Page 343 - Mechanical Behavior of Materials

344 Chapter 8 Fracture of Cracked Members

8.2.5 Internally Flawed Materials

As already discussed in earlier chapters, many brittle materials naturally contain small cracks or
cracklike flaws. This is generally true for glass, natural stone, ceramics, and some cast metals. The
interpretation can be made that these materials have a high yield strength, but this strength can never
be reached under tensile loading because of earlier failure due to small flaws and a low fracture
toughness. Such a viewpoint is supported by the fact that brittle materials have considerably higher
strengths under compression than under tension, because the flaws simply close under compression
and thus have a much reduced effect.
Denoting the inherent flaw size in such a material as a i , Eq. 8.3 gives the ultimate strength in
tension:
K c
σ ut = √ (8.5)
πa i
This situation is illustrated in Fig. 8.6(b). New cracks of size around or below a i have little effect,
as they are no worse than the flaws already present. The material is thus said to be internally
flawed. Also, since the flaws actually present may vary considerably from sample to sample, there
is generally a large statistical scatter in σ ut .

8.3 MATHEMATICAL CONCEPTS

A cracked body can be loaded in any one or a combination of the three displacement modes shown
in Fig. 8.8. Mode I is called the opening mode and consists of the crack faces simply moving apart.
For Mode II, the sliding mode, the crack faces slide relative to one another in a direction normal to
the leading edge of the crack. Mode III, the tearing mode, also involves relative sliding of the crack
faces, but now the direction is parallel to the leading edge. Mode I is caused by tension loading,
whereas the other two are caused by shear loading in different directions, as shown. Most cracking
problems of engineering interest involve primarily Mode I and are due to tension stresses, so we
will limit most of our discussion to this case.
Energy methods were employed in the earliest work on fracture mechanics, reported by
A. A. Griffith in 1920. This approach is expressed by a concept called the strain energy release
rate, G. Later work led to the concept of a stress intensity factor, K, and to the proof that G and K
are directly related.

y y y
Mode I Mode II Mode III
x x
x

z z z

Figure 8.8 The basic modes of crack surface displacement. (Adapted from [Tada 85];
used with permission.)