COSSERAT THEORIES: SHELLS, RODS AND POINTS

COSSERAT THEORIES: SHELLS, RODS AND POINTS

SOLID MECHANICS AND ITS APPLICATIONS Volume 79 Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written bij authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages.

Cosserat Theories: Shells, Rods and Points by M.B. Rubin Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, Israel Springer-Science+Business Media, B.Y.

A C.l.P. C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5531-6 ISBN 978-94-015-9379-3 (ebook) DOI 10.1007/978-94-015-9379-3 Printed on acidjree acid-free paper All Rights Reserved 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000. Softcover reprint ofthe hardcover 1st edition 2000 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, inciuding including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

DEDICATION This book is dedicated to my loving wife Laurel and my sons Adam and Daniel. They have created a beautiful family environment which has given me the necessary peace of mind to concentrate on my research. IN MEMORIAL I have had the great honor and pleasure to have worked for 21 years with Professor Paul M. Naghdi of the Department of Mechanical Engineering at the University of California at Berkeley. I began my work with him in August 1973 during my first quarter at Berkeley as a graduate student and I continued working with him as a colleague until his death on the 9th of July 1994. The idea for this book was conceived after Paul's death as an attempt to preserve his unique approach to the formulation of Cosserat theories. Although the notation and some of the results presented in this book are new, Paul has had such an influence on my thinking that I share all of the originality in this book with him. However, I accept full responsibility for any conceptual or typographical errors.

CONTENTS Preface... xiii CHAPTER 1 Introduction 1 1. 1 The basic idea of a Cosserat model... 1 I.2 A brief outline of the book... 3 1.3 Notation... 9 CHAPTER 2 Basic Tensor Operations in Curvilinear Coordinates 11 2.1 Covariant and contravariant base vectors... 11 2.2 Base tensors and components of tensors... 13 2.3 Basic tensor operations... 15 2.4 Covariant differentiation and Christoffel symbols... 17 CHAPTER 3 Three-Dimensional Continua 19 3. I Configurations and motion... 19 3.2 Balance laws... 21 3.3 Invariance under superposed rigid body motions... 27 3.4 Mechanical power... 34 3.5 An alternative derivation of the balance laws... 35 3.6 An averaged form of the balance of linear momentum... 37 3.7 Anisotropic nonlinear elastic materials... 38 3.8 Constraints....40 3.9 Initial and boundary conditions....43 3.10 Material Symmetry....44 3. 11 Isotropic nonlinear elastic materials....47 3.12 A small strain theory... 51 3.13 Small deformations superimposed on a large deformation... 54 3.14 Pure bending of an orthotropic rectangular parallelepiped... 57 3.15 Torsion of an orthotropic rectangular parallelepiped... 60 3.16 Forced shearing vibrations of an orthotropic rectangular parallelepiped... 63 3.17 Free isochoric vibrations of an isotropic cube... 65 3.18 An orthotropic rectangular parallelepiped loaded by its own weight.... 66 3.19 An isotropic circular cylinder loaded by its own weight.... 67 3.20 Plane strain free vibrations of an isotropic solid circular cylinder.... 68 3.21 Dissipation inequality and material damping... 69 CHAPTER 4 Cosserat Shells 73 4.1 Description of a shell structure... 73 4.2 The Cosserat model of a shell... 77 vii

viii CONTENTS 4.3 Derivation of the balance laws from the three-dimensional theory... 80 4.4 Balance laws by the direct approach... 87 4.5 Invariance under superposed rigid body motions... 92 4.6 Mechanical power... 93 4.7 An alternative derivation of the balance laws... 95 4.8 Anisotropic nonlinear elastic shells... 97 4.9 Constraints... 100 4.10 Initial and boundary conditions........ 106 4.11 Further restrictions on constitutive equations for shells constructed from homogeneous anisotropic nonlinear elastic materials... 108 4.12 A small strain theory... 113 4.13 Small deformations superimposed on a large deformation... 117 4. 14 Pure bending of an orthotropic rectangular plate... 121 4.15 Torsion of an orthotropic rectangular plate... 129 4.16 Forced shearing vibrations of an orthotropic rectangular plate... 134 4.17 Free isochoric vibrations of an isotropic cube... 136 4.18 An orthotropic rectangular plate loaded by its own weight.... 137 4. 19 Elastic shells........................................................................ 140 4.20 Plane strain expansion of an isotropic circular cylindrical shell... 143 4.21 Plane strain free vibrations of an isotropic solid circular cylinder...,... 147 4.22 Expansion of an isotropic spherical shell..... 149 4.23 Free vibrations of an isotropic solid sphere... 156 4.24 An isotropic circular cylindrical shell loaded by its own weight.... 158 4.25 Isotropic nonlinear elastic shells... 161 4.26 A simple derivation of the local equations for shells... 163 4.27 A brief summary of the equations for shells... 165 4.28 Generalized membranes and membrane-like shells... 170 4.29 Simple membranes... 172 4.30 Expansion of an incompressible isotropic spherical shell... 175 4.31 Bending of an orthotropic plate into a circular cylindrical surface... 179 4.32 Linear theory of an isotropic plate... 183 4.33 Dissipation inequality and material damping... 187 CHAPTER 5 Cosserat Rods 191 5. 1 Description of a rod structure... 191 5.2 The Cosserat model of a rod... 194 5.3 Derivation of the balance laws from the three-dimensional theory... 197 5.4 Balance laws by the direct approach... 204 5.5 Invariance under superposed rigid body motions... 207 5.6 Mechanical power... 208 5.7 An alternative derivation of the balance laws... 210

CONTENTS ix IX 5.8 Anisotropic nonlinear elastic rods... 212 5.9 Constraints... 216 5.10 Initial and boundary conditions... " 222 5.11 Further restrictions on constitutive equations for rods constructed from homogeneous anisotropic nonlinear elastic materials... " 224 5. 12 A small strain theory... 229 5.13 Small deformations superimposed on a large deformation... '" 2322 5.14 Pure bending of an orthotropic beam with rectangular cross-section... 235 5.15 Torsion of an orthotropic beam with rectangular cross-section... 243 5. 16 Inhomogeneous shear of an orthotropic beam with rectangular cross-section... '" 245 5. 17 Forced shearing vibrations of an orthotropic beam with rectangular cross-section... 247 5.18 Free isochoric vibrations of an isotropic cube... '"... 250 5.19 An orthotropic beam with rectangular cross-section loaded by its own weight... 251 5.20 Elastic rods... 254 5.21 Plane strain expansion of an isotropic circular cylindrical shell... 256 5.22 Plane strain free vibrations of an isotropic solid circular cylinder... 260 5.23 An isotropic circular cylindrical shell loaded by its own weight.... 262 5.24 Isotropic nonlinear elastic rods... 265 5.25 A simple derivation of the local equations for rods with rectangular cross-sections... 266 5.26 A brief summary of the equations for rods... 270 5.27 Linearized equations for beams with rectangular cross-sections... 275 5.28 Bernoulli-Euler rods... 277 5.29 Timoshenko rods... 283 5.30 Generalized strings... 287 5.31 Simple strings... 288 5.32 Transverse loading of an isotropic beam with a rectangular cross-section.. 290 5.33 Linearized buckling equations... 293 5.34 An intrinsic formulation of Bernoulli-Euler rods with symmetric cross-sections... 303 5.35 Dissipation inequality and material damping... 309 CHAPTER 6 Cosserat Points 311 6.1 Description of a point-like structure... 311 6.2 The Cosserat point model... " '" 313 6.3 Derivation of the balance laws from the three-dimensional theory... 315 6.4 Balance laws by the direct approach... 319 6.5 Invariance under superposed rigid body motions... 321

x CONTENTS 6.6 Mechanical power.... 322 6.7 An alternative derivation of the balance laws... 323 6.8 Anisotropic nonlinear elastic Cosserat points... 325 6.9 Constraints... 328 6.10 Initial Conditions... 333 6.11 Further restrictions on constitutive equations for Cosserat points constructed from homogeneous anisotropic nonlinear elastic materials... 334 6.12 A small strain theory... 336 6.13 Small deformations superimposed on a large deformation... 337 6.14 Forced shearing vibrations of an orthotropic rectangular parallelepiped... 340 6. 15 Free isochoric vibrations of an isotropic cube... 345 6.16 Isotropic nonlinear elastic Cosserat points... 346 6.17 A brief summary of the equations for Cosserat points... 347 6.18 Dissipation inequality and material damping... 351 CHAPTER 7 Numerical Solutions using Cosserat Theories 355 7. 1 The Cosserat approach to numerical solution procedures for problems in continuum mechanics... 355 7.2 Formulation of the numerical solution of spherically symmetric problems using the theory of a Cosserat shell... 357 7.3 Formulation of the numerical solution of string problems using the theory of a Cosserat point... 378 7.4 Formulation of the numerical solution of rod problems using the theory of a Cosserat point... 394 7.5 Formulation of the numerical solution of three-dimensional problems using the theory of a Cosserat point... 410 7.6 Formulation of the numerical solution of two-dimensional problems using the theory of a Cosserat point... 418 APPENDIX A Tensors, Tensor Products and Tensor Operations in Three Dimensions 429 A.l Vectors and vector operations... 429 A.2 Tensors as linear operators... 430 A.3 Tensor products (special case)... 430 A.4 AA Indicial notation... 435 A.5 Tensor products (general case)... 437 A.6 Tensor transformation relations... 440 A.7 Additional definitions and results... 442

CONTENTS xi APPENDIX B Summary of Tensor Operations in Specific Coordinate Systems 447 B.I Cylindrical polar coordinates... 447 B.2 Spherical polar coordinates... 449 EXERCISES 451 ACKNOWLEDGMENTS 467 REFERENCES 467 INDEX 475

PREFACE Continuum mechanics provides a theoretical structure for analyzing the response of materials to mechanical and thermal loads. One of the beauties of continuum mechanics is that the fundamental balance laws (conservation of mass and balances of linear momentum, angular momentum, energy and entropy) are valid for all simple materials. Most of the modern research in continuum mechanics focuses on the development of constitutive equations which are used to characterize the response of a particular class of materials (e.g. invisicid fluids, viscous fluids, elastic solids, viscoelastic solids, elasticplastic solids, elastic-viscoplastic solids, etc.). Within the context of the purely mechanical three-dimensional theory, the conservation of mass and the balance of linear momentum are used to determine the mass density and the position of each material point in the continuum. Whereas, the balance of angular momentum and the notion of invariance under superposed rigid body motions are used to place restrictions on the constitutive equations. In this regard, it should be emphasized that these restrictions help eliminate fundamentals errors in specific constitutive assumptions, but they do not completely characterize the physics of a particular material. Ultimately, the validity of a set of constitutive equations depends on the ability of the person developing the equations to creatively synthesize the experimental data and to propose appropriate history-dependent variables and functional forms that capture the main physics of the material response. The equations characterizing the three-dimensional theory are nonlinear partial differential equations that depend on three spatial coordinates and time. Therefore, in the analysis of a particular structure, it is necessary to satisfy these equations at each point in the structure as well as to satisfy appropriate conditions on the boundary of the structure. However, some structures have special geometrical properties that can be exploited to develop simplified theories. For example, a shell-like structure is "thin" in one of its spatial dimensions, a rod-like structure is "thin" in two of its spatial dimensions, and a point-like structure is "thin" in all three of its spatial dimensions. Consequently, under certain conditions, it is possible to model: a shell-like structure with equations that depend on only two spatial coordinates and time; a rod-like structure with equations that depend on only one spatial coordinate and time; and a point-like structure with equations that depend on time only. In their classical paper (E. and F. Cosserat, 1909), the Cosserat brothers proposed new equations which generalized the notion of a continuum to include a set of director vectors at each material point in addition to the position vector. Although a threedimensional Cosserat continuum is somewhat abstract, the director vectors for shells, rods and points are very physical. Specifically, within the context of the theory of elasticity: a single director is introduced in shell theory to characterize a material fiber through the thickness of the shell; two directors are introduced in rod theory to Xlll

xiv PREFACE characterize two material fibers in the cross-section of the rod; and three directors are introduced in point theory to characterize three material fibers in the point-like structure. Since the directors are general three-dimensional vectors, the Cosserat theories of shell, rods and points allow for more general deformation than the classical theories. In particular, these Cosserat theories are general enough to model all homogeneous deformations exactly. Moreover, they allow for extension and contraction of the material fibers, which can be of great significance in contact problems. Many articles are available in the literature that describe aspects of the Cosserat theories. However, since these articles typically assume familiarity with tensor analysis in general curvilinear coordinates, they are unintelligible by many practicing structural engineers. Moreover, often the articles characterize the constitutive equations in terms of a general strain energy function and they do not propose specific functional forms, especially for nonlinear deformations. One objective of this book is to attempt to remedy both of these problems. Specifically, here the theories are developed using a minimum of mathematics and the essential mathematical preliminaries are included. Also, an attempt has been made to provide specific constitutive equations for nonlinear elastic response which require specification of only typical material and geometric constants. Consequently, these equations can be used directly to formulate and solve nonlinear structural problems. Another objective of this book is to present a unified approach to the development of these Cosserat theories. Specifically, four level of theory are unified: three-dimensional theory; two-dimensional shell theory; one-dimensional rod theory; and zero-dimensional point theory. The Cosserat approach to the development of these structural theories is particularly enlightening because it allows the development of theory that parallels the full three-dimensional development. In particular, the basic balance laws include the conservation of mass and the balance of linear momentum as well as: a single balance of director momentum for shell theory; two balances of director momentum for rod theory; and three balances of director momentum for point theory. Moreover, the balance of angular momentum and the notion of invariance under superposed rigid body motions are used to place restrictions on the constitutive equations. Also, the constitutive equations for elastic structures are developed in terms of a strain energy function so that the resulting equations preserve all of the fundamental properties of the equations of threedimensional elasticity (e.g. an elastic material is an ideal material and the work done on the material from one state to another ispath-independentj. path-independent). The notation used in this book has been modified relative to that which appears in the literature in order to maximize the uniformity of all four levels of theory. This notation makes it easy to recognize similar concepts in each level of theory. Consequently, ideas that are well understood in the three-dimensional theory can be more easily generalized to the structural theories of shells, rods and points. This book is intended for graduate students and researchers in structural mechanics who are unfamiliar with the beauty and power of the Cosserat theories. A number of

PREFACE xv example problems have been analyzed to help evaluate the validity of constitutive assumptions for shells, rods and points. Moreover, exercise problems have been included to help the reader develop essential technical skills that are required to understand and utilize the theory effectively. Also, the book is structured so that the reader can proceed from the preliminary background material directly to the study of either shells, rods or points.