Plastic Deformation Elementary Dislocation Theorydislocations / cm², the total strain produced when...

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


• 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


• 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


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

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


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


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


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


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


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

Plastic Deformation Elementary Dislocation Theorydislocations / cm², the total strain produced when...

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