77 / 352
73
Dutta et al. /20/ proposed another approach - dependence of
q
2
on the distance of the
point from the crack tip,
r
:
(
)
2
/
2
2
1
c
b
r l q
a
q
q e
−
= +
(11)
This dependence is determined from the condition that the parameter set of the GTN
model should be used for determining the crack growth initiation, but also its stable
growth in steel SA333 Gr.6. Here,
q
2
a
and
q
2
b
are two new constants, defining the
variation of
q
2
in the vicinity of the crack tip.
Parameter
f
F
can be determined from microstructural observation of the fracture
surfaces, but this is a very difficult procedure. Zhang et al. /21/ has shown, using the unit
cell model, that
f
F
depends on
f
0
, and can be approximately calculated as:
0
0.15 2
F
f
f
≈ +
(12)
Determination of the critical void volume fraction
f
c
is very important for application
of Eq. (9). Different techniques are proposed in /22 – 24/ for determining its value. Rakin
/25/ and Rakin et al. /15, 26/ have shown an experimental-numerical procedure based on
the examination of the round tensile specimen.
In the initial stage of ductile fracture of steel, the voids nucleate mostly around the
non-metallic inclusions. Therefore, the initial porosity is often assumed to be equal to the
void volume fraction of non-metallic inclusions
f
v
, which can be determined by
quantitative microstructural analysis, briefly described in the next chapter.
The increase of the void volume fraction can be separated into the growth of the
existing voids:
(1 )
p
growth
ii
f
f
ε
= −
(13)
and nucleation of the new (often called secondary) voids under external loading:
p
nucleation N eq
f
A
ε
=
(14)
where
p
ii
ε
is the plastic part of the strain rate tensor.
growth nucleation
f
f
f
=
+
(15)
The increase of the external loading causes further growth of voids and their eventual
coalescence, leading to final fracture.
Chu and Needleman /27/ proposed the following expression for nucleation of
secondary voids, using normal distribution:
2
1 exp
2
2
p
eq N
N
N
N
N
f
A
s
s
ε
ε
π
⎛
⎞
⎛
⎞ −
⎜
⎟
⎜
⎟
=
−⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
(16)
where
f
N
denotes the volume fraction of secondary-void forming particles,
ε
N
is the mean
strain at void nucleation (the strain for which 50% particles are broken or separated from
the matrix) and
s
N
is the standard deviation of this value. Changing the standard deviation
causes the change of the interval width corresponding to the nucleation of most voids.
When applying this approach to quantify the void nucleation, one should have in mind
that it is a simplified procedure, and that variety of sizes and types of particles makes this
mechanism very difficult to be modelled exactly. Therefore, instead of a unique model to