Biomedical Engineering Reference
In-Depth Information
z
reference configuration
V
0
V
(
t
)
P
P
x
0
x
=
x
(
x
0
,
t
)
current configuration, time
t
y
x
Figure 9.1
The position vector of a material point P.
Fig.
9.1
. In an arbitrary current state, at time
t
, the position vector of the point P is
specified by
x
and can be written as
⎡
⎣
⎤
⎦
x
y
z
x
=
x e
x
+
y e
y
+
z e
z
and also
∼
=
.
(9.2)
With the attention focussed on a certain material point, it can be stated that
x
=
x
(
x
0
,
t
) .
(9.3)
This functional relation expresses that the current position
x
of a material point is
determined by the material identification
x
0
in
V
0
of that point and by the current
time
t
. When
x
0
is constant and with
t
passing through a certain time interval,
x
x
0
,
t
) can be considered to be a parameter description (with parameter
t
)of
the trajectory of a material point (defined by
=
x
(
x
0
) through three-dimensional space:
the path of the particle.
Differentiation of the relation
x
=
x
(
x
0
,
t
) to the time
t
, with
x
0
taken constant
(partial differentiation), results in the velocity vector
v
of the material point under
consideration. It can be written:
=
˙
v
x
and also
∼
=˙
∼
,
(9.4)
with
the material time derivative: partial differentiation with respect to the time
t
with constant
(
˙
)
x
0
. For the acceleration vector
a
it follows:
=
˙
=
¨
a
v
x
and also
∼
=˙
∼
=¨
∼
.
(9.5)
