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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 fürMathematik. He is the author of many journal papers and books,including Foliations on Riemannian Manifolds andSubmanifolds (Birkhäuser, 1997), and Geometry of Curves andSurfaces with MAPLE (Birkhäuser, 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.
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Omri Rand Vladimir RovenskiAnalytical Methods inAnisotropic Elasticitywith Symbolic Computational Tools Birkh¨ user a Boston • Basel • Berlin
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Omri Rand Vladimir Rovenski Technion — Israel Institute of Technology Technion — Israel Institute of Technology Faculty of Aerospace Engineering Faculty of Aerospace Engineering Haifa 32000 Haifa 32000 Israel IsraelAMS Subject Classiﬁcations: 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 / Omri Rand, Vladimir Rovenski. p. cm. Includes bibliographical references and index. ISBN 0-8176-4372-2 (alk. paper) 1. Elasticity. 2. Anisotropy. 3. Anisotropy—Mathematical models. 4. Inhomogeneous materials. I. Rovenskii, Vladimir Y, 1953- II. Title QA931.R36 2004 531 .382–dc22 2004054558ISBN-10 0-8176-4272-2 Printed on acid-free paper.ISBN-13 978-0-8176-4372-3 c 2005 Birkh¨ user Boston aAll 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 identiﬁedas 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
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To my family, Ora, Shahar, Tal and Boaz, Omri RandTo my teacher, Professor Victor Toponogov, Vladimir Rovenski
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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 signiﬁcantly to the understanding of physical phenomena. In recent decades, nu-merical computerized techniques have been reﬁned 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 ﬁnite-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 ﬁeld or the stress ﬁeld 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 speciﬁc 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
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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 ﬁeld 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 ﬁrm testimony to the exactness of the presented expressions. For thatreason, the speciﬁc 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 ﬁrst part of the book ( Chapters 1–4) contains the fundamentals of anisotropic elasticity.The second part (Chapters 5–10) 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 deﬁnes 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 deﬁnitions ofgeneralized Neumann/Dirichlet and biharmonic boundary value problems (BVPs). The chapter
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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 deﬁnitions 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 ﬁrst 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. Venant’s 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 ﬁrst 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 deﬁnition 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 simpliﬁed 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).
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x Preface General Style Clariﬁcation 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 clariﬁcation 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( )/dα or ∂( )/∂α, are abbreviated as ( ),α . Similarly, d 2 ( )/dα2or ∂2 ( )/∂α2 , are abbreviated as ( ),αα . (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., ∂Ω Fds, which is written simply as ∂Ω F, and the area integration in the xy-plane,i.e., Ω F dx dy, which is written as Ω F. (4) Within the equation notation, e.g. (1.3), the ﬁrst 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 ﬁrst 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 veriﬁcation 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 Rand Vladimir Rovenski Haifa, Israel
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1Fundamentals of Anisotropic Elasticity andAnalytical MethodologiesThis chapter is devoted to fundamental issues in anisotropic elasticity that should be deﬁnedand reviewed before speciﬁc 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 ﬁnitedeformations. 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-
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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 speciﬁc 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 fulﬁllment 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 inﬂuence on this selection. In what follows, we shall therefore establish andreview the mathematical deﬁnition of deformation for a variety of coordinate systems. For thesake of clarity, we will ﬁrst 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 deﬁnition,the displacements of a material particle are determined by its initial and ﬁnal 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 ∗ (ξ, η, ζ), with position vector x + u, Fig. 1.1(a). Figure 1.1: (a) Deformation of domain Ω into Ω∗ . (b) Vector (small deformation) ﬁeld of Ω.One may therefore write, ξ = x + u, η = y + v, ζ = 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 dξ = (1 + u, x ) dx + u, y dy + u, z dz ,
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1.1 Deformation Measures and Strain 3 dη = v, x dx + (1 + v, y ) dy + v, z dz , dζ = w, x dx + w, y dy + (1 + w, z ) dz . (1.2)A necessary and sufﬁcient 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, y u, z J = v, x 1 + v, y v, z > 0. (1.3) w, x w, y 1 + 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 = ω 0 × x, where ω 0 = (ω0 , ω0 , ¯ ¯ 1 2ω0 ) is a constant vector, see also Remark 1.1). These may be combined to form the basic type 3of 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 ﬁxed (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 = ω0 (z − z0 ) − ω0 (y − y0 ) + u0 , 2 3 vr = ω0 (x − x0 ) − ω0 (z − z0 ) + v0 , 3 1 wr = ω0 (y − y0 ) − ω0 (x − x0 ) + w0 , 1 2 (1.4)i.e., u = ω 0 × (x − x0 ) + u0 . Note that the Jacobian of the above rigid body deformation is ¯equal to or greater than 1, since 1 −ω0 3 ω02 3 Jr = ω03 1 −ω0 = 1 + ∑ (ω0 )2 . 1 i (1.5) −ω0 2 ω01 1 i=1In general, (1.1) in which u, v, w are linear functions of the coordinates constitute an afﬁnedeformation, see S.1.7.1. A common way to present a ﬁeld of small deformation in the elastic media is the so-calledvector ﬁeld 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 = ω 0 × x deﬁnition 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 deﬁne the strain components in a Carte-sian coordinate system x, y, z, in addition to the material points M(x, y, z), M ∗ (ξ, η, ζ) discussedearlier, we assume that a point N, which is inﬁnitesimally close to M and located before defor-mation at (x + dx, y + dy, z + dz) has moved to N ∗ (ξ + dξ, η + dη, ζ + dζ), see Fig. 1.1(a). Letds = |MN| be the distance between the two points before deformation, and ds∗ = |M ∗ N ∗ | be
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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 = (dξ)2 + (dη)2 + (dζ)2 . (1.6)We shall now establish a deﬁnition for suitable measures that will properly describe the defor-mation at the vicinity of the point M. The common measure of deformation is mathematicallydeﬁned as 1 (ds∗ )2 − (ds)2 , and may be expressed as a sum of its components according to 2the resulting six different combinations of products of inﬁnitesimal distances, namely, 1 [(ds∗ )2 − (ds)2 ] = εxx (dx)2 + εyy (dy)2 + εzz (dz)2 + εyz dy dz + εxz dx dz + εxy dx dy . (1.7) 2The coefﬁcients εαβ are traditionally referred to as the engineering strain components at pointM, and are given by 1 2 1 1 εxx = u, x + u +v2 +w2x , εyy = v, y + u2y +v2y +w2y , εzz = w, z + u2z +v2z +w2z , 2 ,x ,x , 2 , , , 2 , , , εyz = v, z + w, y + u, y u, z + v, y v, z + w, y w, z , εxz = u, z + w, x + u, x u, z + v, x v, z + w, x w, z , εxy = 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, eαβ , 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 εαβ (α = β) with γαβ or to use a numerical indexnotation in which the strain vector {εxx , εyy , εzz , εyz , εxz , εxy } is written as {ε1 , ε2 , ε3 , ε4 , ε5 , ε6 }. The nonlinear version of εαβ 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 deﬁne {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 form u = ∑ 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=1 To determine the associated metric tensors, g = {gi j } and g∗ = {g∗j } before and after defor- imation, respectively, given by gi j = r, i · r, j , g∗j = r∗i · r∗j , 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), r∗i = ∑ j=1 (δi j + u j, i ) k j , 3 , (1.12)where δi j is Kronecker’s symbol. Hence, in the Cartesian case under discussion, gi j = δi j ,while g∗j = δi j +ui, j + u j, i + ∑ m=1 um, i um, j . 3 i (1.13)
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1.1 Deformation Measures and Strain 5By deﬁnition, the strain tensor components of the Lagrange-Green deformation measures are 1 εi j = (g∗j − gi j ). (1.14) 2 iThus, the linear (the underlined terms of (1.13)) and nonlinear expressions for strains in tensornotation are 1 ei j = (ui, j + u j, i ) , (1.15a) 2 1 1 3 εi j = (ui, j + u j, i ) + ∑ m=1 um, i um, j . (1.15b) 2 2In particular one obtains eii = ui, i and εii = ui, i + 1 ∑3 (um, i )2 . The resulting nonlinear 2 m=1strain tensor components may be also written as 1 3 2 ∑ m=1 εi j = ei j + (eim + ωim ) (e jm + ω jm ), (1.16)where ωi j are the components of the antisymmetric (ω j i = −ωi j ) rotation tensor, ω , while eand ω are commonly put in a matrix form as ⎡ ⎤ ⎡ ⎤ e11 e12 e13 0 ω12 ω13 e = ⎣ e12 e22 e23 ⎦ , ω = ⎣ −ω12 0 ω23 ⎦ . (1.17) e13 e23 e33 −ω13 −ω23 0The rotation tensor components may also be deﬁned by the rotation vector ω as ¯ 1 ω = ∇ × u = ω23 k1 + ω31 k2 + ω12 k3 . ¯ (1.18) 2 ∂In the above equation, the “nabla” operator is written as ∇ = ∑i ∂x i ki , and the vector cross-product operation yields ω i j = 2 (u j, i − ui, j ), namely, 1 1 1 1 ω23 = (u3, 2 − u2, 3 ) , ω31 = (u1, 3 − u3, 1 ) , ω12 = (u2, 1 − u1, 2 ) . (1.19) 2 2 2By 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 εi j , i, j = 1, 2, 3) and the straincomponents in engineering notation are related as the symmetric matrices ⎡ ⎤ ⎡ ⎤ ε11 ε12 ε13 εx 2 εxy 1 2 εxz 1 ⎣ ε12 ε22 ε23 ⎦ = ⎣ 1 εxy εy ⎦ 2 εyz . 1 2 (1.20) ε13 ε23 ε33 1 2 εxz 2 εyz1 εzThe 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 deﬁnition of the strain is very attractive in many analytical applications, asit only requires the ability to deﬁne the position vectors of a material point before and afterdeformation. Then, (1.11), (1.14) may be directly applied. For the sake of clariﬁcation, 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: ε1 ≡ ε11 ≡ εxx ≡ εx , ε2 ≡ ε22 ≡ εyy ≡ εy , ε3 ≡ ε33 ≡ εzz ≡ εz , ε4 ≡ 2ε23 ≡ εyz ≡ γyz , ε5 ≡ 2ε13 ≡ εxz ≡ γxz , ε6 ≡ 2ε12 ≡ εxy ≡ γxy , (1.21) ωx ≡ ω1 ≡ ω23 , ωy ≡ ω2 ≡ ω31 , ωz ≡ ω3 ≡ ω12 . P.1.1, P.1.2 (with s = 0) demonstrate a derivation of the strain components in Cartesian co-ordinates.
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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 deﬁne the deformation in orthogonalcurvilinear coordinates using three functions {ui (α1 , α2 , α3 )}i=1,2,3 and a local basis {ki }, see ˆ(1.215), as u = ∑ i ui (α1 , α2 , α3 ) ki . ˆ (1.22)We also deﬁne in S.1.7 three functions, f1 , f2 , f3 , that convert curvilinear coordinates of E 3 ˆspace 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 uβ = uβ (u1 , u2 , u3 , α1 , α2 , α3 ), β = x, y, z. Since the displacement components u1 , u2 , u3are functions of α1 , α2 , α3 by themselves, one may also consider the form uβ = uβ (α1 , α2 , α3 ).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, αi · r, α j , g∗j = r∗αi · r∗α j , i , , (1.24)for which (ds)2 = ∑i j gi j dαi dα j and (ds∗ )2 = ∑i j g∗j dαi dα j . The Lagrange-Green deﬁnition iof the strain tensor in this case becomes g∗j − gi j i εi j = . (1.25) 2Hi H j √Note that g is always a diagonal matrix while Lam´ parameters, Hi are deﬁned as Hi = gii . eAs an example, in cylindrical coordinates, where (α1 , α2 , α3 ) = (ρ, θ c , z), in view of (1.218a)one obtains g11 = 1, g22 = ρ2 , g33 = 1, and H1 = 1, H2 = ρ, H3 = 1, while g∗ = 1 + 2uρ, ρ + u2 ρ + u2 ρ + u2 ρ , ρρ ρ, θ, z, g∗ = uθ, ρ (ρ + uθ, θ + uρ ) + uz, ρ uz, θ + (uρ, θ − uθ )(1 + uρ, ρ ), ρθ etc., (1.26)and hence, 1 ε11 = uρ, ρ + (u2 ρ + u2 ρ + u2 ρ ), 2 ρ, θ, z, 1 ε12 = (uρ, θ − uθ + ρ uθ, ρ + uz, ρ uz, θ − uρ, ρ uθ + uρ, ρ uρ, θ + uθ, ρ uθ, θ + uθ, ρ uρ ), (1.27) 2ρ etc.,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 εi j ), we deﬁne 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 ﬁrst equation, namely, i → j,
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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 1 1 e11 = u1, α1 + H1, α2 u2 + H1, α3 u3 , H1 H1 H2 H1 H3 H2 u2 H1 u1 e12 = , α1 + , α2 . (1.28) H1 H2 H2 H1The rotation tensor components are extracted from ˆ H1 k1 ˆ H2 k2 ˆ H3 k3 1 1 ∂ ∂ ∂ ω = ∇×u = ¯ ∂α1 ∂α2 ∂α3 , (1.29) 2 2 H1 H2 H3 H1 u1 H2 u2 H3 u3 1 ∂ ˆ(note that the “nabla” operator in this case is ∇ = ∑i Hi ∂αi ki ), which in view of (1.18) yields(apply cyc-123) 1 ωi j = [(H j u j ), αi − (Hi ui ), α j ]. (1.30) 2 Hi H jIn some applications, it is useful to deﬁne the unit vectors in the coordinate line directions ofthe deformed state, namely k∗ = r∗αi / r∗αi . ˆi , , (1.31)1.1.4 Physical Interpretation of Strain Components1.1.4.1 Relative Extension and Angle ChangeSo far, the deformation measures εi j have been deﬁned mathematically, and were shown toprovide a set of six parameters that reﬂect the deformation at a given material point. To gainsome physical insight and clearer interpretation of the above discussed measures, we will ﬁrstdeﬁne 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 ﬁrst note that (ds∗ )2 − (ds)2 1 2 = ( EMN + EMN )(ds)2 . (1.32) 2 2We subsequently substitute the r.h.s. of (1.7) in the above equation and express the relativeextension as 1 2 E + EMN = εx cos(¯, x)2 + εy cos(¯, y)2 + εz cos(¯, z)2 s s s 2 MN + εyz cos(¯, y) cos(¯, z) + εxz cos(¯, x) cos(¯, z) + εxy 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 deﬁned as cos(¯, α) = dα/ds (α = 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 2 E + Ex = εx or Ex = 1 + 2εx − 1, (1.34) 2 x
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8 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologieswhile as already indicated, for the small strain case, Ex ∼ εx . In addition, one may use (1.12), =(1.15b), (1.34) to show that (apply cyc-xyz) r∗x = 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 speciﬁc conclusion, is that the εαα strain components,may be viewed as the relative extension in α direction for small strain values (εαα ε2 /2), ααwhile 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 εαβ with 2εi 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 + 2εii − 1. (1.36)We continue the physical interpretation of the strain component by looking at the angle change(caused by the deformation), ∆ ϕαβ , between two unit vectors, k∗ and the k∗ , which clearly α βwere perpendicular before deformation (where they were denoted kα and the kβ , respectively).For example, in Cartesian coordinates, the angle between k∗ and k∗ is given by x y π k∗ · k∗ = cos(k∗ , k∗ ) = cos( − ∆ ϕxy ) = sin(∆ ϕxy ). x y x y (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 kz u, y kx + (1 + v, y ) ky + w, y kz k∗ = x √ , k∗ = y . (1.38) 1 + 2εx 1 + 2εyHence, in view of (1.15b) in a tensorial notation we obtain 2εi j sin ∆ ϕi j = √ , (1.39) 1 + 2εii 1 + 2ε j jwhich is applicable for curvilinear coordinates as well. It is therefore shown that for smallstrain ∆ ϕαβ ∼ εαβ , or more generally, ∆ ϕi j ∼ 2εi j . Thus, in the linear case, εi 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 inﬁnitesimal cubic element before deformation is dV = dx dy dz . Dueto the deformation, an inﬁnitesimal 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 write 1 + 2εx εxy εxz dV ∗ 2 = εxy 1 + 2εy εyz = 1 + 2εx + 2εy + 2εz + 4εy εz + 4εx εz dV εxz εyz 1 + 2εz +4εx εy + 8εx εy εz + 2εxy εxz εyz − εyz 2 − 2εyz 2 εx − εxz 2 − 2εxz 2 εy − εxy 2 − 2εxy 2 εz . (1.40)
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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 Ξ1 , Ξ2 , Ξ3 of S.1.3.4, one may write dV ∗ 2 = 1 + 2Ξ1 +4Ξ2 + 8Ξ3 . (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 ∼ = ε x + ε y + ε z = Ξ1 . (1.42) dVFor curvilinear coordinates, (1.40) should be written by replacing εαα with εii and εαβ with2εi j where α, β ∈ {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 ﬁeld. 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 ﬁrst 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 ﬁrstrestrict 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 ε = {εi j } j≥i=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 γxy,xy = εx,yy + εy,xx , (1.45a)
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10 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies γyz,yz = εy,zz + εz,yy , (1.45b) γxz,xz = εx,zz + εz,xx , (1.45c) 2εx,yz = γxz,xy + γxy,xz − γyz,xx , (1.45d) 2εy,xz = γxy,yz + γyz,xy − γxz,yy , (1.45e) 2εz,xy = γyz,xz + γxz,yz − γxy,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 ﬁrst case of (1.44)) as e11, 22 + e22, 11 = 2 e12, 12 . (1.46)To facilitate the discussion of the complete nonlinear strain analysis case, we shall deﬁne 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 conﬁguration, 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 g∗j = gi j + 2εi 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 conﬁguration that occupies a part of a Euclidean space of a given topology while itscurvature tensor vanishes, the curvature tensor of the deformed conﬁguration 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 become R∗ 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 (Γ∗ ,im Γ∗ jn − Γ∗ , ni Γ∗ jm ) and Γ∗ ms = mni 2 m in, mi, jn, f h, f h, p,1 ∗2 (gmp, s + g∗ m − g∗ p ). Note that (1.47) enables us to write the above equations in terms ps, 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 ﬁelds. For example, the compatibility equation for atwo-dimensional case in Cartesian coordinates is2ε12, 12 −ε11, 22 −ε22, 11 = g∗11 [ε11, 1 (ε12, 2 − ε22, 1 ) − ε2 2 ]+g∗22 (ε22, 2 (ε12, 1 − ε11, 2 ) − ε2 1 ) 11, 22, + g∗12 [(ε12, 1 − ε11, 2 )(ε11, 2 − ε22, 1 ) + ε11, 1 ε22, 2 − 2ε11, 2 ε22, 1 ], (1.49)where 1 + 2 ε22 2ε12 1 + 2 ε11 g∗11 = , g∗12 = − , g∗22 = , (1.50) D D Dand D = 1 + 2 ε22 + 2 ε11 + 4 ε11 ε22 − 4 ε2 . 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
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1.2 Displacement by Strain Integration 11be linearized as well, i.e. only the substitution of εi j ≈ ei j would be consistent. As anotherexample, we present the linearized compatibility equation in polar coordinates, (ρ, θ) ερρ, θθ − 2ρ ερθ, ρθ + 2ρ εθθ, ρ + ρ2 εθθ, ρρ − ρ ερρ, ρ − 2ερθ, θ = 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 εi 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 fulﬁllment 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, εx , εy , εz , γyz , γxz and γxy , 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 componentsωx = ω0 , ωy = ω0 and ωz = ω0 at the system origin point, P0 (i.e. at x = y = z = 0). Modifying x y zthe resulting solution for a case where the rigid body components are deﬁned at other locationsis simple. As a ﬁrst step we determine each component of the rotation vector from its three givenpartial derivatives ω i, j = fi j (x, y, z), i, j ∈ {x, y, z}, (1.52)where 1 1 1 fxx = (γxz, y − γxy, z ), fxy = γyz, y − εy, z , fxz = εz, y − γyz, z , (1.53a) 2 2 2 1 1 1 fyx = εx, z − γxz, x , fyy = (γxy, z − γyz, x ), fyz = γxz, z − εz, x , (1.53b) 2 2 2 1 1 1 fzx = γxy, x − εx, y , fzy = εy, x − γxy, y , fzz = (γyz, x − γxz, y ). (1.53c) 2 2 2Hence, since the rotations at P0 (0, 0, 0) are known, those of another point, say P(x, y, z), maybe presented as P ω i = ω0 + 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 of fi j, k = fik, j , i, j, k ∈ {x, y, z}, (1.55)
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