Page 251 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 231 Monday, May 23, 2005 5:47 PM

Fracture Mechanisms in Metals 231

FIGURE 5.12 The limit-load model for void insta-
bility. Failure is assumed to occur when the net sec-
tion stress between voids reaches a critical value.

and fracture occurs when

d
σ = σ (5.16b)
+
nc() db 1
where σ is the maximum remote principal stress. As with Equation (5.15), the Thomason model
1
is of limited practical value. The void interactions leading to ductile failure are far too complex to
be captured by a simple area reduction model. Moreover, the final stage in failure is very abrupt,
as discussed earlier. Once the void fraction reaches 10 to 20%, failure occurs with only a minimal
increase in the nominal strain.

5.1.3 DUCTILE CRACK GROWTH

Figure 5.13 schematically illustrates microvoid initiation, growth, and coalescence at the tip of a
preexisting crack. As the cracked structure is loaded, local strains and stresses at the crack tip
become sufficient to nucleate voids. These voids grow as the crack blunts, and they eventually link
with the main crack. As this process continues, the crack grows.
Figure 5.14 is a plot of stress and strain near the tip of a blunted crack [21]. The strain exhibits
a singularity near the crack tip, but the stress reaches a peak at approximately two times the crack-
1
tip-opening displacement (CTOD). In most materials, the triaxiality ahead of the crack tip provides
sufficient stress elevation for void nucleation; thus the growth and coalescence of microvoids are
usually the critical steps in ductile crack growth. Nucleation typically occurs when a particle is
∼2δ from the crack tip, while most of the void growth occurs much closer to the crack tip, relative
to CTOD. (Note that although a void remains approximately fixed in absolute space, its distance
from the crack tip, relative to CTOD, decreases as the crack blunts; the absolute distance from the
crack tip also decreases as the crack grows.)
Ductile crack growth is usually stable because it produces a rising resistance curve, at least
during the early stages of crack growth. Stable crack growth and R curves are discussed in detail
in Chapter 3 and Chapter 7.
When an edge crack in a plate grows by microvoid coalescence, the crack exhibits a tunneling
effect, where it grows faster at the center of the plate, due to the higher stress triaxiality.

1 Finite element analysis and slip line analysis of blunted crack tips predict a stress singularity very close to the crack tip
(∼0.1 CTOD), but it is not clear whether or not this actually occurs in real materials because the continuum assumptions
break down at such fine scales.