Biomedical Engineering Reference
In-Depth Information
λ
V
L
Figure 7.5
Cross section of a heterogeneous structure.
much smaller than the global dimensions of the considered volume. To support
this visually one might imagine a kind of base material containing very small sub-
structures, that can be visualized under a microscope (individual muscle cells in a
skeletal muscle for example, see Fig.
7.5
).
For the substructures (cells in the example)
may be regarded as a relevant
characteristic length scale for the size, as well as for the mutual stacking. The
characteristic macroscopic length scale for the volume
V
is depicted by
L
.Itis
assumed that the inequality
L
λ
is satisfied, meaning that on a macroscopic
scale the heterogeneity at microscopic level is no longer recognizable. Attention
is now focussed on physical properties, which are coupled to the material. As an
illustration let us take the (mass) density. The local density
λ
ρ
in a spatial point
with position vector
x
is defined by
dm
dV
ρ
=
with:
dV
=
dxdydz
→
0,
(7.4)
where
dm
is the mass of the considered fraction (the cells) in the volume
dV
.
Ignoring some of the mathematical subtleties, a discontinuous field
ρ
(
x
) results
λ
for the local density, with
as the normative measure for the mutual distance of
the discontinuities. On the microscopic level such a variation can be expected, but
on the macroscopic level it is often (not) observable, (often) not interesting and
not manageable.
Let us define the (locally averaged) density
ρ
in the spatial point
x
by
V
ρ
dV
.
ρ
=
m
with:
V
=
x
y
z
and:
m
=
(7.5)
V
