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Multimedia course CONTINUUM MECHANICS FOR ENGINEERS

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By Prof. Xavier Oliver

Technical University of Catalonia (UPC/BarcelonaTech) International Center for Numerical Methods in Engineering (CIMNE) http://oliver.rmee.upc.edu/xo

First edition May 2017 DOI:

10.13140/RG.2.2.22558.95046

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Acknowledgements

• Prof. Carlos Agelet (CIMNE/UPC)
• Dr. Manuel Caicedo (CIMNE/UPC)
• Dr. Eduardo Car (CIMNE)
• Prof. Eduardo Chaves (UCLM)
• Dr. Ester Comellas (CIMNE)
• Dr. Alex Ferrer (CIMNE/UPC)
• Prof. Alfredo Huespe (CIMNE/UNL/UPC)
• Dr. Oriol Lloberas-Valls (CIMNE/UPC)
• Dr. Julio Marti (CIMNE)

… and the past students of my courses on Continuum Mechanics …

Than anks for you

• ur

r con

• ntri

tributi tion !!!! !!!!

TENSOR ALGEBRA

Multimedia Course on Continuum Mechanics

2

Overview

 Introduction to tensors  Indicial or (Index) notation  Vector Operations  Tensor Operations  Differential Operators  Integral Theorems  References

Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 1

Introduction

SCALAR VECTOR MATRIX ?

, , ... σ ε , , ... v f , , ... ρ θ

, ... C

v

4

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.

5

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 α = 1.2 0.3 0.8 vi       =    

0.1 1.3 2.4 0.5 1.3 0.5 5.8

ij

E         =  

6

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 δ =   ⋅ = =   ≠   e e if if

7

Tensor Algebra

Indicial or (Index) Notation

10

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∈

11

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

12

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 ∈

13

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

14

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

15

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 ∈

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

18

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

=

∂ ∂ = = ∂ ∂

∑

19

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 δ =  =  ≠  if if

( ) ( )

11 22 33 12 13 21

1 1 ...

ij

i j i j δ δ δ δ δ δ δ = = = =   =  ≠ = = =   if if

ij ji

δ δ =

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

11 22 33

3

ii

δ δ δ δ = + + =

,

ij j i ij jk ik

u u A A δ δ = =

20

Levi-Civita Epsilon (permutation) e

 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

21

Example

 Prove the following expression is true:

6

ijk ijk =

e e

23

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 =

24

Tensor Algebra

Vector Operations

25

 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 α α α α = = + + a b e e e

i i i i i i

c a b d a b = + = −

i i

b a α =

26

 Scalar or dot product yields a scalar

 In index notation:

 Norm of a vector

Vector Operations

cosθ ⋅ = u v u v

where is the angle between the vectors u and v

θ

2

ˆ ˆ

i i j j i j ij i i

u u u u u u u u e e u δ = ⋅ = ⋅ = =

( ) ( )

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 δ

= =

  ⋅ = ⋅ = ⋅ = = = =    

∑

u v e e e e u v u v

ij

δ

0( ) 1 ( ) i j j i = ≠  = = 

27

 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 α β α β ⋅ = ⋅ ⋅ = ⋅ + = ⋅ + ⋅ ⋅ > ≠ ⋅ = = ⋅ = ≠ ≠ ⊥

Linear operator

28

 Vector product (or cross product ) yields another vector

 In index notation:

Vector Operations

sinθ = × = − × = 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 

( ) ( ) ( )

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

θ

θ π ≤ ≤

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 =

29

 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

30

 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 

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         =            

31

 Some properties of the open product:

Vector Operations

( ) ( ) ( ) ( )

⊗ ⋅ = ⊗ ⋅ = ⋅ = ⋅ u v w u v w u v w v w u

( ) ( )

u v v u ⊗ ≠ ⊗

( )

α β α β ⊗ + = ⊗ + ⊗ 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

32

Example

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

( ) ( )

⋅ ⊗ = ⋅ ⊗ u v w u v w

33

Tensor Algebra

Tensor Operations

39

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

Tensor Operations

+ = + = A B B A C α = A C

ij ij ij

C A B = +

ij ij

C A α =

40

 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

41

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

Tensor Operations

⋅ = ⋅ = 1 u u 1 u

[1]

ij j i i i ij ij

δ δ = ⊗ = ⊗    =   1 e e e e

[ ]

1 1 1

    =       1

( )

α β α β ⋅ + = ⋅ + ⋅ A b c A b A c

Linear operator

42

 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

43

Example

 When does the relation hold true ?

⋅ = ⋅ n T T n

44

 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 α α = 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

α β α β + = +

48

 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

49

 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 = ⇒ =

50

 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)

51

 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

52

Example

 Prove that:

( )

Tr ⋅⋅ = ⋅ A B A B

( )

:

T

Tr = ⋅ A B A B

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 δ

− − − −

 ⋅ = = ⋅   = = ∈   A A 1 A A

( )

3

det det α α = A A

55

Example

 Prove that .

1 2 3

det A

ijk i j k

A A A = e

57

Tensor Algebra

Differential Operators

64

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

65

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   ∂   ∂     ∂ ∇ =   ∂     ∂   ∂  

66

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 ∂ ∂Φ  ∇Φ = ∇ ⊗Φ = ∇ Φ = Φ = ∈  ∂ ∂   ∂Φ ∇Φ = ∇Φ =  ∂  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 ∂Φ ∇Φ = ∂ e v ˆ ˆ

j i j i

x ∂ ∇ = ⊗ ∂ v e e

67

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

68

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

69

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

70

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

71

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

72

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 ∇ ⊗Φ = ∂Φ = ∇Φ = ∂

[ ] [ ]

v

ij j ij i

x ∇ ⊗ = ∂ = ∇ = ∂ v v

[ ] [ ]

A

ijk jk ijk i

x ∇ ⊗ = ∂ = ∇ = ∂ A A

73

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

74

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

75

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

76

Tensor Algebra

Integral Theorems

81

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

V V

dV dS

∂

⋅∇ = ⋅

∫ ∫

n A A

82

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

V V

dV dS

∂

∗∇ = ∗

∫ ∫

n A A A ∗ A

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

Vδ =

∫

j

n

i

x

V V

dS dV

∂

∗ = ∗∇

∫ ∫

n A A

85

Tensor Algebra

References

88

 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

89