Asymptotic Construction of a Generalized Reissner-Mindlin Model for Multilayer Piezoelectric Plates

2007 ◽  
Author(s):  
Lin Liao ◽  
Wenbin Yu
Keyword(s):  
A Priori ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
Single Layer ◽  
Plate And Shell ◽  
Shell Analysis ◽  
Variational Asymptotic ◽  
Variational Asymptotic Method ◽  
Plate Analysis ◽  
Piezoelectric Plates

The variational asymptotic method is used to construct a generalized Reissner-Mindlin model for multilayer piezoelectric plates with faces surfaces or other surfaces parallel to the reference surface coated with electrodes. Without invoking a priori kinematic assumptions, we asymptotically split the original three-dimensional electromechanical problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The through-the-thickness analysis is implemented using the finite element method into the computer program VAPAS (Variational Asymptotic Plate and Shell Analysis). The resulting model is as simple as an equivalent single-layer, first-order shear deformation theory with accuracy comparable to higher-order layerwise theories. Numerical results of cylindrical bending problems for piezoelectric plates have been compared with 3D exact solutions to validate the present model.

Derivation of Plate Theory Accounting Asymptotically Correct Shear Deformation

1997 ◽  
Vol 64 (4) ◽  
pp. 905-915 ◽  
Author(s):  
V. G. Sutyrin
Keyword(s):  
Shear Deformation ◽  
Asymptotic Method ◽  
Plate Theory ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
Optimization Procedure ◽  
Laminated Plates ◽  
Orthotropic Plates ◽  
Variational Asymptotic ◽  
Variational Asymptotic Method

The focus of this paper is the development of linear, asymptotically correct theories for inhomogeneous orthotropic plates, for example, laminated plates with orthotropic laminae. It is noted that the method used can be easily extended to develop nonlinear theories for plates with generally anisotropic inhomogeneity. The development, based on variational-asymptotic method, begins with three-dimensional elasticity and mathematically splits the analysis into two separate problems: a one-dimensional through-the-thickness analysis and a two-dimensional “plate” analysis. The through-the-thickness analysis provides elastic constants for use in the plate theory and approximate closed-form recovering relations for all truly three-dimensional displacements, stresses, and strains expressed in terms of plate variables. In general, the specific type of plate theory that results from variational-asymptotic method is determined by the method itself. However, the procedure does not determine the plate theory uniquely, and one may use the freedom appeared to simplify the plate theory as much as possible. The simplest and the most suitable for engineering purposes plate theory would be a “Reissner-like” plate theory, also called first-order shear deformation theory. However, it is shown that construction of an asymptotically correct Reissner-like theory for laminated plates is not possible in general. A new point of view on the variational-asymptotic method is presented, leading to an optimization procedure that permits a derived theory to be as close to asymptotical correctness as possible while it is a Reissner-like. This uniquely determines the plate theory. Numerical results from such an optimum Reissner-like theory are presented. These results include comparisons of plate displacement as well as of three-dimensional field variables and are the best of all extant Reissner-like theories. Indeed, they even surpass results from theories that carry many more generalized displacement variables. Although the derivation presented herein is inspired by, and completely equivalent to, the well-known variational-asymptotic method, the new procedure looks different. In fact, one does not have to be familiar with the variational-asymptotic method in order to follow the present derivation.

Prediction of Mechanical Properties of Microcellular Plastics Using the Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH) Method

2013 ◽  
Author(s):  
Emily Yu ◽  
Lih-Sheng Turng
Keyword(s):  
Mechanical Properties ◽  
Asymptotic Method ◽  
Thermoplastic Polyurethane ◽  
Three Dimensional ◽  
Unit Cell ◽  
Element Analysis ◽  
Microcellular Plastics ◽  
Variational Asymptotic ◽  
Variational Asymptotic Method ◽  
Injection Molded

This work presents the application of the micromechanical variational asymptotic method for unit cell homogenization (VAMUCH) with a three-dimensional unit cell (UC) structure and a coupled, macroscale finite element analysis for analyzing and predicting the effective elastic properties of microcellular injection molded plastics. A series of injection molded plastic samples — which included polylactic acid (PLA), polypropylene (PP), polystyrene (PS), and thermoplastic polyurethane (TPU) — with microcellular foamed structures were produced and their mechanical properties were compared with predicted values. The results showed that for most material samples, the numerical prediction was in fairly good agreement with experimental results, which demonstrates the applicability and reliability of VAMUCH in analyzing the mechanical properties of porous materials. The study also found that material characteristics such as brittleness or ductility could influence the predicted results and that the VAMUCH prediction could be improved when the UC structure was more representative of the real composition.

Layer-Wise Mixed Models for Accurate Vibrations Analysis of Multilayered Plates

1998 ◽  
Vol 65 (4) ◽  
pp. 820-828 ◽  
Author(s):  
E. Carrera
Keyword(s):  
Mixed Models ◽  
Dynamic Equilibrium ◽  
Variational Equation ◽  
A Priori ◽  
Three Dimensional ◽  
Single Layer ◽  
Multilayered Plates ◽  
Three Dimensional Elasticity ◽  
Free Vibration Response ◽  
Principle Of Virtual Displacements

This paper presents the dynamic analysis of multilayered plates using layer-wise mixed theories. With respect to existing two-dimensional theories at the displacement formulated, the proposed models a priori fulfill the continuity of transverse shear and normal stress components at each interface between two adjacent layers. A Reissner’s mixed variational equation is employed to derive the differential equations, in terms of the introduced stress and displacement variables, that govern the dynamic equilibrium and compatibility of each layer. The continuity conditions at the interfaces are used to write corresponding equations at multilayered level. Related standard displacement formulations, based on the principle of virtual displacements, are given for comparison purposes. Numerical results are presented for the free-vibration response (fundamental and higher order frequencies are calculated) of symmetrically and unsymmetrically laminated cross-ply plates. Several comparisons to three-dimensional elasticity analysis and to some available results, related to both layer-wise and equivalent single-layer theories, have shown that the presented mixed models: (1) match the exact three-dimensional results very well and (2) lead to a better description in comparison to results related to other available analysis.

Asymptotic theory of 3D thermoelastic stress analysis of honeycomb sandwich panels with composite facesheets

2018 ◽  
Vol 22 (6) ◽  
pp. 1952-1982
Author(s):  
MV Peereswara Rao ◽  
K Renji ◽  
Dineshkumar Harursampath
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Present Theory ◽  
Two Dimensional ◽  
Composite Sandwich ◽  
Variational Asymptotic ◽  
Small Parameters ◽  
Dimensional Finite Element ◽  
Variational Asymptotic Method ◽  
Three Dimensional Finite Element

This work presents an asymptotical thermoelastic model for analyzing symmetric composite sandwich plate structures. Use of three-dimensional finite elements to analyze real-life composite sandwich structures is computationally prohibitive, while use of two-dimensional finite element cannot accurately predict the transverse stresses and three-dimensional displacements. Endeavoring to fill this gap, the present theory is developed based on the variational asymptotic method. The unique features of this work are the identification and utilization of small parameters characterizing the geometry and material stiffness coefficients of sandwich structural panels in addition to the small parameters pertaining to any plate-like structure. In this formulation, using variational asymptotic method, the three-dimensional thermoelastic problem is mathematically split into a one-dimensional through-the-thickness analysis, and a two-dimensional reference surface analysis. The through-the-thickness analysis provides the constitutive relation between the generalized two-dimensional strains, and the generalized force resultants for the plate analysis, it also provides a set of closed-form solutions to express the three-dimensional responses in terms of two-dimensional variables, which are determined by solving the equilibrium equations of the plate reference surface. Numerical results are illustrated for a typical composite sandwich panel subjected to a linear-bisinusoidal thermal loading. The three-dimensional responses of the composite sandwich structure from the present theory are compared with the three-dimensional finite element solutions of MSC NASTRAN®. The results from the present theory agree closely with three-dimensional finite element results and yet enable order of magnitude saving in computational resources and time.

Application of Variational Asymptotic Micromechanical Method to Strength Analysis of Composite Materials

2019 ◽  
Vol 801 ◽  
pp. 95-100
Author(s):  
Dileep Kumar ◽  
Dineshkumar Harursampath
Keyword(s):  
A Priori ◽  
Plastic Region ◽  
Mathematical Formulation ◽  
Matrix Material ◽  
Transversely Isotropic ◽  
Constitutive Relationship ◽  
Strength Analysis ◽  
The Matrix ◽  
Variational Asymptotic ◽  
Variational Asymptotic Method

One of the most important features of a material to know before using it is the maximum limit of the load at which it fails. This paper presents a micromechanical strength theory to estimate the tensile strength of the unidirectional fiber reinforced composite. The fibers used can be considered transversely isotropic and elastic till failure, but the matrix material is considered to be Elastic-plastic. The mathematical formulation used is the Variational-Asymptotic Method (VAM), which is used to construct the asymptotically-correct a reduced-dimensional model that is free of a priori assumption regarding the kinematics. The 3-D strain generated in each constituent material is explicitly expressed in 1-D strains and initial curvatures. The advantage of using VAM is that the stress state correlation of constituent materials is taken care of while applying warping constraints. Prandtl-Reuss plasticity theory has been implemented for the plastic region constitutive relationship. The other advantage of this work is that the load-bearing capacity of the composite beyond the elastic region has been considered. Good agreement has been found between experimental data and VAM analysis.

STATIC AND DYNAMIC BEHAVIOUR OF LAMINATED COMPOSITE BEAMS

2001 ◽  
Vol 01 (04) ◽  
pp. 545-560 ◽  
Author(s):  
M. A. RAMOS LOJA ◽  
J. INFANTE BARBOSA ◽  
C. M. MOTA SOARES
Keyword(s):  
Cross Sections ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
Single Layer ◽  
Laminated Composite ◽  
Element Model ◽  
Composite Beams ◽  
Straight Beam ◽  
Transverse Stresses

A higher order shear deformation theory, assuming a non-linear variation for the displacement field, is used to develop a finite element model to predict static and free vibration behaviour of anisotropic multilaminated thick and thin beams. The model is based on a single-layer Lagrangean four-node straight beam element with fourteen degrees of freedom per node. It considers bending into two orthogonal planes, stretching and twisting to enable three-dimensional analysis of frames. The most common cross sections and symmetric and asymmetric lay-ups are studied. The behaviour of the model is tested on thin and thick isotropic and composite beams. Comparisons show that the model is accurate and versatile. The good performance of the present model is evident on the prediction of displacements, normal and transverse stresses and natural frequencies of thin and thick isotropic or anisotropic beam structures.

scholarly journals Analytical Study of Bending Characteristics of an Elastic Rectangular Plate using Direct Variational Energy Approach with Trigonometric Function

2021 ◽  
Vol 5 (6) ◽  
pp. 916-928
Author(s):  
F. C. Onyeka ◽  
B. O. Mama
Keyword(s):  
Deformation Theory ◽  
Thick Plate ◽  
Plate Theory ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
Energy Functional ◽  
Total Potential Energy ◽  
Total Potential ◽  
Plate Analysis ◽  
Potential Energy Functional

In this paper, an analytical three-dimensional (3D) bending characteristic of an isotropic rectangular thick plate with all edges simply supported (SSSS) and carrying uniformly distributed transverse load using the energy technique is presented. The three-dimensional constitutive relations which involves six stress components were used in the established, refined shear deformation theory to obtain a total potential energy functional. This theory obviates application of the shear correction factors for the solution to the problem. The governing equation of a thick plate was obtained by minimizing the total potential energy functional with respect to the out of plane displacement. The deflection functions which are in form of trigonometric were obtained as the solution of the governing equation. These deflection functions which are the product of the coefficient of deflection and shape function of the plate were substituted back into the energy functional, thereafter a realistic formula for calculating the deflection and stresses were obtained through minimizations with respect to the rotations and deflection coefficients. The values of the deflections and stresses obtained herein were tabulated and compared with those of previous 3D plate theory, refined plate theories and, classical plate theory (CPT) accordingly. It was observed that the result obtained herein varied more with those of CPT and RPT by 25.39% and 21.09% for all span-to-thickness ratios respectively. Meanwhile, the recorded percentage differences are as close as 7.17% for all span-to-thickness ratios, when compared with three dimensional plate analysis. This showed that exact 3D plate theory is more reliable than the shear deformation theory which are quite coarse for thick plate analysis. Doi: 10.28991/esj-2021-01320 Full Text: PDF

Three-dimensional analysis of freely vibrating multilayered piezoelectric plates through adaptive global piecewise-smooth functions

2016 ◽  
Vol 27 (20) ◽  
pp. 2862-2876 ◽  
Author(s):  
Arcangelo Messina ◽  
Erasmo Carrera
Keyword(s):  
Dynamical Behavior ◽  
Three Dimensional ◽  
Single Layer ◽  
Computational Effort ◽  
Piecewise Smooth ◽  
Three Dimensional Model ◽  
Smooth Functions ◽  
Piecewise Smooth Functions ◽  
Electric Potentials ◽  
Piezoelectric Plates

In this article, freely vibrating multilayered piezoelectric plates are analyzed through a set of adaptive global piecewise-smooth functions along with governing differential equations and associated boundary conditions, which are consistently derived from the classical theorem of virtual displacements. The analysis demonstrates the capability of the adaptive global piecewise-smooth functions to treat any multilayered plate as if it were made up of a single layer even in the presence of multiphysics analyses such as piezoelectric layers. The relevant model is essentially two-dimensional because it is based on an expansion through the thickness of the plate aimed at modeling a three-dimensional dynamical behavior. In order to demonstrate the effectiveness of the model, all the results are compared to exact three-dimensional results; these latter are extracted through a three-dimensional model based on a transfer matrix technique whose numerical stability is achieved using scaled electric potentials. The exact graphical results are herein illustrated, thus showing both the effectiveness of using weighted electric potentials and the capability of adaptive global piecewise-smooth functions to converge at exact results through a minimum computational effort.

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