INTERMITTENCY AND SCALING OF DISLOCATION FLOW

IN PLASTIC CREEP DEFORMATION

INTERMITTENCY AND SCALING OF DISLOCATION FLOW

IN PLASTIC CREEP DEFORMATION

M. CARMEN MIGUELUNIVERSITAT DE BARCELONA, BARCELONA, SPAIN

ALESSANDRO VESPIGNANI THE ABDUS SALAM ICTP, TRIESTE, ITALY

STEFANO ZAPPERIUNIVERSITA LA SAPIENZA & INFM, ROME, ITALY

JÉROME WEISSLGGE-CNRS, GRENOBLE, FRANCE

JEAN-ROBERT GRASSOLGIT, GRENOBLE, FRANCE

MICHAEL ZAISERTHE UNIVERSITY OF EDINBURG, UK

•INTRODUCTION•DISLOCATIONS:

1.-THEIR DISCOVERY IN CRYSTALS2.-DEFINITION3.-BASIC FEATURES4.-THEIR INTEREST IN STAT. MECHANICS

•CREEP DEFORMATION BY GLIDE5.-GENERAL OBSERVATIONS6.-TIME LAWS OF CREEP7.-ACOUSTIC EMISSION EXPERIMENTS ON ICE SINGLE CRYSTALS8.-DYNAMIC MODEL9.-RESULTS & DISCUSSION10.-CONCLUSIONS & OPEN QUESTIONS

OUTLINE

INTRODUCTIONINTRODUCTION

A.-FERROMAGNETIC PHASE

•Spontaneous magnetization•Breaks the continuous rotationalsymmetry of the disordered phase

B.-SOLID

•Regular arrangement of atoms in a lattice•Breaks the continuous translational symmetry of the liquid phase

DISTORTIONS & DEFECTS

•Goldstone excitations: Spin waves, phonons•Topological excitations: Vortices, dislocations

Generalized elastic theory

End of XIX century: Observation of “slip-bands” in metals (portions of the crystal sheared with respect to each other)

Beginning of XX century: Discovery of metal crystalline structure “Slip-bands” Relative displacement between layers of atoms

DISLOCATIONS: THEIR DISCOVERY IN CRYSTALS

DISLOCATIONS: THEIR DISCOVERY IN CRYSTALS

Theoretical shear strength of a perfect crystal >> Observed one

X-ray diffraction “Grain boundaries”

Slipband

1930’s Orowan, Taylor, Burgers DISLOCATION

POLYCRYSTALLINE ICE

Crystal grains slightly missoriented & separated by grain boundaries: Amorphous material? No.Arrays of dislocation lines !

Linear topological defects in the structure of any crystal

Mostmetals

Abrikosovvortex lattice

Smecticliquid crystals

Colloidalcrystals

ELASTICDEFORMATION el

Reversible changeof shape

PLASTICDEFORMATION Irreversible change

of shape

MECHANICAL PROPERTIES OF CRYSTALSMECHANICAL PROPERTIES OF CRYSTALS

MOTION OF DISLOCATIONS

Releases stress

HIGHERSTRESS

FRACTURE

AND/OR

DUE TO

ELASTICDEFORMATION el

Reversible changeof shape

PLASTICDEFORMATION Irreversible change

of shape

MECHANICAL PROPERTIES OF CRYSTALSMECHANICAL PROPERTIES OF CRYSTALS

MOTION OF DISLOCATIONS

1930’s Orowan, Taylor, Burgers

Releases stress

HIGHERSTRESS

FRACTURE

AND/OR

DUE TO

Linear topological defects in the structure of any crystal (most metals, Abrikosov vortex lattice, colloidal and liquid crystals…)

RELEVANT DISLOCATION FEATURES

Burgers’ vector b = Topological charge c

bdu

•Anisotropic

Elastic stress and strain fields

Low energy cost structures: Walls, dipoles…

Metastability & self-pinning

Long range dislocation interactions

Dislocations annihilation, multiplication...

•Long range 1/r

“Glide” or “slip”: Main type of motion-low energy cost!Involves sequential bond breaking and rebinding

BASIC FEATURES

BURGER’ VECTOR b = TOPOLOGICAL CHARGE

c

bdu

u displacement of atoms from their

ideal position

Boundary conditionfor any circuit around the defect

c

- dislocation axisb invariant

BASIC FEATURES

ELEMENTARY TYPES

Edge b Screw b ||

AT SHORT DISTANCES:

•DISLOCATION CORE-Energy cost E0

•Annihilation of opposite charged dislocation pairs• Cross-slip• Dissociation in partial dislocations, recombination

ELASTIC DEFORMATION AT LONG LENGTH SCALES

Linear elasticity equations & Boundary conditions

BASIC FEATURES

LONG RANGE INTERACTIONS!

0)( 66112

66 uccuc Nii

c

bdui

,...,1,

2

)()(

ln))((ln)()(

ij

ijjjijii

ijjjii

ji

ijjjii

R

RbRbD

a

Rbb

a

RbbDE

ELASTICENERGY

u Displacement field,

Elastic stress tensor

BASIC FEATURES

GENERATE ANISOTROPIC INTERNAL STRESS FIELD

Low energy cost structures: Walls, dipoles, ...

Metastability & self-pinning

BASIC FEATURES

MOTION TYPES

“Glide” or “slip”: Low energy cost!Sequential motion, involves single bond breaking and rebinding

“Climb”: Jump perpendicular to the Burgers’ vector.Involves the presence and/or formation of point defects: Interstitials, vacancies. High energy cost!

•Slip plane:

b

•Slip direction: || b

SLIPSYSTEM,n=1,2,...

MULTIPLICATION

•At various sources activated by the external stress applied.

•Induced by disorder or by cross-slip.

• From the surface

• From “grain-boundaries”

BASIC FEATURES

COMPLEX INTERACTIONS WITH OTHER DEFECTS

Portevin-LeChatelier Effect

FRANK-READ source

Many built-in during the growth process of the crystal

THEIR INTEREST IN EQUILIBRIUM STATISTICAL MECHANICS

Topological Defects in 2D:

Vortices in the XY model

Coulomb gas

Dislocations in crystalsSteps in facets

Phase Transitions

a la Kosterlitz-Thouless: Metal-Insulator (plasma) 2D-melting, Roughening transition

Topological Defects in 3D:

Vortices in superconductorsDislocations in crystals

Quantify & characterizeFLUCTUATIONS!

DISLOCATIONS IN NON-EQUILIBRIUM STATISTICAL MECHANICS

Dynamic Phase Transitions: Induced by their own interesting dynamics

Responsible for:

Plastic Deformation: The result of their time history under the action of external loads e.

• GENERAL LAWS for the temporal evolution of (t)-Creep laws

• COLD HARDENING: Y((t)) - Aging !• FATIGUE FRACTURE: After several cycles of deformation

(Ductile Fragile)

• GENERAL LAWS for the temporal evolution of (t)-Creep laws

• COLD HARDENING: Y((t)) - Aging !• FATIGUE FRACTURE: After several cycles of deformation

(Ductile Fragile)

)()(0

ttdtt

IF e > Y

CONSTANT Stress Plastic deformation

PLASTIC DEFORMATION BY GLIDE: GENERAL EXPERIMENTAL OBSERVATIONS

PLASTIC DEFORMATION BY GLIDE: GENERAL EXPERIMENTAL OBSERVATIONS

Strainrate

•THRESHOLD VALUES of stress: “Yield stress” Y•THRESHOLD VALUES of stress: “Yield stress” Y

TIME LAWS OF CREEP TIME LAWS OF CREEP

UNDER THE ACTION OF CONSTANT STRESS

SECONDARY: StationaryHomogeneous (laminar)

movement of dislocations ?

TERTIARY: Recovery.Usually ends in fracture

PLASTICSTRAIN-RATE

TIME

PRIMARYPower law: t-2/3 “Andrade creep”

Same behavior observed in many different materials!

vb m

Strain Rate

OROWAN´S LAW FOR PLASTIC DEFORMATION

Density of mobile dislocations

Mean velocity

“Macroscopic” constitutive law - Attemps to describe the

average deformation of the crystal due to dislocation glide.

“Macroscopic” constitutive law - Attemps to describe the

average deformation of the crystal due to dislocation glide.

Enormous gap between the theory developed for the interaction

between a few dislocations and the description of macroscopic

deformation Formulation of phenomenological laws based on

empirical observations.

Enormous gap between the theory developed for the interaction

between a few dislocations and the description of macroscopic

deformation Formulation of phenomenological laws based on

empirical observations.

• HOW IS THE LOW-STRESS DRIVEN DYNAMICS AT THE MESOSCOPIC SCALE? (Slightly above the threshold)

How is the creep relaxation?

Are there characteristic time scales?

Does the system reach a stationary state? How is it?

Does the system freeze in metastable configurations?

Are there frustrated dislocations, i.e. trapped for example between dislocation clusters?

• HOW DOES THE SYSTEM RESPOND TO PERTURBATIONS SUCH AS

the annihilation of a pair?

the addition of new dislocations?

VISCOPLASTIC DEFORMATION OF HEXAGONAL ICE SINGLE CRYSTALS

UNDER CREEP

DUE TO MOTION OF A LARGE NUMBER OF DISLOCATIONS

TRANSPARENT Defects

interference Cracks

THE THE EXPERIMENTEXPERIMENT

THE THE EXPERIMENTEXPERIMENT

•CHEAP•EASY GROWTH

SINGLE SLIP

ACOUSTIC EMISSION (AE) FROM COLLECTIVEACOUSTIC EMISSION (AE) FROM COLLECTIVE DISLOCATION MOTIONDISLOCATION MOTION

CREEPCOMPRESSION

Smallshear stress on

the basal planes

Deforms by slip of dislocationson the basal planes along a

preferred direction

ANISOTROPY

ACOUSTICEMISSION

ENERGYDISSIPATION

SUDDEN CHANGES OF

INELASTIC STRAIN

Ice

STATISTICAL ANALYSIS OF THE AE SIGNAL

Energy distribution of acoustic events P(E)

05.06.1

)(

E

EEEP

Power law distributions

Applied Stress0.58 MPa -1.64MPa

Resolved shear stress0.03 MPa - 0.086 MPa

Bursts of activity: Collective dislocation rearrangements

THE MODELTHE MODELTHE MODELTHE MODEL

• CROSS SECTION OF THE REAL SAMPLE

(perpendicular to basal plane)

• INITIAL RANDOM CONFIGURATION OF PARALLEL EDGE DISLOCATIONS

Burgers vectors b or -b (with equal prob.)

( =1 - 5 % )

• LET THE SYSTEM RELAX UNTIL IT REACHES A STILL CONFIGURATION

( s=0.5 - 1 % )

RELAX=NUMERICAL SOLUTION OF THE OVERDAMPED EQUATIONS OF MOTION

Adaptive-Step-Size

Fifth Order Runge-Kutta Method

nm

nmnn bv

IMPLEMENTATION DETAILS

LONG RANGE INTERACTION FORCES & PBC’sEWALD SUMS OVER INFINITE IMAGES

ONE EASY GLIDE DIRECTION (Single slip)PARALLEL TO BURGERS’ VECTOR

ANNIHILATION 2b

MULTIPLICATION MECHANISM FRANK-READ SOURCES (FRS)

IF HIGH STRESS > * Activation threshold value

nm22

nm2nm

2nm

2nmnm

mnm r

1

yx

)y(xxDbyx,σ

APPLY CONSTANT EXTERNAL STRESS

e of the same order of magnitude as the internal stress 1/2

)σσ(μbdt

dxv e

mn

nmnn

n CREEP DYNAMICS CREEP DYNAMICS

Power-law relaxation t-2/3 towards alinear creep regime

nnnvbt)(

PRIMARY

SECONDARY

Peach-Koelher force

IN THE STATIONARY STATE... IN THE STATIONARY STATE...

Formation & Destruction of METASTABLE dislocation CLUSTERS

Stress Shear

high

Dislocation walls...

Dislocation dipoles

low

SLOW FAST Dislocations

Sources ofself-induced

jamming!

I)

ebv External stress-induced velocity

Fast-moving dislocations

vv i Nm

Annihilation Creation of new dislocations

Slow dislocation structuresUndetected background noise!

Single dislocation velocity distribution

Singular response:

2.08.1

)(

E

EEEP

ACOUSTIC EMISSION SIGNAL IN THE MODELACOUSTIC EMISSION SIGNAL IN THE MODEL

“Acoustic” Energy

2VE

In the stationary regime

Mean Velocity vs. time

)(

)(tN

nn

m

vtV

TIME CORRELATIONS OF THE SIGNAL TIME CORRELATIONS OF THE SIGNAL

In the stationary regime

• POWER LAW DISTRIBUTIONS ABSENCE OF

CHARACTERISTIC CORRELATION TIME

•NON-DIFFUSIVE BEHAVIOR

• POWER LAW DISTRIBUTIONS ABSENCE OF

CHARACTERISTIC CORRELATION TIME

•NON-DIFFUSIVE BEHAVIOR

• FORMATION AND DESTRUCTION OF SELF-INDUCED

PINNING SOURCES (Dislocation dipoles, walls, …)• ANNIHILATION OF DISLOCATION PAIRS• CREATION OF NEW DISLOCATIONS IN FRS’s

SINGULAR RESPONSE“AVALANCHES’’

SINGULAR RESPONSE“AVALANCHES’’

• POWER LAW DISTRIBUTIONS FOR INTERMEDIATE

VALUES ABSENCE OF CHARACTERISTIC SIZE

•EXPONENTIAL CUTOFFS FOR LARGE VALUES,

CUTOFF WHEN e

• POWER LAW DISTRIBUTIONS FOR INTERMEDIATE

VALUES ABSENCE OF CHARACTERISTIC SIZE

•EXPONENTIAL CUTOFFS FOR LARGE VALUES,

CUTOFF WHEN e

IN THE STATIONARY STATE... IN THE STATIONARY STATE...I)

LOW STRESS DYNAMICS LOW STRESS DYNAMICS

Without creation of new dislocations

3/1)( tt

ANDRADE´s CREEP

BOX SIZE 100 x 100

II)

Slow power law relaxation of thestrain rate t-2/3

for almost all the time span

t-2/3

Three individual runs e=0.0125

N<v2> ~ Elastic energy at the points where we have dislocations

Red one

Before After

While

BEFORE

Outside the wall

WHILE

Fast dislocations collaborating in the rearrangement

AFTER

Inside the wallN remains constant in this case

BOX SIZE 300 x 300

Same results hold: Without creation of new dislocations For various multiplication rates r

Crossover to linear regime(crossover time gets shorter with r)

3/1)( tt

2)0()()( xtxtX

Subdiffusive behavior

Frustrated dislocations:Dislocations moving inside

traps (i.e. dislocation walls)

MEAN-SQUARE DISPLACEMENTMEAN-SQUARE DISPLACEMENT

Pure metal Temperature ºC Exponent

Cu 685 0.36

295 0.3

Mg 425 0.42-0.45

475 0.75

545 0.39-0.85

295 0.35-0.45

Al 425 0.50-0.55

475 0.18-0.65

Pb 290 0.33

Fe 715 0.33

Fe 1225-1545 0.33

Feltham, 54(Cottrell book)

This law has also been observed in creep experiments performed on polymeric materials such as: celluloid, polyisoprene, polystyrene, methyl methacrylate,...

(J.D. Ferry, Viscoelastic properties of polymers), and other glass-forming materials (see R.H. Colby PRE 61 (2000) 1783 and references therein).

ANDRADE CREEP LAWANDRADE CREEP LAW... ...

Thermal activation of a process that occurs under stress

UNIVERSALITY! Qualitative theories developed by Becker 25, Mott 53, Friedel 64, Cottrel 96, Nabarro 97, ...

1- Strain hardening (linear) raises the yield stress above the applied stress.2- Activation energy E, supplied by thermal fluctuations, to bring the stress in a volume V up to the yield value.3- The same V yields.

e < Y ()

Y() - e = C E

TkVE Be

Y 2)(

Plausible argument (Cottrel 96):

LACK OF CONSENSUS!2 V

III) CREEP LAWS CLASSICAL EXPLANATIONCREEP LAWS CLASSICAL EXPLANATION

A NEW PERSPECTIVE A NEW PERSPECTIVE

SCALING BEHAVIOR PROXIMITY OF AN OUT OF

EQUILIBRIUM CRITICAL POINT (YIELD STRESS VALUE) Y

“NONEQUILIBRIUM PHASE TRANSITION”“NONEQUILIBRIUM PHASE TRANSITION”

ELASTIC PLASTIC

T=0 in our model

Mobile dislocations as t

Stress

Y

JAMMED

MOVING

BOX SIZE 100 x 100

Requires an exhaustivestudy of finite-size effects

Yield threshold value ?

“THERMAL EFFECTS” “THERMAL EFFECTS”

Crossover time fromprimary to secondary creep

decreases with T, but leaves theexponent unchanged!

Andrade’s creep persist up to relatively high temperatures (high enough to destroy the slowly evolving metastable structures)

3/1)( tt

Bond-orientational order

MORE GENERAL FRAMEWORK: DISLOCATION JAMMING MORE GENERAL FRAMEWORK: DISLOCATION JAMMING

Dislocation dynamics shows up other glassy features like:

Aging-like behavior

Waiting time after a sudden quench of random configurations= 100

1000

Creep time

Strain

Metastable pattern formation Kinetic constraints

Broad region of slow dynamics

(recently suggested to refer to a wide variety of physical systems: granular media, colloids, glasses... Liu & Nagel 01)

Loading rate dependence

CONCLUSIONSCONCLUSIONSCONCLUSIONSCONCLUSIONS

EVIDENCE OF COLLECTIVE CRITICAL DYNAMICS

•SLOW DYNAMICSANDRADE´S CREEP•SINGULAR RESPONSE IN THE FORM OF “AVALANCHES’’•AGING

•SLOW DYNAMICSANDRADE´S CREEP•SINGULAR RESPONSE IN THE FORM OF “AVALANCHES’’•AGING

ABSENCE OF

CHARACTERISTIC SCALES

FOR THE SIZE AND TIME-

CORRELATIONS OF THE

REARRANGEMENTS

ABSENCE OF

CHARACTERISTIC SCALES

FOR THE SIZE AND TIME-

CORRELATIONS OF THE

REARRANGEMENTS

INTERMITTENCY AND POWER LAW DISTRIBUTIONS

• ANNIHILATION OF DISLOCATION PAIRS• CREATION OF NEW DISLOCATIONS IN FRS’s• SELF-INDUCED METASTABILITY

Dislocation clusters Dislocation jamming

DIMENSIONS AND SYMMETRIESHigher dimensions and more slip systems

TERTIARY REGIME: RecoveryLonger time spans, higher stress

AGING PHENOMENA: Work-hardening, FatigueMonotonous increase of stress & periodic load cycles

INTERACTION WITH OTHER DEFECTS.Plastic instabilities-Portevin LeChatelier effect.

STOCHASTIC FIELD THEORY.

NON-EQUILIBRIUM CRITICAL SCENARIOCheck robustness and coherence

•“ During creep the rate of flow is limited because of thermal fluctuations are required to bring it about.• Yield stress=Applied stress at which flow can occur without help from thermal fluctuations.• At the beginning of creep, applied stress = “critical” yield stress, so that the activation energy required is small.•As the creep strain the yield stress progressively above the applied stress. Larger thermal fluctuations are then needed which do not occur as frequently, and the rate of flow slows down. •If a stage is reached where the yield stress no longer rises, a steady-state creep is observed.”

•RECENT THEORIES (1990’s) BY THE SAME AND OTHER AUTHORS STILL RELY ON THE SAME “EQUILIBRIUM” IDEAS.• A MAJOR SUBJECT OF DEBATE WITHIN THE DISLOCATION COMMUNITY.

•RECENT THEORIES (1990’s) BY THE SAME AND OTHER AUTHORS STILL RELY ON THE SAME “EQUILIBRIUM” IDEAS.• A MAJOR SUBJECT OF DEBATE WITHIN THE DISLOCATION COMMUNITY.

A NEW PERSPECTIVE A NEW PERSPECTIVEIV)

SCALING BEHAVIOR PROXIMITY OF AN OUT OF

EQUILIBRIUM CRITICAL POINT (YIELD STRESS VALUE) Y

“NONEQUILIBRIUM PHASE TRANSITION”“NONEQUILIBRIUM PHASE TRANSITION”

• UNIVERSALITY CRITICAL EXPONENTS DEPENDING

ON A FEW FUNDAMENTAL PROPERTIES•EXPONENT RELATIONSHIPS & FINITE-SIZE SCALING

ELASTIC PLASTIC

)/()( tftt

)/( Lgs Y

||

T=0 in our model

e

DISLOCATION PILE UPDislocations on separated glide planes trapped in each others’ stress fields

e

•N dislocations of the same sign in 1D•Distribution of static pinning points•Aging

A SIMPLER MODEL A SIMPLER MODELV)

Long range repulsion & Box of finite size & Without pinning Regular lattice minimizes the free energy

Weak pinning Distortions of the latticeUNIVERSALITY CLASS?

WORK IN PROGRESS!