Available online at www.sciencedirect.com

Mechanics of Materials 40 (2008) 401–406

www.elsevier.com/locate/mechmat

An experimental method to determine the contactradius changes during a spherical instrumented indentation

J.-M. Collin *, G. Mauvoisin, R. El Abdi

LARMAUR, FRE-CNRS 2717, Bat 10B, Universite de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France

Received 21 February 2007; received in revised form 25 September 2007

Abstract

An experimental method to determine contact radius changes during a spherical instrumented indentation test is pro-posed in this paper. This method is based on the Hertz theory and only depends on the elastic properties of both the inden-ter and the sample. The programming of several loading, unloading and reloading cycles has allowed the determination ofthe unloading stiffness changes and consequently the contact radius changes. Numerical results have shown that the pro-posed method is sensitive to the piling-up or sinking-in. From the contact radius changes we have proposed a new proce-dure in order to obtain the real indentation load–depth curve from the measured curve. An experimental study has allowedus to validate our procedure during the beginning of the unloading cycle.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Spherical indentation; Contact radius changes; Indenter compliance; Friction influence; Experimental measurement

1. Introduction

Indentation test, historically used to determinethe material hardness, has been developed in orderto determine the material mechanical propertiessince several decades. An instrumented indentationtest consists in measuring simultaneously indenta-tion load P and indentation depth h during the pen-etration of an indenter into a sample. The Meyer’shardness (Meyer, 1908; Tabor, 1951) is then givenby the following relationship, where a is the contactradius:

0167-6636/$ - see front matter � 2007 Elsevier Ltd. All rights reserved

doi:10.1016/j.mechmat.2007.10.002

* Corresponding author. Tel.: +33 2 2323 6282.E-mail address: jean_marc.collin@yahoo.fr (J.-M. Collin).

H ¼ Ppa2

: ð1Þ

Many studies have been conducted in order todetermine the contact radius. We can separate thedifferent methods in two categories. The first oneconcerns the determination of the C2 factor intro-duced by Hill et al. (1989). C2 quantifies the degreeof piling-up (C2 > 1) or sinking-in (C2 < 1) duringthe indentation test. A great number of models havebeen proposed, which are presented in detail in thework of Hernot et al. (2006). The use of C2 isrestricted by the chosen behaviour law, indeed thisfactor depends on the sample plasticity properties.

However, the contact radius can be determinedthrough a different method. Li et al. (1997) suggestedusing the results deduced from the Hertz elastic

.

402 J.-M. Collin et al. / Mechanics of Materials 40 (2008) 401–406

contacts theory (1882) in order to determine the con-tact radius changes during a Berkovich indentation.

Applying the Hertz theory to indentation allowsthe determination of a relationship between theindentation load P and depth h in the elastic regime

P ¼ 4

3

E1� m2

R12h

32; ð2Þ

where m is the Poisson ratio. In the case of an elasticindenter, E/(1 � m2) in Eq. (2) has to be replaced bythe reduced modulus E*

E� ¼1� m2

s

� �Es

þ1� m2

i

� �Ei

� ��1

; ð3Þ

where s stands for ‘‘sample’’ whereas i stands for‘‘indenter’’. Moreover, in the Hertz theory, the con-tact radius value is (Rh)1/2, therefore Eq. (2)becomes

P ¼ 4

3

Eah1� m2

: ð4Þ

Galin (1946) proposed a general solution of elas-tic indentation for every kind of axisymmetric ind-enters. This solution is sometimes wronglyattributed to Sneddon (1965). Then, Bulychevet al. (1975, 1976); Shorshorov et al. (1981) used thissolution in application to indentation test and theinitial unloading stiffness can be defined as follows:

S ¼ dPdh¼ 2E�a: ð5Þ

A complete presentation of these questions isproposed in the paper of Borodich and Keer (2004).

Some methods have been developed to determinethe Young modulus from the Hertz theory. We canquote Doerner and Nix (1986); Loubet et al. (1986);Oliver and Pharr (1992).

Concerning the determination of the Youngmodulus and the hardness, Galanov et al. (1984)proposed another way of determination based ona decomposition of the deformation under theindenter. Models which take into account the plasticdeformations are proposed by this author.

Li et al. (1997) suggested using formula (5) inorder not to determine the Young modulus butthe contact radius. Their method is based on a mod-ulated loading and unloading cycle for the determi-nation of the instantaneous contact stiffness. Mostof the five samples studied were considered as elas-tic-perfectly plastic. Authors compared experimen-tal measurements with numerical results in the

case of a rigid conical indenter. They concluded thatEq. (5) can be applied for the entire indentationcycle. However, comparison made between numeri-cal and experimental results shows that Eq. (5) doesnot lead to a very accurate determination but to atrend of the contact radius changes. This can beexplained by the use of very simple behaviour lawsfor numerical simulations which do not correspondto the real sample’s behaviour. Moreover Eq. (5)does not take into account the radial displacementsof material under the indenter.

This aspect has been studied by Galanov (1983);Galanov et al. (1983) who considers a precise for-mulation of contact problems in order to studythe effect of the tangential displacements in the caseof conical and pyramidal indenters. In order todetermine the contact radius changes during spher-ical indentation, we suggest using the formulationproposed by Hay and Wolff (2001). They intro-duced a factor c and the contact stiffness formula-tion becomes

SðhÞ ¼ dPdh¼ 2cEsa

1� m2s

: ð6Þ

For an elastic spherical indenter, the factor c is de-fined by the following relationship:

c ¼ 1þ 2a3pR

N 1� 2msð Þ 1þ msð ÞN 1� m2

s

� �þ 1� m2

i

� �" #

;

N ¼ Ei

Es

:

ð7Þ

2. Presentation of the proposed method

Eqs. (6) and (7) can be turned into a polynomialequation where a is the unknown parameter. There-fore, the resolution gives the following contactradius

aðhÞ ¼3pR

ffiffiffiffiffiffiffiffiffiffiDðhÞ

p� 1

� �4B

; ð8Þ

where

DðhÞ ¼ 1þ 4Bð1� m2s ÞSðhÞ

3pREs

;

B ¼ Nð1� 2msÞð1þ msÞNð1� m2

s Þ þ ð1� m2i Þ

� �:

ð9Þ

Our method involves several loading, unloading andreloading cycles in order to determine unloadingstiffness changes S(h) during the whole unloading

Fig. 1. Application of the proposed method to the first sample.

Fig. 2. Sink-in for the first sample.

Fig. 3. Application of the proposed method to the secondsample.

J.-M. Collin et al. / Mechanics of Materials 40 (2008) 401–406 403

curves. Lastly, the measurement of the unloadingstiffness changes during each unloading cycle allowsthe determination of contact radius changes withEqs. (8) and (9).

3. Numerical study of the contact radius changes

3.1. Presentation

A numerical study has allowed us to validate theproposed method. This study has been conducted inaxisymmetric mode with the finite element (FE)code Cast3M. The FE mesh requires the use ofseveral areas with 4-nodes elements and linear inter-polation linked with 3-nodes elements areas. Con-cerning the contact area, the FE mesh requires theuse of 8-nodes elements with quadratic interpola-tion. In the contact area itself, the size of the ele-ments is less than 4 lm and the number of theelements used is greater than 4000. The use of verysmall elements in the contact area is necessary inorder to determine the contact radius with a greataccuracy. All samples are simulated with isotropichardening. The study of the contact radius involvesthe programming of several loading–unloading–reloading cycles in order to determine the initialunloading stiffness changes S(h) and the unloadingstiffness changes during the unloading cycles.

3.2. Numerical study of the proposed method

The numerical study of the proposed method issummarized in the case of two representative mate-rials: a steel alloy with high yield stress and a copperalloy with low yield stress. In Figs. 1 and 3 the com-parison is made between the contact radius deter-mined by finite elements (written a(Cast3M)) andthe contact radius determined by the proposedmethod (written a(S)) during loading and unload-ing. In Figs. 2 and 4, the square root of (2Rh) evo-lution (written Sqrt(2Rh)) is drawn in order to showthe sensitivity of our method to sinking-in or piling-up during loading. Moreover, in these two figuresthe contact radius changes calculated with c = 1(written a(gam1)) is drawn in order to show theveracity of the use of Eqs. (8) and (9).

From numerical indentation curves, the pro-posed method allows the determination of the con-tact radius changes during the loading and theunloading cycles (Figs. 1–3), with an average preci-sion greater than 2%. This method is sensitive to thepile-up (Fig. 4) or sink-in (Fig. 2) effect. However,

the method’s application is limited by the a/R value.Indeed, Hay et al. established the limit for a rigidindenter at a value of a/R = 0.2. In the case of anelastic indenter, we have observed that the valueof a/R has to be lower than 0.25. If a/R is higherthan 0.25, Eqs. (8) and (9) will give a worse evalua-tion of a(h).

Fig. 4. Pile-up for the second sample.

404 J.-M. Collin et al. / Mechanics of Materials 40 (2008) 401–406

Concerning the effect of the factor c in the deter-mination of a(h), at the beginning of the indentationtest, a(Cast3M), a(S) and a(gam1) are quite indistin-guishable. However, when the value of a/Rincreases, the use of c = 1 overestimates the contactradius. In this case, we can see that the Hay andWolff correction gives a better evaluation of thecontact radius.

4. Experimental study

4.1. Introduction

Two steel alloys have been selected (a C100 and aC22) because of their microstructure which leads toa good homogeneity. We have developed an exper-imental bench that allows the measurement of theP(h) curve with a very good reproducibility. Theindenter used is made of tungsten carbide. Theexperimental measured displacement (hP0) corre-sponds to the displacement of the center of the

Fig. 5. Difference between measur

indenter, while the required displacement is the dis-placement (hP1) on the surface of the tested sample(Fig. 5). The difference between the two values ofdisplacement (hP0) and (hP1) is not that insignificant.This problem is usually corrected by a calibrationfunction, however, this function can cover differentphenomena. Moreover, our indenter is not exactlya spherical indenter, then, in order to apply ourmethod, we have defined an equivalent radius whichdepends on the contact radius evolution (Req(a)).

4.2. A new procedure to deduce the P(hP1) curve

from the P(hP0) one

Our numerical study has allowed us to study thedeformation of a sphere in contact with a flat sur-face. In this study, the radial displacements of thevertical axe is fixed to 0: u(r) = 0. All the nodes ofthe superior line of the indenter have the same dis-placement and the load is applied on P0. The defor-mation of the sphere leads to the changes ofd = hP0–hP1 during the loading and the unloadingcycles. From our numerical study, the d changescan be described by the following polynomial func-tion (10), depending on the indenter’s radius R, thecontact radius a, the load P and the indenter’sYoung modulus E

d ¼ R Aþ BP

pR2Ei

þ CP

pa2Ei

þ DP

pR2Ei

� �2

þ EP

pa2Ei

� �2

þ FP

pR2Ei

Ppa2Ei

!; ð10Þ

where A = �1.682.10�5, B = 3.298, C = 0.182,D = �6446.562, E = �12.727 and F = 537.889.From Eqs. (8)–(10), we propose a new procedure

ed and wanted displacement.

Fig. 6. Procedure to determine the hP1 values during each unloading.

Fig. 7. Experimental determination of P(hP1) from P(hP0) for aC22 steel.

J.-M. Collin et al. / Mechanics of Materials 40 (2008) 401–406 405

to deduce the P(hP1) curve from the P(hP0) one,which is presented in Fig. 6:

– the input values are P(hP0) and an arbitrary ini-tial a(h) curve;

– from this a(h) curve, we obtain the correspondingradius Req(a);

– these three quantities P(hP0), a(h), Req(a) allow usto calculate the corresponding d(a) curve;

– from the d(a) curve, we can calculate a P(hP1)curve with the relationship hP1 = hP0 � d;

– Eqs. (8) and (9) unable us to deduce the corre-sponding values of Es with S(h) curve calculatedfrom P(hP1);

– the knowledge of the true Es value allows us tofind the correct a(h) curve, which minimizes thedifference between the calculated Es and the truevalue of Es;

– this minimization gives the true a(h) curve andthe true P(hP1) curve;

From this procedure, we deduce the maximal val-ues of hP1 at the beginning of each unloading. Thesevalues are used in Eq. (11) in order to determine thek coefficients for each loading

k ¼ ðhP1 max � hP0 maxÞP max

: ð11Þ

Thus, the hP1 values during each loading are givenby

hP1 ¼ hP0 þ kP : ð12Þ

4.3. Experimental determination of a(h) curve

Our procedure described in Section 4.2 has beenapplied in order to obtain the P(hP1) curve from theP(hP0) curve. As shown in Fig. 7, we obtain a verygood correction of the loading curve, although weobserve some differences between the experimentaland the numerical unloading curves. An explana-tion of these differences may be the kinematic hard-ening influence.

These differences do not allow the comparisonbetween a(h), obtained from Eqs. (8) and (9), andtheoretical a(h), since theoretical a(h) is not knownduring the whole unloading curve. Nevertheless,we can compare theoretical a(h) and the oneobtained from our procedure at the beginning ofthe unloading curve. Figs. 8 and 9 show the applica-

Fig. 8. Determination of a(h) for a C100 steel.

Fig. 9. Determination of a(h) for a C22 steel.

406 J.-M. Collin et al. / Mechanics of Materials 40 (2008) 401–406

tion of our procedure to determine the contactradius’s highest values for the different unloadingcycles. These two experimental examples show thatour procedure allows the determination of the a(h)curve with an average precision better than 5%.

5. Conclusion

A numerical study has allowed us to validate amethod to determine the contact radius changesduring the unloading process in spherical indenta-tion. It is shown that the Hay and Wolff correctionleads to a more accurate determination of the con-tact radius than the existing method.

From our procedure, the knowledge of bothindenter and sample elastic properties leads to therequired load–depth curve and to the contact radiuschanges during a spherical indentation test.

Acknowledgement

This research was sponsored by the French gov-ernment as well as the Brittany Region.

References

Borodich, F.M., Keer, L.M., 2004. Contact problems and depth-sensing nanoindentation for frictionless and frictional bound-ary conditions. Int. J. Solids Struct. 41, 2479–2499.

Bulychev, S.I., Alekhin, V.P., Shorshorov, M.Kh., Ternovskii,A.P., Shnyrev, G.D., 1975. Determining young’s modulusfrom the indenter penetration diagram. Zavod Lab. 41.

Bulychev, S.I., Alekhin, V.P., Shorshorov, M.Kh., Ternovskii,A.P., 1976. Mechanical properties of materials studied fromkinetic diagrams of load versus depth of impression duringmicroimpression. Strength Mater. 8, 1084–1089.

Doerner, M.F., Nix, W.D., 1986. A method for interpreting thedata from depth-sensing indentation instruments. J. Mater.Res. 1.

Galanov, B.A., 1983. Statement and solution of some moreprecise problems of elastic contact of two bodies. Izv. ANSSSR. Mekhanika Tverdogo Tela 18 (6), 53–56.

Galanov, B.A., Grigor’ev, O.N., Mil’man, Yu.V., Ragozin, I.P.,1983. Determination of the Hardness and Young’s modulusfrom the depth of penetration of a pyramidal indentor.Problemy Prochnosti 11, 93–96.

Galanov, B.A., Grigor’ev, O.N., Mil’man, Yu.V., Ragozin, I.P.,Trefilov, V.I., 1984. Determination of the hardness andYoung’s modulus with elastoplastic penetration of indentorsinto materials. Sov. Phys. Dokl. 29 (2), 146–147.

Galin, L.A., 1946. Spatial contact problems of the theory ofelasticity for punches of circular shape in planar projection. J.Appl. Math. Mech. (PMM) 10, 425–448.

Hay, J.C., Wolff, P.J., 2001. Small correction required whenapplying the Hertzian contact model to instrumented inden-tation data. J. Mater. Res. 16 (5), 1280–1286.

Hernot, X., Bartier, O., Berkouche, Y., El Abdi, R., Mau-voisin, G., 2006. Influence of penetration depth andmechanical properties on contact radius determination forspherical indentation. Int. J. Solids Struct. 43 (14–15),4136–4153.

Hertz, H., 1882. Uber die Beruhrung festischer Korper. Z. ReineAngew. Math. 92, 156–171.

Hill, R., Storackers, B., Zdunek, A.B., 1989. A theoretical studyof the Brinell hardness test. Proc. Roy. Soc. Lond. A 423,301–330.

Li, K., Wu, T.W., Li, J.C.M., 1997. Contact area evolutionduring an indentation process. J. Mater. Res. 12 (8), 2064–2071.

Loubet, J.L., Georges, J.M., Meille, G., 1986. Vickers indenta-tion curves of elastoplastic materials. Microindentation tech.In: Mat. Sci. Eng., ASTM STP 889.

Meyer, E., 1908. Untersuchen uber Harteprufung und Harte. Z.des Ver. Deut. Ingenieure 52, 645–654.

Oliver, W.C., Pharr, G.M., 1992. An improved technique fordetermining hardness and elastic modulus using load anddisplacement sensing indentation experiments. J. Mater. Res.7 (6), 1564–1583.

Shorshorov, M.Kh., Bulychev, S.I., Alekhin, V.P., 1981. Work ofplastic and elastic deformation during indentation. Sov. Phys.Dokl. 26, 769–771.

Sneddon, I.N., 1965. The relation between load and penetrationin the axisymmetric Boussinesq problem for a punch ofarbitrary profile. Int. J. Eng. Sci. 3.

Tabor, D., 1951. The Hardness of Metals. Clarendon Press,Oxford, UK.