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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