M Tech Level

1. Stress Strain Relations
(PLASTIC FLOW)
Present By:- Kartik Paliwal
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2. Stress- Strain Relations
(Plastic Flow)
• Plastic Flow takes place when a stress point reaches
the boundary of the elastic zone defined by the yield
locus
• Change in Plastic deformation is discussed here
– stress-strain relations for plastic flow relate strain
increments and so unlike in case of Hooke’s Law
stress-strain components are differential relations
rather than finite relations
• Plastic Flow is irreversible
– Most of deformation work is transformed into heat
– Stresses in final state depend on the strain path
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3. or
iii.
Assumptions
i. The body is isotropic
ii. The volumetric strain is an elastic strain & is
proportional to the mean pressure (  m  p  )
  3k
d  3kd
The total strain increments(𝑑𝜀𝑖𝑗)are made up of elastic strain
increments ( 𝑑𝜀𝑖𝑗
𝑒
) & plastic strain increments ( 𝑑𝜀𝑖𝑗
𝑝
)
𝑑𝜀𝑖𝑗 = 𝑑𝜀𝑖𝑗
𝑒
+ 𝑑𝜀𝑖𝑗
𝑝
3

4. Assumptions(Contd…)
iv. The Elastic Strain Increments are related to stress
components(ij ) through Hooke’s Law
zxzxzx
yzyzyz
xyxyxy
zzz
yyy
x y zxx
G
G
G
E
E
E
de
 d e

1

de
 d e

1

de
 d e

1

 (  )x y
de

1

 (  )z x
de

1

de

1
  (  )
4

5. Assumption (Contd.)
v. The deviatoric components of the plastic strain
increments are proportional to the components of the
deviatoric state of stess
Where is the instantaneous constant of
proportionality
d[ x y zx
p
yy zz
p
xx
p
xx
3
 1
(   )]d (  p
 )] [
3
1
d
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6. Deviatoric Stress
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7. Stress Strain Relations
• But
From (ii), the volumetric strain is purely elastic and hence
Hence we get,
Using this in (v), deviatoric stress components we get
3
y zxxxxd p
 d 
1
(   )]
𝜀 = 𝑒 𝑥𝑥
𝑒
+ 𝑒 𝑦𝑦
𝑒
+ 𝑒 𝑧𝑧
𝑒
+ 𝑒 𝑥𝑥
𝑝
+ 𝑒 𝑦𝑦
𝑝
+ 𝑒 𝑧𝑧
𝑝
𝑒 𝑥𝑥
𝑝
+𝑒 𝑦𝑦
𝑝
+𝑒 𝑧𝑧
𝑝
=0
𝜀 = 𝑒 𝑥𝑥
𝑒 + 𝑒 𝑦𝑦
𝑒 + 𝑒 𝑧𝑧
𝑒
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8. Stress Strain Relations (Contd.)
• Denoting components of stress deviator denoted by sij
the above equations and the remaining ones are
zx
yz
xy
zz
yy
xx
p
xx
p
yy
p
zz
p
xy
p
yz
p
zxd
d
d
d
 ds
 ds
 ds
 ds
d  ds
d  ds
ij
p
ijd s d
Equivalently
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9. PRANDTL-REUSSEQUATIONS
9

10. • Combining
•
•
We get, Prandtl-Reuss Equations
p
ijijijd  d e
 d
ij
ee
ij ij
kji
e
ii
G
E

• d
1
1
d   d
 )  (
dij  dsij
ijijijd  d e
 ds
10

11. 11

12. i.e. p
dW  dT2
d W p  0
d   0
Since
We have
Consider the work done during the plastic strain increment
Hence proved d isnon-negative
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13. • If the von Mises condition is applied
2
pdW  d2s
2
pd  dW / 2sor
i.e. d is proportional to the increment of plastic work.
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14. SAINT VENANT-VONMISES EQUATIONS
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15. • In a fully developed plastic deformation, the
elastic strain components are very small
compared to plastic strain components, so
p
ijijd  d
• This gives Saint Venant-von Mises theory of
plasticity in the form
dij  dsij
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16. • Expanding the equations
3 2
23
23
z x yzz
z xyyy
y zxxx

1
(  )]d 
2
d [

1
(  )]d 
2
d [

1
(  )]d 
2
d[
dyz  dyz
dzx  dzx
dxy  dxy
• The above Equations are also called Levy-Mises
equations.
• It is to be noted that in this case,the principal axes of
strain increments coincide with axes of principal stress
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17. Reference:-
• Book on “ Advanced Mechanics of Solids” By L S Srinath,
Page No. 13
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