Harmonic Lamb Waves Research Articles

Mixed third harmonic shear horizontal wave generation: interaction between primary shear horizontal wave and second harmonic Lamb wave

The third harmonic shear horizontal (SH) waves in a weakly nonlinear plate provide new possibilities for incipient damage detections. The understanding of their generation mechanism, however, is limited to the intuitive process of cubic self-interaction of the primary SH waves. By considering both the third order and fourth order elastic constants, this paper reports the discovery of a new generation mechanism, referred to as mixed generation, which results from the mutual interaction between the primary SH waves and their induced second harmonic Lamb waves. Compared with linearly cumulative third harmonic SH waves induced by the cubic self-interaction of the primary SH waves, the mixed third harmonic SH wave amplitude also increases with the wave propagating distance but in a wavering manner according to the phase velocity matching condition. Upon establishing a complex-domain superposition method which allows a precise extraction of the third harmonic responses, the significance and the propagating characteristics of the mixed third harmonic SH waves are numerically investigated through finite element simulations. Experiments are then conducted with a dedicated subtraction scheme to highlight the material nonlinearity of interest. A gel test is designed and carried out to identify the two types of third harmonic SH wave components from the measured time-domain signals. Both numerical and experimental results confirm the existence and the significance of the mixed third harmonic SH generation mechanism, which may impact on further applications, especially underwater damage inspections.

Application of an Optical Fiber Sensor for Nonlinear Ultrasonic Evaluation of Fatigue Crack

Nonlinear ultrasonic technique that evaluates the growth of the barely visible fatigue crack in metal is difficult to be achieved by conventional optical fiber sensor because of the critical requirements for sensor sensitivity and bandwidth. In this paper, a high-performance optical fiber sensor using phase-shifted fiber Bragg grating and balanced demodulation system was applied to detect the fundamental and second harmonic Lamb waves propagating in an aluminum plate. After the nonlinearity was calculated from the modes of the Lamb waveform detected by the optical fiber sensor, its change due to the second harmonic generation shows an exponential trend in a fatigue test, which can be used to evaluate the crack growth with sufficient reliability. The experimental results from the optical fiber sensor were validated by comparing with results from the PZT sensor. This paper facilitates the development of optics-based nonlinear ultrasonic structural health monitoring for fatigue crack evaluation.

RETRACTED: Closed form analytical solution for dispersion of Lamb waves in functionally graded plates

Abstract Propagation of harmonic Lamb waves in functionally graded (FG) plates with transverse inhomogeneity is analyzed by applying the modified Cauchy six-dimensional formalism. For arbitrary transverse inhomogeneity and arbitrary elastic anisotropy a closed form implicit analytical solution for the dispersion relation is derived. Dispersion relations for materials with exponential inhomogeneity are obtained.

Cauchy formalism for Lamb waves in functionally graded plates

Propagation of harmonic Lamb waves in plates made of functionally graded materials with transverse inhomogeneity is analyzed by applying Cauchy six-dimensional formalism previously developed for the study of Lamb wave propagation in homogeneous or stratified anisotropic plates. For anisotropic plates with arbitrary transverse inhomogeneity a closed form implicit solution for the dispersion equation is derived and analyzed.

Combination of Phase Matching and Phase-Reversal Approaches for Thermal Damage Assessment by Second Harmonic Lamb Waves.

It is known that measurement and extraction of the tiny amplitude of second harmonic Lamb waves are the main difficulties for practical applications of the nonlinear Lamb wave technique. In this study, phase-reversal approaches and phase matching technique are combined to build up the second-harmonic generation (SHG) of Lamb waves. A specific Lamb wave mode pair, which satisfied phase matching conditions, is selected to ensure the generation of cumulative second harmonic waves. Lamb wave signals with the same frequency but in reverse phase, propagating in the given specimen, are added together to counteract the fundamental waves, and simultaneously to enhance the signals of the second harmonic generated. The obtained results show that the phase-reversal approach can enhance the signals of second harmonic Lamb waves, and effectively counteract that of the fundamental waves. The approach is applied to assess the thermal-induced material degradation in the stainless steel plates. Distinctions of the acoustic nonlinearity parameters under different degraded levels are clearly shown in an improved repeatable and reliable manner, while those of linear wave velocity in the specimens are neglectable. The experimental investigations performed indicate that the proposed approach can be taken as a promising alternative for assessment of material degradation in its early stages.

Measuring of the Thickness of Thin Films with the Use of Harmonic Antisymmetric Lamb Waves

We propose a method for measuring the micrometer-range thicknesses of thin films with the use of acoustic waves. The method is based on the use of the nonlinear dependence of antisymmetric harmonic Lamb waves on the thickness of the tested material. The errors of measurements of the thickness of thin films are analyzed with the help of a contactless electromagnetoacoustic transducer of acoustic waves.

Lamb waves in functionally graded plates with transverse inhomogeneity

Propagation of harmonic Lamb waves in plates made of functionally graded materials with transverse inhomogeneity is studied by the modified Cauchy six-dimensional formalism. For arbitrary transverse inhomogeneity, a closed-form dispersion equation is derived. Dispersion relations for materials with different kinds of inhomogeneity are obtained and compared.

Symmetry properties of second harmonics generated by antisymmetric Lamb waves

Symmetry properties of second harmonics generated by antisymmetric primary Lamb waves are systematically studied in this work. In theory, the acoustic field of second harmonic Lamb waves is obtained by using the perturbation approximation and normal modal method, and the energy flux transfer from the primary Lamb waves to second harmonics is mainly explored. Symmetry analyses indicate that either the symmetric or antisymmetric Lamb waves can merely generate the symmetric second harmonics. Finite element simulations are performed on the nonlinear Lamb wave propagation of the antisymmetric A0 mode in the low frequency region. The signals of the second harmonics and the symmetric second harmonic s0 mode are found to be exactly equivalent in the time domain. The relative acoustic nonlinearity parameter A2/A12 oscillates with the propagation distance, and the oscillation amplitude and spatial period are well consistent with the theoretical prediction of the A0-s0 mode pair, which means that only the second harmonic s0 mode is generated by the antisymmetric primary A0 mode. Experiments are further conducted to examine the cumulative generation of symmetric second harmonics for the antisymmetric-symmetric mode pair A3-s6. Results show that A2/A12 increases linearly with the propagation distance, which means that the symmetric second harmonic s6 mode is generated cumulatively by the antisymmetric primary A3 mode. The present investigation systematically corroborates the proposed theory that only symmetric second harmonics can be generated accompanying the propagation of antisymmetric primary Lamb waves in a plate.

A constitutive model for the analysis of second harmonic Lamb waves in unidirectional composites

In recent research on Structural Health Monitoring (SHM) guided waves, especially nonlinear Lamb waves, turned out to be a suitable means for monitoring material deterioration in thin-walled structures. In the corresponding numerical simulations on wave propagation the nonlinear elastic theory by Murnaghan is often implemented, which requires 14 material parameters for transversely isotropic materials. Enhancing an existing linear strain energy potential, a new nonlinear hyperelastic transversely isotropic material model is introduced which reduces the number of independent material parameters to six. In order to verify the applicability of the presented material model with respect to the simulation of nonlinear wave propagation in composite structures, and the generation of higher harmonic wave modes, the existence of a power flux from the fundamental to the higher harmonic mode is investigated analytically and numerically. Analytical considerations show that this power flux exists like in Murnaghan’s theory. For the numerical validation the S0–S0 mode pair in the low frequency range is used. Therefore, the amplitude of the second harmonic wave mode is oscillating with increasing propagation distance. This behavior is in excellent agreement with the theoretical prediction. It is shown further, that even for an oscillating behavior the amplitude of the second harmonic mode can be approximated by a linear curve fit over a considerably propagation distance and hence shows a quasi cumulative behavior. Therefore, the introduced material model is an advantageous alternative to Murnaghan’s theory to simulate the second harmonic Lamb wave generation due to in composite structures.

Characterization of Nonplanar Second Harmonic Lamb Waves With a Refined Nonlinear Parameter

Structural health monitoring (SHM) methods based on the cumulative second harmonic Lamb waves show attractive advantages. An ideal nonlinear parameter should allow precise characterization of the cumulative effects of the distributed nonlinear sources such as the material nonlinearity of a plate (MNP), in the presence of other unavoidable localized nonlinear components. While highlighting the deficiencies of the traditional nonlinear parameter (TNP) in the nonplanar cases, a refined nonlinear parameter (RNP) is proposed. Through compensations for the wave attenuation associated with the wave divergence, the new parameter entails a better characterization and differentiation of the cumulative MNP and other noncumulative localized nonlinear sources. Theoretical findings are ascertained by both finite element (FE) simulations and experiments, through tactically adjusting the dominance level of different nonlinear sources in the system. Results confirm the appealing features of the proposed RNP for SHM applications.

The use of lamb waves for measuring the thicknesses of thin metal films

An experimental device that allows the determination of the thicknesses of thin metal films in the micron range is described. The method is based on the use of the dependence of the phase velocity of harmonic antisymmetric Lamb waves that propagate along a film on its thickness. The measurement accuracy was 8% and was virtually independent of the film thickness.

Adhesive nonlinearity in Lamb-wave-based structural health monitoring systems

Structural health monitoring (SHM) techniques with nonlinear Lamb waves have gained wide popularity due to their high sensitivity to microstructural changes for the detection of damage precursors. Despite the significant progress made, various unavoidable nonlinear sources in a practical SHM system, as well as their impact on the detection, have not been fully assessed and understood. For the real-time and online monitoring, transducers are usually permanently bonded on the structure under inspection. In this case, the inherent material nonlinear properties of the bonding layer, referred to as adhesive nonlinearity (AN), may create undesired interference to the SHM system, or even jeopardize the damage diagnosis if they become serious. In this paper, a nonlinear theoretical framework is developed, covering the process of wave generation, propagation and sensing, with the aim of investigating the mechanism and characteristics of AN-induced Lamb waves in plates, which potentially allows for further system optimization to minimize the influence of AN. The model shows that an equivalent nonlinear normal stress is generated in the bonding layer due to its nonlinear material behavior, which, through its coupling with the system, is responsible for the generation of second harmonic Lamb waves in the plate, subsequently resulting in the nonlinear responses in the captured signals. With the aid of the finite element (FE) modeling and a superposition method for nonlinear feature extraction, the theoretical model is validated in terms of generation mechanism of the AN-induced wave components as well as their propagating characteristics. Meanwhile, the influence of the AN is evaluated by comparing the AN-induced nonlinear responses with those caused by the material nonlinearity of the plate, showing that AN should be considered as a non-negligible nonlinear source in a typical nonlinear Lamb-wave-based SHM system. In addition, the theoretical model is also experimentally validated in terms of the frequency tuning characteristics of the AN-induced wave components. A fairly good agreement is found among the theoretical model, FE model and the experiments, thus confirming the theoretically predicted AN-induced wave generation mechanism and their characteristics.

Second Harmonic Generation of Lamb Wave in Numerical Perspective**Supported by the National Natural Science Foundation of China under Grant Nos 51325504, 11474093, 11622430 and 11474361, the National Key Research and Development Program of China (2016YFC0801903-02), and the Fundamental Research Funds for the Central Universities.

The influences of phase and group velocity matching on cumulative second harmonic generation of Lamb waves are investigated in numerical perspective. Finite element simulations of nonlinear Lamb wave propagation are performed for Lamb wave mode pairs with exact and approximate phase velocity matching, with and without group velocity matching, respectively. The evolution of time-domain second harmonic Lamb waves is analyzed with the propagation distance. The amplitudes of primary and second harmonic waves are calculated to characterize the acoustic nonlinearity. The results verify that phase velocity matching is necessary for generation of the cumulative second harmonic Lamb wave in numerical perspective, while group velocity matching is demonstrated to not be a necessary condition.

A facility for measuring the thickness of thin metal films

A facility is described for measuring the thickness of a thin metal film. A measurement method is proposed that is based on the dependence of the phase velocity of harmonic antisymmetric Lamb waves that propagate along the film on the film thickness. A continuous method for excitation of acoustic waves has been applied to velocity measurements. It makes it possible to eliminate the effect of dispersion of the Lamb-wave velocity on the measurement accuracy.

On the detection of fatigue damage in composites by use of second harmonic guided waves

In composite structures cyclic loading leads to micro-structural damage even at an early stage. This damage grows over lifetime and may finally cause catastrophic failure. In this study nonlinear wave propagation based on the cumulative second harmonic Lamb wave generation is used to detect micro-structural damage and the corresponding material degradation experimentally. It is shown that an increasing relative acoustical nonlinearity parameter indicates a decreasing Young’s modulus. This result is analyzed more in detail by numerical simulations of the nonlinear wave propagation as well as by Young’s modulus determination. Therefore, horizontal cracks, detected by microscopy in damaged material, are added to the waveguide with contact boundary conditions. The numerical analysis shows a higher harmonic mode generation due to the interaction of the waves and the cracks and an increasing nonlinear effect with increasing crack sizes and numbers. Since the stiffness is not affected by horizontal cracks the decreasing Young’s modulus is assumed to be related to non-visible damage like material degradation, leading also to an increasing nonlinear effect of the wave propagation. In conclusion, the relative acoustical nonlinearity parameter can be used as an indicator of the material degradation and, therefore, for monitoring of micro-structural damage.

Numerical simulation of elastic wave propagation in isotropic media considering material and geometrical nonlinearities

In order to detect micro-structural damages accurately new methods are currently developed. A promising tool is the generation of higher harmonic wave modes caused by the nonlinear Lamb wave propagation in plate like structures. Due to the very small amplitudes a cumulative effect is used. To get a better overview of this inspection method numerical simulations are essential. Previous studies have developed the analytical description of this phenomenon which is based on the five-constant nonlinear elastic theory. The analytical solution has been approved by numerical simulations. In this work first the nonlinear cumulative wave propagation is simulated and analyzed considering micro-structural cracks in thin linear elastic isotropic plates. It is shown that there is a cumulative effect considering the S1–S2 mode pair. Furthermore the sensitivity of the relative acoustical nonlinearity parameter regarding those damages is validated. Furthermore, an influence of the crack size and orientation on the nonlinear wave propagation behavior is observed. In a second step the micro-structural cracks are replaced by a nonlinear material model. Instead of the five-constant nonlinear elastic theory hyperelastic material models that are implemented in commonly used FEM software are used to simulate the cumulative effect of the higher harmonic Lamb wave generation. The cumulative effect as well as the different nonlinear behavior of the S1–S2 and S2–S4 mode pairs are found by using these hyperelastic material models.It is shown that, both numerical simulations, which take into account micro-structural cracks on the one hand and nonlinear material on the other hand, lead to comparable results. Furthermore, in comparison to the five-constant nonlinear elastic theory the use of the well established hyperelastic material models like Neo–Hooke and Mooney–Rivlin are a suitable alternative to simulate the cumulative higher harmonic generation.

Third harmonic shear horizontal and Rayleigh Lamb waves in weakly nonlinear plates

The third order harmonic generation (third harmonics as well as cubic sum and difference harmonics) due to the cubic interaction of two collimated elastic waves in a homogeneous, isotropic, weakly nonlinear plate is investigated by using a fourth order expansion of strain energy density to formulate the nonlinear boundary problems. Waves with both shear horizontal (SH) and Rayleigh Lamb (RL) nature are considered as primary or tertiary wave fields. The non-zero power flux condition is evaluated using characteristic parity matrices of the cubic nonlinear forcing terms and third order harmonic mode shapes. Results indicate that waves with either SH or RL nature receive power flux from a specific pattern of primary mode interaction. Further analytical evaluation of the synchronism condition enables identification of primary SH and RL modes that are able to generate cumulative third harmonics. The primary SH modes are shown to be holo-internal-resonant with third harmonic SH fields. This simply means that all points on the primary dispersion curves are internally resonant with third harmonics, which is not the case for second harmonics. Such flexibility will be advantageous for laboratory and field measurements.

Detection of thermal fatigue in composites by second harmonic Lamb waves

Composite materials which are widely used in the aerospace industry, are usually subjected to frequent variation of temperature. Thermal cyclic loading may induce material degradation. Considering the long-term service of aircraft composites and the importance of safety in the aircraft industry, even a little damage that may be accumulative via thermal fatigue is often of great concern. Therefore, there is a demand to develop non-destructive approaches to evaluate thermal fatigue damage in an early stage. Due to the sensitivity of acoustic nonlinearity to micro-damage, the nonlinear ultrasonic technique has been explored as a promising tool for early detection of micro-damage. This paper investigates an experimental scheme for characterizing thermal fatigue damage in composite laminates using second harmonic Lamb waves. The present results show a monotonic increase of acoustic nonlinearity with respect to thermal fatigue cycles. The experimental observation of the correlation between the acoustic nonlinearity and thermal fatigue cycles in carbon/epoxy laminates verifies that nonlinear Lamb waves can be used to assess thermal fatigue damage rendering improved sensitivity over conventional linear feature based non-destructive evaluation techniques. Velocity and attenuation based ultrasonic studies are carried out for comparison with the nonlinear ultrasonic approach and it is found that nonlinear acoustic parameters are more promising indicators of thermal fatigue damage than linear ones.

Experimental investigation of symmetry properties of second harmonic Lamb waves

Recent studies by multiple authors show contradictory results concerning the symmetry properties of second harmonic Lamb waves. This research experimentally investigates this symmetry issue by examining both symmetric (s1-s2) and anti-symmetric (a1-a2 and a2-a4) mode pairs in aluminum plates under the same experimental conditions. The wedge technique is used to generate and detect ultrasonic Lamb wave signals of a specific mode, and the Morlet wavelet is applied to extract the fundamental and second harmonic amplitudes. The measured normalized second harmonic amplitudes of the three different mode pairs all show a linear increase with propagation distance. However, the slopes of the two anti-symmetric mode pairs are smaller by two to three orders of magnitude than that of the symmetric mode pair considered. Further investigations of these two anti-symmetric mode pairs for plates with different levels of material nonlinearity reveal consistently small slopes that are independent of the level of material nonlinearity. Therefore, this research experimentally demonstrates that the second harmonic generation in anti-symmetric Lamb wave modes is extremely inefficient; this result is consistent with some recent theoretical predictions and thus shows that the use of these anti-symmetric modes is not favorable for the experimental characterization of material nonlinearity.

Rayleigh–Lamb wave propagation on a fractional order viscoelastic plate

A previous study of the authors published in this journal focused on mechanical wave motion in a viscoelastic material representative of biological tissue [Meral et al., J. Acoust. Soc. Am. 126, 3278-3285 (2009)]. Compression, shear and surface wave motion in and on a viscoelastic halfspace excited by surface and sub-surface sources were considered. It was shown that a fractional order Voigt model, where the rate-dependent damping component that is dependent on the first derivative of time is replaced with a component that is dependent on a fractional derivative of time, resulted in closer agreement with experiment as compared with conventional (integer order) models, such as those of Voigt and Zener. In the present study, this analysis is extended to another configuration and wave type: out-of-plane response of a viscoelastic plate to harmonic anti-symmetric Lamb wave excitation. Theoretical solutions are compared with experimental measurements for a polymeric tissue mimicking phantom material. As in the previous configurations the fractional order modeling assumption improves the match between theory and experiment over a wider frequency range. Experimental complexities in the present study and the reliability of the different approaches for quantifying the shear viscoelastic properties of the material are discussed.