Sliding of rough surface and energy dissipation with a 3D multiscale approach

8/7/2019 Sliding of rough surface and energy dissipation with a 3D multiscale approach

http://slidepdf.com/reader/full/sliding-of-rough-surface-and-energy-dissipation-with-a-3d-multiscale-approach 1/17

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2010; 83:1255–1271Published online 18 February 2010 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2845

Sliding of rough surfaces and energy dissipation with a 3Dmultiscale approach

G. Anciaux and J. F. Molinari∗,†

Ecole Polytechnique F ed erale de Lausanne ( EPFL), Facult e ENAC-IIS , Laboratoire de Simulation en M ecanique

des Solides, CH-1015 Lausanne, Switzerland

SUMMARY

The purpose of this paper is to present a novel approach to study sliding friction by the use of 3Dmultiscale coupling techniques. We highlight the difficulties of using multiscale schemes that couplemolecular dynamics (MD) with finite elements to perform sliding contact between rough surfaces.The paper is comprised of systematic comparisons between the coupled model with an equivalent MDdomain as well as reduced models. Reduced models appear inadequate for reproducing the time evolutionof frictional forces as the small depths yield a stiffening of the contact region. The overlap domain used inthe multiscale scheme is successful at capturing the complex deformation dynamics beneath the surface.Yet our results call for further developments in multiscale coupling strategies to allow a more rigoroustreatment of heat exchanges through the overlap region. Copyright q 2010 John Wiley & Sons, Ltd.

Received 31 August 2009; Revised 7 December 2009; Accepted 8 December 2009

KEY WORDS: multiscale; concurrent coupling; surface roughness; molecular dynamics; sliding;

dissipation

1. INTRODUCTION

Frictional forces are important to many biological functions and to almost all industrial applications.

With the recent development of nanotechnologies, which implies that surface mechanisms tend to

dominate bulk behavior, contact mechanics and its by-product friction are taking an ever increasing

importance. Perhaps surprisingly, considering that Amonton and Coulomb proposed a model for

friction centuries ago, there is still no consensus on the origins of frictional forces. Different factors

are thought to play a role: these include asperity locking, plastic activity, third body interactions,

∗Correspondence to: J. F. Molinari, Ecole Polytechnique Federale de Lausanne (EPFL), Faculte ENAC-IIS, Laboratoirede Simulation en Mecanique des Solides, CH-1015 Lausanne, Switzerland.

†E-mail: [email protected], URL: http://lsms.epfl.ch

Contract/grant sponsor: Swiss National Foundation; contract/grant number: 200021 122046/1Contract/grant sponsor: European Research Council; contract/grant number: 240332

Copyright q 2010 John Wiley & Sons, Ltd.



1256 G. ANCIAUX AND J. F. MOLINARI

lattice dynamics, heating and melting, and possibly others. Numerical simulations may become,

in the years ahead, a unique tool to understand the parameters that influence friction. One may

envision a virtual testing facility, in which the role played by surface topography and material

parameters on friction and ultimately wear could be investigated.

Recent FEM simulations [1] as well as atomistic studies [2] show that contact mechanics isdominated by nanoscale asperities. Continuum mechanics is unable to capture the details of force

profiles at this scale. In order to represent efficiently the atomic organization and forces at contact

clusters, one can resort to molecular dynamics (MD). However, MD alone is arguably not an

appropriate method as the computational cost is too high to evaluate long-range elastic forces and

heat flow away from the contact interface. The limitations of purely atomistic or purely continuum

simulations have motivated research in multi-scale simulations that bridge atomistic and continuum

models [3–26]. The present paper analyzes the problem of contact mechanics using the overlapping

scheme presented in [26, 27].We begin the paper, in Section 2.1, with a description of the main ingredients of a novel

3D numerical multiscale approach to simulate sliding friction at the nanoscale. The numerical

framework is composed of a Finite Element program, for representing realistic sample sizes, a

MD package, for capturing accurately the atomic interactions at the contacting asperities, and a

coupling scheme between the two physical models. In 3, we then describe the results obtained

for sliding contact between rough surfaces. We discuss the importance of using a large coupling

zone underneath the contact region to obtain the appropriate substrate compliance. Preliminary

measures of friction coefficients are presented. Our simulations also reveal the sensitivity of the

coupling zone to thermal fluxes, which are constantly being generated during sliding contact.

We point out current deficiencies of coupling schemes, which motivates future development of a

fully thermo-mechanical coupling zone.

2. MODEL PREPARATION

2.1. Description of the multiscale model

The purpose of this section is to describe the MD properties and coupling scheme relevant to

the present work. All initial states will consider a perfect copper crystal. The considered lattice

is face-centered cubic (FCC) with dimension a=3.615 A. The mass of each copper atom is

mcu=63.55 g/mol. The inter-atomic potential used to drive atoms through MD is the Lennard

Jones potential [28, 29]. The expression of the associated global potential energy is

E tot =1

2

i, j=i

V (r i j ), (1)

V (r )=

r

12−

r

6, (2)

where and have been computed to fit a Young’s modulus of 106 GPa and a Poisson ratio of

0.3. The cutoff radius has been chosen so that atoms interact only with their first neighbors at a

distance of r 1=a√

2/2=2.55619 A.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2010; 83:1255–1271

DOI: 10.1002/nme



SLIDING OF ROUGH SURFACES AND ENERGY DISSIPATION 1257

Atomic Zone

l(X)

R

Bridging Zone

Continuum Mechanics Zone

Figure 1. Planar coupling zone. Atomic domain is on the left side, whereas continuum mechanics domainis on the right side. The bridging zone is used to define the weighting function ( X )= l( X )/ R.

We use the well-known linear momentum balance expression for a continuum with the

Lagrangian formulation and no body forces:

∇ ·P =0v, (3)

where P is the first Piola–Kirchhoff stress tensor, 0 is the mass per unit of volume at initial time

and v is the velocity field. As a constitutive law for the copper crystal we use the Cauchy–Bornrule (CBR) [6] and linear elasticity. The CBR assumes an homogeneous deformation of the crystal

lattice to compute the Piola–Kirchhoff tensor.

The implemented coupling method is based on an overlapping region R ( Bridging Zone) where

the continuum and the atomistic models are both valid. They are combined by using a weighting

function for the energy. The two models are glued together by constraining the degrees of freedom

in order to have coherent dynamics. Figure 1 shows a typical planar coupling in 2D. The overlap

allows us to change the predominance of each model so that the constraint can take effect gradually

when the overlap is crossed and consequently minimize the effects of wave reflections.

More precisely, a global Hamiltonian is defined as the weighted sum of the two sub-Hamiltonians

referring to the two coupled domains and a constrained MD algorithm is used to ‘glue’ their

displacement fields. If there are L atoms in the Bridging Zone, then we introduce L constraints

{gi } Li=1 as

gi =U(Xi )−di = J

J (Xi )u J −di =0, (4)

where di is the displacement of atom i , U(Xi ) is the macroscopic displacement evaluated by

interpolation at atom i and u J is the nodal displacement at node J .


DOI: 10.1002/nme




The initial method [19] has been modified by Anciaux et al. [27] to improve equilibrium

properties of the coupling model. Concretely, the modified scheme is

M I ¨u

I =−f I +

1

( X I )

L

k =1

kk

*gk

*u I

,

mi di = f i +1

( X i )

Lk =1

kk

*gk

*di

,

where M I is the mass lumped on node I , mi is the mass of atom i . The computation of the

multipliers =(k ) is done by solving the linear system

H =g,

where H is a L× L matrix and g is the vector of dimension L defined by

H i k =t

J J (Xi )( X J ) M J

*gk

*u J

− 1( X i )mi

*gk

*di

,

gi = J

J (Xi )u J −d

i ,

where u,d are the displacements obtained after one time step by the integration scheme when the

constraints are not applied. Actually, if the integration in time of the unknowns follows the above

scheme, no dissipation occurs at all. But then the wave reflection treatment of the method remains

particularly poor as demonstrated in [27]. An option is to use a lumped form of the constraint matrix

H . This brings the coupling interface to reflect much less the incoming high-frequency waves.

But the price to pay stands in an energy dissipative behavior of the coupling interface. Indeed,

a coarse mesh has no possibility to represent high-frequency waves with a lower node density

than the atomic one. A natural way of preventing wave reflections is then to dissipate energy.The exact description of this dissipative effect as well as details on how to set correctly important

parameters—such as element sizes in the coupling interface—can be found in [19, 26, 27]. In the

presented work, this lumping approach has been used.

2.2. Fractal surfaces

To shape two realistic rough surfaces, we used the same approach as in [30]. We constructed two

cubic-like copper crystals from which free surfaces have been cut following a self-affine fractal

[31] generated with a Voss [32, 33] algorithm for a Hurst exponent of 0.7 for each of the surfaces.

A random number generator assures that the two surfaces have non-matching profiles. A schematic

of the domain as well as a 3D representation are shown in Figures 2 and 3. The main sample

length L is 64a, making an apparent contact surface A0 of L× L=4096a2

. We created two atomiczones, initially clearly separated by an important distance. The top one, dedicated to be the rigid

body rough indenter, contains 211 806 atoms. The bottom one, which is the deformable body that

will be coupled to a finite element model, contains 821 318 atoms. In all the following, the atoms

contained in the top region will be recalled as ‘top atoms’, whereas the other ones will be recalled

as bottom atoms. The finite element domain contains 41 472 elements and 7681 nodes and overlaps

the MD region to couple 548 864 atoms. The main purpose of this region is to couple long-range


DOI: 10.1002/nme




Figure 2. Schematic of the coupled model designed for the study of rough surface contact.

elastic fields and damp kinetic surplus to simulate flux away from the contact region as it will be

demonstrated later. The total height of the deformable body is 150a. The simulations at a lower

computational cost of this relatively large depth is made possible, thanks to the coupling region.

Before using the constructed domain, because of the free surface energy, we proceeded to a

relaxation of the global domain, including the top body considered for now as a deformable body.

The boundary conditions during this relaxation were zero displacement for atoms at the top surface

and for the nodes at the bottom surface of the continuum domain.

2.3. Sliding under constant normal pressure

For the rest of the study, the top body is imparted as a rigid body motion. In addition, we wish to

enforce a constant pressure at the contacting asperities. Therefore, the center of mass of the top

body moves according to a force composed of the resultant force due to top atoms’ interactions

with bottom atoms, and of the contribution from the pressure applied on top:

F =at

f at−P · L2, (5)


DOI: 10.1002/nme




Figure 3. 3D View of the mesh and atomic zone coupled together.

where P is the desired pressure, f at is the initial per atom force issued from standard MD force

computation. We then ‘distribute’ such a force over all the atoms so that the resulting atomic

acceleration becomes uniform and follows:

∀i ∈TopAtoms, ai =

at f at−P · L2

mr

, (6)

where mr is the mass assigned to each top atoms. We reduced arbitrarily the mass of all the top

atoms so that the contacting force is transmitted with reduced inertia effects. We used mr =mcu

with mcu being the mass of a single copper atom. With 1 the rigid body is light enough to

move freely, therefore, enforcing an almost constant pressure at the contacting asperities. Results

demonstrating the robustness of this scheme will be shown in the next section. In this paper, threedistinct pressures are investigated: 100, 200 and 300 MPa.

Before proceeding with sliding, we also perform an energy minimization procedure to let the

bottom surface and its asperities deform to sustain the applied pressure. When equilibrium is

found, the sliding motion can be initiated. We then move the rigid body in a mixed pressure–

displacement-controlled manner so that the loaded body does not experience unrealistic stresses

at asperities while controlling the sliding speed. In other words, each top atom sees a force in the


DOI: 10.1002/nme




Z direction, which is controlled by the pressure loading procedure outlined above, and is being

displaced in directions X and Y in accordance to the sliding direction and velocity. The sliding

speed is chosen to be relatively high, at 10 m/s, so that simulation runs are reasonably short.

3. RESULTS

We proceed with the sliding motion at three distinct pressures. Before going deeper in the analysis

of the effect of the coupling zone, we set to check whether or not our strategy to maintain a constant

pressure is successful. In post-treatment, we could extract the mean pressure acting on asperities

(see section 3.2), which was found to match exactly the required values. Also the standard deviation

from the mean values was computed to evaluate the fluctuation of the pressure. Figure 4 presents

these measures for the various cases we computed. The figure exhibits a maximum deviation of

less than 4%, which is acceptable. Also, these simulations can be used to extract the contact area

evolution as presented in Figure 5. In the following, in order to provide an objective characterization

of the coupling method being used for sliding contact, we will be comparing four different models(see also previous work on normal contact [30]):

• Full-MD model (complete model is represented by atoms and assumed as a reference result).

• Coupled model (described above, see Figure 2).

• Reduced model (the sample depth is limited to 56a, i.e. the bottom of the atomistic zone, and

is entirely filled with atoms).

• Reduced with kinetic energy damping (Langevin thermostat at 0 K: no random force).

These various models allow comparisons between cases in which the FE domain has been replaced

by true atoms (Full MD) and in which the FE zone has been eliminated altogether (reduced

models) and replaced by more or less sophisticated boundary conditions on the bottom atoms

(fixed displacement and damping). The goal here is to evaluate, through the various measures and

indicators available, the importance of using a coupling method that extends the compliance of the deformable body and dissipates thermal energy otherwise trapped in the contacting region.

3.1. Kinetic/thermal energy

One important component of concurrent coupling methods in the case of dynamical simulations

is their ability to treat wave reflection problems at the coupling interface [6– 8]. This is clearly

a drawback of these methods, especially in the case of contact problems, since it creates an

overheated situation in the contacting region. As we will see it in the next section, this exerts an

influence on the frictional forces measured at the interface. To quantify completely the treatment

brought by the coupling with regards to high-frequency mechanical waves (which we may loosely

call thermal waves), we measured the kinetic energy of the top zone of the deformable body (e.g.

this contains the energy contained near the atomic asperities, but with no contribution from theatoms of the overlap region).

Figure 6 presents these measures for the Full-MD, coupled and reduced cases. Clearly the

coupled approach always leads to the minimal residual kinetic energy, whereas the reduced cases

tend to store a lot of vibrational energy in the contacting zone. A natural possible overcome, as

presented earlier, is to use an artificial energy damping to recover an energy state sufficiently close

to the full-MD solution. Unfortunately, we will demonstrate that this simple idea is not completely


DOI: 10.1002/nme




Figure 4. Standard deviation of the pressure measurements for the three desiredapplied pressures (100, 200 and 300 MPa).

Figure 5. Evolution of contacting clusters with time. The sliding direction is along the X axis (from left to right). Gray points are contacting atoms at the onset of sliding,

whereas black points are contacting asperities after 20A of sliding.


DOI: 10.1002/nme




Figure 6. Residual kinetic energy in the zone near contacting asperities (thickness 24a) forthe full-MD (plain line), coupled (dashed line) and reduced cases (small dashed line) and for

three different applied pressures.


DOI: 10.1002/nme




Figure 7. Residual kinetic energy in the zone near contacting asperities for the full-MD(plain line), coupled (dashed line), reduced (small dashed line) and damped (gray) cases for

an applied pressure of 100 MPa.

satisfactory, and that a complete domain is needed not to distort the time evolution of the sliding

contact simulation.

For damping, we used a thermostat to constraint some atoms at 0 K but in a non-brutal way. More

precisely a Langevin thermostat at 0 K was applied to the region of the reduced domain overlapping

with the bridging zone. It amounts to introducing an additional force on the corresponding atoms

of the form

F Langevin=−v+(t ), (7)

where is a friction coefficient and where the time dependence of denotes random time fluctu-

ations. Also, is set to have the following proportionality:

∼

k b T m

, (8)

where is a time constant and T the temperature. Since we constrain our domain to 0 K, the

random force vanishes. In Figure 7 we present the results issued with the values 0.5(m/t )

and 0.2(m/t ) for . Perhaps counter intuitively, we observe, Figure 7, that increasing (e.g. the

friction coefficient in the damping algorithm) increases the amount of kinetic energy trapped in

the atomistic domain. This trend has been confirmed for various values of , which are not shown

in Figure 7 for conciseness. A high friction coefficient leads to wave reflections at the frontier of

the damping domain and does not completely solve the problem of overheating observed in the

undamped reduced model. It is interesting to note that at the onset of sliding, the thermostated

models generate a very rapid increase of kinetic energy (see bumps at the dimensionless slidingdistance of 2 in Figure 7). This increase comes from the first reflections of waves at the thermostated

region. Another problem of using our simple thermostat (and especially for the small values of

) is the subsequent flattening of the rate of creation of kinetic energy in the atomistic domain,

whereas the full-MD model displays a linear increase. Therefore, the use of a simple thermostat

seems not sufficient to recover the heat creation and diffusion process that can be observed in the

full-MD case.


DOI: 10.1002/nme




Yet, it is noteworthy that the coupling scheme also fails in recovering this behavior. For the chosen

finite element size in the coupling domain (which is quite large, here h5.3a), the kinetic energy

profile remains almost flat and at a low value in the contacting region, distinctly different from

our reference, pure MD, case. Interestingly, most work in the literature has sought to prevent wave

reflections without necessarily considering that the damping of the problematic (high frequency)waves could impact the uncoupled zones. Indeed, the Bridging Domain method, when handling

properly the undesired high-frequency waves incoming from the molecular domain, is damping

a part of the kinetic energy in an ad hoc way [26, 27]. In previous work [30], we studied the

normal loading of a rigid spherical indenter over a flat deformable surface. The obtained results

showed that the kinetic energy profile of the coupled simulation was quite consistently fitting the

reference full-MD one, and it was postulated that the ad hoc damping in the coupling zone is

convenient for most isothermal processes. The sliding simulations were put to the test coupling

methods in a way that was not foreseen in simpler loading configurations. Here, with an initial

state of 0 K, and with asperities of various sizes and shapes, colliding and scratching at contact

points, thermal vibrations are being generated at an important rate. The resulting heat increase is

an integral part of the contact problem and should not be damped entirely by the coupling zone.

Our current implementation of the coupling zone acts, for large finite elements, as a very efficient

0 K thermostat. This will have to be addressed in future developments of coupling methods as

discussed in the conclusion. An exact match with the full-MD reference domain might call for a

thermo-mechanical multiscale model.

3.2. Frictional forces measurements

In order to evaluate the time evolution of frictional forces, we take advantage of the clear separation

between the rigid body and the deformable body. With both contacting objects being deformable,

the contacting interface would be difficult to determine (due to asperities pull out and materials

mixing occurring at the interface). We have also truncated the adhesive part of the Lennard Jones

potential to prevent adhesion between top and bottom atoms, precluding bottom atoms to remainattached to the rigid body. Therefore, in the present simulations, with the rigid body remaining

fully intact, the forces exerted from the bottom atoms on the top atoms can be easily computed.

The sum of these atomic forces gives the components of the frictional force. The ratio of the total

tangential force in the sliding direction to the normal applied force yields an instantaneous measure

of the friction coefficient exerted by the deformable body on the rigid one. Figure 8 presents the

computed instantaneous friction coefficient in the case of a pure MD simulation (full MD) and a

pressure of 100 MPa. One can see clearly that this measure is quite noisy. A notable point is that

the fluctuations follow a periodicity that can be linked to the lattice spacing.

This noise makes the comparison of the various models difficult. Therefore, we will now consider

the smoothed friction measure by means of a windowed average operator. The size of the window

is arbitrarily fixed to 10 measure points corresponding to a physical time of 100000 time steps

or 10 A of sliding distance. Figure 9 presents the smoothed friction measurements for 0.1 and0.3 GPa imposed pressures. The provided curves show the results obtained for the full MD, the

coupled and the reduced cases. An important point is that the simulations are far from being in a

steady-state sliding motion. The rigid body is more closely performing a scratch test than a sliding

friction test. This can be seen in the large fluctuations of the friction measure, and in the case of the

0.3 GPa applied pressure, in the late stage increase of friction, which we could relate to two large

asperities colliding with one another. If the sliding motion were not strictly enforced, as is the case


DOI: 10.1002/nme




Figure 8. Instantaneous friction coefficient measured during the full-MD simulation. Measurements weremade every 10 000 time steps and the applied pressure was 100 MPa.

Figure 9. Smoothed friction coefficient at pressure 100 and 300 MPa.

in our simulations, these frictional forces fluctuations might lead to stick slip motion. Nonetheless,

attaining a steady-state measure of friction would require a larger sample and a longer simulation

time. It could also be achieved by setting smaller initial roughness. This is out of the scope of

this paper, which is a first attempt to model 3D sliding contact within a multiscale framework.

These aspects are therefore left for future work.


DOI: 10.1002/nme




Figure 10. Mean friction measure for various pressures (0.08, 0.1, 0.2 and 0.3 GPa)and for all the tested methods.

A second interesting observation is that the deviations between the various cases tighten as the

pressure increases. For an applied pressure of 300 MPa, all models generate very similar friction

measurements. While for a pressure of 100 MPa, the three solutions are somewhat distinct, with

an important phase change. Nonetheless, one can observe that for this pressure the shapes of the

curves for the coupled and full-MD cases are more comparable. The curves show three peaks

instead of just two in the reduced case (the thermostat strategies introduced in the previous section

do not improve the reduced model and are not presented here).

A measure of the mean frictional resistance, which will be loosely referred to as mean friction,

is provided in Figure 10. Although the instantaneous friction coefficient varies strongly with time,

the time averages shown in the figure depict interesting trends. The average frictional resistanceappears to be roughly proportional to the amount of kinetic energy stored in the atomistic domain,

see Figure 6. It is remarkable that the mean friction coefficient of the coupled method is always the

highest one, whereas the reduced model with no damping generally generates the lowest resistance

to sliding. The main reason for this trend is thought to be thermal effects. As mentioned earlier,

see also [26, 27], in the coupled model the bridging domain damps waves that are not transferable

to the mesh because of the degrees of freedom elimination. This procedure avoids spurious wave

reflections and is adequate for totally adiabatic problems. In the sliding contact simulations, heat

is being constantly generated. Yet, the coupling acts as a 0 K thermostat, which cools down the

material in the contact zone thereby artificially increasing its stiffness. This explains the shift

in friction forces. It is noteworthy that reduced methods, even with using specific thermostats,

do not provide a good friction coefficient. The reduced models predictions appear further apart

from the full-MD model than the coupled model estimate. Another interesting trend revealed inFigure 10 is that the average resistance to sliding decreases with pressure. This seems again related

to thermal softening but should be taken with care as the simulations have not reached steady state.

Remarkably, at the highest pressures all methods seem to predict almost identical frictional forces.

As we applied a smoothing operator to the friction coefficient, an interesting quantity to examine

is the standard deviation to the smoothed value. Indeed, the level of noise is directly correlated

to the local mechanical properties and dynamics of the deformable body. Figure 11 shows the


DOI: 10.1002/nme




Figure 11. Deviation from windowed average of the instantaneous friction.

mean deviation for all studied cases. Clearly, the coupling method brings out the minimal stan-

dard deviation. It also fits almost exactly to the full-MD measure for pressure 0.08, 0.1 and

0.3 GPa. On the contrary, reduced models, thermostated or not, generate higher standard deviations.

The time fluctuations observed for the instantaneous friction are related to the lattice size. Yet the

magnitude of the measured forces is linked to the stiffness and dynamics of the deformable body.

This explains why the reduced models, which have a reduced compliance due to the removal of

several atomic layers, yield larger contact forces fluctuations.

To quantify this effect, one can measure the standard deviation of the displacement field at

various depths below the surface. This intends to measure the degree of homogeneity of the

deformation. We defined slabs of atoms identified by the Z coordinate. We then computed, during

the whole simulation, the deviation of the displacement field in each slab. Figure 12 presents the

maximal value over all the simulation of the standard deviation herein measured. The couplingregion is indicated as the shaded area, whereas to the right of this area we are approaching the

contact surface. Logically, the standard deviations are highest close to the contact region. Also, the

reduced model displacement field fluctuations decrease to zero for both horizontal ( X ) and vertical

( Z ) displacements when approaching the fixed boundary condition (left side of shaded region).

Remarkably, the coupled and full-MD cases deviations are in very close agreement, and they do

not reach a zero value at the bottom of the overlapping zone. The deviations are the highest for

the X direction, which indicates that shear waves generated by the sliding motion dominate this

problem. These waves create complicated dynamics in regions beneath the surface and as far as

the bottom of the overlap zone. Our results reveal that the coupling method allows the correct

transmission of the dominant waves. The dynamics of too small samples deviate from the expected

full-MD calculation. This highlights the need of pursuing developments in the dynamical behavior

of multiscale methods.

4. CONCLUSION

In this work we presented a 3D multiscale numerical method to treat the sliding contact between

two rough surfaces. We implemented the Bridging Domain method to couple finite elements with


DOI: 10.1002/nme




Figure 12. Maximal standard deviation of the displacement field ( X and Z directions). Theshaded zone is the coupling area. Results are obtained for an applied pressure of 100MPa(other pressures give similar trends).

MD. The approach was analyzed and compared with single-scale methods, including a full-MD

and reduced MD, thermostated or not models. Our results showed that none of the reduced and

the multiscale models were successful in capturing the steady linear increase of kinetic energy

beneath the surface observed in the full-MD calculation. The Bridging Domain method was seen

to be particularly efficient at absorbing high-frequency waves, acting as a 0 K thermostat. This was

argued to be inadequate in sliding contact applications, which are not isothermal processes. Our

observation calls for future challenging development of coupling methods. The actual coupling

formulation may need to be extended to finite temperature in order to allow dissipation, all thewhile taking into account heat storage in the material.

Our results also included preliminary measures of frictional forces. We compared, for various

pressures, the time fluctuations of those forces. We observed that the difference between the

various models tighten with increasing pressure, an indication of the averaging effect of thermal

softening. Coupling methods seem necessary to capture the time fluctuations of the frictional

forces. In particular, the compliance provided by the finite element extension beneath the coupling

zone was shown to reproduce accurately the complex deformation dynamics. On the contrary,

reduced models, with their smaller sampled depths, yield a non-physical stiffening of the contact

response.

In summary, our results demonstrate that the chosen multiscale coupling method is a promising

strategy for the analysis of sliding contacts. Low-frequency mechanical waves are successfully

passed through the coupling zone. Also, with the drastic computational cost reduction brought bythe method, simulation of surfaces with asperities at length scales several orders of magnitude

apart, previously unaffordable with pure MD calculations, are now within reach. Yet, the kinetic

energy time evolution obtained with a coupled simulation do not match pure MD one. In order

to correct this gap, and since simple thermostats inadequately reproduce the kinetic energy time

evolution at the contact region, a rigorous treatment of finite temperature and thermal fluxes seems

to be an important development to undertake.


DOI: 10.1002/nme




ACKNOWLEDGEMENTS

This material is based on the work supported by the Swiss National Foundation under Grant no.200021 122046/1 and the European Research Council Starting Grant no. 240332.

REFERENCES

1. Hyun S, Robbins MO. Elastic contact between rough surfaces: effect of roughness at large and small wavelengths.

Tribology International 2007; 40(10–12):1413–1422.

2. Luan B, Robbins MO. The breakdown of continuum models for mechanical contacts. Nature 2005; 435:929–932.

3. Miller RE, Tadmor E, Ortiz M. Quasicontinuum simulation of fracture at atomic scale. Modelling and Simulation

in Materials Science and Engineering 1998; 6:607–683.

4. Shenoy VB, Tadmor E, Miller RE, Rodney D, Phillips R, Ortiz M. Quasicontinuum models of interfacial structure

and deformation. Physical Review Letters 1998; 80:742–745.

5. Curtin WA, Miller RE. Atomistic/continuum coupling in computational materials science. Modelling and

Simulation in Materials Science and Engineering 2003; 11:R33–R68.

6. Tadmor EB, Ortiz M, Phillips R. Quasicontinuum analysis of defects in solids. Philosophical Magazine 1996;

73(6):1529–1563.

7. Wagner GJ, Liu WK. Coupling of atomistic and continuum simulations using a bridging scale decomposition. Elsevier Science 2003; 190(1):249–274.

8. Xiao SP, Belytschko T. A bridging domain method for coupling continua with molecular dynamics. Computer

Methods in Applied Mechanical Engineering 2004; 193(17–20):1645–1669.

9. Miller RE, Tadmor EB. The quasicontinuum method: overview, applications and current directions. Journal of

Computer-Aided Materials Design 2003; 9:203–239.

10. Broughton JQ, Abraham FF, Bernstein N, Kaxiras E. Concurrent coupling of length scales: methodology and

application. Physical Review B 1999; 60(4):2391–2403.

11. Kohlhoff S, Cumbsch P, Fscchmeister H. Crack propagation in bcc crystal studied with a combined finite-element

and atomistic model. Philosophical Magazine A 1991; 64:851–878.

12. Shenoy VB, Miller RE, Tadmor EB, Rodney D, Phillips R, Ortiz M. An adaptive finite element approach to

atomic-scale mechanics—the quasicontinuum method. Journal of the Mechanics and Physics of Solids 1999;

47:611–642.

13. Sansoz F, Molinari JF. Mechanical behavior of sigma tilt grain boundaries in nanoscale cu and al: a quasicontinuum

study. Acta Materiala 2005; 53(7):1931–1944.14. Tadmor E, Ortiz M, Phillips R, Miller RE. Nanoindentation and incipient plasticity. Journal of Materials Research

1999; 14:2233–2250.

15. Shenoy VB, Rodney D, Phillips R, Tadmor E, Ortiz M. Nucleation of dislocation beneath a plane strain indenter.

Journal of the Mechanics and Physics of Solids 2000; 48:649–673.

16. Shilkrot LE, Miller RE, Curtin WA. Coupled atomistic and discrete dislocation plasticity. Physical Review Letters

2002; 89:025501.1–025501.4.

17. Gracie R, Belytschko T. Concurrently coupled atomistic and XFEM models for dislocations and cracks.

International Journal for Numerical Methods in Engineering 2008; DOI: 10.1002/nme.2488.

18. Liu WK, Karpov EG, Park HS. Nano Mechanics and Materials. Wiley: New York, 2006.

19. Belytschko T, Xiao SP. Coupling methods for continuum model with molecular model. International Journal for

Multiscale Computational Engineering 2003; 1(1):115–126.

20. Anciaux G, Coulaud O, Roman J. High performance multiscale simulation for crack propagation. Proceedings

of HPSEC’2006 , vol. ICPPW’06. IEEE Computer Society Press, August 2006; 473–480.

21. Xiao S, Yang W. Temperature-related Cauchy-born rule for multiscale modeling of crystalline solids.Computational Materials Science 2006; 37:374–379.

22. Luan BQ, Hyun S, Molinari JF, Robbins MO, Bernstein N. Multiscale modeling of two-dimensional contacts.

Physical Review E 2006; 74:046710.1–046710.11.

23. XU M, Belytschko T. Conservation properties of the bridging domain method for coupled molecular/continuum

dynamics. International Journal for Numerical Methods in Engineering 2008; 76(3):278–294.

24. De Borst R. Challenges in computational materials science: multiple scales, multi-physics and evolving

discontinuities. Computational Material Science 2008; 43(1):1–15.


DOI: 10.1002/nme




25. Liu J, Chen SY, Nie XB, Robbins MO. A continuum-atomistic simulation of heat transfer in micro- and

nano-flows. Journal of Computational Physics 2007; 227(1):279–291.

26. Anciaux G. Simulation multi-echelles des solides par une approche couplee dynamique moleculaire/elements

finis. De la modelisation a la simulation haute performances. Ph.D. Thesis, Universite Bordeaux 1, July 2007.

27. Anciaux G, Coulaud O, Roman J, Zerah G. Ghost force reduction and spectral analysis of the 1d bridgingmethod. INRIA Report , HAL, 2008.

28. Allen MP, Tildesley DJ. Computer Simulation of Liquids. Oxford University Press: Oxford, 1989.

29. Rapaport DC. The Art of Molecular Dynamics Simulation. Cambridge University Press: Cambridge, 1995.

30. Anciaux G, Molinari J-F. Contact mechanics at the nanoscale, a 3d multiscale approach. International Journal

for Numerical Methods in Engineering 2009; 79:1041–1067.

31. Persson BNJ, Albohr O, Tartaglino U, Volokitin AI, Tossati E. On the nature of surface roughness with

applications to contact mechanics, sealing, rubber friction and adhesion. Journal of Physics: Condensed Matter

2005; 17:R1–R62.

32. Peitgen HO, Saupe D. The Science of Fractal Image. Springer: New York, 1988.

33. Voss RF. Fundamental Algorithms in Computer Graphics. Springer: Berlin, 1985.


DOI: 10.1002/nme

Date post:	09-Apr-2018
Category:	Documents
Upload:	dang-viet-hung
View:	220 times
Download:	0 times

Sliding of rough surface and energy dissipation with a 3D multiscale approach

Documents