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Transcript of TENSOR ALGEBRA - of Tensor A TENSOR is an algebraic entity with various components which...

• TENSOR ALGEBRAContinuum Mechanics Course (MMC) - ETSECCPB - UPC

• Tensor Algebra

Introduction to Tensors

2

• Introduction

SCALAR

VECTOR

MATRIX

?

, , ...

, , ...v f

, , ...

, ...C

v

3

• Concept of Tensor

A TENSOR is an algebraic entity with various components which generalizes the concepts of scalar, vector and matrix.

Many physical quantities are mathematically represented as tensors.

Tensors are independent of any reference system but, by need, are commonly represented in one by means of their component matrices.

The components of a tensor will depend on the reference system chosen and will vary with it.

4

• Order of a Tensor

The order of a tensor is given by the number of indexes needed to specify without ambiguity a component of a tensor.

Scalar: zero dimension

Vector: 1 dimension

2nd order: 2 dimensions

3rd order: 3 dimensions

4th order

a,a a

,A A

, AA, AA

3.14 1.20.30.8

vi

0.1 0 1.30 2.4 0.5

1.3 0.5 5.8ijE

5

• Cartesian Coordinate System

Given an orthonormal basis formed by three mutually perpendicular unit vectors:

Where:

Note that

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

1 2 3 1 , 1 , 1 e e e

1

0i j iji ji j

e e

if

if

6

• Cylindrical Coordinate System

1 2

1 2

3

cos sin sin cos

r

z

e e ee e ee e

1

2

3

cos ( , , ) sin

x rr z x r

x z

x

1x

2x

3x

7

• Spherical Coordinate System

1 2 3

1 2

1 2 3

sin sin sin cos cos cos sin cos sin cos cos sin

r

e e e ee e ee e e e

1

2

3

sin cos, , sin sin

cos

x rr x r

x r

x

3x

1x

2x

8

• Tensor Algebra

Indicial or (Index) Notation

9

• Tensor Bases VECTOR

A vector can be written as a unique linear combination of the three vector basis for .

In matrix notation:

In index notation:

ie

1 1 2 2 3 3 v v v v e e ev

v

1

2

3

vvv

v

vi ii

v e vii v

tensor as a physical entity

component i of the tensor in the given basis

1v2v

3v

1,2,3i

1,2,3i

10

• Tensor Bases 2nd ORDER TENSOR

A 2nd order tensor can be written as a unique linear combination of the nine dyads for .

Alternatively, this could have been written as:

, 1,2,3i j i j i j e e e eA

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

11 12 13

21 22 23

31 32 33

A A AA A AA A A

A e e e e e ee e e e e ee e e e e e

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

11 12 13

21 22 23

31 32 33

A A A

A A A

A A A

A e e e e e e

e e e e e e

e e e e e e

11

• Tensor Bases 2nd ORDER TENSOR

In matrix notation:

In index notation:

11 12 13

21 22 23

31 32 33

A A AA A AA A A

A

Aij i jij

A e e ijij AA

tensor as a physical entity

component ij of the tensor in the given basis

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

11 12 13

21 22 23

31 32 33

A A A

A A A

A A A

A e e e e e e

e e e e e e

e e e e e e

, 1,2,3i j

12

• Tensor Bases 3rd ORDER TENSOR

A 3rd order tensor can be written as a unique linear combination of the 27 tryads for .

Alternatively, this could have been written as:

i j k i j k e e e e e eA

, , 1,2,3i j k

1 1 1 1 2 1 1 3 1

2 1 1 2 2 1 2 3 1

3 1 1 3 2 1 3 3 1

1 1 2 1 2 2

...

111 121 131

211 221 231

311 321 331

112 122

e e e e e e e e e

e e e e e e e e e

e e e e e e e e e

e e e e e e

A A A A

A A A

A A A

A A

1 1 1 1 2 1 1 3 1

2 1 1 2 2 1 2 3 1

3 1 1 3 2 1 3 3 1

1 1 2 1 2 2

...

111 121 131

211 221 231

311 321 331

112 122

e e e e e e e e ee e e e e e e e ee e e e e e e e ee e e e e e

A A A A

A A A

A A A

A A

13

• Tensor Bases 3rd ORDER TENSOR

In matrix notation:

1 1 1 1 2 1 1 3 1

2 1 1 2 2 1 2 3 1

3 1 1 3 2 1 3 3 1

1 1 2 1 2 2

...

111 121 131

211 221 231

311 321 331

112 122

e e e e e e e e e

e e e e e e e e e

e e e e e e e e e

e e e e e e

A

A A A

A A A

A A A

A A

113 123 133

213 223 233

313 323 333

A A A

A A A

A A A

112 122 132

212 222 232

312 322 332

A A A

A A A

A A A

111 121 131

211 221 231

311 321 331

A A A

A A A

A A A

A

14

• Tensor Bases 3rd ORDER TENSOR

In index notation:

ijk i j kijk

ijk i j k ijk i j k

e e e

e e e e e e

A A

A A

ijkijk A A

1 1 1 1 2 1 1 3 1

2 1 1 2 2 1 2 3 1

3 1 1 3 2 1 3 3 1

1 1 2 1 2 2

...

111 121 131

211 221 231

311 321 331

112 122

e e e e e e e e e

e e e e e e e e e

e e e e e e e e e

e e e e e e

A A A A

A A A

A A A

A A

tensor as a physical entity

component ijk of the tensor in the given basis , , 1,2,3i j k

15

• A tensor of order n is expressed as:

The number of components in a tensor of order n is 3n.

Higher Order Tensors

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

n ni i i i i i iA A e e e e

1 2, ... 1,2,3ni i i where

16

• The Einstein Summation Convention: repeated Roman indices are summed over.

A MUTE (or DUMMY) INDEX is an index that does not appear in a monomial after the summation is carried out (it can be arbitrarily changed of name).

A TALKING INDEX is an index that is not repeated in the same monomial and is transmitted outside of it (it cannot be arbitrarily changed of name).

3

1 1 2 2 3 313

1 1 2 2 3 31

i i i ii

ij j ij j i i ij

a b a b a b a b a b

A b A b A b A b A b

REMARKAn index can only appear up to two times in a monomial.

Repeated-index (or Einsteins) Notation

i is a mute index

i is a talking index and j is a

mute index

17

• Rules of this notation:

1. Sum over all repeated indices.

2. Increment all unique indices fully at least once, covering all combinations.

3. Increment repeated indices first.

4. A comma indicates differentiation, with respect to coordinate xi .

5. The number of talking indices indicates the order of the tensor result

Repeated-index (or Einsteins) Notation

3

,1

i ii i

ii i

u uux x

2 23

, 21

i ii jj

jj j j

u uux x x

3

,1

ij ijij j

jj j

A AA

x x

18

• Kronecker Delta

The Kronecker delta ij is defined as:

Both i and j may take on any value in Only for the three possible cases where i = j is ij non-zero.

10ij

i ji j

if

if

11 22 33

12 13 21

1 10 ... 0ij

i ji j

if

if

ij ji REMARKFollowing Einstens notation: Kronecker delta serves as a replacement operator:

11 22 33 3ii

,ij j i ij jk iku u A A

1,2,3

19

• Levi-Civita Epsilon (permutation)

The Levi-Civita epsilon is defined as:

3 indices 27 possible combinations.

01 123, 231 3121 213,132 321

ijk ijkijk

if there is a repeated index

if or

if or

e

REMARKThe Levi-Civita symbol is also named permutation or alternating symbol.

ijk ikj e e

ijke

20

• Relation between and

1 2 3

1 2 3

1 2 3

deti i i

ijk j j j

k k k

e detip iq ir

ijk pqr jp jq jr

kp kq kr

e e

ijk pqk ip jq iq jp e e

2ijk pjk pie e

6ijk ijk e e

21

• Example

Prove the following expression is true:

6ijk ijk e e

22

• 211 211 212 212 213 213

221 221 222 222 223 223

231 231 232 232 233 233

e e e e e e

e e e e e e

e e e e e e

2i

311 311 312 312 313 313

321 321 322 322 323 323

331 331 332 332 333 333

e e e e e e

e e e e e e

e e e e e e

3i

121 121 122 122 123 123 e e e e e e 2j 131 131 132 132 133 133 e e e e e e 3j

Example - Solution

111 111 112 112 113 113ijk ijk e e e e e e e e1

1

1

1

1

1

6

1i

1j 1k 2k 3k

23

• Tensor Algebra

Vector Operations

24

• Sum and Subtraction. Parallelogram law.

Scalar multiplication

Vector Operations

a b b a ca b d

1 1 2 2 3 3 a a a a b e e e

i i i

i i i

c a bd a b

i ib a

25

• Scalar or dot product yields a scalar

In index notation:

Norm of a vector

Vector Operations

cos u v u vwhere is the angle

between the vectors u and v

2 i i j j i j ij i iu u u u u uu u e eu