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

3

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/

http://www.doitpoms.ac.uk/tlplib/slip/slip_geometry.php�

4



Introduction

DC D D D D D D D D D D D

DC

SITB

CTB


DC


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.

5



Introduction

Van Swygenhoven, Derlet, Frøseth 2006


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

6

Introduction

Kelchner, Zimmerman, et al. 1998




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

7

Introduction

Tschopp, Tucker, McDowell 2008









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

8

Introduction









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

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uniaxial tensile stressσ

11

resolved normal stress

NF NFσ σ= ⋅

11

resolved shear stress

SF SFτ σ= ⋅

11


PF PFτ σ= ⋅

{ }Active slip system

111 110


11


11


NF NFσ σ= ⋅

11


SF SFτ σ= ⋅

11


PF PFτ σ= ⋅


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


11


11


NF NFσ σ= ⋅

11


SF SFτ σ= ⋅

11


PF PFτ σ= ⋅


111 110


11


11


NF NFσ σ= ⋅

11


SF SFτ σ= ⋅

11


PF PFτ σ= ⋅


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)

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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idealnucl

SF NFSF NFτσ

α α=

+

17




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”

http://smartech.library.gatech.edu/dspace/bitstream/1853/16239/1/tschopp_mark_a_200708_phd.pdf�

molecular dynamics simulation of homogeneous nucleation ......nucleation in fcc metals,"...

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