Biomedical Engineering Reference
In-Depth Information
and initial conditions
d
j
(
0. As discussed below, the rate of damage accumu-
lation will be different for different modes. In particular, modes one and two will
defined a functions of the strain energy the material has experienced while the rate
of accumulation in mode three will depend on the history of exposure to hemody-
namic wall shear stress. An example of the functional dependence
d
j
(
α
j
)
0
)=
is,
⎧
⎨
0
,
α
j
<
α
js
D
j
[
α
j
(
t
)]
,
α
js
≤
α
j
<
α
jf
(
α
)=
d
j
and
j
⎩
1
,
α
jf
≤
α
j
e
c
j
(
1
−
α
j
/
α
jf
)
1
−
D
j
[
α
j
(
t
)] =
1
−
e
c
j
(
1
−
α
js
/
α
jf
)
.
(6.52)
1
−
Here,
j
is the damage mode,
α
js
is the critical value of
α
j
for the start damage mode
j
,and
α
jf
is the critical value for complete failure.
Discontinuous damage mode
When a rubber specimen is loaded uniaxially in tension, unloaded and then reloaded,
the applied stress necessary to achieve a given level of strain decreases in the fol-
lowing loading cycles. This phenomena is termed
stress softening
. It is also often
referred to as the
Mullins effect
, so named due to the early studies by Mullins on
stress softening in rubber materials with imbedded particles [82, 83].
Several microstructual explanations have been given to explain this phenomena.
It seems likely that multiple mechanisms are involved, (e.g. [39]). For example, it
has been conjectured that one mechanism for stress softening is the breakage of
bonds between the filler particles and surrounding matrix during previous loading
cycles. Damage of this kind has been modelled by setting the current value of the
accumulation variable
α
1
, equal to the maximum effective strain energy the material
has experienced [107],
W
o
α
1
(
t
)=
max
s
(
s
)
.
(6.53)
∈
[
0
,
t
]
The corresponding evolution equation for
α
1
is then
W
o
if
W
o
W
o
=
α
1
and
>
0
˙
α
1
=
(6.54)
0
otherwise
with the initial condition
0. Namely, damage only accumulates when the
effective strain energy increases beyond the previous maximum.
α
1
(
0
)=
Continuous damage
Damage has also been found to increase during cyclic loading with effective strain
energies below the maximum value obtained during the prior history of loading.
To address this phenomena, Miehe [78] introduced a contribution to damage evo-
lution arising as the arc length of the effective strain energy. Following this work,
