Transcript Chapter 6

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 do dislocations
cause permanent deformation? What materials are
most resistant to permanent deformation?
• Toughness and ductility: What are they and how
do we measure them?
Chapter 6- 1
INTRODUCTION (I)
• The need for
– standardized language for expressing
mechanical properties of materials:
• STRENGTH, HARDNESS, DUCTILITY, and
STIFFNESS
– standardized test methods:
• American Society for Testing and Materials
Standards and others…
Chapter 6-
INTRODUCTION (II)
The result of mechanical testing is
generally a response curve or a (set of)
number(s), in this case a STRESS vs.
STRAIN curve
Courtesy of Plastics Technology Laboratories, Inc 50 Pearl Street, Pittsfield, MA 01201
Chapter 6-
Basic Concepts of Stress and Strain
• Need to compare load on specimens of various size
and shapes:
– For tension and compression
• Engineering Stress, σ = F / A0 , where F is load applied
perpendicular to speciment crosssection and A0 is crosssectional area (perpendicular to the force) before application of
the load.
• Engineering Strain, ε = Δl / l0 ( x 100 %), where Δl change in
length, lo is the original length.
– These definitions of stress and strain allow one to compare
test results for specimens of different cross-sectional area A0
and of different length l0.
Chapter 6-
Basic Concepts of Stress and Strain
• Need to compare load on specimens of various size
and shapes:
– For tension and compression
• Engineering Stress, σ = F / A0 , where F is load applied
perpendicular to speciment crosssection and A0 is crosssectional area (perpendicular to the force) before application of
the load.
• Engineering Strain, ε = Δl / l0 ( x 100 %), where Δl change in
length, lo is the original length.
– For shear
• Shear Stress, τ = F / A0 , where F is load applied parallel to
upper and lower specimen faces of area A0.
• Shear Strain, γ = tan θ ( x 100 %), where θ is the strain angle.
These definitions of stress and strain allow one to compare test results for
specimens of different crosssectional area A0 and of different length l0.
Chapter 6-
ENGINEERING STRESS
• Tensile stress, s:
• Shear stress, t:
Ft
s
Ao
original area
before loading
Stress has units:
N/m2 or lb/in2
Chapter 6- 4
ENGINEERING STRAIN
• Tensile strain:
• Lateral strain:
/2
wo
Applied
• Shear strain:
L/2
Lo
/2
L/2
Resulting
/2
 = tan 
/2 - 
/2
Strain is always
dimensionless.
/2
Chapter 6- 8
COMMON STATES OF STRESS
• Simple tension: cable
F
F
Ao = cross sectional
Area (when unloaded)
F
s
Ao
Note: σ > 0 here !
• Simple shear: drive shaft
Ski lift
(photo courtesy P.M. Anderson)
Fs
t 
Ao
Note: t = M/AcR here.
Chapter 6- 5
OTHER COMMON STRESS STATES (1)
• Simple compression:
Ao
Canyon Bridge, Los Alamos, NM
(photo courtesy P.M. Anderson)
Balanced Rock, Arches
National Park
(photo courtesy P.M. Anderson)
Note: compressive
structure member
(s < 0 here).
Chapter 6- 6
OTHER COMMON STRESS STATES (2)
• Bi-axial tension:
Pressurized tank
(photo courtesy
P.M. Anderson)
• Hydrostatic compression:
Fish under water
s > 0
sz > 0
(photo courtesy
P.M. Anderson)
s h< 0
Chapter 6- 7
OTHER COMMON STRESS STATES (3)
• State of stresses in college life:
σ1, classes
σ2, family
s h< 0
σ4, daily challenges, etc…
σ3, friends, etc…
Chapter 6- 7
SIMPLE STRESS-STRAIN TESTING
Typical tensile specimen
Typical tensile test
machine
Adapted from Fig. 6.2,
Callister 6e.
gauge (portion of sample with
=
length reduced cross section)
Adapted from Fig. 6.3, Callister 6e. (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.)
• Other types of tests:
•compression: brittle materials (e.g.,
concrete)
•torsion: cylindrical tubes, shafts.
Chapter 6- 9
Stress-Strain Testing
• Typical tensile test
machine
extensometer
• Typical tensile
specimen
specimen
Adapted from
Fig. 6.2,
Callister 7e.
gauge
length
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.)
Chapter 6 -
Other Types of Application of Load
Chapter 6-
How does deformation take place in
the material at an atomic scale ?
• Two types of deformation :
– Elastic
• Reversible, no change in the shape and the size of
the specimen when the load is released !
• When under load volume of the material changes !
– Plastic
• Irreversible, dislocations cause slip, bonds are
broken, new bonds are made.
• When load is released, specimen does not return to
original size and shape, but volume is preserved !
Chapter 6-
STRESS-STRAIN CURVE
Necking starts
STRESS
σUTS
REGION I
σYIELD
l0 + le
REGION II
HARDENING OCCURS
DISLOCATION MOTION
AND GENERATION !
E
REGION III
σFAILURE or σFRACTURE
Region I : Elastic Deformation
Hooke’s Law
Region II: Uniform Plastic Deformation
Strain is uniform across material
Region III: Non-uniform Plastic Deformation
Deformation is limited to “neck” region
l0 + l e + lp
STRAIN
l0
εYIELD
εUTS
Chapter 6-
ELASTIC DEFORMATION
1. Initial
2. Small load
3. Unload
bonds
stretch
return to
initial

F
Elastic means reversible!
Bonds stretch and but
recover when load is
released.
Chapter 6- 2
LINEAR ELASTIC PROPERTIES
• Modulus of Elasticity, E:
(also known as Young's modulus)
e
• Hooke's Law (Linear):
F
Under Load
s=Ee
• Poisson's ratio, n:
metals: n ~ 0.33
ceramics: ~0.25
polymers: ~0.40
eL
e
-n
1
eL
No load
F
simple
tension
test
Units:
E: [GPa] or [psi]
n: dimensionless
Chapter 6- 10
NON-LINEAR ELASTIC PROPERTIES
• Some materials will exhibit a non-linear elastic behavior
under stress ! Examples are polymers, gray cast iron,
concrete, etc…
Chapter 6-
Linear Elastic Deformation (Atomic
Scale)
Chapter 2: Inter-atomic Bonding ! Young’s Modulus α (dF/dr) at ro , what else ?
If we increase temperature, how will E behave ?
Chapter 6-
Other Elastic Properties
• Elastic Shear
modulus, G:
t
M
G
t=G
• Elastic Bulk
modulus, K:
DV
P = -K
Vo

M
P
P
K
DV P
Vo
• Special relations for isotropic materials:
E
G
2(1 + n)
simple
torsion
test
E
K
3(1 - 2n)
P
pressure
test: Init.
vol =Vo.
Vol chg.
= DV
Chapter 6 -
YOUNG’S MODULI: COMPARISON
Metals
Alloys
1200
1000
800
600
400
E(GPa)
200
100
80
60
40
109 Pa
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
Aramid fibers only
<100>
AFRE(|| fibers)*
Glass-soda
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 B2,
Callister 6e.
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- 12
USEFUL LINEAR ELASTIC RELATIONS
• Simple tension:
• Simple torsion:
M=moment
 =angle of twist
Lo
2ro
• Material, geometric, and loading parameters all
contribute to deflection.
• Larger elastic moduli minimize elastic deflection.
Chapter 6- 13
PLASTIC DEFORMATION (METALS)
1. Initial
2. Small load
3. Unload
F
Plastic means permanent!
linear
elastic
linear
elastic
plastic

Chapter 6- 3
PLASTIC (PERMANENT) DEFORMATION
(at lower temperatures, T < Tmelt/3)
• Simple tension test:
Chapter 6- 14
YIELD STRENGTH, sy
Some materials do NOT exhibit a distinct transition from elastic to plastic region
under stress, so by convention a straight line is drawn parallel to the stress strain
curve with 0.2 % strain. The stress at the intersection is called the yield stress !
Chapter 6-
HARDENING
• An increase in sy due to plastic deformation.
• Curve fit to the stress-strain response:
Chapter 6- 22
YIELD STRENGTH: COMPARISON
sy(ceramics)
>>sy(metals)
>> sy(polymers)
Room T values
Based on data in Table B4,
Callister 6e.
a = annealed
hr = hot rolled
ag = aged
cd = cold drawn
cw = cold worked
qt = quenched & tempered
Chapter 6- 16
TENSILE STRENGTH, TS
• Maximum possible engineering stress in tension.
NECKING
Adapted from Fig. 6.11,
Callister 6e.
FRACTURE
• Metals: occurs when noticeable necking starts.
• Ceramics: occurs when crack propagation starts.
• Polymers: occurs when polymer backbones are
aligned and about to break.
Chapter 6- 17
TENSILE STRENGTH: COMPARISON
TS(ceram)
~TS(met)
~ TS(comp)
>> TS(poly)
Room T values
Based on data in Table B4,
Callister 6e.
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- 18
DUCTILITY, %EL
• Plastic tensile strain at failure:
Adapted from Fig. 6.13,
Callister 6e.
• Note: %AR and %EL are often comparable.
--Reason: crystal slip does not change material volume.
--%AR > %EL possible if internal voids form in neck.
Chapter 6- 19
Mechanical Strength of Materials
Yield Strength, Tensile Strength and Ductility can be improved by alloying, heat and
mechanical treatment, but Youngs Modulus is rather insensitive to such processing !
Temperature effects : YS, TS and YM decrease with increasing temperature, but
ductility increases with temperature !
Chapter 6-
TOUGHNESS & RESILIENCE
• Energy to break a unit volume of material
• Approximate by the area under the stress-strain
curve.
Engineering
tensile
stress, s
smaller toughness (ceramics)
larger toughness
(metals, PMCs)
smaller toughnessunreinforced
polymers
Engineering tensile strain, e
RESILIENCE is energy stored in the material w/o plastic deformation ! Ur = σy2 / 2 E
TOUGHNESS is total energy stored in the material upon fracture !
Chapter 6- 20
Resilience, Ur
• Ability of a material to store energy
– Energy stored best in elastic region
Ur  
ey
0
sde
If we assume a linear
stress-strain curve this
simplifies to
1
Ur @ sy e y
2
Adapted from Fig. 6.15,
Callister 7e.
Chapter 6 -
TRUE STRESS & STRAIN
σT = σ (1+ ε )
εT = ln (1+ε)
The material does NOT get weaker past M
Chapter 6-
HARDNESS
• Resistance to permanently indenting the surface.
• Large hardness means:
--resistance to plastic deformation or cracking in
compression.
--better wear properties.
Adapted from Fig. 6.18, Callister 6e. (Fig. 6.18 is adapted from G.F. Kinney, Engineering Properties
and Applications of Plastics, p. 202, John Wiley and Sons, 1957.)
Chapter 6- 21
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
Chapter 6 -
Hardness: Measurement
Table 6.5
Chapter 6 -
HARDNESS !!
1.
2.
3.
Relatively simple and cheap
technique
Non-destructive
Related to many other
mechanical properties
Chapter 6-
Variability in Material Properties
• Elastic modulus is material property
• Critical properties depend largely on sample flaws
(defects, etc.). Large sample to sample variability.
• Statistics
n
– Mean
– Standard Deviation
 xn
x
n
2
n
 x i - x  

s
 n -1 


1
2
where n is the number of data points
Chapter 6 -
Design or Safety Factors
• Design uncertainties mean we do not push the limit.
• Factor of safety, N
Often N is
sworking 
sy
between
1.2 and 4
N
• 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.
sworking 
220,000N
 d2 / 4


5
sy
N
d
1045 plain
carbon steel:
sy = 310 MPa
TS = 565 MPa
d = 0.067 m = 6.7 cm
Lo
F = 220,000N
Chapter 6 -
Chapter 6-
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).
• Plastic behavior: This permanent deformation
behavior occurs when the tensile (or compressive)
uniaxial stress reaches sy.
• Toughness: The energy needed to break a unit
volume of material.
• Ductility: The plastic strain at failure.
Note: For materials selection cases related to
mechanical behavior, see slides 22-4 to 22-10.
Chapter 6- 24
ANNOUNCEMENTS
Reading: Chapter 6 and Chapter 7
Homework :
1. 6.4, 6.6, 6.7, 6.20, 6.37, 6.44
2. Start reading and understanding !
Due date: 3-25-2008
Chapter 6- 0