Nanomechanics: Atomistic Modeling

• Introduction• Energy• Kinetics• Kinematics• Damage Mechanics

Outline

Mark F. Horstemeyer, PhDCAVS Chair Professor in Computational Solid MechanicsMechanical EngineeringMississippi State Universitymfhorst@me.msstate.edu

METHODS USED DEPEND ON THEENTITY BEING MODELED

Method Entity Examples

first principles electron density functional theory (DFT)

quantum chemistry (QC)

semi-empirical atom embedded atom method (EAM) modified EAM (MEAM)

N-body, glue

empirical atom, grain, Lennard-Jones (LJ), Pottsdislocation dislocation dynamics (DD)

phenomena continuum Ficks Law

field elasticity/plasticity

Energy: Embedded Atom Method (EAM)

• Total energy E

F i : embedding energy of atom i

i : electronic density of atom i

r ij : separation distance between atom i and j

ij : pair potential of atom i and j

ij

ijij

i ij

ijii rrFE )(2

1)(

)(

1. Molecular Dynamics (f=ma, finite temperatures)2. Molecular Statics (rate independent, absolute zero)3. Monte Carlo Simulations (quasi-static, finite temperatures)

E F 12 r

f E

ij 1

V mvi v j 1

2V ri f j

Determination of Atomic Stress Tensor(Daw and Baskes, 1984, Phys. Rev)

Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials

Local force determined from energy

Dipole Force Tensor (virial stress) is determined from local forces

Note: the difference between EAM and MEAM is an added degree of angular rotations that affect the electron density cloud . For EAM, this quantity is simply a scalar, but for MEAM it includes three terms that are physically motivated:

free surfaces shear crystal asymmetry

Calc. Expt.Lattice Constant (Å) 3.52 3.52Cohesive Energy (eV) 4.45 4.45Vacancy Formation Energy (eV) 1.59 1.6Elastic Constants (GPa)

C11 246.4 246.5

C12 147.3 147.3

C44 124.8 124.7Surface Energy (mJ/m2) 2060 2380Stacking Fault Energy (mJ/m2) 85 125

EAM Represents the properties of Nickel extremely well

CURRENTLY DEVELOPED MEAM FUNCTIONS COVER MOST OF THE PERIODIC TABLE

Mn

Th U

He

B

Sn

Zn

In

Ga

S

Cd

P

HgBa

Sr

Ca

Bi

Be

La

As

Sb

Mg

Li C N

Si

TiSc

H

O

Pr

Na

K

Al

CoV Fe

Zr

Cr Ni

RuY

Hf

GeCu

RhMoNb

PbRe

Nd TbGd

TlAu

Ag

Dy Ho Er

Pt

Pd

IrWTa

Impurities BCC FCC DIA CUB

Pu

HCP

Atomistic StressClausius, Maxwell Viral Theorem, 1870Maxwell Tensor form of Virial, 1874Rayleigh, 1905Irving-Kirkwood, 1949

Generalized Continuum TheoriesCosserat 1909Truesdell, Toupin, Mindlin, Eringen, Green-Naghdi 1960s

Kinetics: Historical Background

• atoms follow Newton’s second law

• temperature maintained at 300K

x

y

z

GEOMETRY FOR ATOMISTIC CALCULATIONS

atoms moved at constant velocity

fixed atoms

periodic in zperiodic,

free surface,or GBin x

0

1

2

3

4

5

0 0.02 0.04 0.06 0.08 0.1

shea

r st

ress

(G

Pa)

shear strain

104 atoms

microyield 1= 4.51 GPa

microyield 2 (0.2% offset)= 4.55 GPa

macroyield = 4.59 GPa

(a) (b)

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

shea

r st

ress

(G

Pa)

shear strain

microyield 1= 2.69 GPa

microyield 2= 3.0 GPa

macroyield= 3.79 GPa

Shear stress-strain curve of material blocks at an applied strain rate of 2.4e8/sec with (a) ten thousand atoms and (b) ten million atoms showing microyield 1 at the proportional limit, microyield 2 at 0.2% offset strain, and macroyield.

107 atoms

yield yield

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2 0.25 0.3

shea

r st

ress

(G

Pa)

shear strain

EVOLUTION OF DISLOCATION SUBSTRUCTURE

Strain rate 108/secT = 300K

106 atoms0.16 m x 0.08 m

107 atoms0.5 m x 0.25 m

shea

r st

ress

(G

Pa)

0

2

4

6

8

10

12

14

0 0.05 0.1 0.15 0.2 0.25

2.4e8/sec6.58e9/sec1.53e10/sec5.26e10/sec

shea

r st

ress

(G

Pa)

shear strain

0

2

4

6

8

10

12

14

0 0.05 0.1 0.15 0.2

2.4e8/sec1.53e10/sec

shear strain

LARGER SAMPLES HAVE CONSIDERABLY LESS STRUCTURE IN THE STRESS/STRAIN

CURVES

0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4 0.5 0.6

single slipdouble slipquadruple slipoctal slippseudopolycrystal

shea

r st

ress

(G

Pa)

shear strain

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6re

solv

ed s

hea

r st

ress

(G

Pa)

shear strain

Schmid type plasticity observed at nanoscaleHorstemeyer, M.F. Baskes, M.I., Hughes, D.A., and Godfrey, A. "Orientation Effects on the Stress State of Molecular Dynamics Large Deformation Simulations," Int. J. Plasticity, Vol. 18, pp.

203-209, 2002.

Strain rate 1010/secT = 300K

Yield Stress Depends on Specimen Size Because of Dislocation Nucleation Dominance

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

100,000 atoms1,000,000 atoms

shea

r st

ress

(G

Pa)

shear strain

Strain rate 2.4 x 108/secT= 300K

Horstemeyer, M.F., Plimpton, S.J., and Baskes, M.I."Size Scale and Strain Rate Effects on Yield and Plasticity of Metals," Acta Mater., Vol. 49, pp. 4363-4374, 2001.

0

2

4

6

8

10

12

0 0.05 0.1 0.15 0.2

20000 atoms1384 atoms196 atoms

shea

r st

ress

(G

Pa)

shear strain

A SIMPLE MODEL TO EXPLAIN STRAIN RATE DEPENDENCE

d

dt

mv

1 v c 2eff b

3kT

10b2

vc

1 v c 2

Dislocation is nucleated at a distance x0 from the free surface at a critical stress *

T

Dislocation accelerates according to classical law

eff 0

M I 0

M I 0 0

M I 0 0

Effective stress depends on image stress, Peierls barrier, and orientation

I x0

x * At nucleation there is no net

stress on the dislocation

Yield is defined when b x = 0.2%

T

10-5

0.0001

0.001

0.01

0.1

10-5 0.001 0.1 10 1000 105 107

yiel

d/el

astic

mod

ulus

strain rate

Follansbee (1988) L=0.1 m

Follansbee (1988) L=1 m

Edington (1969) L=1.25 cm

Edington (1969) L=2 mm

Experimental data examining yield stress versus applied strain rate from Follansbee (1988) and Edington (1969) for copper.

Experiment

0

0.05

0.1

0.15

0.001 0.1 10 1000 10 5 10 7 10 9 10 11

yiel

d s

tres

s/el

asti

c m

odu

lus

strain rate (1/sec)

1332 atoms2e4 atoms1e5 atoms

106 atoms107 atoms108 atoms (1.6 microns)exp (Maloy et al. 1995)

Calculations

PREDICTED STRAIN RATE DEPENDENCE OF YIELD STRENGTH IS CONSISTENT

WITH EXPERIMENTS

0

0.02

0.04

0.06

0.08

0.1

0.001 0.1 10 1000 105 107 109 1011

nanoscalemicroscalemacroscale

yie

ld/e

last

ic m

od

ulu

s

strain rate

a

b

c

d

e

f

g h i

Strain Rate and Size Scale Effects

0

1

2

3

4

5

6

7

8

0 0.05 0.1 0.15 0.2

332 atoms23628 atoms

shea

r st

ress

(G

Pa)

shear strain

Size Scale Dependence Observed in Copper Also

0

5

10

15

20

0 0.05 0.1 0.15 0.2 0.25 0.3

100 atoms324 atoms4356 atoms

true

str

ess

(GP

a)

true strain

Size Scale Effect Observed in Tension Also

10-6

10-5

0.0001

0.001

0.01

0.1

1

10-10 10-8 10-6 0.0001 0.01 1

yiel

d s

tres

s/el

asti

c m

odu

lus

size (m)

Volume averaged stress is a function of volume per surface area

large scale experiments

EAM calculations

indentation and torsion

experiments

interfacial forcemicroscopy experiments Conventional theory

predicts that yield stress is independent of sample size Because of their small size, properties of materials to be used in nano-devices are predicted to be vastly different than the properties of materials used in conventional devices

Horstemeyer, M.F. and Baskes, M.I., “Atomistic Finite Deformation Simulations: A Discussion on Length Scale Effects in Relation to Mechanical Stresses,” J. Eng.Matls. Techn. Trans. ASME, Vol. 121, pp. 114-119, 1998.

Horstemeyer, M.F., Plimpton, S.J., and Baskes, M.I."Size Scale and Strain Rate Effects on Yield and Plasticity of Metals," Acta Mater., Vol. 49, pp. 4363-4374, 2001.

Size Scale is related to Dislocation Nucleation (volume per surface area) and strain gradients

• EAM Ni• Expt.

various fcc metals

• Free surface BC

• T = 300 K

0.001

0.010

0.100

1.E-10 1.E-08 1.E-06 1.E-04

volume/surface area (m)

EAM(presentwork)Michalskeand Houston(1998)McElhaneyet al.(1997)Fleck et al.(1994)

Nix and Gao(1998)modelpower law

norm

aliz

ed r

esol

ved

yi

eld

stre

ss

1x0

xx 0.38

not converged rst strain rate

Gerberich, W.W., Tymak, N.I., Grunlan, J.C., Horstemeyer, M.F., and Baskes, M.I., “Interpretations

of Indentation Size Effects,” J. Applied Mechanics, Vol. 69, No. 4, pp. 443-452, 2002

length scale 1: nanoscale

length scale 2: submicron scale

length scale 3: microscale

length scale 4: macroscale

stress

strain

Schematic showing the stress-strain responses at four different size scales.

Determination of Stress

Dislocations are fundamental defectsrelated to plasticity and damage

Edge dislocation

-10

-5

0

5

10

-30 -20 -10 0 10 20 30

classical continuumnonlocal continuumMEAM s12MEAM s21EAM s12

stre

ss (

MP

a)

distance (Angstroms)

distance for12 stress component

distance for 21stress component

Distance range is 4-48 Angstromsfrom center of dislocation

sig12

sig21

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 10 20 30 40 50

sig12sig21

aver

age

shea

r st

ress

(G

Pa)

length (anstroms)

21

12

Comparison of stress asymmetry in a simple shear simulation at 300K with one dislocation

distance for 12stress component

distance for 21stress component

Distance range is 12-44 Angstromsfrom center of three dislocations

sig12

sig21

-1

-0.5

0

0.5

1

1.5

10 15 20 25 30 35 40 45 50

sig12 sig21

-1

-0.5

0

0.5

1

1.5

aver

age

shea

r st

ress

(G

Pa)

length (anstroms)

Comparison of stress asymmetry in a simple shear simulation at 300K with four dislocations

distance for12 stress component

distance for 21stress component

Distance range is 4-48 Angstromsfrom center of dislocation

sig12

sig21

-0.5

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50

sig12sig21

aver

age

shea

r st

ress

(G

Pa)

length (angstroms)

Comparison of stress asymmetry in a simple shear simulation at 10K with one dislocation

distance for 12stress component

distance for 21stress component

Distance range is 12-44 Angstromsfrom center of three dislocations

sig12

sig21

-0.5

0

0.5

1

1.5

2

10 15 20 25 30 35 40 45 50

sig12sig21

aver

age

shea

r st

ress

(G

Pa)

length (angstroms)

Comparison of stress asymmetry in a simple shear simulation at 10K with four dislocations

distance for 12stress component

distance for 21stress component

Distance range is 12-44 Angstromsfrom center of three dislocations

-sig12

-sig21

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

10 15 20 25 30 35 40 45 50

-12-21

Str

ess

Length (angstroms)

Comparison of stress asymmetry in a pure shear simulation at 10K with four dislocations

distance for 12stress component

distance for 21stress component

Distance range is 12-44 Angstromsfrom center of three dislocations

-sig12

-sig21

-0.03

-0.02

-0.01

0

0.01

0.02

0 10 20 30 40 50

-12-21

Str

ess

Length (angstroms)

Comparison of stress asymmetry in a pure shear simulation at 10K with one dislocation

distance for 12stress component

distance for 21stress component

Distance range is 12-44 Angstromsfrom center of three dislocations

-sig12

-sig21

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

10 15 20 25 30 35 40 45 50

-12-21

Str

ess

Length (angstroms)

Comparison of stress asymmetry in a pure shear simulation at 10K with four dislocations

distance for 12stress component

distance for 21stress component

Distance range is 12-44 Angstromsfrom center of three dislocations

-sig12

-sig21

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 10 20 30 40 50

-12-21

Str

ess

Length (angstroms)

Comparison of stress asymmetry in a pure shear simulation at 10K with one dislocation

Local Continuum Theory Equations

Local Continuum Equations

rr A 1r Br sin

A 1r 3Br sin

r A 1r Br cos

Where constants A, B incorporate elastic constants and boundary conditions

Nonlocal Continuum Theory Equations

Nonlocal Continuum, Eringen 1977

rr bk

2 1 a 1 B 3 1 e

2

sin

bk

2 1 a 1 3B 1 2 2 1 e r2

sin

r bk

2 1 a 1 B 3 1 e

2

cos

where,

B 1 P 2 1 e P2

P 2

P kRa , kra

a: internal characteristic lengthk: constantb: relative radial displacement

EVOLUTION OF MICROSTRUCTURE SHOWSTWIN FORMATION

compression

5% 2.3% 4.8%4.3%

tension

slice through sample

Simulation Setup

Single crystal

5800 atoms

Low angle grain boundary

5860 atoms

High angle grain boundary

5840 atoms

35Å

35Å

35Å

70Å

140Å

[1 0 0][0 1 1]

[1 0 0][0 1 1]

[1 0 0][0 1 1]

[1 0 0][0 1 1]

[0 1 1][1 0 0]

[10 1 1][2 10 10]

Boundary ConditionsFixed

Periodic Periodic

FixedFixed

Prescribed velocity V=0.035 Å/ps

Fixed ends

Forward loading

300 K

Periodic

Flexible

Prescribed velocity V=0.035 Å/ps

Periodic

FixedFixed

Flexible ends

Forward loading

300 K

Single Crystal (Fixed-end

Low Angle Grain Boundary (fixed-end)

High Angle Grain Boundary (fixed-end)

Bauschinger Effect Analysis

• Bauschinger Stress Parameter / Bauschinger Effect Parameter

f: stress in the forward load path at reverse point

r: yield stress in the reverse load path

y: initial yield stress in the forward load path

• Bauschinger effects indicated by BSP and BEP

The larger the BSP and BEP are, the stronger the Bauschinger effect

yf

rf

BEP

f

rf

BSP

• BSP and BEP for fixed ends, reverse load at 9% strain

 

• BSP and BEP for flexible ends, reverse load at 8% strain

Bauschinger Effect Results

f (GPa) r (GPa) y (GPa) BSP BEP

Single Crystal 6.22 3.27 6.92 0.475 2.105Low Angle GB 6.79 2.22 7.61 0.672 2.783High Angle GB 6.89 0.21 7.47 0.969 5.734

f (GPa) r (GPa) y (GPa) BSP BEP

Single Crystal 4.22 2.04 3.34 0.517 1.244Low Angle GB 3.92 -0.88 4.9 0.776 1.559High Angle GB 3.65 0.74 3.2 0.797 3.222

Bauschinger Effects Summary

• Less constrained materials exhibit reduced

yield stresses compared to highly constrained

materials

• Single crystal material has the smallest

Bauschinger effects while high angle GB

material has the strongest for both fixed-end

and flexible-end BC’s

• Dislocation nucleation occurs earlier in low

angle GB than single crystal and high angle

GB cases for both BC’s

Kinematics/Strain

• No size scale effects observed– Macroscale torsion experiments, crystal

plasticity, and atomistics show the same plastic spin independent of size scale

– Macroscale internal state variable theory, crystal plasticity, and atomistics show same strain contours independent of size scale

– High rate plastic collapse of same geometry in macroscale experiments show identical result at atomistic results

Crystal plasticity/finite element simulation of torsion single crystal Cu

peaks

oscillation troughs

max CCW rotation

max CW rotation

Plastic Spin

Displ.

Condition/Method wave amplitude ratio

Torsion

experiment (cm) 0.02

finite elements (cm) 0.05

molecular dynamics (solid cylinder)

0.06

molecular dynamics (hollow cylinder)

0.025

Simple Shear

finite elements (cm) 0.25

molecular dynamics (nm) 0.23

peaks

troughs

Plastic Spin is samethroughout lengthscales

A

L

wave amplitude ratio=A/L

STRAIN CONTOURS AT 30% STRAIN - 8x1 Aspect Ratio

ISV

atomistics

crystal plasticity

orientation angle - crystal plasticity

10-6

10-5

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10

shea

r st

ress

/G

x/y ratio

max crystalline strengthatomistics (8 unit cells)

atomistics (16 unit cells)

atomistics (24 unit cells)

crystal plasticity

macroscale internal state variable theory

Average yield stress for a block of material of varying aspect ratios computed with different modeling methods: a) constant y, varying x; b) constant x, varying y.

(a) (b)

10-5

0.0001

0.001

0.01

0.1

0 2 4 6 8 10

shea

r st

ress

/G

y/x ratio

max crystalline strength

atomistics (8 unit cells)

atomistics (16 unit cells)

atomistics (24 unit cells)

crystal plasticity

macroscale internal state variable theory

60 Å

30 Å

y

xz

13 Å

EAM Model, Single-crystal Copper (1 3 4),

Periodic in z (4 unit cells) constant velocity and strain rate of 109 s-1

FEA40s

112.2 psExperimental

Copper (1 3 4)Partial Collapse

Crack initiation point

EAM

219.8 ps

experimental

8.328s (removed at 5.5s)

FEACopper (1 3 4)Fully Collapse

flow localizationpoint

anisotropic inelastic behavior

central curved linesinner boundary

FEA

8.328s (removed at 5.5s)

EAM

Copper (1 3 4)Fully Collapse

EAMExperimental

Size Scale Effect (30Å/18mm)

Copper (1 3 4)

EAMExperimental

outer boundary inner boundary

10.0 s 12.5 s 15.0 s

FEA FEA FEA

172.8 ps230.0 psCopper (0 0 1)

flow localization

Macroscale Modeling of Size Scale Effects

• Observation 1: we observe a size scale effect on stress state

• Observation 2: we do not observe a size scale effect on the geometric, strain, or plastic spin states

• Question: where should the macroscale repository be for the size scale dependence?

• Answer: constitutive relation in dislocation density, not the strain tensor