One-dimensional theories of motion for beams

Using Gurtin's variational principle, a rational method for deducing the approximate one-dimensional theories of Medick from three-dimensional elasticity is presented.By using suitable unknowns, matrix equations are obtained; these exhibit a hyperbolic structure.

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Résumé Dans le cadre des schématisations de Medick, on présente une méthode permettant de déduire de la théorie tridimensionelle des théories approchées à une dimension de plus en plus fines.Après un choix convenable des inconnues, on obtient les équations matricielles du problème qui mettent en évidence une structure de système hyperbolique.

     

Constraints in linearly elastic materials

The general theory of linear constraints in linear elasticity theory is outlined. For problems that are ill-posed in constraint theory although well-posed in the absence of any constraint, a modified form of constraint theory is proposed.

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Resume Les grandes lignes de la théorie générale des contraints en élasticité linéaire sont exposées. Une forme modifiée de la théorie des contraints est proposée pour les problèmes mal posés en présence de contraints quoique bien posés en l'absence de contraints.

     

Evaluation of aero-optical distortion effects in PIV

Aero-optical distortion effects on the accuracy of particle image velocimetry (PIV) are investigated. When the illuminated particles are observed through a medium that is optically inhomogeneous due to flow compressibility, the resulting particle image pattern is subjected to deformation and blur. In relation to PIV two forms of error can be identified: position error and velocity error. In this paper a model is presented that describes these errors and particle image blur in relation to the refractive index field of the flow. In the case of 2D flows the model equations can be simplified and, furthermore, the background oriented schlieren technique (BOS) can be applied as a means to assess and correct for the optical error in PIV. The model describing the optical distortion is validated by both computer simulation and real experiments of 2D flows. Two flow features are considered: one with optical distortion normal to the velocity (shear layer) and one with optical distortion in the direction of the flow (expansion fan). Both simulation and experiments demonstrate that the major source for the velocity error is the second derivative of the refractive index in the direction of the velocity vector. The aero-optical distortion effect is less critical for shearing interfaces in comparison with compression/expansion fronts, the most critical case being represented by shock waves. Based on the results from the simulated experiments, it is concluded that for the 2D flow case the BOS method allows a measurement of the mean velocity error in PIV and can reduce it to a large extent.     

On the first boundary value problem for the classical theory of elasticity in a three-dimensional domain with a singular boundary

An existence and uniqueness theorem is proved for the first boundary value problem of classical elasticity, relating to a broad class of three-dimensional domains whose boundaries may have edges and vertexes.The qualitative properties, up to the boundary, of the solution are investigated.

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Résumé On démontre un théorème d'existence et unicité pour le premier problème de valeurs au contour de l'élasticité classique, concernant une large classe de domaines de l'éspace à trois dimensions, avec des frontières présentant des arètes et des sommets.On étudie les propriétés qualitatives, jusque'au contour, de la solution.

     

Application of particle image velocimetry to combusting flows: design considerations and uncertainty assessment

Methodological aspects concerning the application of the PIV technique to the study of turbulent flames are discussed in this paper. The physical features of the flow, which have implications for the experimental set-up, image processing and measurement accuracy are identified. Design considerations are developed focusing on several factors: spatial resolution, particle performance, seeding technique, image formation and recording, and image post-processing for the evaluation of the displacement. Relevant uncertainty concerns are related to the effect of the thermophoretic force, acting on a seeding particle while crossing the flame front, and to the non-homogeneity and time-dependence of the refractive index field. The uncertainty due to thermophoresis is assessed by numerically studying the motion of a particle crossing a reference temperature profile. The effect of the refractive index variation is evaluated by means of theoretical analysis of light propagation and image formation, supported by experimental tests designed for this special purpose. Received: 25 November 1999/Accepted: 31 March 2000     

A uniqueness theorem in the theory of Cosserat surface

Within the scope of the non-isothermal theory of an elastic Cosserat surface and for a system of linear equations characterizing the initial mixed boundary-value problems of thermoelastic shells, a uniqueness theorem is obtained without the use of definiteness assumption for the free energy. The theorem holds for nonhomogeneous and anisotropic shells undergoing small motions (and small temperature change) superposed on a large deformation.

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Résumé Dans le cadre de la théorie d'une surface élastique de Cosserat et pour un système d'équations linéaires décrivant les problèmes à conditions initiales et à conditions aux frontières mixtes de coques thermoélastiques, on obtient un théorème d'unicité sans faire appel à une hypothèse de définition pour l'énergie libre. Le théorème est valable pour des coques non-homogènes et anisotropes soumises à de petits mouvements (et petits changements de température) superposés à une grande déformation.

     

Wave propagation in elastic shells

The problem of wave propagation in shells within the framework of a simplified linear shell theory is treated using the method of Hadamard. Speeds of propagation, wave shape and decay, as well as coupling effects, are obtained for longitudinal, transverse and bending waves. The theory is applied to wave propagation on a spherical shell.

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Résumé On traite la problème de la propagation des ondes dans les voiles minces en utilisant la méthode de Hadamard. Les vitesses de la propagation, la forme de l'onde, et aussi les effets d'accouplement sont obtenus pour les onides longitudinales, transversales et fléchissantes. On applique cette théorie à la propagation des ondes dans une coque sphérique.

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This work was supported in part by funds from the National Research Council of Canada, under Grant Number A 3805.     

Flow Characterization Through a Network Cell Using Particle Image Velocimetry

Particle image velocimetry (PIV) with refractive index matching was developed to map pore-scale fluid flow through a clear, acrylic two-dimensional network flow cell. A microscope objective lens was incorporated in the PIV set up so that flow in micro-scale throats could be measured. The flow cell consists of 20 × 20, equal-size cylindrical pore bodies, 2.5mm in diameter and 1.0mm in height, connected on a diamond lattice by 2.5 mm long, square cross-section throats of widths that varied randomly among 0.2, 0.6, and 1.0 mm. Micro-PIV data was used to obtain the two-dimensional streamline pattern of fluid flow and the velocity field over the field of view (FOV) by periodically illuminating seed particles following the flow and cross correlating particle positions to determine displacements over time. Refractive index matching of the flow cell and test fluid minimizes extraneous scattering of light at solid--liquid interfaces improving image resolution. Experimentally determined velocity vectors for single-phase flow through three pore bodies and their adjoining throats as well as for the outlet of the flow cell were compared with numerical simulations of flow through the cell.     

Elasticity,piezoelectricity and crystal lattice dynamics

A review of some recent developments in the area between the dynamical theory of crystal lattices, in the harmonic approximation, and the classical, linear theories of elasticity and piezoelectricity.

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Résumé Un examen de quelques évolutions récentes dans le champ entre la théorie dynamique des réseaux cristallins (upproximation harmonique) et les théories classiques, linéaires, de l'élasticité et de la piézoélectricité.

     

PIV measurements in multiphase flow with nominally high concentration of the solid phase

Optical PIV measurements were aimed at studying fluid mechanical problems in a coal flotation process. Since the associated solid phase (coal/mineral) develops a high degree of light extinction, a solution had to be found for allowing a high concentration of the solid phase without losing the optical access for the PIV light sheet. The solution consisted in simulating the solid phase with glass spheres whose refractive index was matched by an organic liquid in the two-phase system.     

An experimental study of a turbulent vortex ring: a three-dimensional representation

This paper presents a reconstruction of the three-dimensional velocity field of a turbulent vortex ring by means of Taylor’s hypothesis. Stereoscopic PIV is used to acquire three velocity component information on a plane. The accuracy of the Taylor’s hypothesis for this particular flow pattern is first discussed, and the three-dimensional velocity and vorticity information are then presented. This study also introduces an azimuthally averaging method in order to give a mean structure in cylindrical coordinates from a single realization and from which turbulent stresses and production can be estimated. The azimuthally averaged quantities are then compared with the ensemble-averaged results from the previous planar (two-dimensional and stereoscopic) PIV experiments.     

On superposed small deformations on a large deformation of an elastic Cosserat surface

Within the scope of the theory of a Cosserat surface, this paper is concerned with small deformations superposed on a large deformation in elastic shells and plates together with some related aspects of the subject. Special attention is given to problems of stability and vibrations of initially stressed isotropic plates.

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Résumé Dans le cadre de la théorie d'une surface de Cosserat, le présent rapport étudic, pour des plaques et coques élastiques, de petites déformations superposées à une grande déformation, ainsi que certains aspects connexes du sujet. Une attention particulière est accordée aux problèmes de stabilité et de vibrations de plaques isotropiques initialement sous tension.

     

Unobstructed particle image velocimetry measurements within an axial turbo-pump using liquid and blades with matched refractive indices

Performing PIV measurements within complex turbomachinery with multiple blade rows is difficult due to the optical obstruction to the illuminating sheet and to the camera caused by the blades. This paper introduces a refractive index matched facility that overcomes this problem. The rotor and stator blades are made of transparent acrylic, and the working fluid has the same optical refractive index as the blades. A 64% by weight solution of sodium iodide in water is used for this purpose. This liquid has a kinematic viscosity of about 1.1᎒^-6 m²/s, which is almost the same as that of water enabling operation at high Reynolds numbers. Issues related to operating with this fluid such as chemical stability, variations in transmittance and solutions to these problems are discussed. This setup allows full optical access to the entire rotor and stator passages both to the laser sheet and the camera. The experiments are conducted at different streamwise locations covering the entire flow fields around the rotor, the stator, the gap between them, and the wakes behind. Vector maps of the instantaneous and phase-averaged flow fields as well as the distribution of turbulent kinetic energy are obtained. Measurements at different magnifications enable us to obtain an overview of the flow structure, as well as detailed velocity distributions in the boundary layers and in the wakes.     

Optical measurement uncertainties due to refractive index mismatch for flow in porous media

Application of optical techniques such as PIV, PTV, and LDA for velocity field estimation in porous media requires matching of refractive indices of the liquid phase to that of the solid matrix, including the channel walls. The methods most commonly employed to match the refractive indices have been to maximize the transmitted intensity through the bed or to rely on direct refractometer measurements of the indices of the two phases. Mismatch of refractive indices leads to error in estimation of particle position, ε _PD, due to refraction at solid–liquid interfaces. Analytical ray tracing applied to a model of solid beads placed randomly along the optical path is used to estimate ε _PD. The model, after validating against experimental results, is used to generate expression for ε _PD as a function of refractive index mismatch for a range of bead diameters, bed widths, bed porosity, and optical magnification. The estimate of ε _PD, which is found to be unbiased, is connected to errors in PIV measurement using the central limit theorem. Mismatch in refractive indices can also lead to reduction in particle density, N _s, detected light flux, J, and degrade the particle image. The model, verified through experiments, is used to predict the reduction in N _s and J, where it is found that particle defocusing caused by spherical beads in refractive index mismatched porous bed is the primary contributor to reductions of N _s and J. In addition, the magnitude of ε _PD is determined for the use of fluorescent dye emission for particle detection due to wavelength-dependent index of refraction.     

Maximum principles and pointwise error estimates for torsion of shells of revolution

This paper is concerned with the question of assessing the quality of approximate thin shell solutions for the problem of axisymmetric torsion of elastic shells of revolution, an issue previously considered by Ho and Knowles [1]. In both works, the objective is to obtain pointwise estimates, based on the three-dimensional theory of elasticity, for the errors involved in using an approximate shell theory.The mathematical problem is that of obtaining explicit pointwise gradient estimates for the solutions of homogeneous and nonhomogeneous second-order uniformly elliptic equations, an issue of interest beyond the present context. In contrast to arguments using energy inequalities, here we apply a technique based on maximum principles for such equations.The particular examples of cylindrical, spherical and conical shells of revolution are discussed in detail. In the former two cases, it is shown that the shell theory solution is in fact an exact elasticity solution in the Saint-Venant sense. Asymptotic error estimates for general shells of revolution are also given.

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Résumé L'étude çi-dessous traite de la qualité de l'approximation des coques minces appliquée aux problèmes de torsion asymétrique des coques élastiques de révolution, problème déjà considéré par Ho et Knowles [1].L'objectif des deux études est d'évaluer en chaque point, par la théorie de l'élasticité en trois limensions, les erreurs conséquentes à l'utilisation de la théorie de l'approximation des coques.Le problème mathématique consiste à obtenir une estimation explicite en chaque point des gradients pour les solutions d'équations elliptiques uniformes du deuxième ordre, homogènes et non-homogènes, problème d'un interêt qui dépasse le présent contexte. Nous utilisons ici une technique basée sur le principe du maximum pour de telles équations, par contraste avec les raissonnements utilisant des thegalités énergétiques.Les exemples particuliers des coques de révolution cylindriques, sphériques et coniques sont discutés en détail. Dans les deux premiers cas, nous montrons que la solution par la théorie des coques est en fait une solution exacte dans le sens de Saint-Venant. Des estimations asymptotiques des erreurs dans le cas de coques de révolution plus générales sont également données.

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Paper presented at the 6th Canadian Congress of Applied Mechanics (CANCAM 77), The University British Columbia, Vancouver, May 30–June 3, 1977.     

Hierarchy of One-Dimensional Models in Nonlinear Elasticity

By using formal asymptotic expansions, we build one-dimensional models for slender hyperelastic cylinders submitted to conservative loads. According to the order of magnitude of the applied loads, we obtain a hierarchy of models going from the linear theory of flexible bars to the nonlinear theory of extensible strings.

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Résumé On construit, à l'aide de développements asymptotiques formels, des modèles unidimensionnels de cylindres hyperélastiques élancés soumis à des forces conservatives. Suivant l'ordre de grandeur des forces appliquées, on obtient une hiérarchie de modèles allant de la théorie des poutres flexibles jusqu'à la théorie des fils élastiques.

     

Upper and lower bounds for the shear stress in the Saint-Venant theory of flexure

Upper and lower bounds are derived for the shear stress as it is determined by Saint-Venant's theory of flexure, and used to establish the asymptotic character of the classical Strength of Materials formula in the limit of vanishing thickness.

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Résumé On dérive des limites supérieures et inférieures des contraintes tangentielles suivant la théorie de la flexion de Saint-Venant, que l'on utilise aux fins d'établir le caractère asymptotique de la formule de la Résistance des Matériaux dans le cas limite d'une épaisseur extrêmement petite.

     

Wave propagation in elastic plates

The problem of wave propagation within the framework of a complete linear theory of elastic plates is treated using the method of wave curves. A complete classification of the various extensional and bending waves is obtained, along with the corresponding speeds of propagation. These are shown to correspond to the phase velocities of harmonic waves for infinite wave number. The decay and coupling equations are found, and the problem of waves due to a punch applied to the plate surface is treated.

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Résumé On a étudié le problème de la propagation des ondes en utilisant la méthode des courbes ondiales et assumant la théorie linéaire et complète des plaques minces. On obtient une classification totale des ondes diverses, de l'extension et de la flexion, avec des vitesses de la propagation. On prouve aussi, que celles-ci correspondent avec des vitesses des phases de la propagation, en cas de nombre infinite des ondes. Les équations de la décadence et de l'accouplement sont derivées, et on a étudié le problème d'une onde, qui est produite par une emporte-pièce appliquée sur la surface d'une plaque.

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This work was supported in part by funds from the National Research Council of Canada, under Grant Number A3805.     

PIV with volume lighting in a narrow cell: An efficient method to measure large velocity fields of rapidly varying flows

In this work we test a methodology for PIV measurements when a large field of view is required in planar confined geometries. Using a depth of field larger than the channel width, we intend to measure the in-plane variations of the velocity of the fluid averaged through the width of the channel, and we examine in which operating conditions this becomes possible. Measurements of the flow through a narrow channel by PIV are challenging because of the strong velocity gradients that develop between the walls. In particular, all techniques that use small particles as tracers have to deal with the possible migration of the tracers in the direction perpendicular to the walls. Among the complex mechanisms for migration, we focus on the so called Segré-Silberberg effect which can lead to transverse migration of neutrally buoyant tracers of finite size. We report experimental PIV measurements in a Hele-Shaw cell of 1 mm gap, which have been carried out by using neutrally buoyant tracers of size around 10 μm. By considering steady flows, we have observed, in particular flow regimes, the effect of an accumulation of the tracers at a certain distance to the wall due to the so called Segré-Silberberg effect. The particle migration is expected to occur at any Reynolds numbers but the migration velocity depends on the Reynolds number. A significant migration therefore takes place each time the observation duration is large enough compared to the migration time. For a given observation duration, the tracers remain uniformly distributed at low Reynolds numbers whereas they all accumulate at the equilibrium position at large ones. When using volume lighting, the PIV algorithm provides the average velocity of the flow through the gap at low Reynolds number, while it leads to the velocity of the flow at the equilibrium position of the tracers at large Reynolds numbers. By considering unsteady flows, we have observed that the migration does not occur if the timescale of flow variation is short compared to the time required for the parabolic flow to develop across the gap. In this case, there is no transverse velocity gradient and the PIV algorithm provides the fluid velocity. Altogether, these results allow us to propose guidelines for the interpretation of PIV measurements in confined flow, which are based on the theoretical predictions of the tracer migration derived by Asmolov [1].     

Surface and interfacial waves of arbitrary form in isotropic elastic media

A method due to Friedlander of accommodating disturbances of arbitrary form into the theory of surface waves in a semi-infinite isotropic elastic body is extended and shown to yield a simple closed form solution for the displacement field. An analogous treatment of interfacial waves of arbitrary form at a plane contact discontinuity separating different isotropic elastic materials is also given.

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Résumé On développe une méthode, conçue par Friedlander, qui fait entrer les perturbations de forme arbitraire dans la théorie des ondes de surface dans un corps élastique isotropique semi-infini, et on montre qu'elle permet d'obtenir une solution simple et exacte pour le champ de déplacement. Les ondes de forme arbitraire qui existent dans le plan à la frontière de materiaux élastiques isotropiques differents sont traitées de façon analogue.