Biomedical Engineering Reference
In-Depth Information
and using this:
λ
=
λ
(
e
0
)
=
e
0
·
C
·
e
0
.
(10.15)
The tensor
C
is called the
right Cauchy Green deformation tensor
. In compo-
nent form Eq. (
10.15
) can be written as
∼
T
λ
=
λ
(
∼
0
)
=
0
C
∼
0
,
(10.16)
with
C
=
F
T
F
.
(10.17)
The direction change (rotation) of a material line segment can, for the transition
from the reference state to the current state, formally be stated as
e
=
F
·
e
0
d
0
d
=
F
·
e
0
1
F
·
e
0
λ
=
√
e
0
.
(10.18)
e
0
·
C
·
In component form this equation can be written as
∼
=
F
∼
0
d
0
d
=
F
∼
0
1
F
∼
0
λ
=
∼
.
(10.19)
0
C
∼
0
Above, the current state is considered as a 'function' of the reference state: for
a direction
e
0
in the reference configuration, the associated direction
e
and the
stretch ratio
λ
were determined. In the following the 'inverse' procedure is shown.
Based on
F
−
1
·
dx
=
dx
0
,
(10.20)
so
F
−
1
·
ed
=
e
0
d
0
,
(10.21)
and subsequently
F
−
T
F
−
1
2
0
,
e
·
·
·
ed
=
e
0
·
e
0
d
(10.22)
it follows for the stretch ratio
λ
=
d
/
d
0
that
1
λ
=
λ
(
e
)
=
√
.
(10.23)
e
·
F
−
T
·
F
−
1
·
e
This equation can be used to determine the stretch ratio
λ
for a material line ele-
ment with direction
e
in the current configuration (Eulerian description). For this,
the tensor (tensor product)
F
−
T
·
F
−
1
has to be known. The tensor
B
is defined
according to
F
T
,
B
=
F
·
(10.24)
