Oliver Pharr Paper

REVIEWThis section of Journal of Materials Research is reserved for papers that are reviews of literature in a given area.

Measurement of hardness and elastic modulus byinstrumented indentation: Advances in understanding andrefinements to methodologyW.C. OliverMTS Systems Corporation, Oak Ridge, Tennessee, 37830G.M. Pharra)The University of Tennessee and Oak Ridge National Laboratory, Department of Materials Scienceand Engineering, Knoxville, Tennessee 37996

(Received 15 June 2003; accepted 23 September 2003)

The method we introduced in 1992 for measuring hardness and elastic modulus byinstrumented indentation techniques has widely been adopted and used in thecharacterization of small-scale mechanical behavior. Since its original development, themethod has undergone numerous refinements and changes brought about byimprovements to testing equipment and techniques as well as from advances in ourunderstanding of the mechanics of elasticplastic contact. Here, we review our currentunderstanding of the mechanics governing elasticplastic indentation as they pertain toload and depth-sensing indentation testing of monolithic materials and provide anupdate of how we now implement the method to make the most accurate mechanicalproperty measurements. The limitations of the method are also discussed.

I. INTRODUCTIONThe method we introduced in 1992 for measuring

hardness and elastic modulus by instrumented indenta-tion techniques has widely been adopted and used in thecharacterization of mechanical behavior of materials atsmall scales.1,2 Its attractiveness stems largely from thefact that mechanical properties can be determined di-rectly from indentation load and displacement measure-ments without the need to image the hardness impres-sion. With high-resolution testing equipment, this facili-tates the measurement of properties at the micrometerand nanometer scales.35 For this reason, the method hasbecome a primary technique for determining the me-chanical properties of thin films and small structural fea-tures.623 Films with characteristic dimensions of the or-der 1 m are now routinely measured, and with goodtechnique, the method can be used to characterize, atleast in a comparative sense, the properties of films asthin as a few nanometers.

During the past decade, we have made several impor-tant changes to the method that both improve its accuracyand extend its realm of application. These changes havebeen developed both through experience in testing alarge number of materials and by improvements to test-ing equipment and techniques. For example, the meas-urement of contact stiffness by dynamic techniques al-lows for continuous measurement of properties as a func-tion of depth and also facilitates more accurateidentification of the point of first surface contact.24 Wehave also developed improved methods for calibratingindenter area functions and load frame compliances.Thus, the primary purpose of this article is to provide thereader with an update on how we implement the methodin practice now and the improvements we have made toit during the past 10 years.

We have also developed a much better understandingof the contact mechanics on which the method is based,mostly through finite element simulation.2536 Thus, asecond objective of the article is to review what we nowknow about the mechanics of elasticplastic contact andhow this impacts the measurement methods. For in-stance, we now have a much better grasp of the physicalorigin of some of the empirical constants needed in themethod,35 and we also have a much better understandingof the limitations of the method, particularly in materialsthat pile-up.25,34,37

a)This author was an editor of this focus issue during the reviewand decision stage. For the JMR policy on review and publicationof manuscripts authored by editors, please refer to http://www.mrs.org/publications/jmr/policy.html

J. Mater. Res., Vol. 19, No. 1, Jan 2004 2004 Materials Research Society 3

To keep the article to a reasonable length, we haveintentionally limited its focus to the behavior of mono-lithic materials that can be described as a semi-infinite,elasticplastic half spaces. There are, of course, impor-tant issues that must be addressed when applying themethod to thin films to account for the influences of thesubstrate. In fact, much of the recent literature on loadand depth-sensing indentation testing deals with this sub-ject, and there are numerous important unresolved issuesremaining to be explored.1123 We also limit the dis-cussion to materials in which there are no time-dependent deformation mechanisms such as creep or vis-coelasticity. Here, too, there are important issues underinvestigation.3842

To set the stage, we begin with a brief review of themethod as it was originally developed. This is then fol-lowed by a discussion of advances in our understandingof the contact problem and then details of the method aswe now use and apply it.

II. THE METHODA. Basic principles

The method was developed to measure the hardnessand elastic modulus of a material from indentation loaddisplacement data obtained during one cycle of loadingand unloading. Although it was originally intended forapplication with sharp, geometrically self-similar indent-ers like the Berkovich triangular pyramid, we have sincerealized that it is much more general than this and applies

to a variety of axisymmetric indenter geometries includ-ing the sphere. A discussion of why the method works forspherical indentation is given at the end of this section.

A schematic representation of a typical data set ob-tained with a Berkovich indenter is presented in Fig. 1,where the parameter P designates the load and h thedisplacement relative to the initial undeformed surface.For modeling purposes, deformation during loading isassumed to be both elastic and plastic in nature as thepermanent hardness impression forms. During unload-ing, it is assumed that only the elastic displacements arerecovered; it is the elastic nature of the unloading curvethat facilitates the analysis. For this reason, the methoddoes not apply to materials in which plasticity reversesduring unloading. However, finite element simulationshave shown that reverse plastic deformation is usuallynegligible.35

There are three important quantities that must bemeasured from the Ph curves: the maximum load, Pmax,the maximum displacement, hmax, and the elastic unload-ing stiffness, S dP/dh, defined as the slope of theupper portion of the unloading curve during the initialstages of unloading (also called the contact stiffness).The accuracy of hardness and modulus measurement de-pends inherently on how well these parameters can bemeasured experimentally. Another important quantity isthe final depth, hf, the permanent depth of penetrationafter the indenter is fully unloaded.

The analysis used to determine the hardness, H, andelastic modulus, E, is essentially an extension of themethod proposed by Doerner and Nix43 that accounts forthe fact that unloading curves are distinctly curved in amanner that cannot be accounted for by the flat punchapproximation. In the flat punch approximation used byDoerner and Nix, the contact area remains constant as theindenter is withdrawn, and the resulting unloading curveis linear. In contrast, experiments have shown that un-loading curves are distinctly curved and usually well ap-proximated by the power law relation:

P = h hfm , (1)where and m are power law fitting constants.1 Table Isummarizes the values of the constants observed in our

TABLE I. Values of parameters characterizing unloading curves asobserved in nanoindentation experiments with a Berkovich indenter.a

Material

(mN/nmm) mCorrelation

coefficient, R

Aluminum 0.265 1.38 0.999938Soda-lime glass 0.0279 1.37 0.999997Sapphire 0.0435 1.47 0.999998Fused silica 0.0500 1.25 0.999997Tungsten 0.141 1.51 0.999986Silica 0.0215 1.43 0.999985

aData from Ref. 1.FIG. 1. Schematic illustration of indentation loaddisplacement datashowing important measured parameters (after Ref. 1).

W.C. Oliver et al.: Measurement of hardness and elastic modulus by instrumented indentation

J. Mater. Res., Vol. 19, No. 1, Jan 20044

original experiments.1 The variation of the power lawexponents in the range 1.2 m 1.6 demonstrates notonly that the flat punch approximation is inadequate(m 1 for the flat punch), but also that the indenterappears to behave more like a paraboloid of revolution,for which m 1.5.44 This result was somewhat surpris-ing because the axisymmetric equivalent of the Berko-vich indenter is a cone, for which m 2. This discrep-ancy has since been explained by the concept of aneffective indenter shape, which will be discussed inSec. III. A.35,36

The exact procedure used to measure H and E is basedon the unloading processes shown schematically inFig. 2, in which it is assumed that the behavior of theBerkovich indenter can be modeled by a conical indenterwith a half-included angle, , that gives the same depth-to-area relationship, 70.3. The basic assumption isthat the contact periphery sinks in in a manner that can bedescribed by models for indentation of a flat elastic half-space by rigid punches of simple geometry.4448 Thisassumption limits the applicability of the method becauseit does not account for the pile-up of material at thecontact periphery that occurs in some elasticplastic ma-terials. Assuming, however, that pile-up is negligible, theelastic models show that the amount of sink-in, hs, isgiven by:

hs = Pmax

S , (2)

where is a constant that depends on the geometry of theindenter. Important values are: 0.72 for a conicalpunch, 0.75 for a paraboloid of revolution (whichapproximates to a sphere at small depths), and 1.00for a flat punch.44 Based on the empirical observationthat the unloading curves are best approximated by anindenter that behaves like a paraboloid of revolution(m 1.5), the value 0.75 was recommended andhas since become the standard value used for analysis.Recent analytical work has provided a more completepicture of why takes on this value.35,49 This is alsoaddressed in Sec. III. A.

Using Eq. (2) to approximate the vertical displacementof the contact periphery, it follows from the geometry of

Fig. 2 that the depth along which contact is made be-tween the indenter and the specimen, hc hmax hs, is:

hc = hmax Pmax

S . (3)

Letting F(d) be an area function that describes theprojected (or cross sectional) area of the indenter at adistance d back from its tip, the contact area A is then

A = Fhc . (4)The area function, also sometimes called the indenter

shape function, must carefully be calibrated by indepen-dent measurements so that deviations from nonideal in-denter geometry are taken into account. These deviationscan be quite severe near the tip of the Berkovich indenter,where some rounding inevitably occurs during the grind-ing process. Although a basic procedure for determiningthe area function was presented as part of the originalmethod, we have made significant changes to it in recentyears. A complete description of the process we now usein its place is presented in Sec. IV.

Once the contact area is determined, the hardness isestimated from:

H =Pmax

A . (5)

Note that because this definition of hardness is based onthe contact area under load, it may deviate from the tra-ditional hardness measured from the area of the residualhardness impression if there is significant elastic recov-ery during unloading. However, this is generally impor-tant only in materials with extremely small values ofE/H.25

Measurement of the elastic modulus follows from itsrelationship to contact area and the measured unloadingstiffness through the relation

S = 2

Eeff A , (6)

where Eeff is the effective elastic modulus defined by

1Eeff

=

1 2

E +1 i

2

Ei. (7)

The effective elastic modulus takes into account the factthat elastic displacements occur in both the specimen,with Youngs modulus E and Poissons ratio , and theindenter, with elastic constants Ei and i. Note thatEq. (6) is a very general relation that applies to anyaxisymmetric indenter.2,50 It is not limited to a specificsimple geometry, even though it is often associated withflat punch indentation. Although originally derived forelastic contact only,2 it has subsequently been shown toapply equally well to elasticplastic contact50 and that

FIG. 2. Schematic illustration of the unloading process showing pa-rameters characterizing the contact geometry (after Ref. 1).


J. Mater. Res., Vol. 19, No. 1, Jan 2004 5

small perturbations from pure axisymmetry geometry donot effect it either.51 It is also unaffected by pile-up andsink-in.

In the original method for measuring hardness andmodulus, the dimensionless parameter was taken asunity. Traditionally, has been used to account for de-viations in stiffness caused by the lack of axial symmetryfor pyramidal indenters. However, it has been shown thateven for indentation of an elastic half-space by axisym-metric rigid cone, can deviate significantly fromunity.26 In this work, we therefore choose to redefine to account for any and all physical processes that mayaffect the constant in Eq. (6). In Sec. III. E, other valuesfor that may be important in precise measurements willbe considered along with the physical nature of theirorigins.

B. Application to spherical indentationAlthough we have discussed the method primarily as it

pertains to indentation with a sharp Berkovich or conicalindenter, it is not generally recognized that the methodapplies equally well to a sphere and can be used withoutmodification to determine the hardness and modulusfrom spherical indentation. To demonstrate this, it is use-ful to consider the contact depth for spherical indentationdetermined by means of Eq. (3) when applied to Hertzianelastic contact by a spherical indenter of radius R1pressed into a spherical hole of radius R2 that representsthe hardness impression. Note that Hertzian contactanalysis is restricted to the condition that the depth ofpenetration is small relative to the radius of the sphere.Letting R (1/R1 +1/R2)1, the loaddisplacement re-lation is52,53

P = 43REeffh hf32 . (8)The depth that appears in this relation is h hf rather thanh because the displacements are for the elastic unloadingcurve only (Fig. 1). The stiffness during unloading isthen found by differentiating this expression with respectto h, or

S =dPdh = 2REeffh hf

12. (9)

Letting h hmax to evaluate the expressions at themaximum depth and noting that for a sphere is 0.75,substituting Eqs. (8) and (9) into Eq. (3) yields

hc =hmax + hf

2 , (10)

or the contact depth determined from the method is sim-ply the average of the final and maximum depths. This isprecisely the value recommended by Field and Swain intheir method for analyzing load and depth-sensing dataobtained with spherical indenters.54,55 The contact areaand hardness determined by the two methods will thus be

the same as will be the elastic modulus because Eq. (6)applies to any indenter geometry. Note that hardnessesmeasured with a spherical indenter are not necessarily thesame as those for the Berkovich.

III. ADVANCES IN UNDERSTANDINGAlthough the method has extensively been used and

verified for numerous materials, certain aspects of it havealways been of concern to us, either because the physicalprocesses were not completely understood or becausesome of the constants needed to apply it were empiricallyderived. In particular, we have always been interested inunderstanding: (i) why unloading curves are well de-scribed by the power law relation of Eq. (1); (ii) why thepower law exponent in Eq. (1) falls roughly in the range1.2 m 1.6 rather than taking on the value m 2 aswould be expected for a conical indenter; and (iii) whythe best value for the geometric parameter is about0.75. Moreover, from a more fundamental standpoint, itwas not entirely clear to us how the elastic contact solu-tions used to derive Eqs. (3) and (6) could be used tomodel accurately the elastic unloading process as theyapply to the indentation of a flat elastic half-space,whereas the real problem involves a half space whosesurface has severely been distorted by the formation of ahardness impression. These questions have been an-swered through the concept of an effective indentershape.35,36

A. The effective indenter shapeThe ideas underlying the effective indenter shape are

outlined in Fig. 3. The basic principles are derived fromobservations gleaned from finite element simulations ofindentation of elasticplastic materials by a rigid conicalindenter with a half included angle of 70.3.35 During theinitial loading of the indenter [Fig. 3(a)], both elastic andplastic deformation processes occur, and the indenterconforms perfectly to the shape of the hardness impres-

FIG. 3. Concepts used to understand and define the effective indentershape (from Ref. 35).



sion. However, during unloading [Fig. 3(b)], elastic re-covery causes the hardness impression to change itsshape. A key observation is that the unloaded shape is notperfectly conical, but exhibits a subtle convex curvaturethat has been exaggerated in Fig. 3(b) for the purposes ofillustration. The importance of the curvature is that as theindenter is elastically reloaded [Fig. 3(c)], the contactarea increases gradually and continuously until full loadis again achieved, a process which must be the reverse ofwhat happens during unloading because both processesare elastic. It is this continuous change in contact areathat produces the nonlinear unloading curves. Further-more, the relevant elastic contact problem is not that ofconical indenter on a flat surface, but a conical indenterpressed into a surface that has been distorted by the for-mation of the hardness impression.

Experimental evidence for the curvature in the un-loaded hardness impressions is given in Fig. 4, wheretopographical images of a Berkovich indentation in fusedsilica are presented. The images were obtained by scan-ning the specimen laterally under the indenter tip with avery light load applied to the indenter to keep it in con-tact. The three-dimensional (3D) image in Fig. 4(a)

clearly shows the curvature of the indentation faces asdoes the cross section in Fig. 4(b).

The mathematical form of the unloading curve can beunderstood by introducing an effective indenter shape.As shown in the lower portion of Fig. 3, the shape of theeffective indenter is that which produces the same nor-mal surface displacements on a flat surface that would beproduced by the conical indenter on the unloaded, de-formed surface of the hardness impression. As such, theshape is described by a function z u(r), where u(r) isthe distance between the conical indenter and the un-loaded deformed surface and r is the radial distance fromthe center of contact. Thus, provided the shape of thedeformed surface is known, the function u(r) can befound and the effective indenter can be constructed. Fi-nite element simulations have verified that constructingthe effective indenter in this way provides an accuraterepresentation of the unloading data.35,36 Although theexact shape of the effective indenter depends in compli-cated ways on the elastic and plastic deformation char-acteristics of the material, it always has a smooth,rounded profile at its tip. The reason for this is that theslope of the unloaded hardness impression at its centerexactly matches the slope of the conical indenter (i.e., thetwo surfaces conform perfectly at the tip of the indenter).Thus, the plastic deformation producing the hardness im-pression has the interesting effect of removing the elasticsingularity at the tip of the indenter. Finite element stud-ies have also shown that the effective indenter shape iswell approximated by the power-law relation

z = Brn , (11)where B is a fitting constant and the exponent n varies inthe range 26 depending on the material properties.35The lower end of this range corresponds to an indenterwith the shape of a paraboloid of revolution, thus ex-plaining why indentation with a rigid cone is more likethat of a paraboloid of revolution or a sphere.

One can also use the effective shape concept to un-derstand why unloading curves are well described by thepower-law relation of Eq. (1) with power-law exponentsm in the range 1.21.6. Sneddon44 has shown that forindentation of an elastic half-space by an axisymmetricrigid indenter described by the power-law relation ofEq. (11), the loaddisplacement relation is

P =2Eeff

B1n n

n + 1n2 + 12n2 + 1 1nh1+1n ,(12)

where is the factorial or gamma function. Compari-son with Eq. (1) shows that

m = 1 + 1n . (13)Thus, given that n values range from 26, one would

expect ms in the range 1.21.5, in accordance with theexperimental observations in Table I.

FIG. 4. Topographic image of a Berkovich indentation in fused silicaobtained using a quantitative imaging system that uses the indenteritself as the imaging tip: (a) complete 3D image showing the curvatureof the faces; (b) cross section through an edge and the opposing facewith straight lines included to accentuate the curvature.



From a more fundamental standpoint, the shape of theeffective indenter can be estimated by making simpleassumptions about the distribution of pressure under theindenter. As schematically illustrated in the lower portionof Fig. 3, the pressure that develops during initial loadingis determined by complex elastic and plastic deformationprocesses that are generally not amenable to closed-formanalysis. However, during unloading, the pressure de-creases by elastic processes only, and the shape of thehardness impression changes to produce the curved sur-face. Upon reloading, this pressure distribution must berecovered by elastic processes only. Thus, the pressuredistribution at peak load serves to link the elasticplasticprocesses during initial loading to the elastic processesduring unloading and reloading. In this context, the shapeof the effective indenter must be that which produces thissame pressure distribution by elastic deformation of flatelastic half-space.

To implement these ideas, one must have some knowl-edge of what the actual pressure distribution is. Finiteelement simulations of conical indentation of elasticplastic materials have shown that to a first approxima-tion, the pressure is roughly uniform.35 This occurs be-cause plasticity tends to diminish the effects of the elasticsingularity at the tip of the cone and more evenly dis-tribute the pressure. Assuming, then, a perfectly flat pres-sure distribution, elastic contact theory gives the effec-tive indenter shape as

zr =4pamaxEeff

2 Eramax , (14)where p is the pressure, amax is the radius of the contactcircle at peak load, and E(r/amax) is the complete ellipticintegral of the second kind evaluated at r/amax.35,53Curve-fitting procedures show that this shape is verywell approximated by the power-law relation:35

zr = 0.5484pamaxEeff

ramax2.61

. (15)

Note that this equation is exactly of the form ofEq. (11) with n 2.61, for which the correspondingvalue of m according to Eq. (13) is m 1.38. Thus, tothe extent that the pressure distribution is constant, thereis a simple explanation for why the effective indenterbehaves more like a paraboloid of revolution (with n 2.61) than a cone and why unloading curves obtainedwith conical indenters can be described by power-lawrelations like Eq. (1) with m 1.38.

The effective shape concept can also be used to de-velop an equation that describes the entire unloadingcurve in terms of fundamental material properties. Com-bining Eqs. (11), (12), and (15) and noting that the pres-sure p is equivalent to the hardness, H, the unloadingcurve is given by

PPmax

= 0.858 EeffPmaxHh hf1.38

, (16)

which compares relatively well to experimental data.35Note that although the concepts discussed here were de-veloped specifically for conical indentation, they prob-ably apply to other indenter geometries as well, as plas-ticity has the effect of flattening the pressure distribution.Thus, one might expect unloading curves for sphericaland other indenters to be well approximated by this sameequation.

Lastly, the effective indenter shape concept is alsovaluable in the way it provides a physically justifiableprocedure for determining the value of the geometricparameter needed for accurate measurement of H andE.35,49 Using Sneddons method for determining the sur-face displacement at the contact perimeter for indentationwith a rigid punch with profile given by Eq. (11) inconjunction with Eq. (2) yields35:

= m1 2m

2m 1 12m 1

m 1 . (17)This relation is plotted in Fig. 5. Note that over the

range of expected ms (m 1.2 to 1.6), varies onlymildly between 0.74 and 0.79 with an average value of0.76. Thus, the value typically used in experiment, 0.75, is a reasonable estimate, although one could easilyrationalize slightly different values. Alternatively, a bet-ter, more self-consistent approach would be to measureexperimentally the exponent m from the unloading curveand then use Eq. (17) to determine the relevant valueof . This could easily be implemented in experiment.

FIG. 5. The relation between and m given by Eq. (17) (from Ref. 35).



B. Errors due to pile-up for conical andBerkovich indenters

As noted in our discussion of Eq. (2), one significantproblem with the method is that it does not account forpile-up of material around the contact impression, as isobserved in many elasticplastic materials. When pile-upoccurs, the contact area is greater than that predicted bythe method, and both the hardness estimated from Eq. (5)and the modulus from Eq. (6) are overestimated, some-times by as much as 50%.25 This inability to deal withpile-up is a direct consequence of using an elastic contactanalysis to determine the contact depth. Because materi-als deformed elastically always sink-in, pile-up cannotproperly be modeled.

The types of materials and conditions for which pile-up is most likely to occur have been examined by finiteelement simulation.25 The fundamental material proper-ties affecting pile-up are the ratio of the effective modu-lus to the yield stress, Eeff/y, and the work-hardeningbehavior. In general, pile-up is greatest in materials withlarge Eeff/y and little or no capacity for work hardening(i.e., soft metals that have been cold-worked prior toindentation). The ability to work harden inhibits pile-upbecause as material at the surface adjacent to the indenterhardens during deformation, it constrains the upwardflow of material to the surface.

Results of finite element studies that illustrate the in-fluence of material parameters on pile-up and sink-in arepresented in Fig. 6,25 which shows contact profiles underload for indentation by a 70.3 rigid cone. The materialsexamined all had a Youngs modulus E 70 GPa andPoissons ratio 0.25, but the yield stresses weresystematically varied between y 0.114 GPa and

y 26.62 GPa to examine different plastic behaviors.Two separate cases of work hardening were considered:one with no work hardening, that is, a work-hardeningrate d/d 0, (an elasticperfectly plastic mate-rial) and the other with a linear work-hardening rate 10y. These yield strengths and work-hardening ratescover a wide range of metals and ceramics.

In the course of the study, it was found that there is aconvenient, experimentally measurable parameter thatcan be used to identify the expected indentation behaviorof a given material.25 The parameter is the ratio of finalindentation depth, hf, to the depth of the indentation atpeak load, hmax. The ratio hf/hmax can be extracted easilyfrom the unloading curve in a nanoindentation experi-ment. Furthermore, because conical and Berkovich in-denters have self-similar geometries, hf/hmax does notdepend on the depth of indentation. The natural limits forthe parameter are 0 hf/hmax 1. The lower limitcorresponds to fully elastic deformation and the upperlimit to rigidplastic behavior. A similar approach basedon the ratio of the slopes of the loading and unloading

curves, a quantity that can be measured continuouslyduring loading by dynamic stiffness measurement, hasalso been developed.56

The results in Fig. 6 show that the amount of pile-up orsink-in depends on hf/hmax and the work-hardening be-havior. Specifically, pile-up is large only when hf/hmax isclose to 1 and the degree of work hardening is small. Itshould also be noted that when hf/hmax < 0.7, very littlepile-up is observed no matter what the work-hardeningbehavior of the material.

These observations are particularly important whenconsidered in relation to the contact areas shown inFig. 7. This figure includes two separate measures of thecontact area: one obtained by direct examination of thecontact profiles in the finite element mesh, Atrue, and theother by applying Eqs. (3) and (4) to the simulated in-dentation loaddisplacement data to obtain the area de-duced from the data analysis method, Aexpt. To general-ize, each area has been normalized with respect to Aaf,the area given by the indenter area function evaluated atthe maximum indentation depth, hmax. Since Aaf is thearea that would occur in the absence of pile-up or

FIG. 6. Finite element simulation of contact profiles under load forconical indentation of (a) non-work-hardening materials and (b) linearwork-hardening materials with work-hardening rate 10 y (afterRef. 25).



sink-in, values of A/Aaf greater than 1 indicate pile-up,whereas values less than 1 indicate sink-in. From thisfigure it is seen that the data analysis method, because itis based on an elastic analysis, significantly underesti-mates the true contact area for materials in which pile-upis important. Furthermore, when hf/hmax > 0.7, the accu-racy of the method depends on the amount of work hard-ening in the material. If the material is elasticperfectlyplastic, the method underestimates the contact area by asmuch as 50%. On the other hand, contact areas for ma-terials that work harden are predicted very well by themethod.

Note that from an experimental point of view, it is notpossible to predict if a material work hardens basedsolely on the loaddisplacement data. Therefore, in anindentation experiment, care must be exercised whenhf/hmax > 0.7, as use of the method can lead to largeerrors in the contact area. On the other hand, when thepile-up is small (i.e., hf/hmax < 0.7), the contact areasgiven by the method match very well with the true con-tact areas obtained from the finite element analyses, in-dependent of the work-hardening characteristics. Be-cause the hardness of the material is given by H P/A,errors in the contact area will lead to similar errors in thehardness, and according to Eq. (6), the modulus will be inerror by a factor that scales as A.

As a practical matter, if there is suspicion that pile-upmay be important based on the value of hf/hmax and/orother independent knowledge of the properties of thematerial, indentations should be imaged to examine theextent of the pile-up and establish the true area of con-tact. For Berkovich indenters, indentations with a largeamount of pile-up can be identified by the distinctbowing out at the edges of the contact impression.57 Ifpile-up is large, accurate measurements of H and E can-not be obtained using the contact area deduced from the

loaddisplacement data; rather, the area measured fromthe image should be used to compute H and E fromEqs. (5) and (6).

C. Correcting for pile-upDeveloping a method that can be used to correct for

pile-up in a manner that does not involve imaging thecontact impression has been one of the holy grails ofinstrumented indentation research. The basic problem inachieving this goal is that the amount of pile-up andsink-in depends on the work-hardening characteristics ofthe material, so without some independent knowledge ofthe work-hardening behavior, one does not know whatcorrection to apply.

A good example of work in this area is that of Chengand Cheng,2931 who used finite element simulations of awide variety of elasticplastic materials with differentwork-hardening behaviors to examine pile-up during in-dentation with a 68 cone (68 is the centerline-to-faceangle for a Vickers indenter; although a 70.3 cone ismore representative of Berkovich indentation, the basicresults of this work are still relevant). The method theyproposed to account for pile-up is based on the work ofindentation,31 which can be measured from the areas un-der indentation loading and unloading curves. LettingWtot be the total work of indentation (the area under theloading curve) and Wu the work recovered during un-loading (the area under the unloading curve), they foundthat the ratio of the irreversible work to the total work,(Wtot Wu)/Wtot, appears to be a unique function ofEeff/, independent of the work-hardening behavior.31Although their results are presented only in graphicalform, the relation can be approximated as:

Wtot WuWtot

1 5H

Eeff. (18)

Combining Eqs. (5) and (6) and taking 1 leads toanother equation involving H and Eeff:

4

PmaxS2

=

H

Eeff2 . (19)

Because Wtot, Wu, Pmax, and S are all measurable fromloaddisplacement data, Eqs. (18) and (19) represent twoindependent relations that can be solved for H and Eeff ina manner that does not directly involve the contact area.However, the contact area could be computed from thederived value of H by means of Eq. (5). Presumably, thisarea would be the true area including the effects ofpile-up.

To the best of our knowledge, this method has neverbeen tested experimentally. One potential problem is thatalthough pile-up is large and potentially an issue onlywhen Eeff/H and Eeff/y are relatively large (Fig. 6) (i.e.,

FIG. 7. Dependence of normalized contact areas on the experimentalparameter hf/hmax. The areas are normalized with respect to Aaf, thearea given by the area function of the indenter assuming no pile-up orsink-in (after Ref. 25).



in soft metals), close inspection of the data used to deriveEq. (18) shows that (Wtot Wu)/Wtot may not be entirelyindependent of the work-hardening behavior in this re-gime. If this is the case, then the method suffers fromexactly the same problem as others, that is, one musthave an independent knowledge of the work-hardeningbehavior to apply the corrections. Further experimentalexamination of this issue is warranted.

D. Pile-up for spherical indentersWe would also like to comment briefly on the issue of

pile-up during spherical indentation, as the method canbe applied to this important indenter geometry as well.Spherical indentation differs from conical or pyramidalindentation in that there is no elastic singularity at the tipof the indenter to produce large stresses. As a conse-quence, deformation at small loads and displacements isentirely elastic and then transitions to plastic as the in-denter is driven further into the material. This feature ofspherical indentation is what allows stressstrain curvesto be approximated from indentation data using theclassical approach of Tabor.58 Field and Swain54,55 haveapplied Tabors approach to instrumented indentationand have developed a method that uses the indenta-tion loaddisplacement data to approximate the stressstrain curve and the work-hardening exponent. The ap-proach requires that the deformation be fully plastic;that is, beyond the transition from elastically dominatedto plastic-dominated deformation. Also, it is assumedthat in the fully plastic regime, the pile-up geometry isuniquely related to material deformation parameters.

A point we would like to make is that the pile-upgeometry can change considerably during the course ofspherical indentation and it is therefore not possible topredict the pile-up based on the mechanical properties ofthe material alone, even when fully plastic deformationis achieved. The results of a recent finite element studyhelp to elucidate how the pile-up changes with the depthof penetration.34 Figure 8 shows the pile-up profiles foran elasticperfectly plastic material with E/y 200,which is fairly typical for a metal. To present the profileson a single set of axes, the z coordinates have been nor-malized with respect to the depth, h, the r coordinateswith respect to the contact radius, a, and the depth ofpenetration h with respect to the radius of the indenter, R.The pile-up and sink-in behavior are conveniently char-acterized by the parameter s/h, where s is the height ofthe contact periphery relative to the undeformed surface(s/h < 0 for sink-in and s/h > 0 for pile-up).

At small depths, the material deforms only elastically,with the indentation profile corresponding to that ofHertzian contact with sink-in at the contact peripherysuch that s/h 0.5. First yielding occurs at h/R 1.4 104 and a mean pressure pm 1.1y in a region on theaxis of symmetry below the surface at z 0.5a, also in

accordance with Hertzian contact theory.52,53 As the loadon the indenter is increased, the plastic zone grows andspreads upward, and the sink-in diminishes. At h/R ofabout 0.08, material begins to pile-up, and with furtherpenetration, the pile-up grows larger. It should be noted,however, that an upper limiting value for s/h was notreached in the simulations, even at the maximum depthof penetration, h/R 0.34, corresponding to a/R 0.75.This is significant because it shows that the pile-up ge-ometry can continue to change even up to contact radii inexcess of 0.75R, a value near the upper end of mostindentation experiments. Thus, unlike the constraint fac-tor, pm /y, which plateaus at an upper value of approxi-mately 3 at the onset of the fully plastic deformationregime,33,34 pile-up can continue to grow well after fullplasticity is achieved. Therefore, it does not follow thatjust because the constraint factor has become constantthat the pile-up geometry can also be assumed to beconstant. This ever-changing nature of the pile-up geom-etry should carefully be considered when interpretingresults from spherical indentation experiments in mate-rials with a tendency to pile-up.

E. The correction factor The correction factor appearing in Eq. (6) plays a

very important role when accurate property measure-ments are desired. This constant affects not only the elas-tic modulus calculated from the contact stiffness bymeans of Eq. (6), but the hardness as well because pro-cedures for determining the indenter area function are

FIG. 8. Finite element simulation of normalized contact profiles underload for spherical indentation of an elasticperfectly-plastic materialwith E/y 200 showing the influence of the normalized penetrationdepth, h/R, on the pile-up/sink-in behavior (after Ref. 34).



also based on Eq. (6), and area functions can be in errorif the wrong value of is used (area function calibrationmethods are addressed in Sec. IV). Thus, without a goodworking knowledge of , one is limited in the accuracythat can be achieved in the measurement of both H and E.

For the case of small deformation of an elastic materialby a rigid axisymmetric punch of smooth profile, isexactly 1 (note that small deformation is achieved only ifthe half-included angle of the indenter is close to 90).However, because real indentation experiments are con-ducted with non-axisymmetric indenters and involvelarge strains, other values for may be more appropriate.Here, we review the factors that affect for Berkovichindentation and attempt to draw some conclusionsconcerning what value is best. The complex nature ofelasticplastic deformation with pyramidal indentersmakes this a very difficult exercise with results that arenot entirely satisfying.

The importance of correcting Eq. (6) was first recog-nized by King,59 who used numerical methods to explorehow noncircular geometries effect the elastic contactstiffness of materials tested with rigid, flat-endedpunches. He found that that 1.012 for a square-based indenter and 1.034 for a triangular punch. Thelatter value has widely been adopted for instrumentedindentation testing with a Berkovich indenter (a triangu-lar pyramid). Vlassak and Nix later conducted indepen-dent numerical calculations for the flat-ended triangularpunch using a more precise method and found a highervalue, 1.058.60 Hendricks, noting that the pressuredistribution for elastic deformation by a flat punch is notrepresentative of the real elasticplastic problem,adopted another approach to estimate .61 Assuming thepressure profile is perfectly flat, he used simple elasticanalysis procedures to show that 1.0055 for a Vick-ers indenter (square-based pyramid) and 1.0226 forthe Berkovich.

Finite element simulations have also been used toevaluate and explore the factors that influenceit.25,26,29,30,32 Finite element simulations have the advan-tage of including the effects of plasticity, but are subjectto inaccuracies caused by inadequate meshing and con-vergence. Larson et al.32 conducted full 3D finite elementcalculations of true Berkovich indentation for a purelyelastic material and for four different elasticplastic ma-terials that simulate the behavior of aluminum alloys. Forthe purely elastic material, they found that is slightlydependent on Poissons ratio, , through the relation

= 1.2304 1 0.21 0.012 0.413 . (20)Assuming 0.3, this gives 1.14. Note, how-

ever, that this solution applies to indentation of a flat,elastic half-space and thus inherently ignores the fact thatplasticity severely distorts the surface during the forma-tion of the hardness impression. Because of this, one

might expect better results from the elasticplastic simu-lations because they account for plasticity. For the fourelasticplastic materials examined, the value 1.034suggested by King was found to work well, producingerrors of no more than 6.5% in the contact area andhardness.

Recent studies have shown that even for perfectly elas-tic contact by a rigid cone, may deviate from unity dueto departures from the small strain approximation. Hayet al.26 used both finite element simulation and analyticaltechniques to examine indentation of an elastic half-space by a rigid conical indenter. They showed that thederivation used to obtain Eq. (6) with 1 is notstrictly valid for the conical indenter geometry because itimproperly accounts for radial material displacements inthe region of contact (note: Hay et al. originally desig-nated this correction factor as ). They developed anapproximate correction for the effect that gives

=

4 + 0.1548 cot 1 241 2 0.8312 cot 1 241 2

, (21)

where is the half included angle of the indenter. For 0.3 and 70.3, this expression gives 1.067.Although these observations provide physical insightinto another important reason why should be greaterthan unity, it is not clear that values derived fromEq. (21) are particularly meaningful as they apply only toa flat elastic half-space. However, the results do point outthat there may be a dependence of on the cone angle,with smaller cone angles giving greater s.

A thorough finite element study of elasticplastic in-dentation with a 68 cone has been conducted by Chengand Cheng.29,30 Their calculations encompassed a widerange of materials characterized by different E/y ratios,Poissons ratios, , and work-hardening exponents, n.The results are reported in two separate publications.29,30The first, which focuses on nonhardening materials, findsthat 1.05, independent of E/y and .30 In the sec-ond, both hardening and nonhardening materials are con-sidered, and it is reported that 1.085, independent ofE/y and n.29 Note that there is a slight inconsistency inthe two papers in that the non-work-hardening results inthe second paper give a different value of from those inthe first. Therefore, which of these results is better, ifeither, is not clear. We have conducted similar finiteelement simulations ourselves for a 70.3 cone25 and findthat 1.07 for most materials, but may rise to highervalues in materials with large E/y. Also, could beslightly higher in materials that work harden.

The wide range of s reported in these studies makesit difficult to settle in on a single preferred value. Whatcan be concluded, however, is that there are valid reasons



to expect that for the Berkovich indenter should beslightly greater than unity. The deviation from circularcross-section appears to play an important role, as doesthe fact that the original analysis from which Eq. (6) wasderived ignores the radial displacements of the surfacefor a conical indenter. If we discard the values derivedfrom elastic analyses of flat half-spaces, which do notconsider the formation of the hardness impression, theremaining values fall in the range 1.0226 1.085.Thus, 1.05 is probably as good a choice as any, witha potential for error of approximately 0.05. Carefullyperformed experiments and 3D finite element simula-tions that take care to assure good convergence and ac-curacy could help to resolve this issue.

IV. REFINEMENTS TO THE METHODA. Difficulties with the original method

The ease of use of the method in a well-calibratedtesting system has proven to be one of its great advan-tages. On the other hand, the method we originally pro-posed for calibrating the load frame compliance, Cf, andthe area function of the indenter has proven to be cum-bersome and sometimes difficult to implement. We havethus worked over the last several years to develop asimpler, more accurate calibration procedure.

System calibrations are generally based on the relation

C = Cf +2Eeff

1

A, (22)

which simply states that the total measured compliance,C (i.e., the inverse of the measured stiffness), is the sumof the compliance of the load frame (the first term) andthe contact (the second term); that is, the two act likesprings in series. This assumes that the load frame com-pliance is a constant independent of load. If Cf is known,the area function can then be determined from measure-ments of compliance as a function of depth. Alterna-tively, if the area function is known, the load frame com-pliance can be determined as the intercept of a plot of Cversus A1/2. To determine these quantities simulta-neously, we originally suggested an iterative procedurein which simple assumptions are made about the startingvalues of Cf and the area function. Under certain circum-stances, however, problems are encountered with thisprocedure.

At shallow depths, corresponding to small contact ar-eas at the beginning of an indentation experiment, thecontact compliance is high and dominates the total meas-ured compliance (Eq. 22). However, as the contact depth(and area) increases, the contact compliance decreases,and at some point, the load frame compliance becomesthe more dominant factor. Hence, for measurements at

large depths, the load frame compliance must be knownwith great accuracy. The situation is reverse for the areafunction. The macroscopic shape of the indenter at largedepths can be controlled very precisely during its manu-facture. The uncertainty in the area function is due todifficulties in producing a perfectly sharp tip, which aremost important at small depths.

If the load frame compliance is low and the tip isrelatively sharp, there is an intermediate depth range inwhich uncertainties in Cf and the area function are in-consequential. In this case, the iterative process for de-termining the area function and load frame complianceconverges after a few cycles, and our original calibrationmethod works well, albeit through a fairly tedious cal-culation process. However, if the tip is not sharp or theload frame compliance is high, the iterative process con-verges very slowly and sometimes does not converge toa unique solution. Noise in the data compounds this prob-lem. The method we have developed to circumvent theseproblems is detailed in Sec. IV. E.

B. The indenter area functionOne aspect of the original method that has produced a

great deal of confusion concerns the mathematical formof the area function. The form we originally proposedwas

A = n=0

8

Cnhc2n

= C0h2 + C1h + C2h12 + C3h14 + + C8h1128 ,(23)

where C0 . . . C8 are constants determined by curve-fitting procedures. We wish to make it perfectly clearhere that this function was selected strictly for its abilityto fit data over a wide range of depths and not because ithas any physical significance. This being said, the equa-tion is, in fact, quite convenient in describing a number ofimportant indenter geometries. A perfect pyramid orcone is represented by the first term alone. The secondterm describes a paraboloid of revolution, which approxi-mates to a sphere at small penetration depths, and a per-fect sphere of radius R is described by the first two termswith C0 and C1 2R.57 The first two terms alsodescribe a hyperboloid of revolution, a very reasonableshape for a tip-rounded cone or pyramid that approachesa fixed angle at large distances from the tip. One otherform with physical significance is that suggested by Lou-bet et al.,62 which describes a pyramid with a small flatregion on its tip, the so-called tip defect. This geometryis described by the addition of a constant to the first twoterms in Eq. (23). In each case, the experimentally de-termined constants can be compared with the appropriategeometric description to verify the geometry. The higherorder terms in Eq. (23) are generally useful in describing



deviations from perfect geometry near the indenter tipand give the experimenter some flexibility in developingan area function that is accurate over several orders ofmagnitude in depth.

One other important aspect of the area function is thedepth range over which it is to be used. There are anumber of choices that can be made during the curve-fitting process that affect the applicable range of thefunction. If the function is to be applied at depths greaterthan the data used to construct it, one must be concernedabout how the function extrapolates. By assigning theconstant C0 associated with the quadratic term to a valuedetermined by the face angles of the pyramid and re-stricting the remaining constants to positive values, theresulting function will approach the description of theperfect pyramid at large depths and have no localmaxima or minima. This forces the results to be reason-able at all depths. The alternative is to allow all of theparameters to be fit and weight the fitting process to theparticular depth range of interest. This procedure oftenyields a function that fits the specified data better than thefirst approach, but such functions may be highly inaccu-rate outside the depth range used to construct them. Thisapproach may be preferable when the function is to beused for repetitive measurements; for instance, in qualitycontrol applications. Each of these approaches is dem-onstrated in Sec. IV. E.

C. Load divided by stiffness squared, P/S2

One of the most important improvements to the cali-bration procedures is based on an observation originallyreported by Joslin and Oliver that the ratio of the load tothe stiffness squared, P/S2, is a directly measurable ex-perimental parameter that is independent of the penetra-tion depth or contact area provided the hardness andelastic modulus do not vary with depth.63 In a mannersimilar to Eq. (19), this follows by combining Eqs. (5)and (6) to obtain

P

S2=

22H

E2. (24)

The utility of the parameter stems from its indepen-dence of the contact area. Joslin and Oliver took advan-tage of this to evaluate comparatively the mechanicalproperties of materials in which surface roughness led touncertainties in the contact area. The area-independenceof P/S2 also makes it valuable in the determination ofload frame compliances. Stone et al.64 have developedone such procedure, and the procedure we describeshortly is also based on this principle. The basic idea isthat since P/S2 is not influenced by the area, the machinecompliance can be determined in a manner that doesrequire a priori knowledge of the area function. This

decouples the measurement of load frame compliancefrom the area function and eliminates the need for aniterative procedure.

Another useful feature of P/S2 is that it does notdepend on the pile-up or sink-in behavior. Hence, ifthe modulus of the material is known, Eq. (24) canbe used to calculate accurately the hardness, even whenthere is significant pile-up. Conversely, if the hardnessis known (e.g., by direct measurement of the contactarea), then the modulus can be determined. Unfortu-nately, this expression does not allow one to calculate theeffect of pile-up on both the hardness and the modulussimultaneously, unless, perhaps, the method proposed byCheng and Cheng is proven to be experimentally viable31(Sec. III. C).

Several other useful applications of P/S2 as a charac-terizing parameter have been recognized in experimentalstudies. Saha and Nix have shown that Eq. (24) is accu-rate for even films on substrates provided the modulus ofthe film and substrate are similar.20 With a known modu-lus, this facilitates the measurement of film hardness in amanner that fully accounts for pile-up, which can bequite large in many filmsubstrate systems.14 In a similarmanner, Page et al.22 have shown that P/S2 can directlybe correlated to the tribological performance of coatedsystems.

D. Continuous stiffness measurementImprovements to measurement and calibration proce-

dures have also been facilitated by advances in testingtechniques and apparatus. One of the most important ofthese is the continuous stiffness measurement tech-nique (CSM), in which stiffness is measured continu-ously during the loading of the indenter by imposing asmall dynamic oscillation on the force (or displacement)signal and measuring the amplitude and phase of thecorresponding displacement (or force) signal by meansof a frequency-specific amplifier.1,24,65,66 As this tech-nique has matured over the last 10 years, it has dramati-cally reduced our reliance on unloading curves and offersseveral distinct advantages. First, it has the clear advan-tage of providing continuous results as a function ofdepth. Second, the time required for calibration and test-ing procedures is dramatically reduced because there isno need for multiple indentations or unloadings. Third, athigh frequencies, CSM allows one to avoid some of thecomplicating effects of time-dependent plasticity andthermal drift, which caused so much consternation in ouroriginal calibration method. Fourth, the CSM techniqueallows one to measure the effects of stiffness changes aswell as damping changes at the point of initial contact.This facilitates an important new technique for preciselyidentifying the point of initial contact of the indenter withthe sample.



Defining the point of initial contact is the critical start-ing point in the analysis of almost all instrumented in-dentation data. The resolution in the load and displace-ment signals as well as the data acquisition rates clearlyaffect how well one can determine the location of thesurface. However, it is also important to be able to re-solve and detect near-surface forces, as these often havean important bearing on the interpretation of the surfacelocation. Even in the simple case of sharp Berkovichdiamond contact with the surface of fused silica tested inlaboratory air, there is at least one important surfaceforcewater meniscus formation. Another importantsurface effect is adhesion between the tip and the sample.Because of these, techniques that use preloads or back-extrapolation of the data to determine the point of initialcontact can be misleading. The approach we prefer is toobserve the entire mechanical response of the systembefore, during, and after the point of initial contact andidentify the point of contact by examining the overallobserved behavior. As shown in the next section, infor-mation gleaned from CSM measurements has proven in-valuable in this regard and has helped significantly inidentifying the point of first surface contact.

E. Refined measurement andcalibration procedures

The procedure we now use to calibrate the machinecompliance and indenter area function for a Berkovichindenter is based on indentation of fused silica usingcontinuous stiffness measurement. Fused silica is chosenfor many reasons, one of the most important being that itdoes not pile-up because of its small E/H value. Here, wedetail the procedure as we apply it to measurements withthe Nano Indenter XP (MTS Systems Corp., Knoxville,TN). During continuous stiffness measurement with thisinstrument, several parameters can be controlled andmeasured. These include the harmonic load (the ampli-tude of the oscillation in the load signal), the harmonicdisplacement (the amplitude of the oscillation in the dis-placement signal), the harmonic frequency (the fre-quency of the oscillation), the phase angle (the angularphase shift between the load and displacement oscilla-tions), and the harmonic stiffness (which is derived fromthe other quantities).1,65,66 The Nanoindenter XP is in-herently a force-controlled device, but methods havebeen developed to keep the harmonic displacement con-stant by means of feedback control.

The basic experiment begins with a slow approach tothe surface to identify carefully the point of first contactbased on measurements of load, displacement, and stiff-ness. The surface approach velocity is 10 nm/s, with a3-nm oscillation in the harmonic displacement at a fre-quency of 80 Hz. Clearly, a small displacement oscilla-tion is desirable to avoid effects on the total measured

displacement, but larger oscillations improve the signal-to-noise ratio. A 3-nm oscillation was found to be anappropriate compromise for fused silica. Once the sur-face is detected, the indenter is loaded at a constant valueof (dP/dt)/P 0.3 (the loading rate divided by the load),which has the advantage of logarithmically scaling thedata density so that there is just as much data at low loadsas high. Constant (dP/dt)/P tests also have the advantageof producing a constant indentation strain rate, (dh/dt)/h,provided the hardness is not a function of the depth.42For the first 400 nm of displacement, the harmonic dis-placement is kept constant by feedback control at 3 nm.However, when the displacement reaches 400 nm, theexperiment is paused for 10 s during which the harmonicdisplacement is increased and stabilized at 8 nm to re-duce the noise levels in subsequent measurements (Note:The interruption and increase in harmonic displacementhas advantages in hard materials like fused silica, butmay lead to measurement inaccuracies in soft materialslike aluminum). The loading is then continued until themaximum load is achieved (approximately 630 mN), af-ter which the indenter is unloaded at a constant loadingrate dP/dt 19 mN/s to 10% of the maximum load.After holding for 200 s to establish the thermal drift ratefor correction of the displacement data using the tech-nique described in the original method, the indenter iscompletely unloaded. During the experiment, the load onthe sample, the displacement into the surface, the har-monic load, the harmonic displacement, and the phaseangle are all measured at a data acquisition rate of 5 Hz.For the results reported here, the experiment was re-peated 10 times for statistical averaging.

We also generated images of the residual hardnessimpressions used in the calibration experiments to com-pare the areas deduced from the indentation data to theactual contact areas. To do this, the indenter was rasterscanned across the indented area at 2 m/s at a small loadP 5 N to produce a fully quantitative topographicimage of the deformed surface with a resolution of 512 256 pixels. The scanning was accomplished by a mono-lithic, piezoelectrically driven nanopositioning stage with100 100 m of travel and feedback control based oncapacitive displacement sensors. The resolution, accu-racy, linearity, flatness of travel, and settling time for thestage are 2 nm, 2 nm, 10 nm, and 2 s, respectively. Theresulting images, shown in Figs. 4 and 9, do not sufferfrom the distortion typically associated with conven-tional scanning probe microscopy.

A typical load versus displacement curve for the cali-bration experiment is shown in Fig. 10. The curve, whichhas been corrected for thermal drift, exhibits parabolicloading and power-law unloading. Note that neither thepause during loading to increase the harmonic displace-ment amplitude nor the pause during unloading to meas-ure the thermal drift are apparent in the curves, thus



demonstrating that these interruptions do not influencethe data. The stiffnesses measured by CSM during load-ing are shown in Fig. 11. Here again, the brief pause at400 nm is not evident.

Figure 12 shows a plot of load versus displacementenlarged to examine the region of first contact. Figures13 and 14 are the corresponding behavior for the har-monic displacement and phase angle, respectively. In allthree cases, there is a clear change in behavior corre-sponding to contact with the surface. Note, however, thatthe change is smooth and continuous rather than abrupt,resulting in some uncertainty in defining the point of firstcontact. In contrast, the data in Fig. 15 shows that theharmonic stiffness rises rapidly when contact is achieved,thereby providing the best estimate of the surface loca-tion. The effects of a very thin adsorbed layer (mostlikely water) are also apparent; meniscus formation re-sults in a measurable reduction in stiffness just prior tocontact. What actually constitutes the point of first con-tact is debatable when such effects occur; however, it cancertainly be identified to within 2 nm.

Although the harmonic stiffness is the preferred loca-tor of the surface in this particular situation, it should benoted that which of the measurements in Figs. 1215provides the best surface definition depends on the prop-erties of the material and the characteristics of the testingsystem. The resonant frequency of the testing system isparticularly important (12 Hz for the system used here).As the stiffness of the contact increases, the resonantfrequency of the combined instrument and contactsweeps toward higher values. The exact nature of thechanges observed in the dynamic measurements inFigs. 1315 then depends on whether the system movestoward resonance (as in our experiments) or away from itas contact is made. In fact, it is because the system movestoward and through resonance that the harmonic dis-placement in Fig. 13 actually increases rather than de-creasing when contact is made. Often, the most sensitivefrequency for surface detection is the resonant frequencyof the testing system just prior to contact; however, thisis generally also the frequency at which the system ismost vulnerable to noise (vibration, and so forth).

FIG. 9. An indentation in fused silica made by raster scanning with ananopositioning stage. Data obtained from this indentation were usedto calibrate the load frame compliance and area function. (a) Theimage on the left shows the borders defining the two contact areasdiscussed in the text. (b) The image on the right is included for clarity.

FIG. 10. Typical load versus displacement curve for a calibrationexperiment in fused silica.

FIG. 11. A typical stiffness versus displacement curve for the cali-bration experiment. Stiffnesses were determined using the continuousstiffness measurement technique.

FIG. 12. An expanded view of the load versus displacement curve atthe point of contact.



Once the point of contact is determined, the data candirectly be analyzed to determine the load frame compli-ance. The procedure we now prefer is based on a plot ofP/S2 versus indenter displacement like that shown in

Fig. 16. Note that S in this plot represents the stiffness ofthe contact. To determine S, the component due to theload frame must first be removed (Eq. 22). The basicprinciple for the load frame compliance measurement isthat at depths greater than a few hundred nanometerswhere the hardness and modulus of fused silica should beindependent of depth, P/S2 should also be constant ac-cording to Eq. (24). The proper load frame compliance isthen found by changing the value of Cf until one is foundthat produces a flat P/S2 versus h curve at large depths. Itis not necessary to know the area function to implementthis procedure, so an iterative process is not required.

The data at shallow depths must be ignored in thisprocedure because there can be, and usually is, a realvariation in the measured hardness due to the tip imper-fection. Tip blunting results in lower mean pressures be-cause the contact behaves more like a spherical indenteron a flat surface, for which the mean contact pressurefalls to zero at zero depth. This is a real change in themean pressure and not an artifact of an inaccurate areafunction. As the load increases from zero, the contact willat first be elastic; however, at some depth plasticity be-gins, and at a still larger depth, full plasticity is estab-lished. At very large depths, geometric similarity of theBerkovich pyramid together with the homogeneous prop-erties of fused silica assure that the mean pressure (orhardness) remains essentially constant as the depth in-creases. These effects are evident in Fig. 16 as the rise inP/S2 over the approximately first 500 nm of displace-ment. The extent of the effect depends on the conditionof the indenter. For severely worn tips, the region ofnonconstant H and P/S2 can extend to much greaterdepths.

In Fig. 17, we demonstrate the load frame compliancemeasurement procedure for three different assumed val-ues of Cf. The correct value, Cf 9.62 108 m/N,results in a flat curve for the data between 1000 nm and2300 nm where the tip rounding is unimportant. How-ever, increasing or decreasing Cf by as little as 20%

FIG. 13. An expanded view of the harmonic displacement amplitudeversus displacement curve at the point of contact.

FIG. 14. An expanded view of the phase angle versus displacementcurve at the point of contact.

FIG. 15. An expanded view of the harmonic stiffness versus displace-ment curve at the point of contact. Note the minimum observed priorto full contact and the steep rise in stiffness at the point of contact.

FIG. 16. A typical load over stiffness squared versus displacement forthe calibration experiment. Note the constant value beyond 1000 nm.



results in a rising or falling curve. From experience, wehave found that the proper value of P/S2 for fused silicais 0.0015 0.0001 GPa1. This parameter is quite sen-sitive to errors in the calibration of load, displacement,and load frame compliance. Hence, the state of the sys-tem calibration can quickly be assessed by checking thisparameter.

The value of load frame compliance measured in thisway is applicable to any indentation experiment forwhich the contact stiffness does not significantly exceedthe largest stiffness used in the measurements. Note,however, that the region of applicability cannot bejudged through the depth of the indentation or the load onthe indenter; it must be assessed using the stiffness of thecontact.

With the load frame compliance established and themeasured stiffnesses corrected to remove the load framecontribution from the displacements, the loaddisplacementstiffness data can be used to deduce theindenter area function in a fairly straightforward manner.The procedure begins using Eq. (3) with 0.75 todetermine the contact depths corresponding to each datapoint on the loading curve. The corresponding contactareas can be generated in two different ways. If the faceangles of the indenter have precisely been measured, aswe now do routinely, then the constant C0 in the leadterm in the area function can be estimated from the per-fect pyramidal geometry. Provided the indenter is rela-tively sharp, the first term will dominate the area functionat large depths and can be used to determine the large-depth contact area. By means of Eq. (6), the quantityEeff at the largest depth can then be computed from thestiffness and contact area using

Eeff =

2S

A, (25)

which in turn can be used to generate the contact areas atall depths from the measured stiffnesses according to

A =

4S2

Eeff2 . (26)

The area function is then determined by fitting the Aversus hc data to the form of Eq. (23).

Alternatively, if the indenter is not sharp or the faceangles have not been measured, an assumed value for and Eeff for fused silica can be used in Eq. (26) to gen-erate the areas. For fused silica, we normally use E 72.0 GPa and 0.17, which in conjunction with theelastic constants for diamond (E 1141 GPa; 0.07)gives Eeff 69.6 GPa. For , we most often use 1.034, although based on the discussion in Sec. III. E, wemay now reconsider this choice. A spherical indentercould also be calibrated in this way.

For the experimental data presented here (Figs. 10 and11), the first approach was taken to determine the areafunction. A back-reflection laser goniometer was used tomeasure the face angles to within 0.025, giving C0 24.212. In precision work, this must be modified if theindenter is not aligned with its axis perfectly perpendicu-lar to the specimen surface, as such misalignments in-crease the area at a given contact depth. The degree of tiltcan be assessed from measurements of the sides of thehardness impression if a good image is available. Asimple but effective method is based on the ratio of thelong to short side of the hardness impression. Figure 18shows how the contact area at a given depth depends onthis ratio when the indenter is tilted slightly toward anedge or a face. Other tilt directions lie in-between the two

FIG. 17 Load over stiffness squared versus displacement for threedifferent values for the load frame stiffness. The curve with the correctload frame stiffness is flat at large depths.

FIG. 18. The change in the projected area of contact for a tiltedBerkovich indenter versus the ratio of the lengths of the longest to theshortest side.



curves, and the equation shown in the figure is an ap-proximation based on the average. From the image inFig. 9, the ratio of the lengths of the sides is 1.14, re-sulting in a 2% increase in the contact area and a cor-rected lead term of C0 24.65. With this value, the areacalculated at the largest depth is 67.0 0.3 m2, basedon an average of 10 measurements.

It is instructive to compare the calculated areas at thepeak depth with those measured from images of the re-sidual contact impressions. In Fig. 9(a), the inner whiteline traced around the hardness impression encloses ex-actly the predicted area of 67.0 m2. Although there issome ambiguity as to where the true edge of the contactis, it is clear that the predicted area is very close to thatin the image. The outer white line in Fig. 9(a) is thetriangle defined by the corners of the hardness impres-sion. The enclosed corner-to-corner area is considerablylarger, about 97.5 m2, indicating that one must be care-ful to account properly for the bowing in or out of thecontact edges in the measurement of contact areas.

To complete the area function calibration procedure,the other coefficients in Eq. (23) must be determined bycurve-fitting the A versus hc data. If the area function isintended for use over a wide range of depth (e.g., 101500 nm), it is useful to fix the lead term C0 using theaforementioned procedures based on the face angles ofthe indenter and restrict the rest of the coefficients topositive values. Fixing the lead term in this way assuresthat the area function accurately extrapolates at depthslarger than those used in the calibration, and forcing theother terms positive assures a smooth area function. Forthe experimental data in Figs. 10 and 11, the coefficientsfor the area function derived in this way are C0 24.65,C1 202.7, C2 0.03363, C3 0.9318, C4 0.02827, C5 0.03716, C6 1.763, C7 0.04102, andC8 1.881. The errors resulting from the fit are shownin Fig. 19, where the fractional difference between theexperimentally measured area and that predicted by thearea function are plotted as a function of the contactdepth. For depths greater than 200 nm, the area function

predictions are well within 1% of the correct values. Atsmaller depths, the error increases to as much as 4%. Thiscan be improved using the second approach for findingthe area function, limiting the range over which the func-tion is fit, and placing no restrictions on the coefficientsused in the fitting procedures. Letting the depth rangebe 10500 nm, the area function derived in this way as-suming Eeff 69.6 GPa has coefficients C0 24.261849693995, C1 388.715478479561, C2 937.723180561482, C3 251.535343527613, C4 451.330970778406, C5 219.019554856779, C6 157.740285820129, C7 98.1240614964975, andC8 72.6226884095761. More significant digits mustbe included because the mix of positive and negativecoefficients can produce significant round-off errors.Figure 20 shows that this function works well over thecontact depth range 10500 nm, but it should not be usedat larger or smaller depths. In practice, one should care-fully assess the experimental depth range of interest andtailor the area function around it.

Finally, the results obtained here give us an opportu-nity to comment on the value of . In deriving the areafunction by the first method, a value of Eeff 69.5 0.14 GPa was obtained by means of Eq. (25). Assum-ing that Eeff for fused silica indented by diamond is69.64 GPa, then 0.998 0.002. Hence, these experi-ments suggest that is unity for all practical purposes,although the experimental uncertainty is about 5%.

ACKNOWLEDGMENTSResearch sponsored by the Division of Materials Sci-

ences and Engineering, United States Department of En-ergy, under Contract No. DE-AC05-00OR22725 withUT-Battelle, LLC, and by MTS Systems Corporationthrough a research grant to the University of Tennessee.The authors would like to express their sincere gratitudeto Professor W.D. Nix for nearly 30 years of continuousencouragement, support, guidance, and wisdom.

FIG. 19. Difference between the areas predicted by the area functionand the data used to obtain the fit. This area function can be used atlarge depths.

FIG. 20. Difference between the areas predicted by the area functionand the data used to obtain the fit. This area function works well atsmall depths but not at large.



REFERENCES1. W.C. Oliver and G.M. Pharr, J. Mater. Res. 7, 1564 (1992).2. G.M. Pharr, W.C. Oliver, and F.R. Brotzen, J. Mater. Res. 7, 613

(1992).3. J.B. Pethica, R. Hutchings., and W.C. Oliver, Philos. Mag. A 48,

593 (1983).4. F. Frolich, P. Grau, and W. Grellmann, Phys. Status Solidi 42, 79

(1977).5. D. Newey, M.A. Wilkins, and H.M. Pollock, J. Phys. E: Sci. In-

strum. 15, 119 (1982).6. G.M. Pharr and W.C. Oliver, MRS Bull. 17, 28 (1992).7. G.M. Pharr, Mater. Sci. Eng. A 253, 151 (1998).8. T.D. Shen, C.C. Koch, T.Y. Tsui, and G.M. Pharr, J. Mater. Res.

10, 2892 (1995).9. D. Stone, W.R. LaFontaine, P. Alexopoulos, T.W. Wu, and

C-Y. Li, J. Mater. Res. 3, 141 (1988).10. W.D. Nix, Metall. Trans. 20A, 2217 (1989).11. S.V. Hainsworth, H.W. Chandler, and T.F. Page, J. Mater. Res.

11, 1987 (1996).12. M. Tabbal, P. Merel, M. Chaker, M.A. El Khakani, E.G. Herbert,

B.N. Lucas, and M.E. OHern, J. Appl. Phys. 85, 3860 (1999).13. S.V. Hainsworth, M.R. McGurk, and T.F. Page, Surf. Coat. Tech-

nol. 102, 97 (1998).14. T.Y. Tsui and G.M. Pharr, J. Mater. Res. 14, 292 (1999).15. T.Y. Tsui, J.J. Vlassak, and W.D. Nix, J. Mater. Res. 14, 2196

(1999).16. S. Bec, A. Tonck, J-M. Georges, E. Georges, and J-L. Loubet,

Philos. Mag. A 74, 1061 (1996).17. N.X. Randall, Philos. Mag. A 82, 1883 (2002).18. Y.Y. Lim and M.M. Chaudhri, J. Mater. Res. 14, 2314 (1999).19. K. Miyahara, N. Nagashima, and S. Matsuoka, Philos. Mag. A 82,

2149 (2002).20. R. Saha and W.D. Nix, Acta Mater. 50, 23 (2002).21. J.A. Knapp, D.M. Follstaedt, S.M. Myers, J.C. Barbour, and

T.A. Friedman, J. Appl. Phys. 85, 1460 (1999).22. T.F. Page, G.M. Pharr, J.C. Hay, W.C. Oliver, B.N. Lucas,

E. Herbert, and L. Riester, in Fundamentals of Nanoindentationand Nanotribology, edited by N.R. Moody, W.W. Gerberich,N. Burnham, and S.P. Baker (Mater. Res. Soc. Symp. Proc. 522,Warrendale, PA, 1998), p. 53.

23. J. Mencik, D. Munz, E. Quandt, E.R. Weppelmann, andM.V. Swain, J. Mater. Res. 12, 2475 (1997).

24. B.N. Lucas, W.C. Oliver, and J.E. Swindeman, in Fundamentalsof Nanoindentation and Nanotribology, edited by N.R. Moody,W.W. Gerberich, N. Burnham, and S.P. Baker (Mater. Res. Soc.Symp. Proc. 522, Warrendale, PA, 1998), p. 3.

25. A. Bolshakov and G.M. Pharr, J. Mater. Res. 13, 1049 (1998).26. J.C. Hay, A. Bolshakov, and G.M. Pharr, J. Mater. Res. 14, 2296

(1999).27. T.A. Laursen and J.C. Simo, J. Mater. Res. 7, 618 (1992).28. H. Gao, C-H. Chui, and J. Lee, Int. J. Solids Structures 29, 2471

(1992).29. Y-T. Cheng and C-M. Cheng, J. Appl. Phys. 84, 1284 (1998).30. Y-T. Cheng and C-M. Cheng, Int. J. Solids Structures 36, 1231

(1999).31. Y-T. Cheng and C-M. Cheng, Appl. Phys. Lett. 73, 614 (1998).32. P-L. Larsson, A.E. Giannakopoulos, E. Soderlund, D.J. Rowcliffe,

and R. Vestergaard, Int. J. Solids Structures 33, 221 (1996).33. S.D.J. Mesarovic and N.A. Fleck, Proc. R. Soc. London A 455,

2707 (1999).34. B. Taljat, T. Zacharia, and G.M. Pharr, in Fundamentals of Nano-

indentation and Nanotribology, edited by N.R. Moody,W.W. Gerberich, N. Burnham, and S.P. Baker (Mater. Res. Soc.Symp. Proc. 522, Warrendale, PA, 1998), p. 33.

35. G.M. Pharr and A. Bolshakov, J. Mater. Res. 17, 2660 (2002).

36. A. Bolshakov, W.C. Oliver, and G.M. Pharr, in Thin Films:Stresses and Mechanical Properties V, edited by S.P. Baker,C.A. Ross, P.H. Townsend, C.A. Volkert, and P. Brgesen (Mater.Res. Soc. Symp. Proc. 356, Pittsburgh, PA, 1995), p. 675.

37. K.W. McElhaney, J.J. Vlassak, and W.D. Nix, J. Mater. Res. 13,1300 (1998).

38. J-L. Loubet, B.N. Lucas, and W.C. Oliver, NIST Special Publica-tion 896: International Workshop on Instrumented Indentation(National Institute of Standards and Technology, San Diego, CA,1995), pp. 3134.

39. B.N. Lucas, C.T. Rosenmayer, and W.C. Oliver, in Thin FilmsStresses and Mechanical Properties VII, edited by R.C. Camma-rata, M. Nastasi, E.P. Busso, and W.C. Oliver (Mater. Res. Soc.Symp. Proc. 505, Warrebdale, PA, 1998), p. 97.

40. W.H. Poisl, W.C. Oliver, and B.D. Fabes, J. Mater. Res. 10, 2024(1995).

41. B.N. Lucas, W.C. Oliver, J-L. Loubet, and G.M. Pharr, in ThinFilms: Stresses and Mechanical Properties VI, edited byW.W. Gerberich, H. Gao, J.E. Sundgren, and S.P. Baker (Mater.Res. Soc. Symp. Proc. 436, Pittsburgh, PA, 1997), p. 233.

42. B.N. Lucas and W.C. Oliver, Metall. Mater. Trans. 30A, 601(1999).

43. M.F. Doerner and W.D. Nix, J. Mater. Res. 1, 601 (1986).44. I.N. Sneddon, Int. J. Eng. Sci. 3, 47 (1965).45. A.E.H. Love, Q.J. Math. 10, 161 (1939).46. A.E.H. Love, Philos. Trans. A 228, 377 (1929).47. J.W. Harding and I.N. Sneddon, Proc. Cambridge Phil. Soc. 41,

16 (1945).48. I.N. Sneddon, Fourier Transforms (McGraw-Hill, New York,

1951), pp. 450467.49. J. Woirgard and J-C. Dargenton, J. Mater. Res. 12, 2455 (1997).50. C-M. Cheng and Y-T. Cheng, Appl. Phys. Lett. 71, 2623 (1997).51. H. Gao and T-W. Wu, J. Mater. Res. 8, 3229 (1993).52. H. Hertz, Miscellaneous Papers by H. Hertz (Macmillan, London,

1896).53. K.L. Johnson, Contact Mechanics (Cambridge University Press,

Cambridge, 1985).54. J.S. Field and M.V. Swain, J. Mater. Res. 8, 297 (1993).55. J.S. Field and M.V. Swain, J. Mater. Res. 10, 101 (1995).56. J.L. Hay, W.C. Oliver, A. Bolshakov, and G.M. Pharr, in Funda-

mentals of Nanoindentation and Nanotribology, edited byN.R. Moody, W.W. Gerberich, N. Burnham, and S.P. Baker(Mater. Res. Soc. Symp. Proc. 522, Warrendale, PA, 1998),p. 101.

57. J.L. Hay and G.M. Pharr, in ASM Handbook Volume 8: Mechani-cal Testing and Evaluation, 10th ed., edited by H. Kuhn andD. Medlin (ASM International, Materials Park, OH, 2000),pp. 232243.

58. D. Tabor, The Hardness of Metals (Oxford University Press, Lon-don, 1951).

59. R.B. King, Int. J. Solids Struct. 23, 1657 (1987).60. J.J. Vlassak and W.D. Nix, J. Mech. Phys. Solids, 42, 1223

(1994).61. B.C. Hendrix, J. Mater. Res. 10, 255 (1995).62. J.L. Loubet, J.M. Georges, O. Marchesini, and G. Meille, J. Tri-

bology 106, 43 (1984).63. D.L. Joslin and W.C. Oliver, J. Mater. Res. 5, 123 (1990).64. D.S. Stone, K.B. Yoder, and W.D. Sproul, J. Vac. Sci. Technol. A

9, 2543 (1991).65. J.B. Pethica and W.C. Oliver, Phys. Scr. 19, 61 (1987).66. J.B. Pethica and W.C. Oliver, in Thin Films: Stresses and Me-

chanical Properties, edited by J.C. Bravman, W.D. Nix,D.M. Barnett, and D.A. Smith (Mater. Res. Soc. Symp. Proc. 130,Pittsburgh, PA, 1989), p. 13.



Documents

Oliver Pharr Paper