Plastic deformation and

creep in

crystalline materials

Chap. 11

Lec 26 26/03/2014

Mechanical Properties of Materials

Stiffness

Strength

ductility

Toughness

Resistance to elastic deformation

Young’s modulus

Resistance to plastic deformation

Yield stress

Resistance to fracture Energy to fracture

Ability to deform plastically

Strain to fracture

Uniaxial Tensile Test (Experiment 6)

Gaugelength

specimen

Result of a uniaxial tensile test

Slope = Young’s modulus (Y)

UTS

Ultimate tensile strength

yYield strength

(Engineering stress)

(engineering strain)

f (strain to fracture)

necking

Area = Toughness

elas

tic

plastic

break

Yield point

STIFFNESS

STRENGTH

DUCTILITY

If there is a smooth transition from elastic to plastic region (no distinct yield point)

then 0.2 % offset proof stress is used

During uniaxial tensile test the length of the specimen is continually increasing and the cross-sectional area is decreasing.

True stress ≠ Engineering stress (=F/A0)

True strain ≠ Engineering strain (=L/L0)

True stress

iT A

P Ai = instantaneous area

True incremental strain L

dLd T

0

ln0

L

L

L

dLL

L

T True strain

Eqn. 11.3

Eqn. 11.4

nTT K

K Strength coefficient

n work hardening exponent

Eqn. 11.5

What happens during plastic deformation?

• Externally, permanent shape change begins at sy

• Internally, what happens?

What happens to crystal structure after plastic deformation?

?Plastic

Deformation

Some Possible answers

Remains the

same

Changes to

another

crystal

structure

Becomes random

or

amorphous

How Do We Decide?

X-ray diffraction

No change in crystal structure!

No change in internal crystal structure but change in external shape!!

How does the microstructure of polycrystal changes during plastic deformation?

EXPERIMENT 5

Comparison of undeformed Cu and deformed Cu

Slip Lines

Before Deformation After Deformation

Slip lines in the microstructure of

plastically deformed Cu

Callister

Experiment 5

Slip

Slip Planes, Slip Directions, Slip Systems

Slip Plane: Crystallographic planes

Slip Direction: Crystallographic direction

Slip System: A combination of a slip plane and a slip direction

Lec 27 28/03/2014

Slip Systems in Metallic Crystals

Crystal Slip Slip Slip Plane Direction Systems

FCC {111} <110> 4x3=12(4 planes) (3 per plane)

BCC {110} <111> 6x2=12

(6 planes) (2 per plane)

HCP {001} <100> 3x1=3

(1 plane) (3 per plane)

Why slip planes are usually close packed planes?

Why slip directions are close-packed directions?

Slip Systems in FCC Crystal

x

y

z(111)

Tensile vs Shear Stress

• Plastic deformation takes place by slip

• Slip requires shear stress

• Then, how does plastic deformation take place during a tensile test?

s

ND2

1

s

s: Applied tensile stress

N: Slip plane normal

D: Slip direction

F1: angle between s and N

F2 =angle between s and D

Is there any shear stress on the slip plane in the slip direction due to the applied tensile stress?

F

ND2

f1

F

Area=A

= F/ A

FD = F cos 2

Area = As

As = A cos 1

S

DRSS A

F

1

2

cos

cos

A

F

21 coscos A

F

21 coscos RSS

Resolved Shear stress

F

F

F

F

No resolved shear stress on planes parallel or perpendicular to the stress axis

cos 2 = 0 cos 1 = 0

Plastic deformation recap

No change in crystal structure:

slip

twinning

Slip takes place on slip systems (plane + direction)

Slip planes usually close-packed planes

Slip directions usually close-packed direction

Slip requires shear stress

In uniaxial tension there is a shear component of tensile stress on the slip plane in the slip direction:

RESOLVED SHEAR STRESS

21 coscos RSSRSS

21 coscos

y

CRSS

CRITICAL RESOVED SHEAR STRESS

21 coscos yCRSS

s

ND21

s

Lec 28 01.04.2014

21 coscos yCRSS

If we change the direction of stress with respect to the slip plane and the slip direction cos 1 cos 2 will change.

1. CRSS changes.

To maintain the equality which of the following changes takes place?

2. y changes

Schmid’s Law: CRSS is a material constant.

Anisotropy of Yield Stress

21coscos crss

y

Yield stress of a single crystal depends upon the direction of application of load

cos 1 cos 2 is called the Schmid factor

RSS

bb

2coscos 1

y

CRSS

Active slip system

21 coscos yCRSS

aa

2coscos 1

Slip system with highest Schmid factor is the active slip system

Magnitude of

Critical Resolved Shear Stress

Theory (Frenkel 1926)

Experiment

b

d

CRSS

Shear stress

b/2 b

Potential energy

Fe (BCC)

Cu (FCC)

Zn (HCP)

Theory

(GPa)

12

7

5

Experiment

(MPa)

15

0.5

0.3

Ratio

Theory/Exp

800

14,000

17,000

Critical Resolved Shear Stress

?

1934

E. Orowan Michael Polanyi

Geoffrey Ingram Taylor

Solution

Solution

• Not a rigid body slip

• Part slip/ part unslipped

Slip Not-yet-slipped

Boundary between slipped and unslipped parts

on the slip plane

Dislocation Line (One-Dimensional Defect)

Movement of an Edge Dislocation

From

W.D. Callister

Materials Science

and Engineering

Plastic Deformation Summary

• Plastic deformation slip

• Slip dislocations

• Plastic deformation requires movement of dislocations on the slip plane

Recipe for strength?

Remove the dislocation

700

50

Stress, MPa

strain

Cu Whiskers tested in tension

Fig. 11.6

Lec 29 02.04.2014

crystal structure changes?

Mechanism of deformation

No

slip

Nature of stress required for slip Shear stress

Is there shear during tension? Resolved shear stress

Resolved shear stress required for initiating slip

Critical resolved shear stress

CRSS is independent of the direction of application of tensile stress

Lec 29 02.04.2014

Effect of temperature on dislocation motion

Higher temperature makes the dislocation motion easier

WFe S

i

Al2O3

Ni

Cu

18-8 ss

Yie

ld s

tres

s

T/Tm0 0.7

Fig. 11.8

Eqn. 11.14

11.15

11.16

11.17

11.18

Recipe for strength

Remove the dislocation: Possible but Impractical

Alternative:

Make the dislocation motion DIFFICULT

Strengthening Mechanisms

• Strain hardening

• Grain refinement

• Solid solution hardening

• Precipitation hardening

Movement of an Edge Dislocation

A unit slip takesplace only whenthe dislocation

comes out of thecrystal

During plastic deformation dislocation density

of a crystal should go down

Experimental Result

Dislocation Density of a crystal actually goes up

Well-annealed crystal: 1010 m-2

Lightly cold-worked: 1012 m-2

Heavily cold-worked: 1016 m-2?

Dislocation Sources

F.C. Frank and W.T. Read

Symposium on

Plastic Deformation of Crystalline SolidsPittsburgh, 1950

A

B

P

Q

b

b

b

http://zig.onera.fr/~douin/index.html

b

http://zig.onera.fr/~douin/index.html

bb

Fig. 11.9

Problem 11.11

Strain Hardening or Work hardening

Strain, e

sy

sy

During plastic deformation dislocation density increases.

Dislocations are the cause of weakness of real crystals

Thus as a result of plastic deformation the crystal should weaken.

However, plastic deformation increases the yield strength of the crystal: strain hardening or work hardening

?

Dislocation against Dislocation

A dislocation in the path of other dislocation can act as an obstacle to the motion

of the latter

Strain Hardening

]110[2

1

)111(

]110[2

1

)111(

)001(

]011[2

1

]110[2

1

]011[2

1

Sessile dislocation in an FCC crystal

Eqn. 11.20

222

222 aaa

]110[

]110[2

1

(001) not a favourable slip plane (CRSS is high).

The dislocation immobile or sessile.

Energetically favourable reaction

Fig. 11.10

)111(

)111(

Sessile dislocation a barrier to other dislocations creating a dislocation pile-up

Piled up dislocations

Sessile dislocation (barrier)

Fig. 11.10

Lec 30 04.04.2014

Empirical relation for strain hardening or work hardening

A 0

Is the shear stress to move a dislocation in a

crystal with dislocation density

o and A : empirical constants

Eq. 11.21

Fig. 11.11

Dislocation Motion Plastic Deformation

Difficult

Dislocation Motion

Difficult

Plastic Deformation

Strong Crystal

Easy Dislocation Motion Easy Plastic Deformation

Weak Crystal

Grain Boundary

Grain1

Grain 2

Grain boundary

2-D Defect: Grain Boundaries

Single Crystal Polycrystal

No Grain BoundariesGrains of different orientations separated by grain boundaries

Discontinuity of a slip plane across a grain boundary

Disloca-tion

Slip plane

Grain Boundary

Grain Boundary Strengthening

• Slip plane discontinuity at grain boundary

• A dislocation cannot glide across a grain boundary

• Higher stresses required for deformation• Finer the grains, greater the strength

Coarse Grains Fine Grains

Grain Size Strengthening

Hall-Petch Relation

ykD

0

sy: yield strengthD: average grain diameters0, k: constants

Science 5 April 2002: Vol. 296 no. 5565 pp. 66-67 POLYCRYSTALLINE MATERIALSGrain Boundaries and Dislocations

The hardness of coarse-grained materials is inversely proportional to the square root of the grain size. But as Van Swygenhoven explains in her Perspective, at nanometer scale grain sizes this relation no longer holds. Atomistic simulations are providing key insights into the structural and mechanical properties of nanocrystalline metals, shedding light on the distinct mechanism by which these materials deform.

Some recent developments

• Mixture of two or more metals• Solute atoms: a zero dimensional defect or

a point defect• Two types:

– 1. Interstitial solid solution– 2. Substitutional solid solution

Solid Solutions

Interstitial Solid Solution

Perfect CrystalDistortion caused by a

large interstitial atom

Substitutional Solid Solution

Small solute atom

Large solute atom

Solute atom: a zero-dimensional point defect

Solid Solution Strengthening

Strains in the surrounding

crystal

Solute atoms

Obstacle to dislocationmotion

Strongcrystal

Alloys stronger than pure metals

Fig 11.13

Solute Concentration (Atom %) →

50

100

150

10 20 30 40

200

0

Matrix = Cu (r = 1.28 Å)Be (1.12)

Si (1.18)

Sn (1.51)

Ni (1.25)

Zn (1.31)

Al (1.43)

(Values in parenthesis are atomic radius values in Å)

Figure: Anandh Subramaniam

Airbus A380 to be launched on October 2007

A shop inside Airbus A380

Alfred Wilm’s Laboratory 1906-1909

Steels harden by quenching

Why not harden Al alloys also by quenching?

time

Wilm’s Plan for hardening Al-4%Cu alloy

Sorry! No increase in hardness.

550ºC

T

Hea

t

Quench

Hold

Check hardness

Eureka ! Hardness

has Increased !!

One of the greatest technological achievements of 20th century

Hardness increases as a function of time: AGE HARDENING

Property = f (microstructure)

Wilm checked the microstructure of his age-hardened alloys.

Result: NO CHANGE in the microstructure !!

Lec 31 09/04/2014

Creep

Lec 32 11/01.2014

As- quenched hardness

Hardness

time

Peak hardness

Overaging

Hardness initially increases: age hardening

Attains a peak value

Decreases subsequently: Overaging

+: solid solution of Cu in FCC Al

: intermetallic compound CuAl2 4

Tsolvus

supersaturated saturated +

FCC FCC Tetragonal

4 wt%Cu 0.5 wt%Cu 54 wt%Cu

Precipitation of in

Stable

unstable Tsolvus

As-quenched

start finsh

+

Aging

TTT diagram of precipitation of in

A fine distribution of precipitates in matrix causes hardening

Completion of precipitation corresponds to peak hardness

-grains

As quenched

-grains +

Aged Peak aged

Dense distribution of fine

overaged

Sparse distribution of coarse

Driving force for coarsening

/ interfacial energy

0.1 1 10 100

hardness

Aging time

(days)

180ºC

100ºC 20ºC

Aging temperature

Peak hardness is less at higher aging temperature

Peak hardness is obtained in shorter time at higher aging temperature

Fig. 9.15

U

I

T Stable

unstable

As-quenched

start finsh

+

Aging

Tsolvus

1

hardness180ºC

100ºC 20ºC

100 ºC

180 ºC

Hardness increases as as a function of time

No change in microstructure - Wilm!

hardness

time

As-quenched hardness

Guinier-Preston Zones, 1938

Numerous fine precipitates form with time

Not visible in optical micrograph

X-Ray Diffraction (XRD) Transmission Electron Microscopy (TEM)

“It seems justifiable at the moment to conclude that the process of age

hardening in this alloys is associated with the segregation of copper atoms on

the (100) planes of the crystal as suggested by C.H. Desch in The

Chemistry of Solids, 1934”

Preston, 1938, “The Diffraction of X-rays by Age-Hardening Aluminium Copper Alloys

Precipitation Hardening

Precipitates are obstacles to the motion of dislocation

Solute atoms Pebbles

Precipitates boulders

Cake with nuts

Age-hardening = Precipitation hardening

Dislocation-precipitate interaction

Dislocation can

1. Either cut through the precipitate particles (small precipitate)

2. Or they can bypass the precipitates

before after

Precipitate cutting

Fig 11.14 a, c

Dislocation bypassing the precipitate

Fig. 11.14 b and dL

b

Movement of one-dimensional defects called dislocations causes plastic

deformation

Obstacles to the movement of dislocations cause

strengthening

Strengthening Mechanisms

Name Obstacle Type

Solid solution hardening Solute atoms (0-D)

Strain hardening Dislocations (1-D)

Grain refinement Grain boundaries (2-D)

Precipitation hardening Precipitates (3-D)

102

Q1: How do glaciers move?

http://rmkilc.wordpress.com/

105

“Genius is one percent

inspiration and ninety-nine

percent perspiration”

-T.A. Edison

Q2: How do bulbs fuse?

2. Electric Bulb

106

Rolls-Royce Plc

Q3: What does the Rolls-Royce plc make?

107

108

Q: What is common to all the three?

Ans: CREEP

1. Glaciers move due to creep of snow.

2. Bulbs fuse due to creep of W filament.

3. Life of jet engine depends of creep of the turbine blades.

Creep

Creep is time dependent plastic deformation at constant load or stress

It is a “high temperature” deformation

mTT 4.0 Tm is the m.p. in K.

Difference between normal plastic deformation and creep ?

CREEP

Fig. 11.15

CreepDislocation climb

Vacancy diffusion

Cross-slip

Grain boundary sliding

Creep Mechanisms of crystalline materials

Cross-slip

In the low temperature of creep → screw dislocations can cross-slip (by thermal activation) and can give rise to plastic strain [as f(t)]

1 2

3

b Slip plane 1

Slip

plan

e 2

Dislocation climb

Edge dislocations piled up against an obstacle can climb to another slipplane and cause plastic deformation [as f(t), in response to stress]

Rate controlling step is the diffusion of vacancies

Diffusional creep

In response to the applied stress vacancies preferentially move from surfaces/interfaces (GB) of specimen transverse to the stress axis to

surfaces/interfaces parallel to the stress axis→ causing elongation This process like dislocation creep is controlled by the diffusion of

vacancies → but diffusional does not require dislocations to operate

Flow of vacancies

Coble creep → low T → Due to GB diffusion

Nabarro-Herring creep → high T → lattice diffusion

Grain boundary sliding

At low temperatures the grain boundaries are ‘stronger’ than the crystalinterior and impede the motion of dislocations

Being a higher energy region, the grain boundaries melt before the crystalinterior

Above the equicohesive temperature grain boundaries are weaker than grain and slide past one another to cause plastic deformation

117

Starter: initiates columnar grains as in Directional Solidification (DS)

Pigtail: a helical channel which gradually eliminates most columnar grains

Single crystal turbine blade

Single crystal blade: best creep resistance

118

Coarser grains -> Less grain boundaries-> Better for creep application

Single Crystal-> No grain boundaries-> Best for creep application

Nanocrystalline materials -> not good for creep applications!

119

Improvements due to blade manufacturing technique:

Show turbine blades

120

Improvements due to engineering design: Blade cooling

Engineering Materials 1: Ashby and Jones

121

NiCrAlY or NiCoCrAlY

Thermal Barrier Coating (TBC)

Ceramic top coat:

Yittria stabilized Zirconia (YSZ)

1. Low thermal conductivity

2. High thermal expansion

3. High M.Pwww.matsceng.ohio-state.edu

Reduction in surface temp 100-300 oC

Operating temp > M.P. (~1300 oC)

Creep Resistant Materials

Higher operating temperatures gives better efficiency for a heat engine

Creep resistance

Dispersion hardening → ThO2 dispersed Ni (~0.9 Tm)

Solid solution strengthening

High melting point → E.g. Ceramics

Single crystal / aligned (oriented) grains

Cost, fabrication ease, density etc. are other factors which determine the final choice of a material

Commonly used materials → Fe, Ni, Co base alloys

Precipitation hardening (instead of dispersion hardening) is not a good method as particles coarsen (smaller particles dissolve and largerparticles grow interparticle separation ↑)

Ni-base superalloys have Ni3(Ti,Al) precipitates which form a lowenergy interface with the matrix low driving force for coarsening

Cold work cannot be used for increasing creep resistance as recrystallization can occur which will produced strain free crystals

Fine grain size is not desirable for creep resistance → grain boundary sliding can cause creep elongation / cavitation► Single crystals (single crystal Ti turbine blades in gas turbine

engine have been used)► Aligned / oriented polycrystals

No Dislocations

Ultra Strong Crystals

Whiskers

Composite Materials

Various Crystal Defects

Disloca-tions

Grain Boundary

G-P zone

Substitu-tional solute

Interstitial solute

Stacking fault

Vacancy (Diffusion)

Moral of the Story

Strength depends upon defects

Microstructure

• Structural features observed under a microscope– Phases and their distribution– Grains and grain boundaries– Twin boundaries– Stacking faults– Dislocations

Hierarchy of Structures

en g in eerin g s tru c tu re

m ac ros tru c tu re

m icros truc ture

c rys ta l s tru c tu re

a tom ic s tru c tu re

n u c lear s tru c tu re

Physics and chemistry

Metallurgy and Materials Science

Engineering: Civil, Mechanical, etc. 1m

1mm

1mm

1nm

1A0

Real Moral of the Story

Properties depend upon microstructure

Structure Sensitive vs

Structure Insensitive Properties

For true understanding comprehension of detail is imperative. Since such

detail is well nigh infinite our knowledge is always

superficial and imperfect.

Duc Franccois de la Rochefoucald(1613-1680)