tensors.ppt

8/11/2019 Tensors.ppt

1/46

Tensors

References:

P. Chadwick, Contiuum Mechanics, Dover Publications, 1976.

M. E. Gurtin, An Introduction to Continuum Mechanics,

Academic Press, 1981.

Acknowledgements:

H. Garmestani, Georgia Tech

A. Rollett, CMU


2/46

Mappings Sets: Collection of Objects.

Elements: Constituents of a Set.

Mapping (or Transformation):

fmaps D in C

Domain of f: Set D

Codomain of f: Set C

Range of f:

Range is a subset of codomain: rangefmaps D onto C: range

One-to-one mapping:

a e

cbaA ,,

CDf :

})({ Dxxff Cf

Cf

2121 )()( xxxfxf


3/46

Tensors

Tensors: mathematical representation ofphysical entities or phenomena (e.g. Stress,

Strain, Elastic Stiffness)

Tensors of different rank are used to

represent different types of physicalquantities

Tensors of a given rank have certain

common properties and follow certain rules(e.g. coordinate transformation laws)

Tensors are not simply setsor matrices

Tensors do not require definition of acoordinate system


4/46

Scalars: Tensors of Rank Zero

Scalars may be specified completely by

a single number (e.g. density,

temperature, energy)

Scalars are not constants, but may

actually be functions of position and/ortime.


5/46

Points: Rank One Tensors

a

b

cd

ef

g

Euclidean Point Space E

Set of all points in three

dimensional space

Labeling of the points does not

require a coordinate frame!

Operations on points are not

meaningful


6/46

Vectors: Rank One Tensors

Defined as difference for a

given ordered pair (a,b): v=

ba

Euclidean Vector Space V

Still no need of a reference

frame!

a

b

cd

ef

g


7/46

Operations on Vectors:

Dot Product

vu u

v

cosvuvu

uuu Magnitude or Norm

V

vuvuuv , ,,,)( Vwvuwvwuwvu

V uuu 0 0uuu if f0No need for a reference frame yet!

0


8/46

Operations on Vectors:

Dot Product

1nUnit Vector:

Orthogonal Vectors: 0 vuSeveral physical quantities possess thecharacteristics described here for Vectors;therefore they are all classified as Rank OneTensors.

e.g. Force, Displacement, Velocity

These quantities require not only an assignmentof magnitude, but also a specification of direction.


9/46

Coordinate Frames

Right-handed, orthogonal, Cartesian Coordinate system.

Set of orthonormal basis vectors

}{},,{ 321 ieeee

Origin oijji

jiif

jiif

0

1 ee

Kronecker delta ij


10/46

Coordinate Frames

v

iiivevComponents of a vector v:

ve iiv

ii

ii vu evuw Adding Vectors (parallelogram law):

uIndicial Notation:

iii vuw

2/1

iiuu

uuu

Einsteins Summation

Convention

iivu vu

w


11/46

Coordinate Transformation

Matrices: 2-D

Or, inverting

The graph above represents a transformation of coordinates when the system is

rotated at an angle CCW

y

x

y

xor

yxy

yxx

cossin

sincos

sincos

sincos

y

x

y

xor

yxy

yxx

cossin

sincos

cossin

sincos


12/46


Matrices: 3-D

}{},,{ 321 ieeee First Frame:

}{},,{ 321 ieeee Second Frame:

iiii vv eev

jijijiijii vvvv eeee

ijiijij vQvv ee

jiijQvQv ee


13/46


Matrices: 3-D

ijjkikkljlikljlkikji QQQQQQ eeee

Six

equations Three independent parameters in [Q]

ITQQ TQQ 1

1det Q


14/46

Coordinate Transformations:

Euler Angles

A sequence of three rotations (called Euler

angles) are needed to completely specify a

arbitrary transformation.

e1 e2

e3

e1

e

2

e3

Rotation 1: Angle f1about e3so

that the new orientation of e1is

perpendicular to e3.

Rotation 2: Angle Fabout the newe1so that the new orientation of e3is

parallel to e3.

Rotation 3: Angle f2about e3to

bring the two frames in coincidence.


15/46

Bunge Euler Angles

g {1,f,2}

(0 1 2, 0 f , 02 2)

(1))2()3( elj

ekl

eik

eij RRRR

fff

ffffff

cossincossinsin

cossincoscoscossinsincoscossinsincossinsinsincoscoscossinsincossincoscos

11

221212121

221212121e

ijR


16/46

Invariants

kklkkllkiliklilkikii uuuuuuQQuQuQuu

All tensorial quantities possess certain

measures that are independent of the choice of

the coordinate reference frame. For example,

rank one tensors (or vectors) have one

invariant norm or magnitude of the vector.


17/46

Vector (cross) Product

vuw

u

vsinvuw

V vuuvvu ,

0

w

V wvuwvwuwvu ,,)()()(

V vuvuu ,0)(

V vuvuvvuuvuvu ,)())(()()( 2

w is perpendicular to uand v


18/46


,,, 213132321 eeeeeeeee

kijkji eee

Permutation symbol ijk

).,.(

),3,2,1(),,(0

1.,.

:)3,2,1(),,(1

)1.,.(

:),3,2,1(),,(1

213321132

312231123

sametheareindicestwoanyifei

ofnpermutatioanotiskjiif

ei

ofnpermutatiooddaniskjiif

ei

ofnpermutatioevenaniskjiif


19/46


jijijjii

vuvu eeeevu

kijkjivu evu


20/46

Dyadic (Outer) Product

Rank Two Tensor:

jji cba )()( cbacba

ijnjmmnijnmmnijiij WWWW eeeeeeWee

332313

322212

312111

321

3

2

1

bababa

bababa

bababa

bbb

a

a

a

W

)( baW jiij baW

jiijjijijjii Wbaba eeeeeeW


21/46

Rank Two Tensors: Coordinate

Transformations

srijsjrissjrriij

jijisrrsjiij

TQQQQT

QTT

eeee

eeeeeeT

)()(

]][][[][ Tijsjrirs QTQTTQQT

ororder tensn

ororder tensthird

ororder tenssecond

vector

scalar

th

......

nnmnrkrjnimijk

mnrkrjnimijk

mnjnimij

mimi

TQQQT

TQQQT

TQQT

aQa


22/46

Tensor (Inner) Products

3

2

1

333231

232221

131211

3

2

1

a

a

a

TTT

TTT

TTT

b

b

b

bTa

ijijijkkijikjkijkkjiij

aTaTaT

aT

eeeee

eeeTa

ijij baT


23/46


kjkij aSTSaTaTS

STTS

3

2

1

333231

232221

131211

333231

232221

131211

a

a

a

SSS

SSS

SSS

TTT

TTT

TTT


24/46

Rank Two Tensors

Unit (spherical) Tensor or Identity Tensor:

iijiij eeeeI

Isotropic Tensor = iijiij CC eeee

SSSSSSSISS

00A0vAIAvIv

321

0

Transpose of a Tensor:iijjijij

T vSuvuS vSuvSu

SSSTSTTSTS TTTTTTTT


25/46

Rank Two Tensors

Symmetric Tensor:

Tij= Tji T= TT

Skew-symmetric or Antisymmetric:

Wij= -Wji W= -WT

Decomposition of a general tensor: WSL

TT LLWLLS 2

1

2

1


26/46

A linear transformation, from rank one tensor spaceV

into itself defines a rank two tensor

Tensors as Transformations

(Mappings)

cTb

TaaT

TbTab)T(a

)(

RV ,)( ba,TbTabaT

Mapping:

Linear Mapping:


27/46

Tensors

Yestensor?aisthen,3If TaTa

TaaTaaTaaaaT

TbTab)T(abaTbTa

babab)T(a

)(3)3(3)(3)(

33

33)(3


28/46

Tensors

Notensor?aisthen,23If 1 TeaTa

TbTab)T(a

ebeaTbTa

ebaebab)T(a

11

11

2323

2332)(3


29/46

Tensors as MappingsExamples

TQR ][100

0cossin

0sincos

Transform vectors by a right-hand

rotation about the e3axis by an angle .

= 90

33

212

211

cossin

sincos

ab

aab

aab

33

12

21

ab

ab

ab

100

001

010

R

e1

e2

ab

V abRa

e1

e2

ab


30/46

Q: Let Rcorrespond to a 90right-handrigid body rotation about the x3axis.

Find the rotation tensor R.


Q: Let Scorrespond to a 90right-

hand rigid body rotation about the x1

axis. Find the rotation tensor S.

Q: Let Wcorrespond to a 90right-

hand rigid body rotation about thex3axis, then a 90right-hand rigid

body rotation about the x1axis. Find

the rotation tensor W

SRW

S

R

010

100001

100

001010


31/46


TS

ijijSS SSSMagnitude or Norm:

STTS WTWSWTS )(

0SS 0SSS if f0

jiji dcba ))(()( dbcadcba

ijijjlikklijljkiklijlkkljiij

TSTSeeeeTS

eeTeeS

TS


32/46

Trace of a Tensor

baba )(tr

)(

)(

)()()(

332211

T

iiijijjiij

jiijjiij

Ttr

TTT

TTT

trTTtrtr

ee

eeeeT

)( TSTS Ttr )(TTI tr

TT

TT

AWAWWW

ASASSS


33/46

Tensors

SRTTR)(SSTR TT

vuSSvu

321det kjiijk SSSS

11 detdetdetdet

detdetdet

SS

SS

TSST

T

1det R


34/46

Inverse of a Tensor

111 STST

Inverse of a tensor T exists if it is non-singular

ISS 1

11 TT SS


35/46

Positive Definite Tensors

0vvSvv ,0 VPositiveDefinite

0vvSvv ,0 VPositiveSemi-definite


36/46

Dual Vector of an Skew Tensor

0

0

0

12

13

23

ww

ww

ww

W

V vvwWv

3

2

1

w

w

w

w Axial vector of W


37/46

Notations

Lin = the set of all rank two tensors

Lin+= subset of Lin with positive determinants

Sym = subset of Lin that are symmetric

Skw = subset of Lin that are skew

Psym = subset of Sym that are positive definite

Orth = set of all orthogonal tensors

Orth += set of all rotations


38/46

Eigenvalues and Eigenvectors

aTa

Second rank tensors provide a linear transformation of

the vector space into itself. However, there would be

special vectors that satisfy the following condition:

are eigen values and {a} are eigen vectors

and {a} are independent of choice of coordinate frame

Multiple solutions are possible


39/46


0aIT )( aTa

0)det( IT Non-trivial Solutionexists when

0)()()( 32213 TTT iii Characteristicequation


40/46


0)()()( 322

13 TTT iii

ii

Ttri TT)(1

jiijii TTTtrtri 22222

1)]()[(

2

1)( TTT

3213 det)( kjiijk TTTi TT

)(),(),( 321 TTT iii

List of principal

invariants


41/46

Cayley-Hamilton Theorem

0ITTTTTT )()()( 322

1

3 iii

Roots of Characteristic Equation

0)()()( 322

1

3

TTT iii The characteristic equation can produce

up to three real roots.

Multiple vectors are possible for each .

Rotation Tensors have only one real root!

R t f Ch t i ti E ti


42/46

Roots of Characteristic Equation:

Symmetric Tensors

ii

i

ijiijS eeeeS

3

1

A symmetric tensor Spossesses three eigen values(1,2,3) and at least one orthonormal set of eigen

vectors (e1,e2,e3) associated with (1,2,3) respectively.

bSbaSaSS T 0 ababaSbSabba

ijjjjijiijS eeSee


43/46

Symmetric Tensors

ii

i

i eeS 3

1

)(

)(

)(

3213

1332212

3211

S

S

S

i

i

i

3

2

1

00

00

00

iS

e


44/46

Eigen Values and Eigen Vectors

0,3

1

iii

i

i eeC

A positive definite symmetric tensor Cpossessesthree positive eigen values (1,2,3).

Psymiii

i

UeeCU ,3

1

ii

i

i eeC

3

1

11


45/46

Skew Tensors

V vvwWv

0)(1 WW tri

2222 )]()[(

21)( wWWW trtri

0det)(3 WWi

Whas only one real eigenvalue, which is

equal to zero


46/46

Polar Decomposition

1det,

,,

RIRRRR

VU,FFVFFU

VRRUFF

TT

TT Psym

Lin

3

1

113

1

3

1

3

1

3

1

,,

,

i

iii

i

iii

i

ii

ii

i

iii

i

i

lrFrlFrlR

llVrrU

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