SCALING LAWS TO ESTIMATE GRAIN SIZE AND COARSENING IN THE STIR ZONE
description
Transcript of SCALING LAWS TO ESTIMATE GRAIN SIZE AND COARSENING IN THE STIR ZONE
SCALING LAWS TO ESTIMATE GRAIN SIZE AND COARSENING IN THE STIR ZONE
Karem E. TelloColorado School of Mines
Adrian P. GerlichPatricio F. Mendez
Canadian Centre for Welding and JoiningUniversity of Alberta
2
Canadian Centre for Welding and Joining
3
Canadian Centre for Welding and Joining
4
Target Question
• Can we predict grain size in the stir zone?– With insight– Quickly– In a general way– Reliably
• This involves relating processing to microstructure (and readily to properties)
• Test case for scaling laws
5
6
V=6 mm/sTool M5
forall cases
7
8
10
• “Boundary layer” approach– thin region contains complexity and follows tool
geometry– “outer region” involves simpler physics– sticking boundary condition around the pin, mixed
stick and slip under the shoulder
• Focus on deformation around pin– Thin layer surrounding pin (shear layer, “Couette
flow”/extrusion)• Heat Transfer• Deformation
– Base plate• Heat Transfer (preheat from shoulder)
• Hot deformation behavior ~Zener Hollomon
coupled
coup
led
Crawford et al. STWJ 06
11
• “Boundary layer” approach– thin region contains complexity and follows tool
geometry– “outer region” involves simpler physics– sticking boundary condition around the pin, mixed
stick and slip under the shoulder
• Focus on deformation around pin– Thin layer surrounding pin (shear layer, “Couette
flow”/extrusion)• Heat Transfer• Deformation
– Base plate• Heat Transfer (preheat from shoulder)
• Hot deformation behavior ~Zener Hollomon
coupled
coup
led
12
• “Boundary layer” approach– thin region contains complexity and follows tool
geometry– “outer region” involves simpler physics– sticking boundary condition around the pin, mixed
stick and slip under the shoulder
• Focus on deformation around pin– Thin layer surrounding pin (shear layer, “Couette
flow”/extrusion)• Heat Transfer• Deformation
– Base plate• Heat Transfer (preheat from shoulder)
• Hot deformation behavior ~Zener Hollomon
coupled
coup
led
13
14
mechanical energy – stored energy
mechanical energy – stored energy – thermal energy into pin
15
16
17
• “Slow moving heat source” – isotherms near the pin ≈ circular
• “Slow mass input”– deformation around tool has radial symmetry concentric with the tool
• “Thin shear layer”– the shear layer sees a flat (not cylindrical) tool
• “Heat from shoulder results in small T increase”– The heat of the shoulder is distributed over a wide area
Va/a << 1
Va / wad << 1
d/ a << 1
Tp-T∞ / Ts-T∞ << 1
18
19
simplification valid simpl. invalidgray zone
constant, right order of magnitude
20
21
22
23
Can we use scaling laws instead of experiments to predict grain size?
24
Calibration of scaling law
• Need to calibrate T0 and
• For region of valid hypotheses• C1 = 0.835
• C2 = 1.10
€
ΔTs
C1 C2+
25
Calibration of scaling law
26
Prediction of grain size
27
Additional check
28
d=5 mm d=85 mm
d=110 mm d=120 mm
V=0.42 mm/s156 rpm
Tool 6.35 mm
29
Prediction of grain size
30
Discussion
• Grain size during stirring vs. coarsening during cooling cycle
31
Grain size during stirring vs. coarsening during cooling cycle
• During stirringMcQuenn 75, 02
32
Grain size during stirring vs. coarsening during cooling cycle
€
D0 << D
33
Summary
• Simple but accurate expressions for grain size in stir zone– Additional experiment supports calculations
• Scaling law for temperature– Very close to experimental measurements– Easy to couple with empirical correlations of grain growth
• Scaling law for shear– Close to experimental measurements– Supports Sato’s hypothesis that for 6061/3 alloys final
grain growth is mostly due to coarsening
34
Points outside validity of simplification
35
Alternative interpretation of coarsening
• Coarsening: effect of combined time and temperature
• Sato: maximum temperature is dominant• Issues to consider:
– Coarsening happens outside the shear layer– Inside the shear layer we have DRX, not static
coarsening– Maximum temperature is well inside shear layer
36
Integrate from here
37
Goal• Create “textbook” type equations for FSW: Discover Scaling
Laws – e.g. Christensen’s and Rosenthal’s solutions– approximate– use only parameters known a priori– good for process design, control, robotics (fast calculations)– good for analysis of outliers and to extrapolate across alloys– good for reverse problem
– good for summarizing massive amounts of data
– good for meta-models– insightful (explicit variable dependences)
38
Simplified Model of Shear Layer
semi-infinite substrate
shear force from tool
hot and deformed shear layer
d
x
∞
∞
∞Schmidt, Acta Mat. 06
39
Coupling in Shear Layer
xd
T∞
Ts
T0
xd
wa
shear layer
temperature profile
velocity profile
heat is generated by plastic deformation in the shear layer
thickness of shear layer determined by To: “minimum temperature for significant shearing”
heat is dissipated away in the substrate
Decay in velocity is in a distance of the order of the heat penetration.
Shear thinning models: decay in velocity is in smaller distance than heat penetration
40
Scaling Analysis• 4 equations, 4 unknowns• Equations
– shear layer, heat conduction– shear layer, heat generation– constitutive law– base plate, heat conduction
• Unknowns– shear layer thickness– temperature jump inside shear layer– frictional heat generated– flow shear stress
41
Heat Transfer in Shear Layer
xd
T∞
Ts
T0
xd
wa
shear layer
temperature profile
velocity profile
02
2
kq
xT
1D conservation of energy, steady state
0
0
0
TT
xT
x
x
d
little heat lost to tool
T0 : matching parameter
conduction heat transfer
volumetric heat generation
42
Heat Transfer in Shear Layer
*
*0
*
qqq
TTTT
xx
c
S
Δ
dSTΔ
xd
T∞
Ts
T0
normalization of variables
02 **
2
2
2
Δ
qkq
xTT cS
d
normalization of energy equation
OM(1)scaling equation
0ˆ
ˆˆ
2 2 Δ
kqT cS
d1
1 equation3 unknowns
charact value
43
Heat Generation in Shear Layerforce equilibrium (near pin)
tt
shear layer substrate
constantt inertial forces are small relative to flow stress
heat generation
dxdvq ss tt -
t t
44
Heat Generation in Shear Layer
velocity decreases with temperature
no slip condition at pin / substrate interface(potential slip at shoulder!)Schmidt, Modelling Simul. Mater. Sci. Eng. 2004•when tool comes out has aluminum stuck on it• threads and texture help move the metal around•most wear happens during plunging scaling equation
€
ˆ q c = 32
η s ˆ τ ωaˆ δ
normalization of heat generation
€
qcq* = 3
2η sτ ωa
δdvdx ⎛ ⎝ ⎜
⎞ ⎠ ⎟
*
2 2 equations4 unknowns
dxdvq st-
xd
w a
shear layer
45
Constitutive Law in Shear Layer
-
RTQA
n
Rexp
tt
Al 6061
limit of empirical data
extrapolated values
46
Constitutive Law in Shear Layer
)(exp2 ***
xfRTQA
dxdva
s
n
R
-
-
tt
dw
-
RTQA
n
Rexp
tt
-
-
s
n
R TRQAa
ˆexpˆ
ˆ2tt
dw 3 equations
4 unknowns
not a power law
47
Constitutive Law in Shear Layer
TT0
v’c
shear layer
Ts Tm
v’1t1
STΔ
mTΔ
shearing“no shearing” two regimes for Arrhenius-type function
aw
linearized constitutive law
B
33 equations4 unknowns
48
Heat Transfer in Base Plate• Line heat source on a plate
– Low Pe: isotherms ≈ circular– Could be many other temperature distributions
)(ˆ2
00 PefkaTTT ctw
Δ-
)(exp)( 0 PeKPePef -
4 4 equations!4 unknowns
49
Equations• System of 4 equations with 4 unknowns
d, t, ΔTs, qc
1
2
3
4
0ˆ
ˆˆ
2 2 Δ
kqT cS
d
€
ˆ q c = 32
η s ˆ τ ωaˆ δ
)(ˆ2
0 Pefka
T ctwΔ
=
€
50
Solutions• Have the form of power-laws• Use only tabulated parameters
– no need to measure torque or temperatures– involve no empirical factors
51
Comparisons• Solutions should capture
– right order of magnitude– right trends
• Example for temperature
– measurements/numerical solutions are normalized by predictions– should be ~1 in range of hypotheses– should be ~ constant in range of hypotheses
--
TTTT
S
S
ˆ
52
Maximum Temperature
• No calibration factors, only tabulated data• Valid for aluminum and steel• Translation is always slow• Not much variation with Pe
• variation with Pe has been properly captured by scaling law
• Scaling law provides correct order of magnitude• overpredicts temperature
stainless 304 steel 1018
53
Maximum Temperature
• Rotation is typically fast, but can be slow• Not much variation
stainless 304 steel 1018 Ti 6-4
54
Maximum Temperature
• Shear layer is typically thin, but can be thick• For thin shear layer: not much variation• For thick shear layer: consistent deviation
stainless 304 steel 1018
55
Maximum Temperature
• Corrected using trend based on shear layer thickness• Good for aluminums, steels… hopefully for all materials• Good beyond hypotheses (why?)
stainless 304
steel 1018
56
Torque
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
1.00E-02 1.00E-01 1.00E+00Pe
M/M
*
Khandkar etal. AA6061 Liernert etal. AA6061 ExpYang etal Schmidt and Hattel AA2024 2005Lienert etal. 1018 steel Lienert etal. AA7075Long etal. AA5083-O Reynolds etal. AA7050 2003LOng etal. AA7050-T7
.
• No calibration factors, only tabulated data•Valid for aluminum and steel•Not much variation with Pe
• variation with Pe has been properly captured by scaling law
• Scaling law provides correct order of magnitude
• underpredict torque
57
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.01 0.10 1.00 10.00 100.00d*/a
T/T*
Khandkar etal. AA6061 Lienert etal. AA6061 Yang etal. AA2024Schmidt etal. AA2024 Lienert etal. 1018 steel Lienert etal. AA7075Colligan AA5083 Long etal. AA5083-O Reynolds etal. AA7050 2003Long etal. AA7050-T7
Torque• No calibration factors, only tabulated data• Valid for aluminum and steel• Not much variation with relative shear layer thickness
• variation with relative thickness has been properly captured by scaling law
58
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.01 0.10 1.00 10.00 100.00V/wd*
T/T
*
Khandkar etal. AA6061 Lienert etal. AA6061Yang etal. AA2024 Schmidt and Hattel 2005 AA2024Zhu etal. 304SS Lienert etal. 1018 steelLienert etal. AA7075 Long etal. AA5083-OReynolds etal. AA7050 2003 Long etal. AA7050-T7Lienert etal. Ti6Al4V
Torque• No calibration factors, only tabulated data• Valid for aluminum, steel, titanium•High rotation speed: ~ constant•Low rotation speed: consistent deviation
59
Torque
• Corrected using trend based on rotational speed• Good for aluminums, steels, titanium• Good beyond hypotheses
stainless 304
steel 1018
Ti 6-4