Nonlinear Elastic Wave Research Articles

Electro-mechanical coupling diode of elastic wave in nonlinear piezoelectric metamaterials.

In this investigation, the bandgaps and nonreciprocal transmission of the nonlinear piezoelectric phononic crystal and elastic wave metamaterial are studied. Analytical solutions for the wave motion equations with the electro-mechanical coupling are obtained. According to the continuous conditions, the stop bands and transmission coefficients of both fundamental wave and second harmonic are derived by the stiffness matrix method. Some particular examples are presented to show the nonreciprocal transmission of the nonlinear elastic waves. Additionally, nonlinear ultrasonic experiments are applied to verify the theoretical analyses and numerical simulations. This work is intended to be helpful in the design and fabrication of devices of the elastic wave diode with piezoelectric materials.

On an inverse boundary value problem for a nonlinear elastic wave equation

We consider an inverse boundary value problem for a nonlinear elastic wave equation which was studied in [1]. We show that all the parameters appearing in the equation can be uniquely determined from boundary measurements under certain geometric assumptions. The proof is based on second order linearization and Gaussian beams.

Energetic effects analysis of shear waves and their second harmonics in a waveguide with non-ideal component contact

In this article the energetic and kinematic effects that occur in the elastic shear wave and its second harmonics propagation are investigated. The waveguide consists of anisotropic elastic crystal layer of cubic system m3m class enclosed between crystal halfspaces of same anisotropy class. A slipping contact is assumed in the contact zone of waveguide parts. The research is based on a model of general geometrical end physical nonlinearity in dynamic deformation processes. It allows to use elastic potential with the quadratic and cubic deformation components and the deformations with nonlinear terms. The approach of nonlinear elastic wave characteristics expansion into rows of a small parameter is used. Due to this approach at the first stage it's necessary to solve the problem of finding the components of the localized shear wave displacement vector (the problem of the first approximation). In the second stage, using the obtained results of the first approximation problem, the representation of the components of the displacement vector for the second harmonics of the localized elastic wave is solved in analytical form (the problem of the second approximation). By using the obtained kinematic results, the energy effects can be evaluated in the form of a vector of the average for the period of power flow. Specific results of the study of the amplitude-frequency and energy characteristics of the localized shear type elastic waves in the considered waveguide structure were obtained using computer algebra methods.The calculations of cinematic and energetic characteristics (that in contrast to linear SH harmonic are P-SV type waves) have been carried out for NaCl layer and germanium halfspaces waveguide.

Broadband nonlinear elastic wave modulation spectroscopy for damage detection in composites

A non-destructive testing procedure is proposed for damage detection in composites using full wavefield measurement obtained using a scanning laser Doppler vibrometer. Vibrations are excited using two low-power piezoelectric actuators leading to nonlinear elastic wave modulation at the defect. One actuator is supplied with a broadband chirp signal and the other actuator is supplied with a single-frequency sine signal. First, a time–frequency filtering method is proposed to extract specific nonlinear components of interest (e.g. second higher harmonic and first modulation sideband) without the need for multiple excitation sequences. Next, damage maps are constructed using broadband bandpower calculation of the filtered nonlinear components. It is demonstrated that the modulation sidebands provide an exclusive imaging of defect nonlinearity, and are not affected by potential source nonlinearity. The proposed damage map construction procedure is applied for various carbon fiber reinforced polymer test specimens with different damage features: (1) coupon with quasi-static indentation damage, (2) coupon with artificial delaminations, (3) bicycle frame with impact damage, and (4) stiffened aircraft panel with partially debonded stiffener. The obtained results indicate the high performance of the developed procedure for detection of various defect types in curved and/or stiffened carbon fiber reinforced polymer components.

Monitoring the Hydration Process in Carbon Nanotube Reinforced Cement-Based Composites Using Nonlinear Elastic Waves

There is currently an increased need for associating construction material properties and behavior with the nature of their microstructure. One of the major issues in this context is the need for understanding the curing process in freshly poured cement-based materials. This is particularly important when nanoreinforcement materials, such as carbon nanotubes, are used to enhance the mechanical behavior and multifunctionality of the final structure. The solidification point, at which the state of liquid suspension transmutes to the solid state, is of particular interest since it greatly influences the load-bearing capacity of the cement-based material and its structural behavior at the long term. The main purpose of the present work is to develop a reliable method for monitoring the hydration process during the early stages of freshly poured cementitious composites enhanced with carbon nanotubes. This methodology is based on the use of nonlinear elastic waves. To achieve this goal, a combination of contact ultrasonics with noncontact optical detection was used. The detection method for evaluating the setting process is based on the assessment of higher-harmonic amplitudes of an ultrasonic wave, with a given frequency, propagating through the cementitious material. It was observed that the material nonlinearity changes significantly during the hardening process, compared to velocity or attenuation measurements which are based on linear acoustics. These changes were more noticeable as the concentration of carbon nanotubes in the cement matrix increases, indicating that higher harmonics are more susceptible to minute microstructural changes.

Experimental study of thermo-acoustic wave modulation in a cracked plate

This paper presents a study on nonlinear elastic wave modulation due to an applied periodic thermal field. The experimental study is performed with a thermal analogy of the vibro-acoustic wave modulation (VAM) technique. The aim is to identify sources of nonlinear effects appearing in the response signal of a damaged structure. Assigning specific nonlinear effects to their corresponding sources is a key challenge for most nonlinear acoustic methods. Separation of nonlinear effects into those related to damage dynamics and those related to other sources, e.g., material nonlinearity or boundary conditions, is very important for practical applications of nonlinear acoustic methods.In this study, an aluminum plate with a localized fatigue crack is investigated. The plate is excited simultaneously with a high-frequency acoustic wave and a low-frequency thermal field generated by a modulated laser source. The response vibration signal is acquired at different locations (near and far from the damage) using a Laser Doppler Vibrometer (LDV). Signal processing is applied to extract the nonlinear effects in the form of modulation sidebands.

Nonlinear Elastic Waves in Magnetically Ordered Crystals in the Vicinity of Orientational Phase Transitions

A phase diagram of the dynamic magnetoelastic states of an easy-plane antiferromagnet is constructed. A dispersion relation is obtained for nonlinear magnetoelastic eigenwaves. It is shown that, at the point of the orientational phase transition, the dispersion of coupled spin and elastic waves depends only on wave amplitudes and parameters of magnetoelastic coupling.

Propagation of a spherical wave in elastoplastic medium with complex equations of state

The propagation of a spherical wave in the soil is solved in an analytically inverse way for soils with more complex equations of state. The results are obtained to propagate a spherical shock wave in soil with a more complex equation of state for the shape change in the medium. The study shows that taking into account the nonlinear elastic shock waves of the annular stress leads to an increase compared to the elastic medium. Note that in using a complicated equation of state of the soil, a spherical shock wave propagates in the soil, behind the front of which, in the disturbance region, the medium is unloaded.

Mathematical models of the evolution of classical and solitary cylindrical waves

The evolution of nonlinear elastic cylindrical displacement waves for initial profiles in the form of a Hankel and Macdonald functions is analyzed theoretically and numerically. The difference between the two waves is that the MacDonald function has no hump, decreases monotonically and has a concave downward profile, and the Hankel function is a harmonic attenuating wave.
 The main novelty is that the evolution of cylindrical waves is studied for two different approaches to the solution of a nonlinear equation. Some significant differences of these waves are shown. First, the features of the Hankel wave, a harmonic wave (symmetrical profile), are briefly described. Then, theoretically and numerically, a single wave with an initial profile in the form of a MacDonald function is analyzed in more detail. Distortion of the initial profile due to the nonlinear interaction of the wave itself and the increase in the maximum amplitude during wave propagation is common to these profiles. Significant features of the McDonald wave are shown - an uncharacteristic initial profile (a profile without a classical hump) evolves in an uncharacteristic way - the profile becomes much steeper and remains convex downwards.
 Keywords: classical and solitary cylindrical waves; five-constant Murnaghan potential; approximate methods; Hankel and Macdonald initial wave profiles; evolution.

Monitoring the Nonlinear Acoustic Behavior of Fresh Cementitious Materials during the Hardening Process Using Laser Doppler Vibrometry

A challenge in the field of material science and concrete technology is to associate material properties and behavior with changes in internal microstructure. A fundamental issue for planning, implementation, and quality control of construction projects is the fundamental understanding of the hardening process in freshly poured cement-based materials. Shortly after the mixing process, cementitious materials exhibit a substantial damping effect on ultrasonic wave propagation along with low-pulse velocity. The way a wave reflects and diffuses in the volume of a solid material can provide important information about internal heterogeneities. An important application of the wave propagation technique is the determination of layers within the material distinguished by unlike elastic properties. During the hardening process, the ultrasonic wave propagation exhibits a noticeable sensitivity at the point of phase change of the material, that is, the critical point at which the state of the suspension transforms from liquid to solid state. After that point, there is a continuous increase of pulse velocity and ultrasonic signal amplitude. The solidification point is of practical importance since at this point the load-bearing capacity of the composite material and its long-term behavior are based on the connectivity of the solid phase. This study aims to develop a nondestructive tool to monitor freshly poured cement-paste composites at early stages, during the hydration process, by means of nonlinear elastic waves. The measurement setup combined a contact ultrasonic transducer with a noncontact optical detection system, in which the principle of operation is based on the Doppler effect. This methodology enabled us to assess the amplitudes of harmonic vibrations of an acoustic wave propagating through the material with a specific fundamental frequency. This enabled the evaluation of important material properties, such as minute changes in the internal microstructure of fresh concrete during hardening, the evolution of nonlinearity parameters that relate to higher-order elastic constants of the material, as well as the ultrasonic wave velocity.

Dynamics of mechanical metamaterials: A framework to connect phonons, nonlinear periodic waves and solitons

Flexible mechanical metamaterials have been recently shown to support a rich nonlinear dynamic response. In particular, it has been demonstrated that the behavior of rotating-square architected systems in the continuum limit can be described by nonlinear Klein–Gordon equations. Here, we report on a general class of solutions of these nonlinear Klein–Gordon equations, namely cnoidal waves based on the Jacobi elliptic functions sn, cn and dn. By analyzing theoretically and numerically their validity and stability in the design- and wave-parameter space, we show that these cnoidal wave solutions extend from linear waves (or phonons) to solitons, while covering also a wide family of nonlinear periodic waves. The presented results thus reunite under the same framework different concepts of linear and non-linear waves and offer a fertile ground for extending the range of possible control strategies for nonlinear elastic waves and vibrations.

Application of Nonlinear Elastic Wave Spectroscopy in the Field of Nuclear Energy

Abstract In the field of nuclear energy, two particular applications are considered—the measurement of biological shielding concrete and measurement of high-temperature high-pressure piping. For the measurement of biological shielding concrete, a special manipulator was developed, which is applied through the ionization channels of VVER-440 power plants. Biological shielding concrete is covered on all sides with an 11 mm thick steel liner preventing its assessment using any conventional test method. For the measurement of high-temperature high-pressure piping, acoustic emission sensors installed in the LEMOP system (NPP Temelin) are used for both excitation and data acquisition. In operation, this piping is covered with glass wool insulation, but installed sensors on the waveguides make it possible to assess pipelines without removing the insulation. To demonstrate the material degradation, a set of concrete test specimens and a set of steel test specimens were made. Nonlinear elastic wave spectroscopy is a test method based on measuring the elastic–plastic response of the material whose plastic component is caused by microcracks. Part of this response may be caused by inhomogeneity of the material itself or the nonlinearity of the test system used, which is to be minimized in the measurement. Due to degradation, the number of microcracks increases, increasing the total nonlinear response of the material. The aim of this approach is to monitor the degradation of concrete and metallic materials based on the response of the test system to ultrasonic excitation.

Effect of the Third Approximation in the Analysis of the Evolution of a Nonlinear Elastic P-Wave. Part 2*

The propagation of a nonlinearly elastic longitudinal plane displacement wave for a symmetric initial wave profile described by a Gaussian function and an asymmetric initial profile described by a Whittaker function is analyzed. The basic novelty is that the evolution of waves is analyzed by the approximate methods by taking into account the first three approximations. The harmonic wave is analyzed only to compare with the new results for the bell-shaped wave. Some significant differences between the evolutions of waves are shown. Common to these profiles is the distortion of the initial profile due to the nonlinear self-interaction of the wave. The bell-shaped (symmetrical profile) wave retains symmetry when moving in a nonlinearly elastic body. For some initial sets of parameters, this wave does not change the wavelength initially and only shows a tendency towards the formation of two humps instead of one when we take into account the second conditional harmonic (two bell-shaped waves form and adjoin each other and halve wavelength) and the sinking of the left and elevation of the right hump when we take into account the third harmonic. The asymmetric profile described by Whittaker function retains the wavelength and asymmetry when the second approximation of the second harmonic is taken into account, but the amplitude increases rapidly. With additional allowance for the third harmonic, two asymmetric humps are formed, the humps resemble the evolution of the symmetric profile when the second harmonic is taken into account. Common to the evolutions of the symmetric and asymmetric profiles is the distortion of the initial wave profile: the formation of two symmetric humps when the second harmonic is taken into account in the analysis of the Gaussian function and two asymmetric humps when the third harmonic is taken into account in the analysis of the Whittaker function. Taking the third harmonic into account for a symmetric profile makes the two humps asymmetrical for symmetric and asymmetric profiles, only in these two profiles, the asymmetry is different.

Modeling of 3-D elastic waves in nonlinear elastic solids using the iterative nonlinear contrast source method

In nondestructive testing, the generation of higher harmonics and the mixing of elastic waves are used to measure the nonlinearity parameters, which in turn are closely related to the microscopic state of the material. A 3-D numerical tool that can simulate large scale, four-dimensional, nonlinear elastic wavefields would be very useful for the reliable interpretation of experimental results. Here, a method is presented that can efficiently compute the nonlinear displacement fields in a homogeneous, isotropic elastic medium, taking into account its third-order elastic constants (A, B, C). The method is based on the Neumann iterative solution of an integral equation involving a Green's function of the linear `background' medium, and a contrast source representing the nonlinearity of the medium. The integral equation is solved iteratively, and the computations are based on Fast Fourier Transforms using a sampling rate close to the Nyquist limit, i.e., two grid points of the shortest wavelength. The displacement fields are evaluated using the scalar and vector potential functions representing the compressional and shear displacements. In this presentation, the suitability of the INCS method for modeling the harmonics of 3-D nonlinear elastic waves and its validation by comparison with an analytical benchmark solution, will be presented.

Research on perturbation method for nonlinear elastic waves.

This paper explores the propagation of nonlinear elastic waves in a two-dimensional isotropic medium. The analytical expressions of first-order potentials corresponding to second harmonic acoustic components are obtained and discussed by using the perturbation method. Based on the careful theoretical analysis, it is shown that the first-order P wave always has a resonant term, which is proportional to the propagation distance in the condition of simultaneous excitation of the P wave. On the contrary, the first-order SV wave does not have any cumulative effect. Moreover, the nonlinear interactions between the P wave and SV are also presented.

Effect of the Third Approximation in the Analysis of the Evolution of a Nonlinear Elastic P-wave. Part 1*

A nonlinear plane longitudinal elastic displacement wave is studied theoretically and numerically using the Murnaghan model for two forms of initial profile: harmonic and bell-shaped. A major novelty is that the evolution of waves is analyzed by approximate methods taking into account the first three approximations. The harmonic wave is analyzed only to compare with the new results for the bell-shaped wave. Some significant differences between the evolution of waves are shown. The initially symmetric profiles transform differently due to distortion: symmetrically (for the harmonic profile) and asymmetrically (for the bell-shaped profile). The third approximation introduces the fourth harmonic for the harmonic wave when this wave is analyzed by the method of successive approximations, while the bell-shaped wave is characterized in the third approximation differently when using the method of constraints on the displacement gradient. At relatively long distances from the beginning of the propagation, the one-hump bell-shaped wave transforms into a two-hump one. These humps adjoin each other halving their lengths. The third approximation allows us to observe new wave effects: the asymmetry of the left and right humps about their peaks and the asymmetry of the humps about each other; the lowering of the left hump and the rise of the right one. The results obtained are analyzed.

Nonlinear elastic wave propagation and breathing-debond identification in a smart composite structure

This paper addresses a problem on geometric-nonlinearity, due to the occurrence of breathing-debond in a smart composite structure (SCS). In order to solve the problem, elastic wave propagation based numerical simulations and laboratory experiments were carried out on SCS with baseplate – stiffener debonds. The frequency-domain analysis of the numerical and experimental signals shows that the presence of breathing-debond produces a prominent nonlinear response in terms of higher-harmonics. It was also noticed that the magnitudes of those harmonics significantly change with the increase in propagation distance. A baseline-free online breathing-debond source localization strategy is proposed that exclusively uses the changes in second harmonic magnitudes of the elastic wave signals registered at the sensor-networks in SCSs. The study was further extended for variable debond sizes and locations. It is observed that the proposed monitoring strategy is capable of localizing a wide range of breathing-debond sources in the SCSs.

Self-emitted surface corrugations in dynamic fracture of silicon single crystal

When a dynamic crack front travels through material heterogeneities, elastic waves are emitted, which perturb the crack and change the morphology of the fracture surface. For asperity-free crystalline materials, crack propagation along preferential cleavage planes is expected to present a smooth crack front and form a mirror-like fracture surface. Surprisingly, we show here that in single crystalline silicon without material asperities, the crack front presents a local kink during high-speed crack propagation. Meanwhile, local oscillations of the crack front, which can move along the crack front, emerge at the front kink position and generate periodic fracture surface corrugations. They grow from angstrom amplitude to a few hundred nanometers and propagate with a long lifetime at a frequency-dependent speed, while keeping a scale-independent shape. In particular, the local front oscillations collide in a particle-like manner rather than proceeding with a linear superposition upon interaction, which presents the characteristic of solitary waves. We propose that such a propagating mode of the crack front, which results from the fracture energy fluctuation at a critical crack speed in the silicon crystal, can be considered as nonlinear elastic waves that we call "corrugation waves."

Three-dimensional nonreciprocal transmission in a layered nonlinear elastic wave metamaterial

In this investigation, three-dimensional nonreciprocal transmission is realized by the combination of structural asymmetry and material nonlinearity. Analytical solutions of the coupled quasi-longitudinal (qP) and quasi-shear (qS1 and qS2) waves are derived. According to Bloch’s law and the stiffness matrix method, we obtain the band gaps and the transmission coefficients of the fundamental and double frequency waves. We also discuss the effects of the incident azimuthal angle on the propagation behaviors in nonlinear phononic crystals and elastic wave metamaterials. This present work is expected to help design practical devices of nonreciprocal transmission with vector characteristics.

Nonlinear Elastic Wave Energy Imaging for the Detection and Localization of In-Sight and Out-of-Sight Defects in Composites

In this study, both linear and nonlinear vibrational defect imaging is performed for a cross-ply carbon fiber-reinforced polymer (CFRP) plate with artificial delaminations and for a quasi-isotropic CFRP with delaminations at the edge. The measured broadband chirp vibrational response is decomposed into different components: the linear response and the nonlinear response in terms of the higher harmonics. This decomposition is performed using the short-time Fourier transformation combined with bandpass filtering in the time-frequency domain. The linear and nonlinear vibrational response of the defect is analyzed by calculation of the defect-to-background ratio. Damage maps are created using band power calculation, which does not require any user-input nor prior information about the inspected sample. It is shown that the damage map resulting from the linear band power shows high sensitivity to shallow defects, while the damage map associated to the nonlinear band power shows a high sensitivity to both shallow and deep defects. Finally, a baseline-free framework is proposed for the detection and localization of out-of-sight damage. The damage is localized by source localization of the observed nonlinear wave components in the wavenumber domain.