the stretch is written as

» = 1 + µ. (5.14)

Fibres: time-dependent behaviour

74

For suf¬ciently small strain levels, i.e. |µ| 1, and using the notation µ = dµ/dt

™

it can be written:

1 d» 1

D= = µ ≈ ( 1 ’ µ) µ ≈ µ .

™ ™™ (5.15)

» dt 1+µ

Consequently, if |µ| 1 then

1d 1d

D= ≈µ=

™ , (5.16)

dt 0 dt

such that Eq. (5.1) reduces to

F = c· µ .

™ (5.17)

Remark In the literature, the symbol µ is frequently used to denote the elonga-

™

tional rate for large ¬lament stretches instead of D.

5.3 Linear visco-elastic behaviour

5.3.1 Continuous and discrete time models

Biological tissues usually demonstrate a combined viscous-elastic behaviour as

described in the introduction. In the present section we assume geometrically

and physically linear behaviour of the material. This means that the theory leads

to linear relations, expressing the force in the deformation(-rate), and that the

constitutive description satis¬es two conditions:

• superposition The response on combined loading histories can be described as the

summation of the responses on the individual loading histories.

• proportionality When the strain is multiplied by some factor the force is multiplied

by the same factor (in fact proportionality is a consequence of superposition).

To study the effect of these conditions a unit-step function for the force is

introduced, de¬ned as H( t) (Heaviside function):

if t < 0

0

H( t) = . (5.18)

if t ≥ 0

1

Assume that a unit-step in the force F( t) = H( t) is applied to a linear visco-elastic

material. The response µ( t) might have an evolution as given in Fig. 5.7. This

response denoted by J( t) is called the creep compliance or creep function.

75 5.3 Linear visco-elastic behaviour

µ(t )

µ (t ) = J (t )

µ0

t =0 t

Figure 5.7

Typical example of the strain response after a unit-step in the force.

µ

F µ1(t )

F1

µ0(t )

F0

0 t1

0 t1 t t

Figure 5.8

Superposition of responses for a linear visco-elastic material.

Proportionality means, that increasing the force with some factor F0 leads to a

proportional increase in the strain:

F( t) = H( t) F0 ’ µ( t) = J( t) F0 . (5.19)

Superposition implies, that applying a load step F0 at t = 0, with response:

µ0 ( t) = J( t) F0 , (5.20)

followed by a load step F1 at t = t1 with individual response:

µ1 ( t) = J( t ’ t1 ) F1 , (5.21)

leads to a total response, which is a summation of the two:

µ( t) = µ0 ( t) + µ1 ( t) = J( t) F0 + J( t ’ t1 ) F1 . (5.22)

This is graphically shown in Fig. 5.8.

Both principles can be used to derive a more general constitutive equation for

linear visco-elastic materials. Assume, we have an arbitrary excitation as sketched

in Fig. 5.9. This excitation can be considered to be built up by an in¬nite number

of small steps in the force.

Fibres: time-dependent behaviour

76

F

”F ≈ ( dF ) ”ξ

dξ

ξ ξ + ”ξ

t

Figure 5.9

An arbitrary force history in a creep test.

F of the force F between time steps t = ξ and t = ξ + ξ is equal

The increase

to

dF

™

F≈ ξ = F( ξ ) ξ . (5.23)

dξ

The response at time t as a result of this step at time ξ is given by

™

µ( t) = F( ξ ) ξ J( t ’ ξ ) . (5.24)

The time-dependent force F( t) as visualized in Fig. 5.9 can be considered as a

composition of sequential small steps. By using the superposition principle we

are allowed to add the responses on all these steps in the force (for each ξ ).

This will lead to the following integral expression, with all intervals ξ taken

as in¬nitesimally small:

t

™

J( t ’ ξ ) F( ξ ) dξ .

µ( t) = (5.25)

ξ =’∞

This integral was derived ¬rst by Boltzmann in 1876.

In the creep experiment the load is prescribed and the resulting strain is mea-

sured. Often, the experimental set-up is designed to prescribe the strain and to

measure the associated, required force. If the strain is applied as a step, this is

called a relaxation experiment, because after a certain initial increase the force

will gradually decrease in time. The same strategy as used to derive Eq. (5.25) can

be pursued for an imposed strain history, leading to

t

G( t ’ ξ ) µ ( ξ ) dξ ,

™

F( t) = (5.26)

ξ =’∞

with G( t) the relaxation function.

77 5.3 Linear visco-elastic behaviour

The functions J and G have some important physical properties:

• For ordinary materials J increases, G decreases in time (for t > 0):

™

™

J ( t) > 0 G( t) < 0 (5.27)

• The absolute value of the slope of both functions decreases:

d2 J/dt2 < 0 d2 G/dt2 > 0 (5.28)

• In the limiting case for t ’ ∞ the time derivative of G( t) will approach zero:

™

lim G( t) = 0 (5.29)

t’∞

• In the limiting case for t ’ ∞ the time derivative of J( t) will be greater than or equal

zero:

™

lim J ( t) ≥ 0 (5.30)

t’∞

It will be clear that there must be a relation between G( t) and J( t), because

both functions describe the behaviour of the same material. This relationship

can be determined by using Laplace transformation (for a summary of de¬ni-

tions and properties of Laplace transformations see Appendix 5.5). The Laplace

ˆ

transformation x( s) of the time function x( t) is de¬ned as

∞

x( t) e’st dt.

x( s) =

ˆ (5.31)

0

Assuming, that the creep and relaxation functions are zero for t < 0, the Laplace

transforms of Eqs. (5.25) and (5.26) are

ˆ ˆ

µ( s) = J ( s) s F( s)

ˆ (5.32)

ˆ

ˆ

F( s) = G( s) s µ( s) .

ˆ (5.33)

From these equations it is easy to derive that

1

ˆ ˆ

G( s) J ( s) = 2 . (5.34)

s

Back transformation leads to

t

J( t ’ ξ ) G( ξ ) dξ = t. (5.35)

ξ =0

When G( t) or J( t) is known, the material behaviour, at least for one-dimensional

tests, is speci¬ed. There are several reasons why a full explicit description of J

and G is very dif¬cult. It is not possible to enforce in¬nitely fast steps in the

load, so it it not possible to realize a perfect step. Consequently, it is almost not

feasible to determine G( t) and J( t) for very small values of t. At the other side

Fibres: time-dependent behaviour

78

of the time domain the problem is encountered that it is not possible to carry out

measurements for an unlimited (in¬nite) period of time.

Both functions J( t) and G( t) are continuous with respect to time. Often these

functions are approximated by discrete spectra. Examples of such spectra are

N

fk [ 1 ’ e’t/„k ] + t/·,

J( t) = J0 + (5.36)

k=1

with J0 , fk , „k ( k = 1, ..., N) and · material parameters (constants) or

M

gj e’t/„j ,

G( t) = G∞ + (5.37)

j=1

with G∞ , gj , „j ( j = 1, ..., M) material constants.

These discrete descriptions can be derived from spring-dashpot models, which

will be the subject of the next section.

5.3.2 Visco-elastic models based on springs and dashpots: Maxwell model

An alternative way of describing linear visco-elastic materials is by assembling

a model using the elastic and viscous components as discussed before. Two

examples are given, while only small stretches are considered. In that case the

constitutive models for the elastic spring and viscous dashpot are given by

F = c µ, F = c· µ .