Biomedical Engineering Reference
In-Depth Information
Fig. 3.24
a Rheological P
OYNTING
-T
HOMSON
model, b creep behavior and c relaxation behavior
B
c
1
þ
k
1
9
ð
4
Þ
k
9
B
I
ð Þ
2
B
I
3
s
¼
2 p
ð
Þ
e
ð
4
Þ
B
ð
3
:
299
Þ
2 c
1
þ
k
1
9
Þ
e
k
9
B
I
ð Þ
2
B
I
3
ð
where
ð
4
Þ
is given in (
3.261
). With regard to (
3.259
) the combination of (
3.296
) and
(
3.299
) leads to the complete constitutive equation for the K
IRCHHOFF
stress
p
I
þ
2 c
1
þ
k
1
9
s
¼
1
D
ð
4
Þ
k
9
B
I
ð Þ
2
B
J
2
1
B
I
3
ð
Þ
e
ð
3
:
300
Þ
3.2.6.5 Constitutive Stress-Strain Equations for Linear Viscoelasticity
at Finite Deformations
Representing linear viscoelasticity valid for finite strains follows an approach
given by Simo (1987) and is limited, insofar that the material equations imple-
mented in A
BAQUS
F
E
-code are used to model the viscoelastic material behaviour
of human soft tissues and polymeric soft foams. The implemented theory refers to
the rheological P
OYNTING
-T
HOMSON
-model (sometimes also referred to as standard
solid) which consists of a H
OOKE
body and a M
AXWELL
body in parallel, cf.
Fig.
3.24
.
In the small strain linear viscoelastic case, the following set of equations results
for the one-dimensional model (indices ''H'' a n d
'' M'' refer to ''H
OOKE
'' and
'' M
AXWELL
'' )
r
¼
r
H
þ
r
M
;
e
¼
e
H
¼
e
M
r
H
¼
E
1
e
H
r
M
þ
E
ð
3
:
301
Þ
g
r
M
¼
E e
:
Equation (
3.301
)
1
represents the equilibrium state and (
3.301
)
2
is the compat-
ibility condition. Equations (
3.301
)
3
and (
3.301
)
4
are the partial constitutive
equations for the H
OOKE
and the M
AXWELL
model, respectively. In (
3.301
)
1
, r
H
