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Chapter 7: 機械性質 (Mechanical Propertie
s) • Tensile & Hardness Testing;Tensile & Hardness Testing;• Macroscopic Behavior;• Elastic Deformation;• Elastic Modulus;• Proportional Limit; • Poisson's Ratio;• Plastic deformation;• Yielding and Yielding Strength; • Tensile Strength;• Fracture Strength;• Ductility; • Resilience; • Toughness• Property Variability;• Design Safety Factors.
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7.2 應力和應變 (Stress and Strain) 的觀念
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拉力試驗
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• Simple tension: cable
Note: R × ×Ac= M here.
Common States of Stress
o F
A
o
FsA
M
M Ao
2R
FsA
c
• Torsion (a form of shear): drive shaft
Ao = cross sectional
area (when unloaded)
FF
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(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM
o F
A
• Simple compression:
Note: compressivestructure member( < 0 here).(photo courtesy P.M. Anderson)
OTHER COMMON STRESS STATES (i)
Ao
Balanced Rock, Arches National Park
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• Bi-axial tension: • Hydrostatic compression:
Pressurized tank
< 0h
(photo courtesyP.M. Anderson)
(photo courtesyP.M. Anderson)
OTHER COMMON STRESS STATES (ii)
Fish under water
z > 0
> 0
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Stress has units: N/m2 or lbf /in2
Engineering Stress• Shear stress, :
Area, Ao
Ft
Ft
Fs
F
F
Fs
= Fs
Ao
• Tensile stress, :
original area before loading
=Ft
Ao2f
2m
Nor
in
lb=
Area, Ao
Ft
Ft
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• Tensile strain: • Lateral strain:
Strain is alwaysdimensionless.
Engineering Strain
• Shear strain:
90º
90º - y
x = x/y = tan
Lo
Adapted from Fig. 7.1 (a) and (c), Callister & Rethwisch 3e.
/2
Lowo
L L
wo
L/2
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彈性變形 (Elastic Deformation)7.3 應力 - 應變行為 (Stress-Strain behavior)
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Elastic means reversible!
Elastic Deformation2. Small load
F
bonds stretch
1. Initial 3. Unload
return to initial
F
Linear- elastic
Non-Linear-elastic
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Linear Elastic Properties• Modulus of Elasticity, E: (also known as Young's modulus)
• Hooke's Law:
= E
Linear- elastic
E
F
Fsimple tension test
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Plastic means permanent!
Plastic Deformation (Metals)
F
linear elastic
linear elastic
plastic
1. Initial 2. Small load 3. Unload
planes still sheared
F
elastic + plastic
bonds stretch & planes shear
plastic
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• 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.
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MetalsAlloys
GraphiteCeramicsSemicond
PolymersComposites
/fibers
E(GPa)
Based on data in Table B.2,Callister & Rethwisch 3e. Composite data based onreinforced epoxy with 60 vol%of alignedcarbon (CFRE),aramid (AFRE), orglass (GFRE)fibers.
Young’s Moduli: Comparison
109 Pa
0.2
8
0.6
1
Magnesium,Aluminum
Platinum
Silver, Gold
Tantalum
Zinc, Ti
Steel, NiMolybdenum
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)*
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7.4 滯彈性 (Anelasticity)
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7.5 材料的彈性性質 (Poission’s Ratio)
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Poisson's Ratio, • Poisson's ratio, :
Units:E: [GPa] or [psi]: Dimensionless
> 0.50 density increases
< 0.50 density decreases (voids form)
L
-
L
metals: ~ 0.33ceramics: ~ 0.25polymers: ~ 0.40
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• Elastic Shear modulus, G:
G
= G
Other Elastic Properties
simpletorsiontest
M
M
• Special relations for isotropic materials:
2(1 )EG
3(1 2)
EK
• Elastic Bulk modulus, K:
pressuretest: Init.
vol =Vo. Vol chg. = V
P
P PP = -K
VVo
P
V
K Vo
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b
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Mechanical Properties:• Macroscopic Behavior;• Elastic Deformation;• Elastic Modulus;• Proportional Limit; • Poisson's Ratio;• Plastic deformation;• Yielding and Yielding
Strength; • Tensile Strength;• Fracture Strength;• Ductility; • Resilience; • Toughness.
7.6 拉力性質 ( 塑性變形 )
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(at lower temperatures, i.e. T < Tmelt/3)Plastic (Permanent) Deformation
• Simple tension test:
engineering stress,
engineering strain,
Elastic+Plastic at larger stress
p
plastic strain
Elastic initially
Adapted from Fig. 7.10 (a),Callister & Rethwisch 3e.
permanent (plastic) after load is removed
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• Stress at which noticeable plastic deformation has occurred.
when p = 0.002
Yielding and Yield Strength, y
y = yield strength
Note: for 2 inch sample
= 0.002 = z/z
z = 0.004 in
Adapted from Fig. 7.10 (a),Callister & Rethwisch 3e.
tensile stress,
engineering strain,
y
p = 0.002
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Tensile Strength, TS
• Metals: occurs when noticeable necking starts.• Polymers: occurs when polymer backbone chains are aligned and about to break.
Adapted from Fig. 7.11, Callister & Rethwisch 3e.
y
strain
Typical response of a metal
F = fracture or
ultimate
strength
Neck – acts as stress concentrator
eng
inee
ring
TS s
tres
s
engineering strain
• Maximum stress on engineering stress-strain curve.
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Room temperature values
Based on data in Table B.4,Callister & Rethwisch 3e. a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered
Yield Strength : ComparisonGraphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibers
Polymers
Yie
ld s
tren
gth,
y
(MP
a)
PVC
Har
d to
mea
sure
,
sin
ce in
te
nsi
on
, fr
act
ure
usu
ally
occ
urs
be
fore
yie
ld.
Nylon 6,6
LDPE
70
20
40
6050
100
10
30
200
300
400500600700
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
Har
d to
mea
sure
, in
ce
ram
ic m
atr
ix a
nd
ep
oxy
ma
trix
co
mp
osi
tes,
sin
cein
te
nsi
on
, fr
act
ure
usu
ally
occ
urs
be
fore
yie
ld.
HDPEPP
humid
dry
PC
PET
¨
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Tensile Strength: Comparison
Si crystal<100>
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibers
Polymers
Ten
sile
stre
ngth
, TS
(M
Pa)
PVC
Nylon 6,6
10
100
200300
1000
Al (6061) a
Al (6061) agCu (71500) hr
Ta (pure)Ti (pure) aSteel (1020)
Steel (4140) a
Steel (4140) qt
Ti (5Al-2.5Sn) aW (pure)
Cu (71500) cw
LDPE
PP
PC PET
20
3040
20003000
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 B4,Callister & Rethwisch 3e. a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & temperedAFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%fibers.
Room temperature values
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Ductility(延性 )
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• Plastic tensile strain at failure:
Ductility
• Another ductility measure: 100xA
AARA%
o
fo -=
x 100L
LLEL%
o
of
Lf
Ao AfLo
Adapted from Fig. 7.13, Callister & Rethwisch 3e.
Engineering tensile strain,
Engineering tensile stress,
smaller %EL
larger %EL
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Resilience( 彈性能 , 回彈性 ), Ur
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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. 7.15, Callister & Rethwisch 3e.
yyr2
1U
y dUr 0
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• Energy to break a unit volume of material• Approximate by the area under the stress-strain curve.
Toughness( 韌性 )
Brittle fracture: elastic energyDuctile fracture: elastic + plastic energy
Adapted from Fig. 7.13, Callister & Rethwisch 3e.
very small toughness (unreinforced polymers)
Engineering tensile strain,
Engineering tensile stress,
small toughness (ceramics)
large toughness (metals)
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7.7 真應力與真應變
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7.8 塑性變形期間之彈性回復
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Elastic Strain Recovery
Adapted from Fig. 7.17, Callister & Rethwisch 3e.
Str
ess
Strain
3. Reapplyload
2. Unload
D
Elastic strainrecovery
1. Load
yo
yi
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7.10 Fexural Strength of Ceramics)
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Mechanical Properties of Ceramics
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
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• Room T behavior is usually elastic, with brittle failure.• 3-Point Bend Testing often used. -- tensile tests are difficult for brittle materials.
Adapted from Fig. 7.18, Callister & Rethwisch 3e.
Flexural Tests – Measurement of Elastic Modulus
FL/2 L/2
= midpoint deflection
cross section
R
b
d
rect. circ.
• Determine elastic modulus according to:
Fx
linear-elastic behavior
F
slope =
3
3
4bd
LFE
(rect. cross section)
4
3
12 R
LFE
(circ. cross section)
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• 3-point bend test to measure room-T flexural strength.
Adapted from Fig. 7.18, Callister & Rethwisch 3e.
Flexural Tests – Measurement of Flexural Strength
FL/2 L/2
= midpoint deflection
cross section
R
b
d
rect. circ.
location of max tension
• Flexural strength: • Typical values:
Data from Table 7.2, Callister & Rethwisch 3e.
Si nitrideSi carbideAl oxideglass (soda-lime)
250-1000100-820275-700
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30434539369
Material fs (MPa) E(GPa)
22
3
bd
LFffs (rect. cross section)
(circ. cross section)3R
LFffs
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7.16 硬度 (Hardness) 與硬度試驗
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7.19 Property Variability and Design Safety Factors
• Elastic modulus is material property• Critical properties depend largely on
sample flaws (defects, etc.). Large sample to sample variability.
• Statistics
– Mean
– Standard Deviation
21
2
1
n
xxs i
n
n
xx n
n
where n is the number of data points
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• 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 y.