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FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
G – FE FORMULATIONS FOR PLASTICITY
1
These slides are designed based on the book:
Finite Elements in Plasticity – Theory and Practice, D.R.J. Owen and E. Hinton,
1970, Pineridge Press Ltd., Swansea, UK.
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
2
Course Content:
A – INTRODUCTION AND OVERVIEW
Numerical method and Computer-Aided Engineering; Physical
problems; Mathematical models; Finite element method.
B – REVIEW OF 1-D FORMULATIONS
Elements and nodes, natural coordinates, interpolation function, bar
elements, constitutive equations, stiffness matrix, boundary
conditions, applied loads, theory of minimum potential energy; Plane
truss elements; Examples.
C – PLANE ELASTICITY PROBLEM FORMULATIONS
Constant-strain triangular (CST) elements; Plane stress, plane
strain; Axisymmetric elements; Stress calculations; Programming
structure; Numerical examples.
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
3
D – ELASTIC-PLASTIC PROBLEM FORMULATIONS
Iterative solution methods,1-D problems, Mathematical theory of
plasticity, matrix formulation, yield criteria, Equations for plane stress
and plane strain case; Numerical examples
E – PHYSICAL INTERPRETATION OF FE RESULTS
Case studies in solid mechanics; FE simulations in 3-D; Physical
interpretation; FE model validation.
.
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
4
OBJECTIVE
To learn the use of finite element method for
the solution of problems involving elastic-
plastic materials.
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Terikan
0.0 0.1 0.2 0.3 0.4 0.5
Tegasan (
MP
a)
0
200
400
600
Ujikaji A
Ujikaji B
5
Stress-strain diagram
P
P
lo+lloAo
• Effects of high straining rates
• Effects of test temperature
• Foil versus bulk specimens
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
6
Elastic-Plastic Behavior
Characteristics:
(1) An initial elastic material
response onto which a plastic
deformation is superimposed after
a certain level of stress has been
reached
(2) Onset of
yield governed
by a yield
criterion
(3) Irreversible on
unloading
(4) Post-yield deformation occurs
at reduced material stiffness
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
7
Monotonic stress-strain behavior Engineering stress
oA
PS
A
P
oo
o
l
l
l
lle
l
dld
ol
l
l
dlln
eS 1
eA
Ao 1lnln
Typical low carbon steelTrue stress
Engineering strain
True or natural strain
For the assumed constant
volume condition,
Then;
oo lAlA
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
8
Plastic strains
Elastic region – Hooke’s law:
E
Terikan
0.0 0.1 0.2 0.3 0.4 0.5
Tegasan (
MP
a)
0
200
400
600
Ujikaji A
Ujikaji B
Engineering stress-strain diagram
log)1(loglog E
Plastic region:
npK
logloglog nK
True stress-strain diagram
SS316 steel
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
9
PLASTIC STRAIN, p
0.0 0.1 0.2 0.3 0.4 0.5 0.6
ST
RE
SS
,
(M
Pa
)
0
100
200
300
400
500
600
700
PLASTIC STRAIN, p
0.01 0.1 1
ST
RE
SS
,
(M
Pa)
100
1000
= Kn
log K = 2.8735n = 0.1992
r2 = 0.9772
199.03.747 p
Plastic strains - Example
Non-linear /Power-law
SS316 steel npK
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
10
Ideal Materials behavior
Elstic-perfectly plastic
perfectly plastic
Elstic-linear hardening
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
11
Simulation of elastic-plastic problem
Requirements for formulating the theory to model elastic-plastic
material deformation:
• explicit stress-strain
relationship under elastic
condition.
• a yield criterion
indicating the onset
of plastic flow.
• stress-strain relationship
for post yield behavior
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
12
Elastic-plastic problem in 2-D
In elastic region:
klijklij C
jkiljlikklijijklC
, are Lamé constants
jiif
jiifij
0
1
Kronecker’s delta:
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Elastic case for plane stress and plane strain (Review)
13
x
y
xy
u
x
v
y
u v
y x
Strain
Stress
x
y
xy
D
Hooke’s Law
2
12
1 0
1 01
0 0 v
vE
D vv
12
1 0
1 01 1 2
0 0
v vE
D v vv v
v
Elasticity matrix
Plane stress
Plane strain
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
14
Yield criteria:
Stress level at which plastic deformation begins
kf ij
k() is an experimentally determined function of hardening parameter
The material of a component subjected to complex loading will start to
yield when the (parametric stress) reaches the (characteristic stress) in
an identical material during a tensile test.
Other parameters:
Strain
Energy
Specific stress component (shear stress, maximum principal stress)
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
15
Yield criteria:
Stress level at which plastic deformation begins
kf ij
Yield criterion should be independent of coordinate axes orientation,
thus can be expressed in terms of stress invariants:
kijkij
ijij
ii
J
J
J
3
1
2
1
3
2
1
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
16
Plastic deformation is independent of hydrostatic pressure
kJJf 32 ,
kkijijij 3
1
are the second and third invariant of deviatoric stress: 32 , JJ
+
deviatoric hydrostatic
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
17
Yield Criteria
For ductile materials:
•Maximum-distortion (shear) strain energy criterion (von-Mises)
• Maximum-shear-stress criterion (Tresca)
For brittle materials:
• Maximum-principal-stress criterion (Rankine)
• Mohr fracture criterion
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
18
Maximum-distortion-energy theory
(von Mises)
22
13
2
32
2
21 2 Y
22
221
2
1 Y
+
deviatoric hydrostatic
For plane problems:
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
19
Maximum-normal-stress theory
(Rankine)
ult
ult
2
1
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
20
Tresca and von Mises yield criteria
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
21
Tresca and von Mises yield criteria
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
22
Strain Hardening behavior
The dependence of stress level for further plastic deformation,
after initial yielding on the current degree of plastic straining
Perfect plasticity
- Yield stress level does not depend on current
degree of plastic deformation
Strain Hardening models
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
23
Isotropic strain hardening
-Current yield surfaces are a uniform expansion of the original yield curve,
without translation
-For strain soften material, the yield stress level at a point decreases with
increasing plastic strain, thus the original yield curve contracts
progressively without translation
-Consequently, the yield surface becomes a failure criterion
Strain Hardening models
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
24
Kinematic strain hardening
-Subsequent yield surface preserves their shape
and orientation but translates in the stress space.
-This results in Bauschinger effect on cyclic loading
Strain Hardening models
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
25
Bauschinger effect
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
26
Progressive development of yield surface
To relate the yield stress, k to the plastic deformation
by means of the hardening parameter,
Model 1: The degree of work hardening, is
expressed as a function of the total plastic work, Wp
pijijp dW
Increment of plastic strain
Model 2: The strain hardening, is related to a measure
of the total plastic deformation termed the effective,
generalized or equivalent plastic strain:
pp d
pijpijp ddd
3
2
Since yielding is independent of
hydrostatic stress, (dii )p = 0), thus:
pijpijp
pijpij
ddd
dd
3
2
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
27
Isotropic Hardening
For a stress state where f = k represents the plastic state while
elastic behavior is characterized by f < k, an incremental
change in the yield function is:
ij
ij
df
df
Elastic unloading
Neutral loading (plastic behavior for perfectly-plastic material)
Plastic loading (for strain hardening materials)0
0
0
df
df
df
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
28
Elastic-plastic stress-strain relations
In plastic region: pijeijij ddd
Elastic strain increment: kkij
ij
eij dE
dd
21
2
deviatoric hydrostatic
Plastic Flow Rule:
Assumes that the plastic strain increment is proportional to the
stress gradient of the plastic potential, Q:
ij
pij
ijpij
Qdd
Qd
Plastic multiplier
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
29
Associated theory of plasticity
Since the Flow Rule governs the plastic flow after yielding, thus
Q must be a function of and , similar to f, thus:2J 3J
Qf
ij
pij
fdd
Normality condition
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
30
Associated theory of plasticity
Since the Flow Rule governs the plastic flow after yielding, thus
Q must be a function of and , similar to f, thus:2J 3J
Qf
ij
pij
fdd
Normality condition
Particular case of2Jf
ij
ijij
Jf
2
ijpij dd Prandtl-Reuss equations
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
31
The incremental stress-strain relationship for
elastic-plastic deformation:
pijeijij ddd
ij
kkij
ij
eij
fdd
E
dd
21
2
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
32
Uniaxial elastic-plastic strain hardening behavior
Definition of term:
d
dET
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
33
Uniaxial elastic-plastic strain hardening behavior
Yield condition kf ij
Strain hardening hypothesis
pH
Proportional to 2J
p
p
p
Hd
d
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
34
Uniaxial elastic-plastic strain hardening behavior
For uniaxial case 0, 321
ppijpijp
ppp
pp
dd
ddd
dd
3
2
5.032
1
ijij2
3
Plastic straining is
assumed to be
incompressible, = 0.5
E
E
E
d
d
d
dH
T
T
pp
p
1
- The hardening function, H is
determined experimentally from
tension test data.
- H’ is required
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Uniaxial stress-strain curve for elastic-linear hardening
35
Assume
Strain-hardening
parameter:
pe ddd
pd
dH
d
dET
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
36
Numerical solution process for non-linear problems
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Notations:
37
0
FQK
0 FQ K
Nonlinear problem:
Thus, iterative solution is required
Linear elastic problem:
0
fH
0f H
For 1-D (1 variable) problem:
0 fH
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
38
• Direct iteration / successive approximation
• Newton-Raphson method
• Tangential stiffness method
• Initial stiffness method
The problem:
0
fH
0f H
0 fH
Iterative Solution Methods
For single variable
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Iterative Solution Methods
(a) Direct iteration or successive approximation
39
fH-1
H
In each solution step, the previous solution for the
unknowns is used to predict the current values of the
coefficient matrix
For (r+1)th approximation:
fH-11 rr
Convergence is not guaranteed and cannot be predicted at initial
solution stage. Converged iteration when r-1 and r are close.
Initial guess 0 is based on solution for an
average material properties.
0f H
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Direct iteration method
for a single variable
Convex H- relation
40
Task:
to illustrate the process.
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
41
Steps for Direct Iteration Method
Start with {0}
Determine [H(0)]
Calculate
Repeat until {r+1} {r}
fH101
0
H(0)
1 = -H(0) f
r+1 r
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Direct iteration method
for a single variable
42
Concave H- relation
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
43
Assignment # 5
A one degree of freedom problem can be represented by
H + f = 0
where f = 10 and H() = 10 (1+e3)
Design and run a computer algorithm to determine the solution
of the problem using direct iteration method and Newton-
Raphson method.
(a) Show the flow chart of the algorithm and description of the
steps.
(b) Tabulate the iterative values of H and and compare the
converged solution from the two methods. Specify a
convergence tolerance of 1%.
Try with 0 = 0.2Should converge at 7 = -9.444
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
44
Review of Newton-Raphson Method(for finding root of an equation f(x)= 0)
Taylor Series expansion:
...!2
11
2
1111 xfxxxfxxxfxf
Consider only the first 2 terms:
111 xfxxxfxf
Set f(x)=0 to find roots:
1
11
111 0
xf
xfxx
xfxxxf
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
45
Review of Newton-Raphson Method(for finding root of an equation f(x)= 0)
Iterative procedures:
1
11
xf
xfxx ii
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Newton-Raphson Method
46
0
0
fH
fHΨ
rr
The problem
If true solution exists at
Converged iteration when n-1 and n are close
Initial guess is based on 0 is based on solution
for an average material properties.
r
j
iN
1j
r
j
r
i
ψΔψ
rrr J
is the residual force
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Jacobian Matrix
47
HHJ
ψhJ
m
1k
r
k
r
j
ir
ij
j
iij
rrr
rrr
HH
J
1
1
rrr 1
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Newton-Raphson
method for a single
variable
48
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Newton-Raphson
method for a single
variable
49
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
50
Tangential Stiffness Method
(Generalized Newton-Raphson Method)
Assume a trial value {0}
Calculate [H(0)]
Calculate {(0)}
Calculate
Iterate until {r} {0}
0100
H
fH
001
Linearize {} in [H()] such that the term [H’()] can be omitted
Steps:
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Tangential stiffness
method for a single
variable
51
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
52
Initial Stiffness Method
fH rr 11 Recall that in Direct Stiffness Method:
In Tangential Stiffness Method: rrr H 1
This requires complete reduction and solution of the set of
simultaneous equations for each iteration.
Use the initial stiffness [H(0)] for subsequent approximation.
rr H 10
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Initial stiffness method
for a single variable
53
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Uniaxial stress-strain curve for elastic-linear hardening
54
Assume
Strain-hardening
parameter:
pe ddd
pd
dH
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
55
11
11)(
L
EAK
L
EAFK
e
e
e
AdHAddFA
F
LddLddL
p
pe
;
;
HE
E
L
EA
LdE
d
AdH
d
dFK
p
p
ep 1
11
111
)(
HE
E
L
EAK
e
ep
Element stiffness for elastic-plastic material behavior
1 2
L
F
Elastic behavior:
Elastic-plastic behavior:
Element stiffness matrix for elastic-
plastic material behavior
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
56
Numerical singularity issue
Slope, H’ = 0SY
11
111
)(
HE
E
L
EAK
e
ep
After initial yielding, H’=0,
Thus, [Kep](e) = [0]
Use initial stiffness method
to ensure positive definite
[K]
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
57
Program structure
for 1-D problemInput geometry, loading, b.c.
INITIALIZATION Zero all arrays for data storing
INC. LOAD
SET INDICATOR
STIFFNESS
Choose type of solution algorithm – direct iteration,
tangential stiffness, etc
ASSEMBLY
[K](e)
[K] {} = {F}
REDUCTION Solve for {}
RESIDUALCalculate residual force {} for Newton-Raphson,
Initial and Tangential stiffness method
Check for convergence
YN
Output results
ITE
RA
TIO
N L
OO
P
LO
AD
IN
CR
EM
EN
T L
OO
P
FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Incremental stress and strain changes at initial yielding
58
Task:
to illustrate the process.
r
e
rr
e
rr
e
r
rrr
rr
E
H
fH
1
1
11
11 r
pY
r
Y H
Check if the element has
previously yielded: