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

2. 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)

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

4. Momentum:
Energy:
Hooke’s Law:
Rate form

5. 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

6. 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.

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

8. Extension to Inhomogeneous Materials and interfaces1
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)

9. *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

10. 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

11. 3D Discrete Dislocation Dynamics
Equation of Motion
Effective Mass

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

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

14. 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 

15. 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

16. I. Basic Geometry (bcc) Discrete segments of mixed character [001] b
Simulation Cell
( )
(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..

17. 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

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

19. 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

20. 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

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

22. = 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)

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

24. 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

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

26. 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

27. Short Range Interactionsd
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

28. 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

29. Annihilation
Free nodes
Frank-Read source

30. C A D B

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

32. Example:
Jogs: Formation

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

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

35. 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

36. 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.

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

38. 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

39. 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: /s
rdefect = 1022~1024/m3
T = RT ~ 0.5 Tmelt

40. 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)