Guided wave propagation and mode differentiation in hollow cylinders with viscoelastic coatings

Guided wave propagation theories have been widely explored for about one century. Earlier theories on single-layer elastic hollow cylinders have been very beneficial for practical nondestructive testing on piping and tubing systems. Guided wave flexural (nonaxisymmetric) modes in cylinders can be generated by a partial source loading or any nonaxisymmetric discontinuity. They are especially important for guided wave mode control and defect analysis. Previous investigations on guided wave propagation in multilayered hollow cylindrical structures mostly concentrate on the axisymmetric wave mode characteristics. In this paper, the problem of guided wave propagation in free hollow cylinders with viscoelastic coatings is solved by a semianalytical finite element (SAFE) method. Guided wave dispersion curves and attenuation characteristics for both axisymmetric and flexural modes are presented. Due to the fact that dispersion curve modes obtained from SAFE calculations are difficult to differentiate from each other, a mode sorting method is established to distinguish modes by their orthogonality. Theoretical proof of the orthogonality between guided wave modes in a viscoelastic coated hollow cylinder is provided. Wave structures are also calculated and discussed in view of wave mechanics in multilayered cylindrical structures containing viscoelastic materials.     

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.     

无限流体中孔隙介质圆柱周向导波的传播特性

为研究无限大流体约束的孔隙圆柱中周向导波的传播规律,分析孔隙参数对导波传播特性的影响,建立了无限流体中孔隙介质圆柱的理论模型,利用孔隙介质弹性波动理论,建立了周向导波频散方程,通过数值模拟计算得到无限流体中孔隙介质圆柱的频散曲线,探讨了圆柱半径和孔隙参数对导波传播特性的影响,并对导波的衰减特性进行了分析;通过数值计算,得到了周向导波的时域波形,讨论了孔隙参数对波形的影响.结果表明,孔隙介质圆柱半径的改变影响圆柱尺度,孔隙度的改变影响孔隙介质中体声波的波速,都对周向导波频散曲线产生一定的影响,所得到的频散曲线特征及衰减曲线与时域波形吻合.研究结果对开展无限流体中孔隙介质圆柱的超声无损评价提供了一定的理论参考.     

Analysis of wave propagation in cylindrical pipes with local inhomogeneities

In this paper, we present a numerical approach to study the guided elastic wave propagation in cylindrical pipes with local inhomogeneities. A hybrid wave finite element (WFE) and finite element (FE) technique is introduced to investigate the dispersion and wave scattering in pipes by taking full advantage of the existing FE codes. Dynamic reduction technique is employed to improve the computational efficiency, which is particularly suitable for the pipes with standard local features. Numerical examples indicate that the proposed technique provides an effective way to calculate the dispersion relationship and the scattered field. Both the axisymmetric and non-axisymmetric wave scattering problems are considered.     

Simulation of guided waves in complex piping geometries using the elastodynamic finite integration technique

Although many technologies exist for inspecting piping systems, they are most successful on straight pipes and are often unable to accommodate the added complexities of pipe elbows, bends, twists, and branches, particularly if the region of interest is inaccessible. This paper presents a numerical technique based on the elastodynamic finite integration technique for simulating guided elastic wave propagation in piping systems. Comparisons show agreement between experimental and simulated data, and guided wave interaction with flaws, focusing, and propagation in pipe bends are presented. These examples demonstrate the ability of the simulation method to be used to study elastic wave propagation in piping systems which include three-dimensional pipe bends, and suggest its potential as a design tool for designing pipe inspection hardware and ultrasonic signal processing algorithms.     

The effect of pipe bend on low-frequency longitudinal mode guided wave propagation

The pipe bend significantly changes the propagation characteristics of guided wave,and makes the interpretation of the received signals difficult.Therefore,better understanding of guided wave propagating in bended pipe is essential for the inspection of pipeline comprising bends.First of all,the different features of dispersion curves derived with the semi-analytical finite element method for guided wave in bended pipes are summarized.Secondly,based on the dispersion curves for guided wave in bended pipes,experiments are performed to investigate the mode conversions of L(0,1) mode guided wave traveling through pipe bends.It is found that,except for the mode conversion from L(0,1) to F(1,1),the L(0,1) reflections of bends are also observed in some cases,which are proven to be the mode converted negative L(0,1)mode guided wave,and the negative L(0,1) mode guided wave becomes more obvious with the decrease of excitation frequency and bending radius.The findings of this paper will provide some insight for guided wave behavior in bended pipe,and generalize the application of guided wave inspection in practical pipelines.     

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.     

Numerical investigation of elastic modes of propagation in helical waveguides

Steel multi-wire cables are widely employed in civil engineering. They are usually made of a straight core and one layer of helical wires. In order to detect material degradation, nondestructive evaluation methods based on ultrasonics are one of the most promising techniques. However, their use is complicated by the lack of accurate cable models. As a first step, the goal of this paper is to propose a numerical method for the study of elastic guided waves inside a single helical wire. A finite element (FE) technique is used based on the theory of wave propagation inside periodic structures. This method avoids the tedious writing of equilibrium equations in a curvilinear coordinate system yielding translational invariance along the helix centerline. Besides, no specific programming is needed inside a conventional FE code because it can be implemented as a postprocessing step of stiffness, mass and damping matrices. The convergence and accuracy of the proposed method are assessed by comparing FE results with Pochhammer-Chree solutions for the infinite isotropic cylinder. Dispersion curves for a typical helical waveguide are then obtained. In the low-frequency range, results are validated with a helical Timoshenko beam model. Some significant differences with the cylinder are observed.     

Natural beam focusing of non-axisymmetric guided waves in large-diameter pipes

The propagation of non-axisymmetric guided waves in larger diameter pipes is studied in this paper by treating the guided waves as corresponding Lamb waves in an unwrapped plate. This approximation leads to a simpler method for calculating the phase velocities of hollow cylinder guided waves, which reveals a beam focusing nature of non-axisymmetric guided waves generated by a partial source loading. The acoustic fields in a pipe generated by a partial-loading source includes axisymmetric longitudinal modes as well as non-axisymmetric flexural modes. The circumferential distribution of the total acoustic field, also referred as an angular profile, diverges circumferentially while guided waves propagate with dependence on such factors as mode, frequency, cylinder size, propagation distance, etc. Exact prediction of the angular profile of the total field can only be realized by numerical calculations. In particular cases, however, when the wall thickness is far less than the cylinder diameter and the wavelength is smaller than or comparable to the pipe wall thickness, the acoustic field can be analyzed based on the characteristics of Lamb waves that travel along a periodic unwrapped plate. Based on this assumption, a simplified model is derived to calculate the phase velocities of non-axisymmetric flexural mode guided waves. The model is then applied to discussions on some particular characteristics of guided-wave angular profiles generated by a source loading. Some features of flexural modes, such as cutoff frequency values are predicted with the simpler model. The relationship between the angular profiles and other factors such as frequency, propagation distance, and cylinder size is obtained and presented in simple equations. The angular profile rate of change with respect to propagation distance is investigated. In particular, our simplified model for non-axisymmetric guided waves predicts that the wave beam will converge to its original circumferential shape after the wave propagates for a certain distance. A concept of "natural focal point" is introduced and a simple equation is derived to compute the 1st natural focal distance of non-axisymmetric guided waves. The applicable range of the simplified equation is provided. Industrial pipes meet the requirement of wall thickness being far less than the pipe diameter. The approximate analytical algorithms presented in this paper provides a convenient method enabling quick acoustic field analysis on large-diameter industrial pipes for NDE applications.     

Wave propagation in layered piezoelectric rectangular bar: An extended orthogonal polynomial approach

Wave propagation in multilayered piezoelectric structures has received much attention in past forty years. But the research objects of previous research works are only for semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes an extension of the orthogonal polynomial series approach to solve the wave propagation problem in a two-dimensional (2-D) piezoelectric structure, namely, a multilayered piezoelectric bar with a rectangular cross-section. Through numerical comparison with the available reference results for a purely elastic multilayered rectangular bar, the validity of the extended polynomial series approach is illustrated. The dispersion curves and electric potential distributions of various multilayered piezoelectric rectangular bars are calculated to reveal their wave propagation characteristics.     

Analysis of electromagnetic field due to a buried coated PEMC circular cylinder

An analytical solution for the scattering of an electromagnetic plane wave from a coated perfect electromagnetic conducting (PEMC) circular cylinder, buried in the dielectric half space, is presented. Scattering characteristics of a buried PEMC cylinder when coated by double-positive (DPS) or double-negative (DNG) materials is investigated. The cylinder as well as coating layer is of infinite length (2-D problem). Plane wave spectral representations of the fields have been used to solve the problem. Saddle point method is used to solve the integral arising in the analysis. All the multiple interactions between the buried geometry and the dielectric interface separating the two half spaces have been considered in the analysis. The derivation includes both TM and TE polarization cases. It is observed that the response of the coated PEMC cylinder can be used to detect the underground pipes and other buried objects having a cylindrical shape.     

被孔隙介质约束的弹性杆中的导波*

为了研究导波在被孔隙介质约束的弹性杆结构中的传播规律,分析孔隙参数对导波传播特性的影响,本文建立了无限大孔隙介质包裹圆柱体的理论模型,利用孔隙介质弹性波动理论,分析了导波的频散曲线,以及圆柱半径和孔隙参数对于导波传播特性的影响。结果表明,在该结构中传播的纵向导波存在频散特性。内部圆柱半径的改变影响波导结构,从而影响导波传播。外部孔隙介质的渗透率对于导波频散的影响较小,孔隙度的改变影响孔隙介质体波波速,从而影响导波频散曲线的截止频率。同时,导波存在较小的衰减,且衰减随孔隙度增大而增大。这些结果对于后续开展无限大介质包裹弹性杆结构的超声无损评价提供了一定的理论参考。     

Correlation between helical surface waves and guided modes of an infinite immersed elastic cylinder

Scattering of obliquely incident plane acoustic waves from immersed infinite solid elastic cylinders is a complex phenomenon that involves generation of various types of surface waves on the body of the cylinder. Mitri [F.G. Mitri, Acoustic backscattering enhancement resulting from the interaction of an obliquely incident plane wave with an infinite cylinder, Ultrasonics 50 (2010) 675-682] recently showed that for a solid aluminum cylinder, there exist acoustic backscattering enhancements at a normalized frequency of ka?0.1. The incidence angle α_c at which these enhancements are observed lies between the first (longitudinal) and second (shear) coupling angles of the cylinder. He also confirmed the observations previously reported by the authors that there exist backscattering enhancements of the dipole mode at large angles of incidence where no wave penetration into the cylinder is expected. In this paper, physical explanations are provided for the aforementioned observations by establishing a correlation between helical surface waves generated by oblique insonification of an immersed infinite solid elastic cylinder and the longitudinal and flexural guided modes that can propagate along the cylinder. In particular, it is shown that the backscattering enhancement observed at ka?0.1 is due to the excitation of the first longitudinal guided mode travelling at the bar velocity along the cylinder. It is also demonstrated that the dipole resonance mode observed at incidence angles larger than the Rayleigh coupling angle is associated with the first flexural guided mode of the cylinder. The correlation established between the scattering and propagation problems can be used in both numerical and experimental studies of interaction of mechanical waves with cylinders.     

Calculation of leaky Lamb waves with a semi-analytical finite element method

A semi-analytical finite element method (SAFE) has been widely used for calculating dispersion curves and mode shapes of guided waves as well as transient waves in a bar like structures. Although guided wave inspection is often conducted for water-loaded plates and pipes, most of the SAFE techniques have not been extended to a plate with leaky media. This study describes leaky Lamb wave calculation with the SAFE. We formulated a new solution using a feature that a single Lamb wave mode generates a harmonic plane wave in leaky media. Dispersion curves obtained with the SAFE agreed well with the previous theoretical studies, which represents that the SAFE calculation was conducted with sufficient accuracy. Moreover, we discussed dispersion curves, attenuation curves, and displacement distributions for total transmission modes and leaky plate modes in a single side and both two side water-loaded plate.     

Non-Steady Wall-Bounded Flows of Viscoelastic Fluids Under Periodic Forcing

The problem of oscillating flows inside pipes under periodic forcing of viscoelastic fluids is addressed here. Starting from the linear Oldroyd-B model, a generalized Darcy’s law is obtained in the frequency domain and an explicit expression for the dependence of the dynamic permeability on the fluid parameters and forcing frequency is derived. Previous results in both viscoelastic and Newtonian fluids are here shown to be particular cases of our results. On the basis of our calculations, a possible explanation for the observed damping of local dynamic response as the forcing frequency increases is given. Good fitting with recent experimental studies of wave propagation in viscoelastic media is here exhibited. Sound wave propagation in viscoelastic media flowing inside straight pipes is investigated. In particular, we obtain the local dynamic response for weakly compressible flows.     

Excitation and propagation of non-axisymmetric guided waves in a hollow cylinder

Excitation and propagation of non-axisymmetric guided waves in a hollow cylinder is studied by using the normal mode expansion method (NME). Different sources such as angle beam, tube end excitation with normal beam, and comb transducer possibilities are discussed based on the derivations of the NME method. Numerical calculations are focused on the case of angle beam partial loading. Based on the NME method, the amplitude coefficients for all of the harmonic modes are obtained. Due to the difference of phase velocities for different modes, the superimposed total field varies with propagating distances and hence makes particle displacement distribution patterns (angular profile) change with distance. This varying non-axisymmetric angular profile of guided waves represents a nonuniform energy distribution around the hollow cylinder and thus has an impact on the inspection ability of guided waves. The angular profiles of an angle beam source are predicted by theory and then verified by experiments. The predicted angular profiles also provide information for determining the transducer location to find defects in a certain position on the hollow cylinder.     

Coupled flexural,longitudinal and shear wave motion in two- and three-layered damped beams

The purpose of this paper is to show how a ceramic layer attached to a two-layered (elastic/viscoelastic) beam alters the wave propagation mechanism in the beam. The ceramic layer is assumed to possess mass but not longitudinal stiffness. Shear deformation, rotatory, longitudinal and transverse inertia forces are all included in the analysis. The equations of motion of the layered beam are derived by using the virtual work principle. Relevant dispersion curves for an infinite beam are presented and discussed, and are compared with dispersion curves obtained from a number of simplified theories. The influence of the inertia coupling between different wave types, caused by the ceramic layer, has been examined. The loss factors of the different waves are found to be critically dependent upon the inertia coupling and wave-number in particular wave-number regions. This occurs when uncoupled waves of different types have close wave-speeds. Under these circumstances a coupled wave can exist which has much less damping than any of its constituent uncoupled waves.     

Torsional wave dispersion in a finitely pre-strained hollow sandwich circular cylinder

This paper studies torsional wave dispersion in a three-layered (sandwich) hollow cylinder with finite initial strains. The investigations are carried out within the scope of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The mechanical relations of the materials of the cylinders are described through their harmonic potential. The analytical expression is obtained for the low wavenumber limit values of the torsional wave propagation velocity. The numerical results on the influence of the initial stretching or compression of the cylinders along the torsional wave propagation direction are presented and discussed.     

Dynamic viscoelastic effects on sound wave scattering by an eccentric compound circular cylinder

The classical method of separation of variables in conjunction with the translational addition theorem for cylindrical wave functions are employed to obtain an exact solution for two-dimensional interaction of a harmonic plane acoustic wave with an infinitely long (visco)elastic circular cylinder which is eccentrically coated by another (visco)elastic material and is submerged in an ideal unbounded acoustic medium. The novel features of Havriliak-Negami model for dynamic viscoelastic material behaviour are used to take the rheological properties of the coating (and/or core) material into consideration. The analytical results are illustrated with numerical examples in which a steel rod eccentrically coated with (an eccentric steel shell filled with) dissipative materials of distinct viscoelastic properties is insonified by plane sound waves at selected angles of incidence. The effects of incident wave frequency, angle of incidence, core eccentricity and dynamic viscoelastic material properties on the backscattered form function spectra are examined. Limiting cases are considered and fair agreements with available solutions are obtained.     

Transient torsional vibration responses of finite, semi-infinite and infinite hollow cylinders

Torsional guided waves are often used to detect the defects in a hollow cylinder. To realize the excitation of the torsional guided waves with high efficiency, the transient vibration responses of finite, semi-infinite and infinite hollow cylinders to external torsional forces must be clarified theoretically. In this study, the method of eigenfunction expansion is employed to solve the above problems. The exact analytical solutions derived by this method are not only explicit but also concise. Furthermore, the analytical solution of the transient torsional vibration of the finite hollow cylinder is numerically evaluated. The results obtained agree well with those simulated by the finite element method.