ANALYSIS OF A CLASS OF PROBLEMS WITH FRICTIONBY FINITE ELE~ffiNT~ffiTHODS

J. T. ODEN

The university of Texas

1. INTRODUCTION

This year marks the 200-th anniversary of the publication ofthe memoir of the French engineer-scientist A. C. Coulomb,"Theorie des machines simples" in which he presented his law offriction of different bodies slipping on one another. This work,which was based on results of Coulomb's own personal experimentswith simple bodies, earned him the prize of the French Acaden~of Science in 1781 and cast his name indelibly in the pages ofthe history of mechanics. The classical Coulomb law of dry fric-tion, of course, asserts that a relative sliding of two bodiesin contact will occur whenever the net tangential forces on thecontact surface reach a critical value proportional to the forcenormal to the contact surface. The constant of proportionalityis known as the coefficient of friction. There have since beenproposed several modifications of this law to take into accounta distinction between static and dynamic processes, lubricationof the contact surface, etc.

Coulomb himself must have perceived his law as applicable tostatic situations involving bodies which, for all practical pur-poses, are rigid. Indeed, the foundations of continuum mechanicsand its theories of deformable bodies were laid down many decadesafter Coulomb proposed his law. In its origin form, the law wasregarded as describing, for instance, the tendency of a rigidblock of one substance to slide down an inclined plane constructedof another substance as the angle of inclination is increased.It should not be totally surprising, therefore, that when Coulomb'slaw was extended so as to apply to static frictional phenomenain elasticity, some two centuries after it was first proposed,it led to classes of boundary-value problems for which no exis-tence theory is yet available, for which uniqueness of solutions(when they exist) cannot be guaranteed, and which may representpoor mathematical models of dry static friction between deform-able bodies.

One such class of problems, considered by Duvaut and Lions [3],describes the equilibrium of a linearly elastic body in contact

..2 J.T. ODEN

with a rigid foundation on which Coulomb's law holds. A varia-tional principle for problems of this type is given as follows:

Find u E K such that

a(u.v-u) + c(u,v) - c(u,u) > f(v-u) (1. 1)

¥ v E K

Here u is the displacement field in a subset K of the space Vof ad~is5ible displacements,

v = Iv = (v1,v2"" ,vn) € (H"(O})n IIv = 0 a.e. on rD can}

K = {v E V I ~.~ 20 a.e. on fc1

a(' ,.) : vxv .... lR is the bilinear form

a(u,v) = I E. 'ktUk tVi .dx- - . n 1J , ,J

(1.2)

(1.3)

representing the virtual work done by the stress

defaij(~) ==== Eijkt~,t (1.4)

on the virtual strains £ .. (v) = -zl(v.j+ v. i)' 1 < i,j,k,t < N;1] - 1, j, - -

and c:VXV ....lR is the virtual work term due to the Coulomb fric-tional forces:

c(~,~) = If vlan(~) I I~TldsC

(1. 5)

(1.6)

the external forces,

f(v) = r f·v- . rc -

Finally, f E v' represents the work ofassumed here to be given by

dx + J' t •v ds Ir - -

f E (LZ(n))N; :: (L(X)(rF))N

In (1.2), n is a smooth open bounded domain in lRN with boundaryr consisting of three disjoint parts: fD on which displacements

are prescribed, fC which contains the contact surface, and rFon which the prescribed tractions t are applied. The values of

v E V on r are, of course, interpreted in the sense of the trace

PROBLEMS WITH FRICTION 3

1/2 N -;::;-of V onto (H (n) (we assume ID n f c ~) and n is the unit

vector exterior and normal to r. In (1.3) and thereafter, weemploy standard index notations and the summation convention.

defUk,t = a~/axi (~ = (xl,x2, ...,xn) being a point in n with

Cartesian coordinates xi) and the elasticities Eijki are con-

stants with the usual symmetries and positive-definiteness. Thus,the bilinear form a(' ,.) can be assumed to be symmetric, contin-uous, and V-elliptic; i.e., positive constants m and M existsuch that

(1. 7)

for all 1I,V, V, where

(1.8)2 ('II v III = vi' v. . dx- . n ,J 1,J<Xl

In (1.5), v is the coefficient of friction, vEL (fC)' v~vO>O

a.e. on fc a (u)is the normal stress on the contact surface;• n-

cef(J (u) = eJij(u)n.nj , and vT is the tangential component of then _ _ 1 -

trace of the displacement vector ~ on fe'

It is not known whether solutions exist to problem (1.1) ingeneral. The existence of solutions to very special cases ofsuch problems (for n unbounded and N=o2) has been established byN~cas et al [8] and also in Demkowicz and Oden [2], but it isbelieved that there may not, in fact, exist solutions for somemore general situations. Complications arise from the fact thatthe functional c is non-convex and non-differentiable and fromthe fact that if a solution u were to exist, it would necessarily

1 - -1/2.have components in H (m; then a (u) E H (fc) 1S defined byn -

duality and I(J (u) I may have no meaning. We note that if smoothn _

solutions u to (1.1) were to exist, they would be solutions ofthe classi~al elastostatics problem,

I. ai.(u),. + f. =0 0 IJ - J 1 in na ..(u) = E. 'ki~ i1J - 1J ,

U =0 0 on rD

aij(~)ni = ti on rF

I~T(~)I < vlcrn(~) I .--> ~T = ~

I~T(~)I= vlcrn(~)1 > 3)., E lR,

J.T. ODEN4

II. un

III.

u'n < 0 ; a (u)n -

cr (u)un - n

a ..(u)n.n. < 0 I~J - 1 J - on rco

A > 0, s . t. } on f C

u = -Av (u)_T -T -

u = u_T u nn- (1.9)

In the present paper, several special cases of (1.1) [or (l.9)]are considered. These include the Signorini problem without fric-tion, the Coulomb friction problem with prescribed normal pres-sures, a regularized version of this latter problem, and thegeneral problem itself. In addition, the possibility of using anon-local generalization of Coulomb's law is explored. Finiteelement approximations of these problems are described, and sev-eral new algorithms for the numerical solution of certain casesare given. Finally, the results of preliminary numerical exper-iments are presented.

2. VARIOUS CONTACT PROBLEMS IN ELASTOSTATICS

There are certain special cases and modifications of problem(1.1) which are tractable and which can lead to reasonable modelsof certain friction phenomena. We shall now discuss some of thesewhich have been studied recently using finite-element methods.

I. Unilateral Contact without Friction. When the bodies incontact are sufficiently lubricated, frictional forces can beneglected. Then (1.1) reduces to the classical Signorini problemof unilateral contact of an elastic body with a rigid friction-less foundation. This problem is characterized by the variationalproblem of finding u E K such that

a(u,v-u) ~ f(v-u) If v E K (2.1)

It is well known that under the conditions assumed earlier (par-ticularly (l.7»), there exists a unique solution u to (2.1) for

any choice of f E V' (see Fichera [5J or Kikuchi and Oden [7J).Clearly, (2.l) is derived from (l.l) by setting c = O. Moreover,if the solution u to (2.1) is sufficiently smooth, it is also thesolution of the classical problem defined by subsets I and II ofthe system of equations (l.9).

PROBLEMS WITH FRICTION 5

II. Unilateral Contact with Prescribed Normal Pressures. Thisspecial class of contact problems of some importance involvesproblems in which the normal "contact" pressure is prescribedand only the tangential friction forces are unknown. In thiscase, we set

j (v) = J' T IvT Ids- r -

C

2TEL (fC)' T > 0

f(v) f (v) + f F v dsf n n

C}

(2.2)

(2.3)

where T is given data, F is also a prescribed normal force onn

fC' and vn = v·n. We then have a special variational inequality

of the form,

u E K

l.j v E K (2.4)

The functional j in (2.2) is convex and lower semicontinuous,but it is not differentiable (in the Gateaux sense). As a result,it is not difficult to show_that there exists a unique solutionu to (2.3) for each T (and f). Moreover, the solution of u is- -also the unique minimizer of the non-differentiable energy func-tional,

v E K (2.5)

III. A Regularization of Problem (2.4). For v E K, let

and

cp (v)£ -

> £ }

< £

(2.6)

j (v) = ,. T$ (v)ds£- ·r £-

C

(2.7)

where £ is a positive number. Then it can be shown (see Campos,Oden, and Kikuchi [lJ) that the functional .1£ represents a

6 J.T. ODEN

defined in (2.3) which isregularization of the functional jGateaux-differentiable, with

(' acp (u + tv)

<Dj(u),v>= T £~ -£ _ - ot

. rc

dst=Q

(2.8)

and for which j ....j as £ .... O. Here <','> denotes duality£

pairing on H-l/2(rc)xlll/2(fc). We can then consider the regu-larized problem,

Find u E K such that£

a(u ,v-u ) +< Dj (u ),v-u > > f(v-u )-£ - -£ £ -£ - -£ - --£

V;!EK } (2.9)

This problem has a unique solution 1I for each £ > Q.-£

Horeover, if u is the solution of (2.4), then

(2.10)

The regularized problem (2.8) provides a more useful basisfor the construction of approximate methods than does a directuse of (2.4) owing to the non-differentiability of j.

IV. Non-local Friction Laws. As noted earlier, the principalsource of difficulties with the general equibilibrium problem

with Coulomb friction (1.1) is that u,v E K ~ ai" (u) E L2(n).J -

Then a (u) is defined by duality as a member of H-l/2(rc) andn _

and 10 (u) I has no meaninB· Duvaut [3J has observed that thisn _

particular mathematical difficulty is overcome if one replaces10 (u) 1 in the definition of c(',') by a suitable regularization

n -which is a non-negative function in L2(r ). Similar ideas have

Cbeen explored in some detail by Demkowicz and Oden [2]. In cer-tain instances, such regularizations might be interpreted phys-ically as non-local friction laws.

The essential ideas are outline~ AS follows. Let S denote a

linear continuous function from H-l/2(fc) into V which preservesnon-negativeness:

(2.11)

Then we consider the non-local friction problem,

•PROBLEMS WITH FRICTION 7

u E K:

a(u,v-u) + J V Sea (u))(!vTI - luTI)dsf n - - -

C

> f (v-u) v v E K I (2.12)

The resolution of this problem depends intimately on problem(2.4). For each T E L2(fC), we know from (2.4) that there existsa unique u E V such that

-T

u E K : a (u ,v-u ) + J' T ( IuT 1 --T -T - _T f -

C

> f(v-u )- .... ",,1'

I v Dds \-TT

V v E K

(2.13)

This establishes a correspondence B : L2(fc) -+ V defining

u = B(T) (2.14)_T

which is strongly continuous.We will now show that the problem can be reduced to one of

finding a fixed point of the map,

(2.15 )

Indeed, if w* is a fixed point of T and if we set

u* = B(I/I*), then v 5(-0 (u*)) = lj/* (2.16)_ n -

which means that u* is a solution of (2.13) with T = lj/*.But sincelj/*= V s(-an(~*)): we must conclude that ~* is also a solution of

the general problem (2.l2).Duvaut [3] has shown that the nap T, in fact, has a unique

fixed point for coefficients of friction V sufficiently small.Demkowicz and Oden [2] have proved that T always has at least onefixed point for any V and that this fixed point is unique for suf-ficently small v.

3. FINITE ELEMENT APPROXIMATIONS

Let {Vh10<h<1 be a family of finite-dimensional subspaces of

of the space V-constructed using conforming (CO_) piecewise poly-nomial basis functions defined in the usual way over a finite-ele-ment mesh ~ approximating Q. Here h is the mesh paramete~:

h = max h ,h = dia Q Q being a finite element, Q c Qe e e e e On1 < e< E. We suppose that the members of the family {Vh} are

8 J.T. ODEN •constructed using uniform or quasi-uniform refinements of regularmeshes so that interpolation properties of Vh are of the standardform

m NGiven v E (H (n)) n V , m > 0, there

exists ~h E Vh such that

II v - v II < Chmin(k+1-s,m-s) II vii- -h s - _ m

s = 0,1

(3.1)

where k is the degree of the largest complete polynomial contained

in the finite-element shape functions and II vl12 = ,N II vil12 ("\.- m Lj=l m , ..

More specifically, we have in mind finite element approxima-tions of two-dimensional elastostatics problems (N=2) using 9-nodebiquadratic elements (k=2) or 4-node bilinear elements (k=l).

For each choice of Vh' we must also construct an approximation

of the constraint set~. Let 2h denote a finite set of points

on the approximate contact surface r~ - r. Typically, '\ willC Lh

hmerely denote the set of boundary nodal points on fC or the set

of Gaussian quadrature points used in evaluating j(vh,vh) or

j(~t), ~h E Vh. Then we may define - -

(3.2)

In other words, the unilateral contact condition is to be appliedonly at discrete points in our finite-element approximations.Clearly, ~ ¢ K, in general.

A direct approximation of the general problem (11) in ~ isembodied in the discrete problem,

Find uh E K such that- -11

h h h h h h ha(u ,v -u ) + c(u ,v ) - c(u ,u )

vh

E ~ I (3.3)

However, it is seldom advisable to attempt to solve the general

•PROBLENS WITH FRICTION 9

Coulomb friction problem by a direct assault on (3.3). Frequently,the construction of a numerical solution of one or more of thespecial problems described earlier as an intermediate step towardthe analysis of (3.3) is effective. He comment further on onesuch scheme in the next section.

The non-local friction problem (2.12) deserves some comment.Consider a nonlocal friction law of the form,

u (x) = 0~t - -

where, for example, if a (u)n -

v S (a (u)) (x) =9 3 A E]I{ ,n _

A > 0 s.t·~T(:) = -A~T(~(~)

2L (re),

(3.4)

_00

<Xl

where w is a C - mol1ifier kernel of the typep 0

{o , r>pw (r) = 2 2 2 -

P c exp [p /(p -r )] , r 2 p

(3.5)

(3.6)

TIle radius p of the non-local friction law must be regarded as anew mechanical "roughness" property of the surfaces in contactwhich must be determined experimentally along with v.

Suppose that {~ 1L 1 denotes the global finite-element basisa a=

for Vh and that the absolute values of the normal contact pres-

sures are approximated as combinations of functions {~~}~=l'L N

h'Va h'V~vi =~ vi ~a ; p =~ p ~~

a=l ~=l

Set

(3.7)

To obtain a finite element approximation of the non-local problem(2.12), we first solve

10 J.T. aDEN

h~ E~:

( h h h) + J' ph(l)r( h h)da w , v -w v ,IJ S- - - r - -

C

- h h)'> f (v -w Ii vh

E ~

•

(3.8)

where ph(l) is an initial estimate of ph, and then we correct

ph by choosing a ph(2) to satisfy

I h h hV S(a (w )) rev ,w )ds

. f n -C

vhE ~ (3.9)

The corrected ph(2) is introduced into (3.8) and the process isrepeated. At this writing, this algorithm is untested, but codingis underway.

4. ALGORITHMS FOR FRICTION PROBLEMS

We shall nOIJ describe several algori thms for the numericalsolution of various types of friction problems described earlier.Some of these are standard, others are new, but each is suggestedby the structure of the variational problem being analyzed.

1. The Signorini Problem without Friction. One of the simplestalgorithms for solving problems of the form (2.l) is the standarditeration (relaxation) with projection. We have successfullyused algorithms of this type for a wide range of contact problemswithout friction. The basic algorithm is:

1. Select a starting vector uh(l)-ll

o

2. Compute

a (uh(tl)) = uh(t)'nt -u -u-

3. Set13 (uh(t) {a (uh(t))

t+l µ ) = t µo

4. Solve the linear system,

ifa <0t

ifa >0t-

•PROBLEMS WITH FRICTION

h h= f (v ), v E Vh- -

5. Return to step 1 or terminate this process when therelative error

11

is less than a preassigned value. h tHere ]J is an exterior penalty parameter, and u' should

approximate the solution of (2.1) as h,]J....O. This algorithmis used (with several others) in Kikuchi and Oden [7]; see alsoOden, Kikuchi, and Song [9].

II. Problems with Prescribed Normal Pressures. To analyze prob-lem (2.4) numerically, we construct a finite element approxima-tion of the regularized problem (2.9) for small £ > O. This leadsto a system of nonlinear algebraic equations of ~he form

Ku + J (u ) = f-£ -£ -£ -

where K is the usual stiffness matrix of the linearized problem,f is the load vector, and J (u ) is a vector of nonlinear func-- _E -C htions of the vector of nodal values of the approximation u .

-£Standard Newton-Raphson i"teration has proved to be adequate forsolving this system in many numerical experiments. For a varietyof other algorithms for problems of this type, see Glowinski,Lions, and Tremolieres [6]

III. Contact Problems with Coulomb Friction. Campos, Odenand Kikuchi [1] have presented numerical solutions of certaincases of the general problem (l.t) obtained using the followingalgorithm:

l. First consider the case in which no friction is present.We then use algorithm I to obtain a first iterate u(l) of the

hvector of nodal values of u .

2. Using the approximation ul computed in step 1, compute nor-

h(l)mal contact pressures a (u ).

n -3. Treat the computed normal pressures as data in a contact

problem with friction in which the normal pressures are prescribed.Use an algorithm of the type II above to compute tangential fric-

tion forces. By comparing lOT (uh) I - V 10 (uh) I with £, an- n -

estimate of the portions of the contact surface on which full

12 J.T. ODEN •adhesion or sliding can occur.

4. Having calculated tangential frictional forces in step 3,we return to the non-frictional case treated in step 1 and treatthese as prescribed forces. This new Signorini problem is solvedand a second iterate is obtained.

There is no guarantee that this process is convergent, but ithas proved to be convergent and to yield reasonable results incases in which a large portion of the contact area is in fulladhesion (no slipping).

IV. Non-Local Problems. An algorithm for treating finite-ele-ment approximations of the non-local friction problem (2.12) issuggested by the proof of the existence theorem and the develop-ment in Section 3:

1. Choose an arbitrary starting vector p defining a non-nega-

tive contact pressure ph on r~. This beco~es the data T = ph in

a finite-element approximation of problem (2.4) (specifcally,see (3.8)).

2. Use algorithm II above to solve for a corresponding dis-

placement field uh(l) .

h(l) h(l) h3. Compute 0 (u ) and S(a (u )(x), x E fC'n _ n - --4. Use a recurrence formula SUcil as (3.9) to compute a correc-

h(2) htion P of P . h(2)

5. Return to step 2. and compute a corrected ~ . Repeat

this process until the relative error is less than some preassignedtolerance.

5. NUMERICAL EXPERIMENTS

At this writing, extensive numerical experiments have beenperformed using algorithms I, II, and III but coding of algo-rithm IV is yet to be completed. We shall cite one example ofIII, discussed in greater detail in [1].

Consider the problem of indentation of a rigid cylindricalpunch into an elastic half space. This problem, treated as aproblem of plane strain, is modeled by a finite-element mesh of9-node biquadratic elements shown in Fig. lao The material istaken to be isotropic and linearly elastic with modulus E=and Poisson's ratio µ = 0.3. A coefficient of friction ofV = 0.6 was used in our calculations.

The computed deformed shape for a prescribed centerline inden-tation of 6 = 0.8 is indicated in Fig. 2a and the computed tan-gential stresses on the contact surface is plotted in Fig. 2b.A fine mesh was needed in the vicinity of the boundary betweenregions of full adhesion and sliding in order to obtain properresolution of the frictional stresses. The computed results arephysically reasonable: the peak frictional stresses are 0.6 times

•

A

PROBLEMS WITH FRICTION

4

13

c

8

jD

Fig. 1. A finite-element model of ~n elastic foundationindented by a rigid cylindrical punch. The mesh, which modelshalf the body thanks to symmetry, consists of 40 9-node biquadraticelements.

•14 J.T. aDEN

(a) 8=0.8

( b)

-Classical solution withwt friction

A---A - Computed tangential stresses

0-0 - Computed with friction

R64o

50

100

150

250

200

Full adhesion

Fig. 2. a) Computed deformation of body and b) computed stressesat contact surface

•PROBLEMS WITH FRICTION 15

the normal stress at the stick-slide interface. Comparisons ofcomputed normal stresses in the body are in good agreement withanalytical solutions available for the non-friction case.

Acknowledgement. The support of the work described here by theU.S. Air Force Office of Scientific Research under ContractF49620-78-0083 is gratefully acknowledged.

REFERENCES

1. CA}WOS, L., ODEN, J.T. and KIKUCHI, N., A numerical analysisof a class of contact problems with friction in elasto-statics. Compo Math. Appl. Meck. Eng. (to appear).

2. DEMOKOWICZ, L. and ODEN, J.T., On some existence anduniqueness results in contact problems with Coulombfriction. (in preparation)

3. DUVAUT, G., Equilibre d'un solide elastique avec contactunilateral et frottement de Coulomb. Compte-Rendus, Acad.Sc. Paris t.290, 263-265 (1980).

4. DUVAUT, G. and LIONS, J.L., Inequalities in Mechanics andPhysics, Springer-Verlag, N.Y. (1976).

5. FICHERA, L., Boundary-value problems in elasticity \"ithuni latend constraints. Encyclopedia of PhYlJics, Vol. IV a/2-Mechanics of Solids II, edited by C. Truesdell. Springer-Verlag, Berlin, Heidelberg, N.Y. (1972).

6. GLOWINSKI, R., LIONS, J.L. and TREHOLIERES, T., AnalyseNumerique des Inequations VariationneZles, Vols. I and II.Dunod, Paris (1976).

7. KIKUCHI, N. and ODEN, J.T., Contact Problems in Elasticity.SIMI Publications, Philadelphia, Penn. (1981). (to appear)

8. NECAS,J., JARUSEK, J. and HASLINGER, J., On the solution ofthe variational inequality to the Signorini problem withsmall friction. BOLLETJNO U.M.I. (5)17-B, 796-811 (1980).

9. aDEN, J.T., KIKUCHI, N. and SONG, Y.J., Reduced integrationand exterior penalty methods for finite element approx-imations of contact problems in incompressible elasticity.TICOftfReport 80-2, Austin, Texas (1980).