Biomedical Engineering Reference
In-Depth Information
and
tr
3
tr(
B
)
I
D
1
tr(
B
d
B
·
D
)
=
−
·
1
3
J
−
2
/
3
tr(
B
)tr(
D
)
=
J
−
2
/
3
tr(
B
·
D
)
−
2
J
−
2
/
3
tr
F
·
F
T
·
F
T
1
·
F
·
F
−
1
+
F
·
F
T
·
F
−
T
=
1
3
J
−
5
/
3
J
tr(
B
)
−
2
J
−
2
/
3
tr
F
·
F
T
+
F
·
F
T
1
1
2
(
˙
=
+
J
−
2
/
3
)tr(
B
)
2
J
−
2
/
3
tr
B
+
1
1
2
(
˙
=
J
−
2
/
3
)tr(
B
).
(12.45)
With Eq. (
12.45
) the integral expression for
int
0
(
t
) can be elaborated further:
2
GJ
−
2
/
3
tr(
B
)
1
2
K
(
J
−
t
1
1)
2
int
0
=
+
.
(12.46)
τ
=
0
Using
J
=
1 and
B
=
I
for
τ
=
0 results in the current energy density
int
0
, which
only depends on the current left Cauchy Green tensor
B
(
t
), so it can be noted:
2
G
J
−
2
/
3
tr(
B
)
−
3
,
1
2
K
(
J
−
1)
2
1
int
0
(
B
)
=
+
(12.47)
with
1
/
2
.
J
=
(
det(
B
)
)
(12.48)
Again, it can be established, that in
int
0
the volumetric and the deviatoric part are
clearly distinguishable and that both parts deliver an always positive contribution
to the energy density, for every arbitrary deformation process. Finally, it can again
be observed that a cyclic process in the deformation or in the stress will always be
energetically neutral.
More general expressions for (non-linearly) elastic behaviour can formally be
written as
d
d
(
J
,
B
) .
p
=
p
(
J
),
σ
=
σ
(12.49)
For a detailed specification many possibilities exist and have been published in the
scientific literature. An extensive treatment for biological materials is beyond the
scope of the present discussion. For this the reader is referred to more advanced
textbooks on Biomechanics.
