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CHAPTER 7:
MECHANICAL PROPERTIES
CHAPTER 7:
MECHANICAL PROPERTIES
ISSUES TO ADDRESS...
• Stress and strain
• Elastic behavior
• Plastic behavior
• Toughness and ductility
• Ceramic Materials
Stress
Strain
Elasticity
Strength
Tensile
Elongation
Ductile
Fracture
Tension
Flexural
Plasticity
7.2 STRESS & STRAIN
• Tensile stress, s:
• Shear stress, t:
Ft
s
Ao
original area
before loading
Stress has units:
N/m2 or lb/in2
4
Stress (s) for tension and
compression
F
s
Ao
Strain (e) for tension and
compression
Tensile load
Compressive load
Shear stress
Fs
t 
Ao
Shear strain
g = tan q
Torsional deformation
angle of twist, f
7.2 COMMON STATES OF STRESS
• Simple tension: cable
F
s
Ao
• Simple shear: drive shaft
Ski lift
(photo courtesy P.M. Anderson)
Fs
t 
Ao
Note: t = M/Ac
5
OTHER COMMON STRESS STATES
• Simple compression:
Ao
Canyon Bridge, Los Alamos, NM
(photo courtesy P.M. Anderson)
Balanced Rock, Arches
National Park
(photo courtesy P.M. Anderson)
Note: compressive
structure member
(s < 0 here).
6
OTHER COMMON STRESS STATES
• Bi-axial tension:
Pressurized tank
(photo courtesy
P.M. Anderson)
• Hydrostatic compression:
Fish under water
sq > 0
sz > 0
(photo courtesy
P.M. Anderson)
s h< 0
7
ENGINEERING STRAIN
• Tensile strain:
• Lateral strain:
/2
wo
• Shear strain:
L/2
Lo
/2
L/2
q/2
g = tan q
/2 - q
/2
Strain is always
dimensionless.
q/2
8
7.2 STRESS-STRAIN TESTING
• Typical tensile specimen
• Typical tensile
test machine
Adapted from Fig. 6.2,
Callister 6e.
• Other types of tests:
--compression: brittle
materials (e.g., concrete)
--torsion: cylindrical tubes,
shafts.
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.)
9
Normal and shear stresses on an arbitrary plane
Stress is a function of the orientation
On plane p-p’ the stress is not pure tensile
There are two components
Tensile or normal stress s’ (normal to the pp’ plane)
Shear stress t’ (parallel to the pp’ plane)
ELASTIC DEFORMATIONS
7.3 Stress-strain behavior
• Modulus of Elasticity, E:
(also known as Young's modulus)
• Hooke's Law:
s=Ee
• Poisson's ratio, n:
metals: n ~ 0.33
ceramics: ~0.25
polymers: ~0.40
Units:
E: [GPa] or [psi]
n: dimensionless
10
PROPERTIES FROM BONDING: E
• Elastic modulus, E
Elastic modulus
F
L
=E
Ao
Lo
Energy ~ curvature at ro
Energy
E is larger if Eo is larger.
unstretched length
ro
r
smaller Elastic Modulus
larger Elastic Modulus
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7.4 ANESLATICITY
Assumed:
Time-independent elastic deformation
Applied stress produces instantaneous elastic strain
Remains constant while elasticity stress is applied
At release of load, strain is recovered
In real life:
Time-dependent elastic strain component: Anelasticity
Time-dependent microscopic and atomistic processes
For metals is small
Significant for polymeric materials: Viscoelastic behavior
7.5 ELASTIC PROPERTIES OF MATERIALS
Poisson’s ratio
n = -ex/ez = -ey/ez
For isotropic materials
E
G
2(1  n)
YOUNG’S MODULI:
COMPARISON
Metals
Alloys
1200
1000
800
600
400
E(GPa)
200
100
80
60
40
109 Pa
Graphite
Composites
Ceramics Polymers
/fibers
Semicond
Diamond
Tungsten
Molybdenum
Steel, Ni
Tantalum
Platinum
Cu alloys
Zinc, Ti
Silver, Gold
Aluminum
Magnesium,
Tin
Si carbide
Al oxide
Si nitride
Carbon fibers only
CFRE(|| fibers)*
<111>
Si crystal
Aramid fibers only
<100>
AFRE(|| fibers)*
Glass-soda
Glass fibers only
GFRE(|| fibers)*
Concrete
GFRE*
20
10
8
6
4
2
1
0.8
0.6
0.4
0.2
CFRE*
GFRE( fibers)*
Graphite
Polyester
PET
PS
PC
CFRE( fibers)*
AFRE( fibers)*
Epoxy only
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.
PP
HDPE
PTFE
LDPE
Wood( grain)
13
II. MECHANICAL BEHAVIOR—METALS
II. ELASTIC DEFORMATION
1. Initial
2. Small load
3. Unload
bonds
stretch
return to
initial

F
Elastic means reversible!
2
II. PLASTIC (PERMANENT) DEFORMATION
(at lower temperatures, T < Tmelt/3)
• Simple tension test:
15
II. PLASTIC DEFORMATION (METALS)
1. Initial
2. Small load
3. Unload
F
Plastic means permanent!
linear
elastic
linear
elastic
plastic

3
7.6 Tensile properties
• YIELD STRENGTH, sy
Stress at which noticeable plastic deformation has
occurred.
when ep = 0.002
tensile stress, s
sy
engineering strain, e
ep = 0.002
16
7.6 YIELD STRENGTH: COMPARISON
sy(ceramics)
>>sy(metals)
>> sy(polymers)
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
17
7.6 TENSILE STRENGTH, TS
Maximum possible engineering stress in tension
Adapted from Fig. 6.11,
Callister 6e.
• Metals: occurs when noticeable necking starts.
• Ceramics: occurs when crack propagation starts.
• Polymers: occurs when polymer backbones are
aligned and about to break.
18
7.6 TENSILE STRENGTH:
COMPARISON
TS(ceram)
~TS(met)
~ TS(comp)
>> TS(poly)
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.
19
7.6 DUCTILITY, %EL
Degree of plastic deformation at fracture
Brittle, when very little plastic deformation
L f  Lo
x100
• Plastic tensile strain at failure: %EL 
Lo
Adapted from Fig. 6.13,
Callister 6e.
ductility as percent reduction
in area
Ao  A f
%AR 
x100
Ao
• Note: %AR and %EL are often comparable.
--Reason: crystal slip does not change material volume.
--%AR > %EL possible if internal voids form in neck.
20
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Stress-strain of iron at several temperatures
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RESILIENCE
Capacity to absorb energy when deformed elastically and then upon
unloadign, to have this energy recovered
Modulus of Resilience
ey
U r   sde
0
For a linear elastic region:
1
U r  s ye y
2
7.6 TOUGHNESS
• Ability to absorb energy up to fracture
Engineering
tensile
stress, s
smaller toughness (ceramics)
larger toughness
(metals, PMCs)
smaller toughnessunreinforced
polymers
Engineering tensile strain, e
Usually ductile materials are tougher than brittle ones
Areas below the curves
21
7.7 True stress & strain
Decline in stress necessary to continue deformation past M
Looks like metal become weaker
Actually, it is increasing in strength
Cross sectional area decreases rapidly within the neck region
Reduction in the load-bearing capacity of the specimen
Stress should consider deformation
7.7 True stress & strain
HARDENING: An increase in sy due to plastic deformation.
• Curve fit to the stress-strain response:
s T  Ke
n
T
n = hardening exponent
n = 0.15 (some steels)
n = 0.5 (some copper)
22
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7.8 Elastic Recovery After Plastic Deformation
7.9 Compressive, Shear, and
Torsional Deformation
Similar to tensile counterpart
No maximum for compression
Necking does not occur
Mode of fracture different from that of
tension
III. MECHANICAL BEHAVIOR—CERAMICS
Limited applicability, catastrophic fracture in a brittle
manner, little energy absorption
7.10 FLEXURAL STRENGTH
Tensile tests are difficult
difficult to prepare geometry
easy to fracture
ceramics fail at 0.1% strain
bending stress
rod specimen is used
three of four point loading technique
flexure test
7.10 MEASURING STRENGTH
• Flexural strength= modulus of rupture
= fracture strength = bend strength
s fs 
3F f L
s fs 
Ff L
2bd 2
 R3
• Type values:
sfs(MPa)
Si nitride
700-1000
Si carbide
550-860
Al oxide
275-550
glass (soda)
69
Material
Data from Table 12.5, Callister 6e.
E(GPa)
300
430
390
69
7.11 Elastic Behavior (for
ceramics)
Similar to tensile test for metals
Linear stress-strain
Moduli of elasticity for
ceramics are slightly higher
than for metals
No plastic deformation prior
to fracture
7.12 INFLUENCE OF POROSITY ON THE
MECHANICAL PROPERTIES OF CERAMICS
Powder as precursor
Compaction to desire shape
Aluminum oxide
Pores or voids elimination
E = Eo(1 – 1.9P + 0.9P2)
incomplete
Residual porosity remains
Deleterious influence on
elasticity and strength
Volume fraction porosity P
Eo = modulus of elasticity of
the non porous material
-Pores reduce the area
-Pores are stress concentrators
-tensile stress doubles in an
isolated spherical pore
Aluminum oxide
sfs = soe-nP
IV MECHANICAL BEHAVIOR—POLYMERS
7.13 STRESS—STRAIN BEHAVIOR
Stress-strain curves
adapted from Fig.
15.1, Callister 6e.
Inset figures along
elastomer curve
(green) adapted from
Fig. 15.14, Callister
6e. (Fig. 15.14 is from
Z.D. Jastrzebski, The
Nature and Properties
of Engineering
Materials, 3rd ed.,
John Wiley and Sons,
1987.)
• Compare to responses of other polymers:
--brittle response (aligned, cross linked & networked case)
--plastic response (semi-crystalline case)
7.13 T & STRAIN RATE: THERMOPLASTICS
• Decreasing T...
--increases E
--increases TS
--decreases %EL
• Increasing
strain rate...
--same effects
as decreasing T.
26
7.14 Macroscopic Deformation
Semicrystaline polymer
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7.15 Viscoelasticity Deformation
Amorphous polymer:
Glass at low T
Viscous liquid at higher T
Small deformation at low T may be elastic
Hooke’s law
Rubbery solid at intermediate T
A combination of glass and viscous/liquid
Viscoelasticity
Elastic deformation is instantaneous
Upon release, deformation is totally recovered
7.15 Viscoelasticity Deformation
Totally elastic
Load
Viscous
Viscoelastic
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Relaxation Modulus for viscoelastic
polymers:
s (t )
Er (t ) 
eo
Amorphous polystyrene
A viscoelastic polymer
Polystyrene configurations
Almost totally crystalline isotactic
Lightly crosslinked atactic
Viscoelastic creep
Creep modulus Ec(t)
so
Ec (t ) 
e (t )
amorphous
V. Hardness & Other Mechanical Property Considerations
7.16 Hardness
Measure of material resistance to localized plastic deformation
Early tests: Mohs scale 1 for talc and 10 for diamond
Depth or size of an indentation
Tests:
Mohs Hardness
Rockwell Hardness
Brinell Hardness
Knoop & Vickers Microindentation Hardness
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Hardness Conversion
Correlation between Hardness and Tensile Strength
Tensile strength and Hardness
measure metal resistance to plastic
deformation
For example:
TS(Mpa) = 3.45 × HB
or
TS(psi) = 500 × HB
7.17 Hardness of Ceramic Materials
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7.18 Tear Strength & Hardness of Polymers
Thin films in packaging
Tear Strength: Energy required to tear apart a cut specimen of a
standard geometry
VI. Property Variability and Design/Safety Factors
7.19 Variability of Material Properties: Average and
standard deviation
7.20 DESIGN/SAFETY FACTORS
• Design uncertainties mean we do not push the limit.
• Factor of safety, N
Often N is
between
sy
s working 
1.2 and 4
N
• Ex: 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.
s working 
220,000N


2

 d / 4 


sy
N
5
d = 47.5 mm
29
SUMMARY
• 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 sy.
• Toughness: The energy needed to break a unit
volume of material.
• Ductility: The plastic strain at failure.
30