Transcript Fracture -- Griffith Cracks

Griffith Cracks
Flaws Make the World Beautiful!
Were it not for the flaws, rocks and mountains would have
been perfectly boring
Griffith Cracks

Rocks have mechanical inhomogeneities/discontinuities 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
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:
1.
2.
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
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
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:
=Ee

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!)
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
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
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
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
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
Experiment

Assume an elliptical crack with semi-major, c, and semiminor axis, b, with an aspect ratio of c/b

Load the crack with a far remote tensile stress within the
plate (r), normal to c

The local tensile stress perpendicular to the c axis is
magnified several times to C (stress concentration )

Inglis showed that stress, C, 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
Experiment…

The highest tensile stress at the end of crack is
C= r (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

C is approximated by:
C= 2r (c/r)1/2
Note: C 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
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:
C=r (1+2c/b)=r [(1+2(c/r)1/2]

The ratio of stress concentration, C at point c, to the
applied stress, A, is called the elastic stress concentration
factor, which for a thin and long ellipse (b < c) is given
by:
C/r ~ 2c/b = 2(c/r)1/2
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
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
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
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
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
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
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
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., dWR<0),
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)
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
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