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Phononic materials are periodically arranged building blocks in the form of material properties, geometries, and/or boundary conditions. This synthetic architecture makes phononic materials capable of manipulating mechanical waves that have potential applications across multiple disciplines of physics and engineering. Initial studies have been focused on linear phononic materials that assume small-amplitude waves. The incorporation of nonlinearity, however, has been shown to open opportunities for a new realm of dynamic responses valid beyond the small-amplitude regime. Acknowledging this potential, research in the field has undergone a paradigm shift in the last decade or so by exploiting various sources of nonlinearities within phononic materials. A comprehensive overview of the origin of nonlinearities and how they are modeled, solved, and realized in phononic materials, and specifically, what role nonlinearity plays in enabling rich nonlinear wave responses, is crucial for the future advancement of the field. In this review, we discuss recent advances in nonlinear wave propagation in phononic materials and metamaterials by drawing links between different phononic media and their nonlinearity-induced behaviors. We first briefly discuss the analytical methods employed to solve nonlinear wave propagation problems by focusing on foundational models. We then review physics-based sources of nonlinearities, primarily, material, geometric, and contact nonlinearities and elucidate nonlinear wave responses enabled by them in phononic materials and metamaterials. Finally, we outline existing challenges and possible future directions in nonlinear phononics and metamaterials.

International audience; In the current work, we study the role of chirality and non-centrosymmetry on the nonlinear wave propagation characteristics of periodic architectured media. The considered nonlinearities arise from the higher-order inner element kinematics of the periodic media and are therefore directly related to its structural pattern. Regarding centrosymmetric designs, the frequency corrections obtained -in the context of the Lindstedt-Poincare method- suggest that chiral architectures are more sensitive to inner kinematic nonlinearities than well-known, achiral lattice designs. In particular, for hexachiral lattice designs, non-negligible frequency corrections are obtained, not only for the primal eigenmode, but also for higher-order modes, extensively modifying the linear band diagram structure. To the contrary, for achiral, triangular and square lattice designs, inner kinematic nonlinearities mainly influence the primal, lowest eigenmode, with the higher-order modes to remain practically unaffected. Non-centrosymmetric inner designs modify the linear and nonlinear wave propagation material attributes both for chiral and achiral lattice patterns. However, the frequency ranges affected are strongly lattice dependent, with hexachiral and triangular lattices to be primarily influenced in their high frequency range, contrary to square lattices, which are mainly affected in their low frequency region.

We investigate the generalized nonlinear Schrodinger equation (NLSE) of third order analytically, that accept the single-parameter family of one hump embedded soliton. This equation has been utiliz...

In this paper, nonlinearity associated with intense ultrasound is studied by using the one-dimensional motion of nonlinear shock wave in an ideal fluid. In nonlinear acoustics, the wave speed of different segments of a waveform is different, which causes distortion in the waveform and can result in the formation of a shock (discontinuity). Acoustic pressure of high-intensity waves causes particles in the ideal fluid to vibrate forward and backward, and this disturbance is of relatively large magnitude due to high-intensities, which leads to nonlinearity in the waveform. In this research, this vibration of fluid due to the intense ultrasonic wave is modeled as a fluid pushed by one complete cycle of piston. In a piston cycle, as it moves forward, it causes fluid particles to compress, which may lead to the formation of a shock (discontinuity). Then as the piston retracts, a forward-moving rarefaction, a smooth fan zone of continuously changing pressure, density, and velocity is generated. When the piston stops at the end of the cycle, another shock is sent forward into the medium. The variation in wave speed over the entire waveform is calculated by solving a Riemann problem. This study examined the interaction of shocks with a rarefaction. The flow field resulting from these interactions shows that the shock waves are attenuated to a Mach wave, and the pressure distribution within the flow field shows the initial wave is dissipated. The developed theory is applied to waves generated by 20 KHz, 500 KHz, and 2 MHz transducers with 50, 150, 500, and 1500 W power levels to explore the effect of frequency and power on the generation and decay of shock waves. This work enhances the understanding of the interactions of high-intensity ultrasonic waves with fluids.

The most common source of transformer failure is in the insulation, and the most prevalent warning signal for insulation weakness is partial discharge (PD). Locating the positions of these partial discharges would help repair the transformer to prevent failures. This work investigates algorithms that could be deployed to locate the position of a PD event using data from ultra-high frequency (UHF) sensors inside the transformer. These algorithms typically proceed in two steps: first determining the signal arrival time, and then locating the position based on time differences. This paper reviews available methods for each task and then propose new algorithms: a convolutional iterative filter with thresholding (CIFT) to determine the signal arrival time and a reference table of travel times to resolve the source location. The effectiveness of these algorithms are tested with a set of laboratory-triggered PD events and two sets of simulated PD events inside transformers in production use. Tests show the new approach provides more accurate locations than the best-known data analysis algorithms, and the difference is particularly large, 3.7X, when the signal sources are far from sensors.

In this article, an analytical approach is developed to investigate the nonlinear vibrational behavior of an graphene sheet. In order to be a practical instance, the circular nanoplate is a...

The propagation of elastic waves in a circular cylinder and cylindrical annulus for two types of power-law constitutive equations is investigated. These power-law constitutive equations can describe elastic responses where the linearised strain and stress are nonlinearly related. These constitutive equations are a subclass of the more general class of implicit constitutive equations and are characterised by expressing the strain as a non-invertible function of the stress. Pseudo-solitary stress wave solutions for both types of constitutive equations in the circular cylinder and cylindrical annulus are derived. We find that for the power-law constitutive equation of Type I, a shock front will develop at the back of the wave while for the power-law constitutive equation of Type II, a shock front will develop at the front of the wave. Estimates of the times at which the shock front will develop are given. Standing wave solutions for both types of constitutive equations in the circular cylinder and cylindrical annulus are also obtained and the periods of oscillation are compared.

We propose a 3D-printed graded lattice made of hollow elliptical cylinders (HECs) as a new way to design impact mitigation systems. We observe asymmetric dynamics in the graded HEC chains with increasing and decreasing stiffness. Specifically, the increasing stiffness chain shows an acceleration of the propagating waves, while the decreasing stiffness chain shows the opposite. From the standpoint of impact mitigation, the decreasing stiffness chain combined with the strain-softening behavior of HECs results in an order-of-magnitude improvement in force attenuation compared to the increasing stiffness chain. We extend this finding to the graded 2D arrays and demonstrate a similar trend of wave transmission efficiency contrast between the increasing and decreasing stiffness lattices. The 3D-printed HEC lattices shown in this study can lead to the development of a new type of impact mitigating and shock absorbing structures., Comment: 22 pages, 11 figures, 4 appendices

The present work deals with wave generation in fully nonlinear numerical wave tanks (NWT). As an alternative to modelling a moving (physical) wavemaker, a two-dimensional (2D) potential-flow NWT is coupled with an external spectral wave data (SWD) application programming interface (API). The NWT uses the harmonic polynomial cell (HPC) method to solve the governing Laplace equations for the velocity potential and its time derivative, and has previously been extensively validated and verified for numerous nonlinear wave-propagation problems using traditional wave-generation mechanisms. Periodic waves of different steepness generated with a stream-function theory as reference solution in the SWD API are first considered to investigate the method’s numerical accuracy. Thereafter, with a higher-order spectral method (HOSM) as the SWD API solution, irregular waves with different wave heights and water depths relevant for e.g. aquaculture and offshore structures are simulated. Differences between the HPC and HOSM solutions in and near steep crests are investigated. The study aims to demonstrate a robust method to generate and propagate general wave fields for further studies of nonlinear waves and wave-body interaction in both two and three dimensions.

We experimentally demonstrate a simple approach to enhance second-harmonic generation (SHG) from multilayer Gallium Selenide by encapsulating with an optimized thickness of low-index dielectric layers. 46-times enhancement is observed showing good agreement with nonlinear wave propagation simulation.

An advanced fully hydro- and morphodynamic two-dimensional horizontal (2DH) numerical model, describing the processes of nonlinear wave propagation, sediment transport, and morphological c...

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional nonlinear wave propagation. The model involves two small positive parameters measuring the relative strengths of the nonlinear and dispersive effects. We take two different kernel functions that have similar dispersive characteristics in the long-wave limit and compare the corresponding solutions of the Cauchy problems with the same initial data. We prove rigorously that the difference between the two solutions remains small over a long time interval in a suitable Sobolev norm. In particular our results show that, in the long-wave limit, solutions of such nonlocal equations can be well approximated by those of improved Boussinesq-type equations., 13 pages, to appear in Applicable Analysis

This study aims to investigate the capability of two common numerical methods, Homotopy Analysis Method (HAM) and Variational Iteration Method (VIM), and to suggest more efficient approximate solution method to the governing equations of nonlinear surface wave propagation in shallow water. To do so, semi-flat, moderate, and sharp slope of shore which are connected to an open ocean with a uniform depth are exposed to a solitary wave with initial wave height H=2 and stationary elevation d=20. Then, the surface elevation and velocity curves for these profiles are determined and compared by HAM and VIM. To verify the numerical modeling, two slopes i.e. semi-flat and moderate slope are considered and modeled in Flow-3D. Afterwards, the results of surface elevations are compared to each other by using correlation coefficient. The correlation coefficients for the slopes represent that the results coincide well. Ultimately, although the results of both methods are quite similar, using HAM is highly recommend rather than VIM since it makes solution procedure fast-converging and more abridged.

We investigate the existence of solitary waves in a nonlinear square spring-mass lattice. In the lattice, the masses interact with their neighbors through linear springs, and are connected to the ground by a nonlinear spring whose force is expressed as a polynomial function of the masses out-of-plane displacement. The low-order Taylor series expansions of the discrete equations lead to a continuum representation that holds in the long wavelength limit. Under this assumption, solitary wave solutions are sought within the long wavelength approximation, and the subsequent application of multiple scales to the resulting nonlinear continuum equations. The study focuses on weak nonlinearities of the ground stiffness and reveals the existence of 3 types of solitons, namely a bright, a dark, and a vortex soliton. These solitons result from the balance of dispersive and nonlinear effects in the lattice, setting aside other relevant phenomena in 2D waves such as diffraction that may lead to a field that does not change during propagation in nonlinear media. For equal constants of the in-plane springs, the governing equation reduces to the Klein-Gordon type, for which bright and dark solitons replicate solutions for one-dimensional lattices. However, unequal constants of the in-plane springs aligned with the two principal lattice directions lead to conditions in which the soliton propagation direction, defined by the group velocity, differs from the wave vector direction, which is unique to two-dimensional assemblies. Furthermore, vortex solitons are obtained for isotropic lattices, which shows similarities with results previously found in optics, thermal media and quantum plasmas. The paper describes the main parameters defining the existence of these solitary waves, and verifies the analytical predictions through numerical simulations. Results show the validity of obtained solutions and illustrate the main characteristics of the solitary waves found in the considered n M. Ruzzene acknowledges the support of the UC3M-Santander Chairs of Excellence Program during academic year 2016-17. J. Vila Moran acknowledges the support from the US National Science Foundation (Grant number 1332862). The authors from the Universidad Carlos III de Madrid are indebted to the Ministerio de Ciencia e Innovación de España (Project DPI-2014-57989-P) for the financial support.

This present manuscript studies a nonlinear hyperbolic model in fractional form which generalizes the nonlinear Klein–Gordon system. The equation under investigation includes the presence of a time-fractional operator of the Caputo type. A space-fractional form of that equation with integer-order temporal derivative has been previously investigated to elucidate the existence of localized wave transmission in relativistic wave equations. Here, we employ numerical techniques to estimate the solution of the fractional equation. The method has consistency of fourth order in space. Meanwhile, the temporal order of consistency is equal to 3 − α . We considered herein a sinusoidal perturbation of the medium. The simulations show the existence of the transmission of localized nonlinear modes in some complex fractional media governed by hyperbolic models. Physically, the present work investigates the phenomenon of nonlinear supratransmission in a continuous generalization of linear chains of harmonic oscillators with memory effects. In particular, the present work corroborates the presence of this nonlinear phenomenon in chains of pendula with memory and arrays of Josephson junctions attached through superconducting wires and memory effects.

We find that a class of parity-time- ( $$\mathcal {PT}$$ -) symmetric rational potentials can support stable solitons in the defocusing Kerr-nonlinear media, though they may not enjoy entirely real linear spectra. Analytical expressions of spatial solitons are elicited at lots of isolated propagation-constant points, around which several families of numerical fundamental solitons can be found to be stable, which is validated by linear stability analysis and nonlinear wave propagation. Many other intriguing properties of nonlinear localized modes are also discussed in detail, including the interactions, excitations, and transverse power flows. The idea of the $$\mathcal {PT}$$ -symmetric rational potentials can also be extended to other types of nonlinear wave models.

The Haar wavelet method (HWM) is adapted for solving 2D nonlinear wave propagation problems. The 2D Burgers equation is considered as a model equation here. The 2D wavelet expansion is employed for the spatial derivatives and standard ordinary differential equation solvers are used for the temporal derivative. The aim of the study is to validate HWM in multi- dimensional case. The numerical results obtained are found to be in good agreement with analytical solution.

This paper is about the study of space–time fractional Fokas–Lenells equation that describes nonlinear wave propagation in optical fibers. Three prominent schemes are employed for extracting different types of exact soliton solutions. In particular, the [Formula: see text] function method, the hyperbolic function method and the simplest Riccati equation scheme are investigated for the said model. As a sequel, a series of soliton solutions are obtained and verified through MATHEMATICA. The obtained solutions are significant additions in some specific fields of physics and engineering. Furthermore, the 3D graphical descriptions are left to analyze the pulse propagation for the reader.

Various behaviours of nonlinear wave propagation and competition have been discussed and investigated extensively and meticulously, especially when the media are homogeneous. However, corresponding studies in heterogeneous media are much scarcer. In this paper, spontaneously generated waves from one-dimensional heterogeneous oscillatory media, modelled by complex Ginzburg–Landau equations with spatially varied controlling parameters, are investigated. An unexpected homogeneous wave train clearly emerges under certain conditions. With the theory of interface-selected waves, we can theoretically predict the frequencies and wavenumbers under several conditions. This kind of wave train can be found in a wide region of parameter space. These phenomena are robust when parameters are varied nonlinearly or linearly with fluctuation. Moreover, this kind of homogeneous wave plays an important role in wave competition and affects wave propagation in spatially heterogeneous nonlinear systems, which will bring new applications of heterogeneity and provide new ideas for wave control.

The Generalized Nonlinear Schrodinger Equation (GNLSE) finds several applications, especially in describing pulse propagation in nonlinear fiber optics. A well-known and thoroughly studied phenomenon in nonlinear wave propagation is that of modulation instability (MI). MI is approached as a weak perturbation to a pump and the analysis is based on preserving those terms linear on the perturbation and disregarding higher-order terms. In this sense, the linear MI analysis is relevant to the understanding of the onset of many other nonlinear phenomena, but its application is limited to the evolution of the perturbation over short distances. In this work, we propose quasi-analytical approximations to the propagation of a perturbation consisting of additive white noise that go beyond the linear modulation instability analysis. Moreover, we show these approximations to be in excellent agreement with numerical simulations and experimental measurements.

Recently developed soft materials exhibit nonlinear wave propagation with potential applications in energy trapping, shock mitigation and wave focusing. We address finitely deformed materials subject to combined transverse and axial impacts, and study the resultant nonlinear waves. We determine the dependency of the induced motion on the impact characteristics, pre-deformation and the employed constitutive models. When using the neo-Hookean constitutive model, we find it cannot capture shear shocks and tensile-induced shocks, in contrast with experimental results on soft materials. Conversely, we find that the Gent model predicts that compressive impact may not be sufficient to induce a quasi-pressure shock—yet it may induce a quasi-shear shock, where tensile impact can trigger quasi-pressure shock—and may simultaneously trigger a quasi-shear shock. These features are in agreement with experimental data. Further, we show that the tensile impact must be greater than a calculated threshold value to induce shock, and demonstrate that this threshold is lowered by application of pre-shear.

This article presents the experimental work performed as part of the use of the non linear plasma-wave interactions in planar circuits to design microwave power limiters. A first non-resonant plasma-based microwave power limiter composed of a pre-ionized plasma discharge integrated into a 50 Ohms microstrip line has been experimentally characterized with steady-state and transient measurements. These results have been then confronted at each step to a resonant one using the same plasma discharge. This study highlights the effect of the resonance on the non linear plasma-wave interactions and its consequences on the behaviour and the performances of a plasma-based microwave power limiter.

In this paper, we construct a family of wave solutions to the doubly dispersive equation, such as topological, non-topological, singular, compound topological-non-topological bell-type and compound singular, soliton-like, singular periodic wave and exponential function solutions. These analytical solutions are obtained by using the extended sinh-Gordon equation expansion method and the modified $\exp(-\varphi(\zeta))$ -expansion function method. The doubly dispersive equation is an important nonlinear physical model describing the nonlinear wave propagation in the elastic inhomogeneous Murnaghan’s rod. Under a suitable choice of parameters, the 2D, 3D and contour graphics to the reported results are also plotted.

Different from elastic waves in linear periodic structures, those in phononic crystals (PCs) with nonlinear properties can exhibit more interesting phenomena. Linear dispersion relations cannot accurately predict band-gap variations under finite-amplitude wave motions; creating nonlinear PCs remains challenging and few examples have been studied. Recent studies in the literature mainly focus on discrete chain-like systems; most studies only consider weakly nonlinear regimes and cannot accurately obtain some relations between wave propagation characteristics and general nonlinearities. This paper presents propagation characteristics of longitudinal elastic waves in a thin rod and coupled longitudinal and transverse waves in an Euler–Bernoulli beam using their exact Green–Lagrange strain relations. We derive band structure relations for a periodic rod and beam and predict their nonlinear wave propagation characteristics using the B-spline wavelet on the interval (BSWI) finite element method. Influences of nonlinearities on wave propagation characteristics are discussed. Numerical examples show that the proposed method is more effective for nonlinear static and band structure problems than the traditional finite element method and illustrate that nonlinearities can cause band-gap width and location changes, which is similar to results reported in the literature for discrete systems. The proposed methodology is not restricted to weakly nonlinear systems and can be used to accurately predict wave propagation characteristics of nonlinear structures. This study can provide good support for engineering applications, such as sound and vibration control using tunable band gaps of nonlinear PCs.

A landmark of statistical mechanics, spin-glass theory describes critical phenomena in disordered systems that range from condensed matter to biophysics and social dynamics. The most fascinating concept is the breaking of replica symmetry: identical copies of the randomly interacting system that manifest completely different dynamics. Replica symmetry breaking has been predicted in nonlinear wave propagation, including Bose-Einstein condensates and optics, but never observed. Here, we report the experimental evidence of replica symmetry breaking in optical wave propagation The findings demonstrate that nonlinear propagation can manifest features typical of spin-glasses and provide a novel platform for testing so-far unexplored fundamental physical theories for complex systems.

The problem of the propagation of nonlinear waves in complex solids, namely bodies with different internal microstructures, is analyzed. In the first part, we make use of a general model of microstructured solids as introduced by Engelbrechet and Pastrone ( Acc Sc Torino Mem Sc Fis 2011; 35: 23–36) and study two particular relevant models: one-dimensional solid with hierarchical microstructure and with concurrent microstructures. As expected, the hierarchical microstructure leads, with a particular but meaningful choice of the strain energy function, to a sixth-order partial differential equation (PDE) with a characteristic hierarchical structure. Hence, the case of two concurrent microstructures, as introduced by Berezovski, Engelbrecht and Berezovski ( Acta Mech 2011; 220(1–4): 349–363), is studied and again for suitable explicit forms of the energy function we can obtain a fourth-order PDE and actually prove the possibility of propagation of solitary and cnoidal waves.

Gerald Beresford Whitham was one of the leading applied mathematicians of the twentieth century. His original, deep and insightful research into nonlinear wave propagation formed the foundation of and mathematical techniques for much of the current research in this area. Indeed, many of these ideas and techniques have spread beyond wave propagation research into other areas, such as reaction–diffusion, and has influenced research in pure mathematics. His textbook Linear and nonlinear waves , published in 1974, is still the standard reference for the mathematics of wave motion. Whitham was also instrumental in building from scratch the Department of Applied Mathematics at the California Institute of Technology and, through choosing key people in new, promising research areas, in making it into one of the leading centres of applied mathematics in the world, with an influence far beyond its small size. During his academic career, Whitham received major awards and prizes for his research. He was elected a Fellow of the Royal Society in 1965 and a Fellow of the American Academy of Arts and Sciences in 1959, and was awarded the Norbert Wiener Prize for Applied Mathematics in 1980.

This study formulates a frequency-domain computational scheme for simulating nonlinear wave propagation in a homogeneous medium governed by the Westervelt equation. The need for such numeri...

Spatially periodic and stationary localized solutions arising from the dynamics of chains of linearly coupled mechanical oscillators characterized by on site cubic nonlinearity are addressed aiming to explore their relationship with the underlying nonlinear wave propagation regions. Softening and hardening nonlinearities are considered, and regions of occurrence of discrete breathers and multibreathers associated with homoclinic or heteroclinic connections are identified.

The nonlinear (2 + 1)-dimensional Zakharov–Kuznetsov (ZK) equations deal with the nonlinear behavior of waves in collision-less plasma, which contains non-isothermal cold ions and electrons. Two-dimensional dust acoustic solitary waves (DASWs) in magnetized plasma, which consist of trapped electrons and ions are leading to (2 + 1)-dim (ZK) equation by using the perturbation technique. We found the solitary wave solutions of (2 + 1)-dimensional (ZK)-equation, generalized (ZK)-equation and generalized form of modified (ZK)-equation by implementing the modified mathematical method. As a result, we obtained the bright–dark solitons, traveling wave and solitary wave solutions. The physical structure of obtained solutions is represented in 2D and 3D, graphically with the help of Mathematica.

Simulating the propagation of nonlinear ultrasound waves is computationally difficult because of the dense grids needed to capture high-frequency harmonics. Here, a mapped k-space pseudospectral method is presented which allows the use of nonuniform grid spacings. This enables grid points to be clustered around steep regions of the wave field. Compared with using a uniform grid, this significantly reduces the total number of grid points needed for accurate simulations. Two methods for selecting a suitable nonuniform grid mapping are discussed.

Nonlinear wave propagation plays a crucial role in the functioning of many physical and biophysical systems. In the propagation regime, disturbances due to the presence of local external perturbations, such as localised defects or boundary interphase walls have gained great attention. In this article, the complex phenomena that occur when sine-Gordon line solitons collide with localised inhomogeneities are investigated. By a one-dimensional theory, it is shown that internal modes of two-dimensional sine-Gordon solitons can be activated depending on the topological properties of the inhomogeneities. Shape mode instabilities cause the formation of bubble-like and drop-like structures for both stationary and travelling line solitons. It is shown that such structures are formed and stabilised by arrays of localised inhomogeneities distributed in space. Implications of the observed phenomena in physical and biological systems are discussed., Comment: 7 pages, 2 figures. Conference Proceedings

An analytical and simulation based method has been used to exactly solve the nonlinear wave propagation in bulk media exhibiting frequency dependent dielectric susceptibility and magnetic permeability. The method has been further extended to investigate the intensity distribution in a nonlinear meta-material with negative refractive index where both e and μ are dispersive and negative in nature.

[EN] The test for determining the resonance frequencies has traditionally been used to investigate the mechanical integrity of concrete cores, to assess the conformity of concrete constituents in different accelerated durability tests, and to ascertain constitutive properties such as the elastic modulus and the damping factor. This nondestructive technique has been quite appealed for evaluation of mechanical properties in all kinds of durability tests. The damage evolution is commonly assessed from the reduction of dynamic modulus which is produced as a result of any cracking process. However, the mechanical behavior of concrete is intrinsically nonlinear and hysteretic. As a result of a hysteretic stress-strain behavior, the elastic modulus is a function of the strain. In dynamic tests, the nonlinearity of the material is manifested by a decrease of the resonance frequencies, which is inversely proportional to the excitation amplitude. This phenomenon is commonly referred as fast dynamic effect. Once the dynamic excitation ceases, the material undergoes a relaxation process whereby the elastic modulus is restored to that at rest. This phenomenon is termed as slow dynamics. These phenomena (fast and slow dynamics) find their origin in the internal friction of the material. Therefore, in cement-based materials, the presence of microcracks and interfaces between its constituents plays an important role in the material nonlinearity. In the context of the assessment of concrete durability, the damage evolution is based on the increase of hysteresis, as a result of any cracking process. In this thesis three different nondestructive techniques are investigated, which use impacts for exciting the resonant frequencies. The first technique consists in determining the resonance frequencies over a range of impact forces. The technique is termed Nonlinear Impact Resonant Acoustic Spectroscopy (NIRAS). It consists in ascertaining the downward resonant frequency shift that the material undergoes upon increasing excitation amplitude. The second technique consists in investigating the nonlinear behavior by analyzing the signal corresponding to a single impact. This is, to determine the instantaneous frequency, amplitude and attenuation variations corresponding to a single impact event. This technique is termed as Nonlinear Resonant Acoustic Single Impact Spectroscopy (NSIRAS). Two techniques are proposed to extract the nonlinear behavior by analyzing the instantaneous frequency variations and attenuation over the signal ring down. The first technique consists in discretizing the frequency variation with time through a Short-Time Fourier Transform (STFT) based analysis. The second technique consists of a least-squares fit of the vibration signals to a model that considers the frequency and attenuation variations over time. The third technique used in this thesis can be used for on-site evaluation of structures. The technique is based on the Dynamic Acousto- Elastic Test (DAET). The variations of elastic modulus as derived through NIRAS and NSIRAS techniques provide an average behavior and do not allow derivation of the elastic modulus variations over one vibration cycle. Currently, DAET technique is the only one capable to investigate the entire range of nonlinear phenomena in the material. Moreover, unlike other DAET approaches, this study uses a continuous wave source as probe. The use of a continuous wave allows investigation of the relative variations of the elastic modulus, as produced by an impact. Moreover, the experimental configuration allows one-sided inspection., [ES] El ensayo de determinación de las frecuencias de resonancia ha sido tradicionalmente empleado para determinar la integridad mecánica de testigos de hormigón, en la evaluación de la conformidad de mezclas de hormigón en diversos ensayos de durabilidad, y en la terminación de propiedades constitutivas como son el módulo elástico y el factor de amortiguamiento. Esta técnica no destructiva ha sido ampliamente apelada para la evaluación de las propiedades mecánicas en todo tipo de ensayos de durabilidad. La evolución del daño es comúnmente evaluada a partir de la reducción del módulo dinámico, producido como resultado de cualquier proceso de fisuración. Sin embargo, el comportamiento mecánico del hormigón es intrínsecamente no lineal y presenta histéresis. Como resultado de un comportamiento tensión-deformación con histéresis, el módulo elástico depende de la deformación. En ensayos dinámicos, la no linealidad del material se manifiesta por una disminución de las frecuencias de resonancia, la cual es inversamente proporcional a la amplitud de excitación. Este fenómeno es normalmente denominado como dinámica rápida. Una vez la excitación cesa, el material experimenta un proceso de relajación por el cual, el módulo elástico es restaurado a aquel en situación de reposo. Este fenómeno es denominado como dinámica lenta. Estos fenómenos ¿dinámicas rápida y lenta¿ encuentran su origen en la fricción interna del material. Por tanto, en materiales basados en cemento, la presencia de microfisuras y las interfaces entre sus constituyentes juegan un rol importante en la no linealidad mecánica del material. En el contexto de evaluación de la durabilidad del hormigón, la evolución del daño está basada en el incremento de histéresis, como resultado de cualquier proceso de fisuración. En esta tesis se investigan tres técnicas diferentes las cuales utilizan el impacto como medio de excitación de las frecuencias de resonancia. La primera técnica consiste en determinar las frecuencias de resonancia a diferentes energías de impacto. La técnica es denominada en inglés: Nonlinear Impact Resonant Acoustic Spectroscopy (NIRAS). Ésta consiste en relacionar el detrimento que el material experimenta en sus frecuencias de resonancia, con el aumento de la amplitud de la excitación. La segunda técnica consiste en investigar el comportamiento no lineal mediante el análisis de la señal correspondiente a un solo impacto. Ésta consiste en determinar las propiedades instantáneas de frecuencia, atenuación y amplitud. Esta técnica se denomina, en inglés, Nonlinear Single Impact Resonant Acoustic Spectroscopy (NSIRAS). Se proponen dos técnicas de extracción del comportamiento no lineal mediante el análisis de las variaciones instantáneas de frecuencia y atenuación. La primera técnica consiste en la discretización de la variación de la frecuencia con el tiempo, mediante un análisis basado en Short-Time Fourier Transform (STFT). La segunda técnica consiste en un ajuste por mínimos cuadrados de las señales de vibración a un modelo que considera las variaciones de frecuencia y atenuación con el tiempo. La tercera técnica empleada en esta tesis puede ser empleada para la evaluación de estructuras in situ. La técnica se trata de un ensayo acusto-elástico en régimen dinámico. En inglés Dynamic Acousto-Elastic Test (DAET). Las variaciones del módulo elástico obtenidas mediante los métodos NIRAS y NSIRAS proporcionan un comportamiento promedio y no permiten derivar las variaciones del módulo elástico en un solo ciclo de vibración. Actualmente, la técnica DAET es la única que permite investigar todo el rango de fenómenos no lineales en el material. Por otra parte, a diferencia de otras técnicas DAET, en este estudio se emplea como contraste una onda continua. El uso de una onda continua permite investigar las variaciones relativas del módulo elástico, para una señal transito, [CAT] L'assaig de determinació de les freqüències de ressonància ha sigut tradicionalment empleat per a determinar la integritat mecànica de testimonis de formigó, en l'avaluació de la conformitat de mescles de formigó en diversos assajos de durabilitat, i en la terminació de propietats constitutives com són el mòdul elàstic i el factor d'amortiment. Esta tècnica no destructiva ha sigut àmpliament apel·lada per a l'avaluació de les propietats mecàniques en tot tipus d'assajos de durabilitat. L'evolució del dany és comunament avaluada a partir de la reducció del mòdul dinàmic, produït com resultat de qualsevol procés de fisuración. No obstant això, el comportament mecànic del formigó és intrínsecament no lineal i presenta histèresi. Com resultat d'un comportament tensió-deformació amb histèresi, el mòdul elàstic depén de la deformació. En assajos dinàmics, la no linealitat del material es manifesta per una disminució de les freqüències de ressonància, la qual és inversament proporcional a l'amplitud d'excitació. Este fenomen és normalment denominat com a dinàmica ràpida. Una vegada l'excitació cessa, el material experimenta un procés de relaxació pel qual, el mòdul elàstic és restaurat a aquell en situació de repòs. Este fenomen és denominat com a dinàmica lenta. Estos fenòmens --dinámicas ràpida i lenta troben el seu origen en la fricció interna del material. Per tant, en materials basats en ciment, la presència de microfissures i les interfícies entre els seus constituents juguen un rol important en la no linealitat mecànica del material. En el context d'avaluació de la durabilitat del formigó, l'evolució del dany està basada en l'increment d'histèresi, com resultat de qualsevol procés de fisuración. En esta tesi s'investiguen tres tècniques diferents les quals utilitzen l'impacte com a mitjà d'excitació de les freqüències de ressonància. La primera tècnica consistix a determinar les freqüències de ressonància a diferents energies d'impacte. La tècnica és denominada en anglés: Nonlinear Impact Resonant Acoustic Spectroscopy (NIRAS). Esta consistix a relacionar el detriment que el material experimenta en les seues freqüències de ressonància, amb l'augment de l'amplitud de l'excitació. La segona tècnica consistix a investigar el comportament no lineal per mitjà de l'anàlisi del senyal corresponent a un sol impacte. Esta consistix a determinar les propietats instantànies de freqüència, atenuació i amplitud. Esta tècnica es denomina, en anglés, Nonlinear Single Impact Resonant Acoustic Spectroscopy (NSIRAS). Es proposen dos tècniques d'extracció del comportament no lineal per mitjà de l'anàlisi de les variacions instantànies de freqüència i atenuació. La primera tècnica consistix en la discretización de la variació de la freqüència amb el temps, per mitjà d'una anàlisi basat en Short-Time Fourier Transform (STFT). La segona tècnica consistix en un ajust per mínims quadrats dels senyals de vibració a un model que considera les variacions de freqüència i atenuació amb el temps. La tercera tècnica empleada en esta tesi pot ser empleada per a l'avaluació d'estructures in situ. La tècnica es tracta d'un assaig acusto-elástico en règim dinàmic. En anglés Dynamic Acousto-Elastic Test (DAET). Les variacions del mòdul elàstic obtingudes per mitjà dels mètodes NIRAS i NSIRAS proporcionen un comportament mitjà i no permeten derivar les variacions del mòdul elàstic en un sol cicle de vibració. Actualment, la tècnica DAET és l'única que permet investigar tot el rang de fenòmens no lineals en el material. D'altra banda, a diferència d'altres tècniques DAET, en este estudi s'empra com contrast una ona contínua. L'ús d'una ona contínua permet investigar les variacions relatives del mòdul elàstic, per a un senyal transitori. A més, permet la inspecció d'elements per mitjà de l'accés per una sola cara., Eiras Fernández, JN. (2016). Studies on nonlinear mechanical wave behavior to characterize cement based materials and its durability [Tesis doctoral no publicada]. Universitat Politècnica de València. doi:10.4995/Thesis/10251/71439., TESIS

We find and analyze a generalized analytical solution for nonlinear wave propagation in waveguide couplers with opposite signs of the linear refractive index, nonzero phase mismatch between the channels, and arbitrary nonlinear coefficients.

An analytical solution is presented for nonlinear surface wave propagation. A variational iteration method (VIM) was employed and time-dependent profiles of the surface elevation level and velocity obtained analytically for different initial conditions. It is shown that the VIM used here is a flexible and accurate approach and that it can rapidly converge to the same results obtained by the Adomian decomposition method.

Research progress on the regular gaseous detonation initiation and propagation is reviewed,and some fundamental problems are studied based on the recent work on the detonation physics in state key lab of high temperature gas dynamics.These problems constituting the universal framework consist of six issues:one mechanism that is the interaction of nonlinear wave propagation and chemical-reaction,INWPCR;two basic procedures,which are the hot pot ignition and the chemical reaction zone acceleration;three key physical states,which are the equilibrium propagation state,the critical initiation state and the stable cell size.These six issues are investigated through six test cases to ascertain their physical mechanisms,dynamic characteristics,and the existence.Then,the universal framework is applied to explain the recent classic detonation theories,multi-dimensional numerical results,and experimental observations on cellular detonation.Reasonable and universal conclusions can be reached on the base of the universal framework,which provides the physical insight for understanding regular gaseous detonation initiation and propagation.

The applicability of existing nonlinear (triad) spectral models for steep slopes (0.1–0.2) characteristic of reef environments was investigated, using both deterministic (phase-resolving) and stochastic (phased-averaged) formulations. Model performance was tested using laboratory observations of unidirectional wave transformation over steep and smooth bathymetry profiles. The models, developed for mild slopes, were implemented with minimal modifications (the inclusion of breaking parametrizations and linear steep-slope corrections) required by laboratory data. The deterministic model produced typically more accurate predictions than the stochastic one, but the phase averaged formulation proved fast enough to allow for an inverse modeling search for the optimal breaking parametrization. The effects of the additional assumptions of the stochastic approach resulted in a slower than observed evolution of the infragravity band. Despite the challenge posed by the fast wave evolution and energetic breaking characteristic to the steep reef slopes, both formulations performed overall well, and should be considered as good provisional candidates for use in numerical investigation of wave–current interaction processes on steep reefs.

The DORT method is a selective detection and focusing technique originally developed to detect defects and damages which induce linear changes of the elastic moduli. It is based on the time reversal (TR) where a signal collected from an array of transducers is time reversed and then back-propagated into the medium to obtain focusing on selected targets. TR is based on the principle of spatial reciprocity. Attenuation, dispersion, multiple scattering, mode conversion, etc. do not break spatial reciprocity. The presence of defects or damage, may cause materials to show nonlinear elastic wave propagation behavior that will break spacial reciprocity. Therefore the DORT method will not allow focusing on nonlinear elastic scatterers. This paper presents a new method for the detection and identification of multiple linear and nonlinear scatterers by combining nonlinear elastic wave spectroscopy, time reversal and DORT method. In the presence of nonlinear hysteretic elastic scatterers, forcing the solid with a harmonic excitation, the time reversal operator can be obtained not only at the fundamental frequency of excitation, but also at the odd harmonics. At the fundamental harmonic, either inhomogeneities and linear damages can be individually selected but only at odd harmonics nonlinear hysteretic elastic damages exist. A procedure was developed where by decomposing the operator at the odd harmonics, it was possible to focus on nonlinear scatterers and to differentiate them from the linear inhomogeneities. A complete mathematical nonlinear DORT formulation for 1 and 2D structures is presented. To model the presence of nonlinear elastic hysteretic scatterers a Preisach–Mayergoyz (PM) material constitutive model was used. Results relative to 1 and 2 dimensional structures are reported showing the capability of the method to focus and discern selectively linear and nonlinear scatterers. Furthermore, an analysis was conducted to study the influence of the number of sources and their location on the imaging process showing that using a higher numbers of sensors does not automatically bring to a minor uncoupled behaviour between the nonlinear targets.

The computer simulation of the process of nonlinear wave propagation in a structured geophysical medium with elastic-plastic interaction between the structure elements is conducted. It is found that during the wave propagation in a chain of the discrete elements its amplitude decays to the threshold value, when the interaction between the elements becomes elastic, and then the wave is transformed into a soliton-like wave with the threshold amplitude. The deformation diagrams of the massifs at different plastic threshold values are plotted. It is demonstrated that a decrease of the plastic threshold results in an increase of the dissipative properties.

The (1+1)-dimensional F-expansion technique and the homogeneous nonlinear balance principle have been generalized and applied for solving exact solutions to a general (3+1)-dimensional nonlinear Schrodinger equation (NLSE) with varying coefficients and a harmonica potential. We found that there exist two kinds of soliton solutions. The evolution features of exact solutions have been numerically studied. The (3+1)D soliton solutions may help us to understand the nonlinear wave propagation in the nonlinear media such as classical optical waves and the matter waves of the Bose–Einstein condensates.

Recent results on parametric spatial solitary waves arising from multiple resonances in purely nonlinear 2-D lattices are presented. Theory and experiments highlight new possibilities for light self-confinement and steering via engineered planar nonlinear structures in periodically poled materials.

The evolution of concrete behavior in the proximity of a joint under the effect of varying external pressures is studied by means of a novel nonlinear ultrasonic technique denoted as Scaling Subtraction Method. The results obtained show that the proposed method is effective in describing the occurrence of micro-structural changes near the joint and detect potential conditions for crack opening and damage initiation.