Tribology International 117 (2018) 261–271

Contents lists available at ScienceDirect

Tribology International

journal homepage: www.elsevier .com/locate/ tr iboint

Grain-scale mechanics of quartz sand under normal and tangential loading

C.S. Sandeep, K. Senetakis *

Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong Special Administrative Region

A R T I C L E I N F O

Keywords:Micro-mechanicsInter-particle frictionArea of contactPloughing

* Corresponding author.E-mail addresses: sschitta2-c@my.cityu.edu.hk (C.S. Sa

https://doi.org/10.1016/j.triboint.2017.09.014Received 7 May 2017; Received in revised form 10 SepteAvailable online 18 September 20170301-679X/© 2017 Published by Elsevier Ltd.

A B S T R A C T

The study provides some insights into the grain scale mechanical behavior of quartz sand grains at their contacts.It is achieved by conducting laboratory experiments with a custom-built apparatus. A variety of tests were per-formed exploring the effects of pre-loading and pre-shearing on the inter-particle coefficient of friction, thetangential and normal contact behavior. Microscopic observations are made on the grain surfaces before and aftershearing through image processing. Images showed that ploughing dominates the contact behavior at relativelylarger normal loads, but a similar behavior was not apparent at very small loads. The apparent radius of thecontact area of the grains was quantified both experimentally and theoretically using Hertz fitting. It was shownthat theoretical and experimentally derived contact areas matched at relatively greater normal loads but not atsmaller normal loads.

1. Introduction

Recent advances in experimentation by scholars have providedvaluable insights into the tribological behavior of discrete materials[1–4] including geological surfaces in contact [5–9]. The properties ofgeological materials at their contacts are of particular interest in thedisciplines of geotechnical and petroleum engineering, in which appli-cations the sphere-sphere (or particle-particle) configuration must beexplored experimentally [5,6,8]. These properties include the normaland tangential force – displacement relationships and the coefficient offriction which comprise essential parameters to be used in numericalsimulations that use the Discrete Element Method (DEM, after Cundalland Strack [10]) or coupled FEM-DEM analyses.

Based on recent experimental micromechanical studies on the graincontact behavior of geological materials, it may be concluded that theinter-particle friction might depend strongly on the roughness of thesurfaces in contact, since smoother surfaces have been shown to havelower values of friction [5,9,11–13]. For example, Sandeep and Senetakis[12] found much greater friction values at the rough surfaces of volcanicgranules (CDV) which are brittle and soft in nature, in comparison toLeighton Buzzard sand (LBS) quartz grains. Yang et al. [6] found thatploughing mechanism dominates the frictional response of quartz grainsimmersed in water. This ploughing mechanism resulted in greatermagnitude of friction in comparison to surfaces for which no ploughingwas observed, even though the studies by Senetakis et al. [11,14] did notreport notable differences between fairly dry and immersed in liquid

ndeep), ksenetak@cityu.edu.hk (K. S

mber 2017; Accepted 14 September 2

surfaces of geological materials.On the point of view of normal contact tests, Nardelli et al. [9]

compared experimental and Hertzian (theoretical) normalload-displacement behavior of Eglin sand grains. They found an initialplastic response (soft response) during the early stage of contact in theirstudy. They attributed this to asperities deformation which is previouslyobserved by Cavarretta et al. [2]. For relatively crushable and of highroughness carbonate grains, Nardelli and Coop [13] noticed significantdifferences between the apparent Young's moduli, based on Herztianfitting, during the first and second cycle of normal load – displace-ment tests.

To date, there are unexplored areas of research and there is need forfurther systematic works in order to obtain insight into the tribologicalbehavior of granular materials. For example, in nature or man-madeapplications, the grain surfaces may not be subjected to single set ofloading and shearing, but, perhaps, repeated shearing may take place ontheir surfaces as they are subjected to a number of loading and unloadingcycles. However, there have been limited works in the literatureexploring the role of pre-loading and pre-shearing on the micro-mechanical behavior of grain contacts. On the other hand, even thoughthe studies by Cole and Peters [15,16] or Nardelli et al. [9] providedimportant insights into the normal load – displacement behavior ofgeological materials in contact, there are more questions to be answered.For example, it has not been explored in previous works whether thenormal contact loading has an important effect on altering the surfacecharacteristics of geological materials, or shearing must take place to

enetakis).

017

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

obtain measurable changes of the surface properties of grains.In this study, an attempt was made to further explore the tribological

behavior of real sand grains at their contacts and answer some of theaforementioned questions. Particular focus was paid to conduct repeatedshearing tests on the surfaces of quartz grains which were pre-loaded andpre-sheared at increasing or decreasing magnitudes of normal loads.Additionally, cyclic normal loading tests were conducted by loading-unloading the grains and cyclic shearing tests by means of forcereversal under a target displacement amplitude quantifying energy los-ses. Hertz fitting was implemented applying average and local grain radiiand their differences are discussed in the paper. Finally, mechanisms offriction are observed through microscopic image analysis de-couplingnormal and tangential contact behavior.

2. Equipment

The custom-made inter-particle loading apparatus used in the study isshown in Fig. 1. It is capable of conducting normal and tangential loadingtests on geological materials of varying sizes, typically between about 1and 3 mm in size [8,9,11]. Fig. 1 shows the current version of the inter-particle loading apparatus which consists of a stiff loading frame and asled both made of stainless steel. The apparatus also consists of threeloading arms which act in three perpendicular directions, i.e. the direc-tion of shearing (or direction of sliding), the out-of-plane direction andthe vertical (normal to the shearing) direction. Each loading arm consistsof a load cell and a micro-linear actuator (micro-stepping motor). High-resolution load cells of a capacity of 100 N with a precision of 0.02 Nare used to measure the forces in the three directions. Non-contact eddy-current sensors with high resolution (10�5mm) are placed in three di-rections to capture the displacements. The grains are glued to the brassmounts and they are allowed to dry for a minimum of 24 h.

Fig. 2 shows a grain pair of Leighton Buzzard sand (LBS) tested in thestudy; the bottom grain is mounted on the lower brass mount and placedinto the brass well on the sled, where it is firmly fixed using lateralscrews. The top grain is mounted on the upper brass mount and is rigidlyfixed to the vertical arm (upper grain). The normal loading and shearingconditions are shown in Fig. 2(a) and (b), respectively. During theapplication of the normal loading, the bottom grain is fixed and the topgrain is lowered using displacement control mode till the required

Fig. 1. Inter-particle testing apparatus mainly consists of a) Stiff loading frame b) load cell c) lindisplacement sensor g) Digital micro cameras.

262

(nominal) normal load (FN) is reached. During shearing, the normal loadis maintained in a force-controlled manner and the lower grain is movedalong the interface of the two grains using displacement control. Theinter-particle testing apparatus is placed inside a Perspex chamber, whichenables to control the humidity within a range of 15%–85% and tomonitor the temperature [13,17].

3. Materials and testing program

In this study, Leighton Buzzard sand (LBS) grains were examined attheir contacts. LBS majorly consist of fairly rounded quartz grains. Fig. 3gives typical scanning electron microscope (SEM) image for an LBS grainat two different scales. Senetakis et al. [14] conducted white lightinterferometry analyses on LBS grains and found that the averageroughness (Sq) is equal to 0.38 μm with a standard deviation of±0.19 μm. This implies that the surface of the LBS grains is relativelysmooth in comparison, for example, to the rougher surfaces captured byNardelli and Coop [13] on carbonate sand grains (Sq ¼ 0.51 μm) andNardelli et al. [9] for Eglin sand (Sq ¼ 0.53 μm), but relatively rougher incomparison to the average roughness reported by Senetakis et al. [11] forcrushed limestone grains (Sq ¼ 0.31 μm). LBS grains with average size(diameter) of 1–2 mm are used to study their contact properties in thepresent work, conducting a variety of micro-mechanical experiments.

The major micro-mechanical testing program consisted of inter-particle shearing tests on three pairs of LBS grains, named as LBS-A,LBS-B and LBS-C (Table 1). For each pair of grains, a series of shearingtests was conducted at increasing and subsequent decreasing magnitudesof normal loads. For example, as shown in Table 1, for the LBS pair ofgrains LBS-A, the first shearing test was conducted at FN ¼ 1 N and afterthe completion of this test, the normal load was removed, the pair ofgrains was set to their initial position and a second shearing test wasconducted at FN ¼ 3 N. With a similar process, additional shearing testswere conducted at FN equal to 5, 7 and 10 N (or 12 N if applicable) withsubsequent tests at decreasing FN. For these pairs of LBS grains, a total setof twenty-nine shearing tests was conducted as shown in Table 1. All thetests were performed at 60% relative humidity and temperature of22–24�C. Note that previous studies have not demonstrated any notableeffect of the saturation condition (or humidity conditions) of the grains incontact on the frictional response of LBS grains [14] or other types of

ear actuator d) sled placed on chrome steel balls e) soil particles during test f) eddy-current

Fig. 2. Photographs of Quartz grains (a) during the application of normal load. (b) During shearing.

Fig. 3. SEM images LBS showing their spherical nature and smooth surface.

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

geological materials such as natural scaly clays [17]. Additionally, thenormal load – displacement behavior of the three LBS grains (LBS-A,LBS-B and LBS-C) at their contacts was monitored and analyzed duringthe setting of the tests in order to quantify their normal load –

displacement behavior. The tests, as summarized in Table 1, allowed thestudy of the apparent Young's modulus, the tangential stiffness, theinter-particle friction as well as the effect of pre-loading and pre-shearingon the tribological behavior of the grain contacts.

Apart from the major micro-mechanical testing program, additionaltests have also been summarized (Table 2). The majority of these ex-periments in Table 2 are originally presented by Sandeep and Senetakis[12] with new experiments conducted on the course of this study. Theseexperiments consist of repeated shearing on grains for four times i.e.pre-shearing at determined normal load but without insights into thepre-loading effects. Whereas, in the present study (Table 1), the new datacorresponded to pre-shearing at increasing and decreasing magnitudes ofFN thus coupled pre-loading and pre-shearing is examined.

Three more experiments were conducted on different sets of grains byshearing the grains for only once at determined normal load (FN) i.e. 2, 5and 10N. Microscopic examination of the surfaces of the grains wasconducted for these three tests before and after shearing using digitalmicroscope and image analysis. This testing program, as summarized inTables 3 and 4, was conducted at a different normal load for each pair ofgrains. This allowed the observation and quantification of the surfacedamage of the grains at different normal loads and thereby obtainingsome insights into the tribological behavior and micro-mechanisms thattake place at the contacts of geological materials. Note that Tables 3 and4 provide the same set of tests, but there is a difference between them inthe data analysis process as it is explained throughout the paper.

Finally, for a limited number of tests, cyclic normal and tangentialload – displacement tests were conducted on different pairs of LBS grains.The cyclic normal load – displacement tests were conducted to identifyany changes in the normal contact response of the grains subjected torepeated normal loading but without any additional application ofshearing. While, the cyclic tangential load – displacement tests were

263

conducted to quantify energy losses by means of cyclic shearing in theforward and backward direction.

4. Results and discussion

4.1. Normal load-displacement behavior

Fig. 4 shows the normal loading behavior of various quartz grains attheir contacts, which was captured during the setting of the nominalnormal load (FN) in a range of 1–10 N in magnitude. Note that thevariation of the normal force-displacement behavior for the differenttests may be attributed to variabilities in the grain morphologies andasperities geometry for the different pairs of grains. This asperity ge-ometry may be expressed in terms of local radius at the meso-scale e.g.Refs. [18,19], but it is believed that geometry of asperities at the micro-scale which is related to the roughness might have some effect to thenormal contact behavior. For the former (meso-scale), the paper addssome discussion by considering average and local radii on model fittingand derivation of the apparent Young's modulus. For the roughness(micro-scale), even though it is recognized that it may have an importantrole in contact mechanics [20,21] there is not further investigation heresince interferometry analysis has not been conducted in the study.

Based on the results in Fig. 4, all the curves show a non-linear Hert-zian behavior [22,23] which is expected for this material type. Theo-retical Hertzian curves during the application of the normal load forrepresentative curves are plotted to study the range of material apparentYoung's modulus. The Hertzian curves were obtained from Equation(1) [22,23]:

P ¼ 4� ðR � Þ12 � E � � δ32

3(1)

where δ is the normal displacement, P is the normal load correspondingto δ , R* is the equivalent particle radius computed from Equation (2) andE* is the equivalent Young's modulus which was obtained from Equation

Table 1Loading and unloading tangential tests.

No. Pair Test program Code Normal force (N) Horizontal force (N) Inter-particle friction

1 LBS-A Loading ALBS-1L 1 0.28 0.282 ALBS-3L 3 0.71 0.243 ALBS-5L 5 1.05 0.214 ALBS-7L 7 1.39 0.205 ALBS-10L 10 1.88 0.196 Unloading ALBS-7U 7 1.39 0.207 ALBS-5U 5 1.10 0.228 ALBS-3U 3 0.88 0.299 ALBS-1U 1 0.37 0.3710 LBS-B Loading BLBS-1L 1 0.22 0.2211 BLBS-3L 3 0.61 0.2012 BLBS-5L 5 0.92 0.1813 BLBS-7L 7 1.26 0.1814 BLBS-10L 10 1.68 0.1715 Unloading BLBS-7U 7 1.27 0.1816 BLBS-5U 5 1.07 0.2117 BLBS-3U 3 0.71 0.2418 BLBS-1U 1 0.28 0.2819 LBS-C Loading CLBS-1L 1 0.26 0.2620 CLBS-3L 3 0.63 0.2121 CLBS-5L 5 0.93 0.1922 CLBS-7L 7 1.25 0.1823 CLBS-10L 10 1.68 0.1724 CLBS-12L 12 1.96 0.1625 Unloading CLBS-10U 10 1.75 0.1826 CLBS-7U 7 1.35 0.1927 CLBS-5U 5 1.07 0.2128 CLBS-3U 3 0.76 0.2529 CLBS-1U 1 0.33 0.33

Table 2Repeated shearing tests on LBS grains under various normal loads (after [12], updated bythe authors).

No. Test code Normal force Inter-particle friction at a steady state sliding

(N) 1st cycle 2nd cycle 3rd cycle 4th cycle

1 LBS-1 1 0.19 0.18 0.18 0.182 LBS-1B 1 0.25 0.24 0.26 0.243 LBS-2 2 0.26 0.25 – –

4 LBS-2B 2 0.25 0.22 0.22 0.215 LBS-3 3 0.27 0.28 – –

6 LBS-5 5 0.18 0.16 0.16 0.167 LBS-7 7 0.32 0.35 0.36 0.388 LBS-10 10 0.26 0.28 0.29 0.309 LBS-10B 10 0.18 0.20 0.22 0.23

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

(3). In the analysis of the data, E* is used as the fitting parameter to bettermatch the theoretical model to the experimental curves.

1R � ¼ 1

R1þ 1R2

(2)

Table 4Experimental and Hertzian contact radius at various normal loads based on local particle radii.

Local radius (mm) Normal load (FN) E ¼ E1 ¼ E2 E* (GPa) Hertz

LR1 LR2 (N) (GPa)

0.42 0.54 10 65 32.8 0.0380.32 0.57 5 70 35.3 0.0280.53 0.43 2 45 22.7 0.025

Table 3Experimental and Hertzian contact radius at various normal loads based on average particle rad

Radius (mm) Normal load (FN) E ¼ E1 ¼ E2 E* (GPa) Hertz

R1 R2 (N) (GPa)

0.65 0.60 10 55 27.7 0.0440.63 0.50 5 60 30.3 0.0330.60 0.50 2 40 20.0 0.028

264

1E � ¼ 1� ν21

Eþ 1� ν22

E(3)

1 2

In Equation (2) R1 and R2 correspond to the radius of the two particlesin contact. Average radius, R1 and R2 were measured by using Verniercaliper. Even though some grains deviated from being very spherical, themajority of the LBS grains had a fairly spherical shape and a first attemptis conducted here to use the average radius in Equation (2). In Equation(3), E1 and E2 correspond to the apparent Young's modulus of the grainpair in contact (where the subscripts 1 and 2 correspond to the upper andlower grains in contact). In Fig. 4, both the Hertzian curves are plotted forR1¼ R2¼ 0.65mm, which is the average radius of the particles in contactand the Poisson's ratio is taken as ν1 ¼ ν2 ¼ 0.1 [24]. The value of theapparent Young's modulus (E ¼ E1 ¼ E2) using Hertzian fitting andaverage radius is obtained to be 51 and 72 GPa for relatively softer andstiffer curves, respectively. From Fig. 4, the highlighted circle shows thatthe theoretical curves based on Hertzian fit (dot points) are properly ableto fit the trend with the experimental curves, apart from the initialnormal displacement (i.e. initial soft response) during the loading atnormal displacements of less than about 0.002 mm. This observed gap

contact radius ‘α’ (mm) Experimental contact radius (mm) Difference (%)

0.049 22.40.065 56.90.055 54.5

ii.

contact radius ‘α’ (mm) Experimental contact radius (mm) Difference (%)

0.049 10.20.065 49.20.055 49.1

Fig. 4. Normal force vs displacement with Hertzian fitting.

Fig. 6. Variation of apparent Young's modulus (both top and bottom particles) withcycle number.

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

may be attributed to plastic initial response at the contacts due to,perhaps, asperities deformation. For single particle compression testsfrom top and bottom with stiff platens, Cavarretta et al. [2] observed ashort initial regime of plastic deformation which was attributed to somere-arrangement of the particle to find its equilibrium at the early stages ofcompression as well as some plastic deformation of asperities and similartrend was observed by Ref. [9]. Note that in Fig. 4 some minor disparitywas observed between experimental and Hertzian curves even after theinitial displacement. But, the Hertzian curves are able to properly mimicthe trend of the experimental curves after the initial soft response (i.e.initial plastic deformation), even though the apparent Young's modulusin this case was found smaller in magnitude than what would be expectedfor quartz [24,25]. It is needed to be noticed however, that the apparentYoung's modulus is in reality a contact modulus and not the materialelastic property.

As mentioned above, for three pairs of LBS grains, normal contacttests were conducted in a cyclic mode applying determined normal loadsof 1, 5 and 10 N for a total number of four cycles for each pair of grains.Typical FN - δ curves for cyclic normal load tests are given in Fig. 5, wherethe continuous lines corresponded to the first cycle and the dashed linescorresponded to the subsequent second to fourth cycles. Note that theregime of soft response, which was observed during the application of thefirst loading cycle for the experiments presented in Fig. 4 (these testscorresponded to Table 1) as well as during the first loading cycle of the

Fig. 5. Cyclic normal load tests at 5 and 10N normal force.

Fig. 7. Variation of tangential force vs. displacement at various normal loads showingdifferent regions for LBS-A grain pair.

265

experiments of Fig. 5, did not appear during the application of the secondto fourth loading cycle for the cyclic normal load tests. Overall, subse-quent cycles beyond the first, demonstrated predominantly elastic

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

response in the normal direction within the whole range of normal dis-placements. For the LBS pairs of grains subjected to cyclic normal contacttests, notable hysteretic response was not observed beyond the first cycle.But, a minimum energy loss was observed during the first cycle. Asdemonstrated later in this work, microscopic image analyses of grainssubjected to pure normal contact tests did not demonstrate any notablechange in surface characteristics and that shearing must take place toreveal observable damage to the grain surfaces in contact. Thus, it may berevealed that the occurrence of hysteretic response in the normal direc-tion during the first cycle normal loading, for this type of grains incontact, is majorly related to their initial plastic deformation on the onsetof 0.002 mm of normal displacement and perhaps, some asperitiesdamage at the micro-scale.

Hertzian fitting was applied for all the cyclic normal load tests andFig. 6 shows the variation of material apparent Young's modulus(E ¼ E1 ¼ E2) with the number of cycles for the set of the three cyclicnormal contact tests. Note that for the results in Fig. 6, average radius was

Fig. 8. Change in inter-particle friction during loading and unloadin

266

used. It is observed that at the lowest applied normal load, i.e. 1 N, thefour cycles seemed to give the same value of E (i.e. apparent Young'smodulus remained constant). However, at higher loads, i.e. 5 and 10N,the value of E increased slightly from the first cycle to the second one andremained constant thereafter. In Fig. 6, similar to the observations inFig. 4, some variability of the Young's modulus was observed for the threepairs of grains at different normal loads, which is attributed, majorly, tograin morphology and asperity geometry differences between the threepairs of grains.

In DEM simulations, the normal contact response and correspondingnormal stiffness (KN) are important inputs in the study of granular ma-terials [26]. The LBS grains of this study are typical quartz type grains,which are of major interest in geotechnical engineering as well as otherdisciplines and applications. For example they are used as proppant forhydraulic fracturing in petroleum engineering [6]. The test results ofFigs. 4–6 can provide information about the normal contact behavior,thereby, some inputs for DEM analysis for this general type of grains. The

g at various normal loads for (a) LBS-A pair and (b) LBS-C pair.

Fig. 9. Variation of tangential stiffness vs displacement during loading and unloading for (a) LBS-A pair and (b) LBS-C pair.

Fig. 10. Variation of tangential stiffness vs normal force at different displacements duringloading and unloading for LBS-C pair.

Fig. 11. Tangential force vs. displacement for repeating shear tests at constant normalload (after [12], updated by the authors).

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

267

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

normal contact stiffness can be computed as:

KN ¼ Pδ

(4)

Fig. 12. Cyclic shearing tests at various normal loads.

It is important to note that the real normal contact behavior, asderived experimentally approximates the non-linear behavior [26–28]rather than the linear [29]. For example, Yimsiri and Soga [27] high-lighted that the modulus – pressure relationship on the macro-scale forgranular materials is strongly linked to their contact behavior at themicro-level, which includes normal load response at grain contacts andthe presence of asperities. This implies that the deformation character-istics of geo-materials, geophysical analysis and dynamic problems whichall involve the modulus – pressure relationship of particulate media arestrongly linked to the grain scale behavior. Morphological characteristicsat this scale as well as secondary plastic deformation of asperities, mightplay important role in the observed non-linear behavior from themicro-mechanical tests, even though this non-linear behavior did notresult in notable hysteretic response and energy losses and it is majorlyassigned to the change of contact area during the application of thenormal load [23].

4.2. Tangential load-displacement behavior

4.2.1. Behavior from initial asperities deformation to micro-slip and steady-state

All the shearing tests of the major testing program (Table 1) wereconducted at a displacement rate of 0.08 mm/h. Typical plots oftangential force against tangential (horizontal) displacement at variousnormal loads are shown in Fig. 7. In these plots, two different regionsmay be distinguished. In region 1, the tangential load increases at adecreasing rate inferring non-linear behavior. This behavior may bepartly due to the plastic deformation of asperities during tangentialloading. In region 2, micro-slip and steady state are observed at greaterdisplacements. Region 2 is shifting towards greater displacements withthe increase of the normal load. Ni and Zhu [30] stated that micro-slip iscaused due to continuous breakdown of asperities in contact after plasticdeformation has taken place and that the steady state behavior is due tomacro breakdown. For some of the curves the steady state is not perfectlyreached, but, there is a trivial increase of the tangential force withincreasing displacements. This might be because the tangential forces arenot high enough for causing macro breakdown of asperities in contact.Similar behavior was observed by Nardelli et al. [9] and they ascribedthis behavior to surface brittleness for the Eglin sand grains they tested.

4.2.2. Pre-loading and pre-shearing effectsFig. 8 shows the variation of the inter-particle coefficient of friction

for LBS-A (Fig. 8(a)) and LBS-C (Fig. 8(b)) pairs of grains during theloading and unloading program. The data are analyzed in terms of theinter-particle coefficient of friction (μ) which is defined as the ratio of(FT/FN), where FT corresponded to the steady-state or, alternatively, themaximum shearing force if the curve exhibits trivial increase of thetangential force against the displacement. It is observed that during theloading program the value of the inter-particle friction decreased slightlywith the increase in normal load. During the unloading program, theinter-particle friction is increased than that of the loading process. Notethat while in Fig. 8(a), the increase of the coefficient of friction (μ) in theunloading program is more apparent at 3 N of normal load, this thresholdwas observed at FN ¼ 5 N in Fig. 8(b). This is attributed to the greatermaximum normal load that was reached for the LBS-C test in Fig. 8(b),equal to 12 N, while the test for LBS-A was conducted at a maximumnormal load of 10 N. Thus, during the unloading process, the thresholdnormal load below which pre-shearing effects are apparent seems todepend on the maximum previous normal load that shearing took placerather than the number of shearing steps applied. Similar trends areobserved in LBS-B pair also. The inter-particle friction at 1N increasedabout 27–32% after the loading and unloading programs

268

were completed.Fig. 9 shows the variation of the tangential stiffness with displace-

ment for LBS-A and LBS-C pairs of grains, respectively. The tangentialstiffness (KT) was computed similar to the analysis procedure describedby Ref. [11]. It is observed that the value of stiffness increased during theunloading process in comparison to the loading stage and that theresponse is highly non-linear with a substantial drop of KT with verysmall tangential displacements. Fig. 10 shows the variation of thetangential stiffness during loading and unloading at various normal loadsfor the LBS-C pair of grains and at target tangential displacements. It isobserved that the tangential stiffness values increased considerablyduring the unloading process at both observed reference displacements.These differences were more pronounced at the initial stage of shearingi.e. smaller displacements as shown in Fig. 10. Thus, there is an apparentimportant effect of the loading history (pre-loading and pre-shearing) onthe tangential load – displacement response and also on the absolutevalue of KT. Note that apart from the effect of the loading history on theabsolute value of KT, the results of Fig. 9 indicated that the pre-loadingand pre-shearing shifted the displacements beyond which KT started todegrade rapidly and this trend was found more pronounced at smallernormal loads.

4.2.3. Repeating monotonic and cyclic shearing tests at constant normal loadTable 2 and Fig. 11 show the repeated shearing tests conducted at

constant normal loads. It is observed that repeating the shearing test at

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

small to medium normal loads (1–5 N) does not result in any notablechange of the inter-particle friction. But, at higher normal loads (7–10 N),the inter-particle friction is increased with repeated shearing, which wasobserved by Ref. [12] and was confirmed by the newly added data in

Fig. 13. Microscopic images of LBS gains before

269

this study.Fig. 12 shows the two cyclic shearing tests which are conducted on

different pairs of grains at normal loads of 1 and 10N. At each nominalnormal load, a separate pair of grains was tested at a tangential

and after shearing at various normal forces.

Fig. 14. Use of local-radius for interpretation of apparent modulus and Hertz fitting fortypical LBS grains (radius given in mm).

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

displacement amplitude of 0.010 mm. Note the highly non-linearresponse and hysteretic behavior of the grains at their contacts whichwas evident even from the early stages of the shearing process. Energylosses (denoted as D) were calculated for this limited set of tests based onthe second cycle accounting for the area of the closed loop (denoted asΔW) and the elastic energy stored (denoted as W) as:

Dð%Þ ¼ 14� π

� ΔWW

� 100 (5)

Based on this, D (expressed as fraction) was found equal to 0.56 atFN ¼ 1 N for displacement amplitude of 0.01 mm. At the greater normalload FN ¼ 10 N, D was found equal to 0.42, thus energy losses wereslightly reduced for the pair of grains tested at FN ¼ 10 N in comparisonto the experiment conducted at 1 N of normal load. This behavior couldbe explained, partly, by the greater inter-particle friction at low normalloads, which contributed to greater energy losses and vice versa.

4.3. Microscopic observations

Three pairs of grains are taken and sheared once at different normalloads to study the contact radius and the effect of the magnitude ofnormal load on the shearing behavior. Fig. 13 shows the microscopicimages of representative grains from this set series before and aftershearing. The apparent Young's modulus (E ¼ E1 ¼ E2) was obtainedthrough curve fitting. Note that two attempts were made in the study touse the Hertzian model for the purpose of deriving apparent Young'smodulus and they are compared with experimentally measured contactarea for the given tests, which are reported in Tables 3 and 4 The firstattempt was based on average radius of the grains, which approach maybe more rough but it could match to a few tested grains since their shapemight approximate the shape of sphere. In a second attempt, morerigorous approach was implemented using local radii of the asperities(corner radii) of the grains and an example of this approach is shown in

Fig. 15. Microscopic images of LBS grains bef

270

Fig. 14. The local radii of the asperities (LR1 and LR2) of the grains incontact are taken from the image analysis based on each individual set ofgrains. Equations (1)–(3) are used to obtain R* and E*. The Poisson's ratioof the material is taken as 0.1 for all the three pairs. The Hertz contactradius (α) is obtained from Equation (6). Table 3 shows the Hertziancontact radius and experimentally observed contact radius based on themicroscopic images and the consideration of average radius, whilstTable 4 gives a summary of the results on the same pairs of grainsconsidering local radii.

α ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3� P� R �4� E �

3

r(6)

The microscopic images of the grain surfaces after the shearing areshown in Fig. 13(e) and (f), which were used to obtain experimentalcontact radius. Note that for the grain surfaces shown in Fig. 13, thetesting program is conducted by applying normal and tangential loading(shearing) on the grains. Throughout this study, some grains werechecked on the microscope after the application of pure normal load (upto 12N) without additional shearing, but no surface changes werecaptured as shown in Fig. 15. This implies that the sole application of anormal load on the grains in contact may not be sufficient to causenotable surface changes. Pratt and Eisner [31] found that the pureapplication of normal load causes very little mechanical breakdown, andstated that the noticeable breakdown occurs when the tangential force isincreased to a fraction required to cause sliding. This behavior suggeststhat preloading (i.e. application of normal load) without shearing maynot alter the surface characteristics notably, thus, this would not affectsignificantly the inter-particle friction without additional shearing takingplace. Perhaps, these observations also support the minimum energylosses during the cyclic normal load tests, whereas highly hystereticbehavior and energy losses were observed in the shearing tests.

It is observed from Tables 3 and 4 and Fig. 13 that the Hertz contactradius is lesser than that obtained experimentally. Note that the Hertziancontact radius, computed theoretically, and the contact radius obtainedexperimentally correspond to the apparent area of contact [23,32]. Theexperimental contact radius is close to the Hertzian radius at a highernormal load (i.e. FN ¼ 10N). At lower normal loads (i.e. FN ¼ 2 and 5N),the Hertzian contact radius is around half of that obtained experimen-tally. This might be partly because of the real shape of the LBS grainswhich deviates slightly from the shape of a perfect sphere in Table 3 andalso due to the presence of asperities in micro and meso-scale. Similarfindings were reported by Refaie and Halling [33] on metal surfaces.They found that the Hertzian values are valid at higher normal loads andat lower normal loads the contact area is greater than the one predictedtheoretically by the Hertz theory. Note that even by considering localradii, the similar finding were observed, i.e. experimentally obtainedradii are higher than Hertzian contact ones. It worth noticing however,based on the results in Tables 3 and 4, that the consideration of local radiiprovided greater derived apparent modulus (E) in comparison to theconsideration of average radius. But the prediction of the contact

ore and after application of normal load.

C.S. Sandeep, K. Senetakis Tribology International 117 (2018) 261–271

diameter, in comparison to the measured, was relatively better whenaverage radius was used.

Based on the microscopic observations in Fig. 13(f), it is noticed thatat a higher normal load (FN ¼ 10N) ploughing has taken place, whereasthis ploughing is not observed at lower normal loads, as for example atFN ¼ 2N in Fig. 13(e). Bhushan [34] stated that this ploughing causespermanent plastic deformations, while Yang et al. [6] also observed,through image analyses, ploughing to the quartz-type grains they testedin shearing which was observed in their study dominantly for grainsimmersed in water. The ploughing behavior at higher normal loads(FN ¼ 10N) during the shearing tests might have been responsible fordecrease in inter-particle friction. Lower inter-particle friction accountedfor lesser energy loss during cyclic shearing tests at higher normal loads(Fig. 12). But, this ploughing might also cause greater surface roughness,which might have contributed to increase in inter-particle friction duringunloading process as observed in Fig. 8(a) and (b) as well as during therepeated shearing at higher normal loads (Fig. 11).

5. Conclusions

Based on the micromechanical experiments in the study, it wasobserved that the combination of pre-loading and pre-shearing of grainsat higher normal loads led to an increase in friction, which was evidentwhen the pre-loaded and pre-sheared pairs of grains were re-sheared atsmaller normal loads. Repeated shearing of grains at relatively low inmagnitude normal loads did not influence the inter-particle frictiongreatly, whereas at higher normal loads, repeated shearing causes in-crease in friction. Microscopic observations on the grain surfaces after theexperiments showed that inter-particle friction at higher normal loads isgreatly influenced by the ploughing caused during shearing, whereas thisbehavior is not clearly observed at lower normal loads. The apparentradius of contact in quartz sand grains was quantified both experimen-tally and theoretically using Hertz fitting. The apparent diameter ofcontact that was found experimentally at 10N, matches closely withHertzian values, whereas at lower normal loads, i.e 2 N, the experi-mentally obtained contact diameter is higher than the Hertzian value.The use of average radii provided better prediction of the contact radius,but the use of local radii, into the Hertzian model, demonstrated greaterderived apparent moduli. No clear observation of surface damage wasfound after the application of pure normal load, whereas the tracks areclearly observed after the application of shearing. So, it is understoodthat the normal load (up to 12N) alone cannot produce any majorchanges on the surfaces without the application of tangential force.

Acknowledgements

The study was fully supported by the Theme-based research projectScheme (TRS) “Understanding Debris Flow Mechanisms and MitigatingRisks for a Sustainable Hong Kong” Project No. T22-603/15 N (CityU8779012), RGC Hong Kong SAR China. The authors would like to thankthe anonymous reviewers for their constructive comments that helped usto improve the quality of the manuscript. The technicians of the CityUniversity Mr Thomas Tsang and Mr Kian are acknowledged for theircontinuous contribution and help in the lab facilities development andmaintenance.

References

[1] Pohlman NA, Severson BJ, Ottino JM, Lueptow RM. Surface roughness effects ingranular matter: influence on angle of repose and the absence of segregation. PhysRev E 2006;73(031304):1–9.

271

[2] Cavarretta I, Coop MR, O'Sullivan C. The influence of particle characteristics on thebehavior of coarse grained soils. Geotechnique 2010;60(6):413–23.

[3] Hanaor DAH, Gan Y, Einav I. Effects of surface structure deformation on staticfriction at fractal interfaces. Geotech Lett 2013;3(APRIL/JUN):52–8.

[4] Hanaor DAH, Gan Y, Einav I. Static friction at fractal interfaces. Tribol Int 2016;93:229–38.

[5] Cavarretta I, Rocchi I, Coop MR. A new interparticle friction apparatus for granularmaterials. Can Geotechnical J 2011;48(12):1829–40.

[6] Yang L, Wang D, Guo Y, Liu S. Tribological behaviors of quartz sand particles forhydraulic fracturing. Tribol Int 2016;102:485–96.

[7] Cole DM, Hopkins MA. The contact properties of naturally occurring geologicmaterials: experimental observations. Granul Matter 2016;18:61–77.

[8] Senetakis K, Coop MR. The development of a new micro-mechanical inter-particleloading apparatus. Geotechnical Test J 2014;37(6):1028–39.

[9] Nardelli V, Coop MR, Andrade JE, Paccagnella F. An experimental investigation ofthe micromechanics of Eglin sand. Powder Technol 2017;312:166–74.

[10] Cundall PA, Strack ODL. A discrete numerical model for granular assemblies.Geotechnique 1979;29:49–65.

[11] Senetakis K, Sandeep CS, Todisco MC. Dynamic inter-particle friction of crushedlimestone surfaces. Tribol Int 2017;111:1–8.

[12] Sandeep CS, Senetakis K. Exploring the micromechanical sliding behavior of typicalquartz grains and completely decomposed volcanic granules subjected to repeatingshearing. Energies 2017;10(370).

[13] Nardelli V, Coop MR. The micromechanical behaviour of a biogenic carbonate sand.In: proc. Of vi Italian conference of researchers in geotechnical engineering cnrig–geotechnical engineering in multidisciplinary research: from microscale to regionalscale, 22-23 sept., Bologna, Italy158. Procedia Eng; 2016. p. 39–44.

[14] Senetakis K, Coop M, Todisco MC. The inter-particle coefficient of friction at thecontacts of Leighton Buzzard sand quartz minerals. Soils Found 2013a;53(5):746–55.

[15] Cole DM, Peters JF. A physically based approach to granular media mechanics:grain-scale experiments, initial results and implications to numerical modeling.Granul Matter 2007;9:309–21.

[16] Cole DM, Peters JF. Grain-scale mechanics of geologic materials and lunar simulantsunder normal loading. Granul Matter 2008;10:171–85.

[17] Nardelli V, Coop MR, Vitone C, Chen S. The inter-scale behaviour of two naturalscaly clays. Geotech Lett 2016;6(3):1–6.

[18] Bar-Hen M, Etsion I. Experimental study of the effect of coating thickness andsubstrate roughness on tool wear during turning. Tribol Int 2017;110:341–7.

[19] Brzesowsky R.,Spiers C.J., Peach C.J. and Hongz S.J.T. Failure behaviour of singlesand grains: theory versus experiment, J Geophys Res Solid Earth, 116(6), B06205.

[20] Yastrebov V, Anciaux G, Molinaru J-F. On the accurate computation of the truecontact-area in mechanical contact of random rough surfaces. Tribol Int 2017;114:161–71.

[21] Joe J, Scaraggi M, Barber JR. Effect of fine-scale roughness on the traction betweencontacting bodies. Tribol Int 2017;111:52–6.

[22] Hertz H. Uber die Beruhrang fester elastischer Korper (On the contact of elasticsolids). J der rennin und angewandeten Math 1882;92:156–71.

[23] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1985.[24] Heyliger P, Ledbetter H, Kim S. Elastic constants of natural quartz. J Acoust Soc Am

2003;114(2):644–50.[25] Jaeger JC, Cook NGW, Zimmerman R. Fundamentals of rock mechanics. fourth ed.

Oxford: Wiley-Blackwell; 2007.[26] O'Sullivan C. Particulate discrete element modelling: a geomechanics perspective.

Applied Geotechnics, vol. 4. Spon Press; 2011.[27] Yimsiri S, Soga K. Micromechanics-based stress–strain behaviour of soils at small

strains. G�eotechnique 2010;50(5):559–71.[28] Lin X, Ng T-T. A three-dimensional discrete element model using arrays of

ellipsoids. Geotechnique 1997;47(2):319–29.[29] Calvetti F, Nova R, Castellanza R. Modelling the subsidence induced by degradation

of abandoned mines. proceedings of second international symposium on continuousand discontinuous modelling of cohesive-frictional materials (CDM 2004), held instuttgart 27-28 sept. In: Modelling of cohesive-frictional materials. Taylor &Francis; 2004. p. 137.

[30] Ni Jun, Zhu Zhenqi. Experimental study of tangential micro deflection of interfaceof machined surfaces. J Manuf Sci Eng 2001;123(2):365–7.

[31] Courtney-Pratt JS, Eisner E. The effect of a tangential force on the contact ofmetallic bodies. Proc R Soc Lond a Math Phys Eng Sci 1957;238(1215).

[32] Mate M. Tribology on the small scale—a bottom up approach to friction,lubrication, and wear. Oxford, UK: Oxford University Press; 2008.

[33] Refaie M El, Halling J. An experimental study of the apparent area of contact undernominally Hertzian contact of rough surfaces. Proceedings of the Institution ofMechanical Engineers, Conference Proceedings. Sage UK: London, England: SAGEPublications; 1968. 183(16).

[34] Bhushan B. Scanning probe microscopy in nanoscience and nanotechnology 2.Springer Science & Business Media; 2010.