mfs

u a

g P

Un

m 2ne 3

1. Introduction

electrodeposition techniques [3,4] are usually employed.

With FIB-milling, micro/nanopillars are carved out in the

compression tests, the FIB-based microcompression testsattained greater interest and attention in the research com-munity due to their simplicity and ease of operation. Theyhave been applied to materials of various kinds, such assingle crystals [14], bulk-metallic glasses [5,6], shape mem-ory alloys [12] and other materials, which cannot be

* Corresponding authors. Tel.: +852 2766 6652 (Y. Yang), +852 27666665 (J. Lu).

E-mail addresses: mmyyang@polyu.edu.hk (Y. Yang), mmmelu@inet.polyu.edu.hk (J. Lu).

Available online at www.sciencedirect.com

Acta Materialia 57 (2009) 161316Microcompression experiments have recently beenwidely used to study the mechanical behavior of materialsat the micrometer and submicrometer scales [110]. Thematerials investigated displayed distinctly dierentmechanical behavior, relative to their macroscopic counter-parts, with the reduction of their characteristic size to themicron and submicron scales. Thus, microcompressionstudies have recently led to a renewed interest in exploringthe materials size eect on their mechanical properties.

In order to prepare specimens suitable for microcom-pression tests, the focused ion beam (FIB) [1,2,5,6] and

surface of a substrate material; while, with electrodeposit-ion, micro/nanopillars can be grown out of a substrate.Once the specimens, i.e., the micro/nanopillars, are ready,a modied nanoindentation system, tted with a at-enddiamond indenter, is used for carrying out the microcom-pression tests. During the engagement of the indenter, themicropillar protruding from the substrate has the rst con-tact with the indenter tip and is compressed afterwards.The loaddisplacement data are, then, continuouslyrecorded in the same way as in a regular nanoindentationtest [11].

In comparison with the electrodeposition-based micro-Abstract

An investigation of the focused-ion-beam-based microcompression experiments was conducted using metallic-glass micropillars. Theresults displayed an apparent geometry dependence of the measured pillars Youngs modulus if the formula in the literature was used forthe analysis of the experimental data. However, if the eects of the base material and pillar geometry were taken into account with the aidof nite-element simulations, it was shown that the microcompression experiments can reach a resolution similar to that of the nanoin-dentation tests in the measurement of the materials mechanical properties, and therefore provide an alternative to the nanoindentationexperiment in applications that require the characterization of local mechanical properties in multi-structured/multi-phased materialsystems. 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Compression tests; Nanoindentation; Metallic glasses; Finite-element analysisEects of specimen geoon the mechanical behavior o

metallic-glas

Y. Yang a,*, J.C. Ye a, J. LaDepartment of Mechanical Engineering, The Hong KonbDepartment of Materials Science and Engineering, The

Received 14 August 2008; received in revised forAvailable onli1359-6454/$34.00 2008 Acta Materialia Inc. Published by Elsevier Ltd. Alldoi:10.1016/j.actamat.2008.11.043etry and base materialfocused-ion-beam-fabricatedmicropillars

,*, F.X. Liu b, P.K. Liaw b

olytechnic University, Hung Hom, Kowloon, Hong Kong

iversity of Tennessee, Knoxville, TN 37996-2200, USA

5 November 2008; accepted 29 November 2008January 2009

www.elsevier.com/locate/actamat

23rights reserved.

obtained easily in a form suitable for micromechanical test-ing. However, in spite of the successes achieved with themicrocompression tests pertaining to revealing the materi-als size eect, some aspects of the tests still remain elu-sive from an experimental viewpoint.

Due to the limitations in the fabrication process, theshape of the FIB-fabricated micropillars is dierent froma perfect cylinder [2,5,6,8]. Therefore, the stress andstrain in the micro/nanopillars cannot be derived with easeas in macroscopic compression tests; neither can the pillarsYoungs modulus. These geometrical imperfections usuallycomprise taper angles in the pillar diameter and roundingat the pillar/substrate transition due to the ion-beam diver-gence (see the inset of Fig. 1).

As the micro/nanopillar is sitting on a substrate withthe same mechanical properties, the substrate will makethe micro/nanopillar look more compliant under com-

macroscale uniaxial testing or involve the decoupling ofthe structural interactions when the nanoindentation

1614 Y. Yang et al. / Acta Materialia 57 (2009) 16131623pressive loading. In view of these factors, corrections weremade to interpret the experimental data in the elasticregime for the measurements of the materials Youngsmoduli [2,5,6,15,16]. However, these corrections found inthe literature were either based on some simplifyingassumptions [2,5,6] or specic to a certain geometry andmaterial [15,16]. Therefore, the mechanics of how theaforementioned geometrical imperfections and the sub-strate act together to aect the measurements of theYoungs moduli of the micro/nanopillars is still not wellunderstood. This factor is crucial as the precise measure-ment of the sample Youngs modulus is usually the rststep to justify the experimental data for further use. More-over, given the nature of microcompression tests in probingthe materials local mechanical properties, microcompres-sion experiments may have the potential of measuring themechanical properties of an individual constituent in amulti-phased/multi-structured metallic-glass-based mate-rial, such as metallic-glass coatings, metallic-glass-basedlaminates and even plasticity gradient materials, whosemechanical properties are hard to obtain by conventional

Fig. 1. The symbols used in describing the geometry of a micro/nanopillar

(the inset shows the prole of a real metallic glass nanopillar imaged underSEM).method is used. Therefore, it is worthwhile to have a thor-ough investigation of the validity of the microcompressionexperiments that have been used in the mechanical prop-erty measurement at the micrometer and submicrometerscales.

In this paper, the formula that can be found in the liter-ature is rst used to extract the Youngs modulus of the Zr-based metallic-glass micropillars fabricated via FIB. Thedimensional analysis coupled with the nite-element simu-lation is then employed for the systematic investigation ofthe elastic deformation process of the pillar/substrate sys-tem to derive a new formula for the improved measurementof the pillars Youngs modulus. The results are comparedto those obtained from the macroscale compression andnanoindentation tests on the same material. The implica-tions are discussed afterwards for the future applicationsof the microcompression tests on metallic glasses and othermaterials systems suitable for the microcompression study.

2. Analysis and modeling

In the literature, an approximate formula similar towhat follows was usually used for the extraction of the pil-lars Youngs moduli [2,5,6,15,16]. As shown in Fig. 1, set-ting the origin of the coordinate system at the top of thepillar with the positive direction pointing downwards, thetotal displacement, dt, of the pillar upon compressionthrough a at-end rigid punch can be approximated as1:

dt Z H0

4P

pED0 2h tanh2dh ds 1

where P is the total compressive load applied on the pillar;E is the pillars Youngs modulus (the same as that of thesubstrate); H, D0 and h denote, respectively, the pillarheight, top diameter, and taper angle; and ds is the displace-ment of the substrate, which can be approximated withSneddons formula by assuming the pillar penetrating intothe substrate as a rigid body [17]:

ds P 1 m2ED0 2H tanh 2

where m is the Poissons ratio of the substrate (the same asthat of the pillar).

Substituting Eq. (2) into Eq. (1) and rearranging theequation, we can obtain:

E 1 p1 mD08H

4PH

pD0D0 2H tanhdt 3

with Eq. (3), the pillars Youngs modulus can be extractedfrom the initial indentation loaddisplacement curve that

1 It is assumed that the machine compliance eect has already beeneliminated from the acquired displacement data prior to the use of Eq. (1).

For generality, Appendix A details the procedure to calibrate the machinecompliance for a microcompression test.

(Fig. 3a and b).In addition to the geometry and the associated substrate

eects, a multitude of FEM simulations were conducted toinvestigate the Poissons ratio eect. The results showedthat the inuence was negligible for the Poissons ratiosranging from 0.2 to 0.48, which spans the spectrum ofthe magnitude of the Poissons ratio for most bulk-metallicglasses [1823]. As an example, some of the results are plot-ted in Fig. 4. It shows that, although there is a slightincrease in E/E0 with increasing m at the aspect ratio closeto unity, it is still negligibly small, as compared to the inu-ences brought about by the other factors. For the micro/nanopillars of high aspect ratios, the variation in E/E0 withthe increasing Poissons ratio is within the computationalerror.

3. Experiments

In order to justify the above analyses, microcompressionstudies were conducted on a series of Zr50Cu37Al10Pd3micropillars of dierent geometries (Fig. 5), whose top radiiranged from 0.6 to 4 lm and aspect ratios from 1.5 to10. With the conventional elastic buckling analysis, it can

eriacorresponds to the linear elastic deformation of the pillaronce the Poissons ratio is known. However, Eq. (3) is justa crude approximation based on a few simplications: (1)the stresses are assumed to be uniformly distributed overeach cross-section of the pillar; (2) the curvature eect atthe pillar/substrate transition is not considered and (3)the smooth transition of stress and strain from the pillarto the substrate is replaced by a sudden jump caused bythe rigid body assumption. In order to obtain a more accu-rate measurement of the pillars Youngs modulus, a morein-depth analysis is needed regarding the elastic deforma-tion process of the pillar/substrate system.

Based on the dimensional analysis of elastic deforma-tion, the applied load P can be expressed as:

4P

pED20 f H

D0; h;

qD0

; m

dtH

4

where q is the radius of the curvature at the pillar/substratetransition (Fig. 1) and f is a dimensionless function.

Comparing Eqs. (3) and (4), we can obtain

EE0 W H

D0; h;

qD0

; m

5

where E0 is the nominal Youngs modulus extracted basedon Eq. (3) and w is another dimensionless function, whosevalues are to be evaluated.

In order to investigate the variation of E/E0 with the dif-ferent parameters in Eq. (5), nite-element simulationswere carried out using the commercial package ANSYSTM

(ANSYS, Inc., Cannosburg, PA). A suitable substrate sizewas selected through a few simulations of the substrate sizeeect to ensure that the substrate boundary conditions hadno inuence on the simulated forcedisplacement curvesfor the pillar/substrate system. Due to the symmetry, onlyone half of the pillar/substrate geometry was analyzed. Thetwo-dimensional axisymmetrical elements were then usedin meshing the geometry with the highest element densityin the pillar. The mesh gradually coarsened with theincreasing distance from the pillar to optimize the compu-tational cost and accuracy (Fig. 2).

As the rst attempt, the Poissons ratio was xed at aconstant of 0.3. Based on the nite-element simulations,the variation of the ratio, E/E0, with dierent combina-tions of the aspect ratio, H/D0, the taper angle, h, andthe normalized radius of the curvature, q/D0, is shownin Fig. 3ad. As expected, the direct implementation ofEq. (3) led to a lower measurement of the Youngs mod-ulus than the true value and, thus, the ratios of E/E0 wereall greater than unity. As the taper angle increased, theareal fraction of each cross-section along the pillar heightwith nonuniform stresses increased accordingly, indicativeof the increasing deviation from the assumptions that Eq.(3) is based upon. This trend is manifested by the increas-ing E/E0 ratio with the increasing taper angle in Fig. 3ad.On the other hand, as the aspect ratio, H/D decreased,

Y. Yang et al. / Acta Mat0,

the substrate eect became more signicant, resulting inan increasing E/E0 ratio as well. For instance, at an aspectratio around 12, an underestimation of 4080% of themeasured Youngs modulus could be developed fromEq. (3). However, the increase in the radius of the curva-ture at the base of the micro/nanopillar relieved the stressconcentration at the pillar/substrate transition and sub-stantially reduced the overall degree of the underestima-tion when the ratio of q/D0 reached above 0.5 (Fig. 3cand d). In contrast, there was no signicant changeobserved in the E/E0 ratio for 0.01 6 q/D0 6 0.1

Fig. 2. The graded mesh used for the tapered pillar/substrate system witha rounded pillar base in the FEM simulations.

lia 57 (2009) 16131623 1615be shown that the critical aspect ratio was about 11 for the

Fig. 3. Variation of the ratio E/E0 with the combinations of H/

Fig. 4. The eect of the Poissons ratio on E/E0 at dierent aspect ratiosfor a xed radius of curvature and taper angle.

1616 Y. Yang et al. / Acta Materialia 57 (2009) 16131623investigated metallic glass system (Appendix B). Therefore,only a limited number (one out of four microcompressiontests) of useful experimental data can be obtained for thehigh-aspect-ratio micropillars.

In the fabrication process of the bulk-metallic glass,high-purity metals including Zr (99.95%), Cu (99.999%),Al (99.999%), and Pd (99.995%), were used. The masteralloy with a nominal composition of Zr50Al10Cu37Pd3was then obtained by arc melting. To ensure the homoge-neous distribution of the chemical elements, the ingot wasmelted and ipped at least ve times. The glassy alloy wasfabricated by suction-casting the master alloy into a cop-per mold under a Ti-gettered Ar atmosphere.

Following the procedure in Ref. [13], the micropillarswere fabricated on the surface of the bulk-metallic glassusing a dual beam scanning electron microscopy (SEM)/FIB system (Quanta 200 3D, FEI) and the FIB fabricationprocesses were performed by incrementally reducing theion-beam-current density to minimize the potential FIBdamage in the micropillars, which resulted in the appear-ance of a series of concentric rings with the base of themicropillar at the center (Fig. 5).

D0 and h at dierent q/D0 for a xed Poissons ratio of 0.3.

Based on the SEM pictures, the taper angle and aspectratio can be directly measured precisely from the proleof each micropillar. The radius of the curvature at themicropillars base can be obtained through the geometri-cal relations at the pillars base (see Fig. 6), which is givenby:

q cot p4 h2

Db D0 2H tanh2

6

in which Db is the bottom diameter of the micropillar, mea-sured as the distance between the two opposite pointswhere the pillars meridians gradually curve into the sub-strate (Figs. 5 and 6). As shown in Fig. 7a, the measuredtaper angles show a normal distribution in the range of1 to 7 with an average of 3.5, which are close tothe values reported in the literature for FIB-milled micro-pillars [2,14], while the measured values of the radius ofthe curvature for the micropillars are very small(Fig. 7b), most of which are dispersed in the range from2 to 9 nm.

Eq. (3) was rst used to extract the pillars Youngs

Y. Yang et al. / Acta Materialia 57 (2009) 16131623 16174. Results and discussions

Prior to the microcompression tests, quantitativeassessments of the pillar geometries were attempted.

modulus in the same way as in the literature [2,5,6,15,16].

Fig. 6. Symbols used in the derivation of the radius of the curvature at thebase of the micropillar.

Fig. 5. The representative geometries of the FIB-fabricated micropillarswith markings on the pillars outline showing how their geometricalfeatures were measured: the blue arrows are for the top diameter; the redarrows for the bottom diameter and the green dashed lines for the taperangle (note that the vertical length scale diers from the horizontal onedue to the sample stage tilting). (For interpretation of color mentioned inthis gure the reader is referred to the web version of the article.)Fig. 7. Bar charts of the measured values for the taper angle (a) and theradius of the curvature (b) for the Zr50Al10Cu37Pd3 micropillars.

Youngs moduli2 [28], after which the measured Youngsmodulus was reduced to 103 4 GPa. In addition,

erialia 57 (2009) 16131623For the micropillars of approximately the same aspectratios, similar results were obtained for their Youngs mod-uli. However, when putting all the measurements together,a spurious geometry dependence of the micropillarsYoungs modulus on its aspect ratio became visible(Fig. 8). By averaging out the experimental data of allthe micropillars, the measured Youngs modulus was about76 12 GPa.

In contrast, after the correction was made in accord withEq. (5) and Fig. 3, for which the Youngs modulus previ-ously obtained for each micropillar was multiplied by apre-factor, w, that relates to the pillar geometry, a geome-try-independent value for the pillars Youngs modulus wasextracted as shown in Fig. 8, which was about 99 6 GPa.Compared with the scatter in the uncorrected measure-ments, the scatter in the corrected measurements is nowreduced from 16% to 6%, indicative of the improved

Fig. 8. Comparison of the Youngs moduli obtained from the microcom-pression tests before and after correction.

1618 Y. Yang et al. / Acta Matrepeatability after the correction.For the sake of comparison, nanoindentation experi-

ments were conducted on the surface of the same bulk-metallic glass sample using a Berkovich indenter to extractthe Youngs modulus. Only the forcedisplacement curvesthat followed a master loading curve were selected in thedata analysis, and the results were plotted in Fig. 9. Giventhe Poisson ratio of 0.366 for the Zr-based bulk-metallicglass [24], the Youngs modulus of 129 6 GPa wasobtained, which is about 30% higher than that from themicrocompression tests. As pointed out in Refs. [25,26],there is a tendency for the nanoindentation tests, whichare based on the Oliver and Pharr method [11], to overes-timate the Youngs modulus of bulk-metallic glasses. Thisis because the plastic ow in bulk-metallic glasses, havingnearly zero strain hardening, is prone to a pile-up aroundan indent [27], which leads to the underestimation of theindentation area using the Oliver and Pharr method(Fig. 10). In order to compensate for the pile-up eect,an improved indentation method taking into account thematerials pile-up was used to correct the measuredatomic-force microscopy (AFM) was employed to imagethe indent proles for the extraction of the accurate inden-tation contact area for further verication. Following themethod in Ref. [28], the true indentation contact areawas taken as the area encircled by the pile-up materialas apposed to the triangular area related to the indentationcontact depth through the geometrical similarity, whichyielded results similar to those from the improved indenta-tion method (Fig. 11).

Table 1 lists the measurements of the Youngs modulusof the same Zr-based metallic glass from the three dierentmethods. It can be seen that both the microcompressionand nanoindentation tests yielded very similar results.Although the Youngs modulus obtained from the conven-tional compression tests is very close to the lower bound ofthe measurements from the microcompression tests, it is,however, about 10% lower than the average valuesobtained from both the microcompression and nanoinden-tation tests. Although it is still not very convincing aboutthis discrepancy since no experimental scatter was providedfor the Youngs modulus obtained from the bulk samplesin the literature [24], it is not uncommon that the pre-exist-ing casting defects will reduce the eective Youngs modu-lus when a large sample is tested. Given the large dierencein the length scale of the sampling volume among the threemethods (Table 1), the 10% dierence in the measurementsis regarded as acceptable.

Before closing this section, it is worthwhile having a briefdiscussion on the other factors thatmay aect themicrocom-pression tests ofmetallic glasses.When the aspect ratio of themicropillar is over10, it is prone to elastic buckling due tothe misalignment between the indenter and micropillar.Consequently, a quite dierent forcedisplacement curvewas acquired as compared to those from the non-bucklingtests, giving rise to a Youngs modulus measurement muchlower than otherwise (Fig. 12). Given the dramatic changein the forcedisplacement curve, the experimental data cor-responding to elastic buckling can be easily excluded in thesubsequent data analysis. On the other hand, the elasticand geometrical mismatch between the indenter and themicropillar causes a stress concentration at the lateralboundary of the indenter/micropillar contact. In conven-tional compression tests on BMGs [2933], such a kind ofstress concentration at the interface between a platen and acompression specimen leads to an aspect ratio eect on thecompressive behavior of the BMG specimens. In the micro-compression experiments, the aforementioned stress concen-tration still exists and partly contributes to the stress non-uniformity in the micropillars. In the elastic deformationregime, the stress concentration eect was captured by thenite-element simulations for the case of non-frictional

2 The initial yield strength of 1899 MPa and zero strain hardening

exponent, obtained from the macroscale tensile experiments [28], wereused in the correction.

eriaY. Yang et al. / Acta Matindenter/micropillar contact; while in the plastic deforma-tion regime, it is still an unexplored issue to date whetherthe stress concentration can lead to such an aspect ratio eecton the post-yielding behavior of the metallic-glass micropil-lars.As for the yielding point ofmetallic-glassmicropillars, itwas found that, due to the stress concentration, themeasure-ment of the yielding strength was sensitive to the constitutivebehavior assumed, and interested readers are referred to therelated work by Schuster et al. [34] for a detailed discussion.

5. Implications

Based on the above analyses and discussions, we havedemonstrated that, as long as the geometrical features of

Fig. 9. Variation of the metallic glass Youngs modulus with the indentationindentation impression and the inset to the right shows the loaddisplacemen

Fig. 10. The 3-D rendered AFM image of an indentation impressionmade by a Berkovich indenter showing the pile-up around the periphery.lia 57 (2009) 16131623 1619the micropillar are measured from the SEM pictures, themicrocompression tests are capable of measuring the elasticmodulus of the metallic glass with a similar accuracy as thenanoindentation tests. However, in view of the relativelysimple structure of the testing specimen at such a smallscale, a few applications could be envisioned for the micro-compression tests.

Over the past decades, great eorts have been dedicatedto the development of a reliable method to measure themechanical properties of a coating/substrate system, whichrepresents a class of industrial and academic problems ofgreat importance. In the past, the focus has been directedto the use of nanoindentation to study coating/substratesystems [35]. However, the coating/substrate interactions

contact depth (the inset to the left shows the AFM height image of ant curves).

Fig. 11. The true indentation area taking into account the pile-up, asoutlined by the dashed line, vs. the computed indentation area as outlinedby the dotted line in a 2-D AFM image.

complicate the mechanical responses and the interpretationof the experimental data is therefore non-trivial, whichrelies on the prior knowledge of a few key properties thatare related to materials heterogeneity around the interfacebetween the coating and substrate [36]. On the other hand,the microcompression methodology may provide an alter-

substrate by milling out a small-scale specimen in the areaof interest for the microcompression test.

The other type of material systems that attracts the inter-est of many materials scientists and is suitable for micro-compression study now is the so-called plasticity gradientmaterials. One method to realize the plasticity gradient isto introduce a grain size gradient using the well-established

Table 1Summary of the Youngs modulus measurements from the three dierentmethods.

Quantity method Youngs modulus(GPA)

Length-scale of thesampling volume

Macrocompression tests [24] 90 >1 mmMicrocompression tests 99 6 0.53 lmNanoindentation tests 103 4 1.59 lma

a Note that the length scale of the sampling volume in a nanoindentationtest is taken as the plastic zone size, c, approximated by the Johnsindentation model [49] as c = (9P/2pH)0.5, where P is the indentation loadand H is the corresponding indentation hardness.

1620 Y. Yang et al. / Acta Materialia 57 (2009) 16131623native solution to the aforementioned problems by separat-ing the coated material from the substrate in a single test.Since the deformation is limited to the micropillar, the con-cerns related to the interactions between dierent constitu-ents in a multi-structured/multi-phased system can, tosome extent, be relieved. However, given the assumptionsof material homogeneity and isotropy, our method isnow only applicable to a specic type of coating/substratesystems. One recent novel application of metallic glasses isto sputter a metallic-glass thin lm onto the surface of aclassic engineering material to enhance the fatigue resis-tance as shown in Fig. 13a [3740]. In these applications,it is extremely dicult to obtain a reliable assessment ofthe mechanical properties of the coating using the nanoin-dentation method as the confounded mechanicalresponse is site-sensitive due to the material heterogeneityin the underlying structural materials. Now, one plausibleFig. 12. Comparison of the buckling vs. non-buckling forcedisolution to overcome such a confounding eect from thesubstrate is to completely separate the thin lm from the

Fig. 13. FIB images of the complex materials systems suitable formicrocompression tests for the measurement of the mechanical propertiesof an individual constituent: (a) a metallic glass coating on a stainless steelsubstrate and (b) a nanocrystalline nickel iron alloy with a grain sizegradient (the red arrows indicate the cross-section/top surface transition).(For interpretation of color mentioned in this gure the reader is referredto the web version of the article.)splacement data acquired from the microcompression tests.

surface-mechanical-attrition-treatment (SMAT) technique[4145]. After SMAT, a grain size distribution, similar tothat shown in Fig. 13b, can be implemented into a polycrys-talline metal for the optimization of its overall mechanicalproperties. In spite of the overall heterogeneity, some localhomogeneity and isotropy still remain in the areas of similargrain size. Therefore, the local mechanical properties can bemeasured using the micropillars as long as they sample ahigh enough number of similar-sized grains to ensure thehomogeneity and isotropy.

6. Summary and conclusions

In summary, we have demonstrated that the approxima-tion with the Sneddons formula is insucient to obtain anaccurate measurement of the pillars Youngs modulus.

Y. Yang et al. / Acta MateriaBased on the simulations and experiments, an improvedformula can herein be proposed, which is:

E W 1 p1 mD08H

4PH

pD0D0 2H tanhdt 7

where w is the geometry-dependent pre-factor and its valuecan be found in Fig. 3ad according to the aspect ratio, thetaper angle and the radius of the curvature of the micro/nanopillar concerned, which are all measurable quantitiesin a microcompression experiment. Since the taper anglefor the appropriately ion-beam-treated micro/nanopillarside walls ranges from 0 to 6, the aspect ratio mostly usedin the literature is around 23 and the radii of the curvaturemeasured are all within 10% of the corresponding top ra-dius, we would expect that Fig. 3ad has comprised mostof the w values that might be used for future microcom-pression studies.

The advantage of a microcompression test over a nano-indentation test is the relatively simple stress state that islimited within the micropillar. As long as the micropillaris placed far away from the boundary susceptible to aFig. A1. The plot of one of the h P1/2 curves used for the calibration ofthe machine compliance with thermal drift corrected.microstructural change, Eq. (7) is applicable and a reliablemeasurement of the elastic properties should be warranted.

Acknowledgments

Y. Y acknowledges the internal research funding sup-port to this work granted by the Hong Kong PolytechnicUniversity through the Department of Mechanical Engi-neering. The bulk-metallic glass used in the present workis provided by Prof. Y. Yokoyama of Tohoku Universityand Dr. G.Y. Wang of The University of Tennessee.F.X.L. and P.K.L. are grateful to the support of the Na-tional Science Foundation, International Materials Insti-tutes (IMI) Program (DMR-0231320) with Dr C. Huberas the Program Director.

Appendix A

In conventional nanoindentation tests, the machinecompliance is usually calibrated prior to data acquisitionwith its eect being eliminated from the subsequentlyacquired displacement data. However, after many yearsof use, the machine compliance may deviate from thepre-dened or built-in value and, therefore, needs to bere-calibrated. In the case of an unknown machine compli-ance Cf, the measured total displacement, dt, is the sumof the displacements in the material, dm, and the loadingframe, which is:

dt dm Cf P A1To our knowledge, there are two methods available in

the literature to calibrate the machine compliance of a nan-oindentation system [11,46]. For brevity, only the methodproposed by Sun et al. [46] is introduced below for thestudy of our nanoindentation system. Since the shape ofthe diamond nanoindenter only possesses a negligible eecton the loading frame compliance, a conventional Berko-vich nanoindenter can be used to estimate Cf. Withoutprior knowledge of the loading frame compliance Cf andthe indenter tip radius r in a real indentation experimentof isotropic and homogeneous materials, the total indenta-tion displacement h can be expressed as:

h Cf P K1=2P 1=2 g A2where K is a constant that depends on the materialsmechanical properties and g is a geometrical parameterthat is related to the indenter tip radius and thermal drift.

Fitting the Berkovich nanoindentation data obtained onthe Zr50Al10Cu37Pd3 BMG sample with Eq. (A2), themachine compliance was extracted with its value rangingfrom 0.0001 to 0.0007 nm mN1 (one of the extractions isshown in Fig. A1), which conrmed that the eect of themachine compliance had been ruled out in our experimentswith a pre-dened machine compliance of 0.5 nm mN1.Meanwhile, the g value of 15.3 nm corresponds to a tip

lia 57 (2009) 16131623 1621radius of 230 nm for the nanoindenter, which is reason-able for a Berkovich-type indenter already tip-rounded

eriaFig. A2. The 2-D schematics of (a) the original micropillarsubstratesystem, (b) the double tapered straight bar with a uniform mid-section forthe upper bound estimate of the critical buckling load, and (c) the over-constrained micropillar for the lower bound estimate of the critical

1622 Y. Yang et al. / Acta Matafter repeated use (the original tip radius was 150 nm).This implies that the thermal drift was also appropriatelycorrected in the microcompression experiments. Based onthe above discussions, the use of Eq. (1) is justied for dataanalysis.

Appendix B

Due to the complexities in the geometry of the micropil-larsubstrate system, it is non-trivial to obtain a close-formsolution to the critical buckling load. Instead, we are tryingto seek its upper and lower bound estimates by looseningand strengthening the mechanical constraints from the sub-strate, respectively.

To make an upper bound estimate, let us assume that theconstraint from the substrate on the micropillar is restrictedto the materials in a cylindrical volume right below themicropillars base (as shown by the hatched area inFig. A2a). This cylindrical substrate possesses the heightof the original substrate, denoted as a/2 in Fig. A2b, andthe same diameter,Db, as that of themicropillars base. SinceH/a

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Effects of specimen geometry and base material on the mechanical behavior of focused-ion-beam-fabricated metallic-glass micropillarsIntroductionAnalysis and modelingExperimentsResults and discussionsImplicationsSummary and conclusionsAcknowledgmentsAppendix AAppendix BReferences