Dislocations

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal

• Dislocation geometry
• Dislocation motion
• Dislocation mechanics

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal

Consider a single crystal subjected to plastic
deformation in uniaxial direction

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal

 A crystal represented as a rectangular box is plastically deformed in tension*.

The bold lines indicate active slip plane for that particular strain increment.
Macroscopic shape change associated with an ordinary tensile deformation is
actually the cumulative effect of a large number of shear events.
The deformation takes place by sequential slip on various crystal planes!
*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge University Press, 2016, UK Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal
 It is the shear movement of the atoms
with respect to each other that causes
plastic deformation.

But how the movement takes place?

The movement takes place by SLIPPING of the

atom stackings on each other.

https://www.princeton.edu/~maelabs/mae324/07/07mae_16.htm Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal

 Examination of slipped surface under microscope

High magnification micrograph of

surface of specimen subjected to
tensile deformation

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

https://www.princeton.edu/~maelabs/mae324/07/07mae_16.htm Bhopal
Slipping during Bending deformation Slip Plane

Slip surface

www.google.com Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
Yield stress of a metal at
which slip initiates between
adjacent crystallographic
planes will have some
relation with the
interatomic forces between
the atomic planes

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
Approximate theoritical shear stress required for slipping
b
τ Free to slide
a Fixed
τ

D E
A B C
B
τ
2π𝑥
C E τ = K sin
A 𝑏
x
b Deptt. of Mechanical Engineering, MANIT
Dr. Abhinav Varshney,
D Bhopal
 b x

τ
a
τ a
θ

2π𝑥 Comparing (2) and (3)

τ = K sin ……………………(1)
𝑏 𝐺𝑏
K=
For small displacements, 2π𝑎
2π𝑥 Substituting value of K in (1)
τ=K ……………………(2)
𝐺𝑏 2π𝑥
𝑏 τ= sin
Considering the material to deform elastically 2π𝑎 𝑏
𝑥 τ will be maximum at x = b/4
τ=G ……………………(3)
𝑎 𝐺𝑏
τmax =
Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT
Bhopal 2π𝑎
 𝐺𝑏
τmax =
2π𝑎
Material Shear Stress Shear Stress
(Theoretical) (Experimental)
Iron (Fe) 6.6 GPa 310 MPa
Copper (Cu) 1.2 GPa 193 MPa
Gold (Au) 0.74 GPa 100 MPa
Diamond 121 GPa 0.24 MPa
Tungsten 16.5 GPa 0.11 MPa

A lot of discrepancy can be observed in the values of the theoretical and experimental shear strength
of crystals.

It means in the real world it is easy to deform these crystals. This due to the presence of line defects
known as dislocations.

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
Examples of Dislocations y

z
x

b = Burger’s vector

Edge Dislocation Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
Screw Dislocation
*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge University Press, 2016, UK
 Φ is the angle between b and dislocation line.
Φ is called the character angle of the dislocation

A mixed dislocation has both

❑ edge component (b sinΦ)
❑ screw component (b cosΦ).

For edge dislocation Φ = _____

for screw dislocation, Φ = _____

Mixed Dislocation

*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge University Press, 2016, UK Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal
EDGE DISLOCATION Extra half plane

Symbol for edge dislocation

Perfect simple cubic Crystal Crystal with edge dislocation

(Front View)
Edge dislocation is produced by inserting
an extra half plane or removing an extra half plane
Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal
HOW EDGE DISLOCATION MOVES IN CRYSTAL?

Each step in the movement of dislocation requires only a slight rearrangement

of atoms.

Very small force is required

Dr. Abhinavto move
Varshney, Deptt. ofa dislocation.
Mechanical Engineering, MANIT
Bhopal
 Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT
*M.A. Meyers, K.K. Chawla, Mechanical behaviour of materials, Cambridge University
Bhopal Press, 2009, UK
In the absence of dislocations,
movement of each atom is
required and thus large shear
stress is required.

In the presence of dislocations, movement

of only one atom is required. Thus, shear
stress required is very less and plastic
deformation can easily take place.

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
SCREW DISLOCATION

Plane ABCD is the slip plane. Line

DC is the dislocation line
Consider these blocks as one atom. The stacking
of upper set of atoms is displaced towards right
Since on moving from x to y we
and lower stacking to the left.
move in spiral around the
dislocation line, this dislocation is
This leads to distortion or slip of limited set of
called the screw dislocation. One
atoms on another. While there is no such thing
circuit ends at y; continued circuits
that has happened at the other end of the crystal.
end at point z.
Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT
D.R. Askeland, P.P. Fuley, Essentials of Materials Science and Engineering, Bhopal Cengage learning, 2009, Canada
 J I
F E

K
G

Screw dislocation
line

H
Spiral staircase
Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal D
 Edge dislocation in a simple cubic crystal Screw dislocation in a simple cubic crystal

*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge University Press, 2016, UK Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal
• atoms above the slip plane
o atoms below the slip plane

Mixed

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhop

Dislocations are called line defects because they extend along lines (not necessarily straight) in the crystal.

Transmission Electron Microscope (TEM) Micrograph of actual dislocations*

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT Bhopal
* D.B. Williams and C.B. Carters, Transmission Electron Microscopy, A text book for materials science, Springer, 1996
 Dislocation density (ρ)

Dislocations are line objects, the dislocation density is defined as the total
dislocation line length L divided by the volume V of interest,

𝐿
ρ=
𝑉

Dislocation density has the dimensions of inverse area (m-2).

The dislocation density in a well annealed crystal is typically in the range of 109 – 1010 m-2 *

In a cold worked metal, the dislocation density can get in the range of 1014 – 1016 m-2 *

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

*G.E. Deiter, Mechanical Metallurgy, MacGraw-Hill, 3rd Edition, 1986 Bhopal
 Description of a Dislocation

Two vectors are needed to completely specify the type (and sign) of a dislocation:

• Sense vector e (a unit vector parallel to the dislocation line at a particular point)

• Burger’s vector b (defined using Burgers circuit)

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
 Burgers circuit

The Burgers circuit is a sequence of discrete

steps from one lattice point to the next in a
clockwise direction that closes to form a
complete loop when drawn on a perfect crystal.

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
 RH/SF Convention
S F S
F
b

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal CLOSURE FAILURE
 S F S RH/SF Convention
F
b

e e.b = ??

e X b = ??
CLOSURE FAILURE

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
 Use of right hand rule for dislocations
e
F S

e X b = ??

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
 e X b = ??
D.R. Askeland, P.P. Fuley, Essentials of
e.b = ??
Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT
Materials Science and Engineering, Bhopal Cengage learning, 2009, Canada
Positive/negative screw dislocation
e.b > 0 Right hand screw

e.b < 0 Left Hand Screw

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
 Identify the type of screw dislocation (RH/LH) in the following

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

*M.A. Meyers, K.K. Chawla, Mechanical behaviour of materials, Cambridge University
Bhopal Press, 2009, UK
Dislocation Motion

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
 Dislocation motion is always in the direction locally
perpendicular to the dislocation line, expanding or
shrinking the slipped area.

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

Bhopal
*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge University Press, 2016, UK
 Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT
*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge UniversityBhopalPress, 2016, UK
 Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT
*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge UniversityBhopalPress, 2016, UK
 Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT
*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge UniversityBhopalPress, 2016, UK
 Cross Slip

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge UniversityBhopalPress, 2016, UK
 Climb

Dr. Abhinav Varshney, Deptt. of Mechanical Engineering, MANIT

*Wei Cai, William D Nix, Imperfections in crystalline solids, Cambridge UniversityBhopalPress, 2016, UK