strain analysis - imechanica analysis.pdf · strain analysis. r&de (engineers), drdo summary...

R&DE (Engineers), DRDO

rd_

mec

h@

yah

oo

.co

.in

Ramadas Chennamsetti

Strain Analysis


Summary

Deformation gradient

Green-Lagrange strain

Almansi strain

Rotation vector

rd_

mec

h@

yah

oo

.co

.in


2

Rotation vector

Strain transformation

Principal strains

Strain decomposition

Strain compatibility


Introduction

The effect of forces applied to a body – Newton’s second law – Stress analysis

Applied forces => deformations Concerned with study of deformations – geometric

problem – no material properties

rd_

mec

h@

yah

oo

.co

.in


3

problem – no material properties Strains => kinematics of material deformation Change in relative position of any two points in a

continuous body => deformed or strained Distance remains constant – rigid body motion

Rigid body => Translations & Rotations Continuum mechanics approach


Introduction

Deformation – consists of relative motion between particles and rigid body motion

rd_

mec

h@

yah

oo

.co

.in


4

AA’

x

y

A”

Change in position


Introduction

Description of motion Body undergoes macroscopic changes within the body ‘Deformation’

Initial positionFinal position

Particle P occupies (X1, X2, X3)

rd_

mec

h@

yah

oo

.co

.in


5

PP’

X1

x(X1, X2)

X2

O

X(X1, X2)

u(X1, X2, t)

x1

x2

Body ‘B’

Initial position

Co-ordinate system refers to material particles in the body => Material co-ordinate system

Material point P occupies P’ in final config

Deformation csys x1x2x3 => Spatial co-ordinate system


Displacement

Displacement in a body

Application of force

Initial state of body

Mapping from reference csys to deformed csys - function

rd_

mec

h@

yah

oo

.co

.in


6

( )321~~

,, XXXXOA=

AA’

X1

x(X1, X2)

X2

O

X(X1, X2)

u(X1, X2, t)

( )321~~

,,' XXXxOA = ( )321~~

' ,, XXXUAA =

~~~~~~

' ' UXxAAOAOA +==>+=

Initial state of body

Final state of body


Mapping

Material co-ordinate system – spatial co-ordinate system

( )~~

':

Xx

PP

χχ

==>→

rd_

mec

h@

yah

oo

.co

.in


7

( )~~

Substitute material co-ordinates of the particle in mapping function => Final position of the particle

Existence of inverse mapping

)(~

1

~xX −= χ

Substitute spatial co-ordinates => Material co-ordinates – which material occupies the spatial position


Mapping

Variable – velocity, density etc vector / scalar( )

( )( )

Xx

tX

:Mapping

csys materialin Expressed ,~

χ

φφ

=

=

rd_

mec

h@

yah

oo

.co

.in


8

( )( )

( ) ( )( )( )tx

txtX

xX

Xx

,

,,

:Mapping

~

~

1

~

~

1

~

~~

φφ

χφφφ

χ

χ

==>

===>

=

=

−

−

Expressed in spatial co-ordinate system


Displacement

In indices notation

This relates the current deformed geometry with initial undeformed geometry of a point

( )321~~~

,, XXXuXxUXx iii +==>+=

rd_

mec

h@

yah

oo

.co

.in


9

initial undeformed geometry of a point

Displacement field => movement of each point in initial configuration to final configuration

U => rigid body motion and deformation in the body

Deformation – important


Normal strain

Body subjected to loads –AB => A’B’

AB // A’B’ & AB = A’B’

⇒Rigid body translation

rd_

mec

h@

yah

oo

.co

.in


10

A B

A’ B’AB // A’B’ & AB = A’B’

⇒Rigid body rotation

AB // A’B’ & AB ≠ A’B’

⇒Deformation / normal Strain


Shear strain

Shear strain => distortion

C’

Change in angle between two lines => shear strain

rd_

mec

h@

yah

oo

.co

.in


11

A B

A’ B’C

C’

θo

θ

=> shear strain

No change in angle => shear strain = 0

Change in angle –measure of shear strain / distortion


Displacement

General motion of a solid –

y yy

rd_

mec

h@

yah

oo

.co

.in


12x

y

Undeformed statex

y

Rigid body rotation

x

y

Horizontal extensionx

y

Vertical extensionx

y

Shear deformation

Rigid body motion doesn’t effect strain field – no effect on stresses


Strain

Strain – measure of displacement of a body

Various measures of strain - definition of strain is not unique – equally valid definition of strain

Engineering normal strain – change in length/original length

rd_

mec

h@

yah

oo

.co

.in


13

length/original length

Green-Lagrange strain

( ) ( )( )2

22

~~ length initial

length initiallength final −=E


Green – Lagrange strain


X2, x2 ( )321 ,,' xxxP

( )332211 ,,' dxxdxxdxxQ +++

I

II

rd_

mec

h@

yah

oo

.co

.in


14

( )321 ,, XXXP

X1, x1

X3, x3 ( )332211 ,, dXXdXXdXXQ +++

o

I

X i – undeformed configuration, xi – deformed configuration



PQ = dS, P’Q’ = ds

222

~~~~~~

2 ...

dSdSdSdSdSdSPQ

eedSdSedSedSdSdSPQ

iiiijji

jijijjii

=====>

===

δ

rd_

mec

h@

yah

oo

.co

.in


15

Similarly, 222'' dsdsQP i ==

Green – Lagrange strain (E) is defined as,

~~~~

22

~~~~

.....2 TT dXdXdxdxdSdsdXEdX −=−=

~~~dUdXdx += ~

33~

22~

11~

edXedXedXdX ++=

~33

~22

~11

~edxedxedxdx ++=



dU – infinitesimal displacement function of (X1, X2, X3)

( )321~~

vector a isit

,,

edueduedudU

XXXdUdU

++==>

=

rd_

mec

h@

yah

oo

.co

.in


16

( )321

~33

~22

~1

~1

,,

vector a isit

XXXdudu

edueduedudU

ii =

++==>

Use chain rule

~3

32

21

1

dXudXX

udX

X

udX

X

udu i

iiii •∇=

∂∂+

∂∂+

∂∂=



Writing in vector form

( )ii

iiii

eeedXu

eedXu

edU

edXuedudU

•∂=

•∂=

•∇==~~~~

rd_

mec

h@

yah

oo

.co

.in


17

jj

ii

iiijj

iijkk

j

i

ikjkj

iikk

j

ij

dXX

udu

eduedXX

uedX

X

udU

eeedXx

eedXx

edU

∂∂==>

=∂∂=

∂∂==>

•

∂=

•

∂=

~~~~

~~~~~~~

δ



The new position is,iii dudXdx +=

use jj

ii dX

X

udu

∂∂=

∂∂+= dXX

udXdx j

iii

rd_

mec

h@

yah

oo

.co

.in


18

∇+•==>

∂∂+=

∂∂+==>

∂+=

~~~~~

UIdXdx

X

udXdX

X

udXdx

dXX

dXdx

j

iijjj

j

ijiji

jj

ii

δδ



Green-Lagrange strain is defined as,

~~~~~~~~~~~

~~~~~~~~

.......2

.....2

T

T

T

TTT

dXdXUIdXUIdXdXEdX

dXdXdxdxdXEdX

−

∇+

∇+=

−=

rd_

mec

h@

yah

oo

.co

.in


19

( )~~~~

~~

~~~~~~~~~~~

~~~

~~~~~~~

~~~~

~~~

~~~~~~~

~~~~

~~~

.2

1

.......2

.......2

TT

TTTTT

TTTT

UUUUE

dXdXdXUUUUIdXdXEdX

dXdXdXUIUIdXdXEdX

∇∇+∇+∇=

−

∇∇+∇+∇+=

−

∇+

∇+=



∂∂+

∂∂+

∂∂+

∂∂=

222

2

1

X

w

X

v

X

u

X

uXε

Green-Lagrange Strain – Normal strains

rd_

mec

h@

yah

oo

.co

.in


20

∂∂+

∂∂+

∂∂+

∂∂=

222

2

1

Y

w

Y

v

Y

u

Y

vYε

∂∂+

∂∂+

∂∂+

∂∂=

222

2

1

Z

w

Z

v

Z

u

Z

wZε


∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂+

∂∂=

Y

w

X

w

Y

v

X

v

Y

u

X

u

X

v

Y

uXYγ


Green-Lagrange Strain – Shear strains

rd_

mec

h@

yah

oo

.co

.in


21

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂+

∂∂=

Y

w

Z

w

Y

v

Z

v

Y

u

Z

u

Z

v

Y

wYZγ

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂+

∂∂=

Z

w

X

w

Z

v

X

v

Z

u

X

u

X

w

Z

uXZγ



Green-Lagrange strain captures finite deflections, strains and rotations – non-linear terms

Infinitesimal strains, rotations => linear terms –non-linear terms – much smaller than linear terms

∂ u

rd_

mec

h@

yah

oo

.co

.in


22

Small strains known as engineering strains Engineering strains – denoted by => εij

1<<∂∂

j

i

x

u


Engineering strain

dyy

vv

∂∂+

CD

C’D’

x

u

dx

udxx

uu

x ∂∂=

−∂∂+

=ε

vvdy

y

vv

∂−

∂∂+

rd_

mec

h@

yah

oo

.co

.in


23

dxx

uu

∂∂+

dx

dy

A B

A’(u,v) B’ dxx

uu

∂∂+

x

y

y

v

dy

vdyy

v

y ∂∂=

−∂

+=ε


Engineering strain

∂∂+ dyy

vvu,

CD

C’D’

∂∂+

∂∂+ dy

y

vvdy

y

uu , C”

D”

rd_

mec

h@

yah

oo

.co

.in


24

∂∂+ vdxx

uu ,

dx

dy

A B

CD

A’(u,v) B’

x

y

∂∂+

∂∂+ dx

x

vvdx

x

uu ,

B”

θ1

θ2


Engineering strain

Shear strain – Change in the angle Initial angle => 900

Final angle => (900 - θ1- θ2)

Change in angle => θ = θ1+ θ2

v∂ u∂

rd_

mec

h@

yah

oo

.co

.in


25

x

vx

v

dx

vdxx

vv

∂∂≈

∂∂=

−∂∂+

=

1

1tan

θ

θ

y

u

y

u

dy

udyy

uu

∂∂≈

∂∂=

−∂∂+

=

2

2tan

θ

θ

x

v

y

uxy ∂

∂+∂∂≈+= 21 θθγ


Engineering strain

The six strain components–

Normal strains –

Shear strains –x

ux ∂

∂=εy

vy ∂

∂=εz

wz ∂

∂=ε

rd_

mec

h@

yah

oo

.co

.in


26

x

v

y

uxy ∂

∂+∂∂=γ

z

v

y

wyz ∂

∂+∂∂=γ

z

u

x

wzx ∂

∂+∂∂=γ

Indices notation - ( )ijjiij uu ,,2

1 +=ε

Infinitesimal strains => Green-Lagrange strain = Engineering Strain


Engineering strain

Rigid body rotations –y

P(x, y)P’(y, -x)U

~~~

'

~~~

~~~

'

OP

OP

jxiy

jyix

UOPOP

+−=

+=

+=

Linear strains–

rd_

mec

h@

yah

oo

.co

.in


27

xO

Linear strains–

εxx= εyy = -1, γxy = 0

Non-linear strains –

εxx= εyy = 0, γxy = 0Rotated by 900

Linear strains can’t capture large rotations use non-linear strains


Engineering strain

Assuming engineering – linear strains

[ ]

xzxyx

xzxyx

γγεεεε

112

1

2

1Strain tensor

rd_

mec

h@

yah

oo

.co

.in


28

[ ]

=

=

yyzxz

yzyxy

yyzxz

yzyxy

εγγ

γεγεεεεεεε

2

1

2

12

1

2

1

εij – Tensorial strain

γij – Engineering shear strain ijij γε2

1=


Displacement gradient

Displacement gradient tensor – second order tensor

~~~

UHx

uH

j

iij ∇==>

∂∂=

In matrix form

rd_

mec

h@

yah

oo

.co

.in


29

In matrix form

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

==

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

,

x

u

x

u

x

ux

u

x

u

x

ux

u

x

u

x

u

uH jiij



Square matrix – sum of symmetric and skew symmetric matrices

( ) ( )][][][

2

1

2

1][

ASM

MMMMM TT

+=

−++=

rd_

mec

h@

yah

oo

.co

.in


30

( ) ( )][][ ],[][

2

1][ ,

2

1][

][][][

AASS

MMAMMS

ASM

TT

TT

−==

−=+=

+=

[S] – Symmetric tensor, [A] – Skew-symmetric tensor – Main diagonal terms => 0



Decomposing displacement gradient

+

∂∂+

∂∂

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂

=

∂∂∂∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=2

1

2

1

2

1

2

1

2

3

3

2

2

2

1

2

2

1

1

3

3

1

1

2

2

1

1

1

3

2

2

2

1

2

3

1

2

1

1

1

, x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

uuux

u

x

u

x

ux

u

x

u

x

u

u ji

rd_

mec

h@

yah

oo

.co

.in


31

∂∂−

∂∂

∂∂−

∂∂

∂∂−

∂∂

∂∂−

∂∂

∂∂−

∂∂

∂∂−

∂∂

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂

∂∂

∂∂

02

1

2

1

2

10

2

1

2

1

2

10

2

1

2

1

2

3

3

2

1

3

3

1

2

3

3

2

1

2

2

1

1

3

3

1

1

2

2

1

3

3

2

3

3

2

1

3

3

1

23212

3

3

2

3

1

3

321

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

ux

u

x

u

x

u

Zeros in main diagonal



Displacement gradient is sum of infinitesimal strain tensor and rotation tensor

~~

~~

~~

Ω+=

Ω+=

u

H

ε

ε

rd_

mec

h@

yah

oo

.co

.in


32

( ) ( )

( )

( ) tensorsymmetric-Skew 2

1

tensorSymmetric 2

12

1

2

1

,,

,,

,,,,,

,

=>−=Ω

=>+==>

−++=

Ω+=

ijjiij

ijjiij

ijjiijjiji

ijijji

uu

uu

uuuuu

u

ε

ε



If displacement gradient is symmetric => ui,j = uj,i

Rotation tensor becomes zero – only strain tensor exists

Displacement gradient = strain tensor

rd_

mec

h@

yah

oo

.co

.in


33

Displacement gradient = strain tensor

( )ijjiji

ijji

uuu

uu

,,,

,,

2

1 +=

=


Rotation vector

Rotation vector is defined as

×∂∂

=×∂∂=

×∇=

~~~~~

~~

2

1

2

12

1

eex

ueu

xe

U

jii

jjj

iiω

ω

rd_

mec

h@

yah

oo

.co

.in


34

∂∂−

∂∂==>==

∂∂

==>∂∂

=

∂∂

1

3

3

13

~~

~~~~~

2

1 2 1, j i, 3,k Say,

2

1

2

1

22

x

u

x

u

x

ue

x

u

xx

ijki

jkkijk

i

j

ii

ω

εωεω

ω represents components of rigid body rotations


Rotation vector

Relation between rotation vector and tensor

gradient nt Displaceme

2

1

jijij

ijki

jk

u

ex

u

Ω+=∂∂

=>

∂∂

=

ε

ω

rd_

mec

h@

yah

oo

.co

.in


35

( ) ( ) 2

1

2

1

gradient nt Displaceme

jikjijikjiijkjijik

jijii

eee

x

Ω+−=Ω+=

Ω+=∂

=>

εεω

ε

eijk – skew symmetric matrix in ‘i’, ‘j’

lmkjikjiklmkjikjik eeee Ω−==>Ω−=2

1

2

1 ωω


Rotation vector

Multiplication of two permutations

lmkjikjiklmkjikjik eeee

−=

Ω−==>Ω−=

δδδδ

ωω2

1

2

1

rd_

mec

h@

yah

oo

.co

.in


36

( ) ( )

lmklmk

mllmiljmimjljiklmk

iljmimjllmkjik

e

e

ee

Ω=−

Ω−Ω−=−Ω−=

−=

ω

δδδδω

δδδδ

2

1

2

1


Rotation vector

From shear strain

cy

uc

x

v

cy

u

x

v

x

v

y

uxy

−=∂∂=

∂∂=>

==∂∂−=

∂∂=>=

∂∂+

∂∂=

,

constant0γ

rd_

mec

h@

yah

oo

.co

.in


37

yx ∂∂

Integrating these two equations

cxvvcyuu oo +=−= ,

For constant rotation

( ) zz cccy

u

x

v ωω ==>+=>

∂∂−

∂∂=

2

1

2

1

∆x

∆y

∂v/∂x-∂u/∂y


Rotation vector

The displacements are

Extending the same procedure to 3D rigid body rotations – all shear strains => 0

xvvyuu zozo ωω +=−= ,

rd_

mec

h@

yah

oo

.co

.in


38

rotations – all shear strains => 0

Integrate these equations – integration constants

0=∂∂+

∂∂=

x

v

y

uxyγ 0=

∂∂+

∂∂=

z

v

y

wyzγ 0=

∂∂+

∂∂=

z

u

x

wzxγ


Rotation vector

Integration constants – rigid body translations and rotations

yxww

xzvv

zyuu

zxo

yzo

ωωωωωω

+−=+−=

+−=

rd_

mec

h@

yah

oo

.co

.in


39

Rigid body translations and rotations don’t contribute to strains and stresses – may be dropped

Basic assumption – small deformation theory Small deformation theory – Principle of

superposition

yxww xyo ωω +−=


Transformation of strain

Strain – second order tensor

Transformation rules of any second order tensor –applicable – ex. Stress tensor

Transformation of stress tensor

Transformation of strain tensor

mnjnimij ll σσ ='

ll εε ='

rd_

mec

h@

yah

oo

.co

.in


40

Transformation of strain tensor

In matrix notation mnjnimij ll εε ='

[ ] [ ][ ][ ]TTT εε ='

[T] – Transformation matrix => direction cosines


Principal strains

Principal strains – similar to principal stresses –maximum strains acting on three mutually perpendicular planes, where shear stresses => 0

εεεεε lnml =++= Writing in indices notation

rd_

mec

h@

yah

oo

.co

.in


41

εεεεε lnml xzxyxxx =++=

εεεεε mnml yzyyxyy =++=

εεεεε nnml zzyzxzz =++=

Writing in indices notation

( ) 0=− jijjij ll εδεFor non-trivial solution

determinant = 0


Principal strains

Determinant ( )0

0

=−=>

=−

εδε

εδε

ijij

jijij l

Expand this

0322

13 =−+− III εεε

I - Invariant

rd_

mec

h@

yah

oo

.co

.in


42

0321 =−+− III εεεI i - Invariant

z

yzy

xzxyx

zyz

yzy

zxz

xzx

yxy

xyx

zyx

symm

I

I

I

εεεεεε

εεεε

εεεε

εεεε

εεε

=

++=

++=

3

2

1

First invariant I1

known as cubical

dilatation


Strain decomposition

Hydrostatic part represents volume deformation –isotropic tensor – same in all direction

Volumetric strains – accounts for change in size or volume only

Deviatoric strain tensor – accounts for change in

rd_

mec

h@

yah

oo

.co

.in


43

Deviatoric strain tensor – accounts for change in shape – distortion of material

Physically εd => deviation of strain from 1/3rd of pure cubical dilatation

Tr[εd] = 0 => Trace of deviatoric strain tensor is zero



Two dimensional stress analysis – three strain components => εxx, εyy and γxy

Writing in terms of displacements

u∂ v∂ vu ∂+∂=γ

rd_

mec

h@

yah

oo

.co

.in


44

x

uxx ∂

∂=εy

vyy ∂

∂=εx

v

y

uxy ∂

∂+∂∂=γ

∂∂

∂∂∂=

∂∂∂=

∂∂

y

u

xyxy

u

yxx

2

2

3

2

2ε

∂∂

∂∂∂=

∂∂∂=

∂∂

x

v

xyyx

v

xyy

2

2

3

2

2ε

Add these two equations

yxxyxyyyxx

∂∂∂

=∂

∂+

∂∂ γεε 2

2

2

2

2

One compatibility equation



In 3D => six strain compatibility equations

2) ,1(

2

2

2

2

2

====∂∂

∂=

∂∂

+∂

∂

lkji

yxxyxyyyxx

γεε

( )3,2 ,1

22

====∂∂

∂=

∂∂

+∂

∂+∂

∂−

∂∂

lkji

yzzyxxxxxyxzyz εγγγ

rd_

mec

h@

yah

oo

.co

.in


45

( )1 ,3

2

2

2

2

2

====∂∂

∂=∂

∂+∂

∂

lkjizxxzxzzzxx γεε

( )3 ,2

2

2

2

2

2

====∂∂

∂=

∂∂+

∂∂

lkji

zyyzyzzzyy γεε

( )1,3 ,2

22

====∂∂

∂=

∂∂

+∂

∂−∂

∂∂∂

lkji

zxzyxyyyxyxzyz εγγγ

( )2,1 ,3

22

====∂∂

∂=

∂∂

−∂

∂+∂

∂∂∂

lkji

yxzyxzzzxyxzyz εγγγ



Six strain compatibility equations – no displacement term appears

Displacements – integration of strain –displacement relations

Six equations – three displacements => over

rd_

mec

h@

yah

oo

.co

.in


46

Six equations – three displacements => over constraining – can’t get a single valued displacements – ill-conditioned

For well conditioned system => Number of equations = number of unknowns

Three more extra equations required



There exist relations among six strain components => Strain compatibility equations – six equation

Only three of them are independent

rd_

mec

h@

yah

oo

.co

.in


47

Only three of them are independent

Equation -

Displacements => 3

Strain-displacement relations => 6

Independent strain compatibility equations => 3

=> Number of unknowns = number of equations



Satisfy strain compatibility equations => single valued displacement in the domain

Conversely – all six strain compatibility equations satisfied if displacement field is

rd_

mec

h@

yah

oo

.co

.in


48

equations satisfied if displacement field is single valued

Arbitrary selection of strains – fails to satisfy strain compatibility equations



Strains are calculated from displacements

Compatible – no gaps and overlaps

rd_

mec

h@

yah

oo

.co

.in


49

Arbitrary strain field not satisfying compatibility eqns.

Integrated to get displacements

Gaps & overlaps -incompatibility


Polar co-ordinates

Some geometries – cylindrical – easy to handle in cylindrical co-ordinates

Evaluate stress, strains, displacements etc in that co-ordinate system – ex. Radial and

rd_

mec

h@

yah

oo

.co

.in


50

that co-ordinate system – ex. Radial and Hoop stresses

Two different ways – Use strain-displacement relations in cartesian

co-ordinates – convert to polar co-ordinates

Derive from polar co-ordinates


Polar co-ordinates

From cartesian csys –ex, ey – unit vectors in cartesian csys -er, eθ - unit vectors in polar csys

z

Peθ

ez

y eeθey

rd_

mec

h@

yah

oo

.co

.in


51

y

x

P

rθ

er

er

eθ

P(x, y)

x

y ereθ

r

θex

( )Tij UU

~~2

1 ∇+∇=εCartesian strain =>


Polar co-ordinates

Relation between unit vectors –

~~~

~~~

cossin

sincos

ry

rx

eee

eee

+=

−=

θθ

θθ

θ

θ

−=

~

~

~

~

100

0cossin

0sincos r

y

x

e

e

e

e

e

θθθθθ

rd_

mec

h@

yah

oo

.co

.in


52

~~

~

zz ee =

=

~~

~

~

~

~

][

100

zrxyz

zz

eTe

ee

θ

zz

ry

rx

===

θθ

sin

cos

x

yTan

yxr

=

+=

θ

222


Polar co-ordinates

In cartesian csys - gradient

θθθ ∂−∂=∂∂+∂∂=∂∂∂+

∂∂=∇

r

ye

xe yx

sincos

~~

rd_

mec

h@

yah

oo

.co

.in


53

θθθθ

θ

θθθθ

θ

∂∂+

∂∂=

∂∂

∂∂+

∂∂

∂∂=

∂∂

∂∂−

∂∂=

∂∂

∂∂+

∂∂

∂∂=

∂∂

rryy

r

ry

rrxx

r

rxcos

sin

sincos

Substitute all these

ze

re

re zr ∂

∂+∂∂+

∂∂=∇

~~~

1

θθ


Polar co-ordinates

Displacement vector in polar csys –

Gradient of displacement vector –

~~~~zzrr eueueuU ++= θθ

rd_

mec

h@

yah

oo

.co

.in


54

Gradient of displacement vector –

∂∂+

∂∂+

∂∂

+

∂∂+

∂∂++

−∂∂

+

∂∂+

∂∂+

∂∂=∇

~~~~~~

~~~~~~

~~~~~~~

111

zzz

zrzr

zz

rrr

zrz

rrrr

eez

uee

z

uee

z

u

eeu

ree

uu

reeu

u

r

eer

uee

r

uee

r

uU

θθ

θθθθ

θθ

θθ

θθθ


Polar co-ordinates

Strain =>

Stain components in polar co-ordinates

( )[ ]TUU

~~~~ 2

1 ∇+∇=ε

Normal strains

rd_

mec

h@

yah

oo

.co

.in


55

z

uuu

rr

u zr

rrr ∂

∂=

∂∂+=

∂∂= zz ,

1 , ε

θεε θ

θθ

−∂

∂+∂∂=

∂∂+

∂∂=

∂∂+

∂∂=

r

u

r

uu

rr

u

z

uu

rz

u rr

zrzr

zz

θθθ

θθ θ

εεθ

ε 1

2

1 ,

2

1 ,

1

2

1

Normal strains

Shear strains


rd_

mec

h@

yah

oo

.co

.in


56

strain analysis - imechanica analysis.pdf · strain analysis. r&de (engineers), drdo summary...

Documents