Biomedical Engineering Reference
In-Depth Information
Just like in the previous sections, the contribution originating from the volume
change to the stress tensor is considered separately from the distortion (shape
change). For this reason the deformation tensor
F
is decomposed according to:
F
=
J
1
/
3
F
with
J
det(
F
) . (12.36)
The tensor
F
is called the isochoric deformation tensor (because of the equality:
det(
F
)
=
1). The multiplication factor
J
1
/
3
represents the volume change. Depart-
ing from the isochoric deformation tensor
=
F
=
J
−
1
/
3
F
, the associated objective,
B
is defined according to:
isochoric left Cauchy Green tensor
·
F
T
B
=
F
J
−
2
/
3
F
F
T
J
−
2
/
3
B
,
=
·
=
(12.37)
and subsequently the isochoric Finger strain tensor
ε
F
according to:
B
−
I
J
−
2
/
3
B
−
I
.
1
2
1
2
ε
F
=
=
(12.38)
Analogous to the formulation in Section
12.2
linear relations for the hydrostatic
and deviatoric part of the stress tensor can be postulated:
h
σ
=−
p
I
=
K
(
J
−
1)
I
,
(12.39)
F
=
G
B
d
,
d
d
σ
=
2
G
ε
(12.40)
with
K
the compression modulus and
G
the shear modulus of the material.
Summation leads to:
σ
=
K
(
J
−
1)
I
+
G
B
d
.
(12.41)
This coupling between the stress and deformation state is often indicated as
'compressible Neo-Hookean' material behaviour, thus referring to the linearity.
It can be shown that Eq. (
12.41
) does not exactly satisfy the requirement that
in a cyclic process no energy is dissipated. It can be proven that a small (but not
trivial) modification, according to:
G
J
B
d
σ
=
K
(
J
−
1)
I
+
(12.42)
does satisfy this requirement.
Substituting Eq. (
12.42
) into the definition equation for
int
0
(
t
), Eq. (
12.29
)
yields
t
K
(
J
D
)
d
G
tr(
B
d
int
0
(
t
)
=
−
1)
J
tr(
D
)
+
·
τ
.
(12.43)
τ
=
0
Based on the relations given in Chapter
10
the following expressions can be
derived for the first and second term in the integrand:
J
,
J
tr(
D
)
=
(12.44)
