Numerical Methods ill Coupled Systems Edited by R. W. Lewis. r. Bettes,. and E. Hir.tnn © 1984 John Wiley & Sons Ltd

Chapter 7

On the Signorini Problem with Non-local

1. T. Oden and E. B. Pires

Summary The classical model of Coulomb for static dry friction raises several difficulties both from the physical and the mathematical points of view. Physi- cally, Coulomb's law seems to have been devised to describe friction effects between effectively rigid bodies and gross sliding of one body relative to another. Mathematically, the existence of solutions to boundary value problems in elasticity for which Coulomb's law of friction is applied pointwise has been proved only for very special cases. Duvaut, recently, has indicated that the use of a non-local friction law produces a more tractable mathematical theory. Such a non-local model asserts that relative motion at a point occurs when some weighted average of the stresses in the neighbourhood of the point reach a critical value. The present note deals with the characterization of a non-local friction law and with the study of the Signorini problem with non-local friction. Existence and uniqueness results are developed as well as an approximation theory. Finally. the description of the results of a numerical experiment is presented.

7.1 INTRODUCTION We consider in this paper a class of contact problems involving the equilibrium of linearly elastic bodies in contact on surfaces on which a non-local law of friction is assumed to hold. The plausibility of a non-local friction law as an alternative to the classical (local) pointwise Coulomb law of friction is suggested by an examination of both the physics of friction and the mathematics of boundary value problems in elastostatics with local friction. From the purely physical side, it has been recognized for many years that Coulomb's law is, at best, a crude approximation of the actual mechanics of friction. capable of depicting only gross sliding of one effectively on another. The careful examination of the contact surfaces of two metallic bodies pressed together along two apparently nat machined surfaces reveals

217 211-: !\il/II/aiw! .\1/'//'od.\ ill COl/p!cd SYSICIIIS that. at magnilication~ of hetween IOOOX and 5UOOx. the contact surface~ exhihit marked deviations from the surface. These irregularities. which arc large compared with the size of ,I molecule. are referred to as C1sperilies and provide the actual structure through which 10 the apparent surfaces are transmitted. In fact. when two slich bodies are pressed together, real contact only occurs at the peaks of thc asperitics. High stresses will develop. making the asperities yield and fracture to form jUllctions between the bodies in contact. The real cOl/weI area (as opposed 10 thc apparel/t contact area) is. therefore. the summed area of all these flattened surface irregularities which are touching and which support the load. If we now apply a tangential shear forcc of magnitude T parallel to the apparent contact planc. a tendency for the two bodics to slide rclative to each other is creatcd. The normal forces that press the two bodies togethcr are actually transmitted through a thin film of contaminant and metallic oxide a few angstroms thick. and it is the shear strcngth of this film that determincs the coefficient of friction and not that of the parent metals. As T is increased. the junctions are finally fractured and gross sliding of one body relative to another occurs. Thus. the actual variation of normal over the contact surface is not uniformly distributcd. but is concentrated in junctions of crushed asperitics distributed morc or less randomly. On the othcr hand, from the mathematical side. it is known (sec Duvaut [I] and also Duvaut and Lions [2]) that if Coulomb's law is applied pointwisc in contact prohlems involving linearly clastic bodics. the stress component developed normal to the contact surface is ill-defined. Except for some very special cases (sec. for example, Necas et al. [3]), the question of cxistence of solutions to the friction problem is open (see Duvaut and Lions [2]). Duvaut []] observed that the source of difficulty in thc proof of existence is the lack of smoothncss of the normal contact pressure (Tn. By replacing (Tn by a mollified stress. which might be interpreted as assuming a non-local friction law. he was able to develop a complete cxistence and uniqueness theory for certain contact problems. Further results in this direction have been obtained by aden and Pires [4.5] ;lnd Oemkowicz and aden [6]. In the prescnt work wc summarize some results that we have obtained on the analysis of certain contact problems in elasticity in which a non-local law prevails. namely a justification of a specific model of non-local friction. the derivation of a variational principle as well as some results on the existence. uniqueness. and approximation for such problems. Complete proofs of all results are to appear elsewherc [4.5]: see also the unpublished report [7]. Finally. we give numcrical results of somc preliminary calculations.

7.2 A NON-LOCAL FRICTION LAW We consider here the simple physical model shown in Figure 7. I in which a thin weightless strip A of length 2! is pressed against a fjxed mctallic block B 01/ the .t.iiglloril/i Proh!1'1ti "'illr Nfll/-!ow! Frictiol/ 219

"'r 1 vN8t.) I I I I I I I I I t -( J

(0)

---+--- -(

(b)

Figure 7.1 Tangent ial ~trcs~ distrihution al the poinl of impending slipping for (a) primitive Coulomb law and (b) a non-local law of friction.

by a concentratcd normal N applied at its midpoint. At the point or impcnding motion. the tangential stress distribution UT on the contact surface. as predicted by the classical pointwise Coulomb law. is

UT(X) = VND. -! < x< /, (7.1 ) where 8 is the Dirac delta. Of course. (7.1) is merely the symbolic representa- tion of the distribution

(UT. c/J) = ,,{)(o. (M = INc/J(O) (7.2) for all test functions c/J E C; (-!, /). where (' .. ) denotes duality palflng on distributions and test functions (i.e. the action of a distribution q on a test 220 Numerica! Methods ill Coupled Syslems function ¢ is denoted by q(¢)s.(q. ¢». Alternatively. lJ can be inlerpreted as the limit of a lJ-se4ucnce. {W"h •.,,,WI' E c~(-I. /):

(f>{O)=li(¢)=limf' w,,¢dx, VcbEqc(-I./). (7.3) IJ·(I ~l

Here ~(-I. I) denotes the space of test functions defined on the interval (-I, I). Then instead of (7.2) we have

(CTT' ¢) = II ,,-nlim (Nwp, ¢), V¢E ~(-I.l). (7.4)

We see that the classical pointwise version of Coulomb's law must be inter- preted in the sense of distributions. As a typical lJ-sequencc we mention Ixl~p, Ixl>p. (7.5)

A more realistic model of friction is obtained if we take into account the fact that N is not concentrated at a point but is distributed over a deformed asperity. We shall weigh the normal stress distribution over a circular region of radius Po on the contact surface using the o-sequence WI' of (7.5) keeping p = Po. Since N = N(x) is now a function, we have, instead of (7.2).

CTT(X) = vN(y) * WI.,(x- y). (7.6) where * denotes the convolution. Thus, we have arrived at a friction law in which impending motion occurs at a point x on the contact surface when the shear stress at lhat point reaches a value proportional to the weighed average of the normal stress in a neighbourhood of the point. If WPo is used to characterize this weighting function, then the neighbourhood is a circular disc of radius Pu centred at x, the maximum is given to thc stress intensity at the centre of the disc (the deformcd asperity), and exponentially decreasing are assigned to stress intensities as one moves from the centre of the neighbourhood outward to the periphery of the disc. This is shown in Figure 7.1 (b). We can now make the three-dimensional generalization of these results: let UT denote the tangential component of the of a point x = (XI' X2. X3) on the contact surface between two deformable bodies and let CTn(U) and O'T(U) denote the normal and tangential stresses on the contact surface corresponding to the displacement field u. Then we have

IO'T( u)1 < llS( CTII(U)) => UT = 0, (7.7) IU'T(U)I = lIS(CTII(U» => 3.\ > 0 such that 011 lire Sigllorilli Problem with Non-loea! FrictiOIl 221 where S is an operator regularizing the normal stress distribution: e.g.

S(u,,(u»(x) = L. wp

7.3 A VARIATIONAL PRINCIPLE FOR SIGNORINI'S PROBLEM WITH NON-LOCAL FRICfION We consider here a variational formulation of the Signorini problem of contact of a linearly elastic body 0 with a rigid foundation on which the non-local law (7.7) holds. Using standard notations the equilibrium of the body is governed by the following system of equations and inequalities:

(E;lktllk.I).1 +h = 0 lJl 0;

u,=O on rD;

E';klUk,lllj = Ii on rF:

U' 0..;0, un(u),,;O, un(u)u. 0=0 on rc: (7.9)

l(rT(u)l~lJS(un(U»)~lIT=() } , on IC' 1(T1'(u)i==lJS(un(1I))~3A~0 such that u1'=-AO'T

Here r==Ml ==I'"OUI'"FUI'c, E'ikt arc thc elasticities of the material with the usual properties of symmetry and ellipticity, 0 is the unit outward normal to r, and v is the coefficient of friction. a positive constant. A variational principle for problem (7.9) is embodied in the following variational :

Find a displacement field 1I E K such that (7.10) a(u. v - u) + f vS(un(U»(IVTI-luTD ds;;:. f(v- u). 'tnE K. ] rc Here K = the unilateral constraint

={v = (VI. V2, V3)E Vlv, n~O a.e. on fd. with

v = the space of admissible displacements

={vE(H1(0))'lv=o on fD}:

IIvll\' = IIvll, = {L Vi,jVi.; dX} 112, 222 !\iumaim! Atethods in Coupled Systems

a(u. v) = the virtual work of stresses a,,(u) on strains E,,(V)

= f Uij(lI)Ei,(\') dx = f E"k1I1UVi./ dx. II II

(a(· .. ) is a symmetric. continuolls. V-elliptic. bilinear form on V)

f( v) = the virtual work of the extcrnal forccs

= f f· \' dx + ft. v ds. II I,

(f(') is a continuous linear on V.l We assume Ft E L~(H) and Ii E L ~(rF)' A rclated auxiliary problcm consists of finding U E K whenever vS is pre- scribed as a positive function in L~(I'd:

Given TEC~(rcl. T;;!:O a.e., find u,EK such that ) (7.11)

a(u ...V-lI,)+ f" T(/VTI-luTTI> ds> f(v-uT) VVE K.

It can be shown (see aden and Pircs [7]) that

(I) VT. there exists a uniquc solution liT to (7.11) and the correspondence

TI--uT defines a continuous non- B from eO'd into V:

UT =B(T).

1 1 (2) The normal stress (Tn(lIT) is well defincd in the space W' = (H / (rcl)', where H 1/2(1' d is the spacc of normal components of displacement on the contact surface, and an: K -> W' is continuous. (3) Problem (7.11) is equivalent to the system of equations and inequalities (7.9) if we replace the non-local law (7.7) in (7.9) by the following law:

I(TT(UrT)1 < T ~UTT= 0 on (7.12) IUT(UTT)I = T => A> 0 such that

It should be noted that this law of friction is different from Coulomb's law.

Returning to the main variational principle (7.10), we note that the following properties of that problem can be established (see aden and Pires [7] for complete details):

Theorem 1 (1) Every sollllion of the Signorilli problem with nOIl-!ocal friction (7.9) is also a soilltion of the variatiollal illeqllalilY (7.10); conversely. every SOllllioll of (7.10) satisfies (7.9) ill a weak or distriblllio/la! sense. 011 the Sigllorini Prob!em with NOII-loca! Frictioll 223

(2) Lt't meas 1'1» 0 and the smoothing operatorS: W' -+ e(l'd becomp!etely colltinuolls. There exists at least one soilltion u E K of (7.10) for every choice of data f alld t. (3) If the coefficient of friction II is sufficielltly small. then the solutioll of (7.10) is ulliqlle. 0

One method of proof of items (2) and (3) of this theorem was suggested by Duvaut [I]:

Proof A. Let T:L2(rc}-+L2(rd denote the operator formed by the composition

T = liS 0 Un 0 B. where B: L 2(f d....K is the operator inherent in problem (7.11) and S: W· .... L2(r d is the completely continuous smoothing operator in the non-local friction law. B. Let l/f* be a fixed point of T: T(l/f*)=t/J*. C. Let u* be the solution of the auxiliary problem (7.1 ]) corresponding to the choice r = t/J*. D. Then. Vv E K. a(u*. v-u*) + L t/J*(lv·"!-lu.f.1>ds"" f(v-u*) v II

L, T( t/J*)(lv·r1-lu~l) ds

y II

Lc IIS(un(u*»(lvTI-lu~j) ds. i.e. u*= B(t/J*) is a solution of the general problem (7.10). E. Having reduced (7.10) to a fixed point problem. one uses the Schauder- Tychonoff fixed point theorem to show that T has at least one fixed point. For small II. T becomes a contraction map and the fixed point is unique.

7.4 APPROXIMATIONS AND NUMERICAL RESULTS We now consider finite element approximations of the (7.10). For simplicity. we confine ourselves to problems in 1R2 and we assume that n is a polygon. By constructing a regular series of partitions of n into 224 Numerical Methods if/ Coupled Systems finite element meshes over which piecewise polynomial approximations of the displacements are used, it is usually possible to produce a family {V'.}h>ll of finite-dimensional subspaces of the space V of admissible displacements. II being the mcsh parameter (i.e. the largest diametcr of an element in the mesh). Approximations K" of thc unilateral constraint sct must generally be construc- ted so that the contact condition is applied only at nodes on the contact surfacc

~; thus. in general. Kh¢ K. Under these conditions. a Galerkin approximation of (7.10) is characterized as follows:

Find U/. E Kh such that

Vv" E K,l

(7.13 )

By Theorem 1 (if 1/ is small enough) the approximate problem (7.13) possesses a unique solution. The following gencral error estimate is proved in Oden and Pires [5]:

Theorem 2 Let U af/d U" def/ote the solllliOf/S of variatiof/al if/equalities (7.10) and (7.13) af/d let A def/ote the operator (Au,v)=a(u.v). VVE V. Fif/ally, assume that Au - fEU. where U is a Hi/bert space def/sely embedded in the dual V' of V. TJiell, if 1/ is sufficief/tly small, there exists a cOllstallt C> (), independent of 1J. sllch thaI lIu- uhlll ~ e{llu - vhll,+[IIAu - ~lu(lIu - v"lIv' + Ilu" - vllll·]ln II' +[1/111111,(liu - v,,111 +lIu" - vJl'] -} (7.14) for every VE K, v" E Kh' 0

When the solution u of (7.10) is sufficiently smooth and the spaces Vh exhibit the usual interpolation properties. (7.14) can be used to show that the approximation Uh. satisfying (7.13). converges in V to II as h tends to zero. Upon introducing piecewise polynomial approximations U/. and Vh into (7.13). we obtain a system of non-linear algebraic inequalities in the nodal values u~ of u". This system can be solved using a number of techniqucs. We shall now quotc results obtained using the following algorithm:

(1) 02-(nine-node. biquadratic) polynomial approximations of the displace- ment field II are used over each element. (2) An approximation of the auxiliary problem (7.11) is constructed and solved using linear programming techniques (e.g. Uzawa's method, a Newton's On tlte Sigllorini Prob/em witll Non-local Friction 225 method for a regularized problem. standard over-relaxation with projections. etc.). (3) A successive approximation iterative scheme is then utilized which is based on steps A. B. C. D in the fixed point arguments listed in the previous section.

These steps. of course. describe only in general terms a family of algorithms that could be used for problems of this type. We are currently investigating several of these to determine an efficient scheme for solving friction problems of the type in (7.9) or (7.10). Some results recently obtained concerning the approximation of the auxiliary pr6blem (7.11) are. nevertheless. worthwhile mentioning. As pointed out in the previous section, for fixed 7 the variational inequality (7.11) (which is said' to be of the second kind) possesses a unique solution. However. the direct approximation of problem (7.11) by finite elements present several difficulties: one such difficulty is due to the fact that the functional j: V ~ R defined by

(7.15) is non-differentiable. We consider, therefore, a Gateaux-differentiable regu- larization j, ( . ) of j( . ) which is a function of a positive real parameter E. that approximates j(') arbitrarily closely as E tends to zero. The regularized functional j, ( .) is defined by

j. (v) = Lc 71/1, (v) ds. (7.16) where fb, ( . ) is the convex function

forl~I>E. (7.17) f()rl~I~E.

Then. the auxiliary problem (7.11) becomes

, E K Find ur such that (7.18) } a(ur" v-ur,)+(Dj,(ur,), v-ur,):;!: !(v-uT,) 'liVE K.

Here (. , .) denotes duality pairing in V' x V and Dj, the Gateaux derivativ'e of j" It is possible to show that the variational problem (7.18) with the Numerical Methods in Coupleel Systell/s

Figure 7.2 Friction law corresponding to variational problem (7.1 K). rcgularized friction functional is cquivalent to the Signorini prohlem with a law of friction of the typc shown in Figure 7.2. i.e.

if liT < -f'J (TT=-(T/E)II. if IUTI.;; t' (7.19)

if 11-]> f'

It is also possible to show that if u~ is the solution of variational inequality (7.11) and u~. the solution of (7.18) for fixed E > 0, then a constarlt k > () independent of E exists so that

(7.20)

Finally, we mention that one way to analyse problem (7.18) consists of reformulating it by introducing a pcnalty mcthod. For this purpose we have to considcr a penalty functional such as

P(v) =:;(II vn) 'I~• where '·1 dcnotes the on Wand ( . t the positive part of ( . ) in W. This functional is Gateaux-differentiablc with a derivative given by

(DP(u). v)=«unt, vn)\\,. Problem (7.18) then rcduces to

Find u~. E V such that (7.21) VVE V. 1 On the Signorini Probklll with Non-local Frictio/l 227

As a last rcsult it can be shown that if TE L . (1'('), then thcre exists a constant C depending only on F. such that 1111•. -u~.111~ C(f')eS. (7.22) Combination of (7.22) with (7.20) then yields the estimate lIu, - u~.Ih~ kJ~ + C( deS. (7.23) As a rcsprcsentativc numcrical cxample, we prescnt results obtained from a tinite element analysis of a rigid punch problem with non-local friction. The problem is one of plane strain of a hlock of homogeneous isotropic linearly elastic material indented by a rigid cylindrical punch. as shown in Figurc 7.3. The mechanical data of the problem are as follows: E = Young's modulus = 1000 (non-dimensional units). µ =Poisson's ratio=0.3. and the coefficient of friction v = 0.6. The parameter Po in the non-local friction law (7.7) was taken to be 0.1. The rectangular mesh of nine-node, O2 elements shown in the figure was used in the finite element approximation and the system of non-linear algebraic inequalities was solved using an algorithm of the type outlined above for a prescribed indentation of fJ = 0.6. Other are given in the figure.

E: 1,000 µ. : 03 \) : 0.6 p -01

T

4 I

I ·--8----

Figure 7.3 Finitc elemcnt mesh for the prohlem of indentation of an clastic hlock hy a rigid cylinder. 228 Numerica! Met/wds ill Coupled Systems

.-/ -~ - T 4

1'1 1 ~ 8-~---4

--0_ - 0- Normal conloel pressure 200~ 'Q, 0, • tJ.- Tangential !.lress '0, b, 150 b, o, o \ o 100 \ ~, C I 50 4.~ I , H

A'~ .A~ I', ~' . I Ole-A" ,I / 2 A B 4 • f--Shd< II-Slide

Fi~ure 7.4 Computed deformed geometry and stresses on the contact surface.

Computed deformed shape and stress profiles are shown in Figure 7.4. Observe that the tangential friction stresses do not reach a sharp peak. but are smooth at the point of maximum transverse shear stress. The size of this 'boundary-layer' in the neighbOlJthood of the maximum stress, of course. depends upon the magnitude of p. As p ~ 0, a cusp is developed at the peak stress corresponding to the case when the classical pointwise version of Coulomb's law holds. For p> O. this means that there is no sharp dividing linc between full adhcsion (no slip) and sliding areas. but rather a boundary layer over which a transition from no slip to slipping occurs. The existence of this boundary layer is fully consistent with experimental evidence on friction (see Bowden and Tahor [8]) .

• 0" Ihe Sigllorini Prob/elll willi Non-!oca! Friclioll 229

ACKNOWLEDGEMENT

This work was supported by the US Air Force Office of Scientific Research under contract F-49620-78-C-0083.

REFERENCES

I. Duvau!. G. Problemes mathematiques de la mecanique - equilibre d'un solide clas- tique avcc contact unilateral de frottement de Coulomb. C R. Acad. Sci .. Paris, A. 290. 263-265, 1980. 2. Duvaut. G .. and Lions. J. L. Inequalities iI/ Mechanics and PlJysics. Springer- Verlag. New York. 1976. 3. Necas. J.. Jarusek, J.. and Haslinger. J. On the solution of the variational inequality to the Signorini problem with small friction. Boll. UM.I .. (5). 17-R. 796-811. 19HO. 4. Oden. J. T .. and Pires. E. B. Non-local friction in contact problems in elasticity. 5. Pires, E. B.. and Oden. J. T. Error estimates for the approximation of a c1as~ of variational inequalities arising in unilateral problems with friction . .1. Nutl/er. FilII ct. Allal. Optimization. in press. 6. Demkowicz, L., and Oden, J. 1'. On some existence and uniqueness results in contact problems with non-local friction. TICOM Report 81-10. Austin. Texas. 1981. 7. Oden, J. T .. and Pircs, E. B. Contact problems in elastostatics with non-local friction laws. TICOM Report 81-] 2. Austin. Texas. 1981. H. Bowden, F. P., and Tabor, 1'. TlJe Friction and Lubricatioll of Solids. Part II. Clarcndon Press, Oxford. 1964.