REVIEW ARTICLE Viscoelastic measurement techniquessilver.neep.wisc.edu/~lakes/VErsi04.pdf · REVIEW...

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 75, NUMBER 4 APRIL 2004

REVIEW ARTICLE

Viscoelastic measurement techniquesR. S. Lakesa)

Department of Engineering Physics, Engineering Mechanics Program, University of Wisconsin-Madison,147 Engineering Research Building, 1500 Engineering Drive, Madison, Wisconsin 53706-1687

~Received 23 July 2003; accepted 22 December 2003; published 8 March 2004!

Methods for measuring viscoelastic properties of solids are reviewed. The nature of viscoelasticresponse is first presented. This is followed by a survey of time and frequency-domainconsiderations as they apply to mechanical measurements. Subresonant, resonant, and wavemethods are discussed, with applications. ©2004 American Institute of Physics.@DOI: 10.1063/1.1651639#

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I. INTRODUCTION

A. Principles of viscoelasticity

1. Measures of internal friction

Viscoelasticmaterials have a relationship between strand strain which depends on time or frequency.Anelasticsolids represent a subset of viscoelastic materials: they ha unique equilibrium configuration and ultimately recovfully after removal of a transient load.Internal friction refersto the dissipative response of a material when subjectedsinusoidal deformation. All materials exhibit viscoelastic rsponse; elasticity or spring-like behavior does not existreal materials but is an approximate description of materfor which the viscoelastic effects are small enough to igno

Viscoelasticity is of interest in the context of understaning physical processes such as molecular mobilitypolymers,1 and of phase transformations, motion of defecalloying atoms in crystalline solids.2 Viscoelasticity3 is alsoused in the design of materials and devices for a varietypurposes including earplugs, vibration abatement, reducof mechanical shock, instrument mounts, and control ofbound and rolling resistance.

The loss angled is the phase angle between stress astrain during sinusoidal deformation in time. The loss anor the loss tangenttand as a measure of damping or internfriction in a linear material is advantageous in that itclearly defined in terms of observable quantities. Tand de-pends on frequency, and it is customary to plot it on a lorithmic frequency scale. A factor ten ratio in frequencyreferred to as a decade. Tand is also the ratio of the imaginary partG9 to the real partG8 of the complex modulusG* [G81 iG9. The quality factor Q associated with thewidth of resonant peaks@see Eq.~3!# is given for smalld byQ21'tand. If vibration in a resonating system is alloweddecay with time due to material viscoelasticity after remoof the excitation, one may define thelog decrementL interms of amplitudesA1 and A2 of successive cycles asL

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5ln(A1 /A2). For smalld, L'p tand. The specific dampingcapacityC refers to the ratio1,4 of energyDW dissipated fora full cycle to the maximum elastic energyW stored in thematerial. For a linear material,C52p tand; but C is welldefined even for nonlinear materials, consequently it is pferred by some authors. The material stiffness or moduluviscoelastic materials depends on frequency. This is knoas dispersion.

2. Transient properties

Transient properties are defined in terms of responsestep input in time.Creeprefers to the time-dependent straresponse to a step stress. The creep compliance is the sdivided by the constant stress, denotedJE(t) for a tension/compression experiment,JB(t) for a bulk ~volumetric! ex-periment, andJG(t) for a shear experiment.Stress relaxationrefers to the time-dependent stress response to a step sThe relaxation modulus is the stress divided by the consstrain. G(t) is the relaxation modulus in shear,E(t) intension/compression, andB(t) in bulk ~volumetric! deforma-tion. Creep and stress relaxation experiments can be dontension, torsion, bending, bulk~volumetric!, or other defor-mation modes. The time dependence of each of these wigeneral differ. These results are usually presented versustime. The material is linear whenJ(t) is independent ofstress andG(t) is independent of strain, otherwise it is nolinear. Linearity of a material subjected to a series of cretests at different stress can be displayed visually by plottstress versus strain at a given creep time. Such a plocalled an isochronal; it is a straight line if the materiallinearly viscoelastic. If the transient properties depend ontime after formation or transformation of the material as was the timet after load application, the material is saidexhibit physical aging.

3. Significance. Relation to other relaxationprocesses

Viscoelasticity in materials is studied since~i! viscoelas-tic solids are used to absorb vibration,~ii ! viscoelastic effects

© 2004 American Institute of Physics

IP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp

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798 Rev. Sci. Instrum., Vol. 75, No. 4, April 2004 R. S. Lakes

are linked to physical processes such as phase transfotions, diffusion, or motion of dislocations, vacancies aother defects, so that viscoelastic measurements are useprobe into the physics of these processes, and~iii ! viscoelas-ticity is relevant in phenomena such as ball rebound, rollresistance, sag, droop, and spin instability. Also, viscoelarelaxation in the mechanical properties may be relatedrelaxation processes in other material properties such asdielectric permittivity, piezoelectric properties, magneproperties, or thermal expansion.

B. Overview of mechanical test principles

1. Load application and deformation measurement

The simplest method of load application is to use a dweight. This is appropriate for creep tests of moderatelong duration. Dynamic response may be achieved by geating force or torque via electromagnetic interaction betwa coil carrying a controlled electric current and a permanmagnet. Larger forces and torques may be achieved viavocontrolled hydraulic systems. Such systems, typicallypable of large forces, 40 to 400 kN, are commercially avable.

2. Temperature

Since viscoelastic properties depend on temperatsometimes sensitively, environmental control of temperatis usually used. Polymers in the vicinity of the glass trantion and crystalline materials in the vicinity of a phase traformation are very sensitive to temperature, so it is usuacontrol temperature within 0.1 °C or better. In teststension/compression, a length change can arise from creefrom thermal expansion. Therefore excellent temperatcontrol is required in tensile creep or relaxation studiThermal expansion of an isotropic material has no sheatorsional component, therefore torsion tests are less deming of accurate temperature control than are tension or cpression tests. Many polymers as well as concrete andlogical materials are sensitive to hydration, therefohumidity or saturation is monitored or controlled in sutests. Testing at elevated temperatures presents a variechallenges. Since most transducers for load and deformado not survive high-temperature conditions, it is typicalisolate the heated specimen from other parts of the apparExamples are given in Sec. III B.

3. Interpretation

Interpretation of tests is based on use of an analytsolution for the deformation field in the specimen geomein question. If a specimen is gripped at the ends, the deof the grip stress may not be well known, therefore tensitorsion, or bending tests on gripped specimens are usudone on long, slender specimens with length at leasttimes the diameter. The reason is that the nonuniform stat the ends decays over a distance of one or two diameand has minimal influence on the results for a long specimThe slenderness requirement is more severe if the specis anisotropic.


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C. Outline of the review

Within our scope we will consider methods for measuing viscoelastic properties of solid materials. Section II dewith time-domain transient properties: creep and stress reation. Section III treats material response to sinusoidal inunder subresonant~quasistatic! conditions in which inertialterms are negligible. Methods for phase measurementconsidered, as are several kinds of instrument. Sectiondeals with sinusoidal input measurement near the resofrequencies of a specimen. For both resonant and subrnant conditions, methods are presented for handling laand small phase angle between stress and strain. Sectiintroduces variants of the methodology to make measuments on the microscale and nanoscale. Section VI induces wave methods. Section VII deals with the challengeobtaining a wide window on the time or frequency doma

II. TIME DOMAIN MEASUREMENTS

A. Creep

Creep is one of the simplest experimental modalitiescharacterizing viscoelastic behavior. A dead weight can pvide constant stress, provided the deformation is sufficiensmall that the specimen cross section does not change aciably. The load history in time cannot follow a mathematicstep function as one would prefer based on the definitionthe creep compliance. Because the rise time is not zero,ordinarily begins taking data after a period about ten timeslong as the rise time has elapsed.5 It is difficult to apply adead weight in less than 0.5 to 1 s, therefore such creepbegin at 5 to 10 s. A test from 10 s to 3 h covers threedecades; 28 h is required for four decades. Typical crresults6 are shown in Fig. 1. The upper limit on the lengthtime of a creep test is limited by the experimenter’s patienIn some cases in which long-term data are required but cnot be inferred from time-temperature shifts~see Sec. VII!,test time can be up to 20 years.7

Prior to testing, one should monitor deformation at zeload for a period of time to make sure strain from specimpreparation has recovered. Moreover, prior to testing, po

FIG. 1. Creep complianceJ(t) of polyvinyl acetates at various temperatures, adapted from Ref. 6.


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799Rev. Sci. Instrum., Vol. 75, No. 4, April 2004 Viscoelastic measurement techniques

mer specimens are generally preconditioned by applyinfew load cycles consisting of creep and recovery segmeThis is done to achieve reproducible results.8 Biological ma-terials exhibit similar effects.

As for instruments, the torsion-pendulum implementions for subresonant tests~Sec. III B! can be used for creestudies.

B. Stress relaxation

Stress relaxation involves observation of time-dependstress in response to a step strain. The step strain caachieved by using a cam system that is much stiffer thanspecimen to impose a constant deformation following a rtime transient, or by manually advancing a screw up tostop. For example a relaxation instrument for the studyglassy polymers used an eccentric cam system actuatedhandle for load application9 to achieve a range in time othree decades. An axial strain-gauge-type load cell was ufor force measurement. Because in stress relaxationspecimen deformation is to be held constant, it is necesto determine the compliance of the load cell, to verify tha~and other parts of the test frame! was much stiffer than anyspecimen studied. For tension experiments both strinduced deformation and thermal expansion of the loframe can introduce error. The load frame must thereforemade robust, and should be enclosed in a temperatcontrolled chamber independent of the temperature-consystem for the specimen. Since polymers can swell wchanges in humidity, it is necessary to accurately conhumidity in tensile experiments. Results for polymethmethacrylate~PMMA!10 at several temperatures are showin Fig. 2. Alternatively a step strain in time can be applivia a servocontrol device in which strain or displacemenused as a feedback signal.

FIG. 2. Relaxation modulus E~t! of PMMA at various temperatures, adaptefrom McLoughlin and Tobolsky.10


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III. FREQUENCY DOMAIN: SUBRESONANT

A. Phase measurement

For a load history which is sinusoidal in time, the defomation history is also sinusoidal in time, with a phase shprovided the material is linearly viscoelastic and the appatus is linear. This review is for the most part limited to linesystems. For nonlinear materials or devices, overtones~mul-tiples! of the driving frequency appear in the deformation.frequencies sufficiently far below the lowest natural frquency that may be excited in the specimen or in the traducers, or in other parts of the apparatus, the phase abetween load and deformation~or between torque and angular displacement! is essentially equal to the phase angledbetween stress and strain. The driving frequency maytuned or scanned over a range.11 The lower limit of this rangeis limited by drift in the electronics and principally by thexperimenter’s patience. The upper limit of the frequenrange is limited by resonances in the specimen or in partthe apparatus. In the following, the material is assumed tolinearly viscoelastic, so if the stress is sinusoidal in time,strain is also sinusoidal, with a phase shift. As with othmechanical measurements, one must determine the comance of the instrument to make sure the resulting errors incalculated material modulus are sufficiently small. In adtion, phase error due to amplifiers and other electronics mbe quantified so that the investigator is confident the msured phase is representative of the material phased. More-over, instrumental compliance can contribute a phase eeven at frequencies well below any natural frequency.12

1. Time delay

Measurement of the phased can be performed by determining the time delayDt between the sinusoids. The relationship isDt5Td/2p, with T as the period of the sinusoidThe time delay can be determined either on an oscilloscor by detecting the zero-crossing of the load and displament signals and measuring the time delay with a high-sptimer. Noise of electrical or other origin limits the resolutioof the phase.

2. Lissajous (x-y) figures

A graph of load versus deformation~sinusoidal in time!for a linearly viscoelastic material is elliptic in shape~Fig.3!. The phased is given by

sind5A/B ~1!

with A as the horizontal thickness of the ellipse3 andB as thefull width of the figure. To measure small phase angles,middle of the ellipse can be magnified on an oscilloscopecomputer display. Again, the ratio of signal to noise mustsufficiently high to resolve the phase. The area withinfigure is representative of the energy dissipated per cyclthe material.

3. Lock-in measurement

Lock-in amplifiers, given a reference source at a knofrequency, amplify only at that frequency. They are therefcapable of accurate measurements in the presence of n


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which is distributed over a range of frequency. A two-phalock-in amplifier provides phase information as well, andtherefore suitable for viscoelastic measurements. Phasesurement with a lock-in amplifier reduces the scatter inhein the determination of phase. The typical frequency rang1 Hz to 100 kHz, but some models go as low as 0.001 Hzas high as 200 MHz. Phase resolution of some modelclaimed to be 0.001° (d51.831025) however the actuaresolution will depend on the levels of noise and on instment settings, especially the time constant. Automated sof frequency are possible with some lock-in amplifiers.

B. Instruments

Commercial servohydraulic test machines make usean electrical feedback signal from a force transducer,placement transducer, or strain gauge to control a hydravalve. This valve governs the flow of hydraulic fluid that, bpressing on a piston, provides the force. These machinesusually used to perform stress-strain tests to fracture; tcan also be used for creep or stress-relaxation measuremor subresonant tests at low frequency, typically below 10Instrumental resonances limit the frequency range of thdevices.

Commercial ‘‘dynamic mechanical analyzers’’ typicaluse a servocontrolled electromagnetic driver to apply a flural load to a bar specimen. Creep, relaxation, and lowquency subresonant sinusoidal tests can be done with tdevices. Instrumental resonances usually limit the frequerange to below 10 Hz. The range of time or frequency canextended by conducting repeated tests at several temptures and using time-temperature superposition~Sec. VII B!to infer viscoelastic properties over a wider range. Thisproach is applicable only to materials dominated by a sinrelaxation mechanism.

Torsion pendulums~Fig. 4!, which have been made imany variants, typically based on the classical

FIG. 3. Lissajous figure due to strain~e! response to stress~s! that is sinu-soidal in time, in relation to phase angled, between stress and strain aftLakes.3 E* [E81 iE9 is the complex modulus.


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pendulum,13 are described in more detail in Sec. IV A. Thecan be used for creep and for driven sinusoidal tests insubresonant domain provided one has sufficient precisiothe measurement of phase. In a typical embodiment, elecmagnetic drive is used to apply a torque. In earlier instments a moving coil was placed in the field of a permanmagnet. With the advent of better magnet materials, it is nmore practical to use a moving magnet in the field of a cto avoid concern of error caused by stiffness and dampcaused by electrical wires supplying a moving coil. An opcal lever is used to determine angular displacement. Einstruments made use of a beam of light projected on a sand observed visually; more recent ones use a laser beama split silicon sensor. Torsion offers the advantage that smdeformation is readily measured.

A high-resolution inverted torsion pendulum14 ~Fig. 5!provides a six-decade range of frequency from 1025 to 10Hz under subresonant conditions below the natural frequeof 100 to 500 Hz. Phase is determined by using a timermeasure the time delay between the zero-crossing of theand displacement signals. A phase uncertaintydf'631025 at 10 Hz, and better at lower frequencies, wclaimed, based on a time accuracy of 1ms. To achieve suchphase resolution it is necessary to minimize noise sourceelectrical and mechanical origin. The temperature rangeroom temperature to 800 °C, under vacuum. Representaresults are shown in Fig. 6.

Low-temperature variants15 have been used to study superconductors at cryogenic temperature~4.7 K!. As with theabove device, the specimen is kept isolated from the d

FIG. 4. Torsion pendulum of inverted type. Torque is generated byaction of a magnetic field supplied by a permanent magnet~not shown! uponthe coil.


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system and twist-measurement system by an arrangemestalks. These permit the specimen to be cooled in a cryoand also to be subjected to a magnetic field up to 0.2 T~2000G!, but limit the frequency to below 10 Hz. In this systeboth the temperature and magnetic field near the speciare isolated from the magnetic-drive system.

High-temperature variants16 allow measurements to1800 K. In this device, data are obtained by comparisonthe deformation of the specimen~in a furnace! with the de-formation of an elastic standard~outside the furnace!. Aninstrument17 for the study of creep and internal friction ogeological materials up to 1400 °C makes use of a refracmolybdenum alloy in the high-temperature section. Tspecimen is gripped with the aid of a mismatch in thermexpansion between specimen and refractory grips.heated specimen is separated by connecting stalks and aplate from a water-cooled low-temperature zone containthe transducers: a brushless motor to generate the torqcustom torque gage based on strain gauges, and two nontacting inductive displacement transducers to determine

FIG. 5. Driven subresonant torsion pendulum, adapted from Woirget al.14

FIG. 6. Mechanical damping tand of niobium vs frequency in the subresonant regime, showing a Snoek peak due to interstitial oxygen. AdaptedRef. 14.


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angular displacement~Fig. 7!. The stalks and other parts neessary to isolate the heated specimen give rise to an inmental fundamental resonance of 100 Hz. Since the seisfrequency range of interest is 0.001 to 1 Hz, the frequerange in the subresonant regime of the instrument isequate.

A variant suitable for a broad band of frequency, incluing subresonant and resonant, is described in Sec. VII D

C. Treatment of large and small phase

For materials of low loss (tand,1023), resonancemethods~Sec. IV! are the most appropriate because theysensitive. By contrast small phase differences are obscby noise and are difficult to measure directly. If it is necesary to make subresonant measurements on low-loss soexcellent phase resolution is required. Phase resolutionbe limited by the ratio of signal to noise. A lock-in amplifiecan be beneficial in cutting through the noise and improvphase resolution. Moreover one must reduce phase ethat may be introduced by transducers, electronics, andters.

Large phase angles are easy to measure. If, howevermaterial tends to a fluid consistency at sufficiently low frquency or long time, the specimen may not be able to sport its own weight. Methods for viscoelastic fluids are ouside the scope of this review.

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IV. FREQUENCY DOMAIN: RESONANT

A. Lumped resonances: pendulum devices

Resonance methods are appropriate for low-loss maals with tand from 1023 to 1029 or smaller. By contrast, inthe subresonant domain small tand values correspond tosmall phase angles, which are difficult to measure.

The specimen is usually a long bar or rod with antached mass much greater than that of the specimen. Threferred to as a lumped system since virtually all the ineis associated with the attached mass, and virtually allelastic or viscoelastic compliance is associated withspecimen. Analysis of the response of such a systemsimple.

Consider a torsion pendulum with a heavy membermass moment of inertiaI at the end of the specimen whichas shear modulusG and damping~phased between stressand strain!. If the specimen itself supports the inertia member, then it is under a static tension force that could influethe torsion results or cause creep in high temperature tTo avoid this, it is common to use an inverted torsion pedulum with the specimen below the inertia member,weight of which is supported by a fine wire of negligibtorsional stiffness in comparison with the specimen. The sport wire is strong and is sufficiently slender so that its tsional stiffness does not significantly perturb the measument. In these devices the specimen, mounted belowinertia member, experiences pure torsion with no tensiontypical embodiment of the idea is shown in Fig. 4. The tsion pendulum is normally operated at a low resonantquency, e.g., 1 Hz, and the temperature scanned. Theremany examples of this approach.11 The original Kependulum13 used a galvanometer-coil winding in a stamagnetic field to generate the torque and light reflected fra mirror to determine the angular displacement. It was ufor creep and for determination of internal friction by fredecay of vibration~described below!.

Measurement of the natural frequencyn1 allows one tocalculate3,18 the shear modulus as follows:

n15AG8~n1!k

I. ~2!

For a round, straight rod,k5pd4/32L, with d as the roddiameter andL as the rod length;I is the mass moment oinertia of the attached mass. The natural frequency candetermined by observing the period of free vibrations folloing a perturbation or by tuning the frequency in a driv~forced! system.

The damping tand, provided it is small, is calculatedfrom the full width Dn of the resonance curve at halmaximum amplitude

Dn

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This measurement of the peak width gives ‘‘quality factoQ, with Q21'tand for small d.

In a driven system one can make measurements afrom the resonant peak. To interpret, observe that the ph


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The phase anglef is approximately the same asd for suffi-ciently low frequencyn below the natural frequencyn1 , p/2radians atn1 , and p radians for high frequency. The relationship between structural stiffness and material stiffn~modulus! for this lumped system is, withu0 as the angulardisplacement amplitude and M0 as the applied moment amplitude

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For a round, straight rod,k5pd4/32L, with d as the roddiameter andL as the rod length. If the actual experimentinstrument has stalks and heavy attached transducersresonance structure will not be so simple; if instrument renances are at frequencies close to or lower than speciresonances, the instrument is generally suitable only for sresonant studies.

One can also obtain the damping tand, provided it issmall, from the free-decay timet1/e of vibration ~of periodT51/n1) to decrease by a factore52.718 following an im-pulse

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The free-decay approach offers simplicity in that no tunafunction generator is needed; it is sufficient to apply a pturbation. If the torsion pendulum has a sufficiently massinertia member, the natural frequency will be on the ordeHz, and the perturbation can be applied manually, by a mchanical system, or electrically. One may also speak oflog decrementL in terms of amplitudesA1 and A2 of suc-cessive cycles asL5 ln(A1 /A2). For small d, L'p tand.Classic results are shown in Fig. 8.

In lumped-resonance procedures it is usual to scantemperature while observing the behavior near resonaOne can also vary the frequency by varying the attacinertia, but this is cumbersome.

B. Distributed long-bar resonances

Long bars are used for resonant experiments becathey have a relatively simple resonance structure, which splifies interpretation of results. Resonance measuremhave been recognized following Fo¨rster’s 1937 pioneeringwork.19 One may excite vibrations in extension, flexure,torsion. The frequencies are as follows, assuming negligadded mass at the end, withE as Young’s modulus,G as theshear modulus, andr as density, for a bar fixed at one enfree at the other. Here the bar lengthL is one-quarter wave-length. This is in contrast to a bar free at both ends for whthe bar length is one-half wavelength.


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for flexure20,21 of a strip of lengthL and thicknessh, witha151.875, a254.694, anda357.855. HereAE/r is thespeed of longitudinal bar waves andAG/r is the speed oftorsional waves. Material damping is determined via Eq.~3!,which applies to distributed systems as well as lumped oprovided tand is sufficiently small~less than 0.1!. As forflexure, many devices have been designed. The vibratreed approach is particularly convenient. One end of a baclamped and the free end is mechanically excited. Capacexcitation allows one to generate vibration without addany mass to the specimen. Since the coupling is weak,approach is suitable for materials of low damping. Figure

FIG. 8. Tand vs temperature for aluminum13 and brass76 at low frequency,adapted from Keˆ.

FIG. 9. Free-free flexure-resonance apparatus.


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shows a typical setup for free-free vibration in which tspecimen is free at both ends.

C. Piezoelectric ultrasonic oscillator

The piezoelectric ultrasonic composite oscillator tecnique is based on a device that consists of two piezoeleccrystals and a specimen cemented together22–24 ~Fig. 10!.The vibration may be in tension/compression or in torsdepending on the type and orientation of crystal. The renant frequenciesnn5c/l5c(n11/2)/L are governed by thelength L of the crystals and the speedc of the ultrasonicwaves in them. Heren is an integer; the fundamental corresponds ton50, and a wavelength twice the length, corrsponding to the free–free boundary condition of a bar freeboth ends. The specimen is regarded as effectively free eat the glued end because the specimen length is chosennatural frequency matches that of the crystals, so thatstress across the glued interface is minimal. Frequenfrom 20 to about 120 kHz are typical. Quartz is preferredthe crystals since it has a low tand, which can be below 1026

depending on crystal defects. One crystal is driven eleccally to induce vibration. The oscillating voltage inducedstrain in the other crystal is measured. The wires are attacto the crystals’ conducting metal plating at the crystal mpoints, which are nodes for free–free vibration. Becausewire has nonzero size, there is a parasitic damping assocwith it. There is also a parasitic damping due to the cemused to attach crystal and specimen. This damping is smprovided each crystal as well as the specimen has nearlysame natural frequency in which case the stress at thejoint is minimal. The crystal itself has some damping whiis accounted for in the calculation of specimen damping.

Viscoelastic properties of the specimen are inferredfollows from electrical measurements upon the sensor cryand from the dimensions and masses of specimen and ctals:

tand t5~m1a12 tand11m2a2

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The method is usually used to measure modulusdamping at a constant frequency over a range of temperaIf the intended temperature exceeds the Curie point of cryquartz, a long stalk of fused quartz~of length chosen tomatch the crystal frequency! is used between specimen ancrystal so that the crystals can operate at a lower tempture. In addition to the fundamental natural frequency, ocan also excite higher harmonics, however they are weain amplitude than the fundamental.

D. Resonant ultrasound spectroscopy „RUS…

Resonant ultrasound spectroscopy~RUS! measures ofthe resonance structure of a compact specimen suchcube, parallelepiped, sphere, or short cylinder25,26,27to infermaterial properties. Elastic moduli or complex dynamic vcoelastic moduli are determined from the resonant frequcies. Viscoelastic damping is determined from resonant-pwidth following Eq. ~3!. Frequencies are typically from 5kHz to 20 MHz depending on specimen size and propertA cubical specimen is typically held by contact force btween piezoelectric ultrasonic transducers at opposite cor~Fig. 11!. A short cylinder may be held between opposedges. One transducer is a transmitter, and one is a receCorners are used since they provide elastically weak cpling to the transducers, hence minimal perturbation tovibration, so minimum parasitic damping. RUS requires

FIG. 11. Resonant ultrasound spectroscopy~RUS! apparatus.


tof

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dre.al

a-eer

s a

-n-ak

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gluing, clamping, or painstaking alignment of the specimIn comparison with pulsed wave ultrasound~Sec. VI!, RUSrequires no corrections for diffraction or concern aboplane-wave approximations. Although RUS is conceptuaand experimentally simple, determination of anisotropic eltic constants entails elaborate numerical methods.

All the elastic moduliCi jkl of a single specimen can bobtained from a single frequency scan. A numerical algrithm is used to extract the moduli from the resonant fquencies. A rectangular-parallelepiped specimen withaxes aligned with the material symmetry is commonly usFor the higher symmetries it is desirable to have uneqedge lengths to avoid overlap~degeneracy! of differentmodes with the same frequency; the algorithm cannot hanthis overlap. The specimen must be accurately cut. Any ein parallelism or dimensions will translate into a similar sierror in the moduli. A poorly cut specimen can prevent tnumerical inversion from accurately functioning since ththe assumed shape does not correspond to the actual sMost resonant frequencies in RUS depend on combinatof elastic constants.27 Numerical determination of anisotropic moduli using currently available software requiresgood initial guess as input to the algorithm, since it is usuanot straightforward to identify the modes. Since the lowmodes depend primarily on the principal shear modulus,to 40 modes may be used in study of anisotropic soliMissing modes can occur if the specimen displacement atcontact point is zero or is in a direction to which the tranducer is not sensitive. Misidentified modes play havoc wnumerical inversion, therefore another scan may be doneter tilting the specimen. Moreover for anisotropic solids, twidth of resonant peaks, required to infer damping, depeon combinations of complex moduli.28

Modes may be identified by repeating scans for differcube corner-orientations.29 Alternatively one may selectivelyexcite classes of modes by exciting the specimen using ecurrents via a coil of wire surrounding the specimen aembedded in a magnetic field.30 One may also opticallymonitor the vibration using laser interferometry to identithe mode31,32 and avoid the need for a good initial guessproperties. If shear piezoelectric transducers33 are used ratherthan the usual compressional ones, one can identify smodes via the polarization of shear vibration; moreover, snal amplitude for the fundamental is enhanced by onethree orders of magnitude. One can perform RUS studwith off-the-shelf electronics, e.g., a function generatorpable of sweeps, broadband ultrasonic transducers, adigital oscilloscope. Lock-in amplifiers have been succefully used for RUS measurements, but it takes longer to pform a scan than with a heterodyne amplifier used in comercial RUS systems.34

If the specimen is isotropic, analytical formulas aavailable for the lowest mode. For example,35 the lowestmode for an isotropic cube is a torsional mode of frequen

n5A2

pLAG

r. ~12!

HereL is the side length,G is the shear modulus, andr is thedensity. The next higher mode in the cube is about a facto


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1.34 higher in frequency. For a cylinder of lengthL, thetorsional natural frequencies for free-free vibration are

n5n

2LAG

r~13!

with n a positive integer. The next higher mode is aboufactor 1.26 higher. The fundamental torsion mode is the loest mode for a cube and for a short cylinder of length eqto diameter if the Poisson’s ratio is not too close to the lowlimit 21. The mode structure given is for an elastic materFor viscoelastic materials there is frequency depende~dispersion! of the modulus. If tand is relatively large, thedispersion may be sufficient to complicate the analysis. Nmerical algorithms currently used do not account for dispsion, and are appropriate only for low damping anisotromaterials.

Material damping tand is obtained from RUS data bcalculating it from the width of resonance curves via Eq.~3!.Higher modes are closely spaced. Overlap of broad resopeaks in high damping materials renders it impracticaldetermine tand above 0.01 to 0.02 at frequencies other ththe fundamental.36

E. Treatment of large and small phase

For materials of low loss (tand,1024 or equivalentlyQ.104), resonance methods are the most appropriAmong resonance methods, it is easier to measure free dof vibration than to tune a forced-vibration system througvery narrow resonant peak. The decay time of vibration ihigh-Q (Q'109) resonator at acoustic or low ultrasonic frquency can be several hours long. A corresponding timnecessary to set up the vibration. During this period thequency of the electrical source must be sufficiently stablematch that of the resonant specimen. It is necessary in texperiments to minimize all sources of parasitic damping37

Indeed, single crystal sapphire and silicon at low temperahave been reported to exhibit tand as small as 1029. Para-sitic energy dissipation due to air viscosity or radiationsound into the air tends to become important for tad,1024. Bending modes are more vulnerable to such erthan torsion, and slender specimens more so than comones such as those used for RUS. Sound radiation errorRUS38 are more severe for compressional modes thansional modes. To eliminate these errors, tests on low-materials are done under vacuum. Vibration energy canleak out through the support of the specimen, and lead tooverestimate of tand. Such errors are minimized as followIn fixed-free vibration, the support structure must be ordof magnitude more rigid than the specimen, otherwise pasitic dissipation from transmission of waves into the suppwill obtrude in the data. For a fiber specimen, the impedamismatch between specimen and support is very favoraallowing tand as small as 1026 to be measured as describebelow in Sec. V.

In free-free vibration, specimens are usually supportea vibration node, where displacement is minimal. A fithread used for support is sufficiently compliant that therea large mismatch in mechanical impedance, hence mini


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spurious energy loss. The thread is placed near a vibranode. Further reduction in parasitic damping is achievedgreasing the thread.39 If the support thread is to be used fodriving or detection, it must be sufficiently far from thnode.40 In bending experiments, it is better to have the bening motion in the horizontal direction than the vertical diretion because less energy is lost in rotating the support thrthan in stretching it. Measured quality factorQ has beenobserved to vary with suspension length41 for low-dampingcolumnar silicon. In one study,Q measurements in fusesilica42 were limited by losses in suspension wires, asmeasuredQ was found to depend on the shape of each renant mode and the proximity of each mode’s nodal regionthe suspension points. The highestQ measured in this seriewas 1.93107 at 26 kHz.

For low-loss materials a further matter to consider ispossibility of spurious energy loss from the transducers uto induce and detect vibration. Weak coupling is beneficiathis context. Capacitive transducers are favored becausecoupling is intrinsically weak. Even so, it is wise to calculathe error caused by how much energy might be lost inequivalent resistance of the transducer circuit through traducer coupling. Moreover, extra energy dissipation can ocin films adsorbed on the resonator as well as surface damand defects. Consequently, the surface must be kept cand free of damage if the intrinsic damping of the materiato be approached. Specimens are polished and annealreduce surface and internal defects.Q values to 43108 havebeen measured for sapphire at room temperature, and3109 at 4 K.38

For high-loss materials~tand.0.1!, it is easy to measurethe phase directly in the subresonant regime. In resonanperiments, the approximate formulas for half width andfree-decay are not applicable, and the more complex esolution for the vibration must be used for analysis. For eample, large damping can also be measured from resonpeak width, but the analysis is more complicated thanlow tand because the exact solution is necessary. Forample, exact relations betweenQ and tand require a modelof the viscoelastic response of the material.43 If damping isindependent of frequency then

Q215A11tand2A12tand. ~14!

Moreover, resonant peaks in high-loss materials becobroader and of lower amplitude, and signal strength isduced; moreover, nearby modes in a distributed systemoverlap.

As for the decrementL in free-decay of vibration, recalfor small d, L'p tand. The exact expression is44

tand5@12exp~22L!#/2p ~15!

in which these authors define the inverse quality factorQ21

in terms of a ratio of dissipated energyDW to stored energyW in a cycle of deformation, equivalent to tand in the sym-bols used here.~The authors use the symbold for the decre-ment.! The exact solution shows one cannot obtain the daming from the ratio of adjacent oscillations if tand>1/2p'0.16, because for such high damping there areoscillations.


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V. MICROSCALE AND NANOSCALE MEASUREMENTS

Testing of materials on a small scale is done, asmight expect, by scaling down the size of the methodsforce generation and deformation measurement.

As for fibers, at the scale of a few-hundredmm diameter,it is possible to use a miniature torsion pendulum basedelectromagnetic generation of torque, and an optical-lescheme~laser beam reflected from a mirror! to measure an-gular displacement. Such an approach has been used istudy of single osteon fibers~about 0.2 mm diameter! frombone.45 More slender fibers may be excited via a capacitapproach in which an oscillating voltage is applied to a plwithin 1 mm of the fiber, and displacement measuredobserving the shadow of the illuminated fiber with a spsilicon-diode detector.46 The specimens were set into flexurvibration under vacuum. For silica fibers 0.23 to 0.5 mmdiameter in a fixed-free configuration, tand of about 1026

was measured in the frequency range 30 Hz to 1 kHz.observed damping was comparable to predicted damfrom thermoelastic coupling alone; therefore parasitic daming was minimal in this case.

Permanent magnets to generate torque or force cascaled down to magnetic grains47,48 of micrometer-size orbelow. The motion of grains or beads49 embedded in a material may be examined by video microscopy or by lighscattering methods. As with macroscopic experiments, inpretation of results requires a quantitative model ofsystem to relate the raw force-displacement or torque-adata to a stress-strain relationship. The embedded-beadproach has been used mostly for soft materials such asand biological cells. For such materials one need not nesarily excite the bead externally. Thermal motion of the bemay provide enough motion from which one can infer marial properties. This approach has been used for membrand other components of living cells as follows.micrometer-size bead is embedded in the membrane anluminated with an infrared laser.50 The laser light provides anoptical trap. Material properties are inferred from the powspectrum of Brownian motion of the bead as determinfrom an image of the bead projected upon a split silicon-lisensor. The frequency range is from 100 Hz to several khertz.

Thin films may be applied to a substrate of known mterial, typically of low damping. The resulting laminate mabe studied under quasistatic conditions or in lumped renance. Film properties are extracted with the aidcomposite-theory analysis of the laminate. One may alsoflect ultrasonic waves from a surface with a thin film on it51

Material properties are inferred from the amplitude aphase of the obliquely reflected waves. The approachoriginally due to Masonet al.52 who studied viscoelasticproperties of polymer liquids via oblique-wave reflectioFilm properties can also be studied by a variant of resonultrasound spectroscopy in which a film of unknown mateis applied to a block of material of known properties.53

As for properties of materials at and near the surfaindentation creep experiments offer potential for mappproperties or study of small samples. Nanoindentation cr


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be

r-eleap-elss-d-es

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has been conducted with a diamond tip upon polishspecimens.54 Load was ramped to a steady value for cretests. The tip was oscillated to obtain contact stiffness.creep was seen in tungsten as one would expect at modstress levels and at room temperature. Scanning-probecroscopes can be configured to provide phase data as aternative contrast variable in images. Thus far, the chaininference has been too indirect to allow mapping of vcoelastic properties of hard material. The shape of the prtip is often not known with great certainty, and adhesive acontact forces may complicate the analysis.50 Indeed, map-ping of stiffness in a graphite-epoxy composite disclosed rsonable values of moduli55 but damping values for graphitwere too high by orders of magnitude. Soft materials suchgels and biological materials have been studied with mofied scanning-probe microscopes. Modification of the proshape by polystyrene beads allows the experimenter to mreadily infer material properties56 using the Hertz solutionfor a spherical contact problem. Moduli of several kPa atand of about 0.1 have been measured from 20 to 400 H

VI. FREQUENCY DOMAIN: WAVES

A. Shear and longitudinal waves

Wave methods are appropriate at frequencies~typicallymegahertz to tens of megahertz! above those used in resonant and subresonant methods. Measurement of the wspeed allows one to infer a modulus.57 If the wavelength of alongitudinal wave is much longer than the specimen widthdiameter, then the Poisson expansion and contraction isto occur, and the wave speed~bar velocitycB) is governedby Young’s modulusE and densityr:

cB5AE

r. ~16!

If the wavelength of a longitudinal wave is much shortthan the specimen width or diameter, the Poisson effecrestrained. This is the typical situation for most ultrasonstudies at or above 1 MHz. The longitudinal wave speedcL

is given by

cL5AC1111

r, ~17!

with C1111 as the tensorial modulus in the 1 direction. Fisotropic materials58 we have

C11115E12n

~11n!~122n!, ~18!

with n as Poisson’s ratio. For wavelengths which are neitlong nor short, the relationship, as one might expect, is mcomplicated.59

The speedcT of shear or transverse waves in an isotropelastic medium is given by60

cT5AG

r, ~19!

with G as the shear modulus. For viscoelastic materials, emodulus is interpreted as the absolute value of the comp


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modulus, e.g.,uG* u. Wave speed is determined from the timdelay of a pulsed group of waves and the thickness ofspecimen.

For ultrasonic waves~above the upper limit of the human ear, 20 kHz, and most commonly above 0.5 MHz! oneuses piezoelectric transducers to generate and receivewaves. Ultrasonic measurements are commonly performbetween 1 and 20 MHz, but it is possible by special methto go to 1 GHz and beyond.61,62 The electrical wave formsare usually bursts of several cycles at the frequency of inest. The bursts should be short enough in time to discrinate multiple echoes. The transducer may be coupled tospecimen by a thin layer of fluid such as water. This is sficient if longitudinal waves are used but is not approprifor shear waves because fluids do not sustain shear waThe transducer may also be coupled with glue or hiviscosity grease, particularly if quartz crystal plates are uas transducers. Wave velocity is determined from the tdelay of the pulse and from the thickness of the specimThe simplest approach is to find the time delay of the leadedge of the wave form, but that lacks precision becausethe change of pulse shape as it propagates and the difficof identifying an equivalent equivalent point on two leadiedges. A better approach is the pulse-echo-overlap meth63

in which one controls an adjustable delay until the initial afinal wave forms overlap. Precision of a few ppm canachieved.

B. Ultrasonic attenuation

Wave amplitude decays with distance d because of gmetrical spreading in three dimensions, as 1/d2. If the wavesare confined to one dimension, the amplitude also decbecause of absorption in the material itself. This decaycalled attenuation and is given the symbola. In a linearmaterial, the amplitude decays with distance d as exp~2ad!.Attenuation is related to material dampingd as follows:64

a5v

ctan

d

2. ~20!

One way to measure attenuation of ultrasonic waveby comparing the signal amplitude transmitted through tsamples of different length.65 The attenuationa, in units ofnepers per unit length is determined as follows frommagnitudes of the signals:A1 through a specimen of lengtL1 , A2 through a specimen of lengthL2 :

a5ln~A1 /A2!

~L12L2!. ~21!

One neper is a decrease in amplitude of a factor ofSometimes attenuation is expressed in terms of deci~dB!, a logarithmic scale. In this approach it is necessarybe careful that the coupling between transducer and spmen is reproduced for both measurements. Coupling depon the pressure applied to the transducer and the degreflatness of the specimen surface.

Attenuation may be measured by examining the decaamplitudes of echoes of a tone burst reverberating betwparallel surfaces of a specimen.66 A single transducer may b


e

theeds

r-i-hef-ees.-den.goflty

d

e

o-

ysis

iso

e

e.lsoci-dsof

ofen

used to transmit and to receive the waves. The inferencattenuation will be correct provided the energy lost into ttransducer is small enough to neglect. One cannot use cmercial nondestructive test transducers in this approach sthey are designed to have maximal electromechanical cpling and they are damped to achieve well-defined pulsTherefore energy lost into them is substantial. Quartz isvored as a transducer material for the echo method. Quhas low damping and a relatively weak electromechancoupling. Therefore little energy from the sound wave is coverted back into electricity. Even so, the electric circuittached to the transducer is usually designed to minimlosses, which are calculated to aid in interpreting the dThe quartz crystal plate may be glued to the specimenonly longitudinal, not shear, waves will be used, the crysmay be coupled with a thin layer of oil or grease.

For small damping, hence small attenuation, one mconsider the fact that wave propagation in real materialthree dimensional, not one dimensional. If the ultrasotransducer is smaller than the specimen width the wavesdiffract. Diffracted waves lost to the receiving transducgive rise to an overestimate of the true attenuation, therefor low-damping materials, corrections for diffraction are aplied. If the transducer is equal to or larger than the specimwidth, internal reflections from the lateral surfaces may ocur. Longitudinal waves can be converted to slower shwaves by such reflections.

For large damping, there is also considerable dispers~frequency dependence! of the velocity, leading to distortionof the wave form. It may then be difficult to identify corresponding points on wave forms that have passed throdifferent lengths of material. If that is a problem, one mresort to more complex methods based on Fourier transmation.

VII. METHODS FOR BROADBAND RESULTS

A. Motivation

As we have seen, most methods provide results asingle frequency or over a relatively narrow domain, usuathree decades or less, of time or frequency. A wider rangdesirable because even a single-relaxation-time processscribed by an exponential in the time domain, or a Debpeak in the frequency domain, occupies about one decMost viscoelastic properties occur over a broader rangeapplications one may be interested in creep of a materiawell as its ability to absorb vibration at acoustic frequencywaves at ultrasonic frequency.

B. Time-temperature superposition

Viscoelastic behavior depends on temperature. Forample the relaxation modulus function may be writtenE5E(t,T) with t as time andT as temperature. The temperture dependence of properties can be exploited to extendrange of effective time or frequency of measured propertSuppose a series of tests is done at different temperatuOne specimen may be used for each test, or one may aadequate time for recovery between tests. Plot the test reversus log time or log frequency, depending on the type


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test. It may be possible to shift the curves horizontally so tthey coincide. If that is true, the material obeys timtemperature superposition. Such a shift is evidence thattemperature changes are effectively accelerating or retarthe dominant viscoelastic process or processes, becaushift on a log scale is equivalent to multiplying the scalea number. The relaxation function is of the form

E~ t,T!5E~z,T0!, with z5t

aT~T!, ~22!

in which z is called the reduced time,T0 is the referencetemperature, andaT(T) is called the shift factor. The curvobtained by time-temperature shifts is called a master cuMaster curves covering 12 to 15 decades of time orquency have been obtained for amorphous polymers.1 A rep-resentative master curve for polymethyl methacryl~PMMA! is shown in Fig. 12.

Time-temperature superposition is commonly used instudy of polymers. It works well for many materials prvided the widow of time or frequency is relatively narrowthree decades or less. The test of superposition is onlyfinitive by its failure.67 Indeed, materials which appearobey superposition over a three decade window are obseto deviate from it over a five or six decade window.

Time-temperature superposition will fail if the materiundergoes a phase transformation in the temperature raninterest or if it is a composite with multiple processes whgive rise to viscoelasticity, or if a dominant relaxatiomechanism is not thermally activated.

Time-temperature superposition does not imply linearnonlinear behavior. Conversely, a linearly viscoelastic marial may or may not obey time-temperature superposition

FIG. 12. Master curve for PMMA obtained from results at various tempetures via time-temperature superposition, adapted from McLoughlinTobolsky.10


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of

r-

C. Specific forms: Arrhenius and Williams, Landel,and Ferry „WLF…

In some materials, the relaxation is dominated by a thmally activated2 process: the following Arrhenius equatioapplies:

n5n0 exp2U

RT, ~23!

with U as the activation energy,n as frequency for a featuresuch as a peak,n0 as a frequency factor, andR58.32 J/moleK as the gas constant. One may use Bomann’s constantk51.38310223J/K if the energy is so ex-pressed. For such materials, a change in temperaturequivalent to a shift of the behavior on the time or frequenaxis. This is a particular form of time-temperature superpsition, because a temperature change has the effect of string or shrinking the frequency scale. To use this approaone may determine a series of temperature-dependent spat various frequencies, or a series of frequency-depenspectra at a series of temperatures.68 If there is a peak attemperatureTp one may plot lnv versus 1/Tp to determinethe activation energy. It is easier to obtain a reasonable raof frequency in subresonant tests than in resonant ones.

The Arrhenius condition has been used in the interpretion of experiments at a single frequency or a discrete sefrequencies. If the viscoelastic damping is in the form oDebye peak with time constantt0 , the temperature at thepeakTp follows18

~U/k!~1/Tp!52 ln vt0 . ~24!

In this case as well, the activation energy may be extracfrom the slopeU/k of a plot of lnv versus 1/Tp , or a plot oflogv versus 1/Tp for which the slope isU/2.303k. Whendata are taken at only two frequencies in a temperature sof a Debye peak,2

ln~v2 /v1!5~U/k!~T1212T2

21!. ~25!

The time-temperature shift for polymers tends to follothe empirical WLF equation~after Williams, Landel, andFerry!1,69

ln aT 52C1~T2Tref!

C21~T2Tref!, ~26!

in which Tref is a reference temperature. The constantsC1

andC2 depend on the particular polymer.

D. Direct ways to achieve broadband results

Direct measurements over a wide range of frequencypreferable to the more common indirect approach of varythe temperature, for the following reasons.3 Methods basedon temperature shifts are restricted to activated procestherefore viscoelastic spectra due to thermoelasticity or mnetic flux diffusion cannot be obtained by temperature scaSecond, the underlying viscoelastic theory is establishedisothermal conditions and is not directly applicable if temperature is varied continuously. Finally, temperature vations can introduce important structural changes in the spmen during testing.

-d


ethe

as


FIG. 13. Normalized modulus and tand for indium-tinalloy37,75 at room temperature with no appeal to timtemperature-superposition. Shown for comparison isrange of frequency perceived as sound by the ear,well as a Debye peak in damping.

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d,

One may achieve a wide range of time or frequencyconstant temperature by performing experiments with ariety of devices, in which each one covers a portion ofrange.70 This approach is cumbersome. Usually it is necsary to prepare several types of specimen. Six decadefrequency from 1025 to 10 Hz are attainable via subresonaphase measurement as described3 in Sec. III B.

In general one can obtain an extended range of timefrequency in instruments which are capable of both transand dynamic tests, and which have adequately fast stime/high frequency response. For example a commerciastrument or torsion pendulum, limited to 10 Hz~or 0.016 s!by resonances in stalks and transducers, would providedecades if the dynamic results were combined with resultabout five hours of creep. Conversion of results from trsient to dynamic or vice-versa may be done for linear marials by Fourier transformation or by approximate interretions. For example, eight decades from 100 Hz down today of creep were achieved with a biaxial driven torsipendulum71 after modification.72 In this device, torque isgenerated via a moving coil immersed in a static magnfield and is measured by a strain-gauge-type torque cell.gular displacement is measured via a linear variable difential transformer coupled to a rotor. Ten decades from mliseconds to one year are achieved in broadband creepin which load is applied rapidly, and deformation measuelectronically,73 but this requires considerable patience. A11 decade range74 from 100 kHz to subresonant tests in thkHz to mHz range to transient~creep! over several days waobtained as follows. Torque is applied to the specimen etromagnetically by a Helmholtz coil acting on a permanemagnet cemented to the end of the specimen. Deformatiothe specimen end is determined by measurement of dispment of a laser beam using a split silicon-light senmounted upon a fast preamplifier. Resonances are eliminfrom the torque-measuring and angle-measuring devicethis approach. Resonances in the specimen itself arelyzed via an analytical solution that is applicable to homoneous cylindrical specimens, fixed at one end and free a


t-

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t

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other, of any degree of loss. Higher harmonics in torsionreadily generated in this system. Representative results, cpared with those of RUS, are shown in Fig. 13. In furthdevelopment, noise in the tand was reduced by using alock-in amplifier.75

VIII. DISCUSSION

In this review, a variety of experimental methods fviscoelastic solids has been surveyed. Experimental methare available over 20 decades of time and frequency. Mindividual methods permit study at single frequencies ormost of three decades or less for a given specimen. If idesired to conduct experiments over an extended range,may use multiple instruments, time-temperature superption ~which is not appropriate for all materials!, or carefuldesign of a single instrument.

ACKNOWLEDGMENTS

The author acknowledges the support of the NatioScience Foundation through CMS-0136986 and the Matals Research Science and Engineering Center for Nanostured Materials and Interfaces.

1J. D. Ferry,Viscoelastic Properties of Polymers, 2nd ed.~Wiley, NewYork, 1970!.

2A. S. Nowick and B. S. Berry,Anelastic Relaxation in Crystalline Solid~Academic, New York, 1972!.

3R. S. Lakes,Viscoelastic Solids~CRC Press, Boca Raton, FL, 1998!.4A. D. Nashif, D. I. G. Jones, and J. P. Henderson,Vibration Damping~Wiley, New York, 1985!.

5S. Turner,The Physics of Glassy Polymers, edited by R. H. Haward~Wiley, New York, 1973!, Chap. 4.

6K. Ninomiya and J. D. Ferry, J. Phys. Chem.67, 2292~1963!.7G. E. Troxell, J. M. Raphael, and R. W. Davis, ASTM Proc.58, 1 ~1958!.8H. Leaderman,Elastic and Creep Properties of Filamentous Materiaand other High Polymers~Textile Foundation, Washington, DC, 1943!.

9S. S. Sternstein and T. C. Ho, J. Appl. Phys.43, 4370~1972!.10J. R. McLoughlin and A. V. Tobolsky, J. Colloid Sci.7, 555 ~1952!.11A. Riviere, Mechanical Spectroscopy Q21 2001, edited by R. Schaller, G.

Fantozzi, and G. Gremaud~Trans Tech Publications, Zurich, Switzerlan2001!, pp. 366–368.


f

.

l

M

-

f

D.

m.

ys.

,1

.


12S. S. Sternstein, Adv. Chem. Ser.203, 123 ~1983!.13T. S. Ke, Phys. Rev.71, 533 ~1947!.14J. Woirgard, Y. Sarrazin, and H. Chaumet, Rev. Sci. Instrum.48, 1322

~1977!.15G. D’Anna and W. Benoit, Rev. Sci. Instrum.61, 3821~1990!.16P. Gadaud, B. Guisolan, A. Kulik, and R. Schaller, Rev. Sci. Instrum.61,

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