Introduction to Continuum Mechanics 3E pptx

Preface to the Third Edition xii Preface to the First Edition xiii The Authors xiv1.1 Continuum Theory 11.2 Contents of Continuum Mechanics 1 Chapter 2 Tensors 3 Part A The Indicial Nota

Continuum Mechanics

Rosalind and John J Redfern, Jr, Professor of Engineering

Rensselaer Polytechnic Institute, Troy, NY, USA

U T T E R W O R T H

E I N E M A N N

Reprinted in 1999 by Butterworth-Heinemann is an imprint of Elsevier All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical,

photocopying, recording, or otherwise, without the prior written permission of the publisher.

This book is printed on acid-free paper.

Library of Congress Cataloging-in-Publication Data

Lai, W Michael,

1930-Introduction to continuum mechanics/W.Michael Lai,

David Rubin, Erhard Krempl - 3rd ed.

p cm.

ISBN 0 7506 2894 4

1 Contiuum mechanics I Rubin,David,

1942-II.Krempl, Erhard III Title

QA808.2.L3 1993

531-dc20 93-30117

for Library of Congress CIP

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library, The publisher offers special discounts on bulk orders of this book.

For information, please contact:

Manager of Special Sales

Preface to the Third Edition xii Preface to the First Edition xiii The Authors xiv

1.1 Continuum Theory 11.2 Contents of Continuum Mechanics 1

Chapter 2 Tensors 3

Part A The Indicial Notation 32A1 Summation Convention, Dummy Indices 32A2 Free Indices 52A3 Kronecker Delta 62A4 Permutation Symbol 72A5 Manipulations with the Indicial Notation 8Part B Tensors 112B1 Tensor: A Linear Transformation 112B2 Components of a Tensor 132B3 Components of a Transformed Vector 162B4 Sum of Tensors 172B5 Product of Two Tensors 182B6 Transpose of a Tensor 202B7 Dyadic Product of Two Vectors 21

2B8 Trace of a Tensor 222B9 Identity Tensor and Tensor Inverse 232B10 Orthogonal Tensor 242B11 Transformation Matrix Between Two Rectangular

Cartesian Coordinate Systems 262B12 Transformation Laws for Cartesian Components of Vectors 282B13 Transformation Law for Cartesian Components of a Tensor 302B14 Defining Tensors by Transformation Laws 322B15 Symmetric and Antisymmetric Tensors 352B16 The Dual Vector of an Antisymmetric Tensor 362B17 Eigenvalues and Eigenvectors of a Tensor 382B18 Principal Values and Principal Directions of Real Symmetric Tensors 432B19 Matrix of a Tensor with Respect to Principal Directions 442B20 Principal Scalar Invariants of a Tensor 45Part C Tensor Calculus 472C1 Tensor-valued functions of a Scalar 472C2 Scalar Field, Gradient of a Scalar Function 492C3 Vector Field, Gradient of a Vector Field 532C4 Divergence of a Vector Field and Divergence of a Tensor Field 542C5 Curl of a Vector Field 55Part D Curvilinear Coordinates 572D1 Polar Coordinates 572D2 Cylindrical Coordinates 612D3 Spherical Coordinates 63Problems 68

Chapter 3 Kinematics of a Continuum 79

3.1 Description of Motions of a Continuum 793.2 Material Description and Spatial Description 833.3 Material Derivative 853.4 Acceleration of a Particle in a Continuum 873.5 Displacement Field 923.6 Kinematic Equations For Rigid Body Motion 933.7 Infinitesimal Deformations 943.8 Geometrical Meaning of the Components of the Infinitesimal Strain Tensor 99

3.9 Principal Strain 1053.10 Dilatation 1053.11 The Infinitesimal Rotation Tensor 1063.12 Time Rate of Change of a Material Element 1063.13 The Rate of Deformation Tensor 1083.14 The Spin Tensor and the Angular Velocity Vector 1113.15 Equation of Conservation Of Mass 1123.16 Compatibility Conditions for Infinitesimal Strain Components 1143.17 Compatibility Conditions for the Rate of Deformation Components 1193.18 Deformation Gradient 1203.19 Local Rigid Body Displacements 1213.20 Finite Deformation 1213.21 Polar Decomposition Theorem 1243.22 Calculation of the Stretch Tensor from the Deformation Gradient 1263.23 Right Cauchy-Green Deformation Tensor 1283.24 Lagrangian Strain Tensor 1343.25 Left Cauchy-Green Deformation Tensor 1383.26 Eulerian Strain Tensor 1413.27 Compatibility Conditions for Components of Finite Deformation Tensor 1443.28 Change of Area due to Deformation 1453.29 Change of Volume due to Deformation 1463.30 Components of Deformation Tensors in other Coordinates 1493.31 Current Configuration as the Reference Configuration 158Problems 160

Chapter 4 Stress 173

4.1 Stress Vector 1734.2 Stress Tensor 1744.3 Components of Stress Tensor 1764.4 Symmetry of Stress Tensor - Principle of Moment of Momentum 1784.5 Principal Stresses 1824.6 Maximum Shearing Stress 1824.7 Equations of Motion - Principle of Linear Momentum 1874.8 Equations of Motion in Cylindrical and Spherical Coordinates 1904.9 Boundary Condition for the Stress Tensor 1924.10 Piola Kirchhoff Stress Tensors 195

4.11 Equations of Motion Written With Respect to the Reference

Configuration 2014.12 Stress Power 2034.13 Rate of Heat Flow Into an Element by Conduction 2074.14 Energy Equation 2084.15 Entropy Inequality 209Problems 210

Chapter 5 The Elastic Solid 217

5.1 Mechanical Properties 2175.2 Linear Elastic Solid 220Part A Linear Isotropic Elastic Solid 2255.3 Linear Isotropic Elastic Solid 2255.4 Young's Modulus, Poisson's Ratio, Shear Modulus, and Bulk Modulus 2285.5 Equations of the Infinitesimal Theory of Elasticity 2325.6 Navier Equation in Cylindrical and Spherical Coordinates 2365.7 Principle of Superposition 2385.8 Plane Irrotational Wave 2385.9 Plane Equivoluminal Wave 2425.10 Reflection of Plane Elastic Waves 2485.11 Vibration of an Infinite Plate 2515.12 Simple Extension 2545.13 Torsion of a Circular Cylinder 2585.14 Torsion of a Noncircular Cylinder 2665.15 Pure Bending of a Beam 2695.16 Plane Strain 2755.17 Plane Strain Problem in Polar Coordinates 2815.18 Thick-walled Circular Cylinder under Internal and External Pressure 2845.19 Pure Bending of a Curved Beam 2855.20 Stress Concentration due to a Small Circular Hole in a Plate under Tension 2875.21 Hollow Sphere Subjected to Internal and External Pressure 291Part B Linear Anisotropic Elastic Solid 2935.22 Constitutive Equations for Anisotropic Elastic Solid 2935.23 Plane of Material Symmetry 2965.24 Constitutive Equation for a Monoclinic Anisotropic Elastic Solid 299

5.25 Constitutive Equations for an Orthotropic Elastic Solid 3015.26 Constitutive Equation for a Transversely Isotropic Elastic Material 3035.27 Constitutive Equation for Isotropic Elastic Solid 3065.28 Engineering Constants for Isotropic Elastic Solid 3075.29 Engineering Constants for Transversely Isotropic Elastic Solid 3085.30 Engineering Constants for Orthotropic Elastic Solid 3115.31 Engineering Constants for a Monoclinic Elastic Solid 312Part C Constitutive Equation For Isotropic Elastic Solid Under Large Deformation 3145.32 Change of Frame 3145.33 Constitutive Equation for an Elastic Medium under Large Deformation 3195.34 Constitutive Equation for an Isotropic Elastic Medium 3225.35 Simple Extension of an Incompressible Isotropic Elastic Solid 3245.36 Simple Shear of an Incompressible Isotropic Elastic Rectangular Block 3255.37 Bending of a Incompressible Rectangular Bar 3275.38 Torsion and Tension of an Incompressible Solid Cylinder 331Problems 335

Chapter 6 Newtonian Viscous Fluid 348

6.1 Fluids 3486.2 Compressible and Incompressible Fluids 3496.3 Equations of Hydrostatics 3506.4 Newtonian Fluid 3556.5 Interpretation of l and m 3576.6 Incompressible Newtonian Fluid 3596.7 Navier-Stokes Equation for Incompressible Fluids 3606.8 Navier-Stokes Equations for In compressible Fluids in

Cylindrical and Spherical Coordinates 3646.9 Boundary Conditions 3656.10 Streamline, Pathline, Streakline, Steady, Unsteady, Laminar and

Turbulent Flow 3666.11 Plane Couette Flow 3716.12 Plane Poiseuille Flow 3726.13 Hagen Poiseuille Flow 3746.14 Plane Couette Flow of Two Layers of Incompressible Fluids 3776.15 Couette Flow 3806.16 Flow Near an Oscillating Plate 381

6.17 Dissipation Functions for Newtonian Fluids 3836.18 Energy Equation for a Newtonian Fluid 3846.19 Vorticity Vector 3876.20 Irrotational Flow 3906.21 Irrotational Flow of an Inviscid, Incompressible Fluid of

Homogeneous Density 3916.22 Irrotational Flows as Solutions of Navier-Stokes Equation 3946.23 Vorticity Transport Equation for Incompressible Viscous Fluid

with a Constant Density 3966.24 Concept of a Boundary Layer 3996.25 Compressible Newtonian Fluid 4016.26 Energy Equation in Terms of Enthalpy 4026.27 Acoustic Wave 4046.28 Irrotational, Barotropic Flows of Inviscid Compressible Fluid 4086.29 One-Dimensional Flow of a Compressible Fluid 412Problems 419

Chapter7 Integral Formulation of General Principles 427

7.1 Green's Theorem 4277.2 Divergence Theorem 4307.3 Integrals over a Control Volume and Integrals over a Material Volume 4337.4 Reynolds Transport Theorem 4357.5 Principle of Conservation of Mass 4377.6 Principle of Linear Momentum 4407.7 Moving Frames 4477.8 Control Volume Fixed with Respect to a Moving Frame 4497.9 Principle of Moment of Momentum 4517.10 Principle of Conservation of Energy 454Problems 458

Chapter 8 Non-Newtonian Fluids 462

Part A Linear Viscoelastic Fluid 4648.1 Linear Maxwell Fluid 4648.2 Generalized Linear Maxwell Fluid with Discrete Relaxation Spectra 4718.3 Integral Form of the Linear Maxwell Fluid and of the

Generalized Linear Maxwell Fluid with Discrete Relaxation Spectra 4738.4 Generalized Linear Maxwell Fluid with a Continuous Relaxation Spectrum 474

Part B Nonlinear Viscoelastic Fluid 4768.5 Current Configuration as Reference Configuration 4768.6 Relative Deformation Gradient 4778.7 Relative Deformation Tensors 4788.8 Calculations of the Relative Deformation Tensor 4808.9 History of Deformation Tensor Rivlin-Ericksen Tensors 4868.10 Rivlin-Ericksen Tensor in Terms of Velocity Gradients -

The Recursive Formulas 4918.11 Relation Between Velocity Gradient and Deformation Gradient 4938.12 Transformation Laws for the Relative Deformation Tensors under a

Change of Frame 4948.13 Transformation law for the Rivlin-Ericksen Tensors under a

Change of Frame 4968.14 Incompressible Simple Fluid 4978.15 Special Single Integral Type Nonlinear Constitutive Equations 4988.16 General Single Integral Type Nonlinear Constitutive Equations 5038.17 Differential Type Constitutive Equations 5038.18 Objective Rate of Stress 5068.19 The Rate Type Constitutive Equations 511Part C Viscometric Flow Of Simple Fluid 5168.20 Viscometric Flow 5168.21 Stresses in Viscometric Flow of an Incompressible Simple Fluid 5208.22 Channel Flow 5238.23 Couette Flow 526Problems 532

Appendix: Matrices 537 Answer to Problems 543 References 550 Index 552

The first edition of this book was published in 1974, nearly twenty years ago It was written

as a text book for an introductory course in continuum mechanics and aimed specifically at thejunior and senior level of undergraduate engineering curricula which choose to introduce tothe students at the undergraduate level the general approach to the subject matter ofcontinuum mechanics We are pleased that many instructors of continuum mechanics havefound this little book serves that purpose well However, we have also understood that manyinstructors have used this book as one of the texts for a beginning graduate course in continuummechanics It is this latter knowledge that has motivated us to write this new edition In thispresent edition, we have included materials which we feel are suitable for a beginning graduatecourse in continuum mechanics The following are examples of the additions:

1 Am'sotropic elastic solid which includes the concept of material symmetry and theconstitutive equations for monoclinic, orthotropic, transversely isotropic and isotropicmaterials

2 Finite deformation theory which includes derivations of the various finite deformationtensors, the Piola-Kirchhoff stress tensors, the constitutive equations for an incompres-sible nonlinear elastic solid together with some boundary value problems

3 Some solutions of classical elasticity problems such as thick-wailed pressure vessels(cylinders and spheres), stress concentrations and bending of curved bars

4 Objective tensors and objective time derivatives of tensors including corotationalderivative and convected derivatives

5 Differential type, rate type and integral type linear and nonlinear constitutive equationsfor viscoelastic fluids and some solutions for the simple fluid in viscometric flows

6 Equations in cylindrical and spherical coordinates are provided including the use ofdifferent coordinates for the deformed and the undeformed states

We wish to state that notwithstanding the additions, the present edition is still intended to

be "introductory" in nature, so that the coverage is not extensive We hope that this newedition can serve a dual purpose: for an introductory course at the undergraduate level byomitting some of the "intermediate level" material in the book and for a beginning graduatecourse in continuum mechanics at the graduate level

This text is prepared for the purpose of introducing the concept of continuum mechanics

to beginners in the field Special attention and care have been given to the presentation of thesubject matter so that it is within the grasp of those readers who have had a good background

in calculus, some differential equations, and some rigid body mechanics For pedagogicalreasons the coverage of the subject matter is far from being extensive, only enough to providefor a better understanding of later courses in the various branches of continuum mechanicsand related fields The major portion of the material has been successfully class-tested atRensselaer Polytechnic Institute for undergraduate students However, the authors believethe text may also be suitable for a beginning graduate course in continuum mechanics

We take the liberty to say a few words about the second chapter This chapter introducessecond-order tensors as linear transformations of vectors in a three dimensional space Fromour teaching experience, the concept of linear transformation is the most effective way ofintroducing the subject It is a self-contained chapter so that prior knowledge of lineartransformations, though helpful, is not required of the students The third-and higher-ordertensors are introduced through the generalization of the transformation laws for the second-order tensor Indicial notation is employed whenever it economizes the writing of equations.Matrices are also used in order to facilitate computations An appendix on matrices is included

at the end of the text for those who are not familiar with matrices

Also, let us say a few words about the presentation of the basic principles of continuumphysics Both the differential and integral formulation of the principles are presented, thedifferential formulations are given in Chapters 3,4, and 6, at places where quantities needed

in the formulation are defined while the integral formulations are given later in Chapter 7.This is done for a pedagogical reason: the integral formulations as presented required slightlymore mathematical sophistication on the part of a beginner and may be either postponed oromitted without affecting the main part of the text

This text would never have been completed without the constant encouragement and advicefrom Professor F F Ling, Chairman of Mechanics Division at RPI, to whom the authors wish

to express their heartfelt thanks Gratefully acknowledged is the financial support of the FordFoundation under a grant which is directed by Dr S W Yerazunis, Associate Dean ofEngineering The authors also wish to thank Drs V C Mow and W B Browner, Jr for theirmany useful suggestions Special thanks are given to Dr H A Scarton for painstakinglycompiling a list of errata and suggestions on the preliminary edition Finally, they are indebted

to Mrs Geri Frank who typed the entire manuscript

W Michael Lai (Ph.D., University of Michigan) is Professor of Mechanical Engineering and

Orthopaedic Bioengineering at Columbia University, New York, New York He is a member

of ASME (Fellow), AIMBE (Fellow), ASCE, AAM, ASB,ORS, AAAS, Sigma Xi and PhiKappa Phi

David Rubin (Ph.D., Brown University) is a principal at Weidlinger Associates, New York,

New York He is a member of ASME, Sigma Xi, Tau Beta Pi and Chi Epsilon

Erhard Krempl (Dr.-Ing., Technische Hochschule Munchen) is Rosalind and John J Refern

Jr Professor of Engineering at Rensselaer Polytechnic Institute He is a member of ASME(Fellow), AAM (Fellow), ASTM, ASEE, SEM, SES and Sigma Xi

1.1 CONTINUUM THEORY

Matter is formed of molecules which in turn consist of atoms and sub-atomic particles Thusmatter is not continuous However, there are many aspects of everyday experience regardingthe behaviors of materials, such as the deflection of a structure under loads, the rate ofdischarge of water in a pipe under a pressure gradient or the drag force experienced by a bodymoving in the air etc., which can be described and predicted with theories that pay no attention

to the molecular structure of materials The theory which aims at describing relationshipsbetween gross phenomena, neglecting the structure of material on a smaller scale, is known

as continuum theory The continuum theory regards matter as indefinitely divisible Thus,within this theory, one accepts the idea of an infinitesimal volume of materials referred to as

a particle in the continuum, and in every neighborhood of a particle there are always neighborparticles Whether the continuum theory is justified or not depends on the given situation; forexample, while the continuum approach adequately describes the behavior of real materials

in many circumstances, it does not yield results that are in accord with experimental tions in the propagation of waves of extremely small wavelength On the other hand, a rarefiedgas may be adequately described by a continuum in certain circumstances At any case, it ismisleading to justify the continuum approach on the basis of the number of molecules in agiven volume After all, an infinitesimal volume in the limit contains no molecules at all.Neither is it necessary to infer that quantities occurring in continuum theory must be inter-preted as certain particular statistical averages In fact, it has been known that the samecontinuum equation can be arrived at by different hypothesis about the molecular structureand definitions of gross variables While molecular-statistical theory, whenever available, doesenhance the understanding of the continuum theory, the point to be made is simply thatwhether the continuum theory is justified in a given situation is a matter of experimental test,not of philosophy Suffice it to say that more than a hundred years of experience have justifiedsuch a theory in a wide variety of situations

observa-1.2 Contents of Continuum Mechanics

Continuum mechanics studies the response of materials to different loading conditions Itssubject matter can be divided into two main parts: (1) general principles common to all media,

1

and (2) constitutive equations defining idealized materials The general principles are axiomsconsidered to be self-evident from our experience with the physical world, such as conservation

of mass, balance of linear momentum, of moment of momentum, of energy, and the entropyinequality law Mathematically, there are two equivalent forms of the general principles: (1)the integral form, formulated for a finite volume of material in the continuum, and (2) the fieldequations for differential volume of material (particle) at every point of the field of interest.Field equations are often derived from the integral form They can also be derived directlyfrom the free body of a differential volume The latter approach seems to suit beginners Inthis text both approaches are presented, with the integral form given toward the end of thetext Field equations are important wherever the variations of the variables in the field areeither of interest by itself or are needed to get the desired information On the other hand, theintegral forms of conservation laws lend themselves readily to certain approximate solutions

The second major part of the theory of continuum mechanics concerns the "constitutiveequations" which are used to define idealized material Idealized materials represent certainaspects of the mechanical behavior of the natural materials For example, for many materialsunder restricted conditions, the deformation caused by the application of loads disappears withthe removal of the loads This aspect of the material behavior is represented by the constitutiveequation of an elastic body Under even more restricted conditions, the state of stress at a pointdepends linearly on the changes of lengths and mutual angle suffered by elements at the pointmeasured from the state where the external and internal forces vanish The above expressiondefines the linearly elastic solid Another example is supplied by the classical definition ofviscosity which is based on the assumption that the state of stress depends linearly on theinstantaneous rates of change of length and mutual angle Such a constitutive equation definesthe linearly viscous fluid The mechanical behavior of real materials varies not only frommaterial to material but also with different loading conditions for a given material This leads

to the formulation of many constitutive equations defining the many different aspects ofmaterial behavior In this text, we shall present four idealized models and study the behaviorthey represent by means of some solutions of simple boundary-value problems The idealizedmaterials chosen are (1) the linear isotropic and anisotropic elastic solid (2) the incompressiblenonlinear isotropic elastic solid (3) the linearly viscous fluid including the inviscid fluid, and(4) the Non-Newtonian incompressible fluid

One important requirement which must be satisfied by all quantities used in the formulation

of a physical law is that they be coordinate-invariant In the following chapter, we discuss suchquantities

As mentioned in the introduction, all laws of continuum mechanics must be formulated interms of quantities that are independent of coordinates It is the purpose of this chapter tointroduce such mathematical entities We shall begin by introducing a short-hand notation

- the indicial notation - in Part A of this chapter, which will be followed by the concept oftensors introduced as a linear transformation in Part B The basic field operations needed forcontinuum formulations are presented in Part C and their representations in curvilinearcoordinates in Part D

Part A The Indicial Notation

2A1 Summation Convention, Dummy Indices

Consider the sum

We can write the above equation in a compact form by using the summation sign:

It is obvious that the following equations have exactly the same meaning as Eq (2A1.2)

3

etc.

The index i in Eq (2A1.2), or; in Eq (2A1.3), or m in Eq (2A1.4) is a dummy index in the

sense that the sum is independent of the letter used

We can further simplify the writing of Eq.(2Al.l) if we adopt the following convention:Whenever an index is repeated once, it is a dummy index indicating a summation with the

index running through the integers 1,2, , n.

This convention is known as Einstein's summation convention Using the convention,

Eq (2A1.1) shortens to

We also note that

It is emphasized that expressions such as a i b i x i are not defined within this convention That

is, an index should never be repeated more than once when the summation convention is used.

Therefore, an expression of the form

must retain its summation sign

In the following we shall always take n to be 3 so that, for example,

Expanding in full, the expression (2A1.8) gives a sum of nine terms, i.e.,

For beginners, it is probably better to perform the above expansion in two steps, first, sumover i and then sum over j (or vice versa), i.e.,

a x x = a x x + a x x + a x x

etc.

Similarly, the triple sum

will simply be written as

The expression (2A1.11) represents the sum of 27 terms

We emphasize again that expressions such as aii xi xj xj or aijk xixixj xk are not defined in thesummation convention, they do not represent

2A2 Free Indices

Consider the following system of three equations

Using the summation convention, Eqs (2A2.1) can be written as

which can be shortened into

An index which appears only once in each term of an equation such as the index i in

Eq (2A2.3) is called a "free index." A free index takes on the integral number 1, 2, or 3 one

at a time Thus Eq (2A2.3) is short-hand for three equations each having a sum of three terms

on its right-hand side [i.e., Eqs (2A2.1)]

A further example is given by

We note that xj = a jm x m , j= 1,2,3, is the same as Eq (2A2.3) and ej' = Q mj em, j=1,2,3 is the

same as Eq (2A2.4) However,

a i = bj

is a meaningless equation The free index appearing in every term of an equation must be the

same Thus the following equations are meaningful

ai + ki = ci

ai + bicjdj = 0

If there are two free indices appearing in an equation such as

then the equation is a short-hand writing of 9 equations; each has a sum of 3 terms on theright-hand side In fact,

T11 = A 1m A 1m = A 11 A 11 + A12A 12 +A13A l3

T12 =A1mA2m =A11A21 +A12A22 +A13A23

T13 = A1mA3m = A 11 A 31 + A 12 A 32 + A 13 A 33

T33 = A3mA3m = A31A31 + A32A32 + A33A33Again, equations such as

Tij = Tik

have no meaning,

2A3 Kronecker Delta

The Kronecker delta, denoted by dij, is defined as

That is,

d11 = d22 = d33 = 1

d12 =d13 =d21 =d23 =d31 = d32 = 0

In other words, the matrix of the Kronecker delta is the identity matrix, i.e.,

We note the following:

Or, in general

or, in general

In particular

etc

(d) If e1,e2,e3 are unit vectors perpendicular to each other, then

2A4 Permutation Symbol

The permutation symbol, denoted by eijk is defined by

We note that

If e1,e2,e3 form a right-handed triad, then

which can be written for short as

Now, if a = aiei, and b = biei, then

i.e.,

The following useful identity can be proven (see Prob 2A7)

2A5 Manipulations with the Indicial Notation

(a) Substitution

If

and

then, in order to substitute the bi's in (ii) into (i) we first change the free index in (ii) from i to

m and the dummy index m to some other letter, say n so that

Now, (i) and (iii) give

Note (iv) represents three equations each having the sum of nine terms on its right-hand side

(b) Multiplication

If

and

then,

It is important to note that pq # a m b m c m d m In fact, the right hand side of this expression

is not even defined in the summation convention and further it is obvious that

Since the dot product of vectors is distributive, therefore, if a = aiei and b = biei, then

In particular, if e1e2e3 are unit vectors perpendicular to one another, then ei ej = so that

(c) Factoring

If

then, using the Kronecker delta, we can write

so that (i) becomes

Thus,

(d) Contraction

The operation of identifying two indices and so summing on them is known as contraction

For example, Tii is the contraction of Tij,

Part B Tensors

281 Tensor - A Linear Transformation

Let T be a transformation, which transforms any vector into another vector If T transforms

a into c and b into d, we write Ta = c and Tb = d

If T has the following linear properties:

where a and b are two arbitrary vectors and a is an arbitrary scalar then T is called a linear

transformation It is also called a second-order tensor or simply a tensor An alternative and

equivalent definition of a linear transformation is given by the single linear property:

where a and b are two arbitrary vectors and a and ft are arbitrary scalars.

If two tensors, T and S, transform any arbitrary vector a in an identical way, then thesetensors are equal to each other, i.e., Ta=Sa -» T=S

Let T be a transformation which transforms every vector into a fixed vector n Is thistransformation a tensor?

Solution Let a and b be any two vectors, then by the definition of T,

Ta = n, Tb = n and T(a+b) = nClearly,

T(a+b) * Ta+Tb

Thus, T is not a linear transformation In other words, it is not a tensor

t Scalars and vectors are sometimes called the zeroth and first order tensor, respectively Even though they can also be defined algebraically, in terms of certain operational rules, we choose not to do so The geometrical concept of scalars and vectors, which we assume that the students are familiar with, is quite sufficient for our purpose.

Example 2B1.2

Let T be a transformation which transforms every vector into a vector that is k times the

original vector Is this transformation a tensor?

Solution Let a and b be arbitrary vectors and a and ft be arbitrary scalars, then by the

definition of T,

Ta = Jta, Tb = fcb, and T(aa+£b) = fc(aa+/3b)Clearly,

T(aa+£b) = a(ka)+p(kb) = aTa+£Tb

Thus, by Eq (2B1.2), T is a linear transformation In other words, it is a tensor

In the previous example, if fc=0 then the tensor T transforms all vectors into zero Thistensor is the zero tensor and is symbolized by 0

Example 2B1.3Consider a transformation T that transforms every vector into its mirror image with respect

to a fixed plane Is T a tensor?

Solution Consider a parallelogram in space with its sides represented by vectors a and b

and its diagonal represented the resultant a + b Since the parallelogram remains a lelogram after the reflection, the diagonal (the resultant vector) of the reflected parallelogram

paral-is clearly both T(a + b ) , the reflected (a + b), and Ta + Tb, the sum of the reflected a and

the reflected b That is, T(a + b) = Ta + Tb Also, for an arbitrary scalar a, the reflection

of aa is obviously the same as a times the reflection of a (i.e., T(aa )= aTa) because both vectors have the same magnitude given by a times the magnitude of a and the same direction.

Thus, by Eqs (2B1.1), T is a tensor

Example 2B 1.4When a rigid body undergoes a rotation about some axis, vectors drawn in the rigid body ingeneral change their directions That is, the rotation transforms vectors drawn in the rigid bodyinto other vectors Denote this transformation by R Is R a tensor?

Solution Consider a parallelogram embedded in the rigid body with its sides representing

vectors a and b and its diagonal representing the resultant a + b Since the parallelogramremains a parallelogram after a rotation about any axis, the diagonal (the resultant vector) ofthe rotated parallelogram is clearly both R(a + b ) , the rotated (a 4- b), and Ra 4- Rb, thesum of the rotated a and the rotated b That is R(a + b) = Ra + Rb.A similar argument asthat used in the previous example leads to R(aa )= aRa Thus, R is a tensor

Example 2B1.5Let T be a tensor that transforms the specific vectors a and b according to

Ta = a+2b, Tb = a-bGiven a vector c = 2a+b, find Tc

Solution Using the linearity property of tensors

Tc = T(2a+b) = 2Ta+Tb = 2(a+2b)+(a-b) = 3a+3b

2B2 Components of a Tensor

The components of a vector depend on the base vectors used to describe the components

This will also be true for tensors Let ej_, 63, ©3 be unit vectors in the direction of the xi~, X2~,

jt3-axes respectively, of a rectangular Cartesian coordinate system Under a transformation T,these vectors, el5 62, e3 become Tels Te2, and Te3 Each of these Te/ (/= 1,2,3), being a vector,can be written as:

or

It is clear from Eqs (2B2.1a) that

or in general

The components TJJ in the above equations are defined as the components of the tensor T.

These components can be put in a matrix as follows:

TII T n T 13

[T] = 7^! r22 r23

Til 732 ^33

This matrix is called the matrix of the tensor T with respect to the set of base vectors

iei> e2, es} °r le/} f°r short We note that, because of the way we have chosen to denote thecomponents of transformation of the base vectors, the elements of the first column arecomponents of the vector Tej, those in the second column are the components of the vectorTe2, and those in the third column are the components of Te

Example 2B2.1Obtain the matrix for the tensor T which transforms the base vectors as follows:

Example 2B2.2Let T transform every vector into its mirror image with respect to a fixed plane If ej isnormal to the reflection plane (e2 and 63 are parallel to this plane), find a matrix of T

Solution Since the normal to the reflection plane is transformed into its negative and vectors

parallel to the plane are not altered:

Tej = -Cl

Te2 = e2

Te3 = e3

Thus,

"-1 0 0"

[ T ] = 0 1 0

0 0 1

J Cj

We note that this is only one of the infinitely many matrices of the tensor T, each depends

on a particular choice of base vectors In the above matrix, the choice of e, is indicated at thebottom right corner of the matrix If we choose ei and 62 to be on a plane perpendicular tothe mirror with each making 45° with the mirror as shown in Fig 2B.1 and 63 points straightout from the paper Then we have

Tel = «2

Te2 = ejTe3 = e3Thus, with respect to {e/}, the matrix of the tensor is

T ' with respect to the basis e,-

Re 3 = e 3

Thus,

cos# -sin# 0[R] = sinfl cos0 0

0 0 1

L J c j

2B3 Components of a Transformed Vector

Given the vector a and the tensor T, we wish to compute the components of b=Ta from thecomponents of a and the components of T Let the components of a with respect to {61,62,63}

This is the reason we adopted the convention that Tej = T^i+7*2162+ 73163, etc If we had

adopted the convention Te^ = 7ne1+7t1262+^I3e3' etc-' tnen we would have obtained

7*

[b]=[T] [a] for the tensorial equation b = Ta, which would not be as natural

Example 2B3.1Given that a tensor T which transforms the base vectors as follows:

Tej = 2e1-6e2+4e3

T02 = 3ej+462-63

Te3 = -26J+62+263How does this tensor transform the vector a = ej+262+363?

Solution Using Eq (2B3.1b)

It is easily seen that by this definition T + S is indeed a tensor.

To find the components of T + S, let

Using Eqs (2B2.2) and (2B4.1), the components of W are obtained to be

i.e.,

In matrix notation, we have

2B5 Product of Two Tensors

Let T and S be two tensors and a be an arbitrary vector, then TS and ST are defined to bethe transformations (easily seen to be tensors)

and

Thus the components of TS are

i.e.,

Similarly,

In fact, Eq (2B5.3) is equivalent to the matrix equation:

whereas, Eq (2B5.4) is equivalent to the matrix equation:

The two matrix products are in general different Thus, it is clear that in general, the tensorproduct is not commutative (i.e., TS * ST)

If T,S, and V are three tensors, then

(T(SV))a = T((SV)a) = T(S(Va))

(TS)(Va) = T(S(Va))i.e.,

Thus, the tensor product is associative It is, therefore, natural to define the integral positivepowers of a transformation by these simple products, so that

Example 2B5.1(a)Let R correspond to a 90° right-hand rigid body rotation about the^-axis Find the matrixofR

(b)Let S correspond to a 90°right-hand rigid body rotation about thejcj-axis Find the matrixofS

(c)Find the matrix of the tensor that corresponds to the rotation (a) then (b)

(d)Find the matrix of the tensor that corresponds to the rotation (b) then (a)

(e)Consider a point P whose initial coordinates are (1,1,0) Find the new position of thispoint after the rotations of part (c) Also find the new position of this point after the rotations

of part (d)

Solution, (a) For this rotation the transformation of the base vectors is given by

Rej = e2

Re2 = -ejRe3 = e3

so that,

0 -1 0~

[R]= 1 0 0

0 0 1(b)In a similar manner to (a) the transformation of the base vectors is given by

(c)Since S(Ra) = (SR)a, the resultant rotation is given by the single transformation SRwhose components are given by the matrix

"l 0 Ol [0 -1 0] |~0 -1 0"

[SR]= 0 0 - 1 1 0 0 = 0 0 - 1

[0 1 OJ [0 0 IJ [l 0 0(d)In a manner similar to (c) the resultant rotation is given by the single transformation RSwhose components are given by the matrix

"o -i ol fi o ol [ b o i"

[RS]= 1 0 0 0 0 - 1 = 1 0 0

[0 0 IJ [0 1 OJ [0 1 0

(e)Let r be the initial position of the point P Let r* and r** be the rotated position of P

after the rotations of part (c) and part (d) respectively Then

[o ~i ol [i] F-i"

[r*] = [SR][r] = 0 0 - 1 1 = 0

1 0 OJ I OJ [ 1i.e.,

**

r = 62+63This example further illustrates that the order of rotations is important

[T7] = [T]7"

ff>

i.e., the matrix of T is the transpose of the matrix of T

We also note that by Eq (2B6.1), we have

a-T7b = b-(TT)TaThus, b-Ta = b- (T7)ra for any a and b, so that

It can also be established that (see Prob 2B13)

That is, the transpose of a product of the tensors is equal to the product of transposed tensors

in reverse order More generally,

287 Dyadic Product of Two Vectors

The dyadic product of vectors a and b, denoted by ab, is defined to be the transformationwhich transforms an arbitrary vector c according to the rule:

Now, for any c, d, a and/3, we have, from the above definition:

(ab)(ac+£d) = a(b-(ac+0d)) = a((ab-c)+(0b-d)) = a(ab)c+0(ab)d

Thus, ab is a tensor Letting W=ab, then the components of W are:

i.e.,

In matrix notation, Eq (2B7.2a) is

In particular, the components of the dyadic product of the base vectors e, are:

"l 0 0] [b 1 0[e^i] = 0 0 0 , [0^2] = 0 0 0 ,

0 0 0 0 0 0Thus, it is clear that any tensor T can be expressed as:

We note that another commonly used notation for the dyadic product of a and b is a®b

2B8 Trace of a Tensor

The trace of any dyad ab is defined to be a scalar given by a • b That is,

Furthermore, the trace is defined to be a linear operator that satisfies the relation:

Using Eq (2B7.3b), the trace of T can, therefore, be obtained as

that is,

It is obvious that

Show that for any second-order tensor A and B

Solution Let C=AB, then C^-A- im B m y Thus,

Let D=BA, then Dy=B/m/4m/-, and

But Bi nt A m i=B m iAf m (change of dummy indices), that is

2B9 identity Tensor and Tensor Inverse

The linear transformation which transforms every vector into itself is called an identitytensor Denoting this special tensor by I, we have, for any vector a,

and in particular,

Thus, the components of the identity tensor are:

i.e.,

It is obvious that the identity matrix is the matrix of I for all rectangular Cartesian coordinates

and that TI = IT = T for any tensor T We also note that if Ta = a for any arbitrary a, then

T = I

Example 2B9.1

Write the tensor T, defined by the equation Ta = A:a, where k is a constant and a is arbitrary,

in terms of the identity tensor and find its components

Solution Using Eq (2B9.1) we can write A; a as fcla so that Ta = fca becomes

Ta = Maand since a is arbitrary

T = fclThe components of this tensor are clearly,

T • =

kd-1 fj KVy

Given a tensor T, if a tensor S exists such that ST=I then we call S the inverse of T orS=T-1 (Note: With T~1T=T~1+1=T°=I, the zeroth power of a tensor is the identitytensor) To find the components of the inverse of a tensor T is to find the inverse of the matrix

of T From the study of matrices we know that the inverse exists as long as detT^O (that is, T

is non-singular) and in this case, [T]"1 [T] = [T] [T]""1 = [I] Thus, the inverse of a tensorsatisfies the following reciprocal relation:

We can easily show (see Prob 2B15) that for the tensor inverse the following relations aresatisfied,

and

We note that if the inverse exists then we have the reciprocal relation that

This indicates that when a tensor is invertible there is a one to one mapping of vectors

a and b On the other hand, if a tensor T does not have an inverse, then, for a given b, thereare in general more than one a which transforms into b For example, consider the singulartensor T = cd (the dyadic product of c and d , which does not have an inverse because itsdeterminant is zero), we have

Now, let h be any vector perpendicular to d (i.e., d • h = 0), then

That is, all vectors a + h transform under T into the same vector b

2B10 Orthogonal Tensor

An orthogonal tensor is a linear transformation, under which the transformed vectorspreserve their lengths and angles Let Q denote an orthogonal tensor, then by definition,

| Qa | = | a | and cos(a,b) = cos(Qa,Qb) for any a and b, Thus,

for any a and b

Using the definitions of the transpose and the product of tensors:

Therefore,

Since a and b are arbitrary, it follows that

This means that Q~1=Qrand from Eq (2B9.3),

In matrix notation, Eqs (2B10.2a) take the form:

and in subscript notation, these equations take the form:

Example 2B 10.1The tensor given in Example 2B2.2, being a reflection, is obviously an orthogonal tensor.Verify that [T][T] = [I] for the [T] in that example Also, find the determinant of [T]

Solution Using the matrix of Example 2B7.1:

Solution It is clear that

[cos0 -sin<9 ol f cos0 sin0 o] |"l 0 0*

[ 0 0 l l [ 0 0 ij [0 0 1

cos# -sin# 0det[R]s|R| = sin0 cos0 0 = + 1

0 0 1

The determinant of the matrix of any orthogonal tensor Q is easily shown to be equal toeither + 1 or -1 In fact,

[QHQf =[i]