MECHANICAL PROPERTIES II
Mechanical Failure
Photograph of the fuselage of an airplane whose canopy was
fractured during midflight to Hawaii. Investigations showed that the
canopy was weakened as a consequence of corrosion and fatigue.
[Source: © Robert Nichols, Black Star]
S. O. Kasap
University of Saskatchewan
CANADA
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v.1.1 © 1990-2001, S.O. Kasap
Mechanical properties II: Mechanical Failure (© S.O. Kasap, 1990 - 2001: v.1.1)
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CONTENTS
1.
2.
3.
4.
5.
Crack Propagation and Griffith's Theory of Brittle Fracture
Ductile Fracture
Phenomenon of Fatigue
Creep
Impact Energy and Toughness
2
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MECHANICAL PROPERTIES II
Mechanical Failure
Safa Kasap
Department of Electrical Engineering
University of Saskatchewan
Canada
"Very generally there are always two fracture mechanisms competing
to break a material -plastic flow and brittle cracking. The material will
succumb to whichever mechanism is the weaker; if it yields before it
cracks the material is ductile, if it cracks before it yields it is brittle. The
potentiality of both forms of failure is always present in all materials"
J. E. Gordon
The New Science of Strong Materials or Why You Don't Fall Through
the Floor (Penguin Books, London, UK:2nd Ed., 1976; Princeton
University Press, Princeton, NJ, USA, 1984) p.84 .
1 Crack Propagation and Griffith's Theory of Brittle Fracture
Most fractures one way or another involve crack propagation. Normally these cracks are not even visible to
the naked eye though they can be detected under an optical or an electron microscope. Figure 1-1 (a) shows
a perfect crystal with no load. There are no surface imperfections. When a tensile load, F, is applied to this
crystal as in Figure 1-1 (b) then the load is equally divided between each chain of atoms, AA′, BB′ to EE′.
Each bond is strained by an equal amount along the tensile load axis. If we were to view the crystal
macroscopically without the atomic details as in Figure 1-1 (c) we would find that the lines of force are
straight and evenly spaced. Each line of force represents the same amount of force and when we add all the
lines we get the total force F. The same number of lines cross a given area perpendicular to the tensile axis
anywhere in the crystal because the lines are uniformly spaced. The stress, force per unit area, is therefore
uniform across the crystal and is given by σo = F/Ao where Ao is the face area on which the tensile load acts.
Suppose that the surface of the crystal has a crack that penetrates and interrupts the atomic chains
AA′and BB′as depicted in Figure 1-1 (d). When a load is applied, the atoms such as C at the tip of the crack
now have to bear additional loads arising from missing atoms in the crack as shown in Figure 1-1 (e). If we
were to view the crystal macroscopically and draw the lines of force, we observe that the lines that
previously passed through the crack region now have to go around the crack as shown in Figure 1-1 (f). In
the zone around the crack there are more lines of force per unit area and therefore greater stress. This
region around the tip of the crack, shown as hatched in Figure 1-1 (f), has stress concentration. The stress
across the crystal where there is a crack is not uniform and the stress is maximum, due to the largest
number of force lines per unit are, around the tip of the crack. It is apparent that there has been a stress
amplification or concentration at the tip of the crack where the maximum stress is greater than σo . This is
caused because the lines of force are closely spaced and there are more lines per unit area. The increase in
the stress, or stress amplification, depends on the length of the crack c as well as the radius of curvature r of
the crack or its "sharpness". The sharper the crack, the more suddenly the lines of force have to be bent and
more closely separated they become. Far away from the crack, as in the lower portion of the crystal in
Figure 1-1 (f), the lines of force are again evenly spaced and the stress is the applied stress σo.
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Crack
A
A
B
B
C
C
D
D
E
E
A
C
A
Force
F
Force
F
E
C
Force
F
E
Ao
Stress concentration
Force
Force
Force
(a) Perfect crystal with no load. (b) A perfect crystal with an applied load. Each chain of atoms, AA , to
EE , carries an equal amount of force. Solid lines between atoms represent strained bonds which are due
to forces acting on these bonds. (c) Macroscopic representation of (b) without the atomic details. The lines
of force go through the crystal uniformly and in every region the force per unit area is constant. Thus
stress is uniform across the crystal. (d). The surface of the crystal has a crack which penetrates and
interrupts the atomic chains AA and BB . (e) When a load is applied the atoms such as C at the tip of the
crack now have to bear additional loads arising from missing atoms in the crack. Thus there is a stress
concentration around the tip of the crack. (f) Macroscopic representation of (e) without the atomic details.
Lines of force have to go around the crack. In the zone around the crack there are more lines of force per
unit area and therefore greater stress.
Figure 1-1
The local variation of the stress across a specimen containing a surface crack is shown in Figure 1-2
(highly exaggerated).The ratio of the maximum stress σm at the crack tip to the applies stress σo is called the
stress concentration factor, Kt, and represents the amount of stress amplification. Kt depends on the crack
length c and the radius of curvature r of the crack tip. The maximum stress σm at the tip of an elliptical (or
circular) crack is given by

c
σ m = σ 0 1 +
 = Ktσ 0
r

Maximum stress at cracks
(1−1)
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For a sharp crack which has a length much longer than its radius of curvature , c >> r,
c
Maximum stress at cracks
(1−2)
r
Stress concentrations can also occur at the tips of internal cracks just as they do at the tip of a surface crack.
The same equations apply with the same c and r, but the crack length is now 2c.
σ m ≈ 2σ 0
o
= F / Ao
c
r
C
X
Crack
X'
m
o
C
x
The variation of stress across a specimen that has a surface crack of length c and radius of curvature r. The
stress is maximum at the tip C of a crack.
Figure 1-2
Stress amplifications can be appreciable if the cracks are sharp as r would then be small as shown
in Equation (1−2). Suppose that a surface crack is about 0.2 micron long (200 nm). If the crack terminates
sharply at an atomic point ,then its radius of curvature will be of the order of an atomic bond length, say
0.1nm. Such a crack would hardly be visible even under an optical microscope as it is even shorter than the
wavelength of visible light, though it can be probably viewed under an electron microscope. Then the stress
amplification from Equation (1−1), Kt, is about 90. Thus even if the applied stress σo may not very high, the
stress amplification at the tip of this crack may be sufficiently large to rupture the most strained bond at the
tip and hence enlarge the crack. The crack then penetrates or propagates into the bulk by a bond length. Of
course, things get worse when c gets larger and stress amplification becomes greater which causes more
ruptured bonds at the crack tip and a further penetration or propagation of the crack.
At each step of crack propagation the stress amplification ruptures the bond at the crack tip which
allows the crack to spontaneously propagate across the specimen. This is how fracture occurs in such
brittle solids as glasses and ceramics as schematically illustrated in Figure 1-3 (a) and (b). Spontaneous
crack propagation inevitably leads to a dramatic fracture that is typical of brittle solids. Such crack
propagation events almost certainly occur in all brittle materials that do not exhibit any plastic deformation.
The amplified stresses at the crack tip do not plastically deform this region but instead rupture
bonds and spontaneously propagate the crack. By manufacturing crystal whiskers, thin filaments of single
crystals, that are free of any surface cracks, engineers have been able to produce whiskers that approach the
theoretical fracture limit in which nearly all the bonds are ruptured at the fracture plane.
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In ductile metals, if the amplified stresses around the crack tip are sufficiently large then, these
stresses will plastically deform this region as shown in Figure 1-3 (c) which then becomes strain hardened
and more difficult to further deform. If the crack is to propagate spontaneously into the bulk of the material,
then it has to do much work in plastically deform the material that is progressively hardening. Crack
propagation in ductile materials is therefore much more difficult and consequently failure by crack
propagation across the whole component occurs less frequently. It does nonetheless occur given the right
conditions as discussed below.
Brittle
material
Ductile
material
Plastic
work has
hardened
this
region
When the applified stress o is below the critical stress cb required for crack propagation, the crack is
stable. (b) When o, exceeds cb, then the crack propagates across a brittle material. (c) In a ductile
material the crack cannot propagate easily because it needs to do plastic work, plastically deform the tip
region, which requires substantial energy.
Figure 1-3
It is clear that the crack propagation in brittle and ductile materials have different origins. The
propagation of a crack in a brittle material involves no appreciable plastic work at the tip of the crack. For
brittle materials such as glasses and ceramics, the critical applied stress needed to break the bonds at the
crack tip and thereby spontaneously propagate the crack depends not only on the elastic modulus E but
also on the surface tension of the material, γ, which is the surface energy per unit surface area because
new crystal surfaces are created when the crack penetrates into the bulk. This critical stress σcb for brittle
materials was derived by Griffith1 and is given by
2 Eγ
Critical applied stress in brittle materials
(1−3)
πc
When the applied stress exceeds σcb, as shown in Figure 1-3 (b), the amplified stress at the crack tip is able
to rupture the bonds at the tip and thereby create new fracture surfaces. The crack extends further into the
bulk. Its length c becomes longer. This increases the amplified stress σm in Equation (1−1) but lowers the
critical stress σcb in Equation (1−3), that is the applied stress exceeds σcb even more. Greater amplified
stress at the crack tip ruptures even more bonds and the crack propagation becomes even faster. Hence
once crack propagation starts it accelerates and dramatically fractures the specimen. Inasmuch as crack
propagation in brittle materials does not involve plastic work it is said to propagate elastically.
σ cb =
1
Alan Arnold Griffith (1893-1963) was a British engineer who developed our present understanding of brittle fracture.
According to Professor J.E. Gordon, some of the practical work at the time had to be done somewhat discretely by Griffith
and his co-worker Sir Ben Lockspeiser as working on such non-engineering materials as glass was not considered a serious
research activity at the Royal Aircraft Establishment in 1920s. In those days, as one might have guessed, primary
engineering materials were wood and steel. By the way, the derivation of this equation was published by Griffith in the
Philosophical Transactions of the Royal Society of London, A221, 163, 1921. Undoubtedly Griffith's crack theory caused as
much revolution in mechanical engineering as vacuum tubes did in electrical engineering around the same time.
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In ductile materials, the crack propagation inevitably involves plastic deformation around the tip. The critical
applied stress σcd for crack propagation in a ductile metal then depends also on the amount of plastic work
that has to be done to "fracture" the region around the tip for the tip to extend further. The plastic work
done per unit volume to fracture a ductile material in a tensile test is known as toughness which should
therefore influence the critical stress σcd for crack propagation. In general, σcd is given by
2 EGc
Critical applied stress in ductile materials
(1−4)
πc
Where Gc is a material property called the toughness of the material2. Qualitatively it is the plastic
work done per unit surface area of the crack. Effectively it is plastic work done to create a crack with a unit
σ cd =
2
Not
to
be
confused
with
the
modulus
of
toughness
for
tensile fractu
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surface area. For ductile materials Gc is quite large, 10 − 100 kJ m−2, whereas for brittle materials it is small;
for glass it is 0.01 kJ m−2. Thus the critical stress for ductile materials is much greater than that for brittle
materials and therefore ductile materials posses a greater fracture toughness given comparable elastic
moduli.
Example 1-1: Critical stresses for crack propagation
Consider an invisible surface crack on one side (edge) of a thin glass plate and also on one side of a thin
aluminum−alloy plate. Suppose that the crack length is 0.2 µm, and the radius of curvature of the crack is
0.1 nm. Find the stress concentration factor and the critical applied tensile stress for crack propagation in
each plate. What is the maximum stress at the crack tip at the onset of fracture for the thin glass plate?
From materials data books, for glass, E = 70 GPa, γ ≈ 0.51 J m−2 and for the aluminum alloy E = 70 GPa
and Gc = 20 kJ m−2.
Solution
The stress amplification factor Kt is given by
c
0.2 × 10 −6
=2
= 89.4
r
0.1 × 10 −9
For the glass fiber, the critical applied stress σcb is
Kt = 2
σ cb =
2 Eγ
=
πc
2(70 × 10 9 Pa )(0.5 J m -2 )
π (0.2 × 10 -6 m )
= 3.34 × 108 Pa or 334 MPa
The maximum stress at the crack tip for fracture is
σm = Ktσcb = (89.4)(334 MPa) = 29.8 GPa.
We note that the magnified stress at the crack tip is about ~E/3. For the aluminum alloy, any crack
propagation involves plastic work. The critical stress is
σ cd =
2 EGc
=
πc
2(70 × 10 9 Pa )(20 × 10 3 J m -2 )
π (0.2 × 10 -6 m )
= 6.68 × 1010 Pa, 66.8 GPa
The applied stress needed to propagate this crack is 66.8 GPa. This is, in fact, grater than typical
tensile strengths of aluminum alloys, typically below 400−500 MPa. Thus, the fracture is unlikely to
proceed by the propagation of this crack in the case of the aluminum plate.
2 Ductile Fracture
Ductile fracture characteristically involves substantial plastic deformation. As a ductile specimen is loaded
in a tension test and beyond the yield point, dislocation motions give rise to homogeneous plastic strain.
The specimen extends and becomes narrower as shown in Figure 2-1(a). A neck forms when the
engineering stress reaches the tensile strength where local stresses give rise to more plastic deformation in
the neck than other regions as shown in Figure 2-1(b). The true stress in the neck is actually greater than
any other region because the neck is narrower. Large stresses in the neck nucleate microcavities or
microvoids as pictured in Figure 2-1(c). These tiny cavities usually form around particulate inclusions. In
essence they serve as tiny internal cracks. The tips of these cavities experience substantial stress
amplification and therefore propagate the cracks. As the cavities enlarge in directions perpendicular to the
tensile stress axis they gradually coalesce and eventually form an elliptical cavity or crack as in Figure 2-1
(d). In the final stages, the elliptical crack simply propagates to the surfaces by shearing at the maximum
shear stress angle 45° as shown in Figure 2-1(e). The fracture surfaces therefore have a cup and cone type
appearance as indicates in Figure 2-1(f).
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Nucleation
of
microvoids
45°
Cavity due to
Coalesced
microvoids
45°
Necked region
45°
9
Cone
Shearing at 45°
Fibrous structure
45°
45°
Cup
45°
Ductile fracture. (a) dislocation motions cause plastic flow. (b) Necking begins. (d) Microvoids are formed
due to high stresses in the neck region. (e) Microvoids coelesce and give a cavity, and internal crack,
within the neck (e) The material around the crack experiences excessive shear stresses at 45° (maximum
shear stress angle) which propagates the internal crack to the surfaces and causes fracture.
Figure 2-1
3 Phenomenon of Fatigue
Many engineering components experience time varying stresses rather than static loads. Figure 3-1(a) and
(b) show two examples of time dependent stresses. In Figure 3-1(a), the time variation of the stress is a
periodic function of time analogous to an alternating voltage at a power outlet. In Figure 3-1(b), the stress
variation has no periodicity but nonetheless it still exhibits variations and puts a component under a varying
stress. Time varying loads whether periodic cycles or aperiodic fluctuations occur naturally in engineering.
One can think of many examples, for example, the shaft of the electric motor that is carrying a load, as
shown in Figure 3-2, experiences cyclic stresses as the material endures tension and compression during
the rotation. It may be thought that as long as the tensile stress in the shaft does not exceed the tensile
strength, the shaft will not fracture but this is not true.
Failure under time varying stresses is different than that occurring under static tensile loading. A
component that can withstand loads up to its tensile strength under static loading can easily fail under cyclic
conditions with a stress magnitude that is a fraction of its tensile strength. It was thought (long time ago)
that the "specimen got tired of carrying the load and gave up" and the failure was inappropriately named
fatigue. Under a cyclic stress, components usually fail by the nucleation of a surface crack and its
subsequent propagation through the component.
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(t)
10
(t)
max
t
t
min
a
b
Aperiodic stress variations
Alternating stress cycles
Time varying stress can lead to failure even if the maximum stress is below the tensile
strength of the metal.
Figure 3-1
Tension
Compression
Shaft
Electric
motor
Load
Cyclic loads and hence cyclic stresses occur frequently as in this example of an
electric motor driving a load. The shaft experiences cyclic stresses.
Figure 3-2
Fatigue therefore invariably involves failure by crack propagation. The crack however is induced during
load cycling by the changes in the stress direction and the resulting changes in the slip formations on the
surface. The exact details are not treated in this book. Fatigue tests normally employ cyclic stresses with a
well defined period as in Figure 3-1(a). Fatigue is typically characterized by plotting the amplitude of the
cyclic stress, denoted by S, against the number of cycles to failure, N as shown in Figure 3-3. The results
are called S − N plots. If the stress amplitude, S, is equal to the tensile strength, σTS, then obviously the
specimen fractures within quarter of a cycle, time needed to reach σTS. When S is less than σTS it may take
many cycles for the material to fail. The lower the stress amplitude, the greater the number of cycles N to
failure as shown in Figure 3-3. An interesting feature of fatigue is that even when S is less than the yield
strength, the material, nonetheless, still fails provided that it is cycled long enough. Ferrous alloys such as
steels show an endurance limit as identified in Figure 3-3. Endurance limit is the maximum amplitude of
stress that can be cycled infinitely without fracturing the specimen. As long as the stress amplitude is less
than the endurance limit steel components can be cycled by an infinite number of times. Stresses below the
endurance limit therefore identify a safe zone. No such endurance limits exist for aluminum and copper
alloys. They tend to show eventual failure if cycled long enough. Figure 3-4 compares the S-N behavior of
various metals, those that exhibit endurance limit and those that do not.
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Stress Amplitude, S ( max)
TS
11
Time varying stress can
lead to failure even if the
maximum stress is below
the tensile strength of the
metal.
Ferrous alloy
YS
Endurance limit
Safe zone
1/
N
4
1
104
102
106
108
N, number of cycles to failure
(logarithmic scale)
Figure 3-3
Stress (MPa)
450
400
Al alloy (7075-T6)
350
Structural steel rod
Al-4.5%Cu alloy
300
250
200
150
Mild steel (0.2%C)
Copper
100
50
0
104
105
106
107
Cycles to Failure
108
109
Fatigue in ferrous (solid lines) and
non-ferrous (dashed lines) metals
[From: Al alloy (7075-T6) from G.M.
Sinnclair and T.J. Dolan, Trans.
ASME, 75, p.867, 1953; Mild steel
(0.3%C) , Al-4.5%Cu alloy (fully heat
treated) and Cu from O.H. Wyatt and
D. Dew-Hughes, Metals, Ceramics
and Polymers (Cambridge University
Press, 1974), Fig. 5.29, p. 173 and
Fig. 5.28, p. 172 repectively;
Structural steel rod from 1990 Guide
to Selecting Engineering Materials,
Vol. 137, Issue 6, June 1990,
American Society of Metals (ASM
International, 1990), p.40.]
Figure 3-4
4 Creep
When a permanent load is applied to a component, elastic deformation occurs almost instantaneously. At
elevated temperatures, typically above 0.3Tm, where Tm is the melting temperature (in Kelvin), the
component also exhibits slow permanent deformation with time which is called creep. The specimen is said
to creep under a constant load. Figure 4-1 shows what happens when at time t = 0, a constant tensile stress
is applied to a component. It first exhibits instantaneous elastic deformation at t = 0 but then it deforms
plastically and permanently with time. The strain increases with time and shows three characteristic regions
in the ε vs. t behavior as indicated in Figure 4-1. Eventually the specimen fractures after a duration of time
called the rupture time. In the primary creep region, the creep rate continuously slows with time. The
decrease in the creep rate is due the strain hardening of the material. As dislocations become entangled they
cannot move freely and plastic deformation becomes more difficult. Eventually the creep rate reaches a
steady state value which marks the secondary region as shown in Figure 4-1. In this region the creep
continues at a constant but slow rate. Indeed the creep rate is the lowest during this time. The creep process
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during this time normally involves slow atomic and vacancy diffusions to and from dislocations which
disentangles and unpins some of the dislocations. The creep rate, dε /dt, in this regime is therefore
thermally activated. Eventually microvoids or microcracks are formed at grain boundaries by various
processes such as dislocation pileups at grain boundaries or by the relative sliding of neighboring grain
boundaries to name a few. Even vacancies diffusing to grain boundaries can coalesce to form a microvoid.
Formations of microvoids lead to crack propagation across grains and hence to rapid creep rates which
identify the tertiary region in Figure 4-1. Crack propagation eventually lead to failure or rupture.
Stress,
Time
Strain,
Tertiary
Primary
Rupture
Secondary
d = Creep rate
dt
Instantaneous
elastic strain
Time
0
Rupture time, tr
The phenomenon of creep and rupture
Figure 4-1
Strain,
Higher stress or
temperature
B
A
C
T < 0.3Tm
0
Time
Creep at a given temperatureT1 and 1 is the curve A. If temperature or stress is increased then the curve
becomes B. Below 0.3Tm there is significant creep
Figure 4-2
At a higher temperature or greater stress, the creep rate becomes accelerated and the rupture occurs sooner.
Creep is typically characterized by examining the steady state creep rate, the strain rate in the secondary
region, and the rupture time for a given set of variables such as stress and temperature. The dependences of
the rupture time and the creep rate on the stress are shown Figure 4-3 and Figure 4-4. Conventionally the
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stress is plotted along the y−axis and the plots are log−log plots. Temperature has a pronounced effect of
the creep characteristics. Below about 0.3Tm the creep effects are considered negligible.
log(Stress)
T = T1
T2 > T1
T3 > T2
0
log(Rupture time)
Stress and temperature dependence of rupture time tr
Figure 4-3
log(Stress)
T = T1
T2 > T1
T3 > T2
log(Creep time)
0
Stress and temperature dependence of creep rate
Figure 4-4
5 Impact Energy and Toughness
Many materials when in use fail or fracture as a result of impact. Further, some ductile metals, notable BCC
and HCP metals such as iron and all steels, exhibit a change from ductile to brittle failure under fast strain
rates at low temperatures. This behavior is more pronounced if there is a sharp crack or a notch in the
component. It is therefore highly desirable to characterize, in a standard fashion, the failure of materials
under impact and in the presence of a surface notch.
Impact energy is the energy absorbed by a metal specimen to fracture in a standard impact test.
The most common impact test is the Charpy test which involves fracturing a specially shaped metal
specimen by releasing a pendulum hammer from a certain height h as shown in Figure 5-1. After fracturing
the specimen, the hammer rises to a height h′ so that the impact energy absorbed by the specimen in
fracturing is the change in the potential energy of the pendulum, i.e. Mg(h − h′), where M is the mass of the
hammer. The scale can be calibrated to read the impact energy directly. It may be surmised that the impact
energy from the Charpy test correlates to the toughness of the material. Higher impact energies are
recorded for tougher metals. Figure 5-1 illustrates the Charpy test and also shows the special shape of the
test specimen. Notice that the sample is V−notched to ensure that the fracture begins at this point. The
absolute value of the impact energy from Charpy tests depends on the shape and exact dimensions of the
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notch, so that Charpy test results are often useful for comparison purposes only, for example, for the
comparison of impact energy at different temperatures.
Scale
Impact energy, Eimpact
0J
Start
End
Hammer
M
h
h'
V-notch
Notched specimen
Specimen anvil
A schematic illustration of the Charpy impact test
Figure 5-1
Charpy tests are often carried out as a function of temperature to show the ductile-to-brittle transition
exhibited by certain metals, those with BCC and HCP crystal structures, as illustrated in Figure 5-2 for two
carbon steels (steels with 0.11%C and 0.53%C) which have the BCC crystal structure.
Impact Energy (J)
250
200
Steel: 0.11%C (BCC)
150
100
Stainless steeel (FCC)
50
0
-150 -100
Impact energy vs temperature
for two BCC metals, steels with
0.11%C and 0.53%C and one
FCC metal, stainless steel.
Steel: 0.53%C (BCC)
-50
0
50
100
150
200
250
Temperature ( C)
Figure 5-2
An example for an FCC metal, stainless steel, is also shown. Both carbon steels exhibit a ductile to
brittle transition in their fracture behavior which is typical of these BCC structures whereas the stainless
steel exhibits no such transition which exemplifies FCC metals. Comparison of the two steels shows that
the addition of carbon to steel reduces the impact energy and increases the transition temperature in Figure
5-2. Such changes are extremely important in engineering design. For example, a component using
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0.53%C steel will fracture in a brittle fashion and hence dramatically under a high strain rate if it
experiences impact whereas 0.11%C steel can easily absorb large amounts of impact energy and not
fracture. Even 0.11%C steel becomes brittle against impacts at temperatures below –50 °C which, by the
way, are not unusual in northern latitudes (and during some winter days and nights in the author's country).
The different ductile transition behavior of BCC and FCC metals can be intuitively explained by the
temperature dependence of the critical shear stress τcss (see Mechanical Properties I) that is needed to move
dislocations in the crystal and thereby do plastic (ductile) work. In the case of FCC metals τcss is already
smaller than that for BCC crystals and, further, τcss increases more rapidly with decreasing temperature for
BCC metals. Thus, plastic work becomes more difficult in BCC metals at lower temperatures which leads
to the observed brittle impact−fracture.
NOTATION
A
BCC
c
E
ε
F
FCC
Gc
g
γ
HCP
h
Kt
area (m2)
body centered cubic crystal structure
crack length (m)
elastic modulus (Pa or N m-2)
strain (no units)
force (N)
face centered cubic crystal structure
toughness of the material (J m-2)
acceleration due to gravity (9.81 m s-2)
surface tension, surface energy per unit surface
area (J m-2)
hexagonal close-packed crystal structure
height (m)
stress concentration factor (no units)
M
N
r
S
σ
σcd
σcb
σm
σo
σTS
Tm
t
τcss
mass (kg)
number of cycles to failure
radius of curvature of the crack tip (m)
amplitude of the cyclic stress
stress (Pa or N m-2)
critical stress for ductile materials (Pa or N m-2)
critical stress for brittle materials (Pa or N m-2)
maximum stress at the crack tip (Pa or N m-2)
applied stress (Pa or N m-2)
tensile strength (Pa or N m-2)
melting temperature (K)
time
critical shear stress (Pa or N m-2)
Defining Terms
BCC stands for body centered cubic crystal structure.
Brittle materials do not exhibit any marked plastic deformation and their fracture strains are less than a few percent. They
exhibit very limited toughness. Ceramics and glasses are examples of brittle materials.
Dislocation is a line imperfection within a crystal that extends over many atomic distances.
Ductile materials exhibit an ultimate tensile strength point and considerable plastic deformation before fracture. They also
tend to have a high toughness.
Ductile work is the work done during plastic deformation.
Ductile work is the work done during plastic deformation.
Elastic Modulus or Young's modulus (E) is axial stress needed per unit elastic axial strain, and is defined by σ = Eε
where σ is the applied stress and ε is the strain all along the same direction (axis). It gauges the extent to which a
body can be reversibly (and hence elastically) deformed by an applied load in terms of the material properties.
Endurance limit is the maximum amplitude of stress that can be cycled infinitely without fracturing the specimen.
FCC stands for face centered cubic crystal structure.
HCP stands for hexagonal close−packed crystal structure.
Plastic deformation is permanent deformation by material flow as a result of the motion of dislocations.
Plastic deformation is permanent deformation by material flow as a result of the motion of dislocations.
Mechanical properties II: Mechanical Failure (© S.O. Kasap, 1990 - 2001: v.1.1)
An e-Booklet
16
Strain is a measure of the deformation a material exhibits under an applied stress. It is expressed in normalized units. Under
an applied tensile stress, strain (ε) is the change in the length per unit original length, ∆L/Lo. When a shear stress is
applied, the resulting deformation involves a shear angle. Shear strain is defined as the tangent of the shear angle that
is developed by the application of the shearing force.
Strain is a measure of the deformation a material exhibits under an applied stress. It is expressed in normalized units. Under
an applied tensile stress, strain (ε) is the change in the length per unit original length, ∆L / Lo. When a shear stress
is applied, the resulting deformation involves a shear angle. Shear strain is defined as the tangent of the shear angle
that is developed by the application of the shearing force. Volume strain ∆ is the change in the volume per unit
original volume; ∆ = ∆V / V.
Stress is force per unit area, F / A . When the applied force is perpendicular to the area it leads either to a tensile or
compressive stress, σ = F / A. If the applied force is tangential to the area, then it leads to a shear stress, τ = F / A.
Stress is force per unit area, F/A. When the applied force is perpendicular to the area it leads either to a tensile or
compressive stress, σ = F/A. If the applied force is tangential to the area then it leads to a shear stress, τ = F/A.
Surface tension ( γ ) represents the energy required to increase the surface of a body by some unit area keeping the total
number of atoms the same. An atom on the surface of a liquid or a solid has less bonds than an atom within the
bulk of the substance. Thus, if we were to increase the surface area of the body, e.g. changing it from a sphere to a
cube, which has higher surface area, we need to work because we would have less bonds per atom (effectively we
have broken bonds). If energy dE is required to increase the surface area by dA then γ = dE / dA. A hypothetical line
on the surface of a liquid or a body experiences a force per unit length that is equal in magnitude to the surface
tension. Surface tension in a crystal depends on the crystal structure and the surface plane (because the energy
difference per atom on the surface and within the bulk depends on the arrangements of atoms on the crystal surface
and in the crystal bulk).
Toughness is the amount of plastic work done per unit volume to fracture.
Yield Strength is the resistance of the material against plastic deformation.
1.
QUESTIONS AND PROBLEMS
Consider an invisible surface crack on one side (edge) of a thin glass plate. Suppose that the crack
length is 1 µm, and the radius of curvature of the crack is 1 nm. Find the stress concentration factor
and the critical applied tensile stress for crack propagation in this glass plate. What is the maximum
stress at the crack tip at the onset of fracture for the thin glass plate? From materials data books, for
glass, E = 70 GPa, γ ≈ 0.51 J m−2 and for the aluminum alloy E = 70 GPa and Gc = 20 kJ m−2.
[Ans:]
Mechanical properties II: Mechanical Failure (© S.O. Kasap, 1990 - 2001: v.1.1)
An e-Booklet
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© All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
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Permission is granted to individuals for downloading this document from the author’s website or his
CD-ROM for self-study only. Permission is hereby granted to instructors to use this publication as a classhandout if the author’s McGraw-Hill textbook Principles of Electronic Materials and Devices, Second
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Last Updated: 6 January 2002 (v.1.1)
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