Analytical and numerical investigations of one-way mixing of Lamb waves in a thin plate

Experimental observation of zero-group velocity combined harmonic generated by counter-directional Lamb wave mixing

In this paper, we present an experimental observation of the phenomenon known as zero-group velocity (ZGV) combined harmonic generation, which is induced by the mixing of counter-directional Lamb waves. We utilize internal resonant conditions to selectively choose the primary mode pair at specific frequencies for the purpose of combined harmonic generation. To detect the ZGV combined harmonic component, we propose a hybrid system that incorporates dual wedge-transducers for generation and a laser interferometric system for receiving. The appearance of the predicted S1-ZGV combined harmonic at a specific mixing frequency is clearly observed in our experiments. Furthermore, we experimentally verify the controllability of the generated combined harmonics induced by the mixing of Lamb waves.

Investigation of the Zero-Frequency Component of Nonlinear Lamb Waves in a Symmetrical Undulated Plate

When an ultrasonic pulse propagates in a thin plate, nonlinear Lamb waves with higher harmonics and a zero-frequency component (ZFC) will be generated because of the nonlinearity of materials. The ZFC, also known as the static displacement or static component, has its unique application on the evaluation of early-stage damages in the elastic symmetrical undulated plate. In this study, analysis of the excitation mechanism of the ZFC and the second harmonic component (SHC) was theoretically and numerically investigated, and the material early-stage damage of a symmetrical undulated was characterized by studying the propagation of nonlinear Lamb waves. Both the ZFC and SHC can be effectively employed in monitoring the material damages of the undulated plate in its early stage. However, several factors must be considered for the propagation of the SHC in an undulated plate because of the geometric curvature and interference between the second harmonics during propagation, preventing efficient application of this technique. If the fundamental wave can propagate in the plate regardless of the plate boundary conditions, an accumulative effect always exists for the ZFC in a thin plate, indicating that the ZFC is independent of the structural geometry. This study reveals that the ZFC-based inspection technique is more efficient and powerful in characterizing the damages of a symmetrical undulated plate in the early stage of service compared to the second harmonic method.

Modeling and simulation of zero-group velocity combined harmonic generated by guided waves mixing

In this paper, modelling and numerical perspective of zero-group velocity (ZGV) combined harmonic generated by guided waves mixing are investigated. The conditions for the generation of the ZGV combined harmonic are analyzed by S₀-S₀ and SH₀-SH₀ guided waves mixing in an isotropic plate, respectively. The generation of ZGV combined harmonics at sum frequency caused by counter-directional guided waves mixing is observed. It is confirmed that the ZGV combined harmonic with a considerable magnitude can be generated by this counter-directional guided waves mixing when both the internal resonant condition and non-zero power flux are satisfied. The application of generated ZGV combined harmonics for localized material degradation assessment is numerically examined in the given plate. The obtained results indicate that the generated ZGV combined harmonic induced by the counter-directional guided waves mixing can be used to assess the localized material degradation with improved signal-to-noise ratio. This study provides an insight into the physical process of the ZGV combined harmonic generation, and meanwhile offer a promising means for localized material degradation assessment by ZGV combined harmonics generated by guided waves mixing.

Evaluation of Plastic Deformation Considering the Phase-Mismatching Phenomenon of Nonlinear Lamb Wave Mixing

Nonlinear guided elastic waves have attracted extensive attention owing to their high sensitivity to microstructural changes. However, based on the widely used second harmonics, third harmonics and static components, it is still difficult to locate the micro-defects. Perhaps the nonlinear mixing of guided waves can solve these problems since their modes, frequencies and propagation direction can be flexibly selected. Note that the phenomena of phase mismatching usually occur due to the lack of precise acoustic properties for the measured samples, and they may affect the energy transmission from the fundamental waves to second-order harmonics as well as reduce the sensitivity to micro-damage. Therefore, these phenomena are systematically investigated to more accurately assessing the microstructural changes. It is theoretically, numerically, and experimentally found that the cumulative effect of difference- or sum-frequency components will be broken by the phase mismatching, accompanied by the appearance of the beat effect. Meanwhile, their spatial periodicity is inversely proportional to the wavenumber difference between fundamental waves and difference- or sum-frequency components. The sensitivity to micro-damage is compared between two typical mode triplets that approximately and exactly meet the resonance conditions, and the better one is utilized for assessing the accumulated plastic deformations in the thin plates.

One-way Lamb and SH mixing method in thin plates with quadratic nonlinearity: Numerical and experimental studies

This paper numerically and experimentally investigates the resonant behavior of one-way Lamb and SH (shear horizontal) mixing method in thin plates with quadratic nonlinearity. When the primary S₀-mode Lamb waves and SH₀ waves mix in the region with quadratic nonlinearity, both numerical and experimental results verify the generation of the resonant SH₀ waves if the resonance condition [Formula: see text] is satisfied. Meanwhile, we find that the acoustic nonlinear parameter (ANP) increases monotonously with material quadratic nonlinearity, the length of the damage region and the frequency of the resonant wave. Furthermore, the damage region can be located by the time-domain signal of the resonant wave based on one-way S₀-SH₀ mixing method. This study numerically and experimentally reveals that one-way Lamb and SH mixing method is feasible to quantitatively evaluate and locate the damage region of quadratic nonlinearity in thin plates.

Low-frequency Lamb wave mixing for fatigue damage evaluation using phase-reversal approach

Fatigue damage is difficult to detect and evaluate non-destructively, specifically at its early stages (before the macro-crack formation). In this study, fatigue damage is evaluated based on the growth rate of the combinational harmonics generated by mixing of two fundamental symmetric mode (S₀) of Lamb waves in the low frequency range. The incorporation of the phase reversal approach to the wave mixing method could potentially improve the evaluation of the combinational and second harmonics and avoid the influence of other undesirable harmonics. A series of parametric case studies are carried out using the three-dimensional (3D) finite element (FE) method to investigate the effects of the excitation frequencies and time delay of the incident waves in wave mixing on the transient response of a weakly-nonlinear material. The numerical results and experimental results show that the sum combinational harmonic and second harmonics are sensitive to weak material nonlinearities. Further experiments on damaged samples by cyclic loading demonstrate that the sum combinational harmonic has much better sensitivity to the progressive fatigue damage than the the second harmonics. In general, the outcomes of this study indicate that the damage evaluation of early stage fatigue damage is feasible and effective with the wave mixing method using the S₀ waves generated at low frequency, and the phase-reversal approach improves considerably the quality of experimental results in the fatigue damage evaluation.

Cubic nonlinearity parameter measurement and material degradation detection using nonlinear ultrasonic three-wave mixing

Early detection of accumulated damage or material degradation in structures is important to ensure their structural safety. Nonlinear ultrasonic techniques are widely used to measure the quadratic nonlinearity parameter that represents the third-order elastic constants of materials for material degradation detection. In addition, there are ongoing efforts to exploit both the third- and fourth-order elastic constants that describe the cubic nonlinearity parameter to detect material degradation. This study develops a nonlinear ultrasonic three-wave mixing technique to generate and measure third-order combined harmonics (TOCH) in plate-like structures and to measure cubic nonlinearity parameter. The proposed three-wave mixing technique generates three primary Lamb waves in the target structure and measures the TOCH produced by nonlinear cross-interaction of the primary Lamb waves. A theoretical model is developed to describe the generation of TOCH in a nonlinear elastic and homogeneous plate, and the effectiveness of the theoretical model is validated experimentally. Measurements of the TOCH are conducted for intact and degraded aluminum specimens with different degradation levels. Because inherent material nonlinearity and material degradation alter the third- and fourth-order elastic constants of a structure, the three-wave mixing technique for measuring the TOCH can be used to identify the inherent material nonlinearity and material degradation. In particular, the experimental results indicate that the proposed technique is more sensitive to early-stage material degradation than existing nonlinear ultrasonic techniques such as two-wave mixing, and third harmonic generation techniques.

A Feasibility Study for a Nonlinear Guided Wave Mixing Technique

Ultrasonic non-destructive testing is an effective means of examining objects without destroying them. Among such testing, ultrasonic nonlinear evaluation is used to detect micro-damage, such as corrosion or plastic deformation. In terms of micro-damage evaluation, the data that comes from amplitude comparison in the frequency domain plays a significant role. Its technique and parameter are called ultrasonic nonlinear technique and nonlinearity. A certain portion of nonlinearity comes from the equipment system, while the other portion of nonlinearity comes from the material. The former is system nonlinearity, while the latter is material nonlinearity. System nonlinearity interferes with interpretation, because its source is not from the material. In this study, in order to minimize system effects, a mixing technique is implemented. To use the large area inspection ability of the guided wave, the main research issue in this paper is focused on the guided wave mixing technique. Moreover, several bulk wave mixing theory equations become good concepts for guided wave mixing theoretical study, and the conventional nonlinear technique and guided wave mixing experimental results are compared in this study to confirm the reliability. This technique can play an important role in quantitatively discriminating fine damage by minimizing the nonlinearity of the equipment system.

Modeling and simulation of backward combined harmonic generation induced by one-way mixing of longitudinal ultrasonic guided waves in a circular pipe

Nonlinear guided wave has been recognized as a potential means to characterize the state of material microstructures in solids. However, nonlinear guided wave approaches based on second harmonic generation or combined harmonic generation induced by counter-directional wave mixing, are applicable only under the condition that transmitters and receivers are placed at two different ends of the tested structures. These approaches are not effective for characterization of the structures with only one accessible end. In this paper, modeling of the backward combined harmonic at difference frequency induced by one-way mixing of two primary co-directional guided waves is investigated in a circular pipe, where the transmitters and receivers are placed at the same end of the pipe. The backward combined harmonic, generated at difference frequency and propagating in the direction opposite to that of two primary co-directional guided waves, is successfully observed numerically. A strong frequency mixing response characterized by a cumulative growth effect of the generated backward combined harmonic is demonstrated. The use of the generated backward combined harmonic for localized material degradation characterization and location is numerically examined in the given pipe. The obtained results indicate that the use of the backward combined harmonic can locate and characterize the localized material degradations in the given pipe by controlling the mixing zone of two primary co-directional guided waves. This study provides a promising means for characterization of localized degradations in pipes.

Experimental and Numerical Investigation of the Micro-Crack Damage in Elastic Solids by Two-Way Collinear Mixing Method

This study experimentally and numerically investigated the nonlinear behavior of the resonant bulk waves generated by the two-way collinear mixing method in 5052 aluminum alloy with micro-crack damage. When the primary longitudinal and transverse waves mixed in the micro-crack damage region, numerical and experimental results both verified the generation of resonant waves if the resonant condition ωL/ωT=2κ/(κ−1) was satisfied. Meanwhile, we found that the acoustic nonlinearity parameter (ANP) increases monotonously with increases in micro-crack density, the size of the micro-crack region, the frequency of resonant waves and friction coefficient of micro-crack surfaces. Furthermore, the micro-crack damage in a specimen generated by low-temperature fatigue experiment was employed. It was found that the micro-crack damage region can be located by scanning the specimen based on the two-way collinear mixing method.

Forced guided waves in linearly elastic plates (II) - A modified normal-mode expansion method

The classical normal-mode expansion method developed by Auld and Kino in 1973 has been used widely by numerous researchers and practitioners to study forced guided waves in plates and layered media, including both elastic and electromagnetic waves. However, it was shown in Part I of this study that, although this method gives the correct solution when applied to horizontally polarized shear waves and electromagnetic waves, it does not yield the correct elastodynamic solution when applied to Lamb waves. To address this shortcoming, we develop in this paper a modified normal-mode expansion method that yields the correct elastodynamic solution in that the solution satisfies all the elastodynamic governing equations and the boundary conditions for forced Lamb waves in a plate. The efficacy of the modified normal-mode expansion method is further confirmed by comparing its solution with the finite element solution. Further, it is showed that when applied to horizontally polarized shear waves, the modified normal-mode expansion method yields numerically the same result as that of the classical normal-mode expansion method. However, the modified normal-mode expansion solution converges much faster.