Senthil S. Vel

THREE-DIMENSIONAL EXACT SOLUTION FOR THE VIBRATION OF FUNCTIONALLY GRADED RECTANGULAR PLATES

Abstract A three-dimensional exact solution is presented for free and forced vibrations of simply supported functionally graded rectangular plates. Suitable displacement functions that identically satisfy boundary conditions are used to reduce equations governing steady state vibrations of a plate to a set of coupled ordinary differential equations, which are then solved by employing the power series method. The exact solution is valid for thick and thin plates, and for arbitrary variation of material properties in the thickness direction. Results are presented for two-constituent metal–ceramic functionally graded rectangular plates that have a power-law through-the-thickness variation of the volume fractions of the constituents. The effective material properties at a point are estimated by either the Mori–Tanaka or the self-consistent schemes. Exact natural frequencies, displacements and stresses are used to assess the accuracy of the classical plate theory, the first order shear deformation theory and a third order shear deformation theory for functionally graded plates. Parametric studies are performed for varying ceramic volume fractions, volume fraction profiles and length-to-thickness ratios. Results are also computed for a functionally graded plate that has a varying microstructure in the thickness direction using a combination of the Mori–Tanaka and the self-consistent methods. Forced vibrations of a plate with a sinusoidal spatial variation of the pressure applied on its top surface are scrutinized.

Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates

An exact solution is obtained for three-dimensional deformations of a simply supported functionally graded rectangular plate subjected to mechanical and thermal loads on its top and/or bottom surfaces. Suitable temperature and displacement functions that identically satisfy boundary conditions at the edges are used to reduce the partial differential equations governing the thermomechanical deformations to a set of coupled ordinary differential equations in the thickness coordinate, which are then solved by employing the power series method. The exact solution is applicable to both thick and thin plates. Results are presented for two-constituent metal‐ceramic functionally graded rectangular plates that have a power law through-the-thickness variation of the volume fractions of the constituents. The effective material properties at a point are estimated by either the Mori‐Tanaka or the self-consistentschemes. Exact displacementsand stressesatseveral locations for mechanical and thermal loads are used toassess theaccuracyof the classical plate theory, thee rst-ordershear deformation theory, and athird-order shear deformation theory for functionally graded plates. Results are alsocomputed for a functionally graded plate with material properties derived by the Mori‐Tanaka method, the self-consistent scheme, and a combination of these two methods.

Three-dimensional analysis of transient thermal stresses in functionally graded plates

An analytical solution is presented for three-dimensional thermomechanical deformations of a simply supported functionally graded (FG) rectangular plate subjected to time-dependent thermal loads on its top and/or bottom surfaces. Material properties are taken to be analytical functions of the thickness coordinate. The uncoupled quasi-static linear thermoelasticity theory is adopted in which the change in temperature, if any, due to deformations is neglected. A temperature function that identically satisfies thermal boundary conditions at the edges and the Laplace transformation technique are used to reduce equations governing the transient heat conduction to an ordinary differential equation (ODE) in the thickness coordinate which is solved by the power series method. Next, the elasticity problem for the simply supported plate for each instantaneous temperature distribution is analyzed by using displacement functions that identically satisfy boundary conditions at the edges. The resulting coupled ODEs with variable coefficients are also solved by the power series method. The analytical solution is applicable to a plate of arbitrary thickness. Results are given for two-constituent metal-ceramic FG rectangular plates with a power-law through-the-thickness variation of the volume fraction of the constituents. The effective elastic moduli at a point are determined by either the Mori–Tanaka or the self-consistent scheme. The transient temperature, displacements, and thermal stresses at several critical locations are presented for plates subjected to either time-dependent temperature or heat flux prescribed on the top surface. Results are also given for various volume fractions of the two constituents, volume fraction profiles and the two homogenization schemes. 2003 Elsevier Ltd. All rights reserved.

Exact Solution for Rectangular Sandwich Plates with Embedded Piezoelectric Shear Actuators

A liquid crystal display having a wide viewing angle, a large contrast ratio, and little nonuniformity of display, is formed by a pair of substrates, and a liquid crystal layer interposed between the pair of substrates, wherein at least one of the pair of substrates has plural electrodes, a passivation layer formed on the plural electrodes, and an insulating layer formed at concave portions of the passivation layer for making the thickness of the liquid crystal layer uniform.

Analytical Solution for Rectangular Thick Laminated Plates Subjected to Arbitrary Boundary Conditions

Three-dimensional deformations of a multilayered, linear elastic, anisotropic rectangular plate subjected to arbitrary boundary conditions at its edges are analyzed by the generalized Eshelby-Stroh formalism. The rectangular laminate consists of anisotropic and homogeneous laminae of arbitrary thicknesses. Perfect bonding is assumed between the adjoining laminae in the sense that both surface tractions and displacements are assumed to be continuous across their interfaces. The analytical solution is in terms of infinite series, and the effect of truncating the series on the accuracy of the solution is scrutinized. The method is also applicable to rectangular laminated plates, with edges of each lamina subjected to different boundary conditions. Results are presented for thick plates with different sets of edge boundary conditions, e.g., two opposite edges simply supported and the other two subjected to eight different conditions or all four edges clamped.

Exact solution for the cylindrical bending of laminated plates with embedded piezoelectric shear actuators

An exact three-dimensional state space solution is obtained for the static cylindrical bending of simply supported laminated plates with embedded shear mode piezoelectric actuators, and subjected to mechanical and electric loading on the upper and lower surfaces. Each layer of the laminate is made of either an orthotropic elastic material or a piezoelectric material whose poling direction lies in the plane of the plate, with perfect bonding between the adjoining layers. The displacements and stresses for a homogeneous piezoelectric plate for various length-to-thickness ratios are compared with those obtained by the first-order shear deformation theory. Results are also presented for a hybrid laminate with a shear mode piezoelectric core sandwiched between two elastic layers. A comparison of stresses with those in the corresponding surface-mounted extension actuation configuration shows that the longitudinal and shear stresses within the actuator are significantly smaller for the shear actuation mechanism. The analytical results can be used to assess the accuracy of different plate theories and/or validating finite element codes.

Cylindrical Bending of Laminated Plates with Distributed and Segmented Piezoelectric Actuators/Sensors

The generalized plane quasistatic deformations of linear piezoelectric laminated plates are analyzed by the Eshelby‐Stroh formalism. The laminate consists of homogeneous elastic or piezoelectric laminae of arbitrary thickness and width. The three-dimensional differential equations of equilibrium for a piezoelectric body are exactly satiseed at every point in the body. The analytical solution is in terms of an ine nite series; the continuity conditions at the interfaces between adjoining laminae and boundary conditions at the edges are satise ed in the sense of Fourier series. The formulation admits different boundary conditions at the edges and is applicable to thick and thin laminated plates. Results are presented for laminated elastic plates with a distributed piezoelectric actuator on the upper surface and a sensor on the lower surface and subjected to different sets of boundary conditions at the edges. Results are also provided for a piezoelectric bimorph and an elastic plate with segmented piezoelectric actuators bonded to its upper and lower surfaces.

Active Vibration Suppression of Sandwich Beams using Piezoelectric Shear Actuators: Experiments and Numerical Simulations

This article deals with the experimental and numerical assessment of the vibration suppression of smart structures using piezoelectric shear actuators. Experimental results are presented for an adaptive sandwich cantilever beam that consists of aluminum facings and a core composed of two piezoelectric shear actuators and foam. The electric field is applied perpendicular to the poling direction of the piezoelectric actuators to cause transverse shear deformation of the sandwich beam. Active vibration suppression is achieved using either positive position feedback or strain rate feedback. The control system is implemented in real-time using Matlab/Simulink and a dSPACE digital controller. First, the frequency response of the adaptive beam is investigated by using one shear actuator to excite the beam and the other to control its vibration. Parametric studies are conducted to assess the influence of controller parameters on the frequency response of the system. The experimental frequency response function compares well with numerical simulations using the finite element method. Next, the effectiveness of the active vibration suppression system in the time domain is analyzed using a proof-mass actuator that is attached to the tip of the cantilever beam to provide a repeatable vibration input. Experiments and numerical simulations show that the shear actuators can provide significant reduction in tip acceleration and settling time.

The generalized plane strain deformations of thick anisotropic composite laminated plates

We use the Eshelby‐Stroh formalism to analyze the generalized plane strain quasistatic deformations of an anisotropic, linear elastic laminated plate. The laminate consists of homogeneous laminae of arbitrary thicknesses. Computed results are presented for three sample problems to illustrate the eAect of boundary conditions and of the span to height ratio. # 1999 Elsevier Science Ltd. All rights reserved.

Analysis of piezoelectric bimorphs and plates with segmented actuators

Elastic plates with distributed or segmented piezoelectric layers have been analyzed using the classical laminated plate theory, the first-order shear deformation theory, and the results are compared with an analytical solution. The plate theories and the analytical solution take into account both the direct and the converse piezoelectric effects, and assume generalized plane strain deformations. The transverse displacements from both theories are in reasonable agreement. The classical lamination theory gives a discontinuous longitudinal stress at the edges of the segments whereas the analytical solution predicts a continuous curve with steep gradients. Piezoelectric bimorphs with the axis of transverse isotropy inclined at an angle to the thickness direction are also studied using the three formulations. The displacements and stresses obtained from the first-order shear deformation theory are in very good agreement with the analytical solution even for thick plates. It is advantageous to use shear bimorphs since the stresses induced in them are smaller than those in extension bimorphs.