Biomedical Engineering Reference
In-Depth Information
9
Motion: the time as an
extra dimension
9.1
Introduction
Let us consider the geometrical change in time (the deformation and movement
in three-dimensional space) of a coherent amount of material or material fraction,
for which a continuum modelling approach is allowed. In case more fractions
are involved, it is in principle possible to describe the behaviour of each fraction
separately, as if it was isolated from the other fractions (however it wil be neces-
sary to include interactions between fractions). The present chapter is focussed on
a detailed description of motion. In addition, the consequences of configuration
changes for the formulation of physical fields will be discussed. There will be no
attention to the possible causes of the motion. In the present chapter, an approach
will be followed that is common practice in the continuum description of solids
(although it can also be applied to fluids). The specific aspects relevant for fluids
will be treated at the end of the chapter.
9.2
Geometrical description of the material configuration
Consider a coherent amount of material in a completely defined geometrical state
(the reference configuration). From each material point P, that can be allocated,
the position vector
x
0
(with components stored in the column
∼
0
) is known. In
the following, this position vector will be used to identify the material point. The
vector
x
0
is inextricably bound to the material point P, as if it were an attached
label. With respect to a Cartesian
xyz
-coordinate system,
x
0
can be written as
⎡
⎣
⎤
⎦
x
0
y
0
z
0
x
0
=
x
0
e
x
+
y
0
e
y
+
z
0
e
z
and also
∼
0
=
.
(9.1)
Because
x
0
is uniquely coupled to a material point, the components
∼
0
of
x
0
are called material coordinates. The set of position vectors
x
0
that address all
the material points in the configuration comprise the reference volume
V
0
, see
