Remote characterization of defects in plates with viscoelastic coatings using guided waves

The remote inspection for defects in large metallic elements such as pipes, tubes and plates is a field where guided waves are being applied with success. There are a number of situations where a surface coating is added for corrosion protection or insulation purposes. Since the coating materials are usually viscoelastic, the guided wave inspection range may be severely reduced unless a proper mode and an adequate frequency range is selected. Previous authors found the existence of low-attenuated modes at certain frequency ranges, which were used to detect and locate defects at reasonably large distances. This paper studies the potential of guided waves for not only locating but also sizing defects in plates with viscoelastic coatings. A hybrid finite element-boundary element method which explicitly includes the attenuating characteristics of the coating is used to determine Lamb and SH mode conversion factors at corrosion defects under the coating. Through parametric studies and analysis of the numerical results, some trends and features are highlighted that could be used for sizing purposes.     

Reflection and transmission coefficients for diffuse photon density waves

Diffuse photon density waves are proposed for use in characterizing diffuse media, similar to the use of electromagnetic waves to characterize reflection from materials. To this end basic expressions for the reflection and transmission coefficients at interfaces are put forward. Effects such as the existence of modes analogous with Brewster modes are shown. Also, we illustrate through numerical simulations how the diffusion optical parameters are estimated both with and without noise in the data.     

Three-dimensional hybrid model for predicting air-coupled generation of guided waves in composite material plates

For contact-less, non-destructive testing (NDT) purposes using air-coupled ultrasonic transducers, it is often required to numerically simulate the propagation of ultrasonic waves in solid media, and their coupling through air with specific transducers. At that point, one could simulate the propagation in the air and then in the solid component, using a Finite Element (FE) model. However, when three-dimensional (3D) modeling becomes necessary, such a solution reveals to be extremely demanding in terms of number of degrees of freedom and computational time. In this paper, to avoid such difficulties, the propagation in air from an ultrasonic transmitter to a tested solid plate is modeled in 3D using a closed-form solution. The knowledge of the transducer characteristics (diameter, frequency bandwidth, efficiency in Pa/V) allows the spatial distribution and actual pressure (in Pa) of the acoustic field produced in the air to be predicted, for a given input voltage. This pressure field is applied in turn as a boundary condition in a 3D FE model, to predict the plate response (displacement and stress guided beams) for a given distance between the transmitter and the plate, and for a given angle of orientation of the transmitter with respect to the plate. The FE model is so restricted to modeling of the solid structure only, thus reducing very significantly the number of degrees of freedom and computational time. The material constituting the plate is considered to be an anisotropic and viscoelastic medium. To validate the whole modeling process, an air-coupled ultrasonic transducer is used and oriented at a specific angle chosen for generating one specific Lamb mode guided along a composite plate sample, and a laser probe measures the normal velocity at different locations on the surface of the plate. In the field of NDT, it is generally suitable to excite a pure Lamb mode in order to ease the interpretation of received signals that would represent waves scattered by defects. After a validation step, the numerical model is then used to investigate the effect of the material anisotropy on the purity of the incident guided mode.     

Nonlinear guided waves in plates: A numerical perspective

Harmonic generation from non-cumulative fundamental symmetric (_S0$S_0$) and antisymmetric (_A0$A_0$) modes in plate is studied from a numerical standpoint. The contribution to harmonic generation from material nonlinearity is shown to be larger than that from geometric nonlinearity. Also, increasing the magnitude of the higher order elastic constants increases the amplitude of second harmonics. Second harmonic generation from non-phase-matched modes illustrates that group velocity matching is not a necessary condition for harmonic generation. Additionally, harmonic generation from primary mode is continuous and once generated, higher harmonics propagate independently. Lastly, the phenomenon of mode-interaction to generate sum and difference frequencies is demonstrated.     

Modeling laser-generated guided waves in bonded plates by the finite element method

The purpose of this research is to develop an effective model of the transient laser-generated guided waves in bonded plates by the finite element (FE) method. The FE parameters, such as the mesh density and the time step size, which are related to wave propagation, are optimized in order to set up a standard, and the correlation between these parameters is discussed. The waveforms in bonded plates with different bonded interface modeled by the spring model are obtained and then compared. The results show that the transient responses are sensitive to the stiffness coefficients characterizing the cohesive quality. Finally, these FE results are compared with analytical and experimental results. All these comparisons confirm the accuracy of the FE method for modeling laser-generated guided waves in bonded plates.     

The effect on bending waves by defects in pinned elastic plates

This paper presents solutions to a number of problems posed for the out-of-plane displacement of infinite thin elastic plates that are rigidly pinned in periodic configurations, but that possess a finite number of ‘defects’. We begin by considering a single one-dimensional periodic array of pins. We derive an analytic solution for the displacement produced by the forced oscillation of the central pin in the array, and this solution is shown to be closely connected to the problem of scattering of plane waves by an array when a finite number of pins are removed. Attention then focuses on doubly periodic rectangular arrays of pinned points possessing defects. Central to approaching such problems is an understanding of Bloch–Floquet waves in periodic arrays in the absence of defects and a simple method is described for computing the associated dispersion surfaces. The solutions to three problems are then sought: the trapping of localised waves by a finite number of missing pins; trapping of waves by entire rows of missing pins; and the wave radiation pattern due to the forcing of a single pin. All problems are treated analytically using bounded Green's functions for thin elastic plates, a discrete Fourier transform solution method and simple, explicit and rapidly convergent evaluations of the one- and two-dimensional lattice sums that arise.     

分层固体板中导波的激发与频散特性

针对无限大弹性分层固体板,研究了结构中导波的频散和激发特性。首先使用传递矩阵法推导分层板模型中导波的频散方程,然后用二分法求取导波各模式的频散曲线,进而分析结构中导波的频散特性。结果表明:在速度递增或递减的分层板中,基阶模式和高阶模式的高频极限分别等于低速层的瑞利波速和横波波速。对于含低速夹层的分层板,所有模式的高频极限都等于低速层的横波速度。在导波激发特性方面,研究了在具有一定宽度的法向力源作用下的分层板中导波各模式在结构中的法向位移谱。发现在速度递增的分层板结构中基阶模式是主导模式,而对于速度递减和含低速夹层模型,主导模式在不同的频段范围内对应不同的导波模式。     

Research on the propagation of guided waves in multi-layered plates

The dispersion characteristics and excitation mechanisms of the guided waves in multilayered plates are studied in this paper. Firstly, the dispersion equation is obtained by the propagator matrix method. Then, the bisection technique is employed to find all the roots of the dispersion equation. The dispersion characteristics of the guided waves are investigated and analyzed. For the multilayered plates in which the S-wave velocity increases or decreases from top to bottom, it is found that the velocity limits in high frequency of the first and high modes are equal to the Rayleigh wave and S-wave velocities of the low-velocity layer, respectively. It is also found that the velocity limits in the high frequency of all modes are equal to the S-wave velocity of the low-velocity layer for the plate with a low-velocity middle layer. The normal displacement spectrum of all the modes excited by the normal force source with a definite width on the surface of the plate is also investigated. It is proved that the dominant mode is the first mode when the S-wave velocity increases from top to bottom layer and the dominant mode is different in different frequency range for the plate with a low-velocity middle layer.     

Frequency dependent directivity of guided waves excited by circular transducers in anisotropic composite plates

Lamb wave propagation in fiber-reinforced composite plates is featured by a pronounced directivity of wave energy transfer along the fibers from a point surface source. In the case of non-point (sized) source, the main lobe of radiation diagram may turn with frequency up to the orthogonal to the fibers direction. This effect has been theoretically studied and physically explained in the context of semi-analytical integral-equation based mathematical model. The present paper gives its experimental verification.     

Reflection and transmission of waves in pyroelectric and piezoelectric materials

The reflection and transmission theories of waves in pyroelectric and piezoelectric medium are studied in this paper. In general in an infinite homogeneous pyroelectric medium there are four bulk wave modes: quasi-longitudinal, two quasi-transversal and temperature waves. In an infinite homogeneous piezoelectric medium there are three bulk wave modes: quasi-longitudinal and two quasi-transversal waves. In the reflection and transmission problem there are five complex boundary conditions in the pyroelectric medium and four complex boundary conditions for the piezoelectric medium. In this paper, we find that the surface waves will be revealed in the reflection and transmission wave problem. The surface waves have the same wave vector component with the incident waves on the interface plane. The two dimensional reflection problem of waves at the interface between the semi-infinite pyroelectric medium and vacuum is researched in greater detail and a numerical example is given.     

Phased array focusing with guided waves in a viscoelastic coated hollow cylinder

Guided wave phased array focusing has shown many advantages in long-range pipeline inspection, such as, longer inspection distance, greater wave penetration power and higher detection resolution. Viscoelastic coatings applied to a large percentage of pipes for protection purposes created some challenges in terms of focusing feasibility and inspection ability. Previous studies were all based on bare pipe models. In this work, guided wave phased array focusing in viscoelastic coated pipes is studied for the first time. Work was carried out with both numerical and experimental methods. A three-dimensional finite element model was developed for quantitatively and systematically modeling guided waves in pipes with different viscoelastic materials. A method of transforming measured coating properties to finite element method inputs was created in order to create a physically based model of guided waves in coated pipes. Guided wave focusing possibilities in viscoelastic coated pipes and the effects from coatings were comprehensively studied afterwards. A comparison of focusing and nonfocusing inspections was also studied quantitatively in coated pipe showing that focusing increased the wave energy and consequently the inspection ability tremendously. This study provides an important base line and guidance for guided wave propagation and focusing in a real field pipeline under various coating and environmental conditions.     

Reflection and transmission of circularly polarized waves in nonlinear dielectrics

A time-sinusoidal circularly polarized plane standing wave solution is obtained for isotropic lossless dielectric media with arbitrary nonlinearity. The spatial variation of the standing wave depends on the nonlinearity and is found by solving a problem of centralforce motion in which the electric complementary energy density furnishes the potential and the spatial coordinate has the role of time. Two special nonlinear dielectric response laws-cubic and quintic-are treated, and explicit solutions for the spatial variation of the wave amplitude and phase are obtained in terms of elliptic functions and elliptic integrals, respectively. The standing-wave solution is applied to give steady state solutions of two reflection/transmission problems (a) reflection from, or resonance modes between, ideally conducting planes and (b) reflection and transmission at a plane interface between two nonlinear dielectric media.     

Reflection and transmission of waves near the localization threshold

A theory is presented for propagation of waves in bounded media near the mobility edge, based on the self-consistent theory for localization. It predicts a spatially inhomogeneous diffusion constant that leads to scale dependence in enhanced backscattering and transmission.     

Nonlinear strain waves and densities of defects induced in metal plates by external energy fluxes

A model of nonlinear periodic traveling strain waves induced in laser-irradiated metal plates with quadratic nonlinearity is proposed. Interaction between the elastic strain fields and the concentration of point defects is taken into account. The effect of generation-recombination processes on the evolution of nonlinear localized waves is considered. An equation for the amplitudes of the nonlinear waves is derived. It is employed to analyze the attenuation of the waves with allowance for low-and high-frequency losses.     

A study of time harmonic guided Lamb waves and their caustics in composite plates

Spatial steady-state Lamb wave propagation in an anisotropic composite plate excited by harmonic surface sources is modeled using a Green’s matrix representation in a frequency-wavenumber domain. An approach based on a residue integration technique for two dimensional wavenumber integrals for the computation of displacements outside an excitation source is presented in this paper. In the far-field zone of the excitation source, the method of stationary phase is used, which gives an asymptotic expansion of the displacement vector as a sum of cylindrical waves. Near caustic directions, a far-field solution is computed in terms of Airy functions. The results obtained applying residue integration technique and asymptotic expansion are found to be coinciding with the results of the computation by using the adaptive quadratures. Moreover, these approaches agree well with experimental data. Then, the advantages and disadvantages of the various methods applied for modeling of Lamb wave propagation are discussed in this paper. Focussing and other properties of Lamb waves are studied using numerical examples.     

Finite element simulation of the generation and detection by air-coupled transducers of guided waves in viscoelastic and anisotropic materials

The measured characteristics (efficiency and sensitivity) of two air-coupled transducers allow for the prediction of the absolute values of the pressure of the bulk waves generated in air and for the measurement of the pressure of the field radiated in air by guided waves propagating in a structure. With finite element software, the pressure field generated by an air-coupled transducer is simulated by introducing a right-hand side member in the Helmholtz equation, which is used for computing the propagation from the transducer to a plate. The simulated source is rotated in order to impose an angle of incidence with respect to the normal of the plate and generate the corresponding guided mode. Inside the plate, the propagation is simulated with the dynamic equations of equilibrium and a complex stiffness tensor to take into account the viscoelastic anisotropy of the material. For modeling the three-dimensional fields of the guided modes propagating in a two-dimensional non-symmetry plane, a 2.5 dimensional model is introduced. The model computes the value of the pressure field radiated in air by the plates for any guided modes and can predict the detectability of the system for a known defect in a structure. A test bed incorporating two air-coupled transducers is used to generate and receive various guided modes. Two plates made of Perspex and carbon-epoxy composite are tested. The pressure measured by the receiver at various positions is compared to the results of the model to validate it.