Introduction to Tensor Calculus and Continuum Mechanics

374 Pages·2001·3.77 MB·English

by  J. H. Heinbockel

Introduction to
Tensor Calculus
and
Continuum Mechanics
by J.H. Heinbockel
Department of Mathematics and Statistics
Old Dominion University
PREFACE
This is an introductory text which presents fundamental concepts from the subject
areas of tensor calculus, di(cid:11)erential geometry and continuum mechanics. The material
presented is suitable for a two semester course in applied mathematics and is flexible
enough to be presented to either upper level undergraduate or beginning graduate students
majoring in applied mathematics, engineering or physics. The presentation assumes the
students havesomeknowledgefrom theareasofmatrixtheory, linearalgebra andadvanced
calculus. Each section includes many illustrative worked examples. At the end of each
section there is a large collection of exercises which range in di(cid:14)culty. Many new ideas
are presented in the exercises and so the students should be encouraged to read all the
exercises.
The purpose ofpreparing these notes isto condense into anintroductory textthe basic
de(cid:12)nitions and techniques arising in tensor calculus, di(cid:11)erential geometry and continuum
mechanics. In particular, the material is presented to (i) develop a physical understanding
of the mathematical concepts associated with tensor calculus and (ii) develop the basic
equations of tensor calculus, di(cid:11)erential geometry and continuum mechanics which arise
in engineering applications. From these basic equations one can go on to develop more
sophisticated models of applied mathematics. The material is presented in an informal
manner and uses mathematics which minimizes excessive formalism.
The material has been divided into two parts. The (cid:12)rst part deals with an introduc-
tion to tensor calculus and di(cid:11)erential geometry which covers such things as the indicial
notation, tensor algebra, covariant di(cid:11)erentiation, dual tensors, bilinear and multilinear
forms, special tensors, the Riemann Christo(cid:11)el tensor, space curves, surface curves, cur-
vature and fundamental quadratic forms. The second part emphasizes the application of
tensor algebra and calculus to a wide variety of applied areas from engineering and physics.
The selected applications are from the areas of dynamics, elasticity, fluids and electromag-
netic theory. The continuum mechanics portion focuses on an introduction of the basic
concepts from linear elasticity and fluids. The Appendix A contains units of measurements
from the Syst(cid:18)eme International d’Unit(cid:18)es along with some selected physical constants. The
Appendix B contains a listing of Christo(cid:11)el symbols of the second kind associated with
various coordinate systems. The Appendix C is a summary of useful vector identities.
J.H. Heinbockel, 1996
Copyright (cid:13)c 1996 by J.H. Heinbockel. All rights reserved.
Reproduction and distribution of these notes is allowable provided it is for non-pro(cid:12)t
purposes only.
INTRODUCTION TO
TENSOR CALCULUS
AND
CONTINUUM MECHANICS
PART 1: INTRODUCTION TO TENSOR CALCULUS
x1.1 INDEX NOTATION . . . . . . . . . . . . . . . . . . 1
Exercise 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 28
x1.2 TENSOR CONCEPTS AND TRANSFORMATIONS . . . . 35
Exercise 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
x1.3 SPECIAL TENSORS . . . . . . . . . . . . . . . . . . 65
Exercise 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
x1.4 DERIVATIVE OF A TENSOR . . . . . . . . . . . . . . 108
Exercise 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
x1.5 DIFFERENTIAL GEOMETRY AND RELATIVITY . . . . 129
Exercise 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
PART 2: INTRODUCTION TO CONTINUUM MECHANICS
x2.1 TENSOR NOTATION FOR VECTOR QUANTITIES . . . . 171
Exercise 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
x2.2 DYNAMICS . . . . . . . . . . . . . . . . . . . . . . 187
Exercise 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
x2.3 BASIC EQUATIONS OF CONTINUUM MECHANICS . . . 211
Exercise 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
x2.4 CONTINUUM MECHANICS (SOLIDS) . . . . . . . . . 243
Exercise 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
x2.5 CONTINUUM MECHANICS (FLUIDS) . . . . . . . . . 282
Exercise 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
x2.6 ELECTRIC AND MAGNETIC FIELDS . . . . . . . . . . 325
Exercise 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 352
APPENDIX A UNITS OF MEASUREMENT . . . . . . . 353
APPENDIX B CHRISTOFFEL SYMBOLS OF SECOND KIND 355
APPENDIX C VECTOR IDENTITIES . . . . . . . . . . 362
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . 363
1
PART 1: INTRODUCTION TO TENSOR CALCULUS
Ascalarfielddescribesaone-to-onecorrespondencebetweenasinglescalarnumberandapoint. An n-
dimensionalvectorfieldisdescribedbyaone-to-onecorrespondencebetweenn-numbersandapoint. Letus
generalize these concepts by assigning n-squared numbers to a single point or n-cubed numbers to a single
point. When these numbers obey certain transformation laws they become examples of tensor fields. In
general, scalar fields are referred to as tensor fields of rank or order zero whereas vector fields are called
tensor fields of rank or order one.
Closely associated with tensor calculus is the indicial or index notation. In section 1 the indicial
notationis defined andillustrated. We also define and investigatescalar,vector and tensor fields when they
are subjected to various coordinatetransformations. It turns out that tensors have certainproperties which
are independent of the coordinate system used to describe the tensor. Because of these useful properties,
we can use tensors to represent various fundamental laws occurring in physics, engineering, science and
mathematics. These representationsareextremely useful as they are independent of the coordinate systems
considered.
x1.1 INDEX NOTATION
Two vectors A(cid:126) and B(cid:126) can be expressed in the component form
A(cid:126) =A1be1+A2be2+A3be3 and B(cid:126) =B1be1+B2be2+B3be3,
where be1, be2 and be3 are orthogonal unit basis vectors. Often when no confusion arises, the vectors A(cid:126) and
B(cid:126) are expressed for brevity sake as number triples. For example, we can write
A(cid:126) =(A1, A2, A3) and B(cid:126) =(B1, B2, B3)
where it is understood that only the components of the vectors A(cid:126) and B(cid:126) are given. The unit vectors would
be represented
be1 =(1,0,0), be2 =(0,1,0), be3 =(0,0,1).
Astillshorternotation,depictingthevectorsA(cid:126) andB(cid:126) istheindexorindicialnotation. Intheindexnotation,
the quantities
Ai, i=1,2,3 and Bp, p=1,2,3
representthecomponentsofthevectorsA(cid:126) andB(cid:126).Thisnotationfocusesattentiononlyonthecomponentsof
the vectorsandemploysadummy subscriptwhoserangeoverthe integersis specified. ThesymbolAi refers
to allofthe components ofthe vectorA(cid:126) simultaneously. The dummy subscripticanhaveanyofthe integer
values 1,2 or 3. For i = 1 we focus attention on the A1 component of the vector A(cid:126). Setting i = 2 focuses
attention on the second component A2 of the vector A(cid:126) and similarly when i = 3 we can focus attention on
the third componentof A(cid:126). The subscript i is a dummy subscriptandmay be replacedby another letter, say
p, so long as one specifies the integer values that this dummy subscript can have.
2
It is also convenientat this time to mention that higher dimensional vectors may be defined as ordered
n−tuples. For example, the vector
X(cid:126) =(X1,X2,...,XN)
with components Xi, i=1,2,...,N is called a N−dimensional vector. Another notation used to represent
this vector is
X(cid:126) =X1be1+X2be2+(cid:1)(cid:1)(cid:1)+XNbeN
where
be1, be2,..., beN
are linearly independent unit base vectors. Note that many of the operations that occur in the use of the
index notation apply not only for three dimensional vectors, but also for N−dimensional vectors.
Infuturesectionsitisnecessarytodefinequantitieswhichcanberepresentedbyaletterwithsubscripts
or superscripts attached. Such quantities are referred to as systems. When these quantities obey certain
transformation laws they are referred to as tensor systems. For example, quantities like
k ijk j i
Aij e δij δi A Bj aij.
The subscripts or superscripts are referred to as indices or suffixes. When such quantities arise, the indices
must conform to the following rules:
1. They are lower case Latin or Greek letters.
2. The letters at the end of the alphabet (u,v,w,x,y,z) are never employed as indices.
Thenumberofsubscriptsandsuperscriptsdeterminestheorderofthesystem. Asystemwithoneindex
is a first ordersystem. A system with two indices is calleda secondorder system. In general,a system with
N indices is called a Nth order system. A system with no indices is called a scalar or zeroth order system.
The type of system depends upon the number of subscripts or superscripts occurring in an expression.
i m
For example, A and B , (all indices range 1 to N), are of the same type because they have the same
jk st
i mn
number of subscripts and superscripts. In contrast, the systems A and C are not of the same type
jk p
becauseonesystemhastwosuperscriptsandthe othersystemhasonlyonesuperscript. Forcertainsystems
the number of subscripts and superscripts is important. In other systems it is not of importance. The
meaning and importance attached to sub- and superscripts will be addressed later in this section.
In the use of superscripts one must not confuse “powers ”of a quantity with the superscripts. For
1 2 3
example, if we replace the independent variables (x,y,z) by the symbols (x , x , x ), then we are letting
2 2 3
y = x where x is a variable and not x raised to a power. Similarly, the substitution z = x is the
3
replacement of z by the variable x and this should not be confused with x raised to a power. In order to
2 3 2
write a superscript quantity to a power, use parentheses. For example, (x ) is the variable x cubed. One
of the reasons for introducing the superscript variables is that many equations of mathematics and physics
can be made to take on a concise and compact form.
There is a range convention associated with the indices. This convention states that whenever there
is an expression where the indices occur unrepeated it is to be understood that each of the subscripts or
superscripts can take on any of the integer values 1,2,...,N where N is a specified integer. For example,
3
the Kronecker delta symbol δij, defined by δij = 1 if i = j and δij = 0 for i 6= j, with i,j ranging over the
values 1,2,3, represents the 9 quantities
δ11 =1 δ12 =0 δ13 =0
δ21 =0 δ22 =1 δ23 =0
δ31 =0 δ32 =0 δ33 =1.
The symbol δij refers to all of the components of the system simultaneously. As another example, consider
the equation
bem(cid:1) ben =δmn m,n=1,2,3 (1.1.1)
the subscriptsm, noccurunrepeatedontheleftsideofthe equationandhence mustalsooccuronthe right
hand side of the equation. These indices are called “free ”indices and can take on any of the values 1,2 or 3
as specified by the range. Since there are three choices for the value for m and three choices for a value of
n we find that equation (1.1.1) represents nine equations simultaneously. These nine equations are
be1(cid:1) be1 =1 be1(cid:1) be2 =0 be1(cid:1) be3 =0
be2(cid:1) be1 =0 be2(cid:1) be2 =1 be2(cid:1) be3 =0
be3(cid:1) be1 =0 be3(cid:1) be2 =0 be3(cid:1) be3 =1.
Symmetric and Skew-Symmetric Systems
A system defined by subscripts and superscripts ranging over a set of values is said to be symmetric
in two of its indices if the components are unchanged when the indices are interchanged. For example, the
third order system Tijk is symmetric in the indices i and k if
Tijk =Tkji for all values of i,j and k.
A systemdefined by subscriptsandsuperscripts is saidto be skew-symmetricintwoof its indices if the
components change sign when the indices are interchanged. For example, the fourth order system Tijkl is
skew-symmetric in the indices i and l if
Tijkl =−Tljki for all values of ijk and l.
As another example, consider the third order system aprs, p,r,s = 1,2,3 which is completely skew-
symmetric in all of its indices. We would then have
aprs =−apsr =aspr =−asrp =arsp =−arps.
It is left as an exercise to show this completely skew- symmetric systems has 27 elements, 21 of which are
zero. The6nonzeroelementsareallrelatedtooneanotherthrutheaboveequationswhen(p,r,s)=(1,2,3).
This is expressed as saying that the above system has only one independent component.
4
Summation Convention
Thesummationconventionstatesthatwhenevertherearisesanexpressionwherethereisanindexwhich
occurs twice on the same side of any equation, or term within an equation, it is understood to represent a
summationontheserepeatedindices. Thesummationbeingovertheintegervaluesspecifiedbytherange. A
repeatedindexiscalledasummationindex,whileanunrepeatedindexiscalledafreeindex. Thesummation
convention requires that one must never allow a summation index to appear more than twice in any given
expression. Because of this rule it is sometimes necessary to replace one dummy summation symbol by
some other dummy symbol in order to avoid having three or more indices occurring on the same side of
the equation. The index notation is a very powerful notation and can be used to concisely represent many
complex equations. For the remainder of this section there is presented additional definitions and examples
toillustratedthepoweroftheindicialnotation. Thisnotationisthenemployedtodefinetensorcomponents
and associated operations with tensors.
EXAMPLE 1.1-1 The two equations
y1 =a11x1+a12x2
y2 =a21x1+a22x2
canberepresentedasoneequationbyintroducingadummyindex,sayk,andexpressingtheaboveequations
as
yk =ak1x1+ak2x2, k =1,2.
The range convention states that k is free to have any one of the values 1 or 2, (k is a free index). This
equation can now be written in the form
X2
yk = akixi =ak1x1+ak2x2
i=1
whereiisthedummysummationindex. Whenthesummationsignisremovedandthesummationconvention
is adopted we have
yk =akixi i,k =1,2.
Since the subscript i repeats itself, the summation convention requires that a summation be performed by
letting the summation subscript take on the values specified by the range and then summing the results.
The index k which appears only once on the left and only once on the right hand side of the equation is
calledafreeindex. Itshouldbenotedthatbothk andiaredummy subscriptsandcanbe replacedbyother
letters. For example, we can write
yn =anmxm n,m=1,2
where m is the summation index and n is the free index. Summing on m produces
yn =an1x1+an2x2
and letting the free index n take on the values of 1 and 2 we produce the original two equations.
5
EXAMPLE 1.1-2. For yi = aijxj, i,j = 1,2,3 and xi = bijzj, i,j = 1,2,3 solve for the y variables in
terms of the z variables.
Solution: In matrix form the given equations can be expressed:
0 1 0 10 1 0 1 0 10 1
y1 a11 a12 a13 x1 x1 b11 b12 b13 z1
@ A @ A@ A @ A @ A@ A
y2 = a21 a22 a23 x2 and x2 = b21 b22 b23 z2 .
y3 a31 a32 a33 x3 x3 b31 b32 b33 z3
Now solve for the y variables in terms of the z variables and obtain
0 1 0 10 10 1
y1 a11 a12 a13 b11 b12 b13 z1
@ A @ A@ A@ A
y2 = a21 a22 a23 b21 b22 b23 z2 .
y3 a31 a32 a33 b31 b32 b33 z3
The index notation employs indices that are dummy indices and so we can write
yn =anmxm, n,m=1,2,3 and xm =bmjzj, m,j =1,2,3.
Here we have purposely changed the indices so that when we substitute for xm, from one equation into the
other, a summation index does not repeat itself more than twice. Substituting we find the indicial form of
the above matrix equation as
yn =anmbmjzj, m,n,j =1,2,3
where n is the free index and m,j are the dummy summation indices. It is left as an exercise to expand
both the matrix equation and the indicial equation and verify that they are different ways of representing
the same thing.
EXAMPLE 1.1-3. The dot product of two vectors Aq, q = 1,2,3 and Bj, j =1,2,3 can be represented
with the index notation by the product AiBi = ABcosθ i = 1,2,3, A = jA(cid:126)j, B = jB(cid:126)j. Since the
subscript i is repeated it is understood to represent a summation index. Summing on i over the range
specified, there results
A1B1+A2B2+A3B3 =ABcosθ.
Observe that the index notation employs dummy indices. At times these indices are altered in order to
conform to the above summation rules, without attention being brought to the change. As in this example,
the indicesq andj aredummyindices andcanbe changedto otherletters ifonedesires. Also,inthe future,
if the range of the indices is not stated it is assumed that the range is over the integer values 1,2 and 3.
To systems containing subscripts and superscripts one can apply certain algebraic operations. We
present in an informal way the operations of addition, multiplication and contraction.