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at the same time much smaller than L. Provided, that it is possible to specify
such a subvolume V this results in a continuous density ¬eld ρ( x), from which
the microscopic deviations have disappeared, however, which is still containing
macroscopic variations: the real density ¬eld, according to Eq. (7.4), is homoge-
nized. Continuum theory deals with physical properties that, in a way as described
above, are made continuous. The results cannot be used on a microscopic level
with a length scale » , but on a much more global level.
In the above reasoning we talk about two scales: a macroscopic and a micro-
scopic scale. However, in many applications a number of intermediate steps from
large to small can be of interest. This is often seen in technical applications, but is
especially of importance in biological materials. In such a case the characteristic
measure » of the components could suddenly become the relevant macroscopic
length scale!

Example 7.1 Figure 7.6 illustrates some of the scales that can be distinguished in biomechanical
modelling. Figure 7.6(a) could be a model to study a high-jumper. Typically, this
kind of model would be used to study the coordination of muscles, describing the
human body as a whole and examining how it moves. The macroscopic length




(a) (b)




(c) (d)

Figure 7.6
Examples of models at different length scales (a) model of a leg (b) model of a skeletal muscle
(c) representative volume element of a cross section of a muscle (d) model of the cytoskeleton
of a single cell.
Biological materials and continuum mechanics
120

scale L would be in the order of 1 [m]. In this case the user is often interested in
stresses that are found in the bones and joints. The length scale » at which these
stresses and strains are studied is in the order of centimetres. Muscles are often
described with one-dimensional ¬bre-like models as discussed in Chapter 4.
Figure 7.6(b) shows an MRI-image of the tibialis anterior of a rat and a
schematic of the theoretical Finite Element Model that it was based on to describe
its mechanical behaviour ([15]). This was a model designed to study in detail the
force transmission through the tibialis anterior. The muscle was modelled as a
continuum and the ¬bre lay-out was included, as well as a detailed description of
the active and passive mechanical behaviour of the muscle cells. For this model,
L is of the order of a few centimetres and » of the order of 100 [μm].
Figure 7.6(c) shows a microscopic cross section of a skeletal muscle together
with a so-called representative volume element, that is used for a microstructural
model of a skeletal muscle. This model was used to study the transport of oxygen
from the capillaries to the cells and how the oxygen is distributed over cell and
intercellular space. Here each substructure: cells, intercellular spaces were mod-
elled as a continuum. In this case L ≈ 50 [μm] and » ≈ 0.1 [μm]. One could
even go another step further in modelling the single cell as shown in Fig. 7.6(d).
In this case the actin skeleton that is shown could be modelled as a ¬bre-like struc-
ture and the cytoplasm of the cell as a continuum, ¬lled with a liquid or a solid
(L ≈ 10 [μm], » ≈ 10 [nm]).



7.4 Scalar fields

Let us consider the variation of a scalar physical property of the material (or a
fraction of it) in the volume V. An example could be the temperature ¬eld: T =
T( x). Visualization of such a ¬eld in a three-dimensional con¬guration could be
done by drawing (contour) planes with constant temperature. A clearer picture
is obtained, when lines of constant temperature (isotherms or contour lines) are
drawn on a ¬‚at two-dimensional surface of a cross section (see Fig. 7.7). By means
of this ¬gure estimates can be made of the partial derivatives of the temperature
with respect to to x (with y and z constant) and y (with x and z constant). For
example at the lower boundary the partial derivatives could be approximated by
‚T
‚T
C m’1 , C m’1 .
=2 =0
o o
(7.6)
‚x ‚y
In a three-dimensional scalar ¬eld the partial derivatives with respect to the coor-
dinates are assembled in a column, i.e. the column with the components of the
gradient vector associated with the scalar ¬eld. The gradient operator is speci¬ed
121 7.4 Scalar fields

y



18
20
22
3m
24
26
28
30


6m
x

Figure 7.7
Isotherms (—¦ C) in a cross section with constant z.



by ∇ when vector notation is used and with ∇ when components are used. In the

example of the temperature ¬eld this yields
⎡ ‚T ¤ ⎡‚¤
‚x ‚x
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
‚T ‚
∇T = ⎢ ⎥ with ∇ = ⎢ ⎥, (7.7)
‚y ‚y
⎢ ⎢
⎥ ⎥
∼ ∼
⎣ ⎣
¦ ¦
‚T ‚
‚z ‚z
and also
‚T ‚T ‚T ‚ ‚ ‚
+ ey + ez with ∇ = ex + ey + ez .
∇T = ex (7.8)
‚x ‚y ‚z ‚x ‚y ‚z
The gradient of a certain property is often a measure for the intensity of a physi-
cal transport phenomenon; the gradient of the temperature for example is directly
related to the heat ¬‚ux. Having the gradient of a certain property (T), the deriva-
tive of that property along a spatial curve (given in a parameter description, see
Fig. 7.3), x = x( ξ ), can be determined (by using the chain rule for differentiation):
‚T dx ‚T dy ‚T dz
dT
= + +
‚x dξ ‚y dξ ‚z dξ

dx dx
= ( ∇ T)T ∼ = ( ∇ T T) ∼
∼ ∼
dξ dξ
T
dx
= ∇ T.

(7.9)


On the right-hand side of the equation the inner product of the (unnormalized)
tangent vector to the curve and the gradient vector can be recognized. Eq. (7.9)
can also be written as
Biological materials and continuum mechanics
122

dT dx
= · ∇T. (7.10)
dξ dξ
Along a contour line (isotherm) in a two-dimensional cross section dT/dξ = 0,
so the gradient vector ∇T (with components ∇ T) must be perpendicular to the

contour line in each arbitrary point. In a three-dimensional con¬guration it can be
said that the gradient vector of a certain scalar property in a point of V will be
perpendicular to the contour plane through that point. The directional derivative
dT/de of the temperature (the increase of the temperature per unit length along the
curve) in the direction of the unit vector e, with components ∼ can be written as
e
dT dT
= eT ∇ T and also = e · ∇T. (7.11)
∼∼
de de


7.5 Vector fields

In the previous section, as an example, a scalar temperature ¬eld was considered.
Departing from a given temperature ¬eld, the associated (temperature) gradient
¬eld can be determined. This gradient ¬eld can be regarded as a vector ¬eld (in
every point of the volume V the components of the associated vector can be deter-
mined). Considering the special meaning of such a gradient ¬eld, we will not use
that ¬eld as a typical example in the current section. Instead, the (momentary)
velocity ¬eld v = v( x) of the material in the volume V will be used as an illustra-
tion. With respect to the Cartesian xyz-coordinate system the velocity vector v and
the associated column ∼ can be written as
v
⎡ ¤
vx
⎢ ⎥
v = vx ex + vy ey + vz ez , v = ⎣ vy ¦ . (7.12)

vz
The possibilities for a clear graphical illustration of a vector ¬eld (velocity) in a
three-dimensional con¬guration are limited. For a problem with a ¬‚at ¬‚uid ¬‚ow
pattern, for example:
vx = vx ( x, y)
vy = vy ( x, y)
vz = 0, (7.13)
a representation like the one given in Fig. 7.8 can be used. The arrows represent
the velocity vector (magnitude and orientation) of the ¬‚uid at the tail of the arrow.
At a given velocity v in a certain point of V, the velocity component v in
the direction of an arbitrary unit vector e (components in the column e) can be

calculated with
123 7.5 Vector fields

y




1 m/s



x

Figure 7.8
Velocity ¬eld in a cross section with constant z.





n




Figure 7.9
Flux of material through a small surface area.




v = v( e) = v · e and also v = v( e) = vT e. (7.14)
∼ ∼∼



In Fig. 7.9 a small plane is drawn at a certain point in V. The plane has an outward
(unit) normal n. The local velocity of the material is speci¬ed with v. Thus, the
amount of mass per unit of surface and time ¬‚owing out of the plane in the ¬gure
is given by

ρ v · n = ρ vT n. (7.15)
∼∼



This is called the mass ¬‚ux.
Also for vector ¬elds the associated gradient can be determined. The gradient of
a vector ¬eld results in a tensor ¬eld. For example, for the gradient of the velocity
¬eld we ¬nd
Biological materials and continuum mechanics
124

‚ ‚ ‚
∇v = ex + ey + ez ( vx e x + v y e y + v z e z )
‚x ‚y ‚z
‚vy
‚vx ‚vz
= ex ex + ex ey + ex ez
‚x ‚x ‚x
‚vy
‚vx ‚vz
+ ey ex + ey ey + ey e z
‚y ‚y ‚y
‚vy
‚vx ‚vz
+ ez ex + ez e y + ez ez . (7.16)

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