This article was downloaded by: [Memorial University of Newfoundland]On: 01 August 2014, At: 02:40Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Particulate Science and Technology: An InternationalJournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/upst20

Simulation of Adhesive–Dissipative Behavior of aMicroparticle Under the Oblique ImpactRaimondas Jasevičius a b , Jürgen Tomas c , Rimantas Kačianauskas a & Darius Zabulionis a

a Institute of Mechanics , Vilnius Gediminas Technical University , Vilnius , Lithuaniab Department of Software Engineering , Vilnius University , Vilnius , Lithuaniac Institute of Process Engineering , Otto von Guericke University , Magdeburg , GermanyAccepted author version posted online: 01 Apr 2014.Published online: 10 Jul 2014.

To cite this article: Raimondas Jasevičius , Jürgen Tomas , Rimantas Kačianauskas & Darius Zabulionis (2014) Simulationof Adhesive–Dissipative Behavior of a Microparticle Under the Oblique Impact, Particulate Science and Technology: AnInternational Journal, 32:5, 486-497, DOI: 10.1080/02726351.2014.908256

To link to this article: http://dx.doi.org/10.1080/02726351.2014.908256

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Simulation of Adhesive–Dissipative Behavior of aMicroparticle Under the Oblique Impact

RAIMONDAS JASEVICIUS,1,2 JURGEN TOMAS,3 RIMANTAS KACIANAUSKAS,1 and DARIUS ZABULIONIS1

1Institute of Mechanics, Vilnius Gediminas Technical University, Vilnius, Lithuania2Department of Software Engineering, Vilnius University, Vilnius, Lithuania3Institute of Process Engineering, Otto von Guericke University, Magdeburg, Germany

The adhesive–dissipative behavior of a microparticle under the oblique impact is investigated numerically and the new discreteelement method (DEM)-compatible interaction model is elaborated. The modeling approach is based on the Derjaguin–Muller–Toporov model of normal interaction for the adhesive elastic contact. Adhesion hysteresis is specified by the loss of the kineticenergy governed by the fixed amount of the adhesion work, required to separate two adhesive contacting surfaces. This effect iscaptured in the new interaction model by adding an additional dissipative force component to normal contact during unloadingand detachment. The essential feature of this approach, differing from that of the viscous damping model, is that, according tothe proposed method, the amount of the dissipated energy is not influenced by the actual initial velocity during the entire contact.The influence of adhesion on slip friction is reflected by considering the adhesive normal force components in the Coulomb’s lawof friction. The contribution of the adhesion-related dissipation is illustrated by a comparison of the behavior of the attractive–dissipative and attractive–non-dissipative models. The oblique impact of a microparticle on the plane surface at the intermediateimpact angle is also investigated numerically. The link between adhesion and friction is supported by the numerical results.

Keywords: Adhesion hysteresis, contact mechanics, DEM, dissipation, microparticle, oblique impact

Introduction

A collision between two surfaces of microparticles and evenmicron-sized active biological objects such as bacteria ismainly attributed to the attractive van der Waals forces. Inmost cases, the behavior of particles during the contact istreated, however, in a non-unique way, and different attract-ive–non-dissipative or dissipative, models are used in thenumerical analysis. It should be noted that even the simplestadhesion process may be not only reversible, however, it caninvolve the adhesive dissipation of energy. Therefore, theenhancing of the knowledge of the mechanisms of the energydissipation due to adhesion forces in the tangential and nor-mal directions is the main goal of this article. The interactionof microparticles is considered by separating their behaviorin terms of the normal and tangential components.

Historically, two models of the normal single-asperityinteraction are commonly used to adopt the classical Hertzcontact due to the van der Waals attraction. The model sug-gested by Johnson, Kendall, and Roberts (1971), or the JKRmodel, assumes that the attraction between particles is of aninfinitely short range and acts only over the contact area.

The model suggested by Derjaguin, Muller, and Toporov(1975), or the DMT model, predicts that half of the interac-tion force occurs in the outside located annular area, whichis located at the perimeter closed by the contact.

The validity of the both the JKR and DMT models can becharacterized by the so-called Tabor parameter, standing forthe ratio of elastic deformation to the range of the surfaceforce. Based on this statement it was shown that the JKRtheory is suitable for ‘‘soft’’ adhesive particles, while theDMT theory is suitable for smaller and ‘‘stiff’’ adhesive par-ticles, that is, for the ultrafine particles (with the diametersmaller than 10 mm). The DMT model predicts that half ofthe interaction force is developed outside of the annularcontact area, which is located at the perimeter closed by thecontact. The DMT model involves the deformation of theHertz contact and effect of the adhesion, separately.

The earlier developments of the adhesion were of concep-tual character. They were aimed to discover contributionof adhesion in the repulsive contact behavior of particles(see Dominik and Tielens 1997; Johnson 1998; Sridhar andSivashanker 2003; Shi and Polycarpou 2005; Tomas 2007a,2007b; Jasevicius et al. 2009; Severson et al. 2009). Mostof these studies were dealing, however, with the JKR model.An analytical model for the collision dynamics of adhesiveviscoelastic spheres, which accounts for aggregation andrestitution was presented by Brilliantov et al. (2007). Thelatest findings in the van der Waals interaction were alsocomprehensively discussed in the review papers of Adams

Address correspondence to: Raimondas Jasevicius, Institute ofMechanics, Vilnius Gediminas Technical University, SauletekioAve 11, Vilnius, Lithuania. E-mail: raimondas.jasevicius@vgtu.ltColor versions of one or more of the figures in the article can befound online at www.tandfonline.com/upst.

Particulate Science and Technology, 32: 486–497

Copyright # 2014 Taylor & Francis Group, LLC

ISSN: 0272-6351 print=1548-0046 online

DOI: 10.1080/02726351.2014.908256

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

and Nosonovsky (2000), Li et al. (2011), Roa et al. (2011) andreferences herein.

The description of the behavior of a particle motionin the tangential direction is another important issue ofthe discrete element method (DEM) analysis. Adequacy withthe theoretical models and experimental results is still anopen question in DEM because behavior of the hystereticforward–backward slip of particles during contact is non-unique and has not been clearly explained yet. Most of thefriction models of contact preserve concept of the frictionlaws introduced by Amonton in 1699 and Coulomb in 1785(Zmitrowicz 2009) and are described in a framework of thephenomenological approach. Theoretical backgrounds ofthe tangential contact are based on continuum models (seeAdams and Nosonovsky 2000; Wriggers 2009; Zmitrowicz2009). When applied to DEM, the discrete approach pre-sumes classical linear ‘‘spring-slider’’ model with Coulombfriction for contact elasticity. Discussion on non-adhesiveDEM models, including loading–unloading relations withcomputational details, may be found throughout the literat-ture (see Di Maio and Di Renzo 2004; Kruggel-Emden et al.2008; Brilliantov et al. 2008; Zabulionis et al. 2012; Thorntonet al. 2013). The interrelation between viscoelastic andfrictional dissipation in terms of tangential coefficientof restitution and the Coulomb friction was studied byBecker et al. (2008) and Schwager et al. (2008).

Characterization of the tangential behavior in the presenceof van der Waals forces is even more complicated than thenormal interaction. A detailed study of the effect of adhesionon tangential slip was first made by Savkoor and Briggs(1977). There are many theoretical studies and contributionson the attractive interaction during an impact of particles(see Thornton 1991; Brach and Dunn 1998; Dominik andTielens 1997; Kim and Dunn 2007).

Several models of tangential contact have been developedrecently to describe the interrelation between dry friction andnormal attraction forces on particle scale. The application ofthe attractive models to DEM simulations of multi-particlesystems have been demonstrated (Tykhoniuk et al. 2007;Gilabert et al. 2008; Luding 2008; Martin and Bordia 2008;Figueroa et al. 2009). It can be observed that the formulationof physically adjusted criterion of the tangential separation atwhich the slip occurs is a decisive condition of any tangentialmodel in the case of adhesion.

This article addresses an independent dissipation mech-anism of the adhesion hysteresis. Dissipation of energy dueto adhesion, or the so-called adhesion hysteresis, is sucha phenomenon when the amount of energy required to separ-ate two surfaces is greater than the amount of energy gainedby bringing the surfaces together. Physically, hysteresis, orirreversibility, observed in adhesive contact is relevant toprocesses occurring on the microscopic level. More definitely,the adhesive hysteretic behavior of particle can be explainedby the inherent instabilities and irreversibility due to inelasticdeformation (microplasticity) of particle surface asperitiesassociated with the loading and unloading cycles andinterlock occurring at the contact interface (Tomas 2003;Israelachvili 2011).

Physical evidence of the irreversible hysteretic dissipativeforce-displacement characteristics of the interaction of themicroparticles, is basing on the results of physical experi-ments with atomic force measurements (AFM), which wereobtained and described in several investigations (see Wallet al. 1990; Snitka et al. 1997; Cappella and Dietler 1999;Jones 2003; Butt et al. 2005; Tykhoniuk et al. 2007; Zhouand Peukert 2008).

It can be noticed that the origin of the ‘‘adhesion hyster-esis’’ is not unique. Since the surface of the particle is not ide-ally smooth, adhesion hysteresis may be attributed to thesurface effects in the scale of surface asperities. The effectof microplasticity observed in the nanometer range duringcontact of microparticle can be also contribute to theadhesion hysteresis. Thereby various modeling methodsincluding molecular dynamics (MD) and the finite elementmethod (FEM) may be applied on different scales (see Zhouand Peukert 2008; Feng et al. 2009; Sahagun and Saenz 2012;Yang and Martini 2013). Basing on the above review,it could be concluded that numerical analysis of the adhesionhysteresis on the scale of particle is, however, rather scarce(Severson et al. 2009), while applicable models are still underdevelopment.

This article addresses numerical simulation of theadhesion hysteresis during the oblique impact of micro-particles by the DEM. To represent the influence of dissi-pation due to adhesion, the attractive–dissipative modelwas proposed. This model considers the attractive–dissipat-ive van der Waals interaction in terms of the DMT approach.The normal and the tangential interaction as well as compari-son with the non-dissipative behavior is presented in details.Finally, the oblique impact of a microparticle on the planesurface at the intermediate impact angle is also investigatednumerically. The link between adhesion and friction issupported by the numerical results.

Normal Interaction

Attractive–Non-Dissipative Model

An interaction model, describing a collision of the contactingparticles must comprise all forces and torques acting on thecontacting particles during a collision. This constitutiverelationship is illustrated by considering the interactionof the smooth spherical microparticle with a flat surface.A general approach elaborated by Tomas (2004, 2007a)and adopted by Jasevicius et al. (2011) will be applied andthe model of the attractive–non-dissipative particle–planeinteraction denoted as (AnD) model will be introduced.The considered models of the normal interaction aredescribed in terms of the force-displacement relationship.If we define normal orientation of the contact by a unitvector n, the force vector FN¼FNn may be described bythe time-dependent scalar variable FN(t).

Four different stages may be distinguished duringthe normal interaction. They include: the approach, loading,unloading, and detachment. When a particle moves from theinfinity towards the surface, the short range interaction ischaracterized as the approach, while the motion in the

Adhesive–Dissipative Behavior Under the Oblique Impact 487

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

outward direction is characterized as detachment. Thecontact interaction comprises loading and unloading. Fourtime-dependent components of the normal force are used inthe description of the contact. The approach stage is charac-terized by the force FN

appr tð Þ, the loading stage by FNload tð Þ,

unloading by FNunload tð Þ and the detachment stage by

FNdetach tð Þ, respectively. Negative forces are identified as the

attraction forces, while positive as the repulsion forces. Thedistance, or short range interaction, is characterized by thenegative displacement �h(t), indicating the interactionwithout contact. The behavior of the deformation duringthe contact for both loading and unloading forces is charac-terized by the positive displacement h(t) meaning the overlapof particles.

The reversible behavior of the (AnD) model is illustratedin Figure 1 by the graph of typical force-displacementrelationship plotted in the nanoscale, where the total pathof the interaction in the forward direction of the loading isdenoted by a solid curve passing through the points S, L,and U and through the points U, L, and S in the backwarddirection. The zero-point of this diagram h¼ 0 is equivalentto the adhesion separation of a direct contact. It is assumedthat the particle’s interaction with the plane surface occursat the time instant t0. The starting point in Figure 1 is denotedby S. The approach continues until the particle reaches thesurface at the time instant tL with the negative separation dis-placement varying in the range up to zero distance h(tL)¼ 0.During the approach, in time interval (t0� t� tL), the par-ticle is attracted to the plane by the force FN

AnD;appr tð Þ ¼FNadh tð Þ, where a short-range van der Waals adhesion force

is defined as suggested by Tomas (2003, 2007a):

FNAnD;appr tð Þ ¼ � FN

L aF¼02

aF¼0 þ h tð Þj jð Þ2: ð1Þ

Here, decay of the van der Waals interaction between asphere and a plane is defined as inverse square of the sphere-plane distance, while an abrupt at the characteristic adhesiondistance aF¼0 is assumed. Thus, analytical description of the

path of the adhesive contactless approach is relevant to twophysical parameters—the characteristic adhesion distanceaF¼0 and the adhesion force FN

L (the so-called jump-in force)defined at the zero-point of the considered diagram with zerodisplacement h¼ 0.

The force FNL is related to the interfacial energy c of the

interacting bodies. For the case of the in DMT model, see(Derjaguin et al. 1975), it is defined as

FNL ¼ 4 � pcR; ð2Þ

here c is conventional surface energy standing for physicalconstant, while R is radius of the particle.

The contact of particle with the surface begins with theforce FN

L and is denoted by point L on the graph. The endof a loading is characterized by the maximal displacementof contact point hU(tU) reached at the time instant tU. Asa result, the loading stage is defined within the time interval(tL� t� tU). It is characterized with the positive values ofthe contact displacement h(t)> 0.

The contact force FNAnD;load tð Þ of the loading comprising

the repulsive and attractive contributions and is described as

FNAnD;load tð Þ ¼ FN

Hertz tð Þ � FNL ; ð3Þ

where the in time varying the elastic repulsive Hertz contactforce for the sphere–plane contact, reads as follows:

FNHertz tð Þ ¼ 2

3� Eeff

1� n2ffiffiffiffiR

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffih tð Þð Þ3

q: ð4Þ

During the unloading, the contact force follows the reversedloading path between the points U and L (Figure 1). Theunloading force of the (AnD) model FN

AnD;unload tð Þ, varyingin the time interval (tU� t� tLr), inversely replicates theloading force given by Equation (3). Detachment force

FNAnD;detach tð Þ is described in the same manner and follows

the reversed path of the approach between the points L andS. The interaction is assumed to be ended at the point S atthe final time instant tSr.

The above discussion about the attractive–non-dissipativemodel presented here as (AnD) model is summarized inFigure 1, where the approach and the loading path areshown by the solid lines passing through points S and L(line S-L), and L and U (line L-U). The unloading and thedetachment follows simply reversed paths. Equation (1)will be used in the next section for the development of thedissipative model.

Attractive–Dissipative Model

The dissipation of energy is a phenomenon significantlyaffecting the behavior of particle systems. A major motiv-ation for our work is associated with the study of a specifictype of dissipation, the so-called ‘‘adhesion hysteresis,’’ whenthe amount of energy required to separate two surfacesis greater than the amount of energy gained by bringing thesurfaces together. The hysteresis of normal interaction canbe explained due to inelastic deformation (microplasticity)of particle surface asperities and an increase of adhesion

Fig. 1. The attractive–non-dissipative (AnD) and attractive–dissipative (AD) models of normal interaction: the pathsS-L-U-L-S and S-L-U-A-D stand for AnD and AD models,respectively.

488 Jasevicius et al.

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

force, which during deformation is presented as a time depen-dent function.

Empirically, it has been observed that the amount ofenergy WN

attr;diss, dissipated during a complete full approach-separation loop is characterized by the area between theapproach-loading and unloading-detachment branches ofthe hysteretic force-displacement diagram of collision. Theenergy, WN

attr;diss, is related to the critical velocity, tN0;cr, ofnormal adhesive elastic interaction measured in physicalexperiments. If critical velocity tN0;cr is known, as a result,the dissipated energy can be calculated as kinetic energygenerated by the initial critical velocity:

WNattr;diss ¼

meff � tN0;cr2

2; ð5Þ

where meff is the effective mass of the interacting particles.Equation (5) presumes that, actually, adhesion or

cohesion causes the sticking of the particles, and the dissi-pated energy, WN

attr;diss, as well as the critical initial velocity,

tN0;cr, determines the threshold of the behavior of a particle

during its sticking or rebound. A particle does not stick ifthe amount of the initial kinetic energy is higher than theamount of the critical energy dissipated due to the interac-tion. If the initial normal velocity, tN0 , is lower than the criti-cal velocity, tN0;cr, then, after an impact, the particle remainsin the attractive zone and does not rebound. Otherwise, ifthe value of the initial particle velocity is higher than thecritical velocity tN0;cr, then the particle rebounds. It is obviousthat the critical velocity tN0;cr, or the dissipated energyWN

attr;diss

defined by Equation (5), can be regarded as the constants ofinteracting bodies that do not depend on the value of theimpact velocity.

On the other hand, the dissipated energy can be calculatedtheoretically. The amount of dissipated energy duringdeformation may be presented as a product of two physicalconstants, aF¼0 and FN

L , which were discussed earlier. Thisproduct describes the adhesion work which is needed todetach the particle from the surface without any contactdeformation at a distance from zero to infinity. Finally,this work is compared with the dissipated energy related todeformation of asperities. In this case we use a factor kH:

WNattr;diss ¼ kHaF¼0F

NL : ð6Þ

The factor value kH¼ 0 means that contact has nodissipation related to adhesion, while kH¼ 1 means that fullamount of energy aF¼0F

NL is dissipated.

It should be noted that the variation of the dissipativeforce in the contact area due to adhesion hysteresis has notbeen thoroughly investigated and the methodology forevaluation of this variation has not been developed yet. Somelimited contributions could only be mentioned. Feng et al.(2009) have illustrated the phenomenon of the adhesion hys-teresis in the JKR model by FEM simulations. Along withthe JKR model, Severson et al. (2009) have also consideredthe adhesion hysteresis by simulating of the dynamics of aparticle, while the dissipative contribution was characterized

by a sudden jump of the normal force at the beginning of anunloading. Sahagun and Saenz (2012) simulated adhesivehysteresis for measurements of the dynamic force microscopyand presented a model of the unilateral distributed dissipat-ive force. They developed an FN

diss tð Þ attractive–dissipativemodel to describe the reversed process of the removal ofa particle by correcting and modifying the backward behaviorof the non-dissipative model.

To capture the dissipated energy in adhesion hysteresis, weintroduce the attractive–dissipative force FN

attr;diss tð Þ, whichacts during the rebound motion within the entire displace-ment interval between the maximal displacement hU andthe separation, aF¼0. This path is denoted on the graphin Figure 1 by points U, A, and D. To grasp the variationof this force, the additional weighting function, hh tð Þ ¼hU � h tð Þð Þ=hU, linearly increasing from zero at the pointU up to 1 at the point A, is introduced. Consequently, thesuggested variation implies that the maximum value of thedissipation force FN

diss;h0 is reached at the detachment pointA, with the zero displacement h¼ 0. Finally, we relate thisvalue to the amount of dissipated energy:

FNdiss;h0 ¼

2WNattr;diss

aF¼0 þ hU: ð7Þ

Consequently, the variation of the newly introduced force,

FNdiss tð Þ ¼ FN

diss;h0 � �hh tð Þ of the adhesive–dissipative model in

the time interval (tU� t� tA), may be described explicitlyin the same way as in Equation (3):

FNAD;unload tð Þ ¼ FN

Hertz tð Þ � FNL � FN

diss tð Þ: ð8Þ

The detachment force FNAD;detach tð Þ, acting within the time

interval, (tA� t� tD), is the adhesion force which can becalculated by in the same manner as used in the non-dissipativemodel by replacing the force FN

L with FNA ¼ FN

L þ FNdiss;h0

in Equation (1). Explicitly, the detachment force

FNAD;detach tð Þ ¼ �

FNL þ FN

diss;h0

� �a2F¼0

aF¼0 þ h tð Þj jð Þ2: ð9Þ

The above discussion about the attractive–dissipative ADmodel is summarized in Figure 1. Here, the forward motionduring the approach, and the loading path shown by the solidlines passing through points S, L, and U, simply mimics thenon-dissipative (AnD) model. In the irreversible backwardmotion, the unloading and detachment paths are denotedby the dashed lines passing through the points U and A (lineU-A), and A and D (line A-D), respectively. The obtainedEquations (7)–(9) will be used hereafter to evaluate thetangential interaction and for simulation of the oblique impact.

Tangential Interaction

We share a common viewpoint that the tangential interactionduring a collision is coupled with the normal interaction andshould be considered regarding all repulsive and attractivenormal forces. The classical elastic Hertz contact modelwith the Coulomb friction was adopted in the present

Adhesive–Dissipative Behavior Under the Oblique Impact 489

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

investigation. The variables of normal interaction in the caseof repulsion will be considered as positive. If we definethe tangential orientation of the contact by a unit vector t,the force vector FT¼FTt may be described by the time-dependent scalar variable FT(t).

Presumably, the tangential contact may be characterizedby the static friction, or non-slip, and the slip modes.Tangential forces relevant to each of the modes are denotedby vectors FT

st tð Þ and FTsl tð Þ, respectively. A purely static

approach, disregarding kinematic conditions, allows us todistinguish the tangential force components relevant toparticular modes:

FT tð Þ ¼0; no contact;

FTst tð Þ; static friction;

FTsl tð Þ; slip

8><>: ð10Þ

Identification of the contact modes of the particular tangen-tial contact requires the examination of the time-history ofcontact variables. It could be remind that the non-contactmode is defined by the negative values of the normaldisplacement h(t)< 0.

To avoid rigorous analysis of the contact area, which hasalready been analyzed by Thornton (1991), the tangentialmotion of the contacting surfaces is defined by a simple sliprule of the same form as the Amonton=Coulomb laws offriction. This law may be expressed by comparing the moduliof the tangential force FT tð Þ with the limiting critical force

FTcrit tð Þ

FT tð Þ�� �� � FT

crit tð Þ: ð11Þ

The case corresponding to the static friction is defined bythe inequality (11), while the force FT

st obeys the nonlinearelastic constitutive relationship with the elastic tangentialdisplacement

FTst tð Þ ¼ kT tð Þdel tð Þt; ð12Þ

where kT(t) is the variable tangential spring stiffnessdependent on the normal displacement h(t)

kT tð Þ ¼ 8Geff

ffiffiffiffiR

p ffiffiffiffiffiffiffiffih tð Þ

p; ð13Þ

while Geff is the effective shear modulus.The slip mode is defined by the equality

FTsl tð Þ

�� �� ¼ FTcrit tð Þ: ð14Þ

As follows from Equation (11), the behavior of thetangential interaction in the static or slip friction modes ispredefined by the threshold value of the tangential force:

FTcrit tð Þ ¼ l FN

eff tð Þ�� ��; ð15Þ

where l is coefficient of friction.Formally, this equality defines the threshold of forces

in a classical friction problem with the Coulomb limit. Thethreshold value is governed by the effective normal force

FNeff tð Þ, being, however, nonuniquely presented by a combi-

nation of the components of the normal forces. It could be

noted that the Coulomb’s limit may be interpreted asa surface in the space of forces, which, in three-dimensionalproblems, is transformed into the Coulomb’s cone.

Generally, the effective force should reflect variouspossible situations in transition from the unilateral non-adhesive contact interaction to the maximal adhesion (seeWriggers 2009). In the case of adhesion, the adhesion forcecontributes to the limit force (15) and, consequently, to thestick-slip criterion (14). A study of the effect of the vander Waals adhesion on the tangential slip of particles wasgiven by Savkoor and Briggs (1977), while various interpre-tations of the adhesion model were considered by Thornton(1991) and Tomas (2007a, 2007b).

In the simplest case of the non-adhesive repulsive contact,

the effective force is a simple repulsive force FNeff tð Þ ¼

FNHertz tð Þ > 0, while the friction limit is FT

crit tð Þ ¼ l FNHertz tð Þ

�� ��.For the adhesive-non-dissipative (AnD) model, the effective

force denoted hereafter by FTcrit1 tð Þ is a force, meaning the

tangential friction limit within the interval h(t)> 0. It isdescribed in terms of the adhesive-non-dissipative force

FNAnD tð Þ, given in Equation (3), while and the fixed adhesion

force FNL , given in Equation (2), is defined by the friction limit

as follows:

FTcrit1 tð Þ ¼ l FN

AnD tð Þ þ FNL

�� ��: ð16Þ

The effective force FNAnD;eff tð Þ, presented in Equation (16)

as a sum FNAnD tð Þ þ FN

L , gives the final effective normal force

FNAnD;eff tð Þ ¼ FN

Hertz tð Þ. We assume only positive contribution

of the normal force to the tangential contact.Generally, the adhesive–dissipative (AD) behavior in the

tangential contact is considered by including dissipativeforce, FN

diss tð Þ (7). In order to better explain the influence ofnormal hysteresis on the tangential contact, the dissipativecase will be illustrated by two models, (AD1) and (AD2),by introducing two tangential critical forces, FT

crit2 tð Þ and

FTcrit3 tð Þ, respectively.The friction limit for AD1 model is characterized by the

effective force presented as a sum FNAD1;eff tð Þ ¼ FN

AD tð ÞþFNL þ FN

diss tð Þ, which includes additionally the history depen-

dent force FNdiss tð Þ and has final expression FN

AD1;eff tð Þ ¼FNHertz tð Þ. During unloading, the normal force FN

AD tð Þ isdefined by Equation (8). Finally, the friction limit is defined

by FTcrit2 tð Þ ¼ FT

crit;AD tð Þ as follows:

FTcrit2 tð Þ ¼ l FN

AD tð Þ þ FNL þ FN

diss tð Þ�� ��: ð17Þ

The third additional (AD2) model is of transitional charac-ter, where adhesion hysteresis is ignored during the tangentialcontact. As mentioned before, we assume only positivecontribution of the tangential force to the slip criterion.Therefore, when FN

AD2;eff tð Þ � 0, the third criterion becomes0. Finally, the third case will be characterized by the criticalforce FT

crit3 tð Þ

FTcrit3 tð Þ ¼ m �

FNAD tð Þ þ FN

L

�� ��; if FNAD2;eff tð Þ > 0;

0; if FNAD2;eff tð Þ � 0

(; ð18Þ

490 Jasevicius et al.

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

where the effective normal force FNAD2;eff tð Þ for the third

criterion, FTcrit3 tð Þ, is characterized by FN

AD2;eff tð Þ ¼ FNAD tð Þþ

FNL with final expression FN

AD2;eff tð Þ ¼ FNHertz tð Þ � FN

diss tð Þ.Unlike (AD1) model, which has the effective normal force,

FNAD1;eff tð Þ ¼ FN

Hertz tð Þ, in (AD2) model, we take into account

the contribution of the additional adhesive force presented as

the dissipative member, FNdiss tð Þ, which gives hysteresis in

normal force displacement (Figure 1). It should be alsomentioned that the tangential force, during a short rangeinteraction without any contact deformation, is neglected.

Graphical description of all three normal effectiveforces, FN

AnD;eff tð Þ, FNAD1;eff tð Þ, and FN

AD2;eff tð Þ, given above,is presented in Figure 2.

Simulation Methodology

The interaction of the particle is considered in the frameworkof the DEM. Particle’s properties are defined by the prescribedspherical shape, material properties, and the constitutiveinteraction laws. The state of particle i is characterized bythe following vectors attached to the particle mass centre:the position vector xi defining the translational motion withrespect to the fixed Cartesian frame, velocity ti ¼ _xxi ¼dxi=dt and acceleration ai ¼ _tti ¼ €xxi, as well as the vector ofthe angular velocity xi and angular acceleration ei ¼ _xxi.

The moving particle obeys Newton’s second law andits motion in time t is described by a set of second-orderordinary differential equations as follows

mi€xxi tð Þ ¼ Fi tð Þ; ð19ÞIi _xxi tð Þ ¼ Mi tð Þ; ð20Þ

where mi and Ii denote mass and moment of inertia of aparticle, respectively. The vectors Fi(t)¼

PjFi,j(t) and Mi(t)¼P

jMi,j(t) present the resultants of forces and torques added tothe centre of particle the interaction. The force betweenparticle i and collision partner j, that is, a particle or plane,

Fi;j tð Þ ¼ FNi;j tð Þ þ FT

i;j tð Þ is presented by a composition of the

normal and tangential components acting at the contactpoint. These forces are defined by Equations (7)–(9) and(10) and (18), respectively. When rolling friction is neglected,

the particle torque Mi;j tð Þ is caused by the tangentialforce FT

i;j tð Þ.The particle’s motion is obtained by applying an explicit

time integration of the Equations of motion (19) and (20).Thereby, the peculiarities of the model should be taken intoaccount. First, the discontinuity of constitutive models invarious interaction stages, that is, approach-detachmentand=or loading-unloading, should be distinguished;secondly, the Coulomb friction model has the disadvantageof discontinuity at the points at which transition from onemode (static friction) to another (slip) may occur.

It should be noticed that the evaluation of the tangentialcontact requires a rigorous nonlinear incremental time-history analysis. Each of the static and slip contact modesmay be characterized not only by different forces FT

st tð Þdefined by Equation (12) and FT

sl tð Þ defined by Equation(14), respectively, but also by different nature of motion.The reversible motion of the contact point due to the elasticcontact deformation before gross slip is characterized by theelastic displacement del(t), given in Equation (12). When thelimit of the slip region is reached, the contact point movesdue to gross slip accompanied by a slip displacement dsl(t).As a result, the total tangential displacement d(t) consistof two components as

d tð Þ ¼ del tð Þ þ dsl tð Þ: ð21Þ

The incremental methodology for evaluating coupledinteraction of the normal adhesive contact (Jasevicius et al.2011), and the stick-slip of tangential contact with respectto (21) (Zabulionis et al. 2012), compatible with the explicitintegration scheme was elaborated.

Basic Data

Adhesive–dissipative behavior is demonstrated by consi-dering a smooth spherical microparticle with the radiusR¼ 0.6 mm. The oblique impact of a microsphere againsta flat surface in the presence of adhesion is simulated numeri-cally. The basic data of the parameters of the silica particleare selected from the available data sources. The particle’sdensity q¼ 2201 kg=m3. Thus, its moment of inertiaI¼ 2.87 � 10�28 kg �m2 and mass m¼ 2 � 10�15 kg. The proper-ties of elasticity of the particle are defined by the modulusof elasticity E¼ 75GPa and Poisson’s ratio n¼ 0.17. Para-meters of the elasticity of the flat surface are identical tothe particle’s parameters. The coefficient of friction l¼ 0.3.The attractive distance aF¼0¼ 0.336 nm, the attractivejump-in force FN

L ¼ 0:005 mN at the distance h¼ 0 areback-calculated from the results of physical experiment byshear tests of the industrial silica powder. The amount of dis-sipated energy for normal interaction WN

attr;diss ¼ 0:5 � 10�15 Jwhich was taken basing on the data of physical experimentsprovided by Wall et al. (1990).

Numerical Results of the Oblique Impact

Numerical results presented below illustrate performanceof the proposed dissipative model. The simulation of the

Fig. 2. Illustration of the friction limit criterion, the relationshipbetween the effective normal force FN

eff tð Þ and displacement h(t).

Adhesive–Dissipative Behavior Under the Oblique Impact 491

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

experiment of the oblique impact is limited to the impactangle of 45� and equal values of normal and tangential velo-cities, that is, tN0 ¼ tT0 ¼ 1m=s. Thereby, zero initial angularvelocity of the particle x0¼ 0 rad=s is assumed. The freelychosen value of the normal initial velocity tN0 ¼ 1m=s ofthe particle is higher than initial normal critical velocity

tN0;cr ¼ 0:71 m=s required for particle rebound.

Normal Behavior

Both the adhesive–non-dissipative (AnD) and the adhesive–dissipative (AD) constitutive models of the normal interac-tion described by Equations (1)–(9) and shown in Figure 1will be illustrated. It should be reminded that the behaviorof the normal interaction is characterized by the normal forceFN and the normal displacement h. Results of the numericalsimulation during the oblique impact are presented interms of time histories of force FN(t) (Figure 3a) anddisplacement h(t) (Figure 3b). The normal force-displacement relationship is shown in Figure 4.

The non-dissipative (AnD) model is illustrated by a solidthin line (curve 1), while dissipative AD model is presentedby a solid bold line (curve 2). These styles of the lines are usedhereafter in all graphs where they are appropriate. In order tokeep relevance to Figure 1, the entire interaction paths formodels (AnD) and (AD) are denoted by lines S-L-U-L-Sand S-L-U-A-D, respectively.

It is assumed that for both (AnD) and (AD) modelspresented here the interaction starts at the same time instanttS¼ t0¼ 0, when particle reaches interaction range aF¼0, andadhesion begins to act between surfaces of the particle andthe flat surface. During the approach the particle reachesthe surface at time instant tL¼ 0.34 ns with the attractivenormal force FN tLð Þ ¼ �FN

L ¼ �0:005 mN obtained accord-ing to DMT model by Equation (2). The nonlinear loadingpath ends with maximal repulsive force FN

U ¼ 1:9mN andthe maximal overlap hU(tU) ¼1.31 nm at time instanttU¼ 2.27 ns. During the loading, in the time intervaltS� t� tU, variations of the normal force FN(t) and normaldisplacement h(t) are identical for both models. In oursample, the obtained value of maximal force FN

U is muchhigher than the adhesion force FN

L , while the ratio of theseforces is FN

U=FNL ¼ 380.

The unloading process in AnD model ends at the timeinstant tLr while detachment ends at the time instant tSr.In opposite to the force, the contact displacement hU, ismuch shorter in comparison with the attractive distanceshS, the ratio is hU=hS¼ 3.918.

The dissipative behavior of the particle begins to differ atthe time instant tU, which denotes the switch from the loadingto the unloading. The backward motion continues up tothe end of the contact, that is, within tU� t� tD. It shouldbe remarked that due to adhesion hysteresis the duration ofthe contact time increases, that is, tD> tLr (Figure 3).

Qualitatively, the pull-off adhesion force increases duringunloading because of necessity to break increased resistanceof the adhesion of a surfaces due to the high value of surfaceenergy. The normal load-displacement curve (Figure 4)reflects typical hysteretic elastic-dissipative behaviorobserved in AFM experiments (see Wall et al. 1990; Jones

2003). The ratio of the adhesion force FNA ¼ 0:61 mN to the

initial force FNL is extremely large FN

A=FNL ¼ 122. This tend-

ency is observed in known physical experiments that pull-offforce of adhesion of microparticle of 0.6 mm radius canreached more than 0.10 mN (see Tykhoniuk et al. 2007), whilepull-off force of adhesion depends on loading force (see Wallet al. 1990).

The area within the loop of the force-displacement graphof the AD model (Figure 4) presents the amount of thedissipated energy. Interaction energy WN

attr;diss of AD model

Fig. 3. Time histories of the normal force FN(t) (a) and the normal displacement h(t) (b): curve 1 (thin) and curve 2 (bold) denote(AnD) and (AD) models, respectively.

Fig. 4. Normal force and normal displacement relationship:curve 1 denotes (AnD) model and curve 2 denotes (AD) model.

492 Jasevicius et al.

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

was applied in Equations (5) and (6), thus described aboveEquation (2) agrees WN

diss ¼ WNattr;diss ¼ 0:5 � 10�15 J.

The proposed adhesive–dissipative model illustrates howthe dissipative behavior of microparticle during adhesionhysteresis could be captured by introducing dissipative forcedefined by Equations (7) and (9). As a result, the adhesionhysteresis with the fixed amount of dissipated energy may beincorporated into traditional time-history analysis of usingDEM. Contrary to the mostly used viscous damping modelsthe adhesion hysteresis changes the symmetry of contactcontributing mainly the unloading path and the detachment.

Tangential Behavior

Tangential interaction in the presence of adhesion is morecomplicated, but it is much more diverse, if considered fromthe numerical point of view. A characteristic of the inter-action of a particle with a wall depends on the value of theattack angle. Di Maio and Di Renzo (2004) have condition-ally classified the non-dissipative and non-adhesive tangentialinteraction regarding the impact angles, and distinguisheddifferent interaction modes at the small, intermediate, andlarge angles. A similar classification may be conditionallyadopted for the adhesive–dissipative interaction. The adoptedimpact angle of 45� can be interpreted as an intermediateimpact angle, exhibiting slip ! no-slip ! slip behavior.

To exhibit the influence of the adhesion on the behavior ofthe tangential contact, three different cases of the tangentialcontact will be discussed. In the first case, the model of the(AnD) contact together with the critical force FT

crit1 tð Þ definedin Equation (16) was applied. For the dissipative normalcontact, two the (AD1) and (AD2), models will be considered.The values of the Coulomb limit, defined by the critical forces

FTcrit2 tð Þ, presented in Equation (17) in the case of (AD1)model,

and FTcrit3 tð Þ by Equation (18) for (AD2) model, were applied.

At an intermediate angle, the forward Coulomb limit isreached almost at the beginning of the contact. At this mode,the tangential force changes its direction as well. At forwardmoving, the slip mode changes in stick, when the backwardCoulomb limit is reached. At backward moving, the stickmode changes in slip, when the backward Coulomb limit isreached.

The simulation results in terms of time histories of the ofthe tangential contact variables, the tangential force FT(t)and the total tangential displacement dT(t), are presentedin Figure 5. The entire duration of the contact behavior isillustrated in Figures 5a and 5c, while the detailed behaviorof the force and displacement at the end of the contact ispresented in Figures 5b and 5d. The relationship FT(t)�dT(t) is shown in Figure 6.

The behavior of various models in terms of the normal andtangential forces is shown in Figure 7. The total view of therelationship between the normal, FN(t), and tangential,FT(t), forces is given in Figure 7a. Because of the differencesin scale, the vertex of the cone of the Coulomb limit isdepicted separately in Figure 7b. The frictional limit behaviormay be characterized by analogy with plastic yielding, seeWriggers (2009). Generally, the location and the shape ofthe Coulomb cone reflect the influence of the adhesion

on the behavior of the frictional contact. Two positivelyand negatively located edges presented in Figure 7 of theCoulomb cone (Forward CL) and (Backward CL) indicatetwo limits reached in loading and unloading, respectively.The cone of the Coulomb limit is specified by Equation(14), while the tip of the cone is located in the negative areaof the normal force FN tð Þ < 0 and is characterized by thevalue of the normal pull-off force FN

L defined accordingto Equation (2). The inclination angle a reflects the frictioncoefficient m.

The simulation results in Figures 5–7 are denoted in thesame style, where line 1 (a thin line) denotes the non-dissipative (AnD) model, while line 2 (dashed) and line 3(bold solid) denote the dissipative (AD1) and (AD2) models,respectively. Additionally, the characteristic points of thesolution paths are indicated in Figures 6 and 7a, by capitalletters, which are used as subscripts in Figure 5.

In the approach stage between time instants tS and tL(a and c in Figure 5), the particles do not contact, therefore,in this stage the tangential forces of the interacting bodiesare equal to zero. The tangential force arises only at thebeginning of the contact (the point L in Figure 7). In the testsample, the Coulomb limit is reached instantaneously.

During loading, the limit CL is reached in a negligiblyshort time interval and starts at point L with a negative nor-mal force FL. The value of this force, is very small comparedto the values of the normal forces at points U and B. Theforward motion of the particle during the loading followsthe path between points L and U. The tangential force attainsthe maximum at the point B at time instant tB¼ 2.22 ns(Figures 5). In a general case, the switch point B from loadingto unloading of the tangential contact may be reached earlierthan the switch point U of the normal contact, thus tB< tU.In our case, the transition from forward to backward move-ment occurs at the same point, that is, tB¼ tU. The tangentialbehavior during loading is basically predefined by a slidingcriterion (15) with the limit (16), where the contribution ofthe adhesion force FN

L is negligibly small. This motion followsthe forward branch of the cone (Figure 7).

During unloading within the time interval [tB, tC], theparticle follows a static friction mode (Figures 5 and 6).The value of the tangential force is predefined by tangentialstiffness kT(t), depending on normal displacement h(t) asdefined by Equations (12) and (13). The force value dropsto zero at the time instant tF¼ 3.31 ns. The change of the signindicates that the tangential force starts to act against themotion of the contact point, that is, negative unloadingmay be interpreted as reverse loading. When the Coulomblimit is reached on the opposite branch, then, the backwardslip process starts at point C. Because of different interpret-ation of the slip criterion (according to Equations (16)–(18)),the transition point C from stick to slip is different for allthree models (AnD), (AD1), and (AD2), the Coulomb limitwas defined by three different values of the critical force,

FTCi ¼ FT

crit;i, and was reached at three different time instants,

tC1¼ 3.58 ns, tC2¼ 3.59 ns, and tC3¼ 3.51 ns, respectively.Hereby, the values of FT

Ci are as follows: FTC1 ¼ �0:177 mN,

FTC2 ¼ �0:185 mN and FT

C3 ¼ �0:132 mN.

Adhesive–Dissipative Behavior Under the Oblique Impact 493

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

As can be seen from Figures 5a and 5b, tC3< tC1< tC2.The difference between the points tC2 and tC3 indicates thatthe adhesion influences the slip-stick behavior. The (AnD)and (AD1) models which ignores adhesion dissipation effectat slip behavior, effective force has approximately the samemanner in time periods [tC1, tLr] for (AnD) and [tC2, tA]for (AD1) model. In (AD2) model, where adhesion dissi-pation effect is taken into account at tangential contact, thistime period [tC3, tA], has different manner at time periods[tC3, tG] and [tG, tA]. Since, it is a result of the missingnegative attractive force during unloading in (AD2) modelwith the adhesive hysteresis at the slip region. It can beobserved in Figure 5 that the tangential contacts of the

models are interrupted with non-zero displacements. Thetangential interactions of (AD1) and (AD2) models areinterrupted at time instant tA¼ 4.33 nm, while those of(AnD), at tLr¼ 4.21 ns. At the time tA, the tangential displa-cements of the (AD1) and (AD2) models are not equal tozero: dA2¼ 0.205 nm (AD1 model) and dA3¼ 0.159 nm(AD2 model). According to (AnD) model, the tangentialdisplacement dLr¼ 0.269 nm at time instant tLr¼ 4.21 ns.

According to the (AnD) and (AD1) models the areas ofthe triangles in Figure 6 are approximately equal, however,the area of the triangle produced by the (AD2) model is thesmallest. It is due to the fact that, according to the (AD2)model, the negative normal force is ignored, therefore, thebackward slip occurs earlier than it happens accordingto the (AnD) and (AD1) models. It has been found that theslip region is hold during the entire stage of the loading ofthe forward movement of the particle until the beginningof unloading. It can be attributed to the accumulation ofa certain amount of potential energy during the contact,which is released after the detachment. This phenomenonoccurs due to the slip of the contact point. Therefore, thecontact point cannot return to its initial position.

The results obtained show that the area of Figure 6according to (AD2) model is smaller than the areas of the(AD1) and (AnD) models. All models demonstrate the samebehavior during the loading process, while unloading isdifferent. The same values can be expected only in someintervals. According to the (AD1) and (AD2) models, thevalues of the normal and tangential forces coincide between

Fig. 5. Time histories of the tangential force FT(t) (a, b) and the total tangential displacement dT(t) (c, d), a general view (a, c), andthe vicinity of the detachment (c, d): 1 (a thin line) denotes the non-dissipative (AnD) model; 2 (dashed) and 3 (bold solid) linesdenote the dissipative (AD1) and (AD2) models, respectively.

Fig. 6. Tangential force versus tangential displacement: 1, 2, and3 denote (AnD), (AD1), and (AD2) models, respectively.

494 Jasevicius et al.

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

the points B and C3. According to the (AnD) and (AD2)models, the values coincide within the interval bounded bypoints C3 and L.

According to the (AD1) model, the point C2 is thetransition point from no-slip to slip (Figure 7). At this point,the normal force is smaller and the tangential force is biggerthan they are at points C1 and C3, respectively. According tothe (AD1) and (AD2) models, the interaction ends at point A.Unlike the (AD1) model, the (AD2) model, characterized byline AL, is presented with zero tangential force. Presented inFigure 7 a, the curves 1 and 3 of the closed triangle of normaland tangential forces do not exceed the Coulomb limit(models AnD and AD2), while curve 2 has differentCoulomb limit during the backward movement (model AD1).

As mentioned above, the inclination angles of the coneedges indicate the coefficient of friction. Initially, the conewas characterized by the illustrated fixed angle 2a. Theforward edge is specified by the fixed initial angle a, whilethe angles of the backward CL may be different.

For all cases, the loading follows the forward branch.The (AnD) and (AD2) models during unloading follow thebackward paths C1-L and C3-L defined by the same anglea. On the contrary, in the (AD1) model, the unloadingpath ends at point A, while for backward CL this angle is b.The difference between a and b is caused by a differentinterpretation of the slip criterion defined by FT

crit1 tð Þ and

FTcrit2 tð Þ. As a result, the relation between the normal and

tangential forces during the backward movement becomes dif-ferent. The different results obtained according to the methods(AnD) and (AD1) show that adhesive hysteresis is likely toplay a different role in the tangential behavior, and the influ-ence of the adhesion increases during the backward move-ment. For the (AD1) model, the angle decreases, that is, b< a.

Concluding Remarks

The adhesive–dissipative behavior of a microparticle underthe oblique impact is investigated numerically by applyingthe newly developed DEM-compatible attractive dissipativeinteraction model. It has been proved that the suggestedmodel is able to reproduce qualitatively the character ofthe loop of the adhesion hysteresis during the contact of

microparticles, which was observed in physical experimentswith the atomic force microscopy reported by other authors.Actually, the hysteric effect is achieved by introducing theattractive–dissipative component of the normal force, whichis capable of reflecting the loss of the kinetic energy governedby the fixed amount of the adhesion work required to separ-ate two adhesive contacting surfaces. The oblique impact ofa smooth spherical microparticle, with the radius R¼ 0.6 mm,mm, on the plane surface at the intermediate impact angle wasinvestigated numerically to illustrate theoretical considera-tions. The contribution of the adhesion-related dissipationis illustrated by comparing the behavior of the attractive–dissipative and attractive–non-dissipative models. Thebehavior of the tangential contact of the particle and the linkbetween the adhesion and friction during the adhesionhysteresis is illustrated by the behavior of forces with respectto the friction limit described by the Coulomb cone. Thechange of the Coulomb cone during dissipative unloadingwas also demonstrated.

Nomenclature

aF¼0 is a minimum center separation for forceequilibrium of molecular attraction andrepulsion potentials, nm

E modulus of elasticity, GPaF force of single contact, mNF force vector of particle, mNFNL

adhesion force of a rigid contact withoutany contact deformation, mN

FNdiss;h0

maximum value of the dissipation force

FNdiss;h0 is reached at the detachment point

A, with the zero displacement h¼ 0

FNHertz

elastic Hertz force, mNGeff effective shear modulush normal contact displacement (particle

overlap), nm�hh normalized accumulating displacement,

dimensionlessm particle mass, kgR particle radius, mmt, t0 time, initial time, nsx position vector of particle, m

Fig. 7. The behavior modes of various models in terms of the tangential and normal forces at the attack angle of 45� with respect tothe Coulomb limit (CL), a general view (a), and the vicinity of the column cone (b).

Adhesive–Dissipative Behavior Under the Oblique Impact 495

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

WNdiss

energy of dissipation, J

WNattr;diss

energy of dissipation related to attraction, J

Greek Letters

a, b angles of Coulomb cone limit curvel coefficient of friction, dimensionlessn Poisson ratio, dimensionlessq density, kg=m3

t, t0 velocity and initial velocity, m=s

tN0;cr initial critical sticking velocity at normalcontact, ms�1

tN0 , tT0

initial normal and tangential velocity, ms�1

x, x0 angular velocity and initial angular velocity,rad=s

Abbreviation

adh adhesionappr load, unload, detach approaching, loading,

unloading, detachingattr attractivecrit criticaldiss dissipationeff effectiveel, sl, st elastic, slip, staticAnD, AD attractive–non-dissipative model,

attractive–dissipative modelAD1 attractive–dissipative model, where

tangential contact is described with part ofnormal adhesion hysteresis only withpositive values during deformation

AD2 attractive–dissipative model, wheretangential contact is described with wholeof normal adhesion hysteresis with positiveand negative values during deformation

CL Coulomb limitL, C, B, F, G tangential contact starting (L, C) to slip,

start on no-slip contact (B); zero tangentialforce during backward movement at no-slipcontact (F); zero tangential force duringbackward movement at slip contact forAD2 model (G)

N, T normal, tangentialS, L, U, A,Lr, D, Sr

notations of points on the load-displacementcurve: start of interaction (S), loading (L),unloading (U), detachment (A, Lr) and theend of interaction (D, Sr)

Funding

R. Jasevicius is a postdoctoral student at the VilniusUniversity, whose postdoctoral fellowship is being fundedby European Union Structural Funds project ‘‘PostdoctoralFellowship Implementation in Lithuania.’’

ReferencesAdams, G. G., and M. Nosonovsky. 2000. Contact modeling—Forces.

Tribology International 33:431–442.

Amonton, G. 1699. De la resistance causee dans les machines.Memoires de l’Academie Royale A 257–282.

Becker, V., T. Schwager, and T. Poschel. 2008. Coefficient of tangentialrestitution for the linear dashpot model. Physical Review E77:011304.

Brach, R. M., and P. F. Dunn. 1998. Models of rebound and capturefor oblique microparticle impacts. Aerosol Science & Technology20:379–388.

Brilliantov, N. V, N. Albers, F. Spahn, and T. Poschel. 2007. Collisiondynamics of granular particles with adhesion. Physical Review E76:051302.

Butt, H. J., B. Cappell, and M. Kappl. 2005. Force measurementswith the atomic force microscope: Technique, interpretation andapplications. Surface Science Reports 59:1–152.

Cappella, B., and G. Dietler. 1999. Force-distance curves by atomicforce microscopy. Surface Science Reports 34:1–104.

Derjaguin, B. V., V. M. Muller, and Y. P. Toporov. 1975. Effect ofcontact deformations on the adhesion of particles. Journal of Colloidand Interface Science 53:314–325.

Di Maio, F. P., and A. Di Renzo. 2004. Analytical solution forthe problem of frictional-elastic collisions of spherical particlesusing the linear model. Chemical Engineering Science 59:3461–3475.

Dominik, C., and Tielens, A. 1997. The physics of dust coagulation andthe structure of dust aggregates in space. The Astrophysical Journal10:647–673.

Feng, X. Q., H. Li, H.-P. Zhao, and S.-W. Yu. 2009. Numericalsimulations of the normal impact of adhesive microparticles witha rigid substrate. Powder Technology 189:34–41.

Figueroa, I., H. Li, and J. McCarthy. 2009. Predicting the impactof adhesive forces on particle mixing and segregation. PowderTechnology 195:203–212.

Gilabert, F. A., J.-N. Roux, and A. Castellanos. 2008. Computer simu-lation of model cohesive powders: Plastic consolidation, structuralchanges, and elasticity under isotropic loads. Physical Review E78:031305-21.

Israelachvili, J. 2011. Intermolecular and surface forces, 3rd ed.New York: Academic.

Jasevicius, R., R. Kacianauskas, and J. Tomas. 2009. Simulation ofsticking of adhesive particles under normal impact. Vibroengineering11:6–16.

Jasevicius, R., J. Tomas, and R. Kacianauskas. 2011. Simulation ofnormal impact of ultrafine silica particle on substrate. ParticulateScience and Technology 29:107–126.

Johnson, K. L. 1998. Mechanics of adhesion. Tribology International31:413–418.

Johnson, K. L., K. Kendall, and A. D. Roberts. 1971. Surface energyand the contact of elastic solids. Proceedings of the Royal SocietyA 324:301–313.

Jones, R. 2003. From single particle AFM studies of adhesionand friction to bulk flow: Forging the links. Granular Matter4:191–204.

Kim, O. V., and P. F. Dunn. 2007. A microsphere-surface impactmodel for implementation in computational fluid dynamics. AerosolScience 38:532–549.

Kruggel-Emden, H., S. Wirtz, and V. Scherer. 2008. A study ontangential force laws applicable to the discrete element method(DEM) for materials with viscoelastic or plastic behavior. ChemicalEngineering Science 63:1523–1541.

Luding, S. 2008. Cohesive frictional powders: Contact models fortension. Granular Matter 10:235–246.

Martin, C. L., and R. K. Bordia. 2008. Influence of adhesion andfriction on the geometry of packings of spherical particles. PhysicalReview E 77:031307-8.

Sahagun, E., and J. Saenz. 2012. Dissipation by adhesionhysteresis in dynamic force microscopy. Physical Review B 85:235412-5.

496 Jasevicius et al.

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014

Savkoor, A. R., and G. A. D. Briggs. 1977. The effect of tangentialforce on the contact of elastic solids in adhesion. Proceedings ofthe Royal Society A 356:103–114.

Schwager, T., Becker, V., and Poschel, T. 2008. Coefficient of tangen-tial restitution for viscoelastic spheres. European Physical Journal E27:107–114.

Severson, B. L., L. M. Keer, J. M. Ottino, and R. Q. Snurr. 2009.Mechanical damping using adhesive micro or nano powders.Powder Technology 191:143–148.

Shi, X., and A. A. Polycarpou. 2005. Adhesive transition fromnoncontacting to contacting elastic spheres: Extension of theMaugis-Dugdale model. Journal of Colloid and Interface Science281:449–457.

Snitka, V., A. Ulcinas, M. Rackaitis, D. Zukauskas, and M. Fukui.1997. Microtribological properties of diamond-like and hydroge-nated carbon coatings grown by different methods. Diamond andRelated Materials 6:1–5.

Sridhar, I., and S. Sivashanker. 2003. On the adhesion mechanics of multi-layer elastic systems. Surface and Coatings Technology 167:181–187.

Thornton, C. 1991. Interparticle sliding in the presence of adhesion.Journal of Physics D 24:1942–1946.

Thornton, C., S. J. Cummins, and P. W. Cleary. 2013. An investigationof the comparative behaviour of alternative contact force modelsduring inelastic collisions. Powder Technology 233:30–46.

Tomas, J. 2003. Mechanics of nanoparticle adhesion—A continuumapproach. Particles on Surfaces: Detection, Adhesion and Removal8:183–229.

Tomas, J. 2004. Fundamentals of cohesive powder consolidationand flow. Granular Matter 6:75–86.

Tomas, J. 2007a. Adhesion of ultrafine particles-a micromechanicalapproach. Chemical Engineering Science 62:1997–2010.

Tomas, J. 2007b. Adhesion of ultrafine particles-energy absorptionat contact. Chemical Engineering Science 62:5925–5939.

Tykhoniuk, R., J. Tomas, S. Luding,M. Kappl, L. Heim, and H. J. Butt.2007. Ultrafine cohesive powders: From interparticle contacts tocontinuum behaviour. Chemical Engineering Science 62:2843–2864.

Yang, L., and Martini, A. 2013. Nano-scale roughness effectson hysteresis in micro-scale adhesive contact. Tribology International58:40–46.

Wall, S., W. John, H. W. Wang, and S. L. Goren. 1990. Measurementsof kinetic energy loss for particles impacting surfaces. AerosolScience & Technology 12:926–946.

Wriggers, P. 2009. Computational contact mechanics, 2nd ed. NewYork: Springer.

Zabulionis, D., R. Kacianauskas, D. Markauskas, and J. Rojek. 2012.An investigation of nonlinear tangential contact behaviour ofa spherical particle under varying loading. Technical Sciences60:265–278.

Zhou, H., and W. Peukert. 2008. Modeling adhesion forces betweendeformable bodies by FEM Hamaker summation. Langmuir24:1459–1468.

Zmitrowicz, A. 2009. Contact stresses: A short survey of modelsand methods of computations. Archive of Applied Mechanics80:1407–1428.887.

Adhesive–Dissipative Behavior Under the Oblique Impact 497

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 0

2:40

01

Aug

ust 2

014