz-type stress gradient theory

In this study, harmonic wave propagation in elastic solid media is investigated. The accuracy of the proposed nonlocal bi-Helmholtz-type stress gradient theory with two nonlocal parameters is evalu...

Dynamics of ribbed plates with inner resonance: Analytical homogenized models and experimental validation

This paper deals with the theoretical, numerical and experimental behavior of periodic orthogonally ribbed plates. It extends the paper (Fossat et al., 2018) in which a comprehensive homogenized model has been established for flexural and torsional motion of periodic 1D-ribbed plates. New theoretical results describing the out-of-plane behavior of cellular plates involving inner resonance phenomena, are derived using an asymptotic approach. In this aim, the out-of-plane model of beam grids accounting for local bending and torsion is first established through the asymptotic method of homogenization of periodic discrete media. Then, the coupling between the beam grid and the internal plates (fully or partially connected to it) is detailed. This lead to an explicit analytical formulation of the equivalent plate model whose effective parameters arise from the geometry and mechanical properties of the unit cell. The unconventional features of the flexural wave dispersion are shown to be straightforwardly related to inner-resonance phenomena. These theoretical results are successfully compared to numerical computations conducted using WFEM. Furthermore, experiments performed on two prototypes of ribbed plates evidence the ability of the homogenized model to describe their complex dynamic behavior. The latter is characterized by the co-existence of a dynamic regime at both the micro-scale of the period and the macro-scale of the whole structure, that results in an inhomogeneous kinematics where the plate and beam displacements differ at the leading order. These unique features depart from the usual assumptions retained in plate mechanics and generates the observed non-conventional features. In conclusion, it is stressed that the study yields design rules to tailor cellular panels having specific atypical features in a given frequency range.

Stress gradient, strain gradient and inertia gradient beam theories for the simulation of flexural wave dispersion in carbon nanotubes

Flexural wave propagation in carbon nanotubes (CNTs) can be described through higher-order elasticity theories so as to capture the dispersive behavior induced by the inherent nanoscale heterogeneity. Motivated by experimental dispersion characteristics of metal nano-structured crystals, a new three-length-scale gradient formulation has been recently developed by the authors. In addition to the Laplacian of the strain, this model incorporates two higher-order inertia gradients for an improved dispersion behavior. A discrete medium with lumped masses at multiple scales of observation and combination of lumped mass and distributed mass at the macro-scale is introduced here to provide a micro-mechanical background to the proposed three-length-scale gradient model. The next aim of this paper is to assess the ability of this model to simulate flexural wave dispersion occurring in CNTs. We employ gradient-enriched Euler-Bernoulli and Timoshenko beam theories incorporating either stress gradients, or a combination of both strain gradients and inertia gradients – the latter leading to novel gradient-enriched beam theories. It is demonstrated that the proposed three-length-scale gradient elasticity formulation is able to capture the wave dispersion characteristics of armchair single-walled (5,5) and (10,10) CNTs arising from Molecular Dynamics simulations with high accuracy for a wide range of wave numbers. Advantages over alternative formulations of higher-order beam theories with stress gradients or combined strain-inertia gradient enrichments are discussed for comparative purposes.

Wideband dipole logging based on segment linear frequency modulation excitation

A wideband dipole signal is required for dipole dispersion correction and near-borehole imaging. To obtain the broadband flexural wave dispersion, we use a nonlinear frequency modulation (NLFM) signal and propose a segment linear frequency modulation (SLFM) signal as the dipole excitation signal to compensate for the excitation intensity. The signal-to-noise ratio (SNR) of the signal over the entire frequency band is increased. The finite-difference method is used to simulate the responses from a Ricker wavelet, a linear frequency modulation (LFM) signal, an NLFM signal, and an SLFM signal in two borehole models of a homogeneously hard formation and a radially stratified formation. The dispersion and radial tomography results at low SNR of the sound source signals are compared. Numerical modeling suggests that the energy of the flexural waves excited by the Ricker wavelet source is concentrated near the Airy phase. In this case, the dispersion is incomplete and information regarding the formation near or far from the borehole cannot be obtained. The LFM signal yields dispersion information near the Airy phase and the high-frequency range but not in the low-frequency range. Moreover, the information regarding the formation far from the borehole is not accurate. The NLFM signal extends the frequency range of the flexural waves by compensating for the excitation intensity and yields information regarding the formation information, but it is not easy to obtain. The SLFM signal yields the same results as the NLFM signal and is easier to implement. Consequently, the dipole detection range expands and the S-wave velocity calculation accuracy improves.

Determining formation S-wave transverse isotropy from borehole flexural-wave dispersion data

Many earth formations are characterized as transversely isotropic (TI) media. In acoustic logging through a vertical borehole, the S-wave TI property has traditionally been determined from borehole monopole Stoneley-wave measurements, but the feasibility of shear-TI estimation from dipole flexural waves has not been fully investigated. We have developed a methodology to determine the TI parameters from borehole dipole-flexural wave data. Our analysis shows that the Stoneley wave is sensitive to the TI property mainly in an acoustically slow formation, and the sensitivity diminishes when the formation becomes faster. The advantage of the flexural wave over the Stoneley wave is that the former wave is sensitive to the TI property in the slow and fast formations, provided the wave measurement is made in a broad frequency range in which the flexural-wave dispersion characteristics from low to high frequencies can be used. By calculating the theoretical flexural-wave dispersion curve for the TI formation and using it to fit the measured wave dispersion data, we can simultaneously determine the vertical and horizontal S-wave velocities, from which the S-wave TI parameter is obtained. Application of our methodology to field data processing shows that the TI parameter estimated from the flexural wave is almost identical to that from the Stoneley wave for a slow formation. For a fast formation, the flexural-wave result is more accurate and reliable compared with the Stoneley-wave result. Our study, thus, introduces a novel application of dipole acoustic logging.

Imaging Radial Distribution of Water Saturation and Porosity Near the Wellbore by Joint Inversion of Sonic and Resistivity Logging Data

Summary We present a work flow for joint inversion of sonic flexural-wave dispersion data and array-induction resistivity data acquired in a vertical well. The work flow estimates a pixel-based radial distribution of water saturation and porosity extending several feet into the formation at each log depth. Radial changes in saturation and porosity are caused by mud-filtrate invasion and mechanical damage, respectively. The flexural-wave and array-induction data have similar multiple investigation depths extending several feet into the formation. Furthermore, flexural-wave data are sensitive to porosity but have weak sensitivity to saturation, whereas induction data are sensitive to both porosity and saturation. Thus, integration of these data in a joint inversion can help to characterize the formation beyond the altered zone and reduce uncertainty of the interpretation. The work flow is validated on synthetic data for several scenarios of near-wellbore alteration. The work flow is then applied to field data from an offshore well drilled with oil-based mud in a gas-bearing clastic formation. The results are compared with traditional interpretation and core analysis, demonstrating an efficient and accurate inversion-based work flow that can complement traditional formation evaluation in challenging conditions.

Flexural wave band gaps and vibration reduction properties of a locally resonant stiffened plate

A locally resonant stiffened plate is constructed by attaching a two-dimensional periodic array of spring-mass resonators to a traditional periodic stiffened plate. A method based on the finite element method and Bloch theorem is presented for calculating the flexural wave dispersion relation and forced vibration response of the proposed locally resonant stiffened plate. The method is validated by comparing the predictions with simulations by FEM software COMSOL. The effects of the spring-stiffness and mass ratio of local resonators on the flexural wave band gap and vibration reduction performance are analysed, which can facilitate the design of the locally resonant stiffened plate for vibration-reduction applications in engineering. The main findings of this work are as follows. 1) The local resonator can have a significant effect on the propagation of flexural wave in stiffened plate. On the one hand, the local resonator is able to create a low-frequency local resonance band gap; on the other hand, it can enhance the high-frequency Bragg band gap. Within the band gap frequency range, the vibration of the locally resonant stiffened plate can be reduced remarkably. 2) The spring-stiffness of local resonators shows a notable influence on the band gap and vibration reduction performance of the locally resonant stiffened plate. As the spring-stiffness gradually increases, the nature frequency of local resonator is gradually tuned to higher frequency, and the phenomenon of band-gap transition and band-gap near-coupling may arise. Under the near-coupling condition, the pass band between two band gaps turns narrow, and it seems that these two band gaps form a super-wide pseudo-gap (within which only a very narrow pass band exists). This behaviour is of great interest for the broad band vibration reduction applications. Moreover, the complete band gap will disappear if the nature frequency of local resonator is tuned to a higher value than a threshold frequency, which is dependent on the geometrical and material parameters. 3) The influence of the additional mass ratio of local resonator on the band gap behavior is highly relevant to the nature frequency of local resonator. If the nature frequency of resonator is lower than the band-gap near-coupling frequency, both the local resonance band gap and Bragg band gap are broadened with increasing the additional mass ratio of resonator. When the nature frequency of resonator is close to the band-gap near-coupling frequency, the phenomenon of band-gap near coupling and band-gap transition may arise or disappear as the additional mass ratio of resonator gradually changes. When the nature frequency of resonator is higher than the band-gap near-coupling frequency, on the one hand, the lower frequency band gap will disappear rapidly with increasing the mass ratio of resonator. However, it will be present again if the mass ratio of resonator increases up to a large enough value. On the other hand, the higher frequency band gap is broadened with increasing the mass ratio, but if the mass ratio is tuned to a larger value than a specific value, this band gap will transform from local resonance band gap to Bragg band gap, and the normalized gap width of this band gap will be narrowed with increasing the mass ratio.

The dispersion of the flexural waves in a compound hollow cylinder under imperfect contact between layers

The influence of the interface imperfect bonding on the flexural wave dispersion in the bi- layered hollow circular cylinder is studied with utilizing three-dimensional linear theory of elastodynamics. The shear-spring type model is used for describing the imperfect bonding on the interface between the layers and the degree of the imperfectness is estimated through the dimensionless shear-spring parameters which enter the mentioned model. The method for finding the analytical expressions for the sought values and dispersion equation are discussed and detailed. Numerical results on the lowest first and second modes are presented and analyzed. These results are obtained for various values of the shear-spring parameters. According to these results, in particular, it is established that as a results of the imperfection of the bonding between the layers the new branches of the dispersion related the first fundamental mode arise and the character of the dispersion curve related to the second mode becomes more complicated.

Investigation of borehole cross-dipole flexural dispersion crossover through numerical modeling

Crossover of the dispersion of flexural waves recorded in borehole cross-dipole measurements is interpreted as an indicator of stress-induced anisotropy around a circular borehole in formations that are isotropic in the absence of stresses. We have investigated different factors that influence flexural wave dispersion. Through numerical modeling, we determined that for a circular borehole surrounded by an isotropic formation that is subjected to an anisotropic stress field, the dipole flexural dispersion crossover is detectable only when the formation is very compliant. This might happen only in the shallow subsurface or in zones having high pore pressure. However, we found that dipole dispersion crossover can also result from the combined effect of formation intrinsic anisotropy and borehole elongation. We found that a small elongation on the wellbore and very weak intrinsic anisotropy can result in a resolvable crossover in flexural dispersion that might be erroneously interpreted as borehole stress-induced anisotropy. A thorough and correct interpretation of flexural dispersion crossover thus has to take into account the effects of stress-induced and intrinsic anisotropy and borehole cross-sectional geometry.

FlexuralWave Dispersion in Bi-material Compound Solidand Hollow Circular Cylinders

FlexuralWave Dispersion in Bi-material Compound Solidand Hollow Circular Cylinders

A Nonlocal Continuum Mechanics Model for Analyzing the Terahertz Flexural Wave Dispersion Characteristics of a Monolayer Graphene Embedded in Elastic Medium

This article presents the strong, nonlocal scale effect on the terahertz flexural wave dispersion characteristics of a monolayer graphene sheet (GS), which is embedded in elastic medium. The chemical bonds are assumed to be formed between the GS and the elastic medium. From the spectrum analysis, we found that the elastic matrix highly affects the flexural wave mode and rapidly increases the frequency band gap of flexural wave. The nonlocal elasticity calculation shows that the wavenumber escapes to infinite at a certain frequency and the corresponding wave velocity tends to zero at that frequency, indicating localization and stationary behavior.