The nonlinear propagation of acoustic waves in a viscoelastic medium containing cylindrical micropores

Based on an equivalent medium approach, this paper presents a model describing the nonlinear propagation of acoustic waves in a viscoelastic medium containing cylindrical micropores. The influences of pores' nonlinear oscillations on sound attenuation, sound dispersion and an equivalent acoustic nonlinearity parameter are discussed. The calculated results show that the attenuation increases with an increasing volume fraction of micropores. The peak of sound velocity and attenuation occurs at the resonant frequency of the micropores while the peak of the equivalent acoustic nonlinearity parameter occurs at the half of the resonant frequency of the micropores. Furthermore, multiple scattering has been taken into account, which leads to a modification to the effective wave number in the equivalent medium approach. We find that these linear and nonlinear acoustic parameters need to be corrected when the volume fraction of micropores is larger than 0.1%.     

Acoustic nonlinearity of cracks partially filled with a viscous liquid: A quadratic approximation

A mechanism that accounts for the acoustic nonlinearity of cracks partially filled with a viscous liquid is proposed. The mechanism is related to the nonlinear dependences of the capillary and viscous pressures in liquid on the distance between the crack surfaces and on the rate of change of this distance. The nonlinear equation of state is obtained for this type of cracks, and the parameters of this equation are determined. It is shown that the presence of a viscous liquid can lead to a considerable increase in the acoustic nonlinearity of such cracks, as compared to the cracks filled with an ideal liquid.     

声波在有裂纹的固体中的非经典非线性传播

固体材料的无损检测是一个非常重要的课题,带裂纹的固体材料显示非经典非线性声学现象,本文对此现象进行了实验和理论研究。从实验上一维观察到此现象,发现奇次谐波振幅与基波振幅呈平方关系,与理论预计基本吻合;理论上从二维的角度数值模拟了声波在有损耗的带裂纹的固体中的声传播,并讨论了经典非线性和非经典非线性对声传播的影响,发现裂纹的贡献主要体现在非经典非线性上。分析了样品中裂纹的宽度和位置与非线性声参数的关系,在靠近样品中心的两个对称区域以及距离声源较近点,非线性声参数对样品的破损较为敏感,而在中央和距声源最远端敏感性较低;随着裂纹宽度的扩大,非线性声参数也开始变大,但在破损区域蔓延到棒边缘之前,有下降的趋势。      

Nonlinear acoustic spectroscopy of local defects in geomaterials

General approaches to solving the problem of nonlinear acoustic spectroscopy of defects in geomaterials are considered. Expressions that relate the nonlinear response (scattering at combination frequencies) to the position, orientation, and nonlinear characteristics of narrow cracks are obtained. The expressions describe a broad class of nonlinear interactions at a crack. The nonlinearity caused by the contact of uneven rough edges of a crack is analyzed in detail. The results of the analysis are compared with the results obtained earlier from considering micromechanical models and with experimental data. The satisfactory agreement between the theoretical and experimental values of Landau’s moduli suggests that the mechanism of contact nonlinearity may manifest itself in the process of fracture of polycrystalline rock, when narrow cracks with uneven edges are formed. Numerical examples demonstrate the possibility of determining the orientation and position of a narrow crack. The procedure of solving the problem of crack localization is illustrated by the example of a crack in a thin rod. The importance of taking into account the phase data in the determination of the crack coordinate is pointed out.     

All-optical probing of the nonlinear acoustics of a crack

Experiments with an all-optical method for the study of the nonlinear acoustics of cracks in solids are reported. Nonlinear acoustic waves are initiated by the absorption of radiation from a pair of laser beams intensity modulated at two different frequencies. The detection of acoustic waves at mixed frequencies, absent in the frequency spectrum of the heating lasers, by optical interferometry or deflectometry provides unambiguous evidence of the elastic nonlinearity of the crack. The high contrast in crack imaging achieved by remote optical monitoring of the nonlinear acoustic processes is due to the strong dependence of the efficiency of optoacoustic conversion on the state of the crack. The highest acoustic nonlinearity is observed in the transitional state of the crack, which is intermediate between the open and the closed ones.     

Acoustic nonlinearity of cracks partially filled with liquid: cubic approximation.

The theoretical investigation of mechanisms of the acoustic nonlinearity (elastic and inelastic) of cracks partially filled with an ideal and viscous liquid and associated with the nonlinear dependence of the capillary and viscous pressure in the liquid on the distance between the crack surfaces and the velocity of the change of this distance is proposed. The nonlinear (in cubic approximation) equations of the state of these cracks is obtained, and its parameters are defined. It is shown that the presence of the viscous liquid may lead to the considerable increase of the acoustic nonlinearity of such cracks in comparison with cracks filled with the ideal liquid.     

Analytical modeling and vibration analysis of internally cracked rectangular plates

This study proposes an analytical model for nonlinear vibrations in a cracked rectangular isotropic plate containing a single and two perpendicular internal cracks located at the center of the plate. The two cracks are in the form of continuous line with each parallel to one of the edges of the plate. The equation of motion for isotropic cracked plate, based on classical plate theory is modified to accommodate the effect of internal cracks using the Line Spring Model. Berger?s formulation for in-plane forces makes the model nonlinear. Galerkin?s method used with three different boundary conditions transforms the equation into time dependent modal functions. The natural frequencies of the cracked plate are calculated for various crack lengths in case of a single crack and for various crack length ratio for the two cracks. The effect of the location of the part through crack(s) along the thickness of the plate on natural frequencies is studied considering appropriate crack compliance coefficients. It is thus deduced that the natural frequencies are maximally affected when the crack(s) are internal crack(s) symmetric about the mid-plane of the plate and are minimally affected when the crack(s) are surface crack(s), for all the three boundary conditions considered. It is also shown that crack parallel to the longer side of the plate affect the vibration characteristics more as compared to crack parallel to the shorter side. Further the application of method of multiple scales gives the nonlinear amplitudes for different aspect ratios of the cracked plate. The analytical results obtained for surface crack(s) are also assessed with FEM results. The FEM formulation is carried out in ANSYS.     

Modeling nonlinearities of ultrasonic waves for fatigue damage characterization: Theory,simulation, and experimental validation

A dedicated modeling technique for comprehending nonlinear characteristics of ultrasonic waves traversing in a fatigued medium was developed, based on a retrofitted constitutive relation of the medium by considering the nonlinearities originated from material, fatigue damage, as well as the “breathing” motion of fatigue cracks. Piezoelectric wafers, for exciting and acquiring ultrasonic waves, were integrated in the model. The extracted nonlinearities were calibrated by virtue of an acoustic nonlinearity parameter. The modeling technique was validated experimentally, and the results showed satisfactory consistency in between, both revealing: the developed modeling approach is able to faithfully simulate fatigue crack-incurred nonlinearities manifested in ultrasonic waves; a cumulative growth of the acoustic nonlinearity parameter with increasing wave propagation distance exists; such a parameter acquired via a sensing path is nonlinearly related to the offset distance from the fatigue crack to that sensing path; and neither the incidence angle of the probing wave nor the length of the sensing path impacts on the parameter significantly. This study has yielded a quantitative characterization strategy for fatigue cracks using embeddable piezoelectric sensor networks, facilitating deployment of structural health monitoring which is capable of identifying small-scale damage at an embryo stage and surveilling its growth continuously.     

Baseline-free estimation of residual fatigue life using a third order acoustic nonlinear parameter

Prediction of crack growth and fatigue life estimation of metals using linear/nonlinear acousto-ultrasound methods is an ongoing issue. It is known that by measuring nonlinear parameters, the relative accumulated fatigue damage can be evaluated. However, there is still a need to measure two crack propagation states to assess the absolute residual fatigue life. A procedure based on the measurement of a third-order acoustic nonlinear parameter is presented to assess the residual fatigue life of a metallic component without the need of a baseline. The analytical evaluation of how the cubic nonlinear-parameter evolves during crack propagation is presented by combining the Paris law to the Nazarov-Sutin crack equation. Unlike other developed models, the proposed model assumes a crack surface topology with variable geometrical parameters. Measurements of the cubic nonlinearity parameter on AA2024-T351 specimens demonstrated high sensitivity to crack propagation and excellent agreement with the predicted theoretical behavior. The advantages of using the cubic nonlinearity parameter for fatigue cracks on metals are discussed by comparing the relevant results of a quadratic nonlinear parameter. Then the methodology to estimate crack size and residual fatigue life without the need of a baseline is presented, and advantages and limitations are discussed.     

In situ estimation of crack concentration in hard rock

Results of a 2004 field experiment aimed at determining the quadratic nonlinearity parameter in granite that forms the shore of the Ladoga lake are presented. The measurements were based on the observation of the nonlinear interaction between monochromatic waves excited by two hydroacoustic radiators of 1 kW each positioned near the shore. The initial level of nonlinear distortions was much lower than the level of the received difference-frequency signal. The quadratic nonlinearity parameter proved to be higher than that reported in the majority of publications. An assumption was put forward that the high nonlinearity of granite is caused by the high concentration of cracks in it. Pieces of rock were taken from the measurement site, and rectangular samples were prepared from them. The samples were studied by the acoustic spectroscopy method in laboratory conditions. As a result, estimates of crack concentration were obtained, which proved to be consistent with the field measurements of the quadratic nonlinearity parameter. Thus, the possibility of estimating the crack concentration in situ from the measurements of the quadratic nonlinearity parameter was demonstrated.     

Modeling nonlinear processes in electron-optical systems

At sufficiently high energy densities in localized regions, nonlinear effects appear to one extent or another. The field energy and nonlinearity factor of the excited medium define the threshold of the geometry of a localized region in vacuum space. In this work, a method for modeling electron-optical systems with nonlinear elements in the form of nonlinear local physical fields is presented.     

An Intense Wave in Defective Media Containing Both Quadratic and Modular Nonlinearities: Shock Waves,Harmonics, and Nondestructive Testing

The observed nonclassical power-law dependence of the amplitude of the second harmonic wave on the amplitude of a harmonic pump wave is explained as a phenomenon associated with two types of nonlinearity in a structurally inhomogeneous medium. An approach to solving the inverse problem of determining the nonlinearity parameters and the exponent in the above-mentioned dependence is demonstrated. To describe the effects of strongly pronounced nonlinearity, equations containing a double nonlinearity and generalizing the Hopf and Burgers equations are proposed. The possibility of their exact linearization is demonstrated. The profiles, spectral composition, and average wave intensity in such doubly nonlinear media are calculated. The shape of the shock front is found, and its width is estimated. The wave energy losses that depend on both nonlinearity parameters—quadratic and modular—are calculated.     

Nonlinear susceptibility of a metal-dielectric nanocomposite caused by induced dipole-dipole forces between particles

The optical coefficients of a nonlinearity for a macroscopic ensemble of aggregates of metal nanoparticles that is caused by deformation of the spatial structure of clusters in a strong light field are calculated for the first time. For a continuous medium that consists of a nonabsorbing dielectric containing aggregated silver nanoparticles that do not possess an intrinsic optical nonlinearity, coefficients of nonlinear absorption are obtained in relation to the intensity and frequency of incident light. It is shown that, at intensities of up to a few megawatts per square centimeter, the addition to the absorption of a nanocomposite to be modeled is adequately described by a third-order nonlinearity. The magnitude and sign of the nonlinear absorption coefficient agree well with the previously obtained experimental data for aggregated silver and gold colloids under nanosecond excitation.     

Observation of the "Luxemburg-Gorky effect" for elastic waves

An experimental observation of a new nonlinear-modulation effect for longitudinal elastic waves is reported. The phenomenon is a direct elastic wave analogy with the so-called Luxemburg-Gorky (L-G) effect known over 60 years for radio waves propagating in the ionosphere. The effect consists of the appearance of modulation of a weaker initially non-modulated wave propagating in a nonlinear medium in the presence of an amplitude-modulated stronger wave that produces perturbations in the medium properties on the scale of its modulation frequency. The reported transfer of modulation from one elastic wave to another was observed in a resonator cut of a glass rod containing a few small cracks. Presence of such a small damage drastically enhances the material nonlinearity compared to elastic atomic nonlinearity of homogeneous solids, so that the pronounced L-G type cross-modulation could be observed at strain magnitude in the stronger wave down to 10(-7) and smaller. Main features of the effect are pointed out and physical mechanism of the observed phenomena is discussed.     

A novel method for crack detection in beam-like structures by measurements of natural frequencies

A novel method is proposed for calculating the natural frequencies of a multiple cracked beam and detecting unknown number of multiple cracks from the measured natural frequencies. First, an explicit expression of the natural frequencies through crack parameters is derived as a modification of the Rayleigh quotient for the multiple cracked beams that differ from the earlier ones by including nonlinear terms with respect to crack severity. This expression provides a simple tool for calculating the natural frequencies of the beam with arbitrary number of cracks instead of solving the complicated characteristic equation. The obtained nonlinear expression for natural frequencies in combination with the so-called crack scanning method proposed recently by the authors allowed the development of a novel procedure for consistent identification of unknown amount of cracks in the beam with a limited number of measured natural frequencies. The developed theory has been illustrated and validated by both numerical and experimental results.     

Nonlinear acoustic scattering by a partially closed surface-breaking crack

A theoretical model describing the nonlinear scattering of acoustic waves by surface-breaking cracks with faces in partial contact is presented. The nonlinear properties of the crack are accounted for by suitable boundary conditions that are derived from micromechanical models of the dynamics of elastic rough surfaces in contact. Both linear and nonlinear responses of the crack are shown to be largest for a shear vertical wave incident on the surface containing the crack at an angle just above the critical angle for longitudinal waves. These findings question the fitness for the purpose of a conventional inspection method, which utilizes shear vertical waves at 45 degrees of incidence to search for surface-breaking cracks in many engineering components. For angles of incidence proximal to the critical angle of longitudinal waves, the efficiency of the second harmonic's generation appears to be the highest. Thanks to the increased sensitivity to surface-breaking cracks, this configuration seems to offer a solution to the localization problem, a task that has eluded nonlinear techniques operating under other circumstances. Finally, this model suggests a simple interpretation of the highly localized nonlinear response of delaminations in composite materials.     

Local interaction simulation approach to modelling nonclassical,nonlinear elastic behavior in solids

Recent studies show that a broad category of materials share "nonclassical" nonlinear elastic behavior much different from "classical" (Landau-type) nonlinearity. Manifestations of "nonclassical" nonlinearity include stress-strain hysteresis and discrete memory in quasistatic experiments, and specific dependencies of the harmonic amplitudes with respect to the drive amplitude in dynamic wave experiments, which are remarkably different from those predicted by the classical theory. These materials have in common soft "bond" elements, where the elastic nonlinearity originates, contained in hard matter (e.g., a rock sample). The bond system normally comprises a small fraction of the total material volume, and can be localized (e.g., a crack in a solid) or distributed, as in a rock. In this paper a model is presented in which the soft elements are treated as hysteretic or reversible elastic units connected in a one-dimensional lattice to elastic elements (grains), which make up the hard matrix. Calculations are performed in the framework of the local interaction simulation approach (LISA). Experimental observations are well predicted by the model, which is now ready both for basic investigations about the physical origins of nonlinear elasticity and for applications to material damage diagnostics.     

Estimating the sizes of surface cracks based on Hall element measurements of the leakage magnetic field and a dipole model of a crack

A method is proposed for estimating the sizes of surface cracks in magnetic materials. The method is based on applying a magnetic field, then determining the leakage magnetic field in the vicinity of a crack by moving a Hall element on the surface of the material along one or two scanning lines crossing the crack, and measuring the corresponding Hall voltage distribution. A dipole model of a crack is utilized, in which a surface crack is considered as being full of magnetic dipoles aligned parallel to the applied field, and whose density varies linearly along the depth of the crack. Analytical expressions are derived for the z-component of the intensity of the leakage magnetic field, and for the measured Hall voltage in the vicinity of a crack with an arbitrary cross-section along its long axis when it is perpendicular to the applied field. The crack sizes and the parameters of the distribution of magnetic dipoles along the crack depth are computed by crack inversion, which represents a regression for the Hall voltage distribution. A variable theoretical Hall voltage distribution is fitted to the measured Hall voltage distribution by minimizing the corresponding RMS error, which gives the unknown parameters at the end of the minimization. Hall voltage distributions are measured on ferromagnetic steel samples containing one artificial surface crack. Some crack inversions are performed for estimating the maximum crack depth and the crack width of cracks with rectangular and isosceles triangular cross-sections along the long crack axis. The accuracy of these crack inversions increases by utilizing either Hall voltage distributions measured along only one of the scanning lines, instead of along both scanning lines, or by using more precisely measured Hall voltage distributions. The fast and accurate estimation of the maximum crack depth and the crack width by such crack inversions could be important for pipeline inspection. Other crack inversions are performed for determining the cross-section along the long axis of the investigated cracks with satisfactory results. Received: 2 March 2001 / Accepted: 10 April 2001 / Published online: 25 July 2001     

Cascade cross modulation due to the nonlinear interaction of elastic waves in samples with cracks

The phenomenon under study consists in that, in an inhomogeneous material with nonlinearity caused by the presence of soft defects (the so-called “nonclassical” nonlinearity), cascade nonlinear effects are fairly strong and may even become comparable to the first-order effects. Similar cascade effects in media with a common nonlinearity of the crystal lattice are much weaker. This difference can be used as an important diagnostic indicator in nondestructive testing. Experimental data obtained for samples with cracks, which exhibit both ordinary modulation and cross-modulation effects, as well as a cascade cross modulation, are presented. The origin of the enhanced level of cascade effects is explained by modeling with the use of a simple model of nonlinearity of an inhomogeneous material containing soft Hertzian contacts.