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The Hall-Petch Relationship in cast Mg and Mg-Zn Solid Solutions

The Hall-Petch Relationship in cast Mg and Mg-Zn Solid Solutions C.H. Cáceres, Gemma E. Mann, J.R. Griffiths a Co-operative Research Centre CAST Centre of Excellence Design in Light Metals Materials Engineering, School of Engineering, The University of Queensland, Australia

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The Hall-Petch Relationship in cast Mg and Mg-Zn Solid Solutions

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  1. The Hall-Petch Relationship in cast Mg and Mg-Zn Solid Solutions C.H. Cáceres, Gemma E. Mann, J.R. Griffithsa Co-operative Research Centre CAST Centre of Excellence Design in Light Metals Materials Engineering, School of Engineering, The University of Queensland, Australia aCSIRO Materials Science and Engineering PO Box 883, Kenmore, QLD 4069, Australia

  2. k o d-1/2 Hall - Petch Law (1951/1953) • H-P: Strength increases as d-1/2 • Grain boundaries hinder the movement of dislocations. • o relates to the friction stress in single crystal (depends on solute content, crystal structure). • k = stress intensity factor: small for FCC; large and sensitive to temperature for BCC and HCP.

  3. Three Main Discussion Issues re. Mg-Zn alloys Effect on k and o of: • The solute concentration (solid solution softening and hardening effects, and the development of Short Range Order, SRO) • The loading direction (tension or compression) • Pseudoelasticity effects stemming from elastic {10-12} twinning

  4. Materials • Pure Mg, (grain sizes between ~20 μm and 1.5 mm) • Mg-Zn solid solutions (g.s.: 35 to 700 μm) • Zn contents: 0.4at.%; 1at.%; 2.5at% • Grain size refined with Zr to avoid texture effects • Zr content between 0 and 0.34at%.

  5. scatter of data in pure Mg partly connected to columnar grains Alloy 0.8%Zn; . Grain size = 305 m. Pure Mg Grain size (inside the circle) = 747m.

  6. compressive Stress-strain curves for pure Mg, different grain sizes tensile Strength measured at 0.2% plastic strain tension- compression asymmetry: material appears weaker in compression

  7. Ordinary H-P plot for pure Mg using the 0.2% proof stress data 1 mm to 10 μm Note scatter of data compression tension friction stress (intercept): smaller in compression Hauser et al. (1956) k-value (slope) larger in compression

  8. Variable Zn , grain size constant ~75 μm Grain size effects in the alloys Different grain sizes, constant Zn (2.3%) Flow curves Mg-Zn alloys, d=60~90μm and 2.3%at.Zn alloy, different grain sizes

  9. Ordinary H-P plot (0.2% proof stress) for the alloys 2.3%Zn 0.8%Zn k-value larger for the alloys 0.4% Zn Negative friction stress, alloys appear softer than the pure Mg at large grain size Pure Mg Normally the story finishes here

  10. Solute and Crystallographic issues to account for: Solute effects: • Increased k-value with solute content Chapter 2 • Twinning effects: • Pseudoelasticity • Directionality (higher k-value in compression) • Low/negative friction stress in compression for the alloys

  11. Crystallography of Mg, twinning and the tension compression asymmetry S. Graff, W. Brocks, D. Steglich, Int. J. Plasticity 23, (2007) 1957-1978.

  12. {10-12}<10-11> twinningin Mg Prism planes become basal planes and vice verse {10-12} twinning is an “extension” twinning • L. Wu, A. Jain, D.W. Brown, G.M. Stoica, S.R. Agnew, B. Clausen, D.E. Fielden, and P.K. Liaw: Acta Mater. 2008, vol. 56, pp. 688-695.

  13. Examples of twinning in pure Mg Mann, Caceres, Griffiths, Materials Science and Engng. 2006

  14. Twinning + (Prism + Basal & Pyramidal) slip Basal slip Magnesium’s deformation modes Basal slip is the main mode of deformation. The relative activity of twinning, prism and pyramidal slip depends on the texture and loading mode.

  15. Random polycrystals of Mg: tension and compression Compression Tension stress why do you get more {10-12} extension twinning in compression? Profuse twinning in compression creates the tension/compression asymmetry strain

  16. Polar nature of twinning: Random polycrystals=> you get more {10-12} tension twinning in compression. Agnew et al. (2003) (Mann et al, 2006) c-axis extension (some amount of twinning) c-axis extension (lots of twinning)

  17. Pseudoelasticity effects huge loading-unloading hysteresis loops

  18. Micrographs showing reversible twinning (as marked) with loading and unloading in pure Mg. (Caceres-Sumitomo-Veidt, Acta Materialia, 2002)

  19. Pure Mg: at the off-set strain nearly half the strain is pseudoelastic Pseudoelasticity effects The pseudoelastic strain adds to the elastic strain Permanent plastic strain

  20. measurements consistent for all materials Corrected for psudoelasticity Pseudoelasticity effects Correction bigger for small grain size => bigger k after correction Uncorrected (ordinary H-P)

  21. HP- after correcting for pseudoelasticity 2.3%Zn 0.8%Zn similar k-values in tension and compression 0.4% Zn Pure Mg k-values a little bigger than for the Ordinary H-P friction stress still negative for 0.4% Zn

  22. K=values as a function of the Zn content H-P corrected for pseudoelasticity k-value Corrected k-values bigger than for the Ordinary H-P k-value is low for pure Mg, increases rapidly for the alloys Ordinary H-P Zn content

  23. The friction stress as a function of the Zn content Friction stress goes through a minimum at 0.5at.%Zn

  24. Conclusions to the experimental part • The Hall-Petch stress intensity factor, k, is low for pure Mg and increases rapidly for the alloys. • The (ordinary) k-values are larger in compression. • Correcting the strength data for pseudoelasticity ensures consistency in the way the strength is measured. • After correcting for pseudoelasticity the k-values in tension and compression are the same. • The larger k-values in compression of the ordinary Hall-Petch plot are artefacts created by the elastic twinning. • The friction stress goes through a (negative) minimum at 0.5at%Zn. The End to Chapter 2

  25. Chapter 3: Modelling 'Would you tell me, please, which way I ought to go from here?'   'That depends a good deal on where you want to get to,' said the Cat. (Charles Lutwidge Dodgson ) Lewis Carroll, 1865 'What sort of people live about here?'....Said Alice. ‘We're all mad here. I'm mad. You're mad. Said the Cat. 'How do you know I'm mad?' said Alice.  'You must be,' said the Cat, 'or you wouldn't have come here'.

  26. Modelling, (or where we want to get to) • Stepwise increase in k with solute content? • Dip in the friction stress?

  27. Physical meaning of the H-P law for Mg? Armstrong (1968, 1983) Temperature dependence of o and k(for pure Mg) suggests:

  28. Temperature effects on H-P constants Armstrong (1968, 1983) k Hauser et al. 1956 σo Hauser et al. 1956 Solute effects on CRSSbasal: Can calculate from SX data

  29. 120 Solute contributions to the friction stress o (basal slip) Yield strength 90 CRSSbasal pure Mg (~0.5 MPa) SRO ) Random sol. sol. a P RSS M 60 ( Taylor factor (4.5) SRO o s 30 0 Zn at.% 0 1 2 3 Z n c o n t e n t ( a t . % ) Cáceres and Blake (2002) Akhtar and Teghtsoonian, 1969, 1972

  30. Friction stress from first principles Corner

  31. Armstrong’s postulate: does not work Calculated σo Experimental σo

  32. Akhtar and Teghtsoonian, 1969 Ono et al. 2003 Sx and PX values should overlap after correcting by the Taylor factor Hauser et al. 1956 Postulate: The friction stress is related to the CRSS for prism slip. Y-axes scales are related by a Taylor factor of 4.5 Akhtar and Teghtsoonian, 1969

  33. Solute effects on the tensile behaviour of Mg-Zn alloys stress Alloys more ductile than pure Mg (10-30% strain) Pure Mg: low tensile ductility (<10% strain) strain

  34. CRSS prism CRSSprism decreases with increasing solute (solid solution softening) Akhtar and Teghtsoonian, 1969-1972 Solid solution strengthening– Dilute alloys (c < 0.5%Zn) CRSSbasal increases with c2/3 CRSS basal Zn at.% Akhtar and Teghtsoonian, 1969, 1972

  35. Minimum in CRSS prism Effect of solute content, concentrated alloys (c = 0.5~2.6at.%) The athermal character of SRO offsets the solid solution softening CRSS prism (RT) Effect of solute on Prismatic slip? In the concentrated alloys Zn causes extensive hardening by Short Range Order on the Basal planes CRSS basal (RT) Zn at.% Random Sol Solution Akhtar & Teghtsoonian, 1969; Chun & Byrne, 1969; Cáceres & Blake (2002); Blake & Caceres,( 2005)

  36. 80 m s i r p S S R 60 C 40 20 0 0 1 2 3 Z n c o n t e n t ( a t . % ) Solute effects on k The athermal character of SRO offsets the solid solution softening 100 pr ) a P M ( k = α (pr)1/2 (Armstrong) Zn at.% Akhtar and Teghtsoonian, 1969; Chun & Byrne, 1969; Cáceres and Blake (2002)

  37. Calculated and measured k-values k-values corrected for pseudoelasticity Postulate: Model does not account for twinning effects ? Ordinary HP Model: k = α (pr)1/2 Why does pure Mg have a lower than predicted k? Model suggests a dip in k for and a higher value for the concentrated alloys

  38. Examples of twinning in pure Mg Mann, Caceres, Griffiths, Materials Science and Engng. 2006

  39. Twinning, slip flexibility and the k-value • Slip flexibility (Kelly, Strong Solids, 1973). For a polycrystal to be able to undergo arbitrary amounts of plastic deformation, the 5 slip systems must have comparable CRSS’s, and be available at every point across the entire volume of the crystal. • (Kocks and Westlake 1967): Twinning ensures plastic compatibility at the grain boundaries and relaxes the requirement of 5 independent slip systems for the metal to develop full plasticity. • Twinning brings slip flexibility to Mg. • Twinning turns Mg into a ductile metal, and (postulate) lowers k in the process. • Solute interferes with twinning, and the effect of twinning is less for the alloys.

  40. Twinning activation stress and k-values Twinning activation stress (Raeisinia and Agnew, 2010)

  41. Solute effects on the friction stress CRSS prism How do we account for the shortcoming in applied stress ? The behaviour of the friction stress appears consistent with the CRSS for prism slip in both concentration (0.5Zn%) and amplitude (7~10MPa) Calculated CRSS basal does not match the experiments Y-axes scales are related by a Taylor factor of 4.5 Friction stress CRSS for basal slip

  42. Micromechanistic explanation of the solute effects on the friction stress Solid solution softening creates a dip in the stress to the onset of general plasticity at 0.4%Zn The activation of CRSS prism marks the onset of multi-slip, and general plastic strain CRSS prism stress 2.6%Zn Stress strain flow curve on the basal plane of a single grain 1%Zn 0.4%Zn Pure Mg strain

  43. Solute effects on the friction stress CRSS prism Basal slip microplasticity creates stress concentrations which cover the shortcoming of the applied stress to activate prismatic slip. How do we account for the shortcoming in applied stress ? Y-axes scales are related by a Taylor factor of 4.5 Friction stress

  44. Conclusions (modelling) • The stepwise increase in the Hall-Petch stress intensity seems to be related to short range order in the concentrated alloys. • Profuse twinning appears to lower the k-value of pure Mg. • The friction stress appear to be related to the CRSS for prism slip (the onset of multi-slip). • The dip in the friction stress at intermediate concentrations seems related to solid solution softening effects on the prismatic planes.

  45. The End

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