Plasticity Models
Amit Prashant
Indian Institute of Technology Gandhinagar
Short Course on
Finite Element Method, Constitutive Modeling
and Applications
28 Jan – 01 Feb, 2013
Nonlinear Stiffness - Spring model

Linear Spring:
F
F
F
k
F  k.


Non-linear Spring
F
F
F

F=?
2
1
Tensile Test on Metals
3
Compressive and Tensile Response
4
2
Unloading and Reloading Response
s
s
so
so
e
Permanent
Deformation
Permanent
Deformation
e
5
Rheology models for plasticity

Rigid – Perfectly Plastic
s
sy
F
e

Elastic – Perfectly Plastic:
E1
s
F
E1
e
6
3
Rheology models for plasticity, contd…

Rigid – Linear work hardening:
E2
s
E2
F

e
Elastic – Linear work hardening:
E2
s
E1
E2
so
F
E1 + E2
e
7
Some Definitions, contd…

Yield Point:

State of stress at which the material begins to deform
plastically
s
Yield
point
Elastic-perfectly plastic
stress-strain relationship
Perfectly plastic
Linear elastic
e
8
4
Some Definitions, contd…

Yield Criterion:


Failure Criterion:


A law defining the limit of elastic behavior under any
possible combination of stresses
A law defining the limit of stress states achievable within
finite strains.
For elastic-perfectly plastic material:
Yield criterion is the same as failure criterion
2. The yield condition remains unchanged during straining
f(s) = k at yield and df = 0
Possible stress states are limited to f(s) ≤ k
1.
9
Triaxial Compression and Deviatoric Plane
s1
Deviatoric Plane:
Normal to Hydrostatic
axis
s1
s2
s3
s3
s1
s2 = s 3
s2
Triaxial Plane:
Hydrostatic axis in the
plane
2s 2  2s 3
10
5
Tresca Yield/Failure Criteria
 max 
1
s 1  s 3   k , where s 1  s 2  s 3
2
s1
Tresca
s1
2s 2  2s 3
s3
s2
11
von-Mises Yield/Failure Criteria
1
2
2
2
s 1  s 2   s 2  s 3   s 3  s 1   k
3
or, J 2  k
 oct 
Von Mises
s1
Tresca
s2
s3
s1
2s 2  2s 3
12
6
Contact Friction Model
F
F
Tf
W
W
Shear Force, Tf
N
Tf
N
T f   .N
  s n .tan  
Displacement
13
Mohr-Coulomb Yield/Failure Criteria
  s tan 
or:
s 1  s 3   s 1  s 3  sin   0
2
2
Note: Mohr-Coulomb criterion degenerates to Tresca criterion for  = 0.
Von Mises
s1
Mohr-Coulomb
Tresca
s2
s3
s1
2s 2  2s 3
14
7
Drucker-Prager Yield/Failure Criteria
3 2
J 2   oct
2
I1  s 1  s 2  s 3
J 2   I1  k
Drucker Prager
s1
von Mises
s1
2s 2  2s 3
s3
s2
15
Stress induced anisotropy in soils and Concrete
Effect of intermediate principal stress
OCR=10
OCR=5
OCR=1
Failure surface
normalization with third
invariant of stress tensor:
sz
I3
 const
I13
I3  s1s 2s 3
Non-linear Failure
Criteria
n
sx
Note: Superposition of three
different octahedral planes
sy
 I13
 I1 
  27    const
 I3
 pa 
Note: Direct applications of
these models in real
problems are yet to be
16
developed.
8
I3 based failure surface
17
Rigid-plastic material


The yield function for an isotropic material must be
symmetric in the principal stresses  Hence, the
yield criteria can be expressed using stress invariants.
For rigid-plastic material

The state of stress must be such that
f s   k

Strain can only occur if
f s   k

i.e. the stress state is on
the yield surface
Work done (product of stress and strain) by any additional
stress vector is always positive.
18
9
General Stress-strain curve
Peak Shear Strength
q
Softening zone
Zone of instability
Strain
hardening zone
First yield point
Steady State Shear
Strength
eq
Elastic zone
19
Elasto-plasticity

The total strain of an elasto-plastic material may be
considered as the sum of permanent (plastic) and
recoverable (elastic) strains.
de  de e  de p

Elastic part of the incremental stress-strain
relationship can be represented by
d e ije  Cijkl ds ij
20
10
Yield Function

A yield surface is defined as a surface in stress space
such that it bounds stress states which can be reached
without initiating plastic strains. Mathematically, this
surface can be represented by a yield function:

f s ij ,  f   0
s
 f  State Variable, which can be function of
plastic volumetric strain and shear strain,
quantified structure of the soil
21
Hardening Rule

State variable depends on the loading history and its
growth can be related to the plastic strains developed
in the material:

d f  pf d e ijp
e ij



If the small variance in stress is such that df  0 , then this
process is called loading  plastic strains.
If the variance induces df  0, then this process is called
unloading  no plastic strain.
As a limiting case, if the stress increment is along the yield
surface i.e. df  0, then it is defined as neutral loading
 no plastic strain.
22
11
Plastic potential

A function representing the surface in stress space to
which the plastic strain increments are normal.
Plastic strain
increment vector

g s ij ,  g   0
s
 g  State Variable, which can be function of
plastic volumetric strain and shear strain,
quantified structure of the soil
23
Flow Rule

A flow rule is defined to estimate the incremental
plastic strains developed under a loading condition
d e ijp  d 

g s ij ,  g 
s ij
d  is a non-negative constant, which is related to the
hardening modulus H:
d 
1 f
ds ij
H s ij
Macaulay brackets
Macaulay brackets ensures that the function inside will have its
value only if it is positive, otherwise it will remain zero.
24
12
Associative/Non-associative Flow Rule

Associative Flow Rule:

Plastic strain increment
vectors are orthogonal to
the yield surface. Plastic
potential is the same as
yield function.
Plastic strain
increment vector

f s ij ,  f   0
s

Non-associative Flow
Rule:

Plastic strain increment
vectors are orthogonal to
a plastic potential, which
is different from the yield
function.

Plastic strain
increment vector
g s ij ,  g   0
f s ij ,  f   0
s
25
Consistency Condition

The consistency condition defines the stress state to
remain on current yield surface throughout loading
f 

f
f
ds ij 
d f  0
s ij
 f
With flow rule and hardening rule, this consistency
conditions gives the hardening modulus H:
H 
f
 f
  f g
 p
 e p s ij



26
13
Incremental Strains

Incremental plastic strains:
d e ijp 

1
f
g
ds mn
s ij
f   f g  s mn
 p

 f  e kl s kl 
Incremental Total Strain:
d e ij  Cijkl ds ij 
1
f
g
ds mn
s ij
f   f g  s mn


 f  e klp s kl 
27
Stress strain behavior of soils
Drained/Slow
Shearing
Undrained/Rapid Shearing
q
q
eq
eq
ve
V
V
ve
Contractio
eq n
Dilation
ve
u
eq
ve
28
14
Volume Change or Evolution of Pore Water
Pressure During Shearing
Drained
Shearing
Initially loose configuration
Undrained
Shearing
Contractive
Increase in
Pore Water
Pressure
Dilative
Decrease in
Pore Water
Pressure
Initially loose configuration
29
Analysis using Mohr-Coulomb Model

Failure/Yield Function
F

s1  s 3   s1  s 3  sin   c.cos   k
2
2
Principal plastic strains:
e1   .
e2  0
e 3  .
df
1 1
 
    sin    1  sin  
ds 1
2 2
 2
Mohr-Coulomb implies plane strain condition
df

  1  sin  
ds 3
2
e 3  1  sin  

 tan 2  45   2   constant
e1 1  sin  

Hence, strain increment ratio is always constant when
yielding occurs
30
15
Analysis using Mohr-Coulomb Model
Contd…

Maximum shear strain:
 max  e1  e 3 
e

 max
2

2
1  sin   1  sin    
1  sin  ,
0 , 1  sin  
Volumetric strain:
e v  e1  e 2  e 3 



 max
2
1  sin   0  1  sin     max sin 
Dilation
Always dilative for any friction angle is not true for material
like loose sand.
Generally, overestimates the dilation for dense sand too.
A non-associative model may be needed.
31
Analysis using Mohr-Coulomb Model
Contd…

Rate of work per unit volume:
D  s1.e1  s 2 .e 2  s 3.e 3
 max
1  sin  
2
2
1
1

 max  s1  s 3   s 1  s 3  sin  
2
2

s 1.


 max
1  sin    0  s 3.
Hence, D   max .c.cos  and the amount of energy required to
deform the material is proportional to cohesion.
Therefore, a material with zero cohesion, i.e. sand, can be
deformed without supplying energy  Unreasonable
32
16
Non-associative Mohr-Coulomb

Failure criteria
F

s 1  s 3   s 1  s 3  sin   c.cos   0
2
2
Plastic Potential:
G

s1  s 3   s1  s 3  sin  c.cos  0
2
2
Angle of dilation
 
e1  e 3
e1  e 3
33
Non-associative Mohr-Coulomb, contd…
  0 : No volume change
  0 : Contractive or decrease in volume
  0 : Dilative or increase in volume
Plastic strain increment vector



failure surface
Normal to
failure surface
Elastic
region
sm
34
17
Some other issues in soils
 Advanced Models

Effect of porosity, stress history, and hardening
response

Softening in stress-strain relationship and Instability

Dynamic/cyclic loading response


Consolidation process  Diffusion


Visco-elastic models
Visco-elastic-plastic models
and More…
35
Thank You
36
18