ch6 callister
TRANSCRIPT
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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 muchdeformation occurs? What materials deform least?
Plastic behavior: At what point does permanentdeformation 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
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Chapter 6 - 2
Elastic means reversible !
Elastic Deformation1. Initial 2. Small load 3. Unload
F
d
bondsstretch
return toinitial
F
d
Linear-elastic
Non-Linear- elastic
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Chapter 6 - 3
Plastic means permanent !
Plastic Deformation (Metals)
F
d linearelastic
linearelastic
d plastic
1. Initial 2. Small load 3. Unload
p lanesstillsheared
F
d elastic + plastic
bondsstretch& planesshear
d plastic
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Chapter 6 - 4
Stress has units:
N/m2 or lb f /in2
Engineering Stress Shear stress, t :
Area, A
F t
F t
F s
F
F
F s
t = F s A o
Tensile stress, s :
original areabefore loading
Area, A
F t
F t
s = F t A o
2f
2mNor
inlb=
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Chapter 6 - 5
Simple tension: cable
Note: t = M / Ac R here.
Common States of Stress
A o = cross sectionalarea (when unloaded)
F F
o s =
F A
o t =
F s A
s s
M
M A o
2R
F s A c
Torsion (a form of shear): drive shaftSki lift (photo courtesyP.M. Anderson)
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Chapter 6 - 6
(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM
o s = F A
Simple compression:
Note: compressivestructure member(s < 0 here).(photo courtesy P.M. Anderson)
OTHER COMMON STRESS STATES (1)
A o
Balanced Rock, ArchesNational Park
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Chapter 6 - 7
Bi-axial tension: Hydrostatic compression:
Pressurized tank
s < 0 h
(photo courtesyP.M. Anderson)
(photo courtesy
P.M. Anderson)
OTHER COMMON STRESS STATES (2)
Fish under water
s z > 0
s q > 0
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Chapter 6 - 8
Tensile strain: Lateral strain:
Shear strain:
Strain is alwaysdimensionless.
Engineering Strain
q
90
90 - q y
x q g = x /y = tan
e = d
Lo
- d e L = L w o
Adapted from Fig. 6.1 (a) and (c), Callister 7e.
d /2
d L /2
Lowo
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Chapter 6 - 9
Stress-Strain Testing Typical tensile test
machine
Adapted from Fig. 6.3, Callister 7e. (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.)
specimen extensometer
Typical tensile
specimen
Adapted fromFig. 6.2,
Callister 7e.
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 = E e s
Linear-elastic
E
e
F
F simpletensiontest
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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
e n = - L e
metals : n ~ 0.33ceramics : n ~ 0.25polymers : 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 7e.
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Chapter 6 - 13
Elastic Shearmodulus, G :
t G
g t = G g
Other Elastic Properties
simpletorsiontest
M
M
Special relations for isotropic materials:
2(1 n) E
G = 3(1 - 2n)
E K =
Elastic Bulkmodulus, K:
pressure
test: Init.vol = V o.Vol chg.= V
P
P P P = - K
VVo
P V
K V o
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Chapter 6 - 14
Metals
Alloys
GraphiteCeramicsSemicond
Polymers Composites
/fibers
E (GPa)
Based on data in Table B2,Callister 7e .Composite data based onreinforced epoxy with 60 vol%of alignedcarbon (CFRE),aramid (AFRE), orglass (GFRE)fibers.
Youngs Moduli: Comparison
10 9 Pa
0.2
8
0.6
1
Magnesium, Aluminum
Platinum
Silver, Gold
Tantalum
Zinc, Ti
Steel, Ni Molybdenum
G raphite
Si crystal
Glass - soda
Concrete
Si nitride Al oxide
PC
Wood( grain)
AFRE( fibers) *
CFRE *
GFRE*
Glass fibers only
Carbon fibers only
A ramid fibers only
Epoxy only
0.4
0.8
2
46
10
2 0
4 06 08 010 0
200
600800
10 001200
400
Tin
Cu alloys
Tungsten
Si carbide
Diamond
PTF E
HDP E
LDPE
PP
Polyester
PS PET
C FRE( fibers) *
G FRE( fibers)*
G FRE(|| fibers)*
A FRE(|| fibers)*
C FRE(|| fibers)*
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Chapter 6 - 15
Simple tension:
d = FL o E A o
d L = - n Fw o E A o
Material, geometric, and loading parameters all contribute to deflection.
Larger elastic moduli minimize elastic deflection.
Useful Linear Elastic Relationships
F
A o d /2
d L /2
Lo wo
Simple torsion:
a = 2 ML o p r o
4 G
M = momenta = angle of twist
2r o
Lo
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Chapter 6 - 16
(at lower temperatures, i.e. T < T melt /3)Plastic (Permanent) Deformation
Simple tension test:
engineering stress, s
engineering strain, e
Elastic+Plasticat larger stress
permanent (plastic)after load is removed
e p
plastic strain
Elasticinitially
Adapted from Fig. 6.10 (a),Callister 7e.
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Chapter 6 - 17
Stress at which n o t i c e a b l e plastic deformation hasoccurred.
when e p = 0.002
Yield Strength, s y
s y = yield strength
Note: for 2 inch sample
e = 0.002 = z /z
z = 0.004 in
Adapted from Fig. 6.10 (a),Callister 7e.
tensile stress, s
engineering strain, e
s y
e p = 0.002
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Chapter 6 - 18
Room T values
Based on data in Table B4,Callister 7e .a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered
Yield Strength : ComparisonGraphite/Ceramics/Semicond
Metals/ Alloys
Composites/fibers Polymers
Y i e l d s t r e n g
t h ,
s y ( M P a
)
PVC
H
a r d
t o m e a s u r e
,
s i n c e
i n t e n s
i o n , f r a
c t u r e u s u a
l l y o c c u r s
b e f o r e y i e
l d .
Nylon 6,6
LDPE
70
20
40
6050
100
10
30
2 00
3 00
4 005 006 007 00
10 00
2 0 00
Tin (pure)
Al (6061) a
Al (6061) ag
Cu (71500) hr Ta (pure) Ti (pure) a Steel (1020) hr
Steel (1020) cd
Steel (4140) a
Steel (4140) qt
Ti (5Al-2.5Sn) a W (pure)
Mo (pure) Cu (71500) cw
H a r d
t o m e a s u r e ,
i n c e r a m
i c m a
t r i x a n
d e p o x y m a
t r i x c o m p o s
i t e s , s
i n c e
i n t e n s i o n ,
f r a c
t u r e u s u a
l l y o c c u r s
b e f o r e y i e
l d .
H DPE PP
humid
dry
PC PET
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Chapter 6 - 19
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 7e.
s y
strain
Typical response of a metal
F = fracture orultimatestrength
Neck actsas stressconcentrator
e n g
i n e e r i n g
TS
s t r e s s
engineering strain
Maximum stress on engineering stress -strain curve.
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Chapter 6 - 20
Tensile Strength : Comparison
Si crystal
Graphite/Ceramics/Semicond
Metals/ Alloys
Composites/fibers Polymers
T e n s
i l e s t r e n g
t h ,
T S
( M P a
)
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) a W (pure)
Cu (71500) cw
L DPE
PP
PC PET
20
3040
20003000
5000
Graphite
Al oxide
Concrete
Diamond
Glass-soda
Si nitride
H DPE
wood ( fiber)
wood(|| fiber)
1
GFRE (|| fiber)
GFRE ( fiber)
C FRE (|| fiber)
C FRE ( fiber)
A FRE (|| fiber)
A FRE( fiber)
E-glass fib C fibers Aramid fib
Room Temp. valuesBased on data in Table B4,Callister 7e .a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered
AFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%
fibers.
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Chapter 6 - 21
Plastic tensile strain at failure:
Adapted from Fig. 6.13,Callister 7e.
Ductility
Another ductility measure: 100x A
A ARA%
o
fo -=
x 100 L
L L EL %
o
o f - =
Engineering tensile strain, e
E ngineeringtensile
stress, s
smaller % EL
larger % EL Lf Ao A f Lo
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Chapter 6 - 22
Energy to break a unit volume of material Approximate by the area under the stress -strain
curve.
Toughness
Brittle fracture: elastic energyDuctile fracture: elastic + plastic energy
very small toughness(unreinforced polymers)
Engineering tensile strain, e
E ngineeringtensile
stress, s
small toughness (ceramics)
large toughness (metals)
Adapted from Fig. 6.13,Callister 7e.
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Chapter 6 - 23
Resilience, U r
Ability of a material to store energy Energy stored best in elastic region
If we assume a linearstress-strain curve thissimplifies to
Adapted from Fig. 6.15,
Callister 7e.
y y r 2 1
U e s @
ees= y d U r 0
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Chapter 6 - 24
Elastic Strain Recovery
Adapted from Fig. 6.17,Callister 7e.
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Chapter 6 - 25
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
d D Smaller indentsmean largerhardness.
increasing hardness
mostplastics
brasses Al alloys
easy to machinesteels file hard
cuttingtools
nitridedsteels diamond
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Chapter 6 - 26
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 - 27
Hardness: MeasurementTable 6.5
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Chapter 6 - 28
True Stress & StrainNote: S.A. changes when sample stretched
True stress
True Strain
i T AF =s
oi T ln=e e+=e
e+s=s1ln
1
T
T
Adapted from Fig. 6.16,Callister 7e.
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Chapter 6 - 29
Hardening
Curve fit to the stress -strain response:
s T = K e T ntrue stress ( F / A) true strain: ln( L/Lo)
hardening exponent:n = 0.15 (some steels)to n = 0.5 (some coppers)
An increase in s y due to plastic deformation.
s
e
large hardening
small hardening s y 0
s y 1
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Chapter 6 - 30
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 2
1
2
1 --=
n x x s i
n
n x x n
n
=
where n is the number of data points
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Chapter 6 - 31
Design uncertainties mean we do not push the limit.
Factor of safety, N
N y
working
s=s
Often N isbetween1.2 and 4
Example: Calculate a diameter, d , to ensure that yield doesnot occur in the 1045 carbon steel rod below. Use afactor of safety of 5.
Design or Safety Factors
40002202 / d
N ,p
5N
y working
s=s 1045 plain
carbon steel:s y = 310 MPaTS = 565 MPa
F = 220,000N
d
Lo
d = 0.067 m = 6.7 cm
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Chapter 6 - 32
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 ( E or G).
Toughness : The energy needed to break a unitvolume of material.
Ductility : The plastic strain at failure.
Summary
Plastic behavior: This permanent deformationbehavior occurs when the tensile (or compressive)uniaxial stress reaches s y .
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Chapter 6 - 33
Core Problems:
Self-help Problems:
ANNOUNCEMENTS
Reading: