Chapter9 Failure

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Chapter-9 Failure
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Goals Why study Failure?

• Design of a structure should be done with a
  damage tolerance design concept that is a design
  engineer assumes that there are damages (flaws)
  present in the structure and is going to fail by
  one of the failure modes.

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Goals Why study Failure?

• Thus by learning all the failure modes ( i.e
  fracture, fatigue, creep etc) a design engineer
  can design more effectively.
• For example if the fatigue life of an airplane is
  20000 cycles then it is recommended to run that
  airplane for 20000 cycles or less.

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Learning objectives

• Define stress concentration factor, stress
  intensity factor and fracture toughness.
• Define impact testing and ductile to brittle
  transition.

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Learning objectives

• Define fatigue failure, fatigue life time,
  fatigue limit and fatigue strength.
• Define Creep, steady state creep rate and rupture
  life time

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• Fracture
• When a structure breaks at a stress level lower
  than the yield strength of the material that is
  called fracture.

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• Fundamentals of
  
  fracture
• Fracture of a structure could be ductile or
  brittle.
• When the structure undergoes a large amount of
• plastic deformation before fracture that is a
  ductile fracture.

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Fundamentals of fracture

• On the other hand fracture without any plastic
  deformation is a brittle fracture (Figures 9.1,
  9.2, and 9.3).
• Ductile fracture -
• Ductile fracture is explained in Fig 9.2.

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Fig 9.1 (a) Highly ductile fracture (b)
Moderate ductile fracture (c) Brittle
fracture
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Fig 9.2 (a) Initial necking (b) Small cavity
formation (c) Coalescence of cavities to form a
crack (d) Crack propagation (e) Final shear
fracture
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Fig 9.3 (a) Cup and cone fracture in
aluminium(b) Brittle fracture in mild steel
Ductile fracture
Brittle Fracture
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• Fracture Toughness
• The critical stress for crack propagation, ?c, is
  related to crack length through the following
  equation
• KC Y ?c ?(?a)
• Where KC is defined as the fracture toughness and
  it is a measure of materials resistance to
  fracture.

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Fig 9.11 Schematic representation of (a) An
interior crack in a plate of finite width (b) An
edge crack in a plate of semi-finite width
?c
?c
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Fracture Toughness

• The higher the value of KC, greater is the
  materials resistance to fracture.
• Y is a factor that depends on crack and specimen
  size and geometry.

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Fracture Toughness

• When the specimen is thick and pulled under pure
  tension (Fig 9.9(a).) then the corresponding
  fracture toughness is called KIC.

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Fracture Toughness

• Then the critical stress for crack propagation,
  ?c, is related to crack length through the
  following equation
• KIC Y ?c ?(?a)

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Stress Intensity Factor

• If the applied stress s lt ?c , the corresponding
  stress intensity factor, K, is
• K Y ? ?(?a)

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• Design using fracture mechanics
• KIC is a material parameter and is listed in
  design handbooks. Thus the design stress can be
  calculated if the flaw size in the structure is
  known as following
• KIC Y ?f ?(?a)
• Or ?f KIC / (Y ?(?a))
• Y, a, KIC are known

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Design using fracture mechanics

• On the other hand, if the design stress is fixed
  then the allowable flaw size in the structure can
  be calculated as
• KIC Y ?f ?(?ac)
• ?ac KIC/(Y ?f ??)
• or ac 1/? (KIC/ Y ?f )2
• Y, ?f , KIC are known

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Design using fracture mechanics

• Periodic examination of the crack is done by
  using non destructive inspection methods. When a
  ? ac that structure is discarded.

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• Impact Fracture Testing
• Impact testing is performed to find out if the
  fracture is a brittle or ductile fracture.
• Impact testing techniques
• In impact testing a sample with a v shaped
  notch is used. A pendulum is released from a
  height h to strike the sample.

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(a) Specimen used for charpy and Izod impact
tests. (b) A schematic drawing of an impact
testing apparatus.
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Impact Fracture Testing

• The pendulum reaches a final height h after it
  hits the sample.
• Then the impact energy imparted to the sample
• mgh - mgh
• where m is the mass of the pendulum.

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• Ductile to brittle transition
• The impact test is performed at different
  temperatures and impact energy is plotted against
  temperature.
• The low impact energy at low temp indicates a
  brittle fracture and high impact energy at higher
  temperature indicates a ductile fracture.

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Ductile to brittle transition

• The temperature at which the fracture behavior
  changes from ductile to brittle is the ductile
  brittle transition temperature (DBTT).

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Temperature dependence of Charpy V-notch impact
energy (curve A) and percent shear fracture
(curve B) for an A283 steel.
Ductile Brittle Transition Temperature
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Fig 9.21 Photograph of fracture surfaces of A36
steel Charpy V-notch specimens tested at
indicated temperatures.
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Ductile to brittle transition

• Application
• Ships and off shore oil drilling platforms are
  made out of steel.
• The steel used should not be brittle when the
  temperature gets cold.

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Ductile to brittle transition

• Fig 9.22 shows that as the amount of carbon
  increases DBTT increases.
• Thus the steel to be used in the cold weather
  should have as little carbon as possible.

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Fig 9.22 Influence of carbon content on the
charpy V-notch energy-versus-temperature behavior
for steel.
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• Fatigue
• Failure of a machine or a structure under dynamic
  and fluctuating stress is defined as fatigue
  failure.
• The fatigue failure occurs at a stress level much
  lower than the yield strength of materials.

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Variation of stress with time that accounts for
fatigue failures.(a) Reversed stress cycle
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Repeated stress cycle.
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Schematic diagram of fatigue testing for making
rotating-bending tests.
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• Cyclic stress
• The fluctuating stress could be simply sinusoidal
  which is repeated many times.
• The maximum tensile stress is
• defined as ?max
• The minimum stress is ?min
• Then the stress range, ?r, is defined as
• ?r ?max-?min

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Cyclic stress

• Stress amplitude, ?a, is defined as
• ?a ?r /2 (?max-?min) /2
• Stress ratio, R, is defined as
• R (?min/?max)
• The mean stress, ?m, is defined as
• ?m (?max ?min)/2

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• The S-N curve
• Fatigue testing can be done under simple tensile
  compression sinusoidal loads.
• Different samples are subjected to different
  stress levels to failure.

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The S-N curve

• Then the stress against number of cycles to
  failure is plotted.
• The curve is called S-N curve.
• If the S-N curve becomes horizontal at lower
  stress level, that stress is called fatigue
  limit.

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The S-N curve

• The structure would not fail if fatigue limit
  stress is used.
• If the S-N curve does not become horizontal the
  stress corresponding to 107 cycle is defined as
  the fatigue strength.

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The S-N curve

• Fatigue life at a particular stress, S1, is the
  number of cycles to cause failure at
• that stress level.

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Stress amplitude (S) versus logarithm of the
number of cycles to fatigue failure (N) for (a) a
material that displays a fatigue limit.
Fatigue Life
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A material that does not display a fatigue limit.
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• Crack initiation and propagation
• Fatigue failure is characterized by three
  distinct steps crack initiation, crack
  propagation and final failure.
• Crack initiation
• The crack initiates at some point on the surface
  of the structure where there is maximum stress
  concentration.

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• Crack propagation
• Crack propagation forms two types of marking on
  the fracture surface beach markings and
  striations.
• Beach markings are formed when there is
  interruption in the crack propagation stage.
• Striations are formed during the propagation of
  the crack during each stress cycle.

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Fracture surface of a rotating steel shaft that
experienced fatigue failure.
Origin
Final failure
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Crack length versus the number of cycles at
stress levels s1 and s2 for fatigue studies.
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Schematic representation of logarithm fatigue
crack propagation rate da /dN versus logarithm
stress intensity factor range ?K.
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• Surface effects
• The fatigue crack starts on the surface of a
  structure at a stress concentration site.
• Various surfaces treatments are done to improve
  the fatigue life.
• Carburizing treatment (case hardening) you
  learned in chapter 5 is one such treatment.
• Other treatment done is shot peening.

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Demonstration of how design can reduce stress
amplification.(a) poor design sharp corner(b)
Good design
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Schematic S-N fatigue curves for normal and shot-
peened steel.
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• Creep
• Deformation of materials subjected to high
  temperature (0.4 Tm) and stress at the same time
  is called creep.
• Turbine rotors in jet engines and steam generator
  are under that environment.

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• Generalized creep behavior
• A typical creep curve of strain vs. time under
  constant load and constant temperature is shown
  in the next Fig.
• The curve has three distinct regions Primary,
  Secondary and tertiary.
• In the primary region after an instantaneous
  deformation
• the creep rate ?e/ ?t, (Slope of the curve)
  decreases.

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Generalized creep behavior

• In the secondary region the creep rate is
  constant.
• In the tertiary region the creep rate increase
  and then final fracture occurs.

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Typical creep curve of strain versus time at
constant stress and constant elevated
temperature.
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Influence of stress, s and temperature, T on
creep behavior.
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Stress versus rupture life time for a low carbon-
Ni alloy at 3 temperatures.
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Logarithm stress versus the Larson-Miller
parameter for an S-590 iron.