1.About the AuthorsOmri Rand is a Professor of Aerospace Engineering at theTechnion Israel Institute of Technology. He has been involvedin research on theoretical modeling and analysis in the area ofanisotropic elasticity for the last fifteen years, he is the authorof many journal papers and conference presentations in thisarea. Dr. Rand has been extensively active in composite rotorblade analysis, and established many well recognized analyticaland numerical approaches. He teaches graduate courses in thearea of anisotropic elasticity, serves as the Editor-in-Chief ofScience and Engineering of Composite Materials, as a reviewerfor leading professional journals, and as a consultant to variousresearch and development organizations.Vladimir Rovenski is a Professor of Mathematics and a wellknown researcher in the area of Riemannian and computationalgeometry. He is a corresponding member of the Natural ScienceAcademy of Russia, a member of the American MathematicalSociety, and serves as a reviewer of Zentralblatt frMathematik. He is the author of many journal papers and books,including Foliations on Riemannian Manifolds andSubmanifolds (Birkhuser, 1997), and Geometry of Curves andSurfaces with MAPLE (Birkhuser, 2000). Since 1999,Dr. Rovenski has been a senior scientist at the faculty ofAerospace Engineering at the Technion Israel Institute ofTechnology, and a lecturer at Haifa University.

2. Omri Rand Vladimir RovenskiAnalytical Methods inAnisotropic Elasticitywith Symbolic Computational Tools Birkh user aBoston Basel Berlin 3. Omri RandVladimir Rovenski Technion Israel Institute of TechnologyTechnion Israel Institute of Technology Faculty of Aerospace Engineering Faculty of Aerospace Engineering Haifa 32000Haifa 32000 Israel IsraelAMS Subject Classications: 74E10, 74Bxx, 74Sxx, 65C20, 65Z05, 68W30, 74-XX, 74A10, 74A40, 74Axx, 74Fxx,74Gxx, 74H10, 74Kxx, 74N15, 68W05, 65Nxx, 35J55 (Primary); 74-01, 74-04, 65-XX, 68Uxx, 68-XX (Secondary)Library of Congress Cataloging-in-Publication DataRand, Omri. Analytical methods in anisotropic elasticity : with symbolic computational tools / OmriRand, Vladimir Rovenski. p. cm. Includes bibliographical references and index. ISBN 0-8176-4372-2 (alk. paper)1. Elasticity. 2. Anisotropy. 3. AnisotropyMathematical models. 4. Inhomogeneousmaterials. I. Rovenskii, Vladimir Y, 1953- II. Title QA931.R36 2004 531 .382dc22 2004054558ISBN-10 0-8176-4272-2 Printed on acid-free paper.ISBN-13 978-0-8176-4372-3 c 2005 Birkh user BostonaAll rights reserved. This work may not be translated or copied in whole or in part without the written permission of thepublisher (Birkh user Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013,aUSA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form ofinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identiedas such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.Printed in the United States of America.(HP)987654321 SPIN 10855936www.birkhauser.com 4. To my family, Ora, Shahar, Tal and Boaz, Omri RandTo my teacher, Professor Victor Toponogov,Vladimir Rovenski 5. PrefacePrior to the computer era, analytical methods in elasticity had already been developed and im-proved up to impressive levels. Relevant mathematical techniques were extensively exploited,contributing signicantly to the understanding of physical phenomena. In recent decades, nu-merical computerized techniques have been rened and modernized, and have reached highlevels of capabilities, standardization and automation. This trend, accompanied by convenientand high resolution graphical visualization capability, has made analytical methods less attrac-tive, and the amount of effort devoted to them has become substantially smaller. Yet, with sometenacity, the tremendous advances in computerized tools have yielded various mature programsfor symbolic manipulation. Such tools have revived many abandoned analytical methodologiesby easing the tedious effort that was previously required, and by providing additional capabil-ities to perform complex derivation processes that were once considered impractical.Generally speaking, it is well recognized that analytical solutions should be applied to rela-tively simple problems, while numerical techniques may handle more complex cases. However,it is also agreed that analytical solutions provide better insight and improved understanding ofthe involved physical phenomena, and enable a clear representation of the role taken by each ofthe problem parameters. Nowadays, analytical and numerical methods are considered as com-plementary: that is, while analytical methods provide the required understanding, numericalsolutions provide accuracy and the capability to deal with cases where the geometry and othercharacteristics impose relatively complex solutions.Nevertheless, from a practical point of view, analytic solutions are still considered as art,while numerical codes (such as codes that are based on the nite-element method) seem to offera straightforward solution for any type and geometry of a new problem. One of the reasonsfor this view emerges from the variety of techniques that are used for analytical solutions. Forexample, one has the option to select either the deformation eld or the stress eld to constructthe initial solution hypothesis, or, one has the option to formulate the governing equations usingdifferential equilibrium, or by employing more integral energy methodologies for the sametask. Hence, the main obstacle to using analytical approaches seems to be the fact that manyresearchers and engineers tend to believe that, as far as analytic solutions are considered, eachproblem is associated with a specic solution type and that a different solution methodologyhas to be tailored for every new problem. In light of the above, the objective of this book is twofold. First, it brings together andrefreshes the fundamentals of anisotropic elasticity and reviews various mathematical toolsand analytical solution trails that are encountered in this area. Then, it presents a collection 6. viii Prefaceof classical and advanced problems in anisotropic elasticity that encompasses various two-dimensional problems and different types of three-dimensional beam models. The book in-cludes models of various mathematical complexity and physical accuracy levels, and providesthe theoretical background for composite material analysis. One of the most advanced formu-lations presented is a complete analytical model and solution scheme for an arbitrarily loadednon-homogeneous beam structure of generic anisotropy. All classic and modern analytic solutions are derived using symbolic computational tech-niques. Emphasis is put on the basic principles of the analytic approach (problem statement,setting of simplifying assumptions, satisfying the eld and boundary conditions, proof of solu-tion, etc.), and their implementation using symbolic computational tools, so that the reader willbe able to employ the relevant approach to new problems that frequently arise. Discussions aredevoted to the physical interpretation of the presented mathematical solutions. From a format point of view, the book provides the background and mathematical formu-lation for each problem or topic. The main steps of the analytical solution and the graphicalresults are discussed as well, while the complete system of symbolic codes (written in Maple)are available on the enclosed disc. A unique characteristic of this book is the fact that the entire analytical derivation and allsolution expressions are symbolically proved by suitable (computerized) codes. Hence, thechance for (human) error or typographical mistake is eliminated. The symbolic worksheets aretherefore absolute and rm testimony to the exactness of the presented expressions. For thatreason, the specic solutions included in the text should be viewed as illustrative examplesonly, while the solution exactness and its generic applicability are proved symbolically in themost generic manner. The book is aimed at graduate and senior undergraduate students, professors, engineers, ap-plied mathematicians, numerical analysis experts, mechanics researchers and composite mate-rials scientists. Chapter description:The rst part of the book ( Chapters 14) contains the fundamentals of anisotropic elasticity.The second part (Chapters 510) is devoted to various beam analyses and contains recent andadvanced models developed by the authors. Chapter 1 addresses fundamental issues of anisotropic elasticity and analytical methodolo-gies. It provides a review of deformation measures and strain in generic orthogonal curvilinearcoordinates, and reaches the complete nonlinear compatibility equations in such systems. Itthen introduces fundamental stress measures and the associated equilibrium equations. Lateron, energy theorems and variational analyses are derived, followed by a general discussion ofanalytical methodologies and typical solution trails. Chapter 2 reviews the mathematical representation of general anisotropic materials, includ-ing the special cases of Monoclinic, Orthotropic, Tetragonal, Transversely Isotropic, Cubic andIsotropic materials. Later, transformations between coordinate systems of the compliance andstiffness matrices (or tensors) are presented. The chapter also addresses issues such as planesof elastic symmetry, principal directions of anisotropy and non-Cartesian anisotropy. Chapter 3 denes two-dimensional homogeneous and non-homogeneous domain topolo-gies, and presents various plane deformation problems and analyses, including detailed formu-lation of plane-strain/stress and plane-shear states. The derivation yields formal denitions ofgeneralized Neumann/Dirichlet and biharmonic boundary value problems (BVPs). The chapter 7. Preface ixalso addresses Coupled-Plane BVP for materials of general anisotropy. Along the same lines,the classical anisotropic laminated plate theory is then presented. Chapter 4 presents various solution methodologies for the BVPs derived in Chapter 3, andestablishes solution schemes that facilitate applications presented later on. Explicit analyticexpressions for low-order exact/conditional polynomial solutions, and approximate high-orderpolynomial solutions in a homogeneous simply connected domain are derived and illustrated.A formulation based on complex potentials is also thoroughly derived and demonstrated byFourier series solutions.Chapter 5 reviews some basic aspects and general denitions of anisotropic beam analysis,approximate analysis techniques and relevant literature. It discusses the associated couplingcharacteristics at both the material and structural levels. Chapter 6 presents an analysis of general anisotropic beams that may be viewed as a level-based extension of the classical Lekhnitskii formulation, and is capable of handling beamsof general anisotropy and cross-section geometry that undergo generic distribution of surface,body-force and tip loading. The derivation is founded on the BVPs deduced in Chapter 3, anddespite its complexity, it provides a clear insight into the associated structural behavior andcoupling mechanisms.Chapter 7 contains a closed-form formulation for uncoupled monoclinic homogeneousbeams. It rst presents solutions for tip loads, and then a generic formulation for axially non-uniform distribution of surface and body loads. Later on, analysis and examples of beams ofcylindrical anisotropy are presented.The entire reasoning of the approach in this chapter is founded on St. Venants semi-inversemethod of solution and may be considered as dual (though less generic) to the method pre-sented in Chapter 6. Chapter 8 is focused on problems in various non-homogeneous domains. It rst reviewsgeneric formulations of plane BVPs, and then extends the analysis of Chapter 7 to the caseof monoclinic non-homogeneous beams under tip loading, which is founded on extending theclassical denition of the auxiliary problems of plane deformation to the anisotropic case. Thediscussion encompasses the determination of the principal axis of extension, principal planesof bending and shear center. The chapter also presents a generalization of the derivation inChapter 7 to the case of uncoupled non-homogeneous beams that undergo generic distributedloading. Chapter 9 discusses coupled solid monoclinic beams. The analysis presents an approximatemodel that provides insight into and fundamental understanding of the coupling mechanismswithin anisotropic beams at the structural level. The model also supplies a simplied but rela-tively accurate tool for quantitative estimation of coupled beam behavior. In addition, the chapter presents an exact, level-based solution scheme for coupled beams.The derivation employs a series of properly interconnected solution levels and reaches theexact solution in an iterative manner. Chapter 10 handles coupled thin-wall monoclinic beams in a similar (approximate) mannerto Chapter 9. The analysis encompasses beams having either multiply connected domain(closed) or simply connected domain (open) cross-sections.Chapter 11 presents instructions for the symbolic and illustrative programs included in thisbook (implemented in Maple). 8. x PrefaceGeneral Style Clarication Notes: (1) As a general rule, we use a tilde (e.g. A), for temporary variables that have nomeaningful role, and are introduced for the sake of clarication and analytic convenience. Suchvariables are valid locally within the immediate paragraphs in which they appear in. Hence,if such a notation is repeated elsewhere, it stands for a different local meaning; likewise, thesuperscript () may have different meanings in various contexts. (2) Due to the dependency of most involved functions on many parameters, both ordinaryand partial derivatives, say d( )/d or ( )/, are abbreviated as ( ), . Similarly, d 2 ( )/d2or 2 ( )/2 , are abbreviated as ( ), . (3) Integrals appear in a short notation by omitting the explicit indication of the integrationvariables. Two examples are an integration along a closed loop with a circumferential coordi-nate, s, i.e., Fds, which is written simply as F, and the area integration in the xy-plane,i.e., F dx dy, which is written as F. (4) Within the equation notation, e.g. (1.3), the rst digits stand for the chapter in whichit appears, while (1.15a) is an example for an equation in a group of (sub-)equations. By anotation like (1.9a:b) we refer to the second equation of a group of equations that appears inone line that is collectively denoted (1.9a).(5) Within the Section, Program, Remark, Example, Figure and Table notation, e.g. S.1.2,P.1.2, Remark 1.2, Example 1.2, Fig. 1.2, Table 1.2, the rst digit stands for the chapter inwhich it appears.Acknowledgements We wish to acknowledge the great help of Dr. Michael Kazar (Kezerashvili) who had aunique role in exposing us to some great contributions to this science made by the Easternacademia discussed in Chapters 7,8. We are also thankful to the Ph.D. student Michael Grebshtein who made a tremendous con-tribution to the rigor, the analytical uniformity and the symbolic verication of the derivationin Chapters 4,7,8. We warmly thank Ann Kostant, Executive Editor of Mathematics and Physics at Birkh user aBoston, for her support during the publishing process.Omri RandVladimir RovenskiHaifa, Israel 9. ContentsPreface vii1 Fundamentals of Anisotropic Elasticity and Analytical Methodologies11.1 Deformation Measures and Strain . . . . . . . . . . . . . . . . . . . . . . . .11.1.1 Displacements in Cartesian Coordinates . . . . . . . . . . . . . .. . .21.1.2 Strain in Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . .31.1.3 Strain in Orthogonal Curvilinear Coordinates . . . . . . . . . . .. . .61.1.4 Physical Interpretation of Strain Components . . . . . . . . . . .. . .71.2 Displacement by Strain Integration . . . . . . . . . . . . . . . . . . . . .. . .91.2.1 Compatibility Equations . . . . . . . . . . . . . . . . . . . . . . . . .91.2.2 Continuous Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.3 Level Approach . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 121.3 Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.1 Denition of Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.2 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.3 Stress Tensor Transformation due to Coordinate System Rotation. . . 191.3.4 Strain Tensor Transformation due to Coordinate System Rotation. . . 281.4 Energy Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.4.1 The Theorem of Minimum Potential Energy . . . . . . . . . . . . . . . 281.4.2 The Theorem of Minimum Complementary Energy . . . . . . . . . . . 291.4.3 Theorem of Reciprocity . . . . . . . . . . . . . . . . . . . . . .. . . 301.4.4 Castiglianos Theorems . . . . . . . . . . . . . . . . . . . . . . .. . . 311.5 Eulers Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.5.1 Functional Based on Functions of One Variable . . . . . . . . . . . . . 321.5.2 Variational Problems with Constraints . . . . . . . . . . . . . . . . . . 341.5.3 Functional Based on Function of Several Variables . . . . . . . . . . . 361.6 Analytical Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 381.6.1 The Fundamental Problems of Elasticity . . . . . . . . . . . . . .. . . 391.6.2 Fundamental Ingredients of Analytical Solutions . . . . . . . . . . . . 39 10. xii Contents1.6.3 St. Venants Semi-Inverse Method of Solution . . .. . . . . . . . . . . 411.6.4 Variational Analysis of Energy Based Functionals .. . . . . . . . . . . 411.6.5 Typical Solution Trails . . . . . . . . . . . . . . . . . . . . . . . . . . 411.7 Appendix: Coordinate Systems . . . . . . . . . . . . . . .. . . . . . . . . . . 471.7.1 Transformation Between Coordinate Systems . . . . . . . . . . . . . . 471.7.2 Curvilinear Coordinate Systems . . . . . . . . . .. . . . . . . . . . . 492 Anisotropic Materials 532.1 The Generalized Hooks Law . . . . . . . . . . . . . . . . . . . . . . . . .. . 542.2 General Anisotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 552.3 Monoclinic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 562.4 Orthotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.5 Tetragonal Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 582.6 Transversely Isotropic (Hexagonal) Materials . . . . . . . . . . . . . . . .. . 592.7 Cubic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.8 Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.9 Engineering Notation of Composites . . . . . . . . . . . . . . . . . . . . .. . 622.10 Positive-Denite Stress-Strain Law . . . . . . . . . . . . . . . . . . . . . . . . 632.11 Typical Material Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 652.12 Compliance Matrix Transformation . . . . . . . . . . . . . . . . . . . . . . . 662.13 Stiffness Matrix Transformation . . . . . . . . . . . . . . . . . . . . . . .. . 692.14 Compliance and Stiffness Matrix Transformation to Curvilinear Coordinates. . 702.15 Principal Directions of Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . 702.16 Planes of Elastic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.17 Non-Cartesian Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 Plane Deformation Analysis 793.1 Plane Domain Denition and Contour Parametrization . . . . . . . . . .. . .80 3.1.1 Plane Domain Topology . . . . . . . . . . . . . . . . . . . . . .. . .80 3.1.2 Contour Parametrization and Directional Cosines . . . . . . . . .. . .81 3.1.3 Parametrization by Conformal Mapping . . . . . . . . . . . . . . . . .83 3.1.4 Parametrization by Piecewise Linear Functions . . . . . . . . . .. . .853.2 Plane-Strain and Plane-Stress . . . . . . . . . . . . . . . . . . . . . . . . . . .86 3.2.1 Plane-Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87 3.2.2 Plane-Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90 3.2.3 Illustrative Examples of Prescribed Airys Function . . . . . . . .. . .92 3.2.4 The Inuence of Body Forces . . . . . . . . . . . . . . . . . . .. . .93 3.2.5 Boundary and Single-Value Conditions . . . . . . . . . . . . . . . . .94 3.2.6 Plane Stress/Strain Analysis in a Non-Homogeneous Domain . . . . . 1003.3 Plane-Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.3.1 Analysis by Stress Function . . . . . . . . . . . . . . . . . . . .. . . 101 3.3.2 Analysis by Warping Function . . . . . . . . . . . . . . . . . . . . . . 104 3.3.3 Generic Dirichlet/Neumann BVPs on a Homogeneous Domain . . . . . 105 3.3.4 Simplication of Generalized Laplaces and Boundary Operators .. . . 107 3.3.5 Plane-Shear Analysis of Non-Homogeneous Domain . . . . . . . . . . 1083.4 Coupled-Plane BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.5 Analysis of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 110 3.5.1 The Classical Laminated Plate Theory . . . . . . . . . . . . . . . . . . 110 3.5.2 Bending of Anisotropic Plates . . . . . . . . . . . . . . . . . . .. . . 1173.6 Appendix: Differential Operators . . . . . . . . . . . . . . . . . . . . . .. . . 120 11. Contentsxiii 3.6.1Generalized Laplaces Operators . . . . . . . . . . . . . . . . . . . . . 120 3.6.2Biharmonic Operators . . . . . . . . . . . . . . .. . . . . . . . . . . 121 3.6.3Third-Order and Sixth-Order Differential Operators. . . . . . . . . . . 123 3.6.4Generalized Normal Derivative Operators . . . . . . . . . . . . . . . . 123 3.6.5Ellipticity of the Differential Operators . . . . . . . . . . . . . . . . . 1244 Solution Methodologies1254.1 Unied Formulation of Two-Dimensional BVPs . . . . . . . . . . . . . . . . .1264.2 Particular Polynomial Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.2.1 The Biharmonic BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.2.2 The Dirichlet/Neumann BVPs . . . . . . . . . . . . . . . . . . . . . . 130 4.2.3 The Coupled-Plane BVP . . . . . . . . . . . . . . . . . . . . . . . . .1314.3 Homogeneous BVPs Polynomial Solutions . . . . . . . . . . . . . . . . . . . 132 4.3.1 Prescribing the Boundary Functions . . . . . . . . . . . . . . . . . . . 133 4.3.2 Prescribing the Field Equations . . . . . . . . . . . . . . . . . . . . .134 4.3.3 Exact and Conditional Polynomial Solutions . . . . . . . . . . . . . . 135 4.3.4 Approximate Polynomial Solutions . . . . . . . . . . . . . . . . . . . 1414.4 The Method of Complex Potentials . . . . . . . . . . . . . . . . . . . . . . . .149 4.4.1 n-Coupled Dirichlet BVP . . . . . . . . . . . . . . . . . . . . . . . . .150 4.4.2 Application of Complex Potentials to the Dirichlet BVP . . . . . . . . 152 4.4.3 Application of Complex Potentials to the Biharmonic BVP . . . . . . .157 4.4.4 Application of Complex Potentials to a Coupled-Plane BVP . . . . . . 159 4.4.5 Fourier Series Based Solution of a Coupled-Plane BVP . . . . . . . . . 1614.5 Three-Dimensional Prescribed Solutions . . . . . . . . . . . . . . . . . . . . .169 4.5.1 Equilibrium Equations in Terms of Displacements . . . . . . . . . . .169 4.5.2 Fourier Series Based Solutions in an Isotropic Parallelepiped . . . . . .171 4.5.3 Direct Solution in Terms of Displacements for Three-Dimensional Bod-ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1734.6 Closed Form Solutions in Circular and Annular Isotropic Domains . . . . . . . 177 4.6.1 Harmonic and Biharmonic Functions in Polar Coordinates . . . . . . . 177 4.6.2 The Dirichlet BVP in a Circle . . . . . . . . . . . . . . . . . . . . . .178 4.6.3 The Neumann BVP in a Circle . . . . . . . . . . . . . . . . . . . . . .178 4.6.4 The Dirichlet BVP in a Circular Ring . . . . . . . . . . . . . . . . . . 179 4.6.5 The Neumann BVP in a Circular Ring . . . . . . . . . . . . . . . . . . 179 4.6.6 The Biharmonic BVP in a Circle . . . . . . . . . . . . . . . . . . . . . 180 4.6.7 The Biharmonic BVP in a Circular Ring . . . . . . . . . . . . . . . . .1815 Foundations of Anisotropic Beam Analysis1835.1 Notation and Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1845.1.1 Geometrical Degrees of Freedom . . . . . . . .. . . . . . . . . . . . 1845.1.2 Tip and Distributed Loading . . . . . . . . . . . . . . . . . . . . . . . 1875.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.2 Elastic Coupling in General Anisotropic Beams . . . . . . . . . . . . . . . . . 1925.2.1 Coupling at the Material Level . . . . . . . . . .. . . . . . . . . . . . 1925.2.2 Coupling at the Structural Level . . . . . . . . .. . . . . . . . . . . . 1935.3 Simplied Beam Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1985.3.1 Beam-Plate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1995.3.2 Analysis by Cross-Section Stiffness Matrix . . .. . . . . . . . . . . . 2055.3.3 The Inuence of the In-Plane Deformation . . .. . . . . . . . . . . . 2075.3.4 Strength-of-Materials Isotropic Beam Analysis . . . . . . . . . . . . 211 12. xiv Contents5.4 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126 Beams of General Anisotropy2156.1 Stress and Strain . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 2156.1.1 Stress Resultants . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 2176.1.2 Compatibility Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 2196.1.3 Axial Strain Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 2206.1.4 Stress Functions and the Coupled-Plane BVP . . . . . . . . . . . . . . 2216.2 Displacements and Rotations . . . . . . . . . . . . .. . . . . . . . . . . . . . 2246.2.1 Continuous Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 2256.2.2 Level Expressions . . . . . . . . . . . . . .. . . . . . . . . . . . . . 2266.2.3 Axis Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276.2.4 Root Warping Integration . . . . . . . . . . . . . . . . . . . . . . . . . 2296.3 Recurrence Solution Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 2306.3.1 Solution Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2306.3.2 Expressions for the High Solution Levels . . . . . . . . . . . . . . . . 2326.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2346.4.1 Tip Moments and Axial Force . . . . . . . .. . . . . . . . . . . . . . 2346.4.2 Tip Bending Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2386.4.3 Axially Uniform Distributed Loading . . . .. . . . . . . . . . . . . . 2426.4.4 Additional Examples . . . . . . . . . . . . .. . . . . . . . . . . . . . 2456.5 Appendix: Integral Identities . . . . . . . . . . . . .. . . . . . . . . . . . . . 2467 Homogeneous, Uncoupled Monoclinic Beams2497.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2507.2 Tip Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 2507.2.1 Torsional Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2517.2.2 Tip Bending Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.2.3 Summarizing the Tip Loading Effects . . . . . . . . . . . .. . . . . . 2637.3 Axially Distributed Loads . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 2657.3.1 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 2657.3.2 Solution Hypothesis . . . . . . . . . . . . . . . . . . . . .. . . . . . 2667.3.3 The Harmonic Stress Functions . . . . . . . . . . . . . . .. . . . . . 2677.3.4 The Biharmonic and Longitudinal Stress Functions . . . . . . . . . . . 2687.3.5 Verication of Solution Hypothesis . . . . . . . . . . . . . . . . . . . 2687.3.6 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2697.3.7 Detailed Solution Expressions . . . . . . . . . . . . . . . .. . . . . . 2707.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2757.5 Beams of Cylindrical Anisotropy . . . . . . . . . . . . . . . . . . .. . . . . . 2817.5.1 Geometrical Aspects . . . . . . . . . . . . . . . . . . . . .. . . . . . 2837.5.2 Torsion of a Beam of Cylindrical Anisotropy . . . . . . . .. . . . . . 2847.5.3 Extension and Bending of a Beam of Cylindrical Anisotropy. . . . . . 2858 Non-Homogeneous Plane and Beam Analysis2978.1 Plane (Two-Dimensional) BVPs . . . . . . . . . . . . . . . . . . . . . . . . . 2988.1.1 The Neumann BVP . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2988.1.2 The Dirichlet BVP . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2998.1.3 The Biharmonic BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . 3008.1.4 Coupled-Plane BVP . . . . . . .. . . . . . . . . . . . . . . . . . . . 3028.1.5 n-Coupled Dirichlet BVP . . . . .. . . . . . . . . . . . . . . . . . . . 302 13. Contents xv 8.2 Uncoupled Beams Under Tip Loads . . . . . . . . . . . . . . . . .. . . . . . 304 8.2.1 General Aspects and Interlaminar Conditions . . . . . . . .. . . . . . 304 8.2.2 The Auxiliary Problems of Plane Deformation . . . . . . .. . . . . . 305 8.2.3 Tip Axial Force . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 308 8.2.4 Tip Bending Moments . . . . . . . . . . . . . . . . . . . .. . . . . . 310 8.2.5 Tip Bending Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 8.2.6 Torsional Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 8.2.7 Summarizing the Tip Loading Effects . . . . . . . . . . . .. . . . . . 319 8.2.8 Fullling the Tip Conditions . . . . . . . . . . . . . . . . . . . . . . . 321 8.2.9 Principal Axis of Extension and Principal Planes of Bending. . . . . . 322 8.2.10 Shear Center . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 323 8.2.11 Solution Procedure . . . . . . . . . . . . . . . . . . . . . .. . . . . . 323 8.3 Uncoupled Beam Under Axially Distributed Loads . . . . . . . . . . . . . . . 324 8.3.1 The Solution Hypothesis . . . . . . . . . . . . . . . . . . .. . . . . . 324 8.3.2 The Strain Components . . . . . . . . . . . . . . . . . . .. . . . . . 325 8.3.3 Displacements and Rotations . . . . . . . . . . . . . . . . .. . . . . . 326 8.3.4 The Biharmonic Stress Functions . . . . . . . . . . . . . .. . . . . . 326 8.3.5 The Harmonic and Longitudinal Stress Functions . . . . . . . . . . . . 326 8.3.6 The Auxiliary Functions . . . . . . . . . . . . . . . . . . .. . . . . . 327 8.3.7 The Loading Constants . . . . . . . . . . . . . . . . . . . .. . . . . . 327 8.3.8 Verication of Solution Hypothesis . . . . . . . . . . . . . . . . . . . 328 8.3.9 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3299 Solid Coupled Monoclinic Beams3359.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 3359.1.1 Cross-Section Warping . . . . . . . . . . . . . . . . . . . . . . . . . . 3359.1.2 Approximate Analytical Solutions . . . . . . . . . . . . . . . .. . . . 3369.1.3 Coupling Effects in Symmetric and Antisymmetric Solid Beams . . . . 3369.2 An Approximate Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . 3389.2.1 Reduced Stress-Strain Relationships . . . . . . . . . . . . . . . . . . . 3399.2.2 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3419.2.3 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . .. . . . 3449.3 An Exact Multilevel Approach . . . . . . . . . . . . . . . . . . . . . .. . . . 3539.3.1 Displacements and Stress-Strain Relationships . . . . . . . . . . . . . 3549.3.2 Denition of Solution Levels . . . . . . . . . . . . . . . . . . .. . . . 3559.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 36110 Thin-Walled Coupled Monoclinic Beams 369 10.1 Background . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 369 10.2 Multiply Connected Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 37010.2.1 The Elastic Coupling Effects . . . . . . . . . . . . . . . . . . . . . . . 37010.2.2 An Approximate Analytical Model . .. . . . . . . . . . . . . . . . . 371 10.3 Simply Connected Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38010.3.1 The Transverse and Axial Loads Effect. . . . . . . . . . . . . . . . . 38110.3.2 The Torsional Moment Effect . . . . .. . . . . . . . . . . . . . . . . 38911 Program Descriptions 401 P.1 Programs for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402P.1.1 Strain Tensor in Space . . . . . . . . . . . . . . . . . . . . . . . . . . 402 14. xvi ContentsP.1.2Strain Tensor in the Plane . . . . . . . . . . . . . . . . . . . . . . . .402P.1.3Compatibility Equations in Space . . . . . . . . . . . . . . . . . . . .402P.1.4Compatibility Equations in the Plane . . . . . . . . . . . . . . . . . .402P.1.5Displacements by Strain Integration in Space . . . . . . . . . . . . . .403P.1.6Displacements by Strain Integration in the Plane . . . . . . . . . . . .403P.1.7Equilibrium Equations in Space . . . . . . . . . . . . . . . . . . . . .403P.1.8Equilibrium Equations in the Plane . . . . . . . . . . . . . . . . . . .404P.1.9Stress/Strain Tensor Transformation due to Coordinate System Rotation 404P.1.10 Application of Stress/Strain Tensor Transformation . . . . . . . . . . .404P.1.11 Stress/Strain Tensor Transformations from Cartesian to Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404P.1.12 Stress/Strain Visualization . . . . . . . . . . . . . . . . . . . . . . . . 405P.1.13 Eulers Equation for a Functional Based on a Function of One or Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405P.1.14 Elastica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .405P.1.15 Rotation Matrix in Space . . . . . . . . . . . . . . . . . . . . . . . . .405P.1.16 Curvilinear Coordinates in Space . . . . . . . . . . . . . . . . . . . .405P.1.17 Curvilinear Coordinates in the Plane . . . . . . . . . . . . . . . . . . .406P.2 Programs for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406P.2.1 Compliance and Stiffness Matrices Presentation . . . . . . . . . . . . . 406P.2.2 Material Data by Compliance Matrix . . . . . . . . . . . . . . . . . . 406P.2.3 Material Data by Stiffness Matrix . . . . . . . . . . . . . . . . . . . .407P.2.4 Compliance Matrix Positiveness . . . . . . . . . . . . . . . . . . . . . 407P.2.5 Generic Compliance Matrix Transformation due to Coordinate System Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407P.2.6 Application of the Compliance Matrix Transformation . . . . . . . . .407P.2.7 Compliance Matrix Transformation due to Coordinate System Rotation 407P.2.8 Visualization of a Compliance Matrix Transformation . . . . . . . . .408P.2.9 Generic Stiffness Matrix Transformation due to Coordinate System Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .408P.2.10 Stiffness Matrix Transformation due to Coordinate System Rotation . . 408P.2.11 Application of the Stiffness Matrix Transformation . . . . . . . . . . .408P.2.12 Compliance Matrix Transformation from Cartesian to Curvilinear Co- ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408P.2.13 Principal Directions of Anisotropy . . . . . . . . . . . . . . . . . . . .409P.3 Programs for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409P.3.1 Illustrative Parametrizations . . . . . . . . . . . . . . . . . . . . . . .409P.3.2 Regular Polygon Parametrization Using the Schwarz-Christoffel Integral 409P.3.3 Generic Polygon Parametrization Using the Schwarz-Christoffel Integral 410P.3.4 Fourier Series Parametrization of a Polygon . . . . . . . . . . . . . . .410P.3.5 Prescribed Polynomial Solution of the Biharmonic Equation . . . . . .410P.3.6 Application of Prescribed Polynomial Solution of the Biharmonic Equ- ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411P.3.7 Prescribed Polynomial Solution of Laplaces Equation . . . . . . . . . 411P.3.8 Application of Prescribed Polynomial Solution of Laplaces Equation .411P.3.9 Afne Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 412P.3.10 Prescribed Polynomial Solution of Coupled-Plane Equations . . . . . . 412P.3.11 Application of Prescribed Polynomial Solution of Coupled-Plane Equa- tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412P.3.12 Ellipticity of the Differential Operators . . . . . . . . . . . . . . . . . 412 15. Contents xviiP.4 Programs for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413P.4.1 Particular Polynomial Solution of the Biharmonic Equation in an Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413P.4.2 Particular Polynomial Solution of the Biharmonic Equation . . . . . . 414P.4.3 Particular Polynomial Solution of Poissons Equation . . . . . . . . . . 414P.4.4 Particular Polynomial Solution of Poissons Equation in an Ellipse . . . 414P.4.5 Particular Polynomial Solution of Coupled-Plane Equations . . . . . . 415P.4.6 Particular Polynomial Solution of Coupled-Plane Equations in an Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415P.4.7 Prescribed Polynomial Boundary Functions . . . . . . . . . . . . . . . 415P.4.8 Exact/Conditional Polynomial Solution of Dirichlet / Neumann Homo- geneous BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416P.4.9 Symbolic Verication of the Neumann BVP Solution . . . . . . . . . . 416P.4.10 Exact/Conditional Polynomial Solution of Homogeneous Biharmonic BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416P.4.11 Symbolic Verication of the Biharmonic BVP Solution . . . . . . . . . 417P.4.12 Exact/Conditional Polynomial Solution of Homogeneous Coupled-Plane BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417P.4.13 Approximate Polynomial Solution of Homogeneous Dirichlet/Neumann BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417P.4.14 Approximate Polynomial Solution of Homogeneous Biharmonic BVPs 418P.4.15 Approximate Polynomial Solution of Homogeneous Coupled-Plane BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419P.4.16 Rotating Plate: Application of the Biharmonic BVP Solution . . . . . . 419P.4.17 Bending of Thin Plates: Application of the Biharmonic BVP Solution . 419P.4.18 Approximate Polynomial Solution of Homogeneous n-Coupled Dirich- let BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420P.4.19 Fourier Series Solution of Homogeneous Dirichlet BVPs in a Rectangle 420P.4.20 Fourier Series Solution of Homogeneous Coupled-Plane BVPs in a Rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421P.4.21 Equilibrium Equations in Terms of Displacements . . . . . . . . . . . 421P.4.22 Prescribed Solutions in an Isotropic Parallelepiped . . . . . . . . . . . 421P.4.23 Fourier Series Solution of the Dirichlet/Neumann BVPs in an Isotropic Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422P.4.24 Fourier Series Solution of the Dirichlet/Neumann BVPs in an Isotropic Circular Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422P.4.25 Fourier Series Solution of the Biharmonic BVP in an Isotropic Circle . 422P.4.26 Fourier Series Solution of the Biharmonic BVP in an Isotropic Circular Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423P.5 Programs for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423P.5.1 Elementary Strength-of-Materials Isotropic Beam Analysis . . . . . 423P.6 Programs for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423P.6.1 An Anisotropic Beam of Elliptical Cross-Section . . . . . . . . . . . . 423P.7 Programs for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424P.7.1 Tip Loads Effect in a Monoclinic Beam . . . . . . . . . . . . . . . . . 424P.7.2 Auxiliary Harmonic Functions for Elliptical Monoclinic Cross-Sections 424P.7.3 A Monoclinic Beam Under Axially Non-Uniform Distributed Loads (I) 424P.7.4 A Monoclinic Beam Under Axially Non-Uniform Distributed Loads (II) 425P.7.5 Solution for an Elliptical Monoclinic Beam Under Constant Longitu- dinal Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 16. xviii ContentsP.7.6Solution Implementation for an Elliptical Monoclinic Beam Under Con- stant Longitudinal Loading . . . . . . . . . . . . . . . . . . . . . . . .425P.8 Programs for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425P.8.1 Auxiliary Harmonic Functions in a Non-Homogeneous Rectangle . . .425P.8.2 Plane Deformation and the Auxiliary Biharmonic Problems . . . . . .426P.8.3 Fourier Series Based Torsion Function in a Non-Homogeneous Or- thotropic Rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426P.8.4 A Non-Homogeneous Monoclinic Beam Under Tip Loads . . . . . . .426P.8.5 A Non-Homogeneous Beam of Rectangular Cross-Section Under Tip Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427P.8.6 A Monoclinic Non-Homogeneous Beam Under Axially Distributed Non-Uniform Loads (I) . . . . . . . . . . . . . . . . . . . . . . . . . . 427P.8.7 A Monoclinic Non-Homogeneous Beam Under Axially Distributed Non-Uniform Loads (II) . . . . . . . . . . . . . . . . . . . . . . . . .427References 429Index447 17. 1Fundamentals of Anisotropic Elasticity andAnalytical MethodologiesThis chapter is devoted to fundamental issues in anisotropic elasticity that should be denedand reviewed before specic problems are tackled. Hence, the main purpose of this chapter isto provide a common and general background and terminology that will allow further devel-opment of analytical tools. In contrast with traditional and other modern textbooks in the general area of elasticity, seee.g. (Muskhelishvili, 1953), (Milne-Thomson, 1960), (Novozhilov, 1961), (Hearmon, 1961),(Filonenko-Borodich, 1965), (Steeds, 1973), (Sokolnikoff, 1983), (Parton and Perlin, 1984),(Landau and Lifschitz, 1986), (Ciarlet, 1988), (Reismann and Pawlik, 1991), (Barber, 1992),(Green and Zerna, 1992), (Chou and Pagano, 1992), (Wu et al., 1992), (Saada, 1993), (Gould,1994), (Ting, 1996), (Chernykh and Kulman, 1998), (Soutas-Little, 1999), (Boresi and Chong,1999), (Doghri, 2000), (Atanackovic and Guran, 2000), (Slaughter, 2001), the fundamentalissues presented in what follows are fully backed by symbolic codes that testify for the ex-actness of the derivation, and may be employed to produce an enormous amount of additionalinformation and results, in a clear, complete and analytically accurate manner.1.1 Deformation Measures and StrainThe mathematical representation of deformation of an elastic body is under discussion in thissection. Here and throughout this book, the notion elastic stands for a non-rigid solid mediumthat is deformed under external loading and fully recovers its size and shape when loading isremoved. Hence, we shall use the term elastic as equivalent to non-rigid. The change inthe relative position of points is generally termed deformation, and the study of deformationsis the province of the analysis of strain, (Sokolnikoff, 1983). The discussion presented in what follows supplies measures mainly for small and nitedeformations. Such expressions are essential for proper modeling of the linear behavior ofanisotropic elastic media. Under certain assumptions, these expressions may also be interpretedto provide an insight into the physics of the deformation. Unless stated differently, we shall assume that the deformation throughout the elastic bodyis continuous, while mathematically, we presume that all deformation expressions are differ- 18. 2 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologiesentiable with a number of continuous derivatives as required. Cases where the media consistsof different materials, such as laminated composite structures and other non-homogeneous do-mains will be treated as an extension of the homogeneous approach in further chapters. To analytically describe the deformation, we shall adopt a suitable system of curvilinearorthogonal coordinates in a Euclidean space, E 3 , or in a plane, E 2 . The analysis of non-orthogonal curvilinear coordinates is much more complicated, while yielding less practicaladvantage in the context of this book. In essence, the selection of a coordinate system for a specic problem is immaterial, sinceclearly, any given physical deformation may be described in many and various coordinatesystems. However, in the course of any search for analytical solutions, it becomes obvious thatthe selection of a suitable coordinate system may have a tremendous effect on the ability toderive a solution, and on the effort that is required to achieve such a solution. Traditionally, when isotropic elasticity has been under discussion, selection of a proper coor-dinate system was primarily based on the geometry of the problem and the complexity involvedwithin the fulllment of the boundary conditions. As will become clear in further chapters,when an anisotropic elastic body is under discussion, material type and direction should also betaken into account while selecting a coordinate system, and in many cases, material anisotropyhas a predominant inuence on this selection. In what follows, we shall therefore establish andreview the mathematical denition of deformation for a variety of coordinate systems. For thesake of clarity, we will rst deal with Cartesian coordinates, and then generalize the approachfor other orthogonal coordinate systems.1.1.1 Displacements in Cartesian CoordinatesOne of the ways to present deformation in elastic media is based on specifying the displacementcomponents, which for the sake of brevity will also be denoted as displacements. By denition,the displacements of a material particle are determined by its initial and nal locations, whilethe path of the particle between these two points is irrelevant. Suppose now that due to the deformation described by the displacement vector u = (u, v, w),a material particle, M, located at x = (x, y, z) in the elastic media before deformation, has movedto a new location, M (, , ), with position vector x + u, Fig. 1.1(a).Figure 1.1: (a) Deformation of domain into . (b) Vector (small deformation) eld of .One may therefore write, = x + u, = y + v, = z+w, (1.1)where u = u(x, y, z), v = v(x, y, z), w = w(x, y, z). Differentiating the above equations yields d = (1 + u, x ) dx + u, y dy + u, z dz , 19. 1.1 Deformation Measures and Strain3 d = v, x dx + (1 + v, y ) dy + v, z dz , d = w, x dx + w, y dy + (1 + w, z ) dz .(1.2)A necessary and sufcient condition for a continuous deformation u to be physically possible(i.e. with locally single-valued inverse) is that the Jacobian of (1.2) is greater than zero, (Boresiand Chong, 1999),1 + u, x u, yu, zJ = v, x 1 + v, yv, z > 0.(1.3)w, x w, y1 + w, zThe simplest types of displacement vector are translation (i.e. u = u0 , where u0 = (u0 , v0 , w0 )is a constant vector) and small rotation about the origin (i.e. u = 0 x, where 0 = (0 , 0 , 120 ) is a constant vector, see also Remark 1.1). These may be combined to form the basic type3of deformation known as rigid body displacements. In fact, all bodies are to some extent de-formable, and a rigid body deformation (or a non-deformable case) stands for an ideal casewhere the distance between every pair of points of the body remains invariant throughoutthe history of the body. In practice, any rigid deformation (that includes no relative displace-ment of material points) may be composed of three translation components and three rotationcomponents, which are uniformly applied to all material particles. For example, in Cartesiancoordinates where the point x0 = (x0 , y0 , z0 ) is xed (i.e. belongs to the axis of rotation), therigid body displacements ur , vr and wr are expressed using linear functions of x, y, z as ur = 0 (z z0 ) 0 (y y0 ) + u0 , 23 vr = 0 (x x0 ) 0 (z z0 ) + v0 , 31 wr = 0 (y y0 ) 0 (x x0 ) + w0 , 12 (1.4)i.e., u = 0 (x x0 ) + u0 . Note that the Jacobian of the above rigid body deformation isequal to or greater than 1, since 10 3 023 Jr = 0310 = 1 + (0 )2 .1 i (1.5)0 2 011i=1In general, (1.1) in which u, v, w are linear functions of the coordinates constitute an afnedeformation, see S.1.7.1. A common way to present a eld of small deformation in the elastic media is the so-calledvector eld description. Within this technique, at each point a vector that represents the di-rection and relative magnitude of the displacement is drawn, see Fig. 1.1(b) and examples inChapters 7, 8.Remark 1.1 Note that the u = 0 x denition of displacement due to rotation holds for small rotation only. Finite rotation should be treated by transformation matrices as discussedin S.1.7.1.1.1.2 Strain in Cartesian CoordinatesIn order to derive proper deformation measures and dene the strain components in a Carte-sian coordinate system x, y, z, in addition to the material points M(x, y, z), M (, , ) discussedearlier, we assume that a point N, which is innitesimally close to M and located before defor-mation at (x + dx, y + dy, z + dz) has moved to N ( + d, + d, + d), see Fig. 1.1(a). Letds = |MN| be the distance between the two points before deformation, and ds = |M N | be 20. 4 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologiesthe distance between the same material points after deformation. The squares of distances aretherefore given by(ds)2 = (dx)2 + (dy)2 + (dz)2 ,(ds )2 = (d)2 + (d)2 + (d)2 .(1.6)We shall now establish a denition for suitable measures that will properly describe the defor-mation at the vicinity of the point M. The common measure of deformation is mathematicallydened as 1 (ds )2 (ds)2 , and may be expressed as a sum of its components according to2the resulting six different combinations of products of innitesimal distances, namely, 1 [(ds )2 (ds)2 ] = xx (dx)2 + yy (dy)2 + zz (dz)2 + yz dy dz + xz dx dz + xy dx dy . (1.7) 2The coefcients are traditionally referred to as the engineering strain components at pointM, and are given by 1 211xx = u, x + u +v2 +w2x , yy = v, y + u2y +v2y +w2y , zz = w, z + u2z +v2z +w2z , 2 ,x ,x,2 ,,, 2 , ,,yz = v, z + w, y + u, y u, z + v, y v, z + w, y w, z , xz = u, z + w, x + u, x u, z + v, x v, z + w, x w, z ,xy = u, y + v, x + u, y u, x + v, y v, x + w, y w, x . (1.8)A reduction of the above expressions leads to their linear version, e , namely,exx = u, x ,eyy = v, y ,ezz = w, z , (1.9a)eyz = v, z + w, y , exz = u, z + w, x , exy = u, y + v, x .(1.9b)Occasionally, it is convenient to replace ( = ) with or to use a numerical indexnotation in which the strain vector {xx , yy , zz , yz , xz , xy } is written as {1 , 2 , 3 , 4 , 5 , 6 }. The nonlinear version of will be dealt with in what follows as we introduce a more con-sistent analysis of strain using tensor (index) notation. For that purpose, we dene {ki }i=1,2,3as the orthogonal unit vectors of the Cartesian basis (i.e. in the x, y and z directions, respec-tively). Here and in the following derivation, we shall use the index notation x1 x, x2 y, x3 z . The position vector r(x) = x of the point M is written as r = 3 xi ki , while the i=1displacement vector u(x) in the static (i.e. a time independent) case takes the formu = i=1 ui (x1 , x2 , x3 ) ki . 3 (1.10)As already indicated, due to the elastic deformation, the point M is relocated to M , and theposition vector of which r (x) = x + u is expressed as r = 3 (xi + ui ) ki . i=1To determine the associated metric tensors, g = {gi j } and g = {gj } before and after defor- imation, respectively, given bygi j = r, i r, j ,gj = ri rj , i , , (1.11)we expand the partial derivatives of r and r with respect to the coordinate system base r, i = ki ,and similarly to (1.2), ri = j=1 (i j + u j, i ) k j , 3,(1.12)where i j is Kroneckers symbol. Hence, in the Cartesian case under discussion, gi j = i j ,whilegj = i j +ui, j + u j, i + m=1 um, i um, j . 3 i (1.13) 21. 1.1 Deformation Measures and Strain 5By denition, the strain tensor components of the Lagrange-Green deformation measures are1 i j = (gj gi j ). (1.14)2 iThus, the linear (the underlined terms of (1.13)) and nonlinear expressions for strains in tensornotation are1ei j = (ui, j + u j, i ) ,(1.15a)211 3i j = (ui, j + u j, i ) + m=1 um, i um, j .(1.15b)22In particular one obtains eii = ui, i and ii = ui, i + 1 3 (um, i )2 . The resulting nonlinear2 m=1strain tensor components may be also written as1 32 m=1 i j = ei j +(eim + im ) (e jm + jm ),(1.16)where i j are the components of the antisymmetric ( j i = i j ) rotation tensor, , while eand are commonly put in a matrix form as e11 e12 e130 12 13 e = e12 e22 e23 , = 12 0 23 .(1.17) e13 e23 e33 13 230The rotation tensor components may also be dened by the rotation vector as 1 = u = 23 k1 + 31 k2 + 12 k3 . (1.18) 2In the above equation, the nabla operator is written as = i x i ki , and the vector cross-product operation yields i j = 2 (u j, i ui, j ), namely, 1111 23 =(u3, 2 u2, 3 ) , 31 = (u1, 3 u3, 1 ) , 12 = (u2, 1 u1, 2 ) . (1.19)222By comparison with the expressions presented by (1.9a,b), we conclude that the strain com-ponents in tensor notation (which are written by indices as i j , i, j = 1, 2, 3) and the straincomponents in engineering notation are related as the symmetric matrices 11 12 13x2 xy1 2 xz 1 12 22 23 = 1 xy y 2 yz . 1 2 (1.20) 13 23 33 1 2 xz 2 yz1zThe profound advantages of using the tensor notation will become clearer within the coordinatetransformation techniques developed in S.1.3.3, S.1.3.4. In addition, it should be indicated thatthe above tensorial denition of the strain is very attractive in many analytical applications, asit only requires the ability to dene the position vectors of a material point before and afterdeformation. Then, (1.11), (1.14) may be directly applied. For the sake of clarication, we may now summarize all notation forms mentioned above forstrain in Cartesian coordinates. These forms will be exploited as convenience requires through-out this book: 1 11 xx x , 2 22 yy y ,3 33 zz z , 4 223 yz yz , 5 213 xz xz ,6 212 xy xy , (1.21) x 1 23 , y 2 31 ,z 3 12 .P.1.1, P.1.2 (with s = 0) demonstrate a derivation of the strain components in Cartesian co-ordinates. 22. 6 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies1.1.3 Strain in Orthogonal Curvilinear CoordinatesThe derivation presented in this section for deformation measures of orthogonal curvilinearcoordinates in Euclidean space E 3 is founded on the generic discussion of coordinate systemspresented in S.1.7. The reader is therefore advised to become familiar with the mathematicalaspects involved and the notation described in S.1.7. We dene the deformation in orthogonalcurvilinear coordinates using three functions {ui (1 , 2 , 3 )}i=1,2,3 and a local basis {ki }, see (1.215), asu = i ui (1 , 2 , 3 ) ki . (1.22)We also dene in S.1.7 three functions, f1 , f2 , f3 , that convert curvilinear coordinates of E 3space into Cartesian ones, see (1.210). By substituting ki of (1.215), one rewrites (1.22) as u = ux k1 + uy k2 + uz k3 , (1.23)where u = u (u1 , u2 , u3 , 1 , 2 , 3 ), = x, y, z. Since the displacement components u1 , u2 , u3are functions of 1 , 2 , 3 by themselves, one may also consider the form u = u (1 , 2 , 3 ).Subsequently, the position vectors of a point before and after deformation, r and r = r + u,respectively, may be exploited to construct the metric tensors, see (1.11),gi j = r, i r, j ,gj = ri r j ,i ,, (1.24)for which (ds)2 = i j gi j di d j and (ds )2 = i j gj di d j . The Lagrange-Green denition iof the strain tensor in this case becomesgj gi j i i j =. (1.25) 2Hi H j Note that g is always a diagonal matrix while Lam parameters, Hi are dened as Hi = gii .eAs an example, in cylindrical coordinates, where (1 , 2 , 3 ) = (, c , z), in view of (1.218a)one obtains g11 = 1, g22 = 2 , g33 = 1, and H1 = 1, H2 = , H3 = 1, while g = 1 + 2u, + u2 + u2 + u2 ,, , z, g = u, ( + u, + u ) + uz, uz, + (u, u )(1 + u, ), etc., (1.26)and hence,111 = u, + (u2 + u2 + u2 ),2 , , z, 112 = (u, u + u, + uz, uz, u, u + u, u, + u, u, + u, u ), (1.27)2etc.,where the linear terms are underlined. Any orthogonal coordinates may be directly incorpo-rated into P.1.1, P.1.2 to produce the associated strain expressions (in addition to the built-insystems in these programs). Note that although originally derived for the Cartesian system, one may use (1.16) to ex-press the nonlinear strain components in terms of their linear parts and the rotation vectorcomponents in the case under discussion as well. To express ei j for the present case (i.e. the linear parts of i j ), we dene an operation, whichwill be denoted cyc-ijk, as the operation in which we create two additional equations out ofa given equation by forward replacement of the indices for the rst equation, namely, i j, 23. 1.1 Deformation Measures and Strain 7j k, k i, and backwards replacement for the second equation, i k, j i, k j.Hence, by applying cyc-123 to the following two equations, one obtains the required six ei jcomponents 1 11 e11 =u1, 1 +H1, 2 u2 +H1, 3 u3 , H1 H1 H2H1 H3 H2 u2H1 u1 e12 = , 1 + , 2 . (1.28) H1 H2H2 H1The rotation tensor components are extracted from H1 k1 H2 k2 H3 k31 1 = u = 1 2 3, (1.29)22 H1 H2 H3 H1 u1 H2 u2 H3 u31 (note that the nabla operator in this case is = i Hi i ki ), which in view of (1.18) yields(apply cyc-123) 1 i j =[(H j u j ), i (Hi ui ), j ]. (1.30)2 Hi H jIn some applications, it is useful to dene the unit vectors in the coordinate line directions ofthe deformed state, namelyk = ri / ri .i,, (1.31)1.1.4 Physical Interpretation of Strain Components1.1.4.1 Relative Extension and Angle ChangeSo far, the deformation measures i j have been dened mathematically, and were shown toprovide a set of six parameters that reect the deformation at a given material point. To gainsome physical insight and clearer interpretation of the above discussed measures, we will rstdene a more physically-based deformation measure known as the relative extension, whichis essentially the ratio of the change of distances between two adjoint points to the initial dsdistance, namely: EMN = ds ds , see Fig. 1.1(a). While trying to relate the above expression tothe previously derived strain components we rst note that(ds )2 (ds)21 2= ( EMN + EMN )(ds)2 . (1.32)22We subsequently substitute the r.h.s. of (1.7) in the above equation and express the relativeextension as1 2 E + EMN = x cos(, x)2 + y cos(, y)2 + z cos(, z)2ss s2 MN + yz cos(, y) cos(, z) + xz cos(, x) cos(, z) + xy cos(, x) cos(, y) , (1.33) s s s s s swhere the underlined term may be neglected for small strain. Note that the direction cosinesbetween a material element s = MN that is placed at a generic orientation and the (say Carte-sian) axes are dened as cos(, ) = d/ds ( = x, y, z). As a special case, the above result sshows that the relative extensions (elongations at the point M) in the x, y, z directions are givenby (apply cyc-xyz) 1 2E + Ex = x orEx = 1 + 2x 1,(1.34) 2 x 24. 8 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologieswhile as already indicated, for the small strain case, Ex x . In addition, one may use (1.12),=(1.15b), (1.34) to show that (apply cyc-xyz)rx = 1 + Ex . , (1.35)Thus, we may draw two important conclusions from the above discussion. First, as shownby (1.33), all strain components play a role in the determination of the relative extension in anarbitrary direction. Secondly, and a more specic conclusion, is that the strain components,may be viewed as the relative extension in direction for small strain values ( 2 /2), while for the large strain case, the relative extension of the coordinate axes is given by (1.34). The above is completely applicable for curvilinear coordinates as well. Thus, (1.33) may berewritten by replacing x, y, z with 1, 2, 3 and with 2i j . The direction cosines will now repre-sent the angles from the element to the curvilinear directions, namely, cos(s, ki ). Subsequently,the relative extensions along the coordinate lines are Ei = 1 + 2ii 1.(1.36)We continue the physical interpretation of the strain component by looking at the angle change(caused by the deformation), , between two unit vectors, k and the k , which clearly were perpendicular before deformation (where they were denoted k and the k , respectively).For example, in Cartesian coordinates, the angle between k and k is given by x yk k = cos(k , k ) = cos( xy ) = sin( xy ). xyxy(1.37)2Therefore, with the aid of (1.12), (1.34), (1.35) one may write (1 + u, x ) kx + v, x ky + w, x kzu, y kx + (1 + v, y ) ky + w, y kz k =x, k = y.(1.38) 1 + 2x1 + 2yHence, in view of (1.15b) in a tensorial notation we obtain 2i j sin i j = , (1.39) 1 + 2ii 1 + 2 j jwhich is applicable for curvilinear coordinates as well. It is therefore shown that for smallstrain , or more generally, i j 2i j . Thus, in the linear case, i j may be viewed = =as half the change of angle in the i, j-plane caused by the deformation.1.1.4.2 Relative Change in VolumeThe relative change in volume is an additional measure that may be expressed by the straincomponents and therefore serves also as a physical interpretation of these components. The volume of an innitesimal cubic element before deformation is dV = dx dy dz . Dueto the deformation, an innitesimal cubic element is deformed by (1.2) into a parallelepiped,the volume of which is given by dV = J dx dy dz, see (1.3). Using (1.15b) with engineeringnotation for Cartesian coordinates, one may write1 + 2x xy xzdV 2= xy 1 + 2y yz = 1 + 2x + 2y + 2z + 4y z + 4x zdVxz yz 1 + 2z+4x y + 8x y z + 2xy xz yz yz 2 2yz 2 x xz 2 2xz 2 y xy 2 2xy 2 z . (1.40) 25. 1.2 Displacement by Strain Integration 9Clearly, if one uses the above result for the state of principal strain (that will be derived withinS.1.3.4), the underlined terms in (1.40) vanish.Alternatively, by employing the strain tensor invariants 1 , 2 , 3 of S.1.3.4, one may write dV 2 = 1 + 21 +42 + 83 .(1.41) dVHence, for small strain, the underlined terms in the above equation may be neglected, and therelative change in volume, dV /dV 1, becomes a simple invariant of the coordinate systemorientation, namely,dV dV = x + y + z = 1 . (1.42) dVFor curvilinear coordinates, (1.40) should be written by replacing with ii and with2i j where , {x, y, z}, i, j {1, 2, 3}.1.2 Displacement by Strain IntegrationIn many occasions, analytical solution methods yield expressions of the strain distributionswith no previous use or inclusion of the displacement eld. When such distributions are known,various integration steps should be carried out in order to create the displacement compo-nents. Two approaches to the general derivation of these steps will be developed in what fol-lows. Yet, before any integration steps are taken, one needs to verify that the equations areintegrable. For that purpose we shall rst develop the compatibility equations.1.2.1 Compatibility EquationsOne of the fundamental sets of governing equations in the theory of elasticity is known asthe compatibility equations. To clarify the role and origin of these equations, we shall rstrestrict ourselves to small displacements in Cartesian coordinates, where in order to solve agiven problem, one presumes a set of analytical forms for the six strain components. Clearly,the six strain components = {i j } ji=1,2,3 cannot be selected arbitrarily as functions of theCartesian coordinates x = {xi }, as they are determined completely by only three displacementcomponents u = {ui }i=1,2,3 , see (1.15b). The required additional relations are known as thecompatibility equations, and, as already mentioned, in some mathematical contexts they arealso referred to as the integrability conditions. Prior to the general case discussion, we shall present the linear reduction of these equa-tions, which is founded on the six independent components of the linear strain tensor, e ={ei j }i, j=1,2,3 , of (1.15a). By taking second derivatives of e in Cartesian coordinates one maywrite the following identities: emn, i j + ei j, mn = eim, jn + e jn, im , i, j, m, n {1, 2, 3}.(1.43)Only six out of the above 81 equations are independent, for example, those obtained by thefollowing index sets: (mni j) = (1212), (2323), (3131), (1213), (2321), (3132). (1.44)Subsequently, following (1.43), in engineering notation, the (linear) compatibility equations inCartesian coordinates are xy,xy = x,yy + y,xx , (1.45a) 26. 101. Fundamentals of Anisotropic Elasticity and Analytical Methodologies yz,yz = y,zz + z,yy ,(1.45b) xz,xz = x,zz + z,xx ,(1.45c)2x,yz = xz,xy + xy,xz yz,xx ,(1.45d)2y,xz = xy,yz + yz,xy xz,yy ,(1.45e)2z,xy = yz,xz + xz,yz xy,zz .(1.45f)For an x, y-plane deformation, only one independent equation, (1.45a), is obtained and may bewritten in index notation with i = j = 2, m = n = 1 (the rst case of (1.44)) ase11, 22 + e22, 11 = 2 e12, 12 . (1.46)To facilitate the discussion of the complete nonlinear strain analysis case, we shall dene thecompatibility equations as the conditions imposed on the Lagrange-Green tensor, that guaran-tee the existence of a unique displacement solution when the fully nonlinear strain expressionsare utilized. To derive the desired conditions for curvilinear coordinates we shall employ the metric ten-sors in the undeformed and deformed conguration, previously denoted as g and g , respec-tively. We shall also make use of (1.25), which shows that, for generic orthogonal curvilinearcoordinates (in Euclidean space), the metric tensor after deformation is expressed as gj = gi j + 2i j Hi H j .i (1.47)The key to the development of the compatibility conditions is the fact that, similar to the un-deformed conguration that occupies a part of a Euclidean space of a given topology while itscurvature tensor vanishes, the curvature tensor of the deformed conguration must vanish aswell since the deformed body occupies (again) a part of a Euclidean space while preservingthe domain topology (namely, a multiply connected domain of any level will be preserved assuch). We therefore adopt the Riemann-Christoffel curvature tensor of (1.213), see S.1.7.2.2,for g , and the associated 81 complete compatibility equations becomeR j = 0, mni i, j, m, n {1, 2, 3},(1.48)where R j = 1 (g j, ni + g m j g n j g im ) g f h ( ,im jn , ni jm ) and ms =mni 2 m in,mi,jn,fh,f h,p,1 2 (gmp, s + g m g p ). Note that (1.47) enables us to write the above equations in termsps,ms,of strain components. As already indicated, only six of these equations are independent, see(1.44). P.1.3, P.1.4 are capable of producing the fully nonlinear compatibility equations for boththree- and two-dimensional deformation elds. For example, the compatibility equation for atwo-dimensional case in Cartesian coordinates is212, 12 11, 22 22, 11 = g11 [11, 1 (12, 2 22, 1 ) 2 2 ]+g22 (22, 2 (12, 1 11, 2 ) 2 1 )11,22, + g12 [(12, 1 11, 2 )(11, 2 22, 1 ) + 11, 1 22, 2 211, 2 22, 1 ], (1.49)where1 + 2 222121 + 2 11 g11 = , g12 = , g22 = ,(1.50)DDDand D = 1 + 2 22 + 2 11 + 4 11 22 4 2 . Assuming that all strains and their derivatives 12are small compared with unity, one may linearize (1.49) and reach the compatibility equation,(1.46). Note that for this level of accuracy, the expressions for the strain components must 27. 1.2 Displacement by Strain Integration11be linearized as well, i.e. only the substitution of i j ei j would be consistent. As anotherexample, we present the linearized compatibility equation in polar coordinates, (, ) , 2 , + 2 , + 2 , , 2, = 0. (1.51)The fully nonlinear compatibility equations for other cases (including three-dimensionalcases of generic orthogonal curvilinear coordinates) are very lengthy, and as already indicatedmay be obtained in full by activating P.1.3, P.1.4.1.2.2 Continuous ApproachTo derive continuous expressions for the strain components integration, we shall be focusedon the linear strain components in a Cartesian coordinate system (i.e. the case of i j = ei j , seeS.1.1.2), as no general analysis may be drawn for a general nonlinear case. Although similarprocedures may be carried out for any curvilinear coordinates along the same lines, yet inpractice, it is more convenient to transform the strain components into Cartesian coordinates(see S.1.3.4) and integrate them there (and if necessary, transform the resulting displacementsback to the curvilinear coordinates). It is important to reiterate and state that the underlying assumption in the following deriva-tion is the fulllment of the compatibility equations by the strain components. Otherwise, thesystem is not integrable, and the three displacement components u, v and w can not be consis-tently extracted from the six strain components (see discussion in S.1.2.1). In what follows, we shall assume that all strain components, in engineering notation asdescribed by (1.9a,b), namely, x , y , z , yz , xz and xy , are known as general functions of x, yand z. In addition, the rigid body displacements and rotation components are given as the valuesof the three displacement components u = u0 , v = v0 , w = w0 and the three rotation componentsx = 0 , y = 0 and z = 0 at the system origin point, P0 (i.e. at x = y = z = 0). Modifying xyzthe resulting solution for a case where the rigid body components are dened at other locationsis simple. As a rst step we determine each component of the rotation vector from its three givenpartial derivatives i, j = fi j (x, y, z), i, j {x, y, z},(1.52)where111fxx = (xz, y xy, z ), fxy = yz, y y, z , fxz = z, y yz, z , (1.53a)222 11 1fyx = x, z xz, x ,fyy = (xy, z yz, x ), fyz = xz, z z, x ,(1.53b) 22 21 11fzx = xy, x x, y ,fzy = y, x xy, y ,fzz = (yz, x xz, y ). (1.53c)2 22Hence, since the rotations at P0 (0, 0, 0) are known, those of another point, say P(x, y, z), maybe presented asP i = 0 + i( fix dx + fiy dy + fiz dz), i {x, y, z}.(1.54) P0Note that the expressions under the above integrals are complete differentials in view offi j, k = fik, j , i, j, k {x, y, z},(1.55) 28. 121. Fundamentals of Anisotropic Elasticity and Analytical Methodologieswhich are essentially equivalent to the compatibility equations, (1.45af).As a second step we determine each component of displacement from its three given partialderivatives 11u, x = x ,u, y = xy z , u, z = xz + y ,(1.56a) 22 11v, y = y ,v, z = yz x , v, x = xy + z ,(1.56b) 2211 w, z = z , w, x = xz y ,w, y = yz + x . (1.56c)22Similar to the rotation case, when the displacement components at P0 are known, those of pointP may be expressed as P1 1 u = u0 + [x dx + ( xy z ) dy + ( xz + y ) dz],(1.57a)P02 2 P 11 v = v0 + [( xy + z ) dx + y dy + ( yz x ) dz],(1.57b)P0 22P 11 w = w0 +[( xz y ) dx + ( yz + x ) dy + z dz]. (1.57c) P02 2Again, the expressions under the integrals are complete differentials in view of the compat-ibility equations. The above procedure may be executed for a given consistent set of strainfunctions by activating P.1.5.Remark 1.2 The procedure described above may be reduced to the two-dimensional case,where w = 0 and the displacements u, v are functions of x, y only, all strain components thatinclude the z index vanish (namely, xz = yz = z = 0), and the remaining components x (x, y),y (x, y), xy (x, y) satisfy compatibility (1.45a). As a rst step, we determine the z componentof the rotation vector, z (x, y) (obviously, x = y = 0 in this case), from its two given partialderivatives, see (1.54) with i = 3,P z = 0 + zfzx dx + fzy dyP0 x y11= 0 zx, y dx + y, x (0, y) dy + xy xy (0, y) + xy (0, 0).(1.58)0022As a second step, we determine each component of the displacements u(x, y) and v(x, y) fromtheir two given x- and y- partial derivatives, see (1.56a,b), which yields P 1 P 1 u = u0 +[x dx + ( xy z ) dy], v = v0 + [y dy + ( xy + z ) dx].(1.59)P0 2P0 2This procedure may be executed for a given consistent set of strain functions by activatingP.1.6.1.2.3Level ApproachMany problems in elasticity may be analyzed by series expansion of the involved expressionswith respect to one of the coordinate systems. Beam analyses, see Chapters 6 9, are classicexamples for schemes where all quantities may be expanded as Taylor series of the longitu-dinal coordinate, say z. Such representation of the involved expressions may be exploited tosubstantially simplify the integration process described in S.1.2.2. 29. 1.2 Displacement by Strain Integration13 In general, we shall expand all spatial functions as truncated polynomials of degree Km 0of the axial variable, z, and as continuous functions of the cross-section variables x and y. Forany generic function G(x, y, z) written as G(x, y, z) = k=0 G(k) (x, y)zk , mK(1.60)we shall refer to Km as the expansion degree, and to G(k) as the kth (level) component of G.Clearly, G may be integrated and differentiated with respect to z as zzk+1 K 1 G dz = k=0 G(k) G, z = k=0 (k + 1)G(k+1) zk . K m , m(1.61) 0k+1To derive the present approach, we also assume that the strain components are truncated asi = k=0 i (x, y)zk .m K(k)(1.62)In cases where the strain components are not given as in (1.62), one may employ a standardTaylor series expansion, and therefore, in some cases the above truncated form represents anapproximation. We subsequently expect the displacements and rotations to be expressed inlevels as well, namelyu = k=0 u(k) (x, y)zk , v = k=0 v(k) (x, y)zk ,w = k=0 w(k) (x, y)zk , m K mK Km (1.63a)Km i = i (x, y)zk , (k)i = 1, 2, 3. (1.63b)k=0In what follows, Km stands for the maximal expansion degree of all analysis components(clearly, the expansion degree of the strain components will be less than those of the dis-placement components). The fi j functions of (1.53ac) are also written in levels as fi j =Km (k)k=0 fi j (x, y)zk , while (k)1 (k) (k+1)(k) 1 (k) (k+1)(k) (k) 1(k+1)fxx = [xz, y (k + 1)xy ], fxy = yz, y (k + 1)y , fxz = z, y (k + 1)yz ,222 (k)(k+1) 1 (k)(k) 1 (k+1)(k) (k)1 (k+1)(k)fyx = (k + 1)x xz, x , fyy = [(k + 1)xy yz, x ], fyz = (k + 1)xz z, x ,22 2 (k)1 (k) (k)(k) (k)1 (k)(k) 1 (k)(k)fzx = xy, x x, y ,fzy = y, x xy, y ,fzz = (yz, x xz, y ). (1.64)2 22By selecting the polygonal integration trajectory (0, 0, 0) (0, y, 0) (x, y, 0) (x, y, z), wewrite (1.54) as xyzk+1fiy (0, y)dy + k=0 fxz (x, y) (0)(0)K (k)i = 0 +ifix (x, y)dx + m,i {x, y, z}. (1.65) 00k+1Hence, the level components of the rotations are xy1 (k1) (0) (0) (0) (k) i= 0 +ifix (x, y) dx +fiy (0, y) dy, i =f (x, y) (1.66) 00k izwhere k 1, i {x, y, z}. At this stage, (1.56ac) show that (k) (k)1 (k)(k)(k) (k)1 (k)(k) u, x = xx , u, y =xy z ,u, z =xz + y , (1.67a)2 2 (k) (k) (k)1 (k)(k) (k)1 (k)(k) v, y = yy , v, z = yz x , v, x = xy + z ,(1.67b)2 2 (k) (k) (k)1 (k)(k)(k) 1 (k)(k) w, z = zz ,w, x = xz y ,w, y = yz + x ,(1.67c)2 2 30. 14 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologiesand therefore, by analogy with the rotation integration process, one may writex y1 (0)1 1 (k1) (0) (0) (k1) u(0) = u0 +xx dx + u(k) = ( xz + y( xy z )(0, y) dy, ), (1.68) 0 0 2k 2 x 1y 1 1 (k1) (0) (0)(0) (k1) v(0) = v0 + [ xy + z ] dx +yy (0, y) dy, v(k) = ( yz x), 0 20 k 2x 1y 11 (k1)(0) (0) (0)(0) w(0) = w0 +[ xz y ] dx +( yz + x )(0, y) dy, w(k) = zz02 0 2kwhere k 1.Remark 1.3 An alternative way to handle the strain integration for level-based solution callsfor exploiting the strain-displacement relations of (1.9a,b) to show that (k)(k)(k) (k)(k)(k) (k)x = u, x ,y = v, y , xy = u, y + v, x , (1.69a) (k) (k)(k) (k) (k)z = (k + 1)w(k+1) , yz = (k + 1)v(k+1) + w, y ,xz = (k + 1)u(k+1) + w, x .(1.69b)Thus, one can extract the displacement components from (1.69b) as1 (k) w(k+1) =z ,(1.70a) k+11(k+1)1 (k)1(k+1)1 (k)u(k+2) = (xz z, x ), v(k+2) =(yz z, y ), (1.70b) k+2 k+1k+2 k+1(0)(0)(0) (0)u(1) = xz w, x ,v(1) = yz w, y , (1.70c)using the known strain components and the zero level of displacements obtained from (1.69a)with k = 0. In view of (1.19) one may also extract the rotations components as(k) 1 (k)(k) 1(k) (k) 1 (k) (k) 1 = (w, y (k + 1)v(k+1) ), 2 = ((k + 1)u(k+1) w, x ), 3 = (v, x u, y ). (1.71)222To execute the above approach one should pursue the following steps: (1) Calculate w(k) for all k > 0 levels by (1.70a). (2) Calculate u(k) , v(k) for all k > 0 levels by (1.70b). (0) (3) Calculate {i }, i=1,2,3 from (1.66), including the introduction of the rigid body rota-tions {0 }i=1,2,3 . This step is analogous to (1.54) with fxz = fyz = fzz = 0.i (4) Calculate u(0) , v(0) , w(0) from (1.68), including the introduction of the rigid body dis-placements u0 , v0 , w0 . This step is analogous to (1.57ac) with u, z = v, z = w, z = 0. (5) Calculate u(1) , v(1) by (1.70c).(k) (6) Calculate {i }i=1,2,3 for all k > 0 levels from (1.71).1.3Stress MeasuresIn this section we shall introduce the notion of stress and derive different forms of its expres-sion. We shall also discuss the equilibrium equations that are most commonly associated to andwritten by the stress components. The derivation is largely founded on the coordinate systemsanalysis of S.1.7. 31. 1.3 Stress Measures151.3.1 Denition of StressTo dene the stress tensor at a point, it is worthwhile to rst examine the mathematical def-inition of stress in a simple linear case, which may be easily interpreted and associated bycomplementary physical quantities. As will be shown later on, when all nonlinear effects areincluded, one is forced to work with generalized stresses, which are mathematical measuresof the actual stress components and are more difcult to be physically interpreted. We shall rst examine a three-dimensional body by virtually cutting it over an interior planein which we dene a small area A (say, a small circle with a center located at P) as shown inFig. 1.2. The force acting over A normal to the plane will be denoted N while the tangent Figure 1.2: Normal and tangential loads over a small area in an elastic domain.(in-plane) force component will be denoted T. Both N and T are functions of the locationof the circle over the plane (or essentially the location of its center point, P in the plane). Bynarrowing the area A, the point P and the area A collide, and the normal and tangential stresscomponents at P are dened asNTN (P) = lim, T (P) = lim .(1.72)A0 AA0 AHence, the dimension of the stress components is force per unit area.At each point, one may examine an innitesimally small material element, as shown inFig. 1.3. When described in the coordinate space (as opposed to the Euclidean space), be-Figure 1.3: An innitesimal element in its coordinate space. 32. 161. Fundamentals of Anisotropic Elasticity and Analytical Methodologiesfore deformation, the above element may be viewed as an innitesimal cube regardless of thespecic system employed. When described in a Euclidean space in the undeformed state, aninnitesimal material element of general curvilinear coordinates may be described by a cubictopology while all faces are different four-edges-polygons. Hence, its six faces are differentquadrangles (this description is usually termed a rectangular parallelepiped since all con-sidered coordinate systems are orthogonal). For example, the x coordinates of the A and Bvertices (see Fig. 1.3) are, respectively, f1, 1 d1 , f1, 1 d1 + f1, 2 d2 . When the same mate-rial element after deformation is examined, it may be no longer described as a cubic even in itscoordinate space. The material element in the deformed state in both the coordinate space andthe Euclidean space is generally called an oblique angle parallelepiped, and may be gener-ally viewed as a cubic, the corners of which have been displaced differently, so its six faces arenow different quadrangles (in essence, as previously discussed, this general description holdsfor the undeformed case in Euclidean space as well).We initially consider the stress components as an asymmetric second-order tensor. By deni-tion, i j is the stress component in the k j -direction that acts on a plane, which is perpendicularto the ki -direction e.g. 1 j in Fig. 1.3. When the same material element after deformationis examined, one may decompose the stress components by using various coordinate systems.However, we preserve the same notation logic in which i j is dened as the stress componentin the deformed jth direction of a given coordinate system that acts on a plane, which before deformation was perpendicular to the ki -direction.On each of the deformed faces one may dene stress vectors. We will denote by 1 , 2 and 3 the stress vectors over faces #1,2 and 3, see Fig. 1.3. Before deformation the edges lengthsare Hi di (see (1.215)), and their respective areas are Ai = H j Hk d j dk (apply cyc-i jk). Theareas of the corresponding faces after deformation are denoted A .iBased on the above denitions, the forces that act over faces #1,2 and 3 may be written as AA i (i {1, 3}), or, equivalently, Ai i H j Hk d j dk . We will now decompose i along theiideformed and the undeformed directions, respectively, as A i = j=1 i j k , i = j=1 si j k j .3 i3j (1.73) AiIt may be veried that si1 i1 si2 = [I + e + ] i2 ,i = 1, 2, 3. (1.74) si3 i3In the above, I is a unit matrix, e and are given by (1.17), and A i ji j =i, (1.75) Ai 1 + E jwhere E j is the relative extension in the jth direction, see (1.36). i j are usually referredto as thegeneralized stresses and are not stresses in the strict sense. As shown by (1.75),these quantities are based on the element volume before deformation. However, they haveimportant symmetry characteristics that are missing in i j , namely, i j = j i (while i j = j i ).Therefore, under a coordinate system transformation the tensor = {i j } acts like a second-order symmetric tensor. Subsequently, this tensor should be regarded as the stress tensor whena fully nonlinear analysis is employed. Once a solution of a specic problem is carried out,and the components of are derived, the values of i j may be recovered by (1.75). Note thatboundary conditions should be imposed on i j (and not on i j ), as they represent the physicalstress components in the deformed state. 33. 1.3 Stress Measures17At this stage, the only missing component in the scheme is the area ratio A /Ai , which may ibe written as (apply cyc-i jk) Ai= (1 + E j ) (1 + Ek ) sin(k , k ). j k(1.76) AiHere Ei are given in (1.36) and sin(k , k ) may be evaluated using (1.39), which yields (apply j kcyc-i jk)Ai= (1 + 2 j j ) (1 + 2kk ) 42 ,jk i = j = k. (1.77)AiTransformation of stress components between curvilinear and Cartesian coordinates appearsin S.1.3.3. As previously mentioned, when a nonlinear analysis is under discussion, the tensor should be transformed between coordinate systems.Note that in the linear case, we pay no attention to the difference between the element shapebefore and after deformation. Subsequently, A Ai , E j i =1, e I, I, and therefore,si j i j i j , and i j becomes a symmetric tensor as well. ==1.3.2Equilibrium EquationsAn extremely important and necessary ingredient in the set of governing equations for eachproblem in the theory of elasticity are the equilibrium equations. To express these equationsusing the above derived stress denitions, the body forces acting over the unit volume at eachmaterial point should also be considered. The body forces are described by their componentsin the undeformed directions asFb = j=1 Fb j k j ,3(1.78)while the total force action on a volume element is given by Fb dV = Fb H1 H2 H3 d1 d2 d3 .Subsequently, equilibrium may be imposed by equating the resultant vector of all forces actingon the material element to zero, namely, A A A(H2 H31 1 ), 1 + (H1 H3 2 2 ), 2 + (H1 H2 3 3 ), 3 + H1 H2 H3 Fb = 0 .(1.79) A1 A2 A3The above vector equation may be now decomposed into its components. This yields threeequations of equilibrium that may be written as (apply cyc-123)(H2 H3 s11 ), 1 + (H1 H3 s21 ), 2 + (H1 H2 s31 ), 3 + H3 H1, 2 s12 + H2 H1, 3 s13 H3 H2, 1 s22 H2 H3, 1 s33 + H1 H2 H3 Fb1 = 0, (1.80)while si j are given in (1.74). As already indicated, in the linear case, si j i j . =P.1.7, P.1.8 are capable of producing equilibrium equations for various orthogonal coordi-nates in E 3 or E 2 . These equations are written in terms of i j . When nonlinear analysis isrequired, i j should be replaced by si j .We shall present here some illustrative examples of the linear case. For Cartesian coordi-nates we denote the body-force components in the x, y, z directions, as Fb = Xb k1 +Yb k2 + Zb k3 , (1.81)and write x,x + xy,y + xz,z + Xb = 0,(1.82a)xy,x + y,y + yz,z +Yb = 0,(1.82b)xz,x + yz,y + z,z + Zb = 0. (1.82c) 34. 181. Fundamentals of Anisotropic Elasticity and Analytical MethodologiesNote that for the present linear Cartesian case, moment differential equilibrium may be easilyseen as a direct consequence of the stress tensor symmetry. It should be noted that the differential equilibrium equations of (1.82ac) may be derivedfrom an integral (static) equilibrium that is written with the aid of the body and the surfaceloads that act over the volume of each material point, and over the outer surface of the body.Similar to (1.78), surface loads are dened as forces per the unit area at each boundary materialpoint and described by their components in the undeformed directions asFs = j=1 Fs j k j . 3 (1.83)In Cartesian coordinates we write Fs = Xs k1 +Ys k2 + Zs k3 , (1.84)where Xs = x cos(n, x) + xy cos(n, y) + xz cos(n, z), (1.85a) Ys = xy cos(n, x) + y cos(n, y) + yz cos(n, z), (1.85b) Zs = xz cos(n, x) + yz cos(n, y) + z cos(n, z), (1.85c) and cos(n, x), cos(n, y) and cos(n, z) are angle cosines between the normal to the surface andthe x, y, z directions, respectively. At this stage, we express integral force equilibrium asFs + Fb = 0. (1.86)SVHence, by substituting (1.81, 1.84, 1.85a-c) in (1.86) and applying the Divergence Theorem,we reach three integral equations (over the entire body volume) the integrands of which are(1.82ac). Since these equations apply to each inntisimal volume as well, (1.82ac) are re-established. For cylindrical coordinates we denote by Rb , b , Zb the body-force components in the, c , z directions, respectively, and write1 + , + , + z, z + Rb = 0,(1.87a) 1, + 2 + z, z + , + b = 0, (1.87b) 1 z, + z + z, + zz, z + Zb = 0. (1.87c) For spherical coordinates we denote by Rb , b and b the body-force components in the, s , s directions, respectively, and write1 1, + 2 + cot + , + , + Rb = 0, (1.88a) sin 1 1 , + 3 + , + 2 cos + , + b = 0, (1.88b) sin 1 1 , + 3 + , + cos + , + b = 0. (1.88c) sin 35. 1.3 Stress Measures 19For elliptical-cylindrical coordinates (with a = 1, see (1.218b)), one obtains 33(11 22 ) cosh(1 ) sinh(1 )+ A 11, 1 +212 cos(2 ) sin(2 )+ A 12, 2 + A 2 13, 3 + Fb1 A 2 = 0, 33212 cosh(1 ) sinh(1 )+ A 12, 1 +(22 11 ) cos(2 ) sin(2 )+ A 22, 2 + A 2 23, 3 + Fb2 A 2 = 0,3 3 13 cosh(1 ) sinh(1 )+A 13, 1+23 cos(2 ) sin(2 )+A 23, 2+A 2 33, 3+Fb3 A 2 = 0 (1.89)where A = cosh2 (1 ) cos2 (2 ), and Fb1 , Fb2 and Fb3 are the body-force components in the1 , 2 and 3 directions, respectively.1.3.3Stress Tensor Transformation due to Coordinate System RotationWe shall now exploit the general derivation for coordinate systems presented in S.1.7 to trans-form the stress components at a point. More specically, for a given stress tensor in one co-ordinate system, we wish to determine the six independent stress components of the sametensor as seen by another coordinate system. For that purpose, we shall consider the elementsof the transformation matrix T (which are functions of the rotation angles , and , see(1.203), (1.206)) as the Ti j element of the corresponding transformation tensor, and dene the(symmetric) second-order stress tensor as11 12 13 = 12 22 23 .(1.90)13 23 33Hence, the components of the stress tensor in the new system, = {i j }, are obtained by thestandard tensor transformationi j = ab Tia T jb . (1.91)This operation may be expressed using matrix notation as well, as = T TT . (1.92)To simplify the relations between the stress tensor components before and after transforma-tion, we shall look at the above formula using the vectors and , which contain the stresscomponents before and after transformation, namely, = [xx , yy , zz , yz , xz , xy ]T , = [xx , yy , zz , yz , xz , xy ]T . (1.93)These vectors may be related as = M , where M is a (non-symmetric) 6 6 matrix,which is clearly a function of the transformation (Eulers) rotation angles , and . Exampleterms areM (3, 2) = (cos sin sin sin cos )2 , M (2, 3) = sin2 cos2 . (1.94)The reader may activate P.1.9 to generate all terms of M symbolically. Figure 1.4 p