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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Ramadas Chennamsetti2
� 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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Ramadas Chennamsetti3
loading � Ductile materials – exhibit yielding – plastic
deformation before failure � Yield stress – material property� Brittle materials – no yielding – sudden failure � Factor of safety (FS)
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Introduction
� Ductile and brittle materialsσ
σY σF
σ σFσY
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Ramadas Chennamsetti4
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
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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 ??
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Introduction
� Various types of loads acting at the same time
N
M
T
Axial, moment and torque
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Ramadas Chennamsetti6
Axial, moment and torque
p
Internal pressure and external UDL
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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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Ramadas Chennamsetti7
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
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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
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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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Ramadas Chennamsetti12
( ) ( ) ( )
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
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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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Ramadas Chennamsetti13
� 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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Ramadas Chennamsetti15
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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Ramadas Chennamsetti16
-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
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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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Ramadas Chennamsetti19
� 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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Ramadas Chennamsetti20
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
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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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Ramadas Chennamsetti22
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
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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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Ramadas Chennamsetti23
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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Ramadas Chennamsetti26
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’
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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
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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
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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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Ramadas Chennamsetti31
( ) ( ) ( )
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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Ramadas Chennamsetti39
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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Ramadas Chennamsetti42
( )
( )
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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Ramadas Chennamsetti60
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