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