Waveguide metamaterial rod as mechanical acoustic filter for enhancing nonlinear ultrasonic detection

Waveguide ultrasonics is widely used for the measurement of fluid flow,¹ temperature,^2–5 and other material properties.^6–10 Waveguides play a crucial role in applications in which transducers need to be protected from harsh environments, such as extremely high temperatures.¹¹ Here, for the first time, we propose the use of metamaterial concepts for developing a cylindrical wave-guide that leads to an improvement in Nonlinear Ultrasonic (NLU) detection.

Prognosis of material performance using early pre-crack material characterization requires nanoscale sensitivity. Ultrasonic elastic waves have traditionally been employed for measurements, since this modality can penetrate deep into materials while interacting with material features, which provides signatures for early detection of damage. However, the linear ultrasonics approach typically is limited by the wavelength used and, hence, is not sensitive to fine features at the micro/nano-scale unless frequencies in the GHz range can be used. At such high frequencies, due to material attenuation, these elastic waves do not travel deep into materials of interest. In contrast, Nonlinear Ultrasonic (NLU) guided waves^12–15 have been reported extensively to possess the advantages of both the deep penetration and the ability to provide sensitivity to microscopic defects, including precursors to material damage. However, accurate measurements of NLU have traditionally been limited to laboratory demonstrations, and field deployment has been limited due to many factors, including the presence of higher harmonics due to the material and equipment (cables, couplant effects, etc.) that are often difficult to separate from those originating from material damage features. Such higher harmonics would lead to false positives, and it is of much interest to suppress them.

Although filtering options, such as Butterworth and Chebyshev, are available via digital signal processing methods,^16–18 or via analog electronic filters, these have several drawbacks. Digital signal processing is limited by the digitization process wherein the unwanted signals will influence the process and reduce the effectiveness. Electronic filters will end up adding more components, thereby further adding another source of equipment nonlinearity. Here, we propose a novel route to address this issue by using acoustic metamaterials as mechanical filters.

Metamaterials (MM) have attracted much research interest in the last two decades for controlling and manipulating elastic and electromagnetic waves.^19–22 Metamaterials typically consist of features arranged in the subwavelength scale, yielding exotic properties, such as focusing,^23,24 filtering,^25,26 wave manipulation,^27–29 wave guiding,^30–32 wave attenuation,³³ bandgaps,^34–37 subwavelength imaging,^38–40 cloaking,⁴¹ and negative refraction.^42,43 Recently, suppression of the second harmonic using stub-like features on a plate metamaterial was reported.⁴⁴ Plates with sinusoidal corrugations as a part of the transducer can be used for single and multiple stopband filtering of multiple frequencies.⁴⁵ Gliozzi et al. demonstrated the feasibility of an acoustic diode based on alternating nonlinear elastic and metamaterial frequency-filtering regions.⁴⁶ Mostavi et al. showed the suppression of the second harmonic by using a periodic array of composite layers made of glass and water to enhance the damage detection sensitivity of immersion NLU testing.⁴⁷ 

Here, for the first time, we discuss the significant suppression of all higher harmonics in cylindrical circular rod waveguides by carefully creating a bandgap near the fundamental frequency using a ridged metamaterial add-on. The results presented here could offer the exciting possibility of incorporating metamaterials into nonlinear ultrasonic nondestructive evaluation (NDE) and structural health monitoring (SHM), leading to more accurate early stage damage detection that cannot be achieved by conventional ultrasonics. We propose the use of a novel ridged circular rod, as shown in Fig. 1(a), as a waveguide metamaterial to achieve the goal of suppressing NLU higher harmonics.

This paper is organized as follows. First, the optimal design of the geometrical parameters of the metamaterial rod is studied through an analysis of the dispersion curves, yielding values that have bandgaps near the fundamental frequency. Dispersion curves are plotted, and a configuration that yields bandgaps near the fundamental frequency is obtained. Fabrication of the waveguide metamaterial rod with the thus-selected parameters integrated with a bare specimen sample rod and the procedure for experiments are then discussed. Results demonstrating the suppression of higher harmonic components using the proposed concept are then presented and discussed. An application of this concept for the identification of a discontinuity in a cylindrical workpiece through its enhanced nonlinear signature is also demonstrated. Furthermore, three more example cases are presented, showing the selective suppression and enhancement of higher harmonics. This paper concludes with directions for further work.

Figures 1(a)–1(d) present a schematic of the cylindrical wave-guide metamaterial concept proposed here, down to its finite element unit cell. The computer-aided design (CAD) model of the metamaterial rod, along with an enlarged view of the metamaterial section, is presented in Fig. 1(a). A simplified 2D axisymmetric model is shown in Fig. 1(b). The unit cell extracted from the 2D axisymmetric model is shown in Fig. 1(c), marking the dimensional parameters height (h), radius (r), thickness (t), and unit cell length (L). Figure 1(d) depicts the finite element unit cell model showing the corresponding internal mesh of the selected unit cell. The unit cell was assigned the properties of aluminum with Young’s modulus E = 69 GPa, density ρ = 2700 kg/m³, and Poisson’s ratio μ = 0.33. The Bloch–Floquet periodic boundary condition is applied on both lateral faces of the unit cell as the input for the modal analysis using the solid mechanics module in the commercial finite element package.⁴⁸ 

In the demonstration of this concept, as reported here, the aim was to achieve a bandgap near 500 kHz, a frequency that is typically used in waveguide ultrasonics. The radius of the waveguide was taken as 5 mm, based on the practical use cases for such dimensions. The design of the waveguide metamaterial rod involves plotting of dispersion curves by varying the geometrical parameters of the baffles that impact the bandgap, believed to be their height (h), thickness (t), and the periodicity or unit cell length (L), see Fig. 1. These parameters were varied systematically in a matrix fashion as per the operating wavelength, one at a time, as shown in Table I. Dimensional regimes that are small, medium, and matching the order of the wavelength were considered, in line with the physics of wave propagation problems. The dispersion curve shown in Fig. 1(e) is obtained by solving an eigen-frequency problem on a unit cell by applying the periodic boundary conditions at the interfaces between the adjacent units according to the Bloch–Floquet condition⁴⁹ 

where u is the displacement, L is the unit cell length, and k is the wavenumber.

Upon running the models and obtaining the dispersion curves, parameters for the baffles in case (d) in Table I with unit cell length (L) = 4 mm (λ/2), height (h) = 2 mm (λ/4), and thickness (t) = 2 mm (λ/4) showed a bandgap in the required range of 490–550 kHz. Thus, the parameters were obtained and selected for further studies. Essentially, these values mean that moderate periodicity and low dimensions for baffle thickness and height as compared to the operating wavelength lead to the required bandgaps.

It is notable here that the dispersion curves shown in blue in Fig. 1(e) for the case without baffles do not represent Dirac cones since these curves will not intersect. Dirac cone phenomena emerge in two-dimensional guided wave problems and could be interesting for higher order guided modes.

An aluminum waveguide metamaterial rod was fabricated with a base diameter of 10 mm and a length of 1 m, as shown in Fig. 1(a). An aluminum rod without baffles of the same diameter and length was also prepared for comparison. Guided ultrasonic wave experiments were conducted on the sample with and without baffles with the goal of suppressing higher harmonics.

The schematic diagram of the experimental setup is shown in Fig. 2. In this “pitch-catch” arrangement, two contact longitudinal transducers (V101-0.5 MHz Panametrics Inc., Waltham, MA, USA) were placed at the ends of the sample rod with a spring-loaded clamp providing uniform holding conditions.

Ultrasonic guided waves of the longitudinal family were generated using a commercially available transducer with a central frequency of 500 kHz. A ten-cycle tone-burst pulse was generated by the RITEC advanced measurement system RAM-5000 SNAP (RITEC, Warwick, USA) in a pitch-catch configuration as well as a pulse-echo configuration. The signals obtained from the transducer connected to the receiving terminal of SNAP are taken out from the RF monitor terminal and fed to a terminal in a digital storage oscilloscope (DSOX3024T, Keysight Technologies, USA), where signals are displayed and recorded. The pulse-echo signal from the transducer was measured with the help of a signal diplexer (RDX-6, RITEC, Warwick, USA) and fed to another terminal in the same digital signal oscilloscope. The same experiment was repeated on the bare aluminum rod without baffles for comparison.

Figure 3(a) shows the frequency spectrum obtained from the experimental results for with and without metamaterial cases. It is observed that all the higher harmonic components are completely suppressed after using the metamaterial. The application of this technique is demonstrated by introducing a discontinuity between the metamaterial section and the specimen rod. The presence of higher harmonics was observed from the resultant frequency spectrum, as shown in Fig. 3(b). This shows that discontinuities in the rod after the metamaterial section are detectable through higher harmonic components in the received signal. This methodology gives more precise quantification and qualification of the defects in the nonlinear regime, such as micro-cracks, voids, and fatigue cracks.

Short-Time Fourier Transfer (STFT) analysis of the signals reveals more details about the harmonics and the suppression while using the metamaterial structure and again the resurgence of third harmonics in the case with discontinuity. As can be seen in Fig. 3(c), the STFT of a bare rod excited at one end with 500 kHz shows the presence of higher harmonics at 1, 1.5, 2 MHz, etc., at around 290 µs. The same loading conditions were applied on a metamaterial rod designed for a bandgap at 500 kHz, which shows the suppression of all the higher harmonics, as shown in Fig. 3(d). It can be seen that the fundamental frequency is slightly shifted from 500 to 490 kHz, which is the lower bound of the bandgap designed for the metamaterial rod. Even though there is a decrease in the amplitude of waves transmitted, this was sufficient to identify the discontinuity on the rod after the metamaterial, as shown in Fig. 3(e).

Experiments were also carried out with the same settings and measurements were taken at different power levels ranging from 10% to 90% in steps of 10%.

For the case without the metamaterial, it was observed that higher harmonics are available and growing with the increase in the amplitude of the input power level. Cases of 10% and 90% input power levels are plotted in Fig. 4(a) to reduce the cluttering of the graphs. In Fig. 4(b), the results for with the metamaterial case at different power levels (10% and 90%) are plotted. It is observed from the frequency spectrum shown in Fig. 4(b) that no higher harmonic components arise even after raising the power levels to the maximum. This means that the signal with higher harmonic components is not allowed to pass through the metamaterial section of the rod, which acts as a mechanical filter. Thus, the signal passing through the remaining section of the specimen rod will be free from any inherent nonlinearities arising from electronic components (power amplifiers and cables), couplants, and other components at the transduction side. Hence, the signal measured at the receiver side will show only fundamental frequency as the rod is in the pristine condition, free from any kind of defects.

To understand the propagation of higher harmonics clearly, experiments were also performed in the pulse-echo configuration. It is observed from the frequency spectrum shown in Fig. 4(c) that the higher harmonics are reflected back. This means that the signal with unwanted higher harmonic components is not allowed to pass through the metamaterial section, which acts as a mechanical filter by reflecting higher harmonics.

The suppression of higher harmonics is due to the bandgap created near the fundamental frequency [see Fig. 1(e)] by the Bragg scattering phenomenon occurring in the waveguide metamaterial rod. The baffles placed on the surface of the rod strongly reflect the higher harmonics [see Fig. 4(c)], which were verified by the experiments conducted in the pulse-echo configuration. For a structure having periodically repeating geometric features, Bragg scattering occurs around the frequencies governed by the Bragg condition³⁶ 

where L is the lattice constant of the periodic system and λ is the wavelength of the wave in the material. The geometry of the periodically repeating feature can be modified such that the stopbands could be manipulated and engineered for specific purposes. In our case, the design of the unit cell was based on creating a bandgap near the excitation frequency that creates a bandgap near 500 kHz.

In the earlier set of experimental studies, the NLU measurements were enhanced by eliminating the false positives occurring due to the equipment. Next, simulation based studies showing selective suppression of certain harmonics and enhancement of the other harmonics are discussed. The various simulation cases considered for this purpose are mentioned in Table II with their respective baffle parameters used. Time-domain finite element (FE) simulations were done considering the two-dimensional (2D) axisymmetric model shown in Fig. 1(b) using a commercial package (see ABAQUS User Manual version 6.12, Dassault Systemes, Providence, RI, USA 2014, https://www.3ds.com/products-services/simulia/products/abaqus/). At one end of the rod, the harmonics caused by the equipment are introduced in the excitation signal itself. The displacements monitored at the other end were further post-processed to obtain the frequency spectrum. The simulations were also repeated on a plain aluminum rod without baffles for comparison.

Case 1 considers the selective suppression of fundamental harmonics using the metamaterial rod with baffles parameters indicated in Table I. Figure 5(a) shows the frequency spectrum from the simulation study that shows the capability of the MM as a filter suppressing 500 kHz, which is the fundamental frequency, while the second and third harmonics at 1 and 1.5 MHz persist. Simu-lations were carried out using an artificially crafted input signal, which has second and third harmonics along with the fundamental frequency. This was done so as to mimic a scenario where there are unwanted nonlinearities on the input side. From this input signal, the fundamental component alone was filtered out by the MM rod because of the bandgap. However, in the experimental case, as shown in Fig. 5(b), the fundamental and higher harmonics are suppressed. The source of the higher harmonics suppressed here is due to the equipment, but any other sources of nonlinearity would still appear as a higher harmonic component, as discussed in earlier application of detecting discontinuity. When excited with a 250 kHz transducer on a MM rod mentioned in case 2, the selective second harmonic suppression can be seen from the frequency spectrum shown in Fig. 5(c) for simulation and Fig. 5(d) for the experiment.

In the earlier cases, nonlinearity detection can be improved by the selective suppression of fundamental or second harmonics. In case 3, the enhancement of higher harmonics is demonstrated by the MM rod, which can further improve the nonlinearity detection. The frequency spectrum obtained from the experimental result in case 3 is shown in Fig. 6, where it can be observed that there is an enhancement of the fundamental frequency along with other higher harmonics. This enhancement of all the higher harmonics is due to the local resonance in the baffles. NLU is an emerging Non-Destructive Testing (NDT) technique, which is widely accepted to be sensitive to early stage material damage. However, it is important in NLU to suppress unwanted nonlinearity caused due to equipment and factors such as coupling, while it is required to enhance the nonlinearity caused due to the material microstructure, defective features, etc. Today, this can be achieved either by electronic filters or in postprocessing through software algorithms. In order to achieve this, here, we propose two metamaterial based mechanical filters, one on the transduction side to remove unwanted nonlinear signatures and another on the reception side, which will enhance required nonlinearities.

To summarize, the suppression of higher harmonics from the transmission side is demonstrated with experiments and simulations, thereby eliminating the unwanted nonlinearities. The goal is that the higher harmonic components on the transmission side, arising from genuine nonlinearity (e.g., due to a crack/discontinuity) in the wave path, should not be affected when suppressing the fundamental frequency. Accordingly, the simplest case was considered for generating such nonlinearity, which is complete discontinuity. Quantification of the defect signature in the form of nonlinearity is possible, but this is beyond the scope of our work, which is focused on demonstrating the feasibility of our proposed metamaterial approach to enhance the nonlinear response. The defect detection from the higher harmonics with the filtered MM rod case was also demonstrated experimentally. Later from case studies, it was shown that the selective suppression of fundamental harmonics by 99% and second harmonics by 98% was possible, and finally, an enhancement case study showing the enhancement of all the harmonics by more than 100% is discussed. Thus, nonlinear detection capability in waveguide sensing can be improved by using these MM rods as mechanical acoustic filters.

In conclusion, this paper proposed a new concept for suppressing higher harmonics in nonlinear guided wave ultrasonics using a cylindrical ridged rod metamaterial. The concept used the metamaterial as a mechanical filter in between the transducer and the workpiece (bare aluminum rod) to remove the inherent nonlinearity before the high amplitude signal enters the rod to be inspected for defects or defect precursors. This methodology was verified through experiments where longitudinal waveguide ultrasonic modes were generated in a specimen with a metamaterial rod as an add-on filter attached to it. Suppression of higher harmonics was clearly demonstrated from the frequency domain signals for with and without the baffle type metamaterial of the aluminum sample. A discontinuity introduced in the workpiece was experimentally shown to provide a clear indication of a higher harmonic component. This paper also proposes the concept of selective suppression and enhancement of higher harmonics. A similar approach in the waveguide can be used to design an inverse filter that can be used at the reception side to enhance higher harmonics while reducing the fundamental frequency. This could increase the sensitivity of the nonlinear measurements from material nonlinearities to a greater extent. By adjusting the parameters of the baffles, such as height, pitch, and thickness, the range of the bandgap formed can be altered and thereby used for suppressing certain specific harmonics. Further to the initial proof of concept studies presented here, we believe that effective medium models such as those discussed in Ref. 50 could be used to gain physical insight and thus help design more well-tailored metamaterials suited to different frequency regimes. Moreover, effects such as Dirac cones (which could occur for higher order and non-axisymmetric guided wave modes having a two-dimensional wave path in the waveguide) and rainbow trapping⁵¹ could also provide rich avenues for further exploration of methodologies presented here.

The data that support the findings of this study are available from the corresponding author upon reasonable request.