Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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Conclusion

Dislocation motion is quite a complex process spanning several orders of mag-

nitude in length scales. The motion is realized by breaking and re-establishing

bonds on the atomic level. These problems are tackled by molecular dynam-

ics simulations using semi-heuristic methods. Ab initio calculations are still

rare and, for more complex problems, they suﬀer from the still low number

of atoms involved. There is still poor microscopic experimental veriﬁcation

of these dynamic processes. The static core conﬁgurations of dislocations are

studied by high-resolution electron microscopy. The spatial resolution of the

latter techniques has drastically improved by the new aberration-corrected

microscopes. However, the informative value of the micrographs is limited by

the geometrical requirements to the specimens. Simple interpretations are pos-

sible only for edge dislocations arranged end-on in the specimen. Up to now,

there are no TEM in situ loading studies under high-resolution conditions

revealing changes of the core structure of relatively compact cores under load.

This is due partly to experimental diﬃculties and partly to surface eﬀects

from the low specimen thicknesses, which inﬂuence the processes to be stud-

ied. Thus, in understanding the atomic processes of dislocation motion, there

are still some challenges to be respected.

In the materials with low Peierls stress, these atomistic processes do not

control the dislocation dynamics. In addition to the disturbance of the local

coordination of atoms, dislocations carry an elastic stress ﬁeld, which gives

rise to the line contrast of dislocations in diﬀraction contrast TEM. Besides,

the stress ﬁeld comprises the main part of the energy of the dislocation. As,

in a zero-order approximation, the energy is proportional to the length of

the dislocation, many processes can well be described within the framework

of the line tension model. This model and the elastic interactions between

dislocations and other crystal defects as well as with other dislocations form

the bulk of the basic understanding of the dislocation motion. In the 1950s and

60s, many models were formulated on an intuitive way, which are now partly

proven by diﬀraction contrast TEM and other methods. Frequently, these

464 11 Conclusion

processes take place on a scale between about 10 nm and a few micrometers.

They constitute the bulk of the present book.

In the last decades, research has turned to more complex materials like

the intermetallic alloys where the dislocation mobility depends sensitively on

the structure of the dislocation cores. In addition to the nonplanar spreading

of the cores, which directly inﬂuences the glide resistance, nonconservative

processes within the cores seem to play a role in several intermetallics, leading

to a similar macroscopic behavior controlled by diﬀusion, irrespective of the

particular crystal structure. However, there are still no methods for monitoring

the small diﬀusion ﬂuxes in the dislocation cores to prove these models.

In the new ﬁeld of dislocation motion in quasicrystals, the application of

the whole set of methods developed for crystals for over 50 years provided

good insight within the short time of about 20 years, only. Nevertheless, the

present models of climb controlled by the formation of jog pairs on a cluster

level require more direct experimental evidence. Here, it should be possible to

prove the dynamic nonequilibrium point defect concentrations by respective

in situ methods.

Thus, for a number of years, the basic mechanisms controlling the disloca-

tion dynamics have been understood quite well. However, to determine their

actual parameters in the more complex materials requires the combined appli-

cation of macroscopic and microscopic methods as the atomistic interactions,

for example, between dislocations and intrinsic or extrinsic point defects, can-

not be observed directly, and so the interpretation of deformation data on the

basis of microscopic models is frequently still of speculative nature.

It is certainly one aim of the study of dislocation dynamics to establish

phenomenological laws to be incorporated into theories of the collective behav-

ior of many dislocations, which governs macroscopic plasticity. First steps are

here the formulation of constitutive laws combining dislocation dynamics with

the kinetics of dislocation generation and immobilization and annihilation, as

well as the statistical models of plastic instabilities. Other approaches are

the computer simulation of the motion of a greater ensemble of dislocations

or the insertion of dislocation mobility laws into ﬁnite element calculations.

Though these methods have brought some progress, there is, on the other

hand, still no good method to describe the eﬀect of long-range elastic ﬁelds on

the motion of individual dislocations with input parameters that can easily be

measured. In this connection, very little is known about the role of the ather-

mal stress component in the deformation of intermetallic alloys. Besides, the

interplay is not well understood between the eﬀective stress, which inﬂuences

the dislocation multiplication, and the development of the dislocation density

and the resulting athermal stress component. Thus, we may conclude that

the basic knowledge of dislocation dynamics has reached some completion. It

appears, however, challenging and requiring more eﬀorts to uncover and to

design the details of the continuously developing novel materials, and to imple-

ment physics-based dislocation mobility laws into the theories of collective

dislocation behavior.

List of Abbreviations and Symbols

a area, swept area, lattice constant

A area, cross section

APB antiphase boundary

b absolute value of Burgers vector

b Burgers vector

B viscous drag coeﬃcient

B Burgers vector in higher-dimensional hyperspace

edge component of Burgers vector

screw component of Burgers vector



Burgers vector component in physical (parallel) space

⊥

Burgers vector component in perpendicular (orthogonal) space

b.c.c. body centered cubic

c concentration (of defects), crack length

(thermal) equilibrium concentration

−

concentration of positive and negative kinks

concentration of jogs

concentration of kinks

transverse sound velocity

CBED convergent beam electron diﬀraction

CSF complex stacking fault

d width of decomposed dislocations

D diﬀusion coeﬃcient, diameter of particles (extended obstacles), total

density of cross slip events

diﬀusion coeﬃcient of kinks

self-diﬀusion coeﬃcient

E elastic (Young’s) modulus, elastic energy

core energy per unit length of dislocation

energy per unit length of dislocation

energy of edge dislocation

dislocation energy per length in matrix

466 List of Abbreviations and Symbols

dislocation energy per length in particle

energy of screw dislocation

energy per unit length of edge dislocation

energy per unit length of screw dislocation

prelogarithmic factor of energy per unit length of edge dislocation

prelogarithmic factor of energy per unit length of screw dislocation

f normalized (dimensionless) force acting on obstacle, volume fraction

of precipitates

normalized (dimensionless) obstacle strength

f.c.c. face centered cubic

force per unit length acting on dislocation

climb force per unit length resulting from external stress

total climb force per unit length acting on dislocation

glide force per unit length acting on dislocation

osmotic force per unit length due to point defect sub- of super-

saturation

F applied external force, force onadislocationsegment,force

acting on obstacles

obstacle strength

scaling factor in Mott–Labusch statistics

int

interaction force between kinks of double-kink

g diﬀraction vector

G diﬀraction vector in higher-dimensional reciprocal space

crack extension force, energy release rate



parallel component of diﬀraction vector

⊥

perpendicular component of diﬀraction vector

h distance between Peierls valleys, cross slip height

critical height of dislocation dipole

h average cross slip height

H(h) frequency of cross slip of height h

h.c.p. hexagonal close packed

HVEM high-voltage electron microscope/microscopy

J (diﬀusion) ﬂux (of kinks)

J integral

double-kink nucleation rate

k Boltzmann’s constant

K stress intensity factor

critical stress intensity factor

energy factors of screw and edge dislocations

resistance force at particle border

, K

phason elastic constants

l specimen length, length of dislocation segments

square lattice distance

L length of dislocation, segment length, width of double-kink

∗

critical width of double-kink

List of Abbreviations and Symbols 467

LYP lower yield point

m (true) stress exponent



eﬀective stress exponent

orientation factor for climb

eﬀective mass of dislocation per unit length

orientation (Schmid) factor for glide

n coordination number, creep exponent, ageing exponent, reciprocal

exponent of strain rate in equation for yield stress of semiconduc-

tors, strain-softening exponent

n normal vector of specimen, pole near direction of electron beam

p hydrostatic pressure

ext

external pressure

int

internal pressure

probability of unlocking

q dislocation immobilization or annihilation coeﬃcient

experimental activation energy

s slip distance

S elastic stiﬀness

SISF superlattice intrinsic stacking fault

r radius vector, radius of curvature of dislocation segment, strain rate

sensitivity

inner cut-oﬀ radius

experimental strain rate sensitivity

R outer cut-oﬀ radius

R, R

phonon–phason coupling constant

R, R

fault or displacement vector of planar faults



parallel component of displacement (shift) vector

t time

∗

lifetime of kinks

lifetime

waiting time

T absolute temperature

absolute melting temperature

TEM transmission electron microscopy

UYP upper yield point

U displacement in higher-dimensional hyperspace

elastic displacements



elastic displacement in physical (parallel) space

⊥

elastic displacement in perpendicular (orthogonal) space

dislocation velocity

preexponential factor of dislocation velocity

velocity of jog along dislocation

kink velocity

V volume, activation volume

experimental activation volume

468 List of Abbreviations and Symbols

w dislocation multiplication coeﬃcient

W mechanical energy

elastic energy

fdk

formation energy of double-kink

formation energy of kink

int

interaction energy between kinks of double-kink

peierls energy

total mechanical energy

pot

potential energy

x, y, z coordinates (x in forward direction of motion, z along dislocation

line)

area fractions of hard and soft regions in composite model

width of localized obstacles in x direction

z number of atoms in unit cell, atomic number

β angle between Burgers vector and dislocation line direction, size

eﬀect interaction constant between dislocations and spherical inclu-

sions

ΔA activation area

Δd activation distance

ΔF Helmholtz free energy of activation

ΔF

Helmholtz free energy of formation of critical bulge

ΔF

fdk

Helmholtz free energy of formation of double-kink

ΔF

Helmholtz free energy of formation of jog

ΔF

Helmholtz free energy of formation of kink

ΔF

Helmholtz free energy of motion of kinks

ΔG Gibbs free energy of activation

ΔG

ann

Gibbs free energy of dislocation annihilation

ΔG

ﬁ

Gibbs free energy of formation of an interstitial

ΔG

Gibbs free energy of formation of a jog

ΔG

Gibbs free energy of formation of a vacancy

ΔG

Gibbs free energy of self-diﬀusion

ΔH activation enthalpy

ΔS activation entropy

ΔV

ext

external volume change during formation of point defect

ΔW work term

ε plastic (normal) strain

elastic strain

total strain, elastic plus plastic

components of elastic strain tensor

˙ε plastic (normal) strain rate

˙ε

elastic strain rate

˙ε

total strain rate, elastic plus plastic

Δ diﬀerence between longitudinal and transversal elastic strain

γ (plastic) shear strain, energy of planar faults

APB

antiphase boundary energy

List of Abbreviations and Symbols 469

surface energy

stacking fault energy

˙γ plastic shear strain rate

˙γ

preexponential factor of shear strain rate

Γ line tension

Θ work-hardening coeﬃcient

λ displacement of dislocation, sweeping distance of kinks

μ shear modulus, chemical potential

ν Poisson’s ratio

Debye frequency

attempt frequency

ξ line vector of dislocation

normalized width of localized obstacles in x (ξ) direction

 dislocation density



mass density of material



forest dislocation density



mobile dislocation density

σ normal (engineering) stress

∗

eﬀective stress inquasicrystals

internal stress in quasicrystals

components of stress tensor

stress contribution due to the creation of phason walls

yield stress or strength



stress components in parallel space

⊥

stress components in perpendicular space

Σ stress tensor

τ shear stress, value of golden mean

∗

eﬀective shear stress



normalized athermal ﬂow stress of random obstacle array



normalized (dimensionless) stress

friction stress inside particles

critical stress of Frank–Read source

,τ

shear strength in hard and soft regions in composite model

(long-range) internal stress

theoretical shear strength of an ideal crystal

loc

locally acting shear stress

,τ

lower and upper yield stress at yield drop

Orowan stress

Peierls stress

self

self-stress of dislocation

threshold stress, related to internal stress component τ

ω jump frequency of kinks, of point defects

Ω atomic volume, elementary strain if all dislocations overcome an

obstacle

ΔΩ atomic volume diﬀerence

List of Video Clips

Most of the video sequences were produced within the research of the

Plasticity Group in the former Institute of Solid State Physics of the Academy

of Sciences of the GDR and the present Max Planck Institute of Microstruc-

ture Physics, both in Halle (Saale). The main authors are Ulrich Messer-

schmidt and Martin Bartsch. Exceptions are the Videos 8.2 and 9.1, the

authors of which, their aﬃliations, as well as further authors are listed together

with the particular materials.

6 Semiconductors

• Polycrystalline Silicon

Detlef Hoehl and Martina Werner

6.1 Dislocation motion in polycrystalline silicon at about 500 and 550

◦

6.2 Dislocation motion and generation in polycrystalline silicon at about

650

◦

• Ge and Ge-Si Alloys

Martina Werner and Ichiro Yonenaga

6.3 Several slip bands in Ge at about 430

◦

6.4 Motion of dislocations in a dislocation network in Ge at about 610

◦

6.5 Dislocation band in a network in Ge-4 at% Si at about 655

◦

6.6 Dislocation bands in Si-5 at% Ge at about 745

◦

6.7 Dislocation band in Si-5 at% Ge at about 835

◦

7 Ceramic Single Crystals

• Magnesium Oxide

Fritz Appel

7.1 Dislocation motion and generation in MgO single crystals at room

temperature

7.2 Motion of a dislocation in MgO at room temperature

• Zirconia Single Crystals

Dietmar Baither, Bernd Baufeld, and Manfred R¨uhle

472 List of Video Clips

7.3 Dislocation motion in c-ZrO

on a {001} plane during deformation

along [

2] at 1150

◦

7.4 Dislocation processes in c-ZrO

on a (211) plane during in situ strain-

ing along [

2] at 1150

◦

7.5 Dislocation motion in c-ZrO

on a {10

1} plane during deformation

along [100] at 1150

◦

7.6 Domain switching in t



-ZrO

during ferroelastic deformation at 1150

◦

7.7 Domain switching in partially stabilized ZrO

during ferroelastic defor-

mation at 1150

◦

7.8 Dislocation motion in partially stabilized ZrO

after ferroelastic defor-

mation at 1150

◦

8 Metallic Alloys

• Al-1 at % Ag

Rainer Hattenhauer, Peter J. Wilbrandt, and Peter Haasen

8.1 Dislocation motion in an Al-Ag single crystal at room temperature

• Computer Simulation of Dislocation Motion in a Material Hardened by

Precipitates of an Intermetallic Phase

Video by Volker Mohles and Eckhard Nembach, Institute of Materials

Research of M¨unster University

8.2 Simulation of the motion of a dislocation pair in Nimonic PE16

• DislocationGenerationinAl-1at%Ag

Rainer Hattenhauer, Peter J. Wilbrandt, and Peter Haasen

8.3 Loop generation and opening in an Al-Ag single crystal

• DislocationGenerationinTi-6wt%Al

T. Neeraj and Michael J. Mills

8.4 Creation and opening of dislocation loops in air-cooled Ti-6 wt% Al at

room temperature

8.5 Dislocation generation at a jog in step-annealed Ti-6 wt% Al at room

temperature

8.6 Localized (Frank–Read) sources in step-annealed Ti-6 wt% Al at room

temperature

• Dislocation Generation in Duplex Steel

Witold Zielinski

8.7 Dislocation multiplication in a ferrite grain during the deformation of

duplex steel at room temperature

8.8 Frank–Read source in an austenite grain during the deformation of

duplex steel at room temperature

• Oxide Dispersion Strengthened (ODS) Material INCOLOY MA956

Anna Wasilkowska and Aleksandra Czyrska-Filemonowicz

8.9 Dislocation motion in INCOLOY MA956 at room temperature

8.10 Dislocation motion in MA956 at 700

◦

8.11 Dislocation motion in annealed MA956 at 700

◦

8.12 Dislocation motion in MA956 at 1010

◦

8.13 Collective dislocation motion in MA956 at 1010

◦

List of Video Clips 473

• Oxide Dispersion Strengthened (ODS) Material INCONEL MA754

Dietrich H¨aussler and Bernd Reppich

8.14 Dislocation motion in the ODS alloy INCONEL MA754 at 655, 745,

and 790

◦

• Plastic Deformation During Fracture of Al

/Nb Sandwich Specimens

Christian Dietzsch, Christina Scheu, Wolfgang Kurtz, and Manfred R¨uhle

8.15 Dislocation motion in the plastic zone of cracks in Al

/Nb sand-

wiches

9 Intermetallic Alloys

• Ni

Video by G. Mol´enat and Daniel Caillard, Centre d’Elaboration de Mate-

riaux et d’Etudes Structurales, CNRS, Toulouse

9.1 APB jumps in Ni

Al-0.25at%Hf at room temperature

• γ-TiAl

Dietrich H¨aussler, Mark Aindow, and Ian Jones

9.2 Motion of ordinary dislocations in γ-TiAl at room temperature

9.3 Motion of ordinary dislocations in slip bands in γ-TiAl at room tem-

perature

9.4 Motion of 101 superdislocations in γ-TiAl at room temperature

9.5 Motion of twinning dislocations in γ-TiAl at room temperature

9.6 Motion of ordinary dislocations in γ-TiAl at 655

◦

• NiAl

Ralf Haush¨alter and Dietrich H¨aussler

9.7 Motion of dislocations with 100 Burgers vectors in an NiAl single

crystal at room temperature

9.8 Motion of dislocations with 100 Burgers vectors in NiAl-0.2 at% Ta

at room temperature

9.9 Motion of dislocations with 100 Burgers vectors in an NiAl single

crystal at 475

◦

9.10 Motion of dislocations with 100 Burgers vectors in an NiAl single

crystal at 565

◦

9.11 Motion of dislocations with 110 Burgers vectors on {110} planes in

NiAl-0.2 at% Ta at 475

◦

• FeAl

Christian Dietzsch and Easo P. George

9.12 Motion of dislocations with 111 Burgers vectors in Fe-43 at% Al at

room temperature

9.13 Motion of dislocations with 111 Burgers vectors in Fe-43 at% Al at

530

◦

9.14 Motion of dislocations with 100 Burgers vectors in Fe-43 at% Al at

447 and 530

◦

9.15 Dynamic formation and annihilation of dislocations in Fe-43 at% Al

at 700

◦