The rigid punch problem with friction

Inr. 1. Engng Sci. Vol. 29, No. 6, pp. 751-768, 1991 Printed in Great Britain. All rights reserved

THE RIGID PUNCH PROBLEM WITH

~~7225/91 $3.00 + 0.00 Copyright @ 1991 Pergamon Press plc

FRICTION

A. KLARBRING’, A. MIKELIe’ and M. SHILLOR ’ Department of Mechanical Engineering, Linkijping Institute of Technology, S-58183 Linkaping,

Sweden, * Department of Theoretical Physics, Institute “Rudjer Boskovic”, 41001 Zagreb, Yugoslavia and 3 Department of Mathematical Sciences, Oakland University, Rochester, Michigan

48309, U.S.A.

(~mmuni~ated by J. T. ODEN)

Ah-et-The rigid punch problem with Coulomb’s friction and a power law surface normal compliance is considered. A quasistatic variational inequality is derived. When it is discretized in time we obtain, as a step in a sequence, an incremental variational inequality. The existence of a solution of this problem is shown, and some results concerning uniqueness and regularity are considered. The duality theory of Mosco, Capuuo-Dolcetta and Matzeu is used to derive a dual variational inequality where the dual variables may be interpreted as contact tractions. Frictionless problems are briefly discussed.

1. INTRODUCTION

The present work is concerned with fictional punch problems in the framework of linearized elasticity. The physical setting consists of two solid bodies pressed together so that they interact through contact forces. In the problem one of the bodies, the punch, is considered as rigid and the other is deformable. Known forces act on the punch but its placement in space is not prescribed and is part of the solution. We derive for this problem a variational inequality for which we are able to show existence of a solution under appropriate conditions: notably involving compatibility conditions on the applied forces.

In the classical treatises on contact problems (for instance, Galin [l]) the dominant class of problems is punch problems. However, in the modern treatment of contact problems, by variational inequalities, punch problems are rather rare. Exceptional is the presentation of the frictionless punch problem in Kikuchi and Oden [2]. Discussions of punch problems involving friction in the framework of variational inequ~ities seem so far to be completely lacking.

What seems to be of further significance, when comparing the classical and the modern treatments of contact problems, is that the classical solution methods usually treat the traction bn the contact boundary as the primary unknown, while in modern methods the displacement field has that role. The connection between two such different formulations of the same problem is through a theory of duality. It is shown in this paper that the extension given in [3] of the duality theory of Mosco [4] and Capuzzo-Dolcetta and Matzeu [5], furnish the appropriate framework for such a connection.

Generally the dual problems, i.e. problems in terms of tractions on the contact surface, are quasivariational inequalities. These are much more complicated to treat as the convex sets depend on the solutions.

On the other hand, the knowledge of the surface stresses is of major importance in engineering applications, while displa~ments are of less interest.

Following this introduction we give in Section 2 a derivation of the variational inequality. In Subsection 2.1 we give a geometrically exact impenetrability condition for a large displacement situation and then linearize it to obtain a contact condition suitable for use in a small displacement theory. The resulting condition is not new: it was given by Kikuchi and Oden [2].

However, the derivation given here is more general than previous ones. In Subsection 2.2 a classical formulation of the quasistatic rigid punch problem is given. On the contact boundary we use Coulomb’s law of friction and a surface normal compliance contact law. The power law normal compliance for the description of contact interaction was adopted by Oden and Martins [6] based on an impressive review of experimental results. It was further studied in [3,6-181. In

751

752 A. KLARBRING et al,

Subsection 2.3 we first derive a variational inequality formulation of the quasistatic problem. Then we introduce an implicit time discretization and in this way obtain a sequence of static problems, so that the original quasistatic problem is solved approximately in a step by step fashion. In subsequent sections we study one step in the sequence and such an incremental problem is the main concern of this paper.

In Section 3 we give necessary and sufficient conditions for the existence of solutions to the incremental problem. The sufficient conditions are part of an existence theorem, the proof of which is technical and will appear elsewhere [15]. The necessary conditions have a very clear physical meaning, as explained in Remark 3.2.

Section 4 is devoted to further mathematical properties of the problem. First the regularity of the solution is described. It turns out that the displacement and the normal tractions are Holder continuous on the contact surface, while the tangential traction is only in L”(S,). The proof is given in [15].

Then we obtain a uniqueness result for the case when the punch is restricted to move only up and down.

In Section 5 we use a general duality theory, presented in [3], to derive a dual variational inequality problem. This formulations involves the tractions on the contact surface. It is a quasivariational inequality and the existence of a solution follows from the existence of a solution to the primal problem in Section 3.

Finally we consider shortly frictionless contact problems in Section 6. The primal and dual problems are obtained in a straightfo~~d manner from those in Sections 3 and 5, respectively. Then we give a short description of some recent results of Shillor et al. (161, who considered the problem of a flat punch indenting an isotropic homogenous elastic half-plane. The problem can be set as an integral equation, whose properties were investigated in [16] together with some numerical experiments.

2. THE MODEL

We derive the linearized contact conditions and set the problem as a variational inequality.

2.1 The linearized contact condition

Let two domains of R*, denoted as 51’ and Qd, represent the reference configurations of a rigid punch and a deformable body, respectively. Regular mappings

f:Q’+N,

xd: ad-, N,

place the bodies into the ambient space N = R *. To be physically admissible, this placing should be such that

x’(W) n XdfQ”) = 0 (2.1)

The condition (2.1) asserts that the displaced bodies do not occupy the same part of the space N. Assume that to fulfill (2.1) it is only necessary to study parts of the boundaries d&Y and dad, denoted as Sg and S:, respectively. Furthermore, assume that there is an orthogonal coordinate system (0, X, , X2) such that for X = (X,, X2) E Si and some smooth function v, we have

x2 = VVd. (2.2)

Thus the graph of 9 represents the shape of the punch. If the X,-axes is chosen so that it points away from C, then, at least for points sufficiently close to SY, we have that

WX) = @(Xl) -x2, (2.3)

The rigid punch problem with friction 753

satisfies Y(X)>0 if XEW,

Y(X)=0 if XESE,

Y(X)<0 if X$!2’US’.

That is the punch is below the graph of q. The position of the rigid punch can be represented by an orthogonal 2 x 2 matrix Q and a

vector G:

x=x’(X)=Q.X+G. (2.4)

The representation (2.4) extends xr to all of R2 and as such it can be inverted to give

X=Q’.(x-~g).

Thus, we can define a function

Y,(x) = ‘J’(Q’ . (x - xg))

such that Yx(x) > 0 if x e x’(W),

Yx(x) = 0 if x E x’(S3,

W,(x) < 0 if x $ x’(sl’ U Sr).

The mapping xd can be continuously extended, and then restricted to St. Denote this mapping by fd, that is

td : S$+ N. (2.5)

Now, the construction

Yx = Y,(3id(X))

= WQ’ . G”(X) - xg)), (2.6)

defines, for each point X E St, a mapping

Y,:M,X6XR*-+R, (2.7)

where Mx = (2” Ix, X E St} is the set of placings of the point X E S:’ and 6’ is the set of orthogonal 2 x 2 matrices. Then condition (2.1) reads

Y&“(X), Q, xg) 5 0. (2.8)

The linearized contact condition is obtained by the linearization of Yx at the point (id, I, 0), where id is the identity mapping, I the 2 x 2 identity matrix and 0 the zero vector Since 0 is not a vector space, linearization generally requires the techniques of differential geometry. However, we shall see that formal derivations, where we indicate by A a small change of a quantity, will suffice in this particular case. It follows from (2.6)

Aw, = VY(X) - [Q’s (Atd(X) - Ax,,) + AQ’ . (i”(X) - &)I, (2.9)

where X = Q’ - (j”(X) - xg). There holds

which gives in (2.9) Q.AQ’+AQ.Q’=O, (2.10)

AYx = VY(X) * Q’s [At”(X) - (a + (f”(X) - x,,) - A%)] (2.11)

where &3 = AQ * Q’ is a skew-symmetric 2 x 2 matrix. Setting x = f”(X) =X, Q = I and X, = 0 in (2.11) we obtain

AYvx = VY(x) - [Afd(x) - (a . x + A%)].

Next, we introduce displacement fields

ud=xd-id=Axd,

u’=Xr-Id=Ax’,

(2.12)

154 A. KLARBRING et al.

where the right equalities relate to the chosen point of linearization. Furthermore, infinitesimally

u’=Q-x+Ax,,,

and we may write the gradient as

W!(x) = vl + dW/dX, n’,

where n’ is the inward normal unit vector of Sy. In conclusion, we have the linearized version of (2.8):

nr*(ud-u’)-g50, where

g = -Yx(id, I, 0)/d/1 + dW/dX,,

(2.13)

(2.14)

(2.15)

is a measure of the initial gap between the rigid punch and the deformable body (along the normal to Sz).

Note that (2.15) is defined on the boundary 52. The ur appearing in (2.15) is the displacement field of the rigid punch when this is extended to the whole space through (2.4).

The significance of choosing the function Y(X) as in (2.3) is that nr appears in (2.14) and that g has a clear physical interpretation. From the mathematical point of view any smooth function that satisfies the requirements below (2.3) may be used.

The inequality (2.15) was previously derived in Kikuchi and Song [19] and Kikuchi and Oden [2]. However, in these derivations the large displacement contact condition (2.8) is not given explicitly. The present derivation makes clear what exactly is linearized as (2.15) is derived.

A review of the literature on punch problems (see e.g. [2] and references there) shows that, actually, (2.15) is seldom used. The condition more commonly used is

nd.(ud-u’)--50, (2.16)

where nd is the unit outward normal of the deformable body in the reference configuration. This is motivated by the fact that linear elasticity implies small rotations and, thus, the two normals must be approximately equal for the formulation to make physical sense. However, it is clear from the present discussion that (2.16) can be derived as a linearization of a large displacement contact condition: one uses a Y(X) such that Sf is represented by Y(X) = constant < 0. There are certain advantages in using (2.16) and we will do so in the sequel.

2.2 The mathematical problem

In this subsection we give a classical formulation of a quasistatic rigid punch problem with Coulomb’s law of friction and a normal compliance contact law. The deformable body represented by Qd is assumed to be linearly elastic and subjected to body forces f = (fi, fi) and surface tractions t = (tl, t2) on a part S, of EJSZd. It is held fixed on S, c dQd and therefore the displacement u = (ui, UJ (denoted by ud in Subsection 2.1) is prescribed there and equals zero for simplicity. It is assumed that dQd = sq U & U St and that St, S, and S, are disjoint and smooth curves. The following classical equations hold for the displacement u and the stress tensor o = (Oij)

2+&=0 in Qd, I

(2.17)

(2.18)

ui=O on S,, (2.19)

Oijnj = ti on &, (2.20)

where i, j, k, 1 = 1, 2; the summation convention is used; n = (n,, n2) is the outward unit normal to dQd and aijk/ are the elasticity coefficients, assumed to satisfy the usual symmetry and ellipticity conditions.


Fig. 1. The geometric setting. R’ is the rigid punch and Qd the elastic body.

We turn to consider the contact boundary conditions on S,“. It is assumed in the sequel that

s: = f-a, a) x (0).

That is, the contact boundary is flat and of length 2a. The condition (2.16) can then be written as

u,+6x,+/3-g”0, (2.21)

where u,” = u. nd, /3 is the vertical downwards displacement of the punch, i.e. A% = (a; /3), and O=Qz,= -S& represents a rotation of the punch around the origin, see Fig. 1.

Condition (2.21) means that particles of the interior of the bodies cannot occupy the same place in a deformed configuration. Allowing for equality in (2.21) means regarding the contact boundary as a sharply defined nonmaterial boundary. This may not always be a natural concept. Machined metallic surfaces, for instance, possess undulations, called asperities, that are large on a molecular scale. Such surfaces also possess physical ch~acteristi~, due to oxide layers etc., that are different from those of the parent body, see Oden and Martins [6]. If the boundary is considered as a material boundary, allowing for equality in (2.21) means overlapping or interlocking of the material. Of course, if we do not allow equality in (2.21) we have to associate with the contact constraint a different force-displacement characteristic than if we do. The two situations are illustrated in Fig. 2. For light loads the curve in Fig. 2(b) can be approximated by a power law [6]. Modifying slightly the interpretation of g we can assume the following contact law, the so-called “surface normal compliance” (see [lo])

--ON = cN[(uN + @xl+ fi -g)+]““, (2.22)

1 ‘Force” 4 “Force”

(a) (b)

Fig. 2. The relationship between the Signor&i condition and the normal compliance condition.

756 A. KLARBRING et at.

where cN > 0 and m,,, z 1 are two parameters representing the physical characteristics of the interface, (z), = max(O, z), and a, = Uijnini is the normal traction vector (n = 18).

In the tangential direction of S$ we assume Coulomb’s friction law to hold. The relative tangential displacement vector is generally given by

u - ur - nd[nd. (u - d)].

For the case of a flat boundary as defined above, the only non-zero component of this vector is

where uT is the first component of u - u”(n” - II), i.e. the displacement in the x,-direction. The traction vector conjugate to the relative tangential displacement is the tangential traction vector

aT=a.nd-nduN.

The only nonzero component of this vector is ur 3 -ulz. With this notation Coulomb’s law reads:

kJTI s c1 kvlt (2.23)

/o,/<p l%‘I=$&- k=O, (2.24a)

ar=p ~a~~=$&- &SO, (2.24b)

-a,=p @~I=$&- fki,o, (2.24~)

where p is the coefficient of friction and (‘) = a( )/at denotes time rate of change. Now, since a, is directly given by uN, through (2.22), the friction bound p luNl can be replaced by

IuciJ(%J f @x1 f P - g)+lrnN*

It seems apparent that, as suggested by Oden and Martins [6], one can generalize Coulomb’s law by replacing the constant q., by a different constant mT in the above expression for the friction bound. This possibility will be discussed in subsequent sections.

Finally, the setting of the problem involves equilibrium-equations of the rigid punch. By our assumptions external forces T and N, in the coordinate directions, and an external moment M, positive in the right-handed sense, act on the rigid punch. We have,

i

a N=- c~~d.xr, (2.25)

--a

I

(I T= c+dvq, (2.26)

--lx

I

a J/f=-

XlGN &I* (2.27) --a

We derive a variational formulation of the quasistatic problem. That is, given the external loadings, f, t, N, T and M, as functions of time, we want to determine the stress tensor u and the displacements u, LY, /I and 8, as functions of time. To that end, let

be the bilinear form dx = d.~, dxa, and let

a(u, v) = J- C3Ui &Jk -dx (2.28)

sad ““G &.,

that represents the virtual power of the stress tensor field within &,

Vd={v;v=OonSU}


be the set of kinematically admissible displacements within SZd. Then Green’s formula and (2.18) imply that for all v E Vd

a(u,v-i)= - I odz (vi - tit) dx + I, [CTN(ZIN - LiN) + oT(u~ - a,)] do + 1 o&v, - &) h, e St

(2.29)

where u, and ZiT are defined similarly to uN and up Next

crM(t& - z&) = a,(t.Q$ + &, + fi - LiN - ex, - 6) - O&(8 - 4) - u&j - fi).

So by using (2.22), (2.25) and (2.26) we obtain for all 6, /!I and v E Vd

where the normal compliance functional is

Mu1 BP @, (v, s, @I = l,c N K UN + 6x1 + /!I - g)+lm”(u&! + &, + fi> dr c

and it represents the virtual power of the normal stresses. Furthermore,

UT(%- - &) = (JT(ur - 5 - iiT + &) + o,(h - k),

and an equivalent way of writing Coulomb’s law of friction, (2.23) and (2.24), is

-or(2$ - & - (z& - iu)) 1. p ]oNI (]v, - ti] + l&T - iu]) Vt+, &.

Thus, using this and (2.22) we have for all B and v E Vd

I-

- J uT(vT - iT) h qT((u, 6, O), (6 @) -jT((“, 61 @, (u? &)) - T(Lk - k)% (2.31) %

where the friction fMnc~on#~ is

and it represents the virtual power of the tangential stresses. Now, using in (2.29) relations (2.17), (2.20), (2.30) and (2.31) we obtain the following

problem: Z’he quasistatic problem. Find (~,_a,_@, 6): [0, T]+ Vd X R3 such that n(O) = h, d(O) =

@ouo, B(0) = PO, 6(O) = 6 0 and V(v, &, /I, 0) E Vd X R3 there holds

a(u, v - 6) +jN((u, rs, @), (VT 8, @ - (6 b, 6)) +/‘T((@h /% @)I (v, &>) -iT((u, ,@I @), (i, k))

- M(6 - 6) - N@ - 8) - T(ii - &) L F(v - h). (3.32)

Here [0, T] is a time interval of interest, II,,, tvo, & and e. are the initial conditions and

F(v) = _f &Vf dx + 1 tjvi ds, nd s,

represents the virtual power of the external forces acting on the deformable body. In this paper we do not attempt to treat the quasistatic problem directly. Instead we perform

an implicit time discretization of this problem and in that way obtain a sequence of static problems. Nevertheless, if in (2.32) cw, b and 8 are set to zero we obtain the problem first derived in Klarbring et al. [lo]. Existence for this quasistatic problem has since been showed in Klarbring et al. [12] and Andersson 117,181; in [12] by introducing a regularixation in time and in [17] without this regularization. Uniqueness is an open question.


Let the time interval [0, T] be divided into lij subintervals (f,_l, t,) for I= 1, . . . , A and o=t,<* * + < tN= T. The time derivatives appearing in (2.32) are approximated by the backward finite difference, i.e.

ti(tJ = Au’&, - &), (2.33a)

ct(&) = A&& - ti_l), (2.33b)

&6) a A~~/~f~ - tl-l), (2.33~)

e(t,) * A~~/(t~ - s._~). (2.33d)

Here Au’ = u(t,) - I@,-~), etc. Set u’ = u(Q, etc. and substitute

u(tl) = Au’ + u’-‘, a&) = ACY’ + c&l,

/J(tr) = A@’ + @-I, O(t,) = A@ + cj’-‘, (2.34)

into (2.32). Together with (2.33), we then obtain

a(Au’, v - Au’) +j&u’--l + Au’, fl’-’ + A@‘, e’-’ + A#), (v, fi, @) - (Au’, A/3’, As’))

+jT((uf-’ + Au’, B’-’ + Ap’, B’-’ + A@), (v, &))

-jT((uf--l + Au’, p’--” + A@, &’ + A@‘), (Aot, Acr’))

- &#(a - A@‘) - N@ - A/3’) - T(& - Ad)

3 F(V - AU’) - a(U’-‘, V - AU’) v(V, CU, p, 8) E vd X R3. (2.35)

If u, N, /I and 8 are known at time tr-l, (2.35) may be viewed as a problem to be solved for the increments Au’, A(Y’, A/3’ and A@. This is the problem that we are concerned with in the sequel. Before stating this problem formally we introduce additional notations and assumptions on the data. Everywhere below we use

u = Au’, cx = Aa’, P = AS’, o=Ae’

and

f = f@,), t = t(tJ

T = TO,), Iv = I, A4 = M(t,).

(This was already used in (2.35).) Also, we set

F(v) = jad$Uj dr + j+ tiUi dS - U(U’-‘, V), St

and let jN and jT denote the functionals defined previously, but with g replaced with

g, -_ g _ &l_ #-lx1 _ @l-l.

Setting

Vd = {v ~2 (Al)‘; v = 0 on S,}

(2.37)

we have the following problem:

1

find (u, CX, /3, 0) E Vd X R3 such that V(v, &,,& 8) E Vd X It3

46 v - u) + jN((U, B, q, (VT 6, 0) - oh BP 0))

(S) +h(b, it e), (6 5)) -h(h 13, e), (u, 4)

-M(&-8)-N(B--b)-T(&-cu)

rF(v-u). (2.38)

Note that with appropriate regularity assumptions which are stated below, a problem as this one, can be stated for every increment by updating. Indeed if under these assumptions (u, a; /3, 0) is a solution at time step 1 then it is suffi~ently regular and can be considered as data for the 1 + 1 time step.


It turns out that problem (PJ admits an equivalent formulation that is useful in the discussion of duality:

(82)

where

find (II, a, /3, 0) E Vd X R3 such that V(v, h, /?, 6) E Vd X R3

a(u, v - u) + JN(V, B, 6) - JN(U, p, 0)

+jT((u, 6, W, (v, g)) -jT((u, B, e), (u, 4) -M(&-q-N(p+?)-T(&-a)

rF(v-u), (2.39)

JN(V, B> 6 = y& [(UN + ax,+ B - g,)+lmN+l ds. (2.40)

Actually J,,, is related to the Gateaux derivative of jhr. The following result is an obvious extension of Theorem 4.1 in [3]:

THEOREM 2.1. (u, CY, 6, 19) E V” X R3 is a solution of (P2) if and onZy if it is a solution of (PI).

3. NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE

The problem (P2) ( or equivalently (PI)) is noncoercive, therefore its solvability is not guaranteed. In contact mechanics it is usually resolved by imposing compatibility conditions on the data (see e.g. Duvaut and Lions [20], Baiocchi et al. [21] or Shi and Shillor [22]) in addition to the usual requirements.

For simplicity we consider, in this section and in the next one, the case

m=mN=mT, lIm<w, (3.1)

p = const > 0, CNELm(S:), cNZc,> 0 a.e. (3.2)

We remark about the general case below. The following result gives necessary conditions for existence of solutions.

THEOREM 3.1. Assume (3.1) and (3.2). Zf problem (Pz) has a solution then

N 20, (3.3)

IMJ SUN, (3.4)

ITI ‘pN. (35)

PROOF. A simple consequence of the duality theory, given in Section 5.

REMARK 3.2. Conditions (3.3)-(3.5) have very clear and simple interpretation. (3.3) means that the total normal force is pushing the punch against the elastic body. For N < 0 the punch is pushed in the opposite direction and equilibrium is impossible. (3.5) relates to Coulomb’s law of friction for rigid bodies and guarantees that the maximal frictional force of resistance can balance the total applied shear force. (3.4) guarantees that the rigid body does not tople over.

We turn to the existence results. The following assumptions are made on the data

aijk/(x) E L”(Qd) i, j, k, 1~ 1, 2, (3.6)

and

Moreover

We have

aijkl = ujiki = a/&j i, j, k, 1 = 1, 2,

uijklEijckl z A 1 &j 1’ Vcij = Eji some il> 0.

gI E H’V$); f E (P(!Y))*; t E (L*(q)*.

(3.7)

(3-S)

(3.9)

THEOREM 3.3. Assume (3.1), (3.2), (3.6)-(3.9) and in addition that

N>O, IMI<uN and ITI<pN.

Then problem (P2) has a solution.

(3.10)


PROOF. The proof is given in [15] and it is based on the consideration of an auxiliary problem where jr((u, /3, e), (v, (u)) is replaced by

~T(v> a) = P d CNG IVT - PI ds. I c

That is the first argument in jr is “frozen”, G is a given function. Once the existence of a unique solution to the auxiliary problem is established, a solution to (P2) is obtained as a fixed point of an appropriate mapping.

REMARK 3.4. Comparing (3.10) to the necessary conditions (3.3)-(3.5) it is seen that a gap exists between the necessary and sufficient conditions. The case when some of the inequalities are actually equalities requires a separate treatment and additional geometric consideration. For the case with Coulomb’s friction and Signorini’s contact condition see Gastaldi 1231. Physically we require contact, iV > 0, the existence of a stick region on S, (where ur = 0) ITI < @V and that the moment M is balanced.

If we relax (3.1) the necessary conditions cannot be stated as in (3.3)-(3.5) any more. On the other hand concerning sufficient conditions there holds

THEOREM 3.5. Assume (3.2), (3.6)-(3.9) and mr > mN. Zf in addition N > 0, /MI <UN, and

ITI (j& dym”)-’ 4 jwr’y (3.11) c

then Frobiem (P2) has a so~u~on.

PROOF. [1.5].

4. FURTHER MATHEMATICAL RESULTS

Any solution (u, a; /3, 6) to (Pz) is such that u E HI(&) = [H1(Qd)]2. It turns out that actually any such solution has better smoothness properties. This is the content of the regularity result stated below. The interest in it lies in the fact that oT on Sl is actually a bounded function and not only a distribution in H -““(Sf). These regularity properties show the possible behavior of any solution, especially on the contact boundary Sg. Also they open the way to study in some detail the structure of the contact, i.e. the subdivision of Sf into a free part, where a, = 0 (and uN <g,), the stick region, where UT = 0 and the s&p region where UrfO.

The precise statement of the global regularity result is

THEOREM 4.1. Assume that the punch is smooth say g E C’($?), gr E C”(sg) ail I < 4 and dQd is in C’*‘. Let f E L”(Q’), cN E W’sz(S!) and t E W1sm(St). Zf u E H’(SZ’) is a part of a solution

(u, a, B, 0) to (p2) then

u E WySld) vp < 4, (4.1)

and

u E WQq2d) vp < 4. (4.2)

In particular

Here we used the notation H’ = (H’)‘, C” = (C”)‘, etc. The regularity is obtained as follows. Since u E HI(@) then u E Lp(S:) for all p E [2, w), by the imbedding theorem (n = 2). Then CJN E LP(S$), vp < 03, by (2.22) and it follows from (2.23) and the bound

14 5 wrv[(kv + 0x1-t P - gL1”’ (4.4)


that (TT E L’(s,d), VP < ~0. These, together with the assumptions on the data, as well as Shamir’s results [26], imply that u satisfies (4.1) and (4.2). From Sobolev’s theorem (4.3) follows.

To obtain improved local regul~ity let S2& = Bd\NB where N& is an ~d-nei~borhood of the sets .$,, n St, 3” n 3: and $ n 3:. We remove the neighborhoods of the points of dQd where the boundary conditions change type. Then we have the local (in &&) regularity.

THEOREM 4.2. Under the above assumptions

u E w1+1~~--E,p(Q6) ‘d& > 0. (4.5)

In particular

u E @((-a, a)) Vii E [0, l), (4.6)

a, 6 C”((-a, a)) and oT e L”(-a, a). (4.7)

me proof is based on the bootstrap argument. By (4.3) and (4.4) we have that or E I;“($} and a, E L”(S$J by (4.3) and (2.22). Then u E W’+“~-‘,~(~~) Ve >O (small) and Vp < ~0. Then, by the trace theorem u E (?([-a + 6, a - S]) for all 6 >O and all 1 E [0, l), i.e. (4.6) holds and then the first part of (4.7) follows as well.

REMARK 4.3. Problem (Pz) was derived as a step in the time discret~ation of the quasistatic problem in Section 2.3. Therefore it is of interest to notice that since n satisfies (4.1) and (4.2) and u E CA@:) for all A E [0, 4). But then by (2.37), gt+l = g - uN - 8x1 - /3 and g E C’(@, uN E CA(@) and therefore gz+l E CA($) all A E [0, 4). Moreover it follows from the definition of F(v), (2.36), that we have to consider the term a(u, v). We have, using Greens theorem,

a(u, v) =

Now as stated above oN E C”($?) and a, E L”($) for 0 C= 3L < 1 and also u = t E W’,m(S,). We notice that these possess the appropriate regularity as data for the next time step.

These are consistent with having ul+r satisfy (4.1)-(4.3) as well. Thus the regularity (4.1)-(4.3) is preserved as we move from time step I to time step I + 1, and marching in time preserves the smoothness properties of our solutions.

Using the regularity above we have the following uniqueness result for the problem such that the rigid punch is restricted to move up and down easily. The mathematical formulation of such a problem is obtained from (&) by setting (Y = & = 0 and 0 = 6 = 0 as horizontal displacement or inclination are not possible anymore.

THEOREM 4.4. Under the assumptions of Theorem 4.1 if cy, lit, 8 and 6 are jked to zero in (2.38) then there exists ,ul > 0 such that the solution (u, /?) to problem (P2) is unique for all p < pl.

PROOF. Given in [15].

5. DUALITY

In this section we consider the dual problem to (P2). The importance lies in the fact that the problem is formulated in terms of the contact stresses. These are frequently of most interest to the engineer.

5.1 The general duality theory

We review the essential results of the general duality theory developed in [3-51. Let V and Y;: (i = 1,2) be real reflexive Banach spaces. Denote by V* and YF their respective

topological duals and let (a, *)lr or (1, e> y, be the duality pairings, respectively. Let hi : V+ x be two_linear continuous operators and let A: be the transpose of &, i = 1,2.

Finally let Q, : Y, x Y,-, R and g : V x V + R be given mappings (ii = R U (+a~}),

ES 29:6-H


The primal problem, to which we apply duality, is

(P) I

find u E V such that VW E V

q(A1u, As) + g(u, u) 5 &A,4 As) + g(u, w).

We shall need the following assumptions:

WI Y --, V(YO, Y) is, for all y. E Y1, a convex 1.s.c. function on Y2, not identically +m, such that 37 E Y2 so that q(yo, jr) is finite and q(yo, .) is continuous at 8.

[A21 w+g(v, w) is, for each 21 E V, a convex 1.s.c. function on V which is continuous when w = u.

We proceed to obtain an equivalent dual formulation of (P), Let

cp*(YoJY*)=yS=~{(Y*‘Y)4-V(YorY)II

and

be the Fenchel conjugate functions of Q, and g with respect to the second argument and let

ag(v, w) = {w* E v*;g(v, 6) -g(v, w) 2 (w*, 6 - W)“, VG E V}, (5.1)

be the subdifferential of g with respect to the second variable. The dual problem is

1

find u E V, and y * E Yz such that

(p*) - A;Y* E ag(u, u),

q*(Aru, y*) - (y*, A&+ rp*(A,u, z*) - (z*, AZ&,, Vz* E Y;. (5.2)

The relationship between (P) and its dual (P*) was proved in [3]

THEOREM 5.1. Assume that [Al] and [A21 hold. Zf u is a solution of (P) then there exists y * E Yz such that (u, y*) is (I solution of (P*); conversely, if (u, y*) is a solution of (P*) then u solves (P). Moreover the extremality conditions hold

g(u, u) +g*(u, -A;y*) = (-A;y*, u)y, (5.3) and

q(A~u, Azu) + cp*(A~u, Y*) = (Y*, AZ&. (5.4)

5.2 The dual problem

We shall see below that problem (PJ is a particular case of problem (P). To that end, we make the following identifications of spaces and mappings:

V=VdXR3,

Y,=W,XR*,

Y2= WxR3,

A1=AIP:V+Y,,

A2=(y,id,id,id):V+Y2.

Here, ill is the mapping (v, p, O)+ (yNv, /3, O), P is the projection (v, (Y, /3, O)+ (v, 6, e), y : (H1(s2d))2-+ (H”*(S~))* is the trace operator and id :R+ R is the identity mapping. It is known (see e.g. [2]) that we may write

w = (H”*(S$))* = w, x w,

and there are trace operators

yN : (H’(s2d))*-+ w,

yT : (H’(Bd))*-+ w,


such that for any smooth function u in Qd there holds uN = yNu and r.+ = yru. For the flat contact surface treated here it is obvious that yNu and yru are the two components of yu in the coordinate directions. But the above decomposition holds for general ~smooth) bounda~es. Furthermore, we denote by W:, Wg and W* the dual spaces and the duality pairings by

t.9 ‘)W, etc. Any element 5* E W* can be written as a sum of the normal and tangential components, i.e.

for all 5 E W. Next, we define slightly modified forms of the contact functiouaIs by

~N(MV, E2 BP @>I =J*(v, 6, f?l

L(MU, @, B, q, &(V, 6)) =jd(u, BP @>, (VI &I) h,=(yT,id):VdXR4WTXR.

The final identifications which make clear that (P,) is a special case of (P) are:

g((u, a, B, e), (v, 5, B, 8)) = a(u, v - 4 -f(v - 4 - M( e - i3) - Iv@ - j3) - Tf it - a), V(u, a; /I, 8), (v, is; p, 8) E v, (5.7)

fp((&J, B, @)9 (% Ei; P, Q) = UrlN> B, 6)

+h((5N, B, e), h, g)), %!h, 6% 6) E Yr, h c, 8, @) E K. (5.74

To obtain the dual problem we first calculate the conjugate of QI.

LEMMA 5.2. The Fenchel conjugate of cp with respect to the second argument is

(P*(&, 6 @I @), (9*> a*9 B*, a*)) =GJq3 f &,(rl& B*, @) f ~~~~~~,~,~~~~~, g*). (5.8)

Here iK is the indicator function of the set K and

K={(&,8*, ~*)~W~XR~;(rl~,1)~~=8*,(rl~,x,),,=8*};

&*)E W;XR; (r/F, l)w,= -ii*,

Also J& is the conjugate of

fCt)

that is

(5-9)

Here we set

K3 = { qg E Wg; 0,: E L’(Sf) and t&z 0 a.e. on Sli).

PROOF. From the definition we have

To make the maximization problems more manageable, we make the following substitutions:

qj,, = %, - fi - 8x1, ZN E w,

&---ZT + ii, ZTEK


Note that there is a slight abuse of notation, since 0, etc., stand for both a constant element of W, and a real number.

Actually, since q$ E L’(Sg) when .Jg(nG) < 03, the definition of the set K1 simplifies and we may write Jg instead of the duality pairing (a, *)w,. Also it follows from the inequality in K2 that actually n; E Lp(S$ and

In3 % W./[(EN + 6x1+ a - gJ+lrnT9 and therefore we may replace (n;, l)wT= -&* by

i &ds = -Lk*.

s<

Next, we calculate the subdifferential of g with respect to the second argument. g is obviously Gateaux differentiable, and thus, ag = {Dg}, where

Dg((u, (Y, B, e), (v, Cu, Z% 6)) = (Au - L, -T, -N, -W (5.10)

Here, A and L are defined by

a(v, w) = (Av, w)“ci

F(w) = (L, W)“d vv, w E Vd (5.11)

The dual problem then becomes:

(E)

(5.12)

(5.13)

’ find (u, (Y, /3, 0) E V and (g*, (Y*, p*, e*) E Yz such that

(Au-L, -T, -N, -M) = (-y*E*, -a*, -p*, -O*),

.m:) + k,(G P*, e*) + hz~yNu,P.d~;~ (u*)

-(f*, ~~),-c~*~-~*p-e*e~~;fr(~~)+z~,(~~,~*,8*)

+ k(YN".BIe) ( rlL &i*) - (q*, yu), - Ck*cy- a*/3 - e*e

~ v(r]*, C%*, @*, a*) E Y2*.

This problem can be simplified considerably. Since A is invertible, due to Korn’s inequality, (5.12) implies that

u =A-I(-y*c* + L), (5.14)

T=cu*, N=/?*, M = e*. (5.15)

Moreover, if we choose r]L= 5; and &* = (Y* in (5.13) we obtain

Z;(nz) + z,,(n;, 8*, 6*) - ~~(69 - Z&z, p*, e*)

r (q;- 52, yNu)wN + (a* - p*)/3 + (a* - O*)e V(n;, fi*, a*) E W;X R*. (5.16)

This inequality, by definition, implies that (yNu, p, 0) is the subgradient of .Z$ + Z,, at the point (f$, /I*, e*). From the proof of Lemma 5.2 it is apparent that .Zk + ZK, is the Fenchel conjugate of j,. Therefore (Ez, /3*, e*) is also subgradient of .ZN at the point (y,+.,u, /3, 0). Since .ZN is Gataux differentiable we have


This equation gives

51: = cPJ[(&%J + oxI+ B - &)+I”“, (5.18)

a.e. on S,“. Equation (5.18) can be used to express the dependence of K2 on the solution in terms of 5:. Due to this and (5.14) (u, cu, #I, 0) can be eliminated from (P;). Indeed (5.14), (5.15) and (5.18) imply the following problem:

(R:) 1

find E* = (6% 5;) E ki x &(Ez) such that V~I* = (TV&, 9;) E k, x k,(g;)

G(G) - (CY*, yA-‘(-Y*E* + U)W 5JXrl;) - (rl*, YA-‘(-Y*E* + J%V. (5.19)

Here,

R,=[TI~EW;;~);EL’(~~),~~~;~F=N,~ ~rhdF=M], c %

&(EzJ = { 11 ; E w;; 7$E L’(S$, l&l 5 pc~-m”mN(gp’mN, - I

&&=T. $ I

The following theorem summarizes the discussion above and gives the relation between problems (P2) and (R:):

THEOREM 5.3. (u, a; /I, 0) E V is a solution of (P2) if and only if there exists e* E W* which is a solution of (R:) and is such that

- y* f* = Au - L, (5.20)

(YU, a; P, 0) E W45%) + k,(% N, M) + 4~,c,,,s.&X 01. (5.21)

Relations (5.20) and (5.21) are nothing else than restatements of (5.12) and (5.13), defining Problem (P:).

THEOREM 5.4. Problem (R;) has a unique solution if and only if the u part of the solution of problem (P2) is unique.

PROOF. This result follows from (5.20): if u is unique then the uniqueness of E* follows from the injectivity of y*; if E* is unique then u = A-'(- y*g* + L) and the uniqueness follows from the standard linear theory.

5.3 Interpretation of dual variables

Let u be a part of a solution (u, a; p, 0) of (&). Then it follows from Green’s formula that there exists an element s = (oN, oT) E W* such that

a(u, v) - F(v) = (s, yv) w vv E Vd. (5.22)

If u is smooth, oN and ur coincide with the normal and tangential tractions on the contact boundary. From (5.11) we obtain

(Au-L,~)~~=(s,yv)~=(y*s,v)~~Vv~V~,

from which it follows that

(5.23)

Au-L=y*s. (5.24)

Comparing (5.20) and (5.24) we obtain that the dual variables c* are in fact the contact tractions with reversed sign, i.e.

c* = -s = (-ON, .-o& (5.25)

Moreover, we may write

yu = yA-‘(y*s + L)

= G(s) + i, (5.26)

where G is the contact surface Green’s operator and fi = yA-‘L. For some simple geometries and materials, such as the isotropic half-plane, Green’s operators are explicitly known, and


may be found in any treatise on elasticity (see e.g. Johnson [24]). We may write (5.26) as

yr.,u = GAS) + &, (5.27a)

yTu = GT(s) + &. (5.27b)

We use the identification (5.25) and equations (5.27) to rewrite Problem (RZ):

i

find -s = (-rrN, -or) E k, x kZ(-oN) such that

(QZ) Jlt(--tN) --J$(-~T~) I: -(rN - aN, G,(s) + iiN)WN

-(rr - ~~T,GT(S)+~~T)W,~(-%', -~T)~fhx&(-d (5.28)

REMARK 5.5. When c~--* m, formally, ..J$ approaches the indicator function plus a linear term. Problem (Qz) then reduces to the problem derived in Johansson and Klarbring [25], where Signorini’s conditions were assumed instead of normal compliance. In [25] a numerical solution scheme was developed for this problem.

5.4 Proofof Theorem 3.1

We use duality to prove the theorem. Assume that Problem (P2) has a solution. Then it follows from Theorem 5.3 that Problem (R:) (and therefore Problem (Q:)) has a solution. This implies that the set k, il k?Z(-o~), where (- oN, -@T) is a solution to (Q,*), is not empty.

Since .I%- CY~) < m there holds

N 2 0.

Also oN E L’($) hence

I”’ ll =: .2

fJN~1d-G <a JUNldu, I 1 c

=a i

r$,ds =aN. $

Finally, using the definition &(-CX~), there holds

But mr = mN hence

6. THE FRICTIONLESS PROBLEM

The frictionless punch problem with normal compliance is obtained from (P2) by setting p= 0, and in addition we must set T = 0 as equilibrium is impossible with a tangential force. Therefore (Y is set to zero.

The problem obtained from (I$) is

(PO)

(II, /I, 6) E Vd X R2 such that V(v, fi, 6) E Vd X Rz

a(u, v - u) + J&v, j% 8) - &(u, B, 0)

~~~v-u)+N(~-~)+M~~-e). (6-l)


Existence of a solution to (PO) is guaranteed in Theorem 3.3 with the obvious modifications. Following the construction in Section 5 one can see that the dual problem is given by

(PO*)

find

1

E* = (g& 5;) E R, X (0) such that Vq* = (qg, qc) E R, x (0)

G(f3 - (E*, W’(-y*f* x L))w

~JXqrfr) - (q*, W’(--YE* + L)),. (6.2)

As in Section 5 the solvability of (PO) implies that (PO*) has a solution as well. Formally (PO) can be obtained from (P2) by taking the limit ~~-0. It turns out that this

correspondence is not only formal and can be proved rigorously. Frictionless contact problems with normal compliance were considered by Rabier and Oden

[9] where various properties of such problems were investigated. They also show that in the limit cN+03 the frictionless problem reduces to the contact problem with Signorini condition. This can be shown in our punch problem as well.

The frictionless contact problem of a punch in contact with an isotropic homogeneous half-plane, and 0 = 0, was considered recently in Shillor, Spence and Vergottis [16]. There, following Spence [27], the dual problem is obtained directly as an integral equation. Indeed if the elastic half-plane is {(x, y) E R*; y < 0} and the contact with the flat punch is over S, = {(x, y); - 1 IX 5 1, y = 0} then the normalized depth of penetration u = U(X), -1 I x 5 1, is given by

u(x) = 1+ j.4 I

l log 1(x’- +-*1 (U(s)+)m d.Y. 0

(6.3)

The existence and uniqueness of the solution to (6.3) was proved under the restriction that the coefficient p is sufficiently small. Then optimal regularity was derived. It turns out that the solution is HCilder continuous, i.e. u E C*([-1, l]), all 0 ZG h < 1. Whereas the regularity for our problem with friction is u E Cln([-1, 11) n C”((-1, l)), 0 I k < 1 and we do not know if this regularity is optimal.

An iterative method for the calculation of u based on (6.3) was proposed and performed well compared to a method based on an equivalent minimization formulation. Some numerical solutions were presented. For small p both methods converged rapidly but for larger y’s the iterative scheme produced diverging solutions that seem to indicate two different solutions. It is not known whether this is a numerical artifact or that a genuine bifurcation takes place.

Acknowledgemenf-Two of the authors (A. M. and M. S.) are grateful for the financial support and hospitality of the Department of Mechanical Engineering in the LinkBping Institute for Technology during our two weeks stay in June 1989.

REFERENCES

111 L. A. GALIN, Contact P ro bl ems in the Theory of Elasticity (Translation from Russian). North Carolina State . - . _ . . College (1967).

[2] N. KIKUCHI and J. T. ODEN, Conracr Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods SIAM, PhiladelDhia, PA (1988).

[3] A. KLARBRING, A. MiKELIe and h. SgILLbR, Appl. Math. Opt. 22,211-226 (1990). [4] U. MOSCO, J. Math. Anal. Appl. 40, 202-206 (1972). (51 I. CAPUZZO-DOLCETTA and M. MATZEU, Num. Func. Annl. Opt. 2(4), 231-265 (1980). [6] J. T. ODEN and J. A. C. MARTINS, Comput. Meth. Appl. Mech. Engng 52,527-634 (1985). (7) J. A. C. MARTINS and J. T. ODEN, Nonlinear Anol. TMA 11(3), 407-428 (1987). [8] P. J. RABIER, J. A. C. MARTINS, J. T. ODENS and L. CAMPOS, Int. J. Engng Sci. 24(11), 1755-1768

(1986). [9] P. J. RABIER and J. T. ODEN, Nonlinear Anal. TMA 11(12), 1325-1350 (1987); 12(l), 1-17 (1988).

[lo] A. KLARBRING, A. MIKELIC and M. SHILLOR, Int. J. Engng Sci. 26(8), 811-832 (1988). [ll] A. KLARBRING, A. MIKELIG and M. SHILLOR, Nonlinear Anal. TMA W(8), 935-955 (1989). [12] A. KLARBRING, A. MIKELIC and M. SHILLOR, In Uniluteral Problems in Structural Analysis:-3 (Edited by

G. DEL PIER0 and F. MACERI). Springer, Berlin. To appear.


1131 A. KLARBRING, Eur. J. M+. A 9(l), 53-85 (1990). [14] C. M. ELLIOT, A. MIKELIC and M. SHILLOR, Nonfin. Anal. TMA. To appear. [15] A. KLARBRING, A. MIKELIC and M. SHILLOR, In preparation. [16] M. SHILLOR, D. A. SPENCE and P. VERGOITIS, In preparation. [17] L.-E. ANDERSSON, Nonlinear Anal. TMA. To appear. (181 L.-E. ANDERSSON, Report LiTH-MAT-R-89-00, Linkaping University (1989). [19] N. KIKUCHI and Y. T. SONG, Int. J. Engng Sci. 18, 357-377 (1980). [20] G. DUVAUT and J. L. LIONS. Inequalities in Mechanics and Physics. Springer, Berlin (1976). [21] C. BAIOCCHI, F. GASTALDI and F. TOMARELLI, Ann. Scuofa Norm. Sup. Piza 13, 517-659 (1986). [22] P. SHI and M. SHILLOR, R. Sot. Edinburgh. To appear. [23] F. GASTALDI, IAN-CNDR Publication No. 649, Pavia. (241 K. L. JOHNSON, Contact Mechanics. Cambridge University Press (1985). [25] L. JOHANSSON and A. KLARBRING, The rigid punch problem with friction using variational inequalities and

linear complementarity. Preprint of LinkGping University (1990). [26] E. SHAMIR, Israel J. Math. 6, 150 (1968). [27] D. A. SPENCE, Proc. Camb. Phil. Sot. 73, 249 (1973).

(Received 19 September 1990; accepted 15 October 1990)

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The rigid punch problem with friction