Biomedical Engineering Reference
In-Depth Information
10
Deformation and rotation,
deformation rate and spin
10.1
Introduction
Consider the geometrical change of a coherent amount of material or material
fraction, for which modelling as a continuum is assumed to be permitted. The
first part of the present chapter is focussed on the description of the local defor-
mation (generally coupled with rotations of the material) and along with that, the
introduction of a number of different strain measures.
Only after choosing a reference configuration is it possible to define deforma-
tion in a meaningful way (deformation is a relative concept). This implies that
initially the theory and the accompanying application area are related to solids.
When the material is a mixture of several material fractions, each fraction can,
with regard to local geometrical changes, in principle, be isolated from the other
fractions.
The second part of this chapter discusses geometrical changes in time. Cen-
tral concepts in this part are deformation rate and rotation velocity (spin). The
derivations in this part are not only relevant for solids, but even more important
for applications including fluids.
10.2
A material line segment in the reference and current configuration
Consider a coherent amount of material in a fully defined state (the reference
configuration). In a material point P, with position vector
x
0
in the reference con-
figuration, we focus our attention on an arbitrary infinitesimally small material
line segment
dx
0
, see Fig.
10.1
.
With respect to the Cartesian
xyz
-coordinate system the vector
dx
0
can be
written as
⎡
⎣
⎤
⎦
dx
0
dy
0
dz
0
d
x
0
=
dx
0
e
x
+
dy
0
e
y
+
dz
0
e
z
and also
d
∼
0
=
.
(10.1)
