INTERMITTENCY AND SCALING OF D ISLOCATION FLOW IN PLASTIC CREEP DEFORMATION

1. 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

2. OUTLINE • 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

3. 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

4. DISLOCATIONS: THEIR DISCOVERY IN CRYSTALS End of XIX century: Observation of “slip-bands” in metals (portions of the crystal sheared with respect to each other) Slip band Beginning of XX century: Discovery of metal crystalline structure “Slip-bands” Relative displacement between layers of atoms Theoretical shear strength of a perfect crystal >> Observed one X-ray diffraction  “Grain boundaries”

5. POLYCRYSTALLINE ICE Crystal grains slightly missoriented & separated by grain boundaries: Amorphous material? No. Arrays of dislocation lines ! 1930’s Orowan, Taylor, Burgers  DISLOCATION Linear topological defects in the structure of any crystal Abrikosov vortex lattice Smectic liquid crystals Colloidal crystals Most metals

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

7. MECHANICAL PROPERTIES OF CRYSTALS HIGHER STRESS  ELASTIC DEFORMATION el Reversible change of shape PLASTIC DEFORMATION  Irreversible change of shape DUE TO MOTION OF DISLOCATIONS 1930’s Orowan, Taylor, Burgers Releases stress Linear topological defects in the structure of any crystal (most metals, Abrikosov vortex lattice, colloidal and liquid crystals…) AND/OR FRACTURE

8. RELEVANT DISLOCATION FEATURES  Burgers’ vectorb = Topological charge  Elastic stress and strain fields Long range  1/r Long range dislocation interactions Anisotropic Low energy cost structures: Walls, dipoles… Metastability & self-pinning  Dislocations annihilation, multiplication... “Glide” or “slip”: Main type of motion-low energy cost! Involves sequential bond breaking and rebinding

9. BASIC FEATURES BURGER’ VECTORb = TOPOLOGICAL CHARGE u displacement of atoms from their ideal position Boundary condition for any circuit around the defect c  - dislocation axis b invariant

10. 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

11. BASIC FEATURES ELASTIC DEFORMATION AT LONG LENGTH SCALES Linear elasticity equations & Boundary conditions uDisplacement field,  Elastic stress tensor ELASTIC ENERGY LONG RANGE INTERACTIONS!

12. BASIC FEATURES GENERATE ANISOTROPIC INTERNAL STRESS FIELD Low energy cost structures: Walls, dipoles, ... Metastability & self-pinning

13.  b BASIC FEATURES MOTION TYPES “Glide” or “slip”: Low energy cost! Sequential motion, involves single bond breaking andrebinding Slip plane: SLIP SYSTEM, n=1,2,... Slip direction: || b “Climb”: Jump perpendicular to the Burgers’ vector. Involves the presence and/or formation of point defects: Interstitials, vacancies. High energy cost!

14. BASIC FEATURES Many built-in during the growth process of the crystal MULTIPLICATION • At various sources activated by the external stress applied. • Induced by disorder or by cross-slip. • From the surface • From “grain-boundaries” FRANK-READ source COMPLEX INTERACTIONS WITH OTHER DEFECTS Portevin-LeChatelier Effect

15. 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!

16. DISLOCATIONS IN NON-EQUILIBRIUM STATISTICAL MECHANICS Responsible for: Dynamic Phase Transitions: Induced by their own interesting dynamics Plastic Deformation: The result of their time history under the action of external loads e.

17. PLASTIC DEFORMATION BY GLIDE: GENERAL EXPERIMENTAL OBSERVATIONS THRESHOLD VALUES of stress: “Yield stress” Y Plastic deformation CONSTANT Stress IF e > Y Strain rate GENERAL LAWS for the temporal evolution of (t)-Creep laws COLD HARDENING: Y((t)) - Aging ! FATIGUE  FRACTURE: After several cycles of deformation (Ductile Fragile)

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

19. 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. OROWAN´S LAW FOR PLASTIC DEFORMATION Mean velocity Strain Rate Density of mobile dislocations “Macroscopic” constitutive law - Attemps to describe the average deformation of the crystal due to dislocation glide.

20. WE WOULD LIKE TO KNOW... • 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?

21. THE EXPERIMENT VISCOPLASTIC DEFORMATION OF HEXAGONAL ICE SINGLE CRYSTALS UNDER CREEP TRANSPARENT Defects interference Cracks CHEAP EASY GROWTH SINGLE SLIP DUE TO MOTION OF A LARGE NUMBER OF DISLOCATIONS HOW DO WE TRACK THE COLLECTIVE DISLOCATION DYNAMICS?

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

23. 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 Avalanches

24. 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.) ( r0=1 - 5 % ) • LET THE SYSTEM RELAX UNTIL IT REACHES A STILL CONFIGURATION ( rs=0.5 - 1 % )  RELAX=NUMERICAL SOLUTION OF THE OVERDAMPED EQUATIONS OF MOTION Adaptive-Step-Size Fifth Order Runge-Kutta Method

25. 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

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

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

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

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

30. TIME CORRELATIONS OF THE SIGNAL In the stationary regime POWER LAW DISTRIBUTIONS  ABSENCE OF CHARACTERISTIC CORRELATION TIME NON-DIFFUSIVE BEHAVIOR

31. I) IN THE STATIONARY STATE... 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’’ POWER LAW DISTRIBUTIONS FOR INTERMEDIATE VALUES  ABSENCE OF CHARACTERISTIC SIZE EXPONENTIAL CUTOFFS FOR LARGE VALUES, CUTOFF WHEN e 

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

33. Three individual runse=0.0125 While Red one N<v2> ~ Elastic energy at the points where we have dislocations Before After

34. BEFORE Outside the wall

35. WHILE Fast dislocations collaborating in the rearrangement

36. AFTER N remains constant in this case Inside the wall

37. Same results hold: Without creation of new dislocations For various multiplication rates r BOX SIZE 300 x 300 Crossover to linear regime (crossover time gets shorter with r)

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

39. ANDRADE CREEP LAW... 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).

40. CREEP LAWS CLASSICAL EXPLANATION III) UNIVERSALITY! Qualitative theories developed by Becker 25, Mott 53, Friedel 64, Cottrel 96, Nabarro 97, ... Thermal activation of a process that occurs under stress Plausible argument (Cottrel 96): 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 LACK OF CONSENSUS!

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

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

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

44. MORE GENERAL FRAMEWORK: DISLOCATION JAMMING (recently suggested to refer to a wide variety of physical systems: granular media, colloids, glasses... Liu & Nagel 01) • Broad region of slow dynamics • Metastable pattern formation  Kinetic constraints • Dislocation dynamics shows up other glassy features like: • Loading rate dependence • Aging-like behavior Waiting time after a sudden quench of random configurations= 100 Strain 1000 Creep time

45. CONCLUSIONS INTERMITTENCY AND POWER LAW DISTRIBUTIONS • ANNIHILATION OF DISLOCATION PAIRS • CREATION OF NEW DISLOCATIONS IN FRS’s • SELF-INDUCED METASTABILITY • Dislocation clusters • Dislocation jamming ABSENCE OF CHARACTERISTIC SCALES FOR THE SIZE AND TIME-CORRELATIONS OF THE REARRANGEMENTS • SLOW DYNAMICS ANDRADE´S CREEP • SINGULAR RESPONSE IN THE FORM OF “AVALANCHES’’ • AGING EVIDENCE OF COLLECTIVE CRITICAL DYNAMICS

46. Many interesting questions still open... NON-EQUILIBRIUM CRITICAL SCENARIO Check robustness and coherence 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.

47. “ 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.

48. A NEW PERSPECTIVE IV) SCALING BEHAVIOR  PROXIMITY OF AN OUT OF EQUILIBRIUM CRITICAL POINT (YIELD STRESS VALUE) Y ELASTIC PLASTIC T=0 in our model “NONEQUILIBRIUM PHASE TRANSITION” UNIVERSALITY  CRITICAL EXPONENTS DEPENDING ON A FEW FUNDAMENTAL PROPERTIES EXPONENT RELATIONSHIPS & FINITE-SIZE SCALING

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