Page 272 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 252 Monday, May 23, 2005 5:47 PM

252 Fracture Mechanics: Fundamentals and Applications

For plane strain conditions in an edge-cracked test specimen, the volume sampled at a given stress
value is simply BA, where B is the specimen thickness. Therefore, the incremental volume at a
fixed J and σ is given by
o
BJ 2 h ∂
σ
dV() = dσ (A5.6)
1 σ 2 o ∂ σ 1 1

Inserting Equation (A5.1) and Equation (A5.6) into Equation (5.21) gives
 BJ 2 σ max h ∂ 
( )
1
F =− exp  − ∫ ρσ dσ  (A5.7)
 σ 2 o σ u 1 ∂σ 1 1 
here σ max is the peak value of stress that occurs ahead of the crack tip and σ is the threshold
u
fracture stress, which corresponds to the largest fracture-triggering particle the material is likely
to contain.
Note that J appears outside of the integral in Equation (A5.7). By setting J = J in Equation (A5.7),
c
we obtain an expression for the statistical distribution of critical J values, which can be written in
the following form:
 B  J  2 
F =− exp −   c   (A5.8)
1
  B  Θ J   
o
where B is a reference thickness. When B = B , Θ is the 63rd percentile J value. Equation (A5.8)
c
J
o
o
has the form of a two-parameter Weibull distribution, as discussed in Section 5.2.3. Invoking the
relationship between K and J for small-scale yielding gives
 B K  4 

F =− exp −   IC   (A5.9)
1
  B  Θ K   
o
Equation (A5.8) and Equation (A5.9) both predict a thickness effect on toughness. The average
toughness is proportional to 1 B for critical J values and B −0.25 for K data. The average toughness
Ic
does not increase indefinitely with thickness, however. There are limits to the validity of the weakest
link model, as discussed in the next section.
All of the above relationships are valid only when weakest link failure occurs under J-controlled
conditions, i.e., the single-parameter assumption must apply. When the constraint relaxes, critical
J values no longer follow a Weibull distribution with a specific slope, but the effective small-scale
yielding J values, J (see Section 3.6.3), follow Equation (A5.8) if a weakest link mechanism
o
controls failure. Actual J values would be more scattered than J values, however, because the
o
c
ratio J/J increases with J.
o
A5.2 INCORPORATING A CONDITIONAL PROBABILITY
OF PROPAGATION

In many materials, the weakest link initiation of cleavage appears to be necessary but not sufficient.
Figure 5.26 schematically illustrates a probability tree for cleavage initiation and propagation. This
diagram is a slight oversimplification, because the cumulative failure probability must be computed
incrementally.
Modifying the statistical cleavage model to account for propagation requires that the probability
be expressed in terms of a hazard function [52], which defines the instantaneous risk of fracture.