DISLOCATIONSDISLOCATIONS

Edge dislocation

Screw dislocation

Slip(Dislocation

motion)

Plastic Deformation in Crystalline Materials

Twinning Phase Transformation Creep Mechanisms

Grain boundary sliding

Vacancy diffusion

Dislocation climb

Plastic deformation of a crystal by shear

She

arin

g st

ress

()

D isp lace m e n t

Sinusoidal relationship

Realistic curve

a

b

S h ea r s tre ss

m

b

xSinm

2As a first approximation thestress-displacement curve can be written as

a

xGG At small values of displacement

Hooke’s law should apply

b

xm

2 For small values of x/b

a

bGm

2

Hence the maximum shearstress at which slip should occur

2m

G

If b ~ a

2m

G

The shear modulus of metals is in the range 20 – 150 GPa

DISLOCATIONS

Actual shear stress is 0.5 – 10 MPa

I.e. (Shear stress)theoretical > 100 * (Shear stress)experimental !!!!

The theoretical shear stress will be in the range 3 – 30 GPa

Dislocations weaken the crystal

PullCarpet

EDGE

DISLOCATIONS

MIXED SCREW

Random

DISLOCATIONS

Structural Geometrically necessary dislocations

Usually dislocations have a mixed character and Edge and Screw dislocations are the ideal extremes

Slippedpart

of thecrystal

Unslippedpart

of thecrystal

Dislocation is a boundary between the slipped and the unslipped parts of the crystal lying over a slip plane

A dislocation has associated with it two vectors:

linen dislocatio thealongnt vector unit tangeA t

vectorBurgers The b

Burgers Vector

Perfect crystal

Crystal with edge dislocation

Edge dislocation

RHFS: Right Hand Finish to Start convention

Edge dislocation

Direction of t vectordislocation line vector

Direction of b vector

t

b

Dislocation is a boundary between the slipped and the unslipped parts of the crystal lying over a slip plane

The intersection of the extra half-plane of atoms with the slip planedefines the dislocation line (for an edge dislocation)

Direction and magnitude of slip is characterized by the Burgers vectorof the dislocation (A dislocation is born with a Burgers vector and expresses it even in its death!)

The Burgers vector is determined by the Burgers Circuit

Right hand screw (finish to start) convention is used for determiningthe direction of the Burgers vector

As the periodic force field of a crystal requires that atoms must move from one equilibrium position to another b must connect one lattice position to another (for a full dislocation)

Dislocations tend to have as small a Burgers vector as possible

The edge dislocation has compressive stress field above and tensile stress field below the slip plane

Dislocations are non-equilibrium defects and would leave the crystalif given an opportunity

Compressive stress field

Tensile stress field

STRESS FIELD OF A EDGE DISLOCATIONSTRESS FIELD OF A EDGE DISLOCATIONX – FEM SIMULATED CONTOURS

(MPa) (x & y original grid size = b/2 = 1.92 Å)

27 Å

28 Å

FILM

SUBSTRATE

b

Positive edge dislocationNegative edge dislocation

ATTRACTION

REPULSION

Can come together and cancelone another

Motion of Edge

dislocation

Conservative (Glide)

Non-conservative(Climb)

For edge dislocation: as b t → they define a plane → the slip plane

Climb involves addition or subtraction of a row of atoms below the half plane

► +ve climb = climb up → removal of a plane of atoms

► ve climb = climb down → addition of a plane of atoms

Motion of dislocationsOn the slip plane

Motion of dislocation to the slip plane

Edge Dislocation Glide

Shear stress

Surfacestep

Edge Climb

Positive climbRemoval of a row of atoms

Negative climbAddition of a row of atoms

[1] Bryan Baker chemed.chem.purdue.edu/genchem/ topicreview/bp/materials/defects3.html -

[1]

Screw dislocation

Screw dislocation cross-slip

Slip plane 1

Slip plane 2

b

The dislocation is shown cross-slipping from the blue plane to the green plane

The dislocation line ends on: The free surface of the crystal Internal surface or interface Closes on itself to form a loop Ends in a node

A node is the intersection point of more than two dislocations

The vectoral sum of the Burgers vectors of dislocations meeting at anode = 0

Geometric properties of dislocations

Dislocation PropertyType of dislocation

Edge Screw

Relation between dislocation line (t) and b

||

Slip direction || to b || to b

Direction of dislocation line movement relative to b

||

Process by which dislocation may leave slip plane

climb Cross-slip

Mixed dislocations

b

tb

Pure EdgePure screw

[1] http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/backbone/r5_1_2.html

Motion of a mixed dislocation

[1]

We are looking at the plane of the cut (sort of a semicircle centered in the lower left corner). Blue circles denote atoms just below, red circles atoms just above the cut. Up on the right the dislocation is a pure edge dislocation

on the lower left it is pure screw. In between it is mixed. In the link this dislocation is shown moving in ananimated illustration.

Energy of dislocations

Dislocations have distortion energy associated with them

E per unit length

Edge → Compressive and tensile stress fields Screw → Shear strains

Energy of dislocationElastic

Non-elastic (Core)

E

~E/10

2

2

1GbE Energy of a dislocation / unit length

G → () shear modulusb → |b|

2

2

1GbE Dislocations will have as small a b as possible

Dislocations(in terms of lattice translation)

Full

Partial

b → Full lattice translation

b → Fraction of lattice translation

Dissociation of dislocations

Consider the reaction:

2b → b + b

Change in energy:

G(2b)2/2 → 2[G(b)2/2]

G(b)2

The reaction would be favorable

2

1211

6b

3

1121

6b

1

1110

2b

(111)

(111)Slip plane

(111)

1[110]

2 (111)

1[121]

6 (111)

1[211]

6 → +

b12 > (b2

2 + b32)

½ > ⅓

FCC

Shockley Partials

2

1211

6b 3

1121

6b

1

1110

2b

AB

C

(111)

2

1211

6b 3

1121

6b

1

1110

2b

AB

C

(111)

(111)

Some of the atoms are omitted for clarity

(111)

1[121]

6

(111)

1[211]

6

(111)

1[1 1 0]

2

(110)(111),

1[112]

2

(110)

(111)Slip plane

Extra half plane

Burger’s vector

Dislocation line vector

(111)

1[1 1 0]

2

(110)(111),

1[112]

2

(110)

(111)Slip plane

Extra half plane

Burger’s vector

Dislocation line vector

The extra- “half plane” consists of two ‘planes’ of atoms

FCC Pure edge dislocation

BCC Pure edge dislocation

(110)

1[1 1 1]

2

(111)(110),

1[112]

2

(111)

(110)

Slip plane

Extra half plane

Burger’s vector

Dislocation line vector

1111

21112

2

(110)

Dislocations in Ionic crystals

In ionic crystals if there is an extra half-plane of atoms contained only atoms of one type then the charge neutrality condition would be violated unstable condition

Burgers vector has to be a full lattice translationCsCl → b = <100> Cannot be ½<111>NaCl → b = ½ <110> Cannot be ½<100>

This makes Burgers vector large in ionic crystalsCu → |b| = 2.55 ÅNaCl → |b| = 3.95 Å

CsCl

Formation of dislocations (in the bulk of the crystal)

Due to accidents in crystal growth from the melt

Mechanical deformation of the crystal

Annealed crystal: dislocation density () ~ 108 – 1010 /m2

Cold worked crystal: ~ 1012 – 1014 /m2

Burgers vectors of dislocations in cubic crystals

Monoatomic FCC ½<110>

Monoatomic BCC ½<111>

Monoatomic SC <100>

NaCl type structure ½<110>

CsCl type structure <100>

DC type structure ½<110>

Crystallography determines the Burgers vector fundamental lattice translational vector lying on the slip plane

“Close packed volumes tend to remain close packed,close packed areas tend to remain close packed &close packed lines tend to remain close packed”

Slip systems

Crystal Slip plane(s) Slip direction

FCC {111} <110>

HCP (0001) <1120>

BCCNot close packed

{110}, {112}, {123} [111]

No clear choice Wavy slip lines

Anisotropic

Slip

Role of Dislocations

FractureFatigue

Creep Diffusion(Pipe)

Structural

Grain boundary(low angle)

Incoherent Twin

Semicoherent Interfaces

Disc of vacancies ~ edge dislocation