This document discusses constitutive models for elastoplastic materials. It describes how elastoplastic materials behave elastically up to a certain stress limit, after which both elastic and plastic behavior occurs. The document outlines stress-strain curves for hypothetical materials under different loading conditions and discusses yield criteria and hardening rules used in plasticity models to describe the evolution of material behavior. It also summarizes common plasticity models including von Mises and Tresca and the assumptions underlying incremental plasticity theories.

1. UNIVERSITY OF TECHNOLOGY
BUILDING AND CONSTRUCTION DEPARTMENT
GEOTECHNICAL ENGINEERING
Soil Structure Interaction
Soil Constitutive modeling
-2-
Reviewed by
ADNAN NAJEM LAZEM
M.Sc. in Structural Engineering
2012-2013

2. Constitutive models
Part 2
Elastoplastic

3. Elastoplastic material models
• Elastoplastic materials are assumed to
behave elastically up to a certain stress limit
after which combined elastic and plastic
behaviour occurs.
• Plasticity is path dependent – the changes in
the material structure are irreversible

4. Stress-strain curve of a hypothetical material
Idealized results of one-dimensional tension test
Engineering stress
Engineering strain
Yield point
Yield stress
area
initial
force/


l
l
0



Johnson’s limit … 50% of Young modulus value

5. Real life 1D tensile test, cyclic loading
Hysteresis loops move
to the right - racheting
Where is the yield point?
Conventional yield point
Lin. elast. limit

6. Mild carbon steel
before and after heat treatment
Conventional yield point … 0.2%

7. The plasticity theory covers the
following fundamental points
• Yield criteria to define specific stress
combinations that will initiate the non-elastic
response – to define initial yield surface
• Flow rule to relate the plastic strain increments to
the current stress level and stress increments
• Hardening rule to define the evolution of the
yield surface. This depends on stress, strain and
other parameters

8. Yield surface, function
• Yield surface, defined in stress space separates stress states
that give rise to elastic and plastic (irrecoverable) states
• For initially isotropic materials yield function depends on
the yield stress limit and on invariant combinations of
stress components
• As a simple example Von Mises …
• Yield function, say F, is designed in such a way that
plasticity
analytical
for
le
inadmissib
outside,
0
surface
the
on
0
surface
the
within
state
stress
0



F
F
F
0
yield
effective 

 

F
0
...)
,
,
( P

K
F ij
ij 


9. Three kinematic conditions are to be
distinguished
• Small displacements, small strains
– material nonlinearity only (MNO)
• Large displacements and rotations, small strains
– TL formulation, MNO analysis
– 2PK stress and GL strain substituted for engineering
stress and strain
• Large displacements and rotations, large strains
– TL or UL formulation
– Complicated constitutive models

10. Rheology models for plasticity
Ideal or perfect plasticity, no hardening

11. Loading, unloading, reloading and cyclic loading in 1D
stress
strain
+
new
yield
stress
1
-
new
yield
stress
1
initial
yield
stress
Isotropic hardening
new
yield
stress
2
loading
unloading
reloading

12. Isotropic hardening in principal stress space
3
2
1
Y
3
1 ,
0
)
(
tension
in
stress
yield
1D
and
stresses
principal
by
expressed
Tresca





 





F
0
2
]
)
(
)
(
)
[(
tension
in
stress
yield
1D
and
stresses
principal
by
expressed
Mises
von
2
Y
2
1
3
2
3
2
2
2
1 






 






F
 - plane
arccos (2/sqrt(3))

13. stress
strain
+
new
yield
stress
1
initial
yield
stress
Kinematic hardening
loading
unloading
reloading
Loading, unloading, reloading and cyclic loading in 1D

14. Kinematic hardening in principal stress space
constant
...
,
where
,
0
)
(
take
we
hardening)
isotropic
of
case
in
(as
0
)
(
of
instead
P
c
c
F
F
ij
ij
ij
ij
ij










15. Von Mises yield condition, four hardening models
1. Perfect plasticity – no hardening 2. Isotropic hardening
3. Kinematic hardening 4. Isotropic-kinematic

16. Different types of yield functions
)
,
,
(
have
could
we
all,
at
general
not
is
which
Generally,
invariant.
an
usually
,
of
function
scalar
a
is
)
(
e
wher
hardening
isotropic
)
,
(
way
different
a
in
of
component
every
on
depends
hardening
hardening
isotropic
-
non
)
,
(
constant.
a
is
and
e
wher
hardening
kinematic
)
(
.
strain
plastic
permanent
the
on
depends
h
whic
flow)
(free
ns
dislocatio
of
motion
the
blocking
on
depends
hardening
the
Generally,
...
ns
dislocatio
of
nition
Defi
ns.
dislocatio
of
motion
by
caused
is
flow
material
tic
Plas
region.
plasticity
the
around
exists
which
structure
material
healthy'
'
by the
stabilized
is
it
practice
It
forever.
so
do
to
inclided
is
and
flow
to
starts
material
hardening,
no
means
plasticity
perfect
)
(
P
P
P
P
P
P
K
F
F
K
K
K
F
F
F
F
c
c
F
F
F
F
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij



























17. Plasticity models – physical relevance
• Von Mises
- no need to analyze the state of stress
- a smooth yield sufrace
- good agreement with experiments
• Tresca
- simple relations for decisions (advantage for hand calculations)
- yield surface is not smooth (disadvantage for programming,
the normal to yield surface at corners is not uniquely defined)
• Drucker Prager
a more general model

18. 1D example, bilinear characteristics
P
E
T d
d
d 

 


d

plastic
elastic

strain
stress
total
P
T
d
d
d
E
E
E





EP
T d
d
d 

 

E


tan
T
tan E


Strain hardening parameter
Y

H
E
E
E
E
E
E
E
E 






/
1 T
T
T
T
P
… means total or elastoplastic
… elastic modulus
… tangent modulus

19. Strain hardening parameter again
Elastic strains removed
Initial yield
Upon unloading and reloading the effective stress must exceed
Geometrical meaning of the strain hardening parameter is
the slope of the stress vs. plastic strain plot

20. How to remove elastic part
T
T
P
E
E
E
E
E



21. 1D example, bar (rod) element
elastic and tangent stiffness
L

A
F F
Y

 
Y

 
L
EA
F
k 


E
 L
A
E
L
A
F
k
P
E
P
P
T
T
d
d
d
d
d
d
d





 
















P
P
P
P
P
T 1
/
d
/
d
/E
d
E
E
E
L
EA
E
E
L
A
E
k



Elastic stiffness
Tangent stiffness

22. Results of 1D experiments must
be correlated to theories capable
to describe full 3D behaviour of
materials
• Incremental theories relate stress increments to strain increments
• Deformation theories relate total stress to total strain

23. Relations for incremental theories
isotropic hardening example 1/9




 t
t d
d
lim
:
rates
and
increments
between
Relation
0
surface
yield
back to
go
0
0
that
means
it
-
neutral
0
and
0
tic
elastoplas
0
and
0
elastic
0
and
0
elastic
0
if
and
on
depends
n
deformatio
of
increment
0
)
,
(
is
surface
yield
Let the
P
eff
eff
eff
eff
P










F
F
F
F
F
F
F
ij
ij
ij












Parameter only

24. Relations for incremental theories
isotropic hardening example 2/9
Eq. (i) … increment of plastic deformation has a direction
normal to F while its magnitude (length of vector) is not yet known
defines outer normal to F
in six dimensional stress space
0
d
so
ns,
deformatio
plastic
during
zero
be
must
which
d
d
d
al
differenti
total
a
as
expressed
be
can
}
{
and
scalar
unknown
far
so
is
where
1947)
(Drucker,
form
in the
assumed
is
rule
Flow
P
P
P
P
T
31
11
P






















F
F
F
F
F
F
F
F
F
F
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij



















q
q

25. Relations for incremental theories
isotropic hardening example 3/9
elastic total plastic deformations
matrix of elastic moduli
(iii)
eq.
)
(
are
increments
stress
(ii)
eq.
0
d
d
form
the
in
expressed
be
can
0
d
condition
the
}
{
Denoting
P
E
P
T
T
P
T
T
T
P
31
P
11
ε
ε
E
ε
E
σ
ε
p
q
ε
p
q
p

























F
F
F

26. Relations for incremental theories
isotropic hardening example 4/9
q
E
q
q
p
E
q
T
T
T
get
we
(iii)
increments
stress
for
and
(ii)
0
d
(i),
rule
flow
for
relations
the
Combining




F


Dot product and quadratic form … scalar
Row vector
Column vector
Lambda is the scalar quantity determining the magnitude
of plastic strain increment in the flow rule
Still to be determined

27. Relations for incremental theories
isotropic hardening example 5/9



 P
P
E
with
)
(
write
can
we
increment
stress
for the
Now,
ε
ε
ε
E
ε
E
σ 



 q

determined
be
to
has
still
where
)
(
with
form
the
in
increment
strain
total
of
function
a
as
increment
stress
get the
we
for
ng
Substituti
T
T
T
EP
EP
p
Eq
q
q
p
Eq
Eq
E
E
ε
E
σ



 


equal to zero for perfect plasticity
diadic product

28. Relations for incremental theories
isotropic hardening example 6/9
ij
ij
t
t
ij
ij
t
t
t
t
ij
t
ij
ij
ij
ij
t
W
F
A
W
W
W
F
W
W
f
f
F
s
s
J
J
F
F
F



























































P
P
Y
3
2
Y
P
Y
Y
3
2
P
ij
P
P
Y
Y
P
ij
P
ij
P
P
Y
P
Y
P
2
1
D2
2
Y
3
1
D2
T
P
31
P
11
and
using
F
rule
Chain
increments
plastic
by
done
work
d
),
(
at
suggest th
s
Experiment
)
(
need
we
evaluate
to
invariant
deviatoric
second
the
is
where
0
condition
yield
Mises
von
Assume
}
{
of
ion
Determinat


p
A new constant defined
At time t

29. Relations for incremental theories
isotropic hardening example 7/9
 P

E
P
E
T
E
Y
0

Y

t
P
0
 P

t
P
W
T
31
22
11
T
T
P
Y
P
Y
P
Y
Y
P
2
Y
0
2
Y
P
P
P
P
Y
0
Y
t
P
Y
0
Y
2
1
P
}
{
finaly
so
3
2
3
2
3
2
)
(
2
1
)
(
stics
characteri
bilinear
1D
)
(
done
work
elastic
the
1D
in
















A
E
E
EE
E
E
A
E
W
E
W
E
W
t
t
t
t
t
t
t
t
















p
W

30. Relations for incremental theories
isotropic hardening example 8/9
ε
E
σ
bb
E
E
b
q
Eq
q
Eq
b
q
p
p
q
s
σ
ε




EP
T
EP
T
T
T
T
31
23
12
33
22
11
T
T
T
31
23
12
33
22
11
T
31
23
12
m
33
m
22
m
11
T
31
23
12
33
22
11
33
22
11
3
1
m
Y
,
,
}
{
3
2
}
2
2
2
{
}
{
}
{
)
(
follows
as
compute
can
we
and
and
given
For
Summary.




















c
a
c
a
A
E
E
EE
A
s
s
s
s
s
s
s
s
s
s
s
s
ij
ij






















31. J2 theory, perfect plasticity 1/6
alternative notation … example of numerical treatment
)
2
,
2
,
2
,
1
,
1
,
1
(
diag
]
[
with
},
]{
[
}
{
or
)
2
2
2
(
deviator
stress
of
invariant
second
}
{
}
{
deviator
stress
stress
mean
)
(
}
{
}
{
}
{
}
{
law
s
Hooke'
}...
]{
[
}
{
T
2
1
2
2
2
2
2
2
2
2
1
2
D2
T
m
m
m
3
1
m
T
T



















M
s
M
s
J
s
s
s
s
s
s
J
J
s
E
zx
yz
xy
zz
yy
xx
zx
yz
xy
zz
yy
xx
zz
yy
xx
zx
yz
xy
zz
yy
xx
zx
yz
xy
zz
yy
xx






























32. J2 theory, numerical treatment …2/6
Y
eff
T
2
eff
T
T
behaviour
plastic
perfectly
for
criterion
yield
2
/
}
]{
[
}
{
3
3
stress
effective
Mises
von
)
1
/(
with
},
{
2
}
]{
][
[
also
and
0
since
},
]{
[
}
{
}
]{
[
}
{
that
prove
can
one















s
M
s
J
E
G
s
G
s
M
E
s
s
s
s
M
s
M
s zz
yy
xx

33. J2 theory, numerical treatment …3/6
endif
0
else
,
0
then
if
by
expressed
be
can
region
elastic
in
n
deformatio
plastic
no
increment
,
derivative
time
its
...
}
{
2
}
]{
[
}
]{
[
}
]{
[
}
{
law
s
Hooke'
...
}
]{
[
}
]{
[
}
{
parameter
unknown
far
so
is
...
}
]{
[
}
{
}
{
hypothesis
Reuss
-
Prandtl
to
according
rule
Flow
Y
eff
P
P
E
T































s
G
E
E
E
E
E
s
M
F





Six nonlinear differential equations + one algebraic constraint (inequality)
There is exact analytical solution to this. In practice we proceed numerically

34. J2 theory, numerical treatment …4/6
T
2
Y
EP
EP
2
Y
T
2
Y
T
2
Y
2
eff
2
T
T
T
T
T
eff
T
T
eff
eff
Y
eff
3
with
finally
2
3
4
3
get
we
3
/
4
3
/
4
4
2
that
realizing
and
2
for
ng
Substituti
0
also
and
0
0
2
3
condition
plasticity
ating
Differenti
ss
E
E
ε
E
σ
ε
s
ε
ME
s
Ms
s
Ms
s
ε
ME
s
σ
σ
M
s
Ms
s
Ms
s
s
s














G
G
G
G
J
G
G
G



























System of six nonlinear
differential equations
to be integrated

35. J2 theory, numerical treatment …5/6
predictor-corrector method, first part: predictor
1. known stress
2. test stress (elastic shot)
3a. elastic part of increment
T
)
1
( s

 r
T
s

r
3b. plastic part of increment
T
c
.
4 s
s
s 

 r
t
t
σ
)
2
/(
)
1
(
3
.
5 2
Y
T
c 
 ε
s 


 r
ε
E
σ
σ
σ
σ 




 t
t T
T
c
T
'
2
.
6 s
σ
σ 




 G
t
t

36. J2 theory, numerical treatment …6/6
predictor-corrector method, second part: corrector
'
'
'
'
'
eff
Y
Y
'
eff
Y
'
eff
Y
eff
'
)
1
(
have
we
ions
considerat
plasticity
into
enter
not
does
tensor
stress
the
of
part
spherical
the
since
and
)
1
(
)
(
)
(
)
(
)
(
a way that
such
in
find
For
Correction
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t




































s
σ
σ
s
s
s
s
s
s
s
s
s
















37. Secant stiffness method and the method of radial return