Transcript Slide 1

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Mechanical Properties of Materials II
Part II:
(Mechanical & Microstructural Aspect)
S. M. K. Hosseini
[email protected]
[email protected]
[email protected]
References
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Mechanical Behavior of Materials (2000), T. H. Courtney, McGraw-Hill, Boston.
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Principle of Fracture Mechanics in Materials, R.W. Hertzberg, Prentice-Hall.
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Materials Principles & Practice, Butterworth Heinemann, Edited by C. Newey &
G. Weaver.
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Mechanical Metallurgy, McGrawHill, G.E. Dieter, 3rd Ed.
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Light Alloys (1996), I.J. Polmear, Wiley, 3rd Ed.
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Hull, D. and D. J. Bacon (1984). Introduction to Dislocations. Oxford, UK,
Pergamon.
Introduction
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Introduction
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Definition
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Fatigue is the name given to failure in response to alternating loads (as
opposed to monotonic straining).
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Instead of measuring the resistance to fatigue failure through an upper
limit to strain (as in ductility), the typical measure of fatigue resistance
is expressed in terms of numbers of cycles to failure. For a given
number of cycles (required in an application), sometimes the stress (that
can be safely endured by the material) is specified.
Introduction
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Fatigue: general characteristics
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Primary design criterion in rotating parts.
Fatigue as a name for the phenomenon based on the notion of
a material becoming “tired”, i.e. failing at less than its
nominal strength.
Cyclical strain (stress) leads to fatigue failure.
Occurs in metals and polymers but rarely in ceramics.
Also an issue for “static” parts, e.g. bridges.
Cyclic loading stress limit<static stress capability.
Introduction
Fatigue: general characteristics
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Most applications of structural materials involve cyclic
loading; any net tensile stress leads to fatigue.
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Fatigue failure surfaces have three characteristic
features:
– A (near-)surface defect as the origin of the crack
– Striations corresponding to slow, intermittent
crack growth
– Dull, fibrous brittle fracture surface (rapid growth).
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Life of structural components generally limited by
cyclic loading, not static strength.
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Most environmental factors shorten life.
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Fatigue testing, S-N curve
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S-N [stress-number of cycles to failure] curve defines locus of
cycles-to-failure for given cyclic stress.
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Rotating-beam fatigue test is standard; also alternating tensioncompression.
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Plot stress versus the
log(number of cycles
to failure), log(Nf).
[see next slide,
also Courtney figs. 12.8, 12.9]
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For frequencies < 200Hz,
metals are insensitive to
frequency; fatigue life in
polymers is frequency
dependent.
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Fatigue testing, S-N curve
Cyclic Tension-Compression Machine
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Fatigue testing, S-N curve
Rotating Bending Machine
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Fatigue testing, S-N curve
sa
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smean 3 > smean 2 > smean 1
smean 1
smean 2
smean 3
The greater the number of
cycles in the loading history,
the smaller the stress that
the material can withstand
without failure.
log Nf
Note the presence of a
fatigue limit in many
steels and its absence
in aluminum alloys.
[Dieter]
Endurance Limits
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Some materials exhibit endurance limits, i.e. a stress below which
the life is infinite: [fig. 12.8]
– Steels typically show an endurance limit, = 40% of yield; this is
typically associated with the presence of a solute (carbon,
nitrogen) that pines dislocations and prevents dislocation
motion at small displacements or strains (which is apparent in
an upper yield point).
– Aluminum alloys do not show endurance limits; this is related
to the absence of dislocation-pinning solutes. d life.
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At large Nf, the lifetime is dominated by nucleation.
– Therefore strengthening the surface (shot peening) is
beneficial to delay crack nucleation and extend life.
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• Alternating stress  sa = (smax-smin)/2.
• Raising the mean stress (sm) decreases Nf. [see slide
19, also Courtney fig. 12.9]
• Various relations between R = 0 limit and the
ultimate (or yield) stress are known as Soderberg
(linear to yield stress), Goodman (linear to ultimate)
and Gerber (parabolic to ultimate). [Courtney, fig.
12.10, problem 12.3]
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Cyclic strain control complements cyclic stress
characterization: applicable to thermal fatigue, or
fixed displacement conditions.
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Cyclic stress-strain testing defined by a controlled
strain range, ∆pl. [see next slide, Courtney, figs.
12.24,12.25]
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Soft, annealed metals tend to harden; strengthened
metals tend to soften.
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Thus, many materials tend towards a fixed cycle, i.e.
constant stress, strain amplitudes.
Cyclic stress-strain curve
• Large number of cycles typically needed to reach
asymptotic hysteresis loop (~100).
• Softening or hardening possible. [fig. 12.26]
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Wavy-slip materials generally
reach asymptote in cyclic
stress-strain: planar slip
materials (e.g. brass) exhibit
history dependence.
Cyclic stress-strain curve
defined by the extrema, i.e. the
“tips” of the hysteresis loops.
[Courtney fig. 12.27]
Cyclic stress-strain curves tend
to lie below those for
monotonic tensile tests.
Polymers tend to soften in
cyclic straining.
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• Strain is a more logical independent variable for
characterization of fatigue. [fig. 12.11]
• Define an elastic strain range as ∆el = ∆s/E.
• Define a plastic strain range, ∆pl.
• Typically observe a change in slope between the elastic
and plastic regimes. [fig. 12.12]
• Low cycle fatigue (small Nf) dominated by plastic strain:
high cycle fatigue (large Nf) dominated by elastic strain.
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sa := Alternating stress
sm :=
Mean stress
R := Stress ratio
 :=
strain
Nf := number of cycles to failure
A := Amplitude ratio
∆pl :=
Plastic strain amplitude
∆el :=
Elastic strain amplitude
K’ := Proportionality constant, cyclic stress-strain
n’ := Exponent in cyclic stress-strain
c :=
Exponent in Coffin-Manson Eq.;
also, crack length
E :=
Young’s modulus
b :=
exponent in Basquin Eq.
m :=
exponent in Paris Law
K :=
Stress intensity
∆K :=Stress intensity amplitude
a :=
crack length
Low Cycle Fatigue (LCF)
Strain control
of fatigue
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Low Cycle Fatigue (LCF)
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Constitutive relation
for cyclic stress-strain:
n’ ≈ 0.1-0.2
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Fatigue life: Coffin Manson relation:
 p
2
s  K  
  f 2 N f

c
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sf ~ true fracture strain; close to tensile ductility
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c ≈ -0.5 to -0.7
c = -1/(1+5n’); large n’
longer life.
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n
Low Cycle Fatigue (LCF)
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For elastic-dominated strains
at high cycles, adapt
Basquin’s equation:
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Intercept on strain axis of extrapolated elastic line = sf/E.
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High cycle = elastic strain control:
slope (in elastic regime) = b = -n’/(1+5n’) [Courtney, fig.
12.13]
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The high cycle fatigue strength, sf, scales with the yield stress
 high strength good in high-cycle
 e
sa  E
 sf 2N b
2
Low Cycle Fatigue (LCF)
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Low Cycle Fatigue (LCF)
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Low cycle = plastic control: slope = c
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Add the elastic and plastic strains.
  el  pl s
f



2N f
2
2
2
E


b

 
f 2N f
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
c
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Cross-over between elastic and plastic control is typically at
Nf = 103 cycles.
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Ductility useful for low-cycle; strength for high cycle
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Examples of Maraging steel for high cycle endurance,
annealed 4340 for low cycle fatigue strength.
*Variable Stress/Strain Histories
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• When the stress/strain history is stochastically varying, a rule for
combining portions of fatigue life is needed.
• Palmgren-Miner Rule is useful: ni is the number of cycles at each
stress level, and Nfi is the failure point for that stress.
[Ex. Problem 12.2]
Fatigue Fracture Surface (Microstructural Feature)
Crack growth striations in
shaft with key way
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Crack growth striations in
crankshaft
Fatigue Fracture Surface (Microstructural Feature)
Macroscopic Features
Microscopic Features
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Fatigue Fracture Surface (Microstructural Feature)
Concentric pattern is
characteristic of fatigue
failures in which a crack
propagates (grows)
under cyclic loading
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The ripple are called “beach
marks”. This fine markings
represent stepwise advances of the
crack front. These beach mark
shows the progression of a crack
front due to an alternating stress
having different magnitudes.
Fatigue Fracture
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Three stages of fatigue fracture
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Crack Nucleation 
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Stress intensity 
stress intensification at crack tip.
crack propagation (growth);
- stage I growth on shear planes (45°),
strong influence of microstructure [Courtney: fig.12.3a]
- stage II growth normal to tensile load (90°)
weak influence of microstructure [Courtney: fig.12.3b].
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Crack propagation  catastrophic, or ductile failure at crack
length dependent on boundary conditions, fracture toughness.
Fatigue Fracture Surface (Microstructural Feature)
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Stage II. Laird mechanism of crack
propagation (Plastic blunting)
StageI. (Wood’s concepts of fatigue crack formation)
Fatigue Fracture Surface (Microstructural Feature)
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• Flaws, cracks, voids can all act as crack nucleation
sites, especially at the surface.
• Therefore, smooth surfaces increase the time to
nucleation; notches, stress risers decrease fatigue life.
• Dislocation activity (slip) can also nucleate fatigue
cracks.
Fatigue Fracture Surface (Microstructural Feature)
Dislocation Slip
Crack Nucleation
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Dislocation slip -> tendency to localize slip in bands.
[see slide 10, also Courtney fig. 12.3]
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Persistent Slip Bands (PSB’s) characteristic of cyclic
strains.
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Slip Bands -> extrusion at free surface. [see next slide
for fig. from Murakami et al.]
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Extrusions -> intrusions and crack nucleation.
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Slip steps and the
stress-strain loop
Fatigue Crack Propagation
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Crack Opening Displacement Test
COD tests are used to determine the change in crack size in
compact tension specimens subjected to cyclic loads.
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Stage I. Crack initiation
Stage II. Stable crack
growth
Stage III. Unstable crack
growth (Fracture)
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Stage I. Crack initiation
Fatigue Crack Propagation
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• Crack Length := a.
Number of cycles := N
Crack Growth Rate := da/dN
Amplitude of Stress Intensity := ∆K = ∆s√c.
• Define three stages of crack growth, I, II and III,
in a plot of da/dN versus ∆K.
• Stage II crack growth: application of linear elastic
fracture mechanics.
• Can consider the crack growth rate to be related to the
applied stress intensity.
• Crack growth rate somewhat insensitive to R (if R<0)
in Stage II [fig. 12.16, 12.18b]
• Environmental effects can be dramatic, e.g. H in Fe, in
increasing crack growth rates.
Fatigue Crack Propagation
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• Paris Law:
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m ~ 3 (steel); m ~ 4 (aluminum).
Crack nucleation ignored!
Threshold ~ Stage I
The threshold represents an endurance limit.
For ceramics, threshold is close to KIC.
Crack growth rate increases with R (for R>0). [fig.
12.18a]
Fatigue Crack Propagation
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• Striations occur by development of slip bands in each
cycle, followed by tip blunting, followed by closure.
• Can integrate the growth rate to obtain cycles as related
to cyclic stress-strain behavior. [Eqs. 12.6-12.8]
cf
dc
c0 dc/ dN
N II  
cf
N II  
c0
dc
A m s c 
m
Fatigue Crack Propagation
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Provided that m>2 and a is constant, can integrate.
A1s  m 1m / 2  1m / 2 
NII 
c0
cf
(m / 2)  1

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If the initial crack length is much less than the final length,
c0<cf, then approximate thus:
A1s m 1m / 2
NII 
c0
(m / 2) 1
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Can use this to predict fatigue life based on known crack
*Damage Tolerant Design
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Calculate expected growth rates from dc/dN data.
Perform NDE on all flight-critical components.
If crack is found, calculate the expected life of the
component.
Replace, rebuild if too close to life limit.
Endurance limits.
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Geometrical effects
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• Notches decrease fatigue life through stress
concentration.
• Increasing specimen size lowers fatigue life.
• Surface roughness lowers life, again through stress
concentration.
• Moderate compressive stress at the surface
increases life (shot peening); it is harder to nucleate
a crack when the local stress state opposes crack
opening.
• Corrosive environment lowers life; corrosion either
increases the rate at which material is removed from
the crack tip and/or it produces material on the crack
surfaces that forces the crack open (e.g. oxidation).
• Failure mechanisms
Microstructure-Fatigue Relationships
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What are the important issues in microstructure-fatigue
relationships?
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Answer: three major factors.
1: geometry of the specimen (previous slide); anything on
the surface that is a site of stress concentration will
promote crack formation (shorten the time required for
nucleation of cracks).
2: defects in the material; anything inside the material that
can reduce the stress and/or strain required to nucleate a
crack (shorten the time required for nucleation of
cracks).
3: dislocation slip characteristics; if dislocation glide is
confined to particular slip planes (called planar slip) then
dislocations can pile up at any grain boundary or phase
boundary. The head of the pile-up is a stress
concentration which can initiate a crack.
Microstructure affects Crack Nucleation
• The main effect of
microstructure (defects,
surface treatment, etc.)
is almost all in the low
stress intensity regime,
i.e. Stage I. Defects,
for example, make it
easier to nucleate a
crack, which translates
into a lower threshold
for crack propagation
(∆Kth).
• Microstructure also
affects fracture
toughness and
therefore Stage III.
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da/dN
I
II
∆Kc
III
∆Kth
∆K
Defects in Materials
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• Descriptions of defects in materials at the sophomore level
focuses, appropriately on intrinsic defects (vacancies,
dislocations). For the materials engineer, however, defects
include extrinsic defects such as voids, inclusions, grain
boundary films, and other types of undesirable second phases.
• Voids are introduced either by gas evolution in solidification or
by incomplete sintering in powder consolidation.
• Inclusions are second phases entrained in a material during
solidification. In metals, inclusions are generally oxides from
the surface of the metal melt, or a slag.
• Grain boundary films are common in ceramics as glassy films
from impurities.
• In aluminum alloys, there is a hierachy of names for second
phase particles; inclusions are unwanted oxides (e.g. Al2O3);
dispersoids are intermetallic particles that, once precipitated,
are thermodynamically stable (e.g. AlFeSi compounds);
precipitates are intermetallic particles that can be dissolved or
precipiated depending on temperature (e.g. AlCu compounds).
Metallurgical Control: fine particles
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• Tendency to localization of flow is deleterious to the initiation of
fatigue cracks, e.g. Al-7050 with non-shearable vs. shearable
precipitates (Stage I in a da/dN plot). Also Al-Cu-Mg with
shearable precipitates but non-shearable dispersoids, vs. only
shearable ppts.
graph courtesy of J.
Staley, Alcoa
Coarse particle effect on fatigue
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• Inclusions nucleate cracks cleanliness (w.r.t. coarse
particles) improves fatigue life, e.g. 7475 improved by
lower Fe+Si compared to 7075:
0.12Fe in 7475, compared to 0.5Fe in 7075;
0.1Si in 7475, compared to 0.4Si in 7075.
graph courtesy of J.
Staley, Alcoa
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• Increasing hardness tends to raise the endurance limit for high
cycle fatigue. This is largely a function of the resistance to
fatigue crack formation (Stage I in a plot of da/dN).
Mobile solutes that pin
dislocations fatigue
limit, e.g. carbon in steel
Casting porosity affects fatigue
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Gravity cast
versus
squeeze cast
versus
wrought
Al-7010
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Casting tends to result in porosity. Pores are effective sites for nucleation
of fatigue cracks. Castings thus tend to have lower fatigue resistance (as
measured by S-N curves) than wrought materials.
Casting technologies, such as squeeze casting, that reduce porosity tend
to eliminate this difference.
Titanium alloys
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[Polmear]
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For many Ti alloys, the proportion of hcp (alpha) and bcc (beta) phases
depends strongly on the heat treatment. Cooling from the two-phase region
results in a two-phase structure, as Polmear’s example, 6.7a. Rapid cooling
from above the transus in the single phase (beta) region results in a twophase microstructure with Widmanstätten laths of (martensitic) alpha in a
beta matrix, 6.7b.
The fatigue properties of the two-phase structure are significantly better than
the Widmanstätten structure (more resistance to fatigue crack formation).
The alloy in this example is IM834, Ti-5.5Al-4Sn-4Zr-0.3Mo-1Nb-0.35Si-0.6C.
*Design Considerations
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If crack growth rates are normalized by the elastic modulus, then material
dependence is mostly removed! [Courtney fig. 12.20]
Can distinguish between intrinsic fatigue [use Eq. 12.4 for combined
elastic, plastic strain range] for small crack sizes and extrinsic fatigue [use
Eq. 12.6 for crack growth rate controlled] at longer crack lengths. [fig.
12.21….]
Inspection of design charts, fig. 12.22, shows that ceramics sensitive to
crack propagation (high endurance limit in relation to fatigue threshold).
*Design Considerations: 2
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• Metals show a higher fatigue threshold in relation to their
endurance limit. PMMA and Mg are at the lower end of the
toughness range in their class. [Courtney fig. 12.22]
• Also interesting to compare fracture toughness with fatigue
threshold. [Courtney fig. 12.23]
• Note that ceramics are almost on ratio=1 line, whereas metals tend
to lie well below, i.e. fatigue is more significant criterion.
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*Fatigue
property map
[Courtney]
*Fatigue property map
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*Fatigue in Polymers
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Many differences from metals
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Cyclic stress-strain behavior often exhibits softening; also affected
by visco-elastic effects; crazing in the tensile portion produces
asymmetries, figs. 12.34, 12.25.
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S-N curves exhibit three regions, with steeply decreasing region II,
fig. 12.31.
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Nearness to Tg results in strong temperature sensitivity, fig. 12.42
Fatigue: summary
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Critical to practical use of structural materials.
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Fatigue affects most structural components, even apparently
statically loaded ones.
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Well characterized empirically.
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Connection between dislocation behavior and fatigue life offers
exciting research opportunities, i.e. physically based models are
lacking!