1

Part B Tensors

'cos 'sin cos sin '

'sin 'cos sin cos '

x x y x xor

y x y y y

Or, inverting

' cos sin ' cos sin

' sin cos ' sin cos

x x y x xor

y x y y y

The graph above represents a transformation of coordinates when the system is rotated at an angle CCW

2

Let be two vectors in a Cartesian coordinate system. If is a transformation, which transforms any vector into some other vector, we can write

2B1 Tensors ,a b T

Ta c

Tb d

where c and d are two different vectors.

If

T a b Ta Tb

T a Ta

1.1

1.2

a

a

For any arbitrary vector a and b and scalar

T is called a LINEAR TRANSFORMATION and a Second Order Tensor. (1.1a) and (1.2a) can be written as,

T a b Ta Tb 1.3

3

2B1 Tensors

Questions:

If ( ) 3 , then is a tensor?T a a T

( ) 3( ) 3 3

3 3

So it satisfy ( )

( ) 3( ) 3

(3 ) 3

So it satisfy ( )

T a b a b a b

Ta Tb a b

T a b Ta Tb

T a a a

Ta a a

T a Ta

Yes

4

2B1 Tensors

Questions:

1If ( ) 3 2 , then is a tensor?T a a e T

1 1

1 1 1

( ) 3( ) 2 3 3 2

(3 2 ) (3 2 ) 3 3 4

So ( )

T a b a b e a b e

Ta Tb a e b e a b e

T a b Ta Tb

Yes

5

2B2 Components of a Tensor

Scalar has 1 component

Vector has 3 component

Tensor has ? component?

1 1 2 2 3 3

1 1 2 2 3 3

We know

ˆ ˆ ˆ

ˆ ˆ ˆ

Ta b

T a e a e a e b

a Te a Te a Te b

6

2B2 Components of a Tensor

~ ^ ^

i jjiT e T e

11 1 1

12 1 2

ˆ ˆ

ˆ ˆ

T e Te

T e Te

In general ˆ ˆij i jT e Te

The components of Tij can be

written as

11 12 13

21 22 23

31 32 33

T T T

T T T T

T T T

1 1 2 3

2 1 2 3

3 1 2 3

ˆ ˆ ˆ ˆ?? ?? ??

ˆ ˆ ˆ ˆ?? ?? ??

ˆ ˆ ˆ ˆ?? ?? ??

Te e e e

Te e e e

Te e e e

1 11 1 21 2 31 3

2 12 1 22 2 32 3

3 13 1 23 2 33 3

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

Te T e T e T e

Te T e T e T e

Te T e T e T e

7

2B2 Components of a Tensor (cont.)

Example 2B2.3

~Re cos sin ~Re sin cos ~Re

1 1 2

2 1 2

3 3

e e

e e

e

Thus

100

0cossin

0sincos

R

Q: Let R correspond to a righ-hand rotation of a rigid body about the x3 axis by an angle . Find the matrix of R

Q: Let R correspond to a 90° right-hand rigid body rotation about the x3 axis. Find the matrix of R

1 2

2 1

3 3

ˆ ˆRe

ˆ ˆRe

ˆ ˆRe

e

e

e

0 1 0

1 0 0

0 0 1

R

8

2B3 Components of a transformed vector

The transformation ~T is linear if

For any arbitrary vectors aa12, and scalar

is then called a LINEAR TRANSFORMATION and is a second-order tensor.

If bTa~, then

11

2121~

)(~

~~)(

~

aTaT

aTaTaaT

3

2

1

333231

232221

131211

3

2

1

a

a

a

TTT

TTT

TTT

b

b

b

9

2B3 Components of a transformed vector

1 1 2 2 3 3

1 1 2 2 3 3

From the definition of the tensor

We know

ˆ ˆ ˆ

ˆ ˆ ˆ

Ta b

T a e a e a e b

a Te a Te a Te b

Now what is b? what are the components of b?

1 1 1 1 1 2 1 2 3 1 3

2 2 1 2 1 2 2 2 3 2 3

3 3 1 3 1 2 3 2 3 3 3

ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )

b e b a e Te a e Te a e Te

b e b a e Te a e Te a e Te

b e b a e Te a e Te a e Te

1 11 1 12 2 13 3

2 21 1 22 2 23 3

1 31 1 32 2 33 3

b T a T a T a

b T a T a T a

b T a T a T a

3

2

1

333231

232221

131211

3

2

1

a

a

a

TTT

TTT

TTT

b

b

b

10

2B2 Components of a Tensor (cont.)

Vector will have different components in (x,y,z) system and

(x’,y’,z’) system. AB is still the same.

Similarly, by a tensor T at P is the same in the two systems, but, will have different components like

1 2 3

1 2 3

ˆ ˆ ˆ and e ,e ,e

ˆ ˆ ˆ and e ,e ,e

ij

ij

T T

T T

11

2B4 Sum of Tensors

If Tand Sare two tensors, then

( )

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )ij i j i j i j ij ij

T S a Ta Sa

T S e T S e e Te e Se T S

T S T S

Definition:

the sum of and is is also a tensor, defined as:

( )

T S T S

T S a Ta Sa

12

2B5 Product of Two Tensors

Components:

im mjij

im mjij

TS T S

ST S T

or

~~ ~ ~

~~ ~ ~

TS T S

ST S T

In general, the product of two tensors is not commutative:

~~ ~~TS ST

)~

(~

)~~

( aSTaST

13

Q: Let R correspond to a 90° right-hand rigid body rotation about the x3 axis. Find the matrix of R

2B5 Product of Two Tensors

0 1 0

1 0 0

0 0 1

R

1 0 0

0 0 1

0 1 0

S

Q: Let R correspond to a 90° right-hand rigid body rotation about the x1 axis. Find the matrix of S

Q: Let W correspond to a 90° right-hand rigid body rotation about the x3 axis, then a 90° right-hand rigid body rotation about the x1 axis. Find the matrix of W

Ra ( )S Ra

? ( )a S Ra ( )SRa S Ra

W SR

W S R

14

2B6 Transpose of a Tensor

Transpose of ~T is denoted by

~T T

. If aand bare two vectors,

thenTa Tb b T a

In component form, TTij jiT

Note TTT TSST~~

)~~

(

TTTT ABCCBA~~~

)~~~

(

TTT T

15

2B7 Dyadic Product a b W ~

Definition

)()( cbacba

jijiij ebaeeWeW ˆ)(ˆˆ~

ˆ

jijiji aaaaebae )ˆ(ˆ

332313

322212

312111

321

3

2

1~

bababa

bababa

bababa

bbb

a

a

a

W

W , dyadic product of two vectors is also a tensor. It transforms a vector into a vector parallel to a

c

16

2B8 Trace of a Tensor

tr a b a b( )

11 22 33

ˆ ˆ ˆ ˆ( ) ( )

ˆ ˆ

( )

ij i j ij i j

ij i j ij ij ii

T

tr T tr T e e T tr e e

T e e T T

T T T

tr T

17

2B9 Identity Tensor

A linear transformation which transforms every vector

a

into itself is an identify tensor ~I

~I a a

100

010

001~I

If ( ) 3 , we proved is a tensor, what is ?T a a T T

11 1 1 1 1(3 ) 3T e Te e e

(3 ) 3ij i j i j ijT e Te e e

3 0 0

0 3 0

0 0 3ijT

18

2B9 Inverse Tensor

Inverse, if ~Texists if the matrix Tis non-singular,

~ ~T T I 1

Note that

~ ~S T 1

then ~Sis the inverse of ~T, or

~~ST I

Given a tensor ~T, if ~Sexists such that

111 ~~~~ STTS

19

2B10 Orthogonal Tensor

Transformed vectors preserve their lengths and angles, thus if ~Q

is an orthogonal tensor, then ~Qa a

Thus ~ ~Qa Qb a b

for any aand b,

(~

) (~

)~

(~

)Qa Qb b Q QaT

( )( )Tb Q Q a b a b Ia

and

)~

,~

cos(),cos( bQaQba

20

2B10 Orthogonal Tensor (cont.)

Thus ~ ~ ~Q Q IT

Also, we have ~ ~Q Q I1 , So

1TQ Q

Thus for an orthogonal matrix, the transpose is also its inverse.Example:A rigid body rotation is an ORTHOGONAL tensor,

R R I

R

T

det 1

100

0cossin

0sincos

R

21

2B11 Transformation Matrix Between

Two Coordinate Systems Consider two systems 123 123X,, and X,,eee eee ,

ie is obtained from ieusing a rigid body rotation of ie. We are interested to relate the unit vectors iein Xfrom that of X.

ˆ ˆ ˆi i mi me Qe Q e

We see that,

1e

1e

2e2e

3e

3e1 11 1 21 2 31 3

2 12 1 22 1 32 3

3 13 1 23 1 33 3

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

e Q e Q e Q e

e Q e Q e Q e

e Q e Q e Q e

Consider two systems 123 123X,, and X,,eee eee ,

ie is obtained from ieusing a rigid body rotation of ie. We are interested to relate the unit vectors iein Xfrom that of X.

22

2B11 Transformation Matrix Between Two Coordinate Systems (cont.)

Qis the transformation given by We note that forTQQI, Q

is an orthogonal matrix

1 1 1 2

2 2 1 2

3 3 3

ˆ ˆ ˆ ˆe Re cos sin

ˆˆ ˆ ˆRe sin cos

ˆˆ ˆRe

e e

e e e

e e

Thus

100

0cossin

0sincos

R

Q: Let be obtained by rotating the basis about the x3 axis by an angle . Find the transformation matrix of R from to

ie ie

ie ie

23

2B12 Transformation laws for Vectors. Consider one vector a

Let abelong to { }ie . Thus . iia ae in the X frame.

If X is another system, and we wish to express a within X, let

'

i mmie Q e '

'

'

11 21 311 1

2 12 22 32 2

3 313 23 33

. ( . )

[ ] [ ] [ ]

m mi mi mi

i mi m

T

a a Q e Q a e

a Q a

a Q a

Q Q Qa a

a Q Q Q a

a aQ Q Q

Consider one vector a

Let abelong to { }ie . Thus . iia ae in the X frame.

If X is another system, and we wish to express a within X, let

'

i mmie Q e

Thus 123,,aaaare the components of a with respect to the Xsystem.

Thus 123,,aaaare the components of a with respect to the Xsystem.

24

2B12 Transformation laws for Vectors. (cont.)Example:

If '{}ie is obtained by rotating {}ie ccw with respect to

3e-axis find the components of 12aein terms of '{}ie

Answer

0 1 0

1 0 0

0 0 1

Q

'

0 1 0 2 0

1 0 0 0 2

0 0 1 0 0

Ta Q a

25

2B13 Transformation law for tensor

Let and

Recall that the components of a tensor are:

Since i mimeQe, we have

{ }iT e { }iT e

T~

jiij

jiij

eTeT

eTeT~

~

ˆ ˆij mi m nj nT Q e TQ e

)ˆ~

ˆ( nmnjmi eTeQQ

26

2B13 Transformation law for tensor (cont.)

Note that the tensor T is the same, but has different components

Tin the new frame ie compared to in iT e

T~

333231

232221

131211

333231

232221

131211

332313

322212

312111

333231

232221

131211

QQQ

QQQ

QQQ

TTT

TTT

TTT

QQQ

QQQ

QQQ

TTT

TTT

TTT

Thus

QTQT T

27

2B13 Transformation law for tensor (cont.)

Let 010

120

001

T

If Q represents rotation as in the earlier example such that

010

100

001

Q

, then

'

2 1 0

1 0 0

0 0 1

TT Q T Q

Find the matrix of T with respect to the basis ie

28

2B14 Tensors by Transformation laws

{ }ie are the components of the system X, and '

{ }ie are that of

new system X. Q or ijQ define direction cosines tensor

transforming from X toX. '

cos( , )i jijQ e e ; Q is an

orthogonal transformation; T

QQ I

Q~

TQQ~~

29

2B14 Tensors by Transformation laws (cont.)

The components transform as follows:

'

'

'

'

th... ...

Scalar

Vector

Second Order Tensor

Third Order Tensor..... n Order Tensor

nn

i mi m

ij mi nj mn

ijk mi nj rk mnr

ijk mi nj ok sp mno

a Q a

T Q Q T

T Q Q Q T

T Q Q Q Q T

30

2B14 Tensors by Transformation laws (cont.)

a.) If ia are components of vector and ib are components of another vector, then ij i jTabis a second order tensor.

b.) If ia is a vector and ijTare tensor components, then

ijk i jkwaT is a the third order tensor.

c.) The quotient rule

If aiare components of a vector, Tij is the second-order tensor, then a T bi ij j

then is a vectorjb

Similarly Tijand Eij are tensors, then

TCEij ijkl kl

means Cijkl is a fourth-order tensor,