Plastic Deformation &
Elementary Dislocation Theory
Lecture Course for the
Students of Metallurgical Engineering
V V Kutumbarao
LECTURE - 4
2
Brief Review of
Part – 1: The Etchpit
Enter the Dislocation
Dislocation Movement causes Slip
Modes of Deformation - Slip
3
Geometrical properties of dislocations Dislocation always moves in such a way as to increase
the slipped area
Dislocation is coplanar with the slip plane
Dislocation can only move in its own slip plane
Displacement produced when a dislocation moves completely out of the crystal is called the “Burgers Vector” of the dislocation
Dislocation is a line that can have any shape but a dislocation line can not end inside a crystal
Dislocation line perpendicular to b : Edge
Dislocation line parallel to b : Screw
Dislocation line inclined to b : Mixed
4
Burgers
Circuit
• Rules– Circuit traversed in the same manner as a rotating R-H screw
advancing in the direction of the dislocation.
– Circuit must close completely in a perfect crystal and must go completely around the dislocation in the real crystal.
• Vector that closes the circuit in the imperfect crystal is the Burgers Vector
5
6
Part – 2: The Prism Loop
Going in deeper
Stress Fields around Dislocations
Edge Dislocation
• sx is the largest
normal stress
• compressive for y > 0
• tensile for y < 0
• txy is maximum at y=0
222
22
)(
)(
)1(2 yx
yxyGby
s
)(
2
12 22 yx
yGbz
s
222
22
)(
)(
)1(2 yx
yxxGbyx
t
222
22
)(
)3(
)1(2 yx
yxyGbx
s
Screw Dislocation
sx = sy = sz = 0
txy = 0
txz = - Gb y
2 (x²+y²)
tyz = Gb x
2 (x²+y²)
Polar: tqz = Gb/2r
i.e.,independent of q
• No normal stresses
• Only a shear stress field which has complete radial symmetry
Polar coordinates
Edge: sr = sq = - Gb sin q2(1-) r
and trq = tqr = Gb cosq
2(1- ) r
Screw: tqz = Gb/2r
From the above expressions it is seen that
Stress is at r = 0
So for a small cylindrical region r = r0
around dislocation (called the core), the equations are not valid.
r0 is of the order of ~ 0.5 to1nm (b to 4b usually)
7
• Escrew ½ Gb² L
or Energy per unit length ½ Gb²
• Eedge ½ G b² L/(1)
• If = ⅓, Eedge 3/2 Escrew for same length
• Energy length, so dislocations tend to have minimum l - preferred shape is a straight line or a circular loop
• Thus dislocation may bethought of as having a line tension T = dE/dl ½Gb²
• Analogous to surface tension of a liquid.
Strain Energy of a
Dislocation
8
Shear stress required to move a dislocation in
a periodic lattice
where w = ‘width’ of the dislocation dhkl/(1-g)b
Peierls – Nabarro force
b
dGb
wG
hkl
p
)1(
)2(exp
1
2
)2(exp
1
2
t
For metals, d b and ½ , tp 2 x10⁻⁵ G,
close to observed shear strength
For ceramics, w ~ 1b, tp = 4G exp (-2) ~ 7.5x10 G,
i.e, ~10 times larger than in metals!
- 3
9
Other Properties of
Dislocations
• Force on a Dislocation F due to an applied shear stress t is
F = t b
• F is the same at all locations and is always perpendicular to dislocation line
• Shear stress required to bend a dislocation of length l to a radius R
t= Gb/2R = Gb/ l
• Shear strain due to dislocation movement
g = r b x
• Consider a 1cm³ crystal of well annealed copper
• b = 0.256 nm, r = 10⁵/cm² ; x = 0.5 cm
• g = 10⁵ x 0.256 x 10⁻⁹x 0.5 1.28 x10⁻³ 0.128%
• Even if we consider 10⁶ dislocations / cm², the total strain produced when all the dislocations move out of the crystal is < 2.6%.
• Very low compared to observed values: 20 to 50%.
• So a mechanism is required for generating additional dislocations during plastic deformation.
• One such is the Frank-Read source
Dislocation Multiplication
The Frank-Read Source
11
12
Part 3: The Partials
Getting Real
Partial Dislocations in fcc
Shockley
Frank
Lomer-Cotterell
13
Dislocations of like sign on same slip plane:
• Large separation: act independent of each
other
− Energy ∝ 2 X
• Small separation: act like one dislocation:
− Energy ∝
= 2 X above
∴ Dislocations of like sign on the same
slip plane tend to repel each other
2
2Gb
2
)2( 2bG
Dislocation –Dislocation Interactions
14
• Dislocations of like sign
on same slip plane repel
each other
• Dislocations of like sign
on parallel slip planes
form an array
• Line up above one other
cancelling out each
other’s stress fields
• Lattice bending around
the array: Small angle
boundaries
Tilt , if array is of edges
Twist, if array is of
screws15
Dislocation –Dislocation Interactions
Unlike dislocations on same slip plane:
− Attract and annihilate each other forming a perfect
lattice
Dislocation –Dislocation Interactions (contd..)
Unlike dislocations on parallel slip planes:
− Attract each other and form a row of
vacancies or
interstitials
16
b
A
B
P
P’Q
A’
B’
Q’C
• Interaction of dislocations on intersecting slip planes produce jogs in each other
• In general jogs in edge dislocation do not impede their motion
• Jogs in screw dislocations all have edge orientation
• Can only move by climb of the edge segment
Jogs in Dislocations
AP , P’B : screw
segments
PP’ : edge Jog
PP’BC: its slip plane
PP’Q’ Q : extra half
plane
17
Glide and Climb
18
Glide of Edges
Non-conservative motion of edge
jogs in screw dislocations
Willing Surrender
19
Part 4: The Yield Point
Yield Point Phenomenon
• Yield Point: Localized yielding in some
metals, particularly low c steel
• Load increases steadily with elastic
strain up to UYP, drops suddenly,
fluctuates about some approximate
constant value (LYP) producing YPE
and then rises with further strain.
• Lüders Bands: At UYP a discrete band of deformed metal appears on
specimen surface at a stress concn such as a fillet and immediately load drops
to LYP.
• Propagation of the band along the length of the specimen causing YPE.
• Usually several bands at ~45° to tensile axis.
• Also called stretcher strains (Piobert effect)
• Jogs in YPE correspond to formation of several LB.
• YPE ends after LB cover the entire specimen. Flow then follows usual pattern20
Yield Point Phenomenon (contd..)
• Besides low–C steel , YP is observed in other materials as well: Poly xtalline: Mo, Ti, Al alloys and Single xtal: Fe, Cd, Zn, ∝- & β-brass
• YP usually associated with small amounts of interstitials or substitutional impurities e.g., Fe with as low as 0.001 (C+N) shows Y P
• Sharp UYP promoted by elastically rigid machine, very careful specimen alignment, use of specimens free from stress concns, high rate of loading and testing at low temps.
• If first LB forms in middle of specimen, UYP 2 x LYP, otherwise usually UYP 1.1 to 1.2 x LYP
• Onset of general yielding at a stress where average dislocation sources can create slip bands through a good volume of the material
• Thus σ0 = σs + σi , where σs = stress to operate the sources and σi= combined frictional stress due to all obstacles
• Explanation of yield point phenomenon was one of the early triumphs of Dislocation Theory
21
Explanation of yield point phenomenon
• Postulate: Dislocations locked or pinned by solute atom interactions
• C&N readily diffuse to positions of minimum energy around dislocations
e.g., regions just below extra half plane of edge dislocation.
Carbide particles along dislocations in
iron - platelets viewed edge-on
Octahedral interstitial site in a bcc cell
22
Explanation of yield point phenomenon (contd..)
• Strong elastic interaction -
impurities condense into a row
of atoms along the core of the
dislocation.
• Breakaway stress required to
free a dislocation line from a
row of solutes is
where
Ui = interaction energy and
r0 = distance from dislocation
core to line of solute atoms
(0.2nm)
• When pulled free, the
dislocation can move at a
lower stress.
Row of Carbon atoms in maximum
binding position at an edge
dislocation. Applied shear stress
will cause disloc to separate from
C atoms by gliding in the slip plane
22
orb
As
qSin
rUA i
23
• Alternatively, where dislocations are strongly pinned
such as by C & N in Fe, new dislocations must be
generated to allow flow stress to drop (UYP explained)
• Dislocations released pile up at grain boundaries.
• Stress conc at the tip of the pile up combines with
applied stress in the next grain to unlock sources & thus
the Lüders band propagates across the specimen.
Explanation of yield point phenomenon (contd..)
24
• To explain yield drop in several materials
• Special case: impurity locking
where = externally imposed strain rate
= average dislocation velocity
r = mobile dislocation density
• is a strong function of stress
where is resolved shear stress for unit velocity
0t
/
)( 0
mtt
General Theory of Yield Point
25
• If a material has low initial r (high purity material or
material having strongly pinned dislocations) then
has to be high to match
• But for to be high t has to be high
• Once some dislocations move, they begin to multiply
and r increases rapidly, so can drop to maintain
constant and so stress drops
General Theory of Yield Point (contd..)
/
)( 0
mttand
26
General Theory of Yield Point (contd..)
• The conditions at the upper &lower YP can be expressed by
• For small values of m’ (<15),
is large,
so strong yield drop
/1 m
u
l
l
u
r
r
t
t
l
u
t
t
27
General Theory of Yield Point (contd..)
• For iron, m’ = 35, so yield drop substantial if ru <≈10³ /cm²
r of annealed iron is ≈ 106/cm², so most dislocations must be pinned (=99.9%)
• m’ very large for fcc(>100 to 200), so only a small load drop required to cause substantial change in dislocation velocity
/1 m
u
l
l
u
r
r
t
t
28
• YP pronounced if:
1. Low mobile dislocation density at start
2. Potential for rapid dislocation multiplication with increasing strain and
3. Relatively low dislocation velocity - stress sensitivity
• Many ionic (eg LiF) and covalently bonded (eg Si) crystals possess these properties, so exhibit YP
• By contrast, most fcc metals have an initially high rand a very high m’, so an yield drop is an unlikely event in them
General Theory of Yield Point (contd..)
29
Strain Aging
• Reappearance of YP after aging due to re-diffusion
of C&N atoms to dislocations during aging
• Activation energy for return of YP in good agreement
with activation energy for diffusion of C in ∝-iron
• Usually associated with
YP phenomenon
• Strength increases and
ductility decreases after
heating at low temp
following cold working
30
Strain Aging (contd..)
• N plays an important role (more than C) in Fe because
of higher solubility and diffusion coefficient
• Practical importance in forming steel articles by deep
drawing – appearance of undesirable “stretcher
strains”
• Remedy – tie up C and N with V, Ti, Nb, B , Al
(carbide &nitride formers)
• Industry’s usual solution: skin pass rolling and
immediate use, produces sufficient fresh dislocations
31
• Max. velocity at which dislocs can drag along
the atmosphere of impurities is
where D is diffusion coefficient
Protevin - LeChatelier Effect
• Dynamic strain ageing: serrations in stress strain curves, due to successive bouts of yielding and aging
2kTr
DAv
32
Protevin - LeChatelier Effect (contd..)
• At v higher than
dislocation pulls away →
causes yield drop →new
atoms diffuse to lock
dislocations
• Process repeats causing
serrations
• Discontinuous Yielding
• For PC steel, discontinuous yielding in the temp region,
200 to 400 o F, the BLUE BRITTLE REGION: Steel
heated in this temp range (blue oxide coating) shows
decreased ductility and notch- impact resistance.
• Blue brittleness is accelerated strain aging
2kTr
DAv
33
34
The Climax
35
Part 5: Strengthening
Strengthening of Polycrystals
• Strength inversely related to dislocation mobility
• Single crystals rarely used for engineering application (limitations: strength, size, production) (Exceptions: solid state electronic devices, turbine blades)
• Deformation of polycrystals more complex than that of single crystals because of restraining effect of surrounding grains
• Greater complexity required to produce materials of highest strength and usefulness
1. Fine grain size
↑ to increase strength
2. Large additions of solute
↑ to increase strength and bring about new phase relationships
3. Fine particles & phase transformations
↑ to increase strength
36
Grain Size Effects
Fine grain size
• Grain boundaries:
Y regions of disturbed lattice and high surface
energy
Y In order to maintain continuity during deformation
of a polycrystal, it is required to have non
homogeneous deformation and multiple slip
especially near grain boundaries
Y More grain boundaries (finer grain size) will result
in higher strength
37
• Empirical, based on experimental observations
σy= σ0+ ky d-½
σy= yield stress
σ0 = friction stress opposing motion of dislocations
ky = “unpinning” constant measuring extent to
which dislocations are piled up at barriers
d= grain diameter
• This equation is also applicable to flow stress at any
strain e
se = σ0 + ke d- ½
Grain Size Effects (contd..)
Hall – Petch Relationship
38
1. Dislocation Density Model
Flow stress related to dislocation density as
σ0 = σi + 2 ∝ G b r½
(from linear hardening stage of strain hardening
theories)
∝ is a numerical constant = 0.3 to 0.6
Experimental observation: r ∝ 1/d or r = k/d
σ0 = σi + 2 ∝ G b k d-½ = σi + k’ d-½
• Interpretation: influence of grain size on flow stress is
via its influence on dislocation density
Grain Size Effects (contd..)
Hall – Petch Relationship
39
2. Pile up Model
• Dislocations pile up at gbs
• Shear stress at a distance r on either side of the barrier
t = ts (L/r)½
r = distance from head of pile up to nearest dislocation
source in next grain
L = grain dia d
And ts = applied shear stress
Grain Size Effects (contd..)Hall – Petch Relationship
40
ts must first overcome lattice resistance in the first grain, so
when yielding occurs ts= t0 – ti. Therefore
(t0 - ti) (d/r)½ = td
td = shear stress needed to nucleate slip in adjacent grain
• Assuming in general t = σ/2
σ0 = σi + 2τd ( r/d) - ½ = σi + k’ d -½
• k’ : slope of σ0 vs d-½ curve
It does not vary significantly with temperature.
Interpreted as a measure of the stress needed to unpin
dislocation sources locked by solutes
• σi : intercept of σ0 vs d-½ curve
measures lattice friction to unlocked dislocations
strongly temperature, strain and impurity content dependent
• Note: Derivation only applicable to pile-ups of larger than 50
dislocations
Hall – Petch Relationship(contd..)
41
Solid Solution Hardening
• Solid Solutions are either Substitutional or Interstitial
• Addition of solute invariably increases strength
• Factors affecting:
1. Relative size factor
ea = 1/a (da/dc)
where a is the interactomic spacing of the alloy and c is
solute concentration
- leads to elastic interaction between dislocations and
solute atoms
Ui = A Sin q / r
42
2. Relative modulus factor
eG‘ = eG /(1 eG /2)
where eG = 1/G (dG /dc)
o Fleischer: dt /dc varies linearly with eG‘ 3ea
3. Electrical interaction:
o Electron cloud resists compression
o Electrons tend to move from regions of compression to
those of tension at edge dislocations → Electrical dipole
o Monovatent solvent + polyvalent solute: extra electrons
tend to wander away leaving excess positive charge at the
impurity → short range electrostatic interaction between
solute atoms and dislocations
Solid Solution Hardening (contd..)
43
Solid Solution Hardening (contd..)
4. Chemical interaction (Suzuki Locking):
o Dissociation of dislocs affects periodic arrangement of lattice
o Change in free energy with solute conc ( dF/dc) not the same in matrix and failed regions→ interaction between extended dislocations and solute atoms
5. Configurational interaction (Fischer Effect):
o Usually either short range order or clustering in solid solns
(A-B or B-B bonds predominating respectively)
o When dislocation moves through SRO, no. of A-B bonds across the slip plane is reduced → raises the energy of the system
o Similarly, in clustering B-B bonds are disturbed
o Increase in flow stress due to decrease in SRO / clustering :
t = g / b where g is an energy term
44
• From above considerations taken together, we get
t0 = 2.5 G ea4/3 c
for very dilute solid solutions
→ predicts values much lower than found actually
• In solns with long range order (anti-phase domain boundaries) stress required to move a dislocation is
t0 = g/t
where t is anti-phase boundary width and g is its energy
• Ordered alloys with small domain size (≈ 5 nm) are stronger than disordered alloys
Solid Solution Hardening (contd..)
45
Second Phase StrengtheningFine Particles
DS (dispersion
hardening)
PH (precipitation hardening)
Finely dispersed
insoluble second phase
Finely dispersed soluble
second phase
Solubility at HT low Good solubility at HT but
decreasing with decreasing T
No coherency Coherency
Limitless variety by pm Some selected systems only
Resist recovery and
grain growthPpts grow and dissolve at HT
46
Second Phase Strengthening - Fine Particles
• Fine particles act as
barriers to dislocations in
several ways:
• Particles are cut by the
dislocation
• If particles resist cutting,
dislocations have to
bypass them.
• Critical parameter in
deciding this is inter-
particle spacing where f =
vol fraction of particles
and r = radius of particles 47
Cutting Mechanism
• When particles are soft and/or small
• Strengthening occurs due to 5 reasons
1. Strain field due to mismatch between the particle and the matrix (coherency)
where ε is a measure of the strain field
232
1
)(6es
b
rfG
Schematic of Zones
giving rise to coherency
strains: (a) Small solutes
(b) Large solutes
Second Phase Strengthening - Fine Particles
48
2. Difference in SFE
where K(∝) = partial dislocation separating force X
separation
F1 = complex function of w & r
w = width of step formed
3. Chemical hardening: formation of step of width w on
either side of particle increases surface area
where gs = energy of particle–matrix interface
Note:If the particle is ordered, additional hardening occurs,
given by
r
f sg
s
62
21
21
23
)(22
frbEb
f appp g
gs
32
1})ln()(3
{)(
fFE
nmK
bC
pm ggs
Second Phase Strengthening - Fine Particles
49
4. Difference between elastic moduli : (influences line tension of dislocation)
where E1 = Elastic modulus of soft phase; E2 = E of hard phase
5. Difference in Peierls stress between particle and matrix:
where σp & sm are strength of particle & matrix resply.
• Summation of the above five contributions leads to a strength increase that increases with particle size but the exact method of combining the different effects is not very clear.
• Strain softening predominates when this mechanism operates
)(5
221
21
31
mp
bG
rfsss
21
2
2
2
1 )1(8.0
E
EGb
s
Second Phase Strengthening - Fine Particles
50
Bypassing Mechanism
• When the particle is large, difficult for dislocation to cut through. Instead it bypasses the particle by the Orowan mechanism
• Yield strength determined by the strength required to bow the dislocation through the particles.
• When dislocation has reached its minimum curvature,
2R = λ
Thus
Second Phase Strengthening - Fine Particles (contd..)
51
• A dislocation loop is left around each particle.
• Every dislocation passing along will add one loop
• These loops exert a back pressure on dislocation sources which must be overcome for additional slip to take place.
• This causes rapid strain hardening
• Basic Orowan equation has been modified by Ashby as follows:
• Rods and plates strengthen about half as much as spherical particles.
Second Phase Strengthening - Fine Particles (contd..)
52
Strain Hardening
Prediction of strain hardening behavior
• Requires accurate knowledge of how dislocation density
and distribution change with plastic strain
• Parameters extremely sensitive to crystal structure,
stacking fault energy, temperature and strain rate of
deformation etc.
• Thus no unified theory of work hardening!!!
53
Strain Hardening (contd..)
• Shear stress for slip increases with increasing shear
strain
• 100% increase in flow stress from strain hardening not
unusual in single crystals
• Caused by dislocation interactions with other
dislocations and with barriers that impede their motion
• Difficult to mathematically specify group behavior of
dislocations
• Important observation: ρ increases with strain
well annealed 105 to 106/cm²
cold worked 1010 to 1012/cm²
54
1. Pile – Up Theory (Seeger and
Friedel)
• Earliest explanation for strain
hardening
• Rapid hardening in stage II from
dislocations piled up at barriers
in the crystal
• Produce a back stress which
acts opposite to the applied
stress on the slip plane and
opposes motion of additional
dislocations along the slip plane
• Stress concentration on the
leading dislocation in the pile up
• Can cause yielding on the other
side of the barrier or nucleate a
crack at the barrier
Theories of Strain Hardening
• Barriers: grain boundaries, precipitate particles, foreign atoms, sessile dislocations (eg., Lomer -Cottrell barrier)
• If crystal now stressed in opposite direction, flow stress is lowered since back stress will aid the applied stress (Bauschinger effect ), fact verified by experiment
55
2. Dislocation Intersection Mechanism (Mott)
• Dislocations moving in the slip plane cut through other dislocations
intersecting the active slip plane.
• The latter collectively called as a dislocation forest and individually as
trees
• The dislocation intersection results in the formation of jogs.
• Screw dislocations normally can overcome barriers by cross slip.
• However if jogs are present their motion is impeded and can lead to
the formation of vacancies and interstitials if jogs are forced to more
non conservatively.
• Requires increased expenditure of energy, so strain hardening.
• Both the above mechanisms could be operating simultaneously.
• Relative contribution from each can be found by the temperature and
strain rate dependence of strain hardening
• Hardening due to Pile-Ups much less dependent on temperature and
strain rate than that due to dislocation cutting
Theories of Strain Hardening (contd..)
56
57
Part – 6: The Proof
Was it all real?
58
Brief Review of
Part – 1: The Etchpit
Enter the Dislocation
59
Part – 2: The Prism Loop
Going in deeper
60
Part 3: The Partials
Getting Real
Willing Surrender
61
Part 4: The Yield Point
The Climax
62
Part 5: Strengthening
63