Hm

DL

a

ARRA

KMISL

1

odldiiwihrvddtaparnot

0d

Materials Science and Engineering A 499 (2009) 454–461

Contents lists available at ScienceDirect

Materials Science and Engineering A

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

ardness length-scale factor to model nano- andicro-indentation size effects

. Chicot ∗

aboratoire de Mécanique de Lille, LML - UMR 8107, U.S.T. Lille, IUT A GMP, BP 179 - 59 653 Villeneuve d’Ascq, France

r t i c l e i n f o

rticle history:eceived 9 July 2008eceived in revised form 28 August 2008ccepted 7 September 2008

a b s t r a c t

In this paper, we show that nano- and micro-indentation hardness data can be represented adequatelyby the strain gradient plasticity (SGP) theory if the uniformity of the dislocation spacing is taken intoaccount. To give relevant information on the plastic deformation process, we suggest to use a hardness√

eywords:icro- and nano-indentation

ndentation size effectstrain gradient plasticityength-scale factor

length-scale (HLS) factor equal to Ho · h∗, where Ho is the macro-hardness and h* the characteristicscale-length deduced from the hardness–depth relation of the SGP theory. Theoretically, the HLS factoris proportional to both the shear modulus and the Burgers vector, depending on the dislocation spacing.Applied to various crystalline metals, the representation of the experimental HLS factor as a function ofthe theoretical one shows two distinct linear behaviours related to the micrometer and nanometer depthregimes associated with a uniform dislocation organisation beneath the indenter and with dislocations

he ind

CowsrhapcG

srbi(wt

located at the vicinity of t

. Introduction

In indentation, it is well recognized that the hardness valuef materials could be independent of load, it could increase orecrease with load, and it could show a complex variation with

oad changes [1]. At low loads the effect of the surface layers withifferent material properties is more pronounced. However, as the

ndentation depth increases, the bulk effect becomes more dom-nant and eventually there is no change in the hardness value

ith the load. In general, this load-dependence of micro-hardnesss known as the indentation size effect (ISE). This phenomenonas been associated with various causes such as work hardening,oughness, piling-up, sinking-in, shape of indenter, surface energy,arying composition and crystal anisotropy, which have been alliscussed extensively by Cheng and Cheng [2]. Many relationships,ating from 1885 to the present, have been suggested to describehe load-dependence of hardness. In recent years, Ma and Clarke [3]nd Nix and Gao [4] have introduced the concept of strain gradientlasticity (SGP) based on Taylor’s dislocation theory. Besides, Nixnd Gao [4] have shown that the ISE behaviour of crystalline mate-

ials can be accurately modelled using the concept of geometricallyecessary dislocations (GND). The authors based their reasoningn the experimental law needed to advance a mechanism-basedheory of strain gradient plasticity. For a glassy polymer, Lam and

∗ Fax: +33 320 677 321.E-mail address: didier.chicot@univ-lille1.fr.

ldw

r(i1

921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2008.09.040

enter tip in a largest plastic zone, respectively.© 2008 Elsevier B.V. All rights reserved.

hong [5] describe the hardness–depth variation using a SGP the-ry, which differs from that of Nix and Gao [4] in the manner inhich dislocations of the global plastic deformation process are

eparated. Abu Al-Rub and Voyiadjis [6] have developed a similarelationship to determine a material length-scale using micro-ardness results. The relation between micro-indentation hardnessnd indentation depth variables, macro-hardness and length-scalearameters involves an exponent, which can vary between 1 and 2orresponding to the relations of Lam and Chong [5] and Nix andao [4], respectively.

On the other hand, Nix and Gao’s model of GND underneath aharp indenter is well recognized and nowadays largely used to rep-esent micro-indentation hardness data. The relation establishedy Nix and Gao [4] between micro-indentation hardness, H, andndentation depth, h, is given by:

H

Ho

)2= 1 +

(h∗

h

)(1)

here Ho is generally called the macro-hardness and correspondso the hardness that would arise from the statistically stored dis-ocations alone, in the absence of any geometrically necessaryislocations. The distance, h*, is the characteristic scale-lengthhich characterizes the depth-dependence of hardness.

Huang et al. [7], after analysing numerous works [8–16] haveeported that nano-indentation hardness data do not follow Eq.1). For the majority of these authors, nano-indentation and micro-ndentation typically refer to indentation depths below and above00 nm, respectively. On the other hand, in order to explain the dis-

D. Chicot / Materials Science and Engineering A 499 (2009) 454–461 455

Ftt

careFticam{

wai

cttveai[ibnweda

dahh2tdofitrli

Fda

raZmrcdept

mWlbiorhs(ldttIitvittIntciT

ig. 1. Scheme of a conical indenter showing the tip blunt (Rblunt) and the charac-eristic parameters for describing tip radius effects: hblunt to compare a perfect coneo an indenter and hball corresponding to a ball indentation depth.

repancy between Eq. (1) and nano-indentation hardness data, theuthors advanced two factors linked to the effect of the indenter tipadius and the storage volume for the GND. As shown in Fig. 1, theffect of the tip radius greatly depends on the blunt tip indenter.or a new indenter, the tip radius (Rblunt in Fig. 1) is generally closeo 50 nm but it can reach more than 500 nm when the indenters considered as defective. Geometrical considerations allow thealculation of hblunt and hball, equivalent to a ball penetration, asfunction of the tip radius, Rblunt, and the cone semi-angle, ,easured at the tip indenter. The calculations indicate that:

hblunt = Rblunt ·(

1sin

− 1)

/= 0.06Rblunt

hball = Rblunt · (1 − sin ) /= 0.06Rblunt

(2)

here is equal to 70.3◦, which gives a cone area function equiv-lent to those obtained with a Berkovich and a Vickers pyramidndenters.

For a new indenter, hblunt and hball are both equal to 3 nm whichan be often neglected in comparison with the experimental inden-ation depths, h. For a blunted indenter, it is not so obvious since thewo corrected depths are equals to 30 nm which are relatively highalues compared to typical nano-indentation depth values. How-ver, the indenter tip radius is usually taken into account throughcorrected indented area for the calculation of the actual nano-

ndentation depth into the material. According to Huang et al.7], the indenter tip radius effect cannot alone explain the nano-ndentation size effects. As a consequence, the difference shoulde mainly connected to a factor representative of the dislocationsetwork and jointly of the storage volume of the dislocations. That ishy, in order to represent nano-indentation hardness data, Huang

t al. [7] suggest the consideration of a maximum allowable GNDensity allowing the representation of the hardness data below andbove the limit value between nano- and micro-hardness data.

Fig. 2 represents the model of Huang et al. [7] applied to hardnessata resulting from indentation experiments performed by Fengnd Nix [14] on magnesium oxide (MgO). On this figure, the authorsave plotted Nix and Gao’s model which describes adequately theardness data for indentation depths higher than approximately00 nm. We can notice that this limiting indentation depth value iswice the typical value of 100 nm admitted by some authors forelimiting nanometer and micrometer depth regimes. As a firstbservation, the limiting indentation depth does not seem to have a

xed value and depend on the material properties and the indenterip radius. In addition, the model of Huang et al. [7] is not able to rep-esent adequately the hardness–depth variation in the depth-rangeocated between the two indentation scales of measurement, i.e. forndentation depths between 100 and 300 nm for the analysed mate-

dwcaa

ig. 2. Square of indentation hardness, H2, versus the reciprocal of indentationepth, 1/h, given by the finite element analysis for MgO. Nix and Gao’s model islso shown (without accounting for GNDmax) [7].

ial, as shown in Fig. 2. That is why, to overcome such a discrepancyt the transition stage between nanometric and micrometric scales,ong et al. [17] suggested that the hardness–depth data could beodelled in two separate zones by means of two straight lines

epresented by Eq. (1). Subsequently the authors give two sets ofharacteristic indentation parameters (Ho and h*) respectively toescribe nano- and micro-indentation experiments. However Zongt al. [17] do not give additional physical explanations to inter-ret jointly the macro-hardness values (Homicro and Honano) andhe characteristic scale-lengths (h∗

micro and h∗nano).

In the present paper, we have reanalysed various nano- andicro-indentation hardness data obtained on crystalline metals.e show that the hardness–depth relationship, originally estab-

ished by Nix and Gao [4], is in reality well adapted to represent thei-linear indentation behaviour on the two scales of measurement

f some precautions are taken in accordance with the observationsf Durst et al. [15] concerning the size of the plastic zone. In order toender more pertinent the comparison of the two couples (Ho and*) deduced from nano- and micro-indentation experiments, weuggest a new indentation parameter called hardness length-scaleHLS) factor since it is capable to describe the uniformity of the dis-ocation organisation beneath the indenter in the two indentationepth regimes. From an experimental point of view, this parame-er is directly linked to the slope of the straight line representinghe square of the hardness versus the reciprocal indentation depth.n practice, HLS factor is equal to Ho

√h∗. Its experimental value

s then compared to the theoretical product �√b resulting from

he SGP theory, where � is the shear modulus and b the Burgersector. The ratio between these two products is a constant whichs only linked to the dislocation spacing and jointly to the size ofhe plastic zone. In micro-indentation, this factor corresponds tohat of Nix and Gao’s model confirming, therefore, the SGP theory.n nano-indentation, the SGP factor should take into account theon-uniformity of the dislocation spacing underneath the inden-er, or the largest plastic zone, to be valid. As a main result, welearly demonstrate that the difference between nano- and micro-ndentation comes from the dislocation spacing under the indent.hen, to jointly interpret nano- and micro-indentation hardnessata, we suggest the calculation of the hardness length-scale factor

hich gives more information on the plastic deformation pro-

ess itself. Note that this original approach involves the maximumllowable GND density in the same way as the model of Huang etl. [7] and the model of Durst et al. [15].

4 d Engi

2

2

[iafvs

�

w

sow

H

w

e

�

wfstos

hf

H

wld

h

w

gtHt

etsaseon

h

wtNt

canpdiiqzs

h

WtGDtr

vitsdt“wvm

H

a

h

Oczamd�

H

a

h

Fiotit

56 D. Chicot / Materials Science an

. The concept of a plastic deformation toughness

.1. Strain gradient plasticity

The strain gradient plasticity theory developed by Nix and Gao4] assumes for simplicity that the indentation deformation processs accommodated by geometrically necessary dislocations whichre required to account for the permanent shape change at the sur-ace. The GND density, �G, is expressed as a function of the Burgersector, b, the indentation depth, h, and the angle, �, between theurface of the conical indenter and the plane of the surface:

G = 32bh

tan2 � (3)

here � (=19.3◦) corresponds to 90◦- .In indentation, Von Mises criterion relates the equivalent flow

tress and the shear stress, whereas Tabor’s rule (assuming a factorf 3) relates hardness with the equivalent flow stress. Then we mayrite:

= 3� = 3√

3� (4)

here � is the equivalent flow stress and � the shear stress.According to Taylor’s relationship the shear strength can be

xpressed as function of the total dislocation density as:

= ˛�b√�T = ˛�b√�G + �s (5)

here ˛ is a constant depending on the tested metals and rangingrom 0.3 to 0.5 according to Taylor dislocation model [4,7].� is thehear modulus. �T is the total dislocation density in the indenta-ion, which can be separated in two terms, �G being the densityf the geometrically necessary dislocations and �s the density oftatistically stored dislocations.

By combining relations (3)–(5), we may express the macro-ardness, Ho, and the characteristic scale-length, h*, of Eq. (1) as

ollows:

o = 3√

3˛�b√�s (6)

here Ho is the hardness arising from statistically stored dis-ocations alone in the absence of any geometrically necessaryislocations.and:

∗ = 812b˛2 tan2 �

(�

Ho

)2(7)

here h* characterizes the depth-dependence of the hardness.According to Nix and Gao [4], h* is not a constant value for a

iven material and indenter geometry since it depends on the sta-istically stored dislocations density through the macro-hardness,o. Relations (6) and (7) are generally well admitted and often used

o represent the ISE in the micrometer regime of the indentation.In addition to take into account the effect of geometrically nec-

ssary dislocations on the indentation size effect and on the size ofhe plastic zone, Durst et al. [15] propose a correction factor whichhould be introduced into Nix and Gao’s model. Indeed for theseuthors, the size of the storage volume for the geometrically neces-ary dislocations created during the indentation is larger than thosexpected by Nix and Gao’s model. To take into account this effectn the plastic zone, Durst et al. [15] introduce a correcting factor,oted f, into relation (7) as follows:

∗ = 812

13b˛2 tan2 �

(�

Ho

)2(8)

f

here f is chosen by the authors equal to 1.9 in order to approximatehe size of the plastic zone at an equivalent plastic strain of 1.5%.ote that if f is equal to 1, relation (8) is equivalent to relation (7)

hat allows representing Nix and Gao’s model.

sdHsn

neering A 499 (2009) 454–461

Besides, Nix and Gao [4] have also added that an indent withonstant slope is not entirely consistent with the approximation ofconstant strain gradient. This is particularly effective at the begin-ing of the plastic deformation process when the indenter tip justenetrates at the surface of the material. In this case and probablyue by the larger size of the plastic zone and the curvature of the

ndenter tip, the density of geometrically necessary dislocationss higher and located at the vicinity of the indent tip. As a conse-uence, the dislocation spacing becomes non-uniform in the plasticone. In this case, Nix and Gao [4] have expressed the characteristiccale-length similarly to relations (7) and (8):

∗ = 272b˛2 tan2 �

(�

Ho

)2(9)

e can note that relation (9) is equivalent to relation (8) whenhe factor f is equal to 1.44. This value allows reconciling Nix andao’s model considering non-uniform dislocation spacing with theurst et al. model assuming a larger size for the plastic zone under

he indent. In the following we assume that relation (9) is able toepresent nano-indentation results.

Then by applying relations (6) and (7), the GND model pro-ides an excellent description of the depth-dependence of hardnessn the micrometer depth regime. It has been successfully appliedo describe hardness data of various crystalline metals. As a con-equence, to avoid confusion in different Ho and h* calculationsepending on the scale of measurement, nano or micro, we suggesto write the corresponding relations by introducing the subscriptmicro” to indicate that the indentation experiments are performedith a micro-hardness tester. The shear modulus and the Burgers

ector being material properties and independent of the scale ofeasurement, we may write:

omicro = 3√

3˛�b√�s(micro)

nd

∗micro = 81

2b˛2 tan2 �

(�

Homicro

)2(10)

n the basis of Nix and Gao’s analysis on the uniformity of the dislo-ation spacing and the Durst et al. approach on the size of the plasticone, relation (9) is assumed to represent the hardness–depth vari-tion within the nanometer depth regime. For the correspondingacro-hardness, we consider the density of statistically stored

islocations associated to the nano-indentation experiments, i.e.s(nano). As a result, we write:

onano = 3√

3˛�b√�s(nano)

nd

∗nano = 27

2b˛2 tan2 �

(�

Honano

)2(11)

rom a general point of view, the transition between the twondentation behaviours is not fixed in terms of indentation depthr, correlatively, in terms of applied indentation load. Then, thisransition is not directly linked to the use of nano- or micro-ndentation instruments. That is why it would be better to usehe subscripts “UDS” for the description of the uniform dislocation

pacing, instead of “micro”, and “non-UDS” for the non-uniformislocation spacing instead of “nano”. However, since Honano andomicro have been commonly employed in the recent literature, to

eparate the two indentation behaviours, we will also adopt thisomenclature in the forthcoming.

d Engi

2

at(tpitttisitHiftaBhtlethHtem[sctz

atlta

Fdi

l

H

Sttb

H

wo

m

H

ab

H

w

sftiotwlucec(as

D. Chicot / Materials Science an

.2. The hardness length-scale factor

To represent hardness data using the strain gradient plasticitypproach and by applying Eq. (1), either the square of indenta-ion hardness, H2, or the relative square of indentation hardnessH/Ho)2, are generally plotted versus the reciprocal of the indenta-ion depth, 1/h. In this work, the square of indentation hardness isreferred for representing the hardness data because the result-

ng slope is used to calculate the HLS factor. In the first part ofhis paper, we considered the results of Feng and Nix [14] sincehe hardness–depth data clearly show two distinct parts relatedo nano- and micro-indentation experiments. Indeed, as shownn Fig. 3, the experimental data can be adequately fitted by twotraight lines representing respectively the nano and the micro-ndentation hardness data, using the hardness–depth relation ofhe SGP theory. On the two regions of the hardness–depth curve,o and h* are determined by a linear fit regression. For the micro-

ndentation hardness data, the calculation results lead to 9.3 GPaor the macro-hardness, Homicro, and 86 nm for the characteris-ic scale-length, h∗

micro. For the magnesium oxide MgO, Huang etl. [7] give 126 GPa for the shear modulus and 0.298 nm for theurgers vector. Starting from the fit-determined value of the macro-ardness and by applying relation (10), it is possible to calculatehe predicted value of the characteristic scale-length. The calcu-ation gives 71 nm which is in reasonably agreement with thexperimental value of 86 nm. On the other hand, the values ofhe parameters (Honano, h∗

nano) obtained from the nano-indentationardness data are analysed using relation (11). By substitutingonano equal to 10.6 GPa in this relation, the calculated value of

he theoretical h∗nano is equal to 18.2 nm which is very close to the

xperimental one, i.e. 19 nm, indicated on Fig. 3. For this crystallineaterial, the results obtained confirm that Nix and Gao’s model

4] can be validly applied in nano-indentation, if the model con-iders the uniformity of the dislocation spacing. These results alsoonfirm the model of Durst et al. [15] when the correcting fac-or is equal to 1.44 to represent the enlargement of the plasticone.

However, no direct comparison between the two couples (Hond h*)micro and (Ho and h*)nano is obvious. That is why we suggest

o study, in another manner, the slope variations of the two straightines. By using Eq. (1) and rewriting the square of hardness versushe reciprocal of the indentation depth, the slope can be expresseds a function of the macro-hardness and the characteristic scale-

ig. 3. Square of indentation hardness, H2, versus the reciprocal of indentationepth, 1/h, for MgO by the strain gradient plasticity applied to nano- and micro-

ndentation hardness data.

swc

3m

ymmumdmN

TVtm

MN

V

neering A 499 (2009) 454–461 457

ength as follows:

2 = H2o + H2

o · h∗h

(12)

trictly speaking, the slope thus defined is not so pertinent sincehe unit is not conventional. Then, we suggest simply consideringhe square root of the slope to obtain a more usable parameter, i.e.y calculating:

LSF =√H2

o · h∗ = Ho

√h∗ (13)

hich is expressed in MPa m1/2, this unit being equivalent to thatf a toughness.

By using relation (10), we may theoretically express HLSF foricro-indentation:

LSFmicro = 9√2˛ tan � ·�

√b (14)

nd for nano-indentation, i.e. by applying relation (11), HLSFecomes:

LSFnano = 3√

3√2˛ tan � ·�

√b (15)

hich differs from relation (14) only by a factor√

3.Obviously, the slopes of the straight lines only depend on the

hear modulus and Burgers vector of the material. In addition, HLSactor is related to the uniformity of the dislocation spacing throughhe factor of proportionality. Then the hardness length-scale factors proportional to the product �

√b with a factor equal to 2.278 ˛

r 1.315 ˛ for uniform or non-uniform dislocation spacing, respec-ively. The factors depend on the ˛-value. We will see afterwardshich value the coefficient ˛ should take to validate the hardness

ength-scale factor calculation. Applied to the magnesium oxidesing 0.5 for the ˛-value given by Nix and Gao [4], we have cal-ulated the experimental values of HLSF applying relation (13). Thexperimental values are subsequently compared to the theoreti-al ones of HLSFmicro and HLSFnano by applying relations (14) and15), respectively. Table 1 compares the experimental calculationsnd theoretical ones of the hardness length-scale factor and clearlyhows that this approach is very consistent in the two scales of mea-urement. To validate these promising results obtained on MgO,e applied the hardness length-scale factor calculation on various

rystalline metals.

. The hardness length-scale factor applied to variousetals

To discuss about the relevance of the HLS factor, we have reanal-sed various indentation hardness data obtained from nano- andicro-indentation experiments performed on different crystallineaterials (Table 2) drawn from literature. Table 2 collects the val-

es of the shear modulus and the Burgers vector of the studiedaterials on which the SGP theory has been previously applied to

escribe the hardness–depth variations. Table 2 also indicates theacro-hardness and the characteristic scale-length deduced fromix and Gao’s model in nanometer and micrometer depth regimes.

able 1alues of the hardness length-scale factor, HLSF, calculated from experimental inden-

ation analysis and by the application of theoretical calculations using nano- andicro-indentation hardness data performed on the magnesium oxide MgO.

HLSF (theoretical) HLSF (experimental)

icro-indentation 2.48 (Eq. (14)) 2.73 (Eq. (13))ano-indentation 1.43 (Eq. (15)) 1.46 (Eq. (13))

alues given in MPa m1/2.

458 D. Chicot / Materials Science and Engineering A 499 (2009) 454–461

Table 2Values of the shear modulus,�, the Burgers vector, b, the macro-hardness, Ho, and the characteristic scale-length, h*, obtained from nano- and micro-indentation experimentsperformed on various metals.

Metals � (GPa) b (nm) Honano (GPa) h∗nano (�m) Homicro (GPa) h∗

micro(�m) References

Ag (1 0 0)26.4 0.286

– – 0.340 0.757Ma et Clarke [3]

Ag (1 1 0) – – 0.361 0.432Cu (1 1 1)

42 0.256– – 0.581 1.600

Nix and Gao [4]Cu* – – 0.834 0.464MgO 126 0.298 10.6 0.019 9.3 0.086 Huang et al. [7]Ni 76** 0.25 1.24 0.34 0.76 3.05

Zong et al. [17]Ag 33.6 0.29 0.71 0.26 0.25 5.42Au 30.4 0.29 1.02 0.06 0.28 4.66LIGA Ni 73 0.25 – – 2.6 0.34 Lou et al. [18]I

Aifan

Aitca˛ilaUd˛cotiauairit

boe

Ff

mztipaitbbhcp(ilbpIobtp

r 217 0.27 – –

* Cold work hardening copper.** Zong et al. [17] give 94.5 GPa for � but Lou et al. [18], Zhao et al. [20], Zaiser and

ccording to relations (13), (14) and (15), we have plotted the exper-mental hardness length-scale factor, Ho

√h∗, as a function of the

heoretical product �√b (Fig. 4). The values of the HLS factor cal-

ulated using micro-indentation data is represented adequately bystraight line with a slope equal to 1.17. Using the coefficient 2.278corresponding to micro-indentation, the calculated coefficient ˛

s equal to 0.51 which is very close to 0.5 as mentioned by the Tay-or dislocation theory. In nano-indentation, the hardness data arelso aligned on a straight line having a lower slope equal to 0.65.sing relation (15) which represents nano-indentation hardnessata, ˛-calculation deduced from the ratio of proportionality 1.315and the slope 0.65 leads to a value of 0.49 for ˛ which is very

lose to 0.51 obtained in micro-indentation and to the theoreticalne of 0.5. This result is very important since it demonstrates thathe dislocation spacing is uniform in micro-indentation whereas its non-uniform in nano-indentation. This fundamental result thusllows the reconciliation of the two scales of measurement and thenderstanding of the difference between the two sets of data (Hond h*)micro and (Ho and h*)nano resulting from micro- and nano-ndentation hardness data analysis, respectively. In addition, thisesult seems to indicate that the enlargement of the plastic zone,.e. through the value of the correcting factor f, is independent ofhe tested materials since f is equal to 1.44 in all cases.

From a physical point of view, the difference of indentationehaviour at the two scales of experiment is explained in termsf the uniformity of the dislocation spacing. For nano-indentationxperiments corresponding to the beginning of the plastic defor-

ig. 4. Representation of experimental hardness length-scale factor, Ho·h*1/2, as aunction of the theoretical ones, �·b1/2, for various metals.

iwTozctpFoffhrc

naftcdi[Ac

2.47 2.89 Qin et al. [19]

tis [21] and Bonifaz and Richards [22] are agree with 76 GPa.

ation process, dislocations are generated in a non-uniform plasticone where dislocation glide takes place along directions normalo the circular shape of the indenter. Subsequently, when the loadncreases, the plastic zone radius increases and, consequently, thelastic zone boundary enlarges. As a consequence, the dislocationrrangements tend to be homogeneous in the neighbourhood of thendent regarding to the plastic zone extension. To give consistenceo this assumption, we can refer to a similar investigation of shearand evolution in amorphous alloys beneath a Vickers indentationy Zhang et al. [23]. Depending on the applied load, the authorsave illustrated the evolution of the deformation pattern on theross section of a Vickers indenter (Fig. 5). Fig. 5 shows an exam-le of deformation beneath the indent for applied loads of 1000 gFig. 5a) and 10 g (Fig. 5b). In the two situations, a layered bulges located at the indenter tip which is more pronounced for lowoads in comparison to the impression size. In addition for the shearands resulting from dislocations sliding, the authors distinguishrimary and tertiary shear bands according to the indentation load.

n micrometer regime, Zhang et al. [23] note that the lateral sizef the region containing the tertiary shear bands does not extendeyond the diagonal size of the indent on the top surface. Con-rary to the observation of Fig. 5b, we can observe that the serratedrimary shear bands generated under low indentation loads, i.e.

n nanometer depth regime, are located in a semi-circular regionhich reaches the surface beyond the extremities of the indent.

his observation also agrees well with the finite element analysisf Durst et al. [15]. Indeed, Fig. 6 shows the extension of the plasticone for different values of the correcting factor, f. The solid lineorresponds to Nix and Gao’s model in agreement with observa-ions of Fig. 5a. The dotted line represents the enlargement of thelastic zone as it is indicated by the white semi-circular line onig. 5b. Then, we can schematically represent Nix and Gao’s modeln Fig. 7a where the geometrically necessary dislocations are uni-ormly localized beneath the indent in a semi-hemispheric regionor which the size is equivalent to that of the indent. On the otherand, the model of Durst et al. [15] can be represented by a largestegion where the repartition of the dislocations is not uniform as itan be schematically seen in Fig. 7b.

However, it can be seen in Fig. 3 that the transitory state betweenano-indentation and micro-indentation hardness data changesbruptly between hnano and hmicro in the interval of 50 nm and aew microns. Then, Eqs. (14) and (15) are alone able to representhe nano-indentation and the micro-indentation deformation pro-esses and give additional information on the uniformity of the

islocation spacing. Indeed, given as an example, it is clearly seen

n Fig. 4 that the hardness data (black squares) of Ma and Clarke3] are located on the straight line represented by the relation (15).s a consequence, we can suppose that the indentation behaviourorresponds to a plastic deformation mechanism involving the

D. Chicot / Materials Science and Engineering A 499 (2009) 454–461 459

Fig. 5. Evolution of shear band patterns with load beneath the Vickers indentation du

Fig. 6. Axisymmetric geometry of a Berkovich indenter contacting an elastic–plasticmaterial. The equivalent plastic strain ranging from 0.2% to 20% is shown in thecontour plot. The extension of the plastic zone is shown for f = 1 (solid line) andf = 1.9 (dotted line) [15].

npta

4

iIt(TptitnroehsipbnH

ii

Fig. 7. Geometrically necessary dislocations created by a rigid conical indentation wherindentation regime and (b) nano-indentation regime.

e to Zhang et al. [23] (a) under high load and (b) under low load of indentation.

on-uniform dislocation spacing or resulting from an experimentalroblem like the polishing quality. That explains the divergence ofhe results obtained by the SGP theory applied by Nix and Gao [4]nd the indentation experiments carried out by Ma and Clarke [3].

. Nano/micro-hardness–depth conversion

The hardness length-scale factor calculation is interesting sincet allows the calculation of HLSFmicro from HSLFnano, and vice versa.ndeed, the ratio between the two HLS factor is a fixed value equalo

√3 resulting theoretically from the comparison between Eqs.

14) and (15) and experimentally from the slopes ratio 1.17/0.65.his result is of great interest when studying hardness of smallarticles using nano-indentation experiments. Indeed in this situa-ion, hardness determination using micro-indentation experimentss not directly possible because the indented zone can be largerhan the particle size itself. As a consequence, the measured hard-ess value corresponds to a composite hardness which is notepresentative of the hardness behaviour of the particle. Then tobtain micro-indentation information on such a particle, it is nec-ssary to convert nano-indentation results to micro-indentationardness. Nevertheless, the hardness length-scale factor is notufficient to determine the macro-hardness, Ho, and the character-stic scale-length, h*, since HLSF simultaneously involves these twoarameters. To solve this problem, two ways of analysis are possi-le: (i) the determination of the limit indentation depth between

ano- and micro-indentation, i.e. hlim, or (ii) the determination ofomicro from Honano.

By equating the two hardness–depth relations (Eq. (1)) appliedn nanometer and micrometer depth regimes calculated at the limitndentation depth, hlim, corresponding to the passage from micro-

e the dislocation structure is idealized as circular dislocation loops [4]. (a) Micro-

460 D. Chicot / Materials Science and Engineering A 499 (2009) 454–461

Table 3Values of the limit indentation depth, hlim, deduced from the macro-hardness and the characteristic scale-length.

Metals Honano (GPa) h∗nano (�m) Homicro (GPa) h∗

micro(�m) hlim (�m) References

MgO 10.6 0.019 9.3 0.086 0.20 Huang et al. [7]

Ni 1.24 0.34 0.76 3.05 1.30Zong et al. [17]Ag 0.71 0.26 0.25 5.42 0.47

Au 1.02 0.06 0.28 4.66 0.31

Table 4Values of the limit indentation depth, hlim, deduced from dislocation densities.

Metals �snano (m−2) �smicro (m−2) GND (m−2) hlim (�m) References

MgO 1.18 1016 9.09 1015 27.10 1014 0.16 Huang et al. [7]

NAA

t

h

TcTvitamtHtd

h

I

�

wa�h

Adna

h

Ota

�

wiot

fSwTitfameahdvptidt

�

Then, using the macro-hardness obtained in nano-indentation, it ispossible to determine the macro-hardness in the micro-indentationregime. Subsequently, the HLS factor obtained in nanometer depthregime allows the calculation of the HLS factor in the micrometerdepth regime and, consequently, the corresponding characteristic

i 6.31 1014 2.37 1014

g 7.87 1014 0.98 1014

u 1.98 1015 1.49 1014

o nano-indentation, we obtain:

lim = H2LSFnano −H2

LSFmicro

Ho2micro − Ho2

nano(16)

he results of Zong et al. [17] and Huang et al. [7] are used toalculate the limit indentation depth for the materials listed inable 3. The results of hlim are not so interesting since the valuesary greatly between 200 and 1300 nm without any clear tendencyn relation to the intrinsic parameters of the material. In addition,hese values are quite different from that indicated by numerousuthors, around 100 nm, to distinguish the two scales of measure-ent. Nevertheless, by combining relations (10) and (11) related

o the macro-hardness and relations (14) and (15) related to theLS factor, we may express the limit indentation depth as a func-

ion of the Burgers vector and the density of statistically storedislocations in nano- and in micro-indentation ranges as follows:

lim = tan2 �

b · (�s(nano) − �s(micro))(17)

t is interesting to note that by rewriting Eq. (17) we may express:

�s|nanomicro = (�s(nano) − �s(micro)) = tan2 �

bhlim(18)

hich is equivalent to the average value of �G given by Eq. (3)nd �Gmax given by Nix and Gao [4] or Huang et al. [7], i.e.Gmax = (1/2bh)tan2 �, calculated at the limit indentation depthlim.Indeed, we obtain:

GND = �G + �G max

2

∣∣∣h=hlim.

=��s|nanomicro (19)

s a conclusion, the passage between nanometer to micrometerepth regimes occurs when the mean value of the geometricallyecessary dislocations density is calculated using GND and GNDmax

pproaches as follows:

lim = tan2 �

b · GND(20)

n the other hand, application of Eq. (6) allows the calculation ofhe two densities of statistically stored dislocations in micrometernd nanometer depth regimes from the general equation:

s = 4(

Ho)2

(21)

27 �b

here ˛ in Eq. (6) is equal to 0.5.This relation is valid whatever thendentation load, i.e. “Nano” and “Micro” can be used as indexesf �s and Ho independently of the scale of measurement. Thewo dislocations densities, �s(nano) and �s(micro), were determined

Fm

3.94 1014 1.30Zong et al. [17]6.89 1014 0.64

18.31 1014 0.24

rom the macro-hardness and thereafter used to calculate GND.tarting from GND, we recalculate the limit indentation depthsith Eq. (20) in order to compare them to the theoretical values.

able 4 shows that �s is of the order of 1014to 1015 m−2 in micro-ndentation. In nano-indentation, the density is higher accordingo the higher macro-hardness values. The calculated values of hlimrom the macro-hardness and the SGP model (Table 4) are in goodgreement with hlim calculated from experimental values of theacro-hardness and the characteristic scale-length (Table 3). Nev-

rtheless, although this result is interesting since it validates thepproach using the HSL factor, calculation of (Ho, h*)micro from (Ho,*)nano is not yet possible because the density of statistically storedislocations is not known but only deducible from experimentalalues of the macro-hardness. To suggest some ways of study, a sim-le analysis of Table 2 shows that the macro-hardness ratio�, equalo Homicro/Honano, tends to increase when the shear modulus valuencreases. The plot of�2 as a function of� in Fig. 8 confirms this ten-ency. The mathematical expression allowing the representation ofhe data can take the following form:

2 =(

Homicro

Honano

)2=

(�

150

)3/2(22)

ig. 8. Representation of the square of �, i.e. (Homicro/Honano)2, versus the shearodulus �.

d Engi

stf

tHatEitwai0t0

5

(

(

(

(

(

R

[

[

[[

[[[[

[

[

D. Chicot / Materials Science an

cale-length in micro-indentation. Finally, it is possible to constructhe hardness–depth relation of Nix and Gao in micro-indentationrom nano-indentation results.

As an example for the magnesium oxide, the results in nanome-er regime are Honano = 10.6 GPa and h∗

nano = 0.019 �m. From theLS factor in nano-indentation, i.e. HLSFnano = 1.46 MPa m1/2, were able to calculate the HLS factor in micro-indentation usinghe factor

√3. The calculation gives HLSFmicro = 2.53 MPa m1/2.

q. (22) allows then the estimation of the macro-hardnessn micro-indentation from the corresponding one experimen-ally determined in nano-indentation. We obtain Homicro = 9.3 GPahich is the same value than the experimental one. From Homicro

nd HLSFmicro, we calculate the characteristic scale-length in micro-ndentation and we found h∗

micro equals to 0.074 �m instead of.086 �m experimentally found. Note that this value is very closeo the predicted value of the characteristic scale-length equal to.071 �m.

. Concluding remarks

We can give the following conclusions:

1) Nix and Gao’s model is able to represent adequately micro-indentation hardness data as well as nano-indentationhardness data by introducing the notion of uniformity of dis-location spacing in the original model or jointly by taking intoaccount the enlargement of the plastic zone according to themodel of Durst et al. [15]. Here, the correcting factor of 1.44seems to be more appropriate compared to 1.9 given by theseauthors.

2) The hardness length-scale factor is experimentally calculatedusing the macro-hardness and the characteristic scale-lengthdeduced from Nix and Gao’s model. The theoretical factor is

linked to the product of the shear modulus and the square rootof the Burgers vector of the material. In addition, the com-parison of the experimental and theoretical HLS factors givesinformation of the dislocation spacing uniformity through thecoefficient of proportionality.

[[[[

neering A 499 (2009) 454–461 461

3) The main difference between nano-indentation and micro-indentation size effects is linked to the maximum allowableGND density and the associated effect on the size of the plasticzone.

4) The limit indentation depth between the nanometer andmicrometer regimes depends on a GND factor and the Burgersvector.

5) An empirical relation allows the calculation of the macro-hardness in a given depth regime from the other one and theshear modulus of the material.

eferences

[1] D. Chicot, F. Roudet, A. Soom, J. Lesage, Surf. Eng. 23 (2007) 32–39.[2] Y.T. Cheng, C.M. Cheng, Mater. Sci. Eng. R: Rep. 44 (2004) 91–149.[3] Q. Ma, D.R. Clarke, J. Mater. Res. 10 (1995) 853–863.[4] W.D. Nix, H. Gao, J. Mech. Phys. Sol. 46 (1998) 411–425.[5] D.C.C. Lam, A.C.M. Chong, Mater. Sci. Eng. A 281 (2000) 156–161.[6] R.K. Abu Al-Rub, G.Z. Voyiadjis, Int. J. Plasticity 20 (2004) 1139–1182.[7] Y. Huang, F. Zhang, K.C. Hwang, W.D. Nix, G.M. Pharr, G. Feng, J. Mech. Phys. Sol.

54 (2006) 1668–1686.[8] Y.Y. Lim, M.M. Chaudhri, Phil. Magn. A—Phys. Condensed Mat. Struct. Defects

Mech. Properties 79 (1999) 2979–3000.[9] Y. Liu, A.H.W. Ngan, Scripta Mater. 44 (2001) 237–241.10] J.G. Swadener, E.P. George, G.M. Pharr, J. Mech. Phys. Sol. 50 (2002) 681–

694.11] J.G. Swadener, A. Misra, R.G. Hoagland, A. Nastasi, Scripta Mater. 47 (2002)

343–348.12] A.A. Elmustafa, D.S. Stone, J. Mech. Phys. Sol. 51 (2003) 357–381.13] A.A. Elmustafa, A.A. Ananda, W.M. Elmahboub, J. Mater. Res. 19 (2004) 768–

779.14] G. Feng, W.D. Nix, Scripta Mater. 51 (2004) 599–603.15] K. Durst, B. Backes, M. Goken, Scripta Mater. 52 (2005) 1093–1097.16] J.Y. Kim, B.W. Lee, D.T. Read, D. Kwon, Scripta Mater. 52 (2005) 353–358.17] Z. Zong, L. Lou, O.O. Adewoye, A.A. Elmustafa, F. Hammad, W.O. Soboyejo, Mater.

Sci. Eng. A 434 (2006) 178–187.18] J. Lou, P. Shrotriya, S. Allameh, T. Buchheit, W.O. Soboyejo, Mater. Sci. Eng.: A

441 (2006) 299–307.19] J. Qin, Y. Huang, K.C. Hwang, J. Song, G.M. Pharr, Acta Mater. 55 (2007)

6127–6132.20] M. Zhao, W.S. Slaughter, M. Li, S.X. Mao, Acta Mater. 51 (2003) 4461–4469.21] M. Zaiser, E.C. Aifantis, Scripta Mater. 48 (2003) 133–139.22] E.A. Bonifaz, N.L. Richards, Int. J. Plasticity 24 (2008) 289–301.23] H. Zhang, X. Jing, G. Subhash, L.J. Kecskes, R.J. Dowding, Acta Mater. 53 (2005)

3849–3859.