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Mech 473 Lectures
Professor Rodney Herring
Safety Factor for Design
The failure criterion for ductile metals and plastics under tensile or
steady-tensile loading is the yield strength
However, the yield strength is not taken as the design or working
stress for the material.
It is common for engineers to employ a “factor of safety” to ensure
against uncertainties due to variations in the properties of a piece
of material.
In designs using yield strength the safety factor is ……. .
In designs using tensile strength, the safety factor is ……. .
Yield strength is used for ductile materials like structural steels.
Tensile strength is used for brittle materials such as cast iron.
Most brittle materials are used under conditions of compression.
Why then would the tensile strength be used?
Safety Factor for Design
So, for the design stress, sdes,
s des 
s ys
2
or
s des 
s uts
4
Using the design stress, we can obtain the size or area of the material
from the maximum load, Pmax, the structure or component is going
to sustain where
Pmax
A
s des
This safety factor method oversizes the material, which is fine for
static structures, but is a factor for transport structures such as
airplanes and freight trucks, i.e. it’s not appropriate.
Safety Factor for Design
If the mode of loading is shear, we can approximate the maximum
shear strength, tmax, of a material from its yield strength by
s ys
t max 
2
And, the design method is the same as for ductile materials like steel.
For brittle ceramics and glasses, the design for the safety factor is
more complex and Weibull statistics are used, which we’ll discuss
in more detail later.
Safety Factor for Design
Using the yield strength and ultimate tensile strength is sufficient for
most designs, but it’s not a guarantee that catastrophic failure will
not occur, which you’ll be held legally accountable as the
professional engineer responsible for the design.
Current design methods for safety and performance use the fracture
toughness of a material.
The basic premise in using fracture toughness, or fracture mechanics
in design is to assume that materials have defects or cracks in
them.
The material property that resists the propagation of these cracks is
the fracture ……………….. .
Design - Fracture Toughness
What is fracture toughness?
The stress intensity at the crack tip is dependent on both the applied
stress and the length of the crack.
• A mechanical variable, Stress Intensity Factor, KI, is used to
describe the relationship:
KI  fs a
Where,
f is a dimensionless constant (related to geometry of specimen and
flaw)
s is the applied stress
a is the crack length or half the length of an internal crack
KI is a variable but ….. a materials property
KI has unusual unit of Mpa(m)½ or psi(in)½ .
Design - Fracture Toughness
Fracture Toughness
When the stress intensity, KI is increased to a critical value, KIC ,
crack propagation will occur, which will lead to fracture.
It is written as:
K IC  fs a
Where,
KIC is a measure of a materials resistance to crack propagation.
It is a material property.
KIC is dependent on temperature, microstructure, and strain rate.
KIC usually increases with a reduction in grain size.
Fracture Toughness Design
Higher fracture
toughness materials,
such as material B
in the graph, tolerate
higher stresses and
thus larger cracks.
Crack size
Fracture Toughness
How to use KIC ?
• Fracture toughness is most useful in mechanical designs
involving materials with limited toughness or ductility.
• Usually s < syield/n is good enough for ductile materials, which
are statically loaded.
• Design criterion using KIC :
K I  K IC
Taking into account KIC , which is a material property, the allowable
stress (s) and/or the allowable flaw size (a) can be determined.
Material Selection:
• If the maximum applied stress, smax , and maximum crack length
are specified for a certain application, then only the materials
with KIC greater than KI can be used:
K IC  fs max amax
Fracture Toughness
Allowable stress design (if “a” and KIC are specified by application
constraints) then,
K IC
s max 
f amax
Allowable crack size design (if the stress level, smax , and KIC are
specified) then,
amax
 K IC  1

 
 fs max  
Allowable crack size design is possible if the stress level, smax , and
KIC are specified).
Critical Stress-Intensity Factor
A crack propagates when KIc is
attained at the tip of a crack.
Variation of K with thickness of the
material. As the thickness
increases, the stress intensity
becomes the material fracture
toughness, KIc, and independent
of thickness. Thus, objects
break independent of their ……..
Fracture Toughness – Crack Surface Energy
When a material has an applied strain, it undergoes an elastic strain
related to the modulus of elasticity, E, of the material.
When a crack propagates, this strain energy is released, which
reduces the overall energy.
However, two new surfaces are created by the extension of the
crack, which increases the energy associated with the surface.
By balancing the strain energy and the surface energy, g, we find
that the critical stress required to propagate the crack is given by:
Eg
s critical  2s
a
This equation shows that even ……….. cracks can severely limit
the strength of a material.
This equation is particularly applicable to ceramics.
Fatigue
Many components fail by fatigue when subjected to cyclic loads,
which generate nominal stresses below the static ultimate stress of
the material.
Fatigue occurs because each half stress cycle produces ……………
strains, which are not recoverable.
When these minute strains are added, they produce local plastic
strains, which are sufficient to cause submicroscopic cracks.
These small cracks act as stress intensifiers so that the local stress in
the region of the crack can exceed the stress to propagate the
crack.
The crack grows, often over a long period, until the cross sectional
area is lowered below the limit to support a stress that can cause
catastrophic fracture (the Griffith relationship)
The presence of a notch or other stress intensifiers can act as a
starting point for the process of fatigue.
Fatigue
Fatigue is the lowering of strength or failure of a material due
to repetitive stress, which may be above or below the yield
strength.
Many engineering materials such as those used in cars, planes,
turbine engines, machinery, shoes, etc are subjected
constantly to repetitive stresses in the form of tension,
compression, bending, vibration, thermal expansion and
contraction or other stresses.
At a local size scale, the stress intensity exceeds the yield
strength.
For fatigue to occur at least part of the stress in the material
has to be tensile.
Fatigue is most common in metals and plastics, whereas
ceramics fail catastrophically without fatigue because of their
low fracture toughness.
Fatigue – 3 Stages
There are typically three stages to fatigue failure.
First a small crack is initiated or nucleates at the surface and
can include scratches, pits, sharp corners due to poor design
or manufacture, inclusions, grain boundaries or dislocation
concentrations.
Second the crack gradually propagates as the load continues to
cycle.
Thirdly, a sudden fracture of the material occurs when the
remaining cross-section of the material is too small to
support the applied load.
Fatigue
Fatigue failures are often easy to identify.
The fracture surface near the origin is usually …………………. . The
surface becomes rougher as the crack increases in size.
Microscopic and macroscopic examination reveal a beach mark pattern and
striations.
Beach mark patterns indicate that the load is changed during service or the
load is intermittent.
Striations are on a much finer scale and show the position of the crack tip
after each cycle.
Fatigue
The most important fatigue data for engineering designs are the S-N
curves, which is the Stress-Number of Cycles curves.
In a fatigue test, a specimen is subjected to a cyclic stress of a
certain form and amplitude and the number of cycles to failure is
determined.
In a rotating beam fatigue testing machine, the specimen is bent as
it rotates.
The reduced middle section of the specimen alternates between
states of tensile and compressive stress.
The S-N Curve
Results of fatigue tests are presented as plots of nominal cyclic stress,
S, versus number of cycles to failure, N.
At a nominal stress equal to the ultimate stress, the component will
fail after the first half cycle.
At a nominal stress below the yield point, the number of cycles to
failure is relatively large but still finite.
In iron-based materials, there is a nominal stress below which fatigue
does not occur during normal life times, the endurance limit, which
is used as a design parameter.
The S-N curves for a tool steel and an aluminum alloy showing the
number of cycles to failure
Al does not show a fatigue limit but continuously decreases.
Example of Surface Stress Raiser on S-N Curve
The endurance limit is sensitive to the size of the stress raiser that
may exist in the material.
The endurance limit decreases as the size of the stress raiser
decreases, which agrees with the increase in the concentrated stress
as the crack radius decreases.
s  s 1 2 c/r
c
n
(
)
Fatigue
Fatigue Limit:
• For some materials such as steels and Ti alloys, the S-N curves
become horizontal when the stress amplitude is decreased to a
certain level.
• This stress level is called the ……….. Limit, or ………… Limit,
which is typically ~35-60% of the tensile strength for steels.
• In some materials, including steels, the endurance limit is
approximately half the tensile strength given by:
Enduranceratio
endurancelimit
 0.5
tensilestrength
Fatigue Strength:
For materials, which do not show a fatigue limit such as Al, Cu, and
Mg (non-ferrous alloys), ……………………. is specified as the
stress level at which failure will occur for a specified number of
cycles, where 107 cycles is often used.
Fatigue Failures
Types of stresses for fatigue tests include,
axial (tension – compression)
flexural (bending)
torsional (twisting)
From these tests the following data is generated.
s max  s min
Mean St ress, s m 
2
s max  s min
St ress Amplit ude, s a 
2
St ress Range, s r  s max  s min
s min
St ress Rat io, R 
s max
By convention, tensile stresses are positive and compression
stresses are negative.
Fatigue Failures
Examples of stress cycles
where a) shows the stress
in compression and
tension, b) shows there’s
greater tensile stress than
compressive stress and in
c) all of the stress is
tensile.
a
b
c
Fatigue Failures
As the mean stress increases, the stress amplitude must ………….
in order for the material to withstand the applied stress. This
condition is summarized by the Goodman relationship:
  s m 

Stress Amplitude, s a  s fs 1  
  s TS 
Where sfs is the desired fatique strength for zero mean stress and
sTS is the tensile strength of the material.
Example, if an airplane wing is loaded near its yield strength,
vibrations of even a small amplitude may cause a fatigue crack to
initiate and grow. This is why aircraft have a routine inspection
in order to detect the high-stress regions for cracks.
Fatigue Failures
Crack Growth Rate
To estimate whether a crack will grow, the stress intensity factor (DK), which
characterizes the crack geometry and the stress amplitude can be used.
Below a threshold DK a crack doesn’t grow.
For somewhat higher stress intensities, the cracks grow slowly.
For still higher stress-intensities a crack
grows at a rate given by:
da
n
 C (DK )
dN
Where C and n are empirical
constants that depend on
the material.
When DK is high, the cracks
grow in a rapid and
unstable manner until
fracture occurs.
Stress-Corrosion Failure
Stress corrosion happens when a material reacts with corrosive
chemicals in our environment.
• Two good examples are salt on the roads reacting with the
steel in cars causing reduced lifetime of the car’s components
such as its frame and suspension system. Another example is
the salt in the ocean reacting with boats and their moorings
where the corrosion reduces the life of the engine, which is
cooled by the salt water, and the structural integrity of the
boat is jeapodized if salt water sits in the hull or around the
drive shaft.
Stress-Corrosion Failure
• Stress-corrosion will cause failure of materials below their
yield strength because the corrosion will cause cracks to
form, usually along grain boundaries.
• Usually if there is a corrosion product on the surface where a
crack is inside the material.
• The surface flaws themselves can be nucleation sites for
crack growth.
• Usually materials are coated to reduce or prevent corrosion.
The automotive industry has shown excellent results by
applying metal coatings (Sn) and polymer coatings on the
sheet steel used on the body of cars.
Stress-Corrosion Failure
Intergranular cracks near
a stress-corrosion
fracture in a metal.
Note the many
branches where the
corrosion has eaten into
the grain boundaries of
the metal.
On the surface, you’ll see
a corrosion by-product.
The crack inside is
typically much larger
than the surface byproduct.
Stress-Corrosion Failure
Corrosion failures are also strongly affected by the alloying additions
to metals.
The best alloying addition to many metals such as iron and zirconium
is Chromium.
Cr preferentially oxides over Fe and forms a thin stable film, which
substantially reduces further oxidation.
We will see in a Stainless Steel Lecture that the Fe-Cr alloy must be
of sufficient Cr concentration and properly heat treated in order for
the Cr to be effective against corrosion.
Chromia (Cr2O3) blocks …………… diffusion at grain boundaries,
dislocations and defects on the surface of the material so that
oxidation of the host material is substantially reduced.
Corrosion resistant steels, eg., 316 steel, are used for containing
chemicals, such as sulfuric acid and foods, which often contain
organic acids, eg. milk.
Creep
Creep is an important material behaviour at elevated temperature.
At elevated temperatures, (> 0.4 Tm), a material will undergo slow
plastic deformation even under a static stress lower than the yield
strength of the material. This is called creep.
Creep Test: This is subjecting a specimen to a constant load or
stress at a constant temperature and determining the
deformation or strain as a function of time.
Important properties from the creep test include:
creep rate
Dstrain
time to rupture
creep rate 
Dtime
elongation or reduction in area
Creep
The rate and extent of creep is very small at temperatures less than
0.5 Tm so creep is often referred to as a high temperature effect.
Creep occurs at room temperature in lead alloys, e.g., the drain pipes
in very old (Roman) buildings are fatter at the bottom than at the
top since 0.5 Tm for PB is 20 oC.
In the process of creep:
A load is applied which produces a stress less than the yield stress.
The load causes an instantaneous elastic extension as in a tensile
test.
The specimen extends plastically over a relatively long period of
time
The creep rate, i.e., the strain rate of the plastic deformation, varies
with time and temperature.
Creep
The form of the plot of creep strain against time is primarily a
function of temperature – with respect to 0.5 Tm.
There are three distinct stages of creep observed at intermediate
temperature:
The first stage is marked by a relatively rapid initial creep rate,
which decays with time.
The second stage is marked by a constant creep rate, referred t as
steady state creep.
The third stage is marked by an increase in creep rate, which
accelerates as the specimen necks as it approaches rupture.
Stages of Creep
Creep-rupture curve showing three stages.
Creep
If the temperature is well below 0.5 Tm or the stress is well
below the yield point:
- after the first stage, the creep rate decays to a very low
constant value and the specimen does not rupture over
finite times.
If the temperature is above 0.5 Tm and the stress is a
significant fraction of the yield point:
- the first stage is curtailed and a relatively high creep
rate is observed in the second stage, which is followed by
an accelerated creep rate to rupture.
Low temperature creep is usually ignored.
Creep
Creep rates determined in the second stage – under steady
state conditions – are used as design parameters, e.g., for
Al alloys at room temperature.
High temperature creep is a serious problem as it can
significantly shorten the life of a component, which must
be used at high temperatures, e.g., for boiler tubes and
blades for turbine engines.
Creep
In general, the creep rate is given by an Arrhenius relationship
Creep Rate  Ae(Qc
RT )
Where Qc is the activation energy, R is the gas constant (~ 1.987
cal/mol/K), T is the absolute temperature A is the pre-exponential
constant, which is dependent on the applied stress, s, given by,
A  Cs n
where C and n are constants. Qc is related to the activation energy
for self-diffusion when dislocation climb is important.
Creep
The rupture time, tr, due to creep also follows an Arrhenius
relationship
tr  Ks me(Qr RT )
Where Qm is the activation energy for rupture, and K and m are
constants. tr is dependent on the applied stress, s.
Solution Hardening
The stress-time to rupture curves for a heat-resistant alloy. Note the
significant decrease in time for rupture with an increase in stress
and temperature.
Creep
In tensile creep deformation, it can be described by a tensile
viscosity, h,
h
s

Where s is the tensile stress and h is the tensile strain rate.
Deformation mechanisms involved in creep include:
viscous creep: for amorphous solids
vacancies or atoms : diffusion
dislocations : slip
grain boundaries : grain rotation, grain boundary sliding
Viscous Creep
Viscous creep for amorphous solids is a diffusion dependent
process that is enhanced by increasing the temperature, i.e.,
thermally activated process, and follows the Arrhenius equation.
  Ae
Q/RT
Where Q is the activation energy for creep in cal/mol, R is the gas
constant, and T is the absolute temperature in K.
As seen before, during creep A depends on the applied stress.
Creep Mechanisms
for amorphous solids
Creep
In crystalline materials, creep occurs either by diffusional
or dislocation creep.
Diffusional creep involves the motion of vacancies and
this may occur primarily through the grains or along the
grain boundaries.
Vacancy motion through the grains is called the NabarroHerring mechanism.
Vacancy motion along the grain boundaries is called the
Coble mechanism.
Creep
These strain rates are given by
A 2 s (Qv /RT)
  2 e
d T
A 2 s (Qb /RT)
  3 e
dT
Nabarro-Herring
Coble
Where d is the diameter of the grain, Qv is the activation energy of
self or volume diffusion, and Qb is the activation energy for grain
boundary diffusion. A2 is a material constant.
Creep Mechanisms
Note that the vacancies and atoms move in opposite directions.
Creep
In crystalline materials, dislocation creep involves the motion of
dislocations where dislocation climb is an important factor.
Dislocation climb means that the edge of the extra plane of atoms
move to another plane parallel to the previous plane that it was
before.
This dislocation motion also involves the diffusion of vacancies and
thus the strain rate is thermally activated having the form,
  A
sm
T
e (Q/RT)
Dislocation creep
Where m varies from one material to another and is typically on the
order of 5.
Thus creep can become quite complex.
More sophisticated methods are often applied to creep by using the
Sherby-Dorn parameter and Larson-Miller parameter.
Creep Mechanisms
Creep
Only solid solution hardening and precipitation hardening
remain effective at elevated temperatures to help prevent
creep.
Grain boundary sliding
during creep causes a)
the creation of voids at
an inclusion trapped at
the grain boundary and
b) the creation of a void
at a triple point where 3
grains are in contact.
Steady-State Creep
The creep rate at
various
temperatures for
carbon steel used
for pressure
vessels. Note the
logarithic scales,
resulting in the
exponential
dependency of
stress on the
strain rate.
The End