Slides Chapter 1 Mathematical Preliminaries

7/23/2019 Slides Chapter 1 Mathematical Preliminaries

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Common Variable Types in Elasticity

Elasticity theory is a mathematical model of material deformation.Using principles of continuum mechanics, it is formulated in terms ofmany dierent types of eld variables specied at spatial points in thebody under study. Some examples include:

Scalars - Single magnitude

mass density ρ, temperature T , modulus of elasticity E, . . .

Vectors – Three components in three dimensions displacement vector

Matrices – ine components in three dimensions

stress matrix

Other – !ariables "ith more than nine components

Chapter 1 Mathematical Preliminaries

, e#, e$, e% are unit basis

vectors

Elasticity Theory, Applications and Numerics

M.H. Sadd , University of Rhode Island

321eeeu wvu ++=

στττστττσ

=σ

z zy zx

yz y yx

xz xy x

][



IndexTensor !otation&ith the "ide variety of variables, elasticity formulation ma'esuse of a tensor formalism using index notation. This enables

e(cient representation of all variables and governing e)uationsusing a single standardi*ed method.+ndex notation is a shorthand scheme"hereby a "hole set of numbers orcomponents can be represented by asingle symbol "ith subscripts+n general a symbol aij…k "ith N distinct indices represents %N

distinct numbersddition, subtraction, multiplication and e)uality of indexsymbols are dened in the normal fashion- e.g.

Elasticity Theory, Applications and Numerics

M.H. Sadd , University of Rhode Island

=

=

333231

232221

131211

3

2

1

,

aaa

aaa

aaa

a

a

a

a

a iji

±±±±±±±±±

=±

±±±

=±

333332323131

232322222121

131312121111

33

22

11

,

bababa

bababa

bababa

ba

ba

ba

ba

ba ijijii

λλλλλλλλλ

=λ

λλλ

=λ

333231

232221

131211

3

2

1

,

aaa

aaa

aaa

a

a

a

a

a iji

=

332313

322212

312111

bababa

bababa

bababa

ba ji



!otation "#les and $e%nitions

S#mmation Con&ention - if a subscript appears t"ice in

the same term, summation over that subscript from one tothree is implied- for example

symbol aij…m…n…k is said to be symmetric "ith respect toindex pair mn if

symbol aij…m…n…k is said to be antisymmetric "ith respect to

index pair mn if

Useful +dentity

. . . symmetric . . . antisymmetric

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

332211

3

1

332211

3

1

bababababa

aaaaa

iii

j

jij jij

i

iiii

++==

++==

∑

∑

=

=

k mnijk nmij aa .................. =

k mnijk nmij aa .................. −=

][)()(2

1)(

2

1ijij jiij jiijij aaaaaaa +=−++=

)(2

1)( jiijij aaa += )(

2

1][ jiijij aaa −=



The matrix aij and vector b

i are specified by

Determine the following quantities:

Indicate whether they are a scalar, vector or matrix.

Following the standard definitions given in section .!,

Example 1-1: Index Notation Examples


=

=

0

4

2

,

212

340

021

iij ba

][)( ,,,,,,,, ijij jiii jiij jij jk ijijijii aabbbbbbabaaaaaa

(matrix)

000

0168

084

(scalar)200164

(scalar)84

(vector)

8

16

10

(matrix)

7106

18196

6101

(scalar)394149160041

(scalar)7

332211

1221311321121111

332211

332211

333332323131232322222121131312121111

332211

=

=++=++==++++=

=++=

=++=

=++++++++=

++++++++==++=

ji

ii

jiij

iii jij

k ik ik i jk ij

ijij

ii

bb

bbbbbbbb

bbabbabbabbabba

babababa

aaaaaaaa

aaaaaaaaaaaaaaaaaaaa

aaaa

( )

( ) (matrix)

011

101

110

230

142

201

2

1

212

340

021

2

1

2

1

(matrix)

221

241

111

230

142

201

2

1

212

340

021

2

1

2

1

][

)(

−−

−=

−

=−=

=

+

=+=

jiijij

jiijij

aaa

aaa



Special Index Symbols

'ronec(er $elta

roperties:

)lternatin* or Perm#tation Symbol

ε#$% / ε$%# / ε%#$ / #, ε%$# / ε#%$ / ε$#% / 0#, ε##$ / ε#%# /

ε$$$ / . . . / 1Useful in evaluatingdeterminants and vector cross0products


=

≠

==δ

100

010

001

,0

)(,1

jiif

sumno jiif ij

3,

,

,

1,3

=δδ=δ

=δ=δ

=δ=δ

=δ=δ

δ=δ

ijijiiijij

ijik jk ik jk ij

jiiji jij

iiii

jiij

aa

aaaa

aaaa

−+

=εotherwise,0

3,2,1of n erm!tatioo""anisif ,1

3,2,1of n erm!tatioevenanisif ,1

ijk

ijk

ijk

321321

333231

232221

131211

##]"et[ k jiij k k jiijk ijij aaaaaa

aaa

aaa

aaa

aa ε=ε===



Coordinate Trans+ormations

&

e′

#

e′

%

e′

$

e%

e$e#

x 3

x #

x $

x ′

#

x ′

$

x ′

3 To express elasticity variables in

dierent coordinate systems re)uiresdevelopment of transformation rulesfor scalar, vector, matrix and higherorder variables 2 a concept connected"ith basic denitions of tensorvariables. The t"o 3artesian frames4 x #,x $,x %5 and dier only

by orientationUsing 6otation7atrix

transformationla"s for3artesian vectorcomponents


),,( 321 x x x ′′′

),cos( jiij x xQ ′=

3332321313

3232221212

3132121111

eeee

eeee

eeee

QQQ

QQQ

QQQ

++=′

++=′++=′

jiji Q ee =′

j jii Q ee ′=

ii

ii

vvvv

vvvv

eeee

eeeev

′′=′′+′′+′′==++=

332211

332211 jiji vQv =′

j jii vQv ′=



Cartesian Tensorseneral Trans+ormation a.sScalars, vectors, matrices, and higher order )uantities can berepresented by an index notational scheme, and thus all)uantities may then be referred to as tensors of dierent orders. The transformation properties of a vector can be used toestablish the general transformation properties of these tensors.

6estricting the transformations to those only bet"een 3artesiancoordinate systems, the general set of transformation relationsfor various orders are:


or"er %eneral

or"er fo!rth,

or"er thir",

(matrix)or"er secon",

(vector)or"er first,

(scalar)or"er &ero,

...... t pqr mt kr jqipmijk

pqrslskr jqipijkl

pqr kr jqipijk

pq jqipij

pipi

aQQQQa

aQQQQa

aQQQa

aQQa

aQa

aa

⋅⋅⋅=′

=′

=′

=′

=′=′



Example 1-2 Transformation ExamplesThe components of a first and second order tensor in a particular coordinate frame are given by

8etermine the components of each tensor in a ne"coordinate system found through a rotation of 91o 4π9radians5 about the x %0axis. 3hoose a countercloc'"ise

rotation "hen vie"ing do"n the negative x %0axis, see

;igure #0$. The original and primed coordinate systems are sho"n in;igure #0$. The solution starts by determining therotation matrix for this case

The transformation for the vector )uantity follo"s from e)uation4#.<.#5$

and the second order tensor 4matrix5 transformsaccording to 4#.<.#5%

x 3

x #

x $

x ′

#

x ′

$

x ′

3

91o


=

=

423

220

301

,

2

4

1

iji aa

−=

°°°

°°°°°°

=

100

02'12'302'32'1

0cos90cos90cos

90cos60cos10cos90cos30cos60cos

ijQ

−+

=

−==′

2

2'32

322'1

2

4

1

100

02'12'3

02'32'1

jiji aQa

−+−

+=

−

−==′

42'33132'3

2'3314'4'3

32'34'34'7

100

02'12'3

02'32'1

423

220

301

100

02'12'3

02'32'1T

pq jqipij aQQa



Principal Val#es and$irections +or Symmetric

Second Order Tensors The direction determined by unit vector n is said to be a principaldirection or eigenvector of the symmetric second order tensor aij

if there exists a parameter λ 4 principal value or eigenvalue5 suchthat

6elation is a homogeneous system of three linear algebraic

e)uations in the un'no"ns n#, n$, n%. The system possessesnontrivial solution if and only if determinant of coe(cient matrixvanishes

scalars Ia, IIa and IIIa are called the fundamental invariants of

the tensor aij


i jij nna λ= 0)( =λδ− jijij na

0]"et[ 23 =+λ−λ+λ−=λδ− aaaijij III II I a

]"et[

)(2

1

3331

1311

3332

2322

2221

1211

332211

ija

ijij jjiia

iia

a III

aa

aa

aa

aa

aa

aaaaaa II

aaaa I

=

++=−=

++==



Principal )xes o+ Second Order Tensors

+t is al"ays possible to identify a right0handed 3artesian

coordinate system such that each axes lie along principaldirections of any given symmetric second order tensor. Suchaxes are called the principal axes of the tensor, and the basisvectors are the principal directions =n4#5, n4$5 , n4%5>

x 3

x 1

x /

Principal )xesOri*inal i&en )xes


λλ

λ=

3

2

1

00

00

00

ija

=

333231

232221

131211

aaa

aaaaaa

aij



Example 1-0 Principal Val#eProblem

8etermine the invariants, and principal values anddirections of

;irst determine the principal invariants

The characteristic e)uation then becomes

Thus for this case all principal values aredistinct;or the λ# / < root, e)uation 4#.9.#5 gives the

system

"hich gives a normali*ed solution

+n similar fashion the other t"o principal directions arefound to be+t is easily veried that these directions are mutually orthogonal.ote for this case, the transformation matrix Qij dened by 4#.?.#5

becomes


−

=

340

430

002

ija

0)169(2

340

430

002

262630

02

34

43

30

02

,2332

−=−−=−

=

−=−−=−+−+==−+==

a

aiia

III

II a I

,2,

0)2)(2(0022]"et[

321

223

−=λ=λ=λ∴

=−λ−λ⇒=−λ+λ+λ−=λδ− ijija

084

042

03

)1(

3

)1(

2

)1(

3

)1(

2

)1(

1

=−

=+−

=−

nn

nn

n

)2(

132

)1(een +±=

1)2( en ±= )2(

132)3( een −±=

⇒

−

=

'2'10

001

'1'20

ijQ

−=′

00

020

00

ija



Vector, Matrix and Tensor )l*ebra

Scalar or $ot Prod#ct

Vector or Cross Prod#ct

Common Matrix Prod#cts


iibabababa =++=⋅ 332211ba

ik jijk ba

bbb

aaa e

eee

ba ε==×

321

321

321

iijiji

T

ij j jij

a A Aa

Aaa A

===

===

][*

]*[

Aa Aa

a A Aa

T

ijij

jiij

jk ji

kjij

jk ij

B Atr tr

B Atr

B A B A

B A

==

=

==

==

)()(

)(

]][[

B A AB

AB

B A AB

B A AB

T T

T

T

T QaQa =′

⇒=′ pq jqipij aQQa

@a"tion Transforma

ArderSecond



Calc#l#s o+ CartesianTensors

;ield concept for tensor components

Comma notation for partial dierentiation

+f dierentiation index is distinct, order of the tensor "ill beincreased by one- e.g. derivative operation on a vector producesa second order tensor or matrix


)()(),,(

)()(),,(

)()(),,(

321

321

321

x

x

x

ijiijijij

iiiii

i

a xa x x xaa

a xa x x xaa

a xa x x xaa

===

===

===

,,, ,,, ij

k

k iji

j

ji

i

i a x

aa x

aa x

a∂∂

=∂∂

=∂∂

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂ ∂

∂

∂

∂

∂

∂

=

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

,

x

a

x

a

x

a

x

a

x

a

x

a x

a

x

a

x

a

a ji



Vector $ierentialOperations

$irectional $eri&ati&e o+ Scalar 2ield

Common $ierential Operations


f ds

dz

z

f

ds

dy

y

f

ds

dx

x

f

ds

df

∇⋅=∂

∂

+∂

∂

+∂

∂

= n

321 eeends

dz

ds

dy

ds

dx s ++== of directioninvectornormalunit

z y x ∂∂

+∂∂

+∂∂

==∇ 321 eeeoperatoraldierentivector

z

f

y

f

x

f f f f ∂

∂+∂

∂+∂

∂===∇ 321 eee grad functionscalarof gradient

i,

2

i,

,

,

2

+i,

i

e!

e!

!

ee!

e,

kk i

jk ijk

ii

ii

ji

i

uVector aof Laplacian

uVector aof url

uVector aof !iver"ence#calar aof Laplacian

uVector aof $radient

#calar aof $radient

=∇

ε=×∇

=⋅∇ φ=φ∇⋅∇=φ∇

=∇φ=φ∇



Example 1-34 ScalarVector 2ieldExample

Scalar and vector eld functions are given by

3alculate the follo"ing expressions, ∇φ, ∇$φ,∇

B u,

∇u, ∇ × u.

Using the basic relations:

∇φ∇$φ

∇ B u

∇u

∇ × u

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10

3ontours φ/constant and vector distributions of∇φ

vector eld ∇φ is orthogonal to φ0contours 4ture in genera

x

y

0 4satises @aplace e)uation5


22 y x −=φ 321 eeeu xy yz x ++= 32

21

321

21

ee

eee

ee

y y x

xy yz x

z y x

x y

y z u

z z

y x

ji

−−=∂∂∂∂∂∂=

==

+=++=

=−=

−=

)3(

32

'''

0

330

00232032

022

22

,



VectorTensor Inte*ral

Calc#l#s$i&er*ence Theorem

Sto(es Theorem

reen5s Theorem in the Plane

6ero-Val#e Theorem


dV d# # V ∫∫ ∫∫∫ ⋅∇=⋅ unu dV ad# na

# V k k ijk k ij∫∫ ∫∫∫ = ,......

∫∫ ∫ ⋅×∇=⋅#

d# nudr uC

)( ∫∫ ∫ ε=#

r sk ijrst t k ij d# nadxa C

,......

V f dV f k ijV

k ij ∈=⇒=∫∫∫ 00 ......

∫ ∫∫ +=

∂∂

−∂∂

# "dy fdxdxdy

y

f

x

" )( ∫ ∫∫ ∫ ∫∫ =

∂∂

=∂∂

y

# x

# ds fndxdy

y

f ds "ndxdy

x

" ,



Ortho*onal C#r&ilinearCoordinate Systems

Cylindrical CoordinateSystem 7r, , z)

Spherical CoordinateSystem 7R,

,

8

e

0

e

/

e

1

x

3

x

1

x

/

R


,tan

cos

cos,sinsin,sincos

1

21

23

22

21

31

2

3

2

2

2

1

321

x

x

x x x

x x x x %

% x % x % x

−

−

=θ

++=φ

++=

φ=φθ=φθ=

3

1

212

2

2

1

321

,tan,

,sin,cos

x z x

x x xr

z xr xr x

==θ+=

=θ=θ=

−

%e

φe

θe



eneral C#r&ilinearCoordinate Systems

Common $ierential 2orms


233

222

211

2 )()()()( ξ+ξ+ξ= d &d &d &ds

iii &&&& ξ∂

∂=

ξ∂∂

+ξ∂∂

+ξ∂∂

=∇ ∑ 1

1

1

1

33

322

211

1 i eeee

i

ii

f

&

f

&

f

&

f

& f

ξ∂∂

=ξ∂

∂+

ξ∂∂

+ξ∂

∂= ∑ 1

1

1

1

3

3

32

2

21

1

1 i eeee

∑

ξ∂

∂=⋅ ><

i

i

i

i u

&

&&&

&&&

321

321

1u

∑

ξ∂φ∂

ξ∂∂

=φ∇i

i

i

i &

&&&

&&& 2

321

321

2

)(

1

∑∑∑ ><ξ∂∂ε

=×i

i

j k

k k j

k j

ijk &u

&&eu )(

∑∑

ξ∂

∂

+ξ∂

∂

= ><

><

i j ji

j

iu

u

& i

j

j

i e

e

e

u

∇

ξ∂

∂+

ξ∂

∂⋅

ξ∂∂

=∇ ∑∑∑ ><><

j k

jk

j

k ii

i

uu

&& k

j

j

k i e

eee

u

2

),,(,),,( 321321 ξξξ=ξ=ξ mmmm x x x x x



Example 1-94 PolarCoordinates

From relations ".#.$% or simply using the geometry shown in Figure

The basic vector differential operations then follow to be

Elasticity Theory, Applications and NumericsM H Sadd University of Rhode Island

r &&rd dr ds ==⇒θ+= 21

222 ,1)()()(

21

21

cossinsincos

eee

eee

θ+θ−=θ+θ=

θ

r 0

,

,

=∂∂=

∂∂−=

θ∂∂=

θ∂∂ θθ

θr r

r r

r eee

ee

e

θθ

θθ

θθθ

θθθθ

θ

θ

θ

θ

−

θ∂∂

+∇+

−

θ∂∂

−∇=∇

−

θ∂∂+

−

θ∂∂+

∂∂+

∂∂=

θ∂∂

−∂∂

=×

θ∂φ∂

+

∂φ∂

∂∂

=φ∇

θ∂∂

+∂∂

=⋅θ∂

φ∂+

∂

φ∂=φ

θ∂∂

+∂∂

=

eeu

eeeeeeeeu

eu

u

ee

ee

r

r r r r

2

2

2

11

1

)(1

11

1)(

1

1

1

22

2

22

22

2

2

2

r

uu

r u

r

uu

r u

uur

uur r

ur

u

u

r ru

r r

r r r

r r

u

r ru

r r

r r

r r

r r r

r r r

z r

r

r

r

∇

∇

∇

∇

∇

r!reeeeeu , where ×=+= θuur

Documents

Slides Chapter 1 Mathematical Preliminaries