Nonlinear Theory of Elastic Plates

By Anh Le Van

Nonlinear Theory of Elastic Plates provides the theoretical materials necessary for the three plate models—Cosserat plates, Reissner-Mindlin plates and Kirchhoff-Love plates— in the context of finite elastic deformations. One separate chapter is devoted to the linearized theory of Kirchhoff-Love plates, which allows for the study of vibrations of a pre-stressed plate and the static buckling of a plate. All mathematical results in the tensor theory in curvilinear coordinates necessary to investigate the plate theory in finite deformations are provided, making this a self-contained resource.

• Presents the tricky process of linearization, which is rarely dealt with, but explained in detail in a separate chapter
• Organized in a mathematical style, with definitions, hypotheses, theorems and proofs clearly stated
• Presents every theorem with its accompanying hypotheses, enabling the reader to quickly recognize the conditions of validity in results

• Language: English
• Publisher: Elsevier Science
• Release date: May 31, 2017
• ISBN: 9780081023594

Anh Le Van

Anh Le van is Professor at the University of Nantes, France. His research at the GeM (Research Institute in Civil and Mechanical Engineering) includes membrane structures and, more specifically, the problems of contact and buckling of these structures.

Book preview

Nonlinear Theory of Elastic Plates

Anh Le van

Series Editor

Noël Challamel

Table of Contents

Cover

Title page

Dedication

Copyright

Preface

Why the nonlinear framework?

Synopsis of the book

1: Fundamentals of Tensor Theory

Abstract

1.1 Tensor algebra

1.2 Tensor analysis

2: Initial Position of a Plate

Abstract

2.1 Initial position of the mid-surface of the plate

2.2 Initial position of the plate

2.3 Covariant derivative on a surface

2.4 Divergence theorem

3: Cosserat Plate Theory

Abstract

3.1 Current position of the plate mid-surface

3.2 Current position of the plate - Displacement field

3.3 Displacement gradient

3.4 Strain tensor

3.5 Velocity field

3.6 Principle of Virtual Power (PVP)

3.7 Virtual velocity field

3.8 Virtual velocity gradient

3.9 Virtual power of inertia forces

3.10 Virtual power of internal forces

3.11 Virtual power of external forces

3.12 Equations of motion and boundary conditions

3.13 Static problems

3.14 Another method to obtain the equations

3.15 Overview of the equations and unknowns

4: Reissner-Mindlin Plate Theory

Abstract

4.1 Current position of the plate mid-surface

4.2 Current position of the plate - Displacement field

4.3 Gradient of displacement

4.4 Strain tensor

4.5 Velocity field

4.6 Virtual velocity field

4.7 Virtual power of inertia forces

4.8 Virtual power of internal forces

4.9 Virtual power of external forces

4.10 Equations of motion and boundary conditions

4.11 Note on couples

4.12 Static problems

4.13 Overview of equations and unknowns

5: Kirchhoff-Love Plate Theory

Abstract

5.1 Current position of the plate mid-surface

5.2 Current position of the plate - Displacement field

5.3 Strain tensor

5.4 Velocity field

5.5 Virtual velocity field

5.6 Virtual powers of inertia forces

5.7 Virtual power of internal forces

5.8 Virtual power of external forces

5.9 Equations of motion and boundary conditions

5.10 Static problems

5.11 Overview of equations and unknowns

5.12 Example: Kirchhoff-Love plate in cylindrical bending

6: Constitutive Law of Plates

Abstract

6.1 Hyperelastic 3D constitutive law

6.2 Strains in terms of the Z-coordinate

6.3 Stress resultants for Cosserat plates

6.4 Zero normal stress hypothesis σ³³ = 0

6.5 Plane stress state

6.6 Reduced constitutive law

6.7 Stress resultants for Reissner-Mindlin plates

6.8 Stress resultants for Kirchhoff-Love plates

6.9 Review of the hypotheses used

7: Linearized Kirchhoff-Love Plate Theory

Abstract

7.1 Statement of the problem

7.2 Linearization principle

7.3 Linearization of the vectors of the current natural basis

7.4 Linearized current curvatures

7.5 Linearized current Christoffel symbols

7.6 Linearized strain tensor

7.7 Linearized integrated constitutive laws

7.8 Linearized governing equations and boundary conditions - Vibrations of a pre-stressed plate

7.9 Overview of the equations and unknowns

7.10 Displacement equations

7.11 Equilibrium of a pre-stressed plate

7.12 Plate buckling problem

7.13 Example: Buckling of a simply-supported rectangular plate

7.14 Example: Buckling of a circular plate

Appendix: Some Mechanical Relations in 3D Curvilinear Coordinates

Bibliography

Index

Dedication

To my parents

To Nicole and Younnik

To Mai

Copyright

First published 2017 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

For information on all our publications visit our website at http://store.elsevier.com/

© ISTE Press Ltd 2017

The rights of Anh Le van to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-227-4

Printed and bound in the UK and US

Preface

Anh Le van, Nantes, France March 2017

This book is an introduction to nonlinear mechanics for plates. The non-linearities in play may be of geometric origin, due to finite deformations, or of material origin, arising from the hyperelastic nonlinear constitutive laws.

Why the nonlinear framework?

The nonlinear framework applies itself naturally to problems where plates undergo finite strains, finite displacements or finite rotations, for example the bending or forming of a metal sheet.

The nonlinear theory is also found to be necessary to account for the phenomenon of plate buckling (in statics) or of plate instability (in dynamics), even if the pre-critical strains and displacements are not significant. Indeed, we cannot restrict ourselves to a purely linear theory where equations are linearized too soon, but must instead carry out (at least partially) a non-linear analysis prior to linearizing the equations. This is a longer process but it is the only one that makes it possible to obtain the terms governing the buckling.

Synopsis of the book

1. In this book we will study three plate models. From the point of view of the kinematics hypothesis, these range from the most general to the most restrictive; from the point of view of the formulation of the model, these range from the simplest to the most complex:

(a) The Cosserat plate model, whose kinematics is defined by the displacement field of the mid-surface and the field of the director vector, which is a priori arbitrary and independent of the displacement field of the mid-surface.

(b) The Reissner-Mindlin plate model, where the director vector is constrained to be a unit vector.

(c) The Kirchhoff-Love plate model, where the director vector must be both of unit length as well as orthogonal to the deformed mid-surface.

While the explanation clearly demonstrates how the models are related to each other, it also allows the reader to approach each model independently, without referring to the other models.

The governing equations of motion and the force boundary conditions will be obtained by means of the principle of virtual power. Interestingly, it can be seen that the results obtained at this stage are not subject to any hypothesis other than the kinematics assumption inherent to each plate model. Consequently, they are valid regardless of the amplitude of motion or the constitutive material. Results that depend on the material are presented separately.

2. The constitutive laws for plates will be established for hyperelastic materials. We will exclude more complicated behaviors such as elastoplasticity in finite deformations, knowing that the difficulties related to these behaviors already exist in the 3D framework and are not specific to plate models.

Having obtained the constitutive laws, we will survey the whole set of equations and verify that there are as many equations as unknowns. This also is the time for us to review the different hypotheses adopted at different stages in the plate formulations.

3. We will finally study the linearization of the Kirchhoff-Love plate theory. The linear equations that result from this enable the study of the vibration of plates around a reference configuration that may be pre-stressed. Applied to the static case, the linearized equations also enable us to solve the problem of the buckling of plates.

It will be seen that several important subjects are not touched upon in this book. For example, we do not study the constitutive laws for orthotropic plates or stratified plates, finite elastoplastic deformations of plates, or plate finite elements, which are in themselves a large research domain. Nonetheless, the author hopes that this book may offer readers a solid foundation which will allow them to then venture further into the world of more complex nonlinear plate models. Furthermore, this book will also serve as a good introduction for those who wish to study shell models, as the tensor tools used are exactly the same and as the formulation of these theories is similar in all points to plate theory.

Finally, the author would like to thank Ms. AKHILA PHADNIS for her help with the English version of this book.

1

Fundamentals of Tensor Theory

Abstract

This chapter summarizes the definitions and results of the tensor operations that are used in plate theory. It can be divided into two parts:

1. Tensor algebra, where only algebraic operations such as addition and multiplication come into play.

2. Tensor analysis, which also involves the concept of derivatives.

Keywords

Christoffel symbols; Contraction; Covariant derivative; Curvilinear coordinates; Metric tensor; Tensor algebra; Tensor product; Tensor Theory

This chapter summarizes the definitions and results of the tensor operations that are used in plate theory. It can be divided into two parts:

1. Tensor algebra, where only algebraic operations such as addition and multiplication come into play.

2. Tensor analysis, which also involves the concept of derivatives.

The results are reviewed here without the proofs being worked out. For a detailed presentation, the readers are referred to mathematical works dedicated to tensor theory.

1.1 Tensor algebra

Let us consider a 3-dimensional Euclidean vector space E, endowed with the usual scalar product (a, b a.b . A basis (g1, g2, g3), not necessarily orthonormal, is chosen beforehand for E.

1.1.1 Contravariant and covariant components of a vector

Let u be a vector in E. The components of u in the basis (g1, g2, g3) are denoted by u¹, u², u, using the Einstein summation convention over any repeated index; here, the index i varies from 1 to 3. As the basis (g1, g2, g3) is fixed, the vector u is determined by the coefficients u¹, u², u³.

On the other hand, vector u , i ∈ {1, 2, 3}. Indeed, we have

   [1.1]

By writing

   [1.2]

we can rewrite equation[1.1] in matrix form:

   [1.3]

The 3 × 3 matrix [g. .] with components gij, i, j ∈ {1, 2, 3}, is symmetrical. It is invertible because (g1, g2, g3) is a basis and, therefore, either of the triplets (u¹, u², u³) or (u1, u2, u3) allows us to determination of the other one.

Definitions [1.4]

– The contravariant components of the vector u in the basis (g1, g2, g3) are the components u¹,u²,u.

– The covariant components of the vector u in the basis (g1, g2, g3) are the coefficients u1, u2, u.

The notation convention with superscripts and subscripts (upper and lower indices) is systematically adopted in tensor theory. The advantage of this convention, as will be seen later on, is that it allows formulae to be easily read and systematically written.

. We choose a basis (g1, g, formed of two unit gg = 1), and we consider any vector u. In Fig. 1.1:

Figure 1.1 of the contravariant and covariant components of a vector u

– the contravariant components u¹, u² of vector u are the oblique components along g1 and g2.

– the covariant components u1, u2 are the orthogonal projection-value measures for u along g1 and g2.

Using this example, we can see that the contravariant and covariant components are usually distinct. According to Eq. [1.3], the necessary and sufficient condition for them to be identical is that the matrix [g. .] be equal to the identity matrix. That is, (g1, g2, g3) is orthonormal.

Theorem

Let u be a vector with contravariant components ui and covariant components ui; let v be a vector with contravariant components vi and covariant components vi. The scalar product of u and v is expressed by

   [1.5]

1.1.2 Dual basis

Notation

The components of the inverted matrix for [g. .] are designated by gij:

where δji (also written as δij.

Theorem and definition

and defined by

   [1.6]

is a basis of E. It is also called the dual basis of (g1, g2, g3), as opposed to the basis (g1, g2, g3), which is called the primal basis.

It must be pointed out that the dual basis is constructed via the following chain

The following theorem gives another characterization for the dual base in addition to definition [1.6].

Theorem

− The vectors of the primal and dual bases are orthogonal:

− Conversely, any triplet of vectors (a¹, a², a³) which verifies gi.aj = δij is identical to the dual basis: ∀i ∈ {1, 2, 3}, ai = gi.

The following relationship is homologous to [1.2]:

Theorem

In general, the dual basis differs from the primal basis, except for the following special case:

Theorem

The primal basis is orthonormal ⇔ the dual basis is identical to the primal basis.

1.1.3 Different representations of a vector

Theorem

– The following relationships exist between the contravariant and covariant components of a vector u:

We thus lower or raise the indices using the matrices gij and gij.

– The following relationship is homologous to ui ≡ u.gi:

Theorem and definition

A vector can be expressed in either the primal basis or in the dual basis, as follows:

   [1.7]

These two forms are called the contravariant and the covariant representations of u.

From the previous theorem, we can also write u ≡ (u.gi)gi = (u.gi)gi.

Theorem

The scalar product between vectors u and v may be written in different forms

   [1.8]

1.1.4 Results related to the orientation of the 3D space

The earlier results, written in three-dimensional space, may be generalized in the case of a space with n-dimensions (n being finite), using obvious notation changes. On the contrary, the results discussed in this section are only applicable to a 3-dimensional space.

As the space E is 3-dimensional, we can orient it and define a vector product (cross product) in it. We then obtain the following results related to a vector or mixed product.

Theorem

   [1.9]

The vectors g1, g2 are orthogonal to vector g³, but they are not, in general, orthogonal to vector g3, Fig. 1.2

Figure 1.2 Vector product of two vectors of the primal basis

.

Theorem

   [1.10]

Therefore, the primal and dual bases have the same orientation.

Theorem

Hypothesis: the basis (g1, g2, g3) is right-handed (from [1.10], this amounts to assuming that the basis (g¹, g², g³) is right-handed).

Then,

   [1.11]

By combining [1.9] and [1.11], we obtain

Theorem

   [1.12]

1.1.5 Tensor

Definition [1.13]

By definition, a tensor of order p, where p is a nonzero integer, is a multilinear form of order p over Ep. More precisely, if the form T

is a tensor of order p, it satisfies the following p-linearity properties:

   [1.14]

Tensorial algebra is, thus, multilinear algebra.

Let us adopt the following generic system of notations:

– a 1st-order tensor is denoted by a letter with a bar over it, for example ā,

). However, in this book we will use bold-type symbols (as for vectors), T for instance,

.

Definition [1.13] is intrinsic in that it does not call upon the basis of E. In the following section, we will give the image of a tensor by means of the basis (g1, g2, g3) (and its dual basis (g¹, g², g³)).

Theorem and definition

Let ā be a 1st-order tensor. We have

   [1.15]

where

, i ∈ {1, 2, 3}, are called the covariant components of the 1st-order tensor ā,

, i