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This article presents guided-wave scattering by defects in a plate with finite width. The system comprises a homogeneous, isotropic, linearly elastic plate containing defects. The wave scattering is solved by a boundary element method (BEM) with radiation conditions for scattered guided waves. For the radiation conditions, the scattered waves are expressed by modal expansion using the semi-analytical finite-element (SAFE) dispersion analysis method. This study proposes a simple SAFE-BEM formulation based on the Galerkin projection with guided-wave mode structures accommodating beam solutions as the incident wave. The scattered waves are evaluated as far-fields in an infinite plate to consider the scattered far-fields in the framework of two-dimensional (2D) plate-wave theory. Numerical results demonstrate that the proposed SAFE-BEM formulation provides sufficiently accurate solutions. The present treatment of incident and scattered waves suggests that the 3D wave scattering in a plate with finite width can be examined using the framework of 2D plate-wave theory. [Display omitted] • A simple hybrid SAFE-BEM formulation is proposed. • The Gaussian beam of a plate wave is the incidence. • Scattered far-fields are evaluated based on 2D plate-wave theory. • 3D scattering is examined in the context of the 2D theory framework. [ABSTRACT FROM AUTHOR]

In this work, a new approach for solving the inverse problem of the defect location in an elastic plate is developed by combining the artificial neural network (ANN) and the boundary element method (BEM). The inverse problem is formulated as a regression problem, which extracts the solution of highest probability through machine learning (ML) from a large amount of data. The efficiency and accuracy of data generation is guaranteed by using the BEM, which solves for strain values on the boundaries of plates with circular defects and creates the ML datasets. The ANNs, which are constructed by a fully connected multi-layer perceptron, are trained using these datasets to predict the center coordinates and radii of the circular defects and can achieve about 98% accuracy in the detection. Compared with the Lamb wave based structural health monitoring (SHM) techniques, which in general require signal generation, collection and processing, the proposed approach only requires the data of boundary strains of the considered structures as the input data and a simple training process. Therefore, it is much easier to implement and has great potential in applications of SHM. [ABSTRACT FROM AUTHOR]

The concept of odd elasticity was recently introduced to characterize the elastic behavior of solids that consist of active components, exhibiting an asymmetric elastic modulus tensor. In this paper, we propose, for the first time, the microstructure design of an odd plate, which is composed of a lattice plate with a piezoelectric-patch-based sensor–actuator feed-forward system. By leveraging the nonreciprocal coupling between shear forces and bending curvatures, the odd plate constitutive relation is formulated in the low frequency region, which features as four asymmetric coupling parameters known as "odd parameters". We reveal that the two-dimensional (2D) odd plates can perform directional wave energy amplification and the amplification angle can be determined analytically through the rotation of coordinate system. We also numerically demonstrate the directional wave amplification phenomena that arise from the optimal combination of odd parameters. In addition, we analytically uncover the presence of Stoneley-like interfacial waves between two plates with two odd parameters in opposite signs, which is further characterized by the numerical simulation. Unlike interfacial waves between topological structures, the interfacial waves between odd plates can exist for any working frequency, enabling the design of some novel waveguides. This research on the control of flexural waves in odd plates could shed lights on 2D non-Hermitian systems in elasticity. [ABSTRACT FROM AUTHOR]

• Gravity wave interaction with multiple wavy porous barriers is studied. • A quadratic pressure drop is adopted for waves past the porous barriers. • The numerical model is based on an iterative Dual boundary element method (DBEM). • Bragg reflection is observed when the porosity of the multiple barriers is zero. • Increase in the number of porous plates, ripple amplitude and number of ripples reduces the wave transmission appreciably. • The proposed breakwater progressively dissipates the wave energy without increasing the hydrodynamic loads. The gravity wave interaction with horizontally staggered multiple thin wavy porous barriers is analyzed under the assumption of potential flow theory. The dual boundary element method (DBEM) is used to solve the boundary value problem. The accuracy and reliability of the DBEM solutions are ascertained by comparing with known results in the literature and with an independently developed multi-domain boundary element method (MBEM) code. The study is focused on the scattering coefficients and vertical forces on an array of wavy barriers. The numerical results demonstrate that wave transmission by a horizontally staggered porous barrier is much better compared to a single horizontal plate. It is recommended to use plate porosity of 0.05 to 0.20, if transmission coefficient of less than 0.50 is needed, especially for relative water depth k 0 h > 2. With an increase in the number of ripples and ripple amplitude on the porous plates, it is possible to further reduce wave transmission by 5% to 10%. Also, an increase in the inclination of the plate from 0 to 10 degrees can reduce the wave transmission and increases the wave reflection to 5% to 10%. Further, a reduction of about 10% to 30% in the vertical force on the wavy plate is possible for k 0 h ≥ 1 with an increase in the relative ripple amplitude, the number of ripples and increased plate rotation. This essentially will help in the reduction of pile penetration length into the sea bed soil. The results of the present study can be used for the optimal design of wavy plate as a wave barrier for offshore and coastal engineering applications. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive modeling and control method of piezoelectric sensing-actuating system for precise dynamic force output. The complex dynamic behavior over broadband frequencies are modeled by employing multi-field modeling method. Low frequency sensor dynamics and broadband hysteretic dynamics are accurately described by using the established model. One robust repetitive controller, by synthesizing the robust H∞ controller, the repetitive controller and the hysteresis compensator, is proposed for precise dynamic force output. The low frequency sensor dynamics and the inversion error of Bouc–Wen model are both regarded as uncertainty modules in the robust H∞ controller design. The repetitive controller and hysteresis compensator are further employed to enhance the H∞ control performance. Finally, the effectiveness of the proposed modeling and control approaches is validated by using simulation and experimental studies. [ABSTRACT FROM AUTHOR]

The two-dimension problems of the interaction between waves and one, two and a series of deformable plates are discussed, respectively, in this study. The fluid motion is described by the linear wave theory. The deformable plates are simplified as the one-dimensional beams with uniform stiffness and mass distribution. A boundary element method (BEM) is used to calculate the wave field. The Euler–Bernoulli beam theory is used to describe the motion of the deformable plate, which is calculated on the basis of a finite element method (FEM) in combination with the BEM model. The numerical solutions are derived for the velocity potentials together with the displacements of the deformable plates. After the energy conservation is calculated to verity the numerical models, the effects of structural flexibilities and interval between deformable plates on the reflecting waves are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we derive the formulations and carry out the calculation on the wave motion characteristics in layered nanostructures based on the surface/interface elasticity (effect). While the general solution in each bulk layer is obtained in terms of the mathematically elegant and numerically powerful Stroh formalism, the surface/interface constitutive relation is converted into a matrix form which is similar to the bulk layer matrix for easy propagation. To derive a stable global matrix for the layered structures, the dual variable and position method is utilized. Furthermore, the dispersion curves are calculated by a recently presented accurate algorithm. Numerical results show that the surface/interface effect can be significant when the wavenumber is large and the thickness of the layer is small. Besides its effect on the dispersion curves and mode shapes, the nanoscale surface/interface could further change the kinetic and strain energy on the wave. Particularly, a reduction in surface energy, which corresponds to a negative surface energy, could cause the nanostructure unstable. [ABSTRACT FROM AUTHOR]

In the present paper, the dispersion curve of guided modes in a cubic Ni–Mn–Ga thin plate is studied. Exceptional dispersion behavior is reported, with multiple crossings between antisymmetric (A n) and symmetric (S n) Lamb modes. In addition, several dispersion branches contain one or more zero group velocity (ZGV) points. The number of ZGV points per symmetric mode, S n and per antisymmetric mode A n , turns out be 3n-2. • The ordinary differential equation has been proposed for analyzing waves propagating in a single crystal Ni–Mn–Ga plate. • Unusual behavior was found in the dispersion curve of the Lamb wave. • For the first time, multiple ZGV resonances appear in the dispersion curve of all Lamb modes. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the Lamb-type problem of the dynamic response of isotropic half-space subjected to the time-harmonic vertical point load within the simplified strain gradient elasticity theory (SGET). The classical elastodynamics solution of this problem predicts singular displacements and stresses at the point of the load application. In the present study, we use the technique of Lamé potentials and Hankel transform to solve the high-order motion equations of SGET. An explicit solution accounting for traction and double traction boundary conditions is obtained in the transformed domain. The inverse transformation is performed numerically with accurate capturing of the solution behavior around the Rayleigh pole, which position is defined by the modified scale-dependent Rayleigh equation. Based on obtained results, it is shown that SGET solution is bounded at the epicenter of the load, and that it takes into account the spatial dispersion effects for the propagated waves. These effects are important for the analysis of the high-frequency processes in the structured materials. The intensity of non-classical effects in the solution is controlled by the length scale parameters of gradient model. Based on numerical calculations, we derive the universal non-dimensional relations between the amplitudes of the applied load and maximum displacements on the surface, which can be used to identify the length scale parameters of SGET based on the atomistic calculations or experimental data. • Lamb's point-load problem is considered within strain gradient elasticity. • Regularization of the solution at the epicenter of the load is shown. • Spatial dispersion effects make influence on the propagated waves. • Universal relations for identification of the length scale parameters are derived. [ABSTRACT FROM AUTHOR]

It can be difficult to efficiently model ultrasonic waves in 3D structures, especially when the computational model needs to account for complex geometries. This contribution presents a solution based on the Scaled Boundary Finite Element Method (SBFEM). It is a numerical tool suitable for elastodynamic problems. A space-tree discretisation, namely quad-trees, is used. This technique allows the decomposition of an image into quadrilaterals or quads, which are extruded to generate the 3D plate geometry. In particular, small quads resolve regions with discontinuities, allowing them to represent fine details in the structure. Moreover, this meshing technique allows for exploiting cell similarities, making the calculation procedure more efficient. The space-tree discretisations are generated from a high-resolution image containing all the information about damaged regions or boundary conditions. The resulting SBFEM polyhedral domains employ transition elements to ensure correct coupling between cells of different sizes. The analytical solution of a cylindrical scatterer serves as a reference to validate the proposed approach. Other examples also demonstrate the validity of the methodology and its flexibility. • Transition elements couple different cell sizes while retaining high-order. • Quadtrees automate geometry representation and generate 3D plates through extrusion. • Quadtree meshing allows for boosting efficiency by exploiting cell similarities. [ABSTRACT FROM AUTHOR]

This research applies machine and deep learning to nondestructively characterize the thickness and uniformity of a coating in a layered system using dispersion curves. A finite element analysis model is first used to computationally model transient, guided Lamb waves propagating in coated specimens with different coating thicknesses. These time-domain signals are then processed with a two-dimensional Fourier transform to obtain the corresponding frequency–wavenumber relation, which are the dispersion maps of the coated specimen. Dispersion maps are characteristic and depend on both the coating thickness and uniformity, plus its elastic properties (which are taken to be constant). Computationally simulated dispersion maps for a variety of coating properties are obtained and then further processed to extract a feature representation for each dispersion curve. Those extracted features are fed into machine learning classifiers which allow a thickness classification. This machine learning procedure is shown to be effective in classifying the thickness of a uniform coating. However, if the coating thickness is nonuniform, deep learning, specifically a convolutional neural network architecture, is used for classification. The network is evaluated, tested, and recommendations on its use are given. • Using machine learning and deep learning for nondestructive evaluation. • Classify thickness and uniformity of coating in a layered system using dispersion curves. • Calculate dispersion information with a two-dimensional Fourier transform. • Training data is obtained from a finite element analysis. [ABSTRACT FROM AUTHOR]

Propagation of harmonic Lamb waves in functionally graded (FG) plates with transverse inhomogeneity is analyzed by applying the modified Cauchy six-dimensional formalism. For arbitrary transverse inhomogeneity and arbitrary elastic anisotropy a closed form implicit analytical solution for the dispersion relation is derived. Dispersion relations for materials with exponential inhomogeneity are obtained. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the dispersive behaviour of the waves in a traction-free plate, commonly referred to as the Rayleigh–Lamb (R–L) waves, considering the effect of strain gradient elasticity (SGE) and the velocity gradient kinetic energies. The SGE is modelled using the generalized first strain gradient theory (GFSGT) involving three independent material length constants. The transcendental equations in terms of the frequency and wavenumber, for the symmetric and anti-symmetric modes of the shear horizontal (SH) and plane (P & SV) R–L waves are derived here, separately. The dispersion curves for the frequency and the velocity are obtained by solving these equations numerically. This study is also extended for a plate subjected to mixed-boundary conditions. The observations presented here illustrate the significance of gradient elasticity over the behaviour of the elastic wave propagation in micro and nano-plates. Highlights • Effect of gradient elasticity, kinetic energy over dispersion of Rayleigh–Lamb waves studied. • Distinct dispersion behaviour for longitudinal and transverse waves observed here. • Study extended for dispersion of waves in micro-plates subjected to mixed boundary conditions. • Reduction in short-wavelength frequencies upon considering gradient effects noted here. • Significant shift in cut-off frequencies observed in higher branches of dispersion curves. [ABSTRACT FROM AUTHOR]

Abstract Propagation of harmonic Lamb waves in functionally graded (FG) plates with transverse inhomogeneity is analyzed by applying the modified Cauchy six-dimensional formalism. For arbitrary transverse inhomogeneity and arbitrary elastic anisotropy a closed form implicit analytical solution for dispersion equation is derived. Dispersion relations for materials with exponential inhomogeneity are obtained. Highlights • Analytical solution for dispersion of Lamb waves in FG plates is derived. • Modification of Cauchy six-dimensional formalism is constructed. • Numerical examples for plates with exponential inhomogeneity are given. [ABSTRACT FROM AUTHOR]

We analytically investigate the influence of damping on Lamb waves, which are a specific type of guided wave in two-dimensional plates. Considering material attenuation, we suppose that Lamé constants are complex numbers. This leads to the associated wavenumbers being complex, with the imaginary part of the wavenumber being associated with effect of attenuation of the guided wave. In this paper, we show how dispersion curves and attenuation coefficients can be obtained using the self-adjointness and the method of multiple scales (MMS), which is a type of perturbation method. Using the self-adjointness and the MMS, we can calculate the frequency- and wavenumber-dependent attenuation coefficients from the integral values and boundary values of a corresponding eigenfunction with respect to each propagation mode. This analytical method can yield not only dispersion curves but also mode-by-mode attenuation coefficients regardless of the numerical initial values, unlike numerical approaches using the Newton method. Thus, the proposed method can more easily calculate the attenuation coefficients with respect to a particular mode than conventional methods. Furthermore, the results obtained by proposed method were in good agreement with those obtained by the semi-analytical finite element (SAFE) method, which validates the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we investigate the problem of Lamb wave propagation in a layer composed of homogeneous, isotropic material with irregular free surfaces. It is assumed that the irregular free surfaces vary slowly in the direction of wave propagation. By employing an asymptotic perturbation method, a unique asymptotic solution for the amplitude function of Lamb waves is obtained. The dispersion relation is expressed as a function of phase velocity of waves, wave number and the direction of wave propagation. We examine the effect of irregularities of the boundary surfaces on the propagation of waves under the assumption that free surfaces have periodic properties. • Lamb wave propagation in a layer with irregular free surfaces is investigated. • The layer is composed of homogeneous and isotropic material. • A unique asymptotic solution for the amplitude of linear Lamb waves is obtained. • Dispersion relation is analysed. • Effects of periodic surfaces on wave propagation have been demonstrated graphically. [ABSTRACT FROM AUTHOR]

Carbon fiber reinforced polymer (CFRP) could be damaged by low-velocity impacts. Precise localization of impact can increase the efficiency of nondestructive testing by narrowing the scanning in a small area. The conventional localization method is based on the time-of-arrival (ToA) of impact induced guided wave, whose localization accuracy could be influenced by the reflected waves from plate edges. This paper proposes an impact localization method based on time–frequency features of guided wave and convolutional neural network. To test the algorithm, 585 impact experiments are performed at 117 different locations of a CFRP plate. Signals corresponding to 12 randomly selected locations are treated as test set, and the remaining signals are used for training of the network. The results show that the average localization error of the proposed method is 10.2 mm. • The influence of reflected ultrasonic guided wave on the localization of impact is analyzed. • The time–frequency features of guided wave signal is analyzed with wavelet transform. • A convolutional neural network is trained to locate the impacts on a CFRP plate. [ABSTRACT FROM AUTHOR]

This study investigated the thickness mode and thickness shear mode of Lamb waves excited by a surface interdigital transducer and incident on piezoelectric ceramic plates. We numerically calculated the time-domain response, frequency-domain response, and propagation characteristics of these waves. To determine the electric field intensity of a surface interdigital transducer with a semi-infinite domain, the driving potentials of the surface electrodes of the piezoelectric plates were approximated using a series solution and the Legendre function. The wave propagation responses of the plates were determined by considering the electromechanical coupling effect of piezoelectric materials and assuming the electrical boundary conditions of three external excitation sources. A modified Clenshaw–Curtis numerical integration method was used to calculate the frequency-domain response of the Lamb wave displacement of the piezoelectric plates excited by an interdigital electrode. The time-domain response of this displacement was determined using the inverse fast Fourier transform method. This study also examined the near- and far-field Lamb wave responses of interdigital transducers with different electrode numbers and electrode widths. • An interdigital transducer excited Lamb waves on piezoelectric ceramic plates. • Electrode potential was valued with a series solution and Legendre function. • Lamb wave frequency-domain response are computed with a modified Clenshaw–Curtis method. • The inverse fast Fourier transform calculated displacement time-domain response. • Lamb wave response with varied electrode numbers and widths were studied. [ABSTRACT FROM AUTHOR]

The solution of the complete three-dimensional (3D) guided wave propagation problem in infinite laminated composite plates using a semi-analytical method is presented. The proposed method constructs the problem directly from the 3D governing equations, assuming through-the-thickness layerwise laminate mechanics with linear piecewise variation of all displacement components. The solution is then transformed into the frequency–wavenumber domain, applying sequential Fourier transforms in time and two-dimensional space. Finally, a technique for obtaining the 3D solution via the two dimensional (2D) forced problem is presented. Results for various material systems, such as isotropic, cross-ply and quasi-isotropic composite laminate plates, are shown and compared to well-established analytical results. [ABSTRACT FROM AUTHOR]

A generalisation of diffraction tomography that is applicable for mixed-mode Lamb wave imaging is derived, subject to clearly stated simplifying assumptions. The resulting algorithm is further extended to near-field imaging. This algorithm requires a knowledge of the scattering pattern for an appropriate point scatterer, which needs to be determined separately for each type of damage, and for every combination of incident and scattered wave modes. The procedure for determining this point-scatterer pattern is presented for the case of delamination damage modelled as a flexural inhomogeneity. The performance and limit of validity of the resulting algorithm for the A 0 mode are shown to be comparable with those for the acoustic model, but this excellent performance is accompanied by an increased sensitivity to noise relative to simpler algorithms based on beamforming or time reversed imaging. Several strategies for enhancing the noise tolerance are proposed and investigated. Experimental results are presented to demonstrate the practical implementation and imaging performance. These results also illustrate the use of the distorted wave Born approximation for imaging in the presence of known structural complexities, as well as the use of a factorisation approach to minimise the required number of sensors. [ABSTRACT FROM AUTHOR]

Ultrasonic Lamb waves are affected by structural discontinuities (e.g. stiffeners, ribs or joints), and it is necessary to understand the quantitative characteristics of their influence on the waves so that the structural features are not misidentified as defects in nondestructive evaluation (NDE) applications. In this paper, the benchmark problem of Lamb wave reflection at a free end as well as the interaction of Lamb waves with upward and downward step discontinuities is solved using an analytical approach. The free-end problem is considered to show the influence of the decaying waves on the power flow in the plate. In all problems, a near-field solution is obtained, and power flows past the cross-sections of the plate are evaluated. Furthermore, amplitude spectra and results for different step sizes are presented. The analytically derived results are compared with those from transient finite element simulations, showing good agreement. [ABSTRACT FROM AUTHOR]

This paper presents an efficient and accurate analytical method to calculate the scattering of straight-crested Lamb waves from geometric discontinuities. In this method, the scattered field is expanded in terms of complex Lamb wave modes with unknown amplitudes. These unknown amplitudes are obtained from the boundary conditions using a vector projection utilizing the power expression. The process works by projecting the stress conditions onto the displacement eigen-spaces of complex Lamb wave modes and vice-versa. We call this technique “complex modes expansion with vector projection” (CMEP). Unlike other methods, the CMEP approach is versatile and can be readily applied to notches, cracks, or disbonds. In this paper, the methodology is illustrated by applying the CMEP method to a benchmark problem — the geometric discontinuity created by a step in the thickness of a plate. For method verification, the finite element method (FEM) and the axial–flexural analytical model were used. The FEM analysis was conducted in the frequency domain with non-reflecting boundaries. It was found that the CMEP results correspond very well with the FEM results over a wide frequency–thickness range up to 1.5 MHz-mm. The axial–flexural model was used to verify only the CMEP asymptotic behavior toward zero frequency where frequency-domain FEM encounters difficulties. Our study shows that the computational efficiency of CMEP is orders of magnitude higher than FEM. The paper ends with a discussion of how the CMEP method may be extended to the fast and accurate analysis of realistic damage situations. [ABSTRACT FROM AUTHOR]

Monitoring and analysis of acoustic emissions (AEs) generated by material defects is important during nondestructive testing. Here, we characterize the AEs generated during tensile deformation of AISI type 316 stainless steel (SS). The active frequencies of this SS, spanning the 100–200 kHz range, were determined via tensile testing. Fast Fourier Transform (FFT) analysis revealed that the AEs featured four frequency components representative of all signals observed during the strain-hardening stage. For completeness, the propagation of the AE wave, a Lamb wave generated by the point source, was simulated (using the determined frequency components) in a two-dimensional 2 × 80 mm SS plate. The displacement fields that triggered AE of a Lamb wave from a point source were derived by solving the Navier-Lamé equation. This analytical modeling of AE is useful for characterizing emission generated by defects. • Determination of natural frequencies of 316 stainless steel from tensile testing. • Mathematical formulation of Lame wave generated by a point source in two-dimensional plane. • Simulation of the acoustic emission waveforms generated during tensile testing. [ABSTRACT FROM AUTHOR]

The interaction of Rayleigh surface waves with a random arrangement of oscillators on the surface of an elastic half-space is analyzed with the help of the surface elastodynamic Green's function formalism and the effective medium model valid in the long-wavelength limit is derived. It is shown that in order to apply the central assumption of the model, namely, that the local surface displacement at the contact point of an oscillator is equal to the average effective medium displacement, the spring constant of the oscillators must be corrected to include the stiffness of the contact calculated in the quasistatic approximation. The previous results pertaining to the interaction of Rayleigh and Lamb waves with microspheres are shown to be correct despite the apparent inconsistencies in the assumptions. • The interaction of Rayleigh waves with oscillators on an elastic half-space is analyzed in the long-wavelength limit. • The standard effective medium approach is valid if the oscillator frequency is corrected by accounting for contact stiffness. • Previous results on the interaction of Rayleigh waves with microspheres are correct despite inconsistencies in assumptions. • The effective medium model describing the interaction of oscillators with an elastic substrate is put on solid ground. [ABSTRACT FROM AUTHOR]

Phase singularities are generic structures which occur in all wave fields, and they are characterised by an inability to assign a value to the phase. Screw dislocations are a particular kind of phase singularity where the phase possesses a helical structure, with the singularity at the centre of the helix. In this paper we show that it is possible to generate screw dislocations on the surface of elastic isotropic solids by means of the interference of three Rayleigh waves or three Lamb waves. The dispersive character of Lamb waves leads to more complicated behaviour, which may in turn result in greater potential for applications. [ABSTRACT FROM AUTHOR]

This article presents a new approach to designing non-reflective boundary (NRB) for inhibiting Lamb wave reflections at structural boundaries. Our NRB approach can be effectively and conveniently implemented in commercial finite element (FE) codes. The paper starts with a review of the state of the art: (a) the absorbing layers by increasing damping (ALID) approach; and (b) the Lysmer–Kuhlemeyer absorbing boundary conditions (LK ABC) approach is briefly presented and its inadequacy for Lamb wave applications is explained. Hence, we propose a modified Lysmer–Kuhlemeyer approach to be used in the NRB design for Lamb wave problems; we call our approach MLK NRB. The implementation of this MLK NRB was realized using the spring–damper elements which are available in most commercial FE codes. Optimized implementation parameters are developed in order to achieve the best performance for Lamb wave absorption. Our MLK NRB approach is compared with the state of the art ALID and LK ABC methods. Our MLK NRB shows better performance than ALID and LK ABC for all Lamb modes in the thin-plate structures considered in our examples. Our MLK NRB approach is also advantageous at low frequencies and at cut-off frequencies, where extremely long wavelengths exist. A comprehensive study with various design parameters and plate thicknesses which illustrates the advantages and limitations of our MLK NRB approach is presented. MLK NRB applications for both transient analysis in time domain and harmonic analysis in frequency domain are illustrated. The article finishes with conclusions and suggestions for future work. [ABSTRACT FROM AUTHOR]

Fundamental symmetric and anti-symmetric Lamb wave modes are characterised in terms of their velocity and magnitude variation as they encounter a gradual change in the thickness of a composite sandwich plate with a high density foam core (Dyvinicell HP100). Wave mode selection is optimised to enhance the sensitivity of guided waves for debonding inspection of tapered sandwich structures, and the conversion in wave modes after interaction with the taper is also studied. A taper change in the sandwiched structure can induce a clear change in the wave magnitude when the guided waves are excited at relatively low frequency (below around 150 kHz), while there is less effect on the velocity of the propagating waves. The effect of the taper on the signal magnitude and velocity is reduced when the excitation frequency is increased beyond 200 kHz. An imaging algorithm based on time reversal is developed to detect multiple debonding in tapered composite sandwich panels, using guided waves at different excitation frequencies ranging from 150 to 200 kHz, from an active sensor network. Correlation coefficients between the original and reconstructed time reversal signals are calculated to define a damage index for individual sensing paths, which are subsequently used in a fusion process to identify the presence of multiple debonded areas. [ABSTRACT FROM AUTHOR]

A Multiresolution finite wavelet domain meshless approach is presented for the simulation of guided waves in composite beams. The Daubechies wavelet and scaling functions are both employed as basis functions and their remarkable properties for the hierarchical spatial approximation of state variables are explored. The first level of the multiresolution approximation provides the coarse solution, while finer resolution components are sequentially calculated and superimposed on the coarse solution, until the desired level of accuracy is achieved. The development of the multiresolution wavelet-based beam element that encompasses first-order shear deformation theory kinematic assumptions is described, and the multiresolution equations of motion including stiffness and mass matrices are formulated. Numerical results for the simulation of guided wave propagation in aluminum and laminated composite beams are presented, demonstrating substantial reduction in computational effort compared to single resolution approaches and traditional finite elements. Convergence studies using various Daubechies wavelet interpolation functions, are evincing extraordinary convergence rates with very low requirements of grid points per wavelength. Furthermore, additional benefits of the proposed method are encompassed in the analysis of multiple guided wave modes, attributed to the low-pass filter behavior of the scaling functions and the band-pass filter behavior of the wavelet functions that can result in isolation of different wave modes by the individual resolution components of the total solution. • Novel hierarchical refinement type that employs the non-converged solutions to achieve convergence. • Diagonal mass matrices yield uncoupled multiresolution equations of motion using explicit integration. • Rapid simulations of guided wave propagation in aluminum and composite beams. • Remarkable convergence rates compared to single-resolution approaches and finite elements. • Unique capabilities of different wave mode isolation by the individual resolution components of the total solution. [ABSTRACT FROM AUTHOR]

The reciprocity, orthogonality and bi-orthogonality relations are well established for symmetric waveguides, with the most known example being the Rayleigh–Lamb problem of wave propagation in an infinite elastic layer. In this paper, these relations are derived for non-symmetric waveguides using as an example the canonical problem of wave propagation in an infinite elastic plate loaded by a fluid layer of a finite height with uniform mean flow parallel to its surface. The links between formulations of dispersion equation, reciprocity/bi-orthogonality relations and energy flux are demonstrated and discussed. The Green's function for stable regimes of wave motion in the presence of a flow is derived using the bi-orthogonality relation, and partition of the energy flux between transmission paths is analysed. • Reciprocity and bi-orthogonality relations for non-symmetric waveguides. • Relation between dispersion equation and bi-orthogonality. • Modal analysis of wave propagation in the presence of mean flow. • Stability analysis of wave propagation in the presence of mean flow. [ABSTRACT FROM AUTHOR]

We investigate the transmission modes in one-dimensional piezoelectric phononic crystal plate, which is consisted of piezoelectric ceramics placed periodically in epoxy. The influences of piezoelectricity and different electrical boundary conditions on the Lamb waves in the composite plate are studied in detail, and an important elastic wave mode named as piezoelectric-sensitive mode (PSM) is defined. Furthermore, the influences of the piezoelectric constants, the filling fraction and the ratio of thickness to lattice pitch on PSM are given, and the physical mechanism of PSM is also discussed. The numerical results show that PSM has a relatively larger frequency shift and significant change in the displacement distribution when piezoelectricity is considered; the electromechanical coupling coefficient of PSM is much larger than the other modes. It reveals that PSM is an important and useful mode in piezoelectric periodic structures. This investigation is helpful in controlling the band gaps and also gives a potential application in design of sensing system and different piezoelectric devices. [ABSTRACT FROM AUTHOR]

A time domain spectral finite element is developed for improving the efficiency of numerical simulations of guided waves in laminated composite strips. The finite element relies on a new generalized laminate mechanics model formulated to represent symmetric and anti-symmetric Lamb waves. The laminate mechanics incorporate third-order polynomial terms for the approximation of axial and transverse displacement fields through the thickness and consider the displacements of the upper and lower surfaces as degrees of freedom. The laminate theory formulation is easily expanded to a high-order layerwise model. Based on the resultant governing equations of the laminate section, a new finite element with 8 nodal degrees of freedom is formulated; its nodes are collocated with Gauss–Lobatto–Legendre integration points in order to improve computational efficiency. Stiffness and mass matrices are assembled and the transient response is predicted using the explicit central differences time integration scheme. The transient response of Aluminum, Carbon Fiber Reinforced Polymer laminated and sandwich strips is investigated. Numerical results are validated against a semi-analytical solution. The accuracy and computational efficiency of the introduced element regarding the prediction of symmetric and anti-symmetric wave propagation is also quantified. [ABSTRACT FROM AUTHOR]

The scattering problem of a Lamb wave incident on a symmetric pair of surface-breaking transverse cracks in a plate is considered. The Lamb wave is assumed to be obliquely incident on the crack plane. Since the cracks are part-through, the scattered field will contain reflected as well as transmitted waves. The energy of the incoming wave is partitioned into reflected and transmitted wave modes. Energy coefficients of the reflected and transmitted waves are calculated as a function of incident frequency and crack depth. The incidence angle of the incoming wave is also treated as a parameter. Both the reflected and transmitted wave fields are considered as linear superpositions of all real and complex wave modes in the plate. Decomposition of modes is achieved with the help of an orthogonality condition based on the principle of reciprocal work. Continuity of displacement and stress fields is imposed at the crack plane. Energy coefficients for reflection and transmission are obtained from the mode amplitudes. Energy coefficients are shown to be a strong function of incident frequency and crack depth. Experiments are conducted with a PZT transducer network interacting with a symmetric pair of machined cracks in an aluminum plate. Trends predicted by the analysis are reflected in the experimental results. [ABSTRACT FROM AUTHOR]

This paper presents the numerical modeling and simulations of PZT-induced Lamb wave propagation in plate-like structures by using the spectral finite element method. A novel spectral plate finite element, which can efficiently model the three-dimensional (3D) behavior of Lamb waves, is proposed. In the formulation, linear displacement distributions in the thickness direction are assumed for both the PZT layer and the base plate. A way to avoid the thickness locking is proposed and used in the formulations. Two examples, one for the validation of the proposed two-dimensional (2D) spectral finite element and the other for the demonstration of crack detection in plates, are presented and discussed. The contact between the two faces of crack is considered. Numerical results show that (1) only the anti-symmetric mode is prone to thickness locking thus remedy should be made only on this part, (2) the proposed 2D spectral finite element can adequately model the Lamb wave propagation in plate-like structures and the complex scattering for the crack, and (3) crack location can be well determined by a PZT-induced Lamb wave-based diagnosis algorithm. [ABSTRACT FROM AUTHOR]

In the field of structural damage monitoring, sensor arrays are often used to detect damage sites. However, the use of a large number of sensors increases the required amount of data processing, which imposes severe restrictions on effective online monitoring of industrial applications. In this study, an area-division method based on a waveform centroid algorithm is proposed to locate damage sites accurately using a limited number of sensors. Experiments are conducted on a 600 mm × 600 mm × 1.5 mm aluminum alloy plate to evaluate the performance of the proposed method. Four piezoceramics, acting as exciters and receivers, are arranged on the central region of the plate, dividing the entire detection area into eight portions. In order to scan the plate structure, Lamb waves are generated from each of the piezoceramics while the others function as receivers. By performing three successive comparisons after processing the data corresponding to each exciter–receiver​ pair via the waveform centroid algorithm, the specific portion containing the damage site can be identified. Subsequently, two ellipses are drawn using the three piezoceramics neighboring the identified portion, to visually locate the damaged site. The technique is applicable under conditions with low signal-to-noise ratios with an error of 1.5% in this study. It has the potential to be useful in industrial applications, in the context of real-time structural health monitoring. • Damage identification based on lamb guided wave. • Damage site narrowness using the iterative partition algorithm. • TOF assessment via waveform centroid algorithm. [ABSTRACT FROM AUTHOR]

Abstract: Lamb waves propagate over large distances in plate-like thin structures and they have received great attention in the structural health monitoring (SHM) field as an efficient means to inspect a large area of a structure by using only a small number of sensors. The times-of-flight of the Lamb wave modes are useful for detecting damage generated in a structure. However, due to the dispersive and multi-mode nature of Lamb waves, it is very challenging to decompose Lamb wave modes into symmetric and anti-symmetric modes for potential applications to structural health monitoring. Thus, we propose an efficient Lamb wave mode decomposition method based on two fundamental rules: the group velocity ratio rule and the mode amplitude ratio rule. The group velocity ratio rule means that the ratio of the group velocities of and modes must be constant. The mode amplitude ratio rule means that the ratio of the magnitudes of and modes in a measured response signal must be always greater than one once the center frequency of the input signal is determined, such that the magnitude of the mode in the excited signal is larger than that of the mode, and vice versa. The proposed method is verified through experiments conducted for a plate specimen. [Copyright &y& Elsevier]

Abstract: Quantifying wall thickness in the presence of corrosion damage allows the residual life of plate-like structures to be determined. Guided wave tomography is a solution which allows an area to be imaged with data from an array of ultrasonic transducers around the defect, using tomographic methods to establish wall thickness. We exploit the dispersive nature of Lamb waves travelling through the wall to convert the thickness variation into wave velocity variations, providing a mechanism to determine thickness from a velocity reconstruction. HARBUT (the Hybrid Algorithm for Robust Breast Ultrasound Tomography) has been developed to give high-resolution, accurate and robust reconstructions of ultrasonic velocity through the breast; in this paper we extend the method to generate thickness maps for guided wave tomography. By iterating HARBUT we show that we can improve the accuracy of reconstructions of the small, high contrast defects expected from corrosion, allowing estimates of thicknesses within 1 mm for a 10 mm plate from both simulated and experimental data. [Copyright &y& Elsevier]

• The Zener impact was solved by using homotopy analysis method. • Structural compliance breaks contact force symmetry. • Zener energy loss is proportional to the impact velocity to the power of 0.2. Different from the Hertz impact, a sphere impacting on a plate has an extra dissipation by the flexural wave propagating on the plate. This wave dissipation has been noticed by Zener (1941), who introduced this dissipation term into the governing equation. Because of this extra dissipation term by the flexural wave, the Zener equation cannot be treated as the same way as in the Hertz impact problem. As a result, in the past 80 years, except few numerical tries for Zener equation, no analytical solution has been provided for the evolution of compressive displacement or contact force during the Zener impact. In this paper, by using homotopy analysis method, we analytically solved the Zener impact problem. After constructing an auxiliary linear operator, we derived an analytical solution for the evolution of contact force up to the first order of the embedding homotopy parameter. Present model can take full account of the elongation of contact time by plate compliance, which is impossible by using Hertz impact model. Comparisons with finite element simulation show that the prediction of the contact force history is of high accuracy. With the solution of force history, we derived an explicit expression for the energy loss of Zener type, this Zener loss has a different dependence on the impact velocity as the Hunter loss does. Also, both the motions of the sphere and the plate are able to be predicted with high accuracy. In addition, with the zeroth-order solution, we derived explicit expressions for both the coefficient of restitution and the contact duration. [Display omitted] [ABSTRACT FROM AUTHOR]

Abstract: Two analytical modifications of the original viscoelastic time-harmonic Lamb’s problem expressions are presented with the aim of improving their numerical integration efficiency. Firstly, a new change of variable in the Lamb’s problem integrands is proposed, which allows a standardization of the integration sampling vector and a complete spatial-frequency field solution after performing only one numerical integration/transformation. Secondly, the Lamb’s problem static integrands are modified and introduced into the original integrands to reduce their spectral content at high wavenumbers and, therefore, the sampling vector lengths needed to avoid aliasing. [Copyright &y& Elsevier]

Abstract: Lamb waves using surface-bonded piezoelectric transducers (PZTs) have been widely used for nondestructive testing (NDT). However, the identification of individual Lamb wave modes and the subsequent data interpretation are often difficult due to the dispersive and multimodal natures of Lamb waves. To tackle the problem, a Lamb wave mode decomposition technique using concentric ring and circular PZTs is proposed. Its advantages over the conventional approaches are that (1) PZTs need to be placed only a single surface of a specimen and (2) mode decomposition can be performed at any desired frequency without changing the PZT size and/or spacing configuration. The proposed mode decomposition technique is formulated by solving 3D Lamb wave propagation equations considering the PZT size and shape, and this technique requires a specially designed dual PZT composed of concentric ring and circular PZTs. The effectiveness of the proposed technique for the Lamb wave mode decomposition is investigated through numerical simulation and experimental tests performed on an aluminum plate. [Copyright &y& Elsevier]

Abstract: The improved eigenmode matching theory is developed to investigate band structure and transmission spectrum of lamb waves in a periodically stubbed waveguide with two imaginary “vacuum” layers added on the out surfaces of the quasi-one-dimensional system. The numerical results show that the width of some band gap can be changed upon varying the stub depth and width. It is also found that the transmission spectrum of finite system match the band structure very well and it is easier to achieve complete band gaps in the different materials system. [Copyright &y& Elsevier]

Abstract: An analytical investigation of the interaction between piezoelectric wafer active sensor (PWAS), guided Lamb waves, and host structure is presented in this paper, supported with application examples. The analytical investigation assumes a PWAS transducer bonded to the upper surface of an isotropic flat plate. Shear lag transfer of tractions and strains is assumed, and an analytical solution using the space-wise Fourier transform is reviewed, closed-form solutions are presented for the case of both ideal bonding (i.e., load transfer mechanism localized at the PWAS boundary) and not ideal bonding (i.e., load transfer mechanism localized close the PWAS boundary). The analytical solutions are used to derive Lamb wave mode tuning curves which indicate that frequencies exist at which the A0 mode or the S0 mode can be either suppressed or enhanced. The paper further shows that the capability to excite only one desired Lamb wave mode is critical for practical structural health monitoring applications such as PWAS phased array technique (e.g., the embedded ultrasonics structural radar, EUSR) and the sparse array imaging. Extensive experimental tests that verify the tuning mechanism and prediction curves are reported. Examples of correctly tuned EUSR images vs. detuned cases illustrate the paramount importance of Lamb wave mode tuning for the success of PWAS based damage detection. [ABSTRACT FROM AUTHOR]

Abstract: This study investigates the time reversal process (TRP) of Lamb wave signals which are transmitted and received by piezoelectric transducers bonded on plate-like structures. A number of previous studies have paid attention to spatial and temporal refocusing capability of an original excitation through the TRP in highly dispersive and complex media. However, when the TRP is applied to Lamb waves in a homogeneous regular waveguide, the refocusing capability is limited due to permanent residual side bands even if the duration of the time reversed signal increases. Based on the reciprocity of elastodynamics and linear piezoelectricity, theoretical interpretation is conducted for the main and residual side bands of the reconstructed signal in the time domain. In particular, the interpretation includes the temporal effect of velocity and amplitude dispersions, the existence of multimodes, and the reflections from boundaries during the TRP. Then, numerical and experimental tests are conducted to validate the theoretical findings of this paper. Practical issues for the successful implementation of the TRP of Lamb waves are briefly addressed as well. [Copyright &y& Elsevier]

Abstract: In this paper the propagation of plane and circular crested viscothermoelastic waves in a homogeneous isotropic, Kelvin–Voigt type viscoelastic thermally conducting, plate sandwiched between inviscid liquid layers is investigated in the context of classical and non-classical theories of thermoelasticity. The secular equations for the symmetric and skew-symmetric modes of plane and circular crested waves are derived in closed form and isolated mathematical conditions. It is noticed that the motion for both the plane and cylindrical waves in plates is governed by Rayleigh–Lamb-type secular equations. The secular equations for thin plate and short wave length waves are also obtained and discussed. The results in the absence of fluid loading, coupled and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. The dispersion curves, attenuation profiles and specific loss in case of symmetric and skew-symmetric wave modes are also presented graphically for a polymethyl methacrylate material plate under fluid loadings. The effect of dissipation due to viscosity is noticed to be quite significant and clearly visible from various curves in the graphs. [Copyright &y& Elsevier]

The propagation of magnetic-thermoelastic plane wave in an initially unstressed, homogeneous isotropic, conducting plate under uniform static magnetic field has been investigated. The generalized theory of thermoelasticity is employed, by assuming electrical behaviour as quasi-static and the mechanical behaviour as dynamic, to study the problem. The secular equations for both symmetric and skew-symmetric waves have been obtained. The magneto-elastic shear horizontal (SH) mode of wave propagation gets decoupled from rest of the motion and it is not influenced by thermal variations and thermal relaxation times. At short wavelength limits, the secular equations for symmetric and skew-symmetric modes reduce to Rayleigh surface wave frequency equation, because a finite thickness plate in such a situation behaves like a semi-infinite medium. Thin plate results are also deduced at the end. Dispersion curves are represented graphically for various modes of wave propagation in different theories of thermoelasticity. The amplitudes of displacement, perturbed magnetic field and temperature change are also obtained analytically and computed numerically. The result in case of elastokinetics, magneto-elasticity and coupled magneto-elasticity has also been deduced as special cases at appropriate stages of this work. [Copyright &y& Elsevier]

Analysis for the propagation of thermoelastic waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of inviscid liquid or half space of inviscid liquid on both sides, is investigated in the context of coupled theory of thermoelasticity. Secular equations for homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and anti-symmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. It is shown that the purely transverse motion (SH mode), which is not affected by thermal variations, gets decoupled from rest of the motion of wave propagation and occurs along an in-plane axis of symmetry. The special cases, such as short wavelength waves and thin plate waves of the secular equations are also discussed. The secular equations for leaky Lamb waves are also obtained and deduced. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic plate of zinc material bordered with water. The dispersion curves for symmetric and anti-symmetric wave modes, attenuation coefficient and amplitudes of displacement and temperature change in case of fundamental symmetric (S0) and skew symmetric (A0) modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement. [Copyright &y& Elsevier]

Transient Lamb waves in a thin copper plate were generated and measured using a laser ultrasonic setup. A two-dimensional Fourier transform method has been employed to analyse the dispersion curves. This allowed a clear identification of multi-mode Lamb waves. Fitting of the dispersion curve allowed a direct determination of the thickness and bulk velocities, as well as the elastic constants. The obtained experimental results of Lamb waves on a copper plate show excellent agreement with the theoretical model. [Copyright &y& Elsevier]

The scattering problem of an incident plane S0 Lamb wave in a plate with a circular partly through-thickness hole is considered in this paper. A 3D approach is used where the wave fields in the outer part outside the hole and in the inner part beneath the hole are expanded in the possible Lamb modes and the horizontally polarized shear modes. Both propagating modes and evanescent modes are included in the expansions. The expansion coefficients are obtained by utilizing the boundary conditions at the hole boundary and the continuity conditions below the hole. A linear system of equations for the expansion coefficients is obtained by projecting the different boundary conditions onto an orthogonal set of projection functions. A second approach is also presented, mainly for validation of the 3D model. This approach is based on the lowest order plate theories for extensional and flexural motion. Consequently, this model is only valid for very low frequencies. The two plate theories are combined and the wave fields in the different regions of the plate are obtained from the boundary conditions at the hole. Comparisons are made between the two different models for low frequencies and they show good agreement. A few examples are also given from the 3D model for higher frequencies where the plate theory model is not valid. [Copyright &y& Elsevier]

This article deals with the dynamic response of laterally unbounded, horizontally layered plates subjected to dynamic sources applied at arbitrary locations, which is ultimately a classical problem. This response is most often obtained via a modal superposition in terms of the complex Lamb modes by casting the equations in the frequency–space domain, followed by a Fourier inversion into the space–time domain. Then again, a much less often used alternative method relies on formulating the problem directly in the time domain in terms of a modal superposition in the wavenumber domain, which is followed by a Fourier inversion into the space–domain, as considered in further detail herein. This alternative can offer powerful advantages in some cases, such as dealing easily with anisotropy, or with slowly propagating waves even in the absence of damping. At the same time, however, it is beset by difficulties associated with the Fourier inversion into the space–domain. As a matter of fact, during forcing, the truncation of the modal series and of the numerical integrals is hindered by poorly convergent behaviors. Here we overcome both of these difficulties by considering the asymptotic, static behavior of the integrands. We find that by introducing a regularizing term, the response can be effectively separated into near and far fields, despite the fact that these frequency-domain concepts are alien to a wavenumber–time formulation. The so-defined far field is free of sharp variations and can then be computed in a numerical grid that is optimized regarding the propagating wavelengths, which only depend on the time-spectral content of the excitation and not on its space-spectrum. Finally, we also propose a hybrid way to compute the remaining near field by combining with a non-modal formulation, expressed in the wavenumber–Laplace domain. • A regularizing tensor is introduced. • It enables an effective separation into near and far fields in the wavenumber–time domain. • The established regime can be sampled optimally. • The singular (near) field can by computed in a separate and small grid. [ABSTRACT FROM AUTHOR]

This work considers the propagation of bending waves along the free edge of a semi-infinite piezoelectric plate perfectly bonded with a metal strip plate of the same thickness within the framework of the first-order Reissner–Mindlin refined plate theory. The three-dimensional coupled equations for displacement fields are solved with the help of the solution to electric potential for a plate of infinite extent. The exact dispersion relation of bending edge waves is obtained. The propagation of bending edge waves in such a composite plate demonstrates multi-mode due to the presence of the metal strip plate. This wave has properties analogous to the surface wave in an elastic half-space covered by a thin layer within the theory of plane strain. The number of propagation modes depends on the width–thickness ratio of the metal plate and increases as this width–thickness ratio increases. The phase velocity of the bending edge wave in a piezoelectric composite plate is much lower than that of pure piezoelectric plate. Effect of the dimension of the metal strip plate on dispersion curves of bending edge waves is limited. • Bending edge wave in piezoelectric composite plate demonstrates multi-mode propagation. • The number of propagation modes depends on the width-thickness ratio of metal plate. • The edge wave speed is much lower than that of pure piezoelectric plate. • Effect of the dimension of the metal strip plate on phase velocity is limited. [ABSTRACT FROM AUTHOR]