Scale Dependent Plasticity & the
Bauschinger Effect:
C. W. Sinclair
Dept. Materials Engineering, The University of British Columbia, Vancouver Canada
Current Address: IMAP, Université Catholique de Louvain, Louvain la Neuve, Belgium
Mechamat, Aussois: Jan. 12 2010
Acknowledgements:
I
Olivier Bouaziz (ArcelorMittal, France)
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David Embury (McMaster/UBC)
Precipitation + Grain Size Hardening:
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Warren Poole (UBC)
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Yves Brchet
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Henry Proudhon (Ecole des Mines)
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Chris Hutchinson (Monash)
Nanocomposites
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Georges Saada (ONERA)
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Ludovic Thilly (Poitiers)
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Matthew Killick (UBC)
Mechamat, Aussois: Jan. 12 2010
The Bauschinger Effect
“ Pre-straining in any direction,
as defined by the principal axis of
the strain tensor, will introduce
an anisotropy for further
deformation in any other
direction. The intensity of this
prestrain-associated anisotropy is
at maximum when the direction
of further straining is opposite to
that of the prestrain.”
A. Abel, Historical Perspectives and
Some of the Main Features of the
Bauschinger Effect, Mater. Forum,
1987, 10, 11-26.
Mechamat, Aussois: Jan. 12 2010
The Bauschinger Effect
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Apparent Bauschinger
effect from extrinsic
factors:
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non-uniform deformation
residual stresses
Bauschinger effect from
intrinsic factors:
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Pressure dependent
plasticity (e.g. polymers e.g. Liu and Rottler)
Plasticity mechanisms
which are directional (e.g.
twinning)
Jain, Poole, Sinclair and Gharghouri, Scripta Mater., 62
(2010) 301
Here, focus is on Bauschinger effect as it relates to
microstructure
Mechamat, Aussois: Jan. 12 2010
The Bauschinger Effect
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In mechanics anisotropy of Bauschinger effect can be
considered to arise from a backstress opposing the forward
flow: F (σ 0 − X) = σ0
Classical hardening laws can include X but lack a direct link
to underlying microstructure
Mechamat, Aussois: Jan. 12 2010
The Bauschinger Effect
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From perspective of microstructure can view two
complimentary sources† for Bauschinger effect:
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Composite Effect: Macroscopic flow stress is different from
the flow stress of any of the phases
Intrinsic Phase BE Effect: Backstress arising from local (?)
stresses
Unlike continuum view, these are both intrinsically related to
microstructural features (e.g. vol. fr., scale, morphology,
connectivity)
For composite view, we can write:
σ = (1 − f ) σm + f σp
σm = σ − f (σp − σm )
σm = σ − hσm i
†
Allain, Bouaziz, Mater. Sci. Eng. A, 496 (2008) 329
Mechamat, Aussois: Jan. 12 2010
The Bauschinger Effect
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Bauschinger effect can also be strong in single phase materials
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Backstress arising from asymmetric obstacle strengths: Orowan
Model
Backstress arising from local stress fields (e.g. pileups)
Mott-Seeger
µb
σ − σb = σ0 → σb = M n
λ
I Higher order plasticity theories also predict Bauschinger effect
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e.g. Bayley, Brekelmans, Geers, Int. J. Sol. Struct., 43 (2006) 72687286
Queyreau and Devincre, Phil. Mag., 89
(2009) 419 - 430
Mechamat, Aussois: Jan. 12 2010
Scale and The Bauschinger Effect
Why is Scale Important?
The Bauschinger Effect in Polycrystals
The Bauschinger Effect & Work Hardening of Aged Al Alloys
The Bauschinger Effect & Nanocomposite Wires
Summary
Mechamat, Aussois: Jan. 12 2010
Why does scale matter?
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Effect of distributions
becomes more important
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σ
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Griffith, Hall-Petch,
Orowan laws
Processing accentuated
heterogeneity
Discreteness of Plasticity
D
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Strain per grain ≈ b/D →
For D → 0 progressive
yielding
Need to worry about
dislocation - obstacle
interactions specifically
Saada, Verdier, Dirras, Phil. Mag., 87 (2007) 4875 - 4892
Mechamat, Aussois: Jan. 12 2010
Scale and The Bauschinger Effect
Why is Scale Important?
The Bauschinger Effect in Polycrystals
The Bauschinger Effect & Work Hardening of Aged Al Alloys
The Bauschinger Effect & Nanocomposite Wires
Summary
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Polycrystals
Backstress (MPa)
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10
5
0
-5
Npu
Npu
Npu
Npu
-35
-45
0
100
200
300
400
500
=2
=3
=4
=5
600
µb
n
D
Including the probability of
screening: P = n/n∗
σb = M
-15
-25
Grain boundaries a natural
source for backstress
buildup
700
800
(1)
Number of dislocations
Proudhon, Unpublished
√
σ = σ0 + M αµb ρ + σb
Sinclair, Poole, Brechet, Scripta Mater. 2006
Delinc et al., Acta Materialia, 55 (2007) 2337-2350
σb = M
µb n
n 1− ∗
D
n
(2)
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Polycrystals
Sinclair, Poole, Brechet, Scripta Mater. 2006
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Polycrystals
80
3200
2800
60
2400
Back Stress (MPa)
n∗ =15, n∗A =12
θ (MPa)
2000
1600
1200
800
20 µm
2 µm
40
n∗ =15, n∗A =15
20
400
0
0
100
200
300
400
500
0
0
0.02
0.04
0.08
Strain
Stress (MPa)
Shadkam, 2010
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Polycrystals
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Grain boundaries not the only source of heterogeneity twin
boundaries
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Proposed by Bouaziz† that twinning contributes both to isotropic
and kinematic hardening
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Expect twin and grain boundaries to have similar effect.
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Should be able to apply same model to multiple materials
Bouaziz, Allain, Scott, Scripta Mater. 58 (2008) 484
Mechamat, Aussois: Jan. 12 2010
Comparison Model & Experiment
Model Parameters:
Parameter
Fe-Mn
Stainless
µ
65 GPa
65 GPa
b
0.25 nm
0.25nm
M
3.06
3.06
α
0.4
0.4
n0
7.5
7.5
e
30 nm
30 nm
β
3
3
m
2
2
k1
3x10−2 b−1
2x10−2 b−1
k2
4.5
4.0
k3
1.0
1.2
λ
1266
300
F0
0.2
0.13
Mechamat, Aussois: Jan. 12 2010
Comparison Model & Experiment
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Model captures basic
features of experimental
backstress data
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Particularly the rising
importance of kinematic
hardening with strain
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Kinematic hardening, in
the case of Fe-C-Mn,
gives approximately 50%
of the total work
hardening
Mechamat, Aussois: Jan. 12 2010
Scale and The Bauschinger Effect
Why is Scale Important?
The Bauschinger Effect in Polycrystals
The Bauschinger Effect & Work Hardening of Aged Al Alloys
The Bauschinger Effect & Nanocomposite Wires
Summary
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Age Hardened Aluminum Alloys
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Can predict yield strength on aging for aluminum alloys but work
hardening response has only started to be studied recently
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In underaged case, the dominant effect often appears to be
associated with changes in matrix composition
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In overaged case, stronger effect on work hardening rate
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Classic discussion surrounding:
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Geometrically necessary dislocations – Ashby
Internal stresses (Bauschinger effect) – Brown and Stobbs
Isotropic forms (e.g. Kocks-Mecking-Estrin) account for former but
not latter
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Al-Cu-Sn
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Goal: Look at model alloy containing plate-like precipitates
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Complimentary to study by Proudhon et al. Phil. Mag. 2007
J. da Costa Teixeira, L. Bourgeois, C.W. Sinclair, C.R. Hutchinson, Acta Materialia 57 (2009) 6075 – 6089
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Al-Cu-Sn
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large bauschinger effect
but no permanent
softening
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choose an offset - 1%
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can directly observe the
Orowan loops leading
to the elastic straining
of the particles
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Al-Cu-Sn
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large bauschinger effect
but no permanent
softening
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choose an offset - 1%
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can directly observe the
Orowan loops leading
to the elastic straining
of the particles
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Al-Cu-Sn
σf = σ0 + σss + σpn + σdn
1/n
+ σb
Kinematic Hardening:
h
i
σb = M 2 gDµf ∗p 1 − exp −η ∗p
p
Isotropic Hardening:
√
σ = σ0 + M αµb ρ
√
dρ
d = k1 ρ + k3 (p ) − k2 ρ
h
i
k3 = k3max 1 − η exp −η ∗p
p
k2 → follows model of Simar† for
enhanced recovery
Simar A, Brechet Y, de Meester B, Denquin A, Pardoen T. Acta mater. 2007;55:6133
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Al-Cu-Sn
σf = σ0 + σss + σpn + σdn
1/n
+ σb
Kinematic Hardening:
h
i
σb = M 2 gDµf ∗p 1 − exp −η ∗p
p
Isotropic Hardening:
√
σ = σ0 + M αµb ρ
√
dρ
d = k1 ρ + k3 (p ) − k2 ρ
h
i
k3 = k3max 1 − η exp −η ∗p
p
k2 → follows model of Simar† for
enhanced recovery
Simar A, Brechet Y, de Meester B, Denquin A, Pardoen T. Acta mater. 2007;55:6133
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect and Fracture
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Bauschinger effect can be
modified by fracture as it
provides a means of
relaxation of internal stresses
Horstemeyer† has studied
impact of damage on
Bauschinger effect in
Al-Si-Mg
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Proudhon et al. Phil. Mag. 2007
†
‡
Pressure dependence of
damage kinetics induces a
Bauschinger effect
More complex result‡ for
cast iron – increase of BE
with damage
Horstmeyer Scripta Mater. 39 (1998) 1491, Jordon et al Mech. Mater. 39 (2007) 920
Guillemer-Neel et al Metall. Mater. Trans., 31A (2000) 3075
Mechamat, Aussois: Jan. 12 2010
Scale and The Bauschinger Effect
Why is Scale Important?
The Bauschinger Effect in Polycrystals
The Bauschinger Effect & Work Hardening of Aged Al Alloys
The Bauschinger Effect & Nanocomposite Wires
Summary
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
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Classic Example: Pearlitic
Steel†‡
For as transformed material
can have multiple sources of
BE:
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†
‡
Composite effect
(Cementite-Ferrite,
Ferrite-Pearlite)
Intrinsic phase response
(Cementite, Ferrite)
A material where the BE
grows with scale refinement
→ wire drawing
Allain, Bouaziz Mater. Sci. Eng. A496(2008)329
Stobbs, Paetke, Scripta Metall, 33(1985)777
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
I
I
Classic Example: Pearlitic
Steel†‡
For as transformed material
can have multiple sources of
BE:
I
I
I
†
‡
Composite effect
(Cementite-Ferrite,
Ferrite-Pearlite)
Intrinsic phase response
(Cementite, Ferrite)
A material where the BE
grows with scale refinement
→ wire drawing
Allain, Bouaziz Mater. Sci. Eng. A496(2008)329
Stobbs, Paetke, Scripta Metall, 33(1985)777
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
Cu1
0
2
Cu +Nb
Cu
(a)
(b)
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Cu-Nb nanocomposite
wires produced by
“bundle draw”
technique
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Very high strength →
fine scale
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Very large Bauschinger
Effect
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Thermal stability of
microstructure but not
of strength
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
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Cu-Nb nanocomposite
wires produced by
“bundle draw”
technique
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Very high strength →
fine scale
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Very large Bauschinger
Effect
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Thermal stability of
microstructure but not
of strength
Thily et al. Mater. Sci. Eng. A, 309-310 (2001) 510-513
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
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Extended elastic - plastic transition
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Modify on annealing:
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Residual stresses
Strength differential
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
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Surprising stability of Cu0
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Full recrystallization of coarsest Cu at 600°C
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
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Look at relative elastic strain rate in the copper versus reverse strain
xCu =
I
del
Cu
dpl
T
=
θa − fN b EN b
fCu ECu
Expression assumes 1D loading parallel to fibres and elasticity in the
Nb – can check to see when (if) yielding occurs in Nb since above
expression will no long be valid and xCu < 0
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
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Thilly et al.† have used this process coupled with x-ray diffraction to
calibrate the progressive yielding in forward and reverse directions
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Account for the behaviour of Cu having two different
microstructural scales
Thilly et al. Acta Materialia 57 (2009) 31573169
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
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Correlate initial softening with the Cu0 region (softest)
EN b
0
δσ = 2σCu 1 − fN b 1 −
ECu
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In this case (considering texture) EN b ≈ ECu
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect: Transient
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In all samples observe a very large transient softening
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Progressive yielding of the material in the reverse direction – a
continuous range of yield strengths
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The larger the transient the larger the spread in strengths –
ECu ≈ EN b = E
θR = E
dpl
fpl el + fel
dpl
!
dpl
el
<< 1
dpl
θR /E = fel
θR ≈ d (∆σ) /dpl
Mechamat, Aussois: Jan. 12 2010
Bauschinger Effect in Nanocomposites
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Rate of drop higher at intermediate annealing temperatures – wide
spread in flow stress
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At 600 °C effect of spherodization becomes important
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Not enough information – deconvolution of diffraction information?
Mechamat, Aussois: Jan. 12 2010
Alternative Approaches: Dilatometry
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Alternative way to estimate internal stresses is via relaxation
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Mechanisms that lead to relaxation at low temperatures?
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Wires were annealed in a dilatometer under He atmosphere
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Heating rate 10 °C/min to predefined temperature and held followed
by cooling to room temperature
Mechamat, Aussois: Jan. 12 2010
Internal Stress Relaxation
Estimating relaxation due to
plasticity in Cu:
˙Cu,T = ˙N b,T = ˙T
˙i,T = ˙i = ˙i,p + αi Ṫ
fCu ECu ˙Cu + fN b EN b ˙N b = 0
Then:
˙N b = ˙T − αN b Ṫ
i
EN b h
˙Cu = fN b
∆αṪ − ˙Cu,p
E
E
EN b
˙T − αCu + fN b
∆α Ṫ
˙Cu,p =
fCu ECu
E
Mechamat, Aussois: Jan. 12 2010
Alternative Approaches: Rate Sensitivity
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Assume an isotropic (σi ) and kinematic (σb ) that add linearly
d ln σi
mb = 0
d˙
dσ
d ln σi
= σi
= σi mi = (σ − σb ) mi
d ln ˙
d ln ˙
mi =
Mechamat, Aussois: Jan. 12 2010
Alternative Approaches: Rate Sensitivity
I
Assume an isotropic (σi ) and kinematic (σb ) that add linearly
mi =
d ln σi
d˙
mb = 0
dσ
= (σ − σb ) mi
d ln ˙
Mechamat, Aussois: Jan. 12 2010
Scale and The Bauschinger Effect
Why is Scale Important?
The Bauschinger Effect in Polycrystals
The Bauschinger Effect & Work Hardening of Aged Al Alloys
The Bauschinger Effect & Nanocomposite Wires
Summary
Mechamat, Aussois: Jan. 12 2010
Summary
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Bauschinger test a sensitive probe for comparing against
physical understanding of work hardening in a system
I
Microstructural lengthscales play a crucial role → stress
partitioning
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Still difficult to unambiguously capture Bauschinger effect in a
simple model but tells us something important about what
models will not work
Should also consider other methods for examining stress
distributions within materials
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thermal stability
rate sensitivity
others...?
Mechamat, Aussois: Jan. 12 2010