Tribology Letters 8 (2000) 229–232 229

On the interrelation between acoustical processes and materialcharacteristics of the tribojoint surface

Arthur Bereznyakov

St. Kitaenko, 3-A, appt. 27, 601020 Kharkov, Ukraine

Received 5 January 2000; accepted 31 August 2000

An acoustical model of the boundary friction process has been developed where the transformation of surface Rayleigh wavesinto bulk Rayleigh waves is considered, taking dislocation damping into account. A relationship similar to an inverse resonance curvehas been found between the specific friction force and the Rayleigh wave frequency. The specific friction force has been shownto be proportional to the wavelength of the surface microrelief and in inverse proportion to the square of the surface dislocationdensity.

Keywords: flux density, entropy production, absorption coefficient, surface microrelief, Rayleigh waves, dislocation

Friction is a combination of dissipative processes that aredistinct in their nature. Any subdivision of these processesinto different components, e.g., into dry friction and bound-ary friction, is essentially a matter of convention. It is notalways possible to subdivide energy dissipation under ex-perimental conditions into “external” and “internal” fric-tion. Besides, we should bear in mind that in the caseof external friction, several physical mechanisms becomeapparent, one of which under certain circumstances canpredominate. For example, at relatively low loadings andvelocities the contribution of elastic processes can be con-siderable.

While most investigations of friction processes haveadopted a technical, phenomenological approach, only fewworks deal with the physical aspects of the process [1].In that work, a 3D model with infinite periodical relief isconsidered, where acoustic waves emission from the sur-face into the volume is analyzed without accounting for thecrystal lattice defects. This model, however, correlates onlyweakly with experimental data [2–4].

Abbreviations

Ffr friction forcef friction coefficientIR Rayleigh wave intensityps entropy productionS density of entropyvR wave propagation velocityWs acoustic powerX thermodynamic powerx′ thermodynamic coordinateα absorption coefficientγ dislocation densityΛ wavelength of the surface microreliefστ specific friction forceσs scattering cross-sectionω frequency

In this context, a formally simple model is proposed inthis work. In the model, a point indenter is moved over thequasi-periodic surface of a microrelief; a Rayleigh wave isthereby excited. The latter is dissipated on nonuniformitiesand reemitted into the volume of the friction body, whereit is absorbed by dislocations in the surface layer. Thismodel is obviously more realistic than that of [1], since itis associated with the actual material structure. Our calcu-lation procedure is based on the Prigogine theorem, whereit is stated that the entropy production ps = d2S/dV dt in aslightly nonequilibrium system involving linear processes,tends to a minimum for a stationary state characterized bya slight inequilibrium.

Thus, starting with the above-mentioned model, the taskcan be set to find an interrelation between tribological pa-rameters (specific friction force) and the material character-istics (dislocation density, acoustic wave absorption coeffi-cient).

Let us consider the frictional interaction of a point inden-ter with a flat rough friction surface presented schematicallyin figure 1.

Figure 1. Scheme of a triboconjugation surface microrelief with nonuni-formities.

J.C. Baltzer AG, Science Publishers

230 A. Bereznyakov / Triboacoustic processes in the tribojoint

For the indenter moving along the x axis, the sur-face roughness is simulated as a quasi-periodic microreliefshaped along that axis as a cycloid of the arc length Λand the vertex height h. The cycloidal periodicity of therelief is disturbed by randomly arranged surface nonunifor-mities spaced at a mean distance considerably exceedingthe wavelength. The moving indenter acts on the tribojointsurface with a friction force Ffr. The tangential stress σt re-sulting from that force travels along the x axis, generatinga surface Rayleigh wave which would be undamped for thesmooth surface. However, the microrelief nonuniformities(scattering centers) transform the surface wave into a bulk(3D) one. The Rayleigh wave intensity is

IR = στvR, (1)

where vR is the wave propagation velocity.The scattering ability of the surface nonuniformities is

characterized by the scattering cross-section

σs =dWs

IRdΩ, (2)

where Ws is the acoustic power dissipated within the solidangle Ω.

If each surface nonuniformity is considered as anisotropic emitter, which reemits the surface wave into asemispace Ω = 2π, and the acoustic power of a surface el-ement ∆Sxy is presented as Ws = Ivo∆Sxy, then the equa-tion (2) takes the form

σs =Ivo∆Sxy

2πIR,

and thus the intensity (energy flux density) of the bulk waveis

Ivo =2πσsστ IR

∆Sxy=

2πσsστvR

∆Sxy.

If the concentration of surface nonuniformities is ns, theintensity of the resulting incoherent wave reemitted by allnonuniformities from the surface ∆Sxy can be expressed as

IvN = IvoNo = Ivons∆Sxy = 2πσsστvRns.

Accounting for the damping α, the intensity of a waveat the distance z from the surface is

Ivz = 2πσsστvRns exp(−2αz). (3)

The density of each flux Ii is due to the correspondingthermodynamic force Xi. The product of those quantitiesis the entropy production:

ps =∑

~Ii~Xi. (4)

Let the entropy production of a friction site be calculatedusing the method described by us in [5]. The flux densityof a bulk wave can be shown to have the form

dIvz

dz= −dx′i

dt,

where x′i is the volume density of the thermodynamic co-ordinate x′i (x′i = dx′i/dV ).

Substituting equation (3) into this expression, we obtain

4πασsστvRns exp(−2αz) =dx′idt.

Let the specific friction force be determined from thelatter equation:

στ =exp(2αz)

4πασsvRns

dx′idt. (5)

To determine ~Xi, let us write the first law of thermody-namics for a unit volume of the near-surface layer. To thisend, let the equation

TdS = dU − ~Ffrd~l−∑

~Xi d~x′i

be divided by a physically infinitesimal volume ∆V .Dividing the expression under the differential sign by

∆V = ∆Sxyz and denoting volume densities of thermody-namic quantities by the symbol “˜” we obtain

TdS = dU− στdlz−∑

Xi dx′i. (6)

Substitution of equation (5) into (6) gives

TdS = dU− exp(2αz) dl4πασsvRnsz

dx′idt−∑

Xi dx′i,

whence it follows that

∂S∂x′i

=exp(2αz)

4πασsvRnsTz

dldt

and

~Xi = ~∇ (∂S)(∂x′i)

=exp(2αz)

4πασsvRnsTz

(1z

+1T

dTdz− 2α

)dldt~k0, (7)

where ~k0 is a unit vector.Multiplying equations (3) and (7), we can obtain the

entropy production due to the acoustic wave flux:

PSv = IVXV =στ

2αTz

(1z

+1T

dTdz− 2α

)dldt. (8)

Let us transform the derivative dl/dt. According to fig-ure 1, dl = Λ dN , where dN is the number of cycloidvertices which are with the indenter contacted during thetime dt.

Then,

dldt

= ΛdNdt

= Λω

2π, (9)

where ω is the angular frequency of the surface wave.Substituting the equation (9) into (8) we obtain

PSv =ωστΛ

4παTz

(1z

+1T

dTdz− 2α

). (10)

A. Bereznyakov / Triboacoustic processes in the tribojoint 231

The absorption coefficient α in the latter equation is de-fined mainly by dislocations in the metal; according to themodel of dislocational strings [6], it is determined as

α =γ∆0η

2ω2d

2πCτ [(ω20 − ω2)2 + ω2d]

, (11)

where ∆0 ≈ 8Gb2/π3C = 1, η2 = π2C/A, d = B/A, A,Band C are constants of the dislocational strings theory, ω0 isthe eigenfrequency of dislocational strings, d the dampingcoefficient of dislocational strings, γ the surface disloca-tion density, Ct the longitudinal velocity of sound, Cτ thetransversal speed of sound, G the shear modulus and b theBurgers vector.

The contribution of dislocations to the entropy produc-tion is determined [7] as

psd = 10γστb4 Ct

3kT 2. (12)

The total entropy production can be obtained by sum-mation of equations (10) and (12) taking (11) into acount,α and 1/z being neglected as compared to T−1 dT/dz,since grad zT near the contact region is always large:

ps = psv + psd

=ΛστCτ (ω2

0 − ω2)2 + ω2d2

2T 2zγ∆0η2ωd

dTdz

+ 10γστb4 Ct3kT 2

. (13)

It is easy to see from equation (13) that ps has a mini-mum with respect to the dislocation density γ. Accountingfor the condition ∂ps/∂γ = 0, the specific friction forcecorresponding to the minimum entropy production with re-spect to the dislocation density is

στ = 0.15κΛCτ

γ2∆0η2b4Ctz

dTdz

(ω20 − ω2)2 + ω2d2

ωd. (14)

Since the entropy production is determined for a near-surface layer where Rayleigh waves are propagated andwhich has a thickness commensurable to the length λR ofthe Rayleigh wave having the velocity uR = 0.9Ct, it canbe supposed that

z ≈ λR = 2πCτω−1.

Substituting this z value into (14) and taking into accountthat Ct = (E/ρ)1/2, where E is the Young modulus and ρis the material’s density, we obtain

στ =0.075π

kΛγ2∆0b4η2

C1(ω2

0 − ω2)2 + ω2d2

d

dTdz. (15)

Since the tangential stress στ is generated by a pointindenter, it can be considered as the maximum stress in thecontact area [8–10]

στ =3P

2πD2

(13

(1− 2µ) + f(4 + µ)

8

), (16)

where P is the external loading, D the contact area diame-ter, f the friction coefficient, and µ the Poisson coefficient.

Let us transform this equation, taking into account thatthe ratio 4P/πD is equal to the magnitude of the contourpressure and substituting the value of the Poisson coef-ficient, µ = 0.35, that is the most characteristic for themetals,

στ = (0.3 + 1.6f )pc. (17)

Let us determine the friction coefficient from the equa-tion (16),

f = 0.6

(στpc− 0.3

)= 0.6

(0.075π

kΛCl

γ2∆0b4η2pc

(ω20 − ω2)2 + ω2d2

d

dTdz− 0.3

).

(18)

Let us make an approximate estimation of the magni-tudes στ and f for the most characteristic values of theparameters entering in the equations (15) and (18). For allthis we shall be limited by the condition ω ω0 and takeinto account the following relations in conformity with [6]:η2 = ω0L, where L is the length of the dislocational string;A = πρb2.

The characteristic numerical values of the afore-men-tioned parameters are the following: ω0 ≈ 1010 Hz, γ ≈1015 m−2, L ≈ 10−7–10−8 m, Λ ≈ 1×10−6–10×10−6 m,Cl ≈ 5× 103 m/s, dT/dz ≈ 105–106 K/m, pc ≈ 1× 105–3× 105 Pa [14].

With these values στ is in the range 106–107 Pa and thecorresponding values of f lie between 0.5 and 5.

Of course these values are approximate, but can be con-sidered to be in qualitative agreement with the experiment.We should mention that the values στ are more reliablethan the values of f .

The frequency dependence of f (figure 2) is essentiallythat of the friction coefficient on the relative speed v oftribojoint surfaces displacement. The dependence has theshape of an inverted resonance curve. Each of these curveshas a characteristic resonance extremum at the frequency

ωp = (ω20 − d2/2)1/2.

Figure 2. Dependence of the friction coefficient on the Rayleigh wavesfrequency. ω1, ω2 and ω3 – resonance frequencies, d1 > d2 > d3.

232 A. Bereznyakov / Triboacoustic processes in the tribojoint

This extremum is the point of inflexion for curve (1)with a large damping d and a minimum for curves (2)and (3).

Let us compare quantities ω0 and ω. The latter, as isseen from equation (9), is connected with the speed v bythe relationship ω = 2πv/Λ. If Λ is interpreted as themean wavelength of the surface microrelief, then, accordingto [10], this has a value between 1 and 10 µm while themaximum value of v has the order of magnitude 10 m/s.Then, ωmax ≈ 106 Hz while ω0 = 109 Hz [6].

In all cases, the dependence of f on ω becomes appre-ciable at ω > ωmax; thus, this dependence increases withinthe whole frequency range for curve (1) with strong damp-ing (d > ω0) and decreases within the interval [ω, ωp] forcurves (2) and (3) characterized by weak damping. Depen-dence of such a kind is similar qualitatively to that obtainedin [3]. While this dependence can be experimentally ob-served within the frequency range ω ω0 at rather smallω0, it cannot be observed in principle at ω > ω0. Notethat no essential dependence of f on v exists at speedsfor which ω < ωmax. The absence of this dependence inthe acoustic tribojoint model with an aperiodic profile ofthe surface microrelief has already been reported in [11].Thus, formula (18) explains the fact that the dependenceon v is observed in only relatively few experiments. Aninverse proportionality between f and dislocation densityγ was found also in other models [12] and confirmed inexperiments. As to the dependence of f on Λ, there areexperimental data on the relationship between Λ and thedamping coefficient α [13] which of least qualitatively con-firm that dependence.

Additionally, there is a relationship between Λ and themicrorelief irregularity height, h, the latter being consideredas a measure of roughness. This relationship is well knownfor cycloid: Λ = 4h. It is, in fact, the reason for the choiceof the microrelief undulation shape which is close to beingsinusoidal [10]. A similar correlation between Λ and hexists also for a sinusoidal microrelief, but in this case thedependence between these quantities cannot be expressedin terms of elementary functions. Thus, a proportionalitybetween f and the roughness follows from formula (18),albeit indirectly.

It is easy to understand that the real surface microreliefis not a periodic one but rather a superposition of harmon-

ics of different wavelengths, which correspond to differ-ent Rayleigh wave frequencies. However, all the inherentregularities in tribojoint in the stabilization regime underboundary friction which are described above remain validas long as the condition ω ωp is met. The decreaseof f depending on the applied normal load, P , revealed inexperiment [14] also follows from (18).

Finally note that the inverse proportionality between thecoefficient of external friction, f , and the coefficient ofacoustic wave absorption, α, being a measure of internalfriction follows from equations (10)–(14). Such a relation-ship is of course a qualitative one, since it is not only theterms considered here that are included into the total en-tropy production; nevertheless, this relationship allows oneto state that both external and internal friction are compo-nents of the energy dissipation process.

Thus, material parameters of a friction system affect bothexternal and internal friction processes and cause an inter-relation between those processes.

References

[1] E. Adirovich and D.I. Blockhinzev, J. Phys. 7 (1943) 29.[2] A.I. Sviridenok, N.K. Myshkin, T.F. Kalmykova and O.V.

Kholodilov, Acoustic and Electric Methods in Tribotechnique (Naukai Tekhnika, Minsk, 1987) (in Russian).

[3] V.V. Zaporozhets, in: Friction and Wear Problems, 2nd Ed. (Kiev,1972) p. 77 (in Russian).

[4] A.S. Akhmatov, Molecular Physics of Boundary Friction (Fizmatgiz,Moscow, 1963) (in Russian).

[5] A.I. Bereznyakov and E.S. Ventsel, Trenie Iznos 14 (1993) 194.[6] A.V. Granato and K. Lucke, J. Appl. Phys. 27 (1956) 583.[7] J.P. Hirth and J. Lothe, Theory of Dislocations (Mc Graw-Hill, New

York, 1972).[8] V. Hirst, in: Contact Interaction of Solids and Friction and Wear

Forces Calculation (Nauka, Moscow, 1971) p. 23 (in Russian).[9] G.M. Hamilton and L.E. Boodman, J. Appl. Mech. 33 (1966) 183.

[10] I.V. Kragel’sky and N.M. Mikhin, Friction Sites in Machines: A Ref-erence Book (Machinostroenie, Moscow, 1984) (in Russian).

[11] V.M. Baranov, E.M. Kudryavtsev and G.L. Sarychev, Trenie Iznos14 (1994) 989.

[12] A.I. Bereznyakov, E.S. Ventsel and A.V. Evtushenko, Trenie Iznos15 (1995) 181.

[13] N.S. Bykov and Yu.G. Shneider, Akustich. Zh. 6 (1960) 501.[14] N.M. Mikhn and V.S. Kombalov, in: Contact Interaction of Solids

and Friction and Wear Forces Calculation (Nauka, Moscow, 1971)pp. 146–153 (in Russian).