Page 263 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 243 Monday, May 23, 2005 5:47 PM

Fracture Mechanisms in Metals 243

it can be shown (Appendix 5) that the critical values of K and J follow a characteristic distribution
when failure is controlled by a weakest link mechanism: 3
  K  4 
1
F =− exp −   Ic   (5.27a)
   Θ K   
or
  J  2 
F =− exp −   c   (5.27b)
1
   Θ J   
where Θ and Θ are the material properties that depend on microstructure and temperature. Note that
K
J
Equation (5.22a) and Equation (5.22b) have the form of a two-parameter Weibull distribution. The
Weibull-shape parameter, which is sometimes called the Weibull slope, is equal to 4.0 for K data and
Ic
(because of the relationship between K and J) 2.0 for J values for cleavage. The Weibull-scale
c
parameters Θ and Θ are the 63rd percentile values of K and J , respectively. If Θ or Θ is known,
K
Ic
J
c
K
J
the entire fracture toughness distribution can be inferred from Equation (5.27a) or Equation (5.27b).
The prediction of a fracture toughness distribution that follows a two-parameter Weibull
function with a known slope is an important result. The Weibull slope is a measure of the relative
scatter; a prior knowledge of the Weibull slope enables the relative scatter to be predicted a priori,
as Example 5.1 illustrates.
EXAMPLE 5.1
Determine the relative size of the 90% confidence bounds of K Ic and J c data, assuming Equation (5.27a)
and Equation (5.27b) describe the respective distributions.

Solution: The median, 5% lower bound, and 95% upper bound values are obtained by setting F = 0.5,
0.05, and 0.95, respectively, in Equation (5.27a) and Equation (5.27b). Both equations have the form:
λ
F =− exp( − )
1
Solving for λ at each probability level gives

λ 0.50 = 0.693, λ 0.05 = 0.0513, λ 0.95 = 2.996
The width of the 90% confidence band in K Ic data, normalized by the median, is given by
K − K (. 025 − ( 0 0513) 025
.
.
2 996)
.
095 005 = = 0 920
.
.
.
K (. 025
.
0 693)
050
.
and the relative width of the J c scatter band is
J − J 2 996 − 0 0513
.
.
095 005 = = 181
.
.
.
J 0 693
.
050
.
Note that Θ K and Θ J cancel out of the above results and the relative scatter depends only on the Weibull
slope.
3 Equation (5.27a) and Equation (5.27b) apply only when the thickness (i.e., the crack front length) is fixed. The weakest
link model predicts a thickness effect, which is described in Appendix 5.2 but is omitted here for brevity.