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
1 /52
Hall - Petch Law
(1951/1953)
• H-P: Strength increases as d-1/2

   o  kd 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.
k
o
d-1/2
2 /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
3 /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%.
4 /52
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.
5 /52
Stress-strain curves for pure Mg, different grain sizes
compressive
120
tensile
true stress (MPa)
Strength measured
at 0.2% plastic
strain
19
91
0.2%
670
80
tensioncompression
asymmetry:
material appears
weaker in
compression
747
1440
40
0
0
0.01
0.02
0.03
true strain
0.04
0.05
6 /52
Ordinary H-P plot for pure Mg using the 0.2% proof stress data
1 mm to 10 μm
80
d (m)
0 5 5 5 0
0
25 62 27 15 10 70 51 39 30 25 20 17
Note scatter
of data
friction stress
(intercept):
smaller in
compression
(0.2%) yield strength (MPa)
compression
60
tension
Hauser et al.
40
(1956)
20
0
0 20 40 60 80 100120140160180200220240
k-value
(slope)
larger in
compression
d-1/2 (m-1/2)
7 /52
Variable Zn ,
Grain
grain size
constant200
~75 μm
Different grain
size effects in the alloyssizes,
constant
Zn (2.3%)
200
48
160
true stress (MPa)
true stress (MPa)
81
0.2%
120
80
Mg-2.3Zn
Mg-0.8Zn
Mg-0.4Zn
Mg
40
0.2%
150
344
500
100
50
0
0
0
0.01
0.02
0.03
true strain
0.04
0.05
0
0.01
0.02
0.03
true strain
0.04
0.05
Flow curves Mg-Zn alloys, d=60~90μm and 2.3%at.Zn alloy, different grain sizes
8 /52
Ordinary H-P plot (0.2% proof stress) for the alloys
2.3%Zn
d (m)
00 5 5 5 0
25 62 27 15 10 70 51 39 30 25 20 17
(0.2%) yield strength (MPa)
120
Negative
ction stress,
80
loys appear
fter than the
pure Mg at
ge grain size
40
0.8%Zn
k-value
larger for
the Zn
alloys
0.4%
Pure Mg
Normally the story finishes here
Mg-2.3Zn
Mg-0.8Zn
Mg-0.4Zn
Mg
0
0
20 40 60 80 100120140160180200220240
d-1/2 (m-1/2)
9 /52
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
10 /52
Crystallography of Mg, twinning and the tension compression
asymmetry
S. Graff, W. Brocks, D. Steglich, Int. J. Plasticity 23, (2007) 1957-1978.
11 /52
{10-12}<10-11> twinning in Mg
Prism
planes
{10-12}
become is
twinning
basal planes
an
and vice
“extension”
verse
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.
12 /52
Examples of twinning in pure Mg
Mann, Caceres, Griffiths, Materials Science and Engng. 2006
13 /52
Magnesium’s deformation modes
Twinning + (Prism + Basal & Pyramidal) slip
Basal slip
Basal slip is the main mode of deformation.
The relative activity of twinning, prism and pyramidal slip
depends on the texture and loading mode.
14 /52
Random polycrystals of Mg: tension and compression
Compression
Tension
stress
D = 91 m
200
c
 (MPa)
150
why do you get more {10-12} extension
twinning in compression?
100
Profuse twinning in compression creates the
tension/compression asymmetry
50
0
0
0.05
0.1

0.15
strain
0.2
15 /52
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
c-axis extension
(lots of twinning)
c
a)
b)
16 /52
Pseudoelasticity effects
loops
Eap
E
f
true stress, 
huge loading-unloading
hysteresis loops
E
a
p
ao

1
p
e

true strain, 
17 /52
Micrographs showing reversible twinning (as marked) with loading and
unloading in pure Mg. (Caceres-Sumitomo-Veidt, Acta Materialia, 2002)
18 /52
Pure Mg: at the off-set
strain nearly half the
strain is pseudoelastic
Pseudoelasticity effects
loops
0.004
E
E
ao

1
p
true strain, 
Permanent
plastic strain
0.003
 19µm
 91µm
670µm
4
 747µm
0.002
1440µm
 81µm
2.3Zn
2
0.001
a
p
pseudoelastic strain
true stress, 
f
a = p
6
e

0
0.00001
The pseudoelastic
strain adds to the
elastic strain
elastic twinning (vol.%)
Eap
0
0.0001
0.001
0.01
permanent plastic strain
0.1
19 /52
measurements
Correctedconsistent
for
for
all materials
psudoelasticity
Pseudoelasticity effects
0.004
80
pseudoelastic strain
670
747
1440
40
0.003
 19µm
 91µm
670µm
4
 747µm
0.002
120
6
1440µm
 81µm
2.3Zn
2
0.001
80
stress (MPa)
91
0.2%
true stress (MPa)
a = p
19
elastic twinning (vol.%)
120
40
0
0
0
0.01
0.02
0.03
true strain
0.04
0.05
0.00001
0
0.0001
0.001
0.01
permanent plastic strain
0.1
Mg-2.3Zn
Mg
0
0
0.002
0.004
0.006
0.008
0.01
true plastic strain
Correction bigger for small grain size
=> bigger k after correction
Uncorrected
(ordinary H-P)
20 /52
HP- after correcting for pseudoelasticity
(0.2%) yled strength (MPa)
120
d (m)
0 5 5 5 0
0
25 62 27 15 10 70 51 39 30 25 20 17
2.3%Zn
0.8%Zn
80
0.4% Znksimilar
valuesPure
in Mg
tension and
compression
40
k-values a little
bigger than for
the Ordinary H-P
friction stress
still negative
for 0.4% Zn
Mg-2.3Zn
Mg-0.8Zn
Mg-0.4Zn
Mg
0
0 20 40 60 80 100120140160180200220240
d-1/2 (m-1/2)
21 /52
K=values as a function of the Zn content
H-P corrected
for
0.8
pseudoelasticity
k-value
Corrected kvalues bigger
than for the
Ordinary H-P
k-value is low for
pure Mg,
increases rapidly
for the alloys
k (MPa m1/2)
0.6
0.4
Ordinary H-P
perm. set compression
perm. set tension
ordinary compression
ordinary tension
0.2
0
1
Zn content (at.%)
2
3
Zn content
22 /52
The friction stress as a function of the Zn content
20
15
Friction stress
goes through a
minimum at
0.5at.%Zn
 (MPa)
10
5
0
ordinary tension
ordinary compression
perm. set tension
perm. set compression
-5
-10
0
1
2
3
Zn content (at.%)
23 /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
24 /52
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.
Lewis Carroll, 1865
(Charles Lutwidge Dodgson )
'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'.
25 /52
Modelling,
(or where we want to get to)
0.8
•Stepwise increase in k
with solute content?
k (MPa m1/2)
0.6
0.4
perm. set compression
perm. set tension
ordinary compression
ordinary tension
0.2
3
2
1
0
Zn content (at.%)
20
15
•Dip in the friction
stress?
 (MPa)
10
5
0
ordinary tension
ordinary compression
perm. set tension
perm. set compression
-5
-10
0
1
2
3
Zn content (at.%)
26 /52
Physical meaning of the H-P law for Mg?
Armstrong (1968, 1983)
Temperature dependence of o
and k (for pure Mg) suggests:
 y   o  kd
-1/2
 o  CRSSbasal
k  CRSS prism 
1/ 2
27 /52
• Temperature effects on H-P constants
Armstrong
(1968, 1983)
k Hauser et al. 1956
CRSS

1/ 2
prism
( Flynnet al.)
k  CRSS prism 
1/ 2
σo Hauser et al. 1956
Solute effects
on CRSSbasal:
Can calculate
from SX data
 o  CRSSbasal
CRSS basal (Conrad et al.)
28 /52
Solute contributions to the friction stress o (basal slip)
Yield
strength
120
(MPa)
90
 o  M ( o  Br c
CRSSbasal pure
60
o
Taylor
factor (4.5)
Mg (~0.5 MPa)
2/ 3
 BS[c(1- c)] )
2
Random
SRO
RSS
SRO
sol. sol.
30
0
0
1
2
Zn content (at.%)
Cáceres and Blake (2002)
3
Akhtar and Teghtsoonian, 1969, 1972
Zn at.%
29 /52
Friction stress from first principles
 o  M b  M( o  { [Br c ]  [BS (c(1- c)) ] } )
2/3 2
Mg
2 2 1/ 2
o
(MPa)
2.2
Mg-0.4Zn
8.4
Mg-0.8Zn
12
Mg-2.3Zn
27
Corner
30 /52
30
Armstrong’s
postulate:
ordinary tension
ordinary compression
perm. set tension
perm. set compression
calculated
20
does not work
 (MPa)
 o  CRSSbasal
Calculated σo
10
Experimental σo
0
-10
0
1
2
3
Zn content (at.%)
31 /52
Akhtar and Teghtsoonian, 1969
100
CRSS Prism
Ono et al. 2003
Sx and PX values
should overlap
after correcting by
the Taylor factor
Postulate: The friction
stress is related to the
CRSS for prism slip.
o Ono et al.
CRSS (MPa)
Hauser et al. 1956
friction stress (MPa)
100
10
o Hauser et al.
10
1
CRSS basal
Akhtar and Teghtsoonian, 1969
0
100
200
300
T (K)
400
500
Y-axes scales
are related by
a Taylor factor
of 4.5
600
32 /52
Solute effects on the tensile behaviour of Mg-Zn alloys
stress
allTT
2
300
Alloys more ductile than
pure Mg (10-30% strain)
2.4
1
true stress (MPa)
0.4
Pure Mg: low tensile
ductility (<10% strain)
0.2
200
Mg
100
0
0
0.2
0.1
true plastic strain
0.3
strain
33 /52
Solid solution strengthening– Dilute alloys (c < 0.5%Zn)
CRSS
prism
CRSSprism decreases with increasing
solute (solid solution softening)
Akhtar and Teghtsoonian, 1969-1972
40
CRSS basal (MPa)
CRSS prism (MPa)
60
CRSS20
basal
CRSSbasal increases with c2/3
0
0
1
2
Zn content (at.%)
3
Zn at.%
Akhtar and Teghtsoonian, 1969, 1972
34 /52
Effect of solute content, concentrated alloys (c = 0.5~2.6at.%)
Minimum in
The athermal character of SRO
CRSS prism
offsets the solid solution softening
CRSS basal (MPa)
CRSS prism (MPa)
CRSS 60
prism
(RT)
Effect of solute on
Prismatic slip?
40
In the concentrated
alloys Zn causes
extensive hardening
by Short Range Order
on the Basal planes
20
CRSS
basal
(RT)
0
0
1
2
Zn content (at.%)
Zn at.%
3
Akhtar & Teghtsoonian, 1969; Chun & Byrne, 1969;
Cáceres & Blake (2002); Blake & Caceres,( 2005)
Random Sol Solution
35 /52
pr
CRSS prism (MPa)
Solute effects on k
The athermal character of SRO
100
offsets the solid solution softening
80
60
40
20
k = α (pr)1/2
0
0
1
2
Zn content (at.%)
Akhtar and Teghtsoonian, 1969; Chun & Byrne, 1969;
Cáceres and Blake (2002)
3
(Armstrong)
Zn at.%
36 /52
Calculated and measured k-values
k-values
corrected for
pseudoelasticity
0.8
Why does
pure Mg have
a lower than
predicted k?
Ordinary HP
0.6
Model: k = α (pr)1/2
k (MPa m1/2)
Postulate:
Model does
not account
for twinning
effects ?
perm. set compression
perm. set tension
ordinary compression
ordinary tension
0.4
Model suggests a dip
in k for and a higher
value for the
concentrated alloys
0.2
0
1
2
3
Zn content (at.%)
37 /52
Examples of twinning in pure Mg
Mann, Caceres, Griffiths, Materials Science and Engng. 2006
38 /52
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
60
perm. set compression
perm. set tension
ordinary compression
ordinary tension
twinning
40
k (MPa m1/2)
0.6
twinning activation stress (MPa)
0.8
Twinning activation
stress
0.4
20
0.2
0
0
1
2
(Raeisinia and Agnew, 2010)
3
Zn content (at.%)
40 /52
Solute effects on the friction stress
CRSS prism
100
2
4
100
10
1
10
1
Friction stress
0
1
2
Zn content (at.%)
3
CRSS prism slip (MPa)
Calculated CRSS
basal does not match
the experiments
3
o (MPa)
How do we account
for the
The
behaviour
shortcoming
in
of the friction
applied
stress ?
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
CRSS for basal slip
41 /52
Micromechanistic explanation of the solute
Solid
solution stress
effects on the
friction
The activation of CRSS prism
marks the onset of multi-slip,
and general plastic strain
softening creates a
dip in the stress to
the onset of general
plasticity at 0.4%Zn
CRSS prism
stress
2.6%Zn
1%Zn
0.4%Zn
Stress strain
flow curve on
the basal plane
of a single grain
Pure Mg
strain
42 /52
Solute effects on the friction stress
CRSS prism
100
2
100
10
1
10
1
Friction stress
0
1
2
Zn content (at.%)
3
CRSS prism slip (MPa)
3
o (MPa)
How
Basal
doslip
we
account
microplasticity
for the
shortcoming
creates stress
in
applied
concentrations
stress ?
which cover the
shortcoming of the
applied stress to
activate prismatic
slip.
4
Y-axes scales
are related by
a Taylor factor
of 4.5
43 /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.
44 /52
The End
45 /52