(metal plasticity) chapter 15: hardening 10... · 2019. 9. 5. · m.g. lee materials science and...
TRANSCRIPT
M.G. Lee
Materials Science and Engineering, Seoul National University - 1 -
446.631A
소성재료역학(Metal Plasticity)
Chapter 15: Hardening
Myoung-Gyu Lee
Office: Eng building 33-309
Tel. 02-880-1711
TA: Chanyang Kim (30-522)
M.G. Lee
Materials Science and Engineering, Seoul National University - 2 -
Hardening
General principle for strengthening
How to increase strength of metals?- Place obstacles in the path of dislocations, which inhibits the free
movement of dislocations until the stress is increased to move them forward.
2
φcos
L
Gbτ c
L
c
cc φ180φ
Effective particle spacing
Critical angle to which the dislocation bends to breaking away from the obstacle
M.G. Lee
Materials Science and Engineering, Seoul National University - 3 -
Hardening
Work hardening from physical metallurgy
During plastic deformation, there is an increase in dislocation density. It is this increase in dislocation density which ultimately leads to work hardening
Dislocation interact with each other and assume configurations that restrict the movement of other dislocations
The dislocations can be either “strong” or “weak” obstacles depending on the types of interactions that occurs between moving dislocations
M.G. Lee
Materials Science and Engineering, Seoul National University - 4 -
Hardening
Work (strain) hardening of single crystal
Stage I: the stress fields interact during the early stages of deformation resulting in “weak” drag effects
In stage I, dislocations are “weak” obstacles
M.G. Lee
Materials Science and Engineering, Seoul National University - 5 -
Hardening
Work (strain) hardening of single crystal
Stage II: Linear hardening
In stage II, work hardening depends strongly on dislocation density
M.G. Lee
Materials Science and Engineering, Seoul National University - 6 -
Hardening
Work (strain) hardening of single crystal
Stage II: Linear hardening In stage II, work hardening depends
strongly on dislocation density Lomer-Cottrell locks (sessile
dislocation)
M.G. Lee
Materials Science and Engineering, Seoul National University - 7 -
Elastic Plasticity Models
Elastic-plasticity models in sheet metal forming
1) Yield function
2) Hardening model
3) Elastic modulus
Accurate predictions
for stress, deformation
M.G. Lee
Materials Science and Engineering, Seoul National University - 8 -
Yield function
Elasticity models for FEM
Yield functions
• Elastic vs. Plastic
• Rate of plastic strain
pe εεε Yσ1D
σ
fdλdε
p
σf3D
(6D)
w
t
r =de
w
det
R-value
M.G. Lee
Materials Science and Engineering, Seoul National University - 9 -
Yield function
Yield functions
elastic
Plastic
plastic
1D 2D 3D
1-D 2-D 3-D (plane stress)
M.G. Lee
Materials Science and Engineering, Seoul National University - 10 -
Yield function
From Kuwabara
Plane stress
M.G. Lee
Materials Science and Engineering, Seoul National University - 11 -
Yield function
From Kuwabara
Uniaxial tension
Biaxial tension
(small strain range)
Uniaxial
compression
y
x
Von Mises
Hill 1948
Barlat 2000-2d
M.G. Lee
Materials Science and Engineering, Seoul National University - 12 -
Hardening law: importance
Hardening behavior
UpperBottom
Com
pre
ssio
nT
ensio
n
Sidewall region
(Upper) Tension – Compression – Unloading
(Bottom) Compression – Tension – Unloading
Stress-strain behavior under reverse loading
σxx
M.G. Lee
Materials Science and Engineering, Seoul National University - 13 -
Hardening law: importance
Hardening models
CT σσ
CT σσ
Bauschinger effect can be reproduced by kinematic hardening
Isotropic hardening
kinematic hardening
Yσ
Yσ-
Tσ
Cσ
AB
C
Tσ
Cσ
AB
C
σ1
σ2
A
B
C
σ1
σ2
αC B
A
M.G. Lee
Materials Science and Engineering, Seoul National University - 14 -
Hardening law: basic
Uniaxial stress-strain curve of typical metals
M.G. Lee
Materials Science and Engineering, Seoul National University - 15 -
Hardening law: basic
0σf ij
0K,σf iij
Initial yield surface
For perfect plasticity, the yield surface
remains unchanged. But, in general case,
the size, position, shape of the yield
surface change
where K represents hardening
parameters, which evolves during the
plastic deformation
Size
Position Shape
M.G. Lee
Materials Science and Engineering, Seoul National University - 16 -
Hardening law: basic
Isotropic hardening
0KσfK,σf ijiij
Yield function takes the following form: no change in
position, shape, but size changes
0Yσσσσσσ2
1KσfK,σf
2
13
2
32
2
21ijiij
Von Mises
M.G. Lee
Materials Science and Engineering, Seoul National University - 17 -
Hardening law: basic
Isotropic hardening
npεCK
Size of yield surface = from a uniaxial tensile test
npεCYK 0
npεεCK 0
p
10 Cεexp1YYK
Power law hardening
Ludwik model
Swift hardening model
Voce hardening model
M.G. Lee
Materials Science and Engineering, Seoul National University - 18 -
Hardening law: basic
Kinematic hardening
- Isotropic hardening: tensile yield stress = |compressive yield stress|
- No Bauschinger effect
- Kinematic hardening: Softening in the compression direction during the
loading in tensile direction
M.G. Lee
Materials Science and Engineering, Seoul National University - 19 -
Hardening law: basic
Kinematic hardening
0ασfK,σf ijijiij
The hardening parameter is called “back stress”
2)1α1(σ)3α3(σ
2)3α3(σ)2α2(σ
2)2α2(σ)1α1(σ
2
1ijαijσfiK,ijσf
Von Mises
M.G. Lee
Materials Science and Engineering, Seoul National University - 20 -
Hardening law: basic
Flow curve
To model three dimensional deformation behavior, a concept of effective variables,
such as effective stress, effective plastic strain is introduced to control the size or
position of the yield surface
M.G. Lee
Materials Science and Engineering, Seoul National University - 21 -
Hardening law: basic
Kinematic hardening: how to define the movement?
ijijij ασ~dα p
ijij dε~dα
Ziegler model Prager model
M.G. Lee
Materials Science and Engineering, Seoul National University - 22 -
Hardening law: basic
Linear kinematic hardening
p
ijij
p
ijij εC3
2αCdε
3
2dα or
0Kαs:αs2
3f ijijijij
“Size of yield function, K=constant”
“Translation of yield function (back stress) – linearly proportional to the plastic strain rate”
M.G. Lee
Materials Science and Engineering, Seoul National University - 23 -
Hardening law: advanced
Classical isotropic hardening
- Not so effective for non-monotonic straining
- No Bauschinger effect and transient behavior
Classical combination type of iso-kinematic hardening by Prager and
Ziegler
- Bauschinger effect only
- No transient behavior
Combination type of the isotropic and kinematic hardening
- Chaboche, Krieg and Dafalias/Popov
- YU model
- Bauschinger effect and transient behavior
Distortional hardening
- Without kinematic hardening (No back stress)
M.G. Lee
Materials Science and Engineering, Seoul National University - 24 -
Hardening law: advanced
Nonlinear Kinematic Hardening Model (NKH)
αεαpC
3
2 “Recall” term
0Rαs:αs2
3f ijijijij
M.G. Lee
Materials Science and Engineering, Seoul National University - 25 -
Hardening law: advanced
“Chaboche” Model
ii αεαα ipi
n
i
n
1i
iC3
2
1
0Rαs:αs2
3f ijijijij
M.G. Lee
Materials Science and Engineering, Seoul National University - 26 -
Hardening law: advanced
Two surfaces are used to represent Bauschinger and
transient behaviors
- Inner(loading) and outer(bounding) surfaces
( ) 0m
isof
( ) 0m
isoF Α
- Translation of the inner surface
( )iso
dd
: normalized quantity of or pd
M.G. Lee
Materials Science and Engineering, Seoul National University - 27 -
Hardening law: advanced
O
o
A
a
B
b
Α
α
σ
Σ
The hardening curve of the inner
surface is newly updated every time
unloading occurs, considering the
gap
The separation of the isotropic and
kinematic hardening in the inner
surface should be performed
considering the gaps
M.G. Lee
Materials Science and Engineering, Seoul National University - 28 -
Hardening law: advanced
in
d/d
in
M.G. Lee
Materials Science and Engineering, Seoul National University - 29 -
Hardening law: advanced
In order to properly represent the transient behavior, the
hardening behavior of the inner surface is prescribed every
time reloading occurs
The stress relation for the 1-D tension test
p p p
d d d
d d d
Example of gap variation between
inner and outer surfaces
( ( )) ( )inp
in
dA
d
This satisfies the continuous
hardening slope between elastic
and plastic ranges
p
d
d
in
p p
d d
d d
0
for
for
M.G. Lee
Materials Science and Engineering, Seoul National University - 30 -
Hardening model: YU Model
Basic Theory : Yoshida, F. and Uemori, T. , Int. J. Plast. 18 ,661-686, 2002
0
0
p
Bounding surface
Y
B
2Y
Re-yielding
Transient
Behavior
Permanent softening
0
M.G. Lee
Materials Science and Engineering, Seoul National University - 31 -
Hardening model: YU Model
( ) 0F f Y σ α
1 ( ) 0F Bf R σ β
* α α β
** *
2
3
p
e padCa
α n n
sat
pRdR dm R
2
3
p pm bd d d
β ε β
a RB Y
Back stress of “active (real)” yield surface
Back stress of “Bounding” surface
Isotropic part of “Bounding” surfaceControl “global” hardening
M.G. Lee
Materials Science and Engineering, Seoul National University - 32 -
Hardening model: YU Model
Yoshida-Uemori Parameters
Y : Size of loading surface
B : Initial size of bounding surface
C : Parameter for the back-stress evolution
Rsat ,b ,m : Parameters for the size of bounding
surface
h : The parameter for work-hardening stagnation
or cyclic hardening characteristics.
( ) 0F f Y σ α
1 ( ) 0F Bf R σ β
* α α β
** *
2
3
p
e padCa
α n n
sat
pRdR dm R
2
3
p pm bd d d
β ε β
a RB Y
Initial yield surface
Bounding surface
The relative motion law
Isotropic hardening of the bounding surface
Kinematic hardening of the bounding surface
0 0( )[1 exp( )]apE E EE
Plastic strain dependency of Unloading modulus
M.G. Lee
Materials Science and Engineering, Seoul National University - 33 -
Hardening model parameter
Measuring hardening, Bauschinger and transient behaviors
Anti-buckling system
- Prevention of buckling when sheet specimen compressed
- Use of fork-shaped guides along the side of the sheet
- Large clamping force causes the fork to bend
- Difficult to obtain smooth hardening curves
Modified method using simple rigid plates
- Smooth hardening curves were obtained
- Mounted at universal testing machine
M.G. Lee
Materials Science and Engineering, Seoul National University - 34 -
Hardening model parameter
Fork device Modified device
Hydraulic
Specimen
Clamping Plates
Loading
Teflon film
Instron 8516
Instron
Data
M.G. Lee
Materials Science and Engineering, Seoul National University - 35 -
Hardening model parameter
Under-estimate the measured stress value
For the biaxial stress state
Using clamping force and contact area, the biaxial effect can be
corrected
1
11 2 2 1 2 12 2
2
m m m m
mX X X X X X
f
1 2 11 22xx zzX X L L
2 1 21 11 11 222 (2 ) ( 2 )xx zzX X L L L L
2 1 21 11 12 222 ( 2 ) (2 )xx zzX X L L L L
M.G. Lee
Materials Science and Engineering, Seoul National University - 36 -
Hardening model parameter
Tension-compression test
M.G. Lee
Materials Science and Engineering, Seoul National University - 37 -
Hardening model parameter- frictional effect
Over-estimate the measured stress
value
Indirectly evaluated by comparing
the two tensile data: With/without
clamping force
Apparent friction coefficient:
Apparent friction coefficient was
obtained as one average value
since it varies with respect to
engineering strain
( )( )unc stdunc std
oAF F
N N
M.G. Lee
Materials Science and Engineering, Seoul National University - 38 -
Hardening model parameter
1 2 2 1( ) /d d
h h h hd d
in,1
in,2
p
( )( ) ( )( )in inp
in in
dh a b
d
1 ,2 2 ,1
,2 ,1
in in
in in
h ha
1 2
,1 ,2in in
h hb
* *(1 ln( ))in inh
Chaboche model Two-surface model
M.G. Lee
Materials Science and Engineering, Seoul National University - 39 -
Hardening model implementationOutline of implicit integration scheme
Element equilibrium equation for a static state
Global equilibrium equation
: Surface traction + frictional force on element (friction model)
: Element stiffness matrix (constitutive model)
: Element nodal displacement vector
Implicit integration scheme (Newton-Raphson method)
Initially guess
Estimate the residual force
?
Adjust where
no
yesProceed to next increment
M.G. Lee
Materials Science and Engineering, Seoul National University - 40 -
Hardening model implementation
Writing User Subroutine with ABAQUS, ABAQUS
M.G. Lee
Materials Science and Engineering, Seoul National University - 41 -
Hardening model implementation
M.G. Lee
Materials Science and Engineering, Seoul National University - 42 -
Hardening model implementation
Integration of global equilibrium equation (Main code)
Estimate ΔU for new increment.
Calculate strain from ΔU.Call UMAT and update material stress.
Calculate internal force (KU)
Is |Ψ|<Tol ?
Calculate the residual Ψ.
Calculate Jacobian.Accept ΔU as a solution.
Modify ΔU.
Calculate slip and normal pressure from ΔU.Call FRIC and update frictional stress.
Calculate external force (F).
yes no
Material stress update(Subroutine UMAT)
Frictional stress update(Subroutine FRIC)
External data file forcommunication between
the subroutines
M.G. Lee
Materials Science and Engineering, Seoul National University - 43 -
Hardening model implementation
Writing User Subroutine with ABAQUS, ABAQUS
M.G. Lee
Materials Science and Engineering, Seoul National University - 44 -
Hardening model implementation
e p ε ε ε
T C εσ σ
is given.ε
Specific Algorithm
new old
:New old f
Cσ σσ
Stress integration: predictor-corrector
M.G. Lee
Materials Science and Engineering, Seoul National University - 45 -
Hardening model implementation
Writing User Subroutine with ABAQUS, ABAQUS
M.G. Lee
Materials Science and Engineering, Seoul National University - 46 -
Application: springback
M.G. Lee
Materials Science and Engineering, Seoul National University - 47 -
Application: springback
Defect in dimensional accuracy*
Elastic recovery of bending moment by
uneven stress in the thickness direction
after bending
Elastic recovery of bending moment by
uneven stress in the thickness direction
after bending & unbending
Elastic recovery of torsion moment by
uneven stress in plane
Elastic recovery of bending moment along
punch ridgeline by uneven stress in the
thickness direction
* Yoshida & Isogai, Nippon Steel Tech Report, 2013
M.G. Lee
Materials Science and Engineering, Seoul National University - 48 -
Application: springback
Before (selected from literature)
t
RΔθ
Δθ
μΔθ
Δθ
rK,Δθ
Δθn
Kuwabara (1996)
Gerdeen (1994)
Wang (1993)
Wenner (1982)
Wenner (1992)
Lee (1995)Lee (1995)
t
R Tool radius
Thickness
Friction
Hardening
exponent
Strength, anisotropy
M.G. Lee
Materials Science and Engineering, Seoul National University - 49 -
Application: springback
t/2
t/2-σdzwT
22
T14
YtM
2T1Et
3Y
r
1
R
1Δθ
strain
stress
Y
E
Idealized model
M.G. Lee
Materials Science and Engineering, Seoul National University - 50 -
Application: springback
Die
Blank holder
(0.735 kN) Punch
(1m/s)
Punch down
• 2D draw bending simulations
324 mm
360 mm
30 mmGrip Grip
Grip Grip
310 mm
Grip displacement: 24.3
Cutting Cutting
(Stage 1) Pre-tensile strain 8%
(Stage 2) Draw-bend forming and springback
M.G. Lee
Materials Science and Engineering, Seoul National University - 51 -
Application: springback
0 20 40 60 80 100 120 140
0
20
40
60
80
100
Y [m
m]
X [mm]
Experiment
von Mises
Yld2000-2d
DP780 1.4t
Pre-strained
0.0 0.5 1.0 1.50.0
0.5
1.0
1.5DP780 1.4t
yy/
0
xx/0
von Mises
Yld2000-2d
(m=6)
xy=0
M.G. Lee
Materials Science and Engineering, Seoul National University - 52 -
Application: springback
0 20 40 60 80 100 120 140
0
20
40
60
80
100 Experiment
IH
IH-KH
HAH
Y [m
m]
X [mm]
DP780 1.4t
Pre-strained
0.00 0.04 0.08 0.12 0.16
-1200
-800
-400
0
400
800
1200
EXP
IH
IH-KH
HAH
DP780 1.4t
Tru
e s
tre
ss [M
Pa
]
True strain
M.G. Lee
Materials Science and Engineering, Seoul National University - 53 -
Application: issue
Friction coefficient
Top surface
- Punch
0.26
0.26
0.18Top surface
- Holder
Bottom surface
- Die
0.25
0.18
0.17
The friction coefficient is not uniform due to
different contact condition.
A specific constant coefficient may provide similar
predictions, but this value would not work if the forming
condition changes.
M.G. Lee
Materials Science and Engineering, Seoul National University - 54 -
Application: issue
0 20 40 60 80 100 120 140
0
20
40
60
80
100
Y [m
m]
X [mm]
Experiment
Conventional
Advanced
TRIP780 1.2t
BHF: 20 kN
0 20 40 60 800
20
40
60
80
Pu
nch
fo
rce
[kN
]
Punch stroke [mm]
Experiment
Conventional
Advanced
TRIP780 1.2t
BHF: 20 kN
0 20 40 60 800
20
40
60
80
Pu
nch
fo
rce
[kN
]
Punch stroke [mm]
Experiment
Conventional
Advanced
TRIP780 1.2t
BHF: 70 kN
0 20 40 60 80 100 120 140
0
20
40
60
80
100 Experiment
Conventional
Advanced
Y [m
m]
X [mm]
TRIP780 1.2t
BHF: 70 kN
M.G. Lee
Materials Science and Engineering, Seoul National University - 55 -
Application: issue
-60
-40
-20
0
20
40
60
Friction,
0.1
Friction,
0.2
Friction,
0.15
Elastic
modulus
Hardening
law
1 (a)
Ch
an
ge
in
an
gle
[o]
or
cu
rl r
ad
ius [
mm
]
TRIP780 1.2t
BHF: 20 kN
Yield
function
2
-60
-40
-20
0
20
40
60(b)
1
Ch
an
ge
in
an
gle
[o]
or
cu
rl r
ad
ius [
mm
]
TRIP780 1.2t
BHF: 70 kN2
Yield
function
Hardening
law
Elastic
modulus
Friction,
0.1
Friction,
0.15
Friction,
0.2
Sensitivity in springback
At low BHF
Hardening > Elastic
modulus >> friction >
yield function
At high BHF
Hardening > friction>
elastic modulus >> yield
function
Lee et al, IJP, 2015
M.G. Lee
Materials Science and Engineering, Seoul National University - 56 -
Thank you.