l2,E2,A2

l1,E1,A1

(a) Determine the internal force in a cross section between A and B and

in a cross section between B and C.

Determine the stress σ in a cross section between A and B and a cross

(b)

section between B and C.

(c) What happens with the calculated forces and stresses if the Young™s

moduli of both muscle and tendon are reduced to half of their original

value?

(d) Determine the displacements at point B and C as a result of the

applied load F at point C.

7 Biological materials and

continuum mechanics

7.1 Introduction

Up to this point all treated problems were in a certain way one-dimensional.

Indeed, in Chapter 3 we have discussed equilibrium of two- and three-dimensional

bodies and in Chapters 4 the ¬bres were allowed to have some arbitrary orien-

tation in three-dimensional space. But, when deformations were involved, the

focus was on ¬bres and bars, dealing with one-dimensional force/strain rela-

tionships. Only one-dimensional equations have been solved. In the following

chapters, the theory will be extended to the description of three-dimensional

bodies and it is opportune to spend some time looking at the concept of a

continuum.

Consider a certain amount of solid and/or ¬‚uid material in a three-dimensional

space. Although in reality for neighbouring points in space the (physical) charac-

ter and behaviour of the residing material may be completely different (because of

discontinuities at the microscopic level, becoming clearer by reducing the scale of

observation) it is common practice that a less detailed description (at a macro-

scopic level) with a more gradual change of physical properties is used. The

discontinuous heterogeneous reality is homogenized and modelled as a contin-

uum. To make this clearer, consider the bone in Fig. 7.1. Although one might

conceive the bone at a macroscopic level, as depicted in Fig. 7.1(a), as a massive

structure ¬lling all the volume that it occupies in space, it is clear from Fig. 7.1(b)

that at a smaller scale the bone is a discrete structure with open spaces in between

(although the spaces can be ¬lled with a softer material or a liquid). At an even

smaller scale, at the level of one single trabecula individual cells can be recog-

nized. This phenomenon is typical for biological materials at any scale, whether

it is at the organ level, the tissue level or cellular level or even smaller. On each

level a very complex structure can be observed and it is not feasible to take into

account every detail of this structure. That is why a homogenization of the struc-

ture is necessary and depending on the objectives of the modelling (which is the

question that has to be answered in the context of the selection of the theoreti-

cal model) a certain level of homogenization is chosen. It is always necessary to

115 7.2 Orientation in space

Trabecular bone

Marrow cavity

b

Cortical bone

c

a

Figure 7.1

Structure of bone at different length scales (Images courtesy of Mr Bert van Rietbergen).

critically evaluate whether or not the chosen homogenization is allowed and to be

aware of the limitations of the model at hand. In Section 7.3 we will discuss when

a continuum approach is allowed.

To model a continuum, physical variables are formulated as continuous func-

tions of the position in space. Related to this, some attention will be given in

this chapter to the visualization of physical ¬elds. In addition, derivatives of vari-

ables will be discussed. The gradient operator, important in all kinds of theoretical

derivations, constitutes a central part of this. The chapter ends with a section on the

properties of second-order tensors, which form indispensable tools in continuum

mechanics.

7.2 Orientation in space

For an orientation in three-dimensional space a Cartesian xyz-coordinate system

is de¬ned with origin O, see Fig. 7.2. The orientation of the coordinate system is

laid down by means of the Cartesian vector basis {ex , ey , ez }, containing mutually

Biological materials and continuum mechanics

116

z

P

ez x = xex + yey + zez

y

O

ex

ey

x

Figure 7.2

Coordinate system, position of a point P in space.

perpendicular unit vectors. The position of an arbitrary point P in space can be

de¬ned with the position vector x, starting at the origin O and with the end point

in P. This position vector can be speci¬ed by means of components with respect

to the basis, de¬ned earlier:

⎡¤⎡ ¤

x · ex

x

⎢⎥⎢ ⎥

x = x ex + y ey + z ez with x = ⎣ y ¦ = ⎣ x · ey ¦ . (7.1)

∼

x · ez

z

The column x is in fact a representation of the vector x with respect to the chosen

∼

basis vectors ex , ey and ez only. Nevertheless, the position vector at hand will

sometimes also be indicated with x. This will not jeopardize uniqueness, because

∼

in the present context only one single set of basis vectors will be used.

The geometry of a curve (a set of points joined together) in three-dimensional

space can be de¬ned by means of a parameter description x = x( ξ ), in compo-

nents x = x( ξ ), where ξ is speci¬ed within a certain interval, see Fig. 7.3. The

∼ ∼

tangent vector, with unit length, at an arbitrary point of this curve is depicted by

e, satisfying

1 dx dx dx

e= = ·

with (7.2)

dξ dξ dξ

and after a transition to component format:

T

1 dx dx dx

e= =

∼ ∼ ∼

with . (7.3)

∼

dξ dξ dξ

The parameter ξ is distance measuring if equals 1; in that special case the length

of an (in¬nitesimally small) line piece dx (with components dx) of the curve is

∼

exactly the same as the accompanying change dξ .

117 7.3 Mass within the volume V

z

e

x = x(ξ)

ξ

y

O

x

Figure 7.3

A parameterized curve in space.

z

”z

x

y

O

”x

”y

x

Figure 7.4

V=

A subvolume x y z of V in space.

A bound volume in three-dimensional space is denoted by V. That volume can

be regarded as a continuous set of geometric points, of which the position vectors x

have their end points within V. A subvolume is indicated by V. That subvolume

may have the shape of a (rectangular) block, with edges in the same direction

as the coordinate axes, V = x y z (see Fig. 7.4). An in¬nitesimally small

volume element in the shape of a rectangular block is written as dV = dxdydz.

7.3 Mass within the volume V

Assume that the volume V is ¬lled with a material with a certain mass. Sometimes

different fractions of materials can be distinguished, which usually physically

interact with each other. In the following we consider one such fraction, that is

not necessarily homogeneous when observed in detail (at a ˜microscopic™ scale).

Observation at a microscopic scale, in the current context, means a scale that is

Biological materials and continuum mechanics

118

»

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. It is

» is satis¬ed, meaning that on a macroscopic

assumed that the inequality L

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 ρ in a spatial point

with position vector x is de¬ned by

dm

ρ= with: dV = dxdydz ’ 0, (7.4)

dV

where dm is the mass of the considered fraction (the cells) in the volume dV.

Ignoring some of the mathematical subtleties, a discontinuous ¬eld ρ ( x) results

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 de¬ne the (locally averaged) density ρ in the spatial point x by

m

V= m= ρ dV.

ρ= with: x y z and: (7.5)

V V

119 7.3 Mass within the volume V

In this case it is assumed that the dimensions of V are much larger than » and