Fundamentals of Tensor Analysis - RealTechSupportCT/tutorials/tensors/UniColorado... · 2...

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Fundamentals of Tensor Analysis

MCEN 5023/ASEN 5012

Chapter 2

Fall, 2006

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Concepts of Scalar, Vector, and Tensor

Scalar α A physical quantity that can be completely described by a real number.

The expression of its component is independent of the choice of the coordinate system.

Example: Temperature; Mass; Density; Potential….

Vector a A physical quantity that has both direction and length.

The expression of its components is dependent of the choice of the coordinate system.

Example: Displacement; Velocity; Force; Heat flow; ….

Tensor A A 2nd order tensor defines an operation that transforms a vector to another vector

In general, Scalar is a 0th order tensor; Vector is A 1st order tensor; 2nd order tensor; 3rd order tensor…

A tensor contains the information about the directions and the magnitudes in those directions.

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Vectors and Vector Algebra

A vector is a physical quantity that has both direction and length

a

X1

X2

X3What do we mean the two vectors are equal?

What is a unit vector?

The two vectors have the same length and direction

The length of a unit vector is one

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In a Cartesian coordinate, a vector can be expressed by three ordered scalars

ii

iaaaa eeeea ∑=

=++=3

1332211

Summation convention iia ea =


Vectors and Vector Algebraa

X1

X2

X3

1a

2a

3a

Dummy index and free index

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Vector Algebra

Sum:

Scalar Multiplication

Dot Product

Cross Product


( ) iii ba eba +=+

iia ea αα =

( )baba ,cosθba=•

321

321

321

bbbaaaeee

ba =×

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Vector Algebra

Sum:


( ) iii ba eba +=+

X1

X2

ab

ba+b

Parallelogram law of addition

Properties of Sum

abba +=+

( ) ( )cbacba ++=++

( ) oaa =−+

aoa =+

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Vector Algebra


Scalar Multiplication iiaea αα =

X1

X2

aαa (α>0)

αa (α<0)

Properties of Scalar Multiplication

( ) ( )aa βααβ =

( ) aaa βαβα +=+

( ) baba ααα +=+

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Vector Algebra


Properties of Dot Product

Dot Product ( )baba ,cosθba=•

abba •=•

ooa =•

( ) ( ) ( )cabacba •+•=+• βαβα

oaaa ≠⇔>• 0 and oaaa =⇔=• 0

aaa •= norm of a

aaa •=2

oba =• a is orthogonal to b

Direction of a vector:

aaa

aan

•==

9

a

iia e

Symbolic expression

Component expression

X1

X2

X1’

X2’ a

e1

a

1ea •

θ

Vector Algebra


Components of a vector

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Vector Algebra


Dot Product ( )baba ,cosθba=•

Three basis vectors of a Cartesian coordinate ie i=1,2,3

ji ee •⎩⎨⎧

≠=

=jiji

01

ijδ=⎩⎨⎧

≠=

=jiji

ij 01

δ Kronecker delta

Properties of ijδ

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Vector Algebra


Cross Product

321

321

321

bbbaaaeee

ba =×

Permutation symbol

⎪⎩

⎪⎨

⎧−=

indexrepeatednpermutatiooddnpermutatioeven

ijk

01

1ε

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Vector Algebra


Physical Meaning of Cross Product

a

b

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Vector Algebra


Properties of Cross Product

abba ×−=×

321

321

321

bbbaaaeee

ba =×

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Vector Algebra


Triple scalar product ( ) cba •×

a

bc

ba×

( ) ( ) ( ) bacacbcba •×=•×=•×

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Fundamentals of Tensor AnalysisConcept of Tensor

A 2nd order tensor is a linear operator that transforms a vector a into another vector b through a dot product.

A: a b b=Aaor

( ) AbAabaA +=+ αα

Properties due to linear operation

( ) BaAaaBA ±=±

X1

X2

a

b=Aa

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Fundamentals of Tensor AnalysisAnatomy of a Tensor: Concepts of Dyad and Dyadic

ba⊗ ( )abDyad

A dyad is a tensor. It transforms a vector by

A dyadic is also a tensor. It is a linear combination of dyads with scalar coefficients.

dcbaB ⊗+⊗=

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Fundamentals of Tensor AnalysisConcepts of Dyad and Dyadic

Now, consider a special dyad ji ee ⊗

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Fundamentals of Tensor AnalysisConcepts of Dyad and Dyadic

Now, consider a special dyad ji ee ⊗

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Properties of Dyad and Dyadic

( )( ) ( ) ( )dbacbadcba ⊗+⊗=+⊗ αα

( ) ( ) ( )cbcacba ⊗+⊗=⊗+ βαβα

( ) ( ) bAabaA ⊗=⊗

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Special Tensors

Positive semi-definite tensor:

Positive definite tensor:

0≥•Aaa for any 0a ≠

0>•Aaa for any 0a ≠

Unit tensor: I jiij eeI δ=

Transpose of a tensor TA

AAI =

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Fundamentals of Tensor AnalysisDot Product AB

Properties of dot product

( ) ( ) ABCBCACAB ==

AAA =2

AAAA ...=n

BAAB ≠

( ) TTT ABAB =

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Fundamentals of Tensor AnalysisTrace and Contraction

Trace ( ) iiAtr =A

( ) ( )Ttrtr AA =

( ) ( )BAAB trtr =

( ) ( ) ( )BABA trtrtr +=+

( ) ( )AA trtr αα =

Contraction (double dot) BA :

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Tensor Review

⊗

321

321

321

bbbaaaeee

ba =×

Cross Product

Dayd:

ba⊗vector a is sitting in front vector b

( ) ( )cbacba •=⊗

( ) ( )bcabac •=⊗

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Rule of Thumb:

For algebra on vectors and tensors, an index must show up twice and only twice.

If an index shows up once on the left hand side (LHS) of “ = ”sign, it must show up once and only once on the right hand side (RHS) of “ = ” sign. This index is free index.

If an index shows up twice on either LHS or RHS of “ = ”, it does not have to show up on the other side of “ = ”. This index is dummy index.

You are free to change the letters that represent a “free” index or a “dummy” index. But you have to change it in pair.

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Fundamentals of Tensor AnalysisDeterminant and inverse of a tensor

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

333231

232221

131211

detdetAAAAAAAAA

A

( ) BAAB detdetdet =

AA detdet =T

Determinant

A is singular if and only if detA=0Inverse

if detA is not zero AAIAA 11 −− ==

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Properties of inverse tensor

( ) 111 −−− = ABAB

( ) AA =−− 11

( ) 11 1 −− = AAα

α

112 −−− = AAA 111 ... −−−− = AAAA n

( ) ( ) 11 detdet −− = AA

( ) ( ) 11 −− = TT AA ( ) TT −− =AA 1

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Orthogonal tensor

baQbQa •=•

θ

a

b θ

Qa

Qb

IQQ =T

1det ±=Q

1det =Q

1det −=Q

Q is a proper orthogonal tensor and corresponds to a rotation

Q is an improper orthogonal tensor and corresponds to a reflection

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Fundamentals of Tensor AnalysisRotation

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

1000cossin0sincos

θθθθ

Q

X1

X2

X1’

X2’ a

X1

X2a

a’θθ

Rotation of the vector Rotation of the coordinate system

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Symmetric and skew tensor

A=S+W ( )TAAS +=21 ( )TAAW −=

21

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X1

X2

n

m=An

General case: m=An

Eigenvectors and eigenvalues of a tensor A

X1

X2

Eigenvector:

nAn ˆˆ =λ

n̂

nAn ˆˆ =λ

The scalar λ is an eigenvalue of a tensor A if there is a non-zero vector unit eigenvector of A so that n̂

nnA ˆˆ λ=

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Eigenvectors and eigenvalues of a tensor A

nnA ˆˆ λ= ( ) 0nIA =− ˆλ

( ) 0det =− IA λ

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Spectral Decomposition

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