INTERMITTENCY AND SCALING OF D ISLOCATION FLOW IN PLASTIC CREEP DEFORMATION
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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
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•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
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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
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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
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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
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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
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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…)
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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
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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
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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
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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
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BASIC FEATURES
GENERATE ANISOTROPIC INTERNAL STRESS FIELD
Low energy cost structures: Walls, dipoles, ...
Metastability & self-pinning
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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,...
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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
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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!
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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.
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• 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
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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!
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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.
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• 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?
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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
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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
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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
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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
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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
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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
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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)
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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:
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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
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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
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• 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)
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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
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Three individual runs e=0.0125
N<v2> ~ Elastic energy at the points where we have dislocations
Red one
Before After
While
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BEFORE
Outside the wall
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WHILE
Fast dislocations collaborating in the rearrangement
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AFTER
Inside the wallN remains constant in this case
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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
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2)0()()( xtxtX
Subdiffusive behavior
Frustrated dislocations:Dislocations moving inside
traps (i.e. dislocation walls)
MEAN-SQUARE DISPLACEMENTMEAN-SQUARE DISPLACEMENT
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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... ...
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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
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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
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BOX SIZE 100 x 100
Requires an exhaustivestudy of finite-size effects
Yield threshold value ?
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“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
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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
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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
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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
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•“ 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.
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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
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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!