TENSOR ALGEBRA Continuum Mechanics Course (MMC) - ETSECCPB - UPC - - PowerPoint PPT Presentation

TENSOR ALGEBRA

Continuum Mechanics Course (MMC) - ETSECCPB - UPC

Tensor Algebra

Introduction to Tensors

2

Introduction

SCALAR VECTOR MATRIX ?

, , ... σ ε , , ... 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 , A A

, A A

3.14  

1.2 0.3 0.8 vi           

0.1 1.3 2.4 0.5 1.3 0.5 5.8

ij

E           

5

Cartesian Coordinate System

 Given an orthonormal basis formed by three mutually

perpendicular unit vectors:

Where:

 Note that

1 2 2 3 3 1

ˆ ˆ ˆ ˆ ˆ ˆ , ,    e e e e e e

1 2 3

ˆ ˆ ˆ 1 , 1 , 1    e e e 1 ˆ ˆ

i j ij

i j i j             e e if if

6

Cylindrical Coordinate System

1 2 1 2 3

ˆ ˆ ˆ cos sin ˆ ˆ ˆ sin cos ˆ ˆ

r z

θ θ θ θ



      e e e e e e e e

1 2 3

cos ( , , ) sin x r r z x r x z             x

1

x

2

x

3

x

7

Spherical Coordinate System

1 2 3 1 2 1 2 3

ˆ ˆ ˆ ˆ sin sin sin cos cos ˆ ˆ ˆ cos sin ˆ ˆ ˆ ˆ cos sin cos cos sin

r φ

θ φ θ φ θ φ φ θ φ θ φ θ



        e e e e e e e e e e e  

1 2 3

sin cos , , sin sin cos x r r x r x r                 x

3

x

1

x

2

x

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:

ˆi e

1 1 2 2 3 3

ˆ ˆ ˆ v v v    v e e e

v v

 

1 2 3

v v v            v

ˆ vi

i i

  v e

 

vi

i 

v

tensor as a physical entity component i of the tensor in the given basis

1

v

2

v

3

v

 

1,2,3 i

 

1,2,3 i

10

Tensor Bases – 2nd ORDER TENSOR

 A 2nd order tensor can be written as a unique linear combination

• f the nine dyads for .

Alternatively, this could have been written as:

 

, 1,2,3 i j

ˆ ˆ ˆ ˆ

i j i j

  e e e e

A

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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

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 A A A A A A A            A

 

ˆ ˆ Aij

i j ij

 



A e e

 

ij ij

A  A

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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

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

 

, 1,2,3 i j

12

Tensor Bases – 3rd ORDER TENSOR

 A 3rd order tensor can be written as a unique linear combination

• f the 27 tryads for .

Alternatively, this could have been written as:

ˆ ˆ ˆ ˆ ˆ ˆ

i j k i j k

   e e e e e e

A

 

, , 1,2,3 i 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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ...

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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ...

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

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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ...

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:

   

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

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

       



e e e e e e e e e A A A A

 

ijk ijk 

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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ...

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

 

, , 1,2,3 i 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

, ... ˆ

ˆ ˆ ˆ ...

n n

i i i i i i i

A      A e e e e

 

1 2

, ... 1,2,3

n

i 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

• f “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

• f “name”).

3 1 1 2 2 3 3 1 3 1 1 2 2 3 3 1 i i i i i ij j ij j i i i j

a b a b a b a b a b A b A b A b A b A b

 

       

 

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

Repeated-index (or Einstein’s) 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 Einstein’s) Notation

3 , 1 i i i i i i i

u u u x x



     



2 2 3 , 2 1 i i i jj j j j j

u u u x x x



      



3 , 1 ij ij ij j j j j

A A A x x



     



18

Kronecker Delta δ

 The Kronecker delta δij is defined as:

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

1

ij

i j i j        if if

   

11 22 33 12 13 21

1 1 ...

ij

i j i j                      if if

ij ji

  

REMARK Following Einsten’s notation: Kronecker delta serves as a replacement operator:

11 22 33

3

ii

       

,

ij j i ij jk ik

u u A A    

1,2,3

 

19

Levi-Civita Epsilon (permutation) ϵ

 The Levi-Civita epsilon

is defined as:

 3 indices

27 possible combinations.



1 123, 231 312 1 213,132 321

ijk

ijk ijk           if there is a repeated index if

• r

if

• r

e

REMARK The Levi-Civita symbol is also named permutation or alternating symbol.

ijk ikj

  e e

ijk

e

20

Relation between δ and ε

1 2 3 1 2 3 1 2 3

det

i i i ijk j j j k k k

                    e det

ip iq ir ijk pqr jp jq jr kp kq kr

                    e e

ijk pqk ip jq iq jp

      e e 2

ijk pjk pi

  e e 6

ijk ijk 

e e

21

Example

 Prove the following expression is true:

6

ijk 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 2 i 

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 3 i 

121 121 122 122 123 123

    e e e e e e 2 j 

131 131 132 132 133 133

    e e e e e e 3 j 

Example - Solution

111 111 112 112 113 113 ijk ijk 

   e e e e e e e e

1  1  1  1   1   1  

6  1 i  1 j  1 k  2 k  3 k 

23

Tensor Algebra

Vector Operations

24

 Sum and Subtraction. Parallelogram law.  Scalar multiplication

Vector Operations

      a b b a c a b d

1 1 2 2 3 3

ˆ ˆ ˆ a a a         a b e e e

i i i i i i

c a b d a b    

i i

b a  

25

 Scalar or dot product yields a scalar

 In index notation:

 Norm of a vector

Vector Operations

cos   u v u v

where is the angle between the vectors u and v



2

ˆ ˆ

i i j j i j ij i i

u u u u u u u u e e u       

   

1 2 1 2 i i

u u u u u   

   

3 1

ˆ ˆ ˆ ˆ v v v v v

i T i i j j i j i j i j ij i i i i i

u u u u u 

 

              



u v e e e e u v u v

ij



0( ) 1 ( ) i j j i      

26

 Some properties of the scalar or dot product

Vector Operations

     

0, , u v v u u 0 u v w u v u w u u u u u u u v u v u v                          

Linear operator

27

 Vector product (or cross product ) yields another vector

 In index notation:

Vector Operations

sin       c a b b a c a b

i i

ˆ ˆ

i ijk j k i ijk j k

c a b c a b i



    c e e e e {1,2,3}

     

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

ˆ ˆ ˆ a b a b a b a b a b a b       c e e e

where is the angle between the vectors a and b



   

1 2 3 1 2 3 1 2 3

ˆ ˆ ˆ det

symb

a a a b b b            e e e

123 132 1 1

2 3 3 2

a b a b

 



     

e e

1 i 

 

231 3 1 213 1 3 1 1

a b a b

 

 e e 2 i 

 

312 1 2 321 2 1 1 1

a b a b

 

 e e 3 i 

28

 Some properties of the vector or cross product

Vector Operations

   

, , || a b a b               u v v u u v u 0 v u v u v w u v u w

Linear operator

29

 Tensor product (or open or dyadic product) of two vectors:

Also known as the dyad of the vectors u and v, which results in a 2nd

• rder tensor A.

 Deriving the tensor product along an orthonormal basis {êi}:  In matrix notation:

Vector Operations

   A u v uv

   

     

ˆ ˆ ˆ ˆ ˆ ˆ v v

i i j j i j i j ij i j

u u A         A u v e e e e e e

      

v v v

1 T 2 1 2 3 3

u u u               u v u v

   

v ,

ij i j ij ij

A u i j      A u v {1,2,3}

v v v v v v v v v

1 1 1 2 1 3 11 12 13 2 1 2 2 2 3 21 22 23 3 1 3 2 3 3 31 32 33

u u u A A A u u u A A A u u u A A A                     

30

 Some properties of the open product:

Vector Operations

       

         u v w u v w u v w v w u

   

u v v u   

 

          u v w u v u w

       

         u v w u v w u v w w u v

     

     u v w x u x v w

Linear operator

31

Example

 Prove the following property of the tensor product is true:

   

     u v w u v w

32

Example - Solution

   

v v w

k i i i i k k k

c u u           u v w w

       

v w v w

k i i k i i k i k ik

c u u                u v w u v w

   

ˆ ˆ ˆ v w

i j k k k k k k

u                c e u v w e u v w e

scalar vector 1st order tensor (vector) 1st order tensor (vector) 2nd order tensor (matrix)

   

     u v w u v w

vector

c

k-component

• f vector c

k-component

• f vector c

33

Example – Solution

   

     u v w u v w

 

     

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ v w v w v w

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

u u u            u v w e e e e e e e e e

 

     

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ v w v w v w

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

u u u            u v w e e e e e e e e e

34

Vector Operations

 Triple scalar or box product

 In index notation:

     

cos cos sin V                a b c c a b b c a a b c a b c

 

a b c

ijk i j k

V a b c     e

 

1 2 3 1 2 3 1 2 3

det a a a V b b b c c c               a b c

base area height

35

Vector Operations

 Triple vector product

 In index notation:

     

      u v w u w v u v w

     

 

ˆ ˆ ˆ v w v w ˆ ˆ v w v w ˆ ˆ v w v w u v w e e e e e e e

j j klm l m k ijk j klm l m i ijk lmk j l m i il jm im jl j l m i m i m i l l i i

u u u u u u                 e e e e e

ijk pqk ip jq iq jp

      e e

REMARK

36

 Scalar or dot product yields a scalar  Vector or cross product

yields another vector

 Triple scalar or box product yields a scalar  Triple vector product yields another vector

Summary

     c a b b a

 

a b

i ijk j k i

c a b    e

 

cos sin V       a b c a b c

 

a b c

ijk i j k

a b c    e

     

      u v w u w v u v w

 

w v v w

k k i k k i i

u u         u v w

   

cos

T

    u v u v u v v

i i

u   u v

37

Tensor Algebra

Tensor Operations

38

 Summation (only for equal order tensors)  Scalar multiplication (scalar times tensor)

Tensor Operations

    A B B A C   A C

ij ij ij

C A B  

ij ij

C A  

39

 Dot product (.) or single index contraction product

Tensor Operations

REMARK

2

A A A  

   A B B A

Index “j” disappears (index contraction)

i ij j

c A b 

2nd

• rder

1st

• rder

  A b c

1st

• rder

Index “k” disappears (index contraction)

ij i k k j

C b  A   b C A

3rd

• rder

1st

• rder

2nd

• rder

Index “j” disappears (index contraction)

ik i k j j

C A B    A B C

2nd

• rder

2nd

• rder

2nd

• rder

40

 Some properties:  2nd order unit (or identity) tensor

Tensor Operations

    1 u u 1 u

[1]

ij j i i i ij ij

            1 e e e e

 

1 1 1

           1  

          A b c A b A c

Linear operator

41

 Some properties:

2nd Order Tensor Operations

     

                  1 A A A 1 A B C A B A C A B C A B C A B C    A B B A

42

Example

 When does the relation hold true ?

   n T T n

43

Example - Solution

   n T T n

vector 2nd order tensor vector

c

k i ik

c nT 

k ki i i ki

c T n nT

 



  c n T

 

 c T n

k k

c c 

ik ki

T T 

if

 

ˆ ˆ ˆ

k k k i ik k k

c nT     c e n T e e

 

ˆ ˆ ˆ

k k k i ki k k

c nT

 

   c e T n e e

44

Example - Solution

   n T T n

COMPACT NOTATION INDEX NOTATION

      

T T

     n T T n

 

1,2,3

i ik ki i

nT T n k  

MATRIX NOTATION

T

c c

c

45

Example - Solution

      

T T

     n T T n

MATRIX NOTATION

T

c c

c

 

11 12 13 11 12 13 1 1 2 3 21 22 23 21 22 23 2 31 32 33 31 32 33 3

, , T T T T T T n n n n T T T T T T n T T T T T T n                               

 

1 1 2 3 2 3

c c c c c c           

 

11 12 13 11 12 13 1 1 2 3 21 22 23 21 22 23 2 31 32 33 31 32 33 3

, ,

T

T T T T T T n n n n T T T T T T n T T T T T T n                                         

 

1 1 2 3 2 3

T

c c c c c c           

46

 Transpose  Trace yields a scalar

 Some properties:

2nd Order Tensor Operations

 

T ji ij  A

A

 

11 22 33

( )

ii

Tr A A A A A    

 

A A

T

Tr Tr 

   

Tr Tr    A B B A

 

Tr Tr Tr    A B A B

 

Tr Tr    A A

 

 

11 12 13 11 21 31 21 22 23 12 22 32 31 32 33 13 23 33 T

A A A A A A A A A A A A A A A A A A                       A A

 

i j i i

Tr Tr a b a b          a b a b

 

A A

T T



 

T T T

   A B B A

 

u v v u

T

  

 

A B A B

T T T

      

47

 Double index contraction or double (vertical) dot product (:)

 Indices contiguous to the double-dot (:) operator get vertically repeated

(contraction) and they disappear in the resulting tensor (4 order reduction of the sum

• f orders).

2nd Order Tensor Operations

Indices “i,j” disappear (double index contraction)

ij ij

c A B 

2nd

• rder

2nd

• rder

: c  A B

zero

• rder

(scalar)

Indices “j,k” disappear (double index contraction)

jk jk i i B

 c A :  B c A

3rd

• rder

2nd

• rder

1st

• rder

Indices “k,l” disappear (double index contraction)

ij ijkl kl

C B  :  B C A

4th

• rder

2nd

• rder

2nd

• rder

48

 Some properties

2nd Order Tensor Operations

       

: :

T T T T

Tr Tr Tr Tr          A B A B B A A B B A B A

  

  

           

: : : : : : : 1 A A A 1 A B C B A C A C B A u v u A v u v w x u w v x

T T

Tr                 

REMARK

: : A B C B A C   

49

 Double index contraction or double (horizontal) dot product (··)

 Indices contiguous to the double-dot (··) operator get horizontally repeated

(contraction) and they disappear in the resulting tensor (4 orders reduction of the sum

• f orders).

2nd Order Tensor Operations

Indices “i,j” disappear (double index contraction)

ij ji

c A B 

2nd

• rder

2nd

• rder

c   A B

Indices “j,k” disappear (double index contraction)

jk kj i i B

 c A   B c A

3rd

• rder

2nd

• rder

1st

• rder

Indices “k,l” disappear (double index contraction)

ij ijkl lk

C B    B C A

4th

• rder

2nd

• rder

2nd

• rder

zero

• rder

(scalar)

50

 Norm of a tensor is a non-negative real number defined by

Tensor Operations

REMARK Unless one of the two tensors is symmetric.

: A B A B  

Tr     1 A A A 1

 

 

1 2 1 2

:

ij ij

A A    A A A

   

Tr Tr        A B A B B A B A

51

Example

 Prove that:

 

Tr    A B A B

 

:

T

Tr   A B A B

52

 

 

A B A B

T T T ik ik ij ij ik kk ki

c Tr                A B A B A B

Example - Solution

k  j

 

     

A B A B

ki ik ij ji kk ki ik

c Tr        A B A B A B

k  i i  j

53

 Determinant yields a scalar

 Some properties:

 Inverse

There exists a unique inverse A-1 of A when A is nonsingular, which satisfies the reciprocal relation:

2nd Order Tensor Operations

 

11 12 13 21 22 23 1 2 3 31 32 33

1 6

det det det A A

ijk i j k ijk pqr pi qj rk

A A A A A A A A A A A A A A A               e e e

 

det det det    A B A B

det det A A

T 

REMARK The tensor A is SINGULAR if and

• nly if det A = 0.

A is NONSINGULAR if det A ≠ 0.

1 1 1 1

, , {1,2,3}

ik kj ik kj ij

A A A A i j k 

   

            A A 1 A A

 

3

det det    A A

54

 If A and B are invertible, the following properties apply:

2nd Order Tensor Operations

 

 

 

     

 

1 1 1 1 1 1 1 1 1 2 1 1 1 1

1 1 det det det

T T T

 

              

           A B B A A A A A A A A A A A A A A

55

Example

 Prove that .

1 2 3

det A

ijk i j k

A A A  e

56

Example - Solution

11 12 13 21 22 23 31 32 33 11 22 33 21 32 13 31 12 23 13 22 31 23 32 11 33 12 21

det det A A A A A A A A A A A A A A A A A A A A A A A A A A A A                 

57

Example - Solution

111 11 21 31 112 11 21 32 113 11 21 33 121 11 22 31 122 11 22 32 123 11 22 33 131 11 23 31 132 11 23 32 133 11 23 33 211 12 21 31 212 12 21 32 213 12 21 33 221 12 22 31 222 12 22 32

A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A                   e e e e e e e e e e e e e

223 12 22 33 231 12 23 31 232 12 23 32 233 12 23 33 311 13 21 31 312 13 21 32 313 13 21 33 321 13 22 31 322 13 22 32 323 13 22 33 331 13 23 31 332 13 23 32 333 13 23 33

A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A                  e e e e e e e e e e e e e 1  1   1  1  1   1  

11 22 33 12 23 31 13 21 32 13 22 31 12 21 33 11 23 32

A A A A A A A A A A A A A A A A A A      

1 2 3 ijk i j k

A A A  e

1 i  1 j  1 k  2 k  3 k  2 i  3 i  2 j  3 j 

58

 Dot product – contraction of one index:

Summary - Tensor Operations

   

v

i i ij ij ik kj ij

c u A B C        u v C A B

       

i ij j j j ij j i i i

u A c A u d         c u A d A u        

ijk ijk im mjk ijk ijk ijk ijm mk ijk

A A         A A C B C B D B D B B C B D

       

ijkl ijkl ijm mkl ijkl ijkl ijkl ijm mkl ijkl

        C D A B A B B A B A A B B A C D    

ij ij ik kj ij

B A D     D B A

59

 Double dot product – contraction of two indices:

Summary - Tensor Operations

:

ij ij

c A B   A B

   

: :

ikm kmj ij ij ij ij ikm kmj ij ij

C D               C D A B A B B A B A A B B A

       

: :

ijkl ijkl ijmp mpkl ijkl ijkl ijkl ijmp mpij ijkl

      C A C A B D B D B A A B C B A D  

     

: :

ijk ijk ijlm lmk ijk ijk ijk ilm lmjk ijk

      A A C B C B D B D B A A B C B D

       

: :

ij ijk k k k ijk jk i i i

A c A d       c A d A B B B B

60

 Transposed double dot product – contraction of two indexes:

Summary - Tensor Operations

ij ji

c A B    A B

   

ikm mkj ij ij ij ij ikm mkj ij ij

C D                 C D A B A B B A B A A B B A        

ijkl ijkl ijmp pmkl ijkl ijkl ijkl ijmp pmij ijkl

        C A C A B D B D B A A B C B A D        

ijk ijk ijlm mlk ijk ijk ijk ilm mljk ijk

        A A C B C B D B D B B C B D A A

       

ij jik k k k ijk kj i i i

A c A d         c A d A B B B B

61

 Open product – expansion of indexes:

Summary - Tensor Operations

       

v

i j ij ij ij ij kl ijkl ijkl ijkl

u A A B         A u v A B C C

       

i jk ijk ijk ijk ij k ijk ijk ijk

u A A u         u A A u C D C D

       

ijklm ijklm ij klm ijklm ijklm ijklm ijk lm ijklm

A A         A A B B B B C D C D

       

vi

j ij ij ij ij kl ijkl ijkl ijkl

u B B A         B v u B A D D

62

Tensor Algebra

Differential Operators

63

Differential Operators

 A differential operator is a mapping that transforms a field

into another field by means of partial derivatives.

 The mapping is typically understood to be linear.  Examples:

 Nabla operator  Gradient  Divergence  Rotation  …

    , ... v x A x

64

Nabla Operator

 The Nabla operator  is a differential operator

“symbolically” defined as:

 In Cartesian coordinates, it can be used as a (symbolic) vector

• n its own:

.

ˆ

symbolic symb i i

x        e x

 

1 . 2 3 symb

x x x                          

65

Gradient

 The gradient (or open product of Nabla) is a differential

• perator defined as:

 Gradient of a scalar field Φ(x):

 Yields a vector

 Gradient of a vector field v(x):

 Yields a 2nd order tensor

       

.

{1,2,3} ˆ ˆ

symb i i i i i i i i i

i x x x                            e e

       

.

v v , {1,2,3} v ˆ ˆ ˆ ˆ

symb j j ij i j i i j i j i j ij i

i j x x x                              v v v v v e e e e

ˆi

i

x     e v ˆ ˆ

j i j i

x      v e e

66

Gradient

 Gradient of a 2nd order tensor field A(x):

 Yields a 3rd order tensor

         

.

A A , , {1,2,3} A ˆ ˆ ˆ ˆ ˆ ˆ

symb jk jk ijk ijk i jk i i jk i j k i j k ijk i

i j k x x x                                  A A A A A A e e e e e e A ˆ ˆ ˆ

jk i j k i

x       A e e e

67

Divergence

 The divergence (or dot product of Nabla) is a differential

• perator defined as :

 Divergence of a vector field v(x):

 Yields a scalar

 Divergence of a 2nd order tensor A(x):

 Yields a vector

vi

i

x     v

   

.

v v

symb i i i i i i

x x          v v

A ˆ

ij j i

x     A e

 

     

.

A A {1,2,3} A ˆ ˆ

symb ij j ij i ij i i ij j j j i

j x x x                       A A A A e e

68

Divergence

 The divergence can only be performed on tensors of order 1 or

higher.

 If

, the vector field is said to be solenoid (or divergence-free).

  v

 

v x

69

Rotation

 The rotation or curl (or vector product of Nabla) is a differential

• perator defined as:

 Rotation of a vector field v(x):

 Yields a vector

 Rotation of a 2nd order tensor A(x):

 Yields a 2nd order tensor

v ˆ v e

k ijk i j

x     e A ˆ ˆ A e e

kl ijk i l j

x      e

 

   

 

. .

v v {1,2,3} v ˆ ˆ v v v v e e

symb symb k i ijk ijk k ijk j k j j k i i ijk i j

i x x x                       e e e e

   

.

A A , , {1,2,3} A ˆ ˆ ˆ ˆ A A A e e e e

symb kl il ijk kl ijk j j kl il i l ijk i l j

i j k x x x                       e e e

70

Rotation

 The rotation can only be performed on tensors of order 1 or

higher.

 If , the vector field is said to be irrotational (or

curl-free).

  v

 

v x

71

Differential Operators - Summary

scalar field Φ(x) vector field v(x) 2nd order tensor A(x)

GRADIENT DIVERGENCE ROTATION

 

A A

kl il ijk j

x     e

 

v v

k i ijk j

x     e

vi

i

x     v

 

Aij

j i

x     A

   

i i i

x        

   

v

ij j ij i

x         v v

   

A

ijk jk ijk i

x         A A

72

Example

 Given the vector

determine

 

1 2 3 1 1 2 2 1 3

ˆ ˆ ˆ x x x x x x     v v x e e e , , .    v v v

73

Example - Solution

 Divergence:

vi

i

x     v

3 1 2 2 3 1 1 2 3

v v v v

i i

x x x x x x x                v

 

1 2 3 1 2 1

x x x x x x            v

 

1 2 3 1 1 2 2 1 3

ˆ ˆ ˆ x x x x x x     v v x e e e

74

Example - Solution

 Divergence:  In matrix notation:

vi

i

x     v

     

 

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

, ,

T symb symb symb T symb

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x                                                v v

13 31 11

 

1 2 3 1 2 1

x x x x x x            v

75

 In index notation:

Example - Solution

 Rotation:

 

v v

k i ijk j

x     e

 

3 3 2 1 1 2 12 13 21 23 31 32 1 1 2 2 3 3

v v v v v v v v

k i ijk j i i i i i i

x x x x x x x                         e e e e e e e

 

1 2 3 1 2 1

x x x x x x            v

76

 

3 2 123 132 2 3 3 1 213 231 1 2 1 3 2 1 3 2 1 312 321 1 2

v v v v 1 v v v x x x x x x x x x x x                                                 e e e e e e

Example - Solution

 Rotation:  In matrix notation  In compact notation:

 

3 2 1 12 13 21 1 1 2 3 1 2 23 31 32 2 3 3

v v v v v v v i

i i i i i i

x x x x x x                     e e e e e e

1  1  1  1   1   1  

   

1 2 2 2 1 3 3

ˆ ˆ 1 x x x x x      v e e

 

1 2 3 1 2 1

x x x x x x            v

77

Example - Solution

 Rotation:  Calculated directly in matrix notation:

1 1 2 3 2 1 2 3 1

v v v x x x x x x   

   

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

ˆ ˆ ˆ v v det v v v v v v v v v v ˆ ˆ ˆ ˆ ˆ 1

symb

x x x x x x x x x x x x x x x x x                                                                                                     e e e v e e e e e

78

Example - Solution

 Gradient:  In matrix notation  In compact notation:

   

v j

ij ij i

x        v v

1 1 2 3 2 1 2 3 1

v v v x x x x x x   

        

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

1 , ,

symb symb symb T

x x x x x x x x x x x x x x x x x                                          v v v

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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x x x x x x x x              v e e e e e e e e e e e e

13 31 33

79

Tensor Algebra

Integral Theorems

80

Divergence or Gauss Theorem

 Given a field in a volume V with closed boundary surface

∂V and unit outward normal to the boundary n , the Divergence (or Gauss) Theorem states: Where:



represents either a vector field ( v(x) ) or a tensor field ( A(x) ).

A A

V V

dV dS



  

  n

A A A A

V V

dV dS



  

 

n A A A A

81

Generalized Divergence Theorem

 Given a field in a volume V with closed boundary surface

∂V and unit outward normal to the boundary n , the Generalized Divergence Theorem states: Where:



represents either the dot product ( · ), the cross product (  ) or the tensor product (  ).



represents either a scalar field ( ϕ(x) ), a vector field ( v(x) ) or a tensor field ( A(x) ).

V V

dV dS



  

  n

A A A A

V V

dV dS



  

 

n A A A A A  A

82

Curl or Stokes Theorem

 Given a vector field u in a surface S with closed boundary

surface ∂S and unit outward normal to the boundary n , the Curl (or Stokes) Theorem states:

 

S S

dS d



   

 

u n u r

where the curve of the line integral must have positive orientation, such that dr points counter-clockwise when the unit normal points to the viewer, following the right-hand rule.

83

j

n

Example

 Use the Generalized Divergence Theorem to show that

where is the position vector of .

i j ij S x n dS

V 



j

n

i

x

V V

dS dV



  

 

n A A A A

84

Example - Solution

 Applying the Generalized Divergence Theorem:  Applying the definition of gradient of a vector:

V V

dS dV



  

 

x n x

i j S x n dS



S

dS 

 x

n

 

j ij i

x x     x

 

i ij j

x x     x

i j ij S x n dS

V 



85

Example - Solution

 The Generalized Divergence Theorem

in index notation:

 Then, i j ij S x n dS

V 



i i j S V j

x x n dS dV x   

 

i i j ij ij S V V j

x x n dS dV dV V x       

  

86

Tensor Algebra

References

87

 José Mª Goicolea, Mecánica de Medios Continuos: Resumen de Álgebra y

Cálculo Tensorial, UPM.

 Eduardo W. V. Chaves, Mecánica del Medio Continuo,Vol. 1 Conceptos

básicos, Capítulo 1: Tensores de Mecánica del Medio Continuo, CIMNE, 2007.

 L. E. Malvern. Introduction to the mechanics of a continuous medium.

Prentice-Hall, Englewood Clis, NJ, 1969.

 G. A. Holzapfel. Nonlinear solid mechanics: a continuum approach for

• engineering. 2000.

References

88