Common Variable Types in Elasticity

Common Variable Types in ElasticityElasticity theory is a mathematical model of material deformation. Using principles of continuum mechanics, it is formulated in terms of many different types of field variables specified at spatial points in the body under study. Some examples include:

Scalars - Single magnitude mass density , temperature T, modulus of elasticity E, . . .

Vectors – Three components in three dimensions displacement vector

Matrices – Nine components in three dimensions stress matrix

Other – Variables with more than nine components

Chapter 1 Mathematical Preliminaries

321 eeeu wvu

zzyzx

yzyyx

xzxyx

][

, e1, e2, e3 are unit basis vectors

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

Index/Tensor NotationWith the wide variety of variables, elasticity formulation makes use of a tensor formalism using index notation. This enables efficient representation of all variables and governing equations using a single standardized method.

Index notation is a shorthand scheme whereby a whole set of numbers or components can be represented by a single symbol with subscripts

333231

232221

131211

3

2

1

,

aaa

aaa

aaa

a

a

a

a

a iji

In general a symbol aij…k with N distinct indices represents 3N distinct numbers

Addition, subtraction, multiplication and equality of index symbols are defined in the normal fashion; e.g.

333332323131

232322222121

131312121111

33

22

11

,

bababa

bababa

bababa

ba

ba

ba

ba

ba ijijii

333231

232221

131211

3

2

1

,

aaa

aaa

aaa

a

a

a

a

a iji

332313

322212

312111

bababa

bababa

bababa

ba ji


Notation Rules and DefinitionsSummation Convention - if a subscript appears twice in the same term, summation over that subscript from one to three is implied; for example

332211

3

1

332211

3

1

bababababa

aaaaa

iiij

jijjij

iiiii

A symbol aij…m…n…k is said to be symmetric with respect to index pair mn if

kmnijknmij aa ..................

A symbol aij…m…n…k is said to be antisymmetric with respect to index pair mn if

kmnijknmij aa ..................

Useful Identity ][)()(2

1)(

2

1ijijjiijjiijij aaaaaaa

)(2

1)( jiijij aaa )(

2

1][ jiijij aaa . . . symmetric . . . antisymmetric


0

4

2

,

212

340

021

iij ba

][)( ,,,,,,,, ijijjiiijiijjijjkijijijii aabbbbbbabaaaaaa

The matrix aij and vector bi are specified by

Determine the following quantities:

Indicate whether they are a scalar, vector or matrix.

Following the standard definitions given in section 1.2,

(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

iiijij

kikikijkij

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

Example 1-1: Index Notation Examples


Special Index Symbols

Kronecker Delta

100

010

001

,0

)(,1

jiif

sumnojiifij

3,

,

,

1,3

ijijiiijij

ijikjkikjkij

jiijijij

iiii

jiij

aa

aaaa

aaaaProperties:

Alternating or Permutation Symbol

otherwise,0

3,2,1ofnpermutatiooddanisif,1

3,2,1ofnpermutatioevenanisif,1

ijk

ijk

ijk

123 = 231 = 312 = 1, 321 = 132 = 213 = -1, 112 = 131 = 222 = . . . = 0

Useful in evaluating determinants and vector cross-products 321321

333231

232221

131211

||]det[ kjiijkkjiijkijij aaaaaa

aaa

aaa

aaa

aa


Coordinate Transformations

v

e

1

e

3e

2

e3

e2e1

x3

x1

x2

x1

x2

x3 To express elasticity variables in different coordinate systems requires development of transformation rules for scalar, vector, matrix and higher order variables – a concept connected with basic definitions of tensor variables. The two Cartesian frames (x1,x2,x3) and differ only by orientation),,( 321 xxx

),cos( jiij xxQ Using Rotation Matrix

3332321313

3232221212

3132121111

eeee

eeee

eeee

QQQ

QQQ

QQQ

jiji Q ee

jjii Q ee

ii

ii

vvvv

vvvv

eeee

eeeev

332211

332211 jiji vQv

jjii vQv

transformation laws for Cartesian vector components


Cartesian TensorsGeneral Transformation Laws

Scalars, vectors, matrices, and higher order quantities can be represented by an index notational scheme, and thus all quantities may then be referred to as tensors of different orders. The transformation properties of a vector can be used to establish the general transformation properties of these tensors. Restricting the transformations to those only between Cartesian coordinate systems, the general set of transformation relations for various orders are:

order general

order fourth,

orderthird,

(matrix)ordersecond,

(vector)orderfirst,

(scalar)orderzero,

...... tpqrmtkrjqipmijk

pqrslskrjqipijkl

pqrkrjqipijk

pqjqipij

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

423

220

301

,

2

4

1

iji aa

Determine the components of each tensor in a new coordinate system found through a rotation of 60o (/6 radians) about the x3-axis. Choose a counterclockwise rotation when viewing down the negative x3-axis, see Figure 1-2.

The original and primed coordinate systems are shown in Figure 1-2. The solution starts by determining the rotation matrix for this case

100

02/12/3

02/32/1

0cos90cos90cos

90cos60cos150cos

90cos30cos60cos

ijQ

The transformation for the vector quantity follows from equation (1.5.1)2

2

2/32

322/1

2

4

1

100

02/12/3

02/32/1

jiji aQa

and the second order tensor (matrix) transforms according to (1.5.1)3

42/33132/3

2/3314/54/3

32/34/34/7

100

02/12/3

02/32/1

423

220

301

100

02/12/3

02/32/1T

pqjqipij aQQa

x3

x1

x2

x1

x2

x3

60o


Principal Values and Directions for Symmetric Second Order Tensors

ijij nna

The direction determined by unit vector n is said to be a principal direction or eigenvector of the symmetric second order tensor aij if there exists a parameter (principal value or eigenvalue) such that

0)( jijij na

Relation is a homogeneous system of three linear algebraic equations in the unknowns n1, n2, n3. The system possesses nontrivial solution if and only if determinant of coefficient matrix vanishes

0]det[ 23 aaaijij IIIIIIa

]det[

)(2

1

3331

1311

3332

2322

2221

1211

332211

ija

ijijjjiia

iia

aIII

aa

aa

aa

aa

aa

aaaaaaII

aaaaI

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


Principal Axes of Second Order Tensors

3

2

1

00

00

00

ija

It is always possible to identify a right-handed Cartesian coordinate system such that each axes lie along principal directions of any given symmetric second order tensor. Such axes are called the principal axes of the tensor, and the basis vectors are the principal directions {n(1), n(2) , n(3)}

333231

232221

131211

aaa

aaa

aaa

aij

x3

x1

x2

Principal Axes Original Given Axes

n(1)

x1

x2

x3

n(3) n(2)


Example 1-3 Principal Value Problem

Determine the invariants, and principal values and directions of

First determine the principal invariants

340

430

002

ija

50)169(2

340

430

002

25625630

02

34

43

30

02,2332

a

aiia

III

IIaI

The characteristic equation then becomes

5,2,5

0)25)(2(050252]det[

321

223

ijija

Thus for this case all principal values are distinct For the 1 = 5 root, equation (1.6.1) gives the system

084

042

03

)1(3

)1(2

)1(3

)1(2

)1(1

nn

nn

n

which gives a normalized solution )2(5

132

)1( een

In similar fashion the other two principal directions are found to be 1)2( en )2(

5

132

)3( een It is easily verified that these directions are mutually orthogonal. Note for this case, the transformation matrix Qij defined by (1.4.1) becomes

5/25/10

001

5/15/20

ijQ

500

020

005

ija


Vector, Matrix and Tensor Algebra

iibabababa 332211ba

ikjijk ba

bbb

aaa e

eee

ba

321

321

321

iijijiT

ijjjij

aAAa

AaaA

][}{

}]{[

AaAa

aAAaT

ijij

jiij

jkji

kjij

jkij

BAtrtr

BAtr

BA

BA

BA

)()(

)(

]][[

BAAB

AB

BA

AB

BAAB

TT

T

T

Scalar or Dot Product

Vector or Cross Product

Common Matrix Products

TQaQa

pqjqipij aQQa

Law tionTransforma

OrderSecond


Calculus of Cartesian TensorsField concept for tensor components

)()(),,(

)()(),,(

)()(),,(

321

321

321

x

x

x

ijiijijij

iiiii

i

axaxxxaa

axaxxxaa

axaxxxaa

Comma notation for partial differentiation ,,, ,,, ijk

kijij

jii

i ax

aax

aax

a

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

,

x

a

x

a

x

ax

a

x

a

x

ax

a

x

a

x

a

a ji

If differentiation index is distinct, order of the tensor will be increased by one; e.g. derivative operation on a vector produces a second order tensor or matrix


Vector Differential Operations

Directional Derivative of Scalar Field fds

dz

z

f

ds

dy

y

f

ds

dx

x

f

ds

df

n

321 eeends

dz

ds

dy

ds

dxs of direction in vector normal unit

zyx

321 eee operator aldifferenti vector

z

f

y

f

x

ffff

321 eeegrad function scalar of gradient

Common Differential Operations

i,2

i,

,

,2

ji,

i

eu

eu

u

eeu

e,

kki

jkijk

ii

ii

ji

i

uVectoraofLaplacian

uVectoraofCurl

uVectoraofDivergence

ScalaraofLaplacian

uVectoraofGradient

ScalaraofGradient


Example 1-4: Scalar/Vector Field ExampleScalar and vector field functions are given by 22 yx 321 eeeu xyyzx 32

Calculate the following expressions, , 2, ∙ u, u, u.

Using the basic relations:

21

321

21

ee

eee

ee

yyx

xyyzx

zyx

xy

yzu

zz

yx

ji

)3(

32

///

0

330

002

32032

022

22

,

2

∙ u

u

u

Gradient Vector Distribution

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Contours =constant and vector distributions of vector field is orthogonal to -contours (ture in general )

x

y

- (satisfies Laplace equation)


Vector/Tensor Integral Calculus

Divergence Theorem

Stokes Theorem

Green’s Theorem in the Plane

Zero-Value Theorem

dVdSS V u nu dVadSna

S V kkijkkij ,......

S

dS nudruC

)( S rskijrsttkij dSnadxa

C ,......

VfdVf kijV kij 00 ......

CSgdyfdxdxdy

y

f

x

g)(

C ySC xSdsfndxdy

y

fdsgndxdy

x

g,


Orthogonal Curvilinear Coordinate Systems

Cylindrical Coordinate System (r,,z) Spherical Coordinate System (R,,)

e3

e2 e1

x3

x1

x2

r

z

re

ze

e

,tan

cos

cos,sinsin,sincos

1

21

23

22

21

31

23

22

21

321

x

x

xxx

x

xxxR

RxRxRx

31

2122

21

321

,tan,

,sin,cos

xzx

xxxr

zxrxrx

e3

e2e1

x3

x1

x2

R

Re

e

e


General Curvilinear Coordinate Systems

233

222

211

2 )()()()( dhdhdhds

iii hhhh

1ˆ

1ˆ

1ˆ

1ˆ

33

322

211

1 ieeee

iii

f

h

f

h

f

h

f

hf

1ˆ

1ˆ

1ˆ

1ˆ

33

322

211

1 ieeee

i

ii

iu

h

hhh

hhh321

321

1u

i

ii

i h

hhh

hhh 2321

321

2

)(

1

i

ij k

kkjkj

ijk huhh

eu ˆ)(

i jji

j

i

uu

h i

jj

ie

ee

uˆ

ˆˆ

j kjk

j

kii

i

uu

hh k

jj

kie

eee

uˆ

ˆˆˆ2

),,(,),,( 321321 mmmm xxxxx

e3

e2 e1

x3

x1

x2

1

1e

3e

2e

3

2

Common Differential Forms


Example 1-5: Polar Coordinates

rhhrddrds 21222 ,1)()()(

From relations (1.9.5) or simply using the geometry shown in Figure

21

21

cossinˆ

sincosˆ

eee

eee

r0

ˆˆ,ˆ

ˆ,ˆ

ˆ

rrr

rr ee

ee

ee

The basic vector differential operations then follow to be

eeu

eeeeeeeeu

eu

u

ee

ee

r

rrrr

2

ˆ2

ˆ2

ˆˆ1

ˆˆ1

ˆˆˆˆ

ˆ1

)(1

11

1)(

1

1ˆˆ

1ˆˆ

222

2222

2

2

2

r

uu

ru

r

uu

ru

uu

ru

u

rr

u

r

u

u

rru

rr

rrr

rr

u

rru

rr

rr

rr

rrr

rrr

zr

r

r

r

θrzθr eeeeeu ˆˆˆ,ˆˆ where uur

e2

e1 x1

x2

r

e

re


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Common Variable Types in Elasticity