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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
0001-4966/2003/114(3)/1334/12/$19.00
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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
F. M. Guillot and D. H. Trivett: Youngs modulus measurement 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
F. M. Guillot and D. H. Trivett: Youngs modulus measurement 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
F. M. Guillot and D. H. Trivett: Youngs modulus measurement system
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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
F. M. Guillot and D. H. Trivett: Youngs modulus measurement 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
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