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tn x( t) = ( ’1)n . (5.119)
dsn

• If a function x( t) has Laplace transform x( s), then the Laplace transform of x( t) /t,
ˆ
assuming that lim x( t) /t exists, is given by
t’0

ˆ
( x( t) /t) = x( a) da. (5.120)
s

The Fourier transform x— ( t) of time function x( t) is given by


x ( ω) = x( t) eiωt dt, (5.121)
’∞

with i = ’1. The Fourier transform has similar properties to the Laplace
transform. The most important properties of Fourier transforms are:
• Fourier transform is a linear operation.
• When x( t) is a continuous function, the Fourier transform of the time derivative x( t) of

x( t) is given by

x— ( t) = iω x— ( ω) .
™ (5.122)

• Convolution in the time domain is equivalent to a product in the Fourier domain.
Using two time functions x( t) and y( t) with Fourier transforms x— ( ω) and y— ( ω), the
following convolution integral I( t) could be de¬ned:

I( t) = x( „ ) y( t ’ „ ) d„ . (5.123)
’∞

In that case the Fourier transform of this integral can be written as

I — ( ω) = x— ( ω) y— ( ω) . (5.124)

• If a function x( t) has a Fourier transform x— ( ω), then the Fourier transform of the
function tn x( t), with n = 1, 2, 3, . . . can be written as

dn x— ( ω)

= ( i)n
tn x( t) . (5.125)
dωn

• If a function x( t) has Fourier transform x— ( ω), then the Fourier transform of x( t) /t is
given by


(x( t) /t) = iω x( a) da. (5.126)
ω

Finally, Table 5.1 gives some Laplace and Fourier transforms of often used func-
tions. In the table the step function H( t) (Heavyside function) as de¬ned in Eq.
(5.18) is used, as well as the delta function δ( t), de¬ned as
Fibres: time-dependent behaviour
94

Table 5.1 Time functions with their Fourier and Laplace
transforms.
Original function Fourier transform Laplace transform
δ( t) 1 1

H( t) 1/s

1/s2
t H( t)

e’bt H( t) 1/( iω + b) 1/(s+b)

eiω0 δ( ω ’ ω0 )

a/( a2 ’ ω2 ) a/( s2 + a2 )
sin( at) H( t)

iω/( a2 ’ ω2 ) s/( s2 + a2 )
cos( at) H( t)



if: t = 0
0
δ( t) = (5.127)
∞ if: t = 0
and
+∞
δ( t) dt = 1. (5.128)
’∞



Exercises

5.1 A visco-elastic material is described by means of a Maxwell model. The
model consists of a linear dashpot, with damping coef¬cient c· , in series
with a linear spring, with spring constant c (see ¬gure below).

F F



The numerical values for the material properties are:

c = 8 104 [ Nc]
c· = 0.8 104 [ Ncs]


(a) Derive the differential equation for this model.
(b) Give the response for a unit step in the strain. Make a drawing of the
response.
(c) The material is subjected to an harmonic strain excitation with ampli-
tude µ0 and an angular frequency ω. Give the complex modulus E— ( ω)
and the phase shift φ( ω) for this material.
95 Exercises

Give the amplitude of the force for the case µ0 = 0.01 and a frequency
(d)
f = 5 [Hz].
5.2 A material can be characterized with a standard linear model (see ¬gure
below). The material is loaded with a step in the force at time t = 0. At
time t = t1 the material is suddenly unloaded stepwise.

c2
F


µ
· c1




(a) Derive the strain response for this loading history.
(b) Make a drawing of the strain response as a function of time.
(c) Calculate dµ/dt for t = 0.
5.3 In a dynamic experiment a specimen is loaded with a strain as given in the
¬gure below. The strain is de¬ned by:


µ = 0.01 sin( ωt) ,

with ω = 0.1 [rad s’1 ].


0.01

0.005
Strain [“]




tan (δ)
0

“0.005

“0.01
0 10 20 30 40 50 60 70 80 90 100
time [s]
2

1
force [N]




0

“1

“2
0 10 20 30 40 50 60 70 80 90 100
time [s]
Fibres: time-dependent behaviour
96

The response is also given in the ¬gure. Let us assume that we are dealing
with a material that can be described by a Maxwell model (one spring and
dashpot in series).
(a) What is the phase shift if we double the angular frequency of the
load?
(b) What will be the amplitude of the force for that frequency?
(c) Now we do a relaxation experiment, with the same material. We apply
a step strain of µ = 0.02. What is the force at:
t = 0 [s], t = 10 [s] and t = 100 [s]?
5.4 A creep test was performed on some biological material. The applied step-
wise load was F = 20 [N]. Assume that we are dealing with a material that
can be described by a Kelvin“Voigt model. This means that the material
can be modelled as a linear spring and dashpot in parallel.

creep test
0.04

0.035

0.03

0.025
strain [’]




0.02

0.015

0.01

0.005

0
0 20 40 60 80 100
time [s]


(a) Suppose we unloaded the material after 30 [s]. Determine for this case
the strain µ after 60 [s].
(b) After that we dynamically load the material with a force F:

F = F0 sin( ωt) ,

with F0 = 10 [N].
The response is:

µ = µ0 sin( ωt + δ) .

Determine µ0 and δ for the following frequencies f = 0.01 [Hz],
0.1 [Hz], 1 [Hz] and 100 [Hz].
97 Exercises

5.5 For a material the following relaxation and creep function was found,

G( t) = 1 + e’t

1
J( t) = 1 ’ e’t/2 .
2

When we relate a force to strain the unit for G( t) is [N] and the unit for
J( t) is [N’1 ]
(a) Does this material law satisfy Eq. (5.34)?
Test 1: Assume that at t = 0 a step in the force is applied of 1 [N].
(b)
After 1.5 [s] the step is removed. What is the strain after 3 [s]?
Test 2: At t = 0 a step in the strain is applied equal to 0.01. This
(c)
strain is removed at t = 1.5 [s]. What is the value of the force
after 3 [s]?
5.6 A piece of tendon material is subjected to the following strain history. At
t = 0 [s] a step in the strain is applied of 0.01. After 4 [s] the strain is
increased by an extra step of 0.01. After 8 [s] the strain is reduced by 0.005.
The total strain evolution is given in the ¬gure below.

Applied strain as a function of time
0.03


0.025


0.02
total strain [’]


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