This work discusses the deformations of various unsymmetric composite panels due to thermal and mechanical loads. Chapter 2 focuses on the warpage of large unsymmetric curved composite panels due manufacturing anomalies. These panels are subjected to a temperature change of -280°F to simulate the cooling from the autoclave cure temperature. Sixteen layer quasi-isotropic, axial-stiff, and circumferentially-stiff laminates are considered. These panels are intended to be symmetric laminates, but are slightly unsymmetric due to the manufacturing anomalies. Rayleigh-Ritz and finite-element models are developed to predict the deformations. Initially, to serve as a basis for comparison, warpage effects due to orthotropic thermal expansion properties in perfect panels are investigated and are found to produce deformations not captured in two-dimensional theories. This is followed by the investigation of the effects of ply misalignments. Ply misalignments of 5° are incorporated into the laminate, one layer at a time, to produce unsymmetric laminates. It is found that ply misalignments produce warpages much larger than those induced by orthotropic thermal expansion properties. Next, unsymmetric laminates resulting from ply thickness variations are investigated. Layers 10% thicker than nominal are incorporated into the laminate, one layer at a time, while the remaining layers are of uniform thickness. Due to the change in fiber volume fraction of the thicker layers, corresponding material properties are modified to reflect this change. The results show that ply thickness variations cause warpages of about 25-50% of those induced by ply misalignments. Finally, warpage of panels due to nonuniform cooling due to inplane thermal gradients during cure is investigated. A thermal gradient of 0.1°F/in. is used to construct six inplane distributions. It is found that the warpages induced by thermal gradients are very small. The warpages are negligible with respect to those induced by ply thickness variations or ply misalignments. Deformations induced by thermal gradients depend primarily on the magnitude of the thermal gradient, but not on the pattern of distribution. Overall, ply misalignments cause the most warpage, followed by ply thickness variations. Important variables for these imperfections are, the through-thickness location of the imperfections, the orientation of the layer containing the imperfections, and the lamination sequence. All cases show that geometric nonlinearities are important to accurately predict the deformations induced by these imperfections. Chapter 3 discusses the deformations of composite plates that are intentionally fabricated to be unsymmetric. Such plates, if flat, might be considered in applications where bending-stretching coupling effects can be used to advantage. It is assumed the laminates are cured at an elevated temperature and then cooled 280°F. Significant deformations result because of the high level of asymmetry in the laminate construction. Accordingly, geometric nonlinearities are included in the models. Four cross-ply laminates and three angle-ply laminates are considered. Four-term and 14-term Rayleigh-Ritz models are developed, together with finite-element models to model the deformations. Actual specimens were constructed and the deformations measured to compare with predictions. The results show that agreement between predictions and the experimental results are good. The 14-term Rayleigh-Ritz model is found to be the most useful due to its ability to find multiple solutions, its physical basis, and computational efficiency. Chapter 4 discusses the deformations of initially flat aluminum, symmetric, and unsymmetric composite plates due to axial endshortening under various boundary conditions, the aluminum and symmetric plates serving as a baseline. Seven plates are considered, each with three boundary condition combinations, namely, clamped ends and sides (CL-CL), clamped ends with simply-supported sides (CL-SS), and simply-supported ends and sides (SS-SS). Generally, the boundary conditions play a key role in the deformation characteristics of the plates. The aluminum and symmetric cross-ply plates have no out-of-plane deformations until classic buckling, or primary instability, then each exhibits two stable solutions. Each also exhibits secondary instability that results in two stable solutions. The symmetric laminates show less of a dependence on the boundary conditions compared to the unsymmetric laminates. Unsymmetric laminates show a mixture of characteristics. Some cases exhibit primary instability, other cases do not. Some cases exhibit secondary instability, while some case do not. The unsymmetric cross-ply laminates have only one stable solution after secondary buckling, while most other laminates and boundary condition combinations have two stable solutions. It is interesting to note that for the unbalanced unsymmetric [302/90/0]2T laminate, the boundary conditions controlled the sign of the out-of-plane deflection from the onset of axial endshortening. Generally speaking, the CL-CL cases carry the most load, followed by the CL-SS, and then the SS-SS cases. Like all the problems discussed in Chapter 2 and 3, geometric nonlinearities are found to be important for this case as well. / Ph. D.

Composite laminates are increasingly being used as primary load bearing members in<br />structures.  However, because of the directional dependence of the properties of<br />composite materials, additional failure modes appear that are absent in<br />homogeneous, isotropic materials.  Therefore, a stress analysis of a composite<br />laminate is not complete without an accurate representation of the transverse<br />(out-of-plane) stresses.<br /><br />Stress recovery is a common method to estimate the transverse stresses from a<br />plate or shell analysis.  This dissertation extends stress recovery to problems<br />in which geometric nonlinearities, in the sense of von K\\\'rm\\\'{a}n,  are<br />important.  The current work presents a less complex formulation for the stress<br />recovery procedure for plate geometries, compared with other implementations,<br />and results in a post-processing procedure which can be applied to data from<br />any plate analyses; analytical or numerical methods, resulting in continuous or<br />discretized data.<br /><br />Recovered transverse stress results are presented for a variety of<br />geometrically nonlinear example problems: a semi-infinite plate subjected to<br />quasi-static transverse and shear loading, and a finite plate subjected to both<br />quasi-static and dynamic transverse loading.  For all cases, the corresponding<br />results from a fully three-dimensional stress analysis are shown alongside the<br />distributions from the stress recovery procedure.  Good agreement is observed<br />between the stresses obtained from each method for the cases considered.<br />Discussion is included regarding the applicability and accuracy of the<br />technique to varying plate geometries and varying degrees of nonlinearity, as<br />well as the viability of the procedure in replacing a three-dimensional<br />analysis in regard to the time required to obtain a solution.<br /><br />The proposed geometrically nonlinear stress recovery procedure results in<br />estimations for transverse stresses which show good correlation to the<br />three-dimensional finite element solutions.  The procedure is accurate for<br />quasi-static and dynamic loading cases and proves to be a viable replacement<br />for more computationally expensive analyses. / Ph. D.

O objetivo principal deste trabalho é o desenvolvimento de uma formulação e sua implementação computacional para se analisar, via Método dos Elementos Finitos (MEF),o comportamento dinâmico não linear geométrico de sólidos bidimensionais. Trata-se o comportamento geometricamente não linear através de uma formulação posicional classificada como lagrangeana total com cinemática exata. No estudo do comportamento dinâmico utiliza-se um algoritmo de integração temporal baseado na família de integradores temporais de Newmark. Para a consideração do impacto adota-se uma técnica que utiliza como integrador temporal o algoritmo de Newmark, modificado de forma a garantir sua estabilização, e limita-se a posição de cada nó da estrutura que por ventura sofra impacto. O código computacional desenvolvido é validado através de exemplos tradicionais da literatura científica. Analisam-se exemplos com comportamento apenas não linear geométrico e não linear geométrico dinâmico com ou sem impacto / The main goal of this work is the development of a formulation and its computational implementation, based on the finite element method (FEM), to analyze the dynamic geometrically nonlinear behavior of bidimensional solids. The geometrically nonlinear behavior is treated with a positional formulation classified as total Lagrangean with exact kinematics. In the study of the dynamic behavior, a time integration algorithm based on the family of time integrators of Newmark is applied. In order to consider the impact, a technique based on the time integrator of Newmark, modified to assure its stabilization, is used. This technique limits the position of each node that suffers impact. The developed computational code is validated through benchmarks of scientific literature. Examples with static geometrically nonlinear and dynamic geometrically nonlinear behavior, with or without impact, are analyzed

abstract: Recent studies of the occurrence of post-flutter limit cycle oscillations (LCO) of the F-16 have provided good support to the long-standing hypothesis that this phenomenon involves a nonlinear structural damping. A potential mechanism for the appearance of nonlinearity in the damping are the nonlinear geometric effects that arise when the deformations become large enough to exceed the linear regime. In this light, the focus of this investigation is first on extending nonlinear reduced order modeling (ROM) methods to include viscoelasticity which is introduced here through a linear Kelvin-Voigt model in the undeformed configuration. Proceeding with a Galerkin approach, the ROM governing equations of motion are obtained and are found to be of a generalized van der Pol-Duffing form with parameters depending on the structure and the chosen basis functions. An identification approach of the nonlinear damping parameters is next proposed which is applicable to structures modeled within commercial finite element software. The effects of this nonlinear damping mechanism on the post-flutter response is next analyzed on the Goland wing through time-marching of the aeroelastic equations comprising a rational fraction approximation of the linear aerodynamic forces. It is indeed found that the nonlinearity in the damping can stabilize the unstable aerodynamics and lead to finite amplitude limit cycle oscillations even when the stiffness related nonlinear geometric effects are neglected. The incorporation of these latter effects in the model is found to further decrease the amplitude of LCO even though the dominant bending motions do not seem to stiffen as the level of displacements is increased in static analyses. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2015

O objetivo principal deste trabalho é o desenvolvimento de uma formulação e sua implementação computacional para se analisar, via Método dos Elementos Finitos (MEF),o comportamento dinâmico não linear geométrico de sólidos bidimensionais. Trata-se o comportamento geometricamente não linear através de uma formulação posicional classificada como lagrangeana total com cinemática exata. No estudo do comportamento dinâmico utiliza-se um algoritmo de integração temporal baseado na família de integradores temporais de Newmark. Para a consideração do impacto adota-se uma técnica que utiliza como integrador temporal o algoritmo de Newmark, modificado de forma a garantir sua estabilização, e limita-se a posição de cada nó da estrutura que por ventura sofra impacto. O código computacional desenvolvido é validado através de exemplos tradicionais da literatura científica. Analisam-se exemplos com comportamento apenas não linear geométrico e não linear geométrico dinâmico com ou sem impacto / The main goal of this work is the development of a formulation and its computational implementation, based on the finite element method (FEM), to analyze the dynamic geometrically nonlinear behavior of bidimensional solids. The geometrically nonlinear behavior is treated with a positional formulation classified as total Lagrangean with exact kinematics. In the study of the dynamic behavior, a time integration algorithm based on the family of time integrators of Newmark is applied. In order to consider the impact, a technique based on the time integrator of Newmark, modified to assure its stabilization, is used. This technique limits the position of each node that suffers impact. The developed computational code is validated through benchmarks of scientific literature. Examples with static geometrically nonlinear and dynamic geometrically nonlinear behavior, with or without impact, are analyzed

The micro-electromechanical systems (MEMS) industry has grown incredibly fast over the past few years, due to the irresistible character and properties of MEMS. MEMS devices have been widely used in various fields such as aerospace, microelectronics, and the automobile industry. Increasing prominence is given to the development and research of MEMS; this is largely driven by the market requirements. Multi-physics coupled fields are often present in MEMS. This makes the modelling and analysis o such devices difficult and sometimes costly. The coupling between electrostatic and mechanical fields in MEMS is one of the most common and fundamental phenomena in MEMS; it is this configuration that is studied in this thesis. The following issues are addressed: 1. Due to the complexity in the structural geometry, as well as the difficulty to analyze the behaviour in the presence of coupled fields, simple analytical solutions are normally not available for MEMS. The finite element method (FEM) is therefore used to model electrostaticmechanical coupled MEMS. In this thesis, this avenue is followed. 2. In order to capture the configuration of the system accurately, with relatively little computational effort, a geometric non-linear mixed assumed stress element is developed and used in the FE analyses. It is shown that the developed geometrically non-linear mixed assumed stress element can produce an accuracy level comparable to that of the Q8 element, while the number of the degrees of freedom is that of the Q4 element. 3. Selected algorithms for solving highly non-linear coupled systems are evaluated. It is concluded that the simple, accurate and quadratic convergent Newton-Raphson algorithm remains best. To reduce the single most frustrating disadvantage of the Newton method, namely the computational cost of constructing the gradients, analytical gradients are evaluated and implemented. It is shown the CPU time is significantly reduced when the analytical gradients are used. 4. Finally, a practical engineering MEMS problem is studied. The developed geometric nonlinear mixed element is used to model the structural part of a fixed-fixed beam that experiences large axial stress due to an applied electrostatic force. The Newton method with analytical gradients is used to solve this geometrically nonlinear coupled MEMS problem. / Dissertation (MEng (Mechanical))--University of Pretoria, 2007. / Mechanical and Aeronautical Engineering / unrestricted

This study focuses on the response of flat unsymmetric laminates to an inplane compressive loading that for symmetric laminates are of sufficient magnitude to cause bifurcation buckling, postbuckling, and secondary buckling behavior. In particular, the purpose of this study is to investigate whether or not the concept of bifurcation buckling is applicable to unsymmetric laminates. Past work by other researchers has suggested that such a concept is applicable for certain boundary conditions. The study also has as an objective the determination of the response of flat unsymmetric laminates if bifurcation buckling does not occur. The finite-element program ABAQUS is used to obtain results, and a portion of the study is devoted to becoming familiar with the way ABAQUS handles such highly geometrically nonlinear problems, particularly for composite materials and particularly when instabilities and dynamic behavior are involved. Familiarity with the problem, in general, and with the use of ABAQUS, in particular, is partially gained by considering semi-infinite unsymmetrically laminated cross- and angle-ply plates, a one-dimensional problem that can be solve in closed form and with ABAQUS by making the appropriate approximations for the infinite geometry. In this portion of the study it is found that semi-infinite cross-ply laminates with clamped boundary conditions and semi-infinite angle-ply plates with simple-support boundary conditions remain flat under a compressive load until the load magnitude reaches a certain level, at which time the out-of-plane deflection become indeterminate, essentially an eigenvalue problem as encountered with classic bifurcation buckling analyses. Obviously, a linear analysis of such problems would not reveal this behavior and, in fact, there are other revealed significant differences between the predictions of linear and nonlinear analyses. Transversely-loaded and inplane-loaded finite isotropic plates are studied by way of semi-closed form Rayleigh-Ritz-based solutions and ABAQUS in a step to approaching the problem with unsymmetric laminates. A method to investigate the unloading behavior of postbuckled finite isotropic plates is developed that reveal multiple plate configurations in the postbuckled region of the response, and this method is then extended to the study of finite inplane-loaded unsymmetric laminates. To that end, two specific laminates, a symmetric and an unsymmetric cross-ply laminates, and a variety of boundary conditions are used to study the response of inplane-loaded unsymmetric laminates. The symmetric laminate is included to provide a familiar baseline case and a means of comparison. Plates with all four edges clamped and a variety of inplane boundary conditions are studied. Of course the symmetric cross-ply laminate exhibits bifurcation behavior, and when the tangential displacement on the loaded edges and the normal displacement on the unloaded edges are restrained, secondary buckling behavior occurs. For the unsymmetric cross-ply laminate, bifurcation buckling behavior does not occur unless the tangential displacement on the loaded edges and the normal displacement on the unloaded edges are restrained, or the tangential displacement on the loaded edges and the normal displacement on the unloaded edges are free. If either of these conditions are not satisfied, the unsymmetric cross-ply laminate exhibits what could be termed 'near-bifurcation' behavior. In all cases rather complex behavior occurs for high levels of inplane load, including asymmetric postbuckling and secondary buckling behavior. For clamped loaded edges and simply-supported unloaded edges, bifurcation buckling behavior does not occur unless the tangential displacement on the loaded edges and the normal displacement on the unloaded edges are restrained. For this case, rather unusual asymmetric bifurcation and associated limit point behavior occur, as well as secondary buckling. This is a very interesting boundary condition case and is studied further for other unsymmetric cross-ply laminates, including the use of a Rayleigh-Ritz-based solution in attempt to quantify the problem parameters responsible for the asymmetric response. The overall results of the study have led to an increased understanding of the role of laminate asymmetry and boundary conditions on the potential for bifurcation behavior, on the response of the laminate for loads beyond that level. / Ph. D.

This thesis presents a geometrically exact curved beam theory, with the assumption that the cross-section remains rigid, and its finite element formulation/implementation. The theory provides a theoretical view and an exact and efficient means to handle a large range of nonlinear beam problems. A geometrically exact curved/twisted beam theory, which assumes that the beam cross-section remains rigid, is re-examined and extended using orthonormal reference frames starting from a 3-D beam theory. The relevant engineering strain measures at any material point on the current beam cross-section with an initial curvature correction term, which are conjugate to the first Piola-Kirchhoff stresses, are obtained through the deformation gradient tensor of the current beam configuration relative to the initially curved beam configuration. The Green strains and Eulerian strains are explicitly represented in terms of the engineering strain measures while other stresses, such as the Cauchy stresses and second Piola-Kirchhoff stresses, are explicitly represented in terms of the first Piola-Kirchhoff stresses and engineering strains. The stress resultant and couple are defined in the classical sense and the reduced strains are obtained from the three-dimensional beam model, which are the same as obtained from the reduced differential equations of motion. The reduced differential equations of motion are also re-examined for the initially curved/twisted beams. The corresponding equations of motion include additional inertia terms as compared to previous studies. The linear and linearized nonlinear constitutive relations with couplings are considered for the engineering strain and stress conjugate pair at the three-dimensional beam level. The cross-section elasticity constants corresponding to the reduced constitutive relations are obtained with the initial curvature correction term. For the finite element formulation and implementation of the curved beam theory, some basic concepts associated with finite rotations and their parametrizations are first summarized. In terms of a generalized vector-like parametrization of finite rotations under spatial descriptions (i.e., in spatial forms), a unified formulation is given for the virtual work equations that leads to the load residual and tangent stiffness operators. With a proper explanation, the case of the non-vectorial parametrization can be recovered if the incremental rotation is parametrized using the incremental rotation vector. As an example for static problems, taking advantage of the simplicity in formulation and clear classical meanings of rotations and moments, the non-vectorial parametrization is applied to implement a four-noded 3-D curved beam element, in which the compound rotation is represented by the unit quaternion and the incremental rotation is parametrized using the incremental rotation vector. Conventional Lagrangian interpolation functions are adopted to approximate both the reference curve and incremental rotation of the deformed beam. Reduced integration is used to overcome locking problems. The finite element equations are developed for static structural analyses, including deformations, stress resultants/couples, and linearized/nonlinear bifurcation buckling, as well as post-buckling analyses of arches subjected to conservative and non-conservative loads. Several examples are used to test the formulation and the Fortran implementation of the element. / Master of Science

The dissertation “Formation of wrinkles on a coated substate“ treats the finite element simulations of controlled wrinkle formation experiments conducted at the Leibniz Institute for Polymer Research. The systems used for the experiments consist of a soft polydimethylsiloxane (PDMS) layer with a thin, stiff layer on top. The wrinkling process is triggered by a stress mismatch between the bulk and the thin layer. To create the stress mismatch, the bulk material is first uni-axially stretched and then the thin layer is created by a low-pressure plasma treatment of the stretched bulk in a vacuum chamber. Under subsequent relaxation, wrinkles form. Their wavelength depends on the choice of the process gas and the duration of the treatment. The use of thin silicon masks placed directly on the PDMS allows to sharply restrict the plasma-exposed area. Sequential exposures of the same sample to multiple treatment processes with and without a mask allow to locally modify the layer thickness and stiffness. With this, we can locally control the wavelength of the resulting wrinkles and trigger the formation of branches and line defects at the boundary between areas of different wavelengths. The dissertation first covers the mathematical model for the coated substrate, a combination of a hyperelastic material model from three-dimensional elasticity for the bulk (an almost incompressible Mooney–Rivlin material model) and a Cosserat shell model for the film on top. A nonlinear and nonconvex minimization problem is deduced and transferred to a suitable finite element space. Existence of minimizers is proven in the continuous and the discrete case before the discrete problem is solved numerically. The numerical simulations show a good agreement with corresponding physical experiments.

Presented is an overview of a semi-analytical solution which was developed to study the response of internally pressurized elliptical composite cylinders with clamped boundaries. Using a geometrically linear analysis and the solution scheme, the response of a quasi-isotropic elliptical cylinder is compared with the response of a quasi-isotropic circular cylinder in order to study the effects of elliptical geometry. The distinguishing features of the response of an elliptical cylinder are the inward normal displacement of the cross section at the ends of the major diameter that occur despite the outward force of the internal pressure, the presence of circumferential displacements, and the presence of inplane shear strains. These effects lead to spatial variations, including sign reversals, of a number of displacement, strain, and curvature responses. The responses of a quasi-isotropic elliptical cylinder evaluated using a geometrically linear analysis are then compared to the responses evaluated using a geometrically nonlinear analysis. It is shown that geometric nonlinearities tend to flatten certain responses at the ends of the minor diameter, and reduce the magnitude of certain responses in the boundary region. To study the influence of material orthotropy, the responses of axially-stiff and circumferentially-stiff elliptical cylinders evaluated using geometrically nonlinear analyses are examined. It is shown that in some instances material orthotropy can be used to mitigate the influence of the elliptical geometry and make particular responses look like those of a circular cylinder. An evaluation of failure using the maximum stress and Hashin failure criteria and geometrically linear and nonlinear analyses is presented for elliptical cylinders. These failure criteria involve interlaminar shear stresses which are computed by integrating the equilibrium equations of elasticity through the thickness of the cylinder wall. The failure criteria are used to assess the mode of failure (e.g., tensile or compressive fiber or matrix modes), the location of failure, and the pressure at failure. Both criteria predict first failure to occur at the clamped boundaries because of matrix cracking. The predicted failure pressures and circumferential locations are very similar for the two criteria, and the nonlinear analyses predict slightly higher pressures at somewhat different circumferential locations. First fiber failure is also considered. For this failure the two criteria predict similar failure scenarios for the linear analyses, but they differ in their predictions for the nonlinear analyses. Specifically, using the maximum stress criterion, the circumferentially-stiff elliptical cylinder is predicted to fail due to fiber compression, but the Hashin criterion predicts failure to be due to fiber tension, and at a different circumferential location. Also, first fiber failure pressures are at least a factor of two greater than the first matrix failure pressure. / Master of Science