documentss

MM-1

Prof. Dr. Siegfried SchmauderIMWF, Universität Stuttgart

• Lecture 1: Multiscale Modelling (Part I)

• Lecture 2: Multiscale Modelling (Part II)Monte Carlo Method

• Lecture 3: Molecular Dynamics (Part I)

• Lecture 4: Molecular Dynamics (Part II)

Lecture Contents

MM-2

Blank PageBlank Page

MM-3


• Theory

• Different approaches of scale bridging

• Microstructural problem

• Length and time scales

• From atomistic to macroscopic behaviour

• Experimental comparison

Multiscale Modelling (Part I)

MM-4

Theory

Linking relevant scales from atomistics through microscopic to macroscopic length scales in precipitation hardening materials by a multiscalematerials methods (MMM) approach

Phenomena to investigate (at first in Cu containing steels, later in Al-alloys): Formation and growth of precipitates

Interaction of dislocations and precipitates

Macroscopic elastic-plastic and damagebehaviour of bulk material

Identification of the parameters to transport from one method/scale to the next hierarchical level

Component (Integrity)

Wadh, c

-

Jc

Atomistics

Mesostructure

Microstructure(Microcracking)

MM-5

Cover each problem on the appropriate length scale and with the best method (Hierarchical MMM)

Bridge gaps by parameter transfer Precipitate distribution and growth => MC/PFM

Dislocation particle interaction energies for different geometric alignments => MD

Dislocation particle interaction for a wide range of particles => DD

Void formation and growth at inclusions => DM/FEM

F

FMC/PFM MD DD FEM/DM

Theory

MM-6

D a m a g e M e c h a n i c s

10 10-6

10-3

100m

(nm) (m) (mm) (m)Atomistics Materials Science Engineering

Nano Meso I Micro Meso II MacroNano simulation Dislocation Model Microstructure Model Mesoscopic Meso-Macro-

fracture aspects coupling

Atomic bonds Dislocations Microstructure Coating Specimen,Component

Dislocation theory

Dislocation Dynamics

Phase Field Method

Micromechanics Mesomechanics Structural mechanics

Microcrack Delamination Macrocrack

Void formation

FEAt Microstructural model Layered model

Fracture ofatomic bonds

Plasticdeformation

Dislocation/particle-interaction

MolecularDynamics

Monte Carlo

-9

Theory

MM-7

Macro(Mechanics)

Electrons(Bonding)

Atoms(Cohesion)

Microstructure(Micro Cracking)

Specimen(Controlled Failure)

Component(Integrity)

Micro(FEM)

Nano(MD)

Femto(ab initio)

Macro(FEM)

Materials Science(bottom-up-approach)

Theory

MM-8

Different Approaches of Scale Bridging

Fiber

Composite

Matrix

σ

σ

F

F

l

c

(SG) CP-FEM

Experiment

CM

Wadh

c

= ?N

Wad

h, c

, J c

CZM

Wadh

MD

c

c

Wadh

YN Y

Intrinsic mechanisms, relevant scales, constitutive models, sub-modellingScale Appropriate Modelling

(Engineering Approach)

1 model1 scale/method

Relevant Scale Effects

- Local and global effects - Homogenization (Non-)Local Modelling

Integrated Materials Modelling(Submodelling)

1 modelSeveral scales/methods

Boundary Coupling

- Micro/Macro-Coupling - Metal Matrix Composites FE-Atomistic Model

Hierarchical Modelling(New Approach)

Several models Several scales/methods

Parameter Coupling

- Nano/Micro/Macro-Coupling - Interface Fracture Aging of Steel

MM-9

Meso ScaleNano Scale

Macro Scale

Wadh,c

Experimental Macroscopic Behaviour

Jc

ComparisonConvergence?No

Jc Wad

h,

c, J

c

Yes

Wadh

OO

O O O

NbNb Nb

NbNb

AlAl Al Al

Ab-initio/MD

Fro

m L

iter

atu

re

Cohesive Model

c

c

Wadh

Normal separation

Nor

mal

trac

tion

fc , parameter = Wadh, c

Plastic Strain

(SG) CP-FEM+

AluminaNiobium

Plastic Strain

BCC

Scale Bridging Procedure

MM-10

Monte Carlo (MC)

Cu

Mn

Ni

Crystal latticeFixed bcc lattice is occupied with Fe-, Cu-, Ni-, Mn-atoms and 1 vacancy (V). Periodic boundary conditions

Kinetic ModelDiffusion of the atoms works via a thermally activated vacancy mechanism

Jump Rates Mn Ni, Cu, Fe, , kT

ΔEexpυΓ jX

jX

jXjX

MM-11

Interpenetrating microstructure (higher volume fraction of second phase)

Precipitates (low volume fraction of second phase)

Phase Field Method (PFM, 3D)

Shell type precipitate structure

MM-12

Molecular Dynamics (MD)

Frank-Read source

Orowan Cutting

Crack propagation (NiAl)

Nano indentation numerical tensile testsystem: Ni/NiAl, (111)-interfacestrain: 6%development of stacking faults

NiNiAl

MM-13

DD-Simulation of Interaction Between a Dislocation anda Field of Precipitates

(V. Mohles, RWTH Aachen)

Dislocation Dynamics (DD)

MM-14

Finite-Element-Method (FEM, Rousselier)

void nucleation

void growth

void coalescense

01

1 0

feDf

fk

H

kv

v = von Mises stressH = hydrostatic stress0 = yield strengthk = material dependent parameterf = void volume fractionD = material independent parameter = 2

MM-15

material 15MnNi6-3: f0 = 0,05%, lc = 0,05 mm (adjusted)k = 445 MPa, fc = 5.0%, D = 2 (fixed)

notched specimen C(T) specimen

Finite-Element-Method (FEM, Rousselier)

MM-16

MnS-Inclusions

10 µm

Prolate MnS-inclusions Load direction (dz)and direction ofcrack growth (dx)

Load direction (dx)and direction ofcrack growth (dz)

Adjustment of the virtual void volume f0*,e.g. to the load direction (dz) f0* = F f0 with F = (dx dy)0.5 / dz(f0 = measured void volume)

dzdx

dy

f0* = 0.002(adjusted)

f0 = 0.001(measured)

experiment

f0* = 0.002 (adjusted,improved analysis)

MM-17

MC/PFM-MD/DD-FEM

Monte-Carlo (MC)-simulationPhase Field Method (PFM)

Molecular Dynamics (MD)-methodDislocation Dynamics (DD)

Finite-Element-Method (FEM)Damage Mechanics (DM)

0

100

200

300

400

500

600

700

800

0.00 0.05 0.10 0.15 0.20 0.25

A111A112

A113

15 NiCuMoNb 5E60A and E60B

T - Direction

B111B112

B113T= 90°C

E60A

E60B

stre

ss

/ MP

a

strain

Specimen (Macro)Dislocations (Meso I)Atomistic scale (Nano)

initial state – solid solution

numerically derivedJR-curves

mechanical behaviour crack propagation

aged state - precipitates dislocation-precipitate interaction

E

DDMD

PFM

MC

DM

d

d

T t JR

MM-18

Micromechanical Problem

Macro-Meso-Micro Coupling

MM-19

regime 1-9

Notched tensile specimen with elastic-plastic property gradient

Notched Tensile-Specimen,Shot Peening of the Notch Ground

MM-20

Notched Tensile-Specimen, Ti-2.5Cu

MM-21

Fiber Reinforced Tube-Tube-Joint

Das Bild kann zurzeit nicht angezeigt werden.

CUT 2

CUT 1

Conclusion: Reduced Stresses

CUT 1

Tube-Tube-Joint

MM-22

Carbide Rich Layer100 μm

20 μm

Broken Carbides

Austenite Ferrite

Example Meso Simulation: Carbide Seam

Model Carbide Seam

MM-23

Distribution of Effective Strainsat the Austenite/Ferrite Interface

2.54

2.46

2.29

2.11

1.94

1.76

1.58

1.41

1.23

1.06

0.88

0.70

0.53

0.33

0.18

0.00

pl

Meso Simulation: Carbide Seam

MM-24

Coated Hard Metals

CVD Coating Residual Stress Cracks

Cross Section

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

400

500

600

0

E-M

odul

Normierte Rißlänge (a-t)/h

Nor

mie

rte

Ene

rgie

frei

setz

ungs

rate

Conv

CoStriFree

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1Normierte Rißlänge (a-t)/h

Nor

mie

rte

Ene

rgie

frei

setz

ungs

rate

0

20

40

60

80

100

120

0 10 20 30

C oStr iC onv Free

Rißlänge [m]

Riß

wid

erst

and

[N/m

] EC= 400 GPa

C= 9.3e-6 K-1

• Eigenspannungesentwicklung, DT = -400 K• Rißinitiierung in der Oberflächenbeschichtung• Mode I Belastung, Rißfortschrittssimulation

0

2 0

4 0

6 0

8 0

10 0

12 0

14 0

0 10 20 30

Rißlänge [m]

Riß

wid

erst

and

[N/m

]

EC= 250 GPa

EC= 570 GPa

C= 9.3e-6 K-1

da

UWd

Bda

dU

BR elastexterndiss )(11

Rißwiderstand

Crack Resistance CurvesMechanical Loading

Co - Striation

Crack length [m]

Cra

ck g

row

th r

esis

tanc

e [N

/m]

Cra

ck g

row

th r

esis

tanc

e [N

/m]

Crack length [m]

Cra

ck g

row

th r

esis

tanc

e [N

/m]

Nor

mal

ized

cra

ck g

row

th r

esis

tanc

e

Normalized Crack length (a-t)/h

Normalized Crack length (a-t)/h

MM-25

Coating with cracks

Model of coated hardmetaltool with gradient zone

cracks

coatinggradient zone

substrate

CoStri

Convfree

depths in gradient zone x/hE

(x)

/ E

coating

gradient zone

substrate

(a-t)/h

Nor

m.

ener

gy r

elea

se r

ate

CoStri

free

ConvL/H=1

Cracking in WC/Co Hardmetals

coating gradient zone substrate

MM-26

Influence of Fiber Clusters on Damage Propertiesin Short Fiber Reinforced Composites (C5)

B1

B5

B4

B3

B2

100 μm

σ

σ

Mesostructural section

B1

B2 B3

B4

B5

σ

σ

Mesoscopic Model Modeling:

A mesostructural section of a short fiber reinfo-rced composite M124-Saffil (Al/15vol.%Al2O3 -Fibers).

Result: High values of the Damage-parameter are observed in areas with low volume fractions of fibers close to areas of high volume fractions of fibers.

D -P a ra m e te r a t ε g e s .= 0 ,6 %

Damage-Parameter-Distribution

Mesoscopic Damage Simulation

MM-27

Model

Model of the Real Structure CellReal Microstructure

Strain [%]

ExperimentSimulation

%45.0 %337.0

Experiment: Deformation and Fracture

AlSi7Mg3 Cast Alloy (Globular Microstructure): Comparison

Comparison

MM-28

Multiphase Elements

Phase Boundary & Gaussian Points

Integration Point Phase Boundary

MM-29

3D-Reconstruction

AlSi Alloy

Distance BetweenLayers : 10µm

MM-30

2D/3D-Comparison

2D 3D

Designation of the Phases to the Elements

Stress Distribution, Loading Axis Corresponds to x-axis

x Distribution

MM-31

Simulation Results

AlSi Cast Alloy with Lamellar Microstructure

x Distribution

Stress Distribution x, at various total strains

%2.0 %25.0

%26.0 %27.0

MM-32AlSi Cast Alloy with Globular Microstructure

Stress Distribution x, at various total strains

x Distribution

%203.0 %243.0

%277.0 %337.0

Simulation Results

MM-33

Different Microstructures

Randomly distributed fibers Aligned fibers

Al/15vol.%Al2O3 f Al/46vol.%Bf Al/15vol.%Al2O3 p

Particle Reinforced Composites

50 µm

Fe/50vol.%Cu

InterpenetratingMicrostructures

20 µm

MM-34

Al/15vol.%Al2O3 – Real Microstructure

50 m50 µm

Particle Reinforced Composite: Real Microstructure

Cut-out

FE-MeshMicrostructure

MM-35

Al/15vol.%Al2O3 Real Microstructure/Unit Cell:Tensile Test after Thermal Loading

Strain [ ]

Str

ess

[MP

a]

Experiment

Unit Cell

Real Microstruct.

2D-3D – Comparison

MM-36

Al/Al2O3, Experiment and Simulation

tensile test

residual stresses dueto thermal loading,fceramic = 15 %

Comparison: Experiment – Simulation

MM-37

Micromechanical Matricity Model

Sα

Sβ Phase

MM-38

Electron Energy Loss Imaging onExtraction Replicas

Image processing steps for determination of matricity character, demonstrated for the (white) Ag phase in a Ag/Fe composite material

Electron Energy Loss Imaging – Matricity Parameter

MM-39

Independence of f and M

f = variable, M = constant f = constant, M = variable

MM-40

limit curve: pl = Ae-B

= triaxialitypl = plastic strain 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

loka

le p

last

isch

e V

ergl

eich

sdeh

nung

0.9 1.0 1.1 1.2 1.3 1.4 1.5Mehrachsigkeit des Spannungszustandes

Bereich ohne Schädigung

Schädigungsbereich

Schädigungsgrenzkurve

damage region

no damage

limit curve

Experimental Determination of the DamageMaterial Parameters A and B

MM-41

Definition of Damage Parameter Curve

1 > 2

MM-42

Definition of the Damage-Parameter D

Definition of the modified Rice&Tracey Damage-Parameter D

MM-43

.. ( . . )fA e N N

0

0

0 45 10 086 0 2 0 1500

Damage-Parameters A, Bfrom Unit Cell Calculations

pl

Damage-Parameters A, B = f(o, f0, N)Model

R0

r0

L0

0.0 0.5 1.0 1.5 2.0 2.50.0

0.4

0.8

1.2

0.0 0.5 1.0 1.5 2.0 2.50.0

0.4

0.8

1.2

0.0 0.5 1.0 1.5 2.0 2.50.0

0.4

0.8

1.2

f0 = 0 =0.01

0.002

0.001

0.03

210 MPa

70 MPa

481 MPa

20 MPa 0.1

0.15

0.2

N =

pl

pldeD

0

1A

B

N = 0.1 0 = 481 MPa N = 0.1 f0 = 0.01 f0 = 0.01 0 = 210 MPa

ax

r, c

. .. ln (ln ) .NB e fNe

0 0

0

152 2 0 260 082 35B

A

MM-44

Crack Path: Simulation and Experiment

Crack inside an Al/20vol.%SiC-Composite

Simulated Crack Path in the MicrostructureFEM-Model and Calculated Crack Path

Crack Path Simulation

MM-45

Artificial Quasi-Real Microstructures of Materials

Simulation of Crack Growth in Real Microstructure of Tool Steels

Simulation of Crack Growth in the Artificial Microstructures as a Basis to theOptimal Design of Materials (some examples)

Microstructural Design of Tool Steels

Microstructural Design of Tool Steels

CarbidePrecipitates

MM-46

Macroscopic Failure of MMCs

Macroscopic Failure Simulation: Cohesive Surface Model

ExperimentSimulation (T = 460 MPa)n

AI/SiC(10%)2000

1500

1000

500

00 100 200

For

ce [N

]

Displacement [m]

MM-47

CT25 Specimen andDecohesion Model Criteria

(s)T(s)(s)n(s) σn(s)T criticalnjijin

MM-48

CT25 Specimen: Experimentand Simulation (Decohesion Model)

MM-49

TPB specimen: Experimentand Simulation (Decohesion Model)

large specimen: 100mm x 20mm x 20mm small specimen: 50mm x 10mm x 5mm

Al/20vol.%SiC - MMC

MM-50

Matrix

Cu

void with particle

Initial state aged statethermal aging

irradiation

5-15 µm

5-15 nm

Material behaviourStress - Strain curve

Crack growth a

J-In

tegr

al

Material State - Material Behaviour

MM-51

Time and Length Scales for the System Fe/Cu

(nm) (m) (mm) (m)Atomistics Materials Science Engineering

Nano Meso I Micro Meso II Macro

10-9

10-6

10-3

100m

TEMAPFIM/TAPMC

Inverse Time Scale

y

d

h

s

ms

s

ns

ps

fs MS, MD

PrecipitatesInterfaces

Dislocation TheoryRussel&Brown

DamageMechanics

SEM, TEM,EFTEM

J-aF-COD Experiment

Simulation

Strength Increase

MM-53

Experimental and numerical investigations of twomaterial states of the material 15NiCuMoNb5 (WB36)

• Crack growth resistance• Tensile and compressive yield curves• Load-crack opening relation• Notched tensile specimens (2mm and 8mm)• Fracture surface• Inclusions, Particle sizes• Performed FE-calclations

Multiscale Modelling (Part II)Damage Mechanics of WB36

MM-54

Fractography (E60A)

M 50:1 M 3000:1

(15NiCuMoNb5)

MM-55

Tensile and Compressive Yield Curves

0

100

200

300

400

500

600

700

800

0.00 0.05 0.10 0.15 0.20 0.25

A111A112

A113


T - Direction

B111B112

B113T= 90°C

E60A

E60B

Str

ess

/ M

Pa

Strain / m/m

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3


T - Direction

T= 90°C

E60AE60B

Str

eng

th /

MP

aShape Change / m/m

Tensile and Compression Curve of E60A and E60B (15NiCuMoNb5)

MM-56

Notched Tensile Specimens

0

10

20

30

40

50

60

70

80

90

100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

15 NiCuMoNb 5

T= 90°C

T - Direction

E60B

E60A

E60A and E60B

Notch radius 2 mm0

10

20

30

40

50

60

70

80

90

100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

15 NiCuMoNb 5

T= 90°C

T - Direction

E60B

E60A

E60A and E60B

Notch radius 8 mm

Change of Cross SectionChange of Cross Section D / mm

Results of the Notched Tensile Specimen

MM-57

Crack Growth Resistance Curves

0

50

100

150

200

250

300

350

400

450

500

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

15 NiCuMoNb 5E60A and E60BCT25 TL - Specimens

E60B

E60AT=90°C

J-In

teg

ral /

N/m

m

Crack Growth a / mm

Crack Growth Resistance Curves at T = 90 °C

MM-58

Cu-Precipitates

Frequency Distribution of Cu-Precipitates (E60A, E60B)

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30 35 40 45 50

E60B2023 Cu-Particles

Dmax - Distribution15 NiCuMoNb 5

1321 Cu-ParticlesE60A

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30 35 40 45 50

E60B2023 Cu-Particles

Dmin - Distribution 15 NiCuMoNb 5

1321 Cu-ParticlesE60A

Particle Size / nm Particle Size / nm

Fre

qu

ency

of

Par

ticl

es

MM-59

Dmax vs Dmin and Area Distribution of Cu-Precipitates (E60A, E60B)

Dmin / nm Particle Area / nm2

Nu

mb

er o

f P

arti

cles

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40

E60A and E60B2023 Particles

Dmax vs Dmin

15 NiCuMoNb 5

E60A

E60B

1321 Particles

0

20

40

60

80

100

120

140

160

180

0 10 20 30 40 50 60 70 80 90 100

E60A and E60B

E60A: 1321 Particles

Area Distribution

15 NiCuMoNb 5

E60B: 2023 Particles

E60A

E60B

Cu-Precipitates

MM-60

FE-Discretization and Boundary Conditions ofCT-Specimen with Ic = 0.1 mm (Rousselier – Model)

Nodal Points

(QU8)

cl

Elements

MM-61

Performed FE-Calculations

Specimen Shape Calculation Particle Distance Initial Void Volume

Standard Tensile Specimen axial-sym. lc1 = 0,05lc2 = 0,10

f03 = 0,003 1)f01 = 0,001 - f03 = 0,003 2)

Notched Tensile Specimen = 2 mm axial-sym.lc1 = 0,05

lc2 = 0,10

f01 = 0,001 - f05 = 0,003 2) f04 = 0,004 u. f05 = 0,005 1)

f01 = 0,001 - f05 = 0,005 2)

Notched Tensile Specimen = 8 mm axial-sym.lc1 = 0,05

lc2 = 0,10

f01 = 0,001 - f05 = 0,003 2) f04 = 0,004 u. f05 = 0,005 1)

f01 = 0,001 - f05 = 0,005 2)

CT-Specimen 2D / Plane Strainlc1 = 0,05lc2 = 0,10lc2 = 0,15lc2 = 0,20

f01 = 0,001 - f03 = 0,003 2)f01 = 0,001 - f03 = 0,003 2)f03 = 0,003 - f05 = 0,005 2)

f05 = 0,005 2)

TPB-Specimen 2D / Plane Strain lc2 = 0,10 f02 = 0,002 2)

SECT-Specimen 2D / Plane Strain lc2 = 0,10 f02 = 0,002 2)

CCP-Specimen 2D / Plane Strain lc2 = 0,10 f02 = 0,002 2)

1) E60A 2) E60A and E60B

MM-62

Yield Curves

Experimental and Calculated Technical Yield Curves

0

100

200

300

400

500

600

700

800

0.00 0.05 0.10 0.15 0.20 0.25 0.30

T= 90°C

T - Direction

Tensile Specimen A111

Experiment

l c2 = 0,1 mm

fo1 = 0,001

fo2 = 0,002

fo3 = 0,003

FE-Calculations

15 NiCuMoNb 5E60A

0

100

200

300

400

500

600

700

800

0.00 0.05 0.10 0.15 0.20 0.25 0.30

15 NiCuMoNb 5E60B

T - Direction

Tensile Specimen B111

fo1 = 0,001

l c2 = 0,1 mm

Experiment

fo2 = 0,002fo3 = 0,003

T= 90°C

FE-CalculationsStr

ess

/ M

Pa

Strain / m/m Strain / m/m

MM-63

Notch Radius 2 mmL

oad

F /

kN

Change of Cross Section D / mm

Lo

ad /

kN


Comparison of Experimental and Calculated Tensile Specimens, Notch Radius 2 mm (E60A, E60B)

0

10

20

30

40

50

60

70

80

90

100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

15 NiCuMoNb 5

T= 90°C

T - DirectionNotch Radius 2

fo1 = 0,001

lc2 = 0.1 mm

Experiments

Fracturefo2 = 0,002

fo3 = 0,003fo4 = 0,004

fo5 = 0,005 FE-CalculationsE60A

0

10

20

30

40

50

60

70

80

90

100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

15 NiCuMoNb 5

T = 90°C

T - DirectionNotch Radius 2 mmlc2 = 0.1 mm

Experiments

FE-Calculations

fo1 = 0.001

fo2 = 0.002

fo3 = 0.003

Fracture

fo4 = 0.004

fo5 = 0.005

E60B

MM-64

Notch Radius 8 mm

Lo

ad F

/ k

N


Lo

ad /

kN


Comparison of Experimental and Calculated Tensile Specimens, Notch Radius 8 mm (E60A, E60B)

0

10

20

30

40

50

60

70

80

90

100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

15 NiCuMoNb 5

T= 90°C

T - DirectionNotch Radius 8 mm

fo1 = 0.001

Experiments

lc2 = 0.1 mm

FE-Calculations

fo2 = 0.002

fo3 = 0.003

fo4 = 0.004

fo5 = 0.005

E60A

0

10

20

30

40

50

60

70

80

90

100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

15 NiCuMoNb 5

T = 90°C

T - DirectionNotch Radius 8 mmlc2 = 0.1 mm

Experiments

FE-Calculations

fo1 = 0.001fo2 = 0.002

fo3 = 0.003fo4 = 0.004

fo5 = 0.005E60B

MM-65

Load-Crack Opening RelationL

oad

F /

kN

Crack Opening COD / mm

Lo

ad /

kN

Crack Opening COD / mm

Comparison of Experimental and Calculated Crack Opening Behaviourfor the CT-Specimen (E60A, E60B)

0

10

20

30

40

50

60

70

0.0 0.5 1.0 1.5 2.0 2.5

15 NiCuMoNb 5E60A / T = 90°CCT25 TL - Specimen

Experiment

fo1 = 0.001

lc2 = 0.1 mm

fo3 = 0.003

fo2 = 0.002

a = 26.337 mm / W = 50.5 mma/W = 0.522

FE-Calculations

0

10

20

30

40

50

60

70

0.0 0.5 1.0 1.5 2.0 2.5

15 NiCuMoNb 5E60B / T = 90°CCT25 TL - Specimen

Experiment

fo1 = 0.001

fo2 = 0.002

fo3 = 0.003

lc2 = 0.1 mm

FE-Calculations

a = 28.89 mm / W = 50.1 mma/W = 0.577

MM-66

Crack Growth Resistance and Variationof Specimen Geometry

J-In

teg

ral /

N/m

m

Crack Growth a / mm

J-In

teg

ral /

N/m

m

Crack Growth a / mm

Crack Growth Resistances (E60A, E60B)

0

100

200

300

400

500

600

700

800

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

15 NiCuMoNb 5E60A / T = 90°C

a/W = 0.522 / W = 50.5 mm

lc2 = 0.1 mmfo2 = 0.002

CT-Specimen

Experiment (CT-Specimen)

FE-CalculationsSECT-Specimena/W = 0.5 / W = 50 mm

CCP-Specimena/W = 0.5 / 2W=100 mm

TPB-Specimena/W = 0.5 / W = 50 mm

0

100

200

300

400

500

600

700

800

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

15 NiCuMoNb 5

T= 90°C

lc2 = 0.1 mm

fo2 = 0.002

FE-Calculations

SECT-Specimena/W = 0.5 / W = 50 mm

CCP-Specimena/W = 0.5 / 2W = 100 mm

TPB-Specimena/W = 0.5 / W = 50 mm

E60B

Experiment (CT-Specimen)

CT-Specimena/W = 0.577 / W = 50.1 mm

MM-67

Crack Growth Resistance CurvesJ-

Inte

gra

l / N

/mm

Crack Growth a / mm

J-In

etg

ral /

N/m

m

Crack Growth a / mm

Comparison of Experimental and Calculated Crack Growth Resistance Curvesfor the CT-Specimen (E60A, E60B)

0

100

200

300

400

500

600

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

15 NiCuMoNb 5E60A / T = 90°CCT25 TL - Specimen

Experiment

fo1 = 0.001a = 26.337 mm / W = 50.5 mma/W = 0.522

lc2 = 0.1 mm

fo3 = 0.003

fo2 = 0.002FE-Calculations

0

100

200

300

400

500

600

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

15 NiCuMoNb 5E60B / T = 90°CCT25 TL - Specimen

Experiment

fo1 = 0.001

fo3 = 0.003

fo2 = 0.002

FE-Calculations

lc2 = 0.1 mm

a = 28.89 mm / W = 50.1 mma/W = 0.577

MM-68

0

100

200

300

400

500

600

J-In

teg

ral /

N/m

m

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Crack Elongation a / mm

Material 15 NiCuMoNb 5States E60A and E60B

CT25 TL - Specimen

Experiments

FE-Simulations

lc = 0,075 mm

a = 28.89 mm / W = 50,1 mma/W = 0,577

f o2 = 0,002

a = 26,337 mm / W = 50,5 mma/W = 0,522

State E60B

State E60A

T = 90°C

Crack Growth Resistance Curves

MM-69

Volumes and Distances in Caseof Spherical Precipitates: fo, Ic

Plane Dmax [m] Dmin [m]

TSE60A: 1.2 – 56.4E60B: 1.2 – 73.6

E60A: 0.6 – 13.9E60B: 0.6 – 10.0

LTE60A: 1.2 – 60.5E60B: 1.2 – 78.7

E60A: 0.6 – 16.0E60B: 0.6 – 11.4

LSE60A: 1.2 – 130.2E60B: 1.2 – 87.5

E60A: 0.6 – 9.90E60B: 0.6 – 10.3

From these data the initial void volume fraction fo and the mean

particle distance Ic can be determined:

Plane fo [-] lc [mm]

TSE60A: 0.6064·10-3

E60B: 0.6486·10-3E60A: 0.049

E60B: 0.043

LTE60A: 0.6849·10-3

E60B: 0.6121·10-3E60A: 0.050

E60B: 0.044

LSE60A: 0.6703·10-3

E60B: 0.6439·10-3E60A: 0.053

E60B: 0.046

MM-70

Inclusions

Length/Width of Nonmetallic Inclusions (E60A, E60B)

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20

E60A

1291 Particles

Relation Dmax to Dmin

15 NiCuMoNb 5 LS-Plane

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20

E60B

1628 Particles

Relation Dmax to Dmin

15 NiCuMoNb 5 LS-Plane

Dmin / µm

Dm

ax/

µm

Dmin / µm

Dmax

Dm

in

D Dmax D=mean + min)( / 2

Dmax

Dm

in

D Dmax D=mean + min)( / 2

MM-71

MnS-inclusions are not spherical

Frequency Distribution ofNonmetallic Inclusions

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25 30

E60A and E60B

1803 Particles

Dmax - Distribution

15 NiCuMoNb 5 LT-Plane

E60B

E60A

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25 30

E60A and E60B

1628 Particles

Dmin - Distribution

15 NiCuMoNb 5

LS-PlaneE60B

E60A

Particle Size / µm

Fre

qu

ency

Particle Size / µm

Fre

qu

ency

Dm

inDmax

MM-72

Mechanism-based Strain Gradient Plasticity(MSG) - Theory

Dislocation theory according to Taylor

Homogeneous Deformation:

Elasticity

Atomistic

Plasticity

bG

2

Y

l

GS ρραGb

m

fyστ

(1), p

G ηb1rρ

m = Taylor factorp = strain gradientr = Nye Factor0ρG

2222

22y

SbGαm

εfσρ

(in 1)

p2y

2222

yη

σ

bGαmrεfmτ

p

22222y

2222y

2222

22y η

bGαmσ

bGαmσr

bGαm

εfσαGbτ

m = 3,06 (Polycrystal)r = 1,9 (Bending/Torsion)

p2y

lηεfσσ

Steel WB36

Strain Small Inclusions

MM-73

Crack Growth Resistance for E60B

Elongated Particles taken into account, Improved evaluation software

15 NiCuMoNb 5T = 90°C, E60Bf0=0.13%lc=0.1mm

MM-74

Numerically Derived Crack Resistant Curvesfor Different Strength Increases

Crack Resistant Curves – Worst Case

MM-75Presipitation Induced Aging of Steels

From Atomistics to Macroscopic Behaviour

Str

ess/

MP

a

Monte Carlo Simulation Dislocation - Theory Damage Mechanics

Nano

Micro

Macro

Initial State InteractionDislocation – Small Particles

Mechanical Behaviour

Void Formation Large particles

Aged State

Damage BehaviourTemperature, Time

0 0.1Strain / -

0.2 0.3

200

400

600

800Aged State

Initial State

0

100

200

300

400

500

600

700

800

0.00 0.05 0.10 0.15 0.20 0.25

A111A112

A113


T - Direction

B111B112

B113T= 90°C

E60A

E60B

Str

ess

/ M

Pa

Strain / m/m

Micro

15 NiCuMoNb 5T = 90°C, E60Bf0=0.13%lc=0.1mm

MC-76

Blank Page

MC-77


• Theory

• Experimental background (TEM and SANS)

• The System Fe-Cu

• The System Fe-Cu-Ni-Mn

Monte Carlo Method

MC-78

Cu-alloyed steels for pipelines and component steels atservice temperatures of 300 - 340 °C.

Alloying of Cu results in an increase of the yield stress.

After long time service: strengthening and ductility reduction(„embrittlement“).

Reason: Newly formed Cu-precipitates at temperatures of T > 300 °C.

Aim: Improved understanding of the (macroscopic) mechani-cal properties of Cu-alloyed steels.

Methods: Atomistic simulation of the formation and growth ofprecipitates.

Introduction and Motivation

Pipeline, 15NiCuMoNb5

Fracture

[G. Dobmann, 2003, J. Jansky,1993]

MC-79

Initial state Thermally aged state(340 °C, 57000 h)

TEM-Picture [F. Pan, H. Ruoff, 1996]

Results from SANS (Small Angle Neutron Scattering) -investigations [D. Willer, G. Zies, 2001]:

Initial state: Cu-precipitates with a maximum size of R = 2,79 nm.

Thermally aged state: New precipitates with a maxi-mum of R = 1,29 nm are newly formed.

TEM-Pictures of Cu-Precipitates,SANS-Investigations

---- 100 nm ---- 100 nm

Radius R / nm

Rad

ius

dist

ribut

ion

Steel, 15NiCuMoNb5

MC-80

Beginning: appr. 1950, N. Metropolis, USA

Applications today:

Numerical mathematics: ‚random walk‘, Schrödinger-eq.

Statistical mechanics: Master-eq., Transport-eq.

Atomistic Simulation: Gases, Liquids, Solids

Def.: Monte-Carlo-Method: A stochastic process will be

Simulated with the aid of random numbers (computer)

Monte-Carlo-Simulation: Model experiment on computer

Simultaneous example: Calculation of Distribute N random numbers:

Circle area:

Introduction, Monte-Carlo-Method

50 homogeneously distributed random numbers in [0,1]2

1,01,0 yx2rA

N)(insgesamt hlen Zufallsza#

ntKreissegme imlen Zufallszah#

4

ˆ

N10 3,2000000100 2,96000001000 3,204000010000 3,1196000100000 3,13900001000000 3,146044010000000 3,1415962

MC-81

Determinaton of arbitrarily bounded regions (Volumes)

e. g.: Evaluation of metallic sections (microstructure of steel 15NiCuMoNb5, Ferrit [bright] -Bainit [dark])

Question: How large ist the area of each phase? Border is no explicit function.

One solution: Counting equal coloured voxels and comparison with all voxels.

Monte-Carlo solution: Distributed random numbers across the area, counting of random dots of same colour. Comparison with all random dots. Advantage: Exactness can be predetermined, no discretizaion error.

Introduction, Example: Metallic Cross Section

MC-82

Mn Ni, Cu, Fe, , kT

ΔEexpυΓ jX

jX

jXjX

Crystal Lattice

Fixed bcc lattice Periodic Boundary ConditionsLattice occupied with Fe-, Cu-, Ni-, Mn-AtomsAnd one vacancy (V).

Kinetic Model

Diffusion of atoms by thermally activated vacancy mechanism and jump frequencies:

:ΓjX

:υjX

:ΔEjX

:k

:T

Vibration frequencies

Jump frequencies

Activation energies

Boltzmann constant

Temperature

Atomistic Modelling

Ene

rgy

DistanceFe-Atom Cu-Atom Vacancy

FeΔE CuΔE

FeSp,E CuSp,E

=> Kinetic Monte Carlo-Method

MC-83

MnNi,Cu,Fe,

MnNi,Cu,Fe,

εnεnEΔE

kX

jX

(V) Leerstelleder Nachbarn 1.

4

1k

1

kVX(1)

kVX

Atoms-jX desNachbarn 2. und1.

2

1i

4

1k

i

kXjX(i)

kXjXjXSp,jX

Activation energies are strongly dependent on the local atomic arrangements and are calculated separately for each change of atomic positions.

The activation energy is depending on:

• Saddle point energy

• Interatomic energies

• Vacancy interaction energies

• Occupation numbers

Activation Energies

jXΔE

jXSp,E {1,2}i ,ε i

kXjX 1

kVXε {1,2}i ,n i

kXjX

Ene

rgy

DistanceBased on ´broken bond´-Model [F. Soisson, G. Martin, 1996]

Vacancy

1. and 2. neighboursof the Xj-atoms

1. neighbours of the vacancy (V)

MC-84

Contribution of binding energies

on first and second nearest neighbours:

2

εε

1

jXjX2

jXjX

8z1 6z2

First neighbours

Second neighbours

jXcoh,E

,

2

jXjX21

jXjX1

jXcoh, ε2

zε

2

zE

21

jXcoh,2

jXjX

21

jXcoh,1

jXjX

z2z

2Eε

z2z

4Eε

Assumption:

MnNi,Cu,Fe,X j

1

jXjXε 2

jXjXε

Distance

Calculation of interatomic energies i

jXjXε

Binding Energies

MC-85

The mixing energy is defined when taking first and

second nearest neighbours into account:

Calculation of interatomic energies i

kXjXε

:0ωkXjX

:0ωkXjX

:0ωkXjX

System has tendency to from precipitatesIdeal mutual solubility of atoms

and

System possesses tendency to form ordered structures

kXjXω

21

kXjX2

kXkX2

jXjX2

kXjX

21

kXjX1

kXkX1

jXjX1

kXjX

z2z

2ωεε

2

1ε

z2z

4ωεε

2

1ε

2

1i

i

kXjXi

kXkXi

jXjXi

kXjX 2εεε2

zω

Assumption:

2

εε

1

kXjX2

kXjX

jX

kX

Mixing Energies

MC-86

The saddlepoint energy is determined by the vacancy migration energy in

pure metals ( concentration of atom species Fe, Cu, Ni, Mn).

1,2i , εε ijXkX

ikXjX

FeDebye,jX υυ

Calculation of the vacancy interaction energy from the binding

energy and the vacancy formation energy

1

jVXε

„Symmetrical model“:

Vibrational frequencies:

Vacancy interaction energies, saddlepoint energies

jXcoh,E F

jXV,E

Simplyfying assumptions for the system Fe-Cu-Ni-Mn:

(= typ. vibrational frequencies of the atoms)

MnNi,Cu,Fe, , /zE-Eε jX 1jXcoh,F

jXV,1

jVX

M

jXV,E

4

1k

2

kXjX21

kXjX1k1

jVX1M

jXV,jXSp, εzε 1zcεzEE

:ck

jXSp,E

Vacancy Interaction Energy

MC-87

Material Data

Binding energy Fe Ecoh,Fe 4,28 eV [Kittel] Binding energy Cu (in Fe) Ecoh,Cu 4,28 eV as for Fe Binding energy Ni (in Fe) Ecoh,Ni 4,28 eV as for Fe Binding energy Mn (in Fe) Ecoh,Mn 4,28 eV as for Fe Mixing energy Fe-Cu FeCu -0,49 eV calculated with [Liu] Mixing energy Fe-Ni FeNi -0,02 eV calculated with [Liu] Mixing energy Fe-Mn FeMn -0,17 eV calculated with [Liu] Mixing energy Cu-Ni CuNi -0,05 eV calculated with [Liu] Mixing energy Cu-Mn CuMn -0,00 eV calculated with [Liu] Mixing energy Ni-Mn NiMn +0,32 eV calculated with [Liu] Vacancy formation energy Fe EF

V,Fe 1,60 eV [Landolt-Börnstein (LB)] Vacancy formation energy Cu EF

V,Cu 1,60 eV as for Fe Vacancy formation energy Ni EF

V,Ni 1,60 eV as for Fe Vacancy formation energy Mn EF

V,Mn 1,60 eV as for Fe Vacancy migration energy Fe EM

V,Fe 0,90 eV, 1,20 eV calculated with [LB] Vacancy migration energy Cu (in Fe) EM

V,Cu 0,90 eV calculated with [Schick] Vacancy migration energy Ni (in Fe) EM

V,Ni 0,90 eV calculated with [LB] Vacancy migration energy Mn (in Fe) EM

V,Mn 0,90 eV calculated with [LB] Diffusion constant Fe D0,Fe 2,0110-4 m2/s [Landolt-Börnstein] Diffusion constant Cu D0,Cu 2,1610-4 m2/s calculated with [Schick] Diffusion constant Ni D0,Ni 1,4010-4 m2/s [Landolt-Börnstein] Diffusion constant Mn D0,Mn 1,4910-4 m2/s [Landolt-Börnstein] Lattice constant Fe a 0,287 nm [Kittel] Debye-Frequency Fe Debye,Fe 8,701012 1/s [Landolt-Börnstein]

MC-88

Activation Energies for Fe-Cu

Fe

0 1 2 3 4 5 6

0 1,20 1,18 1,16 1,14 1,12 1,09 1,07

1 1,16 1,14 1,12 1,09 1,07 1,05 1,03

2 1,12 1,09 1,07 1,05 1,03 1,00 0,98

3 1,07 1,05 1,03 1,00 0,98 0,96 0,94

4 1,03 1,00 0,98 0,96 0,94 0,92 0,89

5 0,98 0,96 0,94 0,92 0,89 0,87 0,85

6 0,94 0,92 0,89 0,87 0,85 0,83 0,80

7 0,89 0,87 0,85 0,83 0,80 0,78 0,76

2FeCun

Activation energy for one Fe-Atom (in eV) in dependence of the occupation numbers and 1

FeCun 1FeCun

2FeCun

MC-89

Cu

0 1 2 3 4 5 6

0 0,90 0,92 0,94 0,96 0,99 1,01 1,03

1 0,94 0,96 0,99 1,01 1,03 1,05 1,07

2 0,99 1,00 1,03 1,05 1,07 1,10 1,12

3 1,03 1,05 1,07 1,10 1,12 1,14 1,16

4 1,07 1,10 1,12 1,14 1,16 1,19 1,21

5 1,12 1,14 1,16 1,19 1,21 1,23 1,25

6 1,16 1,19 1,21 1,23 1,25 1,27 1,30

7 1,21 1,23 1,25 1,27 1,30 1,32 1,34

2CuCun

1CuCun

Activation energy for one Cu-Atom (in eV) in dependence of the occupation numbers and 1

CuCun

2CuCun

Activation Energies for Fe-Cu

MC-90

18

1iiMC Γt

3simV, L21/c

kTeV/ 1,6exp280c realV,

MCrealV,

simV,real t

c

ct

1. Vacancy is surrounded by 8 neighbouring atoms. For each atom it holds:

• Simulation of activation energy

• Simulation of the jump frequencies

2. Then selection of a jump by a random number

i

Monte Carlo-Time (one vacancy jump):

Vacancy concentration in the simulation:

Vacancy concentration in reality:

Time correction:

Monte-Carlo Simulation,Residence Time Algorithm

3. Performance of the jump and atoms exchange their positions. Vacancy is on new position.

Time scaleMonte Carlo Simulations

Random number in

MC-91

Ternary Systems

t1 t2 t3

t1 t2 t3

Zoom and cut through the precipitates

(a) Superlattice

(b) coated

Variation of the mixing energies enables the formation of

(a) precipitates with superlattices (upper row)

(b) coated precipitates (lower row)

MC-92

t = 0 t = 8 days t = 71 days t = 141 days t = 213 days

Edge length: L = 9 nm (32 a), 65536 lattice positions,

T = 400 °C, 1% Cu (655 atoms), 99% Fe.

Mechanisms of Precipitation,Simulation for T = 673 K (400 °C)

• Diffusing Cu-atoms spontaneously form a nucleus: This nucleus can grow or dissolve.

• Atoms dissolved in the matrix can move to the existing precipitates.

• Cu-atoms dissolve from existing precipitates and move into the matrix.

• Coagulation of precipitates: Two precipitates combine and coagulate to form a larger precipitate.

MC-93

Edge length: 9 nm, 65536 lattice positions,

T = 400 °C, 1% Cu (655 atoms), 99% Fe.

Mechanisms of Precipitation,Simulation for T = 673 K (400 °C)

MC-94

Dodecahedral Precipitates

MC-95

Ostwald-Ripening: An system minimizes ist free energy by interface reduction.

• Dissolution of small particles

• Growth of large particles

Classical description of Ostwald-ripening byLifshitz, Slyozov und Wagner (LSW). Time dependent development of an average particleradius as:

0LSW033 t-tK)(tR(t)R

: Coarsening rate

Simulation for T = 773 K (500 °C),Ostwald-Ripening

Edge length: 36 nm (128 a), 4.194.304 lattice positions, T = 500 °C, 9% Cu (377.487 atoms), 91% Fe.

t = 0 t = 9 hours t = 6 days t = 18 days t = 45 days

LSWK

MC-96

Radii Distribution, ComparisonSimulation - LSW

t = 9 hours t = 6 days

t = 18 days t = 45 days

Radius / aRadius (in units of a = 0,287 nm)

Radius / aRadius / a

Num

ber

of p

reci

pita

tes

Num

ber

or p

reci

pita

tes

Num

ber

of p

reci

pita

tes

Num

ber

of p

reci

pita

tes

Green: LSW-DistributionAssumptions/Prerequisites:• Isothermal diffusion• Cu-conc. very small• Nucleation rate = 0

Red: Simulation

Blue: Average radius

Average radius: 4.331 a = 1.24 nmMaximum radius: 6.9 a = 1.98 nmNumber of precipitates: 476




MC-97

Radii Distributions, Distance Distributions,Pair Correlation Functions

Precipitation states developed in the simulationcan be characterized by (time dependent):

• Radii distributions

• Distance distributions to nearest neighbours

• Pair correlation functions

Radius / a

# P

reci

pita

tes

Distance r / a

Dis

tanc

edi

strib

utio

n

Distance r / a

Pai

r co

rrel

atio

n fu

nctio

n g(

r)

t1 = 9 hourst2 = 6 dayst3 = 18 dayst4 = 45 days

t1 = 9 hourst2 = 6 dayst3 = 18 dayst4 = 45 days

g (r) after 5*109 MCS (t1 = 9 hours)

MC-98

The time law for Ostwald ripening provides coarsening rates KLSW for the precipitates:

/snm101,50K

nm 1,09(0)R36

LSW

Ostwald Ripening

tK(0)R(t)R LSW33

Time t / 106 s

Result: Determination of the coarsening rate KLSW

using Monte Carlo simulations.

R3 (

t) /

a3

for Fe-9%Cu, 500 °C}

Adjusted parameters:(with a=0.287 nm)KLSW=1.5*10-6 nm3/sR(t0)=1.09 nm

MC-99

Section from the Fe-Cu phasediagram. Solubility of Cu in Fe:0,536 at.% at 700 °C

0,023 at.% at 400 °CSimulation: A -> B -> C -> D

Edge length: 36 nm (128 a) 4.194.304 lattice positions, T = 700 °C 1% Cu, 99% Fe

Simulation for 973 K, 673 K (700 °C, 400 °C)

t = 1 s, T = 700 ° C t = 10 s, T = 700 °C

After 10 s: 19 precipitates with radii of 1,5 nm,0,5 % Cu-atoms dissolved in the matrix (marked in green)

Tem

pera

ture

T /

K

Cu-Concentration / at. %

MC-100

t1 = 140 h

• Dissolved Cu-atoms (marked green) form new precipitates.

• Large precipitates grow on cost of small ones (Ostwald ripening).

• Small precipitates dissolve.

Simulation for 973 K, 673 K (700 °C, 400 °C),Radii Distribution

t2 = 1933 h t3 = 4504 h t4 = 9680 h (403 days)

Radius / a

# P

reci

pita

tes

Radius / aRadius / aRadius / a

# ge

n

# ge

n

# ge

n

MC-101

0

c

0

ppt

0

c2

0

ppt1

2

1

3/42

2

1

rr

log

r

rlog

rr

logE

r

rlogE

E

E

mit,E

E1

D

GbΔτ

Δτ2,5Δσ

GPa 83 G

r1000r

b2,5r

0,248nmb

0,6E

E

0c

0

2

1

1/2ppt frπD

Precipitation Strengthening in Fe by Cu-precipi-tates (precipitates are softer than the matrix).

G = Shear modulus of the matrix

E= Energy per length of the dislocation ...

E1,2 = ...in the matrix (1) and in the precipitate, resp. (2)

b = Burgers-vector of the dislocation

rppt = Radius of the precipitate

rc,r0 = External and internal cut-off radius

D = Distance between the precipitates

Assumption: homogeneously distributed precipitates:

f: atom concentration of Cu

For the system Fe-Cu validated values:

Russel-Brown-Model, Strength Increase

(„Taylor-Factor“)

MC-102

Strength increase according to Russel &Brown for f = 0,5 % Cu and f = 1,0 % Cu.The sequence t0 -> t1 -> t2 -> t3 -> t4corresponds to the simulation.

Strength Increase according to Cu Precipitates

Experimental data from ave-raged hardness measurmentsfor steel 15NiCuMoNb5 aftertempering at 400 °C.

Simulation Experiment

• 1 % Cu 0,64 % Cu+foreign atoms• Infinitely fast Finite cooling time

cooling

Result:Experiment and Simulation provide the same valueof maximum strength increase (110 MPa)

Radius rppt / nmTime t / h

/

MP

a

/

MP

a

Simulation (Time scale reduced bya factor of 3)

WB36, experiment

MC-103

The precipitate is nearly spherical (Radius 5,7 nm). A cutthrough the center of the precipitate in the {100}-plane.

Concentration of atom species independence of the distance fromthe center of gravity of theCuMnNi precipitate.

Experimental confirmation by APFIM/TAP (Atom Probe Field Ion Microscopy /Topographic Atom Probe) [R. Kirchheim, Al-Kassab, 2003].

Coated Structure of the Precipitate

Cu

Ni

Mn

Con

cent

ratio

n

Distance from the center of gravity / nm

Result: The precipitate consists of a Cu core, which is coveredby a layer of Ni and Mn.

Simulation:

MC-104

t0 = 0 t1 = 30 y t2 = 60 y t3 = 180 y

t0 = 0 t1 = 30 y t2 = 60 y t3 = 180 y

t0 = 0 t1 = 30 y t2 = 60 y t3 = 180 y

Simulation for 270 °C: Precipitates form very slowly. After 180 years: few, very small nuclei have formed (R = 0.4 nm).

Simulation for 300 °C: Precipitates form more rapid. After 180 years: precipitates with radii of 0.8 nm exist.

Simulation for 330 °C: Precipitates form very fast. After 30 years: precipi-tates with radii of 1.3 nm have formed.

Time Dependence of Precipitation(T = 270 °C, 300 °C and 330 °C)

0,176 % Cu, 1,123 % Ni, 1,016 % Mn

KS02:

MC-105

The function (t) is a measure for the amount of Cu forming precipitates.

1,5n s,08107,5t :K 603T for

1,5n s,09105,8t :K 573T for

1,5n s,10104,1t :K 543T for

n

mCu

mCu

mCu

mCu

τ

texp1tξ

c0tc

tc0tctξ

(t) follows a Johnson-Mehl-Avrami (JMA)-law with an exponent of n = 1,5.

Time Dependence of PrecipitationF

unct

ion (

t)

Time t / s

Result: A temperature increase of 10 K results in twice the velocity for precipitate formation! (van‘t Hoff Law)

MC-106

Simulation for 573 K (300 °C)

Box size: 18 nm (64 a), N = 2 x 643 = 524 288 lattice positions

1,0 at.% Cu 1,0 at.% Ni1,0 at.% Mn

After 3 years: Many small precipitates, then Ostwald ripening.

After 9550 years: Few large precipitates.

Typical computation times:For a 1 GHz-processor: 1010 MCS in 24 h(1 MCS = 1 vacancy jump).Computation time here: 33 days

108 MCS, (t = 3 y)0 MCS, (t = 0) 109 MCS, (t = 25 y)

1010 MCS, (t = 254 y) 1011 MCS, (t = 2754 y) 3,3·1011 MCS, (t = 9550 y)

Time Dependence of Precipitation

MC-107

Time dependent development of the average radius of the precipitates

The function (t) for temperatures between 273 K und 773 K.

Example.: Cu fully precipitated, i.e. (t) = 1:

• at 373 K (100 °C): t = 1012 years

• at 773 K (500 °C): t = 10 hours

Time t / s Zeit t / s

Fun

ctio

n (

t)

Mea

n ra

dius

R(t

) /

a

Mea

n ra

dius

R(t

) /

nm

Age of the univers25 y. 2754 years 9550 years

:

Time Dependence of Precipitation

MC-108

Summary

• The diffusion of atoms was modelled and simulated based on a thermally activated vacancy mechanism for Fe-Cu and Fe-Cu-Ni-Mn.

• Results for the system Fe-Cu:

Detailed insight into the kinetics and mechanisms of precipitation formation.

Simulation of Ostwald ripening: LSW- und simulated radii distributions are similar, simulation of the coarsening rate.

Simulation of the strength increase due to Cu- precipitates according to the theory of Russel&Brown.

• Results for the system Fe-Cu-Ni-Mn:

The shell structure was found for CuNiMn-precipitates as well as the time dependence of precipitation.

Future work

• Simulations of diffusion in further systems (e.g., CuFeMnSi, FeSiCrC).

• Extension of the program for fcc lattices and interstitial diffusion mechanism.

Summary an Future Work

MC-109

t = 0 t = 1 h t = 10 h

Segregation of P- und C-Atoms at a grain boundary in Fe

Simulation of precipitate populations

2*1010 MCS (t = 1 s), T=700°C 7*1010 MCS (t = 10 s), T=400°C

P-Segregation at Grain Boundaries

MC-Applications

MD-111


• Theory

• EAM Potentials

• Frenkel Defects

• Interaction between dislocations and phase boundaries

Molecular Dynamics (Part I)

MD-112

Macro(Mechanics)

Electrons(Bonding)

Atoms(Cohesion)

Microstructure(Micro Cracking)



Micro(FEM)

Nano(MD)

Femto(ab initio)

Macro(FEM)


Theory

MD-113

Molecular Dynamics (MD) Simulations

• Crystal is considered as a system of classical point particles.

• Numerical integration of Newton‘s equationsof motion

ii x

EF

Interatomic Forces:

MD-114

EAM Potentials

)(2

1)(

,

jiijiji

ii rFE

)( ijij

jii r

j

i

ijr

Embedding part Pair potential part

Local electron density

Embedded Atom Method Potentials

MD-115

Ideal LatticeOne Fe atom

replaced by a void

One Fe atom

replaced by a H atomFe-Lattice

(1 1 0)

(1 -1 0)

Not deformed

deformed

(36%)

Hydrogen Embrittlement, Mechanism 1: Weakening of Bonds

MD-116

Edge Dislocation Movement

(I Ī 0) slip plane, Burgers Vector ½ [ I I I ]

MD-117

Dislocation Movement

MD-118

Time [ps]

w/o H

1 H-atom (substitutional)

1 H-atom (interstitial)

4 H-atoms (along dislocation line)

4 H-atoms (distributed at the dislocation core)

H-atoms

Hydrogen Embrittlement: Mechanism 2: Dislocation Pinning

MD-119

H-atoms

4H-atoms 0 K4H-atoms 300 K4H-atoms 600 K

Time [ps]

X-p

ositi

on o

f the

dis

loca

tion

[0,1

nm

]

Time [ps]

w/o H

1 H-atom (substitutional)

1 H-atom (interstitial)

4 H-atoms (along dislocation line)

4 H-atoms (distributed at the dislocation core)

H-atoms

Hydrogen Embrittlement: Mechanism 2: Dislocation Pinning

MD-120

Simulation of Internal Stresses

• Purpose: Simulation of internal stresses based on atomic structural models and interatomic potentials

• Study of influence of atomic defects (voids, dislocations, lattice defects, dopant atoms) on internal stresses

• Chance to predict failure under external load

• Practical application: Ni/Ni3Al-superalloys

MD-121

Internal Stress in a Carbide MultilayerSystem during Indentation


0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

20

X-Achse [nm]

z-A

chse

[nm

]

-2E6-1,7E6-1,4E6-1,1E6-8E5-5E5-2E51E54E57E51E61,3E61,6E61,9E62,2E62,5E62,8E63,1E63,4E63,7E64E6

0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

20

x-Achse [nm]

z-A

chse

[nm

]

-2E6-1,7E6-1,4E6-1,1E6-8E5-5E5-2E51E54E57E51E61,3E61,6E61,9E62,2E62,5E62,8E63,1E63,4E63,7E64E6

MD-122

Internal Stresses in Ni / Ni3AlSuperalloys with Atomic Defects


Bright gray: Ni

Dark gray: Al

MD-123

Molecular Dynamics: Simulation ofInternal Stresses

MD-124

Nanosimulation of the Interaction between

Edge Dislocations and Obstacles (Precipitates)

Nanosimulation

MD-125

EAM Potentials

Fe: G. Simonelli, R. Pasianot, E. Savino: Mat. Res. Soc. Symp. Proc., 291 (1993) 567Cu: A. F. Voter: Los Alamos Unclassified Technical Report #93-3901, 1993Fe-Cu: M. Ludwig, D. Farkas, D. Pedraza, S. Schmauder: Modelling and Simulation in

Material Science and Engineering, 6 (1998) 19Fe-Ni: C. Vailhe, D. Farkas: Mat. Sci. Eng. A 258 (1998) 26Cu-Ni: D. Farkas, J. Clinedist: Mat. Res. Soc. Symp. Proc. 457 (1997) 315

Molecular Dynamics Software

Program „IMD“, developed at the „Institut für Theoretische und Angewandte Physik“ (ITAP) at the University of Stuttgart, Germany.

Capable of Parallel Processing. Set up a world record in 1997 for a model built of1.2*109 atoms, improved to 5.2*109 atoms at a later time

Software and EAM Potentials

MD-126

EAM-Potentials

EAM-Potentials (EAM=Embedded Atom Method)

Fe-Fe EAM-Potential

Cu-Cu EAM-Potential

Fe-Cu EAM-Potential

M. S. Daw, M. I. Baskes, Embedded atom method: Derivation and application to impurities, surfaces and other defects in metals, Phys. Rev. B, Vol. 29, No. 12 (1984), pp. 6443-6453

MD-127

)(:)(

)(:)exp(

)(

212

2102

2

1

4

1

3

21

xxxxhxhhxx

xxx

x

xzHzxax

R

FRE

a

i

i

ii

ij

ij

ai

i

i

ji

ijtot

Fe-Fe EAM-Potenital

Param. Experiment Calculateda0 0.2866 nm 0.2876 nmEcoh 4.28 eV 4.28 eVc11 241 GPa 248 GPac12 143 GPa 152 GPac44 118 GPa 113 GPaEv,for 1.8 eV 1.6 eV

Adjusted to: lattice parameter a0, cohesive energy Ecoh, elastic constants cij, vacancy formation energy EV,for

G. Simonelli, R. Pasianot, E. J. Savino, Mat. Res. Soc. Symp. Proc., 291 (1993) 567

MD-128

rra

MRr

M

ij

ij

ai

i

i

ji

ijtot

eer

DeDr

R

FRE

MM

296

2

21

2

1)(

A. F. Voter, Los Alamos Unclassified Technical Report #93-3901, 1993

Param. Experiment Calculateda0 0.3615 nm 0.3615 nmEcoh 3.54 eV 3.54 eVc11 176 GPa 180 GPac12 125 GPa 122 GPac44 82 GPa 82 GPaEv,for 1.3 eV 1.3 eV

Adjusted to: lattice parameter a0, cohesive energy Ecoh, elastic constants cij, vacancy formation energy EV,for

Cu-Cu EAM-Potential

MD-129

Fe-Cu EAM-Potential

fxedxcbxa

R

FRE

Fwithrandr

CuFeFeCu

ij

ij

ji

i

i

ji

ijtot

lCuFe

21

)(

0)(:

21

0

M. Ludwig, D. Farkas, D. Pedraza, S. Schmauder, Modelling and Simulation in Material Science and Engineering, 6, pp. 19-28 (1998)

Point defects in Fe, periodic boundary conditions, constant volume, 686 atoms

x=0...1. Parameter: a=1.0, b=4.7, c=0.99, d=4.095, e=1.0, f=4.961

Param. Experiment CalculatedEV 1.8 eV 1.6 eVECu 1.233 eV 1.236 eVEV-Cu 0.14 eV 0.18 eVE2Cu 0.19 eVEk(2) 0.05 eV 0.01 eV

b

b

b

MD-130

Small Cu precipitate in Fe matrix,coherent

Large Cu precipitate in Fe matrix,

becoming unstable

Cu Precipitates in Fe Matrix

MD-131High Resolution Electron Microscopy: Cu Precipitate in Fe Matrix

HRTEM image of a twinned 9R copper precipitate in an Fe-Cu specimen. The angle between the (009)9R basal and (114)9R twin planes is 61º ;

R. Monzen, M. Iguchi, M.L. Jenkins, Phil. Mag. Let. 80 (2000) 137.

Cu Precipitate

MD-132

Simulation Model

Simulation model of an Fe single crystal with uniaxial tensile load

MD-133

The Xi - Yi (i=1,2,...,5) planes for five crystal orientations in bcc-Fe

Crystal Orientations

MD-134

Atomic configuration for orientation No. 1 with free boundary conditions at strain=0.24

Plastic Deformation during External Straining

MD-135Atomic Arrangement, Orientation 3

Atomic arrangement for crystal orientation No. 3 with free boundary conditions

(a) strain = 0.136; (b) strain = 0.48

Twinning

MD-136Example for Void Formation (Other Direction)

(a) (b)Atomic configurations for orientation No. 5 with periodic boundary conditions:

(a) strain=0.144; (b) strain=0.16

Void Formation

MD-137

Stress - Strain curves for D=20, H=26 and different crystal orientations, periodic boundary conditions

Stress-Strain-Curves

Orientation 1

Orientation 2

Orientation 4

Orientation 5

Orientation 3

MD-138

Strain

Stress-Strain-Diagram

Voided Material

MD-139

Y

O X

X

X

X

XX

X X

X

XX

XXX

XXX

XX

XX

Atomic Structure during Deformation; 20 Frenkel Defects

Frenkel Defects

MD-140

Dependence of () on Frenkel Defects, Orientation 1

Stress-Strain-Curves

ideal lattice

1 Frenkeldefect

20 Frenkeldefects

5 Frenkeldefects

MD-141

Hideyuki INOUE. Yasuhiro Akahoshi and Shoji Harada

Fig. 1: Initial configuration of single crystals with random orientation.

Fig. 2: Energy distribution of analysed body. (The lightand the shaded positions show atoms with low and highpotential energy, respectively. The encircled numberindicates consecutive number of sub-grains)

Fig. 3: Tensile stress versus total strain up to 2%.

Model Fe polycrystal

Temperature RelaxationTensile problem

300K 100K.300K 500K.700K

Number of atoms RelaxationTensile problem

794S7306

Mass of atom 9.273588x10-26kgTime step 1 .0 fsNumerical integrationof equation

Verlet's method

Potential Morse typepotentialStain rate 1.0x l.0-1/ step

Step number forcalculation

Relaxation

Tensile problem

4.0x104

2.0xl02

Total strain 20.0%

Boundary conditions Periodic B. C.

Fixed B. C.

X1 direction

X2 direction

Strength of Nanocrystalline versus Single Crystalline Metal

Nanocrystalline versus Single Crystalline Metal

Analysis conditions

SC

PC

MD-142

Simulation of Cycle- and Temperature-Dependence of Failure

Comparison of deformation stateat several temperatures.

Relation between stress amplitude and number of cycles to failure.

Process of crack initiation and growth at 300 KRelationship between tensile stress and

total strain at several temperatures.

700K

100K

Initial state

2 = 6.2% (defect generation)

2 = 6.4%

100 K

300 K

500 K

700 K

2 = 6.2%

MD-143

Schematic representation of a section through the sample, showing the initial position of the edge dislocations and the Cu atoms (grey).

The interaction between a moving edge dislocation in an Fe crystal and a Cu-precipitate is investigated by molecular dynamics (MD) calculations.

In the absence of external stresses, two edge dislocations with the same slip plane and opposite Burgers vectors within a perfect Fe crystal lattice are investigated.

Initial Positions of the Edge Dislocations

Nanosimulation of the Interaction between Edge Dislocations and Obstacles (Precipitates)

MD-144

Detailed Structure of one ofthe Dislocation Cores

Detailed structure of one of the dislocation cores during dislocation migration through the obstacle.

Fe atoms yellow, Cu atoms grey.

The distance along the z-axis between the upper and the bottom plane is 6 x 0.176 nm = 1.056 nm

MD-145

Interactions between precipitates and dislocations are investigated using atomistic computer simulations. In particular, the effect of Cu-precipitates on the core structures, slipping behaviour, and Critical Resolved Shear Stress (CRSS) of an edge dislocation in a bcc Fe single crystal is considered.

Model of a bcc Fe single crystal with an edge dislocation and a Cu precipitate under shear deformation.

Interaction between Precipitates and Dislocations

MD-146

Profiles of dislocation lines on the slip plane (1 -1 0):(a) equilibrium dislocation(b) slipping dislocation

Dislocation Cutting a Cu-Precipitate

MD-147

Profiles of the dislocation lines on the slip plane (1 -1 0) at different deformation stages (strains: 0.5%, 2.0%, 2.3%, 2.35%)

Profiles of Dislocation Lines

MD-148

Shear stress (a) and normal stress (b) distribution for the pure edge dislocation along Burgers vector (1 1 1) on the slip plane (1 -1 0).

She

ar s

tres

s (G

Pa)

Nor

mal

str

ess

(GP

a)

Initial equilibrium dislocation

----- Slipping dislocation

Dislocation Stress Distribution

MD-149

Stress distribution atthe arm part (z=10 a0 ) (a, b) and at the middlepart (c, d)

She

ar s

tres

s (G

Pa)

Nor

mal

str

ess

(GP

a)

She

ar s

tres

s (G

Pa)

Nor

mal

str

ess

(GP

a)

(a)

(c)

(b)

(d)

Å

ÅÅ

Å

Dislocation Stress Distribution

MD-150

Average shear stress - strain curve for the bcc Fe single crystal with a Cu-precipitate and a single edge dislocation under external shear deformation

(Insert: pure iron).

Shear stress - strain curve with and without a Cu precipitate

Shear Stress-Strain-Curve

She

ar s

tres

s (G

Pa)

Strain

without inclusion(12 MPa)

with inclusion(500 MPa)

MD-151

3-dim. atomistic simulation of dislocation bending and cutting of Cu-cluster in Fe

Scheme of a Dislocation, Blocked byor Cutting a Precipitate

Blocked

Cutting

MD-152

Schematic of a dislocationcutting a precipitate to explainthe definition of the criticalangle.

The angle between the arms of a dislocation, together with the distance between the obstacles, is the key parameter to calculate the increase in matrix strength due to precipitation hardening. The shear stress is given by:

23

2cos

L

GbBrown, Ham

= Shear stress

G = Shear modulus of the matrix

b = Burgers vector in the matrix

L = Obstacle spacing in the slip plane

= Critical angle between the dislocation arms

Shear Stress and Critical Angle

MD-153

23

2cos

L

GbBrown, HamThe shear stress is given by:

Russel&Brown derived the shear stress from a relationship between the energies of thedislocation per unit length inside (E1) and outside (E2) the precipitate as (overaged state):

43

22

211

E

E

L

Gb

Therefore,4

3

22

21

23

12

cos

E

E

where the ratio E1 / E2 depends on the precipitate radius as

002

01

2

1

log

log

log

log

rRrR

rR

E

rr

E

E

E

r = Precipitate radius R = Outer cut off radiusr0 = Inner cut off radius

Shear Stress, Russel&Brown-Theory

MD-154

Russel&Brown adopted the following values for the Fe/Cu system based on experimental strengthening data from literature:

r0 = 2.5 b with b = 0.248 nm (Burgers vector)

R = 1000 r0

Two examples for precipitate diameters d=1.3 nm and d=3.2 nm:

d=1.3 nm : critical angle = 171° (small precipitate)

d=3.2 nm : critical angle = 140° (larger precipitate)

K.C. Russell, L.M. Brown, Acta Met. 20 (1972) 969-974

0.6EE 21 /

Shear Stress, Russel&Brown-Theory

MD-155

Larger Precipitate, Diameter 3.04 nm

Larger precipitate, diameter 3.04 nm

In the case of the 3.04 nm diameter Cu precipitate passing does not happen and the dislocation line is pinned by the precipitate, with free ends oscillating.

The dislocation is not able to cut the obstacle. It can only pass through the precipitate completely as soon as an external shear stress is applied to increase the stress beyond the Peierls stress.

= 140°

MD-156

Small Precipitate, Diameter 1.32 nm

Small precipitate, diameter 1.32 nm

Starting from the initial position, the movement ofthe dislocation line takes place such that it is curvedtowards the precipitate (see Fig. b in comparison toFig. a).

Furtheron, the edge dislocation passes through theprecipitate and after passing, a backward bowing canbe recognized (see Fig. h), indicating the persistingattractive force between the precipitate and thedislocation line.

Altogether, the movement of the dislocation takes placealmost without impedement.

= 170°

MD-157


• Interaction between dislocations and phase boundaries

• Inverse Hall-Petch effect

• Solid solution hardening

Molecular Dynamics (Part II)

MD-158

Progress in used Software

Aim:Looking into the effects of

- interaction of particles- influence of differently sized particles

MD-159

Strength Increase in Cu-alloyed Steels due to Precipitates after Anealing (57000h, 340°C)

MD-Simulation

0

100

200

300

400

500

600

700

800

0.00 0.05 0.10 0.15 0.20 0.25Strain / m/m

A111A112

A113

Material 15 NiCuMoNb 5States E60A and E60B B111B112

B113T= 90°C

Zustand E60A

Zustand E60B

Str

ess

/ M

Pa

Stress-Strain-Curces of Cu-Alloyed Steels

MD-160

Characterization of a Precipitate by APFIM / TAP

MD-161

Characterization of a Precipitate by APFIM / TAP

MD-162

Atom Probe Field Ion Microscope / Topographic Atom Probe

Research Group R. Kirchheim / T. Al-Kassab, University of Göttingen, Germany

APFIM / TAP

MD-163

„Cu“-Precipitates: More Realistic Model

Cu

Mn

Ni

Con

cent

ratio

n

Distance (center of gravity, nm)

MD-164

Temperature Dependence ofCritical Resolved Shear Stress

MD-165

Cu-Precipitates

• Size of precipitate (radius)• Distance between precipitates (box length)• Shape (spherical, ellipsoidal)• Position of glide plane (central, marginal)• Composition (Fe atoms)

Free parameters:

MD-166

Different Radii and Distancesof Spherical Precipitates

Lc

1

Spherical Precipitates

MD-167

Precipitates of Different Shape: Ellipsoids

2.5 nm

2b

Ellipsoidal Precipitates

bCu [nm]

MD-168

Different Positions of Glide Plane

2.5 nm

Different Positions of Glide Plane

MD-169

Repulsion

+: Positive Pressure-: Negative Pressure

Repulsion and Attraction of Dislocations

+

-

+Attraction

++

-

MD-170

24 MPa

80 MPa

Repulsion and Attraction of Dislocations

MD-171

Different Cu Concentrations

Influence of Cu-Concentration

MD-172

Cu/Ni-Precipitates

• Radius (Ni, CuNi)• Composition (Fe, Cu atoms)• Ni precipitates with Cu core

Free parameters:

Cu/Ni-Precipitates

MD-173

Important Physical Data for Fe, Cu, Ni

Fe Cu Nibcc fcc bcc fcc bcc

a0 2.866 Å 3.615 Å 2.881 Å 3.520 Å 2.812 Å

Ecoh 4.28 eV 3.54 eV 3.49 eV 4.45 eV 4.37 eV

Bulkmodulus

179.97 GPa 141.03 GPa 127.29 GPa 180.19 GPa 143.73 GPa

c11 243.73 GPa 179.34 GPa 109.43 GPa 244.01 GPa 101.62 GPa



ShearModulus

G[111]

69.76 GPa 21.84 GPa 24.110 GPa

Derived from nanosimulation

Important Physical Data for Fe, Cu, Ni

MD-174

Spherical Cu and Ni Precipitates of Different Radii

Spherical Cu and Ni Precipitates

MD-175

Different Fe-Concentrations

Different Fe-Concentrations

MD-176

Spherical Cu/Ni-Precipitates

Ordered Cu/Ni-precipitate

B2-structure

NiCu

Ordered and Random Spherical Cu/Ni-Precipitates

MD-177

Spherical Cu-Precipitates with Ni-Shell

NiCu

Cu-Precipitates with Ni-Shell

MD-178

Maximal density

Zero density

Minimal density

Burgers Vector Density within Glide Plane

12.5 Å Ni 12.5 Å Ni / 4 Å Cu 12.5 Å Ni / 6 Å Cu

12.5 Å Ni / 10 Å Cu 12.5 Å Cu

MD-179

NiCu

Spherical Cu-Precipitates with Ni-Shell

MD-180

Critical Resolved Shear Stress:from idealized Model to Reality

Overview on the numerical correction factors of the critical resolved shear stress versus the idealized simulation configuration:

1.) Temperature: temperature of mechanical exp. 90°C vs. 0K (in basic simulation):Reduction by ca. 33%

2.) Nickel-shell (chemical inhomogeneity): Reduction by ca. 55%3.) Presence of iron in the precipitate: Reduction by ca. 5%4.) Scatter of precipitate position parallel to dislocation movement:

Reduction by ca. 50%5.) Scatter of precipitate sizes: Reduction by ca. 20%6.) Scatter of precipitate distances: Reduction by ca. 20%

Idealized simulation result for precipitates, aligned on linear chains, withidentical distances and sizes according to the mean sizes and distances: Critical resolved shear stress: 300 MPaTaking into account the reducing effecs ( 1 to 6 ), 300 MPa shrink to 35 MPa. The critical tensile stress is calculated from the critical shear stress byMultiplying with the Schmid factor (~ 3.05), resulting in an increase in tensile tress by

100 MPaIn agreement with the experimental observation due to thermal load.

MD-181Interaction of a Dislocation with a Fe/Cu-Interface

Molecular Dynamics Simulation

MD-182

Dislocation Movement underexternal Shear Loading

• Ni3Al-Precipitate in Ni• System size: 24.8 nm x 9.75 nm x 14.7 nm (325 000 Atoms)• Diameter of precipitate: 5 nm• Maximal Shear deformation: = 0.95 %• Real Time: 37.5 ps

Partial Dislocations

Stacking faultAlNi

Glide Plane ofDislocation

MD-183

Inverse Hall-Petch Effect

Simulating nanocrystalline copper The smallest grain sizes. Larger grains. Flow stress: an optimal grain size. Dislocation structure.

Conclusions.

MD-184

Dislocations and Grain Boundaries

Dislocations carry the plastic deformation.

Grain boundaries hinder the motion of dislocations.

MD-185



When grains become smaller, the material becomes harder(Hall-Petch effect)

y

d1

Hall (1952)


MD-186



When grains become smaller, the material becomes harder(Hall-Petch effect)

d

kyy ,

d1


MD-187

The Hardness of N.C. Metals

S. Takeuchi, Scripta Mater. 44, 1483 (2001).

MD-188

Simulations of N.C. Copper

Set up the system in the computer. Do Molecular Dynamics while

deforming the sample. Interpret the results.

MD-189

Set up the system in the computer. Do Molecular Dynamics while

deforming the sample. Interpret the results.

Material: copper. No texture. Strain rate: 5108 s-1. Temperature: 300 K.

Simulations of N.C. Copper

MD-190

Results – Small Grains

380000 atoms – 7 nm grains

Structure:

Blue atoms: f.c.c. structure, this is inside the grains.

Yellow atoms: h.c.p. structure, this is stacking faults etc.

Red atoms: irregular structure, this is grain boundaries and dislocation cores.

MD-191


Plastic deformation:

The dislocation activity cannot account for the observed plastic deformation.

Something else is happening, perhaps the grain boundaries.

Results – Small Grains

MD-192

Deformation Map, Small Grains

The main deformation is in the grain boundaries. Little “conventional” dislocation activity.


MD-193

Stress vs. Strain, Small Grains

The hardness increases with the grain size.(reverse Hall-Petch effect)

• Nature 391, 561 (1998).• Phys. Rev. B 60, 11971 (1999).

MD-194

Deformation Map, Large Grains

The main deformation is inside the grains. Dislocations carry the deformation.

101 million atoms – 49 nm grains

MD-195

What happens in the Grains?

50 million atoms.20 grains.Grain size: 39 nm.

Blue atoms:perfect crystal

Yellow atoms:stacking faults

Red atoms:grain boundariesdislocation cores

MD-196

A Change in Deformation Mode

Small grains (d < 10 nm) Deformation is in the grain boundaries. Smaller grains more grain boundaries

easier deformation.

Larger grains (d > 15 nm) Dislocations carry the deformation. Grain boundaries hinder the dislocation motion. Smaller grains more grain boundaries

harder material.

MD-197

An optimal Grain Size

For small grains the strength increase with increasing grain size.

For large grains the strength decrease with increasing grain size.

MD-198

What happens inside the Grains?

MD-199

Dislocation Structures (pile-ups)

Dislocations queued up on the same glide plane.

Pressed towards a grain boundary by the external stress.

Held apart by their mutual repulsion.

The stress concentration from the pile-up cause dislocation activity in the next grain.

MD-200

Summary – Optimal Grain Size

Using parallel computers, molecular dynamics simulations (MD) with 107 – 108 atoms are possible with realistic interatomic forces. It is possible to simulate the plastic deformation of

polycrystalline metals with realistic grain sizes.

Nanocrystalline copper has an optimal grain size at 10 – 15 nm, where the hardness is maximal. In smaller grains, grain boundary sliding is the dominant

deformation mechanism, and a reverse Hall-Petch effect is seen.

In larger grains, dislocations carry the deformation. Grain boundaries cause pile-ups. The Hall-Petch effect is seen.

MD-201

dissolved atoms

initial configuration

-Fe/C

experiment(literature)

Concentration / %

Moleculardynamic (MD)-Simulation is adequate to simulate the solid solution hardening in Fe (and other metals). For this purpose, foreign atoms are distributed statistically in a simulation box and their resistance against the movement of an edge dislocation on a low level energetic glide system is calculated.

Solid Solution Hardening

Fe/

Experiment (Literature)

Concentration / %

Cri

tical

she

arst

ress

c

/ MP

a

MD-202

dislocation and dissolved atoms

dissolved atoms

initial configuration

experimental resultssimulation

Concent / %

Incr

ease

inY

ield

stre

ss /

MP

a

Fe/

Experiment (Literature)

Concentration / %

Cri

tica

l str

ess c

/ M

Pa

Solid Solution Hardening

MM-203

Macro(Mechanics)

Electrons(Bonding)

Atoms(Interaction)

Microstructure(Localisation)



Micro(FEM)

Nano(MD)

Femto(ab initio)

Macro(FEM)


Conclusion

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