Thomas Stoughton Feb 8-9, 2012 AHSS Workshop

MSR Lab Meeting 2012, Jan 31• TBS • 1

Plasticity

Thomas Stoughton

Feb 8-9, 2012

AHSS Workshop


Objective

Presentation on the current state of knowledge of plasticity, constitutive behavior, and forming

limits with a focus on opportunities, roadblocks, threats and requirements for use of AHSS in

automotive applications.


Outline

Microstructure/Polycrystalline vs. ContinuumApplication Needs; Texture & High Exponent Yield Functions; Forming Limits of AHSS; Lessons from Metallic Glass

Elasto-plasticityYoung’s Modulus variation, quasiplastic strain

Distortional Hardening BehaviorIsotropic, kinematic, distortional hardening

Forming LimitsNonlinear Strain Path Effects, Curvature Effects, Necking vs. Fracture, Heightened importance for AHSS

Challenges


Microstructure vs. Continuum Approach

Phenomenon suggesting use of Micro-level ModelTripping and/or Twinning mechanismsDual and Complex phasesHighly textured alloys

Limited Slip Systems (FCC & HCP)Elongated Grain ShapesLarge Grains and/or Ultra-thin sheet

Unusual Hardening or Failure Behaviors


Perceived Characteristics of the Two Approaches to Modeling

Continuum Approach

Low Accuracy High

Cost

High

Low

Microstructure Approach


Reliability is the Primary Driver For Alloy Development

Continuum Approach

Low Accuracy High

Cost

High

LowCost is not so important as an

understanding and opportunity for optimization using a

mechanism-based approach.

Alloy Development



Microstructure Approach is Ideal For Alloy Development

Continuum Approach

Low Accuracy High

Cost

High

Low


Alloy Development


Cost is the Primary Driver For Virtual Die Tryout


Continuum Approach

Low Accuracy High

Virtual Die Tryout

Cost

High

Low

Alloy DevelopmentTwo reasons… Still have Physical Tryout as a backup to oversights

missed by Virtual Tryout… and…


Why COST is so Important in Virtual Tryout

Strain FLC

Finite Element Simulation of the Forming Process

Approve for Die Build

Finding the right forming conditions for a given panel requires SCORES of iterations on blank size, restraining forces, and tool/product shape to get it right…

Multiply this by the 100’s of dies necessary to form the panels of a

vehicle… the need for minimizing cost per analysis is clear.


Can the Micro Approach become more efficient to handle Virtual Die Tryout?


Continuum Approach

Low Accuracy High

Virtual Die Tryout

Cost

High

Low

No Texture Evolution

Weaken Grain Coupling

Ignore Grain Boundaries

Reduce Grain Orientations

These and other modifications can be applied individually or in

combination…


Can the Macro Approach become sufficiently reliable to satisfy the needs?


Continuum Approach

Low Accuracy High

Virtual Die Tryout

Cost

High

LowVon Mises Yield

Hill Yield

ChaboucheHardening Yoshida

Hardening

Non-Associated Flow Barlat Yield

Anisotropic Hardening

These and other modifications can be applied individually or for many cases, in combination…

Multi-phase Hardening


Synergy Between Approaches


Continuum Approach

Low Accuracy High

Virtual Die Tryout

Cost

High

Low

Advanced Material Model

Numerical Experiments

Model Calibration


Alloy Development

Synergy Between Approaches


Continuum Approach

Low Accuracy High

Virtual Die Tryout

Cost

High

Low

Numerical Experiments

Model Calibration

Advanced Model Development


Simplified View of Application Areas

Alloy Development Virtual Die Tryout

Advanced Continuum Model

Development


Continuum Approach


Calibration


Outline


Elasto-plasticity HysterisisYoung’s Modulus variation, quasiplastic strain

Distortional HardeningIsotropic, kinematic, distortional hardening


Challenges


Hysterisis of loading/unloading

1

Complex Unloading Model for Springback Prediction

Dissertation CommitteeDr. Robert H. Wagoner, Advisor

Dr. June Key LeeDr. Stephen Eric BechtelDr. Rebecca B. Dupaix

Dept. of Mechanical EngineeringThe Ohio State University

Oral Examination for the Degree of Doctor PhilosophyFeb 23, 2011

Li Sun

Uniaxial Loading-Unloading Test

8

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08 0.1

True

Str

ess

(MPa

)

True Strain

DP780-1.4

0

200

400

600

800

1000

1200

0 0.025 0.05 0.075 0.1

True

Str

ess

(MPa

)

True Strain

DP980 -1.43

See Close-Up

DP 780 DP 980


Three ways to model the behaviorExpanded View of Loading-Unloading Test

9

0

200

400

600

800

1000

1200

0.069 0.072 0.075 0.078

True

str

ess

(MPa

)

True strain

DP980 -1.43

Measuredunload-load loop

208 GPa

208 GPa

145 GPa

QPE ep

recov

c1c2

1) Ignore hysterisis and treat it as a change in Elastic Modulus (GREEN Line)

2) Define yield stress near to the proportional limit and treat the nonlinear post-yield behavior as a micro-plasticity domain of conventional plasticity,

3) Leave elasticity and plasticity the same, but include a new type of quasi-plastic strain, QPE.


2 Surface Framework of QPE Model

Elastic state

:d d0 eC

QPE state

0 : :

/ /

e

e

e e

d d dd d d

d d d dQPE

QPE QPE

C

Plastic state0 : : ( )

/ /

e

e

e e

d d d dd d d d

d d d d

p

QPE p

QPE QPE

C CApparent Young’s Modulus

-30

-20

-10

0

10

20

30

1

2

*

nn*

f1

f2

0 1 1 exp pE E E b d d

Elastic + QPE State

Elastic + QPE +Plastic State

Elastic State


Advantages of QPE Model

Unfinished Cycles of Loading-Unloading Test

18

0

200

400

600

800

1000

1200

0.03 0.04 0.05 0.06

True

Str

ess

(MPa

)

True Strain

DP980-1.43

QPE model

Measured (loading)

Measured(unloading)

Partial Unloading of Forming Stresses is Common in Curved Areas of the Product


Outline





Challenges


Nature of Distortional Hardening

Deformation Behavior Under Conditions of

Combined Stress

- Prof. Y. Tozawa

1977 GMR Symposium

Experimental Probing of the Yield Surface Evolution

70/30 Brass

-250

-200

-150

-100

-50

0

50

100

150

200

250

-250 -200 -150 -100 -50 0 50 100 150 200 250

Yield Work Contour0.002 0.0050.010 0.0200.030 0.0500.070 0.100


Proportional Loading Tests Suggest Isotropic Hardening

70/30 Brass

Sigma-Z

Sigma-Y

Sigma-X

-200

-150

-100

-50

0

50

100

150

200

-200 -150 -100 -50 0 50 100 150 200

Yield Work0.002 0.0050.010 0.0200.030 0.0500.070 0.100

Pi-plane view shows a von Mises behavior for brass


Complete Non-Proportional Loading Tests Show Complex Hardening Behavior

70/30 Brass

Sigma-Z

Sigma-Y

Sigma-X

-200

-150

-100

-50

0

50

100

150

200

-200 -150 -100 -50 0 50 100 150 200

Yield Work0.002 0.0050.010 0.0200.030 0.0500.070 0.100

70/30 Brass30% Uniaxial

Sigma-Y Sigma-Z

Sigma-X

-600

-400

-200

0

200

400

600

-600 -400 -200 0 200 400 600

Yield Work0.302 0.3100.350 0.400

30% Strain




Sigma-Y Sigma-Z

Sigma-X

-600

-400

-200

0

200

400

600

-600 -400 -200 0 200 400 600

Yield Work0.302 0.3100.350 0.400

1

1) Distortion of the Yield Surface




Sigma-Y Sigma-Z

Sigma-X

-600

-400

-200

0

200

400

600

-600 -400 -200 0 200 400 600

Yield Work0.302 0.3100.350 0.400

2

2


2) Anisotropic hardening




Sigma-Y Sigma-Z

Sigma-X

-600

-400

-200

0

200

400

600

-600 -400 -200 0 200 400 600

Yield Work0.302 0.3100.350 0.400



3) Shape stabilizes after 1% and before 5% strain

1%5%

10%




Sigma-Y Sigma-Z

Sigma-X

-600

-400

-200

0

200

400

600

-600 -400 -200 0 200 400 600

Yield Work0.302 0.3100.350 0.400



3) Shape stabilizes after 1% and before 5% strain

4) Stabilized shape is different from the Initial Yield Surface

initial“final”


Normalized Yield Behavior to Unit Circle


Sigma-Y

Sigma-Z

Sigma-X

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Yield Work0.302 0.3100.350 0.400

Initial Yield (0.2%)

Yield (0.2%)After 30% Strain In Uniaxial Tension

Recovery After 1% Strain



Interesting Question: Can one model this with kinematic hardening?


Similar Distortion Observed In Steel

Proportional Loading Tests

Suggest Isotropic Hardening

Uniaxial Prestrain to 5%, 10%, and 20%

Show Distortion of the Subsequent Yield

Anisotropic Hardening After 20% Strain


Advanced Kinematic Hardening Models

Proportional Loading Tests

Suggest Isotropic Hardening

Uniaxial Prestrain to 5%, 10%, and 20%

Show Distortion of the Subsequent Yield

Anisotropic Hardening After 20% Strain


Advanced Kinematic Hardening Models

Initial Yield

Mixed Kinematic Model(Errors up to 15%)

Actual Yield After 20% Uniaxial Prestrain

Prestrain Loading

Direction

Questions:

How do we accurately model this behavior?

What happens under non-linear loading?


Continuum Approach

Characterizing Distortional Hardening is a prime example to benefit from this plan


Development



Outline





Challenges


Effect of Bending On Forming Limits

Punch Tip Radius, R

Lock Bead Radius6.35 mm

PunchTravel

Die Profile Radius 6.88 mm

Specimen

Punch

38.10 mm

MaterialThickness

63.50 mm

95.25 mm

When does necking occur if the sheet metal is curved?

OK for Necking

Causes Neck to Form


Suppression of Necking Can Lead to Fracture Without Necking

Fracture is not considered in traditional manufacturing

… now recognized as a problem with AHSS

Fracture Limit ?

Necking Limit

0.00

0.10

0.20

0.30

0.40

0.50

0.60

-0.25 -0.15 -0.05 0.05 0.15 0.25

Necking Limit

Fracture Limit

Safe

Neck before

Fracture

Fracture without Necking


Why Fracture is More Important for AHSS

0.00

0.10

0.20

0.30

0.40

0.50

0.60

-0.25 -0.15 -0.05 0.05 0.15 0.25

Necking Limit

Fracture Limit

Fracture without Necking

The possibility of Fracture Without Necking

depends only on geometry, which defines the strain difference through the curved

sheet {=ln(1+t/R) } ,and its relation to the strain gap

between the Necking and Fracture Limits.


The Challenge of Nonlinear Paths

What is SAFE?

2

1


What does this data mean for Linear Paths?


Uniaxial Tension Strain History

0% Strain



5% Strain



12% Strain



17% Strain


The Strain-Based FLC is DYNAMIC


How can we reliably assess formability?


Strain FLC

Strain Analysis +8.7% Margin of Safety

Ignoring the DYNAMIC nature of the FLC has costly consequences

Tryout Result:Splits at “Zone 4”

in an area with 17% thinning

Splits In Zone 4Complex parts & processes designed

base solely on net strain and the strain FLC, even with what seems to be high

margin of safety…

…may still fail in tryout and require additional changes to product and tooling shape or

processing conditions.


Paradigm Change: a new perspective

No assumptions, just a simple question…

Are these experimental results LESS complex in stress-space?


Transformation equations for an arbitrary time record of plastic strain

tr

rt

tKt

np

121 2

01

ttt 12

Normal Anisotropic Hill Model

tt

2

1 t

p ttr

rtttdr

rt0 21

22

21 1

221

1

ttt

1

2

trrtrrtt

1

tt

2

1

Arrieux, et al., 1982

First Proposed By Arrieux in 1982 Based On Forming Limit

Behavior of Steel


Strain FLC to Stress FLC For Linear Strain Paths

2

1

2897.0p

MPa 5.3721

MPa 2.102

02730.0

0994.02

2896.01

Numerical Example

2

1


Strain FLC to Stress FLC For Bi-Linear Strain Paths

2905.0p

MPa 4.3761

MPa 1.212

05578.0

Numerical Example

1244.0pA

070.0070.0

1

2

A

A

2358.00173.0

1

2

1661.0pB1658.00527.0

2

1

B

B

Input

2

1

Does this FLC Curve Look Familiar?


Observation Leads to New Solution

Stress-Based FLC’s do not appear to be sensitive to changes in strain path


Stress Based FLC’s are not Sensitive to Path


Same FEA

Body Side Component

Stress FLC

Stress Analysis-20.1% Margin of Safety

Strain FLC

Strain Analysis Expected +8.7% Margin of Safety

Tryout ResultSplits at “Zone

4”

Splits In Zone 4

Used to Approve Die Build

Stress Map of Critical Areas

Same FE Simulation


Other Stress-Equivalent Solutions

tr

rt

tKt

np

121 2

01

ttt 12

Normal Anisotropic Hill Model

tt

2

1

ttt

1

2

trrtrrtt

1

2

11-to-1p p 1-to-1

t

p ttr

rtttdr

rt0 21

22

21 1

221

1

Arrieux, et al., 1982

Proposed FLC metrics insensitive to path change:

Zeng, et al., 2008

Yoshida, et al., 2007

1-to-1

Stoughton & Yoon, 2011

1tanp


Polar Diagram of the EPS

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30

Long

itudi

nalT

rue

Stra

in

TransverseTrue Strain

Experimental FLC's in Conventional Strain Diagram

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30EP

S*C

OS(

)

EPS*SIN( )

Experimental FLC's in Polar EPS Diagram


Illustration of Similarity & Differences

Path Sensitive Strain FLD Polar EPS Diagram

ttv p cos2

ttv p sin1

A

1

ttp

21, AA

A

eefEPS21, AA ee

A

1

tt

2

1 ttt 1211 ,tantan


Illustration of Similarity & Differences


2 2

3

21, BB ee

2211 , BBBB eeee

B

B

Bo

Bo

21, BBBB eefEPSEPSo

3

21, BB

B

eef

EPSo


The Reason


A

1

2 2

3

A

B

B

Bo

Bo

3

1

Apriori Unknown Evolution of the Strain Limit Static EPS Limit


Importance of Nonlinear Path for AHSS

In the past, anomalous failures caused by ignoring the effect of nonlinear paths on formability… i.e., treating strain limits as static limits… have led industry to limit strains in future applications to even lower limits…Industry cannot afford this solution using AHHS with lower ductility than low carbon steel… not when a solution is available to maximize the use of the available ductility.


Thank you for your attention.

Questions?

Documents

Thomas Stoughton Feb 8-9, 2012 AHSS Workshop