Page 264 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 244 Monday, May 23, 2005 5:47 PM

244 Fracture Mechanics: Fundamentals and Applications

There are two major problems with the weakest link model that leads to Equation (5.27a) and
Equation (5.27b). First, these equations predict zero as the minimum toughness in the distribution.
Intuition suggests that such a prediction is incorrect, and more formal arguments can be made for
a nonzero threshold toughness. A crack cannot propagate in a material unless there is sufficient
energy available to break bonds and perform plastic work. If the material is a polycrystal, additional
work must be performed when the crack crosses randomly oriented grains. Thus, one can make an
estimate of threshold toughness in terms of energy release rate:

G ≈ 2γφ (5.28)
c(min) p

where φ is a grain misorientation factor. If the global driving force is less than G c(min) , the crack
cannot propagate. The threshold toughness can also be viewed as a crack arrest value: a crack
cannot propagate if K < K .
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A second problem with Equation (5.27a) and Equation (5.27b) is that they tend to overpredict
the experimental scatter. That is, scatter in experimental cleavage fracture toughness data is usually
less severe than predicted by the weakest link model.
According to the weakest link model, failure is controlled by the initiation of cleavage in the
ferrite as the result of the cracking of a critical particle, i.e., a particle that satisfies Equation (5.17)
or Equation (5.18). While weakest link initiation is necessary, it is apparently not sufficient for
total failure. A cleavage crack, once initiated, must have a sufficient driving force to propagate.
Recall Figure 5.22, which gives examples of unsuccessful cleavage events.
Both problems, threshold toughness and scatter, can be addressed by incorporating a conditional
probability of propagation into the statistical model [42, 43]. Figure 5.26 is a probability tree for
cleavage initiation and propagation. When a flawed structure is subject to an applied K, a microcrack
may or may not initiate, depending on the temperature as well as the location of the eligible cleavage
triggers. The initiation of cleavage cracks should be governed by a weakest link mechanism, because the
process involves searching for a large enough trigger to propagate a microcrack into the first ferrite grain.

FIGURE 5.26 Probability tree for cleavage initiation and propagation.