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Griffith Cracks

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Title: Griffith Cracks


1
Griffith Cracks
  • Flaws Make the World Beautiful!
  • Were it not for the flaws, rocks and mountains
    would have been perfectly boring

2
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

3
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

4
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

5
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!)

6
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

7
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 micro-cracks, grain
    boundaries, and pores, control the mechanical
    behavior of imperfect rocks

8
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 micro-crack
    geometry and fracture toughness

9
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

10
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

11
Experiment
  • Assume an elliptical crack with semi-major, c,
    and semi-minor axis, b, with an aspect ratio of
    c/b
  • Load the crack with a far 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

12
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13
Experiment
  • The highest tensile stress at the end of crack is
  • sC sr (1 2 c/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)
    and not on the size of the elliptical cavity
  • When b 0 (i.e., r b2/c 0), the stress
    concentration at the crack tip is infinity

14
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
  • In Fig. 1.1, note that r is zero at tip of sharp
    crack, where stress concentration is maximum, and
    is given by
  • sCsr (12c/b)sr (12(c/r)1/2
  • 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

15
Griffith (1920)
  • Griffith (1920), realized the significance of
    micro-cracks 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

16
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

17
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

18
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

19
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

20
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

21
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

22
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)

23
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

24
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
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