Contact Acoustic Nonlinearity Research Articles

Time reversal invariance for a nonlinear scatterer exhibiting contact acoustic nonlinearity

The time reversal invariance of an ultrasonic plane wave interacting with a contact interface characterized by a unilateral contact law is investigated analytically and numerically. It is shown analytically that despite the contact nonlinearity, the re-emission of a time reversed version of the reflected and transmitted waves can perfectly recover the original pulse shape, thereby demonstrating time reversal invariance for this type of contact acoustic nonlinearity. With the aid of finite element modelling, the time-reversal analysis is extended to finite-size nonlinear scatterers such as closed cracks. The results show that time reversal invariance holds provided that all the additional frequencies generated during the forward propagation, such as higher harmonics, sub-harmonics and zero-frequency component, are fully included in the retro-propagation. If the scattered waves are frequency filtered during receiving or transmitting, such as through the use of narrowband transducers, the recombination of the time-reversed waves will not exactly recover the original incident wave. This discrepancy due to incomplete time invariance can be exploited as a new method for characterizing damage by defining damage indices that quantify the departure from time reversal invariance. The sensitivity of these damage indices for various crack lengths and contact stress levels is investigated computationally, indicating some advantages of this narrowband approach relative to the more conventional measurement of higher harmonic amplitude, which requires broadband transducers.

Contact acoustic nonlinearity (CAN)-based continuous monitoring of bolt loosening: Hybrid use of high-order harmonics and spectral sidebands

Abstract The significance of evaluating bolt tightness in engineering structures, preferably in a continuous manner, cannot be overemphasized. With hybrid use of high-order harmonics (HOH) and spectral sidebands, a contact acoustic nonlinearity (CAN)-based monitoring framework is developed for detecting bolt loosening and subsequently evaluating the residual torque on a loose bolt. Low-frequency pumping vibration is introduced into the bolted joint to produce a “breathing” effect at the joining interface that modulates the propagation characteristics of a high-frequency probing wave when it traverses the bolt, leading to the generation of HOH and vibro-acoustic nonlinear distortions (manifested as sidebands in the signal spectrum). To gain insight into the mechanism of CAN generation and to correlate the acquired nonlinear responses of a loose joint with the residual torque remaining on the bolt, an analytical model based on micro-contact theory is established. Two types of nonlinear index, respectively exploiting the induced HOH and spectral sidebands, are defined without dependence on excitation intensity and are experimentally demonstrated to be effective in continuously monitoring bolt loosening in both aluminum-aluminum and composite-composite bolted joints. Taking a step further, variation of the index pair is quantitatively associated with the residual torque on a loose bolt. The approach developed provides a reliable method of continuous evaluation of bolt tightness in both composite and metallic joints, regardless of their working conditions, from early awareness of bolt loosening at an embryonic stage to quantitative estimation of residual torque.

Numerical study on sub-harmonic generation due to interior and surface breaking cracks with contact boundary conditions using time-domain boundary element method

Recently, the ultrasonic testing (UT) based on the contact acoustic nonlinearity (CAN) has attracted notice as a new technique for detection of closed cracks which cannot be detected by using the linear UT (LUT). In the nonlinear UT (NLUT), detection and size measurement of flaws are conducted via the frequency spectrum analysis of nonlinear ultrasonic waves which consist of higher- and sub-harmonic waves. Although the mechanism of higher-harmonic generation due to CAN has been understood mostly from theoretical and experimental points of view, that of sub-harmonic generation has not been revealed yet. In addition, there are few numerical and theoretical studies on the phenomenon of sub-harmonic generation due to the CAN. In this paper, the boundary integral equation for two-dimensional (2-D) elastic wave scattering by cracks is formulated and numerically solved to investigate the behavior of the sub-harmonic waves. The clapping motion and dynamic friction on crack faces are modeled by considering contact boundary conditions, and their interaction with the characteristic of frequency response in the corresponding linear system is investigated. In this study, time-domain numerical simulations are implemented for both an interior crack in an unbounded elastic solid and a surface breaking crack in an elastic half-space. Some frequency response analyses for a linear system corresponding to the time-domain nonlinear simulations are also performed by using the frequency-domain boundary element method. From obtained numerical results, the causes of sub-harmonic generation are discussed.

Vibro-acoustic modulation (VAM)-inspired structural integrity monitoring and its applications to bolted composite joints

Vibro-acoustic modulation (VAM) – one of the prevailing nonlinear methods for material characterization and structural damage evaluation – is based on the effect of modulation of a low-frequency vibration (pumping vibration) on a high-frequency acoustic wave (probing wave). In this study, the contact acoustic nonlinearity (CAN), associated with changes in the solid-solid interface of a bolted joint under VAM, due to bolt loosening, is explored analytically and experimentally, on which basis a VAM-inspired approach is developed for monitoring structural integrity of bolted joints. Numerical simulation based on a theoretical model with structural nonlinear contact stiffness is implemented, to achieve insight into CAN induced by a loose bolt. A quantitative correlation between vibro-acoustic nonlinear distortion and degree of bolt loosening is ascertained. The developed approach is applied to detect bolt loosening in a composite bolted joint and to evaluate the residual torque of the loose bolt quantitatively. Take a step further, the VAM-based nonlinear approach is compared against an elastic wave-based linear method, underscoring a higher sensitivity to bolt loosening. From early awareness of bolt loosening to continuous evaluation of the bolt loosening progress, this VAM-based approach has provided a cost-effective framework for monitoring the health and integrity of a composite bolted joint.

Nonlinear Lamb waves in plate/shell structures

In view of the fact that conventional ultrasonic detection technology is not effective for the detection of distributed fne damage and contact-type structural damage in materials, nonlinear ultrasound technology has been attracting more and more attention in recent years. Ultrasonic waves propagate in the form of Lamb wave modes in the structure of plates and shells. However, due to the dispersion and multimode characteristics of Lamb waves, the progress of theoretical and experimental research on nonlinear Lamb waves is slow. Based on the classical nonlinear theory, this paper summarizes the theoretical and experimental research progress of nonlinear Lamb waves induced by the inherent nonlinear-ity of material, and summarizes the application of the second harmonics of Lamb waves in damage evaluation of material. The current status of the research on nonlinear Lamb waves induced by contact-type structural damage is discussed based on contact acoustic nonlin-earity. At the end, this paper envisions the future research topics and trends in nonlinear Lamb waves.

Time reversal invariance for a one-dimensional model of contact acoustic nonlinearity

The interaction of a one-dimensional (1D) wave packet with a contact interface characterized by a unilateral contact law is investigated analytically and through a finite difference model. It is shown that this interaction leads to the generation of higher harmonic, sub-harmonic and zero-frequency components in the reflected wave, resulting in a pulse distortion that is attributable to contact acoustic nonlinearity. However, the results also show that the re-emission of a time reversed version of this distorted first reflection results in a healing of the distortions and a perfect recovery of the original pulse shape, thereby demonstrating time reversal invariance for this type of contact acoustic nonlinearity. A step-by-step analysis of the contact interaction provides insights into both the distortion arising from the first interaction and the subsequent healing during the second interaction. These findings suggest that time reversal invariance should also apply more generally for scatterers exhibiting non-dissipative contact acoustic nonlinearity.

Artificial neural network model using ultrasonic test results to predict compressive stress in concrete

This study focused on modeling the behavior of the compressive stress using the average strain and ultrasonic test results in concrete. Feed-forward backpropagation artificial neural network (ANN) models were used to compare four types of concrete mixtures with varying water cement ratio (WC), ordinary concrete (ORC) and concrete with short steel fiber-reinforcement (FRC). Sixteen (16) <TEX>$150mm{\times}150mm{\times}150mm$</TEX> concrete cubes were used; each contained eighteen (18) data sets. Ultrasonic test with pitch-catch configuration was conducted at each loading state to record linear and nonlinear test response with multiple step loads. Statistical Spearman's rank correlation was used to reduce the input parameters. Different types of concrete produced similar top five input parameters that had high correlation to compressive stress: average strain (<TEX>${\varepsilon}$</TEX>), fundamental harmonic amplitude (A1), <TEX>$2^{nd}$</TEX> harmonic amplitude (A2), <TEX>$3^{rd}$</TEX> harmonic amplitude (A3), and peak to peak amplitude (PPA). Twenty-eight ANN models were trained, validated and tested. A model was chosen for each WC with the highest Pearson correlation coefficient (R) in testing, and the soundness of the behavior for the input parameters in relation to the compressive stress. The ANN model showed increasing WC produced delayed response to stress at initial stages, abruptly responding after 40%. This was due to the presence of more voids for high water cement ratio that activated Contact Acoustic Nonlinearity (CAN) at the latter stage of the loading path. FRC showed slow response to stress than ORC, indicating the resistance of short steel fiber that delayed stress increase against the loading path.

Continuous Monitoring of Residual Torque of Loose Bolt in a Bolted Joint

Bolted joints are ubiquitous in engineering assets, performing a critical role of transferring loads among interconnecting components. Effectual inspection and monitoring of loosening of the bolts throughout their intended service life warrants a reliable service of the joints. In this study, identification of bolt loosening and quantitative estimate of the residual torque for three types of joints was attempted, theoretically and experimentally, using two approaches in a comparative manner: a wave energy dissipation (WED)-based linear acoustic approach and a contact acoustic nonlinearity (CAN)-based vibro-acoustic modulation (VM) method. In the former, an analytical model, residing on the Hertzian contact theory, was established to link the WED to the residual torque of a loose bolt. In the latter, the contact stiffness at the interconnecting interface, when a pumping vibration introducing perturbation to probing waves, was calibrated in terms of a Tayler series, on which basis a modulation index was constructed. With the index, nonlinear signal spectral features were quantitatively associated with residual torque. The two approaches were validated experimentally, to find a good agreement between theoretical prediction and experimental estimate. The comparison between two approaches reveals that the nonlinear VM-based method offers enhanced accuracy and sensitivity to bolt loosening than a linear WED-based approach, with additional capacity of detecting bolt loosening in multi-type joints.

Efficient Simulations of the Nonlinear Wave Modulation Induced by a Closed Crack Using Local Contact Modelling

Nonlinear wave modulation methods have been used for the detection of small defects such as closed cracks or delaminations. These methods rely on the nonlinear interaction of a low frequency pumping wave and a high frequency probing wave, which produces sideband frequencies equal to the sum and difference of the input frequencies. Since two very different frequencies are involved, numerical modelling approaches generally require to run the simulations for a long time in order to achieve a steady state regarding the low frequency excitation. In this work, we propose a numerical model where the low frequency excitation is accounted by a time dependent stress state at the contact interface. This local stress state is the prospective stress in the un-cracked structure due to the pumping wave. This is performed by implementing an additional stress parameter directly in the contact laws used to model the contact dynamics occurring at the crack. The high frequency excitation is generated in the solid by imposing displacements on the boundary of the domain, as done in previous work. This concept is first demonstrated through a 1D Finite Difference model with a contact interface between a semi-infinite solid and a rigid boundary. Secondly, this approach is implemented in a 2D Finite Element model where a closed crack of finite length is considered. The results demonstrate the interest of the method for investigating nonlinear wave modulation, and they provide a benchmark for understanding experimental results that may involve more complicated manifestations of contact acoustic nonlinearity.

Locating delaminations in laminated composite beams using nonlinear guided waves

This paper proposes a new method for detecting and locating delaminations in laminated composite beams using nonlinear guided wave. It is shown that when incident wave interacts at the delamination, the nonlinear effect of wave interaction with contact interfaces at the delamination generates higher harmonic guided waves due to contact acoustic nonlinearity (CAN). The proposed method employs a transducer network to detect and locate the delamination using the higher harmonic guided waves. A sequential scan is used to inspect the laminated composite beams by actuating the fundamental anti-symmetric mode (A0) of guided wave at one of the transducers while the rest of the transducers are used for measuring the impinging waves. A series of numerical case studies are performed using three-dimensional explicit finite element simulations, which consider different delamination locations, lengths and through-thickness locations. Experimental case studies are carried out to further validate and demonstrate the capability of the proposed method. The results show that the proposed method is able to accurately detect and locate the delamination in the laminated composite beams using the higher harmonic guided waves. One of the advantages of the proposed method is that it does not rely on baseline data to detect and locate the delamination, and hence, it has less influence by varying operational and environmental conditions.

Higher harmonic generation of guided waves at delaminations in laminated composite beams

Detection and characterization of delamination damage are of great importance to the assurance of structural safety. This work investigates the potential of a baseline-free structural health monitoring technique based on higher harmonics resulting from the nonlinear interaction of guided wave and a delamination. The nonlinearity considered in this study arises from the clapping of the sub-laminates in the delaminated region, which generates contact acoustic nonlinearity. Both explicit finite element simulations and experimental tests are conducted on composite laminates containing a delamination of different sizes and at different through-thickness locations. The results show that the interaction between the fundamental asymmetric mode (A0) of guided wave and a delamination generates contact acoustic nonlinearity in the form of higher harmonics, which provides a good measure for identifying the existence of delaminations and determining their sizes in laminated composite beams. This new insight into the generation mechanisms of nonlinear higher order harmonics in composite laminates will enhance the detection and monitoring of damage in composite structures.

Resonant acoustic nonlinearity of defects: Non-classical manifestations and applications for diagnostic imaging

Acoustic nonlinearity of non-bonded interface (Contact Acoustic Nonlinearity (CAN)) is a type of non-classical nonlinearity perspective for applications in non-destructive evaluation of cracked defects. The presence of such defects also leads to a local decrease in rigidity and manifests in a characteristic frequency of the defect. A frequency match between the driving acoustic wave and this frequency provides a Local Defect Resonance (LDR) and an efficient energy delivery from the wave into the defect. A combination of CAN and LDR, therefore, manifests a profound nonlinearity even at moderate acoustic excitation level. Under resonance conditions, a strong enhancement of the higher harmonic (HH) amplitudes is observed. A high quality factor of LDR is also used as a filter/amplifier/resonator for efficient frequency mixing. These “conventional” nonlinear effects are not the only dynamic scenario of nonlinear phenomena for resonant defects. A combined effect of LDR and CAN also results in qualitatively new features characteristic of nonlinear and parametric resonances. Due to parametric instability, nearly total input energy at fundamental frequency can be converted into HH or subharmonic vibrations of the defects. Both super- and subharmonic LDR are strongly localized in the defect area that provides a background for highly-sensitive defect-selective imaging.

Steady-state analysis of nonlinear ultrasonic waves caused by crack face friction

In general, the detection of closed cracks is difficult by using the conventional linear ultrasonic testing. Therefore, the nonlinear ultrasonic testing (NLUT) based on the contact acoustic nonlinearity (CAN) is expected to be an effective technique for the problem. Several studies have reported the presence of nonlinear ultrasonic waves (higher- and sub-harmonic waves) due to the CAN. However, the theoretical explanation of the physical phenomena is not sufficient at present. For accurate NLUT, theoretical investigation of the nonlinear vibration phenomena of crack faces is desirable as a basic research. It is true that the time-domain analysis is becoming possible for the purpose. However, it is difficult to investigate in detail steady-state vibration of crack faces, which is supposed to affect behavior of nonlinear ultrasonic waves significantly. In this study, a novel numerical method is developed for the NLUT by coupling the boundary element method (BEM) and harmonic balance method (HBM) in order to investigate the above steady-state vibration. The scattering model is for the 2-D SH wave field that causes the crack face friction. Several numerical examples are presented in order to demonstrate validity and effectiveness of the proposed method.

Second harmonic ultrasonic waves detection using a double-layered piezoelectric transducer

A real-time detection system for second harmonic ultrasonic pulse waves using a double-layered piezoelectric transducer (DLPT) has been constructed. We also analyzed the effective detection of the second-harmonic ultrasonic pulse waves generated by closed-cracks and/or contact surfaces of solids (CAN: contact acoustic nonlinearity) using DLPT and the pulse inversion averaging (PIA) method. Lamb waves were also detected in a double-layered piezoelectric transducer (DLPT). The second harmonic components generated from a closed crack in the glass plate were detected in the pulse-echo method. The second harmonic components in the received waveform in the closed crack area were approximately 6 dB higher than that in the crack-free area. To confirm the origin of the second harmonic components, vibration velocities were detected by using laser Doppler vibrometry. As a result, the threshold amplitude for the generation of the second harmonic components of the Lamb wave was confirmed. This result indicates the existence of the closed cracks in the glass plate.

Ultrasonic array imaging of contact-acoustic nonlinearity

Classically, ultrasonic arrays achieved physical beam-forming through the application of phase delays to the parallel transmission of elements. Alternatively, beam-forming may be emulated through post-processing of sequentially transmitted array data. Though exactly equivalent for linear systems, these parallel and sequential fields differ in their nonlinear propagation. Information pertaining to elastic nonlinearity is encoded onto the relative parallel/sequential field by interference between the field components of individual element transmissions. A family of nonlinear imaging techniques has been developed which exploit differences between parallel and sequential fields in order to image elastic nonlinearity. Diffuse energy and coherent scattering variants of the technique are demonstrated for the imaging and characterisation of contact-acoustic nonlinearity in application to the detection and sizing of closed fatigue cracks.

Quantitative evaluation of residual torque of a loose bolt based on wave energy dissipation and vibro-acoustic modulation: A comparative study

A wave energy dissipation (WED)-based linear acoustic approach and a vibro-acoustic modulation (VM)-based nonlinear method were developed comparatively, for detecting bolt loosening in bolted joints and subsequently evaluating the residual torque of the loose bolt. For WED-based, an analytical model residing on the Hertzian contact theory was established, whereby WED was linked to the residual torque of a loose bolt, contributing to a linear index. For VM-based, contact acoustic nonlinearity (CAN) engendered at the joining interface, when a pumping vibration perturbs a probing wave, was interrogated, and the nonlinear contact stiffness was described in terms of a Taylor series, on which basis a nonlinear index was constructed to associate spectral features with the residual torque. Based respectively on a linear and a nonlinear premise, the two indices were validated experimentally, and the results well coincided with theoretical predication. Quantitative comparison of the two indices surmises that the VM-based nonlinear method outperforms the WED-based linear approach in terms of sensitivity and accuracy, and particularly when the bolt loosening is in its embryo stage. In addition, the detectability of the nonlinear index is not restricted by the type of the joint, against a high dependence of its linear counterpart on the joint type.

Closed crack imaging using time reversal method based on fundamental and second harmonic scattering

A recent variant of time reversal imaging is employed for reconstructing images of a closed crack, based on both the fundamental and the second harmonic components of the longitudinal scattered field due to an incident longitudinal wave. The scattered field data are generated by a finite element model that includes unilateral contact with Coulomb friction between the crack faces to account for the Contact Acoustic Nonlinearity. The closure state of the crack is controlled by specifying a pre-stress between the crack faces. The knowledge of the scattered field at the fundamental (incident) frequency and the second harmonic frequency for multiple incident angles provides the required inputs for the imaging algorithm. It is shown that the image reconstructed from the fundamental harmonic closely matches the image that is obtained from scattering data in the absence of contact, although contact between the crack faces reduces the amplitude of the scattered field in the former case. The fundamental harmonic image is shown to provide very accurate estimates of crack length for low to moderate levels of pre-stress. The second harmonic image is also shown to provide acceptable estimates of crack length and the image is shown to correlate with the source location of second harmonic along the crack, which becomes increasingly localized near the crack tips for decreasing levels of pre-stress. The influence of the number of sensors on the image quality is also discussed in order to identify the minimum sensors number requirement. Finally, multiple frequency imaging is performed over a fixed bandwidth to assess the potential improvement of the imaging algorithm when considering broadband information.

Analytical extension of Finite Element solution for computing the nonlinear far field of ultrasonic waves scattered by a closed crack

Nonlinear scattering of ultrasonic waves by closed cracks subject to contact acoustic nonlinearity (CAN) is determined using a 2D Finite Element (FE) coupled with an analytical approach. The FE model, which includes unilateral contact with Coulomb friction to account for contact between crack faces, provides the near-field solution for the interaction between in-plane elastic waves and a crack of different orientations. The numerical solution is then analytically extended in the far-field based on a frequency domain near-to-far field transformation technique, yielding directivity patterns for all linear and nonlinear components of the scattered waves. The proposed method is demonstrated by application to two nonlinear acoustic problems in the case of tone-burst excitations: first, the scattering of higher harmonics resulting from the interaction with a closed crack of various orientations, and second, the scattering of the longitudinal wave resulting from the nonlinear interaction between two shear waves and a closed crack. The analysis of the directivity patterns enables us to identify the characteristics of the nonlinear scattering from a closed crack, which provides essential understanding in order to optimize and apply nonlinear acoustic NDT methods.

Nonlinear ultrasonic technique for closed crack detection

The detection of cracks at the early stage of fracture is important in industrial structures in order to guarantee their structural safety. Ultrasound has been widely utilized in the field of nondestructive testing of materials. However, most of these conventional methods using ultrasonic characteristics in the linear elastic region are mostly sensitive to opened cracks but much less sensitive to such closed cracks. The nonlinear ultrasonic technique (NUT) based on the contact acoustic nonlinearity (CAN) has been considered as a promising method for the closed crack detection. However, most of the previous studies were limited to the modeling of the second-order harmonic wave generation at contacted interfaces and its verification by testing artificially contacted interfaces in the through-transmission method. In this study, we investigated experimentally the contact acoustic nonlinearity at a real crack by using the measurement system constructed in the reflection mode that permits the transducers to access the only single side of a test structure. Results showed that the magnitude of the second-order harmonic wave represented the existence of the closed area clearly and that the crack sizing performance was greatly improved by the combination of the linear and nonlinear ultrasonic techniques.

2D finite element modeling of the non-collinear mixing method for detection and characterization of closed cracks

The non-collinear mixing technique is applied for detection and characterization of closed cracks. The method is based on the nonlinear interaction of two shear waves generated with an oblique incidence, which leads to the scattering of a longitudinal wave. A Finite Element model is used to demonstrate its application to a closed crack. Contact acoustic nonlinearity is modeled using unilateral contact law with Coulomb׳s friction. The method is shown to be effective and promising when applied to a closed crack. Scattering of the longitudinal wave also enables us to image the crack, giving its position and size.