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researchers and makes it freely available over the web where possible

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To cite this version

Mohamed BENTOUMI D BOUZID H BENZAAMA Alberto MEJIAS Steacutephania KOSSMAN AlexMONTAGNE Alain IOST Didier CHICOT - Multiscale and multicycle instrumented indentation todetermine mechanical properties Application to the BK7 crown borosilicate - Journal of Materialsresearch p1-12 - 2017

Any correspondence concerning this service should be sent to the repository

Administrator archiveouverteensameu

Science Arts amp Meacutetiers (SAM)is an open access repository that collects the work of Arts et Meacutetiers ParisTech

researchers and makes it freely available over the web where possible

This is an author-deposited version published in httpsamensameuHandle ID httphdlhandlenetnull

To cite this version

Mohamed BENTOUMI D BOUZID H BENZAAMA Alberto MEJIAS Steacutephania KOSSMAN AlexMONTAGNE Alain IOST Didier CHICOT - Multiscale and multicycle instrumented indentation todetermine mechanical properties Application to the BK7 crown borosilicate - Journal of Materialsresearch p1-12 - 2017

Any correspondence concerning this service should be sent to the repository

Administrator archiveouverteensameu

ARTICLES

Multiscale and multicycle instrumented indentation to determinemechanical properties Application to the BK7 crown borosilicate

M Bentoumia) and D BouzidInstitut Optique et Meacutecanique de Preacutecision LOA Ferhat Abbas Seacutetif 1900 Algeacuterie

H BenzaamaEacutecole Nationale Polytechnique drsquoOran ENPO Oran 31000 Algeacuterie

A MejiasArts et Meacutetiers ParisTech MSMP ENSAM 8 Boulevard Louis XIV Lille 59046 France and Facultad deIngenieriacutea CIMEC Universidad de Carabobo Valencia 2005 Venezuela

S KossmanUniv Lille FRE 3723-LML-Laboratoire de Meacutecanique de Lille Lille 59000 France

A Montagne and A IostArts et Meacutetiers ParisTech MSMP ENSAM 8 Lille 59046 France

D ChicotUniv Lille FRE 3723-LML-Laboratoire de Meacutecanique de Lille Lille 59000 France

(Received 2 September 2016 accepted 22 December 2016)

In this work nano micro and macro-indentation tests under standard or multicycle loadingconditions were performed for studying the mechanical behavior of a crown borosilicate glasssample with the objective to study the scale effect in indentation and the influence of cracksformation on the assessment of mechanical properties When no cracks were initiated during theindenter penetration especially for low indentation loads the mechanical properties were deducedby applying different methodologies (i) Standard (or monocyclic) loading (ii) ContinuousStiffness Measurement mode (iii) Constant and progressive multicycle loading and (iv) Dynamichardness computation It has been found independently of the loading conditions Martenshardness and elastic modulus are approximately 33 and 70 GPa respectively However whencracking and chipping are produced during the indentation test two damage parameters related tohardness and elastic modulus can be used for representing the decrease of the mechanicalproperties as a function of the relative penetration depth

I INTRODUCTION

Optical glasses are mainly used for lenses and mirrorsin optical devices which are the most often intendedfor telescopes microscopes and photographic targets(sights collimators and eye pieces) The main typesof glasses belong to the crown and flint familiesThese materials are appropriate for optical transmissionin a range between 380 and 2100 nm due to theirhomogeneous microstructure their low porosity andtheir easy machinability As an example a concave lensof crown combined with another convex lens in flintis known to correct chromatic aberration of an opticaldevice12

To optimize their performance in service and forobtaining high quality optical devices the reflective

surface must be carefully prepared to have a smoothsurface and ldquozerordquo defects like pores or nano-cracks Thesurface must also be flat to avoid any optical waversquosdeflections Additionally the surface preparation processmust minimize the introduction of residual stresses at thesurface Within this compulsory objective the surfacepreparation process must follow specific steps The firstpre-polishing step consists of a lapping phase withagglomerate tools or abrasive particles in suspension3ndash7

This stage is used to eliminate the macro-geometricaldefects of the as-prepared specimen and to ensure thesurface flatness and the parallelism between the twoopposite faces After lapping the optical surface isgenerally opaque Consequently the succeeding polish-ing step consists of mechanical polishing with sandpapersof different grades to eliminate the micro-geometricaldefects and to produce a high quality surfaceNevertheless a variety of polishing conditions is pro-posed in literature to achieve optical devices after a lowrate of material removal and roughness reduced to onlyfew nanometers8ndash13

Contributing Editor George M Pharra)Address all correspondence to this authore-mail hamoudi_10yahoofr

DOI 101557jmr2016523

J Mater Res 2017 Materials Research Society 2017 1

To study the mechanical behavior of this kind of brittlematerial instrumented indentation test is one of the mostconvenient method for determining hardness and elasticmodulus However under certain loading conditionscracks can appear along the indent diagonals of a sharpindenter and their effects on the mechanical propertiesmeasurement are not clearly demonstrated In thiswork it was studied that the mechanical behavior ofthe crown BK7 glass using nano- micro- and macro-indentation loadings applying standard (monocyclic)tests Continuous Stiffness Measurement (CSM) modeat nano scale range multicycle indentation with constantand progressive loading at micro and macro scale rangesThe main purpose of this work is to obtain a multiscaleapproach of the mechanical properties of the crown glassand to establish relationships between standard tests andmulticycle tests at a multiscale approach

II MATERIAL AND EXPERIMENTS

A Crown borosilicate BK7 sample and surfacepreparation

The crown borosilicate BK7 (BS-BK7) glass is a brittlemineral glass containing mainly silica Its composition(in wt) is presented in Table I

The crown borosilicate glass (BK7 glass) is the mostused glass in optical design glasses its compositionremains close to that of soda lime glass with a highcontent of silica soda and calcium The used sample hada parallelepiped shape of 28 24 6 mm dimensionsThe main physical properties ie density reflective indexAbbe number and Poisson ratio are collected in Table II

The glass was lapped and afterward polished withspecific pellets For the lapping the pellets werecomposed of 98 of aluminum oxide Al2O3 browncorundum type having an index of 9 following theMohs hardness scale and 389 gcm3 of density Thepreparation consists of mixing 70 of alumina and30 of epoxy resin during 10 min for obtaining ahomogeneous mixture Different grain sizes for thepellets (40 30 15 7 and 3 lm) were elaborated underthe same conditions of compaction load (10 KN) heating

temperature (150 degC) time-duration treatment (30 s) andabrasive-binder ratio for preparing the different succes-sive lapping steps

For the final polishing the pellets of 10 mm ofdiameter having a mean grain size of 05 lm arecomposed of cerium oxide CeO2 type white CERI100VO produced by Piplow and Brandt societyTo optimize the pellets efficiency three types of binderhave been tested (i) polyurethane (ii) silica glasspowder and (iii) epoxy resin Preliminary tests havebeen performed on the glass by measuring the amount ofmass loss after lapping with grain sizes of 40 and 30 lmusing the three types of pellets The best results wereobtained with the epoxy resin binder This is probablydue to a better bind between the abrasive grains Thishigh quality connection is associated with the meltingtemperature and the viscosity coefficient of the resinwhich allows a better wrapping of the grains togetherduring the cooking The low efficiency of the two othertypes of binder is associated to the low wettability of thepolyurethane leading to a poor adhesion of the hardparticles The use of the silica glass powder is probablydue to its solid form as binder Finally the epoxy resinhas been selected to produce the pellets for lapping andpolishing For lapping and polishing the samples bypellets the rotation speed of the sample holder is of60 rpm During the lapping a damaged subsurface(DSS) is formed due to the destructive contact betweenthe abrasive grains of the pellets and the glass samplesThe abrasive particles provoke the delamination of theDSS by cracking visible at a macroscopic scale whereassmall stripes can be undetectable even by optical micro-scopes1415 The material removal leads to the formationof peaks and cavities whose magnitudes depend on thepellets grain size Note that this grain size can be mea-sured by two techniques ie transverse and angularpolishing proposed by Esmaeilzare et al16 The severityof the lapping stage can also cause apparition of residualstresses which modify directly the surface integrity andconsequently the global behavior of the material17ndash20

B Specimen roughness

The surface roughness was measured by means ofoptical profilometry (Veeco Wyko NT9300 VeecoEdina Minnesota) The initial surface roughness beforelapping is shown in Fig 1(a) Previously to lapping andpolishing the roughness Ra and Rq is equal to 093 and174 lm respectively At the end of the lapping stageusing different grain sizes the roughness decreases untilRa 5 009 and Rq 5 012 lm [Fig 1(b)] The polishingusing cerium oxide pellets with a mean grain size of05 lm decreases rapidly the roughness value after only10 min of polishing giving a Ra 5 001 lm and Rq 5002 lm [Fig 1(c)]

TABLE I Composition of the BK7 glass under study

SiO2 Na2O CaO K2O B2O3 BaO TiO2

BK7 glass 688 92 09 73 107 23 08

TABLE II Main physical properties of the BK7 glass used to performthe instrumented indentation tests

Density(gcm3)

Refractionindex

Abbenumber

Poissonratio

BK7 glass BK7 244 152 642 020

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

J Mater Res 20172

C Indentation experiments

Nano-indentation experiments were performed with aNano Indenter XPtrade (MTS Nano Instruments OakRidge Tennessee) employing a Berkovich diamond in-denter The load range of the instrument is set between 10mN and 10 N The samples were fixed on a metallicsupport using a thermoplastic adhesive (Crystal bond590) Two types of test were performed in thisinstrument First 9 standard tests (load-unload)forming a grid (3 3) were performed with the sametesting conditions maximum load 500 mN strainrate 005 s1 time to loadunload 30 s and holdingtime 15 s The standard tests allow computing thehardness and the elastic modulus at the maximum in-dentation load Second 9 tests using the ContinuousStiffness Measurement (CSM) mode the testing param-eters were harmonic displacement 2 nm acquisitionfrequency 45 Hz maximum penetration depth 2500 nmThis method allows the description of hardness andelastic modulus variation with the indentation depth

Micro-indentation experiments were carried out usinga micro-hardness tester CSM 2ndash107 provided with aBerkovich pyramidal diamond indenter The load rangeof this equipment goes from 100 mN to 20 N Theresolution for load measurement is 100 lN and for depthis 03 nm In this work a continuous multicycle protocolwas executed this protocol allows the computation ofthe mechanical properties (hardness and elastic modulus)after each cycle The testing conditions were 100 cyclesper test loading and unloading last 30 s each to guaranteethe same time for all the cycles The minimum loadapplied at the first cycle was 200 mN and the maximumload at the last cycle was 2000 mN the unloading limitwas set to 20 of the maximum load at each cycle Thedwell-time of 15 s was imposed at the maximum loadand additionally a dwell-time of 15 s was set at theminimum load of each cycle

Macro-indentation experiments were performed witha Macro Indenter Zwick ZHU 25 The standard reso-lution for depth measurement system is 002 lm

FIG 1 Roughness parameters obtained from the BK7 glass sample (a) initial state (b) after lapping and (c) after cerium oxide pelletpolishing

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

3J Mater Res 2017

In macro-indentation two kinds of multicycle tests wereperformed using progressive load and constant load byvarying the number of cycles and the range of loads Thedifferent indentation testing conditions under micro- andmacro-indentation have been collected in Table III

III ELASTIC MODULUS AND HARDNESSDETERMINATION BY INSTRUMENTEDINDENTATION TEST

The main mechanical properties determined by instru-mented indentation are the hardness usually called HITand the reduced elastic modulus ER Their calculation isbased on the two following relations which are derivedfrom the methodology originally proposed by Oliver andPharr21

HIT frac14 Pmax

AC

and ER frac14 1b

ffiffiffip

p2

SffiffiffiffiffiffiAC

p eth1THORN

Where AC is the projected contact area corresponding tothe maximum applied load Pmax S de notes the contactstiffness b is a geometric correction factor equals to105 according to Antunes et al22 who obtained thisvalue independently of the material by using a three-dimensional simulation of the Vickers indentation

The reduced elastic modulus ER involves the elasticproperties of the material and the indenter as a functionof their elastic modulus and Poisson ratios defined asfollows

1ER

frac14 1 m2

Ethorn 1 m2i

Ei

eth2THORN

Where the elastic modulus of the diamond indenterEi and the Poisson ratio mi are equal to 1140 GPa and

007 respectively23 E and m denote the elastic propertiesof the tested material

It can be noted that to obtain accurate values for thehardness HIT and the reduced elastic modulus ER thecontact area must be properly determined by taking intoaccount the influence of the tip defect of the indenterUnfortunately its consideration into the calculationespecially for very low indenter displacements is abso-lutely necessary since the use of indenters inevitablyleads to a bluntness of the indenter tip which can beassumed to have a spherical shape With this objectivein nano-indentation Oliver and Pharr21 proposed thefollowing complex area function

AC frac14 245 h2c thorn C1 hc thorn C2 h1=2c

thorn C3 h1=4c thorn C8 h1=128c eth3THORN

Where the coefficients C1 through C8 are constants andthe leading term describes a perfect pyramidal indenterThe others describe deviations from the conical geometrydue to blunting at the tip hc is the contact depth which isequivalent to the indenter-depth along which the diamondindenter rests in contact with the material according tothe deformation around the indent Its calculation ispresented here-after

The fitting coefficients are obtained on a calibrationsample often fused silica necessitating the use of theContinuous Stiffness Measurement (CSM) mode whichallows collecting numerous indentation data at the sameindentation point and consequently a precise determi-nation of the values of these coefficients Howeverwhen CSM mode is not available which is often thecase on several commercial indentation instruments inthe micro and macro ranges Chicot et al24 proposed analternative method by applying a model constructed onthe base of the functions developed by Antunes et al25

TABLE III Multicycle indentation testing conditions applied in micro- and macro-indentation

Test Loading type Load range (N) Maximum load (N) Unloading (N) Dwell-time (s) Number of cycles

No 1 Progressive 02ndash2 (micro) 2 20 Pmax 15 100

No 2 Progressive 7ndash30 (micromacro)

7

2 10

710 1015 1525 2530 30

No 3 Progressive 50ndash300 (macro) 300 1015

630

No 4 Constant 5ndash50 (micromacro)

5 05

15 5

10 115 1520 530 540 550 5

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

J Mater Res 20174

and by Berla et al26 Chicot et alrsquos model is expressedas follows

AC frac14 245 hc thorn hb 1 exp 2hchb

3=2 2

eth4THORN

Where the effective truncation lengths of the indenter tiphb can be determined by regression analysis on a known-material or estimated from microscopic observations27

In micro-indentation the model of Troyon andHuang28 which consists in adding the effective truncationlength of the indenter tip hb to the contact depth hc isenough accurate This amounts to neglect the exponentialterm in Eq (4) In this condition the contact area iscalculated by the following relation

AC frac14 245 hc thorn hbeth THORN2 eth5THORNNote that this relation is only valid for indenter

displacements higher than approximately 100ndash150 lmdepending on the magnitude of the tip defect dimensionIn macro-indentation for indenter displacements muchbigger than the dimension of the tip defect the relationconsidering a perfect pyramidal indenter can be reason-ably applicable

AC frac14 245 h2c eth6THORN

On the other hand as it has been demonstrated recentlyby Yetna NrsquoJock et al29 the computation of the contactdepth hc depends on the deformation mode around theindent ie sink-in or pile-up phenomena The distinctioncan be done by comparing the remnant indentation depthto the maximum indentation displacement ratio to thelimit value of 083 For glasses for which this ratiois lower than this critical value sink-in phenomenonoccurs during the indentation process Consequentlythe contact depth is calculated by means of the meth-odology developed by Oliver and Pharr21

hc frac14 hmax ePmax

S eth7THORN

Where hmax is the indenter displacement reached by theindenter at the maximum applied load and e equals to075 for sharp indenters

The typical loading and unloading curve indicatingthe parameters involved into the computation of thecontact area used for the determination of the mechan-ical properties can be found21

On the other hand depending on the displacementmeasurement system of the instrument the deflections ofthe load-frame can be included in the depth measurement

especially for the micro-indentation instrument usedin this study In this case the displacement into thespecimen can be accessible from the total displacementgiven by the instrument and the frame compliance ofthe instrument Cf deriving from Eq (1) as follows

1S

frac14 Cf thornffiffiffip

p2

1b ER

1ffiffiffiffiffiffiAC

p eth8THORN

Note that the plot of the inverse of the unloading slopeas a function of the inverse of the square root of the contactarea allows the determination of the reduced elasticmodulus directly from the slope of the straight line

On the other hand HIT cannot be calculated froma standard loading curve because the contact depthinvolved in the contact indentation depth calculation isnot accessible Consequently the dynamic hardness whichis usually calculated from a loading curve is representedby the Martens hardness (calculated with the total(or instantaneous) penetration depth hi correspondingto the load Pi) similiar to the Vickers hardness(computed with the indent diagonal) as follows

HM frac14 Pmax

AR

with AR frac14 2643 hmax thorn hbeth THORN2 eth9THORN

Where AR is the actual contact area that takes intoaccount the influence of the tip defect Note that for thelowest indenter displacements Eq (3) can be adapted bymultiplying all the coefficients by 1079 (ie the quotient2643245) and in Eq (4) by changing 245 to 2643 andalso for both equations hc should be replaced by hmax tocalculate the actual contact area considering the tip defectfor very low indenter displacements

IV RESULTS AND DISCUSSION

Indentation experiments have been performed on aBK7 glass sample to determine the elastic modulus andthe hardness before and after damage resulting fromcracking during indentation tests at high loads Indeeddepending on the indentation scale of measurementcracking and chipping have been observed around theindent for high indentation loads in the macro-indentationdomain Therefore when no damage is observed innano- and micro-indentation load ranges the mechani-cal properties were calculated

A NANO-indentation

1 Standard indentation

Figure 2(a) shows all the 9 loadndashdisplacement curvesobtained by nano-indentation performed randomly at thesurface of the specimen

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

5J Mater Res 2017

Even if some of them are very well super imposed allthe curves presented in Fig 2(a) are not perfectly superimposed representing some dissimilarities at the surfaceof the tested material Probably this is related to theformation of a very thin layer on the surface sample afterpolishing or to the presence of heterogeneities in thematerial considering that the maximum indentation depthwas 25 lm To calculate the mechanical properties acalibration of the Berkovich indenter tip has been per-formed on fused silica to determine the fitting coefficientsof the contact area function expressed in Eq (3) In thiswork it was limited as the number of coefficients of three(C0 5 245 C1 5 429 nm and C2 5 1292 nm32)On the other hand in spite of the existing differencesbetween the indentation curves the obtained values forMartens hardness and elastic modulus were pretty pre-cise 328 6 013 GPa and 75 6 2 GPa respectively

2 Dynamic hardness

To artificially increase the number of data someauthors30 suggested the computation of a dynamic orcontinuous hardness over all the loading curve by com-puting the hardness from each point of the indentationdata (hi Pi) corresponding to the instantaneous penetra-tion depth and load respectively This method whichleads only to the calculation of the Martens hardnessis interesting but usually overestimates the hardnessnumber because of dwell-time (15 s) which is appliedat the maximum load not considered into the compu-tation Depending on the creep behavior of the mate-rial the depth continues to increase thus increasing thecontact area

Implementing the dynamic analysis of the loadingcurve the calculated dynamic hardness is supposed tobe equivalent to the Martens hardness Subsequently tostudy the hardness behavior at the nanoscale range it iscomputed by the Martens hardness along all the loadingcurve using Eq (8) replacing (hmax Pmax) by each

(hi Pi) point of the loading data and where AR isthe contact area calculated by Eq (3) Therefore theMartens hardness is plotted versus the indenter displace-ment [Fig 2(b)] The curve in this figure represents themean dynamic hardness value and its standard deviationis calculated by considering the 9 indentation curves ofthe standard tests

The high standard deviation in Fig 2(b) is due to thedifferences in the loadndashdisplacement curves illustrated inFig 2(a) However when the indenter reaches a deptharound 1 lm the Martens hardness becomes ratherconstant equal to 345 6 022 GPa which is slightlyhigher than the Martens hardness determined from thestandard tests at the maximum indentation load As it waspreviously mentioned this difference is due to the effectof the dwell-time at the maximum load in the standardtests which tends to increase the penetration depth owingto the creep indentation phenomenon Assuming that theadditional depth due to creep is directly proportional tothe applied load we can write the following relation

DhiDhmax

frac14 Pi

Pmax

eth10THORN

Where Pmax is the maximum indentation load at whichthe dwell-time is applied and which provokes an increasein depth of Dhmax Consequently the instantaneous gapin depth Dhi is computed from the corresponding appliedload Pi

Following this reasoning the gap in depth can beadded to the actual indentation depth to calculate thecorrected contact area In this condition the recalculatedMartens hardness leads to a mean value of 330 GPawhich is similar to the Martens hardness computed at themaximum load

3 CSM mode

One of the most reliable methods for determininghardness and elastic modulus is the Continuous Stiffness

FIG 2 (a) Loadndashdisplacement curves obtained from a monocycle test applied on the surface and (b) Variation of the dynamic Martens hardness asfunction of the indenter displacement obtained by nano-indentation from the BK7 glass sample

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

J Mater Res 20176

Measurement (CSM) mode which consists to impose aharmonic oscillation of the force during loading In thisregard the mechanical properties are obtained as afunction of the indenter displacement Usually theinstrumented hardness HIT is computed using the CSMmode instead of Martens hardness To validly comparethe different approaches to calculate the mechanicalproperties the Martens hardness is computed ratherthan the HIT by applying Eq (8) Figure 3 presents theMartens hardness [Fig 3(a)] and the elastic modulus[Fig 3(b)] versus the indenter displacement into thesample of the surface

In Figs 3(a) and 3(b) the Martens hardness and theelastic modulus are constant independently on the dis-placement except for the first nanometers where the tipblunting affects the measurement usually the bounds todelimitate reliable results are taken from the calibrationperformed on the fused silica sample Therefore theMartens hardness and the elastic modulus are equal to330 6 016 GPa and 715 6 29 GPa respectivelyThese values are comparable to the results obtained bythe standard method and the dynamic approach

B MICRO-indentation

The micro-indenter CSM 2ndash107 does not dispose ofthe CSM mode instead multi-cycle tests were performedand therefore Eq (4) is applied to compute the contactarea In a previous investigation27 the tip defect for theBerkovich indenter was estimated by using a scanningelectronic microscope at very high magnification toaccurately determine the tip defect dimension hb Thelength of the truncated indenter tip was estimated at50 nm (Fig 4)

Figure 5(a) shows an example of a loadingndashunloadingcurve for multicycle tests on the BK7 glass sample5 tests using the multi-cycle protocol described before

(No 1 Table III) were conducted randomly on the surfacesample

To determine the elastic modulus of the materialaccording to Eq (8) the inverse of the total contactstiffness is plotted against the inverse of the square rootof the contact area to determine the frame compliance CfAfter wards the value of the frame compliance is usedinto the computation of the reduced modulus [Eq (1)] torecalculate the elastic modulus [Eq (2)] Finally follow-ing this procedure the variations of hardness and elasticmodulus as a function of displacement into the surfaceare represented in Figs 6(a) and 6(b) respectively

Remarkably in Fig 6 the hardness and the elasticmodulus do not vary with displacement Obtaining thehardness value equals to 35 6 01 GPa and the elasticmodulus equals to 68 6 1 GPa These results approvethat multi-cycle indentation can be validly used to deter-mine mechanical properties at least for this material andthis range of loads where damage was not optically

FIG 3 (a) Martens hardness and (b) elastic modulus variations versus the indenter displacement obtained by means of CSM mode in nano-indentation on the BK7 glass sample

FIG 4 Evaluation of the truncated tip length of the bluntedBerkovich indenter used in micro-indentation tests by SEM analysisat high magnification

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

7J Mater Res 2017

visible after the indentation tests It can be noted thatthese values agree with those obtained at the nano-scalerange

C MACRO-indentation

1 Progressive multi-cyclic loading

An example of the loadndashdisplacement curve for thetest condition No 2 (Table III) is shown in Fig 5(b) atthe maximum load of 25 N Similarly the micro-indentation tests to evaluate the mechanical behaviorof the BK7 glass sample under progressive multi-cycleloading the Martens hardness and the elastic modulusare calculated at the end of each cycle Howevercompared to the results obtained in micro-indentation atdifferent ranges of load the macro-indentation resultsshow that the Martens hardness and the elastic moduluscontinuously decrease with the number of cyclesTo represent this behavior we plotted the damage

coefficients defined by Lemaitre and Dufailly31 DH andDE versus the relative penetration depth of the indenterDhh0 In a general way these different parameters can beexpressed as

DH frac14 1 HM

HM0 DE frac14 1 E

E0and

Dhh0

frac14 h h0h0

eth11THORN

Where HM0 E0 and h0 are the corresponding values ofthe parameters reached after the first cycle

After the first cycle at 5 N both Martens hardnessand elastic modulus have a constant value indepen-dently of the maximum test load at the last cycle ieHM0 5 4 6 05 GPa and E0 5 66 6 3 GPa Thesevalues agree with the values obtained at the othersscales of measurement Besides the penetration depthreached after the first cycle is constant for all the testsh0 5 67 6 02 lm Figures 7(a) and 7(b) represent

FIG 5 (a) Loadndashdisplacement curve for a multicycle test in micro-indentation performed on the surface of the BK7 glass sample and(b) Multicycle macro-indentation curve performed on the BK7 glass sample under the following conditions 100 cycles in progressive loading modebetween 5 and 25 N with a holding time of 15 s at the maximum load of each cycle and between cycles

FIG 6 (a) Martens hardness HM and (b) elastic modulus E as function of the indenter displacement obtained by multicycle indentation on theBK7 glass sample

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

J Mater Res 20178

DH and DE as a function of the relative penetrationdepth Dhh0

According to Fig 7 the damage parameters increaseafter each cycle as a function of the relative penetra-tion depth Dhh0 demonstrating the occurrence of aphenomenon during the progressive loading and probablycracking initiation and propagation Also the example ofloading-displacement curve illustrated in Fig 5(b) showssome slight discontinuities corresponding to crackingobserved by optical microscopy as reported by Malzbenderet al32 On the contrary after micro-indentation tests wedid not observe cracks in the imprints

It is noticeable that the two damage parameters relatedto the Martens hardness and to the elastic modulus areclose to a directly proportional relation with the relativepenetration depth Dhh0 expressed by the same propor-tionality factor of 022

Different tests according to the test conditionsNo 3 (Table III) at a maximum load of 300 N and

two dwell-times (15 and 30 s) have been performed atthe surface of the BK7 glass sample Figure 8 showsmulti-cycle indentation experiments performed between50 and 300 N with a holding time of 15 s [Figs 8(a)]and 30 s [Fig 8(b)] the limit unloading load was fixedto 10 N

Figure 8 shows that at higher loads abrupt disconti-nuities are registered in the loading-displacement curvesProbably the cracks initiating at lower forces propagateand lead to chipping which is visible in the profile ofthe loading curves as a horizontal gap during loadingChipping phenomenon starts at a certain load whichchanges with the test conditions in this case with theholding time at maximum load This result corroboratesthe fact that no simple generalization may be made con-cerning crack initiation sequences33 It is also remarkablethat the damage caused by chipping seems to be lesspronounced for the lower dwell-time Indeed only onechip scar has been observed probably corresponding

FIG 7 Damage coefficients (a) DH and (b) DE as a function of the relative penetration depth of the indenter Dhh0 The parameters are calculatedafter each cycle corresponding to multicycle progressive tests in macro-indentation (test conditions No 2) The loads in the legend correspond to themaximum applied load at each test

FIG 8 Multicycle macro-indentation tests performed on the BK7 glasssample with 6 cycles in progressive loading mode between 50 and 300 Nwith a holding time of 15 s (a) and 30 s (b)

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

9J Mater Res 2017

to the pop-in visible around a value close to 170 NContrarily for a dwell-time of 30 s several chip scars areobservable along the different edges of the imprint whichhave been occurred at different indentation loads ie120 190 and 280 N corresponding to visible plates

From a mathematical point of view the samemethodology described before was adopted to representthe influence of chipping on the measurement of themechanical properties Figures 9(a) and 9(b) showthe damage parameters DH and DE versus the relativepenetration depth Dhh0 For comparison we representin these figures the damage parameters obtained atlowest loads [Figs 7(a) and 7(b)] For tests made underconditions No 3 (Table III) HM0 5 35 6 01 GPaE0 5 69 6 1 GPa and h0 5 23 6 07 lm Similarly tothe results for lower indentation loads the proportion-ality factor is approximately 02 except for the elasticmodulus for which this factor decreases until 012 forthe higher damage

2 Constant multi-cycle loading

To evaluate the damage caused by multi-cycle inden-tation process it was studied that the effect of a constantmulti-cycle indentation mode (test conditions No 4)which consists of applying 5 cycles under the samemaximum load Figure 10 shows a good reproducibilityin the loadndashdisplacement curves regardless the maximumload

Contrary to the results obtained by applying pro-gressive multi-cycle loading with the constant multi-cycle loading the elastic modulus remains constant(696 6 09 GPa) along the whole test [Fig 11(b)]However the Martens hardness value decreases since themaximum displacement increase slightly after each cycleat the maximum applied load (Fig 10) To show thisvariation the Martens hardness damage parameter (DH)

is represented by the relative indenter displacementDhh0 [Fig 11(a)]

Cleary a more detailed investigation is needed tounderstand the behavior of the BK7 glass because inmulti-cycle progressive loading tests at loads in themacro range the formation of cracks was observedwhere the elastic modulus decreases as the load increasesowing to the damage becomes more severe (crackspropagation chipping etc) However with multi-cycletests at constant load this behavior is not present evenin the same range of loads which was unexpectedThen this means that damage recognized by the initiationand propagation of cracks at loads under 50 N is possiblyrelated to the number of cycles and consequentlycracking initiates under certain critical stressndashstrain fieldreached at a specific number of cycles

FIG 9 Damage coefficients (a) DH and (b) DE as a function of the relative penetration depth of the indenter Dhh0 The parametersare calculated after each cycle corresponding to multicycle progressive tests in macro-indentation with a maximum load of 300 N(test conditions No 3)

FIG 10 Multicycle macro-indentation performed on the BK7glass sample with 5 cycles performed at constant loads between5 and 50 N

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

J Mater Res 201710

V CONCLUSIONS

A multiscale analysis by instrumented indentationwas performed on the BK7 glass sample to determinethe hardness and the elastic modulus using standardand multicycle indentation tests By means of nano-indentation and micro-indentation tests Martens hardnessand elastic modulus were estimated to be approximately35 and 70 GPa respectively and they are apparentlyindependent of the testing conditions No visible crackswere found at these scales of measurement These resultsdemonstrate that multicycle tests in micro-indentation arean alternative to the CSM method

On the contrary the initiation of radial cracksis observed after multicycle macro-indentation tests(progressive loading) these cracks propagate and theyare transformed in chipping at higher loads The forma-tion and propagation of cracks are represented by smalldiscontinuities in the loading curves chipping phenom-enon is observed as abrupt changes in the penetrationdepth Nevertheless Martens hardness and elastic mod-ulus were computed for each cycle showing a decreaseof these two properties with the increasing of loadTherefore it was studied that the variation of the damageparameters related to the actual mechanical properties(Martens hardness and elastic modulus) with the relativepenetration depth these parameters describe a linearvariation thus permitting a connection with the relatedtheory of the cracks formation during the indentationprocess

The elastic modulus by means of macro-indentationmulticycle tests at constant load is around 70 GPa similarto the values in nano- and micro-indentation Howeverthe Martens hardness slightly decreases and the damageparameter varies linearly with the relative indentationdepth At the macroscale range of loads the responseof the material is affected by the number of cycles in

relation to the damage caused by the repeated indenterpenetration at each cycle

REFERENCES

1 B Balland Optique geacuteomeacutetrique Imagerie et instruments(Geometrical Optics Imaging and Instruments) (PPUR PressesPolytechniques Lausanne 2007) p 860

2 J Phalippou Verres Proprieacuteteacutes et applications (GlassesProperties and Applications) Techniques de lrsquoingeacutenieurAF3601 July 10 2001

3 GF VanderVoort Metallography Principles and Practice(McGraw-Hill New York 1984) p 752

4 WI Rupp Loose abrasive grinding of optical surface Appl Opt11(12) 2797ndash2810 (1972)

5 AA Tesar and BA Fuchs Removal rates of fused silicawith cerium oxide and pitch polishing Proc Soc Photo-OptInstr Eng 1531 80ndash90 (1992)

6 HH Karow Fabrication Methods for Precision Optics(Wiley-Interscience Hoboken 2004) p 768

7 E Brinksmeier O Riemer and A Gessenharter Finishing ofstructured surfaces by abrasive polishing Precis Eng 30(3)325ndash336 (2006)

8 HH Pollicove and DT Moore Optics manufacturing technologymoves toward automation Laser Focus World 27 145ndash149(1991)

9 D Golini and W Czajkowski Micro grinding makes ultra-smoothoptics fast Laser Focus World 28 146ndash152 (1992)

10 D Golini Influence of process parameters in deterministic micro-grinding OSA 13 28ndash31 (1994)

11 HH Pollicove Computer aided optics manufacturing OptPhotonics News 6 15ndash19 (1994)

12 RL Aghan and LE Samuels Mechanisms of abrasive polishingWear 16(4) 293ndash301 (1970)

13 Y Xie and B Bushan Effects of particle size polishing padand contact pressure in free abrasive polishing Wear 200(1ndash2)281ndash295 (1996)

14 JC Lambropoulos S Xu and T Fang Loose abrasive lappinghardness of optical glasses and its interpretation Appl Opt 36(7)1501ndash1516 (1997)

15 T Suratwala P Davis L Wong P Miller M Feit J Menapaceand R Steele Sub-surface mechanical damage distributions

FIG 11 (a) Martens hardness damage factor versus the relative indenter displacement and (b) elastic modulus versus the indentation depth at thefirst cycle The indentation tests are performed under the test conditions No 4

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

11J Mater Res 2017

during grinding of fused silica J Non-Cryst Solids 352(52ndash54)5601ndash5617 (2006)

16 A Esmaeilzare A Rahimi and SM Rezaei Investigation ofsubsurface damage and surface roughness in grinding process ofZerodur glass-ceramic App Surf Sci 313 67ndash75 (2014)

17 C Anunmana KJ Ausavice and JJ Mecholsky Jr Residualstress in glass Indentation crack and fractography approachesDental Mater 25 1453ndash1458 (2009)

18 R Komanduri DA Lucca and Y Tani Technological advancesin fine abrasive processes Annals of the CIRP 46(2) 545ndash596(1997)

19 SD Jacobs SR Arrasmith IA Kozhinova LL GreggAB Shorey HJ Romanofsky D Golini WI KordonskiP Dumas and S Hogan MRF Computer-controlled opticsmanufacturing Am Ceram Soc Bull 78 42ndash48 (1999)

20 JP Marioge Surface optique Meacutethodes de fabrication et decontrocircle recherches (Optical Surface Production and ControlMethods Researches) (EDP Sciences France Les Ulis 2000)pp 26ndash33

21 W Oliver and G Pharr An improved technique for determininghardness and elastic modulus using load and displacementsensing indentation experiments J Mater Res 7(6) 1564ndash1583 (1982)

22 JM Antunes LF Menezes and JV Fernandes Three-dimensional numerical simulation of Vickers indentation testsInt J Sol Struct 43(13ndash4) 784ndash806 (2006)

23 JTR Field and RH Telling The Elastic Modulus and PoissonRatio of Diamond Research Note (Cavendish LaboratoryCambridge 1999)

24 D Chicot M Yetna NrsquoJock ES Puchi-Cabrera A IostMH Staia G Louis G Bouscarrat and R Aumaitre A contactarea function for Berkovich nanoindentation Application to

hardness determination of a TiHfCN thin film Thin Solid Films558(2) 259ndash266 (2014)

25 JM Antunes A Cavaleiro LF Menezes MI Simoes andJV Fernandes Ultra-microhardness testing procedure withVickers indenter Surf Coat Technol 149 27ndash35 (2002)

26 LA Berla AM Allen SM Han and WD Nix A physicallybased model for indenter tip shape calibration for nanoindentationJ Mater Res 25 735ndash745 (2010)

27 D Chicot P De Baets M Staia E Puchi-Cabrera G LouisYP Delgado and J Vleugels Influence of tip defect and indentershape on the mechanical properties determination by indentationof a TiB2ndash60 B4C ceramic composite Int J Refract Met HardMater 38 102ndash110 (2013)

28 M Troyon and L Huang Correction factor for contact area innanoindentation measurements J Mater Res 20 610ndash617(2005)

29 MY Nrsquojock D Chicot J Ndjaka J Lesage X DecoopmanF Roudet and A Mejias A criterion to identify sinking-in andpiling-up in indentation of materials Int J Mech Sci 90145ndash150 (2015)

30 D Chicot and D Mercier Improvement in depth-sensingindentation to calculate the universal hardness on the entireloading curve Mech Mater 40(4ndash5) 171ndash182 (2008)

31 J Lemaitre and J Dufailly Damage measurements Eng FractMech 28 643ndash661 (1987)

32 J Malzbender JMJ den Toonder AR Balkenende andG de With Measuring mechanical properties of coatingsA methodology applied to nano-particle fille sol-gel coatings onglass Mater Sci Eng R 36 47ndash103 (2002)

33 RF Cook and GM Pharr Direct observation and analysis ofindentation cracking in glasses and ceramics J Am Ceram Soc73 787ndash817 (1990)

M Bentoumi et al Multiscale and multicycle instrumented indentation to determine mechanical properties

J Mater Res 201712