1. SCALING LAWS TO ESTIMATE GRAIN SIZE AND COARSENING IN THE STIR ZONE Karem E. Tello Colorado School of Mines Adrian P. Gerlich Patricio F. Mendez Canadian Centre for Welding and Joining University of Alberta

2. Canadian Centre for Welding and Joining 2

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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 4

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6. 6 V=6 mm/s Tool M5 for all cases

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10. 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 Crawford et al. STWJ 06

11. 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

12. 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

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14. 14 mechanical energy – stored energy mechanical energy – stored energy – thermal energy into pin

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17. 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/  << 1 Va   a  << 1  a << 1 T p  ∞ / T s  ∞  << 1

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19. 19 simplification validsimpl. invalid gray zone constant, right order of magnitude

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23. Can we use scaling laws instead of experiments to predict grain size? 23

24. Calibration of scaling law Need to calibrate T 0 and For region of valid hypotheses C 1 = 0.835 C 2 = 1.10 24 C1C1 C2C2 +

25. Calibration of scaling law 25

26. Prediction of grain size 26

27. Additional check 27

28. 28 d=5  md=85  m d=110  md=120  m V=0.42 mm/s 156 rpm Tool 6.35 mm

29. Prediction of grain size 29

30. Discussion Grain size during stirring vs. coarsening during cooling cycle 30

31. Grain size during stirring vs. coarsening during cooling cycle During stirring 31 McQuenn 75, 02

32. Grain size during stirring vs. coarsening during cooling cycle 32

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 33

34. Points outside validity of simplification 34

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 35

36. 36 Integrate from here

37. 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. 38 Simplified Model of Shear Layer semi-infinite substrate shear force from tool hot and deformed shear layer  x ∞ ∞ ∞ Schmidt, Acta Mat. 06

39. 39 Coupling in Shear Layer heat is generated by plastic deformation in the shear layer thickness of shear layer determined by T o : “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. 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. 41 Heat Transfer in Shear Layer 1D conservation of energy, steady state little heat lost to tool T 0 : matching parameter conduction heat transfer volumetric heat generation

42. 42 Heat Transfer in Shear Layer normalization of variables normalization of energy equation OM(1) scaling equation 1 1 equation 3 unknowns charact value

43. 43 Heat Generation in Shear Layer force equilibrium (near pin)   shear layer substrate inertial forces are small relative to flow stress heat generation  

44. 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 normalization of heat generation 2 2 equations 4 unknowns x   a shear layer

45. 45 Constitutive Law in Shear Layer Al 6061 limit of empirical data extrapolated values

46. 46 Constitutive Law in Shear Layer 3 equations 4 unknowns not a power law

47. 47 Constitutive Law in Shear Layer shearing “no shearing” two regimes for Arrhenius-type function linearized constitutive law B 3 3 equations 4 unknowns

48. 48 Heat Transfer in Base Plate Line heat source on a plate – Low Pe: isotherms ≈ circular – Could be many other temperature distributions 4 4 equations! 4 unknowns

49. 49 Equations System of 4 equations with 4 unknowns , ,  T s, q c 1 2 3 4 =

50. 50 Solutions Have the form of power-laws Use only tabulated parameters – no need to measure torque or temperatures – involve no empirical factors

51. 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

52. 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. 53 Maximum Temperature Rotation is typically fast, but can be slow Not much variation stainless 304steel 1018Ti 6-4

54. 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 304steel 1018

55. 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. 56 Torque 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. 57 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. 58 Torque No calibration factors, only tabulated data Valid for aluminum, steel, titanium High rotation speed: ~ constant Low rotation speed: consistent deviation

59. 59 Torque Corrected using trend based on rotational speed Good for aluminums, steels, titanium Good beyond hypotheses stainless 304 steel 1018 Ti 6-4