Dissertations / Theses on the topic 'Plates theory'

The paper treats the theory of thin coated plates under a variety of load and deposition conditions. In addition to some bending problems caused by external load the so called Stoney-equation is considered.

This thesis presents two studies on elastic plates. In the first study, we discuss the choice of elastic energies for thin plates and shells, an unsettled issue with consequences for much recent modeling of soft matter. Through consideration of simple deformations of a thin body in the plane, we demonstrate that four bulk isotropic quadratic elastic theories have fundamentally different predictions with regard to bending behavior. At finite thickness, these qualitative effects persist near the limit of mid-surface isometry, and not all theories predict an isometric ground state. We discuss how certain kinematic measures that arose in early studies of rod mechanics lead to coherent definitions of stretching and bending, and promote the adoption of these quantities in the development of a covariant theory based on stretches rather than metrics. In the second work, the effects of in-plane swelling gradients on thin, anisotropic plates are investigated. We study systems with a separation of scales between bending energy terms. Warped equilibrium shapes are described by two parameters controlling the spatial "rolling up'' and twisting of the surface. Shapes within this two-parameter space are explored, and it is shown that shapes will either be axisymmetric or twisted depending on swelling function parameters and material anisotropy. In some axisymmetric shapes, pitchfork bifurcations to twisted solutions are observed by varying these parameters. We also show that a familiar soft mode of the catenoid to helicoid transformation of an isotropic material no longer exists with material anisotropy.<br>Master of Science<br>This thesis presents two studies on the subject of thin, elastic bodies, otherwise known as plates. Plate theory has important applications in many areas of life, ranging from the design and construction of civil structures to the mechanics of wrinkling sheets. In the first work, we discuss how different elastic plate theories have qualitatively different predictions on how a plate will behave when bent. We discuss the different physical implications of each model, and relate our findings to previous studies. Additionally, we promote the use of certain technical measures in the study of plates corresponding to the most coherent definitions of bending and stretching. In the second work, we study the effects of in-plane swelling gradients on elastic plates whose material stiffnesses vary with direction. Inspired by wood panels that warp when exposed to moisture, we model elastic plates exposed to various swelling patterns and determine the resulting warped shapes. We find that some shapes are axisymmetric, while others prefer to twist when exposed to moisture-induced swelling. By varying certain parameters of the swelling functions, or by increasing the material fiber stiffness, we also find a qualitative change in shape from an axisymmetric to a twisted surface.

Creep deformation in welded pipes and plates is of particular importance in the power industries. Most failures of welded pipes occur in the type IV region at the boundary with the parent material, which is relatively much harder. This thesis extends the work of Nicol (1985), Hawkes (1989) and Newman (1993) on the Cosserat theory of plates and shells, and has two major aims. The first is to develop the work of Hawkes to model successfully the strain rates in four-zone, thickwalled welded pipes. It is possible to determine the effects that the thickness of the pipe wall, the radius of the pipe and the creep index, n, in Norton's law have on the strain-rate distribution throughout the pipe. Using Continuum Damage Mechanics and this Cosserat model, the position and time to rupture in welded pipes is then calculated. The second aim of this thesis is to develop further the initial modelling work of Nicol (1985) and Hawkes (1989) by obtaining some simple perturbation models for both welded pipes and plates. The results obtained with the perturbation solutions are then compared with the numerical solutions of Newman (1993) for a plate and the numerical solutions derived in this thesis for a pipe.

In recent years advances in technology have allowed the transition of composite structures from secondary to primary structural components. Consequently, a lot of applications demand development of thicker composite structures to sustain heavier loads. Typical sandwich panels consist of two thin metallic or composite face sheets separated by a honeycomb or foam core. This configuration gives the sandwich panel high stiffness and strength and enables excellent energy absorption capabilities with little resultant weight penalty. This makes sandwich structures a preferred design for a lot of applications including aerospace, naval, wind turbines and civil industries. Most aerospace structures can be analyzed using shell and plate models and many such structures are modeled as composite sandwich plates and shells. Accurate theoretical formulations that minimize the CPU time without penalties on the quality of the results are thus of fundamental importance. The classical plate theory (CPT) and the first order shear deformation theory (FSDT) are the simplest equivalent single-layer models, and they adequately describe the kinematic behavior of most laminates where the difference between the stiffnesses of the respective phases is not huge. However, in the case of sandwich structures where the core is a much more compliant and softer material as compared to the face sheets the results from CPT and FSDT becomes highly inaccurate. Higher order theories in such cases can represent the kinematics better, may not require shear correction factors, and can yield much more accurate results. An advanced Extended Higher-order Sandwich Panel Theory (EHSAPT) which is a two-dimensional extension of the EHSAPT beam model that Phan presented is developed. Phan had extended the HSAPT theory for beams that allows for the transverse shear distribution in the core to acquire the proper distribution as the core stiffness increases as a result of non-negligible in-plane stresses. The HSAPT model is incapable of capturing the in-plane stresses and assumes negligible in-plane rigidity. The current research extends that concept and applies it to two-dimensional plate structures with variable aspect ratios. The theory assumes a transverse displacement in the core that varies as a second order equation in z and the in-plane displacements that are of third order in z, the transverse coordinate. This approach allows for five generalized coordinates in the core (the in-plane and transverse displacements and two rotations about the x and y-axes respectively). The major assumptions of the theory are as follows: 1) The face sheets satisfy the Euler-Bernoulli assumptions, and their thicknesses are small compared to the overall thickness of the sandwich section; they undergo small strains with moderate rotations. 2) The core is compressible in the transverse and axial directions; it has in-plane, transverse and shear rigidities. 3) The bonding between the face sheets and the core is assumed to be perfect. The kinematic model is developed by assuming a displacement field for the soft core and then enforcing continuity of the displacement field across the interface between the core and facesheets. The constitutive relations are then defined, and variational and energy techniques are employed to develop the governing equations and associated boundary conditions. A static loading case for a simply supported sandwich plate is first considered, and the results are compared to existing solutions from Elasticity theory, Classical Plate Theory (CPT) and First-Order Shear Deformation Plate Theory (FSDT). Subsequently, the governing equations for a dynamic analysis are developed for a laminated sandwich plate. A free vibration problem is analyzed for a simply supported laminated sandwich plate, and the results for the fundamental natural frequency are compared to benchmark elasticity solutions provided by Noor. After validation of the new Extended Higher Order Sandwich Panel Theory (EHSAPT), a parametric study is carried out to analyze the effect of variation of various geometric and material properties on the fundamental natural frequency of the structure. After the necessary verification and validation of the theory by comparing static and free vibration results to elasticity solutions, a nonlinear static analysis for square and rectangular plates is carried out under various sets of boundary conditions. The analysis was carried out using variational techniques, and the Ritz method was used to find an approximate solution. The kinematics were developed for a sandwich plate undergoing small strain and moderate rotations and nonlinear strain displacement relations were evaluated. Approximate and assumed solutions satisfying the geometric boundary conditions were developed and substituted in the total potential energy relations. After carrying out the spatial integrations, the total potential energy was then minimized with respect to the unknown coefficients in the assumed solution resulting in nonlinear simultaneous algebraic equations for the unknown coefficients. The simultaneous nonlinear equations were then solved using the Newton-Raphson method. A convergence study was carried out to study the effect of varying the number of terms in the approximate solution on the overall result and rapid convergence was observed. The rapid convergence can be attributed to the fact that the assumed approximate solution not only satisfies the geometric boundary conditions of the problem but also the natural boundary conditions. During calculations four cases of boundary conditions were considered 1) Simply Supported with moveable edges. 2) Simply Supported with fixed edges. 3) Clamped with moveable edges. 4) Clamped with fixed edges. For movable boundary conditions, in-plane displacements along the normal direction to the supported edges are allowed whereas the out-of-plane displacement is fixed. For the immovable boundary condition cases, the plate is prevented from both in-plane and out-of-plane displacements along the edges. For the simply supported cases rotations about the tangential direction are allowed, and for the clamped cases no rotations are allowed.In recent years advances in technology have allowed the transition of composite structures from secondary to primary structural components. Consequently, a lot of applications demand development of thicker composite structures to sustain heavier loads. Typical sandwich panels consist of two thin metallic or composite face sheets separated by a honeycomb or foam core. This configuration gives the sandwich panel high stiffness and strength and enables excellent energy absorption capabilities with little resultant weight penalty. This makes sandwich structures a preferred design for a lot of applications including aerospace, naval, wind turbines and civil industries. Most aerospace structures can be analyzed using shell and plate models and many such structures are modeled as composite sandwich plates and shells. Accurate theoretical formulations that minimize the CPU time without penalties on the quality of the results are thus of fundamental importance. The classical plate theory (CPT) and the first order shear deformation theory (FSDT) are the simplest equivalent single-layer models, and they adequately describe the kinematic behavior of most laminates where the difference between the stiffnesses of the respective phases is not huge. However, in the case of sandwich structures where the core is a much more compliant and softer material as compared to the face sheets the results from CPT and FSDT becomes highly inaccurate. Higher order theories in such cases can represent the kinematics better, may not require shear correction factors, and can yield much more accurate results. An advanced Extended Higher-order Sandwich Panel Theory (EHSAPT) which is a two-dimensional extension of the EHSAPT beam model that Phan presented is developed. Phan had extended the HSAPT theory for beams that allows for the transverse shear distribution in the core to acquire the proper distribution as the core stiffness increases as a result of non-negligible in-plane stresses. The HSAPT model is incapable of capturing the in-plane stresses and assumes negligible in-plane rigidity. The current research extends that concept and applies it to two-dimensional plate structures with variable aspect ratios. The theory assumes a transverse displacement in the core that varies as a second order equation in z and the in-plane displacements that are of third order in z, the transverse coordinate. This approach allows for five generalized coordinates in the core (the in-plane and transverse displacements and two rotations about the x and y-axes respectively). The major assumptions of the theory are as follows: 1) The face sheets satisfy the Euler-Bernoulli assumptions, and their thicknesses are small compared to the overall thickness of the sandwich section; they undergo small strains with moderate rotations. 2) The core is compressible in the transverse and axial directions; it has in-plane, transverse and shear rigidities. 3) The bonding between the face sheets and the core is assumed to be perfect. The kinematic model is developed by assuming a displacement field for the soft core and then enforcing continuity of the displacement field across the interface between the core and facesheets. The constitutive relations are then defined, and variational and energy techniques are employed to develop the governing equations and associated boundary conditions. A static loading case for a simply supported sandwich plate is first considered, and the results are compared to existing solutions from Elasticity theory, Classical Plate Theory (CPT) and First-Order Shear Deformation Plate Theory (FSDT). Subsequently, the governing equations for a dynamic analysis are developed for a laminated sandwich plate. A free vibration problem is analyzed for a simply supported laminated sandwich plate, and the results for the fundamental natural frequency are compared to benchmark elasticity solutions provided by Noor. After validation of the new Extended Higher Order Sandwich Panel Theory (EHSAPT), a parametric study is carried out to analyze the effect of variation of various geometric and material properties on the fundamental natural frequency of the structure. After the necessary verification and validation of the theory by comparing static and free vibration results to elasticity solutions, a nonlinear static analysis for square and rectangular plates is carried out under various sets of boundary conditions. The analysis was carried out using variational techniques, and the Ritz method was used to find an approximate solution. The kinematics were developed for a sandwich plate undergoing small strain and moderate rotations and nonlinear strain displacement relations were evaluated. Approximate and assumed solutions satisfying the geometric boundary conditions were developed and substituted in the total potential energy relations. After carrying out the spatial integrations, the total potential energy was then minimized with respect to the unknown coefficients in the assumed solution resulting in nonlinear simultaneous algebraic equations for the unknown coefficients. The simultaneous nonlinear equations were then solved using the Newton-Raphson method. A convergence study was carried out to study the effect of varying the number of terms in the approximate solution on the overall result and rapid convergence was observed. The rapid convergence can be attributed to the fact that the assumed approximate solution not only satisfies the geometric boundary conditions of the problem but also the natural boundary conditions. During calculations four cases of boundary conditions were considered 1) Simply Supported with moveable edges. 2) Simply Supported with fixed edges. 3) Clamped with moveable edges. 4) Clamped with fixed edges. For movable boundary conditions, in-plane displacements along the normal direction to the supported edges are allowed whereas the out-of-plane displacement is fixed. For the immovable boundary condition cases, the plate is prevented from both in-plane and out-of-plane displacements along the edges. For the simply supported cases rotations about the tangential direction are allowed, and for the clamped cases no rotations are allowed.

The bifurcation phenomenon occurring in twisted square plates with free edges subject to contrary self-equilibrating corner loading was examined. In order to eliminate lateral deflection of the test plates due to their own weight, a special loading apparatus was constructed which held the plates in a vertical plane. The complete strain field occurring at the plate centre was measured using two strain gauge rosettes mounted on opposing sides of the plate at the centre. Principal curvatures were calculated and related to corner load for several plates with differing edge length/thickness ratios. A Southwell plot was used relating mean curvature to the ratio mean curvature/Gaussian curvature, from which the Gaussian curvature occurring at bifurcation was determined. The critical dimensionless twist ka was then calculated for each plate size. It was found that there is a linear relation between the critical dimensionless twist ka occurring at bifurcation, and the thickness to edge length ratio h/a ratio, specifically: ka = 10.8h/a.<br>Applied Science, Faculty of<br>Mechanical Engineering, Department of<br>Graduate

One of the most important branches of applied mechanics is the theory of plates - defined to be plane structural elements whose thickness is very small when compared to the two planar dimensions. There is an abundance of plate theories in the literature modeling classical elastic solids that fit this description. Recently, however, there has been a steady growth of interest in modeling materials with microstructures that exhibit length-scale dependent behavior, generally known as Cosserat elastic materials. Concurrently, there has also been an increased interest in the construction of reduced dimensional models of such materials owing to advantages like reduced computational effort and a simpler, yet elegant, resulting mathematical formulation. The objective of this work is the formulation and implementation of a theory of elastic plates with microstructure. The mathematical underpinning of the approach used is the Variational Asymptotic Method (VAM), a powerful tool used to construct asymptotically correct plate models. Unlike existing Cosserat plate models in the literature, the VAM allows for a plate formulation that is free of a priori assumptions regarding the kinematics. The result is a systematic derivation of the two-dimensional constitutive relations and a set of geometrically-exact, fully intrinsic equations gov- erning the motion of a plate. An important consequence is the extraction of the drilling degree of freedom and the associated stiffness. Finally, a Galerkin approach for the solution of the fully-intrinsic formulation will be developed for a Cosserat sur- face analysis which will also be compatible with more traditional plate solvers based on the classical theory of elasticity. Results and validation are presented from linear static and dynamic analyses, along with a discussion on some challenges and solution techniques for nonlinear problems.One of the most important branches of applied mechanics is the theory of plates - defined to be plane structural elements whose thickness is very small when compared to the two planar dimensions. There is an abundance of plate theories in the literature modeling classical elastic solids that fit this description. Recently, however, there has been a steady growth of interest in modeling materials with microstructures that exhibit length-scale dependent behavior, generally known as Cosserat elastic materials. Concurrently, there has also been an increased interest in the construction of reduced dimensional models of such materials owing to advantages like reduced computational effort and a simpler, yet elegant, resulting mathematical formulation. The objective of this work is the formulation and implementation of a theory of elastic plates with microstructure. The mathematical underpinning of the approach used is the Variational Asymptotic Method (VAM), a powerful tool used to construct asymptotically correct plate models. Unlike existing Cosserat plate models in the literature, the VAM allows for a plate formulation that is free of a priori assumptions regarding the kinematics. The result is a systematic derivation of the two-dimensional constitutive relations and a set of geometrically-exact, fully intrinsic equations gov- erning the motion of a plate. An important consequence is the extraction of the drilling degree of freedom and the associated stiffness. Finally, a Galerkin approach for the solution of the fully-intrinsic formulation will be developed for a Cosserat sur- face analysis which will also be compatible with more traditional plate solvers based on the classical theory of elasticity. Results and validation are presented from linear static and dynamic analyses, along with a discussion on some challenges and solution techniques for nonlinear problems.

This thesis presents the first-known exact solutions for vibration of stepped circular Mindlin plates. The considered circular plate is of several step-wise variation in thickness in the radial direction. The Mindlin first order shear deformable plate theory is employed to derive the governing differential equations for the annular and circular segments. The exact solutions to these differential equations may be expressed in terms of the Bessel functions of the first and second kinds and the modified Bessel functions of the first and second kinds. The governing homogenous system of equations is assembled by implementing the essential and natural boundary conditions and the segment interface conditions. Vibration solutions are presented for circular Mindlin plates of different edge support conditions and various combinations of step-wise thickness variations. These exact vibration results may serve as important benchmark values for researchers to validate their numerical methods for such circular plate problems<br>Master of Engineering (Civil)

We present an elastic orthotropic plate theory in plane strain and axisym-metric deformations by first developing their uniform asymptotic expansions of the exact solutions for the basic governing boundary value problems. Then, the establishment of the necessary conditions for decaying states, both explicitly and asymptotically, enables us to determine the outer solution without reference to the inner solution and clarify the precise meaning of the well known St.Venant's principle under the circumstances considered here. The possible existence of corner stress singularities was examined by establishing and solving three transcendental governing equations. By developing a generalized Cauchy type singular integral equation for the plane strain deformation and an integral equation of the second kind for the axi-symmetric deformation and taking the corner stress singularities into consideration, we obtained accurate numerical solutions for all canonical boundary value problems which are needed in the asymptotic necessary conditions for decaying states. Finally, the accuracy of the numerical solutions of canonical boundary value problems and the efficiency of the plate theory were confirmed through the applications of solving two physical problems and comparing with the existing results.<br>Science, Faculty of<br>Mathematics, Department of<br>Graduate

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 59-62).<br>This thesis is divided into two main sections: the first containing the analysis of the broadband vertical coupler, and the second involving the theory and design of the integrated optical isolators. In the first part we propose, theoretically investigate, and numerically demonstrate a compact (less than 10[mu]m) broadband (more than 300nm) fiber-chip vertical coupler. The structure utilizes a Fresnel lens, or more advanced integrated optics, placed above a short, ridge and deep etched, vertical coupler in a Si waveguide. This optics is placed in order to match the radiating fields to the fiber mode. We use semivectorial simulations with a simple stochastic optimization to design a good integrated optics without cylindrical constrains. Three-dimensional Finite-Difference Time-Domain (FDTD) simulations reveal ~ 50% fiber coupling efficiency and a bandwidth of 200nm. In the second part we propose, theoretically investigate, and numerically demonstrate six designs of integrated optical isolators. We first derive analytically the value of the off-diagonal gyrotropic permittivity tensor element, Eg. We then use this value to calculate a non-reciprocal phase shift in a Manganese, and a N/P doped silicon waveguide using analytic, perturbation, and a novel mode numeric approach. Finally, using the obtained magnitudes of the nonreciprocal phase shifts six integrated optical isolator designs are proposed.<br>by Brad Gilbert Cordova.<br>S.M.

Cette thèse est faite de deux parties. La première partie est un article rédigé conjointementavec Martin Puchol et Jialin Zhu. La deuxième partie est une série de résultats obtenus par moi-même liés au théorème de Riemann-Roch-Grothendieck pour les fibrés vectoriels plats. Dans la première partie, nous donnons une preuve analytique d'un résultat décrivant le comportement de la torsion analytique en théorie de de Rham lorsque la variété considérée est séparée en deux par une hypersurface. Plus précisément, nous donnons une formule liant la torsion analytique de la variété entière aux torsions analytiques associées aux variétés à bord avec des conditions limites relative ou absolue le long de l'hypersurface. Dans la deuxième partie de cette thèse, nous raffinons les résultats de Bismut-Lott pour les images directes des fibrés vectoriels plats au cas où le fibré vectoriel plat en question est lui-même la cohomologie holomorphe d'un fibré vectoriel le long d'une fibration plate à fibres complexes. Dans ce contexte, nous donnons une formule de Riemann-Roch-Grothendieck dans laquelle la classe de Todd du fibré tangent relatif apparaît explicitement. En remplaçant les classes de cohomologie par des formes explicites qui les représentent en théorie de Chern-Weil, nous généralisons ainsi des constructions de Bismut-Lott<br>This thesis consists of two parts. The first part is an article written jointly with Martin Puchol and Jialin Zhu, the second part is a series of results obtained by myself in connection with the Riemann-Roch-Grothendieck theorem for flat vector bundles. In the first part, we give an analytic approach to the behavior of classical Ray-Singer analytic torsion in de Rham theory when a manifold is separated along a hypersurface. More precisely, we give a formula relating the analytic torsion of the full manifold, and the analytic torsion associated with relative or absolute boundary conditions along the hypersurface. In the second part of this thesis, we refine the results of Bismut-Lott on direct images of flat vector bundles to the case where the considered flat vector bundle is itself the fiberwise holomorphic cohomology of a vector bundle along a flat fibration by complex manifolds. In this context, we give a formula of Riemann-Roch-Grothendieck in which the Todd class of the relative holomorphic tangent bundle appears explicitly. By replacing cohomology classes by explicit differential forms in Chern-Weil theory, we extend the constructions of Bismut-Lott in this context

The problem of nonlinear parametric vibrations and stability analysis of the symmetric laminated plates is considered. The proposed method is based on multimode approximation of displacements and solving series auxiliary linear tasks. The main feature of the work is the application of the R-functions theory, which allows investigating parametric vibrations of plates with complex shape and different boundary conditions.

A mechanical wave is generated as a result of an external force interacting with the well-defined medium and it propagates through that medium transferring energy from one location to another. The ability to generate and control the motion of the mechanical waves through the finite medium opens up the opportunities for creating novel actuation mechanisms not possible before. However, any impedance to the path of these waves, especially in the form of finite boundaries, disperses this energy in the form of reflections. Therefore, it is impractical to achieve steady state traveling waves in finite structures without any reflections. In-spite of all these conditions, is it possible to generate waveforms that travel despite reflections at the boundaries? The work presented in this thesis develops a framework to answer this question by leveraging the dynamics of the finite structures without any active control. Therefore, this work investigates how mechanical waves are developed in finite structures and identifies the factors that influence steady state wave characteristics. Theoretical and experimental analysis is conducted on 1D and 2D structures to realize different type of traveling waves. Owing to the robust characteristics of the piezo-ceramics (PZTs) in vibrational studies, we developed piezo-coupled structures to develop traveling waves through experiments.The results from this study provided the fundamental physics behind the generation of mechanical waves and their propagation through finite mediums. This research will consolidate the outcomes and develop a structural framework that will aid with the design of adaptable structural systems built for the purpose. The present work aims to generate and harness structural traveling waves for various applications.<br>Ph. D.

This thesis presents new boundary element formulations for solution of bending problems in plates and shells. Also presented are the dual boundary element formulations for analysis of crack problems in plates and shells. Reissner plate theory is adopted to represent the bending and shear, and two dimensional (2-D) plane stress is used to model the membrane behaviour of the plate. New set of boundary element formulations to solve bending problems of shear deformable shallow shells having quadratic mid-surface is derived based on the modified Reissner plate and two dimensional plane stress governing equations which are now coupled due to the curvature of the shell. Dual Boundary Element Methods (DBEM) for plates and shells are developed for fracture mechanics analysis of structures loaded in combine bending and tension. Five stress intensity factors, that is, two for membrane and three for bending and shear are computed. The JIntegral technique and Crack Surface Displacements Extrapolation (CSDE) technique are used to compute the stress intensity factors. Special shape functions for crack tip elements are implemented to represent mom accurately displacement fields close to the crack tip. Crack growth processes are simulated with an incremental crack extension analysis. During the simulation, crack growth direction is determined using the maximum principal stress criterion. The crack extension is modelled by adding new boundary elements to the previous crack boundaries. As a consequence remeshing of existing boundaries is not required, and using this method the simulation can be effectively performed. Finally, a multi-region boundary element formulation is presented for modelling assembled plate-structures. The formulation enforces the compatibility of translations and rotations as well as equilibrium of membrane, bending and shear tractions. Examples are presented for plate and shell structures with different geometry, loading and boundar-y conditions to demonstrate the accuracy of the proposed formulations. The results obtained are shown to be in good agreement with analytical and other numerical results. Also presented are crack growth simulations of flat and curved panels loaded in combine bending and tension. The DBEM results are in good agreement with existing numerical and experimental results. Assembled plate-structure and a non-shallow shell bending problems are also analysed using a multi-region formulation developed in this thesis.

This thesis presents the first-known exact solutions for vibration of stepped circular Mindlin plates. The considered circular plate is of several step-wise variation in thickness in the radial direction. The Mindlin first order shear deformable plate theory is employed to derive the governing differential equations for the annular and circular segments. The exact solutions to these differential equations may be expressed in terms of the Bessel functions of the first and second kinds and the modified Bessel functions of the first and second kinds. The governing homogenous system of equations is assembled by implementing the essential and natural boundary conditions and the segment interface conditions. Vibration solutions are presented for circular Mindlin plates of different edge support conditions and various combinations of step-wise thickness variations. These exact vibration results may serve as important benchmark values for researchers to validate their numerical methods for such circular plate problems

A control metric, the weighted sum of spatial gradients (WSSG), has been developed for use in active structural acoustic control (ASAC). Previous development of WSSG [1] showed that it was an effective control metric on simply supported plates, while being simpler to measure than other control metrics, such as volume velocity. The purpose of the current work is to demonstrate that the previous research can be generalized to plates with a wider variety of boundary conditions and on less ideal plates. Two classes of plates have been considered: clamped flat plates, and ribbed plates. On clamped flat plates an analytical model has been developed for use in WSSG that assumes the mode shapes are the product of clamped-clamped beam mode shapes. The boundary condition specific weights for use in WSSG have been derived from this formulation and provide a relatively uniform measurement field, as in the case of the simply supported plate. Using this control metric, control of radiated sound power has been simulated. The results show that WSSG provides comparable control to volume velocity on the clamped plate. Results also show, through random placement of the sensors on the plate, that similar control can be achieved regardless of sensor location. This demonstrates that WSSG is an effective control metric on a variety of boundary conditions. Ribbed plates were considered because of their wide use in aircraft and ships. In this case, a finite-element model of the plate has been used to obtain the displacement field on the plate under a variety of boundary conditions. Due to the discretized model involved, a numerical, as opposed to analytical, formulation for WSSG has been developed. Simulations using this model show that ASAC can be performed effectively on ribbed plates. In particular WSSG was found to perform comparable to or better than volume velocity on all boundary conditions examined. The sensor insensitivity property was found to hold within each section (divided by the ribs) of the plate, a slightly modified form of the flat plate insensitivity property where the plates have been shown to be relatively insensitive to sensor location over the entire surface of the plate. Improved control at natural frequencies can be achieved by applying a second control force. This confirms that ASAC is a viable option for the control of radiated sound power on non-ideal physical systems similar to ribbed plates.

The purpose of the work described in this thesis is to decrease the computational cost of refined plate/shell models without loosing accuracy in the plate/shell response analysis. The axiomatic/asymptotic technique has been employed for this purpose. The refined models have been obtained by means of the Carrera Unified Formulation and both the Principle of Virtual Displacement (PVD) and the Reissner Mixed Variational Theorem (RMVT) have been employed to derive the governing equations. Mechanical and multifield problems are considered. Navier-like closed form solutions has been employed, and for this reason simply supported isotropic and orthotropic plates and shells has been considered. The refined models analyzed have been implemented according to the schemes known as Equivalent Single Layer and Layer Wise. In some case, the accuracy of the reduced refined models have been compared to the refined models available in the open scientific literature.

In the thesis explicit dual parabolic-elliptic models are constructed for the Konenkov flexural edge wave and the Stoneley-type flexural interfacial wave in case of thin linearly elastic plates. These waves do not appear in an explicit form in the original equations of motion within the framework of the classical Kirchhoff plate theory. The thesis is aimed to highlight the contribution of the edge and interfacial waves into the overall displacement field by deriving specialised equations oriented to aforementioned waves only. The proposed models consist of a parabolic equation governing the wave propagation along a plate edge or plate junction along with an elliptic equation over the interior describing decay in depth. In this case the parabolicity of the one-dimensional edge and interfacial equations supports flexural wave dispersion. The methodology presented in the thesis reveals a dual nature of edge and interfacial plate waves contrasting them to bulk-type wave propagating in thin elastic structures. The thesis tackles a number of important examples of the edge and interfacial wave propagation. First, it addresses the propagation of Konenkov flexural wave in an elastic isotropic plate under prescribed edge loading. For the latter, parabolic-elliptic explicit models were constructed and thoroughly investigated. A similar problem for a semi-infinite orthotropic plate resulted in a more general dual parabolic-elliptic model. Finally, an anal- ogous model was derived and analysed for two isotropic semi-infinite Kirchhoff plates under perfect contact conditions.

The theory is based on an assumed displacement field, in which the surface displacements are expanded in powers of the thickness coordinate up to the third order. The theory allows parabolic description of the transverse shear stresses, and therefore the shear correction factors of the usual shear deformation theory are not required in the present theory. The theory accounts for small strains but moderately large displacements (i.e., von Karman strains). Exact solutions for certain cross-ply shells and finite-element models of the theory are also developed. The finite-element model is based on independent approximations of the displacements and bending moments (i.e., mixed formulation), and therefore only C°-approximations are required. Further, the mixed variational formulations developed herein suggest that the bending moments can be interpolated using discontinuous approximations (across inter-element boundaries). The finite element is used to analyze cross-ply and angle-ply laminated shells for bending, vibration, and transient response. Numerical results are presented to show the effects of boundary conditions, lamination scheme (i.e., bending-stretching coupling and material anisotropy) shear deformation, and geometric nonlinearity on deflections and frequencies. Many of the numerical results presented here for laminated shells should serve as references for future investigations.<br>Ph. D.

Dans ce mémoire nous étudions les hypersurfaces Levi-plates analytiques dans les surfaces algébriques complexes. Il s'agit des hypersurfaces réelles qui admettent un feuilletage par des courbes holomorphes, appelé le feuilletage de Cauchy Riemann (CR). Dans un premier temps nous montrons que si ce dernier admet une dynamique chaotique (i.e. s'il n'admet pas de mesure transverse invariante) alors les composantes connexes de l'extérieur de l'hypersurface sont des modifications de domaines de Stein. Ceci permet d'étendre le feuilletage CR en un feuilletage algébrique singulier sur la surface complexe ambiante. Nous appliquons ce résultat pour montrer, par l'absurde, qu'une hypersurface Levi-plate analytique qui admet une structure affine transverse dans une surface algébrique complexe possède une mesure transverse invariante. Ceci nous amène à conjecturer que les hypersurfaces Levi-plates dans les surfaces algébriques complexes qui sont difféomorphes à un fibré hyperbolique en tores sur le cercle sont des fibrations par courbes algébriques<br>In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. These are real hypersurfaces that admit a foliation by holomorphic curves, called Cauchy Riemann foliation (CR). First, we show that if this foliation admits chaotic dynamics (i.e. if it doesn't admit an invariant transverse measure), then the connected components of the complement of the hypersurface are Stein. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse invariant measure. This leads us to conjecture that Levi-flat hypersurfaces in complex algebraic surfaces that are diffeomorphic to a hyperbolic tori bundle over the circle are fibrations by algebraic curves

Employing the Finite Element Method (FEM), we numerically study three problems involving free and forced vibrations of linearly and nonlinearly elastic plates with a third order shear and normal deformable theory (TSNDT) and the three dimensional (3D) elasticity theory. We used the commercial software ABAQUS for analyzing 3D deformations, and an in-house developed and verified software for solving the plate theory equations. In the first problem, we consider trapezoidal load-time pulses with linearly increasing and affinely decreasing loads of total durations equal to integer multiples of the time period of the first bending mode of vibration of a plate. For arbitrary spatial distributions of loads applied to monolithic and laminated orthotropic plates, we show that plates' vibrations become miniscule after the load is removed. We call this phenomenon as vibration attenuation. It is independent of the dwell time during which the load is a constant. We hypothesize that plates exhibit this phenomenon because nearly all of plate's strain energy is due to deformations corresponding to the fundamental bending mode of vibration. Thus taking the 1st bending mode shape of the plate vibration as the basis function, we reduce the problem to that of solving a single second-order ordinary differential equation. We show that this reduced-order model gives excellent results for monolithic and composite plates subjected to different loads. Rectangular plates studied in the 2nd problem have points on either one or two normals to their midsurface constrained from translating in all three directions. We find that deformations corresponding to several modes of vibration are annulled in a region of the plate divided by a plane through the constraining points; this phenomenon is termed mode localization. New results include: (i) the localization of both in-plane and out-of-plane modes of vibration, (ii) increase in the mode localization intensity with an increase in the length/width ratio of a rectangular plate, (iii) change in the mode localization characteristics with the fiber orientation angle in unidirectional fiber- reinforced laminae, (iv) mode localization due to points on two normals constrained, and (iv) the exchange of energy during forced harmonic vibrations between two regions separated by the line of nearly stationary points that results in a beating-like phenomenon in a sub-region of the plate. This technique can help design a structure with vibrations limited to its small sub-region, and harvesting energy of vibrations of the sub-region. In the third problem, we study finite transient deformations of rectangular plates using the TSNDT. The mathematical model includes all geometric and material nonlinearities. We compare the results of linear and nonlinear TSNDT FEM with the corresponding 3D FEM results from ABAQUS and note that the TSNDT is capable of predicting reasonably accurate results of displacements and in-plane stresses. However, the errors in computing transverse stresses are larger and the use of a two point stress recovery scheme improves their accuracy. We delineate the effects of nonlinearities by comparing results from the linear and the nonlinear theories. We observe that the linear theory over-predicts the deformations of a plate as compared to those obtained with the inclusion of geometric and material nonlinearities. We hypothesize that this is an effect of stiffening of the material due to the nonlinearity, analogous to the strain hardening phenomenon in plasticity. Based on this observation, we propose that the consideration of nonlinearities is essential in modeling plates undergoing large deformations as linear model over-predicts the deformation resulting in conservative design criteria. We also notice that unlike linear elastic plate bending, the neutral surface of a nonlinearly elastic bending plate, defined as the plane unstretched after the deformation, does not coincide with the mid-surface of the plate. Due to this effect, use of nonlinear models may be of useful in design of sandwich structures where a soft core near the mid-surface will be subjected to large in-plane stresses.<br>Doctor of Philosophy

L’utilisation des multicouches prend de plus en plus d’ampleur dans le domaine des sciences de l’ingénieur, tout d’abord dans l’industrie, et plus récemment de plus en plus en Génie Civil. Qu’il s’agisse de complexes mêlant des polymères, du bois ou du béton, des efforts importants sont nécessaires pour la modélisation fine de ce type de matériaux. En effet, des phénomènes induits par l’anisotropie et l’hétérogénéité sont associés à ces multi-matériaux : effets de bords, dilatations thermiques différentielles, délaminages/décollements ou non linéarités de type viscosité, endommagement, plasticité dans les couches ou aux interfaces. Parmi les modèles proposés dans la littérature, on trouve par exemple des modèles monocouche équivalente ou de type "Layerwise" (une cinématique par couche). Appartenant à cette deuxième catégorie, des modèles ont été développés depuis quelques années dans le laboratoire Navier et permettent une description suffisamment fine pour aborder les problématiques spécifiques citées plus haut tout en conservant un caractère opératoire certain. En introduisant des efforts d’interfaces comme des efforts généralisés du modèle, ces approches ont montré leur efficacité vis-à-vis de la représentation des détails au niveau inter- et intra-couches. Il est alors aisé de proposer des comportements et des critères d’interfaces et d’être efficace pour la modélisation du délaminage ou décollement, phénomène très présent dans les composites multicouches assemblés et collés. Par conséquent, un programme éléments finis MPFEAP a été développé dans le laboratoire Navier. Le modèle a également été introduit sous la forme d’un User Element dans ABAQUS, dans sa forme la plus simple (interfaces parfaites).Un nouveau model layerwise est proposé dans ce mémoire pour les plaques multicouches, appelé "Statically Compatible Layerwise Stresses with first-order membrane stress approximations per layer in thickness direction" SCLS1. Le modèle est conforme aux équations d’équilibre 3D ainsi qu’aux conditions aux limites de bord libre. En outre, une version raffinée du nouveau modèle est obtenu en introduisant plusieurs couches mathématiques par couche physique. Le nouveau modèle a été mis en œuvre dans une nouvelle version du code éléments finis MPFEAP.En parallèle, un programme d’éléments finis basé sur la théorie Bending-Gradient développée dans le laboratoire Navier est proposé ici. Le modèle est une nouvelle théorie de plaque épaisse chargée hors-plan où les inconnues statiques sont celles de la théorie Love-Kirchhoff, à laquelle six composantes sont ajoutées représentant le gradient du moment de flexion. La théorie Bending- Gradient est obtenue à partir de la théorie Generalized-Reissner: cette dernière implique quinze degrés de liberté cinématiques, huit d’entre eux étant lié uniquement à la déformation de Poisson hors-plan, et donc l’idée principale de la théorie de plaque Bending-Gradient est de simplifier la théorie Generalized-Reissner en réglant ces huit d.o.f. à zéro et de négliger la contribution de la contrainte normale σ33 dans l’équation constitutive du modèle de plaque. Un programme éléments finis appelé BGFEAP a été développé pour la mise en œuvre de l’élément de Bending-Gradient. Un User Element dans Abaqus a été aussi développé pour la théorie Bending-Gradient<br>The use of multilayer is becoming increasingly important in the field of engineering, first in the industry, and more recently more and more in Civil Engineering. Whether complex blend of polymers, wood or concrete, significant efforts are required for accurate modeling of such materials. Indeed, phenomena induced anisotropy and heterogeneity are associated with these multi-material: edge effects, differential thermal expansion, delamination/detachment or nonlinearities viscosity type damage, plasticity in layers or interfaces. Among the models proposed in the literature, we found for example equivalent monolayer model or of "LayerWise" type (a kinematic per layer). Belonging to the second category, models have been developed in recent years in Navier allow a sufficiently detailed description to address specific issues mentioned above while maintaining a surgical nature. By introducing interface forces as generalized forces of the model, these approaches have demonstrated their effectiveness vis-à-vis the representation of details at inter- and intra-layers. It is then easy to offer behaviors and interfaces criteria and to be effective for modeling delamination or detachment, phenomenom very present in multilayered composites assembled and glued together. Therefore, a finite element program MPFEAP was developed in Navier laboratory. The model was also introduced as a User Element in ABAQUS, in its simplest form (perfect interfaces).A new layerwise model for multilayered plates is proposed in this dissertation, named Statically Compatible Layerwise Stresses with first-order membrane stress approximations per layer in thickness direction SCLS1. The model complies exactly with the 3D equilibrium equations and the free-edge boundary conditions. Also, a refined version of the new model is obtained by introducing several mathematical layers per physical layer. The new model has been implemented in a new version of the in-house finite element code MPFEAP.In parallel, a finite element program based on the Bending-Gradient theory which was developed in Navier laboratory, is proposed here. The model is a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Love-Kirchhoff theory, to which six components are added representing the gradient of the bending moment. The Bending-Gradient theory is obtained from the Generalized-Reissner theory: the Generalized-Reissner theory involves fifteen kinematic degrees of freedom, eight of them being related only to out-of-plane Poisson’s distortion and thus, the main idea of the Bending-Gradient plate theory is to simplify the Generalized-Reissner theory by setting these eight d.o.f. to zero and to neglect the contribution of the normal stress σ33 in the plate model constitutive equation. A finite element program called BGFEAP has been developed for the implementation of the Bending-Gradient element. A User Element in Abaqus was also developed for the Bending-Gradient theory

Failure criteria play a vital role in the numerical analysis of reinforced concrete structures. The current failure criteria can be classified into two types, namely the empirical and theoretical failure criteria. Empirical failure criteria normally lack reasonable theoretical backgrounds, while theoretical ones either involve too many parameters or ignore the effects of intermediate principal stress on the concrete strength. Based on the octahedral shear stress model and the concrete tensile strength under the state of triaxial and uniaxial stress, a new failure criterion, that is, the simplified unified strength theory (UST), is developed by simplifiing the five-parameter UST for the analysis of reinforced concrete structures. According to the simplified UST failure criterion, the concrete strength is influenced by the maximum and intermediate principal shear stresses together with the corresponding normal stresses. Moreover, the effect of hydrostatic pressure on the concrete strength is also taken into account. The failure criterion involves three concrete strengths, namely the uniaxial tensile and compressive strengths and the equal biaxial compressive strength. In the numerical analysis, a degenerated shell element with the layered approach is adopted for the simulation of concrete structures. In the layered approach, concrete is divided into several layers over the thickness of the elements and reinforcing steel is smeared into the corresponding number of layers of equivalent thickness. In each concrete layer, three-dimensional stresses are calculated at the integration points. For the material modelling, concrete is treated as isotropic material until cracking occurs. Cracked concrete is treated as an orthotropic material incorporating tension stiffening and the reduction of cracked shear stiffness. Meanwhile, the smeared craclc model is employed. The bending reinforcements and the stirrups are simulated using a trilinear material model. To verify the correctness of the simplified UST failure criterion, comparisons are made with concrete triaxial empirical results as well as with the Kupfer and the Ottosen failure criteria. Finally, the proposed failure criterion is used for the flexural analysis of simply supported reinforced concrete beams. Also conducted are the punching shear analyses of single- and multi-column-slab connections and of half-scale flat plate models. In view of its accuracy and capabilities, the simplified UST failure criterion may be used to analyse beam- and slab-type reinforced concrete structures.

Failure criteria play a vital role in the numerical analysis of reinforced concrete structures. The current failure criteria can be classified into two types, namely the empirical and theoretical failure criteria. Empirical failure criteria normally lack reasonable theoretical backgrounds, while theoretical ones either involve too many parameters or ignore the effects of intermediate principal stress on the concrete strength. Based on the octahedral shear stress model and the concrete tensile strength under the state of triaxial and uniaxial stress, a new failure criterion, that is, the simplified unified strength theory (UST), is developed by simplifiing the five-parameter UST for the analysis of reinforced concrete structures. According to the simplified UST failure criterion, the concrete strength is influenced by the maximum and intermediate principal shear stresses together with the corresponding normal stresses. Moreover, the effect of hydrostatic pressure on the concrete strength is also taken into account. The failure criterion involves three concrete strengths, namely the uniaxial tensile and compressive strengths and the equal biaxial compressive strength. In the numerical analysis, a degenerated shell element with the layered approach is adopted for the simulation of concrete structures. In the layered approach, concrete is divided into several layers over the thickness of the elements and reinforcing steel is smeared into the corresponding number of layers of equivalent thickness. In each concrete layer, three-dimensional stresses are calculated at the integration points. For the material modelling, concrete is treated as isotropic material until cracking occurs. Cracked concrete is treated as an orthotropic material incorporating tension stiffening and the reduction of cracked shear stiffness. Meanwhile, the smeared craclc model is employed. The bending reinforcements and the stirrups are simulated using a trilinear material model. To verify the correctness of the simplified UST failure criterion, comparisons are made with concrete triaxial empirical results as well as with the Kupfer and the Ottosen failure criteria. Finally, the proposed failure criterion is used for the flexural analysis of simply supported reinforced concrete beams. Also conducted are the punching shear analyses of single- and multi-column-slab connections and of half-scale flat plate models. In view of its accuracy and capabilities, the simplified UST failure criterion may be used to analyse beam- and slab-type reinforced concrete structures.<br>Thesis (Masters)<br>Master of Philosophy (MPhil)<br>School of Engineering<br>Full Text

Made available in DSpace on 2014-06-11T19:25:21Z (GMT). No. of bitstreams: 0 Previous issue date: 2004-01-23Bitstream added on 2014-06-13T19:53:02Z : No. of bitstreams: 1 waidemam_l_me_ilha.pdf: 2247815 bytes, checksum: aab885504632d6169da23c5ac9796141 (MD5)<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)<br>Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)<br>Este trabalho tem como objetivo principal analisar o comportamento dinâmico de estruturas laminares planas com carregamento perpendicular ao plano médio, em particular as placas delgadas, utilizando-se, para isso, a teoria clássica de flexão de placas e a discretização estrutural feita com os elementos finitos triangulares e retangulares trabalhando em conjunto e em separado. Na dedução das matrizes de rigidez e de massas dos elementos finitos em questão utiliza-se a formulação com parâmetros generalizados e com coordenadas homogêneas, cujas funções aproximadoras contêm nove e doze monômios, respectivamente, extraídos do polinômio algébrico cúbico em “x” e “y”. Para a consideração do amortecimento utiliza-se o Método de Rayleigh e para a integração numérica ao longo do tempo utiliza-se o Método de Newmark, via algoritmo previsor / corretor. Ao final deste são elaborados vários exemplos elucidativos visando uma análise quantitativa e qualitativa dos resultados obtidos.<br>In this work the dynamic behavior of plane laminate structures, with load applied perpendicularly to the middle plan, has been analyzed. The classic theory of bending plates and structural subdivision - done with triangular and rectangular finite elements working together as well as in separate – are used to study thin plates. The formulation employing generalized parameters and homogeneous coordinates, using approximating functions containing nine and twelve terms starting from the cubic algebraic polynomial in Cartesian coordinates x and y, is used to obtain the stiffness and mass matrices for the triangular and rectangular finite element, respectively. The Rayleigh Method is used to take into account the structural dumping while the Newmark Method is used to perform the numeric integration in the time, by means of predictor / corrector scheme. Additionally, several elucidating examples are elaborated in order to analyze the final results.

O objetivo principal deste trabalho é apresentar uma associação dos cálculos elástico e plástico, para análise de lajes-cogumelo. Foram considerados painéis retangulares, com carregamento uniformemente distribuído. O cálculo elástico é utilizado como pré-dimensionamento, fornecendo a razão entre os momentos de vão e permitindo a fixação dos momentos de plastificação negativos. O cálculo plástico é feito através da Teoria das Charneiras Plásticas, que é utilizada na obtenção dos momentos de plastificação, com os quais é feito o dimensionamento. Também se apresentam dois exemplos completos, para comparar os resultados do cálculo elástico com aqueles relativos ao procedimento ora proposto. O cálculo elástico foi feito através do Processo dos Pórticos Equivalentes. Não houve diferença significativa entre os resultados obtidos com os dois processos, para os exemplos considerados. Novas pesquisas são sugeridas para aprimorar o procedimento proposto.<br>The main aim of this work is to present an association of elastic plate theory and plastic analysis, to beamless slabs (flat slabs and flat plates) analysis. Rectangular panels uniformly loaded were considered. Elastic plate theory is used to predimensioning, determining the relation between positive bending moments and permiting to fix the ultimate negative moments of resistance. Plastic analysis is made by Yield Line Theory, that is used to obtain ultimate moments of resistance, which are used in dimensioning. Two complete examples are also presented, to compare results of elastic plate theory with those relative to proposed proceeding. Elastic calculus is made by Equivalent Frame Method. There was not significant difference between results obtained with the two processes in the considered examples. New researches are suggested in order to perfect proposed proceeding.

L'informatique a changé profondément les aspects méthodologiques du processus de découverte dans les différents domaines du savoir. Les chercheurs ont à leur disposition aujourd'hui de nouvelles capacités qui permettent d'envisager la résolution de nouveaux problèmes. Les plates-formes parallèles et distribués composées de ressources partagés entre différents participants peuvent rendre ces nouvelles capacités accessibles à tout chercheur et offre une puissance de calcul qui a été limitée jusqu'à présent, aux projets scientifiques les plus grands (et les plus riches). Dans ce document qui regroupe les résultats obtenus pendant mon doctorat, nous explorons quatre facettes différentes de la façon dont les organisations s'engagent dans une collaboration sur de plates-formes parallèles et distribuées. En utilisant des outils classiques de l'analyse combinatoire, de l'ordonnancement multi-objectif et de la théorie des jeux, nous avons montré comment calculer des ordonnancements avec un bon compromis entre les résultats obtenu par les participants et la performance globale de la plate-forme. En assurant des résultats justes et en garantissant des améliorations de performance pour les différents participants, nous pouvons créer une plate-forme efficace où chacun se sent toujours encourager à collaborer et à partager ses ressources. Tout d'abord, nous étudions la collaboration entre organisations égoïstes. Nous montrons que le comportement égoïste entre les participants impose une borne inférieure sur le makespan global. Nous présentons des algorithmes qui font face à l'égoïsme des organisations et qui présentent des résultats équitables. La seconde étude porte sur la collaboration entre les organisations qui peuvent tolérer une dégradation limitée de leur performance si cela peut aider à améliorer le makespan global. Nous améliorons les bornes d'inapproximabilité connues sur ce problème et nous présentons de nouveaux algorithmes dont les garanties sont proches de l'ensemble de Pareto (qui regroupe les meilleures solutions possibles). La troisième forme de collaboration étudiée est celle entre des participants rationnels qui peuvent choisir la meilleure stratégie pour leur tâches. Nous présentons un modèle de jeu non coopératif pour le problème et nous montrons comment l'utilisation de "coordination mechanisms" permet la création d'équilibres approchés avec un prix de l'anarchie borné. Finalement, nous étudions la collaboration entre utilisateurs partageant un ensemble de ressources communes. Nous présentons une méthode qui énumère la frontière des solutions avec des meilleurs compromis pour les utilisateurs et sélectionne la solution qui apporte la meilleure performance globale<br>Computer science is deeply changing methodological aspects of the discovery process in different areas of knowledge. Researchers have at their disposal new capabilities that can create novel research opportunities. Parallel and distributed platforms composed of resources shared between different participants can make these new capabilities accessible to every researcher at every level, delivering computational power that was restricted before to bigger (and wealthy) scientific projects. This work explores four different facets of the rules that govern how organizations engage in collaboration on modern parallel and distributed platforms. Using classical combinatorial tools, multi-objective scheduling and game-theory, we showed how to compute schedules with good trade-offs between the results got by the participants and the global performance of the platform. By ensuring fair results and guaranteeing performance improvements for the participants, we can create an efficient platform where everyone always feels encouraged to collaborate and to share its resources. First, we study the collaboration between selfish organizations. We show how the selfish behavior between the participants imposes a lower bound on the global makespan. We present algorithms that cope with the selfishness of the organizations and that achieve good fairness in practice. The second study is about collaboration between organizations that can tolerate a limited degradation on their performance if this can help ameliorate the global makespan. We improve the existing inapproximation bounds for this problem and present new algorithms whose guarantees are close to the Pareto set. The third form of collaboration studied is between rational participants that can independently choose the best strategy for their jobs. We present a non-cooperative game-theoretic model for the problem and show how coordination mechanisms allow the creation of approximate pure equilibria with bounded price of anarchy. Finally, we study collaboration between users sharing a set of common resources. We present a method that enumerates the frontier of best compromise solutions for the users and selects the solution that brings the best value for the global performance function

We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solidand fluidmechanics.For manyof these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model.Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite elementmodelalways yields a discrete system ofequations witha symmetric positive definite coeffcientmatrix.These attributes, amongst manyothers highlightedand detailed in this work, allow the developmentofrobust andeffcient finite elementmodels for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures.

Although laminated materials have been used for decades, their employment has increased nowadays in the last years as a result of the gained confidence of the industry on these materials. This has provided the scientific community many reasons to dedicate considerable amount of time and efforts to address a better understanding of their mechanical behavior. With this objective both, experimental and numerical simulation have been working together to give response to a variety of problems related with these materials. Regarding numerical simulation, a correct modeling of the kinematics of laminated materials is essential to capture the real behavior of the structure. Moreover, once the kinematics of the structure has been accurately predicted other non-linear phenomena such as damage and/or plasticity process could be also studied. In consequence, in order to contribute to the constant development of simpler and more efficient numerical tools to model laminated materials, a numerical method for modeling mode II/III delamination in advanced composite materials using one- and two-dimensional finite elements is proposed in this work. In addition, two finite elements base on a zigzag theory for simulating highly heterogeneous multilayered beams and plates structures are developed here. The document is written based on results of four papers published in indexed journals. Copies of all these papers are included in Appendix. The main body of this thesis is constituted by Chapters 2 to 4. Chapter 2 deals with the numerical treatment of laminated beams and plates. Chapter 3 presents the formulation of the LRZ beam and the QLRZ plate finite elements based on the Refined Zigzag Theory. Finally, the main contribution of this thesis, the LRZ/QLRZ delamination model, is developed in Chapter 4.<br>Aunque los materiales laminados se han utilizado durante décadas, su uso ha aumentado en los últimos años como resultado de una mayor confianza por parte de la industria. Esto ha proporcionado a la comunidad científica muchas razones para dedicar una considerable cantidad de tiempo y esfuerzos en aras de una mejor comprensión de su comportamiento mecánico. Con este objetivo tanto la simulación experimental como numérica han estado trabajando juntos para dar respuesta a una variedad de problemas relacionados con estos materiales. En cuanto a la simulación numérica, un correcto modelado de la cinemática de los materiales laminados es esencial para capturar el comportamiento real de la estructura. Por otra parte, una vez que la cinemática de la estructura se ha predicho con precisión otros fenómenos no lineales como los proceso de daño y/o plasticidad podrían ser también estudiados. En consecuencia, con el fin de contribuir al constante desarrollo de herramientas numéricas más simples y eficaces para modelar materiales laminados, un método numérico para el modelado de la delaminación (modo II/III) en materiales compuestos avanzados utilizando elementos finitos de una y dos dimensiones es propuesto en este trabajo. Además, dos elementos finitos para la simulación de vigas y placas de varias capas altamente heterogéneos son desarrollados aquí. El documento está escrito en base a los resultados de cuatro artículos publicados en revistas indexadas. Copias de estos artículos se incluyen en el Apéndice. El cuerpo principal de esta tesis está constituido por los Capítulos 2-4. El Capítulo 2 aborda el tratamiento numérico de vigas y placas laminadas. El capítulo 3 presenta la formulación de los elementos finitos de viga LRZ y placa QLRZ basados en la Teoría Zigzag Refinada. Finalmente, la principal contribución de esta tesis, el modelo de delaminación LRZ/QLRZ, se desarrolla en el capítulo 4.

Este trabalho aborda os conceitos básicos da teoria de segunda ordem das placas elásticas delgadas, utilizando o Método dos Elementos Finitos (introduzido através do Método dos Resíduos Ponderados, na variante de Galerkin). São deduzidas as matrizes de rigidez geométrica, de rigidez secante e de rigidez tangente, relativas ao problema em consideração. É proposta ainda uma conduta notavelmente simplificada, que facilita sobremaneira a construção da matriz de rigidez tangente.<br>This paper delas with the basic concepts of the secondf order theory of thin elastic plates, through the use of the Finite Element Method 9introcuced through the Weighted Residual Method, in Galerkin\'s approach). The matrices of geometric stiffness, secant stiffness, and tangent stiffness for the problem under consideration are deduced. It is also proposed an outstandingly simplified conduct, which will greatly easen the construction of the tangent stiffness matrix.

Cette thèse est consacrée à la modélisation de la propagation des ondes planes dans les plaques multicouches infinies, dans le cadre de l'élasticité linéaire. L’objet du travail est de trouver une approximation analytique ou semi-analytique des relations de dispersion des ondes lorsque le rapport de l'épaisseur de la plaque sur la longueur d'onde est petit. Ces relations de dispersion, liant la fréquence angulaire et le nombre d'onde, fournissent des informations clés sur les caractéristiques de propagation des différents modes. On propose dans cette thèse deux modélisations : le modèle du Bending-Gradient et la méthode des développements asymptotiques. La pertinence de ces méthodes est testée en comparant leurs prédictions à celles des théories de plaques bien connues, et à des résultats de référence obtenus par la méthode des éléments finis. Au préalable, dans la première partie de la thèse, une justification mathématique de la théorie du Bending-Gradient dans le cadre statique est réalisée à l’aide des méthodes variationnelles. Il s'agit d'abord d'identifier les espaces mathématiques dans lesquels les problèmes variationnels du Bending-Gradient sont bien posés. Puis, des théorèmes d'existence et d'unicité des solutions correspondantes sont ensuite formulés et prouvés. La deuxième partie est consacrée à la formulation des équations du mouvement du Bending-Gradient. Des simulations numériques sont effectuées pour plusieurs types d'empilements, permettant ainsi de tester la validité du modèle pour la modélisation de la propagation des ondes de flexion. La troisième partie est dédiée à l'analyse asymptotique des équations tridimensionnelles du mouvement, menée à bien grâce à la méthode des développements asymptotiques, le petit paramètre étant le rapport de l'épaisseur sur la longueur d'onde. En supposant que les champs tridimensionnels s'écrivent comme des séries en puissance du petit paramètre, on obtient une succession de problèmes à résoudre en cascade. La validité de cette méthode est évaluée par comparaison avec la méthode des éléments finis<br>This thesis is dedicated to the modelling of plane wave propagation in infinite multilayered plates, in the context of linear elasticity. The aim of this work is to find an analytical or semi-analytical approximation of the wave dispersion relations when the ratio of the thickness to the wavelength is small. The dispersion relations, linking the angular frequency and the wave number, provide key information about the propagation characteristics of the wave modes. Two methods are proposed in this thesis: the Bending-Gradient model and the asymptotic expansion method. The relevance of these methods is tested by comparing their predictions to those of well-known plate theories, and to reference results computed using the finite element method. Preliminarily, the first part of the thesis is devoted to the mathematical justification of the Bending-Gradient theory in the static framework using variational methods. The first step is to identify the mathematical spaces in which the variational problems of the Bending-Gradient are well posed. A series of existence and uniqueness theorems of the corresponding solutions are then formulated and proved. The second part is dedicated to the formulation of the equations of motion of the Bending-Gradient theory. Numerical simulations are realized for different types of layer stacks to assess the ability of this model to correctly predict the propagation of flexural waves. The third part is concerned with the asymptotic analysis of the three-dimensional equations of motion, carried out using the asymptotic expansion method, the small parameter being the ratio of the thickness to the wavelength. Assuming that the three-dimensional fields can be written as expansions in power of the small parameter, a series of problems which can be solved recursively is obtained. The validity of this method is evaluated by comparison with the finite element method

Nesta pesquisa, estuda-se o fenômeno de localização de modos de vibração em estruturas moduladas quase periódicas de comportamento linear e não-linear. Em particular, contempla-se uma aplicação na Engenharia Civil, os painéis de placas periódicos com pequenas imperfeições fracamente acoplados entre si através de viga de grande rigidez, e principalmente submetidos à variação de forças de membrana introduzidas por meio da protensão, o que só pode ser levado em conta introduzindo a rigidez geométrica no modelo matemático. No caso de sistemas lineares, a presença de pequenas desordens nas características de rigidez ou massa de subsistemas fracamente acoplados pode causar confinamento espacial nas vibrações livres, conhecido como Localização de Modos, e pode inibir a propagação da resposta forçada. È o que tem sido mostrado na literatura técnica, em especial nas áreas de Engenharia Mecânica e Aeroespacial. Os efeitos de localização serão obtidos numa perspectiva modal. O programa de elementos finitos DYMPLATE implementado pelo autor para análise dinâmica não-linear de estruturas de placas, será utilizado para modelar estruturas periódicas (ordenadas) e quase periódicas (desordenadas). Os modelos são linearizados em torno de configurações deformadas de referência. O problema algébrico de autovalores é resolvido para obter as freqüências naturais e correspondentes modos de vibração. Estruturas planas constituídas por placas protendidas com módulos repetidos, pequenas imperfeições e diferentes condições de apoio e de carregamento, serão utilizadas na investigação numérica da influência de diversos fatores na Localização de Modos, em especial as forças de membrana.<br>In this research, the phenomenon of vibration modes localization in nearly periodic modular structures of linear and nonlinear behavior is studied. Of special interest is an application in Civil Engineering, lightly coupled periodic plate panels with small imperfections, mainly submitted to the variation of membrane forces introduced by prestress forces, which can be only considered by introducing geometric stiffness in the mathematical model. In the linear case, the presence of small disorders in the stiffness or mass characteristics of lightly coupled sub-systems can cause spatial confinement of free vibrations, known as Mode Localization, and can inhibit the propagation of the forced response. That is what has been shown in the literature, especially in the areas of Mechanical and Aerospace Engineering. The effects of localization are viewed from a modal perspective. DYMPLATE, a finite element software implemented by the author for nonlinear dynamic analysis of plates, will be utilized to model ordered and disordered plate periodic structures. The models are linearized about a deformed reference configuration. The algebraic eigenvalue problem is solved to obtain the natural frequencies and corresponding modes shapes. Plane structures constituted by prestressed plates with repetitive dynamic characteristics, small imperfections and different boundary conditions and loads, will be utilized in the numerical investigation of the influence of numerous factors in the Mode Localization.

Orientador: Rogério de Oliveira Rodrigues<br>Banca: Wilson Sergio Venturini<br>Banca: Mônica Pinto Barbosa<br>Resumo: Este trabalho tem como objetivo principal analisar o comportamento dinâmico de estruturas laminares planas com carregamento perpendicular ao plano médio, em particular as placas delgadas, utilizando-se, para isso, a teoria clássica de flexão de placas e a discretização estrutural feita com os elementos finitos triangulares e retangulares trabalhando em conjunto e em separado. Na dedução das matrizes de rigidez e de massas dos elementos finitos em questão utiliza-se a formulação com parâmetros generalizados e com coordenadas homogêneas, cujas funções aproximadoras contêm nove e doze monômios, respectivamente, extraídos do polinômio algébrico cúbico em "x" e "y". Para a consideração do amortecimento utiliza-se o Método de Rayleigh e para a integração numérica ao longo do tempo utiliza-se o Método de Newmark, via algoritmo previsor / corretor. Ao final deste são elaborados vários exemplos elucidativos visando uma análise quantitativa e qualitativa dos resultados obtidos.<br>Abstract: In this work the dynamic behavior of plane laminate structures, with load applied perpendicularly to the middle plan, has been analyzed. The classic theory of bending plates and structural subdivision - done with triangular and rectangular finite elements working together as well as in separate - are used to study thin plates. The formulation employing generalized parameters and homogeneous coordinates, using approximating functions containing nine and twelve terms starting from the cubic algebraic polynomial in Cartesian coordinates "x" and "y", is used to obtain the stiffness and mass matrices for the triangular and rectangular finite element, respectively. The Rayleigh Method is used to take into account the structural dumping while the Newmark Method is used to perform the numeric integration in the time, by means of predictor / corrector scheme. Additionally, several elucidating examples are elaborated in order to analyze the final results.<br>Mestre

This paper studies the geometrically non-linear bending behavior of functionally graded beams subjected to buckling loads using the finite element method. The computational model is based on an improved first-order shear deformation theory for beams with five independent variables. The abstract finite element formulation is derived by means of the principle of virtual work. High-order nodal-spectral interpolation functions were utilized to approximate the field variables which minimizes the locking problem. The incremental/iterative solution technique of Newton's type is implemented to solve the nonlinear equations. The model is verified with benchmark problems available in the literature. The objective is to investigate the effect of volume fraction variation in the response of functionally graded beams made of ceramics and metals. As expected, the results show that transverse deflections vary significantly depending on the ceramic and metal combination.<br>Revisión por pares