/

Journal de ~lecaniquc theorique cl appliquec.Journal /If theoretical and applied mechanics

Special issue. supplemcnt n' I 10 vol. 7 - J 988

New models of frictionnonlinear elastodynamics problems

J. T. ODENTexas Institute of Computational Mechanics. The University of Texas. Austin. Texas 78712, U.S.A.

Resume Cette note presente Ie developpement de nouveaux modeles de plusieurs phenomenes

de contact avec froctement. Une theorie de frottement statique pour des surfaces metalliques

seches est construite sur l'hypothese que les asperites elastoplastiques sont distribuees

aleatoirement sur la surface de contact. Dans cette theorie, lea forces d'adhesion sont prises

en compte et Ie glissement commence soit quand la force limite purement statique cst atteinte,

soit quand II y a des liaisons par asperites ou fractures. 11 esC possible de deduire de ces

theories des estimations du coefficient de frottement statique.

Abstroct - This note outlines development of new models of severol contact and friction phenomena. A

theory of static friction for dry metallic surfaces is derived on the assumption that elasto-plastic asperities

are rondomly distributed over the contact surface. Tn this theory, adhesion forces are taken into accOlmt

and sliding begins when either a purely static limit load is reached or there are fracture or asperity junctions.

It is possible to derive estimates of the static coefficient of friction from these theories.

INTRODUCTION

Over the last three years, considerable interest has arisen in new models of compliant interfaces of metallic

contact surfaces and of associated models of friction. In panicular. the model of Oden and Martins [1,2), which is based

on extensive experimental and theoretical evidence, is characterized by a constitutive law for normal compliance which

gives the normal contact stress (l'n as a power-law function of the approach of a. Thus, if lin is the nonnal

component of (relative) displaccment of particles on the contact surface rc of an elastic body n, and g denotes the

initial "gap" between opposing contact surfaces. then one has

IIln(l'n= cn (lin - g)+ (I)

where cn and mn are material constants and (.)+ dcnotes the positive part of the argument ('). The sliding

resistance is lhen characterizcd by a function with a similar form.

mT(l'T = cT (un - g)+

JOURNAL 1>E MECANIQUE THEORIQUE ET APPLIQUEE/JOURNAL OF THEORETICAL AND APPLIED MECHANICS

0750-7240/1988/47 8/S 2.801 © Gauthier-Villars

(2)

48

so that thc instantaneous coefficient of friction is

J. T. ODEN

(3)

The use of these simplc ftiction laws in applications 10 problems in elaslodynamics has been remarkably

successful. They can be lISed 10 resolve long-slanding paradoxes between experimental reslilts on dynamic friction

(secll]): they Icad 10 models of sliding friclion in excellent agreement with expcriment: lhey cnable one 10 calculate

acollstical frequcm;ies of sliding friction forces that are in good agreement with observed frequencies (a propeny not

exhibited by modcls based on classical friction laws): they lead 10 a tractable and useful mathcmatical theory of friction

[3,4,5J; and they provide the basis for the first mathematical rcsults on comroltheory for the motion of deformable

bodiesresisted by frictional forces (6J.

To datc, these friction laws are phcnomenological iu character, there being no results on what micromechanical

mechanisms are responsible for these paniclilar constitlltive laws. Thus, it is difficliit to develop extensions of these

models to friction problems in which viscoelaslic effects, plasticity. or wear of Ihe interfaces are present. In the prcsem

note, random microtopography models are proposed which are capable of capmring micromechanics of asperities and, by

statistical methods, producing macromodels of interface response. Only the formaJ apparatus for constructing lhese

models is discussed in this notc. Full details and applications are to be the sllbject of a laler paper.

RANDOM MICROTOP(){jHAPHY MODEL

We consider two such opposing surfaces I and 2 which are to ultimatcly come in COIll:lCt.Reference planes

defining thc mean asperity height of each sllrf,lce profile are eSl<lblished,and we characterize the shape of each proli lc by

introdllcing fllnctions zl and ~, given the height of asperities above the respective refcrcnce planes. i.e. the functions Zj

= zi (x.y). i = 1,2, with (x,y) a point in the parallel mean-height reference planes. define the prof1les of the rough

material surfaces I and 2, respectively. The distance h between planes is the separation of the surfaces, and the

dislance between actual opposing material points is denoted s. Thus. at a point (x.y) on the reference plane, we have

s = h - Z (4)

whcre Z is the slim surface (Cf Francis [7]).

Clearly, the lIndefonned surf:lces overlap whenevcr s (x.y) < 0

As the nonna! load pressing the surfaces together increases, thc approa\:h a decn:ases and at each minimum of the

function s <I microcontact nuclcatcs and expands due to local defom1ation of the surfaces.

It is well known in tribology that techniques used to produce engineering surfaces usually produce a Gaussian

distribUlion of the surface heights Zj' Moreover. the sum Z of lwo Gaussian surfaces is also Gaussian: indeed. if zJ and

z2 arc not exactly Gaussian. their sum surface will hc closer 10 Gaussian lhan cither surface. If lhe shape of an asperity

NUMERO SPECIAL, 1988

NEW MODELS OF FRICTION fOR NONLINEAR ELASTODYNAMICS PROBLEMS 49

is assumed 10 be paraboloidaJ, as have been done by several authors, then lhe pcak heights and curvatllres are correlated

random variables. with the result that a Gallssian dislribution of heights and cllrvatures may lead to a cllmliialive

probabilily distribution of surface heights which is non-Gaussian.

If the standard deviations of z, a1/dx, a271iJx2 are a, <1', <1", respectively, then the spectral width parameter

a is defined by a 1/2 = (l' (l' " I (l" 2 .

With these notations in hand, we can define the joint probability density function for random surface peak

heights and curvatures as:

(5)

where P = (3n.a)ll2, and ~ and T1 are non-dimensionalized peak heights and curvatures (~= 1>eak / (l' : T1= JC13 I

(J"--I2 ).

LeI Np = the number of sum peaks wilhin a nominal contact area AO:

v = area density of asperity peaks.

Then lhc number of peaks in AU is

Let t denote the non-dimensional separation.

t = h / (l'

Then, if X is a generic variable of interest associaled with the micromcchanics of an asperity, X being a fllnction of ~

and 11. then its expected va/lie is

E(X) (6)

and the macrocontact expectatioll of X is obtained by summing X over all microcontacts:

(7)

The notation T1' in (6) and (7) is inlended to indicate lhal if the approach of [7] is to be used, in which the shape of an

asperilY is paraboloidal only at its venex. then, instead of writing X is a function of peak heights and radii R of

JOURNAL DE MECANIQUE THEORtQUE ET APPlIQUEE/JOURNAl OF THEORETICAL AND APPLIED MECHANICS

50 J. T aDEN

rl' = [I + w (~ - t) ] TJ

w being a non-analytic function of ~ - t.

CONTACT MECHANICS OF AN ASPERITY

We now focus on the analysis of a typical asperity in contact with rigid flat. The asperity is a body of

revolution, symmetric about its z = x3 - a,xis, and subjected to adhesion pressures q on its exterior surfaces that are not

in contact Wilh a rigid flat, and to contact prcssures due to its indentation into the rigid flat. We shall assume that thc

asperity is composed of an isotropic viscopJastic materia) characterized by a collection of constitutive equations of the

internal state-variable type. The governing equalions in cartcsian coordinates are listcd as follows:

J. Momenmm Equations. Considering qllasi·static defonllations for now. we have

where

~j.j = 0

~j = lhc Cauchy stress tensor at a point x = (xl' x2' x3) e Q, nbeing lhe openmaterial domain of the asperity

and ~j,j is the divergcnce of the stress rate qj'

(8)

2. Strain-Rates. 'Ine rates £ij of the total infinilesimal strains are givcn in tcrms of the velocily gradients

Uj,j by thc lIslial relation

.E..

IJ1

2( u . + u· . )

I,J J.1

£e + i?ij ij

where €i~ and Ei; are the elastic and inelastic strain rates, rcspectively.

(9)

3. Constitutive Relations. A quite broad class of malerials is charactcrized by conslilutive equations of the

foml

0'.. = E"klcckJIJ IJ'.nEij = fij ( 0', Z )

zk = gk (0', z)

(10)

where Eijkl is I Iooke's lensor of elasticities, fij and gk are smooth functions of the stress 0' and of z. and z = (zk) is

a vector- or scalar-valucd internal state variable characlerizing various micromechanical changes in the material.

NtJMERO SPECIAL. 1988

NEW MODELS OF FRICTION FOR NONLINEAR ELASTODYNAMICS PROBLEMS S1

Constilutive models of this type can be used to characterize viscoelastic. elasti-plastic, and viscoplastic response of

metals by appropriate choices of the constitutive fllnclionals: eighl lheories of this form, proposed by theoretical and

experimental studies of different aUlhors on viscoplasticity of metals, are summarized in Bass and Oden [8]. As one

example. we mention the theory of [9] and 110], for which

ZI

(II)

Here qj = qj - 8ij O'kk/ 3 is the deviatoric stress, J2 is the second principal invariant of qj , DO and m are

material parameters and Hkl is :1 hardness tensor. In lhis case. 2 is scalar-valued and coincides Wilhthe "plaslic

work": Z = Eij qj' The parameler m is related to the strain·rate sensitivity of the materia\. If the material

n nresponse is such that the nonelaslic strains £ij are not associaled with volume changes. £ij can be replaced by its

ndeviatoric pan. Ejj'. This type of lhcory C:ID be IIsed 10 model strain hardening, cyclic stress-strain relations for

rale-dependent plasticity. and olher inelastic phenomena.

4. Boundary and Unilmera/ COll1act Conditions. The asperity C:1Il be viewed as a paraboloidal prolllrbance of

a defomlable half-spacc. The condilion that it cannOl penetrate the rigid flal is

on r 12)

where s = h - 7. is the distance from the aspcrily surface r to the flat. The contact region rc is defined by

{ X E r I u3 (x) s (x) }

£ 9(-)]

s

where q

OUlside fc we have the traction bollndary condilion

<>.3jnj = q : O'lj nj = ~j nj = 0 on r -fc

= q (s) is the adhesion pressure defined by

My £ 3q(s) = - [(-)

3£ S

(13)

(14)

JOURNAL DE MECANIQUE THF.ORIQUE ET APPLIQUEE/JOURNAL OF THEORETICAL AND APPLIED MECHANICS

52

where

Elsewhere, we have

J. T. ODEN

fiy = surface energy (y = Y1+ Y2- Y12= lotal surface energy of conmct surfaces)

E = molecular spacing

uj (x) ~ 0 as Ixl ~ + co

S. Initial Conditions. Smoolh functions uO(x) and zO(x) are prescribed such thaI for x en,

(IS)

SLIDING RESISTANCE

Extcnsive seqllences of experiments on sliding resistance of lhin films. involving 27 different materials and a

wide range of nonnalloads. are described in the papers of [11]. [12J, 113], and [14]. In all of thcse studies, il was

discovered lhat (on a microcontact interface) the imerfacial shear stress ts during sliding was a funclion of the normal

stress On = ~j nj ni' and, according 10 [71. their cmpirica1 findings suggest that

for light loads

(0.6 < m < 1.4) for intennediatc loads

0.0 ~ C (O'n) < 1.9;d (log ts)

d (log O'n)o for heavy loads (16)

wherein loads were varied over a factor of 10 or more and starting from IS MPa (light), 40 MPa (intcrnlediate) and 200

MPa (heavy). [7] points out that a good approximation to all of these cases is the simple quadratic function.

(17)

where cO,c l' c2 are material constants.

SAMPLE CALCULATION: DERIVATION OF THE COEFFICIENT OF FRICTION

Once a micro-shear resislance is characterized, lhe macro sliding resistance can be computed using the

slatis\ical summation procedures described earlier. To cite one interesling case, if the asperity is a pllrely elastic sphere

of radius R and modulus E, if (17) holds, and if the true microcolllact area is given by the Hertz sollition, e.g.

NUMERO SPECIAL, 1988

NEW MODELS OF FRICTION FOR NONLINEAR ELASTODYNAMICS PROBLEMS 53

hA(-)

0'(18)

then the lime-averaged expecled value of the coefficient of fIiclion µ can be calculated; vis.

wherein

hµ(-)

0'

(19)

hW(- )

0'macrocontact normal force

0" E Ao, p3/2 L\~Tl

6rr-J3

Here Ajj is the rnicrocontact area at grid point (ij) and O'n is Ihc microcontact normal stress given by Ihe classical

Henz theory.

The establishmelll of Ihe micromechanical normal response due 10 adhesion and 10 changes in lhe approach and

lhc establishmcnt of the micromechanical sliding resistance by (16) or (17) completes the dcscription of lhc principal

eomponenls of the frielionmodel. Specific cases must now be defined and analyzed before thc study can conlinlle.

Space limitations prevent further discussion of this approach 10 friction modeling. However. il should be clear

thaI it provides a unified approach toward incorporating micromechanical effects into a realistic model of interface

response. Applications of this theory and more-detailed discussions are to be given in a fonhcoming paper.

Acknowledgement

This work was supported by Contract F49602-86-C-0051 fromlhe Air Force Office of Scientific Rescarch.

REFERE~CES

Martins, J.A.C. and Oden. LT.. "Existence and Uniqueness Resliits for Dynamic ContactProblems with Nonlinear Normal and Friction Interface Laws. Journal of NonlinearAnalysis, Theory. Melhods and Applicatio1l. II. (3). 623-653, 1987.

2 Martins, LA.C. and Oden, J.T ..Metallic Surfaces, ConstitutivcApplications, lJni\'crsity of Arizona.

"A Simple Method for Dynamic Friction Effects onLaws for Enginccring Materials Thcory andJanuary. 1987.

JOURNAL DE MECANIQUE THEORIQUE ET APPLIQUEE/JOURNAL OF THEORETICAL AND APPLIED MECHANICS

54 J. T. aDEN

3 Rabier, P., Martins, LA.C., Oden. LT.. and Campos, L., "Existence and LocalUniqueness of Solutions to Contact Problems in Elasticity with Nonlinear Friction Laws".International JOl/rnal of Engineering Science, Vol. 24, No. II, P. 1755 - 1768, 1986.

4 Kikuchi, K. and Oden. J.T., Contact Problems in Elasticity.Philadelphia, 1987.

SIAM Publications.

5 Rabier, P. J. and Oden, J.T., "Solution to Signorini-Like Contact Problems ThroughInterface Models (pan I and II)", Journal Nonlinear Analysis. (to appear.)

6 White, L. and Oden, J.T.. "Dynamics and Control of Viscoelastic Solids with Contact andFriction Effects", (sllbmiued for publication.)

7 Francis, II.A., "Application of Spherical Indcntalion Mechanics to Reversible andIrreversible Contact between Rough Surfaces", Wear, 45, p. 261 - 269, 1977.

8 Bass, J.M. and Oden, J.T ..Problems in ViscoplasticilY",1987.

"Adaptive Finite Element Methods for a Class of EvolutionInternational Journal of Engineering Science. 25 (6) 623-653.

9 Bodner, S.R.. and Partom, Y.. "A Largc Deformation Elastic- Viscoplastic Analysis of aThick Walled Sphcrical Shcll". JOl/mal of Applied Mechanics. 39, p.751. 1972.

10 Bodner, S.R. and Stouffer, D.C.,Anisotropic Plastic Flow of Melals".757-764, 1979.

"Constitutive Modcl for the Deformation InducedInternational Journal of Engineering Science. 17, p.

II Boyd. 1. and Robertson, B.P.. "The Friction Propertics of Various Lubricants at HighPressures". Trar~~.ASME. 67. pp.51-59. 1945.

12 Briscoe. B.Joo SCTliton. B.. and Willis. F. Roo "The Shcar Strenglh of Thin LubricantFilms", Proc. Roy. Soc. (London), A333, p. 99·114. 1973.

13 Towle, L.Coo "Shear Strength and Polymer Friction", Am. Chem. Soc. Polymer Scienceand Technology, Vol. 5A, pp. 179-189, 1974.

14 Briscoe, B.J., and Tabor, D.. "The Effect of Pressllre on the Frictional Properties ofPolymers", Wear. 34, p. 29 - 38, 1975.

NUMERO SPECIAl., 1988