Biomedical Engineering Reference
In-Depth Information
where
D
is the symmetric part of the velocity gradient and the over dot signifies the
material derivative. It then follows from (6.41)
F
0
∂
C
0
F
0
:
D
W
o
∂
W
dW
o
=
−
+
2
(
1
−
d
)
(6.44)
and therefore, from (6.43),
C
0
F
0
:
D
W
o
∂
F
0
∂
W
o
d
+
σ
−
2
(
1
−
d
)
≥
0
(6.45)
so that,
2
F
0
∂
C
0
F
0
W
o
∂
o
o
σ
=(
1
−
d
)
σ
,
σ
=
(6.46)
with the additional requirement that,
W
o
d
D
in
=
≥
0
.
(6.47)
o
is the effective Cauchy stress tensor for the hypothetical undamaged
material. Defining
f
as the thermodynamic conjugate to
d
through
The quantity
σ
d
D
in
=
f
(6.48)
we see immediately from (6.41), (6.43), (6.47) and (6.48) that,
=
−
∂
W
∂
d
W
o
f
=
and
f
≥
0
.
(6.49)
d
The quantity
f
is termed the thermodynamic force and drives the damage evolution,
[78]. The evolution equation for
f
(or equivalently
W
o
) can then be calculated from
(6.49),
f
o
:
D
(6.50)
The quantity
f
is therefore the rate of work by the effective stresses per unit volume
of the body, (
stress power
).
=
σ
.
Damage modes
We now turn attention to the cause of damage and in particular to the definition
of a damage evolution equation. As in [74], we consider three modes of damage
with damage variables denoted as
d
j
with
j
3. Two of these modes are purely
mechanical following earlier work [78, 107]. As elaborated on below, the third mode
of damage arises due to enzymatic degradation of the wall, arising for example from
elastase activity [74]. We quantify damage accumulation through the variable
=
1
,
2
,
α
j
and assume
d
j
(
α
)
is a monotonically increasing and smooth function with
j
d
j
(
α
j
)
∈
[
0
,
1
]
for all
α
j
∈
[
α
js
,
α
jf
]
(6.51)
