Mechanics of Materials II

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Mechanics of Materials II

• UET, Taxila
• Lecture No. (45)

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Mechanisms of material failure

•  In order to understand the various approaches to
  modeling fracture, fatigue and failure, it is
  helpful to review briefly the features and
  mechanisms of failure in solids.
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Failure under monotonic loading

• If you test a sample of any material under
  uni-axial tension it will eventually fail.  The
  features of the failure depend on several
  factors, including
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• 1-The materials involved and their
  micro-sctructure
•    2-The applied stress state
•    3- Loading rate
•    4-Temperature
•    5-Ambient environment (water vapor or
  presence of corrosive environments)

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Brittle and Ductile failures

• Materials are normally classified loosely as
  either brittle  or ductile depending on the
  characteristic features of the failure. 

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Brittle Materials

• Examples of brittle materials include
• glasses,
• ceramics (Oxides, Carbides Nitrides)
• and Cast Iron

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Features of a brittle material are

• 1-Very little plastic flow occurs in the specimen
  prior to failure
• 2-The two sides of the fracture surface fit
  together very well after failure. 
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• 3- The fracture surface appears faceted you can
  make out individual grains and atomic planes.
• 4- In many materials, fracture occurs along
  certain crystallographic planes.  In other
  materials, fracture occurs along grain boundaries
  

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Ductile Materials

• Examples of ductile
• Tin and lead
• FCC metals at all temperatures
• BCC metals at high temperatures
• polymers at relatively high temperature.   

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• Features of a ductile fracture are
• Extensive plastic flow occurs in the material
  prior to fracture.
• There is usually evidence of considerable necking
  in the specimen

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• Fracture surfaces dont fit together.
• The fracture surface has a dimpled appearance
  you can see little holes

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• Complex Materials
• Of course, some materials have such a complex
  microstructure (especially composites) that its
  hard to classify them as entirely brittle or
  entirely ductile.
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• How Brittle Fracture occurs
• Brittle fracture occurs as a result of a single
  crack, propagating through the specimen.  Most
  materials contain pre-existing cracks, in which
  case fracture is initiated when a large crack in
  a region of high tensile stress starts to grow. 

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How ductile fracture occurs

• Ductile fracture occurs as a result of the
  nucleation, growth and coalescence of voids in
  the material
• Failure is controlled by the rate of nucleation
  of the voids and their rate of growth.

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Schematic Representation for Ductile Brittle
Fracture
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• Does material fail under stresses lower than
  yield strength?

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• The answer is yes

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• This chapter would not be complete, therefore,
  without reference to certain loading conditions
  under which materials can fail at stresses much
  less than the yield stress, namely creep and
  fatigue.

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Creep and Fatigue

• In the preceding sections it has been suggested
  that failure of materials occurs when the
  ultimate strengths have been exceeded.

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Definition of creep

• Creep is the gradual increase of plastic strain
  in a material with time at constant load.

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Creep at high temperature

• At elevated temperatures some materials (most
  metals) are susceptible to this phenomenon and
  even under the constant load mentioned, strains
  can increase continually until fracture.

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Applications of creep

• This form of fracture is particularly relevant to
  turbine blades, nuclear reactors, furnaces,
  rocket motors, etc.

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Typical creep curve.
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In previous figure

• The general form of the strain versus time graph
  or creep curve is shown for two typical operating
  conditions.

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Four features

• In each case the curve can be considered to
  exhibit four principal features

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• (a) An initial strain, due to the initial
  application of load. In most cases this would be
  an elastic strain.
• (b) A primary creep region, during which the
  creep rate (slope of the graph) diminishes.

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• (c) A secondary creep region, when the creep rate
  is sensibly low (Constant).
• (d) A tertiary creep region, during which the
  creep rate accelerates to final fracture.

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Design for creep

• It is clearly vital that a material which is
  susceptible to creep effects should only be
  subjected to stresses which keep it in the
  secondary (straight line) region throughout its
  service life.

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• This enables the amount of creep extension to be
  estimated and allowed in design.

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Fatigue

• Definition of Fatigue
• Fatigue is the failure of a material under
  fluctuating stresses each of which is believed to
  produce minute amounts of plastic strain.

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Applications of fatigue

• Fatigue is particularly important in components
  subjected to repeated and often rapid load
  fluctuations, e.g. aircraft components, turbine
  blades, vehicle suspensions, etc.

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Representation of Fatigue

• Fatigue behaviour of materials is usually
  described by a
• fatigue life or S-N curve
• in which the number of stress cycles N to produce
  failure
• is plotted against S.

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• A typical S-N curve for mild steel is shown in
  next figure

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S-N Curve
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Fatigue limit

• The particularly relevant feature of this curve
  is the limiting stress Sn since it is assumed
  that stresses below this value will not produce
  fatigue failure however many cycles are applied,
  i.e. there is infinite life.

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Design for fatigue

• In the simplest design cases, therefore, there is
  an aim to keep all stresses below this limiting
  level.

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Solved

• Examples

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• Example 1
• Determine the stress in each section of the bar
  shown in next Figure when subjected to an axial
  tensile load of 20 kN. The central section is 30
  mm square cross-section the other portions are
  of circular section, their diameters being
  indicated.
• What will be the total extension of the bar? For
  the bar material E 210GN/m2.

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Figure (1)
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Solution
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Stress in section (1)
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Stress in section 2
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Stress in section 3
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• Now the extension of a bar can always be written
  in terms of the stress in the bar since

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Example 2

• (a) A 25 mm diameter bar is subjected to an axial
  tensile load of 100 kN. Under the action of this
  load a 200mm gauge length is found to extend by
  the distance
• 0.19 x 10-3 mm.
• Determine the modulus of elasticity for the bar
  material.

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Solution (a)
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• (b) If, in order to reduce weight whilst keeping
  the external diameter constant, the bar is bored
  axially to produce a cylinder of uniform
  thickness, what is the maximum diameter of bore
  possible, given that the maximum allowable stress
  is 240MN/m2?
• The load can be assumed to remain constant at 100
  kN.

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Solution (b)

• Let the required bore diameter be d mm the
  cross-sectional area of the bar will then be
  reduced to

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• But this stress is restricted to a maximum
  allowable value of 240 MN/m2.

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• The maximum bore possible is thus 9.72 mm.

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• (c) What will be the change in the outside
  diameter of the bar under the limiting stress
  quoted in (b)?
• Where E 210 GN/m2
• and ? 0.3

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Solution (C)

• The change in the outside diameter of the bar
  will be obtained from the lateral strain,

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?d/d
s/E
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• Example 3
• The coupling shown in next Figure is constructed
  from steel of rectangular cross-section and is
  designed to transmit a tensile force of 50 kN. If
  the bolt is of 15 mm diameter calculate

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• (a) the shear stress in the bolt
• (b) the direct stress in the plate
• (c) the direct stress in the forked end of the
  coupling.

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Double Shear
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Solution

• (a) The bolt is subjected to double shear,
  tending to shear it as shown in Figure. There is
  thus twice the area of the bolt resisting the
  shear and from equation

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v
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• (c) The force in the coupling is shared by the
  forked end pieces, each being subjected to a
  direct stress

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Problems