The majority of commercially available ultrasonic transducers used to excite and measure wave propagation in structures can be coupled only to a free surface. While convenient, this method is likely to excite multiple structural modes, making data interpretation difficult. Furthermore, the many modes excited make predicting the structure’s response a computationally intensive task. Here the dynamic radial displacement induced by a transient radial point load is calculated at more than 230,000 points on the outer surface of a virgin steel pipe to simulate a typical experiment. The radial component of the displacement field is calculated by convolving the Green’s functions of the pipe with the transient load. These functions are calculated on personal computers (in a distributed arrangement) by employing modal summation. The mode shapes are obtained from a Semi-Analytical Finite Element formulation used in conjunction with a separation of variables. The results are presented in a four dimensional animation, providing easier interpretations and insight into how to best select observation points for the detection of defects. The accuracy of the calculated displacements is verified experimentally. Agreement is good when magnitude and phase corrections are incorporated from the frequency response curves of the transducers used. / February 2005

This thesis presents a theoretical study of thermoelastic guided waves in cylinders in the context of Lord-Shulman generalized theory of thermoelasticity. Two different methods were formulated to study dispersion relations in infinite cylinders. One of them is a Semi Analytical Finite Element (SAFE) method and the other is an analytical method. In the SAFE method, the dispersion equation has been formulated as a generalized eigenvalue problem by treating radial displacement and temperature with a one dimensional finite element model through the thickness of the cylinder. In the analytical method, displacement potentials are introduced to obtain the dispersion relations of guided wave modes. This method is applicable to isotropic cylinders and has been developed primarily to cross check the SAFE formulation. Frequency spectra obtained by both methods for an isotropic cylinder have shown excellent agreement with each other. Since the SAFE method can be used for an anisotropic composite cylinder, guided wave modes for anisotropic and composite cylinders are presented. Transient analysis of ultrasonic guided waves generated by concentrated heating of the outer surface of an infinite anisotropic cylinder has also been studied. The SAFE method is employed to model the response of a cylinder due to a pulsed laser focused on its surface. Green’s functions were constructed numerically by superposition of guided wave modes in frequency and wave number domains. Time histories of the propagating modes are then calculated by applying an inverse Fourier transformation in the time domain. Transient radial displacements of longitudinal and flexural modes of a silicon nitride cylinder are presented. Propagation of thermoelastic waves in finite length circular cylinders have also been investigated. The SAFE method is used to simulate the guided wave modes in the cylinder. Frequency spectra obtained by the SAFE formulation, for a finite length transversely isotropic cylinder, are validated by comparing the numerical results with relevant publications. Frequency spectra for axisymmetric and asymmetric modes in a silicon nitride finite cylinder with both ends insulated and restrained by frictionless rigid walls are presented. The plain strain problem of circumferential guided waves is also studied and the results are validated for an isothermal case.

The majority of commercially available ultrasonic transducers used to excite and measure wave propagation in structures can be coupled only to a free surface. While convenient, this method is likely to excite multiple structural modes, making data interpretation difficult. Furthermore, the many modes excited make predicting the structure’s response a computationally intensive task. Here the dynamic radial displacement induced by a transient radial point load is calculated at more than 230,000 points on the outer surface of a virgin steel pipe to simulate a typical experiment. The radial component of the displacement field is calculated by convolving the Green’s functions of the pipe with the transient load. These functions are calculated on personal computers (in a distributed arrangement) by employing modal summation. The mode shapes are obtained from a Semi-Analytical Finite Element formulation used in conjunction with a separation of variables. The results are presented in a four dimensional animation, providing easier interpretations and insight into how to best select observation points for the detection of defects. The accuracy of the calculated displacements is verified experimentally. Agreement is good when magnitude and phase corrections are incorporated from the frequency response curves of the transducers used.

This thesis presents a theoretical study of thermoelastic guided waves in cylinders in the context of Lord-Shulman generalized theory of thermoelasticity. Two different methods were formulated to study dispersion relations in infinite cylinders. One of them is a Semi Analytical Finite Element (SAFE) method and the other is an analytical method. In the SAFE method, the dispersion equation has been formulated as a generalized eigenvalue problem by treating radial displacement and temperature with a one dimensional finite element model through the thickness of the cylinder. In the analytical method, displacement potentials are introduced to obtain the dispersion relations of guided wave modes. This method is applicable to isotropic cylinders and has been developed primarily to cross check the SAFE formulation. Frequency spectra obtained by both methods for an isotropic cylinder have shown excellent agreement with each other. Since the SAFE method can be used for an anisotropic composite cylinder, guided wave modes for anisotropic and composite cylinders are presented. Transient analysis of ultrasonic guided waves generated by concentrated heating of the outer surface of an infinite anisotropic cylinder has also been studied. The SAFE method is employed to model the response of a cylinder due to a pulsed laser focused on its surface. Green’s functions were constructed numerically by superposition of guided wave modes in frequency and wave number domains. Time histories of the propagating modes are then calculated by applying an inverse Fourier transformation in the time domain. Transient radial displacements of longitudinal and flexural modes of a silicon nitride cylinder are presented. Propagation of thermoelastic waves in finite length circular cylinders have also been investigated. The SAFE method is used to simulate the guided wave modes in the cylinder. Frequency spectra obtained by the SAFE formulation, for a finite length transversely isotropic cylinder, are validated by comparing the numerical results with relevant publications. Frequency spectra for axisymmetric and asymmetric modes in a silicon nitride finite cylinder with both ends insulated and restrained by frictionless rigid walls are presented. The plain strain problem of circumferential guided waves is also studied and the results are validated for an isothermal case.

The majority of commercially available ultrasonic transducers used to excite and measure wave propagation in structures can be coupled only to a free surface. While convenient, this method is likely to excite multiple structural modes, making data interpretation difficult. Furthermore, the many modes excited make predicting the structure’s response a computationally intensive task. Here the dynamic radial displacement induced by a transient radial point load is calculated at more than 230,000 points on the outer surface of a virgin steel pipe to simulate a typical experiment. The radial component of the displacement field is calculated by convolving the Green’s functions of the pipe with the transient load. These functions are calculated on personal computers (in a distributed arrangement) by employing modal summation. The mode shapes are obtained from a Semi-Analytical Finite Element formulation used in conjunction with a separation of variables. The results are presented in a four dimensional animation, providing easier interpretations and insight into how to best select observation points for the detection of defects. The accuracy of the calculated displacements is verified experimentally. Agreement is good when magnitude and phase corrections are incorporated from the frequency response curves of the transducers used.

A survey of relevant literature on the topic of wave propagation and scattering in pipes is given first. This review is followed by a theoretical framework which is pertinent to wave propagation in homogeneous, isotropic, pipes. Emphasis is placed on approximate solutions stemming from a computer based, Semi-Analytical Finite Element (SAFE) formulation. A modal analysis of the dynamic response of homogeneous, isotropic pipes, when subjected to a transient ultrasonic excitation, demonstrates that dominant features, i.e., singularities in an unblemished pipe’s displacement Frequency Response Function (FRF) coincide with its cutoff frequencies. This behaviour is confirmed experimentally. A novel technique is developed to deduce such a pipe’s wall thickness and elastic properties from three cutoff frequencies. The resulting procedure is simulated numerically and verified experimentally. Agreement between the new ultrasonic procedure and traditional destructive tests is within experimental uncertainty. Then a hybrid-SAFE technique is used to simulate waves scattered by various open rectangular notches. The simulations show, for the first time, that singularities distinct from the unblemished pipe’s cutoff frequencies arise in a displacement FRF when an axisymmetric notch is introduced. They also suggest that the new singularities depend on the properties of the parent pipe and the finite element region but effects are local to a notch. It is demonstrated further that the difference between the frequency at which a singularity introduced by a notch occurs and the nearest corresponding unblemished pipe’s cutoff frequency is a function of the notch’s dimensions. By plotting contours of constant frequency differences, it is shown that it is usually possible to characterize the notch’s dimensions by using two modes. However, the frequency difference for a third mode may be also needed occasionally. The more general case of nonaxisymmetric notches is shown to be a straightforward extension of the axisymmetric case.

A survey of relevant literature on the topic of wave propagation and scattering in pipes is given first. This review is followed by a theoretical framework which is pertinent to wave propagation in homogeneous, isotropic, pipes. Emphasis is placed on approximate solutions stemming from a computer based, Semi-Analytical Finite Element (SAFE) formulation. A modal analysis of the dynamic response of homogeneous, isotropic pipes, when subjected to a transient ultrasonic excitation, demonstrates that dominant features, i.e., singularities in an unblemished pipe’s displacement Frequency Response Function (FRF) coincide with its cutoff frequencies. This behaviour is confirmed experimentally. A novel technique is developed to deduce such a pipe’s wall thickness and elastic properties from three cutoff frequencies. The resulting procedure is simulated numerically and verified experimentally. Agreement between the new ultrasonic procedure and traditional destructive tests is within experimental uncertainty. Then a hybrid-SAFE technique is used to simulate waves scattered by various open rectangular notches. The simulations show, for the first time, that singularities distinct from the unblemished pipe’s cutoff frequencies arise in a displacement FRF when an axisymmetric notch is introduced. They also suggest that the new singularities depend on the properties of the parent pipe and the finite element region but effects are local to a notch. It is demonstrated further that the difference between the frequency at which a singularity introduced by a notch occurs and the nearest corresponding unblemished pipe’s cutoff frequency is a function of the notch’s dimensions. By plotting contours of constant frequency differences, it is shown that it is usually possible to characterize the notch’s dimensions by using two modes. However, the frequency difference for a third mode may be also needed occasionally. The more general case of nonaxisymmetric notches is shown to be a straightforward extension of the axisymmetric case.

Dans ce travail de thèse, nous nous sommes intéressés à la modélisation et la simulation de la propagation d'ondes ultrasonores dans l'os cortical. Plus précisément, nous avons étudié et analysé la technique dite des ultrasons quantitatifs (Quantitative Ultrasound, QUS) pour l'évaluation de la qualité du tissu osseux. Il s'agit d'une technique émergente dont l'application aux tissus osseux suscite un intérêt particulier dans la communauté scientifique. Le tissu osseux étant un tissu vivant, il est sujet au vieillissement et à divers pathologies parmi lesquelles on peut citer ostéoporose, ostéomalacie, ostéoporomalacie, ou encore, la maladie dite de Paget. Pour accompagner les soins à prodiguer au tissu osseux, une surveillance de sa qualité s'avère indispensable. Dans ce contexte, les méthodes ultrasonores sont réputées être intéressantes, de par leurs caractères non-invasif, peu coûteux, portable et non-ionisant. Cependant, utiliser des ultrasons dans le cadre de la caractérisation du tissu osseux, suppose une compréhension profonde des différents phénomènes physiques mis en jeu lors de leur propagation. Dans cette optique, notre travail est développé dans la thématique de la modélisation dédiée à la propagation des ondes ultrasonores dans des guides d'ondes multidimensionnels, hétérogènes, anisotropes, et composés de matériaux dont l'hétérogénéité peut être qualifiée d'aléatoire. Une des originalités de cette thèse concerne l'étude des coefficients de réflexion et de transmission et des courbes de dispersion en présence d'incertitudes dues aux propriétés matérielles. Dans une première partie, nous étudions les phénomènes de réflexion/transmission via un modèle tri-couches bidimensionnels prenant en compte les tissus mous et l'hétérogénéité aléatoire du tissu osseux. Nous avons pu analyser l'impact de ces caractéristiques sur les coefficients de réflexion et de transmission. Un gradient de propriétés matérielles de l'os est introduit, et son impact sur les coefficients d'intérêt est examiné. L'aspect modal des ondes est exploré, en étudiant la dispersion des ondes de Lamb. Les résultats obtenus dans une configuration géométrique bidimensionnelle ont permis de discuter l'influence des divers paramètres, en terme de propriétés mécaniques et/ou géométriques, sur la propagation des ondes ultrasonores dans le tissu cortical. Dans une deuxième partie, le modèle est étendu pour une configuration géométrique cylindrique. La discussion est menée afin d'analyser l'influence de la géométrie tridimensionnelle de l'os sur les phénomènes de propagation / In this thesis, we are interested in the modeling and simulation of the propagation of ultrasonic waves in the cortical bone. Precisely, we have studied and analyzed the Quantitative Ultrasound (QUS) technique for the evaluation of the quality of bone tissue. It is an emerging technique those the application to bone tissue arouses particular interest in the scientific community. Since bone tissue is a living tissue, it is subject to aging and various pathologies, such osteoporosis, osteomalacia, osteoporomalacia, or the so-called Paget disease. To assist in therapeutic follow-up of the bone, monitoring of quality of bone tissue is essential. In this context, methods based on QUS technique are deemed to be interesting, due of their non-invasive, inexpensive, portable and non-ionizing characteristics. However, use the ultrasound in the context of characterization of bone tissue, requires a deep understanding of the different physical phenomena involved in their propagation. In this perspective, our work is developed in the modeling theme dedicated to the propagation of ultrasonic waves in multidimensional, heterogeneous, anisotropic waveguides, constituted of materials whose heterogeneity can be qualified as random. One of the originalities of this thesis concerns the study of the reflection and transmission coefficients and the dispersion curves in the presence of uncertainties in the material properties. In a first part, we study the reflection/transmission phenomena via a two-dimensional tri-layer model taking into account the soft tissues and the random heterogeneity of the bone tissue. We analyzed the impact of these characteristics on the reflection and transmission coefficients. A gradient of material properties is introduced, and its effect on the coefficients of interest is examined. The modal aspect of the waves is explored, by studying the dispersion of Lamb waves. The results obtained in a two-dimensional geometrical configuration made it possible to discuss the influence of the various parameters, in terms of mechanical and/or geometric properties, on the propagation of the ultrasonic waves in the cortical tissue. In a second part, the proposed model is extended for a cylindrical geometric configuration. The discussion is carried out in order to analyze the influence of the three-dimensional geometry of the bone on the phenomena of propagation

Dans un contexte de densification des villes et de leurs réseaux de transport, les gens sont de plus en plus exposés au bruit. Ainsi, le résultat de l'étude d'impact vibro-acoustique joue un rôle primordial dans l'expansion du réseau ferroviaire. L'une des principales sources est le bruit de roulement : La rugosité de la surface de la roue et du rail produit un déplacement imposé sur ces derniers. Ce déplacement entraine une réponse vibratoire des roues et de la voie ferrée et leurs rayonnements acoustiques. Cette thèse propose une amélioration de la modélisation vibro-acoustique de la voie ferrée.Pour la réponse vibratoire, le coté infini de la voie et sa déformation dans les 3 dimensions rendent les modèles analytiques et les éléments finis non-optimales dans la gamme de fréquence de l’audible. La méthode élément fini semi-périodique (SAFEM) est utilisée dans cette thèse pour modéliser une voie à support continue. Elle est ensuite couplée au théorème de Floquet pour modéliser une voie à support périodique. Cependant, cette technique génère des problèmes numériques qui ont imposé un algorithme adapté. La méthode d'Arnoldi du second ordre (SOAR) est utilisée avant de résoudre l'équation SAFEM permet de résoudre ces problèmes ainsi qu’apporter la stabilité requise. Des comparaisons avec d’autres modèles et des données expérimentales permettent de valider la méthode.Pour le rayonnement acoustique, la simulation de grand domaine en haute fréquence rendent inadapté l'utilisation de techniques conventionnelles (FEM, BEM, ...). La méthode proposée ici : la théorie variationnelle du rayon complexe est particulièrement bien adaptée à ce cas. Les principales caractéristiques de l'approche VTCR sont l'utilisation d'une formulation faible du problème acoustique, qui permet de considérer automatiquement les conditions limites entre sous-domaines. Ensuite, l'utilisation d'une répartition intégrale des ondes planes dans toutes les directions permet de simuler le champ acoustique. Les inconnues du problème sont leurs amplitudes. Cette méthode qui a déjà montré son efficacité pour les domaines fermés a été étendue au domaine ouvert et couplée à la réponse vibratoire. Des comparaisons avec des solutions analytiques et des simulations FEM à basse fréquence permettent de valider la méthode. / In a context of urban and transport network densification, people are increasingly exposed to noise. Consequently, the result of vibro-acoustic impact assessment has a pivotal role in rail network expansion. One of the main sources is the rolling noise: Roughness on the wheel and rail surface produce an imposed displacement one the both. This last, generates vibrational response of wheels and the railway track and their acoustic radiation. This PhD thesis presents some improvements of the vibro-acoustic railway track modelling.Concerning vibrational response, the infinite dimension in the longitudinal direction of the track and its deformation in the 3 dimensions, make the analytical models and finite elements non-optimal. The Semi-analytical finite element method (SAFEM), used in this thesis, is particularly well adapted in this case. Firstly, it is used to model railway track on a continuous support. Then, it is coupled with Floquet theorem to model tracks with a periodic support. However, this technique suffers from numerical problems that imposed an adapted algorithm. The second-order Arnoldi method (SOAR) is used to tackle them. This reduction allows to eliminate critical values improving the robustness of the method. Comparison with existing techniques and experimental results validate this model.Concerning acoustic radiation, big domains simulations at high frequency are almost unfeasible when using conventional techniques (FEM, BEM,…). The method used in this thesis, the Variational theory of complex ray (VTCR) is particularly well adapted to these cases. The principal features of VTCR approach are the use of a weak formulation of the acoustic problem, which allows to consider automatically boundary conditions between sub-domains. Then, the use of an integral repartition of plane waves in all the direction allow to simulate the acoustic field. The unknowns of the problem are their amplitudes. This method well assessed for closed domain, has been extended to open domain and coupled to vibrational response of the rail. Comparison with analytic solution and FEM simulation at low frequency allow to validate the method.Coupling these both methods allowed to simulate complex real life vibro-acoustic scenarios. Result of different railway tracks are presented and validated