Mechanical Properties of Materials
1
Chapter 6 -
2
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 -
Stress-Strain Testing
3
• Typical tensile test � machine
specimen
extensometer
• Typical tensile � specimen
Chapter 6 -
Engineering Stress
4
Units for stress:� MPa = 106 Pa = 106 N/m2 or lbf /in2
• Shear stress, τ:
Area, Ao
F
F
τ
=
F
A
o
• Tensile stress, σ:
original cross-sectional
area before loading
σ
=
F
A
o
Area, Ao
F
F
Chapter 6 -
Engineering Strain
5
• Tensile strain (εz):
• Lateral strain (εx):
Both tensile and shear strain �are dimensionless
• Shear strain (γ):
θ
y
Δx
γ = Δx/y = tan θ
εz
=
Δl
lo
Δl/2
lo
do
-
Δd
ε
x
=
d0
Δd/2
Chapter 6 -
Common States of Stress
6
• Simple tension:
cable
Ski lift (photo courtesy P.M. Anderson)
σ
=
F
A
o
= cross-sectional
area of cable (with no load)
F = force
F
Tensile stress = σ
A0
Chapter 6 -
Common States of Stress (cont.)
7
• Torsion (a form of shear): drive shaft
Ski lift (photo courtesy P.M. Anderson)
s
τ
=
F
A
M
M = moment
2R
A
c
A
s
F
M
AcR
=
AC = cross-sectional � area of drive shaft � (with no load)
Chapter 6 -
OTHER COMMON STRESS STATES (i)
8
(photo courtesy P.M. Anderson)
Canyon Bridge, Los Alamos, NM
o
σ
=
F
A
• Simple compression:
Note: structure members�are under compression
(F < 0 and σ < 0).
(photo courtesy P.M. Anderson)
A
o
Balanced Rock, Arches
National Park
Chapter 6 -
OTHER COMMON STRESS STATES (ii)
9
• Bi-axial tension:
• Hydrostatic compression:
Pressurized tank
σ < 0
h
(photo courtesy
P.M. Anderson)
(photo courtesy
P.M. Anderson)
Fish under water
σ
z
> 0
σ
θ
> 0
Chapter 6 -
True Stress & Strain
10
Adapted from Fig. 6.16, Callister & Rethwisch 10e.
where Ai = instantaneous � cross-sectional � area
Conversion Equations: � valid only to the onset � of necking
Chapter 6 -
True Stress-True Strain Relationship
11
• Most alloys, between point of yielding and onset of necking
−− n and K values depend on alloy and treatment
−− n = strain-hardening exponent
−− n < 1.0
• σT vs. εT -- influence of n.
σ
T
=
K
ε
T
(
)
n
σT
εT
larger n
small n
Chapter 6 -
Linear Elastic Properties
12
• Hooke's Law:
σ = E ε
σ
Linear-
elastic
• Modulus of Elasticity, E:
(also known as Young's modulus)
E
ε
• Elastic deformation is nonpermanent and reversible!
– generally valid at small deformations
– linear stress strain curve
compression
tension
Units:
E: [GPa] or [psi]
1 GPa = 109 Pa
Chapter 6 -
Useful Linear Elastic Relationships
13
• Simple tension:
Δl
=
Fl
o
E
A
o
Δd
=
-
ν
Fd
o
E
A
o
• Deflection is dependent on � material, geometric, and � loading parameters.
• Materials with large elastic � moduli deform less
A
o
Chapter 6 -
Useful Linear Elastic Relationships (cont.)
14
• Simple torsion
α
=
32
Ml
o
π
d
o
4
G
M = moment
α
= angle of twist
do
lo
Chapter 6 -
Elastic Modulus – Comparison of Material Types
15
Metals
Alloys
Graphite
Ceramics
Semicond
Polymers
Composites
/fibers
E(GPa)
Composite data based on
reinforced epoxy with 60 vol%
of aligned
carbon (CFRE),
aramid (AFRE), or
glass (GFRE)
fibers.
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
4
6
10
2
0
4
0
6
0
8
0
10
0
2
00
6
00
8
00
10
00
1200
4
00
Tin
Cu alloys
Tungsten
<100>
<111>
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)*
Chapter 6 -
Elastic Deformation
16
Elastic deformation is �nonpermanent and reversible!
2. Small load
Force, F
Δl
bonds
stretch
1. Initial
3. Unload
return to
initial
F
Δl
Linear-
elastic
Non-Linear-
elastic
Atomic configurations—before, during, after load (force) application
= metal atom
Chapter 6 -
Influence of Bonding Forces
17
Fig. 6.7, Callister & Rethwisch 10e.
Interatomic
Separation r
Interatomic
Force F
Stongly bonded – larger E
Weakly bonded – smaller E
Chapter 6 -
Poisson's ratio
18
• Poisson's ratio, ν:
Units:
ν: dimensionless
For most metals, ceramics and polymers:
0.15 < ν ≤ 0.50
metals: ν ~ 0.33�ceramics: ν ~ 0.25�polymers: ν ~ 0.40
εz
εx
-ν
ε
ν
=
-
z
εx
compression
tension
Chapter 6 -
Other Elastic Properties
19
• Elastic Shear
modulus, G:
τ
G
γ
τ = G γ
simple
torsion
test
M
M
• Elastic constant relationships for isotropic materials:
2(1 + ν)
E
G
=
3(1 - 2ν)
E
K
=
= moment
0
• Elastic Bulk
modulus, K:
Pressure test: �Init. vol. = Vo
Vol. chg. = ΔV
P =
P
P
P = -
K
Δ
V
V
o
P
-Δ
V
K
V
o
hydrostatic
pressure
0
Chapter 6 -
Plastic Deformation (Metals)
20
Plastic deformation is permanent and nonrecoverable.
F
Δl
linear
elastic
linear
elastic
3. Unload
atoms
remain
displaced
Δl
plastic
1. Initial
= metal atom
2. Apply load
F
Δl
elastic +
bonds
stretch
& atoms
displaced
Δl
plastic
Δl
plastic
Chapter 6 -
Plastic Deformation
21
• Stress-strain plot for simple tension test:
Adapted from Fig. 6.10 (a),
Callister & Rethwisch 10e.
• Plastic Deformation is permanent and nonrecoverable
stress, σ
strain, ε
Stressed into �Plastic Region,�Elastic + Plastic
εp
plastic strain
Elastic�Deformation
Stress Removed,
Plastic Deformation
Remains
Chapter 6 -
Yield Strength
22
• Yield strength = stress at which noticeable plastic deformation � has occurred
Adapted from Fig. 6.10 (a),
Callister & Rethwisch 10e.
• Transition from elastic to plastic deformation is gradual
σy = yield strength
Note: for 5 cm sample
ε = 0.002 = Δz/z
Δz = 0.01 cm
when εp = 0.002
σ (stress)
ε (strain)
σy
ε
p
= 0.002
Chapter 6 -
Yield Strength – Comparison of Material Types
23
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
Graphite/
Ceramics/
Semicond
Metals/
Alloys
Composites/
fibers
Polymers
Yield strength,
σ
y
(MPa)
PVC
Hard to measure
,
since in tension, fracture usually occurs before yield.
Nylon 6,6
LDPE
70
20
40
60
50
100
10
30
200
300
400
500
600
700
1000
2000
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
Hard to measure,
in ceramic matrix and epoxy matrix composites, since
in tension, fracture usually occurs before yield.
H
DPE
PP
humid
dry
PC
PET
¨
Chapter 6 -
VMSE: Virtual Tensile Testing
24
Chapter 6 -
Tensile Strength
25
• Metals: Maximum on stress-strain curve appears at the onset � of noticeable necking
Adapted from Fig. 6.11, Callister & Rethwisch 10e.
σy
strain
Typical response of a metal
Fracture strength
Neck – acts �as stress concentrator
engineering
TS
stress
engineering strain
• Tensile strength (TS) = maximum stress on engineering � stress-strain curve.
Chapter 6 -
Tensile Strength: Comparison of Material Types
26
Si crystal
<100>
Graphite/
Ceramics/
Semicond
Metals/
Alloys
Composites/
fibers
Polymers
Tensile
strength, TS
(MPa)
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)
a
W
(pure)
Cu
(71500)
cw
L
DPE
PP
PC
PET
20
30
40
2000
3000
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
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.
Room temperature� values
Chapter 6 -
Ductility
27
• Ductility = amount of plastic deformation at failure:
• Specification of ductility
-- Percent elongation:
-- Percent reduction in area:
lf
Ao
Af
lo
Adapted from Fig. 6.13, Callister & Rethwisch 10e.
tensile strain, ε
tensile
stress, σ
low ductility
high ductility
Chapter 6 -
Resilience
28
Ur = Area under stress-strain curve
�to yielding
If assume a linear stress-strain curve this simplifies to
y
y
r
2
1
U
ε
σ
≅
εy
Fig. 6.15, Callister & Rethwisch 10e.
Chapter 6 -
Toughness
29
• 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
Brittle fracture: small toughness�Ductile fracture: large toughness
very small toughness
(unreinforced polymers)
tensile strain, ε
tensile
stress, σ
small toughness (ceramics)
large toughness (metals)
Chapter 6 -
Elastic Strain Recovery
30
Fig. 6.17, Callister & Rethwisch 10e.
Stress
Strain
3. Reapply
load
2. Unload
D
Elastic strain
recovery
1. Load
initial yield strength = σyo
yield strength for 2nd � deformation = σyi
Chapter 6 -
Hardness
31
• 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 type-
10 mm sphere
apply known force
measure size
of indent after
removing load
d
D
Smaller indents
mean larger
hardness.
increasing hardness
most
plastics
brasses
Al alloys
easy to machine
steels
file hard
cutting
tools
nitrided
steels
diamond
Chapter 6 -
Measurement of Hardness
• 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
32
Rockwell Hardness
• Several scales—combination of load magnitude, indenter size
Chapter 6 -
Measurement of Hardness (cont.)
33
Brinell Hardness
Chapter 6 -
Variability of Material Properties
34
n = number of measurements
xi = specific measured value
i = 1
n
i = 1
n
Chapter 6 -
Design/Safety Factors
35
• 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
Depending on application, �N is between 1.2 and 4
yield strength
Chapter 6 -
Design/Safety Factors (cont.)
36
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.
4
Steel rod:
σ
y
= 310 MPa
F = 220,000 N
d
d = 0.060 m = 60 mm
Solving for the rod diameter d yields
Chapter 6 -
Summary
37
• Applied mechanical force—normalized to stress
• 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
• Degree of deformation—normalized to strain
• Stiffness—a material's resistance to elastic deformation
−−elastic (or Young's) modulus
Chapter 6 -
Summary (cont.)
38
• 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 -
ANNOUNCEMENTS
39
Core Problems:
Self-help Problems:
Reading:
Chapter 6 -