The Fundamentals of Materials Science
An Introduction to Materials Science
School of Materials Science and Engineering
Shengjuan Li
Email:[email protected]
Chapter 6: Mechanical Properties of Metals
2
Chapter 6: Mechanical Properties
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?
School of Materials Science and Engineering
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Materials Testing MachineMaterials Tester with Temperature
Control to 100kN
HIT pendulum impact testers Micro Vickers hardness tester
USST-ZWICK Joint Laboratory
4
Learning Objectives1. Define engineering stress and engineering strain.
2. State Hooke’s law, and note the conditions under which it
is valid.
3. Define Poisson’s ratio.
4. Given an engineering stress–strain diagram, determine
(a) the modulus of elasticity,
(b) the yield strength (0.002 strain offset),
(c) the tensile strength,
(d) Estimate the percent elongation.
5. For the tensile deformation of a ductile cylindrical
specimen, describe changes in specimen profile to the point
of fracture.
6. Compute ductility in terms of both percent elongation and
percent reduction of area for a material that is loaded in
tension to fracture.
School of Materials Science and Engineering
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7. Give brief definitions of and the units for modulus of
resilience回弹模量 and modulus of toughness韧性模量(static).
8. For a specimen being loaded in tension, given the applied
load, the instantaneous cross-sectional dimensions, as well
as original and instantaneous lengths, be able to compute
true stress and true strain values.
9. Name the two most common hardness-testing
techniques; note two differences between them.
10. (a) Name and briefly describe the two different
Micro-indentation hardness testing techniques, and
(b) cite situations for which these techniques are
generally used.
11. Compute the working stress for a ductile material.
Learning Objectives
School of Materials Science and Engineering
6
Elastic Deformation
Elastic means reversible!
2. Small load
F
d
bonds
stretch
1. Initial 3. Unload
return to
initial
F
d
Linear-elastic
Non-Linear-elastic
School of Materials Science and Engineering
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Plastic Deformation (Metals)
Plastic means permanent!
F
d
linear elastic
linear elastic
dplastic
1. Initial 2. Small load 3. Unload
planesstill sheared
F
delastic + plastic
bonds stretch & planes shear
dplastic
8
6.1 Introduction
The load is static or changes
relatively slowly with time and
is applied uniformly over a
cross section or surface of the
metal materials.
(a) Tension
(b) Compression
(c) Shear
(d) Torsion
Three factors: load; load duration; environmental conditions
(a)Tensile deformation
(b)Compressive deformation
(c)Shear deformation
(d)Torsional deformation
9
Engineering Stress
Stress has units:
N/m2 or lbf /in2
• 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
10
Common States of Stress
• Simple tension: cable
o
s =F
A
Note: t = M/AcR here.
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
11
OTHER COMMON STRESS STATES (i)
(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)
Ao
Balanced Rock, Arches National Park
12
OTHER COMMON STRESS STATES (ii)
• Bi-axial tension: • Hydrostatic compression:
Pressurized tank
s < 0h
(photo courtesy
P.M. Anderson)
(photo courtesy
P.M. Anderson)Fish under water
sz > 0
sq > 0
6.2 Stress and Strain
Stress-Strain Testing
13
• Typical tensile test
machine
A standard tensile specimen with
circular cross section.
specimenExtensometer 引伸计
• Typical tensile
specimen
Adapted from
Fig. 6.2,
Callister &
Rethwisch 8e.
gauge length
The standard diameter (D): 12.8 mm (0.5 in)
The reduced section length: > 4D; 60 mm (2 1/4 in.)
Gauge length: for ductility computations, 50 mm (2.0 in.).
Engineering stress and engineering strain
14
Load or Force versus Elongation:
These load–deformation characteristics are dependent on the
specimen size.
For example, it will require twice the load to produce the same
elongation if the cross-sectional area of the specimen is doubled.
To minimize these geometrical factors, load and elongation are
normalized to the respective parameters of engineering stress
and engineering strain.
engineering stress MPa
engineering strain unitless
15
Engineering Strain
• Tensile strain: • Lateral strain:
Strain is always
dimensionless.
• 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
16
Compression tests
The force is compressive and the specimen contracts
along the direction of the stress.
engineering stress < 0
engineering strain < 0
Compressive tests are used when a material’s behavior
under large and permanent (i.e., plastic) strains is desired,
as in manufacturing applications, or when the material is
brittle in tension.
17
Shear and Torsional tests
Shear stress
• Shear strain:
q
90º
90º - qy
x qg = x/y = tan
pure shear
18
Torsion
The torsional forces produce a rotational motion about the
longitudinal axis of one end of the member relative to the
other end.
Ex. machine axles, drive shafts, twist drills.
Torsional tests: cylindrical solid shafts or
tubes.
A shear stress is a function of the applied
torque T, whereas shear strain is related to
the angle of twist.
19
Geometric Considerations of the Stress State
The stress state is a function of the orientations
of the planes upon which the stresses are taken
to act.
The plane p-p’ that is oriented at some
arbitrary angle.
6.3 Stress-strain behavior
20
• Modulus of Elasticity, E:
(also known as Young's modulus)
• Hooke's Law:
s = E e s
Linear-
elastic
E
e
F
Fsimple tension test
Linear Elastic Properties
(GPa or psi)
Hooke’s law—relationship between engineering stress
and engineering strain for elastic deformation (tension
and compression)
E—thought of as stiffness, or a material’s resistance to
elastic deformation.
The greater the modulus, the stiffer the material, or
the smaller the elastic strain that results from the application
of a given stress.
There are some materials (e.g., gray cast iron, concrete, and many
polymers) for which this elastic portion of the stress–strain curve
is not linear.
21
Tangent modulus
切线模量Secant modulus
割线模量
On an atomic scale, macroscopic
elastic strain is manifested as small
changes in the interatomic
spacing and the stretching of
interatomic bonds.
The magnitude of the modulus
of elasticity is a measure of the
resistance to separation of adjacent
atoms, that is, the interatomic
bonding forces.
This modulus is proportional to the
slope of the interatomic force–
separation curve at the equilibrium
spacing:
22
Mechanical Properties
Mechanical Properties
23
Slope of stress strain plot (which is proportional to the elastic
modulus) depends on bond strength of metal.
Adapted from Fig. 6.7,
Callister & Rethwisch 8e.
结合强度，键强度
24
Other Elastic Properties
• Elastic Shear modulus, G: tG
gt = G gsimple
torsion
test
M
M
• Special relations for isotropic materials:
2(1+n)
EG =
3(1-2n)
EK =
• Elastic Bulk modulus, K:
Pressure test:
Init. vol =Vo.
Vol chg. = V
P
P PP = -K
VVo
P
V
KVo
Poisson's ratio, n
25
• 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
(ex ,ey)
(ez)
26
Young’s Moduli: Comparison
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.
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)*
27
Electron configuration—atomic structure---bonding---properties----applications
28
Useful Linear Elastic Relationships
• Simple tension:
d = FLo
EAo
dL= -nFw o
EAo
• Material, geometric, and loading parameters all
contribute to deflection.挠度
• Larger elastic moduli minimize elastic deflection.
F
Aod/2
dL/2
Lowo
• Simple torsion:
a=2MLo
ro4G
M = moment a = angle of twist
2ro
Lo
6.4 Anelasticity弹性后效，滞弹性
Assumption: the elastic deformation is time independent.
In most engineering materials, time-dependent elastic strain
component.
This time-dependent elastic behavior is known as anelasticity,
and it is due to time-dependent microscopic and atomistic
processes that are attendant to the deformation.
Metallic materials----neglected
Polymeric materials—significant
viscoelastic behavior
29
EXAMPLE PROBLEM 6.1, 6.2
viscous and elastic characteristics when
undergoing deformation
30
Plastic (Permanent) Deformation
(at lower temperatures, i.e. T < Tmelt/3)
• 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
31
Yield Strength, sy
• Stress at which noticeable plastic deformation has
occurred.when ep = 0.002
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 = 0.0020.2% permanent deformation
32
33
Yield Strength : Comparison
Room temperature
values
Based on data in Table B.4,
Callister & Rethwisch 8e.
a = annealed 退火hr = hot rolled 热轧ag = aged 时效cd = cold drawn 冷拔cw = cold worked 冷作qt = quenched & tempered 调质
（淬火） （回火）
Graphite/ 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
te
nsio
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
¨
Tensile Strength, TS
34
• 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.
35
Tensile Strength: Comparison
36
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
37
Ductility
• Plastic tensile strain at failure:
• Another ductility measure:Percent reduction in area
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
A material that experiences very little or no plastic deformation
upon fracture is termed brittle.
Brittle materials are approximately considered to be those
having a fracture strain of less than about 5%.
断裂应变
Percent elongation
38
These properties are sensitive to any prior deformation,
the presence of impurities, and/or any heat treatment to
which the metal has been subjected.
The modulus of elasticity is one mechanical parameter
that is insensitive to these treatments.
钼
Resilience, Ur （回弹性）
Ability of a material to store energy
Energy stored best in elastic region
e
es=y
dUr 0
39
If we assume a linear stress-
strain curve this simplifies to
Adapted from Fig. 6.15,
Callister & Rethwisch 8e.
yyr2
1U es@
modulus of resilience
The modulus of resilience is defined as the maximum
energy that can be absorbed per unit volume without
creating a permanent distortion.
40
yyr21
U es@
Thus, resilient materials are those having high yield
strengths and low moduli of elasticity; such alloys
would be used in spring applications.
Resilience is the ability of a material to absorb
energy when it is deformed elastically, and
release that energy upon unloading
Applications—materials--stress-strain curves--property
41
Toughness 韧性
• Energy to break a unit volume of material
• Approximate by the area under the stress-strain curve.
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)
Elastic Strain Recovery
42
Adapted from Fig. 6.17,
Callister & Rethwisch 8e.
Str
ess
Strain
3. Reapplyload
2. Unload
D
Elastic strain
recovery
1. Load
syo
syi
Process
Property
Hardness
43
• 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
localized plastic deformation (e.g., a
small dent or a scratch)
Hardness tests are performed
more frequently.
1. Simple and inexpensive—
No special specimen, and the
testing apparatus is relatively
inexpensive.
2. Nondestructive—
Neither fractured nor excessively
deformed; a small indentation.
3. Other mechanical
properties often may be
estimated from hardness
data, such as tensile strength.
44
Hardness: Measurement
45
Rockwell Hardness tests (HR)
--depth of indentation
No major sample damage
Each scale runs to 130 but only useful in range 20-100.
Two types of tests：
1）Rockwell
Minor load 10 kg （enhance test accuracy）
Major load 60 (A), 100 (B) & 150 (C) kg
Indenter：A = diamond, B = 1/16 in. ball, C = diamond
2）superficial Rockwell （for thin specimens） Minor load 3 kg
Major load 15 , 30 & 45 kg
HB = Brinell Hardness (HB)
Load/area of indentation
Indenter：10mm in diameter
hardened steel淬火钢，硬化钢;
tungsten carbide碳化钨，硬质合金
Standard loads range：500-3000 kg in 500-kg increment
Correlation between Hardness and Tensile strength
resistance to plastic deformation
TS (psia) = 500 x HB
TS (MPa) = 3.45 x HB
46
Hardness: Measurement
NanoIndentor
Hardness: Measurement
47
Table 6.5
布氏硬度Load/ area of indentation
维氏硬度Load/ area of indentation
努氏硬度Load/ area of indentation
洛氏硬度Depth of indentation
True Stress & Strain
48
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.
Hardening
49
• Curve fit to the stress-strain response:
sT = K eT( )n
“true” stress (F/A) “true” strain: ln(L/Lo)
Strain-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
EXAMPLE PROBLEM 6.4, 6.5
Variability in Material Properties
50
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
The factors lead to uncertainties in measured data:
test method, variations in specimen fabrication
procedures, operator bias, and apparatus calibration.
EXAMPLE PROBLEM 6.6
51
Design or Safety Factors
• 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.
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
52
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).
• Toughness: The energy needed to break a unit
volume of material.
• Ductility: The plastic strain at failure.
• Plastic behavior: This permanent deformation
behavior occurs when the tensile (or compressive)
uniaxial stress reaches sy.
That’s all for
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