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Materials Chemistry and Physics 119 (2010) 75–81

Contents lists available at ScienceDirect

Materials Chemistry and Physics

journa l homepage: www.e lsev ier .com/ locate /matchemphys

nfluence of visco-elasto-plastic properties of magnetite on the elastic modulus:ulticyclic indentation and theoretical studies

. Chicota,∗, F. Roudeta, A. Zaouib, G. Louisc, V. Lepinglec

Laboratoire de Mécanique de Lille, LML - UMR 8107, U.S.T. Lille, IUT A GMP, BP 179, 59653 Villeneuve d’Ascq, FranceLaboratoire de Mécanique de Lille, LML - UMR 8107, U.S.T. Lille, Polytech-Lille, 59653 Villeneuve d’Ascq, FranceEcole des Mines de Douai, BP 10838, 59508 Douai Cedex, France

r t i c l e i n f o

rticle history:eceived 5 March 2009eceived in revised form 15 July 2009

a b s t r a c t

In the present work, we study the indentation behaviour of the magnetite coexisting with hematitein a natural dual-phase crystal. In particular we show the influence of cycling indentation conditionson the elastic modulus measurement in relation to the visco-elasto-plastic properties of the material.

ccepted 19 July 2009

eywords:xideslastic propertiesolecular dynamics

ndentation (normal

Elastic properties of Fe3O4 are investigated using Oliver and Pharr’s technique, which is based on depth-sensing indentation (DSI) analysis. Depending on the visco-elasto-plastic properties of the material, theindentation test conditions (monotonic, cyclic, loading and unloading rates, dwell time at peak load, . . .) canmodify the shape of the load–depth curve and, subsequently, the results. Molecular dynamics simulationbased on shell model potential, is used to determine elastic quantities including elastic modulus, bulkmodulus and Young modulus.

icro and nano)

. Introduction

The elastic modulus of materials can be estimated by depth-ensing indentation (DSI). The most conventional DSI test producesload–depth curve for a monotonic loading which can be used

o determine mechanical properties such as the elastic modulus.ased on the analysis of the unloading part of a load–depth curve,liver and Pharr [1] provide a methodology to calculate a suitablealue of the reduced modulus which takes into account the elas-ic properties of both the material and the indenter. This can bechieved by plotting the inverse of the contact stiffness as a func-ion of the inverse of the indentation contact depth. Otherwise, ifhe specimen is loaded up to a specific value, unloaded and imme-iately reloaded, a cyclic indentation curve is produced [2]. In thisesting situation, numerous authors have mentioned anomalousehaviours such as pop-ins, hysteresis loops formation, disparityetween two consecutive cycles and depth increasing during thenloading usually called “bowing out” [3]. On the other hand, visco-lasto-plastic properties can have some influence on the behaviour

f the material under indentation [4]. Usually, difference observedetween monocyclic and multicyclic indentation is mainly relatedo cumulative plasticity [2]. In spite of such possible effects depend-ng on the material, Richter et al. [5] have successfully applied

∗ Corresponding author.E-mail address: didier.chicot@univ-lille1.fr (D. Chicot).

254-0584/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.matchemphys.2009.07.033

© 2009 Elsevier B.V. All rights reserved.

multicyclic nanoindentation to estimate the elastic modulus ofsuperhard materials.

Nevertheless some questions arise concerning the accuracy ofthe elastic modulus since it is determined from the unloading curvewhich is influenced by the different effects, especially by the visco-elasto-plastic behaviour of the material. That is why the presentwork attempts to provide additional explanations in the elasticmodulus determination. Various cycle indentation tests were car-ried out on a magnetite coexisting with hematite in a naturalcrystal. But since the material is a dual phase no conventional tech-niques, tensile test or ultrasonic method for instance, are valid.

To complete the study we have performed molecular dynam-ics simulation of magnetite to provide elastic quantities whichare afterwards used to support our discussion. This method waspreviously used for simulating nanoindentation process and forextracting elastic properties of other class of material [6]. We willfocus here only on the evaluation of elastic quantities of magnetitefrom molecular dynamics method based on shell model, withoutsimulating the nanoindentation or the interface between indenterand material.

2. Materials

The natural oxide studied in the present contribution has been found in Russia. Itlooks like a crystal with a perfect octahedron of 2.5 cm in length. The sample was pre-viously cut in two perpendicular directions to analyse the effect of crystallographicorientations. The two sections obtained were mounted in epoxy resin and groundedconsecutively with 600 and 1200 grit SiC papers. The specular aspect is obtained bypolishing the sample with alumina powder of 1 �m particle size. After polishing, the

76 D. Chicot et al. / Materials Chemistry and Physics 119 (2010) 75–81

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Table 2Calculated elastic matrix (in GPa) of Fe3O4.

Indices 1 2 3 4 5 6

1 311.56 184.37 184.37 0.0 0.0 0.02 184.37 311.56 184.37 0.0 0.0 0.0

ig. 1. Optical observation of the natural oxide showing two phases, hematite (clearone) and magnetite (dark zone), and some Vickers indents performed in the mag-etite.

ample is cleaned in pure ethanol. Several observations are performed on the twoamples using an optical microscope (G×500). Fig. 1 clearly shows that two typesf oxide, magnetite (Fe3O4) and hematite (Fe2O3), coexist in this natural oxide. Noignificant difference was noticed on the two samples concerning microstructurespect and experimental indentation data. As a consequence, the mechanical studys made on any section of the natural crystal.

. Molecular dynamics analysis of the elasticity ofagnetite

In the present atomistic description, the crystal lattice is builtrom ions interacting via pair potentials and polarisable by means ofhe shell model [7] in the NPT ensemble. The interatomic potentialnergy is then the sum of the long-range Coulombic and short-ange non-Coulombic contributions. The simulations of the ironxide lattices were obtained using two-body potentials for Fe3+–O,e2+–O, and O2−–O2− interactomic interactions that have beeneveloped by Lewis and Catlow [8] and Woodley et al. [9]. The shellodel parameters interactions were refined to achieve the best fit

o the experimental data for the lattice structure of Fe3O4. The fitf the potential parameters is done using the relaxed fitting algo-ithm of Gale [10] in which the changes in the structural parametersn optimization are used to calculate the residual error. The short-ange potential parameters for each ion–ion interaction and shellodel parameters are listed in Table 1. The calculations were per-

ormed using energy minimization procedure. This potential modelas successfully used of other class of materials in previous studies

11,12].The elastic constants collected in Table 2 represent the second

erivatives of the energy density with respect to strain [13]:

ij = 1V

(∂2U

∂εi∂εj

)(1)

here V is the volume, U is the energy and ε is the displacement.The elastic constant tensor is a 6 × 6 symmetric matrix. The

1 potentially independent matrix elements are usually reduced

able 1hort-range potential and shell model parameters for each ion–ion interaction.

Short range Shell model

Interaction A (eV) ˚ (Å) C (eV Å6) Y (e) k (eV Å−2)

Fe3+–O2− 1102.4 0.33 0 Fe3+ 4.97 304.7Fe2+–O2− 694.1 0.34 0 Fe2+ 2.00 10.9O2−–O2− 22764.0 0.15 27.9 O2− −2.21 27.3

3 184.37 184.37 311.56 0.0 0.0 0.04 0.0 0.0 0.0 12.20 0.0 0.05 0.0 0.0 0.0 0.0 12.20 0.06 0.0 0.0 0.0 0.0 0.0 12.20

considerably by symmetry according to Nye [14] as follows:

C = 1V

(Dεε − DεiD−1ij

Djε) (2)

where Dεε = (∂2U/∂ε∂ε)int., Dεi = (∂2U/∂ε∂˛i)ε, Dij = (∂2U/∂˛i∂ˇj)ε,Djε = (∂2U/∂ˇj∂ε)ε. The elastic compliances, S, can be readily cal-culated from the above expression by inverting the matrix, i.e.S = C−1.

The bulk modulus B0 is also related to the components of theelastic constant. According to Voight’s definition the obtained bulkmodulus B0 and shear modulus G0 values are respectively 226.78and 32.76 GPa. Concerning the Young modulus, the obtained valueis 174.48 GPa; the corresponding value of Poisson ratio is 0.372.

4. Experimental indentation tests

DSI were performed with a Micro-hardness Tester 2-107 fromCSM instruments. The indenter used is a Vickers pyramidal. Allindentation tests were carried out under the constant room tem-perature of 20 ◦C. Three different cyclic indentation tests have beenchosen in order to evaluate the influence of the loading/unloadingrates, the dwell times and the indentation history (or elasto-plastic deformation) on the elastic modulus (Fig. 2). Indentationno. 1 (Fig. 2a) is a monocycle where the loading and unload-ing rates, expressed in N min−1, are chosen equal to twice thevalue of the peak load expressed in Newton [15] and the dwelltime at the maximum load is equal to 15 s. For indentation no.2 (Fig. 2b), the loading/unloading rate is constant and equal to5000 mN min−1. The first cycle is immediately followed by fouradditional cycles repeated between 0 mN and the same ultimateapplied load as the previous cycle. Indentation no. 3 (Fig. 2c) isa multicyclic indentation corresponding to unloading–reloadingrepeated cycles performed by using an incremental loading mode.We have imposed a loading/unloading rate identical to the one ofindentation no. 2 and we have tested various dwell times rang-ing from 0 to 60 s. Multicyclic indentation presented Fig. 2c showthat the loading curve of cycle indexed i follows the loading curveplotted at the cycle (i − 1), even if disparities exist between twoconsecutive cycles in relation to the dwell load.

For the three indentation conditions, the dwell time at peak loadresults in a horizontal plate. This indicates that the crystal of mag-netite exhibits creep behaviour. In some cases, depending or noton the creep response of the material, the unloading portion of theload–displacement curve shows “bowing out”. This phenomenoncan modify or make the measurements of the reduced modulusimpossible. A study of creep behaviour is then necessary in orderto evaluate its influence on the modulus calculation. But it is ren-dered difficult due to the fact that dwell times and “bowing out”are probably two phenomena in competition with each other: lowdwell times may reasonably involve pronounced “bowing out” andthe inverse for the highest ones. To separate their contributions, we

studied the effect of different dwell times (or creep conditions) withthe objective of reducing the “bowing out” of the higher durationof the dwell load. To estimate the creep effect on the indenta-tion depth as a time-function, numerous authors propose differentmodels implying fitting parameters [16–20]. But numerous authors

D. Chicot et al. / Materials Chemistry

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ig. 2. Load–displacement curves obtained on the crystal of magnetite: (a) indenta-ion no. 1, under different maximum indentation loads applied, (b) indentation no., with five cycles performed under Pmax = 7000 mN and (c) indentation no. 3, with0 cycles with different dwell loads between 100 and 10 000 mN.

21–24] analysed the creep curves by associating elementary springnd dashpot mechanical elements in rheological models. In a recentork, Chicot and Mercier [25] have shown the possibility to repre-

ent creep behaviour using a rheological model where the reduced

odulus is explicitly introduced, the objective being to represent

epth increment (�h) as a function of time (t):

hcreep = P

26.43h(t = 0)C

[1ER

(1 − exp

[−t

ER

�R

])+ 1

�It]

(3)

and Physics 119 (2010) 75–81 77

where P is the applied load, h(t = 0) is the indentation depth mea-sured at the beginning of the horizontal plate, C is the constant ofTabor [26], ER is the reduced modulus, �R and �I are the coefficientsof viscosity of the material and the instrument respectively.

This rheological model can be compared to the four-elementMaxwell–Voigt combination which also introduces two viscositycoefficients. But in addition, since the index ‘R’ refers to the param-eters linked to the materials, we suggest that the index ‘I’ refers tothe instrument influence which can interfere in the creep behaviourof the material/instrument couple.

In the present study, we have selected 0, 5, 10, 20, 40 and 60 s asdwell time since the indentation depth slowly increases after 15 safter a rapid increase between 0 and 15 s.

5. Some recent corrections in the reduced moduluscalculation

Oliver and Pharr [1], based on the original work of Doerner andNix [27], proposed to calculate the elastic modulus from the totalcompliance of the specimen and of the instrument. From a gen-eral point of view, the total compliance is the contribution to thedepth-measurement deflections of the load frame added to thedisplacement into the material. Numerous authors calculate thereduced modulus from a load–depth curve after introducing thecalibrated value of the frame compliance (Cf) in the software. Thiscalibrated value being determined by performing indentations ona sapphire for example. Fischer-Cripps [28] adds that the framecompliance term takes into account the indenter shaft and the spec-imen mounting. In order to avoid additional error in the reducedmodulus calculation, we introduced none value of Cf in the soft-ware. However, since all the indentation tests are performed onthe same sample, the compliance calculation should theoreticallygive the same value independently of the indentation testing. Toseparate the two contributions, the deflections of the instrumentand displacement into the material respectively, Oliver and Pharr[1] propose the following relation since the compliance is given bythe inverse of the contact stiffness (1/S = dh/dP) calculated on theunloading curve:

1S

=(

dh

dP

)= Ct = Cf +

√�

21

ˇ�ER

√AC

(4)

where Ct is the total compliance, Cf is the frame compliance, ˇ is acorrection factor depending on the shape of the indenter and equalsto 1.05 [29]. AC is the projected contact area of the elastic contactmeasured from the indentation hardness impression. � is a cor-rection factor depending on Poisson’s ratio [30,31]. In the presentstudy, � = 1.0418 with � = 0.37. ER is the reduced modulus definedas:

ER =(

1 − �2m

Em+ 1 − �2

i

Ei

)−1

(5)

where Em, �m and Ei, �i represent the elastic modulus and Poisson’sratio of the material and of the indenter, respectively.

For the diamond indenter, an elastic modulus and a Poisson’sratio of 1140 GPa and 0.07 were respectively used [32]. For theelastic properties of magnetite, numerous works were devoted tothe determination of Poisson’s ratio and the elastic modulus of thisoxide. For Poisson’s ratio, we considered in the following the valueof 0.37 corresponding to the value deduced from molecular dynam-ics analysis. This value differs from usual values even if it is of the

same order than the average, i.e. 0.31, calculated using Poisson’sratio given in literature [33–36]. For additional evidence, the elas-tic modulus of magnetite is found to be equal to 150 GPa by Woodet al. [37], or 175 GPa by Seo and Chiba [38] for a bulk magnetite.From a general insight, these values are very close to the theoreti-

78 D. Chicot et al. / Materials Chemistry and Physics 119 (2010) 75–81

Ft

cp

6

l((idotcoTtmfiYo2tlsbuto

swsnsettbviiiH

ig. 3. Inverse of the contact stiffness versus the inverse of the contact depth for thewo situations of loading rates applied to the crystal of magnetite.

al one, i.e. 174.48 GPa, resulting from molecular dynamics analysiserformed on the magnetite crystal.

. Discussions

To analyse the influence viscosity through the variation of theoading rate, we have plotted the inverse of the contact stiffness1/S) as a function of the inverse of the contact indentation depth1/hC). These two parameters are deduced on the one hand fromndentation no. 1 where the loading rate takes different valuesepending on the peak load, on the other hand from the first cyclef indentation no. 2 where the loading rate is constant. Fig. 3 showshe straight lines resulting from the analysis of the two indentationonditions and the application of relation (4). The calculated slopesf the two straight lines are similar and very close to 0.77 �m2 N−1.he value of the reduced modulus, ER, calculated by applying rela-ion (4) is 211 GPa and, consequently, the elastic modulus of the

aterial deduced from the relation (5) is equal to 223 GPa. As arst conclusion, we can note that this value is not equals to theoung modulus (174.48 GPa) but that it is very close to the valuef the bulk modulus calculated by molecular dynamics method, i.e.26.78 GPa. This result is very interesting and confirms here thathe depth-sensing indentation allows calculating the bulk modu-us instead of the Young modulus. This result is however expectedince the bulk modulus of a material determines how much it wille compress under a given amount of external pressure in vol-me; whereas Young’s modulus describes tensile elasticity or theendency of a material to be deformed along a given axis whenpposing forces are applied.

As further conclusion, the frame compliance is not a fixed valueince two different indentation modes lead to different values for Cfhile all the indentation tests are performed on the same mounted

ample, using the same instrument and the same indenter. Inumerous studies, the frame compliance appears to be importantince an incorrect value can lead to erroneous calculations of thelastic modulus. In nanoindentation, the influence of the tip inden-er and the frame compliance is probably more important due tohe measurement scale. Nevertheless, the observed variation of Cfrings some questions. Since the frame compliance is not a constantalue whereas the environmental testing conditions (instrument,

ndenter, specimen) are the same, the explanation should be foundn the different applied loading rates and, consequently, in the abil-ty of the material to plasticize itself during the indentation process.owever, it is not possible to prove which of the instrument or

Fig. 4. Inverse of the contact stiffness as a function of the inverse of the contactdepth (indentation no. 2) in relation with the number of cycles.

the material has an impact on the measurement because of thetwo possibilities of interpretation: (1) modification of the elasticresponse of the instrument due to the loading mode and (2) accom-modation changes of the material under the indenter penetration.In fact, by applying low loading rates, the material accommodatesitself by plastic deformation and, consequently, the creep responseof the material can lead to an increase of the maximum indenta-tion depth. Then, the load–depth curve is moved towards the rightof the graph and the calculated value of the contact depth has ahigher value and, as a result, its inverse decreases. The outcome isthat the straight line corresponding to the indentation no. 1 in Fig. 3is shifted towards the lowest values. As a consequence, the framecompliance, Cf, differs from zero.

Furthermore, in multicyclic indentation (indentation no. 2), wehave studied the effect of the number of cycles, all performed byapplying the same maximum peak load as in the previous cycles.Fig. 4 presents the inverse of the contact stiffness as a function ofthe inverse of the contact depth calculated after each cycle. Fig. 4shows no significant influence of the number of cycles since all thedata collection can be represented by the same straight line. Then,the bulk elastic modulus (i.e. 223 GPa previously calculated withthe same slope value) is independent of the number of the firstfive cycles. It is noticeable that the bulk modulus is the same asthe one obtained by monotonic indentation. Indeed, Fig. 2b clearlyshows the visco-elasto-plastic behaviour of the magnetite becausethe indentation depth increases at each new cycle under the dwellload. Nevertheless, it is surprising that no modification of the elas-tic properties measurement is observable. This indicates that thevisco-elasto-plastic behaviour of the material has no effect on theelastic properties measurement.

Fig. 5 collects the graphs related to the application of the methodof Oliver and Pharr, which is carried out by DSI under differentdwell times performed in indentation no. 3. Before discussing theobtained results, it seems important to separate DSI analysis usinglower (1000 mN—black symbols) and higher (10 000 mN—open sym-bols) load values. Indeed, it is clearly shown on this figure thatthe data range related to the lowest loads well covers the x-axiswhereas the data corresponding to the highest loads are mainlylocated near the zero data point. In addition, the highest loadshighlight disparities due to the indentation location at the sur-

face, since the three data collections are not superimposed. Thisproblem probably comes from the non-homogeneities of the testedmaterial since the material is a dual phase. As a consequence,indentation location has a major role depending on the phases

D. Chicot et al. / Materials Chemistry and Physics 119 (2010) 75–81 79

ct dep

alceiiat

Fig. 5. Inverse of the contact stiffness as a function of the inverse of the conta

ffected by the indentation. Indeed, depending on the applied load,arger plastic zone can be developed under the indent and theyan involve the two phases having different mechanical prop-

rties. To take into account this problem, we only analysed thendentation data obtained with the low applied loads after repeat-ng three indentation tests. Fig. 5 shows that experimental datare adequately represented by the same straight line accordingo Oliver and Pharr’s method for multicyclic indentation. Conse-

th on the crystal of magnetite with various dwell times for indentation no. 3.

quently, two main conclusions related to the compliance and tothe elastic modulus were drawn. In all situations, the compliancetakes values lower than those obtained by applying indentation no.

1 (i.e. Cf < 0.159 �m N−1). In addition, Cf decreases with the dwelltime, t, and remains constant and equal to 0.05 �m N−1, for dura-tions higher than 15 s. On the other hand, we note from Fig. 5 thatthe slopes of the straight lines increases as a function of time. Fromthe different slopes, we calculated the bulk elastic modulus (Fig. 5)

80 D. Chicot et al. / Materials Chemistry and Physics 119 (2010) 75–81

Fc

w(oeiTmspobctiibbtidgdd

1

stage to secondary stage after 20 s. This second stage is usuallycharacterized by a linear representation of the deformation as atime-function whereas the first stage is represented by a powerlaw [40]. For this reason, the rheological law is applied only on

ig. 6. Superposition of load–depth curves obtained under monocyclic and multi-yclic indentations with 60 s of dwell times.

hich are located between 250 GPa (high dwell times) and 400 GPalow dwell times). To explain the difference observed with previ-us indentation no. 1 and no. 2 (i.e. 223 GPa), we correlated thelastic withdrawal of the material during the indenter removal tots necessary adaptation or accommodation during the dwell time.he effect of such a phenomenon is clearly shown in Fig. 6 whereonocyclic and multicyclic indentations are represented on the

ame graph. For the first cycle, the two curves are well superim-osed showing thus the good reproducibility of the indentation testf this material. For the other cycles, the indentation depth reachedy the indenter is lower in the multicyclic test than in the mono-yclic test. This confirms that the depth increasing is impeded byhe plasticity generated by dislocations movements occurring dur-ng the dwell load of the indentation process. Finally, the multicyclicndentation leads to a cumulative plasticity as already mentionedy Komvopoulos and Yang [2]. Nevertheless, we noted that theulk elastic moduli obtained under multicyclic indentation tendowards the theoretical value when the dwell time increases. Thisndicates that the dislocation network reorganizes itself during thewell time allowing thus to annihilate some dislocations and toradually decrease the level of plasticity. In addition to explain theisparities obtained for the lowest dwell time, it is necessary toiscuss both on the “bowing out” and on the creep behaviour:

) Effect of “bowing out”The “bowing out” manifests itself by an increase of the con-

tact indentation depth whereas the indenter is withdrawn. Fig. 7presents an enlargement of the load–depth curves at the peakload of 10 000 mN. This figure shows that the “bowing out” ispronounced for the lowest dwell times and becomes quasi-nullafter 10 s. This phenomenon could be linked to the initial disloca-tions network. Indeed, due to the cumulative plasticity occurringin the preceding cycles, the number of dislocations likely to par-ticipate to the hardening, or be generated during the indenterwithdrawal, and diminishes drastically the higher cumulativeplasticity. Then, when the hardening tends to a limit value, the“bowing out” cannot take place because no dislocations can con-tribute to this process. To take into account this phenomenon,Li et al. [39] modified the power law connecting the load to

the indentation depth, and found different ratios, 1.1 and 1.5,between the two moduli calculated by applying the modifiedmodel and Oliver and Pharr’ original model, these ratios depend-ing on the unloading rate. Then values 1.5/1.7 calculated here arein good agreement with the results of Li et al. [39].

Fig. 7. “Bowing out” as a function of the dwell time.

2) Effect of creepThe dwell load induces necessarily creep by indentation

depending on the viscosity of the tested material. To study creep,it is possible to represent the depth increasing as a function oftime by means of rheological model [25]. The reduced modu-lus determined by Oliver and Pharr’ method allows applyingthe relation (3) to determine the constant of Tabor, C, and thetwo coefficients of viscosity, �R and �I. For multicyclic indenta-tions, the dwell load is studied by plotting the relative variationof recorded indentation depth (�h) as a function of the dwelltime (t) according to the relation (3). The constant of Tabor C,the reduced coefficient of viscosity of the material, �R, and ofthe instrument, �I, respectively, are determined as a functionof the dwell time. In practice, an important problem of fittingappears, due to the number of data considered for the fittingand to the extent of the x-axis. For example, Fig. 8 shows thecalculation of the rheological parameters fitted by using differ-ent lengths of segment on the x-axis, ranging between 5 and60 s and indicated by the arrows. This figure clearly shows thatthe fitting parameters depend on the length of the x-axis con-sidered. In addition, the creep behaviour changes from primary

Fig. 8. Rheological parameters obtained by data fitting covering different portionsof x-axis indicated by the arrows.

istry

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[[[[[[[[[[[[[[[[[

[[[

[38] M. Seo, M. Chiba, Electrochim. Acta 47 (2001) 319.

D. Chicot et al. / Materials Chem

the first twenty seconds of the test. However, to compare theinfluence of various dwell times, the fitting parameter shouldbe determined using the same extent of the x-axis, i.e. at least5 s, and on the last cycle. After calculation we show that the rhe-ological parameters are quasi-constant except for 5 s for whichthe constant C is twice higher and the two viscosity coefficientsare lower than the other values (Fig. 8). In addition we verifiedhere that the constant C is close to 3 according to Tabor’s value[26]. Concerning the value found for the viscosity coefficient,Cseh et al. [41] showed that indentation creep tests performedwith both flat ended cylindrical and hemispherical indenters aresuitable for viscosity measurements in the viscosity range of 0.1and 100 GPa s. Nevertheless, few studies have been performedto study viscosity of metallic materials. However, Fischer-Cripps[22] has found values ranging between 1 and 73 GPa s dependingon the rheological model applied for representing creep nanoin-dentation results obtained on a fused silica with a cube cornerindenter and on a 1 �m Al film on silicon with a spherical inden-ter. In a previous work [25], similar values of viscosity coefficientare obtained for metallic materials.

The main information is that the creep response does not seemo be affected by the hardening history of the material, i.e. by theumulative plasticity. This result agrees with the fact that the elas-ic modulus is different for the dwell time of 5 s and it becomesuasi-constant for dwell times higher than 10 s. Then, the change ofhe elastic modulus calculated by multicyclic indentation is mainlyttributed to the “bowing out” and the manner to considerate thenloading curve to calculate both the slope and the indentationepth.

. Conclusion

The elastic modulus under various multicyclic indentations ondual-phase crystal of magnetite and hematite has been inves-

igated by depth-sensing indentation at room temperature. Fromgeneral point of view, the comparison with molecular dynamics

esults clearly shows that monotonic DSI test leads to the bulk elas-ic modulus instead of the Young modulus. In addition, we observedhat the loading/unloading rates and the number of cycles, in a mul-icyclic indentation using the same maximum applied load, have noignificant influence on the mechanical response since we obtainhe same theoretical bulk elastic modulus. Then, in this condition,isco-elasto-plastic properties of the material have no visible effect

n the elastic measurement even if the indentation depth increasest each addition cycle. However, we have unexpectedly observedariations of the elastic response as a function of the dwell timen multicyclic indentation using incremental loading. In this case,he indentation at cycle (i + 1) corresponds to the indentation of

[

[[

and Physics 119 (2010) 75–81 81

the plastically deformed material resulting of the indentation atcycle (i). This result can be explained by reorganisation of the dis-locations network during creep of the material and by emergenceof dislocations immediately after the withdrawal of the indenter,both at origin of the cumulative plasticity. However, we noticed thatthe elastic modulus tends to be the theoretical one deduced frommolecular dynamics analysis when increasing dwell time which isassimilated to an accommodation time for the dislocation network.

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