Page 265 - Shigley's Mechanical Engineering Design

bud29281_ch05_212-264.qxd 11/27/2009 6:46 pm Page 240 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas:

240 Mechanical Engineering Design
The use of elastic stress-concentration factors provides an indication of the average
load required on a part for the onset of plastic deformation, or yielding; these factors
are also useful for analysis of the loads on a part that will cause fatigue fracture.
However, stress-concentration factors are limited to structures for which all dimensions
are precisely known, particularly the radius of curvature in regions of high stress con-
centration. When there exists a crack, flaw, inclusion, or defect of unknown small radius
in a part, the elastic stress-concentration factor approaches infinity as the root radius
approaches zero, thus rendering the stress-concentration factor approach useless.
Furthermore, even if the radius of curvature of the flaw tip is known, the high local
stresses there will lead to local plastic deformation surrounded by a region of elastic
deformation. Elastic stress-concentration factors are no longer valid for this situation,
so analysis from the point of view of stress-concentration factors does not lead to cri-
teria useful for design when very sharp cracks are present.
By combining analysis of the gross elastic changes in a structure or part that occur
as a sharp brittle crack grows with measurements of the energy required to produce new
fracture surfaces, it is possible to calculate the average stress (if no crack were present)
that will cause crack growth in a part. Such calculation is possible only for parts with
cracks for which the elastic analysis has been completed, and for materials that crack in a
relatively brittle manner and for which the fracture energy has been carefully measured.
10
The term relatively brittle is rigorously defined in the test procedures, but it means,
roughly, fracture without yielding occurring throughout the fractured cross section.
Thus glass, hard steels, strong aluminum alloys, and even low-carbon steel below
the ductile-to-brittle transition temperature can be analyzed in this way. Fortunately,
ductile materials blunt sharp cracks, as we have previously discovered, so that fracture
occurs at average stresses of the order of the yield strength, and the designer is prepared
for this condition. The middle ground of materials that lie between “relatively brittle”
and “ductile” is now being actively analyzed, but exact design criteria for these materi-
als are not yet available.

Quasi-Static Fracture
Many of us have had the experience of observing brittle fracture, whether it is the break-
ing of a cast-iron specimen in a tensile test or the twist fracture of a piece of blackboard
chalk. It happens so rapidly that we think of it as instantaneous, that is, the cross section
simply parting. Fewer of us have skated on a frozen pond in the spring, with no one near
us, heard a cracking noise, and stopped to observe. The noise is due to cracking. The
cracks move slowly enough for us to see them run. The phenomenon is not instantaneous,
since some time is necessary to feed the crack energy from the stress field to the crack for
propagation. Quantifying these things is important to understanding the phenomenon “in
the small.” In the large, a static crack may be stable and will not propagate. Some level of
loading can render the crack unstable, and the crack propagates to fracture.
The foundation of fracture mechanics was first established by Griffith in 1921
using the stress field calculations for an elliptical flaw in a plate developed by Inglis in
1913. For the infinite plate loaded by an applied uniaxial stress σ in Fig. 5–22, the max-
imum stress occurs at (±a, 0) and is given by

a

(σ y ) max = 1 + 2 σ (5–33)
b

10
BS 5447:1977 and ASTM E399-78.