Callister - An introduction - 8th edition

8.5 Principles of Fracture Mechanics • 243
stress raiser
by the specimen cross-sectional area (perpendicular to this load). Because of
their ability to amplify an applied stress in their locale, these flaws are sometimes
called stress raisers.
If it is assumed that a crack is similar to an elliptical hole through a plate and
is oriented perpendicular to the applied stress, the maximum stress, m , occurs at
the crack tip and may be approximated by
For tensile loading,
computation of
maximum stress at
a crack tip
s m 2s 0 a a 1/2
b
r t
(8.1)
where 0 is the magnitude of the nominal applied tensile stress, t is the radius of
curvature of the crack tip (Figure 8.8a), and a represents the length of a surface
crack, or half of the length of an internal crack. For a relatively long microcrack
that has a small tip radius of curvature, the factor (a/ t ) 1/2 may be very large. This
will yield a value of m that is many times the value of 0 .
Sometimes the ratio m / 0 is denoted as the stress concentration factor K t :
K t s m
2 a a 1/2
b
s 0 r t
(8.2)
which is simply a measure of the degree to which an external stress is amplified at
the tip of a crack.
By way of comment, it should be said that stress amplification is not restricted
to these microscopic defects; it may occur at macroscopic internal discontinuities
(e.g., voids or inclusions), at sharp corners, scratches, and notches.
Furthermore, the effect of a stress raiser is more significant in brittle than in
ductile materials. For a ductile metal, plastic deformation ensues when the maximum
stress exceeds the yield strength. This leads to a more uniform distribution
of stress in the vicinity of the stress raiser and to the development of a maximum
stress concentration factor less than the theoretical value. Such yielding and stress
redistribution do not occur to any appreciable extent around flaws and discontinuities
in brittle materials; therefore, essentially the theoretical stress concentration
will result.
Using principles of fracture mechanics, it is possible to show that the critical stress
c required for crack propagation in a brittle material is described by the expression
Critical stress for
crack propagation in
a brittle material
s c a 2Eg s
pa b 1/2
(8.3)
where
E modulus of elasticity
s specific surface energy
a one-half the length of an internal crack
All brittle materials contain a population of small cracks and flaws that have a
variety of sizes, geometries, and orientations. When the magnitude of a tensile stress
at the tip of one of these flaws exceeds the value of this critical stress, a crack forms
and then propagates, which results in fracture. Very small and virtually defect-free
metallic and ceramic whiskers have been grown with fracture strengths that approach
their theoretical values.