Chapter 6: Mechanical Properties of Metals
ISSUES TO ADDRESS...
• When a metal is exposed to mechanical forces, what
parameters are used to express force magnitude and
degree of deformation?
• What is the distinction between elastic and plastic
deformations?
• How are the following mechanical characteristics of
metals measured?
(a) Stiffness
(b) Strength
(c) Ductility
(d) Hardness
• What parameters are used to quantify these properties?
Chapter 6 -
1
Common States of Stress
• Simple tension:
cable
F
F = force
A o = cross-sectional
area of cable (with no load)
Tensile stress = σ
Ski lift (photo courtesy
P.M. Anderson)
F
σ=
A0
Chapter 6 -
2
• Simple compression:
Ao
Canyon Bridge, Los Alamos, NM
(photo courtesy P.M. Anderson)
Balanced Rock, Arches
National Park
(photo courtesy P.M. Anderson)
σ=
F
Ao
Note: structure members
are under compression
(F < 0 and σ < 0).
Chapter 6 -
3
OTHER COMMON STRESS STATES (ii)
• Bi-axial tension:
Pressurized tank
(photo courtesy
P.M. Anderson)
• Hydrostatic compression:
Fish under water
(photo courtesy
P.M. Anderson)
σθ > 0
σz > 0
σh < 0
Chapter 6 -
4
Stress-Strain Testing
• Typical tensile test
machine
extensometer
• Typical tensile
specimen
specimen
Fig. 6.2, Callister & Rethwisch 10e.
Fig. 6.3, Callister & Rethwisch 10e.
(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.)
https://youtu.be/D8U4G5kcpcM
Chapter 6 -
5
Engineering Stress
• Tensile stress, σ:
F
• Shear stress, τ:
F
Area, Ao
Area, Ao
F
F
σ=
Ao
original cross-sectional
area before loading
F
τ =
Ao
F
Units for stress:
MPa = 106 Pa = 106 N/m2 or lbf /in2
Chapter 6 -
6
Engineering Strain
• Tensile strain (εz):
• Lateral strain (εx):
Δl/2
Δl
εz =
lo
• Shear strain (γ):
do
lo
ε x =- Δd
d0
Δd/2
θ
Dx
y
γ = Δx/y = tan θ
Both tensile and shear strain
are dimensionless
Chapter 6 -
7
Useful Linear Elastic Relationships
• Simple tension:
Δl = Fl o
EA o
Δd = - ν Fd o
EA o
Ao
• Deflection is dependent on
material, geometric, and
loading parameters.
• Materials with large elastic
moduli deform less
Chapter 6 -
8
Linear Elastic Properties
• Elastic deformation is nonpermanent and reversible!
– generally valid at small deformations
– linear stress strain curve
• Modulus of Elasticity, E:
(also known as Young's modulus)
σ
tension
• Hooke's Law:
E
σ=Eε
Units:
E: [GPa] or [psi]
1 GPa = 109 Pa
compression
Linearelastic
ε
Chapter 6 -
9
Elastic Modulus – Comparison of
Material Types
Metals
Alloys
1200
1000
800
600
400
E(GPa) 200
100
80
60
40
Graphite
Composites
Ceramics Polymers
/fibers
Semicond
Diamond
Tungsten
Molybdenum
Steel, Ni
Tantalum
Platinum
Cu alloys
Zinc, Ti
Silver, Gold
Aluminum
Magnesium,
Tin
Si carbide
Al oxide
Si nitride
Carbon fibers only
CFRE(|| fibers)*
<111>
Si crystal
<100>
Aramid fibers only
Glass -soda
AFRE(|| fibers)*
Glass fibers only
GFRE(|| fibers)*
Concrete
GFRE*
20
10
8
6
4
2
1
0.8
0.6
0.4
0.2
CFRE*
GFRE( fibers)*
Graphite
Polyester
PET
PS
PC
CFRE( fibers) *
AFRE( fibers) *
Epoxy only
Based on data in Table B.2,
Callister & Rethwisch 10e.
Composite data based on
reinforced epoxy with 60 vol%
of aligned
carbon (CFRE),
aramid (AFRE), or
glass (GFRE)
fibers.
PP
HDPE
PTFE
LDPE
Wood(
grain)
Chapter 6 - 10
Elastic Deformation
Atomic configurations—before, during, after load (force) application
1. Initial
2. Small load
3. Unload
bonds
stretch
return to
initial
Δl
= metal atom
Force, F
Elastic deformation is
nonpermanent and reversible!
F
Linearelastic
Non-Linearelastic
Δl
Chapter 6 - 11
Poisson's ratio
• Poisson's ratio, ν:
εz
compression
εz
ν =εx
metals: ν ~ 0.33
ceramics: ν ~ 0.25
polymers: ν ~ 0.40
Units:
ν: dimensionless
εx
-ν
tension
For most metals, ceramics and
polymers:
0.15 < ν ≤ 0.50
https://youtu.be/tuOlM3P7ygA
Chapter 6 - 12
Other Elastic Properties
• Elastic Shear
modulus, G:
M = moment
τ
G
τ=Gγ
γ
0
• Elastic Bulk
modulus, K:
simple
torsion
test
M
P
ΔV
P = -K
Vo
P
K
-ΔV
Vo
0
P = hydrostatic
pressure
• Elastic constant relationships for isotropic materials:
E
G=
2(1 + ν)
P
Pressure test:
Init. vol. = Vo
Vol. chg. = ΔV
E
K=
3(1 - 2ν)
Chapter 6 - 13
Plastic Deformation (Metals)
1. Initial
= metal atom
2. Apply load
bonds
stretch
& atoms
displaced
3. Unload
Δl elastic + Δl plastic
Δl plastic
atoms
remain
displaced
F
F
Plastic deformation is permanent
and nonrecoverable.
linear
elastic
linear
elastic
Δl plastic
Δl
Chapter 6 - 14
Plastic Deformation
• Plastic Deformation is permanent and nonrecoverable
• Stress-strain plot for simple tension test:
Stressed into
Plastic Region,
Elastic + Plastic
stress, σ
Elastic
Deformation
Stress Removed,
Plastic Deformation
Remains
εp
strain, ε
plastic strain
Adapted from Fig. 6.10 (a),
Callister & Rethwisch 10e.
Chapter 6 - 15
Yield Strength
• Transition from elastic to plastic deformation is gradual
• Yield strength = stress at which noticeable plastic deformation
has occurred
when εp = 0.002
σ (stress)
σy = yield strength
σy
Note: for 5 cm sample
ε = 0.002 = Δz/z
Δz = 0.01 cm
ε p = 0.002
ε (strain)
Adapted from Fig. 6.10 (a),
Callister & Rethwisch 10e.
Chapter 6 - 16
Yield Strength – Comparison of
Material Types
Metals/
Alloys
2000
Graphite/
Composites/
Ceramics/ Polymers
fibers
Semicond
200
Al (6061) ag
Steel (1020) hr
Ti (pure) a
Ta (pure)
Cu (71500) hr
100
70
60
50
40
Al (6061) a
30
20
10
Tin (pure)
¨
dry
PC
Nylon 6,6
PET
PVC humid
PP
HDPE
in ceramic matrix and epoxy matrix composites, since
in tension, fracture usually occurs before yield.
300
Hard to measure,
700
600
500
400
Ti (5Al-2.5Sn) a
W (pure)
Cu (71500) cw
Mo (pure)
Steel (4140) a
Steel (1020) cd
since in tension, fracture usually occurs before yield.
1000
Hard to measure ,
Yield strength,σ y (MPa)
Steel (4140) qt
Room temperature
values
Based on data in Table B.4,
Callister & Rethwisch 10e.
a = annealed
hr = hot rolled
ag = aged
cd = cold drawn
cw = cold worked
qt = quenched & tempered
LDPE
Chapter 6 - 17
VMSE: Virtual Tensile Testing
Chapter 6 - 18
Tensile Strength
• Tensile strength (TS) = maximum stress on engineering
stress-strain curve.
Adapted from Fig. 6.11,
Callister & Rethwisch 10e.
TS
Fracture
strength
engineering
stress
sy
Typical response of a metal
Neck – acts
as stress
concentrator
strain
engineering strain
• Metals: Maximum on stress-strain curve appears at the onset
of noticeable necking
Chapter 6 - 19
Tensile Strength: Comparison of
Material Types
Metals/
Alloys
Tensile strength, TS (MPa)
5000
3000
2000
1000
300
200
100
40
30
20
Graphite/
Composites/
Ceramics/ Polymers
fibers
Semicond
C fibers
Aramid fib
E-glass fib
Steel (4140) qt
Diamond
W (pure)
Ti (5Al-2.5Sn)aa
Steel (4140)
Si nitride
Cu (71500) cw
hr
Cu (71500)
Al oxide
Steel (1020)
Al (6061) ag
Ti (pure) a
Ta (pure)
Al (6061) a
Si crystal
<100>
Glass-soda
Concrete
Graphite
AFRE(|| fiber)
GFRE(|| fiber)
CFRE(|| fiber)
Nylon 6,6
PC PET
PVC
PP
HDPE
wood(|| fiber)
GFRE( fiber)
CFRE( fiber)
AFRE( fiber)
LDPE
10
wood (
1
fiber)
Room temperature
values
Based on data in Table B4,
Callister & Rethwisch 10e.
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.
Chapter 6 - 20
Ductility
• Ductility = amount of plastic deformation at failure:
• Specification of ductility
lf − l0
-- Percent elongation:
%EL =
x 100
l0
-- Percent reduction in area:
A0 − A f
%RA =
x 100
A0
low ductility
tensile
stress, σ
high ductility
lo
Ao
Af
lf
Adapted from Fig. 6.13,
Callister & Rethwisch 10e.
tensile strain, ε
Chapter 6 - 21
Resilience
• Resilience—ability of a material to absorb energy
during elastic deformation
• Energy recovered when load released
• Resilience specified by modulus of resilience, Ur
Ur = Area under stress-strain curve
to yielding =
εy
∫ 0 σ dε
If assume a linear stress-strain
curve this simplifies to
εy
Fig. 6.15, Callister & Rethwisch 10e.
1 ε
Ur ≅ σy y
2
Chapter 6 - 22
Toughness
• Toughness of a material is expressed in several contexts
• For this chapter, toughness = amount of energy absorbed
before fracture
• Approximate by area under the stress-strain curve—units
of energy per unit volume
small toughness (ceramics)
tensile
stress, σ
large toughness (metals)
very small toughness
(unreinforced polymers)
tensile strain, ε
Brittle fracture: small toughness
Ductile fracture: large toughness
Chapter 6 - 23
True Stress & Strain
• True stress
σ T = F Ai
• True strain
εT = ln ( ℓ i ℓ o )
where Ai = instantaneous
cross-sectional
area
Conversion Equations:
valid only to the onset
of necking
σ T = σ (1+ ε )
εT = ln (1+ ε )
Adapted from Fig. 6.16,
Callister & Rethwisch 10e.
Chapter 6 - 24
True Stress-True Strain Relationship
• Most alloys, between point of yielding and onset of necking
n
σT = K εT
( )
-- n and K values depend on alloy and treatment
-- n = strain-hardening exponent
-- n < 1.0
• σT vs. εT -- influence of n.
σT
larger n
small n
εT
Chapter 6 - 25
Elastic Strain Recovery
yield strength for 2nd
deformation = σyi
D
initial yield strength = σyo
Stress
2. Unload
1. Load
3. Reapply
load
Strain
Fig. 6.17, Callister &
Rethwisch 10e.
Elastic strain
recovery
Chapter 6 - 26
Hardness
• Measure of resistance to surface plastic deformation—
dent or scratch.
• Large hardness means:
-- high resistance to deformation from compressive loads.
-- better wear properties.
one indenter type10 mm sphere
apply known force
brasses
Al alloys
Smaller indents
mean larger
hardness.
d
D
most
plastics
measure size
of indent after
removing load
easy to machine
steels
file hard
cutting
tools
nitrided
steels
diamond
increasing hardness
https://youtu.be/7Z90OZ7C2jI
Chapter 6 - 27
Measurement of Hardness
Rockwell Hardness
• Several scales—combination of load magnitude, indenter size
• Examples:
– Rockwell A Scale – 60 kg load/diamond indenter
– Superficial Rockwell 15T Scale – 15 kg load/ 1/16 in. indenter
• Rockwell hardness designation: (hardness reading) HR
• Examples: 57 HRA; 63 HR15T
• Hardness range for each scale: 0-130 HR;
useful range: 20-100 HR
Chapter 6 - 28
Measurement of Hardness (cont.)
Brinell Hardness
• Single scale
• Brinell hardness designation: (hardness
reading) HB
– P = load (kg)
– 500 kg £ P £ 3000 kg (500 kg increments)
• Relationships—Brinell hardness & tensile strength
– TS (psia) = 500 x HB
– TS (MPa) = 3.45 x HB
Chapter 6 - 29
Design/Safety Factors
• Because of design uncertainties allowances must
be made to protect against unanticipated failure
• For structural applications, to protect against possibility
of failure—use working stress, σw, and a
factor of safety, N
σw =
σy
yield strength
N
Depending on application,
N is between 1.2 and 4
Chapter 6 - 30
Design/Safety Factors (cont.)
Example Problem: A cylindrical rod, to be constructed from
a steel that has a yield strength of 310 MPa, is to withstand
a load of 220,000 N without yielding. Assuming a value of 4
for N, specify a suitable bar diameter.
σw =
220,000 N
2
⎛d ⎞
π⎜ ⎟
⎝2⎠
d
σy
N
Steel rod:
σy = 310 MPa
4
F = 220,000 N
Solving for the rod diameter d yields
d = 0.060 m = 60 mm
Chapter 6 - 31
Summary
• Applied mechanical force—normalized to stress
• Degree of deformation—normalized to strain
• Elastic deformation:
--non-permanent; occurs at low levels of stress
--stress-strain behavior is linear
• Plastic deformation
--permanent; occurs at higher levels of stress
--stress-strain behavior is nonlinear
• Stiffness—a material's resistance to elastic deformation
--elastic (or Young's) modulus
Chapter 6 - 32
Summary (cont.)
• Strength—a material's resistance to plastic deformation
--yield and tensile strengths
• Ductility—amount of plastic deformation at failure
--percents elongation, reduction in area
• Hardness—resistance to localized surface deformation
& compressive stresses
--Rockwell, Brinell hardnesses
Chapter 6 - 33