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Chapter 6 - 2
Elastic means reversible!
Elastic Deformation
2. Small load
F
d
bonds
stretch
1. Initial 3. Unload
return to
initial
F
d
Linear-
elastic
Non-Linear-elastic
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Chapter 6 - 3
Plastic means permanent!
Plastic Deformation (Metals)
F
d
linearelastic
linearelastic
dplastic
1. Initial 2. Small load 3. Unload
planesstillsheared
F
delastic + plastic
bondsstretch& planesshear
dplastic
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Chapter 6 - 4
Stress has units:
N/m2 or lbf/in2
Engineering Stress
Shearstress, t:
Area,Ao
Ft
Ft
Fs
F
F
Fs
t =Fs
Ao
Tensile stress, s:
original area
before loading
s = FtAo
2f
2mNor
inlb=
Area,Ao
Ft
Ft
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Chapter 6 - 5
Simple tension: cable
Note: t = M/AcRhere.
Common States of Stress
os=
F
A
o
t =Fs
A
ss
M
M Ao
2R
FsAc
Torsion (a form of shear): drive shaftSki lift (photo courtesyP.M. Anderson)
Ao = cross sectional
area (when unloaded)
FF
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Chapter 6 - 6
(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM
os= F
A
Simple compression:
Note: compressivestructure member
(s < 0 here).(photo courtesy P.M. Anderson)
OTHER COMMON STRESS STATES (i)
Ao
Balanced Rock, ArchesNational Park
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Chapter 6 - 8
Tensile strain: Lateral strain:
Strain is always
dimensionless.
Engineering Strain
Shearstrain:
q
90
90 -qy
x qg = x/y= tan
e = d
Lo
Adapted from Fig. 6.1(a) and (c), Callister & Rethwisch 8e.
d/2
Lowo
-deL=
L
wo
dL/2
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Chapter 6 - 9
Stress-Strain Testing
Typical tensile test
machine
Adapted from Fig. 6.3, Callister & Rethwisch 8e. (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.)
specimenextensometer
Typical tensile
specimen
Adapted from
Fig. 6.2,
Callister &Rethwisch 8e.
gauge
length
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Chapter 6 - 10
Linear Elastic Properties
Modulus of Elasticity, E:(also known as Young's modulus)
Hooke's Law:
s = Ee s
Linear-elastic
E
e
F
Fsimpletensiontest
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Chapter 6 - 11
Poisson's ratio, n
Poisson's ratio, n:
Units:
E: [GPa] or [psi]
n: dimensionless
n > 0.50 density increases
n < 0.50 density decreases(voids form)
eL
e
-n
en= - L
e
metals: n ~ 0.33
ceramics: n ~ 0.25
polymers: n ~ 0.40
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Chapter 6 - 12
Mechanical Properties
Slope of stress strain plot (which is
proportional to the elastic modulus) dependson bond strength of metal
Adapted from Fig. 6.7,
Callister & Rethwisch 8e.
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Chapter 6 - 13
Elastic Shear
modulus, G:
t
Gg
t = Gg
Other Elastic Properties
simple
torsion
test
M
M
Special relations for isotropic materials:
2(1n)
EG=
3(12n)
EK =
Elastic Bulk
modulus, K:
pressure
test: Init.vol =Vo.
Vol chg.
= V
P
P PP= -K
VVo
P
V
KVo
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Chapter 6 - 14
Metals
Alloys
Graphite
Ceramics
Semicond
PolymersComposites
/fibers
E(GPa)
Based on data in Table B.2,
Callister & Rethwisch 8e.
Composite data based on
reinforced epoxy with 60 vol%
of aligned
carbon (CFRE),
aramid (AFRE), or
glass (GFRE)
fibers.
Youngs Moduli: Comparison
109 Pa
0.2
8
0.6
1
Magnesium,
Aluminum
Platinum
Silver, Gold
Tantalum
Zinc, Ti
Steel, Ni
Molybdenum
Graphite
Si crystal
Glass -soda
Concrete
Si nitrideAl 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
6080
100
200
600800
10001200
400
Tin
Cu alloys
Tungsten
Si carbide
Diamond
PTFE
HDPE
LDPE
PP
Polyester
PSPET
CFRE( fibers) *
GFRE( fibers)*
GFRE(|| fibers)*
AFRE(|| fibers)*
CFRE(|| fibers)*
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Chapter 6 - 15
Simple tension:
d = FLoEAo
dL
= - nFwoEAo
Material, geometric, and loading parameters all
contribute to deflection.
Larger elastic moduli minimize elastic deflection.
Useful Linear Elastic Relationships
F
Aod/2
dL/2
Lowo
Simple torsion:
a = 2MLoro
4G
M = momenta = angle of twist
2ro
Lo
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Chapter 6 - 16
(at lower temperatures, i.e. T< Tmelt/3)
Plastic (Permanent) Deformation
Simple tension test:
engineering stress, s
engineering strain, e
Elastic+Plasticat larger stress
ep
plastic strain
Elasticinitially
Adapted from Fig. 6.10(a),
Callister & Rethwisch 8e.
permanent (plastic)after load is removed
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Chapter 6 - 17
Stress at which not iceable plastic deformation has
occurred.when ep = 0.002
Yield Strength,sy
sy= yield strength
Note: for 2 inch sample
e = 0.002 = z/z
z= 0.004 in
Adapted from Fig. 6.10(a),
Callister & Rethwisch 8e.
tensile stress, s
engineering strain, e
sy
ep = 0.002
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Chapter 6 - 18
Room temperaturevalues
Based on data in Table B.4,
Callister & Rethwisch 8e.
a = annealed
hr = hot rolled
ag = aged
cd = cold drawn
cw = cold worked
qt = quenched & tempered
Yield Strength : ComparisonGraphite/Ceramics/Semicond
Metals/Alloys
Composites/fibers
Polymers
Yieldstreng
th,
sy
(MPa
)
PVC
H
ard
tomeasure
,
since
intens
ion,
fra
ctureusua
llyoccurs
beforey
ield
.
Nylon 6,6
LDPE
70
20
40
6050
100
10
30
200
300
400
500600700
1000
2000
Tin (pure)
Al (6061) a
Al (6061) ag
Cu (71500) hrTa (pure)Ti (pure) aSteel (1020) hr
Steel (1020) cd
Steel (4140) a
Steel (4140) qt
Ti (5Al-2.5Sn) aW (pure)
Mo (pure)Cu (71500) cw
Hard
tomeasure,
inceram
icma
trixan
depoxyma
trixcompo
sites,
since
intens
ion,
frac
tureusua
llyoccurs
befo
rey
ield
.
HDPEPP
humid
dry
PC
PET
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Chapter 6 -
VMSE: Virtual Tensile Testing
19
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Chapter 6 - 20
Tensile Strength, TS
Metals: occurs when noticeable necking starts.
Polymers: occurs when polymer backbone chains are
aligned and about to break.
Adapted from Fig. 6.11,Callister & Rethwisch 8e.
sy
strain
Typical response of a metal
F= fracture or
ultimate
strength
Neck acts
as stress
concentratorenginee
ring
TS
stres
s
engineering strain
Maximum stress on engineering stress-strain curve.
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Chapter 6 - 21
Tensile Strength: Comparison
Si crystal
Graphite/Ceramics/Semicond
Metals/Alloys
Composites/fibers
Polymers
Tens
ilest
reng
th,
TS
(MPa
)
PVC
Nylon 6,6
10
100
200300
1000
Al (6061) a
Al (6061) ag
Cu (71500) hr
Ta (pure)Ti (pure) a
Steel (1020)
Steel (4140) a
Steel (4140) qt
Ti (5Al-2.5Sn) aW (pure)
Cu (71500) cw
LDPE
PP
PC PET
20
3040
2000
3000
5000
Graphite
Al oxide
Concrete
Diamond
Glass-soda
Si nitride
HDPE
wood ( fiber)
wood(|| fiber)
1
GFRE(|| fiber)
GFRE( fiber)
CFRE(|| fiber)
CFRE( fiber)
AFRE(|| fiber)
AFRE( fiber)
E-glass fibC fibersAramid fib
Based on data in Table B.4,
Callister & Rethwisch 8e.
a = annealed
hr = hot rolled
ag = aged
cd = cold drawncw = cold worked
qt = quenched & tempered
AFRE, GFRE, & CFRE =
aramid, glass, & carbon
fiber-reinforced epoxy
composites, with 60 vol%
fibers.
Room temperaturevalues
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Chapter 6 - 22
Plastic tensile strain at failure:
Ductility
Another ductility measure: 100xA
AARA%
o
fo-
=
x 100
L
LLEL%
o
of -
=
LfAo AfLo
Adapted from Fig. 6.13,
Callister & Rethwisch 8e.
Engineering tensile strain, e
Engineering
tensile
stress, s
smaller %EL
larger %EL
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Chapter 6 - 23
Energy to break a unit volume of material
Approximate by the area under the stress-strain curve.
Toughness
Brittle fracture: elastic energy
Ductile fracture: elastic + plastic energy
Adapted from Fig. 6.13,
Callister & Rethwisch 8e.
very small toughness(unreinforced polymers)
Engineering tensile strain, e
Engineeringtensile
stress, s
small toughness (ceramics)
large toughness (metals)
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Chapter 6 - 24
Resilience, Ur
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 & Rethwisch 8e.
yyr2
1U es@
e
es= y
dUr0
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Chapter 6 - 25
Elastic Strain Recovery
Adapted from Fig. 6.17,
Callister & Rethwisch 8e.
Stre
ss
Strain
3. Reapplyload
2. Unload
D
Elastic strain
recovery
1. Load
syo
syi
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Chapter 6 - 26
Hardness
Resistance to permanently indenting the surface.
Large hardness means:-- resistance to plastic deformation or cracking in
compression.
-- better wear properties.
e.g.,10 mm sphere
apply known force measure sizeof indent afterremoving load
dDSmaller indentsmean largerhardness.
increasing hardness
mostplastics
brassesAl alloys
easy to machinesteels file hard
cuttingtools
nitridedsteels diamond
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Chapter 6 - 27
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
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Chapter 6 - 28
Hardness: MeasurementTable 6.5
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Chapter 6 - 29
True Stress & Strain
Note: S.A. changes when sample stretched
True stress
True strain
iT AF=s
oiT ln=e
e+=e
e+s=s
1ln
1
T
T
Adapted from Fig. 6.16,
Callister & Rethwisch 8e.
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Chapter 6 - 31
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
s =
n
xi-x
2
n- 1
1
2
n
xx
n
n
=
where n is the number of data points
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Chapter 6 - 32
Design uncertainties mean we do not push the limit.
Factor of safety, N
N
y
working
s=s
Often Nisbetween
1.2 and 4
Example: Calculate a diameter, d, to ensure that yield doesnot occur in the 1045 carbon steel rod below. Use a
factor of safety of 5.
Design or Safety Factors
220 ,000 N
d2/ 4
5N
y
working
s=s 1045 plain
carbon steel:
sy= 310 MPa
TS = 565 MPa
F= 220,000N
d
Lo
d= 0.067 m = 6.7 cm
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Chapter 6 - 33
Stress and strain: These are size-independentmeasures 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 (Eor G).
Toughness: The energy needed to break a unit
volume of material.
Ductility: The plastic strain at failure.
Summary
Plastic behavior: This permanent deformation
behavior occurs when the tensile (or compressive)
uniaxial stress reaches sy.
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