6
A criterion to identify sinking-in and piling-up in indentation of materials M. Yetna Njock a,b,n , D. Chicot a , J.M. Ndjaka b , J. Lesage a , X. Decoopman a , F. Roudet a , A. Mejias a,c a Université Lille Nord de France, Lille1, LML, UMR 8107, F-59650 Villeneuve d'Ascq, France b Université de Yaoundé 1, Faculté des Sciences, Département de Physique, BP 812 Yaoundé, Cameroon c Universidad de Carabobo, Facultad de Ingeniería, Laboratorio de Materiales, Valencia, Venezuela article info Article history: Received 11 September 2014 Received in revised form 23 October 2014 Accepted 10 November 2014 Available online 15 November 2014 Keywords: Indentation Elastic modulus Hardness Piling-up Sinking-in abstract The instrumented indentation test is usually used to determine the mechanical properties of materials. Depending on the nature of the material, the way the matter ows under the indenter by piling-up or sinking-in affects the calculation of these mechanical properties. Consequently, corrections proposed by Oliver and Pharr and Loubet et al. should be done according to these two behaviors in addition to other corrections associated with the indenter tip defect as well as the compliance of the instrument. In this work we tested different materials having supposedly piling-up or sinking-in behavior: low-carbon steel, aluminum, brass, copper, beta tricalcium phosphate (β-TCP) bioceramic, rolled or sintered stainless steel and ceramic composite TiB 2 60% B 4 C by using two types of indenter, i.e. Vickers and Berkovich ones. From the corrected loadindenter displacement curve, we showed that a criterion, dened as the ratio between the residual indentation depth and the maximum indentation depth reached at the maximum load, is able to identify the predominant deformation mode. For materials for which this ratio is higher than 0.83 piling-up prevails while it is sinking-in when it is lower than 0.83. When the ratio equals 0.83, the two modes of deformation should coexist since the calculations made using either correction of Oliver and Pharr or Loubet et al. give the same results. This novel way of considering the instrumented indentation measurements renders more accurate the determination of the hardness and the elastic modulus since the observation of the indent is then not required for identifying the deformation mode which affects the contact area calculation. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction During the indentation of a material by a very hard indenter, the matter may ow differently depending on the mechanical properties of the material, the nature and the shape of the indenter. Usually two distinct modes of deformation are consid- ered: i) "Sinking-in" when the material is pulled down toward the tip of the indent and ii) "Piling-up" when the material is pushed away from the center of the indent. For classical indentation tests where the diagonal of the indent is measured optically, these modes of deformation have little effect on the hardness measure- ment since it is recognized that the diagonal length of the indent remains constant under the maximum loading and after the withdrawal of the indenter. For instrumented indentation tests, the mechanical properties, both hardness and elastic modulus, are calculated with a precise value of the contact area which is related to the contact depth. It is clear that whatever the two modes sinking-in and piling-up affect its calculation [1]. For example, Alcala et al. [2] mentioned that errors up to 30% can be introduced in the computation of the contact area if the deformation mode is not taken into account. Since the determination of true hardness and elastic modulus requires the knowledge of the contact depth as precisely as possible, numerous studies have been performed on the conditions of its determination and give the corrections that have to be applied in order to take into account the bluntness of the indenter tip [3], the frame compliance [4] and the two modes of deformation [5]. For these latter it was observed that for soft materials with low values of both hardness to elastic modulus ratio (H/E) and strain hardening exponent, n, to elastic modulus ratio (n/E), the piling-up mode predominates [6]. Similarly Cheng and Cheng [7], Xu and Rowcliffe [8], found that, for a given indenter, piling-up or sinking- in behavior are associated to the ratio of the yield stress Y to elastic modulus E. For high values of Y/E only sinking-in occurs while for small values piling-up or sinking-in may occur [9]. When E is not Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijmecsci International Journal of Mechanical Sciences http://dx.doi.org/10.1016/j.ijmecsci.2014.11.008 0020-7403/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author at:Université Lille Nord de France, Lille1, LML, UMR 8107, F-59650 Villeneuve d'Ascq, France. E-mail address: [email protected] (M.Y. Njock). International Journal of Mechanical Sciences 90 (2015) 145150

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Page 1: A criterion to identify sinking-in and piling-up in indentation of materials

A criterion to identify sinking-in and piling-up in indentationof materials

M. Yetna N’jock a,b,n, D. Chicot a, J.M. Ndjaka b, J. Lesage a, X. Decoopman a,F. Roudet a, A. Mejias a,c

a Université Lille Nord de France, Lille1, LML, UMR 8107, F-59650 Villeneuve d'Ascq, Franceb Université de Yaoundé 1, Faculté des Sciences, Département de Physique, BP 812 Yaoundé, Cameroonc Universidad de Carabobo, Facultad de Ingeniería, Laboratorio de Materiales, Valencia, Venezuela

a r t i c l e i n f o

Article history:Received 11 September 2014Received in revised form23 October 2014Accepted 10 November 2014Available online 15 November 2014

Keywords:IndentationElastic modulusHardnessPiling-upSinking-in

a b s t r a c t

The instrumented indentation test is usually used to determine the mechanical properties of materials.Depending on the nature of the material, the way the matter flows under the indenter by piling-up orsinking-in affects the calculation of these mechanical properties. Consequently, corrections proposed byOliver and Pharr and Loubet et al. should be done according to these two behaviors in addition to othercorrections associated with the indenter tip defect as well as the compliance of the instrument. In thiswork we tested different materials having supposedly piling-up or sinking-in behavior: low-carbon steel,aluminum, brass, copper, beta tricalcium phosphate (β-TCP) bioceramic, rolled or sintered stainless steeland ceramic composite TiB2–60% B4C by using two types of indenter, i.e. Vickers and Berkovich ones.From the corrected load–indenter displacement curve, we showed that a criterion, defined as the ratiobetween the residual indentation depth and the maximum indentation depth reached at the maximumload, is able to identify the predominant deformation mode. For materials for which this ratio is higherthan 0.83 piling-up prevails while it is sinking-in when it is lower than 0.83. When the ratio equals 0.83,the two modes of deformation should coexist since the calculations made using either correction ofOliver and Pharr or Loubet et al. give the same results. This novel way of considering the instrumentedindentation measurements renders more accurate the determination of the hardness and the elasticmodulus since the observation of the indent is then not required for identifying the deformation modewhich affects the contact area calculation.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

During the indentation of a material by a very hard indenter,the matter may flow differently depending on the mechanicalproperties of the material, the nature and the shape of theindenter. Usually two distinct modes of deformation are consid-ered: i) "Sinking-in" when the material is pulled down toward thetip of the indent and ii) "Piling-up" when the material is pushedaway from the center of the indent. For classical indentation testswhere the diagonal of the indent is measured optically, thesemodes of deformation have little effect on the hardness measure-ment since it is recognized that the diagonal length of the indentremains constant under the maximum loading and after thewithdrawal of the indenter. For instrumented indentation tests,the mechanical properties, both hardness and elastic modulus, are

calculated with a precise value of the contact area which is relatedto the contact depth. It is clear that whatever the two modessinking-in and piling-up affect its calculation [1]. For example,Alcala et al. [2] mentioned that errors up to 30% can be introducedin the computation of the contact area if the deformation mode isnot taken into account. Since the determination of true hardnessand elastic modulus requires the knowledge of the contact depthas precisely as possible, numerous studies have been performed onthe conditions of its determination and give the corrections thathave to be applied in order to take into account the bluntness ofthe indenter tip [3], the frame compliance [4] and the two modesof deformation [5].

For these latter it was observed that for soft materials with lowvalues of both hardness to elastic modulus ratio (H/E) and strainhardening exponent, n, to elastic modulus ratio (n/E), the piling-upmode predominates [6]. Similarly Cheng and Cheng [7], Xu andRowcliffe [8], found that, for a given indenter, piling-up or sinking-in behavior are associated to the ratio of the yield stress Y to elasticmodulus E. For high values of Y/E only sinking-in occurs while forsmall values piling-up or sinking-in may occur [9]. When E is not

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ijmecsci

International Journal of Mechanical Sciences

http://dx.doi.org/10.1016/j.ijmecsci.2014.11.0080020-7403/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author at: Université Lille Nord de France, Lille1, LML, UMR 8107,F-59650 Villeneuve d'Ascq, France.

E-mail address: [email protected] (M.Y. N’jock).

International Journal of Mechanical Sciences 90 (2015) 145–150

Page 2: A criterion to identify sinking-in and piling-up in indentation of materials

known, some information about the deformation mode can bedrawn from the knowledge of the indentation depth h. A systema-tic study of piling-up and sinking-in modes as well as theirinfluence on the determination of h has been performed byGiannakopoulos and Suresh [10] using finite-element simulationson elastic–plastic materials. It was found that the ratio of theresidual depth hf to the maximal depth of penetration hmax

obtained directly from the load–displacement curve allows iden-tifying the sinking-in or piling-up modes. For materials havinghf /hmax40.875 piling-up is likely to occur while it is sinking-in forhf /hmaxo0.875. The lower limit hf /hmax¼0 corresponds to fullyelastic deformation and the upper limit hf /hmax¼1 corresponds torigid-plastic behavior. In the case of pyramidal indenter also,Alcala et al. [2] have rewritten the relation between the contactarea and depth penetration in terms of a factor α which takesaccount of the surface deformation. It has been found that α is 41for piling-up and o1 for sinking-in.

The methodologies described above are all of interest but they arebased on measurements that have to be as precise as possible. Sincethese measurements are affected by experimental biases such as theindenter tip defect [3], and the compliance of the experimental set up,frame and sample dimensions and mounting [4], some correctionshave to be done to the measurements according to these biases inorder to obtain valid results [2–4]. The usual methods of calculationwhich take into account the surface deformation modes are those ofOliver and Pharr [11] for sinking-in and Loubet et al. [12] for piling-up.Although the effect can be considered as negligible for nano indenta-tion measurements, a compliance correction is necessary in micro-indentation tests since the measured indentation depth is sensible tothe sample mounting and the indentation testing conditions [13].

In this paper, a variety of materials that are likely to exhibit one orthe other mode of deformation are studied. The frame compliance isdetermined for correcting the indentation depth. Two types of indenter,Berkovich and Vickers, are used to examine their effect on thedeformation mode and, consequently, on the mechanical behavior.Corrected instrumented microindentation measurements are studiedin order to confirm the literature assessments and to define aparameter able to identify the deformation mode of the differentmaterials without any other observations of the indent or measure-ments than the values obtained from the standard instrumentedindentation test. The elastic modulus determined by using Vickersand Berkovich indenters is afterwards compared and discussed afterOliver and Pharr [11] and Loubet et al. [12] corrections according to thedeformationmode of the indent to validate the proposedmethodology.

2. Theoretical background

2.1. Hardness and young modulus

During the two last decades, the instrumented indentation test(IIT) has been developed. It allows determining some moremechanical properties of materials than the sole conventionalhardness, such as Young's modulus [11,14], the work-hardeningcoefficient [15–18] and the yield stress [19] as well as the fracturetoughness [20]. From IIT measurements leading to a load (P)–indenter displacement (h) curve (Fig. 1), hardness, H, is defined asthe ratio between the maximum load Pmax and the projectedcontact area Ac:

H¼ Pmax

Acð1Þ

By analyzing the unloading part of a load–depth curve obtainedby instrumented indentation, Oliver and Pharr [11] used thefollowing expression that relates the slope S at the origin of the

unloading curve to the reduced modulus ER:

ER ¼S2

ffiffiffiffiffiπAc

rð2Þ

where ER includes the material parameters of the indenter (Ei, νi)and of the investigated material (E, ν) in the relation:

1ER

¼ 1�υ2

Eþ1�υ2i

Eið3Þ

The slope of the curve upon unloading is indicative of thestiffness S of the contact. This value generally is the inverse of thetotal compliance CT which includes a contribution from both thecompliance of the sample being tested and the load framecompliance of instrument. For the calculation of the slope at theorigin of the unloading part of the IIT curve, Oliver and Pharr [11]suggested to fit the curve by a power law relating the indentationload P to the difference between the indentation depth h and theresidual indentation depth:

P ¼ Bðh�hf Þm ð4Þwhere B, m and hf are values determined by a step by step best fitanalysis. In practice, only data in the range 40–98% of themaximum load are used for the fitting.

2.2. The contact area

From Eqs. (1) and (2) it is seen that the projected contact area,Ac, is a key factor in the calculation of mechanical parameters. Fora perfect geometry of the indenter, three sides for a pyramidBerkovich indenter and four sides for a pyramid Vickers indenter,the projected contact area is proportional to the square of thecontact depth hC by the following relation:

AC ¼ 24:56 h2c ð5ÞDepending on the deformation and mechanical properties of

the tested material, we have mentioned that piling-up or sinking-in may occur during the indentation process. A schematic repre-sentation of the two modes is presented in Fig. 2 for a Vickersindenter.

It is clear that both modes of deformation render difficult aprecise determination of the penetration depth and consequentlyof the contact area. Methods have been proposed by variousauthors. For sinking-in, Oliver and Pharr [11] expressed the contactdepth hC by a function of the maximum indentation depth, hmax,the maximum load, Pmax, and the elastic unloading stiffness, S, asfollows:

hCs ¼ hmax�Pmax

Sð6Þ

hf hmax

Pmax

dP/dh

SUnloading

Loading

Load

, P (m

N)

depth, h (nm)

Fig. 1. Schematic load–indenter displacement curve obtained from instrumentedindentation test using a Vickers indenter.

M.Y. N’jock et al. / International Journal of Mechanical Sciences 90 (2015) 145–150146

Page 3: A criterion to identify sinking-in and piling-up in indentation of materials

The coefficient ε is a constant whose value depends on thegeometry of the indenter. For a conical punch ε¼ 0:72, for aparaboloid of revolution which approximates to a sphere ε¼ 0:75and for a flat punch ε¼ 1:00 [11].

In the case of piling-up, Loubet et al. [12] have suggested thefollowing expression for the contact depth:

hCp ¼ αðhmax�Pmax

SÞ ð7Þ

where α is a constant equal to 1.2.When the indenter has not a perfect shape, i.e. when there is a

bluntness of the indenter tip, the contact depth is modified.Following Troyon and Huang [21] the tip defect can be taken intoaccount by the following relation which is adequatly consistent forindenter displacement higher than a value around 200 nm whichis often the case in microindentation:

AC ¼ 24:56 ðhcþhbÞ2 ð8Þwhere hb is the truncation length of the tip defect.

Introducing the different contact depths, Eqs. (6) and (7)according to Oliver and Pharr [11] and Loubet et al. [12] modelsrespectively, in this contact area function, Eq. (8), we obtained thefollowing expressions:

ACOP ¼ 24:56ðhmax�εPmax

SþhbÞ2 ð9Þ

ACLA¼ 24:56α2ðhmax�

Pmax

SþhbÞ2 ð10Þ

Some investigations have been made for the determination ofhb using SEM direct observations [13]. Values of 50 nm forBerkovich indenter and 150 nm for Vickers indenter were deter-mined. These values are far above the values given by Hochstetteret al. [3], namely 5 nm for fused silica and 25 nm for polymers,obtained by extrapolation of the linear part of the stiffness–depthcurve to the zero value of contact stiffness. It is clear, though, thatthe tip defect should be the same regardless of the tested material.Detection of the surface for the polymers is put forward by theauthor to explain this discrepancy.

2.3. Frame compliance

Apart from the tip defect, another element to take into account isthe rigidity of the experimental equipment which takes part in themeasurement of the contact depth. The inverse of the rigidity is thecontact stiffness of the instrument frame, Cf , which has beendescribed and discussed in detail in André et al. [22] and Fischer-Cripps [23] for example. A common approach for determining Cf is tocalibrate the instrument using standard calibration samples generallymade of fused silica [23]. This approach is not error-free since theapparent frame compliance of the instrument does not have aconstant value depending on the specimen material, its mountingand its dimensions. To solve this problem Chicot et al. [24] proposed

to determine directly the apparent frame compliance of each seriesof indentation experiments from the plot of the total compliance CT

versus the square root of the reciprocal contact area 1=ffiffiffiffiffiAc

pas the

work by Doener and Nix [14] suggested in the expression:

CT ¼1S¼

ffiffiffiffiπ

p

21

βErffiffiffiffiffiffiAC

p þCf ð11Þ

In this relation, β is a constant associated to the indentergeometry. β¼1 for a conical indenter, 1.034 for a Berkovichindenter and 1.012 for a Vickers one [25]. However, these resultsare debatable, as Woirgard et al. [26] demonstrated analyticallythat β¼1.061 and β¼1.023 for triangular and square-basedindenters, respectively. In addition to β, Hay et al. [27] consideredthe elastic radial displacement neglected in Sneddon's formulationand proposed a complementary correction factor γ function of theindenter's half-angle, ψ and Poisson's ratio, ν:

CT ¼1S¼

ffiffiffiffiπ

p

21

βγErffiffiffiffiffiffiAC

p þCf ð12Þ

In the following the term β has been assumed to be equal to 1.05while parameter γ is calculated using the following equation [27]:

γ ¼ πðπ=4Þþ0:1548ð1�2υÞ=ð4ð1�υÞÞcot Ψ� �π=2�0:8312ð1�2υÞ=ð4ð1�υÞÞcot Ψ� �2 ð13Þ

We have seen here above that the frame compliance Cf can bedetermined by plotting CT as a function of ðACÞ�1=2. The coordinateat the origin gives Cf while the slope of the straight line allowsdetermining ER using Eqs. (11) or (12). Afterwards, compliancecorrection is applied to the contact depth using the methodologyproposed by Fischer-Cripps [23]:

h0 ¼ hmes�Cf P ð14Þ

This equation can be used now to rebuild the true loading–unloading curve and consequently to recalculate the contact area,using Eq. (8), that will be used to determine hardness and elasticmodulus of tested materials.

3. Materials and experimental methods

3.1. Materials

Eight different materials were prepared for the experiments. 1) ATiB2–60% B4C ceramic composite made by pulsed electric currentsintering. The sintered sample was 4 mm in thickness and 30mm indiameter. 2) A beta tricalcium phosphate (β-TCP) bioceramic wassynthesized by co-precipitation of a mixture of diammonium phos-phate solution NH4 (HPO4)2 and a calcium nitrate solution Ca(NO3)2,4H2O using aqueous precipitation technique. A ratio Ca/P¼1.52 waschosen in order to obtain stoichiometric powders [28]. 3) A series ofsamples were prepared from commercial bars of 13 mm in diametermade of low-carbon steel (ASTM A36), aluminum (6061-T6), brass(C22000) and copper (99% of purity). After sectioning, the sampleswere ground down to a thickness of 8 mm. One face of each samplewas polishedmechanically using abrasive sandpapers until 1200 gradeand then polished to a mirror finish with diamond polishing materials.4) Another set of samples of 10 mm in diameter and 25mm in lengthwere taken from bars made of a 316 stainless steel and of a stainlesssteel obtained by direct metal laser sintering (DMLS). These sampleswill be referred to as SS for Stainless Steel and RPSS for RapidPrototyping Stainless Steel respectively in the following. The manu-facturing conditions and other details on these materials can be foundin [29]. A careful metallographic preparation of the samples allowed usto obtain an average roughness of Ra¼0.2 mm.

Sp

h hhcphcs

Ss

P

Piling-up Sinking-in

Fig. 2. Representation of piling-up and sinking-in during an instrumented inden-tation test with a Vickers indenter.4.

M.Y. N’jock et al. / International Journal of Mechanical Sciences 90 (2015) 145–150 147

Page 4: A criterion to identify sinking-in and piling-up in indentation of materials

4. Experiments

Instrumented indentation tests have been performed using aCSM2-107 microhardness tester. Vickers and Berkovich indenterswere used for each sample analysis with maximum loads rangingfrom 50 mN to 15 N. At least 20 tests were performed in this range.In addition, a dwell-time of 15 s was imposed at the maximumapplied load according to the standard CSM test procedure. Loadingand unloading rates (in mN/min) has been set up at twice the valueof the maximum applied load according to Quinn et al. [30]. The tipdefects of each indenter were evaluated using an Hitachi fieldemission scanning electron microscope of type S-4300 SE/N. Themeasurements gave 150 nm for the Vickers indenter and 50 nm forthe Berkovich [13]. Note that Berkovich indentations were notperformed on SS, RPSS, β-TCP and copper samples.

5. Results and discussion

The instrumented indentation tests have been performed on thedifferent materials using a variety of maximum loads. In order to showthe repeatability of the loading part, we present, as an examplein Fig. 3, the loading–unloading curves obtained using the Vickersindenter applied to the sample aluminum 6061-T6. It is important tonote that for the other materials and by applying the two indentertypes, i.e. the Vickers and the Berkovich ones, the loading–unloadingcurves show also a very good repeatability comparable to thatpresented in Fig. 3.

Using raw data from the P�h curves (example Fig. 3) given bythe instrument, Ac values were calculated using Eq. (9) for sinking-in and (10) for piling-up. In these equations the tip defect is alsotaken into account. The corrective factor γ has been calculated andpresented in Table 1 and subsequently the total compliance, CT ,was then plotted versus the inverse of the square root of theindentation contact area, 1=

ffiffiffiffiffiffiAC

p(Fig. 4). At this stage no informa-

tion about the mode of deformation is available. For that reasonwe applied both Oliver and Pharr and Loubet et al. methods to thesame sets of measurements.

As an example, data obtained for the Brass samples are presentedin Fig. 4 to show the linear relation between the total compliance andthe inverse of the square root of the contact area during theindentation test with a Vickers and Berkovich indenter. A linearregression has been used to fit the data points. The intercept of thestraight line with the compliance axis gives the frame compliance ofthe instrument, Cf .

One remark can be drawn from the observation of Fig. 4. First, it isnoticeable that Cf is the same using Oliver and Pharr (Cf ¼0.042) orLoubet et al. (Cf ¼0.044) corrections on Vickers measurements andon Berkovich measurements (respectively 0.140 and 0.141). Linearregression applied to the data allows the determination of Cf . Table 2

summarizes the Cf values obtained for all the situations of materialsand indenter types. For each material, Oliver and Pharr or Loubet et al.equations give the same Cf . This is reasonable since the two equationsare applied to the same set of results for a given material. Cf varieswith the materials, indenter, sample dimensions and mounting. Thevalue of row for low-carbon steel and column for Berkovich indenterwas set to zero since the value is very close to zero and moreover anegative value would have no physical meaning.

Using these values of Cf the penetration depth h0 is calculatedfollowing Eq. (14) and subsequently a new value of the slope, S0, canbe calculated starting from Eq. (4). Ultimately for calculating thecontact area, we used formula that takes into account both tip defectand compliance corrections applying Eqs. (9) and (10). From theserelations, the reduced modulus ER is then deduced from the slope ofthe straight lines by plotting the total measured compliance versusthe inverse of square root of contact area. The results are presentedin Table 3. It shows, for the two indenters, the different value of thereducedmodulus determined using themethods of Oliver and Pharr [11]and Loubet et al. [12]. For comparison Table 3 also mentions the theo-retical value found in literature, ERth

[28,29,31–34].For aluminum, low-carbon steel, commercial brass, copper and

SS, it is observed that Loubet et al. formula gives the closest values toliterature data, ERth

, while it is Oliver and Pharr that gives a consistentvalue to the literature for the TiB2–60% B4C ceramic composite andthe beta tricalcium phosphate (β-TCP) bioceramic. Considering thesefirst results, we may suppose that aluminum, low-carbon steel, brass,copper and SS present pilling-up deformation mode while theceramic composite TiB2–60% B4C and the bioceramic β-TCP presentsinking-in. The RPSS material is particular since the two correctionformulas give close values to the theoretical one. This is clearlyvisible in Fig. 5 which represents ACLA

versus ACOP .For the RPSS materials, the calculated points are situated on the

identity line. This result is very interesting since it suggests thatthe two behaviors could coexist for some materials. In suchsituations then, neither piling-up nor sinking-in should predomi-nate. Fig. 6 gives a schematic representation of simultaneouspiling-up and sinking-in.

We have mentioned in the introduction that Giannakopoulosand Suresh [10] have suggested that the ratio of the residual depthhf to the maximal depth penetration hmax could be used to identifythe mode of deformation around the indent: hf /hmax40.875 forpiling-up and hf /hmaxo0.875 for sinking-in. These values wereobtained by finite element calculations under the assumptions ofperfectly rigid indenter and equipment. It is reasonable to thinkthat a different value could be found in practice when taking intoaccount all the corrections that we have mentioned.

5.1. A criterion for deformation mode identification

Following a similar reasoning as in Giannakopoulos and Suresh [10],let Δ be the ratio of the final depth hf' and the corrected maximumindentation depth hmax' calculated using Eq. (14):

Δ¼ hf'hmax'

ð15Þ

0 5000 10000 15000 200000

5000

10000

15000

Load

, P (m

N)

Displacement indenter, h (nm)

Fig. 3. Loading–unloading curves obtained from instrumented indentation testusing a Vickers indenter for aluminum 6061-T6.

Table 1Corrective factor of our samples.

Materials Aluminum SS Brass Low-carbonsteel

RPSS TiB2–60%B4C

Copper β-TCP

Poisson'sratio

0.34 0.30 0.33 0.30 0.30 0.156 0.355 0.3

Correctivefactor γ

1.056 1.067 1.059 1.067 1.067 1.097 1.05 1.067

M.Y. N’jock et al. / International Journal of Mechanical Sciences 90 (2015) 145–150148

Page 5: A criterion to identify sinking-in and piling-up in indentation of materials

where hf' is one of the fitting parameters obtained from the followingequation:

S'¼ dPdh'

� �h' ¼ hmax'

¼mBðh'�hf'Þm�1 ð16Þ

A summary of the values of Δ obtained for both Berkovich andVickers indenters, is presented in Table 4.We note thatΔ is greater than0.83 for materials mentioned in literature as having only the piling-updeformation mode while it is lower than 0.83 for the one known for

0.0000 0.0001 0.00020.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

Experimental data with Vickers tipLinear regression

CT

(µm

/N)

CT

(µm

/N)

CT

(µm

/N)

CT

(µm

/N)

0.0000 0.0001 0.0002

Experimental data with Vickers tipLinear regression

0.00000 0.00005 0.00010 0.00015 0.000200.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5Experimental data with Berkovich tipLinear regression

A (nm) A (nm)

0.00000 0.00005 0.00010 0.00015

A (nm)

Experimental data with Berkovich tipLinear regression

A (nm)

d

Fig. 4. Contact stiffness of Brass in function of the inverse of square root of the contact area for Vickers (a, b) and Berkovich (c, d) indenter following the both methodologiesof Oliver and Pharr [11] and Loubet et al. [12].

Table 2Frame-compliance, Cf in mm/N, obtained from instrumented indentation test byusing a Berkovich and Vickers indenters, assuming the presence of the two surfacedeformation modes.

Vickers indenter Berkovich indenter

Low-carbon steel 0.00370.006 �0.00770.005Brass 0.0470.015 0.1470.01Aluminum 0.270.01 0.0370.01copper 0.0870.01 –

SS 0.4970.02 –

RPSS 0.1270.01 –

β-TCP 0.2670.03 –

TiB2–60% B4C 0.1170.01 0.0870.01

Table 3Reduced Young's modulus EROP and ERlA

calculated according to Oliver and Pharr[11] and Loubet et al. [12] methods using Vickers and Berkovich indenters,comparison with the theoretical value ERth

.

Reduced modulus Vickers indenter Berkovich indenter ERth

EROP ERLAEROP ERLA

Low-carbon steel 227 191 237 199 200Brass 113 95 125 106 86Aluminum 102 85 89 76 72copper 154 129 – – 131SS 228 194 – – 188RPSS 177 150 – – 165β-TCP 154 142 – – 160TiB2–60% B4C 370 345 333 300 365

Fig. 5. ACLAversus ACOP

for all the materials.

M.Y. N’jock et al. / International Journal of Mechanical Sciences 90 (2015) 145–150 149

Page 6: A criterion to identify sinking-in and piling-up in indentation of materials

sinking-in. The RPSS for which we have suggested that neither piling-up nor sinking-in is preponderant has a value of Δ equal to 0.83.Δ¼0.83 could then be considered as a criterion to precise the limitingsituation between predominant sinking-in and predominant piling-up.

6. Conclusion

By analyzing eight different materials and Vickers and Berko-vich indenters, instrumented micro-indentation tests have beenperformed at various maximum indentation loads. The experi-mental results were corrected afterwards following a step by stepprocedure that we developed in order to take into account the tipdefect, the frame compliance and the deformation mode. The mainconclusions that we can draw from this work are the following:

a. It is possible to use either Vickers or Berkovich indenters sincethe calculations of the elastic modulus give predictions of thesame order within the natural standard error.

b. A criterionΔ representing the ratio of the final depth hf' to thecorrected maximum indentation depth hmax' was proposed inorder to identify the piling-up or sinking-in deformation modeof the material without any additional observations or mea-surements other than the standard indentation data.

c. It is possible to determine a limiting value of 0.83 for Δ toseparate the preponderant deformation mode; Δ40.83 forpiling-up and Δo0.83 for sinking-in.

d. For a material having Δ¼0.83 no deformation mode is pre-ponderant and either Loubet et al. or Oliver and Pharr correc-tions can be used to calculate the true contact area.

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Fig. 6. Schematic representation of simultaneous piling-up and sinking-in.

Table 4The different mean values of the experimental criterion, Δ, during an instrumentedindentation test with Vickers and Berkovich indenters.

Vickers indenter Berkovich indenter

Low-carbon steel 0.9470.02 0.9470.01Brass 0.9070.03 0.9170.02Aluminum 0.9070.02 0.8870.02Copper 0.9270.01 –

SS 0.8870.02 –

RPSS 0.8370.02 –

β-TCP 0.7270.01 –

TiB2–60% B4C 0.4470.1 0.5570.05

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