A seminumerical method for three-dimensional frictionless contact problems

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Mathl. Comput. Modelling Vol. 28, No. 4-8, pp, 413-425. 1998 @ 1998 Elsevier Science Ltd. AH rights reserved Printed in Great Britain 08957177/98 $19.00 + 0.00 PII:

SO8957177(98)00131-9

A Seminumerical Method for Three-Dimensional F’rictionless Contact Problems M. SASS1 Laboratoire M&zaSurfCER ENSAM Aix-en-Provence 2, COWSdes Arts et MBtiers, 13617 Aix-en-Provence, France and Laboratoire LMBS URA CNRS 1776 ENSAM, Paris, fiance M. DESVIGNES Laboratoire M&Surf CER ENSAM Aix-en-Provence 2, tours dea Arts et MBtiers, 13617 Aix-en-Provence, France Oaix.ensam.fr Abstract-A seminumericalmethod of solving three-dimensionalfriction&s contact problems betweentwo elasticbodiessubjectedto a normalloading is presented. This methodiaa generalization of an existing one, the generality is relating to the contact type (conforms1 or antiformal) and to the contact geometry (arbitrary shape of the bodies). Based on the use of influence functions, this method requires the discretization of the different contact criteria over a candidate contact surface and the optimization of the solution to fulfill these criteria. It allows us the determination of the true contact surface, the pressure distribution, and the approach between the two bodies. The method has been implemented in a computer program called “CONTELA”. and has been applied for antiformal geometric (our results are in close agreement with Hertz analytical model) and for closely conformal geometries (our calculations confirm the Hertz theory limitation to deal with this kind of contact geometry). @ 1998 Elsevier Science Ltd. All rights reserved.

Keywords-Antiformal contact, Conformalcontact, Seminumerical method, Contact surfaceop timiation, Contact pressure. 1. INTRODUCTION Contact is very frequent in mechanisms like gears, rails, etc.

It is considered as one of the

main causes of failure (contact fatigue), and consequently, the major factor in mechanism’s life prediction. Friction phenomenon which occurs with the contact is generally harmful. It can be attenuated by lubrication, by a smooth body surface, by a good choice of contact bodies material, etc. In this paper, frictionless elastostatic contact problems are investigated. More specifically, we take an interest in the determination of the contact surface, the pressure distribution, and the rigid-body approach between two contact bodies subjected to a normal loading. Various techniques are usually used to resolve such problems. One can distinguish the analytical technique, the numerical technique, and the seminumerical technique. In the analytical approach, the solution is expressed in closed form making fast the calculations. Despite this advantage, this approach is overstrained to a luck of generality, only a few elementary problems like the famous Hertz problem (1,2] has been treated. A wide bibliography regarding this approach is reported in [3]. The numerical approach aims to find a solution for the problem in a discrete finite space. Discretization can be total (Finite-Element Method [4-6]), or partial (Boundcrry Element Met-

413

414

M. SASSIAND M. DESVIGNES

hod [7,8]). In spite of it’s universality, this method is not convenient for a class of contact problems requiring the optimization of the shapes and/or the dimensions of the contact bodies. The seminumerical approach only necessitates the discretization of the integral equation governing the local contact problem. The development of computer science during the last decades has favoured the use of this technique to solve a wide range of contact problems with sheep cost (CPU time). Conrey and Seireg [9] were among the first to employ thii technique to solve a contact problem by means of a simplex-type algorithm to minimize the potential energy of contact. Later, Singh and Paul [lo] utilized the flexibility method of structural analysis to solve the same problem. By discretizing a candidate contact surface into rectangular cells on which the contact pressure is assumed to be constant, they derived a linear system of equations in terms or the unknowns contact pressure. The system obtained proved to be ill-conditioned and had required the development of specific algorithms. To overcome this difficulty, Hartnett [ll] has suggested the substitution of the global ill-conditioned linear system of equations for several coupled linear systems of small order. Ahmadi [12] generalized Hartnet’s method to solve the friction contact problem between two elastic bodies with edge effects. The above studies deal with antiformal contact problems (the dimensions of the plane contact surface are small compared to those of the contact bodies): all these studies use Boussinesq’s influence functions. For conformal contact between two bodies of arbitrary shape, the bodies dimensions and the contact surface are comparable, the Boussinesq’s influence functions are not appropriate. It is generally necessary to find more individualized influence functions for each of the two bodies. Paul and Hashemi [13] employed a modified form of Boussinesq’s influence functions for the conformal contact of rail and wheel. A study of various exact and approximate influence functions for simple contact geometries (sphere, cylinder, etc.) has been described by Woodward [14], where numerical influence functions has been generated by a three-dimensional Finite-Element program. The present work represents an extension of the previously described method. It can be applied for either antiformal or conformal frictionless contact problems between two elastic bodies of arbitrary shape. In Section 2, we reformulate the contact criteria governing the problem. Section 3 takes an interest in the discretization of the contact criteria over a candidate contact surface. Section 4 briefly explains the resolution method and the optimization technique used. Two numerical examples are presented in Section 5, one is relating to an antiformal contact, the other to a conformal contact, and conclusions are presented in Section 6. Note that throughout this paper, Q Greek index takes values in {1,2} and denotes bodies; hence, Latin subscripts i, j, k, I show values in {1,2,3,. . . }. Einstein’s summation convention is only assumed on repeated Latin indices; vectors and tensors are denoted by bold characters.

2. FORMULATION

OF THE CONTACT

PROBLEM

Two three-dimensional linearly elastic bodies, Ra (a = 1,2), are pressed together as in Figure 1. Bodies fi2” are characterized by their Young’s modulus E, and their Poisson’s ratio v,. Let us denote I’” = aQQ, the boundary of Ra, I’:, the part of l? susceptible to come in contact with the other part (contact region), and Ic the common part of l? and F2 after deformation (contact stcrface). The outward normal of l?: is denoted no. The initial contact between sZ1 and R2 is assumed to occur at a single point 0 origin of the right-hand coordinate system (0, x, y, z). Frictionless contact dictates that the resultant applied force RQ (by equilibrium R’ = R2) passes through point 0. The coordinate system (0, x, y, z) is constructed so that R” will be supported by z axis. In this coordinate system, the boundary profile surface FQ is defined by an explicit equation 2, = fa(z, y).

A Seminumerical Method

415

0

-5,.z I

4z

I

“1

Figure 1. Contact geometry.

In the framework of a seminumerical technique, the resolution of the contact problem requires the use of two kinds of equations.

(4

Static equilibrium. Bodies RQ are in equilibrium under the resultant external force RQ and the resultant contact force due to the pressure distribution p over I’,:

Rff+

dr,=0. JJrc(-pxP)

G)

(b) Elastic

equilibrium. These equations, also called contact criteria, express the fact that two points A41 and A&, respectively, of I’: and I’: candidate for the contact will merge after deformation. Initial separation sin (labelled h, before deformation) and final separation sj (after deformation) between the points A4, are (Figure 1)

sin = MiM2 = h

and

Sf =

MYM;.

By definition of the contact, within IC, the components of sj in a base (n, t, w) bound to the contact point (Figure 2) vanish, one can write the three criteria sj . n = 0, sj . t = 0,

(W on rc,

sj *w = 0,

Figure 2. Definition of the local beae (n, t, w).

G) w

416

M. SASSIAND M. DEWIGNES

2.1.Final Separation

sf

When the bodies are pressed together, the displacement of the point M,

is the sum of two

displacements (Figure 1) [13,14]: - a rigid body motion 6, along the z axis: M,M& - an elastic displacement uQ : MdMz

= -6,

. z (6, > 0 and 62 < 0),

= uQ.

Consequently, the final separation can be expressed as follows: sf=M:‘M;=u2-u’+&z+h,

with 6 = 61 - 62

and

6 > 0,

6 is the rigid-body approach between 0’ and R2. This above expression introduces the surface displacements ua and the initial separation h between two candidate contact points.

These two quantities are investigated in the two next

sections. 2.2. Surface

Displacements

u”

The surface displacements are given by influence functions (or Green’s functions). These functions relate the elastic displacements of a given point M(x) (field point) of R” to an applied unit force at some other point M/(x’) ( source point). Note, that x (respectively, x’) locates the vector position of M (respectively, M’) and subsequently, source variable will be marked by primes. Assume that influence functions labelled Ga(x, x’) are known, then the displacement of the point M(x), due to the load -p(x’).rP(x’) applied at M/(x’) is G”(x,x’).(-p(x’).rP(x’)). Consequently, the displacement field arising from the pressure distribution p(x’) acting on rc is UQ(X) = On lTC: n2(x’) = -n/(x’), the following expressions:

u’(x) =

JJ rc

Ga(x, x’) . p(x’) . nO(x’) . dI’,,

let us denote: Vx’ E lYc, n(x’)

JJ JJ

= n2(x’)

= -nl(x’),

one can obtain

Gr(x, x’) . p(x’) . n(x’) . cdl?,,

rc

u"(x) = -

r.

2.3. Approximation

vx E v.

VXE~,.

G2(x, x’) . p(x’) . n(x’) . dr,,

of the initial

Separation

h

Initially, the positions of points Ml (E I’:) and A42 (E I’:) candidate for the contact at the point M of lYeare unknown. For antiformal contacts, MI and M2 are approximated by the projections of M along the z axis on the surfaces r: (h separation is supported by the z axis) [10,13,15]. In wnformal contacts, this assumption is not appropriate. It introduces some disparities in the results because it cannot take into account the tangential and lateral displacements of the point M. A better method for this class of contact problems carried out in [14] suggests as a first approximation to take MI and M2 at equal distance from 0 (distance measured along the contour curves I’: and I’:). Thii method is convenient for simple contact shape (for example, sphere and cylinder) but it is not easy to use for more general contact surface. We propose herein an approximation independent of the contact shape: points MI and M2 are taken as the orthogonal projection of the point M, respectively, on l?: and I’: (Figure 3). Uniqueness of the projections is insured by the regularity of the profile functions fa(z, y). An approximate position of the point M, can be determined by the intersection of the contact surface’s normal at M and the contact region I’:.

A Seminumerical Method

417

Figure 3. Candidate contact points.

2.4. Contact Criteria From the expressionsof u1 and u2 previously calculated in Section 2.2, the final separation sf becomes Sf(X) = -

JJ

rc

G(x, x’) . p(x’) . n(x’) . dI’, + 6 . z + h(x),

with G = G’ + G2.

Using this upper expression of sf, the three criteria (C,), (C,), (C,) can be written as follows: s,(x) = sf(x) f n(x) = -

JJ JJ . JJ w(x)

n(x) . G(x, x’) . p(x’) - n(x’) . dI’, + b . n(x) . z

rc

+ n(x) . h(x) = 0,

St(X) = q(x).

t(x)

= -

(C?J

t(x) . G(x, x’) . p(x’) . n(x’) . cdl?,+ 6 . t(x) aII

rc

+ t(x) . h(x) = 0,

s,(x) = Sf(X) *w(x)

= -

(Ct)

G(x, x’) . p(x’) - n(x’) . dI’, + 6 - w(x) az

rc

+ w(x)

. h(x) = 0,

(0

rc,Sn, st, and sw are the components of sf in (n, t, w). To abbreviate notations, these three criteria can be grouped in one equation

VX E

s,(x) = Sf(X) . v(x) = -

. JJ v(x)

G(x, x’) . p(x’) - n(x’) . dI‘, + 6. v(x) +z

rc

+ v(x)

. h = 0,

(G)

vx E rc,

where v = n, t, w. 2.5. Approximation

of the Candidate Contact Surface Profile PC

In antiformal contact, the contact surface is assumed to be plane. The surface contour is preliminarilyoverestimated by a Hertzian calculation [11,12,16]. Whereas, in conformal contact thii contact surface is initially unknown, some hypotheses are required to suggest a candidate contact surface. For example [13], suppose that this surface lies on one of the body’s surfaces. (This is equivalentto assumingthat one of the two contact bodies is rigid.) The effective contact surface is an intermediate surface between the profilesI’: and I’:. In the case of Hertzian profiles (paraboloids), one can establish [15]that the equation of the intermediatesurface profile is defined by

z = fb, Y)= K ’fl(TY)

+

(1 -

K)

* fi(GY),

with

Kc-&L_kl + kz’ 1 - Y2 ka=-&% -

K is a sharing coefficient.

a

(K E P,

ll),

418

M. SASSI AND M. DEWIGNES

We suggest employing this approximation in the gene& case (arbitrary profiles). One can easily find two particular cases: the first occurs when one of the two bodies is rigid (let us assume, for example, that R2 is rigid); then K = 0 and f(z, y) = fz(z, 9). The second case occurs when the two bodies have the same material, then K = l/2 and f(z, y) = [f~(z, y) + fs(z,y)]/2. An example later will justify the use of this approximation.

3. DISCRETIZATION

OF THE CONTACT

PROBLEM

The purpose is to find the contact surface rc, the pressure distribution p(x’) over lYc,and the approach 6 between the bodies 0’ and R2. For a given candidate contact surface F’c (for example, the intermediate surface), the trite rion (C,) becomes an integral equation of first kind, which we shall solve by an iterative numerical scheme. The projection T’c of rc, (x, y) plane is overestimated by a Hertzian calculation. This allows the determination of a first candidate contact region I’=. Let us discretize re into N rectangular cell l?j, (J’ = 1, N) (Figure 4), so that the pressure on the elementary cell l?i, at xi is assumed to be constant (p(xi) = pj on I’$).

Figure 4. Discretization of the contact surface rC.

The contact criteria (C,) can be expressed as follows: v = n,t,w,

i=l,N,

s&i)

= - 5

v(xi)

’ G(Xi,

Xj> ’ pj . n(Xj)

. dr;

j=l

1

i- 6 . V(Xi)

. z + V(xi)

. h = 0,

(G-1)

where indices i, j represent, respectively, the field and source points. Over cell rj, : n(xj) is constant, denoting h, (v = n, t, w) the components of h in the (n, t, w) base, q = v aa, and using contract indices k, 1, one can write i=l,N,

v = n,t,w,

k,l E {1,2,3},

Gkl (xi, xj) - dI’z

otherwise

i=l,N,

v = n,t,w,

1

+ 6 . VS(X~) + h,,(xi)

j E {l,...,N},

sV(X~) = -FG ‘pj + 6. Q(x~)

+ h,(xi)

= 0,

with i=l,N,

j=l,N,

v = n,t,w,

FG = nl(xj) *vk(xi)

JJc

k,l E {1,2,3],

yj Gkl(xi,Xj)

. dr$.

= 0,

(c’-2)

(G-3)

A Seminumerical

However, taking in consideration

419

that R* = RQ . z, equation (C,) becomes

R” -

This equation is then diicretized

Method

1.7rc

p(x’) . nT(x’) . dr, = 0.

(G-l)

as follows:

(G-2) Let l?i denote the projection of I’{ on the (x, y) plane (Figure 5), then dl?{ = ns(xj) consequently, (C&2) becomes Aj

’ pj

=

P,

j E {l,...,N},

.

(C,-3)

with Aj, the area of the cell I’{ and P verify P = R’ = R2. Equations (C,-3),(C,-3) boundary conditions

allow the determination FG ‘pj

Pj 2 0,

on ro,

pj = 0,

on

. cdl?:,

and the

(Cd

ar,,

of the unknowns pj (j = 1, N), 6, and lYcby solving the system

-b.ns(Xi)

= h,(~i),

(sd-1)

Aj *pj = P, Pj 2 0, pj

=

@d-2) (Sd)

on ro,

0, on

dro,

@d-3)

i=l,N,

j E (1,. . . ) N}.

@d-4)

Figure 5. Integration area dI$.

4. RESOLUTION

PROCEDURE

In this section, we present the method used to solve the linear system (Sd), and the optimization scheme based on the criteria (C,),(C,). 4.1. RX?solUtion of (!&) Contact region I’0 is estimated as explained in Section 3, the contact surface rc is then divided into N cells. Influence functions FG can also be evaluated for each cell I?j, (N2 terms). In a 6rst step, the resolution by a Gauss elimination scheme of the linear system formed by equations (S&i) and (S&2) gives the (N + 1) unknowns (pressures pj and rigid-body approach 6). In a second step, taking into account the criteria (S&3) and (S&4), cells I$ with negative pressure are eliminated. So, one can obtain a novel contact region (rc)i, i.e., a novel cells number Ni. The procedure is repeated m times until we have N,,, = Nm+i (so (IO),,, = (Is)m+i). The pressure pj (j = 1, Nf, where Nf is the final cell number) and the rigid-body approach 6 calculated in the last iteration are the solution of system (Sd). For a given initial separation, this procedure based on the criteria (C,), (C,), (Cb) only guarantees the vanishing of the normal separation sn between Ml and Ms. In the next section, an optimization procedure for the initial separation calculation is presented. The criteria (C,),(C,) will be used to minimize the tangential and the lateral components of the separation.

420

M.

SASSI AND

M.

DJBVIGNJLS

4.2. Solution Optimization A first approximation of the initial separation estimated by the projection method has allowed the determination of a pseudo solution of the contact problem. This separation is considered as a bound of the exact separation; it will be labelled sin = MTMg = hr, (Figure 6). Let us define an other bound by assuming the separation sin parallel to the z axis: sin = MfMZ = h, (this approximation is very common for antiformal contacts).

Figure 6. Contact points optimization.

For a given contact surface profile (i.e., for a given value of the sharing coefficient K E [0, l]), let us consider the set Dh=

h,h=-

MrMz Mu

E arc (kf,ZMr) and MZ E arc (kf,tM,P) , 1 { for h E Dh, problem (Sd) has a solution S(h). For each solution S(h), using the criteria (C,),(C,), one can calculate the separations st(xi) and sW(xi) for each cell I’: by St(Xt) SO

=

-F:j *Fiji+ 6.

tg(Xi)

=

-F$

Ws(Xi)

.pj + 6.

+ ht(Xt),

(G-4) i=

+ h,(&),

1,...)

Nf,

j E {Wf).

(Cul-4)

Let us define on Dh the two cost functions

CFst(h)

=

J

&t(xi)12, i=l

CFs,(h)

=

&Jx~)]~.

J i=l Taking into account the choice of the base (n, t, w), vector w is nearly perpendicular to z (Figure 2), consequently s,,,(xi) is small compared to st (xi), then CFst(h)

>> CFs,(h),

Vh E Dr.,.

The privileged criterion is then (C,), the approximate problem becomes: find h, E Dh to fulfil CFst(h,)

= hT& [CFst

WI.

An iterative optimization scheme of this above cost function is performed, it allows us to converge to the solution h,, and consequently to the final solution S(h,) of the contact problem. Concerning the criterion (C,), calculations released with the solution S(h,) for several values of K (K E (0, l]), show that the value K = kz/(k~ +k 2) minimize approximately the cost function CFs,(h,) (cf. example in Section 5.2). This justifies subsequently the use of the intermediate contact profile surface. Nevertheless, the adjunction of the criterion (C,) in the optimization scheme is possible. However, to have a middle compromise accuracy/time calculation, we have merely used the proposed approximation. The time calculation for an initial discretization into N = 200 cells is about 4 to 6 minutes on a HP 9000-720 workstation.

A Seminumerical Method

5.

NUMERICAL

421

EXAMPLES

From the discretized procedure presented, a FORTRAN computer program called “CONTELA” is developed. To illustrate the method, in the two next sections, it is applied first to an antiformal contact geometry, and second to a conformal contact geometry. 5.1. Antiformal

Contact

Geometry

For antiformal geometry, the dimensions of the deformed contact patch remain small compared to the principal radii of the undeformed surfaces. The Hertz analytical model gives the solution of the contact problem. This allows us to validate the results given by “CONTELA”. In this case of antiformal contact, the projection method presented in Section 4.2 gives a separation supported by the z axis (hp = h,). Influence functions used are the half-space Boussinesq’s functions k,l E {1,2,3}.

G:l = Bkq,

The components of the tensor B” expressed in a local coordinate system (0, x~, ya, a,) bound to the half space z, 2 0 occupied by Ra can be found in [17]. We report here the three components of B” which enter in the calculations when the load is normal to the contact surface (n = z)

with rk

=3&-X;,

r =

(r? + r; +ry2

xi = z’ =

(

0,

ECX kX = 2( 1 + Va) ’ Table 1 gives the indentation of a half space by a sphere (radius &, = 20 mm, E, = 210000 MPa, and ua = 0.3) with a normal load P = lOOON, the results and the relative deviation between the calculations of Hertz and “CONTELA” (for N = 200). Table 1. Comparison between the results of “CONTELA” and Hertz. Major Axis

Minor Axis

Approach

Max. Preesure

a (mm)

b (mm)

6 (mm)

pmax (MPa)

Hertz

0.50658

0.50658

0.01283

1860

“CONTELA”

0.50436

0.50436

0.01280

1862

0.43%

0.43%

0.23%

0.10%

Deviation

Figures 7 and 8 show the results of the two models concerning the contact region and the pressure distribution along the x axis. One can observe the good correlation between the two models.

M. SASSI AND M. DEWICNES

422

0.6~ y(mm)

.6

-0.6 Figure 7. Contact region.

5.2. Conformal

-0.4

-0.2

0

0.2

0.4

0.6

Figure 8. Pressure distribution p(z, 0).

Contact Problem

One of the major difficulties in the case of conformal contact problems is the determination of suitable influence functions. No exact solution is known for general contact geometry. We utilize here an approximate model derived from the Kelvin’s fundamental functions used in the Boundary Element Method. In this model, an approximate expression of the influence can be written as follows: Gkl=G~l+G~l=+6k1+kd.~],

k,l

E

{WA},

with bkl, the Kronecker symbol kd=k;+k;, kf =k;+kj,

1 16?r(l - V&J= ’ “7 = (3 - 4v,). k;, k; =

and Tk T=

=xk-x;, (t~+~Fj+$)l’~,

& PLQ= 2( 1 + Va) * Among several influence functions tested for conformal contact problems, these functions are proved to model correctly highly conformal geometry. In fact, the last model has been compared with Finite Element results obtained using ABAQUS software for the 3D contact problem between a sphere (radius R& = 5.556mm) indenting a cylindrical cavity (radius Rg = 5.64mm). The two bodies are made of the same material as the previous example. Figure 9 shows the variation of the maximum pressure versus the applied load P. The results prove the efficiency of the model to treat this kind of contact problem. For a normal load P = lOOON, Table 2 groups the results of “Model B” using Boussinesq’s in@bence functions, “Model K” proposed for the conformal contact problem, FE model and the relative deviation between the two first models and the FE results. Note, that because of the mesh’s size, FE calculations give approximate values for the major axis (a) and the minor axis (b).

A Seminumerical

Method

423

1800 . . PIIUX

1400 a.

-“CONTELA” -

FE

t 200

P 0

I 0

500

1000

1so0

2WO

Figure 9. Variation of maximal surface pressure as a function of the applied Table 2. “CONTELA”

I

results-comparison

of the models.

“Model K”

Except for these values, one can observe that “Model K” agrees better with FE calculations than “Model B”. Figures 10 and 11 show the results of the two models concerning the contact region and the pressure distribution along x axis. One can observe a disparity between results of the two models. This disparity due to the conformity effects was also reported in [13,14].This confirms the Hertz theory (based on the “Model B”) limitation to treat conformal contacts. In particular, our results confirm that for conformal contact, the Hertz theory overestimates the contact patch (major axis, minor axis) and underestimates the maximum pressure p,,, at the contact center.

o,2

y(mm

1

Figure 10. Contact

region.

Figure 11. Pressure distribution

p(z9

0).

For this last example, Figure 12 shows the variation of the cost functions CFst and CFs, versus the sharing coefficient K. In this figure, one can show that the cost function CFs, is minimum for a value of K close to 0.5 which is the value for the intermediate surface (here ki = kz). Besides, a note that has been mentioned above, CFst is ten as great as CFs,.

424

M. SASSIAND M. DEWIGNES

z ’

0

0.2

CIA



0.6

0.8

1

Figure 12. Variation of CF ss function of K.

6. CONCLUSIONS A seminumerical method for the resolution of three-dimensional elastostatic frictionless contact problems is presented. Based on the use of the influence functions, this method necessitates the discretization of the contact criteria over a candidate contact surface, and the optimization of the solution to fulfill these criteria. It allows the determination of the contact surface, the pressure distribution, and the rigid-body approach between the contact bodies. The method has been implemented in a computer program called “CONTELA” and allows the resolution with sheep cost (CPU time of 4 to 6 minutes on HP 9000-720 workstation) of 3D contact problems between two bodies of arbitrary shape. For conformal contact geometry, a particular emphasis has been reserved to the optimization of the contact surface shape; the intermediate contact surface is proved to be a good approximation for the contact surface. Calculations confirm the Hertz theory limitation to deal with conformal contact geometry. This method will allow the determination of the contact stress field which is useful for a better prediction of the life duration of some mechanism using conformal contacts.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

K.L. Johnson, On hundred years of Hertz contact, Proc. Instn. Mech. Engrs. 196, 363-378 (1982). K.L. Johnson, Contact Mechanics, Cambridge University Press, (1995). G.M.L. Gladwell, Contact Problems in the Classical Theory of Elasticity, Sijhoff & Noordhoff, (1980). Z.H. Zhong, Finite Element Procedures for Contact Impact Problems, Oxford University Press, (1993). S. Vijayaker, H. Busby and L. Wilcox, Finite element analysis of three-dimensional conformal contact with friction, Computers d Structures 33 (l), 49-61 (1989). G. Inglebert, C. Kallala and Y. Pinto, Methode des elements finks appliqk aux contacts, M&unique MatMauz Electricit 46 (2), 80-83 (June 1993). CA. Brebbia, J.C.F. Telles and L.C. Wrobel, Boundary Element Techniques, Theory and Application, pp. 184-204, Springer-Verlag, Berlin, (1984). G. Karami, A Boundary Element Method for Two-Dimensional Contact Problem, Lecture Notes in Engineering, (Edited by Brebbia and Orszag), Springer-Verlag, Berlin, (1989). T.F. Conrey and A. Seireg, A mathematical programming method for design of elastic bodies in contact, Journal of Applied Mechanics (??ans. ASME), 387-392 (June 1971). K.P. Singh and B. Paul, Numerical solution of non-Hertzian elastic contact problems, Journal of Applied Mechanics (tins ASME) 41, 484-490 (June 1974). M.J. Hartnett, A general numericalsolution for elastic body contact problems Solid Contact and Lubrication (Edited by H.S. Cheng and L.M. Keer), ASME AMD 39, 51-66 (1980). N. Ahmadi, L.M. Keer and T. Mura, Non-Hertzian contact stress analysis for an elastic half space-Normal and sliding contact, Int. J. of Solids and Structures 19 (4), 357-373 (1983). B. Paul and J. Haahemi, Contact pressures on closely conforming elastic bodies, Journal of Applied Mechanics (tins. ASME) 48, 543-548 (September 1981). W. Woodward and B. Paul, Contact stresses for closely conforming bodies-application to cylinders and spheres, Technical Report No. 2, DOT-TST-77-48. M. Foulon and A. Hey, Sur les contacts ponctuels, Reuue Fkncaise de Mkanique 4, 223-233 (1985).

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16. R.L. Munisamy, D.A. Hills and D. Nowell, A numerical analysis of an elasticity dissimilar three-dimensional sliding contact, Proc. Instn. Mech. Engrs. 206, 203-211 (1992). 17. T.W. Wu and M. Stern, Boundary integral equations in three-dimensional elest&atics using the Boussinesq-Cerruti fundamental solution. Eng. Analysis with Boundary Elements 8 (2), 94-102 (1991).

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