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An Introduction to Differential Geometry with Applications to Elasticity an Introduction to Differential Geometry with Applications to Elasticity

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An Introduction to Differential Geometry with Applications to Elasticity an Introduction to Differential Geometry with Applications to Elasticity AN INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH APPLICATIONS TO ELASTICITY AN INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH APPLICATIONS TO ELASTICITY Philippe G. CIARLET City University of Hong Kong Reprinted from Journal of Elasticity, Vol. 78–79 (2005) Library of Congress Cataloging-in-Publication Data A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-10 1-4020-4247-7 (HB) ISBN-13 987-1-4020-4247-8 (HB) ISBN-10 1-4020-4248-5 (e-book) ISBN-13 978-1-4020-4248-5 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com Printed on acid-free paper All Rights Reserved © 2005 Springer No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in the Netherlands Contents Preface 1 1 Three-dimensional differential geometry Introduction.............................. 3 1.1 Curvilinear coordinates . 5 1.2Metrictensor............................. 7 1.3 Volumes, areas, and lengths in curvilinear coordinates . 10 1.4Covariantderivativesofavectorfield................ 13 1.5 Necessary conditions satisfied by the metric tensor; the Riemann curvaturetensor........................... 18 1.6 ExistenceofanimmersiondefinedonanopensetinR3 with a prescribedmetrictensor....................... 19 1.7 Uniqueness up to isometries of immersions with the same metric tensor................................. 30 1.8 Continuity of an immersion as a function of its metric tensor . 38 2 Differential geometry of surfaces Introduction.............................. 53 2.1 Curvilinear coordinates on a surface . 55 2.2 First fundamental form . 59 2.3 Areas and lengths on a surface . 61 2.4 Second fundamental form; curvature on a surface . 63 2.5 Principal curvatures; Gaussian curvature . 67 2.6 Covariant derivatives of a vector field defined on a surface; the GaußandWeingartenformulas................... 73 2.7 Necessary conditions satisfied by the first and second fundamen- tal forms: the Gauß and Codazzi-Mainardi equations; Gauß’ TheoremaEgregium......................... 76 2.8 Existence of a surface with prescribed first and second fundamen- talforms................................ 79 2.9 Uniqueness up to proper isometries of surfaces with the same fundamental forms . 89 2.10 Continuity of a surface as a function of its fundamental forms . 94 Contents 3 Applications to three-dimensional elasticity in curvilinear coordinates Introduction.............................. 103 3.1 The equations of nonlinear elasticity in Cartesian coordinates . 106 3.2 Principle of virtual work in curvilinear coordinates . 113 3.3 Equations of equilibrium in curvilinear coordinates; covariant derivatives of a tensor field ..................... 121 3.4 Constitutive equation in curvilinear coordinates . 123 3.5 The equations of nonlinear elasticity in curvilinear coordinates . 124 3.6 The equations of linearized elasticity in curvilinear coordinates . 126 3.7 A fundamental lemma of J.L. Lions . 129 3.8 Korn’s inequalities in curvilinear coordinates . 131 3.9 Existence and uniqueness theorems in linearized elasticity in curvi- linearcoordinates........................... 138 4 Applications to shell theory Introduction.............................. 147 4.1ThenonlinearKoitershellequations................ 149 4.2ThelinearKoitershellequations.................. 158 4.3 Korn’sinequalitiesonasurface................... 166 4.4 Existence and uniqueness theorems for the linear Koiter shell equations; covariant derivatives of a tensor field defined on a surface................................. 179 4.5 A brief review of linear shell theories . 187 References 195 Index 203 # Springer 2005 PREFACE This book is based on lectures delivered over the years by the author at the Universit´e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at City University of Hong Kong. Its two-fold aim is to give thorough introduc- tions to the basic theorems of differential geometry and to elasticity theory in curvilinear coordinates. The treatment is essentially self-contained and proofs are complete. The prerequisites essentially consist in a working knowledge of basic notions of anal- ysis and functional analysis, such as differential calculus, integration theory and Sobolev spaces, and some familiarity with ordinary and partial differential equations. In particular, no aprioriknowledge of differential geometry or of elasticity theory is assumed. In the first chapter, we review the basic notions, such as the metric tensor and covariant derivatives, arising when a three-dimensional open set is equipped with curvilinear coordinates. We then prove that the vanishing of the Riemann curvature tensor is sufficient for the existence of isometric immersions from a simply-connected open subset of Rn equipped with a Riemannian metric into a Euclidean space of the same dimension. We also prove the corresponding uniqueness theorem, also called rigidity theorem. In the second chapter, we study basic notions about surfaces, such as their two fundamental forms, the Gaussian curvature and covariant derivatives. We then prove the fundamental theorem of surface theory, which asserts that the Gauß and Codazzi-Mainardi equations constitute sufficient conditions for two matrix fields defined in a simply-connected open subset of R2 to be the two fundamental forms of a surface in a three-dimensional Euclidean space. We also prove the corresponding rigidity theorem. In addition to such “classical” theorems, which constitute special cases of the fundamental theorem of Riemannian geometry, we also include in both chapters recent results which have not yet appeared in book form, such as the continuity of a surface as a function of its fundamental forms. The third chapter, which heavily relies on Chapter 1, begins by a detailed derivation of the equations of nonlinear and linearized three-dimensional elastic- ity in terms of arbitrary curvilinear coordinates. This derivation is then followed by a detailed mathematical treatment of the existence, uniqueness, and regu- larity of solutions to the equations of linearized three-dimensional elasticity in 1 2 Preface curvilinear coordinates. This treatment includes in particular a direct proof of the three-dimensional Korn inequality in curvilinear coordinates. The fourth and last chapter, which heavily relies on Chapter 2, begins by a detailed description of the nonlinear and linear equations proposed by W.T. Koiter for modeling thin elastic shells. These equations are “two-dimensional”, in the sense that they are expressed in terms of two curvilinear coordinates used for defining the middle surface of the shell. The existence, uniqueness, and regularity of solutions to the linear Koiter equations is then established, thanks this time to a fundamental “Korn inequality on a surface” and to an “infinites- imal rigid displacement lemma on a surface”. This chapter also includes a brief introduction to other two-dimensional shell equations. Interestingly, notions that pertain to differential geometry per se,suchas covariant derivatives of tensor fields, are also introduced in Chapters 3 and 4, where they appear most naturally in the derivation of the basic boundary value problems of three-dimensional elasticity and shell theory. Occasionally, portions of the material covered here are adapted from ex- cerpts from my book “Mathematical Elasticity, Volume III: Theory of Shells”, published in 2000 by North-Holland, Amsterdam; in this respect, I am indebted to Arjen Sevenster for his kind permission to rely on such excerpts. Other- wise, the bulk of this work was substantially supported by two grants from the Research Grants Council of Hong Kong Special Administrative Region, China [Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604]. Last but not least, I am greatly indebted to Roger Fosdick for his kind suggestion some years ago to write such a book, for his permanent support since then, and for his many valuable suggestions after he carefully read the entire manuscript. Hong Kong, July 2005 Philippe G. Ciarlet Department of Mathematics and Liu Bie Ju Centre for Mathematical Sciences City University of Hong Kong.
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