Biomedical Engineering Reference
In-Depth Information
an infinitesimal volume element in referential and spatial setting. Accordingly, the
Jacobian
J
=
v/
=
(
)>
d
d
V
det
F
0 is the ratio of the deformed to the undeformed
volume. Finally, let d
A
=
N
d
A
and d
a
=
n
d
a
define the referential and spatial
area normals. Then, Nanson's formula
n
d
a
N
d
A
describes the trans-
formation of infinitesimal area elements between the reference and the spatial con-
figuration with the co-factor of
F
defined as cof
=
cof
(
F
)
·
J
F
−
t
. Fibre-reinforcement
of the material is incorporated by assuming two families of fibres to be embedded
in the continuum. Their orientation is characterised by referential unit vectors
a
0
i
,
i
(
F
)
=
=
1
,
2 with
||
a
0
i
|| =
1.
2.2 General Gradient-Enhanced Format of the Free Energy
We assume the local free energy
ʨ
loc
(
F
,
a
0
i
,ʺ)
=
ʨ
iso
(
F
)
+
f
d
(ʺ) ʨ
ani
(
F
,
a
0
i
)
(1)
to account for anisotropic non-linear elastic material response under the influence
of scalar damage. Basically, we additively compose the effective free energy of
the undamaged material of an isotropic contribution
ʨ
iso
representing the ground
substance and of an anisotropic contribution
ʨ
ani
associated with
i
fibre families.
We assume only the anisotropic part to be affected by the damage, whereas the
isotropic matrix material remains elastic. In Eq. (
1
),
ʺ
∈[
,
∞
)
is a scalar internal
damage variable, characterising a material stiffness loss of the fibres, while
f
d
(ʺ)
=
1
0
represents an appropriate damage function that is at least twice
differentiable and satisfies
f
d
(
−
d
∈
(
0
,
1
]
0. This ensures purely
elastic behaviour of the undamaged material and complete loss of the related material
stiffness for
0
)
=
1 and
f
d
(ʺ
ₒ∞
)
ₒ
. Conceptually following the approach byDimitrijevic andHackl
[
2
], we introduce a gradient-enhanced non-local free energy
ʺ
ₒ∞
ʨ
nloc
as
ʨ
nloc
(ˆ,
∇
X
ˆ,ʺ
;
F
)
=
ʨ
grd
(
∇
X
ˆ
;
F
)
+
ʨ
plty
(ˆ, ʺ) .
(2)
Here,
ʨ
grd
contains the referential gradient of the non-local damage field variable
ˆ
while
ʨ
plty
incorporates a penalisation term which links the non-local damage vari-
able
ˆ
to the local damage variable
ʺ
. Consequently, we obtain an enhanced free
energy as
ʨ
int
(
F
,ˆ,
∇
X
ˆ,
a
0
i
,ʺ)
=
ʨ
loc
(
F
,
a
0
i
,ʺ)
+
ʨ
nloc
(ˆ,
∇
X
ˆ,ʺ
;
F
).
(3)
Provided that the external load can be derived from a potential, we can specify the
local external energy function as
. In summary, the total potential energy
function is additively composed of the internal and external contribution so that its
ʨ
ext
(
˕
)
