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.

Enhanced First-Order Shear Deformation Theory for Laminated and Sandwich Plates

2005 ◽  
Vol 72 (6) ◽  
pp. 809-817 ◽  
Author(s):  
Jun-Sik Kim ◽  
Maenghyo Cho
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Plate Theory ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Laminated Plates ◽  
Sandwich Plates ◽  
First Order ◽  
Transverse Shear Strains

A new first-order shear deformation theory (FSDT) has been developed and verified for laminated plates and sandwich plates. Based on the definition of Reissener–Mindlin’s plate theory, the average transverse shear strains, which are constant through the thickness, are improved to vary through the thickness. It is assumed that the displacement and in-plane strain fields of FSDT can approximate, in an average sense, those of three-dimensional theory. Relationship between FSDT and three-dimensional theory has been systematically established in the averaged least-square sense. This relationship provides the closed-form recovering relations for three-dimensional variables expressed in terms of FSDT variables as well as the improved transverse shear strains. This paper makes two main contributions. First an enhanced first-order shear deformation theory (EFSDT) has been developed using an available higher-order plate theory. Second, it is shown that the displacement fields of any higher-order plate theories can be recovered by EFSDT variables. The present approach is applied to an efficient higher-order plate theory. Comparisons of deflection and stresses of the laminated plates and sandwich plates using present theory are made with the original FSDT and three-dimensional exact solutions.

An Improved Reissner-Mindlin Plate Theory for Composite Laminates

Aerospace ◽  
2004 ◽  
Author(s):  
Wenbin Yu
Keyword(s):  
Exact Solution ◽  
Asymptotic Method ◽  
Composite Laminates ◽  
Plate Theory ◽  
Three Dimensional ◽  
Mindlin Plate ◽  
Two Dimensional ◽  
Previous Theory ◽  
Variational Asymptotic ◽  
Variational Asymptotic Method

An improved Reissner-Mindlin theory for composite laminates without invoking ad hoc kinematic assumptions is constructed using the variational-asymptotic method. Instead of assuming a priori the distribution of three-dimensional displacements in terms of two-dimensional plate displacements as what is usually done in typical plate theories, an exact intrinsic formulation has been achieved by introducing unknown three-dimensional warping functions. Then the variational-asymptotic method is applied to systematically decouple the original three-dimensional problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The resulting theory is an equivalent single-layer Reissner-Mindlin theory with an excellent accuracy comparable to that of higher-order, layerwise theories. The present work is extended from the previous theory developed by the writer and his co-workers with two sizable contributions: (a) six more constants (33 in total) are introduced to allow maximum freedom to transform the asymptotically correct energy into a Reissner-Mindlin model; and (b) the semi-definite programming technique is used to seek the optimum Reissner-Mindlin model. Furthermore, it is proved the first time that the recovered three-dimensional quantities exactly satisfy the continuity conditions on the interface between different layers and traction boundary conditions on the bottom and top surfaces. It is also shown that that two of the equilibrium equations of three-dimensional elasticity can be satisfied asymptotically, and the third one can be satisfied approximately so that the difference between the Reissner-Mindlin model and the second order asymptotical energy can be minimized. Numerical examples are presented to compare with the exact solution as well as the classical lamination theory and first-order shear-deformation, demonstrating that the present theory has an excellent agreement with the exact solution.

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.

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.

Effect of Stress Concentration on Laminated Plates

2012 ◽  
Vol 29 (2) ◽  
pp. 241-252 ◽  
Author(s):  
A. S. Sayyad ◽  
Y. M. Ghugal
Keyword(s):  
Stress Concentration ◽  
Shear Deformation ◽  
Transverse Shear ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
Laminated Plates ◽  
Shear Stresses ◽  
Normal Strain ◽  
Concentrated Load ◽  
Transverse Shear Stresses

AbstractThis paper deals with the problem of stress distribution in orthotropic and laminated plates subjected to central concentrated load. An equivalent single layer trigonometric shear deformation theory taking into account transverse shear deformation effect as well as transverse normal strain effect is used to obtain in-plane normal and transverse shear stresses through the thickness of plate. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. A simply supported plate with central concentrated load is considered for the numerical analysis. Anomalous behavior of inplane normal and transverse shear stresses is observed due to effect of stress concentration compared to classical plate theory and first order shear deformation theory.

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