Materials Science and Engineering A307 (2001) 172–181

Nanoindentation hardness of submicrometer alumina ceramics

A. Krell a,*, S. Schadlich b

a Fraunhofer-Institute for Ceramic Technologies and Sintered Materials (IKTS), Winterbergstr. 28, D-01277 Dresden, Germanyb Fraunhofer-Institute for Material and Beam Technology (IWS), D-01277 Dresden, Germany

Received 27 March 2000; received in revised form 29 August 2000

Abstract

Nanoindentations at loads of 20–200 mN were performed on submicrometer sintered alumina ceramics of different residualporosity. The indentation size effect of the hardness can be obtained without variation of the maximum load directly from loadingcurves, if the Young’s modulus of the material is known and if the shape of the penetration curve is analyzed carefully. Similarinfluences of testing load (indentation size effect), grain size and porosity are observed as known from conventional measurements.At small testing loads of 25–50 mN, the Vickers hardness of dense sub-mm corundum ceramics rises to 25–30 GPa depending onthe microstructure. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Hardness; Grain size effect; Load influence; Ceramics; Alumina

www.elsevier.com/locate/msea

1. Introduction

Depth-sensing indentation with loads \1 N wasproposed to avoid the inaccuracy of optical measure-ments and, possibly, the indentation size (load) effect[1,2]. Recently, displacement sensing indentation at-tracts most interest where the optical assessment ofextremely small indentations in hard materials isdifficult, e.g. in thin coatings where the plastic zone ofthe indent must not exceed the thickness of the layer.Surprisingly, even for technically important ‘standard’ceramics like alumina, few reliable penetration datahave been published and these data suffer from funda-mental shortcomings:1. Most of the original data give different penetrations

at same load in curves that run to different maxi-mum loads Fmax: examining nominally identical(0001) surfaces of sapphire up to 120 mN [3] and upto 3 N [4], a group of authors reported results thatat an equal load of 120 mN differed by 160%. Apossible reason may be that standards encourageloading to different Fmax within a constant period oftime (e.g. German DIN 50359-1). Such approaches

imply different deformation rates up to differentFmax with the consequence of different penetrationsin materials that exhibit indentation creep.

2. The analysis of the penetration data seems sodifficult that most of the (few) authors who pub-lished depth sensing experiments for Al2O3 did notderive any hardness values [4–7].

One exception is the analysis by Oliver and Pharr [3]for sapphire faces characterized as (0001). Their analy-sis of the tangents to the unloading curve (at differentpeak loads) gives a significant indentation size effect atsmall testing loads (:28 GPa at 120 mN, 31 at 10 mNand 35 GPa at 1.5 mN). The hardness appears high for(0001) which is one of the softer planes (prismatic andpyramidal planes are harder and more wear resistant[8–11]).

Considering the importance of alumina ceramics forboth fundamental studies and for technical applica-tions, it was the objective of the present work to derivereliable hardness data for corundum (a-Al2O3) at smallindentation sizes. With our observation of extendedcreep in (1210) sapphire at loads B200 mN even onvery slow penetration (0.44 mN s−1) [12], the investiga-tions were performed with polycrystals where creep issmall. The experiments focused on samples with asub-mm microstructure and with a minimum of flawsbecause of the challenging properties of these ceramics

* Corresponding author. Tel.: +49-351-2553538; fax: +49-351-2553600.

E-mail address: krell@ikts.fhg.de (A. Krell).

0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0921 -5093 (00 )01818 -9

A. Krell, S. Schadlich / Materials Science and Engineering A307 (2001) 172–181 173

and the chance to associate a polycrystalline state of amaterial with a high homogeneity on the scale ofnanoindents.

To test the reliability of the depth sensing equipmentand analysis, the results are compared with conven-tional (light or electron optical) measurementswhenever possible.

2. Experimental

2.1. Materials and testing conditions

Preparation and flaw populations of highly perfectsintered sub-mm alumina samples have been describedpreviously [13,14], (1210) sapphire was used as a refer-ence material. Table 1 summarizes important data. Thesurfaces were diamond polished to mirror quality.

A Vickers indenter was used in all investigations.Without a reliable procedure to eliminate the influenceof cracking on the increasing compliance and penetra-tion, hardness data must not be derived from penetra-tion curves at testing loads F\Fc (cracking threshold).Therefore, all depth sensing investigations were per-formed at FB200 mN, where according to our obser-vation on (1210) sapphire [12] and literature reports,cracking does not occur (Fc in alumina is 250–500 mN[15,16]). All hardness data derived here at F\Fc wereobtained by conventional electron or light-optical mea-surements of the diagonal sizes of the indents, thusavoiding any influences of cracking on these data.

Depth sensing experiments were performed with thenanoindentation device DUH202 (Shimadzu) withnominal peak loads of 25–1960 mN (depth of penetra-tion 0.4–4 mm). The equipment was programmed togive a cycle with the same rates on loading and unload-ing. A dwell time at maximum load was to check forresidual creep. The same dwell was repeated after un-loading to 2% of the maximum load to compensate forthe thermal drift of the system [3,5]. Ten curves wereobtained on polycrystalline surfaces to account for thescatter caused by residual porosity and otherheterogeneities.

Without an international standard for hardness test-ing of ceramics, first penetration experiments with peakloads of 50 mN were performed with a loading rate of4.4 mN s−1 which raises the load to the final peakwithin :10 s, as required by standards for testingmetals (DIN 50359-1 for depth sensing measurements,EN ISO 6507-1 for conventional Vickers testing). Thedwell time at maximum load was 5 s in these experi-ments. However, this procedure associates differentmaximum loads Fmax with different deformation rates,rendering experiments with different Fmax incompara-ble. Also, high deformation rates may inhibit full plas-tic deformation during loading (indicated by largertime-dependent creep at Fmax). Therefore, comparativemeasurements were performed at Fmax=50 mN at areduced rate of 0.44 mN s−1 and a longer dwell of 30s.

According to the above standards, hardness resultshave to be given with GPa units for nanoindentation(testing loads B200 mN in the present work) butexpressed as kgf mm−2 (usually given without the unit)for optically determined hardness values measured atlarger testing loads. Therefore, both presentations willbe given here.

To evaluate possible effects from imperfections of thediamond, the tip was displayed by its imprint onsmooth sapphire (Fig. 1). As evident from the kink inone of the diagonals, the lateral width of this ‘roofedge’ is 0.2090.05 mm. The influence of this imperfec-tion on the recorded penetration curves will be ad-dressed below in Section 2.2.1.3.

Reference data for sapphire at larger testing loadshave been published previously [2,17]. With light-opti-cal measurement at ]3 N, the typical S.D. in sapphireis 590.5, . . . , 1 GPa except for F=98.1 N (10 kgf),where extended cracking increases the scatter to 91.5, . . . , 2 GPa. This light-optical approach is increas-ingly unreliable at loads 53 N on testing hardmaterials (small indents), and scanning electron micro-scopic measurements were performed here for peakloads between 50 mN and 2 N. The accuracy of thesehardness results is better than 91 GPa for testingloads above 200 mN.

Table 1Investigated materials (cp. Section 2.2.1/Eq. (2b) and Eq. (2c) for elastic property data)

Residual porosity P Grain size (mm)Method of preparation

0.1% ( [ E:400 GPa; E*=309 GPa)Pressure filtration of a powder \99.99% Al2O3 0.63Sintered aluminadoped with 0.1 wt.% MgO

0.471.2% ( [ E:390 GPa; E*=303 GPa)4.8% ( [ E:363 GPa; E*=287 GPa) 0.31

Verneuil grown (undoped) 0 ( [ E:400 GPa; E*=309 GPa) –(1210) Sapphire

A. Krell, S. Schadlich / Materials Science and Engineering A307 (2001) 172–181174

Fig. 1. Indentation on (1210) sapphire produced at a peak loadF=200 mN. One of the diagonals of the indent is parallel to the traceof (0001) marked by the basal twins. The kinked line of one of thediagonals marks a ‘roof edge’ that increases the area exposed topressure.

of elastic deformation, converting Hu into the conven-tional hardness Hv (related to the plastic indent size)[18]:

Hv=4Hu/{1+(1−12 Hu/E*)}2; (2a)

E* is the effective contact stiffness. With Young’smodulus E and Poisson’s ratio n of the sample (sub-script ‘s’) and the diamond indenter (subscript ‘i’), E* isgiven by

E*={(1−n s2)/Es+ (1−n i

2)/Ei}−1. (2b)

For our analysis, E*=309 GPa was derived fromEAl2O3

=400 GPa, nAl2O3=0.235, Ei=1140 GPa and

ni=0.07 as well substantiated data for sapphire and forthe diamond indenter (the averages also consider thecrystallographic anisotropy). The porosity P of poly-crystalline samples (Table 1) was taken into account bya porosity dependent modulus E(P) (upper Hashinbound) related to E(P=0)=Eo=400 GPa

E(P)=Eo(1−P)/(1+P), (2c)

Table 1 gives estimates of E and E* for the threealumina microstructures and for sapphire.

2.2.1.2. Con6entional determination of origin from shapeanalysis. The evaluation of penetration h is complicatedby the difficult determination of the origin of the curveswhich start smoothly from the h-axis at F=0. Onrecording, an uncertain apparent origin is automaticallyset (determined by the sensitivity of hardware andsoftware), but the real origin and the difference ho

between the apparently recorded depth happ and thetrue penetration h are unknown.

To find ho, a careful definition of the typical averageshape of the penetration curve is required. The origi-nally recorded penetration data represent the typicalbehavior of the microstructure, the more grains of themicrostructure are subjected to the indentation, i.e. thelarger the load. Therefore, the original curves wereshifted to give one common apparent happ,max at Fmax

and the resulting display was addressed to define the‘typical average shape’ used for the subsequent analysis.Fig. 2 gives an example. Note that the origin of thecurves is still arbitrary in this stage of the presentation.

Assuming the most simple power law, such curves aredescribed by

F(h)=A(happ−ho)n, (3a)

and a set of three data points (happ,i, Fi) gives the threeparameters A, ho and n. The evaluation of the penetra-tion curves requires knowledge of ho which is deter-mined from three points (happ,i, Fi) with F1\F2\F3 ifthese are selected according to F1/F2=F2/F3 (e.g. 200,100 and 50 mN):

Fig. 2. Penetration in dense submicrometer alumina (average grainsize 0.63 mm, residual porosity 0.1%; loading rate 0.44 mN s−1 up toFmax=196 mN, 30 s dwell). Pool of originally recorded curvesrepresenting locations of a similar median hardness.

2.2. Methods of analysis

2.2.1. Analysis of loading cur6es

2.2.1.1. Vickers hardness from penetration. With theusual relationships for the Vickers geometry, the totalpenetration depth h (including elastic deformation)gives an apparent hardness that is sometimes called‘universal hardness’ Hu (LVH in some early papers)

Hu=F/26.43 h2, (1)

where F is the applied testing load. It has been shownthat with a known Young’s modulus E of the testedmaterial, an analytic solution separates the contribution

A. Krell, S. Schadlich / Materials Science and Engineering A307 (2001) 172–181 175

ho= (happ,1happ,3−happ,22 )/(happ,1+happ,2−2happ,2).

(3b)

It is worth emphasizing that hoBhapp,3(Bhapp,2Bhapp,1): due to the smooth approach of the penetrationto the axis at F�0, Eq. (3b) is very sensitive requiringcareful smoothing of the curves (local heterogeneities inthe material may intermediately affect the shape ofsections of the curve). Otherwise, local deviations of therecorded happ,i from the power law may give unrealisticresults ho\h3 associated with a singularity in the de-nominator of Eq. (3b). Several precautions were takento minimize these inaccuracies:� On measuring happ,i a first formal zero-point has to

be used, the position of which is arbitrary. Thisformal zero-point is defined far on the negative sideof the recorded original curves with the objective toobtain larger calculated ho values (avoiding a calcu-lation with ho in Eq. (3b) near zero).

� To minimize problems with local irregularities in thecurves, large distances between the three required Fi

were used to minimize the error in calculating ho

from the apparent penetration depths (e.g. F1=196,F2=44.3 and F3=10 mN).

2.2.1.3. Shape analysis assuming a penetration differentfrom F� (h−ho)n. Whereas the foregoing section con-siders a shift of the penetration curve along the x-axis,a small ‘roof edge’ at the tip of the indenter (Fig. 1)requires the introduction of an additional term intoEqs. (1) and (3a). For Vickers indenters with c as thelength of the roof edge, Dengel has described theadditional area at penetration h as 2·c ·h/cos 68° [19],such that Eq. (1) is replaced by

Hu=F/(26.43 h2+2·c ·h/cos 68°) (4a)

As a consequence, the recorded penetration F(h) willnot obey the relationship F� (h−ho)n because thecurve is shifted in the x and y directions. Comparedwith an ideal Vickers pyramid, the roof edge of animperfect indenter increases the area exposed to pres-sure, reducing the penetration h(F). Hence, an erro-neously increased universal hardness Hu results whenEq. (1) is applied to penetration curves recorded withsuch an imperfect indenter.

The size of the roof edge effect on Hu can be derivedfrom Eqs. (1) and (4a). With the dependence of theincrease in the exposed area on h, the effect of the roofedge also depends on the penetration and is given by[20]

DHU

HU

=2·c ·h/cos 68°

26.43 h2 =0.202ch

, (4b)

where the positive sign describes the deviation of theerroneously determined (too high) hardness from theexact result. With Eq. (4b), the deviations in Hu in-crease at h�0.

Similarly, the effect of a rounded tip of the indenter(radius R) is [20]

DHU

HU

=0.157Rh

+0.00057R2

h2 . (4c)

As to the consequences for the qualitative shape ofthe penetration curves, Eq. (4a) can be rewritten as

F=26.43 Hu[(h-ho)2−ho2]. (5)

with ho= −c/(26.43 cos 68°). Compared with Eqs. (1)and (3a), the curve passes (h, F)= (0, 0) with an apexshifted to (ho, −26.43 Hu·ho

2). Then, the effect of thekink at the indenter tip (Fig. 1) is an only slightlychanged curvature of the quadratic curve at F\0(h\0). Figs. 3 and 4 display these influences of theindenter on penetration and derived hardness Hu as-suming also an additive incidence of roof edge and tiprounding. Note in Fig. 4 that the error in Hu is as largeas 50% at F=10 mN (h:0.1 mm) but rather small at50 mN (h:0.3 mm).

Whereas the real situation may encounter some su-perposition of both effects described by Eqs. (4a–c)and Eq. (5), no information other than c:0.15–0.25mm is obtained from Fig. 1. Also, nothing is knownabout other influences from the equipment (e.g. elasticdeformation of the hard-metal fixation of the diamond[20]). A more fundamental difficulty comes from ourobservations addressed in Section 3.1: in a mathemati-cal treatment such as Eq. (3a), a ‘true’ indentation sizeeffect means an exponent n"2 (usually nB2, e.g.n:1.90–1.95 for Al2O3). It is, by no means obviousthat such a behavior should follow a power law. There-fore, it does not appear straightforward to consider

Fig. 3. Deviation of the penetration curve from power law (Eqs. (4a)and (4c) and Eq. (5)) caused by a roof edge (Fig. 1) and a rounded tipof the diamond. The experimentally recorded (solid) curve exhibitsless penetration with an imperfect tip. Dashed lines give the correctedpenetration.

A. Krell, S. Schadlich / Materials Science and Engineering A307 (2001) 172–181176

Fig. 4. Tip imperfections of the indenter increase the (apparent)universal hardness Hu for h�0 (cp. Eq. (4b)). Using indenters withknown imperfections, the ‘real’ hardness would be obtained if thecorrecting Eqs. (4a), (4b) and (4c) fitted the shape of the tip exactly.

As to this latter advantage, however, the smallestvariations in the shape of the curve cause large changesin E*, usually with a trend that Young’s moduli aredetermined too high: similar to the results of Oliver andPharr who derived E=441 GPa from their experimentswith sapphire, the analysis of the unloading curves forthe dense polycrystals in the present investigation re-sulted in E=445–460 GPa (the true value for thesesamples is 380–400 GPa). The difference between theincorrect E:450 GPa derived from the unloadingcurves and a more realistic value of E=400 GPacorresponds to a difference in Hv of 1.5–2 GPa.

To avoid such inaccuracies, all of the present resultsfrom depth sensing tests were derived directly from thepenetration depth and calculated using Eqs. (1), (2a),(2b) and (2c) and Eq. (3b).

3. Results

3.1. Kinetic effects

When, in agreement with standards, test are run upto maximum loads of 50 and 200 mN within the sametime of loading, the different rates are associated withdifferent penetration depths (at the same load) andwould give different hardness values with all of theapproaches discussed above. Therefore, all final investi-gations were performed with one loading rate appliedwith all Fmax. As expected, curves running with anequal loading rate to different Fmax give identical pene-trations when compared at one load. This finding — astrivial as fundamental — is unaffected by the ap-proaches of data processing. Different results publishedin the literature are caused by different loading condi-tions used in experiments with different Fmax.

In the present experiments with sub-mm Al2O3, noneof the polycrystalline materials with different residualporosity exhibited larger time dependent penetrationeffects. However, creep during dwell was small butdifferent: 0.3–2% of the total penetration within 5 safter loading with 4.4 mN s−1 and 50.5% within 30 safter slow loading with 0.44 mN s−1. The final testswere performed with the rate of 0.44 mN s−1 and with30 s dwell at Fmax to incorporate most of this (small)creep into the measured hardness.

3.2. Shape of penetration cur6es and related indentationsize effect in sub-mm Al2O3

The originally recorded penetration curves were eval-uated with respect to their curvatures and preliminarilypooled as apparently ‘softer’, ‘median’ and ‘harder’locations in the microstructures. Fig. 2 displays theresult for the group representing locations of a ‘median’hardness in the dense sample.

explicitly one or another special effect, for example theinfluence of the roof edge on the penetration curves.Instead, an approach is proposed here to analyze thecurve deformations introduced by all these (more orless unknown) influences in a more general, formal waythat also takes into account the deviations from apower law when Hu in Eq. (1), Eq. (3a) or Eq. (5)depends on the penetration.

When Eq. (3b) (assuming a power law) is applied todifferent parts of the measured (original) loading curve(which deviates from an F�hn shape), the analysis will,formally, give different ho(F) for the different sectionsof the penetration curve. Investigating this feature, alarger ratio F1/F2=F2/F3=1/0.7 was chosen to avoidthe introduction of additional errors from the localscatter of h along the curves, when Fi/Fi+1 is close to 1.Thus, the curves running up to 196 mN were analyzedin four sections between :9–25, 25–50, 50–100 and100–196 mN. The results are presented in Section 3.2.2.2.2. Analysis of the tangent to the unloading cur6es

Another approach uses the information about E* inthe unloading part of the penetration curve to provideHv and the Young’s modulus of the sample withoutseparate evaluation of Hu [3]. The construction of atangent to the upper (rather straight) part of the un-loading curve implies similar fundamental accuracyproblems, as discussed above for the determination ofthe origin of the loading curves (smooth approach tothe tangential line). However, on analyzing unloadingcurves, the input of this kind of error occurs twice(originating from the uncertain slope of the tangent andthe absolute position of the origin of the loading curve),and there is some trade-off between this increasedinaccuracy and the advantage to obtaining an estimateof a local Young’s modulus.

A. Krell, S. Schadlich / Materials Science and Engineering A307 (2001) 172–181 177

In a first evaluation of the shape of these curves, itwas assumed that they obey the power law (Eq. (3a))up to 196 mN. The solid lines in Fig. 5 display theresulting penetration curve (derived from the data inFig. 2 using Eq. (3b)) together with the associatedHu(h) (Eq. (1)) and Hv(h) (Eq. (2a)). The ‘scatter’ givenas an example at Fmax is the difference between theaverages of the pools of ‘softer’, ‘median’ and ‘harder’locations.

The most stringent result in Fig. 5 is the obviouslywrong indentation size effect (ISE) in the hardness ascalculated from penetration by Eq. (1), Eq. (2a) andEq. (3b): Fig. 5 gives Hv(FB50 mN)\30 GPa andHv(F\200 mN)�15 GPa, whereas the conventionallymeasured microhardness Hv of such microstructures is20–22 GPa at F=1000–5000 mN [21]. Note that thisunusual ISE appears also in the universal hardness Hu,i.e. it is independent of elastic recovery effects that mayaffect the ratio between Hu and Hv.

With the imperfect indenter (Fig. 1) and the uncer-tain assumption of a power law like F� (h−ho)n, Eq.(4a) and Fig. 4 predict an exaggerated (wrong) ISE forHu derived from penetration data which were processedin the way performed for the solid curve in Fig. 5.However, contrary to the prediction of Figs. 3 and 4,the hardness in Fig. 5 is not only too high at very smallloads and penetrations, it is also surprisingly small at

larger loads \100 mN. Therefore, other (material)effects beyond the tip shape influence have to beconsidered.

With a power law such as Eq. (3a) and with ahardness defined by the ratio of force and square ofindentation size, the formal description of a strong ISEis an exponent B2 (replotting the solid curve of Fig. 5in a double logarithmic presentation gives n:1.65).Our general knowledge of the ISE in alumina ceramics,however, provides little support for such an interpreta-tion: compared with the extreme load dependence re-ported in Fig. 5, the ‘true’ ISE in sintered sub-mmalumina is very small with an increase of the hardnessof no more than 3 GPa between 10 and 0.3 kgf of load[17,21]. To check the consequences of the erroneouslyenlarged ISE in Fig. 5 for the shape of the penetrationcurve, an additional (dashed) reference curve is intro-duced in Fig. 5 to display the difference between� the penetration expected for a constant hardness

throughout the whole load range, and� the original penetration observed here (solid line in

Fig. 5).By arbitrary assessment, at :75 mN the (fictitious)

reference curve exhibits the same penetration (the samehardness) as the originally derived curve. The shapes ofboth curves exhibit a fundamental difference that is incontrast to the slight effect of the imperfection of the

Fig. 5. Penetration and hardness in dense sub-mm alumina (average grain size 0.63 mm, residual porosity 0.1%), derived from the originallyrecorded data (Fig. 2) with ho from Eq. (3b) assuming a power law F� (h−ho)n (solid lines); the locally observed scatter of penetration andhardness is given at Fmax=196 mN. In contrast, the dashed curves show the fictitious behavior expected for a constant hardness that does notdepend on penetration.

A. Krell, S. Schadlich / Materials Science and Engineering A307 (2001) 172–181178

Fig. 6. Logarithmic plot of penetration data in dense submicronalumina after correction for the origin of the originally recordedcurves (Fig. 2) assuming a load dependent ho(F).

90.001 mm means a rather large hardness difference of0.8 GPa. Considering also the scatter in the recordeddata (Fig. 2), the final accuracy of the derived hardnessvalues is no better than :92–3 GPa.

3.3. Comparison with a single crystalline reference(sapphire)

As reference, supplementary conventional hardnessmeasurements were performed on (1210) sapphire andare included in a comprehensive presentation of presentindentation derived and previous conventional hardnessdata of sintered sub-mm alumina (Fig. 7). The figure isa qualitative confirmation of the absolute levels and theload influence obtained here from nanoindentation forpolycrystalline sub-mm Al2O3.

Fig. 8 gives a closer inspection of the right border ofFig. 7 with few supplementing data at slightly higherporosity of the polycrystals (hardness HV10 at a loadof 10 kgf). The graph displays a relationship betweenthe porosity dependent hardness of sintered aluminaand of sapphire that, according to the results of Fig. 7,seems to remain the same down to smaller loads of:200 mN or less.

4. Discussion

Apparently none of the previously suggested effectsexplain the obviously incorrect hardness derived fromthe loading curve in Fig. 5, assuming a power law suchas Eq. (3a). It is, therefore, a conclusion from theexperimental results in Fig. 5 that a different relation-ship is appropriate. As mentioned in Section 2.2.1.3,both shape imperfections of the diamond tip (Eq. (4a))and the deformation performance of the indented mate-rial can cause deviations of the recorded penetrationfrom F� (h−ho)n. With the results of the foregoingsection it seems probable that several mechanisms arecontributing.

If the hardness is to be expressed by the usual ratioof testing load and the square of h, with a penetration

indenter displayed in Fig. 3. Obviously, with penetra-tion data derived by assuming a power law such as Eq.(3a) or Eq. (5), it is this shape of the penetration curvewhich implements the improper indentation size effectof the hardness in Fig. 5.

This problem cannot be solved by shifts in the originas described by ho in Eq. (3a) or Eq. (5). Instead, Fig.5 appears as an experimental support for the assump-tion in Section 2.2.1.3 that penetration curves F(h) maynot at all exhibit power law characteristics.

To test this hypothesis further, the original data fromFig. 2 were re-examined by a section-wise applicationof Eq. (3b) to investigate the constancy or a suspectedformal load dependence of ho. As a result, for theloading part of the curves in Fig. 2 the procedure givesho(F) values which, for increasing penetration, areshifted into the positive direction of the h-axis. Fig. 6 isa double-logarithmic plot of the corrected penetrationcurve F=F(happ−ho(F)) of the dense microstructure,Table 2 gives the derived hardness results depending onthe porosity of the different samples. Note that at 10mN, a small difference in the penetration depth of

Table 2Nanoindentation derived hardness of sintered submicrometer alumina ceramics with different residual porositya

Testing load Residual porosity 1.2% Residual porosity 0.1%Residual porosity 4.8%(grain size 0.47 mm) (grain size 0.63 mm)(grain size 0.31 mm)

h (mm)/Hv (GPa)h (mm)/Hv (GPa)h (mm)/Hv (GPa)

196 mN 0.654/28.1100 mN 0.464/28.8

0.352/23.850 mN 0.345/24.6 0.329/28.525 mN 0.240/25.90.243/25.1 0.231/29.310 mN 0.1445/30.40.150/27.00.157/25.9

a The data for different loads were derived from the continuous course of average loading curves.

A. Krell, S. Schadlich / Materials Science and Engineering A307 (2001) 172–181 179

Fig. 7. Indentation size effect (load influence) in the Vickers hardness of submicrometer sintered alumina ceramics with different residual porosity.The graph compares conventionally measured data and present results from nanoindentation. As a materials reference, conventional hardnessmeasurements were performed on (1210) sapphire and are included.

that does not follow power law characteristics, therecan only be a formal solution to this problem. Whenthe shapes of such recorded penetration curves areanalyzed by equations which, like Eq. (3b), erroneouslyassume a power law, a fictitious load dependent originho(F) as introduced in Section 2.2.1.3 is the purelyformal consequence. However, as a practical result ofsuch an evaluation, Fig. 6 appears as a reliable repre-sentation of the penetration in polycrystalline aluminaat loads below the threshold of cracking. With fewexceptions at loads \100 mN, all recorded penetrationcurves are very similar for the dense submicrometermaterial confirming its high degree of homogeneitywith a low frequency of small flaws [13,14]. Thesepolycrystalline materials do not exhibit significantcreep.

Fig. 9 compares the corrected penetration curve ofthe dense sub-mm microstructure with the curve ob-tained by Zeng and Rowcliffe for a dense alumina withan average grain size of 5 mm [5]. The nanoindentationmeasured by Zeng and Rowcliffe [5] appears as themost reliable previously published dataset on sinteredalumina. To enable comparison, their original datawere re-evaluated here in the same manner as outlinedin Section 2.2.1.3, but the deviation of their curve from

a power law was small with all ho(F) close to their zeropoint. Obviously, deviations of measured penetrationcurves from a power law may be different, dependingfor example on the different qualities of the indentersand on a different performance of the indented mate-

Fig. 8. Trade-off between minimum grain size and increasing porositywith respect to the macrohardness HV10 compared with the hardnessof (1210) sapphire. All grain sizes 0.3–0.6 mm (given as notes).

A. Krell, S. Schadlich / Materials Science and Engineering A307 (2001) 172–181180

Fig. 9. Comparison of nanopenetration in different sintered aluminamicrostructures observed here and previously. Both curves are dis-played after the same evaluation of the origin of the curves asperformed for Fig. 6.

— is higher than the hardness of the single crystal asfar as this advantage is not offset by an increasingporosity of the sintered material.

When, on the other hand, the recorded penetrationcurves of the dense sub-mm Al2O3 (Fig. 2) are evaluatedassuming a power law like F� (h−ho)n with one con-stant value of ho, the result would be a hardness as lowas Hv=22 GPa at a load of 100 mN (Fig. 5). At thisload (penetration h:0.5 mm), an individual correctionfor the kinked roof edge of the indenter (Fig. 1) by Eqs.(4a), (4b) and (4c) would reduce Hu further by 10%(Fig. 4) with a resulting decrease of Hv to 19 GPawhich is, obviously, too low compared with conven-tional measurements.

5. Conclusions

The few published depth sensing hardness datarecorded for alumina at small loads below the thresholdof cracking (5200 mN) exhibit a large scatter of theoriginal penetration curves and are further complicatedby the lack of a unified analysis. The present investiga-tion addresses several possible reasons for these devia-tions and studies some effects of the microstructure.The conclusions can be summarized in the followingparagraphs.

In depth sensing indentation testing, the comparisonof loading curves that run up to different Fmax requiresequal loading rates (contrary to recommendations insome standards). Without a reliable approach to ac-count for cracking, hardness measurements by depthsensing approaches have to be restricted to testing loadsbelow the threshold of cracking (Fmax5200 mN inAl2O3 ceramics).

In polycrystalline sub-mm Al2O3, creep at the inden-tation site is small (at least for slow loading of 0.44 mNs−1 in the present investigations).

Penetration cannot be described generally by a powerlaw such as F� (h−ho)n because both imperfections ofthe indenter tip (Eq. (4a)) and the performance of theindented material may induce substantial deviations.The shape of recorded penetration curves can, however,be formally evaluated by power law equations with aload dependent correction ho(F) for the origin of thecurves.

The specific shape of penetration curves startingsmoothly from zero introduces a considerable inaccu-racy in any mathematical evaluation of the real positionof the origin. It is demonstrated here that the problemis even greater in approaches that analyze the upperpart of the unloading curve. Therefore, with the restric-tion of unloading analyses to the penetration at maxi-mum load, the present investigation gives preference tothe determination of the load dependence of the hard-ness from the continuous course of loading curvesprovided a reliable Young’s modulus is available.

rial. Whereas the similar loading curves of the twomaterials may appear as a qualitative mutual confirma-tion, unloading is very different, indicating a muchlower Young’s modulus for the coarser ceramic. Unfor-tunately, the authors did not derive hardness valuesfrom their penetration data and a present re-evaluationseems difficult, regarding the different elastic contribu-tions to the deformation.

Fig. 7 compares the hardness results for the loadinterval between 10 mN (1 g) and 98.1 N (10 kg). Theindentation size effect observed here for nanoindenta-tion in sub-mm sintered Al2O3 ceramics (Table 2: 1–2GPa through a decrease in F of one order of magni-tude) agrees well with data reported previously forlarger testing loads (:+3 GPa when F decreases bytwo orders of magnitude [17,21]); it is slightly smallerthan reported by Oliver and Pharr [3] for nanoindenta-tion on (0001) sapphire (:93 GPa on decreasing theload from 120 to :10 mN).

The influence of residual porosity on the hardness israther large. Pores would hardly influence the penetra-tion at the small sites affected by nanoindentation whenmost of the porosity in these submicrometer ceramicswere concentrated in few larger pores as those thatinitiate macroscopic fracture [14]. Instead, the steadydecrease of the hardness with rising porosity indicates ahomogenous spatial distribution of 6ery small pores.

The comparison of nanoindentation results for sin-tered sub-mm Al2O3 ceramics with conventionallyderived hardness data of sapphire in Fig. 7 gives addi-tional support for the relevance of the present analysis:as it is common experience, very fine grained corundumpolycrystals exhibit a hardness which — due to thelimitation of plastic deformability by grain boundaries

A. Krell, S. Schadlich / Materials Science and Engineering A307 (2001) 172–181 181

At the typical loads in nanoindentation (10–200mN), the Vickers hardness of sintered submicrometeralumina rises up to :30 GPa for dense microstructuresand up to :25 GPa with 5% of residual porosity. Theinfluence of the porosity, the indentation size effect(influence of the testing load) and the relationship withthe hardness of single crystals (sapphire) are essentiallythe same as observed at larger loads by conventionalmeasuring approaches.

Acknowledgements

Part of this work has been supported by the SaxonState Ministry of Science and Art under contract 4-7533-70-5140-98/4. The authors also acknowledge stim-ulating discussion with Dr H.-J. Weiss from FraunhoferIWS Dresden on the evaluation of penetration curves.

References

[1] F. Frohlich, P. Grau, W. Grellmann, Phys. Stat. Sol. A. 42(1977) 79–89.

[2] A. Krell, Kristall Technik/Cryst. Res. Technol. 15 (1980) 1467–1474.

[3] W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564–1583.

[4] R.F. Cook, G.M. Pharr, J. Hard Mater. 5 (1994) 179–190.[5] K. Zeng, D. Rowcliffe, Philos. Mag. A. 74 (1996) 1107–1116.[6] R.F. Cook, G.M. Pharr, J. Am. Ceram. Soc. 73 (1990) 787–817.[7] C.J. McHargue, M.E. O’Hern, D.L. Joslin, The influence of ion

implantation on the near-surface mechanical properties of ce-ramics, in: M.F. Doerner, W.C. Oliver, G.M. Pharr, F.R.Brotzen (Eds.), Thin Films: Stresses and Mechanical PropertiesII — Material Research Society Symposium Proceedings, vol.188, Materials Research Society, Pittsburgh, 1990, pp. 111–120.

[8] R.P. Steijn, J. Appl. Phys. 32 (1961) 1951–1958.[9] E.J. Duwel, J. Appl. Phys. 33 (1962) 2691–2698.

[10] C.A. Brookes, R.P. Burnand, Hardness anisotropy in crystallinesolids, in: J.H. Westbrook, H. Conrad (Eds.), The Science ofHardness Testing and its Research Applications, The AmericanSociety for Metals, Metals Park, OH, 1973, pp. 119–211.

[11] T. Goscinska, J. Auleytner, Kristall Tech. 14 (1979) 783–795.[12] A. Krell, S. Schadlich, Proc. 7th Int. Conf. Sci. Hard Materials

(ICSHM7), Ixtapa, March 5-9, 2001, in press.[13] A. Krell, P. Blank, J. Eur. Ceram. Soc. 76 (1996) 1189–1200.[14] A. Krell, Am. Ceram. Soc. 81 (1998) 1900–1906.[15] J. Lankford, D.L. Davidson, J. Mater. Sci. 14 (1979) 1662–1668.[16] J. Lankford, Compressive microfracture and indentation damage

in Al2O3, in: R.C. Bradt, D.P.H. Hasselman, F.F. Lange (Eds.),Fracture Mechanics of Ceramics, vol. 3, Plenum, New York,1978, pp. 245–255.

[17] A. Krell, Mater. Sci. Eng. A. 245 (1998) 277–284.[18] H.-J. Weiss, A. Krell, B. Muller, Phys. Stat. Sol. A. 93 (1986)

509–514.[19] D. Dengel, Materialprufung 30 (1988) 184–186.[20] W. Weiler, H.-H. Behncke, Materialprufung 32 (1990) 301–303.[21] A. Krell, Int. J. Refract. Met. Hard Mater. 16 (1998) 331–335.

.