A modified Follansbee-Kocks model for 6061-T6 aluminum · A modiﬁed Follansbee-Kocks model for 6061-T6 aluminum Anup Bhawalkar and Biswajit Banerjee Department of Mechanical Engineering,

A modified Follansbee-Kocks modelfor 6061-T6 aluminum

Anup Bhawalkar and Biswajit Banerjee

Department of Mechanical Engineering, University of Utah

ASME Applied Mechanics and Materials Conference, 2007

Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 1 / 33

Outline

1 Mechanical Behavior of 6061-T6 Aluminum

2 Models

3 Modified Follansbee-Kocks Model

4 How well does our model do?

5 Some numerical simulations

6 Summary


Mechanical Behavior of 6061-T6 Aluminum

Temperature DependenceStrain-Rate = 0.001 /s.

200 300 400 500 600 700 8000

100

200

300

400

Temperature (K)

Flow

Str

ess

(MPa

)

Strain-Rate = 1000 /s.

200 300 400 500 600 700 8000

100

200

300

400

Temperature (K)

Flow

Str

ess

(MPa

)Sigmoidal curves?

For sources of data see Anup Bhawalkar’s M.S. Thesis.



Strain-Rate Dependence

Strain = 0.2; Temperature = 300K

10−5

10−1

103

107

101110

2

103

104

Strain Rate (/s)

Flo

w S

tres

s (M

Pa)

Recent Data

Older Data

?

High temperature data?For sources of data see http://www.eng.utah.edu/ banerjee/Papers/MTS6061T6Al.pdf/Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 4 / 33


Pressure Dependence

Strain Rate = 0.001/s; Plastic Strain = 0.05

0 100 200 300 400200

250

300

350

400

450

500

Pressure (MPa)

Flow

Str

ess

(MPa

)300 K

367 K

422 K

478 K

High strain rate data?Experimental data from Davidson, 1973



Can a single flow stress model predict all these behaviors?


Models

Older Models

Steinberg, Cochran, Guinan (1980), Steinberg and Lund(1989).Johnson and Cook (1983, 1985), Johnson and Holmquist(1988).Zerilli and Armstrong (1987, 1993), Abed and Voyidajis (2005).

Different regimes need different sets of parameters.


Models

More Recent Models

Mechanical Threshold Stress Model - Follansbee and Kocks(1988), Goto et al. (2000).Preston, Tonks, Wallace (2003).

Physically based to some extent. May be possible to extend sothat the same parameters can be used for a large domain ofregimes.


Models

Original Follansbee-Kocks Model

σy(σe, ε̇, p, T ) = [τa + τi(ε̇, T ) + τe(σe, ε̇, T )]µ(p, T )

µ0(1)

where

σe =an evolving internal variable that has units of stress (also called the mechanical threshold stress)

ε̇ =the strain rate

p =the pressure

T =the temperature

τa =the athermal component of the flow stress

τi =the intrinsic component of the flow stress due to barriers to thermally activated dislocation motion

τe =the component of the flow stress due to structure evolution (e.g., strain hardening)

µ =the shear modulus

µ0 =a reference shear modulus at 0 K and ambient pressure.


Models

Assumptions in Original Model

Thermally activated dislocation motion dominant andviscous drag effects on dislocation motion are small.

This assumption restricts the model to strain rates of 104 s−1

and less.

High temperature diffusion effects (such as solute diffusionfrom inside grains to grain boundaries) are absent.

This assumption limits the range of applicability of the modelto temperatures less than around 0.6 Tm. For 6061-T6aluminum alloy this temperature is approximately 450 - 500 K.


Modified Follansbee-Kocks Model

The Modified Follansbee-Kocks Model



A Simple Modification

σy(σe, ε̇, p, T ) = [τa + τi(ε̇, T ) + τe(σe, ε̇, T )]µ(p, T )

µ0

Since hardening is relatively small we can

Try to get the correct temperature dependence of µ and τi .Add a viscous terms that can account for viscous drag.Include a modification for overdriven shocks a laPreston-Tonks-Wallace.

to get

σy =

min

{[τv + (τa + τi + τe)

µ

µ0

], σys

}for T < Tm

µv ε̇ for T ≥ Tm

(2)



A Model for the Shear Modulus

Temperature dependence from Nadal and LePoac (2003) andpressure dependence from Burakovsky and Preston (2005).

µ(p, T ) =1

J (T̂ , ζ)

[{µ0 + p

∂µ

∂p

(a1

η1/3 +a2

η2/3 +a3

η

)}(1− T̂ )+

ρ

C Mkb T

] (3)

η :=ρ

ρ0; C :=

(6π2)2/3

3f 2 ; bT :=

T

Tm

J (bT , ζ) := 1 + exp

"−

1 + 1/ζ

1 + ζ/(1 − bT )

#for bT ∈ [0, 1 + ζ] .



A Model for the Melt Temperature

The Burakovsky-Greeff-Preston model (2003):

Tm(ρ) = Tm0 η1/3 exp

{6Γ1

(1

ρ1/30

−1

ρ1/3

)+

2Γ2

q

(1

ρq0

−1ρq

)}. (4)

η :=ρ

ρ0

Γ(ρ) =1

2+

Γ1

ρ1/3+

Γ2

ρq



Model Checks

Shear Modulus

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

T/Tm

Shea

r M

odul

us (

GPa

)

η=0.95

η=1.05

Melt Temperature

0 25 50 75 100 125 1500

1000

2000

3000

4000

5000

6000

Pressure (GPa) T

m (

K)

Older data

Recent data

Models do reasonably well.



A Model for τi

Use a quadratic model to allow for rapid decrease in τi at hightemperatures:

τi = σi

1−

(

kb Tg0i b3 µ(p, T )

lnε̇0i

ε̇

)1/qi

2

1/pi

. (5)



Fit Parameters for τi

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

pi

y =

( σ y

)µ/

ε0i

/ ε)]1/qix = [(k T/ µ b3) ln(b

σi = 375 MPa, g = 0.61

0i

IntervalConfidence95%



A Model for the Viscous Drag

Use ideas from Kumar and Kumble (1969) and Frost and Ashby(1971).

τv =2√

3

Bρm b2 ε̇ (6)

Need to find drag coefficient B and the density of mobiledislocations ρm.



The Drag Coefficient

Assume thatB = Be + Bp

where Be = electron drag, Bp = phonon drag. Neglect Be fortemperatures greater than 50 K.

B ≈ λp Bp =λp q10 cs

〈E〉 (7)

where q = cross-section of dislocation core, cs = shear wavespeed, and

〈E〉 =3 kb T ρ

MD3

θD

T

!; θD =

h c̄

kb

3 ρ

4 π M

!1/3



Mobile Dislocation Density

Use simple model developed by Estrin and Kubin (1986) ?

dρm

dεp=

M1

b2

(ρf

ρm

)− I2(ε̇, T ) ρm −

I3b√

ρf

dρf

dεp= I2(ε̇, T ) ρm +

I3b√

ρf −A4(ε̇, T ) ρf

(8)

Stiff differential equations!A model that works for our purposes is

ρm ≈ ρm0(1 + T̂ )m . (9)



Check Viscous Drag Model

10−5 10−3 10−1 101 103 105

Strain Rate (s)−1

τ v (M

Pa)

510

100

10−5

10−15

−1010

900 K300 K

50 K


How well does our model do?

How well does our model do ?



Temperature Dependence

Strain-Rate = 0.001 /s.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

350

400

450

500

T/Tm

σ y (M

Pa)

ε = 0.001 s−1 εp = 0.02

Strain-Rate = 1000 /s.

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

T/Tm

σ y (M

Pa)

ε = 1000 s−1 εp = 0.10



Strain Rate Dependence

10−5

10−1

103

107

1011

102

103

104

ε (s−1)

σ y (M

Pa)



Pressure Dependence

0 100 200 300 400200

250

300

350

400

450

500

Pressure (MPa)

Flow

Str

ess

(MPa

)


Some numerical simulations

Numerical validation of the model



Flyer Plate Impact

Gas gun

flyer plate

Target Specimen

VISAR

Flyer Plate

Free Surface withmirror finish

Evacuated Targetchamber

Direction of



Flyer Plate Simulations

hi = 1.879 mm, ht = 3.124 mm,v0 = 270 m/s.

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

50

100

150

200

250

300

Time (µ sec)

Fre

e S

urfa

ce v

eloc

ity (

m/s

)

MPM

Exp. Data

FEM

hi = 1.600 mm, ht = 3.073 mm,v0 = 265 m/s.

−0.5 0 0.5 1 1.5 20

50

100

150

200

250

300

Time (µ sec) F

ree

Sur

face

Vel

ocity

(m

/s)

Expt. DataSimulation

Experimental data from Isbell (2005).



Taylor Impact Tests

��

��

Cxf

Cyf

D f

L f

Xf

L af

Wf

0.2 L 0

Centroid

X

Y

A f



Comparison of Metrics

Final Length

0.3 0.4 0.5 0.6 0.7 0.8 0.90.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Experimental Lf/L

0

Sim

ulat

ed L

f/L0

X = Y

Mushroom Diameter

1 1.2 1.4 1.6 1.8 2 2.21

1.2

1.4

1.6

1.8

2

2.2

2.4

Experimental Df/D

0Si

mul

ated

Df/D

0

X = Y



Comparison of Profiles

l0 = 30.00 mm, l0 = 6.00 mm, l0= 358 m/s, T0 = 295 K

−15 −10 −5 0 5 10 150

5

10

15

20

25

30

35

40

mm

mm

Expt.MTS

l0 = 30.00 mm, d0 = 6.00 mm,v0 = 194 m/s, T0 = 635 K

−15 −10 −5 0 5 10 150

5

10

15

20

25

30

mm

mm

Expt.MTS

Experimental data from Gust (1982).


Summary

Summary

Improved high temperature prediction.Improved strain rate dependence at high rates.Pressure dependence cannot be solely from shear modulus.


Appendix For Further Reading

For Further Reading I

B. Banerjee and A. Bhawalkar.An extended Mechanical Threshold Stress plasticity model:modeling 6061-T6 aluminum alloy.under review, 2007.

A. Bhawalkar,The Mechanical Threshold Stress Plasticity Model for 6061-T6aluminum alloy and its numerical validationM.S. Thesis, University of Utah, 2006.


Documents

A modified Follansbee-Kocks model for 6061-T6 aluminum · A modiﬁed Follansbee-Kocks model for 6061-T6 aluminum Anup Bhawalkar and Biswajit Banerjee Department of Mechanical Engineering,