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Los Alamos N A T I O N A L L A B O R A T O R Y

SLIDING FRICTION IN COMPRESSED METALS AT HIGH VELOCITIES

J. E. Hammerberg B. L. Holian S. J. Zhou

Shock Waves in Con&nsec St. Petersburg September 1996

Matter

A

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.

i

Sliding Friction in Compressed Metals at High Velocities J.E.Hammerberg

Applied Theoretical and Computational Physics Division B.L.Holian

Theoretical Division S.J.Zhou

Center for Non-Linear Studies Los Alamos National Laboratory

Los Alamos, NM USA 87501

We present the results of massively parallel simulations of sliding friction for ductile metals at high velocities (of order significant fractions of the transverse sound velocity) and kbar pressures. Our initial studies have been in two dimensions for smooth and roughened interfaces and typical system sizes have been 65,000 atoms. We have used embedded atom model (EAM) potentials to account for the density dependence of atomic potentials in defective environments and have integrated the classical equations of motion using the method of molecular dynamics. To measure the tangential force in these numerical experiments we have studied systems with upper and lower reservoir regions connected to workpiece regions. The workpieces are constrained by external forces in the reservoir regions to move relative to one another. This is accomplished by adding an external force in the reservoir region (typically 20 atomic layers in height) such that the average velocities in these regions are fixed a t Tfru . Initially, the upper workpiece plus reservoir has tangential P velocity up and the lower workpiece plus reservoir velocity -up’ The normal force in the reservoir regions is such that the load corresponds to a given initial pressure. The reservoir regions also incorporate a ramped viscous damping force which eliminates spurious reflections a t the reservoir-workpiece boundary. The simulations for flat interfaces have verified some of the predictions of the adhesive model of friction, showing an increase of the tangential force with pressure (but a decrease of the coefficient of friction with pressure). A new feature is a velocity weakening of the tangential force with increasing velocity. We discuss the morphological changes which occur in the surface region for copper-copper interfacpa that %re nccnr in td with thic frirtinnal weakening at high velocities.

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

I. Introduction

The properties of sliding metal interfaces under dynamic loading conditions are poorly known.

For regimes of sliding speeds of order 0.1 the transverse sound speed and pressures of order shock

pressures in metals, the following are essentially open questions from both a theoretical and

experimental perspective: the velocity dependence of the tangential force, the nature of surface

and subsurface micro structure and dislocation structure and evolution.

Experimentally, there has been some recent progress in pressure-shear geometries by Prakash and

Clifton (l) for elastic materials. Theoretically, Sokoloff (2) has treated a simplified model for

which an inverse velocity dependence for p, the coefficient of sliding friction, has been predicted.

In lieu of experimental work and as an incentive to perform relevant experiments it is of interest to

perform numerical experiments. Of particular relevance are molecular dynamics (MD) experi-

ments for materials with well characterized density dependent interatomic interaction potentials.

For ductile metals (e.g., copper, nickel, aluminum), such potentials exist and are capable of repro-

ducing equations of state and defect properties as Holian et al. (3) have shown. We shall present

here the results of a series of atomistic simulations for copper-copper interfaces in a two dimen-

sional geometry, focusing mainly on the density and velocity dependence of the coefficient of

friction.

The velocities now attainable with MD simulations are limited by simulation time constraints. For

the problems of sliding interfaces, realistic steady states necessitate interfacial relative displace-

ments of the order of tens of interatomic distances. Massively parallel simulations are now capa-

ble of times of order lo3 zo where T~ is an optical phonon period. This limits sliding speeds to v,l

> 0.1 &psec, i.e

the transverse sound speed, ct. The density, however, is limited only by the accuracy of the inter-

atomic potentials. We discuss in the following sections the dynamics of copper interfaces for slid-

ing speeds in excess of 0.01 ct and densities in the kbar range. Section I1 gives some general

cm/psec or roughly speaking to relative velocities greater than 0.01 times

arguments concerning the density dependence of the coefficient of friction for ductile metals. Sec-

tion I11 describes the MD simulation method for calculating the coefficient of friction. In section

IV we discuss the results of simulations for copper interfaces and in section V draw conclusions.

II. Density dependence of the coefficient of friction

It is thought that for ductile metals with pristine surfaces, i.e. chemically uncontaminated inter-

faces, the frictional properties are determined by the microscopic interactions of the surface inho-

mogeneities (asperities). For two such contacting surfaces in relative motion, local stresses are

sufficiently great that the majority of interacting asperities are flowing plastically so that the evo-

lution of the surface distribution toward a steady state is determined ultimately by the dislocation

dynamics and microstructure, at least for ductile metals. A gross mean field picture of such a situ-

ation is the so-called adhesive model of friction for which the tangential force is taken to be

Ftan, =

where A,E is the effective area of contact for the asperities and z is the shear flow stress for plastic

flow of the weaker of the two interfacial materials. Thus as shown in figure 1, rather than increas-

ing linearly, as a function of normal force the tangential force is bounded by the flow stress times

the nominal surface area A, Fmg -> AT. The transition between these two behaviors depends on

the plastic contact problem whose solution gives the dependence of AeE on Fnomd. Indeed,

behavior of this kind is observed e~perimentally(~) with the transition Fnod/A of order the flow

stress. Rate dependent effects are subsumed in the flow stress ~ ( p , T , w,v), a function of pressure

p, temperature T, plastic strain w and plastic strain rate \ir . At high densities, then, for which

A e r A the density dependence of the tangential force is determined by the density dependence of

2.

Generally, for a ductile metal, the main density dependence of z comes from the density depen-

dence of the shear modulus, i.e.,

where Q is a scaled flow stress whose density dependence is weaker and related to the dislocation

dynamics determining plastic strain and strain rate. For moderate pressures that are small relative

to the shear modulus itself, a linear expansion of G in pressure like that of the bulk modulus in the

Murnaghan equation of state, is usually rather accurate:

G ( p , T ) = G(0, T ) + a p . (3)

Thus from eqn. 1 the adhesive model would predict that Fmg is an increasing function of pressure

and that the coefficient of friction, g= Fmg/Fnoml, is a decreasing function of p since Fnod/A

initial density is po and a constant normal load is applied to each particle in the reservoir region

consistent with <v,>=O at the initial density and temperature. A transverse external force F,,(t)

is also applied uniformly to the atoms in the reservoir regions so that <v,>=fup there for all

times. The interatomic potentials are the same for all atoms. Those that are not in the reservoir

regions evolve according to the inter particle interactions only, with no external forces imposed.

Thus we have for reservoir atoms (i in the reservoir),

2 d x i (tangential)

m t 7 = C f j i + mig , dt j

(5)

where mi is the particle mass and fji is the force on i due to particle j (which may be either inside

or outside the reservoir). In order that the tangential component of the velocity be constant, we

require that

and therefore that

iweservorr C . L C f j i ] J

g = - M r eservoir

The time dependent tangential force (per atom) is then mig(t). The normal load for a given density

po and temperature T is computed from the load required such that <vy>=O with up&.

In simulations such as these, elastic and inelastic excitations are generated at the interface and

propagate to the reservoir region. In order to avoid reflections of excitations back to the interface

we have used a ramped viscous damping technique introduced by Holian and Rave10(~~~) which

provides an impedance matched acoustic absorber in the reservoir region. By this mechanism we

avoid spurious finite size reflections from the boundary which would otherwise interact with the

interfacial dynamics . The initial interface is specified by a longer wavelength modulation upon which is imposed a ran-

dom disorder. We define

yl(x) = ( h + Gh,(x))sin(kx + Ql)

y2(x) = ( h + Gh2(x))sin(kx + Q2).

The upper region is defined as

and the lower region by

We have typically taken for the simulations presented here 6 h a . 5 ro and 6h/h=O.O1 where the

zero pressure nearest neighbor distance for Cu is ro = 2.5681. For the initial configuration in the

two-dimensional simulations which we discuss here, we have taken a triangular lattice with the

interface along a { 1 1 1 } fcc plane. For the total potential energy of the system, we use an embed-

ded atom method (EAM) many-body potential of the form

Q, = pi, 1

where the local density at atom i is

The specific forms in eqn.s (10)-(12) for q ( r i j ) , the Lennard-Jones spline pair potential, the

embedding function F, and the weighting function w are given in reference [3] for Cu. This EAM

potential gives reasonable agreement in three dimensions with the vacancy formation energies and

compressive properties of copper. For studying the dislocation dependent properties such as duc-

tile interfacial slip, it is essential to include the local density-dependence of the many-body inter-

action in a metal.

IX Velocity and density dependence of the coefficient of friction

We have studied the velocity and density dependence of interfacial slip for a homogeneous inter-

face with EAM copper as a prototypical ductile metal. The results presented here are all for two-

dimensional systems of 65,000 atoms. In the computational units we use (nearest neighbor dis-

tance ro, atomic mass m and cohesive bond energy E: the unit of time is to = ro - ) the equilib- t 2 rium copper density is - = 1.15 and the transverse sound speed is 3. The EAM potential

A b

described in the previous section has been used to describe the density-dependent particle-particle

interactions. Initial temperatures were essentially zero and the reservoir region was thermostated

to maintain this temperature there. The reservoir region consisted of twenty atomic layers. The

non-reservoir atoms were not thermostated and evolved according to the system interactions as

described in the preceding section.

The general behavior that we find for the tangential force is of two types, as is illustrated in fig. 3,

where the tangential force as a function of time is plotted for two velocities ~ ~ 4 . 4 and up=l.O at

density p0=1.19. This general transition in behavior occurs at imposed velocities of order 0.1 ct.,

and is mirrored in the magnitude of the coefficient of friction (see fig. 4 where p as a function of

up is shown for three densities p0=1.19, 1.23 and 1.27 corresponding to pressures of p=8,18 and

30 GPa).

The nature of this transition is related to the stick-slip behavior of the roughened interface. For

velocities below u p , the interface is 1ocked.Since the system is continuously driven, slip occurs

is shown in figures 7a-7e for p0=1.27 and up=0.49, slightly above u p ~ t , the transition velocity

shown in fig. 4. In this figure times range from 100 to to 500 to and the grey scale corresponds to

different crystalline orientations from -30' to +30° from the initial vertical hexagonal axis. Black

(+30°) and white (-30') are equivalent. The coarsening away from the interface at longer times is

very similar to that observed by Rigney et al.(7) at much lower velocities. At higher velocities the

grain structure is finer, as shown in fig. 8 (up=l.O, p0'1.27).

V.Discussion

These simulations have verified some of the general features of the adhesive model of tangential

slip in ductile metals. The tangential force in these two-dimensional simulations has been found to

be an increasing function of the pressure, and the coefficient of friction a decreasing function of

pressure, in agreement with the simple arguments of section IT. The velocity dependence, how-

ever, has shown a transition to a lower friction state accompanied by a homogeneously nucleated

nano-structure (cf., micro-structure) which coarsens away from the interface. The friction coeffi-

cient weakens with velocity in this regime. It is not possible for us to say at the moment what the

functional form of this velocity dependence is, but it is weaker than '/up.

Typical expehenta l surfaces have asperity size distributions that are micron in scale, whereas

the present simulations are of order 200 8, in the normal dimension. Larger MD systems of order

10' atoms are now possible in two and three dimensions, allowing one to reach the micron scale

in two dimensions. However, simulations such as ours are more appropriately viewed as providing

small-scale input at the level of individual asperity contact for more mesoscopic treatments. In sit-

uations at high densities where the asperity distribution gives flattened interfaces, these simula-

tions describe the actual state of matter at rapidly sliding interfaces. To determine the density at

which this flattening occurs requires a more detailed treatment of the meso-physics of the rough-

ened interface.It would be of interest to test some of the results of these simulations by appeal to

experiment. Pressure-shear experiments on pristine flat copper interfaces would reach velocities

and densities relevant to these simulations. Dynamic neutron resonant absorption experiments to

measure simultaneously the interfacial temperature and velocities have also been proposed(*) We

hope that the present work will act as an incentive to perform such dynamic experiments.

Figure captions:

Figure 1. Characteristic behavior of the tangential force as a function of load for a ductile metal.

Figure 2. Computational configuration for MD simulations. Periodic boundary conditions are

applied at the left and right boundaries.

voir region.

Figure 3. Tangential force as a function of time in units of to (2.3 psec.) for up=0.4 and u,=l.O.

Figure 4. Coefficient of friction for 2d EAM copper as a function of velocity up. Squares:po=1.19,

open circles:po=1.23, and dots:po=1.27. Points are connected with lines to guide the eye.

Figure 5. Distribution of particles with potential energy $>-2.6s as a function of distance from the

interface for times 16 to, 1 0 0 to and 400 to. Bin size is Ax=l.

Figure 6. Tangential force as a function of time for ~ ~ ' 1 . 0 . Upper curves (from top) po=1.27,

1.23, and 1.19. Lower curves: scaled curves as described in text. Curves have been smoothed by

is the magnitude of the transverse velocity in the reser-

averaging over 10 to.

Figure 7.Grain structure as a function of time for p0=1.27 and up=0.49 at times 100 to, 200

to, 400 to, and 500 to.

Figure 8. Grain structure for po=1.27 and up=l.O, t=500 to. Same scale as figure 7.

300

Figure 1.

Figure 2.

L a L-

@

Aeff fr

r' X -

Figure 3.

Figure 4.

0.1 0

0.05

0.00

-0.05 1 1 1 1 J 0 100 200 300 400 500

U c ( l .19) = 0.35 f 0.05 uC(1 .23) = 0.45 -t 0.05

uJI 2 7 ) = 0.485 0.005

0.5

0.4

0.3

1

0.2

0.1

0.0 0.0 0.2 0.4 0.6 0.8 1 0

I

Fig u re

160

140

120

Lo Gi- 100 5 I . A 80 -e-

5 60

Figure 6.

40

20

-1 00 -50 0 50 100 Distance from interface

8 E3, I= (d

1-

0.08

0.04

0.00

-0.04 I I I I I I

0 200 400 600 8

4

0

-4

c

I PI ’ , I I I -

0 20 40

Figure 7.

Figure 8.

I I Sliding Friction in Compressed

Metals at High Velocities I I

J. E. Hammerberg Applied Theoretical & Computational Physics Division

6. L. Holian Theoret i ca I Division

S. J. Zhou Center for Non-Linear Studies

Los Alamos National Laboratory

Summary

Compressed flat interfaces, generalities

MD simulations with absorbing reservoirs

Tangential force, coefficient of friction and velocity weakening, microstructures

Experi ment

Compressed flat interfaces, generalities For two ductile metals in contact and flowing plastically expect the average tangential force to be given by:

where (FtangentiaP = Aeff T

Aeff = effective area of contact T = flow stress of the weaker of

the two materials

- m E aa cn E co

.I c.

I-

7 = TIP, T, +, d+/dt, vrell d*/dt = plastic strain rate p = pressure T = temperature

h e 1 = relative tangential velocity of the i terfaces

t.p = plastic strain

Pressure dependence of the coefficient of friction

The main pressure dependence of the flow stress enters through the shear modulus G:

For pressures t0.5 G a reasonable approximation to G is:

Coefficient of friction:

Therefore, expect that <Ftangential> is an increasing function of pressure at fixed velocity and that the coefficient of friction is a decreasing function of pressure

Molecular dynamics simulations We define a reservoir region, typically consisting of 20 atomic 1 , 4 . iw- up layers in which a force is applied such that the average tangential velocity is constant:

1 + mig 0 0 (tang entia I) I I (z f(tangential)

ji m iXi

mi$ tangential) I

1 ieRes

(i in reservoir) I .rY

A similar equation for the normal direction applies a normal load consistent with the pressure at the density of interest SO that Vnormal= 0

A ramped viscous damping algorithm is applied in the reservoir region to damp elastic waves and propagating dislocations so that there are no reflections from the reservoir region (Holian, B. L. and Ravelo, R., Phys. Rev. 851,11275 (1995); Zhou, S. J., et.al.,

< n = o

EL) - V

I CD

0

1 I_

U CD a CD P)

T

a 0 U

U bo L

iu

c) 0 3 U e. 0 3 v)

I.

9)

0" 3 v)

cn

S sa 0

I.

3 - rc I.

9) 3

sa s v)

c 9)

0 3 v)

I.

3 - rc I.

N Q) ul cn n A

m . S v) 3 m 55)

c) S

I. a 0 =r CD

S 3 c) 0 3 v)

rc

z

rc

;3 I.

3 CD U .

3 v) m 0 3 2 m ;3 I.

3 CD U

0 rc

CD CD

I. s 'E3 CD =. 0 U c) I.

3 v)

A L L . I.

3 s I. rc =r

U 0 3 CD 0 3

s 2 0

CD CD m S 9) -

II U

CD 3 v) 0 3 sa

I.

3

I.

CT 0

N Y 0 sa

z 0 S 3

3 0

CD

I.

3 Er s 7

c) CD cn

9) 3 U

B P w

Y

U 9) z

Q) cn sa S rc

'E)

0 = = 0 v) I.

G 3

U I.

0 3

0 00 b

0 iu

0 .

0 A

0 iu

0 b

0 .

iu W

II 0

cn

0

cn

- b I+

0

0 b,

0 bo

. 0

0 0 01

0

0 0

1v 0 0

rc

0

cd 0 0

P 0 Y

01 0 0

Ftang 0 0 0 0 0

. 0 UI

I

E =

I

. 0

I I 1 - 1 7

S

II i;'- 0

0 . &

v) s P v)

5 CD

- I.

CD U

I.

Y

0

N 0

izl 0 P 00

n

z I.

0 S

N

0

v)

CD

3 m

s s G

0

P 0 0

Q, 0 0

0 0 0 0 P O

0 0 . 0 00

x . 2 3 ? n 3 > 3

Microstructure

For the velocities greater than approximately u, = 0.5, the applied tangential force scales as:

I

(F(tangentia1) ( t ) = F ,f(Xt)

where h is the distance from the interface to the reservoir region

For the lowest velocities, the surfaces stick and slip occurs at the boundary between the reservoir and a work hardened medium with a large delocation density

For high velocities for which the above scaling applies, a region of diordered nanocrystallites is formed which diffuses inward A

Los Alamos

The figures that follow illustrate for the case of up = 1.0 and p = 1.19. The figures show the potential energy of each atom for a time sequence o f t = 10,50,100,200, and 400 to. The final microstructure is shown for t = 400 to in a full view and a zoomed view.

up = 1.0, p - - 1.19 and t = 10

Figure 3.

II

II

up = 1.0, p - - 1.19 and t = 100

Figure 5.

up = 1.0, p - - 1.19 and t = 400 to zoomed

Figure 9. Applied Theoretical & Computational Physics Dio ision

The diffusive nature of this evolution is seen in this plot of the integrated number of atoms with potential energy greater than a cutoff in strips of width unity in the direction normal to the interface

160

140

120 L

80 -8 w

60

40

20

I I I I I

tt

P * a58

98 58

I) 58

0 e

os 98

-1 00 -50 0 50 100 Distance from interface

Figure 10

c

V

Figures 11-15 show the differences in grain structure for two cases with po = 1.19; figures 11 through 13 are for up = 0.4; figures 11 and 12 show the grain map at 2004, and 400t0, respectively; figure 13 is an enlarged view of the central region; figures 14 and 15 are for

= 1 m 0 at time 400to, with figure 15 an enlarged view of the central region on the same scale as figure 13

Figure 11 Figure 12 Figure 13

Figure 14 Figure 15 Applied Theoretical & Computational Phrrsics Diuision Los Alamos

Figure C shows the grain structure for po = 1.27 and up = 0.49

t = 100 to t = 200 to t = 300 to

Figure C. L

Figures A and B show time sequences for po = 1.27 above (up = 0.49) and below (up = 0.47) the transition Figure A shows the number of particles with potential energy 2 - 5.2 in strips of width as a function of distance from the interface

Figure A

-1 N I

+ t

up = 0.47

0 Los Alamos

Figure B shows the average tangential velocity for the same bin size

Conclusions We have found qualitative agreement with the density dependence of the adhesive model of friction for flat interfaces at high velocities

The velocity dependence for flat Cu interfaces is a decreasing function of ve I oci ty (ve I oci ty weakening )

Above a critical velocity we have observed the formation of a nonocrystalline region which grows diffusively away from the interface

+;+ For nonuniform interfaces, which have not been discussed here, the dynamics is more complicated and interesting showing material transfer, debris formation, fracture and healing, void nucleation and spall

&

Experiments *@ Neutron resonant absorption experiment at the Los Alamos Neutron Scattering Center

(LANSCE)

cu Ta '7

Flyer plate

I " Be

V Detector

- cu

-.-

E * line shape allows interfacial temperature determination * energy shift allows interfacial velocity determination

U

Experiments +:+ Pressure shear experiments