part b tensors
DESCRIPTION
Part B Tensors. Or, inverting. The graph above represents a transformation of coordinates when the system is rotated at an angle CCW. 2B1 Tensors. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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.
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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
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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
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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
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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
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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~~
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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
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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,