molecular dynamics simulation of homogeneous nucleation ......nucleation in fcc metals,"...
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Molecular Dynamics Simulation of Homogeneous Nucleation of Dislocations in Single Crystals
M.A. Tschopp1, D.E. Spearot2, D.L. McDowell3
1Center for Advanced Vehicular Systems (CAVS), Mississippi State University
2School of Mechanical EngineeringUniversity of Arkansas
3School of Materials Science and EngineeringGWW School of Mechanical Engineering
Georgia Institute of Technology
ASME IMECE 2009-1135612-6 Symposium on Modeling at Atomic/Molecular Scale
Thursday, November 19, 2009, 11:30-13:00
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Outline
• Introduction
• Simulation Methodology
• Atomistic Simulations Results
• Strain Rate Dependence of Dislocation Nucleation
• Orientation Dependence of Dislocation Nucleation
• Importance of non-Schmid stresses (i.e., normal stress)
• Homogeneous dislocation nucleation model• Activation energy, activation volume
• Dislocation nucleation mechanisms
• Partial vs Full dislocation nucleation
• Stable/Unstable stacking fault energy
• Conclusion
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Introduction
Dislocation motion in FCC crystals is well known to be governed by the critical resolved shear stress via Schmid’s law.
Factors controlling dislocation nucleation is not as well understood.
http://www.doitpoms.ac.uk/
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Dislocation motion in FCC crystals is well known to be governed by the critical resolved shear stress via Schmid’s law.
Factors controlling dislocation nucleation is not as well understood.
Introduction
DC D D D D D D D D D D D
DC
SITB
CTB
DC D D D D D D D D D D D
DC
DC D D D D D D D D D D D
DC
SITB
CTB
SITB
CTB
SITB
CTB
Strain (%)
0.00 0.02 0.04 0.06 0.08
Stre
ss (G
Pa)
0
2
4
6
8
10
12
14
Increasing Inclination Angle, Φ
Φ=0ο
Φ=10.02ο
Φ=19.47ο
Φ=29.50ο
Φ=35.26ο
Φ=43.31ο
e.g., σbcmax
Energy Structure Dislocation Nucleation
Tschopp, Spearot, McDowell, “Influence of Grain Boundary Structure on Dislocation Nucleation in FCC Metals," Dislocations in Solids, Vol. 14, pp. 43-139 (2008)
Many recent simulations have focused on atomic mechanisms associated with
heterogeneous dislocation nucleation related to plasticity at the nanoscale, e.g., nanocrystalline materials, nanowires, etc.
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Dislocation motion in FCC crystals is well known to be governed by the critical resolved shear stress via Schmid’s law.
Factors controlling dislocation nucleation is not as well understood.
Introduction
Van Swygenhoven, Derlet, Frøseth 2006
Many recent simulations have focused on atomic mechanisms associated with
heterogeneous dislocation nucleation related to plasticity at the nanoscale, e.g., nanocrystalline materials, nanowires, etc.
Leach, Gall, et al. 2007 T. Zhu, Ju Li, et al. 2008
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Introduction
Kelchner, Zimmerman, et al. 1998
Dislocation motion in FCC crystals is well known to be governed by the critical resolved shear stress via Schmid’s law.
Factors controlling dislocation nucleation is not as well understood.
Many recent simulations have focused on atomic mechanisms associated with
dislocation nucleation related to plasticity at the nanoscale, e.g., nanocrystalline
materials, nanowires, etc.
Uchic, Dimiduk 2005
Improved understanding of how dislocations nucleate homogeneously within a perfect
single crystal is also relevant, especially as the length scale, and the probability of
dislocation sources, decreases.
Important at smaller scalesNano-indentation experiments
Schuh, Mason, Lund 2005
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Introduction
Tschopp, Tucker, McDowell 2008
Dislocation motion in FCC crystals is well known to be governed by the critical resolved shear stress via Schmid’s law.
Factors controlling dislocation nucleation is not as well understood.
Many recent simulations have focused on atomic mechanisms associated with
dislocation nucleation related to plasticity at the nanoscale, e.g., nanocrystalline
materials, nanowires, etc.
Improved understanding of how dislocations nucleate homogeneously within a perfect
single crystal is also relevant, especially as the length scale, and the probability of
dislocation sources, decreases.
Xu, Yip, et al. 2006
Atomistic simulations show that the localized stress state (non-Schmid stresses) plays an
important role in dislocation nucleation. How does the localized stress state affect dislocation nucleation in single crystals?
Non-glide stress on GSF curve
Ito & Vitek, 2001
Importance of normal stress in plasticity
Fatemie & Socie, 1988Qin & Bassani, 1992Lund & Schuh, 2003
Non-glide stress on coreNormal stress on GB nucleation
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Introduction
Dislocation motion in FCC crystals is well known to be governed by the critical resolved shear stress via Schmid’s law.
Factors controlling dislocation nucleation is not as well understood.
Many recent simulations have focused on atomic mechanisms associated with
dislocation nucleation related to plasticity at the nanoscale, e.g., nanocrystalline
materials, nanowires, etc.
Improved understanding of how dislocations nucleate homogeneously within a perfect
single crystal is also relevant, especially as the length scale, and the probability of
dislocation sources, decreases.
Atomistic simulations show that the resolved normal stress plays an important role in
dislocation nucleation. How does loading axis orientation under tension and
compression affect dislocation nucleation?
Tschopp, Spearot, McDowell, “Atomistic simulations of homogeneous dislocation nucleation in single crystal copper,” MSMSE 15 (2007) 693-709.
The objective of this research is to investigate how the stress required for
homogeneous dislocation nucleation in single crystal Cu changes as a function of
crystallographic orientation.
loading axisorientation
11
uniaxial tensile stressσ
11
resolved normal stress
NF NFσ σ= ⋅
11
resolved shear stress
SF SFτ σ= ⋅
11
resolved shear stress
PF PFτ σ= ⋅
{ }Active slip system
111 110
loading axisorientation
11
uniaxial tensile stressσ
11
resolved normal stress
NF NFσ σ= ⋅
11
resolved shear stress
SF SFτ σ= ⋅
11
resolved shear stress
PF PFτ σ= ⋅
{ }Active slip system
111 110
[ ]110 loading axis
[ ]112
( )111 Slip Plane
910ε =
[ ]110 loading axis
[ ]112
( )111 Slip Plane
910ε = 910ε =
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Simulation MethodologySingle Crystal Configurations
3D periodic simulation cell• Minimum 163 nm3 cell size to avoid spurious boundary effects on dislocation nucleation
Mishin et al. (2001) Cu EAM potential• Relatively accurate description of unstable and stable stacking fault energy
Uniaxial Tensile Deformation• Equilibrate at temperature 10 K and 300 K• Uniaxial strain of 109 s-1 in the loading direction• Modified NPT equations of motion (Spearot et al. 2005) are used for the lateral boundaries• Strain until dislocation nucleation • Maximum stress corresponds to nucleation stress
Tschopp, Spearot, McDowell, MSMSE (2007) 693; Tschopp, McDowell JMPS (2008); Tschopp, McDowell APL (2007)
Stereographic triangle showing the 47 uniaxial loading axis orientations investigated in the single
crystal deformation simulations.
910=ε
0=σ
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Homogeneous Dislocation Nucleation Strain Rate Dependence of Stress
Spearot, Tschopp, McDowell, Scripta Materialia (2009)
Strain, εYY
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Stre
ss, σ
YY (G
Pa)
0
2
4
6
8
10
12
14
[321] 107 s-1 strain rate [321] 108 s-1 strain rate [321] 109 s-1 strain rate
Strain Rate (s-1)
1e+6 1e+7 1e+8 1e+9
Nor
mal
ized
Ten
sile
Stre
ss R
equi
red
for D
islo
catio
n N
ucle
atio
n, σ
YY
,max
/ σ Y
Y,1
0^9
0.95
0.96
0.97
0.98
0.99
1.00
1.01
[100] [110] [111] [210] [221] [311] [321]
“Orientation and Rate Dependence of Dislocation Nucleation Stress Computed using Molecular Dynamics”
As the strain rate is reduced from 109 to 107 s-1, the tensile stress required for dislocation nucleation is reduced by at most 4%.
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Homogeneous Dislocation Nucleation Calculated Stresses
Tschopp, Spearot, McDowell, MSMSE (2007) 693
Homogeneous dislocation nucleation stress is computed for each of the 47 loading orientations.
Strain, εYY
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Stre
ss, σ
YY (G
Pa)
0
2
4
6
8
10
12
14
[321] 107 s-1 strain rate [321] 108 s-1 strain rate [321] 109 s-1 strain rate
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Homogeneous Dislocation Nucleation Slip Systems
Tschopp, Spearot, McDowell, MSMSE (2007) 693
Homogeneous dislocation nucleation for (a) [110], (b) [111], (c) [221] and (d) [321] loading orientations in single crystal Cu at 10K
under uniaxial tension. Inset shows the {111} slip plane.
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Resolved Stress ComponentsOrientation Dependence
Spearot, Tschopp, Jacob, McDowell, Acta Materialia (2007) 705; Tschopp, Spearot, McDowell, MSMSE (2007) 693
loading axisorientation
11
uniaxial tensile stressσ
11
resolved normal stress
NF NFσ σ= ⋅
11
resolved shear stress
SF SFτ σ= ⋅
11
resolved shear stress
PF PFτ σ= ⋅
{ }Active slip system
111 110
loading axisorientation
11
uniaxial tensile stressσ
11
resolved normal stress
NF NFσ σ= ⋅
11
resolved shear stress
SF SFτ σ= ⋅
11
resolved shear stress
PF PFτ σ= ⋅
{ }Active slip system
111 110
Schmid Factor
0.500
0.475
0.450
0.4250.400
0.3750.350
0.325
[ ]111
[ ]110[ ]100
0.650.600.550.500.450.400.350.30
0.250.20
0.15
[ ]111
[ ]110[ ]100
Normal Factor
Dislocation nucleation tends to follow both the Schmid resolved shear stress and the
non-Schmid resolved normal stress.
Schmid factor(SF)
Normal factor(NF)
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14Tschopp, McDowell, JMPS (2008) 1806
Tension
For dislocation nucleation in tension, the following trends emerge:
• the higher the Schmid factor, the lower the dislocation nucleation stress• the higher the “normal” factor, the
lower the dislocation nucleation stress
idealnucl SF
τσ =
idealnucl NF
τσ =
Resolved Stress ComponentsTensile Nucleation Stresses
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Dislocation Nucleation Stress Model
Spearot, Tschopp, McDowell, Acta Materialia (2007) 705
idealnucl
SF NFSF NFτσ
α α=
+
idealmotion SF
τσ =Classical Schmid Law
w/ non-Schmid normal stress component
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Dislocation Nucleation Stress Model
Spearot, Tschopp, McDowell, Acta Materialia (2007) 705
w/ non-Schmid normal stress component
idealnucl
SF NFSF NFτσ
α α=
+
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Dislocation Nucleation Stress Model
Spearot, Tschopp, McDowell, Acta Materialia (2007) 705
w/ non-Schmid normal stress component
idealnucl
SF NFSF NFτσ
α α=
+
Predicted Stress Required for Dislocation Nucleation (GPa)
2 4 6 8 10 12 14 16 18 20 22
Cal
cula
ted
Stre
ss R
equi
red
for
Dis
loca
tion
Nuc
leat
ion
(GPa
)
2
4
6
8
10
12
14
16
18
20
22Cu Single Crystal @ 10 KCu Single Crystal @ 300 K
In single crystal FCC Cu, the non-Schmid normal stress
plays an important role in the dislocation nucleation stress.
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Influence of Normal Stress on Ideal Shear Strength in FCC metals
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Dislocation nucleation using IPFEM
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Conclusion
Bicrystal Simulations of
Dislocation Nucleation
Single Crystal Simulations of
Dislocation Nucleation
NORMAL STRESS
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Thank you!
Questions?
Dissertation (free online):
Tschopp, M.A., “Atomistic simulations of dislocation nucleation in single crystals and grain boundaries”http://smartech.library.gatech.edu/dspace/bitstream/1853/16239/1/tschopp_mark_a_200708_phd.pdf
GOOGLE: “tschopp” and “ETD”