Lecture Notes on
The Mechanics of Elastic Solids
Volume II:Continuum Mechanics
Version 1.0
Rohan Abeyaratne
Quentin Berg Professor of Mechanics
MIT Department of Mechanical Engineering
and
Director { SMART Center
Singapore MIT Alliance for Research and Technology
Copyright
c
Rohan Abeyaratne,1988
All rights reserved.
http://web.mit.edu/abeyaratne/lecture
notes.html
11 May 2012
3
Electronic Publication
Rohan Abeyaratne
Quentin Berg Professor of Mechanics
Department of Mechanical Engineering
77 Massachusetts Institute of Technology
Cambridge,MA 021394307,USA
Copyright
c
by Rohan Abeyaratne,1988
All rights reserved
Abeyaratne,Rohan,1952
Lecture Notes on The Mechanics of Elastic Solids.Volume II:Continuum Mechanics/
Rohan Abeyaratne { 1st Edition { Cambridge,MA and Singapore:
ISBN13:9780979186509
ISBN10:0979186501
QC
Please send corrections,suggestions and comments to abeyaratne.vol.2@gmail.com
Updated 5 May 2013,20 July 2012.
4
i
Dedicated to
Pods and Nangi
for their gifts of love and presence.
iii
NOTE TO READER
I had hoped to nalize this second set of notes an year or two after publishing Volume I
of this series back in 2007.However I have been distracted by various other interesting tasks
and it has sat on a backburner.Since I continue to receive email requests for this second
set of notes,I am now making Volume II available even though it is not as yet complete.In
addition,it has been\cleanedup"at a far more rushed pace than I would have liked.
In the future,I hope to suciently edit my notes on Viscoelastic Fluids and Microme
chanical Models of Viscoelastic Fluids so that they may be added to this volume;and if I
ever get around to it,a chapter on the mechanical response of materials that are aected by
electromagnetic elds.
I would be most grateful if the reader would please inform me of any errors in the notes
by emailing me at abeyaratne.vol.2@gmail.com.
v
PREFACE
During the period 1986  2008,the Department of Mechanical Engineering at MIT oered
a series of graduate level subjects on the Mechanics of Solids and Structures that included:
2.071:Mechanics of Solid Materials,
2.072:Mechanics of Continuous Media,
2.074:Solid Mechanics:Elasticity,
2.073:Solid Mechanics:Plasticity and Inelastic Deformation,
2.075:Advanced Mechanical Behavior of Materials,
2.080:Structural Mechanics,
2.094:Finite Element Analysis of Solids and Fluids,
2.095:Molecular Modeling and Simulation for Mechanics,and
2.099:Computational Mechanics of Materials.
Over the years,I have had the opportunity to regularly teach the second and third of
these subjects,2.072 and 2.074 (formerly known as 2.083),and the current three volumes
are comprised of the lecture notes I developed for them.First drafts of these notes were
produced in 1987 (Volumes I and III) and 1988 (Volume II) and they have been corrected,
rened and expanded on every subsequent occasion that I taught these classes.The material
in the current presentation is still meant to be a set of lecture notes,not a text book.It has
been organized as follows:
Volume I:A Brief Review of Some Mathematical Preliminaries
Volume II:Continuum Mechanics
Volume III:Elasticity
This is Volume II.
My appreciation for mechanics was nucleated by Professors Douglas Amarasekara and
Munidasa Ranaweera of the (then) University of Ceylon,and was subsequently shaped and
grew substantially under the in uence of Professors James K.Knowles and Eli Sternberg
of the California Institute of Technology.I have been most fortunate to have had the
opportunity to apprentice under these inspiring and distinctive scholars.
I would especially like to acknowledge the inumerable illuminating and stimulating in
teractions with my mentor,colleague and friend the late Jim Knowles.His in uence on me
cannot be overstated.
vi
I am also indebted to the many MIT students who have given me enormous fulllment
and joy to be part of their education.
I am deeply grateful for,and to,Curtis Almquist SSJE,friend and companion.
My understanding of elasticity as well as these notes have benetted greatly from many
useful conversations with Kaushik Bhattacharya,Janet Blume,Eliot Fried,Morton E.
Gurtin,Richard D.James,Stelios Kyriakides,David M.Parks,Phoebus Rosakis,Stewart
Silling and Nicolas Triantafyllidis,which I gratefully acknowledge.
Volume I of these notes provides a collection of essential denitions,results,and illus
trative examples,designed to review those aspects of mathematics that will be encountered
in the subsequent volumes.It is most certainly not meant to be a source for learning these
topics for the rst time.The treatment is concise,selective and limited in scope.For exam
ple,Linear Algebra is a far richer subject than the treatment in Volume I,which is limited
to real 3dimensional Euclidean vector spaces.
The topics covered in Volumes II and III are largely those one would expect to see covered
in such a set of lecture notes.Personal taste has led me to include a few special (but still
wellknown) topics.Examples of these include sections on the statistical mechanical theory
of polymer chains and the lattice theory of crystalline solids in the discussion of constitutive
relations in Volume II,as well as several initialboundary value problems designed to illustrate
various nonlinear phenomena also in Volume II;and sections on the socalled Eshelby problem
and the eective behavior of twophase materials in Volume III.
There are a number of Worked Examples and Exercises at the end of each chapter which
are an essential part of the notes.Many of these examples provide more details;or the proof
of a result that had been quoted previously in the text;or illustrates a general concept;or
establishes a result that will be used subsequently (possibly in a later volume).
The content of these notes are entirely classical,in the best sense of the word,and none
of the material here is original.I have drawn on a number of sources over the years as I
prepared my lectures.I cannot recall every source I have used but certainly they include
those listed at the end of each chapter.In a more general sense the broad approach and
philosophy taken has been in uenced by:
Volume I:A Brief Review of Some Mathematical Preliminaries
I.M.Gelfand and S.V.Fomin,Calculus of Variations,Prentice Hall,1963.
J.K.Knowles,Linear Vector Spaces and Cartesian Tensors,Oxford University Press,
New York,1997.
vii
Volume II:Continuum Mechanics
P.Chadwick,Continuum Mechanics:Concise Theory and Problems,Dover,1999.
J.L.Ericksen,Introduction to the Thermodynamics of Solids,Chapman and Hall,1991.
M.E.Gurtin,An Introduction to Continuum Mechanics,Academic Press,1981.
M.E.Gurtin,E.Fried and L.Anand,The Mechanics and Thermodynamics of Con
tinua,Cambridge University Press,2010.
J.K.Knowles and E.Sternberg,(Unpublished) Lecture Notes for AM136:Finite Elas
ticity,California Institute of Technology,Pasadena,CA 1978.
C.Truesdell and W.Noll,The nonlinear eld theories of mechanics,in Handbuch der
Physik,Edited by S.Flugge,Volume III/3,Springer,1965.
Volume III:Elasticity
M.E.Gurtin,The linear theory of elasticity,in Mechanics of Solids  Volume II,edited
by C.Truesdell,SpringerVerlag,1984.
J.K.Knowles,(Unpublished) Lecture Notes for AM135:Elasticity,California Institute
of Technology,Pasadena,CA,1976.
A.E.H.Love,A Treatise on the Mathematical Theory of Elasticity,Dover,1944.
S.P.Timoshenko and J.N.Goodier,Theory of Elasticity,McGrawHill,1987.
The following notation will be used in Volume II though there will be some lapses (for
reasons of tradition):Greek letters will denote real numbers;lowercase boldface Latin letters
will denote vectors;and uppercase boldface Latin letters will denote linear transformations.
Thus,for example,;; :::will denote scalars (real numbers);x;y;z;:::will denote vectors;
and X;Y;Z;:::will denote linear transformations.In particular,\o"will denote the null
vector while\0"will denote the null linear transformation.
One result of this notational convention is that we will not use the uppercase bold letter
X to denote the position vector of a particle in the reference conguration.Instead we use
the lowercase boldface letters x and y to denote the positions of a particle in the reference
and current congurations.
viii
Contents
1 Some Preliminary Notions 1
1.1 Bodies and Congurations.............................2
1.2 Reference Conguration..............................4
1.3 Description of Physical Quantities:Spatial and Referential (or Eulerian and
Lagrangian) forms.................................6
1.4 Eulerian and Lagrangian Spatial Derivatives...................7
1.5 Motion of a Body..................................9
1.6 Eulerian and Lagrangian Time Derivatives....................10
1.7 A Part of a Body..................................11
1.8 Extensive Properties and their Densities.....................12
2 Kinematics:Deformation 15
2.1 Deformation....................................16
2.2 Deformation Gradient Tensor.Deformation in the Neighborhood of a Particle.18
2.3 Some Special Deformations............................21
2.4 Transformation of Length,Orientation,Angle,Volume and Area.......25
2.4.1 Change of Length and Orientation....................26
2.4.2 Change of Angle..............................27
2.4.3 Change of Volume.............................28
ix
x CONTENTS
2.4.4 Change of Area...............................29
2.5 Rigid Deformation.................................30
2.6 Decomposition of Deformation Gradient Tensor into a Rotation and a Stretch.32
2.7 Strain........................................36
2.8 Linearization....................................39
2.9 Worked Examples and Exercises.........................42
3 Kinematics:Motion 63
3.1 Motion.......................................64
3.2 Rigid Motions...................................65
3.3 Velocity Gradient,Stretching and Spin Tensors.................66
3.4 Rate of Change of Length,Orientation,and Volume..............68
3.4.1 Rate of Change of Length and Orientation................68
3.4.2 Rate of Change of Angle..........................70
3.4.3 Rate of Change of Volume.........................71
3.4.4 Rate of Change of Area and Orientation.................71
3.5 Current Conguration as Reference Conguration................74
3.6 Worked Examples and Exercises.........................78
3.7 Transport Equations................................86
3.8 Change of Observer.Objective Physical Quantities...............89
3.9 Convecting and CoRotating Bases and Rates..................93
3.10 Linearization....................................95
3.11 Worked Examples and Exercises.........................96
4 Mechanical Balance Laws and Field Equations 101
4.1 Introduction....................................102
CONTENTS xi
4.2 Conservation of Mass...............................104
4.3 Force........................................105
4.4 The Balance of Momentum Principles......................110
4.5 A Consequence of Linear Momentum Balance:Stress..............110
4.6 Field Equations Associated with the Momentum Balance Principles......114
4.7 Principal Stresses..................................116
4.8 Formulation of Mechanical Principles with Respect to a Reference Conguration.118
4.9 Stress Power....................................125
4.9.1 A WorkEnergy Identity..........................125
4.9.2 Work Conjugate StressStrain Pairs...................126
4.10 Linearization....................................127
4.11 Objectivity of Mechanical Quantities.......................128
4.12 Worked Examples and Exercises.........................129
5 Thermodynamic Balance Laws and Field Equations 147
5.1 The First Law of Thermodynamics........................147
5.2 The Second Law of Thermodynamics.......................149
5.3 Formulation of Thermodynamic Principles with Respect to a Reference Con
guration......................................152
5.4 Summary......................................153
5.5 Objectivity of Thermomechanical Quantities...................154
5.6 Worked Examples and Exercises.........................158
6 Singular Surfaces and Jump Conditions 165
6.1 Introduction....................................165
6.2 Jump Conditions in 1D Theory..........................167
xii CONTENTS
6.3 Worked Examples and Exercises.........................171
6.4 Kinematic Jump Conditions in 3D........................173
6.5 Momentum,Energy and Entropy Jump Conditions in 3D...........176
6.5.1 Linear Momentum Balance Jump Condition...............176
6.5.2 Summary:Jump Conditions in Lagrangian Formulation........179
6.5.3 Jump Conditions in Eulerian Formulation................180
6.6 Worked Examples and Exercises.........................181
7 Constitutive Principles 193
7.1 Dierent Functional Forms of Constitutive Response Functions.Some Exam
ples.........................................196
7.2 Illustration.....................................198
8 Thermoelastic Materials 203
8.1 Constitutive Characterization in Primitive Form................203
8.2 Implications of the Entropy Inequality......................204
8.3 Implications of Material Frame Indierence...................207
8.4 Discussion......................................209
8.5 Material Symmetry.................................213
8.5.1 Some Examples of Material Symmetry Groups.............217
8.5.2 Imposing Symmetry Requirements on Constitutive Response Functions.218
8.6 Materials with Internal Constraints........................220
8.7 Some Models of Elastic Materials.........................223
8.7.1 A Compressible Fluid...........................223
8.7.2 NeoHookean Model............................224
8.7.3 BlatzKo Model..............................227
CONTENTS xiii
8.7.4 Gent Model.Limited Extensibility....................228
8.7.5 Fung Model for Soft Biological Tissue..................229
8.8 An Elastic Body with One Preferred Direction.................230
8.9 Linearized Thermoelasticity............................233
8.9.1 Linearized Isotropic Thermoelastic Material...............238
8.10 Worked Examples and Exercises.........................238
9 Elastic Materials:Micromechanical Models 253
9.1 Example:Rubber Elasticity...........................254
9.1.1 A Single Long Chain Molecule:A OneDimensional Toy Model....255
9.1.2 A Special Case of the Preceding OneDimensional Long Chain Molecule.258
9.1.3 A Single Long Chain Molecule in Three Dimensions..........261
9.1.4 A Single Long Chain Molecule:Langevin Statistics...........263
9.1.5 A Molecular Model for a Generalized NeoHookean Material......264
9.2 Example:Lattice Theory of Elasticity......................266
9.2.1 A Bravais Lattice.Pair Potential.....................267
9.2.2 Homogenous Deformation of a Bravais Lattice..............269
9.2.3 Traction and Stress............................270
9.2.4 Energy...................................273
9.2.5 Material Frame Indierence........................275
9.2.6 Linearized Elastic Moduli.Cauchy Relations..............275
9.2.7 Lattice and Continuum Symmetry....................276
9.2.8 Worked Examples and Exercises.....................280
10 Some Nonlinear Eects:Illustrative Examples 285
10.1 Example (1):Simple Shear............................285
xiv CONTENTS
10.2 Example (2):Deformation of an Incompressible Cube Under Prescribed Ten
sile Forces......................................288
10.3 Example (3):Growth of a Cavity.........................297
10.4 Example (4):In ation of a ThinWalled Tube..................301
10.5 Example (5):Nonlinear Wave Propagation....................310
10.6 Worked Examples and Exercises.........................317
11 Linearized (Thermo)Elasticity 319
11.1 Linearized Thermoelasticity...........................319
11.1.1 Worked Examples and Exercises.....................323
11.2 Linearized Elasticity:The Purely Mechanical Theory.............326
11.2.1 Worked Examples and Exercises.....................328
12 Compressible Fluids.Viscous Fluids.339
12.1 Compressible Inviscid Fluids (Elastic Fluids)..................340
12.1.1 Worked Examples and Exercises.....................344
12.1.2 Adiabatic Flows..............................346
12.1.3 Worked Examples and Exercises.....................347
12.2 Incompressible Viscous Fluids...........................350
12.2.1 Example:A Newtonian Fluid.......................354
12.2.2 Example:A Generalized Newtonian Fluid................355
12.2.3 Worked Examples and Exercises.....................356
12.2.4 An Important Remark:..........................364
12.3 Incompressible Inviscid Fluids...........................364
12.3.1 Worked Examples and Exercises.....................365
13 Liquid Crystals 371
CONTENTS xv
13.1 Introduction....................................371
13.2 Formulation of basic concepts...........................374
13.3 Reduced Constitutive Relations..........................381
13.3.1 Restrictions due to dissipation inequality.................381
13.3.2 Restrictions due to material frame indierence.............384
13.3.3 Summary..................................386
13.4 A Particular Constitutive Model.........................387
13.4.1 A Free Energy Function :the Frank Energy.............387
13.4.2 An Extra Stress .............................388
13.5 Boundary Conditions:Anchoring.........................389
13.6 Worked Examples and Exercises.........................389
Chapter 1
Some Preliminary Notions
In this preliminary chapter we introduce certain basic notions that underly the continuum
theory of materials.These concepts are essential ingredients of continuum modeling,though
sometimes they are used implicitly without much discussion.We shall devote some attention
to these notions in this chapter since that will allow for greater clarity in subsequent chapters.
For example we frequently speak of an\isotropic material".Does this mean that the
material copper,for example,is isotropic?Suppose we have a particular piece of copper
that is isotropic in a given conguration,and we deform it,will it still be isotropic?What
is isotropy a property of?The material,the body,or the conguration?Speaking of which,
what is the dierence between a body,a conguration,and a region of space occupied by a
body (and is it important to distinguish between them)?...Often we will want to consider
some physical property (e.g.the internal energy) associated with a part of a body (i.e.a
denite set of particles of the body).As the body moves through space and this part occupies
dierent regions of space at dierent times and the value of this property changes with time,
it may be important to be precise about the fact that this property is assigned to a xed
set of particles comprising the part of the body and not the changing region of space that it
occupies....Or,consider the propagation of a wavefront.Consider a point on the wavefront
and a particle of the body,both of which happen to be located at the same point in space
at a given instant.However these are distinct entities and at the next instant of time this
same point on the wavefront and particle of the body would no longer be colocated in space.
Thus in particular,the velocity of the point of the wavefront is dierent from the velocity
of the particle of the body,even though they are located at the same point in space at the
current instant.
1
2 CHAPTER 1.SOME PRELIMINARY NOTIONS
The concepts introduced in this chapter aim to clarify such issues.We will not be pedan
tic about these subtleties.Rather,we shall make use of the framework and terminology
introduced in this chapter only when it helps avoid confusion.The reader is encouraged
to pay special attention to the distinctions between the dierent concepts introduced here.
These concepts include the notions of
a body,
a conguration of the body,
a reference conguration of the body,
the region occupied by the body in some conguration,
a particle (or material point),
the location of a particle in some conguration,
a deformation,
a motion,
Eulerian and Lagrangian descriptions of a physical quantity,
Eulerian and Lagrangian spatial derivatives,and
Eulerian and Lagrangian time derivatives (including the material time derivative).
1.1 Bodies and Congurations.
Our aim is to develop a framework for studying how\objects that occur in nature"respond
to the application of forces or other external stimuli.In order to do this,we must construct
mathematical idealizations (i.e.mathematical models) of the\objects"and the\stimuli".
Specically,with regard to the\objects",we must model their geometric and constitutive
character.
We shall use the term\body"to be a mathematical abstraction of an\object that
occurs in nature".A body B is composed of a set of particles
1
p (or material points).In a
given conguration of the body,each particle is located at some denite point y in three
dimensional space.The set of all the points in space,corresponding to the locations of all
the particles,is the region R occupied by the body in that conguration.A particular body,
composed of a particular set of particles,can adopt dierent congurations under the action
1
A particle in continuum mechanics is dierent to what we refer to as a particle in classical mechanics.
For example,a particle in classical mechanics has a mass m> 0,while a particle in continuum mechanics is
not endowed with a property called mass.
1.1.BODIES AND CONFIGURATIONS.3
of dierent stimuli (forces,heating etc.) and therefore occupy dierent regions of space under
dierent conditions.Note the distinction between the body,a conguration of the body,and
the region the body occupies in that conguration;we make these distinctions rigorous in
what follows.Similarly note the distinction between a particle and the position in space it
occupies in some conguration.
In order to appreciate the dierence between a conguration and the region occupied
in that conguration,consider the following example:suppose that a body,in a certain
conguration,occupies a circular cylindrical region of space.If the object is\twisted"about
its axis (as in torsion),it continues to occupy this same (circular cylindrical) region of space.
Thus the region occupied by the body has not changed even though we would say that the
body is in a dierent\conguration".
More formally,in continuum mechanics a body B is a collection of elements which can
be put into onetoone correspondence with some region R of Euclidean point space
2
.An
element p 2 B is called a particle (or material point).Thus,given a body B;there is
necessarily a mapping that takes particles p 2 B into their geometric locations y 2 R in
threedimensional Euclidean space:
y = (p) where p 2 B;y 2 R:(1.1)
The mapping is called a conguration of the body B;y is the position occupied by the
particle p in the conguration ;and Ris the region occupied by the body in the conguration
.Often,we write R= (B):
Figure 1.1:A body B that occupies a region R in a conguration .A particle p 2 B is located at
the position y 2 R where y = (p).B is a mathematical abstraction.R is a region in threedimensional
Euclidean space.
2
Recall that a\region"is an open connected set.Thus a single particle p does not constitute a body.
4 CHAPTER 1.SOME PRELIMINARY NOTIONS
Since a conguration provides a onetoone mapping between the particles p and posi
tions y,there is necessarily an inverse mapping
1
from R!B:
p =
1
(y) where p 2 B;y 2 R:(1.2)
Observe that bodies and particles,in the terminology used here,refer to\abstract"enti
ties.Bodies are available to us through their congurations.Actual geometric measurements
can be made on the place occupied by a particle or the region occupied by a body.
1.2 Reference Conguration.
In order to identify a particle of a body,we must label the particles.The abstract particle
label p,while perfectly acceptable in principle and intuitively clear,is not convenient for
carrying out calculations.It is more convenient to pick some arbitrary conguration of the
body,say
ref
,and use the (unique) position x =
ref
(p) of a particle in that conguration
to label it instead.Such a conguration
ref
is called a reference conguration of the body.
It simply provides a convenient way in which to label the particles of a body.The particles
are now labeled by x instead of p.
A second reason for considering a reference conguration is the following:we can study
the geometric characteristics of a conguration by studying the geometric properties of
the points occupying the region R = (B).This is adequate for modeling certain materials
(such as many uids) where the behavior of the material depends only on the characteristics
of the conguration currently occupied by the body.In describing most solids however one
often needs to know the changes in geometric characteristics between one conguration and
another conguration (e.g.the change in length,the change in angle etc.).In order to
describe the change in a geometric quantity one must necessarily consider (at least) two
congurations of the body:the conguration that one wishes to analyze,and a reference
conguration relative to which the changes are to be measured.
Let
ref
and be two congurations of a body B and let R
ref
and R denote the regions
occupied by the body B in these two conguration;see Figure 1.2.The mappings
ref
and
take p!x and p!y,and likewise B!R
ref
and B!R:
x =
ref
(p);y = (p):(1.3)
Here p 2 B;x 2 R
ref
and y 2 R:Thus x and y are the positions of particle p in the two
congurations under consideration.
1.2.REFERENCE CONFIGURATION.5
Figure 1.2:A body B that occupies a region R in a conguration ,and another region R
ref
in a second
conguration
ref
.A particle p 2 B is located at y = (p) 2 R in conguration ,and at x =
ref
(p) 2 R
ref
in conguration
ref
.The mapping of R
ref
!R is described by the deformation y =
b
y(x) = (
1
ref
(x)).
This induces a mapping y = by(x) from R
ref
!R:
y =
b
y(x)
def
= (
1
ref
(x));x 2 R
ref
;y 2 R;(1.4)
by is called a deformation of the body from the reference connguration
ref
.
Frequently one picks a particular convenient (usually xed) reference conguration
ref
and studies deformations of the body relative to that conguration.This particular con
guration need only be one that the body can sustain,not necessarily one that is actually
sustained in the setting being analyzed.The choice of reference conguration is arbitrary
in principle (and is usually chosen for reasons of convenience).Note that the function by in
(1.4) depends on the choice of reference conguration.
When working with a single xed reference conguration,as we will most often do,one
can dispense with talking about the body B,a conguration and the particle p,and work
directly with the region R
ref
,the deformation y(x) and the position x.
However,even when working with a single xed reference conguration,sometimes,when
introducing a new concept,for reasons of clarity we shall start by using p,B etc.before
switching to x,R
ref
etc.
6 CHAPTER 1.SOME PRELIMINARY NOTIONS
There will be occasions when we must consider more than one reference conguration;
an example of this will be our analysis of material symmetry.In such circumstances one can
avoid confusion by framing the analysis in terms the body B,the reference congurations
1
;
2
etc.
1.3 Description of Physical Quantities:Spatial and
Referential (or Eulerian and Lagrangian) forms.
There are essentially two types of physical characteristics associated with a body.The rst,
such as temperature,is associated with individual particles of the body;the second,such
as mass and energy,are associated with\parts of the body".One sometimes refers to these
as intensive and extensive characteristics respectively.In this and the next few sections we
will be concerned with properties of the former type;we shall consider the latter type of
properties in Section 1.8.
First consider a characteristic such as the temperature of a particle.The temperature
of particle p in the conguration is given by
3
=
(p) (1.5)
where the function
(p) is dened for all p 2 B.Such a description,though completely
rigorous and welldened,is not especially useful for carrying out calculations since a particle
is an abstract entity.It is more useful to describe the temperature by a function of particle
position by trading p for y by using y = (p):
=
(y)
def
=
1
(y)
:(1.6)
The function
(y) is dened for all y 2 R.The functions
and
both describe temperature:
(p) is the temperature of the particle p while
(y) is the temperature of the particle located
at y.When p and y are related by y = (p),the two functions
and
have the same value
since they both refer to the temperature of the same particle in the same conguration and
they are related by (1.6)
2
.One usually refers to the representation (1.5) which deals directly
with the abstract particles as a material description;the representation (1.6) which deals
3
Even though it is cumbersome to do so,in order to clearly distinguish three dierent characterizations
of temperature from each other,we use the notation
();
() and
b
() to describe three distinct but related
functions dened on B;R and R
ref
respectively.
1.4.EULERIAN AND LAGRANGIAN SPATIAL DERIVATIVES.7
with the positions of the particles in the deformed conguration,(the conguration in which
the physical quantity is being characterized,) is called the Eulerian or spatial description.
If a reference conguration has been introduced we can label a particle by its position
x =
ref
(p) in that conguration,and this in turn allows us to describe physical quantities
in Lagrangian form.Consider again the temperature of a particle as given in (1.5).We can
trade p for x using x =
ref
(p) to describe the temperature in Lagrangian or referential form
by
=
b
(x)
def
=
(
1
ref
(x)):(1.7)
The function
b
is dened for all x 2 R
ref
:The referential description
b
(x) can also be
generated from the spatial description through
b
(x) =
(by(x)):(1.8)
It is essential to emphasize that the function
b
does not give the temperature of a particle
in the reference conguration;rather,
b
(x) is the temperature in the deformed conguration
of the particle located at x in the reference conguration.
A physical eld that is,for example,described by a function dened on R and expressed
as a function of y,can just as easily be expressed through a function dened on R
ref
and
expressed as a function of x;and vice versa.For example in the chapter on stress we will
encounter two 2tensors Tand S called the Cauchy stress and the rst PiolaKirchho stress.
It is customary to express T as a function of y 2 R and S as a function of x 2 R
ref
:T(y)
and S(x).This is because certain calculations simplify when done in this way.However they
both refer to stress at a particle in a deformed conguration where in one case the particle is
labeled by its position in the deformed conguration and in the other by its position in the
reference conguration.In fact,by making use of the deformation y = by(x) we can write T
as a function of x:
b
T(x) = T(by(x)),and likewise S as a function of y:
S(y) = S(by
1
(y)),if
we so need to.
1.4 Eulerian and Lagrangian Spatial Derivatives.
To be specic,consider again the temperature eld in the body in a conguration .We
can express this either in Lagrangian form
=
b
(x);x 2 R
ref
;(1.9)
8 CHAPTER 1.SOME PRELIMINARY NOTIONS
or in Eulerian form
=
(y);y 2 R:(1.10)
Both of these expressions give the temperature of a particle in the deformed conguration
where the only distinction is in the labeling of the particle.These two functions are related
by (1.8).
It is cumbersome to write the decorative symbols,i.e.,the\hats"and the\bars",all the
time and we would prefer to write (x) and (y).If such a notation is adopted one must be
particularly attentive and continuously use the context to decide which function one means.
Suppose,for example,that we wish to compute the gradient of the temperature eld.
If we write this as r we would not know if we were referring to the Lagrangian spatial
gradient
r
b
(x) which has components
@
b
@x
i
(x);(1.11)
or to the Eulerian spatial gradient
r
(y) which has components
@
@y
i
(y):(1.12)
In order to avoid this confusion we use the notation Grad and grad instead of r where
Grad = r
b
(x);and grad = r
(y):(1.13)
The gradient of the particular vector eld by(x),the deformation,is denoted by F(x) and
is known as the deformation gradient tensor:
F(x) = Grad
b
y(x) with components F
ij
=
@by
i
@x
j
(x):(1.14)
It plays a central role in describing the kinematics of a body.
The symbols Div and div,and Curl and curl are used similarly.
In order to relate Grad to grad we merely need to dierentiate (1.8) with respect to
x using the chain rule.This gives
@
b
@x
i
=
@
@y
j
@by
j
@x
i
=
@
@y
j
F
ji
= F
ji
@
@y
j
(1.15)
where summation over the repeated index j is taken for granted.This can be written as
Grad = F
T
grad :(1.16)
1.5.MOTION OF A BODY.9
Similarly,if w is any vector eld,one can show that
Grad w = (grad w)F;(1.17)
and for any tensor eld T,that
Div T = J div (J
1
FT) where J = det F:(1.18)
1.5 Motion of a Body.
A motion of a body is a oneparameter family of congurations (p;t) where the parameter
t is time:
y = (p;t);p 2 B;t
0
t t
1
:(1.19)
This motion takes place over the time interval [t
0
;t
1
].The body occupies a timedependent
region R
t
= (B;t) during the motion,and the vector y 2 R
t
is the position occupied by the
particle p at time t during the motion :For each particle p;(1.19) describes the equation
of a curve in threedimensional space which is the path of this particle.
Next consider the velocity and acceleration of a particle,dened as the rate of change of
position and velocity respectively of that particular particle:
v = v
(p;t) =
@
@t
(p;t) and a = a
(p;t) =
@
2
@t
2
(p;t):(1.20)
Since a particle p is only available to us through its location y,it is convenient to express
the velocity and acceleration as functions of y and t (rather than p and t).This is readily
done by using p =
1
(y;t) to eliminate p in favor of y in (1.20) leading to the velocity and
acceleration elds
v(y;t) and
a(y;t):
v =
v(y;t) = v
(p;t)
p=
1
(y;t)
= v
(
1
(y;t);t);
a =
a(y;t) = a
(p;t)
p=
1
(y;t)
= a
(
1
(y;t);t);
(1.21)
where v
(p;t) and a
(p;t) are given by (1.20).
It is worth emphasizing that the velocity and acceleration of a particle can be dened
without the need to speak of a reference conguration.
10 CHAPTER 1.SOME PRELIMINARY NOTIONS
If a reference conguration
ref
has been introduced and x =
ref
(p) is the position of a
particle in that conguration,we can describe the motion alternatively by
y =
b
y(x;t)
def
= (
1
ref
(x);t):(1.22)
The particle velocity in Lagrangian form is given by
v =
b
v(x;t) = v
(
1
ref
(x);t) (1.23)
or equivalently by
v = bv(x;t) =
v(by(x;t);t):(1.24)
Similar expressions for the acceleration can be written.The function bv(x;t) does not of
course give the velocity of a particle in the reference conguration but rather the velocity at
time t of the particle which is associated with the position x in the reference conguration.
Sometimes,the reference conguration is chosen to be the conguration of the body at
the initial instant,i.e.,
ref
(p) = (p;t
0
),in which case x = by(x;t
0
).
1.6 Eulerian and Lagrangian Time Derivatives.
To be specic,consider again the temperature eld in the body at time t:As noted
previously,it is cumbersome to write the decorative symbols,i.e.,the\hats"and the\bars"
over the Eulerian and Lagrangian representations
(y;t) and
b
(x;t) and so we sometimes
write both these functions as (x;t) and (y;t) being attentive when we do so.
For example consider the time derivative of .If we write this simply as @=@t we would
not know whether we were referring to the Lagrangian or the Eulerian derivatives
@
b
@t
(x;t) or
@
@t
(y;t) (1.25)
respectively.To avoid confusion we therefore use the notation
_
and
0
instead of @=@t
where
_
=
@
b
@t
(x;t) and
0
=
@
@t
(y;t):(1.26)
_
is called the material time derivative
4
(since in calculating the time derivative we are
keeping the particle,identied by x,xed).
4
In uid mechanics this is often denoted by D=Dt.
1.7.A PART OF A BODY.11
We can relate
_
to
0
by dierentiating
b
(x;t) =
(by(x;t);t) with respect to t and using
the chain rule.This gives
@
b
@t
=
@
@y
i
@by
i
@t
+
@
@t
(1.27)
or
_
=
0
+(grad ) v (1.28)
where v is the velocity.Similarly,for any vector eld w one can show that
_w = w
0
+(grad w)v (1.29)
where gradw is a tensor.Unless explicitly stated otherwise,we shall always use an over dot
to denote the material time derivative.
1.7 A Part of a Body.
We say that P is a part of the body B if (i) P B and (ii) P itself is a body,i.e.there is
a conguration of B such that (P) is a region.(Note therefore that a single particle p
does not constitute a part of the body.)
If R
t
and D
t
are the respective regions occupied at time t by a body B and a part of it
P during a motion,then D
t
R
t
.
As the body undergoes a motion,the region D
t
= (P;t) that is occupied by a part of the
body will evolve with time.Note that even though the region D
t
changes with time,the set
of particles associated with it does not change with time.The region D
t
is always associated
with the same part P of the body.Such a region,which is always associated with the same
set of particles,is called a material region.In subsequent chapters when we consider the
\global balance principles of continuum thermomechanics",such as momentum or energy
balance,they will always be applied to a material region (or equivalently to a part of the
body).Note that the region occupied by P in the reference conguration,D
ref
=
ref
(P),
does not vary with time.
Next consider a surface S
t
that moves in space through R
t
.One possibility is that this
surface,even though it moves,is always associated with the same set of particles (so that it
\moves with the body".) This would be the case for example of the surface corresponding to
the interface between two perfectly bonded parts in a composite material.Such a surface is
called a material surface since it is associated with the same particles at all times.A second
12 CHAPTER 1.SOME PRELIMINARY NOTIONS
possibility is that the surface is not associated with the same set of particles,as is the case
for example for a wave front propagating through the material.The wave front is associated
with dierent particles at dierent times as it sweeps through R
t
.Such a surface is not a
material surface.Note that the surface S
ref
,which is the preimage of S
t
in the reference
conguration,does not vary with time for a material surface but does vary with time for a
nonmaterial surface.
In general,a time dependent family of curves C
t
,surfaces S
t
and regions D
t
are said to
represent,respectively,a material curve,a material surface and a material region if they are
associated with the same set of particles at all times.
1.8 Extensive Properties and their Densities.
In the previous sections we considered physical properties such as temperature that were
associated with individual particles of the body.Certain other physical properties in con
tinuum physics (such as for example mass,energy and entropy) are associated with parts of
the body and not with individual particles.
Consider an arbitrary part P of a body B that undergoes a motion .As usual,the
regions of space occupied by P and B at time t during this motion are denoted by (P;t)
and (B;t) respectively,and the location of the particle p is y = (p;t).
We say that
is an extensive physical property of the body if there is a function
(;t;)
dened on the set of all parts P of B which is such that
(i)
(P
1
[ P
2
;t;) =
(P
1
;t;) +
(P
2
;t;) (1.30)
for all arbitrary disjoint parts P
1
and P
2
(which simply states that the value of the
property
associated with two disjoint parts is the sum of the individual values for
each of those parts),and
(ii)
(P;t;)!0 as the volume of (P;t)!0:(1.31)
Under these circumstance there exists a density!(p;t;) such that
(P;t;) =
Z
P
!(p;t;) dp:(1.32)
1.8.EXTENSIVE PROPERTIES AND THEIR DENSITIES.13
Thus,we have the property
(P;t;) associated with parts P of the body and its density
!(p;t;) associated with particles p of the body,e.g.the energy of P and the energy density
at p.
It is more convenient to trade the particle p for its position y using p =
1
(y;t) and
work with the (Eulerian or spatial) density function
!(y;t;) in terms of which
(P;t;) =
Z
D
t
!(y;t;) dV
y
:
Any physical property associated in such a way with all parts of a body has an associated
density;for example the mass m,internal energy e,and the entropy H have corresponding
mass
5
,internal energy and entropy densities which we will denote by ;"and .
References:
1.C.Truesdell,The Elements of Continuum Mechanics,Lecture 1,SpringerVerlag,
NewYork,1966.
2.R.W.Ogden,NonLinear Elastic Deformations,xx2.1.2 and 2.1.3,Dover,1997.
3.C.Truesdell,A First Course in Rational Continuum Mechanics,xx1 to 4 and x7 of
Chapter 1 and xx1{4 of Chapter 2,Academic Press,New York 1977.
5
In the particular case of mass,one has the added feature that m(P) > 0 whence (y;t) > 0.
Chapter 2
Kinematics:Deformation
In this chapter we shall consider various geometric issues concerning the deformation of a
body.At this stage we will not address the causes of the deformation,such as the applied
loading or the temperature changes,nor will we discuss the characteristics of the material of
which the body is composed,assuming only that it can be described as a continuum.Our
focus will be on purely geometric issues
1
.
A roadmap of this chapter is as follows:in Section 2.1 we describe the notion of a
deformation.In Section 2.2 we introduce the central ingredient needed for describing the
deformation of an entire neighborhood of a particle { the deformation gradient tensor.Some
particular homogeneous deformations such as pure stretch,uniaxial extension and simple
shear are presented in Section 2.3.We then consider in Section 2.4 an innitesimal curve,
surface and region in the reference conguration and examine their images in the deformed
conguration where the image and preimage in each case is associated with the same set of
particles.A rigid deformation is described in Section 2.5.The decomposition of a general
deformation gradient tensor into the product of a rigid rotation and a pure stretch is pre
sented in Section 2.6.Section 2.7 introduces the notion of strain,and nally we consider the
linearization of the prior results in Section 2.8.
1
It is worth mentioning that in developing a continuum theory for a material,the appropriate kinematic
description of the body is not totally independent of,say,the nature of the forces.For example,in describing
the interaction between particles in a dielectric material subjected to an electric eld,one has to allow for
internal forces and internal couples between every pair of points in the body.This in turn requires that the
kinematics allow for independent displacement and rotation elds in the body.In general,the kinematics
and the forces must be conjugate to each other in order to construct a selfconsistent theory.This will be
made more clear in subsequent chapters.
15
16 CHAPTER 2.KINEMATICS:DEFORMATION
2.1 Deformation
In this chapter we will primarily be concerned with how the geometric characteristics of one
conguration of the body (the\deformed"or\current"conguration) dier from those of
some other conguration of the body (an\undeformed"or\reference"conguration).Thus
we consider two congurations in which the body occupies the respective regions
2
R and
R
0
.The corresponding position vectors of a generic particle are y 2 R and x 2 R
0
.In this
chapter we shall consider one xed reference conguration and therefore we can uniquely
identify a particle by its position x in that conguration.The deformation of the body from
the reference conguration to the deformed conguration is described by a mapping
y = by(x) (2.1)
which takes R
0
!R.We use the\hat"over y in order to distinguish the function by() from
its value y.As we progress through these notes,we will most often omit the\hat"unless
the context does not make clear whether we are referring to
b
y or y,and/or it is essential to
emphasize the distinction.
The displacement vector eld bu(x) is dened on R
0
by
bu(x) = by(x) x;(2.2)
see Figure 10.2.In order to fully characterize the deformed conguration of the body one
must specify the function by (or equivalently bu) at every particle of the body,i.e.on the
entire domain R
0
.
We impose the physical requirements that (a) a single particle
3
x will not split into two
particles and occupy two locations y
(1)
and y
(2)
,and that (b) two particles x
(1)
and x
(2)
will
not coalesce into a single particle and occupy one location y.This implies that (2.1) must
be a onetoone mapping.Consequently there exists a onetoone inverse deformation
x = bx(y) (2.3)
that carries R!R
0
.Since (2.3) is the inverse of (2.1),it follows that
bx(by(x)) = x for all x 2 R
0
;by(bx(y)) = y for all y 2 R:(2.4)
2
In Chapter 1 we denoted the region occupied by the body in the reference conguration by R
ref
.Here,
we call it R
0
.
3
Whenever there is no confusion in doing so,we shall use more convenient but less precise language such
as\the particle x"rather than\the particle p located at x in the reference conguration".
2.1.DEFORMATION 17
Figure 2.1:The respective regions R
0
and R occupied by a body in a reference conguration and a
deformed conguration;the position vectors of a generic particle in these two congurations are denoted by
x and y.The displacement of this particle is u.
Unless explicitly stated otherwise,we will assume that by(x) and bx(y) are\smooth",
or more specically that they may each be dierentiated at least twice,and that these
derivatives are continuous on the relevant regions:
by 2 C
2
(R
0
);bx 2 C
2
(R):(2.5)
We will relax these requirements occasionally.For example,if we consider a\dislocation"it
will be necessary to allow the displacement eld to be discontinuous across a surface in the
body;and if we consider a\twophase composite material",we must allow the gradient of the
displacement eld to be discontinuous across the interface between the dierent materials.
Finally,consider a xed righthanded orthonormal basis fe
1
;e
2
;e
3
g.When we refer to
components of vector and tensor quantities,it will always be with respect to this basis.In
particular,the components of x and y in this basis are x
i
= x e
i
and y
i
= y e
i
;i = 1;2;3.
In terms of its components,equation (2.1) reads
y
i
= y
i
(x
1
;x
2
;x
3
):(2.6)
See Problems 2.1 and 2.2.
18 CHAPTER 2.KINEMATICS:DEFORMATION
2.2 Deformation Gradient Tensor.Deformation in the
Neighborhood of a Particle.
Let x denote the position of a generic particle of the body in the reference conguration.
Questions that we may want to ask,such as what is the state of stress at this particle?will
the material fracture at this particle?and so on,depend not only on the deformation at x but
also on the deformation of all particles in a small neighborhood of x.Thus,the deformation
in the entire neighborhood of a generic particle plays a crucial role in this subject and we now
focus on this.Thus we imagine a small ball of material centered at x and ask what happens
to this ball as a result of the deformation.Intuitively,we expect the deformation of the ball
(i.e.the local deformation near x,) to consist of a combination of a rigid translation,a rigid
rotation and a\straining",notions that we shall make precise in what follows.The socalled
deformation gradient tensor at a generic particle x is dened by
F(x) = Grad y(x):(2.7)
This is the principal entity used to study the deformation in the immediate neighborhood of
x.The deformation gradient F(x) is a 2tensor eld and its components
F
ij
(x) =
@y
i
(x)
@x
j
(2.8)
correspond to the elements of a 3 3 matrix eld [F(x)].
Figure 2.2:An innitesimal material ber in the reference and deformed congurations.
Consider two particles p and q located at x and x+dx in the reference conguration;their
2.2.DEFORMATION GRADIENT TENSOR 19
locations are depicted by P and Q in Figure 2.2.The innitesimal material ber
4
joining
these two particles is dx.In the deformed conguration these two particles are located at
y(x) and y(x+dx) respectively,and the deformed image of this innitesimal material ber
is described by the vector
dy = y(x +dx) y(x):(2.9)
Since p and q are neighboring particles we can approximate this expression for small jdxj by
the Taylor expansion
dy =
Grad y
dx +O(jdxj
2
) = F dx +O(jdxj
2
);(2.10)
which we can formally write as
dy = Fdx;(2.11)
or in terms of components as
dy
i
= F
ij
dx
j
or fyg = [F] fxg:(2.12)
Note that this approximation does not assume that the deformation or deformation gradient
is small;only that the two particles p and q are close to each other.
Thus F carries an innitesimal undeformed material ber dx into its location dy in the
deformed conguration.
In physically realizable deformations we expect that (a) a single ber dx will not split
into two bers dy
(1)
and dy
(2)
,and (b) that two bers dx
(1)
and dx
(2)
will not coalesce into
a single ber dy.This means that (2.11) must be a onetoone relation between dx and
dy and thus that F must be nonsingular.Thus in particular the Jacobian determinant,J,
must not vanish:
J = det F 6= 0:(2.13)
Next,consider three linearly independent material bers dx
(i)
;i = 1;2;3,as shown in
Figure 2.3.The deformation carries these bers into the three locations dy
(i)
= Fdx
(i)
;i =
1;2;3.A deformation preserves orientation if every righthanded triplet of bers fdx
(1)
;
dx
(2)
;dx
(3)
g is carried into a righthanded triplet of bers fdy
(1)
;dy
(2)
;dy
(3)
g,i.e.the defor
mation is orientation preserving if every triplet of bers for which (dx
(1)
dx
(2)
) dx
(3)
> 0
is carried into a triplet of bers for which (dy
(1)
dy
(2)
) dy
(3)
> 0.By using an iden
tity established in one of the worked examples in Chapter 3 of Volume I,it follows that
4
The notion of a material curve was explained at the end of Section 1.7:the ber here being a material
ber implies that PQ and P
0
Q
0
are associated with the same set of particles.
20 CHAPTER 2.KINEMATICS:DEFORMATION
Figure 2.3:An orientation preserving deformation:the righthanded triplet of innitesimal material bers
dx
(1)
;dx
(2)
;dx
(3)
are carried into a righthanded triplet of bers dy
(1)
;dy
(2)
;dy
(3)
.
(dy
(1)
dy
(2)
) dy
(3)
= (Fdx
(1)
Fdx
(2)
) Fdx
(3)
= (det F) (dx
(1)
dx
(2)
) dx
(3)
.Conse
quently orientation is preserved if and only if
J = det F > 0:(2.14)
In these notes we will only consider orientationpreserving deformations
5
.
The deformation of a generic particle x + dx in the neighborhood of particle x can be
written formally as
y(x +dx) = y(x) +Fdx:(2.15)
Therefore in order to characterize the deformation of the entire neighborhood of x we must
know both the deformation y(x) and the deformation gradient tensor F(x) at x;y(x) char
acterizes the translation of that neighborhood while F(x) characterizes both the rotation
and the\strain"at x as we shall see below.
A deformation y(x) is said to be homogeneous if the deformation gradient tensor is
constant on the entire region R
0
.Thus,a homogeneous deformation is characterized by
y(x) = Fx +b (2.16)
where F is a constant tensor and b is a constant vector.It is easy to verify that a set of
points which lie on a straight line/plane/ellipsoid in the reference conguration will continue
5
Some deformations that do not preserve orientation are of physical interest,e.g.the turning of a tennis
ball inside out.
2.3.SOME SPECIAL DEFORMATIONS.21
to lie on a straight line/plane/ellipsoid in the deformed conguration if the deformation is
homogeneous.
2.3 Some Special Deformations.
Figure 2.4:Pure homogeneous stretching of a cube.A unit cube in the reference conguration is carried
into an orthorhombic region of dimensions
1
2
3
.
Consider a body that occupies a unit cube in a reference conguration.Let fe
1
;e
2
;e
3
g
be a xed orthonormal basis with the basis vectors aligned with the edges of the cube;see
Figure 2.4.Consider a pure homogeneous stretching of the cube,
y = Fx where F =
1
e
1
e
1
+
2
e
2
e
2
+
3
e
3
e
3
;(2.17)
where the three
0
i
s are positive constants.In terms of components in the basis fe
1
;e
2
;e
3
g,
this deformation reads
0
B
B
@
y
1
y
2
y
3
1
C
C
A
=
0
B
B
@
1
0 0
0
2
0
0 0
3
1
C
C
A
0
B
B
@
x
1
x
2
x
3
1
C
C
A
:(2.18)
The 1 1 1 undeformed cube is mapped by this deformation into a
1
2
3
orthorhombic region R as shown in Figure 2.4.The volume of the deformed region is
1
2
3
.The positive constants
1
;
2
and
3
here represent the ratios by which the three
edges of the cube stretch in the respective directions e
1
;e
2
;e
3
.Any material ber that was
22 CHAPTER 2.KINEMATICS:DEFORMATION
parallel to an edge of the cube in the reference conguration simply undergoes a stretch and
no rotation under this deformation.However this is not in general true of any other material
ber { e.g.one oriented along a diagonal of a face of the cube { which will undergo both a
length change and a rotation.
The deformation (2.17) is a pure dilatation in the special case
1
=
2
=
3
in which event F =
1
I.The volume of the deformed region is
3
1
.
If the deformation is isochoric,i.e.if the volume does not change,then
1
;
2
;
3
must
be such that
1
2
3
= 1:(2.19)
Figure 2.5:Uniaxial stretch in the e
1
direction.A unit cube in the reference conguration is carried into
a
1
1 1 tetragonal region R in the deformed conguration.
If
2
=
3
= 1,then the body undergoes a uniaxial stretch in the e
1
direction (and no
stretch in the e
2
and e
3
directions);see Figure 2.5.In this case
F =
1
e
1
e
1
+ e
2
e
2
+ e
3
e
3
;= I +(
1
1)e
1
e
1
:
If
1
> 1 the deformation is an elongation,whereas if
1
< 1 it is a contraction.(The terms
\tensile"and\compressive"refer to stress not deformation.) More generally the deformation
y = Fx where
F = I +( 1)n
n;jnj = 1;(2.20)
2.3.SOME SPECIAL DEFORMATIONS.23
represents a uniaxial stretch in the direction n.
The cube is said to be subjected to a simple shearing deformation if
y = Fx where F = I +k e
1
e
2
and k is a constant.In terms of components in the basis fe
1
;e
2
;e
3
g,this deformation reads
0
B
B
@
y
1
y
2
y
3
1
C
C
A
=
0
B
B
@
1 k 0
0 1 0
0 0 1
1
C
C
A
0
B
B
@
x
1
x
2
x
3
1
C
C
A
:(2.21)
The simple shear deformation carries the cube into the sheared region R as shown in Figure
2.6.Observe that the displacement eld here is given by u(x) = y(x) x = Fx x =
k(e
1
e
2
)x = kx
2
e
1
.Thus each plane x
2
= constant is displaced rigidly in the x
1
direction,
the amount of the displacement depending linearly on the value of x
2
.One refers to a plane
x
2
= constant as a shearing (or glide) plane,the x
1
direction as the shearing direction and k
is called the amount of shear.One can readily verify that det
I +k e
1
e
2
= 1 wherefore
a simple shear automatically preserves volume.
More generally the deformation y = Fx where
F = I +km
n;jmj = jnj = 1;m n = 0;(2.22)
represents a simple shear whose glide plane normal and shear direction are n and mrespec
tively.
If
3
= 1;
equation (2.17) describes a plane deformation in the 1;2plane (i.e.stretching occurs only in
the 1;2plane;bers in the e
3
direction remain unstretched);and a plane equibiaxial stretch
in the 1;2plane if
1
=
2
;
3
= 1:
If the material bers in the direction dened by some unit vector m
0
in the reference
conguration remain inextensible,then m
0
and its deformed image Fm
0
must have the same
length:jFm
0
j = jm
0
j = 1 which holds if and only if
Fm
0
Fm
0
= F
T
Fm
0
m
0
= 1:
24 CHAPTER 2.KINEMATICS:DEFORMATION
Figure 2.6:Simple shear of a cube.Each plane x
2
= constant undergoes a displacement in the x
1
direction
by the amount kx
2
.
For example,if m
0
= cos e
1
+sin e
2
,we see by direct substitution that
1
;
2
must obey
the constraint equation
1
cos
2
+
2
sin
2
= 1:
Given m
0
,this restricts F.
We can now consider combinations of deformations,each of which is homogeneous.For
example consider a deformation y = F
1
F
2
x where F
1
= I + a
a,F
2
= I + km
n,
the vectors a;m;n have unit length,and m n = 0.This represents a simple shearing of
the body (with amount of shear k,glide plane normal n and shear direction m) in which
x!F
2
x,followed by a uniaxial stretching (in the a direction) in which F
2
x!F
1
(F
2
x);
see Figure 2.7 for an illustration of the case a = n = e
2
;n = e
1
.
The preceding deformations were all homogeneous in the sense that they were all of the
special form y = Fx where F was a constant tensor.Most deformations y = y(x) are not
of this form.A simple example of an inhomogeneous deformation is
y
1
= x
1
cos x
3
x
2
sinx
3
;
y
2
= x
1
sinx
3
+x
2
cos x
3
;
y
3
= x
3
:
9
>
>
=
>
>
;
This can be shown to represent a torsional deformation about the e
3
axis in which each
plane x
3
= constant rotates by an angle x
3
.The matrix of components of the deformation
2.4.LENGTH,ORIENTATION,ANGLE,VOLUME AND AREA 25
Figure 2.7:A unit cube subjected to a simple shear (with glide plane normal e
2
) followed by a uniaxial
stretch in the direction e
2
.
gradient tensor associated with this deformation is
F
=
@y
i
@x
j
=
0
B
B
B
B
@
cos x
3
sinx
3
x
1
sinx
3
x
2
cos x
3
sinx
3
cos x
3
x
1
cos x
3
x
2
sinx
3
0 0 1
1
C
C
C
C
A
;
observe that the components F
ij
of the deformation gradient tensor here depend of (x
1
;x
2
;x
3
).
See Problems 2.3 and 2.4.
2.4 Transformation of Length,Orientation,Angle,Vol
ume and Area.
As shown by (2.15),the deformation gradient tensor F(x) characterizes all geometric changes
in the neighborhood of the particle x.We now examine the deformation of an innitesimal
material ber,innitesimal material surface and an innitesimal material region.Specically,
we calculate quantities such as the local
6
change in length,angle,volume and area in terms
of F(x).The change in length is related to the notion of ber stretch (or strain),the change
in angle is related to the notion of shear strain and the change in volume is related to the
6
i.e.the geometric changes of innitesimally small line,area and volume elements at x.
26 CHAPTER 2.KINEMATICS:DEFORMATION
notion of volumetric (or dilatational) strain { notions that we will encounter shortly and
play an important role in this subject.The change in area is indispensable when calculating
the true stress on a surface.
2.4.1 Change of Length and Orientation.
Figure 2.8:An innitesimal material ber:in the reference conguration it has length ds
x
and orientation
n
0
;in the deformed conguration it has length ds
y
and orientation n.
Suppose that we are given a material ber that has length ds
x
and orientation n
0
in the
reference conguration:dx = (ds
x
)n
0
.We want to calculate its length and orientation in
the deformed conguration.
If the image of this ber in the deformed conguration has length ds
y
and orientation n,
then dy = (ds
y
)n.Since dy and dx are related by dy = Fdx,it follows that
(ds
y
)n = (ds
x
)Fn
0
:(2.23)
Thus the deformed length of the ber is
ds
y
= jdyj = jFdxj = ds
x
jFn
0
j:(2.24)
The stretch ratio at the particle x in the direction n
0
is dened as the ratio
= ds
y
=ds
x
(2.25)
and so
= jFn
0
j:(2.26)
2.4.LENGTH,ORIENTATION,ANGLE,VOLUME AND AREA 27
This gives the stretch ratio = (n
0
) = jFn
0
j of any ber that was in the n
0
direction
in the reference conguration.You might want to ask the question,among all bers of all
orientations at x,which has the maximum stretch ratio?
The orientation n of this ber in the deformed conguration is found from (2.23) to be
n =
Fn
0
jFn
0
j
:(2.27)
2.4.2 Change of Angle.
Figure 2.9:Two innitesimal material bers.In the reference congurations they have equal length ds
x
and directions n
(1)
0
and n
(2)
0
.
Suppose that we are given two bers dx
(1)
and dx
(2)
in the reference conguration as
shown in Figure 2.9.They both have the same length ds
x
and they are oriented in the
respective directions n
(1)
0
and n
(2)
0
where n
(1)
0
and n
(2)
0
are unit vectors:dx
(1)
= ds
x
n
(1)
0
and
dx
(2)
= ds
x
n
(2)
0
.Let
x
denote the angle between them:cos
x
= n
(1)
0
n
(2)
0
.We want to
determine the angle between them in the deformed conguration.
In the deformed conguration these two bers are characterized by Fdx
(1)
and Fdx
(2)
.
By denition of the scalar product of two vectors Fdx
(1)
Fdx
(2)
= jFdx
(1)
jjFdx
(2)
j cos
y
and so the angle
y
between them is found from
cos
y
=
Fdx
(1)
jFdx
(1)
j
Fdx
(2)
jFdx
(2)
j
=
Fn
(1)
0
Fn
(2)
0
jFn
(1)
0
jjFn
(2)
0
j
:(2.28)
28 CHAPTER 2.KINEMATICS:DEFORMATION
The decrease in angle =
x
y
is the shear associated with the directions n
(1)
0
;n
(2)
0
:
= (n
(1)
0
;n
(2)
0
).One can show that 6= =2;(see Section 25 of Truesdell and Toupin).
You might want to ask the question,among all pairs of bers at x,which pair suers the
maximum change in angle,i.e.maximum shear?
2.4.3 Change of Volume.
Figure 2.10:Three innitesimal material bers dening a tetrahedral region.The volumes of the tetrahe
drons in the reference and deformed congurations are dV
x
and dV
y
respectively.
Next,consider three linearly independent material bers dx
(i)
;i = 1;2;3,as shown in
Figure 2.10.By geometry,the volume of the tetrahedron formed by these three bers is
dV
x
=
1
6
(dx
(1)
dx
(2)
) dx
(3)
;
see the related worked example in Chapter 2 of Volume I.The deformation carries these
bers into the three bers dy
(i)
= Fdx
(i)
.The volume of the deformed tetrahedron is
dV
y
= j
1
6
(dy
(1)
dy
(2)
) dy
(3)
j = j
1
6
(Fdx
(1)
Fdx
(2)
) Fdx
(3)
j
= j det Fj j
1
6
(dx
(1)
dx
(2)
) dx
(3)
j = det F dV
x
;
where in the penultimate step we have used the identity noted just above (2.14) and the
fact that det F > 0.Thus the volumes of a dierential volume element in the reference and
deformed congurations are related by
dV
y
= J dV
x
where J = det F:(2.29)
2.4.LENGTH,ORIENTATION,ANGLE,VOLUME AND AREA 29
Observe from this that a deformation preserves the volume of every innitesimal volume
element if and only if
J(x) = 1 at all x 2 R
0
:(2.30)
Such a deformation is said to be isochoric or locally volume preserving.
An incompressible material is a material that can only undergo isochoric deformations.
2.4.4 Change of Area.
Figure 2.11:Two innitesimal material bers dening a parallelogram.
Next we consider the relationship between two area elements in the reference and de
formed congurations.Consider the area element in the reference conguration dened by
the bers dx
(1)
and dx
(2)
as shown in Figure 2.11.Supppose that its area is dA
x
and that
n
0
is a unit normal to this plane.Then,from the denition of the vector product,
dx
(1)
dx
(2)
= dA
x
n
0
:(2.31)
Similarly if dA
y
and n are the area and the unit normal,respectively,to the surface dened
in the deformed conguration by dy
(1)
and dy
(2)
,then
dy
(1)
dy
(2)
= dA
y
n:(2.32)
It is worth emphasizing that the surfaces under consideration (shown shaded in Figure 2.11)
are composed of the same particles,i.e.they are\material"surfaces.Note that n
0
and
30 CHAPTER 2.KINEMATICS:DEFORMATION
n are dened by the fact that they are normal to these material surface elements.Since
dy
(i)
= Fdx
(i)
,(2.32) can be written as
Fdx
(1)
Fdx
(2)
= dA
y
n:(2.33)
Then,by using an algebraic result from the relevant worked example in Chapter 3 of Volume
I,and combining (2.31) with (2.33) we nd that
dA
y
n = dA
x
J F
T
n
0
:(2.34)
This relates the vector areas dA
y
n and dA
x
n
0
.By taking the magnitude of this vector
equation we nd that the areas dA
y
and dA
x
are related by
dA
y
= dA
x
J jF
T
n
0
j;(2.35)
On using (2.35) in (2.34) we nd that the unit normal vectors n
0
and n are related by
n =
F
T
n
0
jF
T
n
0
j
:(2.36)
Observe that n is not in general parallel to Fn
0
indicating that a material ber in the
direction characterized by n
0
is not mapped into a ber in the direction n.As noted previ
ously,n
0
and n are dened by the fact that they are normal to the material surface elements
being considered;not by the fact that one is the image of the other under the deformation.
The particles that lie along the ber n
0
are mapped by F into a ber that is in the direction
of Fn
0
which is not generally perpendicular to the plane dened by dy
(1)
and dy
(2)
.
See Problems 2.5  2.10.
2.5 Rigid Deformation.
We now consider the special case of a rigid deformation.A deformation is said to be rigid
if the distance between all pairs of particles is preserved under the deformation,i.e.if the
distance jz xj between any two particles x and z in the reference conguration equals the
distance jy(z) y(x)j between them in the deformed conguration:
jy(z)y(x)j
2
=
h
y
i
(z)y
i
(x)
ih
y
i
(z)y
i
(x)
i
= (z
i
x
i
)(z
i
x
i
) for all x;z 2 R
0
:(2.37)
2.5.RIGID DEFORMATION.31
Since (2.37) holds for all x,we may take its derivative with respect to x
j
to get
2F
ij
(x) (y
i
(z) y
i
(x)) = 2(z
j
x
j
) for all x;z 2 R
0
;(2.38)
where F
ij
(x) = @y
i
(x)=@x
j
are the components of the deformation gradient tensor.Since
(2.38) holds for all z we may take its derivative with respect to z
k
to obtain F
ij
(x)F
ik
(z) =
jk
,
i.e.
F
T
(x)F(z) = 1 for all x;z 2 R
0
:(2.39)
Finally,since (2.39) holds for all x and all z;we can take x = z in (2.39) to get
F
T
(x)F(x) = I for all x 2 R
0
:(2.40)
Thus we conclude that F(x) is an orthogonal tensor at each x:In fact,since det F > 0;it is
proper orthogonal and therefore represents a rotation.
The (possible) dependence of Fon x implies that Fmight be a dierent proper orthogonal
tensor at dierent points x in the body.However,returning to (2.39),multiplying both sides
of it by F(x) and recalling that F is orthogonal gives
F(z) = F(x) at all x;z 2 R
0
;(2.41)
(2.41) implies that F(x) is a constant tensor.
In conclusion,the deformation gradient tensor associated with a rigid deformation is a
constant rotation tensor.Thus at all x 2 R
0
we can denote F(x) = Qwhere Qis a constant
proper orthogonal tensor.Thus necessarily a rigid deformation has the form
y = y(x) = Qx +b (2.42)
where Q is a constant rotation tensor and b is a constant vector.Conversely it is easy to
verify that (2.42) satises (2.37).
A rigid material (or rigid body) is a material that can only undergo rigid deformations.
One can readily verify from (2.42) and the results of the previous section that in a rigid
deformation the length of every ber remains unchanged;the angle between every two bers
remains unchanged;the volume of any dierential element remains unchanged;and the unit
vectors n
0
and n normal to a surface in the reference and deformed congurations are simply
related by n = Qn
0
.
32 CHAPTER 2.KINEMATICS:DEFORMATION
2.6 Decomposition of Deformation Gradient Tensor into
a Rotation and a Stretch.
As mentioned repeatedly above,the deformation gradient tensor F(x) completely charac
terizes the deformation in the vicinity of the particle x.Part of this deformation is a rigid
rotation,the rest is a\distorsion"or\strain".The central question is\which part of F
is the rotation and which part is the strain?"The answer to this is provided by the polar
decomposition theoremdiscussed in Chapter 2 of Volume I.According to this theorem,every
nonsingular tensor F with positive determinant can be written uniquely as the product of a
proper orthogonal tensor R and a symmetric positive denite tensor U as
F = RU;(2.43)
R represents the rotational part of F while U represents the part that is not a rotation.It
is readily seen from (2.43) that U is given by the positive denite square root
U=
p
F
T
F (2.44)
so that R is then given by
R= FU
1
:(2.45)
Since a generic undeformed material ber is carried by the deformation from dx!dy =
Fdx,we can write the relationship between the two bers as
dy = R(Udx):(2.46)
This allows us to view the deformation of the ber in twosteps:rst,the ber dx is taken by
the deformation to Udx,and then,it is rotated rigidly by R:dx!Udx!R(Udx) = dy.
The essential property of U is that it is symmetric and positive denite.This allows
us to physically interpret U as follows:since U is symmetric,it has three real eigenvalues
1
;
2
and
3
,and a corresponding triplet of orthonormal eigenvectors r
1
;r
2
and r
3
.Since
U is positive denite,all three eigenvalues are positive.Thus the matrix of components of
U in the principal basis fr
1
;r
2
;r
3
g is
[U] =
0
B
@
1
0 0
0
2
0
0 0
3
1
C
A
;
i
> 0:(2.47)
2.6.ROTATION AND STRETCH 33
If the components of dx in this principal basis are
fdxg =
0
B
@
dx
1
dx
2
dx
3
1
C
A
then [U] fdxg =
0
B
@
1
dx
1
2
dx
2
3
dx
3
1
C
A
:
Thus when dx!Udx,the ber dx is stretched by the tensor U in the principal directions
of U by amounts given by the corresponding eigenvalues of U.The tensor U is called the
right stretch tensor.
The stretched ber Udx is now taken by the rigid rotation R from Udx!R(Udx).
Note that in general,the ber dx will rotate while it undergoes the stretching deformation
dx!Udx,since dx is not necessarily parallel to Udx;however this is not a rigid rotation
since the length of the ber also changes.
The alternative version of the polar decomposition theorem (Chapter 2 of Volume I) pro
vides a second representation for F.According to this part of the theorem,every nonsingular
tensor F with positive determinant can be written uniquely as the product of a symmetric
positive denite tensor V with a proper orthogonal tensor R as
F = VR;(2.48)
the tensor R here is identical to that in the preceding representation and represents the
rotational part of F.It is readily seen from (2.48) that V is given by
V =
p
FF
T
(2.49)
and that R is given by
R= V
1
F:(2.50)
Since R= V
1
F = FU
1
it follows that V = FUF
1
.
A generic undeformed ber dx can therefore alternatively be related to its image dy in
the deformed conguration by
dy = V(Rdx);(2.51)
and so we can view the deformation of the ber as rst,a rigid rotation from dx to Rdx,
followed by a stretching by V.Since V is symmetric and positive denite,all three of its
eigenvalues,
1
;
2
and
3
are real and positive;moreover the corresponding eigenvectors
form an orthonormal basis f`
1
;`
2
;`
3
g { a principal basis of V.Thus the deformation can
alternatively be viewed as,rst,a rigid rotation of the ber by the tensor R followed by
34 CHAPTER 2.KINEMATICS:DEFORMATION
stretching in the principal directions of V:dx!Rdx!V(Rdx).The tensor V is called
the left stretch tensor.
It is easy to show that the eigenvalues
1
;
2
and
3
of U are identical to those of
V.Moreover one can show that the eigenvectors by fr
1
;r
2
;r
3
g of U are related to the
eigenvectors f`
1
;`
2
;`
3
g of V by`
i
= Rr
i
;i = 1;2;3.The common eigenvalues of U and
V,are known as the principal stretches associated with the deformation (at x).The stretch
tensors U and V can be expressed in terms of their eigenvectors and eigenvalues as
U=
3
X
i=1
i
r
i
r
i
;V =
3
X
i=1
i
`
i
`
i
;(2.52)
see Section 2.2 of Volume I.As shown in one of the worked examples in Chapter 2 of Volume
I,we also have the representations
F =
3
X
i=1
i
`
i
r
i
;R=
3
X
i=1
`
i
r
i
:(2.53)
The expressions (2.24),(2.27),(2.28),(2.29) and (2.35) describe changes in length,ori
entation,angle,volume and area in terms of the deformation gradient tensor F.Since a
rotation does not change length,angle,area and volume we expect that these equations
(except for the one for orientation) should be independent of the rotation tensor R in the
polar decomposition.By using F = RU in (2.24),(2.28),(2.29) and (2.35) it is readily seen
that they can be expressed in terms of U as
ds
y
= ds
x
p
U
2
n
0
n
0
;
cos
y
=
U
2
n
(1)
0
n
(2)
0
q
U
2
n
(1)
0
n
(1)
0
q
U
2
n
(2)
0
n
(2)
0
;
dV
y
= dV
x
det U;
dA
y
= dA
x
(det U)jU
1
n
0
j;
9
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
;
(2.54)
which emphasizes the fact that these changes depend only on the stretch tensor U and not
the rotational part Rof the deformation gradient tensor
7
.The formula (2.27) for the change
of orientation of a ber takes the form
n = R
Un
0
jUn
0
j
;(2.55)
7
Recall that for any tensor A and any two vectors x and y,we have Ax y = x A
T
y.
2.6.ROTATION AND STRETCH 35
which shows that the orientation of a ber changes due to both stretching and rotation.
Observe that the expressions in (2.54) give us information about the deformed images of
various geometric entities,given their preimages in the reference conguration;for exam
ple,the right hand side of (2.54)
1
involves the orientation n
0
of the ber in the reference
conguration;the right hand side of (2.54)
2
involves the orientations n
(1)
0
and n
(2)
0
of the two
bers in the reference conguration;and so on.
If instead,the geometric entities are given in the deformed conguration,and we want to
determine the geometric properties of their preimages in the reference conguration,these
can be readily calculated in terms of the left stretch tensor V.Consider,for example,a ber
which in the deformed conguration has length ds
y
and orientation n.Then its length ds
x
in the reference conguration can be calculated as follows:
ds
x
= jdxj = jF
1
yj = jR
1
V
1
yj = jV
1
yj = ds
y
jV
1
nj:(2.56)
Similarly,if two bers ds
y
n
(1)
and ds
y
n
(2)
in the deformed conguration are given and
they subtend an angle
y
,then the angle
x
that their preimages subtend in the reference
conguration is given by
cos
x
=
V
2
n
(1)
n
(2)
p
V
2
n
(1)
n
(1)
p
V
2
n
(2)
n
(2)
:(2.57)
Similarly an expression for the volume dV
x
in the reference conguration of a dierential
volume element can be calculated in terms of the volume dV
y
in the deformed conguration
and the stretch tensor V;and likewise an expression for the area dA
x
in the reference
conguration of a dierential area element can be calculated in terms of the area dA
y
and
unit normal n in the deformed conguration and the stretch tensor V.
Thus we see that the left stretch tensor V allows us to compute geometric quantities
in the reference conguration in terms of their images in the deformed conguration;and
that similarly the right stretch tensor U allows us to compute geometric quantities in the
deformed conguration in terms of their preimages in the reference conguration.In this
sense we can view U and V as,respectively,Lagrangian and Eulerian stretch tensors.
Remark:It is quite tedious to calculate the tensors U = (F
T
F)
1=2
and V = (FF
T
)
1=2
.
However,since there is a onetoone relation between U and U
2
,and similarly between V
and V
2
,we can just as well use U
2
and V
2
as our measures of stretch;these are usually
denoted by C and B:
C = F
T
F = U
2
;B = FF
T
= V
2
;(2.58)
36 CHAPTER 2.KINEMATICS:DEFORMATION
and are referred to as the right and left Cavchy{Green deformation tensors respectively.Note
that the eigenvalues of C and B are
2
1
;
2
2
and
2
3
;where
i
are the principal stretches,and
that the eigenvectors of C and B are the same as those of U and V respectively.The two
CauchyGreen tensors admit the spectral representations
C =
3
X
i=1
2
i
(r
i
r
i
);B =
3
X
i=1
2
i
(`
i
`
i
):(2.59)
The particular scalarvalued functions of C
I
1
(C) = tr C;I
2
(C) =
1
2
h
tr C
2
tr C
2
i
;I
3
(C) = det C;(2.60)
are called the principal scalar invariants of C.It can be readily veried that these functions
have the property that for each symmetric tensor C,
I
i
(C) = I
i
(QCQ
T
);i = 1;2;3;(2.61)
for all orthogonal tensors Q.They are invariant scalarvalued functions in this sense.Finally,
it can be shown that they satisfy the identity
det(CI) =
3
+I
1
(C)
2
I
2
(C) +I
3
(C)
for all scalars .
The principal scalar invariants can be written in terms of the principal stretches as
I
1
(C) =
2
1
+
2
2
+
2
3
;I
2
(C) =
2
1
2
2
+
2
2
2
3
+
2
3
2
1
;I
3
(C) =
2
1
2
2
2
3
:(2.62)
The principal scalar invariants of B and C coincide:
I
i
(C) = I
i
(B);i = 1;2;3:
See Problem 2.11.
2.7 Strain.
It is clear that U and V are the essential ingredients that characterize the nonrigid part
of the deformation.If\the body is not deformed",i.e.the deformed conguration happens
2.7.STRAIN.37
to coincide with the reference conguration,the deformation is given by y(x) = x for all
x 2 R
0
,and therefore F = I and U = V = I.Thus the stretch equals the identity I in the
reference conguration.\Strain"on the other hand customarily vanishes in the reference
conguration.Thus strain is simply an alternative measure for the nonrigid part of the
deformation chosen such that it vanishes in the reference conguration.This is the only
essential dierence between stretch and strain.Thus for example we could take UI for
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