Chapter 8: Deformation & Strengthening

Mechanisms

ISSUES TO ADDRESS...
• How are dislocations involved in the plastic deformation
of materials?
• Does the crystal structure of a material affect its mechanical
characteristics? If so, how and why?
• How are mechanical properties affected by
dislocation mobilities?
• What techniques are used to increase the
strength/hardness of metals/alloys?
• How are mechanical characteristics of deformed
metal specimens altered by heat treatments?

Chapter 8 - 1
 Plastic Deformation by Dislocation
Motion
• Plastic deformation occurs by motion of dislocations
(edge, screw, mixed) – process called slip
• Applied shear stress can cause extra half-plane of atoms
[and edge dislocation line ( )] to move as follows: Fig. 8.1, Callister &
Rethwisch 5e.

• Atomic bonds broken and reformed along slip plane as

dislocation (extra half plane) moves.
Chapter 8 - 2
 Analogy Between Dislocation
Motion and Caterpillar Locomotion
• Caterpillar locomotion – hump formed and propelled by
lifting and shifting of leg pairs
• Dislocation motion – movement of extra half-plane of
atoms by breaking and reforming of interatomic bonds

Fig. 8.3, Callister & Rethwisch 5e.

Chapter 8 - 3
 Motion of Edge and Screw Dislocations
• Direction of edge disl. line ( ) motion—in direction of
applied shear stress .

Edge dislocation

• Direction of screw disl. line ( ) motion—perpendicular to

direction of applied shear stress.

Screw dislocation

Fig. 8.2, Callister & Rethwisch 5e.

Chapter 8 - 4
 Dislocation Characteristics
Metals

+ + + + + + + +
+ + + + + + + +
+ + + + + + + +
electron cloud ion cores

• Metals:
- Examples: copper, aluminum, iron
- Dislocation motion—relatively easy
- Metallic bonding—non-directional
- Close-packed planes and directions for slip

Chapter 8 - 5
 Dislocation Characteristics
Ceramics

• Ceramics—Covalently Bonded
- Examples: silicon, diamond
- Dislocation motion—relatively
difficult
- Covalent bonding—directional

• Ceramics—Ionically Bonded
+ - + - + - +
- Examples: NaCl, MgO
- Dislocation motion—relatively - + - + - + -
difficult + - + - + - +
- Few slip systems
 motion of nearby ions of like
charge (+ and -) restricted by
electrostatic repulsive forces

Chapter 8 - 6
 Slip Systems
Slip System—Combination of slip plane and slip direction
– Slip Plane
• Crystallographic plane on which slip occurs most
easily
• Plane with high planar density
– Slip Direction
• Crystallographic direction along which slip occurs
most easily
• Direction with high linear density

Chapter 8 - 7
 Slip Systems (cont.)
• For FCC crystal structure – slip system is
– Dislocation motion on planes
– Dislocation motion in directions
– A total of 12 independent slip systems for FCC

direction
Fig. 8.6, Callister &
Rethwisch 5e.

plane

• For BCC and HCP— other slip systems

Chapter 8 - 8
 Slip in Single Crystals
Resolved Shear Stress
• Applied tensile stress—shear stress component
when slip plane oriented neither perpendicular nor
parallel to stress direction ϕ
-- From figure, resolved shear stress, τR λ

• τR depends on orientation of normal to

slip plane and slip direction with direction
of tensile force F:

Fig. 8.7, Callister &

Rethwisch 5e.

Chapter 8 - 9
 Slip in Single Crystals
Resolved Shear Stress (cont.)

• Relationship between tensile stress, σ,

and τR: ϕ
λ

Fig. 8.7, Callister &

Rethwisch 5e.

Chapter 8 - 10
 Slip in Single Crystals:
Critical Resolved Shear Stress
• Dislocation motion—on specific slip system—when R reaches
critical value:
-- “Critical resolved shear stress”, CRSS
-- Slip occurs when R > CRSS
-- Typically 0.1 MPa < CRSS < 10 MPa
• In a single crystal there are
-- multiple slip systems
-- a variety of orientations
• One slip system for which R is highest: R(max) > ⌠ (cos cosϕ
)max
-- Most
• Yield favorably
strength oriented
of single ⌠y, when
slip system
crystal,

Chapter 8 - 11
 Single Crystals
Slip—Macroscopic Scale
• Parallel slip steps form on
surface of single crystal

• Steps result from motion of

large numbers of dislocations
on same slip plane

• Sometimes on single crystals

appear as "slip lines”

Fig. 8.8, Callister &

Rethwisch 5e.

Chapter 8 - 12
 Deformation of Single Crystals
Example Problem
A single crystal of some metal has a crss of 20.7 MPa and is
exposed to a tensile stress of 45 MPa.
(a) Will yielding occur when ϕ = 60° and  = 35°?
(b) If not, what stress is necessary?

Solution:
(a) First calculate τR

Chapter 8 - 13
 Deformation of Single Crystals
Example Problem (cont.)
(b) To calculate the required tensile stress to
cause yielding use the equation:

With specified values

Therefore, to cause yielding,

Chapter 8 - 14
 Slip in Polycrystalline Materials
σ
• Polycrystalline materials—
many grains, often random
crystallographic orientations

• Orientation of slip planes, slip Adapted from Fig.

directions (ϕ, λ)—vary from 8.10, Callister &
Rethwisch 5e.
grain to grain. (Photomicrograph
courtesy of C. Brady,
National Bureau of
• On application of stress—slip Standards [now the
National Institute of
in each grain on most favorable Standards and
Technology,
slip system. Gaithersburg, MD].)

- with largest τR
- when τR > τcrss

• In photomicrograph—note slip
lines in grains have different
orientations. 300 μm

σ
Chapter 8 - 15
 Slip in Polycrystalline Materials (cont.)
• Grains change shape (become distorted)—during plastic
deformation—due to slip
• Manner of grain distortion similar to gross plastic deformation
- Grain structures before and after deformation (from rolling)
- Before rolling—grains equiaxed & randomly oriented
 Properties isotropic
- After rolling (deformation)—grains elongated in rolling direction
 Also preferred crystallographic orientation of grains
 Properties become somewhat anisotropic
- before rolling
- after rolling Adapted from Fig. 8.11,
Callister & Rethwisch 5e.
(from W.G. Moffatt, G.W. Pearsall,
and J. Wulff, The Structure and
Properties of Materials, Vol. I,
Structure, p. 140, John Wiley and
Sons, New York, 1964.)

235 μm rolling direction

Chapter 8 - 16
 Strengthening Mechanisms for Metals
• For a metal to plastically deform—dislocations must move
• Strength and hardness—related to mobility of dislocations
-- Reduce disl. mobility—metal strengthens/hardens
-- Greater forces necessary to cause disl. motion
-- Increase disl. mobility—metal becomes weaker/softer

• Mechanisms for strengthening/hardening metals—

decrease disl. mobility

• 3 mechanisms discussed
-- Grain size reduction
-- Solid solution strengthening
-- Strain hardening (cold working)

Chapter 8 - 17
 Strengthening Mechanisms for Metals
Mechanism I – Reduce Grain Size
• Grain boundaries act as barriers
to dislocation motion
• At boundary
— Slip planes change directions
(note in illustration)
— Discontinuity of slip planes
• Reduce grain size
— increase grain boundary area Fig. 8.14, Callister & Rethwisch 5e.
(From L. H. Van Vlack, A Textbook of Materials
— more barriers to dislocation motion Technology, Addison-Wesley Publishing Co., 1973.
Reproduced with the permission of the Estate of
— increase yield strength, tensile Lawrence H. Van Vlack.)

strength & hardness

• Dependence of σy on average grain diameter, d:

—σ0, ky = material constants

Chapter 8 - 18
Strengthening Mechanisms for Metals
Mechanism II – Solid-Solution Strengthening
• Lattice strains around dislocations
– Illustration notes locations of tensile, compressive
strains around an edge dislocation

Fig. 8.4, Callister

& Rethwisch 5e.
(Adapted from W.G.
Moffatt, G.W. Pearsall,
and J. Wulff, The
Structure and
Properties of
Materials, Vol. I,
Structure, p. 140,
John Wiley and Sons,
New York, 1964.)

Chapter 8 - 19
 Solid Solution Strengthening (cont.)
• Lattice strain interactions with strains introduced by impurity atoms
• Small substitutional impurities introduce tensile strains
• When located above slip line for edge dislocation as shown:
– partial cancellation of impurity (tensile) and disl. (compressive) strains
– higher shear stress required to cause disl. motion

Fig. 8.17, Callister &

Rethwisch 5e.
Chapter 8 - 20
 Solid Solution Strengthening (cont.)
• Large substitutional impurities introduce compressive strains
• When located below slip line for edge dislocation as shown:
– partial cancellation of impurity (compressive) and disl. (tensile) strains
– higher shear stress required to cause disl. motion

Fig. 8.18, Callister &

Rethwisch 5e.

Chapter 8 - 21
VMSE Solid-Solution Strengthening Tutorial

Chapter 8 - 22
 Solid Solution Strengthening (cont.)
• Alloying Cu with Ni increases σy and TS.
• Tensile strength & yield strength increase with wt% Ni.
Tensile strength (MPa)

180

Yield strength (MPa)

Adapted from Fig.
400 8.16 (a) and (b),
Callister &
120 Rethwisch 5e.
300

200 60
0 10 20 30 40 50 0 10 20 30 40 50
wt.% Ni, (Concentration C) wt.%Ni, (Concentration C)

• Empirically,

Chapter 8 - 23
 Strengthening Mechanisms for Metals
Mechanism III – Strain Hardening
• Plastically deforming most metals at room temp. makes
them harder and stronger
• Phenomenon called "Strain hardening (or cold working)”
• Deformation—often reduction in cross-sectional area.
-Rolling
roll
Ad
Ao
roll

• Deformation amt. = percent coldwork (%CW)

Chapter 8 - 24
 Strain Hardening (cont.)
As %CW increases
• Yield strength (σy) increases.
• Tensile strength (TS) increases.
• Ductility (%EL or %AR) decreases.

Adapted from Fig. 8.20,

Callister & Rethwisch 5e.

low carbon steel

Chapter 8 - 25
 Strain Hardening (cont.)
Lattice strain interactions between dislocations

Fig. 8.5, Callister &

Rethwisch 5e.

Chapter 8 - 26
 Strain Hardening (cont.)
Dislocation Density and Cold Working

total dislocation length

Dislocation density =
unit volume
– Dislocation density in undeformed metal
 105-106 mm-2
– Dislocation density increases with increasing deformation
– Dislocation density in deformed (cold-worked) metal
 109-1010 mm-2

Chapter 8 - 27
 Strain Hardening (cont.)
Mechanism of Strain Hardening
• Dislocation structure in Ti after cold working.
• Dislocation density increases
with deformation (cold work) by
formation of new dislocations
• As dislocation density
increases, distance between
dislocations decreases
• On average, disl.-disl. strain
interactions are repulsive
• Dislocation motion hindered by
presence of other dislocations

Fig. 5.12, Callister & Rethwisch 5e.

(Courtesy of M.R. Plichta, Michigan
Technological University.) Chapter 8 - 28
 Affect of Cold Work on Mechanical
Properties
Example Problem:
Compute the yield and tensile strengths, and ductility for a
cylindrical Cu specimen that has been cold worked by
reducing its diameter from 15.2 mm to 12.2 mm.

Copper
Cold
Work

Do = 15.2 mm Dd = 12.2 mm

Chapter 8 - 29
 Example Problem (cont.)

• Solution:

Chapter 8 - 30
 Example Problem (cont.)
• Yield and tensile strength, and ductility (%EL) are
determined graphically as shown below for %CW = 35.6%

60

tensile strength (MPa)

yield strength (MPa)

700 800

ductility (%EL)
40
500 600
300 MPa Cu
300 Cu 400 340 MPa 20
Cu 7%
100 200 00
0 20 40 60 0 20 40 60 20 40 60
% Cold Work % Cold Work % Cold Work

σy = 300 MPa TS = 340 MPa %EL = 7%

Fig. 8.19, Callister & Rethwisch 5e. [Adapted from Metals Handbook: Properties and Selection: Irons
and Steels, Vol. 1, 9th edition, B. Bardes (Editor), 1978; and Metals Handbook: Properties and Selection: Nonferrous
Alloys and Pure Metals, Vol. 2, 9th edition, H. Baker (Managing Editor), 1979. Reproduced by permission of ASM
International, Materials Park, OH.]
Chapter 8 - 31
 Heat Treatment of Cold-Worked
Metal Alloys
• Heat treating cold worked metals brings about changes in structure
and properties
• As a result, effects of cold work are nullified!
• This type of heat treatment sometimes termed “annealing”
• 1 hour treatment at Tanneal decreases tensile strength & increases %EL

Three Annealing stages:

1. Recovery (100-200°C)
600 60 2. Recrystallization (200-
tensile strength (MPa)

tensile strength
500°C)

ductility (%EL)
50
500
3. Grain Growth (> 500°C)
40
Fig. 8.22, Callister & Rethwisch 5e.
(Adapted from G. Sachs and K. R. Van Horn,
400 30 Practical Metallurgy, Applied Metallurgy
and the Industrial Processing of Ferrous and
Nonferrous Metals and Alloys, 1940.
ductility 20 Reproduced by permission of ASM
International, Materials Park, OH.)
300
100 200 300 400 500 600 700
annealing temperature (°C) Chapter 8 - 32
 Recovery
During recovery – reduction in disl. density – annihilation of disl.
• Scenario 1 extra half-plane
of atoms
Dislocation
atoms annihilation-
diffuse half-planes
to regions come together
of tension
extra half-plane
of atoms
• Scenario 2
3. “Climbed” disl. can now
move on new slip plane
2. grey atoms leave by
4. dislocations of opposite
vacancy diffusion
sign meet and annihilate
allowing disl. to “climb”
1. dislocation blocked; Obstacle dislocation
can’t move to the right

Chapter 8 - 33
 Recrystallization
• New grains form that:
-- have low dislocation densities
-- are small in size
-- consume and replace parent cold-worked grains.

Recrystallized grains

Fig. 8.21 (a),(c),

Callister &
Rethwisch 5e.
(Photomicrographs
courtesy of J.E.
Burke, General
Electric Company.)

33%CW brass before heat treatment After 4 sec. at 580°C

Chapter 8 - 34
 Recrystallization (cont.)
• All grains in cold-worked material have been consumed/replaced.

Fig. 8.21 (d),

Callister &
Rethwisch 5e.
(Photomicrograph
courtesy of J.E. Burke,
General Electric
Company.)

After 8 sec. at 580°C

Chapter 8 - 35
 Recrystallization Temperature
TR = recrystallization temperature = temperature
at which recrystallization just reaches
completion in 1 h.
0.3Tm < TR < 0.6Tm

For a specific metal/alloy, TR depends on:

• %CW -- TR decreases with increasing %CW
• Purity of metal -- TR decreases with
increasing purity

Chapter 8 - 36
 Cold Working vs. Hot Working

• Hot working  deformation above TR

• Cold working  deformation below TR

Chapter 8 - 37
 Grain Growth
• Grain growth occurs as heat treatment continues.
-- Average grain size increases
-- Small grains shrink (and ultimately disappear)
-- Large grains continue to grow

Fig. 8.21 (d),(e), Callister

& Rethwisch 5e.
(Photomicrographs courtesy of
J.E. Burke, General Electric
Company.)

After 8 sec. at 580°C After 15 min. at

580°C Chapter 8 - 38
 Grain Size Influences Properties

• Metals having small grains – relatively strong

and tough at low temperatures

• Metals having large grains – good creep

resistance at relatively high temperatures

Chapter 8 - 39
 Grain Growth (cont.)
• Empirical relationship—dependence of average grain
size (d) on heat treating time (t):

material constant
exponent typ. ~ 2 —depends on T
—independent of t

Initial average grain

diam. before heat
treatment

Chapter 8 - 40
Recovery, Recrystallization, & Grain Growth
Summary

TR = recrystallization
temperature
TR annealing time = 1 h

Fig. 8.22, Callister & Rethwisch 8e.

(Adapted from G. Sachs and K. R. Van Horn,
Practical Metallurgy, Applied Metallurgy
and the Industrial Processing of Ferrous and
Nonferrous Metals and Alloys, 1940.
Reproduced by permission of ASM
International, Materials Park, OH.)

º
Chapter 8 - 41
 Design Problem
Description of Diameter Reduction
Procedure
A cylindrical rod of brass originally 10 mm in diameter
is to be cold worked by drawing. The circular cross
section will be maintained during deformation. A cold-
worked tensile strength in excess of 380 MPa and a
ductility of at least 15 %EL are desired. Furthermore,
the final diameter must be 7.5 mm. Explain how this
may be accomplished.

Chapter 8 - 42
 Design Problem (cont.)
Solution:
Brass
First compute the %CW. Cold
Work

D o = 10 mm D d = 7.5 mm

Chapter 8 - 43
 Design Problem Solution (cont.)
60
800

40
600
540

400 20

6
200 0
0 20 40 60 0 20 40 60
% Cold Work % Cold Work Fig. 8.19, Callister
& Rethwisch 5e.

• For %CW = 43.8%

– TS = 540 MPa > 380 MPa
– %EL = 6 < 15
• This doesn’t satisfy criteria… what other options are
possible?
Chapter 8 - 44
 Design Problem Solution (cont.)
60
800

40
600

400 20
380 15

200 0
0 1220 40 60 0 20 27 40 60
% Cold Work % Cold Work Fig. 8.19, Callister
& Rethwisch 5e.

For TS > 380 MPa > 12 %CW

For %EL > 15 < 27 %CW

To meet criteria
—deformation requirement 12 < %CW < 27
Chapter 8 - 45
 Design Problem Solution (cont.)
Procedure: Cold work, anneal, then cold work again.
• To meet criteria, for 2nd deformation step: 12 < %CW < 27
– We will deform to 20%CW
• Diameter after first cold work stage (but before 2nd cold work
stage), Di, calculated as follows:

Intermediate diameter =

Chapter 8 - 46
 Design Problem Summary

Stage 1: Cold work – reduce diameter from 10 mm to 8.39 mm

Stage 2: Heat treat (allow recrystallization)

Stage 3: Cold work – reduce diameter from 8.39 mm to 7.5 mm

Therefore, all criteria satisfied

Chapter 8 - 47
Mechanical Properties of Polymers –
Stress-Strain Behavior
brittle polymer

plastic
elastomer
elastic moduli
– less than for metals Adapted from Fig. 7.22,
Callister & Rethwisch 5e.

• Fracture strengths of polymers ~ 10% of those for metals

• Deformation strains for polymers > 1000%
– for most metals, deformation strains < 10%
Chapter 8 - 48
 Mechanisms of Deformation—Brittle
Crosslinked and Network Polymers
Near σ(MPa) Near
Initial Failure
Initial Failure x brittle failure

x plastic failure

aligned, crosslinked ε network polymer

polymer

Chapter 8 - 49
 Mechanisms of Deformation —
Semicrystalline (Plastic) Polymers
σ(MPa) fibrillar
structure
Inset figures along plastic
x brittle failure
response curve adapted from near
Figs. 8.27& 8.28, Callister & onset of
failure
Rethwisch 5e. necking plastic failure
x

unload/reload

e
crystalline
block segments
separate
undeformed
structure amorphous
crystalline
regions
regions align
elongate
Chapter 8 - 50
 Predeformation by Drawing
• Drawing…(ex: monofilament fishline)
-- stretches the polymer prior to use
-- aligns chains in the stretching direction
• Results of drawing:
-- increases the elastic modulus (E) in the
stretching direction
-- increases the tensile strength (TS) in the
stretching direction Adapted from Fig. 8.28,
-- decreases ductility (%EL) Callister & Rethwisch 5e.

• Annealing after drawing...

-- decreases chain alignment
-- reverses effects of drawing (reduces E and
TS, enhances %EL)
• Contrast to effects of cold working in metals!

Chapter 8 - 51
 Mechanisms of Deformation—
Elastomers
σ(MPa)
x brittle failure Stress-strain curves
adapted from Fig. 7.22,
Callister & Rethwisch 5e.
Inset figures along
elastomer curve (green)
adapted from Fig. 8.30,
plastic failure
x Callister & Rethwisch 5e.
(Fig. 15.15 adapted from Z. D.
Jastrzebski, The Nature and
x Properties of Engineering
Materials, 3rd edition.
elastomer Copyright © 1987 by John
Wiley & Sons, New York.
final: chains Reprinted by permission of
John Wiley & Sons, Inc.)

ε
are straighter,
still
cross-linked
initial: amorphous chains are deformation
kinked, cross-linked. is reversible (elastic)!

• Compare elastic behavior of elastomers with the:

-- brittle behavior (of aligned, crosslinked & network polymers), and
-- plastic behavior (of semicrystalline polymers)
(as shown on previous slides)
Chapter 8 - 52
 Summary
• Plastic deformation occurs by motion of dislocations
• Crystallographic considerations:
-- Minimum atomic distortion from dislocation motion
- in slip planes
- along slip directions

• Deformation of polycrystals—change of grain shapes

• Strength is increased by decreasing dislocation
mobility.
• Strengthening techniques for metals:
-- grain size reduction
-- solid solution strengthening
-- strain hardening (cold working)

Chapter 8 - 53
 Summary (cont.)

• Heat treatment of deformed metal specimens:

-- Processes
- Recovery
- Recrystallization
- Grain growth
-- Consequences—property alterations
- Softer and weaker
- More ductile

Chapter 8 - 54
 ANNOUNCEMENTS
Reading:

Core Problems:

Self-help Problems:

Chapter 8 - 55