Physical metallurgy principles

Chapter 5 Dislocations and Plastic Deformation

Frank – read source* A dislocation generator

The dislocation segment xy may

move in plane ABCD under the

applied stress. Its ends, x and y,

however, are fixed.

Various stages in the generation

of a dislocation loop at a Frank-

Read source.

Nucleation of dislocations If dislocations are not formed by dislocation

generators (Frank-Read source), then they must be created by a nucleation process.

Nucleation process created in two ways:

Homogeneous nucleation: formed in a perfect lattice by the action of a simple stress, no agency other than stress being required.

Heterogeneous nucleation: the dislocations are formed with the help of defects present in the crystal, perhaps impurity particles.

Defects make the formation of dislocations easier by lowering the applied stress required to form dislocations.

That homogeneous nucleation of dislocation requires extremely high stresses.

If dislocations are not formed by Frank-Read sources, then they must be nucleated heterogeneously.

Bend gliding

The bending of crystals can be explained in terms of Frank-Read or other sources.

Elastic deformation: (before the yield point of crystal)

the stress distribution

section-cross theof inertia ofmoment the:

axis neutral thefrom measured distance verticalthe:

moment bending the:

I

y

MI

Myx

The shear stress component parallel to the slip plane.

The sense of the shear stress changes its sign as it crosses the neutral axis.

The shear stress is zero at the neutral axisand a maximum at the extreme ends of the slip plane.

upper: compressive stress; lower: tensile stress

The stress distribution on slip planes corresponding to

the elastic deformation.

positive edge : move toward the surface (high stress region) and disappear

negative edge : move toward the specimen’s neutral

axis(decreasing shear stress)

neutral axis : free of dislocations, not be stressed above the elastic limit, deformation will be elastic and not plastic

Rotational slip

Type of deformation due to dislocations: simple shear, bending, rotational slip.

Rotational slip required more than one set of dislocations(slip plane must contain more than one slip direction)

Ex. FCC , and HCP

An array of parallel screw dislocations A double array of screw dislocations

Critical resolved shear stress Critical resolved shear stress: the yield

stress for crystals of a shear stress resolved on the slip plane and in the slip direction.

plane slip on the resolved stressshear :

plane slip on the stress :

coscoscos

cos

cos

A

n

nA

n

n

sp

nA

sp

n

A

f

A

f

A

f

A

A

•When a crystal possesses several

crystallographically equivalent slip systems,

slip will start first on the system having the

highest resolved shear stress.

FCC slip systems

* Close-packed plane: {111}

* Close-packed directions:<110>

* Slip systems: 4 x 3=12

* Large number of equivalent slip system

a: several slip systems have nearly equal resolved shear stresses.

b: 1: only one slip plane

2: multiple glide on intersecting slip planes

3: decreasing the rate of increase of the dislocation density

---Flow curve for fcc single crystal.

• Stage ⅠPredominantly primary slip (easy slip, easy glide)

Some second slip and ⊥ interaction ⇒ very low strain hardening rate ( 𝑑𝜏𝑑𝛾).

⊥𝑠 glide for large distance with little hinderance.

• Stage Ⅱmultiple slipCottrell- Lomer locks, Scew jogs, and other barriers⊥ pile ups⊥ tangles, formation of ⊥ cells, 𝜌~1012𝑐𝑚−2 at failure.

High 𝑑𝜏𝑑𝛾

• Stage ⅢDecreasing 𝑑𝜏

𝑑𝛾

Cross-slip of pinned ⊥𝑠

For high 𝛾𝑆.𝐹 metals⇒cross-slip is easier.⇒earlier onset of stage Ⅲ

HCP slip system

* Close-packed plane: basal plane (0001)

* Close-packed direction: <11-20>

Titanium (0001) 110

Titanium (10-10) 49

Beryllium (0001) 39

Zirconium (10-10) 6.2

Zinc, cadmium and magnesium possess both a low critical resolved

shear stress and a single primary slip plane (basal plane)

Just for Zn and Cd

BCC slip system

Close-packed direction: <111>

Slip plane (contains a close-packed <111> direction) : {110},{112},{123}

The lack of close-packed plane causes high critical resolved shear stress for slip.

Ex. Fe: 28 MPa

Cross slip Cross-slip: there are two or more slip planes with a

common slip direction occur in crystals.

Only screw dislocation can shift of the dislocation from one plane to another in the cross-slip.

Cross slip for total and extended dislocations

A. a total dislocation can readily cross-slip between a pair of octahedral planes.

B. ½ [-110]1/6[-211] +1/6[-12-1]

C. cross-slip of the leading partial , 1/6[-211] 1/6[-121]+1/6[-1-10]

D. cross-slip of the trailing partial, 1/6[-12-1]+1/6[-1-10]1/6[-21-1]

stair-rod

cross-slip for a total is much easier than extended s due to the formation of

stair-rod .

Work hardening

a point: elastic limit, begin to deform plastically and neck.

The true strength of the metal normally increases with increasing strain until it fractures.

strain gengineerin :

strain true:

stress gengineerin :

stress true:

)1ln(ln

)1(

0

t

t

t

t

l

lA

P

Geometrical instability

<u,

Due to large work hardening capacity,

instability cannot continue.

Localized strain at “weak line”

occurs regularly and repeatly.

>u,

work hardening rate decrease to

a point that instability once formed

continuous to develop.

40

41

42

43

44

Instability in tension

-Necking generally begins at max. load during the tensile deformation of a

ductile metal. An idea plastic material in which no strain hardening occurs

would become unstable in tension and begin to neck just as soon as

yielding took place.

-a real metal undergoes strain hardening, which tends to increase the

load-carrying capacity of the specimen as deformation increases. This

effect is opposed by the gradual decrease in the cross-sectional area of

the specimen as it elongates.

-Necking or localized deformation begins at max load, where the increase

in stress due to decrease in the cross-sectional area becomes greater

than the increase in the load-carrying ability due to strain hardening.

-The condition of instability leading to localized deformation is defined by

the condition dP=0.

45

46

t

t

t

t

tt

Necking criterion (Considère’s criterion)

47

Relationship between dislocation density and the stress

r is the measured dislocation density, k is a constant , and 0 is the stress obtain when is extrapolated to zero

The relationship in terms of the resolved stress on the active slip plane,

Taylor’s relationthe dislocation interaction must be overcome in order to allow the dislocations to continue to glide

k = amb

Observing dislocation in crystal by etching reagent, which forms an etch pit on the surface of a crystal at each point where a dislocation intersects the surface.

The velocity of a dislocation moving under a fixed applied stress = distance that a dislocation moved by the time

Strain and dislocation velocity

V=d/t , v: dislocation velocity ; d: dislocation motion distance ; t: time of application of the stress.

Edge dislocation would normally move 50 times faster than screw dislocation.

kTE

m

efv

Tv

scmv

ex

cm/

DD

v

vv

/

55.16

)(

1

ln :stressconstant

/101.13.5

65.2 :n velocitydislocatio

MPa 2.65 16.5,m MPa, 5.3D:crystal LiF .

sec 1 of

n velocitydislocatio a yields that stress:

stressshear : )(

n velocitydislocatio: lnln

-

•The movement of dislocation is not smooth and continuous,

but rather it occurs in steps.

•Thermal vibrations aid the applied stress to overcome these

these obstacles to dislocation motion.

obstaclean at n waitsdislocatio the timeaverage:

obstaclesbetween flight of time:

obstaclesbetween distance average the:

:n velocitydislocatio

w

f

wwf

t

t

l

t

l

tt

lv

V

nldensity n dislocatio:

V

xbnl

crystal theofheight :z ; :strain shear Az

Ab

passedn dislocatio offraction the:A

movesn dislocatio of distance:x tor;burger vec:

)/()/(:amount sheared

ex.

rate.strain applied and

ndislocatio of velocity ebetween th iprelationsh :equationOrowan

vbt

xb

t

xb

V

Ab

b

AAbxxb

rr

r

r