Bachelors Thesis

Free Vibration Analysis of Functionally Graded Plate Using Higher Order Shear Deformation Theory

Summary:

The present work focuses on free vibration analysis of plate resting on Winkler-Pasternak model foundation with simply supported edges. The considering material plate is made up of functionally graded material (FGM).FGM’s are the advancement of composites which was used for the first time in aerospace industry as a thermal barrier during space plane project in Japan in late 1980’s.Pure metals were scarcely used for engineering applications while composites are burdensome to manufacture without sharp interfaces therefore functionally graded materials are used. Functionally graded materials are the heterogeneous materials whose mechanical properties are assumed to be varying exponentially with changing in composition, micro-structure or porosity in the thickness direction according to the power law distribution, it also have a smooth variation in parameters at the interface unlike composites. Comparatively FGM have an excellent bonding capacity with the least risk of de-lamination. The applications of functionally graded materials are seen in the fields of aerospace, aeronautics, computer circuits industries, marine, automobile, and in other industries. The back bone of this paper is to do analytical study on the free vibration analysis of functionally graded material by employing higher order shear deformation theory. The accuracy of the results depends on our choice of field variables, evaluation tool and the theory which can shape the structure of our model. The classical Kirchhoff theory (CLKT) fails to include out plane stresses and over estimates the vibration analysis. The first order shear deformation theory (FSDT) by Reissner and Mindlin encounters this problem by introducing a shear correction factor which can conquer with the constant transverse shear stress distribution. The inaccuracy occurs due to this constant transverse shear stress distribution was hence forth, an artificial correction factor must be considered. The above discussed boundaries can be overcome by Higher order shear deformation theory (HSDT) assumes non-linear variation of displacement and approximately parabolic variation of shear stresses and shear correction factors are not required for calculating shear stresses. When polynomial higher order shear deformation theory (PHSDT) is applied as number of unknowns increases the accuracy also increases, but at the cost of complexity, to reduce this complexity Non- polynomial shear deformation theory (NPSDT) is applied by maintaining the accuracy. The function in the NPSDT is responsible for the accuracy. By employing Hamilton principle the governing differential equation is obtained. Implementing Navier closed type solution technique the vibration analysis is carried out. NPSDT generates efficient results than available numerical techniques like FEM, FDM etc.

A rectangular plate made of Al/Al203 which is commonly being used in fields such as aerospace, electrical, mechanical and marine industry. Since the FGM plate is symmetrical in geometry, solving it becomes less complicated. This FGM plate is rested on the Winkler foundation with simply supported edges. One of the famous methods for modeling is to describe the interaction between the plate and foundation is Winkler-Pasternak model. In the early stages, Winkler replaced the elastic foundation with the help of linear elastic springs but there was no interactions between the springs and was only depended on stiffness k and was popularly known as one- parameter model. This model was firstly used in analyzing the deflections and the resultant stresses in rail road and tracks. Later to conquer the problem of discontinuity of shear deformation between the springs Pasternak in 1954 developed the Winkler model by connecting the end of the springs to the plate having incompressible, vertical elements which will deform in the lateral direction. In the latter we willconsider Winkler constant Ko and shear stiffness K1. This is popularly known as two-parameter model. The study focuses on the wet natural frequency of functionally graded plate which is placed in hydro-thermal environment. The agenda is to maintain the continuum in the case study and to carried out a detailed analysis of wet natural frequency of Functionally graded plate which is scarcely find in the literature survey and compare it with the dry natural frequency of Functionally graded plate. The effect of this wet natural frequency on aspect ratio, power index (gradient index) and thermal loading has been investigated. Using MATLAB platform the coding has been generated. Moreover the governing equations exactly satisfy the Navier solution. The analytical formulation of natural frequencies and their respective mode shapes on bottom and top surfaces of the FGM plate has a good replica with the computational results. This work will stands as an excellent reference for further investigations in FGM plate. Evaluated results are validated with the existing results those are well agreement with the available advanced theories and stands better than conventional theories