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

Lecture 2: The Theory of Curved Dislocations

The Burgers Displacement Equation

Stress Field of a curved dislocation

Volterra Dislocations

• Circular Dislocation Loop

• Rectangular Dislocation Loop

• The stress filed about a straight segment dislocation

The Somigliana Ring Dislocation

Referecences

• I. Demir, J.P. Hirth and H.H. Zbib, The Somigliana Ring Dislocation, J. Elasticity, 28, 223-246, 1992.

• Khraishi, T.A., Zbib, H.M., Hirth, J.P. and de La Rubia, T.D., “The stress Field of a General Volterra Dislocation

Loop: Analytical and Numerical Approaches”, Philosophical Magazine, 80, 95-105, 2000.

• Khraishi, T. and Zbib, H.M., The Displacement Field of a Rectangular Volterra Dfislocation Loop, Phil Mag,82,

265-277, 2002.

Mixed Dislocation

The Peach-Koehler equation for the self-stresses of any curved, closed dislocation loop, is given by the following line integral (see Hirth and Lothe 1982):

k

C ii

imkm

iC

imm

iC

imm

xdRxxxx

Rb

G

xdRx

bG

xdRx

bG

23

22

14

88

)(

The self-stress

“1” P

Rjp

dl’

C1

Field point

b

k

C jm

ijkik

C

mikij

A j

mm xdxx

RbxRdbRdA

xbu

222

)1(8

1

8

1

8

1)(

r

The Burgers equation for displacements,

for any closed curved dislocation loop,

is given in terms of line and area integrals as (see Hirth and Lothe [21]):

Alternatively, Burgers equation has the following vector form:

CCR

dgrad

R

d lRblbbru

)(

)()(

18

1

4

1

4

A

R

d3

AR

is the solid angle through which the positive side of A is seen from is seen from r , and is defined as

The displacement field is important in interaction problems between dislocations and

embedded particles in a softer metallic matrix, e.g. in metal-matrix composites (MMCs), the

boundary condition in the problem is that of zero displacement at a set of collocation points on the

interface. The condition is enforced by annulling the undesired displacements caused by the crystal

or matrix dislocations with the displacements coming from the distribution of rectangular

dislocation loops. In order for this to happen, a quantification of the displacement field of each of

the rectangular dislocation loops is necessary and is therefore developed in this paper.

The integration of the Burgers equation to obtain the displacements turns out to be rather

involved. It is carried out with respect to the global xyz system. To perform the integration, the

indices in (1) are first expanded. This gives three independent equations, one for each displacement

component ),and,,( wvu corresponding to subscript or index m values of 1, 2 and 3, respectively.

Note that u is the displacement in the x-direction, v in the y-directions and w in the z-direction.

Before continuing along with the integration, few steps are in order. First, note that the elevation of

the dislocation loop is fixed with respect to the global coordinate system xyz. This means that the

dislocation loop lies in a plane parallel to the xy planes such that z is a constant. Second, note that

the following holds: RR /22 . Third, along segment 1, ax and 0xd , along segment 3,

ax and 0xd , along segment 2, by and 0yd , and along segment 4, by and

0yd .

Rectangular Dislocation Loop

A rectangular dislocation loop is composed of four linear dislocation segments (see figure 1). For this work, the Burgers vector b is assumed to be constant, in magnitude and direction, from one point to another along the loop.

x

z

yO

2a

2b

bz

bx

by

r´

rR

1

2

pt P = (x,y,z)

3

4

z´

dl´

Khraishi, T. and Zbib, H.M., The Displacement Field of a Rectangular Volterra Dfislocation Loop, Phil

Mag,82, 265-277, 2002.

The displacement field of a rectangular dislocation loop

bu

z

y

x

b

b

b

w

v

u

][

333231

232221

131211

K

KKK

KKK

KKK

22

222222

22

222222

11

)()(

)()()()()()(

)(

)()(

)()()()()()(

)(

)1(8

)(

4

),,(

zzxa

zzybxa

yb

zzybxa

ybxa

zzxa

zzybxa

yb

zzybxa

ybxa

zz

zyx

K

222222

222222

12

)()()(

1

)()()(

1

)()()(

1

)()()(

1

)1(8

)(

zzybxazzybxa

zzybxazzybxazz

K

32

10

12

xa 0

0.5

1

1.5

2

ya

0.060.040.02

00.020.040.06

ubz

32

10

12

xa

Figure 3. Three-dimensional carpet plot of u(x,y,z) normalized by the Burgers vector magnitude |b|

, where b=(bx=0,by=0,bz0). Here, =1/3, a=b=100bz, z=0, and z=20bz.

Khraishi, T. and Zbib, H.M., Dislocation Dynamics Simulations of the Interaction Between a Short Rigid Fiber and a Glide Dislocation Pile-up. Comp. Mater. Sci. 24, 310-322 2002.

Surface pt, P

B

C

B. C.:

u=0 A

Surface pt, P

B

C

B. C.:

u=0 A x'

z'

y'

Loop i

Oz

Ox

z

y

i

R

Loop A

pt P (x,y,z)

by'

bx'

bz'

x'

z'

y'

Loop i

O

x'

z'

y'

Loop i

Oz

Ox

z

y

i

R

Loop A

pt P (x,y,z)

z

Ox

z

y

Ox

z

y

ii

R

Loop A

R

Loop A

pt P (x,y,z)

pt P (x,y,z)

by'

bx'

bz'

by'

bx'

bz'

A rigid fiber surrounded by a glide

dislocation loop. The fiber surface is

meshed with

rectangular elements representing

rectangular dislocation loops (see

inset). Point P is just a field point.

A schematic of a rigid

fiber close to elastic

displacement sources

(e.g. a dislocation loop

and/or segment). The

correct boundary

condition to satisfy is

zero elastic

displacement on the

fiber's surface.

R

r

r

pt. P

x

loop A

zy

x

y

zz ,

OrA

C

xb

yb

zz bb ,Rr

r

pt. P

x

loop A

z

y

x

y

zz ,

OrA

C

dA

Circular Dislocation loop

Prismatic Dislocation Loops

• Prismatic dislocation loops with a Burgers vector normal to the plane of the loop can form in a material subject to irradiation or quenching by the precipitation of vacancies or interstitial atoms.

• Glide or shear loops can form readily, e.g., in the glide plane of a Frank-Read source as it continues to emit propagating dislocations. Hence, it is important to quantify the stress field of these loops in order to account for their interaction with other dislocation curves or defects in the crystal.

b

b

The stress field of a circular Volterra dislocation loop

zyzxyz

zyzxxz

yxyyxxxy

zz

yyyxxxyy

yyyxxxxx

xy

yx

yxxy

xxyy

yxyx

zz

2

22

2

2

2

22

2

2

2

22

2

2

2

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

kCkCGb

yy

yx

zxxzx

yy

zy

zxzz

zx

zxyy

zxzxx

KEKE

KEKEKE

KEKE

KEKE

KEKE

KEKE

KEKEKE

32313029

282726252423

22212019

18171615

14131211

10987

654321

122

12142

122

1212

1212

)4(1212

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x/R

zz/(

Gb

z/2

(1

- ))

N=6

N=12

N=18

N=24

szz, analy.

x

y

(a) (b) (c)

bz

RR

31 r/

The stress field of a circular loop, and straight segments approximations

Interaction between a defect (Frank sessile loop; defect

clusters in irradiated materials) and a glide edge

dislocation

d=2R

ly

b z

Fig. 5. (a) A TEM picture showing dislocation pinning by the dispersed hardening

mechanism. (b) A DD picture showing the same mechanism at work as captured in the

simulations. The dots in the figure represent defect clusters.

Interaction is

weak31 r/

Somigliana Dislocation

Many internal defect, such as broken fiber, debonding etc., can be

modeled as a set of discontinuities In the continuum resulting in

perturbations (of small or large magnitudes) in an otherwise simple

elastic field. The Somigliana dislocation can be considers as one

of the most general kins of such discontinuities an, thus, can be

used to model many imperfections.

Burgers vector is NOT conserved along the dislocation line

The Somigliana Ring Dislocation

used to model cylindrical cracks

432422412

332322312

232222212

132122112

2

431421411

331321311

231221211

131121111

1

1212

,,,

,,,

,,,

,,,

,,,

,,,

,,,

,,,

)(

)(

)(

AAA

AAA

AAA

AAA

Gb

k

kE

kK

AAA

AAA

AAA

AAA

Gb

drz

z

r

Rr

R

Gb

for

0

2

1

1

12

1

Rr

r

RGb

for

0

0

1

1

12 2

1

,,

22222222

32111

2bRrRrbbpg

pRgr

bA

,,

2222222

32

24222222

212144

2223 pgRrRrb

gp

bRpgbRrRrb

Rpgr

bA

,,

22222

2211232 rRbrb

Rgr

bA

Etc…

1. Demir,I., Hirth,J.P. and Zbib,H.M., "The Extended Stress Field Around the

Cylindrical Crack Using the Theory of Dislocation Pile-ups", Int. J. Engrg

Sci., 30, 829-845, 1992.

2. Zbib, H.M., Hirth, J.P. and Demir, I., ``The Stress Intensity Factor of

Cylindrical Cracks", Int. J. Engrg. Sci., 33,247-253, 1995.

3. Zbib, H.M. and Demir, I., "On the Micromechanics of Defects in

Fiber-Matrix Composites", review article, in: Damage in Composite

Materials, Voyiadjis, G.Z., ed., pp. 103-133, Elsevier Science Publisher,

N.Y., 1993.

4. Zbib, H.M. and Demir. I.,``The Interface Ring Dislocation in Fiber-Matrix

Composites: Approximate Analytical Solution", ASME Journal of

Engineering Materials and Technology, 116, 279-285, 1994.

5. Close, S. and Zbib, H.M., “The Stress Intensity Factors and Interactions

Between Cylindrical Cracks in Fiber Matrix Composites”, an invited

article, in: Damage and Interfacial Debonding in Composites, ed.

Voyiadjis,G., Z., and Allen, D.H., Elsevier Science, The Netherlands,

1996, pp. 3-27.

1. Demir,I., Hirth,J.P. and Zbib,H.M., "The Extended Stress Field Around the Cylindrical Crack Using the Theory of

Dislocation Pile-ups", Int. J. Engrg Sci., 30, 829-845, 1992.

2. Close, S. and Zbib, H.M., “The Stress Intensity Factors and Interactions Between Cylindrical Cracks in Fiber Matrix

Composites”, an invited article, in: Damage and Interfacial Debonding in Composites, ed. Voyiadjis,G., Z., and Allen,

D.H., Elsevier Science, The Netherlands, 1996, pp. 3-27.

Modeling the cylindrical crack around a fiber as a pileup of Somigliana ring dislocations

A curved dislocation can be approximated as a set of straight dislocation segments

x

y

(a) (b) (c)

bz

RR

Arbitrary shape

circular shape

Discrete segments

Continuous curve

N

j

D

jjp1

1,)(

J+1J

Nodes and collocation points on dislocation

loops and curves“1”

“3”

P

Rjpj

j+1

j-1

dl’

C1

C2

C3

i

i+1

i-1i-2

jj+1

“2”

vj+1

vj

v

Field point

Velocity

vector

kC

2

i

αβ

βαi

3

imkm

α

2

iC

imβmβ

2

iC

imαmαβ

xdRx

δxxx

Rb

ν)π(14

G

xdRx

bπ8

GxdR

xb

π8

G)(σ

p

Arbitrary shape

z

x

y A

B

b

x,y,zp

)()()( ABP ijijij

zzzRyx

RR

xb

R

xb

R

y

Rb

RR

yb

R

x

Rb

R

xb

x

R

x

R

yb

x

R

x

R

xb

R

x

R

xb

R

y

R

yb

y

R

y

R

xb

y

R

y

R

yb

x

R

x

R

xb

x

R

x

R

yb

zyx

yz

zyxxz

yx

xy

yxzz

yx

yy

yxxx

,,222222

33

2

0

3

2

30

2

2

2

2

22

2

2

2

20

32320

2

2

2

2

22

2

2

2

20

2

2

2

2

22

2

2

2

20

1

1

21

21

22

21

21

21

21

Other forms are given in

Hirth and Lothe (1982, p. 134)

This form is most convenient to use

RA

RB

The stress field of a dislocation segment

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x/R

zz/(

Gb

z/2

(1

- ))

N=6

N=12

N=18

N=24

szz, analy.

x

y

(a) (b) (c)

bz

RR

Discrete Circular Volterra Dislocation Loop

Stacking-Fault Tetrahedra

A

B

C

D

SFT

Observed in quenched metals and metals and alloys of low stacking-fault energy. It consists of a tetahedron of intrinsic staking faults on {111} planes with 1/6<110> type stair-rod dislocations along the edges of the tetrahedron. A dislocation loop (of Frank partial dislocation formed by the collapse of vacancies), may dissociate into a low-energy stair-rod dislocation and a Shockly partial on an intersecting slip plane, leading to the formation of a SFT.

R=10b, y=5

-8.00E+08

-6.00E+08

-4.00E+08

-2.00E+08

0.00E+00

2.00E+08

4.00E+08

6.00E+08

75 80 85 90 95 100 105 110 115 120 125

x (b)

glid

e P

F-fo

rce (

N/m

)

SFT

FS loop

Defect:

a)Loop,

b) dissociate

into SFT

Edge

disloc

ation

Xy

.Comparison between the Stress Fields of Unit Defects

To contrast the differing stress fields of a FS loop and a SFT, we calculate the Peach-Koehler force that both exert at

a point on a straight edge dislocation situated an x-distance from the loop plane (or correspondingly, from the

tetrahedron's base plane). An edge view of the loop and its possible SFT offspring (shown in dashed line) is exhibited

in this figure along with a nearby edge dislocation. In this figure, the glide force (in the direction of the dislocation's

Burgers vector) is plotted versus the y-coordinate of the dislocation at a given x. Note that in the force exerted by the

SFT is smaller than that caused by the FS loop. An exception to this is the case when x=9b where the edge

dislocation almost cuts the tip of the tetrahedron in question. In such a case, the SFT force is stronger than the

loop's. Moreover, if the dislocation line collides with a SFT (a more probable event than a loop collision, considering

volume differences), then their inelastic interaction is also expected to be strong. In this figure, the edge length of the

triangular FS loop, or correspondingly the SFT, is equal to 10b, which is a typical defect dimension as measured

experimentally. Now, if one fixes the y value and varies the x-coordinate instead, it would still be observed that

overall, and as long as elastic interactions are concerned, the FS loop causes a stronger glide force on the

dislocation. The results demonstrate a higher stress field associated with the FS loop, which should be in-line with

physical reasoning. Since the SFT forms from a FS loop (i.e. it’s a preferred lower energy state), it is then expected

that a more relaxed strain-field accompany such geometry over that of a loop.

Dislocation multiplication

Frank-Read sources