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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