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

Transcription

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, 007 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 4: Dislocations in Crystal Structures Dislocation Dynamics Crystals is a solid Closed Packed Crystal structure Slip systems in F.C.C. Crystal Slip systems in B.C.C. Crystal Basic Geometry (bcc) Microscopic strain and its relation to dislocation motion Peierl s stress

4 Macroscopic experiment, Macroscopic Scale representative homogeneous element Continuum Plasticity Mesoscopic Scale Polycrystalline plasticity Microscopic Scale dislocations in single crystal

5 Dislocation Dynamics 1m Dislocation structure in a high purity copper single crystal deformed in tension (Hughes) DD from In-situ Exp.

6 Shock recovery experiments M. Schneider et al., Acta Mater. (003) TEM: 0.5 mm slices are cut from the recovered sample <134> Cu 05 J pulse P max (GPa) ~50-5 ~5-15 ~8- ~-0 Dislocation density increases with pressure

7 Shock in Cu, P H =50 GPa Dislocation dynamics.5 μm MD by E. M. Bringa (LLNL)

8 Crystals is a solid in which atoms are periodically arranged in a regular pattern. The periodicity can be described by a space lattice which is a regular, 3-dimensional arrangement of lattice points. The space lattice can be generated by a simple transformation of a unit cell. A unit cell is completely characterized by 3 lattice vectors a, b, c (or a, b, c, and three angles α, β,γ). When a unit cell contains only one atom within it, it is called a primitive cell. There are only 7 possible (geometrically) crystal system and 14 Bravais Lattice (space lattice ) to characterize the crystal structures, 1. Cubic: a=b=c, α=β=γ= 90 o P (primitive), I (Body centered), F (Face centered), 4-3-fold symmetry. Tetragonal: a=b#c, α=β=γ= 90 0 P, I. 1-4 fold symmetry 3. Orthorhombic: a#b#c, α=β=γ#90 0 P, I, F, B (Base centered), 3- fold symmetry 4. Rhombohedral: a=b=c, α=β=γ#90 0, P, 1-3 fold symmetry 5. Hexagonal: a=b#c, α=β γ= 10 0, P,1-6 fold symmetry 6. Monoclinic: a=b=c, α=γ= 90 0 #β, P. B, 1- Fold symmetry 7. Triclinic: a#b#c, α#β#γ, P, NONE α β γ b a c

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10 Miller indices Represents the orientation of a plane or a direction in crystal in relation to the unit cell axes. i) Direction; [h, k, l] h, k, l are the smallest integers in x, y, z axes <h, k, l> - family of [h, k, l] ii) Plane; (h,, l) The intercept distance of plane with x, y, a, axes are a, b, c. Take the reciprocal of a, b, c, then make them the smallest integer numbers taking out a common factor (i.e vector normal to plane) x h z k z l [h, k, l] y 1 ( h, k, l) ( bc, ac, ab) abc a c b y x

11 Closed Packed Crystal structure 1) Closed-Packed plane has the largest possible number of atoms per area or the highest atomic density. There are only two ways to accomplish this. A B C A BC.. ; (F.C. C.) the closed-packed plane is {111} A B A B ; (H.C. P.) {0001} the closed packed plane is Since the Burger s vectors are restricted to a perfect lattice vector and the energy of a dislocation line is proportional to b, the slip planes are naturally defined as these close-packed plane.

12 Slip system in F.C.C. Crystal In F.C.C crystal, the shortest lattice vector in {111} is <011> Therefore b a 011 Hence the slip system of F.C.C crystal is given as {111} - <011> and there are twelve combinations

13 Designation of Slip Systems in FCC ( 111) Critical plane A ( 111) Primary plane B ( 111) conjugate plane C ( 111) Cross slip plane D I Six Slip Directions [011] [011 [101 [101 II ] III ] IV ] V [110] VI [110] Each slip plane contains three slip directions

14 Slip system in F.C.C. Crystal& Cross-slip planes

15 I. Basic Geometry (bcc) Simulation Cell (5-0 m ) [001] [100] Continuum crystal Initial Condition: *Random distribution (dislocation, Frank-Read Pinning points (particles) *Dislocation structures Discrete segments of mixed character [010] (101) Expected outcome! Mechanical properties (yield stress, hardening, etc..). Evolution of dislocation structures Strength, model parameters, etc.. b Slip plane

16 Microscopic strain and its relation to dislocation motion (Hull and Bacon 1984)

17 3D Discrete Dislocation Dynamics p W N l ivgi i1 V N p l iv V i1 ( ni bi bi ni gi ( n i b i b i n i ) ) Dislocation velocity? Dislocation length? Dislocation Burgers Vector? Equation of Motion F m* v i i Effective Mass m* 1 v dw dv v t

18 The macroscopic strain and its relation to dislocation motion Can also be derived from energy argument du Work done by externally applied shear stress W e Adu dx work by dislocation motion: W d ( b ) Ldx W e =W d Adu bv bldx du bldx h Ah d bdx bldx V bdx Dislocation density: ρ = l/v Dislocation length per unit volume

19 Peierl s stress The stress field was determiend by treating the material as an elastic continuum, yielding a stress singularity at the dislocation core. The Peirel s stress introduces the effect of Lattice periodicity. xy b d sin 4u x b b x u tan 1 d x (1 ν) The width of the dislocation d (1 ν)

20 The Peierls Stress field for edge dislocation: xy xx yy zz b (1 ) b (1 ) b (1 ) ( xx yy (3y ) x (y ζ) x ) x x (y ζ) y (y ζ) xy(y ζ) x (y ζ) y(y ζ) x (y ζ) x x y (y ζ) The Peierls Stress field for edge dislocation reduces to the Volterra dislocation for r (x The parameter δ removes the singularity at the origin r=0 that is present for the Volterra dislocation y ) 1/

21 Elastic Energy of Peierls dislocation Consists of two parts: 1) Elastic strain energy stored in two half crystal ) Misfit energy due to the distorted bonds Elastic strain energy stored in two half crystal μb ln 4π (1 ν) EEl R ζ Misfit energy due to the distorted bonds E M μb 4(1- )

22 Peierls energy During the dislocation glide, the misfit energy changes periodically, but the elastic energy (Volterrs s dislocation) does not change. However, the equation given in previous slide does not contain periodic form. This is because we assumed a continuous displacement field. It is shown that when one accounts for atomic periodicity one gets: E M μb exp (1- ) μb 4(1- ) 4 cos(4 ) b Then, the Peierls force Then, the Peierls stress Note: as In general p 10 4 p ~10 E M p max exp 1- b exp 1-4 b (very sensitive) 4 b

23 Dislocation dynamics in BCC system