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.

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)

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.

<p>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.</p><p>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.</p><p>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. </p>

Several problems must be solved in the construction, design, and operation of a nuclear fusion reactor. One of the chief problems in the manufacture of high-powered copper/polymer composite magnets is the difficulty to precisely control the fiber volume fraction. In this thesis, the effect of variations in fiber volume fraction on thermal stresses in copper/cyanate ester composite cylinders is investigated. The cylinder is a composite that uses copper wires that run longitudinally in a cyanate ester resin specifically developed by Composite Technology Development, Inc. This composite cylinder design is commonly used in magnets for nuclear fusion reactors. The application of this research is for magnets that use cylindrical coil geometry such as the Mega Amp Spherical Tokamak (MAST) in the UK. However, most stellarator magnet designs use complex geometries including the National Compact Stellarator Experiment (NCSX), and the Quasi-Poloidal Stellarator (QPS). Even though the actual stresses calculated for the cylindrical geometry may not be directly applicable to these projects, the relationship between fiber volume fraction and stresses will be useful for any geometry. The effect of fiber volume fraction on stresses produced by mechanical, thermal and magnetic loads on cylindrical magnet coils is studied using micromechanics with laminate plate theory (LPT) and finite element analysis (FEA). Based on the findings of this research, variations in volume fraction do significantly affect the stress experienced by the composite cylinder. Over a range of volume fractions from 0.3 to 0.5, the LPT results demonstrate that thermally induced stresses vary approximately 30% while stresses due to pressure vary negligibly. The FEA shows that magnetic stresses vary much less at around only 5%. FEA results seem to confirm the LPT model. It was also concluded that the stress in the insulation layers due to all types of loadings is significant and must be considered when using this system in fusion applications.

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.

Several problems must be solved in the construction, design, and operation of a nuclear fusion reactor. One of the chief problems in the manufacture of high-powered copper/polymer composite magnets is the difficulty to precisely control the fiber volume fraction. In this thesis, the effect of variations in fiber volume fraction on thermal stresses in copper/cyanate ester composite cylinders is investigated. The cylinder is a composite that uses copper wires that run longitudinally in a cyanate ester resin specifically developed by Composite Technology Development, Inc. This composite cylinder design is commonly used in magnets for nuclear fusion reactors. The application of this research is for magnets that use cylindrical coil geometry such as the Mega Amp Spherical Tokamak (MAST) in the UK. However, most stellarator magnet designs use complex geometries including the National Compact Stellarator Experiment (NCSX), and the Quasi-Poloidal Stellarator (QPS). Even though the actual stresses calculated for the cylindrical geometry may not be directly applicable to these projects, the relationship between fiber volume fraction and stresses will be useful for any geometry. The effect of fiber volume fraction on stresses produced by mechanical, thermal and magnetic loads on cylindrical magnet coils is studied using micromechanics with laminate plate theory (LPT) and finite element analysis (FEA).Based on the findings of this research, variations in volume fraction do significantly affect the stress experienced by the composite cylinder. Over a range of volume fractions from 0.3 to 0.5, the LPT results demonstrate that thermally induced stresses vary approximately 30% while stresses due to pressure vary negligibly. The FEA shows that magnetic stresses vary much less at around only 5%. FEA results seem to confirm the LPT model. It was also concluded that the stress in the insulation layers due to all types of loadings is significant and must be considered when using this system in fusion applications.

Aluminum hexagonal honeycombs loaded in their tubular direction have extremely good mechanical properties, including high stiffness to weight and energy absorption capacities. The corresponding load-displacement curve exhibits a long plateau accompanied by small fluctuations. These fluctuations are due to the propagation of a folding front through the studied sample, which is due to the creation of folds. This plateau load makes honeycombs the perfect candidates for use as energy-dissipative devices such as bumpers. Previous studies have largely focused on the study of the plateau load with less attention given to the length of the folds. However, it will be seen that this parameter is crucial for both the complete understanding of the mechanics of the folding and the derivation of the plateau load. We present first an introduction to the thematic of honeycomb. Then, the first model focuses precisely on the fold length. Two hypotheses are considered, a correlation between elastic buckling and folding first and a local propagation of the existing fold secondly. The second hypothesis is found to be correct, and the results are good for thin foils. For thick foils, a geometric limitation occurs, which makes the results less precise. Then, we are able to use the previous kinematics for the folding and derive a new set of formulas for the plateau load. The results are compared with experimental results and past formulas, and are found to be good, especially for thin foils, where our results for the fold length are more precise.

No description available.

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