Page 265 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 245 Monday, May 23, 2005 5:47 PM

Fracture Mechanisms in Metals 245

Once cleavage initiates, the crack may either propagate in an unstable fashion or arrest, as in Figure 5.23.
Initiation is governed by the local stress at the critical particle, while propagation is controlled by the
orientation of the neighboring grains and the global driving force. The overall probability of failure is
equal to the probability of initiation times the conditional probability of propagation.
This model assumes that if a microcrack arrests, it does not contribute to subsequent failure. This
is a reasonable assumption, since only a rapidly propagating crack is sufficiently sharp to give the stress
intensification necessary to break bonds. Once a microcrack arrests, it is blunted by local plastic flow.
Consider the case where the conditional probability of propagation is a step function:

K 0, K <
P =  I o
pr
K 1, I  K ≥ o
That is, assume that all cracks arrest when K < K and that a crack propagates if K ≥ K at the
o
I
I
o
time of initiation. This assumption implies that the material has a crack-arrest toughness that is
single valued. It can be shown (see Appendix 5.2) that such a material exhibits the following fracture
toughness distribution on K values:

   K  4  K  
4


F =− exp  −  IC  −  o   for K > K o (5.29a)
1
I
    Θ K   Θ K   


F = 0 for K ≤ K o (5.29b)
I
Equation (5.24) is a truncated Weibull distribution; Θ can no longer be interpreted as the 63rd
K
percentile K value. Note that a threshold has been introduced, which removes one of the
Ic
shortcomings of the weakest link model. Equation (5.24) also exhibits less scatter than the two-
parameter distribution (Equation 5.22a), thereby removing the other objection to the weakest
link model.
The threshold is obvious in Equation (5.29), but the reduction in relative scatter is less so. The
latter effect can be understood by considering the limiting cases of Equation (5.24). If K /Θ >> 1,
K
o
there are ample initiation sites for cleavage, but the microcracks cannot propagate unless K > K .
I
o
Once K exceeds K , the next microcrack to initiate will cause total failure. Since initiation events
o
I
are frequent in this case, K values will be clustered near K , and the scatter will be minimal. On
Ic
o
the other hand, if K /Θ << 1, Equation (5.29) reduces to the weakest link case. Thus the relative
o
K
scatter decreases as K /Θ increases.
o
K
Equation (5.24) is an oversimplification, because it assumes a single-valued crack-arrest tough-
ness. In reality, there is undoubtedly some degree of randomness associated with microscopic crack
arrest. When a cleavage crack initiates in a single ferrite grain, the probability of propagation into
the surrounding grains depends in part on their relative orientation; a high degree of mismatch
increases the likelihood of arrest at the grain boundary. Anderson et al. [43] performed a probabilistic
simulation of microcrack propagation and arrest in a polycrystalline solid. Initiation in a single
grain ahead of the crack tip was assumed, and the tilt and twist angles at the surrounding grains
were allowed to vary randomly (within the geometric constraints imposed by assuming {100}
cleavage planes). An energy-based propagation criterion, suggested by the work of Gell and Smith
[44], was applied. The conditional probability of propagation was estimated over a range of applied
K values. The results fit an offset power law expression:
I
P pr K = I − K α( o ) β (5.30)