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On the Signorini Problem with Non-Local Friction

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On the Signorini Problem with Non-Local Friction 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 Friction 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 rigid body 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 plane 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 forces normal 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 stress 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 force 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 weight is given to thc stress intensity at the centre of the disc (the deformcd asperity), and exponentially decreasing weights 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 displacement 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<!x-yl)(-un(u(y))) dy, (7.8) where x and yare points on the contact surface re. 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.
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