NUMERICAL SOLUTION OF SOME VIBRATION PROBLEMS OF FGM CIRCULAR PLATES UNDER THERMAL ENVIRONMENT

Abstract

The phenomenon of vibration occurs widely in nature and may also be observed in all fields of modern science and technology ranging from space shuttles, missiles and aircraft to household goods, such as washing machines, grinders and juicers etc. The static and dynamic behavior of plates and plate-type structures has been the subject of study to many researchers for more than two centuries due to their practical importance in various fields of engineering. Now-a-days, technologists are always been in search of stronger and stronger materials for building machines and structures. A specific class of such materials is functionally graded materials (FGMs) which are tailored by mixing two or more materials to achieve the desired mechanical properties along one or more direction(s). Due to the distinguish ability of protecting to the surrounding materials under elevated temperature without losing their structural integrity, FGMs found their wide applications in a variety of practical situations, particularly, in nuclear energy reactors, energy conservation devices, armor protections for military applications, etc. The requirement of use them efficiently, a good knowledge of their construction and dynamic behavior is essential with a fair amount of accuracy. The present thesis comprises a theoretical study on the vibration characteristics of one-directional / two-directional functionally graded (FG) circular plates of uniform / non-uniform thickness together with or without the effect of Pasternak foundation and in-plane force on the basis of classical plate theory / first-order shear deformation theory under temperature field. The effect of thermal environment has been examined by assuming temperature-independent / temperature dependent mechanical properties of the plate material and one / two-dimensional temperature variation(s) while the concept of physical neutral surface has also been introduced for moderately thick plates. The thesis contains ten chapters. Chapter I presents an up-to-date survey of the literature on the static / dynamic behavior of FG structures with other complicating effects such as elastic foundation, in-plane force and thickness variation etc. which motivates the work of the thesis. In Chapters II to IX, the analyses of such plates of circular geometry with different considerations have been presented by employing two different numerical techniques, namely, generalized differential quadrature rule (GDQR) and generalized differential quadrature method (GDQM). The detailed analysis of the results in the form of conclusions together with future scope of the present work is given in Chapter X and at the end, Bibliography is appended. i A brief discussion of individual chapters is given below: Chapter II attempts to investigate the vibration behaviour of one-directional FG circular plates subjected to a non-linear temperature variation in the thickness direction. The constituent materials of the plate are Titanium alloy-TC4 (Ti-6Al-4V) for metal and Zirconia (ZrO) for ceramic. The temperature-dependent mechanical properties of the plate material are varying as a function of volume fraction index across the thickness of the plate. Keeping the uniform temperature over the top and bottom surfaces, the equations of thermoelastic equilibrium and axisymmetric motion for such a plate model have been derived by Hamilton’s energy principle on the basis of classical plate theory. GDQR has been applied to obtain the thermal displacements and characteristic equations for clamped and simply-supported plates. The lowest three roots of these equations have been computed by MATLAB and reported as the values of frequency parameter for the first three modes of vibration. The zeros of the Chebyshev-Gauss-Lobatto polynomials have been used as the grid points. The influence of volume fraction index, plate thickness and thermal environment on the frequencies of the plate has been investigated. Three dimensional mode shapes have been illustrated for a specified plate. In Chapter III, the analysis presented in Chapter II has been extended to study the effect of Pasternak foundation on the free axisymmetric vibrations of thin circular plates subjected to mechanical in-plane force. Effects of temperature difference, volume fraction index, in-plane force, foundation parameters and boundary conditions have been analyzed on the natural frequencies of the plates. Compressive in-plane loads for which the plate ceases to vibrate have been reported as critical buckling loads. Bisection method has been used in obtaining the values of critical buckling load. In Chapter IV, GDQM has been applied to analyze the axisymmetric vibrations of thin FG circular plates of linearly varying thickness. The equations for thermoelastic equilibrium and axisymmetric motion, associated with thermal and vibration states of the plate have been extracted by Hamilton’s energy principle. The resulting equations have been discretized by GDQM to obtain the characteristic equations for clamped and simply-supported plates. The effect of plate parameters i.e. volume fraction index and taper parameter with the variation in temperature profile, on the vibration characteristics of the plate has been analyzed for the first three modes. Three-dimensional plate configurations have been shown. ii Chapter V is an extension of Chapter IV and concerns with the thermally induced transverse vibrations of circular plates with quadratic thickness variation along the radial direction. The numerical values of thermal displacements from the thermoelastic equilibrium equation and frequencies from the equation of motion for the first three modes of vibration have been obtained by using GDQM for clamped and simply-supported plates. Effects of taper parameters and temperature elevation at ceramic surface together with volume fraction index on these frequencies have been investigated. Three-dimensional mode shapes have been plotted for the specified plate. Chapter VI provides an analysis for the radially symmetric vibrations of thin circular plates with exponentially varying thickness under uniform peripheral loading. The equations of thermoelastic equilibrium and axisymmetric motion for such a plate model have been derived by Hamilton’s energy principle. Employing GDQM, the frequency equations for clamped and simply-supported plates have been obtained and solved numerically by MATLAB to compute the frequencies for the first three modes of vibration. The influence of various parameters such as volume fraction index, in-plane force parameter and temperature profile with varying values of taper parameter, has been analyzed on the natural frequencies of the plate. By allowing the frequency to approach zero, the critical buckling loads with varying values of other parameters have been computed. Three-dimensional mode shapes have been plotted. In Chapter VII, the effects of physical neutral surface and two-dimensional temperature distribution on the free axisymmetric vibrations of bi-directional FG circular plates have been analyzed by employing first-order shear deformation theory. The temperature-dependent mechanical properties of the plate material are assumed to vary according to a power-law distribution along the thickness direction and exponential distribution in radial direction. The exact solution of two-dimensional heat conduction equation has been obtained from separation of variables method and imposing the thermal boundary conditions which has been used in extracting the coupled differential equations for thermoelastic equilibrium as well as axisymmetric motion by Hamilton’s energy principle. The GDQM has been applied to compute the numerical values for thermally induced displacements and natural frequencies for clamped and simply-supported plates for the first three modes. The effects of various parameters such as thermal boundary conditions, volume fraction index, heterogeneity parameter and density parameter on the vibration characteristics have been presented. Three-dimensional mode shapes have been presented for the specific plate. iii In Chapter VIII, the analysis of Chapter VII has been extended for the bi-directional FGM circular plates with linearly varying thickness under two-dimensional temperature variation. Employing GDQM, the frequency equations for clamped and simply-supported plates have been obtained and solved numerally to retain their lowest three roots as the natural frequencies for the first three modes. The impact of physical neutral plane, volume fraction index, taper parameter, heterogeneity parameter, density parameter and temperature profile on the vibration characteristics has been captured. Three-dimensional mode shapes have been illustrated for a specified plate. Chapter IX presents the effects of quadratically varying thickness, physical neutral surface, transverse shear and rotatory inertia on the radially symmetric vibrations of moderately thick circular plates under two-dimensional temperature variation. The temperature-dependent mechanical properties of the plate material are graded in the thickness as well as in radial directions. Hamilton’s energy principle has been used to deduce the coupled differential equations for thermoelastic equilibrium and axisymmetric motion of clamped and simply-supported plates. The lowest three roots of the frequency equations have been computed and reported as the natural frequencies for the first three modes. The parametric varying studies have been performed to analyze the effect of volume fraction index, taper parameters, heterogeneity parameter, density parameter and thermal boundary conditions on the natural frequencies of the plates. Three-dimensional mode shapes have been plotted. Chapter X gives a summary of the significant contribution / outcomes based upon individual chapters from II to IX. The further extensions of the present work have also been suggested.