Biomedical Engineering Reference
In-Depth Information
and using this:
1
λ
=
λ
(
e
)
=
√
.
(10.25)
B
−
1
e
·
·
e
The tensor
B
is called the
left Cauchy Green deformation tensor
. In component
form Eq. (
10.25
) can be written as
1
λ
=
λ
(
∼
)
=
∼
,
(10.26)
T
¸
−
1
∼
with
B
=
F F
T
.
(10.27)
The direction change (rotation) of a material line segment with direction
e
in the
current configuration with respect to the reference configuration can formally be
calculated with
F
−
1
d
d
·
e
e
0
=
F
−
1
0
=
F
−
1
·
e
·
e
λ
=
√
,
(10.28)
e
·
B
−
1
·
e
and alternatively, using components:
F
−
1
∼
d
d
∼
0
=
F
−
1
∼
0
=
F
−
1
∼
λ
=
∼
.
(10.29)
T
B
−
1
∼
At the end of the present section we will investigate the influence of an extra
displacement as a rigid body of the current configuration for the tensors that were
introduced above, see Section
9.7
. Properties with respect to the extra rotated (via
the rotation tensor
P
) and translated (via the translation vector
λ
) virtual state are
denoted by the superscript
∗
. Because
F
∗
=
P
·
F
,
(10.30)
it can immediately be verified that
C
∗
=
F
∗
T
F
∗
=
F
T
P
T
F
T
P
−
1
·
·
·
P
·
F
=
·
·
P
·
F
F
T
F
T
=
·
I
·
F
=
·
F
=
C
,
(10.31)
and
B
∗
=
F
∗
·
F
∗
T
=
P
·
F
·
F
T
·
P
T
=
P
·
B
·
P
T
.
(10.32)
Based on Eq. (
10.31
) the right Cauchy Green tensor
C
is called
invariant
for extra
displacements of the current state as a rigid body. This invariance is completely
trivial if the Lagrange description
λ
=
λ
(
e
0
) for the stretch ratio is taken into
consideration.
