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INTERMITTENCY AND SCALING OF DISLOCATION FLOW IN PLASTIC CREEP DEFORMATION M. CARMEN MIGUEL UNIVERSITAT DE BARCELONA, BARCELONA, SPAIN ALESSANDRO VESPIGNANI THE ABDUS SALAM ICTP, TRIESTE, ITALY STEFANO ZAPPERI UNIVERSITA LA SAPIENZA & INFM, ROME, ITALY JÉROME WEISS LGGE-CNRS, GRENOBLE, FRANCE JEAN-ROBERT GRASSO LGIT, GRENOBLE, FRANCE MICHAEL ZAISER THE UNIVERSITY OF EDINBURG, UK

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INTRODUCTION DISLOCATIONS: 1.-THEIR DISCOVERY IN CRYSTALS 2.-DEFINITION 3.-BASIC FEATURES 4.-THEIR INTEREST IN STAT. MECHANICS CREEP DEFORMATION BY GLIDE 5.-GENERAL OBSERVATIONS 6.-TIME LAWS OF CREEP 7.-ACOUSTIC EMISSION EXPERIMENTS ON ICE SINGLE CRYSTALS 8.-DYNAMIC MODEL 9.-RESULTS & DISCUSSION 10.-CONCLUSIONS & OPEN QUESTIONS OUTLINE

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INTRODUCTION A.-FERROMAGNETIC PHASE Spontaneous magnetization Breaks the continuous rotational symmetry 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 Theoretical shear strength of a perfect crystal >> Observed one X-ray diffraction “Grain boundaries” Slip band

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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 Most metals Abrikosov vortex lattice Smectic liquid crystals Colloidal crystals

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ELASTIC DEFORMATION el Reversible change of shape PLASTIC DEFORMATION Irreversible change of shape MECHANICAL PROPERTIES OF CRYSTALS MOTION OF DISLOCATIONS Releases stress HIGHER STRESS FRACTURE AND/OR DUE TO

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ELASTIC DEFORMATION el Reversible change of shape PLASTIC DEFORMATION Irreversible change of shape MECHANICAL PROPERTIES OF CRYSTALS MOTION OF DISLOCATIONS 1930’s Orowan, Taylor, Burgers Releases stress HIGHER STRESS 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 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 u displacement of atoms from their ideal position Boundary condition for any circuit around the defect c - dislocation axis b invariant

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BASIC FEATURES ELEMENTARY TYPES Edge b Screw b || AT SHORT DISTANCES: DISLOCATION CORE-Energy cost E 0 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! ELASTIC ENERGY 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 SLIP SYSTEM, 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 crystals Steps in facets Phase Transitions a la Kosterlitz-Thouless: Metal-Insulator (plasma) 2D-melting, Roughening transition Topological Defects in 3D: Vortices in superconductors Dislocations in crystals Quantify & characterize FLUCTUATIONS!

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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) IF e > Y CONSTANT Stress Plastic deformation PLASTIC DEFORMATION BY GLIDE: GENERAL EXPERIMENTAL OBSERVATIONS Strain rate THRESHOLD VALUES of stress: “Yield stress” Y

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TIME LAWS OF CREEP UNDER THE ACTION OF CONSTANT STRESS SECONDARY: Stationary Homogeneous (laminar) movement of dislocations ? TERTIARY: Recovery. Usually ends in fracture PLASTIC STRAIN-RATE TIME PRIMARY Power law: t -2/3 “Andrade creep” Same behavior observed in many different materials!

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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. 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 EXPERIMENT CHEAP EASY GROWTH SINGLE SLIP

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ACOUSTIC EMISSION (AE) FROM COLLECTIVE DISLOCATION MOTION DISLOCATION MOTION CREEP COMPRESSION Small shear stress on the basal planes Deforms by slip of dislocations on the basal planes along a preferred direction ANISOTROPY ACOUSTIC EMISSION ENERGY DISSIPATION SUDDEN CHANGES OF INELASTIC STRAIN Ice

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STATISTICAL ANALYSIS OF THE AE SIGNAL Energy distribution of acoustic events P(E) Power law distributions Applied Stress 0.58 MPa -1.64MPa Resolved shear stress 0.03 MPa - 0.086 MPa Bursts of activity: Collective dislocation rearrangements

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THE 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

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IMPLEMENTATION DETAILS LONG RANGE INTERACTION FORCES & PBC’s EWALD 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

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APPLY CONSTANT EXTERNAL STRESS e of the same order of magnitude as the internal stress 1/2 CREEP DYNAMICS Power-law relaxation t - 2/3 towards a linear creep regime PRIMARY SECONDARY Peach-Koelher force

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IN THE STATIONARY STATE... Formation & Destruction of METASTABLE dislocation CLUSTERS Stress Shear high Dislocation walls... Dislocation dipoles low SLOWFASTDislocations Sources of self-induced jamming! I)

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External stress-induced velocity Fast-moving dislocations NmNm Annihilation Creation of new dislocations Slow dislocation structures Undetected background noise! Single dislocation velocity distribution Singular response:

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ACOUSTIC EMISSION SIGNAL IN THE MODEL “Acoustic” Energy In the stationary regime Mean Velocity vs. time

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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... I)

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LOW STRESS DYNAMICS Without creation of new dislocations ANDRADE´s CREEP BOX SIZE 100 x 100 II)II) Slow power law relaxation of the strain rate t -2/3 for almost all the time span t -2/3

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Three individual runs e =0.0125 N ~ 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 wall N 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)

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Subdiffusive behavior Frustrated dislocations: Dislocations moving inside traps (i.e. dislocation walls) MEAN-SQUARE DISPLACEMENT

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Pure metalTemperature ºCExponent Cu6850.36 2950.3 Mg4250.42-0.45 4750.75 5450.39-0.85 2950.35-0.45 Al4250.50-0.55 4750.18-0.65 Pb2900.33 Fe 7150.33 Fe 1225-15450.33 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 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 Plausible argument (Cottrel 96): LACK OF CONSENSUS! III) CREEP LAWS CLASSICAL EXPLANATION

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A NEW PERSPECTIVE SCALING BEHAVIOR PROXIMITY OF AN OUT OF EQUILIBRIUM CRITICAL POINT (YIELD STRESS VALUE) Y “NONEQUILIBRIUM PHASE TRANSITION” ELASTIC PLASTIC T=0 in our model Mobile dislocations as t Stress YY JAMMED MOVING

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BOX SIZE 100 x 100 Requires an exhaustive study of finite-size effects Yield threshold value ?

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“THERMAL EFFECTS” Crossover time from primary to secondary creep decreases with T, but leaves the exponent unchanged! Andrade’s creep persist up to relatively high temperatures (high enough to destroy the slowly evolving metastable structures) Bond-orientational order

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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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CONCLUSIONSCONCLUSIONS EVIDENCE OF COLLECTIVE CRITICAL DYNAMICS SLOW DYNAMICS ANDRADE´S CREEP SINGULAR RESPONSE IN THE FORM OF “AVALANCHES’’ AGING SLOW DYNAMICS ANDRADE´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 SYMMETRIES Higher dimensions and more slip systems TERTIARY REGIME: Recovery Longer time spans, higher stress AGING PHENOMENA: Work-hardening, Fatigue Monotonous increase of stress & periodic load cycles INTERACTION WITH OTHER DEFECTS. Plastic instabilities-Portevin LeChatelier effect. STOCHASTIC FIELD THEORY. NON-EQUILIBRIUM CRITICAL SCENARIO Check 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 IV) SCALING BEHAVIOR PROXIMITY OF AN OUT OF EQUILIBRIUM CRITICAL POINT (YIELD STRESS VALUE) Y “NONEQUILIBRIUM PHASE TRANSITION” UNIVERSALITY CRITICAL EXPONENTS DEPENDING ON A FEW FUNDAMENTAL PROPERTIES EXPONENT RELATIONSHIPS & FINITE-SIZE SCALING ELASTIC PLASTIC T=0 in our model

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ee DISLOCATION PILE UP Dislocations on separated glide planes trapped in each others’ stress fields ee N dislocations of the same sign in 1D Distribution of static pinning points Aging A SIMPLER MODEL V) Long range repulsion & Box of finite size & Without pinning Regular lattice minimizes the free energy Weak pinning Distortions of the lattice UNIVERSALITY CLASS? WORK IN PROGRESS!

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4.2.3. Time-Dependent Properties (1) Creep plastic deformation under constant load over time at specified temp. strain vs. time curve a) primary creep:

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