Transcript No Slide Title
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 two.
In designs using tensile strength, the safety factor is four.
Yield strength is used for ductile materials
like structural steels.
Tensile strength is used for brittle materials
such as cast iron.
Since 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, s des , s
des
s 2
ys
or s
des
s
uts
4 Using the design stress, we can obtain the size or area of the material from the maximum load, P max , the structure or component is going to sustain where A P s max
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, t max , of a material from its yield strength by t s
ys
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 the
“
fracture toughness”
.
of these cracks is
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
, K I , is used to describe the relationship: Where,
K I
f
s
a 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
K I is a variable but NOT a material property
K I has unusual unit of Mpa(m) ½ or psi(in) ½ .
Design - Fracture Toughness
Fracture Toughness When the stress intensity,
K I is increased to a critical value
the
critical stress intensity
,
K
IC which will lead to fracture.
, , i.e.,
crack propagation will occur
, It is written as:
K IC
f
s
a
Where,
K
IC is a measure of a materials resistance to crack propagation.
It is a material property.
K
IC is dependent on
temperature, microstructure, and strain rate
.
K
IC 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 K IC ?
Fracture toughness is most useful in mechanical designs involving materials with limited toughness or ductility
.
• Usually s < s
yield /n is good enough for ductile materials
, which are statically loaded, e.g., soft aluminum.
• The design criterion that should be used is
K
IC , where
K I
K IC
Taking into account
K
IC stress , which is a material property, the allowable (s) and/or the allowable flaw size (a) can be determined.
Fracture Toughness
• Material Selection:
If the maximum applied stress,
s
max , and maximum crack length are specified for a certain application, then only the materials with K IC greater than K I can be used
:
K IC
f
s max
a
max
As a professional engineer
, if you are tasked with applying materials for a design requiring structural integrity, you should
request from the designer the maximum applied stress and maximum crack length
.
Fracture Toughness
Allowable
stress design
(if “a max ” and
K
IC are specified by the application constraints) then the allowed stress is, s max
f K IC
a
max Allowable
crack size design
(if the stress level, s max specified) then,
a
max
f K
s
IC
max 1 , and
K
IC are
Critical Stress-Intensity Factor
A crack propagates when K Ic 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, K Ic , and independent of thickness.
Thus, objects break independent of their size
.
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
increase 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 crac k
is given by: s
critical
2 s
E
g
a
This equation shows that even small cracks can severely limit the strength of a material
.
This equation is particularly applicable to ceramics.
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
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
minute 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 – 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 smooth . 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
.
How is this misalignment compensated in a car’s driveshaft?
This often happens in rotating shafts used in motors.
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 strength, 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.
For materials, which do not show a endurance limit such as
Al, Cu, and Mg (non-ferrous alloys)
,
fatigue strength is specified
as the stress level at which
failure will occur for a specified number of cycles
,
where 10 7 cycles is often used
.
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
Fatigue Limit
, or
Endurance 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: Endurance ratio endurance tensile limit strength 0 .
5
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 (radius of crack), which agrees with the
increase in the concentrated stress as the crack radius decreases
.
s c n 1 ( 2 c/r )
Fatigue Failures
Types of stresses for fatigue tests include, 1) axial (tension – compression) 2) flexural (bending) 3) torsional (twisting) From these tests the following data is generated.
Mean Stress, Stress Range, s s m Stress Amplitude, r s s s a max max s 2 max s s min 2 min s min Stress Ratio,
R
s s min 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 decrease
in order for the material to withstand the applied stress. This condition is summarized by the
Goodman relationship
: Where s
fs
and s TS Stress Amplitude, s a s fs 1 s s m TS
is the desired fatique strength for zero mean stress,
s
m
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 (
D
K), can determine the crack geometry and the stress amplitude to be used.
Below a threshold
D
K 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 dN
C
( )
n
Where C and n are empirical constants that depend on the material.
When
D
K is high, the cracks grow in a rapid and unstable manner until fracture occurs.
Fatigue Failures
From the steady state crack growth relationship of
da dN
C
( )
n
if we integrate between the initial size of a crack and the crack size required for fracture to occur, we find that the number of cycles to failure, N, is given by
N
2 (
a c
( 2 ) ( 2
n
) /
n
)
Cf
2
n
D (
a i
s ) (
n
2
n
/
n
2 ) / 2
where C and n are empirical constants that depend on the material.
Stress-Corrosion Failure
• •
Stress corrosion happens when a material reacts with corrosive chemicals in our environment.
A good example is 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, which is jeopardized 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.
Why? because the corrosion will cause cracks to form, usually along grain boundaries.
Usually if there is a corrosion product on the surface there is a crack inside the material.
The surface flaws themselves can be nucleation sites for crack growth during service of the material .
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 by product.
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 our Stainless Steel Lectures 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.
Corrosion resistant steels, eg., 316 steel, are used for containing chemicals, such as sulfuric acid and foods, which often contain organic acids, eg. milk.
Stress-Corrosion Failure
Chromia (Cr 2 O 3 ) blocks oxygen diffusion at grain boundaries, dislocations and defects on the surface of the material so that oxidation of the host material is substantially reduced.
We will also see in our discussion of dislocations that the crystal structure plays a big role in determining whether a material will form cracks or a surface layer of rust during corrosion.
So, we haven’t finished with corrosion.
Creep
Creep is an important material behaviour at elevated temperature.
At elevated temperatures , ( > 0.5 T m ), 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: 1) creep rate 2) time to rupture
creep rate
D
strain
D
time
3) elongation or reduction in area
Creep
The rate and extent of creep is very small at temperatures less than 0.5 T m 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 T m for PB is 20 o C.
In the process of creep: • A
load is applied
which produces a
stress less than the yield stres
s.
• 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 T m .
• • •
There are three distinct stages of creep observed at intermediate temperature: The first stage is marked by a rate, which decays with time.
relatively rapid initial creep The second stage is marked by a as steady state creep.
constant creep rate, referred t 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 stag
e, 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 o
f 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
Q c
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
C
s
n
where C and
n
are constants.
Q
c is related to the activation energy for self-diffusion when dislocation climb is important
.
Creep - Rupture Time
The rupture time, t r , relationship due to creep also follows an Arrhenius
t
r
K
s
m
e
(
Q r RT
)
Where Q
r
constants.
is the activation energy for rupture, and K and m are
t r
is dependent on the applied stress,
s
m
.
Creep – Rupture Time
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 e Where s is the tensile stress and e 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
such as many types of plastics is a
diffusion dependent process
that is enhanced by increasing the temperature, i.e., thermally activated process, and follows the
Arrhenius equation
.
e 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
Nabarro Herring mechanism
.
Vacancy motion
along
the grain boundaries is called the
Coble mechanism
.
Creep Mechanisms
Note that the vacancies and atoms move in opposite directions.
Creep
These strain rates are given by
e A 2 s d 2 T e (Q
v
/RT) e A 2 s d 3 T e (Q
b
/RT)
Nabarro-Herring Coble
Where
d is the diameter of the grain
, Q v self or volume diffusion, and Q b is the activation energy of is the activation energy for grain boundary diffusion, which is usually half that of self or volume diffusion. A 2 is a material constant.
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, e A s T
m
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.
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.