Page 285 - Handbook of Materials Failure Analysis

2 Nonlocal Rousselier’s Damage Model 281

Table 12.1 Summary of Different Models Used for Analysis of Ductile and
Cleavage Fracture
Applicability For Details,
(Ductile/ Refer
Type of Cleavage Section/
Model Fracture) Yield Criteria/Failure Probability References

Rousselier’s Ductile q 2,[7]
ϕ ¼
damage fracture 1 d
model p
+ Dσ k d exp R ε eq ¼ 0
ð 1 dÞσ k
Gurson- Ductile ! 2 [8,9]
q
Tvergaard- fracture ϕ ¼
R ε eq
Needleman’s
model !
p
2
1 q 3 f ¼ 0
+2q 1 f cosh 1:5q 2
R ε eq
Beremin’s Cleavage σ w m 3,[20]
model fracture P f ¼ 1 exp ,
σ u
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
X m V i
m i
σ w ¼ σ I
V 0
i¼1
1 ð
_ ! ! ! _ ! !
dx ¼ Ψ y ; x fy dΩ y (12.1)
!
Ψ x Ω

! ! !
where y is the position vector of the infinitesimally small volume dΩ and Ψ y ; x
is the Gaussian weight function given by
0 1
2
!
!
1 x y
! ! B C
Ψ y ; x ¼ exp A (12.2)
8π 3=2 3 4l 2
l
@
The length parameter l in Equation 12.2 determines the size of the volume, which
effectively contributes to the nonlocal quantity and is related to the scale of the
microstructure. The above integral nonlocal kernel holds the property that the local

!
continuum is retrieved if l ! 0. By expanding fy in Taylor’s series and substitut-
ing in Equation 12.1, one can obtain the damage diffusion equation as
2 _
_
_
d f C length r d ¼ 0 (12.3)
where C length is the characteristic length parameter of the material [17–19]. The yield
function ϕ of the Rousselier’s model [7] is modified by substituting the nonlocal
damage d in place of the local ductile void volume fraction f as [17] follows.
q p
ϕ ¼ + Dσ k d exp R ε eq ¼ 0 (12.4)
1 d ð 1 dÞσ k