# CHAPTER 6: MECHANICAL PROPERTIES

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Chapter 6
-

Stress

and
strain
: What are they and why are

Elastic

behavior: When loads are small, how much

deformation occurs? What materials deform least?

Plastic

behavior: At what point do dislocations

cause permanent deformation? What materials are

most resistant to permanent deformation?

1

Toughness

and
ductility
: What are they and how

do we measure them?

CHAPTER 6:

MECHANICAL PROPERTIES

Chapter 6
-

INTRODUCTION (I)

The need for

standardized language for expressing
mechanical properties of materials:

STRENGTH, HARDNESS, DUCTILITY, and
STIFFNESS

standardized test methods:

American Society for Testing and Materials
Standards and others…

Chapter 6
-

INTRODUCTION (II)

Courtesy of Plastics Technology Laboratories, Inc
50 Pearl Street, Pittsfield, MA 01201

The result of mechanical testing is
generally a response curve or a (set of)
number(s), in this case a STRESS vs.
STRAIN curve

Chapter 6
-

Basic Concepts of Stress and Strain

Need to compare load on specimens of various size
and shapes:

For tension and compression

Engineering Stress,
σ

= F / A
0

, where
perpendicular to speciment crosssection

and
A
0

is cross
-
sectional area (perpendicular to

the force)
before
application of

Engineering Strain,
ε

=
Δ
l

/ l
0

( x 100 %), where
Δ
l

change in
length, lo is the original length.

These definitions of stress and strain allow one to

compare
test results for specimens of different cross
-
sectional

area A
0

and of different length l
0
.

Chapter 6
-

Basic Concepts of Stress and Strain

Need to compare load on specimens of various size
and shapes:

For tension and compression

Engineering Stress,
σ

= F / A
0

, where
perpendicular to speciment crosssection

and
A
0

is cross
-
sectional area (perpendicular to

the force)
before
application of

Engineering Strain,
ε

=
Δ
l

/ l
0

( x 100 %), where
Δ
l

change in
length, lo is the original length.

For shear

Shear Stress,
τ

= F / A
0

, where
parallel

to
upper and lower
specimen
faces of area
A
0
.

Shear Strain,
γ

=
tan
θ

( x 100 %), where
θ

is the strain angle
.

These definitions of stress and strain allow one to

compare test results for
specimens of different crosssectional

area A
0

and of different length l
0
.

Chapter 6
-

4

Tensile

stress,
s
:

Shear

stress,
t
:

s

F
t
A
o
original area
Stress has units:

N/m
2

or lb/in
2

ENGINEERING STRESS

Chapter 6
-

8

Tensile

strain:

Lateral

strain:

Shear

strain:

/2

/2

/2 -

/2

/2

/2

L
/2

L
/2
L
o
w
o

= tan

Strain is always

dimensionless.

ENGINEERING STRAIN

Applied

Resulting

Chapter 6
-

5

Simple

tension: cable

Simple

shear: drive shaft

o
t

F
s
A
Note:
t

=
M
/
A
c
R

here.

Ski lift

(photo courtesy P.M. Anderson)

COMMON STATES OF STRESS

o
s

F
A
A

o

= cross sectional

F

F

Note:
σ

> 0 here !

Chapter 6
-

Canyon Bridge, Los Alamos, NM
6

Simple

compression:

Note: compressive

structure member

(
s

< 0 here).

(photo courtesy P.M. Anderson)

(photo courtesy P.M. Anderson)

OTHER COMMON STRESS STATES (1)

A

o

Balanced Rock, Arches

National Park

Chapter 6
-

7

Bi
-
axial

tension:

Hydrostatic

compression:

Fish under water
Pressurized tank

s
z
> 0
s

> 0
s

< 0

h

(photo courtesy

P.M. Anderson)

(photo courtesy

P.M. Anderson)

OTHER COMMON STRESS STATES (2)

Chapter 6
-

7

State of stresses in college life
:

s

< 0

h

OTHER COMMON STRESS STATES (3)

σ
1
, classes

σ
2
, family

σ
3
, friends, etc…

σ
4
, daily challenges, etc…

Chapter 6
-

Typical tensile specimen

9

Other types of tests:

compression: brittle materials (e.g.,
concrete)

torsion: cylindrical tubes, shafts.

hardness: surfaces of metals, ceramics

Typical tensile test
machine

Callister 6e.

Callister 6e.

(Fig. 6.3 is taken
from H.W. Hayden, W.G. Moffatt, and J. Wulff,
The
Structure and Properties of Materials
, Vol. III,
Mechanical Behavior
, p. 2, John Wiley and Sons, New
York, 1965.)

SIMPLE STRESS
-
STRAIN TESTING

gauge

length

(portion of sample with

reduced cross section)

=

Chapter 6
-

Stress
-
Strain Testing

• Typical tensile test

machine

Callister 7e.

(Fig. 6.3 is taken from H.W.
Hayden, W.G. Moffatt, and J. Wulff,
The Structure and Properties of
Materials
, Vol. III,
Mechanical Behavior
, p. 2, John Wiley and Sons,
New York, 1965.)

specimen

extensometer

• Typical tensile

specimen

Fig. 6.2,

Callister 7e.

gauge

length

Chapter 6
-

Other Types of Application of Load

Chapter 6
-

How does deformation take place in
the material at an atomic scale ?

Two types of deformation :

Elastic

Reversible, no change in the shape and the size of
the specimen when the load is released !

When under load volume of the material changes !

Plastic

Irreversible, dislocations cause slip, bonds are

original size and shape, but volume is preserved !

Chapter 6
-

STRESS
-
STRAIN CURVE

STRESS

STRAIN

REGION I

REGION II

HARDENING OCCURS

DISLOCATION MOTION

AND GENERATION !

REGION III

Region I : Elastic Deformation

Hooke’s Law

Region II: Uniform Plastic Deformation

Strain is uniform across material

Region III: Non
-
uniform Plastic Deformation

Deformation is limited to “neck” region

σ
YIELD

σ
UTS

E

ε
YIELD

ε
UTS

σ
FAILURE
or
σ
FRACTURE

l
0

l
0
+ l
e

l
0
+ l
e

+ l
p

Necking starts

Chapter 6
-

F

bonds
stretch
initial
2

1. Initial

Elastic means
reversible
!

Bonds stretch and but
released.

ELASTIC DEFORMATION

Chapter 6
-

Modulus of Elasticity, E
:

(also known as Young's modulus)

10

Hooke's Law (Linear)
:

s

=
E

e

Poisson's ratio,
n
:

metals:
n

~ 0.33

ceramics: ~0.25

polymers: ~0.40

Units:

E: [GPa] or [psi]

n
: dimensionless

LINEAR

ELASTIC PROPERTIES

e

L

e

1

-

n

e

e

L

F

F

simple

tension

test

Chapter 6
-

NON
-
LINEAR

ELASTIC PROPERTIES

Some materials will exhibit a non
-
linear elastic behavior
under stress ! Examples are polymers, gray cast iron,
concrete, etc…

Chapter 6
-

Linear Elastic Deformation (Atomic
Scale)

Chapter 2: Inter
-
atomic Bonding ! Young’s Modulus
α

(dF/dr) at r
o

, what else ?

If we increase temperature, how will E behave ?

Chapter 6
-

12

0.2
8
0.6
1
Magnesium,
Aluminum
Platinum
Silver, Gold
Tantalum
Zinc, Ti
Steel, Ni
Molybdenum
G
raphite
Si crystal
Glass
-
soda
Concrete
Si nitride
Al oxide
PC
Wood( grain)
AFRE( fibers)
*
CFRE
*
GFRE*
Glass fibers only
Carbon
fibers only
A
ramid fibers only
Epoxy only
0.4
0.8
2
4
6
10
2
0
4
0
6
0
8
0
10
0
2
00
6
00
8
00
10
00
1200
4
00
Tin
Cu alloys
Tungsten
<100>
<111>
Si carbide
Diamond
PTF
E
HDP
E
LDPE
PP
Polyester
PS
PET
C
FRE( fibers)
*
G
FRE( fibers)*
G
FRE(|| fibers)*
A
FRE(|| fibers)*
C
FRE(|| fibers)*
Metals

Alloys

Graphite

Ceramics

Semicond

Polymers

Composites

/fibers

E(GPa)

10
9

Pa
Based on data in Table B2,

Callister 6e
.

Composite data based on

reinforced epoxy with 60 vol%

of aligned

carbon (CFRE),

aramid (AFRE), or

glass (GFRE)

fibers.

YOUNG’S MODULI: COMPARISON

Chapter 6
-

3

1. Initial

Plastic means
permanent
!

F

linear
elastic
linear
elastic

plastic
PLASTIC DEFORMATION (METALS)

Chapter 6
-

14

• Simple tension test:

(at lower temperatures, T < T
melt
/3)

PLASTIC (PERMANENT) DEFORMATION

Chapter 6
-

YIELD STRENGTH,
s
y

Some materials do NOT exhibit a distinct transition from elastic to plastic region
under stress, so by convention a straight line is drawn parallel to the stress strain
curve with 0.2 % strain. The stress at the intersection is called the yield stress !

Chapter 6
-

• An increase in
s
y

due to plastic deformation.

22

• Curve fit to the stress
-
strain response:

HARDENING

Chapter 6
-

16

Room T values

s
y(ceramics)

>>
s
y(metals)

>>
s
y(polymers)
Based on data in Table B4,

Callister 6e
.

a = annealed

hr = hot rolled

ag = aged

cd = cold drawn

cw = cold worked

qt = quenched & tempered

YIELD STRENGTH: COMPARISON

Chapter 6
-

17

• Maximum possible engineering stress in tension.

• Metals:

occurs when noticeable
necking

starts.

• Ceramics:

occurs when
crack propagation

starts.

• Polymers:

occurs when
polymer backbones

are

Callister 6e.

TENSILE STRENGTH, TS

NECKING

FRACTURE

Chapter 6
-

18

Room T values

TS
(ceram)

~
TS
(met)

~
TS
(comp)
>>
TS
(poly)
Based on data in Table B4,

Callister 6e
.

a = annealed

hr = hot rolled

ag = aged

cd = cold drawn

cw = cold worked

qt = quenched & tempered

AFRE, GFRE, & CFRE =

aramid, glass, & carbon

fiber
-
reinforced epoxy

composites, with 60 vol%

fibers.

TENSILE STRENGTH: COMPARISON

Chapter 6
-

• Plastic tensile strain at failure:

19

Note: %AR and %EL are often comparable.

--
Reason: crystal slip does not change material volume.

--
%AR > %EL possible if internal voids form in neck.

Callister 6e.

DUCTILITY, %EL

Chapter 6
-

Mechanical Strength of Materials

Yield Strength, Tensile Strength and Ductility can be improved by alloying, heat and
mechanical treatment, but Youngs Modulus is rather
insensitive

to such processing !

Temperature effects : YS, TS and YM decrease with increasing temperature, but
ductility increases with temperature !

Chapter 6
-

• Energy to break a unit volume of material

• Approximate by the area under the stress
-
strain

curve.

20

smaller toughness-
unreinforced
polymers
Engineering tensile strain,
e
E
ngineering
tensile
stress,
s
smaller toughness (ceramics)
larg
er toughness
(metals, PMCs)
TOUGHNESS & RESILIENCE

RESILIENCE is energy stored in the material w/o plastic deformation ! U
r

=
σ
y
2

/ 2 E

TOUGHNESS is total energy stored in the material upon fracture !

Chapter 6
-

Resilience,
U
r

Ability of a material to store energy

Energy stored best in elastic region

If we assume a linear
stress
-
strain curve this
simplifies to

Callister 7e.

y

y

r

2

1

U

e

s

@

e
e
s

y
d
U
r
0
Chapter 6
-

TRUE STRESS & STRAIN

σ
T

=
σ

(1+
ε

)

ε
T

= ln (1+
ε
)

The material does NOT get weaker past M

Chapter 6
-

• Resistance to permanently indenting the surface.

• Large hardness means:

--
resistance to plastic deformation or cracking in

compression.

--
better wear properties.

21

Callister 6e.

(Fig. 6.18 is adapted from G.F. Kinney,
Engineering Properties

and Applications of Plastics
, p. 202, John Wiley and Sons, 1957.)

HARDNESS

Chapter 6
-

Hardness: Measurement

Rockwell

No major sample damage

Each scale runs to 130 but only useful in range
20
-
100.

Major load 60 (A), 100 (B) & 150 (C) kg

A = diamond, B = 1/16 in. ball, C = diamond

HB = Brinell Hardness

TS

(psia) = 500 x HB

TS
(MPa) = 3.45 x HB

Chapter 6
-

Hardness: Measurement

Table 6.5

Chapter 6
-

HARDNESS !!

1.
Relatively simple and cheap
technique

2.
Non
-
destructive

3.
Related to many other
mechanical properties

Chapter 6
-

Variability in Material Properties

Elastic modulus is material property

Critical properties depend largely on sample flaws
(defects, etc.). Large sample to sample variability.

Statistics

Mean

Standard Deviation

2
1
2
1

n
x
x
s
i
n
n
x
x
n
n

where
n

is the number of data points

Chapter 6
-

• Design uncertainties mean we do not push the limit.

Factor of safety,
N

N
y
working
s

s
Often
N

is

between

1.2 and 4

• Example:

Calculate a diameter,
d
, to ensure that yield does

not occur in the 1045 carbon steel rod below. Use a

factor of safety of 5.

Design or Safety Factors

4
000
220
2
/
d
N
,

5

N
y
working
s

s
1045 plain

carbon steel:

s

y

= 310 MPa

TS
= 565 MPa

F

= 220,000N

d

L

o

d

= 0.067 m = 6.7 cm

Chapter 6
-

Chapter 6
-

Stress

and
strain
: These are size
-
independent

measures of load and displacement, respectively.

Elastic

behavior: This reversible behavior often

shows a linear relation between stress and strain.

To minimize deformation, select a material with a

large elastic modulus (E or G).

Plastic

behavior: This permanent deformation

behavior occurs when the tensile (or compressive)

uniaxial stress reaches
s
y
.

24

Toughness
: The energy needed to break a unit

volume of material.

Ductility
: The plastic strain at failure.

Note: For materials selection cases related to
mechanical behavior, see slides 22
-
4 to 22
-
10.

SUMMARY

Chapter 6
-

Reading: Chapter 6 and Chapter 7

Homework :

Example problems: 6.1, 6.2, 6.3

Due date:
27
-
04
-
20
11

0

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