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A dynamic Young’s modulus measurement system for highlycompliant polymers

Francois M. Guillota) and D. H. TrivettGeorge 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 Young’s modulus of highly compliantelastomers at elevated hydrostatic pressures and as a function of temperature is presented. A samplecut in the shape of a bar is adhered to a piezoelectric ceramic shaker and mounted vertically insidea pressure vessel equipped with glass windows. Two independent measurement methods are thenused: a resonant technique, to obtain low-frequency data, and a wave propagation technique, toobtain 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 pressurevessel, and a set of two separate vibrometers monitors the speed of longitudinal waves propagatingin the sample, through windows located on the sides of the vessel. The apparatus is contained insidean environmental chamber for temperature control. Using this approach, Young’s modulus data canbe obtained at frequencies typically ranging from 100 Hz to 5 kHz, under hydrostatic pressuresranging from 0 to 2.07 MPa~300 psi!, and at temperatures between22 °C and 50 °C. Experimentalresults obtained on two commercial materials, Rubatex® R451N and Goodrich Thorodin™ AQ21,are presented. The effects of lateral inertia, resulting in dispersive wave propagation, are discussedand their impacts on Young’s modulus measurements are examined. ©2003 Acoustical Society ofAmerica. @DOI: 10.1121/1.1604121#

PACS numbers: 43.20.Ye, 43.20.Jr, 43.20.Ks@YHB#

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I. INTRODUCTION

Compliant polymers, such as voided polyurethanes,materials of interest because of their vibration isolation adamping properties. Knowledge of the dynamic elastic prerties of such elastomers, as a function of temperaturehydrostatic pressure, is essential to predict their performato validate numerical models, and to design sonar systeAs is the case for any homogeneous, isotropic material, otwo elastic constants~along with the density! are needed tocompletely characterize them. Among these elastic constathe bulk and Young’s moduli are the easiest to obtain expmentally, and also constitute the best pair, in terms of coputational error minimization, from which to compute othmoduli. In the Acoustic Material Laboratory of the GeorgInstitute of Technology, two independent experimental stems have been developed to perform dynamic measments of these two moduli. The present article describesdynamic Young’s modulus apparatus and presents dataillustrate a portion of the capabilities of the measuremsystem. These data were obtained on a commercial clocell foam neoprene, Rubatex® R451N, and on a commerciaelastomer, Goodrich Thorodin™ AQ21.

Several methods have been used to dynamically meathe Young’s modulus of viscoelastic materials. The mwidely employed approach is the resonant bar techniquetroduced by Norris and Young,1,2 where a sample in theshape of a bar is excited harmonically at one end and

a!Author to whom correspondence should be addressed. [email protected]

1334 J. Acoust. Soc. Am. 114 (3), September 2003 0001-4966/2003/1

ed 28 Sep 2010 to 202.117.81.6. Redistribution subject to ASA license or c

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ratio of the end accelerations is measured. At resonanceresponse of the sample can be used to obtain values ocomplex modulus. Madigoski and Lee3,4 utilized the tech-nique extensively to characterize a number of materials afunction of temperature, and used the time–temperatureperposition principle5 ~or WLF shift procedure, named afteWilliams, Landel, and Ferry! to obtain data over extendefrequency ranges. Recently, the resonance method hasadopted as a standard by the American National StandInstitute.6 Garrett7 used an electrodynamic transductioscheme to excite torsional, longitudinal, and flexural modin rods of circular or elliptical cross section. Both the longtudinal and flexural modes yield values for the Youngmodulus. Garrett measured the resonance frequenciethese modes to compute values of the magnitude ofmodulus. Guo and Brown8 extended this method to measuthe complex Young’s modulus by fitting the analytical soltion of the longitudinal wave equation to experimental daTheir approach allows the determination of the modulusfrequencies that are not restricted to the resonance freqcies of the modes. Measurements as a function of both tperature and hydrostatic pressure were reported by Wet al.,9 between 7 °C and 40 °C and over a 0- to 3.45-M~500-psi! range. Their approach consisted of measuring,laser vibrometry, the dynamic response of a rectangublock excited by a shaker, and matching the data with pdictions from a finite-element code in which the complelastic moduli were the adjustable parameters. Their samwas inside a pressure chamber submerged in a water battemperature control. Although this method allows one to otain data over relatively large frequency ranges withoutil:

14(3)/1334/12/$19.00 © 2003 Acoustical Society of America

opyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp

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need for time–temperature shifts, it suffers from occasioconvergence problems associated with the inversion offinite-element code.

The method described in this article uses an improversion of the resonant bar method and combines it witwave-speed measurement technique. These approawhich are implemented using laser Doppler vibrometehave the advantage of providingdirectly measuredcomplexwave-speed data~which are converted to Young’s moduludata, using the density of the material! over a much broadefrequency range than that typically afforded by the resonbar method alone. Therefore, fewer temperature data setneeded in order to generate master curves, reducingamount of uncertainty associated with multiple WLF shifFurthermore, since all the sample displacements are msured with noncontact vibrometers, no mass corrections~dueto the presence of accelerometers or electromagnetic trducers! need to be taken into account. Samples are plainside a stainless-steel vessel which is itself contained inan environmental chamber, providing pressure and tempture capabilities over a 0- to 2.07-MPa~300-psi! range and a22 °C to 50 °C range, respectively.

II. THEORETICAL FRAMEWORK

The theoretical background of the techniques emploin the system can be compiled from several articles,1,3,4,8

technical reports,2,10 and textbooks.11–14 However, becausesome of these references are not readily available andcause none of them provides, in one single convenient plall the information relevant to this work, we chose to presbelow the most important theoretical points of the measuments.

A. Resonance measurements

Consider the forced vibrations of a homogeneous badensityr and lengthL, with a constant cross section and nmass attached to its free end~Fig. 1!. One end of the bar isdriven with a harmonic displacementu0(t)5U0eivt, and theresulting axial displacement~relative to the driven end! at adistancex from the driven end isu(x,t)5U(x)eivt. Assum-ing a uniform, uniaxial stress distribution inside the bar, aneglecting the effects of lateral inertia, the equation of mtion can be written as

]sx

]x5r

]2

]t2 ~u1u0!, ~1!

wheresx is the uniaxial stress in the bar in the longitudin(x) direction. In this case, Hooke’s law states thatsx

5E* «x , whereE* is the complex Young’s modulus and«x

is the axial strain experienced by the material, defined«x[]u/]x. Substituting these two relations into Eq.~1!gives

E*]2u

]x2 5r]2

]t2 ~u1u0!, ~2!

which, in turn, leads to the following ordinary differentiaequation:

J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003 F. M. G


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where k is the complex wave number, defined ask[v(r/E* )1/25v/cb* , and cb* 5(E* /r)1/2 is the complexbar wave speed. The boundary conditions for the problemzero relative displacement atx50, i.e., u(0,t)50 or U(0)50, and zero stress atx5L, i.e., sx(L,t)50 ordU/dxat x5L50, which lead to the following solution for Eq~3!:

U~x!5U0@cos~kx!1tan~kL!sin~kx!21#. ~4!

Now that the equation of motion has been solved,resonances of the sample~bar! can be studied. To do this, leus consider the complex ratio of the free-end displacementhe driven end displacement,Q* , which, from Eq.~4!, canbe expressed as

Q* 5U~L !1U0

U05

1

cos~kL!. ~5!

On the other hand, using complex notation, Young’s moducan be written as

E* 5E81 iE95E@cos~d!1 i sin~d!#, ~6!

whereE8 is the elastic~or storage! modulus,E9 is the loss~or viscous! modulus,E is the magnitude, and tan (d) is theloss factor. Using these relations, Eq.~5! can then be ex-pressed as

FIG. 1. Forced vibrations of a polymer sample in the form of a bar of lenL with uniform cross section. The ceramic shaker and the vibratitransmitting metal rod correspond to the particular experimental setupin this work. The sample excitation is measured at the bottom surface ometal rod.

1335uillot and D. H. Trivett: Young’s modulus measurement system


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Q*5cos~kL!5cosFvLS r

E@cos~d!1 i sin~d!# D1/2G , ~7!

which can be separated into real and imaginary parts infollowing manner:

1

Q*5cosFvLS r

ED 1/2

cosS d

2D GcoshFvLS r

ED 1/2

sinS d

2D G1 i sinFvLS r

ED 1/2

cosS d

2D GsinhFvLS r

ED 1/2

sinS d

2D G .~8!

If the complex ratio is written asQ* 5Qeiu, then at reso-nance, whenu5(2n23)(p/2) andn51,2,3,... is the resonance number, the left-hand side of Eq.~8! is

1

Q*5 i

~21!n11

Q~n51,2,3,...!. ~9!

Combining Eqs.~8! and~9! leads to the following system otwo equations in two unknowns~E andd!:

cosFvLS r

ED 1/2

cosS d

2D GcoshFvLS r

ED 1/2

sinS d

2D G50,

~10!

sinFvLS r

ED 1/2

cosS d

2D GsinhFvLS r

ED 1/2

sinS d

2D G5~21!n11

Q.

One can easily show that the solutions to Eq.~10! are

d52 tan21F sinh21~1/Q!

~2n21!p

2G , ~11!

and

E5rF 4L f res

~2n21!cosS d

2D G2

. ~12!

One should again emphasize that Eqs.~11! and~12! are validat resonance only, thatf res is the resonance frequency, anthat Q is the amplitude of the displacement ratio. Theequations are in agreement with the theory presented in2, for the case of an end mass equal to zero. One canfrom the above analysis that measuringf res and Q at reso-nance, and using Eqs.~11! and~12! provides a direct methodfor determining the complex Young’s modulus of a samwith no mass attached to its free end. Once the moduamplitude and the loss factor are known, the elastic andmoduli can be computed using

1336 J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003


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B. Wave-speed measurements

Let us again consider the bar shown in Fig. 1. Insteada continuous harmonic signal, the shaker is excited witshort burst consisting of a gated sinusoidal signal. In tcase, monitoring the propagation of this burst allows onemeasure both the wave speed and the attenuation in theterial. The wave-speed magnitude,cb , is obtained in astraightforward manner by measuring the time necessarythe burst to travel a distanced along the sample. The signaattenuation~due to the material’s internal damping! is ob-tained from the change in signal amplitude over the sadistance. Assuming that the displacement perturbation cosponding to the traveling burst can be written as

u~x,t !5U0ei ~vt2kx!, for t1<t<t2 , ~14!

and recalling from the previous section that the compwave numberk is given by

k5vS r

E* D 1/2

, ~15!

then, using Eq.~6!, one obtains

k5vS r

E@cos~d!1 i sin~d!# D1/2

5vS r

ED 1/2FcosS d

2D2 i sinS d

2D G5k82 ia, ~16!

and

u~x,t !5U0e2ax•ei ~vt2k8x!, ~17!

wherea is the attenuation coefficient. Thus, as can be sfrom Eq. ~17!, a can be computed from the amplitude of thdisplacement signal measured at two locations spaced aby a distanced, according to

a51

dlnS U u~x,t !

u~x1d,t !U D . ~18!

The magnitude of Young’s modulus and its loss factor tand)can then be obtained from

E5rcb2, ~19!

and, from Eq.~16!

F. M. Guillot and D. H. Trivett: Young’s modulus measurement system


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d52 sin21S acb

v D . ~20!

Finally, the elastic and loss moduli are given by

E85E cos~d!5rcb2 cosF2 sin21S acb

v D G ,~21!

E95E sin~d!5rcb2 sinF2 sin21S acb

v D G .C. Dispersive effects

It has been assumed so far that the effects of latinertia in the sample were negligible. The purpose of tsection is to reexamine this assumption and to determunder which conditions these effects need to be takenaccount in the measurement of Young’s modulus. Letagain consider the sample in the form of a bar depictedFig. 1. Taking into account the lateral displacements resing from the Poisson coupling of the longitudinal wave, akeeping the assumption of uniaxial stress in the bar, the geral Hooke’s law can be written as

«x51

E*sx , and «y5«z52

h

E*sx , ~22!

whereh is the material’s Poisson’s ratio and the subscridenote the spatial directions~see Fig. 1!. If u(x,t), v(x,t),andw(x,t) denote the displacements in thex, y, andz direc-tions, respectively, then Eq.~22! can be used to compute thlateral displacements in terms of the longitudinal displament as

v52hy]u

]xand w52hz

]u

]x, ~23!

wherey and z are the coordinates of a point in the sampcross section~the center of the coordinate system coincidwith the center of the cross section!. Using Eq.~23!, it ispossible to formulate the potential and kinetic energies ofbar, and to apply Hamilton’s principle to derive the equatiof motion corresponding to longitudinal vibrations with laeral inertia effects, known as the Love equation.11 The detailsof this derivation are somewhat lengthy and can be foundRef. 12. Love’s equation of motion is

E*]2u

]x2 1rh2K2]4u

]x2]t2 5r]2u

]t2 , ~24!

whereK2 is the polar radius of gyration of the cross sectioConsidering a solution to Eq.~24! of the form

u~x,t !5Ueiv~ t2@x/c* # !, ~25!

wherec* is the velocity of the longitudinal waves, and sustituting ~25! into ~24!, yields

2E*v2

c* 2 1rh2K2v4

c* 2 52rv2. ~26!

Equation~26! can be rearranged to yield



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5cb* 5c* F11S hKv

c* D 2G1/2

. ~27!

Equation~27! indicates that elastic wave propagation in tbar is dispersive and relates the complex bar speedcb* to thecomplex longitudinal wave speedc* at a given frequency,v.The same result could have been obtained from the esolution, derived by Pochhammer, to the wave propagaproblem in a circular rod of radiusa, in the low-frequencyapproximation13 ~i.e., when the radiusa is much smaller thanthe wavelength, and when, consequently, the uniaxial stassumption holds!. This was also observed by Rayleigh,14

who derived Eq.~27! in the case of a circular rod. Forsquare cross section of side lengths, as is the case in oumeasurements,K25s2/6, and Eq.~27! becomes

S E*

r D 1/2

5cb* 5c* F111

6 S hsv

c* D 2G1/2

. ~28!

As can be seen from Eq.~28!, the lateral inertia effects in-crease with increasing values of Poisson’s ratio, frequenand sample cross section; they are also more substantilower wave speeds. Therefore, when the experimental cotions are such that these effects are significant, the measwave speedc* differs from the bar wave speedcb* , and Eq.~28! must be used to compute Young’s modulus. Practicait is assumed that the correction factor~i.e., the second terminside the brackets! is small enough so that Eq.~28! appliesto themagnitudeof the complex speeds. The experimentalneeds to be especially cautious when measuring Youmodulus on materials with large values of Poisson’s ratiohigh frequency. In such cases, or in cases where Poissratio is not known, it is advisable to minimize the laterdimensions of the sample, in order to minimize the dispsive effects.

III. SAMPLE PREPARATION AND EXPERIMENTALAPPARATUS DESCRIPTION

A. Sample preparation

Samples are cut in the shape of a bar with a squcross-sectional area. The maximum sample length is limby the pressure vessel dimensions and is about 27 cmexplained above, it is desirable to use a sample with a mmal cross section. In the case of Rubatex® R451N, the ma-terial comes in the form of 6.35-nm~0.25-in.!-thick sheetsthat were cut in 0.25-in.-wide strips using a mat cutter. TGoodrich Thorodin™ AQ21 sample used in this study wcut into a bar with a 5- by 5-mm cross-sectional area. Ttop and bottom ends of each sample were then cut witrazor blade in order to get plane and smooth surfaces, ware necessary for a good bond between the sample andmetal of the excitation-transmitting rod shown in Fig. 1. Tend surfaces also need to be perpendicular to the axis osample so that the latter can be mounted in a perfectlytical position inside the apparatus, and therefore be excitea primarily longitudinal mode, minimizing spurious flexurmotions.

For materials whose color and texture make them golight scatterers, no further preparation is needed. Howe



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other materials, such as Rubatex® R451N ~which is blackand has a matte finish!, and Goodrich Thorodin™ AQ21~which is yellow and somewhat translucent!, do not scattersufficient amounts of light and therefore require a surfatreatment in order to be measured with a laser vibrometerobvious requirement is that the surface treatment not chathe elastic properties of the sample. With this in mind,following treatments have been used. In the caseRubatex® R451N, for the measurement of the motion of tbottom end of the sample, a small piece of metallized t~about the same dimensions as the cross-sectional are! isglued ~using a cyanoacrylate adhesive! to the end of thesample. This piece of tape does not constrain the longitudmotion, and its mass is negligible with respect to that ofsample. For measurement of the motion of the lateral sidethe sample~resulting from the Poisson-coupled longitudinmotion!, a layer of talc powder is deposited on these sfaces. The load resulting from the presence of the talcnegligible and its powdery nature allows the sample to min the lateral direction without any constraint. In the caseGoodrich Thorodin™ AQ21, a thin layer of white correctiofluid ~‘‘white-out’’ ! was applied at the bottom end of thsample. The same treatment as for the Rubatex® R451N ma-terial ~metallized tape! could have been used, but the movation for trying the correction fluid was to compare its peformance to that of the tape. The white fluid was found toa slightly better light scatterer than the tape; however, bare perfectly adequate for the resonance measurementsthe side measurements on Goodrich Thorodin™ AQ21thin layer of gold-colored ink was used, which proved toa better reflector than the talc powder. The absence ofverse effects from the tape, the powder, and the ink onsample motion was verified experimentally by measuringvibrations of samples with and without these surface trements: the measured displacements were identical incases.

As a final note specific to Rubatex® R451N, before cut-ting a sample, the actual first preparatory step consistefollowing the manufacturer’s recommendation of placing tsheet of foam in an oven, set at 71 °C~160 °F!, for 24 h, inorder to completely cure the material and insure stable perties. It was found, however, that even after 4 days oftreatment, measured properties had a tendency to chwith time. Therefore, the data that are presented beshould be considered a ‘‘snapshot’’ of the Rubatex® R451Nmaterial constants.

B. Experimental apparatus

The experimental apparatus is shown in Fig. 2. Tsample is glued to a 2.5-cm-long vibration-transmittimetal rod~with approximately the same cross-sectional aas the sample!, which is itself glued to a shaker, usingcyanoacrylate adhesive. The shaker is made of ten piezotric ceramic discs@EC64 from EDO Corporation: 1.78 cm~0.700-in.! outer diameter30.22-cm~0.085-in.!-thick# plusone depolarized and unelectroded disk at each each enthe stack, for shielding purposes; the disks are glued togeusing an epoxy resin. The shaker, in turn, is epoxied tthreaded mounting piece that attaches to the top of the p



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sure vessel. To prevent delaminations caused by temperacycling, the vibration-transmitting rod and the mountinpiece are made of Invar, which has a very low coefficientthermal expansion, comparable to that of the ceramic usethe shaker.

The pressure vessel@inside dimensions: 38.72 cm(15.25 in.)L310.2 cm (4 in.)W37.6 cm (3 in.)D] is madeof 303 stainless steel and it is designed to be used up toMPa ~500 psi!. It features three 1.9-cm~0.75 in.!-thick glasswindows for optical access to the sample: one round wind@Ø 3.5 cm ~1.375 in.!# located at the bottom of thevessel and two long [email protected] cm (11.35 in.)L33.4 cm (1.35 in.)W# located along its sides. They arsealed inside the vessel with a 1.59-mm~0.0625 in.!-thickAramid gasket used in conjunction with a silicone RTV seant. An RTD sensor~Sensor Scientific PT1000! is used tomeasure the temperature inside the vessel. Pressure is crwith a compressed air cylinder connected to the vessel~notshown in Fig. 2! and both pressure and temperaturemonitored using a Heise PM Indicator gauge.

Outside the pressure vessel, three laser Doppler vibmeter sensor heads~Polytec CLV-800-ff!, capable of measuring out-of-plane~normal! vibrations, are mounted on postioning slides~two sets of 5-in. and 15-in. Velmex Bislidefor the side lasers; two 25-mm and one 110-mm NatioAperture Mini Stages for the bottom one!. The entire system~vessel, lasers, and slides! is placed inside an environmentachamber~Thermotron S-27C! for temperature control, andneeds to be operated from outside that chamber. Forpurpose, the slides are remotely driven by two controll~Velmex VP 9000 and National Aperture Servo 1000!, whichallow precise positioning of the vibrometers. Also, a minture video camera connected to a television monitormounted next to each sensor head; it is focused on thewhere the laser beam illuminates the surface on the samand provides a direct visual check of the beam positionwell as a view of the sample from three different anglesneutral density filter is placed in front of each cameraorder to reduce the light intensity and to improve the laspot resolution on the monitor. Two small flashlights are usto enhance the image of the sample. Because of the saperture afforded by the bottom window and lack of spacethat area of the setup, the corresponding camera is positioat a right angle and a small mirror is employed to obtainview of the bottom of the sample.

The shaker is excited by two types of signal, dependon which measurement is performed. For the resonance msurements, a continuous signal with varying frequencyused, provided by a lock-in amplifier~Stanford ResearchSystems model SR850!. For the wave-speed measuremena burst signal composed of six cycles at a given frequencused, provided by a function generator~Wavetek model 80!.In both cases, signals are amplified by a power ampli~Krohn-Hite model 7500! before being sent to the shakeSurface motion signals measured by the sensor headselectronically processed by the vibrometer controller: CL1000~bottom laser! or CLV-2000~side lasers!. These signalsare filtered~Krohn-Hite filter model 3382! before being dis-played on an oscilloscope~Tektronix model TDS 3014!. In



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FIG. 2. Picture of the experimental setup.~a! General view.~b! Close-up.

1339J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003 F. M. Guillot and D. H. Trivett: Young’s modulus measurement system

ed 28 Sep 2010 to 202.117.81.6. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp

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the case of the resonance measurements, the velocity siare also input to the lock-in amplifier. Both the oscilloscoand the lock-in amplifier are equipped with a GPIB bus ulized to transfer velocity signals to a notebook computerdata processing.

IV. MEASUREMENT PROCEDURE AND DATAPROCESSING

A. Resonance measurements

Based on the theory described in Sec. II A, the produre to implement the resonance measurement techniqas follows. These measurements are obtained using thetom laser only. The lock-in amplifier, in the frequency swemode, excites the shaker with a continuous sine wave whfrequency ranges from a lower limit to an upper limit. Thamplifier also records the amplitude and phase of the sigoutput by the laser vibrometer, the latter being employedmeasure both the velocity of the driven end~the metal rodwithout the sample adhered to it! and the velocity of the freeend. These signals, digitized with 401 points by the lockamplifier, are combined numerically inMATLAB to produceamplitude and phase plots of the sample frequency respoFirst, a ‘‘global’’ sweep is performed over an extended frquency range in order to identify all the measurable renances of the sample. The resulting plots are shown in Fiover a 50–2400-Hz range, which was the typical range ufor the measurements on the Rubatex® R451N sample. Eachresonance frequency is approximately determined from

FIG. 3. Resonance plots obtained on a Rubatex® R451N sample at 30 °Cand ambient pressure, displaying four measurable resonances.~a! Amplitudeplot of the ratio of the velocity of the free end to that of the driven end.~b!Corresponding phase plots.



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phase plot at points where the phasef is equal to (2n23)390. A second sweep is then performed over a narrowquency range centered around each of these frequenciesthe resulting plots are used to obtain more accurate valuef and Q at resonance. The narrow range has a span ofHz, which, combined with the 401 points of digitizationallows for the determination of the resonance frequency wa resolution of 0.25 Hz. These data, along with the lenand density of the sample, are then input into Eq.~13! tocompute the elastic and loss moduli. The maximum measment errors associated with these two quantities can bemated as follows. The 0.25-Hz resolution results in a fquency uncertainty of 0.5% at 50 Hz; assuming a60.5-mmerror on the sample length measurement results in a 0length uncertainty for a 15-cm-long sample~the smallestsample used in the system!. The systematic errors from thlaser vibrometer decoder module and from the lock-in aplifier do not affect the measurement ofQ, the amplituderatio, because it is a relative measurement. Therefore, therrors are not taken into account. This analysis resultstotal uncertainty of 1.6% for both the elastic and loss modOne should note that this does not include the uncertarelated to the density, which affects all measurements~elasticand loss moduli, from both resonance and wave-speed tniques! in an identical manner.

B. Wave-speed measurements

In the method described above, internal damping in mterials always limits the number of identifiable resonancand, therefore, sets an upper frequency limit on the metfor a given sample. One way to extend measurementshigher frequency regions is to reduce the length ofsample, which, as Eq.~12! reveals, would shift the value oresonance frequencies higher. In theory, successive cutsbe made to produce increasingly shorter samples to covefrequency range of interest, as long as the sample maintan appropriate dimension ratio~i.e., L@ lateral dimensions!.However, this practice possesses two disadvantages. Firthe sample under study is inhomogeneous, then cuttingsample into smaller lengths effectively results in measurindifferent sample after each cut, introducing additional msurement errors. Second, each sample shortening reqopening the pressure vessel and the environmental chamand the subsequent readjusting of the necessary pressurtemperature conditions, rendering the process quite time csuming. A better way to extend the experimental Younmodulus frequency range is to measure the wave speedthe attenuation of longitudinal waves propagating insidesample. In our apparatus, this technique is implemented wthe two side lasers measuring the lateral component olongitudinal excitation propagating along the sample. A shburst is used to excite the shaker, which excites a longitunal wave in the sample and the resulting lateral motion~dueto the Poisson’s ratio effect! is then recorded at two differenlocations along the length of the sample. Two laser vibromters located on opposite sides of the sample are needeorder to insure that only the lateral motion~which is a sym-metrical motion! produced by the longitudinal wave is mesured. Any other spurious vibrations~due to mounting im-



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perfections or outside vibrations! that the sample mayexperience create bending motions, which manifest thselves in the form of antisymmetrical displacements. Thefore, these spurious vibrations can be systematically elinated by adding the signals from each vibrometer.

The amount of attenuation~which, above the glass transition temperature, increases with frequency! in the materialsets both the lower- and the upper-frequency limits ofwave-speed method. Indeed, it is imperative that the lavibrometers measure only the direct signal produced byshaker without any superimposed reflections from the eof the sample. Thus, sufficient attenuation needs topresent in order to eliminate these reflections before treach the measurement locations; this determines the lofrequency limit~for a given sample length!. Conversely, theupper limit is attained when attenuation is so large asprevent accurate measurement of the signal amplitude alocation furthest away from the shaker. Thus, the actualquency range of the measurements described in this ardepends only on the attenuation of the particular mateunder study, and will therefore vary from one material to tother. For the Rubatex® R451N sample, combining the resonance and the wave-speed techniques, the range was foube approximately 200–8000 Hz. These values, of coulike the attenuation, also depend on temperature and psure.

Practically, the wave-speed technique is implementedfollows. The function generator excites the shaker withburst signal composed of 6 cycles at a given frequency, wa 1-s repetition rate. For example, frequencies ranging f4500 to 8000 Hz in steps of 500 Hz were used forRubatex® R451N sample. Both laser sensor heads are ptioned at the same height, about 2 cm below the shaker;constitutes the top location. At each frequency, signals freach vibrometer are displayed on the oscilloscope, whdigitizes them using 10 000 points. Sixty-four averagestaken, and the signals from each side are added; the resusignal is transferred to a computer for further processiThis is repeated at the distant location, 10 cm below theone. Typical signals measured at both locations and usedwave speed and attenuation computation are shown in FiIn MATLAB , signals are windowed and zero padded befbeing subjected to a fast Fourier transform. The centerquency of each Fourier-transformed signal is determinand, at that frequency, the amplitudeA and the phasew of theratio of the top location signal to that of the distant locatiare computed. These yield the attenuation and the wspeed, respectively, according to

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These data, along with the density of the sample, are tinput into Eq.~21! to compute the elastic and loss moduThe uncertainties associated with these measurements asessed as follows. Assuming a60.5-mm error on distancemeasurement results in a 0.5% uncertainty ford; the system-



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V. EXPERIMENTAL RESULTS

A. Rubatex ® R451N

The following data were obtained on a Rubatex® R451Nneoprene sample, with a density of 579 kg/m3, and cut tlength of 23.3 cm. These two values were measured at rotemperature and ambient pressure. The subsequent disional changes resulting from varying the hydrostatic prsure and the temperature were assessed using one of thelasers and the positioning slide controller: the laser beamfocused at the bottom of the sample whose change of ption was recorded for the various pressures and temperaused. This yielded the length of the sample, whose valueused to estimate the density of the material. One should nthat, for samples that have been measured in the bulk molus system of the Acoustic Material Laboratory, the depedence of density on pressure and temperature is knoHowever, Rubatex® R451N is too soft to be measured in thsystem. The data shown below are not corrected for dissive effects, as Poisson’s ratio of Rubatex® R451N~which ison the order of 0.25! is small enough to render the corretions negligible.

1. Young’s modulus as a function of hydrostaticpressure

Figures 5 and 6 display resonance and wave-speed msurement results in the form of the real and imaginary paof the modulus, as a function of pressure. Data were obtaat 30 °C, and 0~i.e., ambient!, 69 and 138 kPa~0, 10, and 20psi!. At least 30 min were allowed to elapse after each npressure setting, and the pressure was always increa

FIG. 4. Velocity signals measured at 5500 Hz by the side laser vibromeat two locations spaced 10 cm apart on the surface of a Rubatex® R451Nsample~30 °C and ambient pressure!. The relative phase and amplitude othese signals are used to compute, respectively, the speed and attenualongitudinal waves propagating inside the sample.



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FIG. 6. Loss Young’s modulus of Rubatex® R451N measured at 30 °C, asfunction of hydrostatic pressure. The solid lines are curve fits.

FIG. 7. Elastic Young’s modulus of Rubatex® R451N measured at ambienpressure, as a function of temperature. The solid lines are curve fits.

FIG. 5. Elastic Young’s modulus of Rubatex® R451N measured at 30 °C, aa function of hydrostatic pressure. The solid lines are curve fits.



These plots clearly show that the two techniques provresults that are compatible with each other, and thatwave-speed approach allows one to substantially extendmeasurement frequency range beyond what the clasresonant bar method can offer, without having to cutsample.

Figure 5 indicates that the material gets softer withcreasing hydrostatic pressure, as the walls of the closedbuckle and become more compliant. The loss modulus, hever, is relatively unaffected by pressure changes, as shin Fig. 6.

2. Young’s modulus as a function of temperature

Figures 7 and 8 display resonance and wave-speed msurement results in the form of the real and imaginary paof the modulus, respectively, as a function of temperatu

FIG. 8. Loss Young’s modulus of Rubatex® R451N measured at ambienpressure, as a function of temperature. The solid lines are curve fits.

FIG. 9. Loss Young’s modulus of Rubatex® R451N at ambient pressure anat 30 °C, resulting from horizontal shifts applied to the 20 °C and 10 °C dsets. The 20 °C and 10 °C data have been corrected by multiplicative faof T30r30 /T20r20 andT30r30 /T10r10 , respectively. The solid line is a curvefit.



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Data were obtained at ambient pressure, and at 30 °C, 20and 10 °C. The sample was allowed to equilibrate overnibetween each temperature. Figures 7 and 8 indicate thatthe elastic and the loss moduli increase significantly wdecreasing temperature, as expected above the glass ttion temperature. Only three resonances were measurabthe two lowest temperatures, due to the increased loss.

Finally, the time–temperature superposition principcan be applied to these data in the following manner. Fthe loss modulus temperature curves are shifted horizonto get the best possible fit to a smooth curve, as illustrateFig. 9. In that figure, the reference temperature is 30 °C,the horizontal shift factors applied to the 20 °C and 10data sets area2054.3 anda10519.0, respectively. These lastwo data sets have been corrected by multiplicative factor

FIG. 10. Elastic Young’s modulus of Rubatex® R451N at ambient pressurand at 30 °C, resulting from horizontal shifts applied to the 20 °C and 10data sets. The shift factors are the same as the ones used for the losslus. The 20 °C and 10 °C data have been corrected by multiplicative facof T30r30 /T20r20 andT30r30 /T10r10 , respectively. The solid lines are curvfits.

FIG. 11. Elastic Young’s modulus of Rubatex® R451N at ambient pressurand at 30 °C, resulting from both horizontal and vertical shifts applied to20 °C and 10 °C data sets. The solid line is a curve fit.



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T30r30/T20r20 andT30r30/T10r10, respectively~where theTsare temperatures~in Kelvin! and ther’s are the corresponding densities!, as specified by Ferry.15 The shift factors arethen used to shift the elastic modulus horizontally by tsame amount, producing the curves shown in Fig. 10. Infigure, the 20 °C and 10 °C data have also been correctemultiplicative factors ofT30r30/T20r20 and T30r30/T10r10,respectively. Finally, a vertical shift is applied to the temperature segments of Fig. 10, in order to obtain the smocurve displayed in Fig. 11. The additive vertical shift factoare n20525.03106 Pa and n105212.43106 Pa for the20 °C and 10 °C data sets, respectively. The physical meing of these vertical shifts may be related to the static molus dependence on temperature. However, in the case oRubatex® R451N sample, their significance is not entireclear, as the elastic modulus of this material has been

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FIG. 12. Wave speed of Goodrich Thorodin™ AQ21 measured at 50 psa function of temperature. The dots represent wave-speed values uncorrfor dispersion. In the case of the resonance data, the correction is negligand the white dots~uncorrected values! appear in the center of the blackfilled markers~corrected values!. In the case of the wave-speed data, tcorrection is more significant, and the black dots are distinct from the blafilled markers.

FIG. 13. Elastic Young’s modulus of Goodrich Thorodin™ AQ21 measuat 50 psi, as a function of temperature. The solid lines are curve fits.



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served to change with time~by repeating measurements osubsequent days!, and the data used to generate Fig. 11 wcollected over a period of 3 days~one temperature per day!.

One should note that some authors16 use the loss factoinstead of the loss modulus to obtain the horizontal sfactors. Other authors6 even perform the horizontal shift firson the modulus magnitude and use the resulting factorshift the loss factor data~their reasoning is that the magntude is more accurately measured than the loss factor!. It hasbeen found, for the 13 samples of various materials msured to date in our system, that the best results are obtausing the approach described in the preceding paragrThe reason for choosing the loss modulus to determineinitial horizontal shift is that it does not require any verticcorrection: indeed, it can be observed in Fig. 9~where only ahorizontal shift has been applied!, that the curve fit to theloss modulus data tends to a value of zero in the zero

FIG. 14. Loss Young’s modulus of Goodrich Thorodin™ AQ21 measure50 psi, as a function of temperature. The solid lines are curve fits.

FIG. 15. Loss Young’s modulus of Goodrich Thorodin™ AQ21 at 50and at 30 °C, resulting from horizontal shifts applied to the 10 °C and22 °Cdata sets. The 10 °C and22 °C data have been corrected by multiplicativfactors ofT30r30 /T10r10 andT30r30 /T22r22 , respectively. The solid line isa curve fit.



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quency limit ~i.e., the dc limit!, as must be the case for anviscoelastic material. All samples measured to date havequired the adjustment oftwo parameters for the elastimodulus, the horizontal and the vertical shift factors. Tbeing the case, the absence ofa priori knowledge of thehorizontal factors would result in nonunique curves, eacorresponding to a different combination of horizontal avertical shifts.

B. Goodrich Thorodin™ AQ21

The following data were obtained on a Goodrich Thordin™ AQ21 sample, with a density of 1050 kg/m3, and cto a length of 26.0 cm. This material is a solid, nearly incopressible polyurethane, with a Poisson’s ratio value close0.5. Consequently, its Young’s modulus does not exhibit a

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FIG. 16. Elastic Young’s modulus of Goodrich Thorodin™ AQ21 at 50 pand at 30 °C, resulting from horizontal shifts applied to the 10 °C and22 °Cdata sets. The shift factors are the same as the ones used for the losslus. The 10 °C and22 °C data have been corrected by multiplicative factoof T30r30 /T10r10 and T30r30 /T22r22 , respectively. The solid lines arecurve fits.

FIG. 17. Elastic Young’s modulus of Goodrich Thorodin™ AQ21 at 50 pand at 30 °C, resulting from both horizontal and vertical shifts applied to10 °C and22 °C data sets. The solid line is a curve fit.



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hydrostatic pressure dependence, as was verified experitally by measuring its elastic and loss moduli at 0.35, 1.and 2.07 MPa~50, 150, and 300 psi, respectively!, and atthree temperatures~30 °C, 10 °C, and22 °C!. At each tem-perature, the modulus values for all three pressures widentical ~within the measurement error bounds!. On theother hand, the large value of Poisson’s ratio is responsfor substantial dispersive effects that need to be takenaccount when converting the wave speed to Young’s molus.

1. Wave speed as a function of temperature

Figure 12 shows the magnitude of the wave speed, msured using both methods, at 30 °C, 10 °C, and22 °C. Thedots represent the values directly obtained from the measments, without taking into account the dispersion givenEq. ~28!. One can see that, even though the sample usethis study has a smaller cross section than that ofRubatex® R451N sample, its large Poisson’s ratio makesnecessary to correct for the dispersion, in order to get acrate values of the modulus. This is especially evident at hfrequencies. One should also note that, at a given frequethe effect is more pronounced at higher temperatures, wthe wave speed is lower. The subsequent values of Youmodulus shown in the next section are computed fromcorrected wave-speed values.

2. Young’s modulus as a function of temperature

Figures 13 and 14 show the elastic and loss modurespectively, as a function of temperature. These dataused to perform a WLF shift as explained in Sec. V Aresulting in the curves depicted in Figs. 15, 16, and 17,reference temperature of 30 °C. The horizontal shift factare a1059.3 anda22545.0, and the vertical ones aren10

511.83106 Pa andn10512.23106 Pa. Again, the lossmodulus is observed to be zero in the dc limit.

VI. CONCLUDING REMARKS

This article describes a new system for measuringcomplex Young’s modulus of compliant polymers. The sytem combines a new approach to the resonant bar technwhere noncontact laser measurements are performedsamples without end mass, with a wave-speed techniquesignificantly extends the frequency range of the experimeinvestigation without requiring any sample modification. Tapparatus is designed for pressure measurements ranfrom 0 to 2.07 MPa~300 psi! and for temperature measurments ranging from22 °C to 50 °C. Data obtained oRubatex® R451N and on Goodrich Thorodin™ AQ21 dem



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onstrated the capabilities of this new method for measurboth the elastic and the loss components of Young’s moduas a function of pressure and temperature. The GoodThorodin™ AQ21 data also illustrated the fact that, for mterials with a large Poisson’s ratio and measured at highquencies, dispersive effects might be present and need ttaken into account. Finally, it was shown that the temperatdata collected with the system can be shifted according totime–temperature superposition principle, to produce macurves over extended frequency ranges.

ACKNOWLEDGMENTS

This work was supported by the Office of Naval Rsearch, Code 334, Stephen Schreppler. RBX Industriesis gratefully acknowledged for providing the Rubatex®

R451N material.

1D. M. Norris, Jr. and W. C. Young, ‘‘Complex-modulus measurementlongitudinal vibration testing,’’ Exp. Mech.10, 93–96~1970!.

2D. M. Norris, Jr. and W. C. Young, ‘‘Longitudinal forced vibration oviscoelastic bars with end mass,’’ U.S. Army Cold Regions ResearchEngineering Laboratory, Hanover, NH 03775, Spec. Rep. 135~1970!.

3W. M. Madigoski and G. Lee, ‘‘Automated dynamic Young’s modulus aloss factor measurements,’’ J. Acoust. Soc. Am.66, 345–349~1979!.

4W. M. Madigoski and G. F. Lee, ‘‘Improved resonance technique for mterials characterization,’’ J. Acoust. Soc. Am.73, 1374–1377~1983!.

5M. L. Williams, R. F. Landel, and J. D. Ferry, ‘‘The temperature depedence of relaxation mechanisms in amorphous polymers and other gforming liquids,’’ J. Am. Chem. Soc.77, 3701–3706~1955!.

6ANSI S2.22-1998, ‘‘Resonance method for measuring the dynamicchanical properties of viscoelastic materials,’’American National StandInstitute, published through the Acoustical Society of America, New YoNY ~1998!.

7S. L. Garrett, ‘‘Resonant acoustic determination of elastic moduli,’’Acoust. Soc. Am.88, 210–221~1990!.

8Q. Guo and D. A. Brown, ‘‘Determination of the dynamic elastic moduand internal friction using thin resonant bars,’’ J. Acoust. Soc. Am.108,167–174~2000!.

9R. L. Willis, L. Wu, and Y. H. Berthelot, ‘‘Determination of the compleYoung and shear dynamic moduli of viscoelastic materials,’’ J. AcouSoc. Am.109, 611–621~2001!.

10R. N. Capps, ‘‘Elastomeric materials for acoustical applications,’’ NaResearch Laboratory, Underwater Sound Reference Detachment, OrlFL 32856~1989!.

11A. E. H. Love,A Treatise on the Mathematical Theory of Elasticity~Do-ver, New York, 1944!, p. 428.

12K. F. Graff, Wave Motion in Elastic Solids~Dover, New York, 1991!, pp.116–120.

13J. D. Achenbach,Wave Propagation in Elastic Solids~North-Holland,Amsterdam, 1990!, pp. 242–246.

14J. W. S. Rayleigh,The Theory of Sound~Dover, New York, 1945!, Vol. 1,pp. 251–252.

15J. D. Ferry,Viscoelastic Properties of Polymers, 3rd ed. ~Wiley, NewYork, 1980!, pp. 266–270.

16P. H. Mott, C. M. Roland, and R. D. Corsaro, ‘‘Acoustic and dynammechanical properties of a polyurethane rubber,’’ J. Acoust. Soc. Am.111,1782–1790~2002!.

