Interaction of elastic waves in solids with quadratic and cubic nonlinearity.

This article investigates the interactions of two-plane waves in weakly nonlinear elastic solids containing quadratic and cubic nonlinearity. The analytical solutions for generated combined harmonic waves are derived using the Green's function approach applied to a generated system of quasi-linear equations of motion. Wave mixing solutions are obtained and include shape functions that permit closed-form solutions for a variety of interaction geometries. An explicit example is highlighted for a spherical interaction volume assuming isotropic elastic constants. Several parameters of the generated field after mixing are analyzed including resonant and nonresonant mixing, the role of interaction angle, and the frequencies of the two incident waves. Wave mixing offers the potential for sensing localized elastic nonlinearity and the present model can be used to help design experimental configurations.

Theoretical and numerical modeling of Rayleigh wave scattering by an elastic inclusion.

This work presents theoretical and numerical models for the backscattering of two-dimensional Rayleigh waves by an elastic inclusion, with the host material being isotropic and the inclusion having an arbitrary shape and crystallographic symmetry. The theoretical model is developed based on the reciprocity theorem using the far-field Green's function and the Born approximation, assuming a small acoustic impedance difference between the host and inclusion materials. The numerical finite element (FE) model is established to deliver a relatively accurate simulation of the scattering problem and to evaluate the approximations of the theoretical model. Quantitative agreement is observed between the theoretical model and the FE results for arbitrarily shaped surface/subsurface inclusions with isotropic/anisotropic properties. The agreement is excellent when the wavelength of the Rayleigh wave is larger than, or comparable to, the size of the inclusion, but it deteriorates as the wavelength gets smaller. Also, the agreement decreases with the anisotropy index for inclusions of anisotropic symmetry. The results lay the foundation for using Rayleigh waves for quantitative characterization of surface/subsurface inclusions, while also demonstrating its limitations.

Evaluating elongated grains with diffuse ultrasonic double scattering and rectangular transducers.

Diffuse scattering of ultrasound by the microstructure of polycrystal specimens can be used to evaluate grain size and grain elongation. The existing diffuse scattering models mostly dealt with circular transducers whose symmetrical sound field is insensitive to the asymmetric elongated grain. The sound field of a rectangular transducer provides a new perspective for acquiring additional information. First, the existing single scattering response (SSR) and double scattering response (DSR) models are modified for a rectangular transducer, where the sound field of a rectangular transducer is equivalent to that of an elliptical transducer in the far-field. Therefore, an equivalent single Gaussian beam model is derived using amplitude-equivalent and beamwidth-equivalent coefficients. Then, the spatial correlation function of elongated grains is transformed into the wavenumber domain, giving rise to the SSR and DSR of a rectangular transducer that reveals the interaction effect of an asymmetric sound field and elongated grains on ultrasonic backscattering. The experimental results show that the sizes of elongated grains in a cold-rolled aluminum are evaluated as 1086 ± 8, 90 ± 4, and 10 ± 1 µm in the x, y, and z directions, where the exact values are 1184.2 ± 11.9, 80.7 ± 5.2, and 8.3 ± 0.5 µm according to metallographic measurements.

Attenuation and dispersion of leaky Rayleigh wave in polycrystals.

In this work, we use the characteristic equation of leaky Rayleigh waves (LRWs) and a unified approach of bulk waves proposed by Stanke and Kino [J. Acoust. Soc. Am. 75, 665-681 (1984)] to calculate the attenuation and velocity dispersion of LRWs in polycrystals. Numerical results demonstrate that the total attenuation including the leakage attenuation and scattering attenuation is proportional to frequency and independent of grain size in the Rayleigh scattering regime. Meanwhile, the variation of phase velocity in all scattering regimes remains at ∼0.7% according to the theoretical expectation; this means that the velocity dispersion of the LRWs can be ignored, consistent with the conventional viewpoint. Measurements are conducted on stainless steel at different ultrasonic frequencies (all in the Rayleigh scattering regime). The non-paraxial sound field model is used here to eliminate the diffraction loss and to obtain the total attenuation. Experimental results verify that LRWs have very little velocity dispersion. Meanwhile, experimental fitting data reveal that the modified theoretical model can be used to evaluate the total attenuation (only ∼2% discrepancies) of LRWs under the consideration of the diffraction effect. The relative errors between experimental scattering attenuation and theoretical value ranged from 11% to 18%, mainly owing to the effect of surface roughness and measurement inaccuracy.

Sideband peak count-index technique for monitoring multiple cracks in plate structures using ordinary state-based peri-ultrasound theory.

This work presents a peri-ultrasound theory based on ordinary state-based peridynamics for modeling elastic waves propagating in three-dimensional (3-D) plate structures and interacting with multiple cracks. A recently developed nonlinear ultrasonic technique called sideband peak count-index (or SPC-I) is adopted for monitoring one or more cracks with thickness values equal to 0 mm (crack-free), 1, 2, and 4 mm. Three separate scenarios-one crack, two cracks, and four cracks in 3-D plate structures-are investigated. These cracks can be classified as thin and thick cracks depending on the horizon size, which is mentioned in peri-ultrasound theory. Computed results for all three cases show larger SPC-I values for thin cracks than for thick cracks and the case of no cracks. This observation is in line with the previously reported results in the literature and proves that the state-based peri-ultrasound theory can capture the expected nonlinear response of elastic waves interacting with multiple cracks without changing the cracks' surface locations artificially, and this is always needed in most of the other numerical methods. The proposed state-based peri-ultrasound theory is more flexible and reliable for solving 3-D problems, and the out-of-plane wave field can be obtained for engineering analysis.

Modeling ultrasonic wave fields using a Quasi-Monte Carlo method: Wave transmission through complicated interfaces.

The sound fields generated by ultrasonic transducers can be modeled using the Quasi-Monte Carlo (QMC) method with a high level of accuracy and efficiency from Zhang [J. Acoust. Soc. Am. 149(1), 7-15 (2021)]. In this work, this method is extended to simulate transmitted wave fields through complicated interfaces. When a wave propagates in two-layer media, the vibrating waves over the interface radiated by the transducer can be treated as the source for generating waves in the second medium, thus, a nested-form Rayleigh integral expression can be used as a model equation for the transmitted wave calculation. When the QMC method is used to solve the nested integral, pseudo-random samples for constructing the transducer and the interface are sampled separately and the transmitted wave fields are obtained using the final sample mean. Numerical examples and results are presented when the wave transmits normally or obliquely through planar or curved interfaces. The results indicate that the high level of accuracy and efficiency remains when the QMC method is used to model the transmitted wave fields. One important advantage is that wave fields can be well simulated using the QMC method when the wave transmits through a complicated interface as long as the interface can be constructed using pseudo-random samples.

Modeling of wave fields generated by ultrasonic transducers using a quasi-Monte Carlo method.

The sound fields generated by ultrasonic transducers are modeled using the quasi-Monte Carlo (QMC) method, which is found to overcome the conflict between accuracy and efficiency that occurs in existing wave field calculation methods. The RI equation, which is frequently used as a model equation in ultrasonic field calculation, is used here as an exact method and for comparison purposes. In the QMC method, the judgment sampling method and Halton sequence are used for pseudo-random sampling from the sound source, and then the sound field distributions are found by solving the integral solution using the sample mean. Numerical examples and results are presented when modeling unfocused, focused, and steered and focused beam fields. The accuracy and efficiency of the QMC method are discussed by comparing the results obtained using different modeling methods. The results show that the proposed method has a high level of efficiency due to the nature of the QMC algorithm and a high level of accuracy because no approximation is required. In addition, wave fields can be modeled with the QMC method as long as sound sources can be effectively pseudo-randomly sampled, allowing the proposed method to be applied to various types of transducers.

Propagation of leaky Rayleigh waves along a curved fluid-solid interface.

A characteristic equation is derived for a leaky Rayleigh wave (LRW), propagating on a curved fluid-solid interface. The equations of motion for the curved solid and fluid are formulated using the constitutive equations of a homogenous isotropic curved solid and an inviscid fluid, respectively. The displacement potential functions are used to simplify the derivation. The interface conditions are used to ensure continuities of the mass, momentum, and energy across the interface. Then, with the consideration of the interface radius of the curvature, the characteristic equation for the LRW is established and solved numerically by Muller's method. One important outcome is that there is weaker directional dependence for the velocity of the LRWs on the radius of curvature in comparison with the Rayleigh waves at an air-solid interface. However, the numerical results show a strong directional dependence for the attenuation due to the LRW leakage on the complex curvatures. Moreover, a quantitative relation between the curvature and attenuation caused by the leakage for different materials is shown. The results are significant especially with respect to relevant future applications of ultrasonic testing.

Higher-order spatial correlation coefficients of ultrasonic backscattering signals using partial cross-correlation analysis.

The spatial correlation properties of ultrasonic backscattering signals in random media have important implications. For example, they can be used for microstructural characterization and flaw detection in engineering materials. However, the traditional spatial correlation coefficient (SCC) is only a leading order quantity that does not capture the true spatial correlations of random media. This is caused by neglecting confounding variables such as non-zero means or other non-zero odd-order moments. Here, the SCC is generalized from zeroth- to general-order through partial cross-correlation analysis. A series of indicators are defined to quantify the SCC curve at zero time lag, and the maximum time shift curve, which are both functions of lateral separation between two sensor positions. A stainless-steel specimen and a focused ultrasonic transducer are used to verify the method. Scattering measurements show that the higher-order SCC can consistently capture spatial correlations whereas the zeroth-order SCC is inadequate. The zeroth-order SCC is shown to predict a step size that can be more than six times too large. Thus, the present method can provide better understanding of statistical correlations and conditions to measure uncorrelated backscattering signals.

Acoustic-resolution-based photoacoustic microscopy with non-coaxial arrangements and a multiple vertical scan for high lateral resolution in-depth.

In conventional acoustic-resolution-based photoacoustic microscopy (ARPAM), a focused ultrasound transducer is placed coaxially with the laser beam to obtain the generated ultrasound signals. The information from deep regions can be greatly affected by the shallow targets. More importantly, in ARPAM the irreconcilable conflict between the lateral resolution and depth of fields has always been a major factor that lowers the imaging quality. In this work, an ARPAM system was developed, in which a non-coaxial arrangement of light illumination and acoustic detection was adopted to alleviate the influence of the tissue surface on the deep targets, and a focal zone integral algorithm was applied with a multiple scanning scheme to improve the lateral resolution. The system can achieve a consistent high lateral resolution of 0.5 mm over a large range in the axial direction. Both the phantom experiment and the chicken embryo in vivo results indicate that the proposed method can provide more in-depth information compared with the conventional ARPAM method. With the development of high repetition lasers and the advancement of image scanning technologies, the proposed method may play an important role in cerebral vascular imaging, superficial tumor imaging, and other related biomedical imaging applications.

Flaw detection with ultrasonic backscatter signal envelopes.

Ultrasound is a prominent nondestructive testing modality for the detection, localization, and sizing of defects in engineering materials. Often, inspectors analyze ultrasonic waveforms to determine if echoes, which stem from the scattering of ultrasound from a defect, exceed a threshold value. In turn, the initial selection of the threshold value is critical. In this letter, a time-dependent threshold or upper bound for the signal envelope is developed based on the statistics governing the scattering of ultrasound from microstructure. The utility of the time-dependent threshold is demonstrated using experiments conducted on sub-wavelength artificial defects. The results are shown to enhance current nondestructive inspection practices.

Application of Fresnel Zone Plate Focused Beam to Optimized Sensor Design for Pulse-Echo Harmonic Generation Measurements.

In nonlinear acoustic measurements involving reflection from the stress-free boundary, the pulse-echo method could not be used because such a boundary is known to destructively change the second harmonic generation (SHG) process. The use of a focusing acoustic beam, however, can improve SHG after reflection from the specimen boundary, and nonlinear pulse-echo methods can be implemented as a practical means of measuring the acoustic nonlinear parameter (ß) of solid specimens. This paper investigates the optimal sensor design for pulse-echo SHG and ß measurements using Fresnel zone plate (FZP) focused beams. The conceptual design of a sensor configuration uses separate transmission and reception, where a broadband receiver is located at the center and a four-element FZP transmitter is positioned outside the receiver to create a focused beam at the specified position in a solid sample. Comprehensive simulations are performed for focused beam fields analysis and to determine the optimal sensor design using various combinations of focal length, receiver size and frequency. It is shown that the optimally designed sensors for 1 cm thick aluminum can produce the second harmonic amplitude and the uncorrected nonlinear parameter corresponding to the through-transmission method. The sensitivity of the optimal sensors to the changes in the designed sound velocity is analyzed and compared between the odd- and even-type FZPs.

Modeling Flaw Pulse-Echo Signals in Cylindrical Components Using an Ultrasonic Line-Focused Transducer with Consideration of Wave Mode Conversion.

Investigations on flaw responses can benefit the nondestructive testing of cylinders using line-focused transducers. In this work, the system function, the wave beam model, and a flaw scattering model are combined to develop an ultrasonic measurement model for line-focused transducers to predict flaw responses in cylindrical components. The system function is characterized using reference signals by developing an acoustic transfer function for line-focused transducers, which works at different distances for both planar and curved surfaces. The wave beams in cylindrical components are modeled using a multi-Gaussian beam model, where the effects of wave mode conversion and curvatures of cylinders are considered. Simulation results of wave beams are provided to analyze their propagation behaviors. The proposed ultrasonic measurement model is certified from good agreement between the experimental and predicted signals of side-drilled holes. This work provides guidance for evaluating the detection ability of line-focused transducers in cylindrical component testing applications.

Acoustic nonlinearity parameter measurements in a pulse-echo setup with the stress-free reflection boundary.

This paper describes the acoustic nonlinearity parameter (ß) determination for fluids using a pulse-echo method with the stress-free boundary. A newly derived ß formula requires the measurement of the fundamental and second harmonic displacements with appropriate corrections for attenuation, diffraction, and boundary reflection. Measurements are composed of two steps: receiver calibration and harmonic generation. The ß values calculated for water at several distances between the planar transducer and the water-air interface are in good agreement with literature, providing a validation for the method.

Ultrasonic Phased Array Sparse-TFM Imaging Based on Sparse Array Optimization and New Edge-Directed Interpolation.

The ultrasonic phased array total focusing method (TFM) has the advantages of full-range dynamic focusing and high imaging resolution, but the problem of long imaging time limits its practically industrial applications. To reduce the imaging calculation demand of TFM, the locations of active array elements in the sparse array are optimized by combining almost different sets with the genetic algorithm (ADSGA), and corrected based on the consistency of the effective aperture with the equivalent point diffusion function. At the same time, to further increase the imaging efficiency, a sparse-TFM image with lower resolution is obtained by reducing the number of focus points and then interpolated by the new edge-directed interpolation algorithm (NEDI) to obtain a high quality sparse-TFM image. Compared with TFM, the experimental results show that the quantitative accuracy of the proposed method is only decreased by 1.09% when the number of sparse transmitting elements reaches 8 for a 32-element transducer, and the imaging speed is improved by about 16 times with the same final pixel resolution.

Diffuse ultrasonic backscatter using a multi-Gaussian beam model.

Diffuse ultrasonic backscatter is widely used to evaluate microstructural parameters of heterogeneous materials. Recent singly scattered response (SSR) models utilize a single-Gaussian beam (SGB) assumption which is expected to have limitations. Following a similar formalism, a model is presented using a multi-Gaussian beam (MGB) assumption to characterize the transducer beam for longitudinal-to-longitudinal scattering at normal incidence through an interface with arbitrary curvature. First, the Wigner transform of the transducer field is defined using conjugate double-layer MGB expressions. The theoretical analysis shows that ten groups of Gaussian beams are sufficient for convergence. Compared with the SGB-SSR curve, the shape of MGB-SSR curve is positive skewed. Differences between the MGB-SSR model and the SGB-SSR model are quantified and shown to be complex functions of frequency, sample curvature, transducer parameters, and focal depth in the material. Finally, both models are used to fit experimental spatial variance data from a 304 stainless steel pipe with planar, convex, and concave surfaces. The results show that the MGB-SSR has some characteristics suggesting a better fit to the experiments. However, both models result in grain size estimates within the uncertainty of the optical microscopy suggesting that the SGB is sufficient for normal incidence pulse-echo measurements.

Calibration of focused ultrasonic transducers and absolute measurements of fluid nonlinearity with diffraction and attenuation corrections.

This paper presents analytical and experimental techniques for absolute determination of the acoustic nonlinearity parameter (ß) in fluids using focused transducers. When focused transducers are used for ß measurements, the geometrical and mechanical calibrations are generally required for accurate determination of the receiver transfer function from which the absolute pressure can be calculated. The fundamental and second harmonic wave amplitudes in harmonic generation measurements should be modified to account for beam diffraction and material absorption. All these issues are resolved in this study and the proposed technique is validated through the ß measurement in water. An experimental method is developed to determine the effective radius and focal length of focused transducers. A simplified self-reciprocity calibration procedure for a broadband focused receiver is described. The diffraction and attenuation corrections for the fundamental and second harmonic waves are explicitly derived using the multi-Gaussian beam model, and the effects on the ß determination are discussed. When the diffraction and attenuation corrections are all properly made, the measurement of ß over a large range of propagation distances is possible with errors less than 8%.

Measurement of Rayleigh Wave Beams Using Angle Beam Wedge Transducers as the Transmitter and Receiver with Consideration of Beam Spreading.

A theoretical model, along with experimental verification, is developed to describe the generation, propagation and reception of a Rayleigh wave using angle beam wedge transducers. The Rayleigh wave generation process using an angle beam wedge transducer is analyzed, and the actual Rayleigh wave sound source distributions are evaluated numerically. Based on the reciprocity theorem and considering the actual sound source, the Rayleigh wave beams are modeled using an area integral method. The leaky Rayleigh wave theory is introduced to investigate the reception of the Rayleigh wave using the angle beam wedge transducers, and the effects of the wave spreading in the wedge and transducer size are considered in the reception process. The effects of attenuations of the Rayleigh wave and leaky Rayleigh wave are discussed, and the received wave results with different sizes of receivers are compared. The experiments are conducted using two angle beam wedge transducers to measure the Rayleigh wave, and the measurement results are compared with the predictions using different theoretical models. It is shown that the proposed model which considers the wave spreading in both the sample and wedges can be used to interpret the measurements reasonably.

A self-reciprocity calibration method for broadband focused transducers.

A procedure is developed for self-calibration of broadband, spherically focused ultrasonic transducers based on reciprocity. The input and received signals are measured in a pulse-echo configuration. These signals are used in conjunction with a multi-Gaussian beam model to obtain the electromechanical transfer function of the transducer. This calibration procedure is advantageous because it reduces the experimental configuration to a single transducer and a reflector. Experimental results indicate that the transfer function is insensitive to on-axis reflector placement. This result supports the feasibility of integrating the calibration procedure into actual testing in some situations.

Ultrasonic backscattering model for Rayleigh waves in polycrystals with Born and independent scattering approximations.

This paper presents theoretical and numerical models for the backscattering of 2D Rayleigh waves in single-phase, untextured polycrystalline materials with statistically equiaxed grains. The theoretical model, based on our prior inclusion-induced Rayleigh wave scattering model and the independent scattering approximation, considers single scattering of Rayleigh-to-Rayleigh (R-R) waves. The numerical finite element model is established to accurately simulate the scattering problem and evaluate the theoretical model. Good quantitative agreement is observed between the theoretical model and the finite element results, especially for weakly scattering materials. The agreement decreases with the increase of the anisotropy index, owing to the reduced applicability of the Born approximation. However, the agreement remains generally good when weak multiple scattering is involved. In addition, the R-R backscattering behaviour of 2D Rayleigh waves is similar to the longitudinal-to-longitudinal and transverse-to-transverse backscattering of bulk waves, with the former exhibiting stronger scattering. These findings establish a foundation for using Rayleigh waves in the quantitative characterisation of polycrystalline materials.