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 zbib@wsu.edu 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
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
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
Macroscopic experiment, Macroscopic Scale representative homogeneous element Continuum Plasticity Mesoscopic Scale Polycrystalline plasticity Microscopic Scale dislocations in single crystal
Dislocation Dynamics 1m Dislocation structure in a high purity copper single crystal deformed in tension (Hughes) DD from In-situ Exp.
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
Shock in Cu, P H =50 GPa Dislocation dynamics.5 μm MD by E. M. Bringa (LLNL)
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
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
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.
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
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
Slip system in F.C.C. Crystal& Cross-slip planes
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
Microscopic strain and its relation to dislocation motion (Hull and Bacon 1984)
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
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
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)
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/
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- )
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
Dislocation dynamics in BCC system