Tesi sul tema "Plate theory"

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.

Crack propagation and branching are modeled using nonlocal peridynamic theory. One major advantage of this nonlocal theory based analysis tool is the unifying approach towards material behavior modeling- irrespective of whether the crack is formed in the material or not. No separate damage law is needed for crack initiation and propagation. This theory overcomes the weaknesses of existing continuum mechanics based numerical tools (e.g. FEM, XFEM etc.) for identifying fracture modes and does not require any simplifying assumptions. Cracks grow autonomously and not necessarily along a prescribed path. However, in some special situations such as in case of ductile fracture, the damage evolution and failure depend on parameters characterizing the local stress state instead of peridynamic damage modeling technique developed for brittle fracture. For brittle fracture modeling the bond is simply broken when the failure criterion is satisfied. This simulation helps us to design more reliable modeling tool for crack propagation and branching in both brittle and ductile materials. Peridynamic analysis has been found to be very demanding computationally, particularly for real-world structures (e.g. vehicles, aircrafts, etc.). It also requires a very expensive visualization process. The goal of this paper is to bring awareness to researchers the impact of this cutting-edge simulation tool for a better understanding of the cracked material response. A computer code has been developed to implement the peridynamic theory based modeling tool for two-dimensional analysis. A good agreement between our predictions and previously published results is observed. Some interesting new results that have not been reported earlier by others are also obtained and presented in this paper. The final objective of this investigation is to increase the mechanics knowledge of self-similar and self-affine cracks.

Sandwich plates and layered composites are common in many structural applications because of their combination of high stiffness and low weight. These plates combine top and bottom layers of high Young's modulus with intermediate layers of material carrying predominantly shear loads. Finite elements developed for the analysis of sandwich plates need to accurately model transverse shear stresses through the plate thickness. This study was inspired by an Office of Naval Research project to investigate the suitability of steel sandwich plates as ship hulls. A finite element implementation based on a third-order shear deformation element was used in a standard finite element program to model transverse shear stresses in a simply supported plate. Four elements based on third-order theory are developed and tested. Using static condensation to reduce the number of degrees of freedom required by a third-order plate element does not preserve the element's accuracy in either displacements or stresses, and stresses do not converge with refinement of the mesh. For the thin isotropic plate case, some condensed elements give reasonable displacement and stress results, but only for certain choices of mesh and the element is less versatile than one based on first order plate theory. None of the condensed elements give good results for composite plates of any thickness.
Master of Science

Circular voids are often incorporated into concrete bridge decks to reduce their self-weight without greatly reducing the flexural stiffness. Incorporating voids within a slab offers many advantages over a conventional solid concrete slab, for example a lower total cost of construction, reduced material use, and enhanced structural efficiency. The advantages of this topology are obvious, however the voids within the slab complicate the analysis of the structure. The incorporation of the voids within the slab results in different flexural stiffness in the longitudinal and transverse directions, resulting in an orthotropic slab. Another feature which distinguishes voided slabs from other common bridge types is the deformable nature of their cross-sections, which influences the load distribution of the structure. The need for a method of analysis which accounts for the orthotropic behaviour and deformable nature of the cross-section has been suggested by many in the past. The idealisation of voided slabs as an orthotropic plate has been the subject of extensive research. When modelling a voided slab as an orthotropic plate, it is necessary to calculate the reduction in the longitudinal and transverse stiffness due to the presence of the voids. Several equivalent plate parameters, which take on numerous forms, have been suggested by various authors to account for the effect of the voids. No research has been reported in technical literature to compare these different methods employing orthotropic plate parameters. Key shortcomings to these methods include the lack of definition of the suggested equivalent plate parameters, and the geometrical parameters of voided slabs which influence their behaviour. In an attempt to address these limitations, the aim of this study is to verify and validate the effect of the void diameter ratio and void spacing on the structural behaviour of voided slabs, which are the main influences on the orthotropic behaviour and cross-section deformation. The different methods of analysis using orthotropic plate theory suggested by various authors employing equivalent plate parameters are compared and discussed. The objectives of the study were achieved based on the finite element approach using ABAQUS. Finite element models with a void diameter to slab depth ratio range of 0.5 to 0.9, and a void spacing range of 0.9m to 2.7m were considered and analysed. Comparisons were made of the longitudinal and transverse stress distribution results from these models in order to draw conclusions. Results from solid models using both isotropic and orthotropic materials based on the equivalent plate parameters suggested from literature are also presented for comparison in order to verify the methods suggested by the technical literature for the analysis of voided slab bridge decks. Results of the finite element modelling show that the addition of voids causes large variations to the transverse stress distribution from the typical parabolic transverse stress distribution shape, leading to large peak transverse stresses in the flanges above and below the voids. These variations are due to the deformable nature of the cross-section. The voids also lead to a stress raising effect on the longitudinal stresses. It was found that an increase in void diameter to slab thickness ratio results in a rapid increase in both the longitudinal and transverse stresses, which shows that there is an increase in orthotropic behaviour and deformation of the cross-section with an increase in void diameter ratio. From the results, it can be concluded that the optimal void diameter ratio is between 0.6 and 0.8. This range of void diameter ratios allow for greater efficiency due to reduced dead load and material use, without generating excessive stresses due to cellular distortion resulting from excessively thin and flexible flanges above and below the voids. The spacing of the voids was found to have minimal effect on the stress distributions for a logical void spacing. These results show that the orthotropic behaviour and deformation of the cross-section are more sensitive to variations in void diameter ratio than the spacing of the voids. The void diameter ratio should therefore form the basis of the equivalent plate parameters for the use of orthotropic plate theory. The use of a solid orthotropic plate to idealise a voided slab showed reasonable agreement with the results from three-dimensional models, with some discrepancies in the different authors' methods noted. The net effect of using a two-dimensional analysis is the averaging out of the stress transverse distribution, which cannot predict the peak stresses around the voids. The orthotropic models compared more favourably with the 3D model than the isotropic models with increasing void diameter ratio. The results presented have shown that the incorporation of voids begins to affect the structural behaviour of the slab once the void diameter ratio exceeds 0.6, and the orthotropic behaviour becomes considerable. The stress raising effect of the voids should therefore be accounted for in the analysis of a voided slab once the void diameter ratio exceeds 0.6. It is recommended that a solid isotropic slab can be used to idealise a voided slab when the void diameter to slab depth ratio is less than 0.6. When the void diameter ratio is greater than 0.6, the transverse stiffness should be evaluated independently from the longitudinal stiffness, and orthotropic models are more suitable. For higher void diameter ratios, the method employed by Sen et al. (1994), which employs a reduced depth solid orthotropic slab in conjunction with stress multipliers, was found to be the most accurate method for idealising voided slabs. It is evident from this study, that while a three-dimensional finite element model may be too complex for everyday use, it may be extremely valuable for determining the local effects due to the presence of the voids.

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.

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.
Applied Science, Faculty of
Mechanical Engineering, Department of
Graduate

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.
Doctor of Philosophy

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.
Master of Science
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.

Granular media continue to be among the most manipulated materials found in various industries. Particle interactions in granular flow has fundamental importance in analyzing the performance of a wide range of key engineering applications such as hoppers, tumblers, and mixers etc. In spite of such ubiquitous presence, till date, our understanding of the granular flow is very limited. This restricts our ability to design efficient and optimal granular processing equipment. Additionally, the existing design abilities are also constrained by the number of particles to be analyzed, where, a typical industrial application involves millions of particles. This motivated the current research where investigations on the above limitations are pursued from three different angles: experimental, theoretical, and simulation. More specifically, this work aims to study particle-wall interaction and developing a computationally efficient cellular automata simulation framework for industrial granular applications. Towards this end, the current research is divided into two part: (I) energy dissipation during particle-wall interaction (II) cellular automata modeling. In part I, detailed experiments are performed on various sphere-thin plate combinations to measure the coefficient of restitution (COR) which is a measure of energy dissipation and it is one of the most important input parameters in any granular simulation. Alternatively, the energy dissipation measure also used to evaluate the elastic impact performance of superelastic Nitinol 60 material. Explicit finite element simulations are performed to gain detail understanding of the contact process and underlying parameters such as contact forces, stress-strain fields, and energy dissipation modes. A parametric study reveals a critical value of plate thickness above which the effect of plate thickness on the energy dissipation can be eliminated in the equipment design. It is found that the existing analytical expressions has limited applicability in predicting the above experimental and numerical results. Therefore, a new theoretical model for the coefficient of restitution is proposed which combines the effect of plastic deformation and plate thickness (i.e. flexural vibrations). In part II, in order to advance the existing granular flow modeling capabilities for the industry (dry and slurry flows) a cellular automata (CA) modeling framework is developed which can supplement the physically rigorous but computationally demanding discrete element method (DEM). These include a three-dimensional model which takes into account particle friction and spin during collision processing, which provides the ability to handle flows beyond solely the kinetic regime, and a multiphase framework which combines computational fluid dynamics (CFD) with CA to model multi-million particle count applications such as particle-laden flows and slurry flows.

In this study, numerical investigations of stability and transition problems were performed for 2D compressible boundary layers over a flat plate in adiabatic wall condition. Emphasis was placed on linear stability theory. The mathematical formulation for 3D boundary layers with oblique waves including detailed theoretical information was followed by use of the numerical techniques for the solution of resulting differential system of the instability problem, consequently an eigenvalue problem. First, two-dimensional sinusoidal disturbances were analyzed at various Mach numbers including the subsonic, transonic, supersonic and even hypersonic flow speeds. In this case, the second mode (acoustic mode), namely the Mack mode, and its behavior with the increasing Mach number were visualized. The results were then compared with the available data in literature concluding with good agreements. Secondly, similar analysis was carried out for oblique waves. Here, not only the effect of flow speed but also the effect of wave orientation was demonstrated. For this purpose, instability problem was solved for several wave angles at each Mach number in the range of M=0 and M=5. In this respect, the angle at which the waves were most unstable was also obtained at each investigated flow speed. The resultant stability diagrams corresponding to M=4 and higher Mach numbers for which both first and the second modes appear revealed that plane waves were more stable than oblique waves for the Tollmien-Schlichting mode, however, this was the opposite for the acoustic mode where oblique waves were observed to be more stable. As a final step, estimation of the transition location was handled for the most unstable wave condition. Smith-Van Ingen transition method was applied as the prediction device. The results representing the influence of Mach number on transition Reynolds number were then compared with the experimental data as well as the numerical ones in literature ending up with very good agreements.

Buckling and postbuckling of plates and stiffened panels are considered. Computational models for direct calculation of the response are developed using large deflection plate theory and energy principles. Deflections are represented by trigonometric functions. All combinations of biaxial in-plane compression or tension, shear, and lateral pressure are included in the formulations. The procedure is semi-analytical in the sense that the incremental equilibrium equations are derived analytically, while a numerical method is used for solving the equation systems, and for incrementation of the solution.

Unstiffened plate models are developed both for the simply supported case and for the clamped case. For the simply supported case the material types considered are isotropic elastic, orthotropic elastic, and elastic-plastic. Two models are developed for analysis of local buckling of stiffened plates, one for open profiles and one for closed profiles. A global buckling model for stiffened panels is developed by considering the panel as a plate with general anisotropic stiffness. The stiffness coefficients are input from the local analysis. Two models are developed for combined local and global buckling, in order to account for interaction between local and global deflection. The first is for a single stiffened plate, and uses a column approach. The second is for a stiffened panel with several stiffeners.

Numerical results are calculated for a variety of plate and stiffener geometries for verification of the proposed model, and comparison is made with nonlinear finite element methods. Some examples are presented. For all models, the response in the elastic region is well predicted compared with the finite element method results. Also, the efficiency of the calculations is very high. Estimates of ultimate strength are found using first yield as a collapse criterion. In most cases, this leads to conservative results compared to predictions from finite element calculations.

Theoretical formulations, analytical solutions, and finite element solutions for laminated composite plate and shell structures with smart material laminae are presented in the study. A unified third-order shear deformation theory is formulated and used to study vibration/deflection suppression characteristics of plate and shell structures. The von K??rm??n type geometric nonlinearity is included in the formulation. Third-order shear deformation theory based on Donnell and Sanders nonlinear shell theories is chosen for the shell formulation. The smart material used in this study to achieve damping of transverse deflection is the Terfenol-D magnetostrictive material. A negative velocity feedback control is used to control the structural system with the constant control gain. The Navier solutions of laminated composite plates and shells of rectangular planeform are obtained for the simply supported boundary conditions using the linear theories. Displacement finite element models that account for the geometric nonlinearity and dynamic response are developed. The conforming element which has eight degrees of freedom per node is used to develop the finite element model. Newmark's time integration scheme is used to reduce the ordinary differential equations in time to algebraic equations. Newton-Raphson iteration scheme is used to solve the resulting nonlinear finite element equations. A number of parametric studies are carried out to understand the damping characteristics of laminated composites with embedded smart material layers.

The Mindlin-Reissner plate model is widely used for the elastic deformation simulation of moderately thick plates. Shear locking occurs in the case of thin plates, which means slow convergence with respect to the mesh size. The Kirchhoff plate model does not show locking effects, but is valid only for thin plates. One would like to have a method suitable for both thick and thin plates. Several approaches are known to deal with the shear locking in the Mindlin-Reissner plate model. In addition to the well-known MITC elements and other approaches based on a mixed formulation, hierarchical methods have been developed in the recent years. These are based on the Kirchhoff model and add terms to account for shear deformations. We present some of these methods and develop a new hierarchic plate formulation. This new model can be discretised by a combination of C0 and C1 finite elements. Numerical tests show that the new formulation is locking-free and numerically efficient. We also give an extension of the model to a hierarchical Naghdi shell based on a Koiter shell formulation with unknowns in Cartesian coordinates.

This thesis presents a multi-purpose sensor concept viable for the simultaneous measurement of pressure, temperature and thickness of plate structures. It also establishes the knowledge base necessary for future sensor design. Thermal-Acousto Photonic Non-Destructive Evaluation (TAP-NDE) is employed to remotely initiate and acquire interrogating ultrasonic waves. Parameters including pressure, temperature and plate thickness are determined through exploring the dispersion features of the interrogating waves. A theoretical study is performed, through which a modified wave propagation theory applicable to homogeneous, isotropic, linear elastic materials is formulated along with an associated numerical model. A numerical scheme for solving the model is also developed using FEMLAB, a finite element based PDE solver. Gabor Wavelet Transform (GWT) is employed to map numerical time waveforms into the joint time-frequency domain. Wave time-frequency information enables dispersion curves to be extracted and material pressure, temperature and thickness to be determined. A sensor configuration design integrating the wave generation and sensing components of the proven TAP-NDE technology is also developed. Conclusions of the research are drawn from wave dispersion obtained corresponding to the following ranges of parameters: 300-500kHz for frequency, 25-300oC for temperature, 1-3mm for plate thickness, and 6 10 1?? - 7 1 10 ?? N/m for pressure. Each of the three parameters considered in the study has a different level of impact on plate wave dispersion. Plate thickness is found to have the most impact on wave dispersion, followed by temperature of the plate. The effect attributable to pressure is the least prominent among the three parameters considered. Plate thickness and temperature can be readily measured while simultaneously resolved using dispersion curves. However, pressure variation can only be differentiated when the plate is smaller than 1mm in thickness. It is observed that the thicker the plate, the faster the frequency group velocity. Also, the group velocities of all frequency components considered are seen to increase with increasing temperature, but decrease with increasing pressure.

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
Master of Engineering (Civil)

The aerospace industry is increasingly coming to the conclusion that physics-based high-fidelity models need to be used as early as possible in the design of its products. At the preliminary design stage of wing structures, though highly desirable for its high accuracy, a detailed finite element analysis(FEA) is often not feasible due to the prohibitive preparation time for the FE model data and high computation cost caused by large degrees of freedom. In view of this situation, often equivalent beam models are used for the purpose of obtaining global solutions. However, for wings with low aspect ratio, the use of equivalent beam models is questionable, and using an equivalent plate model would be more promising. An efficient method, Equivalent Plate Analysis or simply EPA, using an equivalent plate model, is developed in the present work for studying the static and free-vibration problems of built-up wing structures composed of skins, spars, and ribs. The model includes the transverse shear effects by treating the built-up wing as a plate following the Reissner-Mindlin theory (FSDT). The Ritz method is used with the Legendre polynomials being employed as the trial functions. Formulations are such that there is no limitation on the wing thickness distribution. This method is evaluated, by comparing the results with those obtained using MSC/NASTRAN, for a set of examples including both static and dynamic problems. The Equivalent Plate Analysis (EPA) as explained above is also used as a basis for generating other efficient methods for the early design stage of wing structures, such that they can be incorporated with optimization tools into the process of searching for an optimal design. In the search for an optimal design, it is essential to assess the structural responses quickly at any design space point. For such purpose, the FEA or even the above EPA, which establishes the stiffness and mass matrices by integrating contributions spar by spar, rib by rib, are not efficient enough. One approach is to use the Artificial Neural Network (ANN), or simply called Neural Network (NN) as a means of simulating the structural responses of wings. Upon an investigation of applications of NN in structural engineering, methods of using NN for the present purpose are explored in two directions, i.e. the direct application and the indirect application. The direct method uses FEA or EPA generated results directly as the output. In the indirect method, the wing inner-structure is combined with the skins to form an "equivalent" material. The constitutive matrix, which relates the stress vector to the strain vector, and the density of the equivalent material are obtained by enforcing mass and stiffness matrix equities with regard to the EPA in a least-square sense. Neural networks for these material properties are trained in terms of the design variables of the wing structure. It is shown that this EPA with indirect application of Neural Networks, or simply called an Equivalent Skin Analysis (ESA) of the wing structure, is more efficient than the EPA and still fairly good results can be obtained. Another approach is to use the sensitivity techniques. Sensitivity techniques are frequently used in structural design practices for searching the optimal solutions near a baseline design. In the present work, the modal response of general trapezoidal wing structures is approximated using shape sensitivities up to the second order, and the use of second order sensitivities proved to be yielding much better results than the case where only first order sensitivities are used. Also different approaches of computing the derivatives are investigated. In a design space with a lot of design points, when sensitivities at each design point are obtained, it is shown that the global variation in the design space can be readily given based on these sensitivities.
Ph. D.

The Mindlin-Reissner plate model is widely used for the elastic deformation simulation of moderately thick plates. Shear locking occurs in the case of thin plates, which means slow convergence with respect to the mesh size. The Kirchhoff plate model does not show locking effects, but is valid only for thin plates. One would like to have a method suitable for both thick and thin plates. Several approaches are known to deal with the shear locking in the Mindlin-Reissner plate model. In addition to the well-known MITC elements and other approaches based on a mixed formulation, hierarchical methods have been developed in the recent years. These are based on the Kirchhoff model and add terms to account for shear deformations. We present some of these methods and develop a new hierarchic plate formulation. This new model can be discretised by a combination of C0 and C1 finite elements. Numerical tests show that the new formulation is locking-free and numerically efficient. We also give an extension of the model to a hierarchical Naghdi shell based on a Koiter shell formulation with unknowns in Cartesian coordinates.:1 Introduction 2 Plate theory 3 Shell theory 4 Conclusion

Boundary-layer transition from a laminar to a turbulent regime is a critical driver in the design of high-speed vehicles. The aerothermodynamic loads associated with transitional or fully turbulent hypersonic boundary layers are several times higher than those associated with laminar flow. The presence of isolated roughness elements on the surface of a body can accelerate the growth of incoming disturbances and introduce additional instability mechanisms in the flow field, eventually leading to a premature occurrence of transition. This dissertation studies the instabilities induced by three-dimensional discrete roughness elements located inside a high-speed boundary layer developing on a flat plate. Two-dimensional local linear stability theory (2D-LST) is employed to identify the instabilities evolving in the three-dimensional flow field that characterizes the wake induced by the roughness elements and to investigate their evolution downstream. A formulation of the disturbance energy evolution equation available for base flows depending on a single spatial direction is generalized for the first time to base flows featuring two inhomogeneous directions and perturbations depending on three spatial directions. This generalization allows to obtain a decomposition of the temporal growth rate of 2D-LST instabilities into the different contributions that lead to the production and dissipation of the total disturbance energy. This novel extension of the formulation provides an additional layer of information for understanding the energy exchange mechanisms between a three-dimensional base flow and the perturbations resulting from 2D-LST. Stability computations for a calorically perfect gas illustrate that the wake induced by the roughness elements supports the growth of different sinuous and varicose instabilities which coexist together with the Mack-mode perturbations that evolve in the flat-plate boundary layer, and which become modulated by the roughness-element wake. A single pair of sinuous and varicose disturbances is found to dominate the wake instability in the vicinity of the obstacles. The application of the newly developed decomposition of the temporal growth rate reveals that the roughness-induced wake modes extract most of their potential energy from the transport of entropy fluctuations across the base-flow temperature gradients and most of their kinetic energy from the work of the disturbance Reynolds stresses against the base-flow velocity gradients. Further downstream, the growth rate of the wake instabilities is found to be influenced by the presence of Mack-mode disturbances developing on the flat plate. Strong evidence is observed of a continuous synchronization mechanism between the wake instabilities and the Mack-mode perturbations. This phenomenon leads to an enhancement of the amplification rate of the wake modes far downstream of the roughness element, ultimately increasing the associated integrated amplification factors for some of the investigated conditions. The effects of vibrational molecular excitation and chemical non-equilibrium on the instabilities induced by a roughness element are studied for the case of a high-temperature boundary layer developing on a sharp wedge configuration. For this purpose, a 2D-LST solver for chemical non-equilibrium flows is developed for the first time, featuring a fully consistent implementation of the thermal and transport models employed for the base flow and the perturbation fields. This is achieved thanks to the automatic derivation and implementation tool (ADIT) available within the von Karman Institute extensible stability and transition analysis (VESTA) tool-kit, which enables an automatic derivation and implementation of the 2D-LST governing equations for different thermodynamic flow assumptions and models. The stability computations for this configuration show that sinuous and varicose disturbances also dominate the wake instability in the presence of vibrational molecular energy mode excitation and chemical reactions. The resulting base-flow cooling associated with the modeling of such high-temperature phenomena is found to have opposite stabilizing and destabilizing effects on the streamwise evolution of the sinuous and varicose instabilities. The modeling of vibrational excitation and chemical non-equilibrium acting exclusively on the perturbations is found to have a stabilizing influence in all cases.
Doctorat en Sciences de l'ingénieur et technologie
info:eu-repo/semantics/nonPublished

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.

Thesis (M. S.)--Civil and Environmental Engineering, Georgia Institute of Technology, 2007.
Committee Chair: Mulalo Doyoyo; Committee Co-Chair: Reginald Desroches; Committee Member: Laurence J. Jacobs.

La methode d'integration asymptotique de gol'denveizer et le formalisme variationnel de ciarlet et destuynder (connu aussi comme la technique du zoom) ont montre leur efficacite pour etablir des theories de corps elastiques elances tels que les plaques minces. La meme idee est utilisee ici pour obtenir differentes approximations a la theorie des plaques electro ou magnetoelastiques. La metode et les resultats critiques sont illustres sur trois exemples. Le premier, traite en quasi-electrostatiqus, concerne les plaques piezoelectriques. Le second, traite en quasi-magnetostatique, concerne la theorie des plaques elastiques ferromagnetiques (non conductrices). Le troisieme cas s'applique aux plaques conductrices de courant electrique

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

La théorie de la tectonique des plaques permet de décrire les mouvements de premier ordre qui opèrent à la surface de la Terre. S'il est acquis que la convection dans le manteau terrestre en est le moteur, les liens entre les phénomènes profonds et les caractéristiques tectoniques de la surface restent largement méconnus. Jusqu'à très récemment, les modèles de convection du manteau terrestre ne produisaient pas de tectonique de surface pouvant être comparée à celle de la Terre. Récemment, des modèles globaux de convection qui reproduisent une tectonique de surface comparable à la Terre au premier ordre ont été mis au point. Ces modèles produisent des courants mantelliques ascendants et descendants de grande échelle et des déformations localisées en surface dans les zones de divergence et les zones de convergence. Ils génèrent une expansion des fonds océaniques de manière auto-cohérente proche de celle reconstruite pour les 200 derniers millions d'années de l'histoire de la Terre et une dérive de continents similaire à celle observée grâce au paléomagnétisme. Cette thèse s'inscrit parmi les premières tentatives d'utilisation de modèles de convection sphériques auto-organisés à des fins de compréhension de la tectonique de surface. La tectonique produite dans ce type de modèles de convection sera caractérisée finement à travers l'étude des limites de plaques, de leur agencement et de leurs vitesses de déplacement. L'objectif est de pouvoir comparer qualitativement et quantitativement les résultats des calculs de convection avec les reconstructions des mouvements de la surface terrestre grâce à la tectonique des plaques et aux observations de terrain. Dans cette optique, les limites tectoniques ont été définies à la main dans un premier temps afin de comprendre la physique qui gouverne l'agencement caractéristique des plaques tectoniques terrestres. En effet, celle-ci est composée de sept grandes plaques et plusieurs petites dont la répartition statistique indique deux processus de mise en place distincts. Nous avons déterminé les processus responsables de la mise en place de l'agencement caractéristique des plaques tectoniques en surface en faisant varier la résistance de la lithosphère. Plus la lithosphère est résistante, plus la longueur totale et la courbure des zones de subduction diminue à la surface des modèles. Cela s'accompagne également d'une diminution du nombre de petites plaques. En étudiant la fragmentation au niveau des jonctions triples, nous avons montré que les petites plaques étaient associées aux géométries courbées des fosses océaniques. En revanche, les grandes plaques sont contrôlées par les grandes longueurs d'onde de la convection mantellique. Ces deux processus impliquent deux temps de réorganisation, c'est-à-dire l'apparition et la disparition d'une plaque plongeante dans le manteau terrestre (environ 100 millions d'années) pour les grandes plaques, alors que l'échelle de temps de réorganisation des petites plaques dépend des mouvements des fosses et est ainsi plus rapide d'un ordre de grandeur. Afin d'effectuer des analyses quantitatives rapides, des méthodes d'analyse automatique de la surface et de l'intérieur des modèles ont été développées. La première technique concerne la détection automatique des plaques tectoniques à la surface des modèles (ADOPT). ADOPT est un outil de détection basé sur une technique de segmentation d'images utilisée pour détecter des bassins versants. Les champs à la surface des modèles sont transformés en reliefs, soit directement, soit après un processus de filtrage. Cette détection permet d'obtenir des polygones de plaques comparable aux analyses réalisées à la main. Une autre technique de détection a été mise au point pour étudier les panaches mantelliques [etc...]
Plate tectonics theory describes first order surface motions at the surface of the Earth. Although it is agreed upon that convection in the mantle drives the plates, the relationships between deep dynamics and surface tectonics are still largely unknown. Until recently, mantle convection models could not produce surface tectonics that could be compared to that of the Earth. New global models are able to form large-scale ascending and descending mantle currents, as well as narrow regions of localized deformation at the surface where convergence and divergence occur. These models selfconsistently generate an expansion of the oceanic floor similar to that of the last 200 million years on Earth, and continental drift similar to what can be reconstructed with palaeomagnetism. This Ph.D. thesis constitutes one of the first attempts to use self-organised, spherical convection models in order to better understand surface tectonics. Here, the tectonics produced by the models is finely charaterized through the study of plate boundaries, their organisation and their velocities. The goal is to be able to compare qualitatively and quantitatively the results of convection computations with surface motions, as reconstructed using the rules of plate tectonics and field observations. Plate boundaries emerging from the models were first traced and analyzed by hand so as to understand the physics that govern the typical organization of the tectonics plates on Earth. It is characterised by seven large plates and several smaller ones, following a statistical distribution that suggests that two distinct physical processes control the plates’ layout. We have determined the processes responsible for this distribution while varying the strength of the lithosphere (the yield stress). In our models, the stronger the lithosphere, the greater the total subduction length and their curvature, and the fewer the small plates. By studying surface fragmentation with triple junctions, we showed that the formation of small plates is associated with oceanic trench curvature. Large plates, however, are controlled by the long wavelengths of the convection cells. These two processes involve two different reorganisation times, controlled either by the accretion and the subduction of the large plates (about 100 Myrs), or by trench motions for the smaller plates. In order to improve the efficiency of our analysis, we have developed automated methods to study the surface and the interior of the models. The first technique is about detecting the tectonic plates automatically at the surface of the models. It is called ADOPT. It is a tool based on image segmentation technique to detect the watersheds. The surface fields of the convection models are converted into a relief field, either directly or using a distance method. This automatic detection allows to obtain plates polygons similar to the hand analysis. Another technique of detection has been developed to study mantle plumes. These analyzes were used to determine the driving forces behind the plates layout, to quantify the timing of reorganizations and to evaluate the implication of the models rheology on the surface distribution. These new analytical tools and the constant evolution of the quality of mantle convection models allow us to improve our understanding of the link between mantle dynamics and surface tectonics, but also to target necessary improvements in the convection models used

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.

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 this essay is to study the evolution and differences of motifs on bronze age objects associated with men and women, the differences in plastic ornamentation and the differences in grave goods in five danish oakcoffin burials from II-III, and fourtythree swedish cremation graves from period IV-VI. The methods used for the motifs and plastic ornamentation are an iconographical one and for the grave goods I have chosen to make a chart. The spiral motifs on beltplates and the ships on the bronze razors, for example, have played a significant part in bronze age life. The same could be said of the plastic ornamentation on the horse headed razors from period II-III. The grave goods from the danish oakcoffin burials could possibly be seen as something connected with sunpriestesses. The few gravegoods coming from the swedish cremation graves could be viewed as a consequence of larger villagers appearing during the late bronze age.

Les panneaux sandwichs sont des éléments de structure omniprésents au quotidien. Leur efficacité structurelle n'est plus à démontrer. Elle est même un élément déterminant dans le marché qui leur est associé. Ce mémoire de doctorat s'intéresse à un nouveau type d 'âme de panneau sandwich qui pourrait être amené à supplanter le nid d'abeilles dans certaines applications, le module à chevrons. L'objectif est donc de pouvoir faire une estimation précise du comportement de ces nouvelles âmes. Cependant le gain en efficacité structurelle des panneaux sandwichs se paye par une augmentation considérable de la complexité de leur comportement mécanique. C'est en particulier le cas de la raideur à l'effort tranchant qui est déterminante pour estimer l'efficacité d'une âme de panneau sandwich. Ainsi, ce travail nous a amené à reconsidérer en profondeur les méthodes pour calculer le comportement à l'effort tranchant des plaques en général. Il nous a conduit à proposer une nouvelle théorie des plaques ainsi qu'une méthode d'homogénéisation associée dans le cas périodique. Cette théorie peut être considérée comme l'extension de la théorie bien connue de Reissner-Mindlin au cas des plaques hétérogènes. Elle ne peut cependant pas être réduite au mod èle de Reissner-Mindlin dans le cas général. Dans le cas particulier des panneaux sandwichs incluant le module à chevrons, l'application de cette méthode d'homogénéisation permet de mettre en évidence un phénomène de distorsion des peaux qui affecte de façon notable la raideur à l'effort tranchant de ces panneaux

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.
Science, Faculty of
Mathematics, Department of
Graduate

Experimental observations indicate the presence of attached, gravity induced, horizontal buoyant currents above large area fires. Their driving mechanism is indirect and resembles the one observed above heated horizontal plates. Classic plume modelling is satisfactory for providing information for the flow far from the source. In dealing with large areas and directing attention to the flow close to the source, the classic plume theory should fail because the radial pressure gradient that is responsible for the driving of the flow is squeezed in the long and thin classic plume assumption. For this we propose a new plume structure for the description of the buoyant flow above a circular region of large radius L as “The flow field must be divided into three regions. A region where the flow is predominantly horizontal and attached to the surface, a transition region from horizontal to vertical where separation of the attached current takes place, and a region where vertical flow is established and classic plume theory can be applied”. A model for the description of the gross properties of the horizontal currents is developed under the term “horizontal plume”. The modified Richardson number for the horizontal plume a, being analogous to the radius of the large area, is studied asymptotically in the limit a → ∞ and second order uniformly valid semi-analytical solutions are obtained. The hot plate experiment was set up in order to test the model and facilitate its improvement. A chapter is dedicated to the data analysis coming from thermocouple readings and visualisation of the flow using particle image velocimetry.In the remainder of this thesis two classic problems of laminar natural convection are revisited. That of the first order laminar boundary layer above an isothermal circular plate of radius a and the first order laminar boundary layer above the semi- infinite plate inclined to horizontal. In both cases allowances to variable property effects were made through the introduction of a nondimensional parameter λT, with its value set to zero implying the assumption of the Boussinesq approximation. For the circular plate, fourth order series solutions were obtained valid at the edge of the plate where the effects of λT and Prandtl number Pr are studied. Furthermore a finite difference scheme for the numerical solution of the nonsimilar partial integro- differential equation was developed using the Keller Box method and compared with results obtained from the commercial finite element software COMSOL Multiphysics 3.5a. For the semi-infinite plate, fourth order series approximations valid at the edge of the plate were obtained, while an extensive analysis for the effect of λT, Pr and inclination parameter σ was performed on the flow. Positions of the separation points when the inclination is negative (σ < 0) as a function of Pr and λT were recovered.

This thesis is about using structural-dynamics based methods to address the existing challenges in the field of Structural Health Monitoring (SHM). Particularly, new structural-dynamics based methods are presented, to model areas of damage, to do damage diagnosis and to estimate and predict the sensitivity of structural vibration properties like natural frequencies to the presence of damage. Towards these objectives, a general analytical procedure, which yields nth-order expressions governing mode shapes and natural frequencies and for damaged elastic structures such as rods, beams, plates and shells of any shape is presented. Features of the procedure include the following: 1. Rather than modeling the damage as a fictitious elastic element or localized or global change in constitutive properties, it is modeled in a mathematically rigorous manner as a geometric discontinuity. 2. The inertia effect (kinetic energy), which, unlike the stiffness effect (strain energy), of the damage has been neglected by researchers, is included in it. 3. The framework is generic and is applicable to wide variety of engineering structures of different shapes with arbitrary boundary conditions which constitute self adjoint systems and also to a wide variety of damage profiles and even multiple areas of damage. To illustrate the ability of the procedure to effectively model the damage, it is applied to beams using Euler-Bernoulli and Timoshenko theories and to plates using Kirchhoff's theory, supported on different types of boundary conditions. Analytical results are compared with experiments using piezoelectric actuators and non-contact Laser-Doppler Vibrometer sensors. Next, the step of damage diagnosis is approached. Damage diagnosis is done using two methodologies. One, the modes and natural frequencies that are determined are used to formulate analytical expressions for a strain energy based damage index. Two, a new damage detection parameter are identified. Assuming the damaged structure to be a linear system, the response is expressed as the summation of the responses of the corresponding undamaged structure and the response (negative response) of the damage alone. If the second part of the response is isolated, it forms what can be regarded as the damage signature. The damage signature gives a clear indication of the damage. In this thesis, the existence of the damage signature is investigated when the damaged structure is excited at one of its natural frequencies and therefore it is called ``partial mode contribution". The second damage detection method is based on this new physical parameter as determined using the partial mode contribution. The physical reasoning is verified analytically, thereupon it is verified using finite element models and experiments. The limits of damage size that can be determined using the method are also investigated. There is no requirement of having a baseline data with this damage detection method. Since the partial mode contribution is a local parameter, it is thus very sensitive to the presence of damage. The parameter is also shown to be not affected by noise in the detection ambience.

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
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

This master's thesis aims to make the BRIO Labyrinth Game autonomous and the main focus is on the development of a path following controller. A test-bench system is built using a modern edition of the classic game with the addition of a Raspberry Pi, a camera and two servos. A mathematical model of the ball and plate system is derived to be used in model based controllers. A method of using path projection on a cubic spline interpolated path to derive the reference states is explained. After that, three path following controllers are presented, a modified LQR, a Gain Scheduled LQR and a Gain Scheduled LQR with obstacle avoidance. The performances of these controllers are compared on an easy and a hard labyrinth level, both with respect to the ability of following the reference path and with respect to success rate of controlling the ball from start to finish without falling into any hole. All three controllers achieved a success rate over 90 % on the easy level. On the hard level the Gain Scheduled LQR achieved the highest success rate, 78.7 %, while the modified LQR achieved the lowest deviation from the reference path. The Gain Scheduled LQR with obstacle avoidance performed the worst in both regards. Overall, the results are promising and some insights gained when designing the controllers can possibly be useful for development of controllers in other applications as well.

The manufacture, laminate design, and modeling of a part with complex geometry are explored. The ultimate goal of the research is to produce a model that accurately predicts part stiffness. This is validated with experimental results of composite parts, which refine material properties for use in a final prototype part model. The secondary goal of this project is to explore manufacturing methods for improved manufacturability of the complex part. The manufacturing portion of the thesis and feedback into material model has incorporated a senior project team to perform research on manufacturing and create composite part to be used for experimental testing. The senior project was designed, led, and managed by the author with support from the committee chair. Finite element modeling was refined using data from coupon 3-point bend testing to improve estimates on material properties. These properties were fed into a prototype part model which predicted deflection of composite parts with different layups and materials. The results of the model were compared to experimental results from prototype part testing and 3rd party analysis. The results showed that an accurate mid-plane shell element model could be used to accurately predict deflection for 2 of 3 experimental parts. There are recommendations in the thesis to further validate the models and experimental testing.

The topic of this thesis is the numerical simulation of transversely isotropic 3D structures and laminates by means of the adaptive finite element method. To achieve this goal, the theoretical background of elastic deformation problems, transverse isotropy, plate theory, and the classical laminate theory is recapitulated. The classical laminate theory implies a combination of the membrane problem and the plate problem with additional coupling terms. The focus of this work is the adjustment of two integral parts of the adaptive FE algorithm according to the classical laminate theory. One of these parts is the solution of the FE system; a good preconditioner is needed in order to use the conjugate gradient method efficiently. It is shown via a spectral equivalence bound that the combination of existing preconditioners for the membrane and plate problems poses a capable preconditioner for the combined laminate problem. The other part is the error estimation process; the error estimator determines where the current mesh has to be refined for the next step. Existing results on residual error estimators for the elasticity problem, the biharmonic problem, and the plate problem are combined and extended to obtain a posteriori local residual error indicators for the classical laminate theory problem. The effectiveness of both results is demonstrated by numerical examples.

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.