Topic 17
Modeling of
Elasta-Plastic and
Creep
Response Part I
Contents:
• Basic considerations in modeling inelastic response
• A schematic review of laboratory test results, effects of
stress level, temperature, strain rate
• One-dimensional stress-strain laws for elasto-plasticity,
creep, and viscoplasticity
• Isotropic and kinematic hardening in plasticity
• General equations of multiaxial plasticity based on a
yield condition, flow rule, and hardening rule
• Example of von Mises yield condition and isotropic
hardening, evaluation of stress-strain law for general
analysis
• Use of plastic work, effective stress, effective plastic
strain
• Integration of stresses with subincrementation
• Example analysis: Plane strain punch problem
• Example analysis: Elasto-plastic response up to ultimate
load of a plate with a hole
• Computer-plotted animation: Plate with a hole
Textbook:
Section 6.4.2
Example:
6.20
References:
The plasticity computations are discussed in
Bathe, K. J., M. D. Snyder, A. P. Cimento, and W. D. Rolph III, "On Some
Current Procedures and Difficulties in Finite Element Analysis of Elas­
tic-Plastic Response," Computers & Structures, 12, 607-624, 1980.
17-2 Elasto-Plastic and Creep Response - Part I
References:
(continued)
Snyder, M. D., and K. J. Bathe, "A Solution Procedure for Thermo-Elas­
tic-Plastic and Creep Problems," Nuclear Engineering and Design, 64,
49-80, 1981.
The plane strain punch problem is also considered in
Sussman, T., and K. J. Bathe, "Finite Elements Based on Mixed Inter­
polation for Incompressible Elastic and Inelastic Analysis," Computers
& Structures, to appear.
Topic Seventeen 17-3
. WE
"D/SC.L{ SS"E D
THE "PRE\llO\A<;
IN
LEe­
. WE Now
WANT TO
.:DIS CIA':,'> TI1 E:
TU'?ES THE MODELING
MO"DE LINlQ
OF ELASTIC.
IN ELASTIC
-
l'NEA'R
STRAIN
-
MA1E'RIALS
S~E-S<;­
-
0 F
HATER,ALS
ELASTO-1>LASTICIT'f
ANb CREEP
LA~
NONLINEAR S>T'RESc:,­
WE T'ROc.EEtl
STRAIN LAw
FCLLoWS :
THE T. L.
ANt> lA.L.
FO~t-IULAT10NS.
- WE
AS
- WE J:> \ SC \A S S
MObELlNE;
DISC\i\S, 50 TS'RIEH'i
~'R\E
Fl'l
OF SUCH
~ES'?ONse IN 1-1>
ANAl'lSIS
INELASTIC HATc"RIAL
~EI-\A'J\o'RSJ AS
SE'R'lJE'b
IN
O'Rt\T01<.'(
0\3,-
LAB,TES-TS,
- WE
GENE~AU2E
OuR
HObEUN6 CONSlbERA.lloNS
10 2-1) ANb3.-1)
STRES,S
5\1v..A.TI\lNS
Markerboard
17-1
17-4 Elasto-Plastic and Creep Response - Part I
Transparency
17-1
MODELING OF INELASTIC
RESPONSE:
ELASTO-PLASTICITY, CREEP
AND VISCOPLASTICITY
• The total stress is not uniquely
related to the current total strain.
Hence, to calculate the response
history, stress increments must be
evaluated for each time (load) step
and added to the previous total
stress.
Transparency
17-2
• The differential stress increment is
obtained as - assuming infinitesimally
small displacement conditionsdO'ij- = C~s (de rs
where
-
de~~)
C~s = components of the elasticity
tensor
de rs = total differential strain increment
de~~ = inelastic differential strain
increment
Topic Seventeen 17-5
The inelastic response may occur
rapidly or slowly in time, depending on
the problem of nature considered.
Modeling:
• In ~Iasticity, the model assumes that
de~s occurs instantaneously with the
load application.
• In creep, the model assumes that
de~~ occurs as a function of time.
• The actual response in nature can be
modeled using plasticity and creep
together, or alternatively using a
viscoplastic material model.
-
In the following discussion we
assume small strain conditions,
hence
• we have either a materially­
nonlinear-only analysis
• or a large displacement/large
rotation but small strain
analysis
Transparency
17-3
Transparency
17-4
17-6 Elasto-Plastic and Creep Response - Part I
Transparency
17-5
• As pointed out earlier, for the large
displacement solution we would use
the total Lagrangian formulation and
in the evaluation of the stress-strain
laws simply use
-
Green-Lagrange strain component
for the engineering strain compo­
nents
and
-
Transparency
17-6
2nd Piola-Kirchhoff stress compo­
nents for the engineering stress
components
Consider a brief summary of some
observations regarding material
response measured in the laboratory
• We only consider schematically what
approximate response is observed; no details
are given.
• Note that, regarding the notation, no time, t,
superscript is used on the stress and strain
variables describing the material behavior.
Topic Seventeen 17-7
MATERIAL BEHAVIOR,
"INSTANTANEOUS"
RESPONSE
Transparency
17-7
Tensile Test: Assume
• small strain conditions
• behavior in compression
is the same as in tension
Hence
Cross­
sec:tional
area Ao-z...
e
e - eo
eo
= --::---=-
p
(J'
=
-A
o
engineering
stress, (J
fracture
x ultimate
strain
5'
assumed
I
I
I
I
I
test
ngineering
strain, e
I
I
I
~,,
.......
_-----
,.",;'
'"
Constant temperature
Transparency
17-8
17-8 Elasto-Plastic and Creep Response - Part I
Transparency
17-9
Effect of strain rate:
(J'
de.
.
dt Increasing
e
Effect of temperature
Transparency
17-10
(J'
temperature is increasing
e
Topic Seventeen 17-9
MATERIAL BEHAVIOR, TIME­
DEPENDENT RESPONSE
Transparency
17-11
• Now, at constant stress, inelastic
strains develop.
• Important effect for materials when
temperatures are high
Typical creep curve
Transparency
17-12
Engineering strain, e
fracture
a = constant
temperature = constant
Instantaneous
strain
{
(elastic and
elasto-plastic)
Primary
range
Secondary
range
Tertiary
range
--:..+----=--+------'--~--+------l-
time
17-10 Elasto-Plastic and Creep Response - Part I
Transparency
17-13
Effect of stress level on creep strain
temperature = constant
x
e
x ---.s---::fracture
I
time
Transparency
17-14
Effect of temperature on creep strain
-S" fracture
e
x
u=constant
""x
time
Topic Seventeen 17-11
MODELING OF RESPONSE
Transparency
17-15
Consider a one-dimensional situation:
~
~tu
h=
p
jt.---t
L
'e
tp
A
~ ~L
• We assume that the load is increased
monotonically to its final value, P*.
• We assume that the time is "long" so
that inertia effects are negligible
(static analysis).
plasticity
effects
P*:..-....r_--,pt-redominate
Load
.creep effects
Transparency
f
17-16
time-dependent inelastic strains
are accumulated - modeled as
creep strains
'-"
time inte~al t* (small)
without time-dependent
inelastic strains
time
17-12 Elasto-Plastic and Creep Response - Part I
Transparency
17-17
Plasticity, uniaxial, bilinear
material model stress
rry- - - - -
strain
·1·
tOe
= E teE
telN = teP
t rr
Creep, power law material model:
eC
Transparency
17-18
=
d
t*
(small)
t
ao
a
<T ,
t a2
t rr
=
E teE
telN
=
teC
+ teP
time
• The elastic strain is the same as in
the plastic analysis (this follows from
equilibrium).
• The inelastic strain is time-dependent
and time is now an actual variable.
'Ibpic Seventeen 17-13
Viscoplasticity:
• Time-dependent response is modeled
using a fluidity parameter 'Y:
e = <1E + 'Y (~
cry
.
.
-
Transparency
17-19
1)
/
VP
e
where
(cr _ cry) -_ {
0 I cr -< cry
cr - cry I cr > cry
Typical solutions (1-0 specimen):
steady~state depends on
total
strain.
.
Increasing 'Y
elastic
strain
solution
total
strain \
increase in
.
fry as function
VP
of e
elastic
strain
time
non-hardening material
time
hardening material
Transparency
17·20
17-14 Elasto-Plastic and Creep Response - Part I
Transparency
17-21
PLASTICITY
• So far we considered only loading
conditions.
• Before we discuss more general
multiaxial plasticity relations, consider
unloading and cyclic loading
assuming uniaxial stress conditions.
Transparency
17-22
• Consider that the load increases in
tension, causes plastic deformation,
reverses elastically, and again causes
plastic deformation in compression.
load
elastic
plastic
time
Topic Seventeen 17-15
Bilinear material assumption, isotropic
hardening
ET
Transparency
(T
17-23
(Ty
e
plastic strain
e~
(T~
plastic strain
I---+-l
e~
Bilinear material assumption, kinematic
hardening
(J
ET
Transparency
17-24
(Ty
e
plastic strain
e~
17-16 Elasto-Plastic and Creep Response - Part I
Transparency
17-25
MULTIAXIAL PLASTICITY
To describe the plastic behavior in
multiaxial stress conditions, we use
• A yield condition
• A flow rule
• A hardening rule
In the following, we consider isothermal
(constant temperature) conditions.
Transparency
17-26
These conditions are expressed using a
stress function tF.
Two widely used stress functions are the
von Mises function
Drucker-Prager function
Topic Seventeen 17-17
von Mises
tF
= ~ ts
2
I}
Transparency
17-27
ts .. _ tK
II'
Drucker-Prager
t a.
_II ;
t(j
3
= ~ -1 ts.. ts .
2
If
If
We use both matrix notation and index
notation:
de~1
de~2
de~3
de~2 + de~1
de~3 + de~2
de~3 + de~1
matrix notation
dCT
=
dCT11
dCT22
dCT33
dCT12
dCT23
dCT31
note that both de~2
and de~1 are added
Transparency
17-28
17-18 Elasto-Plastic and Creep Response - Part I
Transparency
17-29
deijP--
[
de~,
de~1
de~1
de~2
de~2
de~2
de~3 ]
de~3
de~3
index
notation
d<J~
Transparency
17-30
=
[ dcrl1
d<J21
d<J31
d<J12 dcr'3 ]
d<J22 d<J23
d<J32 d<J33
The basic equations are then (von Mises tF):
1) Yield condition
tF C<Jij, tK) = 0
current
str~sses~ function of
plastic strains
tF is zero throughout the plastic response
• 1-D equivalent:
(uniaxial stress)
~ C<J2 - t<J~) = 0
7
current stresses
~
function of
plastic strains.
Thpic Seventeen 17-19
2) Flow rule (associated rule):
P
deii. =
I
where
tx.
t
atF
X. -at
Transparency
17-31
ITi}
is a positive scalar.
• 1-D equivalent:
P
2 t\ t
de11 = 3 I\. IT
P
1
d e22 = - 3
t\ t
1
t\ t
P
d e33
= -
3
I\.
I\.
IT
IT
Transparency
17-32
3) Stress-strain relationship:
dIT = C E (d~ - d~P)
• 1-D equivalent:
dIT
= E (de11
- de~1)
17-20 Elasto-Plastic and Creep Response - Part I
Transparency
17-33
Our goal is to determine CEP such that
EP
dO"
=
C
de
~-
\
instantaneous elastic-plastic stress-strain matrix
General derivation of CEP :
Transparency
17-34
Define
Topic Seventeen 17-21
Transparency
17-35
Using matrix notation,
:: tq22:: tq33::
tgT = [tq11
't. t
i
P22:
I
We now determine
=0
Using tF
= - t-
a(Ji}
=
tx.
tIt
P12
i
P23
It]
i
P31
in terms of de:
during plastic deformations,
atF
t
dF
i
P33
d(Jii. +
I
'gT do- -
atF
--:::t=Pat
ei}
p
deii.
I
'ET~
tx.tg
=0
Transparency
17-36
17-22 Elasto-Plastic and Creep RespoDse - Part I
Transparency
Also
17-37
tg TdO" = tg T (QE (d~ - d!t))
L
.
t
The flow rule assumption may be
written as
Hence
tgT dO" = tgT (C E (d~ _ tA tg ))= tA tQT tg
-
Transparency
Solving the boxed equation for t A gives
17-38
Hence we can determine the plastic
strain increment from the total strain
increment:
I
..
tota strain Increment
p
~ de
-
plastic strain
increment
=
tgTCEd~~t
( tQTt + tgT CEtg) 9
g
Topic Seventeen 17-23
Transparency
17-39
Example: Von Mises yield condition,
isotropic hardening
Two equivalent equations:
t
~
O"Y=T
(trr •
t
)2
(t
t)2
- 2:
t
t)2
~~\+t-t~/?~
principal stresses
tF _ 1
(t
Transparency
17-40
t
t. t _
sij. ~ K , K -
~5
·
. stresses: "Sij- =
deVlatonc
1
3
t 2
0" Y
t
(Tij- -
(Tmm
-3-
~
Uij
17-24 Elasto-Plastic and Creep Response - Part I
Transparency
17-41
Transparency
17-42
End view of
yield surface
Yield surface
for plane stress
We now compute the derivatives of the
yield function.
First consider tpy.:
_
a (1 t t
1t 2
- - atef! 2 sy. SiJ- - 3 O'y)
t _
atF
py. - - atef!
II'
t(J"it
II'
fixed
(t(J"ij-
fixed implies tSij- is fixed)
'Ibpic Seventeen 17-25
What is the relationship between t cry
and the plastic strains?
Transparency
17-43
We answer this question using the
concept of "plastic work".
• The plastic work (per unit volume) is
the amount of energy that is
unrecoverable when the material is
unloaded.
• This energy has been used in
creating the plastic deformations
within the material.
• Pictorially: 1·0 example
Transparency
stress
17-44
~et
Shaded area equals
plastic work t wp :
' - - - - - - - - - - strain
=Jor
tePIjo
• In general, 'W
p
Tay.de~
17-26 Elasto-Plastic and Creep Response - Part I
Transparency
17-45
Consider 1-0 test results: the current
yield stress may be written in terms of
the plastic work.
Shaded area =:
Wp = ~ (~T
- i) (t(1~
-
(1~)
' - - - - - - - - - - strain
Transparency
17-46
We can now evaluate tPij- - which
corresponds to a generalization of the
1-D test results to multiaxial conditions.
t
t
(1Y
t _ 2 t (d\JY a W p
Pij- - 3 (]'Y dtW p ater
ate~
\
.
)/a
t /
'Ibpic Seventeen 17-27
Alternatively, we could have used that
dtW p = tcr dtet
where
(effective stress)
Transparency
17-47
(increment iii
effective
t plastic strain)
and then the same result is obtained
using
P
t .. _ 2 t (d\ry ate )
Pit - 3 (J'y dte P ate~
Next consider tq~:
Transparency
17-48
17-28 Elasto·Plastic and Creep Response - Part 1
Transparency
17-49
We can now evaluate gEP:
de11
I
1 -V - ~ (511
' ) 'I -V- -
1-2v
2 de12
de22
2
11-2v
I
I
I522 I •.• I - P
Q. 1
~ '
511
511 , 512 I •.•
I
I
I
- 1 - I" - - - - - I .......
- - - - - - - - - - - - - - - - -:2:
I
I
cEP
-
=
L
_E_
1+v
1- V
(')
1 _ 2v - ~ 522
I
I
I···
I
1_
-
I .•• I I
Q. '
I
,
P 522 512 I···
I" - - - - - - I I
~ '533 1512 I ...
-'1-----
I
2'­
symmetric
I I
2
-~('512)
I •.•
I
-------1I •••
I
where
13
= -3 f1
2
2
O"y
(
1
+ g E 1ET 1 + v )
3 E - ET
E
Evaluation of the stresses at time t+ ~t:
Transparency
17-50
The stress integration
must be performed at
each Gauss integration
point.
Topic Seventeen 17-29
Transparency
17-51
We can approximate the evaluation of
this integral using the Euler forward
method.
• Without subincrementation:
t+~le
I+~I
- EP
e
EP
Ie
C d~ . C
,d~"/
J:
-
I
- - -e
t
• With n subincrements:
Transparency
17-52
de
n
~t
t+~--s-­
n
+ ...
de
t+(n-1)Lh
n
17-30 Elasto-Plastic and Creep Response - Part I
Transparency
17-53
Pictorially:
t
t+8t
subincrements
Transparency
17-54
Summary of the procedure used to
calculate the total stresses at time
t+dt.
Given:
STRAIN = Total strains at time t+dt
SIG = Total stresses at time t
EPS = Total strains at time t
(a) Calculate the strain increment
DELEPS:
DELEPS = STRAIN - EPS
Topic Seventeen 17-31
(b) Calculate the stress increment
DELSIG, assuming elastic behavior:
E
DELSIG = C * DELEPS
(c) Calculate TAU, assuming elastic
behavior:
TAU = SIG + DELSIG
Transparency
17-55
(d) With TAU as the state of stress,
calculate the value of the yield
function F.
(e) If F(TAU) < 0, the strain increment
is elastic. In this case, TAU is
correct; we return.
(f) If the previous state of stress was
plastic, set RATIO to zero and go
to (g). Otherwise, there is a
transition from elastic to plastic and
RATIO (the portion of incremental
strain taken elastically) has to be
determined. RATIO is determined
from
F (SIG + RATIO
* DELSIG)
=
0
since F = 0 signals the initiation of
yielding.
Transparency
17-56
17-32 Elasto·Plastic and Creep Response - Part I
Transparency
17-57
(9) Redefine TAU as the stress at start
of yield
TAU = SIG + RATIO * DELSIG
and calculate the elastic-plastic
strain increment
DEPS = (1 - RATIO) * DELEPS
(h) Divide DEPS into subincrements
DDEPS and calculate
TAU ~ TAU + CEP * DDEPS
for all elastic-plastic strain
subincrements.
Topic Seventeen 17-33
1.... ....---
2b
Slide
17-1
---I.~I
p
PLASTIC ZONE
Plane strain punch problem
P/l
I
z
lOl-
10-
A
Ii.
.....
I
,,"~.
Finite element model of punch problem
Slide
17-2
17-34 Elasto-Plastic and Creep Response - Part I
'CW..
"200 Ib.
Slide
17-3
h
••
.. .
\
"
.. ""
T,.
STRESS, 0' lplll
- - -
THEOR£TICAL
'00
-- - -----0
I 0
2.0
30
DISTANCE FROM CENTERLINE, b (inl
Solution of Boussinesq problem-2 pt. integration
Slide
17-4
200
•
VIZ
• er"
..
STRESS, 0' lpoi)
- - -
T,Z
TH(ORETlCAl
'00
--- ---- ---10
2.0
30
DISTANCE FROM CENTERLINE, b (in)
Solution of Boussinesq, problem-3 pt. Integration
Topic Seventeen 17-35
Slide
17-5
p
~
APPLIED
LOAD,
P/2kb
0.05
QIQ
DISPLACEMENT,
• 2 pi In,
a S"pI inl
0.'5
0.20
,,/b
Load-displacement curves for punch problem
17-36 Elasto-Plastic and Creep Response - Part I
Transparency
17-58
Limit load calculations:
p
rTTTTl
o
• Plate is elasto-plastic.
Elasto-plastic analysis:
Transparency
17-59
Material properties (steel)
a
740
-I--
a
(MPa)
~
Er = 2070 MPa, isotropic hardening
'-E=207000 MPa, v=0.3
e
• This is an idealization, probably
inaccurate for large strain conditions
(e > 2%).
Topic Seventeen 17-37
TIME =
LOAD =
0
0.0 MPA
Computer
Animation
Plate with hole
TIME - 41
LOAD - 512.5 MPA
TIME - 62
LOAD - 660.0 MPA
--.,.,......
----..
--....
-----,
..
_.
......
=====..:=r~
~"
~
,,
-.~
...........,
• •• •1
• •• •1
!!!!
~
\
..-
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Finite Element Procedures for Solids and Structures
Klaus-Jürgen Bathe
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
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