Theory of Shells

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ISBN: 9780444828910 | 0444828915
Cover: Hardcover
Copyright: 5/11/2000

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The objective of Volume III is to lay down the proper mathematical foundations of the two-dimensional theory of shells. To this end, it provides, without any recourse to any a priori assumptions of a geometrical or mechanical nature, a mathematical justification of two-dimensional nonlinear and linear shell theories, by means of asymptotic methods, with the thickness as the "small" parameter.

Mathematical Elasticity: General plan

Mathematical Elasticity: General preface

Preface to Volume I

Preface to Volume II

Preface to Volume III

xxvii

Differential geometry at a glance

xxxix

Three-dimensional elasticity in curvilinear coordinates at a glance

xlvii

Two-dimensional linear shell equations at a glance

Two-dimensional nonlinear shell equations at a glance

lvii

PART A. LINEAR SHELL THEORY

Three-dimensional linearized elasticity and Korn's inequalities in curvilinear coordinates

(58)

Introduction

(2)

Three-dimensional linearized elasticity in Cartesian coordinates

(7)

Curvilinear coordinates and metric tensor in a three-dimensional domain

(10)

The variational equations of three-dimensional linearized elasticity in curvilinear coordinates

(10)

Covariant derivatives and Christoffel symbols in a three-dimensional domain

(3)

Linearized change of metric tensor in curvilinear coordinates

(2)

The boundary value problem of three-dimensional linearized elasticity in curvilinear coordinates

(3)

A lemma of J. L. Lions; three-dimensional Korn's inequalities and infinitesimal rigid displacement lemma in curvilinear coordinates

(10)

Existence and uniqueness theorem in curvilinear coordinates

(4)

Complement: Recovery of a three-dimensional manifold from its metric tensor field

(7)

Exercises

(5)

Inequalities of Korn's type on surfaces

(76)

Introduction

(2)

Curvilinear coordinates and metric tensor on a surface

(10)

Curvature tensor on a surface

(12)

Covariant derivatives and Christoffel symbols on a surface

(5)

Linearized change of metric tensor on a surface

(3)

Linearized change of curvature tensor on a surface

(7)

Inequalities of Korn's type and infinitesimal rigid displacement lemma on a general surface

100

(17)

Inequality of Korn's type and infinitesimal rigid displacement lemma on an elliptic surface

117

(13)

Complement: Recovery of a surface from its metric and curvature tensor fields

130

(7)

Exercises

132

(5)

Asymptotic analysis of linearly elastic shells: Preliminaries and outline

137

(56)

Introduction

137

(4)

The three-dimensional equations of a linearly elastic shell

141

(8)

The three-dimensional equations over a domain independent of ε

149

(5)

Geometrical and mechanical preliminaries

154

(7)

The two-dimensional equations of linearly elastic ``membrane'' and ``flexural'' shells derived by means of a formal asymptotic analysis

161

(22)

Summary of the convergence theorems

183

(10)

Exercises

190

(3)

Linearly elastic elliptic membrace shells

193

(48)

Introduction

193

(3)

Linearly elastic elliptic membrance shells: Definition, example, and assumptions on the data; the three-dimensional equations over a domain independent of ε

196

(5)

Averages with respect to the transverse variable

201

(4)

A three-dimensional inequality of Korn's type for a family of linearly elastic elliptic membrane shells

205

(4)

Convergence of the scaled displacements as ε → 0

209

(14)

The two-dimensional equations of a linearly elastic elliptic membrane shell; existence, uniqueness, and regularity of solutions; formulation as a boundary value problem

223

(7)

Justification of the two-dimensional equations of a linearly elastic elliptic membrane shell; commentary and refinements

230

(11)

Exercises

235

(6)

Linearly elastic generalized membrane shells

241

Introduction

241

(4)

Linearly elastic generalized membrane shells: Definition and assumptions on the data; the three-dimensional equations over a domain independent of ε

245

(3)

Analytical preliminaries

248

(10)

A three-dimensional inequality of Korn's type for a family of linearly elastic shells

258

(3)

Generalized membrane shells of the first and second kinds

261

(3)

Admissible applied forces

264

(2)

Convergence of the scaled displacements as ε → 0

266

(21)

The two-dimensional equations of a linearly elastic generalized membrane shell; existence and uniqueness of solutions

287

(4)

Justification of the two-dimensional equations of a linearly elastic generalized membrane shell; examples, commentary, and refinements

291

(6)

Exercises

297

Linearly elastic flexural shells

209

(124)

Introduction

299

(3)

Linearly elastic flexural shells: Definition, examples, and assumptions on the data; the three-dimensional equations over a domain independent of ε

302

(6)

Convergence of the scaled displacements as ε → 0

308

(9)

The two-dimensional equations of a linearly elastic flexural shell; existence and uniqueness of solutions

317

(6)

Justification of the two-dimensional equations of a linearly elastic flexural shell; commentary and refinements

323

(10)

Exercises

328

(5)

Koiter's equations and other two-dimensional linear shell theories

333

(48)

Introduction

333

(2)

The two-dimensional Koiter equations for a linearly elastic shell: Existence, uniqueness, and regularity of solutions; formulation as a boundary value problem

335

(10)

Justification of Koiter's equations for all types of linearly elastic shells

345

(15)

Koiter's equations: Additional commentary and bibliographical notes

360

(3)

The two-dimensional Naghdi equations for a linearly elastic shell; existence and uniqueness of solutions

363

(4)

Other linear shell theories

367

(2)

Linear shallow shell theories

369

(12)

Exercises

372

(9)

PART B. NONLINEAR SHELL THEORY

Asymptotic analysis of nonlinearly elastic shells: Preliminaries

381

(52)

Introduction

381

(5)

Three-dimensional nonlinear elasticity in Cartesian coordinates

386

(6)

Three-dimensional nonlinear elasticity in curvilinear coordinates

392

(11)

The three-dimensional equations of a nonlinearly elastic shell

403

(4)

The three-dimensional equations over a domain independent of ε

407

(4)

Geometrical and mechanical preliminaries

411

(2)

The method of formal asymptotic expansions

413

(2)

The leading term is of order zero

415

(9)

Identification of a two-dimensional variational problem satisfied by the leading term

424

(9)

Exercises

430

(3)

Nonlinearly elastic membrane shells

433

(36)

Introduction

433

(3)

Nonlinearly elastic membrane shells: Definition, examples, and assumptions on the data

436

(7)

The two-dimensional equations as a variational problem

443

(2)

The two-dimensional equations as a minimization problem

445

(2)

The two-dimensional equations of a nonliearly elastic membrane shell derived by means of a formal asymptotic analysis; commentary

447

(6)

The two-dimensional equations of a nonlinearly elastic membrane shell derived by means of Γ-convergence theory; commentary

453

(16)

Exercises

465

(4)

Nonlinearly elastic flexural shells

469

(76)

Introduction

469

(3)

Identification of a two-dimensional variational problem satisfied by the leading term when there are nonzero admissible inextensional displacements

472

(30)

Nonlinearly elastic flexural shells: Definition, examples, and assumptions on the data

502

(5)

The two-dimensional equations as a variational problem

507

(12)

The two-dimensional equations as a minimization problem

519

(2)

The two-dimensional equations of a nonlinearly elastic flexural shell derived by means of a formal asymptotic analysis; commentary

521

(5)

Existence of solutions to the minimization problem

526

(19)

Exercises

539

(6)

Koiter's equations and other two-dimensional nonlinear shell theories

545

(12)

Introduction

545

(1)

The two-dimensional Koiter equations for a nonlinearly elastic shell

545

(4)

Other nonlinear shell theories

549

(3)

Nonlinear shallow shell theories

552

(5)

References

557

(26)

Index

583

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Theory of Shells

Summary

Table of Contents

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