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Chapter 6 - 1
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 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 6:
Mechanical Properties
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
Chapter 6 - 3
Plastic means permanent!
Plastic Deformation (Metals)
F
d
linear elastic
linear elastic
dplastic
1. Initial 2. Small load 3. Unload
planesstill sheared
F
delastic + plastic
bonds stretch & planes shear
dplastic
Chapter 6 - 4
Stress has units:
N/m2 or lbf /in2
Engineering Stress
• Shear stress, t:
Area, Ao
Ft
Ft
Fs
F
F
Fs
t =Fs
Ao
• Tensile stress, s:
original area
before loading
s =Ft
Ao2
f
2m
Nor
in
lb=
Area, Ao
Ft
Ft
Chapter 6 - 5
• Simple tension: cable
Note: t = M/AcR here.
Common States of Stress
o
s =F
A
o
t =Fs
A
ss
M
M Ao
2R
FsAc
• Torsion (a form of shear): drive shaftSki lift (photo courtesy
P.M. Anderson)
Ao = cross sectional
area (when unloaded)
FF
Chapter 6 - 6
Tensile Compressive
ShearTorsion
Chapter 6 - 7
(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM
o
s =F
A
• Simple compression:
Note: compressive
structure member
(s < 0 here).(photo courtesy P.M. Anderson)
OTHER COMMON STRESS STATES (i)
Ao
Balanced Rock, Arches National Park
Chapter 6 - 8
• Bi-axial tension: • Hydrostatic compression:
Pressurized tank
s < 0h
(photo courtesy
P.M. Anderson)
(photo courtesy
P.M. Anderson)
OTHER COMMON STRESS STATES (ii)
Fish under water
sz > 0
sq > 0
Chapter 6 - 9
• Tensile strain: • Lateral strain:
Strain is always
dimensionless.
Engineering Strain
• Shear strain:
q
90º
90º - 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
Chapter 6 -
Poisson’s Ratio
10
Chapter 6 - 11
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
Chapter 6 - 12
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.
Young’s 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
<100>
<111>
Si carbide
Diamond
PTFE
HDPE
LDPE
PP
Polyester
PSPET
CFRE( fibers) *
GFRE( fibers)*
GFRE(|| fibers)*
AFRE(|| fibers)*
CFRE(|| fibers)*
Chapter 6 - 13
(at lower temperatures, i.e. T < Tmelt/3)
Plastic (Permanent) Deformation
• Simple tension test:
engineering stress, s
engineering strain, e
Elastic+Plastic at larger stress
ep
plastic strain
Elastic initially
Adapted from Fig. 6.10(a),
Callister & Rethwisch 8e.
permanent (plastic) after load is removed
Chapter 6 - 14
• Stress at which noticeable 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
Chapter 6 - 15
Room temperature
values
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
Yie
ld s
tre
ng
th,s
y(M
Pa)
PVC
Ha
rd to
me
asu
re,
sin
ce
in t
en
sio
n, fr
actu
re u
su
ally
occu
rs b
efo
re y
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) cdSteel (4140) a
Steel (4140) qt
Ti (5Al-2.5Sn) aW (pure)
Mo (pure)Cu (71500) cw
Ha
rd to
me
asu
re,
in c
era
mic
ma
trix
an
d e
po
xy m
atr
ix c
om
po
sites, sin
ce
in te
nsio
n, fr
actu
re u
su
ally
occu
rs b
efo
re y
ield
.
HDPEPP
humid
dry
PC
PET
¨
Chapter 6 - 16
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
concentrator
en
gin
eering
TSstr
ess
engineering strain
• Maximum stress on engineering stress-strain curve.
Chapter 6 - 17
Tensile Strength: Comparison
Si crystal<100>
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibers
Polymers
Ten
sile
str
en
gth
, T
S(M
Pa)
PVC
Nylon 6,6
10
100
200
300
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 fib
C fibersAramid fib
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
AFRE, GFRE, & CFRE =
aramid, glass, & carbon
fiber-reinforced epoxy
composites, with 60 vol%
fibers.
Room temperature
values
Chapter 6 - 18
• Plastic tensile strain at failure:
Ductility
• Another ductility measure: 100xA
AARA%
o
fo-
=
x 100L
LLEL%
o
of-
=
Lf
AoAf
Lo
Adapted from Fig. 6.13,
Callister & Rethwisch 8e.
Engineering tensile strain, e
Engineering
tensile
stress, s
smaller %EL
larger %EL
Chapter 6 - 19
• 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
Engineering
tensile
stress, s
small toughness (ceramics)
large toughness (metals)
Chapter 6 - 20
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
dUr 0
Chapter 6 - 21
Elastic Strain Recovery
Adapted from Fig. 6.17,
Callister & Rethwisch 8e.
Str
ess
Strain
3. Reapplyload
2. Unload
D
Elastic strain
recovery
1. Load
syo
syi
Chapter 6 - 22
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 size of indent after removing load
dDSmaller indents mean larger hardness.
increasing hardness
most plastics
brasses Al alloys
easy to machine steels file hard
cutting tools
nitrided steels diamond
Chapter 6 - 23
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 - 24
Hardness: MeasurementTable 6.5
Chapter 6 - 25
True Stress & Strain
Note: S.A. changes when sample stretched
• True stress
• True strain
iT AF=s
oiT ln=e
e=e
es=s
1ln
1
T
T
Adapted from Fig. 6.16,
Callister & Rethwisch 8e.
If no volume change
Chapter 6 - 26
Hardening
• Curve fit to the stress-strain response:
sT = K eT n
“true” stress (F/A) “true” strain: ln(L/Lo)
hardening exponent:n = 0.15 (some steels) to n = 0.5 (some coppers)
• An increase in sy due to plastic deformation.s
e
large hardening
small hardeningsy0
sy1
Chapter 6 - 27
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
Chapter 6 - 28
• 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
220,000N
d2 / 4 5
N
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
Chapter 6 - 29
• 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).
• 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.
Chapter 6 - 30
Core Problems: 6.4, 6.8,6.17, 6.22, 6.26,
6.37, 6.45, 6.53, 6.55
Self-help Problems:6.13, 6.50, 6.62
ANNOUNCEMENTS
Reading:6.1-6.12