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Guided time-harmonic shear horizontal SH waves propagating in the circumferential direction of an isotropic hollow cylinder are studied. The dispersion equation as well as the displacement and stress field across the wall thickness is derived analytically. Compared with the SH waves in a plate, a quantitative guideline of how well a plate model can...
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... cracks and corrosion defects are often found in both industrial and military hollow cylindrical structures like pipelines and cylindrical containers. Reliable and easy-to-use inspection systems are in great need to locate the defects and to be able to characterize and size them efficiently. Hirao and Ogi 1 proposed a circumferential SH-wave Electromagnetic Acoustic Transducer ͑ EMAT ͒ technique for detecting corrosion defects on the outer surface of steel pipelines with and without protective resin coating. Gauthier 2 used multimode SH waves generated by EMATs to form B-scan images of a defect on a pipe. The reflection and transmission coefficients of SH waves passing through a two-dimensional surface- breaking defect or a stringer-like internal inclusion in a pipe was reported by the present authors. 3 However, all of these publications were based on an empirical plate-model approximation for a pipe of large diameter-to-wall-thickness ratio. A rigorous theory of guided SH waves propagating in the circumferential direction of a hollow cylinder needs to be established. 4 Gazis 5,6 theoretically investigated guided waves that propagate in the axial direction and are resonant in the circumferential direction of a hollow cylinder. The case when the axial wave number is zero decouples into axially motion- independent plane–strain vibration and longitudinal shear vibration, both of which are standing waves in the circumferential direction. Liu and Qu 7 developed the model of guided plain–strain waves propagating in the circumferential direction of a hollow cylinder, with emphasis on the dispersion relation and displacement profile derivation and discussion. In this paper, the guided time-harmonic SH wave propagating in the circumferential direction of a hollow cylinder is studied. The dispersion equation as well as the displacement and stress distribution across the wall of the hollow cylinder is derived analytically. They are compared with that of the SH waves in a plate of the same thickness numerically. Rela- tive errors of the phase and group velocities when using a plate model to approximate a pipe are given as a quantitative measure of whether that approximation is valid. With the approximated plate model, the guided wave interaction and scattering from a three-dimensional defect in a hollow cylindrical structure of large diameter-to-wall-thickness ratio could be tackled without extensive eigenmode and wave structure calculations in each scattering direction. 8 Consider steady-state time-harmonic waves propagating in the circumferential direction of a hollow cylinder of inner radius a and outer radius b , as shown in Fig. 1. Assume the material is linearly elastic and isotropic, and the wave motion is independent of z . Two types of guided waves are possible in this hollow cylinder: one is the plane–strain vibration wave similar to the Lamb wave in a plate; 7 the other is the longitudinal shear wave, which will be studied in detail in this paper. The analytical derivation of the frequency equation of the guided circumferential SH waves begins with Navier’s equation of motion as Eq. ͑ 1 ͒ , where u is the displacement vector, is the density, and are Lam ́ ’s constants, respectively ٌ 2 u ϩ ͑ ϩ ͒ ٌٌ • u ϭ ͑ ץ 2 u / ץ t 2 ͒ . ͑ 1 ͒ Consider only the displacement in the z direction in the cylindrical coordinates, i.e., u r ϭ u ϭ 0 and u z 0; Eq. ͑ 1 ͒ can be written ...
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Citations
... This analytical expression is consistent with the SH wave dispersion equation for a plate. [22] It should be noted that the pipe-to-plate approximation is reasonable only when the thickness to outer diameter ratio of the pipe, i.e., the standard dimension ratio (SDR), is less than 10%-15%. [23] 3. Inversion formulas for thickness ...
Thickness measurement plays an important role in the monitoring of pipeline corrosion damage. However, the requirement for prior knowledge of the shear wave velocity in the pipeline material for popular ultrasonic thickness measurement limits its widespread application. This paper proposes a method that utilizes cylindrical shear horizontal (SH) guided waves to estimate pipeline thickness without prior knowledge of shear wave velocity. The inversion formulas are derived from the dispersion of higher-order modes with the high-frequency approximation. The waveform of the example problems is simulated using the real-axis integral method. The data points on the dispersion curves are processed in the frequency domain using the wave-number method. These extracted data are then substituted into the derived formulas. The results verify that employing higher-order SH guided waves for the evaluation of thickness and shear wave velocity yields less than 1% error. This method can be applied to both metallic and non-metallic pipelines, thus opening new possibilities for health monitoring of pipeline structures.
... hollow cylinders the wave equation of which were given by Liu [22] and Zhao [23], respectively. Dispersion and multimodality are key characteristics of wave propagation in pipelines, governed by the frequency-dependent nature of guided waves in waveguides. ...
... Monitoring of mechanical changes due to pitting, cracking, material conversion from corrosion and/or erosion, and material addition from material migration and accumulation in pipes, especially in hard-to-access environments [1], such as under insulation or paint, is important in many industries that involve liquid or gas storage and flow [2,3]. Commercialguided wave systems show a sensitivity to changes of 5% or above of cross-sections [4,5]. ...
Ultrasonic testing (UT) has been widely used for the Nondestructive Evaluation (NDE) of pipes due to its many favorable characteristics. However, one of the main challenges in the general use of UT for real-world pipelines is the sensitivity of this method to environmental and operational condition changes. This paper proposes a new UT method with enhanced compensation for environmental effects and operational condition changes. In particular, the effectiveness of the new method is tested in the presence of temperature variations, and changes in water flow rate inside a stainless-steel pipe. The proposed UT method uses multi-mode and broadband guided ultrasonic waves in the pipe walls, excited and received by single-element ultrasonic sensors that are spatially separated, forming a measurement zone between any pair of such transmit and receive sensors. Amplitude changes, time shifts, and frequency content variations in the ultrasonic signal due to temperature changes and water flow are evaluated and compensated for reliable UT of mechanical changes in the pipe. It is observed that spurious effects of water flow on ultrasonic response, if not properly compensated, can dominate over effects due to actual mechanical changes, but such liquid-boundary effects can be compensated effectively by the proposed time- and frequency-filtering method.
... Circumferential guided-wave-based method is especially suitable for monitoring large-diameter pipes such as pressure vessels and oil tanks, while for long range pipes, it can be used to inspect the risk areas such as the welding zones and pipe support regions. Among the circumferential guided wave modes in pipes, the fundamental circumferential shear horizontal (CSH0) wave is an appealing wave mode to develop phased array systems, since it is quasi-nondispersive [35]. However, there appears no attempt to develop a CSH0 wave phased array, which is mainly attributed to the difficulty of generating pure CSH0 wave. ...
Torsional or longitudinal wave-based inspection is an effective method for rapid defect screening in pipes, but it has difficulty to locate the circumferential position of a defect due to the axially symmetric properties of torsional and longitudinal modes. In order to overcome this problem, several flexural-wave-based focusing methods have been proposed while multiple mode propagation associated with dispersion has to be taken into account in them, which increases the complexity of identifying defects in practical applications. In this work, based on our recently developed bidirectional SH wave piezoelectric transducer (BSH-PT), a fundamental circumferential shear horizontal (CSH0) wave linear phased array was proposed for in-situ defect localization in large-diameter pipes. Firstly, the working principle of the BSH-PT array was presented and its performance on defects detection was assessed by simulations using the total focusing method (TFM). Then, experiments were conducted to examine the array’s performance. Results showed that the proposed BSH-PT array can detect multiple defects simultaneously at varied frequencies. Finally, a method for eliminating the mirrored artifact was proposed and then validated by both simulations and experiments on pipes. Considering the quasi-nondispersive character of CSH0 wave and the desirable performance of the BSH-PT on exciting and receiving pure CSH0 wave, the proposed CSH0 wave phased array may be useful in structural health monitoring (SHM) of large-diameter pipes such as pressure vessels and oil tanks.
... where p is the angular wavenumber with respect to the propagation direction, θ, ω is the angular frequency, n is the mode order, A n is the amplitude of each mode, and U n is the displacement profile along the r-coordinate for each mode. The exact dispersion relationship and mode-profile for CSH waves depend on the pipe's thickness and outer to inner radius ratio [13].The propagation of CSH guided waves is more complex than in the plate, due to its solutions containing Bessel functions. The proof for these solutions will not be presented in this work in detail and can be found in the literature [7], [12], [13], but in general the characteristic equation for a single layer annulus is given the frequencywavenumber pairs that nullify the determinant of the following matrix: ...
... The exact dispersion relationship and mode-profile for CSH waves depend on the pipe's thickness and outer to inner radius ratio [13].The propagation of CSH guided waves is more complex than in the plate, due to its solutions containing Bessel functions. The proof for these solutions will not be presented in this work in detail and can be found in the literature [7], [12], [13], but in general the characteristic equation for a single layer annulus is given the frequencywavenumber pairs that nullify the determinant of the following matrix: ...
... In spite of difficulties in generating the SH wave [11], theoretical studies and SH guided wave applications have received increasing interest. Various analytical and numerical studies about the characteristics of SH guided waves have been reported [12][13][14][15][16], and most of them search for numerical solutions. Based on the Peano-series method, a theoretical formulation for shear-horizontal waves in the PZT-5H piezoelectric plate has been presented [17]. ...
In this paper, the propagation of transversally horizontal waves (SH) propagating in a piezoelectric plate is investigated. The dispersion curve is obtained numerically in the frequency domain by employing the power series technique (PST). In order to verify the sensitivity of the SH wave characteristics to different piezoelectric plate properties, the effect of each plate property is studied separately. Through different plates, the influence of each plate property on dispersion and SH wave structure is discussed. The electrically open and shorted conditions are considered. The study is focused on the first three SH modes. It is shown that the plate properties can be classified into two categories. Further, there is compensation between their effects on the phase velocity, electromechanical coupling coefficient, and SH wave structure. These results can be useful for the designs of acoustic wave devices.
... Guided circumferential waves, which can be effectively used to detect and characterize structural defects or fatigue cracks in cylindrical structures, have been an active research topic because of their pivotal significance not only in NDE (Valle et al., 2001;Luo et al., 2005) but also in the application realm of EA devices (White, 1970). For isotropic linear elastic materials, the investigation of Lamb-type or SH-type circumferential waves (hereafter abbreviated as Lamb waves or SH waves) propagating along the cylindrical surface have been conducted by Liu and Qu (1998), Gridin et al. (2003), and Zhao and Rose (2004), just to name a few. The guided circumferential waves propagating in anisotropic cylindrical curved plates were studied numerically by the Fourier series expansion technique (Towfighi et al., 2002). ...
... Since guided circumferential waves can be readily utilized to detect and characterize structural defects or fatigue damages in cylindrical structures (Liu and Qu, 1998;Valle et al., 2001;Zhao and Rose, 2004;Luo et al., 2005), in this section, we will consider a small-amplitude circumferential wave motion in an EA tube subjected to the biasing fields determined in Section 3. Based on the state-space formalism obtained in the previous section, the approximate laminate technique will be used here to derive the dispersion relations of the circumferential waves. ...
... In this section, the SSM will be utilized to obtain the frequency spectra and the phase velocity spectra for SH waves propagating in EA tubes subjected to an axial pre-stretch and a radial electric voltage. The guided SH wave technique provides a reliable and efficient route to locating and sizing the axial defects or cracks in tubes (Zhao and Rose, 2004;Luo et al., 2005). Before developing a self-sensing EA actuator based on guided SH waves under biasing fields, the effects of the biasing fields on the wave propagation characteristics must be also understood and explored for both thin and thick tubes. ...
In this paper, we present an analysis of the guided circumferential elastic waves in soft EA tube actuators, which has potential applications in the in-situ non-destructive evaluation or online structural health monitoring (SHM) to detect structural defects or fatigue cracks in soft EA tube actuators and in the self-sensing of soft EA tube actuators based on the concept of guided circumferential elastic waves. Both circumferential SH and Lamb-type waves in an incompressible soft EA cylindrical tube under inhomogeneous biasing fields are considered. The biasing fields, induced by the application of an electric voltage difference to the electrodes on the inner and outer cylindrical surfaces of the EA tube in addition to an axial pre-stretch, are inhomogeneous in the radial direction. Dorfmann and Ogden's theory of nonlinear electroelasticity and the associated linear theory for small incremental motion constitute the basis of our analysis. By means of the state-space formalism for the incremental wave motion along with the approximate laminate technique, dispersion relations are derived in a particularly efficient way. For a neo-Hookean ideal dielectric model, the proposed approach is first validated numerically. Numerical examples are then given to show that the guided circumferential wave propagation characteristics are significantly affected by the inhomogeneous biasing fields and the geometrical parameters. Some particular phenomena such as the frequency veering and the nonlinear dependence of the phase velocity on the radial electric voltage are discussed. Our numerical findings demonstrate that it is feasible to use guided circumferential elastic waves for the ultrasonic non-destructive online SHM to detect interior structural defects or fatigue cracks and for the self-sensing of the actual state of the soft EA tube actuator.
... The theory of guided waves plays a critical role in development of guided wave transducers, since the phase and group velocity dispersion curves for a given waveguide and the wave structures must be calculated to understand the excitation mechanism. Although there exist considerable amount of literatures and books addressing SH waves in plate-like structures [10,47], CSH waves [48]and torsional waves [49][50][51] in pipelike structures and Love waves [52] in layered structures, here it is still necessary to recapitulate the basic theories of these waves and discuss their similarities and excitation mechanisms. On the other hand, understanding the working principle of piezoelectric and magnetostrictive materials is also critical in development and application of guided wave transducers. ...
... Viktorov [54] first studied the problem of wave propagation in cylindrical layers and showed that waves propagation in the circumferential direction of cylindrical structures are quite different from those propagation in planar structures. Qu and his collaborators [55,56] presented detailed derivations of the dispersion equations and wave structures of CLamb waves, while the problem of CSH waves propagation in cylindrical structures was firstly addressed by Zhao and Rose [48]. ...
Shear horizontal (SH) waves are of great importance in structural health monitoring (SHM) and nondestructive testing (NDT), since the lowest order SH wave in isotropic plates is non-dispersive. The SH waves in plates, circumferential SH waves and torsional waves in pipes have remarkable resemblances in dispersion characteristics and wave structures, so the latter two can also be called as SH waves in pipes. This paper reviews the state-of-the-art research on SH wave transducers for SHM and NDT. These transducers are grouped into the following categories: Lorentz-force-based electromagnetic acoustic transducers (EMATs), magnetostrictive EMATs, shear wave piezoelectric wedge transducers, thickness-shear piezoelectric transducers and face-shear piezoelectric transducers. The working principles, applications, merits and limitations of different kinds of SH wave transducers are summarized, with a focus on discussing the various configurations for exciting and receiving directional, omnidirectional SH waves in plates and torsional waves in pipes. This paper is expected to greatly promote the applications of SH waves in SHM, NDT and the related areas such as elastic metamaterials.
... Liu et al. presented the first numerical examples of the dispersion curves for circumferential Lamb waves propagating in the isotropic circular annulus [3]. A nearly decade later, the first results for the circumferential SH waves were introduced in the reference [5,6]. In the above study, they use Bessel functions to expand the displacement field. ...
... They all asymptotically approach the Rayleigh wave velocity, which is nondispersive. These characteristics much resemble the circumferential SH modes in the isotropic hollow cylinder [6]. ...
A new spectral method based on the hybrid formula system of the state-vector and Legendre polynomials are used to derive the explicit expressions of circumferential guided waves in the anisotropic multilayer composite cylinders with an arbitrary lay-up. In this hybrid method, the state vector was utilized to transform the wave equation, boundary conditions, and interface continuous conditions in the cylinder coordinate system to form the dispersion equation in terms of the state matrix. Then, following the principle of the Galerkin method, projecting the dispersion state matrix equations onto the weight functions consisting of Legendre polynomials, the system of algebraic equation is obtained. The dispersion curves and mode shape can be obtained by solving the eigenvalues and eigenvectors of the system of algebraic equations. This hybrid method uses the orthogonal and recursive properties of the Legendre polynomial to simplify the integral expressions involved in all the matrices and yields the closed-form solutions. Numerical verification, including the cases of isotropic and anisotropic viscoelastic/elastic multilayer cylinders, was conducted to evaluate the performance of the proposed method. The results show that the proposed method overcomes the cumbersome of the traditional Legendre polynomial method to treat the interface displacement and stress continuity. The results also confirm that the hybrid method has the exponential convergence.
... Early researchers often use plate-wave solutions to model the circumferential guided waves. Liu and Qu [19] and Zhao and Rose [20] developed detailed theoretical models for circumferential guided waves. Their results show that the dispersion curves and wave structures of circumferential guided waves are quite different from that of plate waves especially in thick wall pipes. ...
The fundamental circumferential shear horizontal (CSH0) wave is of practical importance in monitoring corrosion defects in large-diameter pipes due to its virtually non-dispersive characteristics. However, so far, there have been limited CSH0 wave transducers which can be used to constitute a structural health monitoring (SHM) system for pipes. Moreover, the CSH0 wave’s capability of sizing the corrosion-like defect has not yet been confirmed by experiments. In this work, firstly, the mechanism of exciting CSH waves was analyzed. A method based on our previously developed bidirectional SH wave piezoelectric transducers was then proposed to excite the pure CSH0 mode and first order circumferential shear horizontal (CSH1) mode. Both finite element simulations and experiments show that the bidirectional transducer is capable of exciting pure CSH0 mode traveling in both circumferential directions of a 1 - mm thick steel pipe from 100 to 300 kHz. Moreover, this transducer can also serve a sensor to detect CSH0 mode only by filtering circumferential Lamb waves over a wide frequency range from 100 to 450 kHz. After that, a method of sizing a rectangular notch defect by using CSH0 wave was proposed. Experiments on an 11 - mm thick steel pipe show that the depth and circumferential extent of a notch can be accurately determined by using the proposed method. Finally, experiments were performed to investigate the reflection and transmission characteristics of CSH0 and CSH1 waves from notches with different depths. It was found that transmission coefficients of CSH0 mode decrease with the increasing of notch depth, which indicates that it is possible to monitor the depth change of corrosion defects by using CSH0 wave.
