Fracture Mechanics

1
Fracture Mechanics

• Types of fracture
• Sudden fracture suffered by brittle materials at
  ordinary temperatures, structural steels at low
  temperature.
• Fracture initiated by pre-existing flaws
  (cracks). In this case, the design strategy
  adopted is for non-propagation or controlled
  propagation.
• Fatigue-initiated fracture.
• Creep rupture at elevated temperatures.
• Linear Elastic Fracture Mechanics (L.E.F.M.)
• The 2nd type can be studied by assessing the
  stresses around a crack tip using Elasticity
  theory

2
L. E. F. M.

• Griffiths theory for an elastic solid under
  stress (strain energy stored)
• During the formation of a crack of length a
• Ue(a) is elastic strain energy released.
• For a below a critical value, this is
  compensated by energy stored on the surfaces of
  the crack. Define
• Us(a) material capacity for surface energy
  storage
• For a small crack growth, the material behaviour
  is stable (crack is stationary) if
• dUs gt dUe
• This is satisfied up to a certain ac above which
  the crack propagates. Therefore, the critical
  condition is
• dUs dUe (1)
• This leads to a relation between critical design
  parameters (fracture load, maximum crack length)
  and material fracture properties.

3
Stress intensity factor

• Stress near the crack tip
• The solution for the stresses around an
  elliptical hole can lead to that for a crack by
  letting b/a ?0. Then the ellipse tends to become
  a crack of length 2a.

• Stress close to the crack tip

4
Stress intensity factor

• Moving the origin of the co-ordinate system to
  the tip of the crack and changing the horizontal
  co-ordinate to
• r x a
• transforms the formula for the stress near the
  crack tip to

• Very near the tip, that is, for r ltlt a, the above
  reduces to

• Stress intensity factor

5
Stress intensity factor

• General form
• where W is the critical solid dimension. Function
  f is often represented by Y in the technical
  literature.
• Modes of crack displacement
• I opening (the more significant)
• II sliding
• III tearing

6
L. E. F. M.

• Consider a crack of length 2a in an infinite
  plate (of unit thickness) under uniform tension s
  in the direction perpendicular to the crack. Then
  (accounting for symmetry with respect to y axis
  and considering only half of the plate)
• dUs Gda (2)
• where G is the energy absorption capacity per
  unit area of crack and da an infinitesimal crack
  growth.
• Respective energy release

7
L. E. F. M.

• Opening displacement
• Substituting the above into the integral for dUe

• Critical condition (1) gives

• GC critical strain energy release rate or crack
  extension force
• KIC fracture toughness (material properties )

8
L. E. F. M.

• Relation between GC and KIC

• Theory limitation due to metal plasticity
  this must be restricted to a minimum around the
  crack tip.
• Plane strain conditions must be satisfied.
• Criterion of validity both B and a should be
  greater than approximately 25 times the plane
  strain plastic zone size, that is,