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Chapter :6
MECHANICAL
PROPERTIES

Chapter Outline
 Terminology for Mechanical Properties
 The Tensile Test: Stress-Strain Diagram
 Properties Obtained from a Tensile Test
 True Stress and True Strain
 The Bend Test for Brittle Materials
 Hardness of Materials

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Questions to Think About
• 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?
• Toughness and ductility: What are they and how
do we measure them?
• Ceramic Materials: What special provisions/tests
are made for ceramic materials?

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Stress-Strain Test
specimen
machine

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Tensile Test

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Important Mechanical Properties
from a Tensile Test
• Young's Modulus: This is the slope of the linear
portion of the stress-strain curve, it is usually
specific to each material; a constant, known value.
• Yield Strength: This is the value of stress at the
yield point, calculated by plotting young's modulus
at a specified percent of offset (usually offset =
0.2%).
• Ultimate Tensile Strength: This is the highest
value of stress on the stress-strain curve.
• Percent Elongation: This is the change in gauge
length divided by the original gauge length.

Terminology
 Load - The force applied to a material during
testing.
 Strain gage or Extensometer - A device used for
measuring change in length (strain).
 Engineering stress - The applied load, or force,
divided by the original cross-sectional area of the
material.
 Engineering strain - The amount that a material
deforms per unit length in a tensile test.

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F

bonds
stretch
return to
initial
1. Initial 2. Small load 3. Unload
Elastic means reversible.
Elastic Deformation

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1. Initial 2. Small load 3. Unload
Plastic means permanent.
F

linear
elastic
linear
elastic
plastic
Plastic Deformation (Metals)

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Typical stress-strain
behavior for a metal
showing elastic and
plastic deformations,
the proportional limit P
and the yield strength
σy, as determined
using the 0.002 strain
offset method (where there
is noticeable plastic
deformation). P is the
gradual elastic to
plastic transition.

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Plastic Deformation (permanent)
• From an atomic perspective, plastic
deformation corresponds to the breaking of
bonds with original atom neighbors and
then reforming bonds with new neighbors.
• After removal of the stress, the large
number of atoms that have relocated, do
not return to original position.
• Yield strength is a measure of resistance
to plastic deformation.

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(c)2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
• Localized deformation of a ductile material during a tensile
test produces a necked region.
• The image shows necked region in a fractured sample

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Permanent Deformation
• Permanent deformation for metals is
accomplished by means of a process called
slip, which involves the motion of
dislocations.
• Most structures are designed to ensure that
only elastic deformation results when stress
is applied.
• A structure that has plastically deformed, or
experienced a permanent change in shape,
may not be capable of functioning as
intended.

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tensile stress, 
engineering strain, 
y
p = 0.002
Yield Strength, y

Stress-Strain Diagram
Strain ( ) (e/Lo)
4
1
2
3
5
Elastic
Region
Plastic
Region
Strain
Hardening Fracture
ultimate
tensile
strength
Elastic region
slope=Young’s(elastic) modulus
yield strength
Plastic region
ultimate tensile strength
strain hardening
fracture
necking
yield
strength
UTS

y

ε
E
σ 
ε
σ
E 

1
2
y
ε
ε
σ
E



Stress-Strain Diagram (cont)
• Elastic Region (Point 1 –2)
- The material will return to its original shape
after the material is unloaded( like a rubber band).
- The stress is linearly proportional to the strain in
this region.
ε
E
σ 
: Stress(psi)
E : Elastic modulus (Young’s Modulus) (psi)
: Strain (in/in)
σ
ε
- Point 2 : Yield Strength : a point where permanent
deformation occurs. ( If it is passed, the material will
no longer return to its original length.)
ε
σ
E 
or

• Strain Hardening
- If the material is loaded again from Point 4, the
curve will follow back to Point 3 with the same
Elastic Modulus(slope).
- The material now has a higher yield strength of
Point 4.
- Raising the yield strength by permanently straining
the material is called Strain Hardening.
Stress-Strain Diagram (cont)

• Tensile Strength (Point 3)
- The largest value of stress on the diagram is called
Tensile Strength(TS) or Ultimate Tensile Strength
(UTS)
- It is the maximum stress which the material can
support without breaking.
• Fracture (Point 5)
- If the material is stretched beyond Point 3, the stress
decreases as necking and non-uniform deformation
occur.
- Fracture will finally occur at Point 5.
Stress-Strain Diagram (cont)

(c)2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
The stress-strain curve for an aluminum alloy.

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T
E
N
S
I
L
E
P
R
O
P
E
R
T
I
E
S

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Room T values
a = annealed
hr = hot rolled
ag = aged
cd = cold drawn
cw = cold worked
qt = quenched & tempered
Yield Strength: Comparison

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• After yielding, the stress necessary to
continue plastic deformation in metals
increases to a maximum point (M) and
then decreases to the eventual fracture
point (F).
• All deformation up to the maximum
stress is uniform throughout the tensile
sample.
• However, at max stress, a small
constriction or neck begins to form.
• Subsequent deformation will be
confined to this neck area.
• Fracture strength corresponds to the
stress at fracture.
Region between M and F:
• Metals: occurs when noticeable necking starts.
• Ceramics: occurs when crack propagation starts.
• Polymers: occurs when polymer backbones are aligned and about to break.
Tensile Strength, TS

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In an undeformed
thermoplastic polymer
tensile sample,
(a)the polymer chains
are randomly
oriented.
(b)When a stress is
applied, a neck
develops as chains
become aligned
locally. The neck
continues to grow
until the chains in the
entire gage length
have aligned.
(c) The strength of the
polymer is increased

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Room T values
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

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• Tensile stress, : • Shear stress, t:
 
Ft
Ao
original area
before loading
Stress has units: N/m2 or lb/in2
Engineering Stress

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VMSE
http://www.wiley.com/college/callister/0470125373/vmse/strstr.htm
http://www.wiley.com/college/callister/0470125373/vmse/index.htm

Example 1 SOLUTION

Example 1
Tensile Testing of Aluminum Alloy
Convert the change in length data in the table to engineering
stress and strain and plot a stress-strain curve.

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• Another ductility measure: 100
% x
A
A
A
AR
o
f
o 

• Ductility may be expressed as either percent elongation (%
plastic strain at fracture) or percent reduction in area.
• %AR > %EL is possible if internal voids form in neck.
100
% x
l
l
l
EL
o
o
f 

Ductility, %EL
Ductility is a measure of the
plastic deformation that has
been sustained at fracture:
A material that
suffers very
little plastic
deformation is
brittle.

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Toughness
Lower toughness: ceramics
Higher toughness: metals
Toughness is
the ability to
absorb
energy up to
fracture (energy
per unit volume of
material).
A “tough”
material has
strength and
ductility.
Approximated
by the area
under the
stress-strain
curve.

• Energy to break a unit volume of material
• Approximate by the area under the stress-strain
curve.
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smaller toughness-
unreinforced
polymers
Engineering tensile strain, 
Engineering
tensile
stress, 
smaller toughness (ceramics)
larger toughness
(metals, PMCs)
Toughness

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Linear Elastic Properties
Modulus of Elasticity, E:
(Young's modulus)
• Hooke's Law:  = E 
• Poisson's ratio:
metals: n ~ 0.33
ceramics: n ~0.25
polymers: n ~0.40
Units:
E: [GPa] or [psi]
n: dimensionless
n  x/y

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Engineering Strain
Strain is dimensionless.

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Axial (z) elongation (positive strain) and lateral (x and y)
contractions (negative strains) in response to an imposed
tensile stress.

True Stress and True Strain
 True stress The load divided by the actual cross-sectional
area of the specimen at that load.
 True strain The strain calculated using actual and not
original dimensions, given by εt ln(l/l0).
•The relation between the true stress-
true strain diagram and engineering
stress-engineering strain diagram.
•The curves are identical to the yield
point.

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Stress-Strain Results for Steel Sample

Example 2:
Young’s Modulus - Aluminum Alloy
From the data in Example 1, calculate the modulus of
elasticity of the aluminum alloy.

• Use the modulus to determine the length after
deformation of a bar of initial length of 50 in.
• Assume that a level of stress of 30,000 psi is applied.
Example 2: Young’s Modulus - Aluminum Alloy - continued

40
0.2
8
0.6
1
Magnesium,
Aluminum
Platinum
Silver, Gold
Tantalum
Zinc, Ti
Steel, Ni
Molybdenum
Graphite
Si crystal
Glass-soda
Concrete
Si nitride
Al oxide
PC
Wood( grain)
AFRE( fibers)*
CFRE*
GFRE*
Glass fibers only
Carbon fibers only
Aramid fibers only
Epoxy only
0.4
0.8
2
4
6
10
20
40
60
80
100
200
600
800
1000
1200
400
Tin
Cu alloys
Tungsten
<100>
<111>
Si carbide
Diamond
PTFE
HDPE
LDPE
PP
Polyester
PS
PET
CFRE( fibers)*
GFRE( fibers)*
GFRE(|| fibers)*
AFRE(|| fibers)*
CFRE(|| fibers)*
Metals
Alloys
Graphite
Ceramics
Semicond
Polymers
Composites
/fibers
E(GPa)
109 Pa Composite data based on
reinforced epoxy with 60 vol%
of aligned carbon (CFRE),
aramid (AFRE), or glass (GFRE)
fibers.
Young’s Moduli: Comparison

Example 3: True Stress and True Strain
Calculation
Compare engineering stress and strain with true stress and
strain for the aluminum alloy in Example 1 at (a) the
maximum load. The diameter at maximum load is 0.497
in. and at fracture is 0.398 in.
Example 3 SOLUTION

Strain Hardening
An increase in y due to
plastic deformation.

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Strain Hardening (n, K or C values)

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Mechanical Behavior - Ceramics
• The stress-strain behavior of brittle
ceramics is not usually obtained by a
tensile test.
1. It is difficult to prepare and test
specimens with specific geometry.
2. It is difficult to grip brittle materials without
fracturing them.
3. Ceramics fail after roughly 0.1% strain;
specimen have to be perfectly aligned.

The Bend Test for Brittle Materials
 Bend test - Application of a force to the center of a bar
that is supported on each end to determine the
resistance of the material to a static or slowly applied
load.
 Flexural strength or modulus of rupture -The stress
required to fracture a specimen in a bend test.
 Flexural modulus - The modulus of elasticity calculated
from the results of a bend test, giving the slope of the
stress-deflection curve.

(c)2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
The stress-strain behavior of brittle materials compared with
that of more ductile materials

(c)2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
(a) The bend test often used for measuring the strength
of brittle materials, and (b) the deflection δ obtained by
bending

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• Schematic for a 3-
point bending test.
• Able to measure the
stress-strain behavior
and flexural strength
of brittle ceramics.
• Flexural strength
(modulus of rupture or
bend strength) is the
stress at fracture.
Flexural Strength
See Table 7.2 for more values.

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• Room T behavior is usually elastic, with brittle failure.
• 3-Point Bend Testing often used.
--tensile tests are difficult for brittle materials.
• Determine elastic modulus according to:
E 
F

L3
4bd3

F

L3
12R4
rect.
cross
section
circ.
cross
section
MEASURING ELASTIC MODULUS

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• 3-point bend test to measure room T strength.
F
L/2 L/2
cross section
R
b
d
rect. circ.
location of max tension
• Flexural strength:
rect.
fs  m
fail

1.5FmaxL
bd2

FmaxL
R3
• Typ. values:
Material fs(MPa) E(GPa)
Si nitride
Si carbide
Al oxide
glass (soda)
700-1000
550-860
275-550
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300
430
390
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Data from Table 12.5, Callister 6e.
MEASURING STRENGTH

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--brittle response (aligned chain, cross linked & networked case)
--plastic response (semi-crystalline case)
Stress-Strain Behavior: Elastomers
3 different responses:
A – brittle failure
B – plastic failure
C - highly elastic (elastomer)

Hardness of Materials
 Hardness test - Measures the resistance of a material to
penetration by a sharp object.
 Macrohardness - Overall bulk hardness of materials
measured using loads >2 N.
 Microhardness Hardness of materials typically measured
using loads less than 2 N using such test as Knoop
(HK).
 Nano-hardness - Hardness of materials measured at 1–
10 nm length scale using extremely small (~100 µN)
forces.

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Hardness
• Hardness is a measure of a material’s resistance
to localized plastic deformation (a small dent or
scratch).
• Quantitative hardness techniques have been
developed where a small indenter is forced into
the surface of a material.
• The depth or size of the indentation is measured,
and corresponds to a hardness number.
• The softer the material, the larger and deeper the
indentation (and lower hardness number).

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• Resistance to permanently indenting the surface.
• Large hardness means:
--resistance to plastic deformation or cracking in
compression.
--better wear properties.
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

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Hardness Testers

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Conversion of
Hardness
Scales
Also see: ASTM E140 - 07
Volume 03.01
Standard Hardness Conversion
Tables for Metals Relationship
Among Brinell Hardness, Vickers
Hardness, Rockwell Hardness,
Superficial Hardness, Knoop
Hardness, and Scleroscope
Hardness

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Correlation
between
Hardness and
Tensile
Strength
• Both hardness and tensile
strength are indicators of a
metal’s resistance to plastic
deformation.
• For cast iron, steel and
brass, the two are roughly
proportional.
• Tensile strength (psi) =
500*BHR

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• 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 y.
• Toughness: The energy needed to break a unit
volume of material.
• Ductility: The plastic strain at failure.
Summary