Post on 09-Apr-2018
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HSM 2010
Metz, December 8-10, 2010
Modelling of high strain rate behaviour of
metals accounting for micro-shear banding
Ryszard B. PCHERSKI
Institute of Fundamental Technological Research
Polish Academy of Sciences, Warsaw, Poland
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Outline
1. Shear banding:
- one of the mechanisms of plastic flow in metallic
materials (also in the case of HSM processes, e.g.:
A. Molinari, C. Musquar and G. Sutter, [2002], P. Chevrier et al.)
- the dominant mechanism of plastic deformation ofufg, nano-metals and glassy metals.
2. Constitutive relations.
3. Identification of the viscoplasticity model forufg and nano-crystalline Fe.
4. Effects of strength differential and Lode angle
5. Concluding remarks.
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Observations of shear banding - ufg Fe
d = 980 nm d = 268 nm
quasistatic high-strain rate deformations
Change in deformation mode of ultrafine grained consolidated iron under
uniaxial compression: (a) uniform low-rate deformation with d = 980 nm; (b)
non-uniform low-rate deformation with d = 268 nm and (c) non-uniform high-rate
deformation with d = 268 nm (Jia, Ramesh and Ma [2003]) .
d = 980 nm d = 268 nm
crystallographic slip shear banding
d = 268 nm
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Experimental results of Jia, Ramesh and Ma,Acta Materialia, 51 (2003)
Deformation of nano - and ufg metals
SBf - shear bandingcontribution
function
Z. Nowak, P. Perzyna
R.B. Pecherski,Archives of Metallurgy and
Materials 52, 2007 freepdf available on the Journal
website.
Typical stress-strain curves
obtained for the consolidated
iron under quasi-static and
high-strain-rate uni-axial
compression.
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Trace of
the cluster
of MSB
CSB
[Dziado, 1993]
(courtesy of profesor Andrzej Korbel)
Multiscale hierarchy of shear bands in
polycrystalline metals
Micro-shear band - longand very thin (ca. 0.1 )sheet-like region ofconcentrated andintensive plastic shearcrossing grain boundarieswithout deviation
Particular MSB operatesonly once and develops
fully in very short time
m
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Literature
1. R.B. Pcherski, Modelling of large plasticdeformations based on the mechanism of micro-shearbanding. Physical foundations and theoreticaldescription, Arch. Mech. (1992).
2. R.B. Pcherski, Macroscopic measure of the rate ofdeformation produced by micro-shear banding, Arch.Mech. (1997).
3. R.B. Pcherski, Macroscopic effects of micro-shear
banding in plasticity of metals, Acta Mechanica (1998)
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Multiscale hierarchy of shear bands
in polycrystalline metals
Schematic illustrationof multi-level hierarchy
of micro-shear bands:
a) polycrystalline RVE
with the increasingzone of shear banding,
b) cluster of active
micro-shear bands,
c) a single micro-shear
band (comp. of CSB)
R.B. Pcherski, Macroscopic measure of the rate of deformationproduced by micro-shear banding, Arch. Mech. [1997]
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Macroscopic measure of the rate of deformationaccounting for shear banding
R.B. Pcherski, Arch. Mech. 49, (1997)
, 2
p p p
S SB
p
S
SB
p
SB
p pSB SB
SB
rateof plastic deformation by slip
rate of plastic deformation by shear banding
d d d shear strain ra
d f instantaneous shear banding contribution
d
te
D D D
D
D
D D
(1 )
S SB
SB s
f
f
SBSBf
0
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Account for the change of deformation path (Lode angle )
K. Kowalczyk Gajewska, R.B. Pcherski (2005)
( ( ))1
oMS
ffa b
e
p
p
pp
D
Ddet
2
36)cos(
))cos(1()(
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V SBSB
Vf
V
RepresentativeVolume Element
traversed by shear
bands
V
VSB
VsViscoplastic flow law
accounting for shear
banding in application for
ufg metals(Z. Nowak, P. Perzyna, R.B.
Pecherski, Arch. Metall.
Materials, 2007)
S SB
s SBV V V
SBSBf
Volume fraction
of shear banding
Inst. contr.
of shear
banding
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,
, ( ),
(1 )
yield stren
(
gth at sh
(1 )(
ea
1 ) ,
r
) 01
s SB
s s s SB SB SB SB SB
V SB s SB V SB SB V SB
V
s S S SBB B
P P P
P k V P k V V P k V
Vk k f f k f f
k k
k
f
V
for kf
Balance of plastic deformation power in RVE
assumption - no hardening
0
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Viscoplastic flow law accounting for shear banding
12
0
2 2
0
2
(1 )(1 )
(1 ) , 0
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Viscoplasticity model accounting
for SDE
1
2 2 2
,
( )
1 3( 1) 9( 1) 4
2
p
G G
T
Y
m m e
G G
G F
F
D
=
G
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Identification procedure:
if d > 300 nm ( , , , )
if d < 300 nm ( , , , )
cal
eq
0SB
f 0 2n 3 1
3 0x10vp s
0 08D
0 1SBf 0B 5 0a 0 0 95SBf
0( ) ( )( ) 1 ( )
vpvp n
caleq A d B d d
D
0
0
0
121 exp( ( ) )
1 ( ) 1( )1 exp( ( ) )
SB vpvp
cal SB
vpeq
f
a b dfA d
da b d
D
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Grain diameter
d
A(d) [MPa]
A*
(d) [MPa]
b (d) B(d)[ MPa ]
80 nm 1205.0 98.28 0.0
138 nm 1120.0 94.95 0.0
268 nm 710.0 28.59 0.0
980 nm 106.78 ---- 602.25
20m 100.28 ---- 269.41
Identified parametersDynamic compression
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S ( )
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2 22 2
0 0
' '23
39 3 0
3
13
0 1
,
,
:
T C T C
C T T C Y Y Y Y
e m Y Y m Y Y
m etr
f m v cr + =
W. BURZYSKI (1900-1970) Study on Material Effort Hypotheses,1928 , PhD Thesis (in Pol.)
Ueber die Anstrengungshypothesen, Schweiz. Bauzeitung,1929.
Theoretical foundations of the hypotheses of material effort,
Engineering Transactions, 56, No. 3, 269305, 2008 the English translation of the paper published in Polish
in Czasopismo Techniczne, 47, 1929, 1-41, Lww.
A new, well founded physically, energy-based criterion was proposed:
f
v
- density of elastic energy of distortion
- density of elastic energy of volume change
2 23
Y Y
T C
Y Y Y
Huber Mises Hencky
condition
1
0
Beltrami
Huber
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3 T CY Y Y
The depiction of particular cases ofBurzyski yield condition inthe pressure - equivalent stress coordinates
(W. Burzyski [1928], M. yczkowski [1999]).
3 T CY Y Y
23 T CY Y Y
,e m ,e m
Paraboloid limit surfaces according to Burzyski criterion
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Paraboloid limit surfaces according to Burzyski criterion(T. Fr, Z. Kowalewski, R. B. Pcherski, A. Rusinek
Engineering Transactions [2010])
75%Cr - 25% Al2O3 composite 6061+2Zr+20Al2O3 composite
3
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Comparison of the formulations proposed by
F. Schleicher [1926] and W. Burzyski [1928]
According to W.B.
According to F.S. Poisson ratio
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Yield condition accounting for SDE and Lode angle(R.B. Pcherski, P. Szeptyski, M. Nowak [2010])
W. Burzyski [1928](cf. English translation in
Engineering Transactions, 2010
available on internet)
f
v
- density of elastic energy of distortion
- density of elastic energy of volume change
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Huber-Mises-Hencky cylinder
Burzynski-Drucker-Prager cone
Burzynski-Torreparaboloid
ellipse
m
e
YC
T
Y3
C
Y3
YT
The Burzyski yield condition and its particular cases:
ellipsoid, paraboloid, cone and cylinder.
SDE
SDE Strength Differential EffectC
Y
T
Y
k
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Examples of yield surfaces accounting to the
effect of Lode angle
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Examples how to account for Lode angle
Raniecki&Mrz [2009]
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Our identification for INCONEL 718
Empirical relaton:
Energy-based criterion
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Concluding remarks1. The plasticity flow law accounting for shear
banding was applied for modelling evolution oftexture and metal forming processes (K. Kowalczyk-Gajewska, Z. Mrz, R.B. Pcherski [2005], [2007], [2009]).
2. Viscoplasticity constitutive relations were
developed for modelling the behaviour of ufg andnc-metals (Z. Nowak, P. Perzyna, R.B. Pcherski [2007])
3. The effect ofLode angle was described in the
shear banding contribution in the rate ofdeformation accounting for the influence of the
change of deformation path and also in the
formulation of yield criterion.
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Thank you for your attention