summer school

Generalized Continua

and Dislocation Theory

Theoretical Concepts,

Computational Methods

And Experimental Verification

July 9-13, 2007

International Centre for Mechanical Science

Udine, Italy

Lectures on:

Introduction to and fundamentals of

discrete dislocations and dislocation

dynamics. Theoretical concepts and

computational methods

Hussein M. ZbibSchool of Mechanical and Materials Engineering

Washington State University

Pullman, WA

zbib@wsu.edu

Contents

Lecture 1: The Theory of Straight Dislocations – Zbib

Lecture 2: 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

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

Lecture 4: Dislocations in Crystal Structures

Macroscopic experiment

“Macroscopic Scale”

representative

“homogeneous” element

Continuum Plasticity

“Mesoscopic Scale”

Polycrystalline

plasticity

“Microscopic Scale”

dislocations in single

crystal

,

DD from In-situ Exp.

Dislocation Dynamics

1m

Dislocation structure in a high purity copper

single crystal deformed in tension (Hughes)

Shock recovery experiments M. Schneider et al., Acta Mater. (2003)

TEM:

0.25 mm slices are

cut from the

recovered sample

~50-25 ~25-15 ~8-2 ~2-0Pmax (GPa)

Dislocation density increases with pressure

<134> Cu

205 J pulse

Dislocation dynamics

Shock in Cu, PH=50 GPa

2.5 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,

αβγ

ab

c

1. Cubic: a=b=c, α=β=γ= 90o

P (primitive), I (Body centered), F (Face centered),

4-3-fold symmetry

2. Tetragonal: a=b#c, α=β=γ= 900

P , I. 1-4 fold symmetry

3. Orthorhombic: a#b#c, α=β=γ#900

P, I, F, B (Base centered), 3-2 fold symmetry

4. Rhombohedral: a=b=c, α=β=γ#900, P, 1-3 fold symmetry

5. Hexagonal: a=b#c, α=β γ= 1200, P ,1-6 fold symmetry

6. Monoclinic: a=b=c, α=γ= 900 #β, P. B, 1-2 Fold symmetry

7. Triclinic: a#b#c, α#β#γ, P, NONE

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)

y

k

hl

x

z

[h, k, l]

x

z

ab

c

y

),,(1

),,( abacbcabc

lkh

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.)

the closed packed plane is {0001}

Since the Burger’s vectors are restricted to a perfect lattice vector and the energy of a dislocation line is proportional to b2, 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

0112

ab

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

Six Slip Directions

]011[I ]011[V

II ]110[

III ]101[

IV ]011[

]110[VI

Each slip plane contains three slip directions

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

I. Basic Geometry (bcc)

Simulation Cell

(5-20 )

[100][010]

[001]

Slip plane

(101)

m b

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..

Discrete segments

of mixed character

“Continuum” crystal

(Hull and Bacon 1984)

Microscopic strain and its relation to dislocation motion

3D Discrete Dislocation Dynamics

)nbbn( iii

p

i

N

i

gii

V

vl

1 2

)nbbn(W iii

p

i

N

i

gii

V

vl

1 2

ii vm *F

t

v

dv

dW

vm

1*

Equation of Motion

Effective Mass

Dislocation

velocity?

Dislocation

length?

Dislocation

Burgers

Vector?

The macroscopic strain and its relation to dislocation motion

Can also be derived from energy argument

Work done by externally

applied shear stressdu AduWe

work by dislocation motion: LdxbWd )( dx

We=Wd

vb

dxbd

dxbV

bldx

Ah

bldx

h

du

bldxAdu

Dislocation density:

ρ = l/V

Dislocation length per

unit volume

Peierl’s stressThe 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.

b

ux

4sin

2 d

bxy

x1tan2

bux

ν)2(1

d

The width of the dislocation

ν)(1

d

2

The Peierls Stress field for edge dislocation:

)(

)1(2

)1(2

)1(2

yyxxzz

yy

xx

xy

b

b

b

22

2

22

22

2

22

2222

ζ)(yx

y2x

ζ)(yx

y

ζ)(yx

ζ)2y(y

ζ)(yx

)2(3y

ζ)(yx

ζ)2xy(y

ζ)(yx

x

The Peierls Stress field for edge dislocation reduces to the Volterra dislocation for

1/222 )y(xr

The parameter δ removes the singularity at the origin r=0 that is present for the Volterra dislocation

Elastic Energy of Peierls dislocation

Consists of two parts:

1) Elastic strain energy stored in two half crystal

2) Misfit energy due to the distorted bonds

Elastic strain energy stored in two half crystal

2ζ

Rln

ν)(14π

μbE

2El

2

Misfit energy due to the distorted bonds

)-(14

μbE

2

M

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:

)4cos(4

exp

b)-(12

μb

)-(14

μbE

22

M

Then, the Peierls force

b

EM

4expmax

-1

b2 2

Then, the Peierls stress

bp

4exp

-1

2

Note: as p (very sensitive)

In general 24 10~10 p

Dislocation dynamics in BCC system