(Metal Plasticity) Chapter 15: Hardening 10... · 2019. 9. 5. · M.G. Lee Materials Science and...

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

[email protected]

TA: Chanyang Kim (30-522)

mailto:[email protected]

M.G. Lee


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


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


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


Hardening


Stage II: Linear hardening

In stage II, work hardening depends strongly on dislocation density

M.G. Lee


Hardening


Stage II: Linear hardening In stage II, work hardening depends

strongly on dislocation density Lomer-Cottrell locks (sessile

dislocation)

M.G. Lee


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


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


Yield function

Yield functions

elastic

Plastic

plastic

1D 2D 3D

1-D 2-D 3-D (plane stress)

M.G. Lee


Yield function

From Kuwabara

Plane stress

M.G. Lee


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


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


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


Hardening law: basic

Uniaxial stress-strain curve of typical metals

M.G. Lee



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



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



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



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

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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



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



Kinematic hardening: how to define the movement?

ijijij ασ~dα p

ijij dε~dα

Ziegler model Prager model

M.G. Lee



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


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



Nonlinear Kinematic Hardening Model (NKH)

αεαpC

3

2 “Recall” term

0Rαs:αs2

3f ijijijij

M.G. Lee



“Chaboche” Model

ii αεαα ipi

n

i

n

1i

iC3

2

1

0Rαs:αs2

3f ijijijij

M.G. Lee



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



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



in

d/d

in

M.G. Lee



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


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



( ) 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



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


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



Fork device Modified device

Hydraulic

Specimen

Clamping Plates

Loading

Teflon film

Instron 8516

Instron

Data

M.G. Lee



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



Tension-compression test

M.G. Lee


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



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


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


Hardening model implementation

Writing User Subroutine with ABAQUS, ABAQUS

M.G. Lee



M.G. Lee



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




M.G. Lee



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




M.G. Lee


Application: springback

M.G. Lee



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



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



t/2

t/2-σdzwT

22

T14

YtM

2T1Et

3Y

r

1

R

1Δθ

strain

stress

Y

E

Idealized model

M.G. Lee



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



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



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


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


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


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


Thank you.

[email protected]

(Metal Plasticity) Chapter 15: Hardening 10... · 2019. 9. 5. · M.G. Lee Materials Science and...

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Transcript of (Metal Plasticity) Chapter 15: Hardening 10... · 2019. 9. 5. · M.G. Lee Materials Science and...