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

dξ

dx dx

= ( ∇ T)T ∼ = ( ∇ T T) ∼

∼ ∼

dξ dξ

T

dx

= ∇ T.

∼

(7.9)

∼

dξ

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)