NUMERICAL ANALYSIS OF CERTAIN CONTACT PROBLEMS IN ELASTICITY WITH NON-CLASSICAL FRICTION LAWS

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1 Comp.lm &. Sr., Vol. 16. No pp J98l. Printed in Greal Britain.,..z3 Z 004S-79.\9J83JOOOJ~SSOJ.OOO Pergamon Prels Ltd. NUMERCAL ANALYSS OF CERTAN CONTACT PROBLEMS N ELASTCTY WTH NON-CLASSCAL FRCTON LAWS 1. T. ODEN and E. B. PRES Texas nstitute for Computational Mechanics. Department of Aerospace Engineering and Engineering Mechanics. The University of Texas at Austin. TX 78712, U.S.A. Abstract-An algorithm for the numerical analysis of a highly nonlinear variational inequality encountered in the study of contact problems with non-classical friction laws is described. Numerical results obtained for a representative example problem are given NTRODUCTON n a recent paper[l]. we introduced new non-c1assiclll Jaws of friction for contact problems involving linearly elastic bodies together with variational principles for contact problems in elastostatics in which these laws are assumed to hold. We also gave a rather detailed series of physical and mathematical arguments as to why a departure from the classical pointwise version of Coulomb's law is called for if a sound and tractable analysis of most friction problems is to be obtained. The classical Signorini problem in elastostatics involves the equilibrium of an elastic body n C R 3 in contact with a rigid foundation within a portion rc of its boundary. The variational principle for Signorini's problem for the case in which a general non-classical friction law holds on the contact surface rc is derived in [) and takes on the following form: Find a displacement field u belonging to a unilateral constraint set K such that t(u. \' - u) + j,,(u, v) - jp.,(u. u) ~ f(v - u) () for all virtual displacements v in K. Here the following notations and conventions are employed: t(u. v) = ( uij(u)eli(v) dx n = the virtual work produced by the stresses Ui/(U) = EijklUk.1 corresponding to displacement u on the strains eij(v) = ~(V,j + Vi.i) due to the virtual displacement v [E 1ikl = Hooke's tensor of elastic constants satisfying the usual ellipticity and symmetry conditions, dx = dx, dx2 dx3' Uk.1 = auklax x = (X. X2, X3) E n. etc.] jp,(u. \') = ( Sp(U (u»)qlvt) ds Jle = virtual work on contact surface due to frictional stresses. f(v) = ( f v dx +1 t v ds. Ju 'F Here the boundary r of the body is assumed to be divided into three parts: f v. where the displacements are prescribed as zero, f F, where the boundary tractions t are prescribed, and fe, the candidate contact area. n the expression for jp. (-, '), Sp( T) = the coefficient of friction. 0 ~ v = a stress smoothing operator depending on the real parameter p >0: if T is a normal stress on rc (T ~ 0), then O~r~p r~p U,,(u) = normal contact pressure = <Ti/(U)lillj, = components of unit outward normal to f: q,,(r) = continuous piecewise differentiable, monotone-increasing function of r, r> 0, depending on a parameter e > 0, such that lim t/j:(r) = 1. lim t/j'.(r) =. C'-o r- n the expression for f(-). f = the body force per unit volume, assumed to be given as a smooth (e.g. L 2 _) function of x; t = the surface traction prescribed on f F, also assumed to be smooth; U;/(U)lj(X) = ti!x), x Ef F. The virtual displacements for this problem produce finite energy if they belong to the space v = {v = (Vb V2, V3)!Vj E H(n), v;=oa.e.onf D,i=,2,3} (2) where H(f),) is the usual Sobolev space of order 1. f the unloaded body is initially a distance g from the rigid foundation, the admissible displacements are required to 481

2 482 J. T. ODEN and E. B. PRES remain in the constraint set K = {v E V/v. n S g a.e. on fd, (3) This is the constraint set appearing in (). We note that the space V is a Banach space when equipped with the energy norm. We give mote detailed interpretations of some of the quantities introduced above in subsequent discussions. Our purpose in the present paper is to describe finite element methods for solving the highly nonlinear variational inequality characterizing the non-classical friction problem (). 2. PROPERTES OF THE NON-C.ASSCAL FRCTON PROBLEM Before describing a numerical approach to contact problems with friction, it is informative to list certain properties of the general variational principle embodied in (). () Exislellce alld ulliquellcss Under the conditions listed earlier. there exists at least one solution to the variational problem () for any v ~ 0, p > 0, > O. f is sufficiently small, this solution is unique. (2) Churaclerizalioll f the solution u to () is sufficiently smooth, then it is also a solution to the following Signorini-type contact problem: (ElikJU1.l).1 + fl = 0 in n U =0 on r0 E1iklUklli = lion r F Ulllj < g~u,,(u) = 0 ] Ujllj = g:u,,(u) soon un(u) = Ujj(U)ljllj UT(u) E ltij(u)lj - Un (U)llj f c UTi(u)=-v S,.(T,, () U )q,.'(url- U'/'1 U,. on fc. Conversely, any solution of (5) is also a solution of (). (3) llerprelalioll The friction law in (5) is 1l01l-/oca/ and nol-lillear. The parameter p is a measure of the characteristic radius of deformed asperities on the contact surface. For p > 0, the shear stress at impending sliding on rc is proportional to the weighted average Sp(u n ) of the normal stresses U in a 2p-neighborhood of points on the contact surface. As p-+o a purely local friction law is obtained. On the other hand, - is a measure of the stiffness of the elastic junctions formed on the contact surfaces. f no load reversal takes place, we may take, for example, r - /2 if r > q,,(r) = { r/2 if r ~. (4) (5) (6) Then any shear stress UT will produce a small tangential displacement 0T. and large tangential motions corresponding to relative sliding of the surfaces in contact will occur when UT reaches (perhaps asymptotically) a critical value U or, Sp(lT,,). (4) Special cases A number of important special cases can be derived from (). (a) p -+ 0, > O. This yields a nonlinear friction law with (for U = g) (b) p > 0, -+ O. This yields the non-local friction Jaw developed by Oden and Pires [2]. JUT(u)(x)1 < Sp(u,,(U»~UT = 0 = Sp(u,,(U))~UT2 O. (c) p This case corresponds Coulomb law of friction. to the classical (d) p-+o, -+O, u,,(u) prescribed on re. n this case, the contact area is known in advance, and the friction functionals in () reduce to j(u) = f. TUT ds rc with T > 0 given in, say, eo'c). (e) ' = O. This is the classical unilateral contact problem without friction studied by Signorini. Problem () reduces to: Find U E K such that } a(u. v - u) 2 J(v - u) for all v in K. (7) J. FNTE ELEMENT APPROXL\lATONS We can consider finite element approximations of the nonlinear variational inequality (\) for the case of twodimensional problems. Our method involves partitioning n into a mesh of Q2-finite elements (9-node quadrilaterals which are tensor products of quadratics) as indicated in Fig.. n this way we construct a finitedimensional subspace V of the space V of (2) spanned by piecewise biquadratic functions of X and x~: h h - V = {(V, V2 ) = v/.lv,,; ), E Q~(nc)' n = un,. i= 1.2; ~e ~ El. To handle the unilateral constraint. we introduce the penalty term ilc (U' n-g).v ds where ) is the penalty parameter. a positive constant representing the spring stiffness of an elastic foundation, which can be made to approximate the rigid foundation arbitrarily closely as ) tends to zero. (U n - g)... represents the positive part of U. n - g (which should be zero). and v = v. n. (8) (9)

3 Numerical analysis of certain contact problems in elasticity 483 ~ r:---- Fig.. Finite element discretization employing Ql,biquadratic elements. where t, are the quadrature points used in 1. We choose here Simpson's rule for evaluating ( ). We next compute a regularization of these stresses by sctting (13) at each nodal point on rc. Thus, both the norma) contact pressure tth and its regularization Th = Sp( Uh,,) are continuous. piecewise quadratic polynomials on the contact surface rc h. (3) Next. we use thc approximate regularized normal contact pressures Tt[l as data to solve a problem with nonlinear friction but with prescribed normal contact pressures. Thus we set e> 0 and definc (V ( ) ) 1 '.J.'(/ )W,T' VhTd rc Win 1,.1, W,Vt = 1'1 '1'. WhT S (14) for any Wh' Vh E Vit. where q,:o is given in (6). Then a correction w,,' of Uh(l, for frictional effects is obtaincd by solving the linear problcm: a(wh(ll. V,,)+ (Vj,.,(Wl), V,,) =!(V,) VVhE V it (S) Our finite element approximation of () then assumes the following form;. where For given. p. e. h, and ~. find a displacement field u = U,,( '. p. e., i) such that (16) a(u. Vh - Uh) + jp.,(u. v,,) - jp,,(uh' Uh)+ i- ( (U,,' n - g)+(vh ,,) ds ~ (10) J'C ~ f(vh - Uh) for all v in Vh. We have derived general error estimates for approximations such as (10) in [3]. A detailed study of the behavior of the error as the various parametcrs are varied is to be the subject of a separate paper. The main objective of the present notc is to describe one algorithm based on (10) which has led to the successful solution of some representative problems. ~. AN ALGORTH~t FOR SOS.cLASSCAL FRlCTtON PROBl.H1S We now describe an algorithm essentially based on the discrete variational inequality (10) which can be used in the numerical analysis of problems of this typc. () (/ = 0). To obtain starting values for an iterative scheme, we first analyze the approximate problem for the frictionless case. For this purpose. we make use of an existing code which employs a penalty method. reduced integration. and a standard Newton-Raphson scheme for handling the unilateral constraint (see Oden and Kikuchi[4] and Campos et al.[5]). n effect, we calculate as a first iterate the solution of the equality, a!uh lll. v,,)+ ~p-ll(ll~,~- g)+vh) =!(Vh) lv E V h () where ~ > 0 is a penalty parameter and T(-) is a quadrature rule for integrating numerically the penalty terms. (2) Having calculated Uh(, for a specified ~ and h, we compute nodal values of the normal contact pressurc by setting (4) Having obtained the solution Wh(l) of (14) we now calculale nodal values of the tangential stress by selling We then return to the first step. i.e. to the problem without friction and solve it again treating UhT(Wt[l)) as given tangential forces applied at the contact surface. This leads to new iterates Ut[2,. cr\;,: and T,,t of the displacement field. normal contact pressure and its regularization. We repeat step 3 replacing in (14)-(16) 710(1) by 11,m thus obtaining a second approximation of UhT. We continue this process until successive solutions do not differ by a preassigned tolerance. 5. NUMERCAl. RESULTS The algorithm described in the preceding section was applied to a representative contact problem with nonlocal and nonlinear friction laws in force. The problem considered is that of a square rigid punch indented into a lin~arly elastic body, as shown in Fig. 2. The geometry and data for the problem are indicated in the figure. Owing to symmetry, it was necessary to consider only one half of the body. A rather fine mesh of Q2-elements was used in the analysis of this particular example, and it is also indicated in the figure. The computed deformed shape of the mesh is indicated in Fig. 3 and the computed stresses on the contact surface are shown in Fig. 4. We observe that in this case the effects of the microstructure of the contact surface (p = 0.1) have a dramatic influence on the stress distribution. Had Coulomb's law been used. stress singularities in both the normal and the shear stress on the contact surface could be expected.

4 484 J. T. ODEN and E. B. PRES L j E = 1, = 0.3 v = 0.3 p = 0.01 = = 10-8 T f 2 ~'l 1 Fig. 2. ndentation of an elastic half-space by a rigid rectangular punch: undeformed configuration. $ Applied Force Fig. 3. Compuled deformed configuration.

5 Numerical analysis of certain contact problems in elasticity t1r_.b-..o- P J Fig. 4. Normal and tangential stress profiles on the contact surlace (mollified). A ',,. ' ' -o~ Nomal Contact Pressure -,6- Tangent ial Stress ~, ' ' ~:,, ~, A' 'i',....o-p 11 ' ' ~' t,' ':,.11 '1.' \~ ~ Hid.1 -- slide Here we obtain instead a finite stress owing to the finite limit of the diameter of deformed asperities in the microstructure. The maximum magnitudes of these contact stresses do not change appreciably with a refinement of the mesh, indicating that the finite values are a result of the choice of a non-local law and not discretization error. We note also that there is not a sharp front separating the full-stick and sliding regions on the contact surface. Again. this is due to the non-local character of the assumed friction law. This result is in agreement with physical experiments on friction between metallic bodies. The influence of non-zero stiffness of the junctions between the punch and the elastic body is not appreciable in the present example since e was taken to be quite small 00-4 ). However, we expect this parameter to be critical in calculating friction effects in many practical situations. We hope to explore the effects of varying this parameter in later numerical experiments. One serious shortcoming of the algorithm described here is its inability to easily accommodate the effects of load histories. This, of course, is of critical importance in friction problems since final stresses depend strongly on the loading history. We are presently working on a new family of algorithms that will allow us to take into account quasi-static loading histories. This work shall also be the subject of a forthcoming paper. AcknoK'ledgement- The support of this work by the U.S. Air Force Office of Scientific Research through contract F C-0083 is gratefully acknowledged. REFERENCES. 1. T. Oden and E. B. Pires, Nonlocal and nonlinear friction laws and variational principles for contact problems in elasti city. J. Appl. Mech. (to appear). 2. J. T. Oden and E. B. Pires, Contact problems with non-local friction. Finite Element: Special Problems in Solid Mechanics (Edited by 1. T. Oden and G. F. Carey). Prentice Hall. Englewood Cliffs (to appear). 3. E.. Pires and J. T, Oden, Error eslimates for the approximation of a class of variational inequalities arising in unilateral problems with friction. J. Num. Functional Anal. Optimization (to appear). 4. N. Kikuchi and 1. T. Oden. Contact Problems in Elas/icity. SAM (to appear). 5. L. Campos, 1. T. Oden and N. Kikuchi, Analysis of a class of contact problems with friction. Comput. Me/h. App/. Mech. Engng (to appear).