Biomedical Engineering Reference
In-Depth Information
Chapter 2
Basic Continuum Kinematics
The theme of this chapter was stated with exuberance and in an idealistic deterministic
extreme by Marquis Pierre-Simon de Laplace (1759-1827): “Thus, we must con-
sider the present state of the universe as the effect of its previous state and as the
cause of those states to follow. An intelligent being which, for a given point in time,
knows all the forces acting upon the universe and the positions of the objects of
which it is composed, supplied with facilities large enough to submit these data to
numerical analysis, would include in the same formula the movements of the largest
bodies of the universe and those of the lightest atom. Nothing would be uncertain for
it, and the past and future would be known to it.”
1
2.1 The Deformable Material Model, the Continuum
In the deformable material model all types of motion are permitted, but the
deformational motions are usually the major concern. Consider the image
O
of an
object in Euclidean space. The object is in a configuration
O
(0) at
t
0 and in a
configuration
O
(
t
) at time
t
(Fig.
2.1
). The mathematical representation of the
motion of a three-dimensional deformable continuum gives a complete history of
the motion of each point
P
on the object
O
(0),
P
¼
O
(0); in words,
P
O
(0)
means all points
P
contained in (
0. In order to
identify each point
P
in the object
O
(0) and to follow the movement of that point in
subsequent configurations of the object
O
(
t
), each point on an object is given a
reference location in a particular coordinate system, called the reference coordinate
system. The selection of the reference configuration is the choice of the modeler;
here the reference configuration is taken as the configuration of the object at time
t
) the image of the object,
O
,at
t
¼
¼
0. To distinguish between the reference location of a point on an object and a
location of the same point at a later time, the terminology of “particle” and “place”
1
Translated by John H. Van Drie (
http://www.johnvandrie.com
).
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