Tribology International 120 (2018) 398–410

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Tribology International

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Scratch-tip-size effect and change of friction coefficient in nano / microscratch tests using XFEM

Kwangmin Lee a, Karuppasamy Pandian Marimuthu a, Chang-Lae Kim b, Hyungyil Lee a,*

a Department of Mechanical Engineering, Sogang University, Seoul 04107, Republic of Koreab Department of Mechanical Engineering, Yonsei University, Seoul 03722, Republic of Korea

A R T I C L E I N F O

Keywords:Scratch-tip-size effectCoefficient of frictionScratch testXFEM

* Corresponding author.E-mail address: hylee@sogang.ac.kr (H. Lee).

https://doi.org/10.1016/j.triboint.2018.01.003Received 11 September 2017; Received in revised form 2Available online 3 January 20180301-679X/© 2018 Elsevier Ltd. All rights reserved.

A B S T R A C T

Compared to the experimental studies, numerical studies on scratch test have been limited due to the difficulty ofnumerical modeling. Here, we experimentally investigate the scratch-tip-size effect (STSE) based on coefficient offriction (COF) change to make extended finite element (XFE) models considering STSE, COF change, failuremodes, yielding strength, shear strength, elastic recovery and pile-up. XFE models are then validated throughnano/micro scratch tests on soda-lime glass. Finally, components of COF are extracted and then effects of tensilecracks and parameters of Drucker-Prager model on COF are investigated. We expect that proposed XFE modelscontribute to the understanding of damage characteristics during the scratch test.

1. Introduction

Scratch test is a widely used method for evaluating tribologicalproperties of materials. The test is simple to perform; a scratch-tip isdrawn across the surface under increasing load until some well-definedfailure occurs at a certain load, which is called the critical load [1].However, due to the complexity of analysis, the scratch test has beendeveloped slowly; on the other hand, indentation test, which is a similartest with the scratch test, has gained popularity over years for evaluatingmaterial properties. In analyzing the scratch test results, understandingfriction mechanism i.e. coefficient of friction (COF) change during thescratch test is an especially challenging task as friction is affected bymany parameters such as failure modes, plastic deformation, surfaceroughness, physicochemical interactions, material properties and envi-ronment. It has been shown that humidity affects the critical loads anddeformation regime in scratching [2], and temperature affects hardnessand the length of the cracks in the glass [3].

Along with acoustic emission and contact resistance signals, COF isone of the measurements describing failures during the scratch test [4].By observing the change of COF, the tribological behaviors such asstick-slip phenomenon, adhesion, plastic deformations, and fracture canbe analyzed. For example, Li et al. [5] found the occurrence of stick-slipbehavior during the scratch test on soda-lime glass by observingsaw-teeth wave in COF curve. Yonghui and Xiang [6] observed the crackpropagation in diamond like carbon (DLC) film when the COF increased

December 2017; Accepted 1 January

unexpectedly. Several researchers investigated failure and deformationof materials during the scratch test by understating each component ofCOF [7,8], where overall COF is composed of asperity deformation,adhesion and plowing COF [9]. Dividing overall COF into each compo-nent of COF is useful to comprehend damages during the scratch test.

Many experimental variables influence the COF, and among them,scratch-tip size is an important factor as failure mode in scratch testvaries with scratch-tip sizes, so called scratch-tip size effect (STSE). TheSTSE is an extension of indentation size effect (ISE); if indenter size issmall, yielding occurs instead of cracking in brittle materials [10]. Inanother way, ISE phenomenon can be defined as change in hardness ofmaterials with indentation size or indentation load. To simulate ISE orSTSE in amorphous materials, the exact yield model, which is limited inthe literature, is required to describe the elasto-plastic behavior ofamorphous materials by considering ISE. Expanding cavity models(ECMs) were developed using nano indentation [11]; however, discrep-ancy occurred between experiment and ECMs since ECMs were madebased on von Mises and Tresca yield criterion. Although Drucker-Prager(DP) yield model and Mohr-Coulomb model were also used to describeplastic deformation of glass in nano indentation test [12,13] andcompressive test [14,15], ISE has not been considered along withindentation load-depth curve. In other words, hardness-depth curve hasnot been taken into account with load-depth curve at the same time.

The aim of this work is to develop an extended finite element (XFE)model for scratch test simulations to study the scratch behavior of

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K. Lee et al. Tribology International 120 (2018) 398–410

materials. STSE, COF change, failure modes, yielding strength, shearstrength, elastic recovery and pile-up are considered while developingthe XFE model. Yielding is governed with DP yield models during nano/micro scratch simulations. Numerical results are validated by conductingnano and micro scratch test on soda-lime glass. Finally, components ofCOF are extracted and then effects of tensile cracks and parameters ofDrucker-Prager model on COF are investigated by using proposed XFEmodels. We expect that our research contributes to the understanding ofdamage characteristics of coated materials along with bulk amorphousmaterials.

2. Background theory

In amorphous materials, indentation size effect has been observed[16–18]; yet existing yield models do not reflect the variance of hardnessalong with the indentation depth i.e. ISE [12–15]. Analytical solutions forcomponents of the COF have been established along with the develop-ment of contact mechanics [19–24]. However, those cannot be practi-cally used as parameters such as asperity angle and contact area are hardto be measured from the experiments. On the other hand, by combiningtheoretical and numerical approaches based on experimental data, eachcomponent of COF can be extracted to some extent as explained in thissection.

2.1. Indentation size effect (ISE)

The ISE is a trend wherein hardness decreases with increasingindenter size or indentation load [25]. This phenomenon occurs in anyshapes of the indenter-tip [26]. As hardness is closely related to strengthof materials [27], the change of yield strength due to ISE has to beconsidered in nano/micro scratch tests.

Besides the change of hardness due to ISE, brittle materials showdifferent deformation behavior depending on the indenter size. Mouginot[10] found that in soda-lime glass, when size of spherical and cylindricalindenter-tip is relatively small, yielding occurs instead of crack propa-gation (Fig. 1); and suggested that there is a transition radius Rt ofindenter-tip for each material. If the indenter-tip radius R is larger thanRt, cracking occurs. In contrast, if R is smaller than Rt, yielding occurs. Adatabase of Rt for several materials is provided in reference [10].

2.2. Yielding of soda-lime glass in indentation

Based on the indentation theory of a rigid perfectly plastic solid [28],Tabor [29] showed that the hardness H is related to the yield strength Yof the material in simple compression as follows

H ¼ pm ¼ C Y (1)

where pm is the mean contact pressure and C is a constant whose value isabout 3, depending to some extent upon the geometry of the indenter.Tabor [29] assumed that the material follows the Tresca yield criterion;however, amorphous materials do not follow the Tresca criterion [11]

Fig. 1. Deformation behavior depending on indenter-tip size [10].

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due to substantial material densification caused by the hydrostaticpressure [12]. Therefore, a pressure dependent yield model has to beused instead of Tresca criterion for describing the yielding of amorphousmaterials. In addition, amorphous materials are not perfectly plastic buthighly elastic materials.

Drucker-Prager (DP) model has been suggested to describe plasticdeformation of glass in nano indentation test [12] and compressive test[14,15]. Rather than a pressure independent yield model, a linear DPmodel describes the plastic behavior of amorphous materials; however,ISE cannot be considered with the linear DP model. In this work, pa-rameters of linear DP model are calibrated to describe plastic deforma-tion of soda-lime glass by using nano/micro indentation. DP modelreflects the plastic deformation characteristics of the glass at some extent.To the authors best knowledge, this is the first attempt to make yieldmodel by considering ISE, in which load-depth and hardness-depthcurves are considered at the same time.

The DP yield model is a pressure dependent model as follows [30].

Y ¼ σo þ p tan β (2)

where σo is the zero-pressure yield strength, β is the material angle offriction and p is the hydrostatic pressure. The σo is related to uniaxialcompression yield strength σc, if hardening is defined by the σc [31], as

σo ¼�1� 1

3tan β

�σc (3)

2.3. Coefficient of friction components extraction method

The overall COF μt is expressed as [9].

μt ¼ μd þ μa þ μp (4)

where μd is asperity deformation, μa is adhesion and μp is plowing COF.These components of COF relatively contribute to friction mechanismdepending on the condition of the sliding interface, which can be affectedby the materials used, history of sliding, surface topography, and envi-ronment [9]. To explain the above postulate more clearly, Suh and Sin[9] subdivided friction mechanism into six stages considering the timedependent behavior of friction; more details are given in references [9,32].

Analytical solutions of each component of μt cannot be practicallyused due to the following reasons: (i) The solutions of μd and μa need anaccurate value of asperity angle [7], which cannot be measured directlyfrom any test and (ii) measurement of μp value from the experiments isdifficult due to insufficient studies about predicting elastic recovery [23,24,33–35], which occurs behind the scratch-tip during the scratch test(Fig. 2) [36].

On the other hand, components of COF can be extracted fromexperimental measured μt by using finite element method (FEM) withanalytical solutions of COF; μt can be divided into interfacial COF μs, sum

Fig. 2. Photograph of contact area during scratch test [34].

Table 2Scratch test parameters for linearly increasing load.

parameters value

nano scratch test micro scratch test

type progressive progressivebegin load (mN) 1 1000

K. Lee et al. Tribology International 120 (2018) 398–410

of μd and μa, and μp

μt ¼ μs þ μp (5)

The extraction procedure is as follows: (i) adjusting Coulomb's fric-tion in the FE model, which has the same μt changes with the experiment,(ii) obtaining μp from the FE results by setting zero Coulomb's friction,thus μs theoretically also becomes zero and (iii) subtracting μp from theexperimental value of μt to obtain μs from Eq. (5). This procedure is onlyapplicable when the deformations of the specimen are the same in the FEanalysis with and without Coulomb's friction. This means that elasticrecovery and pile-up in those models during the simulation has to beanalogous, thus, the presence of Coulomb's friction in FE model shouldnot affect μp value significantly. Detailed explanations are provided inAppendices A–B.

3. Experiments

3.1. Material property evaluation

Nano indentation tests (UNHT3, Anton Paar, Switzerland) were con-ducted with Berkovich tip to measure the mechanical properties of soda-lime glass (Seoul special glass Ind., South Korea) at nano scale. Byassuming Poisson's ratio ν¼ 0.23 [37], Young's modulus E of soda-limeglass at nano scale is obtained from load-depth curve at the maximumload, 50mN, by using Oliver and Pharr method [38]. The hardness andfracture toughness of soda-lime glass in nano indentation are taken fromliterature (Table 1) [39].

Vickers hardness tests (ZHV U-S, Zwick Roell, Germany) were con-ducted to measure the mechanical properties of soda-lime glass at microscale. Hardness and fracture toughness of soda-lime glass are obtained byusing the methods described in literatures [16,40,41]. Literature data areused for micro scale Young's modulus and Poisson's ratio (Table 1) [37].

The notable thing is that the hardness of soda-lime glass obtainedfrom nanoindentation test H¼ 7.8 GPa is larger than that from microindentation test H¼ 5.4 GPa. This represents that hardness varies withindentation depth, which is so called indentation size effect (ISE) [25].

3.2. Nano/micro scratch test

Nano scratch tester (NST3, Anton Paar, Switzerland) and microscratch tester (KK 01/KatoTech, Japan) were chosen for the experiments.Each nano/micro scratch experiments on soda-lime glass was performedten times under room temperature and humidity conditions. To investi-gate STSE, sphero-conical scratch-tip with R¼ 2 μm (<Rt¼ 0.218mm)was used in the nano scratch test and spherical scratch-tip withR¼ 0.5mm (>Rt¼ 0.218mm) was used in the micro scratch. Table 2shows the scratch test parameters.

3.3. Experimental observation

The scratch-tips of different size cause various failure modes (chip-ping in nano scratch test, tensile cracking in micro scratch test), whichlead to different COF changes. Failure modes were observed by usingoptical microscope (BX51, Olympus, Japan) and scanning electron mi-croscope (SEM) (JEOL-7800F, JEOL Ltd., Japan). Overall COFs μt wereobtained by dividing tangential load by normal load.

SEM images of nano scratch path in soda-lime glass specimen areshown in Fig. 3. Yielding occurs at 30 μm (Fig. 3a) whereas ductilefracture starts at 60 μm (Fig. 3b) in front of the scratch-tip; this shows that

Table 1Properties of soda-lime glass obtained from experiments.

material E (GPa) ν Н (GPa) KIC (MPa m1/2)

soda-lime glass (nano scale) 76 0.23 7.8 0.75soda-lime glass (micro scale) 70 0.23 5.4 0.75

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there is a gap between yield strength and ultimate strength. Right afterductile fracture, chipping or spallation occurs unceasingly (Fig. 3c andd).

When an abrupt change of μt occurs in the COF curves of nano scratchtests, a critical load of 18mN is obtained as shown in Fig. 4 [42]. μtslightly increases at yielding and rapidly increases when ductile fractureoccurs; wear particles were observed. As the normal load increases, μtcontinues to increase slowly.

In the micro scratch test, unlike in the nano scratch test, tensile cracksoccur behind the scratch-tip instead of yielding or chipping due to STSEat normal load of 50mN (Fig. 5a). As the normal load increases, tensilecracks become larger (Fig. 5c and d). Since there is no critical point inCOF curve (Fig. 6), brittle fracture occurring behind the scratch-tip doesnot significantly affect the COF change.

μt increases slightly due to plastic deformation. At early stages, μp isalmost zero as plastic deformation did not occur significantly. Then μt isapproximately 0.12 and μs has the same value as μt according to Eq. (5).Note that μs, 0.12, at the early stage in the micro scratch test is differentfrom that, 0.1, in the nano scratch test since different materials are usedfor scratch-tips. The explanation of μs between the diamond-tip and thesoda-lime glass specimen under nano scratch test is given in section 5.1.

4. XFE modeling of scratch test and calibration of linear Drucker-Prager model

4.1. XFE modeling of scratch test

FE models for nano/micro scratch simulation (Fig. 7) are made assimilar to the experiments. The displacements ux, uy and uz in þx, – y, – zdirections are applied on the scratch tip instead of loadings on thescratch-tip. The commercial software Abaqus 6.13 program (2013) [31]is used for analysis. Three-dimensional (3D) FE model consists of about110,000 nodes and 90,000 elements for nano scratch test and 380,000nodes and 330,000 elements for micro scratch test. One-half of FE modelsare used and boundary conditions about xy-plane, xz-plane and yz-planeare applied to fix the specimen. A rigid surface with sphero-conical ge-ometry (R¼ 2 μm and 45� cone semi-angle) and hemi-spherical geometry(R¼ 0.5mm) is used as a scratch tip for nano and micro scratch simu-lations, respectively. To obtain better convergence, exponentialpressure-overclosure relationship is applied instead of hardpressure-overclosure relationship at the contact between the scratch-tipand specimen [31].

Material properties of soda-lime glass are assigned to the FE models.The linear DP model is used for governing the plastic deformation of thespecimen. Formation of tensile cracks is modelled with extended finiteelement method (XFEM) to investigate effect of tensile cracks (brittlefracture) on COF in micro scratch test. Since XFEM is based on linearelastic fracture mechanics [31], only the elastic material properties areused. Cohesive zone model (CZM) is used to govern the damage initiationand propagation.

Friction model with a limit on the critical shear stress (Fig. 8) is

end load (mN) 100 100000loading rate (mN/min) 99 72000speed (mm/min) 0.3 14.4length (mm) 0.3 20scratch-tip material diamond SS 440scratch-tip size sphero-conical geometry

(R¼ 2 μm and 45� cone ofsemi-angle)

hemispherical geometry(R¼ 0.5 mm)

Fig. 3. SEM photomicrographs of nano scratch on soda-lime glass made with sphero-conical scratch tip (R¼ 2 μm and 45� cone of semi-angle) at 99mN/minloading rate and 0.3mm/min speed: (a) overall top view, (b) initial part, (c) middle part, (d) end part.

Fig. 4. Normal load, tangential load and COF change during nano scratch test.

K. Lee et al. Tribology International 120 (2018) 398–410

implemented for considering gross sliding; if shear stress in materialreaches the shear strength of material, gross sliding occurs [45]. The COFchange is also implemented in FE models by using user subroutine(FRIC_COEF). In order to obtain decreasing Coulomb's friction in FEanalyzes, the friction between contact surfaces must be defined usinguser subroutine (FRIC_COEF) as built-in friction definition in Abaqus(2003) allows only monotonically increasing Coulomb's friction. Byadjusting Coulomb's friction, μs is controlled; detailed explanations ofrelation between Coulomb's friction and μs are given in Appendix A.

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4.2. Calibration of linear Drucker-Prager models

To understand the elasto-plastic constitutive behavior of material innano/micro scratch tests, the parameters of linear DP models are cali-brated by using FE analysis of nano/micro indentation along withexperimental data in literatures [16–18,38]. Unlike previous studies [13,43], hardness-depth curve as well as the load-depth curve is consideredto account for the ISE.

FE models for nano/micro indentation (Fig. 9) are similar to experi-mental one in Refs. [14–16,36]. Three-dimensional (3D) FE model con-sists of about 45,000 nodes and 38,000 elements. One-sixth FE model isused in nano indentation, and one-fourth FE model is used in microindentation for reducing computational time. Boundary conditions areapplied to fix the specimen. Berkovich and Vickers tips are considered innano and micro indentation test, respectively. Material properties ofdiamond (Young's modulus¼ 1140 GPa and Poisson's ratio¼ 0.07) areassigned to the indenter tip.

Experimental load-depth curve and hardness-depth curve of nanoindentation (with Berkovich indenter) are taken from literatures [17,18,38]. Peng et al. [17] made a Pmax-hc regression curve by using theapproach of Hays and Kendall [44] where Pmax is the maximum inden-tation load and hc is the contact depth. Then maximum indentation depthhmax is estimated from the Pmax-hc regression curve by using

hmax ¼ hc þ εPmax

S(6)

where ε is the constant dependent on the indenter geometry and S is theinitial unloading stiffness in indentation load-depth curve. For Berkovich

Fig. 5. (a) Optical photograph and (b–d) SEM pho-tographs of micro scratch on soda-lime glass madewith hemispherical geometry (R¼ 0.5 mm) at72000mN/min loading rate and 14.4 mm/min speed:(a) starting point where tensile crack occurs, (b)initial part, (c) middle part, (d) end part.

K. Lee et al. Tribology International 120 (2018) 398–410

indenter, it has been shown that ε has an empirical value of 0.75 [38]. Liet al. [18] found that S is a function of hc as

S ¼ 3:96� 10�4hc (7)

By using the maximum indentation load Pmax and projected area Ac ofthe hardness impression, the hardness from nano indentation can beobtained

Hb ¼ Pmax

Ac¼ Pmax

24:5h2c(8)

Then, the nanoindentation load-depth and hardness-depth curves areobtained from Eqs. (6)–(8) as shown in Fig. 10.

Experimental load-depth curve and hardness-depth curve ofmicro indentation (with Vickers indenter) are also taken fromliterature [16]. Gong et al. [16] made a P-d regression curve byusing the Mayer's law where P is an applied indentation load and

Fig. 6. Normal load and COF change during micro scratch test.

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d is a length of the diagonal of the resultant impression. Based on Pand the pyramidal contact area A of indentation, micro indentationhardness can be expressed as

Hv ¼ P=A ¼ αP�d2 (9)

where α¼ 1.8544 is a constant for Vickers indenter. Then micro inden-tation load-depth and hardness-depth curves are obtained as shown inFig. 11.

The parameters σc and β in linear DP model are calibrated for soda-lime glass by using nano/micro indentation simulation. In fact, bothindentation load-depth and hardness-depth curve could not be fittedaccurately at the same time when linear DP model is applied, i.e., linearDP model is insufficient to consider ISE. Whatever value of materialconstants σc and β are, hardness does not decrease, but constant withindentation depth. Henceforth, the linear DP model is calibrated (i) byobtaining experimental hardness values from nanoindentation, in whichhardness converges to a certain value from a specified depth, and then(ii) fitting FE indentation load-depth and hardness-depth curves with theexperimental data. By comparing the load and hardness data for variousvalues of σc and β at certain depth with experimental data (Tables 3 and4), an appropriate linear DP model for soda-lime glass under nano/microindentation is calibrated as

Y ¼ 1:757þ p tan 20∘ ½GPa� (10)

By applying calibrated linear DP model, Figs. 10 and 11 compare theFE load-displacement curve with those from experiments.

5. XFEA results

5.1. Components of COF

During the nano scratch test on soda-lime glass, COF alters by plasticdeformation and ductile fracture. Only with μt, it is difficult to analyzethe deformation and failure mode. For that reason, μt is divided intocomponents of COF based on experimental and FEA data by using the

Fig. 7. FE models for scratch test simulation at nano/micro scales.

Fig. 8. Slip regions for the friction model with limit on shear strength [29].

Fig. 9. FE models for indentation test simulation at nano/micro scales; pa-rameters calibration of Drucker-Prager model considering ISE.

K. Lee et al. Tribology International 120 (2018) 398–410

COF components extraction method. To clarify the distance dependentfriction behavior of soda-lime glass in the nano scratch test, the plot offriction versus distance slid with components of COF is subdivided asdescribed in the following stages (Fig. 12).

Stage I: Since the scratch-tip is very small, μp increases due tothe occurrence of plastic deformation even in this first stage; onlythe spherical part of the scratch-tip is in contact with the specimen.Significant fracture does not occur, thus, μs is constant with thevalue 0.1.

Stage II: In this second stage, ductile fracture starts to occur. It isassumed that this failure mode does not affect the plastic deformation ofthe specimen, hence, μp increase without abrupt change. On the otherhand, chipping from the ductile fracture in front of the scratch-tip mayincrease μs rapidly.

Stage III: This stage begins right after the sharp increase in μt resultingfrom ductile fracture. Like in the stage II, μp increases without abruptchange; whereas μs decreases continuously due to the decrease inasperity deformation, as the asperities of surface are gradually removedcreating a mirror finishes [9]. The same phenomena are reflected in FEmodel for nano scratch test with a decrease in Coulomb's friction (Fig.17).

Stage IV: At this stage, effect of conical part of scratch-tip on μp be-comes dominant. This changes the slope of μp. However, μs decreasescontinuously; to implement this change of μs, Coulomb's friction indeveloped FRIC_COEF subroutine increases slightly unlike decrease of μs(Fig. 17).

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5.2. Elastic recovery and pile-up

In the previous section, components of COF are extracted from μtwithout considering the elastic recovery and pile-up by using FEA.However, it is assumed that failure mode does not affect μp; in otherwords, Coulomb's friction change in FEmodel does not affect μp. To checkthis assumption is valid, elastic recovery and pile up, which are the mainfactor affecting μp, are investigated in this section.

FE models with and without Coulomb's friction are used for a com-parison. From now on, for convenience, we denominate the formal modelas an ITF (interfacial COF) and the letter one as a NITF (no interfacialCOF). To observe the effect of elastic recovery (occurring behind thescratch-tip) on COF, a contact area angleω resulting from elastic recoveryis checked between ITF and NITF; detailed explanation is given in Ap-pendix B. We observe almost the same ω in both ITF and NITF from the

Fig. 10. Comparison of indentation load and hardness variations withindentation depth during nanoindentation between experiment and FEA.

Fig. 11. Comparison of indentation load and hardness variation with inden-tation depth during micro indentation between experiment and FEA.

Table 3Comparison of load and hardness data at hmax¼ 1 μm with different values of σc and β innano indentation FEA with those in experiment (Pmax¼ 93mN and Hb¼ 6.6 GPa at thehmax¼ 1 μm in the experiment [17,18]).

materialproperties

FEA

Pmax (mN) error (%) Hb (GPa) error (%)σc (GPa) β

0.5 10 42.3 54.5 1.64 75.220 45.7 50.8 2.00 69.730 50.6 45.6 2.54 61.540 58.5 37.1 3.39 48.650 70.0 24.7 4.74 28.1

1.0 10 65.3 29.8 3.28 50.320 70.3 24.4 3.80 42.430 76.3 18.0 4.53 31.440 83.4 10.4 5.59 15.350 92.2 0.9 6.99 5.9

1.5 10 81.9 11.9 4.63 29.920 86.8 6.7 5.26 20.330 91.0 1.1 6.10 7.640 97.6 4.9 7.18 8.850 104.3 12.1 8.42 27.6

2.0 10 93.8 0.8 5.88 10.920 97.5 4.9 6.55 0.830 101.5 9.18 7.33 11.140 106.2 14.2 8.33 26.250 111.9 20.3 9.80 48.5

Bold values are calibrated ones.

Table 4Comparison of load and hardness data at the d¼ 0.15mm with different values of σc and β

in micro indentation FEA and those in experiment (Pmax¼ 50mN and Hv¼ 5.3 GPa whend¼ 0.15mm in the experiment [16]).

materialproperties

FEA

Pmax (N) error (%) Hv (GPa) error (%)σc (GPa) β

0.5 10 19.9 60.2 1.68 68.320 23.7 52.6 1.94 63.430 27.7 44.5 2.29 56.840 33.0 34.1 2.74 48.350 40.1 19.7 3.35 36.8

1.0 10 35.7 28.7 2.92 44.920 38.7 22.7 3.18 40.030 43.5 13.0 3.60 32.140 43.5 13.0 3.62 31.750 58.6 17.2 4.75 10.4

1.5 10 44.9 10.2 3.70 30.220 51.0 2.1 4.20 20.830 56.3 12.5 4.66 12.140 62.4 24.8 5.19 2.150 72.1 44.3 5.85 10.4

2.0 10 57.2 14.3 4.69 11.120 60.3 20.6 4.98 6.030 65.8 31.5 5.45 2.840 69.8 39.7 5.82 9.850 79.3 58.7 6.65 25.5

Bold values are calibrated ones.

Fig. 12. Components of coefficient of friction during nano scratch test.

K. Lee et al. Tribology International 120 (2018) 398–410

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simulations. In addition, ω in both ITF and NITF are nearly 32.8� whentips are reached at 150 μm in x-direction.

To investigate the effect of pile-up (occurs front half of the scratch-tip)due to μs on μp change, ITF and NITF are compared by using analyticalsolutions. As the effect of elastic recovery on μp is negligible, μp caused byfront half of the scratch-tip where pile-up occurs is compared only.Analytical solutions for μp for sphero-conical scratch-tip on the front halfof the scratch-tip can be derived as [21].

μp ¼2πa�2

�sin�1 a� a

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

p �; a ¼ w

2rfor w � wc (11)

μp ¼2πa�2

�sin�1 a� a

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

p � þ 2π

�w2 � w2

c

w2

�tan θ ;

a ¼ wc

2rfor w > wc

(12)

where r is a radius of scratch-tip and w is a groove width, θ is an angle

Fig. 13. Schematic image of sphero-conical scratch-tip.

K. Lee et al. Tribology International 120 (2018) 398–410

between cone and the sliding direction and wc is critical diameter ofscratch tip on which spherical shape ends and conical shape starts (Fig.13). When w�wc, μp of sphero-conical scratch-tip is the same with μp ofspherical scratch-tip like Eq. (11). When w>wc, μp of sphero-conicalscratch-tip changes like Eq. (12). By using Eqs. 11 and 12 based onheight of mound in the ITF and NITF during the nano scratch test, μp isobtained (Fig. 14).

The result shows that pile-up increases when μs is applied (Fig. 14a).However, this information does not directly show how μp increases due toμs change. Instead, the μp change obtained from Eqs. 11 and 12 showshow μp increases due to μs change (Fig. 14b). As value of μp in betweenITF and NITF are not much different, effect of μs caused by pile-up on μp isnegligible. Therefore, μp obtained from NITF can be used instead of μp,which is required in ITF.

5.3. Validation of FE models for nano/micro scratch test simulations

The material properties of soda-lime glass obtained from nano Ber-kovich indentation tests, Vickers hardness tests and literature data(Table 1) are assigned to the FE models. The material is assumed to beisotropic and follows the calibrated linear DP yield model in Eq. (10). Ashigh compressive load is applied on the soda-lime glass during the nano/micro scratch tests, Mohr-Coulomb criterion should be used to obtainshear strength of soda-lime glass. The Mohr-Coulomb criterion describesthe shear strength of a brittle material as a function of applied pressure

τ ¼ τo þ αp (13)

where τo is the shear strength at pure shear, α is the coefficient of internal

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friction and p is the normal pressure. At high-pressure, Chocron et al.[46] argued that shear strength of glass is

τ ¼ 0:495þ 0:652p ½GPa� (14)

based on hydraulic pressure tests results.The FEA results show that normal load and COF change in FEA is

almost the same with that in the experiment (Figs. 15 and 16). Thisvalidates the FE models for nano/micro scratch test as it describes COFchange and deformation as in the experiments. In the case of nano scratchsimulation, Coulomb's friction is implemented as shown in Fig. 17. Thisone is deduced from μs in the nano scratch experiment (Fig. 12). Like μs inthe experiment (Fig. 12), Coulomb's friction also fluctuates (Fig. 17). Inthe case of micro scratch simulation, Coulomb's friction is set to 0.12based on the initial part of experimental result (Fig. 6), as plastic defor-mation is negligible in the initial part under micro scratch test. The FEAresult shows that μt increase is almost the same with the experiment (Fig.16b), since calibrated DP model describes the effect of plastic deforma-tion on μt precisely. As μt during the nano/micro scratch test is smallerthan 0.652 (Fig. 16), gross sliding does not occur during the nano/microscratch test FEA simulation.

5.4. Tensile crack

In micro scratch tests, tensile cracks, which are one feature of brittlefracture, occur behind the scratch-tip. For the maximum normal load of50 N, maximum principal stress of 0.54 GPa is observed in the FE ana-lyses; thus, damage initiation stress is chosen as 0.54 GPa for damageinitiation during XFE analysis. In fact, this value is dependent on elementsize; if element size is changed, damage initiation stress also has tochange. Fracture energy Γ is obtained from the fracture toughness KIC(Table 1); under plane strain condition, Γ is

Γ ¼ ð1� ν2ÞE

K2IC (15)

In XFEA, tensile crack occurs behind the scratch-tip when appliednormal load reaches 50 N as shown in Fig. 18. The tensile cracks changemaximum principal stress in the model, but they do not affect COFchange in XFEA (Fig. 19).

5.5. Effect of linear DP model parameters on COF

σc and β are the parameters in linear DPmodel, which determine yieldstrength of material. In scratch test, starting point and aspect of theplastic deformation vary depending on the values of σc and β. Thus, toinvestigate effect of parameters of DP model on COF, FE analyses areperformed by changing the parameters values as listed in Table 5, whichshows the representative material properties and scratch test parameters

Fig. 14. Comparison of pile-up and plowing coeffi-cient of friction between ITF and NITF.

Fig. 15. Comparison of normal load between exper-iments and FEA.

Fig. 16. Comparison of coefficient of friction be-tween experiment and FEA.

Fig. 17. Adjustment of Coulomb's friction in nano scratch test FE model basedon experimental result.

Fig. 18. Tensile cracks occurring behind scratch-tipin XFE model.

Fig. 19. Variance of COF in XFEA; tensile crack does not affect COF change.

K. Lee et al. Tribology International 120 (2018) 398–410

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Table 5Material properties and scratch test parameters for FE analyses.

materialproperties

value parameters value

Young'smodulus

100 GPa scratch test type progressive

Poisson's ratio 0.3 horizontal tipspeed

0.6 mm/s

shear strength ∞ vertical tipspeed

0.002mm/s

Coulomb'sfriction

0.1 scratch-tipmaterial

rigid body

β 10, 20, 30,40, 50

scratch-tip size hemispherical geometry(R¼ 10 μm)

σc 1, 2, 3, 4,5 GPa

Fig. 21. Coefficient of friction values with variance of σc and β at the end ofscratch test in FEA.

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for the analyses. These values cover the material property range of gen-eral amorphous materials. In this section, the effect of plastic deforma-tion on COF change is mainly studied by assuming constant values ofCoulomb's friction and shear strength.

The results show that the smaller σc and β, the larger the final COFvalue (Fig. 20). This is because when σc is small, plastic deformationstarts to occur at earlier stage and if β is small, plastic deformation occursrapidly. COF values with variance of σc and β at the end of scratch show asimilar tendency (Fig. 21), since it is an extension of the effect of pa-rameters in DP model on COF result (Fig. 20).

6. Discussion

The proposed COF components extraction method is used foranalyzing deformation and failure modes of material during thescratch test (Fig. 12). This method is meaningful and more accurate inthat it uses FEM and theoretical solutions together to analyze COFunlike using only theoretical solutions in the conventional method.However, in this method, μp is obtained from simulation where idealenvironment is assumed. Hence, there is small but difference betweenμp in ITF and that in NITF. Therefore, this method should be used forqualitative analysis of μp; for quantitative analysis, additional work isrequired.

On the other hand, proposed XFE model can be used quantitativelyin some measure since normal load and COF change between FEA andexperiment are almost the same (Figs. 15 and 16). In XFEA results,formation of tensile cracks alters the maximum principal stress dis-tribution around the scratch path without affecting COF change (Fig.19). This is because maximum principal stress is not a parameter of μt(Eq. (A.3)). Hence, tensile crack has negligible effect in friction model;it is unnecessary to use XFEM in friction related research. Besides, the

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result of parameter studies of DP model can be applied to evaluatecoated materials as well as bulk materials. If two different materialsare used as the coating materials that have the same interfacialtoughness but follow disparate DP models then the larger σc and β, themore difficult it delaminates from the substrate.

The advantage of present XFE models for the nano/macro scratch testis that it takes into account various variables – STSE, COF change, failuremodes, yielding strength, shear strength, and elastic recovery behind thescratch-tip – at the same time when compared with prior studies [47–49].Furthermore, this model can be especially useful for coating simulationsas it considered COF carefully, since the delamination of the coatingusually occurs due to mode II fracture at the interface between coatingand substrate caused by shear stress. Compared with prior coatingmodels [50–53], proposed model also considers various variablessimultaneously.

7. Conclusions and future work

Based on nano/micro scratch experiments, XFE models weredeveloped by considering STSE and COF change. A good agreementbetween the experiment and FEA results of normal load and COFchange was obtained for both nano/micro scratch tests. COF compo-nents extraction method was developed based on theoretical,

Fig. 20. Effect of Drucker-Prager model parameterson COF.

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numerical approaches and experimental data. Requirements aboutelastic recovery and pile-up to use COF components extraction methodwere introduced. In case of nano scratch test on soda-lime glass,satisfied requirements and components of COF were analyzed. Byobserving μs change, Coulomb's friction change was estimated formaking FRIC_COEF subroutine. We observed negligible effect of ten-sile cracks on COF change using XFEA. Therefore, it is suggested not touse XFEM for analyzing COF change in scratch test. The XFE resultscan be used to understand the friction mechanism and STSE of ma-terials during the scratch test.

The present work is useful for modelling amorphous materialsunder scratch test. However, chipping, loading rate, scratch speed,fracture criterion, relative humidity and temperature were notimplemented in the XFE model. In addition, DP model cannot reflect

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ISE completely. Thus, further studies on ductile fracture criterion forbrittle materials and effect of loading rate and scratch speed on criticalload will improve the practical applicability of 3D XFE model. Futurework will involve establishing failure criterion and yield model forexplaining ISE in amorphous materials, in order to describe a scratchbehavior of amorphous materials more precisely, and improving XFEmodel by considering scratch speed and environment as additionalparameters.

Acknowledgments

This research was supported by the Basic Science Research Programsthrough the National Research Foundation of Korea (NRF-2015R1A2A1A15056163 and NRF-2017R1A2B3009706).

Appendix

A Analytical solutions for components of coefficient of friction

The overall coefficient of friction (COF) μt is expressed as [9]

μt ¼ μd þ μa þ μp (A.1)

where μd is asperity deformation, μa is adhesion and μp is plowing COF. If we define interfacial COF μs as the sum of μd and μa, the μt has the form like

μt ¼ μs þ μp (A.2)

The theoretical solution for μt for a spherical tip under the scratch test is [20–21]

μ ¼ 2πa�2

hsin�1 a� a

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pþ 2

�sp

��1�

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

p ; a ¼ w

2r(A.3)

where r is the radius of scratch-tip,w is the groove width, s is the shear pressure and p is the applied normal pressure (Fig. A.1). If the shear strength of amaterial is so high that gloss sliding does not occur, Coulomb's friction is equal to s/p. Then, μs and μp are

μs ¼2πa�2

�2�sp

��1�

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

p ; a ¼ w

2r(A.4)

μp ¼2πa�2

hsin�1 a� a

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

p i; a ¼ w

2r(A.5)

Fig. A.1. Idealized models of sphere wear debris [21].

By considering elastic recovery behind the scratch-tip, Lafaye et al. [24] developed μp as

μp ¼2a2

(ρ2 sin�1½ða=ρÞcos ω� � a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ2 � a2 cos2 ω

pcos ω

π þ 2ωþ sin 2 ω

)(A.6)

where ω is elastic recovery contact area angle and ρ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � a2 sin2 ω

p(Fig. A.2). The theoretical solutions for components of COF when scratch-tip is

conical and sphero-conical shape can be found in references [21–24]. In those shape of scratch-tips, components of COF are also expressed similarly toEqs. (A.4)�(A.6).

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K. Lee et al.

409

Fig. A.2. (a) Schematization of the normal sectionshowing the elastic recovery represented by the rear angleω. (b) Definition of the cross section in the case of aspherical tip, and (c) front view of the cross section in (b)[24].

Strictly speaking, note that μs and μp in Eqs. (A.4)�(A.6) do not purely represent the real meaning of the μs and μp. In real meaning of μs, it has to onlyconsider interfacial characteristics, not plastic deformation. Conversely, in the case of μp, it has to only consider plastic deformation not interfacialcharacteristics. However, μs in Eq. (A.4) contains the term a, and this means that it considers the deformation of specimen during the scratch test.Accordingly, μp also cannot imply all friction force caused by plastic deformation. In this way, although the above expressions do not represent the realmeaning of μs and μp, we use them for the convenience of recognition.

B Components of coefficient of friction in finite element analysis

In the scratch experiment, we can obtain μt from a ratio of applied normal load to tangential load. However, just analysis of μt is not enough tounderstand friction mechanisms such as failure modes or plastic deformation during the scratch test. Thus, a method that divides μt obtained from theexperiment into μs and μp is needed; we developed the method by using FEM with theoretical solutions of COF.

As explained in Appendix A, μt is the sum of μs and μp. Since Coulomb's friction corresponds to s/p in μs in Eq. (A.4), provided Coulomb's friction iszero, μs is also zero. Accordingly, if the Coulomb's friction is set to zero in the FEA under the same conditions as the experiment, we can consider μp as μt.Then, μs is obtained by subtracting acquired μp from μt, measured from the experiment.

However, it is not always possible to use above method. To use the method, elastic recovery and pile-up during the scratch test have to be ignored. Inother words, deformation between FE model with Coulomb's friction adjusted with experiment and that with no Coulomb's friction has to be similar.From now on, for brevity, we denominate the formal model as an ITF (interfacial COF) and the letter one as a NITF (no interfacial COF). In case of elasticrecovery, μp depends on ω as shown in Eq. (A.6). Therefore, if ω is similar in those FE models, the effect of elastic recovery on COF change is also similar,thus, can be ignored. When conical scratch-tip is used and no strain hardening occurs in thematerial, the value of ω is related to rheological factor X [47]

X ¼ Eσo

tan θ (B.1)

where E is Young's modulus, σo is yielding strength and θ is the angle between cone and the sliding direction. The values of ω according to X obtainedfrom numerical simulations is given in reference [47]. In case of pile-up, Prasad et al. [54] found that an increase in μs causes pile-up during the scratchtest, thus, the plastic deformation of FE model with Coulomb's friction is larger than that of plastic deformation when Coulomb's friction is zero. If theeffect of elastic recovery on COF change between ITF and NITF is the same, thus, can be ignored, it is enough to compare the only μp caused by the fronthalf of the scratch-tip where pile-up occurs. This theoretical solution of μp caused by the front half of the scratch-tip for a spherical tip under the scratchtest is Eq. (A.5) and that for another tip can be found in and derived from reference [21]. Before using this method, it has to be checked whether thepile-up due to μs increases the COF significantly.

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