determination of the mechanical properties of metallic...

PHILOSOPHICAL MAGAZINE A 2002 VOL 82 NO 10 2013plusmn2029

Determination of the mechanical properties of metallicthin reglms and substrates from indentation tests

K Tunvisut E P Busso N P OrsquoDowdyDepartment of Mechanical Engineering Imperial College London SW7 2BX

UK

and H P Brantner

Montanuniversitaet Leoben Leoben Austria

[Received 16 July 2001 and accepted in revised form 1 March 2002]

AbstractA procedure to obtain the elasto-plastic mechanical properties of strain-

hardening materials from indentation tests based on dimensional analysis andregnite-element techniques is proposed The method is applicable to homogeneousmaterials and to coatings deposited on substrates of known mechanicalproperties The Youngrsquos modulus of the material is extracted from the initialslope of the unloading indentation curve and the yield strength and strain-hardening exponent are obtained from the maximum indentation load and thecontact area after unloading The method is used to obtain the properties of ahigh-alloy steel and Mo and AlSi coatings deposited on a steel substrate byplasma spraying The sensitivity of the measurement to the depth ofindentation is discussed

1 IntroductionIn recent years indentation tests have been used to determine elasto-plastic

properties such as Youngrsquos modulus yield strength and strain-hardening exponent(for example Doerner and Nix (1986) Oliver and Pharr (1992) and Cheng and Cheng(1999)) For instance Youngrsquos modulus may be inferred from the unloading inden-tation loadplusmndepth curve and the yield strength from the maximum indentation loadIn addition a method to extract the macrow stress and the strain-hardening exponentusing indentation data has been proposed by Giannakopoulos and Suresh (1999)However a common limitation to these approaches when applied to coatings is thatthe indentation test must be performed at relatively shallow depths where the inmacru-ence of the substrate is negligible In this way the coating can be treated as ahomogeneous material and the e ect of the substrate is ignored However as illu-strated here the prediction of bulk coating properties from indentation tests carriedout at very small indentation depths is di cult and may lead to an overestimation ofthe mechanical properties

More recently an alternative method has been proposed by Tunvisut et al (20002001) to estimate the elasto-plastic behaviour of thin reglms from indentation tests

Philosophical Magazine A ISSN 0141plusmn8610 printISSN 1460-6992 online 2002 Taylor amp Francis Ltd

httpwwwtandfcoukjournals

DOI 10108001418610210134332

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The method takes into account the e ect of the substrate on the indentation responseand therefore can be used to analyse indentation tests on thin reglms performed atboth shallow and deep depths Tunvisut et al (2001) carried out dimensional analysisand regnite element studies of indentation of coated substrates and showed that theelasto-plastic mechanical properties (Youngrsquos modulus yield strength and strain-hardening behaviour) could be uniquely identireged from measurement of the peakload and unloading slope of the indentation curve and the contact area of indenta-tion (the region where permanent deformation was observed after removal of theindenter) Tunvisut et al (2000) showed that the method could also be applied touncoated substrates and comparison was made with existing and proposed techni-ques for determining coating properties from indentation tests (for example Oliverand Pharr (1992) and Giannakopoulos and Suresh (1999)) It was shown that the useof the contact area avoided the problem of uniqueness which may arise if the inden-tation curve alone is used to predict the material properties

In this work a procedure is outlined on the basis of these numerical studies todetermine directly from indentation measurements the mechanical properties ofcoatings and homogeneous substrates The use of the method is illustrated for thecase of a high-chromium steel (AISI D2) and for plasma-sprayed Mo and AlSicoatings deposited on a steel substrate The latter coatings have been used toimprove the wear behaviour of synchronizers used in automotive gear boxes Inorder to predict the in-service behaviour of such coatings an accurate descriptionof the elasto-plastic mechanical behaviour of the coating is required (for exampleYan et al (2000))

The sensitivity of the predicted elasto-plastic properties of the coatings to theindentation depth is also discussed

2 Indentation methods

21 Indentation of bulk or uncoated substrates (method A)In previous work (Tunvisut et al 2000 2001) the indentation of an elasto-plastic

solid by an ideally sharp rigid frictionless conical indenter of half-angle sup3 ˆ 70deg wasstudied (reggure 1 (a)) The stressplusmnstrain behaviour of the material was assumed to belinear in the elastic regime with Youngrsquos modulus E and in the plastic regime thestressplusmnstrain behaviour is described by a power-law relation

y

ˆfrac14

frac14y

hellipfrac14=frac14ydaggern

forfrac14 4 frac14y

frac14 gt frac14y

8gtlt

gt

8gtlt

gt

where frac14 and are the stress and (total) strain respectively frac14y is the yield strength

y ˆ frac14y=E is the yield strain and n is the strain-hardening exponent The material isassumed to obey the von Mises yield criterion and under multiaxial loading thestress and strain in equation (2) are replaced by the equivalent von Mises stress andstrain respectively

It was found that conical indentation of such a material is described by thefollowing dimensionless relationships

2014 K Tunvisut et al

(1)

(2)

Fm

Eh2m

ˆ PPbhellipy n cedil sup3dagger hellip3dagger

Af

h2m

ˆ PPghellipy n cedil sup3dagger hellip4dagger

where Fm and hm are the maximum indentation load and depth respectively (seereggure 1 (b)) and Af is the area of the indentation after unloading A similar relation-ship to equation (3) has been provided by Cheng and Cheng (1999) for perfectlyplastic (non-hardening) materials and more recently for power-law-hardeningmaterials by Cheng and Cheng (2000)

The functional relations given in equations (3) and (4) have been determined fora range of material behaviours and typical results from parametric regnite-element(FE) studies are shown in reggure 2 Here the maximum normalized indentation loadand regnal contact area are given in terms of the yield strain for di erent strainhardening exponents The value of Poissonrsquos ratio was taken to be 03 in all cases

Equations (3) and (4) in conjunction with data of the type illustrated in reggure 2can then be solved to evaluate the yield strain and strain-hardening exponent of thematerial To allow the method to be applied directly accurate regts to the data of thetype shown in reggure 2 have been obtained using the least-squares method It hasbeen found that

Fm

Eh2m

ˆ 73082y 873098

y 024 ln y Dagger 036iexcl

n026 ln yDagger010 hellip5dagger

Af

h2m

ˆ 6 ln y 178013y Dagger 454 ln y Dagger 586

iexcln01 ln y 01 Dagger 1557 hellip6dagger

The implicit dependence in equations (5) and (6) of the two unknown propertiesnamely the yield strain y and the strain-hardening exponent n on the known quan-tities requires that the equations be solved numerically using for instance an itera-tive Newton-type method

If Youngrsquos modulus E of the material is unknown it was shown by Tunvisut etal (2000) that this may be obtained directly from the initial unloading slope at theonset of unloading (ie at h ˆ hm) This method generalizes the approach of Oliverand Pharr (1992) which may be inaccurate for soft materials that is frac14y=E lt 0005

Mechanical properties of thin reglms and substrates 2015

Figure 1 (a) Schematic diagram of an indentation test using a conical indenter (b) Typicalindentation loadplusmndisplacement curve

(Bolshakov et al 1997) The dependence of unloading slope on the normalized yieldstrength is illustrated in reggure 3 (a) Although not shown it has been found fromFE studies that the initial unloading slope is almost independent of the hardeningexponent n

For low-strength materials frac14y=E lt 001 the normalized initial unloading slopedepends strongly on frac14y=E while for high-strength materials frac14y=E 5 001 thedependence is weaker and the unloading slope can be assumed to be independentof frac14y=E For the latter case we can then write

1

Ehm

dF

dh hˆhm

ˆ 68 forhf

hm

4 0875 hellip7dagger

The range of applicability of equation (7) is given in terms of a directly measurablequantity hf =hm the ratio of the unloaded depth to the maximum depth rather than


Figure 2 Results of FE parametric studies for the uncoated substrate showing (a) themaximum normalized indentation load and (b) the normalized regnal contact area interms of the normalized yield strength for di erent strain-hardening exponents

Figure 3 Results of FE parametric studies for the uncoated substrate showing relationshipsbetween (a) the initial unloading slope normalized using hm and frac14y=E and (b) theinitial unloading slope normalized using hf and frac14y=E

the ratio frac14y=E which is not known a priori For the case where hf=hm gt 0875 (ie alow-strength material) a di erent normalizing parameter was used the initialunloading slope is divided by hf E rather than by hmE where hf is the regnal depthafter unloading (see reggure 1 (b)) Figure 3 (b) shows that when this normalization isused the dependence on frac14y=E is weak for frac14y=E lt 001 leading to the followingexpression

1

Ehf

dF

dh hˆhm

ˆ 78 forhf

hm

gt 0875 hellip8dagger

Equations (7) and (8) allow direct identiregcation of Youngrsquos modulus of the materialfrom the initial unloading slope of the indentation curve for high- and low-strengthmaterials respectively

It is worth noting that the applicability of the method depends strongly on theaccuracy associated with the experimental indentation data For instance a 5 errorin the unloading slope may result in a maximum error of 5 30 and 50 in thepredicted values of E frac14y and n respectively (Tunvisut et al 2000)

22 Indentation of a coated substrate (method B)Consider a system similar to that discussed in sect 21 except that the indented solid

is now replaced with a coating of thickness h0 Youngrsquos modulus Ec yield strength

frac14yc Poissonrsquos ratio cedilc and strain-hardening exponent nc deposited on an elasticsubstrate of known Youngrsquos modulus Es and Poissonrsquos ratio cedils The relevant func-tional dependences of the geometrical and material parameters for the thin reglmplusmnsubstrate bimaterial system have been presented by Tunvisut et al (2001) and aregiven as

1

h0Es

dF

dh hˆhm

ˆ PPdhm

h0

Ec

Es

cedilc cedils sup3 hellip9dagger

Fm

h20Es

ˆ PPahm

h0

Ec

Es

frac14yc

Es

cedilc cedils nc sup3 hellip10dagger

Af

h20

ˆ PPlhm

h0

Ec

Es

frac14yc

Es


Parametric FE analyses of the indentation of a range of coatingplusmnsubstrate systemsfor a maximum indentation depth hm ˆ 033h0 were performed to calibrate thefunctions given in equations (9)plusmn(11) Typical results are shown in reggure 4 Figure4 (a) can be used to determine Youngrsquos modulus of the coating from the initialunloading slope and the yield strength and strain-hardening exponent can beobtained from reggures 4 (b) and (c) An accurate calibration of the relationshipderegned in equation (9) from the type of FE results shown in reggure 4 gives

1

h0Es

dF

dh hˆhm

ˆ21EE086 EE 4 1 hellip12dagger

7EE 032 271EE001 Dagger 139EE0002 Dagger 223 EE gt 1 hellip13dagger

8lt

where EE=Ec=EsIt proves convenient to separate each of the dimensionless functions deregned by

equations (10) and (11) into a term that is independent of the strain-hardening



Figure 4 Typical results from the FE parametric study of indentation of the coatingsubstrate system showing (a) (dF=dhdaggerhˆhm

versus Ec=Es (b) Fm=h20Es versus frac14yc=Es

and (c) Af=h20 versus frac14yc=Es

exponent nc and another term that depends on nc that is we obtain for a regxed valueof hm=h0

Fm

h20Es

ˆ PPa EE frac14frac14hellip dagger Dagger PPb EE frac14frac14 nchellip dagger hellip14dagger

Af

h20

ˆ PPc EE frac14frac14hellip dagger Dagger PPd EE frac14frac14 nchellip dagger hellip15dagger

where frac14frac14=frac14yc=Es is the normalized coating yield stress In the above PPa and PPc

provide the solution for a non-hardening material (nc ˆ 1) and the functions PPb

and PPd provide the corrections to PPa and PPc respectively to account for strainhardening

The dimensionless functions PPa PPb PPc and PPd used in equations (14) and (15)obtained by regtting to FE data are given in appendix A Thus equations (14) and (15)constitute a set of simultaneous nonlinear equations from which the two unknowncoating parameters namely frac14yc and nc can be obtained The implicit nature of theequations require that equations (14) and (15) must be solved numerically for frac14 andnc using for example an iterative Newton-type scheme

As discussed in the previous section the method proposed in this work reliesstrongly on accurate measurements of the maximum indentation load initial unload-ing slope and regnal contact area A sensitivity analysis showed that a 5 error in themeasurement of the unloading slope could result in errors of up to 40 in thepredicted value of frac14yc

The method for uncoated materials discussed in sect 21 method A can also be usedto determine the properties of a thin reglm provided that the indentation depth doesnot exceed a certain fraction of its thickness to ensure that the presence of theunderlying substrate does not a ect the indentation curve The indentation depthunder which the e ect of the substrate is negligible depends on the mismatch in themechanical properties of the coating and the substrate Appendix B provides theconditions under which method A may be used to predict coating behaviour for awide range of coatingsubstrate systems Further discussion of these analyses hasbeen provided by Tunvisut (2002)

3 Experimental workIndentation tests have been carried out on an alloy steel (AISI D2) and on AlSi

and Mo coatings deposited on a case hardened steel (with Es=194 GPa) by plasmaspraying The test on the alloy steel was used as a validation of the procedure as themechanical properties of the steel were also measured in uniaxial tensile tests

Prior to performing the tests all samples were polished and the coating thick-nesses were measured The mean thickness of the AlSi and Mo coatings were foundto be 295 and 270mm respectively Indentation tests were carried out using Vickersand Berkovich indenters up to di erent depths the details of which will be givenbelow and at least two tests were performed for the same maximum indentationdepth on each material Details of the experimental data obtained for each materialare presented in appendix C

The results presented in sect 2 which can be used to extract the mechanical proper-ties of a material from an indentation test are for a conical indenter It has beenshown in numerical studies (for example Cheng and Cheng (1998)) that indentationcurves for a Berkovich indenter a Vickers indenter and a 70deg conical indenter are


almost identical (they di er by less than 5 for the wide range of cases examined)Furthermore based on the approach of Oliver and Pharr (1992) it can be inferredthat the contact areas for the 70deg conical indenter the Berkovich indenter and theVickers indenter at the same indentation depth di er by less than 5 (ie thegeometrical constant used to obtain the contact area from the slope of the unloadingcurve di ers by less than 5 for these indenters)

Thus the relations presented in sect 2 may be used to analyse indentation tests usingBerkovich and Vickers indenters as well as conical indenters

31 Shallow indentations (hm lt 05 middotm)Indentation tests up to a maximum depth of less than 05 mm were carried out

using a nanoindenter instrument with a diamond Berkovich indenter A typicalindentation curve measured in the AlSi coating system is shown in reggure 5 (a)The image of the indentation after unloading was obtained using a nanoscopeThe shape of a typical impression is shown in reggure 5 (b) The regnal contact areaAf after unloading for a perfectly sharp indenter can be calculated directly from theimage assuming a triangular contact region

Af ˆ dd 2

31=2 sin shy hellip16dagger

where shy is the angle between the central line and the face of the indenter hereshy ˆ 653deg and

dd ˆ d1 Dagger d2 Dagger d3

3 hellip17dagger

The dimensions d1 d2 and d3 in equation (17) are the diagonals of the indentation areaafter unloading (see reggure 5 (b)) In practice owing to the e ect of indenter tip round-ing equation (16) will not precisely describe the relationship between indentationdepth and contact area for an actual indenter To determine this relationship an experi-


Figure 5 Typical results from nanoindentation tests on the AlSi coating (a) indentationcurve (b) image of indentation after unloading

mental calibration is carried out closely following the procedure of Oliver and Pharr(1992) The contact area shape function has been calibrated in the following form

Af ˆ dd 2

31=2 sin shyDagger C1dd Dagger C2dd 1=2 Dagger C3dd

1=4 Dagger Dagger C8dd 1=128 hellip18dagger

where C1 C2 C3 C4 C5 C6 C7 and C8 are constants here found to be 416 1038159 175 177 156 107 and 87 respectively The lead term in equation (18)describes a perfect Berkovich indenter the others describe deviations from thisidealized geometry (eg tip rounding) The indenter geometry is precisely deregnedthrough the use of equation (18) However for certain materials the shape of thecontact region may deviate considerably from a triangle depending on the materialproperties and indentation depth invalidating the use of equation (18) However forthe range of the materials and indentation depths studied in the current workinsigniregcant deviation has been observed and the assumption of a triangular impres-sion is acceptable

32 Deep indentations (hm gt 05 middotm)For 05 mm lt hm 4 10 mm tests were performed using a Fisherscope dynamic

hardness tester with a Vickers indenter The maximum load available was 1 Nthus for larger indentations a macrohardness tester was used A typical macro-indentation curve for the AlSi system is shown in reggure 6 (a) together with a typicalindentation impression in reggure 6 (b)

For the macrohardness measurements only the indentation load and the imageof the indented region could be recorded Thus for these measurements the indenta-tion depth h in reggure 6 (a) was estimated from the mean value of the two impressiondiagonals d1 and d2 in reggure 6 (b) Then

hdd

7ˆ 1

7

d1 Dagger d2

2 hellip19dagger


Figure 6 Typical results from macroindentation tests on the AlSi coating (a) indentationcurve (b) image of indentation after unloading

Note that the value of h calculated via equation (19) is the contact depth whichwill include the e ects of piling-up and sinking-in and thus may overestimate orunderestimate the actual indentation depth Figure 7 provides values of d=hobtained from FE calculations for a range of materials that is a range of frac14y=EWhen d=h 7 the e ect of piling-up or sinking-in is negligible and equation (19) isappropriate in the determination of the indentation depth from the impression diag-onal For hard materials (ie high frac14y=E) sinking-in is observed d=h lt 7 and equa-tion (19) overestimates the indentation depth On the other hand when piling-uptakes place (materials with low frac14y=E) equation (19) underestimates the indentationdepth (d=h gt 7) However for the range of materials being studied in this paper0004 lt frac14y=E lt 001 it may be seen in reggure 7 that d=h 7 and thus the indentationdepth estimated from equation (19) is acceptable equation (19) was found to over-estimate h by approximately 3 for the Mo coating and to underestimate h by lt8for the AlSi coating which will result in less than a 15 error in the material yieldstrength using the current method (see also Tunvisut (2002))

The contact area Af for the Vickers macroindentation can be obtained directlyfrom the indentation impression by assuming that the indenter is perfectly sharpETHanacceptable approximation for deep indentations The contact area can then beapproximated from

Af ˆ dd2

185 hellip20dagger

Again the assumption inherent in equation (20) is that the indentation impression isof diamond shape We have observed this shape of impression for all the indentationtests carried out (see for example reggure 6 (b))


Figure 7 Ratio of the impression diagonal to the indentation depth for a Vickers indenterfor di erent materials

4 Results and discussion

41 Uncoated substrateThe methodology proposed here will regrst be used to determine the mechanical

properties of the alloy steel Method A is used and the resulting predictions for thematerialrsquos yield stress Youngrsquos modulus hardening coe cient and 02 yieldstrength frac1402 using two di erent indenter types and three maximum indentationdepths are presented in table 1 As seen in table C1 in appendix C di erent resultsare obtained from the nanoindentation measurements (hm lt 035 mm) depending onwhether the indentation is located in the carbide or martensite phases To obtain themacroscopic elasticplusmnplastic properties the values for the individual phases wereaveraged The use of an unweighted average to calculate the material propertiesmay be open to question but the result is included in table 1 for completeness Inaddition to the indentation tests uniaxial tensile tests were carried out and these dataare also given

The measured stressplusmnstrain uniaxial response of the AISI D2 steel is shown inreggure 8 together with the predicted responses calculated using the material para-meters obtained from indentation at di erent indentation depths The result for thelowest indentation depth is not included It may be seen that the results are similarwithin the experimental scatter however on the basis of the mean result there is asmall depth dependence of the result with strength increasing somewhat withdecreasing indentation depth Overall there is good agreement with the tensiledata particularly at the deepest indentation depth

42 Mo and AlSi coatingsThe results from indentation tests on the Mo and AlSi coatings have been

analysed using the proposed methods For the tests with maximum indentationdepths less than 05 mm which is relatively shallow compared with the coating thick-ness method A can be used For AlSi and Mo coatings indented to the depth of 97and 89 mm respectively (hm=h0=033) the substrate has a strong e ect on theindentation test results and therefore method B is used For the latter dataYoungrsquos modulus cannot be determined because the unloading curve is not avail-able Therefore the average values of Youngrsquos modulus obtained from the shallowindentation tests were used to interpret these data The calculated Youngrsquos modulusyield strength and hardening exponent of the coatings at di erent depths are pre-sented in tables 2 and 3 It can be seen that as for the steel substrate there is againgood agreement in the predicted Youngrsquos moduli but in this case there is a strongdependence of the yield strength values on depth It should be pointed out that thisdepth dependence is also evident in the raw experimental data for example the


Table 1 Predicted and measured elasto-plastic properties for the AISI D2 steel

hm Indenter E frac14y frac1402

Method (mm) type (GPa) (MPa) n (MPa)

Uniaxial test plusmn plusmn 217 30 plusmn 3 05 1786 3A lt035 Berkovich 264 58 2413 290 376 68 2750 275A 0544 Vickers 240 8 2120 215 7 25 2182 252A 2434 Vickers 210 8 1970 115 10 13 2008 121

normalized maximum load as specireged by equation (3) is depth dependent for thelowest depths with the same indenter type even taking the experimental scatter intoconsideration (see tables C 2 and C 3 in appendix C)

43 Indentation size e ectsGood agreement between the measured properties and that determined from the

tensile curve has been obtained for the steel substrate with a small depth dependenceof the elasticplusmnplastic properties However our results suggest that for the AlSi and


Figure 8 Stressplusmnstrain curve for AISI D2 steel

Table 2 Predicted elasto-plastic properties of Mo coating using indentation test data

Indenter hm E frac14y frac1402

Method type (mm) (GPa) (MPa) n (MPa)

A Berkovich 017 220 12 5900 700 2 02 6116 724A Berkovich 02 242 10 4800 550 5 05 4916 751A Vickers 16 194 50 3400 600 1 05 3788 1025B Vickers 89 plusmn 2700 300 19 5 2712 308

Table 3 Predicted elasto-plastic properties of AlSi coating using indentation test data




Mo coatings the properties depend on the depth of indentation Similar observationshave been reported for example by Iost and Bigot (1996) and Rother et al (1998)This `size e ectrsquo is also seen here in the raw data as discussed in the previous sectionand so cannot be associated with uncertainties in the proposed method Theobserved depth dependence may be due to a number of factors for example thee ect of inhomogeneities in the coatings (there is known to be porosity in the Mocoatings and some unmelted particles in the AlSi coating which may have a largere ect at small indentation depth) or the e ect of tip rounding which is more impor-tant at smaller indentation depths (for example Cheng and Cheng (1998)) or experi-mental uncertainties at shallow depths (for example Blau (1983)) Alternatively suchsize e ects may be due to the nature of the plastic deformation at small size scalesPlastic deformation under small indentation loads leads to large local strain gradi-ents which can cause enhanced strain hardening due to the additional contributionfrom geometrically necessary dislocations (for example De Guzman et al (1993)Fleck and Hutchinson (1993) and Busso et al (2000)) The incorporation of thesee ects will lead naturally to an indentation size e ect (hardness and apparent yieldstrength increasing with decreasing depth)

Based on this concept Nix and Gao (1998) studied indentation size e ects forductile materials By taking into account the e ect of the geometrically necessarydislocations generated near the indenter surface and using Taylorrsquos relation for themacrow stress (for example Hull and Bacon (1984)) they obtained a simple relationshipfor the depth dependence of the hardness H (indentation force divided by contactarea)

H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger

In equation (21) h is the indentation depth H0 the hardness in the limit of inregnitedepth and h is a characteristic length that depends on the shape of the indenterthe shear modulus and H0 It has been shown by Tunvisut (2002) using similararguments that the expression for the depth dependence of the yield strength maybe written as

frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger

where frac14y is the yield strength obtained for a given depth h of indentation frac14y0

represents the value obtained in the limit of inregnite depth and k is a constantwhich depends only on material properties that is Youngrsquos modulus yield strengthhardening behaviour and indentation depth The relationship obtained from equa-tion (22) is plotted in reggure 9 for the AlSi and Mo coatings together with themeasured data It may be seen that the data seem to follow the trend predicted byequation (22) Thus there is some support for the use of strain gradient approachesto account for depth dependence of predicted material properties from indentationtests However further studies are needed to validate the approach fully In any casethe observed results highlight the fact that care needs to be taken if material proper-ties are obtained from indentations at very small length scales using for examplenanoindentation techniques


5 ConclusionsMethods to determine the mechanical properties of materials from indentation

tests have been described and these methods have been used to extract Youngrsquosmodulus yield strength and hardening exponent of an AISI D2 steel and of AlSiand Mo coatings The values of Youngrsquos modulus determined from the indentationtests are relatively independent of indentation depth and for the steel substrate thedata are consistent with uniaxial tensile data Quite a strong depth dependence of theestimated values of yield strength has been found for the AlSi and Mo coating thatis increasing yield strength with decreasing indentation depth This dependence maybe due to the indentation size e ect which can be signiregcant when the indentationdepth is small relative to intrinsic material length scales Such size e ects should betaken into account if nanoindentation measurements are to be used to predict bulkmechanical properties

ACKNOWLEDGEMENTS

Support for this work has been provided by the government of Thailand and bythe European Union through Brite EUram project BE97-4283 The authors aregrateful to Dr N Renevier from Teer Coatings Ltd UK for providing us withthe microindentation test results on the AISI D2 steel Helpful comments from oneof the reviewers are acknowledged

APPENDIX AD i m e ns i onl e s s f u nc t i ons f or m e tho d B

The dimensionless functions which appear in equations (14) and (15) are givenbelow In the equations EE ˆ Ec=Es and frac14frac14 ˆ frac14yc=Es


Figure 9 Illustration of indentation depth dependence of the yield strength for Mo and AlSicoating

For 001lt EE 4 01

PPa ˆ hellip06EE2 Dagger 077EE 0003daggerfrac14frac14 17EE2Dagger38EEDagger001 hellipA 1dagger

PPb ˆ 0014 ln frac14frac14EE084nhellip01 ln frac14frac14 009daggerEE03

c hellipA 2dagger

PPc ˆ 07EE042frac14frac14 01EE 0586

hellipA 3dagger

PPd ˆ hellip 027 ln frac14frac14EE 05 07EE044daggern13EE01frac14frac14014EE02

c hellipA 4dagger

For 01lt EE 4 1

PPa ˆ hellip 032EE2 Dagger 091EE 0003daggerfrac14frac14 033EE2Dagger057EEDagger02 hellipA 5dagger


c hellipA 6dagger


hellipA 7dagger

PPd ˆ hellip 027 ln frac14frac14EE 05 07EE04daggern 13EE01frac14frac14014EE02

c hellipA 8dagger

For 1lt EE 4 10

PPa ˆ hellip 002EE2 Dagger 026EE Dagger 038daggerfrac14frac14 0004EE2Dagger005EEDagger0412 hellipA 9dagger


c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger

APPENDIX BSel ect i on of i n de ntat i on m e th od

Tables B 1plusmnB 3 can be used as guidelines in selecting an appropriate indentationmethod to be used in determining the mechanical properties of coatings deposited ona substrate from indentation tests carried out at di erent thicknesses


Table B 1 Indentation depth of one tenth of thecoating thickness

Method for following valuesof frac14yc=Es

Ec=Es 0001 0005 001 005

001 A A A A01 A A B B1 A A A A2 A A A A

10 A A A A

APPENDIX CIn d e ntat i on t e st m e as u re m e nt s

Experimental data obtained from indentation tests for the high-Cr steel (AISID2) and plasma-sprayed Mo and AlSi coatings deposited on a steel substrate arereported in tables C 1plusmnC 3


Table B 2 Indentation depth of one sixth of thecoating thickness


Ec=Es 0001 0005 001 005

001 B B B B01 A B B B1 A A A A2 A A A B

10 A A B B

Table B 3 Indentation depth of one third of thecoating thickness


Ec=Es 0001 0005 001 005

001 B B B B01 B B B B1 A A A A2 B B B B

10 B B B B

Table C 1 Measurements for indentation tests on AISI D2 steel

hm hf dF=dhhˆhmAf

Fm (mm) (mm) (mNmm) (mm2)

12996 4 mN (carbides) 0183 0005 0085 0008 480 50 062 00212985 15 mN (martensite) 034 002 022 003 340 20 34 01

505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3

Table C 2 Measurements for indentation tests on the Mo coating

hm hf dF=dhhˆhmAf Fm=h2

m

Fm (mm) (mm) (mNmm) (mm2) (Nmm2)

8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004

ReferencesBlau P J 1983 Metallography 16 1Bolshakov A Oliver W C and Pharr G M 1997 Mater Res Soc Symp Proc 436

141Busso E P Meissonnier F and Orsquo Dowd N P 2000 J Mech Phys Solids 48 2333

Cheng Y T and Cheng C M 1998 J Mater Res 13 1059 1999 Int J Solids Structs36 1231 2000 Surf Coating Technol 133plusmn134 417

De Gusman M S Neubauer G Flinn P and Nix W D 1993 Mater Res SympProc 308 613

Doerner M F and Nix W D 1986 J Mater Res 1 601

Fleck N A and Hutchinson J W 1993 Adv appl Mech 33 295

Giannakopoulos A E and Suresh S 1999 Scripta mater 40 1191Hull D and Bacon D J 1984 Introduction to Dislocations (Oxford Butterworth-

Heinemann)Iost A and Bigot R 1996 J Mater Sci 31 3573

Nix W D and Gao H 1998 J Mech Phys solids 46 411

Oliver W C and Pharr G M 1992 J Mater Res 7 1564Rother B Steiner A Dietrich D A Jehn H A Haupt J and Gissler W 1998

J Mater Res 13 2071

Tunvisut K 2002 PhD Thesis University of LondonTunvisut K OrsquoDowd N P and Busso E P 2000 Proceedings of the Fourth International

Conference on Modern Practice in Stress and Vibration Analysis Nottingham 2000edited by A A Becker (EMAS Limited) p 89 2001 Int J Solids Structs 38 335

Yan W Busso E P and OrsquoDowd N P 2000 Proc R Soc A 456 2387


Table C 3 Measurements for indentation tests on the AlSi coating


m


985 2 mN 0115 0006 0089 0003 557 14 032 002 0074 00074938 6 mN 032 002 027 004 140 14 23 06 0048 0006500 8 mN 35 04 315 023 1700 50 400 90 0041 0009400 7 N 97 8 plusmn plusmn 170000 1800 0042 0007

The method takes into account the e ect of the substrate on the indentation responseand therefore can be used to analyse indentation tests on thin reglms performed atboth shallow and deep depths Tunvisut et al (2001) carried out dimensional analysisand regnite element studies of indentation of coated substrates and showed that theelasto-plastic mechanical properties (Youngrsquos modulus yield strength and strain-hardening behaviour) could be uniquely identireged from measurement of the peakload and unloading slope of the indentation curve and the contact area of indenta-tion (the region where permanent deformation was observed after removal of theindenter) Tunvisut et al (2000) showed that the method could also be applied touncoated substrates and comparison was made with existing and proposed techni-ques for determining coating properties from indentation tests (for example Oliverand Pharr (1992) and Giannakopoulos and Suresh (1999)) It was shown that the useof the contact area avoided the problem of uniqueness which may arise if the inden-tation curve alone is used to predict the material properties

In this work a procedure is outlined on the basis of these numerical studies todetermine directly from indentation measurements the mechanical properties ofcoatings and homogeneous substrates The use of the method is illustrated for thecase of a high-chromium steel (AISI D2) and for plasma-sprayed Mo and AlSicoatings deposited on a steel substrate The latter coatings have been used toimprove the wear behaviour of synchronizers used in automotive gear boxes Inorder to predict the in-service behaviour of such coatings an accurate descriptionof the elasto-plastic mechanical behaviour of the coating is required (for exampleYan et al (2000))

The sensitivity of the predicted elasto-plastic properties of the coatings to theindentation depth is also discussed

2 Indentation methods

21 Indentation of bulk or uncoated substrates (method A)In previous work (Tunvisut et al 2000 2001) the indentation of an elasto-plastic

solid by an ideally sharp rigid frictionless conical indenter of half-angle sup3 ˆ 70deg wasstudied (reggure 1 (a)) The stressplusmnstrain behaviour of the material was assumed to belinear in the elastic regime with Youngrsquos modulus E and in the plastic regime thestressplusmnstrain behaviour is described by a power-law relation

y

ˆfrac14

frac14y

hellipfrac14=frac14ydaggern

forfrac14 4 frac14y

frac14 gt frac14y

8gtlt

gt

8gtlt

gt

where frac14 and are the stress and (total) strain respectively frac14y is the yield strength

y ˆ frac14y=E is the yield strain and n is the strain-hardening exponent The material isassumed to obey the von Mises yield criterion and under multiaxial loading thestress and strain in equation (2) are replaced by the equivalent von Mises stress andstrain respectively

It was found that conical indentation of such a material is described by thefollowing dimensionless relationships


(1)

(2)

Fm

Eh2m


Af

h2m





Fm

Eh2m

ˆ 73082y 873098



Af

h2m









1

Ehm

dF

dh hˆhm

ˆ 68 forhf

hm







1

Ehf

dF

dh hˆhm

ˆ 78 forhf

hm







1

h0Es

dF

dh hˆhm

ˆ PPdhm

h0

Ec

Es


Fm

h20Es

ˆ PPahm

h0

Ec

Es

frac14yc

Es


Af

h20

ˆ PPlhm

h0

Ec

Es

frac14yc

Es



1

h0Es

dF

dh hˆhm



8lt









Fm

h20Es


Af

h20

















Af ˆ dd 2




3 hellip17dagger





Af ˆ dd 2







hdd

7ˆ 1

7

d1 Dagger d2

2 hellip19dagger





Af ˆ dd2

185 hellip20dagger






























H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger


frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger






ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m



Fm

Eh2m


Af

h2m





Fm

Eh2m

ˆ 73082y 873098



Af

h2m









1

Ehm

dF

dh hˆhm

ˆ 68 forhf

hm







1

Ehf

dF

dh hˆhm

ˆ 78 forhf

hm







1

h0Es

dF

dh hˆhm

ˆ PPdhm

h0

Ec

Es


Fm

h20Es

ˆ PPahm

h0

Ec

Es

frac14yc

Es


Af

h20

ˆ PPlhm

h0

Ec

Es

frac14yc

Es



1

h0Es

dF

dh hˆhm



8lt









Fm

h20Es


Af

h20

















Af ˆ dd 2




3 hellip17dagger





Af ˆ dd 2







hdd

7ˆ 1

7

d1 Dagger d2

2 hellip19dagger





Af ˆ dd2

185 hellip20dagger






























H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger


frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger






ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m





1

Ehm

dF

dh hˆhm

ˆ 68 forhf

hm







1

Ehf

dF

dh hˆhm

ˆ 78 forhf

hm







1

h0Es

dF

dh hˆhm

ˆ PPdhm

h0

Ec

Es


Fm

h20Es

ˆ PPahm

h0

Ec

Es

frac14yc

Es


Af

h20

ˆ PPlhm

h0

Ec

Es

frac14yc

Es



1

h0Es

dF

dh hˆhm



8lt









Fm

h20Es


Af

h20

















Af ˆ dd 2




3 hellip17dagger





Af ˆ dd 2







hdd

7ˆ 1

7

d1 Dagger d2

2 hellip19dagger





Af ˆ dd2

185 hellip20dagger






























H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger


frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger






ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m




1

Ehf

dF

dh hˆhm

ˆ 78 forhf

hm







1

h0Es

dF

dh hˆhm

ˆ PPdhm

h0

Ec

Es


Fm

h20Es

ˆ PPahm

h0

Ec

Es

frac14yc

Es


Af

h20

ˆ PPlhm

h0

Ec

Es

frac14yc

Es



1

h0Es

dF

dh hˆhm



8lt









Fm

h20Es


Af

h20

















Af ˆ dd 2




3 hellip17dagger





Af ˆ dd 2







hdd

7ˆ 1

7

d1 Dagger d2

2 hellip19dagger





Af ˆ dd2

185 hellip20dagger






























H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger


frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger






ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m








Fm

h20Es


Af

h20

















Af ˆ dd 2




3 hellip17dagger





Af ˆ dd 2







hdd

7ˆ 1

7

d1 Dagger d2

2 hellip19dagger





Af ˆ dd2

185 hellip20dagger






























H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger


frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger






ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m







Af ˆ dd 2




3 hellip17dagger





Af ˆ dd 2







hdd

7ˆ 1

7

d1 Dagger d2

2 hellip19dagger





Af ˆ dd2

185 hellip20dagger






























H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger


frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger






ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m




Af ˆ dd 2







hdd

7ˆ 1

7

d1 Dagger d2

2 hellip19dagger





Af ˆ dd2

185 hellip20dagger






























H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger


frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger






ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m





Af ˆ dd2

185 hellip20dagger






























H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger


frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger






ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m





























H

H0

ˆ 1 Daggerh

h

1=2

hellip21dagger


frac14y

frac14y0

2

ˆ k 1 Dagger h

h hellip22dagger






ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m





ACKNOWLEDGEMENTS






For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m



For 001lt EE 4 01



c hellipA 2dagger


hellipA 3dagger


c hellipA 4dagger

For 01lt EE 4 1



c hellipA 6dagger


hellipA 7dagger


c hellipA 8dagger

For 1lt EE 4 10



c hellipA 10dagger


hellipA 11dagger


c hellipA 12dagger






Ec=Es 0001 0005 001 005


10 A A A A






Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m








Ec=Es 0001 0005 001 005


10 A A B B



Ec=Es 0001 0005 001 005


10 B B B B


hm hf dF=dhhˆhmAf



505 09 mN 054 001 0437 0004 890 30 4684 00051000 20 mN 243 002 204 002 3500 150 102 3



m


8900 26 mN 0172 0005 0119 0009 240 7 064 003 0301 00188947 5 mN 0212 0006 0127 0002 290 8 077 013 0199 0011500 10 mN 164 015 105 023 2000 322 35 5 019 004

1500 70 N 89 8 plusmn plusmn 230000 7000 019 004











J Mater Res 13 2071







m













J Mater Res 13 2071







m



determination of the mechanical properties of metallic...

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