Elasticity theory applications and numerics solution manualElasticity theory applications and numerics solution manual
Table of contents : Part 1: Foundations and elementary applications1. Mathematical Preliminaries2. Deformation: Displacements and Strains3. Stress and Equilibrium4. Material Behavior – Linear Elastic Solids5. Formulation and Solution Strategies6. Strain Energy and Related Principles7. Two-Dimensional Formulation8. Two-Dimensional Problem Solution9. Extension, Torsion, and Flexure of Elastic CylindersPart 2: Advanced applications10. Complex Variable Methods11. Anisotropic Elasticity12. Thermoelasticity13. Displacement Potentials and Stress Functions: Applications to Three-Dimensional Problems14. Nonhomogeneous Elasticity15. Micromechanics Applications16. Numerical Finite and Boundary Element Methods Citation preview Elasticity Theory, Applications, and Numerics Fourth Edition Martin H. Sadd Professor Emeritus, University of Rhode Island Department of Mechanical Engineering and Applied Mechanics Kingston, Rhode Island Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). MATLABÒ is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLABÒ software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLABÒ software. Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-815987-3 For information on all Academic Press publications visit our website at Publisher: Katey Birtcher Acquisition Editor: Steve Merken Editorial Project Manager: Susan Ikeda Production Project Manager: Sruthi Satheesh Cover Designer: Greg Harris Typeset by TNQ Technologies Preface As with the previous works, this fourth edition continues the author’s efforts to present linear elasticity with complete and concise theoretical development, numerous and contemporary applications, and enriching numerics to aid in problem solution and understanding. Over the years the author has given much thought on what should be taught to students in this field and what educational outcomes would be expected. Theoretical topics that are related to the foundations of elasticity should be presented in sufficient detail that will allow students to read and generally understand contemporary research papers. Related to this idea, students should acquire necessary vector and tensor notational skills and understand fundamental development of the basic field equations. Students should also have a solid understanding of the formulation and solution of various elasticity boundary-value problems that include a variety of domain and loading geometries. Finally, students should be able to apply modern engineering software (MATLAB, Maple or Mathematica) to aid in the solution, evaluation and graphical display of various elasticity problem solutions. These points are all emphasized in this text. In addition to making numerous small corrections and clarifications, several new items have been added. A new section in Chapter 5 on singular elasticity solutions has been introduced to generally acquaint students with this type of behavior. Cubic anisotropy has now been presented in Chapter 11 as another particular form of elastic anisotropy. Inequality elastic moduli restrictions for various anisotropic material models have been better organized in a new table in Chapter 11. The general Naghdi-Hsu solution has now been introduced in Chapter 13. An additional micromechanical model of gradient elasticity has been added in Chapter 15. A couple of new MATLAB codes in Appendix C have been added and all codes are now referenced in the text where they are used. With the addition of 31 new exercises, the fourth edition now has 441 total exercises. These problems should provide instructors with many new and previous options for homework, exams, or material for in-class presentations or discussions. The online Solutions Manual has been updated and corrected and includes solutions to all exercises in the new edition. All text editions follow the original lineage as an outgrowth of lecture notes that I have used in teaching a two-course sequence in the theory of elasticity. Part I of the text is designed primarily for the first course, normally taken by beginning graduate students from a variety of engineering disciplines. The purpose of the first course is to introduce students to theory and formulation, and to present solutions to some basic problems. In this fashion students see how and why the more fundamental elasticity model of deformation should replace elementary strength of materials analysis. The first course also provides foundation for more advanced study in related areas of solid mechanics. Although the more advanced material included in Part II has normally been used for a second course, I often borrow selected topics for use in the first course. The elasticity presentation in this book reflects the words used in the title - theory, applications, and numerics. Because theory provides the fundamental cornerstone of this field, it is important to first provide a sound theoretical development of elasticity with sufficient rigor to give students a good foundation for the development of solutions to a broad class of problems. The theoretical development is carried out in an organized and concise manner in order to not lose the attention of the less mathematically inclined students or the focus of applications. With a primary goal of solving problems of engineering interest, the text offers numerous applications in contemporary areas, including anisotropic composite and functionally graded materials, fracture mechanics, micromechanics modeling, thermoelastic problems, and computational finite and boundary element methods. Numerous solved example problems and exercises are included in all chapters. xi xii Preface The new edition continues the special use of integrated numerics. By taking the approach that applications of theory need to be observed through calculation and graphical display, numerics is accomplished through the use of MATLAB, one of the most popular engineering software packages. This software is used throughout the text for applications such as stress and strain transformation, evaluation and plotting of stress and displacement distributions, finite element calculations, and comparisons between strength of materials and analytical and numerical elasticity solutions. With numerical and graphical evaluations, application problems become more interesting and useful for student learning. Other software such as Maple or Mathematica could also be used. Contents summary Part I of the book emphasizes formulation details and elementary applications. Chapter 1 provides a mathematical background for the formulation of elasticity through a review of scalar, vector, and tensor field theory. Cartesian tensor notation is introduced and is used throughout the book’s formulation sections. Chapter 2 covers the analysis of strain and displacement within the context of small deformation theory. The concept of strain compatibility is also presented in this chapter. Forces, stresses, the equilibrium concept and various stress contour lines are developed in Chapter 3. Linear elastic material behavior leading to the generalized Hooke’s law is discussed in Chapter 4, which also briefly presents nonhomogeneous, anisotropic, and thermoelastic constitutive forms. Later chapters more fully investigate these types of applications. Chapter 5 collects the previously derived equations and formulates the basic boundary value problems of elasticity theory. Displacement and stress formulations are constructed and general solution strategies are identified. This is an important chapter for students to put the theory together. Chapter 6 presents strain energy and related principles, including the reciprocal theorem, virtual work, and minimum potential and complementary energy. Twodimensional formulations of plane strain, plane stress, and antiplane strain are given in Chapter 7. An extensive set of solutions for specific two dimensional problems is then presented in Chapter 8, and many applications employing MATLAB are used to demonstrate the results. Analytical solutions are continued in Chapter 9 for the Saint-Venant extension, torsion, and flexure problems. The material in Part I provides a logical and orderly basis for a sound one-semester beginning course in elasticity. Selected portions of the text’s second part could also be incorporated into such a course. Part II delves into more advanced topics normally covered in a second course. The powerful method of complex variables for the plane problem is presented in Chapter 10, and several applications to fracture mechanics are given. Chapter 11 extends the previous isotropic theory into the behavior of anisotropic solids with emphasis on composite materials. This is an important application and, again, examples related to fracture mechanics are provided. Curvilinear anisotropy including both cylindrical and spherical orthotropy is included in this chapter to explore some basic problem solutions with this type of material structure. An introduction to thermoelasticity is developed in Chapter 12, and several specific application problems are discussed, including stress concentration and crack problems. Potential methods, including both displacement potentials and stress functions, are presented in Chapter 13. These methods are used to develop several three-dimensional elasticity solutions. Chapter 14 covers nonhomogeneous elasticity, and this material is unique among current standard elasticity texts. After briefly covering theoretical formulations, several two-dimensional solutions are generated along with comparison field plots with the corresponding homogeneous cases. Chapter 15 Preface xiii presents a collection of elasticity applications to problems involving micromechanics modeling. Included are applications for dislocation modeling, singular stress states, solids with distributed cracks, micropolar, distributed voids, doublet mechanics and higher gradient theories. Chapter 16 provides a brief introduction to the powerful numerical methods of finite and boundary element techniques. Although only two-dimensional theory is developed, the numerical results in the example problems provide interesting comparisons with previously generated analytical solutions from earlier chapters. This fourth edition of Elasticity concludes with four appendices that contain a concise summary listing of basic field equations; transformation relations between Cartesian, cylindrical, and spherical coordinate systems; a MATLAB primer; and a self-contained review of mechanics of materials. The subject Elasticity is an elegant and fascinating subject that deals with determination of the stress, strain, and displacement distribution in an elastic solid under the influence of external forces. Following the usual assumptions of linear, small-deformation theory, the formulation establishes a mathematical model that allows solutions to problems that have applications in many engineering and scientific fields such as: • Civil engineering applications include important contributions to stress and deflection analysis of structures, such as rods, beams, plates, and shells. Additional applications lie in geomechanics involving the stresses in materials such as soil, rock, concrete, and asphalt. • Mechanical engineering uses elasticity in numerous problems in analysis and design of machine elements. Such applications include general stress analysis, contact stresses, thermal stress analysis, fracture mechanics, and fatigue. • Materials engineering uses elasticity to determine the stress fields in crystalline solids, around dislocations, and in materials with microstructure. • Applications in aeronautical and aerospace engineering typically include stress, fracture, and fatigue analysis in aerostructures. • Biomechanical engineering uses elasticity to study the mechanics of bone and various types of soft tissue. The subject also provides the basis for more advanced work in inelastic material behavior, including plasticity and viscoelasticity, and the study of computational stress analysis employing finite and boundary element methods. Since elasticity establishes a mathematical model of the deformation problem, it requires mathematical knowledge to understand formulation and solution procedures. Governing partial differential field equations are developed using basic principles of continuum mechanics commonly formulated in vector and tensor language. Techniques used to solve these field equations can encompass Fourier methods, variational calculus, integral transforms, complex variables, potential theory, finite differences, finite elements, and so forth. To prepare students for this subject, the text provides reviews of many mathematical topics, and additional references are given for further study. It is important for students to be adequately prepared for the theoretical developments, or else they will not be able to understand necessary formulation details. Of course, with emphasis on applications, the text concentrates on theory that is most useful for problem solution. xiv Preface The concept of the elastic forceedeformation relation was first proposed by Robert Hooke in 1678. However, the major formulation of the mathematical theory of elasticity was not developed until the nineteenth century. In 1821 Navier presented his investigations on the general equations of equilibrium; this was quickly followed by Cauchy, who studied the basic elasticity equations and developed the concept of stress at a point. A long list of prominent scientists and mathematicians continued development of the theory, including the Bernoullis, Lord Kelvin, Poisson, Lame’, Green, Saint-Venant, Betti, Airy, Kirchhoff, Rayleigh, Love, Timoshenko, Kolosoff, Muskhelishvilli, and others. During the two decades after World War II, elasticity research produced a large number of analytical solutions to specific problems of engineering interest. The 1970s and 1980s included considerable work on numerical methods using finite and boundary element theory. Also during this period, elasticity applications were directed at anisotropic materials for applications to composites. More recently, elasticity has been used in modeling materials with internal microstructures or heterogeneity and in inhomogeneous, graded materials. The rebirth of modern continuum mechanics in the 1960s led to a review of the foundations of elasticity and established a rational place for the theory within the general framework. Historical details can be found in the texts by Todhunter and Pearson, History of the Theory of Elasticity; Love, A Treatise on the Mathematical Theory of Elasticity; and Timoshenko, A History of Strength of Materials. Exercises and web support Of special note in regard to this text is the use of exercises and the publisher’s website, www.textbooks. elsevier.com. Numerous exercises are provided at the end of each chapter for homework assignments to engage students with the subject matter. The exercises also provide an ideal tool for the instructor to present additional application examples during class lectures. Many places in the text make reference to specific exercises that work out details to a particular topic. Exercises marked with an asterisk (*) indicate problems that require numerical and plotting methods using the suggested MATLAB software. Solutions to all exercises are provided to registered instructors online at the publisher’s website, thereby providing instructors with considerable help in using this material. In addition, downloadable MATLAB software is available to aid both students and instructors in developing codes for their own particular use to allow easy integration of the numerics. As with the previous edition, an on-line collection of PowerPoint slides is available for Chapters 1-9. This material includes graphical figures and summaries of basic equations that have proven to be useful during class presentations. Feedback The author is strongly interested in continual improvement of engineering education and welcomes feedback from users of the book. Please feel free to send comments concerning suggested improvements or corrections via surface mail or email ([email protected]). It is likely that such feedback will be shared with the text’s user community via the publisher’s website. Acknowledgments Many individuals deserve acknowledgment for aiding in the development of this textbook. First, I would like to recognize the many graduate students who sat in my elasticity classes. They have been a continual source of challenge and inspiration, and certainly influenced my efforts to find more effective ways to present this material. A special recognition goes to one particular student, Qingli Dai, who developed most of the original exercise solutions and did considerable proofreading. . I would also like to acknowledge the support of my institution, the University of Rhode Island, for providing me time, resources and the intellectual climate to complete this and previous editions. Several photoelastic pictures have been graciously provided by the URI Dynamic Photomechanics Laboratory (Professor Arun Shukla, director). Development and production support from several Elsevier staff was greatly appreciated. As with the previous editions, this book is dedicated to the late Professor Marvin Stippes of the University of Illinois; he was the first to show me the elegance and beauty of the subject. His neatness, clarity, and apparently infinite understanding of elasticity will never be forgotten by his students. Martin H. Sadd xv About the Author Martin H. Sadd is Professor Emeritus of Mechanical Engineering at the University of Rhode Island. He received his Ph.D. in mechanics from the Illinois Institute of Technology in 1971 and began his academic career at Mississippi State University. In 1979 he joined the faculty at Rhode Island and served as department chair from 1991 to 2000. He is a member of Phi Kappa Phi, Pi Tau Sigma, Tau Beta Pi, Sigma Xi, and is a Fellow of ASME. Professor Sadd’s teaching background has been in the area of solid mechanics with emphasis in elasticity, continuum mechanics, wave propagation, and computational methods. His research has included analytical and computational modeling of materials under static and dynamic loading conditions. He has authored over 75 publications and is the author of Continuum Mechanics Modeling of Material Behavior (Elsevier, 2019). xvii CHAPTER Mathematical preliminaries 1 Similar to other field theories such as fluid mechanics, heat conduction, and electromagnetics, the study and application of elasticity theory requires knowledge of several areas of applied mathematics. The theory is formulated in terms of a variety of variables including scalar, vector, and tensor fields, and this calls for the use of tensor notation along with tensor algebra and calculus. Through the use of particular principles from continuum mechanics, the theory is developed as a system of partial differential field equations that are to be solved in a region of space coinciding with the body under study. Solution techniques used on these field equations commonly employ Fourier methods, variational techniques, integral transforms, complex variables, potential theory, finite differences, and finite and boundary elements. Therefore, to develop proper formulation methods and solution techniques for elasticity problems, it is necessary to have an appropriate mathematical background. The purpose of this initial chapter is to provide a background primarily for the formulation part of our study. Additional review of other mathematical topics related to problem solution technique is provided in later chapters where they are to be applied. 1.1 Scalar, vector, matrix, and tensor definitions Elasticity theory is formulated in terms of many different types of variables that are either specified or sought at spatial points in the body under study. Some of these variables are scalar quantities, representing a single magnitude at each point in space. Common examples include the material density r and temperature T. Other variables of interest are vector quantities that are expressible in terms of components in a two- or three-dimensional coordinate system. Examples of vector variables are the displacement and rotation of material points in the elastic continuum. Formulations within the theory also require the need for matrix variables, which commonly require more than three components to quantify. Examples of such variables include stress and strain. As shown in subsequent chapters, a three-dimensional formulation requires nine components (only six are independent) to quantify the stress or strain at a point. For this case, the variable is normally expressed in a matrix format with three rows and three columns. To summarize this discussion, in a three-dimensional Cartesian coordinate system, scalar, vector, and matrix variables can thus be written as follows mass density scalar ¼ r displacement vector ¼ u ¼ ue1 þ ve2 þ we3 3 2 sx sxy sxz 7 6 stress matrix ¼ ½s ¼ 4 syx sy syz 5 szx szy sz Elasticity. Copyright © 2021 Elsevier Inc. All rights reserved. 3 4 Chapter 1 Mathematical preliminaries where e1, e2, e3 are the usual unit basis vectors in the coordinate directions. Thus, scalars, vectors, and matrices are specified by one, three, and nine components respectively. The formulation of elasticity problems not only involves these types of variables, but also incorporates additional quantities that require even more components to characterize. Because of this, most field theories such as elasticity make use of a tensor formalism using index notation. This enables efficient representation of all variables and governing equations using a single standardized scheme. The tensor concept is defined more precisely in a later section, but for now we can simply say that scalars, vectors, matrices, and other higher-order variables can all be represented by tensors of various orders. We now proceed to a discussion on the notational rules of order for the tensor formalism. Additional information on tensors and index notation can be found in many texts such as Goodbody (1982), Simmons (1994), Itskov (2015) and Sadd (2019). 1.2 Index notation Index notation is a shorthand scheme whereby a whole set of numbers (elements or components) is represented by a single symbol with subscripts. For example, the three numbers a1, a2, a3 are denoted by the symbol ai, where index i has range 1, 2, 3. In a similar fashion, aij represents the nine numbers a11, a12, a13, a21, a22, a23, a31, a32, a33. Although these representations can be written in any manner, it is common to use a scheme related to vector and matrix formats such that 3 2 3 2 a1 a11 a12 a13 7 6 7 6 ai ¼ 4 a2 5; aij ¼ 4 a21 a22 a23 5 (1.2.1) a3 a31 a32 a33 In the matrix format, a1j represents the first row, while ai1 indicates the first column. Other columns and rows are indicated in similar fashion, and thus the first index represents the row, while the second index denotes the column. In general a symbol aij.k with N distinct indices represents 3N distinct numbers. It should be apparent that ai and aj represent the same three numbers, and likewise aij and amn signify the same matrix. Addition, subtraction, multiplication, and equality of index symbols are defined in the normal fashion. For example, addition and subtraction are given by 3 3 2 2 a1 b1 a11 b11 a12 b12 a13 b13 7 7 6 6 ai bi ¼ 4 a2 b2 5; aij bij ¼ 4 a21 b21 a22 b22 a23 b23 5 (1.2.2) a3 b3 a31 b31 and scalar multiplication is specified as 3 2 2 la1 la11 7 6 6 lai ¼ 4 la2 5; laij ¼ 4 la21 la3 la31 a32 b32 la12 la22 la32 la13 a33 b33 3 7 la23 5 la33 (1.2.3) The multiplication of two symbols with different indices is called outer multiplication, and a simple example is given by 3 2 a1 b1 a1 b2 a1 b3 7 6 ai bj ¼ 4 a2 b1 a2 b2 a2 b3 5 (1.2.4) a3 b1 a3 b2 a3 b3 1.2 Index notation 5 The previous operations obey usual commutative, associative, and distributive laws, for example ai þ bi ¼ b i þ ai aij bk ¼ bk aij ai þ ðbi þ ci Þ ¼ ðai þ bi Þ þ ci (1.2.5) ai ðbjk cl Þ ¼ ðai bjk Þcl aij ðbk þ ck Þ ¼ aij bk þ aij ck Note that the simple relations ai ¼ bi and aij ¼ bij imply that a1 ¼ b1, a2 ¼ b2, . and a11 ¼ b11, a12 ¼ b12, . However, relations of the form ai ¼ bj or aij ¼ bkl have ambiguous meaning because the distinct indices on each term are not the same, and these types of expressions are to be avoided in this notational scheme. In general, the distinct subscripts on all individual terms in an equation should match. It is convenient to adopt the convention that if a subscript appears twice in the same term, then summation over that subscript from one to three is implied, for example 3 X aii ¼ aii ¼ a11 þ a22 þ a33 i¼1 3 X aij bj ¼ (1.2.6) aij bj ¼ ai1 b1 þ ai2 b2 þ ai3 b3 j¼1 It should be apparent that aii ¼ ajj ¼ akk ¼ ., and therefore the repeated subscripts or indices are sometimes called dummy subscripts. Unspecified indices that are not repeated are called free or distinct subscripts. The summation convention may be suspended by underlining one of the repeated indices or by writing no sum. The use of three or more repeated indices in the same term (e.g., aiii or aiijbij) has ambiguous meaning and is to be avoided. On a given symbol, the process of setting two free indices equal is called contraction. For example, aii is obtained from aij by contraction on i and j. The operation of outer multiplication of two indexed symbols followed by contraction with respect to one index from each symbol generates an inner multiplication; for example, aijbjk is an inner product obtained from the outer product aijbmk by contraction on indices j and m. A symbol aij.m.n.k is said to be symmetric with respect to index pair mn if aij.m.n.k ¼ aij.n.m.k (1.2.7) while it is antisymmetric or skewsymmetric if aij.m.n.k ¼ aij.n.m.k (1.2.8) Note that if aij.m.n.k is symmetric in mn while bpq.m.n.r is antisymmetric in mn, then the product is zero (1.2.9) aij.m.n.k bpq.m.n.r ¼ 0 A useful identity may be written as 1 1 aij ¼ ðaij þ aji Þ þ ðaij aji Þ ¼ aðijÞ þ a½ij (1.2.10) 2 2 The first term a(ij) ¼ 1/2(aij þ aji) is symmetric, while the second term a[ij] ¼ 1/2(aij aji) is antisymmetric, and thus an arbitrary symbol aij can be expressed as the sum of symmetric and 6 Chapter 1 Mathematical preliminaries antisymmetric pieces. Note that if aij is symmetric, it has only six independent components. On the other hand, if aij is antisymmetric, its diagonal terms aii (no sum on i) must be zero, and it has only three independent components. Since a[ij] has only three independent components, it can be related to a quantity with a single index, for example ai (see Exercise 1.15). Example 1.1 Index notation examples The matrix aij and vector bi are specified by 2 1 6 aij ¼ 4 0 2 3 2 0 7 4 3 5; 1 2 2 3 2 6 7 bi ¼ 4 4 5 0 Determine the following quantities: aii, aijaij, aijajk, aijbj, aijbibj, bibi, bibj, a(ij), a[ij], and indicate whether they are a scalar, vector, or matrix. Following the standard definitions given in Section 1.2 aii ¼ a11 þ a22 þ a33 ¼ 7 ðscalarÞ aij aij ¼ a11 a11 þ a12 a12 þ a13 a13 þ a21 a21 þ a22 a22 þ a23 a23 þ a31 a31 þ a32 a32 þ a33 a33 ¼ 1 þ 4 þ 0 þ 0 þ 16 þ 9 þ 4 þ 1 þ 4 ¼ 39 ðscalarÞ 3 2 1 10 6 7 6 7 aij ajk ¼ ai1 a1k þ ai2 a2k þ ai3 a3k ¼ 6 4 6 19 18 5ðmatrixÞ 6 10 7 2 10 3 6 7 7 aij bj ¼ ai1 b1 þ ai2 b2 þ ai3 b3 ¼ 6 4 16 5ðvectorÞ 8 aij bi bj ¼ a11 b1 b1 þ a12 b1 b2 þ a13 b1 b3 þ a21 b2 b1 þ . ¼ 84 ðscalarÞ bi bi ¼ b1 b1 þ b2 b2 þ b3 b3 ¼ 4 þ 16 þ 0 ¼ 20 ðscalarÞ 2 3 4 8 0 6 7 6 7 bi bj ¼ 6 8 16 0 7ðmatrixÞ 4 5 0 0 0 2 1 2 0 3 7 1 16 7 6 aðijÞ ¼ ðaij þ aji Þ ¼ 6 0 4 3 7 þ 5 4 2 2 2 1 2 2 a½ij 3 1 2 0 6 7 1 16 7 ¼ ðaij aji Þ ¼ 6 0 4 3 7 5 2 24 2 1 2 2 1 0 16 6 62 4 24 0 3 2 2 3 2 1 1 1 3 7 7 6 7 7 6 1 7 ¼ 6 1 4 2 7ðmatrixÞ 5 5 4 1 2 2 2 3 2 3 1 0 2 0 1 1 6 7 6 7 16 7 6 7 1 7ðmatrixÞ 6 2 4 1 7 ¼ 6 1 0 5 4 5 24 0 3 2 1 1 0 1.3 Kronecker delta and alternating symbol 7 1.3 Kronecker delta and alternating symbol A useful special symbol commonly used in index notational schemes is the Kronecker delta defined by 8 2 3 > 1 0 0 > < 1; if i ¼ j ðno sumÞ 6 7 ¼ 40 1 05 (1.3.1) dij ¼ > 0; if isj > : 0 0 1 Within usual matrix theory, it is observed that this symbol is simply the unit matrix. Note that the Kronecker delta is a symmetric symbol. Particularly useful properties of the Kronecker delta include the following dij ¼ dji dii ¼ 3; dii ¼ 1 dij aj ¼ ai ; dij ai ¼ aj (1.3.2) dij ajk ¼ aik ; djk aik ¼ aij dij aij ¼ aii ; dij dij ¼ 3 Another useful special symbol is the 8 > < þ1; εijk ¼ 1; > : 0; alternating or permutation symbol defined by if ijk is an even permutation of 1; 2; 3 if ijk is an odd permutation of 1; 2; 3 (1.3.3) otherwise Consequently, ε123 ¼ ε231 ¼ ε312 ¼ 1, ε321 ¼ ε132 ¼ ε213 ¼ 1, ε112 ¼ ε131 ¼ ε222 ¼ . ¼ 0. Therefore, of the 27 possible terms for the alternating symbol, three are equal to þ1, three are equal to 1, and all others are 0. The alternating symbol is antisymmetric with respect to any pair of its indices. This particular symbol is useful in evaluating determinants and vector cross products, and the determinant of an array aij can be written in two equivalent forms a11 a12 a13 (1.3.4) det½aij ¼ jaij j ¼ a21 a22 a23 ¼ εijk a1i a2j a3k ¼ εijk ai1 aj2 ak3 a31 a32 a33 where the first index expression represents the row expansion, while the second form is the column expansion. Using the property dip diq dir (1.3.5) εijk εpqr ¼ djp djq djr dkp dkq dkr another form of the determinant of a matrix can be written as 1 det½aij ¼ εijk εpqr aip ajq akr 6 (1.3.6) 8 Chapter 1 Mathematical preliminaries 1.4 Coordinate transformations It is convenient and in fact necessary to express elasticity variables and field equations in several different coordinate systems (see Appendix A). This situation requires the development of particular transformation rules for scalar, vector, matrix, and higher-order variables. This concept is fundamentally connected with the basic definitions of tensor variables and their related tensor transformation laws. We restrict our discussion to transformations only between Cartesian coordinate systems, 0 0 0and thus consider the two systems shown in Fig. 1.1. The two Cartesian frames (x1, x2, x3) and x1 ; x2 ; x3 differ only by orientation, and the unit basis vectors for each frame are {ei} ¼ {e1, e2, e3} and e0i ¼ e01 ; e02 ; e03 . Let Qij denote the cosine of the angle between the x0i -axis and the xj-axis (1.4.1) Qij ¼ cos x0i ; xj Using this definition, the basis vectors in the primed coordinate frame can be easily expressed in terms of those in the unprimed frame by the relations e01 ¼ Q11 e1 þ Q12 e2 þ Q13 e3 e02 ¼ Q21 e1 þ Q22 e2 þ Q23 e3 (1.4.2) e03 ¼ Q31 e1 þ Q32 e2 þ Q33 e3 or in index notation e0i ¼ Qij ej (1.4.3) x3 x3 v e3 e3 e2 e2 e1 e1 x1 x2 x1 FIG. 1.1 Change of Cartesian coordinate frames. x2 1.4 Coordinate transformations 9 Likewise, the opposite transformation can be written using the same format as ei ¼ Qji e0j (1.4.4) Now an arbitrary vector v can be written in either of the two coordinate systems as v ¼ v1 e1 þ v2 e2 þ v3 e3 ¼ vi ei (1.4.5) ¼ v01 e01 þ v02 e02 þ v03 e03 ¼ v0i e0i Substituting form (1.4.4) into (1.4.5)1 gives v ¼ vi Qji e0j but from (1.4.5)2, v ¼ v0j e0j ; and so we find that v0i ¼ Qij vj (1.4.6) In similar fashion, using (1.4.3) in (1.4.5)2 gives vi ¼ Qji v0j (1.4.7) Relations (1.4.6) and (1.4.7) constitute the transformation laws for the Cartesian components of a vector under a change of rectangular Cartesian coordinate frame. It should be understood that under such transformations, the vector is unaltered (retaining original length and orientation), and only its components are changed. Consequently, if we know the components of a vector in one frame, relation (1.4.6) and/or relation (1.4.7) can be used to calculate components in any other frame. The fact that transformations are being made only between orthogonal coordinate systems places some particular restrictions on the transformation or direction cosine matrix Qij. These can be determined by using (1.4.6) and (1.4.7) together to get vi ¼ Qji v0j ¼ Qji Qjk vk (1.4.8) From the properties of the Kronecker delta, this expression can be written as dik vk ¼ Qji Qjk vk or Qji Qjk dik vk ¼ 0 and since this relation is true for all vectors vk, the expression in parentheses must be zero, giving the result Qji Qjk ¼ dik In similar fashion, relations (1.4.6) and (1.4.7) can be used to eliminate vi (instead of Qij Qkj ¼ dik (1.4.9) v0i ) to get (1.4.10) Relations (1.4.9) and (1.4.10) comprise the orthogonality conditions that Qij must satisfy. Taking the determinant of either relation gives another related result (1.4.11) det Qij ¼ 1 Matrices that satisfy these relations are called orthogonal, and the transformations given by (1.4.6) and (1.4.7) are therefore referred to as orthogonal transformations. 10 Chapter 1 Mathematical preliminaries 1.5 Cartesian tensors Scalars, vectors, matrices, and higher-order quantities can be represented by a general index notational scheme. Using this approach, all quantities may then be referred to as tensors of different orders. The previously presented transformation properties of a vector can be used to establish the general transformation properties of these tensors. Restricting the transformations to those only between Cartesian coordinate systems, the general set of transformation relations for various orders can be written as a0 ¼ a; zero order ðscalarÞ a0i ¼ Qip ap ; first order ðvectorÞ a0ij ¼ Qip Qjq apq ; second order ðmatrixÞ a0ijk ¼ Qip Qjp Qkr apqr ; third order a0ijkl (1.5.1) ¼ Qip Qjq Qkr Qls apqrs ; fourth order « a0ijk.m ¼ Qip Qjq Qkr /Qmt apqr.t ; general order Note that, according to these definitions, a scalar is a zero-order tensor, a vector is a tensor of order one, and a matrix is a tensor of order two. Relations (1.5.1) then specify the transformation rules for the components of Cartesian tensors of any order under the rotation Qij. This transformation theory proves to be very valuable in determining the displacement, stress, and strain in different coordinate directions. Some tensors are of a special form in which their components remain the same under all transformations, and these are referred to as isotropic tensors. It can be easily verified (see Exercise 1.8) that the Kronecker delta dij has such a property and is therefore a second-order isotropic tensor. The alternating symbol εijk is found to be the third-order isotropic form. The fourth-order case (Exercise 1.9) can be expressed in terms of products of Kronecker deltas, and this has important applications in formulating isotropic elastic constitutive relations in Section 4.2. The distinction between the components and the tensor should be understood. Recall that a vector v can be expressed as v ¼ v1 e1 þ v2 e2 þ v3 e3 ¼ vi ei ¼ v01 e01 þ v02 e02 þ v03 e03 ¼ v0i e0i (1.5.2) In a similar fashion, a second-order tensor A can be written A ¼ A11 e1 e1 þ A12 e1 e2 þ A13 e1 e3 þ A21 e2 e1 þ A22 e2 e2 þ A23 e2 e3 þ A31 e3 e1 þ A32 e3 e2 þ A33 e3 e3 (1.5.3) ¼ Aij ei ej ¼ A0ij e0i e0j and similar schemes can be used to represent tensors of higher order. The representation used in Eq. (1.5.3) is commonly called dyadic notation, and some authors write the dyadic products eiej using a tensor product notation ei 5 ej. Additional information on dyadic notation can be found in Weatherburn (1948) and Chou and Pagano (1967). Relations (1.5.2) and (1.5.3) indicate that any tensor can be expressed in terms of components in any coordinate system, and it is only the components that change under coordinate transformation. For example, the state of stress at a point in an elastic solid depends on the problem geometry and applied 1.5 Cartesian tensors 11 loadings. As is shown later, these stress components are those of a second-order tensor and therefore obey transformation law (1.5.1)3. Although the components of the stress tensor change with choice of coordinates, the stress tensor (representing the state of stress) does not. An important property of a tensor is that if we know its components in one coordinate system, we can find them in any other coordinate frame by using the appropriate transformation law. Because the components of Cartesian tensors are representable by indexed symbols, the operations of equality, addition, subtraction, multiplication, and so forth are defined in a manner consistent with the indicial notation procedures previously discussed. The terminology tensor without the adjective Cartesian usually refers to a more general scheme in which the coordinates are not necessarily rectangular Cartesian and the transformations between coordinates are not always orthogonal. Such general tensor theory is not discussed or used in this text. Example 1.2 Transformation examples The components of a first- and second-order tensor in a particular coordinate frame are given by 2 3 2 1 1 6 7 6 ai ¼ 4 4 5; aij ¼ 4 0 2 3 3 0 3 7 2 25 2 4 Determine the components of each tensor in a new coordinate system found through a rotation of 60 (p/3 radians) about the x3-axis. Choose a counterclockwise rotation when viewing down the negative x3-axis (see Fig. 1.2). The original and primed coordinate systems shown in Fig. 1.2 establish the angles between the various axes. The solution starts by determining the rotation matrix for this case 2 cos 60 cos 30 6 Qij ¼ 4 cos 150 cos 90 cos 60 cos 90 cos 90 3 2 1=2 7 6 pffiffiffi cos 90 5 ¼ 4 3=2 cos 0 0 pffiffiffi 3 3=2 0 7 1=2 0 5 0 1 The transformation for the vector quantity follows from Eq. (1.5.1)2 2 1=2 6 pffiffiffi a0i ¼ Qij aj ¼ 4 3=2 0 pffiffiffi 3=2 1=2 0 pffiffiffi 3 32 3 2 1 1=2 þ 2 3 pffiffiffi 7 76 7 6 0 54 4 5 ¼ 4 2 3=2 5 2 1 2 0 and the second-order tensor (matrix) transforms according to (1.5.1)3 2 a0ij ¼ Qip Qjq apq 1=2 6 pffiffiffi 6 ¼ 6 3=2 4 2 6 6 ¼6 4 0 pffiffiffi 32 1 3=2 0 76 76 1=2 0 76 0 54 3 0 1 pffiffiffi 3=4 7=4 pffiffiffi 3=4 5=4 pffiffiffi pffiffiffi 3=2 þ 3 1 3 3=2 0 3 32 1=2 76 pffiffiffi 76 2 2 76 3=2 54 2 4 0 pffiffiffi 3 3=2 þ 3 pffiffiffi 7 7 1 3 3 27 5 4 pffiffiffi 3T 3=2 0 7 7 1=2 0 7 5 0 1 12 Chapter 1 Mathematical preliminaries where [ ]T indicates transpose (defined in Section 1.7). Although simple transformations can be worked out by hand, for more general cases it is more convenient to use a computational scheme to evaluate the necessary matrix multiplications required in the transformation laws (1.5.1). MATLAB software is ideally suited to carry out such calculations, and an example program to evaluate the transformation of second-order tensors is given in Example C.1 in Appendix C. 1.6 Principal values and directions for symmetric second-order tensors Considering the tensor transformation concept previously discussed, it should be apparent that there might exist particular coordinate systems in which the components of a tensor take on maximum or minimum values. This concept is easily visualized when we consider the components of a vector as shown in Fig. 1.1. If we choose a particular coordinate system that has been rotated so that the x3-axis lies along the direction of the vector, then the vector will have components v ¼ {0, 0, jvj}. For this case, two of the components have been reduced to zero, while the remaining component becomes the largest possible (the total magnitude). This situation is most useful for symmetric second-order tensors that eventually represent the stress and/or strain at a point in an elastic solid. The direction determined by the unit vector n is said to be a principal direction or eigenvector of the symmetric second-order tensor aij if there exists a parameter l such that aij nj ¼ lni a11 n1 þ a12 n2 þ a13 n3 ¼ ln1 (1.6.1) a21 n1 þ a22 n2 þ a23 n3 ¼ ln2 a31 n1 þ a32 n2 þ a33 n3 ¼ ln3 x3 x3 x2 x2 60 x1 x1 FIG. 1.2 Coordinate transformation. 1.6 Principal values and directions for symmetric second-order tensors 13 where l is called the principal value or eigenvalue of the tensor. Relation (1.6.1) can be rewritten as ðaij ldij Þnj ¼ 0 and this expression is simply a homogeneous system of three linear algebraic equations in the unknowns n1, n2, n3. The system possesses a nontrivial solution if and only if the determinant of its coefficient matrix vanishes det½aij ldij ¼ 0 Expanding the determinant produces a cubic equation in terms of l det½aij ldij ¼ l3 þ Ia l2 IIa l þ IIIa ¼ 0 (1.6.2) where Ia ¼ aii ¼ a11 þ a22 þ a33 a11 1 IIa ¼ ðaii ajj aij aij Þ ¼ 2 a21 a12 a22 þ a22 a32 a23 a11 þ a33 a31 a13 a33 (1.6.3) IIIa ¼ det½aij The scalars Ia, IIa, and IIIa are called the fundamental invariants of the tensor aij, and relation (1.6.2) is known as the characteristic equation. As indicated by their name, the three invariants do not change value under coordinate transformation. The roots of the characteristic equation determine the allowable values for l, and each of these may be back-substituted into relation (1.6.1) to solve for the associated principal direction n. Usually these principal directions are normalized using nn ¼ 1. Under the condition that the components aij are real, it can be shown that all three roots l1, l2, l3 of the cubic equation (1.6.2) must be real. Furthermore, if these roots are distinct, the principal directions associated with each principal value are orthogonal. Thus, we can conclude that every symmetric second-order tensor has at least three mutually perpendicular principal directions and at most three distinct principal values that are the roots of the characteristic equation. By denoting the principal directions n(1), n(2), n(3) corresponding to the principal values l1, l2, l3, three possibilities arise: 1. All three principal values are distinct; the three corresponding principal directions are unique (except for sense). 2. Two principal values are equal (l1 s l2 ¼ l3); the principal direction n(1) is unique (except for sense), and every direction perpendicular to n(1) is a principal direction associated with l2, l3. 3. All three principal values are equal; every direction is principal, and the tensor is isotropic, as per discussion in the previous section. Therefore, according to what we have presented, it is always possible to identify a right-handed Cartesian coordinate system such that each axis lies along the principal directions of any given 14 Chapter 1 Mathematical preliminaries symmetric second-order tensor. Such axes are called the principal axes of the tensor. For this case, the basis vectors are actually the unit principal directions {n(1), n(2), n(3)}, and it can be shown that with respect to principal axes the tensor reduces to the diagonal form 3 2 l1 0 0 7 6 a0ij ¼ 4 0 l2 0 5 (1.6.4) 0 0 l3 Note that the fundamental invariants defined by relations (1.6.3) can be expressed in terms of the principal values as Ia ¼ l1 þ l2 þ l3 IIa ¼ l1 l2 þ l2 l3 þ l3 l1 (1.6.5) IIIa ¼ l1 l2 l3 The eigenvalues have important extremal properties. If we arbitrarily rank the principal values such that l1 > l2 > l3, then l1 will be the largest of all possible diagonal elements, while l3 will be the smallest diagonal element possible. This theory is applied in elasticity as we seek the largest stress or strain components in an elastic solid. Example 1.3 Principal value problem Determine the invariants and principal values and directions of the following symmetric secondorder tensor 3 2 2 0 0 7 6 aij ¼ 4 0 3 4 5 0 4 3 The invariants follow from relations (1.6.3) Ia ¼ aii ¼ 2 þ 3 3 ¼ 2 2 0 3 4 2 0 IIa ¼ þ þ ¼ 6 25 6 ¼ 25 0 3 4 3 0 3 2 IIIa ¼ 0 0 0 3 4 ¼ 2ð9 16Þ ¼ 50 4 3 0 The characteristic equation then becomes det½aij ldij ¼ l3 þ 2l2 þ 25l 50 ¼ 0 0ðl 2Þ l2 25 ¼ 0 rl1 ¼ 5; l2 ¼ 2; l3 ¼ 5 Thus, for this case all principal values are distinct. 1.6 Principal values and directions for symmetric second-order tensors 15 For the l1 ¼ 5 root, Eq. (1.6.1) gives the system ð1Þ 3n1 ¼ 0 ð1Þ ð1Þ 2n2 þ 4n3 ¼ 0 ð1Þ ð1Þ 4n2 8n3 ¼ 0 pffiffiffi which gives a normalized solution n(1) ¼ (2e2 þ e3)/ 5. In similar fashion, the other two prinpffiffiffi cipal directions are found to be n(2) ¼ e1, n(3) ¼ (e2 2e3)/ 5. It is easily verified that these directions are mutually orthogonal. Fig. 1.3 illustrates their directions with respect to the given coordinate system, and this establishes the right-handed principal coordinate axes x01 ; x02 ; x03 For this case, the transformation matrix Qij defined by (1.4.1) becomes 2 pffiffiffi pffiffiffi 3 0 2= 5 1= 5 6 7 Qij ¼ 4 1 0 0 5 pffiffiffi pffiffiffi 0 1= 5 2= 5 Notice the eigenvectors actually form the rows of the Q matrix. Using this in the transformation law (1.5.1)3, the components of the given second-order tensor become 2 5 0 0 3 7 6 a0ij ¼ 4 0 2 0 5 0 0 5 This result then validates the general theory given by relation (1.6.4), indicating that the tensor should take on diagonal form with the principal values as the elements. Only simple second-order tensors lead to a characteristic equation that is factorable, thus allowing solution by hand calculation. Most other cases normally develop a general cubic equation and a more complicated system to solve for the principal directions. Again, particular routines within the MATLAB package offer convenient tools to solve these more general problems. Example C.2 in Appendix C provides a simple code to determine the principal values and directions for symmetric second-order tensors. x3 x1′ n (1) n (2) x2 n (3) x1 x2′ x3′ FIG. 1.3 Principal axes for Example 1.3. 16 Chapter 1 Mathematical preliminaries 1.7 Vector, matrix, and tensor algebra Elasticity theory requires the use of many standard algebraic operations among vector, matrix, and tensor variables. These operations include dot and cross products of vectors and numerous matrix/ tensor products. All of these operations can be expressed efficiently using compact tensor index notation. First, consider some particular vector products. Given two vectors a and b, with Cartesian components ai and bi, the scalar or dot product is defined by a $ b ¼ a1 b1 þ a2 b2 þ a 3 b3 ¼ ai bi (1.7.1) Because all indices in this expression are repeated, the quantity must be a scalar, that is, a tensor of order zero. The magnitude of a vector can then be expressed as a ¼ ða$aÞ1=2 ¼ ðai ai Þ1=2 (1.7.2) The vector or cross product between two vectors a and b can be written as e1 a b ¼ a1 b1 e2 a2 b2 e3 a3 ¼ εijk aj bk ei b3 (1.7.3) where ei are the unit basis vectors for the coordinate system. Note that the cross product gives a vector resultant whose components are εijkajbk. Another common vector product is the scalar triple product defined by a1 a 2 a3 (1.7.4) a $ b c ¼ b1 b2 b3 ¼ εijk ai bj ck c1 c2 c3 Next consider some common matrix products. Using the usual direct notation for matrices and vectors, common products between a matrix A ¼ [A] with a vector a can be written as Aa ¼ ½Afag ¼ Aij aj ¼ aj Aij aT A ¼ fagT ½A ¼ ai Aij ¼ Aij ai (1.7.5) where aT denotes the transpose, and for a vector quantity this simply changes the column matrix (3 1) into a row matrix (1 3). Note that each of these products results in a vector resultant. These types of expressions generally involve various inner products within the index notational scheme and, as noted, once the summation index is properly specified, the order of listing the product terms does not change the result. We will encounter several different combinations of products between two matrices A and B AB ¼ ½A½B ¼ Aij Bjk ABT ¼ Aij Bkj AT B ¼ Aji Bjk trðABÞ ¼ Aij Bji tr ABT ¼ tr AT B ¼ Aij Bij (1.7.6) 1.8 Calculus of Cartesian tensors 17 where AT indicates the transpose and tr (A) is the trace of the matrix defined by ATij ¼ Aji (1.7.7) tr ðAÞ ¼ Aii ¼ A11 þ A22 þ A33 Similar to vector products, once the summation index is properly specified, the results in (1.7.6) do not depend on the order of listing the product terms. Note that this does not imply that AB ¼ BA, which is certainly not true. 1.8 Calculus of Cartesian tensors Most variables within elasticity theory are field variables, that is, functions depending on the spatial coordinates used to formulate the problem under study. For time-dependent problems, these variables could also have temporal variation. Thus, our scalar, vector, matrix, and general tensor variables are functions of the spatial coordinates (x1, x2, x3). Because many elasticity equations involve differential and integral operations, it is necessary to have an understanding of the calculus of Cartesian tensor fields. Further information on vector differential and integral calculus can be found in Hildebrand (1976) and Kreyszig (2011). The field concept for tensor components can be expressed as a ¼ aðx1 ; x2 ; x3 Þ ¼ aðxi Þ ¼ aðxÞ ai ¼ ai ðx1 ; x2 ; x3 Þ ¼ ai ðxi Þ ¼ ai ðxÞ aij ¼ aij ðx1 ; x2 ; x3 Þ ¼ aij ðxi Þ ¼ aij ðxÞ « It is convenient to introduce the comma notation for partial differentiation a;i ¼ v a; vxi ai;j ¼ v ai ; vxj aij;k ¼ v aij ; vxk / It can be shown that if the differentiation index is distinct, the order of the tensor is increased by one. For example, the derivative operation on a vector ai, j produces a second-order tensor or matrix given by 3 2 va1 va1 va1 6 vx vx vx 7 2 37 6 1 7 6 6 va2 va2 va2 7 7 6 ai;j ¼ 6 7 6 vx1 vx2 vx3 7 7 6 4 va3 va3 va3 5 vx1 vx2 vx3 Using Cartesian coordinates (x, y, z), consider the directional derivative of a scalar field function f with respect to a direction s df vf dx vf dy vf dz ¼ þ þ ds vx ds vy ds vz ds 18 Chapter 1 Mathematical preliminaries Note that the unit vector in the direction of s can be written as dx dy dz e1 þ e2 þ e3 ds ds ds Therefore, the directional derivative can be expressed as the following scalar product n¼ df ¼ n$Vf ds (1.8.1) where Vf is called the gradient of the scalar function f and is defined by Vf ¼ grad f ¼ e1 vf vf vf þ e2 þ e3 vx vy vz (1.8.2) and the symbolic vector operator V is called the del operator V ¼ e1 v v v þ e2 þ e3 vx vy vz (1.8.3) These and other useful operations can be expressed in Cartesian tensor notation. Given the scalar field f and vector field u, the following common differential operations can be written in index notation Gradient of a Scalar Vf ¼ f;i ei Gradient of a Vector Vu ¼ ui;j ei ej Laplacian of a Scalar V2 f ¼ V$Vf ¼ f;ii Divergence of a Vector V$u ¼ ui;i Curl of a Vector (1.8.4) V u ¼ εijk uk;j ei Laplacian of a Vector V2 u ¼ ui;kk ei If f and j are scalar fields and u and v are vector fields, several useful identities exist VðfjÞ ¼ ðVfÞj þ fðVjÞ V2 ðfjÞ ¼ V2 f j þ f V2 j þ 2Vf$Vj V$ðfuÞ ¼ Vf$u þ fðV$uÞ V ðfuÞ ¼ Vf u þ fðV uÞ V$ðu vÞ ¼ v$ðV uÞ u$ðV vÞ V Vf ¼ 0 (1.8.5) V$Vf ¼ V2 f V$V u ¼ 0 V ðV uÞ ¼ VðV$uÞ V2 u 1 u ðV uÞ ¼ Vðu$uÞ u$Vu 2 Each of these identities can be easily justified by using index notation from definition relations (1.8.4). 1.8 Calculus of Cartesian tensors 19 Example 1.4 Scalar and vector field examples Scalar and vector field functions are given by f ¼ x2 y2 and u ¼ 2xe1 þ 3yze2 þ xye3. Calculate the following expressions, Vf, V2f, V$u, Vu, V u. Using the basic relations (1.8.4) Vf ¼ 2xe1 2ye2 V2 f ¼ 2 2 ¼ 0 V$u ¼ 2 þ 3z þ 0 ¼ 2 þ 3z 2 3 2 0 0 6 7 6 7 Vu ¼ ui;j ¼ 6 0 3z 3y 7 4 5 y x 0 e1 e2 e3 V u ¼ v=vx v=vy v=vz ¼ ðx 3yÞe1 ye2 2x 3yz xy Using numerical methods, some of these variables can be conveniently computed and plotted in order to visualize the nature of the field distribution. For example, contours of f ¼ constant can easily be plotted using MATLAB software, and vector distributions of Vf can be shown as plots of vectors properly scaled in magnitude and orientation. Fig. 1.4 shows these two types of plots, and it is observed that the vector field Vf is orthogonal to the f-contours, a result that is true in general for all scalar fields. Numerically generated plots such as these will prove to be useful in understanding displacement, strain, and/or stress distributions found in the solution to a variety of problems in elasticity. y x FIG. 1.4 Contours of f [ constant and vector distributions of Vf. 20 Chapter 1 Mathematical preliminaries Next consider some results from vector/tensor integral calculus. We simply list some theorems that have later use in the development of elasticity theory. 1.8.1 Divergence or Gauss theorem Let S be a piecewise continuous surface bounding the region of space V. If a vector field u is continuous and has continuous first derivatives in V, then ZZ ZZZ u n dS ¼ V u dV (1.8.6) S V where n is the outer unit normal vector to surface S. This result is also true for tensors of any order ZZ ZZZ aij .k nk dS ¼ aij .k; k dV (1.8.7) S V 1.8.2 Stokes theorem Let S be an open two-sided surface bounded by a piecewise continuous simple closed curve C. If u is continuous and has continuous first derivatives on S, then I ZZ u dr ¼ ðV uÞ n dS (1.8.8) C S where the positive sense for the line integral is for the region S to lie to the left as one traverses curve C and n is the unit normal vector to S. Again, this result is also valid for tensors of arbitrary order, and so I ZZ aij.k dxk ¼ εrsk aij.k;s nr dS (1.8.9) C S It can be shown that both divergence and Stokes theorems can be generalized so that the dot product in (1.8.6) and/or (1.8.8) can be replaced with a cross product. 1.8.3 Green’s theorem in the plane Applying Stokes theorem to a planar domain S with the vector field selected as u ¼ fe1 þ ge2 gives the result Z ZZ vg vf dxdy ¼ ðfdx þ gdyÞ (1.8.10) vy S vx C Further, special choices with either f ¼ 0 or g ¼ 0 imply Z Z ZZ ZZ vg vf dxdy ¼ gnx ds; dxdy ¼ fny ds S vx C S vy C (1.8.11) 1.9 Orthogonal curvilinear coordinates 21 1.8.4 Zero-value or localization theorem Let fij.k be a continuous tensor field of any order defined in an arbitrary region V. If the integral of fij.k over V vanishes, then fij.k must vanish in V ZZZ fij.k dV ¼ 0 0 fij.k ¼ 0 V (1.8.12) V 1.9 Orthogonal curvilinear coordinates Many applications in elasticity theory involve domains that have curved boundary surfaces, commonly including circular, cylindrical, and spherical surfaces. To formulate and develop solutions for such problems, it is necessary to use curvilinear coordinate systems. This requires redevelopment of some previous results in orthogonal curvilinear coordinates. Before pursuing these general steps, we review the two most common curvilinear systems, cylindrical and spherical coordinates. The cylindrical coordinate system shown in Fig. 1.5 uses (r, q, z) coordinates to describe spatial geometry. Relations between the Cartesian and cylindrical systems are given by x1 ¼ r cos q; x2 ¼ r sin q; x3 ¼ z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 x2 r ¼ x1 þ x2 ; q ¼ tan ; z ¼ x3 x1 (1.9.1) x3 z êz êθ êr e3 e1 x2 e2 θ r x1 FIG. 1.5 Cylindrical coordinate system. The spherical coordinate system is shown in Fig. 1.6 and uses (R, f, q) coordinates to describe geometry. The relations between Cartesian and spherical coordinates are x1 ¼ R cos q sin f; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ x21 þ x22 þ x23 ; x2 ¼ R sin q sin f; f ¼ cos 1 x3 ¼ R cos f x3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; x21 þ x22 þ x23 q ¼ tan 1 x2 x1 (1.9.2) 22 Chapter 1 Mathematical preliminaries x3 êR êθ φ êφ e3 e1 R θ e2 x2 x1 FIG. 1.6 Spherical coordinate system. The unit basis vectors for each of these curvilinear systems are illustrated in Figs. 1.5 and 1.6. These represent unit tangent vectors along each of the three orthogonal coordinate curves. Although primary use of curvilinear systems employs cylindrical and spherical coordinates, we briefly present a discussion valid for arbitrary coordinate systems. Consider the general case in which three orthogonal curvilinear coordinates are denoted by x1, x2, x3, while the Cartesian coordinates are defined by x1, x2, x3 (see Fig. 1.7). We assume invertible coordinate transformations exist between these systems specified by xm ¼ xm x1 ; x2 ; x3 ; xm ¼ xm x1 ; x2 ; x3 (1.9.3) In the curvilinear system, an arbitrary differential length in space can be expressed by 2 2 2 ðdsÞ2 ¼ h1 dx1 þ h2 dx2 þ h3 dx3 (1.9.4) where h1, h2, h3 are called scale factors that are in general non-negative functions of position. Let ek be the fixed Cartesian basis vectors and b e k the curvilinear basis (see Fig. 1.7). By using similar concepts from the transformations discussed in Section 1.4, the curvilinear basis can be expressed in terms of the Cartesian basis as b e1 ¼ dxk 1 vxk ek ¼ ek h1 vx1 ds1 b e2 ¼ dxk 1 vxk ek ¼ ek h2 vx2 ds2 b e3 ¼ dxk 1 vxk ek ¼ ek h3 vx3 ds3 (1.9.5) 1.9 Orthogonal curvilinear coordinates 23 x3 ξ3 ξ2 ê3 ê2 ê1 e3 ξ1 e2 e1 x2 x1 FIG. 1.7 Curvilinear coordinates. where we have used (1.9.4). By using the fact that be i $b e j ¼ dij ; relation (1.9.5) gives ðh1 Þ2 ¼ vxk vxk vx1 vx1 ðh2 Þ2 ¼ vxk vxk vx2 vx2 ðh3 Þ2 ¼ vxk vxk vx3 vx3 (1.9.6) It follows from (1.9.5) that the quantity Qkr ¼ 1 vxk ðno sum on rÞ hr vxr (1.9.7) represents the transformation tensor giving the curvilinear basis in terms of the Cartesian basis. This concept is similar to the transformation tensor Qij defined by (1.4.1) that is used between Cartesian systems. The physical components of a vector or tensor are simply the components in a local set of Cartesian axes tangent to the curvilinear coordinate curves at any point in space. Thus, by using transformation relation (1.9.7), the physical components of a tensor a in a general curvilinear system are given by a ¼ Qpi Qqj /Qsk apq.s (1.9.8) where apq.s are the components in a fixed Cartesian frame. Note that the tensor can be expressed in either system as a ¼ aij.k ei ej /ek ¼ a b ek eib e j /b (1.9.9) 24 Chapter 1 Mathematical preliminaries Because many applications involve differentiation of tensors, we must consider the differentiation of the curvilinear basis vectors. The Cartesian basis system ek is fixed in orientation and therefore vek / vxi ¼ vek/vxj ¼ 0. However, derivatives of the curvilinear basis do not in general vanish, and differentiation of relations (1.9.5) gives the following results vb em 1 vhm 1 vhm b b en er ; ¼ hn vxn hr vxr vxm vb em 1 vhn b en; ¼ hm vxm vxn msn; msnsr (1.9.10) no sum on repeated indices Using these results, the derivative of any tensor can be evaluated. Consider the first derivative of a vector u v v vu vb em b emÞ ¼ e m þ u n u ¼ n ðu b vxn vx vxn vx (1.9.11) The last term can be evaluated using (1.9.10), and thus the derivative of u can be expressed in terms of curvilinear components. Similar patterns follow for derivatives of higher-order tensors. All vector differential operators of gradient, divergence, curl, and so forth can be expressed in any general curvilinear system by using these techniques. For example, the vector differential operator previously defined in Cartesian coordinates in (1.8.3) is given by X 1 v 1 v 1 v 1 v b b b V¼b e1 ei þ e þ e ¼ (1.9.12) 2 3 h1 vx1 h2 vx2 h3 vx3 hi vxi i and this leads to the construction of the other common forms X 1 vf 1 vf 1 vf 1 vf b b b ei þ e þ e ¼ 2 3 h1 vx1 h2 vx2 h3 vx3 hi vxi i 1 X v h1 h2 h3 u Divergence of a Vector V $ u ¼ h1 h2 h3 i vxi hi ! 1 X v h1 h2 h3 vf 2 Laplacian of a Scalar V f ¼ h1 h2 h3 i vxi ðhi Þ2 vxi Gradient of a Scalar Vf ¼ b e1 XXX εijk v ðu h Þb e j k i hh i j k j k vx XXbe i vu vb ej b Gradient of a Vector Vu ¼ e j þ u i hi vxi vx i j 0 1 ! Xb X X b vu vb e v e e j i k b e j þ u k A [email protected] Laplacian of a Vector V2 u ¼ i k h h vx vx vx i k i j k Curl of a Vector Vu ¼ (1.9.13) (1.9.14) (1.9.15) (1.9.16) (1.9.17) (1.9.18) It should be noted that these forms are significantly different from those previously given in relations (1.8.4) for Cartesian coordinates. Curvilinear systems add additional terms not found in rectangular 1.9 Orthogonal curvilinear coordinates 25 coordinates. Other operations on higher-order tensors can be developed in a similar fashion (see Malvern, 1969, app. II). Specific transformation relations and field equations in cylindrical and spherical coordinate systems are given in Appendices A and B. Further discussion of these results is taken up in later chapters. Example 1.5 Polar coordinates Consider the two-dimensional case of a polar coordinate system as shown in Fig. 1.8. The differential length relation (1.9.4) for this case can be written as ðdsÞ2 ¼ ðdrÞ2 þ ðrdqÞ2 and thus h1 ¼ 1 and h2 ¼ r. By using relations (1.9.5) or simply by using the geometry shown in Fig. 1.8 b e r ¼ cos qe1 þ sin qe2 b e q ¼ sin qe1 þ cos qe2 (1.9.19) and so vb er ¼b eq; vq vb eq ¼ be r ; vq vb e r vbe q ¼ ¼0 vr vr (1.9.20) The basic vector differential operations then follow to be V¼b er v 1 v þb eq vr r vq Vf ¼ be r vf 1 vf þb eq vr r vq 1 v 1 vuq ðrur Þ þ r vr r vq 1 v vf 1 v2 f r þ 2 2 V2 f ¼ r vr vr r vq 1 v 1 vur b ez Vu¼ ðruq Þ r vr r vq vur vuq 1 vur 1 vuq be r b b uq b ur b er b e q be r þ eqb er þ eq þ eq Vu ¼ vr vr r vq r vq 2 vuq ur 2 vur uq 2 b 2 b e r þ V2 uq þ 2 eq V2 u ¼ V2 ur 2 r vq r r vq r V$u ¼ (1.9.21) where u ¼ ur b ez ¼ b e r be q . Notice that the Laplacian of a vector does not simply pass e r þ uq b eq, b through and operate on each of the individual components as in the Cartesian case. Additional terms are generated because of the curvature of the particular coordinate system. Similar relations can be developed for cylindrical and spherical coordinate systems (see Exercises 1.17 and 1.18). 26 Chapter 1 Mathematical preliminaries x2 êθ êr r θ e2 e1 x1 FIG. 1.8 Polar coordinate system. The material reviewed in this chapter is used in many places for formulation development of elasticity theory. Throughout the text, notation uses scalar, vector, and tensor formats depending on the appropriateness to the topic under discussion. Most of the general formulation procedures in Chapters 2e5 use tensor index notation, while later chapters commonly use vector and scalar notation. Additional review of mathematical procedures for problem solution is supplied in chapter locations where they are applied. References Chou PC, Pagano NJ: Elasticitydtensor, dyadic and engineering approaches, Princeton, NJ, 1967, D. Van Nostrand. Goodbody AM: Cartesian tensors: with applications to mechanics, fluid mechanics and elasticity, New York, 1982, Ellis Horwood. Hildebrand FB: Advanced calculus for applications, ed 2, Englewood Cliffs, NJ, 1976, Prentice Hall. Itskov M: Tensor algebra and tensor analysis for engineers, ed 4, New York, 2015, Springer. Kreyszig E: Advanced engineering mathematics, ed 10, New York, 2011, John Wiley. Malvern LE: Introduction to the mechanics of a continuous medium, Englewood Cliffs, NJ, 1969, Prentice Hall. Sadd MH: Continuum mechanics modeling of material behavior, 2019, Elsevier. Simmons JG: A brief on tensor analysis, New York, 1994, Springer. Weatherburn CE: Advanced vector analysis, LaSalle, IL, 1948, Open Court. Exercises 1.1 For the given matrix/vector pairs, compute the following quantities: aii, aijaij, aijajk, aijbj, aijbibj, bibj, bibi. For each case, point out whether the result is a scalar, vector or matrix. Note that aijbj is actually the matrix product [a]{b}, while aijajk is the product [a][a]. Exercises 2 2 3 0 7 6 7 7 6 7 2 7 ; bi ¼ 6 1 7 5 4 5 1 1 6 6 ðaÞ aij ¼ 6 0 4 4 0 1 1 1 0 1 6 6 ðcÞ aij ¼ 6 1 4 0 0 1 2 0 2 3 2 7 6 7 7 6 7 1 7 ; bi ¼ 6 1 7 5 4 5 1 2 6 6 ðbÞ aij ¼ 6 0 4 1 0 4 2 1 0 0 6 6 ðdÞ aij ¼ 6 0 4 2 0 2 2 2 3 1 6 7 7 7 6 7 2 7 ; bi ¼ 6 0 7 5 4 5 3 2 3 3 2 1 27 0 3 0 3 2 3 2 6 7 7 7 6 7 1 7 ; bi ¼ 6 0 7 5 4 5 0 1 1.2 Use the decomposition result (1.2.10) to express aij from Exercise 1.1 in terms of the sum of symmetric and antisymmetric matrices. Verify that a(ij) and a[ij] satisfy the conditions given in the last paragraph of Section 1.2. 1.3 If aij is symmetric and bij is antisymmetric, prove in general that the product aijbij is zero. Verify this result for the specific case by using the symmetric and antisymmetric terms from Exercise 1.2. 1.4 Explicitly verify the following properties of the Kronecker delta dij aj ¼ ai dij ajk ¼ aik 1.5 Formally expand the expression (1.3.4) for the determinant and justify that either index notation form yields a result that matches the traditional form for det[aij]. 1.6 Determine the components of the vector bi and matrix aij given in Exercise 1.1 in a new coordinate system found through a rotation of 45 (p/4 radians) about the x1-axis. The rotation direction follows the positive sense presented in Example 1.2. 1.7 Consider the two-dimensional coordinate transformation shown in Fig. 1.8. Through the counterclockwise rotation q, a new polar coordinate system is created. Show that the transformation matrix for this case is given by cos q sin q Qij ¼ sin q cos q b1 a11 a21 If bi ¼ , aij ¼ are the components of a first- and second-order tensor in the x1, b2 a12 a22 x2 system, calculate their components in the rotated polar coordinate system. 1.8 Show that the second-order tensor adij, where a is an arbitrary constant, retains its form under any transformation Qij. This form is then an isotropic second-order tensor. 1.9 The most general form of a fourth-order isotropic tensor can be expressed by adij dkl þ bdik djl þ gdil djk 28 Chapter 1 Mathematical preliminaries where a, b, and g are arbitrary constants. Verify that this form remains the same under the general transformation given by (1.5.1)5. 1.10 For the fourth-order isotropic tensor given in Exercise 1.9, show that if b ¼ g, then the tensor will have the following symmetry Cijkl ¼ Cklij. 1.11 Show that the fundamental invariants can be expressed in terms of the principal values as given by relations (1.6.5). 1.12 Determine the invariants, and principal values and directions of the following matrices. Use the determined principal directions to establish a principal coordinate system, and following the procedures in Example 1.3, formally transform (rotate) the given matrix into the principal system to arrive at the appropriate diagonal form. 3 2 2 3 1 1 0 2 1 0 7 6 6 7 7 6 7 ðaÞ6 4 1 1 0 5ðAnswer : li ¼ 2; 0; 1Þ ðbÞ4 1 2 0 5ðAnswer : li ¼ 3; 1; 0Þ 2 0 0 1 1 1 0 6 ðcÞ6 4 1 0 1 0 0 3 2 6 7 6 6 07 5ðAnswer : li ¼ 2; 0; 0Þ ðdÞ4 3 0 0 0 0 3 0 3 6 7 07 5ðAnswer : 0 6 li ¼ 3; 6; 9Þ 1.13 A second-order symmetric tensor field is given by 3 2 x1 0 2x1 7 6 aij ¼ 4 x1 6x21 0 5 0 0 5x1 Using MATLAB (or similar software), investigate the nature of the variation of the principal values and directions over the interval 1 x1 2. Formally plot the variation of the absolute value of each principal value over the range 1 x1 2. 1.14 Calculate the quantities V$u, V u, V2u, Vu, tr(Vu) for the following Cartesian vector fields: ðaÞ u ¼ 2x1 e1 þ x1 x2 e2 þ 2x1 x3 e3 ðbÞ u ¼ x21 e1 þ 2x1 x2 x3 e2 þ 2x23 e3 ðcÞ u ¼ x2 e1 þ 2x2 x3 e2 þ 4x21 x3 e3 1.15 The dual vector ai of an antisymmetric second-order tensor aij is defined by ai ¼ 1/2εijkajk. Show that this expression can be inverted to get ajk ¼ εijkai. Exercises 29 1.16 Using index notation, explicitly verify the vector identities: ðaÞ ð1:8:5Þ1;2;3 ðbÞ ð1:8:5Þ4;5;6;7 ðcÞ ð1:8:5Þ8;9;10 1.17 Extend the results found in Example 1.5, and determine the forms of Vf, V$u, V2f, and V u for a three-dimensional cylindrical coordinate system (see Fig. 1.5). 1.18 For the spherical coordinate system (R, f, q) in Fig. 1.6, show that h1 ¼ 1; h2 ¼ R; h3 ¼ R sin f and the standard vector operations are given by Vf ¼ b eR vf 1 vf 1 vf þb ef þb eq vR R vf R sin f vq V$u ¼ 1 v 2 1 v 1 vuq R uR þ sin fuf þ 2 vR R sin f vf R sin f vq R V2 f ¼ 1 v 1 v vf 1 v2 f 2 vf R þ sin f þ vR vf R2 vR R2 sin f vf R2 sin2 f vq2 1 v vuf 1 vuR 1 v ðsin fuq Þ ðRuq Þ þb ef R sin f vf R sin f vq R vR vq Vu¼b eR þb eq vuR 1 v Ruf R vR vf
CHAPTER Deformation: displacements and strains 2 We begin development of the basic field equations of elasticity theory by first investigating the kinematics of material deformation. As a result of applied loadings, elastic solids will change shape or deform, and these deformations can be quantified by knowing the displacements of material points in the body. The continuum hypothesis establishes a displacement field at all points within the elastic solid. Using appropriate geometry, particular measures of deformation can be constructed leading to the development of the strain tensor. As expected, the strain components are related to the displacement field. The purpose of this chapter is to introduce the basic definitions of displacement and strain, establish relations between these two field quantities, and finally investigate requirements to ensure single-valued, continuous displacement fields. As appropriate for linear elasticity, these kinematical results are developed under the conditions of small deformation theory. Developments in this chapter lead to two fundamental sets of field equations: the strainedisplacement relations and the compatibility equations. Further field equation development, including internal force and stress distribution, equilibrium, and elastic constitutive behavior, occurs in subsequent chapters. 2.1 General deformations Under the application of external loading, elastic solids deform. A simple two-dimensional cantilever beam example is shown in Fig. 2.1. The undeformed configuration is taken with the rectangular beam in the vertical position, and the end loading displaces material points to the deformed shape as shown. As is typical in most problems, the deformation varies from point to point and is thus said to be nonhomogeneous. A superimposed square mesh is shown in the two configurations, and this indicates how elements within the material deform locally. It is apparent that elements within the mesh undergo extensional and shearing deformation. An elastic solid is said to be deformed or strained when the relative displacements between points in the body are changed. This is in contrast to rigid-body motion where the distance between points remains the same. In order to quantify deformation, consider the general example shown in Fig. 2.2. In the undeformed configuration, we identify two neighboring material points Po and P connected with the relative position vector r as shown. Through a general deformation, these points are mapped to locations P0o and P0 in the deformed configuration. For finite or large deformation theory, the undeformed and deformed configurations can be significantly different, and a distinction between these two configurations must be maintained leading to Lagrangian and Eulerian descriptions; see, for example, Sadd (2019). However, since we are developing linear elasticity, which uses only small deformation theory, the distinction between undeformed and deformed configurations can be dropped. Elasticity. Copyright © 2021 Elsevier Inc. All rights reserved. 31 32 Chapter 2 Deformation: displacements and strains (Undeformed) (Deformed) FIG. 2.1 Two-dimensional deformation example. Using Cartesian coordinates, define the displacement vectors of points Po and P to be uo and u respectively. Since P and Po are neighboring points, we can use a Taylor series expansion around point Po to express the components of u as u ¼ uo þ vu vu vu rx þ ry þ rz vx vy vz v ¼ vo þ vv vv vv rx þ ry þ rz vx vy vz w ¼ wo þ (2.1.1) vw vw vw rx þ ry þ rz vx vy vz where u, v, w are the Cartesian components of the displacement vector. P′ P r Po (Undeformed) FIG. 2.2 General deformation between two neighboring points. r′ P′o (Deformed) 2.1 General deformations 33 Note that the higher-order terms of the expansion have been dropped since the components of r are small. The change in the relative position vector r can be written as Dr ¼ r0 r ¼ u uo (2.1.2) and using (2.1.1) gives Drx ¼ vu vu vu rx þ ry þ rz vx vy vz Dry ¼ vv vv vv rx þ ry þ rz vx vy vz Drz ¼ vw vw vw rx þ ry þ rz vx vy vz (2.1.3) or in index notation Dri ¼ ui; j rj The tensor ui, j is called the displacement gradient tensor, and may be written out as 2 3 vu vv vu 6 7 6 vx vy vz 7 6 7 6 vv vv vv 7 6 7 ui; j ¼ 6 7 6 vx vy vz 7 6 7 6 vw vv vw 7 4 5 vx vy vz (2.1.4) (2.1.5) From relation (1.2.10), this tensor can be decomposed into symmetric and antisymmetric parts as ui; j ¼ eij þ uij (2.1.6) where 1 ui; j þ uj; i 2 (2.1.7) 1 uij ¼ ui; j uj; i 2 The tensor eij is called the strain tensor, while uij is referred to as the rotation tensor. Relations (2.1.4) and (2.1.6) thus imply that for small deformation theory, the change in the relative position vector between neighboring points can be expressed in terms of a sum of strain and rotation components. Combining relations (2.1.2), (2.1.4), and (2.1.6), and choosing ri ¼ dxi, we can also write the general result in the form eij ¼ ui ¼ uoi þ eij dxj þ uij dxj (2.1.8) 34 Chapter 2 Deformation: displacements and strains Because we are considering a general displacement field, these results include both strain deformation and rigid-body motion. Recall from Exercise 1.15 that a dual vector ui can be associated with the rotation tensor such that ui ¼ 1/2εijkujk. Using this definition, it is found that 1 vu3 vu2 u1 ¼ u32 ¼ 2 vx2 vx3 1 vu1 vu3 u2 ¼ u13 ¼ (2.1.9) 2 vx3 vx1 1 vu2 vu1 u3 ¼ u21 ¼ 2 vx1 vx2 which can be expressed collectively in vector format as u ¼ ð1=2ÞðV uÞ: As is shown in the next section, these components represent rigid-body rotation of material elements about the coordinate axes. These general results indicate that the strain deformation is related to the strain tensor eij, which in turn is related to the displacement gradients. We next pursue a more geometric approach and determine specific connections between the strain tensor components and geometric deformation of material elements. 2.2 Geometric construction of small deformation theory Although the previous section developed general relations for small deformation theory, we now wish to establish a more geometrical interpretation of these results. Typically, elasticity variables and equations are field quantities defined at each point in the material continuum. However, particular field equations are often developed by first investigating the behavior of infinitesimal elements (with coordinate boundaries), and then a limiting process is invoked that allows the element to shrink to a point. Thus, consider the common deformational behavior of a rectangular element as shown in Fig. 2.3. The usual types of motion include rigid-body rotation and extensional and shearing deformations as (Undeformed Element) (Rigid-Body Rotation) (Horizontal Extension) (Vertical Extension) FIG. 2.3 Typical deformations of a rectangular element. (Shearing Deformation) 2.2 Geometric construction of small deformation theory 35 illustrated. Rigid-body motion does not contribute to the strain field, and thus also does not affect the stresses. We therefore focus our study primarily on the extensional and shearing deformation. Fig. 2.4 illustrates the two-dimensional deformation of a rectangular element with original dimensions dx by dy. After deformation, the element takes a parallelogram form as shown in the dotted outline. The displacements of various corner reference points are indicated in the figure. Reference point A is taken at location (x,y), and the displacement components of this point are thus u(x,y) and v(x,y). The corresponding displacements of point B are u(x þ dx,y) and v(x þ dx,y), and the displacements of the other corner points are defined in an analogous manner. According to small deformation theory, u(x þ dx,y) z u(x,y)þ(vu/vx)dx, with similar expansions for all other terms. The normal or extensional strain component in a direction n is defined as the change in length per unit length of fibers oriented in the n-direction. Normal strain is positive if fibers increase in length and negative if the fiber is shortened. In Fig. 2.4, the normal strain in the x-direction can thus be defined by εx ¼ A0 B0 AB AB From the geometry in Fig. 2.4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 vu vv vu vu vv vu 0 0 dx ¼ 1 þ 2 þ AB ¼ þ dx dx z 1 þ dx þ dx þ vx vx vx vx vx vx where, consistent with small deformation theory, we have dropped the higher-order terms. Using these results and the fact that AB ¼ dx, the normal strain in the x-direction reduces to εx ¼ vu vx (2.2.1) u dy y D′ C′ β v(x,y+dy) y D C B′ A′ α dy v dx x v(x,y) A dx B u(x+dx,y) u(x,y) x FIG. 2.4 Two-dimensional geometric strain deformation. 36 Chapter 2 Deformation: displacements and strains In similar fashion, the normal strain in the y-direction becomes εy ¼ vv vy (2.2.2) A second type of strain is shearing deformation, which involves angle changes (see Fig. 2.3). Shear strain is defined as the change in angle between two originally orthogonal directions in the continuum material. This definition is actually referred to as the engineering shear strain. Theory of elasticity applications generally use a tensor formalism that requires a shear strain definition corresponding to one-half the angle change between orthogonal axes; see previous relation (2.1.7)1. Measured in radians, shear strain is positive if the right angle between the positive directions of the two axes decreases. Thus, the sign of the shear strain depends on the coordinate system. In Fig. 2.4, the engineering shear strain with respect to the x- and y-directions can be defined as p :C 0 A0 B0 ¼ a þ b 2 For small deformations, a z tan a and b z tan b, and the shear strain can then be expressed as gxy ¼ gxy vu vv dy dx vu vv vy þ þ ¼ ¼ vx vu vv vy vx dx þ dx dy þ dy dx vy (2.2.3) where we have again neglected higher-order terms in the displacement gradients. Note that each derivative term is positive if lines AB and AC rotate inward as shown in the figure. By simple interchange of x and y and u and v, it is apparent that gxy ¼ gyx. By considering similar behaviors in the y-z and x-z planes, these results can be easily extended to the general three-dimensional case, giving the results εx ¼ vu vv vw ; εy ¼ ; εz ¼ vx vy vz (2.2.4) gxy vu vv vv vw vw vu þ ; gyz ¼ þ ; gzx ¼ þ ¼ vy vx vz vy vx vz Thus, we define three normal and three shearing strain components, leading to a total of six independent components that completely describe small deformation theory. This set of equations is normally referred to as the strainedisplacement relations. However, these results are written in terms of the engineering strain components, and tensorial elasticity theory prefers to use the strain tensor eij defined by (2.1.7)1. This represents only a minor change because the normal strains are identical and shearing strains differ by a factor of one-half; for example, e11 ¼ ex ¼ εx and e12 ¼ exy ¼ 1/2gxy, and so forth. Therefore, using the strain tensor eij, the strainedisplacement relations can be expressed in component form as vu vv vw ; ey ¼ ; ez ¼ vx vy vz 1 vu vv 1 vv vw 1 vw vu þ þ þ exy ¼ ; eyz ¼ ; ezx ¼ 2 vy vx 2 vz vy 2 vx vz ex ¼ (2.2.5) 2.2 Geometric construction of small deformation theory 37 Using the more compact tensor notation, these relations are written as eij ¼ 1 ui; j þ uj;i 2 (2.2.6) while in direct vector/matrix notation the form reads e¼ 1 ½Vu þ ðVuÞT 2 (2.2.7) where e is the strain matrix, Vu is the displacement gradient matrix, and ðVuÞT is its transpose. The strain is a symmetric second-order tensor (eij ¼ eji) and is commonly written in matrix format 2 3 ex exy exz 6 7 e ¼ ½e ¼ 4 exy ey eyz 5 (2.2.8) exz eyz ez Before we conclude this geometric presentation, consider the rigid-body rotation of our twodimensional element in the x-y plane, as shown in Fig. 2.5. If the element is rotated through a small rigid-body angular displacement about the z-axis, using the bottom element edge, the rotation angle is determined as vv/vx, while using the left edge, the angle is given by vu/vy. These two expressions are of course the same; that is, vv/vx ¼ vu/vy and note that this would imply exy ¼ 0. The rotation can then be expressed as uz ¼ [(vv/vx) (vu/vy)]/2, which matches with the expression given earlier in (2.1.9)3. The other components of rotation follow in an analogous manner. Relations for the constant rotation uz can be integrated to give the result u ¼ u o uz y (2.2.9) v ¼ vo þ u z x where uo and vo are arbitrary constant translations in the x- and y-directions. This result then specifies the general form of the displacement field for two-dimensional rigid-body motion. We can easily − u y y dy dx v x x FIG. 2.5 Two-dimensional rigid-body rotation. 38 Chapter 2 Deformation: displacements and strains verify that the displacement field given by (2.2.9) yields zero strain. For the three-dimensional case, the most general form of rigid-body displacement can be expressed as u ¼ u o uz y þ uy z v ¼ v o ux z þ uz x (2.2.10) w ¼ w o u y x þ ux y As shown later, integrating the strainedisplacement relations to determine the displacement field produces arbitrary constants and functions of integration, which are related to rigid-body motion terms of the form given by (2.2.9) or (2.2.10). Thus, it is importan