Chapter 1. Tensor Algebra

8/10/2019 Chapter 1. Tensor Algebra

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Continuum Mechanics

C. Agelet de SaracibarETS Ingenieros de Caminos, Canales y Puertos, Universidad Politcnica de Catalua (UPC), Barcelona, Spain

International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain

Chapter 1Introduction to Vectors and Tensors


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Contents

1. Introduction

2. Algebra of vectors

3. Algebra of tensors

4. Higher order tensors5. Differential operators

6. Integral theorems

Introduction to Vectors and Tensors > Contents

Contents

October 20, 2012 Carlos Agelet de Saracibar 2


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Tensors

Continuum mechanics deals withphysical propertiesof

materials, such as solids, liquids, gases or intermediate states,

which are independent of any particular coordinate system in

which they are observed.

Thosephysical properties are mathematically represented by

tensors.

Tensorsare mathematical objectswhich have the required

property of being independentof any particular coordinate

system.

Introduction to Vectors and Tensors > Introduction

Introduction



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Tensors

According to their tensorial order, tensors may be classified as:

Zero-order tensors: scalars, i.e. density, temperature, pressure

First-order tensors: vectors, i.e. velocity, acceleration, force

Second-order tensors, i.e. stress, strain

Third-order tensors, i.e. permutation

Fourth-order tensors, i.e. elastic constitutive tensor

First- and higher-order tensors may be expressed in terms of

their componentsin a given coordinate system.

Introduction to Vectors and Tensors > Introduction

Introduction



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Scalar

A physical quantity, completely described by a single real

number, such as the temperature, densityorpressure, is called a

scalarand is designated by .

A scalarmay be viewed as a zero-order tensor.

Introduction to Vectors and Tensors > Algebra of Vectors

Algebra of Vectors


Vector

A vectoris a directed line element in space. A model for physical

quantities having both direction and length, such as velocity,

accelerationorforce, is called a vectorand is designated by

A vectormay be viewed as afirst-order tensor.

, , .u v w

, ,


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Sum of Vectors

The sumof vectors yields a new vector, based on the parallelo-

gram law of addition.

The sumof vectors has the following properties,

where denotes the unique zero vectorwith unspecified

direction and zero length.


Algebra of Vectors


u v v uu v w u v w

u 0 u

u u 00


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Scalar Multiplication

Let be a vector and be a scalar. The scalar multiplication

gives a new vector with the same direction as if or with

the opposite direction to if .

The scalar multiplicationhas the following properties,


Algebra of Vectors


u u

u u u

u v u v

u u

u 00u


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Dot Product

The dot (or scalaror inner) productof two vectors and ,

denoted by , is a scalar given by,

where is the norm(or lengthor magnitude)of the vector ,

which is a non-negative real number defined as,

and is the anglebetween the non-zero vectors and

when their origins coincide.


Algebra of Vectors


cos , , 0 , u v u v u v u v

u

u v

u v

u

1 2

0 u u u

, u v u v


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Dot Product

The dot (or scalaror inner) productof vectors has the following

properties,


Algebra of Vectors


0

0

00, ,

u v v u

u 0

u v w u v u w

u u u 0

u u u 0

u v u 0 v 0 u v


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Orthogonal Vectors

A non-zero vector is said to be orthogonal(or perpendicular)

to a non-zero vector if,

Unit Vectors

A vector is called a unit vectorif its norm (or lenght or

magnitude) is equal to 1,


Algebra of Vectors


u

0, , , 2 u v u 0 v 0 u v u v

v

e

1e


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Projection

The projectionof a vector along the direction of a unit vector

is defined as,


Algebra of Vectors


u e

cos , , 0 , u e u u e u e


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Cartesian Basis

Let us consider a three-dimensional Euclidean spacewith a fixed

set of three right-handed orthonormal basis vectors ,

denoted as Cartesian basis, satisfying the following properties,

A fixed set of three unit vectors which are mutually orthogonalform a so called orthonormal basis vectors, collectively denoted

as ,and define an orthonormal system.


Algebra of Vectors


1 2 3, ,e e e

1 2 1 3 2 3

1 2 3

0

1

e e e e e e

e e e

ie

1e

2e

3e

1,x

2,x

3,x

O


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Cartesian Components

Any vector in the three-dimensional Euclidean spaceis

represented uniquelyby a linear combination of the basis

vectors , i.e.

where the three real numbers are the uniquely

determined Cartesian components of vector along the given

directions , respectively.

Using matrix notation, the Cartesian components of the vectorcan be collected into a vector of components given by,


Algebra of Vectors


u

1 2 3, ,e e e

1 1 2 2 3 3u u u

u e e e

1 2 3, ,u u uu

1 2 3, ,e e e

u

1 2 3T

u u uu

3u


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Index Notation

Using index notationthe vector can be written as,

or in short form, leaving out the summation symbol, simply as

where we have adopted the summation convention introduced

by Einstein.

The index that is summed over is said to be a dummy(or

summation) index. An index that is not summed over in a given

term is called a free(or live) index.


Algebra of Vectors


u

i iuu e

3

1 i ii u

u e

d d l b f


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Kronecker Delta

The Kronecker delta is defined as,


Algebra of Vectors


ij

1 if

0 ifij

i j

i j

The Kronecker delta satisfies the following properties,

Note that the Kronecker delta also serves as a replacement

operator.

3, ,ii ij jk ik ij j iu u

I d i V d T Al b f V


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Dot Product

The dot (or scalaror inner) product of two orthonormal basis

vectors and , denoted as , is a scalar quantity and can

be conveniently written in terms of the Kronecker delta as,


Algebra of Vectors


ie

i j ij e e

je i je e

I t d ti t V t d T Al b f V t


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Components of a Vector

Taking the basis , the projectionof a vector onto the basis

vector yields the i-th componentof ,


Algebra of Vectors


u

ie

i j j i j ji iu u u u e e e

u

ie

I t d ti t V t d T Al b f V t


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Dot Product

Taking the basis , the dot(or scalaror inner) productof two

vectors and , denoted as , is a scalar quantity and using

index notationcan be written as,


Algebra of Vectors


u

ie

1 1 2 2 3 3

i i j j i j i j i j ij i iu v u v u v u vu v u v u v

u v e e e e

v u v

I t d ti t V t d T > Al b f V t


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Euclidean Norm

Taking the basis , the Euclidean norm(or lengthor

magnitude) of a vector , denoted as , is a scalar quantity

and using index notationcan be written as,


Algebra of Vectors


ie

1 2 1 21 2 1 2

1 22 2 2

1 2 3

i i j j i j ij i iu u u u u u

u u u

u u u e e

u u



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Assignment 1.1

Write in index form the following expressions,


Assignment 1.1


v

v a b c

v a b c

a b c d

v a b c c d



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Assignment 1.1

Write in index form the following expressions,


Assignment 1.1


,

,

,

,

j j i i i i i j j i

i j j i i i i i j j

i i j j i i j j

i j j k k i i i i i j j k k

a b c v v a b c

a b c v v a b c

v a b c d v a b c d

a b c c d v v a b c c d

v a b c e e

v a b c e e

a b c d

v a b c c d e e



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Cross Product

The cross(or vector) productof two vectors and , denoted

as , is another vector which is perpendicular to the plane

defined by the two vectors and its norm (or length) is given by

the area of the parallelogram spanned by the two vectors,

where is a unit vector perpendicular to the plane defined by

the two vectors, , in the direction given by the

right-hand rule, .


Algebra of Vectors


u v

u v

sin , , 0 , u v u v u v e u v

e

0, 0 u e v e



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Norm of the Cross Product of two Vectors

The norm (or magnitudeor length) of the cross (or vector)

product of two vectors and , denoted as , measures

the area spanned by the two vectors and is given by,


Algebra of Vectors


sin , , 0 , u v u v u v u v

u v u v

u v

u v

u

v



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Permutation Symbol

The permutation (or alternatingor Levi-Civita) symbol is

defined as,


Algebra of Vectors


ijk

1 if 123, 231 312

1 if 213,132 3210 if there is a repeated index

ijk

ijk or

ijk or



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Permutation Symbol

The permutation (or alternatingor Levi-Civita) symbol may

be written in terms of the Kronecker deltaand the following

expressions hold,


Algebra of Vectors


ijk

2 2 6, ,ijp pqk ip jq iq jp ijk pjk ip ijk ijk ii

1 2 3

1 2 3

1 2 3

det , det

i i i ip iq ir

ijk j j j ijk pqr jp jq jr

k k k kp kq kr



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Cross Product

Taking the right-handed orthonormal basis , the cross (or

vector)product of two basis vectors and , denoted as

is defined as,


Algebra of Vectors


jeie

i j ijk k e e e

1 2 3 2 3 1 3 1 2

2 1 3 3 2 1 1 3 2

1 1 2 2 3 3 .

, , ,

, , ,

0

e e e e e e e e e

e e e e e e e e e

e e e e e e

The cross (or vector)product of two right-handed orthonormal

basis vectorssatisfies the following properties,

ie,i je e



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Cross Product

Taking the basis , the cross(or vector) productof two

vectors and , denoted as , is another vector and can be

written in index form as,


Algebra of Vectors


2 3 3 2 1 3 1 1 3 2 1 2 2 1 3

1 2 3

1 2 3

1 2 3

det

i i j j i j i j i j ijk k ijk i j k u v u v u v u vu v u v u v u v u v u v

u u uv v v

u v e e e e e e

e e e

e e e

ieu v u v



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Cross Product

The cross (or vector) product of vectors has the following

properties,


Algebra of Vectors


, , ||

u v v u

u v 0 u 0 v 0 u v

u v u v u v

u v w u v u w



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Assignment 1.2

Using the expression,

obtain the following relationships between the permutationsymbol and the Kronecker delta,

t oduct o to ecto s a d e so s geb a o ecto s

Assignment 1.2


2 2 6, ,ijp klp ik jl il jk ipq jpq ij ijk ijk ii

2 2 2 2sin , , 0 , u v u v u v u v



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Box Product

The box (or triple scalar) product of the three right-handed

orthonormal basis vectors , denoted as , is a

scalar quantity and can be conveniently written in terms of the

permutation symbol as,

g

Algebra of Vectors


i j k i j k ijk

ikj i k j i k j

e e e e e e

e e e e e e

, ,i j ke e e i j k e e e



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Box Product

Taking the orthonormal basis , the box(or triple scalar)

productof three vectors , denoted as , is a

scalar quantity and using index notation can be written as,

g

Algebra of Vectors


1 2 3

1 2 3

1 2 3

det

i i j j k k i j k i j k

ijk i j k

u v w u v wu v w

u u u

v v vw w w

u v w e e e e e e

ie, ,u v w u v w



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Box Product

The box (or triple scalar) product of three vectors

represents the volumeof the parallelepiped spanned by the

three vectors forming a right-handedtriad,

g

Algebra of Vectors


V

u v w

, ,u v w

V u v w

v

w

u

v w



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Box Product

The box (or triple scalar) product has the following properties,

Algebra of Vectors


0

u v w w u v v w u

u v w u w v

u u v v u v



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Triple Vector Product

Taking the orthonormal basis , the triple vectorproductof

three vectors , denoted as , is a vector and

using index notation can be written as,

Algebra of Vectors


ijk i pqj p q k kij pqj i p q k

kp iq kq ip i p q k

i k i k i i k k

u v w u v w

u v w

u v w u v w

u v w e e

e

e e

u w v u v w

ie, ,u v w u v w



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Taking the orthonomal basis , the triple vectorproductof

three vectors , denoted as , is a vector and

using index notation can be written as,

Algebra of Vectors


ijk pqi p q j k jki pqi p q j k

jp kq jq kp p q j k

j j k k j j k k

u v w u v w

u v w

u w v v w u

u v w e e

e

e e

u w v v w u

ie, ,u v w u v w



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The triple vectorproducthas the following properties,

Algebra of Vectors


u v w u w v u v w

u v w u w v v w u

u v w u v w

Introduction to Vectors and Tensors > Algebra of Tensors


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Second-order Tensors

A second-order tensor may be thought of as a linear operator

that acts on a vector generating a vector , defining a linear

transformationthat assigns a vector to a vector ,

The second-order unit tensor, denoted as , satisfies the

following identity,

Algebra of Tensors


A

u v

v Au

v u

1

u 1u



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As linear operatorsdefining linear transformations, second-

order tensors have the following properties,

Algebra of Tensors


A u v Au Av

A u Au

A B u Au Bu



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Tensor Product

The tensor product(or dyad)of two vectors and , denoted

as , is a second-order tensor which linearly transformsa

vector into a vector with the direction of following the rule,

A dyadicis a linear combination of dyads.

Algebra of Tensors


u vu v

u

u v w u v w v w u

w



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Tensor Product

The tensor product (or dyad) has the following linear properties,

Generally, the tensor product (or dyad) is notcommutative, i.e.

Algebra of Tensors


u v w x u v w u v x

u v w u w v w

u v w v w u u v w

u v w x v w u x u x v w

A u v Au v

u v v u

u v w x w x u v



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Tensor Product

The tensor product(or dyad) of two orthonormal basis vectors

and , denoted as , is a second-order tensor and can be

thought as a linear operatorsuch that,

Algebra of Tensors


i je ei

e

je

i j k jk i e e e e



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Any second-order tensor may be represented by a linear

combination of dyads formed by the Cartesian basis

where the nine real numbers, represented by , are theuniquely determined Cartesian components of the tensor

with respect to the dyads formed by the Cartesian basis ,

represented by , which constitute a basis second-order

tensor for .The second-order unit tensormay be written as,

Algebra of Tensors


A

ij i jA A e e

ijAA

ie

i je e ie

A

ij i j i i 1 e e e e



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Cartesian Components

The ij-th Cartesian component of the second-order tensor ,

denoted as , can be written as,

Algebra of Tensors


A

i j i kl k l j i kl lj k

i kj k kj i k kj ik ij

A A

A A A A

e Ae e e e e e e

e e e e

ijA

The ij-th Cartesian component of the second-order unit tensor

is the Kronecker delta,

1

i j i j ij e 1e e e



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Matrix of Components

Using matrix notation, the Cartesian components of any second-

order tensor may be collected into a 3 x 3 matrix of compo-

nents, denoted by , given by,

Algebra of Tensors


A

A

11 12 13

21 22 23

31 32 33

A A AA A A

A A A

A



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Using matrix notation, the Cartesian components of the second-

order unit tensor may be collected into a 3 x 3 matrix of

components, denoted as , given by,

Algebra of Tensors


1 0 00 1 0

0 0 1

1

1

1



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The tensor product (or dyad) of the vectors and , denoted as

, may be represented by a linear combination of dyads

formed by the Cartesian basis

where the nine real numbers, represented by , are the

uniquely determined Cartesian components of the tensor

with respect to the dyads formed by the Cartesian basis ,

represented by .

Algebra of Tensors


i j i ju v u v e e

i ju v

ie

i je e ie

u v

u v

u v



47/122


Using matrix notation, the Cartesian components of any second-

order tensor may be collected into a 3 x 3 matrix of

components, denoted as , given by,

Algebra of Tensors


1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

u v u v u vu v u v u v

u v u v u v

u v

u v

u v



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Using index notation, the linear transformation can be

written as,

yielding

Algebra of Tensors


v Au

ij i j k k ij k i j k ij k jk iij j i i i

A u A u A u

A u v

v Au e e e e e e e

e e

, i ij jv A u v Au



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Positive Semi-definite Second-order Tensor

Asecond-order tensor is said to be a positive semi-definite

tensor if the following relation holds for any non-zero vector

Algebra of Tensors


A

0 u Au u 0

Positive Definite Second-order Tensor

Asecond-order tensor is said to be a positive-definite tensor

if the following relation holds for any non-zero vector

0 u Au u 0

u 0

A

u 0



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Algebra of Tensors


Transpose of a Second-order Tensor

The unique transpose of a second-order tensor , denoted as

is governed by the identity,

A

,

,

T

T

i ij j i ji j j ji i j iv A u v A u u A v u v

v A u u Av u v

,TA

The Cartesian componentsof the transpose of a a second-order

tensor satisfy,

Using index notationthe transpose of may be written as, :

T T T

ij i j j i jiijA A

A e A e e Ae

A

A

T

ji i jA A e e



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Algebra of Tensors


Transpose of a Second-order Tensor

The matrix of components of the transpose of a second-order

tensor is equal to the transpose of the matrix of components

of and takes the form,

A

11 21 31

12 22 32

13 23 33

TTA A AA A A

A A A

A A

A



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Algebra of Tensors


Symmetric Second-order Tensor

A second-order tensor is said to be symmetricif the following

relation holds,

Using index notation, the Cartesian components of a symmetricsecond-order tensor satisfy,

Using matrix notation, the matrix of components of a symmetricsecond-order tensor satisfies,

A

TA A

A

,Tij i j ji i j ij jiA A A A A e e e e A

A

TT A A A



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Algebra of Tensors


Dot Product

Thedot product of two second-order tensors and , denoted

as , is a second-order tensor such that,

A B

AB u A Bu u

AB



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Algebra of Tensors


Dot Product

Thecomponents of the dot product along an orthonormal

basis read,

i j i jij

i kj k i k kj

ik kj

B B

A B

AB e AB e e A Be

e A e e Ae

AB

ie



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Algebra of Tensors


Dot Product

Thedot product of second-order tensors has the following

properties,

2

AB C A BC ABC

A AA

AB BA



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Algebra of Tensors


Trace

The traceof a dyad , denoted as , is a scalar

quantity given by,u v

tr i iu v u v u v

tr u v

The traceof a second-order tensor , denoted as , is a scalarquantity given by,

11 22 33

tr tr tr

tr

ij i j ij i j

ij i j ij ij

ii

A A

A A

A

A A A

A e e e e

e e

A

A trA



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Algebra of Tensors


Trace

The traceof a second-order tensor has the following properties,

tr tr

tr tr

tr tr tr

tr tr

T

A A

AB BA

A B A B

A A



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Algebra of Tensors


Double Dot Product

Thedouble dot product of two second-order tensors and ,

denoted as , is defined as,

or using index notation,

: tr tr

tr tr

:

T T

T T

A B A B B A

AB BA

B A

A B

:A B

: :ij ij ij ijA B B A A B B A


l b f


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Algebra of Tensors


Double Dot Product

Thedouble dot product has the following properties,

: : tr

: :

: : :

: :

:

:

T T

i j k l i k j l ik jl

A 1 1 A A

A B B A

A BC B A C AC B

A u v u Av u v A

u v w x u w v x

e e e e e e e e


l b f


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Algebra of Tensors


Euclidean Norm

Theeuclidean norm of a second-order tensor is a non-

negative scalar quantity, denoted as , given by,

A

1 21 2

: 0ij ijA A A A A

A


Al b f T


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Algebra of Tensors


Determinant

The determinant of a second-order tensor is a scalar quantity,

denoted as , given by,

A

11 12 13

21 22 23

31 32 33

1 2 3

det det

det

16

ijk i j k ijk pqr pi qj rk

A A AA A A

A A A

A A A A A A

A A

det A


Al b f T


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Algebra of Tensors


Determinant

The determinant of a second-order tensor has the following

properties,

3

det det

det det

det det det

T

A A

A A

AB A B


Al b f T


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Algebra of Tensors


Singular Tensors

A second-order tensor is said to be singular if and only if its

determinant is equal to zero, i.e.

is singular det 0 A A

A


Al b f T


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Algebra of Tensors


Inverse of a Second-order Tensor

For anon-singular second-order tensor , i.e. , there

exista unique non-singular inverse second-order tensor,

denoted as ,satisfying,

yielding,

A

1 1

AA u A A u 1u u u

det 0A

1A

1 1 1 1, ik kj ik kj ijA A A A AA A A 1


Al b f T


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Algebra of Tensors


Inverse of a Second-order Tensor

The inverse second-order tensor satisfies the following

properties,

11

1 1

1 1 1

1 1 1

2 1 1

11det det

TT T

A A

A A A

A A

AB B A

A A A

A A


Al b f T


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Algebra of Tensors


Orthogonal Second Order Tensor

Anorthogonal second-order tensor is a linear operator that

preserves the norms and the angles given by two vectors,

and the following relations hold,

Q

,

T

T

Qu Qv u Q Qv u vu v

Qv Qu v Q Qu v u

1 2 1 2T Qu u Q Qu u u u u

1, , det 1T T T Q Q QQ 1 Q Q Q


Al b f T


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Algebra of Tensors


Rotation Second Order Tensor

A rotation tensor is a proper orthogonal second-order tensor,

i.e., is a linear operator that preserves the lengths (norms) and

the angles given by two vectors,

and satisfies the following expressions,

R

1, , det 1T T T R R RR 1 R R R

,

T

T

Ru Rv u R Rv u vu v

Rv Ru v R Ru v u

1 2 1 2T

Ru u R Ru u u u u


Algebra of Tensors


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Algebra of Tensors


Symmetric/Skew-symmetric Additive Split

Asecond-order tensor can be uniquely additively split into a

symmetricsecond-order tensor and a skew-symmetric

second-order tensor , such that,

A

, ij ij ijA S W A S W

1 1

2 2

1 1

2 2

,

,

T T

ij ij ji ji

T T

ij ij ji ji

S A A S

W A A W

S A A S

W A A W

S

W


Algebra of Tensors


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Algebra of Tensors


Symmetric/Skew-symmetric Second-order Tensors

The matrices of components of a symmetricsecond-order

tensor and a skew-symmetric second-order tensor , are

given by,

W

11 12 13 12 13

12 22 23 12 23

13 23 33 13 23

0, 0

0

S S S W W S S S W W

S S S W W

S W

S


Algebra of Tensors


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Algebra of Tensors


Skew-symmetric Second-order Tensor

Askew-symmetric second-order tensor can be defined by

means of an axial (or dual) vector , such that,

W

and using index notation the following relations hold,

1 1

2 2

,

,

ij i j ijk k i j ij ijk k

k k ijk ij k k ijk ij

W W

W W

W e e e e

e e

2

Wu u u

W


Algebra of Tensors


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Algebra of Tensors


Skew-symmetric Second-order Tensor

Using matrix notationand index notationthe following relations

hold,

12 13 3 2

12 23 3 1

13 23 2 1

1 2 3 23 13 12

0 0

0 00 0

, , , ,T T

W W

W WW W

W W W

W

2 2 2 2 2 212 13 23 1 3 32 2 2W W W W


Algebra of Tensors


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Algebra of Tensors


Spherical Second-order Tensor

Asecond-order tensor is said to be spherical if the following

relations hold,

A

0 0

0 0

0 0

A

,

tr tr 3 , 3

ij ij

ii ii

A

A

A 1

A 1


Algebra of Tensors


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Algebra of Tensors


Deviatoric Second-order Tensor

Asecond-order tensor is said to be deviatoricif the following

condition holds,

A

tr 0, 0iiA A


Algebra of Tensors


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Algebra of Tensors


Spherical/Deviatoric Additive Split

Asecond-order tensor can be uniquely additively split into a

spherical second-order tensor and a deviatoric second-

order tensor , such that,

A

1 1

3 3

1 1

3 3

dev , dev

tr ,

dev tr , dev

esf esf

ij ij ij

esf esf

ij kk ij

ij kk ijij

A A A

A A

A A A

A A A

A A 1

A A A 1

esfA

dev A


Algebra of Tensors


75/122

Eigenvalues/Eigenvectors

Given a symmetric second-order tensor , its eigenvalues

and associated eigenvectors , which define an

orthonormal basis along the principal directions, satisfy the

following equation (Einstein notation does not applies here),

Algebra of Tensors


A

in

,i i i i i An n A 1 n 0

In order to get a non-trivial solution, the tensor has to

be singular, i.e., its determinant, defining the characteristic

polynomial, has to be equal to zero, yielding,

det 0i ip A 1

iA 1

i


Algebra of Tensors


76/122


Given a symmetric second-order tensor , its eigenvalues

and associated eigenvectors , which define an

orthonormal basis along the principal directions, characterize

the physical nature of the tensor.

The eigenvaluesdo not depend on the particular orthonormal

basis chosen to define the components of the tensor and,

therefore, the solution of the characteristic polynomial do not

depends on the system of reference in which the components ofthe tensor are given.

Algebra of Tensors


A

i in


Algebra of Tensors


77/122

Characteristic Polynomial and Principal Invariants

The characteristic polynomialof a second-order tensor takes

the form,

Algebra of Tensors


where the coefficients of the polynomial are the principal scalarinvariantsof the tensor (invariants in front of an arbitrary

rotation of the orthonormal basis) given by,

3 21 2 3det 0i ip I I I A 1

A

1 1 2 3

2 2

2 1 2 2 3 3 1

3 1 2 3

1

2

tr

tr tr

det

I

I

I

A A

A A A

A A


Algebra of Tensors


78/122


Given a deviatoric part of a symmetric second-order tensor

, its eigenvalues and asociated eigenvectors ,

which define an orthonormal basis along the principal directions,

satisfy the following equation (Einstein notation does not applies

here),

Algebra of Tensors


dev Ai in

dev , devi i i i i An n A 1 n 0

In order to get a non-trivial solution, the tensor has

to be singular, i.e., its determinant, defining the characteristic

polynomial, has to be equal to zero, yielding,

det dev 0i ip A 1

dev iA 1


Algebra of Tensors


79/122

Characteristic Polynomial and Principal Invariants

The characteristic polynomialof the deviatoric part of a second-order tensor takes the form,

Algebra of Tensors


where the coefficients of the polynomial are the pincipal scalarinvariantsof the deviatoric part of the tensor (invariants in front

of an arbitrary rotation of the orthonormal basis) given by,

3 21 2 3det dev 0i ip I I I A 1

1

2

2 1 1 2 2 3 3

3 1 2 3

1 1

2 2

dev tr dev 0

dev tr dev

dev det dev

I

I

I

A A

A A

A A

dev A


Algebra of Tensors


80/122


The eigenvalues/eigenvectorsproblem for a symmetric second-order tensor and its the deviatoric part satisfy the

following relations (Einstein notation does not applies here),

Algebra of Tensors


dev A

dev , devi i i i i An n A 1 n 0

1

3

dev tr i i i i

A 1 n A A 1 n 0

,i i i i i An n A 1 n 0

A

Then,

1

3tr ,i i i i A n n

yielding,


Algebra of Tensors


81/122

For a second-order unit tensor, all three eigenvalues are equal toone, , and any direction is a principal direction.

Then we may take any orthonormal basis as principal directions

and the spectral decomposition can be written,

Spectral Decomposition

The spectral decomposition of the secod-order unit tensor takesthe form,

Algebra of Tensors


ij i j i i 1 e e e e

1 2 3 1

1,3 i i i ii 1 n n n n


Algebra of Tensors


82/122

Spectral Decomposition

The spectral decomposition of a symmetric second-order tensortakes the form (Einstein notation does not applies here),

Algebra of Tensors


A

1,3 i i ii

A n n

If there are two equal eigenvalues , the spectraldecomposition takes the form,

1 2

3 3 3 3 3 A 1 n n n n

1 2 3

1,3 i i i ii

A 1 n n n n

If the three eigenvalues are equal , the spectraldecomposition takes the form,

Introduction to Vectors and Tensors > Higher-order Tensors

Third-order Tensors


83/122

Third-order Tensors

A third-order tensor, denoted as , may be written as a linearcombination of tensor products of three orthonormal basis

vectors, denoted as , such that,

A third-order tensor has components.

Third order Tensors


3 3 3

1 1 1 ijk i j k ijk i j k i j k

e e e e e e

i j k e e e

33 27


Fourth-order Tensors


84/122


A fourth-order tensor, denoted as , may be written as a linearcombination of tensor products of four orthonormal basis

vectors, denoted as , such that,

A fourth-order tensor has components.

Fourth order Tensors


3 3 3 3

1 1 1 1 ijkl i j k l ijkl i j k l i j k l e e e e e e e e

i j k l e e e e

43 81




85/122

Double Dot Product with Second-order Tensors

The double dot productof a fourth-order tensor with asecond-order tensor is another second-order tensor given

by,



: ,ijkl kl i j ij i j ij ijkl kl

A B B A B A e e e e

A B




86/122

Transpose of a Fourth-order Tensor

The transpose of a fourth-order tensor is uniquely defined asa fourth-order tensor such that for arbitrary second-order

tensors and the following relationship holds,



BA

T

: : : : , ,T T Tij ijkl kl kl klij ij ijkl klij

A B B A A B B A

The transpose of the transpose of a fourth-order tensor is

the same fourth-order tensor ,

T

T




87/122

Tensor Product of two Second-order Tensors

The tensor product of two second-order tensors and is afourth order tensor given by,



,ij kl i j k l ijkl i j k l

ijkl ij kl ijkl

A B

A B

A B e e e e e e e e

A B

BA

The transpose of the fourth-order tensor is given by, A B

,

TT

ij kl i j k l

T

ijkl i j k l

TT

ijkl ij kl ijkl ijkl

B A

B A

A B B A e e e e

e e e e

A B B A




88/122

Fourth-order Identity Tensors

Let us consider the following fourth-order identity tensors,



1 1

2 2

ik jl i j k l

il jk i j k l

ik jl il jk i j k l

e e e e

e e e e

e e e e




89/122


The fourth-order identity tensor satisfies the followingexpressions,



: ,

:

ik jl kl i j ij i j

ij ik jl k l kl k l

A A

A A

A e e e e A

A e e e e A

: : : : : :

: : : :

T

T

A B A B B A B A

A B B A

The fourth-order identity tensor is symmetric, satisfying,




90/122





:

:

T

il jk kl i j ji i j

Tij il jk kl k l lk k l

A A

A A A

A e e e e A

A e e e e A

: : : : : :

: : : :

T TT

T

A B A B B A B A

A B B A





91/122




1 1 : :

2 21 1

: :2 2

T

T

A A A A

A A A A

1 1 : : : : : : 2 2

: : : :

T T

T

T

A B A B B B A A B A

A B B A





92/122

Fourth-order Deviatoric Projection Operator Tensor

The fourth-order deviatoric projection operator tensor isdefined as,


dev

1 1

3 3,dev dev ik jl ij kl ijkl

1 1

The deviatoric part of a second-order tensor can be obtained

as,

A

1 1

3 3

1 1

3 3

dev : : tr ,

dev

dev

dev kl ik jl ij kl kl ij kk ijij ijklA A A A

A A 1 1 A A A 1

A

Introduction to Vectors and Tensors > Differential Operators

Differential Operators


93/122

Nabla

The nabla vector differential operator, denoted as , is definedas,

Using Einstein notation,

p


1,3 ii

ix

e

iix

e




94/122

Laplacian

The laplacian scalar differential operator, denoted as , isdefined as,

p


2

2

ix

Hessian

The hessian symmetric second-order tensor differential

operator, denoted as ,is defined as,

2

i j

i jx x

e e




95/122

Divergence, Curl and Gradient

The divergence differential operator div () is defined as,

p


div iix

e

The curl differential operator curl () is defined as,

curl iix

e

The gradient differential operator grad () is defined as,

grad i ii ix x

e e




96/122

Laplacian

The laplacian scalar differential operator can be expressed as thediv (grad ()) operator,

p


2

2 divgrad

ix

Hessian

The hessian symmetric second order-tensor differential operator

can be expressed as the grad (grad ()) operator,

2

gradgradi ji jx x

e e




97/122

Gradient of a Scalar Field


p


The gradient of a scalar fieldis a vector field defined as,

grad i ii ix x

e e

,grad i i iix

e e




98/122

Laplacian of a Scalar Field

The laplacian differential operator is defined as,

p


The laplacian of a scalar fieldis a scalar field defined as,

2

,2 ii

ix

2

2

ix




99/122

Hessian of a Scalar Field

The hessian differential operator is defined as,


The hessian of a scalar fieldis a symmetric second-order tensor

field defined as,

2

,i j ij i ji jx x

e e e e

2

i j

i jx x

e e




100/122

Gradient of a Vector Field



The gradient of a vector fieldis a second-order tensor fielddefined as,

grad i ii ix x

e e

,grad i

j i j i j i j

j j

uu

x x

uu u e e e e e




101/122

Curl of a Vector Field



The curl of a vector fieldis a vector field defined as,

,curl k k

j j k ijk i ijk k j i

j j j

u uu

x x x

uu u e e e e e

curl iix

e

If the curl of a vector fieldis zero, the vector field is said to becurl-freeand there is a scalar field such that,

curl 0 | grad u u




102/122

Curl of the Gradient of a Scalar Field

The curl differential operator curl () and gradient differentialoperator grad () are defined, respectively, as,


The gradient of a scalar fieldis a vector field defined as,

,grad i i i

i

x

e e

curl , gradi ii ix x

e e

The curl of the gradient of a vector fieldis a null vector,

,curlgrad ijk kj i e 0




103/122

Divergence of a Vector Field



The divergence of a vector fieldis a scalar field defined as,

,div i i i

j i j ij i i

j j j i

u u uu

x x x x

uu u e e e

div iix

e

If the divergence of a vector fieldis zero, the vector field is said

to be solenoidalor div-freeand there is a vector field such that,

div 0 | curl u v u v




104/122

Divergence of the Curl of a Vector Field

The divergence differential operator div () and curl differentialoperator curl () are, respectively, defined as,


The curl of a vector fieldis a vector field defined as,

,curl k k

j j k ijk i ijk k j i

j j j

u uu

x x x

uu u e e e e e

div , curli ii ix x

e e

The divergence of a curl of a vector fieldis a null scalar,

,divcurl 0ijk k jiu u u




105/122

Laplacian of a Vector Field

The laplacian differential operator is defined as,


The laplacian of a vector fieldis a vector field defined as,

2

,2

ii i jj i

j

uu

x

u u e e

2

2

ix




106/122

Divergence of a Second Order Tensor Field



The divergence of a second-order tensor fieldis a vector field

defined as,

,

div

ik ik j i k j kj i

j j j

ij

i ij j i

j

A A

x x x

AA

x

A A

Ae e e e e

e e

div iix

e




107/122

Curl of a Second-order Tensor Field



The curl of a second-order tensor fieldis a second-order tensorfield defined as,

,

curl

kj

l l k jl l

kj

lki i j lki kj l i j

l

A

x x

AA

x

A A

Ae e e e

e e e e

curl iix

e




108/122

Gradient of a Second-order Tensor Field



The gradient of a second-order tensor fieldis a third-ordertensor field defined as,

grad i ii ix x

e e

,

grad

kk

ij

i j k ij k i j k

k

x

AA

x

A A

Ae

e e e e e e


Assignments


109/122

Assignment 1.3

Establish the following identities involving a smooth scalar fieldand a smooth vector field ,


(1)

(2)

div div grad

grad grad grad

v v v

v v v

v


Assignments


110/122

Assignment 1.3

Establish the following identities involving a smooth scalar fieldand a smooth vector field ,


(1)

(2)

div div grad

grad grad grad

v v v

v v v

v

, ,,

(1) div div grad

div div gradi i i i iiv v v

v v v

v v v

, ,,

(2) grad grad grad

grad grad gradi i j i jj ij ijijv v v

v v v

v v v


Assignments


111/122

Assignment 1.4 [Classwork]

Establish the following identities involving the smooth scalarfields and , smooth vector fields and , and a smooth

second order tensor field ,


u

(1)

(2)

(3)

(4)

(5)

div div grad

div div : grad

div curl curl

div grad divgrad grad grad

T

A A A

A v A v A v

u v v u u v

u v u v u v

vA


Assignments


112/122


Establish the following identities involving the smooth scalarfields and , smooth vector fields and , and a smooth

second order tensor field ,


u vA

, ,,

(1) div div grad

div div gradij ij j ij j i ii jA A A

A A A

A A A

, ,,

(2) div div : grad

div div : grad

T

T

ji j ji i j ji j iiA v A v A v

A v A v A v

A v A v A v


Assignments


113/122


, ,,, ,

(3) div curl curl

div

curl curl

ijk j k ijk j i k ijk j k ii

k kij j i j jik k i

u v u v u v

v u u v

u v v u u v

u v

v u u v

, ,,

(4) div grad div

div grad divi j i j j i j j ii iju v u v u v u

u v u v u v

u v u v v

, ,,

(5) grad grad gradgrad grad gradi ii i ii


Assignments


114/122

Assignment 1.5 [Homework]

Establish the following identities involving the smooth scalarfield , and smooth vector fields and ,


u v

(1)

(2)

(3)

(4)

(5)

grad grad grad

curl grad curl

curl div div grad grad

grad div curl curl

2grad : grad

T T

u v u v v u

v v v

u v u v v u u v v u

v v v

u v u v u v u v


Assignments


115/122

Assignment 1.5 [Homework]

Establish the following identities involving the smooth scalarfield , and smooth vector fields and ,


u v

, ,,

(1) grad grad grad

grad

grad grad

grad grad

grad grad

T T

j j j i j j j ii i

j jji ji

T T

j jij ij

T T

i

u v u v u v

v u

v u

u v u v v u

u v

u v

u v

u v v u


Assignments


116/122


, ,,

(2) curl grad curl

curl

grad curl

ijk k ijk j k ijk k jji

i

v v v

v v v

v

v v

, ,

, ,

, ,

, , , ,

(3) curl div div grad grad

curl

grad div div grad

ijk ijk klm l mk j ji

ijk klm l j m ijk klm l m j

il jm im jl l j m il jm im jl l m j

i j j j j i i j j j i j

i ii i

u v

u v u v

u v u v

u v u v u v u v

v u

u v u v v u u v v u

u v u v

u v u v v u


Assignments


117/122


, ,

, ,, ,

, , , , ,

(4) grad div curl curl

, grad div

curl curl curl

i i jj j jii i

ijk ijk klm m l ijk klm m ljk ji j

il jm im jl m lj j ij i jj j ji i jj

v v v

v v

v v v v v

v v v

v v

v v

, ,, , ,

, , , ,

(5) 2grad : grad

2

2grad : grad

i i i i i j i i i jjj j j

i jj i i j i j i i jj

u v u v u v u v

u v u v u v

u v u v u v u v

u v

u v u v u v


Assignments


118/122


Given the vector determine


div ,curl ,grad , .v v v v 1 2 3 1 1 2 2 1 3x x x x x x v v x e e e


Assignments


119/122





, 2 3 1div i iv x x x v

1 2 3

1 2 3

1 2 3

3,2 2,3 1 1,3 3,1 2 2,1 1,2 3

1 2 2 2 1 3 3

curl det

1

v v v

v v v v v v

x x x x x

e e e

v

e e e

e e


Assignments


120/122





,

2 3 1 3 1 2

2 1

grad

grad 0

1 0 0

i j i jv

x x x x x xx x

v e e

v

,i i i jj iv v v e e 0

Introduction to Vectors and Tensors > Integral Theorems

Integral Theorems


121/122


Divergence Theorem

Given a vector field in a volume V with closed boundarysurface and outward unit normal to the boundary , the

divergence(or Gauss) theoremreads,

Given a second-order tensor field in a volume V with closed

boundary surface and outward unit normal to the boundary

the divergence(or Gauss) theoremreads,

uV n

divV V V

dS dV dV

u n u u

V nA

divV V V

dS dV dV

A n A A

Introduction to Vectors and Tensors > Integral Theorems

Integral Theorems


122/122

Curl Theorem

Given a vector field in a surface S with closed boundaryand outward unit normal to the surface , the curl(or Stokes)

theoremreads,

where the curve of the line integral must have positive

orientation, such that drpoints counter-clockwise when the unit

normal points to the viewer, following the right-hand rule.

u Sn

rotS S S

d dS dS

u r u n u n

Documents

Chapter 1. Tensor Algebra