### Chapter 7: Mechanical Properties

```Chapter 7:
Mechanical Properties
• 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 does permanent
deformation occur? What materials are most
resistant to permanent deformation?
• Toughness and ductility: What are they and how
do we measure them?
Chapter 7 - 1
Elastic Deformation
1. Initial
bonds
stretch
initial
d
F
F
Linearelastic
Elastic means reversible!
d
Non-Linearelastic
Chapter 7 - 2
Plastic Deformation (Metals)
1. Initial
bonds
stretch
& planes
shear
delastic + plastic
planes
still
sheared
dplastic
F
F
Plastic means permanent!
linear
elastic
linear
elastic
dplastic
d
Chapter 7 - 3
Engineering Stress
• Tensile stress, s:
• Shear stress, t:
Ft
F
Area, Ao
Area, Ao
Ft
Ft
lb f
N
= 2 or
s=
2
in
m
Ao
original area
Ft
Fs
Fs
Fs
t=
Ao
Ft
F
 Stress has units:
N/m2 or lbf /in2
Chapter 7 - 4
Common States of Stress
• Simple tension: cable
F
F
A o = cross sectional
F
s=
s
Ao
s
• Torsion (a form of shear): drive shaft
M
Ac
M
Fs
Ski lift
(photo courtesy
P.M. Anderson)
Ao
Fs
t =
Ao
2R
Note: t = M/AcR here.
Chapter 7 - 5
OTHER COMMON STRESS STATES (i)
• Simple compression:
Ao
Canyon Bridge, Los Alamos, NM
(photo courtesy P.M. Anderson)
Balanced Rock, Arches
National Park
(photo courtesy P.M. Anderson)
F
s=
Ao
Note: compressive
structure member
(s < 0 here).
Chapter 7 - 6
OTHER COMMON STRESS STATES (ii)
• Bi-axial tension:
Pressurized tank
(photo courtesy
P.M. Anderson)
• Hydrostatic compression:
Fish under water
sq > 0
sz > 0
(photo courtesy
P.M. Anderson)
sh< 0
Chapter 7 - 7
Engineering Strain
• Tensile strain:
• Lateral strain:
d/2
e = d
Lo
wo
• Shear strain:
-dL
eL =
wo
Lo
dL /2
q
g = x/y = tan q
x
90º - q
y
90º
Strain is always
dimensionless.
Adapted from Fig. 7.1 (a) and (c), Callister & Rethwisch 3e.
Chapter 7 - 8
Stress-Strain Testing
• Typical tensile test
machine
extensometer
• Typical tensile
specimen
specimen
Fig. 7.2,
Callister &
Rethwisch 3e.
gauge
length
Adapted from Fig. 7.3, Callister & Rethwisch 3e. (Fig. 7.3 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.)
Chapter 7 - 9
Linear Elastic Properties
• Modulus of Elasticity, E:
(also known as Young's modulus)
• Hooke's Law:
s=Ee
s
F
E
e
Linearelastic
F
simple
tension
test
Chapter 7 - 10
Poisson's ratio, n
• Poisson's ratio, n:
eL
eL
n=e
metals: n ~ 0.33
ceramics: n ~ 0.25
polymers: n ~ 0.40
Units:
E: [GPa] or [psi]
n: dimensionless
e
-n
n > 0.50 density increases
n < 0.50 density decreases
(voids form)
Chapter 7 - 11
Mechanical Properties
• Slope of stress strain plot (which is
proportional to the elastic modulus) depends
on bond strength of metal
Adapted from Fig. 7.7,
Callister & Rethwisch 3e.
Chapter 7 - 12
Other Elastic Properties
• Elastic Shear
modulus, G:
t
M
G
t=Gg
• Elastic Bulk
modulus, K:
V
P = -K
Vo
g
M
P
P
K
V P
Vo
• Special relations for isotropic materials:
E
G=
2(1 + n)
simple
torsion
test
E
K=
3(1 - 2n)
P
pressure
test: Init.
vol =Vo.
Vol chg.
= V
Chapter 7 - 13
Young’s Moduli: Comparison
Metals
Alloys
1200
1000
800
600
400
E(GPa)
200
100
80
60
40
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
109 Pa
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 B.2,
Callister & Rethwisch 3e.
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)
Chapter 7 - 14
Useful Linear Elastic Relationships
• Simple tension:
d = FL o d = -n Fw o
L
EA o
EA o
F
a=
wo
2ML o

r o4 G
M = moment
a = angle of twist
d/2
Ao
dL /2
• Simple torsion:
Lo
Lo
2ro
• Material, geometric, and loading parameters all
contribute to deflection.
• Larger elastic moduli minimize elastic deflection.
Chapter 7 - 15
Plastic (Permanent) Deformation
(at lower temperatures, i.e. T < Tmelt/3)
• Simple tension test:
Elastic+Plastic
at larger stress
engineering stress, s
Elastic
initially
permanent (plastic)
after load is removed
ep
engineering strain, e
plastic strain
Adapted from Fig. 7.10 (a),
Callister & Rethwisch 3e.
Chapter 7 - 16
Yield Strength, sy
• Stress at which noticeable plastic deformation has
occurred.
when ep = 0.002
tensile stress, s
sy
sy = yield strength
Note: for 2 inch sample
e = 0.002 = z/z
 z = 0.004 in
engineering strain, e
ep = 0.002
Adapted from Fig. 7.10 (a),
Callister & Rethwisch 3e.
Chapter 7 - 17
Yield Strength : Comparison
Metals/
Alloys
2000
Graphite/
Ceramics/
Semicond
Polymers
Composites/
fibers
200
Al (6061) ag
Steel (1020) hr
Ti (pure) a
Ta (pure)
Cu (71500) hr
100
70
60
50
40
Al (6061) a
30
20
10
Tin (pure)
¨
dry
PC
Nylon 6,6
PET
PVC humid
PP
HDPE
LDPE
Hard to measure,
300
in ceramic matrix and epoxy matrix composites, since
in tension, fracture usually occurs before yield.
700
600
500
400
Ti (5Al-2.5Sn) a
W (pure)
Cu (71500) cw
Mo (pure)
Steel (4140) a
Steel (1020) cd
since in tension, fracture usually occurs before yield.
1000
Hard to measure ,
Yield strength, sy (MPa)
Steel (4140) qt
Room temperature
values
Based on data in Table B.4,
Callister & Rethwisch 3e.
a = annealed
hr = hot rolled
ag = aged
cd = cold drawn
cw = cold worked
qt = quenched & tempered
Chapter 7 - 18
Tensile Strength, TS
• Maximum stress on engineering stress-strain curve.
Adapted from Fig. 7.11,
Callister & Rethwisch 3e.
TS
F = fracture or
ultimate
strength
engineering
stress
sy
Typical response of a metal
Neck – acts
as stress
concentrator
strain
engineering strain
• Metals: occurs when noticeable necking starts.
• Polymers: occurs when polymer backbone chains are
aligned and about to break.
Chapter 7 - 19
Tensile Strength: Comparison
Metals/
Alloys
Tensile strength, TS (MPa)
5000
3000
2000
1000
300
200
100
40
30
Graphite/
Ceramics/
Semicond
Polymers
C fibers
Aramid fib
E-glass fib
Steel (4140) qt
W (pure)
Ti (5Al-2.5Sn)aa
Steel (4140)
Cu (71500) cw
Cu (71500) hr
Steel (1020)
Al (6061) ag
Ti (pure) a
Ta (pure)
Al (6061) a
AFRE(|| fiber)
GFRE(|| fiber)
CFRE(|| fiber)
Diamond
Si nitride
Al oxide
Room temperature
values
Si crystal
<100>
Glass-soda
Concrete
Nylon 6,6
PC PET
PVC
PP
HDPE
20
Composites/
fibers
Graphite
wood(|| fiber)
GFRE( fiber)
CFRE( fiber)
AFRE( fiber)
LDPE
10
wood (
1
fiber)
Based on data in Table B4,
Callister & Rethwisch 3e.
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.
Chapter 7 - 20
Ductility
• Plastic tensile strain at failure:
Lf - Lo
x 100
%EL =
Lo
smaller %EL
Engineering
tensile
stress, s
larger %EL
Lo
Ao
Af
Lf
Adapted from Fig. 7.13,
Callister & Rethwisch 3e.
Engineering tensile strain, e
• Another ductility measure:
%RA =
Ao - Af
x 100
Ao
Chapter 7 - 21
Toughness
• Energy to break a unit volume of material
• Approximate by the area under the stress-strain curve.
Engineering
tensile
stress, s
small toughness (ceramics)
large toughness (metals)
very small toughness
(unreinforced polymers)
Adapted from Fig. 7.13,
Callister & Rethwisch 3e.
Engineering tensile strain, e
Brittle fracture: elastic energy
Ductile fracture: elastic + plastic energy
Chapter 7 - 22
Resilience, Ur
• Ability of a material to store energy
– Energy stored best in elastic region
Ur =
ey
0
sde
If we assume a linear
stress-strain curve this
simplifies to
1
Ur @ sy e y
2
Adapted from Fig. 7.15,
Callister & Rethwisch 3e.
Chapter 7 - 23
Elastic Strain Recovery
sy i
D
syo
Stress
3. Reapply
Strain
Adapted from Fig. 7.17,
Callister & Rethwisch 3e.
Elastic strain
recovery
Chapter 7 - 24
Mechanical Properties
Ceramic materials are more brittle than metals.
Why is this so?
• Consider mechanism of deformation
– In crystalline, by dislocation motion
– In highly ionic solids, dislocation motion is difficult
• few slip systems
• resistance to motion of ions of like charge (e.g., anions)
past one another
Chapter 7 - 25
Flexural Tests – Measurement of
Elastic Modulus
• Room T behavior is usually elastic, with brittle failure.
• 3-Point Bend Testing often used.
-- tensile tests are difficult for brittle materials.
F
cross section
L/2
d
b
rect.
L/2
Adapted from Fig. 7.18,
Callister & Rethwisch 3e.
R
d = midpoint
circ.
deflection
• Determine elastic modulus according to:
F
E =
F
d 4 bd
d
E =
F
x
slope =
3
F
d
linear-elastic behavior
L
(rect. cross section)
3
3
L
d 12  R
4
(circ. cross section)
Chapter 7 - 26
Flexural Tests – Measurement of
Flexural Strength
• 3-point bend test to measure room-T flexural strength.
cross section
d
b
rect.
L/2
F
L/2
Adapted from Fig. 7.18,
Callister & Rethwisch 3e.
R
d = midpoint
circ.
deflection
location of max tension
• Flexural strength:
s fs =
s fs =
3 Ff L
2 bd
Ff L
R
3
2
• Typical values:
sfs (MPa) E(GPa)
Si nitride
250-1000 304
Si carbide
100-820 345
Al oxide
275-700 393
glass (soda-lime) 69
69
Material
(rect. cross section)
(circ. cross section)
Data from Table 7.2, Callister & Rethwisch 3e.
Chapter 7 - 27
Mechanical Properties of Polymers –
Stress-Strain Behavior
brittle polymer
plastic
elastomer
elastic moduli
– less than for metals
Adapted from Fig. 7.22,
Callister & Rethwisch 3e.
• Fracture strengths of polymers ~ 10% of those for metals
• Deformation strains for polymers > 1000%
– for most metals, deformation strains < 10%
Chapter 7 - 28
Influence of T and Strain Rate on Thermoplastics
• Decreasing T...
-- increases E
-- increases TS
-- decreases %EL
• Increasing
strain rate...
-- same effects
as decreasing T.
s(MPa)
80 4°C
60
20°C
40
Plots for
semicrystalline
PMMA (Plexiglas)
40°C
20
0
60°C
0
0.1
0.2
e
to 1.3
0.3
Adapted from Fig. 7.24, Callister & Rethwisch 3e. (Fig. 7.24 is from T.S.
Carswell and J.K. Nason, 'Effect of Environmental Conditions on the
Mechanical Properties of Organic Plastics", Symposium on Plastics,
American Society for Testing and Materials, Philadelphia, PA, 1944.)
Chapter 7 - 29
Time-Dependent Deformation
• Stress relaxation test:
-- strain in tension to eo
and hold.
-- observe decrease in
stress with time.
tensile test
eo
s(t)
time
• Relaxation modulus:
E r (t ) =
eo
for T > Tg.
5
10
Er (10 s) 3
in MPa 10
rigid solid
(small relax)
transition
region
1
10
10-1
strain
s(t )
• There is a large decrease in Er
viscous liquid
10-3 (large relax)
(amorphous
polystyrene)
7.28, Callister &
Rethwisch 3e. (Fig.
7.28 is from A.V.
Tobolsky, Properties
and Structures of
Polymers, John
Wiley and Sons, Inc.,
1960.)
60 100 140 180 T(°C)
Tg
• Representative Tg values (C):
PE (low density)
PE (high density)
PVC
PS
PC
- 110
- 90
+ 87
+100
+150
Selected values from
Table 11.3, Callister
& Rethwisch 3e.
Chapter 7 - 30
Hardness
• Resistance to permanently indenting the surface.
• Large hardness means:
-- resistance to plastic deformation or cracking in
compression.
-- better wear properties.
apply known force
measure size
of indent after
e.g.,
10 mm sphere
D
most
plastics
brasses
Al alloys
Smaller indents
mean larger
hardness.
d
easy to machine
steels
file hard
cutting
tools
nitrided
steels
diamond
increasing hardness
Chapter 7 - 31
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 7 - 32
Hardness: Measurement
Table 7.5
Chapter 7 - 33
True Stress & Strain
Note: S.A. changes when sample stretched
• True stress
• True strain
sT = F Ai
e T = ln  i  o 
s T = s 1 + e 
e T = ln 1 + e 
Adapted from Fig. 7.16,
Callister & Rethwisch 3e.
Chapter 7 - 34
Hardening
• An increase in sy due to plastic deformation.
s
large hardening
sy
1
sy
small hardening
0
e
• Curve fit to the stress-strain response:
 
sT = K eT
“true” stress (F/A)
n
hardening exponent:
n = 0.15 (some steels)
to n = 0.5 (some coppers)
“true” strain: ln(L/Lo)
Chapter 7 - 35
Variability in Material Properties
• Elastic modulus is material property
• Critical properties depend largely on sample flaws
(defects, etc.). Large sample to sample variability.
• Statistics
n
– Mean
x =
 xn
n
1
– Standard Deviation
n
2



x
x
i
s = 

n -1

2



where n is the number of data points
Chapter 7 - 36
Design or 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
• 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.
s working =
220 , 000 N

 d /4
2

5
sy
N
d
1045 plain
carbon steel:
sy = 310 MPa
TS = 565 MPa
d = 0.067 m = 6.7 cm
Lo
F = 220,000N
Chapter 7 - 37
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.
Chapter 7 - 38
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