nanoindentation techniques materials

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Micro- and nanoindentation techniques formechanical characterisation of materials

N K Mukhopadhyay1 and P Paufler2

Indentation techniques have been extensively used for mechanical characterisation of materials

Development of instrumented indentation techniques at low and ultra low load levels has further

improved their utility for understanding the mechanical responses of solids at micro and nano

scales The variation of hardness with the loaddepth of indentation known as indentation size

effect has led to difficulties in using hardness as a fundamental or characteristic mechanical

property of materials Detailed discussions are focused on the issue of indentation size effect in

brittle and ductile solids Various theoretical models accounting for the indentation mechanics are

highlighted The results obtained from these techniques with quasicrystals bulk metallic glasses

and nanomaterials are reviewed The issues related to phase transformation during indentation

tests are briefly discussed The industrial use of the indentation technique has been pointed out

Some of the current issues and directions for future research in this field are mentioned

Keywords Microindentation Nanoindentation Indentation mechanics Hardness Indentation size effect

Historical development of the concept ofhardnessThere have been various approaches to the conceptof hardness It was Aristotle (384ndash322 BC Bekker1829) who called something lsquohardrsquo if the surface didnot move under external pressure When subdividinghardness tests into cutting and non-cutting methodsof deformation early attempts belong to the formerCutting of the material was mainly the consequence ofscratching the specimen by needles or corners of certainreference materials Non-cutting methods use an inden-tation perpendicular to the surface of the specimen by anindenter fixed to a load cell or by a bullet of knownmomentum It is not surprising that hardness valuesderived from these methods either relative or absolutedid not agree satisfactorily because of the differentphysical processes involved in testing Much work hasbeen done in the past to understand these processes ata microscopic scale to make hardness values morecomparable and to find a hardness concept whichgenuinely reflects a materialrsquos property (Turner 1886Poschl 1909 Tertsch 1949 Tabor 1951 Mott 1956Glazov and Vigdorovic 1962 1971 Juskin 1971 Petty1971 Grigorovic 1976 Fischer-Cripps 2004)

To recognise hardness as a quantity characteristic of amaterial and to use it for analysis several stages had to bepassed through Some are just mentioned in the present

study Apart from Aristotlersquos general statement men-tioned above at the end of the 17th century people werealready aware of the varying resistance of minerals andother materials against mechanical loading in particularagainst scratching Huygens (1690) noticed the fascinatingfact that the cleavage plane of calcite may be scratchedwhen moving the knife forward and may not whenmoving it in the opposite direction Reaumur (1722)differentiated the quality of steel using a bar withincreasing hardness from one end to the other The degreeof hardness was attributed to the specimen according tothe position on the bar which the specimen could scratchfirst Linne (1768 1793) listed a number of terms relevantfor an appropriate terminology of the mechanics of solidswhich has been adopted by other writers Werner (1774)introduced four categories of hardness (hard semihardsoft very soft) defined by the behaviour of the materialwhen scratched by a knife or a file using gauge mineralsfor comparison Then Hauy (1801) classified mineralsaccording to their ability to scratch each other assigningfour grades to the materials scratching quartz glass orcalcite and those not scratching calcite respectively Mohs(1812) proposed to use hardness (combined with severalother properties) as a distinguishing feature of solidsLater on on the basis of the definition of hardness heintroduced a ten-stage hardness scale (now named afterhim) and instructions for its realisation (Mohs 1822) Theanisotropy of the depths of scratches along differentdirections in a plane as well as on crystallographicallydifferent planes at constant load was measured thor-oughly by Frankenheim (1829 1831) Performingscratches by hand he noticed that features of the scratchalso depend on the topology of the contact and on thespeed of scratching Thus he recommended using relativemeasures of scratching hardness

1Centre of Advanced Studies Department of Metallurgical EngineeringInstitute of Technology Banaras Hindu University Varanasi 221 005 India2Institute for Structural Physics Dresden University of TechnologyD-01062 Dresden Germany

Corresponding author email mukhobhuacin or mukho_nkrediffmailcom

2006 Institute of Materials Minerals and Mining and ASM InternationalPublished by Maney for the Institute and ASM InternationalDOI 101179174328006X102475 International Materials Reviews 2006 VOL 51 NO 4 209

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So far the force applied was either not determined atall or defined only qualitatively It was Seebeck (1833)who built an instrument enabling well defined loads tobe applied Hardness was taken proportional to theminimum load applied perpendicular to the surfacewhich gave just a noticeable scratch when the surfacewas shifted laterally Franz (1850) introduced a similarmethod independently of Seebeck enabling the indenterto be moved across the fixed crystal The methods ofscratching were subsequently made on a more quanti-tative basis by evaluating the scratch geometry Usingessentially Seebeckrsquos device Grailich and Pekarek (1854)improved the mechanical parts and the optical inspec-tion of the scratches They called their instrument asclerometer and in doing so launched the branch ofsclerometry (eg Sklerometrija 1968) With the aid ofthis instrument Exner (1873) supplied data on theanisotropy of hardness in great detail Sometimesalternative definitions were applied as for examplehardness being inversely proportional to the lateral forcewhich is needed to move the specimen under a givennormal load (Franz 1850 Grailich and Pekarek 1854)In any case hardness values obtained this way wererather upper limits of this entity because they weretaken from the minimum load giving rise to a visiblescratch The visibility however strongly depends on thesurface state of the specimen andor the optical methodof recording the scratch To reduce this uncertaintyPoschl (1909) proposed to measure the change of thescratch width versus the change of load instead of thewidth itself Martens (1898) introduced a scratchhardness tester which has been widely used by mechan-ical engineers to measure the scratch width and thenormal load He took as hardness the normal loadwhich caused a given scratch width Recent work onnanoscratching shows a proportionality between lateralforce and (normal force)32 for a number of materials(Kaupp and Naimi-Jamal 2004) Further developmentof this idea will not be followed here because it leads tothe concept of wear which is not in the focus of thepresent review

In addition to scratching which is still used today tostudy wear resistance other approaches to hardnessassessment in the 19th century should be mentionedbriefly Examples are planing drilling and grindingPlaning was put forward by Pfaff (1883) who improvedthe accuracy of measuring the depth of a scratch byrepeating the procedure n times along the same scratchand moreover employing several diamonds side-by-sideat the same time Knowing the mass density of thespecimen and the area scanned the scratch depth couldbe calculated from the weight loss after planing Relativehardness values of various minerals or various planesand directions of the same mineral were then obtainedfrom the proportion of inverse planing depths keepingparameters such as load and speed to be constantDrilling with the aid of a cleaved diamond tetrahedronhas been recommended for hardness measurement byPfaff (1884) and Jaggar (1898) taking for hardness thenumber of rotations needed to penetrate the surface to agiven depth When grinding with a standardised powderhardness had been taken proportional to the timeneeded for the removal of a certain layer or inverselyproportional to the loss of volume during grinding(Jannettaz and Goldberg 1895 Rosiwal 1896) Later

on von Engelhardt and Haussuhl (1960) showed thatresistance against grinding varies rather with the specificfree surfaceinterface energy

Coming back to non-cutting methods an early attemptis mentioned first to find numerical values for the relativehardness of metals due to officers of the US OrdnanceDepartment who in 1856 pressed a pyramid on the metalunder test recording the volume of the indent and usingbronze as a standard (Turner 1886) Then Calvert andJohnson (1859) applied a steel cone to penetrate thematerial under investigation until a predeterminedindentation depth was achieved The load necessary wasused as a measure of hardness relative to the value of pigiron The subsequent development of indentation hard-ness testing methods was mostly stimulated by the rapidlygrowing demand of metallurgy

A milestone towards a quantitative concept ofhardness was the approach published by Hertz (1882)who defined the indentation hardness of solids ascontact pressure in a small circular area at the elasticitylimit This definition was applied to brittle as well as toductile solids However experimental verification latterproved difficult whereas the elasticity limit of brittlesolids could be determined from the pressure at the onsetof cracking This was the advent of contact mechanicsan important tool of present-day hardness testing (cfeg Johnson 1996) In order to take the onset of plasticdeformation explicitly into account Prandtl (19201921) introduced as decisive criterion the critical shearstress as the difference of two principal stresses underthe indenter using a proposal made by von Mises (1913)

On the basis of Hertzrsquos definition Auerbach (18901891) determined the hardness of brittle solids byoptically in situ monitoring the nucleation of cracksunder load in glasses and crystalline quartz Herecognised that the hardness defined by the firstappearance of a crack was subject to variation withthe curvature of the indenter Since cracks opened undertensile stress he concluded that the surface conditionmight also have influenced crack nucleation Extendingthe in situ measurements to ductile transparent solidssuch as rock salt calcite fluorite and others Auerbach(1892 1896) proposed that hardness be taken as Hertzrsquospressure at the onset of plastic flow indicated by analmost constant contact pressure When applying to(opaque) metals the area of contact was determinedemploying a blackened indenter (Auerbach 1900)

Using loads below 2 N hardness was termed micro-hardness though the distinction between macro- andmicrohardness is rather arbitrary Nevertheless a micro-hardness tester looked different mainly because of thehigher optical resolution required Various devices havebeen developed Lips and Sack (1936) and Lips (1937)introduced a small attachment to the light microscopewhich enabled indentations into small grains of multi-phase alloys to be made by moving the specimentowards a Vickers diamond pyramid coupled to aspring In the Vickers indenter the angle betweenopposite faces is 136u This angle results from thecondition that from the tangents drawn from the pointsof contact of an impression of diameter 0375D (where Dis the ball diameter as used in the Brinell test) to thecircumference of the indent the included angle be 136uSmith and Sandland (1925) found that the hardnessvalues obtained with the Vickers indenter of that angle

Mukhopadhyay and Paufler Micro- and nanoindentation techniques for mechanical characterisation

210 International Materials Reviews 2006 VOL 51 NO 4

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and with the Brinell ball are nearly equal when the loadhas been adjusted to produce a Brinell impression of thediameter 0375D which is considered the ideal impres-sion diameter halfway between 025D and 05D Unlikeother test methods Vickers hardness is named after thecompany Vickers Ltd which designed the first hardnesstesting machine with this indenter After replacing thismicrohardness tester by the objective of the microscopethe diameter of the indents (of the order of 10 mm) couldbe measured (for Russian devices cf Glazov andVigdorovic 1962 1971) While in a conventional micro-or macrohardness test penetration depths h of the orderof 101ndash103 mm and forces F in the range 1022ndash102 Noccur in a nanohardness test both h and F aresignificantly smaller (F10 mN h10 mm) and ndash inthis case compulsory ndash the dependence F(h) is recordedthroughout the loadingndashdeloading cycle (cf Fischer-Cripps 2004)

Indentation methods experienced another diversifica-tion Starting with spherical indenters (Hertz 1882

Brinell 1900) other shapes had been found usefulfor various purposes such as cones (Ludwik 1908Rockwell 1922) double cones (Grodzinski 1951)trigonal prisms (Attinger 1947) or pyramids (tetragonalSmith and Sandland 1925 orthorhombic Knoop et al1939 trigonal Berkovic 1951) It was already noticedby Auerbach (1891) that the exact shape of the indentermust have an impact upon the hardness value Theshape of the indents as well as the relation betweenhardness values and indentation geometry is displayed inTable 1

This brings us to the evaluation of indentationhardness H as it is used today Let an indenter underload F penetrate the surface of a solid so that the contactarea between indenter and surface is A Then thehardness H of the solid can be defined as the ratio

H~force F (perpendicular to the surface)

contact area A(1)

If A is taken as the curved area of the indent H depends

Table 1 Geometrical shape of indenter and projected indentation along with hardness equation and penetration depthsdisplayed for various indenters

Testingmethod

Shape ofindenter

Shape of impressionprotection Hardness value Penetration depth

Brinell He~2F

pD2 1 1(d=D)2frac12 1=2

Brinell hardness number BHN[F]5kg [D] [d]5mm

t5D1ndash[1ndash(dD)2]12ltd22D

VickersHV~

2F sin 680

d2~

18544F

d2

Vickers hardness number VHN

frac12F~kg frac12d~mm

tltd7

KnoopHK~

F

A~

1440F

d2

Knoop hardness number KHN


tltd306

LudwikHL~

4F sin450

pd2~

09F

d2

t5d2

GrodzinskiHG~F=At~

6rF

pd2~

09F

tan(a=2)d3

Double cone number a~1540 r~2 mm

HGN~277F

d3106

frac12D~mm a~1540 frac12F~kg

tltd80 for a5154u r52 mm

Berkovich HB~F=A~2F=a2ffiffiffi3p

tlt019a


International Materials Reviews 2006 VOL 51 NO 4 211

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strongly on the shape of the indenter Hence BrinellLudwik Grodzinski Rockwell Vickers or Knoophardness numbers are differentiated A more physicalmeaning is assigned to H when A is taken as theprojection of the area of contact Ap between indenterand surface onto the surface This definition has becomeknown as Meyer hardness Hmeyer (Meyer 1908) thephysical meaning of which is the mean pressure p overthe surface of the indentation if the friction between thesurface of the indenter and the sample can be neglectedThen the pressure is normal to the surface of the indentFor symmetry reasons the horizontal component of theresultant force upon the specimen is zero The verticalcomponent of that force is equal to the total force Fwhich amounts to

ethr0

p 2pxdx~ppr2~pAp~F (2)

x being the radius of an annulus of the contact surface2r is the chordal diameter and pr2 the projected area ofthe indentation Thus Hmeyer5p Knowing the geometryof the contact surface of any test method thecorresponding hardness values may be readily convertedinto values of Hmeyer Recently Pohlenz et al (2001)studied the different hardness definitions usable formicro and nanoindentation in order to describe thehardness of substrates and layers by relational coeffi-cients Malzbender (2003) discussed in detail the variousdefinitions of hardness and elastic modulus as obtainedwith conical and spherical indenters and made acomparison in order to obtain a relationship thatpermits a conversion and assessment of their differences

Macro- micro- and nanoindentation hardness testingcontinue to be used extensively in materials evaluationand various aspects of this subject have been reviewedearlier [Westbrook and Conrad 1973 Blau and Lawn1986 Biswas et al 1996 Fischer-Cripps 2004 specialissues in Philosophical Magazine A (1996 2002) andJournal of Materials Research (1999 2004) MaterialsResearch Society Symposium Proceedings (Baker et al1995 2000 Drory et al 1995 Gerberich et al 1996Cammarata et al 1997 Moody et al 1998 Vinci et al1999 Ozakan et al 2001)] It is interesting to note thatvarious techniques have been developed to study thematerials response during and after the indentation Inaddition to conventional optical microscopy thesetechniques include cathodoluminescence scanning elec-tron microscopy transmission electron microscopy(TEM) focused ion beam techniques (FIB) interfaceforce microscopy (IFM) atomic force microscopy(AFM) acoustic microscopy acoustic emission techni-que Raman spectroscopy (see for example Brown et al1988 Boldt et al 1992 Muraki et al 1997 Ray et al1999 Wolf and Paufler 1999abc Kiely et al 1999Mukhopadhyay et al 2001 Tymiak et al 2004)Recently Bhushan and Li (2003) have reviewed in detailthe various aspects of nanoindentation test apparatusthe data analysis and the application of nanoindentationtechniques for determination of mechanical propertieswith special emphasis on thin films Here the origin ofthe variation of hardness with the loaddepth ofindentation is discussed in detail The use of indentationtechniques and results on new materials such asquasicrystals bulk metallic glasses and nanomaterials

are clearly illustrated The issues related to phasetransformation during indentation tests are discussedCurrent industrial practices employing indentationtechniques are mentioned

Indentation size effectThe hardness obtained from the geometrically similarindenters ie conical or pyramidal (eg Knoop VickersBerkovich) at various loads is expected to remainunchanged as the strain during the indentation is constantunlike the spherical indenter However in a practicalsituation the hardness using similar indentations is foundto vary with the load The increase in hardness withdecreasing load which is known as indentation size effect(ISE) is often observed in metallic ceramic and inter-metallic materials (Fig 1a) (see for example Mott 1956Gane 1970 Gane and Cox 1971 Chen andHendrickson 1973 Boldt et al 1992 Poole et al 1996Grau et al 1998 Murthy et al 1999 Mukhopadhyayet al 2001 Gong and Li 2000 Ma and Clarke 1995Gao et al 1999a Elmustafa and Stone 2002 2003Swadener et al 2002 Paufler and Wolf 2003) There isalso a report of a decrease in the hardness with thedecrease in the load known as reverse ISE (RISE)(Sargent 1986 Upit and Varchenya 1973 Lim andChaudhuri 1999 Sangwal 2000) Earlier variousexplanations were offered to account for the ISE andRISE All these interpretations to some extent had someutility but they failed to account for this effectuniversally Turley and Samuels (1981) earlier suggestedthat ISE is due to abraded surface layers and oxides onthe indented surface Pethica and Taylor (1979) inter-preted that the ISE is due to chemical contaminationHowever Samuels (1986) proposed that the ISE is aresult of inadequate measurement capability of smallareas of indents and elastic recovery of indents Li et al(1993) suggested that ISE can be attributed to theindenterndashspecimen friction These variations of thehardness lead to uncertainty in the determination ofthe characteristic mechanical properties of materialsTherefore it is important to understand the mechanismsresponsible for the variations in the hardness There areseveral empirical and micro-mechanism based theoriesbased on elastic recovery energy balance strain gradientplasticity (SGP) surface roughness and friction toaccount for the ISE However it must be emphasisedthat the origin of the ISE and the RISE are not yet clearlyunderstood and it is still a debatable and controversialsubject A full hardness characterisation with a hardnessload curve is necessary to attach true significance tothe hardness value as a characteristic property of thematerial Various approaches to account for the origin ofthe ISE are discussed in the following sections

Meyerrsquos power lawPlots of hardness versus loadindentation data aresometimes fitted to Meyerrsquos law which was originallydeveloped to determine the work hardening capacity ofmetals in Brinell hardness tests The power law equationgenerally employed to analyse the loadndashindentation data(eg Mott 1956 Buckle 1965) can be expressed as

P~k1dn (3)

where P is the load k1 and n are materials constants and



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d is the indentation (diagonaldiameter) size This powerlaw equation is also referred to in the literature asMeyerrsquos law (Hays and Kendall 1973 Li and Bradt1991 Gong and Li 2000 Murthy et al 1999 Liu et al2003) It should be noted that Meyerrsquos equation wasoriginally developed for a spherical indenter where n isdirectly related to the strain hardening coefficient of thematerial (Meyer 1908) Onitsch (1947) extendedMeyerrsquos power law equation for nonspherical indentersand observed that in a macrohardness range n is 20whereas in a microhardness range n is less than 2

irrespective of the type of material When n is equal to 2the above power law equation is also quoted as Kickrsquoslaw in literature (Kick 1885) Figure 1a shows theindentation size effect while plotting the Vickers hard-ness of quasicrystalline material at various loads Thesedata were further analysed using Meyerrsquos power law(equation (3)) in a logndashlog plot (Fig 1b) from which thepower law index n is obtained as 1919 the index ofindentation size effect It has been pointed out earlier byseveral workers that Meyerrsquos constant k1 has a strangedimension of force(length)n which is dependent on thevalue of n (Li and Bradt 1991 Sargent 1986 Ghoshet al 2003) In order to resolve this problem Li andBradt (1991) introduced the reference indentation sizecorresponding to a load independent hardness whereasSargent (1986) suggested the use of 10 mm indentationsize corresponding to the standard hardness as areference Meyerrsquos power law is unable to determinethe true or load-independent hardness as it is continu-ously decreasing with load In fact in an extendedload range Meyerrsquos power law may not be able togive a good correlation with the experimental dataNevertheless it is convenient to handle the indentationdata Grau et al (1998) studied the strain ratedependence of the hardness of glass and found a goodcorrelation with Meyerrsquos law by analysing the depthsensing Vickers hardness data on glass for variousloading regimes However the differences in Meyerrsquosparameters under different loading regimes requirefurther investigation for better understanding of theimplication of Meyerrsquos law

Minimum resistance modelHays and Kendall (1973) were the first to suggest thatthere is a minimum resistance on the surface whichleads to the violation of Meyerrsquos law and hence the ISEis observed This is significant at lower load Thereforethe actual force acting during indentation is lower thanthat According to this model the forcendashdisplacementrelation can be expressed as

P~wzkhd2 (4a)

One can determine w (minimum resistance againstplastic deformation) and constant kh from the intersec-tion of the linear plot of P versus d or P versus d2Therefore the load-independent hardness can beexpressed as

HT~kPw

d2~kkh (4b)

where k is the indenter shape factor (eg 1854 forVickers hardness) This approach has been verified invarious metallic materials such as Al Cu and mild steelusing the Vickers and Knoop indenter However theminimum load obtained by this analysis was higher thanany practical value It was also found that for crystalsshowing a RISE the plot of experimental data does notgive a good fit Furthermore negative values of w for thedata at low loads do not give rise to any meaningfulinterpretation

Polynomial seriesBernhardt (1941) Mitsche (1948) Buckle (1965) andother workers (Frohlich et al 1977 Babini et al 1987)have approached the relation between P and d from a

1 a Variation of microhardness obtained from AlndashCondash

CundashSi single quasicrystals with load establishing the

indentation size effect error bar in each measurement

can be seen (after Mukhopadhyay et al 2001) b

variation of microhardness with load from AlndashCundashCo

poly-quasicrystals showing indentation size effect c

corresponding logarithmic plot from which the Meyerrsquos

index n is obtained as 1919 (after Murthy et al 1999)



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different point of view They have proposed a poly-nomial relation to correlate P and d as

P~X

i

aidi (5)

where i is a series of integers By limiting the number ofterms and assuming ao is zero for P50 one gets

P~a1dza2d2 By considering the contracted series asan energy balance because Pd is equivalent to theenergy or work so that a1d may be related to the surfaceenergy (a1d2) and the a2d2 term may be related to volumeenergy of deformation (a2d3) some form of physicalsignificance has been attributed to both a1 and a2 Basedon the consideration of energyndashbalance Bernhardt firstsuggested that the first term of the equation representsthe surface energy term and the second term representsthe volume energy term As a result of this seriesapproach the a1 represents the energy per indentationsurface area and the a2 represents the energy perindentation induced volume deformation From variousanalyses it appears that attaching true significance to a1

in terms of surface energy was not quite successful Itcan be pointed out that the ao term has been neglectedwithout sufficient justification However according tothe Hays and Kendall model ao is w (minimumresistance) and therefore it should be related to theload for initiation of permanent deformation Becauseao is so small it can be neglected in microindentationHowever to understand the nanoindentation processthe load for indentation of permanent deformation mayneed to be included which necessitates incorporation of

the effects of the indenter tip radius effect on the elasticlimit during indentation

Proportional specimen resistance (PSR)Li and Bradt (1991 1993) have extended the idea thatthe resistance offered by the surface is not constantthroughout the indentation size but rather dependent onthe indentation size They proposed a model based on aproportional surface resistance (PSR) model Accordingto this model load is expressed as

P~a1dza2d2 (6)

where a1 is related to the proportional resistance offeredby the surface of the specimen and a2 is related to thevolume Therefore the hardness can be expressed as ka2The effect of the first term is larger as the load isdecreased However at higher load the effect of the firstterm is negligible The plot of Pd versus d should yield alinear plot from which one can determine a1 and a2 andthus the true hardness This is exactly the same as thetruncated polynomial series indicated above Howeverthe physical interpretation of the two constants a1 and a2

is distinctly different In the Li and Bradt model (1991)a1 describes the Newtonian like specimen resistanceof the test specimen and a2 is Kickrsquos law coefficientrelating to the true hardness Ren et al (2002) haveinvestigated microhardness indentations on single-crystal MgO (001) along 110 in air for loads between0125 and 1001 kg and temperatures between 20 and600uC (Fig 2a) and also the influence of interfacialcoatings and lubricants They observed that the ISEdecreased with increasing test temperature and it wasunaffected by coating or lubrication The experimentaldata fitted equally well either the Meyerrsquos power-law andthe proportional specimen resistance (PSR) models(Fig 2ab) They have proposed qualitatively that theISE is controlled by the extent of elastic recoveryoccurring on removal of the load and shown thecorrelation of n (as in equation (1)) the index of ISEwith the hardness-to-elastic modulus ratio HE (Fig 3)

Energy balance approachQuinn and Quinn (1997) have investigated the varia-tion of Vickers hardness with indentation load for avariety of ceramic materials They observed that such

2 a Indentation data for MgO (001) along 110 plotted

according to Meyerrsquos power law in logarithmic form b

indentation data for MgO (001) along 110 plotted

according to the PSR model (after Ren et al 2002)

3 Correlation of power-law index n with hardness-

to-modulus ratio HE for a range of single-crystal cera-

mics (after Ren et al 2002)



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hardnessndashload curves exhibit a distinct transition to aplateau with constant hardness level and concluded thatthe transition in such curves corresponds to the intrinsichardness values of the materials These investigatorssuggested an energy balance model for the Vickersindentation process The model considers that theexternal work applied by the indenter is consumed inthe deformation and fracture process in the material Theload dependence of hardness has been considered by Liand Bradt (1996) and Dutta et al (2001) According totheir approach the measured diagonal of an indentationat a particular load is an apparent value which remainsassociated with an uncertain amount of relaxation Theextent of relaxation in the indentation diagonal occursdue to several possibilities such as crack formationdislocation activity elastic recovery at the tip of theindentation The true hardness can be evaluated as

H~Ho 1zde

d

2

(7)

where d5dozde is the apparent diagonal do is the truediagonal and de the relaxation in the diagonal afterremoval of the indenter

However recent works by several researchers haveshown that the linear relationship between Pd versus dmay only be valid in a narrow range of applied loadsWhen a relatively wider range of applied loads isconsidered the above equation was found insufficientA modification of the above equation has beensuggested (Gong and Li 2000 Quinn and Quinn1997) The above equation can be written as

Po~adozbHTd2o (8)

where Po and do are the load and indentation sizewithout any experimental error a is constant related tothe surface energy of the material and b is a constantrelated to the indenter Gong and Li (2000) have arguedthat experimental errors are usually inevitable inconventional hardness testing and therefore it shouldbe considered In general the experimental error in thetest has been divided into two classes (i) measurement ofindentation size and (ii) indentation load Consideringthe experimental error in both the load and theindentation size as r and d respectively ie by insertion

of P05Pzr and do5dzd into equation (6) equa-tion (7) may be rearranged in the following form

P~aoza1dza2d2 (9a)

where ao~bHTdzadr a1~2bHTdza a2~bHT Phas now been split into three parts It is important to notethat all of the parameters are functions of the experi-mental error and the true hardness However a2 is onlydependent on the true hardness Equation (9a) can beused for estimating the true hardness ie the energyneeded to produce the permanent deformation of a unitvolume Now a reasonable explanation can be offered forthe size effect in low hardness testing The first parameterao is related to the surface residual stress depending onthe surface preparation and is not really a materialsproperty The second parameter is due to the creation of anew surface by indentation and cracking the third isdependent on the volume of the indentation The aboveequation is easy to appreciate from the energy point ofview The energy applied is related to Pd and the energiesrelated to the surface phenomena and the volume arebasically a1d2 and a2d3 The argument is if the effectiveenergy or the force can be determined then the truehardness which is a characteristic property of thematerial can be easily determined and in that sense theISE can be tackled Here a2 is related to the true hardness

HA~k(aoza1dza2d2)

d2~k

ao

d2z

a1

d

zHT (9b)

In many cases it is found that the value of ao is so lowthat the above equation without ao will fit theexperimental data very well The physical significanceof the above equation lies in splitting the energy or theforce into the surface and volume related terms Thisalso takes care of the effect of microcracking The ratioof a2a1 has been suggested to be related to HEparameter of the materials It is important to mentionthat the above model is able to explain the ISE as well asthe RISE whereas the model by Bradt and Li will not beable to explain the RISE (Figs 4 and 5)

4 Indentation size-dependence of apparent hardness for

annealed mullite sample (after Gong and Li 2000)

5 Load dependence of apparent hardness for sample Ti

(CN) based cermet square symbols represent experi-

mentally measured data solid line represents predic-

tion of equation (8) and dashed line represents

prediction of equation (8) assuming a050 (after Gong

et al 2001)



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Combined approach Meyerrsquos power law andenergy balance modelRecently Mukhopadhyay (2005) combined both theMeyerrsquos law approach and the energy balance modeland analysed the indentation data The analysis has beendiscussed there The hardness for an indentation usingload P the corresponding indentation area A and theindentation size s can be written using equation (1) interms of indentation size Meyerrsquos constant and theindenter shape factor as

Hs~P

A~k

P

s2~

kk1sn

s2~kk1sn2 (10a)

From the above equation Meyerrsquos constant k1 can bewritten in terms of hardness as

k1~Hs

ksn2(10b)

Now inserting the expression for constant k1 inequation (1)

P~Hs

ksn2dn~

Hss2

k

d

s

n

(11)

or simply as P~KmDn where Km~ Hss2

k~Ps is defined

as the normalised Meyerrsquos constant and D~ ds

is defined

as the normalised indentation size By this transfor-mation one can overcome the dimensional problemencountered in the classical Meyerrsquos equation Now thisnew Meyerrsquos constant can be related to hardness or loadfor the indentation which can be defined at any lengthscale However as a natural choice an indentation sizes51 mm will be assumed so that one can recover Meyerrsquosequation in the sense of parameters but not in terms ofexact units and dimensions of Meyerrsquos constants Thenormalised Meyerrsquos constant will have a force dimen-sion So equation (9) can be transformed to a hardnessequation and can be written as (using s51 mm)

H~H1mDn2~kKm

s2Dn2~kKmDn2 (12)

Now one can summarise the relations obtained bynormalising the classical Meyerrsquos power law equationas

P~KmDn or P~PsDn (13a)

and

H~H1mDn2 (13b)

Using the notation of Li and Bradt (1991) for the criticalindentation size as do and critical load Pc fromequation (6a)

P~Pcd

d0

n

(14)

While comparing this equation with equation (17) of Liand Bradt (1991) it can be noted that both equations areidentical except for an extra (2n) factor which isassociated with the right-hand side of their equa-tion (17) After analysing their approach it can be seenthat this factor was erroneously incorporated in thenormalised Meyerrsquos equation proposed by them thoughthe actual numerical value may not significantly affectthe analysis However the normalised Meyerrsquos equation

in terms of critical load and indentation size should berepresented correctly by equation (14) above

The nature of equation (14) suggests that hardnesscontinuously decreases with the increase in loadsizeTherefore it cannot predict the transition from ISEregime to non-ISE regime In order to determine thetransition the true hardness HT based on the energybalance model is incorporated Initially at lower loadthe apparent hardness will give rise to ISE But after acritical load or indentation size HA will be equal to HTThe apparent hardness obtained from normalisedMeyerrsquos equation (13b) can be equated with the truehardness corresponding to the critical indentation sized Therefore from the above argument Meyerrsquosequation can be correlated with the energy balancemodel The condition of equality is as follows

HA~H1Dn2 HT~kc

therefore

H1Dn2~kc (15)

Now from the above condition the critical indentationsize d can be obtained after which ISE should cease toexist Using the value of H15kKms2 and the aboveequation has been rearranged as

d~Kms2

c

12n

~Km

c

12n

in mm as s~1 mm (16)

This is an important relation which correlates thenormalised Meyerrsquos equation and the energy balancemodel The implication of this equation suggests theexistence of a critical length scale related to the upperbound of the ISE Similarly the corresponding criticalload can also be determined

The indentation data obtained from decagonalquasicrystals AlCoCu (Murthy et al 1999) andAlCoNi (Liu et al 2003) and the intermetalliccompound Mg32(AlZn)49 (Mukhopadhyay et al 2004)have been analysed (Mukhopadhyay 2005) The truehardness and critical indentation size have been deter-mined Figure 6 shows the plot of the loadndashindentationand hardnessndashindentation data from Vickers microin-dentation experiments Both the Meyerrsquos equation andthe energy balance model are fit with the experimentaldata satisfactorily (Fig 6a) with a regression coefficient099 The true hardness (HT) was obtained from theenergy balance model in each case The details of thecoefficients are summarised as follows

(a) For the decagonal quasicrystalline materialAlCoCu (Murthy et al 1999)

Meyerrsquos law P~000631d19409 energy modelP~000358z00096dz00049d2

HT~18544|00049N

mm2~908 GPa

(b) For the decagonal quasicrystal AlCoNi (Liu et al2003)

Meyerrsquos law P~00061d19059 energy modelP~005800011dz00044d2

HT~18544|00044N

mm2~816 GPa

(c) For the intermetallic compound Mg32(AlZn)49

(Mukhopadhyay et al 2004)



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HT~18544|00016N

mm2~297 GPa

Now using equation (12) d values are 70 27 and36 mm for the AlCoCu and AlCoNi quasicrystals andfor the Mg32(AlZn)49 phase respectively This appearsto be reasonable and consistent with the trend of thehardness plot with the indentation size which can beseen in Fig 6b The intersection between Meyerrsquos curveof hardness and the true hardness line can clearly beseen The true hardnesses for the AlCoCu AlCoNi andMg32(AlZn)49 phases were found to be 901 816 and297 GPa respectively which also seem to be consistentThe present analysis lends strong support to theproposed approach for determining the critical indenta-tion diagonal by involving both Meyerrsquos power lawequation and the energy balance model Thus thenormalised Meyerrsquos equation proposed here can give riseto a better understanding of Meyerrsquos constant (Km) andits exponent (n) These parameters combined with thecoefficient of the energy balance model can predict thecritical indentation size after which ISE does not exist

Elasticplastic deformation (EPD) modelBull et al (1989) proposed a quantitative model toexplain the ISE often observed in the hardness responseof hard brittle material which is based on mixedelastic and plastic deformation whereas the plastic

deformation occurs progressively in a discrete manner torelieve stresses created by the elastic flexure of thesurface at the edges of the deformation Duringunloading of the indenter recovery of the elasticincrement of the deformation which proceeds each newband of plastic deformation results in an indentationappearing smaller than expected particularly as theindentation size decreases to approach the scale ofplastic deformation band spacing The model fitsobserved experimental data well and the analysis ofhardnesssize data in this way is shown to allow both fora bulk hardness value and a characteristic deformationband scale to be calculated for a given sample

It is proposed that this model is applicable to hardmaterials where the elastic deformation effects aresignificant and the yielding or cracking occurs atintervals forming visible lines at an average spacingcharacteristic of the material the grain size and thesurface finish As the contact area extends further underthe increasing load yielding occurs at the outer edgeswhere the tensile stresses of the surrounding elastic fieldcombine with the stretching imposed by the sloping facesof the indenter It seems reasonable to assume theaverage elastic recovery to be d and thus

dm~did (17)

The hardness Ho of the ideal plastic material where thedeformation is completely continuous is defined as

Ho~kPd2i (18)

where k is a constant and P is load For the proposedsystem of non-continuous deformation the aboveequation may be used to derive Hm from dm

Hm~kPd2m (19a)

Also equation (18) can be rewritten as

Ho~kP dmzdeth THORN2(19b)

Dividing equation (19b) by equation (19a) gives (byrearrangement)

Hm~Ho 1zd

dm

2

(20)

At high load dmampd Hm tends to Ho At lower load as dis the more significant fraction of dm the measuredhardness will increase The values of Ho and d may bedetermined by fitting experimentally determined valuesof Hm and dm Table 1 shows fitting parameters

Indentation-induced crackingLi and Bradt (1991) proposed that during loading thetest load is balanced by the total specimen resistancecomposed of four components due to (i) friction at theindenterspecimen facet interface (frictional component)(ii) elastic deformation (iii) plastic deformation and (iv)specimen cracking According to these authors fric-tional and elastic effects lead to the normal ISE whileindentation cracking contributes to the apparent hard-ness HA measured by a Vickers diamond indenter Itmay be written as

HA~l1K1P

d2

zK2

P5=3

d3

(21)

6 a Plot of variation of load versus indentation diagonal

obtained from Vickers microindentation experiment

(Mukhopadhyay et al 2004) of Mg32(AlZn)49 intermetal-

lic phase b experimental hardness data hardness

curve from Meyerrsquos equation and true hardness line

obtained from energy balance model are plotted

against indentation diagonal (after Mukhopadhyay

2005)



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where d is the indentation diagonal l1 K1 and K2 areconstants The constant K2 depends on the applied loadP while K1 is a geometrical conversion factor whosevalue depends on the indenter geometry For an ideallyplastic body HA is equal to the first part whereas forbrittle solids the second part is related to HA

Peng et al (2004) investigated the nanoindentationhardness of a commercially available soda-lime glass atetragonal ZrO2 polycrystal and a hot-pressed Si3N4 inthe peak load range from 75 to 500 mN and observedthe ISE This was further analysed using Meyerrsquos lawthe HaysndashKendall approach the proportional specimenresistance (PSR) model the elastic recovery model andthe modified PSR model It was established that (i)Meyerrsquos law provides a satisfactory description for theexperimental data for each material but cannot accountfor the origin of ISE (ii) the HaysndashKendall approachthe elastic recovery model the PSR model and modifiedPSR model yield meaningless values of the parameters inthe corresponding equations For each material the truehardness was also determined based on the PSR modelthe elastic recovery model and the modified PSR modelrespectively It was found that the true hardness valuesobtained from these different models are similar to eachother This similarity can be attributed to the similaritybetween the empirical equations employed in thesemodels A similar type of analysis and conclusions wasalso made by Sangwal (2000)

Friction and surface effectsInitial elastic resistance which is characterised by asharp fall in hardness above a very small indentation sizehas been identified in low load indentation tests of non-metals Such behaviour is consistent with a requirementfor a critical strain energy to trigger permanentdeformation or more probably cracking Atkinson(1995ab) has shown that in many cases friction hasbeen responsible for a marked indentation size effect inlow load testing of some metals and the magnitude ofthe effect has been associated with stain hardening Thisintrinsic form of a size effect has been related to thespecial deformation conditions of a plastic hinge at theperimeter of the indentation The principal factor inthe size effect in low load testing of Fe and Al has beenidentified as due to friction It is therefore reasonable tosuppose that the minimal size effects in these ultra microindentation tests could be a consequence of a particu-larly low friction condition It has been shown by Bobjiand Biswas (1999) that the surface roughness has asubstantial influence on the nanohardness irrespectiveof whether the bulk and surface mechanical propertiesare the same Y Wei et al (2004) have studied the ISEand attributed it to dislocation density theory as well assome environmental effects such as indenter tip curva-ture and surface roughness Zhang et al (2004) discussedthe role of plastic deformation of rough surfaces in thesize-dependent hardness They proposed a bearing ratiofor nanoindentation of rough surfaces During anindentation the work done can be separated into bulkwork and surface work The surface work causes theplastic deformation of an indented rough surface andthus dissipates energy which is necessary to form theimpression of the solids The energy dissipation occur-ring at the indented surface is among the factors thatcause the ISE at the micronanoscales The surface effect

predominates when the indentation depth is shallowThey have found good agreement between theoreticaland experimental results of the size-dependent hardnessindicating that the surface effect plays an important rolein size-dependent hardness

Gerberich et al (2002) proposed that the ISE can belinked to energy for the newly created surface andthe plastic strain energy dissipation and estimated thesurface work and volume work associated with theindentation Their analysis indicates that the totalsurface work is given by the product of the contact areaand the surface energy They observed that the ratio ofsurface work to plastic volume work is nearly constantfor a wide range of shallow depths and decreases rapidlywith increasing depth of penetration and consequentlygiving rise to ISE Zhang and Xu (2002) have studied thesurface effects on nanoindentation and introduced anapparent surface stress that represents the energydissipated per unit area of a solid surface in nanoinden-tation tests

Strain gradient plasticity geometricallynecessary dislocationsStelmashenko et al (1993) De Guzman et al (1993)Fleck et al (1994) Ma and Clarke (1995) Poole et al(1996) Nix and Gao (1998) McElhaney et al (1998)Gao et al (1999ab) Acharya and Bassani (2000) andHuang et al (2001) advocated strain gradient theory toaccount for ISE This theory assumes that the flow stressis related to the statistically stored dislocations andgeometrically necessary dislocations (Fig 7) Accordingto the strain gradient plasticity (SGP) model the straingradient plays an important role in plastic deformationIn the formulation of plasticity theory the constitutivelaw contains strain gradient as a variable and hencethere is an intrinsic length scale This theory is based onthe observation that gradients of plastic shear result inthe storage of the so-called geometrically necessarydislocations (GND) which affect the yield stress in a

7 Schematic diagram of geometrically necessary disloca-

tions created by rigid conical indenter dislocation

structure is idealised as circular dislocation loops

angle between surface of conical indenter and surface

plane of indented material is q and indentation depth

is denoted by h (after Qui et al 2001)



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similar manner as the common statistically storeddislocations The densities of the geometrically necessarydislocations being proportional to the strain gradientbecome appreciable when the relevant size of indenta-tion is small The indentation at low load means that thestrain gradient is higher as the strain under the indent isconstant (for example 8 in Vickers geometry) Thestrain gradient is directly proportional to the geome-trically necessary dislocation The shear stress t isrelated to dislocation densities as follows

t~CGbffiffiffiffiffiffiffiffiffiffiffiffiffiffirszrg

p(22a)

where G is the shear modulus b is the magnitude ofBurgers vector and C is a constant taken to be 13 byAshby (1970) rs and rg are the densities of thestatistically stored dislocations and GNDs Ma andClarke (1995) estimated the density of GNDs of theindentation of diagonal length d as

rgamp4c

bd(22b)

For metals the hardness is three times the flow stress sothe hardness can be approximately written in terms ofdislocation as

HampGbrs 1z4c

rsbd

12

(23a)

Obviously when rgamprs the above equation is dominantto control the hardness and it explains the increase inhardness as the load decreases However at higher load orindentation diameter the densities of GNDs may benegligible compared to the statistically stored dislocationsThe above equations can be written in the following formso that it can be tested by fitting the experimental data

H2~H2o 1z

a

d

(23b)

where a~ 4crsb

is constant for a certain material One can

plot H2 versus 1d and from the intercept the sizeindependent plastic hardness can be obtained and fromthe slope the dislocation densities can be determinedMa and Clarke (1995) have done this experiment on asilver single crystal in order to test the strain gradientplasticity theory GNDs are the dislocations which arenecessary to accommodate the geometry of plasticdeformation Such dislocations are required to createthe plastic indent in a microindentation process Thesedislocations act as obstacles to the statistically storeddislocations and cause additional work hardening of thematerial It is also important to mention that Ma andClarke (1995) developed a geometrical scaling model forISE They partitioned the applied indentation force intothe force on the flat surfaces and the force over the edgesand finally derived size-dependent hardness for variousindentation tips The fitting of the experimental datawith this geometric scaling model appears to be as goodas with the strain gradient plasticity (SGP) model Forsimplicity in the case of the SGP model it can beassumed that the indent is accommodated by thecircular loops of GNDs with the Burgers vectors normalto the plane of the surface The presence of these GNDscauses storage of additional defects and increases thedeformation resistance by acting as obstacles to thestatistically stored dislocations

Nix and Gao (1998) expressed the SGP model usingthe GNDs and Taylorrsquos dislocation work hardeningtheory for a geometrically similar indenter with thesize-dependent hardness in terms of indentation depth as

H2~Ho

ffiffiffiffiffiffiffiffiffiffiffiffiffi1z

h

h

r(24a)

where h is a characteristic length that depends on theindenter shape the Burgers vector and statisticallystored dislocation Using this model Nix and Gao(1998) developed a law for strain gradient plasticitywhich became the theoretical basis of the mechanismbased strain gradient (MSG) plasticity to explain theISE (Fig 8) Qui et al (2001) considered the intrinsiclattice resistance (Wo) which varies with lattice orienta-tion and modified the above equation as

H2~3sozHo

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

3s0

H0

2

zh

h

s(24b)

With this modified model Qui et al (2001) explained thedependence of ISE on the crystalline orientation whichwas reported by Stelmashenko et al (1993) in a Wcrystal Swadener et al (2002) have modified the aboveequation by adding a contact depth-dependent constantH1 as

H2~Ho


h

h

rzH1 (24c)

where H1 is a work hardening component representingthe increase in hardness from the onset of yielding to aneffective strain The modified equation was employed forstudying ISE in NaCl and LiF single crystals Elmustafaand Stone (2002 2003) have added a contact depth-dependent term Hf (which is similar to H1) representingthe hardening mechanism other than dislocations Theyhave observed that for indents shallower than 150 nm

8 Microindentation hardness data for single-crystal and

polycrystalline copper as well as for single crystal sil-

ver h is indentation depth H is microindentation hard-

ness and H0 is indentation hardness for large depths

of indentation NixndashGao relation is also shown for

each set of experimental data and it agrees well with

the microindentation hardness data (after Qui et al

2001)



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the slope of the plot decreased abruptly by a factor of10 in comparison with the microhardness and deepnanoindentation data resulting a bilinear behaviour Asmore and more experimental results were reported itwas found that the equation could not satisfactorily fitall the experimental results (Mukhopadhyay et al2005) There are attempts to add an extra term to takecare of extra terms due to hardening or softeningdepending on the material (Fig 9)

Dislocation mechanics classicalIn the MSG model Gao et al (1999a) extended Taylorrsquosdislocation model to include the strain gradient plasti-city In Taylorrsquos dislocation model the dislocations areconsidered as statistically stored As the size of theindent becomes smaller the gradient becomes steeperTaylorrsquos dislocation hardening model is modified toaccommodate GNDs Gao et al (1999ab) indicated thatthe strain gradient plasticity is applicable between 01and 10 mm When the indent is less than 01 mmdislocation mechanics dominates Several workers haveextended the classical theory of dislocations withoutinvoking the strain gradient plasticity theory to explainthe ISE and the RISE (Mott 1956 Gane 1970 Chenand Hendrickson 1973 Upit and Varchenya 1973Sargent 1986 Vitovec 1986 Sangwal 1989 Lim andChaudhuri 1999 Sangwal 2000) In the case of anindentation experiment with sharp indenter stressesalways build up under an indenter with a pointed tipsufficient for homogeneous nucleation of dislocation in aperfect crystal Therefore a change in the length ofdislocation sources and in the concentration of defects instressed volumes of different size cannot play thedetermining role compared to the conditions of disloca-tion movement under the indenter Upit and Varchenya(1973) suggested that the peculiarity in the dislocationmovement in the case of indentation is that dislocationloops nucleated under the indenter near the surface

expand and propagate into the crystal thereby causingthe ISE Chen and Hendrickson (1973) studied theindentation of Ag single crystals with a wide range ofloads and found both ISE and RISE below a certainload They observed the distinct differences in terms ofappearances of dislocation rosettes around the indentsThey concluded that the interaction of the dislocationpattern is different and can be held responsible for ISEand RISE Lim and Chaudhuri (1999) made anexperimental investigation of nanohardness of thepolycrystalline work-hardened and annealed oxygenfree copper (OFC) for different indenter loads Thework-hardened sample shows a three-stage behaviour ndashonce H decreases (I) and increases (II) and thendecreases again (III) This phenomenon cannot beobviously explained by the strain gradient plasticitymodel A three-stage qualitative model has beenproposed In stage I ndash at low penetration depth(150 nm) ndash dislocation loops are nucleated at a relativelyhigh shear stress value of about (G75) According to thismodel at relatively low penetration depths the nano-hardness reflects the shear stress value required fornucleation and expansion of dislocation loops Also inthis stage the earlier dislocation densities have littleeffect on nanohardness values The indenter penetrationincreases and consequently the numbers and diametersof dislocation loops become significant This is thebeginning of the second stage when the nanohardness ofthe work-hardened material may even increase withincreasing indenter penetration owing to dislocationinteractions For still larger indentations the third stagebegins when a sufficiently large number of dislocations iscreated around the indentation After the nucleation ofinitial dislocation loops under the indenter tip as theload is increased more and more dislocation loops willbe created the diameter of which correspond to thecurrent size of the indentation contact Then to increasethe indentation size further sufficiently high shear stress

9 Plot of experimental H versus d (mm) data obtained from microindentation experiment along with those predicted

based on SGP and modified SGP model including the extra softening term the modified SGP model shows a better

agreement compared to the SGP model (after Mukhopadhyay et al 2005)



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values will be required to cause the dislocation loops togrow in size and to glide It was suggested that the shearstress required to propagate a dislocation loop in copperis related to its radius L by

t~Gb

L(25)

where G544 GPa is the shear modulus of copper andb50225 nm is the Burgers vector of dislocationsTherefore with increasing radii of the dislocation loopsand with increasing indenter depth the shear stressrequired to glide them will decrease and consequentlyhardness will decrease with load and then finally will beindependent of load at larger load

It can be indicated that it is difficult to explainmaxima due to the RISE using the PSR elasticplasticdeformation (EPD) and SGP models It is understoodthat the average hardness increases with the dislocationdensity The appearance of maxima suggests that theindentation process is intimately connected with nuclea-tion and motion of dislocations The inherent defectstructures would obviously act as obstacles for themotion of the defects produced by nucleation It is clearfrom the literature that the RISE occurs only in thesingle crystals undergoing plastic deformationTherefore it may be concluded that the RISE occursonly in materials where plastic deformation is dominantIt is well known that in the initial stage (ie at low loadsor strain) plastic deformation of crystals mainlyinvolves the nucleation of dislocations in a particularslip system but with increasing strain dislocationmultiplication (by processes like cross slip and activityof Frank read source) takes place The latter process isaccompanied by work hardening The RISE is alsoexplained based on the effects of vibration and indenterbluntness at low load (Westbrook 1967 Hanneman andWestbrook 1986) the applied energy loss as a result ofspecimen chipping around the indentation (Banerjee andFeltham 1974 Feltham and Banerjee 1992) and thegeneration of radial or median cracks (Li and Bradt1996)

Sangwal (2000) suggested that at low loads when onlyone slip system is active active parallel glide planes arefew Therefore the nucleation of dislocations rapidlypropagates into the material without substantiallyexperiencing mutual interaction stress between themHowever with increase in load when the number ofparallel glide planes is increased the motion ofdislocations gliding along them slows down due to themutual interaction stresses between dislocations Thisleads to a slow increase in indentation depth withincreasing indentation pressure Consequently the

increase in H with h steadily becomes weaker until Hattains a steady value after some particular value ofload The RISE phenomena essentially take place whichreadily undergo plastic deformation It appears that theRISE can be caused by (i) the relative predominance ofnucleation and multiplication of dislocations and (ii) therelative predominance of the activity of either two sets ofslip planes of particular slip systems or two slip systemsbelow and above a particular load

Mechanics of indentationAmong the several theoretical models the three mostextensively discussed models will be highlighted hereThese are (i) the elastic indentation model (ii) the rigidperfectly plastic model and (iii) the spherical cavityexpansion model The behaviour of these three solidscan be realised from the stressndashstrain diagram in Fig 10

Elastic modelThe elastic indentation model was originally developedby Hertz (1882) Boussinesq (1885) and Sneddon (1965)independently In this model it is assumed that theindenter is rigid and the material of the specimensatisfies the linear elasticity theories (Fig 11) Thepressure distribution p(r) for the cone can be expressedas

p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)

where E is the elastic modulus v is Poissonrsquos ratio h isthe semiapical angle of the cone and a is indentationdiameter The mean contact pressure can be written as

pm~H~E cot h

2(1n2)(26b)

The solution for other wedge sphere and flat-endsolutions can be found in the work of Sneddon (1965)

10 Stressndashstrain diagram for a perfectly plastic b elasticndashplastic c real elasticndashplastic solids

11 Indentation of ideally elastic solids by conical indenter



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and Timoshenko and Goodier (1951) With the elasticrubber the indentation pressure is a direct measure of theelastic properties of the material One can see that thehardness is independent of indentation size However fora spherical indenter H will be a function of the load andthe radius of the curvature R of the sphere

H~P1=3

p

4E

3R(1n2

2=3

(26c)

It has been assumed that friction and the elasticdeformation of the indenter are negligible For elasticmaterial the indentation disappears after unloadingTherefore the depth can be measured from theinstrumented hardness testing equipment so that thearea can be found from the depth measurement

Rigid perfectly plastic model (slip-line fieldapproach)This model is based on the slip-line field theoryadvocated by Prandtl (1920) and later generalised byHill et al (1947) In this model the material isconsidered as rigid perfectly plastic ie no plasticdeformation occurs until a stress Y is reached Thehardness obtained by this theory is the upper bound ofthe hardness These solutions successfully predict thehardness for materials with high values of EY usingsharp indenters but not so successfully for materialswith low values of EY or for blunt indenters Thepressure p across the punch is uniform and has the value

p~H~cY~2k 1zp

2

(27)

Thus H526Y (Tresca criterion) and H53Y (von Misescriterion) In general one can write H5cY where c isthe appropriate constraint factor which will depend onthe geometry of the indenter and the interfacial frictionThe slip-line field for a two-dimensional wedge ofsemiapical angle h is shown in Fig 12 and the pressureacross the face of the indenter for frictionless indenta-tions is p5H52k(1za) where cos(2hndasha)5cos a(1zsin a) The slip-line field approach to plasticdeformation has been widely applied to metal formingmetal working and indentation hardness It was pointedout by Tabor (1986) that deformation patterns predictedby slip-line theory were not observed in indentationexperiments in metals However it was later recognisedthat the mode of deformation observed by Tabor (1986)closely agrees with that proposed by Mulhearn (1959)and Samuels and Mulhearn (1957) as compression mode

(Fig 13) This is perhaps because the metals are notrigid but elasticndashplastic materials

Spherical cavity expansion modelThe spherical cavity model (Fig 13) was first advocatedby Bishop et al (1945) and later developed by Marsh(1964) Hirst and Howse (1969) and Johnson (1970)Among all these Johnsonrsquos analysis has been the mostrecognised analysis for indentation of elasticndashplasticsolids The results are reasonably close to the experi-mental observations for materials with either very high orvery low values of EY An improved analysis comes fromJohnson (1970) The radial expansion model is thebottom half of the spherical cavity and therefore doesnot provide for any piling up of any displaced materialoutside the indentation Evidently the volume of theindentation is ultimately taken up in the elastic hinter-land so that the shape of the indenter must be involvedFor a cone of semiapical angle h the critical parameternow becomes (EY)cot h while for a spherical indenterthe corresponding quantity is (EY)(aR) The finalrelation for cones or pyramids is then

H

Y~

2

31z ln

E cot h

3Y

(28)

while for spherical indenters cot h is replaced by aR Thistreatment assumes that the material has a constant yieldstress Y and that there is no work hardening produced bythe indentation process itself According to Hillrsquos originalsolution the pressure in the cavity when the plasticndashelasticboundary is at a distance c from the centre is

P

Y~

2

3z2 ln

c

a(29)

This implies that the elasticndashplastic boundary coincideswith the boundary of the cavity itself (c5a) at P523Yand below this contact pressure the analysis fails and noplastic flow can occur However there is no case of plasticindentation occurring in any system for an indentationpressure less than Y On the other hand in the regionwhere p53Y equation (27) shows that c532 and it mustbe assumed that the elastic yielding of the hinterland nolonger influences the plastic flow of the material Thecontact pressure now corresponds to the classic theory fora rigid plastic solid However there is nothing in theexpanding cavity model to indicate that the indentationpressure has an upper limit of 3Y As noted by Taborthe expanding cavity model is a helpful and fairlyrealistic description of the indentation process in the

12 Slip-line field solution for indentation of rigid plastic solid by frictionless wedge of semiapical angle h



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lsquocompressionrsquo or radial flow mode Attempts have beenmade to improve it to give better numerical agreementwith experiment A more satisfactory approach involvesfinite element analysis and the newer technique ofboundary element analysis but even these techniqueshave limitations imposed by the theoretical assumptionsinvolved

Elastic and perfectly plastic modelRecently Yu and Blanchard (1996) developed ananalytical model of hardness for four major indentationtests They have determined the analytical relationshipfor calculating hardness from the uniaxial materialproperties and indenter geometry for a wide variety ofelastic and plastic materials The models are presentedhere in order to understand the mechanics of theindentation process These models provide a simplebut powerful method for relating hardness from onetype of hardness test with that of a different test orevaluating uniaxial properties from hardness measure-ments For example the hardness of a conical indenta-tion can be directly calculated by the followingprocedure if the uniaxial properties of the material andthe angle b of the indenter are known

H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)

where Cb~2ffiffi3p (284523757|102b) and 0ub(

375u To evaluate the uniaxial properties of a materialfrom the hardness measurements the yield stress can beevaluated from the following relation

Y~E tan b

2(1n2)Cbtanh1 2(1n2)H

E tan b(30b)

Determination of elastoplastic propertiesrelationship between hardness and elasticmodulusThe analysis of loadndashindentation data followingthe technique of Oliver and Pharr (1992) has been

extensively discussed by Bhushan and Li (2003) andOliver and Pharr (2004) and therefore it will not bediscussed here further Some other approaches toanalysing the loadndashindentation data will be discussedIt is important to mention that still the problem ofactual contact area in case pile-up and sinking-in has notyet been sorted out In order to obtain a more accuratevalue of hardness and Youngrsquos modulus there aresuggestions to use the actual contact area by imaging theindents using scanning probe microscopy (Lim andChaudhuri 1999) The total work (Wtot) of indentationand the reversible work Wp of indentation definedrespectively as the area under the loading curve and thatbetween the loading and unloading curves (Fig 14)have also been used for materials characterisation(Sakai 1993 Hainsworth et al 1996 Rother 1995Rother and Dietrich 1994 Cheng and Cheng 1998e1999 Faulkner et al 1998 Giannakopoulos andSuresh 1999) It is therefore relevant to review thework investigating the relationships between thesedifferent materials properties and the parameters obtainedfrom the indentation experiments Giannakopoulos andSuresh (1999) discussed three-dimensional finite elementsimulations of elastoplastic indentation along withVickers and Berkovich indentation experiments andprovided the following results (assuming the loadndashdepthrelation as P5Ch2 where C is the indentation curvature)

C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)

In the above equation sy and s029 are the yield strengthand stress corresponding to the characteristic plasticstrain of 029 for the indented material in uniaxialcompression The constants M157143 and M2521 forthe Vickers pyramid indenter with an included tip angleof 136u The corresponding values for the Berkovichindenter are M56618 and M2520875 with an included

13 Compression mechanism of indentation proposed by Mulhearn (1959) showing elasticndashplastic boundary and defor-

mation resembling expansion of spherical cavity into elasticndashplastic solid by internal hydrostatic pressure



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tip angle of 1306u The circular conical indenter alsofollows the same results as the Vickers or Berkovichdepending on the apex angle of the cone If PavWy fallsoutside the bounds of the equation the indenter responseis either elastic or elasticndashperfectly plastic By accountingfor the effects of strain hardening on pile-up and sinking-in and on the true contact area through three-dimensionalsimulations the following relationship between the Amax

and hmax has been derived for elastoplastic materials

Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)

This equation is a polynomial fit to the computationallydetermined values of Amaxh2 The effective Youngrsquosmodulus of the indenterndashspecimen system is defined as

E~1n2

Ez

1n2in

Ein

1

~1

cffiffiffiffiffiffiffiffiffiffiAmax

p dP

dh

(33)

The subscripts lsquoinrsquo (equation (33)) represent propertiesof the indenter and dPdh is the slope of the Pndashh curve ofthe initial stages of unloading from Pmax The constantc51142 for the Vickers pyramid indenter 1167 for theBerkovich indenter and 1128 for the circular indenter ofany included apex angle The ratio of the penetration hr

upon complete unloading to the maximum penetrationdepth hmax before unloading is indicative of the extentof plastic deformation and strain hardening such that

s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)

Elastoplastic finite element analysis of the sharpindenter also reveals that

hr

hmax

~1dpav

E(35)

where d55 for a Vickers pyramid indenter andd54678 for the Berkovich indenter the conicalindenter results are similar to the Vickers or Berkovichindenters depending on the included apex angle Thestep-by-step method has been suggested extractingthe materials properties Following this approachMukhopadhyay et al (2001) determined the Youngrsquosmodulus of quasicrystalline materials which matcheswell the data obtained by the ultrasonic technique

Energy-based approaches to the indentation of brittlematerials are proposed by Sakai (1993) analysing theinstrumented load indentation data The hysteresis loopenergy Ur which is dissipated during the indentationloadingndashunloading cycle is related to the true hardness Happarent hardness HA and the work of indentation c1The true hardness has its energy-derived meaning of theirreversible energy consumption to create a unit volumeof indentation of ideally plastic materials The relation-ship between Ur and the three half powers of indentationload P32 and between Ur and the volume of theindentation impression VI are used to separate the plasticcontribution from the complicated plasticelastic surfacedeformation processes in the indentation tests The linearrelationship of Ur versus P32 provides an importantexperimental technique for determination of true hard-ness H of the brittle materials The linear relationship ofUr versus V1 is available to the experimental determina-tion of c1 The relationship has been developed as follows

Ur~1

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

a0 tan2 y

s 1ffiffiffiffiffiHp P3=2 (36)

which enables one to evaluate the true hardness of thebrittle materials from an experimental relationshipTheoretical predictions obtained through this formalismwere experimentally confirmed The energy-basedapproach and its application to brittle materials canprovide significant and potential directions for studies ofthe loadndashindentation size effect machining-induceddamage brittleductile transition behaviour at elevatedtemperatures of ceramic materials as well as of thedislocation process and plasticity of ceramic singlecrystals Hv Vickers hardness with Meyer definition(indentation load divided by the projected area ofindentation impression) j ratio of residual indentationdepth and the depth at maximum load finite elementanalysis for elastoplastic Vickers indentation are con-ducted in which the effect of strain hardening onindentation behaviour is intensively examined A novelprocedure of graphical superposition is proposed todetermine the representative yield stress YR It isconfirmed that the concept of YR applied to elasticndashfully plastic solids is sufficient to describe the indenta-tion behaviour of elastoplastic solids with strain hard-ening However Taborrsquos representative strain (8) atwhich YR is described is only applicable to elastoplasticsolids with their ratio of Youngrsquos modulus E to yieldratio Y (EY) ranging from about 400 to 1000 The truehardness H as a measure of plasticity is estimated fromthe Meyer hardness HM and then successfully related toyield stress Y and the strain-hardening modulus Ep aswell as YR True hardnesses obtained from thisformalism have been displayed for materials inTable 3 The true hardness is always found to be muchhigher than the conventional hardness especially in the

14 Illustration of a conical indentation and b loading and

unloading curves where Wtot is total work Wp is irre-

versible work and We is reversible work Wtot5

WpzWe (after Cheng et al 2002)



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case of brittle materials However this approach has yetto be adopted in real practice

Using a scaling approach to the indentation problemsan approximate relationship between the ratio ofhardness to elastic modulus and the ratio of irreversiblework to total work has been proposed The work ofindentation and the geometry of indentation can bereferred in Fig 14 (Cheng and Cheng 1998andashe Chenget al 2002) The ratio of hardness and Youngrsquos moduluscan be obtained from measuring the work of indenta-tion Together with a well known relationship betweenthe elastic modulus initial unloading slope and contactarea a new method is then suggested for estimating thehardness and the modulus of a solid using instrumentedindentation with conical or pyramidal indenters In thefollowing the relationships are shown

dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot

is a dimensionless function Several

experimental data points from the literature usingBerkovich diamond indenters on Cu W Al fused silicaand sapphire are also shown to follow this closerelationship The finite elastic constants of the diamondindenter are taken into account using the reducedmodulus It has been further emphasised that furtherexperiments and modelling efforts will ascertain theaccuracy of this proposed method

Mencik and Swain (1994) have explored the relation-ship between WpWtot and hfhm By assuming that

respective loading and unloading curves are given byF~ahm and F~b(hhf )

l where a and b are functionsof materials properties and indenter geometry arelationship between WpWtot and hfhm is then given by

Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)

This discussion obviously suggests that a generalrelationship exists between WpWtot and hfhm and thisrelationship is explicitly independent of indenter geo-metry This relationship is also explicitly independent ofthe details of the material properties and the stress anddistribution with similar indenters

Using dimensional analysis and finite element calcula-tions several relationships that relate the features ofindentation loading and unloading curves to the hard-ness elastic modulus and work of indentation areproposed for elasticndashplastic solids These relationshipsprovide new insights into indentation measurements(Cheng et al 2002) Earlier Lawn and Howes (1981)studied the elastic recovery effect in the indentation ofseveral ceramic materials and steels By assuming thatthe respective loading and unloading curves are given byF~ah2 and F~B(h2h2

f ) where A and B are functionsof materials properties and indenter geometry arelationship between WpWtot and hfhm was obtained

F~B(h2h2f ) (39)

The results from this equation are plotted in Fig 15The finite element calculations and equation agree wellwith each other Finite element calculations reveal therelationships between final depth hardness and elasticmodulus and they are shown in Fig 16 It is obviousthat an approximately linear function exists between hfhm and HE for each indenter angle The relationshipscan be summarised as

hf

hm

~1lH

E where l~150 tan (h)z0327

for 600iexclhiexcl800 (40)

Table 2 Fitting parameters for observed scale dependence of hardness for various ceramic materials

Material Ho kg mm22 d mm q kg mm22 n

Sapphire (1012) 2129 134 4265 181Deranox (alumina pc) 1642 205 3839 179MgO (001) (sc) 791 258 2374 173Silicon (001) (pc) 699 397 3004 165SiC (001) (sc) 2513 192 6972 172REFEL SiC (pc) 2461 110 3764 190ZrO2 1216 158 2374 183

Parameters q and n refer to the usual ISE model of eqn (Meyerrsquos law) while Ho and d are derived from the new model described byequation (18) It should be noted that while q in the ISE model is hardness value standardised at unit indentation size Ho in the newmodel is large-scale macroscopic hardness (sc5single crystal pc5polycrystal) (Bull et al 1989)

Table 3 Density elastic modulus and hardness parameters (Sakai 1993)

MaterialBulk densityg cmndash3 E GPa HV GPa

True hardnessH GPa

Work of indentationc GPa

Apparent hardnessHA GPa

Al 271 747 039 040 061 065Cu 890 139 101 105 140 156MgO 352 309 491 746 490 681Si3N4 322 331 151 197 930 172SiC 322 448 237 540 970 221Glassy carbon 150 299 ndash 202 025 250



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It is important to point out that Marx and Blake (1997)have shown a linear relationship for conical indentationwith a semicone angle of 703u in elasticndashplastic solidswith bilinear uniaxial stressndashstrain relationships Chengand Cheng (1998e 1999) have shown a similar relation-ship for conical indentation in elasticndashplastic solids withpower law work hardening for a particular semiconeangle of 68u Several other workers have proposedmodels that link the two quantities (Lawn and Howes1981 Sakai 1993) In these models the degrees ofpiling-up and sinking-in of surface profiles were treatedas adjustable parameters The relationships between hfhm and HE are influenced by these parameters Lawnand Howes (1981) proposed the following relationship

hf

hn

2

~1 2cE

cH

tan h

H

E(41)

where cE and cH are parameters that are affected by thedegree of surface sinking-in or piling-up A reasonableagreement between the LawnndashHowes model and thefinite element results was found when cEcH is about118 However the above equation does not predict theapproximately linear relationship Future research isneeded to understand better the relationship between hfhm and HE From the studies of scaling and finiteelement analysis Cheng et al (2002) have obtained thefollowing relation using We5WtotndashWp

H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)

However the value for k obtained using this expressioncannot be applied to sharp cones (45u) owing to thevalidity range of l From this relationship the ratio ofHE can be obtained readily by measuring the workof indentation to obtain WeWtot Furthermore sincethe ratio of HE2 can be obtained from the initialslopes of the unloading curves (Joslin and Oliver1990 Hainsworth et al 1996 Cheng and Cheng1998abcde) the values of H and E can in principlebe obtained from the work of indentation and the initialunloading slope

Indentation studies of quasicrystalsQuasicrystalline phases (QC) of either icosahedral ordecagonal symmetry share an homologous brittle-to-ductile transition temperature (BDT) of TTm07 (Tm

melting temperature) (Bresson 1994) with the majorityof crystalline intermetallic compounds (Fleischer 1994Sauthoff 1995) This suggests that diffusion processesare strongly involved in plastic deformation It alsomeans that nucleation and movement of cracks shouldbecome a predominant mechanism in most of the QCphases when subjected to shear stress at room tempera-ture However due to the small ratio of indent-to-sample diameter indentation experiments are infact performed under confining pressure conditions Ithas been known for a long time that confining pres-sure techniques where uniaxial stress states are

16 Relationship between hf5hm and H5E (after Cheng

et al 2002)

15 Relationship between hf5hm and Wp5Wtot (after Cheng et al 2002)



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superimposed by hydrostatic pressure may well shiftthe BDT of crystalline materials down to even roomtemperature by suppressing crack growth (eg Castainget al 1981 Francois et al 1988 Androussi et al1989) Indeed this principle also seems to workwith QC Recently a permanent macroscopic plasticstrain of almost 20 has been achieved by Fikaret al (2001) after deformation of poly-quasicrystallinei-Al635Cu240Fe125 under isostatic pressure of 5 GPaand at a strain rate of 510 s21 at 300 K Importantlytheir TEM investigation of the microstructure did notreveal significant differences between room temperaturehigh confining pressure and high temperaturelowconfining pressure specimens The microstructure wasrather found to be characterised by either a tweedlike or a platelet like contrast and isolated dislocationsThe platelets could be shown to correspond to twopentagonal approximant phases (Fikar et al 2001)Indication for phase transitions as the vehicle ofpermanent deformation at room temperature were alsoreported by other workers For example Wu et al(2000) and Dong et al (2001) found a transition ini-Al62Cu255Fe125 to a bcc structure in the regionbelow scratches whereas Kang and Dubois (1992a)observed transitions under uniaxial compression ofi-Al635Cu245Fe12 and d-Al63Cu175Co175Si2

Another mechanism of plastic deformation at roomtemperature has been proposed by Wollgarten and Saka(1997) and Wollgarten et al (1999) Examining the areabelow a Vickers indent in i-AlPdMn by TEM theyconcluded contrary to Fikar et al (2001) that a poly-grained material with grain sizes starting from 10 nm upto 500 nm had been formed and deformation occurredvia grain boundary sliding At about 200uC dislocationgeneration was found However the mechanism of thisfragmentation process was not specified

Elastic propertiesThe reduced modulus Er has been derived for variousQCs from a number of indentation tests using thenanoindenter (Table 4) Examples of corresponding F(h)curves are displayed in Fig 17 Aluminium-based alloysexhibit higher moduli than others The surprisinglysmall modulus of dodecagonal NbndashTandashTe is of the sameorder of magnitude as elemental tellurium (471 GPaBrandes and Brook 1992) rather than of tantalum or

niobium (1857 GPa or 1049 GPa Brandes and Brook1992) Apart from the small effect in d-AlndashCundashCondashSi nopronounced elastic anisotropy has been found in any ofthe QC investigated in agreement with the conclusionsof Chernikov et al (1998) Reynolds et al (1990) andSpoor and Maynard (2001) Note that Er increases whenthe contact depth hc approaches the surface region(Fig 18) It becomes fairly constant for contact depths200 nm This obvious surface influence may arisefrom intrinsic as well as from extrinsic causes and is stillnot understood Future databases will have to considerthat

Hardness indentation depthDespite its physical foundation Meyer hardness H is nota simple material property Figure 19 displays values forH(F) over four orders of magnitude of the load F whichshow that hardness more than doubles when loweringthe load by a factor of 10 Though this phenomenon hasprincipally been known for quite a long time (cf Mott1956) hardness numbers are even today taken ascharacteristic of the material at least when specifyingthe (large) load and the shape of the indenter appliedDespite the problems connected with the significance ofhardness values as a material property the morepragmatic view is taken to use hardness as a quantityfor comparison and as a feature which ndash at sufficientlylow loads ndash will enable elementary processes of plasticdeformation to be detected For the sake of comparisonwith other data microhardness results will first bepresented (load 1 N) along with nanohardness data(load 01 mN) in Table 5 Al-based QC proved harderthan Zn-based The nanohardness is considerably higherthan the microhardness (see below) Anisotropies aregenerally small but reproducible Unresolved systematicdeviations seem to prevent results of different authorsfrom coincidence

The increase in hardness for FR0 or hR0 has becomeknown as a (positive) indentation size effect (ISE) (cfFig 19) It has been reported to occur in a wide range ofmaterials (eg Si TiB2 (Brookes 1983) MgO (Ren et al2002) Al (Atkinson 1995ab) and has been attributed toseveral mechanisms such as eg elastic recovery (Mott1956) indenterndashspecimen friction (Li et al 1993) orgeometrically necessary dislocations (Fleck et al 1994)ISE and its RISE are discussed in more detail The latter

Table 4 Reduced modulus Er at 300 K as derived from indentation experiments on surfaces with normal parallel torotation axes indicated

QC Er GPa E GPa

d-AlndashCondashCundashSi 164iexcl15 10 87 (poly QC) (Kang and Dubois 1992b)175iexcl15 2

d-AlndashCondashNi 195iexcl15 2 195 (Chernikov et al 1998) 177 10 (Li et al 2004)i-AlndashCundashFe (poly QC) 180iexcl20 172 (Vanderwal et al 1992) 168 (Tanaka et al 1996)

1402 (Fleury et al 2001) 131 (Lee et al 2001) 61268 (Kang and Dubois 1992b)i-AlndashMnndashPd 190iexcl12 5 182 (Tanaka et al 1996)

200 (Yokoyama et al 1993)i-DyndashMgndashZn 128iexcl15 5i-HondashMgndashZn 136iexcl15 5i-MgndashYndashZn 125iexcl15 5 62 (Edagawa et al 1998)

126 (Sterzel et al 2000)dd-NbndashTandashTe 35iexcl10

Paufler and Wolf (2003)2 5 10 surface normal || to two- five- or 10-fold symmetry axis



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model leads to the following form of the depthdependence of hardness H

H~H0


h

hc

s(43)

where hc is the contact depth H05H(hcRlsquo)52712 Gba(rs)

12 and h5(812)b a2 tan2 q (GH0)2 G is the shearmodulus b the Burgers vector modulus rs density ofstatistically stored dislocations q the angle between thesurface of the indenter and the surface of the specimen aconstant lt05 and h the length that characterises thedepth dependence of the hardness (Nix and Gao 1998)There is certainly a lower boundary of validity ofequation (43) due to the atomic structure ie hc1 nmAn upper limit for hc will be due to the formation ofcracks etc hence dependent on both the material and

the testing conditions Equation (43) may be rewrittenas

H~frac12H20 z(405ba2tan2 qG2=hc)1=2

(44)

From measurements of H(hc) parameters of equa-tion (44) may be obtained Fitting contact depths in therange 100 nmhc250 nm to equation (44) Paufler andWolf (2003) found H05113 GPa and h5408 nm forunimplanted d-AlCoNi (Fig 20) If b5b||50377 nm(Yan et al 1994 Kupsch et al 2001) then a shearmodulus G5478 GPa results (with a505 and q5377u)These values are consistent with those known from otherauthors The model of geometrically necessary disloca-tions fits well for contact depths hch At smaller hc

values (ie higher dislocation densities) H increases moreslowly than predicted contrary to metals (Lorenz 2001)Here the validity of the Taylor model may becomequestionable apart from probable surface influences

18 Reduced Youngrsquos modulus of dodecagonal

(Nb015Ta085)181Te112 at room temperature as a func-

tion of contact depth hc different point sets corre-

spond to different sites (after Paufler and Wolf 2003)

19 Meyer hardness versus load (indentation size effect ISE)

for i-Al70 Pd21Mn9 and i-Y10Mg30Zn60 at room tempera-

ture as derived from Vickers (F10 mN) and Berkovich

(F10 mN) indentation tests sample surface perpendi-

cular to fivefold symmetry axis (Wolf et al 2001)

17 F(h) curves obtained with Berkovich indenter for QC based on MgndashZn with Y or rare-earths as third component sur-

face normal fivefold reduced Youngrsquos moduli were derived from the unloading parts (after Paufler and Wolf 2003)



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Hardness H versus contact depth hc for unimplanted andfor implanted i-HoMgZn (Fig 21) and dodecagonal (dd)NbTaTe (Fig 22) showed similar behaviour

Time dependenceTo assess the influence of loading time t upon thehardness H measurements were performed with i-YMgZn varying t in the range 261022ndash102 s by a pulseindentation method (see Golovin et al 2002)

According to Fig 23 the change of almost four ordersof magnitude in t leads to a decrease of only 10 in HHence the contact area approaches its equilibrium valuepretty rapidly ie the dominating deformation does notrequire much time

Effect of implantationSpecimens of i-AlndashPdndashMn i-HoMgZn and d-AlndashCondashNihave been implanted using Bz ions (ion dose 1016 cm22

Table 5 Hardness at room temperature

QC HMeyer GPa HV GPa HMeyer GPa

d-AlCoCu 1017iexcl030 (Murthy et al 1999)d-AlCoCuSi 97iexcl03 2 (Paufler and Wolf 2003) 81iexcl04 (polyQC) (Kang and Dubois

1992b)168iexcl12 2 (Wolf andPaufler 1999b)

102iexcl04 10 (Paufler and Wolf 2003) 782iexcl010 10 (Mukhopadhyay et al2001)

146iexcl15 10 (Wolf andPaufler 1999b)

96 (Wittmann et al 1991)d-AlCoNi 109iexcl03 (Paufler and Wolf 2003) 87 10 94 2 (Takeuchi et al 1991) 114 10 (Li et al 2004)

864iexcl028 10 (Liu et al 2003)d-AlCrCuFe 83iexcl07 (Michel 1992)i-AlCuFe 785iexcl060 (Giacometti et al 1999)

98iexcl10 (Koster et al 1993)70iexcl04 (Kang and Dubois 1992b)

i-AlCuLi 51 (Takeuchi et al 1991)382iexcl016 (Bhaduri Sekhar 1987)410iexcl015 (Sainfort and Dubost 1988)

R-AlCuLi 48 (Takeuchi et al 1991)i-AlCuRu (poly-QC) 105 (Takeuchi et al 1991)i-AlPdMn 93iexcl03 (Paufler and Wolf 2003) 84iexcl04 5 (Deus et al 1997) 216iexcl20 (Wolf and Paufler

1999b)82iexcl03 2 (Deus et al 1997)69 2 71 5 (Yokoyama et al 1993)80 3 (Yokoyama et al 1993)(poly-QC) 93 (Takeuchi et al 1991)594iexcl002 (Wollgarten and Saka 1997)

i-YMgZn 57iexcl02 (Paufler and Wolf 2003) 43 (Edagawa et al 1998) 123iexcl12 2 (Wolf and Paufler1999b)

i-DyMgZn 66iexcl02 (Paufler and Wolf 2003)i-HoMgZn 54iexcl02 (Paufler and Wolf 2003)Al73Co27 105iexcl03 (Paufler and Wolf 2003)Al5Co2 106iexcl03 (Paufler and Wolf 2003)Al3Co 90iexcl03 (Paufler and Wolf 2003)Al13Co4 94iexcl03 (Paufler and Wolf 2003)Al9Co2 88iexcl02 (Paufler and Wolf 2003)

Meyer hardness from microhardness Load 1 NMicrohardness various loads of the order of 1 NMeyer hardness deduced from nanohardness data using a corner-of-a-cube indenter Load 01 mN23] 5105surface of two- three- five- or 10-fold symmetry

20 Hardness H versus contact depth hc for d-AlCoNi at

room temperature squares experimental data (after

Paufler and Wolf 2003)solid line model according to

Fleck et al (1994)

21 Hardness H versus contact depth hc for unimplanted and

for implanted i-HoMgZn (after Paufler and Wolf 2003)



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ion energy 180 keV) This treatment is known to giverise to atomic disorder Measurements by these authorsshowed that the hardness of all QC investigated at roomtemperature decreased due to irradiation This is similarto the observations of Burnett and Page (1984) with SiAccording to Monte-Carlo simulation the maximumpenetration depth of Bz ions is 500 nm for i-AlndashPdndashMnand 600 nm for i-HoMgZn (Paufler and Wolf 2003)Thus the indentation depth was well below the range ofimplantation Figure 24 shows the impact of implanta-tion upon the F(h) curve of i-HoMgZn The indenterpenetrated deeper into the implanted QC ie thehardness decreased For hardness values see Fig 21Moreover the ISE practically vanished The latter resultresembles the absence of ISE due to the free volumeeffect in amorphous materials which facilitates atomictransport Also the crack formation probability nearindents was found to be reduced and pile-ups werefound to be less pronounced (Fig 25) ie decreasinghardness seems to be accompanied by increasingtoughness

Pop-inLooking more closely at Fig 17 sudden displacementdiscontinuities during loading can be observed One in i-HoMgZn is more pronounced at about 2200 N othersare less well developed This phenomenon now knownas a lsquopop-inrsquo event has been found in various crystallinematerials and was attributed to the nucleation ofdislocation loops or cracks or to phase transitions underthe indentation stress (cf eg Page et al 1992) lsquoPop-insrsquo were observed in QC for the first time by Wolf et al(2001) Unloadingreloading cycles were performed(Fig 26) to check the first appearance of inelasticdeformation

Hardness decreased due to ion implantation howeverpop-ins continued to occur Having undertaken pureelastic cycles a further increase in load initiated the firstpop-in indicating the transition to an inelastic deforma-tion mode (cf Fig 27) As the penetration depth wasincreasing discontinuously the contact pressure pm

(hardness) decreased in the same way (for quantitativedetails cf Fig 27) The sharper the indenter the deeperwas the pop-in Figure 28 illustrates a typical series of

22 Hardness H versus contact depth hc for dd-NbTaTe

different point sets correspond to different sites

along the surface (after Paufler and Wolf 2003)

23 Dependence of hardness H of i-YMgZn on the loading

time t using a Berkovich indenter (load 40 mN) H is

the lsquouniversal hardnessrsquo which includes elastic defor-

mation thus H values are about 20 smaller than

the Meyer nanohardness used elsewhere in this work

(after Paufler and Wolf 2003)

24 Comparison of multi-indentation forcendashdisplacement

curves (Berkovich indenter) for unimplanted and B

implanted i-HoMgZn (fivefold surface) six unloading

reloading cycles have been performed for the same

indent before final unloading was done staircase like

multiple pop-ins were often found in QC (after Paufler

and Wolf 2003)

25 Comparison of Berkovich indents at room tempera-

ture in i-AlPdMn implanted (left) and unimplanted

(right) surface normal of fivefold symmetry lateral

scan width 4 mm height scale 200 nm no cracks

could be observed near the left indent and pile-ups

were found to be less pronounced (after Paufler and

Wolf 2003)



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pop-ins when a cube corner was used Figure 29indicates the corresponding pressure drops versuscontact depth While the force was increasing thecontact pressure decreased (ie the contact areaincreased) during loading Drops per pop-in amountedtypically to 1 GPa

Adopting the model of dislocation nucleation andormotion facilitated by hydrostatic stress it is concludedthat the first pop-in corresponds to the maximum shearstress to activate slip It is approximately given by theTresca stress (Johnson 1996) tTresca5046pm which islt58 GPa for i-YMgZn at room temperature As aresult of the absence of strain hardening (at elevatedtemperature) (cf Brunner et al 2000 Urban et al1999 2002) subsequent bunches of dislocations can beactivated at lower shear stress One of the consequencesof dislocation movement could be deformation bytwinning martensitic or related phase transitions tomeet the demands of volume change as observed byFikar et al (2001)

Influence of temperatureMeasurements of microhardness of d-AlCoNi d-AlCuCoSi and i-AlPdMn as a function of temperaturehave been performed up to lt800 K (Fig 30) HardnessH decreased with increasing temperature T in two stagessimilar to the hardness and yield stress of manycrystalline intermetallic compounds (Kirsten et al1964 Schulze and Paufler 1972) Obviously twodistinct mechanisms predominate at lower (l) and higher(h) temperatures The transition in H(T) occurs athomologous temperatures (TTm)0lt06 (Wolf andPaufler 1999ab 2001) Assuming thermally activatedprocesses below and above (TTm)0 the ratio of slopes[dHdT]l[dHdT]h gives an indication of the ratio ofactivation volumes v For i-AlPdMn it was found that[dHdT]l[dHdT]h5vhvllt012 If the low temperatureprocess was governed by a cluster friction mechanismand the high temperature process by recovery-controlleddeformation (Messerschmidt et al 2000) then a factorof 10 in the activation volumes seems reasonableUniaxial macroscopic deformation experiments on d-AlCoNi (Feuerbacher et al 1997) and i-AlPdMn (Geyeret al 2000) without hydrostatic pressure however havenot yet been done at TTm(07

26 Loadndashdisplacement curve of a multi-indentation test

on a fivefold surface of i-YndashMgndashZn using a sphero-

cone indenter loadingunloadingreloading cycles

with increasing maximum force lead to the occur-

rence of pop-ins and simultaneously inelastic hys-

teresis (after Paufler and Wolf 2003)

27 Contact pressure pm5FAc versus contact depth hc

derived from Fig 26 near first pop-in note that

hctotal displacement as a result of the pop-in pres-

sure drops from 13 to 8 GPa (after Paufler and Wolf

2003)

28 Loadndashdisplacement curve for i-YndashMgndashZn when a

cube-corner indenter is used a series of pop-in

events has developed with increasing load (after

Paufler and Wolf 2003)

29 Contact pressure versus contact depth calculated

from Fig 28 (after Paufler and Wolf 2003)



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Crack formationWhen indented by a microhardness tester at roomtemperature most quasicrystals like other brittlematerials develop cracks starting at the corners of theimpression pyramid A statistical analysis of crackdirections near indents on surfaces of both two- andfivefold rotational symmetry of Al70Pd23Mn7 revealedthat cracks exhibit a shear like habit and prefer anoverall propagation along planes of two- three- andfivefold symmetry rather than following the plane ofmaximum stress (Deus et al 1997) However lookingmore closely at the crack paths meandering phenomenaappeared (Wolf and Paufler 1999a) They are particu-larly frequent in the case of cracks with cleavage planesof twofold symmetry perpendicular to the surface Theirpropagation direction suddenly changed after typicallengths of 50ndash100 nm The angular alteration proved tobe a multiple of 72u thus permitting to continuepropagation along a plane of the same symmetry(Wolf et al 1997 Wolf and Paufler 1999b) Smallerlateral changes in the order of 1 nm could not beexcluded due to the limited resolving power of themethod The rough appearance of cleavage planes hasbeen predicted in principle by Trebin (1999) from two-dimensional numerical simulations They showed thatcracks can propagate by emitting dislocations whichleave a phason wall of reduced surface energy in theirtrail This trail avoids structural clusters

The occurrence of Palmquist type microcracks at thecorners of microindents has been used to evaluate themode I critical stress intensity factor (fracture tough-ness) of i-AlPdMn The value KIc5126 MPa m12 (Deuset al 1997) is an order of magnitude smaller than that ofconventional aluminium alloys (eg KIc517 MPa m12

for AlndashLindashCundashMgndashZr (Brandes and Brook 1992) butcomparable to other crystalline intermetallic andquasicrystalline phases such as eg KIc51 MPa m12

(Wittmann et al 1991) and 14iexcl01 (Mukhopadhyayet al 2001) for d-AlCuCoSi KIc5101iexcl009 MPa m12

(Liu et al 2003) 080iexcl008 MPa m12 (Murthy et al1999) and 081 MPa m12 (Li et al 2004) for d-AlCoNiA strong influence of surface symmetry upon themorphology of impressions has been reported by Wolf

et al (1997 2000) and Wolf and Paufler (1999bc)Moreover while the surface near an indent on anicosahedral QC showed typical pie chart like pile-ups itwas found rather smooth on decagonal QC In the caseof microhardness indents the apparent volume of thepiled-up elevations was about 30ndash40 larger than thevolume of the impression This is obviously due tolateral cracking and subsurface defect formationContrary to that the volume balance for sphericalindentation under the same conditions proved negativeie about 70 of the displaced material was lsquomissingrsquo(Wolf et al 2000) There is evidence for long-rangetransport of matter from the indent towards the outerregions thus elevating the overall surface level of theimpression environment and making precise AFMtopology difficult A proof of the conservation ofvolume after plastic deformation however could notyet be given Violations if any might be due to phasetransitions It is interesting to point out that Reiboldet al (2005) have demonstrated the existence ofnanocrystals near the nanoindent through high resolu-tion electron microscopy Mukhopadhyay et al (2006a)have discussed that in the absence of active dislocationsin quasicrystals the plasticity during nanoindentationcan be related to the nucleation and growth of shearbands (ie localised deformation) which is observed inAFM images as well as reflected in a loadndashdisplacementdiagram in the form of discontinuities (ie pop-in effect)Thus it is reasonable to argue that deformation duringroom temperature nanoindentation proceeds via shearband formation and subsequent phase transformationHowever more detailed studies in this direction arerequired to understand this aspect from the stability andcrystallography of this complex structure

Bulk metallic glassesIndentation techniques have been employed to deter-mine the mechanical properties of thin ribbons ofmetallic glasses (Sargent and Donovan 1982) Now anew generation of metallic glasses unlike thin ribbonscan be fabricated in bulk form by conventional foundrypractices (Inoue 2000 Johnson 2002) Extensiveindentation studies on various types of bulk metallicglasses (BMG) have been pursued in order to exploretheir structural applications Like crystalline materialsthe constraint factor for correlating hardness with yieldstress also seems to be interesting in the case of BMGThe applicability of the MohrndashCoulomb versus the vonMises criteria for plastic deformation during indenta-tion is discussed by several workers In the loadndashdisplacement indentation curve many interestingfeatures in the form of serrations ripples displacementbursts discontinuities pop-ins (all these features areidentical but differ in terms of scaling) have beenreported The importance of shear bands for plasticdeformation are emphasised Several studies also indi-cate the possibility of phase transformation duringindentation

Recently Schuh and Nieh (2004) have surveyed theindentation tests on bulk metallic glasses and reportedon the hardness measurements as well as the onset ofplasticity the role of shear banding structural changesbeneath the indenter and the rate dependent effectsmeasured by nanoindentation It has been reported thatyield strength Wy can be approximated following Taborrsquos

30 Meyer hardness versus temperature of d-Al73Co13Ni14

and i-Al70Pd21Mn9 between room temperature and

600uC (load F51 N) as derived from Vickers tests

(surface orientation in parentheses) (after Paufler and

Wolf 2003)



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relation (Tabor 1986)

H~Ksy (45)

where K is dependent on the indenter shape as well as onthe mechanical properties of the material beingindented especially on the strain hardening character-istics The value of K is found to be close to 3 in most ofthe amorphous materials including glassy materials andbulk metallic glasses (Fig 31) From Fig 31 a simplelinear relationship can be justified though there is ascatter which needs to be understood from themechanics of plastic deformation of glassy materials Itcan be mentioned that metallic glasses are non-strainhardenable materials and therefore they can be regardedas rigid perfectly plastic material The slip-line fieldtheory applicable for perfectly rigid plastic materialsuggests a simple relation between hardness and yieldstrength The experimental results shown in Fig 31 arebased on the Vickers or Berkovich indenters and therelevant numerical slip-line field analysis for variousconical indenter can be expressed as (Lockett 1963)

H~syffiffiffi

3p 141z272heth THORN (46)

where h is the half angle of the cone-shaped indenter inradians Introducing the value of 2570u for Vickers orBerkovich indenters in equation (2) and comparing withequation (45) the value K527 is obtained which isclose to the empirical value 3 Dao et al (2001) haveconsidered this problem in much more detail usingfinite element simulations and obtained the followingrelationship

H~2N1

Bsy N2z ln

Er

sy

(47)

where B N1 N2 are numerical constants and Er is thereduced modulus It appears that for metallic glassesErWy can be approximated to 45 Setting N1 and N2

equal to 94509 and 212433 respectively B can beobtained as 245 Considering all constants of equa-tion (47) and comparing with equation (45) the value ofK is obtained as 21 which is a lower bound of theexperimental data It is observed from experimental andvarious theoretical work that the constraint factor islower than the value for crystalline material Thereforethe material is harder than the crystalline material withthe same yield strength The reason has been argued onthe basis that the glassy material is pressure-sensitiveand the normal stress on a shear plane plays animportant role and it delays the yielding which causesthe shallower indentation compared to that predicted bythe von Mises criterion Therefore the MohrndashCoulombcriterion appears to be more applicable for the glassymaterial This can be expressed as

ty~kasn (48)

where ty is the shear stress on the slip plane at yield sn isthe normal stress acting on the shear plane and k and aare system-specific constants that define the shearstrength and the atomistic lsquofriction coefficientrsquo of theglass respectively Vaidyanathan et al (2001) performedthe simulations run for the von Mises and MohrndashCoulomb criteria and showed the comparison with theexperimental data (Fig 32) It is clear from this workthat the von Mises criterion predicts greater depth at aparticular load whereas the MohrndashCoulomb criterionpredicts a lower depth which agrees well with theexperimental data with a5013 Though these results aresupporting the MohrndashCoulomb criterion still morestudies are required in this direction RecentlyRamamurty and co-workers (Patnaik et al 2004 Janaet al 2004ab Ramamurty et al 2005) have studied thedeformation characteristics and recognised with the help

31 Plot of hardness and yield stress for various types of

metallic glasses most of the data can be seen to lie

closely near the slope K53 though some theoretical

values of K are somewhat lower than the trend data

from various metallic glasses are compiled and

plotted by Schuh and Nieh (2004)

32 Comparison of theoretical and experimental data from

the work of Vaidyanathan et al (2001) for Berkovich

indentation on Zr-based bulk metallic glass experi-

mental data are found to agree well with those calcu-

lated by the MohrndashCoulomb yield criterion (after

Schuh and Nieh 2004)



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of their experimental evidence that the von Misescriterion is inappropriate for metallic glasses eventhough some earlier reports supported this model(Kimura and Masumoto 1983 Bruck et al 1994) Ithas been suggested that a MohrndashCoulomb type criterionwhich takes into account the normal stress across theslip plane in addition to the local shear stress is bettersuited They have attributed this aspect to the highconstraint factor of metallic glasses such as Zr-basedglass They have also extended the expansion cavitymodel and found that this model agrees well with theirFEM results It is shown that the high plastic constraintfactor exhibited by metallic glasses at large indentationstrains is an outcome of their pressure-sensitive plasticdeformation

In metallic glasses a few very small and isolated pop-in events were observed by Wang et al (2000) in a Zr-based glass after which the phenomenon was reportedin many other alloys (Golovin et al 2001 Schuh et al2002 Schuh and Nieh 2003 Nieh et al 2002 Greeret al 2004 Greer and Walker 2002 Wright et al2001) In all these alloys the displacement bursts arevery small in the range of 1ndash25 nm and appear to occurwith higher frequencies at a low indentation rate(Fig 33) However Vaidyanathan et al (2001) did notobserve the displacement burst which could be due tothe high loading rate during indentation Various BMGssuch as Cu-based (Cu60Zr20Hf10Ti10) Zr-based(Zr65Al10Ni10Cu15) Pd-based (Pd40Ni10Cu30P20)and La-based (La55Al25Cu10Ni5Co5) were studied(Schuh and Nieh 2003)

It is now understood that the plastic deformation ofBMG is dependent upon structural dynamics Homo-geneous and heterogeneous deformation have beendiscussed in Zr and La based BMG in terms of thekinetic aspects of plastic deformation Heterogeneousdeformation (localised deformation) occurs at roomtemperature by the formation of localised shear bands

followed by rapid propagation of these bands andcatastrophic fracture It is known that in compressiontests the BMGs exhibit essentially no strain hardeningand plastic flow is serrated with many small load dropsFigure 34 shows the formation of shear bands which arecommonly observed on the surface and beneath thesurface while indenting bulk metallic glasses (Jana et al2004a) The nature of shear bands also depends on theindentation rate which can be clearly seen in Fig 35(Jiang and Atzmon 2003) Wright et al (2001) andGolovin et al (2001) used nanoindentation for the studyof serrated flow in BMGs and observed discretedisplacement burst (lsquopop-insrsquo) which they attributed tothe emission of shear bands It is also observed that alower strain rate promotes more prominent serrations ordisplacement bursts Chinh et al (2004) observed theplastic instability as serrated flow in bulk metallicglasses They noted that the phenomenon is similar tothat in crystalline alloys and manifested itself as discretesteps in the loadndashdepth indentation curve (Schuh andNieh 2003 Dao et al 2001 Vaidyanathan et al 2001Kim et al 2002 Wright et al 2001 Jiang and Atzmon2003 Benameur et al 2002 Nieh et al 2002 Golovinet al 2001) In crystalline solids the physical basis forthe appearance of plastic instabilities is the negativestrain rate sensitivity originating mainly from theinteraction with the precipitates

As loading rate is increased by several orders ofmagnitude the nature of serrated flow changes sub-stantially from step like Pndashh curves at the lowest rates toa very smooth parabolic curve at the highest rates(Fig 36) At rates in between these extremes Pndashh curvesexhibit serrations that appear more as fluctuations orripples than as discrete horizontal displacement bursts

33 Example of indentation loading (Pndashh) curves for a

variety of metallic glasses illustrating that discrete

pop-ins or flow serrations are common to many

amorphous alloys origin of each curve has been off-

set for clarity and several pop-in events have been

denoted by arrowheads (after Schuh and Nieh 2004) a As-cast alloy subjected to indentation load of 2500 gb As cast alloy subjected to indentation load of 2000 gexhibiting semi-circular shear band morphology

34 Morphology of subsurface deformation zones under-

neath Vickers indenter (after Jana et al 2004a)



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During nanoindentation with a constant loading ratethe displacement rate is a non-linear function of time tas is the indentation strain rate defined by 1h(dhdt) Atthe outset of each experiment the strain rate is very highbut decreases with depth as 1h approaches an approxi-mately constant value at very large depths The trends instrain rate are not smooth and monotonic but exhibitmany short peaks that appear to increase in size as theindentation proceeds These short bursts of rapiddisplacement correspond to the pop-in events exhibitedin the Pndashh curve Furthermore the magnitude of thepeaks is strongly affected by the indentation strain rateThese effects are shown for other types of BMGs such asZr-based and Pd-based systems

Golovin et al (2001) have earlier observed thatlocalised (inhomogeneous) plastic flow of metallicglasses can be accompanied by noticeable serrations inthe stressndashstrain curves They have reported the resultsof nanoindentation tests of bulky glassy PdCuNiP Thestrain serrations of the indenter can occur both towardsand against the applied force It is known thatnanocrystallisation can form in shear bands producedduring severe bending or high energy ball milling of thinribbons of a metallic glass Kim et al (2002) havedemonstrated experimentally that highly confined andcontrolled local contact at the ultrafine scale in the formof quasi-static nanoindentation of a bulk glassy metalalloy at room temperature can cause nanocrystallisationHowever other studies did not show any phasetransformation except for a subtle change in the TEM

contrast which may be due to the lower strain rate or tohydrostatic pressure Recently Mukhopadhyay et al(2006b) showed the evolution of nanocrystals duringindentation of Cu-based metallic glasses It was alsoreported that the propensity of nanocrystallisation ismuch more prominent near the indent region comparedto that of regions far from the indent However thisaspect of phase transformation from the kinetic point ofview for nucleation and growth of nanocrystals duringthe indentation period requires further theoretical andexperimental investigation

NanomaterialsSome interesting work on nanomaterials employingmicronanoindentation techniques is discussedAttempts are not made to discuss nanostructured thinfilm studies except to cite a few examples RecentlyMirshams and Parakala (2004) reported nanoindenta-tion experiments on electrodeposited nanocrystalline(size 19 nm) and commercially produced microcrystal-line (21 mm) Ni with three geometrically differentindenters The highest value was obtained from theconical indenter and the lowest from the Berkovichindenter The hardness measurements obtained fromBerkovich and cube-corner indenters showed a goodcorrelation with the strain gradient plasticity model TheISE was found to be strongly dependent on the indentergeometry and less on indentation depth There wasnegligible strain rate dependence (within the range of005ndash015 s21) of hardness to deeper depths and asignificant increase in the hardness due to the decreasein grain size The hardness obtained by Berkovich andcube-corner indenters has been shown to justify themechanism based strain gradient plasticity approachTheir results also indicated that the microstructurallength scale parameter is small in nanocrystallinematerial and therefore the strain gradient length ishigher in nanocrystalline nickel to produce the samestress in microcrystalline material The present studysuggested that a more detailed study is needed for theanalysis of deformation mechanism for conical indenter

35 AFM illumination image of indents produced at pene-

tration rates of a 100 nm sndash1 and b 1 nm sndash1 on amor-

phous Al90Fe5Gd5 (after Jiang and Atzmon 2003)

36 Indentationndashdepth load curve of a Pdndash40Nindash20P

BMG measured at different loading rates (after Schuh

et al 2002)



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Schwaiger et al (2003) examined strain rate sensitivity ina nanocrystalline Ni sample (grain size 40 nm) usingthe nanoindentation technique with different loadingrates and observed that nanocrystalline (nc) pure nickelwas found to exhibit a positive strain-rate sensitivity inflow stress an effect that was not observed in ultrafinecrystalline (ufc) and microcrystalline (mc) nickel Thisrate sensitivity was confirmed by the depth sensingnanoindentation and tensile testing The strain ratesensitivity was observed to be related to the grain sizeThe indentation hardness at 1 mm depth increased from57 GPa at a strain rate of 001 sndash1 to 64 GPa at a strainrate of 01 sndash1 A similar trend was observed at higherloads the hardness increased by almost 10 when theloading rate was increased from 38 to 1861 mN sndash1

Shaz et al (2002) have investigated the microhardnessmeasurement of TindashZrndashNi nanoquasicrystalline (40 nm)materials The indentation experiments were carried outat 25 50 100 and 200 g load which show the reverseindentation size unlike the micrometre size materialSEM observation clearly brings out the presence ofshear bands but no cracking was found (Fig 37) As hasbeen shown earlier microquasicrystalline and singlequasicrystalline samples always show microcracking(Murty et al 1999 Mukhopadhyay et al 2001) Inthe present system because of the formation ofnanoquasicrystalline phases the hardness as well as

ductility has increased The microhardness of nanoqua-sicrystalline alloy shows several interesting features Theaverage hardness of the alloy was found to be around36 GPa The absence of cracks around the indentedarea suggests a superior toughness of the material Thisis further supported by the evolution of shear bands asignature of localised deformation The hardness wasfound to increase with load initially and then todecrease This may be due to the easy flow of materialsin nanocrystallinenanoquasicrystalline material bygrain boundary sliding or to phase transformationLocalised flow indicates the evolution of shear bandsduring indentation loading analogous to deformationof metallic glasses (Sargent and Donovan 1982)However this is an interesting feature and may requirefurther study Mukhopadhyay et al (2004) havesynthesised nanoquasicrystalline phases in theMg32(AlZn)49 intermetallic alloy and compared themicrohardness data obtained from the nanoquasi-crystalline and microcrystalline phases of the samecomposition The microhardness values of the nano-quasicrystalline phases are reported to be more thanthose of the microcrystalline phases They found that theindentation size effect is more prominent in nanophasematerial compared to that of microphase material(Fig 38) The propensity of cracking is also less innanomaterials

The applicability of the HallndashPetch equation tomaterials with grain sizes smaller than 1 mm is aquestion of significant technological importance in viewof the recent advances in materials processing techniquessuch as rapid solidification vapour deposition andsputtering producing ultrafine grain sizes Experi-mental studies have indicated that the refining of grainsizes to the nanometre scale can improve mechanicalproperties For example an increase from 09 GPa inthe hardness of nickel was observed when the grain sizewas decreased from 125 mm to 12 nm (Hughes et al1986) Similarly an increase from 05 to 25 GPa wasachieved in nanocrystalline copper by reducing the grainsize from 50 mm to 6 nm (Nieman et al 1989) Theobserved increase in hardness with grain size which can

37 a Microindentation at 200 g absence of cracks shows

better toughness of material b magnified image of

shear bands developed during indentation c SEM of

microindentation of microcrystalline version of TindashZrndash

Ni alloy showing no shear bands (after Shaz et al

2002)

38 Plot showing microhardness varying with load in

Mg32(AlZn)49 for a RSP foil b as cast variation of

hardness with load (indentation size effect) is greater

in RSP foils containing nanophase materials (after

Mukhopadhyay et al 2004)



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be explained by the HallndashPetch equation has also beenreported by other workers (Valiev et al 1992 Jang andKoch 1999) An important aspect of strengthening bymicrostructural refinement is that the strength increaseneed not be at the expense of ductility However it isalso well documented in the literature that below acertain grain size the HallndashPetch slope may decreaseand even become negative (Fig 39) (Chokshi et al1989 Lu et al 1990 Fougere et al 1992) The type ofprocessing method used for varying the grain size suchas heat treatment as well as the presence of imperfec-tions such as triple junctions or porosity (Palumbo et al1990 Wang et al 1995) have been cited among thepossible causes for observed inverse HallndashPetch beha-viour Recently Farhat et al (1996) investigated hard-ness of nanocrystalline aluminium of various sizesranging from 15 to 100 nm using the ultra-microhard-ness indenter and observed that the experimental dataare well represented by the HallndashPetch relationshipFrom the present study it can be concluded that thegrain boundaries may be providing strengthening downto grain sizes as small as 15 nm

Nanophase TiAl with grain sizes in the range of 10ndash20 nm was synthesised by magnetron sputtering in aninert gas atmosphere and consolidated in situ undervacuum and Vickers microindentation tests werecarried out by Chang et al (1993) The Vickers micro-hardness of these samples at room temperature and at230uC revealed an inverse HallndashPetch relationship atsmall grain sizes 10ndash30 nm and the usual HallndashPetchbehaviour at larger grain sizes A small component ofindentation creep was also observed The maximumhardness is four times larger than that of a cast TiAlspecimen of the same composition The Vickers hard-ness was also observed to decrease rapidly withtemperature above 200uC Schuh and Nieh (2003) haveinvestigated the HallndashPetch breakdown regime byconducting hardness testing in nanocrystalline pure Niand NindashW alloys They observed the breakdown of theHallndashPetch relationship near d514 nm and d57 nm inthe case of nanocrystalline Ni and NindashW alloy andsuggested a diffusional creep mechanism is responsiblefor this inverse relationship

Recently Veprek et al (2000) developed multiphasenanocomposite coatings (3ndash20 mm thick) consisting ofnanocrystalline TiN amorphous Si3Ni4 and amorphous

and nanocrystalline TiSi2 nc-TiNa-SiNxa- and nc-TiSi2 on steel substrate by the chemical vapourdeposition (CVD) technique The load independentVickers microhardness from 80 to 105 GPa wasmeasured by the loadndashdepth sensing techniques forapplied loads between 30 and 200 mN and verified bymeasuring the size of the remaining plastic indentationusing SEM It has been observed that the ultrahardnessof 80ndash100 GPa depends on the amount of a- and nc-TiSi2 phases Veprek (1999) has reported that when thegrain size decreases below 10 nm the inverse HallndashPetchrelation is observed It has been concluded that a highhardness with a high fracture toughness and elasticrecovery is a simple consequence of such nanostructure

As pointed out by Koch and Narayan (2001) thereare many problems associated with measurement ofhardness as a function of grain size for nanocrystallinematerials If the grain size is varied by annealing thefinest grain samples for grain growth it is possible thatother structural andor compositional effects may occuron annealing Most of the experiments that report theinverse HallndashPetch effect have samples which exhibitclear artefacts or are at least questionable Howeverthere are reports where no obvious artefacts or otherproblems exist They have identified at least three suchapparently artefact free examples of the inverse HallndashPetch effect in hardness tests (Figs 40 and 41)Computer simulations also predict softening at grainsizes below some critical value Both the simulationsshow this critical grain size to be of the order of 10 nmor smaller While it has been difficult to assess thehardness of the smallest nanocrystalline samples unam-biguously it appears that the inverse HallndashPetch effect isreal Models which describe the deformation of nanos-cale materials should use only the few experimental setsof data that are clearly artefact free to test theirpredictions (Fig 42) (Koch and Narayan 2001)Obviously many more investigations are required toestablish the inverse HallndashPetch behaviour in nanocrys-talline materials and its mechanism

Phase transitions during hardness testsEarly observations of morphologic features of scratcheson glass by Klemm and Smekal (1941) led Madelung(1942) to conclude that local melting could arise due tothe highly localised introduction of mechanical energy

39 Hardness versus grain size dndash05 for nanocrystalline

Cu and Pd (after Chokshi et al 1989)


(FeCo)33Zr67 alloys (after Alves et al 1996)



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into the solid surface Although first-order phasetransitions of this type will probably not occur duringindentation at conventional speed pressure-inducedstructural phase transitions which have become knownfor numerous materials should nevertheless play a rolein hardness tests in particular when loadndashdisplacementcurves are considered In order to initiate a certain phasetransition the (hydrostatic or deviatoric) pressurebeneath the indenter should exceed the critical valuefor the phase transition but should not do so for thespontaneous nucleation of dislocations in the parentphase If the latter condition is violated plastic strainwill diminish the stress quickly below the critical valuewhich in turn may be lowered under deviatoric stresses(Gilman 1993) This implies that the presence ofdislocations and their ability to move under the stressfield of the indent will have an impact upon theoccurrence of phase transitions too Since the hardnessis of the order of the average contact pressure beneaththe indent materials that undergo a phase transitionduring indentation might be anticipated knowing theirphase transition pressures

It is therefore not surprising that semiconductorswere the materials where genuine phase transitions hadbeen directly observed for the first time These solidsexhibit both a very low initial dislocation density and ahigh Peierls stress for moving dislocations IndeedGridneva et al (1972) concluded from an athermalregion of the temperature dependence of hardness of Siand Ge that a phase transformation had occurred andEremenko and Nikitenko (1972) reported on theformation of the high pressure phase VII in siliconwhen indented at temperatures of 400ndash700uC Accordingto Hu et al (1986) this phase appeared underhydrostatic pressures of 40 GPa Comparing averagepressures achieved during hardness measurements withphase transition pressures Gerk and Tabor (1978) drewthe conclusion that a metallic phase of Ge Si anddiamond must have been formed beneath the indenterA phase transition of single-crystalline Si and Gethrough an electrically conducting state to a metastableamorphous phase after unloading Vickers andKnoop indentations (loading-unloading rate

1667 N s21 maximum loads 01ndash05 N) has beendetected by Clarke et al (1988) using electron diffractionand electrical conductivity measurements The authorsargued that remanent stresses inside the indentedmaterial give rise to this amorphous phase Afterscratching a small amorphous region was developedsurrounded by a region of high dislocation density(Minowa and Sumino 1992) Using a spherical indenterWilliams et al (1999) observed the transition of Si to aszlig-tin structure at about 133 GPa which comparesfavourably with 125 GPa found under hydrostaticpressure (Hu et al 1986) It seems worth mentioningthat transition pressures observed with indentationexperiments are not to be equated with equilibriumpressures because of kinetic effects They may beoverestimated on increase in Si for example by aminimum of 2 GPa (Hu et al 1986)

Phase transformations alter the shape of the loadndashpenetration depth curve They may show up in a lsquopop-inrsquoalong the loading curve (Williams et al 1999 Bradbyet al 2000 2001 2003) andor a lsquopop-outrsquo andor anlsquoelbowrsquo (Page et al 1992 Novikov et al 1996 Williamset al 1999 Domnich et al 2000 Bradby et al 20012003 Zarudi et al 2003 Ho et al 2004) on theunloading branch Whereas lsquopop-insrsquo often also indicatethe nucleation of dislocations (eg Lorenz 2001 Lorenzet al 2003) lsquopop-outsrsquo and lsquoelbowsrsquo seem to be morespecific of a phase transition They occur due to asudden expansion of the volume which has no counter-part in plastic slip Figure 43 gives an example indicat-ing that loadingunloading rates and maximum loadsdetermine which of these features will appear

Because of the small volumes involved with indentationvarious techniques have been applied to prove that a phasetransformation is correlated with those mechanical fea-tures These are mainly transmission electron microscopyelectron diffraction studies (Eremenko and Nikitenko1972 Clarke et al 1988 Wu et al 1999 Bradby et al2000 2001 Mann et al 2000 2002 Ge et al 2003 Zarudiet al 2003 Haberl et al 2004) Raman microspectro-metry (Kailer et al 1997 Lucazeau and Abello 1997Domnich et al 2000 Bradby et al 2001 Mann et al2002 Ge et al 2003 Zarudi et al 2003) differentialscanning calorimetry (Riontino and Massazza 2004)acoustic emission (Mann et al 2000) or electricalresistance measurements (Clarke et al 1988 Mann et al2000 2002 Bradby et al 2003 Ho et al 2004)

While silicon is by far the dominant goal of phasetransformation studies connected with indentation

41 Hardness versus grain size dndash05 for electrodeposited

Ni (after Erb 1995)


Zn made by laser ablation or mechanical attrition

(after Koch and Narayan 2001)



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other materials have been studied lsquoPop-inrsquo and lsquoelbowrsquoshaped parts of the loadndashpenetration curve along withhysteresis have been observed with quasicrystallinei-Y10Mg30Zn60 by Wolf and Paufler (2001) Wolf et al(2001) and Paufler and Wolf (2003) (Fig 26) Also

other authors reported on lsquopop-insrsquo with quasicrystals(AlndashCundashFe Dub et al 2001 2002 TindashZrndashNi Azhazhaet al 2004) Although the deformation behaviour ofquasicrystals is generally less well understood than thatof crystalline materials there are direct and indirectindications of phase transformations occurring in thistype of solid under localised load The former refer to i-AlndashCundashFe where a scratch-induced phase transitionfrom the quasicrystalline to a bcc phase was detectedusing electron diffraction (Wu et al 2000 Dong et al2001) Nanocrystallisation in quasicrystals has beenreported by Reibold et al (2005) while indenting theAlndashNindashCo decagonal single quasicrystals They foundthat the phase transformation is quite likely due to thevolume reduction during this indentation Phase trans-formations of the martensitic type giving rise to theshape-memory effect have been examined using bothspherical and pyramidal indenters The deformation ofspherical indents (tip radius 213 mm indents (8 mm) onNiTi was almost completely reversed by heating (Niet al 2002) At indentation depth less than 100 nmrecovery was found to be almost complete (Shaw et al2003)

Automated ball indentation (ABI) andfield indentation microprobe (FIM)techniqueThe industrial applications of nanohardness measure-ments were reviewed by Michler and Dommann (2001)Here in situ indentation techniques are discussed forevaluating mechanical properties in order to assess thedegradation and hence the remaining life of hightemperature components in a thermal power plant theoilgas industry nuclear industry aerospace and chemi-cal industries Monitoring the progressive changes inmechanical properties has been mandatory from thepoint of view of ensuring the structural integrity of thecomponents and safety of the operation Thereforenondestructive methods for mechanical characterisationare required for any materials aging and life manage-ment programme

A field indentation microprobe (FIM) apparatus wasdeveloped and patented by Haggag and Nanstad (1989)and Haggag et al (1989 1990) to evaluate nondestruc-tively the mechanical properties The FIM consists oftwo main units an automated ball indentation (ABI)unit for measuring the mechanical properties and anondestructive evaluation (NDE) unit (consisting ofultrasonic transducers and a video camera) for deter-mining the physical properties such as crack sizematerial pile-up around indentation residual stresspresence and orientation The main components of theFIM apparatus are shown schematically in Fig 44 Thetripod arrangement adjusts the ball indenter to beperpendicular to the surface of the structure or testspecimen The load can be applied by hydraulicpneumatic mechanical or any other means The ABItest is based on multiple indentations (at the samepenetration location) of a polished metallic surface by aspherical indenter (in the order of 1 mm diameter) Thedata are collected during the test and these are analysedautomatically by the computer attached to the equip-ment The applied loads and associated displacements(depth of penetration of the indenter into the test

a lsquoPop-outrsquo appears in unloading curve at slower load-ingunloading rates (1 mN sndash1) and higher maximumloads (50 mN) b lsquoElbowrsquo is formed at faster rates(3 mN sndash1) and lower loads (30 mN) c Mixed behaviouris also possible

43 Impact of phase transitions upon shape of loadndash

displacement curves as obtained by nanoindentation

of Si (after Domnich et al 2000)



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ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd

specimen) are measured using a load cell and a linearvariable differential transducer (LVDT) The loaddisplacement data (Fig 45) from each unloadingsequence are fitted with a first-degree polynomial andthe fit extrapolated to obtain the displacement corre-sponding to zero load These displacements and themaximum cycle load and displacement values from eachindentation sequence are used to determine the hardnessand yield strength to produce the true-stresstrue-plastic-strain curve and to estimate fracture toughnessThe ABI analyses are based primarily on elasticity andplasticity theories and some empirical correlations asdescribed by Haggag and Nanstad (1989) and Haggaget al (1989 1990) The agreement between ABI-deriveddata and those from standard ASTM uniaxial tensileand fracture toughness tests was demonstrated to besatisfactory The authors have suggested that thistechnique could also be extended for high temperaturestudies as well as indentation fatigue fracture and creepThe mechanical properties determined by the FIMapparatus include elastic modulus yield strengthLuders strain strain-hardening exponent Brinell hard-ness true-stresstrue-plastic strain curve up to 20strain and presence of residual stress The fracturetoughness is also estimated using the ABI measured flowproperties and a modified critical fracture strain modelFurthermore the shift in the ductile-to-brittle transitiontemperature for steel plates and welds for example dueto neutron irradiation embrittlement can be estimatedfrom the ABI measured changes in the materialrsquos yieldstrength and flow properties

Zarzour et al (1996) carried out microindentationhardness tests on HY-100 weldments by automated ballindentation on the weldments and obtained informationfrom the ABI tests in various regions of the heat affectedzone (HAZ) The results provide a relationship betweengrain size and corresponding stressndashstrain data acrossthe HAZ Byum et al (1997) investigated the through-the-thickness variations of mechanical properties inSA508 Gr3 pressure vessel steels using the automatedball indentation (ABI) test technique and evaluatedmechanical properties such as the yield strengthultimate strength flow curve and hardness fromindentation loadndashdepth curves Malow et al (1998)conducted ABI tests on nanocrystalline (nc) iron whichwas produced by mechanical attrition and compactedinto near fully dense samples followed by isothermalannealing at 800 K resulting in grain sizes between 15and 24 nm The ABI method proved useful in examiningthe mechanical properties of nc iron compared to that ofconventional hardness testing methods Stressndashstraincurves were obtained which indicated a low strainhardening at high flow stresses around 3 GPa and aroom temperature strain-rate sensitivity The deforma-tion pile-up around the indentations exhibited intenseplastic deformation in localised shear bands

Murty et al (1998) investigated the tensile andfracture properties of ASTM grade A36 steel using anondestructive StressndashStrain Microprobe system (SSM)developed on the basis of automated ball indentation(ABI) technique on as-received and cold workedmaterials at several temperatures in the range and at aconstant strain rate From their study they observed aclose agreement with the expected tensile and fractureproperties and concluded that ABI is a reliablenondestructive technique for determining tensile andfracture properties of materials Murty et al (1999) havestudied the gradients in mechanical and fracture proper-ties of SA-533B steel welds using the ball indentationtechnique The local stressndashstrain behaviours of differentmicrostructural zones of the weld were determinedGradients in the strength of the base metal weld metaland the different positions in the heat affected zone wereobserved to be consistent with the changes in themicrostructure Mathew et al (1999) have carried outABI tests on service aged cast stainless steel componentsand identified the embrittlement as well as increase in

44 Schematic diagram showing basic components of

field indentation microprobe (FIM) apparatus (after

Haggag et al 1989)

45 Samples of ABI test results (load versus depth using

a 076 mm diameter ball indenter) on 316L stainless

steel (SS) base metal (after Haggag et al 1989)



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lishe

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Man

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ublis

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(c)

IOM

Com

mun

icat

ions

Ltd

strength and decrease in toughness due to aging Theyhave further investigated the effect of aging on mechan-ical behaviour of Alloy 625 using the non-destructiveStressndashStrain Microprobe (SSM) system based on anautomated ball indentation (ABI) technique and invol-ving multiple indentations by a small spherical indenterat a single penetration location under strain-controlledconditions and evaluated mechanical properties such asyield strength ultimate tensile strength strength coeffi-cient and strain hardening exponent These studiesdemonstrate that ABI can be used as a non-destructivetechnique to determine changes in mechanical propertiesof nickel base alloy components due to aging Seok andMurty (2000) using ABI tests have explained thedecrease in fracture resistance (JndashR) curves in SA516steel due to increased strain hardening and generation oftensile residual stress at the crack tip during cyclicloading Murty and Mathew (2004) have studied theapplicability of an automated technique based on ballindentation for laboratory and field applications fordetermining the mechanical and fracture properties ofmaterials Das et al (2004 2005ab) have exploited thisball indentation technique extensively to study themechanical properties of many engineering and serviceaged materials They have observed that the pile-uparound the indent is an important parameter Based onthe pile-up height they have proposed a uniqueclassification scheme for materials

From all these reports it is becoming clear that ballindentation techniques can be used as a NDT tool forevaluating the mechanical properties of aged compo-nents so that the extension of their life can be assessedFrom these studies it should be emphasised that the ballindentation technique has good potential for directindustrial application and hence research in this direc-tion should be pursued

Concluding remarksIndentation hardness obtained at micro and nano leveltesting exhibits a large mismatch The differences inhardness are generally attributed to the indenter shapeeffect indentation size effect as well as to the pile-upeffect and hence it deserves further study for quantita-tive understanding The instrumentation for collectingthe data without visual observation through AFM iseasy to use and automated but the property evaluatedfrom such data may not be sufficiently accurateTherefore the need for visual observation has beenemphasised to have a better correlation between themicrohardness and nanohardness The strain gradientplasticity theory is quite successful for explaining theISE in ductile materials but there is an inherentdifficulty to deal with nanoscale deformation TheRISE which cannot be explained using SGP theoryneeds to be investigated in a more analytical mannerThe issue of indentation size effect will continue todominate the field of indentation until the mechanicsand mechanism of the deformation under indentation atvarious length scales are fully understood Theoreticalunderstanding of the material flow in quasicrystals bulkmetallic glasses and nanomaterials must first be devel-oped Therefore the issue of mechanical deformationusing indentation techniques in new materials is worthpursuing for resolving many unsolved problems Theindustrial use of indentation techniques for determining

the remaining life of components in service has provedto be indispensable compared to other conventionaltechniques However semiempirical approaches for theassessment of brittleness need to be established on thebasis of more fundamental analysis

Acknowledgements

The authors would like to thank ProfessorsS Ranganathan P Ramachandra Rao S LeleK Chattopadhyay SN Ojha GVS SastryU Ramamurty N Chakraborty I Manna BSMurty and Dr T Sudarshan and Dr RK MandalDr M Chandrasekhar and Dr VS Sarma for their keeninterest and useful discussions The authors would alsolike to thank Dr VC Srivastava Mr G NarayanaThakur Prasad Sunil Pal G Subba Rao Andre Belgerfor supplying useful references and stimulating discus-sions One of the authors (NKM) thanks the Alexandervon Humboldt Foundation Germany for a researchfellowship during which period a part of the work wascompleted Partial financial support from theDepartment of Science and Technology (DST) Indiais also gratefully acknowledged

References1 Acharya A Bassani JL (2000) J Mech Phys Solids 48 1565ndash

1595

2 Alves H Ferreira M Koster U Muller B (1996) Mater Sci

Forum 225ndash226 769ndash774

3 Androussi Y Vanderschaeve G Lefebvre A (1989) Philos

Mag A 59 1189ndash1202

4 Ashby MF (1970) Philos Mag 21 399ndash424

5 Atkinson M (1995a) J Mater Sci 30 1728ndash1732

6 Atkinson M (1995b) J Mater Res 10 2908ndash2915

7 Attinger C (1947) Ind Diam Rev 7 264

8 Auerbach F (1890) Nachr v d Konigl Gesellsch dWiss zu

Gottingen No16 518ndash541

9 Auerbach F (1891) Ann Phys Chem 43 61ndash100



12 Auerbach F (1900) Ann Phys 4F 3 108ndash115

13 Azhazha V Dub S Khadzhay G Merisov B Malykhin S

Pugachov A (2004) Philos Mag 84 983ndash990

14 Babini GN Bessosi A Glassi C (1987) J Mater Sci 22

1687ndash1693

15 Baker SP Ross CA Townsend PH Volkert CA

Boegesen P (eds) (1995) lsquoThin films stresses and mechanical

properties Vrsquo MRS Symp Proc 356 Pittsburgh PA Materials

Research Society

16 Baker SP Cook RF Coecoran SG Moody NR (eds)

(2000) lsquoFundamentals of nanoindentation and nanotribology IIrsquo

MRS Symp Proc 649 Pittsburgh PA Materials Research

Society

17 Banerjee R Feltham P (1974) J Mater Sci 9 1478ndash1482

18 Bekker I (1829) lsquoAristotelis meteorologicarsquo Berolini Typis

Academicis

19 Benameur T Hajlaoui K Yavari AR Inoue A Rezgui B

(2002) Mater Trans JIM 43 2617

20 Berkovic ES (1951) Ind Diam Rev 11 129ndash132

21 Bernhardt EO (1941) Z Metallkd 33 135

22 Bhaduri SB Sekhar JA (1987) Nature 327 609ndash610

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24 Bishop RF Hill R Mott NF (1945) Proc Phys Soc 57

149

25 Biswas SK Venkatesh K Bobji MS Sebastian KS (1996)

Trans Indian Inst Met 49 725ndash738

26 Blau PJ Lawn BR (eds) (1986) lsquoMicroindentation techniques

in materials science and engineeringrsquo ASTM STP 889

Philadelphia PA USA ASTM

27 Bobji MS Biswas SK (1999) J Mater Res 14 2259ndash2268

28 Boldt PH Weatherly GC Embury JD (1992) Mater Sci

Eng A155 251ndash255



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lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

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Villars

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31 Bradby JE Williams JS Wong-Leung J Swain MV

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34 Bresson L (1994) in lsquoLectures on quasicrystalsrsquo (ed F Hippert

D Gratias) 549ndash559 Les Ulis Les editions de physique

35 Brinell JA (1900) Baumaterialienkunde 5 276ndash280 294ndash297

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37 Brown LM Khan MY Chaudhri MM (1988) Philos Mag

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38 Bruck HA Christman T Rosakis AJ Johnson WL (1994)

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39 Brunner D Plachke D Carstanjen HD (2000) Phys Status

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40 Buckle H (1965) lsquoMikroharteprufung und ihre anwendungrsquo

Stuttgart Berliner Union Verlag

41 Bull SJ Page TF Yoffe EH (1989) Philos Mag Lett 59

281ndash288

42 Burnett PJ Page TF (1984) J Mater Sci 19 845ndash860

43 Byum TS Hong JH Haggag FM Farrel K Lee EH

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44 Calvert FC Johnson R (1859) Ann Phys Chem 108 575ndash

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45 Cammarata RC Nastasi MA Busso EP Oliver WC (eds)

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Symp Proc 505 Pittsburgh PA Materials Research Society

46 Castaing J Veyssiere P Kubin LP Rabier J (1981) Philos


47 Chang H Altstetter CJ Averback RS (1993) J Mater Res

7 2962ndash2990

48 Chen CC Hendrickson AA (1973) in lsquoThe science of hardness

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Metals

49 Cheng Y-T Cheng C-M (1998a) Philos Mag Lett 77 39

50 Cheng Y-T Cheng C-M (1998b) J Mater Res 13 1059

51 Cheng Y-T Cheng C-M (1998c) J Appl Phys 84 1284ndash

1291

52 Cheng Y-T Cheng C-M (1998d) Philos Mag Lett 78 115

53 Cheng Y-T Cheng C-M (1998e) Appl Phys Lett 73 614ndash

616

54 Cheng Y-T Cheng C-M (1999) J Mater Res 14 3493

55 Cheng Y-T Li Z Cheng C-M (2002) Philos Mag A 82

1821ndash1829

56 Chernikov MA Ott HR Bianchi A Darling TW (1998)

Phys Rev Lett 80 321ndash324

57 Chinh NQ Gubicza J Kovacs Z Lendvi J (2004) J Mater

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59 Clarke DR Kroll MC Kirchner PD Cook RF Hockey

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60 Dao M Chollacoop N VanVlieti KJ Venkatesh TA

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62 Das G Ghosh S Ghosh S Sahay SK (2005a) Mater Lett

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63 Das G Ghosh S Ghosh S Ghosh RN (2005b) Mater Sci

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66 Domnich V Gogotsi Y Dub S (2000) Appl Phys Lett 76

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67 Dong C Wu J Zhang L Dubois J-M Brunet P Zhou Q

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Society

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70 Dub SN Novikov N Milman YuV (2002) Philos Mag 82

2161ndash2172

71 Dutta AK Chattopadhyay AB Ray KK (2001) J Mater

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73 Elmustafa AA Stone DS (2002) Acta Mater 50 3641ndash3650

74 Elmustafa AA Stone DS (2003) J Mech Phys Sol 51 357ndash

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75 von Engelhardt W Haussuhl S (1960) Kolloid-Zeitschr 173

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76 Erb U (1995) Nanostructured Mater 6 533ndash538

77 Eremenko VG Nikitenko VI (1972) Phys Status Solidi (a)

14 317ndash330

78 Exner F (1873) Untersuchungen uber die Harte an

Krystallflachen Kk Hof- u Staatsdruckerei Vienna

79 Farhat ZN Ding Y Northwood DO Alpas AT (1996)

Mater Sci Eng A206 302ndash313

80 Faulkner A Tang KC Sen S Arnel RD (1998) J Strain

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81 Feltham P Banerjee R (1992) J Mater Sci 27 1626

82 Feuerbacher M Bartsch M Grushko B Messerschmidt U

Urban K (1997) Philos Mag Lett 76 369ndash375

83 Fikar J Bonneville J Rabier J Baluc N Proult A Cordier

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K746

84 Fischer-Cripps AC (2004) lsquoNanoindentationrsquo 2nd edn New

York Springer

85 Fleck NA Muller GM Ashby MF Hutchinson JW

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86 Fleischer RL (1994) in lsquoIntermetallic compounds Principles

and practicersquo (ed JH Westbrook RL Fleischer) Vol 2 237ndash

256 John Wiley amp Sons

87 Fleury E Lee JH Kim SH Song GS Kim JS Kim

WT Kim DH (2001) Mater Res Soc Symp Proc 643

K211ndash216

88 Fougere GE Weertman JR Weigel RW (1992) Scripta


89 Francois P Lefebvre A Vanderschaeve G (1988) Phys Status


90 Frankenheim ML (1829) lsquoDe crystallorum cohaesionersquo Diss

Vratislaviae

91 Frankenheim ML (1831) Z Phys Math 9 94ndash106 194ndash208

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methodorsquo Diss inaug Bonnae 1850 Also in Annd Physik u

Chemie 80 37ndash55

93 Frohlich F Grau P Grellmann W (1977) Phys Status Solidi

42 79ndash89

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95 Gane N (1970) Proc R Soc A 317 367ndash391

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97 Gao H Huang Y Nix WD Hutchinson JW (1999b)

Theory J Mech Phys Solids 47 1239ndash1263

98 Ge D Domnich V Gogotsi Y (2003) J Appl Phys 93 2418ndash

2423

99 Gerberich WW Gao H Sundgren J-E Baker SP (eds)

(1996) lsquoThin films stresses and mechanical properties VIrsquo MRS


100 Gerberich WW Tymiak NI Grunlan JC Horstemeyer

MF (2002) J Appl Mech Trans ASME 69 433ndash442

101 Gerk AP Tabor D (1978) Nature 271 732ndash733

102 Geyer B Bartsch M Feuerbacher M Urban K

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103 Ghosh S Das S Bandyopadhayay TK Bandyopadhyay

PP Chattopadhyay AB (2003) J Mater Sci 38 1565ndash1572

104 Giacometti E Baluc N Bonneville J Rabier J (1999) Scripta

Mater 41 989ndash994

105 Giannakopoulos AE Suresh S (1999) Scripta Mater 40

1191ndash1198

106 Gilman JJ (1993) Philos Mag B 67 207ndash214



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and semiconductorsrsquo New York Consultants Bureau

109 Golovin YuI Ivolgin VI Khonik VA Kitagawa K

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110 Golovin YuI Tyurin AI Farber BYa (2002) Philos Mag

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111 Gong J Li Y (2000) J Mater Sci 35 209ndash213

112 Gong J Miao H Zhao Z Guan Z (2001) Mater Sci Eng

A 303 179ndash186

113 Grailich J Pekarek F (1854) Sitzungsber d kaiserl Akademie

d Wiss Math-Naturw Cl Vienna 13 410ndash436

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115 Greer AL Walker IT (2002) Mater Sci Forum 386ndash388 77

116 Greer AL Castellero A Magde SV Walker IT Wilde

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117 Gridneva IV Milman YuV Trefilov VI (1972) Phys Status


118 Grigorovic VK (1976) Tverdostrsquo i mikrotverdostrsquo metallov Izd

Nauka Moscow

119 Grodzinski P (1951) Mikroskopie 6 118ndash120 J Sci Instrum

28 117ndash121

120 Haberl B Bradby JE Swain MV Williams JS Munroe P

(2004) Appl Phys Lett 85 5559ndash5561

121 Haggag FM Nanstad RK (1989) in lsquoInnovative approaches

to irradiation damage and failure analysisrsquo (ed DL Marriott

TR Mager WH Bamford) PVP Vol 170 41ndash46 New York

American Society of Mechanical Engineers

122 Haggag FM Nanstad RK Braski DN (1989) lsquoInnovative

approaches to irradiation damage and failure analysisrsquo (ed DL

Marriott TR Mager WH Bamford) PVP Vol 170 101ndash107

New York American Society of Mechanical Engineers

123 Haggag FM Nanstad RK Hutton JT Thomas DL

Swain RL (1990) in lsquoApplications of automation technology to

fatigue and fracture testingrsquo (ed AA Braun NE Ashbaugh

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American Society for Testing and Materials

124 Hainsworth SV Chandler HW Page TF (1996) J Mater

Res 14 2283ndash2295

125 Hanneman RE Westbrook JW (1986) Philos Mag 18 73

126 Hauy RJ (1801) lsquoTraite de mineralogiersquo Tome Premier 268ndash

271 Paris Louis

127 Hays C Kendall EG (1973) Metallography 6 275ndash282

128 Hertz H (1882) Verh d phys Gesellsch in Berlin 1 67ndash69

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129 Hill R Lee EH Tupper SJ (1947) Proc R Soc A 188 273ndash

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444

131 Ho S-T Chang Y-H Lin H-N (2004) J Appl Phys 96

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132 Hu JZ Merkle LD Menoni CS Spain IL (1986) Phys

Rev B 34 4679ndash4684

133 Huang Y Gao H Nix WD Hutchinson JW (2001) Anal J

Mech Phys Solids 48 99ndash128

134 Hughes GD Smith SD Pande CS Johnson HR

Armstrong RW (1986) Scripta Metall 20 93ndash97

135 Huygens CHR (1690) lsquoTraite de la lumierersquo 70 Gressner amp

Schramm Lipsiae

136 Inoue A (2000) Acta Mater 48 279ndash306

137 Jaggar TA (1898) Z Kristallogr 29 262ndash275

138 Jana S Bhowmick R Kawamura Y Chattopadhyay K

Ramamurty U (2004a) Intermetallics 12 1097ndash1102

139 Jana S Ramamurthy U Chattopadhyay K Kawamura Y

(2004b) Mater Sci Eng A 375ndash77 1191ndash1195

140 Jang JSC Koch CC (1999) Scripta Mater 24 599

141 Jannettaz P Goldberg M (1895) Assoc franc p lrsquoavanc d sc

9 Aug Abstract in Z Kristallogr 28 (1897) 103

142 Jiang WH Atzmon M (2003) J Mater Res 18 755ndash757

143 Johnson WL (2002) JOM 54 40ndash43

144 Johnson KL (1996) lsquoContact mechanicsrsquo Cambridge

Cambridge University Press

145 Johnson KL (1970) J Mech Phys Solids 18 115ndash126

146 Joslin DL Oliver WC (1990) J Mater Res 5 123ndash126

147 Journal of Materials Research (special issue) (1999) 14 2196ndash

2350 (2004) 19 1ndash396

148 Juskin NP (1971) lsquoMechaniceskie svojstva mineralovrsquo Izd

Nauka Leningrad

149 Kailer A Gogotsi YG Nickel KG (1997) J Appl Phys 81

3057ndash3063

150 Kang SS Dubois JM (1992a) Europhys Lett 18 45ndash51

151 Kang SS Dubois JM (1992b) Philos Mag A 66 151ndash163

152 Kaupp G Naimi-Jamal MR (2004) Z Metallkd 95 297ndash305

153 Kick F (1885) lsquoDas Gesetz der proportonalen Winderstande und

Wissenschaftsanwendungrsquo Leipzig Felix referred to in C Hays

and EG Kendall Metallography 6 (1973) 275ndash282

154 Kiely JD Jarausch KF Houston JE Russel PE (1999)

J Mater Res 14 2219ndash2227

155 Kim J-J Choi Y Suresh S Argon AS (2002) Science 295

654ndash657

156 Kimura H Masumoto T (1983) in lsquoAmorphous metallic

alloysrsquo (ed FE Luborsky) 187 London Butterworth amp Co

Ltd

157 Kirsten CH Paufler P Schulze GER (1964) Mtber Dt

Akad Wiss Math-Naturw Kl 51 Heft 5 Akademie-Verlag

Berlin 1ndash24

158 Klemm W Smekal A (1941) Naturwissenschaften 29 688ndash690

159 Knoop F Peters CG Emerson WB (1939) Natl Bur Stand

23(1) 39

160 Koch CC Narayan J (2001) Mater Res Soc Symp Proc

634 B511ndashB5111

161 Koster U Liu W Liebertz H Michel M (1993) J Non-

Cryst Solids 153amp154 446ndash452

162 Kupsch A Meyer DC Gille P Paufler P (2001)

Z Kristallogr 216 607ndash610

163 Lawn BR Howes VR (1981) J Mater Sci 16 2745ndash2752

164 Lee SM Kim BH Kim WT Kim DH (2001) Mater Res

Soc Symp Proc 643 K261ndashK266

165 Li H Bradt RC (1991) Mater Sci Eng A 142 51ndash61

166 Li H Bradt RC (1993) J Mater Sci 28 917ndash926


168 Li H Ghosh A Han VH Bradt RC (1993) J Mater Res

8 1028ndash1032

169 Li X Zhang L Gao H (2004) J Phys D Appl Phys 37

735ndash757

170 Lim YY Chaudhuri MM (1999) Philos Mag A 79 2979ndash

3000

171 Linne C (1768) lsquoSystema naturaersquo Tom III Holmiae

172 Linne C (1793) lsquoSystema naturaersquo Tom III 13th edn JF

Gmelin Lipsiae

173 Lips EMH (1937) Z Metallkd 29 339ndash340

174 Lips EMH Sack J (1936) Nature 138 328ndash329

175 Liu XB Yang GC Fan P (2003) J Mater Sci Lett 22

611ndash613

176 Lockett FJ (1963) J Mech Phys Solids 11 345

177 Lorenz D (2001) Thesis Martin-Luther Universitat Halle-

Wittenberg

178 Lorenz D Zeckzer A Hilpert U Grau P Johansen H

Leipner HS (2003) Phys Rev B 67 172101-1ndash4

179 Lu K Wei WD Wang JT (1990) Scripta Metall Mater 24

2319ndash2323

180 Lucazeau G Abello L (1997) J Mater Res 12 2262ndash2273

181 Ludwik P (1908) lsquoDie Kegelprobersquo Berlin Springer

182 Ma Q Clarke DR (1995) J Mater Res 10 853ndash863

183 Madelung E (1942) Naturwissenschaften 30 223ndash224

184 Malow TR Koch CC Miraglia PQ Murty KL (1998)

Mater Sci Eng 252 36ndash43

185 Malzbender J (2003) J Eur Ceram Soc 23 1355ndash1359

186 Mann AB Van Heerden D Pethica JB Bowes P Weihs

TP (2002) Philos Mag A 82 1921ndash1929

187 Mann AB Van Heerden D Pethica JB Weihs TP (2000)


188 Marsh DM (1964) Proc R Soc Lond A 279 420

189 Martens A (1898) lsquoHandbuch der Materialienkunde fur den

Maschinenbaursquo Vol I 241 Berlin Springer

190 Marx V Blake H (1997) Acta Mater 45 3791ndash3800

191 Mathew MD Murty KL Rao KBS Mannan SL (1999)

Mater Sci Eng A 24 159ndash166

192 Mcelhaney KW Vlassak JJ Nix WD (1998) J Mater Res

13 1300ndash1306

193 Mencik J Swain MV (1994) Mater Forum 18 277ndash288

194 Messerschmidt U Bartsch M Geyer B Feuerbacher M

Urban K (2000) Philos Mag A 80 1165ndash1181

195 Meyer E (1908) Z d Ver Deutscher Ingenieure 52 645ndash654

740ndash748 835ndash844

196 Michel M (1992) Diploma thesis University of Dortmund

197 Michler M Dommann A (2001) Z Metallkd 92 1035ndash1039

198 Minowa K Sumino K (1992) Phys Rev Lett 69 320ndash322



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd

199 von Mises R (1913) Nachrv d Konigl GesellschdWiss zu

Gottingen math-phys Klasse 582ndash592

200 Mirshams RA Parakala P (2004) Mater Sci Eng A 372

252ndash260

201 Mitsche R (1948) Osterr Chem Z 49 186

202 Mohs F (1812) lsquoVersuch einer Elementar-Methode zur natur-

historischen Bestimmung und Erkennung der Foszligilienrsquo Erster

Theil 9ndash10 Vienna Camesinische Buchhandlung

203 Mohs F (1822) lsquoGrund-Riszlig der Mineralogiersquo Erster Theil 374ndash

382 Dresden Arnoldische Buchhandlung

204 Moody NR Gerberich WW Burnhan N Baker SP (eds)

(1998) lsquoFundamentals of nanoindentation and nanotribologyrsquo

MRS Symposium Proc 522 Pittsburgh PA Materials Research

Society

205 Mott BW (1956) lsquoMicro-indentation hardness testingrsquo London

Butterworth Scientific Publications

206 Mukhopadhyay NK (2005) J Mater Sci 40 241ndash244

207 Mukhopadhyay NK Bhatt J Pramanick AK Murty BS

Paufler P (2004) J Mater Sci 39 5155ndash5159

208 Mukhopadhyay NK Sarma VS Sankaran S (2005) Trans

Ind Inst Metals 58 809ndash818

209 Mukhopadhyay NK Weatherly GC Embury JD (2001)


210 Mukhopadhyay NK Belger A Paufler P Gille P (2006a)

Philos Mag 86 999ndash1006

211 Mukhopadhyay NK Belger A Paufler P Kim DH (2006b)

Mater Sci Eng A to be published

212 Mulhearn TO (1959) J Mech Phys Solids 7 85ndash96

213 Muraki N Katagiri G Sergo V Pezzotti G Nishida T

(1997) J Mater Sci 32 5419ndash5423

214 Murthy GVS Ray AK Minz RK Mukhopadhyay NK

(1999) J Mater Sci Lett 18 255ndash258

215 Murty KL Mathew MD (2004) Nuclear Eng Design 328

81ndash86

216 Murty KL Miraglia PQ Mathew MD Shah VN

Haggag FM (1999) Int J Pressure Vessels Piping 76 361ndash369

217 Murty KL Mathew MD Wang Y Shah VN Haggag

FM (1998) Int J Pressure Vessels Piping 75 831ndash840

218 Ni W Cheng Y-T Grummon DS (2002) Appl Phys Lett

80 3310ndash3312

219 Nieh TG Schuh CA Wadsworth J Li Y (2002)

Intermetallics 10 1177ndash1182

220 Nieman GW Weertman JR Siegel RW (1989) Scripta

Metall 23 2013ndash2018

221 Nix WD Gao H (1998) J Mech Phys Solids 46 411ndash425

222 Novikov NV Dub SN Milman YuI Gridneva IV (1996)

J Superhard Mater 18 32

223 Oliver WC Pharr GM (1992) J Mater Res 7 1564ndash1583


225 Onitsch EM (1947) Mikroskopie 2 131

226 Ozakan CS Cammarata RC Freund LB Gao H (eds)

(2001) lsquoThin films stresses and mechanical properties IXrsquo MRS


227 Page TF Oliver WC McHargue CJ (1992) J Mater Res

7 450ndash473

228 Palumbo G Erb U Aust KT (1990) Scripta Metall Mater

24 2347ndash2350

229 Patnaik MNM Narasimhan R Ramamurty U (2004) Acta


230 Paufler P Wolf B (2003) in lsquoQuasicrystalsrsquo (ed H-R Trebin)

501ndash522 WileyndashVCH

231 Peng Z Gong J Miao H (2004) J Eur Ceram Soc 24

2193ndash2201

232 Pethica JB Taylor D (1979) Surf Sci 89 182

233 Petty ER (1971) in lsquoTechniques of metals researchrsquo (ed RF

Bunshah )Vol 5 Part 2 157ndash221 John Wiley

234 Pfaff F (1883) Sitzungsber d math-phys Classe d kb

Akademie d Wiss Munchen 13 55ndash68 and 372 Abstract in

Z Kristallogr 1885 10 528ndash531


Akademie d Wiss Munchen 14 255 Abstract in Z Kristallogr

1885 10 531ndash532

236 Philosophical Magazine A (special issue) (1996) 74 (2002) 82

1807ndash2231

237 Pohlenz F Hermann K Seemann R Menelao F (2001)

Z Metallkd 92 9ndash31

238 Poole WJ Ashby MF Fleck NA (1996) Scripta Mater 34

559ndash564

239 Poschl V (1909) lsquoDie Harte der festen Korper und ihre

physikalisch-chemische Bedeutungrsquo Dresden Steinkopff

240 Prandtl L (1920) Gottinger Nachr Math Phys KL 74 37

(1920) Nachr vd Konigl GesellschdWiss zu Gottingen

math-phys Klasse 74ndash85

241 Prandtl L (1921) Z Angew Math Mech 1 15ndash20

242 Qui X Huang Y Nix WD Hwang KC Gao H (2001)

Acta Mater 49 3949ndash3958

243 Quinn JB Quinn GD (1997) J Mater Sci 32 4331ndash4346

244 Ramamurty U Jana S Kawamura Y Chattopadhyay K

(2005) Acta Mater 53 705ndash717

245 Ray AK Das G Mukhopadhyay NK Bhattacharya DK

Dwarakadasa ES Parida N (1999) Bull Mater Sci 22 25ndash

32

246 Reaumur RAF de (1722) lsquoLrsquoart de convertir le fer forge en

acier et lrsquoart drsquoadoucir le fer fondursquo Paris Michel Brunet

247 Reibold M Belger A Mukhopadhyay NK Gille P Paufler

P (2005) Phys Status Solidi (a) 202 2267ndash2276

248 Ren XJ Hooper RM Griffiths C Henshall JL (2002)

Philos Mag A 82 2113ndash2120

249 Reynolds GAM Golding B Kortan AR Parsey JM

(1990) Phys Rev B41 1194ndash1195

250 Riontino G Massazza M (2004) Philos Mag 84 967ndash981

251 Rockwell SR (1922) Trans Am Soc Steel Treat 2 1013

252 Rosiwal A (1896) Verhandl d k k geolog Reichsanst Wien

17 475

253 Rother B Dietrich DA (1994) Phys Status Solidi 142 389ndash

407

254 Rother B (1995) J Mater Sci 30 5394ndash5398

255 Sainfort P Dubost B (1988) in lsquoQuasicrystalline materialsrsquo (ed

Ch Janot and J-M Dubois) Singapore World Scientific 361ndash

371

256 Sakai M (1993) Acta Metal Mater 41 1751ndash1758

257 Samuels LE (1986) in lsquoMicroindentation techniques in materi-

als science and engineeringrsquo ASTM STP 889 (ed PJ Blau BR

Lawn) Philadelphia PA USA ASTM

258 Samuels LE Mulhearn TO (1957) J Mech Phys Solids 5

125

259 Sangwal K (2000) Met Chem Phys 63 145ndash152

260 Sangwal K (1989) J Mater Sci 24 1128

261 Sargent PM (1986) in lsquoMicroindentation techniques in materi-

als science and engineeringrsquo AST STP 889 (ed PJ Blau BR

Lawn) 160ndash174 Philadelphia PA ASTM

262 Sargent PM Donovan PE (1982) Scripta Metall 16 1207ndash

1212

263 Sauthoff G (1995) lsquoIntermetallicsrsquo 501 Weinheim VCH

264 Schuh CA Nieh TG (2003) Acta Mater 51 87ndash99

265 Schuh CA Nieh TG (2004) J Mater Res 19 46ndash57

266 Schuh CA Nieh TG Kawamura Y (2002) J Mater Res

17 1651ndash1654

267 Schulze GER Paufler P (1972) Abhandlg Sachs Akad Wiss

Math-Naturwiss Kl 51 Heft 5 Akademie-Verlag Berlin 1ndash24

268 Schwaiger R Moser B Dao M Chollacoop N Suresh S


269 Seebeck A (1833) lsquoPrufungs-Programm des Colnischen

Realgymnasiumsrsquo Berlin

270 Seok CS Murty KL (2000) Int J Pressure Vessels Piping 77

303ndash311

271 Shaw GA Stone DS Johnson AD Ellis AB Crone WC


272 Shaz MA Mukhopadhyay NK Mandal RK Srivastava

ON (2002) J Alloys Comp 342 49ndash52

273 Sklerometrija (1968) Izd Nauka Moscow

274 Smith R Sandland G (1925) J Iron Steel Inst 111 285ndash294

275 Sneddon IN (1965) Int J Eng Sci 3 47ndash57

276 Spoor PS Maynard JD (2001) in lsquoHandbook of elastic

properties of solids liquids gasesrsquo Vol II (ed M Levy) 125ndash141

San Diego CA Academic Press

277 Stelmashenko NA Walls MG Brown LM Milman YV


278 Sterzel R Hinkel C Haas A Langsdorf A Bruls G

Assmus W (2000) Europhys Lett 49 742ndash747

279 Swadener JG Misra A Hoagland RG Nastasi M (2002)

Scripta Mater 47 343ndash348

280 Tabor D (1951) lsquoThe hardness of metalsrsquo Oxford Clarendon

Press

281 Tabor D (1986) in lsquoMicroindentation techniques in materials

science and engineeringrsquo ASTM STP 889 (ed PJ Blau BR

Lawn) 129ndash159 Philadelphia PA USA ASTM

282 Takeuchi S Iwanaga H Shibuya T (1991) Jpn J Appl Phys

30 561ndash562



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lishe

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ey P

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hing

(c)

IOM

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mun

icat

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283 Tanaka K Mitarai Y Koiwa M (1996) Philos Mag A73

1715ndash1723

284 Tertsch H (1949) lsquoDie Festigkeitserscheinungen der Kristallersquo

171ndash257 Vienna Springer-Verlag

285 Timoshenko S Goodier JN (1951) lsquoTheory of elasticityrsquo 2nd

edn New York McGraw Hill

286 Trebin H-R (1999) Mater Res Soc Symp Proc 553 337ndash343

287 Tymiak NI Daugela A Wyrobek TJ Warren OL (2004)


288 Turley DM Samuels LE (1981) Metallography 14b 275ndash294

289 Turner T (1886) Proc Philos Soc Birmingham 5 282ndash312

290 Upit GP Varchenya SA (1973) in lsquoThe science of hardness


Conrad) 135ndash146 Metals Park OH ASTM

291 Urban K Feuerbacher M Wollgarten M (2002) In

lsquoQuasicrystalsrsquo (ed J-B Suck M Schreiber P Haussler) 305ndash

318 Berlin Springer-Verlag

292 Urban K Feuerbacher M Wollgarten M Bartsch M

Messerschmidt U (1999) In lsquoPhysical properties of quasicrys-

talsrsquo (ed ZM Stadnik) 361ndash401 Berlin Springer-Verlag

293 Vaidyanathan R Dao M Ravichandran G Suresh S (2001)

Acta Mater49 3781ndash3789

294 Valiev RJ Chmelik F Bordeaux F Kapeiski G Bandelet

B (1992) Scripta Metall Mater 24 855

295 Vanderwal JJ Zhao P Walton D (1992) Phys Rev B46

501ndash502

296 Veprek S (1999) J Vac Sci Technol A 17 2401

297 Veprek S Niederhofer A Moto K Bolom T Mannling

HD Nesladek P Dollinger G Bergmaier A (2000) Surf

Coatings Technol 133ndash134 152ndash159

298 Vitovec FH (1986) in lsquoMicroindentation techniques in materials



299 Vinci R Kraft O Moody N Besser P Shaffer E (eds)

(1999) lsquoThin films stresses and mechanical properties VIIIrsquo MRS


300 Wang JG Choi BW Nieh TG Liu CT (2000) J Mater

Res 15 798ndash807

301 Wang N Wang Z Aust KT Erb U (1995) Acta Metall


302 Wei Y Wang X Zhao M (2004) J Mater Res 19 208ndash217

303 Werner GA (1774) lsquoVon den auszligerlichen Kennzeichen der

Foszligilienrsquo Leipzig SL Crusius

304 Westbrook JH (1967) in lsquoEnvironment-sensitive mechanical

behaviourrsquo (ed ARC Westwood NS Stoloff) 247 New York

Gordon and Breach

305 Westbrook JH Conrad H (eds) (1973) lsquoThe science of

hardness and its research applicationsrsquo Metals Park OH ASTM

306 Williams JS Chen Y Wong-Leung J Kerr A Swain MV

(1999) J Mater Res 14 2338ndash2343

307 Wittmann R Urban K Schandl M Hornbogen E (1991)


308 Wolf B Paufler P (1999a) Phys Status Solidi (a) 172 341ndash

361

309 Wolf B Paufler P (1999b) Surf Interface Anal 27 592ndash

599

310 Wolf B Paufler P (1999c) Microsc Anal Nov 25ndash27

311 Wolf B Paufler P (2001) Proceedings of the NATO Advanced

Study Institute on lsquoFundamentals of tribology and bridging

the gap between the macro- and micronanoscalesrsquo vol 10 (ed

B Bhushan) Keszthely Hungary August 2000 Academic

Publishers Dordrecht 549ndash556

312 Wolf B Bambauer K-O Paufler P 2001 Mater Sci Eng

A298 284ndash295

313 Wolf B Deus C Paufler P (1997) Surf Interface Anal 25

561ndash568

314 Wolf B Swain M Kempf M Paufler P (2000) J Mater Sci

35 723ndash734

315 Wollgarten M Saka H (1997) in lsquoNew horizons in quasicrys-

talsrsquo (ed AI Goldman DJ Sordelet PA Thiel JM Dubois)

World Scientific

316 Wollgarten M Saka H Inoue A (1999) Philos Mag A 79

2195ndash2208

317 Wright WJ Saha R Nix WD (2001) Mater Trans JIM 42

642ndash649

318 Wu JS Brien V Brunet P Dong C Dubois JM (2000)


319 Wu YQ Yang XY Xu YB (1999) Acta Mater 47 2431ndash

2436

320 Yan Y Zhang Z Wang R (1994) Philos Mag Lett 69 123ndash

130

321 Yokoyama Y Inoue A Masumoto T (1993) Mater Trans

JIM 34 135ndash145

322 Yu W Blanchard JP (1996) J Mater Res 11 2358ndash2367

323 Zarudi I Zhang LC Swain MV (2003) Appl Phys Lett 82

1027ndash1029

324 Zarzour JF Konkel PJ Dong H (1996) Mater Character

37 195ndash209

325 Zhang TY Xu WH (2002) J Mater Res 17 1715

326 Zhang TY Xu WH Zhao MH (2004) Acta Mater 52 57ndash

68



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So far the force applied was either not determined atall or defined only qualitatively It was Seebeck (1833)who built an instrument enabling well defined loads tobe applied Hardness was taken proportional to theminimum load applied perpendicular to the surfacewhich gave just a noticeable scratch when the surfacewas shifted laterally Franz (1850) introduced a similarmethod independently of Seebeck enabling the indenterto be moved across the fixed crystal The methods ofscratching were subsequently made on a more quanti-tative basis by evaluating the scratch geometry Usingessentially Seebeckrsquos device Grailich and Pekarek (1854)improved the mechanical parts and the optical inspec-tion of the scratches They called their instrument asclerometer and in doing so launched the branch ofsclerometry (eg Sklerometrija 1968) With the aid ofthis instrument Exner (1873) supplied data on theanisotropy of hardness in great detail Sometimesalternative definitions were applied as for examplehardness being inversely proportional to the lateral forcewhich is needed to move the specimen under a givennormal load (Franz 1850 Grailich and Pekarek 1854)In any case hardness values obtained this way wererather upper limits of this entity because they weretaken from the minimum load giving rise to a visiblescratch The visibility however strongly depends on thesurface state of the specimen andor the optical methodof recording the scratch To reduce this uncertaintyPoschl (1909) proposed to measure the change of thescratch width versus the change of load instead of thewidth itself Martens (1898) introduced a scratchhardness tester which has been widely used by mechan-ical engineers to measure the scratch width and thenormal load He took as hardness the normal loadwhich caused a given scratch width Recent work onnanoscratching shows a proportionality between lateralforce and (normal force)32 for a number of materials(Kaupp and Naimi-Jamal 2004) Further developmentof this idea will not be followed here because it leads tothe concept of wear which is not in the focus of thepresent review

In addition to scratching which is still used today tostudy wear resistance other approaches to hardnessassessment in the 19th century should be mentionedbriefly Examples are planing drilling and grindingPlaning was put forward by Pfaff (1883) who improvedthe accuracy of measuring the depth of a scratch byrepeating the procedure n times along the same scratchand moreover employing several diamonds side-by-sideat the same time Knowing the mass density of thespecimen and the area scanned the scratch depth couldbe calculated from the weight loss after planing Relativehardness values of various minerals or various planesand directions of the same mineral were then obtainedfrom the proportion of inverse planing depths keepingparameters such as load and speed to be constantDrilling with the aid of a cleaved diamond tetrahedronhas been recommended for hardness measurement byPfaff (1884) and Jaggar (1898) taking for hardness thenumber of rotations needed to penetrate the surface to agiven depth When grinding with a standardised powderhardness had been taken proportional to the timeneeded for the removal of a certain layer or inverselyproportional to the loss of volume during grinding(Jannettaz and Goldberg 1895 Rosiwal 1896) Later

on von Engelhardt and Haussuhl (1960) showed thatresistance against grinding varies rather with the specificfree surfaceinterface energy

Coming back to non-cutting methods an early attemptis mentioned first to find numerical values for the relativehardness of metals due to officers of the US OrdnanceDepartment who in 1856 pressed a pyramid on the metalunder test recording the volume of the indent and usingbronze as a standard (Turner 1886) Then Calvert andJohnson (1859) applied a steel cone to penetrate thematerial under investigation until a predeterminedindentation depth was achieved The load necessary wasused as a measure of hardness relative to the value of pigiron The subsequent development of indentation hard-ness testing methods was mostly stimulated by the rapidlygrowing demand of metallurgy

A milestone towards a quantitative concept ofhardness was the approach published by Hertz (1882)who defined the indentation hardness of solids ascontact pressure in a small circular area at the elasticitylimit This definition was applied to brittle as well as toductile solids However experimental verification latterproved difficult whereas the elasticity limit of brittlesolids could be determined from the pressure at the onsetof cracking This was the advent of contact mechanicsan important tool of present-day hardness testing (cfeg Johnson 1996) In order to take the onset of plasticdeformation explicitly into account Prandtl (19201921) introduced as decisive criterion the critical shearstress as the difference of two principal stresses underthe indenter using a proposal made by von Mises (1913)

On the basis of Hertzrsquos definition Auerbach (18901891) determined the hardness of brittle solids byoptically in situ monitoring the nucleation of cracksunder load in glasses and crystalline quartz Herecognised that the hardness defined by the firstappearance of a crack was subject to variation withthe curvature of the indenter Since cracks opened undertensile stress he concluded that the surface conditionmight also have influenced crack nucleation Extendingthe in situ measurements to ductile transparent solidssuch as rock salt calcite fluorite and others Auerbach(1892 1896) proposed that hardness be taken as Hertzrsquospressure at the onset of plastic flow indicated by analmost constant contact pressure When applying to(opaque) metals the area of contact was determinedemploying a blackened indenter (Auerbach 1900)

Using loads below 2 N hardness was termed micro-hardness though the distinction between macro- andmicrohardness is rather arbitrary Nevertheless a micro-hardness tester looked different mainly because of thehigher optical resolution required Various devices havebeen developed Lips and Sack (1936) and Lips (1937)introduced a small attachment to the light microscopewhich enabled indentations into small grains of multi-phase alloys to be made by moving the specimentowards a Vickers diamond pyramid coupled to aspring In the Vickers indenter the angle betweenopposite faces is 136u This angle results from thecondition that from the tangents drawn from the pointsof contact of an impression of diameter 0375D (where Dis the ball diameter as used in the Brinell test) to thecircumference of the indent the included angle be 136uSmith and Sandland (1925) found that the hardnessvalues obtained with the Vickers indenter of that angle



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contact area A(1)



Testingmethod

Shape ofindenter


Brinell He~2F

pD2 1 1(d=D)2frac12 1=2



VickersHV~

2F sin 680

d2~

18544F

d2



tltd7

KnoopHK~

F

A~

1440F

d2



tltd306

LudwikHL~

4F sin450

pd2~

09F

d2

t5d2

GrodzinskiHG~F=At~

6rF

pd2~

09F

tan(a=2)d3


HGN~277F

d3106




tlt019a



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ethr0







P~k1dn (3)




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P~wzkhd2 (4a)


HT~kPw

d2~kkh (4b)













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P~X

i

aidi (5)






P~a1dza2d2 (6)













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H~Ho 1zde

d

2

(7)



Po~adozbHTd2o (8)



P~aoza1dza2d2 (9a)


HA~k(aoza1dza2d2)

d2~k

ao

d2z

a1

d

zHT (9b)









et al 2001)



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Hs~P

A~k

P

s2~

kk1sn

s2~kk1sn2 (10a)


k1~Hs

ksn2(10b)


P~Hs

ksn2dn~

Hss2

k

d

s

n

(11)


k~Ps is defined


is defined


H~H1mDn2~kKm

s2Dn2~kKmDn2 (12)



and

H~H1mDn2 (13b)


P~Pcd

d0

n

(14)




HA~H1Dn2 HT~kc

therefore

H1Dn2~kc (15)


d~Kms2

c

12n

~Km

c

12n






HT~18544|00049N

mm2~908 GPa



HT~18544|00044N

mm2~816 GPa





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HT~18544|00016N

mm2~297 GPa





dm~did (17)


Ho~kPd2i (18)


Hm~kPd2m (19a)




Hm~Ho 1zd

dm

2

(20)



HA~l1K1P

d2

zK2

P5=3

d3

(21)








2005)



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lishe

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Man

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ublis

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(c)

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mun

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ions

Ltd















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lishe

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ey P

ublis

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(c)

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mun

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ions

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p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



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t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






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H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





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H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






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Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









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dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






contact area A(1)



Testingmethod

Shape ofindenter


Brinell He~2F

pD2 1 1(d=D)2frac12 1=2



VickersHV~

2F sin 680

d2~

18544F

d2



tltd7

KnoopHK~

F

A~

1440F

d2



tltd306

LudwikHL~

4F sin450

pd2~

09F

d2

t5d2

GrodzinskiHG~F=At~

6rF

pd2~

09F

tan(a=2)d3


HGN~277F

d3106




tlt019a



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


ethr0







P~k1dn (3)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




P~wzkhd2 (4a)


HT~kPw

d2~kkh (4b)













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


P~X

i

aidi (5)






P~a1dza2d2 (6)













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ho 1zde

d

2

(7)



Po~adozbHTd2o (8)



P~aoza1dza2d2 (9a)


HA~k(aoza1dza2d2)

d2~k

ao

d2z

a1

d

zHT (9b)









et al 2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


Hs~P

A~k

P

s2~

kk1sn

s2~kk1sn2 (10a)


k1~Hs

ksn2(10b)


P~Hs

ksn2dn~

Hss2

k

d

s

n

(11)


k~Ps is defined


is defined


H~H1mDn2~kKm

s2Dn2~kKmDn2 (12)



and

H~H1mDn2 (13b)


P~Pcd

d0

n

(14)




HA~H1Dn2 HT~kc

therefore

H1Dn2~kc (15)


d~Kms2

c

12n

~Km

c

12n






HT~18544|00049N

mm2~908 GPa



HT~18544|00044N

mm2~816 GPa





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


HT~18544|00016N

mm2~297 GPa





dm~did (17)


Ho~kPd2i (18)


Hm~kPd2m (19a)




Hm~Ho 1zd

dm

2

(20)



HA~l1K1P

d2

zK2

P5=3

d3

(21)








2005)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


ethr0







P~k1dn (3)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




P~wzkhd2 (4a)


HT~kPw

d2~kkh (4b)













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


P~X

i

aidi (5)






P~a1dza2d2 (6)













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ho 1zde

d

2

(7)



Po~adozbHTd2o (8)



P~aoza1dza2d2 (9a)


HA~k(aoza1dza2d2)

d2~k

ao

d2z

a1

d

zHT (9b)









et al 2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


Hs~P

A~k

P

s2~

kk1sn

s2~kk1sn2 (10a)


k1~Hs

ksn2(10b)


P~Hs

ksn2dn~

Hss2

k

d

s

n

(11)


k~Ps is defined


is defined


H~H1mDn2~kKm

s2Dn2~kKmDn2 (12)



and

H~H1mDn2 (13b)


P~Pcd

d0

n

(14)




HA~H1Dn2 HT~kc

therefore

H1Dn2~kc (15)


d~Kms2

c

12n

~Km

c

12n






HT~18544|00049N

mm2~908 GPa



HT~18544|00044N

mm2~816 GPa





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


HT~18544|00016N

mm2~297 GPa





dm~did (17)


Ho~kPd2i (18)


Hm~kPd2m (19a)




Hm~Ho 1zd

dm

2

(20)



HA~l1K1P

d2

zK2

P5=3

d3

(21)








2005)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




P~wzkhd2 (4a)


HT~kPw

d2~kkh (4b)













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


P~X

i

aidi (5)






P~a1dza2d2 (6)













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ho 1zde

d

2

(7)



Po~adozbHTd2o (8)



P~aoza1dza2d2 (9a)


HA~k(aoza1dza2d2)

d2~k

ao

d2z

a1

d

zHT (9b)









et al 2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


Hs~P

A~k

P

s2~

kk1sn

s2~kk1sn2 (10a)


k1~Hs

ksn2(10b)


P~Hs

ksn2dn~

Hss2

k

d

s

n

(11)


k~Ps is defined


is defined


H~H1mDn2~kKm

s2Dn2~kKmDn2 (12)



and

H~H1mDn2 (13b)


P~Pcd

d0

n

(14)




HA~H1Dn2 HT~kc

therefore

H1Dn2~kc (15)


d~Kms2

c

12n

~Km

c

12n






HT~18544|00049N

mm2~908 GPa



HT~18544|00044N

mm2~816 GPa





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


HT~18544|00016N

mm2~297 GPa





dm~did (17)


Ho~kPd2i (18)


Hm~kPd2m (19a)




Hm~Ho 1zd

dm

2

(20)



HA~l1K1P

d2

zK2

P5=3

d3

(21)








2005)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


P~X

i

aidi (5)






P~a1dza2d2 (6)













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ho 1zde

d

2

(7)



Po~adozbHTd2o (8)



P~aoza1dza2d2 (9a)


HA~k(aoza1dza2d2)

d2~k

ao

d2z

a1

d

zHT (9b)









et al 2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


Hs~P

A~k

P

s2~

kk1sn

s2~kk1sn2 (10a)


k1~Hs

ksn2(10b)


P~Hs

ksn2dn~

Hss2

k

d

s

n

(11)


k~Ps is defined


is defined


H~H1mDn2~kKm

s2Dn2~kKmDn2 (12)



and

H~H1mDn2 (13b)


P~Pcd

d0

n

(14)




HA~H1Dn2 HT~kc

therefore

H1Dn2~kc (15)


d~Kms2

c

12n

~Km

c

12n






HT~18544|00049N

mm2~908 GPa



HT~18544|00044N

mm2~816 GPa





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


HT~18544|00016N

mm2~297 GPa





dm~did (17)


Ho~kPd2i (18)


Hm~kPd2m (19a)




Hm~Ho 1zd

dm

2

(20)



HA~l1K1P

d2

zK2

P5=3

d3

(21)








2005)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ho 1zde

d

2

(7)



Po~adozbHTd2o (8)



P~aoza1dza2d2 (9a)


HA~k(aoza1dza2d2)

d2~k

ao

d2z

a1

d

zHT (9b)









et al 2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


Hs~P

A~k

P

s2~

kk1sn

s2~kk1sn2 (10a)


k1~Hs

ksn2(10b)


P~Hs

ksn2dn~

Hss2

k

d

s

n

(11)


k~Ps is defined


is defined


H~H1mDn2~kKm

s2Dn2~kKmDn2 (12)



and

H~H1mDn2 (13b)


P~Pcd

d0

n

(14)




HA~H1Dn2 HT~kc

therefore

H1Dn2~kc (15)


d~Kms2

c

12n

~Km

c

12n






HT~18544|00049N

mm2~908 GPa



HT~18544|00044N

mm2~816 GPa





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


HT~18544|00016N

mm2~297 GPa





dm~did (17)


Ho~kPd2i (18)


Hm~kPd2m (19a)




Hm~Ho 1zd

dm

2

(20)



HA~l1K1P

d2

zK2

P5=3

d3

(21)








2005)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


Hs~P

A~k

P

s2~

kk1sn

s2~kk1sn2 (10a)


k1~Hs

ksn2(10b)


P~Hs

ksn2dn~

Hss2

k

d

s

n

(11)


k~Ps is defined


is defined


H~H1mDn2~kKm

s2Dn2~kKmDn2 (12)



and

H~H1mDn2 (13b)


P~Pcd

d0

n

(14)




HA~H1Dn2 HT~kc

therefore

H1Dn2~kc (15)


d~Kms2

c

12n

~Km

c

12n






HT~18544|00049N

mm2~908 GPa



HT~18544|00044N

mm2~816 GPa





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


HT~18544|00016N

mm2~297 GPa





dm~did (17)


Ho~kPd2i (18)


Hm~kPd2m (19a)




Hm~Ho 1zd

dm

2

(20)



HA~l1K1P

d2

zK2

P5=3

d3

(21)








2005)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


HT~18544|00016N

mm2~297 GPa





dm~did (17)


Ho~kPd2i (18)


Hm~kPd2m (19a)




Hm~Ho 1zd

dm

2

(20)



HA~l1K1P

d2

zK2

P5=3

d3

(21)








2005)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



p(22a)


rgamp4c

bd(22b)


HampGbrs 1z4c

rsbd

12

(23a)


H2~H2o 1z

a

d

(23b)

where a~ 4crsb




H2~Ho


h

h

r(24a)


H2~3sozHo


3s0

H0

2

zh

h

s(24b)


H2~Ho


h

h

rzH1 (24c)









2001)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


t~Gb

L(25)







p(r)~E cot h

2(1n2)cosh1 (a=r) (26a)


pm~H~E cot h

2(1n2)(26b)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~P1=3

p

4E

3R(1n2

2=3

(26c)



p~H~cY~2k 1zp

2

(27)




H

Y~

2

31z ln

E cot h

3Y

(28)


P

Y~

2

3z2 ln

c

a(29)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



H~E tan b

2(1n2)tanh

2(1n2)YCb

E tan b

(30a)



Y~E tan b


E tan b(30b)



C~P

h2~M1s029 1z

sy

s029

M2z ln (

E

sy)

for 05cent

pav

syiexcl30 (31)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Amax

h2~9961264(1S)z10542(1S)2

22957(1S)3z15767(1S)4 with S~Pav

E

(32)


E~1n2

Ez

1n2in

Ein

1

~1


p dP

dh

(33)



s029sy

029E~10142

hr

hmax

0957hr

hmax

2

(34)


hr

hmax

~1dpav

E(35)



Ur~1

3


a0 tan2 y









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



dF

dh

h~hm

~2ffiffiffipp E

ffiffiffiffiAp

(37a)

H

E2~

4

p

F

(dF=dh)2(37b)

H~p

4P2

h

WtotWu

Wtot

dF=dheth THORN2

F(37c)

E~p

4Ph

WtotWu

Wtot

dF=dheth THORN2

F(37d)

where PhWtotWu

Wtot






Wp

Wtot~

mz1

lz1

hf

hm

ml

lz1(38)




F~B(h2h2f ) (39)


hf

hm

~1lH









True hardnessH GPa






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


hf

hn

2

~1 2cE

cH

tan h

H

E(41)


H

E~k

We

Wtotwhere k~

1

l(1zc)for 600

iexclhiexcl800 (42)





et al 2002)




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








QC Er GPa E GPa









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~H0


h

hc

s(43)





(44)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd























Fleck et al (1994)





Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




















and Wolf 2003)







Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd














2003)









Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Wolf 2003)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


H~Ksy (45)


H~syffiffiffi



H~2N1

Bsy N2z ln

Er

sy

(47)



ty~kasn (48)
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd
















Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










et al 2002)



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd










2002)








Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd













Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








Ni (after Erb 1995)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd






Haggag et al 1989)






Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd





Acknowledgements



1595


















1687ndash1693




Research Society




Society



Academicis








149











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




Res 19 31ndash45










59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd



Villars


085205-1ndash9













Plenum Press


A 57 187ndash196








281ndash288





582







7 2962ndash2990




Metals




1291



616



1821ndash1829




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59 2246ndash2251






1183


2214ndash2216



K751ndashK7511




Society




2161ndash2172








381


20ndash35



14 317ndash330






Anal 33 411ndash418






K746


York Springer








K211ndash216






Vratislaviae


332ndash357



Chemie 80 37ndash55


42 79ndash89




507ndash515




2423














1191ndash1198




Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd








A 82 1857ndash1864



A 303 179ndash186











Nauka Moscow


28 117ndash121




















271 Paris Louis





289


444


3562ndash3564








Schramm Lipsiae

















2350 (2004) 19 1ndash396


Nauka Leningrad


3057ndash3063










654ndash657



Ltd



Berlin 1ndash24



23(1) 39


634 B511ndashB5111












8 1028ndash1032


735ndash757


3000



Gmelin Lipsiae




611ndash613



Wittenberg




2319ndash2323



















13 1300ndash1306











Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd




252ndash260










Society




















81ndash86






80 3310ndash3312















7 450ndash473


24 2347ndash2350






2193ndash2201









1885 10 531ndash532


1807ndash2231




559ndash564














32








(1990) Phys Rev B41 1194ndash1195




17 475


407




371






125







1212





17 1651ndash1654








303ndash311


















Press





30 561ndash562



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



Pub

lishe

d by

Man

ey P

ublis

hing

(c)

IOM

Com

mun

icat

ions

Ltd


1715ndash1723
























501ndash502












Res 15 798ndash807








Gordon and Breach








361


599








A298 284ndash295


561ndash568


35 723ndash734



World Scientific


2195ndash2208


642ndash649




2436


130


JIM 34 135ndash145



1027ndash1029


37 195ndash209



68



nanoindentation techniques materials

Documents

hardness concept

indentation mechanics

conceptof hardness

variation of hardness

loaddepth of indentation

bar withincreasing hardness

hardness values morecomparable

paufler2indentation