Scratching the surface: Elastic rotations beneath nanoscratch and nanoindentationtests

A.Kareera, E. Tarletonb,a, C. Hardiec,a, S.V.Hainsworthd, A.J.Wilkinsona

aDepartment of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, United KingdombDepartment of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United KingdomcCulham Centre for Fusion Energy, UK Atomic Energy Authority, Abingdon, OX14 3DB, United Kingdom

dAston Institute of Materials Research, School of Engineering and Applied Science, Aston University, Aston, Birmingham, B47ET, United Kingdom

Abstract

In this paper, we investigate the residual deformation field in the vicinity of nanoscratch tests using twoorientations of a Berkovich tip on an (001) Cu single crystal. We compare the deformation with that fromindentation, in an attempt to understand the mechanisms of deformation in tangential sliding. The latticerotation fields are mapped experimentally using high-resolution electron backscatter diffraction (HR-EBSD)on cross-sections prepared using focused ion beam (FIB). A physically-based crystal plasticity finite elementmodel (CPFEM) is used to simulate the lattice rotation fields, and provide insight into the 3D rotation fieldsurrounding a nano-scratch experiment, as it transitions from an initial static indentation to a steady-statescratch. The CPFEM simulations capture the experimental rotation fields with good fidelity, and show howthe rotations about the scratch direction are reversed as the indenter moves away from the initial indentation.

Keywords: Nanoindentation, Nanoscratch, HR-EBSD, CPFEM

1. Introduction

The extensive development of nanomechanical test-ing instruments has expanded the capabilities of nano-scale measurement beyond basic indentation hard-ness. Nanoscratch has generated significant inter-est and can be performed on commercial nanoinden-ters, requiring only minor adaptations to the soft-ware. When used in combination, nanoscratch andnanoindentation provide a powerful means to inves-tigate the near surface mechanical and tribologicalproperties of small volumes of material [1, 2, 3, 4];it enables the study of nanoscale friction [5, 6] andallows the adhesive strength and fracture propertiesof coated systems to be characterised [7, 8, 9].

Macroscopically, Tabor showed that on the ba-sis of plastic deformation, there is a strong correla-tion between the indentation and scratch hardnesswhen the measurement is based on a mean pressure[10]. As a consequence, the indentation hardness isthe key metric used to define wear resistance [11].The complex surface interactions involved in wearprocesses are extremely difficult to understand and

simplified models such as this do not completely takeinto account the physical mechanics that are occur-ring. Nanoscratch testing has the advantage, in thatit provides an experimental platform to reproduce asingle point, sliding asperity contact, that is believedto control the process of abrasive wear [4, 12, 13].Hence, it is becoming increasingly common to usethis technique to study the near surface mechanicalproperties via determination of the scratch hardness,from which the wear resistance of material systemscan be inferred [14, 15, 16, 17]. The remaining is-sue resides in the definition of scratch hardness; thesimplest and most frequently used measurement ofscratch hardness is analogous to indentation hardnessand is defined as the ratio between the normal loadand the projected load bearing area. The studies thathave used this definition, show substantial differencesbetween the measured indentation and scratch hard-ness [18, 19, 20]. By further incorporating the lateralforce into the measurement, it is possible to obtain ascratch hardness measurement that is in closer agree-ment to the indentation hardness, on isotropic mate-rials [21, 22]. This however, is not applicable to all

Preprint submitted to Acta Materialia July 22, 2020

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material classes, particularly metallic samples thatexhibit work hardening and anisotropy, as shown in[23, 24]. A number of critical considerations must ad-ditionally be accounted for; namely the effect of fric-tion, plasticity size effects, the resultant strain on thematerial, work hardening as a result of evolving dislo-cation structure and the direction of flow of displacedmaterial in each loading direction [15]. In order tointerpret the differences measured in indentation andscratch hardness, it is important to develop a deeperunderstanding of the mechanics of nanoscratch for-mation.

Experimental observations of the deformation fieldbeneath indentation experiments, have been used tointerpret the hardening behaviour in various materi-als. These studies have revealed that the plastic zoneis extremely complex and there is a continued effort torelate the deformation field to the measured mechan-ical properties [25, 26, 27, 28, 29]. Several studieshave reported that for indentation with a geomet-rically self-similar indenter, the geometrically neces-sary dislocation (GND) structure does not develop ina self-similar, hemispherical way as often assumed insimplified explanations [30, 31, 32]. The lattice ro-tation fields below indentation experiments with var-ious tip geometries, have revealed distinct pattern-ing within the plastic zone that exhibit well definedboundaries and a steep orientation gradient where achange in the sign of the rotation direction is observed[25, 33, 34]. The investigation of plastic deformationand induced lattice rotations is of great interest foran improved micromechanical understanding of in-dentation experiments owing to the close connectionbetween crystallographic shear and the resulting lat-tice rotation.

Simulation methods, such as the Crystal Plastic-ity Finite Element Method (CPFEM), provide fur-ther insight into the mechanics of the deformationfield, when applied in conjunction with indentationexperiments [35]. Through incorporation of an ap-propriate, physically based, constitutive model alongwith details of the microstructure and constitutiveparameters, CPFEM enables the effect of grain size,crystallographic orientation [36] and plasticity size ef-fects [37] to be studied. Zaafarani and co-workerssimulated the lattice rotation field below sphericalindentations in copper using CPFEM, and directlycompared the simulated rotation fields with that ob-tained from EBSD [34]. Simulating the lattice rota-tion field facilitates the interpretation of the deforma-

tion mechanisms, by separating the crystallographicshear occurring on individual slip systems, and di-rectly relating it to the patterns observed in the lat-tice rotation fields [38]. CPFEM has the added ben-efit in that it offers information on the spatial 3Ddistribution of the deformation field, in real time, asit evolves throughout the experiment. This is neces-sary to interpret dynamic experiments, whereby ex-perimentally it is only possible to study the final, de-formed state via postmortem analysis.

In comparison to nanoindentation experiments,the study of deformation below nanoscratch exper-iments is still in its infancy; the deformation field isfurther complicated due to the lateral force. Macro-scopic scratch experiments show that plastic defor-mation induces changes in the microstructure of thematerial, resulting in a distinct discontinuity betweena surface layer and the underlying bulk material [39,40, 41, 42, 43, 44]. This physical boundary has alsobeen identified in TEM studies around nanoscratchexperiments in Ni3Al where the plastic zone consistsof a core region with high dislocation density sur-rounded by an outer region with lower dislocationdensity [45].

Simplified mechanistic models have been proposedto simulate the plasticity dominated deformation fieldaround nanoscratch experiments similar to those usedfor nanoindentation that assume the plastic zone isproportional to the scratch width [20, 46] but thesemodels are purely theoretical and, in most cases, arenot validated. Isotropic Finite Element models havebeen used to simulate the strain field around nano-scratch experiments in an attempt to validate analyt-ical models [47, 48] and to describe the strain field inboth bulk and coated systems [49, 50]. A model byHolmberg and co-workers found that the stress fieldunder scratch experiments in coated systems is differ-ent to that under bulk samples. This was attributedto the mismatch between material properties in thecoating and substrate, which restricted the ability forthe residual stresses to elastically recover. In the ab-sence of a coating, elastic recovery is accommodated,resulting in a different stress field [51]. However, a de-tailed study of the effect of crystallographic orienta-tion and the resulting lattice rotation field surround-ing nanoscratch experiments remains unexplored.

In this paper, we use High Resolution EBSD (HR-EBSD) to experimentally map the lattice rotationfield in the vicinity of nanoindentation and nano-scratch experiments [32, 52, 53, 54, 55]. Scratch and

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indentations were generated under the same normalforce (3mN) using a Berkovich indenter, in singlecrystal copper. A physically based CPFE model isused to simulate the scratch experiment. Lattice ro-tation fields from the simulation are directly com-pared with the experimental results and provide aninsight into the three-dimensional mechanisms thatoccur during deformation beneath a sliding contactwhich can help understand the quantitative differ-ences observed between indentation and scratch hard-ness.

2. Methods

2.1. Nanoscratch and nanoindentation

Nanoscratch and nanoindentation experiments werecarried out on a sample of single-crystal, oxygen-free pure copper with 99.9% purity, oriented in the(001) crystallographic plane (obtained from Good-fellow UK). The sample was annealed in air for 4hours at 600◦C, followed by a mechanical and elec-trolytic polish in order to obtain a smooth flat sur-face, with negligible residual stresses. The indenta-tion and scratches were made using a Keysight (for-merly Agilent, formerly MTS) G200 instrumented in-dentation system, fitted with a lateral force measure-ment probe and a Berkovich diamond tip. The tipradius of 20nm was measured using AFM. The in-dentation was made using a quasi-static loading func-tion to a maximum normal force of 3 mN and theindenter was aligned such that one facet of the in-denter was perpendicular to the [100] direction. Thescratches were formed in both the edge forward (EF)and face forward (FF) tip orientation, parallel to the[100] direction (corresponding to the x1 direction inFigure 1(b)) at a velocity of 10 µm s−1. A scratchlength of 100 µm and constant normal force of 3 mN,were used for the scratches and a three-pass scratchmethod was implemented to correct for surface rough-ness and sample tilt, by subtracting the surface pro-file displacement (pass 1) from the scratch displace-ment (pass 2). The final profiling pass (pass 3), of-fers insights into elastic recovery. Full details of thethree-pass scratch method are provided in [24]. Thecorrected penetration depth channel is plotted as afunction of scratch distance in Figure 1(a).

2.2. Cross-sectioning

Cross-sections through the nanoindent and thescratches were prepared using a Zeiss Auriga FIB-SEM. Cross-sections were taken from two locations in

the scratch; scratch section A is cut across the centreof the scratch and scratch section B is cut at the endof the scratch (these locations are shown in Figure1(b)). The indentation section was taken from thecentre of the indentation, in the same orientation asthe scratch such that the cross-sectioned surface wasoriented in the (100) crystallographic plane. Cross-sectional slices, of approximately 20µm × 10µm ×3µm in size, were lifted from the sample in-situ andmounted on an Omniprobe TEM copper grid. SEMimages of the prepared cross-sections, prior to lift-out, are given in Figures 1(c) and 1(d).

2.3. High Resolution Electron Backscatter Diffraction

EBSD measurements of the cross-sections and sur-face of the scratches were made in a Zeiss Merlin FEGSEM equipped with a Bruker e−FlashHR EBSD de-tector operated by Esprit 2.0 software. The EBSDpatterns (EBSPs) were acquired using an electronbeam energy of 20kV and a probe current of 5nA; astep size of 50nm was used and EBSPs were collectedand saved at a resolution of 800× 600 pixels.

HR-EBSD is used to map the lattice rotation fieldin the vicinity of the indentation and scratch experi-ments. The technique uses an image cross-correlationbased analysis to measure the lattice curvature withinthe crystal. A reference pattern is selected from a lo-cation within the grain far from the regions of high de-formation (in the case of the single crystal used in thisstudy, the reference pattern is selected far from theindented/scratched region of the orientation map).Each test pattern within the map is cross-correlatedwith respect to the reference pattern; the patternshifts of a number of regions of interest (ROIs) aremeasured and related to the crystal lattice rotation.A full description of the mathematics describing themethod can be found in references [56, 32, 57]. Lat-tice rotations between test and reference pattern canbe used to estimate the GND density based on Nye’sframework [58, 52]. The total dislocation densitycomprises individual dislocation densities from eachdislocation slip system. For an FCC crystal, it isassumed that the GNDs are either pure screw dislo-cations, or pure edge dislocations with 〈110〉 Burgersvectors giving 18 types of unknown dislocation den-sities. 2D maps allow the lattice rotation gradientsalong two orthogonal axes within the surface to bedetermined giving six out of the nine lattice rota-tion tensor components. At each point in the map, aset of possible GND combinations that satisfy the six

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Figure 1: a) Raw experimental penetration depth vs. scratch distance for FF and EF scratch. b) Schematic of the scratchdirections with respect to the crystallographic orientation and location of sections lifted for EBSD analysis. c) SEM image of EFscratch and FIB prepared scratch section A d) SEM image of FIB prepared indentation cross-section.

measured lattice curvatures are found and the com-bination that gives the minimum total line energyis chosen. Further information on the calculation ofGND density can be found in [32]. For the experi-mental conditions used, this method measures latticerotations with an angular sensitivity of 3× 10−4 radwhich corresponds to a lower bound GND densitynoise floor of 1.5× 1013m−2.

2.4. Crystal plasticity finite element modelling

Finite element simulations were performed usingAbaqus 2016 to investigate the influence of deforma-tion due to scratching with the EF tip geometry, insingle crystal copper. A crystal plasticity user mate-rial (UMAT) for Abaqus was used based on the userelement (UEL) by Dunne et al. [59, 35].

The deformation is decomposed multiplicativelyinto a plastic, F p, and elastic, F e, deformation gra-dient

Fij = F eikF

pkj (1)

the flow rule has the form

F pij = Lp

ikFpkj . (2)

Where the plastic velocity gradient, Lp, is given bythe crystallographic strain rate resulting from dislo-cation glide on the active slip systems with slip direc-tion sk and slip plane normal nk

Lpij =

k=12∑k=1

γk(τ)ski nkj (3)

The crystallographic slip rate γ is given by

γk(τ) = A sinh(B(|τk| − τc)

)sgn(τk) (4)

for |τk| > τc and γk = 0 otherwise, where the resolvedshear stress on slip system k is τk = σijn

ki s

kj . The

critically resolved shear stress is assumed the samefor each slip system,

τc(ρ) = τ0c + CGb√ρ. (5)

For simplicity we assume that the dislocation density,ρ is proportional to the plastic strain

ρ = D

√2

3εpij ε

pij (6)

4

where the plastic strain rate εpij is the symmetric part

of Lpij . The fitting constants were A = 10−6 s−1,

B = 0.1 MPa−1, an obstacle strength term C = 0.05and D = 2.45 × 104 µm−2. Initial values were takenfrom [60] and calibrated to match the normal forceduring the scratch. Consequently plastic deformationinduces a dislocation density which hardens the slipsystems via an increase in the CRSS, τc(ρ).

Lattice Rotation: The elastic distortion and rigidbody rotation of the lattice is found by computing theelastic part of the deformation gradient, from (1):

F eij = FikF

pkj

−1(7)

and from this the elastic part of the velocity gradi-ent and its anti-symmetric spin component are calcu-lated:

Leij = F e

ikFe−1kj (8)

W eij =

1

2(Le

ij − Leji) (9)

Finally the crystal orientation matrix R is calcu-lated using an implicit integration of the elastic spin,which is updated at the end of each increment:

Rij(t+ ∆t) = [I −∆tW e(t+ ∆t)]−1ik Rkj(t) (10)

Rij rotates a vector expressed in the crystal frame toa vector in the deformed material frame. The crys-tal frame in this study is initially aligned with thereference frame of the model and so Rij(0) = δij .

The rotation matrix at the end of the simulationwas used to define the elastic rotations about the 3axes in the reference frame x3, x2 and x1 for compar-ison with experimental HR-EBSD data: [61, 62]:

ω32 = tan−1(R32/R33) (11)

ω13 = − sin−1 (R31) (12)

ω21 = tan−1(R21/R11) (13)

A total of 4950 linear hexahedral elements with 8Gauss points per element (C3D8) were used to repre-sent a block of copper with dimensions of L = 25×15×7.5 µm, with symmetry boundary conditions appliedalong the (010) mid plane allowing only half of thedomain to be simulated. The scratch test was simu-lated by modelling an indentation followed by an edgeforward (EF) scratch step with a constant displace-ment of u3 = −247 nm and applied lateral displace-ment of u1 = 10 µm at a rate of 10 µm/s, these pa-rameters were chosen to match the experiment. Dis-placement boundary conditions were used to improve

the stability and efficiency of the implicit solution forglobal equilibrium of nodal forces. Force boundaryconditions are particularly unstable for highly non-linear problems such as contact. A scratch length of10 µm was found to be sufficient to reach a steadystate scratch formation. A biased mesh under thescratch was used for improved accuracy and com-putational efficiency, with an approximate elementsize of w1 = 0.4 µm along the scratch direction x1,w2 = 0.2 µm along x2, increasing up to w2 = 1 µmfar from the scratch. The indenter tip was mod-elled as a rigid part with a perfect Berkovich ge-ometry. The finite sliding, node to surface, Abaquscontact algorithm was used with the default hardcontact property. The absolute values for the lat-eral and normal force are determined by the mate-rial model however, their ratio is governed entirelyby the friction behaviour. A friction coefficient of0.15 was used to specify tangential behaviour betweenthe surfaces in contact. This value was calculatedfrom the experimental data by resolving the normal,FN , and tangential forces, FT , on the indenter tipfaces during sliding; where the friction coefficient isµ = FT /FR. The result is the true friction coeffi-cient between the contact surfaces which is indepen-dent of geometry, scratch speed or scratch directionin this case. Therefore values for both tip orienta-tions studied were identical within the experimentalscatter. In contrast calculating the friction coefficientwithout accounting for the geometry, i.e. simply tak-ing the ratio of the applied lateral and normal forces,µ = FL/FN , is no longer fully defined by the twomaterials but is dependent on the stress state andthe resultant plastic deformation imposed by the tiporientation/geometry/speed. Consequently FL/FN issignificantly different for the EF and FF orientationsin this study; 0.4 and 0.6 respectively.

The new friction coefficient described in AppendixA, µ = FT /FR where FT and FR are the tangentialand resolved orthogonal forces acting on the facet(s)of the indenter, depends only on the interaction be-tween the two material surfaces in contact and doesnot vary with tip orientation. The friction coefficientwas found to be 0.15 in both scratch directions andwas used as the friction coefficent in the Abaqus sur-face contact algorithm in the FE model. The sym-metry plane was fixed from translation in the normaldirection, x2 = 0, the nodes on the top surface weretraction free, while the remaining four surfaces of thecube were fixed. Elastic anisotropy was used with the

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following elastic constants for copper: c11 = 168.4,c12 = 121.4, c44 = 75.4 or E = 66.7 GPa, G = 75.4GPa, and ν = 0.419. 12 〈110〉{111} fcc slip sys-tems were included with an initial CRSS of τ0c = 1MPa. Further details on the UMAT can be found in[60, 63, 64, 65, 62, 66]. Simulations provide directcomparison with the EF scratch test.

3. Results

The experimentally measured and simulated ver-tical displacement, normal force and lateral force aregiven in Figure 2 for the EF tip orientation. Fig-ure 2(a) shows the vertical displacement of the tipfor the full 100 µm scratch length measured exper-imentally. The simulated scratch data is presentedfor comparison. The vertical dashed lines representthe indenter displacement from zero to the maximumdisplacement of 247 nm, with no lateral movement,where the tip was loaded and unloaded respectively.Data from a scratch distance of 40-50 µm are shownfor the forces (Figure 2(b)), where the experimentalscratch had reached a steady state and the normalforce was maintained at 3 mN without any influenceof loading and unloading of the indenter. As the sim-ulated scratch was only 10 µm long, this is comparedwith the steady state region, between 40-50 µm, ofthe experimental scratch data. The oscillations thatappear in the simulated data are an artefact with awavelength defined by the the node spacing w1 andthe initial 2 µm can be interpreted as the settlingin portion of the scratch. Once steady state scratchdeformation is achieved, there is excellent agreementbetween the experimental and simulated normal andlateral forces.

Experimental and simulated lattice rotation fields,ω21, ω13 and ω32 are shown in Figures 3-5. Thesefields correspond to rotations about the x3, x2 and x1

axis respectively (the crystallographic orientation isshown in Figure 1(b)). The colour code representsthe lattice rotation in radians; the scale has beenconfined to a magnitude of 0.03 radians (1.7◦) to en-able a clearer visualisation of the shape and senseof the rotation fields. Owing to a combination ofedge effects, highly localised deformation and milling-induced curvature, there was insufficient overlap be-tween the captured EBSD patterns and the referencepattern in regions closest to the indenter. As a result,HR-EBSD was unable to compute the lattice rota-tion fields in these regions. To indicate the actual

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Figure 2: a) EF experimental and simulated penetration depthvs. scratch distance. b) corresponding normal force and lateralforce vs. scratch distance between 40−50 µm for experimentaland simulated EF scratch.

surface of the sample, lattice rotation maps are over-laid on the greyscale image quality map from EBSD.Throughout this work, we refer to a positive latticerotation as an anticlockwise rotation about an axiswhen looking down the axis towards the origin.

Figures 3(a)-(c) show the lattice rotation fields forthe indentation cross section, measured experimen-tally using HR-EBSD. Figure 3(d)-(f) are the equiv-alent fields for the FF scratch (100) cross section (in-dicated A in Figure 1) and Figures 3(g)-(i) are thecorresponding fields from the EF scratch. Finally,Figures 3(j)-(k) show the corresponding simulated ro-tation fields for the EF scratch which was taken froma scratch distance of 5 µm.

From the experimental maps, it is clear that thelattice rotation field below a scratch extends signif-icantly further than beneath an indentation createdunder the same normal force and that the FF scratchhas a larger deformed region (plastic zone) than theEF scratch. The faceted Berkovich indenter createsa deformation field that takes a distinctive double-lobed form, with a steep maxima in the rotations inregions tangent to the indenter facets - the orienta-tion of these facets with respect to the loading axis is

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ω13 ω32

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ntFF

scr

atch

EF

scra

tch

Sim

ulat

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EF

scra

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+ve

+ve+ve

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Figure 3: Lattice rotation field maps for the indentation section and the scratch sections A (see Figure 1) about the x3, x2 and x1

axis; ω21, ω13 and ω32 respectively. a, b and c are rotation fields from the indentation, d, e and f are the FF scratch (100) crosssection rotation field, g, h and i are the corresponding EF scratch rotation field. j, k and l show the simulated rotation fields ofthe EF scratch. The colour code shows the lattice rotation in radians. Lattice rotation maps are overlaid on the greyscale imagequality maps from EBSD. Scaling is identical for all diagrams.

different for each experiment and this is reflected inthe shape of the plastic zone. In the indentation andEF oriented scratch, the double-lobe shape is moreprominent. In the EF scratch the tip is oriented suchthat a sharp edge leads the deformation, whereas inthe FF oriented scratch a flat facet drives the de-formation. For the indentation alone, the very sharppoint of the indenter is driving deformation vertically.

In order to describe the rotation fields, it is as-sumed that material rotates about the apex of theindenter tip. The indentation field can be interpretedas the lattice rotating toward the central loading axisof the indentation for ω21 and ω13 (Figure 3(a) and(b)); ω32 shows the lattice rotates towards the freesurface and towards the centre of the indentation

(Figure 3(c)). Note that for the indentation, ω21 andω13 (Figure 3(a) and (b)) are dependent on the loca-tion of the section. Although every effort was madeto prepare this section across the centre of the inden-tation, experimentally this is challenging when usinga FIB to prepare cross-sections, and it is likely that itis slightly off centre (by approximately 50 - 150 nm).

The ω21 and ω13 rotation fields in both scratchexperiments show that the lattice rotates about theindenter apex, with a positive rotation about the x2

axis perpendicular to the scratch direction. The in-plane rotation fields ω32 (Figure 3(f) and (i)) showthat in scratch, the lattice rotates in the opposite di-rection to that observed for indentation. For the FFcase (Figure 3(f)) there is an inner region of counter

7

rotation. A region of counter rotation can also beidentified in the EF rotation field (Figure 3(i)) al-though it is less pronounced. The model is able toaccurately simulate then sign of the experimentallyobserved deformation-induced rotation field, includ-ing the inner zone of counter rotation. Although thesimulated plastic zone size is smaller than the exper-iment, the double-lobed shape is reproduced. Thesimulation also provides additional information re-garding the lattice rotations close to the indenterapex, where experimental data could not be obtained.

Similar rotation field maps of the scratch surfaceare given in Figure 4 about the same axes. On thefree surface, the rotation fields correspond with thatobserved subsurface with a change in rotation senseeither side of the scratch track (in ω21 and ω32) and apositive lattice rotation about the axis perpendicularto the direction of travel, [100], in ω13. The experi-mental scratch width is approximately 4 µm, whichis in accordance with that produced from the simula-tion. Note that in Figure 4, the experimental scratchwidth appears wider due to the lack of experimen-tal HR-EBSD data close to the scratch edges, dueto the poor quality EBSD patterns. At the end ofthe scratch track there is considerable deformationfrom piled-up material. For the EF case, the signof the in-plane rotation ω21 changes at the end ofthe scratch (Figure 4(d) and (g)). This can be moreclearly seen in Figure 5, which shows the experimen-tal and simulated rotation fields from scratch sectionB, the (010) plane. As before with indentation, therotation fields shown for scratch section B are depen-dent on the precise location of where the cross sectionwas prepared, and although the aim was to target thevery centre of the scratch track, experimentally thisis challenging and it is likely that the slice was takenslightly off centre. The exact offset for the experimen-tal cross-section is unknown however it is assumed tobe within the range of 50 nm-200 nm. Figures 5(g)-(i) show the simulated cross-sections offset by 100 nmfrom the centre of the scratch for comparison. Thediscrepancy between the experimental and simulatedω21 rotation field is likely due to this uncertainty.

Figure 6 compares the experimental GND den-sity field measured using HR-EBSD, with that of thedislocation density calculated by the model, by nu-merical integration of ρ (defined in equation (6)) overtime. The magnitude and distribution of dislocationdensity calculated by the model is representative ofthat measured for the experiment, which is remark-

able given the simple form of the dislocation densityevolution and hardening laws used in the model. Thequalitative agreement between simulated and experi-mentally measured dislocation density plots supportsboth the assumption that dislocation multiplication isapproximately proportional to effective plastic strain,and the use of the Taylor hardening law in the simu-lation.

4. Discussion

This work uses HR-EBSD to study the local de-formation field around nanoscale experiments. Themultiple views of the scratch rotation field, presentedin the experimental results enables a more compre-hensive investigation of the volume of deformed ma-terial surrounding the scratch. However, it remainslimited to a snapshot, postmortem analysis and isnot sufficient to measure deformation in the regionswhere the largest rotations occur, close to the inden-ter apex. The 3-D rotation fields predicted using acrystal plasticity simulation provide a real-time visu-alisation of the deformation field during scratch for-mation and are able to predict the deformation closeto the indenter, where experimental data is missing.Hence in this work, the experiments and simulationsare complementary to each other, and enable a morecomplete investigation. The CPFE model is able toaccurately predict the rotation field for the EF tiporientation in terms of sense and axis. The accuratesimulation of the normal and lateral forces, relies ona correct coefficient of friction which is determinedfrom the experimental data as outlined in AppendixA. The most commonly reported coefficient of frictionfor scratch tests uses the ratio between the lateraland normal force, however, this has been found to benon-physical when used in the model. A new coeffi-cient of friction, based on the resolved forces actingon the facets of the tip (see Appendix A) is used inthe model. This approach is more physically appro-priate as it is based on the geometry of the tip andthe surfaces in contact and is able to accurately pre-dict the experimental forces. Figure A.1 (b) showsthe experimentally measured friction coefficients, de-fined by the conventional method and the new pro-posed method; incorporating the resolved forces onthe facets of the indenter reduces the friction coef-ficient and produces a coefficient independent of tiporientation. This method can be applied to any pyra-midal indenter with a known geometry. Compared

8

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crat

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Figure 4: Lattice rotation field maps, experimental and simulated, of the free (001) surface about the x3, x2 and x1 axis. Colourcode shows the lattice rotation in radians. Lattice rotation maps are overlaid on the greyscale image quality maps from EBSD.Scaling is different between experiment and simulation, refer to scale bars.

to the model, a larger plastic zone is observed in theexperimental measurements of rotations and disloca-tion density fields surrounding the scratch. This dif-ference may be the effect of the indenter roundingin the experiment (20 nm tip radius) which woulddisplace more material than the perfect tip used inthe model and/or heterogeneous material propertiesin the surface layer of the sample as a result of samplepreparation induced damage.

Using the results obtained experimentally and bycrystal plasticity, the rotation fields can be summarisedas follows. For indentation, the lattice rotates aboutthe indenter apex towards the central indenter load-ing axis. This can be qualitatively understood interms of the material that must be displaced by theindenter immediately below it, towards the surface tocreate pile-up around the indentation. The rotationfields ω21 and ω13 measured experimentally, suggestthat the centre of the indentation would be furtheralong the x1 axis, into the page of Figure 3(a),(b)and (c).

The sign of the rotation fields around the scratchexperiments broadly follow a similar pattern for both

the EF and FF tip orientations and can be describedby two simultaneous mechanisms. Firstly, the latticerotates anticlockwise about the x2, as the indenter‘pulls’ the surrounding lattice along with it in the di-rection it is traversing. The second mechanism causesthe lattice to rotate about the indenter away fromthe centre of the normal loading axis of the indenter,represented in the ω32 rotation field, which is the op-posite sense to that observed in the indentation. Thisis the most striking difference observed between theindentation and scratch rotation fields in the experi-mental data.

The differences in the shape of the plastic zonebetween the two scratch tip geometries might be ex-plained in terms of the way the material ahead of theindenter is displaced around the tip for each orienta-tion. In the FF tip orientation, the most efficient waywould be for material to move underneath the tip,whereas in the EF orientation the angled facets wouldassist displacement of material laterally around thetip. The exact displacements are more complex andwould involve multiple directions for both tip orienta-tions however this simple analogy could describe the

9

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

2um

ω32ω13ω21

ω32ω13ω21

x1x2 in to page

x3

ω32ω13ω21

Sim

ulat

ed

EF

scra

tch

EF

scra

tch

FF s

crat

ch

+ve

+ve

+ve

a. c.

d. e. f.

g. h. i.

b.

Figure 5: Lattice rotation field maps for scratch section B about the x3, x2 and x1 axis. The colour code shows the latticerotation in radians. Lattice rotation maps are overlaid on the greyscale image quality maps from EBSD. Scaling is identical forall diagrams.

differences between the two scratch tip geometries.Additionally, the FF tip orientation has a higher lat-eral component in the resolved force, compared tothe EF tip orientation (see equations in AppendixA) which would result in more work being done onthe material, and as a result more plasticity.

As the simulation modelled an indentation stepfollowed by the scratch step, analogous to the ex-periment, it is possible to investigate the mechanicsof scratch formation, as the loading state transitionsfrom a static indentation to a steady state scratch.This will aid the interpretation of differences observedbetween the indentation and scratch experiments. InFigure 7, the simulated ω32 rotation field, for a setof successive planes throughout the scratch, paral-lel to the (100) plane (i.e. parallel to scratch sec-tion A) are given. Figure 7(a), where the indenter issolely under a normal load (i.e. static indentation),shows a rotation field with four lobes, where the zonesclose to the surface have the same rotation sense tothat observed in the experimental indentation rota-tion (Figure 3(c)). This four-lobed rotation field hasbeen observed for static indentations using a wedgeindenter [67] and spherical indentations [68, 60]. Asthe scratch begins to traverse laterally (Figure 7(b)),an additional outer region of counter rotation begins

to develop at the surface, whilst the four-lobed in-dentation rotation zone becomes further confined. Asthe scratch progresses, (Figure 7(c)-(f)) the outer ro-tation zone expands further and dominates the plas-tic deformation field. This abrupt change in rota-tion direction, could correspond with the tribologi-cally induced discontinuity, known as a dislocationtrace line, ubiquitously observed in various materialsystems. The origin of this has been attributed to thereorganisation of dislocations beneath a sliding con-tact as a result of the in-plane shear component of thestress field [69, 39]. It appears that by incorporatinga lateral component of force, a strongly inhomege-neous stress state subsurface, different to that undera statically loaded indent, activates slip systems thatcause an outer rotation field to be formed whilst con-straining the inner rotation field. Referring back tothe scratch displacement profile in Figure 1(a) it canbe seen that in the early stages of the scratch forma-tion, the displacement initially reaches a maximum ofapproximately ∼400 nm, where the maximum normalforce of 3 mN is supported by the indentation alone.When the tip begins to traverse laterally, and theouter deformation field begins to form, the 3 mN isonly enough force to produce a displacement of ∼250nm. Hence the indenter rises up until it reaches a

10

2um

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

51015

EF

scra

tch

Inde

ntExperiment Simulation

a.

c. d.

b.

Figure 6: GND density map for the indentation (a) and EF scratch section A (b) calculated experimentally using HR-EBSD.Corresponding total dislocation density from the CPFE simulation for the indentation step (b) and the steady state scratchcross-section (d). Colour code shows the dislocation density per m−2.

steady state penetration depth. Although the modeluses a displacement controlled scratch step to aid nu-merical stability, the same mechanisms are observedin Figure 2. The initial indentation to a target depthof 247 nm requires a normal force of 1.5 mN, as thescratch progresses laterally, and the outer deforma-tion field is formed, the normal force must increasein line with the experiment to maintain the constantpenetration depth.

Hence this work highlights that with a reasonablesimple variation in the test, i.e. incorporating lateralmovement of the tip, the deformation field is signifi-cantly different and more plasticity is induced in thematerial. An outer rotation zone, of opposite sense,forms in addition to the rotation field created by in-dentation alone. This implies that an indentationprocess does not have all of the components to de-scribe the deformation associated with a sliding con-tact. As a result, in terms of predicting the wear resis-tance of materials, indentation hardness may not bethe most suitable measure. Scratch hardness, basedon scratch testing will provide a significantly closerrepresentation of the deformation associated with asliding contact.

5. Conclusions

We present an investigation of the deformationfield in the vicinity of a controlled scratch using asharp (EF) and relatively blunt (FF) indenter orien-tation, in a Cu single crystal, for direct comparisonwith a static indentation. CPFE was calibrated withonly 3 parameters and reproduced the normal andlateral forces, as well as all 3 rotation fields on all 3principal planes. A simple hardening law was foundsufficient and gave remarkable agreement with themeasured GND density. This combined experimental-modelling approach provides a more complete under-standing of the nanoscratch formation. The mainconclusions are as follows:

• By applying the same normal force, the threeexperiments show different lattice rotation fieldsin terms of the morphology of all three rotationsbased on the direction of loading with respectto the tip orientation. The biggest difference inthe lattice rotation is observed in the sign of theω32 rotation, which is reversed for indentationand scratch.

• By simply incorporating lateral movement ofthe tip, the deformation field is distinctively dif-ferent to indentation. This implies that inden-tation, and therefore hardness calculated from

11

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

2um

a.

b.

c.

d.

e.

f.

x = 0

x = 300 nm

x = 600 nm

x = 1200 nm

x = 2000 nm

x = 5000 nm

x1

x3

x2

+ve

+ve+ve

Figure 7: Simulated ω32 rotation field, for a set of successive planes through the scratch, parallel to the (100) plane (i.e. parallelto scratch section A). The colour code shows the lattice rotation in radians.

indentation alone, cannot fully capture the de-formation associated with a sliding contact. Scratchhardness may provide a more appropriate pre-dictor of wear, as it is able to capture all compo-nents of deformation associated with a slidingcontact.

• The commonly reported friction coefficient, FL/FN ,is not specific to the geometry of the contactand therefore is not sufficient to use in the CPFEmodel to accurately predict the normal and lat-eral forces. A more appropriate methodologyfor calculating the friction coefficient, from nano-scratch, is proposed.

Acknowledgements

AK and ET acknowledges support from the En-gineering and Physical Sciences Research Council un-der Fellowship grants EP/R030537/1 and EP/N007239/1and Platform grant EP/P001645/1. CH also acknowl-edges funding by the UKRI Energy Programme (GrantNo. EP/T012250/1). We are grateful for use of char-acterisation facilities within the David Cockayne Cen-tre for Electron Microscopy, Department of Materials

which has benefited from financial support providedby the Henry Royce Institute (Grant ref EP/R010145/1)

Author Contributions

AK undertook the experimental work (indenta-tion/scratch experiments, FIB and HR-EBSD). ETand CH performed the CPFEM simulations. SVHand AJW advised on the indentation/scratch testingand EBSD respectively. All authors contributed tointerpretation of the results. AK led drafting of themanuscript, with all authors contributing and agree-ing this final version.

Appendix A. Calculating the friction coeffi-cient from experimental data

The coefficient of friction used in the model isgiven by Equation (A.1) where FT is the tangentialforce parallel to the indenter facets and FR is the re-solved force perpendicular to the indenter facets (seeFigure A.1). FR and FT are given by Equations (A.3)and (A.2) respectively for the EF tip geometry. Theauthors note an error in the equation for FR presented

12

in the appendix of [24] and (A.3) is the correct ex-pression.

µ = FT /FR (A.1)

FT =FL

2cosφ− FN

2sinφ (A.2)

FR = FX cos θ +FN

2sin θ

=FL

4cos θ +

FN

2sin θ (A.3)

where FX = (FL/2) cos(60◦) = FL/4, θ = 65.3◦ andφ = 12.95◦ for a Berkovich indenter.

60°

A

B

A

B

a.

b.

0 20 40 60 80 100Scratch distance / m

0

0.2

0.4

0.6

0.8

1

fric

tion

EF FN/FLEF FT/FRFF FN/FLFF FT/FR

Figure A.1: (a) Schematic of indenter indicating forces actingon the facets. (b) Friction coefficient as a function of scratchdistance using traditional and new definition for both the EFand FF tip orientation

In the above equations, it is assumed that thedirection of sliding in the EF case is along the scratchdirection, however in reality a small component ofsliding may be normal to the scratch direction.

In the face forward case (FF) the reaction andtangential forces are

FR = FL cos θ + FN sin θ (A.4)

FT = FL sin θ − FN cos θ (A.5)

and so the friction coefficient (A.1) becomes

µ =

FLFN

tan θ − 1FLFN

+ tan θ= tan(θ − α) (A.6)

where α = tan−1(FN/FL).In the FF scratch orientation, the lateral force is

along FX in Figure A.1, with both FT and FR on thesame plane shown as section B in the figure.

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