Page 289 - Handbook of Materials Failure Analysis

4 Types of Fracture Mechanics Specimens Analyzed in Upper Shelf 285

dV
ð
P f ¼ 1 exp g σ i I (12.8)
V V ref
where V is the volume of the plastically deformed zone (an essential condition for
slip induced nucleation of cleavage micro-cracks) in the component, V ref is the ref-
3 i
erence volume which is usually taken as 0.001 mm . The function g(σ I ) expresses the
probability of failure of an infinitesimal volume i (having volume dV) according to
the expression
dV
dP f ¼ g σ i I (12.9)
V ref
i
where σ I is the maximum principal stress acting at a material point i in the plastically
deformed region having volume dV. According to Beremin’s model [20] (which uses
Weibull’s statistics for distribution of defects responsible for triggering cleavage
i
according to Griffith’s theory), the form of g(σ I ) is defined as
m
i
i I
σ
g σ ¼ (12.10)
I
σ u
where m and σ u are the Weibull’s shape and size parameters, respectively. Using
Equation 12.10 in Equation 12.8, one can obtain the probability of cleavage fracture
P f at any given loading as [20]
m

σ w
P f ¼ 1 exp (12.11)
σ u
where the loading parameter is defined as the Weibull stress σ w and is expressed
as [20]
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
X m V i
m i
σ w ¼ σ (12.12)
I
i¼1 V 0
In this chapter, Equation 12.11 is used to calculate the probability of cleavage frac-
ture of the CT specimen at different loading levels and at different temperatures,
where the Weibull stress will be calculated using the stress field ahead of the growing
crack using Equation 12.12. The nonlocal Rousselier’s damage model has been used
to calculate the crack-tip stress field. The effect of evolution of damage (ductile void
volume fraction) on the stress field is taken care of by this model and hence, the effect
of prior stable crack growth before initiation of cleavage fracture is modeled implic-
itly through this framework.

4 TYPES OF FRACTURE MECHANICS SPECIMENS
ANALYZED IN UPPER SHELF AND DBTT REGIME
The nonlocal formulation of the Rousselier’s damage model was used for simulation
of fracture resistance behavior of various types of specimens in the upper-shelf as
well as the DBTT regime. There are various advantages of the nonlocal formulation