Download - FY Lecture4
Constitutive modeling of large-strain cyclic plasticity
for anisotropic metals
Fusahito YoshidaDepartment of Mechanical Science and Engineering
Hiroshima University, JAPAN
1: Basic framework of modeling2: Models of orthotropic anisotropy3: Cyclic plasticity – Kinematic hardening model4: Applications to sheet metal forming and some
topics on material modeling
1. Springback simulation
2. Springback compensation based on optimization technique
3. Some topics on material modeling- Modeling of yield point phenomena- Multi-scale modeling- Material database
Lecture 4: Contents
Springback Simulation
• Hat-type draw bending• S-rail forming• Bumper beam• B-pillar etc.
Isotropic hardening model
Yoshida –Uemori model
Accuracy of springback analysis strongly depends on material models.
Experiment on 980HSS sheet
(by LS-DYNA)Yoshida-Uemori
Accurate description of the Bauschingereffect
Experiment
Isotropic hardening
CASE 1: Hat draw-bending
◆ by PAM-Stamp 2G with Yoshida model
2
X=115X=0X=-115X= -115
X= 115
X= 0980MPa HSS
CASE 2: S-rail forming
Selection of a material model is of vital importance foraccurate simulation of springback
Yoshida-Uemori model
IH model by PAM-STAMP 2G
Bumper beam
Experiment
Yoshida-Uemori model by PAM-STAMP 2G
CASE 3: Bumper beam
Application of Yoshida model for massApplication of Yoshida model for mass--production partsproduction parts
MAZDA-5(2005 model)’s B-pillar rein
Section cut
1st forming (completed)
Blank
Material; SPHN590R-DS t1.6(Red)SPCN780Y-N-E t1.8(Green)SPCN590R-N t1.4(Blue)
Yoshida(Blue line)
Real(Red line)
CASE 4:
Comparison between FE simulation (Pam-Stamp2G) and experimental results
Isotropic hardening Yoshida-Uemori Kinematic hardening
Simulation error less than ±1.0mm75.81% 91.53%
CASE 5: B-pillar (780+980 MPa HSS tailored blank)
After Holding Process After Stamping Process After Trimming ProcessInitial Blank
Holding Process Calculation
Stamping Process Calculation
1st SpringbackCalculation
2ndSpringback Calculation
Trimming
CASE 6: L-shaped beam
980HSS sheet
Simulation of wrinklesby PAM-STAMP 2G: Yoshida-Uemori Model
Photos
3D measurement
Simulation
No drawbead 2-mm drawbeads 4-mm drawbeads
Springback Compensationbased on Optimization Technique
• Drawbeads for S-rail forming • Tool shape design for bumper beam
Optimum Optimum DrawbeadDrawbead Setting Setting for for SpringbackSpringback CompensationCompensation
in Sin S--rail Formingrail Forming
Twisting springback
Optimum drawbead
Effect of Drawbead on Springback
Remove a part of draw bead line
sec-1
sec-2
sec-1 sec-2
Counterclockwise direction
sec-1
sec-2
sec-1 sec-2
Clockwise direction◆ Full drawbead setting
◆ Partial drawbead settingsec-1 sec-2
12
Design variables x = Drawead heights No. Drawbead height mm
No.1 H1 = x1
No.2 H2 = x2
No.3 H3 = x3 +H1
No.4 H4 = x4 +H2
No.5 H5 = ( H1 + H7 ) / 2
No.6 H6 = ( H2 + H8 ) / 2
No.7 H7 = 2.0
No.8 H8 = 2.0
Where design variables x = x1, x2, x3, x4 are 0.0 ≦ x1 ≦2.00.0 ≦ x2 ≦2.0
-0.4 ≦ x3 ≦0.4-0.4 ≦ x4 ≦0.4
Springback Control by Drawbead as a Problem of Optimization
Dra
wbe
adhe
ight
Hi
Objective function to be minimizedF(x)= twising angle
◆ No drawbead
◆ Optimum drawbeads
Tortional angle
3.5 degree
Section-1 Section-2
Section-1 Section-2
Section-1
Section-2Result of FE simulation based Optimization
0.4 degree
Torsional springback is successfullysuppressed by optimum drawbead setting
Experimental Verification
Blank holdingDrawing Triminng
Optimum drawbead
◆ No drawbead
◆ Optimum drawbeads
Tortional angle
4.0 degree
Section-1 Section-2
Section-1 Section-2
Section-1
Section-2Experimental Verification
0.5 degree
Torsional springback is successfullysuppressed by optimum drawbead setting –Verified!
Determination of optimum tool shapes for bumper beam
B.Longitudinal springback
A. Cross-section opening springback
Springback compensation for A (cross section) and B (longitudinal) types were treated separately.
⊿di
f(x) =∑⊿ di
Target shape
x1x2
x3
Punch
Die
PadY
Z
g1(x)
g2(x)
g3(x)=min⊿ di
Y
Z
⊿d1⊿d2FE simulation Result after springback calculation
Y
Z
Die design as an optimization problem(Cross section)
Minimize objective function f(x)-Subject to
1 1 2 2 3 3( ) , ( ) , ( )g C g C g C≤ ≤ ≤x x x
Design variables
Objective function
Constraints
Before springback
X
Z After springback Target shape
8.1mm
Die design as an optimization problem(Longitudinal direction)
x r=Design variable
Objective function
53.4%28.4%
Result of optimization(Final shape of the beam after springback)
Some topics on material modeling
• Yield-point phenomena• Multi-scale modeling
Yield point
Yield plateau
..
Workhardening
0
100
200
300
400
500
0 0.02 0.04 0.06 0.08 0.1
Crosshead speed 0.5(mm/s)Crosshead speed 0.005(mm/s)Crosshead speed 0.0005(mm/s)
Stress (MPa)
Strain
Rate-dependent Yield-Point Phenomena
Non-uniform plastic deformation due to Luders band propagation..
Modeling and Simulations of Yield-Point Phenomena (Overview)
• Metal physics: Cottrell & Bibly (1949); Lomer (1952); …;Stein and Low (1966); …; Fujita & Miyazaki (1978); …; Neuhauser & Hampel (1993)
• Constitutive modeling: Jhonston & Gilman (1959); Hahn (1962); Shioya & Shioiri (1976); Yoshida (IJP 16 (2000) 359)
• FE simulation of Luders-band propagation:Itoh-Yoshida et al. (1992); Tsukahara (1998); Kyriakides (2001); Sun-Yoshida et al. (2003)
• Polycrystal plasticity simulation: Ghosh et al. (2004)
Model of cyclic plasticity
..
pmbγ ρ ν=&
neff
Dτ
τν
⎛ ⎞= ⎜ ⎟
⎝ ⎠
eff cτ τ τ= −
np c
mbDτ
τ τγ ρ −=&
bmρeffτ
: Burgers Vector: mobile dislocation density: effective resolved shear stress
ν : velocity of dislocationsDτ : drag stress
Framework of Constitutive Modeling(1) Single crystal
cτ : interaction stress acting on moving dislocations
Yield-point phenomena result from rapid dislocation multiplication and the stress-dependence of dislocation velocity.
( ) n
m Y RbM D
σρε− +
=&( )32
p εσ−
=s α
ε &&
( ) ( )3 :2
σ = − −s α s α
Framework of Constitutive Modeling(2) Polycrystals
: stress deviator,: backstress deviator,: isotropic hardening stress,: initial yield stress, : Taylor factor
RY M
sα
−s α
α
O
p&ε
s
Yield surface
Yoshida, F, Int. J. Plasticity 16 (2000) 359-380
Y R+
3, :2
n
mLBF
LBF
b YM Dρ σε σ−
= =& s s
..
Plastic deformatonat workhardening region:
( ) ,
3 ( - ) : ( - )2
n
mWH
WH
b Y RM Dρ σε
σ
− +=
=
&
s sα α
Plastic deformation at Luders-band front:
Yoshida, F.: Int. J. Plasticity 16 (2000) ,359Yoshida, F. et al.: Int. J. Plasticity 24 (2008),1792
A Model of Yield Point Phenomena
Rapid dislocation multipicationmρ
ρ
{ }
0
0 0( ) 1 exp( )
ma
asy
f
C
f f f f
ρ ρ
ρ ρ ε
λε
=
= +
= + − − −
: mobile dislocation density: total dislocation density
Initial value of mobile dislocation density is very small because of
the Cottrell atmosphere.
(Hahn 1962; Kohda 1973; Hull & Bacon 1984)
Model of rapid dislocation multiplicationmρ
ρ
{ }
0
0 0( ) 1 exp( )
ma
asy
f
C
f f f f
ρ ρ
ρ ρ ε
λε
=
= +
= + − − −
: mobile dislocation density: total dislocation density
Very low mobile dislocation density because of Cottrell locking
(Hahn 1962; Kohda 1973; Hull & Bacon 1984)
A sharp yield point and the subsequent abrupt yield drop is a consequence of rapid dislocation multiplication and strong stress dependency of dislocation velocity.
0
100
200
300
400
500
0 0.02 0.04 0.06 0.08 0.1
Crosshead speed 0.5(mm/s)Crosshead speed 0.005(mm/s)Crosshead speed 0.0005(mm/s)
Stre
ss(M
Pa)
Strain
0
100
200
300
400
500
0 0.02 0.04 0.06 0.08 0.1
Crosshead speed 0.5(mm/s)Crosshead speed 0.005(mm/s)Crosshead speed 0.0005(mm/s)
Stre
ss(M
Pa)
Strain
Experiment Simulation
F. Yoshida, Int. J. Plasticity, 16 (2000), pp.359-380
Uniaxial tension
Elimination of Yield-Point by temper rolling
解析結果 実際の圧延
0
100
200
300
400
500
0 0.05 0.1 0.15 0.2
Stress (MPa)
Strain
Before skin-pass rolled
0.5% 1.0% 2.0% rolled
FE simulation of temper rolling
Yoshida, F. et al.: A plasticity model describing yield-point phenomena of steels and its application to FE simulation of temper rolling, Int J. Plasticity 24 (2008) pp.1792-1818.
FE simulation Experiment
Α model for β-Ti (Ti-20V-4Al-1Sn) at elevated temperature
⎟⎠⎞
⎜⎝⎛−
−−=
RTQ
DRY
Mb n
isomp exp0σρε&
( ) isop
isoisoiso aRRQBR −−= ε&&
Strain hardening Dynamic recovery
X.T. Wang, F. Yoshida et al.: Mat Trans 50-9 (2009), pp.1576
specimen
Servo-controlled testing machine
Laser displacement
FurnaceExtensometer
Continuum mechanics Crystal plasticity DD,MD
Modeling of single crystal for each phase
Modeling for multi-phase & polycrystalmaterials
Macro modeling
HomogenizationDislocation motion,
accumulation and D-structure formation
Volume fraction of each phase and texture
Models of obstacles (G-boundaries,
precipitates, etc.)
Material parameters associated with micro structures
シミュレーションへの応用
FE forming simulation
Multi-scale modeling for prediction of macro elasto-plasticity behavior of materials
Material tests & Parameter identificationMaterial Database
Cyclic plasticity Yield function, material parameters
Database
SPCN780Y
Material parameter identification
Automatic idetificationsoftware Forming limit criteria & material parameters
Sheet metal forming simulation