Evaluation of Material Integrity Using Higher-Order Harmonic Generation in Propagating Shear Horizontal Ultrasonic Waves

Material nonlinearity is explored for the assessment of structural integrity. Crack-wave interactions are of particular interest. The major focus is on higher-order harmonics, generated in propagating shear horizontal (SH) waves. These harmonics are generated due to global material nonlinearity and local effects such as fatigue cracks. The theoretical background of the proposed method is explained. The method is examined using numerical simulations and experimental tests. The former involves the Local Interaction Simulation Approach (LISA), implemented for the nonlinear shear horizontal wavefield. The latter is based on a high-frequency shear excitation approach. Experimental tests are conducted using a series of beam specimens with fatigue cracks. Low-profile, surface-bonded piezoceramic shear actuators are used for excitation. The excitation frequency is selected to minimize the number of generated modes in the examined specimens. Nonlinear ultrasonic responses are collected using a non-contact laser vibrometer. The results show that higher-order harmonic generation-based on shear horizontal wave propagation-can be used for crack detection in the presence of global material nonlinearity.

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.

An iterative method to evaluate one-dimensional pulsed nonlinear elastic wavefields and mixing of elastic waves in solids

Over the last 15 years, literature on nondestructive testing has shown that the generation of higher harmonics and nonlinear mixing of waves could be used to obtain the nonlinearity parameters of an elastic medium and thereby gather information about its state, e.g., aging and fatigue. To design ultrasound measurement setups based on these phenomena, efficient numerical modeling tools are needed. In this paper, the iterative nonlinear contrast source method for numerical modeling of nonlinear acoustic waves is extended to the one-dimensional elastic case. In particular, nonlinear mixing of two collinear bulk waves (one compressional, one shear) in a homogeneous, isotropic medium is considered, taking into account its third-order elastic constants ( A,  B,  and  C). The obtained results for nonlinear propagation are in good agreement with a benchmark solution based on the modified Burgers equation. The results for the resonant waves that are caused by the one-way and two-way mixing of primary waves are in quantitative agreement with the results in the literature [Chen, Tang, Zhao, Jacobs, and Qu, J. Acoust. Soc. Am. 136(5), 2389-2404 (2014)]. The contrast source approach allows the identification of the propagating and evanescent components of the scattered wavefield in the wavenumber-frequency domain, which provides physical insight into the mixing process and explains the propagation direction of the resonant wave.

Observing dual-mode second-harmonic generation in an isotropic nonlinear elastic plate

In this paper, we discuss the observation of simultaneously propagating dual-mode second-harmonic fundamental symmetric (s₀) mode and a vertical-shear horizontal (sh₀^⊥) mode generated on an aluminium plate of 1 mm thickness while excited with a fundamental shear horizontal (SH₀) mode in a large domain. The experimental observations, and numerical models, were used to confirm the presence of dual-mode second-harmonics. The two wavemodes would separate from each other in the time-domain, wherein the travel length exceeded a minimum cut-off length of 40λ. This approach can provide important implications for more versatile early state defect detection.

Nonlinear mixing of non-collinear guided waves at a contact interface

Wave mixing offers several practical benefits relative to harmonic generation for detecting both distributed and localised damage. An analytical model is proposed for predicting the frequency, mode and direction of propagation of the mixed modes arising from the nonlinear mixing of two incident guided-wave modes at an interface exhibiting contact acoustic nonlinearity. These predictions are validated by a finite-element (FE) analysis involving a unilateral contact law that models both clapping and frictional sliding at the interface. This analysis also provides quantitative insights regarding the optimal interaction angle between the incident waves in the case of shear-horizontal SH₀ modes, and the dependence of the mixed-wave amplitude on the load ratio (i.e. ratio of contact stress to normal stress across the interface due to both incident waves). The non-collinear mixing of guided waves is also investigated experimentally for various values of contact stress and incident stress amplitude, demonstrating the existence of mixed waves, as well as showing that interface mixing leads to a higher amplitude mixed wave than that due to the background material nonlinearity. This higher amplitude combined with great flexibility in the choice of incident wave parameters (frequency, mode and direction) makes wave mixing an attractive practical approach for detecting contact acoustic nonlinearity at crack-like defects and interfaces. Experimentally, the maximum value of mixed-wave amplitude is obtained at the same value of contact stress for the various incident stress amplitudes, whereas the FE model shows a maximum at a unique value of the load ratio. This difference may be a consequence of surface roughness, which is not included in the FE model.

Ultrasonic frequency mixing via local defect resonance for defect imaging in composites

A new approach to nonlinear frequency mixing based on local damage resonance is proposed, analysed and tested experimentally for flexural waves in composites. The method is free from stringent requirements on the mode types and frequencies for interacting waves. The resonance of damage enhances strongly its higher-order nonlinear response and boosts the efficiency of generation for numerous-order combination frequencies. The damage resonance combined with its strong nonlinearity also provides locality of nonlinear interaction even for continuous wave operation. The combination frequencies generated locally in the damaged area are the footprints of damage and used for its detection, location and visualization. A single C-scan yields a number of images of the defect corresponding to various nonlinearly generated frequencies. Various versions of the resonant frequency mixing are considered and applied to nonlinear imaging of defects in composite materials.

Isolated frequencies at which nonlinear materials behave linearly

In this Rapid Communication, we demonstrate that specific frequencies in weakly nonlinear lattices avoid the generation of higher harmonics, and thus the lattices behave linearly. Using a multiple scales analysis, we present plane-wave solutions that persist at only a single frequency and wave number; i.e., whose spatiotemporal production of higher harmonics is remarkably small. We study monatomic and diatomic chains with quadratic and cubic stiffness nonlinearities as example systems. Direct numerical integration of the equations of motion confirms that finite amplitude plane waves assigned to these special frequencies produce negligible higher harmonics when injected into the lattices. Such findings provide new considerations for the operating frequency of nonlinear communications devices, sensors, and transducers for enhanced signal-to-noise ratios.

Linearly polarized waves of finite amplitude in pre-strained elastic materials

We study the propagation of linearly polarized transverse waves in a pre-strained incompressible isotropic elastic solid. Both finite and small-but-finite amplitude waves are examined. Irrespective of the magnitude of the wave amplitude, these waves may propagate only if the (unit) normal to the plane spanned by the directions of propagation and polarization is a principal direction of the left Cauchy-Green deformation tensor associated with the pre-strained state. A rigorous asymptotic analysis of the equations governing the propagation of waves of small but finite amplitude reveals that the time scale over which the nonlinear effects become significant depends on the direction along which the wave travels. Moreover, we design theoretically an experimental procedure to determine the Landau constants of the fourth-order weakly nonlinear theory of elasticity.

Doubly nonlinear waveguides with self-switching functionality selection capabilities

In this article, we investigate the effects of the interplay between quadratic and cubic nonlinearities on the propagation of elastic waves in periodic waveguides. Through this framework, we unveil an array of wave control strategies that are intrinsically available in the response of doubly nonlinear systems and we infer some basic design principles for tunable elastic metamaterials. The objective is to simultaneously account for two sources of nonlinearity that are responsible for distinct and complementary phenomena and whose effects are therefore typically discussed separately in the literature. Our study explicitly targets the intertwined effects that the two types of nonlinearity exert on each other, which modify the way in which their respective signatures are observed in the dynamic response. Through two illustrative examples we show how the dispersion correction caused by cubic nonlinearity can be used as an internal switch, or mode selector, capable of tuning on or off certain high-frequency response features that are generated through quadratic mechanisms. To this end, a multiple scale analysis is employed to obtain a full analytical solution for the nonlinear response that includes a complete description of the dual frequency-wave number dispersion correction shifts induced on all the branches, and elucidates the conditions necessary for the establishment of phase matching conditions.

Monitoring of Thermal Aging of Aluminum Alloy via Nonlinear Propagation of Acoustic Pulses Generated and Detected by Lasers

Nonlinear acoustic techniques are established tools for the characterization of micro-inhomogeneous materials with higher sensitivity, compared to linear ultrasonic techniques. In particular, the evaluation of material elastic quadratic nonlinearity via the detection of the second harmonic generation by acoustic waves is known to provide an assessment of the state variation of heat treated micro-structured materials. We report on the first application for non-destructive diagnostics of material thermal aging of finite-amplitude longitudinal acoustic pulses generated and detected by lasers. Finite-amplitude longitudinal pulses were launched in aluminum alloy samples by deposited liquid-suspended carbon particles layer irradiated by a nanosecond laser source. An out-of-plane displacement at the epicenter of the opposite sample surface was measured by an interferometer. This laser ultrasonic technique provided an opportunity to study the propagation in aluminum alloys of finite-amplitude acoustic pulses with a strain up to 5 × 10−3. The experiments revealed a signature of the hysteretic quadratic nonlinearity of micro-structured material manifested in an increase of the duration of detected acoustic pulses with an increase of their amplitude. The parameter of the hysteretic quadratic nonlinearity of the aluminum alloy (Al6061) was found to be of the order of 100 and to exhibit more than 50% variations in the process of the alloy thermal aging. By comparing the measured parameter of the hysteretic quadratic nonlinearity in aluminum alloys that were subjected to heat-treatment at 220 °C for different times (0 min, 20 min, 40 min, 1 h, 2 h, 10 h, 100 h, and 1000 h), with measurements of yield strength in same samples, it was established that the extrema in the dependence of the hysteretic nonlinearity and of the yield strength of this alloy on heat treatment time are correlated. This experimental observation provides the background for future research with the application goal of suggested nonlinear laser ultrasonic techniques for non-destructive evaluation of alloys’ strength and rigidity in the process of their heat treatment.

Experimental Configuration to Determine the Nonlinear Parameter β in PMMA and CFRP with the Finite Amplitude Method

Parameters to measure nonlinearity in polymethylmethacrylate (PMMA) and carbon fiber reinforced polymer (CFRP) materials have been determined with nonlinear ultrasound (NLUS). The nonlinear parameter β has been determined using the variation of the Finite Amplitude Method (FAM) with harmonic generation. Using this as a reference, the first contribution of this work consists of deducting the experimental configuration necessary to measure this nonlinear parameter in a correct and feasible way. Excitation level, frequency of the wave generated, number of cycles analysed and the distances transducer-specimen and specimen-hydrophone have been determined in both materials. The second contribution is a semi-analytical model that allows to obtain the nonlinear parameter in materials by removing water contribution and considering geometric and viscous attenuation, using the data obtained in an immersion tank. Finally, an application of this model has been carried out in PMMA in order to determinate the nonlinear parameter in this material. From the results, we confirm that the configuration determined in this paper to obtain the parameter β decreases the noise in the measurements.

Breaking the Symmetry of Momentum Conservation Using Evanescent Acoustic Fields

Although the conservation of momentum is a fundamental law in physics, its constraints are not fulfilled for wave propagation at material boundaries, where incident waves give rise to evanescent field distributions. While nonlinear susceptibility tensor terms can provide solutions in the optical regime, this framework cannot be applied directly to acoustic waves. Now, by considering a complete representation of wave interactions and scattering at boundaries, we are able to show a generic formalism of sum-frequency mixing for the whole scattering field including all evanescent waves. This general case was studied analytically and verified both numerically and experimentally for ultrasonic waves, showing that considering evanescent waves leads to an anomalous nonlinear interaction which enhances sum-frequency generation. This new interpretation not only provides a deeper understanding of the momentum conservation laws in acoustics but also promises translation of this new understanding into optics and photonics, to enhance nonlinear interactions.

Finite-element analysis of non-collinear mixing of two lowest-order antisymmetric Rayleigh-Lamb waves

Recently, a perturbation approach was adopted for the non-collinear mixing of plate waves in an isotropic elastic plate with quadratic nonlinearity by Ishii, Biwa, and Adachi [J. Sound Vib. 419, 390-404 (2018)] and a resonance condition was derived theoretically, namely, that a significant scattered plate wave is generated when its frequency and wavevector coincide with the sum/difference of those of its primary waves. However, that analysis assumed monochromatic plane waves that interacted everywhere in the plate. To apply the non-collinear mixing of plate waves to nondestructive evaluation of plate-like structures, the influence of the spatial and temporal finiteness of the primary waves on the generation of the scattered wave must be elucidated. To that end, the present study conducts three-dimensional dynamic finite-element analyses on the mixing of lowest-order antisymmetric Rayleigh-Lamb waves that have finite beam widths and time durations. The generation of scattered lowest-order symmetric Rayleigh-Lamb and shear horizontal waves with sum frequencies is discussed for various beam widths, intersection angles, and primary frequencies. The resonance condition derived originally for monochromatic plane waves in the aforementioned study is found to be valid even when the wave interaction is allowed only within limited space and time.

Receiver calibration and the nonlinearity parameter measurement of thick solid samples with diffraction and attenuation corrections

This paper presents analytical and experimental techniques for accurate determination of the nonlinearity parameter (β) in thick solid samples. When piezoelectric transducers are used for β measurements, the receiver calibration is required to determine the transfer function from which the absolute displacement can be calculated. The measured fundamental and second harmonic displacement amplitudes should be modified to account for beam diffraction and material absorption. All these issues are addressed in this study and the proposed technique is validated through the β measurements of thick solid samples. A simplified self-reciprocity calibration procedure for a broadband receiver is described. The diffraction and attenuation corrections for the fundamental and second harmonics are explicitly derived. Aluminum alloy samples in five different thicknesses (4, 6, 8, 10, 12cm) are prepared and β measurements are made using the finite amplitude, through-transmission method. The effects of diffraction and attenuation corrections on β measurements are systematically investigated. When diffraction and attenuation corrections are all properly made, the variation of β between different thickness samples is found to be less than 3.2%.

Self-action of propagating and standing Lamb waves in the plates exhibiting hysteretic nonlinearity: Nonlinear zero-group velocity modes

An analytical theory accounting for the influence of hysteretic nonlinearity of micro-inhomogeneous plate material on the Lamb waves near the S₁ zero group velocity point is developed. The theory predicts that the main effect of the hysteretic quadratic nonlinearity consists in the modification of the frequency and the induced absorption of the Lamb modes. The effects of the nonlinear self-action in the propagating and standing Lamb waves are expected to be, respectively, nearly twice and three times stronger than those in the plane propagating acoustic waves. The theory is restricted to the simplest hysteretic nonlinearity, which is influencing only one of the Lamé moduli of the materials. However, possible extensions of the theory to the cases of more general hysteretic nonlinearities are discussed as well as the perspectives of its experimental testing. Applications include nondestructive evaluation of micro-inhomogeneous and cracked plates.

Nonlinear effect of elastic vortexlike motion on the dynamic stress state of solids

We present a theoretical analysis of the dynamic stress-strain state of regions in a solid body that are involved in a collective elastic vortexlike motion. It is shown that the initiation of elastic vortexlike motion in the material is accompanied by the appearance of dilatancy and equivalent strain, the magnitudes of which are proportional to the square of the ratio of linear velocity on the periphery of the elastic vortex to the velocity of longitudinal elastic waves (P wave). Under conditions of dynamic loading the described dynamic effects are able to initiate inelastic deformation or destruction of the material at loading speeds of a few percent of the P-wave speed. The obtained analytical estimates suggest that dynamic nonlinear strains can make a significant contribution in a number of widely studied nonlinear dynamic phenomena in solids. Among them are the effect of acoustic (dynamic) dilatancy in solids and granular media, which leads to the generation of longitudinal elastic waves by transverse waves [V. Tournat et al., Phys. Rev. Lett. 92, 085502 (2004)10.1103/PhysRevLett.92.085502] and the formation of an array of intense "hot spots" (reminiscent of shear-induced hydrodynamic instabilities in fluids) in adiabatic shear bands [P. R. Guduru et al., Phys. Rev. E 64, 036128 (2001)10.1103/PhysRevE.64.036128].

Dynamic acousto-elastic testing applied to a highly dispersive medium and evidence of shell buckling of lipid-coated gas microbubbles

Dynamic acousto-elastic testing is applied to a mixture of lipid-coated microbubbles in water. A dynamic change of ambient pressure is produced by a 16 kHz pressure wave having a peak pressure amplitude of 28 kPa. The induced changes of phase velocity and attenuation are captured by a sequence of short ultrasound pulses with a center frequency of 4 MHz. As a consequence of the dispersion brought about by the resonance of microbubbles at a frequency close to 2 MHz, time-domain approaches like the cross-correlation method are shown to be unsuited to determine the variation in ultrasound wavespeed. A frequency-domain analysis shows that the acousto-elastic effect (first order pressure derivative of ultrasound phase velocity) depends on the ultrasound frequency. The acousto-elastic effect tends to that measured in water for an ultrasound frequency above the resonance frequency of microbubbles, while it is two orders of magnitude larger for an ultrasound frequency close to or below the resonance frequency of microbubbles. Besides the large magnitude of the acousto-elastic effect observed for an ultrasound frequency below the resonance frequency of microbubbles, the first order pressure derivative of ultrasound phase velocity is negative. This supports the occurrence of shell buckling of lipid-coated microbubbles induced by the 16 kHz pressure wave.

Acoustic nonlinearity parameters for transversely isotropic polycrystalline materials

This article considers polycrystalline materials with macroscopic elastic anisotropy and the effect of the anisotropy on the quadratic nonlinearity parameter used to describe second harmonic generation in solids. The polycrystal is assumed to have transversely isotropic elastic symmetry, which leads to a directional dependence of the nonlinearity parameters. Additionally, the anisotropy leads to second harmonic generation from an input shear wave. Estimates of the longitudinal and shear wave nonlinearity parameters are given as a function of single-crystal elastic constants, macroscopic anisotropy constants, and propagation direction. An inverse model is presented that relates measured nonlinearity parameters to the macroscopic anisotropy constants. The estimates of the nonlinearity parameters can be used to approximate the damage-free or baseline nonlinearity parameter of structural components, which helps the effort toward absolute measures of material damage.

Modelling harmonic generation measurements in solids

Harmonic generation measurements typically make use of the plane wave result when extracting values for the nonlinearity parameter, β, from experimental measurements. This approach, however, ignores the effects of diffraction, attenuation, and receiver integration which are common features in a typical experiment. Our aim is to determine the importance of these effects when making measurements of β over different sample dimensions, or using different input frequencies. We describe a three-dimensional numerical model designed to accurately predict the results of a typical experiment, based on a quasi-linear assumption. An experiment is designed to measure the axial variation of the fundamental and second harmonic amplitude components in an ultrasonic beam, and the results are compared with those predicted by the model. The absolute β values are then extracted from the experimental data using both the simulation and the standard plane wave result. A difference is observed between the values returned by the two methods, which varies with axial range and input frequency.

Noncollinear wave mixing for measurement of dynamic processes in polymers: physical ageing in thermoplastics and epoxy cure

Elastic wave mixing using an immersion method has shown effective monitoring and scanning capabilities when applied to thermoplastic ageing, epoxy curing, and non-destructive testing. In water, excitation and reception of waves do not require physical contact between the tools and the specimen, making the acquisition of high-resolution C-scans possible. The nonlinear material parameters exhibit a much higher sensitivity to the specimen state compared to linear ones. Thus, the nonlinear data for polymethyl methacrylate (PMMA) have a 40% difference between zones of "young" and "aged" material, while the linear data show no difference at all. Methodology and logistics of the immersion wave-mixing method are discussed in detail. Monitoring of epoxy curing has also revealed a good sensitivity of the method to this complex process including several characteristic stages, such as the time of maximal viscosity, the gel time, and the vitrification time. These stages are independently verified in separate rheometry measurements. The presented method allows for a number of possibilities: wave-mode and frequency separations, elimination of surrounding medium influence, "steering" (scanning) a scattered wave, controlling the location of the intersection volume, single-sided or double-sided measurements, and operation in detector mode.

Possible second-order nonlinear interactions of plane waves in an elastic solid

There exist ten possible nonlinear elastic wave interactions for an isotropic solid described by three constants of the third order. All other possible interactions out of 54 combinations (triplets) of interacting and resulting waves are prohibited, because of restrictions of various kinds. The considered waves include longitudinal and two shear waves polarized in the interacting plane and orthogonal to it. The amplitudes of scattered waves have simple analytical forms, which can be used for experimental setup and design. The analytic results are verified by comparison with numerical solutions of initial equations. Amplitude coefficients for all ten interactions are computed as functions of frequency for polyvinyl chloride, together with interaction and scattering angles. The nonlinear equation of motion is put into a general vector form and can be used for any coordinate system.

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

There is an unresolved debate in the scientific community about the shape of the quasistatic displacement pulse produced by nonlinear acoustic wave propagation in an elastic solid with quadratic nonlinearity. Early analytical and experimental studies suggested that the quasistatic pulse exhibits a right-triangular shape with the peak displacement of the leading edge being proportional to the length of the tone burst. In contrast, more recent theoretical, analytical, numerical, and experimental studies suggested that the quasistatic displacement pulse has a flat-top shape where the peak displacement is proportional to the propagation distance. This study presents rigorous mathematical analyses and numerical simulations of the quasistatic displacement pulse. In the case of semi-infinite solids, it is confirmed that the time-domain shape of the quasistatic pulse generated by a longitudinal plane wave is not a right-angle triangle. In the case of finite-size solids, the finite axial dimension of the specimen cannot simply be modeled with a linear reflection coefficient that neglects the nonlinear interaction between the combined incident and reflected fields. More profoundly, the quasistatic pulse generated by a transducer of finite aperture suffers more severe divergence than both the fundamental and second order harmonic pulses generated by the same transducer.

Anisotropy of dynamic acoustoelasticity in limestone, influence of conditioning, and comparison with nonlinear resonance spectroscopy

Anisotropy of wave velocity and attenuation induced by a dynamic uniaxial strain is investigated by dynamic acoustoelastic testing in limestone. Nonlinear resonance spectroscopy is performed simultaneously for comparison. A compressional resonance of the sample at 6.8 kHz is excited to produce a dynamic strain with an amplitude varied from 10(-7) to 10(-5). A sequence of ultrasound pulses tracks variations in ultrasonic velocity and attenuation. Variations measured when the ultrasound pulses propagate in the direction of the uniaxial strain are 10 times larger than when the ultrasound propagation occurs perpendicularly. Variations consist of a "fast" variation at 6.8 kHz and an offset. Acoustically induced conditioning is found to reduce wave velocity and enhance attenuation (offset). It also modifies "fast" nonlinear elastodynamics, i.e., wave amplitude dependencies of ultrasonic velocity and attenuation. At the onset of conditioning and beyond, different excitation amplitudes bring the material to non-equilibrium states. After conversion of velocity-strain dynamic relations into elastic modulus-strain dynamic relations and integration with respect to strain, the dynamic stress-strain relation is obtained. Analysis of stress-strain hysteresis shows that hysteretic nonlinear elasticity is not a significant source of the amplitude-dependent dissipation measured by nonlinear resonance spectroscopy. Mechanisms causing conditioning are likely producing amplitude-dependent dissipation as well.

Amplitude-dependent internal friction, hysteretic nonlinearity, and nonlinear oscillations in a magnesite resonator

The results of experimental and theoretical studies of low-frequency nonlinear acoustics phenomena (amplitude-dependent loss, resonance frequency shifts, and a generation of second and third harmonics) in a magnesite rod resonator are presented. Acceleration and velocity oscillograms of vibrations of the free boundary of the resonator caused by harmonic excitations were measured and analyzed. A theoretical description of the observed amplitude dependences was carried out within the framework of the phenomenological state equations that contain either of the two types of hysteretic nonlinearity (elastic and inelastic). The type of hysteresis and parameters of acoustic nonlinearity of magnesite were established from comparing the experimental measurements with the theoretical dependences. The values of the parameters were anomalously high even when compared to those of other strongly nonlinear polycrystalline materials such as granite, marble, limestone, sandstone, etc.

Reflection of compressional and Rayleigh waves on the edges of an elastic plate with quadratic nonlinearity

Previous ultrasonic studies have demonstrated that measurements of material nonlinearities can provide a means for detecting early signs of fatigue damage using both compressional (P) and Rayleigh (R) surface waves. However, these experimental studies have typically been limited to the direct wave arrival between the source and receiver in simple geometries where no reflection occurs. In particular, the degree of material nonlinearity is often quantified by the ratio of the cumulative amplitude of the first harmonic to that of the fundamental for the direct arrival only. Hence a practical question arises over the interpretation of ultrasonic measurements of material nonlinearities in the presence of reflected nonlinear waves. Thus, this article investigates the reflection problem of P or R waves at the edge of a homogeneous plate with quadratic nonlinearity using both a theoretical formulation, based on perturbation analysis, and direct numerical simulations using a Cellular Automata formulation. The numerical approach is first validated against an existing theoretical formulation for reflecting nonlinear P waves. It is then used to simulate the nonlinear reflection of R waves at a plate's edge for which no closed-form formulation is presently available.

Nonlinear effects for coda-type elastic waves in stressed granular media

Experimental results and their interpretations are presented on the nonlinear acoustic effects of multiple scattered elastic waves in unconsolidated granular media. Short wave packets with a central frequency higher than the so-called cutoff frequency of the medium are emitted at one side of the statically stressed slab of glass beads, and received at the other side after multiple-scattering and nonlinear elastic effects. Typical signals are strongly distorted compared to their initially radiated shape both due to nonlinearity and scattering. It is shown that acoustic waves with a deformation amplitude much lower than the mean static deformation of the contacts in the medium can modify the elastic properties of the medium. This addresses the problem of acoustic wave action on granular matter during and after acoustic excitation, which is necessary to understand in the nondestructive testing of the elastic properties of granular media by acoustic methods. Coda signal analysis is shown to be a powerful time-resolved tool in monitoring slight modifications in the elastic response of an unconsolidated granular structure.

Nonlinear shear wave interaction in soft solids

This paper describes nonlinear shear wave experiments conducted in soft solids with transient elastography technique. The nonlinear solutions that theoretically account for plane and nonplane shear wave propagation are compared with experimental results. It is observed that the cubic nonlinearity implied in high amplitude transverse waves at f(0)=100 Hz results in the generation of odd harmonics 3f(0), 5f(0). In the case of the nonlinear interaction between two transverse waves at frequencies f(1) and f(2), the resulting harmonics are f(i)+/-2f(j)(i,j=1,2). Experimental data are compared to numerical solutions of the modified Burgers equation, allowing an estimation of the nonlinear parameter relative to shear waves. The definition of this combination of elastic moduli (up to fourth order) can be obtained using an energy development adapted to soft solid. In the more complex situation of nonplane shear waves, the quadratic nonlinearity gives rise to more usual harmonics, at sum and difference frequencies, f(i)+/-f(j). All components of the field have to be taken into account.

Probing weak forces in granular media through nonlinear dynamic dilatancy: clapping contacts and polarization anisotropy

Rectification (demodulation) of high-frequency shear acoustic bursts is applied to probe the distribution of contact forces in 3D granular media. Symmetry principles allow for rectification of the shear waves only with their conversion into longitudinal mode. The rectification is due to nonlinear dynamic dilatancy, which is found to follow a quadratic or Hertzian power law in the shear wave amplitude. Evidence is given that a significant portion of weak contact forces is localized below 10(-2) of the mean force-a range previously inaccessible by experiment. Strong anisotropy of nonlinearity for shear waves with different polarization is observed.

Effects of harmonic phase on nonlinear surface acoustic waves in the (111) plane of cubic crystals

Spectral evolution equations are used to perform numerical studies of nonlinear surface acoustic waves in the (111) plane of several nonpiezoelectric cubic crystals. Nonlinearity matrix elements which describe the coupling of harmonic interactions are used to characterize velocity waveform distortion. In contrast to isotropic solids and the (001) plane of cubic crystals, the nonlinearity matrix elements usually cannot be written in a real-valued form. As a result, the harmonic components are not necessarily in phase, and dramatic variations in waveforms and propagation curves can be observed. Simulations are performed for initially monofrequency surface waves. In some directions the waveforms distort in a manner similar to nonlinear Rayleigh waves, while in other directions the velocity waveforms distort asymmetrically and the formation of shocks and cusped peaks is less distinct. In some cases, oscillations occur near the shocks and peaks because of phase differences between harmonics. A mathematical transformation based on the phase of the matrix elements is shown to provide a reasonable approximation of asymmetric waveform distortion in cases where the matrix elements have similar phase.

Strain wave evolution equation for nonlinear propagation in materials with mesoscopic mechanical elements

Nonlinear wave propagation in materials, where distribution function of mesoscopic mechanical elements has very different scales of variation along and normally to diagonal of Preisach-Mayergoyz space, is analyzed. An evolution equation for strain wave, which takes into account localization of element distribution near the diagonal and its slow variation along the diagonal, is proposed. The evolution equation provides opportunity to model propagation of elastic waves with strain amplitudes comparable to and even higher than characteristic scale of element localization near Preisach-Mayergoyz space diagonal. Analytical solutions of evolution equation predict nonmonotonous dependence of wave absorption on its amplitude in a particular regime. The regime of self-induced absorption for small-amplitude nonlinear waves is followed by the regime of self-induced transparency for high-amplitude waves. The developed theory might be useful in seismology, in high-pressure nonlinear acoustics, and in nonlinear acoustic diagnostics of damaged and fatigued materials.

Theory of non-collinear interactions of acoustic waves in an isotropic material with hysteretic quadratic nonlinearity

A particular form of the energy potential (cubic in strains) is proposed, which leads to the bow-tie behavior of the nonlinear modulus in an isotropic material with hysteresis of quadratic nonlinearity. The nonlinear scattering of a weak probe wave in the field of a strong pump wave is analyzed. It is demonstrated that collinear interactions of the shear waves are allowed in materials with nonlinearity hysteresis. Both in collinear and non-collinear frequency-mixing processes the combination frequency is composed of the probe wave frequency and one of the even harmonics of the pump wave. In general, the developed theory predicts that in the presence of the hysteretic nonlinearity the number of possible resonant scattering processes increases. In particular, if frequency-mixing processes are forbidden in the material with the elastic quadratic nonlinearity (for a fixed ratio of primary frequencies), they may be allowed in the materials with hysteretic quadratic nonlinearity. Moreover, in materials with hysteresis of the nonlinearity the resonant frequency mixing for a fixed ratio of primary frequencies may be allowed for multiple mutual orientations of the primary wave vectors.

Second-harmonic generation in sound beams reflected from, and transmitted through, immersed elastic solids

Second-harmonic generation in sound beams reflected from and transmitted through thick isotropic elastic solids is investigated experimentally. Measurements of diffraction patterns are compared with a theoretical model based on integral solutions for harmonic generation in sound beams. The solutions are connected by classical linear theory for reflection and transmission at fluid-solid interfaces. Nonspecular phenomena associated with rapid phase variations near critical angles are accurately described. The principal restriction is that the solid is sufficiently thick that internal reflections may be ignored.

Solitons in DNA

DNA is modeled as a homogeneous, cylindrical rod with nonlinear elasticity using the Ostrovskii-Sutin equation (OSE) with periodic boundary conditions. This equation predicts that longitudinal sound waves will be concentrated into packets called solitons. From a study of the damped OSE, we conclude that decay time is almost independent of the solitonic character of the solution. For the damped, driven OSE, on the other hand, we find that spectral features (such as absorption line widths and fine structure) are strongly influenced by the presence of anharmonicity. This effect is enhanced as the length of the DNA is increased.