Showing papers on "Plate theory published in 2002"

Linear and nonlinear structural mechanics

Abstract: Preface. 1. Introduction. 1.1 Structural Elements. 1.2 Nonlinearities. 1.3 Composite Materials. 1.4 Damping. 1.5 Dynamic Characteristics of Linear Discrete Systems. 1.6 Dynamic Characteristics of Nonlinear Discrete Systems 1.7 Analyses of Linear Continuous Systems. 1.8 Analyses of Nonlinear Continuous Systems. 2. Elasticity. 2.1 Principles of Dynamics. 2.2 Strain--Displacement Relations. 2.3 Transformation of Strains and Stresses. 2.4 Stress--Strain Relations. 2.5 Governing Equations. 3. Strings and Cables. 3.1 Modeling of Taut Strings. 3.2 Reduction of String Model to Two Equations. 3.3 Nonlinear Response of Strings. 3.4 Modeling of Cables. 3.5 Reduction of Cable Model to Two Equations. 3.6 Natural Frequencies and Modes of Cables. 3.7 Discretization of the Cable Equations. 3.8 Single--Mode Response with Direct Approach. 3.9 Single--Mode Response with Discretization Approach. 3.10 Extensional Bars. 4. Beams. 4.1 Introduction. 4.2 Linear Euler--Bernoulli Beam Theory. 4.3 Linear Shear--Deformable Beam Theories. 4.4 Mathematics for Nonlinear Modeling. 4.5 Nonlinear 2--D Euler--Bernoulli Beam Theory. 4.6 Nonlinear 3--D Euler--Bernoulli Beam Theory. 4.7 Nonlinear 3--D Curved Beam Theory Accounting for Warpings. 5. Dynamics of Beams. 5.1 Parametrically Excited Cantilever Beams. 5.2 Transversely Excited Cantilever Beams. 5.3 Clamped--Clamped Buckled Beams. 5.4 Microbeams. 6. Surface Analysis. 6.1 Initial Curvatures. 6.2 Inplane Strains and Deformed Curvatures. 6.3 Orthogonal Virtual Rotations. 6.4 Variation of Curvatures. 6.5 Local Displacements and Jaumann Strains. 7. Plates. 7.1 Introduction. 7.2 Linear Classical Plate Theory. 7.3 Linear Shear--Deformable Plate Theories. 7.4 Nonlinear Classical Plate Theory. 7.5 Nonlinear Modeling of Rectangular Surfaces. 7.6 General Nonlinear Classical Plate Theory. 7.7 Nonlinear Shear--Deformable Plate Theory. 7.8 Nonlinear Layerwise Shear--Deformable Plate Theory. 8. Dynamics of Plates. 8.1 Linear Vibrations of Rectangular Plates. 8.2 Linear Vibrations of Membranes. 8.3 Linear Vibrations of Circular and Annular Plates. 8.4 Nonlinear Vibrations of Circular and Annular Plates. 8.5 Nonlinear Vibrations of Rotating Disks. 8.6 Nonlinear Vibrations of Near--Square Plates. 8.7 Micropumps. 8.8 Thermally Loaded Plates. 9. Shells. 9.1 Introduction. 9.2 Linear Classical Shell Theory. 9.3 Linear Shear--Deformable Shell Theories. 9.4 Nonlinear Classical Theory for Double--Curved Shells. 9.5 Nonlinear Shear--Deformable Theories for Circular Cylindrical Shells. 9.6 Nonlinear Layerwise Shear--Deformable Shell Theory. 9.7 Nonlinear Dynamics of Infinitely Long Circular Cylindrical Shells. 9.8 Nonlinear Dynamics of Axisymmetric Motion of Closed Spherical Shells. Bibliography. Subject Index.

A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity

Abstract: The energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a 0-limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v V U ! R n , U R n . We show that the L 2 -distance of rv from a single rotation matrix is bounded by a multiple of the L 2 -distance from

Modeling MEMS and NEMS

Abstract: INTRODUCTION MEMS and NEMS A Capsule History of MEMS and NEMS Dimensional Analysis and Scaling Exercises A REFRESHER ON CONTINUUM MECHANICS Introduction The Continuum Hypothesis Heat Conduction Elasticity Linear Thermoelasticity Fluid Dynamics Electromagnetism Numerical Methods for Continuum Mechanics SMALL IS DIFFERENT The Backyard Scaling Systems Exercises THERMALLY DRIVEN SYSTEMS Introduction Thermally Driven Devices From PDE to ODE: Lumped Models Joule Heating of a Cylinder Analysis of Thermal Data Storage Exercises MODELING ELASTIC STRUCTURES Introduction Examples of Elastic Structures in MEMS/NEMS The Mass on a Spring Membranes Beams Plates The Capacitive Pressure Sensor Exercises MODELING COUPLED THERMAL-ELASTIC SYSTEMS Introduction Devices and Phenomena in Thermal-Elastic Systems Modeling Thermopneumatic Systems The Thermoelastic Rod Revisited Modeling Thermoelastic V-Beam Actuators Modeling Thermal Bimorph Actuators Modeling Bimetallic Thermal Actuators Exercises MODELING ELECTROSTATIC-ELASTIC SYSTEMS Introduction Devices Using Electrostatic Actuation The Mass-Spring Model Modeling General Electrostatic-Elastic Systems Electrostatic-Elastic Systems - Membrane Theory Electrostatic-Elastic Systems - Beam and Plate Theory Analysis of Capacitive Control Schemes Exercises MODELING MAGNETICALLY ACTUATED SYSTEMS Introduction Magnetically Driven Devices Mass-Spring Models A Simple Membrane Micropump Model A Small-Aspect Ratio Model Exercises MICROFLUIDICS Introduction Microfluidic Devices More Fluidic Scaling Modeling Squeeze Film Damping Exercises BEYOND CONTINUUM THEORY Introduction Limits of Contiuum Mechanics Devices and Systems Beyond Continuum Theory Exercises REFERENCES APPENDICES Mathematical Results Physical Constants INDEX Each chapter also contains Related Reading and Notes sections.

Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates

Abstract: 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.

Thermal Buckling of Functionally Graded Plates

Abstract: Equilibrium and stability equations of a rectangular plate made of functionally graded material under thermal loads are derived, based on the classical plate theory. When it is assumed that the material properties vary as a power form of thicknesscoordinate variable z and when the variational method is used, the system of fundamental differential equations isestablished. Thederived equilibrium and stability equationsforfunctionally graded plates areidenticalwith theequationsforhomogeneousplates. Bucklinganalysisoffunctionally graded platesunderfour typesofthermalloadsiscarriedoutresultinginclosed-formsolutions.Thebucklingloadsarereducedtothecritical buckling temperature relationsfor functionally graded plates with linearcomposition of constituent materials and homogeneous plates. The results are validated with the reduction of the buckling relations for functionally graded plates to those of isotropic homogeneous plates given in the literature.

Thermal buckling of functionally graded plates based on higher order theory

Abstract: Equilibrium and stability equations of a rectangular plate made of functionally graded material (FGM) under thermal loads are derived, based on the higher order shear deformation plate theory. Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. The derived equilibrium and stability equations for functionally graded plates (FGPs) are identical to the equations for laminated composite plates. A buckling analysis of a functionally graded plate under four types of thermal loads is carried out and results in closed-form solutions. The critical buckling temperature relations are reduced to the respective relations for functionally graded plates with a linear composition of constituent materials and homogeneous plates. The results are compared with the critical buckling temperatures obtained for functionally graded plates based on classical plate theory given in...

Refined plate theory and its variants

Abstract: Thedevelopmentofa newree ned platetheoryand its two simplevariantsisgiven. Thetheorieshavestrongcommonality withtheequationsofclassicalplatetheory (CPT).However,unlikeCPT,thesetheoriesassumethatlateral and axial displacements have bending and shear components such that bending components do not contribute toward shearforces and, likewise, shearing components do not contribute toward bending moments. The theory and one of its variants are variationally consistent, whereasthesecond variant isvariationally inconsistent and usesthe relationships between moments, shear forces, and loading. It should be noted that, unlike any other ree ned plate theory, thegoverning equation as well as the expressions for moments and shear forces associated with thisvariant areidentical tothoseassociated withtheCPT,savefortheappearanceofasubscript. Theeffectivenessofthetheory and itsvariantsisdemonstratedthroughanexample. Surprisingly,theanswersobtained by boththevariantsofthe theory, one of which is variationally consistent and the other one is inconsistent, are same. The numerical example studied, therefore, not only brings out the effectiveness of the theories presented, but also, albeit unintentionally, supports the doubts, e rst raised by Levinson, about the so called superiority of variationally consistent methods.

Buckling of Functionally Graded Plates under In-plane Compressive Loading

Abstract: In the present article, equilibrium and stability equations of rectangular functionally graded plates (FGPs) are determined using the variational approach. Derivation of equations are based on the classical plate theory. It is assumed that the material properties vary with the power product form of thickness coordinate variable z. Equilibrium and stability equations for FGPs are the same as the equations for homogeneous plates. The equilibrium and stability equations are employed to study the buckling behaviour of functionally graded plates with all edges simply supported and subjected to in-plane loading conditions. By equating power law index to zero, predicted relation is reduced to the buckling equation of homogeneous plates which is available in the literature.

Thermal Stresses in Functionally Graded Beams

Abstract: Thermoelastic equilibrium equations for a functionally graded beam are solved in closed-form to obtain the axial stress distribution. The thermoelastic constants of the beam and the temperature were assumed to vary exponentially through the thickness. The Poisson ratio was held constant. The exponential variation of the elastic constants and the temperature allow exact solution for the plane thermoelasticity equations. A simple Euler ‐ Bernoulli-type beam theory is also developed based on the assumption that plane sections remain plane and normal to the beam axis. The stresses were calculated for cases for which the elastic constants vary in the same manner as the temperature and vice versa. The residual thermal stresses are greatly reduced, when the variation of thermoelastic constants are opposite to that of the temperature distribution. When both elastic constants and temperature increasethrough the thickness in the samedirection, they causea signie cant raise in thermal stresses. For the case of nearly uniform temperature along the length of the beam, beam theory is adequate in predicting thermal residual stresses.

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

Abstract: 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.

Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical implementations

Abstract: SUMMARY This paper presents numerical evaluations related to the multilayered plate elements which were proposed in the companion paper (Part 1). Two-dimensional modellings with linear and higher-order (up to fourth order) expansion in the z-plate=layer thickness direction have been implemented for both displacements and transverse stresses. Layer-wise as well as equivalent single-layer modellings are considered on both frameworks of the principle of virtual displacements and Reissner mixed variational theorem. Such a variety has led to the implementation of 22 plate theories. As far asnite element approximation is concerned, three quadrilaters have been considered (four-, eight- and nine-noded plate elements). As a result, 22×3 dierentnite plate elements have been compared in the present analysis. The automatic procedure described in Part 1, which made extensive use of indicial notations, has herein been referred to in the considered computer implementations. An assessment has been made as far as convergence rates, numerical integrations and comparison to correspondent closed-form solutions are concerned. Extensive comparison to early and recently available results has been made for sample problems related to laminated and sandwich structures. Classical formulations, full mixed, hybrid, as well as three-dimensional solutions have been considered in such a comparison. Numerical substantiation of the importance of the fullment of zig-zag eects and interlaminar equilibria is given. The superiority of RMVT formulatednite elements over those related to PVD has been concluded. Two test cases are proposed as 'desk-beds' to establish the accuracy of the several theories. Results related to all the developed theories are presented for therst test case. The second test case, which is related to sandwich plates, restricts the comparison to the most signicant implementednite elements. It is proposed to refer to these test cases to establish the accuracy of existing or new higher-order, rened or improvednite elements for multilayered plate analyses. Copyright ? 2002 John Wiley & Sons, Ltd.

First-order-theory-based thermoelastic stability of functionally graded material circular plates

Abstract: Thermal buckling of circular plates made of functionally graded material is discussed. The nonlinear equilibrium and linear stability equations are derived using variational formulations. The thermal buckling of solid circular plates under uniform temperature rise, gradient through the thickness, and linear temperature variation along the radius are considered, and the buckling temperatures are derived. The buckling temperatures are derived for simply supported and clamped edges. The results are verified with known results in the literature.

Measurement of the scattering of a Lamb wave by a through hole in a plate

Abstract: Flexural waves propagating in an aluminum plate containing a circular hole are studied. In the experiments the first antisymmetric Lamb wave mode A0 is excited selectively by a piezoelectric transducer. The scattered field around a circular cavity is measured pointwise using a heterodyne laser interferometer. The measurements are compared with theoretical calculations. Different approximate analytical approaches, employing Kirchhoff and Mindlin types of plate theories to describe the scattered field, are used. Good agreement between the experimental data and the analytical solutions is found within the ranges of validity of the different models. Introduction of a small imperfection, like a notch, at the boundary of the cavity changes the measured scattered field significantly. The approach allows a fast measurement of large surfaces and might be useful for nondestructive testing purposes, e.g., the detection of cracks at fastener holes in airplane fuselage.

Temperature Profile Influence on Layered Plates Response Considering Classical and Advanced Theories

Abstract: A study on the influence of the through-the-thickness temperature profile T(z) on the thermomechanical response of multilayered anisotropic thick and thin plates has been conducted. The heat conduction problem is solved, and the temperature variation T c (z) is then calculated. The governing thermomechanical equations of multilayered plates are written considering a large variety of classical and advanced or zigzag theories into account. The principle of virtual displacement and the Reissner mixed variational theorem are employed. Linear, up to fourth-order expansions in z are retained for the assumed transverse stress and displacement fields. As a result, more than 20 plate theories are compared. The numerical investigation is restricted to orthotropic layered plates with harmonic in-plane distribution of both thermal loadings and unknown variables. Four sample plate problems are treated that are related to plates made of isotropic and/or orthotropic layers that are loaded by different top-bottom plate surface temperature conditions. Comparison is made to results related to a linear profile T a (z), which is usually assumed in open literature. The following is concluded: Thick plates could exhibit a layerwise form temperature profile T c (z). T a (z) case is approached for thin plate geometries. The use of linear temperature profile leads to large errors in tracing the response of thick plate geometries. The accuracy of plate theories is affected to great extent by the form of temperature variation T(z). Refinements of classical plate theories can be meaningless unless the calculated T c (z) is introduced. The layerwise form of T c (z) would require layerwise assumptions for stresses and/or displacements. Plate theories that neglect transverse normal strains lead to very inaccurate results in both thick and thin plates analysis. At least a parabolic expansion for transverse displacement is required to capture transverse normal thermal strains that vary linearly along the plate thickness.

On consistent plate theories

Abstract: Applying the uniform-approximation technique, consistent plate theories of different orders are derived from the basic equations of the three-dimensional linear theory of elasticity. The zeroth-order approximation allows only for rigid-body motions of the plate. The first-order approximation is identical to the classical Poisson-Kirchhoff plate theory, whereas the second-order approximation leads to a Reissner-type theory. The proposed analysis does not require any a priori assumptions regarding the distribution of either displacements or stresses in thickness direction.

Buckling of symmetrically laminated composite plates using the element-free galerkin method

Abstract: An element free Galerkin (EFG) method is presented for buckling analyses of isotropic and symmetrically laminated composite plates using the classical plate theory. The shape functions are constructed using the moving least squares (MLS) approximation, and no element connectivity among nodes is required. The deflection can be easily approximated with higher-order polynomials as desired. The discrete eigenvalue problem is derived using the principle of minimum total potential energy of the system. The essential boundary conditions are introduced into the formulation through the use of the Lagrange multiplier method and the orthogonal transformation techniques. Since the dimension of the eigenvalue problem obtained by the present method is only one third of that in the conventional finite element method (FEM), solving the eigenvalue problem in the EFG is computationally more efficient compared to the FEM. Buckling load param-eters of isotropic and symmetrically laminated composite plates for different boundary conditions are calculated to demonstrate the efficiency of the present method.

Piezoelectrically actuated flextensional micromachined ultrasound transducers. I. Theory

Abstract: This series of two papers considers piezoelectrically actuated flextensional micromachined ultrasound transducers (PAFMUTs) and consists of theory, fabrication, and experimental parts. The theory presented in this paper is developed for an ultrasound transducer application presented in the second part. In the absence of analytical expressions for the equivalent circuit parameters of a flextensional transducer, it is difficult to calculate its optimal parameters and dimensions and difficult to choose suitable materials. The influence of coupling between flexural and extensional deformation and that of coupling between the structure and the acoustic volume on the dynamic response of piezoelectrically actuated flextensional transducer are analyzed using two analytical methods: classical thin (Kirchhoff) plate theory and Mindlin plate theory. Classical thin plate theory and Mindlin plate theory are applied to derive two-dimensional plate equations for the transducer and to calculate the coupled electromechanical field variables such as mechanical displacement and electrical input impedance. In these methods, the variations across the thickness direction vanish by using the bending moments per unit length or stress resultants. Thus, two-dimensional plate equations for a step-wise laminated circular plate are obtained as well as two different solutions to the corresponding systems. An equivalent circuit of the transducer is also obtained from these solutions.