viscoelastic characterization of polymers using instrumented indentation

Viscoelastic Characterization of Polymers UsingInstrumented Indentation. I. Quasi-Static Testing

M. R. VANLANDINGHAM,1 N.-K. CHANG,2 P. L. DRZAL,1 C. C. WHITE,1 S.-H. CHANG2

1National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8615

2Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan

Received 4 October 2004; revised 12 January 2005; accepted 20 February 2005DOI: 10.1002/polb.20454Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: The use of instrumented indentation to characterize the mechanicalresponse of polymeric materials was studied. A model based on contact between arigid probe and a linear viscoelastic material was used to calculate values for thecreep compliance and stress relaxation modulus for two glassy polymeric materials,epoxy and poly(methyl methacrylate), and two poly(dimethyl siloxane) (PDMS) elasto-mers. Results from bulk rheometry studies were used for comparison with the inden-tation stress relaxation results. For the two glassy polymers, the use of sharp pyrami-dal tips produced responses that were considerably more compliant (less stiff) thanthe rheometry values. Additional study of the deformation remaining in epoxy afterindentation creep testing as a function of the creep hold time revealed that a largeportion of the creep displacement measured was due to postyield flow. Indentationcreep measurements of the epoxy with a rounded conical tip also produced nonlinearresponses, but the creep compliance values appeared to approach linear viscoelasticvalues with decreasing creep force. Responses measured for the unfilled PDMS weremainly linear elastic, with the filled PDMS exhibiting some time-dependent andslight nonlinear responses in both rheometry and indentation measurements. VVC 2005

Wiley Periodicals, Inc.* J Polym Sci Part B: Polym Phys 43: 1794–1811, 2005

Keywords: creep; indentation; modulus; relaxation; rheology; strain; stress; vis-coelastic properties

INTRODUCTION

Instrumented indentation is increasingly beingused to probe the mechanical response of poly-meric and biological materials. These types of

materials behave in a viscoelastic fashion, andthus their mechanical behavior is dependent onthe test conditions, including the amount ofstrain, the strain rate, and the temperature.Often in instrumented indentation, however,properties are measured with loading historiesand analysis developed for elastic and elasto-plastic materials, for which time-dependentbehavior is normally neglected. In studies inwhich attempts have been made to characterizeviscoelastic behavior with indentation,1,2 limit-ing and sometimes invalid assumptions havebeen made, and linear viscoelasticity has beenapplied despite the intense strains local to theindenter tip. Normally linear viscoelastic behav-

Certain commercial instruments and materials are identi-fied in this article to adequately describe the experimentalprocedure. In no case does such identification imply recom-mendation or endorsement by the U.S. Army Research Labo-ratory or the National Institute of Standards and Technol-ogy, nor does it imply that the instruments or materials arenecessarily the best available for the purpose.

Correspondence to: M. R. VanLandingham (E-mail:[email protected])

Journal of Polymer Science: Part B: Polymer Physics, Vol. 43, 1794–1811 (2005)VVC 2005 Wiley Periodicals, Inc. *This article is a US Government work and, as

such, is in the public domain in the United States of America.

1794

ior is measured at small strain levels3 in muchthe same way as an elastic modulus of a metalor ceramic would be measured at small strainsat which linear elastic behavior dominates.However, unlike metals and ceramics, for whichthe elastic modulus can be estimated fromindentation measurements with the unloadingdata, polymers exhibit viscoelastic behavior dur-ing indentation measurements such that similarcalculations of the elastic modulus are inaccu-rate.

The mechanical response of viscoelastic mate-rials is highly dependent on the types and lev-els of stress and strain as well as the strainrate. An analysis of indentation data, however,is typically based on force and displacementmeasurements, and stress and strain values areoften estimated only in a nominal sense. Forexample, the mean stress, often used as ameasure of the hardness, H, is the ratio of theforce, P, to the contact area, A, where A is ingeneral related to the displacement, h, by thetip geometry. Only in the case of a flat punch,for which A is constant with h, is H a functionof force only. A general indentation strain ratecan be calculated for any tip geometry from theratio _h/h, where _h is the rate of change of hwith time t, or _h is equal to dh/dt. For a para-boloidal tip, a representative measure of theindentation strain is proportional to the ratio ofthe contact radius, r, to the tip radius, R, wherer is a function of h.4–6 A nominal indentationstrain for a conical or pyramidal tip is relatedto the characteristic included angle or angles ofthe tip and is not a function of h. For Vickersand Berkovich pyramidal indenters, whichideally have the same area function, A(h),empirically based analyses attributed to Tabor4

yield an estimated representative strain forthese self-similar tips of 8–10%. However,Chaudhri7 estimated that the representativestrain ranged from 25 to 36% for a Vickersindentation of polycrystalline copper, and finiteelement analysis has been used to estimatestrains local to a Berkovich indentation tip tobe in excess of 100%, with a large volume ofmaterial subjected to at least 15% strain.8 Asdiscussed by Dao et al.,8 values for a represen-tative indentation strain will depend on therelationships between the indentation parame-ters and mechanical properties upon which itsdefinition is based. Thus, for the indentation ofpolymeric materials, the representative strains

may be different from that defined previouslyin other studies and may differ between differ-ent viscoelastic materials. Also, viscoelasticmaterials might be much more sensitive to thelocal variations in the strain and strain ratethan metals and ceramics.

For polymeric and biological materials, a widerange of mechanical behavior can result fromthe imposition of finite strains. Analytical solu-tions of quasi-static contact between a rigidindenter and a linear viscoelastic solid9–14 canbe used to account for the viscoelastic constitu-tive behavior when indentation data are ana-lyzed for these types of materials. These analy-ses are based on the development of an appro-priate boundary-value problem that satisfies theequations of equilibrium. The stress–strain rela-tions for linear, isotropic viscoelasticity aregiven by12

eijðtÞ ¼ 1

2

Z t

0

Jðt� �Þ @

@�sijð�Þd�

sijðtÞ ¼ 2

Z t

0

Gðt� �Þ @

@�eijð�Þd� ð1Þ

"ijðtÞ ¼ 1

3

Z t

0

Bðt� �Þ @

@��ijð�Þd�

�ijðtÞ ¼ 3

Z t

0

Kðt� �Þ @

@�"ijð�Þd� ð2Þ

B(t) and K(t) are the dilatational creep andrelaxation functions, respectively, relating thestress and strain invariants, rii and "ii, � is a‘‘dummy’’ variable for time in the integralnotation, and t is time. J(t) and G(t) are theshear creep and relaxation functions, respec-tively, relating the deviatoric stress tensor, sij,and the deviatoric strain tensor, eij.

These expressions are formulated in terms ofintegral operators associated with the heredi-tary function, so that relaxation times are givenby a continuous spectrum. Alternatively, theseexpressions can be restated in terms of differen-tial operators with a viscoelastic model ofsprings and dashpots, corresponding to a dis-crete spectrum of relaxation times, or with otherequivalent ways of expressing linear viscoelasticbehavior, including direct measurements.9,10,12

The following single-integral constitutive equa-tion has also been used:11,13

VISCOELASTIC CHARACTERIZATION. I 1795

�ij ¼Z t

0�2Gðt� �Þ @"ijð�Þ

@�þ �ij�ðt� �Þ @"kkð�Þ

@�

� �d�

ð3Þ

G(t) is again the relaxation modulus in shear,and Lame’s constant, �(t), is related to the relax-ation modulus in extension, E(t), and Poisson’sratio, v(t). The lower limit of 0� is used in caseof a jump in stresses and strains at t ¼ 0. Forhomogeneous, isotropic, elastic materials, E, G,and v are related by

G ¼ E

2ð1þ vÞ ð4Þ

This equation holds for a viscoelastic solid onlyat equilibrium. Also, for viscoelastic materials,the shear, bulk, and extensional creep complian-ces, J(t), B(t), and D(t), respectively, are not sim-ple inverse functions of their respective relaxa-tion moduli, G(t), K(t), and E(t), as they are forelastic materials, and v is, in general, a functionof time,3 although it is often taken to be con-stant for simplicity. Finally, creep compliance isgenerally determined as the ratio of an appliedconstant stress, r0, and the resulting time-dependent strain, "(t), and the stress relaxationmodulus is normally determined as the ratio ofan applied constant strain, "0, and the resultingtime-dependent stress, r(t), in both cases underuniaxial loading conditions.

Viscoelastic solutions to boundary value prob-lems can often be solved by the application of theLaplace transform to remove the variable, t, fromthe system of equations. This approach yields anelastic problem in the transformed variables.With the elastic solution to the transformed prob-lem, a viscoelastic solution is achieved by thereplacement of the elastic constants with theappropriate viscoelastic operators; that is, theelastic–viscoelastic correspondence principle isinvoked, and then the inverse Laplace transformis performed. In the case of the given contactproblem, however, the boundary conditions arenormally taken from the compatibility betweendisplacements and stresses with the prescribedsurface displacements and surface tractions,respectively, and these conditions are a functionof time.9–11 Therefore, in general, the transformapproach is not applicable, although a solutionby Lee and Radok10 of this type for an incom-pressible material (v ¼ 0.5) has been shown to bevalid for a monotonically increasing contact

radius. A slightly different approach by Ting11

yields separate solutions for increasing anddecreasing contact radius.

Indentation analogues to creep and relaxationmeasurements have been suggested. For indenta-tion creep, a constant force P0 is applied att ¼ 0 and held. During such a test, the penetrationdepth, h(t), and hence the contact area, A(t)¼ �r2(t), increase with time such that the stress isnot constant but rather decreases with time. Addi-tionally, the stress state below the indenter tip willbe multiaxial rather than uniaxial. Despite thesedepartures from traditional creep testing, viscoe-lastic contact models have been applied to theindention creep problem to yield the followingrelationship between J(t), P0, A(t), and h(t) for aparaboloidal indenter of radius R:

JðtÞ ¼ 8hðtÞ ffiffiffiffiffiffiffiffiffiAðtÞp

3ffiffiffi�

p ð1� vÞP0ð5Þ

Simplifications related to eq 5 include theassumptions that v is constant and that J(t) isequal to 1/G(t). The indentation analogue tostress relaxation involves conditions in which aconstant penetration depth, h0 (with a corre-sponding contact area, A0), is applied at t ¼ 0and held. Although the state of stress is againcomplex, the nominal indentation strain shouldremain relatively constant. For a paraboloidalindenter of radius R, viscoelastic contact modelsgive the following relationship between G(t), h0,A0, and the force, P(t):

GðtÞ ¼ 3ffiffiffi�

p ð1� vÞPðtÞ8h0

ffiffiffiffiffiffiA0

p ð6Þ

Again, v is assumed to be constant. Additionally,the Ting model yields equations for constant-force indentation creep and constant-depthstress relaxation for conical tips:

JðtÞ ¼ AðtÞð1� vÞP0 tan �

ð7Þ

GðtÞ ¼ ð1� vÞPðtÞ tan �

A0ð8Þ

In these equations, � is the cone semiapicalangle, and v is assumed constant in both equa-tions. The additional assumption that J(t) isequal to 1/G(t) is made in eq 7. Also, eqs 7 and 8were recently derived for pyramidal indenters6

directly from Hooke’s law under the assumption

1796 VANLANDINGHAM ET AL.

that a representative stress is given by P/A anda representative strain is given by (cot �)(dh/h).Thus, the creep compliance and relaxation mod-ulus in eqs 5–8 appear to be ratios of these sim-ple representations for stress and strain. Inindentation analysis, pyramidal tips are oftenmodeled as perfect conical tips, often with anadditional geometric factor that is related tononcircular contact areas of pyramidal tips withrespect to conical tips.8 Although eqs 7 and 8were applied to pyramidal tip indentation meas-urements in this study, this additional geometricfactor, normally a 3–5% correction of the contactarea, was not used.

In recent efforts to measure the viscoelasticbehavior of polymers with instrumented indenta-tion, linear viscoelastic models, including modelssimilar to the Ting model and the Lee–Radokmodel and/or simple spring–dashpot models(e.g., the standard linear solid), have been ap-plied.1,2,6,15,16 Whether or not linear viscoelas-ticity is obeyed during instrumented indentationmeasurements, however, is difficult to ascertain.The intense stresses and strains expected local tothe tip–sample contact as well as the mixedstress conditions suggest that simple linear vis-coelastic relations may not be applicable. Onetest of linear viscoelasticity is the lack of depend-ence of the creep compliance (or stress relaxationmodulus) on the magnitude of the stress (orstrain). Although this condition is necessary butnot sufficient proof of linear viscoelasticity, eqs5–8 present an opportunity to determine for par-aboloidal and conical (and presumably pyrami-dal) indentation probes if linear viscoelasticity isnot obeyed under particular test conditions. Addi-tionally, these equations provide a means ofmeasuring viscoelastic behavior with instru-mented indentation that can be compared toother types of rheological measurements.

In this article, analyses based on contactbetween a rigid indenter and a linear viscoelas-tic material are used to calculate J(t) and G(t)from instrumented indentation testing. Four dif-ferent polymers are characterized with tradi-tional rheometry and with measurements of theconstant-force indentation creep and constant-depth stress relaxation. For the indentationmeasurements, checks of linear viscoelasticityare performed by the study of the dependence ofthe creep compliance on the indentation force.The resulting measurements of the creep com-pliance and relaxation modulus are comparedwith traditional rheometry measurements.

EXPERIMENTAL

Materials

The materials used in this study included anamine-cured epoxy, poly(methyl methacrylate)(PMMA), and two poly(dimethyl siloxane)(PDMS) materials. Epoxy films approximately190 �m thick were cast onto silicon wafers in aCO2-free and H2O-free glovebox with a draw-down technique. Highly pure diglycidyl ether ofbisphenol A with a mass per epoxy equivalent of172 g (DER 332, Dow Chemical) and 1,3-bis(a-minomethyl)cyclohexane (Aldrich) were mixedat the stoichiometric ratio. All samples werecured at room temperature for 48 h, and thiswas followed by postcuring at 130 8C for 2 h.The films were then removed from the siliconsubstrates by immersion in warm water andthen peeling with tweezers. The PMMA samplewas obtained from a commercial Plexiglasacrylic sheet from AtoHaas North America, Inc.The PDMS samples were obtained from DowCorning Corp. The first sample (called filledPDMS) was a general-purpose (GP-50) silica-filled crosslinked PDMS with a thickness of3.2 mm. The second, unfilled PDMS sample wasmade from Sylgard 184 mixed at a 10:1 (resin/crosslinker) ratio. This PDMS material was castonto a glass plate, degassed for 30 min at 20 inHg,and cured for 4 h at 65 8C; this resulted in asample thickness of 2.7 mm. The glass-transi-tion temperature (Tg) for epoxy and PMMA wasestimated with differential scanning calorimetrywith heating and cooling rates of 10 8C/min; thevalues were determined from cooling. The val-ues of Tg for the PDMS materials were deter-mined as the peak in the loss modulus withshear rheometry with a heating rate of 2 8C/min,0.05% strain, and a frequency of 1 Hz. Theresulting values of Tg were 112 6 1 8C for theepoxy films, 106 6 1 8C for PMMA, �67 6 3 8Cfor the filled PDMS, and �120 6 2 8C for theunfilled PDMS. Because all other rheologicaltesting was performed at room temperature, thevalues for Poisson’s ratio were assumed to be0.3 for the epoxy and PMMA samples and 0.5for the PDMS samples. For each materialstudied, rheometry testing and indentation test-ing were performed concurrently over a span ofapproximately 3 days, and so physical agingtimes for the epoxy and PMMA were assumed tobe the same for all tests.


Solid Rheometry Measurements

Stress relaxation measurements were made intorsion on the filled PDMS material with anadvanced rheometric expansion system (ARES,Rheometrics Scientific, Inc.) and in tension onthe PMMA and epoxy samples with a rheologi-cal solids analyzer (RSA II, Rheometrics Scien-tific, Inc.). The unfilled PDMS material wastested in compression with the RSA II. Instru-mental capabilities limited the amount of strainapplied to PDMS samples in the ARES to 0.08(8%) and in the RSA II to 0.01 (1%). Equation 4was used to convert measurements of the stressrelaxation modulus in tension or compression,E(t), to that in shear, G(t).

Instrumented Indentation Measurements

Instrumented indentation was performed with aNanoIndenter XP and a NanoIndenter DCM(MTS Systems, Inc.). The XP system, in general,was used for applied forces from 100 to 0.2 mN,whereas the DCM system was used for appliedforces from 10 to 0.01 mN. Both systems couldbe used to characterize the epoxy and PMMAsamples, but only the DCM system could beused to characterize the much more compliantPDMS samples because of the force resolutionlimitations of the XP system. For measurementsmade with the XP system, two different probetip shapes were used, a Berkovich pyramidal tipand a rounded 908 conical tip with a tip radiusof approximately 10 �m (manufacturer specifica-tion). Only a Berkovich tip was available fortesting with the DCM system. The tip shape hasbeen measured for these probes by the indenta-tion of reference samples and by the directimaging of the probes with an atomic forcemicroscope, as detailed elsewhere.17 For theindentation creep and stress relaxation tests, atechnique often called the continuous stiffnessmethod was employed, in which a small har-monic oscillation was superposed over the con-stant force (creep) or displacement (relaxation)with a frequency of 45 Hz and an amplitude ofapproximately 5 nm, as controlled with signalfeedback. An appropriate dynamic model of thesystem18 with an assumption of negligibledamping was then used to calculate values of Aas a function of h for use in eq 5–8.

The indentation creep response was measuredwith step loading to a prescribed force, P0, whichwas then held for 100 s. For a given test, P0 wasreached in less than 0.1 s and was maintained

within 62 �N for the XP system and 61 �N forthe DCM system. After this near-step loading,however, the harmonic oscillation superposedover the constant force required time to stabilizebefore measurements of the contact area wereconsidered to be accurate. This time lapse wasapproximately 10–12 s for the XP system and 4–5 s for the DCM system. To monitor the thermaldrift rate of the system, the same constant forcetest method was performed on fused silica beforeand after each set of tests on a polymer sample.Because the system thermal drift could be posi-tive or negative, no attempt was made to removethe system drift from the creep measurements.Rather, the hold time was limited to approxi-

Figure 1. Examples of system performance for theindentation creep tests for (a) the XP system with a200 �N force and (b) the DCM system with a 10 �Nforce. Both examples are for a Berkovich tip indentingepoxy. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com]


mately 100 s, at which point the creep rates forthe two glassy polymers became of the sameorder as the drift rate (ca. 60.02 nm/s). For a setof four or more tests at the same nominal force,the difference between the lowest and highestvalues of P0 was less than 50 �N for the XP sys-tem and 3 �N for the DCM system, regardless ofthe magnitude of the prescribed force. Examplesof system performance at the lowest appliedcreep forces are shown in Figure 1(a) for the XPsystem and in Figure 1(b) for the DCM system.Creep compliance was calculated with eq 5 or 7according to the tip geometry.

The indentation relaxation response wasmeasured with a step displacement to a pre-scribed depth, h0. As for the indentation creeptests, the length of the relaxation tests was lim-ited to 100 s to minimize uncertainty due to sys-tem thermal drift. For a given test, h0 wasreached in approximately 6 s, and this was fol-lowed by a period of roughly 6–8 s during whichthe system feedback attempted to control the dis-placement at the prescribed constant value. Aslight overshoot of approximately 5–10% of h0

was observed for the XP system, and an over-shoot of 1–3% of h0 was observed for the DCM

Figure 2. Examples of system performance for the indentation stress relaxationtests for (a) the XP system with a 500 nm displacement and (b) the DCM systemwith a 100 nm displacement. Both examples are for a Berkovich tip indenting epoxy.

Figure 3. Displacement plotted as a function of time for indentation creep testswith and without a superposed 5 nm amplitude dynamic oscillation at 45 Hz for sev-eral quasi-static creep force levels: (a) epoxy and (b) PDMS. All data are for the DCMsystem and a Berkovich tip.


system. After the initial overshoot, displacementwas maintained within 61–2 nm for both the XPand DCM systems, except for very large displace-ments with the XP system, for which displace-ment variations were as much as 65 nm for anominal displacement of 4000 nm. For a set of 10tests at the same nominal depth, the differencebetween the lowest and highest values of h0 wasless than 5 nm for both systems for target depthsof 1500 nm or less and 10 nm for target depthsgreater than 1500 nm. However, the repeatabilityof the force values was better for the DCM sys-tem than for the XP system. Examples of systemperformance at the lowest applied depth levelsare shown in Figure 2(a) for the XP system andin Figure 2(b) for the DCM system. The relaxa-tion modulus was calculated with eq 6 or 8according to the tip geometry.

Residual Depth (RD) Measurements

An additional study of RD as a function of timefor indentation creep measurements on epoxywas made for a Berkovich tip and the DCM sys-tem. The same type of indentation creep testdescribed in the previous subsection was used.However, the creep force was removed after 20 sfor five tests, after 50 s for five tests, and after100 s for five tests for each of three force levels:

0.05, 0.5, and 5 mN. The samples were stored ina laboratory environment overnight and thenscanned in the tapping mode with a DigitalInstruments Dimension 3100 atomic force micro-scope and a NanoScope 3a controller (Veeco Met-rology). One image of each residual impressionwas captured, and cross-section and bearinganalyses provided in the offline atomic forcemicroscopy (AFM) software were used to deter-mine the depth of the impression.

RESULTS AND DISCUSSION

Effect of Harmonic Oscillation

As detailed in the Experimental section, aharmonic oscillation was superposed over thequasi-static forces in the indentation creep andstress relaxation measurements; a controlleddisplacement amplitude of approximately 5 nmand a frequency of 45 Hz was used for both theXP and DCM systems. Although the harmonicdisplacement was generally small with respectto the quasi-static displacements, the dissipatedenergy could have caused an increase in thetemperature that would have altered the meas-ured response in comparison with tests madewithout this harmonic oscillation. In Figure 3,

Figure 4. Log–log plot of the creep compliance, J(t), for an indentation creepexperiment on epoxy with a rounded conical tip (manufacturer-determined tip radiusof 10 �m) and the XP system. Each data point represents an average value from aminimum of eight experiments (two to three sets of four to six tests), with the errorbars representing an estimated standard deviation (k ¼ 1). In some cases, the errorbars are smaller than the data point symbols. [Color figure can be viewed in theonline issue, which is available at www.interscience.wiley.com]


the displacement responses to several creepforces are shown as a function of time for boththe epoxy and filled PDMS. In both cases, theevolution of the displacement with time wasidentical within the experimental uncertaintyfor the measurements made with and withoutthe harmonic oscillation component.

Creep and Stress Relaxation of Epoxy and PMMA

An example of indentation creep compliance

determined for an epoxy sample with a rounded

conical tip (manufacturer-determined tip radius

of 10 �m) is shown in Figure 4. The creep com-

pliance is clearly dependent on the indentation

Figure 5. Log–log plot of the creep compliance, J(t), for an indentation creep experi-ment on epoxy with a Berkovich tip and the XP system. Each data point represents anaverage value from a minimum of eight experiments (two to three sets of four to six tests),with the error bars representing an estimated standard deviation (k ¼ 1). [Color figurecan be viewed in the online issue, which is available at www.interscience.wiley.com]

Figure 6. Log–log plot of the creep compliance, J(t), for an indentation creep experi-ment on epoxy with a Berkovich tip and the DCM system. Each data point represents anaverage value from a minimum of eight experiments (two to three sets of four to six tests),with the error bars representing an estimated standard deviation (k ¼ 1). [Color figurecan be viewed in the online issue, which is available at www.interscience.wiley.com]


forces between 0.2 and 20 mN, and this is anindication of nonlinear behavior. The creep com-pliance was also observed to be a function offorce for the epoxy sample with Berkovichindentation tips for both the XP (Fig. 5) and theDCM (Fig. 6). Furthermore, the compliance val-ues for the two sets of Berkovich indentationtests were similar for similar applied force lev-els, with differences likely due to slight differen-ces in the tip geometry17 leading to differencesin the applied stress field.

Qualitatively, the data appear to be consistentwith the behavior expected of a glassy epoxy poly-mer:3,19 compliance values are on the order of10�9 Pa and trend higher with increasing creeptime and increasing force. Additionally, thetrends in the compliance values with time appearto be similar for each of the force levels studiedwith the rounded cone and for the higher forcelevels with the Berkovich tips, and this suggestsseparability of the time-dependent behavior fromthe stress-dependent behavior.3 The differencesin the slope at the lower indentation creep forcesfor the Berkovich tips (Figs. 5 and 6) could haveresulted from uncertainties in the tip shaperelated to indentation depths less than 100 nm.Normally, the tip shape is determined for a large

range of contact depths, and the curve fits usedto represent the area function tend to match thetip shape data best at larger depths at which thepyramidal angle dominates. The error percentagein the contact area at shallow depths was esti-mated to be at most 10%, which would shift theJ(t) data up by less than 0.05 log units.

The results of stress relaxation testing arepresented for epoxy in Figure 7, includingBerkovich tip indentation and rheometry data.For the data measured with the RSA II rheome-ter in tension on the epoxy material, the stressrelaxation modulus values were similar forstrain levels of 0.01 and 0.1%. The application ofa 1% strain, however, resulted in lower modulusvalues and a slight increase in the time depend-ence in comparison with the two lower strainlevels; this is typical of nonlinear viscoelasticbehavior of glassy polymers.3 Similar behaviorwas observed for the stress relaxation modulusvalues measured with indentation with a Berko-vich tip: increases in the applied constant dis-placement resulted in lower relaxation modulusvalues. Although relatively small for a glassypolymer at room temperature, the time depend-ence measured with indentation was similar tothat measured with rheometry.

Figure 7. Log–log plot of the stress relaxation modulus, G(t), for an indentation relax-ation experiment on epoxy with a Berkovich tip and both the XP and DCM systems.Each data point represents an average value from a minimum of eight experiments(two to three sets of four to six tests). Superposed on the plot are rheometry results fromthe RSA II system in tension [G(t) was converted from E(t) with eq 4], for which eachdata point represents an average of three experiments. For all data shown, the errorbars represent an estimated standard deviation (k ¼ 1). The percentages for the tensilerheometry data represent the percentage of the strain applied to the samples. [Color fig-ure can be viewed in the online issue, which is available at www.interscience.wiley.com]


In Figure 8, log–log plots of the creep compli-ance and stress relaxation modulus are shownfor PMMA. The indentation data in these plotswere collected with Berkovich tips with both the

XP and DCM systems. The data are again quali-tatively consistent with behavior expected of aglassy polymer, and the magnitudes of the creepcompliance and relaxation modulus values are

Figure 8. Log–log plots of (a) the creep compliance, J(t), and (b) the stress relaxa-tion modulus, G(t), for indentation creep and stress relaxation experiments, respec-tively, on PMMA with a Berkovich tip and both the XP and DCM systems. Each datapoint represents an average value from a minimum of eight experiments (two tothree sets of four to six tests), with the error bars representing an estimated stand-ard deviation (k ¼ 1). Superposed on part b are rheometry results from the RSA IIsystem in tension [G(t) was converted from E(t) with eq 4], for which each data pointrepresents an average of three experiments. For all data shown, the error bars repre-sent an estimated standard deviation (k ¼ 1). The percentages for the tensile rheom-etry data represent the percentage of the strain applied to the samples. [Color figurecan be viewed in the online issue, which is available at www.interscience.wiley.com]


very similar to values determined for epoxy (seeFigs. 5–7). However, the indentation valuesappear to be less dependent on the indentationforce (creep) or indentation displacement (relax-ation). This observation seems to indicate lessstrain sensitivity for PMMA than for epoxy atthe large strain levels imposed during Berkovichtip indentation.

Comparing the Berkovich tip indentationmeasurements of J(t) and G(t) for epoxy andPMMA (see Figs. 5–8), we found that anincrease in the applied force caused J(t) toincrease, and an increase in the displacementcaused a decrease in G(t); this is consistent withthe qualitatively reciprocal nature of these func-tions. Furthermore, the assumption that J(t)G(t)is equal to 1 appears to hold for tests in whichthe force and displacement levels were similar.For example, with the DCM data from Figures 6and 7, respectively, J(2.5 mN, 10 s) was 1.74� 10�9 Pa�1 and G(1000 nm, 10 s) was 0.56 � 109

Pa; therefore, J(t)G(t) was 0.97. However, the val-ues of G(t) measured via indentation with a Ber-kovich tip were much lower than the G(t) valuescalculated from tensile rheometry, by 70–80% forepoxy and by 80% for PMMA. For many poly-mers, the stress relaxation modulus decreasessignificantly with increasing strain, and so thetrends with time are similar and the data can besuperposed by vertical shifts.3 Thus, this discrep-ancy could be related to the much larger strainlevels related to indentation tests with a Berko-vich tip in comparison with the rheometry tests.

To explore whether the measured creepresponse was mainly viscoelastic or included asignificant contribution from yield behavior, theresidual indentation depth was studied for theBerkovich indentation of epoxy with the DCMsystem; the results are shown in Figure 9. Thedifferences between the measured creep dis-placement, h, and the corresponding RD valuesmeasured with AFM are shown for differentcreep hold periods. For example, h(100 s) � h(20 s)represents the difference in the creep displace-ment measured for a hold time of 100 s andthat measured for a hold time of 20 s, as meas-ured with the DCM system; the correspondingdifference in RD between an impression pro-duced for a hold time of 100 s and that producedfor a hold time of 20 s is designated RD(100 s)� RD(20 s). Examples of RD measurements withAFM are shown in Figure 9(b,c) for a 5 mNcreep force and hold times of 20 s and 100 s,respectively. On the basis of the vertical dis-

tance measurements shown, the correspondingRD values were RD(20 s) ¼ 304.2 nm andRD(100 s) ¼ 337.9 nm. After the averaging offive such measurements for each creep forceand hold time, the average values were sub-tracted to yield the data in Figure 9(a). For eachof the three force levels, the increase in h withtime is associated with a significant increase inRD. Thus, for the Berkovich indentation ofepoxy, the measured change in the creep dis-placement over a give period appears to largelyconsist of material flow characteristic of yield orpostyield conditions rather than a purely viscoe-lastic creep response, even at very low force lev-els (50 �N). The corresponding measurements ofJ(t) and G(t) thus contain a substantial contribu-tion from yield phenomena, and this explainsthe large differences in comparison with the val-ues associated with viscoelastic behavior. Thisconclusion likely also applies for the Berkovichindentation of PMMA and the indentationresults for epoxy with the rounded conical tip.

To estimate the levels of strain applied duringthe indentation measurements, effective strainswere calculated for the Berkovich and roundedconical tip shapes used with the XP system.These calculations were made with empiricallybased analyses attributed to Tabor4 in whichideal plastic behavior was assumed. Althoughthe postyield flow significantly affected themeasured indentation response for epoxy andPMMA, these materials are far from ideal plas-tic materials. Thus, this analysis is at best qual-itative. The following two equations were usedto estimate the effective strain, �", for ideal para-boloidal and conical tip geometries, respectively:

�" ¼ 0:2r=R ð9Þ

�" ¼ 0:25 cot � ð10Þ

For the Berkovich tip, an effective conical angle,�, was determined to be approximately 70.458.For the rounded cone, an effective radius, Reff,was determined from tip shape analysis to rangefrom 4 �m at shallow depths to 10 �m at largerdepths.17 In Figure 10, data from Figures 4 and5 are combined to show that the correspondingJ(t) values are lower for the rounded conical tip,except for 10 and 20 mN. Additionally, the J(t)values for the rounded conical tip at 2.5 and5 mN creep forces are similar to values for theBerkovich tip and the DCM system at the lowercreep force levels (see Fig. 6). The relative effec-


Figure 9. (a) Bar chart representing the differences between the measured creepdisplacement, h, for the Berkovich indentation of epoxy with the DCM system andthe corresponding RD values measured with AFM. Each data point represents anaverage of five experiments, and the error bars represent the corresponding esti-mated standard deviation (k ¼ 1). (b,c) Examples of cross sections from AFM imagesfor a 5 mN creep force after 20 and 100 s hold periods, respectively. [Color figure canbe viewed in the online issue, which is available at www.interscience.wiley.com]

tive strain, shown in Figure 11, is also similarfor the rounded conical tip in comparison with aBerkovich tip for creep forces of 2.5 mN andhigher. Thus, at these higher creep forces, theeffective strains and the corresponding J(t)values for the rounded conical tip are similar tothose for the Berkovich tips. Also, the largervariation in the creep compliance for therounded conical tip with force reflects the largerexpected changes in the associated effectivestrain in comparison with the Berkovich tip.Although the plots in Figure 11 indicate noexpected variation in strain imposed by an idealBerkovich tip, a small variation in the effectivestrain related to deviations in the actual tipgeometry from the ideal case could have affectedthe measurements. More likely, however, theepoxy is sensitive not only to the nominal strainlevel, such as that based on eq 10, but also tolocal strain levels that perhaps are mildlydependent on displacement.

In Figure 11(b), a snapshot of J(t) at t ¼ 50 sfor epoxy is shown as a function of the esti-mated effective strain for the rounded conicaltip (from Fig. 4) and the Berkovich tip (fromFig. 5) used with the XP system. The dashedline represents a linear extrapolation of thestrain dependence of J(t ¼ 50 s) to small strainsfor the rounded conical tip data. This extrapola-tion appears to indicate that, at strain levelsbelow 1%, J(t) values should approach values

typically measured for epoxy under conditions oflinear viscoelastic behavior. Thus, achievingsuch small strain values for indentation meas-urements will likely require the use of largerounded tips and small penetration depths toreduce the r/R ratio to appropriate levels.

PDMS Creep and Stress Relaxation

Indentation creep compliance results for thefilled PDMS material with a Berkovich tip andthe DCM system are shown in Figure 12. Again,the data appear, at least qualitatively, to be con-sistent with the expected behavior:3 the compli-ance values are on the order of 10�6 Pa andtrend slightly higher with increasing creep time.Also, the compliance values generally tend todecrease with increasing force. However, thedata scatter was significant, and, for much ofthe force range, that is, between 100 �N and2 mN, J(t) appears to be similar.

The results of stress relaxation testing arepresented for both PDMS materials in Figure13, including indentation and rheometry data.Nonlinear behavior was observed for the filledPDMS in the rheometry measurements in tor-sion, as G(t) values decreased with increasingstrain levels. A small amount of time depend-ence, which decreased with increasing strain,was also observed. In comparison, the relaxationbehavior of the unfilled PDMS [Fig. 13(b)] meas-

Figure 10. Log–log plot of the creep compliance, J(t), as a function of time t, compar-ing the indentation creep data for epoxy from Figures 4 (rounded conical tip and XP)and 5 (Berkovich tip and XP). The error bars were removed for clarity. [Color figurecan be viewed in the online issue, which is available at www.interscience.wiley.com]


ured in compression was linear elastic; althoughthe rheometry data is shown only for a strain of0.05%, the response measured over a range ofstrain levels (data not shown) was the samewithin the experimental uncertainty as thatshown. Both the nonlinear response at relativelylow strain levels and the slight time dependenceobserved in Figure 13(a) are probably related tothe presence of the silica filler. Unfilled PDMSis linear elastic to large strain levels, whereaslightly filled PDMS and other rubbery cross-

linked polymers often behave in a nonlinearelastic fashion, sometimes with a slight timedependence; this is consistent with the observedbehavior.

For the filled PDMS, the indentation relaxa-tion modulus values were independent of thepenetration depth for depths from 1 to 5 �mwithin the data scatter. For penetration depthsof 10 and 15 �m, the relaxation modulus valuesare again similar to but slightly higher thanthose for the three smaller penetration depths.

Figure 11. (a) Plot of the effective strain estimates (after Tabor;4 predictions of theeffective strain, �", were made with eqs 9 and 10 with tip shape information)measured for the Berkovich and rounded conical tips used in this study and(b) plot of the creep compliance, J(t), at time t ¼ 50 s for epoxy as a function ofthe estimated effective strain for the rounded conical tip and the Berkovich tipused with the XP system. The dashed line represents a linear extrapolation ofthe strain dependence of J(t ¼ 50 s) to small strains. [Color figure can beviewed in the online issue, which is available at www.interscience.wiley.com]


Similarly for the unfilled PDMS, the indentationrelaxation modulus values were independent ofthe penetration depth for depths from 3.5 to20 �m within the data scatter, with values cor-responding to depths of 2.5 and 0.75 �m beingslightly lower. The correspondence of larger val-ues of G(t) to larger penetration depths could berelated to several factors. First, this trend couldbe the result of a stiffening effect due to arestriction of network motion with increasedlocal strain levels. Second, the tip shape wascharacterized only for contact depths less than500 nm because of the maximum force limita-tions of the DCM system when fused silica glassis indented. The curve fit to this tip shape datawas then extrapolated to the much larger con-tact depths reached during the indentation ofPDMS samples, and thus the error in the valuesof the area used in the calculations likelyincreased with increasing contact depth. Addi-tionally, because the PDMS materials areextremely compliant, the identification of theinitial point of tip–sample contact is difficult. Infact, penetration depths of hundreds of nano-meters or more can occur at force levels on theorder of 1 �N, largely because of the competingtip–sample adhesion forces. This problem ofdetecting initial contact combined with the lim-ited tip shape information available could havecaused the trend in G(t) values with depth, as

uncertainties in the initial contact point willhave a larger influence on data taken at smallerdepths. Until these measurement issues aresolved, understanding whether such measuredtrends are real or artificial will be difficult.

The difference in the PDMS relaxation modu-lus values between the rheometry measure-ments and indentation measurements is largefor both PDMS materials. Comparing G(t) val-ues from torsion measurements at 8% strainwith the indentation G(t) values for the filledPDMS, we found that both sets of data exhibiteda similar lack of time dependence, but theindentation values were lower by 65%. The dif-ference in the indentation values in comparisonwith the rheometry values at 1% strainincreased to 70%. For the unfilled PDMS, thedifference was even larger, ranging from 85%for the larger penetration depths to 95% for adepth of 0.75 �m. The larger difference for theunfilled PDMS with respect to the filled PDMScould be related to the larger influence of uncer-tainties related to detecting the initial contactpoint and the larger influence of tip–sampleadhesion on the contact mechanics for the morecompliant unfilled material.

For the stress relaxation modulus data forepoxy, PMMA and PDMS [Figs. 7, 8(b), and 13,respectively], any potential vertical shifting ofthe curves as a function of the strain level

Figure 12. Log–log plot of the creep compliance, J(t), for an indentation creep experi-ment on PDMS with a Berkovich tip and the DCM system. Each data point represents anaverage value from a minimum of eight experiments (two to three sets of four to six tests),with the error bars representing an estimated standard deviation (k ¼ 1). [Color figurecan be viewed in the online issue, which is available at www.interscience.wiley.com]


would tend to indicate much larger effectivestrains in the indentation measurements withrespect to the rheological measurements.However, a number of complicating factors existregarding the indentation creep and stress

relaxation measurements. First, for Berkovichindentation and likely for indentation with therounded conical tip, the strain levels were largeenough to induce yielding of the epoxy andPMMA, potentially causing the much lower

Figure 13. Log–log plots of the stress relaxation modulus, G(t), for relaxationexperiments on the two PDMS materials: (a) filled PDMS and (b) unfilled PDMS.Indentation relaxation measurements were made with a Berkovich tip and the DCMsystem. Each data point represents an average value from a minimum of eightexperiments (two to three sets of four to six tests). Superposed on the plots are rhe-ometry results. In part a, data are shown for the ARES torsional rheometer; in partb, data from the RSA II in compression are shown [G(t) was converted from E(t) witheq 4]. For all rheometry data shown, each data point represents an average of threeexperiments, and the error bars represent an estimated standard deviation (k ¼ 1).The percentages given in part a for the torsional rheometry data represent the per-centage of the strain applied to the samples. [Color figure can be viewed in the onlineissue, which is available at www.interscience.wiley.com]


relaxation modulus values and much highercreep compliance values with respect to tradi-tional viscoelastic measurements. For the PDMSmaterials, uncertainties associated with a lackof tip shape information at large depths, prob-lems detecting the point of initial tip–samplecontact, and tip–sample adhesion also presentproblems regarding the quantitative nature ofthe indentation measurements. Second, the cal-culations of the indentation creep complianceand stress relaxation modulus were based on alinear viscoelastic model, whereas the measure-ments indicated nonlinear viscoelastic and yieldbehavior for the epoxy and PMMA and nonlin-ear elastic behavior for the filled PDMS. Also,the model does not include effects of tip–sampleadhesion, which likely played a significant rolein the PDMS measurements. Finally, the meas-urements of J(t) and G(t) were based on adynamic model of the indentation system inwhich damping at the tip–sample junction isassumed to be negligible; this assumption waspotentially violated in these measurements.Thus, the absolute magnitudes of the indenta-tion values of J(t) and G(t) plotted in Figures 4–10, 12, and 13 are not without significant uncer-tainty. However, the exhibited behavior appearsto be consistent with rheometry measurementsand with the known bulk rheology of these poly-mers at the high stress and strain levelsexpected under the indenter tip.

CONCLUSIONS

The use of instrumented indentation to charac-terize the mechanical response of polymericmaterials was studied. A model based on contactbetween a rigid probe and a linear viscoelasticmaterial was used to calculate values for thecreep compliance and stress relaxation modulusfor epoxy, PMMA, and two PDMS materials.Bulk rheometry studies were used for compari-son. Unfortunately, the magnitudes of the inden-tation and rheometric values of the creep com-pliance or stress relaxation modulus are difficultto compare directly. This deficiency is related tothe following factors:

1. The indentation values were calculatedwith an analytical model based on linearviscoelastic behavior that does not includetip–sample adhesion. However, the major-ity of the measured responses were nonlin-

ear, particular for the Berkovich tip inden-tation of glassy epoxy and PMMA samples,for which a significant fraction of theresponse was due to yield behavior. Also,tip–sample adhesion likely affected theresults for the PDMS materials.

2. For compliant materials, such as PDMS,detecting the point of initial tip–samplecontact is challenging and can lead to arti-ficial trends in modulus and compliancevalues. Additionally, the large penetrationdepths that must be used to overcome lowforce limitations of the instrumentedindentation system require large extrapo-lations of tip geometry data that can leadto errors in the magnitudes of the modulusand compliance.

3. Although the use of harmonic oscillationsuperposed over the quasi-static forces didnot alter the measured creep and stressrelaxation responses, the determination ofthe contact area, which was used to calcu-late the creep compliance and relaxationmodulus, was based on a dynamic model ofthe indentation system in which dampingat the tip–sample junction is assumed to benegligible. For viscoelastic materials, thisassumption might be violated, and in fact, aseparate assumption of infinite load-framestiffness can be used to extract energy stor-age and loss characteristics of polymers.

4. Often in the analysis of instrumentedindentation data, factors are applied tocorrect for differences in experimentalcontact conditions and model contact con-ditions. However, these correction factors,which were not used in this study, havebeen determined for linear elastic andelastoplastic constitutive behavior, andthe appropriateness of their use for vis-coelastic behavior is unknown.

Despite these issues, the trends in the indenta-tion data are similar to those in the rheometrydata, and this suggests that the measurementsmay, in certain cases, have sufficient physicalsimilarity. However, care must be taken tounderstand the behaviors being measured withrespect to the stress and strain levels appliedunder the indentation tip. With such under-standing, the large magnitude and nonuniform-ity of the strains and the mixed deformationmodes associated with indentation measure-ments can be used potentially to access a wide


range of viscoelastic behavior by variations ofthe tip shape and force (depth) levels. This capa-bility would include the potential to use largerounded tips to measure linear viscoelastic char-acteristics, with the understanding that tip–sample adhesion plays a more prominent role asthe tip size increases. New analyses and measure-ment protocols are currently being explored fordeveloping a more complete understanding of therelationships between instrumented indentationdata and viscoelastic behavior.

The authors gratefully acknowledge fruitful collabora-tions with John Villarrubia of the National Institute ofStandards and Technology regarding tip shape determi-nation with atomic force microscopy, Sheng Lin-Gibsonof the National Institute of Standards and Technologyregarding torsional rheometry measurements, Xiao-hong Gu of the National Institute of Standards andTechnology for epoxy sample preparation, and VincentJardret, formerly of MTS Systems Corp., for his helpwith the test method development. The authors alsoacknowledge helpful discussions with Don Hunston ofthe National Institute of Standards and Technologyand Greg McKenna of Texas Tech University.

REFERENCES AND NOTES

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3. Ferry, J. D. Viscoelastic Properties of Polymers;Wiley: New York, 1980.

4. Tabor, D. Hardness of Metals; Clarendon: Oxford,1951.

5. Briscoe, B. J.; Fiori, L.; Pelillo, E. J Phys D: ApplPhys 1998, 31, 2395.

6. Shimizu, S.; Yanagimoto, T.; Sakai, M. J Mater

Res 1999, 14, 4075.7. Chaudhri, M. M. Acta Mater 1998, 46, 3047.8. Dao, M.; Chollacoop, N.; Van Vliet, K. J.; Venka-

tesh, T. A.; Suresh, S. Acta Mater 2001, 49, 3899.9. Lee, E. H. Q. Appl Math 1955, 13, 183.

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27, 438.11. Ting, T. C. T. J Appl Mech 1966, 33, 845.12. Yang, W. H. J Appl Mech 1966, 33, 395.13. Hunter, S. C. J Mech Phys Solids 1960, 8, 219.14. Graham, G. A. C. Int J Eng Sci 1965, 3, 27.15. Lu, H.; Wang, B.; Ma, J.; Huang, G.; Viswanathan,

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