A modified Follansbee-Kocks model
for 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
Modified
of Utah)
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
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
2 / 33
Mechanical Behavior of 6061-T6 Aluminum
Temperature Dependence
Strain-Rate = 0.001 /s.
Strain-Rate = 1000 /s.
400
Flow Stress (MPa)
Flow Stress (MPa)
400
300
200
100
0
200 300 400 500 600 700 800
Temperature (K)
300
200
100
0
200 300 400 500 600 700 800
Temperature (K)
Sigmoidal curves?
For sources of data see Anup Bhawalkar’s M.S. Thesis.
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
3 / 33
Mechanical Behavior of 6061-T6 Aluminum
Strain-Rate Dependence
Strain = 0.2; Temperature = 300K
4
10
Flow Stress (MPa)
?
Older Data
3
10
Recent Data
2
10
−5
10
−1
10
3
10
Strain Rate (/s)
7
10
11
10
High temperature data?
ForBiswajit
sources
of data (University
see http://www.eng.utah.edu/
banerjee/Papers/MTS6061T6Al.pdf/
Anup Bhawalkar and
Banerjee
Modified
of Utah)
Follansbee-Kocks
Model
McMat07
4 / 33
Mechanical Behavior of 6061-T6 Aluminum
Pressure Dependence
Strain Rate = 0.001/s; Plastic Strain = 0.05
Flow Stress (MPa)
500
450
300 K
400
367 K
350
422 K
300
478 K
250
200
0
100 200 300 400
Pressure (MPa)
High strain rate data?
Experimental data from Davidson, 1973
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
5 / 33
Mechanical Behavior of 6061-T6 Aluminum
Can a single flow stress model predict all these behaviors?
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
6 / 33
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.
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
7 / 33
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 so
that the same parameters can be used for a large domain of
regimes.
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
8 / 33
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.
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
9 / 33
Models
Assumptions in Original Model
Thermally activated dislocation motion dominant and
viscous 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 diffusion
from inside grains to grain boundaries) are absent.
This assumption limits the range of applicability of the model
to temperatures less than around 0.6 Tm . For 6061-T6
aluminum alloy this temperature is approximately 450 - 500 K.
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
10 / 33
Modified Follansbee-Kocks Model
The Modified Follansbee-Kocks Model
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
11 / 33
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 la
Preston-Tonks-Wallace.
to get
("
#
)

µ

min
τv + (τa + τi + τe )
, σys
for T < Tm
µ0
σy =
(2)


µv ε̇
for T ≥ Tm
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
12 / 33
Modified Follansbee-Kocks Model
A Model for the Shear Modulus
Temperature dependence from Nadal and LePoac (2003) and
pressure dependence from Burakovsky and Preston (2005).
!)
"(
∂µ a1
a2
a3
1
µ(p, T ) =
(1 − Tb)+
µ0 + p
+ 2/3 +
1/3
b
∂p
η
η
η
J (T , ζ)
(3)
i
ρ
k T
CM b
(6π 2 )2/3 2 b
T
f ; T :=
3
Tm
"
#
1 + 1/ζ
J (b
T , ζ) := 1 + exp −
for b
T ∈ [0, 1 + ζ] .
1 + ζ/(1 − b
T)
η :=
ρ
ρ0
; C :=
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
13 / 33
Modified Follansbee-Kocks Model
A Model for the Melt Temperature
The Burakovsky-Greeff-Preston model (2003):
!
(
2Γ2
1
1
Tm (ρ) = Tm0 η 1/3 exp 6Γ1
− 1/3 +
1/3
q
ρ
ρ
0
η :=
Γ(ρ) =
1
1
q − q
ρ0 ρ
!)
. (4)
ρ
ρ0
1
2
+
Γ1
ρ1/3
+
Γ2
ρq
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
14 / 33
Modified Follansbee-Kocks Model
Model Checks
Melt Temperature
Shear Modulus
6000
5000
η=1.05
40
Recent data
4000
30
Tm (K)
Shear Modulus (GPa)
50
20
η=0.95
10
3000
2000
0
0
0
Older data
1000
0.2 0.4 0.6 0.8
T/Tm
1
0
25 50 75 100 125 150
Pressure (GPa)
Models do reasonably well.
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
15 / 33
Modified Follansbee-Kocks Model
A Model for τi
Use a quadratic model to allow for rapid decrease in τi at high
temperatures:

!1/qi 2 1/pi

kb T
ε̇0i


τi = σi 1 −
ln
.

3
 g0i b µ(p, T )

ε̇



Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
(5)
16 / 33
Modified Follansbee-Kocks Model
Fit Parameters for τi
0.05
σi = 375 MPa, g = 0.61
0.045
0i
0.04
95%
Confidence
Interval
y = (σy / µ)
pi
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x = [(kbT/ µ b3) ln(ε0i/ ε)]1/qi
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
17 / 33
Modified Follansbee-Kocks Model
A Model for the Viscous Drag
Use ideas from Kumar and Kumble (1969) and Frost and Ashby
(1971).
2
B
τv = √
ε̇
(6)
3 ρm b 2
Need to find drag coefficient B and the density of mobile
dislocations ρm .
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
18 / 33
Modified Follansbee-Kocks Model
The Drag Coefficient
Assume that
B = Be + Bp
where Be = electron drag, Bp = phonon drag. Neglect Be for
temperatures greater than 50 K.
B ≈ λp Bp =
λp q
hEi
10 cs
(7)
where q = cross-section of dislocation core, cs = shear wave
speed, and
!
!
hEi =
3 kb T ρ
M
D3
θD
T
; θD =
h c̄
3ρ
kb
4πM
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
1/3
McMat07
19 / 33
Modified Follansbee-Kocks Model
Mobile Dislocation Density
Use simple model developed by Estrin and Kubin (1986) ?
!
I3 √
dρm M1 ρf
= 2
− I2 (ε̇, T ) ρm −
ρf
dεp
ρm
b
b
(8)
I3 √
dρf
= I2 (ε̇, T ) ρm +
ρf − A4 (ε̇, T ) ρf
dεp
b
Stiff differential equations!
A model that works for our purposes is
ρm ≈ ρm0 (1 + Tb)m .
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
(9)
McMat07
20 / 33
Modified Follansbee-Kocks Model
Check Viscous Drag Model
5
10
0
10
900 K
τv (MPa)
300 K
−5
10
50 K
−10
10
−15
10
−5
10
−3
10
−1
1
3
10
10
10
Strain Rate (s−1)
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
5
10
McMat07
21 / 33
How well does our model do?
How well does our model do ?
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
22 / 33
How well does our model do?
Temperature Dependence
Strain-Rate = 0.001 /s.
Strain-Rate = 1000 /s.
ε = 0.001 s−1 εp = 0.02
ε = 1000 s−1 εp = 0.10
500
600
450
500
400
400
300
σy (MPa)
σy (MPa)
350
250
200
300
200
150
100
100
50
0
0
0.2
0.4
0.6
T/Tm
0.8
1
0
0
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
0.2
0.4
0.6
T/Tm
0.8
1
McMat07
23 / 33
How well does our model do?
Strain Rate Dependence
4
σy (MPa)
10
3
10
2
10 −5
10
−1
10
3
10
−1
ε (s )
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
10
7
11
10
McMat07
24 / 33
How well does our model do?
Pressure Dependence
500
Flow Stress (MPa)
450
400
350
300
250
200
0
100
200
300
Pressure (MPa)
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
400
McMat07
25 / 33
Some numerical simulations
Numerical validation of the model
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
26 / 33
Some numerical simulations
Flyer Plate Impact
Evacuated Target
chamber
Target Specimen
Gas gun
Direction of
flyer plate
VISAR
Free Surface with
mirror finish
Flyer Plate
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
27 / 33
Some numerical simulations
Flyer Plate Simulations
hi = 1.879 mm, ht = 3.124 mm,
v0 = 270 m/s.
hi = 1.600 mm, ht = 3.073 mm,
v0 = 265 m/s.
300
300
Expt. Data
Simulation
Exp. Data
FEM
250
Free Surface Velocity (m/s)
Free Surface velocity (m/s)
250
200
MPM
150
100
50
0
−0.2
200
150
100
50
0
0.2
0.4
0.6
0.8
Time (µ sec)
1
1.2
1.4
0
−0.5
0
0.5
1
Time (µ sec)
1.5
2
Experimental data from Isbell (2005).
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
28 / 33
Some numerical simulations
Taylor Impact Tests
Y
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Xf
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L af
00000000000
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00000000000
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00000000000
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00000000000
C 11111111111
xf 11111111111
00000000000
00000000000
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00000000000
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Af
00000000000
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00000000000
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00000000000
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00000000000
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Wf
00000000000
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00000000000
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00000000000
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00000000000
11111111111
Centroid
C
yf
0.2 L 0
Lf
X
Df
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
29 / 33
Some numerical simulations
Comparison of Metrics
Final Length
Mushroom Diameter
2.4
0.9
2.2
0.8
2
f
Simulated L /L
0
Simulated Df/D0
1
0.7
0.6
0.5
1.6
1.4
X=Y
0.4
0.3
0.3
X=Y
1.8
0.4
0.5
0.6
0.7
0.8
Experimental Lf/L0
1.2
0.9
1
1
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
1.2
1.4
1.6
1.8
2
Experimental Df/D0
2.2
McMat07
30 / 33
Some numerical simulations
Comparison of Profiles
l0 = 30.00 mm, l0 = 6.00 mm, l0
= 358 m/s, T0 = 295 K
l0 = 30.00 mm, d0 = 6.00 mm,
v0 = 194 m/s, T0 = 635 K
40
30
Expt.
MTS
35
Expt.
MTS
25
30
20
mm
mm
25
20
15
15
10
10
5
5
0
−15
−10
−5
0
mm
5
10
15
0
−15
−10
−5
0
mm
5
10
15
Experimental data from Gust (1982).
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
31 / 33
Summary
Summary
Improved high temperature prediction.
Improved strain rate dependence at high rates.
Pressure dependence cannot be solely from shear modulus.
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
32 / 33
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-T6
aluminum alloy and its numerical validation
M.S. Thesis, University of Utah, 2006.
Anup Bhawalkar and Biswajit Banerjee (University
Modified
of Utah)
Follansbee-Kocks Model
McMat07
33 / 33