Transcript Fracture-Ch7

Fracture Overview
Fall 2011
Rectangular Plate with Hole
Figure 6.1 Rectangular plate with hole subjected to axial load. (a)
Plate with cross-sectional plane; (b) one-half of plate with stress
distribution; (c) plate with elliptical hole subjected to axial load.
• Thermal Stress:
• Stress = E.a. dT
Flaws Make the World Beautiful!
Were it not for the flaws, rocks and mountains would have been perfectly boring
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
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
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:
dUT = dUs - dWR + dUE
• 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