Modal displacement analyses of Lamb waves in micro/nano-plates based on the consistent couple stress theory

In micro-scale structures, the influence of scale effects on the mechanical properties cannot be ignored, and the continuum mechanics theory cannot effectively describe guided wave propagation. This paper studies modal displacements in an isotropic micro/nano-scale plate - an essential factor for the mode selection of Lamb waves in non-destructive testing technology. To achieve this, the consistent couple stress theory, which additionally accounts for the impact of scale effects on wave propagation, has to be employed. A mathematical derivation method with singular value decomposition is proposed, and the excitation amplitude is computed by applying a normal stress source to evaluate the in-plane and out-of-plane displacements. The results indicate that the in-plane displacement vanishes on the plate surface for anti-symmetric modes when the phase velocity equals the velocity of shear waves. Moreover, the intersections of symmetric and anti-symmetric dispersion curves identify specific frequencies at which the out-of-plane displacement vanishes on the free surface. These findings highlight the difference in modal displacement characteristics between micro-scale and macro-scale structures. The results from this study may furnish theoretical guidance for ultrasonic non-destructive testing of microstructures.

Notes on osculations and mode tracing in semi-analytical waveguide modeling

The dispersion curves of (elastic) waveguides frequently exhibit crossings and osculations (also known as veering, repulsion, or avoided crossing). Osculations are regions in the dispersion diagram where curves approach each other arbitrarily closely without ever crossing before veering apart. In semi-analytical (undamped) waveguide models, dispersion curves are obtained as solutions to discretized parameterized Hermitian eigenvalue problems. In the mathematical literature, it is known that such eigencurves can exhibit crossing points only if the corresponding matrix flow (parameter-dependent matrix) is uniformly decomposable. We discuss the implications for the solution of the waveguide problem. In particular, we make use of a simple algorithm recently suggested in the literature for decomposing matrix flows. We also employ a method for mode tracing based on approximating the eigenvalue problem for individual modes by an ordinary differential equation that can be solved by standard procedures.

Ultrasonic guided wave measurement in a wooden rod using shear transducer arrays

Research related to acoustic/ultrasonic guided wave testing in wood is still at an early stage. This paper describes the first study to perform ultrasonic guided wave measurements in a wooden rod using arrays of shear transducers. Enhancement of either longitudinal L(0,1) or torsional T(0,1) wave modes and suppression of other modes was able to be achieved using these arrays. At low frequencies, it was found that the L(0,1) wave mode had a similar speed to that obtained using the traditional resonance and time of flight methods. The torsional T(0,1) wave mode has not been used before for non-destructive testing of wood. Since it is non-dispersive, it would appear to be suitable for wood property estimation and structural health monitoring of wooden structures. These results indicate that ultrasonic guided wave testing techniques have strong potential to be used to provide improved measurement of wood properties and structural health monitoring of wooden structures.

Identification of long-range ultrasonic guided wave characteristics in cortical bone by modelling

The propagation of ultrasonic guided waves in cortical bone has potential to inform medical caregivers about the condition of the bone structure. However, as waveguides, human long bones such as the tibia are complex in terms of their material behavior and their geometric features. They exhibit anisotropic elasticity and internal damping. For the first time, wave propagation is modelled in the irregular hollow tibial cross-section, which varies along its long axis. Semi-analytical, frequency domain, and time domain finite element analyses providing complimentary information about long-range wave propagation characteristics in such a waveguide are applied to the mid-diaphyseal region of a human tibia. Simulating the guided waves generated by a contact transducer, the signals received in axial transmission indicate the consistent presence of low phase velocity non-dispersive propagating modes. The guided waves capable of traveling long distances have strong potential for diagnosis of fracture healing.

Modeling and numerical study of the influence of imperfect interface properties on the reflection coefficient for isotropic multilayered structures

The microelectronics industry is expressing an increased demand for the development of non-destructive tools and methods for health control and diagnostics in multilayered structures. The purpose of these tools is to detect problems such as delaminations, inclusions and microcracks. The aim of this paper is to study the effect of imperfect interfaces on the wave propagation in multilayered structures. This type of structure represents the typical architecture of many microelectronic components. This study will be based on the calculation of the reflection coefficient and the guided waves dispersion curves. The investigated structure is an isotropic trilayer where two metallic layers are bonded together by an adhesive layer made of an epoxy resin. Comparisons were performed in order to evaluate numerically the influence of several properties of the adhesive layer on the guided waves behavior. In addition, an imperfect viscoelastic interface layer model [1] has been implemented in order to simulate different adherence qualities between the metallic layers.

Modeling guided wave propagation in functionally graded plates by state-vector formalism and the Legendre polynomial method

A numerical method is presented for the investigation of the propagation characteristic of guided waves in functionally gradient material (FGM) plates. Based on the State-vector formalism and Legendre polynomial method, the typical non-stratified computing of dispersion curves of FGMs is realized, by introducing the univariate nonlinear regression to optimize the arbitrary gradient distribution of material component. Comparing with the conventional Matrix method, the proposed method avoids the exhausting root-locating algorithm of solving the transcendental equation by a single-variable scanning process. This method turns it into an algebraic eigenvalue problem, which mainly depends on the orthogonal completeness and strong recursive property of Legendre polynomial series. It provides a fast and flexible approach to extracting the dispersion curves, displacement distribution and stress profile, simultaneously. Results from chrome-ceramic FGM plate are compared with those from the previous articles to confirm the feasibility and accuracy of the proposed method. Then, this approach is further applied to iron based alumina FGM. The dispersion curves with different gradient function are calculated to illustrate the influence of the gradient variation. Moreover, the influence of the cut-off order of Legendre orthogonal polynomials on the convergence of dispersion curves is also revealed through numerical examples. Utilizing the mapping relationship between the gradient distribution and the propagation characteristics, it gives theoretical support for nondestructive evaluation and quantitative estimation of the structural characteristics of FGM plates.

Dispersion curve estimation via a spatial covariance method with ultrasonic wavefield imaging

Numerous Lamb wave dispersion curve estimation methods have been developed to support damage detection and localization strategies in non-destructive evaluation/structural health monitoring (NDE/SHM) applications. In this paper, the covariance matrix is used to extract features from an ultrasonic wavefield imaging (UWI) scan in order to estimate the phase and group velocities of S0 and A0 modes. A laser ultrasonic interrogation method based on a Q-switched laser scanning system was used to interrogate full-field ultrasonic signals in a 2-mm aluminum plate at five different frequencies. These full-field ultrasonic signals were processed in three-dimensional space-time domain. Then, the time-dependent covariance matrices of the UWI were obtained based on the vector variables in Cartesian and polar coordinate spaces for all time samples. A spatial covariance map was constructed to show spatial correlations within the full wavefield. It was observed that the variances may be used as a feature for S0 and A0 mode properties. The phase velocity and the group velocity were found using a variance map and an enveloped variance map, respectively, at five different frequencies. This facilitated the estimation of Lamb wave dispersion curves. The estimated dispersion curves of the S0 and A0 modes showed good agreement with the theoretical dispersion curves.

Extraction of guided wave dispersion curve in isotropic and anisotropic materials by Matrix Pencil method

Guided wave dispersion curves in isotropic and anisotropic materials are extracted automatically from measured data by Matrix Pencil (MP) method investigating through k-t or x-ω domain with a broadband signal. A piezoelectric wafer emits a broadband excitation, linear chirp signal to generate guided waves in the plate. The propagating waves are measured at discrete locations along the lines for one-dimensional laser Doppler vibrometer (1-D LDV). Measurements are first Fourier transformed into either wavenumber-time k-t domain or space-frequency x-ω domain. MP method is then employed to extract the dispersion curves explicitly associated with different wave modes. In addition, the phase and group velocity are deduced by the relations between wavenumbers and frequencies. In this research, the inspections for dispersion relations on an aluminum plate by MP method from k-t or x-ω domain are demonstrated and compared with two-dimensional Fourier transform (2-D FFT). Other experiments on a thicker aluminum plate for higher modes and a composite plate are analyzed by MP method. Extracted relations of composite plate are confirmed by three-dimensional (3-D) theoretical curves computed numerically. The results explain that the MP method not only shows more accuracy for distinguishing the dispersion curves on isotropic material, but also obtains good agreements with theoretical curves on anisotropic and laminated materials.

Non-contact ultrasonic technique for Lamb wave characterization in composite plates

A fully non-contact single-sided air-coupled and laser ultrasonic non-destructive system based on the generation and detection of Lamb waves is implemented for the characterization of A0 Lamb wave mode dispersion in a composite plate. An air-coupled transducer (ACT) radiates acoustic pressure on the surface of the composite and generates Lamb waves within the structure. The out-of-plane velocity of the propagating wave is measured using a laser Doppler vibrometer (LDV). In this study, the non-contact automated system focuses on measuring A0 mode frequency-wavenumber, phase velocity dispersion curves using Snell's law and group velocity dispersion curves using Morlet wavelet transform (MWT) based on time-of-flight along different wave propagation directions. It is theoretically demonstrated that Snell's law represents a direct link between the phase velocity of the generated Lamb wave mode and the coincidence angle of the ACT. Using Snell's law and MWT, the former three dispersion curves of the A0 mode are easily and promptly generated from a set of measurements obtained from a rapid ACT angle scan experiment. In addition, the phase velocity and group velocity polar characteristic wave curves are also computed to analyze experimentally the angular dependency of Lamb wave propagation. In comparison with the results from the theory, it is confirmed that using the ACT/LDV system and implementing simple Snell's law method is highly sensitive and effective in characterizing the dispersion curves of Lamb waves in composite structures as well as its angular dependency.

The properties of thickness-twist (TT) wave modes in a rotated Y-cut quartz plate with a functionally graded material top layer

We propose the use of thickness-twist (TT) wave modes of an AT-cut quartz crystal plate resonator for measurement of material parameters, such as stiffness, density and material gradient, of a functionally graded material (FGM) layer on its surface, whose material property varies exponentially in thickness direction. A theoretical analysis of dispersion relations for TT waves is presented using Mindlin's plate theory, with displacement mode shapes plotted, and the existence of face-shear (FS) wave modes discussed. Through numerical examples, the effects of material parameters (stiffness, density and material gradient) on dispersion curves, cutoff frequencies and mode shapes are thoroughly examined, which can act as a theoretical reference for measurements of unknown properties of FGM layer.

Continuous leakage location in noisy environment using modal and wavelet analysis with one AE sensor

A noble method for continuous leakage source location with one sensor in gas-filled pipe and noisy environment is proposed based on wavelet analysis and modal location theory. The leakage signals were analyzed into high and low frequencies by wavelet decomposition and noises and reflected waves were omitted. Then the processed signals were reconstructed. A frequency range of 0-250kHz was selected using wavelet packet decomposition to analyze the flexural and extensional modes of leakage signals. In this frequency range, the peak magnitudes of wavelet transform could determine the arrival times of different modes. The frequencies which the peak magnitudes of wavelet transform occurred per them were chosen to calculate the group velocities. Experiments were carried out and the locating results show the high precision of this algorithm.

A rapid, fully non-contact, hybrid system for generating Lamb wave dispersion curves

A rapid, fully non-contact, hybrid system which encompasses an air-coupled transducer (ACT) and a laser Doppler vibrometer (LDV) is presented for profiling A0 Lamb wave dispersion of an isotropic aluminum plate. The ACT generates ultrasonic pressure incident upon the surface of the plate. The pressure waves are partially refracted into the plate. The LDV is employed to measure the out-of-plane velocity of the excited Lamb wave mode at some distances where the Lamb waves are formed in the plate. The influence of the ACT angle of incidence on Lamb wave excitation is investigated and Snell's law is used to directly compute Lamb wave dispersion curves including phase and group velocity dispersion curves in aluminum plates from incident angles found to generate optimal A0 Lamb wave mode. The measured curves are compared to results obtained from a two-dimensional (2-D) Fast Fourier transform (FFT), Morlet wavelet transform (MWT) and theoretical predictions. It was concluded that the experimental results obtained using Snell's law concept are well in accordance with the theoretical solutions. The high degree of accuracy in the measured data with the theoretical results proved a high sensitivity of the air-coupled and laser ultrasound in characterizing Lamb wave dispersion in plate-like structures. The proposed non-contact hybrid system can effectively characterize the dispersive relation without knowledge of neither the materials characteristics nor the mathematical model.

Guided waves dispersion equations for orthotropic multilayered pipes solved using standard finite elements code

The dispersion curves for hollow multilayered cylinders are prerequisites in any practical guided waves application on such structures. The equations for homogeneous isotropic materials have been established more than 120 years ago. The difficulties in finding numerical solutions to analytic expressions remain considerable, especially if the materials are orthotropic visco-elastic as in the composites used for pipes in the last decades. Among other numerical techniques, the semi-analytical finite elements method has proven its capability of solving this problem. Two possibilities exist to model a finite elements eigenvalue problem: a two-dimensional cross-section model of the pipe or a radial segment model, intersecting the layers between the inner and the outer radius of the pipe. The last possibility is here adopted and distinct differential problems are deduced for longitudinal L(0,n), torsional T(0,n) and flexural F(m,n) modes. Eigenvalue problems are deduced for the three modes classes, offering explicit forms of each coefficient for the matrices used in an available general purpose finite elements code. Comparisons with existing solutions for pipes filled with non-linear viscoelastic fluid or visco-elastic coatings as well as for a fully orthotropic hollow cylinder are all proving the reliability and ease of use of this method.