Page 273 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 253 Monday, May 23, 2005 5:47 PM

Fracture Mechanisms in Metals 253

For a random variable T, the hazard function H(T ), and the cumulative probability are related as
follows:
 T 
F  − ∫ H =− exp ( ) d T  (A5.10)
T 1
 T o 
where T is the minimum value of T. By comparing Equation (A5.17) and Equation (A5.18), it can
o
easily be shown that the hazard function for the weakest link initiation, in terms of stress intensity,
is given by

HK() = 4 K 3 (A5.11)
Θ 4 K

assuming B = B . The hazard function for total failure is equal to Equation (A5.11) times the
o
conditional probability of failure:

HK() = P pr 4 Θ K 4 K 3 (A5.12)

Thus, the overall probability of failure is given by

 K K 3 
F 1   − ∫ 0 P =− exp pr 4 Θ 4 K dK   (A5.13)

Consider the case where P is a constant, i.e., it does not depend on the applied K. Suppose,
pr
for example, that half of the carbides of a critical size have a favorable orientation with respect to
a cleavage plane in a ferrite grain. The failure probability becomes

  K  4 
F =− exp −  0 5 .    (A5.14)
1
   Θ K   

In this instance, the finite propagation probability merely shifts the 63rd percentile toughness to a
higher value:
Θ = . 025 Θ = 2 . 1 19 Θ
K K * K
The shape of the distribution is unchanged, and the fracture process still follows a weakest link
model. In this case, the weak link is defined as a particle that is greater than the critical size that
is also oriented favorably.
Deviations from the weakest link distribution occur when P depends on the applied K. If the
pr
conditional probability of propagation is a step function
 K 0, I K < o
P = 
pr
K 1, I  K ≥ o

the fracture-toughness distribution becomes a truncated Weibull (Equation (5.24)); failure can occur
only when K > K . The introduction of a threshold toughness also reduces the relative scatter, as
o
discussed in Section 5.2.3.