CHAPTER 6: MECHANICAL PROPERTIES

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Chapter 6
-

ISSUES TO ADDRESS...


Stress

and
strain
: What are they and why are


they used instead of load and deformation?


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
F is load applied
perpendicular to speciment crosssection

and
A
0

is cross
-
sectional area (perpendicular to

the force)
before
application of
the load.


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
F is load applied
perpendicular to speciment crosssection

and
A
0

is cross
-
sectional area (perpendicular to

the force)
before
application of
the load.


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
F is load applied
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
before loading
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

Area (when unloaded)

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

Adapted from Fig. 6.2,


Callister 6e.


Adapted from Fig. 6.3,
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

Adapted from Fig. 6.3,
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

Adapted from
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
broken, new bonds are made.


When load is released, specimen does not return to
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
return to
initial
2

1. Initial

2. Small load

3. Unload

Elastic means
reversible
!

Bonds stretch and but
recover when load is
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

No load

Under Load

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

2. Small load

3. Unload

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


aligned and about to break.

Adapted from Fig. 6.11,
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.


Adapted from Fig. 6.13,
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

Adapted from Fig. 6.15,
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

Adapted from Fig. 6.18,
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.


Minor load 10 kg


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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