Lectures on: Introduction to and fundamentals of discrete dislocations and dislocation dynamics. Theoretical concepts and computational methods

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1 Lectures on: Introduction to and fundamentals of discrete dislocations and dislocation dynamics. Theoretical concepts and computational methods Hussein M. Zbib School of Mechanical and Materials Engineering Washington State University Pullman, WA summer school Generalized Continua and Dislocation Theory Theoretical Concepts, Computational Methods And Experimental Verification July 9-13, 7 International Centre for Mechanical Science Udine, Italy

2 Contents Lecture 1: The Theory of Straight Dislocations Zbib Lecture : The Theory of Curved Dislocations Zbib Lecture 3: Dislocation-Dislocation & Dislocation-Defect Interactions -Zbib Lecture 4: Dislocations in Crystal Structures - Zbib Lecture 5: Dislocation Dynamics - I: Equation of Motion, effective mass - Zbib Lecture 6: Dislocation Dynamics - II: Computational Methods - Zbib Lecture 7 : Dislocation Dynamics - Classes of Problems Zbib

3 Lecture 5: Dislocation Dynamics - I: Equation of Motion, effective mass Equation of Motion Dislocation Mobility Driving Force Effective mass

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5 Z Stress Frame 1 13 Mar Z X Y 1 Frame 16 Mar -1 1 X -1 E E+8-1 X 1E+8 Y 1 1 5E Strain

6 Dislocation equation of motion m* v Bv F Effective mass Inertia force B m* 1 v B electron, phonon dw dv Drag force v t Total Peach- Koehler Force + self-force on Dislocation segment i i b i F i

7 Typical Stress Dependence of Dislocation Velocity Log V 1 C S V 3 1 m/s 1m /s 1. Drag controlled viscous motion at high V and : V ~ b/b.thermally activated motion at low V and : V ~ V exp [- G/kT] T1 T C waiting stage + running stage Jerky dislocation motion

8 Phonon wind contribution Phonons are scattered by the dislocation strain field. Phonon flutter contribution Phonons are scattered by the oscillating dislocation (reradiated phonons). Phonon viscosity contribution Dissipation mechanism through slow phonons. Raman scattering Configurational oscillations of the core at Peierls relief. Bph =!!!! Dislocation Phonon Drag Bph For typical parameters, Bph.1 Pas at T < 1 K Bph = ~ 5 Pas at T = q D (Brailsford, 197; Alshits, 1975)

9 Electron Drag Be ELECTRON DRAG IN A NONMAL CONDUCTOR Electrons are scattered by the dislocation strain field Be = Cn e bp F C. ~.1, n e :electron density, p F : Fermi momentum For typical parameters, B e.5 Pas~5 Pas. ELECTRON DRAG IN A SUPERCONDUCTOR Below the critical temperature, Cooper Pairs are formed from individual electrons, which does not contributes to additional electron scattering, so that where BeSuper(T) = BeNormal G (T,V) G (T,V) =/(1+ Exp[/kT]) is the BCS temperature-dependent energy gap For slow dislocations in Nb, BeSuper (4.K) ~ BeNormal /4

10 r(ps) B(T)/B total ( q D ) Btotal Be Tq D Bw Bvs Bfl BRam 15 1 Drag Coefficients vs Temperature Drags in Cu Bw: phonon wind effect Bvs: viscous effect Bfl: fluttering effect Bram: Ramman scattering Be: electron drag Btotal: phonon + electron Based on Alshit s formulae for edge dislocation (1975) Relaxation Time vs Temperature 5 T/q D

11 M Double-Kink Theory (bcc: Low mobility) g bh a Dk kt - Fk exp kt Viscous Phonon and Electron damping Mg 1 B ( T ) fcc: High mobility 4 e.g AL: 1 1 Pa. s Urabe and Weertman (1975)

12 Driving force F F Peirels F Disl Disl F Self F External F Obstacle F Im age F Osmotic F Thermal

13 Osmotic Force: Due to non-conservative motion of edge dislocation (climb) that results in the production of intrinsic point defects F Osmotic b b [ b k b T ln( c c o )]

14 Thermal Force - SDD: Stochastic Dislocation Dynamics Equation of motion of a dislocation segment of length l with an effective mass density m * (Ronnpagel et al 1993, Raabe et al 1998): FThermal τ bi i the stochastic stress component satisfies the conditions of ensemble averages τ( t) τ( t) τ( t * kt B b lm t t) The strength of is chosen from a Gaussian distribution with the standard deviation of T B l 1b kt B b l t Pa.s l 5b 1K = 8.11MPa = 11.5 MPa For Cu with t = 5 fs

15 T k (K) 7 6 Mv SDD: Fluctuation of Kinetic Temperature k B T DOF K Pinned dislocation with initial velocity = m/s Total dislocation length = b & segment length =1b System temperature = 3 K, w/o applied stress t (ps)

16 Stochastic Dislocation Dynamics (SDD) Assumptions 1. Generated heat dissipates immediately into heat bath.. Lost kinetic energy is feedbacked into dislocation system as stochastic thermal agitation (DD Brownian dynamics) 3. Properties in dissipation process are known in advance based on quantum mechanical calculation/ atomistic simulations. 4. Dislocation motion is Markov process under constant stress or constant volume. B interstitial B climb B screw B mix B jog B edge Thermal stress pulses isothermal Elastic media (Thermal bath) Energy loss by drags (Phonon, Electron, Magnon, etc)

17 SDD: Dislocation Percolation among Random Obstacle Arrays CRSS ~ 14MP, No activation observed below 8 MP at 1K Average obstacle Spacing = 1b Hiratani and Zbib,

18 Concept of Effective mass Velocity-stress field of a moving edge dislocation Consider a uniformly moving screw dislocation with a velocity V, then it can be shown that the field equations have a traveling wave solution, and the coordinate transformations is the relativistic one x t ' ' xvt (1V /C (1V t-vx/c /C t ) 1/ t 1/ t ) ;, y ' C t y, z μ ρ ' 1/ z (transverse sound wavevelocity) Therefore, the solution is the same as of that of stationary dislocation but written out in terms of the new coordinates, i.e u z (x, y,t) γ 1 V C t b π 1/ tan 1 γy x Vt ; Hirth and Lothe, 198)

19 The stress are xy xz bμ π bμ π γy (x Vt) γ γ(x -Vt) (x Vt) γ y y The stress field is contracted toward the core, and intensified in that region Similar expression can be derived for the edge dislocation

20 Forces on High Velocity Dislocations Using the results for moving dislocations, one can derive the strain energy (W s ) 1 W dv ij s ij the kinetic energy (W k ) 1 W k v. vdv the total energy W W W s W k Hirth, Zbib and Lothe, Forces on High Velocity Dislocations, Modeling & Simulations in Maters. Sci. & Enger, 6, , 1998

21 C l For the moving screw dislocation For the moving edge dislocation W W W C 16 l 8 l 14 1 v W C v k l l W W W 1 W W k 1 W W s 1 W C v s l l l 1 v / C 1/ t 1 / / l 1 v C C l is the longitudinal sound velocity and C t is the transverse sound velocity Hirth, Zbib and Lothe, Forces on High Velocity Dislocations, Modeling & Simulations in Maters. Sci. & Enger, 6, , 1998

22 Equation of Motion An expression for the equation of motion in a moving frame of reference at velocity v is F( v) Bv F F i a F(v) B F i F a is the inertial force, is the mobility, is the internal force arising from interactions with other defects and from the Peierls barrier if present, and is the force produced by applied stresses. All of the forces other than Bv are thermodynamic forces [1], that is they represent virtual changes in free energy with respect to infinitesimal displacements in dislocation position. Hence dw Fdx Fvdt

23 and Thus Also the Lagrangian L = L(v) is given by With the definition of quasimomentum p dw v dw dt dv t 1 dw v F v dv t Definition: effective mass L p m * W k W s dl dv 1 v dw dv And W v dl k dv we have p d L v F t dv t Definition: effective mass m * d L dv

24 With the use of W = W k + W s, it can be shown that * d L m dv 1 v dw dv

25 Effective mass Effective mass m*= 1 v dw dv For screw dislocation: 1 3 m * W v v For edge dislocation: W C m * l l l v W 1/ b R n 4 r / 1 v / C 1 1 v In the limit of small velocity, they reduce to standard forms, e.g. Gilman(1997), Beltz (1968), Weertman (1961) l 3 / Cl 5

26 Molecular dynamica simulation of the motion of an edge dislocation Shastry and Diaz de la Rubia, LLNl, Rise-time to for V to reach steady state MD calculations by Shastry, 1998 Inertia effect is very small for V<.5 C

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29 Shock in Cu, P H =5 GPa Dislocation dynamics.5 μm MD by E. M. Bringa (LLNL)