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1996 | Book

Mathematical Analysis of Thin Plate Models

Authors: Philippe Destuynder, Michel Salaun

Publisher: Springer Berlin Heidelberg

Book Series : Mathématiques et Applications

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About this book

Shells and plates have been widely studied by engineers during the last fifty years. As a matter of fact an important number of papers have been based on analytical calculations. More recently numerical simulations have been extensively used. for instance for large displacement analysis. for shape optimization or even -in linear analysis -for composite material understanding. But all these works lie on a choice of a finite element scheme which contains usually three kinds of approximations: 1. a plate or shell mndel including smnll parameters associated to the thickness, 2. an approximntion of the geometry (the medium sUrface of a shell and its boundary), 3. afinite element scheme in order to solve the mndel chosen. VI Obviously the conclusions that we can draw are very much depending on the quality of the three previous choices. For instance composite laminated plates with damage like a delamination is still an open problem even if interesting papers have already been published and based on numerical simulation using existing fmite element and even plate models. • In our opinion the understanding of plate modelling is still an area of interest. Furthermore the links between the various models have to be handled with care. The certainly best understood model is the Kirchhoff-Love model which was completely justified by P. O. Ciarlet and Ph. Destuynder in linear analysis using asymptotic method. But the conclusion is not so clear as far as large displacements are to be taken into account.

Table of Contents

Frontmatter
Chapter 1. Plate Models for Thin Structures
Abstract
In order to simplify three-dimensional equations, mechanical assumptions can be used in order to derive a simpler model for thin structures. The Kirchhoff-Love model is obtained by this method. But the Naghdi formulation is also an interesting plate model. In this chapter, both are derived from the three-dimensional theory, using the Hellinger-Reissner mixed formulation.
Philippe Destuynder, Michel Salaun
Chapter 2. Variational Formulations for Bending Plates
Abstract
From the continuous point of view, the bending plate model involves a fourth-order operator. The goal of this chapter is to give several variational formulations of the plate model which only involve second order operators. First of all, we shall derive simple penalty methods based on the introduction of the rotation of the unit normal to the medium surface of the plate. But a major difficulty appears due to the boundary conditions on the edges of the plate. Then a more general mixed formulation is derived, for which several numerical applications are given in the other chapters. Let us go further in the details of the formulations which are explored. After a brief recall of the classical primal formulation, we focus historically on the first attempt to derive a mixed formulation. The idea is due to R. Glowinski. We extend it to more general boundary conditions.
Philippe Destuynder, Michel Salaun
Chapter 3. Finite Element Approximations for Several Plate Models
Abstract
After a brief reminder of finite elements methods we discuss the possible approximations of the penalty or modified penalty model which has been studied in Chapter II. The connections with the famous QUAD 4 element (and its numerous variants) are examined. This permits one to give a partial justification of this well-known element and an extension to general cases including triangles. Then the natural duality technique is used to construct a new kind of structural finite elements which are analyzed from the error point of view.
Philippe Destuynder, Michel Salaun
Chapter 4. Numerical Tests for the Mixed Finite Element Schemes for Bending Plates
Abstract
The numerical performances of the finite element schemes discussed in the previous chapters are presented. The results obtained are compared to the ones of the QUAD 4 element of MacNeal. Additionnally the solution methods are evaluated, and the vector and parallel optimization has been used in order to deliver the best computational time.
Philippe Destuynder, Michel Salaun
Chapter 5. A Numerical Model for Delamination of Composite Multilayered Plates
Abstract
After a brief recall on the laminated plate theory, a computational model for studying the delamination is presented. In order to avoid a too complicated mathematical justification (which is included in [1]), we base our developments on physical feelings. Then a mechanical example including comparison with experiments is discussed. All the chapter is a survey of a study performed five years ago with our friend Thierry Nevers. The extended versions can be found in the references [2], [3], [4], [5], [6].
Philippe Destuynder, Michel Salaun
Backmatter
Metadata
Title
Mathematical Analysis of Thin Plate Models
Authors
Philippe Destuynder
Michel Salaun
Copyright Year
1996
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-51761-7
Print ISBN
978-3-540-61167-7
DOI
https://doi.org/10.1007/978-3-642-51761-7