Mechanics of Materials 69 (2014) 146–158

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Mechanics of Materials

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Correction of indentation load–depth curve based on elasticdeformation of sharp indenter

0167-6636/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.mechmat.2013.10.002

⇑ Corresponding author. Tel.: +82 2 705 8636; fax: +82 2 712 0799.E-mail address: minsist@sogang.ac.kr (M. Kim).

Minsoo Kim ⇑, Sungsik Bang, Felix Rickhey, Hyungyil LeeSogang University, Department of Mechanical Engineering, Seoul 121-742, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 June 2012Received in revised form 30 August 2013Available online 11 October 2013

Keywords:Sharp indenterLoad–depth curveKick’s law coefficientRigid indenterElastic indenterHalf-included angle

Indenters in numerical analyses are usually assumed rigid. For rigid indenters, the ratios ofKick’s law coefficient to elastic modulus (C/E) are equal for materials with equal values ofyield strain (eo) and strain hardening coefficient (n). However, this is not the case for elasticindenters as indenter deformation leads to a shift of the load–depth (P–h) curve.

In this study, we employ conical indenters with half-included angle h = 70.3�, tip radiusR = 0.025 mm to derive a corrected P–h curve; maximum indentation depth is set tohmax = 0.05 mm. Applying finite element analysis (FEA), we show that materials with equaleo and n give equal C/E-values for rigid indenters but different C/E-values for elastic inden-ters. To account for the elastic deformation of the indenter, we establish a method by intro-ducing a correction factor j so that equal C/E-values are obtained. It is found that for agiven h, j does not change when varying indenter size, indentation depth and indentertip-radius. For a different indenter angle h, a different value of j is obtained, yet by follow-ing the same correction procedure.

We further demonstrate that corresponding j values for triangular pyramidal indentersare different from those of conical ones, due to the difference in actual projected contactareas. Consequently, the proposed method is applicable to any indenter shape.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

By indentation, material properties (such as Young’smodulus E, yield strength ro, strain hardening exponentn, hardness, fracture toughness, creep coefficient) orresidual stresses can be derived from the load–depthcurves obtained by micro-indentation of micro-specimensor parts in use.

In most of the early studies on indentation, the theoret-ical solution to the Boussinesq problem proposed by Sned-don (1965) was applied to determine Young’s modulus.However nowadays, the majority of researchers use theOliver and Pharr (1992) method. Recently, efforts havebeen made not only to obtain elastic properties but to ex-tend the frame to elasto-plastic properties. Lee et al. (2005)simulated spherical indentation using FEA and suggested

dimensionless indentation variables to quantify the influ-ence of material properties; based on this, material proper-ties were determined. Sharp indenters (conical, triangularpyramidal, quadrangular pyramidal) are preferred to per-fectly spherical indenters due to their relatively easy man-ufacturing, and consequently research on sharp indentersis more advanced (Bucaille et al., 2003; Chen et al., 2007;Chollacoop et al., 2003; Dao et al., 2001; Hyun et al.,2011; Le, 2008, 2009; Lee et al., 2008, 2010; Ogasawaraet al., 2005; Swaddiwudhipong et al., 2005; Tho et al.,2004).

A schematic load–depth curve for indentation with aself-similar sharp indenter is depicted in Fig. 1. The curvefollows Kick’s law

P ¼ Ch2 ð1Þ

where P, h and C are indentation load, indentation depth(0 6 h 6 hmax) and Kick’s law coefficient, respectively. Pmax

denotes the maximum load and the corresponding depth is

Fig. 1. Schematic illustration of a P–h curve of elastic–plastic materialindented with self-similar sharp indenter.

Fig. 2. Schematic of indentation profiles with finite tip-radius.

M. Kim et al. / Mechanics of Materials 69 (2014) 146–158 147

hmax. The slope S is defined as S � dP/dh at h = hmax (Fig. 1).Since for indentation with self-similar sharp indenters, theself-similarity vanishes due to the tip radius R, a method is

Fig. 3. Overall mesh design using axisymmetric conical indenter [25,

Table 1Material properties for FE analyses.

Material properties of indenters Material propert

Rigid (EI =1) Elastic modulusPoisson’s ratio (m

Tungsten carbide WC(EI = 537 GPa, mI = 0.24)

Yield strain (eo(�Strain hardening

required to eliminate the influence of the tip radius. Asshown in Fig. 2, Lee et al. (2008) consider the gap hg be-tween sharp indenters with zero and finite tip radius.The meanings of subscripts are self-evident. Indentergeometry gives the relation between R and hg

hg ¼ R1

sin h� 1

� �ð2Þ

Denoting the half-included angle by h, Lee et al. (2008)eliminated the influence of the tip radius by allowing forthe difference in indentation depths (=hg) between a per-fectly conical indenter and an indenter with finite radiusso that Eq. (1) becomes

P ¼ C hþ hg� �2 ð3Þ

In case of sharp indenters, materials with differentmaterial properties may produce equal load–depth curvesdue to self-similarity. Chen et al. (2007) and Lee et al.(2008) for conical indenters and Hyun et al. (2011) andKim et al. (2013) for sharp indenters studied the relationbetween the strain hardening exponent and the Kick’slaw coefficient C for different yield strains (eo � ro/E) andshowed that there are many materials with equal C-values.Therefore, when evaluating material properties with self-similar indenters, at least two indenters with differenthalf-included angles need to be utilized. Bucaille et al.(2003) and Chollacoop et al. (2003) suggested a methodto evaluate indentation properties by employing two con-ical indenters with different half-included angles. In theseand other studies, representative strain was used to deter-mine material properties. Swaddiwudhipong et al. (2005)

400 nodes and 25,000 4-node axisymmetric elements (CAX4)].

ies of specimens Values used in FEA

(E) 70, 200, 300 (GPa)) 0.3ro/E)) 0.001, 0.004, 0.006, 0.010exponent (n) 3, 5, 10

(h +hg)2

0.000 0.001 0.002 0.003

P

0

50

100

150

200 θ = 70.3o, εo = 0.001, n = 10

70200300

70200300

E (GPa) [rigid indenter]

E (GPa) [WC indenter]

70

200

300E (GPa)

hmax = 0.05 mm, R = 0.025 mm(R/hmax = 0.5)

Fig. 4. P vs. (h + hg)2 curves for various indenters.

(h +hg)2/hmax

2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

P/Eh

max

2

0.00

0.05

0.10

0.15

0.20 θ = 70.3o, R/hmax = 0.5, εo = 0.001, n = 10

70200300

70200300

E (GPa) [rigid indenter]

E (GPa) [WC indenter]

Fig. 5. Normalized P vs. (h + hg)2 curves for various indenters (eo = 0.001,n = 10).

(h +hg)2/hmax

2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

P/Eh

max

2

0.0

0.3

0.6

0.9

1.2

1.5θ = 70.3o, R/hmax = 0.5, εo = 0.010, n = 3

E (GPa)70 70

200300WC indenter

E (GPa) [rigid indenter]

300200

(a)

(h +hg)2/hmax

2

0.90 0.95 1.00 1.05 1.10

P/Eh

max

2

0.4

0.8

1.2θ = 70.3o, R/hmax = 0.5, εo = 0.010, n = 3

E (GPa)70

200300

70200300WC indenter

E (GPa) [rigid indenter]

(b)

Fig. 6. Normalized P vs. (h + hg)2 curves for various indenters (a)normalized P vs. (h + hg)2 curves (b) an enlargement of the0.9 < (h + hg)2/hmax

2 < 1.1 region [eo = 0.010, n = 3].

1/n0.0 0.1 0.2 0.3 0.4 0.5 0.6

C/E

0.0

0.3

0.6

0.9

1.2

70200300

θ = 70.3oεo

0.010

0.0060.004

0.001E (GPa)

WC indenter

Fig. 7. C/E vs. 1/n curves for various values of eo (WC indenter).

148 M. Kim et al. / Mechanics of Materials 69 (2014) 146–158

and Le (2008, 2009) took an energy-based approach, andobtained elasto-plastic material properties for severalmaterials by defining the coefficients of curvature as afunctions of material properties for two indenters with dif-ferent half-included angles. Hyun et al. (2011) and Kimet al. (2013) expressed C as a function of eo and n and pro-posed an algorithm through which the stress–strain curvecan be derived from the load–depth curve for indentationwith two self-similar indenters with different half-included angles.

While for rigid indenters, the dimensionless ratio C/E isthe same for materials with identical eo and n, this is notthe case for elastic indenters as here the elasticity of the in-denter causes a shift of the load–depth curve so that C/E-values (for materials with equal eo and n-values) differ.

Fig. 8. (a) Schematic of conical indenter composed of cylindrical and tipparts and (b) an enlargement of the part shown dotted box (the vicinity ofindenter tip).

Table 2Values of Cf from various j for h = 70.3�.

zT (mm) Conical indenter h = 70.3�, Cf (nm/N)

j (�zT1/hmax)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 15.165 7.582 5.055 3.791 3.033 2.528 2.166 1.896 1.685

Table 3Comparison of [C/E]c values for different elastic moduli (eo = 0.010, n = 3).

Elastic modulus (C/E) [C/E]c (h = 70.3�, eo = 0.010, n = 3)

j (�zT1/hmax)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

E70 (0.937) 1.033 0.996 0.976 0.966 0.960 0.956 0.954 0.952 0.950E200 (0.869) 1.231 1.026 0.969 0.942 0.927 0.917 0.910 0.905 0.901E300 (0.822) 1.379 1.045 0.961 0.923 0.901 0.887 0.877 0.870 0.864Gap70–200 (%) 16.1 2.8 0.7 2.6 3.6 4.4 4.8 5.2 5.5Gap200–300 (%) 12.0 1.9 0.8 2.0 2.8 3.2 3.5 3.8 4.0

κ ( / )

0.0 0.2 0.4 0.6 0.8 1.0 1.2

[C/E

] c

0.8

0.9

1.0

1.1

1.2

1.3

1.4

E (GPa)

300

70200

0.937

0.8690.822

C/E

θ = 70.3o, εo = 0.010, n = 3

1 maxTL h≡

reg. lineκ = 0.27

(a)

κ ( / )0.0 0.2 0.4 0.6 0.8 1.0

Gap

of [C

/E] c (

%)

0

3

6

9

12

15

18

1 maxTL h≡

[C/E70]c : [C/E200]c

[C/E200]c : [C/E300]c

θ = 70.3o, εo = 0.010, n = 3(b)

Fig. 9. (a) [C/E]c with j for three E values and (b) gap of [C/E]c with j forthree E values (h = 70.3�, eo = 0.010, n = 3).

n2 4 6 8 10 12 14

Gap

of C

/E (%

)

0

3

6

9

12

15

εo

0.010

C/E70 > C/E300

0.0060.004

0.001

θ = 70.3 o(a)

n0 2 4 6 8 10 12

Gap

of [C

/E] c (

%)

0.0

0.5

1.0

1.5

2.0

εo0.010

0.0060.004

0.001

θ = 70.3o, κ ( / ) = 0.31 maxTz h≡(b)

Fig. 10. (a) Gap of C/E70–300 and (b) Gap of [C/E]c70–300 with n for variousvalues eo.

M. Kim et al. / Mechanics of Materials 69 (2014) 146–158 149

Although indenters used in testing are made of elasto-plas-tic materials such as tungsten–carbide or diamond, in mostof the studies on indentation they are assumed to be rigid(Bucaille et al., 2003; Chen et al., 2007; Cheng and Cheng,1999; Chollacoop et al., 2003; Dao et al., 2001; Fischer-Cripps, 2003; Le, 2008, 2009; Ogasawara et al., 2006;Swaddiwudhipong et al., 2005). Further, in studies whereelastic indenters were used (Hyun et al., 2011; Kim et al.,

2013), the change of C/E with E could not be explained.For real elastic indenters, a correction of the C/E-value istherefore necessary so as to determine a representativeC/E-value regardless of elastic modulus of indenter.

This paper is organized as follows. In chapter 2, weshow that the P–h curves from both rigid and elastic conicalindenters follow Kick’s law. We then demonstrate that formaterials of equal eo and n, C/E-values are equal for a rigidindenter but different for an elastic indenter. In chapter 3,for a given indenter angle, we find representative C/E-val-

150 M. Kim et al. / Mechanics of Materials 69 (2014) 146–158

ues by introducing a correction factor j. We also check theinfluence of indentation variables (indenter size, indenta-tion depth, tip radius) on j. In chapter 4, the variation ofj with indenter angle h is dealt with. In chapter 5, we showthat the j values of triangular pyramidal indenters are dif-ferent from those of conical ones, and the reason for this isbriefly discussed.

2. Conical indenters and Kick’s law

As mentioned above, the loading curves for self-similarsharp indenters, e.g. conical and pyramidal indenters, gen-erally follow Kick’s law (Eq. (1)). In this study, we give con-sideration to the elastic deformation of conical indenterswith h = 70.3� based on the study by Hyun et al. (2011).We use the same values for indentation depth and tip ra-dius as in Hyun et al. (2011), that is hmax = 0.05 mm andR = 0.025 mm. The corrections for various indenter sizes,indentation depths and tip radii are described in theAppendices.

Fig. 3 shows the 2D FE model of the conical indentationand it consists of about 25,400 nodes and 25,000 4-nodeaxisymmetric elements (CAX4) (ABAQUS, 2010). Axisym-metric boundary conditions are imposed on the nodes onthe axis of symmetry. The indenter moves down to pene-trate the material the bottom of which is fixed. The inden-ter is assumed rigid (EI =1) or made of tungsten carbide(WC; EI = 537 GPa, mI = 0.24). In simulations with WC ind-enters, we consider only elastic indenter deformation. Ta-ble 1 shows 36 different materials (E = 70200, 300 GPa;eo = 0.001, 0.004, 0.006, 0.010; n = 3, 5, 10) used to analyzethe change of C/E. The values cover the range of generalmetallic materials.

P � (h + hg)2-curves are obtained by conical indentationof materials with eo = 0.001 and n = 10 but different elasticmoduli E = 70, 200, 300 GPa, using rigid as well as WC-ind-

Table 4.1Comparison of [C/E]c values for different elastic moduli with h = 70.3� indenter(n

n = 3 Elastic modulus (C/E) j (�zT1/h

0.2

eo = 0.001 [C/E]c E70 (0.286) 0.291E200 (0.281) 0.297E300 (0.277) 0.299Gap70–200 (%) 1.9Gap200–300 (%) 0.7

eo = 0.004 [C/E]c E70 (0.614) 0.639E200 (0.587) 0.656E300 (0.566) 0.664Gap70–200 (%) 2.5Gap200–300 (%) 1.3

eo = 0.006 [C/E]c E70 (0.747) 0.785E200 (0.706) 0.807E300 (0.676) 0.821Gap70–200 (%) 2.8Gap200-300 (%) 1.8

eo = 0.010 [C/E]c E70 (0.937) 0.996E200 (0.869) 1.026E300 (0.822) 1.045Gap70–200 (%) 2.8Gap200–300 (%) 1.9

enters, with h = 70.3� and R = 0.025 mm (Fig. 4). Indepen-dent of material properties of specimen and indenter, thecurve follows Kick’s law. Normalizing ordinate and abscis-sa of the P � (h + hg)2-diagram by Eh2

max and h2max, respec-

tively, we obtain coinciding load–depth curves formaterials with equal eo and n, independent of E (Fig. 5).However, when increasing eo to eo = 0.010 while decreasingn to n = 3, i.e. pronouncing the influence of material’s elas-ticity, the normalized P � (h + hg)2-curves for materialswith equal eo and n do not coincide. In Fig. 6, Kick’s lawis applied but the difference between C/E-values in therange of C/E70 > C/E200 > C/E300 amounts to 13%. The devia-tion becomes clear in Fig. 6(b), which is an enlargement ofthe red box in Fig. 6(a).

The values of C/E and Pmax from indentation with rigidindenters are higher than those from indentation withelastic indenters. The shift of P–h curve can be explainedby the elastic deformation of indenter. Hyun et al. (2011)and Kim et al. (2013) established a function of eo and nto determine C/E and suggested a dual indentation methodfor material property evaluation requiring two self-similarindenters with different angles. However, since C/Echanges with E, one has to use a different regression for-mula for each value of E. Therefore, to make C/E a represen-tative value independent of the elastic modulus of thematerial, a correction of the C/E-value is necessary. In thenext chapter, we propose a correction formula giving rep-resentative C/E-values independent of E.

3. Correction formula for elastic conical indenter

As shown in Fig. 6, for elastic indenters, C/E-values formaterials with equal eo and n differ from each other dueto deformation of the indenter resulting in a downwardshift of P–h curves. Based on the finding that the differencein C/E increases the higher eo and the lower n is (Fig. 7), we

= 3).

max)

0.3 0.4 0.5 0.6

0.289 0.288 0.288 0.2870.291 0.289 0.287 0.2860.291 0.287 0.285 0.284

0.7 0.1 0.2 0.50.1 0.5 0.7 0.8

0.630 0.626 0.624 0.6220.632 0.620 0.613 0.6090.628 0.612 0.602 0.596

0.2 1.1 1.7 2.20.5 1.3 1.8 2.1

0.772 0.766 0.762 0.7590.771 0.754 0.744 0.7370.768 0.743 0.729 0.719

0.2 1.6 2.5 3.00.4 1.4 2.0 2.4

0.976 0.966 0.960 0.9560.969 0.942 0.927 0.9170.961 0.923 0.901 0.887

0.7 2.6 3.6 4.40.8 2.0 2.8 3.2

Table 4.2Comparison of [C/E]c values for different elastic moduli with h = 70.3� indenter(n = 5).

n = 5 Elastic modulus (C/E) j (�zT1/hmax)

0.2 0.3 0.4 0.5 0.6

eo = 0.001 [C/E]c E70 (0.190) 0.193 0.192 0.192 0.191 0.191E200 (0.188) 0.195 0.193 0.192 0.191 0.190E300 (0.187) 0.197 0.193 0.191 0.190 0.190Gap70–200 (%) 1.1 0.3 0.1 0.2 0.4Gap200–300 (%) 0.8 0.2 0.1 0.3 0.4

eo = 0.004 [C/E]c E70 (0.484) 0.500 0.494 0.492 0.490 0.489E200 (0.469) 0.512 0.497 0.489 0.485 0.482E300 (0.457) 0.520 0.497 0.486 0.480 0.476Gap70–200 (%) 2.3 0.4 0.5 1.1 1.4Gap200–300 (%) 1.6 0.1 0.6 1.0 1.2

eo = 0.006 [C/E]c E70 (0.616) 0.642 0.633 0.629 0.626 0.625E200 (0.590) 0.660 0.635 0.623 0.617 0.612E300 (0.571) 0.673 0.636 0.619 0.609 0.602Gap70–200 (%) 2.7 0.3 0.9 1.6 21.Gap200–300 (%) 2.1 0.2 0.7 1.3 1.6

eo = 0.010 [C/E]c E70 (0.813) 0.857 0.842 0.835 0.830 0.827E200 (0.764) 0.883 0.841 0.821 0.809 0.801E300 (0.730) 0.902 0.838 0.809 0.792 0.781Gap70–200 (%) 2.9 0.2 1.7 2.7 3.3Gap200–300 (%) 2.1 0.3 1.4 2.0 2.5

Table 4.3Comparison of [C/E]c values for different elastic moduli with h = 70.3� indenter(n = 10).

n = 10 Elastic modulus (C/E) j (�zT1/hmax)

0.2 0.3 0.4 0.5 0.6

eo = 0.001 [C/E]c E70 (0.141) 0.142 0.142 0.142 0.142 0.142E200 (0.140) 0.144 0.142 0.142 0.141 0.141E300 (0.139) 0.145 0.143 0.142 0.141 0.141Gap70–200 (%) 0.8 0.2 0.1 0.2 0.3Gap200–300 (%) 0.8 0.4 0.2 0.1 0.1

eo = 0.004 [C/E]c E70 (0.407) 0.418 0.415 0.413 0.412 0.411E200 (0.396) 0.426 0.416 0.411 0.408 0.406E300 (0.389) 0.433 0.418 0.410 0.406 0.403Gap70–200 (%) 1.9 0.3 0.4 0.9 1.2Gap200–300 (%) 1.6 0.4 0.2 0.5 0.8

eo = 0.006 [C/E]c E70 (0.535) 0.554 0.547 0.544 0.542 0.541E200 (0.516) 0.569 0.550 0.541 0.536 0.533E300 (0.502) 0.580 0.552 0.539 0.532 0.526Gap70–200 (%) 2.6 0.5 0.6 1.1 1.6Gap200–300 (%) 2.1 0.4 0.3 0.8 1.2

eo = 0.010 [C/E]c E70 (0.731) 0.767 0.754 0.748 0.745 0.742E200 (0.693) 0.791 0.757 0.740 0.730 0.724E300 (0.667) 0.812 0.759 0.734 0.720 0.711Gap70–200 (%) 3.1 0.3 1.2 2.0 2.6Gap200–300 (%) 2.7 0.3 0.7 1.4 1.7

M. Kim et al. / Mechanics of Materials 69 (2014) 146–158 151

suggest the following correction formula by consideringshape and material properties of the indenter

hc ¼ h� Cf P ð4Þ

Here hc is the corrected indentation depth, Cf (=CCf + CT

f )(mm/N) the correction coefficient, and P the load at inden-tation depth h. Cf is the sum of the C-value for the cylindri-cal part (CC

f ) and that for the tip part (CTf ); both are

independent of the material and functions of the indentergeometry. CC

f and CTf are defined as follows:

CCf ¼

zC

ACEI

; CTf ¼

1EI

Z zT1þzT

2

zT1

1

AT dzT

¼ 1EI

Z zT1þzT

2

zT1

1LT2p tan2 h

dzT ð5Þ

Here EI is the indenter’s elastic modulus, zC and zT theheight of cylindrical part and tip, respectively, and AC andAT the corresponding cross-sections. Note that unlike realindenters, a mathematically sharp elastic indenter causesa stress singularity at the tip. To avoid the associated

1/n0.0 0.1 0.2 0.3 0.4 0.5 0.6

C/E

0.0

0.1

0.2

0.3

70200300

θ = 45o

εo

0.010

0.0060.004

0.001E (GPa)

WC indenter

Fig. 11. C/E vs. 1/n curves for various values of yield strain (h = 45�)(cf. Fig. 7 (h = 70.3�)).

κ ( / )

0.0 0.2 0.4 0.6 0.8 1.0 1.2

[C/E

] c

0.18

0.24

0.30

0.36E (GPa)

300

70

200

0.2110.1980.189

C/E

θ = 45o

1 maxTz h≡

εo = 0.010, n = 3

reg. line

κ = 0.72

(a)

κ ( / )0.0 0.2 0.4 0.6 0.8 1.0

Gap

of [C

/E] c (

%)

0

5

10

15

20

1 maxTz h≡

[C/E70]c : [C/E200]c

[C/E200]c : [C/E300]c

θ = 45o, εo = 0.010, n = 3(b)

Fig. 12. (a) [C/E]c with j for three E values and (b) Gap of [C/E]c with j forthree E values (h = 45�, eo = 0.010, n = 3) (cf. Fig. 9 (a),(b):h = 70.3�).

152 M. Kim et al. / Mechanics of Materials 69 (2014) 146–158

numerical problem, we set the lower limit of integration toa finite value z1

T rather than zero. In this study we assumea value for zT

1; further, h = 70.3�, zC = 0.25, and zT = 0.25 mm(Fig. 8). Inserting Eq. (5) into Eq. (4), we get

hc ¼ h� 1EI

LC

AC þ1

p tan2 h� 1

zT1 þ zT

2

þ 1zT

1

� �" #P ð6Þ

Except for zT1, as zT

1 + zT2 is a given constant (zT), all values

in Eq. (6) are known constants. Then, defining zT1 � j hmax,

j 2 [0,1], we can rewrite Eq. (6) analogous to Eq. (4) as

hc ¼ h� aþ bj

� �P ð7Þ

The constants a and b in Eq. (7) depend on indentergeometry and indentation depth. For j, values between0.1 and 0.9 are assumed; the corresponding Cf values arelisted in Table 2. For materials where the difference be-tween C/E-values is high (eo = 0.010, n = 3), we calculatethe [C/E]c-values and compared their relative deviationfrom the referential [C/E200]c-value (Table 3).

In Fig. 9(a), the [C/E]c-values are shown for different val-ues of C/E (square symbols) and j. Independent of j, wefind C/E < [C/E]c. Solid lines are the regression lines forthe value of [C/E]c obtained with j values in the range[0.1,0.9]. The j value at the intersection of the threeregression lines is 0.27. However, j = 0.3 can be consideredas a correction factor, because the difference among the [C/E]c values is smallest (less than 1%) (Fig. 9(b)). It can fur-ther be noted that for higher E the difference between [C/E]c-values is higher. The j-value is independent of thelengths zC and zT, i.e. independent of the indenter size, asshown in Appendix A In conclusion, j = 0.3 provides thea representative value to correct C/E-values for conical ind-enters with h = 70.3�. It is yet necessary to check whetherthe value of j is applicable to a wide range of materials.

Table 5Values of Cf from various j for h = 45� [cf. Table 2 (h = 70.3�)].

zT (mm) Conical indenter h = 45�, Cf (nm/N)

j (�zT1/hmax)

0.2 0.3 0.4 0.5

0.50 59.276 39.517 29.638 23.710

We thus investigate the representativeness of j = 0.3for various materials (Table 1). The gap between valuesof [C/E]c changes depending on j but is smaller than thegap between C/E-values. Albeit different in magnitude,the average difference for C/E70 and C/E300 is 6.6% but de-creases to 0.7% after correction with j = 0.3 (Fig. 10). Ifthere is negligible difference between the [C/E]c-values,we can consider the [C/E]c-values to be independent ofthe material’s E. Therefore, j = 0.3 applies also to othermaterials for conical indentation with h = 70.3� (Tables 4).For brevity here, we note that regardless of indentationdepth and tip radius, the proposed correction method isapplicable (Appendix B and C). When expanding the scopefurther and changing the indenter angle, it is necessary tocheck again whether there is a constant j value. This willbe discussed in the next chapter.

4. Variation of j with indenter angle

Hyun et al. (2011) proposed a method for evaluatingmaterial properties by dual indentation. This method basesupon the C/E-values obtained from indentation with twoconical indenters with different h (h = 70.3� and 45�). For

0.6 0.7 0.8 0.9

19.759 16.936 14.819 13.172

Fig. 13. Schematic of two indenters with same projected contact area.

Fig. 15. Schematic of Berkovi

1/n0.0 0.1 0.2 0.3 0.4 0.5 0.6

C/E

0.0

0.3

0.6

0.9

1.2

70200300

α = 65.3o εo

0.010

0.0060.004

0.001 E (GPa)

WC indenter

(a)

1/n0.0 0.1 0.2 0.3 0.4 0.5 0.6

C/ E

0.0

0.1

0.2

0.3

70200300

α = 37.9o

εo

0.010

0.0060.004

0.001E (GPa)

WC indenter

(b)

Fig. 14. C/E vs. 1/n curves for various values of eo with triangularpyramidal indenter ((a) a = 65.3�: Berkovich, (b) a = 37.9�).

M. Kim et al. / Mechanics of Materials 69 (2014) 146–158 153

conical indenters with h = 70.3�, we have shown in chapter3 that j = 0.3. Despite the different indenter angle, bystudying the change of C/E with j, we could confirm therepresentativeness of j = 0.3 even for materials with highgaps between C/E-values. As for h = 70.3�, we take {zC,zT}= {0.25 mm, 0.25 mm} for h = 45� (Fig. 8). For indenterswith h = 45�, C/E-values differ for materials with equal eo

and n (Fig. 11), as was the case for h = 70.3�. ApplyingEqs. (6) and (7), we obtain a = 1.963 � 10�7 andb = 4.573 � 10�5. Cf-values corresponding to each j in therange j 2 [0.2,0.9] – the same set that was taken forh = 70.3� – are listed in Table 5. j = 0.1 is excluded fromthe range of j because the gap between [C/E]c-values ishigher than 40%. Similar to the case h = 70.3�, we find thatregardless of j, C/E < [C/E]c) Fig. 12(a)); however, forh = 45�, the gap between [C/E]c-values is smallest whenj = 0.7 (Fig. 12(b)). This value holds for a wide range ofmaterials as shown in (Tables 6). Therefore, for conical ind-enters with h = 45�, the correction factor is j = 0.7.

5. Values of j for triangular pyramidal indenters

The representative j value is now determined for trian-gular pyramidal indenters. Fig. 13 shows the schematicdiagram of a triangular pyramidal indenter, and its equiv-

ch indenter (cf. Fig. 8).

κ ( / )0.0 0.2 0.4 0.6 0.8 1.0 1.2

[C/E

] c

0.8

1.0

1.2

1.4

1.6 E (GPa)300

70

200

0.9890.9350.897

C/E

α = 65.3o, εo = 0.010, n = 3

1 maxTz h≡

reg. line

κ = 0.35

(a)

κ ( / )0.0 0.2 0.4 0.6 0.8 1.0 1.2

[C/E

] c

0.18

0.24

0.30

0.36

E (GPa)300

70

200

0.2130.2000.192

C/E

α = 37.9o

1 maxTz h≡

εo = 0.010, n = 3

reg. line

κ = 0.53

(b)

Fig. 16. Gap of [C/E]c with j for three E values and optimum j (eo = 0.010,n = 3) (cf. Figs. 10 and 14 (conical indenter)).

154 M. Kim et al. / Mechanics of Materials 69 (2014) 146–158

alent axisymmetric conical indenter with an indenterangle giving the same cross-sectional area. If the center-line-to-face angle of the triangular pyramidal indenter (a)is given, the corresponding half-included angle of theconical indenter (h) can be calculated with Eq. (8).

Table 6.1Comparison of [C/E]c values for different elastic moduli with h = 45 indenter (n =

n = 3 Elastic modulus (C/E) j (�LT1/hmax)

0.2 0.3eo = 0.001 [C/E]c E70 (0.056) 0.058 0.057

E200 (0.055) 0.061 0.059E300 (0.055) 0.063 0.060Gap70–200 (%) 4.9 2.8Gap200–300 (%) 3.7 2.0

eo = 0.004 [C/E]c E70 (0.129) 0.140 0.136E200 (0.124) 0.156 0.144E300 (0.120) 0.170 0.151Gap70-200 (%) 10.5 5.7Gap200-300 (%) 9.1 4.6

eo = 0.006 [C/E]c E70 (0.161) 0.179 0.173E200 (0.154) 0.206 0.186E300 (0.148) 0.230 0.197Gap70–200 (%) 13.1 7.0Gap200-300 (%) 11.8 5.8

eo = 0.010 [C/E]c E70 (0.211) 0.241 0.231E200 (0.198) 0.291 0.254E300 (0.189) 0.337 0.272Gap70–200 (%) 16.9 9.1Gap200–300 (%) 15.8 7.3

h ¼ tan�1

ffiffiffiffiffiffiffiffiffiffi3ffiffiffi3p

p

stan a

0@

1A ð8Þ

At equal indentation depths, the half-included angleh = 70.3� of a conical indenter gives the same projectedcontact area as the triangular pyramidal indenter witha = 65.3� (Berkovich) (Fig. 13). Equally, inserting the valuesof conical indenter h = 45� into Eq. (8), we arrive ata = 37.9�. Again, C/E-values differ for materials with equaleo and n (Fig. 14). We therefore assume the correction todepend on the indenter shape. Considering geometricalsymmetry, we make the 1/6 triangular pyramidal indentermodel using 45,000 nodes and 36,000 8-node brick ele-ments (C3D8) (ABAQUS, 2010). Due to the absence of acylindrical part in our triangular pyramidal indenter model(Fig. 15), Eq. (6) can be transformed into Eq. (9)

hc ¼ h� 1EI

1p tan2 h

� 1zT

1 þ zT2

þ 1zT

1

� �� �P ð9Þ

We take the same values for zT, hmax and the same rangeof j for the triangular pyramidal indenter as were taken forthe conical indenter, i.e. zT = 0.25 mm, hmax = 0.05 mm,j65.3 2 [0.1,0.9], j37.9 2 [0.2,0.9]. We perform above cor-rection to a material having a large gap between C/E-values(eo = 0.010 and n = 3), and obtain j65.3 = 0.35 andj37.9 = 0.53 as representative values (Fig. 16).

Invoking that the optimum j-values for conical inden-ters were j70.3 = 0.3, j45 = 0.7, we find that the optimumj also depends on the indenter shape. The gap betweenC/E-values increases with increasing indenter angle andelastic modulus (Table 7). Shim et al. (2007) and Kimet al. (2013) reported that conical and triangular pyramidalindenters related by Eq. (8) produce different contact areas

3).

0.4 0.5 0.6 0.7 0.8 0.90.057 0.057 0.057 0.057 0.057 0.0560.058 0.057 0.057 0.057 0.057 0.0560.059 0.058 0.057 0.057 0.057 0.0561.7 1.1 0.7 0.3 0.1 0.11.2 0.7 0.3 0.1 0.1 0.1

0.134 0.133 0.132 0.132 0.132 0.1310.139 0.136 0.134 0.132 0.131 0.1300.142 0.138 0.135 0.132 0.131 0.1303.3 1.9 0.9 0.3 0.2 0.62.5 1.4 0.6 0.1 0.3 0.5

0.170 0.168 0.167 0.166 0.166 0.1650.177 0.172 0.169 0.167 0.165 0.1640.183 0.175 0.170 0.167 0.164 0.1634.0 2.3 1.1 0.2 0.5 0.93.2 1.7 0.8 0.2 0.3 0.7

0.226 0.223 0.221 0.219 0.218 0.2180.238 0.229 0.224 0.220 0.217 0.2150.247 0.234 0.225 0.220 0.216 0.2125.1 2.8 1.3 0.2 0.7 1.33.8 1.9 0.7 0.1 0.7 1.1

Table 6.2Comparison of [C/E]c values for different elastic moduli with h = 45 indenter (n = 5).

n = 5 Elastic modulus (C/E) j (�LT1/hmax)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9eo = 0.001 [C / E]c E70 (0.032) 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033

E200 (0.032) 0.034 0.033 0.033 0.033 0.033 0.033 0.033 0.032E300 (0.032) 0.035 0.034 0.03 0.033 0.033 0.033 0.032 0.032Gap70-200 (%) 2.4 1.3 0.6 0.3 0.1 0.1 0.2 0.3Gap200-300 (%) 2.1 1.2 0.6 0.4 0.2 0.1 0.1 0.1

eo = 0.004 [C / E]c E70 (0.090) 0.095 0.093 0.092 0.092 0.091 0.091 0.091 0.091E200 (0.087) 0.102 0.097 0.094 0.093 0.092 0.091 0.091 0.090E300 (0.086) 0.109 0.100 0.096 0.094 0.093 0.092 0.091 0.090Gap70-200 (%) 7.2 3.8 2.1 1.0 0.5 0.1 0.3 0.6Gap200-300 (%) 6.2 3.3 2.0 1.4 0.7 0.3 0.1 0.1

eo = 0.006 [C / E]c E70 (0.118) 0.127 0.124 0.122 0.122 0.121 0.121 0.120 0.120E200 (0.114) 0.141 0.131 0.127 0.124 0.122 0.121 0.120 0.120E300 (0.111) 0.154 0.138 0.130 0.126 0.124 0.122 0.121 0.120Gap70-200 (%) 10.1 5.6 3.4 2.1 1.2 0.6 0.1 0.3Gap200-300 (%) 8.9 4.7 2.8 1.7 1.0 0.6 0.2 0.1

eo = 0.010 [C / E]c E70 (0.164) 0.182 0.175 0.173 0.171 0.170 0.169 0.168 0.168E200 (0.156) 0.211 0.190 0.181 0.176 0.172 0.170 0.168 0.167E300 (0.151) 0.236 0.201 0.187 0.179 0.174 0.170 0.168 0.166Gap70-200 (%) 13.8 7.7 4.5 2.8 1.6 0.6 0.1 0.5Gap200-300 (%) 12.0 5.8 3.2 1.7 0.8 0.2 0.3 0.7

Table 6.3Comparison of [C/E]c values for different elastic moduli with h = 45 indenter (n = 10).

n = 10 Elastic modulus (C/E) j (�LT1/hmax)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9eo = 0.001 [C / E]c E70 (0.021) 0.022 0.022 0.021 0.021 0.021 0.021 0.021 0.021

E200 (0.021) 0.022 0.022 0.022 0.022 0.021 0.021 0.021 0.021E300 (0.021) 0.022 0.022 0.022 0.022 0.022 0.021 0.021 0.021Gap70-200 (%) 1.8 1.0 0.6 0.5 0.1 0.1 0.1 0.1Gap200-300 (%) 1.6 0.9 0.6 0.3 0.3 0.3 0.1 0.1

eo = 0.004 [C / E]c E70 (0.068) 0.071 0.070 0.069 0.069 0.069 0.069 0.069 0.069E200 (0.067) 0.076 0.072 0.071 0.070 0.070 0.069 0.069 0.069E300 (0.066) 0.079 0.074 0.072 0.071 0.070 0.069 0.069 0.069Gap70-200 (%) 6.2 3.6 2.4 1.6 1.1 0.8 0.5 0.2Gap200-300 (%) 4.8 2.6 1.6 1.0 0.6 0.3 0.1 0.1

eo = 0.006 [C / E]c E70 (0.093) 0.099 0.097 0.096 0.096 0.095 0.095 0.095 0.095E200 (0.091) 0.108 0.102 0.099 0.097 0.096 0.096 0.095 0.095E300 (0.089) 0.115 0.106 0.101 0.099 0.097 0.096 0.095 0.095Gap70-200 (%) 8.1 4.7 2.9 1.9 1.2 0.7 0.3 0.1Gap200-300 (%) 7.2 3.8 2.4 1.5 0.9 0.5 0.2 0.1

eo = 0.010 [C / E]c E70 (0.136) 0.148 0.144 0.142 0.141 0.140 0.139 0.139 0.139E200 (0.131) 0.167 0.154 0.148 0.144 0.142 0.140 0.139 0.138E300 (0.127) 0.184 0.161 0.152 0.146 0.143 0.141 0.139 0.138Gap70-200 (%) 11.5 6.4 3.9 2.4 1.4 0.7 0.2 0.3Gap200-300 (%) 9.8 4.9 2.8 1.6 0.8 0.3 0.2 0.5

Table 7Comparison of C/E values for different indenter shape and elastic moduli (eo = 0.010, n = 3).

Angle E (GPa) Angle E (GPa)

70 200 300 70 200 300

C/E h = 70.3� 0.937 0.869 0.822 h = 45.0� 0.211 0.198 0.189a = 65.3� 0.989 0.935 0.897 a = 37.9� 0.213 0.200 0.192Gap (%) 5.5 7.6 9.1 Gap (%) 0.9 1.0 1.6

M. Kim et al. / Mechanics of Materials 69 (2014) 146–158 155

156 M. Kim et al. / Mechanics of Materials 69 (2014) 146–158

at the same equal indentation depth. In consequence, therepresentative j depends on the indenter shape.

Table B.1Values of C/E for various values of hmax and fixed R = 0.025 mm.

R/hmax (R = 0.025 mm) C/E Gap (%)

70 200 300

1/4 0.821 0.785 0.757 7.81/3 0.817 0.775 0.745 8.91/2 0.813 0.764 0.730 10.2

6. Conclusion

The P–h curves of rigid and elastic sharp indenters fol-low Kick’s law. C/E-values and Pmax of rigid indenter arehigher than those of elastic indenter. For rigid indenter,the C/E-values are the same for different elastic moduli ofthe indented material, while C/E-values vary with elasticmodulus for elastic indenters.

To consider the elastic indenter deformation, we intro-duced a correction factor j, and found that j = 0.3 is theoptimum value to correct P–h curves for conical indenterswith h = 70.3�. For a given h, the optimum j is independentof indenter size, indentation depth or indenter tip-radius.Changing h to 45�, a different value of j is obtained bythe same procedure (j = 0.7). The corresponding j valuesfor triangular pyramidal indenters are different from thosefor conical ones, due to the difference in actual projectedcontact areas. Overall, the proposed correction methodcan be readily extended to other indenter shapes.

Acknowledgments

This work was supported by the National ResearchFoundation of Korea (NRF) Grant funded by the Korea gov-ernment (MEST) (No. NRF-2012 R1A2A2A 01046480).

1/1 0.808 0.754 0.716 11.42/1 0.808 0.750 0.712 11.9

Table B.2Values of [C/E]c with j = 0.3 for various values of hmax and fixedR = 0.025 mm.

R/hmax

(R = 0.025 mm)Cf (nm/N)

[C/E]c Gap(%)

70 200 300

1/4 2.572 0.851 0.864 0.872 2.51/3 3.415 0.846 0.852 0.857 1.31/2 5.055 0.842 0.841 0.839 0.41/1 10.155 0.838 0.832 0.825 1.52/1 20.265 0.839 0.833 0.828 1.4

Appendix A. Invariance of j with indenter size

For z C = zT = 0.25 mm, j = 0.3 represents [C/E]c-values.We confirmed for materials with a high gap between C/E-values (eo = 0.010, n = 3) that the indenter geometry influ-ences the representative j. When z C = zT = 0.25 mm andj = 0.3, the correction parameter Cf is 5.055 � 10�6 mm/N. Comparing Cf-values for the cylindrical part(CC

f = 3.240 � 10�7 mm/N) and the tip part((CT

f = 4.731 � 10�6�mm/N), we find that (CCf � CT

f and thisholds even if j changes; CT

f dominates Cf. We thus in-creased the length of indenter tip part, zT, to 0.5,0.75,1 mm (Fig. A.1), and checked the change of j with in-denter shape. The resulting values of [C/E]c, the corre-

Fig. A.1. Schematic of conical indenter for fixed cylindrical p

sponding j, the gap between [C/E70]c and [C/E200]c arelisted in Table A.1. As the length of indenter tip part (zT)increases, j decreases.

For longer zT, the optimum j value seems to be j = 0.2(Table A.1). However, zC is generally much longer than zT inreal indenters, so that results for zC < 0.5 zT are of hardlyany practical use. The value that best represents the [C/E]c-values for all real indenters is therefore j = 0.3.

Appendix B. Invariance of j with indentation depth

Using an indenter with h = 70.3� and zC = zT = 0.25 mm(Fig. 8), the change of the correction factor is examinedfor the material that exhibits the high gap between C/E-values (eo = 0.010, n = 5), by varying the indentation depth.Note that we used the same values for indentation depthand tip radius as those of Hyun et al. (2011), that is‘‘hmax = 0.05 mm and R = 0.025 mm’’. We now vary theindentation depth as hmax = 0.0125, 0.025, 0.050, 0.075,

art (zC = 0.25 mm) with various lengths of tip part (zT).

Table C.1Values of C/E for various values of R and fixed hmax = 0.05 mm.

R/hmax (hmax = 0.05 mm) C/E Gap (%)

70 200 300

0.0 0.937 0.868 0.820 13.50.5 0.937 0.869 0.822 13.31.0 0.938 0.870 0.823 13.31.5 0.939 0.872 0.825 13.1

R/hmax

0.0 0.5 1.0 1.5 2.0 2.5

C/E

or [C

/E] c

0.7

0.8

0.9

1.0

1.1

1.2E (GPa)εo = 0.01, n = 3

hmax = 0.05 mm

C/E [C/E]c70

200300

0.0(mm) 0.025 0.050 0.075R

Fig. C.1. C/E and [C/E]c vs. R/hmax curves for three values of E.

R/hmax

0.0 0.5 1.0 1.5 2.0 2.5

C/E

or [C

/E] c

0.70

0.75

0.80

0.85

0.90

0.95

1.00

E (GPa)εo = 0.01, n = 5R = 0.025 mm C/E [C/E]c

70200300

Fig. B.1. C/E and [C/E]c vs. R/hmax curves for three values of E.

(h+hg)2 or (hc+hg)

2 (mm2)0.000 0.003 0.006 0.009 0.012

P (k

N)

0.0

0.5

1.0

1.5

2.0

(h + hg)2

(hc + hg)2

E = 200 GPa, εo = 0.01, n = 5

R/hmax

R = 0.025 mm 1/4

1/3

1/2

1/1

Fig. B.2. P vs. (h + hg)2 and (hc + hg)2 curves for various values of R/hmax.

(h+hg)2 or (hc+hg)

2 (mm2)

0.0045 0.0060

P (k

N)

0.8

0.9

1.0

1.1

1.2

(h + hg)2

(hc + hg)2

E = 200 GPa, εo = 0.01, n = 5R = 0.025 mm

1/4

1/3

R/hmax

Fig. B.3. An enlargement of the circled region in Fig. B.2.

M. Kim et al. / Mechanics of Materials 69 (2014) 146–158 157

0.100 mm (R/hmax = 2,1,1/2,1/3,1/4). For all the materials,the C/E value increases with hmax while the deviationbetween C/E-values decreases (Table B.1). Although relatedfigures are omitted here for brevity, we confirmed that theoptimum zT

1 is 0.3 hmax (j = 0.3) for all cases examined. Notethe substantial variation of Cf and slight decrease of [C/E]c

with R/hmax in Table B.2. But the gaps between [C/E70]c and[C/E300]c are sufficiently small (Table B.2, Fig. B.1). InFig. B.2, gray lines imply uncorrected P � (h + hg)2 curves,and black lines corrected P � (hc + hg)2 curves. All curvesfollow Kick’s law. However, due to the variation of Cf withR/hmax, the loading slopes of corrected curves slightly vary(Fig. B.3).

Appendix C. Invariance of j with tip radius

We used the same values for indentation depth and tipradius as those of Hyun et al. (2011), that is ‘‘hmax = 0.05 -mm and R = 0.025 mm’’. To study whether R has any influ-ence on the correction method for hmax = 0.05 mm andh = 70.3�, R is now varied as R = 0, 0.025, 0.050, 0.075 mm(R/hmax = 0, 0.5, 1.0, 1.5). A material with eo = 0.01 andn = 3 is chosen again. Tables C.1 and C.2 and Fig. C.1 clearlyshows that R has a negligible influence on the C/E–value,and j = 0.3 is still valid regardless of the tip-radius ofindenter.

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