A dynamic Youngs modulus measurement system for highly compliant
polymersFrancois M. Guillota) and D. H. Trivett George W. Woodruff
School of Mechanical Engineering, Georgia Institute of Technology,
Atlanta, Georgia 30332-0405
Received 6 May 2003; accepted for publication 7 July 2003 A new
system to determine experimentally the complex Youngs modulus of
highly compliant elastomers at elevated hydrostatic pressures and
as a function of temperature is presented. A sample cut in the
shape of a bar is adhered to a piezoelectric ceramic shaker and
mounted vertically inside a pressure vessel equipped with glass
windows. Two independent measurement methods are then used: a
resonant technique, to obtain low-frequency data, and a wave
propagation technique, to obtain higher-frequency data. Both
techniques are implemented utilizing laser Doppler vibrometers. One
vibrometer detects sample resonances through a window located at
the bottom of the pressure vessel, and a set of two separate
vibrometers monitors the speed of longitudinal waves propagating in
the sample, through windows located on the sides of the vessel. The
apparatus is contained inside an environmental chamber for
temperature control. Using this approach, Youngs modulus data can
be obtained at frequencies typically ranging from 100 Hz to 5 kHz,
under hydrostatic pressures ranging from 0 to 2.07 MPa 300 psi ,
and at temperatures between 2 C and 50 C. Experimental results
obtained on two commercial materials, Rubatex R451N and Goodrich
Thorodin AQ21, are presented. The effects of lateral inertia,
resulting in dispersive wave propagation, are discussed and their
impacts on Youngs modulus measurements are examined. 2003
Acoustical Society of America. DOI: 10.1121/1.1604121 PACS numbers:
43.20.Ye, 43.20.Jr, 43.20.Ks YHB
I. INTRODUCTION
Compliant polymers, such as voided polyurethanes, are materials
of interest because of their vibration isolation and damping
properties. Knowledge of the dynamic elastic properties of such
elastomers, as a function of temperature and hydrostatic pressure,
is essential to predict their performance, to validate numerical
models, and to design sonar systems. As is the case for any
homogeneous, isotropic material, only two elastic constants along
with the density are needed to completely characterize them. Among
these elastic constants, the bulk and Youngs moduli are the easiest
to obtain experimentally, and also constitute the best pair, in
terms of computational error minimization, from which to compute
other moduli. In the Acoustic Material Laboratory of the Georgia
Institute of Technology, two independent experimental systems have
been developed to perform dynamic measurements of these two moduli.
The present article describes the dynamic Youngs modulus apparatus
and presents data that illustrate a portion of the capabilities of
the measurement system. These data were obtained on a commercial
closedcell foam neoprene, Rubatex R451N, and on a commercial
elastomer, Goodrich Thorodin AQ21. Several methods have been used
to dynamically measure the Youngs modulus of viscoelastic
materials. The most widely employed approach is the resonant bar
technique introduced by Norris and Young,1,2 where a sample in the
shape of a bar is excited harmonically at one end and thea
Author to whom correspondence should be addressed. Electronic
mail: [email protected] J. Acoust. Soc. Am. 114 (3),
September 2003
ratio of the end accelerations is measured. At resonance, the
response of the sample can be used to obtain values of the complex
modulus. Madigoski and Lee3,4 utilized the technique extensively to
characterize a number of materials as a function of temperature,
and used the timetemperature superposition principle5 or WLF shift
procedure, named after Williams, Landel, and Ferry to obtain data
over extended frequency ranges. Recently, the resonance method has
been adopted as a standard by the American National Standard
Institute.6 Garrett7 used an electrodynamic transduction scheme to
excite torsional, longitudinal, and exural modes in rods of
circular or elliptical cross section. Both the longitudinal and
exural modes yield values for the Youngs modulus. Garrett measured
the resonance frequencies of these modes to compute values of the
magnitude of the modulus. Guo and Brown8 extended this method to
measure the complex Youngs modulus by tting the analytical solution
of the longitudinal wave equation to experimental data. Their
approach allows the determination of the modulus at frequencies
that are not restricted to the resonance frequencies of the modes.
Measurements as a function of both temperature and hydrostatic
pressure were reported by Willis et al.,9 between 7 C and 40 C and
over a 0- to 3.45-MPa 500-psi range. Their approach consisted of
measuring, by laser vibrometry, the dynamic response of a
rectangular block excited by a shaker, and matching the data with
predictions from a nite-element code in which the complex elastic
moduli were the adjustable parameters. Their sample was inside a
pressure chamber submerged in a water bath for temperature control.
Although this method allows one to obtain data over relatively
large frequency ranges without the 2003 Acoustical Society of
America
1334
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need for timetemperature shifts, it suffers from occasional
convergence problems associated with the inversion of the
nite-element code. The method described in this article uses an
improved version of the resonant bar method and combines it with a
wave-speed measurement technique. These approaches, which are
implemented using laser Doppler vibrometers, have the advantage of
providing directly measured complex wave-speed data which are
converted to Youngs modulus data, using the density of the material
over a much broader frequency range than that typically afforded by
the resonant bar method alone. Therefore, fewer temperature data
sets are needed in order to generate master curves, reducing the
amount of uncertainty associated with multiple WLF shifts.
Furthermore, since all the sample displacements are measured with
noncontact vibrometers, no mass corrections due to the presence of
accelerometers or electromagnetic transducers need to be taken into
account. Samples are placed inside a stainless-steel vessel which
is itself contained inside an environmental chamber, providing
pressure and temperature capabilities over a 0- to 2.07-MPa 300-psi
range and a 2 C to 50 C range, respectively.
II. THEORETICAL FRAMEWORK
The theoretical background of the techniques employed in the
system can be compiled from several articles,1,3,4,8 technical
reports,2,10 and textbooks.1114 However, because some of these
references are not readily available and because none of them
provides, in one single convenient place, all the information
relevant to this work, we chose to present below the most important
theoretical points of the measurements.A. Resonance
measurements
FIG. 1. Forced vibrations of a polymer sample in the form of a
bar of length L with uniform cross section. The ceramic shaker and
the vibrationtransmitting metal rod correspond to the particular
experimental setup used in this work. The sample excitation is
measured at the bottom surface of the metal rod.
d 2U dx 2
k 2U
k 2U 0 ,
3
Consider the forced vibrations of a homogeneous bar of density
and length L, with a constant cross section and no mass attached to
its free end Fig. 1 . One end of the bar is driven with a harmonic
displacement u 0 (t) U 0 e i t , and the resulting axial
displacement relative to the driven end at a distance x from the
driven end is u(x,t) U(x)e i t . Assuming a uniform, uniaxial
stress distribution inside the bar, and neglecting the effects of
lateral inertia, the equation of motion can be written asx 2
where k is the complex wave number, dened as k * * ( /E * ) 1/2
/c b , and c b (E * / ) 1/2 is the complex bar wave speed. The
boundary conditions for the problem are zero relative displacement
at x 0, i.e., u(0,t) 0 or U(0) 0, and zero stress at x L, i.e., x
(L,t) 0 or dU/dx at x L 0, which lead to the following solution for
Eq. 3: U x U 0 cos kx tan kL sin kx 1 . 4
x
t2
u u0 ,
1
Now that the equation of motion has been solved, the resonances
of the sample bar can be studied. To do this, let us consider the
complex ratio of the free-end displacement to the driven end
displacement, Q * , which, from Eq. 4 , can be expressed as Q* U L
U0 U0 1 . cos kL 5
where x is the uniaxial stress in the bar in the longitudinal
(x) direction. In this case, Hookes law states that x E * x , where
E * is the complex Youngs modulus and x is the axial strain
experienced by the material, dened by u/ x. Substituting these two
relations into Eq. 1 x gives2
On the other hand, using complex notation, Youngs modulus can be
written as E* E iE E cos i sin , 6
E*
u
2
x2
t2
u u0 ,
2
which, in turn, leads to the following ordinary differential
equation:J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
where E is the elastic or storage modulus, E is the loss or
viscous modulus, E is the magnitude, and tan ( ) is the loss
factor. Using these relations, Eq. 5 can then be expressed
as1335
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system
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1 Q*
1/2
cos kL
cos
L
E cos
i sin
,
7
E
E cos 4L f res cos 2n 1 22
cos 2 tan
1
sinh
1
1/Q 2
,
which can be separated into real and imaginary parts in the
following manner: E 1 Q*1/2 1/2
2n 1
13 E sin 4L f res cos 2n 1 22
cos
L
E L E
cos1/2
2
cosh sinh
L
E L E
sin1/2
2 2 . 8
sin 2 tan
1
sinh
1
1/Q 2
.
2n 1
i sin
cos
2
sin
B. Wave-speed measurements
If the complex ratio is written as Q * Qe , then at reso(2n 3)(
/2) and n 1,2,3,... is the resonance, when nance number, the
left-hand side of Eq. 8 is 1 Q* 1 n Q1
i
i
n 1,2,3,... .
9
Combining Eqs. 8 and 9 leads to the following system of two
equations in two unknowns E and :1/2 1/2
Let us again consider the bar shown in Fig. 1. Instead of a
continuous harmonic signal, the shaker is excited with a short
burst consisting of a gated sinusoidal signal. In this case,
monitoring the propagation of this burst allows one to measure both
the wave speed and the attenuation in the material. The wave-speed
magnitude, c b , is obtained in a straightforward manner by
measuring the time necessary for the burst to travel a distance d
along the sample. The signal attenuation due to the materials
internal damping is obtained from the change in signal amplitude
over the same distance. Assuming that the displacement perturbation
corresponding to the traveling burst can be written as u x,t U 0e
it kx
, for t 1 t t 2 ,
14
cos
L
E
cos
2
cosh
L
E
sin
2
0, 10 and recalling from the previous section that the complex
wave number k is given by1/2
1/2
1/2
sin
L
E
cos
2
sinh
L
E
sin
2
1 n Q
1
.
k
E*
,
15
then, using Eq. 6 , one obtains One can easily show that the
solutions to Eq. 10 are1/2
2 tan
1
sinh
1
k 1/Q 2 and u x,t , 11
E cos1/2
i sin cos 2 i sin 2 k i , 16
2n 1
E
and 4L f res cos 2n 1 22
U 0e
x
e i
t k x
,
17
E
.
12
One should again emphasize that Eqs. 11 and 12 are valid at
resonance only, that f res is the resonance frequency, and that Q
is the amplitude of the displacement ratio. These equations are in
agreement with the theory presented in Ref. 2, for the case of an
end mass equal to zero. One can see from the above analysis that
measuring f res and Q at resonance, and using Eqs. 11 and 12
provides a direct method for determining the complex Youngs modulus
of a sample with no mass attached to its free end. Once the modulus
amplitude and the loss factor are known, the elastic and loss
moduli can be computed using1336 J. Acoust. Soc. Am., Vol. 114, No.
3, September 2003
where is the attenuation coefcient. Thus, as can be seen from
Eq. 17 , can be computed from the amplitude of the displacement
signal measured at two locations spaced apart by a distance d,
according to u x,t 1 ln d u x d,t
.
18
The magnitude of Youngs modulus and its loss factor tan ( ) can
then be obtained from E c 2, b 19
and, from Eq. 16F. M. Guillot and D. H. Trivett: Youngs modulus
measurement system
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2 sin
1
cb
.
20
E*
1/2
c* c* 1 b
K c*
2 1/2
.
27
Finally, the elastic and loss moduli are given by E E cos c 2
cos 2 sin b c2 b sin 2 sin1
cb cb
, 21 .
E
E sin
1
C. Dispersive effects
It has been assumed so far that the effects of lateral inertia
in the sample were negligible. The purpose of this section is to
reexamine this assumption and to determine under which conditions
these effects need to be taken into account in the measurement of
Youngs modulus. Let us again consider the sample in the form of a
bar depicted in Fig. 1. Taking into account the lateral
displacements resulting from the Poisson coupling of the
longitudinal wave, and keeping the assumption of uniaxial stress in
the bar, the general Hookes law can be written asx
Equation 27 indicates that elastic wave propagation in the bar
is dispersive and relates the complex bar speed c * to the b
complex longitudinal wave speed c * at a given frequency, . The
same result could have been obtained from the exact solution,
derived by Pochhammer, to the wave propagation problem in a
circular rod of radius a, in the low-frequency approximation13
i.e., when the radius a is much smaller than the wavelength, and
when, consequently, the uniaxial stress assumption holds . This was
also observed by Rayleigh,14 who derived Eq. 27 in the case of a
circular rod. For a square cross section of side length s, as is
the case in our measurements, K 2 s 2 /6, and Eq. 27 becomes
E*1/2
c* c* 1 b
1 6
s c*
2 1/2
.
28
1 E*
x,
and
y
z
E*
x,
22
where is the materials Poissons ratio and the subscripts denote
the spatial directions see Fig. 1 . If u(x,t), v (x,t), and w(x,t)
denote the displacements in the x, y, and z directions,
respectively, then Eq. 22 can be used to compute the lateral
displacements in terms of the longitudinal displacement asv
y
u and x
w
z
u , x
23
As can be seen from Eq. 28 , the lateral inertia effects
increase with increasing values of Poissons ratio, frequency, and
sample cross section; they are also more substantial at lower wave
speeds. Therefore, when the experimental conditions are such that
these effects are signicant, the measured wave speed c * differs
from the bar wave speed c * , and Eq. b 28 must be used to compute
Youngs modulus. Practically, it is assumed that the correction
factor i.e., the second term inside the brackets is small enough so
that Eq. 28 applies to the magnitude of the complex speeds. The
experimentalist needs to be especially cautious when measuring
Youngs modulus on materials with large values of Poissons ratio, at
high frequency. In such cases, or in cases where Poissons ratio is
not known, it is advisable to minimize the lateral dimensions of
the sample, in order to minimize the dispersive effects.III. SAMPLE
PREPARATION AND EXPERIMENTAL APPARATUS DESCRIPTION A. Sample
preparation
where y and z are the coordinates of a point in the sample cross
section the center of the coordinate system coincides with the
center of the cross section . Using Eq. 23 , it is possible to
formulate the potential and kinetic energies of the bar, and to
apply Hamiltons principle to derive the equation of motion
corresponding to longitudinal vibrations with lateral inertia
effects, known as the Love equation.11 The details of this
derivation are somewhat lengthy and can be found in Ref. 12. Loves
equation of motion is2
E*
u2
2
x
K2
u x t22
4
2
u , t2
24
where K 2 is the polar radius of gyration of the cross section.
Considering a solution to Eq. 24 of the form u x,t Ue it x/c *
,
25
where c * is the velocity of the longitudinal waves, and
substituting 25 into 24 , yields2 4 2
E*
c *2
K2
2
c *2
.
26
Equation 26 can be rearranged to yieldJ. Acoust. Soc. Am., Vol.
114, No. 3, September 2003
Samples are cut in the shape of a bar with a square
cross-sectional area. The maximum sample length is limited by the
pressure vessel dimensions and is about 27 cm. As explained above,
it is desirable to use a sample with a minimal cross section. In
the case of Rubatex R451N, the material comes in the form of
6.35-nm 0.25-in. -thick sheets that were cut in 0.25-in.-wide
strips using a mat cutter. The Goodrich Thorodin AQ21 sample used
in this study was cut into a bar with a 5- by 5-mm cross-sectional
area. The top and bottom ends of each sample were then cut with a
razor blade in order to get plane and smooth surfaces, which are
necessary for a good bond between the sample and the metal of the
excitation-transmitting rod shown in Fig. 1. The end surfaces also
need to be perpendicular to the axis of the sample so that the
latter can be mounted in a perfectly vertical position inside the
apparatus, and therefore be excited in a primarily longitudinal
mode, minimizing spurious exural motions. For materials whose color
and texture make them good light scatterers, no further preparation
is needed. However,1337
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system
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other materials, such as Rubatex R451N which is black and has a
matte nish , and Goodrich Thorodin AQ21 which is yellow and
somewhat translucent , do not scatter sufcient amounts of light and
therefore require a surface treatment in order to be measured with
a laser vibrometer. An obvious requirement is that the surface
treatment not change the elastic properties of the sample. With
this in mind, the following treatments have been used. In the case
of Rubatex R451N, for the measurement of the motion of the bottom
end of the sample, a small piece of metallized tape about the same
dimensions as the cross-sectional area is glued using a
cyanoacrylate adhesive to the end of the sample. This piece of tape
does not constrain the longitudinal motion, and its mass is
negligible with respect to that of the sample. For measurement of
the motion of the lateral sides of the sample resulting from the
Poisson-coupled longitudinal motion , a layer of talc powder is
deposited on these surfaces. The load resulting from the presence
of the talc is negligible and its powdery nature allows the sample
to move in the lateral direction without any constraint. In the
case of Goodrich Thorodin AQ21, a thin layer of white correction
uid white-out was applied at the bottom end of the sample. The same
treatment as for the Rubatex R451N material metallized tape could
have been used, but the motivation for trying the correction uid
was to compare its performance to that of the tape. The white uid
was found to be a slightly better light scatterer than the tape;
however, both are perfectly adequate for the resonance
measurements. For the side measurements on Goodrich Thorodin AQ21,
a thin layer of gold-colored ink was used, which proved to be a
better reector than the talc powder. The absence of adverse effects
from the tape, the powder, and the ink on the sample motion was
veried experimentally by measuring the vibrations of samples with
and without these surface treatments: the measured displacements
were identical in both cases. As a nal note specic to Rubatex
R451N, before cutting a sample, the actual rst preparatory step
consisted of following the manufacturers recommendation of placing
the sheet of foam in an oven, set at 71 C 160 F , for 24 h, in
order to completely cure the material and insure stable properties.
It was found, however, that even after 4 days of this treatment,
measured properties had a tendency to change with time. Therefore,
the data that are presented below should be considered a snapshot
of the Rubatex R451N material constants.B. Experimental
apparatus
The experimental apparatus is shown in Fig. 2. The sample is
glued to a 2.5-cm-long vibration-transmitting metal rod with
approximately the same cross-sectional area as the sample , which
is itself glued to a shaker, using a cyanoacrylate adhesive. The
shaker is made of ten piezoelectric ceramic discs EC64 from EDO
Corporation: 1.78 cm 0.700-in. outer diameter 0.22-cm 0.085-in.
-thick plus one depolarized and unelectroded disk at each each end
of the stack, for shielding purposes; the disks are glued together
using an epoxy resin. The shaker, in turn, is epoxied to a threaded
mounting piece that attaches to the top of the pres1338 J. Acoust.
Soc. Am., Vol. 114, No. 3, September 2003
sure vessel. To prevent delaminations caused by temperature
cycling, the vibration-transmitting rod and the mounting piece are
made of Invar, which has a very low coefcient of thermal expansion,
comparable to that of the ceramic used in the shaker. The pressure
vessel inside dimensions: 38.72 cm (15.25 in.)L 10.2 cm (4 in.)W
7.6 cm (3 in.)D] is made of 303 stainless steel and it is designed
to be used up to 3.45 MPa 500 psi . It features three 1.9-cm 0.75
in. -thick glass windows for optical access to the sample: one
round window 3.5 cm 1.375 in. located at the bottom of the vessel
and two long windows 28.8 cm (11.35 in.)L 3.4 cm (1.35 in.)W
located along its sides. They are sealed inside the vessel with a
1.59-mm 0.0625 in. -thick Aramid gasket used in conjunction with a
silicone RTV sealant. An RTD sensor Sensor Scientic PT1000 is used
to measure the temperature inside the vessel. Pressure is created
with a compressed air cylinder connected to the vessel not shown in
Fig. 2 and both pressure and temperature are monitored using a
Heise PM Indicator gauge. Outside the pressure vessel, three laser
Doppler vibrometer sensor heads Polytec CLV-800-ff , capable of
measuring out-of-plane normal vibrations, are mounted on
positioning slides two sets of 5-in. and 15-in. Velmex Bislides for
the side lasers; two 25-mm and one 110-mm National Aperture Mini
Stages for the bottom one . The entire system vessel, lasers, and
slides is placed inside an environmental chamber Thermotron S-27C
for temperature control, and needs to be operated from outside that
chamber. For that purpose, the slides are remotely driven by two
controllers Velmex VP 9000 and National Aperture Servo 1000 , which
allow precise positioning of the vibrometers. Also, a miniature
video camera connected to a television monitor is mounted next to
each sensor head; it is focused on the area where the laser beam
illuminates the surface on the sample, and provides a direct visual
check of the beam position as well as a view of the sample from
three different angles. A neutral density lter is placed in front
of each camera in order to reduce the light intensity and to
improve the laser spot resolution on the monitor. Two small
ashlights are used to enhance the image of the sample. Because of
the small aperture afforded by the bottom window and lack of space
in that area of the setup, the corresponding camera is positioned
at a right angle and a small mirror is employed to obtain a view of
the bottom of the sample. The shaker is excited by two types of
signal, depending on which measurement is performed. For the
resonance measurements, a continuous signal with varying frequency
is used, provided by a lock-in amplier Stanford Research Systems
model SR850 . For the wave-speed measurements, a burst signal
composed of six cycles at a given frequency is used, provided by a
function generator Wavetek model 80 . In both cases, signals are
amplied by a power amplier Krohn-Hite model 7500 before being sent
to the shaker. Surface motion signals measured by the sensor heads
are electronically processed by the vibrometer controller: CLV1000
bottom laser or CLV-2000 side lasers . These signals are ltered
Krohn-Hite lter model 3382 before being displayed on an
oscilloscope Tektronix model TDS 3014 . InF. M. Guillot and D. H.
Trivett: Youngs modulus measurement system
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FIG. 2. Picture of the experimental setup. a General view. b
Close-up.
J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
F. M. Guillot and D. H. Trivett: Youngs modulus measurement
system
1339
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phase plot at points where the phase is equal to (2n 3) 90. A
second sweep is then performed over a narrow frequency range
centered around each of these frequencies, and the resulting plots
are used to obtain more accurate values of f and Q at resonance.
The narrow range has a span of 100 Hz, which, combined with the 401
points of digitization, allows for the determination of the
resonance frequency with a resolution of 0.25 Hz. These data, along
with the length and density of the sample, are then input into Eq.
13 to compute the elastic and loss moduli. The maximum measurement
errors associated with these two quantities can be estimated as
follows. The 0.25-Hz resolution results in a frequency uncertainty
of 0.5% at 50 Hz; assuming a 0.5-mm error on the sample length
measurement results in a 0.3% length uncertainty for a 15-cm-long
sample the smallest sample used in the system . The systematic
errors from the laser vibrometer decoder module and from the
lock-in amplier do not affect the measurement of Q, the amplitude
ratio, because it is a relative measurement. Therefore, these
errors are not taken into account. This analysis results in a total
uncertainty of 1.6% for both the elastic and loss moduli. One
should note that this does not include the uncertainty related to
the density, which affects all measurements elastic and loss
moduli, from both resonance and wave-speed techniques in an
identical manner.FIG. 3. Resonance plots obtained on a Rubatex
R451N sample at 30 C and ambient pressure, displaying four
measurable resonances. a Amplitude plot of the ratio of the
velocity of the free end to that of the driven end. b Corresponding
phase plots.
B. Wave-speed measurements
the case of the resonance measurements, the velocity signals are
also input to the lock-in amplier. Both the oscilloscope and the
lock-in amplier are equipped with a GPIB bus utilized to transfer
velocity signals to a notebook computer for data processing.IV.
MEASUREMENT PROCEDURE AND DATA PROCESSING A. Resonance
measurements
Based on the theory described in Sec. II A, the procedure to
implement the resonance measurement technique is as follows. These
measurements are obtained using the bottom laser only. The lock-in
amplier, in the frequency sweep mode, excites the shaker with a
continuous sine wave whose frequency ranges from a lower limit to
an upper limit. The amplier also records the amplitude and phase of
the signals output by the laser vibrometer, the latter being
employed to measure both the velocity of the driven end the metal
rod without the sample adhered to it and the velocity of the free
end. These signals, digitized with 401 points by the lock-in
amplier, are combined numerically in MATLAB to produce amplitude
and phase plots of the sample frequency response. First, a global
sweep is performed over an extended frequency range in order to
identify all the measurable resonances of the sample. The resulting
plots are shown in Fig. 3 over a 502400-Hz range, which was the
typical range used for the measurements on the Rubatex R451N
sample. Each resonance frequency is approximately determined from
the1340 J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
In the method described above, internal damping in materials
always limits the number of identiable resonances, and, therefore,
sets an upper frequency limit on the method for a given sample. One
way to extend measurements to higher frequency regions is to reduce
the length of the sample, which, as Eq. 12 reveals, would shift the
value of resonance frequencies higher. In theory, successive cuts
can be made to produce increasingly shorter samples to cover the
frequency range of interest, as long as the sample maintains an
appropriate dimension ratio i.e., L lateral dimensions . However,
this practice possesses two disadvantages. First, if the sample
under study is inhomogeneous, then cutting the sample into smaller
lengths effectively results in measuring a different sample after
each cut, introducing additional measurement errors. Second, each
sample shortening requires opening the pressure vessel and the
environmental chamber, and the subsequent readjusting of the
necessary pressure and temperature conditions, rendering the
process quite time consuming. A better way to extend the
experimental Youngs modulus frequency range is to measure the wave
speed and the attenuation of longitudinal waves propagating inside
a sample. In our apparatus, this technique is implemented with the
two side lasers measuring the lateral component of a longitudinal
excitation propagating along the sample. A short burst is used to
excite the shaker, which excites a longitudinal wave in the sample
and the resulting lateral motion due to the Poissons ratio effect
is then recorded at two different locations along the length of the
sample. Two laser vibrometers located on opposite sides of the
sample are needed in order to insure that only the lateral motion
which is a symmetrical motion produced by the longitudinal wave is
measured. Any other spurious vibrations due to mounting imF. M.
Guillot and D. H. Trivett: Youngs modulus measurement system
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perfections or outside vibrations that the sample may experience
create bending motions, which manifest themselves in the form of
antisymmetrical displacements. Therefore, these spurious vibrations
can be systematically eliminated by adding the signals from each
vibrometer. The amount of attenuation which, above the glass
transition temperature, increases with frequency in the material
sets both the lower- and the upper-frequency limits of the
wave-speed method. Indeed, it is imperative that the laser
vibrometers measure only the direct signal produced by the shaker
without any superimposed reections from the ends of the sample.
Thus, sufcient attenuation needs to be present in order to
eliminate these reections before they reach the measurement
locations; this determines the lowerfrequency limit for a given
sample length . Conversely, the upper limit is attained when
attenuation is so large as to prevent accurate measurement of the
signal amplitude at the location furthest away from the shaker.
Thus, the actual frequency range of the measurements described in
this article depends only on the attenuation of the particular
material under study, and will therefore vary from one material to
the other. For the Rubatex R451N sample, combining the resonance
and the wave-speed techniques, the range was found to be
approximately 200 8000 Hz. These values, of course, like the
attenuation, also depend on temperature and pressure. Practically,
the wave-speed technique is implemented as follows. The function
generator excites the shaker with a burst signal composed of 6
cycles at a given frequency, with a 1-s repetition rate. For
example, frequencies ranging from 4500 to 8000 Hz in steps of 500
Hz were used for the Rubatex R451N sample. Both laser sensor heads
are positioned at the same height, about 2 cm below the shaker;
this constitutes the top location. At each frequency, signals from
each vibrometer are displayed on the oscilloscope, which digitizes
them using 10 000 points. Sixty-four averages are taken, and the
signals from each side are added; the resulting signal is
transferred to a computer for further processing. This is repeated
at the distant location, 10 cm below the top one. Typical signals
measured at both locations and used for wave speed and attenuation
computation are shown in Fig. 4. In MATLAB, signals are windowed
and zero padded before being subjected to a fast Fourier transform.
The center frequency of each Fourier-transformed signal is
determined, and, at that frequency, the amplitude A and the phase
of the ratio of the top location signal to that of the distant
location are computed. These yield the attenuation and the wave
speed, respectively, according to ln A , d c d. 29
FIG. 4. Velocity signals measured at 5500 Hz by the side laser
vibrometers at two locations spaced 10 cm apart on the surface of a
Rubatex R451N sample 30 C and ambient pressure . The relative phase
and amplitude of these signals are used to compute, respectively,
the speed and attenuation of longitudinal waves propagating inside
the sample.
atic errors associated with the vibrometers and the oscilloscope
do not affect the measurement of the ratio A; this results in a
maximum total error on the order of 1.0% for the elastic modulus
and on the order of 2.0% for the loss modulus. As previously
mentioned, these values do not include the error associated with
density measurements.V. EXPERIMENTAL RESULTS A. Rubatex R451N
The following data were obtained on a Rubatex R451N neoprene
sample, with a density of 579 kg/m3, and cut to a length of 23.3
cm. These two values were measured at room temperature and ambient
pressure. The subsequent dimensional changes resulting from varying
the hydrostatic pressure and the temperature were assessed using
one of the side lasers and the positioning slide controller: the
laser beam was focused at the bottom of the sample whose change of
position was recorded for the various pressures and temperatures
used. This yielded the length of the sample, whose value was used
to estimate the density of the material. One should note that, for
samples that have been measured in the bulk modulus system of the
Acoustic Material Laboratory, the dependence of density on pressure
and temperature is known. However, Rubatex R451N is too soft to be
measured in that system. The data shown below are not corrected for
dispersive effects, as Poissons ratio of Rubatex R451N which is on
the order of 0.25 is small enough to render the corrections
negligible.1. Youngs modulus as a function of hydrostatic
pressure
30
These data, along with the density of the sample, are then input
into Eq. 21 to compute the elastic and loss moduli. The
uncertainties associated with these measurements are assessed as
follows. Assuming a 0.5-mm error on distance measurement results in
a 0.5% uncertainty for d; the systemJ. Acoust. Soc. Am., Vol. 114,
No. 3, September 2003
Figures 5 and 6 display resonance and wave-speed measurement
results in the form of the real and imaginary parts of the modulus,
as a function of pressure. Data were obtained at 30 C, and 0 i.e.,
ambient , 69 and 138 kPa 0, 10, and 20 psi . At least 30 min were
allowed to elapse after each new pressure setting, and the pressure
was always increased.1341
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FIG. 5. Elastic Youngs modulus of Rubatex R451N measured at 30
C, as a function of hydrostatic pressure. The solid lines are curve
ts.
FIG. 8. Loss Youngs modulus of Rubatex R451N measured at ambient
pressure, as a function of temperature. The solid lines are curve
ts.
These plots clearly show that the two techniques provide results
that are compatible with each other, and that the wave-speed
approach allows one to substantially extend the measurement
frequency range beyond what the classical resonant bar method can
offer, without having to cut the sample. Figure 5 indicates that
the material gets softer with increasing hydrostatic pressure, as
the walls of the closed cells buckle and become more compliant. The
loss modulus, however, is relatively unaffected by pressure
changes, as shown in Fig. 6.2. Youngs modulus as a function of
temperature
Figures 7 and 8 display resonance and wave-speed measurement
results in the form of the real and imaginary parts of the modulus,
respectively, as a function of temperature.FIG. 6. Loss Youngs
modulus of Rubatex R451N measured at 30 C, as a function of
hydrostatic pressure. The solid lines are curve ts.
FIG. 7. Elastic Youngs modulus of Rubatex R451N measured at
ambient pressure, as a function of temperature. The solid lines are
curve ts. 1342 J. Acoust. Soc. Am., Vol. 114, No. 3, September
2003
FIG. 9. Loss Youngs modulus of Rubatex R451N at ambient pressure
and at 30 C, resulting from horizontal shifts applied to the 20 C
and 10 C data sets. The 20 C and 10 C data have been corrected by
multiplicative factors of T 30 30 /T 20 20 and T 30 30 /T 10 10 ,
respectively. The solid line is a curve t. F. M. Guillot and D. H.
Trivett: Youngs modulus measurement system
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FIG. 10. Elastic Youngs modulus of Rubatex R451N at ambient
pressure and at 30 C, resulting from horizontal shifts applied to
the 20 C and 10 C data sets. The shift factors are the same as the
ones used for the loss modulus. The 20 C and 10 C data have been
corrected by multiplicative factors of T 30 30 /T 20 20 and T 30 30
/T 10 10 , respectively. The solid lines are curve ts.
FIG. 12. Wave speed of Goodrich Thorodin AQ21 measured at 50
psi, as a function of temperature. The dots represent wave-speed
values uncorrected for dispersion. In the case of the resonance
data, the correction is negligible, and the white dots uncorrected
values appear in the center of the blacklled markers corrected
values . In the case of the wave-speed data, the correction is more
signicant, and the black dots are distinct from the blacklled
markers.
Data were obtained at ambient pressure, and at 30 C, 20 C, and
10 C. The sample was allowed to equilibrate overnight between each
temperature. Figures 7 and 8 indicate that both the elastic and the
loss moduli increase signicantly with decreasing temperature, as
expected above the glass transition temperature. Only three
resonances were measurable at the two lowest temperatures, due to
the increased loss. Finally, the timetemperature superposition
principle can be applied to these data in the following manner.
First, the loss modulus temperature curves are shifted horizontally
to get the best possible t to a smooth curve, as illustrated in
Fig. 9. In that gure, the reference temperature is 30 C, and the
horizontal shift factors applied to the 20 C and 10 C data sets are
a 20 4.3 and a 10 19.0, respectively. These last two data sets have
been corrected by multiplicative factors of
T 30 30 /T 20 20 and T 30 30 /T 10 10 , respectively where the
Ts are temperatures in Kelvin and the s are the corresponding
densities , as specied by Ferry.15 The shift factors are then used
to shift the elastic modulus horizontally by the same amount,
producing the curves shown in Fig. 10. In that gure, the 20 C and
10 C data have also been corrected by multiplicative factors of T
30 30 /T 20 20 and T 30 30 /T 10 10 , respectively. Finally, a
vertical shift is applied to the temperature segments of Fig. 10,
in order to obtain the smooth curve displayed in Fig. 11. The
additive vertical shift factors are 20 5.0 106 Pa and 10 12.4 106
Pa for the 20 C and 10 C data sets, respectively. The physical
meaning of these vertical shifts may be related to the static
modulus dependence on temperature. However, in the case of the
Rubatex R451N sample, their signicance is not entirely clear, as
the elastic modulus of this material has been ob-
FIG. 11. Elastic Youngs modulus of Rubatex R451N at ambient
pressure and at 30 C, resulting from both horizontal and vertical
shifts applied to the 20 C and 10 C data sets. The solid line is a
curve t. J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
FIG. 13. Elastic Youngs modulus of Goodrich Thorodin AQ21
measured at 50 psi, as a function of temperature. The solid lines
are curve ts. 1343
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system
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FIG. 14. Loss Youngs modulus of Goodrich Thorodin AQ21 measured
at 50 psi, as a function of temperature. The solid lines are curve
ts.
served to change with time by repeating measurements on
subsequent days , and the data used to generate Fig. 11 were
collected over a period of 3 days one temperature per day . One
should note that some authors16 use the loss factor instead of the
loss modulus to obtain the horizontal shift factors. Other authors6
even perform the horizontal shift rst on the modulus magnitude and
use the resulting factors to shift the loss factor data their
reasoning is that the magnitude is more accurately measured than
the loss factor . It has been found, for the 13 samples of various
materials measured to date in our system, that the best results are
obtained using the approach described in the preceding paragraph.
The reason for choosing the loss modulus to determine the initial
horizontal shift is that it does not require any vertical
correction: indeed, it can be observed in Fig. 9 where only a
horizontal shift has been applied , that the curve t to the loss
modulus data tends to a value of zero in the zero fre-
FIG. 16. Elastic Youngs modulus of Goodrich Thorodin AQ21 at 50
psi and at 30 C, resulting from horizontal shifts applied to the 10
C and 2 C data sets. The shift factors are the same as the ones
used for the loss modulus. The 10 C and 2 C data have been
corrected by multiplicative factors of T 30 30 /T 10 10 and T 30 30
/T 2 2 , respectively. The solid lines are curve ts.
quency limit i.e., the dc limit , as must be the case for any
viscoelastic material. All samples measured to date have required
the adjustment of two parameters for the elastic modulus, the
horizontal and the vertical shift factors. This being the case, the
absence of a priori knowledge of the horizontal factors would
result in nonunique curves, each corresponding to a different
combination of horizontal and vertical shifts.B. Goodrich Thorodin
AQ21
The following data were obtained on a Goodrich Thorodin AQ21
sample, with a density of 1050 kg/m3, and cut to a length of 26.0
cm. This material is a solid, nearly incompressible polyurethane,
with a Poissons ratio value close to 0.5. Consequently, its Youngs
modulus does not exhibit any
FIG. 15. Loss Youngs modulus of Goodrich Thorodin AQ21 at 50 psi
and at 30 C, resulting from horizontal shifts applied to the 10 C
and 2 C data sets. The 10 C and 2 C data have been corrected by
multiplicative factors of T 30 30 /T 10 10 and T 30 30 /T 2 2 ,
respectively. The solid line is a curve t. 1344 J. Acoust. Soc.
Am., Vol. 114, No. 3, September 2003
FIG. 17. Elastic Youngs modulus of Goodrich Thorodin AQ21 at 50
psi and at 30 C, resulting from both horizontal and vertical shifts
applied to the 10 C and 2 C data sets. The solid line is a curve t.
F. M. Guillot and D. H. Trivett: Youngs modulus measurement
system
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hydrostatic pressure dependence, as was veried experimentally by
measuring its elastic and loss moduli at 0.35, 1.04, and 2.07 MPa
50, 150, and 300 psi, respectively , and at three temperatures 30
C, 10 C, and 2 C . At each temperature, the modulus values for all
three pressures were identical within the measurement error bounds
. On the other hand, the large value of Poissons ratio is
responsible for substantial dispersive effects that need to be
taken into account when converting the wave speed to Youngs
modulus.1. Wave speed as a function of temperature
onstrated the capabilities of this new method for measuring both
the elastic and the loss components of Youngs modulus as a function
of pressure and temperature. The Goodrich Thorodin AQ21 data also
illustrated the fact that, for materials with a large Poissons
ratio and measured at high frequencies, dispersive effects might be
present and need to be taken into account. Finally, it was shown
that the temperature data collected with the system can be shifted
according to the timetemperature superposition principle, to
produce master curves over extended frequency
ranges.ACKNOWLEDGMENTS
Figure 12 shows the magnitude of the wave speed, measured using
both methods, at 30 C, 10 C, and 2 C. The dots represent the values
directly obtained from the measurements, without taking into
account the dispersion given by Eq. 28 . One can see that, even
though the sample used in this study has a smaller cross section
than that of the Rubatex R451N sample, its large Poissons ratio
makes it necessary to correct for the dispersion, in order to get
accurate values of the modulus. This is especially evident at high
frequencies. One should also note that, at a given frequency, the
effect is more pronounced at higher temperatures, where the wave
speed is lower. The subsequent values of Youngs modulus shown in
the next section are computed from the corrected wave-speed
values.2. Youngs modulus as a function of temperature
This work was supported by the Ofce of Naval Research, Code 334,
Stephen Schreppler. RBX Industries Inc. is gratefully acknowledged
for providing the Rubatex R451N material.D. M. Norris, Jr. and W.
C. Young, Complex-modulus measurement by longitudinal vibration
testing, Exp. Mech. 10, 9396 1970 . 2 D. M. Norris, Jr. and W. C.
Young, Longitudinal forced vibration of viscoelastic bars with end
mass, U.S. Army Cold Regions Research and Engineering Laboratory,
Hanover, NH 03775, Spec. Rep. 135 1970 . 3 W. M. Madigoski and G.
Lee, Automated dynamic Youngs modulus and loss factor measurements,
J. Acoust. Soc. Am. 66, 345349 1979 . 4 W. M. Madigoski and G. F.
Lee, Improved resonance technique for materials characterization,
J. Acoust. Soc. Am. 73, 1374 1377 1983 . 5 M. L. Williams, R. F.
Landel, and J. D. Ferry, The temperature dependence of relaxation
mechanisms in amorphous polymers and other glassforming liquids, J.
Am. Chem. Soc. 77, 37013706 1955 . 6 ANSI S2.22-1998, Resonance
method for measuring the dynamic mechanical properties of
viscoelastic materials, American National Standard Institute,
published through the Acoustical Society of America, New York, NY
1998 . 7 S. L. Garrett, Resonant acoustic determination of elastic
moduli, J. Acoust. Soc. Am. 88, 210221 1990 . 8 Q. Guo and D. A.
Brown, Determination of the dynamic elastic moduli and internal
friction using thin resonant bars, J. Acoust. Soc. Am. 108, 167174
2000 . 9 R. L. Willis, L. Wu, and Y. H. Berthelot, Determination of
the complex Young and shear dynamic moduli of viscoelastic
materials, J. Acoust. Soc. Am. 109, 611 621 2001 . 10 R. N. Capps,
Elastomeric materials for acoustical applications, Naval Research
Laboratory, Underwater Sound Reference Detachment, Orlando, FL
32856 1989 . 11 A. E. H. Love, A Treatise on the Mathematical
Theory of Elasticity Dover, New York, 1944 , p. 428. 12 K. F.
Graff, Wave Motion in Elastic Solids Dover, New York, 1991 , pp.
116 120. 13 J. D. Achenbach, Wave Propagation in Elastic Solids
North-Holland, Amsterdam, 1990 , pp. 242246. 14 J. W. S. Rayleigh,
The Theory of Sound Dover, New York, 1945 , Vol. 1, pp. 251252. 15
J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed. Wiley,
New York, 1980 , pp. 266 270. 16 P. H. Mott, C. M. Roland, and R.
D. Corsaro, Acoustic and dynamic mechanical properties of a
polyurethane rubber, J. Acoust. Soc. Am. 111, 17821790 2002 .1
Figures 13 and 14 show the elastic and loss modulus,
respectively, as a function of temperature. These data are used to
perform a WLF shift as explained in Sec. V A 2, resulting in the
curves depicted in Figs. 15, 16, and 17, at a reference temperature
of 30 C. The horizontal shift factors are a 10 9.3 and a 2 45.0,
and the vertical ones are 10 1.8 106 Pa and 10 2.2 106 Pa. Again,
the loss modulus is observed to be zero in the dc limit.VI.
CONCLUDING REMARKS
This article describes a new system for measuring the complex
Youngs modulus of compliant polymers. The system combines a new
approach to the resonant bar technique, where noncontact laser
measurements are performed on samples without end mass, with a
wave-speed technique that signicantly extends the frequency range
of the experimental investigation without requiring any sample
modication. The apparatus is designed for pressure measurements
ranging from 0 to 2.07 MPa 300 psi and for temperature measurements
ranging from 2 C to 50 C. Data obtained on Rubatex R451N and on
Goodrich Thorodin AQ21 dem-
J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
F. M. Guillot and D. H. Trivett: Youngs modulus measurement
system
1345
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