V Multiscale Plasticity : Dislocation Dynamics ta H.M. Zbib, Washington State University\ ta dislocations cracks penny-shaped cracks Ua V

Contents A. Basic Structure of The multi-scale model of Plasticity 1. Representative Volume Element: Inhomogeneous; homogeneous 2. Basic laws of continuum mechanics 3. Constitutive Laws: Connection with Dislocation Dynamics 4. The Finite Element formulation 5. Boundary Conditions: Dislocation-Boundary Interaction 6. Extension to Heterogeneous materials B. Basic Structure of Dislocations Dynamics Basic Geometry: Identification of Slip geometry (bcc), Discretization of dislocation curves Equation of Motion: Glide, climb, cross-slip, multiplication Long Range Stress Field and Driving Force Short-range interactions Boundary Conditions: periodic, reflected, free, etc.. C. Numerical issues: Long-range Interactions: Superdislocations vs FEM Time step and Segment length D. Critical Issues _Examples E. Movie (Typical simulations)

Inelastic strain Internal stress (caused by defects) Elastic strain caused by external stress External Stress Elastic stress caused by internal defects

Momentum: Energy: Hooke’s Law: Rate form

ta-t t V ta Interaction with External free surfaces: Ua The solution for the stress field of a dislocation segment is known for the case of infinite domain and homogeneous materials, which is used in DD codes. Therefore, the principle of superposition is employed to correct for the actual boundary conditions, for both finite domain and homogenous materials. Interaction with External free surfaces: ta dislocations cracks penny-shaped cracks Ua V t ta-t

Internal surfaces: Internal surfaces such as micro-cracks and rigid surfaces around fibers, say, are treated within the dislocation theory framework, whereby each surface is modeled as a pile- up of infinitesimal dislocation loops. Hence defects of these types are all represented as dislocation segments and loops, and there interaction with external free surfaces follows the method discussed above.

FE Formulations: Stiffness Matrix Applied stress Image stresses Internal stress: Long range interactions Shape Change

Extension to Inhomogeneous Materials and interfaces 1 Infinite domain with properties of material“1” 1 2 T 1 1 2 “Eigenstress” Elastic Strain field in domain “2” caused by dislocations in Domain “1” (infinite-domain solution)

*Dislocation image stress/ boundaries The “msm3d” Model micro3d (micro-scale) Discrete Dislocation dynamics fea3d (macro-scale) Continuum Solid mechanics *Dislocation image stress/ boundaries * Local (non-uniform) stress from: image stress, applied loads, and long- range dislocation stress *Shape changes *Tempertaure field Plastic strain increment ht3d Heat Transfer

x y A b B z p x,y,z Remote stress Field Other forms are given in Hirth and Lothe (1982, p. 134) This form is most convenient to use

3D Discrete Dislocation Dynamics Equation of Motion Effective Mass

Osmotic Force: Due to non-conservative motion of edge dislocation (climb) that results in the production of intrinsic point defects

Double-Kink Theory (bcc: Low mobility) Viscous Phonon and Electron damping fcc: High mobility e.g AL: Urabe and Weertman (1975)

Thermal Force - SDD: Stochastic Dislocation Dynamics Equation of motion of a dislocation segment of length Dl with an effective mass density m* (Ronnpagel et al 1993, Raabe et al 1998): the stochastic stress component t satisfies the conditions of ensemble averages The strength of t is chosen from a Gaussian distribution with the standard deviation of T 10K 2 = 8.11MPa = 11.5 MPa 50 5 28.7 40.6 100 10 57.4 81.1 300 30 181 256 For Cu with = 50 fs 

SDD: Fluctuation of Kinetic Temperature Pinned dislocation with initial velocity = 0 m/s Total dislocation length = 2000 b & segment length =10b System temperature = 300 K, w/o applied stress

I. Basic Geometry (bcc) Discrete segments of mixed character [001] b Simulation Cell (5-20 ) (101) [100] [010] “Continuum” crystal Slip plane Initial Condition: Expected outcome! *Random distribution Mechanical properties (yield stress, (dislocation, Frank-Read hardening, etc..). Pinning points (particles) Evolution of dislocation structures *Dislocation structures Strength, model parameters, etc..

Nodes and collocation points on dislocation loops and curves Long-range Stress field “1” “3” P Rjp j j+1 j-1 dl’ C1 C2 C3 i i+1 i-1 i-2 “2” vj+1 vj v Field point Velocity vector   Nodes and collocation points on dislocation loops and curves

Discretization Stress Field of a 3D Straight dislocation segment is known explicitly (Hirth & Lothe, 1982). Discretize each curve into a set of mixed segments.

x y A b B z p x,y,z Remote stress Field Other forms are given in Hirth and Lothe (1982, p. 134) This form is most convenient to use

Identification of basic geometry For each node identify: Coordinates, Burgers vector slip plane index neighboring nodes (k & j) Node type (free, fixed, junction, jog, boundary,etc. i(x,yz) j k b “Basic Unit” y x z

  Self-force and Peach-Koehler Force at Dislocation Nodes bi i Fi

= Force from segment CA+ Force from segment BD+ Force from segment AB Self-Force 2 A 1 C Force at sub-segment B = Force from segment CA+ Force from segment BD+ Force from segment AB D (see Hirth and Lothe, 1982, p. 131)

Self-Force per unit length Explicit expression x z A C B D

e.g. Average force per unit length : Similar expressions are obtained for the normal force. These expressions reduce to those given in Hirth and Lothe (1982, p.138) for Cut-off parameter; numerical parameter which can be adjusted to account for core energy

e.g. For pure edge pure screw dislocations, this reduces to A b b B (force per unit length)

V b  II. Equation of Motion i=1,…N y x z Nonlinear system of ODE’s Initial Configuration V Nonlinear system of ODE’s b  y x z

Short Range Interactions d A short-range interaction occurs when the distance “d” between two dislocations becomes comparable to the size of the core *annihilation, *formation of dipoles, *jogs, and *junctions. BCC: 4 Burgers vectors, without regard to slip planes  8 possible distinct reactions, -4 repulsive, 3 attractive, and one annihilation. When slip planes are considered: For the {110} and {112} planes  420 attractive reactions all of which are sessile (Baird and Gale, 1965) Detailed investigation of each possible interaction can become very cumbersome

C A D B Criteria for determining interaction: 1. Critical distance criterion (Essman and Mughrabi, 1979) 2. Force-based criterion: Implicitly takes into account the effect of the local fields arising from all surrounding dislocations. C A D B

Annihilation Free nodes Frank-Read source

C A D B

Example: Junction Junction node [010] glide plane (100) In a crude approximation, this implies that this reaction is energetically favorable.

Example: Jogs: Formation

Vacancy or interstitial generating jogs Line tension approximation formation energy Jogs: Motion Vacancy or interstitial generating jogs Line tension approximation For Ta:

Dipole Dipoles form naturally (numerically) without a need for a “Rule”.

Cross-Slip Example: Model Cross-slip node Initial dislocation source Activation energy Fundamental frequency of vibrating dislocation of length L Initial dislocation source on (011) with

FEA- Treatment of long-range stress (Finite domains) described above using the internal stress concept (SD) results into the body-force vector {fB} evaluated for each element. Thus, the resulting stress field in each element includes stress from all external agencies and all dislocations. And therefore, the driving force on each dislocation is readily evaluated from this stress field. This approximation works well for dislocation-dislocation interaction that do not reside in the same element (and surrounding elements). The interaction of dislocations belonging to the same element must be computed one-to-one (M2, M=number of dislocation segments in a given element). For example, consider dislocation “i” in an element containing M dislocations, then equation (12) is replaced by with S being the total stress in the element computed from the FEA described above and SD is the internal stress from dislocations within the element.

Infinite domains Boundary conditions: Periodic, reflected Superdislocation (Long Range stresses) Finite Domains: Boundary condition: Free, rigid, interfaces, mixed Image stresses: FEA

Main computaional Cell A is surrounded by MxM Cells (41x41) C, or D Cell with N Dislocations y P: Dislocation in Cell A Q P Zo Z A: Main Computational Cell(Grain) B,C, D.: Cells/Grains x

Elements Involved 1.Defects & Materials 3.Simulation/ Coding Continuum/ Finite Element Discrete Dislocation Dynamics Thermal Fluctuations Nonlinearities Geometric Distortions 2.Sample Conditions Creep – 10-4/s Quasi-static Strain Rate High Strain Rate: 103 108/s rdefect = 1022~1024/m3 T = RT ~ 0.5 Tmelt

Numerical Parameters: Material parameters   Elastic properties Dislocation Mobility law Core size = 1b Stacking-fault energy, & activation energy for cross-slip Double-Kink Theory (bcc: Low mobility) Viscous Phonon & Electron damping Numerical Parameters: DD Cell size (= FE mesh size)   Dislocation segment length (min (3b) and max) (variable), number of integration points (10) Number of cells (infinite domain; 3x3x3) Number of sub-cells (elements for FE) time step for DD; Max flight distance (variable time step ) Time step for FE (dynamics) ~ (smallest dimension/shear speed)