theories of - iMechanica

62
R&DE (Engineers), DRDO Theories of Failure [email protected] Ramadas Chennamsetti Theories of Failure

Transcript of theories of - iMechanica

R&DE (Engineers), DRDO

Theories of Failure

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Ramadas Chennamsetti

Theories of Failure

R&DE (Engineers), DRDO

Summary

� Maximum principal stress theory

� Maximum principal strain theory

� Maximum strain energy theory

� Distortion energy theory

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� Distortion energy theory

� Maximum shear stress theory

� Octahedral stress theory

R&DE (Engineers), DRDO

Introduction� Failure occurs when material starts exhibiting

inelastic behavior � Brittle and ductile materials – different modes

of failures – mode of failure – depends on loading

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loading � Ductile materials – exhibit yielding – plastic

deformation before failure � Yield stress – material property� Brittle materials – no yielding – sudden failure � Factor of safety (FS)

R&DE (Engineers), DRDO

Introduction

� Ductile and brittle materialsσ

σY σF

σ σFσY

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εεe εp

Ductile material

Well – defined yield point in ductile materials – FS on yielding

No yield point in brittle materials sudden failure – FS on failure load

0.2% εε

Brittle material

R&DE (Engineers), DRDO

Introduction

� Stress developed in the material < yield stress

� Simple axial load

σxσx

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If σx = σY => yielding starts – failure

Yielding is governed by single stress component, σx

Similarly in pure shear – only shear stress.

If τmax = τY => Yielding in shear

σxσx

Multi-axial stress state ??

R&DE (Engineers), DRDO

Introduction

� Various types of loads acting at the same time

N

M

T

Axial, moment and torque

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Axial, moment and torque

p

Internal pressure and external UDL

R&DE (Engineers), DRDO

Introduction� Multiaxial stress state – six stress components – one

representative value � Define effective / equivalent stress – combination of

components of multiaxial stress state� Equivalents stress reaching a limiting value – property

of material – yielding occurs – Yield criteria

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of material – yielding occurs – Yield criteria� Yield criteria define conditions under which yielding

occurs� Single yield criteria – doesn’t cater for all materials� Selection of yield criteria � Material yielding depends on rate of loading – static &

dynamic

R&DE (Engineers), DRDO

Introduction

� Yield criteria expressed in terms of quantities like stress state, strain state, strain energy etc.

� Yield function => f(σij, Y),σij = stress state� If f(σij, Y)<0 => No yielding takes place – no

failure of the material

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failure of the material� If f(σij, Y) = 0 – starts yielding – onset of yield

If f(σij, Y) > 0 - ?? � Yield function developed by combining stress

components into a single quantity – effective stress => σe

R&DE (Engineers), DRDO

Introduction

� Equivalent stress depends on stress state and yield criteria – not a property

� Compare σe with yield stress of material

� Yield surface – graphical representation of σ

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yield function, f(σij, Y) = 0

� Yield surface is plotted in principal stress space – Haiagh – Westergaard stress space

� Yield surface – closed curve

R&DE (Engineers), DRDO

Parameters in uniaxial tension

� Maximum principal stress

� Maximum shear stress

Applied stress => Y

σ1 = Y, σ2 = 0, σ3 = 0Y Y

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� Maximum shear stress

� Maximum principal strain

2231

max

Y=−= σστ

( )E

Y

EEY =+−= 321 σσυσεσ1 = Y, σ2 = 0, σ3 = 0

R&DE (Engineers), DRDO

Parameters in uniaxial tension

� Total strain energy density

� Distortional energy

Linear elastic material E

YYU Y

2

2

1

2

1 == ε

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� Distortional energy

[ ]

+

−−

−=

=p

p

p

p

p

pYY

00

00

00

00

00

00

000

000

00

σ

First invariant = 0 for deviatoric part => p = Y/3

U = UD + UV

R&DE (Engineers), DRDO

Parameters in uniaxial tension

Volumetric strain energy density, UV = p2/2K

( )

UUU

YEK

Y

K

pU

VD

V 6

21

1822

22

−=

−=== υ

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( ) ( ) ( )

G

YU

E

Y

E

Y

E

Y

E

YU

UUU

D

D

VD

6

13

21366

21

22

2222

=

+=+−=−−=

−=

υυυ

Similarly for pure shear also

R&DE (Engineers), DRDO

Failure theories� Failure mode –

� Mild steel (M. S) subjected to pure tension� M. S subjected to pure torsion� Cast iron subjected to pure tension � Cast iron subjected to pure torsion

� Theories of failure

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� Theories of failure � Max. principal stress theory – Rankine� Max. principal strain theory – St. Venants� Max. strain energy – Beltrami � Distortional energy – von Mises� Max. shear stress theory – Tresca� Octahedral shear stress theory

R&DE (Engineers), DRDO

Max. principal stress theory

� Maximum principal stress reaches tensile yield stress (Y)

� For a given stress state, calculate principle stresses, σ1, σ2 and σ3

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� Yield function

( )

definednot 0

yielding ofonset 0

yielding no 0 If,

, ,max 321

>=

<−=

f

f

f

Yf σσσ

R&DE (Engineers), DRDO

Max. principal stress theory

� Yield surface –

0 ,0 Y

0 ,0 Y

0 ,0 Y

222

111

=−=+=>±==−=+=>±==−=+=>±=

YY

YY

YY

σσσσσσσσσ

Represent six surfaces

σ

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0 ,0 Y 333 =−=+=>±= YY σσσ

σ1

σ2

σ3

Y

Y

Y

Yield surface

Yield strength – same in tension and compression

R&DE (Engineers), DRDO

Max. principal stress theory

� In 2D case, σ3 = 0 – equations become

0 ,0 Y

0 ,0 Y

222

111

=−=+=>±==−=+=>±=

YY

YY

σσσσσσ

σ1

σ2

Y

Y

-YClosed curve

Stress state inside – elastic, outside => Yielding

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-Y

Stress state inside – elastic, outside => Yielding

Pure shear test => σ1 = + τY, σ2 = - τY

For tension => σ1 = + σY

From the above => σY = τY

Experimental results – Yield stress in shear

is less than yield stress in tension

Predicts well, if all principal stresses are tensile

τY τy

τyPure shear

R&DE (Engineers), DRDO

Max. principal strain theory

� “Failure occurs at a point in a body when the maximum strain at that point exceeds the value of the maximum strain in a uniaxial test of the material at yield point”

� ‘Y’ – yield stress in uniaxial tension, yield

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� ‘Y’ – yield stress in uniaxial tension, yield strain, εy = Y/E

� The maximum strain developed in the body due to external loading should be less than this

� Principal stresses => σ1, σ2 and σ3 strains corresponding to these stress => ε1, ε2 and ε3

R&DE (Engineers), DRDO

Max. principal strain theory

Strains corresponding to principal stresses -

( )

( )2

321

1

σσυσε

σσυσε

+−=

+−=EE

For onset of yielding

( )

( )Y

YE

Y ±=+−=>= 2211 σσυσε

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( )

( )123

3

312

2

σσυσε

σσυσε

+−=

+−=

EE

EE

Maximum of this should be less than εy

( )

( ) YE

Y

YE

Y

±=+−=>=

±=+−=>=

2133

1322

σσυσε

σσυσε

There are six equations –each equation represents a plane

R&DE (Engineers), DRDO

Max. principal strain theory

� Yield function

e

kjikji

Yf

kjiYf

υσυσσσσ

υσυσσ

−−=

−=

=−−−=≠≠

max

3 ,2 ,1,, ,max

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� For 2D casekji

kjie υσυσσσ −−=

≠≠max

YY

YY

±=−=>=−

±=−=>=−

1212

2121

υσσυσσυσσυσσ

There are four equations, each equation represents a straight line in 2D stress space

R&DE (Engineers), DRDO

Max. principal strain theory

Equations –

Plotting in stress space

YY

YY

−=−=−−=−=−

1212

2121

,

,

υσσυσσυσσυσσ

σνσσ =−σ

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Yσνσσ =− 12

Yσνσσ =− 21

σ1

σ2

Yσνσσ −=− 12

Yσνσσ −=− 21 σy

σy

Failure – equivalent stress falls outside yield surface

R&DE (Engineers), DRDO

Max. principal strain theory

� Biaxial loading

σ1

-σ2

For onset of yielding –

( )( )υσ

υσυσσ+=

+=−=1

121

Y

Y

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σ1 = | σ2 |= σMaximum principal stress theory –

Y = σ

Max. principal strain theory predicts smaller value of stress

than max. principal stress theory

Conservative design

R&DE (Engineers), DRDO

Max. principal strain theory

� Pure shearτyPrincipal stresses corresponding to

shear yield stress

σ1 = +τy , σ2 = -τy

For onset of yielding – max. principal

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For onset of yielding – max. principal strain theory

Y = τy + υ τy = τy (1 + υ)

Relation between yield stress in tension and shear

τy = Y/ (1 + υ) for υ = 0.25

ττττy = 0.8Y Not supported by experiments

R&DE (Engineers), DRDO

Strain energy theory

� “Failure at any point in a body subjected to a state of stress begins only when the energy density absorbed at that point is equal to the energy density absorbed by the material when subjected to elastic limit in a uniaxial stress

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subjected to elastic limit in a uniaxial stress state”

� In uniaxial stress (yielding)

σ = Eε => Hooke’s law

Strain energy density,

E

YU

dUdUy

ijij

2

0

2

1=

==>= ∫∫ε

εσεσ

R&DE (Engineers), DRDO

Strain energy theory

� Body subjected to external loads => principal stresses

σ1

σ2

Strain energy associated with principal stresses

1

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σ3( )

( )

( )

( )212

3

132

2

321

1

3322112

1

σσυσε

σσυσε

σσυσε

εσεσεσ

+−=

+−=

+−=

++=

EE

EE

EE

U

( )[ ]32132123

22

21 2

2

1 σσσσσσυσσσ ++−++=E

U

For onset of yielding,

( )[ ]32132123

22

21

2

22

1

2σσσσσσυσσσ ++−++=

EE

Y

R&DE (Engineers), DRDO

Strain energy theory

� Yield function –

( )

( ) stress Equivalent 13322123

22

21

2

22

2133221

23

22

21

++−++==>

−=

−++−++=

Yf

Yf

e

e

σσσσσσυσσσσσ

σσσσσσυσσσ

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� For 2D stress state => σ3 = 0 – Yield function becomes

( )0 safe ,0 Yielding

stress Equivalent 133221321

<==>++−++==>

f f e σσσσσσυσσσσ

221

22

21 Yf −−+= συσσσ

For onset of yielding => f = 0 0221

22

21 =−−+ Yσυσσσ

Plotting this in principal stress space

R&DE (Engineers), DRDO

Strain energy theory

Rearrange the terms –12 21

3

2

2

1 =

+

YYYY

σσυσσ

This represents an ellipse –Transform to ζ-η csys

σ

σ2

Y ζ

η

45o

1

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σ1Y

45o

( )

( )ηζηζσ

ηζηζσ

+=+=

−=−=

2

145cos45sin

2

145sin45cos

2

1

Equivalent stress inside – no failure

Substitute these in the above expression

R&DE (Engineers), DRDO

Strain energy theory

Simplifying,

( ) ( )11

11

2

2

2

2

2

2

2

2

=+=>=

+

+

−baYY

ηζ

υ

η

υ

ζ

Semi major axis – OA => ( )υ−=

1

Ya

σ2

Y ζ

ηA

B

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Semi major axis – OA =>

Semi minor axis – OB =>

( )υ−1

( )υ+=

1

Yb

σ1Y

45o

o

B

Higher Poisson ratio – bigger major axis, smaller minor axis

If υ = 0 => circle of radius ‘Y’

R&DE (Engineers), DRDO

Strain energy theory

� Pure shear τy

Principal stresses corresponding to shear yield stress

σ1 = +τy , σ2 = -τy

ττ

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( ) ( )υτ

ευτ

ε +−=+= 1 ,1 21 EEyy

Strain energy, ( ) ( ) yy YYEE

U τυτυτ +==>=+= 12

2

12

2

1 22

τy = 0.632 Y

R&DE (Engineers), DRDO

Distortional energy theory (von-Mises)

� Hydrostatic loading � applying uniform stress from all the

directions on a body

� Large amount of strain

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energy can be stored

� Experimentally verified

� Pressures beyond yield

stress – no failure of material

� Hydrostatic loading – change in size – volume

Pressure ‘p’ applied from all sides

R&DE (Engineers), DRDO

von-Mises theory� Energy associated with volumetric change –

volumetric strain energy� Volumetric strain energy – no failure of

material� Strain energy causing material failure –

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� Strain energy causing material failure –distortion energy – associated with shear –First invariant of deviatoric stress = 0

� For a given stress state estimate distortion energy – this should be less than distortion energy due to uniaxial tensile – safe

R&DE (Engineers), DRDO

von-Mises theory

� Given stress state referred to principal co-ordinate system –

[ ]p

p

p

p

p

p

σσ

σ

σσ

σσ

00

00

00

00

00

00

00

00

00

2

1

2

1

+

−−

−=

=

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( ) ( ) ( )

iip

ppp

J

pp

σ

σσσ

σσ

3

1

0

0 invariant,First

000000

321

1

33

==>

=−+−+−=

Principal strains => ε1, ε2, ε3

Volumetric strain => εV = ε1+ ε2 + ε3

R&DE (Engineers), DRDO

von-Mises theoryThis gives –

( ) ( ){ }( ) ( ) ( )

321

321321321

21321

21

υσσσυε

σσσυσσσεεεε

−=++−=

++−++=++=

pEE

E

V

V

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( ) ( ) ( ) ( )

( ) ( )13322123

22

21

2321

2

2

1

strains & stresses principal todueenergy strain 6

21

2

213213

2

12

1 energy,strain Volumetric

σσσσσσυσσσ

σσσυυυ

ε

++−++=

=

++−=−=−=

=

EEU

UE

pE

pE

pU

pU

V

VV

R&DE (Engineers), DRDO

von-Mises theory

� Distortional energy –σm

σm

σm

σ2 - σm

σ1 - σm

σ - σ= +

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σm σ3 - σm

UD = U - UV

( ) ( )[ ] ( )2321133212

23

22

21 6

212

2

1 σσσυσσσσσσυσσσ ++

−−++−++=EE

UD

Simplifying this

( ) ( ) ( )[ ]213

232

22112

1 σσσσσσ −+−+−=G

UD

R&DE (Engineers), DRDO

von-Mises theory

� Compare this with distortion in uniaxial tensile stress

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

213

232

221

2

12

1

6σσσσσσ −+−+−===

GG

YUD

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( ) ( ) ( )213

232

221

22 σσσσσσ −+−+−==> Y

Yield function,

( ) ( ) ( )[ ]213

232

221

2

22

2

1 stress, Equivalent σσσσσσσ

σ

−+−+−=

−=

e

e Yf

R&DE (Engineers), DRDO

von-Mises theory

Principal stresses of deviatoric shear stress, Sii

[ ]

+

−−

−=

=

pS

p

p

p

p

p

p

0000

00

00

00

00

00

00

00

00

00

3

2

1

3

2

1

σσ

σ

σσ

σσ

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+

=p

p

p

S

S

S

00

00

00

00

00

00

][

3

2

1

σ

Sii = σii – p => σii = Sii + p

( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( )2

132

322

212

213

232

221

2

2

2

pSpSpSpSpSpSY

Y

+−+++−+++−+=

−+−+−= σσσσσσ

R&DE (Engineers), DRDO

von-Mises theory

Simplifying this expression –

Hydrostatic pressure does not appear in the expression

( ) ( ) ( )213

232

221

22 SSSSSSY −+−+−=

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� von-Mises criteria has square terms – result independent of signs of individual stress components

� Von-Mises equivalent stress => +ve stress

R&DE (Engineers), DRDO

von-Mises theory

� 2D stress state => σ3 = 0

Yield function, 221

22

21 Yf −−+= σσσσ

Onset of yielding, 221

22

21 Y=−+ σσσσ

σ2Re-arrange the terms –

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σ1Y

Y ζ

η

45o

o

A

B

Re-arrange the terms –

12

21

2

2

2

1 =

+

YYY

σσσσ

This represents an ellipse

Y

Y

3

2 OB axis,minor - Semi

2 OA axis,major -Semi

=

=

R&DE (Engineers), DRDO

von-Mises theory

� Pure shear –τy

Principal stresses corresponding to shear yield stress

σ1 = +τy , σ2 = -τy

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YY yy 577.03 221

22

21

2 ==>=−+= ττσσσσ

Suitable for ductile materials

Shear yield = 0.577 * Tensile yield

R&DE (Engineers), DRDO

von-Mises theory

� Plot yield function in 3D principal stress space

σ2

σ1= σ2= σ3 σ1= σ2= σ3

σ2

A

B=++ σσσ

~n

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σ1

σ3

( ) ( ) ( ) 02 2213

232

221 =−−+−+−= Yf σσσσσσ

Cylinder, with hydrostatic stress as axis

Axis makes equal DCs with all axes

σ1

σ3

o

B0321 =++ σσσDeviatoric plane

~3

~2

~1

~

~~~~ 3

1

kjiOA

kjin

σσσ ++=

++=

R&DE (Engineers), DRDO

von-Mises theory

� Projection of OA on hydrostatic axis

++

++

=

===>=

~~~32

~1

~~

~~

.

.cos cos.

kjikji

OBn

nOAOAnOAnOA

σσσ

θθ

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( )

( )

( )

++=

++++=

++++===>

++=

=

~~~~~~

321

~

~~~321

~~

321

~~~~3

~2

~1

3

3

1

3

1

3

13

kjipkjiOB

kjinOBOB

OB

OB

σσσ

σσσ

σσσ~~~

~~~

OBOABA

BAOBOA

−=

+=

BA = r = radius of cylinder

R&DE (Engineers), DRDO

von-Mises theory

� Radius of cylinder

~

213

~

132

~

321

~

~~~~3

~2

~1

~~~~

3

2

3

2

3

2kjiR

kjipkjiOBOARBA

−−+

−−+

−−=

++−

++=−==

σσσσσσσσσ

σσσ

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( ) ( )( ) ( ) ( ) 22

132

322

21

13322123

22

21

2321

3211

23

22

21

~3

~2

~1

~

~~~~

2 criteria Yield

20

00 tensor,stress deviatoris ofinvariant First

R Radius

333

YSSSSSS

SSSSSSSSSSSS

SSSJ

SSS

kSjSiSR

=−+−+−=>

++−=++==++

=++=>=++==>

++=

R&DE (Engineers), DRDO

von-Mises theoryYield criteria,

( )

( )SSSSSSY

SSSSSSSSS

SSSSSSSSSY

2

1

2 Use,

23

22

21

23

22

21

2

13322123

22

21

13322123

22

21

2

+++++=

++−=++

−−−++=

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( )

( )

RY

RSSSY

SSSSSSY

2

3

2

3

2

32

223

22

21

2

321321

=

=++=

+++++=

Yielding depends on deviatoric stresses Hydrostatic stress has no role in yielding

R&DE (Engineers), DRDO

von-Mises theory� Second invariant of deviatoric stress

[ ]

1332212

3

1

3

2

2

12

3

2

1

0

0

0

0

0

0

00

00

00

SSSSSSJ

S

S

S

S

S

SJ

S

S

S

S

++=

++==>

=

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Ramadas Chennamsetti43

( )

( ) ( )

22

2222

223

22

212

23

22

212

23

22

21133221

3 function yield Redfining

32

322

1

2

1

2

1

YJf

JYRY

RSSSJSSSJ

SSSSSSSSS

−==>

==>=

=++==>++−=

++−=++

J2 Materials

R&DE (Engineers), DRDO

Max. shear stress theory (Tresca)

� “Yielding begins when the maximum shear stress at a point equals the maximum shear stress at yield in a uniaxial tension”

TT KY

K ===>−== max21

max τσστ Y Y

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Ramadas Chennamsetti44

TT KK ===>==22 max

21max ττ

If maximum shear stress < Y/2 => No failure occursτyFor pure shear, σ1 = +τy , σ2 = -τy

Y Y

TyT KK ===>−== ττσστ max21

max 2

Shear yield = 0.5 Tensile yield

R&DE (Engineers), DRDO

Tresca theory

� In 3D stress state – principal stresses => σ1, σ2

and σ3

� Maximum shear stress

−−−

, ,.max 133221 σσσσσσ

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Ramadas Chennamsetti45

� Yield function

2 ,

2 ,

2.max 133221

yielding ofOnset 0

yielding No 0

22 ,

2 ,

2.max 133221

=>==><

=−

−−−=

f

f

YKf T

σσσσσσ

R&DE (Engineers), DRDO

Tresca theory

� Following equations are obtained

( ) ( ) TT

TT

KfKf

KK

2 , ;2 ,

22

2121221211

2121

+−=−−=

±=−=>=−

σσσσσσσσ

σσσσ

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Ramadas Chennamsetti46

( ) ( )

( ) ( ) TT

TT

TT

TT

KfKf

KK

KfKf

KK

2 , ;2 ,

22

2 , ;2 ,

22

1313613135

1313

3232432323

3232

+−=−−=

±=−=>=−+−=−−=

±=−=>=−

σσσσσσσσ

σσσσσσσσσσσσ

σσσσ

R&DE (Engineers), DRDO

Tresca theory

� Redefining yield function as,

( ) ( )( )( )( )

TT KKf 22 , , 2121321 +−−−= σσσσσσσ

( ) ( ) ( ) ( ) ( ) ( )211324323212211321 ,.,.,.,., , , σσσσσσσσσσσσσ ffffff =

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Ramadas Chennamsetti47

( ) ( )( )( )( )( )( )TT

TT

TT

KK

KK

KKf

22

22

22 , ,

1313

3232

2121321

+−−−+−−−

+−−−=

σσσσσσσσ

σσσσσσσ

Each function represents a plane in 3D principal stress space

( ) ( )( ) ( )( ) ( )( )2213

2232

2221321 444 , , TTT KKKf −−−−−−= σσσσσσσσσ

No effect of hydrostatic pressure in Tresca criteria

R&DE (Engineers), DRDO

Tresca theory

� Yield function in principal stress space

σ3σ3 A

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Ramadas Chennamsetti48

σ2σ1

O σ2 - σ1 = 2KTσ1 - σ2 = 2KT

σ1

σ2

Hydrostatic axis

Tresca yield surface View ‘A’ – along hydrostatic axis

R&DE (Engineers), DRDO

Tresca theory

� Yield surface intersects principal axes at 2KT

73.543

1cos

axis cHydrostati

0

321

==>=

===>

αα

σσσ

A

Wall/plane of hexagon

σ3 - σ1 = 2K

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Ramadas Chennamsetti49

3

22cos

2

26.3590

300

T

T

KOAOB

KOA

==

===>=+

θ

θθα

OB – projection of OA on deviatoric plane

O

Hydrostatic axis

A

B α

Deviatoric plane, σ1 + σ2 + σ3 = 0

θC

R&DE (Engineers), DRDO

Tresca theory� Tresca hexagon

σ3A, B

DO C

D

3

22 TK

300

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Ramadas Chennamsetti50

σ2σ1

O σ2 - σ1 = 2KTσ1 - σ2 = 2KTC

DO C

T

T

KOC

KOC

ODOC

2

2

3

3

22

30cos

==>

==>

=

R&DE (Engineers), DRDO

Tresca theory

� 2D stress state -σ3 = 0

T

T

T

K

K

K

2

2

2

1

2

21

±=±=

±=−

σσ

σσ

Each equation represents two lines in 2D stress space

σ2 A2K

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Ramadas Chennamsetti51

σ1

σ2

O

2KT

- 2KT

σ1 - σ2 = 2KT

- σ1 + σ2 = 2KT σ1 = 2KT

σ1 = - 2KT

σ2 = - 2KT

σ2 = 2KT

A

B

450

O A

B

450

2KT

T

TT

KOA

OC

KKOAOB

2245cos

22

1245cos

==

===

Yield curve – elongated hexagon

C

R&DE (Engineers), DRDO

Tresca theory

� 2D stress state -σ3 = 0

TK221 ±=− σσ

Each equation represents two lines in 2D stress space

σ2

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Ramadas Chennamsetti52

T

T

T

K

K

K

2

2

2

1

2

21

±=±=

±=−

σσ

σσ

σ1

σ2

O

2KT

- 2KT

σ1 - σ2 = 2KT

- σ1 + σ2 = 2KT σ1 = 2KT

σ1 = - 2KT

σ2 = - 2KT

σ2 = 2KT

A

B

450

Yield curve – elongated hexagon

C

R&DE (Engineers), DRDO

von-Mises – Tresca theories

� Pure tension –

( ) ( ) ( )[ ]213

232

221

22 6

1 σσσσσσ −+−+−== MKJvon-Mises criteria =>

Tresca’s criteria =>

−−−=

2 ,

2 ,

2max 133221 σσσσσσ

TK

σ1 = Y, σ2 = 0, σ3 = 0

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Ramadas Chennamsetti53

YKYK TM 2

1 ,

3

1 ==

222

Pure shear=> σ1 = +τy, σ2 = -τy => TyM KK == τ

( ) Tresca 5.0 Mises),(von577.0

2

1 ,

3

1

yy YY

YKYK yTyM

=−=

====

ττ

ττ

von-Mises criteria predicts 15% higher shear stress than Tresca

R&DE (Engineers), DRDO

von-Mises – Tresca theories

� 2D stress space – von-Mises and Tresca

σ1

σ2

Y ζ

η

45o

A

B

σ2

σ1

2τy

ζ

η

45o

A

B

2τy

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Ramadas Chennamsetti54

σ1Yo

{ }2121

2122

21

2

, ,.max σσσσσσσσ

−=−+=

Y

Y

Yielding in uniaxial tension

Tresca – conservative

−=

−+=

2 ,

2 ,

2.max

2

3

2121

2122

21

2

σσσστσσσστ

y

y

σ1o

yτ3

Yielding in shear

von-Mises – conservative

R&DE (Engineers), DRDO

von-Mises – Tresca theories

� Experiments by Taylor & Quinney**

� Thin walled tube subjected to axial and torsional loads

T 22

1 xyxxxx τσσσ +

+=

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Ramadas Chennamsetti55

P P

T

A

σxx

Aσxx

τxy

** Taylor and Quinney “Plastic deformation of metals”, Phil. Trans. Roy. Soc.A230, 323-362, 1931

22

2

1

22

22

xyxxxx

xy

τσσσ

τσ

+

−=

+

+=

R&DE (Engineers), DRDO

von-Mises – Tresca theories

� Tresca criteria

12/

if, yielding No

42

2

22

22222

21

<

+

+==>+

=−=

YY

YY

xyxx

xyxxxyxx

τσ

τστσσσ

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Ramadas Chennamsetti56

� von-Mises criteria2/

YY

13/

if, yielding No

322

222

2122

21

2

<

+

+=

−+=

YY

Y

Y

xyxx

xyxx

τσ

τσσσσσ

R&DE (Engineers), DRDO

von-Mises – Tresca theories

� Plotting these two criteria –

τxy/ Y

0.6

Experimental data shows good agreement with von-Mises theory.

Tresca – conservative

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Ramadas Chennamsetti57

σxx/ Y1

von-Mises

Tresca

Aluminium

Mild steel

Copper

Tresca – conservative

von-Mises theory more accurate – generally used in design

Experiments show that for ductile materials yield in shear is 0.5 to 0.6 times of yield in tensile

R&DE (Engineers), DRDO

Octahedral shear stress theory

� Octahedral plane – makes equal angles with all principal stress axes – direction cosines same

� Shear stress acting on this plane – octahedral shear

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Ramadas Chennamsetti58

( ) ( ) ( )[ ]213

232

221

2

9

1 σσσσσστ −+−+−=oct

Body subjected to pure tension, σ1 = Y, σ2 = σ3 = 0

( ) ( ) ( )[ ]213

232

221

2

22

2

9

2

σσσσσσ

τ

−+−+−=

=

Y

Yoct

R&DE (Engineers), DRDO

Octahedral shear stress theory

� Comparing this with von-Mises theory => both are same

� Pure shear

τy

σ1 = τy, σ2 = - τy, σ3 = 0

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Ramadas Chennamsetti59

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]213

232

221

2

22

2213

232

221

2

6

3

29

6

9

1

σσσσσστ

ττ

τσσσσσστ

−+−+−=

=

=−+−+−=

y

yoct

yoctSame as von-Mises theory in pure shear

Octahedral shear stress theory => von-Mises theory

R&DE (Engineers), DRDO

Tensile & shear yield strengths

� Each failure theory gives a relation between yielding in tension and shear (υ = 0.25)

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R&DE (Engineers), DRDO

Failure theories in a nut shell

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Ramadas Chennamsetti61

R&DE (Engineers), DRDO

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Ramadas Chennamsetti62