Biomedical Engineering Reference
In-Depth Information
3.2 Gradient-Enhanced Part of the Free Energy
According to Eq. (
2
), we assume the non-local part of the free energy to be additively
composed of a gradient-related term
ʨ
grd
and of a penalty term
ʨ
plty
and specify
these terms as
c
d
2
||∇
x
ˆ
||
ʨ
plty
(ˆ, ʺ)
=
ʲ
d
2
2
ʨ
grd
(
∇
X
ˆ
;
F
)
=
and
2
[
ˆ
−
ʺ
]
.
(20)
The energy-related penalty parameter
ʲ
d
approximately enforces the local damage
field
to coincide. Furthermore, the gradient parameter
c
d
controls the quasi-non-local character of the formulation and characterises the degree
of gradient regularisation:
c
d
ʺ
and the non-local field
ˆ
0 leads to
the regularised gradient-enhanced model. The damage-related parameters included
in constitutive Eq. (
20
) together with their units are summarised in Table
1
.
=
0 results in a local model, while
c
d
>
3.3 Gradient-Enhanced Damage Model
In order to obtain the stress-like thermodynamic forces driving the local dissipative
damage process, we follow the standard Coleman-Noll procedure. Differentiation
of the general format of the free energy (
3
) with respect to time and application of
the Clausius-Planck inequality yields, amongst others, a contribution including the
thermodynamic force
q
=
q
loc
+
q
nloc
=−
∂
d
ʨ
conjugate to the damage variable
d
,
i.e.
q
loc
=
ʨ
ani
and
q
nloc
=
ʲ
d
[
ˆ
−
ʺ
]
∂
d
ʺ.
(21)
Furthermore, we adopt the damage condition
ʦ
d
=
−
ʺ
≤
q
0
(22)
where
0 to damage evolution.
Based on the postulate of maximum dissipation, we construct a constrained opti-
misation problem involving the Lagrange multiplier
ʦ
d
<
0 refers to the purely elastic loading and
ʦ
d
=
ʻ
. This results in the following
associated evolution equation for the damage variable
ʺ
=
ʻ
∂ʦ
d
∂
q
=
ʻ
with
ʺ
|
t
=
0
=
ʺ
d
.
(23)
where initiation and termination of damage are governed by the Karush-Kuhn-Tucker
complementary conditions
ʻ
≥
,ʦ
d
≤
,ʻʦ
d
=
.
0
0
0
(24)
