Nonlinear Elastic Wave Research Articles

Novel results from quadratically nonlinear elastic wave models using Murnaghan’s potential

AbstractIn this article, we study one, two and three-dimensional nonlinear elastic wave equations using quadratically nonlinear Murnaghan potential. We employ two effective methods for obtaining approximate series solutions the Adomian decomposition and the variational iteration method. These methods have the advantage of not requiring any physical parametric assumptions in the problem. Finally, these methods can generate expansion solutions for linear and nonlinear differential equations without perturbation, linearization, or discretization. The results obtained using the adopted methods along various initial and boundary conditions are in excellent agreement with the numerical results on MATLAB, which show the reliability of our methods to these problems. We came to the conclusion that our methods are accurate and simple to use.

Seismic scattering inversion for multiple parameters of overburden-stressed isotropic media

Subsurface reservoirs are compressed by overburden stress resulting from the gravity of overlying masses, and the resulting stress changes significantly affect the seismic reflection responses generated at the reservoir interfaces. Several exact reflection coefficient equations have been well established to delineate the role that in situ stress plays in altering the energy transition and amplitude of seismic reflection responses. These exact equations, however, cannot be effectively used in practice due to their intricate formulations and the difficult geophysical estimation for the third-order elastic constants (3oECs) embedded in reflection coefficients. Based on the theories of nonlinear elasticity and elastic wave inverse scattering, we derive an approximate seismic reflection coefficient equation for overburden-stressed isotropic media in terms of the P-wave modulus, shear modulus, density, and a defined stress-related parameter (SRP). The SRP is a combined quantity of elastic moduli, 3oECs, and overburden stress, which can be naturally treated as a dimensionless stress-induced anisotropy parameter. Its inclusion effectively eliminates the need for 3oECs information when using our equation to estimate the desired reservoir properties from seismic observations. By comparing our equation to the exact one, we confirm its validity within the moderate-stress range. Then, we introduce a Bayesian inversion approach incorporating the new reflection coefficient equation to estimate four model parameters. In our approach, the Cauchy and Gaussian distribution functions are used for the a priori probability and the likelihood distributions, respectively. The synthetic tests from two well-log data sets and a field example demonstrate that four parameters can be reasonably inverted using our approach with rather smooth initial models, which illustrates the feasibility of our inversion approach.

Nonlinear soliton spiral induces coupled multimode dynamics in multi-stable dissipative metamaterials

With the robust and self-trapped properties, recent advances about soliton dynamics in multi-stable mechanical metamaterials have led to many innovative techniques from signal processing to robotics. This work proposes a multi-stable mechanical metamaterial driven by nonlinear dissipative solitons, in which the coupling and decoupling of multiple locomotion modes can be achieved. Based on a cylinder network with asymmetric energy landscape, the uniform field model of Landau theory is developed. During the theoretical calculation, the analytical solutions of several dissipative solitons are derived, which allow multiple special behaviors of solitary waves, such as wave velocity gaps, directional propagation and spiral phase transition. By incorporating such effects into robotic designs, a variety of complex movements can be achieved by a single structure, including hopping, rolling, rotating, swinging, bending and translational components. In particular, as excitation positions change, the mechanical metamaterial can flexibly switch multiple locomotion modes without changing configurations, e.g., spinning and spin-less, straight and oblique as well as coupled multimode movements. This work wishes to provide some new inspirations for the applications of nonlinear elastic wave metamaterials and phase transition theory in robotics.

Mathematical framework of nonlinear elastic waves propagating in pre-stressed media

Mathematical framework of nonlinear elastic waves propagating in pre-stressed media

Modelling of non-linear elastic constitutive relationship and numerical simulation of rocks based on the Preisach–Mayergoyz space model

SUMMARY The existence of pores, cracks and cleavage in rocks results in significant non-linear elastic phenomena. One important non-linear elastic characteristic is the deviation of the stress–strain curve from the linear path predicted by Hooke's law. To provide a more accurate description of the non-linear elastic characteristics of rocks and to characterize the propagation of non-linear elastic waves, we introduce the Preisach–Mayergoyz space model. This model effectively captures the non-linear mesoscopic elasticity of rocks, allowing us to observe the stress–strain and modulus–stress relationships under different stress protocols. Additionally, we analyse the discrete memory characteristics of rocks subjected to cyclic loading. Based on the Preisach–Mayergoyz space model, we develop a new non-linear elastic constitutive relationship in the form of an exponential function. The new constitutive relationship is validated through copropagating acousto-elastic testing, and the experimental result is highly consistent with the data predicted by the theoretical non-linear elastic constitutive relationship. By combining the new non-linear elastic constitutive relationship with the strain–displacement formula and the differential equation of motion, we derive the non-linear elastic wave equation. We numerically solve the non-linear elastic wave equation with the finite difference method and observe two important deformations during the propagation of non-linear elastic waves: amplitude attenuation and dispersion. We also observe wave front discontinuities and uneven energy distribution in the 2D wavefield snapshot, which are different from those of linear elastic waves. We qualitatively explain these special manifestations of non-linear elastic wave propagation.

Non-contact Non-linear Impact Resonance Acoustic Spectroscopy for thermal damage characterization on mortar samples

Abstract The high sensitivity of the nonlinear terms of the elastic response of materials to the incipient appearance of damage has led to the appearance of the Non-linear Elastic Wave Spectroscopy methods (NEWS). Particularly, the Non-linear Impact Resonance Acoustic Spectroscopy (NIRAS) technique detects changes in the resonance of a material (frequency, damping factor, etc.) as a function of the intensity of the impact. Traditionally, mechanical waves have been monitored using an external sensor in contact with the material. Currently, alternative technologies are capable of capturing mechanical waves without direct contact or with embedded sensors in the material itself from its manufacture. This helps the automation process or the continuous monitoring of the material. Mortar is the most used composite material in construction. In this work, thermal damage on mortar samples is characterized by NIRAS tests using different sensing techniques. Accelerometers have been used as the reference technique, while laser interferometer and Fiber Bragg Grating (FBG) have been used as non-contact and embedded sensor techniques respectively. Two levels of damage are presented: sound samples and 400ºC damage. The different techniques offer similar results, showing the capability of the proposed techniques (FBG and laser interferometry) to be an accurate alternative to traditional contact techniques for NIRAS testing.

Nonlinear elastic waves generated by surface wave interaction with local plasticity in thick-walled structures

Thick-walled structures (TWSs) are frequently utilized as primary load-bearing structures across diverse industrial sectors, illustrated by train axles. Proper health monitoring, particularly for early damage detection, is required to ensure the operational safety and structural integrity of such structures. Nonlinear elastic-wave-based structural health monitoring techniques present a potentially efficacious methodology, which relies heavily upon the understanding of nonlinear wave generation and propagation characteristics. Unfortunately, due to the complexity introduced by various wave modes in TWSs, a comprehensive understanding of nonlinear elastic waves is still lacking. Considering practical local plasticity on the surface of a TWS, this study investigates the characteristics of nonlinear elastic wave generation and propagation resulting from the fundamental surface wave-local plasticity interaction. Theoretical analyses were carried out to examine the scattering features of the generated nonlinear bulk waves based on Snell’s law. Finite element simulations were then performed to confirm the predicted scattering characteristics of the nonlinear bulk waves, with the introduction of a local material nonlinearity to replicate the plasticity in a TWS. Using the extrusion method, a local plastic zone was produced on an aluminum thick-walled beam. Experiments were finally carried out using a laser vibrometer to confirm the scattering features of nonlinear elastic waves. It was found that the nonlinear shear bulk waves are mainly generated due to the fundamental surface interaction with local plasticity. Furthermore, the generated nonlinear shear bulk waves propagate at a fixed angle towards the interior of the TWS, which can be measured on the opposite surface. The elucidation of the nonlinear elastic wave generation and propagation characteristics paves the way for an effective sensor arrangement in early damage detection applications.

In vivo imaging and computational modeling of nonlinear shear waves in living skeletal muscles

How the state of living muscles modulates the features of nonlinear elastic waves generated by external dynamic loads remains unclear because of the challenge of directly observing and modeling nonlinear elastic waves in skeletal muscles in vivo, considering their active deformation behavior. Here, this important issue is addressed by combining experiments performed with an ultrafast ultrasound imaging system to track nonlinear shear waves (shear shock waves) in muscles in vivo and finite element analysis relying on a physically motivated constitutive model to study the effect of muscle activation level. Skeletal muscle was loaded with a deep muscle stimulator to generate shear shock waves (SSWs). The particle velocities, second and third harmonics, and group velocities of the SSWs in living muscles under both passive and active states were measured in vivo. Our experimental results reveal, for the first time, that muscle states have a pronounced effect on wave features; a low level of activation may facilitate the occurrence of both the second and third harmonics, whereas a high level of activation may inhibit the third harmonic. Finite element analysis was further carried out to quantitatively explore the effect of active muscle deformation behavior on the generation and propagation of SSWs. The simulation results at different muscle activation levels confirmed the experimental findings. The ability to reveal the effects of muscle state on the features of SSWs may be helpful in elucidating the unique dynamic deformation mechanism of living skeletal muscles, quantitatively characterizing diverse shock wave-based therapy instruments, and guiding the design of muscle-mimicking soft materials.

Global existence of nonlinear elastic waves: Non-hyperelastic case

For the Cauchy problem of nonlinear elastic wave equations for three-dimensional isotropic, homogeneous and hyperelastic materials with null condition, global existence of classical solutions with small initial data was proved in Agemi (2000) [1] and Sideris (2000) [17] independently. How to treat the more general non-hyperelastic case is suggested as an open problem in Agemi (2005) [2]. The aim of this paper is to resolve this problem.

Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe

Abstract This research article delves into the intricate domain of nonlinear wave dynamics within the framework of a Murnaghan hyperelastic circular pipe. Thus, the current study makes use of some powerful analytical approaches to examine the propagation of nonlinear elastic waves on a Murnaghan hyperelastic circular pipe. The work is exceptional since it allows for the incorporation of double dispersion terms and material nonlinearity in the controlling nonlinear mode. The study entails a thorough examination of the propagation and interaction of solitons within the Murnaghan hyperelastic medium, providing insights into the distinctive nonlinear wave phenomena manifested by circular pipe configurations. Theoretical insights are substantiated by numerical simulations, presenting a comprehensive understanding of the dynamic responses within these elastic structures. In the end, graphical representations of some of the derived solutions have been provided for clarification. In addition, the reported solutions in the study help researchers working in modern fields of engineering and materials science to obtain valuable insights that can inform the design, analysis, and optimization of materials and structures in contemporary applications.

Chirp-coded ultrasound for localization of linear and nonlinear scatterers in isotropic media

Chirp-coded ultrasound has been under active research for reciprocal Time Reversal with Nonlinear Elastic Wave Spectroscopy for ultrasonic wave focusing and additionally detection and analysis of nonlinear defects. Localization and imaging of defects under these principles, without scanning or use of other complicated methods, is an open problem. Frequency Modulated Constant Wave technology is widely known in radar applications where the distance to multiple reflecting targets, their velocities and angles of return are found from the chirp signal. These principles are not yet widely used in ultrasound technology. Chirp-coded ultrasound is common between the two methods and enable to maximize total power on target due to longer transmitted signals (as opposed to pulsed operation). This work proposes to unite the Frequency-Modulated Constant-Wave scatterer localization with reciprocal Time Reversal – Nonlinear Elastic Wave Spectroscopy principles to enable the localization of scatterers distant from the transducers and the analysis of their nonclassical nonlinearity which often can be caused by microdamage or cracking. In this paper only the first part (localization of scatterers using frequency-modulated constant wave ultrasound) of the problem is investigated. The main goal is to verify the feasibility of the method for ultrasound in solids and identify the main problems to be solved. Physical experiments and simulations are used to ascertain the detectability of linear and nonlinear scatterers in an isotropic solid medium (acrylic) using ultrasound. Potential problems with this approach are identified, to be solved in future work. In the future, this approach could enable to spatially filter the return signals, enabling to use reciprocal time reversal for focusing the wave energy in space directly onto selected scatterers, using the simplest of test setups: one transmitting and two receiving transducers.

Local plasticity-induced nonlinear surface wave scattering in thick-walled structures

Thick-walled structures (TWSs) are frequently utilized as primary load-bearing structures. The early detection of damage is crucial to ensure the operational safety and integrity of these structures. Nonlinear elastic-wave-based techniques provide a potential solution to the problem, which heavily relies on the good understanding of nonlinear wave generation and propagation characteristics. The task, however, is technically challenging due to the complex wave modes existing in a TWS. Targeting a meticulously manufactured local plasticized damage region on the surface of a TWS, which can be regarded as the precursor of incipient damage, this study investigates the scattering features of nonlinear elastic waves resulting from the interaction between the fundamental quasi-surface wave and the local plasticity. Finite element simulations and experiments show that, upon interacting with the nonlinear material zone, both nonlinear quasi-surface waves and shear bulk waves can be generated, with the latter scattered towards the opposite surface of the TWS. The scattered nonlinear shear waves exhibit a strong and predictable directivity with high nonlinear energy content, which is conducive to the detection and localization of the incipient surface damage of the TWS.

Scattering and rigidity for nonlinear elastic waves

Scattering and rigidity for nonlinear elastic waves

Propagation of solitary waves in origami-inspired metamaterials

We propose a design strategy for creating origami-like mechanical metamaterials with diverse non-linear mechanical properties and capable of remote actuation. The proposed triangulated cylindrical origami (TCO)-inspired metamaterials enable the highly desirable strain-softening/hardening and snap-through behaviors via a multi-material and highly deformable hinge design. Moreover, we couple these novel non-linear mechanical properties of the TCO origami-inspired metamaterials with the transformative ability of hard-magnetic active materials, allowing for untethered shape- and property-actuation in the developed metamaterials. We develop a mathematical modeling framework for the proposed TCO origami-inspired metamaterials, building on approximating the highly deformable hinges as a combination of longitudinal and rotational springs. We validate the accuracy of the developed mathematical modeling approach by comparing the analytically predicted compressive response of a unit cell structure with the corresponding numerical and experimental results. Using the developed mathematical modeling framework, we investigate the magnetic field-induced large deformation and superimposed solitary wave propagation in the TCO origami-inspired metamaterial system. We show that the proposed metamaterial allows us to tune the key characteristics of the enabled non-linear solitary waves, including their characteristic width and amplitude. The proposed design strategy for readily manufacturable origami-inspired metamaterial systems paves a novel path for practical engineering applications. Our studies also underscore the potential of magneto-mechanical interaction in the design of reconfigurable metamaterial systems with superior non-linear mechanical and elastic wave properties.

Negative Refraction of Mixing Waves in Nonlinear Elastic Wave Metamaterials

Negative Refraction of Mixing Waves in Nonlinear Elastic Wave Metamaterials

Determination of the density in a nonlinear elastic wave equation

Determination of the density in a nonlinear elastic wave equation

Nonlinear elastic wave dispersions of solar cells strengthened by advanced functionally graded materials via both mathematical modeling and deep neural networks technique

From claims of roughly 3 % efficiency in 2009 to over 25 % efficiency now, perovskite solar cells have made amazing developments in recent years. Perovskite solar cells have quickly increased their efficiency, but there are still a number of obstacles to overcome before they can be considered a viable commercial technology. So, improving the stability of the perovskite solar cells and obtaining the mechanical properties of this kind of structure using computer simulation to decrease the computational cost is a challenging issue for designers. In this work, instead of a metal layer in the fabrication of solar cells, a functionally graded (FG) layer in which the material properties in this layer change in each direction. For the first time in the current report, the nonlinear phase velocity of the 3D-FG solar cells via mathematical simulation is presented. For solving the nonlinear partial differential equations (PDEs) of the perovskite solar cells, an analytical method is used. After doing some mathematical manipulation using the analytical method, a multiple scales method for solving the nonlinear equations in the time domain is presented. In order to compensate for the absence of a dataset suitable for deep neural networks (DNN), a mathematical simulation is conducted to acquire the nonlinear phase velocity characteristics of enhanced solar cells. The DNN method is used to forecast the nonlinear phase velocity characteristics of the present study, after the completion of training, testing, and validation processes. This approach has the advantage of cheap computing expense. Finally, the results show that when the panels tend to be too concave or convex, the nonlinear phase velocity is unstable. As an amazing outcome for related industries, the nonlinear phase velocity is very sensitive to the values of Kx, Ky, and Rx/a, and all designers according to their purposes should consider the amount of these parameters. The current results’ outcomes can be used in future building of the perovskite solar cell knowing about nonlinear dynamics features of 3D-FG perovskite solar cells.

Numerical model of nonlinear elastic bulk wave propagation in solids for non-destructive evaluation

Nonlinear ultrasonic techniques can be difficult and non-intuitive to understand due to the range of wave mixing combinations available between similar or different wave modes. To overcome this, a numerical model that uses a Finite Difference Time Domain (FDTD) scheme, without a staggered grid system, to solve the nonlinear elastic bulk wave equations in two dimensions is proposed in this paper, with the purpose of better understanding nonlinear ultrasonic techniques. Both material and geometrical nonlinearities are considered and a stress-type boundary condition is used to model the excitation. The energy transfer between frequency bands is shown and the amplitude trend of the harmonics are validated against available theories. It is then used to simulate the nonlinear ultrasonic field generated by a finite-size element of an array in a solid to better understand its behaviour. An array-element directivity at the second harmonic frequency is shown. Since the FDTD model does not account for attenuation, a correction using a ray-based approach is applied to the simulated result for direct comparison against experimental measurements. The field obtained from a typical array used in non-destructive testing is then simulated to characterise its behaviour. Experimental comparison of the nonlinear displacement fields for different focal positions showed good agreement, further validating the FDTD model. This FDTD model opens opportunities to virtually experiment and design appropriate nonlinear ultrasonic array inspections whilst isolating and understanding contribution from material nonlinearity.

Complementary Methods for the Assessment of the Porosity of Laser Additive-Manufactured Titanium Alloy.

The method of making parts through additive manufacturing (AM) is becoming more and more widespread due to the possibility of the direct manufacturing of components with complex geometries. However, the technology's capacity is limited by the appearance of micro-cracks/discontinuities during the layer-by-layer thermal process. The ultrasonic (US) method is often applied to detect and estimate the location and size of discontinuities in the metallic parts obtained by AM as well as to identify local deterioration in structures. The Ti6Al4V (Ti64) alloy prepared by AM needed to acquire a high-quality densification if remarkable mechanical properties were to be pursued. Ultrasonic instruments employ a different type of scanning for the studied samples, resulting in extremely detailed images comparable to X-rays. Automated non-destructive testing with special algorithms is widely used in the industry today. In general, this means that there is a trend towards automation and data sharing in various technological and production sectors, including the use of intelligent systems at the initial stage of production that can exclude defective construction materials, prevent the spread of defective products, and identify the causes of certain instances of damage. Placing the non-destructive testing on a completely new basis will create the possibility for a broader analysis of the primary data and thus will contribute to the improvement of both inspection reliability and consistency of the results. The paper aims to present the C-scan method, using ultrasonic images in amplitude or time-of-flight to emphasize discontinuities of Ti64 samples realized by laser powder-bed fusion (L-PBF) technology. The analysis of US maps offers the possibility of information correlation, mainly as to flaws in certain areas, as well as distribution of a specific flaw in the volume of the sample (flaws and pores). Final users can import C-scan results as ASCII files for further processing and comparison with other methods of analysis (e.g., non-linear elastic wave spectroscopy (NEWS), multi-frequency eddy current, and computer tomography), leading to specific results. The precision of the flight time measurement ensures the possibility of estimating the types of discontinuities, including volumetric ones, offering immediate results of the inspection. In situ monitoring allows the detection, characterization, and prediction of defects, which is suitable for robotics. Detailing the level of discontinuities at a certain location is extremely valuable for making maintenance and management decisions.

Localization and classification of scattered nonlinear ultrasonic signatures in bio-mechanical media using time reversal approach.

This paper deals with the time reversal approach along with signal classification using ϕ-divergences in biomedical applications for localization and statistical classification of ultrasonic nonlinearities. The time reversal (TR) approach in combination with nonlinear elastic wave spectroscopy (NEWS) is used to obtain the nonlinear signature of air bubbles with different sizes and ultrasound contrast agents in a liquid. An optimized chirp-coded signal in the range of 0.6-3 MHz is used as a compression coding. The signal classification is performed using the fuzzy classification method and the divergence decision tree algorithm using specific ϕ-divergence spectral measures extracted from the received ultrasonic response containing acoustic nonlinearities. The classification results prove that different types of nonlinearities extracted with classical "pulse inversion" based coding methods can be identified. Simultaneously, the different positions of scattered sources are distinguished by ϕ-divergence methods. The potential of time reversal nonlinear elastic wave spectroscopy methods for understanding of ultrasonic wave propagation in complex media is clearly exhibited.