Milestone L3:MPO.GTRF.P8.01 Microscale Model for Simulating

Fretting Wear of Brittle Materials with Third-body Wear Debris

Interaction

A. V. Hayrapetian and M. J. Demkowicz

Massachusetts Institute of Technology (MIT)

May 8, 2014

CASL-U-2014-0139-000

Microscale model for simulating fretting wear of brittle materials withthird-body wear debris interaction

A.V. Hayrapetiana, M.J. Demkowiczb,∗

aDepartment of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USAbDepartment of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract

Fretting wear models typically estimate wear rates using some form of a work-rate wear model with constant wearcoefficients, but experimental results show that using a constant wear coefficient makes the wear model insufficient atmodeling the complex dependence of the wear rate on various operating conditions of fretting wear. A new modelingframework is discussed that is designed with the goal of simulating abrasive/adhesive fretting wear mechanisms ofbrittle materials at the micron length-scale with the presence and interaction of third-body wear debris in order tostudy the effects of fretting operating conditions on wear rates. The capabilities of the model include the ability togenerate new wear debris particles through brittle fracture of elastically-deforming asperities and modeling contactwith friction and adhesion between any bodies in the simulation. These capabilities are demonstrated through twosmall simulations, and a validation study on the friction and sliding capabilities of the model is discussed through thesimulation of a block sliding under gravity down an inclined plane.

Keywords: fretting wear, third body, wear debris, discrete element, contact

1. Introduction

Fretting wear is a common issue in many engineer-ing applications. One particular application where itplays an important role is grid-to-rod fretting (GTRF)in nuclear reactors; GTRF is the cause of about 74%of leaking fuel incidents in light water-cooled nuclearfission reactors [1]. In GTRF, the turbulent coolant vi-brates fuel rods against supporting grids which resultsin the wear of the thin cladding walls of the rod. Cur-rent attempts to model this process and estimate the rodlifetime, such as VITRAN (Vibration Transient Anal-ysis Nonlinear) [2, 3], involve using work-rate wearmodels [4, 5] with constant empirical wear coefficients,in which the wear rate is considered to be proportionalonly to the rate of mechanical work done in the rubbingprocess. However, several experimental results haveshown that a work-rate wear model with constant wearcoefficients cannot properly model the complex depen-dency of the wear rate on various operating conditions.

In order to develop a better model to estimate wearrates under the variety of different possible operating

∗Corresponding authorEmail addresses: areg@mit.edu (A.V. Hayrapetian),

demkowicz@mit.edu (M.J. Demkowicz )

conditions relevant in fretting wear, it is necessary tounderstand the mechanisms through which the changingoperating conditions affect the rate of wear debris gen-eration from the surface of the fretting materials. Wehave developed a modeling framework to dynamicallysimulate fretting wear mechanisms at the micron length-scale which can be used to study how wear debris is gen-erated through colliding asperities on the surface of thetwo rough, brittle fretting materials, how accumulatingthird-body wear debris generated from wear play a rolein the further evolution of the wear process, and howthese processes vary with different operating conditionssuch as normal load, slip amplitude, material properties,type of fretting motion (pure sliding or a hybrid motionof sliding and impact), and other properties.

Currently, we have developed the framework and cor-responding code for critical features for modeling wearof rough brittle materials, such as elasticity, contact,fracture, friction, and sliding. These capabilities aredemonstrated through a simulation of two elastic cir-cles pushed together in contact and sliding against eachother with friction and also through a simulation of thefracture of a single asperity due to a collision with anobstacle. We have also partially validated the frictionand sliding capabilities of the model through the case

Preprint submitted to Computational Methods in Applied Mechanics and Engineering May 8, 2014

CASL-U-2014-0139-000

study of a block sliding down an inclined plane undergravity.

Many simulation techniques focused on modelingand understanding wear accurately model the forces,traction and slip between the geometries sliding againsteach other and then reduce the information from thesesimulations to a work rate to be used in a simple work-rate wear model with a constant wear coefficient. Typ-ically the forces and slip calculations used to estimatethe wear come from analytical expression from elasticsolid mechanics [6, 7], or from finite element simula-tions [8, 9, 10]. In these simulations, progressive wear-ing of the surface is modeled, but wear debris is not in-cluded in the simulations. Attempts have been made tomodel the third-body wear debris in fretting wear sim-ulations. The wear debris layer has been modeled asan anisotropic elastic-plastic material using finite ele-ments in a fretting wear simulation using Archard’s lawin [11]; the model is capable enough to track the growthand shrinkage of the debris layer thickness through ejec-tion of wear debris from the contact interface, but thereare limitations in the model due to the requirement ofusing a continuum simplification of the discrete weardebris particles. A third-body layer of individual rigiddebris particles modeled using discrete elements wasused by [12] to study the effect of the third-body layeron the macroscopic friction behavior of two sliding bod-ies; however, the wear debris particles are included inthe beginning of the simulation and cannot be generatedthrough wear of surfaces.

Alternatives to fretting wear models based on work-rate in the literature appears to be limited to only mod-eling fatigue-based mechanisms of wear. These mecha-nisms have been studied in a 2-D numerical model [13]and in 3-D using the finite element method [14] to findthe likely location of crack initiation and the number ofcycles for initiation and propagation of the crack. Ananalytical model of fretting wear that is informed by fi-nite element analysis but based solely on the fatigue-based mechanism of fretting wear has also been devel-oped [15]. While these studies and models are usefulfor fatigue-based mechanisms, they do not address thechallenge of modeling abrasive/adhesive wear mecha-nisms in a way that does not rely on the limited work-rate model.

Experiments by [16] and [17] of dry fretting wear be-tween two metals show how a work-rate wear modelwith constant wear coefficients fail to predict the depen-dence of the wear rate on parameters such as normalload, slip amplitude of the oscillatory motion, and onthe number of cycles. The results show that the wearcoefficient in the work-rate model increases with slip

amplitude rather than remaining constant. In particu-lar, there is a large sudden increase in the wear coeffi-cient at a particular slip amplitude which represents thetransition from partial slip, during which fatigue-basedmechanisms of wear like delamination [18] are domi-nant, to gross slip, in which the wear rate is accountedfor mostly by abrasive and adhesive wear. The exper-imental results also show a decrease in the wear coef-ficient as the number of cycles increase, which can beexplained by the increasing true area of contact betweenasperities (and thus decrease in local pressure) becauseasperities deform and surface roughness changes as thefretting process continues. These are just some of manyexperimental results that demonstrate the need to modelwear rates through an understanding the micromechani-cal state of contact and not only a work-rate wear modelwith constant empirical wear coefficients. Other exper-iments [19, 20, 21, 22] show how the grid support ge-ometry in the case of GTRF, further complicate wearrate predictions, and experiments on the dependence ofthe wear rate on a hybrid sliding/impact fretting motionby [23, 24] show that the wear coefficient can changeby an order of magnitude simply by adjusting the typeof fretting motion. These experimental results motivateour desire to understand how the wear coefficient varieswith certain important operating conditions in frettingwear. And to do this, we plan to study the wear processusing the modeling framework we developed which cansimulate fretting wear mechanisms such as brittle abra-sive wear of colliding asperities, interaction of third-body wear debris, and contact with adhesion and slidingfriction of the fretting surfaces and wear debris.

2. Requirements for wear model

Experimental results have shown that a work-ratewear model with a constant wear coefficient is unableto model the complex dependence of the wear rate onoperating conditions such as contact load, slip displace-ment, number of cycles, type of fretting motion, andmore. By allowing the wear coefficient to not be con-stant but instead vary with these operating conditions,the work-rate wear model can become more useful formodeling fretting wear under more complicated scenar-ios. Understanding the dependence of the wear coeffi-cient on the relevant operating conditions through ex-periments alone would require a huge combination ofoperating condition values which quickly become un-feasible. Instead, simulations providing the ability tostudy the mechanism through which changes in the op-erating conditions influence the wear process offer atractable solution to creating a fretting wear mechanism

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map and models of how the wear coefficient changeswith operating conditions.

Different mechanisms of wear are already wellknown in the literature, such as ductile/brittle abrasiveand adhesive wear, and fatigue-based mechanisms ofwear. However, though fretting wear is often charac-terized by its tendency to collect wear debris in the con-tact region, a survey of the literature shows there hasbeen limited analysis on the effect of the third-bodylayer on the wear rate. Experiments discussed in [25]showed that the third-body wear debris layer may re-duce the wear rate by acting as a compacted lubricatinglayer or alternatively may increase the wear rate due tothe abrasive behavior of the hard oxide debris particles;and, whether the wear debris is beneficial or harmfulfor wear depends on parameters such as the normal loadand slip amplitude. Therefore, the main focus of ourmodel is to understand the behavior of wear debris onwear rates and how changes in the operating conditionsaffect this behavior. For the model to be capable enoughto allow for studying this behavior, we require a micronlength-scale model, because that is the length scale ofthe wear debris, which can simulate the interaction andcontact of individually tracked third-body wear debrisparticles with each other and with asperities on the sur-face of the two fretting materials, and which can alsotrack the amount, size and shape of wear debris gener-ated through the wearing process.

To simplify the requirements of the model while stillremaining capable of handling our main focus, we re-strict the model to brittle abrasive and adhesive wearthrough the brittle fracture of elastically deformed as-perities. Ductility and plastic deformation can be safelyignored when modeling the wear of brittle materials,such as the zirconia oxide film that grows on the gridsupports and fuel rod cladding in the case of GTRF.Also, abrasive and adhesive mechanism of wear domi-nate the influence on the wear rate compared to fatigue-based mechanisms of wear in the operating conditionsof gross slip which is our primary regime of interest.

The strict requirements on the model to be incred-ibly flexible with regards to the possibility of contactbetween any surfaces in the simulations, including sur-faces that are dynamically formed when new wear de-bris are generated, require a new method designed forthis style of contact from the beginning. Inspiration forthe model was taken from both molecular dynamics andmore appropriately the discrete element method (DEM)in terms of the ability for discretely tracked elements inthe system to interact with their neighbors which changeas the simulation evolves. Unlike DEM, however, rigidparticles could not be used because elastic strain is nec-

essary to model the fracture of asperities and generatenew wear debris. The elastic deformation of materialsimulated in the model is inspired by the finite elementmethod, and in fact the model uses something similarto constant strain triangular finite elements in 2-D tomodel elasticity. Beyond contact, elastic deformation,and fracture, further requirements for the model includethe need to properly model friction and adhesion be-tween contacting surfaces. These requirements are sat-isfied in our model through modified dynamics of thecontact mechanism. While it is ideal for the model tocapable of working in 3-D, a 2-D model can still pro-vide very useful results; our model is currently limitedto 2-D as a simplification, although there is no foresee-able limitation in extending it to 3-D.

3. Methods

In this model, all elastic bodies are comprised ofsmaller bodies called discrete elements. Fig. 1 showsa typical discretization of two circular bodies into dis-crete elements. The discrete elements are not actuallyshown in that image because the discrete elements donot have a specified shape. The only state tracked foreach discrete element e is its mass me, position pe, ve-locity ve, and effective size re. The discrete elements arerepresented by each of the vertices of the triangles in thefigure. The discrete elements at the boundary of a bodyare called surface elements. Although all discrete ele-ments have an effective size, in Fig. 1 the effective sizeof only the surface elements is visualized by the bluecircles centered on the position of the elements; the ra-dius of each circle in the visualization is set equal to theeffective size of its corresponding element.

Neighboring discrete elements can join together toform a joint; the discrete elements forming the jointare called the members of the joint. Joints can be oneof two types: interior joints or contact joints. Interiorjoints represent a solid connection between the joint’smember elements; within an interior joint, there is con-tiguous matter bonded together that is shared betweenthe member elements. Conversely, contact joints repre-sent two independent solids that are merely in contactwith one another. The space within a contact joint is di-vided into the two solids by a partition called the contactplane. In Fig. 1, both joints are visualized: the discreteelements in each of the two bodies are held together byinterior joints, which are visualized in the figure by thegreen triangles (in 2-D) with black edges; also, the con-tact between the two bodies is represented through con-tact joints, which are visualized by the yellow triangles

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Fig. 1. Two circular bodies, which are each discretizedby discrete elements held together with interior joints,are in a state of contact. The contact between the bodiesis modeled using the contact joints.

with orange edges. All joints are able to model the elas-tic deformation of the material within the domain of thejoint. Interior joints also provide the capability of frac-ture, and contact joints provide the capabilities of fric-tion and adhesion.

3.1. Elasticity of joints

Elasticity is modeled within the joints. Both interiorand contact joints can be deformed elastically, but thedynamics of contact joints are slightly more involvedbecause the contact joints need to model the differencesin the stress carrying capacity of contact joints, which isdiscussed in sections 3.3 and 3.4.

The displacement field u(X) of the matter within ajoint is assumed to be fully described by a single de-formation gradient tensor F for the joint such that thepartial derivative of the displacement field with respectto material coordinates is ∂u

∂X = F − I, where I is theidentity tensor. Then, the displacement field within ajoint can be determined if the reference configuration ofthe joint and the current positions of the joint’s mem-ber elements are known. The reference configuration isthe set of independent displacement vectors between thepositions of the joint’s member elements when the jointis in a stress-free state. For interior joints, the referenceconfiguration is initially determined at the beginning ofthe simulation: it is either directly prescribed for eachinterior joint, or more likely it can be easily calculatedfrom the initial positions of the elements if the interiorjoints at the start of the simulation are taken to be in astress-free state.

In 2-D, exactly three discrete elements can form ajoint, which has the shape of a triangle. The three mem-ber elements of joint j are e j

1, e j2, and e j

3, which are or-dered such that the path drawn going from e j

1 to e j2 to

e j3 and back to e j

1 turns in a counter-clockwise orienta-tion. The vector s j

i for i ∈ 1, 2, 3 is the directed edge

of the triangle associated with joint j that is opposite tothe member element e j

i . For example, s j1 = pe j

3−pe j

2and

s j2 = pe j

1− pe j

3. The reference configuration of joint j

in 2-D is given by w j1,w

j2. If a joint j is in the stress-

free state at time t = 0, then the vectors of the referenceconfiguration of joint j are given by w j

1 = s j1(t = 0) and

w j2 = s j

2(t = 0), or some rotation of those vectors. Ifthe current positions of the member elements of joint jat time t are known, and the reference configuration ofjoint j is also known, then the deformation gradient ten-sor F j(t) that defines the displacement field within thejoint j at time t can be calculated as follows:

F j(t) =[s j

1(t) s j2(t)] [

w j1 w j

2

]−1

=[(pe j

3(t) − pe j

2(t)) (pe j

1(t) − pe j

3(t))] [

w j1 w j

2

]−1.

(1)

While elasticity is modeled in the joints, the elas-tic response of a body under loads and deformationultimately needs to come from the motion of the dis-crete elements that make up the body. The elementsare just treated as point masses that move according tofe = meae = me

∂2pe∂t2 . The net force on the element fe

needs to be selected to appropriately model a realisticelastic response. The net force is determined by con-serving the energy of the system (sum of kinetic andpotential energy). If the potential energy of the systemis Π, then the net force on each element e is

fe = −∂Π

∂pe. (2)

The potential energy of the elastic body with domainΩ and boundary Γ that has body forces b acting on thebody and traction forces t acting on part of its boundaryΓσ is given by

Π =

∫Ω

UdA −∫Ω

b · udA −∫Γσ

t · udS , (3)

where U is the elastic strain energy density. Becausethe joints define where matter exists within a body andthe joints do not overlap, the domain of a body Ω is theunion of the disjoint domains of joints. If the domain ofan joint j is Ω j, then Ω =

⋃∀ j Ω j and

Π =∑∀ j

∫Ω j

UdA −∫Ω j

b · udA

, (4)

assuming there are no tractions t. Tractions are not typ-ically useful for the wear simulations intended to be run

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with this model. Tractions are not needed for contact be-cause the contact joints directly model the contact, andprescribed tractions at the boundary of a body can besimulated by simply prescribing equivalent body forceson the surface elements of the body.

Combining (2) and (4) gives

fe =∑∀ j

−∫Ω j

∂U

∂pedA +

∫Ω j

∂(b · u)∂pe

dA

. (5)

Je is defined as the set of joints that the discrete elemente belongs to as a member. By recognizing that the strainenergy densityU(x), body force b(x), and displacementu(x) only depend on pe when x ∈

⋃j∈Je

Ω j, (5) can bewritten as

fe = felastice + fbody

e

= −∑j∈Je

∫Ω j

∂U

∂pedA +

∑j∈Je

∫Ω j

∂(b · u)∂pe

dA, (6)

wherefelastic

e = −∑j∈Je

∫Ω j

∂U

∂pedA, (7)

andfbody

e =∑j∈Je

∫Ω j

∂(b · u)∂pe

dA. (8)

Under the approximating assumption that the bodyforce b(x) is homogeneous over the domain of a singlejoint j, or b(x) = b j for x ∈ Ω j, the contribution to thenet force due to body forces, equation (8), reduces in the2-D model to

fbodye =

13

∑j∈Je

A jb j, (9)

where A j is the area of the joint j. Usually the bodyforce is due to an acceleration, such as the accelerationdue to gravity g. In that case, the body force is b = ρg,where ρ is the density of the material, which is assumedto be homogeneous over the domain of a single joint.Applying gravity as a body force can be greatly sim-plified by selecting appropriate masses for the discreteelements. By selecting the mass of the elements at thestart of the simulation according to

me =13

∑j∈Je

ρ jA j, (10)

where ρ j is the homogeneous density of the materialwithin the joint j, (9) is further simplified to fbody

e = meg

for the case of the body force being due to an accelera-tion g.

The general form of the potential energy in (3) allowsfor any hyperelastic material to be used in the model.However, to actually calculate felastic

e in (7), a specificmaterial model must be chosen to get an expression forthe strain energy density. The Saint Venant-Kirchoff

material [26] was chosen for the current implementa-tion of the model, which is an isotropic material withstrain energy density

U =12λ(tr E)2 + µ tr (E2), (11)

where λ and µ are the Lame constants, and E is the La-grangian Green strain given by

E =12

(FᵀF − I). (12)

With the selection of the Saint Venant-Kirchoff mate-rial model, the contribution to the net force due to elas-ticity, equation (7), reduces in 2-D model to

felastice =

∑j∈Je

[12σ jR π

2s j

M je

]. (13)

M je ∈ 1, 2, 3 is an index map such that M j

e ji

= i ∀i ∈

1, 2, 3. R π2

is a rotation matrix that rotates vectorscounter-clockwise by 90 and is defined by

R π2

=

[0 −11 0

].

σ j is the Cauchy stress in the joint j and is given by

σ j =1

det F j F jS j(F j)ᵀ, (14)

where S j is the second Piola-Kirchoff stress in jointj and relates to the Lagrangian Green strain E j =12 ((F j)ᵀF j − I) in the joint j according to

S j = λ(tr E j)I + 2µE j. (15)

3.2. Fracture of interior jointsFracture is modeled by simply removing interior

joints when a fracture condition has been met for theinterior joint. Because the mass and velocity is trackedwith the discrete elements and not the joints, mass andlinear/angular momentum is conserved in the fractureprocess.

In general, the fracture condition can come from a co-hesive zone model. In the current implementation of the

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model, a very simple fracture algorithm is used in whichthe interior joint is removed if the maximum principalCauchy stress in the interior joint exceeds some thresh-old value σ f t.

To roughly estimate that threshold value the fracturetoughness KIc and linear elastic fracture mechanics areused. Assuming a symmetric crack of width 2a orientedin the worse possible direction, and assuming the crackcan be approximately modeled as existing in an infinitebody with a far field stress σ∞, the stress intensity factorfrom linear elastic fracture mechanics is KI = σ∞

√πa.

Fracture occurs when KI = KIc, so KIc = σ f t√πa.

Therefore, the fracture threshold is estimated as

σ f t =KIc√πa, (16)

where a is taken to be some fraction of the length scaleof the interior joint.

When an interior joint is fractured, the member ele-ments of the joint become surface elements. Becauseof this, they can later join together into contact joints tomodel contact. Discrete elements from a freshly frac-tured interior joint can immediately come together toform a stress-free contact joint, in which case, the modelis simulating the process of the material fracturing butnot actually separating from each other by a significantdistance. The material can then later separate as the dis-crete elements are pulled apart, eventually causing thecontact joint to be removed. The details of this proces-sare discussed in section 3.3.2.

3.3. Contact jointsThere are some important differences between con-

tact joints and interior joints. While the matter withinthe domain of an interior joint is supposed to be bondedtogether as part of a solid body, the matter in contactjoints consists of two independent solids that are just incontact with one another. The domain of the contactjoint is divided by a partition called the contact plane.In 2-D, this contact plane is actually just a line, andit is always parallel to the edge from the joint’s mem-ber elements e j

1 to e j2 and located in between the edge

and the third member element e j3. Within the domain

of the contact joint, the matter on one side of the con-tact plane belongs to one body while the matter on theother side of the contact plane belongs to another body.It may be possible that the two solids are actually partsof a larger single body, but at the scale of an individualcontact joint, they appear to be from different bodies.Since within the domain of the contact joint, there aretwo bodies that are in contact, the ability to transfer ten-sion and shear between the two bodies is limited. In

particular, a tensile load above some adhesive thresholdwill cause the two bodies to separate, thus removing thecontact joint, and a shear load above a frictional limitwill cause bodies to slip along the contact plane, thusseverely deforming the contact joint. Severe deforma-tion due to slipping can also cause the bodies to slideoff one another which also leads to the removal of thecontact joint. Finally, the contact joints need to be cre-ated in the first place to represent contact between twobodies that were originally not in contact and preventthem from passing through each other. All of these dif-ferences between contact joints and interior joints arediscussed in detail in the remainder of section 3.

3.3.1. Forming contact jointsThere are restrictions on how contact joints can be

formed. First, only surface element can join together toform a contact joint. In particular, the first two surfaceelements of a contact joint, e j

1 and e j2, need to be belong

to a directed edge from e j2 to e j

1 that is a boundary edge.A boundary edge is any directed edge between surfaceelements that belongs to any interior joint. Second, acontact joint cannot be created that uses a directed edgethat already exists (an edge between identical discreteelements but in the opposite direction is fine). Thesetwo constraints prevent contact joints from forming thatcould overlap with each other or an interior joint. Fi-nally, the most important constraint on forming contactjoints is the proximity constraint which only allows sur-face discrete elements to form a contact joint if they are“close enough” to one another.

The proximity constraint determines when the sur-face of two bodies are close enough to be consideredin contact. The actual surface of a body discretized withdiscrete elements is not tracked in this model becausethe discrete elements making up the body do not have adefined shape. Instead the surface of a body can be ap-proximated by the location of the surface elements andeach of their effective sizes. For example, in Fig. 1, al-though the black edges of the triangles near the surfacein each of the bodies are far apart from one another, thetwo bodies in that figure are actually touching (althoughjust barely). The extent of the blue circles, which rep-resent the effective size of the discrete elements, givesa vague approximation of the actual surface of the bod-ies. The condition used in the current model to deter-mine whether the proximity constraint is satisfied for acandidate contact joint j is to check if both e j

1 is closeenough to e j

3 and if e j2 is close enough to e j

3. Checkingwhether e j

1 is close enough to e j2 is unnecessary because

they must be part of a boundary edge and are therefore

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CASL-U-2014-0139-000

always assumed to be close enough. For two elements tobe considered “close enough” to one another, we requirethe distance between the two elements to be less thanthe sum of their effective sizes. Therefore, the proxim-ity constraint of a candidate contact joint j is satisfied if‖pe j

3− pe j

1‖ < re j

3+ re j

1and ‖pe j

3− pe j

2‖ < re j

3+ re j

2.

The effective size of each of the elements is deter-mined before the start of the simulation. It is desirableto choose an effective size such that circles centered onthe elements with radii set to the effective size of theelements would have some overlap with adjacent cir-cles, but at the same time not be too big. Each joint hassome characteristic length which in 2-D is defined as thesquare root of the area of the joint. Choosing the effec-tive size of an element through a weighted average ofthe characteristic lengths of the joints that the elementbelongs to as a member, where the weights are deter-mined from the areas of the joints, was found to providethe best results. Mathematically, the effective size of anelement e is

re = c∑j∈Je

a je

√A j, (17)

where the weights a je are given by

a je =

A j ∑j∈Je

A j

, (18)

and c is a constant to universally adjust the weights. Aconstant of c = 0.9 was used to provide good results.

If the constraints for forming a contact joint are allsatisfied, the contact joint must be created. Currently theimplementation of the model iterates though the pairs ofelements of all boundary edges and for each pair iteratesthrough all surface elements in the simulation lookingfor a third element that can form a contact joint that sat-isfy all the constraints. If there are more than one validcandidates for the third element, the element closest tothe boundary edge is chosen. The computational com-plexity of this naive algorithm is of order N2, where Nis the number of surface elements at any given time inthe simulation, but restricting the search space to onlynearby surface elements by using data structures likequadtrees can reduce the computational complexity ofthe algorithm to be linear with N.

3.3.2. Breaking contact jointsContact joints must be removed when the joints mem-

ber elements should no longer be considered to be incontact. This can happen if the surfaces are pulled apartfrom one another, but it can also happen if one surface

slides away from another surface. In reality, if there isany amount of slipping that occurs at a point, the twomaterial points that were originally coincident prior toslipping will no longer be in contact after slipping oc-curs. In the model, this is approximated by consider-ing the two surfaces represented by a contact joint to bein contact even with some amount of sliding (which isdiscussed in section 3.4) until the sliding becomes toogreat that the surfaces cannot reasonably be consideredin contact anymore. These two different ways of end-ing contact are modeled by removing the contact jointsif either of two conditions are satisfied: the pull apartcondition or the slide off condition.

The slide off condition is satisfied for contact joint jif the projection of e j

3 onto the boundary edge from e j1 to

e j2 is far away from the boundary edge. Mathematically,

this means the slide off condition is satisfied if either (s j1 ·

s j3)/‖s j

3‖2 or (s j

2 · sj3)/‖s j

3‖2 is greater than some positive

threshold value. We have currently selected a thresholdof 0.50, which represents the slide off condition beingmet if the projection of e j

3 extends beyond the nearestvertex of the boundary edge by an amount greater than50% of the length of the boundary edge.

The pull apart condition is determined from the stressstate in the contact joint. In a non-adhesive contact, thecontact joint could not take any tensile load normal tothe contact plane. We extend this concept in order toprovide a very simple model of adhesion by consider-ing the contact to be pulled apart only if the normal ten-sile stress σn exceeds some threshold value σat. So, thepull apart condition is satisfied if σn > σat. The pullapart condition can be updated in the future with moresophisticated models of adhesion.

3.4. Friction

In this model, friction is modeled through the evo-lution of the reference configuration of contact joints.With a given reference configuration, the stress statein a contact joint changes as the current configurationevolves due to the moving member discrete elements.The stress state can be used to determine the tangentialload T and normal load N acting on the slip plane of thecontact joint. So, in general the tangential and normalloads of contact joint j are functions of the referenceand current configurations: T j = T (s j

1, sj2,w

j1,w

j2) and

N j = N(s j1, s

j2,w

j1,w

j2), where the non-linear functions

T and N are known and depend on the elastic proper-ties of the material within a contact joint which does notchange throughout the lifetime of that contact joint.

To model friction, restrictions are placed on the tan-gential force. The contact joints cannot carry a tangen-

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CASL-U-2014-0139-000

tial force that is greater in magnitude than the frictionlimit µ|N|, where the coefficient of friction µ is the coef-ficient of static friction µs under the conditions of stickor is the coefficient of kinetic friction µk under the con-ditions of slip.

During each iteration in a simulation, the discrete el-ement positions are updated and the deformation andstress state within each joint is calculated. Duringthe updating stage, each contact joint j must satisfythe restriction on the tangential force described above:|T j(tn)| ≤ µ|N j(tn)|. To satisfy this requirement, the ref-erence state is adjusted in a constrained manner to sim-ulate the slipping phenomena. In 2-D, the contact planeis an line, so a unit vector t j

parallel to that line repre-senting the contact plane of contact joint j is used; thevector t j

is actually defined in the reference space so thatit is parallel to the vector −(w j

1 + w j2). After the discrete

element positions have been updated, the vectors s j1(tn)

and s j2(tn) are known. The reference configuration of the

contact joint in the prior iteration, w j1(tn−1),w j

2(tn−1),is also known. If |T (s j

1(tn), s j2(tn),w j

1(tn−1),w j2(tn−1))| ≤

µ|N(s j1(tn), s j

2(tn),w j1(tn−1),w j

2(tn−1))|, then the conditionis satisfied with the current reference configuration, andthus w j

1(tn) = w j1(tn−1) and w j

2(tn) = w j2(tn−1). On the

other hand, if that inequality is not satisfied, then thefollowing non-linear equation,

|T (s j1(tn), s j

2(tn),w j1(tn−1) + s j

n t j,w j

2(tn−1) + s jn t j

)| =

µ|N(s j1(tn), s j

2(tn),w j1(tn−1) + s j

n t j,w j

2(tn−1) + s jn t j

)|,(19)

needs to be solved for the single degree of freedom s jn,

which is the amount of slipping that occurs in contactjoint j from time tn−1 to tn. After solving for s j

n, the newreference configuration for the contact joint j is givenby:

w j1(tn) = w j

1(tn−1) + s jn t j,

andw j

2(tn) = w j2(tn−1) + s j

n t j.

3.5. Remeshing of contact joints

3.5.1. Reason for remeshingFig. 2 shows a sequence of steps in a simulation of

two surfaces in sliding contact. The domain of eachof the joints visualized in the figure are uniquely col-ored to aid in understanding what is occurring duringthese steps. Also, the edges of the interior joints arecolored black, while the edges of the contact joints arecolored orange. As the sliding progresses, the contact

joints are deforming under shear considerably becausethey are undergoing slip. If the joints were to continueto deform, eventually the slide off condition discussedin section 3.3.2 would be satisfied and the contact jointswould be removed. It is not ideal for contact joints to beremoved and recreated from one time step to another inthe simulations when in reality there is always a smoothgradual change in the contact conditions; the stress stateof the newly formed contact joints would not carry overthe elastic shear deformation and normal loading of thepreceding contact joints, and thus the total normal andtangential loads would periodically fluctuate with dis-continuous jumps in the loads instead of remaining sta-ble. To overcome these issues, the concept of remeshingadjacent contact joints is introduced.

An example of the remeshing process can be seen inFig. 2 in the transition from step 2 to step 3. The yel-low and orange contact joints in step 2 are replaced withtwo new contact joints of a slightly different, desatu-rated, yellow and orange color. Similarly, the dark greenand blue contact joints in step 2 are replaced in step 3with the light green and light blue contact joints. Thechanges from step 2 to step 3 does not simply consist ofremoving the old contact joints and forming new onesaccording to the procedures in sections 3.3.1 and 3.3.2.Instead, the adjacent contact joints are instantaneouslyreplaced within a single time step and the reference con-figuration of the new contact joints is carefully selectedto maintain certain invariants which is discussed in sec-tion 3.5.3.

Fig. 2 also demonstrates how the remeshing proce-dure allows for long-distance sliding. The quadrilateralformed by the desaturated orange contact joint and lightblue contact joint in step 4, is roughly the same shape asthe quadrilateral formed by the dark green contact jointand dark blue contact joint in step 1. The two quadri-laterals also involve the same surface elements of thebottom surface. However, the surface elements of theorange-blue quadrilateral of step 4 involves a differentpair of top surface elements than the pair of top surfaceelements taking part in the green-blue quadrilateral ofstep 1. In fact, the top surface elements involved havebeen shifted over by one element to the left along theboundary edge due to the fact that the top surface issliding towards the right. This process could continue,leading to discrete elements in contact several steps laterthat were originally a very large distance apart. Andwhile the long-distance sliding takes place, the contactjoints could be continuously updated through remeshingso that at any given time there is always stable contact(represented through the contact joints) between the twobodies.

8

CASL-U-2014-0139-000

Fig. 2. Two surfaces in sliding contact causing the contact joints to be remeshed.

3.5.2. Remeshing conditionsRemeshing of skewed adjacent contact joints needs

to occur before the slide off condition has been satis-fied causing the contact joints to be removed. Of coursesometimes, it is desirable to allow the slide off condi-tion to be satisfied for a contact joint. For example,in Fig. 2, the red contact joint is adjacent to the yel-low contact joint in step 3. Then, from step 3 to step4, the red contact joint is removed due to the slide off

mechanism. In this case, this is desired behavior andit would not be desirable to attempt remeshing the redand yellow contact joints. Another example of whenremeshing would be undesirable is in step 1 of Fig. 2.The orange and yellow contact joints will eventually beremeshed right after step 2, but it is not desirable to pre-maturely remesh the orange and yellow contact joints instep 1 because they would become even more skewedthan they already. The desired behavior for remeshingcan be achieved by only remeshing if two conditions aresatisfied.

The first condition is a check to make sure thatremeshing would not create contact joints that overlap.For example, if an attempt was made to remesh the redand yellow contact joints in step 3 of Fig. 2, the result-ing new contact joints would actually overlap with eachother and with some of the interior joints of the bot-tom body. This is clearly unacceptable, so this potentialproblem is avoided by verifying that none of the inte-rior angles of the quadrilateral formed by the adjacentcontact joints have an angle greater than 180.

The second condition that must be satisfied is calledthe skewness condition, which requires that the shapeof the new contact joints after remeshing be less skewedthan the shape of the contact joints before remeshing.More precisely, the skewness of a contact joint is mea-sured by a metric called the shape factor. For a 2-D con-tact joint, the shape factor is the minimum of the cosinesof the angles of the contact joint. Less skewed contactjoints have larger shape factors. The shape factor has amaximum value of 0.5 for the least skewed contact joint,which has the shape of an equilateral triangle. So, if ad-jacent joints j1 and j2 are being considered for remesh-ing into new adjacent contact joints j3 and j4, the skew-ness condition is satisfied if SF j1 + SF j2 < SF j3 + SF j4 ,where SF j is the shape factor of contact joint j.

3.5.3. Invariants

When it is appropriate to remesh adjacent contactjoints, the new contact joints that replace the previousones each require a reference configuration. The refer-ence configurations are selected to maintain certain in-variants that must hold throughout the remeshing pro-cess. These invariants come from the conservation oflinear/angular momentum and the conservation of en-ergy; the conservation of mass is automatically satis-fied because the discrete elements are not created or de-stroyed.

Remeshing can be better understood by consideringthe example of remeshing adjacent contact joints j1 andj2 into new contact joints j3 and j4. Contact joints j1and j2 form a quadrilateral in 2-D with elements e1, e2,e3, and e4 at the vertices of the quadrilateral. The pathfrom e1 to e2 to e3 to e4 and back to e1 turns in a counter-clockwise orientation. The edge vectors of the quadri-lateral are known and can be calculated from the knownvectors s j1

1 , s j12 , s j2

1 , s j22 . The four forces acting on the

elements of the quadrilateral due to the elastic contri-butions of contact joints j1 and j2 only are also knownfrom (13). For linear momentum to be conserved, thenet force on each element after remeshing should re-main the same as it was prior to remeshing. This meansthat the four forces acting on the elements of the quadri-lateral due to elastic contributions of only contact jointsj3 and j4 should be the same as the four forces acting onthe elements due to elastic contributions of only contactjoints j1 and j2. This imposes 8 constraints (four 2-Dforces) on the reference configuration vectors w j3

1 , w j32 ,

w j41 , and w j4

2 because the edges of joints j3 and j4 (s j31 ,

s j32 , s j4

1 , s j42 ) are fixed to satisfy the known edge vectors of

the quadrilateral. Two of those 8 constraints are redun-dant due to the conservation of linear momentum andone additional constraint is redundant due to the con-servation of angular momentum. So, the forces on theelements of the quadrilateral impose only 5 constraintson the reference configuration vectors that need to besolved. The reference configuration vectors have a totalof 6 degrees of freedom in 2-D (one of the four refer-ence configuration vectors is redundant because two ofthe four reference configuration vectors will always beequal and opposite). The additional constraint needed

9

CASL-U-2014-0139-000

to solve the problem comes from the conservation of en-ergy. The elastic strain energy in the quadrilateral priorto remeshing must equal the elastic strain energy in thequadrilateral after remeshing, or U j1 + U j2 = U j3 + U j4 ,where U j is the strain energy of joint j.

So, satisfying the invariants during the remeshingprocedure requires solving a system of six non-linearequations for six unknowns. The unknowns are the in-dependent variables defining the referencing configura-tions of the two new contact joints j3 and j4. Five of thesix equations satisfy conservation of momentum, andthe sixth equation satisfies conservation of energy. Theequations are non-linear because of the non-linear de-pendence of the forces and strain energy density to thereference configuration, as can be seen through equa-tions (1), (11), (12), (13), (14), and (15).

3.6. Time integration scheme and boundary conditions

With the forces on the elements known from (6),(9), and (13), the positions of the elements pe(t) canbe evolved using explicit time integration according toNewton’s equation of motion fe = me

∂2pe∂t2 . The current

implementation of the model uses an explicit linear mul-tistep method for the time integration, in particular thefourth-order Adams–Bashforth method.

As mentioned before, traction boundary conditionsare simulated by assigning an appropriate body force tothe surface discrete elements that are part of the bound-ary that the traction would be applied to. For example,a uniform traction of t applied to an edge of length Lconnecting elements e1 and e2 is satisfied by applying abody force of 1

2 Lt to each of the elements e1 and e2.Displacement boundary conditions are easy to con-

trol because the simulation tracks the positions of thediscrete elements. The simulation allows for varioustypes of displacement boundary conditions. Some ofthe discrete elements can be selected to be restrained intheir motion in some way. For example, a roller supportcan be simulated by preventing a change in the positionof an element in one direction while allowing for move-ment in the perpendicular direction. This boundary con-dition is simulated in simulations by modifying the ve-locity vector to be consistent with the boundary condi-tion. Another way the simulation allows handling dis-placement boundary conditions is by directly control-ling the position of some discrete elements over time.This boundary condition is simulated by modifying thecurrent configuration of discrete elements at each timestep to be consistent with the imposed displacements.

4. Applications/Validation

The capabilities of this model, such as elasticity, frac-ture, contact, friction, adhesion, long-distance sliding,and tracking third-bodies, are demonstrated with twosimple simulations. The first simulation consists of twoelastic circles coming into contact and sliding againsteach other to demonstrate the capabilities of contact,friction, and long distance-sliding through the use ofremeshing. The second simulation demonstrates a typ-ical wear mechanism: an asperity coming into contactand breaking off due to fracture from the elastic strain-ing in the asperity and becoming a new wear debris par-ticle that is properly tracked in the simulation. Then, thecase of a block sliding down an inclined plane is studiedto act as a partial validation on some of the capabilitiesof the model. For all of these simulations, the materialparameters (which are typical values for zirconia oxide)are: Young’s modulus E = 200 GPa, Poisson’s ratio ν =

0.3, fracture toughness KIc = 13 MPa m1/2, and densityρ = 6 g cm−3.

In the visualizations of the simulations that are shownin the figures, the discrete elements themselves, whichare located at the vertices of the triangles representingthe joints, are not shown. However, only for discrete el-ements at the surface, the effective sizes of the elementsare shown with blue circles centered at the location ofthe elements. Triangles with black edges represent theinterior joints, and triangles with orange edges representthe contact joints. The contact joints have a preferredorientation that defines the contact plane, but the con-tact planes are not shown in the visualizations below.The σyy component of the Cauchy stress within eachjoint (both interior and contact) are visualized throughcolor plots.

4.1. Circles in contact

In the first simulation, two circular bodies are broughtinto contact with each other by holding the bottom halfof the bottom body fixed and driving the top half ofthe top body downward until a compressive contact isformed between the two elastic circular bodies, shownin Fig. 3a. The orange-bordered contact joint in thatfigure show the state of elastic contact between the twobodies. The number of contact joints increase as thecompression grows, which demonstrate the increasingarea of contact between the two elastic circles in con-tact.

The top half of the top circular body is then movedrigidly to the right to allow the circles to slide againsteach other with friction. Fig. 3b show how the contactjoints have deformed under shear to represent sliding

10

CASL-U-2014-0139-000

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σy

y (GP

a)

−30

−20

−10

0

10

20

(a) Snapshot at t = 1.488 ns

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σ

yy (G

Pa)

−30

−20

−10

0

10

20

(b) Snapshot at t = 3.537 ns

Fig. 3. Snapshots of the simulation of circles in contact.

under friction. Some of the shear is carried elasticallyby the contact joint until a limit, after which point thefurther deformation of the triangle is due to slipping. Alittle later in the simulation, see Fig. 3c, the deforma-tion of the contact joints has become so severe that anautomatic remesh occurred to allow for further sliding.

As sliding continues, the contact joints move alongthe surface of the circular body but because the two bod-ies are getting further apart with additional sliding, thecompressive loading in the contact joints is reduced, asseen in Fig. 3d, until eventually tensile loading exceed-ing the adhesion threshold develops normal to the con-tact plane of the contact joints resulting in the contactjoints disappear and thus causing the two circular bod-ies to pull apart, as seen in Fig. 3e.

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σy

y (GP

a)

−30

−20

−10

0

10

20

(c) Snapshot at t = 4.363 ns

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x−coordinate (µm)

y−

coo

rdin

ate

(µm

)

σyy (G

Pa)

−30

−20

−10

0

10

20

(d) Snapshot at t = 6.578 ns

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σy

y (GP

a)

−30

−20

−10

0

10

20

(e) Snapshot at t = 7.999 ns

Fig. 3. Snapshots of the simulation of circles in contact.11

CASL-U-2014-0139-000

4.2. Asperity contact and break-off

In the second simulation, a moving asperity of arough surface encounters an obstacle which causes theasperity to break off the surface. At the start of the sim-ulation, the top body is brought down to come into con-tact with the circular obstacle, which is held in placeduring the simulation, and then it begins moving right.In Fig. 4a, the contact between the asperity and the ob-stacle is modeled by the orange-bordered contact jointsthat form between the elements within the top body andthe element within the circular obstacle.

As the top body moves to the right, shown in Fig. 4,the impeding obstacle causes interior joints in the topbody to progressively fracture from elastic strain untileventually the entire left asperity breaks off from therest of the body. The fragmented asperity is then a third-body wear debris particle that is free to move within thesimulation under its dynamic forces. In fact, there arefive third-body particles in Fig. 4c: one is the large as-perity fragment, the second is a small triangular bodymade from three discrete elements, and the remainingthree third-body particles are each single discrete ele-ments that are adhering to the surface of the top body.

4.3. Block sliding down inclined plane

The simulation of a block sliding down an inclinedplane under the force of gravity provides a partial val-idation of some of the capabilities of the model. Inparticular, the simulation needs to accurately model thecontact and friction between the block and surface onwhich the block is sliding. Long-distance sliding re-quires the remeshing capabilities of the model to main-tain a smooth always present contact between the blockand plane. The friction model of the contact joints needsto be capable of accurately modeling the transition fromstick to slip so that the block does not start sliding untilthe force pulling on the block exceeds the static frictionforce.

In order to provide the validations desired, we firstset up the geometry of a block resting on an inclinedplane. A union of the geometries of the plane inclinedat an angle of 15° and the block on top of it, is first usedin a energy minimization simulation to find the staticequilibrium state of the body under the force of gravity(g = 98 nm ns−2). The minimization of the potential en-ergy is done through steepest descent using the gradientof the energy (the forces on the elements) until the crite-ria on the forces is satisfied such that the system reachesa state close to static equilibrium (at least enough to re-duce severe bounces in the later dynamic simulation).

−2 −1 0 1 2−1

−0.5

0

0.5

1

1.5

2

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σyy (G

Pa)

−50

−40

−30

−20

−10

0

10

20

30

(a) Snapshot at t = 4.098 ns

−2 −1 0 1 2−1

−0.5

0

0.5

1

1.5

2

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σy

y (GP

a)

−50

−40

−30

−20

−10

0

10

20

30

(b) Snapshot at t = 5.356 ns

−2 −1 0 1 2−1

−0.5

0

0.5

1

1.5

2

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σyy (G

Pa)

−50

−40

−30

−20

−10

0

10

20

30

(c) Snapshot at t = 6.988 ns

Fig. 4. Snapshots of the simulation of asperity contactand break-off.

12

CASL-U-2014-0139-000

0 1 2 3 4

−0.5

0

0.5

1

1.5

2

2.5

3

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σyy (G

Pa)

−12

−10

−8

−6

−4

−2

0

Fig. 5. State of block and inclined plane geometries un-der gravity after relaxing with energy minimization.

Fig. 5 shows the state of the system after the minimiza-tion procedure. The interior joints in the interface be-tween the block and the plane are then converted intocontact joints with the same reference configuration tokeep the same state of stress, which separates the blockand the inclined plane into two different bodies readyfor the dynamic simulation.

Simulations are run in order to identify two regimesof sliding: stick and slip. The block is expected toslide when the static coefficient of friction is less thantangent of the angle of the incline. Instead of alteringthe angle of the inclined plane, which would require are-minimization procedure, the coefficients of frictionare modified for various dynamic simulations. Throughthese simulations we were able to identify that the tran-sition between stick to slip occurred when the static co-efficient of friction µs < tan(15) ≈ 0.268, as expected.The details of two particular simulations, one for stickand one for slip, are included below.

The dynamic simulation of the block on an inclinedplane with a static and kinetic coefficient of friction ofµs = µk = 0.28 is used to demonstrate sticking. Atthe start of the simulation, there is a very slight down-ward movement in which a new leading contact joint iscreated, but then the block stops moving and remainsfixed for the rest of the simulation, see Fig. 6. Thetangential and normal forces on the contact surface ofthe block, seen in Fig. 7, show how the forces matchwhat is expected from a simple free-body diagram anal-ysis of a rigid block on an inclined plane. The tan-gential force of friction actually increases from a lowervalue to the expected result during the beginning of thesimulation which coincides with the initial downwardmovement noticed in the dynamic simulation. The nor-

0 1 2 3 4

−0.5

0

0.5

1

1.5

2

2.5

3

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σyy (G

Pa)

−20

−15

−10

−5

0

Fig. 6. Block on inclined plane under a state of stick.

0 2 4 6 8 10 121

2

3

4

5

6

7

8

9

time (ns)

forc

e per

unit

dep

th (

mN

/µm

)

Normal force (from simulation)N = mg cos(θ) (expected normal force)Tangential friction (from simulation)µ

k N (with µ

k = 0.28)

Fig. 7. Forces acting on the contact surface of the non-sliding block on an inclined plane.

mal force slightly oscillates around the expected normalforce, most likely due to the initial configuration not be-ing perfectly in the static equilibrium state.

Another dynamic simulation, this time with a staticcoefficient of friction µs = 0.25 < tan(15) and a ki-netic coefficient of friction µk = 0.20, is used to demon-strate the case of the block sliding down the inclinedplane. The block immediately begins sliding down atthe start of the simulation. A snapshot of the simulationas the block is sliding down the plane can be seen inFig. 8. The tangential and normal forces on the contactsurface of the sliding block are similar to that of the non-sliding block, in that they both slightly oscillate aroundthe forces expected from a free-body diagram analysis.One important difference, however, is that the frictionforce on the sliding block is µkN as expected, not µsN,where N is the normal force. By calculating the timederivative of the position of the center of mass of theblock, the speed at which the block was sliding down

13

CASL-U-2014-0139-000

0 1 2 3 4

−0.5

0

0.5

1

1.5

2

2.5

3

x−coordinate (µm)

y−

coord

inat

e (µ

m)

σy

y (GP

a)

−20

−15

−10

−5

0

Fig. 8. Block sliding down an inclined plane under grav-ity.

the inclined plane was plotted versus time. The plot wasvery close to linear, and fitting a line to the data gave aslope of approximately 6.1 nm ns−2 for the line, which isthe acceleration of the sliding block. From a free-bodyanalysis of a rigid block on the inclined path, the net ac-celeration of the block is given by g(sin(θ)−µk cos(θ)) ≈6.4 nm ns−2, which is less than a 5% difference from themeasured acceleration from the simulation results.

5. Discussion

As discussed in the introduction, finite element sim-ulations of fretting have been coupled with Archard’slaw or other work-rate type models to estimate the wearrates over the contact interfaces. Some of these ap-proaches have also adjusted the geometry of the wearingsurfaces according to the estimated wear rates by chang-ing finite element node positions and using adaptiveremeshing. While these approaches work fine, they donot consider the presence of discrete third body wear de-bris at the contact interface. Adding in additional bodiesmeshed using finite elements to interact through contactin the wear process presents challenges and enormouscomplexity for the typical contact algorithms used inthe finite element method (FEM). All surfaces that maypossibly come into contact need to be defined prior tostarting or continuing FEM simulations. However, ina wear problem, new surfaces will be generated due towear as the simulation progresses. So, it is not knowna priori which finite elements will be participating incontact with other finite elements.

The model presented in this paper is able to dynami-cally update the set of entities that can be in contact over

time. Contact is handled using the dynamically createdand destroyed contact joints between discrete elements.There is no master/slave contact algorithm between tri-angular (in 2-D) geometries which need to be marked ascapable of contact with other triangular geometries aswould be the case in FEM; instead all surface elementshave the potential to join together into a contact jointwith nearby neighboring surface elements. This flexi-bility allows the model to handle contact between thetwo fretting surfaces, between third bodies and the fret-ting surfaces, and between any number of third bodies,regardless of whether the third body or fretting surfaceexisted in its current form at the start of the simulationor if it was generated over time through the wear pro-cess.

This sort of flexibility in contact is also present in thediscrete element method (DEM). An easy, scalable con-tact algorithm is important in the types of problems typi-cally used in DEM simulations, such as the flow of gran-ular materials. However, there are considerable limita-tions with DEM that make it unsuitable for fretting wearsimulations. The most important is the fact that the par-ticles modeled by DEM are rigid. Without the abilityto model stress state in say a third-body modeled usingthe discrete elements of DEM, there is no accurate wayof modeling the fracture of the wear debris particle intomultiple smaller particles. This capability is immenselyuseful in simulating problems like comminution, but itsalso very important in fretting wear. Furthermore, if thefretting surfaces were also modeled using rigid DEMelements then fracture of asperities would also not bepossible. The other major issue with the capabilities ofDEM is that its contact model is too simple. Contactin DEM between the rigid particles is typically mod-eled using a penalty method or Hertz contact and mayeven include enhancements such as adhesion, see [12].However, there is no clear way of adding friction be-tween contacting elements in DEM that could allow themethod to accurately simulate the problem of the blocksliding down an inclined plane discussed in this paper.

A hybrid approach between DEM and FEM may helpresolve some these issues. In fact, the method presentedin this paper might be considered to be a conceptualhybrid between DEM and FEM. However, the hybridmethod typically used in the literature is the combinedfinite-discrete element method developed by Munjiza[27], in which the discrete particles typically modeledin DEM are discretized by a finite element mesh. Whilethis method would allow for accurately computing thestress state in both the fretting bodies and the third bod-ies so that the fracture of wear debris and asperitieswould be possible, it still shares the same problems of

14

CASL-U-2014-0139-000

DEM in terms of having difficulty accurately modelingfriction between the discrete particles. Furthermore, us-ing traditional FEM contact algorithms in this hybridmethod would remove the advantages of an easy DEM-like contact algorithm, resulting in the previously dis-cussed problems of FEM contact modeling of a third-body layer with many dynamically-generated discretewear debris particles.

The small simulations in this paper merely providea demonstration of the capabilities of the model. Themodel is flexible enough to allow asperities on the roughsurface of a fretting body to fracture and become third-body wear debris which can continue to interact throughcontact and friction with all of the other bodies in thesimulation. Also, the problem of the block sliding downthe inclined plane shows how the same contact and fric-tion algorithm in the model is capable of simulating theblock sliding down the plane that is consistent with re-sults expected from a simple free-body diagram analysisof a rigid sliding block with a Coulomb friction model.The acceleration from the simulation is within 5% ofthe acceleration calculated from the analytical result de-rived from the simplified free-body diagram analysis. Inparticular, the acceleration from the simulation resultsare less than analytically calculated acceleration, whichis expected because there are slight vibrations duringsliding and that vibrational energy is transferred into thebulk of elastic block, whereas in the free-body analysisthe block is rigid and the sliding is considered to be per-fectly smooth.

While the results from sliding block problem are con-sistent, further friction validation studies are needed tobetter validate the friction and contact capabilities ofthis model. The study of Hertz contact between an elas-tic circular body and a flat with an oscillating slidingmotion can be another great validation study becauseanalytical expressions are known for parameters like thesize of the stick and slip region, the pressure and sheardistribution over the contact surface, and the frictionalwork done per fretting cycle [28].

The other capabilities of the model, such as fracture,need validation as well. A simulation of comminution,in which brittle materials are successively crushed andgrinded together, can provide some validation for thefracture and contact capabilities of this model. Resultsfrom the simulation such as the size, shape, and num-ber of particles from the process can be compared withresults from comminution experiments.

Finally, the purpose of this model is to apply it to fret-ting wear simulations to better understand the frettingmechanisms and how they change with different operat-ing conditions. So, even in 2-D, simulations of fretting

wear between two rough surfaces, similar in concept tothe asperity fracture simulations presented in this paperbut at much larger size, can be conducted to study themechanisms of fretting wear of brittle materials. Param-eters that can be varied in the set of simulations includenormal load, slip amplitude, surface roughness, pres-ence of an initial third-body wear debris layer of differ-ent compositions (number of particles, and size/shape ofparticles), oscillation frequency, and variants in the fret-ting motion such as a combined sliding and impact mo-tion. Through qualitative analysis of the wear processand quantitative measurements of the amount of wear inthe simulations, an understanding of the dependence offretting wear behavior on the operating conditions canbe obtained and a fretting wear mechanism map can becreated.

6. Conclusion

A model for simulating fretting wear of brittle materi-als at the micron length-scale is proposed which has thecapabilities of generating wear debris and handling con-tact between the any bodies present in the dynamic sim-ulation, including new bodies generated during the evo-lution of the simulation due to fracture of asperities. Ex-perimental results have shown work-rate wear modelswith constant wear coefficients are insufficient for mod-eling fretting wear under the variety of operating condi-tions that may occur. The model proposed in this papercan be used to better understand how the wear coeffi-cient changes with some relevant operating conditionsby studying how changes in the operating conditions af-fect the mechanisms of wear in the simulations. Resultsand analysis from future simulations using this modelcan provide tools, such as a wear mechanism map, toimprove wear rate estimates in fretting wear problemslike the case of grid-to-rod fretting in nuclear fission re-actors.

Acknowledgements

We acknowledge Professors Ken Kamrin and DavidParks at the Massachusetts Institute of Technology forhelpful discussions and guidance. Also we gratefullyacknowledge the Consortium for Advanced Simulationof Light Water Reactors (CASL), a US Department ofEnergy innovation hub, for sponsorship of this effort.

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