tutorial i: mechanics of ductile crystalline solids

IHPC-IMS Program onAdvances & Mathematical Issues

in Large Scale Simulation(Dec 2002 - Mar 2003 & Oct - Nov 2003)

Tutorial I:Mechanics of Ductile Crystalline Solids

Alberto M. CuitiñoMechanical and Aerospace Engineering

Rutgers UniversityPiscataway, New [email protected]

Institute of High Performance Computing Institute for Mathematical Sciences, NUS

Singapore 2003 cuitiño@rutgers

Collaborators

• Bill Goddard • Marisol Koslowski• Michael Ortiz• Alejandro Strachan• Habin Su• Shanfu Zheng


Hierarchy of Scales

length

time

mmnm µm

ms

µs

ns

Phase stability, elasticityEnergy barriers, pathsPhase-boundary mobility

Microstructures

Grains

Direct FE simulationPolycrystals

Single crystals

SCS test

Force Field


The Role of Dislocations

InitialELASTICITY

PLASTICITY

DislocationMotion

DislocationGeneration

area swept by dislocation


Anatomy of a Dislocation Loop

Animations from: http://uet.edu.pk/dmems/

b : burgers vector

Screw Segment

Edge Segment

Mixed Segment

Area swept by dislocation loop

Glide Plane

Mixed Screw

Edge


Dislocation Motion and Arrest

OBSTACLES

DISLOCATION FOREST

Animation from:

zig.onera.fr/DisGallery/ junction.html


Tracking Dislocation in ONE Plane

(Ortiz, 1999)


Overview

Effective Dislocation Energy • Core Energy

• Dislocation Interaction

• Irreversible Obstacle Interaction

Macroscopic Averages

Equilibrium configurations

• Closed form solution at zero temperature.

• Metropolis Monte Carlo algorithm and mean field approximation at finite temperatures.


Effective Energy

where with

S

ii

S

ekl

eijijkl dStdSxdcuE )(

2

1][ 3

pij

eijjiu ,

)(Sm Djip

ij Elastic interaction Core energy External field

Slip Surface

S

Displacement jump across Sm


Elastic interaction

j

pmnijmnkik cGu

, Displacement field:

Elastic distortion:

Elastic interaction:

Green function for an isotropic crystal:

pklj

pmnijmnlki

ekl cG

,,

kdAE p

uvp

mnmnuv3*

21

3int ˆˆˆ

2

1

ki

kiki xx

rrG

22

28

1


Elastic Interaction

kdkkKb

E 22

21

2

2int ˆ),(

4)2(

1][

bxxxx /),(),( 2121 with

22

21

22

22

21

21

21 1

1),(

kk

k

kk

kkkK

b

RA2

A1

21

2121

1

1

2int

10

01

8 AA

dSdSR

bE

(Hirth and Lothe,1969)


External Field

Ndis

n n

n

xx

Cxs

1

)(

applied

shear stressforest dislocations

S

ext dSbsE

S

iiext dStE

with: ii bxxxx ),(),( 2121

||/),(),( 2121 bbxxtxxs ii


Core Energy

ZS :

0 b/2 b-b/2-b-3b/2

() jjiiijZ

bbC

2

1min)(

ijij dC

2

22

4inf)(

d

bZ

S

core dSE )(

Ortiz and Phillips, 1999

d: inter-planar distance


Phase-Field Energy

kdsbK

b

d

bE 2*

2222

2ˆˆˆ

4ˆˆ

22

1,

2/1

ˆ

2/1

ˆˆKdKd

s

b

d

02

*2

22

2 2/1

ˆˆˆ2/14)2(

1][ Ekd

Kd

sbkd

Kd

KbE

Minimization with respect to gives:

core regularization factor

elastic energy


Phase-Field Energy

d

Peierls

kd

K

sdE

d

2

2

2

20 1

ˆ

22

1

Core regularization

Elastic energy

d0

Core regularization factor 21

1)(ˆ

dd Kk

d

xbPeierls

)1(2tan

21


Energy minimizing phase-field

][inf

EY

0s CsG

Unconstrained minimization problem: sKb

ˆˆ2

if

with )1/(

1

2

2)(

22

21

22

212

2

xx

xx

bkd

K

e

bxG

xik

CsGPXd PX

d


Irreversible Process and Kinetics

xdxfEEW nnnnnn 2111 )(][][]|[

Irreversible dislocation-obstacle interaction may be built into a variational framework, we introduce the incremental work function:

incremental work dissipated at the obstacles

4

bf Primary and forest dislocations react to form a jog:

Updated phase-field follows from: ]|[inf 1

1

nn

XW

n

Short range obstacles:

N

iidi

P xxfbxf1

)()(


Irreversible Process and Kinetics

g g

Kuhn-Tucker optimality conditions: ii

ni

ni 1

01 i

ni fg

0i

01 i

ni fg

0i

01 ii

ni fg 01

iini fg

Equilibrium condition:

N

i

ni

n gb1

11 1

xdxgxdxf nnn

fg

nn

n

21121 )(sup)(1


Solution Procedure

1. Stick predictor. Set

2. Reaction projection

3. Phase-field evaluation

ni

ni 1~ and compute the reactions:

)~(~ 1

1

11

ni

N

iij

nj CGg

ini

ni fgg 11 ~~

1

1

11

nN

j

njij

ni CgG


Closed-form solution

Dislocation loops

1

1

11

nN

j

njij

ni CgG

)( jiij xxGG

db

Gii

1

2

122

Calculations are gridless and scale with the number

of obstacles


Macroscopic averages

Slip

Dislocation density

Obstacle concentration

Shear stress

01

N

iilN

b

b0

c

f

sb0 42

0 1010 cF obs

bl 310

216

0

110

1

mbl

21712 1

1010m

c


Forest Hardening

Obstacle distribution

BOUNDARY CONDITIONSPeriodic

OBSTACLE STRENGTH Uniform, f = 10 G b2

PEIERLS STRESS p= 0

Parameters


Monotonic loading

Stress-strain curve. Evolution of dislocation density with strain.


Dislocation Patterns

Evolution of dislocation patternas a function of slip strain


Interaction with Obstacles

Detail of the evolution of the dislocation patternshowing dislocations bypassing a pair of obstacles


Fading memory

a

c d

b

e f

Stress-strain curve.

Three dimensional view of the evolution of the phase-field, showing the the switching of the cusps.


Stress-strain curve.

a

Cyclic loading

b c

d e f

g h iEvolution of dislocation density with strain.


Irreversibility/Cyclic Loading


Poisson ratio effects

b

-50 0 50-1

-0.5

0

0.5

1/0

/o

= 0.0 = 0.3 = 0.5

-50 0 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

/0

/o

= 0.0 = 0.3 = 0.5

Stress-strain curve Dislocation density vs.strain

0.0 3.0 5.0

Stress-strain curve. Evolution of dislocation density with strain.


Obstacle density

22 10bc

82 10bc62 10bc

42 10bc


Multiple Glide


Physics Capturing Capabilities

Ortiz,1999

The aim of this study is to develop a phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals.

This representation enables to identify individual dislocation lines and arbitrary dislocation geometries, including tracking intricate topological transitions such as loop nucleation, pinching and the formation of Orowan loops.

This theory permits the coupling between slip systems, consideration of obstacles of varying strength, anisotropy, thermal and strain rate effects.


Summary

1. Phase-field model.

2. Closed form solution at zero temperature.

3. Temperature effects.

4. Strain rate effects.

5. Dislocation structures in grain boundaries.


Another Study Case: FerroElectrics

ab initio QMEoS of various phasesTorsional barriersVibrational frequencies

Force Fields and MDElastic, dielectric constantsNucleation BarrierDomain wall and interface mobilityPhase transitionsAnisotropic Viscosity

Meso- Macro-scaleNanostructure-properties relationshipsConstitutive Laws

Direct problem

Inverse problem


MesoscaleTi = 0

En = 0

u

Mechanical and Dielectric constants from atomistics G0 assumed equal to Gm


Evolution of Polarized Region

• Nucleate at end boundary

• Propagate along chain

• Loop movie

Evolution of interface

• Nucleate at side boundary

• Propagate along chain


Mesoscale Simulations: Stress

Evolution of stress


Polarization

Evolution of polarization


/E

-0.05 0 0.05 0.1

-0.08

-0.04

0

0.04

0.08

0.12

0.16

1

1

2

2

3

3

Increasing G0

G0 - const (109 J/m3)

G

(1

09

J/m

3)

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1

2

3

Hys

tere

sis

Loop

Ene

rgy

Hysteresis Loop


Towards material design: sensitivity analysis and inverse problem

Sensitivity analysis: “taking derivatives across the scales”

(%)1

(%)3

(%)1

(%)2

Cl

nucl

nucl

hyst

Cl

hyst E

E

WW

Derivative of the area in the hysteresis loop (loss) with respect to percentage of CTFE

Nucleation energy

CTFE%

%6

From atomistics: Nucleation energy decreases 3% per 1 mol % CTFE

From mesoscale: Hysteresis decreases 2% per 1% reduction in nucleation energy

Multi-scale: Hysteresis decreases XX % per 1 mol % CTFE

tutorial i: mechanics of ductile crystalline solids

Documents

arresttracking dislocation

dislocation loopanimations

updated phasefield

mean field approximation

elastic distortion

incremental work function

core energyortiz

green function