Biomedical Engineering Reference
In-Depth Information
This relation enables, to formally calculate the displacement (in the current con-
figuration at time
t
with respect to the reference configuration) of a material point,
defined in the reference configuration with material identification
x
0
. For the use
of Eq. (
9.25
), it is assumed that
x
(
x
0
,
t
) is available.
In the Eulerian description
u
is considered to be a function of
x
in
V
(
t
) and
t
and thus
u
=
u
(
x
,
t
)
=
x
−
x
0
(
x
,
t
) .
(9.26)
With Eq. (
9.26
), it is possible to formally calculate the displacement (in the cur-
rent configuration with respect to the reference configuration) of a material point,
which is actually (at time
t
) at position
x
in the three-dimensional space. Neces-
sary for this is, that
x
0
(
x
,
t
) is known, expressing which material point
x
0
at time
t
is present at the spatial point
x
, in other words, the inverse relation of
x
(
x
0
,
t
).
9.6
The gradient operator
In Chapter
7
the gradient operator with respect to the current configuration was
treated extensively. In fact, the current field of a physical variable (for example the
temperature
T
) was considered in the current configuration with domain
V
(
t
) and
as such defined according to an Eulerian description. The gradient of such a vari-
able is built up from the partial derivatives with respect to the spatial coordinates,
for example:
⎡
⎣
⎤
⎦
∂
T
∂
x
∇
T
=
e
x
∂
T
∂
x
+
e
y
∂
T
∂
y
+
e
z
∂
T
∂
∂
T
∂
y
and also
∼
T
=
.
(9.27)
z
∂
T
∂
z
The current field (with respect to time
t
) can also be mapped onto the reference
configuration with volume
V
0
and thus formulated by means of a Lagrangian
description. In this formulation the gradient can also be defined and is built up
from partial derivatives with respect to the material coordinates:
⎡
⎣
⎤
⎦
∂
T
∂
x
0
∇
0
T
=
e
x
∂
x
0
+
e
y
∂
T
y
0
+
e
z
∂
T
T
∂
T
∂
y
0
∼
0
T
=
and also
.
(9.28)
∂
∂
∂
z
0
∂
T
∂
z
0
