reversible and partly irreversible (for visco-elastic behaviour) or fully irreversible

(for viscous behaviour). In the latter case all externally applied mechanical energy

to the material is dissipated and converted into other forms of energy (in general

a large part is converted into heat).

11.5 Lagrangian and Eulerian description of the balance equations

In summary, the balance equations of mass and momentum as derived in the

sections 11.2 and 11.3 can be written as

ρ = ’ρtr( D) = ’ρ ∇ · v,

™

balance of mass

™

balance of momentum ∇ · σ + ρq = ρa = ρ v,

191 11.5 Lagrangian and Eulerian balance equations

and in column/matrix notation:

ρ = ’ρtr( D) = ’ρ∇ T v,

™

balance of mass ∼∼

T

∇ Tσ + ρq = ρa = ρ ∼.

™

balance of momentum v

∼ ∼

∼

In a typical Lagrangian description, the ¬eld variables are considered to be a func-

tion of the material coordinates x0 , being de¬ned in the reference con¬guration,

and time t. It can be stated that the balance laws have to be satis¬ed for all x0

within the domain V0 and for all points in time.

Because of the physical relevance (for solids) of the deformation tensor F the

balance of mass will usually not be used in the differential form as given above,

but rather as

ρ0 ρ0

ρ= and also ρ = . (11.21)

det( F) det( F)

The gradient operator ∇ (and also ∇ ) in the balance of momentum equation, built

∼

up from derivatives with respect to the spatial coordinates, can be transformed

into the gradient operator ∇ 0 (and ∇ 0 ) with respect to the material coordinates,

∼

see Section 9.6. The balance of momentum can then be formulated according to

™

F’T · ∇ 0 · σ + ρq = ρa = ρ v, (11.22)

and also

T

T

’T

∇0 σ + ρq = ρa = ρ ∼.

™

F v (11.23)

∼ ∼

∼

In a typical Eulerian description, the ¬eld properties are considered to be a func-

tion of the spatial coordinates x, indicating locations in the current con¬guration,

and time t. It can be stated that the balance laws have to be satis¬ed for all x within

the domain V and for all points in time.

To reformulate the balance laws the material time derivative is ˜replaced™ by the

spatial time derivative, see Section 9.4. For the mass balance this yields

δρ

+ v · ∇ρ = ’ρtr( D) = ’ρ ∇ · v, (11.24)

δt

and so

δρ

+ ∇ · ( ρv) = 0. (11.25)

δt

In column/matrix notation:

δρ

+ ∼T ∇ ρ = ’ρtr( D) = ’ρ∇ T v,

v∼ (11.26)

δt ∼ ∼

Local balance of mass, momentum and energy

192

and so

δρ

+ ∇ T ( ρv) = 0. (11.27)

δt ∼ ∼

For the balance of momentum this yields

δv

T

∇ · σ + ρq = ρ ∇v ·v+ , (11.28)

δt

and in column/matrix notation:

δv

T T

∇ Tσ + ρq = ρ ∇ vT v+ ∼

. (11.29)

δt

∼ ∼∼ ∼

∼

For the special case of a stationary ¬‚ow of a material, the following balance

equation results for the mass balance:

∇ · ( ρv) = 0, (11.30)

and also

∇ T ( ρv) = 0. (11.31)

∼ ∼

The momentum equation reduces to:

T

∇ · σ + ρ q = ρ ∇v · v, (11.32)

and also

T T

∇ Tσ + ρ q = ρ ∇ vT v. (11.33)

∼ ∼∼ ∼

∼

Exercises

11.1 Compressible air is ¬‚owing through the bronchi. A bronchus is modelled

as a straight cylindrical tube. We consider a stationary ¬‚ow. For each cross

section of the tube the velocity of the air V and the density ρ is constant

over the cross section. In the direction of the ¬‚ow, the velocity of the air and

the density vary, because of temperature differences. Two cross sections A

and B at a distance L are considered. See the ¬gure below.

193 Exercises

B

A

ρB

ρA VB

VA

L

Which relation can be derived for the variables indicated in the ¬gure?

11.2 For a solid element the deformation matrix F is given as a function of time

t, with reference to the undeformed con¬guration at time t = 0:

¤

⎡

1 + ±t 0 ±t

⎥

⎢

¦ with ± = 0.01 [s’1 ].

F=⎣ 0 1 0

±t 0 1 + ±t

In the reference con¬guration the density ρ0 equals 1500 [kg m’3 ]. As a

result of the deformation process the density ρ of the material will change.

Determine the density as a function of the time.

12 Constitutive modelling of solids

and fluids

12.1 Introduction

In the ¬rst section of this chapter the (biological) material under consideration

can be regarded as a solid. This implies that it is possible to de¬ne a reference

con¬guration and local deformations can be related to this reference con¬guration

(see Chapter 10). It is assumed that the deformations (or more precisely, the path

along which the actual deformations are reached: the deformation history) fully

determine the current stress state (see Chapter 8), except for the special case when

we deal with incompressible material behaviour. A number of non-mechanical

phenomena are not included, such as the in¬‚uence of temperature variations.

The constitutive equations discussed in the present chapter address a relation

that can be formally written as

σ ( t) = F{F( „ ) ; „ ¤ t}, (12.1)

with σ ( t) the current Cauchy stress tensor at time t and with F( „ ) the deformation

tensor at the relevant (previous) times „ up to the time t, assuming compressible

material behaviour. A speci¬cation of this relation (the material behaviour) can

only be obtained by means of experimental studies. In the current chapter we

restrict our attention to elastic behaviour. In that case the deformation history is

not relevant and we can formally write

σ = σ ( F) . (12.2)

Further elaboration of this relation will initially be done for the case of small

deformations and rotations, so under the condition F ≈ I. After that, the conse-

quences of including large deformations (and large rotations) will be discussed.

In that case it is important to account for the fact that large additional rigid body

motions are not allowed to induce extra stresses in the material.

In the second part of this chapter we focus on ¬‚uid behaviour. Because in a

¬‚uid the reference con¬guration is usually not de¬ned, an Eulerian approach will

be chosen and the velocity gradient tensor L and the deformation rate tensor D

play a central role.

195 12.2 Elastic behaviour at small deformations

12.2 Elastic behaviour at small deformations and rotations

To describe the current deformation state of the material a reference con¬guration

has to be speci¬ed. Although, in principle, the choice for the reference con¬gu-

ration is free, in this case a ¬xed, stress free state is chosen. This is not as trivial

as it may seem, because for many biological materials a zero-stress state does not

exist in vivo, however, this discussion is not within the scope of the current book.

Interested readers are referred to e.g. [7]. So

σ ( F = I) = 0. (12.3)

It should be noted that this statement only applies to purely elastic material

behaviour. If the deformation history is important for the current stress state, the

relation given above is generally certainly not valid.

The current deformation tensor F fully describes the local deformations with

respect to the reference con¬guration. This also holds for the linear strain tensor µ

that was introduced in Section 10.4 under the strict condition that F ≈ I. Because

rotations can be neglected under these conditions, the Cauchy stress tensor σ can

be coupled directly to the linear strain tensor µ via an expression according to the

format:

σ = σ ( µ) with σ ( µ = 0) = 0. (12.4)

The exact speci¬cation of Eq. (12.4) has to be derived from experimental work

and is a major issue in biomechanics. In the current section we will explore the

commonly used Hooke™s law, which is often adopted as the ¬rst approximation

to describe the material behaviour. Hooke™s law supplies a linear relation between

the components of the stress tensor σ , stored in the (3 — 3) matrix σ and the

components of the linear strain tensor µ, stored in the (3 — 3) matrix µ. In addition,

it is assumed that the material behaviour is isotropic (the behaviour is identical in

all directions).

Hooke™s law will be speci¬ed, using the decomposition of the stress tensor in a

hydrostatic and a deviatoric part, as described in Section 8.7:

σ = σ h + σ d = ’pI + σ d , (12.5)

with p the hydrostatic pressure, de¬ned as

tr( σ )

p=’ . (12.6)

3

The linear strain tensor µ can be split in a similar way:

µv

µ = µ + µ = I + µd ,

h d

(12.7)

3

Constitutive modelling of solids and fluids

196

with µv the (relative) volume change:

µv = tr(µ) . (12.8)

This requires some extra elucidation. In general, the relative volume change is

de¬ned as (see Sections 10.4 and 10.5)

dV ’ dV0 T

= J ’ 1 = det( F) ’1 = det I + ∇ 0 u

µv = ’ 1. (12.9)

dV0

However, because the components of ∇ 0 u can be neglected with respect to 1 it

can be written (after linearization):

T

µ v = 1 + tr ∇ 0u ’1

1 T

= tr ∇ 0u + ∇ 0u = tr(µ) . (12.10)

2