The propagation of waves through nonlinear media is of interest here, namely as it pertains to two specific examples, a nonlinear dielectric and a hyperelastic solid. In both cases, we examine the propagation of two-dimensional, weakly nonlinear, quasi-planar waves. It is found that such systems will have a nonlinearity that is intrinsically cubic, and therefore, a classical Zabolotskaya-Khokhlov equation cannot give an accurate description of the wave evolution. To determine the general evolution equation in such systems, a multi-timing technique developed by Kluwick and Cox (1998) and Cramer and Webb (1998) will be employed. The resultant evolution equations are seen to involve only one new nonlinearity coefficient rather than the three coefficients found in other studies of cubically nonlinear systems. After determining the general evolution equation, inclusion of relaxation, dispersion and dissipation effects can be easily incorporated. / Master of Science

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This thesis has two parts: In the first part, we study the dispersion characteristics of structural-acoustic waveguides by obtaining closed-form solutions for the coupled wave numbers. Two representative systems are considered for the above study: an infinite two-dimensional rectangular waveguide and an infinite fluid- filled orthotropic circular cylindrical shell. In the second part, these asymptotic expressions are used to study the nonlinear wave propagation in the same two systems. The first part involves obtaining asymptotic expansions for the fluid-structure coupled wave numbers in both the systems. Certain expansions are already available in the literature. Hence, the gaps in the literature are filled. Thus, for cylindrical shells even in vacuo wavenumbers are obtained as part of the objective. Here, singular and regular perturbation methods are used by taking the thickness parameter as the asymptotic parameter. Valid wavenumber expressions are obtained at all the frequencies. A transition in the behavior of the flexural wavenumbers occurs in the neighborhood of the ring frequency. This frequency of transition is identified for the orthotropic shells also. The closed-form expressions for the orthotropic shells are obtained in the limit of slight orthotropy for the circumferential orders n > 0 at all the frequency ranges. Following this, we derive the coupled wavenumber expressions for the two systems for an arbitrary fluid loading. Here, the two-dimensional rectangular waveguide is considered first. This rectangular waveguide has a one-dimensional plate and a rigid surface as its lateral boundaries. The effects due to the structural boundary are studied by analyzing the phase change due to the structure on an incident plane wave. The complications due to the cross-sectional modes are eliminated by ignoring the presence of the other rigid boundary. Dispersion characteristics are predicted at various regions of the dispersion diagram based on the phase change. Moreover, the also identified. Next, the rigid boundary is considered and the coupled dispersion relation for the waveguide is solved for the wavenumber expressions. The coupled wavenumbers are obtained as the coupled rigid-duct, the coupled structural and the coupled pressure-release wavenumbers. Next, based on the above asymptotic analysis on a two-dimensional rectangular waveguide, the asymptotic expansions are obtained for the coupled wavenumbers in isotropic and orthotropic fluid- filled cylindrical shells. The asymptotic expansions of the wavenumbers are obtained without any restriction on the fluid loading. They are compared with the numerical solutions and a good match is obtained. In the second part or the nonlinear section of the thesis, the coupled wavenumber expressions are used to study the propagation of small but a finite amplitude acoustic potential in the above structural-acoustic waveguides. It must be mentioned here that for the rst time in the literature, for a structural-acoustic system having a contained fluid, both the structure and the acoustic fluid are nonlinear. Standard nonlinear equations are used. The focus is restricted to non-planar modes. The study of the cylindrical shell parallels that of the 2-D rectangular waveguide, except in that the former is more practical and complicated due to the curvature. Thus, with regard to both systems, a narrow-band wavepacket of the acoustic potential centered around a frequency is considered. The approximate solution of the acoustic velocity potential is found using the method of multiple scales (MMS) involving both space and time. The calculations are presented up to the third order of the small parameter. It is found that the amplitude modulation is governed by the Nonlinear Schr•odinger equation (NLSE). The nonlinear term in the NLSE is analyzed, since the sign of the nonlinear term in the NLSE plays a role in determining the stability of the amplitude modulation. This sign change is predicted using the coupled wavenumber expressions. Secondly, at specific frequencies, the primary pulse interacts with its higher harmonics, as do two or more primary pulses with their resultant higher harmonic. This happens when the phase speeds of the waves match. The frequencies of such interactions are identified, again using the coupled wavenumber expressions. The novelty of this work lies firstly in considering nonlinear acoustic wave prop-agation in nonlinear structural waveguides. Secondly, in deriving the asymptotic expansions for the coupled wavenumbers for both the two-dimensional rectangular waveguide and the fluid- filled circular cylindrical shell. Then in using the same to study the behavior of the nonlinear term in NLSE. And lastly in identifying the frequencies of nonlinear interactions in the respective waveguides.

A periodic structure consists of spatially repeating unit cells. From man-made multi-span bridges to naturally occurring atomic lattices, periodic structures are ubiquitous. The periodicity can be exploited to generate frequency bands within which elastic wave propagation is impeded. A limitation to the linear periodic structure is that the filtering properties depend only on the structural design and periodicity which implies that the dispersion characteristics are fixed unless the overall structure or the periodicity is altered. The current research focuses on wave propagation in nonlinear periodic structures to explore tunability in filtering properties such as bandgaps, cut-off frequencies and response directionality. The first part of the research documents amplitude-dependent dispersion properties of weakly nonlinear periodic media through a general perturbation approach. The perturbation approach allows closed-form estimation of the effects of weak nonlinearities on wave propagation. Variation in bandstructure and bandgaps lead to tunable filtering and directional behavior. The latter is due to anisotropy in nonlinear interaction that generates low response regions, or "dead zones," within the structure.The general perturbation approach developed has also been applied to evaluate dispersion in a complex nonlinear periodic structure which is discretized using Finite Elements. The second part of the research focuses on wave dispersion in strongly nonlinear periodic structures which includes pre-compressed granular media as an example. Plane wave dispersion is studied through the harmonic balance method and it is shown that the cut-off frequencies and bandgaps vary significantly with wave amplitude. Acoustic wave beaming phenomenon is also observed in pre-compressed two-dimensional hexagonally packed granular media. Numerical simulations of wave propagation in finite lattices also demonstrated amplitude-dependent bandstructures and directional behavior so far observed.

This research investigates wave propagation in an elastic half-space with a quadratic nonlinearity in its stress-strain relationship. Different boundary conditions on the surface are considered that result in both one- and two-dimensional wave propagation problems. The goal of the research is to examine the generation of second-order frequency effects and static effects which may be used to determine the nonlinearity present in the material. This is accomplished by extracting the amplitudes of those effects in the frequency domain and analyzing their dependency on the third-order elastic constants (TOEC). For the one-dimensional problems, both analytical approximate solutions as well as numerical simulations are presented. For the two-dimensional problems, numerical solutions are presented whose dependency on the material's nonlinearity is compared to the one-dimensional problems. The numerical solutions are obtained by first formulating the problem as a hyperbolic system of conservation laws, which is then solved numerically using a semi-discrete central scheme. The numerical method is implemented using the package CentPack. In the one-dimensional cases, it is shown that the analytical and numerical solutions are in good agreement with each other, as well as how different boundary conditions may be used to measure the TOEC. In the two-dimensional cases, it is shown that there exist comparable dependencies of the second-order frequency effects and static effects on the TOEC. Finally, it is analytically and numerically investigated how multiple reflections in a plate can be used to simplify measurements of the material nonlinearity in an experiment.

The analysis of wave propagation in hyperelastic waveguides has significant applications in various branches of engineering like Non-Destructive Testing and Evaluation, impact analysis, material characterization and damage detection. Linear elastic models are typically used for wave analysis since they are sufficient for many applications. However, certain solids exhibit inherent nonlinear material properties that cannot be adequately described with linear models. In the presence of large deformations, geometric nonlinearity also needs to be incorporated in the analysis. These two forms of nonlinearity can have significant consequences on the propagation of stress waves in solids. A detailed analysis of nonlinear wave propagation in solids is thus necessary for a proper understanding of these phenomena. The current research focuses on the development of novel algorithms for nonlinear finite element analysis of stress wave propagation in hyperelastic waveguides. A full three-dimensional(3D) finite element analysis of stress wave propagation in waveguides is both computationally difficult and expensive, especially in the presence of nonlinearities. By definition, waveguides are solids with special geometric features that channel the propagation of stress waves along certain preferred directions. This suggests the use of kinematic waveguide models that take advantage of the special geometric features of the waveguide. The primary advantage of using waveguide models is the reduction of the problem dimension and hence the associated computational cost. Elementary waveguide models like the Euler-Bernoulli beam model, Kirchoff plate model etc., which are developed primarily within the context of linear elasticity, need to be modified appropriately in the presence of material/geometric nonlinearities and/or loads with high frequency content. This modification, besides being non-trivial, may be inadequate for studying nonlinear wave propagation and higher order waveguide models need to be developed. However, higher order models are difficult to formulate and typically have complex governing equations for the kinematic modes. This reflects in the relatively scarce research on the development of higher order waveguide models for studying nonlinear wave propagation. The formulation is difficult primarily because of the complexity of both the governing equations and their linearization, which is required as part of a nonlinear finite element analysis. One of the primary contributions of this thesis is the development and implementation of a general, flexible and efficient framework for automating the finite element analysis of higher order kinematic models for nonlinear waveguides. A hierarchic set of higher order waveguide models that are compatible with this formulation are proposed for this purpose. This hierarchic series of waveguide models are similar in form to the kinematic assumptions associated with standard waveguide models, but are different in the sense that no conditions related to the stress distribution specific to a waveguide are imposed since that is automatically handled by the proposed algorithm. The automation of the finite element analysis is accomplished with a dexterous combination of a nodal degrees-of-freedom based assembly algorithm, automatic differentiation and a novel procedure for numerically computing the finite element matrices directly from a given waveguide model. The algorithm, however, is quite general and is also developed for studying nonlinear plane stress configurations and inhomogeneous structures that require a coupling of continuum and waveguide elements. Significant features of the algorithm are the automatic numerical derivation of the finite element matrices for both linear and nonlinear problems, especially in the context of nonlinear plane stress and higher order waveguide models, without requiring an explicit derivation of their algebraic forms, automatic assembly of finite element matrices and the automatic handling of natural boundary conditions. Full geometric nonlinearity and the hyperelastic form of material nonlinearity are considered in this thesis. The procedures developed here are however quite general and can be extended for other types of material nonlinearities. Throughout this thesis, It is assumed that the solids under investigation are homogeneous and isotropic. The subject matter of the research is developed in four stages: First, a comparison of different finite element discretization schemes is carried out using a simple rod model to choose the most efficient computational scheme to study nonlinear wave propagation. As part of this, the frequency domain Fourier spectral finite element method is extended for a special class of weakly nonlinear problems. Based on this comparative study, the Legendre spectral element method is identified as the most efficient computational tool. The efficiency of the Legendre spectral element is also illustrated in the context of a nonlinear Timoshenko beam model. Since the spectral element method is a special case of the standard nonlinear finite element Method, differing primarily in the choice of the element basis functions and quadrature rules, the automation of the standard nonlinear finite element method is undertaken next. The automatic finite element formulation and assembly algorithm that constitutes the most significant contribution of this thesis is developed as an efficient numerical alternative to study the physics of wave propagation in nonlinear higher order structural models. The development of this algorithm and its extension to a general automatic framework for studying a large class of problems in nonlinear solid mechanics forms the second part of this research. Of special importance are the automatic handling of nonlinear plane stress configurations, hierarchic higher order waveguide models and the automatic coupling of continuum and higher order structural elements using specially designed transition elements that enable an efficient means to study waveguides with local inhomogeneities. In the third stage, the automatic algorithm is used to study wave propagation in hyperelastic waveguides using a few higher order 1D kinematic models. Two variants of a particular hyperelastic constitutive law – the6-constantMurnaghanmodel(for rock like solids) and the 9-constant Murnaghan model(for metallic solids) –are chosen for modeling the material nonlinearity in the analysis. Finally, the algorithm is extended to study energy-momentum conserving time integrators that are derived within a Hamiltonian framework, thus illustrating the extensibility of the algorithm for more complex finite element formulations. In short, the current research deals primarily with the identification and automation of finite element schemes that are most suited for studying wave propagation in hyper-elastic waveguides. Of special mention is the development of a novel unified computational framework that automates the finite element analysis of a large class of problems involving nonlinear plane stress/plane strain, higher order waveguide models and coupling of both continuum and waveguide elements. The thesis, which comprises of 10 chapters, provides a detailed account of various aspects of hyperelastic wave propagation, primarily for 1D waveguides.

[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 / [CA] 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]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/71439 / Premios Extraordinarios de tesis doctorales