Biomedical Engineering Reference
In-Depth Information
;
¼
g
1
þ
X
i
1
X
N
G
N
G
G
ðÞ
G
0
t
s
G
t
s
G
i
g
i
e
g
i
1
e
G
0
¼
G
ðÞ ð
3
:
330
Þ
i
¼
1
i
¼
1
and interchanging the time arguments in the integrand of (
3.327
) (note that t
0
!
t
t
0
and dt
0
!
dt
0
and d
=
dt
0
!
d
=
dt
0
and the interchange of the integration
limits!), the following material equation form is obtained
s
ðÞ¼
s
H
ðÞþ
s
D
ðÞ
s
H
ðÞ
s
0
with
ðÞ
ð
3
:
331
Þ
Z
t
_
Gt
ðÞ
G
0
ð
4
Þ
s
D
ðÞ
s
0
ðÞþ
p
F
1
t
t
t
ð Þ
s
0
t
t
ð Þ
F
T
t
t
ð Þ
dt
0
t
t
0
¼
0
In (
3.330
) and (
3.331
), G(t) is the time-dependent small-strain shear modulus,
G
0
is the instantaneous shear modulus, g
?
is the long-term shear modulus, the g
i
are relative moduli, the s
i
are relaxation times and the N
G
are model parameters.
Equations (
3.331
) are identical to the form provided in Abaqus FE-code
(Abaqus 2010) except for the volumetric part s
H
ðÞ
s
0
ðÞ
. In contrast to the
theory outlined previously, where the dissipative parts of the material equation
arise exclusively from the deviation terms, in A
BAQUS
, in addition, the volumetric
part s
H
ðÞ
is extended by a dissipative contribution. The contribution is in the form
of a hereditary integral with a kernel identical to (
3.330
) given by (the transfor-
mation of the second identity in (
3.332
) is analogue to (
3.330
))
;
¼
k
1
þ
X
i
1
X
N
K
N
j
K
ðÞ
K
0
t
s
K
t
s
K
i
1
e
K
0
¼
K
ðÞ: ð
3
:
332
Þ
k
i
e
k
i
i
¼
1
i
¼
1
Using (
3.324
) and (
3.325
) and considering (
3.262
), the following conversion
holds for the tensorial structure under the integral in (
3.331
)
3
ð
4
Þ
F
1
t
t
t
ð Þ
s
0
t
t
ð Þ
F
T
t
t
ð Þ
p
t
F
T
ð
3
:
333
Þ
ð
4
Þ
ðÞ
F
1
t
t
ð Þ
s
0
t
t
ð Þ
F
T
t
t
ð Þ
F
ðÞ
P
ðÞ
such that, finally, using (
3.330
)-(
3.333
), the following alternative forms to the
constitutive equation provided in A
BAQUS
yields:
