Damage detection with the fundamental mode of edge waves

Abstract

Detection of mechanical damage using Lamb or Rayleigh waves is limited to relatively simple geometries, yet real structures often incorporate features such as free or clamped edges, welds, rivets, ribs and holes. All these features are potential sources of wave reflections and scattering, which make the application of these types of guided waves for damage detection difficult. However, these features can themselves generate so-called ‘feature-guided’ waves. This article details the first application of the fundamental mode of transient edge waves for detection of mechanical damage. The fundamental edge wave mode (ES₀) – a natural analogue to Rayleigh waves – is weakly dispersive and may decay with propagation distance. The phase and group velocities of the ES₀ wave mode are close to the fundamental shear horizontal (SH₀) and symmetric Lamb (S₀) wave modes, at low and high frequencies, respectively. It is therefore quite challenging to excite a single ES₀ mode and avoid wave coupling. However, it was found experimentally that at medium range frequencies the ES₀ mode can be decoupled from SH₀ and S₀ modes, and its decay is small, allowing for distant detection of defects and damage along free edges of slender structural components. This article provides a brief theory of edge waves, excitation methodology and successful examples of distant detection of crack-like and corrosion damage in I-beam sections, which are widely applied in engineering and construction.

Keywords

Feature-guided waves edge waves damage distant monitoring laser vibrometry

Introduction

The application of Rayleigh and specifically Lamb waves for damage identification and distant monitoring of inaccessible locations (e.g. below ground or hard to access areas) has received a great deal of attention over the past decades.^1–4 Several damage detection methodologies (e.g. baseline subtraction and reverse-time migration) have been previously developed, which are based on various guided wave phenomena associated with the presence of damage, such as scattering and attenuation, harmonic generation, or mode conversion.^1–7 The fundamental symmetric Lamb wave mode (S₀) and the fundamental and the first-order antisymmetric Lamb wave modes (A₀ and A₁, respectively) were widely investigated and applied for detection of defects and mechanical damage in plate and shell structures at low and high ultrasonic frequencies, correspondingly.⁵ Theoretical and experimental investigations focusing on propagation of Lamb waves and detection of defects in complex geometries have also been presented in a number of recent papers.^8–11 Many of these papers highlight the difficulties in detecting damage due to wave reflection, conversion and other wave phenomena associated with structural features.

The S₀ wave mode has traditionally been more popular in experimental studies as it is relatively easy to excite, and it is fairly non-dispersive in the low-frequency range. The latter is important as, for example, the use of non-dispersive or weakly dispersive wave modes simplifies signal processing, allowing for identification of defect locations using a simple time-of-flight approach. However, defect size resolution may be relatively low as it is largely determined by the wavelength, which can be relatively large in the low-frequency range. Several studies have indicated that antisymmetric Lamb wave modes may be used in order to achieve better defect size resolution and probability of damage detection, as these modes have shorter wavelengths in the same frequencies as S₀ mode .^1–4 Motivated by this advantage, a new technique to suppress the symmetric Lamb wave mode during signal excitation was suggested by Nagy et al.,⁶ thus allowing for distant damage (i.e. wall thickness) inspections/monitoring based on the A₀ wave mode.

Detection of defects and mechanical damage using Rayleigh and Lamb waves is still largely limited to relatively simple geometries, yet real structures often incorporate features such as free or clamped edges, welds, ribs, rivets and holes.^1–4 All these features are potential sources of wave dispersion and scattering as well as other complex wave phenomena, which make the application of these types of guided waves for damage detection difficult.¹¹ However, these features, as well as irregularly shaped waveguides (e.g. rail lines, stiffened plates), can themselves generate so-called ‘feature-guided’ waves, in which the energy is concentrated in local structural variations. Subsequently, several damage detection methods, which utilise feature-guided wave modes, such as circumferential creeping, weld-guided or wedge wave modes, have been suggested and explored, with some of these methods exhibiting encouraging results in a laboratory enviroment.^12–16

The present article is focused on the fundamental mode of edge waves (or ES₀ mode), which is very promising for distant damage identification. Edge waves are a natural analogue of classical surface waves, first studied by Rayleigh.¹⁷ Edge waves are guided by the apex of a plate and are concentrated near the free edge and are therefore not affected by the interior of the structure. In addition, edge waves are not subject to spatial dispersion as is often present with Lamb or Rayleigh waves. However, edge waves can decay with propagation distance, and the rate of the decay is strongly dependent on the product of the excitation frequency and plate thickness, as well as Poisson’s ratio of the material,^18–21 where a larger Poisson’s ratio generally leads to stronger decay. This decay can be attributed to leakage of the wave energy into the surrounding material through conversion to other wave modes, such as the S₀ and SH₀ modes.²²

Edge waves were first experimentally observed by Shaw²³ on vibrating thick barium titanate discs. The first theoretical explanation of this phenomenon was given by Gazis and Mindlin,²⁴ who demonstrated that there exists an infinite spectrum of edge wave modes. There were several detailed theoretical and experimental studies on edge resonance, which is a special case of edge waves with zero wavenumber along the edge direction.^18,19 Relatively fewer investigations, mostly theoretical, however, have been focused on behaviour of transient edge waves. Despite previous studies which have focused on feature-guided wave modes such as the circumferential creeping wave,¹² edge-guided wave^14–16 and weld-guided waves,¹³ it seems that there are no studies which comprehensively explore the possible applications of the fundamental edge wave mode for distant damage detection in slender structures with free edge.

In a recent publication, several transient edge wave modes were excited by a flat piezoelectric transducer bonded to the edge of a thick elastic plate.²⁰ The wave excitation and propagation were analysed using a Fourier transform, revealing a very complicated system of propagating waves associated with this method of excitation and selected excitation frequencies, thus making it difficult to implement the outcomes of this work for practical purposes of non-destructive defect detection.

In this study, the solitary ES₀ wave mode was generated using the wedge excitation method to avoid the generation of multiple wave modes, which are detrimental for signal processing, defect sensitivity and damage detection. A range of excitation frequencies were investigated, demonstrating that distant detection of damage is feasible at some specific range (2–5) of the product of the excitation angular frequency, $ω$ , and the plate thickness, $2 h$ , divided by the fundamental shear wave speed, $c_{2}$ . The wave dispersion and decay were found to be almost negligible in this range allowing the application of various standard approaches for signal processing and detection of damage, such as time-of-flight or signal subtraction. The article is concluded with practical recommendations for the development of distant damage detection and health monitoring systems based on the fundamental mode of edge waves.

This article is structured as follows: the following section provides a brief summary of relevant theoretical results in the frame of three-dimensional linear elasticity. The wedge excitation method and details of the experimental set up and measurement procedures are then discussed. The results of the experimental study are presented and discussed, with specific applications to the detection of damage. The article is concluded with possible future developments.

Brief edge wave theory

Theoretical solutions for edge waves can be obtained within the three-dimensional linear theory of elasticity. The governing equations can be expressed in terms of the displacement vector $u = (u_{1}, u_{2}, u_{3})$

(λ + μ) grad div u + μ Δ u = ρ \overset{\cdot\cdot}{u}

(1)

in a domain, $V = (| x_{1} | < \infty, - \infty < x_{2} \leq 0 | x_{3} | \leq h)$ , which represents a semi-infinite plate (see Figure 1). Here, $λ$ , $μ$ and $ρ$ are Lame constants and the mass density, $Δ$ is the three-dimensional Laplace operator and $\overset{\cdot\cdot}{u} = \partial^{2} u / \partial t^{2}$ .

The components of the Cauchy stress tensor, $σ_{ij}$ , can then be expressed as

σ_{ij} = λ δ_{ij} div u + μ (u_{i, j} + u_{j, i})

(2)

where $δ_{ij}$ is the Kronecker delta and $u_{i, j} = \partial u_{i} / \partial x_{j}$ .

Traction-free boundary conditions are imposed on the two side faces of the plate $(x_{3} = \pm h)$ . The excitation signal can be modelled as surface tractions applied at the plate edge $(x_{2} = 0)$

σ_{ij} n_{j} = q_{i} (x_{1}, x_{3}, t)

(3)

The Laplace transform with respect to time $t$ and Fourier transform with respect to the spatial coordinate $x_{1}$ can then be applied to represent the solution of the problem in an integral form

U (x_{2}, x_{3}, ω, ξ) = \int_{- \infty}^{\infty} \int_{0}^{\infty} u (x_{1}, x_{2}, x_{3}, t) e^{- i (ω t - ξ x_{1})} dtd x_{1}

(4)

It was demonstrated in several theoretical studies that the solution of equations (1) to (3) can also be represented as an infinite series of wave modes which satisfy homogeneous boundary conditions, equation (2), on the faces of the plate. In the special case of symmetric excitation, only symmetric wave modes can be excited, and the solution can therefore be represented as a superposition of horizontally polarised shear wave modes (upper index H) and Lamb wave modes (upper index L)^18–20

\begin{matrix} U (x_{2}, x_{3}, ω, ξ) = \sum_{n = 0}^{\infty} C_{n}^{H} U_{n}^{H} (x_{3}) e^{- i ξ_{n}^{H} x_{2}} \\ + \sum_{n = 0}^{\infty} C_{n}^{L} U_{n}^{L} (x_{3}) e^{- i ξ_{n}^{L} x_{2}} \end{matrix}

(5)

where $U_{n}^{H}$ and $U_{n}^{L}$ are eigenfunctions, which are also related to the wavenumber, for horizontally polarised shear and symmetric Lamb wave modes. The coefficients $C_{n}^{H}$ and $C_{n}^{L}$ are unknown and must be found by enforcing the boundary conditions on the plate surfaces and edges as previously described. Various collocation techniques have previously been applied by Feng et al.²¹ and Wilde et al.²⁰ to determine these coefficients. Equation (5) shows that the edge wave mode is essentially a superposition of an infinite number of two propagating wave modes: shear horizontal (SH_n) and symmetric Lamb (S_n).

Figure 2 shows dispersion curves for the fundamental wave modes ES₀, SH₀ and S₀ (solid lines) as obtained in several papers,^18–21 and Figure 3 shows the experimental dependence of the phase velocity of the fundamental edge wave mode as a function of the normalised product of the excitation frequency, $ω$ , and plate thickness, $2 h$ , divided by the shear wave speed, $c_{2}$ (hereafter called the normalised frequency–thickness product). The latter was obtained by the present authors²² and in a general agreement with theoretical predictions.¹⁸

It can be seen from Figure 3 that the dispersion of the fundamental mode is very small over the wide range of excitation frequencies. It can be noted that in accordance with theoretical studies at $ω \times 2 h \to 0$ , the phase speed of ES₀ mode limits to the Rayleigh wave speed as obtained under plane stress conditions. In experimental studies, this zero-limit is not achievable due to sample size limitations. The similar conclusion is not true for $ω \times 2 h \to \infty$ , and the phase speed does not converge to the corresponding speed of Rayleigh waves under plane strain conditions as it also follows from theoretical results.¹⁸

From analysis of dispersion curves in Figure 2, the suitable frequency range, which can be utilised for distant damage detections, is $ω 2 h / c_{2}$ < 5. This upper limit prevents the excitation of multiple modes, complex interference and coupling between the wave modes. Excitation at higher frequencies can lead to a number of challenges related to the coupling of ES₀ and S₀, such as a modulation phenomenon, which was recently demonstrated experimentally for high frequencies-thickness values of around $ω 2 h / c_{2}$ = 10. This phenomenon is due to the proximity of the ES₀ and S₀ wave velocities, which have the similar displacement components.²² In addition, a much stronger attenuation was found by the present authors at higher frequencies, which is in general agreement with past theoretical studies.^18–21 However, there were no systematic theoretical studies focusing on the decaying properties of edge waves, which are required for the development of damage detection techniques in order to evaluate the signal attenuation and inspection range. Another factor, which may significantly affect the decay of edge waves and inspection range, is the radius of the curvature of the free edge, which is always present in real structural components and is disregarded in theoretical studies. Therefore, one of the objectives of the current study is to evaluate the decaying properties of ES₀ waves experimentally, which is important in order to identify the possible range of distant damage inspections.

The lower limit of the normalised frequency–thickness range (in this study ∼2) is associated with the size of the tested sample or the necessity to avoid the interference with the interior of the structure (in the current study, with the web of as I-beam). Like Rayleigh waves, the fundamental edge wave mode exponentially decays with distance from the edge, becoming negligible within 1–2 wavelengths. Another consideration affecting the upper and lower bounds is the defect resolution. For reliable detection, the wavelength of guided or bulk waves is normally selected to be comparable with the characteristic size of defects.^1–4

Edge wave excitation and measurement

Selection of the excitation frequency with guided wave techniques is largely related to the detectable size of defects; as mentioned in the previous section, the size of the defect detected is usually of the same order as the wavelength. In the case of decaying edge waves, selection of the excitation frequency is also dependent on the required inspection range. The current study is focused on the detection of damage in flanges of I-beam sections, which are very common across many industries and applications. Therefore, an additional requirement for the frequency selection was limiting interactions between the excited ES₀ wave mode and the web of the I-beam section, which provides the lower bound for the wave excitation frequencies. Throughout this study, we utilised the following normalised frequency–thickness values: $ω 2 h / c_{2}$ = 1.9, 3.4 and 4.8 (or 200, 350 and 500 kHz excitation frequencies, respectively; see Figure 2). A Tektronix arbitrary function generator was used to generate a Hann windowed sinusoidal toneburst at the selected excitation frequency, which was amplified to ±200 V using a Krohn-Hite 7500 amplifier.

To maximise the amplitude of the ES₀ wave mode and avoid multiple mode generation, the wedge excitation technique was applied.²² The utilisation of the wedge excitation method allows the limitation of excitation energy which is transferred to other wave modes, in particular, the S₀ and SH₀ modes at high and low excitation frequencies, respectively. The selected wedge angle essentially promotes energy conversation of the longitudinal wave generated by the PZT to the fundamental edge wave mode.

When selecting material for the wedge, it is very desirable to have a low longitudinal wave speed to decrease the foot area of the wedge, as well as low attenuation to maximise the amplitude of the ES₀ wave mode. Several materials (such as Perspex and different polyethylene grades) were investigated and tested, with the final wedge design manufactured from an ultrahigh molecular weight polyethylene product called ‘Polystone’ by manufacturer ‘Dotmar’. This material was selected due to its low longitudinal wave speed of approximately 2300 m/s as well as low attenuation characteristics in comparison with other tested materials. The final wedge arrangement used for generation of the ES₀ wave mode in the I-beam flange is shown in Figure 4. Based on Snell’s law, the wedge angle $(θ_{w})$ for the fundamental edge wave mode in the specified normalised frequency–thickness range (2–5) is approximately 53° (see Figure 4). Additional fixtures were fabricated (three-dimensional (3D) printed) as shown in Figure 4 to align the excitation wedge along the I-beam flange and provide a consistent pressure between the contact surfaces, which strongly affects the coupling conditions.

The experimental measurement of the out-of-plane surface displacements was conducted using a Polytec PSV-400-M2-20 scanning laser Vibrometer operating in one-dimensional (1D) mode, which measures the out-of-plane speed using the Doppler effect at selected measurement points, and the corresponding displacement history which is determined by simple integration of the speed function over time. A band pass filter appropriate for the specific excitation frequency was applied to reduce the noise. Multiple vibrometer scans of 500 averages were taken to additionally reduce the effect of stochastic noise and generation bias produced by the transducer. The laser vibrometer, experimental set up and signal flow are schematically illustrated in Figure 5. In practical applications, a PZT sensor(s) can provide the same or similar sensitivity as the laser Vibrometer; therefore, the outcomes of this study can also be readily applied to practical design of damage detection systems.

Damage detection

Two examples of the application of the fundamental edge wave mode to damage identification which are presented in this section utilise real structural components, that is, I-beam sections, which are very common in engineering and construction. The specific manufacturing regulations for I-beams, along with other hot-rolled steel beam products, can be found in Australian Standards AS3679.1:2010. In particular, these beams are used for fabrication of Stobie Poles, which form the main part of the power transmission infrastructure in South Australia.²⁵ There are more than 500,000 Stobie poles in South Australia alone, many of them are older than 50 years and some of them are subjected to severe damage (see Figure 8). Therefore, the evaluation of damage of these poles including the corrosion damage in inaccessible (or below the ground) I-beam sections is of significant importance in order to manage the ageing critical infrastructure and maintain their safety and reliability of operation.

Application of time of flight to damage detection

As the fundamental edge wave mode is only weakly dispersive with the phase speed changing typically between 0.934 $c_{2}$ and 0.924 $c_{2}$ , where $c_{2}$ is the speed of the shear wave, it is feasible to detect and locate a single defect within the flange of an I-beam section using a simple echo-pulse method and time-of-flight (TOF) approach for identification of the defect location. Here, the TOF between the incident and the reflected waveforms are determined using the autocorrelation algorithm, allowing the distance to the defect to be calculated. A schematic of the experimental set up is shown in Figure 6 and an example of typical results of distant inspections with the fundamental mode of edge waves is provided in Figure 7 for a range of excitation frequencies.

The experimental values of the TOF were determined using the autocorrelation algorithm, which identifies finite duration of waveforms within the recorded signal and scales these to determine repetitions of the initial waveform within the same signal at different time lags. The autocorrelation algorithm returns a value between 0 and 1, which represents the signal similarity, with 0 being no similarity and 1 being perfect correlation with the initial waveform. Using this method, it was possible to calculate the TOF of signal reflections more accurately than with the conventional peak-to-peak measurement approaches.

It can be seen from Figure 7 that the damage and its location can be reliably identified with all excitation frequencies. While the lowest frequency (200 kHz) has the lowest decay, the signal is significantly affected by noise, which is likely due to interference between the ES₀ wave mode and the I-beam web section. It is also possible that the lowest frequency is influenced by the mode coupling due to the proximity of the characteristics of the SH₀ wave mode to ES₀; see Figure 2 or the internal structure of the I-beam. Higher frequency signals are less noisy; however, a larger decay can be noted, which can also be a main limitation for the inspection range.

Application of signal subtraction method for detection of distributed damage

Distant detection of distributed damage, such as corrosion (see Figure 8), with the echo-pulse method in this case is quite challenging due to multiple weak reflections from the damage and the interior of the structure, decay, conversion and dispersion of guided waves as well as possible transducer/wedge misalignment and change of the coupling conditions during different signal measurements. Figure 9 shows typical signals obtained from testing undamaged and damaged samples using the similar arrangement as shown in Figures 5 and 6. A standard method for damage identification in the case of multiple reflections normally utilises signal subtraction. The realisation of this method can be different. Below we describe a possible signal processing procedure, which is based on signal subtraction.

In this procedure, $N$ baseline signals, $X b_{i} (t)$ , within the time interval $0 < t < t_{\max}$ are recorded by interrogating the healthy structure, and $t_{\max}$ largely determines the range of the distant inspections. The signals are first normalised using the maximum amplitude value of each individual $i$ -signal as follows

X b'_{i} (t) = \frac{X b_{i} (t)}{| max (X b_{i} (t)) |}

(6)

After that, the normalised signals are synchronised using the standard least squares regression. An average baseline signal for all measurements is calculated by averaging all signals recorded in the healthy structure

\bar{Xb} (t) = \frac{1}{N} \sum_{i = 1}^{N} X b'_{i} (t)

(7)

where $N$ is the number of baseline measurements recorded in the healthy structure. The residual of the baseline signals, $R b_{i} (t)$ , can then be calculated as

R b_{i} (t) = \bar{Xb} (t) - X b'_{i} (t)

(8)

The average residual signal from the healthy flange is as follows

\frac{1}{N} \sum_{i = 1}^{N} R b_{i} (t)

(9)

As well as the residual signal and envelope

max_{i = 1, \dots, N} | R b_{i} (t) |

(10)

can both be used to identify damage.

Figure 10(a) shows a typical baseline residual signal and envelope, and Figure 10(b) shows the averaged residual envelope (solid line), and the maximum residual envelope, $E (t)$ (dotted line). A total of $N$ = 5 measurements with 500 averages for every measurement were recorded for the healthy flanges of I-beams. The difference between these two lines indicates the influence of many factors, which are difficult to control, such as liquid coupling between contact surfaces, pressure, and misalignment.

Figure 11 shows the comparison of the average residual signal and the envelope signal from the healthy flange and residual signals from corroded flanges of A and B I-beams shown in Figure 8. The exceedance of residual signals from corroded sections is evident and can serve as an indication of damage in damage detection techniques. However, the severity of the damage cannot be directly estimated from this comparison. Moreover, the severity of the damage of corroded flanges of I-beams A and B is very similar but the shapes of the residual signals are quite different. This is attributed to the complex wave interactions and wave pattern associated with the multiple reflections, which makes the distant detection of distributed damage with the echo-pulse method a very challenging task.

Further laboratory tests were completed on I-beam sections surrounded by soil, which demonstrated no discernable differences between the above-ground and below-ground signals, enabling the current result to be applied for in situ testing. These tests, as well as the sensitivity studies, are not discussed in this article, which is intended as a feasibility study and is largely focused on the practical application of the fundamental edge wave mode for distant detection of damage.

Conclusion

The application of the fundamental mode of edge waves for distant damage inspections along free edges of slender structural components represents a great interest due to the ability of this guided wave mode to propagate long distances without significant decay and the absence of spatial dispersion, which is often the main limitation for guided wave inspections with Rayleigh or Lamb waves. Furthermore, the fundamental edge wave mode is almost nondispersive, which allows defect identification using a simple time-of-flight approach.

In this initial study, the fundamental edge wave mode (ES₀) was successfully generated and utilised to detect cracks and distributed corrosion damage in I-beam flanges. Selection of a suitable excitation frequency range was based on several conflicting considerations: avoidance of potential edge wave interactions with the web of the I-beam (lower frequency bound) from one hand and multiple mode generation, significant decay and interaction with other guided wave modes (higher frequency bound) from the other hand. The wedge excitation technique was applied to generate the ES₀ wave mode and limit the amplitude of the SH₀ and S₀ guided wave modes, which have very similar wave velocities. As a wedge material, ultrahigh molecular weight polyethylene demonstrated a good potential for excitation of the ES₀ wave mode due to its low longitudinal wave speed of approximately 2300 m/s as well as low attenuation in comparison with other tested materials. The selected normalised frequency–thickness values (2–5) provided a suitable practical range for distant defect and damage detection, which was demonstrated in the current study for real damaged structural components. A simple signal subtraction technique was also adopted in order to detect distributed (corrosion) damage in I-beam flanges. However, this technique is only one possible example of a signal processing methodology for damage detection, and it may be modified or replaced with more suitable methods if required.

The effects of the edge radius or edge shape, transducer misalignment and contact conditions were beyond the scope of this initial study. Finally, the obtained results are very encouraging for the application and development of practical distant inspection methods to detect fatigue cracks as well as distributed (corrosion) damage in inaccessible locations using transient fundamental mode of edge waves.

Footnotes

Acknowledgements

The authors would also to thank SA Power Networks for their valued support, guidance and donation of sample specimens.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Australian Research Council through Discovery Project DP160102233, LE170100079, DP200102300 and the Australian Government Research Training Programme Scholarship. Their support is greatly appreciated.

ORCID iDs

James M Hughes

Aditya Khanna

Ching-Tai Ng

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