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

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Canadian Centre for Welding and Joining

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Canadian Centre for Welding and Joining

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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

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8

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• “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

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mechanical energy – stored energy

mechanical energy – stored energy – thermal energy into pin

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• “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

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simplification valid simpl. invalidgray zone

constant, right order of magnitude

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Can we use scaling laws instead of experiments to predict grain size?

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Calibration of scaling law

• Need to calibrate T0 and

• For region of valid hypotheses• C1 = 0.835

• C2 = 1.10

€

ΔTs

C1 C2+

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Calibration of scaling law

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Prediction of grain size

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Additional check

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d=5 mm d=85 mm

d=110 mm d=120 mm

V=0.42 mm/s156 rpm

Tool 6.35 mm

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Prediction of grain size

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Discussion

• Grain size during stirring vs. coarsening during cooling cycle

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Grain size during stirring vs. coarsening during cooling cycle

• During stirringMcQuenn 75, 02

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Grain size during stirring vs. coarsening during cooling cycle

€

D0 << D

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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

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Points outside validity of simplification

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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

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Integrate from here

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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)

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Simplified Model of Shear Layer

semi-infinite substrate

shear force from tool

hot and deformed shear layer

d

x

∞

∞

∞Schmidt, Acta Mat. 06

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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

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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

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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

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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

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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

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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

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Constitutive Law in Shear Layer

-

RTQA

n

Rexp

tt

Al 6061

limit of empirical data

extrapolated values

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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

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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

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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

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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

=

€

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Solutions• Have the form of power-laws• Use only tabulated parameters

– no need to measure torque or temperatures– involve no empirical factors

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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

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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

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Maximum Temperature

• Rotation is typically fast, but can be slow• Not much variation

stainless 304 steel 1018 Ti 6-4

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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

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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

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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

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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

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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

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Torque

• Corrected using trend based on rotational speed• Good for aluminums, steels, titanium• Good beyond hypotheses

stainless 304

steel 1018

Ti 6-4