A modified Follansbee-Kocks model for 6061-T6 aluminum · A modified Follansbee-Kocks model for...
Transcript of A modified Follansbee-Kocks model for 6061-T6 aluminum · A modified Follansbee-Kocks model for...
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
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 2 / 33
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.
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 3 / 33
Mechanical Behavior of 6061-T6 Aluminum
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
Mechanical Behavior of 6061-T6 Aluminum
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
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified 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 of Utah)Modified 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 of Utah)Modified 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 sothat the same parameters can be used for a large domain ofregimes.
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified 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 of Utah)Modified Follansbee-Kocks Model McMat07 9 / 33
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.
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 10 / 33
Modified Follansbee-Kocks Model
The Modified Follansbee-Kocks Model
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified 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 laPreston-Tonks-Wallace.
to get
σy =
min
{[τv + (τa + τi + τe)
µ
µ0
], σys
}for T < Tm
µv ε̇ for T ≥ Tm
(2)
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 12 / 33
Modified Follansbee-Kocks Model
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] .
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 13 / 33
Modified Follansbee-Kocks Model
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
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 14 / 33
Modified Follansbee-Kocks Model
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.
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified 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 hightemperatures:
τi = σi
1−
(
kb Tg0i b3 µ(p, T )
lnε̇0i
ε̇
)1/qi
2
1/pi
. (5)
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 16 / 33
Modified Follansbee-Kocks Model
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%
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified 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).
τv =2√
3
Bρm b2 ε̇ (6)
Need to find drag coefficient B and the density of mobiledislocations ρm.
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 18 / 33
Modified Follansbee-Kocks Model
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
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 19 / 33
Modified Follansbee-Kocks Model
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)
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 20 / 33
Modified Follansbee-Kocks Model
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
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 21 / 33
How well does our model do?
How well does our model do ?
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 22 / 33
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
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 23 / 33
How well does our model do?
Strain Rate Dependence
10−5
10−1
103
107
1011
102
103
104
ε (s−1)
σ y (M
Pa)
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 24 / 33
How well does our model do?
Pressure Dependence
0 100 200 300 400200
250
300
350
400
450
500
Pressure (MPa)
Flow
Str
ess
(MPa
)
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 25 / 33
Some numerical simulations
Numerical validation of the model
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 26 / 33
Some numerical simulations
Flyer Plate Impact
Gas gun
flyer plate
Target Specimen
VISAR
Flyer Plate
Free Surface withmirror finish
Evacuated Targetchamber
Direction of
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 27 / 33
Some numerical simulations
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).
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 28 / 33
Some numerical simulations
Taylor Impact Tests
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Cxf
Cyf
D f
L f
Xf
L af
Wf
0.2 L 0
Centroid
X
Y
A f
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 29 / 33
Some numerical simulations
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
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 30 / 33
Some numerical simulations
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).
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified 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 of Utah)Modified 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-T6aluminum alloy and its numerical validationM.S. Thesis, University of Utah, 2006.
Anup Bhawalkar and Biswajit Banerjee (University of Utah)Modified Follansbee-Kocks Model McMat07 33 / 33