Biomedical Engineering Reference
In-Depth Information
Fig. 23.5
Normalized
distribution of the exponential
cohesive law illustrated for
shearing related traction
forces and displacement
previous, then unloading takes place according to
τ
0
κ
0
[
t
s
=
u
]
s
.
(23.35)
Damage is defined as
−
|
t
s
|
τ
ult
.
D
=
1
(23.36)
Macro crack is developed when the local damage approaches maximum (
D
1)
and therefore when the exponential approaches zero. When locally the opening de-
creases compared to previous time step unloading takes place. The cohesive law
parameters
=
G
c
and
τ
ult
can be obtained from experimental data.
23.2.3.3 Yield Criterion
Crack growth is determined by damage in the solid matrix. Therefore, the yield
criterion, next to the cohesive zone, is related to the effective stress. The effective
stress at the crack tip varies locally, therefore the critical effective stress state is
calculated non-locally using Gaussian functions following Wells and Sluys (
2001
),
i.e.
r
i
2
l
a
,
(23.37)
n
tot
n
tot
(
2
π)
2
/
3
l
a
w
i
w
tot
e
−
σ
tip
=
σ
e1
,
2
,
tot
=
w
j
,
i
=
i
=
1
j
=
1
with
r
i
the distance between integration point
i
and the crack tip and
l
a
is a length
scale parameter which determines the influence of a sample point.
σ
e1
,
2
and associ-
ated angle
α
n
is evaluated either from Camacho and Ortiz (
1996
) (mode I):
σ
e
x
−
2
σ
e
x
−
σ
e
y
σ
e
y
σ
e
xy
,
σ
e1
,
2
=
±
+
(23.38)
2
2
