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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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12 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologieswhich are essentially equivalent to the compatibility equations, (1.45a–f). As a second step we determine each component of displacement from its three given partialderivatives 1 1 u, x = εx , u, y = γxy − ωz , u, z = γxz + ωy , (1.56a) 2 2 1 1 v, y = εy , v, z = γyz − ωx , v, x = γxy + ωz , (1.56b) 2 2 1 1 w, z = εz , w, x = γxz − ωy , w, y = γyz + ωx . (1.56c) 2 2Similar to the rotation case, when the displacement components at P0 are known, those of pointP may be expressed as P 1 1 u = u0 + [εx dx + ( γxy − ωz ) dy + ( γxz + ωy ) dz], (1.57a) P0 2 2 P 1 1 v = v0 + [( γxy + ωz ) dx + εy dy + ( γyz − ωx ) dz], (1.57b) P0 2 2 P 1 1 w = w0 + [( γxz − ωy ) dx + ( γyz + ωx ) dy + εz dz]. (1.57c) P0 2 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, γxz = γyz = εz = 0), and the remaining components εx (x, y),εy (x, y), γxy (x, y) satisfy compatibility (1.45a). As a ﬁrst step, we determine the z componentof the rotation vector, ωz (x, y) (obviously, ωx = ωy = 0 in this case), from its two given partialderivatives, see (1.54) with i = 3, P ωz = ω0 + z fzx dx + fzy dy P0 x y 1 1 = ω0 − z εx, y dx + εy, x (0, y) dy + γxy − γxy (0, y) + γxy (0, 0). (1.58) 0 0 2 2As 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 + [εx dx + ( γxy − ωz ) dy], v = v0 + [εy dy + ( γxy + ωz ) dx]. (1.59) P0 2 P0 2This procedure may be executed for a given consistent set of strain functions by activatingP.1.6.1.2.3 Level 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.
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1.2 Displacement by Strain Integration 13 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 , m K (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 z zk+1 K −1 G dz = ∑ k=0 G(k) G, z = ∑ k=0 (k + 1)G(k+1) zk . K m , m (1.61) 0 k+1To derive the present approach, we also assume that the strain components are truncated as εi = ∑ k=0 εi (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, namely u = ∑ 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 m K K m (1.63a) Km ωi = ∑ ωi (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 (1.53a–c) 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 = [γxz, y − (k + 1)γxy ], fxy = γyz, y − (k + 1)εy , fxz = εz, y − (k + 1)γyz , 2 2 2 (k) (k+1) 1 (k) (k) 1 (k+1) (k) (k) 1 (k+1) (k)fyx = (k + 1)εx − γxz, x , fyy = [(k + 1)γxy − γyz, x ], fyz = (k + 1)γxz − εz, x , 2 2 2 (k) 1 (k) (k) (k) (k) 1 (k) (k) 1 (k) (k)fzx = γxy, x − εx, y , fzy = εy, x − γxy, y , fzz = (γyz, x − γxz, y ). (1.64) 2 2 2By selecting the polygonal integration trajectory (0, 0, 0) → (0, y, 0) → (x, y, 0) → (x, y, z), wewrite (1.54) as x y zk+1 fiy (0, y)dy + ∑ k=0 fxz (x, y) (0) (0) K (k)ωi = ω0 + i fix (x, y)dx + m , i ∈ {x, y, z}. (1.65) 0 0 k+1Hence, the level components of the rotations are x y 1 (k−1) (0) (0) (0) (k) ωi = ω0 + i fix (x, y) dx + fiy (0, y) dy, ωi = f (x, y) (1.66) 0 0 k izwhere k ≥ 1, i ∈ {x, y, z}. At this stage, (1.56a–c) show that (k) (k) 1 (k) (k) (k) (k) 1 (k) (k) u, x = εxx , u, y = γxy − ωz , u, z = γxz + ωy , (1.67a) 2 2 (k) (k) (k) 1 (k) (k) (k) 1 (k) (k) v, y = εyy , v, z = γyz − ωx , v, x = γxy + ωz , (1.67b) 2 2 (k) (k) (k) 1 (k) (k) (k) 1 (k) (k) w, z = εzz , w, x = γxz − ωy , w, y = γyz + ωx , (1.67c) 2 2
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14 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologiesand therefore, by analogy with the rotation integration process, one may write x y 1 (0) 1 1 (k−1) (0) (0) (k−1) u(0) = u0 + εxx dx + u(k) = ( γxz + ωy ( γxy − ωz )(0, y) dy, ), (1.68) 0 0 2 k 2 x 1 y 1 1 (k−1) (0) (0) (0) (k−1) v(0) = v0 + [ γxy + ωz ] dx + εyy (0, y) dy, v(k) = ( γyz − ωx ), 0 2 0 k 2 x 1 y 1 1 (k−1) (0) (0) (0) (0) w(0) = w0 + [ γxz − ωy ] dx + ( γyz + ωx )(0, y) dy, w(k) = εzz 0 2 0 2 kwhere 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) εx = u, x , εy = v, y , γxy = u, y + v, x , (1.69a) (k) (k) (k) (k) (k) εz = (k + 1)w(k+1) , γyz = (k + 1)v(k+1) + w, y , γxz = (k + 1)u(k+1) + w, x . (1.69b)Thus, one can extract the displacement components from (1.69b) as 1 (k) w(k+1) = εz , (1.70a) k+1 1 (k+1) 1 (k) 1 (k+1) 1 (k) u(k+2) = (γxz − εz, x ), v(k+2) = (γyz − εz, y ), (1.70b) k+2 k+1 k+2 k+1 (0) (0) (0) (0) u(1) = γxz − w, x , v(1) = γyz − 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) ω1 = (w, y −(k + 1)v(k+1) ), ω2 = ((k + 1)u(k+1) −w, x ), ω3 = (v, x −u, y ). (1.71) 2 2 2To 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 {ωi }, i=1,2,3 from (1.66), including the introduction of the rigid body rota-tions {ω0 }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 (1.57a–c) with u, z = v, z = w, z = 0. (5) Calculate u(1) , v(1) by (1.70c). (k) (6) Calculate {ωi }i=1,2,3 for all k > 0 levels from (1.71).1.3 Stress 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.
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1.3 Stress Measures 151.3.1 Deﬁnition of StressTo deﬁne the stress tensor at a point, it is worthwhile to ﬁrst 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 difﬁcult to be physically interpreted. We shall ﬁrst examine a three-dimensional body by virtually cutting it over an interior planein which we deﬁne 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 deﬁned as N T σN (P) = lim , σT (P) = lim . (1.72) A→0 A A→0 AHence, the dimension of the stress components is force per unit area. At each point, one may examine an inﬁnitesimally 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 inﬁnitesimal element in its coordinate space.
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16 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologiesfore deformation, the above element may be viewed as an inﬁnitesimal cube regardless of thespeciﬁc system employed. When described in a Euclidean space in the undeformed state, aninﬁnitesimal 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, α1 dα1 , f1, α1 dα1 + f1, α2 dα2 . 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 deﬁni-tion, σi j is the stress component in the k j -direction that acts on a plane, which is perpendicular ˆto the ki -direction — e.g. σ1 j in Fig. 1.3. When the same material element after deformation ˆis examined, one may decompose the stress components by using various coordinate systems.However, we preserve the same notation logic in which σi j is deﬁned as the stress componentin the deformed jth direction of a given coordinate system that acts on a plane, which before ˆdeformation was perpendicular to the ki -direction. On each of the deformed faces one may deﬁne stress vectors. We will denote by σ 1 , σ 2 andσ 3 the stress vectors over faces #1,2 and 3, see Fig. 1.3. Before deformation the edges’ lengthsare Hi dαi (see (1.215)), and their respective areas are Ai = H j Hk dα j dαk (apply cyc-i jk). Theareas of the corresponding faces after deformation are denoted A∗ . i Based on the above deﬁnitions, the forces that act over faces #1,2 and 3 may be written as A∗A∗ σ i (i ∈ {1, 3}), or, equivalently, Ai σ i H j Hk dα j dαk . We will now decompose σ i along the i ideformed and the undeformed directions, respectively, as A∗ σ i = ∑ j=1 σi j k∗ , σ i = ∑ j=1 si j k j . 3 i 3 ˆj ˆ (1.73) AiIt may be veriﬁed that ⎧ ⎫ ⎧ ⎫ ⎨ si1 ⎬ ⎨ σi1 ⎬ si2 = [I + e + ω ] σi2 , i = 1, 2, 3. (1.74) ⎩ ⎭ ⎩ ⎭ si3 σi3In the above, I is a unit matrix, e and ω are given by (1.17), and A∗ σi j σi j = i , (1.75) Ai 1 + E jwhere E j is the relative extension in the jth direction, see (1.36). σi j are usually referredto as the“generalized 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 σi j , namely, σi j = σ j i (while σi j = σ j i ).Therefore, under a coordinate system transformation the tensor σ = {σi 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 speciﬁc problem is carried out,and the components of σ are derived, the values of σi j may be recovered by (1.75). Note thatboundary conditions should be imposed on σi j (and not on σi j ), as they represent the physicalstress components in the deformed state.
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1.3 Stress Measures 17 At this stage, the only missing component in the scheme is the area ratio A∗ /Ai , which may ibe written as (apply cyc-i jk) A∗ i = (1 + E j ) (1 + Ek ) sin(k∗ , k∗ ). ˆ j ˆk (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) A∗i = (1 + 2ε j j ) (1 + 2εkk ) − 4ε2 , 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 tensorσ 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, ω I, and therefore,si j ∼ σi j ∼ σi j , and σi j becomes a symmetric tensor as well. = =1.3.2 Equilibrium 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 deﬁnitions, 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 as Fb = ∑ 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 dα1 dα2 dα3 .Subsequently, equilibrium may be imposed by equating the resultant vector of all forces actingon the material element to zero, namely, A∗ A∗ A∗ (H2 H3 1 σ 1 ), α1 + (H1 H3 2 σ 2 ), α2 + (H1 H2 3 σ 3 ), α3 + 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 ), α1 + (H1 H3 s21 ), α2 + (H1 H2 s31 ), α3 + H3 H1, α2 s12 + H2 H1, α3 s13 − H3 H2, α1 s22 − H2 H3, α1 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 ∼ σi 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 σi j . When nonlinear analysis isrequired, σi 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 σx,x + τxy,y + τxz,z + Xb = 0, (1.82a) τxy,x + σy,y + τyz,z +Yb = 0, (1.82b) τxz,x + τyz,y + σz,z + Zb = 0. (1.82c)
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18 1. 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 (1.82a–c) 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 deﬁned as forces per the unit area at each boundary materialpoint and described by their components in the undeformed directions as Fs = ∑ 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 = σx cos(n, x) + τxy cos(n, y) + τxz cos(n, z), ¯ ¯ ¯ (1.85a) Ys = τxy cos(n, x) + σy cos(n, y) + τyz cos(n, z), ¯ ¯ ¯ (1.85b) Zs = τxz cos(n, x) + τyz cos(n, y) + σz 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 as Fs + Fb = 0. (1.86) S VHence, 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(1.82a–c). Since these equations apply to each inﬁntisimal volume as well, (1.82a–c) are re-established. For cylindrical coordinates we denote by Rb , Θb , Zb the body-force components in theρ, θc , z directions, respectively, and write 1 σρρ − σθθ + σρθ, θ + σρρ, ρ + σρz, z + Rb = 0, (1.87a) ρ 1 σθθ, θ + 2σρθ + σθz, z + σρθ, ρ + Θb = 0, (1.87b) ρ 1 σθz, θ + σρz + σρz, ρ + σzz, z + Zb = 0. (1.87c) ρ For spherical coordinates we denote by Rb , Θb and Φb the body-force components in theρ, θs , φs directions, respectively, and write 1 1 σρφ, φ + 2σρρ − σθθ − σφφ + cot φ σρφ + σρθ, θ + σρρ, ρ + Rb = 0, (1.88a) ρ ρ sin φ 1 1 σθφ,φ + 3σρθ + σθθ, θ + 2 cos φ σθφ + σρθ, ρ + Θb = 0, (1.88b) ρ ρ sin φ 1 1 σφφ,φ + 3σρφ + σθφ, θ + cos φ σφφ − σθθ + σρφ, ρ + Φb = 0. (1.88c) ρ ρ sin φ
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1.3 Stress Measures 19For elliptical-cylindrical coordinates (with a = 1, see (1.218b)), one obtains 3 3(σ11 −σ22 ) cosh(α1 ) sinh(α1 )+ A σ11, 1 +2σ12 cos(α2 ) sin(α2 )+ A σ12, 2 + A 2 σ13, 3 + Fb1 A 2 = 0, 3 32σ12 cosh(α1 ) sinh(α1 )+ A σ12, 1 +(σ22 −σ11 ) cos(α2 ) sin(α2 )+ A σ22, 2 + A 2 σ23, 3 + Fb2 A 2 = 0, 3 3 σ13 cosh(α1 ) sinh(α1 )+A σ13, 1+σ23 cos(α2 ) sin(α2 )+A σ23, 2+A 2 σ33, 3+Fb3 A 2 = 0 (1.89)where A = cosh2 (α1 ) − cos2 (α2 ), and Fb1 , Fb2 and Fb3 are the body-force components in theα1 , α2 and α3 directions, respectively.1.3.3 Stress 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 speciﬁcally, 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 ψ, θ and φ, see(1.203), (1.206)) as the Ti j element of the corresponding transformation tensor, and deﬁne the(symmetric) second-order stress tensor as ⎡ ⎤ σ11 σ12 σ13 σ = ⎣ σ12 σ22 σ23 ⎦ . (1.90) σ13 σ23 σ33Hence, the components of the stress tensor in the new system, σ = {σi j }, are obtained by thestandard tensor transformation σi j = σab Tia T jb . (1.91)This operation may be expressed using matrix notation as well, as σ = T · σ · 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 σ and σ , which contain the stresscomponents before and after transformation, namely, σ = [σxx , σyy , σzz , σyz , σxz , σxy ]T , σ = [σxx , σyy , σzz , σyz , σxz , σxy ]T . (1.93)These vectors may be related as σ = Mσ · σ , where Mσ is a (non-symmetric) 6 × 6 matrix,which is clearly a function of the transformation (Euler’s) rotation angles ψ, θ and φ. Exampleterms are Mσ (3, 2) = (cos φ sin θ sin ψ − sin φ cos ψ)2 , Mσ (2, 3) = sin2 φ cos2 θ. (1.94)The reader may activate P.1.9 to generate all terms of Mσ symbolically. Figure 1.4 presentsan example of axis rotation and the associated Mσ matrix. Such a view for different sets ofrotation angles may be obtained by activating P.1.10. To transform a stress tensor given in orthogonal curvilinear coordinates to Cartesian coor-dinates, we select the transformation tensor of (1.216). The result is a function of the spe-ciﬁc point under discussion, and may be written as a function of the curvilinear coordinatesα1 , α2 , α3 . The corresponding transformation angles may be determined by (1.208). P.1.11 executes stress tensor transformations between curvilinear and Cartesian coordinates.
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20 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies z z_new 1 0.8 0.6 0.4 0.2 y_new value 0 –0.2 –0.4 y –0.6 2 –0.8 1 4 rowx 2 3 6 4 5 x_new column 6 (a) Axis orientation. (b) The matrix Mσ . Figure 1.4: Example of axis orientation and the resulting Mσ for ψ = 30◦ , θ = φ = 0.1.3.3.1 Principal StressesWe shall now seek expressions for the principal stresses at a point. In essence, we are lookingfor a set of rotation angles that will deﬁne a new system orientation, in which only the normalstress components σα (α = x, y, z) are nonzero at a point. Clearly, these values of rotation angleswill ensure that the vector σL = [σyz , σxz , σxy ]T (1.95)will vanish, or in other words, the state where σ is diagonal, and its elements are called theprincipal stresses at a point. One way to carry out this task is to set the following set of threeequations: σL = Lσ · σ = 0, (1.96)where Lσ is a 3 × 6 matrix (essentially the last three lines of the matrix Mσ ). For a givenstress tensor, these are three equations in the three unknown orientation angles ψ, θ, and φ. Thesolution of such a system is not trivial, and it is much more convenient to adopt a more standardway. In this new course, we carry out an eigenvalue analysis for σ and obtain its eigenvalues{σP }i=1,2,3 and eigenvectors {vi }i=1,2,3 so that i σ · vi = σP · vi . (1.97)Hence, in the principal stress state, σii = σP and σi j = 0 (i = j). i Implementation of the above eigenvalue analysis requires that the system matrix determinantwill vanish, namely, σ11 − σP σ12 σ13 σ12 σ22 − σP σ23 = 0, (1.98) σ13 σ23 σ33 − σPwhich leads to the cubic polynomial equation σ3 − Θ1 σ2 + Θ2 σ − Θ3 = 0. (1.99)Θi are usually referred to as the stress invariants and are given by the matrix minors as Θ1 = tr σ = σ11 + σ22 + σ33 , (1.100a) 1 Θ2 = (tr σ )2 − tr σ 2 = σ11 σ22 + σ22 σ33 + σ33 σ11 −σ2 − σ2 − σ2 , 12 13 23 (1.100b) 2 Θ3 = det σ = σ11 σ22 σ33 +2σ12 σ13 σ23 − σ11 σ2 − σ22 σ2 − σ33 σ2 . 23 13 12 (1.100c)
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1.3 Stress Measures 21Since Θi are invariants, the resulting eigenvalues are invariants as well. Therefore, the aboveinvariants may be expressed in a more compact form using the principal stresses by ignoring theunderlined terms (1.100a–c) and replacing σii with σP . From this point on, we shall assume that ithe three eigenvalues obtained by solving (1.99) are put in a decreasing order, σP ≥ σP ≥ σP , 1 2 3and their eigenvectors vi = {vi (1), vi (2), vi (3)} are put in the same order. To determine the above stress eigenvalues, it is convenient to deﬁne the stress deviator tensorσD = σi j − 1 Θ1 δi j , which is a second-order tensor as well, and its eigenvalues will be denoted ij 3σDP ≥ σDP ≥ σDP . It is therefore clear that (apply cyc-123) 1 2 3 1 σDP = 1 2σP − σP − σP . 1 2 3 (1.101) 3In addition, the invariants of the stress deviator tensor may be expressed as functions of thestress tensor invariants, namely, ΘD = σDP + σDP + σDP = 0, 1 1 2 3 (1.102a) 1 ΘD = σDP σDP + σDP σDP + σDP σDP = Θ2 − Θ2 , 2 1 2 1 3 2 3 (1.102b) 3 1 2 1 ΘD = σDP σDP σDP = Θ3 − Θ1 Θ2 + Θ3 . 3 1 2 3 1 (1.102c) 27 3Hence, to determine σDP , one needs to solve the cubic equation i (σD )3 + ΘD σD − ΘD = 0, 2 3 (1.103)which is simpler than (1.99). Once σDP are known, σP are directly obtained via σP = σDP + i i i i3 Θ1 , see also Remark 1.4.1 To determine the transformation matrix, TE , from a given state to the principal directionsalong with the corresponding transformation angles ψ, θ and φ, we use the above derivedeigenvectors vi as ⎡ ⎤ v1 (1) v1 (2) v1 (3) TE = ⎣ v2 (1) v2 (2) v2 (3) ⎦ . (1.104) v3 (1) v3 (2) v3 (3)To preserve orientation (the transformation matrix determinant must be equal to a unit), oneshould change the sign of one of the eigenvectors if required. Also, by simple eigenvalueanalysis argumentation we obtain ⎡ P ⎤ σ1 0 0 TE · σ · (TE )T = ⎣ 0 σP 0 ⎦ . 2 (1.105) 0 0 σP 3Example 1.1 A Generic Stress State. To demonstrate the above derivation, we shall examine a generic stress state as shown inFig. 1.5(a). This state serves as an example only (while units will not be indicated). The aboveprocedure shows that the eigenvalues are σP = .0823, σP = −.0187, σP = −.0937 and that 1 2 3 ⎡ ⎤ −0.138 0.982 −0.194 ⎢ ⎥ TE = ⎢ −0.446 −0.237 −0.860 ⎥ . ⎣ ⎦ (1.106) 0.882 0.0288 −0.464Thus, according to (1.208), the rotation angles are ψ = −81.99◦ , θ = −11.19◦ and φ = 61.65◦ .The corresponding principal stress state is shown in Fig. 1.5(b). The principal stress directionsat a point may be derived by P.1.12.
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22 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies 0.05 0.05 value 0value 0 –0.05 –0.05 1 3.5 1 3.5 1.5 3 1.5 3 2 2.5 2 2.5 2.5 column 2.5 2 column row 3 2 row 3 1.5 1.5 3.5 3.5 1 1 (a) A generic stress tensor. (b) The principal stress tensor. Figure 1.5: Example of a stress tensor and its principal state.Remark 1.4 Two additional stress invariants, Θ4 and Θ5 (which are identical for both thestress and the stress deviator tensors) may be deﬁned. The ﬁrst one is 1 2 2 2 Θ4 = √ σP − σP 1 2 + σP − σP 2 3 + σP − σP , 3 1 (1.107) 6while it is easy to verify that Θ4 = 3 Θ1 − Θ2 1 2 = −ΘD . The second invariant is deﬁned by 2 1 2σP − σP − σP √ σDP tan(Θ5 ) = √ 2 1 3 = 3 DP 2 DP . (1.108) 3 σP − σP 1 3 σ1 − σ3Due to the order of the principal stresses, i.e. σP ≥ σP ≥ σP , this invariant is bounded as 1 2 3|Θ5 | ≤ π/6. The above deﬁnitions enable us to write 2Θ4 2π 2Θ4 σDP , σDP = √ sin(Θ5 ± ), 1 3 σDP = √ sin(Θ5 ), 2 (1.109) 3 3 3which allows graphical interpretation of Θ4 and Θ5 as presented in Fig. 1.6(a).1.3.3.2 Visualizing the State of Stress at a PointMany visualization methods of the state of stress at a point have been discussed extensively inthe literature. In view of the powerful modern visualization tools, the classical methods seemless attractive and important. We will describe the main ideas in this area brieﬂy. A good starting point is the examination of the stresses over a face of an inﬁnitesimal cubehaving general orientation so that the normal to the face under discussion is oriented at the xdirection as shown in Fig. 1.6(b). Also, the normal stress and the resultant shear stress over thisface are denoted as σN = σ11 , (1.110a) σT = σ2 + σ2 . 12 13 (1.110b)We shall now assume that the original x-, y-, z-axes are the directions of the principal stressstate, and write the above two equations using (1.91), in addition to (1.204) (with i = 1). This
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1.3 Stress Measures 23 (a) Geometrical relations between the Θ4 , Θ5 in- (b) An inﬁnitesimal cube at an ar- variants and the principal σDP eigenvalue. i bitrary orientation. Figure 1.6: Stress invariants and transformation notation.procedure constitutes the following set of three equations: σN = σP T11 + σP T12 + σP T13 , 1 2 2 2 3 2 (1.111a) σ2 = (σP )2 T11 + (σP )2 T12 + (σP )2 T13 − σ2 , T 1 2 2 2 3 2 N (1.111b) 1 = T11 + T12 + T13 , 2 2 2 (1.111c) 2 2 2which may be solved for T11 , T12 and T13 as σ2 + σN − σP σN − σP T 2 3 T11 = 2 , (1.112a) σP − σP σP − σP 2 1 3 1 σ2 + σN − σP σN − σP T 3 1 T12 = 2 , (1.112b) σP − σP σP − σP 3 2 1 2 σ2 + σN − σP σN − σP T 1 2 T13 = 2 . (1.112c) σP − σP σP − σP 1 3 2 3In view of the eigenvalues’ order (σP ≥ σP ≥ σP ) and the fact that the l.h.s. of the above 1 2 3equations are non-negative, the following inequalities are obtained: σ2 + σN − σP T 2 σN − σP ≥ 0, 3 (1.113a) σ2 + T σN − σP 3 σN − σP 1 ≤ 0, (1.113b) σ2 + T σN − σP 1 σN − σP 2 ≥ 0. (1.113c)We may now plot the above conditions in the σN − σT plane, in order to ﬁnd a region where allinequalities are satisﬁed, which will yield the valid combinations of normal and shear stress atthe point under discussion. This graphical analysis is shown in Fig. 1.7(a) and is traditionallyreferred to as Mohr’s diagram. The ﬁrst inequality shows a region outside a circle, the diameterof which is located over the σN -axis between σN = σP and σN = σP . The second inequality 2 3shows a region inside a circle, the diameter of which is located over the σN -axis between
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24 1. Fundamentals of Anisotropic Elasticity and Analytical MethodologiesσN = σP and σN = σP . The third inequality shows a region outside a circle, the diameter of 3 1which is located over the σN -axis between σN = σP and σN = σP . Note that only the upper 1 2part of the σN -σT plane is of interest in this case since by deﬁnition σT ≥ 0, see (1.110b).An immediate and clear result of Mohr’s diagram is that the maximal shear stress is given by σP −σP σP +σPσm = 1 2 3 , which occurs at σm = 1 2 3 . For the state of stress described in Example 1.1, T NP.1.12 has produced Mohr’s diagram shown in Fig. 1.7(b). σT 0.05 σN –0.095 –0.058 –0.019 0 0.082 (a) General notation. (b) The values of Example 1.1. Figure 1.7: Mohr’s diagram. For each combination of σN and σT in Mohr’s diagram corresponds a pair of angles (ψ, θ).Note that the value of φ has no importance in this case, since it represents a rotation about thex-direction (as it will change neither σN nor σT ). To determine the rotation angles between theprincipal directions and the coordinate systems of the point under discussion for a given set ofσN and σT , we evaluate T11 , T12 by (1.112a,b) and use (1.208:a,b). The inverse problem is ofcourse much easier to solve since for each set of two rotation angles ψ and θ, one can calculateσN and σT directly from (1.111a,b), (1.207), see also Remark 1.5. Suppose now that we wish to calculate the angles ψ and θ that are required to reach a givenset of σN and σT from a state of stress, which is not principal. To do that, we substitute σNand σT in the l.h.s. of (1.110a,b) while in the r.h.s. we substitute the expressions obtained from(1.92) with φ = 0, namely σ11 = cos2 θ cos2 ψ σ11 + 2 cos2 θ cos ψ sin ψ σ12 − 2 cos θ cos ψ sin θ σ13 + cos2 θ sin2 ψ σ22 − 2 cos θ sin ψ sin θ σ23 + sin2 θ σ33 , (1.114a) σ12 = − cos θ cos ψ sin ψ σ11 + cos θ(cos2 ψ − sin2 ψ) σ12 + sin θ sin ψ σ13 + cos θ sin ψ cos ψ σ22 − sin θ cos ψ σ23 , (1.114b) σ13 = sin θ cos ψ cos θ σ11 + 2 sin θ cos ψ cos θ sin ψ σ12 + (cos θ − sin θ) cos ψ σ13 2 2 2 + sin θ sin2 ψ cos θ σ22 + (cos2 θ − sin2 θ) sin ψ σ23 − cos θ sin θ σ33 . (1.114c)Substitutions of the above in (1.110a,b) converts these equations into a system of two equationsthat should be simultaneously solved for ψ and θ, Since in the general case, the expressions are not simple enough for analytic solution, agraphical representation of these equations is given in Fig. 1.8 for two cases (produced byP.1.12). In Fig. 1.8(a), the angles are measured from the non-principal stress state of Exam-ple 1.1, while in Fig. 1.8(b), the angles are measured from the corresponding principal stressstate. In these ﬁgures, the thick line shows solutions for (1.110a), while the thin line shows so-lutions for (1.110b) (angles are given in radians). The desired solutions are therefore the pointswhere the above lines coincide.
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1.3 Stress Measures 25 6 6 5 5θ 4 θ4 3 3 2 2 1 1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 (a) Non-principal stress state. ψ (b) Principal stress state. ψ Figure 1.8: Graphical solution of (1.110a,b) for σN = 0, τT = 0.045. As shown, there are a number of solutions in the region under discussion, and the picture has a period of π in θ and 2π in ψ. To visualize the above solution, one may also plot a graph 0.08 0.08 0.04 0.04 0 z 0 –0.04 –0.04 –0.08 0 –0.08 0.4 0.8 0 –0.04 –0.08 psi 0.4 0.2 0 0 –0.04 1.2 0.8 0.6 y x 1.6 1.6 1.4 1.2 1 theta 0.04 0.08 0.04 (a) |σN | over ψ − θ plane. (b) 3D surface of |σN |. Figure 1.9: σN as function of ψ and θ measured from the (principal) state of stress, see Example 1.1. of σN = σ11 over the ψ-θ plane, see Fig. 1.9(a), and similarly, by substituting (1.114b,c) in (1.110b), σT as described by the surface in Fig. 1.10(a) is obtained. One may also create a three-dimensional (spherical) surface of σN = σ11 (θ, ψ), and σT = σ2 (θ, ψ) + σ2 (θ, ψ). To do that we use the results of Remark 1.12 and replace ψ by θs 12 13 and θ by φs − π , respectively, where θs and φs are spherical angles, see Fig. 1.20(b). These 2 spherical plots are shown in Figs. 1.9(b), 1.10(b) where each point on the surface represents an orientation of the x-axis of the transformed system (by connecting the origin with it). Thus, σN and σT are directly proportional to the distance that is measured along the x-axis between the origin and the corresponding surfaces.
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26 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies0.08 0.06 0.040.06 0.020.04 z 0 –0.020.02 –0.04 0 –0.06 0 0.4 –0.04 0.8 0 –0.06 psi 0.4 0.2 y 0 –0.02 1.2 0.8 0.6 0.02 x 1.6 1.6 1.4 1.2 1 theta 0.04 0.06 (a) σT over ψ − θ plane. (b) 3D surface of σT .Figure 1.10: σT as a function of ψ and θ measured from the (principal) state of stress, see Example 1.1. Further on, we may also plot graphs of ψ and θ as functions of σN and σT over the validregion of Mohr’s diagram. From (1.112a–c), for each (σN , σT ) point we calculate T11 , T12 andT13 , and then use (1.208) to determine ψ and θ, respectively. The resulting diagrams are shownin Fig. 1.11 produced by P.1.12. Note that each surface is plotted over the valid area of Mohr’sdiagram only. See also Remark 1.6. 8080 6060 4040 20 0.0820 0.06 0 0.04 0 0 0.02 –0.08 0.02 0.04 –0.08 –0.04 –0.04 0 0.02 0.04 0 0.06 0.02 0 0.06 0.08 0.08 0.06 (a) The angle ψ (deg.). (b) The angle θ (deg.). Figure 1.11: The angles ψ and θ that transform a principal stress state for each σN , σT point.Remark 1.5 One may deﬁne the mean shear stress, σ0 , by integrating τT over the surface, ω, Tof an inﬁnitesimally small sphere, the center of which is located at the point under discussion.This operation may be expressed as 1 2 σ0 = lim T σ2 = T Θ4 . (1.115) ω→0 S(ω) ω 5The above relation connects the mean shear stress with the principal stresses. According to(1.109), the maximal shear stress may be written as σm = 2 (σP − σP ) = 1 (σDP − σDP ) = 1 √ T 1 3 2 1 3 2/52Θ4 cos(Θ5 ), and thus, σT T 0 /σm = cos(Θ ) which, by taking into account the valid range of Θ5 5
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1.3 Stress Measures 27in (1.109), yields 8 σ0 2 ≥ m≥ T . (1.116) 15 σT 5Remark 1.6 An additional, rather classical visualization tool is based on the well-knownStress Quadric of Cauchy. To derive the related equations, we start from the principal stressdirections (namely, assume that the x, y, z system origin is located at the point under discussionwhile the coordinate lines coincide with the principal directions at that point). As a ﬁrst step,we look at a point located at a distance A and arbitrary direction, which may be expressed byψ and θ. The coordinates of this point are, see Remark 1.12, x = A cos θ cos ψ, y = A cos θ sin ψ, z = −A sin θ. (1.117)As a second step, we deﬁne σN = σ11 as the normal stress obtained in the x-direction of thenew system by rotating the coordinate system with the above angles. This stress componentmay be easily obtained from (1.114a) by setting σii = σP and σi j = 0 (i = j), as i σ11 = cos2 θ cos2 ψ σP + cos2 θ sin2 ψ σP + sin2 θ σP , 1 2 3 (1.118)and by using (1.117) A2 σ11 = σP x2 + σP y2 + σP z2 . 1 2 3 (1.119)We shall now call for σ11 to be proportional to the inverse of A, namely, σ11 = c/A2 where c isa normalization constant that may take both positive and negative values. Such a requirementyields a relatively simple quadratic surface, which is given by σP x2 + σP y2 + σP z2 = c. 1 2 3 (1.120)The above surface may be classiﬁed by the four types: (1) ellipsoid, (2) unparted and bipartedhyperboloids, (3) cylinder over ellipse, (4) hyperbola and parallel planes. Figure 1.12 (pro-duced by P.1.12) presents an illustrative case where the value c = 1 yields the unparted hyper- 1 z 0 –1 –0.5 y 0 0.5 1.5 1 0.5 0 x–0.5 –1 –1.5 Figure 1.12: Three-dimensional plot of (1.120) for the state of stress described in Example 1.1.boloid (i.e. the outer surface), and the value c = −1 yields the biparted hyperboloid (i.e. theinner surface). Hence, when applying stress transformation from a given system to a “new”system (denoted by a bar), the interpretation of the surface in Fig. 1.12 is as follows: the valueof σN is inversely proportional to the square of the distance that is measured along the x-axisbetween the origin and the surface.
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28 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies1.3.4 Strain Tensor Transformation due to Coordinate System RotationSince both the strain and the stress at a point are described as symmetric second-order tensors,their transformation, invariants and principal axes are derived by the same rules. Hence, similar to the discussion of S.1.3.3, one may transform the strain components at apoint by applying the tensor transformation or by using the matrix notation εab = εab Tia T jb , ⇔ ε = T · ε · TT . (1.121) Compared with S.1.3.3, the discussion regarding the strain tensor transformation will bebrief in view of the above similarities between the stress and the strain tensor transformation,the associated eigenvalue analysis, the principal axis determination, etc., Hence, the discussionregarding the stress tensor is applicable in a direct manner by replacing σi j with εi j , etc. Sub-sequently, a matrix Mε = Mσ may be deﬁned, and the strain tensor invariants Ξ1 , Ξ2 , Ξ3 , theeigenvalues εP and the principal axes may be calculated analogously. i One may also draw a Mohr’s diagram in the εN -εT plane (where εN = ε11 , εT = ε2 + ε2 )12 13using the same technique that has been described for the stress description in the σN -σT plane(see S.1.3.3), and similarly, all other visualization methods are applicable in this case as well,see also discussion in Remark 2.3. The reader may activate P.1.9, P.1.10 to transform strain coefﬁcients by coordinate systemrotation and P.1.12 for visualization of a state of strain at a point. P.1.11 executes transforma-tions of a strain tensor between curvilinear and Cartesian coordinates.1.4 Energy TheoremsIn this section we shall review some work- and energy-based measures that are encountered inthe theory of elasticity. These measures may be further exploited to create functionals that maybe combined with variational analysis to derive the associated Euler’s equations, see S.1.5,and other solutions in this area. In general, two main measures, the “Potential Energy” and the “Complementary Energy”,should be addressed. These measures serve as the basis to the well known Theorem of Mini-mum Potential Energy, the Theorem of Minimum Complementary Energy and the Theorem ofReciprocity.1.4.1 The Theorem of Minimum Potential EnergySuppose that a body of volume B and surface S = SL + SD , has reached equilibrium under theaction of distributed load Fs over the SL part of its surface, and body force, Fb , that appliesat each material point. The deformation u = (u1 , u2 , u3 ) over the remaining SD part of thesurface is known as well. We introduce a variation of them, δ u (i.e. “virtual displacements”),that vanish over SD . Since the body and the surface loads are ﬁxed during the application ofthe virtual displacements, and δ u = 0 on SD , one can write the “virtual work” as δU = Fs · δ u + Fb · δ u = δ ( Fs · u + Fb · u) (1.122) SL B SL Bwhere “ · ” is a scalar product of vectors. Further on, the strain energy is given by U≡ W, (1.123) B
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1.4 Energy Theorems 29where W is the volume density of the strain energy, see S.2.10. Equating the variation of theexternal load potential and the strain energy, may be written as δV = 0, where V= W− Fs · u − Fb · u. (1.124) B SL BThis enables us to conclude that the potential energy, V , has a stationary value among alladmissible variations of the displacements ui from the equilibrium state. Since V (u + δ u) ≥ 0 for any variation δ u, the Theorem of Minimum Potential Energy maybe expressed as: Among all displacements that satisfy the boundary conditions, those that satisfy the equilib-rium equations as well make the potential energy an absolute minimum. The converse theorem is true as well, see (Sokolnikoff, 1983).Remark 1.7 In the time-dependent case, the above statement may be extended to the well-known Hamilton’s principle for the time-dependent virtual displacements that satisfy all geo-metrical boundary conditions at all times, see e.g. (Langhaar, 1962): t2 δ (T −V ) dt = 0, (1.125) t1where T stands for the total kinetic energy stored in the system, and t1 and t2 are times wherethe deformation is known. This principle may be invoked to derive the equations of motionof an elastic body, or in other words, the resulting Euler’s equations of such functional aretime-dependent. For time-dependent cases, one should also consider employing Ritz’s method(see (Sokolnikoff, 1983) and further on in S.1.6.5.4), which leads to Lagrange’s equations, seeExamples 1.6, 1.8.1.4.2 The Theorem of Minimum Complementary EnergyIn contrast with the previous case where displacement components variations were dealt with,we shall now employ small variations of the stress distribution. Let σi j be a known set of stress distributions satisfying the equilibrium and compatibilityequations, the natural boundary conditions on SL , and the geometrical boundary conditions onSD as required. We introduce a perturbation, or a “virtual” (small) system of stress δσi j , see(1.85a–c), as δXs = δσx cos(n, x) + δτxy cos(n, y) + δτxz cos(n, z), ¯ ¯ ¯ (1.126a) δYs = δτxy cos(n, x) + δσy cos(n, y) + δτyz cos(n, z), ¯ ¯ ¯ (1.126b) δZs = δτxz cos(n, x) + δτyz cos(n, y) + δσzz cos(n, z). ¯ ¯ ¯ (1.126c)Further on, we assume that: (a) δσi j satisfy equilibrium (when body forces are ignored); (b) δ Fs vanish on SL ; (c) δσi j are arbitrary on SD (i.e. not constrained). Note that δσi j do not necessary satisfy the compatibility equations. Subsequently, for a givenδσi j , the variation of the strain energy is deﬁned as δU = δW = u · δ Fs + W (δσi j ), (1.127) B SD B
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30 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologieswhere u are the displacements associated with σi j . The once underlined term on the r.h.s. of(1.127) represents the work done by the virtual stresses on the displacements over SD . Thetwice underlined term is the strain energy associated with δσi j (which is a function of speciﬁcconstitutive relations) and is always positive (see (2.25) and the discussion in S.2.10 regardingthe positive-deﬁnite stress-strain law). Thus, one may deﬁne a quantity called “complementaryenergy” as V∗ = U − u · Fs , (1.128) SDand constitute the Theorem of Minimum Complementary Energy by stating that: Among all stress distributions that satisfy the equilibrium equations and the boundary con-ditions, those that satisfy the compatibility equations as well make the complementary energyan absolute minimum. The converse theorem is true as well, see also (Sokolnikoff, 1983) and Example 7.2. We may now utilize the Theorem of Minimum Complementary Energy and conﬁne thediscussion to the case of SD = 0 (i.e. SL = S) or the case where u vanish on SD , which formallyyields V ∗ = U. In words, in the case where no geometrical constraints exist (i.e. only naturalboundary conditions are imposed), or in the case where zero displacements are forced over SD ,one may state that: Among all stress distributions that satisfy the equilibrium equations and the natural bound-ary condition, those that satisfy the compatibility equations as well make the strain energy anabsolute minimum. This theorem is frequently referred to as The Principle of Least Work.Remark 1.8 In a state of equilibrium under the action of surface loads Fs and body force Fb ,the strain energy, U, is equal to half of the work that would be done by surface and body loads(of magnitude equal to their values in the equilibrium state) by acting through the displace-ments from the unstressed state to the equilibrium state, namely, 2U = Fs · u + Fb · u. (1.129) SL BThis Clapeyron’s Theorem does not conﬂict with (1.124), which is set for a small virtual per-turbation of the displacements and not their magnitude in the equilibrium state.1.4.3 Theorem of ReciprocityConsider two observations of the response of a given elastic body to two different systems of (1) (1)loads. First, the loading system Fs , Fb has been applied and created a deformation u(1) andstrain energy U (1) . Then, (after the ﬁrst loading system has been removed) a loading system (2) (2)Fs , Fb has created the deformation u(2) and the associated strain energy U (2) . Applying nowthe ﬁrst system loading and then the second one (without removing the ﬁrst one), we get thedeformation u(1) + u(2) and strain energy U (1) +U (2) +U (12) . Inverting the order of loading weget the deformation u(2) + u(1) and strain energy U (2) +U (1) +U (21) . Since we are reaching thesame state in both experiments, U (21) = U (12) . This result constitutes the Reciprocal Theoremof Betti and Rayleigh: If an elastic body is subjected to two systems of body and surface loads, the work that wouldbe done by the ﬁrst system in acting through the displacements that were created by the actionof the second system is equal to the work that would be done by the second system in actingthrough the displacements that were created by the action of the ﬁrst system.
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1.5 Euler’s Equations 31This result may be written as (1) (2) (1) (2) (2) (1) (2) (1) Fs · ui + Fb · ui = Fs · ui + Fb · ui . (1.130) S B S B As a special case of the above theorem, we may consider two discrete surface loads actingover points A and B of an elastic body, and suppose that no body forces are applied. When theforce Fs (A) (at A) is applied, the displacements uA at A and uA at B are obtained. When the A Bforce Fs (B) (at B) is applied, the displacements uB at A and uB at B are obtained. For this case A B(1.130) yields Fs (A) · uB = Fs (B) · uA . A B (1.131)If we now adopt the notation uB = αAB Fs (B) and uA = αBA Fs (A) (where the matrices αAB and A BαBA are usually referred to as the inﬂuence coefﬁcients), (1.131) shows that αAB = αBA .1.4.4 Castigliano’s TheoremsWe shall now examine the case of the Theorem of Minimum Potential Energy, where bodyforces are absent and the surface loading Fs k (k = 1, . . . , K) are discrete. For such a case W (u1 , . . . , uK ) − ∑ k=1 Fs k · uk , K V= (1.132) Bwhere Fs k are surface loading and uk are displacements of the kth point (of the application of ∂Vthe force Fs k ). Variation of V leads to ∑K ∂ u δ uk = 0, while (1.123) together with the fact k=1 k ∂Uthat the variations δ uk are arbitrary, yield the system Fs k = ∂ u . This result, widely known as kCastigliano’s First Theorem, may be interpreted as follows: the partial derivative of the strainenergy with respect to generalized displacements is equal to the corresponding generalizedforce. Note that this result stands for small perturbations, and thus should be applied either to thecase of relatively small loads, or to the case where the system is linear in the sense that theresulting displacements vary linearly with the loads (in such a case, U, uk is a linear function ofuk ). To develop Castigliano’s Second Theorem we write the potential energy as W (Fs 1 , . . . , Fs K ) − ∑ k=1 Fs k · uk , K V= (1.133) Band again the minimization leads to equations ∑K ∂∂V δ Fs k = 0, which for arbitrary varia- k=1 F sktions δ Fs k yield uk = ∂∂U . The above may put in words as: Fs k The partial derivative of the strain energy with respect to a generalized force is equal to thecorresponding generalized displacements. The reservation mentioned above regarding the system linearity holds in this case as well.1.5 Euler’s EquationsIn this section we shall employ variational analysis (or calculus of variations) techniques thatdeal with minimization of functionals. In the present context, a functional is an operator thatconverts a set of functions to a number. A fundamental result of the calculus of variations is thatthe extreme values of a functional must satisfy an associated differential equation (or a set ofdifferential equations) over the discussed domain that are generally termed Euler’s equations.
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32 1. Fundamentals of Anisotropic Elasticity and Analytical MethodologiesThe notion “extreme values” stands for local minima, maxima or inﬂection points. Hence,the underlying idea is founded on the existence of a physical global quantity that remainsmaximal or minimal at all times, regardless of the nature (i.e. stationary or time-dependent) ofthe problem. Note that variational calculus is a fundamental analytical tool in many other areasof general physics and engineering, and a review of the mathematical methods associated withit may be found, e.g., in (Sagan, 1969). Two general assumptions are typically associated with the analytical methodologies appliedin calculus of variations. First, we generally assume that all functions and functionals are con-tinuous and have continuous derivatives as required. In addition, the functional values are as-sumed to be positive. This section contains a brief survey of the techniques associated with deriving Euler’s equa-tions out of a given functional, and examples for use of these equations in the area of elasticity. For the sake of abbreviating, in this section, derivatives of functions of one variable, for d2y dmyexample y(x), are denoted as dx = y , dx2 = y , dxm = y(m) . dy1.5.1 Functional Based on Functions of One VariableWe shall ﬁrst calculate extreme values of integral functional, J, whose integrand, F, containsone or several functions associated with the admissible function y(x) of the C2 class on theinterval [x0 , x1 ]. As an example, consider the problem x1 J(y) = F(x, y, y ) dx → min, (1.134) x0where F is a continuous function of three arguments (the problem of determining a maximummay be dispensed with F replaced by −F). The boundary values of y(x) are generally given as y(x0 ) = y0 , y(x1 ) = y1 . (1.135)The minimization of the functional J(y) leads to the Euler’s equation for its integrand, see(Sokolnikoff, 1983), d F, y − (F ) = 0, (1.136) dx , ywhere, obviously, d (F ) = F, y y + F, y y y + F, y x . (1.137) dx , y yEquation (1.136) is a necessary condition that J(y) possess a stationary value. A similar derivation may be carrying out for the extreme problem x1 J(y) = F(x, y, y , y , . . . , y(m) ) dx → min, (1.138) x0where m ≥ 1, and the admissible function y(x) belongs to the Cm+1 class on the interval [x0 , x1 ],and satisﬁes the boundary conditions y(x0 ) = y0 , y (x0 ) = y0 , ... y(m) (x0 ) = y0 , (1.139) (m) y(x1 ) = y1 , y (x1 ) = y1 , ... y (x1 ) = y1 .The minimization in this case leads to the following Euler’s equation: d d2 dm F, y − (F, y ) + 2 (F, y )−, · · · , +(−1)m m (F, y(m) ) = 0, (1.140) dx dx dx
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1.5 Euler’s Equations 33 diwhile F dxi , y(i) are derived analogously to (1.137). To generalize the previous case of (1.134), consider the problem deﬁned in a different wayby the functional x1 J(y) = F(x, y1 , . . . , yn , y1 , . . . , yn ) dx → min, (1.141) x0where n ≥ 1, and F is a continuous function of 2n + 1 arguments. We suppose that the admis-sible functions yi (x) of one variable belong to the C2 class on the interval [x0 , x1 ], and that theboundary values are deﬁned as yi (x0 ) = yi0 , yi (x1 ) = yi1 , 1 ≤ i ≤ n. (1.142)The minimization leads to the following system of Euler’s equations: d F, yi − (F ) = 0, 1 ≤ i ≤ n. (1.143) dx , yi To generalize the case of (1.141) furthermore, we note that similar calculations performedon the extreme problem x1 (m) (m) J(y) = F(x, y1 , . . . , yn , y1 . . . , yn , . . . , y1 , . . . , yn ) dx → min (1.144) x0lead to the following system of Euler’s equations: d d2 dm F, yi − (F, yi ) + 2 (F, yi )−, · · · , +(−1)m m (F (m) ) = 0, 1 ≤ i ≤ n. (1.145) dx dx dx , yi The Euler Lagrange( F, x, y(x) ) and Euler Lagrange( F, x, [y1 (x), . . . , yn (x)] ) commandsfrom the Variational Calculus package of (Maple, 2003), compute the Euler’s equations of thefunctionals (1.134), (1.141). The higher-order functionals (1.138), (1.144) may be reduced tothese forms as well.Remark 1.9 Consider the variational problem (1.141) when for each function, only one ofthe boundary conditions (1.142) is given. For example, yi (x0 ) = yi0 , 1 ≤ i ≤ n. Clearly, theadmissible functions yi (x) in this case form a larger class. The minimizing process showsthat Euler’s equations given in (1.143) are obtained only if the following boundary conditions(usually called “natural”) are satisﬁed: F, yi (x1 ) = 0, 1 ≤ i ≤ n. (1.146)This characteristic of the variational process is one of its profound advantages in the area ofelasticity, as it is capable of providing part of the boundary conditions as well.Example 1.2 Rotating Beam. This example demonstrates the application of the Theorem of Minimum Potential Energyto derive Euler’s equation associated with rotating isotropic beam. In this case, there are threecomponents that contribute to the potential energy: the strain energy, the rotational tension,and the surface loads. Let z be a coordinate along the beam axis, while we are looking forthe beam axis deﬂection in the y direction v(z), 0 ≤ z ≤ l, see Fig. 1.13. The (bending) strainenergy UB in this example will be expressed by the bending curvature v and Young’s modulus,E (see (5.44) with 1/a33 = E), namely 1 1 l 2 UB = W= σz εz = EIx v , (1.147) B 2 B 2 0
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34 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies Figure 1.13: Rotating beam notation in Example 1.2.where Ix = Ω y2 is the cross-section moment (of inertia) about the x-axis. The rotationaltension may be treated in two different ways. We ﬁrst may consider this effect as a contributorto the strain energy. Its contribution equals to the product of the (given) tensile force, T (z), and √the extension created by the bending v(z), which is written as 1 + v 2 − 1 ∼ 1 (v )2 . Hence, =2 l 1 2 UT = T (z) v dz. (1.148) 0 2Alternatively, the tension effect may be viewed as a transverse distributed loading of (T (z) v ) .Since this loading is not constant but depends on the displacements (unlike the body loads in(1.124)), its potential should be therefore written as 1 l VT = − (T (z) v ) v dz. (1.149) 2 0The given surface loads in the y direction, fy (z), do not depend on the deformation. Therefore,their potential is − 0l fy (z)y dz. Overall, the functional F of (1.138) may be written as 1 2 1 F = EIx v − (T (z) v ) v − fy (z)v . (1.150) 2 2By employing the minimization results of (1.140) with F = F(z, v, v , v ), see also P.1.13, weobtain the Euler’s equation (EIx v ) − (T (z)v ) − fy (z) = 0. (1.151)For a rotation about the y-axis as shown in Fig. 1.13, the tension force is T = 1 m(z)Ω2 (l 2 − z2 ), 2where m(z) is the mass distribution per unit length and Ω is the rotational velocity.1.5.2 Variational Problems with ConstraintsConsider a case where Lagrange multipliers should be employed within the variational prob-lem (1.134) with the isoperimeter conditions x1 Jk (y) = Fk (x, y, y ) dx = gk , 1 ≤ k ≤ n, (1.152) x0where gk are constants and Fk are continuous functions of three arguments. The correspondingLagrange functional has the form x1 [F(x, y, y ) + ∑ k=1 λk Fk (x, y, y )] dx. n JL (y) = (1.153) x0
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1.5 Euler’s Equations 35In this case we solve the variation problem JL (y) → min, by considering Lagrange multipliers,λk , as constants.Example 1.3 Elastica. The analysis described in this example deals with large deformations of an elastic rod.This kind of problems are traditionally termed “Elastica”. For further reading see (Frisch-Fay,1962), (Stronge and Yu, 1993). The results presented below are documented in P.1.14. The bending energy of an elastic rod, which is deformed as the plane curve r(s) = [x(s), y(s)]of length l is assumed, for simplicity, to be proportional to the integral of the squared curva-ture over the length of the curve, i.e. 0l κ2 (s) ds, see Example 1.2. Here s ∈ [0, l] is the naturallength parameter of the curve. Recall that the curvature κ(s) of a plane curve is given as dθ dswhere θ(s) is the angle between the local tangent line and the x-axis. One may therefore askthe following question: what shape will the curve take if the total turning of its tangent is givenand the turning is zero and θe at the endpoints. As a constrained variational problem we write l dθ 2 l J= ( ) ds → min, J1 = θ ds = g, with θ(0) = 0, θ(l) = θe . (1.154) 0 ds 0The Lagrange functional of (1.153) takes the form JL (y) = 0l [( dθ )2 + λθ] ds and the function dsF depends on two of the three variables s, θ, dθ and on the parameter λ. The Euler’s equation dsfor the constrained problem is of the form of (1.136), d2θ λ−2 = 0. (1.155) ds2Using the boundary conditions one may ﬁnd the solution θ = λ s2 − λl 2 −4θ e 4 4l s, that depends onthe parameter λ, which with the constraint J1 = g gives λ 3 λl 2 − 4θe 2 l g=( s − s )|l = 12θe − λl 2 . (1.156) 12 8l 0 24Hence, λ = 12 l3 (θe l − 2g) and 6g s 2 6g s θ = (3θe − ) + (−2θe + ) . (1.157) l l l lTo present the above result we note that a plane curve r(s) = [x(s), y(s)] with a given curvaturefunction κ(s) = dθ , may be reconstructed in view of ds dx dy (s) = cos θ, (s) = sin θ, (1.158) ds dsby the formulas s s s x(s) = x0 + cos θ ds, y(s) = y0 + sin θ ds, θ= κ ds. (1.159) 0 0 0The resulting curve is known as Euler’s spiral (or the spiral of Cornu), and an example of it isshown in Fig. 1.14. Another example in the same class deals with an elastic rod, the shape of which is expressedas x = x(s), y = y(s) while its angles at the end points (x1 , y1 ) and (x2 , y2 ) are θ1 and θ2 ,respectively. The constraints of this problem are written by the curve projections onto the x-and y- axes as, see (1.158), l l cos θ ds = x2 − x1 , sin θ ds = y2 − y1 . (1.160) 0 0
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36 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies 0.15 y 0.1 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 –0.05 x –0.1 Figure 1.14: Euler’s spiral for l = 1, θe = 3 π, g = 1. 2Using two Lagrange multipliers, we write the (1.153) type integral l dθ 2 JL = [( ) + λ1 cos θ + λ2 sin θ] ds → min. (1.161) 0 dsThe resulting Euler’s equation (1.136) depends on the parameters λ1 , λ2 and has the form of d2θ −2 + λ2 cos θ − λ1 sin θ = 0. (1.162) ds2One may now move to a new coordinate system, say x, y, by rotating the plane by an angle,say ∆θ, so that the curve of (1.162) will be described as d2θ = µ cos θ, (1.163) ds2where θ = θ + ∆θ and µ = 2 λ2 + λ2 . This new problem is now associated with the modiﬁed 1 1 2boundary conditions and constraints, which are written with the transformed values (x1 , y1 ),(x2 , y2 ). Subsequently, the above problem should be solved under the conditions of θ1 = θ1 +∆θ at (x1 , y1 ) and θ2 = θ2 + ∆θ at (x2 , y2 ). The solution, namely, θ = θ(s, λ1 , λ2 ), should alsofulﬁll the modiﬁed constraints l l cos θ ds = x2 − x1 , sin θ ds = y2 − y1 , (1.164) 0 0which enables the determination of λ1 and λ2 . For more details regarding the above solution,see (Oprea, 1997).1.5.3 Functional Based on Function of Several VariablesWe next consider the problem J(u) = F(x, y, u, u x , u y ) → min, (1.165) Ωwhere F is a continuous function of ﬁve arguments. We suppose that the admissible functionsu(x, y) belong to the C2 class on the two-dimensional domain Ω, while the boundary conditionis formulated in terms of a given continuous function ϕ(x, y) along the contour u=ϕ on ∂Ω. (1.166)The minimization leads to the Euler’s equation ∂ ∂ F, u − (F, u x ) − (F, u y ) = 0, (1.167) ∂x ∂y
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1.5 Euler’s Equations 37where, obviously, ∂ ∂ (F, u x ) = F, u x u x u, xx + F, u x u u, x + F, u x x , (F, u y ) = F, u y u y u, yy + F, u y u u, y + F, u y y . (1.168) ∂x ∂yMore generally, if the functional F(x1 , . . . , xn , u, u x1 , . . . , u xn ) depends on 2n + 1 (n > 1) vari-ables including the function u(x1 , . . . , xn ) of n variables and its ﬁrst derivatives, then the result-ing Euler’s equation takes the form ∂ F, u − ∑ i=1 n (F, u xi ) = 0. (1.169) ∂xi Similar to (1.165) case, when higher (second-order) derivatives are included the extremeproblem becomes J(u) = F(x, y, u, u x , u y , u xx , u xy , u yy ) → min, (1.170) Gwhere the admissible functions u(x, y) belong to the C3 class on the domain Ω, and take speci-ﬁed continuous values on the boundary. These calculations lead to the following Euler’s equa-tion ∂ ∂ ∂2 ∂2 ∂2 F, u − (F, u x ) − (F, u y ) + 2 (F, u xx ) + (F, u xy ) + 2 (F, u yy ) = 0. (1.171) ∂x ∂y ∂x ∂x∂y ∂yWe shall not treat here explicitly the general Euler’s equation, where the functional F dependson the function u(x1 , . . . , xn ) (of n variables) and its higher derivatives (of order ≤ m). Suchderivation is implemented in P.1.13, which was used to create the following examples.Example 1.4 Variational Problem Related to Poisson’s Equation. Of a particular interest is the special class of Euler’s equations that emerge from the func-tional F = a44 (u x )2 − 2a45 u x u y + a55 (u y )2 + 2u f (x, y), (1.172)where f is a given function, and the real numbers ai j , i, j = 4, 5 satisfy a44 > 0, a55 >0, a44 a55 − a2 > 0. In this particular case of (1.165), the minimizing process yields the Euler’s 45equation known as the generalized Poisson’s equation in Ω: a44 u, xx − 2a45 u, xy + a55 u, yy = f (x, y). (1.173)In the isotropic case, where a44 = a55 , a45 = 0, the differential operator of (1.173) is propor- ∂2 ∂2tional to the classic Laplacian ∇(2) = ∂x2 + ∂y2 (which is a scalar square of the “nabla” operator ∂ ∂∇ = { ∂x , ∂y }), see also (Sokolnikoff, 1983). More general versions of this operator appear inChapter 3.Example 1.5 Variational Problem Related to the Biharmonic problem. One may examine the variational problem of the functional F = (u xx + u yy )2 − 2 f (x, y)u, (1.174)where f is a given function, and the admissible functions, u(x, y), belong to the C4 class on thetwo-dimensional domain, Ω, and are subjected to the boundary conditions on the contour, ∂Ω,given by d u = ϕ, u=h on ∂Ω, (1.175) dn
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38 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologieswhere ddn is a normal derivative and ϕ(x, y), h(x, y) are given functions on ∂Ω. In this caseof (1.170), we ﬁnd that the minimization process yields the Euler’s equation known as thefundamental form of the biharmonic equation in Ω: ∇(4) u = f , (1.176)where ∇(4) = ∂x4 + 2 ∂x∂∂y2 + ∂x4 is the biharmonic operator, see (Sokolnikoff, 1983). Replac- ∂ ∂ 4 4 4 2ing the functional of (1.174), for example, by F = c0 (u xx )2 + c1 u xx u xy + c2 u xx u yy + c3 u xy u yy + c4 (u yy )2 + 2 f (x, y)u, (1.177)where ci are real numbers satisfying certain “positiveness” conditions, one obtains (1.176) witha more general version of the biharmonic operator ∇(4) , see P.1.13, ∂4 ∂4 ∂4 ∂4 ∂4 c0 + c1 3 + c2 2 2 + c3 + c4 4 . (1.178) ∂x 4 ∂x ∂y ∂x ∂y ∂x∂y3 ∂xThis operator is extensively treated in Chapter 3.Example 1.6 Dynamic Torsion of a Beam. Consider an isotropic beam of circular cross-section, the axis of which is stretched along thez-axis and is acted upon by a torsional moment distribution mz (z,t). The beam is characterizedby a torsional rigidity, D(z), and a polar moment of inertia, I p (z). The twist angle is denotedφ(z,t). In this case V and T may be simpliﬁed to (see also Remark 7.5) 1 l l 1 l V= D(z) (φ, z )2 dz − mz (z,t) φ dz, T= I p (z) (φ,t )2 dz, (1.179) 2 0 0 2 0and therefore, according to (1.125) t2 J(φ) = (T −V ) dt. (1.180) t1We then arrive at a functional F(z,t, φ, φ z , φ t ) deﬁned in (1.165) that takes the form 1 1 F = I p (z) (φ t )2 − D(z) (φ z )2 + mz (z,t) φ. (1.181) 2 2Subsequently, (1.167) yields with the aid of (1.168) the governing Euler’s equation, which maybe titled as the “equation of motion” in this case, see P.1.13, [D(z) φ, z ], z − I p (z) φ,tt + mz (z,t) = 0. (1.182)1.6 Analytical MethodologiesStrictly speaking, there is no rigorous set of rules that allows one to follow and create analyticalsolutions for any new problem. For that reason, analytic solutions are still considered as “art”,while numerical codes (such as the ﬁnite-element method, etc.) seem to offer a more “straight-forward” solution for any type and geometry of a new problem. This fact emerges from thevariety of techniques that are used for analytical solutions, and from the almost ”uniform” ap-proach that may be exploited when numerical tools are utilized (see also the preface to thisbook). Hence, with that respect, analytical methods are less “direct” compared with numerical
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1.6 Analytical Methodologies 39methods, and a bit of guessing and intuition are always helpful. Yet, there are few standardpaths that are commonly used to create analytical solutions, and it may be stated that most ofthe existing analytical solutions make use of one of these paths. It is therefore beneﬁcial toclassify and discuss these groups of methodologies. In what follows, we shall present the most useful range of analytical methodologies inanisotropic elasticity. Bearing in mind that each of the relevant methodologies is a disciplineby itself, we shall conﬁne the discussion here to the fundamentals of each method. Prior to the description of the above paths, we shall summarize and review some basic def-initions and the various systems of governing equations that are encountered in the ﬁeld ofelasticity, and the speciﬁc role of each one of them. Similar to the previous section, we shall adopt here the notation for derivatives of functions d2y dmyof one variable as dx = y , dx2 = y , dxm = y(m) . dy1.6.1 The Fundamental Problems of ElasticityWe shall ﬁrst review the three basic fundamental problems that are generally treated within thetheory of elasticity, while it is a common practice to try and describe all problems encounteredas special cases of these three problems. We shall present here only the steady state version ofthese problems and for simplicity, work in Cartesian coordinates. The ﬁrst fundamental problem considers a body that undergoes a given distribution of loadsover its boundary surface, S. This problem may be expressed as: ﬁnd the deformation (displace-ment) functions u(x, y, z), v(x, y, z) and w(x, y, z) that satisfy the equilibrium equations (1.82a–c)and the boundary conditions Xs = fx , Ys = fy , Zs = fz on S, (1.183)where Xs ,Ys , Zs are the stress resultants of (1.85a–c), and fx , fy , fz are known functions on S. The second fundamental problem considers a body that undergoes a given distribution ofboundary surface deformation. It may be expressed as: ﬁnd the deformation (displacement)functions u(x, y, z), v(x, y, z) and w(x, y, z) that satisfy the equilibrium equations (1.82a–c) andthe boundary conditions u = u∗ , v = v∗ , w = w∗ on S, (1.184)where u∗ , v∗ , w∗ are known functions on S. The third fundamental problem (sometimes referred to as the mixed problem, see (Muskhe-lishvili, 1953)), considers a body that undergoes a given distribution of surface loads on partof its boundary surface, say SL , and a given distribution of surface deformation on the remain-ing boundary surface, say SD (i.e. S = SL SD ). It may be expressed as: ﬁnd the deformation(displacement) functions u(x, y, z), v(x, y, z) and w(x, y, z) that satisfy the equilibrium equations(1.82a–c) and the boundary conditions Xs = fx , Ys = fy , Zs = fz on SL , (1.185a) ∗ ∗ ∗ u=u , v=v , w=w on SD , (1.185b) u∗ , v∗ , w∗where fx , fy , fz and are known functions on SL and SD , respectively. Note that in all cases, the equilibrium equations of (1.82a–c) contain the body-force compo-nents as well.1.6.2 Fundamental Ingredients of Analytical SolutionsEquilibrium Equations: The ﬁrst system of equations that should be discussed are the equi-librium equations in their generic form (1.79), or in their linear version for Cartesian coor-
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40 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologiesdinates as presented by (1.82a–c). These equations were generally derived for elastic mediaregardless of its characteristics (i.e., isotropic or anisotropic) in terms of the stress componentsand the body force at a point. By employing the appropriate stress-strain relations, one mayexpress these equations in terms of the strain components, which will make the resulting equa-tions material-dependent (and may complicate their exploitation in many cases). Therefore,the equilibrium equations are usually kept in terms of stresses. When applied, the differentialequilibrium equations are also referred to as ﬁeld equations as they should be fulﬁlled at eachmaterial point. Note that some analytical solutions also utilize various integral forms of the equilibriumequations, which are obtained by integrating the stresses over some deﬁnite volume portion,V , bounded by a surface S, of the elastic body. Adopting the notation of (1.81), (1.84) for theCartesian case, simple static force and moment equilibrium show that {Xs , Ys , Zs , xYs − yXs , xZs − zXs , yZs − zYs } S + {Xb , Yb , Zb , xYb − y Xb , x Zb − z Xb , y Zb − zYb } = {0, 0, 0, 0, 0, 0}. (1.186) VThe ﬁrst three of the above equations are identical to (1.86). Stress-Strain Relationships: For materials that exhibit linear stress-strain relationships, thissystem is basically the generalized Hook’s law, that will be derived in S.2.1. For nonlinearmaterials, such relationships is typically based on power or exponential expressions. Mostof the existing analytical solutions for elastic media (as opposed to solutions in plasticity),are derived for linear stress-strain relationships. This is also true for solutions that includegeometrically nonlinear effects. Compatibility Equations: The system of six compatibility equations derived in S.1.2.1 andrepresented by (1.48) (while the linear version for Cartesian coordinates is given by (1.45a–f)),is initially written in terms of the strain components. Clearly, using the appropriate stress-strainrelations, the compatibility equations may be expressed in terms of stresses as well, but theywill include material properties, which will make them less attractive in generic analyticalsolutions for anisotropic elasticity. The linear and nonlinear version of the compatibility equations shown in S.1.2.1 shouldnot be confused with linear and nonlinear expressions of displacement derivatives in the straincomponents, see S.1.1.2. However, for consistency, linear strain-displacements relations shouldbe invoked when linear compatibility equations are employed, while nonlinear strain-displace-ments relations should be invoked when nonlinear compatibility equations are employed. Nev-ertheless, nonlinear analyses typically bypass the need for nonlinear compatibility equationsby selecting solution trail “A” discussed in what follows within S.1.6.5.1. Boundary Conditions: There are two main types of boundary conditions that should bediscussed here, and they are traditionally referred to as “natural” and “geometrical” (or “es-sential”) boundary conditions. The natural boundary conditions are those of (1.85a–c), and are based on equating the in-ternal stress components over the boundaries to the external surface load that acts there. Theseboundary conditions may sometimes be applied in an integral form. For example, at the “free-end” of a slender beam, one may require that instead of demanding that all six σi componentswill vanish at each point over the end cross-section, only the net (integral) resultant loads willvanish there (see further discussion in S.5.1.3). The geometrical boundary conditions are restrictions posed on the displacements (and/or itsderivatives) at some portion of the outer surface of the elastic media as shown by (1.184) or
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1.6 Analytical Methodologies 41(1.185b). This type of boundary conditions may also have an integral version. For example, fora clamp slender beam (cantilever), one may require that the mean displacements over the rootcross-section will vanish as opposed to the requirement of zero displacements at each pointover that cross-section. In this case, the requirement of zero displacement ﬁeld yields threeintegral equations. A similar approach may be adopted for the root rotations (see details inS.5.1.3).1.6.3 St. Venant’s Semi-Inverse Method of SolutionThe so-called “St. Venant’s Semi-Inverse Method of Solution” serves as a basic solution phi-losophy in the theory of elasticity although, fundamentally, it is a general method that has nospeciﬁc relation to a speciﬁc ﬁeld or theory, and it may be applied elsewhere. Therefore, themethod deserves special attention. The St. Venant’s Semi-Inverse Method of Solution consists of a set of equations that repre-sent the solution “mathematical structure” or the “solution hypothesis”. These are sometimescalled “assumptions”, but not in the sense that some parts of the solution or some details of theproblem’s physics are ignored or overlooked. Essentially, these “assumptions” are just a pre-sumed structure of the ﬁnal mathematical solution, and typically include various parameters tobe determined. Hence, the initial solution structure is an assumption, which may be based ona “guess”, “intuition”, “past experience”, etc. It is important to stress the point that the sourceof these preliminary assumptions need not be proved as long as they fulﬁll all relevant govern-ing conditions once all unknown parameters are determined. Thus, once a solution has beensuccessfully reached, the above “assumptions” are converted into a valid solution. When nonlinear problems are under discussion, one may raise the question of solutionuniqueness. Typically, this question remains open, and the solution justiﬁcation should bebased on additional physical arguments.1.6.4 Variational Analysis of Energy Based FunctionalsFollowing the discussion of S.1.5, it is worth reiterating here that within the context of energyconsideration in the theory of elasticity, the notion variational analysis stands for the class ofmethodologies that determine, reach and view the state of equilibrium in elastic media from aunique point of view. Such methods, utilizing the well-established theorems derived in S.1.4that employ energy-related measures, show that equilibrium in elastic bodies is governed byglobal minimum principles, and combine it with the Calculus of variations as demonstratedby the elementary examples of S.1.5. The introduction of these methods into the theory ofelasticity was ﬁrst proposed by Ritz and Rayleigh, see (Sokolnikoff, 1983), and developedfurther by many others, see e.g. (Courant and Hilbert, 1989). When variational tools are employed, two types of outcome should be expected. The ﬁrst oneincludes the differential governing equations of the physical process, and is generally termed asthe corresponding “Euler’s Equations”, already discussed in S.1.5. However, the second typeof outcome is the actual (static or time-dependent) response of the elastic body. This type ofoutcome will be discussed further on within S.1.6.5.4.1.6.5 Typical Solution TrailsAs a general conclusion from the discussion so far, it may be stated that the governing equa-tions of elasticity for an anisotropic body may be expressed using two different approaches.First, one may employ the approach that is generally referred to as “Differential Equilibrium”,
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42 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies“Newton’s laws” or “Vector Mechanics”, which all stand for the creation of a set of equa-tions that will assure the fulﬁllment of the differential equilibrium equations, the differentialcompatibility equations, and the local (natural and geometrical) boundary conditions. Alterna-tively, one may employ the variational analysis of energy based functionals discussed in S.1.5,S.1.6.4 to either derive the governing equations and boundary conditions, or to directly derivea speciﬁc solution. Strictly speaking, the two methods described above are equivalent. One may claim that theformer approach that utilizes equilibrium considerations is more “physical oriented”, while theconcept of the latter is less intuitive in that sense. Generally speaking, while both approachesare suitable to accommodate closed-form or numerical solutions, variational analysis is some-times superior as it is capable of supplying, in addition to a speciﬁc solution, the governingequations, and sometimes part of the boundary conditions (see Remark 1.9), as well. In what follows, we shall review the basic solution trails that are typically encountered withinthe theory of anisotropic elasticity.1.6.5.1 Solution Trail “A”: Deformation HypothesisIn view of S.1.6.3, Trail “A” consists of prescribing the expressions for the deformation com-ponents (or their measures) as presented by Fig. 1.15. Subsequently, the strain components Figure 1.15: Solution Trail “A”.are determined, see S.1.1, and the stress components are derived using the constitutive rela-tions thereafter, see S.2.1. The stress components enable the construction of the equilibriumequations, see S.1.3.2, and the natural boundary conditions, (1.185a), while the geometricalboundary conditions, (1.185b), are directly expressed in terms of the assumed deformation.1.6.5.2 Solution Trail “B”: Stress/Strain HypothesisTrail “B” consists of prescribing the expressions for the stress or the strain components aspresented by Fig. 1.16. Since the strain components may always be obtained by the stresscomponents via the constitutive relations, see S.2.1, and vice versa, the following procedure isidentical to both cases. Subsequently, the strain components enable us to construct the compati-bility equations, see S.1.2.1, while the stress components enable the construction of the equilib-rium equations, see S.1.3.2, and the natural boundary conditions (1.185a). The displacements
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1.6 Analytical Methodologies 43are then derived by strain integration (see S.1.2) and so, the geometrical boundary conditions(1.185b) may also be expressed. Figure 1.16: Solution Trail “B”.1.6.5.3 Solution Trail “C”: Stress Functions HypothesisTrail “C” consists of prescribing the stress functions expressions as presented by Fig. 1.17.In this case, the stress components are determined according to a speciﬁc kind of the stress Figure 1.17: Solution Trail “C”.function/s employed and therefore satisfy equilibrium by deﬁnition. The strain componentsare then derived using the constitutive relations, see S.2.1. The strain components enable usto construct the compatibility equations, see S.1.2.1, while the stress components enable theconstruction of the natural boundary conditions (1.185a). In addition, the displacements arederived by strain integration (see S.1.2) so that the geometrical boundary conditions (1.185b)may also be expressed.
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44 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies1.6.5.4 Solution Trails “D” and “E”: Energy ConsiderationsIn this section, a generic methodology that employs energy theorems and provides the actualresponse without explicit formulation of Euler’s equations will be discussed. In trail “D” we apply the Theorem of Minimum Potential Energy. In such a case, one isrequired to prescribe the displacements (in such a way that the geometrical boundary conditionsare satisﬁed), see Fig. 1.18(a), and to minimize V of (1.124). In trail “E” we apply the Theorem of Minimum Complementary Energy, and one is requiredto prescribe the stresses (in such a way that the equilibrium and the natural boundary conditionsare satisﬁed), see Fig. 1.18(b), and to minimize V ∗ of (1.128). In the latter case, displacementsmay be obtained by suitable integration, see S.1.2. The variational minimization process takescare of the missing part, i.e. the equilibrium and natural boundary conditions in the ﬁrst case,and the compatibility conditions in the second case. (a) Solution Trail “D”. (b) Solution Trail “E”. Figure 1.18: Energy based solution trails “D” and “E” . Within the methodologies in this class of trails, we shall provide some more attention toRitz’s Method and conﬁne the discussion to the one-dimensional case where we seek an ap-proximate solution of a governing equation that may be extracted from a functional, J. To execute Ritz’s Method for a generic J > 0 functional, see e.g. (1.134), one ﬁrst needs toadopt a relatively complete set of functions ηi (see Remark 1.10), each of which satisﬁes theboundary conditions (1.135). The approximate solution (the exact one is denoted by y∗ (x)) istherefore expressed as y∗ (x) = y(x) + ∑ i=1 ai ηi N N (1.187)where y(x) is an auxiliary function and ai are unknown (real) coefﬁcients (which actuallydepend on N) that should be determined by the method. One may therefore examine the valueof J when N terms are employed in (1.187), which will be denoted JN . Subsequently, onemay substitute y∗ (x) in the functional of (1.124) and express it as JN (a1 , a2 , . . . , aN ) = J(y∗ ). N NMinimization of JN with respect to {ai } yields the following linear system of N unknowns andequations: ∂JN = 0, i = 1, . . . , N. (1.188) ∂aiEach approximation level includes the previous one, and a non-increasing series min JN ≤min JN−1 ≤ · · · ≤ min J1 is obtained. Thus, this approximation yields an upper bound for min J.Since F is a continuous function, the condition F(x, y∗ , y∗ x ) − F(x, y∗ , y∗x ) < ε1 may be N N, ,satisﬁed for any (small) ε1 by a appropriate value of N, and since J = x1 x0 F dx, it will be also
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1.6 Analytical Methodologies 45bounded by (small) ε2 = (x1 − x0 )ε1 as x1 ∗ |JN (y∗ ) − J(y)| = N |F ∗ (x, y∗ , y∗ x ) − F(x, y∗ , y∗x )| dx < ε2 . N N, , (1.189) x0In two (or more) dimensions we analogously write u∗ (x, y) = u(x, y) + ∑ i=1 ai ηi (x, y), N (1.190)where u(x, y) is an auxiliary function and ai are unknown (real) coefﬁcients that should be de-termined by the method. In some, mainly “dynamics oriented” analyses, the coefﬁcients ai arereferred to as “Generalized Coordinates”. As shown in Example 1.8, when a time-dependentproblem is under discussion, this concept assigns ai to be time-dependent while the “shapefunctions” ηi remain spatial functions only.Remark 1.10 In the one-dimensional case, a set of functions {ηi (x)} on [x0 , x1 ] is said to berelatively complete if for every ε > 0 and y(x) there are N and y∗ as deﬁned by (1.187) such Nthat |y∗ − y| < ε and y∗ x − y, x < ε for all x ∈ [x0 , x1 ]. In two dimensions, a set of functions N N,{ηi (x, y)} on Ω is said to be relatively complete if for every ε > 0 and u(x, y) there are N andu∗ as deﬁned by (1.190) such that |u∗ − u| < ε, u∗ x − u, x < ε and u∗ y − u, y < ε for all N N N, N,(x, y) ∈ Ω. The extension of the above for higher dimensions is clear.Example 1.7 Bending of a Beam. We shall demonstrate here the usage of Ritz’s Method with the Theorem of Minimum Poten-tial Energy, i.e. J = V , see (1.124) for a “simply-supported” uniform isotropic beam of lengthl, and Young’s modulus E that undergoes a transverse loading fy (z) as shown in Fig. 1.19(a).In the absence of body loads, (1.124) yields for this case (see also Example 1.2) l 1 [ EIx v∗ − fy (z)v∗ ]dz . 2 V= (1.191) 0 2To employ Ritz’s method for determining v(z) we substitute v = v∗ in (1.191), where i πz v∗ = ∑ i=1 ai sin( N ), (1.192) lwhich clearly satisﬁes the geometrical boundary conditions v∗ (0) = v∗ (l) = 0 as required. For uniform loading, i.e. fy = f0 = const. and N = 3, one obtains 2 l f0 1 EIx π4 1 81 V =− (a1 + a3 ) + 3 ( a2 + 4 a2 + a2 ) . (1.193) π 3 l 4 1 2 4 3 4 l 4 f0 4 4The solution of the three equations, V, ai = 0, yields a1 = EIx π5 , a2 = 0 and a3 = 243lEIf0π5 . x l4 0Even with only the ﬁrst term, the above approximation, i.e. v = 4 EI fπ5 sin (πζ) where ζ = z , is l x l 4 f0in a satisfactory agreement with the exact one given by v = 24EIx ζ(ζ3 − 2ζ2 + 1). In addition, for linear loading distribution of fy = f0 (z − 2 ) we obtain l l 2 f0 EIx π4 1 2 81 V = a2 + 3 ( a1 + 4 a2 + a2 ) . 2 (1.194) 2π l 4 4 3 5The solution of the three equations, V, ai = 0, in this case yields a1 = a3 = 0, a2 = − 16EIf0π5 . l x
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46 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies When the above beam is “clamped-free” (cantilever) as shown in Fig. 1.19(b), one mayalternatively use the approximation v∗ = ∑ i=2 ai zi , N (1.195)which satisﬁes the geometrical boundary conditions v = v = 0 at z = 0. For fy = f0 = const. l 4 f0 2such approximations yield the exact one, v = 24EIx ζ ζ2 − 4ζ + 6 , for N = 4 (i.e. three terms). Figure 1.19: Isotropic beam under transverse loading: (a) Simply-supported, (b) Cantilever.Example 1.8 Lagrange’s Equations. We adopt here the concept of expanding each displacement component as a relatively com-plete set of shape functions as in Example 1.7, but we consider a time-dependent problem bysetting the generalized coordinates of the problem, a = (a1 , . . . , a3N ), to be functions of time,t. Hence, in the general case u∗ (x, y, z,t) = ui (x, y, z,t) + ∑ j=1 am (t)η j (x, y, z), N i i = 1, 2, 3, (1.196)where ui are auxiliary (known) functions and m = (i − 1)N + j. We now employ the Theoremof Minimum Potential Energy by writing the variation of V in (1.124) as δV = (grada U − QS − QB ) · δ a = 0 (1.197)where grada = ( ∂a , . . . , ∂a∂ ). In the above we have employed the notation ∂ 1 3N grada U = grada ( W ), QS = (QS , . . . , QS ), 1 N QS = j Fs · u , a j , B S QB = (QB , . . . , QB ), 1 N QB = j Fb · u , a j . (1.198) BBy assuming that in a time-dependent case, the dominant portion of the body force is Fb = −ρu ¨(where ρ is the material density and a dot stands for a time derivative) we write ∂ QB = − ρ u · u,aj = − ¨ ( ρ u · u,aj ) + ˙ ρ u · u,aj . ˙ ˙ (1.199) j B ∂t B BWe further note that the kinetic energy is given by T = 1 2 (ρ ∑3 u2 ), and therefore, the i=1 ˙i Bonce underlined term in (1.199) is ∂T /∂a j (note that ˙ ∂ui /∂a j = ˙ ˙ ∂Ui /∂a j in view of (1.196)),
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1.7 Appendix: Coordinate Systems 47and the twice underlined term is ∂T /∂a j . Hence, (1.197) become ∂ ∂T ∂T ∂U − + − QS = 0, j = 1, . . . , 3N, (1.200) ∂t ∂a j ˙ ∂a j ∂a j jwhich are well known as Lagrange’s equations.1.6.5.5 Solution Trail “F”: Galerkin’s MethodGalerkin’s method is a general powerful method for solving boundary value problems, andthere are many applications related to it in the theory of elasticity. It resembles Ritz’s methodin the sense that it is also based on expanding an approximate solution into a series of rela-tively complete sets of “shape” functions, but it is founded on a different functional type. InGalerkin’s method, we select the functional to be the error L(u∗ ), but instead of employing theoperation J= |L(u∗ )|2 → min, (1.201) Ωwe set the requirement Ji = L(u∗ )ηi = 0, (1.202) Ωby which we force the error function to be orthogonal to each one of the shape functions. Since J is based on the differential operator and not on the potential energy, each of theshape functions must satisfy all boundary conditions (natural and geometrical). This may posesubstantial constraints on the admissible families of shape functions. For example, the seriesused in (1.195) for a cantilever beam is not applicable here since the ηi functions do not satisfythe natural boundary conditions (one by one) at the free tip.1.7 Appendix: Coordinate Systems1.7.1 Transformation Between Coordinate SystemsThere are many occasions when one wishes to change the coordinate system for which theproblem is expressed. As already discussed in the introduction of S.1.1, from an analyticalpoint of view, such a change may be crucial for the existence and complexity of a closed-formsolution. In this book we deal with transformations of three types of mathematical identities.The ﬁrst one is the vector space transformation (which is equivalent to coordinate transforma-tion of a point in Euclidean space). The second one is the (second-order) tensor transformation,which should be applied to the strain and stress components, see S.1.3.3, S.1.3.4, and the thirdone is the (fourth-order) tensor transformation of the constitutive relations, namely, the com-pliance and the stiffness matrices, which are dealt with in Chapter 2. As a ﬁrst step we shall deﬁne a generic linear transformation in Euclidean space E 3 (withCartesian metric), by a transformation matrix, T, written as ⎡ ⎤ T11 T12 T13 ⎢ ⎥ T = ⎢ T21 T22 T23 ⎥ . ⎣ ⎦ (1.203) T31 T32 T33T is a 3 × 3 matrix that transforms a position vector v = (v1 , v2 , v3 ) given in a certain coordinatesystem, S, to another vector v = (v1 , v2 , v3 ), which deﬁnes the same point in a new system, S,
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48 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologiesby v = T · v. The matrix T is orthogonal, and its transposed and inverse matrices coincide, i.e.,TT = T−1 , or ∑ k=1 Tik Tjk = δi j , i, j ∈ {1, 2, 3}. 3 (1.204)Hence, det(T) = ±1 (while in order to preserve orientation, the +1 is selected). The orthogonal matrix, T, may be composed in various ways. Here, we shall adopt the usageof three Euler’s angles φ, θ, ψ, that represent rotations about the x, y, z directions, respectively.For any single rotation about the x, y or z directions, one may assign as T the following Tx (φ),Ty (θ) or Tz (ψ) matrices, respectively: ⎡ ⎤ ⎡ ⎤ 1 0 0 cos θ 0 − sin θ ⎢ ⎥ ⎢ ⎥ Tx (φ) = ⎢ 0 cos φ sin φ ⎥ , ⎣ ⎦ Ty (θ) = ⎢ 0 ⎣ 1 0 ⎥, ⎦ 0 − sin φ cos φ sin θ 0 cos θ ⎡ ⎤ cos ψ sin ψ 0 ⎢ ⎥ Tz (ψ) = ⎢ − sin ψ cos ψ 0 ⎥ . ⎣ ⎦ (1.205) 0 0 1However, a generic orthogonal transformation will be composed as a rotation by the angle ψabout the z-axis, followed by a rotation by the angle θ about the y-axis (in its new positionx , y , z ), followed by a rotation by the angle φ about the x-axis (in its new position x , y , z ),see Fig. 1.20(a). Thus, the ﬁnal resulting system is x, y, z. In this case, the orthogonal matrix (a) Generic Euler’s angles. (b) Spherical angles. Figure 1.20: Deﬁnition of Euler’s and spherical angles.T = {Ti j } takes the form T = Tx (φ) · Ty (θ) · Tz (ψ), (1.206)where its elements are, see P.1.15,T11 = cos θ cos ψ, T12 = cos θ sin ψ, T13 = − sin θ, (1.207)
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1.7 Appendix: Coordinate Systems 49T21 = sin φ sin θ cos ψ − cos φ sin ψ, T22 = sin φ sin θ sin ψ + cos φ cos ψ, T23 = sin φ cos θ,T31 = cos φ sin θ cos ψ + sin φ sin ψ, T32 = cos φ sin θ sin ψ − sin φ cos ψ, T33 = cos φ cos θ.As will be discussed further on, T will also be utilized as a second-order tensor. As a generalrule, we shall carry on the notation ( ) for the value of ( ) in the transformed (“new”) coordinatesystem. Note that for a given transformation tensor, the corresponding transformation anglesmay be determined by ﬁve terms of (1.207), as T12 T23 tan ψ = , sin θ = −T13 , tan φ = . (1.208) T11 T33Special attention should be devoted to the angles’ sign in the case where T11 = 0 and/or T33 = 0.Remark 1.11 While utilizing the procedure described above for constructing the transforma-tion matrix by Euler’s angles, it should be noted that the order of rotation angles has a crucialimportance. For example, the reader may verify that a transformation based on ψ = π/2, θ =π/2, φ = 0, is inverted by ψ = −π/2, θ = 0, φ = π/2. There are six different ways of select-ing the order of the three different axes. The selection made above (z, y, x) is convenient forstress and strain analysis (see S.1.3.3). However, a different order may be easily constructedby changing the order in the matrix product of (1.206) (see also P.1.15). Note that there aresimilar methods where the transformation may be deﬁned by rotation about repeated axes likeT = Tz (φ) · Ty (θ) · Tz (ψ).Remark 1.12 It is important to distinguish between the above Euler’s angles, φ, θ and ψ andthe “standard” spherical angles, θs = ∠(OP , x) ∈ [0, 2π] and φs = ∠(OP, z) ∈ [0, π] used (inanalytical geometry) to locate a point in space. As shown in Fig. 1.20(b), one may deﬁne thex, y, z coordinates of a point, P, in space using the standard spherical angles θs and φs (P is theorthogonal projection of P onto the x, y-plane). By setting the distance of P from the origin tobe a unit (|OP| = 1) we write x = sin φs cos θs , y = sin φs sin θs , z = cos φs . (1.209)However, using Euler’s angles with φ = 0 (see Fig. 1.20(a)), we may write the coordinatesof point P assuming that it is placed on the x coordinate line (of the new system) by settingv = (1, 0, 0), and calculating v = T−1 · v, which yields x = cos θ cos ψ, y = cos θ sin ψ, z =− sin θ. Therefore, θs = ψ, φs = π + θ. Note that while the spherical angles are commutative, 2as indicated in Remark 1.11, Euler’s angles are not, and the above holds for the order of anglesadopted in this section only.1.7.2 Curvilinear Coordinate Systems1.7.2.1 DeﬁnitionWe shall assign (α1 , α2 , α3 ) ∈ A to be the curvilinear coordinates in a domain C of Euclideanspace E 3 (with orthonormal basis {ki }), see Fig. 1.21, and therefore, the position vector ofeach point may be expressed by three functions that relate αi to the Cartesian coordinates by atopological transformation, r : A → C, as r = ∑ i=1 fi (α1 , α2 , α3 ) ki . 3 xi = fi (α1 , α2 , α3 ), i = 1, 2, 3, ⇔ (1.210)Similarly, two functions of α1 , α2 are used for curvilinear coordinates in a plane domain. Curvi-linear coordinates are called regular if the Jacobian, Jr , of (1.210) is nonzero, namely, ⎡ ⎤ f1, α1 f1, α2 f1, α3 Jr = det(dr) = 0, dr = ⎣ f2, α1 f2, α2 f2, α3 ⎦ . (1.211) f3, α1 f3, α2 f3, α3
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50 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologies Figure 1.21: Orthogonal curvilinear coordinates in domain C of E 3 .In the general case of curvilinear coordinates in E 3 as described by (1.210), one may constructthe metric differential quadratic form and the metric tensor ds2 = ∑ i j gi j dαi dα j , g = {gi j }, (1.212)whose coefﬁcients gi j = r, αi · r, α j (i, j = 1, 2, 3) are functions of αi . In the particular case of Cartesian coordinates, gi j = δi j and ds2 = dx2 + dy2 + dz2 . Anotherspecial case is the afﬁne coordinate system which is deﬁned by linear transformation functionsfi = f i1 α1 + f i2 α2 + f i3 α3 (i = 1, 2, 3), where f i are constants. For this kind of coordinates,both the matrix dr of (1.211) and the metric tensor g = {gi j } are constants.1.7.2.2 The Riemann-Christoffel Curvature TensorThe following two problems, that were ﬁrst formulated and solved by Lam´ for a particular ecase, see (Cartan, 1946), are essential in the theory of curvilinear coordinates. First: for a given metric form ds2 or metric tensor g = {gi j } (see (1.212)), ﬁnd conditionson the coefﬁcients gi j so they may represent the metric of some curvilinear coordinates α j in adomain A of E 3 (or E 2 ). Second: for a given metric form ds2 or metric tensor g = {gi j } (see (1.212)), of somecurvilinear coordinates α j in a domain A of E 3 (or E 2 ), ﬁnd a transformation in the form of(1.210), i.e. ﬁnd the associated functions fi (α1 , α2 , α3 ). We shall elaborate here on the ﬁrst problem, as the second one ﬁnds its answer in the strainintegration schemes presented within S.1.2.2, S.1.2.3. The discussion below is founded on the well-known Riemann Theorem, see (Oprea, 1997),that may be put as: A symmetric tensor g = {gi j }, i, j = 1, 2, 3 is a metric tensor for a Euclideanspace if it is non-singular (det(g) = 0), positive-deﬁnite (g > 0), and the Riemann-Christoffelcurvature tensor R = {Rmni j } that is formed from it, vanishes identically. The components of the Riemann-Christoffel curvature tensor are known to be 1 Rmni j = (gm j, ni + gin, m j − gmi, n j − g jn, im ) − g f h (Γ f ,im Γh, jn − Γ f , ni Γh, jm ), (1.213) 2while g−1 = {gqs } is the inverse matrix of g. Here we have used the notation Γ p,ms for Christof-fel symbols of the ﬁrst kind, (not a tensor), which are based on the ﬁrst-order derivatives of the
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1.7 Appendix: Coordinate Systems 51metric gi j , namely, 1 Γ p, ms = (gmp, s + g ps, m − gms, p ) . (1.214) 2Only six of the 81 curvature tensor R components are independent, and are obtained by thesame set of (1.44). In a two-dimensional case, there is only one (out of 16) independent com-ponents of curvature tensor, for example, R1221 .1.7.2.3 Orthogonal Curvilinear Coordinate SystemsWhen the matrix g is diagonal (i.e. gi j = 0 for i = j), the coordinates (1.210) are said to beorthogonal, and one may deﬁne the well-known Lam´ ’s parameters, which are essentially ethe absolute values of r, αi , namely, Hi = |r, αi |. Subsequently, as shown in Fig. 1.21, we maydeﬁne unit vectors k i in the αi directions as ˆ 1 ki = ˆ r, α , i = 1, 2, 3. (1.215) Hi iThe arc-lengths of elements in the k i directions are dsi = Hi dαi . Hence, in view of (1.211), a ˆtransformation matrix that converts a vector in Cartesian coordinates into a vector in orthogonalcurvilinear coordinates is given by ⎡ 1 1 1 ⎤ H1 f 1, α1 H1 f 2, α1 H1 f 3, α1 ⎢ 1 1 1 ⎥ T = ⎣ H2 f1, α2 H2 f2, α2 H2 f3, α2 ⎦ . (1.216) 1 1 1 H3 f 1, α3 H3 f 2, α3 H3 f 3, α3 ˆThe derivatives of the unit vectors ki with respect to the curvilinear coordinates are, see(Novozhilov, 1961), ⎡ ⎤ ⎧ ⎫ 0 − H2 H1, α2 − H3 H1, α3 1 1 ⎪ ˆ k1, α1 ⎪ ⎢ ⎪ ⎪ˆ ⎪ 1 ⎥ ⎪ k1, α ⎪ ⎢ ⎪ ⎪ 0 H1 H2, α1 0 ⎥ ⎪ 2⎪⎪ ⎢ ⎥ ⎪ˆ ⎪ k1, α ⎪ ⎪ ⎢ 0 0 1 H3, α1 ⎥ ⎪ ⎪ 3⎪ ⎪ ⎢ H1 ⎥⎧ ⎫ ⎪ ⎪k ⎪ ⎢ 1H ⎪ ˆ 2, α1 ⎪ ⎢ H2 1, α2 ⎨ ⎪ ⎬ 0 0 ⎥ ˆ ⎥ ⎨ k1 ⎬ ⎢− 1 H − H3 H2, α3 ⎥ k2 . 1 k2, α2 = ⎢ H1 2, α1 ˆ 0 ⎥ ˆ (1.217) ⎪ˆ ⎪k ⎪ ⎢ ⎪ ⎢ ⎥⎩ ˆ ⎭ ⎪ ⎪ ⎪ 2, α3 ⎪ ⎢ 0 0 1 H3, α2 ⎥ k3 ⎪ ⎪k ⎪ ⎥ ⎪ ˆ 3, α1 ⎪ ⎢ 1 H H2 ⎪ ⎪ ⎪ ⎢ H 1, α3 ⎥ ⎪ ⎪ˆ ⎪ 0 0 ⎥ ⎪ k3, α ⎪ ⎢ 3 ⎪ ⎪ ⎪ ⎣ ⎥ ⎪ ⎩ˆ 2⎪ ⎭ 0 1 H3 H2, α3 0 ⎦ k3, α3 − H1 H3, α1 − H2 H3, α2 1 1 0Example 1.9 Standard Orthogonal Coordinate Systems. In the few illustrative orthogonal coordinate systems listed below the coordinate parame-ters are denoted by α1 , α2 , α3 . The expressions for fi (see (1.210)) that appear in (1.218a–d)are for: (a) Cylindrical coordinates (α1 = ρ, α2 = θc , α3 = z), (b) Elliptical-cylindrical coor-dinates (α3 = z), (c) Spherical coordinates (α1 = ρ, α2 = θs , α3 = φs ), (d) Bipolar-cylindricalcoordinates (α3 = z), f1 = α1 cos α2 , f2 = α1 sin α2 , f3 = α3 , (1.218a) f1 = a cosh α1 cos α2 , f2 = a sinh α1 sin α2 , f3 = α3 , (1.218b) f1 = α1 cos α2 sin α3 , f2 = α1 sin α2 sin α3 , f3 = α1 cos α3 , (1.218c) a sinh α2 a sin α1 f1 = , f2 = , f3 = α3 (1.218d) cosh α2 − cos α1 cosh α2 − cos α1
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52 1. Fundamentals of Anisotropic Elasticity and Analytical Methodologieswhere a > 0. Additional example is the Ellipsoidal coordinate system, which is deﬁned by α1 α2 α3 (α2 − b2 )(α2 − b2 )(b2 −α2 ) 1 2 3 (α2 −a2 )(a2 −α2 )(a2 −α2 ) 1 2 3f1 = , f2 = , f3 = ,(1.219) ab b a2 − b2 a a2 − b2where a ≥ b > 0. Symbolic computational tools (e.g. (Maple, 2003)) support many other three- (a) Cylindrical. (b) Spherical. (c) Ellipsoidal. Figure 1.22: Image of various curvilinear coordinate systems.dimensional orthogonal systems. By activating P.1.16, P.1.17, the reader may examine addi-tional curvilinear coordinates, see Fig. 1.22 for illustrative examples. It is interesting to note that for orthogonal curvilinear coordinates, one may write the sixindependent components of (1.213) as the following six differential relations between Lam´ eparameters (apply cyc-123): H2, α1 H1, α2 H1, α2 H1, α3 H1, α3 H1, α2 H2, α3 ( ), α1 + ( ), α2 + 2 = 0, ( ), α2 − = 0. (1.220) H1 H2 H2 H3 H2 H3The above six conditions may therefore be titled as the integrability conditions of a givenmetric tensor g = ∑3 Hi2 (dαi )2 . One may also reach (1.220) by considering the identities i=1 (k i, αi ), α j = (k j, α j ), αi , ˆ ˆ (1.221)and substitute in it the k i, α j of (1.217). Then, each one of the three basic sets (i, j) = (1, 2), ˆ(1, 3), (2, 3) yields two equations. These are obtained by the corresponding (two) k i -compo- ˆnents that appear in each equation. For orthogonal coordinates in plane (polar, bipolar, elliptical, etc.) the equality R1221 = 0coincides with the underlined terms in (1.220:a).
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2Anisotropic MaterialsThis chapter reviews the mathematical representation of anisotropic materials and their charac-teristics as observed by various coordinate systems. Generally speaking, the elastic propertiesare characterized by certain functional relationships between loads and deformation, and thenature of these relationships is in the focus of this chapter. Prior to the detailed consideration, it is worth clarifying the notions “anisotropy” and “non-homogeneous”, which are highly relevant for the type of boundary value problems discussedin this book. For that purpose, suppose that we wish to measure the properties of a given elasticmedium using a given coordinate system. We shall deﬁne the medium as “non-homogeneous”if we ﬁnd that the material properties are functions of the coordinate system location. On theother hand, we shall deﬁne the medium as “anisotropic” if we ﬁnd that the material propertiesare functions of the coordinate system orientation. This dependence of properties on theirmeasuring axes orientation enables us to deﬁne the notion “anisotropy” also as the tendency ofthe material to react differently to stresses applied in different directions. It is therefore possible to have all four combinations of “isotropic and homogeneous”,“anisotropic and homogeneous”, “isotropic and non-homogeneous” and “anisotropic and non-homogeneous” materials (or elastic domains). This chapter is devoted to the anisotropic material properties, while the homogeneity issuewill be considered within speciﬁc applications in further chapters. The mathematical repre-sentation of anisotropic materials is closely related to the type of analysis and the class ofproblems that are under discussion. Within this book, we will be focused on the class of prob-lems in anisotropic elasticity, which are characterized by typical physical dimensions that jus-tify the employment of effective elastic properties. In general, the concept of working witheffective elastic properties implies that emphasis is put on the macro-mechanical behavior ofthe involved phenomena, while paying much less attention to the exact micro-structure of thematerial. From an analytical methodologies point of view, this enables the utilization of con-tinuous functions for describing the deformation (and likewise other physical measures) overﬁnite volumes, the typical dimensions of which are on the order of those of the structure underdiscussion. Although the above approach applies to isotropic materials as well, it is of special impor-tance for anisotropic materials due to their special micro-structure. For anisotropic materials,
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54 2. Anisotropic Materialsthe micro-structure usually consists of different components or layers that are tied together in“natural” materials, e.g. crystals, rocks, wood, etc., or a number of ingredients that are moldedor bonded together in “man-made” materials, such as composite materials that typically con-sist of a matrix, which is reinforced by high-strength ﬁbers. In the latter case, the concept ofworking with effective elastic properties, which are determined by various “homogenizationmethods”, completely bypasses the physical mechanisms that describe the ﬁber-matrix rela-tions, and treats the effective combination as if the material is homogeneous. The above way of working with effective material properties should not be confused with“smearing” and “averaging” methods that are frequently encountered in laminated compositeanalysis, see S.3.5.2, S.5.3.1. In such cases, laminated stacks of composite plies are handled ina simpler way by treating them as an “averaged” homogeneous material, see e.g. (Ochoa andReddy, 1992). For further reading on anisotropic and composite materials systems see (Vinson and Chou,1975), (Halpin, 1992), (Mallick, 1993), (Parton and Kudryavtsev, 1993), (Daniel and Ishai,1994), (Gibson, 1994), (Bull, 1995), (Hull and Clyne, 1996), (Mallick, 1997), (Jones, 1999),(Kelly and Zweben, 2000), (Vinson and Sierakowski, 2002), (Reifsnider and Case, 2002),(Koll´ r and Springer, 2003). a For further reading on homogenization methods and effective properties see (Devries et al.,1989), (Tarnopol’skii et al., 1992), (Ashbee, 1993), (Ghosh et al., 1995), (Ponte-Castaneda andWillis, 1995), (Ponte-Castaneda, 1996), (Kalamkarov and Kolpakov, 1997), (Kolpakov, 2004).2.1 The Generalized Hook’s LawThe deﬁnition of a generic anisotropic material is founded on the so-called “GeneralizedHooke’s Law” (GHL), which in fact establishes the material “constitutive relations” or the“stress-strain relations”. At this stage we shall be focused on the case of Cartesian anisotropywhere the GHL linearly connects the engineering strain and stress vectors, ε , and, σ , respec-tively. These vectors are written as (see S.1.1.2) ε = [εxx , εyy , εzz , γyz , γxz , γxy ]T , σ = [σxx , σyy , σzz , τyz , τxz , τxy ]T . (2.1)Subsequently, the GHL is typically written in one of the following forms: ε = a·σ , σ = A·ε , (2.2)where clearly, the above sixth-order symmetric matrices are related by A = a−1 . We shalldenote a and A as the compliance matrix and the stiffness matrix, respectively. In what follows the discussion is conﬁned to the case of linear (“engineering”) strain ex-pressions, which are based on derivatives of the displacements as shown by (1.9a,b), namely,εαα = uα, α and γαβ = uα, β + uβ, α for α, β = x, y, z. For the present purpose of deﬁning the material properties, we shall make two additionalassumptions. First, we shall restrict our attention to the range of relatively small strain wheremost materials exhibit linear stress-strain relations, and completely disregard materials that donot exhibit linear stress-strain relations at all. In addition, we shall ignore the fact that manymaterials exhibit linear behavior that should be quantiﬁed by different elastic moduli over thetwo sides of the zero-strain point (e.g., various Graphite fabrics show different elastic modulifor tension and compression, see (Chen and Saleeb, 1994)). Programs P.2.1, P.2.2, P.2.3 derivethe compliance and stiffness matrices in both symbolic and numerical manners.
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2.2 General Anisotropic Materials 55Remark 2.1 The strain vector of (2.2), i.e., ε = [ε1 , . . . , ε6 ]T (see notation in (1.21)), is themechanical strain vector that is induced by the stress vector. When the body is subjected to achange of temperature, this strain vector should be replaced by ε − ε◦ where ε will now standfor the total strain vector and ε◦ = [ε◦ , . . . , ε◦ ]T is the thermal strain vector, which is induced 1 6by the change in temperature. For general anisotropic materials (see S.2.2) one may writeε◦ = α◦ (T − Tr ), where α◦ are thermal expansion coefﬁcients (of dimension 1/per degree), i i iT is the actual temperature and Tr is a reference temperature for which the thermal straincomponents are deﬁned as zero. Equation (2.2) is therefore converted to ε − ε◦ = a · σ andσ = A · (ε − ε◦ ) . For isotropic materials (see S.2.8) it is customary to write ε◦ = α◦ (T − Tr ) ifor i = 1, 2, 3 and ε◦ = 0 for i = 4, 5, 6, where α◦ is a constant. i2.2 General Anisotropic MaterialsFor the most general anisotropic materials, which are also referred to as triclinic materials, thecompliance matrix a = {ai j } contains 21 independent elastic coefﬁcients. For further use andfor obvious reasons, this matrix will be denoted GEN21 and is written as (note that as a generalrule, symmetric matrices will be left blank in their lower left part) ⎡ ⎤ a11 a12 a13 a14 a15 a16 ⎢ a22 a23 a24 a25 a26 ⎥ ⎢ ⎥ ⎢ a33 a34 a35 a36 ⎥ a=⎢ ⎥. (2.3) ⎢ a44 a45 a46 ⎥ ⎣ ⎦ Sym. a55 a56 a66Similarly, the stiffness matrix of this general anisotropic material, A = {Ai j }, may be writtenby its 21 components as ⎡ ⎤ A11 A12 A13 A14 A15 A16 ⎢ A22 A23 A24 A25 A26 ⎥ ⎢ ⎥ ⎢ A33 A34 A35 A36 ⎥ A=⎢ ⎥. (2.4) ⎢ A44 A45 A46 ⎥ ⎣ ⎦ Sym. A55 A56 A66Hence, by adopting the above GHL representation of the material constitutive relations, underthe above mentioned provisions, all materials may be deﬁned by numerical assignment of allterms in their matrices ai j and/or Ai j . These terms will be further referred to as “elastic moduli”.Later on, when the transformation of these matrices due to a rotation of the coordinate systemwill be discussed, we shall be obliged to deﬁne the compliance and stiffness tensors as well. Man-made materials and many natural materials exhibit some symmetry and micro-structuresimpliﬁcation that lead to GHL that consists of less than 21 independent elastic moduli. Forexample, (Lekhnitskii, 1981) presents the properties of quartz (rock crystal) in which nine ofthe ai j moduli vanish, and there are six additional relations between the remaining moduli, andhence, overall, there are only six independent moduli that characterize the GHL in that case.Subsequently, the following sections will be devoted to materials that are less general thanthe above GEN21 material. Most of the materials below are named by their internal micro-structure, and we shall present their effective elastic properties only as reﬂected by the aboveGHL.
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56 2. Anisotropic Materials2.3 Monoclinic MaterialsA ﬁrst type of materials that may be deﬁned by less than 21 independent parameters are mon-oclinic materials that are deﬁned by 13 independent coefﬁcients and, as will become clearerlater on (see S.2.16), contain one plane of elastic symmetry. There are three main kinds ofmaterials in this category, and their classiﬁcation is based on the type of population of theircompliance and stiffness matrices. The ﬁrst kind is the material described by ⎡ ⎤ a11 a12 a13 0 0 a16 ⎢ a22 a23 0 0 a26 ⎥ ⎢ ⎥ ⎢ a33 0 0 a36 ⎥ a=⎢ ⎥, (2.5) ⎢ a44 a45 0 ⎥ ⎣ ⎦ Sym. a55 0 a66which for future use will be denoted MON13z. The numerical part of this notation stands forthe number of independent coefﬁcients, and the index “z” is added to indicate that the x, y-planeis a plane of elastic symmetry. A similar population of the compliance (or the stiffness) matrixmay be obtained when orthotropic material is rotated about the z-axis, as will be shown lateron within S.2.4. The second and the third kinds of materials in this family are described by thecompliance matrices ⎡ ⎤ ⎡ ⎤ a11 a12 a13 a14 0 0 a11 a12 a13 0 a15 0 ⎢ a22 a23 a24 0 0 ⎥ ⎢ a22 a23 0 a25 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ a33 a34 0 0 ⎥ ⎢ a33 0 a35 0 ⎥ ⎢ ⎥, ⎢ ⎥, (2.6) ⎢ a44 0 0 ⎥ ⎢ a44 0 a46 ⎥ ⎣ ⎦ ⎣ ⎦ Sym. a55 a56 Sym. a55 0 a66 a66and for similar reasons will be denoted MON13x and MON13y materials, respectively. Thesame matrix population is obtained for the corresponding stiffness matrices (A) of monoclinicmaterials.2.4 Orthotropic MaterialsA simpler class of materials known as orthotropic materials, is deﬁned by nine independentmoduli only. These materials have three orthogonal planes of elastic symmetry and are usuallydeﬁned by the so-called engineering constants Eii , Gi j and νi j , as ⎡ ν ν ⎤ 1 E11 − E21 22 − E31 33 0 0 0 ⎢ − ν12 1 ν − E32 ⎥ ⎢ E11 E22 0 0 0 ⎥ ⎢ ν13 ν 33 ⎥ ⎢−E − E23 1 0 0 0 ⎥ a=⎢⎢ 11 22 E33 ⎥. (2.7) 1 ⎥ ⎢ 0 0 0 G23 0 0 ⎥ ⎢ ⎥ ⎣ 0 0 0 0 1 G13 0 ⎦ 1 0 0 0 0 0 G12For further use, this matrix form will also be denoted as ORT9. The above matrix is written ν ν ν ν ν νin full to emphasize its symmetry properties E12 = E21 , E13 = E31 , E23 = E32 . One may gain 11 22 11 33 22 33some physical interpretation of the above notation by noticing ﬁrst that Eii is the amount of εii 1
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2.4 Orthotropic Materials 57per unit σii . Also, −ν12 is the ratio εyy /εxx when σxx is applied, −ν13 is the ratio εzz /εxx whenσxx is applied, etc. In addition, Gi j is the amount of γi j per unit σi j for (i, j) = (1, 2), (1, 3) 1and (2, 3). This physical interpretation serves also as the basis for experimental proceduresthat establish the physical magnitude of the elastic moduli for materials in this class, see e.g.(Jenkins, 1998), (Jones, 1999). As already indicated, simple inversion of the compliance ma-trix, a, yields the stiffness matrix A that exhibits population and symmetry, which are identicalto those of a. The non-vanishing terms of the stiffness matrix are given by E11 E11 E11 A11 = (1 − ν23 ν32 ) , A12 = (ν21 + ν23 ν31 ) , A13 = (ν21 ν32 + ν31 ) , D0 D0 D0 E22 E22 E33 A22 = (1 − ν13 ν31 ) , A23 = (ν32 + ν12 ν31 ) , A33 = (1 − ν12 ν21 ) , D0 D0 D0 A44 = G23 , A55 = G13 , A66 = G12 , (2.8)where D0 /(E11 E22 E33 ) is the determinant of the 3 × 3 upper-left minor of the a matrix of (2.7).D0 may be written as D0 = 1 − ν12 ν21 − ν13 ν31 − ν23 ν32 − ν12 ν23 ν31 − ν13 ν21 ν32 . (2.9)The same stiffness matrix may be written using the nine parameters of the lower part of a only(i.e. Eii , ν12 , ν13 , ν23 and Gi j ) as E11 E33 E22 E33 E33 A11 = (1 − ν2 23 ), A12 = (ν12 + ν23 ν13 ), A13 = (ν12 ν23 + ν13 ), D0 E22 D0 E22 D0 E22 E33 E33 E22 E33 E22 A22 = (1 − ν2 13 ), A23 = (ν23 + ν12 ν13 ), A33 = (1 − ν212 ), (2.10) D0 E11 D0 E11 D0 E11and E33 E22 E33 E33 D0 = 1 − ν2 23 − ν212 − 2ν12 ν23 ν13 − ν2 13 . (2.11) E22 E11 E11 E11Clearly, if required, the stiffness matrix may also be expressed using the nine coefﬁcientsE11 , E22 , E33 , ν21 , ν31 , ν32 and G12 , G13 , G23 only. It is sometimes useful to visualize the population and the relative magnitude of the vari-ous coefﬁcients in the compliance and stiffness matrices, as shown in Fig. 2.1 for a typicalorthotropic material.0.4 1000.2 2 50 2 0 4 4 0 1 row 1 row 2 3 6 2 3 6 4 5 4 5 column 6 column 6 (a) The compliance matrix a (1/GPa). (b) The stiffness matrix A (GPa). Figure 2.1: The compliance and stiffness matrices for typical orthotropic Graphite/Epoxy material. As already indicated in S.2.3, when an orthotropic material is rotated about the x-, y- orz- directions, the resulting compliance/stiffness matrix population is identical to the form of
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58 2. Anisotropic MaterialsMON13x, MON13y and MON13z, respectively. Such materials are sometimes referred to asGenerally Orthotropic. However, it should be noted that their matrices are based on 10 inde-pendent parameters only (nine of the orthotropic material plus one angle of rotation), whilegeneric monoclinic materials are based on 13 independent parameters. For example, Fig. 2.2, presents a beam of a MON13y material that was created from or-thotropic material by rotating it by the angle −θy about the y-axis (x, y, z are the beam axes,x , y , z are the orthotropic material principal axes). As shown, this rotation may also be viewed Figure 2.2: A MON13y homogeneous beam made of rotated orthotropic material.as a rotation of the x , y , z coordinate system by the angle θy .2.5 Tetragonal MaterialsTetragonal material is an orthotropic one that has a pair of coordinate directions along whichthe elastic properties are identical, and different properties along the third coordinate axis.Hence, there are three fundamental types of this material. For example, when the (x, y) axesare selected, the material remains unchanged under the transformation x → −y, y → x (wherethe “minus” has been introduced just for preserving the system orientation). Such a materialmay therefore be deﬁned by the six independent parameters E, E , G, G , ν, ν , so that E11 = E22 = E, E33 = E , G23 = G13 = G , G12 = G, E ν12 = ν21 = ν, ν13 = ν23 = ν , ν31 = ν32 = ν , (2.12) Ewhich yields the following compliance matrix: ⎡1 ν ν ⎤ E −E −E 0 0 0 ⎢ 1 −E ν 0⎥ ⎢ 0 0 ⎥ ⎢ E ⎥ ⎢ 1 0 0 0⎥ a=⎢ E ⎥. (2.13) ⎢ Sym. 1 0 0⎥ ⎢ G ⎥ ⎣ 1 0⎦ G 1 GFor further use, tetragonal materials where x − y directions are interchangeable will be denotedTRG6z, while similarly, the other two kinds will be denoted TRG6x and TRG6y. The popula-tion of the above compliance matrix for all three pairs of identical directions is summarized in
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2.6 Transversely Isotropic (Hexagonal) Materials 59 Material → TRG6x TRG6y TRG6z Interchangeable Directions → y−z x−z x−y Module ↓ 1 1 1 a11 E E E 1 1 1 a22 E E E 1 1 1 a33 E E E ν ν ν a12 −E −E −E ν ν ν a13 −E −E −E ν ν ν a23 −E −E −E 1 1 1 a44 G G G 1 1 1 a55 G G G 1 1 1 a66 G G G Table 2.1: The compliance matrix moduli for the three fundamental tetragonal materials.Table 2.1. The stiffness matrix of tetragonal materials may be deﬁned using the following Qiquantities: E 1−ν 2E E E (1 − ν) Q1 = · , Q2 = , 1 − ν − 2ν 2 E E 1+ν 1 − ν − 2ν 2 E E E ν+ν 2E E Eν Q3 = · , Q4 = , (2.14) 1 − ν − 2ν 2 E E 1+ν 1 − ν − 2ν 2 E Ewhile for the case where the x − y directions are interchangeable, ⎡ ⎤ Q1 Q3 Q4 0 0 0 ⎢ Q1 Q4 0 0 0⎥ ⎢ ⎥ ⎢ Q2 0 0 0⎥ A=⎢ ⎥. (2.15) ⎢ G 0 0⎥ ⎣ ⎦ Sym. G 0 GThe population of the above stiffness matrix for all three types of tetragonal materials is sum-marized in Table 2.2. Note that Qi depend only on the four parameters E, E , ν, and ν , whichtogether with G , G constitute a set of six independent parameters.2.6 Transversely Isotropic (Hexagonal) MaterialsMany natural and man-made materials are classiﬁed as transversely isotropic (transtropic forshort, or hexagonal). This deﬁnition stands for the case when one can ﬁnd a line that allows arotation of the material about it without changing its properties. The plane, which is perpen-dicular to this line (which is in fact an axis of rotational symmetry) is called a plane of elastic
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60 2. Anisotropic Materials Material → TRG6x TRG6y TRG6z Interchangeable Directions → y−z x−z x−y Module ↓ A11 Q2 Q1 Q1 A22 Q1 Q2 Q1 A33 Q1 Q1 Q2 A12 Q4 Q4 Q3 A13 Q4 Q3 Q4 A23 Q3 Q4 Q4 A44 G G G A55 G G G A66 G G G Table 2.2: The stiffness matrix moduli for the three fundamental tetragonal materials.symmetry or plane of isotropy. Hence, each plane that contains the axis of rotation is a planeof symmetry, and therefore, transversely isotropic material admits an inﬁnite number of elasticsymmetries. A modern example for such a material are laminates made of randomly orientedchopped ﬁbers that are in general placed in a certain plane. The effective material propertieshave no profound direction in that plane, which then becomes a plane of elastic symmetry, see(Giurgiutiu and Reifsnider, 1994). Thus, transversely isotropic materials have one isotropic plane, and are deﬁned by ﬁve in-dependent parameters: E, E — Young’s moduli for tension or compression in the plane ofisotropy and in a direction normal to it, respectively, ν — Poisson’s ratio characterizingtransverse contraction in the plane of isotropy when tension is applied in a direction normal toit, respectively, G , G — shear moduli for the plane of isotropy and any plane perpendicularto it, respectively, see (Lekhnitskii, 1981). Note the notion “Young’s modulus” and “Poisson’sratio” are adopted from pure isotropic terminology, and used here as analogy only. Consider for example, the case when the x, y-plane is isotropic, namely, where a rotation ofthe material about the z-axis has no inﬂuence on its properties. The compliance matrix of thistransversely isotropic material is identical to that of the tetragonal material TRG6z, while inaddition, a dependency of G on E and ν should be introduced. The source of this dependencyis the “isotropic relation” A66 = (A11 − A12 )/2 or a66 = 2 (a11 − a12 ) that will be clariﬁed bythe discussion of isotropic materials within S.2.8. Hence, in tetragonal material terminologyfor TRG6z, the above isotropic relation yields G = (Q1 − Q3 )/2, which may be simpliﬁed as E G= . (2.16) 2 (1 + ν)An identical expression is deduced for the other two material versions. Subsequently, to obtainthe three kinds of transversely isotropic materials, TI5x, TI5y, TI5z, from the tetragonal ma-terials TRG6x, TRG6y, TRG6z, respectively, one should introduce ν = ν(G, E) as shown by(2.16) directly into Table 2.1, and into the Qi terms in Table 2.2.
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2.7 Cubic Materials 612.7 Cubic MaterialsA sub-group of the above discussed materials are the so-called cubic materials which are basedon three independent parameters only, and may be deﬁned as orthotropic materials with E11 =E22 = E33 = E, ν12 = ν13 = ν23 = ν, and G12 = G13 = G23 = G. Hence, the compliance andthe stiffness matrices for cubic material may be written as ⎡ ⎤⎡1 ν −E ν −E 0 0 0⎤ E(1−ν) Eν Eν 0 0 0 (1+ν)(1−2ν) (1+ν)(1−2ν) (1+ν)(1−2ν) E ν ⎢ ⎥⎢ 1 −E 0 0 0⎥ ⎢ E(1−ν) Eν 0 0 0⎥⎢ E ⎥ ⎢ (1+ν)(1−2ν) (1+ν)(1−2ν) ⎥⎢ 1 0 0 0⎥ ⎢ ⎥⎢ E ⎥, ⎢ E(1−ν) 0 0 0⎥.(2.17)⎢ 1 0 0⎥ ⎢ (1+ν)(1−2ν) ⎥⎢ G ⎥ ⎢ G 0 0⎥⎣ 1 0⎦ ⎢ ⎥ Sym. G ⎣ Sym. G 0⎦ 1 G GFor further use, these materials will be denoted CUB3, and clearly, they may also be deﬁnedas special transversely isotropic or tetragonal materials with E = E, G = G and ν = ν.2.8 Isotropic MaterialsTo derive the stress-strain relationships for isotropic materials, we make use of the stress andstrain deviator tensors σD = σi j − 3 Θ1 δi j and εD = εi j − 1 Ξ1 δi j . Here we have also invoked ij 1 ij 3the stress and strain ﬁrst linear invariants, namely, the sum of normal stresses Θ1 = σ11 + σ22 +σ33 , see (1.100a), and the sum of normal strains Ξ1 = ε11 + ε22 + ε33 , which is the relativechange in volume for the linear theory, see S.1.1.4.2. The uniqueness of isotropic stress-strain relationships emerges from the fact that for thesematerials, the components of the stress deviator are proportional to the components of thestrain deviator. For the moment, we shall denote this proportion constant (which is clearly amaterial property) as 2G, and therefore σD = 2G εD . Thus, by substituting the above men- ij ijtioned stress and strain deviators one obtains 2G σi j = 2G εi j + (K − )Ξ1 δi j , (2.18) 3where K = Θ1 /3Ξ1 is another independent material property, the bulk modulus. The physicalinterpretation of K is clear as it is the ratio of the mean normal stress (Θ1 /3) to the relativechange in volume (Ξ1 ), and therefore may be viewed as a measure or the material resistanceto a change in volume. Hence, overall, we have now two material independent properties, Kand G, which for the sake of analytical uniformity will be replaced by two other independentconstants, the modulus of elasticity E, and the non-dimensional ratio ν, so that E E K= , G= . (2.19) 3 (1 − 2ν) 2 (1 + ν)For ν = 1 we have K = ∞, which characterizes an incompressible elastic material (i.e. a ma- 2terial that does not change in volume when subjected to tensile or compressive strains). Theabove notation allows one to consider isotropic material as a special orthotropic material withE11 = E22 = E33 = E, ν12 = ν13 = ν23 = ν and G12 = G13 = G23 = 2(1+ν) (see (2.7)), and E
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62 2. Anisotropic Materialstherefore ⎡ ⎤ 1 −ν −ν 0 0 0 ⎢ 1 −ν 0 0 0 ⎥ ⎢ ⎥ 1 ⎢ 1 0 0 0 ⎥ a= ⎢ ⎥, (2.20a) E ⎢ 2 (1 + ν) 0 0 ⎥ ⎣ ⎦ 2 (1 + ν) 0 2 (1 + ν) ⎡ ⎤ 1−ν ν ν 0 0 0 ⎢ 1−ν ν 0 0 0 ⎥ ⎢ ⎥ E ⎢ 1−ν 0 0 0 ⎥ A= ⎢ ⎥ (1 + ν)(1 − 2ν) ⎢ ⎢ 1 2 −ν 0 0 ⎥.⎥ (2.20b) ⎣ 0 ⎦ 2 −ν 1 2 −ν 1The physical interpretation of G as the elastic shear modulus becomes now clear as it relatesthe shear and strain components σi j = 2Gεi j (i j = 12, 13, 23) or σαβ = Gεαβ (αβ = xy, xz, yz).Traditionally, E is referred to as Young’s modulus, and ν as Poisson’s Ratio. Note that forν = 1/2, the r.h.s. of (1.42) vanishes, which again testiﬁes for the material incompressibility. For further use, isotropic materials will be denoted ISO2 which clearly reﬂects the fact thatthey are fully characterized by two independent parameters only.Remark 2.2 It is worth mentioning some classical notation used for isotropic materials. By νEdeﬁning Lam´ ’s constants as µ = G, λ = (1+ν)(1−2ν) , one may write the constitutive relations eusing Kronecker’s symbol δi j as λ 1 σi j = λδi j Ξ1 + 2µ εi j , εi j = − δi j Ξ1 + σi j . (2.21) 2µ 2µ2.9 Engineering Notation of CompositesEngineering notation of (anisotropic) composite materials is usually founded on slightly differ-ent (and approximate) convention. Typically, four parameters, E1 , E2 , ν12 and G12 are deﬁned,where the subscript 1 stands for the “ﬁber” (or the “stiff”) direction, and the subscript 2 standsfor the “perpendicular to the ﬁber” (or the “soft”) direction. In some cases, 1 and 2 are re-placed by L (“Longitudinal”) and T (“Transverse”), respectively. This notation is used mainlyfor unidirectional materials, or for materials that have a profound stiff direction. Such materialsmay be viewed as transversely isotropic materials (that should be deﬁned by ﬁve independentparameters, see S.2.6), and in order to examine these engineering parameters the reader is re-ferred to Fig. 2.3. In this ﬁgure, the material is transversely isotropic where the x, y-plane is anisotropic plane, and the stiff direction is stretched along the z-axis. Hence, in engineering no-tation, the x1 - and x2 - directions coincide with the z- and x- directions, respectively. E1 and E2are the moduli in these directions and therefore, based on (2.12), (2.16), E1 = E and E2 = E.ν12 is the major Poisson’s ratio deﬁned as a contraction in the x-direction due to unit extensionin the z-direction, and therefore ν12 = ν . G12 is the shear modulus in the x1 , x2 -plane, andtherefore, G12 = G . Hence, data regarding G (the shear module in the x, y-plane) or ν (theminor Poisson’s ratio that represents, for example, a contraction in the y-direction due to unitextension in the x-direction) is usually not measured nor reported. Note that only one of theseparameters is required as they are related by (2.16). It should also be noted that for typicaltransversely isotropic materials, the variation of ν will have only negligible inﬂuence on A33
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2.10 Positive-Deﬁnite Stress-Strain Law 63 Figure 2.3: Engineering notation of transversely isotropic material (TI5z).which is the main driver of the structural behavior in slender structures, where unidirectionalmaterials are typically placed (beams, slender plates, etc., see Chapter 5). To show that, wemay expand A33 (Q2 in this case, see Table 2.2 and (2.14)) into Taylor series around ν = 0, andexamine the case of ν < 1 and E/E 1, namely E2 2ν 2 E A33 = [1 + ν + O ν2 ] or lim A33 = E . (2.22) E − 2ν 2E E − 2ν 2 E E →0 EHence, the above underlined term is negligible compared with the unit. Thus, A33 is only aweak function of ν, and lack of data in this case is of minor importance. On the other hand,expanding A12 (Q3 in this case, see Table 2.2 and (2.14)) yields ν 2E 2 E 2 − 2ν 2 EE + 2ν 4 E 2 E A12 = [1 + ν + O ν2 ] or lim A12 = ν . (2.23) E − 2ν 2E Eν 2 (E − 2ν 2 E) E →0 1 − ν2 EExamination of the underlined term under the same arguments shows that A12 is a strong func-tion of ν, and in cases where this module is required and important, knowledge of the value ofν is inevitable.Remark 2.3 The reader should be aware of the fact that unlike the isotropic case, for aniso-tropic materials, the stress principal directions are not identical to the strain principal directions,see S.1.3.3, S.1.3.4. This fact becomes clearer if one examines the normal strain, εN = ε11 , andthe shear strain, εT = ε2 + ε2 , in the stress principal directions 12 13 εN = a11 σP + a12 σP + a13 σP , 1 2 3 (2.24a) 2 2 ε2 T = a51 σP + a52 σP + a53 σP + 1 2 3 a61 σP + a62 σP + a63 σP , 1 2 3 (2.24b)from which follows that in the general case, εT is not zero. However, orthotropic (and simpler)materials do show coincidence of stress and strain principal directions.2.10 Positive-Deﬁnite Stress-Strain LawSo far, the above discussed stress-strain laws were presented as real symmetric 6 × 6 matricesof different types of population. However, energy considerations show that there are certain
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64 2. Anisotropic Materialspositive deﬁniteness constraints on the coefﬁcients in these matrices, which mathematicallymay be put as a system of inequalities. To derive the above inequalities, we should ﬁrst review the expression for W , the volumedensity of the strain energy or the elastic potential that is stored in an elastic body, which isdeformed by external loads. For a linear, anisotropic elastic media, W is put in the followingquadratic form: 1 W = ∑ i, j αi j εi ε j , (2.25) 2where αi j are related to the elements of the stiffness matrix as Ai j = 1 (αi j + α ji ). The above 2expression for W emerges from various constraints. The ﬁrst one is the requirement that W willbe a function of the strain components that vanishes for εi = 0, and thus, a constant value (thatreﬂects a pre-stressed state) should not be incorporated. In addition, W should reﬂect a linearstress-strain relation, namely, 1 2∑j σi = W, εi = (αi j + α ji ) ε j . (2.26)The above shows that the stress-strain matrix is symmetric (i.e. Ai j = (αi j + α ji )/2). Higher-order dependency of W in εi would have caused a nonlinear stress-strain relation. Subsequently,W may also be written as 1 W = ∑ i σi εi . (2.27) 2Equation (2.27) shows that in order to ensure that W will attain positive values for all nonzerostrain states (i.e. energy will not be created), the stress-strain relations matrix should bepositive-deﬁnite. Recall that a matrix, M, is said to be positive-deﬁnite if the inequality z T ·M · z > 0 holds for any nonzero “test vector” z. The actual implementation of a positive-deﬁnitecheck of a matrix may be easily performed via its eigenvalue analysis. Thus, in the general case, there are six inequalities that should be simultaneously fulﬁlled,see also P.2.4. For example, examination of the compliance matrix of a MON13z material, thenumber of inequalities may be reduced by the obvious preliminary assumption that considersall the diagonal terms, aii , as positive (as clearly demanded by the requirement εi /σi > 0). Insuch a case, one obtains an additional four inequalities, the simplest two of which are a11 a22 −a2 > 0 and a44 a55 − a2 > 0. For orthotropic materials (see (2.7)), by considering the stiffness 12 45matrix and assuming positive E11 , E22 , E33 , G23 , G13 , G12 , the stress-strain matrix becomespositive-deﬁnite if E11 > ν2 and D0 > 0, 12 (2.28) E22where D0 is given by (2.9) or (2.11). Simple argumentation of coordinate interchanging revealsthat the ﬁrst inequality of (2.28) may be replaced by one of the following inequalities: E11 E22 > ν2 , 13 > ν2 . 23 (2.29) E33 E33For transversely isotropic materials we assume that E, E and G are all positive. Hence, for thetypical case where the x, y-plane is isotropic, one obtains the inequalities (ν − 1) E + 2ν 2 E <0, ν > −1, which may also be written as E ν2< , |ν| < 1. (2.30) EFor isotropic materials, by assuming positive E, one obtains the conditions (2ν − 1)(ν + 1) <0, ν > −1, which may be solved for ν as 1 −1 < ν < . (2.31) 2
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2.11 Typical Material Characteristics 652.11 Typical Material CharacteristicsThere is a wide range of properties for anisotropic/composite materials, and the tables in thissection are provided for orientation purposes only, while exact values should be carefully ex-tracted from proper sources, see e.g. (Dostal, 1987), (Stuart, 1996), (Akovali, 2001), (Adamset al., 2002). The tables below are for typical properties of unidirectional composites (mainly commer-cial fabrics), and isotropic materials. Yet, it should be noted that characterization of compositematerials encompasses, in addition to the classical strength values, many other considerations,such as speciﬁc strength (i.e. strength to weight ratio), thermal expansion coefﬁcients, dam-age tolerance and fatigue characteristics, production aspects, etc., which are not treated in thepresent context. Anisotropic natural materials are mainly various kinds of rocks and woods. In some cases,naturally laminated rock structures are found, and for the purpose of macro-analysis thesemay be viewed as transversely isotropic materials. In addition, many kinds of woods maybe regarded as reinforced material as they are characterized by a natural lay-up where therelatively stiff ﬁbers are oriented in a certain direction, and may be considered as transverselyisotropic materials as well. Man-made materials are usually produced by various mixture rules of stiff reinforced ﬁberslayered in a relatively soft “matrix”, see e.g. (Mallick, 1993). In such cases, the material prop-erties strongly depend on the volume fracture of each component and on the ratio of the amountof ﬁbers layered in the principal direction and perpendicular to it. When the ﬁbers are placedalong the principal direction only, a pure unidirectional lamina is obtained. In practices, composite materials are typically used as “prepregs”. The engineering term“prepreg” stands for reinforced fabrics that are pre-impregnated with resin and partially cured.Such a material may be used by manufacturers to mold various parts, even without adding anyresin, which allows avoiding of wet lay-up. Table 2.3 presents typical properties of unidirec-tional prepregs, where ρ stands for the laminate density, and V f is the ﬁber volume fraction. Property Graphite/ Boron/ Carbon/ Kevlar/ E-Glass/ S-Glass/ Epoxy Epoxy Epoxy Epoxy Epoxy Epoxy E (“E11 ”) (GPa) 290. 204. 140.–180. 88. 39. 43. E (“E22 ”) (GPa) 6. 20. 9.–10. 5.6 8.6 8.9 G (“G12 ”) (GPa) 5. 9. 4.–7. 2.2 3.8 4.5 ν (“ν12 ”) 0.23 0.23 0.28–0.30 0.34 0.28 0.27 ρ (g/cm3 ) 1.6 2.0 1.5–1.6 1.4 2.1 2.0 Vf 0.57 0.67 0.6–0.7 0.60 0.55 0.50 Table 2.3: Typical properties of unidirectional prepregs. For reference purposes, Table 2.4 provides typical properties of three isotropic metals. Property Aluminum Steel Titanium E (GPa) 73. 210. 110. ν 0.33 0.30 0.30 Table 2.4: Typical properties of isotropic metals.
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66 2. Anisotropic MaterialsRemark 2.4 As shown by the above tables, Poisson’s ratios, ν (or ν ), are typically positive.Rubber-type materials are examples of relatively high Poisson’s ratio that is close to 0.5. Yet,some materials of internal “foam structures” exhibit negative Poisson’s ratios (i.e. expand lat-erally when stretched), see e.g. (Lakes, 1987).2.12 Compliance Matrix TransformationTo create a consistent transformation for the compliance matrix due to spatial rotation, oneshould ﬁrst develop the following fourth-order compliance tensor, C, that plays the role ofthe constitutive relations and consistently connects the strain and stress tensors as (note thatsummation is carried out for identical indices) εab = Cabcd σcd . (2.32)Since the tensor C plays the role of the matrix a in the generalized Hooke’s law discussed inS.2.1, it exhibits triple symmetry, namely Cabcd = Cbacd = Cabdc = Ccdab . (2.33)To support the discussion regarding the transformation of C, two typical strain components,εxx and γxz are examined. We ﬁrst write them in the matrix notation of (2.3) εxx = a11 σxx + a12 σyy + a13 σzz + a14 σyz + a15 σxz + a16 σxy , (2.34a) γxz = a51 σxx + a52 σyy + a53 σzz + a54 σyz + a55 σxz + a56 σxy . (2.34b)Subsequently, we express the same terms in tensor notation. In view of εxx = ε11 we write εxx = C1111 σ11 +C1122 σ22 +C1133 σ33 + 2C1123 σ23 + 2C1113 σ13 + 2C1112 σ12 , (2.35)and in view of γxz = 2ε13 we write γxz = 2C1311 σ11 + 2C1322 σ22 + 2C1333 σ33 + 4C1323 σ23 + 4C1313 σ13 + 4C1312 σ12 . (2.36)The above form is obtained due to the double summation in (2.32). As an example, for c = 2and d = 3, one gets the term Cab23 σ23 , while for c = 3 and d = 2, one gets the term Cab32 σ32 ,which is identical to the previous one. Therefore, it may concluded that C1111 = a11 , 2C1123 =a14 , 2C1311 = a15 , 4C1323 = a45 , etc. To take the above considerations into account, we deﬁnea modiﬁed compliance matrix that connects the vectors ε = [ε11 , ε22 , ε33 , ε23 , ε13 , ε12 ]T andσ = [σ11 , σ22 , σ33 , σ23 , σ13 , σ12 ]T , as ε = C · σ . Hence, the matrix C = {ci j } is given by ⎡ ⎤ a11 a12 a13 a14 /2 a15 /2 a16 /2 ⎢ a22 a23 a24 /2 a25 /2 a26 /2 ⎥ ⎢ ⎥ ⎢ a33 a34 /2 a35 /2 a36 /2 ⎥ C=⎢ ⎥. (2.37) ⎢ a44 /4 a45 /4 a46 /4 ⎥ ⎣ ⎦ a55 /4 a56 /4 a66 /4Analogously to the notation in (1.21), we also establish the following transformation (knownas Voigt’s contracted indicial notation) between pairs of indices, say (i, j) (i, j = 1, 2, 3) and asingle index, say k (k = 1, . . . , 6) (1, 1) → 1, (2, 2) → 2, (3, 3) → 3; (2, 3) → 4, (1, 3) → 5, (1, 2) → 6, (2.38)
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2.12 Compliance Matrix Transformation 67while an (i, j) pair is treated as a ( j, i) pair. Subsequently, we use the above index transfor-mation for the pair of indices (i1 , i2 ) to determine a single index i, and for the pair of indices( j1 , j2 ) to determine a single j, and construct the tensor C out of the matrix C as Ci1 i2 j1 j2 = ci j . (2.39)We now apply a tensor transformation to C based on the transformation tensor T developed inS.1.7.1 and obtain the corresponding tensor in the new system Cklst = Cmnpr Tkm Tln Tsp Ttr . (2.40)Based on the indices notation of (2.38), the modiﬁed compliance matrix c = {ci j } in the newsystem becomes c i j = C i1 i2 j1 j2 . (2.41)Finally, one may construct the desired new compliance matrix a = {ai j } according to the rela-tions shown in (2.37) (namely, the relations between ci j and ai j are identical to those betweenci j and ai j ) ⎡ ⎤ c11 c12 c13 2c14 2c15 2c16 ⎢ c22 c23 2c24 2c25 2c26 ⎥ ⎢ ⎥ ⎢ c33 2c34 2c35 2c36 ⎥ a=⎢ ⎥. (2.42) ⎢ 4c44 4c45 4c46 ⎥ ⎣ ⎦ 4c55 4c56 4c66The expressions for generic rotation that include all three angles are huge, and the reader mayexamine all of them symbolically by activating P.2.5. In what follows, only representativeexamples are given. For that purpose, we shall deﬁne a 21 terms vector aV as aV = [a11 , . . . , a16 , a22 , . . . , a26 , a33 , . . . , a36 , a44 , . . . , a46 , a55 , a56 , a66 ]T , (2.43)while its version after rotation will be denoted analogously as aV = {ai j }. These vectors arerelated by aV = Ma · aV , where Ma is a (non-symmetric) 21 × 21 matrix. We shall now look atthe coefﬁcient of a12 in the expression for a16 , which according to the above notation is givenby Ma (6, 2). To clarify that, we write Ma (6, 2) = 2A1 B1 , where A1 = cos φ cos2 ψ + 2 sin θ sin φ sin ψ cos ψ − cos φ sin2 ψ, B1 = cos ψ cos3 θ sin ψ.As another example, we examine the coefﬁcient of a44 in the expression for a25 , which is givenby Ma (10, 16) = A2 B2C2 , where A2 = sin φ cos θ, B2 = sin φ sin θ sin ψ + cos φ cos ψ, C2 = sin φ sin θ cos ψ − sin ψ cos φ sin2 θ + cos2 θ cos φ sin ψ. (2.44)Again, all 441(= 212 ) terms of Ma may be easily produced by P.2.5, and the reader may createcompliance matrices for speciﬁc materials and rotation angles by activating P.2.6. Due to the symmetry characteristics of the tensor C, see (2.33), there are two quantities thatare invariants under the most general transformation (that includes all three rotation angles φ,θ and ψ). These are deﬁned as (see linear invariants in P.2.5) J1 =Ciklm δik δlm = C1111 +C2222 +C3333 + 2C1122 + 2C1133 + 2C2233 , a (2.45a) J2 =Ciklm δil δkm = C1111 +C2222 +C3333 + 2C1212 + 2C2323 + 2C1313 , a (2.45b)
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68 2. Anisotropic Materialswhere δab is Kronecker’s symbol. In terms of ai j , these invariants become a44 + a55 + a66 J1 = a11 + a22 + a33 + 2(a12 + a13 + a23 ), a J2 = a11 + a22 + a33 + a . (2.46) 2In addition, there are invariants for each direction, namely, quantities that remain unchangedwhen the rotation is carried out about a single axis only. Four of these invariants, which aredenoted Iia , are given in Table 2.5 (may also be produced by P.2.5 by setting any two rotationangles to zero). The role played by those and other invariants will be discussed later on withinS.2.15. As indicated, the explicit expressions for ai j of GEN21 materials are lengthy. Yet, for Rotation Axis → x y z Invariant ↓ a I1 a33 + a22 + 2a32 a11 + a33 + 2a13 a11 + a22 + 2a12 a I2 a44 − 4a32 a55 − 4a13 a66 − 4a12 a I3 a66 + a55 a44 + a66 a44 + a55 a I4 a13 + a21 a12 + a23 a13 + a23 A I1 A33 + A22 + 2A32 A11 + A33 + 2A13 A11 + A22 + 2A12 A I2 A44 − A32 A55 − A13 A66 − A12 A I3 A66 + A55 A44 + A66 A44 + A55 A I4 A13 + A21 A12 + A23 A13 + A23 Table 2.5: Invariants for a single axis transformation.orthotropic materials one may use P.2.7, which yields, for example, 1 1 a11 = cos4 θ sin4 ψ + (A sin2 θ + B cos2 θ cos2 ψ) cos2 θ sin2 ψ + sin4 θ (2.47) E22 E33 1 + C cos2 θ cos2 ψ sin2 θ + cos4 θ cos4 ψ, E11which as expected, does not depend on φ, while ν23 1 ν12 1 ν13 1 A = −2 + , B = −2 + , C = −2 + . (2.48) E22 G23 E11 G12 E11 G13The general transformation invariants in this case take the form ν12 ν13 ν23 1 1 1 J1 = −2 a −2 −2 + + + , (2.49a) E11 E11 E22 E22 E33 E11 1 1 1 1 1 1 J2 = a + + + + + . (2.49b) E22 E33 2G13 2G23 E11 2G12P.2.8 provides two kinds of visualization techniques. First, as shown in Fig. 2.4, each termof the compliance matrix may be plotted over the ψ, θ-plane, while for the terms that arefunctions of φ, the latter should be ﬁxed. In addition, spherical plots of each ai j term may beof interest. To do that, we use the results of Remark 1.12 and replace ψ by θ s and θ by φ s − π , 2respectively (where θ s and φ s are spherical angles, see Fig. 1.20(b)), and set a value for φ when
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2.13 Stiffness Matrix Transformation 690.12 0.1 –0.0050.08 –0.010.06 –0.0150.04 –0.020.02 0 –0.025 0 0.4 –0.03 0.4 1.6 1.4 0.8 1.6 1.4 0.8 1.2 1 theta 1.2 1 0.8 0.6 1.2 0.8 0.6 1.2 theta psi 0.4 0.2 1.6 psi 0.4 0.2 0 0 1.6 (a) a11 . (b) a13 for φ = 0. Figure 2.4: Two terms of the compliance matrix over the ψ − θ plane (×109 1/GPa).required. We then create a surface where each point of it represents an orientation of the x-axisof the transformed system (by connecting the origin with it), and the ai j under discussion aredirectly proportional to the distance that is measured along the x-axis between the origin andthe surface. Examples of two terms of the compliance matrix are given in Fig. 2.5. 0.04 0.02z 0 0.01 –0.04 z 0 0.1 –0.01 0.05 0 0.1 –0.02 y 0.05 –0.05 0 0.01 0.02 x 0.01 –0.1 –0.1 –0.05 y 0 0 x –0.01 –0.02 (a) a11 . (b) a13 for φ = 0. Figure 2.5: Spherical plot of two terms of the compliance matrix (×109 1/GPa).2.13 Stiffness Matrix TransformationThe derivation of the stiffness matrix transformation, A, see (2.4), is similar to the one pre-sented in S.2.12 for the compliance matrix, but involves some very important modiﬁcationsdue to the different matrix deﬁnitions in this case. Again, we ﬁrst write two typical stress-strain relations using the stiffness matrix σxx = A11 εxx + A12 εyy + A13 εzz + A14 εyz + A15 εxz + A16 εxy , (2.50a) σxz = A51 εxx + A52 εyy + A53 εzz + A54 εyz + A55 εxz + A56 εxy . (2.50b)In tensor notation, the stiffness tensor is deﬁned as σab = Sabcd εcd , (2.51)
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70 2. Anisotropic Materialswhere S is a fourth-order tensor with triple symmetry, Sabcd = Sbacd = Sabdc = Sbadc . (2.52)In view of σxx = σ11 , σxz = σ13 , the above two representative relations may be written as σxx = S1111 ε11 + S1122 ε22 + S1133 ε33 + 2S1123 ε13 + 2S1113 ε13 + 2S1112 ε12 , (2.53a) σxz = S1311 ε11 + S1322 ε22 + S1333 ε33 + 2S1323 ε23 + 2S1313 ε13 + 2S1312 ε12 . (2.53b)Similar to the previous case of the compliance matrix, due to the double summation, someterms appear twice (for example, for c = 2, d = 3 one gets the term Sab23 ε23 and for c = 3,d = 2 one gets the term Sab32 ε32 , which are identical). Noting that ε13 = γxz /2, εxx = ε11 ,etc., one may conclude that S1111 = A11 , S1123 = A14 , S1311 = A51 , S1323 = A54 , etc. Hence,in this case, a modiﬁed stiffness matrix is not required, and using the index transformationof (2.38), we determine the tensor S as Si1 i2 j1 j2 = Ai j , while the corresponding stress tensortransformation yields Sklst = Smnpr Tkm Tln Tsp Ttr . (2.54)The modiﬁed compliance matrix, A, in the new coordinate system is extracted from S, asAi j = Si1 i2 j1 j2 . To present some examples in this case, we may deﬁne, analogously to (2.43), twovectors AV and AV , and a matrix MA . It turns out that the coefﬁcient of A12 in the expressionfor A16 is given by MA (6, 2) = Ma (6, 2)/2, while the coefﬁcient of A44 in the expression for A25is MA (10, 16) = 2Ma (10, 16) (see S.2.12). In general, it is interesting to note that MA (i, j) =Da (i, j)Ma (i, j) where the elements of the matrix Da (i, j) may attain only ﬁve discrete real A Avalues (which are not functions of the rotation angles), 1 1 Da (i, j) ∈ { , , 1, 2, 4}. A (2.55) 4 2The reader may examine all the terms of A using the symbolic results of P.2.9 (see also P.2.10),and produce stiffness matrices for speciﬁc materials and rotation angles by P.2.11. While gen-eral invariants for the stiffness transformation matrix will be dealt with later on, the single axistransformation invariants for the present case are denoted IiA and appear in Table 2.5, see P.2.9.2.14 Compliance and Stiffness Matrix Transformation to Curvilinear CoordinatesTo transform the constitutive relations from Cartesian coordinates into orthogonal curvilinearcoordinates, one may use the techniques presented in S.2.12, S.2.13, but replace the transfor-mation matrix T with the one given by (1.216). In the general case, the resulting complianceand stiffness matrices become functions of the α1 , α2 and α3 coordinates. Such a case mayalso be termed as a case of non-uniform constitutive relations, i.e. where the material proper-ties depend on the coordinates. P.2.12 executes transformations of a compliance matrix given in Cartesian coordinates intoits form in various orthogonal curvilinear coordinates.2.15 Principal Directions of AnisotropyIn this section we shall develop the analytical background required for the determination of theprincipal directions of anisotropy. These directions are useful for general material classiﬁca-
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2.15 Principal Directions of Anisotropy 71tion, and in particular for the task of comparing anisotropic materials. Conceptually, the follow-ing discussion is of “local nature” and deals with the “local principal directions of anisotropy”,i.e. at a given point. To illustrate the main issue under discussion, suppose that two GEN21 anisotropic materialsgiven by two sets of 21 independent coefﬁcients, need to be compared. Clearly, two materialsare identical if all their 21 coefﬁcients are equal, see also Remark 2.5, but there is a possibility,that two materials that apparently seem different, differ only by rigid body rotation (i.e. a rota-tion that may depend on one, two or three Euler angles). To clarify and study this possibility,there is a need to deﬁne unique canonical principal directions of anisotropy, in which thereis only one way to present a given material regardless of its orientation with respect to thecoordinate system. The common criteria for deﬁning the principal directions of anisotropy isbased on the following statement: A coordinate system is said to coincide with the principaldirections of anisotropy if the material, when subjected to “all-around uniform pure extensionstate”, forms a “pure tension state”. To develop the required transformation of a given stiffness matrix to the principal directionsof anisotropy, we ﬁrst introduce the symmetric bulk modulus tensor ⎡ ⎤ K11 K12 K13 ⎢ ⎥ K = ⎢ K12 K22 K23 ⎥ . ⎣ ⎦ (2.56) K13 K23 K33K is one of two tensors that may be obtained by contraction of the stiffness tensor S, and itselements are written as Ki j = ∑ k=1 Si jkk = ∑ k=1 Amk , 3 3 (2.57)where the index m corresponds to the pair (i, j) by the index transformation (2.38). We may now examine the special case of strain principal directions and write the state of“all-round uniform extension” as εi j = εδi j (i, j = 1, 2, 3), where ε is a reference (constant)strain. According to (2.57), (2.2:b), the stress components are given by σi j = Ki j ε. Hence,the principal directions of the tensor K coincide with the stress principal directions for thecase under discussion, which as stated above, constitute the principal directions of anisotropy.Quantitatively, in these principal directions, σi j = KiP εδi j , where {KiP } are the three eigenval-ues of the tensor K. Extracting the rotation angles out of the eigenvalue analysis of the bulkmodulus tensor K is identical to that discussed for the eigenvalue analysis of the stress tensorσ — see Example 1.1 of S.1.3.3.1. Strictly speaking, to ensure a canonical (unique) principal system of anisotropy, the eigen-values of K should always be put in a certain order before calculating the resulting transforma-tion matrix. In the present case we select the order K3 ≥ K2 ≥ K1 and for the moment assume P P Pthat all eigenvalues of K are different (cases with multiple eigenvalues are discussed in Re-mark 2.6). This order force the material to orient its “strongest” direction in the z-direction andthe “weakest” one in the x-direction. In such a case, the transformation is completely uniquesince the eigenvalue analysis itself is unique. To illustrate the above concept, suppose that we deal with an orthotropic material that hasbeen rotated so its stiffness matrix is populated as MON13z material (see Fig. 2.8), and we wishto restore the material principal directions of anisotropy, in which A16 = A26 = A36 = A45 = 0.The condition K12 = 0 only guarantees that A16 + A26 + A36 = 0, see (2.57) (also note thatK13 = K23 = 0 in this case). However, since the transformation is unique, there is no more thanone solution to the equation K12 = 0, and thus, this solution must yield a rotation that willbring the material back to its orthotropic population. Furthermore, in this case we employ theangle ψ only. The relevant terms in the transformed system may be written as functions of the
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72 2. Anisotropic Materialscoefﬁcients in the original system as A16 = −A26 sin4 ψ + (A22 − 2 A66 − A12 ) cos ψ sin3 ψ + 3 (A26 − A16 ) cos2 ψ sin2 ψ + (2 A66 − A11 + A12 ) cos3 ψ sin ψ + A16 cos4 ψ, A26 = −A16 sin4 ψ + (2 A66 − A11 + A12 ) cos ψ sin3 ψ + 3 (A16 − A26 ) cos2 ψ sin2 ψ + (A22 − 2 A66 − A12 ) cos3 ψ sin ψ + A26 cos4 ψ, A36 = A36 cos 2ψ+(A23 −A13 ) cos ψ sin ψ, A45 = A45 cos 2ψ+(A44 −A55 ) cos ψ sin ψ. (2.58)Thus, by substitution of the expressions of (2.58) in K12 = A16 + A26 + A36 and utilization ofelementary trigonometric identities, one obtains 1 K12 = (A16 + A26 + A36 ) cos 2ψ + (A22 − A11 − A13 + A23 ) sin 2ψ, (2.59) 2and the condition K12 (ψ) = 0 yields 1 2(A36 + A26 + A16 ) ψ= arctan . (2.60) 2 A13 − A23 − A22 + A11Hence, the coordinate system should be rotated about the z-axis in the amount given abovein order to convert the coordinate system into its principal directions of anisotropy, see alsoRemark 2.7. Note that a generic MON13z material will remain of MON13z-type despite thefulﬁllment of the condition K12 = 0. As already indicated, due to its symmetry properties, the fourth-order stiffness tensor S hasanother (in addition to K) symmetric second-order contracted tensor which is given by ⎡ ⎤ L11 L12 L13 ⎢ ⎥ L = ⎢ L21 L22 L23 ⎥ , ⎣ ⎦ (2.61) L31 L32 L33where Li j = ∑ k=1 Sik jk = ∑ k=1 Amn , 3 3 (2.62)and again, the indices m and n correspond to the pairs (i, k) and ( j, k), respectively, by the indextransformation of (2.38). K and L, and hence their six invariants may be used for “fast” comparisons between two setsof stiffness matrices. Identical invariants is of course a necessary (but not sufﬁcient) conditionfor declaring identity between two matrices. The invariants may be expressed (analogously tothe stress tensor invariants) by their tensor trace and determinant as 1 I1 = tr K, K I2 = K (tr K)2 − tr K2 , I3 = det K, K (2.63) 2while those of L are obtained analogously. Hence IiK , IiL (i = 1, 2, 3), are the six invariants ofthe stiffness matrix, while explicit expressions for the ﬁrst two linear invariants are I1 = A11 +A22 +A33 +2(A12 +A13 +A23 ), K I1 = A11 +A22 +A33 +2(A44 +A55 + A66 ). (2.64) LNote that in the isotropic case the bulk modulus, see S.2.8, is K = 9 I1 or 1 I2 /I1 or I3 /I2 . 1 K 3 K K K KThe above invariants could also be obtained by replacing C with S in (2.45a,b), see also P.2.9.
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2.15 Principal Directions of Anisotropy 73Remark 2.5 Strictly speaking, two materials behave in an identical manner if the expressionfor W , the volume density of the strain energy or the elastic potential, that is stored in the elasticbody as expressed by the strain component (see S.2.10) is identical for both materials. In viewof (2.25), this requirement is equivalent to equality of all their 21 elastic components.Example 2.1 Principal Directions of Anisotropy. The illustrative transformation of the stiffness matrix to its principal directions of anisotropy,shown in Fig. 2.6(a), may be produced by P.2.13. The K tensor eigenvalues in this case are0.225 × 1011 , 0.234 × 1011 and 1.386 × 1011 , and the resulting transformation matrix is (notethat each line of T is an eigenvector of K) ⎡ ⎤ −0.612 −0.148 −0.777 ⎢ ⎥ T = ⎢ −0.354 −0.828 0.436 ⎥ . ⎣ ⎦ (2.65) −0.707 0.542 0.455As demonstrated by Fig. 2.6(b), and although not seen from Fig. 2.6(a), the original matrixhas three planes of elastic symmetry since its population is identical to that of an orthotropic(ORT9) material. 120 40 100 30 80 20 60 1 10 2 40 0 2 3 20–10 4 row–20 4 row 0 5 1 2 6 1 2 6 3 4 3 4 5 6 5 6 column column (a) A generic stiffness matrix. (b) Canonical-principal system. Figure 2.6: A stiffness matrix and its image in a canonical-principal coordinate system (GPa).Remark 2.6 When the material poses a state of multiple eigenvalues of K, the canonical trans-formation might not be unique, since there are an inﬁnite number of sets of principal directionsof anisotropy (trivial example for that are isotropic materials for which K is diagonal with equalelements Kii = E/(1 − 2ν), but there may be other special cases of stiffness matrices of multi-ple eigenvalues of K). In most cases, multiple eigenvalues indicate equivalence of stiffness intwo or more directions and pose no difﬁculty. It should be noted that such situations are not “stable” in the sense that even a slightly smallchange in the numerical value of the material characteristics will destroy this type of symmetryand yield different eigenvalues of K.Remark 2.7 In addition to K12 discussed above, the remaining non-diagonal terms of K, K13 = A15 + A25 + A35 , K23 = A14 + A24 + A34 , (2.66)enable us to determine the following additional single angle transformations: 1 2(A25 + A15 + A35 ) 1 2(A14 + A34 + A24 )θ= arctan , φ= arctan . (2.67) 2 A33 − A11 + A23 − A12 2 A12 − A13 − A33 + A22
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74 2. Anisotropic MaterialsAnalogously to (2.60), the above may be used to determine rotations required to set to zero therespective terms of the bulk modulus tensor, and are valid for single angle rotation only.2.16 Planes of Elastic SymmetryOne of the interesting characteristics of anisotropic materials is the existence and the number ofplanes of elastic symmetry created by their micro-structure. This notion may be examined fromboth mathematical and physical points of view. Similar to the case of local principal directionsof anisotropy, planes of elastic symmetry are deﬁned locally, i.e. at a given point. The transformation techniques of S.2.15 will be adopted for the task of proving the existenceof a symmetry of a given material with respect to a given (or presumed) plane. From its consti-tutive relations point of view, a material is said to be symmetric with respect to the xi −x j plane,if its stiffness (or compliance) matrix remains unchanged (invariant) under a transformation, inwhich the sign of the xk -direction (k = i, j) is changed. For example, a material that remains unaltered by the coordinate transformation matrix ⎡ ⎤ 1 0 0 ⎢ ⎥ Txy = ⎢ 0 1 0 ⎥ ⎣ ⎦ (2.68) 0 0 −1is symmetric with respect to the x, y-plane, and an analogous technique may be used for otherplanes (including those that are not perpendicular to one of the coordinate lines). Another mathematical point of view is based on the transformation techniques discussed inS.2.13, which clarify the fact that a stiffness matrix transformation is not sensitive to a changeof sign of the transformation matrix, see (2.54). Hence, the above Txy could be replaced withT xy = −Txy , which is the transformation matrix obtained for ψ = π and θ = φ = 0 (the sameis of course true for the compliance matrix). From a physical point of view, the above transfor-mation shows that one may rotate the material by 180◦ about an axis that is perpendicular toa plane of elastic symmetry without changing the material characteristics such as the transfor-mation of the x − y system into the x − y system shown in Fig. 2.7. This kind of symmetry isusually termed a “symmetry of the second-order” (as a symmetry of the nth -order is the casewhere a rotation of 2π/n does not alter the material).Figure 2.7: An example of a beam with x, y-plane of elastic symmetry. Rotating the coordinate systemby 180◦ about the z-axis does not alter the stiffness and compliance matrices. As far as the number of planes of elastic symmetry is concerned, for a given GEN21 aniso-tropic material (see S.2.1), we may set the following question: are there one or more planes thatcould be virtually drawn inside the material with respect to which the material is symmetric ?
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2.16 Planes of Elastic Symmetry 75In principle, one may adopt the transformation techniques of S.2.15 to rotate the system ofcoordinates to a presumed new spatial state, and then check if one of the new coordinate planesis a plane of elastic symmetry. In the general case, where the existence and number of the planesof elastic symmetry is not known, this process becomes quite tedious. A much more efﬁcientway is offered in what follows. The reasoning of the following method is founded on the fact that a transformation to theprincipal system of anisotropy ensures that materials with one plane of elastic symmetry willbe transformed into one of the monoclinic forms, and materials with three orthogonal planesof elastic symmetry will be transformed into the orthotropic form, see S.2.15. Hence, to evaluate the existence and number of the planes of elastic symmetry, we ﬁrst rotatethe material to its principal directions of anisotropy xA , yP , zP . Once the stiffness matrix in the P A Aprincipal directions of anisotropy is obtained, one may examine the matrix population, andcompare it to that of orthotropic and (the three fundamental kinds of) monoclinic materials asshown in Fig. 2.8. This comparison indicates that if the matrix is populated as the MON13xmaterial, the yP − zP plane is an elastic symmetry plane. Similarly, if the matrix is populated A Aas the MON13y or MON13z materials, the planes of elastic symmetry are xA − zP or xA − yP , P A P Arespectively. If the matrix is populated as the ORT9 material, all three yA P − zP , xP − yP and A A AxA − zP planes are elastic symmetry planes. From the above, it is clear that no case of (only) P Atwo planes of elastic symmetry exists. Figure 2.8: Complience/Stiffness matrix populations of orthotropic and monoclinic materials. We may now summarize our ﬁndings regarding the number of planes of elastic symmetry invarious material types in a form of (Ni , Np ) pairs where Ni stands for the minimal number ofnonzero independent elastic moduli, and Np stands for the maximal number of planes of sym-metry. For that purpose, we ﬁrst observe all materials in their principal directions of anisotropy,see S.2.15. We subsequently denote the case of general anisotropic material as (18, 0) since itmay be rotated by three Euler angles that will zero out (up to) three of its elastic moduli. Similarly, monoclinic material will be denoted (12, 1) as it may be rotated by one angle tozero out (at most) one of its elastic moduli (see also Remark 2.8). Orthotropic material will bedenoted as (9, 3), and tetragonal materials as (6, ∞) (both may not be rotated to zero out one ormore elastic moduli). Heuristically, and without a proof, it is interesting to note that all of theabove cases obey a simple rule given by 18 − Ni Np = (18 ≥ Ni ≥ 6). (2.69) Ni − 6All cases of Ni < 6, such as the transversely isotropic materials (5, ∞), cubic materials (3, ∞),and the isotropic materials (2, ∞), yield inﬁnite N p as well. Equation (2.69) is presented inFig. 2.9, which generally illustrates the increasing number of planes of symmetry with thedecreasing number of independent unknowns used to characterize the material. It should be noted again that from the (2.69) and Fig. 2.9 point of view, Np stands for themaximal number of planes of elastic symmetry of each material type, since clearly, for a givenNi , examples of materials with fewer planes of elastic symmetry may be easily created.
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76 2. Anisotropic Materials Np 3 1 0 Ni 23 56 9 12 18 Figure 2.9: Number of symmetry planes vs. number of independent unknowns.Remark 2.8 In rare and special cases, the procedure discussed in this section may mislead byits conclusion regarding the number of planes of elastic symmetry. An example is the stiffnessmatrix populated as ORT9 in addition to A16 = A, A26 = −A, while A = 0 and A11 = A22 ,A13 = A23 , A44 = A55 . In such a case, the criteria for principal directions are fulﬁlled and (2.59)shows that no rotation is required (and the material seems to be of MON13z-type that has onlyone plane of elastic symmetry). Yet, ψ may be selected so that both A16 and A26 vanish (whichwill yield an ORT9 material that has three planes of elastic symmetry). Such cases do not poseany difﬁculty as they are not “stable” in the sense discussed in Remark 2.6.2.17 Non-Cartesian AnisotropyIn S.2.1, we have presented the generalized Hook’s law for the case where the strain and thestress components are described in Cartesian coordinates and for material that poses Cartesiananisotropy. Yet, one also ought to deal with cases where the material directions are not parallel(or perpendicular) to the axes of a Cartesian system. Such cases are generally referred to asa non-Cartesian anisotropy. A common example for a non-Cartesian anisotropy is the so-called cylindrical anisotropy, where two of the material axes coincide with the tangent and thenormal to concentric circles in a given plane, and the third one coincides with the directionperpendicular to that plane. In composite material terminology, this is the type of material lay-up produced by ﬁlament winding of a circular shell, and it may also be viewed locally as athin layer of Cartesian anisotropy materials that are laid over the surface of a prismatic circulartube. To study the case of cylindrical anisotropy, we note that α1 = r (= ρ), α2 = θ c , α3 = z (seeExample 1.9) and the corresponding Lam´ ’s parameters are H1 = 1, H2 = r, H3 = 1. Thus, T eof (1.216) becomes a function of θ c only, ⎡ ⎤ cos θ c sin θ c 0 T = ⎣ − sin θ c cos θ c 0 ⎦ . (2.70) 0 0 1We now deﬁne the cylindrical anisotropy as the case where at each point, the material hasthe properties of a reference material of Cartesian anisotropy so that its coordinate lines x, yand z coincide with the local r, θ c and z directions, respectively — see Fig. 2.10. Hence, thematerial properties become independent of θ c , and we therefore conclude that the stiffness orcompliance matrices of a material of cylindrical anisotropy are constants. By selecting θ c =0, T becomes a unit matrix and the above matrices become identical to those of Cartesiananisotropy. To demonstrate that, consider Fig. 2.11(a) that presents a cylinder with orthotropic
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2.17 Non-Cartesian Anisotropy 77 Figure 2.10: A thin “material strip” of (a) Cartesian anisotropy, (b) cylindrical anisotropy. Figure 2.11: Circular domains of Cartesian, (a), (c), and cylindrical, (b), (d), anisotropy.laminae having the compliance matrix, a, of (2.7). When these laminae are wrapped over acircle, see Fig. 2.11(b), a material of cylindrical anisotropy is obtained and may be written as ⎡ 1 ν ν ⎤ ⎧ ⎫ − E21 − E31 0 0 0 ⎧ ⎫ ⎪ ⎪εr ⎪ ⎢ Eν12 11 22 ν 33 ⎥ ⎪ σr ⎪ ⎪ ⎪ ⎢−E ⎪ 1 − E32 0 0 0 ⎥ ⎪ σθ ⎪ ⎪ ⎪ ⎪ εθ ⎪ ⎢ ν 11 ⎪ ⎪ E22 33 ⎥⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ − 13 ν − E23 1 ⎥ ⎨ σz ⎬ εz 0 0 0 = ⎢ E11 22 E33 ⎥ ⎥ ⎪ τθz ⎪ . (2.71) ⎪ γ θz ⎪ ⎢ 0 ⎪ ⎪ ⎢ 0 0 1 0 0 ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎪ γ rz ⎪ G23 ⎥ ⎪ τrz ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 ⎩ ⎭ 0 0 0 1 0 ⎦⎪ ⎪ ⎩ ⎭ G13 γ rθ 1 τrθ 0 0 0 0 0 G12Similarly, when the above orthotropic material is rotated by an angle θz and yields a MON13zas shown in Fig. 2.11(c), one may visualize placing the laminae as shown in Fig. 2.11(d) toform the following constitutive relations: ⎧ ⎫ ⎡ ⎤⎧ ⎫ ⎪ εr ⎪ ⎪ ⎪ a11 a12 a13 0 0 a16 ⎪ σr ⎪ ⎪ ⎪ ⎪ εθ ⎪ ⎢ ⎪ ⎪ a22 a23 0 0 a26 ⎥ ⎪ σθ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎨ ⎬ ⎥⎪ ⎪ ⎨ ⎬ εz ⎢ a33 0 0 a36 ⎥ σz =⎢ ⎥ . (2.72) ⎪ γ θz ⎪ ⎢ ⎪ ⎪ ⎣ a44 a45 0 ⎥ ⎪ τθz ⎪ ⎪ ⎪ ⎪ γ rz ⎪ ⎦⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ Sym. a55 0 ⎪ τrz ⎪ ⎪ ⎪ ⎩ ⎭ γ rθ a66 τrθSee further use of cylindrical anisotropy in S.7.5 and Example 10.1. Along the same lines,other cases of non-Cartesian anisotropy may be deﬁned. To form a material with a sphericalanisotropy we examine the transformation matrix T for a spherical coordinate system (see
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78 2. Anisotropic Materials(1.206), Remark 1.12 and Example 1.9), which is a function of θ s and φ s , namely, ⎡ ⎤ sin (φ s ) cos (θ s ) sin (θ s ) sin (φ s ) cos (φ s ) ⎢ ⎥ T=⎢ ⎣ − sin (θ s ) cos (θ s ) 0 ⎥. ⎦ (2.73) cos (φ s ) cos (θ s ) sin (θ s ) cos (φ s ) − sin (φ s )Here, we deﬁne a material with a spherical anisotropy, as the case where at each point, thematerial has the properties of a reference material of Cartesian anisotropy so that its coordinatelines x, y and z are taken to coincide with the local r, θ s and −φ s directions, respectively. Hence, the transformation becomes independent of θ s and φ s in this case as well, and T maybe taken as the constant matrix that corresponds to θ s = 0, φ s = π . 2 As indicated in S.1.1, S.1.7, the description of the strain measures and the stress componentsin generic curvilinear coordinates might have a tremendous inﬂuence on the complexity ofthe analytical solution. Typically, for isotropic materials, speciﬁc curvilinear coordinates areselected in a way that simpliﬁes the boundary conditions. Yet, as shown in S.2.17, the directionsof anisotropy might have a crucial inﬂuence on the resulting constitutive relations. To clarify the above, consider a given homogeneous domain (of uniform elastic proper-ties), which is described by uniform (constant) constitutive relations in Cartesian coordinates.When the same domain is described by another coordinate system, the constitutive relationsbecome coordinate-dependent. As examples, a homogeneous domain of Cartesian anisotropywill be described as a coordinate-dependent material in cylindrical coordinates, and homoge-neous domain of cylindrical anisotropy will be described as a coordinate-dependent materialin Cartesian coordinates. This phenomena may cause considerable complexity that should beconsidered against the simpliﬁcation gained by using the curvilinear coordinates. Figure 2.12: A rectangle of cylindrical anisotropy and a circle of Cartesian anisotropy. As a result of the above discussed phenomena, in most cases, the selection of the coordinatesystem in anisotropic elasticity is determined by the material properties solely and not by theboundary shape. For example, Fig. 2.12 shows a rectangular domain of cylindrical anisotropy(a “wood-type” problem), which should be analyzed by cylindrical coordinates to prevent anon-uniform constitutive relation, and a circular domain of Cartesian anisotropy that should beanalyzed using Cartesian coordinates for the same reason. It is therefore clear that employing Cartesian coordinates for Cartesian anisotropy is idealfor “rectangular” domains, and likewise, employing cylindrical coordinates for cylindricalanisotropy is ideal for tubes and shells (provided that the origin of the cylindrical anisotropycoincides with the origin of the cylindrical coordinates, in which the problem is solved, see alsoS.7.5 and Fig. 7.14). In the above cases, the presentation of both the material and the boundaryconditions is “natural”.
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3Plane Deformation AnalysisThis chapter reviews various types of plane deformation analyses encountered in diversiﬁedproblems within the area of anisotropic elasticity, and presents them in a uniform mathematicalformat. The associated boundary value problems (BVPs) are classiﬁed and reduced to certaintypes, which are commonly expressed using the generalized Laplace, the biharmonic and thethird-order differential operators. Needless to say, any kind of a three-dimensional problem reduction into a two-dimensionalone is accompanied by appropriate approximation level, which reﬂects a neglect of some com-ponents of the strain and/or stress tensors. The problems derived in this chapter are classiﬁedaccording to the components that are neglected in each case. Table 3.1 presents a summary ofthe discussed cases where neglected components are denoted by “0” (while ∼ 0 stands for acase where consistent formulation yields a zero value, but a suitable approach may yield anestimation of this small quantity). Plane Plane Plane Coupled Plates under Strain Stress Shear plane problem Bending εx , εy , γxy σx , σy , τxy 0 εz 0 0 0 γyz , γxz , τyz , τxz 0 0 ∼0 σz 0 0 0 Material MON13z MON13z MON13z GEN21 MON13z Derived in S.3.2.1 S.3.2.2 S.3.3 S.3.4 S.3.5.2 Table 3.1: Classiﬁcation of plane problems. One should realize that various problems encountered in the mechanics of anisotropic elas-ticity may pose identical (or similar) BVP. As an example, it will be shown that the threeproblems of plane-strain, plane-stress and bending of plates are mathematically identical asthey require a solution of a biharmonic BVP of the same type.
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80 3. Plane Deformation Analysis For each BVP, we shall be focused mainly on the necessary existence conditions associatedwith its mathematical presentation correctness, while assuming that their sufﬁciency (similarto the solution uniqueness) emerges from the mechanical sense of the (linear) problems.3.1 Plane Domain Deﬁnition and Contour Parametrization3.1.1 Plane Domain TopologyFigure 3.1(a) presents a simply connected plane domain, Ω, its circumference ∂Ω, and thecircumferential arc-length coordinate, s. The entire circumference length is denoted by L, andtherefore L ≥ s ≥ 0 (a domain, Ω, is said to be simply connected, if any closed curve insideit may be shrunk into a point by means of continuous deformation during which it alwaysremains in Ω. Otherwise, Ω is said to be multiply connected). (a) Simply connected. (b) Multiply connected and (c) Laminated. non-homogeneous. Figure 3.1: Plane domain geometry notation. A non-homogeneous domain consists of several simply connected domains of different “ma-terials”, Ω[1] , . . . , Ω[N] , see Fig. 3.1(b), while the exterior domain, Ω[0] , serves as a “surroundingmaterial”: Ω = N Ω[ j] . The boundary contours of the domains are denoted by ∂Ω j , while j=0the contour ∂Ω0 contains all those with j > 0, see Fig. 3.1(b). Topologies where more closedboundaries encircle other closed boundaries are also possible (see e.g. Fig. 3.2). Multiply con-nected domains are similar in essence, and may be generally described by Fig. 3.1(b) wherethe material of one (or more) of the domains Ω[1] , . . . , Ω[N] is removed. Many applications pose non-homogeneous laminated domains, and so many analyses arefocused on modern composites. In such cases, Ω is divided into N > 1 (rectangular) domainsof different materials Ω[ j] ( j = 1, . . . , N) as shown in Fig. 3.1(c), and hence, Ω[0] = 0. / In both Figs. 3.1(b),(c), the boundaries of the domains Ω[ j] ( j = 1, . . . , N) are separatedinto the “dividing curves” and the “free curves”. The dividing curves between two differentdomains of materials Ω[i] and Ω[ j] are denoted by ∂Ωi j (or ∂Ω j i ). The “free” curves of eachdomain (i.e. the curves that are free of contact with other domains) are denoted by ∂Ω j . Asshown, there are cases where the free curve of a domain consists of different segments. Yet,∂Ω j serves as a common indication for all these free segments of the Ω[ j] domain. The above general topologies may be formatted as shown in Fig. 3.2. In these symmetrictables, we indicate on the diagonal the number of free surface segments of each domain, while
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3.1 Plane Domain Deﬁnition and Contour Parametrization 81 Figure 3.2: Topology tables of two non-homogeneous cross-sections: (a) Generic, (b) Laminated.for the off diagonal terms, “x” stands for a mutual dividing curve between the “row region” andthe “column region”. The overall number of boundary contours in a given domain is thereforeequal to the sum of the diagonal terms and the number of “x” signs over the upper (or lower)triangles of these topology tables. A few words should be devoted to the notation convention that applies in what follows. Asuperscript in square brackets, say H[ j] , indicates that the value of H is calculated for (or istaken for) the material of domain Ω[ j] . In what follows, the reader will also ﬁnd functionsof x, y that contain characteristics (such as elastic moduli, or generalized harmonic functions)that appear without domain index. The value of these characteristics is determined locallyaccording to the domain in the (x, y) location under discussion. For future purposes (see S.3.2.6, S.3.3.5, S.3.4, Chapter 8, etc.) we deﬁne for a generalquantity H, [ j] H[i] = H[ j] − H[i] . (3.1)Hence, the above equation may be interpreted as subtraction of H that has been calculated fordomain Ω[i] from H that has been calculated for domain Ω[ j] .Remark 3.1 The nature of the non-homogeneity presented above may be deﬁned as piecewiseconstant non-homogeneity since it basically assumes that each region is homogeneous withinitself and material properties “jump” between regions. Such a non-homogeneity dominatesmany applications of man-made materials and structures, Yet, non-homogeneity may be alsodeﬁned by materials with continuously varying elastic moduli. Only a few examples will bedevoted to such cases (see e.g. Remark 8.3). The underlying assumption that remains valid for all two-dimensional analyses and all sub-sequent beam analyses, is that all sub-domains involved are perfectly bonded such that noslippage occurs, and all in-plane and out-of-plane displacements at the interfaces are continu-ous.3.1.2 Contour Parametrization and Directional CosinesMany solution methods require parametrizations of plane region contours. The contour of asimply connected shape (see e.g. Fig. 3.1(a)) is a simple closed curve, that may be generallydeﬁned by periodic functions of parameter t as x = x(t), y = y(t), t ∈ [0, T ]. (3.2)
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82 3. Plane Deformation AnalysisDeriving the functions x(t) and y(t) for general shapes of the domain Ω may be carried out invarious ways as will be described shortly. P.3.1 enables the examination of some prescribedparametrization functions. Once the parametrization functions are determined, the arc-lengthfunction, s(t), is derived by the following integration: t dx 2 dy s(t) = ( ) + ( )2 dt. (3.3) 0 dt dtFrom an analytical point of view, the above integral may impose a relatively difﬁcult task.Therefore, ways to minimize the participation of an explicit s(t) expression will be exploredwithin the solution methodologies described further on in Chapter 4. Note that the above parametrization technique may be also applied to any other non-Carte-sian coordinates, where αi of S.1.7.2 are deﬁned as functions of t. Figure 3.3: Directional cosine notation: α ≡ (n, x), β ≡ (n, y). ¯ ¯ To deﬁne the directional cosines, we ﬁrst examine Figs. 3.3(a),(b), where the x, y coordi- ¯nates, the normal to the contour, n, and the circumferential direction, s, are shown. Accord-ingly, the angle cosines between the normal to the contour and the x- and y- axes, see alsoFig. 3.3(c), are given by dy dx dx dy cos(n, x) = ¯ = , cos(n, y) = − ¯ = (3.4) ds dn ds dnand the following identity holds: cos(n, x)2 + cos(n, y)2 = 1. ¯ ¯ (3.5)When the domain contour is presented as a C1 -regular function y(x), see Fig. 3.3(d), cos(n, x) = ¯−√ dx/dt 2 , and cos(n, y) = √ ¯ 1 2 . Note that signs should be changed if the direction 1+(dy/dt) 1+(dy/dt)of s in Fig. 3.3(d) is reversed.Example 3.1 Directional Cosines in an Ellipse. For the case of an ellipse of semi-axes a ≥ b > 0, one may adopt the geometrical anglet ∈ [0, 2π] as the contour’s parameter and write x = a cos t, y = b sin t. Hence the cosines of
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3.1 Plane Domain Deﬁnition and Contour Parametrization 83the angles between the normal to the ellipse and the x- and y- axes are given as (see (3.4)) b bx a ay cos(n, x) = ¯ cos t = , cos(n, y) = ¯ sin t = , (3.6) λ λa λ λb b2 2 2where λ = ds dt = a2 sin2 t + b2 cos2 t = a2 x + a2 y2 b is the “parametrization velocity”.Remark 3.2 The above deﬁnitions are valuable for common integral evaluations, which arefounded on Green’s Theorem, Ω (N, x + M, y ) = ∂Ω N cos(n, x) + M cos(n, y), as ¯ ¯ Λ, y = Λ cos(n, y), ¯ Λ, x = Λ cos(n, x), ¯ (3.7) Ω ∂Ω Ω ∂Ωor the corollaries when Λ = 1, Λ = x or Λ = y {1, x, y} cos(n, x) = {0, SΩ , 0}, ¯ {1, x, y} cos(n, y) = {0, 0, SΩ }. ¯ (3.8) ∂Ω ∂Ω For further use, we shall document here some integral identities that contain the functionΛ of a constant value Λ0 over a contour ∂Ω. Using identities (x Λ), x = Λ + x Λ, x , (y Λ), y =Λ + y Λ, y and (3.7), (3.8) we obtain {x, y}Λ, y = {0, SΩ Λ0 − Λ}, (3.9a) Ω Ω {x, y}Λ, x = {SΩ Λ0 − Λ, 0}. (3.9b) Ω Ω3.1.3 Parametrization by Conformal MappingThe conformal mapping technique is a powerful analytical tool that may be invoked for contourparametrization. Within√ approach, we look at each point on the contour as a complex thisnumber z = x + iy (i = −1), and employ the appropriate conformal mapping to transform adomain of a unit circle to the simple connected domain under discussion. The reader should also be aware of the fact that unlike the classical techniques (see e.g.(Sokolnikoff, 1983)), we exploit the conformal mapping only for deriving the parametrizationfunctions x(t) and y(t), and hence, within the present context, we care for the contour pointsonly. Transformations of a unit circle to various simply connected domains are widely discussed inthe literature. P.3.1 enables the examination of some prescribed w(z) transformation functions. The Schwarz-Christoffel transformation methodology may be further invoked to map a unitcircle exterior onto the exterior of a general polygon of n vertices, see (Muskhelishvili, 1953).By denoting the complex location at the origin plane, where the unit circle is deﬁned, as ζ( = ξ + iη), this transformation is given by z 1 ζ2 ∏ k=1 (ζ − Zk )αk , n w(z) = A P(ζ) dζ + B, P(ζ) = n ≥ 3, |ζ| ≥ 1, (3.10) 1where the polygon vertices wk correspond to the points Zk , k = 1, . . . , n on the unit circle. Theinterior angles at the polygon vertices, ϕk , are determined by the parameters αk ∈ (−1, 1) sothat ϕk = π(1 − αk ). The complex constants A and B represent “stretching” and “rigid” rotationand translation of the domain. In the simple case of a regular n-polygon, αk = 2 , one may select nZk = exp( 2π(k−1) i), and the transformation (3.10) is written as, see Example 3.2, n z 2 dζ wn = A (1 − ζn ) n + B, n ≥ 3. (3.11) 1 ζ2
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84 3. Plane Deformation AnalysisFor a generic n-vertices polygon, the sum of internal angles is given by ∑k ϕk = π(n − 2),and therefore ∑n αk = 2. Hence, P(ζ) of (3.10) may be written as P(ζ) = ∏n (1 − Zk )αk . k=1 k=1 ζUsing the product of a binomial z-series (1 − Zk )αk = 1 − αk Zk + · · · , one may write P(ζ) = ζ ζ1 − 1 ∑n αk Zk + · · · (i.e. a series expansion about inﬁnity). To ensure that w(z) is a single- ζ k=1valued function on the unit circle exterior (i.e. contains no logarithmic singularity at the origin),one must require vanishing of the 1 coefﬁcient in this series, i.e., ζ ∑ k=1 αk Zk = 0. n (3.12)Obviously, the position of a point, Zk , on a unit circle is deﬁned by the value of its central angle(i.e. a scalar). Therefore, for a given n-vertices polygon one needs to determine n such centralangles and the complex constants A and B, i.e. a total of n + 4 scalars. These have to satisfythe following n + 4 equations/conditions: given location of one vertix, given orientation of oneedge, n − 1 given edge lengths, and (3.12). This constitutes an implicit procedure that may bederived in a closed form only for a few simple cases. Since in the general case, the integrals of (3.10) may not be carried out analytically in aclosed form, we expand them into Laurent-Taylor power series and practically use only a trun- kpcated version of it, namely, w ≈ C0 + ∑n=1 Cn ζn . Here, Cn are known constants (Ck p = 0), ζand k p > 0 will be referred to as the parametrization level. The parametrization of an ellipse inExample 3.2 is therefore characterized by k p = 1, as the corresponding conformal mapping isw = a−b 1 + a+b ζ . The image of the unit circle |ζ| = 1 by such truncated parametrization is a 2 ζ 2simple closed curve that approximates the polygon, given accurately by (3.10).Example 3.2 Regular n-Polygon Parametrization. To clarify the discussion in this section regarding a regular n-polygon parametrization, threetransformations, for a triangle, w3 (z), a square, w4 (z), and a pentagon w5 (z), will be presentedhere. These, and additional results for any regular polygon may be derived by activating P.3.2.The transformations map the circumference of a unit circle (i.e., |z| = 1) directly onto thedesired contour and are given using (3.11) as 2 1 2 z 1 − ζ3 3 z 1 − ζ4 2 z 1 − ζ5 5 w3 = − dζ , w4 = − dζ , w5 = − dζ . (3.13) 1 ζ2 1 ζ2 1 ζ2By expanding each one of (3.13) into a series, we get 1 z2 z5 1 z3 z7 1 z4 z9 w3 ≈ − − − +···, w4 ≈ − − − +···, w5 ≈ − − − + · · · .(3.14) z 3 45 z 6 56 z 10 75While transforming a unit circle circumference, we usually assign the parameter t ∈ [0, 2π]to be the central angle of this circle, namely, z = cos t + i sin t, and thus, the contours wn (t) =xn (t) + iyn (t) are approximated by polynomials of cos t and sin t of order k p . Note that thereare minimal values of k p > 1 for each conﬁguration (except an ellipse with k p = 1) that makethe truncated series valid. These are k p = 2 for a triangle, k p = 3 for a square and k p = 4 for apentagon, etc. For a triangle, a square and a pentagon, xn (t) and yn (t), are approximated as 1 2 2 x3 (t) ≈ − cos t − cos2 t, y3 (t) ≈ sin t − cos t sin t, 3 3 3 1 2 7 2 x4 (t) ≈ − cos t − cos3 t, y4 (t) ≈ sin t − cos2 t sin t, (3.15) 2 3 6 3 1 4 4 2 4 x5 (t) ≈ − cos t − cos4 t + cos2 t, y5 (t) ≈ sin t + cos t sin t − cos3 t sin t. 10 5 5 5 5
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3.1 Plane Domain Deﬁnition and Contour Parametrization 85One may easily convert the above expressions into Fourier series form as 1 1 x3 (t) ≈ − cos t − cos(2t), y3 (t) ≈ sin t − sin(2t), 3 3 1 1 x4 (t) ≈ − cos t − cos(3t), y4 (t) ≈ sin t − sin(3t), (3.16) 6 6 1 1 x5 (t) ≈ − cos t − cos(4t), y5 (t) ≈ sin t − sin(4t). 10 10P.3.3 produces generic polygon parametrizations. Figure 3.4 presents an example with verticesZi = [1, .306 + .952i, −.807 + .590i, −.508 − .862i, .187 − .982i] where the corresponding an-gles are ϕi = [98◦ , 118◦ , 88◦ , 128◦ , 108◦ ]. Z[3] 1 1 Z[2] 0.5 0.5 0 1 2 3 4 5 6 –0.5 0.5 1 t –0.5 –0.5 Z[1] Z[4] –1 –1 Z[5] (b) The parametrization functions x(t) (thin) (a) The polygon. and y(t) (thick). Figure 3.4: Parametrization by conformal mapping of a generic pentagon.3.1.4 Parametrization by Piecewise Linear FunctionsAn alternative and very robust method to parameterize an n–vertices polygon by trigonometricpolynomials is founded on expressing the functions x(t) and y(t) in a piecewise linear manner.Within this method, 0 ≤ t ≤ 2π is assumed, and the parametrization functions are written as ⎧ ⎧ ⎪ a1 t + b1 , ⎨ t ∈ [0,t1 ], ⎪ c1 t + d1 , ⎨ t ∈ [0,t1 ], a2 t + b2 , t ∈ [t1 ,t2 ], c2 t + d2 , t ∈ [t1 ,t2 ], x(t) = y(t) = (3.17) ⎪ ⎩ ··· ⎪ ⎩ ··· an t + bn , t ∈ [tn−1 , 2π], cn t + dn , t ∈ [tn−1 , 2π].Two trails to employ the above piecewise linear functions x(t) and y(t) may be applied. The ﬁrst one (denoted “Method A”) uses a non-dimensional parameter deﬁned as t = 2π s , lwhere l is the circumferential length. In this case, the integration of (3.3) is not required, andthe “length” devoted in terms of t for each edge is proportional to its actual length, namely,∆t = 2π ∆l s . Another way (denoted “Method B”) is based on dividing the range of the parameter 0 ≤ t ≤2π into n equal segments for a generic polygon of n edges, so that each segment, regardless ofits length, occupies an equal “length” in terms of t, namely, ∆t = 2π . n
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86 3. Plane Deformation Analysis If necessary, a Fourier series expansion is then carried out for x(t) and y(t) separately, andtheir truncated versions, up to k p harmonics, are obtained as the parametrization functions. P.3.4 describes various polygon parametrizations by the above two methods, see also Exam-ple 3.3. Method B has some advantages over Method A when a polygon with large edge lengthdiversity is under discussion. Qualitatively, in such a case, when Method A is employed, thefunctions x(t) and y(t) contain sharp turning points and abrupt changes that result in “Gibbs-type” phenomena. These phenomena are much less sharp when Method B is used, see Exam-ple 3.3.Example 3.3 Generic (Non-Convex) Polygon Parametrization. By activating P.3.4, a series of parametrizations for a non-convex polygon were obtained,see Fig. 3.5(a). The thin and thick lines represent the (actual) piecewise and the trigonometric 1 5 5 5 2 3 4 5 0 4 4 4 –1 3 3 3 –2 2 2 2 1 1 1 –3 0 0.5 1 1.5 2 0 0 –4 0.5 1 1.5 2 0.5 1 1.5 2 (a) k p = 4. (b) k p = 3. (c) k p = 5. (d) k p = 7.Figure 3.5: Parametrization by a piecewise linear and trigonometric functions (Method B). (a): a genericpentagon, (b)–(d): A “C” domain approximated by various parametrization levels.(approximate) curves, respectively. For illustration, the constant and the ﬁrst four harmonics ofy(t) of Fig. 3.5(a) are written as y ∼ − 1.800 − 1.225 cos t + 0.802 cos(2t) − 0.356 cos(3t) + 0.0766 cos(4t) = + 0.793 sin t − 0.622 sin(2t) − 0.276 sin(3t) + 0.0496 sin(4t) + · · · . (3.18)In addition, Figs. 3.5(b)–(d) show a series of approximations of a “C”–domain (with eightvertices). To demonstrate the difference between Methods A and B, Fig. 3.6 presents a slender triangleand the quality of approximation obtained by k p = 3 for its x(t) function.3.2 Plane-Strain and Plane-StressThere are plenty of physical situations, where complete three-dimensional considerations arenot necessary, and various kinds of two-dimensional reductions of the problem are (fully orpartially) acceptable. The fundamental two-dimensional states are widely known as “Plane-Strain” and the “Plane-Stress”. Each of the above cases leads to a different analysis, which isformulated by a different version of the biharmonic BVP. We shall ﬁrst consider the plane-
103.
3.2 Plane-Strain and Plane-Stress 87 1 1 9 6 –3 –2 –1 1 2 3 –3 –2 –1 1 2 3 3 t t –1 1 –1 –1 (a) (b) Method A. (c) Method B. Figure 3.6: The function x(t) for the slender triangle shown in (a) as approximated by Methods A, B.strain case, since, in contrast to only an approximate reduction in the plane-stress case, it doesconstitute a complete and exact presentation of a special three-dimensional state. To support the following discussion, it is convenient to express the generalized Hook’s law(GHL) for a linear material (see S.2.1) as ε = Aε · σ + Bε · σ, ε = Cε · σ + Dε · σ, (3.19)where ε = [εx , εy , γxy ], ε = [εz , γyz , γxz ], σ = [σx , σy , τxy ], σ = [σz , τyz , τxz ]. The involved matri-ces are ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ a11 a12 a16 a13 a14 a15 a33 a34 a35 Aε = ⎣ a12 a22 a26 ⎦ , Bε = ⎣ a23 a24 a25 ⎦ , Dε = ⎣ a34 a44 a45 ⎦ , (3.20) a16 a26 a66 a36 a46 a56 a35 a45 a55and Cε = BT . Similarly, the GHL may be inverted and expressed by the stiffness matrix as ε σ = Aσ · ε + Bσ · ε, σ = Cσ · ε + Dσ · ε, (3.21)where ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ A11 A12 A16 A13 A14 A15 A33 A34 A35 Aσ = ⎣ A12 A22 A26 ⎦ , Bσ = ⎣ A23 A24 A25 ⎦ , Dσ = ⎣ A34 A44 A45 ⎦ , (3.22) A16 A26 A66 A36 A46 A56 A35 A45 A55and Cσ = Bσ T . Note that the matrices Aε , Dε , Aσ , Dσ are symmetric.3.2.1 Plane-StrainThe approximation known as “Plane-Strain” is founded on the assumption that the elastic bodydeformation is conﬁned to the plane deformation components, while all strain components thatinclude indices of the direction normal to the plane vanish. This simulates well the state inthe inner sections of “long” elastic bodies, such as prismatic bodies that undergo uniformloading along their outer surface, see Figs. 3.7(a),(b). In such a case, each cross-section maybe analyzed under the assumption that the displacements u, v are functions of x, y (only) andw = 0. Therefore, the associated strain components are εx = u, x , εy = v, y , εz = γyz = γxz = 0, γxy = u, y + v, x , (3.23)
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88 3. Plane Deformation AnalysisFigure 3.7: A slender cylindrical body in a state of Plane-Strain. The arrows in (b) represent the surfaceloading components.or, according to (3.19) and (3.21), ε = 0, σ = −D−1 · Cε · σ, ε ε = (Aε − Bε · D−1 · Cε ) σ, ε (3.24a) σ = Cσ ε, ε = A−1 · σ . σ (3.24b)We shall now restrict the discussion to the MON13z-type material (see S.2.3), where a14 =a15 = a24 = a25 = a34 = a35 = a46 = a56 = 0, and similarly, A14 = A15 = A24 = A25 = A34 =A35 = A46 = A56 = 0. Then, the matrices Bε , Dε and Bσ , Dσ of (3.20) become ⎡ ⎤ ⎡ ⎤ a13 0 0 a33 0 0 Bε = ⎣ a23 0 0 ⎦ , Dε = ⎣ 0 a44 a45 ⎦ , (3.25a) a36 0 0 0 a45 a55 ⎡ ⎤ ⎡ ⎤ A13 0 0 A33 0 0 Bσ = ⎣ A23 0 0 ⎦ , Dσ = ⎣ 0 A44 A45 ⎦ . (3.25b) A36 0 0 0 A45 A55The stress component σz may be presented by two forms, using (3.24a) or (3.24b), respec-tively, 1 σz = − (a13 σx + a23 σy + a36 τxy ) , (3.26a) a33 σz = A13 εx + A23 εy + A36 γxy . (3.26b)In both cases, τyz = τxz = 0. Equations (3.24a,b) also show that ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎨ εx ⎬ b11 b12 b16 ⎨ σx ⎬ ε = b·σ ⇔ εy = ⎣ b12 b22 b26 ⎦ · σy . (3.27) ⎩ ⎭ ⎩ ⎭ γxy b16 b26 b66 τxyHere the symmetric matrix b = {bi j } (i, j = 1, 2, 6) is deﬁned by its coefﬁcients ai3 a j3 bi j = ai j − , i, j = 1, 2, 6 (3.28) a33referred to as the reduced compliance elastic coefﬁcients. The inverse relation may be writtenas σ = Aσ · ε, see (3.21), and hence, Aσ = b−1 . At this stage it is convenient to introduce the classical Airy’s stress function, Φ(x, y), by itsderivatives as σx = Φ, yy , σy = Φ, xx , τxy = −Φ, xy . (3.29)
105.
3.2 Plane-Strain and Plane-Stress 89Reviewing the equilibrium equations (1.82a–c), and the compatibility equations (1.45a–f),shows that the deﬁnitions and assumptions described above identically satisfy all the abovetwo sets of equations (for vanishing body forces the inﬂuence of which will be discussed lateron), with the exception of the compatibility equation (1.45a). In terms of the stress function Φand using (3.27), (3.29), the above compatibility equation turns to be the following homoge-neous biharmonic equation: (4) ∇1 Φ = 0, (3.30) (4)where the generalized biharmonic operator ∇1 is given by (4) ∂4 ∂4 ∂4 ∂4 ∂4 ∇1 = b22 − 2b26 3 + (2b12 + b66 ) 2 2 − 2b16 + b11 4 . (3.31) ∂x4 ∂x ∂y ∂x ∂y ∂x∂y3 ∂yIn this operator notation, the superscript stands for the order of derivatives in the operator, andthe subscript stands for its version number, which depend on the material and assumptionsinvolved (see also Example 1.5). Throughout the derivation, we shall make use of additional (4)versions of the biharmonic operator ∇n documented in S.3.6.2. Correspondingly, Φ, is said tobe a generalized biharmonic function, see e.g. (Lekhnitskii, 1981), (Lurie and Vasiliev, 1995). For orthotropic materials, b16 = b26 = 0, since a16 = a26 = a36 = 0, and (3.31) is simpliﬁed (4)to ∇2 given by (3.203). For this case, (3.26a) yields 1 σz = − (a13 σx + a23 σy ) . (3.32) a33In the isotropic case (where E stands for Young’s modulus and ν for Poisson’s ratio, see S.2.8),one may substitute in (3.28) the values 1 ν 2 (1 + ν) a11 = a22 = a33 = , a12 = a13 = a23 = − , a66 = , (3.33) E E Eto obtain 1 − ν2 1+ν 2 (1 + ν) b11 = b22 = , b12 = −ν , b66 = (3.34) E E E(also note that in this case, 2 a12 + a66 = 2 a11 and 2 b12 + b66 = 2 b11 ). Thus, the operator 2(3.31) is further streamlined to 1−ν ∇(4) , where we use the classical biharmonic operator of E(3.205), while σz of (3.32) becomes σz = ν (σx + σy ) . (3.35)Remark 3.3 As shown by (3.26a), (3.32), (3.35), to create a state of plane-strain in a “long”cylindrical body as shown in Fig. 3.7(a) (where cross-sections are parallel to the x, y-plane),a speciﬁc distribution of σz must be maintained throughout the body and over the end cross-sections. Equivalently to this requirement one may supply an axial force, Pz , and two transversemoments Mx and My at the cylinder ends, see Fig. 3.7(c), which are given by {Pz , Mx , My } = σz { 1, y, −x}. (3.36) ΩAccording to St. Venant’s Principle (see S.5.1.3.1), the exact distribution of σz over the endsis immaterial, since the difference between the applied distribution and the one described by(3.26a) (for the general case), will constitute a self-equilibrated system the inﬂuence of whichwill diminish with the distance from the ends.
106.
90 3. Plane Deformation Analysis To eliminate the above need to supply tip resultants, the literature (see e.g. (Lekhnitskii,1981)) offers an additional solution, which is superimposed on the discussed one. In this ad-ditional solution, we introduce a stress component σz = ∆ σ(x, y) only. Clearly, such state ofstress satisﬁes both the equilibrium and the compatibility equations. When the origin of thecoordinate system is placed at the cross-section centroid, ∆ σ(x, y) may be selected based on M(3.36) as ∆ σ = − SΩ − Mxx y + Iyy x, where Pz I {SΩ , Ix , Iy } = {1, y2 , x2 } (3.37) Ωare the cross-section area and the corresponding moments (of inertia) about the x- and y- axes. Indeed, a superposition of the above solution on the previously mentioned solution for plane-strain, eliminates the required end loads since (σz + ∆ σ){ 1, y, −x} = {0, 0, 0}, (3.38) Ωand again, by invoking the St. Venant’s principle, one may argue that the effect of the σz + ∆ σdistribution and the exact one (σz = 0 at the tip) will decay far from the end cross-sections.However, one should note that according to (3.19), (3.25a), the above additional solution in-duces for the MON13z material under discussion the additional strain components ∆ εx = a13 ∆ σ , ∆ εy = a23 ∆ σ , ∆ εz = a33 ∆ σ , ∆ γxy = a36 ∆ σ . (3.39)These strain components will induce additional plane deformation (which may be easily ob-tained by the procedure described in S.1.2 and P.1.5). In particular, it should be noted that εzdoes not vanish anymore. Hence, the above superimposed solution replaces the residual axialstress with a residual axial strain, see also Example 3.5.3.2.2 Plane-StressThe state of “Plane-Stress” is suitable for analyzing thin plates that are subjected to in-planeloading, see Fig. 3.8, and will be further discussed within the laminated plates theory of S.3.5.1.The notion “thin” means that the (constant) thickness of the plate, h, is much smaller than itstypical in-plane dimension — see Fig. 3.8, where the x, y-plane coincides with the plate’s mid-dle plane. In this kind of analysis, we assume that all stress components that include the z indexvanish, and therefore, set the assumption σ = 0 (namely, σz = τyz = τxz = 0) in (3.19), (3.25a),and consider σx , σy , and τxy as functions of x and y only. We carry out an analysis that is anal-ogous to the plane-strain case, while replacing σi with εi and ai j with Ai j . Hence, for generalMON13z material (see S.2.3 ), γyz = γxz = 0 and εz = a13 σx + a23 σy + a36 τxy , (3.40)or εz = − A1 (A13 εx + A23 εy + A36 γxy ). According to the above described analogy, 33 ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎨ σx ⎬ B11 B12 B16 ⎨ εx ⎬ σ = B·ε ⇔ σy = ⎣ B12 B22 B26 ⎦ · εy , (3.41) ⎩ ⎭ ⎩ ⎭ τxy B16 B26 B66 γxywhere the symmetric matrix B = {Bi j }, i, j = 1, 2, 3 is deﬁned by its components Ai3 A j3 Bi j = Ai j − , (3.42) A33
107.
3.2 Plane-Strain and Plane-Stress 91Figure 3.8: A thin plate in a state of plane-stress. The arrows represent the in-plane loading components.referred as the reduced stiffness elastic coefﬁcients. Since B−1 = Aε , the inverse relation isgiven by ε = Aε · σ. (3.43)Employing Airy ’s stress function for the present case, see (3.29), shows that the equilibriumequations are satisﬁed, and the compatibility equation (1.45a) is fulﬁlled provided that (4) ∇3 Φ = 0, (3.44)which constitute the general biharmonic operator deﬁnition for plane-stress (4) ∂4 ∂4 ∂4 ∂4 ∂4 ∇3 = a22 − 2a26 3 + (2a12 + a66 ) 2 2 − 2a16 + a11 4 . (3.45) ∂x4 ∂x ∂y ∂x ∂y ∂x∂y3 ∂yAs already indicated, the state of plane-stress is only an approximate reduction of the generalthree-dimensional formulation. To show that, one should notice that the above derivation leavesthree unsatisﬁed compatibility equations (1.45b,c,f), namely, εz, yy = εz, xx = εz, xy = 0. Hence,unless εz (see (3.40)) is linear, i.e. may be written as εz = κ1 x + κ2 y+ ε0 , the above plane-stressanalysis is inaccurate. For orthotropic materials, a16 = a26 = a36 = 0, and (3.40) becomes εz = a13 σx + a23 σy , (3.46) (4)and the biharmonic operator of (3.45) is simpliﬁed to ∇4 given by (3.207). For the isotropic case the elastic moduli are reduced to (1 − ν)E νE E A11 = A22 = A33 = , A12 = A13 = A23 = , A66 = , (3.47) (1+ν)(1−2ν) (1+ν)(1−2ν) 2(1+ν)and therefore E νE E B11 = B22 = , B12 = , B66 = . (3.48) 1 − ν2 1 − ν2 2 (1 + ν)In addition ν εz = − (σx + σy ), (3.49) Eand the operator of (3.45) is reduced to E ∇(4) , where ∇(4) is the classical biharmonic operator 1(3.205).
108.
92 3. Plane Deformation Analysis3.2.3 Illustrative Examples of Prescribed Airy’s FunctionAs a ﬁrst study of a homogeneous body plane deformation, one may use a technique in whichAiry ’s stress function, Φ, is prescribed. This enables an initial and simple examination of thestress function and the resulting stress distribution properties. For that purpose, we expand Φas a homogeneous polynomial of a given order kΦ ≥ 2, as Φ(kΦ ) = ∑ i=0 Bi, kΦ −i xi ykΦ −i , Φk (3.50)which yields an expression with kΦ + 1 undetermined coefﬁcients. Since we are dealing withthe biharmonic equation where stresses are obtained by second-order derivatives of Φ(kΦ ) ,values of kΦ < 2 are meaningless. Superposition of homogeneous polynomials of different kΦdegree is clearly possible as shown further on in S.4.3. Substitution of (3.50) in one of the above discussed versions of the biharmonic equation, (4) (kΦ )∇n Φ = 0 , yields an equation where a homogeneous polynomial of order k = max(0, kΦ −4) must vanish, namely ∑k Di, k−i xi yk−i = 0, where the coefﬁcients Di,k−i are linear func- i=0tions of Bi, kΦ −i and the elastic moduli. According to a known algebraic lemma, to satisfy theabove equation identically for all x and y, one has to set Di, k−i = 0 (0 ≤ i ≤ k). For kΦ > 3 thisconstitutes a system of kΦ − 3 linear equations that reﬂects dependencies between the coefﬁ-cients Bi,kΦ −i (0 ≤ i ≤ kΦ ). For kΦ ≤ 3 no equations should be solved. Hence, for any value ofkΦ ≥ 3 there are four undetermined coefﬁcients in (3.50), while for kΦ = 2 there are only three.Example 3.4 Prescribed Stress Function on a Rectangle. Illustrative application of a prescribed stress function may be created by activating P.3.5,which for kΦ = 4 (i.e., k = 0) yields Φ(4) = B04 y4 +B13 xy3 +B22 x2 y2 +B31 x3 y+B40 x4 . Using,for example, the biharmonic operator for plane-strain, (3.31), the only condition in this caseturns to be D00 = 24 b22 B40 − 12 b26 B31 + 4 (2 b12 + b66 ) B22 − 12 b16 B13 + 24 b11 B04 = 0, (3.51)which is a single equation with ﬁve unknown coefﬁcients. Thus, one may arbitrarily choosethe four coefﬁcients B31 , B22 , B13 and B04 . By eliminating B40 , the biharmonic function maybe written as b11 4 b16 4 2 b12+b66 4 b26 4 3 Φ(4)=(y4− x )B04+( x +xy3 )B13+(x2 y2− x )B22+( x +x y)B31 . (3.52) b22 2b22 6b22 2b22The corresponding stress components (3.29) are homogeneous polynomials of order 2: σx = 12 B04 y2 + 6 B13 xy + 2 B22 x2 , 12b11 6b16 2 b12 + b66 6b26 σy = (− B04 + B13 − 2 B22 + B31 )x2 + 6 B31 xy + 2 B22 y2 , b22 b22 b22 b22 τxy = −3 B13 y2 − 4 B22 xy − 3 B31 x2 . (3.53)According to (3.43), the strain components are also homogeneous polynomials of order 2.The displacement components u(x, y), v(x, y) in this case may be obtained by the integrationprocedure of S.1.2 and P.1.6 as third-order homogeneous polynomials. To further illustrate this example we activate P.3.6 for kΦ = 4, and examine the rectangulardomain |x| ≤ 1, |y| ≤ 1. We select all Bi j to vanish except for, B22 = 1. Figure 3.9 presents theabove stress function and the resulting deformation ﬁeld for the MON13z material obtained bya typical Graphite/Epoxy orthotropic material, which is rotated by 30◦ about the z-axis.
109.
3.2 Plane-Strain and Plane-Stress 93 1 0.6 0.5 0.4 0.2 0 0–0.2 –0.5 1 0.5 –1 0 –0.5 x 0 y –1 –0.5 0.5 –1 –1 –0.5 0 0.5 1 (a) Φ(x, y). (b) Displacement ﬁeld. Figure 3.9: Stress function and displacement vector ﬁeld over a rectangle (Example 3.4).3.2.4 The Inﬂuence of Body ForcesAs shown in previous sections, the plane-strain and plane-stress analyses yield different kindsof homogeneous biharmonic ﬁeld equations. These results will be extended in this section toaccount for a generic distribution of body forces, Xb (x, y), Yb (x, y). For that purpose, we replacethe stresses of (3.29) by σx = Φ, yy +U 1 , σy = Φ, xx +U 2 , τxy = −Φ, xy , (3.54)which contains two separate potential functions U 1 (x, y) and U 2 (x, y) so that Xb = −U 1, x , Yb = −U 2, y . (3.55)It is a simple task to verify that the equilibrium equations are satisﬁed in this case as well. How-ever, the compatibility equation (1.45a) (with the new strain expressions obtained by substitut-ing (3.54) in (3.27)) may be written in terms of the stress function, Φ, as the non-homogeneousbiharmonic equation (4) ∇1 Φ = F0 (x, y), (3.56)where F0 = −b12U 1,xx + b16U 1,xy − b11U 1,yy − b22U 2,xx + b26U 2,xy − b12U 2,yy . (3.57)Equation (3.56) replaces (3.30) in the case of plane-strain. Similarly, the above may apply to(3.44) for the plane-stress, which yields (4) ∇3 Φ = F0 (x, y), (3.58)where F0 is identical to the one given by (3.57) except for bi j that are replaced by ai j . In theisotropic case, (3.56) takes the form (see (3.205)) ν ∇(4) Φ = (U 1,xx +U 2,yy ) − (U 1,yy +U 2,xx ), (3.59) 1−ν
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94 3. Plane Deformation Analysiswhile (3.58) becomes ∇(4) Φ = ν(U 1,xx +U 2,yy ) − (U 1,yy +U 2,xx ). (3.60)When both Xb , Yb emerge from the same potential, U (e.g. the case of gravity loads), the r.h.s. of(3.59), (3.60) are simpliﬁed to 2ν−1 ∇(2)U for plane-strain and (ν − 1) ∇(2)U for plane-stress, 1−νwhere ∇(2) is the simplest Laplace’s operator, see (3.198).3.2.5 Boundary and Single-Value ConditionsLet Ω be a plane domain with boundary ∂Ω, along which a circumferential coordinate s isdeﬁned, see Fig. 3.10. The domain boundary is subjected to a distribution of forces per unit Figure 3.10: Notation for a simply connected domain.length Xs and Ys in the x- and y- directions, respectively. Hence cos(n, z) = 0, and the boundary ¯conditions (1.85a,b) may be written as σx cos(n, x) + τxy cos(n, y) = Xs , ¯ ¯ τxy cos(n, x) + σy cos(n, y) = Ys , ¯ ¯ (3.61)which with the aid of (3.4) become dy dx dy dx σx − τxy = Xs , τxy − σy = Ys . (3.62) ds ds ds dsTo enable further handling of the above boundary conditions, we shall also use the followinggeneral identities for C2 -differentiable functions: d dx dy Φ, y = Φ, yx + Φ, yy = Φ, yy cos(n, x) − Φ, yx cos(n, y), ¯ ¯ (3.63a) ds ds ds d dx dy Φ, x = Φ, xx + Φ, xy = Φ, xy cos(n, x) − Φ, xx cos(n, y). ¯ ¯ (3.63b) ds ds dsBy utilizing the above, one may express the boundary conditions (3.61) using (3.54) as d {Φ,x , Φ,y } = {−F1 , F2 } on ∂Ω, (3.64) dswhere the boundary functions, F1 (x, y) and F2 (x, y) are deﬁned on ∂Ω as dx dy F1 = Ys +U 2 = Ys −U 2 cos(n, y), ¯ F2 = Xs −U 1 = Xs −U 1 cos(n, x). ¯ (3.65) ds ds
111.
3.2 Plane-Strain and Plane-Stress 95There are certain conditions that the functions F1 and F2 should satisfy. In what follows theseconditions will be derived independently from a pure mathematical point of view while physi-cal argumentation will follow. The mathematical approach is founded on the requirement thatΦ and its ﬁrst partial derivatives will be single-valued, namely, d {Φ, x , Φ, y , Φ} = {0, 0, 0}. (3.66) ∂Ω dsUsing (3.64), the ﬁrst two integral conditions of (3.66) may be written as {−F1 , F2 } = {0, 0}. (3.67) ∂ΩIn addition, the single-valued property for Φ (the third integral condition of (3.66)) may betransformed to d dx dy ( Φ) = (Φ, x + Φ, y ) = 0. (3.68) ∂Ω ds ∂Ω ds dsThe above equations may be further derived via integration by parts as: dx d d d Φ, x = [x Φ, x ] s=L − s=0 x Φ,x = x Φ,x − x Φ,x (3.69a) ∂Ω ds ∂Ω ds ∂Ω ds ∂Ω ds dy d d d Φ, y = [y Φ, y ] s=L − s=0 y Φ,y = y Φ,y − y Φ,y . (3.69b) ∂Ω ds ∂Ω ds ∂Ω ds ∂Ω dsThe equality of the once underlined terms and the twice underlined terms in each of the aboveequations deserves a careful explanation. We ﬁrst note that the x value is identical in bothboundary points as s = L (where L is the length of ∂Ω) and s = 0 represent the same point.In addition, the differences in Φ, x and Φ, y is independent of the point that we select as thestarting/end of the closed integral. Hence, (3.68) turns to be d d d d x Φ,x + y Φ,y − (x Φ,x + y Φ,y ) = 0, (3.70) ∂Ω ds ∂Ω ds ∂Ω ds dswhich may also be written as −x F1 + y F2 − (F2 y − F1 x) = 0. (3.71) ∂Ω ∂Ω ∂ΩThe above should be valid for each x and y and therefore takes the form {F1 , F2 , F2 y − F1 x} = {0, 0, 0}. (3.72) ∂Ω The conditions of (3.72) may also be examined from a physical point of view. To show thatand following Green’s Theorem, we derive the ﬁrst two conditions as dx (Ys +U 2 )= Ys − U 2,y = Ys + Yb = 0, (3.73a) ∂Ω ds ∂Ω Ω ∂Ω Ω dy (Xs −U 1 )= Xs − U 1,x = Xs + Xb = 0, (3.73b) ∂Ω ds ∂Ω Ω ∂Ω Ω
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96 3. Plane Deformation Analysiswhile the underlined part of (3.71) becomes dx dy [(Ys +U 2 )x − (Xs −U 1 )y] = (Ys x − Xs y) − U 2,y x −U1,x y = ∂Ω ds ds ∂Ω Ω (Ys x − Xs y) + (Yb x − Xb y) = 0. (3.74) ∂Ω ΩHence, physical considerations show that (3.73a,b) are essentially the equilibrium equationsfor forces in the x- and y- directions, respectively. Similarly, (3.74) sets the condition for “zeronet moment” (in the z-direction). Solution methodologies and symbolic algorithms for solving the above discussed BVPs forgeneral geometries, body-force and surface loads distributions are presented in Chapter 4. To summarize the above, we should now distinguish between three cases. In the ﬁrst one,both Φ and its derivatives Φ, x , Φ, y are single-valued. The conditions for that are (3.72), whichis the only meaningful case for a simply connected domain, as they are essential for the exis-tence and uniqueness of the solution, see Example 3.7. In a multiply connected domain, the stress function Φ is said to be single-valued with single-valued derivatives, if the surface loading acting on the contours ∂Ω j satisﬁes (3.72) for eachboundary component separately, namely, ( j) ( j) ( j) ( j) {F1 , F2 , xF1 − yF2 } = {0, 0, 0}. (3.75) ∂Ω jYet, the fact that Φ and its derivatives are not single-valued on some of the boundary compo-nents does not disqualify it as a solution as long as the sum of the above integral vanishes, ∑j ( j) ( j) ( j) ( j) {F1 , F2 , xF1 − yF2 } = {0, 0, 0}. (3.76) ∂Ω jWe therefore deﬁne a second case, which may occur in multiply connected domains, whereonly the derivatives Φ, x , Φ, y are single-valued functions. The conditions for that are (3.67), seealso Example 3.8. In the third case, both Φ and its derivatives Φ, x , Φ, y are not single-valued,see Example 3.9. In both cases, all conditions of (3.76) are satisﬁed.Remark 3.4 In a state of plane-stress, where Xs and Ys are also functions of z, one may replacethem with their average (over the thickness, h) values X s and Y s depending on x, y, namely, 1 h/2 1 h/2 Xs = Xs dz, Ys = Ys dz. (3.77) h −h/2 h −h/2The resulting stress and strain distribution will obviously be independent of z, and in thin plates,this operation, which “smears” all variations in the z-direction yields acceptable approximation.Further derivation of laminated (non-homogeneous) plates appears in S.3.5.1.Example 3.5 Prismatic Body Under Hydrostatic Load. Consider a “long” prismatic body of generic cross-section that undergoes a pure hydro-static pressure, P. The surface loads are therefore Xs = −P cos(n, x) and Ys = −P cos(n, y), see ¯ ¯Fig. 3.11(a) (the coordinate system origin is assumed to coincide with the cross-section areacentroid). The stress function in this simple plane-strain case is independent of the materialproperties or domain shape and is given by P Φ = − (x2 + y2 ). (3.78) 2
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3.2 Plane-Strain and Plane-Stress 97 Figure 3.11: A generic domain under hydrostatic pressure in Example 3.5.One may verify that both (3.56) and (3.64) are satisﬁed in this case (where no body forcesare included). The resulting nonzero stress components are σx = σy = −P and σz = P(a13 +a23 )/a33 . The nonzero strain components are (see (3.27)) εx = −P(b11 + b12 ), εy = −P(b22 +b12 ) and γxy = −P(b16 + b26 ), and their integration by the procedure of S.1.2 yields b16 + b26 b16 + b26 u = −P[(b11 + b12 ) x + y], v = −P[(b22 + b12 ) y + x], w = 0. (3.79) 2 2To create “free tips” of this long prismatic body, according to Remark 3.3, one needs to add tothese tips a uniform stress distribution in the amount of ∆σz = −P(a13 + a23 )/a33 . Accordingto the analysis that will be derived within Chapter 5 for a material of generic anisotropy (see(5.29a–c) and note that for the MON13z material under discussion we set a34 = a35 = 0), thedisplacements of such an additional solution would be 1 1 ∆u = ∆σz (a13 x + a36 y), ∆v = ∆σz ( a36 x + a23 y), ∆w = ∆σz a33 z. (3.80) 2 2Superposition of ∆u, ∆v, ∆w and u, v, w, respectively, yields 1 1 u f = −P[(a11 + a12 ) x+ (a16 + a26 )y], v f = −P[(a12 + a22 ) y+ (a16 + a26 )x], (3.81a) 2 2 w f = −P(a13 + a23 )z. (3.81b)Clearly, (3.81a,b) create a state of plane-stress, see S.3.2.2. If we further wish to simulate theeffect of “all-around uniform pressure” on the prismatic body we need to superimpose on u f ,v f , w f an additional solution of (3.80) with ∆σz = −P, which yields 1 u p = −P[(a11 + a12 + a13 ) x+ (a16 + a26 + a36 )y], (3.82a) 2 1 v p = −P[(a12 + a22 + a23 ) y+ (a16 + a26 + a36 )x], (3.82b) 2 w p = −P(a13 + a23 + a33 )z, (3.82c)that obviously would emerge from a simple solution for which σx = σy = σz = −P. Note thatthe deformation described above was derived for zero displacements and rotations at x = y =z = 0. For a generic MON13z material it is clear that unlike the isotropic case, the in-plane defor-mation of both solutions of (3.81a,b) and (3.82a–c) is not “radial”, namely, the displacements
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98 3. Plane Deformation Analysiscan not be written as u = αx and v = αy, where α is a constant. Thus, even for symmetric cross-section geometry, the deformation ﬁeld will not exhibit symmetry. As an example, in the case offree tips, the displacements in the x direction of the two points (x0 , 0) and (−x0 , 0) will be sym-metric with respect to the y-axis, i.e., u+ = −u− = −P (a11 + a12 ) x0 , but the displacements inthe y direction of the same points will be opposite as well, i.e., v+ = −v− = − 2 P (a16 + a26 ) x0 , 1which results in a deformation as shown in Fig. 3.11(b). In the special case where a MON13zhas been created by rotating an orthotropic material about the z-axis by an angle θz as shownin Fig. 3.11(c) (see also S.2.3, S.2.4), P.2.7 shows that E11 − E22 a16 + a26 = 2 cos θz sin θz , (3.83a) E11 E22 E11 − E22 + ν13 E11 − ν23 E22 a16 + a26 + a36 = 2 cos θz sin θz . (3.83b) E11 E22Hence, by setting the system of coordinates to coincide with the orthotropic material principaldirections (of anisotropy) as shown in Fig. 3.11(c), θz = 0 should be taken, and the underlinedterms in (3.81a), (3.82a,b) vanish. The deformation of cross-sections in such a case will beradial only over the axes (i.e. along the x = 0, y = 0 lines).Example 3.6 Rectangular Domain Under Low-Order body-force and Surface Loads. To illustrate a few elements of the above derivation, consider a rectangular domain (of di-mensions 2a, d), which is acted upon by a given uniform body force Xb , see Fig. 3.12. Thestress solution is derived under the assumption Ys = U 2 = σy = 0 and U 1 = σx = Ax + B,where A and B are arbitrary constants. Equations (3.55) show that Yb = 0, and A = −Xb . Inaddition, according to (3.57), F0 = 0, and subsequently the particular solution Φ p = 0 as well.To satisfy the boundary conditions (3.62) on the edges x = −a (where dy = −1) and x = a ds(where dy = 1), one should set Xs + = Aa + B, Xs − = Aa − B. Thus, (3.65) show that both F1 , dsand F2 vanish along all edges as well, and therefore Φ = Φ0 = 0. This solution includes thecase of symmetric loading where B = 0 and Xs + = Xs − = Aa and σx = Ax, and the case ofB = Aa where Xs + = 2Aa, Xs − = 0 and σx = A(x + a). In both cases the body-force resultant(−2Aad) is cancelled out by the surface load resultant. Figure 3.12: A rectangle subjected to a constant body force in the x-direction.Example 3.7 Single-Valued Stress Function with Single-Valued Derivatives. Consider, via polar coordinates (r, θ) notation, an isotropic circular ring with radii R2 > R1 >0 loaded by normal stress σrr = R2 on the inner boundary and σrr = R2 on the outer boundary 1 1 1 2(i.e. the stresses are not functions of θ). The well-known solution in this case is Φ = ln r, and
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3.2 Plane-Strain and Plane-Stress 99it yields the stresses σrr = −σθθ = and σrθ = 0 or in Cartesian coordinates σx = −σy = 1 r2cos(2θ) sin(2θ) r2 , τxy = r2 , see P.1.11. The reader may verify that all single-value conditions of (3.75)are satisﬁed for each of the two boundary components of the circular ring.Example 3.8 Non Single-Valued Stress Function with Single-Valued Derivatives. Consider the ring of Example 3.7 loaded by tangential shear stress τrθ = R2 over the in- 1 1ner boundary and τrθ = 1 R2 over the outer boundary. In polar coordinates, the solution of 2this simple problem is Φ = θ = arctan(y/x), which satisﬁes the biharmonic equation forisotropic materials, ∇(4) Φ = 0 (see (3.213)), and all boundary conditions. Note that deriva-tives, Φ, x = x2 +y2 , Φ, y = x2 +y2 , are single-valued in the ring. y x This solution yields the stresses σrr = σθθ = 0, τrθ = 1 r2 , see Fig. 3.13(a) for R1 = 1, R2 = 2.For Cartesian coordinates, the above is translated into σx = −σy = − sin(2θ) , τxy = cos(2θ) . r2 r2For the inner boundary cos(n, x) = − cos θ and cos(n, y) = − sin θ, and therefore (3.61) shows ¯ ¯that Xs = sin2θ , Ys = − cos2 θ . For the outer boundary cos(n, x) = cos θ, cos(n, y) = sin θ, and R R ¯ ¯ 1 1therefore (3.61) shows that Xs = − sin2θ , Ys = R cos θ R2 . Hence, (3.73a,b) are satisﬁed for each 2 2boundary component, namely, {Xs , Ys } = {0, 0}, i = 1, 2. (3.84) ∂ΩiHowever, (3.74) is not satisﬁed for each boundary component separately, but for the entiredomain only as required by (3.76), since (Ys x − Xs y) = 2π, (Ys x − Xs y) = −2π, (3.85) ∂Ω1 ∂Ω2which indicates again that Φ is not a single-valued function. 1 10.9 0.50.80.7 00.60.5 –0.50.4 –20.3 –2 –1 –1 –1 1.5 1 0 1.5 1 0.5 0 y 0.5 0 –0.5 1 y 0 –0.5 x –1 –1.5 1 x –1 –1.5 –2 –2 (a) σrθ of Example 3.8. Φ is not single- (b) σrr of Example 3.9. Both Φ and its valued but its derivatives are. derivatives are not single-valued. Figure 3.13: Examples of non single-valued stress functions.Example 3.9 Non Single-Valued Stress Function with non Single-Valued Derivatives. In this example, we re-employ the circular ring of Example 3.7 while it is loaded by anormal force σrr = (c1 cos θ + c2 sin θ)/R1 on the inner boundary and a normal force σrr =(c1 cos θ + c2 sin θ)/R2 on the outer boundary, where c1 and c2 are constants. The solution of
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100 3. Plane Deformation Analysisthis problem is Φ = 1 rθ(c1 sin θ− c2 cos θ). It yields the stresses σrr = (c1 cos θ+ c2 sin θ)/r, 2σθθ = τrθ = 0, see Fig. 3.13(b) (for c1 = 1, c2 = 0, R1 = 1, R2 = 2), or in Cartesian coordinates 1 σx = (c1 cos θ + c2 sin θ) cos2 θ, (3.86a) r 1 σy = (c1 cos θ + c2 sin θ) sin2 θ, (3.86b) r 1 τxy = (c1 cos θ + c2 sin θ) cos θ sin θ. (3.86c) rIt is easy to show that (3.73a,b) become {Xs ,Ys } = −π{c1 , c2 }, {Xs ,Ys } = π{c1 , c2 }, (3.87) ∂Ω1 ∂Ω2and thus Φ’s derivatives are not single-valued. In addition, despite the fact that the conditionof (3.74) is satisﬁed for each contour, namely, ∂Ωi (Ys x − Xs y) = 0, i = 1, 2, Φ is not single-valued on each boundary component since as shown by (3.71), for Φ to be single-valued, both(3.73a,b) and (3.74) should be satisﬁed. Yet, all conditions of (3.76) are satisﬁed in this case.3.2.6 Plane Stress/Strain Analysis in a Non-Homogeneous DomainLet Φ be an unknown stress function on a non-homogeneous plane domain, Ω, so that Φ[ j] =Φ|Ω[ j] are C4 -continuous functions. We employ the basic relations of (3.54) for the stress com-ponents and require that the in-plane resultants, F1 = − ds Φ, x , F2 = ds Φ, y , and the displace- d dments u, v will remain continuous over dividing surfaces. Subsequently, using the notation of(3.1), we formulate the biharmonic BVP for plane-stress/strain in a non-homogeneous domain,see Fig. 3.1(b,c), as (4) ∇n Φ = F0 over Ω, n ∈ {1, 3}, (3.88a) d {Φ, x , Φ, y } = {−F1 , F2 } on ∂Ω, (3.88b) ds d d [ j] { Φ, x , Φ, y , u, v}[i] = {0, 0, 0, 0} on ∂Ωi j (3.88c) ds dswhile the functions Fi (i = 0, 1, 2), are taken from (3.57), (3.65). Note that for n = 1, F0 iswritten with the bi j coefﬁcients, while for n = 3, F0 is written with the ai j coefﬁcients. Formore details see S.8.1.3.3.3 Plane-ShearThis section is focused on a two-dimensional problem, which differs from the problems dis-cussed in S.3.2. As done previously, the x, y-plane contains the domain Ω, however, we assumethat the only stress components considered are τyz (x, y) and τxz (x, y), see Fig. 3.14(a), and allother stress components are assumed to be known or to vanish. This type of analysis is classi-ﬁed as a “plane-shear” problem, since in MON13z material it includes the strain componentsγyz (x, y) and γxz (x, y) only. In other words, the stresses τyz (x, y) and τxz (x, y) and the strainsγyz (x, y) and γxz (x, y) do not depend on the other stress or strain components. The deformationconsists mainly of the axial component w(x, y), with typically small participation of u(x, y) andv(x, y).
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3.3 Plane-Shear 101 Figure 3.14: Plane-shear problem notation: (a) Two-dimensional domain. (b) A prismatic body. The above kind of two-dimensional loading is useful in slender bodies analysis, when theloading over the end cross-sections consists of the τyz (x, y) and τxz (x, y) components only, inaddition to the outer (“axial”) surface loading component, Zs (s), see Fig. 3.14(b), e.g. torsionproblems discussed further on within Chapters 4 and S.7.2.1. We will separately consider two different analyses of the same problem — by stress function,and by warping function (both functions are conjugate in a certain sense).3.3.1 Analysis by Stress FunctionTo carry out the analysis we examine the differential equilibrium equation in the z-directiononly (as the remaining two equations are satisﬁed identically). To extend the scope and appli-cability of the present analysis, we shall write this equation as τyz, y + τxz, x = Z(x, y), (3.89)where Z is a function that may include a generic body-force distribution, i.e., Z = −Zb , or anyother given function. The boundary condition for this case, see (1.85c) with cos(n, z) = 0, is ¯ τxz cos(n, x) + τyz cos(n, y) = Zs . ¯ ¯ (3.90)For reasons that will become clearer later on, we divide the compatibility equations (1.45a–f) into two sets: Set “A” that have a direct relation to the present two-dimensional analysisincludes the compatibility equations (1.45d, e). Set “B” consists of the remaining four compat-ibility equations, (1.45a–c, f). p p Suppose now that an auxiliary stress solution τyz (x, y) and τxz (x, y) of (3.89) with Z = 0, p p p ptogether with other components τxy , σx , σy , σz , has been found. The notion “stress solution”means that its components may not be legitimate ones for the theory of elasticity, since the pcorresponding strains εi (x, y) (i = 1, . . . , 6) do not necessarily satisfy the compatibility equa-tions. Yet, the above stress solution has to satisfy the ﬁrst two equilibrium equations, (1.82a,b)(with Xb , Yb ), the compatibility equations in set B, and the other two boundary conditions, see(1.85a,b), σx cos(n, x) + τxy cos(n, y) = Xs , p ¯ p ¯ τxy cos(n, x) + σy cos(n, y) = Ys . p ¯ p ¯ (3.91)
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102 3. Plane Deformation Analysis pWhen no stress solution is available, the following derivation remains valid with σi = 0 (i =1, . . . , 6) as well. We now introduce the stress function ψ(x, y) by assuming that τyz = −ψ, x + τyz +U 4 , p τxz = ψ, y + τxz +U 3 , p (3.92)while the remaining stress components are τxy = τxy , p σx = σx , p σy = σy , p σz = σz . p (3.93)In the above, U 3 , U 4 are the body-force potentials, namely, U 3, x +U 4, y = Z. (3.94)Note that (3.92) identically satisfy (3.89). Our task is therefore to determine a BVP for ψ(x, y)that will make τyz and τxz , together with other stress components, a valid solution. For the MON13z material (2.5) under discussion, from (1.82b) follow γyz = −a44 ψ, x + a45 ψ, y + a44 (τyz +U 4 ) + a45 (τxz +U 3 ), p p (3.95a) γxz = −a45 ψ, x + a55 ψ, y + a45 (τyz +U 4 ) + a55 (τxz +U 3 ), p p (3.95b) p p p pwhile the other strain components are given as γxy = γxy , εx = εx , εy = εy , εz = εz . Since according to (1.52) with i = j = 3 and f33 from (1.53c), γxz, y − γyz, x = −2 ωz, z , whereωz is the rotation about the z-axis, one may use (3.95a,b) and write (2)∇3 ψ = [a44 (τyz +U 4 ) + a45 (τxz +U 3 )], x − [a45 (τyz +U 4 ) + a55 (τxz +U 3 )], y − 2 ωz, z . (3.96) p p p pTo remain within a two-dimensional analysis, ωz should be (at most) a linear function of z. (2)Equation (1.9a) constitutes the governing equation for the stress function, ψ. The operator ∇3is referred to as the generalized Laplace’s operator (see also Example 1.4) and is given as (2) ∂2 ∂2 ∂2 ∇3 = a44 − 2a45 + a55 2 . (3.97) ∂x2 ∂x ∂y ∂yFor orthotropic materials, a45 = 0, and the r.h.s. of (3.97) is simpliﬁed to another version ofthe Laplace’s operator, given in (3.196). For isotropic materials, a44 = a55 = 2(1+ν) , and the Eoperator of (3.97) becomes 2(1+ν) ∇(2) , i.e. proportional to the simplest version of the Laplace’s E ∂ 2 ∂ 2operator ∇(2) = ∂x2 + ∂y2 (see also Remark 3.5). Note that for MON13z material a34 = a35 = 0,and one may replace a44 , a45 and a55 with b44 , b45 and b55 , respectively, see (3.28) and S.3.6.1. To determine ωz, z of (3.96), we now write the compatibility equations of Set A as −2 (ωz, z ), x = (γxz, y − γyz, x ), x = 2εx, yz − γxy, xz , p p (3.98a) −2 (ωz, z ), y = (γxz, y − γyz, x ), y = −2εy, xz + γxy, yz , p p (3.98b)which may be viewed as a set of two partial differential equations that determine ωz, z of x, y.Suitable solutions for this system should be adapted for each speciﬁc case. To formulate the boundary condition for (3.96) we use the surface loading Zs = Zs cos(n, x) x ¯ y+Zs cos(n, y) and employ the identity (see (3.4)) ¯ d ψ = ψ, y cos(n, x) − ψ, x cos(n, y). ¯ ¯ (3.99) ds
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3.3 Plane-Shear 103This enables us to write the boundary conditions for stress function ψ as d ψ = Zs − τxz −U 3 cos(n, x) + Zs − τyz −U 4 cos(n, y) x p ¯ y p ¯ on ∂Ω. (3.100) dsTherefore, (3.96), (3.100) constitute the Dirichlet BVP. The single-valued requirement for ψis written as Zs − τxz −U 3 cos(n, x) + Zs − τyz −U 4 cos(n, y) = 0 . x p ¯ y p ¯ (3.101) ∂ΩHence, ψ is given on ∂Ω by ψ(s) = ψ0 + 0s F3 ds, where ψ0 is an arbitrary constant. Theresultant loads over Ω may be determined by the following integration scheme, Fig. 3.14(b): {Px , Py , M z } = {τxz , τyz , xτyz − yτxz }, (3.102) Ωin which (3.92) should be substituted. In some circumstances, the Green’s Theorem version of(3.7) may be helpful in evaluating the integrals of (3.102). p p p p For the simplest assumption 2εx ,yz − γxy ,xz = h1 and −2εy ,xz + γxy ,yz = h2 , where h1 , h2 areconstants, one may ﬁnd from (3.98a, b) −2 ωz, z = h1 x + h2 y − 2 θ, (3.103)where the twist angle (per unit length) θ = ωz 0, z , (3.104)may be interpreted as the rotation of the origin x = y = 0 about the z-axis per unit length (notethat as already indicated, ωz 0, z is independent of z). The actual value of θ is out of the scope ofthe present two-dimensional discussion, see Remark 3.6.Remark 3.5 Note that any arbitrary isotropic biharmonic function Φ(x, y) may be expressedin terms of three harmonic functions (λi , i = 1, 3), as Φ(x, y) = x λ1 (x, y) + y λ2 (x, y) + λ3 (x, y). (3.105)The validity of λ3 is trivial in view of ∇(4) = ∇(2) · ∇(2) , while for λ1 , λ2 , it is easy to see that ∂4 Φ ∂4 Φ ∂4 Φ +2 2 2 + 4 = (3.106) ∂x4 ∂x ∂y ∂y x [(∇(2) λ1 ) + (∇(2) λ1 ) ] + y [(∇(2) λ2 ) + (∇(2) λ2 ) ] + 4[(∇(2) λ1 ) + (∇(2) λ2 ) ]. ,xx ,yy ,xx ,yy ,x ,yRemark 3.6 The torsion problem, see e.g. S.7.2.1.2, is a special case of the present two-dimensional analysis, when Zs = Zb = 0, no body-force loads or auxiliary solution are included p p p p p p(i.e., σxx = σyy = σzz = τxy = τxz = τyz = U 3 = U 4 = h1 = h2 = 0). So, (3.96), (3.100) become (2) d ∇3 ψ = −2θ over Ω,ψ = 0 on ∂Ω. (3.107) dsFor a simply connected cross-section, ψ is constant over the contour, say ψ0 , while for conve-nience, ψ0 = 0 may always be selected. The integration results of Remark 3.2 with Λ = ψ show that the resultant loads are {Px , Py , M z } = {0, 0, −2SΩ ψ0 + 2 ψ}, (3.108) Ωwhich also clariﬁes the fact that the parts of torsional moment contributed by τyz and τxz areequal. The BVP under discussion may be derived for θ = 1 and considering the M z moment asthe “resultant moment per unit twist”, which essentially deﬁnes the torsional rigidity.
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104 3. Plane Deformation AnalysisExample 3.10 Isotropic Beam Under Tip Bending. This example presents a stress solution that satisﬁes the equilibrium equations and the com-patibility equations of Set B (but not those of Set A). Consider an isotropic “clamped-free” beam of a rectangular cross-section (that occupies thedomain |x| < a/2, |y| < b/2, and has a cross-sectional moment of inertia Ix = 12 a b3 ) subjected 1to a tip load Py . The “stress solution” of this case will be 2 yPy Py 2 b σzz = − p (l − z) , τyz = − p [y − ], σxx = σyy = τxy = τxz = 0, p p p p (3.109) Ix 2Ix 2while the corresponding strain components are 1 p ν p 2(1 + ν) p εzz = p σ , εxx = εyy = − σzz , p p γyz = p τyz , γxz = γxy = 0. p p (3.110) E zz E EOne may verify that all equilibrium equations and the compatibility equations in Set B are iden-tically satisﬁed. However, those of Set A are violated, and hence, this solution is not “exact”.Yet, by employing the analysis in this section, one may determine the τyz (x, y) and τxz (x, y)stress distributions that will correct the above deﬁciency by setting 2ν Py 2(1+ ν) Z = −σzz, z , h1 = − p , a44 = a55 = , h2 =a45 = U 3 =U 4 = θ= Zs = 0. (3.111) E Ix E 2(1+ν)Substituting these values in (3.96) and dividing by E leads to the governing equation ν Py ∇(2) ψ = − · x, (3.112) 1 + ν Ixwhile (3.100) shows that ds ψ = 0 on ∂Ω. Hence, by solving this BVP one obtains the desired dcorrection to the above stress solution that will satisfy all requirements.3.3.2 Analysis by Warping FunctionIn order to repeat the analysis of S.3.3.1 from the warping point of view, (i.e. using out-of-plane warping, w(x, y), in the z-direction), we assume that the rotation of the domain about thez-axis does not depend on x, y, i.e., ωz ≡ ωz 0 (z), and we write the displacements as u = −y ωz 0 , v = x ωz 0 , w = w(x, y). (3.113)For simplicity, rigid body components of u, v are not placed here. This deformation ﬁeld cre-ates zero-strain components except for γxz = w, x − y θ, γyz = w, y + x θ, (3.114)where the notation (3.104) for θ is used again. The above equations show that in order toremain within a two-dimensional analysis, ωz 0 (z) should be (at most) a linear function of z,so that the twist, θ, is constant. This set of strains (that satisﬁes compatibility equation bydeﬁnition) should now be substituted in the equilibrium equations, for which we ﬁrst write thecorresponding non-vanishing stress components in MON13z material as a55 a45 a45 a44 τyz = (w, y + x θ) − (w, x − y θ) , τxz = − (w, y + x θ) + (w, x − y θ) , (3.115) a0 a0 a0 a0
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3.3 Plane-Shear 105where a0 = a44 a55 − a2 . Similar to S.3.3.1, we may extend the scope of the present derivation 45by superimposing the present solution with a given auxiliary two-dimensional deformationsystem that satisfy all compatibility equations (for example, a deformation system that hasoriginated from a continuous set of displacements), the differential equilibrium equation in thex- and y- directions and the boundary conditions, but does not necessarily satisfy the equilib-rium equation in the z-direction, (1.82c), and/or the boundary conditions associated with it,(1.85c). All quantities related to the above auxiliary solution will be denoted by a superscript“p”. Subsequently, we reach the following Neumann BVP for w: (2) ∇3 w = F0w over Ω, D1 w = F3w n on ∂Ω, (3.116)where F0w = −Zb − σzz, z + τyz, y + τxz, x , p p p (3.117a) F3w = [a0 Zs −τxz x p +θ(a45 x+a44 y)] cos(n, x)+[a0 Zs −τyz ¯ y p −θ(a55 x+a45 y)] cos(n, y).(3.117b) ¯In (3.116) we have used the Laplace’s operator (3.97), and introduced the Neumann-type nboundary operator, D1 , as ∂ ∂ ∂ ∂ D1 = (a44 n − a45 ) cos(n, x) + (−a45 + a55 ) cos(n, y). ¯ ¯ (3.118) ∂x ∂y ∂x ∂yFor the sake of convenience, the solution of (3.116) is carried out under the integral condition Ω w = 0 (i.e. zero average out-of-plane deformation) or, alternatively, w(0, 0) = 0. As will be shown within S.3.3.3, the existence and uniqueness of solution of the BVP (3.116)requires F3w = F0w . (3.119) ∂Ω ΩIn evaluating the above integrals, the terms containing θ in F3w may be neglected by virtue ofGreen’s Theorem, see (3.8). This reduces the existence and uniqueness condition to 1 Zs − τxz cos(n, x) + τyz cos(n, y) = − p ¯ p ¯ [Zb + σzz, z + τyz, y + τxz, x ]. (3.120) p p p ∂Ω ∂Ω a0 Ω The resultant loads may be derived using (3.102) and (3.7) in this case as well.3.3.3 Generic Dirichlet/Neumann BVPs on a Homogeneous DomainThis section summarizes some basic properties of the Dirichlet/Neumann BVPs. Note thatP.3.7, P.3.8 illustrate prescribed polynomial solutions of Laplace’s equation for MON13z andisotropic materials (i.e. prescribed solution of the ﬁeld equation without the boundary condi-tions, analogously to P.3.5, P.3.6 of Example 3.4).3.3.3.1 The Neumann BVP (2)We shall keep the previous notation of the differential operators ∇3 of (3.97), and the Neu- nmann-type boundary operator, D1 , (3.118), while for the moment, ai j will be considered asknown coefﬁcients, the physical meaning of which is immaterial. Let F0Λ and F3Λ be given functions on the homogeneous domain Ω and its boundary ∂Ω,as shown in Fig. 3.1(a). We also deﬁne the function Λ, which is said to be a generalized(Neumann) harmonic function if it satisﬁes the following ﬁeld equation and the Neumann-typeboundary condition: (2) ∇3 Λ = F0Λ over Ω, D1 Λ = F3Λ n on ∂Ω. (3.121)
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106 3. Plane Deformation AnalysisThe solution of the BVP (3.121) is carried out under the condition Λ(0, 0) = 0. We shall now document a few useful identities for generalized harmonic functions. First,substituting (3.7) in the expression of Dn in (3.118) yields 1 yDn Λ = 1 [y(a44 Λ, xx − 2a45 Λ, xy + a55 Λ, yy ) + (−a45 Λ, x + a55 Λ, y )], (3.122) ∂Ω Ω (2)which, with the deﬁnition of ∇3 in (3.97), yields the ﬁrst of the following equations: (2) (−a45 Λ, x + a55 Λ, y ) = yDn Λ − 1 y∇3 Λ, (3.123a) Ω ∂Ω Ω (2) (a44 Λ, x − a45 Λ, y ) = xDn Λ − 1 x∇3 Λ. (3.123b) Ω ∂Ω ΩAnalogously, the evaluation of ∂Ω (x Dn Λ), yields the second equation. 1 Moreover, for any generalized harmonic function, Λ, see (3.4), (3.118), dy dx Dn Λ = (a44 Λ,x − a45 Λ,y ) 1 − (−a45 Λ,x + a55 Λ,y ) on ∂Ω, (3.124) ds dswhich, by virtue of Green’s Theorem, becomes Dn Λ = 1 [(a44 Λ,x − a45 Λ,y ) dy − (−a45 Λ,x + a55 Λ,y ) dx ] ∂Ω ∂Ω (2) = (a44 Λ,xx − 2a45 Λ,xy + a55 Λ,yy ) = ∇3 Λ, (3.125) Ω Ωor in terms of the given functions in (3.121) F3Λ = F0Λ . (3.126) ∂Ω ΩEquation (3.126) is further used and denoted as the solution existence condition of the Neu-mann BVP, (3.121), in a simply connected domain. It is also customary to employ the notationF3Λ = PΛ cos(n, x) + QΛ cos(n, y) which yields ¯ ¯ (−a45 Λ, x + a55 Λ, y ) = [QΛ + y(P,Λ + QΛ − F0Λ )], x ,y (3.127a) Ω Ω (a44 Λ, x − a45 Λ, y ) = [PΛ + x(P,Λ + QΛ − F0Λ )]. x ,y (3.127b) Ω Ω3.3.3.2 The Dirichlet BVPWe shall present here the Dirichlet BVP (2) d ∇3 Λ = F0Λ Λ = F3 over Ω, on ∂Ω. (3.128) dsThe following condition is necessary for Λ to be single-valued, and thus for the existence anduniqueness of the BVP (3.128) in a homogeneous simply connected domain: F3 = 0. (3.129) ∂ΩRemark 3.7 For the Dirichlet/Neumann BVPs in multiply connected homogeneous domain,Ω, with boundary ∂Ω = j ∂Ω j , conditions (3.126), (3.129) are transformed using the orienta-tion of the boundary components shown in Fig. 3.15, to the more complicated form Λ ( j) ∑j ∑j ( j) F3 = F0Λ , F3 = 0. (3.130) ∂Ω j Ω ∂Ω j
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3.3 Plane-Shear 107 Figure 3.15: A multiply connected domain boundary.3.3.4 Simpliﬁcation of Generalized Laplace’s and Boundary OperatorsEmploying the afﬁne transformation x = ax, y = y + bx, (3.131)where a and b are nonzero constants, the domain contour in the x, y-plane is transformed intoa different contour in the x y -plane as schematically shown in Fig. 3.16. Figure 3.16: The “original” x, y-plane and the “transformed” x y -plane. We parameterize the contour by two functions x(t) and y(t), and therefore the transformedparametrization functions x (t) = a x(t) and y (t) = y(t) + b x(t) of S.3.1.2, while a speciﬁcvalue of t relates the same point on the contours in both planes. Suppose that a generalized harmonic function, Λ(x, y), is deﬁned on Ω. Using the inversetransformation x = 1 x , y = y − b x , one may present the following derivatives of Λ(x, y): a aΛ, x = a Λ, x + b Λ, y , Λ, y = Λ, y , (3.132a)Λ, xx = a 2 Λ, x x + 2 a b Λ, x y + b 2 Λ, y y , Λ, xy = a Λ, x y + b Λ, y y , Λ, yy = Λ, y y . (3.132b) (2)One may reduce the operation ∇3 (see (3.97)) to a simpler one using derivatives with re- ∂2 ∂2spect to x and y , by requiring that the coefﬁcients of (∂x )2 and (∂y )2 will be equal while the ∂2coefﬁcient of ∂x ∂y vanishes. This procedure yields 1 a45 a= a44 a55 − a2 , 45 b= . (3.133) a44 a44Therefore, the generalized Poisson’s equation (3.128 :a) may be simpliﬁed to 1 Λ, x x + Λ, y y = F Λ (x , y ) on ∂Ω . (3.134) a44 a2 0
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108 3. Plane Deformation AnalysisHence, by applying the transformation (3.131) to
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