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.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/102661 |
| Date | 18 September 2019 |
| Creators | Chattopadhyay, Arka Prabha |
| Contributors | Engineering Science and Mechanics, Batra, Romesh C., Hanna, James, Thangjitham, Surot, Sun, Shu Ming, Cramer, Mark S. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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