Finite-size effects on the quasistatic displacement pulse in a solid specimen with quadratic nonlinearity

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.

Application of Odd Harmonic Resonances of a Single Crystal to Generation and Reception of Superharmonic Waves for Sensitive Monitoring of Heat-Treated Materials

In nonlinear ultrasonic testing, the quadratic and more recently cubic nonlinearity parameters are frequently measured as a quantitative indicator of damaged material state. Application of higher-order harmonics can improve the sensitivity of detection and monitoring for damages and microstructures due to their higher values of nonlinearity parameters. The excitation and reception of higher-order harmonics, so-called superharmonics, which use the third to fifth harmonics arising from nonlinear wave propagation, is not sufficiently investigated and applied. The purpose of this communication is to develop a highly sensitive superharmonic nondestructive technique that efficiently generates and receives third- and fifth-order harmonics using the odd harmonic resonances of a single piezoelectric crystal. The method focuses on the measurement of fifth harmonic generation and reception, and the calculation of the relative quintic nonlinearity parameter (δ'). The method also addresses the issue of source nonlinearity that may be contained in the measured fifth harmonic amplitude. The measurement results of δ' for a series of precipitation heat-treated samples clearly show a much better sensitivity than the results of the cubic nonlinearity parameter (γ'). The proposed method enables a highly sensitive and true pulse-echo mode nonlinear ultrasound testing.

Microcrack localization based on static component induced by a primary A0 Lamb wave in a thin plate

A microcrack localization method based on a static component (SC) induced by a primary A0 Lamb wave is proposed. Based on the bilinear stress-strain constitutive model, a two-dimensional finite element model is built to investigate the interaction between microcracks and Lamb waves. The A0 Lamb wave at low frequency is selected to be the primary Lamb wave, which is beneficial to microcracks localization. Based on the time of flight of the generated SC pulse, an indicator named normalized amplitude index is defined for finding the location and number of microcracks. Simulation results show that one or multiple microcracks can be effectively located.

Modeling and simulation of static component generation of Lamb wave propagation in a layered plate

Modeling of the acoustic-radiation-induced static component (SC) generation of primary Lamb wave tone burst propagating in a layered plate is conducted. Accompanying the propagation of primary Lamb wave tone burst, there are the finite-duration SC bulk driving force in the interior of the layered plate, and the finite-duration SC traction stress on each surface/interface. According to the modal analysis approach for waveguide excitation, the function of the finite-duration SC bulk driving force and traction stress is to generate the finite-duration SC of primary Lamb wave tone burst. Compared with the second harmonic (SH) generation of Lamb wave propagation in a layered plate, the phase velocity matching is no longer required for the generation of the SC with a cumulative growth effect. Based on modeling of the finite-duration SC generation, it is found that the integrated amplitude of the finite-duration SC generated by propagation of the primary Lamb wave tone burst does grow with propagation distance. Meanwhile, the numerical analyses and the finite element (FE) simulations are conducted to investigate the effect of the said SC generation. It is found that, although the interfacial layer of the layered plate considered is quite thin compared with the upper and lower layers, the numerical analyses indicate that the influence of the interfacial property on the cumulative growth effect of the SC of primary Lamb wave is significant. Furthermore, the FE simulations demonstrate that the cumulative SC of primary Lamb wave tone burst will exhibit a monotonic and sensitive response to the degree of interfacial degradation. This investigation provides an physical insight not previously available into the process of the SC generation of Lamb wave propagation in a layered plate.

Static component generation and measurement of nonlinear guided waves with group velocity mismatch

This study focuses on static component generation (SCG) and its measurement wherein a group velocity mismatch (GVM) exists between the primary guided wave and the generated static component (SC). The SCGs by primary S0, A0, and SH0 waves are investigated. It is confirmed that the SCs are S0 mode. The GVM causes the temporal waveforms of the SCs to tend to increase in width with propagation distance. A feasible method is proposed accordingly for measurement of SCG with GVM using only lead zirconic titanate based transducers, wherein the SCs generated by two counter-propagating primary waves are modulated and superposed on each other.

Optimal Design of Annular Phased Array Transducers for Material Nonlinearity Determination in Pulse-Echo Ultrasonic Testing

Nonlinear ultrasound has been proven to be a useful nondestructive testing tool for micro-damage inspection of materials and structures operating in harsh environment. When measuring the nonlinear second harmonic wave in a solid specimen in the pulse-echo (PE) testing mode, the stress-free boundary characteristics brings the received second harmonic component close to zero. Therefore, the PE method has never been employed to measure the so-called "nonlinear parameter (β)", which is used to quantify the degree of micro-damage. When there are stress-free boundaries, a focused beam is known to improve the PE reception of the second harmonic wave, so phased-array (PA) transducers can be used to generate the focused beam. For the practical application of PE nonlinear ultrasonic testing, however, it is necessary to develop a new type of PA transducer that is completely different from conventional ones. In this paper, we propose a new annular PA transducer capable of measuring β with improved second harmonic reception in the PE mode. Basically, the annular PA transducer (APAT) consists of four external ring transmitters and an internal disk receiver at the center. The focused beam properties of the transducers are analyzed using a nonlinear sound beam model which incorporates the effects of beam diffraction, material attenuation, and boundary reflection. The optimal design of the APAT is performed in terms of the maximum second harmonic reception and the total correction close to one, and the results are presented in detail.

The zero-frequency component of bulk waves in solids with randomly distributed micro-cracks

When a longitudinal wave (bulk wave) propagates in elastic solids with randomly distributed micro-cracks, the acoustic nonlinear behavior including the zero-frequency component and higher harmonics can be generated due to the clapping and slipping behavior of micro-cracks. In this paper, the analytical solution based on the bi-linear stiffness model of micro-cracks and the numerical simulation with random micro-crack modeling are implemented to investigate the behavior of the zero-frequency component. The theoretical and numerical results both show that the zero-frequency component of bulk waves can be generated by the micro-cracks, which is more sensitive than the conventional second harmonics. Meanwhile, we find that the acoustic nonlinearity parameter based on the zero-frequency component increases linearly with the crack density, the length of the micro-crack region and the fundamental frequency in the low-frequency region. Moreover, the zero-frequency component of the reflected waves is also investigated, indicating it can be used to locate the micro-crack region.

Ultrasonic nonlinearity parameter in uniaxial stress condition

The ultrasonic nonlinearity parameter derived for one-dimensional propagation of a longitudinal wave in an isotropic material has been considered useful in the evaluation of material degradation. To demonstrate this, many researchers have reported on the correlation with the yield strength obtained from a tensile test. However, there is an essential issue with this procedure - which is that the ultrasonic nonlinearity parameter is derived in a state where the lateral strain is restrained, whereas the tensile test to measure the yield strength is carried out under uniaxial stress conditions, where lateral deformation is free. In this study, to address this issue, the authors have defined the ultrasonic nonlinearity parameter under uniaxial stress conditions which were the same as the tensile test, and showed that the correlation with the yield strength was higher than the currently used ultrasonic nonlinearity parameter. To verify the validity of the proposed ultrasonic nonlinearity parameter, experiments were carried out for Al6061-T6 alloy specimens heat-treated with different aging times. Results showed that the proposed ultrasonic nonlinearity parameter exhibited a much higher correlation with yield strength than the currently used nonlinearity parameter.

Interaction of Lamb Wave Modes with Weak Material Nonlinearity: Generation of Symmetric Zero-Frequency Mode

The symmetric zero-frequency mode induced by weak material nonlinearity during Lamb wave propagation is explored for the first time. We theoretically confirm that, unlike the second harmonic, phase-velocity matching is not required to generate the zero-frequency mode and its signal is stronger than those of the nonlinear harmonics conventionally used, for example, the second harmonic. Experimental and numerical verifications of this theoretical analysis are conducted for the primary S₀ mode wave propagating in an aluminum plate. The existence of a symmetric zero-frequency mode is of great significance, probably triggering a revolutionary progress in the field of non-destructive evaluation and structural health monitoring of the early-stage material nonlinearity based on the ultrasonic Lamb waves.

Second harmonic reflection and transmission from primary S0 mode Lamb wave interacting with a localized microscale damage in a plate: A numerical perspective

Second harmonic generation has been widely used in characterizing microstructural changes which are evenly distributed in a whole structure. However, few attention has been paid to evaluating localized micro-scale damages. In this paper, second harmonic reflection and transmission from the primary S0 mode Lamb wave interacting with a localized microstructural damage is numerically discussed. Schematic diagram for deriving fundamental temporal waveform and reconstructing the second harmonic temporal waveform based on Morlet wavelet transform is presented. Second harmonic reflection and transmission from an interface between the zones of linear elastic and nonlinear materials is firstly studied to verify the existence of interfacial nonlinearity. Compositions contributing to second harmonic components in the reflected and transmitted waves are analyzed. Amplitudes of the reflected and transmitted second harmonic components generated at an interface due to the interfacial nonlinearity are quantitatively evaluated. Then, second harmonic reflection and transmission from a localized microscale damage is investigated. The effects of the length and width of a microscale damage on WCPA (wavelet coefficient profile area) of the reflected and transmitted second harmonic components are studied respectively. It is found that the second harmonic component in the reflected waves mainly reflects the interfacial nonlinearity while second harmonic in the transmitted waves reflects the material nonlinearity. These findings provide some basis on using second harmonic generation for characterization and detection of localized microstructural changes.