Mechanics of Material Interfaces, edited by A.P.S. Selvadurai and G.Z. VoyiadjisElsevier Scit>llcePuhlishers B.V.. Amsterdam, 1986 - Printed in The Netherlands

INTERFACE MODELS, VARIATIONAL PRINCIPLES AND NUMERICAL SOLUTIONS FOR OYNANIC

FRICTION PROBLEMS

J.A.C. MARTINS and J.T. ODEN

Texas Institute for Computational Mechanics, Aerospace Engineering and

\ Engineering Mechanics Department, The University of Texas at Austin

J..ABSTRACT

The present paper summarizes several results of our current research ondynamic friction prOblems involving dry metallic bodies. We develop a modelof interface response which incorporates a constitutive equation for the nor-mal deformabi1ity of the interface and the Coulomb's law of friction. Wedevelop continuum models, variational principles and finite elements approxi-mations for e1astodynamics and steady sliding frictional problems. Existenceand uniqueness for the latter hold under appropriate assumptions on thedata. The dynamic stability of steady frictional sliding is numericallystudied and, assuming no distinction between static and kinetic coefficientsof friction. we demonstrate numerically the possibility of the occurrence ofapparent distinctions between those coefficients and the occurrence of stick-slip relaxation oscillations.

HlTRODUCTION. AN INTERFACE t()OEL

Stick-slip motion vs. steady sliding

Let us consider a body (slider) in frictional contact with a fixed surface.

as depicted in Fig. 1a. The slider is intended to move along the surface with

some (small) tangential velocity UT, this velocity being impressed upon it by

some driving mechanism which itself possesses specific elastic and damping

properties. As many examples in our daily life teach us, what frequently

happens is that the desired steady sliding at the prescribed velocity UT is

not achieved. Instead, the body advances in an intermittent fashion with suc-

cessive alternate periods of rest and sudden sliding (see Fig. 2a). These

relaxation type oscillations are usually known as stick-slip motion (after

Bowden and Leben (ref.2».

Since the work of Blok (ref.3) in 1940, it is known that the essential con-

dition for the'occurrence of stick-slip oscillations in sliding motion is a

decrease of the friction force with the increase, from zero, of the sliding

velocity. Since then, most of the experimental studies of the stick-slip

oscillations have concentrated on the determination of the functional depend-

ence of the coefficient of friction with the sliding velocity. The overwhelm-

ing majority of such experimental studies involve experimental methods in

3

4

which the tangential motion and the tangential forces on a slider, such as that

in Fig. 1. are carefully monitored while the net normal force is assumed to be

constant. Then, all the variations in the observed friction force are assumed

to be a consequence of corresponding variations of the coefficient of friction

with sliding velocity.

Unfortunately, the resul ts of 1iterally hundreds of experimental tests have

little, if any, agreement. While it may be accepted that different pairs of

metals in contact and different lubrication conditions produce characteristics

(coefficient of friction vs. sliding velocity plots) with clearly distinct

shapes, it is known that even for the same combination of metals and lubrica-

tion conditions, changes on only the dynamic properties of the experimental

set up (e.g. stiffness) or on the driving velocity, produce dramatic changes

in the observed characteristics.

It thus appears that these experimental characteristics are not at all an

intrinsic property of the pairs of surfaces in contact. Such a point of view

has been expressed by several authors who attribute apparent decreases of the

coefficient of friction with the increase in sliding velocity to the occur-

rence of micro-vibrations of the slider in the direction normal to the contact

surface. In fact, To 1stoi observed in (ref.4) that the forward movements of a

slider during stick-slip motion invariably occur in strict syncronism with

upward jumps. Oscillograms of the normal contact oscillations which occur

during the sliding portions of the stick-slip cycles were obtained by Tolstoi,

Borisova and Grigorova (ref.5). An examination of these oscillations revealed

a fundamental frequency consistent with the normal interface stiffness pro-

perties (see Budanov, Kudinov and Tolstoi (ref.G». Especially interesting

is the experimental observation of Tolstoi (ref.4) that external normal

damping of the normal oscillations of a slider can eliminate its stick-slip

motion, and produce a smooth sliding with no quantitative difference between

the so-called static and kinetic coefficients of friction.

These observations (and several others described in (ref.1» led us to the

conc 1us ion tha t all appIWp1L.ULte model. oo~. &Udillg oJUc..ti.o 11 mu.6t 'ulCOltpolULte

pll!Jl>.<£,aU1j ltea.6onabte nolltrKll contact CDIu:UUOIIJ.. A constitutive equation for

the normal deformability of the interface and the Coulomb's law of friction

will be thus the essential ingredients of the interface model to be developed

in the following.

A model of interface responseConsider a continuous material body B, in contact with another material

body B1 over a contact surface rC c aB.

The contact surface rC represents the boundary of the parent bulk materialof which the body B is composed. One can regard it as parallel to a surfacerepresenting the average surface height of the asperities of the physical bodyB. We suppose that rC has a well defined exterior normal vector ~.

For simplicity of presentation, but with easy generalization, we assumethat the body B1 is rigid and ideally flat. In the spirit of Fig. Ib, we alsoassume that the body B1 does not mo~e on the direction of ~, but that it canmove with some prescribed velocity ~~ parallel to rC'

We suppose that the actual interface (asperities, oxide film, adsorbe~ gas,work-hardened material, etc.) is initially of thickness to as shown in Fig. 3.The initial gap g between Band Bl is defined as the distance, along thedirection of the normal vector ~, between the highest asperities of the body Band the flat surface of Bl in the reference (undeformed) configuration. Theinterface thickness after deformation is denoted by t in Fig. 3, and the actualdisplacement of rC in the direction of ~ is un = ~ .~. Thus, the approach ofthe material contact surfaces is a = to - t = (un - g)+ where (.)+ = max{O.·}.

On the other hand, if ~T = ~ - un ~ denotes the tangential velocity of thepoints on fe' ~hen the relative sliding velocity between bodies Band Bl isequal to ~T - ~~. Here (") denotes partial differentiation with respect totime a~ ( ).

Denoting by a and aT the normal and tangential (frictional) stresses onn -rC' respectively, the constitutive relations for the interface adopted hereare the following:

Normal interface response

5

Friction conditions

mTIqT1 ~ cT(un-g)+

andm .

u > g ===> ~ I a I < c (u -g) T ==> U - UC = 0n \ -T T n + -T -T -and

Here cn' mn, bn, in' cT' mr are material parameters characterizing theinterface and are to be determined experiwental1y.

(1)

(2)

6

The following remarks provide an explanation and interpretation of theserelations:

1. The interface constitutive equation (1) combines a nonlinear power-lawm

elastic contribution -a~ ; cn(un-g)+n with a nonlinear dissipative component

given by -a~ = bn(Un-g)~n un' The form of the nonlinearly elastic contribu-tion a~ is consistent with experimental observations summarized in (ref.1,7)for the case of interfaces subjected to low nominal pressures, characteristicof sliding interfaces:

edan I = 0 (a = (un-g)+)da a=O

m-a~ .. a n with 2 ::.mn ::.3.33.

I

Tables with experimental values of the constants cn and mn for severalcombinations of materials and surface finishes can be found in (ref.7).

We also observe that, for metallic surfaces, after a period of rubbing andsmoothing of the interface, its normal response is essentially elastic, ifsevere wear is prevented during the process of sliding (ref.l).

2. The nonlinear dissipative term o~ is designed to model, only in anapproximate manner, the hysteresis loops which are known to result from theactual e1asto-plastic behavior of the interface asperities. Indeed, the con-stitutive equation (1) allows for the approximation of the loading paths ofthe form presented in Fig. 4a, which are known to occur if the surfaces areallowed to unload completely (Thornley et al. (ref.8)) by loops of the formpresented in Fig. 4b. The idea of a similar approximation was proposed byHunt and Crossley (ref.g) for vibroimpact phenomena involving macroscopicHertzian contacts. For small energy losses, the correlation between thedamping coefficient bn and the energy loss per cycle of contact is readilyobtainable (ref.g).

3. The friction law (2) is a slight generalization of Coulomb's frictionlaw, which is recovered if mn = ffir and'normal dissipative effects are negli-gible (ad: 0). In that case, the maximum value of the modulus of the fric-ntion stress is equal to the product of ~(= cT/cn) and the normal pressure(Iq~axi = µIanl), where µ is the usual coefficient of friction. The law (2)allows for a dependence of the coefficient of friction on the normal contactpressure. The general form for this dependence, consistent with (2) (againfor negligible ad), is the following:

n

m 1mClonlQ with Q = (mT/mn)-l and C = CT/cn

T n

4. In using the friction law (2), we aMwne .tha.t 110 d.i.6UIlc.-ti.oll be.tweell

coeUic.i.eltth 00 l>.ta..tic. and tU.1'le.U.c. 6.>Uc..tion ex..u.t6. Here we fo11ow, with avery simple model, the essential ideas of Tolstoi and co-workers (ref.4,5.6).In the next sections, we show that apparent decreases of measurable coeffi-cients of friction can be the result of dynamic instabilities which are aconsequence of the inherent non-symmetry of operators governing problems withfriction.

SOME FRICTIONAL CONTACT PROBLEMS IN ELASTICITY THEORYA problem in elastodynamics

We begin by considering a metallic body (Fig. 5) the interior of which isan open bounded domain in RN (N=2 or 3). Points (particles) in n withcartesian coordinates xi' 1 ~ i ~ N, relative to a fixed coordinate frame aredenoted by ~ (= (xl x2"'" xN» and the volume measure by dx (= dXl dx2 ...dxn)· The smooth boundary f of n contains four open subsets fD, rF, fE' fC'such that

7

f = uf ,r n r. = fJa a a 8 Va F 8, a,B E {O,F,E,C} (3)

and we denote by r. the following open subset of f

r = int(f-fD) (4 )

Points on r with cartesian coordinates si' 1 ~ i ~N, are denoted by ~ and thesurface measure is denoted by ds.

We assume that the metallic body has a linearly elastic behavior character-ized by the generalized Hooke's law,

Dij(~) = Eijkt uk,l' 1 ~ i, j, k, 1 ~ N (5)

where the Dij denote the components of the Cauchy stress tensor, ~ =(u1' u2' ..., uN) = ~(~,t) is the vector field of displacements, the componentsof which are sufficiently smooth functions of position (~) and time (t) andEijkl = Eijkl(~) are the usual elasticities of the material. In (5) andthroughout this work ( )'l denotes partial differentiation with respect to xland the summation convention is used.

We suppose that body forces with compone~ts of force per unit volume bi =bi (~,t), 1 ~ i ~ N, act in the body. We denote by p = p(~) the mass densityof the material of which the body is composed. We also suppose that the bodyis fixed on rD and that tractions ti = ti(~,t), 1 ~ i ~ N, are prescribed onfF' The body may be elastically supported on fE by linear springs of moduliK .. = K ..(s). The undeformed configuration of those springs corresponds tolJ lJ - E E

the displacements U. = U. (s) on fE'1 1_

8

We also suppose that the body may come in contact along a (candidate) con-tact surface fC with a rough foundation which slides by the material contact

'C 'Csurface with a velocity ~T = ~T(~) tangent to fC' We assume that g, cn' cT'b are given as sufficiently smooth functions of s, have the physical meaningn -mentioned earlier, and that the real numbers mn, mT, ~n are also specified.

Assuming sufficient smoothness for all the functions involved, the equa-tions governing this elastodynamics problem, for a time interval IO,Tl, aregrouped as follows:

Linear momentum equations

(J •• (u).. + b. = pu. in n x (O,T)lJ - J 1 1

where the (Jij satisfy the constitutive equations (5).

Boundary conditions

ui = 0 on fO x (O,T)

(G)

(7)

I

cr .. (u)n.1J - J

(J •• (u) n·1J - J

ti on rF x (O,T)

E-K ..(u.-U.) on fE x (O,T)1J J J

(8 )

(9)

(I) and (2) on fC x (O,T)

Initial conditions

(10)

(11 )

A weak formulation for the elastodynamics problem above leads to a varia-tional statement in the form of an inequality, which is a version of theprinciple of virtual power. Taking, for Simplicity p = 1, bn = 0, we havethus the following

Problem 1. Find a function u such that

<~(t). y-~(tt> + a(~(t). y-~(t» + «p(u(t»), v-u(tf> + j(~(t),,!)

- j(~(t), ~(t» ;:. <!(t). y-~(ti> \Iv € V (12)

with the initial conditions:

~(O) = iTo ~(O) = ~1 (13)

1 NIn the above V = {y € (H (n») Iy(y) = Q a.e. on fO} denotes the space ofadmissable displacements (velocities) and y denotes the trace operator (ref.10); «.;> denotes duality pairing on V'xV where V' is the dual space of V;

9

(14 )

v E V

€ V

u,v E V

u,v

~,!) E y(V)

a(~,y) = ao(~'Y) + aE(Y(~)'Y(Y» denotes the virtual work (power) produced

by the deformdtion of the linearly elastic body n plus the linear springs on

f E; <P (~),i> denotes the vit·tua1 work (power) produced by the norma 1

deformation of the contact interface; j(~,y) denotes the virtual power pro-

duced by the frictional sliding and ~(t),~ denotes the virtual work

(power) produced by the external forces on the displacement (velocity) y.Assuming that the data satisfy suitable smoothness requirements, the defini-

tions of these forms are the following

a (u,v) = J E. "" uk 0 v. J' dx u ,v E Vo - - Q 1J...... ," 1, - -

aE(f;,n) = J K.. f;. n· ds- - 1J J 1fE

Jmn

~(~),~ = cn(un-g)+ vndsfC

. J mT·CJ(~,~) = cT(un-g)+ I~T-~TldSrC

~(t),~ = f b(t) ·vdx + f t(t)·y(v)ds + J K ..UEv.ds~;",.r - - - - 1J J 1Il fF fE

..

We nOI'/turn our attention to the approximation of Problem 1 by a family

of regularized problems more suitable for computational purposes.

Toward this end, we approximate the friction functional j: V x V -> IR,

which is nondifferentiab1e in the second argument (velocity), by a family of

regularized functionals j which are convex and differentiable in the secondcargument:

(15 )u,v E Vj (u, v)c - - J

mT·CcT(un-g)+ ~c(~T-~T) ds

rC

Here the function ~c: (Lq(rc»N -> Lq(fC) is a continuously differentiable

approximation of the function I· I: (Lq(fC»N -> Lq(fC) (ref.l,11,12).

The partial derivative of jc with respect to the second argument, at

(~,~) in the direction of v, is then given by:

u,v,w E V

<ic(~'~)' ? ;:q2 jc(~'~)' Y>J

mT ·C= cT(un-g)+ (~c(~T-~T) (~T)]dS

rcwhere ~c(~)(~) ;:~~(~)(~), ~,~ E (Lq(fC»N, is the directional

~c at ~ on the direction of ').

We now define the regularized form of Problem 1:

(16)

derivative of

10

Problem 1. Find a function u such thatE -E

<::~E(t), ~ + a(~E(t), y) + ~(~E(t», ~

+ ~E(~t(t), ~E(t», 9- = <!(t), ? Vv e V

with the initial conditions

(17)

~E(O) = ~l (18)

The steady frictional sliding of a metallic bodyWe consider here a metallic body satisfying the same constitutive equation

and submitted to the same type of forces and boundary conditions as in theprevious section. The essential difference is that now we seek an equili-brium position of the body in unilateral contact with the rough moving founda-tion. Denoting by ! = !(~)the unit tangent ve~~or at each point of rCparallel to the prescribed (non-zero) velocity ~T of the rough foundation,the friction conditions (2) simplify to the

Steady sliding friction equation:

I

(1g)

This is the essential modification to be introduced in the governingequations (6-10). Obviously we have ~ = ~ = 9 and all the functions areassumed independent of time. Also, the domains n x (O,T), rO x (O,T), etc.in (6-10) should be replaced by n, fD, etc.

Let us now assume that the domain Q is such that

n € CO,l (20)

and that the data satisfy the following conditions (with 1 ~ i ,j,k,t ~ N)

EijkR. E L"'(fl);EijkR. = EjikR. = Eijtk = Ektij a.e. in fl

3Mo > D such that . ".lax II EijkR."""n~ Mol~l.J ,k,R.~N

3ClO> 0 such that EijkiAkiAij ::..ClOAilij a.e. in \!, for every

symmetri c array A ..lJ

K .. E L"'(1:);K .. = K .. a.e. on 1:; K .. = 0 a.e. on fF U fClJ lJ Jl lJ

3~\E > 0 such that ".la~ \I Kijll'"1:~ ME1~1,J~N '

ibE> 0 such that K. ·a.a. > ClEa.a. a.e. on fE for every vector a,lJ J 1 - 1 1 1

(21)

(22)

bi E L2(n); ti e Lq' (E), ti = 0 a.e. on fE U fC; u; € Lq' (E);

N NT

1' € L~(E), E T? = 1, E T.n. = 0 a.e. on r.; g E Lq(E);

i=l 1 i=l 1 1

cn'CT E L~(r.), cn = cT = 0 a.e. on fF U fE

In the above q and q' satisfy

q = l+mo; me = max {mn,mT}; l/q+l/q'

where mn,'"1 € R sati sfy

(23)

(24 )

(25)

11

The virtual work produced by the friction stresses on the displacement ~

is now given by

fmT

q(~),~ :: .q£(~,q),~ = - cT(un-g)+ ~'~Tds ~,~ € V (26)fC

It is then possible to show that the classical formulation of the steady

sliding problem is equivalent, in an appropriate sense (ref.!3), to the

following variational formulation.

Problem 2. Find a function ~O € V such that

a(~O'~) + q(~O),t> + .q(~o),t> = <f,f> 'Iv E V (27)

If the coefficient of friction or the applied forces and initial gap are

"sufficiently small" it is also possible to prove (ref. 13) existence and

uniqueness for the steady sliding problem:

Pruposition. Let assumptions (3), (4) and (20) on n. rand E hold

'together with assumptions (21-25) on the data. Let meas fO > 0 or meas fE> O.Let the following assumptions also hold:

There exists ~* E V such that y(~*) . ~ = g a.e. E

mn,rur < 3 if N = 3

(28)

(29)

(30)

Then there exists a constant C = C(mT,q,N,n,E) > 0, such that if, in addition,

m -1IIcT"""fc (Ilfllv' + MII~*llv) T

a~

with i>l dud a denoting tile continuity and coercivity constants for the bi-

linear form a(·,.), then there exists a R> 0 such that Problem 2 has a unique

solution in the closed ball K = {~ G V II~ - ~*lIv ~ R}.

12

Dynamic stability of the steady frictional slidingWe are interested here in the analysis of the dynamic stability of the

steady sliding equilibrium position ~O. We restrict ourselves to two dimen-sional problems (N=2).

A natural idea is to study the behavior of a linearized version of (12,17)in the neighborhood of ~O' Such linearization is possible because the non-linear operator

A : V ---> V'

<li(~),?= a(~,~) + <1'(~),? + q(~).? ~.~E V

emerging from (27) and in (12,17) for ~T(~) = g, I~i(~)l > £ > 0 a.e. ~ E fC'is continuously differentiable in V (ref.13).

A first step toward the stability analysis of the equilibrium position ~Ois then the formulation of the eigenvalue problem associated with thesesguilinear form a(·,·) defined by the derivative of A at ~ = ~O (fixed):

a(w,v) = f E"I<.Wk .v .. dx + J Ki .w.Vjds + J K w v ds + J KT W v ds-- fl 1J .. , .. 1,J f JJ f nnn r nnT

E C C

where superimposed bars denote complex conjugation, vT = ~T . : = y(~) , ~,

K denotes the linearized normal stiffness of the foundation at the equili-nbruim position ~O and KTn denotes the coupling stiffness coefficient betweennormal displacements and tangential stresses, also at the equilibruim position

~O' i.e.,

m -1K = m c (u _ g) nn n n on +

m -1KTn = -mTcT(uon - g)+T

The eigenvalue problem associated with the non-symmetric form a(, •. ) isthen:

Problem 3. Find the values A € ~ for which there exists w E V, wi 0,such that

a(~,~) + A2(~,y) = 0 Vy E V

Here,·the L2~ii1nerproduct is (~,'{)= L uividx, \!,!' E (L2(Q»)N.1/

(31)

FINITE ELEMENT APPROXIMATIONS AND ALGORITHMSThe discrete problems

Using standard finite element procedures, approximate versions of Problems1£. 2 and 3 can be constructed in finite dimensional subspaces Vh(C: VC:V').Here, we shall restrict ourselves to present the form of the resulting systemsof equations for those discrete problems.

The finite elements approximation of Problem 1£ leads to the following

system of nonlinear ordinary differential equations,

~ r (t) + ~ r(t) + ~(r(t) + ~(r(t),t(t) = f(t)

with the initial conditions,

(32)

13

(33 )

Here r(t), r(t) and r(t) denote the column vectors of nodal displacements,

velocities and accelerations, respectively; M(K) is the standard mass (stiff-

ness) matrix; f(t) the vector of consistent nodal forces; ~(r)and ~(r,~)are

the vectors of nodal normal and friction forces on fC and rO(rl) is the vector

of initial nodal displacements (velocities).

Using the same notations above, the steady sliding Problem 2 leads to the

following system of nonlinear algebraic equations (set ~ = i = 0 in (32»,

~ ~O + P(~O)+ ~(~O,Q)= ~ (34)

and Problem 3 leads to the following algebraic eigenvalue problem

(35)

where

n TnK = K + ~ (~O)+ ~ (~O'~)

n def a ()K (r) --= -- P r-. ar - .KTn(r,r) ~~!~ J(r,r)- - - ar - - -

We stress the fact that the matrix ~Tn is not symmetric. This results

from the inherent non-symmetry of the coupling between normal and tangential

variables on the interface: normal stresses depend only upon the magnitude

of the normal penetration, but the friction stresses depend not only upon

the tangential velocities but also upon the normal penetration. It is pre-

cisely this non-symmetry that makes the eigenvalue problem (35) different

from standard structural dynamics eigenvalue problems. Effects of this lack

of symmetry on the dynamic response of the elastic body are discussed in the

nUlnerical examples.

Finally we observe that disSipative effects other than dry friction (linear

viscous terms, normal contact damping of the form (1» can be easily added to

the developments of the previous sections. The major consequence for the

eigenvalue problem (35) is then the introduction of additional terms, linear

in ~.

14

AlgorithmsThe algorithms that we have used for solving the discrete dynamic system

(32) involve variants of standard schemes in use in nonlinear structuralmechanics calculations: Newmark's method and explicit central-differencetechnique, both associated with Newton-Raphson iterations within each timestep. For further details see (ref.l).

In order to compute the solutions of (34) for a certain range of values ofcT' we subdivide the interval [O,cTJ into a prescribed number NINCT of incre-ments t.cT =cT/NINCT and, for the Kth increment, K=O, ...,NINCT, we again usethe Newton-Raphson method to solve the nonlinear system (34) at each value

of cT'For the computed equilibrium positi01 at each increment K, the nonsym-

metric eigenvalue problem (35) is solved using standard eigenvalue routines.

NUMERICAL EXAMPLESThe steady sliding of a block on a moving foundation

We consider here a block sliding, with friction, on a moving foundation(see Fig, 6). We assume that the block has a linearly elastic behavior with

6 3 -1-2a Young's modulus E = 1.4 x 10 (10 Kg cm s ) and a Poisson's ratiov = 0.25. For simplicity, we assume that the body is in a state of planestrain. The geometry, total mass (M), total weight (W), total tangentialstiffness (Kx) and contact properties are given as follows:

L=48.8cm M=450Kg }3 -2H=30.5cm W=45010Kgcms (36)

10 3 -3.5-2cn = 10 10 Kg cm s, mn = ~ = 2.5, cT = µcn

BO = 30.5 cm (37)

The finite elements model consists of a 4x3 mesh of nine-node isoparametricelements as depicted in Fig. 7.

A simple calculation reveals that for the given geometry a necessary con-dition for equilibrium is that µ < L/H = 1.6; for µ = 1.6 the body will beginto tumble (with unbounded rotations). In this section we compute the steadysliding equilibrium positions of the block for several values of µ in theadmissable range [0,1.6) and for each of those configurations we solve theeigenvalue problem (35).

In Fig. 7, we show the deformed mesh configurations at several values of µ.The most important information however comes from the corresponding eigen-values. In the absence of damping, all the eigenvalues are pure imaginaryfor small vaiues of µ. For µ ~ 0.32 (see Fig. 8) the occurrence of eigen-

values with positive real part is observed. This implies that for such values

of µ the steady sliding is dynamically unstable.

As might be expected, for the small magnitude of the applied forces, the

block behaves much like a rigid body (see Fig. 7). For this reason, we have

performed also the same computations, for the same block of Fig. 6 and with

the same data (36,37), assuming that the block is a rigid body in plane

motion. The corresponding three degrees of freedom are indicated in Fig. 6:

the sliding and penetrating (normal) displacements of the center of mass,

uxG and uyG' and the rotation, ue.

The eigenvalues associated with the tangential displacement are a conjugate

imaginary pair (±VVK 1M) which does not vary with µ. In Fig. 9 we show thexevolution of the four eigenvalues of (37) associated with the normal and

rotational freedoms. As expected all of these eigenvalues compare favorably

with the eigenvalues associated with similar modes obtained with the finite

elements model (ref. 1).

Perturbation of the steady sliding of a block on a moving foundation

In this section we show the effects of the eigenvalue structure of (35) on

the motion of the block of Fig. 6. For each set of data (in particular some

value of µ) we solve numerically the nonlinear equations of motion (32) with

the following initial conditions: the initial displacements are prescribed

as those corresponding to the steady sliding equilibrium configuration appro-

priate for the value of µ considered; initial velocities represent a small

perturbation (upwards) of the translational velocity (uxG = 0.0,

U G = -O.Olcm s-l, U = 0.0). Various cases were run assuming either ay e

linearly elastic block (the finite element model of Fig. 7) or a rigid block

(the three degrees of freedom model of Fig. 6). Other than friction, the

only dissipative effect considered with the first model was a normal contact

dissipation of the forol in (1) with b = 0.381 x 1010 (103Kg.cm-4 . s-I) andntn = 2.5; with the second model the results shown here were obtained with

linear viscous damping characterized by diagonal coefficients Cx' Cy' CeoOther data is given in the figures. The finite element results were obtained

using the central-difference algorithm with 6tmax = 3 x 10-6 s and a

diagonalized mass matrix. The rigid body motions were calculated usingNewmark's method with 6t = 10-5 s.max

We first consider the small friction case, i.e., when (in the absence of

damping) the eigenvalues A of (35) are all imaginary. In this case, starting

from the initial conditions mentioned above an apparently stable small-

amplitude oscillation is obtained (ref.l).

15

16

More interesting facts occur if ~ is sufficiently large that some of the

eigenvalues of (35) have (even in the presence of some damping) positive real

parts. The following remarks summarize our interpretation of the numerical

results in this situation:

Due to the instability associated with normal and rotational modes, the

normal and rotational oscillations grow (Fig. 10).

The variation of the normal force on the contact produces changes in the

sliding friction force which in turn produce a tangential oscillation.

The tangential oscillation may then become suffiCiently large that, for

small values of the belt velocity uC the points of the body on the contact

surface attain the velocity UC and ~he body sticks for short intervals ofxtime (see Figs. 11, 15).

With the increase in magnitude of the normal oscillations, actual normal

jumps of the body may occur (see Figs. 10, 14).

The repeated periods of adhesion have the result of decreasing the average

value of the friction force on the contact and, due to the absence of equil i-brium with the restoring force on the tangential spring, the tangential

displacement of the center of mass decreases (see Figs. 12, 13).

It follows that for given geometry and material data, one of two following

situations may occur: (a) for values of u; larger than some critical value.

the normal, rotational and tangential oscillations evolve to what appears to

be a steady oscillation with successive periods of adhesion and sliding, the

average values of the friction force and of the spring elongation being

smaller than those corresponding to the steady sliding equilibrium position'C -1 . C

(see Fig. 12, U = 0.287 cm s ). (b) for values of U lower than the criti-x xcal value, and at a sufficiently small value of the spring elongation, the

normal (and rotational) damping is able to damp out the normal (and rotation-

a1) oscillation and the body sticks (see Figs. 12, 13) since the restoring

force of the spring is then smaller than the maximum available friction force.

Thw., mOlU-toiUltg the .&pIt.Olg dOH9aWlM, M .W o6telt done .til 6iUc.ti.ollex.peJL.UtlentA, ea..6e (a.) loouf.d be pelt.ce.tved M all appalten.tttj &mooth .&liding (uUh

a coe66.i.ci.ent 06 k..i/leU.c 6tic.ti.olt 6maUVt thall the CDe66.i.ci.ent 06 &.tiLti.c

6tic.ti.oll altd Ca6e (b) teJOuf.dbe peltCe.tved M &t.ic.k-6Up mo.t'..wll.

Acknowledgement. The authors gratefully acknowledge support of this work by

the Air Force Office of Scientific Research under Contract F4950-84-0024.

REFERENCESJ.T. Oden and J.A.C. Martins, "Models and Computational Methods forDynamic Friction Phenomena," Computer Methods in App1i. Mech. and Eng.(to appear)

2 F.P. Bowden and L. Leben, "The Nature of Sliding and the Analysis ofFriction," Proc. Roy. Soc, Lond., A169, pp. 371-391, 1939.

17

3 H. B10k, "Fundamental Mechanical Aspects of Boundary Lubrication,S.A.E. Journal, 46(2), pp. 54-68, 1940.

4 O.M. Tolstoi, "Significance of the Normal Degree of Freedom and Ilatura1Normal Vibrations Contact Friction. Wear, 10, pp. 199-213, 1967.

5 D.M. To1stoi. G.A. Borisova and S.R. Grigorova, "Role of Intrinsic ContactOscillations in Normal Direction During Friction," Nature of the Frictionof Solids, Nauka i Tekhnika, p. 116, Minsk, 1971.

6 B.V. Budanov, V.A. Kudinov and a.M. Tolstoi, "Interaction of Friction andVibration," Trenie i Iznos, vol. 1. No.!, pp. 79-89, 1980.

7 N. Back, M. Burdekin and A. Cowley, "Review of the research on Fixed andSliding Joints," Proc. 13th International Machine Tool, Des. and Res. Conf ..ed. by S.A. Tobias and F. Koenigsberger. MacMillan, London, 1973,

8 R.H. Thornley, R. Connolly, M.M. Barash and F. Koenigsberger, "The Effectof Surface Topography Upon the Static Stiffness of Machine Tool Joints,"Int. J. Mach. Tool Des. Res., 5, pp. 57-74, 1965.

9 K.H. Hunt and F.R.E. Crossley, "Coefficient of Restitution Interpreted asDamping in Vibroimpact," Journal of Applied Mechanics, pp. 440-445, June,1975.

10 Adams, Sobolev Spaces, Academic Press, N.Y., 1977.11 G. Duvaut and J.L. Lions, Inegualities in Mechanics and Physics Springer-

Verlag, Berlin, Heidelberg, New York, 1976.12 J.A.C. Martins and J.T. Oden, "A Numerical Analysis of a Class of Problems

in Elastodynamics with Friction," Comput. Meths. Appl. Mech. Engrg., 40,pp. 327-360,1983.

13 J.A.C. Martins, Some Dynamic Frictional Contact Problems involvin Metal-lic Bodies, Ph.D. dissertation, Unlverslty of exas at Austin, 1985.-----

N

~Iider

(0) ~~- rl-JJ.»»»»>m7JI

driver fixed

N

slider't:l

(b).. M~ r+-J.L

-'

UT

driver

Fi g. 1. Models of two (equivalent)sliding systems which may have stick-slip oscillations.

10J

lbl

slip

stlck

slip

stick

Fig. 2. Typical traces of stick-slip motion for systems (a) and(b) in Fig. 1.

Un / rc (initial contact)

-~-::~,~-----------------t--¥=-~~-~-~~~~~~=~~-,~~nlfi:u]rat ion)"......,i \ ..,'''__j " ........... ....._.. _,'" t

~~ ", " --'-.'/ ',/' _,/ ~' t 0

;;jJ)/j/f~/7/777fiiimWmJ777/7Body B1

18

T

tg

~

Body B (reference )r: configuration

Tn

Fig. 3. Initial gap, nonna1 displacement and penetrating approach atthe contact surface rC'

a(e) (b)

Fig. 4. Hysteresis loops for the normal deformation of the interface(schematic). (a) Experimentally observed loop, under quasi-staticloading conditions. (b) Hysteresis loop modelled by the constitutiveequation (1 ) under dynamic loading conditions.

Fig. 5. Geometry and notation used inelastodynamics problem with slidingfriction.

Fig. 6. A block sliding with frictionon a moving belt.

19

NOO£ 29r./c

0000 a

0000 0000 CCO

• 18m COOO0

COoo OCCO 000

cooO a0000 OCCO 000

--

Fig. 7. Oeformed configurations of a linearly elastic blockfor the steady sliding equilibrium configurations at severalvalues of u .

II. [0.32.0.64)

;.( ;.( x ':[1.Z8.1.~2]

[1.44.1.44]

o .,..., [1.28.1.52]

.. _ 0[1.20.'.52]

o o o o o o o

PEAL(0.40.0.88 )~

Fig. 8. Eigenvalues with positive real parts for a finite elementsdiscretization of a block in steady sliding on a moving belt.

20

• REAl.••++•++++t

(f;+i

FiG. 9. Evolution of the Eigenvalues on the complex planefor the successive equil ibrium positions obtained '/lithin-creasing coefficient of friction.

Fig. l~. Phase plane plot of the initial partof the normal oscillation of the center of mass.Rigid body model \1ith .the data (36) and Bo= 14 cm,]=70(103 Kg cmZ), K =2388 (103 Kg s-/) C /2 M w =0e005, C /2 M w O=O~04. C /2 I we =U.04;xµ=1; xo =O.127Ycm s-l~ e 0

x

z§1it..r

U: -0.l27 em ,-I

r21

-e.500

...v·z~'":J'">-VoJ:~~

-0.750 ~ .:.I>-

-1.000

0.0025 oJlloso .0')75~ ~110101('T~rr,EIII~L 01 PLll· I 1T1 0TIlE COJITACT , Rf A' , .m

Fig. 11, Phase plane plot of the tangential motion of the points of theblock on the contact surface: Rigid body model with the data of Fig. 10except for Kx = 4441(103 Kg S-2).

~o0.10 E...

>:...... '"a: '"-' ,.a. ,.,- a.00

-' '"o.os :5 ~~ 5w,-'~:! ~>- ....

TII'£ (.)0.\ 0.3 o.~ 0.5 O.G

Fig. 12. Influence of the velocity of the support on the tan-gential motion of the center of mass. Rigid body model withdata of Fig. 10.

22

0.013'

-t-0.00 T1UEhl

+0.Z4

~lg. 13. Evolution of the spring elongation for two velocitiesU. Finite element model with the data (36) and K = 11100 KgS~2, Bo=30.5 cm, µ= 0.6. x

-0.5

Fig. 14. Evolution of the normal contact stress at node 29during the contact portion of one cycle of normal o~Eillation.Finity elements model with the data of Fig. 13 and Ux = 0.08cm s- .

I..no

contact •0.0

0.6

a.~

o.z

'" '""'.OJ ~a: _... ,U'I E_ u

S .~... -v ...- Q~ ::

slip

'SLIP'

• stick • noslip -+-- contact

23

Fig. 15. Evolution of the friction stress at node 29 duringtile cOOltact portion of one cycle of normal oscillation. Finiteelements model with the data of FiQ. 13 and OC=O.DB cm s-1- x