R.C. Batra

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.

Some basic fracture mechanics concepts in functionally graded materials

In this paper, the crack-tip fields in a general nonhomogeneous material are summarized. The fracture toughness and R-curve of functionally graded materials (FGMs) are studied based on the crack-bridging concept and a rule of mixtures. It is shown that the fracture toughness is significantly increased when a crack grows from the ceramic-rich region into the metal-rich region in an alumina-nickel FGM. By applying the concept of the toughening mechanism to the study of the strength behavior of FGMs, it is found that the residual strength of the alumina-nickel FGM with an edge crack on the ceramic side is quite notch insensitive.

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.

Three-dimensional thermoelastic deformations of a functionally graded elliptic plate

Abstract A new solution in closed form is obtained for the thermomechanical deformations of an isotropic linear thermoelastic functionally graded elliptic plate rigidly clamped at the edges. The through-thickness variation of the volume fraction of the ceramic phase in a metal–ceramic plate is assumed to be given by a power-law type function. The effective material properties at a point are computed by the Mori–Tanaka scheme. It is found that the through-thickness distributions of the in-plane displacements and transverse shear stresses in a functionally graded plate do not agree with those assumed in classical and shear deformation plate theories.

Review of modeling electrostatically actuated microelectromechanical systems

A wide range of microelectromechanical systems (MEMSs) and devices are actuated using electrostatic forces. Multiphysics modeling is required, since coupling among different fields such as solid and fluid mechanics, thermomechanics and electromagnetism is involved. This work presents an overview of models for electrostatically actuated MEMSs. Three-dimensional nonlinear formulations for the coupled electromechanical fluid–structure interaction problem are outlined. Simplified reduced-order models are illustrated along with assumptions that define their range of applicability. Theoretical, numerical and experimental works are classified according to the mechanical model used in the analysis.

Electromechanical Model of Electrically Actuated Narrow Microbeams

A consistent one-dimensional distributed electromechanical model of an electrically actuated narrow microbeam with width/height between 0.5–2.0 is derived, and the needed pull-in parameters are extracted with different methods. The model accounts for the position-dependent electrostatic loading, the fringing field effects due to both the finite width and the finite thickness of a microbeam, the mid-plane stretching, the mechanical distributed stiffness, and the residual axial load. Both clamped–clamped and clamped-free (cantilever) microbeams are considered. The method of moments is used to estimate the electrostatic load. The resulting nonlinear fourth-order differential equation under appropriate boundary conditions is solved by two methods. Initially, a one-degree-of-freedom model is proposed to find an approximate solution of the problem. Subsequently, the meshless local Petrov–Galerkin (MLPG) and the finite-element (FE) methods are used, and results from the three methods are compared. For the MLPG method, the kinematic boundary conditions are enforced by introducing a set of Lagrange multipliers, and the trial and the test functions are constructed using the generalized moving least-squares approximation. The nonlinear system of algebraic equations arising from the MLPG and the FE methods are solved by using the displacement iteration pull-in extraction (DIPIE) algorithm. Three-dimensional FE simulations of narrow cantilever and clamped–clamped microbeams are also performed with the commercial code ANSYS. Furthermore, computed results are compared with those arising from other distributed models available in the literature, and it is shown that improper fringing fields give inaccurate estimations of the pull-in voltages and of the pull-in deflections. 1641

Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle

A three-dimensional mixed variational principle is used toderive a Kth-order two-dimensional lineartheory for an anisotropichomogeneouspiezoelectric(PZT)plate.Themechanicaldisplacements, theelectricpotential,theinplane components of the stress tensor, and the in-plane components of the electric displacement are expressed as a e niteseries of orderK inthethickness coordinate bytakingLegendrepolynomialsas thebasisfunctions. However, the transverse shear stress, the transverse normal stress, and the transverse electric displacement are expressed as a e nite series of order (K +2) in the thickness coordinate. The formulation accounts for the double forces without moments that may change the thickness of the plate. Results obtained by using the plate theory are given for the bending of a cantilever thick plate loaded on the top and the bottom surfaces by uniformly distributed 1) normal tractions and 2) tangential tractions. Results are also computed for the bending of a cantilever thick PZT beam loaded by 1) a uniformly distributed charge density on the top and the bottom surfaces and 2) equal and opposite normal tractions distributed uniformly only on a part of the beam. The seventh-order plate theory captures well the boundary-layer effects near the clamped and the free edges and adjacent to the top and the bottom surfaces of a thick orthotropic cantilever beam with the span to the thickness ratio of two. Also, through-the-thickness variation of the transverse shear and the transverse normal stresses agree well with those computed from the analytical solution of the three-dimensional elasticity equations. The governing partial differential equations are second order, so that Lagrange basis functions can be used to solve the problem by the e nite element method.

STRESS INTENSITY RELAXATION AT THE TIP OF AN EDGE CRACK IN A FUNCTIONALLY GRADED MATERIAL SUBJECTED TO A THERMAL SHOCK

We analyze thermal stresses and the stress intensity factor in an edge-cracked strip of a functionally graded material (FGM) subjected to sudden cooling at the cracked surface. It is assumed that the shear modulus of the material decreases hyperbolically with the higher value at the surface exposed to the thermal shock and that the thermal conductivity varies exponentially. Volume fractions of the constituents in a ceramic-metal FGM are then determined with the assumed shear modulus gradient using a three-phase model of conventional composites. The differences between the other assumed material properties and those predicted by the three-phase model are delineated and the applicability of the assumed FGM is discussed. It is shown that the maximum tensile thermal stress in the strip without cracks is substantially reduced by the assumed thermal conductivity gradient and that the magnitude of the compressive stress is increased. A strong compressive zone just away from the thermally shocked surface is devel...

TRANSIENT THERMOELASTIC DEFORMATIONS OF A THICK FUNCTIONALLY GRADED PLATE

We study transient thermoelastic deformations of a thick functionally graded plate with edges held at a uniform temperature and either simply supported or clamped. Either the temperature or the heat flux is prescribed on the top surface of the plate with the bottom surface of the plate kept at either a uniform temperature or thermally insulated. Stresses and deformations induced due to the simultaneous application of the transient thermal and mechanical loads are also computed. The problem is solved by using a higher order shear and normal deformable plate theory and a meshless local Petrov–Galerkin method. Only nodal coordinates are needed, and neither nodal connectivity nor a background mesh is employed. The validity of the method and of the computer code is established by comparing computed results with the analytical solution of the three-dimensional thermoelasticity equations for a simply supported plate. Results are then computed for clamped plates. It is found that the centroidal deflection and the axial stress induced at the centroid of the top surface of the plate are significantly influenced by boundary conditions at the plate edges.