1 /52 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 a CSIRO Materials Science and Engineering PO Box 883, Kenmore, QLD 4069, Australia

2 /52 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.  d -1/2 k oo

3 /52 Three Main Discussion Issues re. Mg-Zn alloys Effect on k and  o of: 1. The solute concentration (solid solution softening and hardening effects, and the development of Short Range Order, SRO) 2.The loading direction (tension or compression) 3. Pseudoelasticity effects stemming from elastic {10-12} twinning

4 /52 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 /52 Alloy 0.8%Zn;. Grain size = 305  m. Pure Mg Grain size (inside the circle) = 747  m. scatter of data in pure Mg partly connected to columnar grains

6 /52 Stress-strain curves for pure Mg, different grain sizes Strength measured at 0.2% plastic strain tension- compression asymmetry: material appears weaker in compression tensile compressive

7 /52 friction stress (intercept): smaller in compression k-value (slope) larger in compression Ordinary H-P plot for pure Mg using the 0.2% proof stress data Note scatter of data compression tension 1 mm to 10 μm Hauser et al. (1956)

8 /52 Flow curves Mg-Zn alloys, d=60~90μm and 2.3%at.Zn alloy, different grain sizes Grain size effects in the alloys Different grain sizes, constant Zn (2.3%) Variable Zn, grain size constant ~75 μm

9 /52 Ordinary H-P plot (0.2% proof stress) for the alloys Negative friction stress, alloys appear softer than the pure Mg at large grain size k-value larger for the alloys Pure Mg 0.4% Zn 2.3%Zn 0.8%Zn Normally the story finishes here

10 /52 Solute and Crystallographic issues to account for: Solute effects: Increased k-value with solute content Twinning effects: Pseudoelasticity Directionality (higher k-value in compression) Low/negative friction stress in compression for the alloys Chapter 2

11 /52 Crystallography of Mg, twinning and the tension compression asymmetry S. Graff, W. Brocks, D. Steglich, Int. J. Plasticity 23, (2007)

12 /52 {10-12} twinning in Mg 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 Prism planes become basal planes and vice verse {10-12} twinning is an “extension” twinning

13 /52 Examples of twinning in pure Mg Mann, Caceres, Griffiths, Materials Science and Engng. 2006

14 /52 Basal slip Twinning + (Prism + Basal & Pyramidal) 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 /52 stress strain Random polycrystals of Mg: tension and compression Tension Compression Profuse twinning in compression creates the tension/compression asymmetry why do you get more {10-12} extension twinning in compression?

16 /52 Polar nature of twinning: Random polycrystals=> you get more {10-12} tension twinning in compression. c-axis extension (some amount of twinning) Agnew et al. (2003) (Mann et al, 2006) c-axis extension (lots of twinning)

17 /52 Pseudoelasticity effects huge loading-unloading hysteresis loops

18 /52 Micrographs showing reversible twinning (as marked) with loading and unloading in pure Mg. (Caceres-Sumitomo-Veidt, Acta Materialia, 2002)

19 /52 Pseudoelasticity effects The pseudoelastic strain adds to the elastic strain Pure Mg: at the off-set strain nearly half the strain is pseudoelastic Permanent plastic strain

20 /52 Pseudoelasticity effects Uncorrected (ordinary H-P) Corrected for psudoelasticity Correction bigger for small grain size => bigger k after correction measurements consistent for all materials

21 /52 HP- after correcting for pseudoelasticity Pure Mg 0.4% Zn 2.3%Zn 0.8%Zn similar k- values in tension and compression k-values a little bigger than for the Ordinary H-P friction stress still negative for 0.4% Zn

22 /52 K=values as a function of the Zn content Ordinary H-P H-P corrected for pseudoelasticity k-value is low for pure Mg, increases rapidly for the alloys k-value Zn content Corrected k- values bigger than for the Ordinary H-P

The friction stress as a function of the Zn content 23 /52 Friction stress goes through a minimum at 0.5at.%Zn

24 /52 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

Chapter 3: Modelling 25 /52 '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 /52 Stepwise increase in k with solute content? Dip in the friction stress? Modelling, (or where we want to get to)

27 /52 Physical meaning of the H-P law for Mg? Armstrong (1968, 1983) Temperature dependence of  o and k (for pure Mg) suggests:

28 /52 Temperature effects on H-P constants Armstrong (1968, 1983) σ o Hauser et al k Hauser et al Solute effects on CRSS basal : Can calculate from SX data

29 /52 Solute contributions to the friction stress  o (basal slip) Yield strength 0123 Zn content (at.%)  o ( M P a ) Zn at.% SRO RSS Cáceres and Blake (2002) Taylor factor (4.5) CRSS basal pure Mg (~0.5 MPa) Akhtar and Teghtsoonian, 1969, 1972 SRO Random sol. sol.

30 /52 Friction stress from first principles Corner

31 /52 Calculated σ o Armstrong’s postulate: does not work Experimental σ o

32 /52 Akhtar and Teghtsoonian, 1969 Hauser et al Ono et al Postulate: The friction stress is related to the CRSS for prism slip. Y-axes scales are related by a Taylor factor of 4.5 Sx and PX values should overlap after correcting by the Taylor factor

33 /52 Solute effects on the tensile behaviour of Mg-Zn alloys Alloys more ductile than pure Mg (10-30% strain) Pure Mg: low tensile ductility (<10% strain) stress strain

34 /52 Solid solution strengthening– Dilute alloys (c < 0.5%Zn) CRSS prism decreases with increasing solute (solid solution softening) Akhtar and Teghtsoonian, 1969, 1972 CRSS prism Zn at.% CRSS basal increases with c 2/3 CRSS basal Akhtar and Teghtsoonian,

35 /52 Effect of solute content, concentrated alloys (c = 0.5~2.6at.%) In the concentrated alloys Zn causes extensive hardening by Short Range Order on the Basal planes Effect of solute on Prismatic slip? CRSS prism (RT) Zn at.% CRSS basal (RT) Akhtar & Teghtsoonian, 1969; Chun & Byrne, 1969; Cáceres & Blake (2002); Blake & Caceres,( 2005) The athermal character of SRO offsets the solid solution softening Random Sol Solution Minimum in CRSS prism

36 / ( M P a ) 0123 Zn content (at.%) C R S S p r i s m  pr Zn at.% Akhtar and Teghtsoonian, 1969; Chun & Byrne, 1969; Cáceres and Blake (2002) Solute effects on k k = α (  pr ) 1/2 (Armstrong) The athermal character of SRO offsets the solid solution softening

37 /52 Calculated and measured k-values k-values corrected for pseudoelasticity Model: k = α (  pr ) 1/2 Model suggests a dip in k for and a higher value for the concentrated alloys Ordinary HP Why does pure Mg have a lower than predicted k? Postulate: Model does not account for twinning effects ?

38 /52 Examples of twinning in pure Mg Mann, Caceres, Griffiths, Materials Science and Engng. 2006

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. 39 /52

Twinning activation stress and k-values 40 /52 Twinning activation stress (Raeisinia and Agnew, 2010)

41 /52 Solute effects on the friction stress The behaviour of the friction stress appears consistent with the CRSS for prism slip in both concentration (0.5Zn%) and amplitude (7~10MPa) Y-axes scales are related by a Taylor factor of 4.5 Friction stress CRSS prism CRSS for basal slip How do we account for the shortcoming in applied stress ? Calculated CRSS basal does not match the experiments

Micromechanistic explanation of the solute effects on the friction stress 42 /52 CRSS prism stress strain The activation of CRSS prism marks the onset of multi-slip, and general plastic strain Stress strain flow curve on the basal plane of a single grain Pure Mg 0.4%Zn 1%Zn 2.6%Zn Solid solution softening creates a dip in the stress to the onset of general plasticity at 0.4%Zn

43 /52 Solute effects on the friction stress Y-axes scales are related by a Taylor factor of 4.5 Friction stress CRSS prism How do we account for the shortcoming in applied stress ? Basal slip microplasticity creates stress concentrations which cover the shortcoming of the applied stress to activate prismatic slip.

44 /52 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 /52 The End