Griffith Cracks

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Griffith Cracks

• Flaws Make the World Beautiful!
• Were it not for the flaws, rocks and mountains
  would have been perfectly boring

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Griffith Cracks

• Rocks have mechanical inhomogeneities/discontinuit
  ies e.g.
• Flaws, fossils, inclusions, cavities, grain
  boundaries, and microcracks
• These inhomogeneities have different elastic
  properties compared to the surrounding rock
• Their presence perturbs the otherwise
  homogeneous, mechanically- or thermally-induced
  remote stress field
• This leads to an inhomogeneous stress field that
  initiates joints when the concentrated local
  tensile stress exceeds the tensile strength of
  the rock

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Micro-Flaws

• Micro-flaws are the main factor in structural
  failure in man-made structures (e.g., ship,
  bridge, dam), by producing stress concentration
• Facts
• Failures in high strength material commonly occur
  under low stress
• Brittle solid materials are much weaker (i.e.,
  have lower fracture strength) under tension than
  under compression

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Micro-Flaws - Facts

• 3. The fracture strength, which is an inherent
  property for an ideal, continuous brittle solid,
  and represents the critical stress needed to
  fracture, is not highly reproducible
• Testing methods, dimensions of test specimens,
  environmental conditions, and intrinsic
  structural characteristics are but a few factors
  influencing the variation of the fracture
  strength of brittle solid material

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Strength

• Resistance of a rock to fracture
• Is a critical value of stress at which fracture
  occurs and rock fails
• The theoretical tensile strength, which is the
  stress needed to break atomic bonds of an ideal
  brittle material, is about one tenth of its
  Young's modulus (E/10). Recall that
• ? E e
• The Young's modulus, E, for most rocks is
  commonly of the order of 105 or 106 bars,
  implying great strengths for these rocks (i.e.,
  104 or 105 bar or 10-100 kbar!)

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Strength

• However, for real brittle material, the measured
  tensile strength is 1 to 2 orders of magnitude
  less than the theoretical tensile strength (i.e.,
  E/1000-E/100)
• i.e., rock strength is in the order of 102-103
  bars
• This indicates that the fracture strength in such
  solids is not an intrinsic material property
• The discrepancy between the molecular cohesive
  forces and the observed tensile strength of real
  material solids has been attributed by Griffith
  (1920) to the presence of planar defects or
  cracks that are since referred to as the Griffith
  cracks

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Micro-Flaws

• Structural Definition of Rock
• Polycrystalline aggregate that commonly has a
  random population of mostly inhomogeneous and
  anisotropic pre-existing or mechanically-induced
  micro-flaws
• These flaws, that include microcracks, grain
  boundaries, and pores, control the mechanical
  behavior of imperfect rocks

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Micro-Flaws

• Problems dealing with crack initiation are
  concerned with how and where cracks start,
  whereas those dealing with propagation study the
  path that the cracks take, and the extent to
  which they grow
• Fracture mechanics established a relationship
  between fracture strength and microcrack geometry
  and fracture toughness

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Inglis (1913)

• Recognized the destructive influence of cracks in
  brittle material
• Determined stresses around an elliptical
  stress-free hole and an extreme case of a fine
  straight crack
• He examined a brittle, homogeneous, isotropic,
  plate under tension using a mathematical approach

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Inglis (1913)

• Showed that a pull applied to the ends of an
  elastic plate would produce tensile stresses at
  the tip of a crack, that may exceed the elastic
  limit of the material and lead to the propagation
  of the crack
• Showed that the increase in the length of the
  crack exaggerates the stress even more, such that
  the crack would continue to spread

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Experiment

• Assume an elliptical crack with semi-major axis,
  c, and semi-minor axis, b, with an aspect ratio
  of c/b
• Load the crack with a remote tensile stress
  within the plate (sr), normal to c
• The local tensile stress perpendicular to the c
  axis is magnified several times to sC (stress
  concentration )
• Inglis showed that stress, sC, at the tip of the
  crack,
• varies with the length and radius of curvature(r
  b2/c) at the apices of the crack,
• and is proportional to the square root of length
  (c), and inversely proportional to the r of the
  crack sC 2sr (c/r)1/2

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Experiment

• The highest tensile stress at the end of crack is
• sC sr (1 2c/b)
• The stress concentration is maximum at the crack
  tip (where r is minimum), and rapidly decreases
  within a short distance from the crack tip
• sC is approximated by
• sC 2sr (c/r)1/2
• Note sC depends on shape (i.e., aspect ratio,
  c/b) and not on the size of the elliptical cavity
• When b 0 (i.e., radius of curvature, r b2/c
  0), the stress concentration (sC sr (1 2c/b)
  at the crack tip is infinity

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Stress Concentration Factor

• The minimum value of the radius of curvature at
  the end of major axis of the elliptical cavity,
  r, is at point C and is given by r b2/c
• The ratio of stress concentration, sC at point c,
  to the applied stress, sA, is called the elastic
  stress concentration factor, which for a thin and
  long ellipse (b lt c) is given by
• sC/sr 2c/b 2(c/r)1/2
• Note that r is zero at tip of sharp crack, where
  stress concentration is maximum, and is given by
• sC sr (12c/b) sr (1 2(c/r)1/2

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Griffith (1920)

• Griffith (1920), realized the significance of
  microcracks in reducing the fracture strength
• Applied the mathematical work of Inglis (1913)
  for an elliptical hole, and developed a
  theoretical criterion of rupture based on the
  concept of minimum potential energy of classical
  mechanics and thermodynamics which seeks a
  minimum total free energy of a system

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Griffith Theory

• In the Griffith theory, the theoretical strength
  is the microscopic fracture stress which is
  actually reached in a very small volume of the
  rock while the mean stress may remain very low
• Griffith's work, which has since been known as
  the Griffith energy balance approach, and has
  served as a foundation for fracture mechanics,
  deals with the equilibrium state of an elastic,
  solid body, deformed by specified surface forces

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Griffith (1920)

• Griffith extended the theorem of minimum energy
  by accounting for the increase of surface energy
  which occurs during formation of cracks
• He assumed that the equilibrium position is one
  in which rupture of the solid occurs if the
  system is allowed to pass from an unbroken to a
  broken state through a process involving
  continuous reduction of potential energy

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Griffith (1920)

• Griffith (1920) argued that brittle solids fail
  by incremental propagation of a multitude of
  randomly-oriented, small pre-existing cracks
• Griffith cracks are common in rocks and include
  intragranular and intergranular microcracks
  (grain boundaries) and larger transgranular
  cracks
• In a larger scale, the Griffith flaws include
  joints, faults, and bedding planes

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Fracture Strength

• Brittle fracture strength depends largely on the
  elastic properties of the elastic rock and the
  length and sharpness of the flaws
• Stress concentrators such as low aspect ratio
  cavities, inclusions, material property
  mismatches, and fossils, give rise to tensile
  stresses that may fracture rocks even when
  applied stresses are compressive provided they
  are non-hydrostatic

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Griffith (1920)

• The intensification of stress depends on the
• Length and orientation of the crack with respect
  to the applied stress
• Radius of curvature at their tips, rendering
  certain cracks more vulnerable than others
• The propagation of these Griffith cracks involves
  the creation of two new incremental crack
  surfaces which is a process that absorbs energy

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Griffith (1920)

• The creation of these new surfaces in the
  interior of a solid (by crack propagation) leads
  to an increase in potential energy as work must
  be done against the cohesive forces of the
  molecules on either side of the crack
• The work is part of the potential surface energy
  of the system. Thus bounding surfaces posses a
  surface tension and a corresponding amount of
  potential energy

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Griffith Energy-Balance Concept

• If we subject the outer boundary of a rock to
  tension (such that boundary moves out)
• This decreases the potential energy (i.e.,
  dWRlt0), of the loading device (Fig. 3.2
  Engelder). R designates rock
• The work to propagate the crack is positive, and
  is defined as an increase in surface energy (dUs)

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Griffith Energy-Balance Concept

• As the crack propagates, the rock undergoes a
  change in strain energy (dUE).
• The total change in energy for crack propagation
  is
• ?UT ?Us - ?WR ?UE
• Griffith energy balance concept
• Propagation occurs without changing the total
  energy of the rock-crack system.
• i.e., for an increment of crack extension (dc),
• d UT /dc 0

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Griffith Energy-Balance Concept

• This means that the mechanical and surface energy
  terms within the rock-crack system must balance
• The motion of the crack walls represents a
  decrease in mechanical energy
• While work (as surface energy) must be done to
  remove the restraints across crack increment