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 / Plates and shells are defined as structures which have thickness much smaller as compared to their length and width. These structures are extensively used in many fields of engineering such as, designing ship hulls, airplane wings and fuselage, bodies of automobile, etc. Depending on the complexity of a plate/shell deformation problem, deriving analytical solutions is not always viable and one relies on computational methods to obtain numerical solutions of the problem. However, obtaining 3-dimensional (3D) numerical solutions of deforming plates/shells often require high computational effort. To avoid this, plate/shell theories are used for modeling these structures, which, based on certain assumptions, reduce the 3D problem into an equivalent 2-dimensional (2D) problem. However, quality of the solution obtained from such a theory depends on how suitable the assumptions are for the specific problem being studied. In this work, one such plate theory called as the Third Order Shear and Normal Deformable Theory (TSNDT) is used to model the mechanics of deforming rectangular plates under different boundary conditions (constraint conditions for the boundaries of the plate) and loading conditions (conditions of applied loads on the plate). We develop the TSNDT mathematical model of plate deformations and solve it using a computational technique called as the Finite Element Method (FEM) to analyze three different problems of mechanics of rectangular plates. These problems are briefly described below. vi In the first problem, we study vibrations of rectangular plates under time dependent (dynamic) loads. When a dynamic load acts on a plate, due to the effects of inertia, the plate continues to vibrate after the removal of the load. This is analogous to ringing of a bell long after the strike of the hammer on the bell. In this study we show that such vibrations of a rectangular plate can be varied by changing time dependencies of the applied load. We observe that under certain particular loading conditions, vibrations of the plate becomes miniscule after the load removal. We call this phenomenon as Vibration Attenuation and investigate this computationally in different problems of plate deformation using FEM solutions. In the second problem, we computationally investigate the effects of presence of internal fixed points (points within the volume of the plate restricted of motion) on the vibration characteristics of rectangular plate using TSNDT FEM solutions. We observe that when one or more points at locations inside a rectangular plate are fixed, vibration behavior of the plate significantly changes and the deformations are localized in certain regions of the plate. This phenomenon is called as Mode Localization. We study mode localization in rectangular plates under different boundary and loading conditions and analyze the effects of plate dimensions, locations of the internal fixed points and dynamic load characteristics on mode localization. In the third problem, we investigate the effects of introduction of nonlinearities into the TSNDT mathematical model of plate deformations. Simple models in mechanics consider materials to be linearly elastic, which means that the deformations of a body are proportional to the applied loads in a linear relation. However, most materials in nature undergoing large deformations (human tissues, rubbers, and polymers, for example) do not behave in this fashion and their deformation depends nonlinearly to applied loads. To investigate the effects of such nonlinearities, we study the behavior of nonlinearly elastic plates under different boundary and loading conditions and delineate the differences in the results of linearly elastic and nonlinearly elastic plates using the TSNDT FEM solutions. Findings of this study establishes that linear models overestimate the plate deformation under given boundary and loading conditions as compared to nonlinear models. This understanding may help in developing better design criteria for plates undergoing large deformations.

This Thesis comprises of two parts containing similar studies of Nonlinear Localization induced Base Isolation of structural systems. The present method of base isolation,like other nonlinear vibration isolation methods, enjoys certain merits like capability of absorbing broad band vibrations, attenuating heavy shocks etc. The research in this thesis is an extension of this base isolation strategy first proposed by Vakakis and co-author. The strategy involves augmenting an appendage referred to as the secondary system with the main structural unit or the primary system, which we want to isolate from disturbances at the base. The primary system is coupled to the secondary system through a stiffness element. Both the primary and secondary systems have nonlinear dynamic behavior. It is seen that for certain choice of values of the coupling element, steady state vibration of very small magnitude is induced in the primary system. This result was established by considering a general discrete nonlinear system with viscous damping. Now it is a well known fact that viscous damping, though being widely used in literature as well as in practice doesn't turn out to be accurate enough to capture structural damping behaviors. Moreover, the actual damping mechanism if governed by some nonlinear function of the system variables, may influence the physics governing the nonlinear localization phenomenon in a manner rendering the present method not suitable for structural systems at the very outset. So in the present study we focus our attention in establishing the robustness and hence utility of the method by considering technically more defensible models of structural damping. These models efficiently capture certain complex phenomena which structures are known to exhibit. The occurrence of localization induced vibration isolation in structural systems in the presence of these damping models is taken as a proof of the efficacy of the method and its applicability to a wide range of situations. The present study establishes existence of localization through relevant analytical and numerical exercises. In the first part of the thesis we take up the study of nonlinear localization induced base isolation of a three degrees of freedom system having cubic nonlinearities under sinusoidal base excitation. The damping forces in the system are hysteretic in nature. In the present setting this is captured by Bouc-Wen model of hysteresis. Bouc-Wen model is one of the most widely used phenomenological model of hysteresis to have a ready-to-use mathematical description of hysteretic patterns appearing in structural engineering systems. The nature of responses of the different degrees of freedom as excitation frequency varies is a better way of analyzing the performance of the vibration isolation system. We adopt this line of approach for the present study. Normally Harmonic Balance Method (HBM) serves this purpose very well but in the present case as the hysteretic variable is not explicitly related to the system variables, HBM cannot be straightway implemented. Moreover, the hysteretic variable is related to other state variables through a relation which contains non-smooth terms. As a result, Incremental Harmonic Balance (IHB) method is used to obtain amplitude frequency relationship of the system response. The stability analysis of the solution branches is done by using Floquet Theory. Direct numerical simulation is then made use of to support our results that are obtained from this approximate numeric-analytic estimate of the amplitudefrequency relationships of the system, which helps us to analyze the efficacy of this method of base isolation for a broad class of systems. In the next part we consider a similar system where the damping forces in the system are described by functions of fractional derivative of the instantaneous displacements. Fractional Derivative based damping model has been found to be very effective in describing structural damping. We adopt half-order fractional derivative for our study, which can capture damping behavior of polymeric material very well. Typically linear and quadratic damping is considered separately as these are the two most relevant representations of structural damping. Under the assumption of smallness of certain system parameters and nonlinear terms an approximate estimate of the response at each degree of freedom of the system is obtained using Method of Multiple Scales. We then consider a situation where the nonlinear terms and certain other system parameters are no longer small. For the case where asymptotic methods are no longer valid, the assessment of performance of the vibration isolation system is made from amplitude-frequency relations. As a result, we take recourse to the Harmonic Balance Method in conjunction with arc length based continuation technique for obtaining the frequency amplitude plot for linear damping and Incremental Harmonic Balance method for quadratic damping, each of which is validated against results obtained from direct numerical simulation of the system. It needs to be appreciated that base isolation obtained this way has no counterpart in the linear theory.