Symmetric low-frequency feature-guided ultrasonic waves in thin plates with transverse bends
Introduction
Ultrasonic guided waves are attractive for rapid inspection of large structures, due to their ability to travel long distances along uncoated structures without much attenuation [1], [2]. Due to their through-thickness modal structures, they offer the possibility to inspect for both surface and internal defects. Thus today guided waves are well-established in the long-range inspection of simple structures such as annular pipes [3], [4] and are also being widely considered for inspection and monitoring of plate-like structures [5], [6]. The potential of guided waves to inspect more complex structures is also now being studied, and this has been enriched by novel feature-guided waves that can exist in special topographical or geometric features in waveguides. Guided wave modes can be trapped in the local vicinity of features such as cross-section changes or curvature present in the structure along the wave propagation (or ‘axial’) direction: see for example [7], [8], [9]. Similar mode-localization has also been proposed for the general three-dimensional problem where ‘transverse’ features are located perpendicular to the wave propagation direction [10]. For example, recently guided waves localized in transverse cross-section changes [11] have been demonstrated and are attractive for the inspection of long weld segments [12], [13].
The authors of the present paper are interested in developing techniques for fast inspection of polygonal metallic pipe structures as encountered for example in hexagonal canisters used for storage of fuel rods in the nuclear industry. Considering a single bend of the polygonal pipe, recent work from our group [14] showed that transverse bends of smooth curvature in thin plates can support ‘bend-guided’ waves. Three-dimensional finite element (FE) simulations validated by experiments showed that bent plates could support two low-frequency bend-guided waves when subjected to uniform in-plane or axial excitation. Of these, the faster travelling mode has properties similar to, but travels at group velocities lower than, the S0 (fundamental symmetric) Lamb mode in normal (flat) plates. The slower bend-guided mode is surprisingly similar to the A0 (fundamental antisymmetric) Lamb mode in normal plates, somewhat unexpectedly for a thickness-wise symmetric excitation. Both bend-guided modes diminished in strength for shallower bends. The antisymmetric bend-guided mode exists for a very small number of low bend angles, and it is also highly dispersive at lower frequencies. Our preliminary studies showed that the symmetric bend-guided mode on the other hand can exist in a larger range of low bend angles, and it is also relatively non-dispersive at low frequencies. Moreover for guided wave inspection, in-plane modes are of interest, as they can traverse a structure without coupling energy into any surrounding medium. In view of this, here we study the low-frequency symmetric bend-guided mode in more detail.
In this process, we draw upon research in the broad area of ‘feature-guided waves (FGW)’ [15] dealing with waves guided by local features in waveguides, which are related to the historical studies on waves in features such as wedges and ridges [16], [17]. FGW studies consider modes that travel confined in an extended local feature, while having an un-coupled modal structure in the plane perpendicular to the wave propagation axis. It is often difficult to obtain analytical solutions to FGW problems and numerical solution methods are necessary. Commonly used techniques include perturbation methods [18], [19], normal mode expansion [20] and high frequency (ray) asymptotics [21]. Interestingly some of the first demonstrations of the application of what is now commonly known as the semi-analytical finite element (SAFE) method were in the context of wedge acoustic waves [22], [23]. Recently the SAFE method is increasingly popular for studying the modal properties of waves in structures with complex sections that are uniform in the wave propagation direction (e.g., rail [24], [25], welds [12], [13]). The SAFE method uses a finite element representation of the waveguide’s cross section, with the assumption of harmonic fields along the wave propagation direction. Solutions to governing wave equations are found by expressing them as a linear eigenvalue problem in terms of wave numbers, and evaluating at several frequencies of interest. This method provides a strong advantage over conventional 3D FE modeling as simulation of only the cross-section of a geometry significantly reduces the computational cost. More recently several developments [26], [27], [28] have made the SAFE technique widely accessible to researchers who are now able to implement it in popular commercial FE packages.
The paper is organized as follows. We begin with a brief description of guided waves in plates and features such as wedges and ridges, after which define our problem of interest. This is followed by a description of numerical (3D FE) studies of guided waves in plates bent at various angles. The experimental procedure for validating these trends, and the SAFE modeling approach used for obtaining further physical insights on the problem, are then presented. FE results and trends validated by experiments are used to demonstrate that transverse plate bends can focus and guide ultrasonic waves, leading to low-frequency ‘bend-guided’ modes. The symmetric bend-guided mode is then studied in more detail, considering its generation, velocity, attenuation and displacement mode shapes for several bend angles. Finally, an analysis of the physics of guided waves in bent plates using the SAFE method is used to discuss the results, after which we conclude with an outlook on implications and further work.
Section snippets
Ultrasonic guided waves in plates and topographic features
The characteristics of ultrasonic guided waves in plates have been widely studied (see for example [29]). Dispersion curves that plot modal features such as phase or group velocity as a function of the frequency describe their propagation characteristics (see [1], [2], [29] and also, ‘DISPERSE’, a software package developed at Imperial College London that helps trace the dispersion curves for simple waveguides [30]). Here we consider only the low-frequency range below 500 kHz-mm, since this is
Procedure for finite element simulations
Numerical simulation techniques such as the finite element (FE) method are attractive for solving elastic wave problems involving structures or features with complex geometry which are otherwise difficult to solve analytically [39], [40], [41]. Here we make use of three dimensional FE simulation implemented in a commercial package [42] to model wave propagation in the bent plates. For stability of explicit FE of wave propagation problems, the step time Δt < Δx/Cmax where Δx is the size of the
Low-frequency feature guided modes in plate bends
Symmetric plate modes usually arise when a symmetric or ‘axial’ excitation is applied on a vertical nodal line across the section of a flat plate [43]. Fig. 5 shows a snapshot of the contour of the total displacement magnitude from a 3D FE simulation of a 3 mm thick flat (180° bend angle) plate subjected to a symmetric axial line excitation with a center-frequency of 100 kHz. S0 and SH0 modes along and perpendicular to the direction of excitation can be observed; surface waves at the plate edges
Discussion
The studies presented thus far in this paper were at a single frequency and did not consider how the ‘bend-guided’ modes arise. A more detailed understanding of this phenomenon and an appreciation of the influence of frequency and bend angle is important for consideration of these modes in the practical NDE of bent structures.
Conclusions
This paper is set in the context of the application of ultrasonic guided waves for fast inspection of structures having complex cross-sections where feature-guided waves are emerging as attractive tools for practical deployment. 3D finite element (FE) simulation validated with experiments show that bends in plates can act as features that can concentrate and guide ultrasonic energy along their axis. At low frequencies, two feature-guided modes are identified when the bent plate is subjected to
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