Experimental determination of dynamic Modulus of Elasticity of cork
composites
Luís Miguel Ferreira dos Santos Príncipe Correia
Instituto Superior Técnico
Portugal, June 2012
ABSTRACT
This paper presents a study of the dynamic behavior of different materials, including
composite cork and rubber, cork agglomerate and a type of rubber. Because of the importance
of these materials as passive damping elements and a more precise characterization, it is
important to model to quantify and to understand how to dissipate energy.
The performance of passive dampers comes from the use of its own constituent materials
in order to dissipate the energy from vibration or shock. The capacity to absorb vibration or
shock is determined by its damping capacity, which is measured by the damping factor.
This paper presents a study based on frequency analysis, in order to determine the
dynamic modulus of elasticity of materials. To achieve this, we designed a measurement chain
suitable for measuring the dynamic modulus of elasticity in its two components: a real or
conservative and imaginary or dissipative.
Experimental tests were performed at low frequencies (quasi-static) in a servo-hydraulic
machine and high frequency (vibration) in a electrodynamics vibrator.
The results showed regularly, but also the need for additional data obtained from testing
frequency and amplitude ranges other than those used.
States of America and they were initially
1. INTRODUCTION
used in oil sealing applications in the
Composite materials are engineered
automotive industry. The company Amorim
materials from two or more materials with
Cork Composites (ACC), is a leading global
significantly different physical, chemical and
producer of
mechanical properties.
different compositions of this material for
The cork
and
commercially by
designations
rubber,
known
the Anglo-Saxon
corkrubber or
rubbercork,
was developed in the 1960’s in the United
rubber-cork, and
various
applications, such as
material
damping and sound
produces
gaskets,
insulation,
paving material, footwear material, fists of
tools
and fishing
rods,
miscellaneous
household and office supplies. Gaskets of
this type of material are used in gearboxes,
the amplitude of movement through
transmissions, valve covers, oil tanks, gas
factor and modifying the phase angle by
meters, transformers, pump oil and water,
adding it 90º
static seals oils, solvents, greases and
relation to the displacement [4]. Thus, the
sealing of water, air and other fluids [1], [2],
relationship between
[3].
displacement in simple
and 180º
respectively,
in
acceleration and
harmonic
motion
can be obtained by combining the eqs. 2.1
This paper presents the study of
and 2.3:
dynamic behavior for materials like rubbercork with
different
quantitative
(2.4)
compositions, through the characterization
of this behavior by the complex modulus.
2.2.
2. THEORETICAL ANALYSIS
2.1.
Hysteresis loop
As a result of hysteresis, when the
Simple harmonic motion (S. H. M.)
materials are subjected to an extension or
deformation alternating with any frequency,
The simple harmonic motion is the
stress versus strain curves response show
least complex of periodic motion. It is an
a phase lag or phase angle ( ) between
alternate
stress and strain.
movement and
can
be
represented by the circular functions sine or
cosine. In this case represented by x (t)
displacement, velocity and acceleration will
be the
first
and second
derivative of
displacement, respectively, over
time
[4].
Thus are obtained:
 Displacement:
(2.1)
Figure 2.1 – Delay in phase between the stress
( ) and strain ( ) [5]
The
 Velocity:
tests possible
(2.2)
to
of several
establish, by empirical
evidence, for most structural materials as
well
 Acceleration:
performance
as the
connections
between
the components, the area of the hysteresis
(2.3)
loop is not directly dependent on the rate of
change
in force,
that is, independent
of
the eqs. 2.2 and 2.3 show that the velocity
frequency of the charging process but is
and
proportional
acceleration of
a
harmonic
to
the square
of
the
displacement functions are also the same
amplitude of the force ( ) versus deflection
harmonic frequency. Each derived modifies
( ) [6].
Thus, a complex component has
stiffness mechanical behavior equation can
be written as:
(2.8)
Applying
this
equation
to
the
relationship between stress and strain
(Hooke’s law), these quantities can be
Figure 2.2 – Hysteresis loop in a process of
deformation [6]
related by a complex Young’s modulus,
The loss factor η, also known as
such that:
damping factor is proportional to the ratio of
energy
dissipated
per
cycle and
(2.9)
the
maximum elastic energy (or the maximum
in which:
amount of potential energy released during
(2.10)
the deformation cycle) as shown in eq. 2.5:
where
(2.5)
represents the elastic energy
stored in the material and
the energy
dissipated. This module is represented in
where
is the energy dissipated per
cycle,
is the maximum elastic energy
and
the complex Argand plane in Fig. 2.3:
is the loss factor.
(2.6)
2.3.
Complex stiffness modulus
To model the hysteresis of materials
involved, the stiffness and damping can be
represented
by
the
stiffness
Figure 2.3 – Representation of the storage
modulus and loss [7] (edited)
is the stiffness
For the damping factor noted that if
is the loss factor of the material.
the dissipated energy can calculate a cycle
, where
and
complex
through the area comprised within the
Alternatively,
hysteresis loop. This area, as known as, the
(2.7)
where
is the constant elastc stiffness of
the spring (elastic component) and
spring
energy dissipated in one cycle, can be
constant
component).
of
loss
a
(dissipative
obtained by numerical integration. From
eqs. 2.5 and 2.6 and the definition given in
2.8, the variation of energy in each cycle
(dissipation) is given by:
To
(2.11)
calculate
the
imaginary
component of complex modulus of stiffness
where
is the amplitude of displacement
performed.
is first necessary to calculate the damping
factor by eq. 2.13. After this calculation, the
Thus, the hysteretic damping factor
constant loss of the spring is given by:
can be written from the combination of eq.
(2.14)
2.6 in eq. 2.5, resulting in:
2.4
(2.12)
recapitulating
that
for
very
narrow
hysteresis loops, one can assume that
and the damping factor hysteresis
System of one degree of freedom
with hysteretic damping
Figure 2.5 represents a system of
one degree of freedom with hysteretic
damping.
is defined as:
(2.13)
where
is the magnitude of the applied
force.
Whereas a hysteresis loop as shown
in figure 2.4, can be represent a straight
line passing through the points of greatest
deformation. The slope of this line is the
elastic stiffness constant ( ).
Figure 2.5 – Representation a system of one
degree of freedom with hysteretic damping
The differential equation of simple
harmonic
motion
for
a
system
with
hysteretic damping is given by:
(2.15)
and substituting eq. 2.1 and 2.3 into eq.
2.15 yields:
(2.16)
Thus, the relation between response
and strength, designated receptance is
given by:
Figure 2.4 – Elastic stiffness constant
hysteresis loop [8]
from a
(2.17)
2.5.
System of two degrees of freedom
mathematical inverse of eq. 2.19 and
whose real and imaginary parts are given
by:
with hysteretic damping
(2.20)
A system of two degrees of freedom
with hysteretic damping is shown in figure
(2.21)
2.6.
from which follows:
(2.22)
(2.23)
3. MATERIALS
AND
EXPERIMENTAL
PROCEDURE
Figure 2.6 – Representation a system of two
degree of freedom with hysteretic damping
3.1.
All the materials involved in this work
The system of figure 2.6 is defined by
were
the following differential equations:
Materials
provided
by
the
Amorim
Cork
Composites, the world leading company in
the production of cork. The materials had
the
(2.18)
commercial
VC2100,
designations:
VC6400,
VC1001,
VC5200,
polychloroprene rubber (neoprene) and
NL20 (natural cork).
2.6.
Transmissibility
3.2
The transmissibility of responses or
Experimental Procedure
In
order
to
characterize
the
the relationship of responses between the
rubbercork
two degrees of freedom can be obtained
could better describe the Single Degree of
through the system eq. 2.18, particularly
Freedom modal system started to be
from the second equation, resulting in the
constructed for the vibrations tests taking
following transfer function:
into account the ASTM 5992-96 and the
agglomerates,
models
that
ISO 9052-1 standards, which were the main
(2.19)
standards found that could be helpful in this
work.
For
convenience
we
used
the
3.3.
Specimens construction
Considering
the
standards,
and
regarding the materials thickness provided
by Amorim, all the rubbercork specimens
(29 specimens) were cut in disk plates with
80 mm of diameter and later bounded
between two Ck45 steel disk plates with the
same diameter. The Ck45 steel has a very
high Young modulus (190-210GPa) and
therefore a high stiffness value. The choice
of this material for the disks was particularly
important because it should prevent steel
resonance from being nearby the frequency
range tests and should also guarantee
significant
stiffness
that
shall
prevent
bending and assure that won’t be any extra
vibration mode.
Figure 3.2 – Photograph of the machine where
the tests were carried out at low frequency
3.5
Description
of
tests
at
high
frequency
The high frequency tests (200 Hz to
1200 Hz) were performed in Vibration
Laboratory, located in the Department of
Figure 3.1 – Measuring chain a) Ck45 steel disk
plate after faced in lathe; b) Rubbercork
specimen; c) Final measuring chain ready for
tests [9]
3.4.
Description
of
tests
at
low
Mechanical
Engineering
at
Instituto
Superior Técnico.
The configuration adopted in this
study is shown in figure 3.3. The objective
frequency
pursued is to acquire the accelerations the
The low frequency tests (0.1 Hz to 10
disks on above and below.
Hz) were performed on a servo-hydraulic
machine INSTRON 8802, available at the
Mechanical Laboratory with a 100 kN load
cell capacity, figure 3.2.
For the tests, the specimens were
mounted on the machine and the variables
necessary
for
the
test,
including
the
excitation frequency and the maximum
allowable
displacement,
were
inserted
through the controller which is coupled to
the machine.
Figure 3.3 – Experimental setup
4. RESULTS
Figure 4.3 – Accelerations of specimens in the
low and high frequency tests
Figure 4.1 – Experimental results for the
stiffness elastic component
Figure 4.4 – Results of Young modulus for the
tests at low and high frequency
The results of the Young's modulus
for the
higher
variation
for
frequencies show
each specimen. But
little
these
values are significantly different from the
Figure 4.2 – Experimental results for the
stiffness dissipative component
values obtained
for low
frequencies.
Generally speaking, the Young's modulus
(real
part
of
the complex modulus
of
elasticity), displays a sharp rise between 10
Hz and 200 Hz, which stabilizes after a
much
smoother curve which depends
on
results
of
the
tests
for
specimens
the specimen. Since the amplitudes are
compounds of cork and rubber (samples
also very different before and after the rise,
1001, 5200 and 6400) are in the range of
this result may indicate a non-linearity to
results for the cork and rubber. The small
the amplitude or frequency. Only the tests,
difference in values of the damping factor
for the same frequency but with significantly
between 5200 and 6400 samples may be
different amplitudes and others in the range
related to the very accented difference in
between 10 Hz and 200 Hz (not performed
the percentage of cork in each of the
for the reasons mentioned above) would
clusters.
allow clarify which of these causes should
be considered.
5. CONCLUSIONS
Were performed tests on specimens
of cork, rubber and rubbercork at different
frequencies, in particular in the ranges
of 0.1 Hz to 10 Hz and 200 Hz to 1200
Hz. The results show some regularity for
each
range but, generally,
have
a
significant evolution. Since it has not been
studied the dependence of amplitude in the
results, it was not possible to conclude
whether this
evolution depends
on
the
amplitude or the existence of a transition
zone accented with frequency.
In any event, since the amplitude of
deformation for the range of 0.1 Hz to 10
Hz were the same, the evolutions (very
slight but regular and should be considered
significant) in this range can be attributed
Figure 4.5 – Results of damping factor for the
tests at low and high frequency
to a dependency of frequency, supporting
the hypothesis that has a important role in
For damping factor, it is observed
from
the
figure
4.5
that there
is
gap values obtained at low frequencies in
relation to
the
significant than
others but
for
much
the
less
Young's
modulus, once again, this may be due to
the
different amplitudes used in
the transition frequency.
a
the
tests or the frequency dependence. Here,
too, lacks confirmation tests.
In figure 4.6 we can conclude that the
In the range of 200 Hz to 1200 Hz,
we have a smooth transition whereby even
considered mean values without great error.
Note also the case of the specimen 1001
that
has
the rubber
a
gas
which
phase embedded
in
decisively affects the
Young modulus but not the damping factor.
For damping factor, there is a trend fairly
regular with the percentage of cork.
6. REFERENCES
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Alguns
Parâmetros
de
Projecto
na
Funcionalidade de Juntas de Vedação de
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FCT/UNL; Lisboa; 2003.
[2] http://www.corkcomposites.amorim.com/
(consultado em Março de 2012).
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Materiais Compósitos de Borracha Natural
Para Aplicações Anti-Vibráticas; Relatório
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M.
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