Biomedical Engineering Reference
In-Depth Information
ð
4
Þ
occurs in quadratic form in the modified right G
REEN
strain tensor G.In(
3.227
) A
is a fourth order (material) tensor (material tetrad) and, c
2
and D are material
parameters.
H
OLZAPFEL
-W
EIZSÄCKER
-Model.
According
to
Holzapfel
and
Weizsäcker
(1998), the strain energy function
Þ
c
2
e
Q
1
w
ðÞ¼
c
1
C
I
3
ð
ð
3
:
228
Þ
represents an extension of (
3.226
) in the form of an isotropic-anisotropic split and
is intended for modelling of arteries where the first term in (
3.228
) represents the
Neo-H
OOKE
term which is linear in the invariant C
I
and is used for description of
the isotropic material behaviour of elastin fibres. The second term represents the
anisotropic material behaviour of collagen fibres (cf. also (
3.226
) and (
3.227
)).
H
OLZAPFEL
-G
ASSER
-O
GDEN
-Model. Especially for modelling of blood vessel
wall material, the following strain energy function was proposed by Gasser et al.
(2006)
C
;
H
ð Þþ
f
ðÞ
w C
;
H
ð Þ¼
w
ð
3
:
229
Þ
with
w C
;
H
ð Þ¼
c
1
C
I
3
C
;
H
ð Þ
and
f
ðÞ
:
¼
D
1
2
J
2
1
ð
Þ
w
f
ð
Þ
ln J
ð
3
:
230
Þ
where the isotropic term c
1
ð
C
I
3
Þ
(also given in (
3.228
) is intended to describe a
collagen-free ground matrix and the following term given in (
3.230
)
X
N
C
;
H
ð Þ
:
¼
k
1
2k
2
e
k
2
E
i
1
w
f
ð
3
:
231
Þ
i
¼
1
with
E
i
:
¼
H
i
C
1
j C
I
3
Þ
C
IV
i
1
ð
Þþ
1
3j
ð
ð
Þ
ð
3
:
232
Þ
C
IV
i
:
¼
K
0i
C
trK
0i
C
;
H
i
:
¼
jI
þ
1
3j
ð
Þ
K
0i
;
K
0i
:
¼
a
0i
a
0i
is intended to define the anisotropic material behaviour of embedded collagen fibre
families (N B 3).
In addition, H
i
represents a generalized structure tensor describing the fibre
dispersion with the direction dyad (direction tensor) K
0i
composed of the stan-
dardized principal fibre directions a
0i
(in the ICFG). C
IVi
is a ''mixed'' invariant
composed of C and K
0i
and the D, k
i
(i = 1, 2) and j are material parameters where
the latter represents a measure for the fibre dispersion of one fibre family about the
principal direction. The fibres run parallel to the principal direction if j = 0
(perfect alignment) and are randomized about the principal direction if j = 1/3
(isotropic case). Hence, for the isotropic case (
3.229
) takes the following form
