(Metal Plasticity)

446.631A

소성재료역학

(Metal Plasticity)

Chapter 15: Hardening

Myoung-Gyu Lee

Office: Eng building 33-309 Tel. 02-880-1711

myounglee@snu.ac.kr

TA: Chanyang Kim (30-522)

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 cos φ L

τ Gb ^c

L

 c ^

Effective particle spacing

Critical angle to which the dislocation bends to breaking away from the obstacle

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

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

Hardening

Work (strain) hardening of single crystal

 Stage II: Linear hardening

 In stage II, work hardening depends strongly on dislocation density

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)

Elastic Plasticity Models

Elastic-plasticity models in sheet metal forming 1) Yield function

2) Hardening model 3) Elastic modulus

Accurate predictions

for stress, deformation

Yield function

Elasticity models for FEM

 Yield functions

• Elastic vs. Plastic

• Rate of plastic strain

e ε p

ε

ε ^Y  

1D σ

σ dλ f

dε ^p



 

  ^σ

f

3D (6D)





r = d e

d e

R-value

Yield function

Yield functions

elastic

Plastic

plastic

1D 2D 3D

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

Yield function

Plane stress

Yield function

From Kuwabara

Uniaxial tension

Biaxial tension (small strain range)

Uniaxial

compression

 _y

 _x

Von Mises Hill 1948

Barlat 2000-2d

Hardening law: importance

Hardening behavior

Upper Bottom

Compressio n Tension

Sidewall region

(Upper) Tension – Compression – Unloading (Bottom) Compression – Tension – Unloading

σ_xx

Hardening law: importance

Hardening models

C

T σ

σ 

C

T σ

σ 

Bauschinger effect can be reproduced by kinematic hardening Isotropic hardening

kinematic hardening

σ

σ

-

σ

σ

A B

C

σ

σ

A B

C

σ

σ

A B C

σ

σ

C α B

A

Hardening law: basic

Uniaxial stress-strain curve of typical metals

Hardening law: basic

  ^{σ 0}

f



 ^{σ , K}  ⁰

f



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

Hardening law: basic

Isotropic hardening

 ^{σ , K}    ^{f σ K 0}

f



 

Yield function takes the following form: no change in position, shape, but size changes

      ^{σ σ}   ^{σ σ}   ^{σ σ}   ^{Y 0}

2 K 1

σ f K

σ ,

f



 











 

Von Mises

Hardening law: basic

Isotropic hardening

  ^ε

C K 

Size of yield surface = from a uniaxial tensile test

  ^ε

C Y

K 



 ^{ε ε}



C

K 



 





1

0

Y 1 exp Cε

Y

K    

Power law hardening Ludwik model

Swift hardening model

Voce hardening model

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

Hardening law: basic

Kinematic hardening

 ^{σ , K}   ^{f σ α}  ⁰

f







The hardening parameter is called “back stress”

   ^{ }  ^ _ ^{  (σ} ₁ ^{ α} ₁ ^{)  (σ} ₂ ^{ α} ₂ ^{)  2   (σ} ₂ ^{ α} ₂ ^{)  (σ} ₃ ^{ α} ₃ ^{)   2  (σ} ₃ ^{ α} ₃ ^{)  (σ} ₁ ^{ α} ₁ ^{)  2} _ ^

2 1 α ij

σ ij i f

K ij , f σ

Von Mises

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

Hardening law: basic

Kinematic hardening: how to define the movement?

ij ij

ij

~ σ α

dα  dα

~ dε

Ziegler model Prager model

Hardening law: basic

Linear kinematic hardening

p ij ij

p ij

ij

C ε

3 α 2

3 Cdε

dα  2 or   

 ^{s α}   ^{: s α}  ^{K 0}

2

f  3





  “Size of yield function, K=constant”

“Translation of yield function (back stress) – linearly proportional to the plastic

strain rate”

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)

Hardening law: advanced

Nonlinear Kinematic Hardening Model (NKH)



 







 α C ε ^p α 3

2 “Recall” term

 ^{s α}   ^{: s α}  ^{R 0}

2

f  3





 

Hardening law: advanced

“Chaboche” Model



 











  



 α i ε α i

α _i ^p _i

n i

n

1 i

C i

3 2

1

 ^{s α}   ^{: s α}  ^{R 0}

2

f  3





 

Hardening law: advanced

 Two surfaces are used to represent Bauschinger and transient behaviors

- Inner(loading) and outer(bounding) surfaces

( )

0 f      

( )

0 F   Α   

- Translation of the inner surface

iso

( ) d d 

 

 

  : normalized quantity of or d

  

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



Hardening law: advanced



d/d









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

( ( )) (

)

p

in

d A

d

    

  

 

       

This satisfies the continuous hardening slope between elastic and plastic ranges

p

d d



 ^{ }   

p p

d d

d d



 

    0

for

for

Hardening model: YU Model

0

0

^p



Bounding surface

Y B

2Y

Re-yielding

Transient Behavior

Permanent softening

0

Hardening model: YU Model

( ) 0

F  f σ α    Y

 

1

( ) 0

F  f σ β   B  R 

*

 

α α β

*

* *

2 3

p

e p

ad

C a

 

 

 

 

 

α  n  n

 sat 

R p

dR  m  R d 

2 3

p p

m b

d β     d ε  β d    

a    B R Y

Back stress of “active (real)” yield surface Back stress of “Bounding” surface

Isotropic part of “Bounding” surface

Control “global” hardening

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

R

,b ,m : Parameters for the size of bounding surface

h : The parameter for work-hardening stagnation or cyclic hardening characteristics.

( ) 0

F  f σ α    Y

 

1

( ) 0

F  f σ β   B  R 

*

 

α α β

*

* *

2 3

p

e p

ad

C a

 

 

 

 

 

α  n  n





R

dR  m  R d  2

3

p p

m b

d β     d ε  β d    

a    B R Y

Initial yield surface

Bounding surface

The relative motion law

Isotropic hardening of the bounding surface

Kinematic hardening of the bounding surface

0

(

)[1 exp(

)]

E E   E  E   

Plastic strain dependency of Unloading modulus

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

Hardening model parameter

Fork device Modified device

Hydraulic Specimen

Clamping Plates Loading

Teflon film

Instron 8516 Instron

Data

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 1

1 2

2

2

2

m m m m

m

X X X X X X

  f  ^{              }

 

 

1 2 11 xx 22 zz

X ^  X ^  L ^   L ^ 

2 1 21 11 11 22

2 X   X   (2 L   L  ) 

 ( L   2 L  ) 

2

2

(

2

)

(2

)

X ^  X ^  L ^  L ^   L ^  L ^ 

Hardening model parameter

Tension-compression test

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

( )

( F

F

)

A

N N

 

  ^  ^

Hardening model parameter

1 2

d

(

d ) /

h h h h

d d

 

 

 

    

_in,1

_in,2

_



^



_p

(

)( ) (

)( )

p

in in

d h a b

d

    

 ^    ^{  }   

h   h  h  h

*

(1 ln( ))

in in

h   

  

  

Chaboche model Two-surface model

Hardening model implementation

Outline 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

yes

Proceed to next increment

Hardening model implementation

Hardening model implementation

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.

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 for communication between

the subroutines

Hardening model implementation

Hardening model implementation

e p

     ε ε ε

T

   C ε

σ σ

is given.

ε

Specific Algorithm

 

new old

   

    

 

  

old

f



     

σ σ

Stress integration: predictor-corrector

Hardening model implementation

Writing User Subroutine with ABAQUS, ABAQUS

Application: springback

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

Application: springback

Before (selected from literature)

t

R Δθ

Δθ

μ Δθ

Δθ

r K,

Δθ Δθ n

Kuwabara (1996)

Gerdeen (1994) Wang (1993) Wenner (1982)

Wenner (1992)

t

R

Tool radius Thickness

Friction

Strength, anisotropy

Application: springback



 ^t/2

t/2

- σdz

w T

 ² 

2

T 4 1

M  Yt 

 ^{1 T 2} 

Et 3Y r

1 R

Δθ  1   

strain stress

Y

E

Idealized model

Application: springback

Die

Blank holder

(0.735 kN) Punch

(1m/s)

Punch down

• 2D draw bending simulations

324 mm 360 mm

30 mm

Grip Grip

Grip Grip

310 mm

Grip displacement: 24.3

Cutting Cutting

(Stage 1) Pre-tensile strain 8%

(Stage 2) Draw-bend forming and springback

Application: springback

0 20 40 60 80 100 120 140 0

20 40 60 80 100

Y [mm]

X [mm]

Experiment von Mises Yld2000-2d DP780 1.4t

Pre-strained

0.0 0.5 1.0 1.5

0.0 0.5 1.0 1.5

DP780 1.4t



/ 



/ 

von Mises Yld2000-2d (m=6)



=0

Application: springback

0 20 40 60 80 100 120 140

0 20 40 60 80 100

Experiment IH

IH-KH HAH

Y [mm]

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

True stress [MPa]

True strain

Application: issue

Friction coefficient

Top surface - Punch

0.26

0.26 Top surface 0.18

- 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.

Application: issue

0 20 40 60 80 100 120 140 0

20 40 60 80 100

Y [mm]

X [mm]

Experiment Conventional Advanced TRIP780 1.2t

BHF: 20 kN

20 40 60 80

Punch force [kN]

Experiment Conventional Advanced TRIP780 1.2t

BHF: 20 kN

20 40 60 80

Punch force [kN]

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 [mm]

X [mm]

TRIP780 1.2t BHF: 70 kN

Application: issue

-60 -40 -20 0 20 40 60

Friction, 0.1

Friction, 0.2 Friction,

0.15 Elastic

modulus Hardening

law

1



(a)

Change in angle [o ] or curl radius [mm]

TRIP780 1.2t BHF: 20 kN

Yield function

2

-60 -40 -20 0 20 40 (b) 60

1



Change in angle [o ] or curl radius [mm]

TRIP780 1.2t BHF: 70 kN

2

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

Thank you.

myounglee@snu.ac.kr