Existence and local uniqueness of solutions to contact problems in elasticity with nonlinear friction laws

ht. J. Engng Sci. Vol. 24, No. 11, pp. 1755-1768, Printed in Great Britain.

1986 0

EXISTENCE AND LOCAL UNIQUENESS CONTACT PROBLEMS IN ELASTICITY FRICTION LAWS

002&7225/86 $3.00 + .%I 1986 Pergamon Journals Ltd

OF SOLUTIONS TO WITH NONLINEAR

P. RABIER Southern Methodist University, Dallas, Texas, U.S.A.

J. A. C. MARTINS and J. T. ODEN Texas Institute for Computational

Mechanics, ASE/ME Dept., The University of Texas at Austin, U.S.A.

and L. CAMPOS Dept. of Mechanical Engineering, Technical University of Lisbon, Portugal

(Communicated

by J. T. ODEN)

Abstract--In this paper, we show how the use of interface models greatly facilitates the mathematical analysis of static friction phenomena. Such models have already found extensive arguments in their support, from both the experimental and numerical sides. They are then not conveniently introduced for purely mathematical purposes. Nevertheless, they also appear to provide a satisfactory substitute to the classical approach in terms of a variational problem with unilateral constraint. Indeed, we have been able to prove, with relatively simple arguments, the existence and uniqueness results that the classical theory has been unable to generate despite numerous attempts. In addition, in the case with no friction, the solution obtained through the classical theory is shown to be recovered in the limiting case of an infinite normal stiffness, thus giving one more justification of the validity of such models. 1. INTRODUCTION

The classical theory of contact in elastostatics with Coulomb’s law of friction in force on the contact surface has not been a successful theory, from either the physical or the mathematical point of view. Physically, it is known that standard contact laws provide a crude and imperfect model of actual contact phenomena on dry metallic surfaces. On the mathematical side, the significant difficulties inherent in proving the existence of solutions for contact problems with friction are well-known. The question of existence of solutions to the general Signorini problem with friction was put forth as an open problem by Duvaut and Lions [S], although NeEas, JaruSek and Haslinger [8,15] have recently managed to state existence and uniqueness results under special hypotheses. Duvaut [4] pointed out that a mollification of the contact pressure would provide sufficient regularity for the establishment of existence of solutions to Signorini-type problems, and this led to several studies of non-local friction laws (e.g. [3,18-201). While physical arguments can be given that support such laws, it is not known how adequate they are for modeling many common frictional phenomena. In a recent memoir [17], Oden and Martins presented extensive arguments in support of phenomenological laws governing the contact interface of the form an = -cc,a”n CT== cTamTz(if sliding occurs),

(1.1)

on the contact surface, where gn and uT are normal and tangential stresses, z is a unit tangent vector field parallel to the sliding velocity, c., cT, m, and mT are material parameters and a is the penetration approach of the contact surfaces. These laws assert that the contact interface has a normal compliance characterized by a power law relation between the normal stress and a, the change in the distance between surfaces due to deformation, while the frictional stresses are also defined by a power law in a, which reduces to the usual Coulomb’s law of friction gT = ~[G,[z if mT = m, and ~1= c=/c, is the coefficient of friction. Arguments based on experimental, theoretical and computational results that the above 1755

1756

P. RABIER et al.

relations do, in fact, provide a reasonable model for a wide range of frictional phenomena are supplied in [ 171. For representative physical experiments supporting the normal contact power law above, we refer to the paper by Back et al. [2]. A question that naturally arises at this point is whether or not the use of these new and more sound physical models of friction in a theory of contact problems in elastostatics will provide a mathematical framework that admits the establishment of existence and uniqueness results under appropriate hypotheses. We show in the present paper that the answer to this question is affirmative. In the present work, we formulate a class of contact problems in elastostatics with nonlinear friction laws of the form (1.1) acting on the contact surface. We prove that solutions exist in appropriate Sobolev spaces under reasonable hypotheses on the regularity and smallness of the data, in particular the smallness of the coefficient of friction for prescribed external loads. These results are established through the implicit function theorem which also yields local uniqueness with no additional assumptions and, in some respects, are reminiscent of those in [3] and [4]. 2. THE

STEADY

FRICTIONAL

SLIDING

OF A METALLIC

BODY

We begin by considering a metallic body, the interior of which is an open bounded domain R in RN (N = 2 or 3). Points (particles) in n with Cartesian coordinates xi, 1 d i G N, relative to a fixed coordinate frame are denoted by x = (xi,. . . , xN) and the volume measure by dx. The smooth boundary I of R (in practice, Lipschitz continuous in the sense of [14]) contains four disjoint open subsets I,, In, I, and I, such that

r=UTn meas(r=\r,)

= 01

(2.1)

with UE (C, D, E, F}. The surface measure (cf. [14, pp. 119-1231) on I or any of its measurable subsets is denoted by ds and we shall set C = int(r\rn).

(2.2)

We assume that the metallic body has a linearly elastic behavior characterized generalized Hooke’s law

by the

aij(U) = EijklUk,l inR, (2.3)

1 d i,j, k, 1 < N,

whereu = (q,..., uN)is a sufficiently smooth field of displacements in fi, Eijkl are the usual elasticity coefficients of the material satisfying Eijkr = Eijlk = Ejikl = Ekrij,

1 6 i,j, k, 1 < N

(2.4)

and (Tijare the Cartesian components of the Cauchy stress tensor verifying aij

=

Cji,

1 d i,j < N.

In (2.3) and throughout this work, standard indicial notation and the summation convention are employed (e.g. uk,, 3 &@x,). We suppose that body forces with components per unit volume bi, 1 < i < N, act in s1. We also suppose that the body is fixed on I’, and tractions t are prescribed on I,. The body may be elastically supported on I, by distributed springs of moduli Kij satisfying the symmetry conditions Kij = Kji,

1 d i,j d N.

(2.5)

The undeformed configuration of these springs corresponds to the displacement UE on rE. Finally, we assume that the body may come in contact with a rough foundation along a candidate contact surface I,. The foundation slides by the material contact surface with a velocity parallel to a prescribed unit vector field z tangent to I,. The initial gap (in the reference confirguration u = 0) between the body and the foundation is denoted by g 3 0.

Existence and local uniqueness of solutions to contact problems in elasticity

1151

Of course, since g is measured in the reference configuration, physical significance requires g to be “small” along l-c. As mentioned earlier, c,, cT, m, and tnT will denote material parameters of the contact interface. We shall denote by n = (nl,. . . , nN) the outward normal unit vector along the boundary r, by on and uT the normal stress and the tangent vector stress on the boundary, whose values at a displacement u are, respectively oiA”)ninj,

OJ”) = CT(U) = C(U) *n

- (T,n

(i.e. bTi(U)

= OiXU)tIj -

B,(U)ni,

1 < i,j < N).

Similarly, the displacement u on the boundary r will be decomposed into normal and tangential components u, and uT respectively, according to U, = u-n = Uini, UT = u -

(i.e. UTi= Ui - U”ni,1 < i < N).

u,n

Assuming sufficient smoothness for all the functions involved, the steady sliding equilibrium position of the metallic body is characterized by the following system of equations: (1) Equilibrium equations o~~(u),~ +

bi = 0

in

Q,

l
(2.6)

where .,~U) is defined by the constitutive relation (2.3). (2) Boundary conditions (2a) Prescribed displacements Ui=O

on

l
Fn,

(2.7)

(2b) Prescribed tractions tTiAU)nj =

r,,

on

ti

l
(2.8)

(2~) Linear elastic constraint Ui,(U)nj =

-

Kil(Ul

-

UF)on

r,,

1 < i,j, I < N.

(2.9)

(2d) Normal contact constitutive equation a,(u)

=

-c,[(u,

-

g)+]Q

on

r,.

(2.10)

on

r,.

(2.11)

(2e) Frictional contact condition bT(u) = c&4,

-

g)+]V

3. VARIATIONAL FORMULATION In order to establish a variational principle for the steady sliding equilibrium problem introduced in the previous section, we shall now make precise the assumed smoothness of the domain R and of the functions involved. Additional properties of these functions, relevant for later purposes, will also be specified. For a domain Q with a Lipschitz continuous boundary, it is well-known that the unit outer normal n is defined almost everywhere on I’ and is a measurable (and bounded) function (see e.g. [ll, 141). In other words ni E qr),

l
fl ni’= 1 a.e. on r.

(3.1)

We shall denote by H’(a) the usual Sobolev space. Restrictions to the boundary r of elements of H’(Q) will always be understood in the sense of traces [thus, as elements of H”‘(r)]. With this convention, we define the space I/ of admissible displacements by

1758

P. RABIER

V=

et al.

{vE [ZYI’(Q)]~:v =

0 a.e. on l-n}.

(3.2)

As a closed subspace of [H’(Q)]“, the space V is a Hilbert space for the usual [II’@)]“inner product inducing the norm

l/Z (UiDi + llvllv = 1 vi,jUi,j)dx

UR

(3.3)

The trace operator maps the space V linearly and continuously onto the subspace W of [H”2(IJ]N defined by W = {c E [H’/“(IJ]“: 5 = 0 a.e. on l-u}.

(3.4)

As a closed subspace of [H’12(r)JN, the space W is a Hilbert space. A possible choice for the Hilbertian norm on W (which amounts to identify W with the orthogonal of [H&I)]” in v) is llgllw = inf(llvllv:v =

(onr}.

(3.5)

The (topological) dual spaces of I/ and W will be denoted by I/’ and w’ respectively. The notations (0, 0)” and (0, l)~ will be used for the corresponding duality pairings. From the Sobolev’s embedding theorems, the space P2(r) is continuously embedded inLq(r)forl
(3.6)

with l, = c-n = (fliELP(r)

[rev.LpWI,

(3.7)

rT = 5 - 5, n E cLp(r)]N

(resp. [L”(IZ)]“).

(3.8)

Note that CT- n = 0 a.e. on

r (resp. IS).

(3.9)

So as to allow the greatest generality in our later purposes, we shall somewhat specify the value of q by setting

‘=

any convenient real number > 2 4 ifN =3.

if N = 2,

(3.10)

We now turn to listing the assumptions that will henceforth be in force on the data of the problem. First, we assume that the elasticity coefficients Eijkl verify Eijkl

(3.11)

E L”(R)

and that the uniform ellipticity condition Eij,,(x)A,[Aij

>,

cloAijAij

(3.12)

holds for almost all x ~0 and every N x N symmetric array A,,, with some constant cl,, > 0. Analogous conditions are imposed on the springs moduli Kij, namely Kij

E

wb),

(3.13)

Existence and local uniqueness of solutions to contact problems in elasticity Kij(x)a@j

2

&$$j,

1759

(3.14)

for almost all x E r, and every vector a = (a,, . . . , uN) where c1r > 0 is a constant. As for the displacement UE (cf. Section 2) we assume us E [L”*(l-,)]?

(3.15)

The prescribed tractions t on r, verify t E [P*(l-,)-JN*

(3.16)

The exponents m, E R and mT E R are submitted to the conditionst 1
l
+coifN=2;

Note from (3.10) that this assumption can be rewritten as mT < 4 - 1,

1 cm,,

(3.17)

after increasing the value of q when N = 2 if necessary. This modification does not affect (3.15) or (3.16) since q* decreases as q increases. The vector field z tangent to r along rc is taken unitary, i.e. r d7a(rc),

2 7: = 1a.e. on r,,

i=l

z-n =

(3.18)

Oa.e.onrc.

For consistency in our assumptions, we assume that the initial gap g verifies g E Lq(rc),

(3.19)

although g is always continuous in practice, so that condition (3.19) is not affected by possibly increasing q as above if necessary. The material parameters c, and cT characterizing the normal and tangential response along l-c are supposed to verify c,,c+Lm(rc)

(3.20)

and, indeed, they are constant in most applications. Finally, the body forces b satisfy the condition b E [LZ(s2)]N.

(3.21)

At this stage, we introduce (i) The continuous bilinear from a(~, l) over V defined by

4wv) = ao(wv)+ ~E(U,V),

(3.22)

where u,(u,v) = virtual work produced by the action of the stresses CJ&I) on the strains &i,(v) =

ai,(u)sij(v)dx = Eijk,u,,,vi,jdx, sn sCl

(3.23)

U&I, v) = virtual work produced by the deformation of the linear springs on r,

=

srE

KijUjUids.

(ii) The nonlinear operators P and Q from I/ to V t Very little restrictive in the applications (see Cl73 for details).

(3.24)

P. RABIER et al.

1760

(P(u),v),

= virtual work produced by the normal stresses on the displacement v

=

c,C(a, - g)+lmnu, ds, src

(3.25)

(@(I&V) = virtual work produced by the friction stresses on the displacement v =-

cJ(u, - g)+lm’z*vds. src

(3.26)

(iii) The external forces fe I/’ (body forces b, prescribed tractions t, initial deflection U” of the linear springs) (f,v),

= Lb*vdx

+ LFtvds + ~~ijLJ~~ids.

(3.27)

Variational statement. With the above definitions and notations established, we can now state a weak formulation for the steady sliding problem (2.6)-(2.11): Find a function u E I/ such that

a(u,v)+ (PW,vh +
(3.28)

for every v E I/ 4. RELATIONSHIP BETWEEN THE CLASSICAL THE VARIATIONAL FORMULATIONS

AND

We now show in what sense the solutions of the variational problem (3.28) can be interpreted for the solutions to problem (2.6)-(2.11). Our approach here is quite standard and relies on a generalized Green’s formula, given below in Lemma 4.1. In what follows, &@(!A) denotes the space of indefinitely differentiable functions with compact support in R equipped with the usual inductive limit topology and by &V(O)its topological dual, the space of distributions over R. We introduce the operator div 6: V-+ [g(Q)]” Cdiv4v)li = oij(v),j= (Eijkluk,J,j,

(4.1)

with Eijkl satisfying the symmetry conditions (2.4) and (3.11))(3.12). Let V0 denote the subspace of V defined by y = (v E V;diva(v) E [L’(Q)]“}. Lemma 4.1 (generalized Green’s formula): There is a unique linear continuous

(4.2) mapping

71:vu3 w verifying

Cn(u)li = ~ij~u)nj on I-, for every

uE 5

such that ai,

(4.3)

= Eijklnk,lE C’(n), 1 < i,j, k, 1 6 N, and

a&, v) +

sn

[div u(u)] - v dx = (n(u), v),,

(4.4)

for every pair (u, v) E Vv x V. Proof: The proof of this result follows standard arguments and is omitted. See e.g. [9, Theorems 5.8 and 5.91 for a similar statement. On the basis of the above lemma, we can now prove

Existence

and local uniqueness

of solutions

to contact

problems

in elasticity

1761

Theorem 4.1: An element UE V is a solution of the variational problem (3.28) if and only if (i) u E V0and diva(u) + b = Oa.e.inQ,

(4.5)

(ii) 7c(u)E [Lq*(E)]” and [recall the notations (3.7)-(3.8)] 7c(u)= t a.e. on r,,

(4.6)

U7)a.e. on L-E,

(4.7)

[n(u)]. = - c,[(u, - g),]“” a.e. on rc,

(4.8)

[n(u)IT = cJ(u, - g)+lmTz a.e. on IYc.

(4.9)

[7T(U)]i

=

-Kii(uj

-

Proof. Let u E V be a solution of the variational problem (3.28) and let v E [g(Q)]” c V. Since all the boundary terms vanish for such a choice of v, one has bmvdx, sR

a,(u,v) = i.e.

(TiXn)Di,j dx =

bivi dx.

sn

sn

As the above relation holds for every VE [g(n)]“, -diva(u)

= b

it is equivalent to

in [CZ(Q)lN,

which proves (i). Next, as we have just seen that u E Vband for any given element v E V, the generalized Green’s formula of Lemma 4.1 and the definitions (3.22)-(3.27) yield. -

(diva(u))*vdx

+ (rr(u),v)w = JR

JR

+

s

+ cT[(u, - d+l m~u,,

1-c&

bmvdx

- g)+lm? .v}ds.

rc

With (4.5) and since the trace map is onto from V to W (cf. Section 3), this reduces to

<4u),Ow =

s

Ilr.Cds

(4.10)

Z

for every r E W where I,% E [L9*(C)IN is (uniquely) defined by t $ =

onr,,

CeKiJIUj -Mu.

-

u~)I1 d+l”nn


On Mu.

k -

d+lmrT on&.

From the density of the space W in the space [L9(E)IN (cf. Section 3), it follows that Z(U) can be uniquely extended (by ti) as a linear continuous form on [L9(X)]“. Hence R(U)= @E [Lq*(X)IN. Boundary conditions (4.6)-(4.9) are immediate from this result and the notations (3.7)-(3.8). To prove that conditions (i) and (ii) of Theorem 4.1 imply UE V and u is a solution to the variational problem (3.28), we need only to reverse the steps of the above proof.

1762

P. RABIER 5.

EXISTENCE

et al

AND LOCAL UNIQUENESS OF STEADY EQUILIBRIUM SOLUTIONS

SLIDING

In this section we study questions of existence and uniqueness of solutions to the variational problem (3.28). Results will be established either for arbitrary coefficient of friction and sufficiently small external forces? or for arbitrary external forces and sufficiently small coefficient of friction. It will be convenient to rewrite the variational problem (3.28) as a functional equation between I/ and I/‘. To do this, we introduce the operator A E .Z(I’, V’) defined by a(u, v) = (4 v>,, for every pair (II,v) E V x I/ so that the problem becomes: Find u E V such that Au + P(u) + Q(u) = f.

(5.1) (5.2)

We now proceed to review a few mathematical concepts and related results. Given an open subset o of IW”,let us first recall that a mapping

is said to be a Carathtodory function if (i) for every t E R, the function F(o, t): co + R is measurable, (ii) for almost all x E o, the function F(x, l): [w-+ [wis continuous. With the Carathtodory function F, we associate the operator P from the space of measurable real-valued functions on o into itself by F”(Y)(X) = ~CWb)l~ for almost all x E w (the measurability of P(y) for a measurable y is not an obvious result; see e.g. [6, Proposition 1.1, p. 2181). The operator P is usually referred to as the Nemytskii operator associated with F. A key property, established by Krasnoselskii [lo], is that the operator F acts from Lp(w) into L’(w), 1 d p,r < + co if and only if there is a function CIE L’(o) and a constant /I > 0 such that, for almost all x E o and every t E R IF@, t)l d a(x) + BIdPi’.

if so, F is then automatically continuous from Lp(w) into L’(o). Differentiability of the Nemytskii operator can also be expressed in terms of the mapping F under additional hypotheses. We shall make use of the following result (cf. Rabier [21, Theorem 3.21). Proposition 5.1: Let F: co x R -+ R be a Caretheodory function and suppose that the function F(x, 0) is continuously differentiable for almost all x E o. Suppose further that for some 2 < p < + 00 there is a function MEL(Pi2)*(o) and a constant /3 > 0 such that for almost all x E o and every t E R IF,@, 0

d 4x1 + /4tlp-2,

(5.3)

where we have set Ft = iJF/i%. Then F, is a Carathtodory function and Ft~ C”[Lp(w), L(P’2)*(w)]. Besides, if th e function F(o,O) belongs to the space Lp*(o), one has FE C’[L@), zY*(w)] and the derivative of F is given by DQy) z = Qy)z

E Lp*(o)

(5.4)

for every pair (y, z)E [tp(o)12. With an appropriate modification on the growth of F,, the result stated in Proposition 5.1 remains true for p = + 00 (cf. [21]) but it is known to be always false when p = 2, except if F(x, t) is linear in t and F(x, 1) E L”(o). In other words, restriction to spaces L’(O) with p > 2 is indispensable to have differentiability properties of nonlinear Nemytskii’s operators. The previous definitions and properties easily carry on to the case when the open set w is replaced by the Lipschitz continuous boundary I- (in the sense of [141) of t This condition

can however

be partly

released;

see Remark

5.1.

Existence and local uniqueness of solutions to contact problems

in elasticity

1763

an open subset 0 of RM+ l . Indeed, it suffices to use a system of local charts and an associated partition of the unity to reduce the problem on a collection of open subsets w of W. Note that by definition of the measure on r and since r is Lipschitz continuous, such a transformation introduces L”(o)-weights only, which does not affect the value of the exponents p and r. Taking M = N - 1 and R as in Sections 1-4, we deduce Proposition

5.2: Let m > 1 be a given real number. Then, the mapping yEL”+‘(l-)

-+ (y+)mnE(m+l)*(r),

is of class C’ and its derivative is given by

2ELm+l(r)~~(y+)m-lz~~(m+1)*(r). Proof: For (x, t) E r x R, let us set F(x, t) = (t +)“, where t+ = t for t > 0 and t+ = 0 otherwise. It is obvious that F is a CarathCodory function and that F,(x, t) = m(t+)“-’ is continuous. The growth condition (5.3) is satisfied with p=m+ 1, a=0 and J?=m. As F(o,O)=OEL (m+l)*(r), the result follows from the preceeding comments and Proposition 5.1. n Corollary 5.1: Under the standing assumptions, the operators P and 0 defined by (3.25) and (3.26) respectively are of class C’ from V to V’. Besides, for every pair (u, h) E V2, (DP(u). h,v),,

=

m,c,[(u, - g)+]mn-‘h,u,ds,

(5.5)

s rc for every v E V and (DO(u).h,v), =

m+,[(u,

-

g)+lmr-lh,z.vds,

(5.6)

s rc for every v E V. Proof: Denoting again by c, and g the extensions by 0 of c, and g on r\rc definition of the operator P (3.25) is

”=

an equivalent

J r

Mu, - g)+Imnu, ds,

(5.7)

for every pair (u, V)E V x V. In this form, it is clear that P is composed with the following C’ mappings: the trace u E V + u I,-E W [cf. (3.4)], ii!) the linear continuous embedding W G [Lq(r)lN (cf. Section 3), (iii) the linear continuous inner product with the normal vector n E [L”(r)]N uE

(iv)

p(r)jNH U,= U.n E qr),

the translation [cf. (3.25)] y E Lyr)H y -

gEqr),

(v) the linear continuous embedding Lq(IJ C Lmn+l(r) [cf. (3.17)] (vi)

the mapping [cf. (3.17) and Proposition yeLmn+l

(vii)

5.2 with m = m,]

(r)H (y+p E~(mfl+l)*(r),

the linear continuous multiplication by the function C,E L”(T)

yEL(mn+l)*(r)Hc,yEL(mn+l)*(r), (viii) the continuous embedding L(‘“n+l)*(r) C Lq*(r), (ix) the transpose of (iii), (ii), and (i) above. Formula (5.5) is then a simple application of the chain rule.

1764

P. RABIER

et al.

A similar method establishes the continuous differentiability of the operator Q(3.26) after writing (DQ(u).h,v),

=

s l-

mTcT[(~, - g)+-JmT-’h,r.vds,

where cT, g and r have been extended by 0 on I\I,. The steps (i)-(viii) are the same, except for changing m, into mr and c, into cr. The step (ix) must be modified into “the transpose of the linear continuous inner product with the [L”(I)lN vector field z: v E [~yr)lN

+

Z.

v E

u(r),

and the transpose of (ii) and (i)“. n We are now in position to prove our first existence and local uniqueness result. Theorem 5.1: Assume that meas > 0 or meas > 0. Then, there are neighborhoods U and U’ of the origin in I/ and I/’ respectively such that for every fez U’ the functional equation (5.2) has one and only one solution UE U. Proof: If meas > 0, it is well-known from Korn’s inequality and by a simple contradiction argument that the bilinear continuous form ae(o, 0) is also V-elliptic (see e.g. [9]) and, due to (3.14) and (3.24), the same is true with the bilinear continuous form a(~, l) (3.22). If meas > 0, the same arguments as above joined to (3.14) prove the I/-ellipticity of a(~, l). As a result, the operator A (5.1) verifies A E Zsom( V, v).

(5.8)

Let us denote by G: V-, ‘v’ the mapping G(u) = Au + P(u) -t Q(u). Since g 2 0 on rc, one has G(0) = 0. Besides, from Corollary 5.1, the mapping G is continuously differentiable and (using g 2 0 again) DG(0) = A. From (5.8) and the inverse mapping theorem, G is a Cl-diffeomorphism between two neighborhoods of the origin in T/ and v’ respectively and our assertion follows. H Remark 5.1: Existence and uniqueness for “small” right-hand sides f e v’ can be partly released as follows: set v,= (vEI/;u,(=v.n)=Oonr,}.

(5.9)

An immediate verification shows that V, is a closed subspace of I/ so that I/= v,o

(5.10)

vi,

where the orthogonal is taken in the sense of the equivalent inner product a(o, l) (assuming of course meas > 0 or meas > 0 as in Theorem 5.1). For each v E V, let us write v=v,+v;, according to the decomposition (5.10). The variational problem (3.28) can then be rewritten as: Find u = u,, + ui: such that a(uo, vLJ + a(& >6, + (mcl

+ ui>,hJ + vii>, +
+ UOI),vo + Vi>” =
+ (f,Vii)”

(5.11)

for every VE r Now, from the definition (5.9) of V, and that of the operators P and CD[cf. (3.25)-(3.26)] it is clear that P(u, + u&, = P(uA),

aqu, + II,‘) = a@).

In addition, it is also immediate that (P(u), v,,)” = 0 for every u E I/ and every v. E Vo. It follows that (5.11) reduces to the system: Find ui E VA such that

Existence

and local uniqueness

44,vk)

of solutions

+
to contact

problems

+ <~,(u~),v~h

in elasticity

= 03 Vk>”

1765

(5.12)

for every V$E V& and, next, find USE V, such that (5.13)

a(uo,vcl) + <@(ui&vcJ, = (f,vJ,

for every v,, E V,. Equation (5.12) has the same form as the original problem but is posed on the space V$ instead of V. The same arguments as in Theorem 5.1 thus yield existence and uniqueness of u& for sufficiently small right-hand sides, namely provided that the action of f on the elements of Vi: (and not of the whole space I’) is small enough. Once II&has been determined from (5.12), existence and uniqueness of u0 solution to (5.13) follows with no further assumption since equation (5.13) is linear in ue. H The second part of our results consists in proving existence and local uniqueness of a solution to the variational problem (3.28) [or to the equivalent functional equation (5.2)] for arbitrary external forces fE I/’ but sufficiently small coefficient of friction. To do this we need the following preliminary lemma Lemma 5.1: Assume that meas@‘,) > 0 or meas > 0. Then, for every fe V’ the equation (5.14)

Au + P(u) = f

has a unique solution u and the operator A + DP(u) is an is an isomorphism of V to V’. Proof: We begin with the observation that, for every UE V, P(u)E V’ is the derivative at u of the functional (well-defined from the admissible values of m, given in (3.17))

s

“[(u,

UEV+

-g)+lmn+lds

> 0.

(5.15)

rem, + 1

Using a system of local charts and an associated partition of the unity this follows from a well-known result by Krasnoselskii [lo, Lemma 5.1, p. 701 when I, is replaced by an open subset of R N- ’ . From Corollary 5.1, one has (DP(u).v,v),

=

m,c,[(u,

- g)+]“n-‘u,Zds 2 0

(5.16)

src for every VE I’. From (5.16) and the V-ellipticity of a(~, 0) under either assumption meas > 0 or meas > 0 as recalled in Theorem 5.1, the continuous functional J

s

-..&--[(u, uEV+J(U)=~u(u,u)+

- g)+]“n+’

ds - (f, u)”

rem, + 1

(5.17)

is strictly convex and coercive. As solutions to equation (5.14) are precisely the critical points of the functional J, existence and uniqueness follows. For every u E V, the I/-ellipticity of the bilinear form a(~,.) also yields the V-ellipticity of the bilinear form [c.f. (5.16)] (V,W)EV x V+a(v,w)

+ (DP(u)*v,w)v

= ([A + DP(u)]*v,w),

and hence, A + DP(u) E Zsom( V, V). Theorem 5.2: Assume that meas > 0 or meas the unique solution of the equation

n

> 0. For every f E I/’ denote by u” E V

Au0 + P(uO) = f

(5.18)

(cf. Lemma 5.1). Then, there is a neighborhood A of the origin in the space L”(T,) and there is a neighborhood U” of u” in V such that equation (5.2) has a unique solution u E U” for every cT E A. Proof: In this proof we shall make explicit the dependence of the operator 0 on the coefficient cT by setting 0 = Q(c,,u). For a given coefficient CUEL”(T,-) we proved in Corollary 5.1 that Q is continuously differentiable with respect to UE V with the formula

P. RABIER

I766

g&,u).h,v= > V

et al.

s

mTcT[(un - g)+lmT-‘h,r.vds

(5.19)

rc

holding for every (cT, II, h, v)~L”(r~) x V3. The definition (3.26) and formula (5.19) make it clear that both mappings CDand &D/&I are continuous with respect to the pair (c~,u)EL”(TJ x V. Now define the operator G:L”(TJ x I/-+ I/’ by G(c,, II) = Au + P(u) + @(c,, u), for (c,, u)~L~(rc)

x V. As @(O,l) = 0, equation (5.18) reads G(0, u”) = f.

On the other hand, as (X@I)(O,O) = 0 [cf. (5.19)] one has g

(0,u”) = A + DP(u’).

(5.20)

From the above properties of a’, both G and dG/au are continuous with respect to the pair (~~,u)~L”(rc) x I/ and, from (5.20) and Lemma 5.1, ~(O,u”)EIsom(V,

V’).

With the so-called “strong version” of the implicit function theorem (see e.g. Lyusternik and Sobolev [13]) there are neighborhoods A of the origin in L”(T,) and U” of u” in I/ such that the equation G(c,,u) = f [which is nothing but equation (5.2)] has exactly one solution u E I/ for every given cT E A and the proof is complete. n 6. THE PROBLEM WITH NO FRICTION; RELATIONSHIP TO SIGNORINI’S PROBLEM We shall complete this paper by showing how the problem with no friction (i.e. cT = 0) relates to the classical Signorini’s problem: Minimize ;u(v,v) -
(6.1)

K = {vE V;u,d Oonr,}.

(6.2)

over the closed convex set

It is known (see e.g. [S]) that this problem has a unique solution u” verifying (formally at least) diva(u’) + b = 0 ~c(uO)= t

inR,

on r,,

(6.4)

- Uj”) on

ECU = -K&y U: d 0

=

r,,

(6.5)

onr,,

(6.6)

[~c(u~)], < 0, [74u”)],u,0 = 0 [x(uO)]~

(6.3)

0

on

on rc,

r,.

(6.7)

(6.8)

In this case, we have assumed for simplicity that g = 0. From Lemma 5.1, the problem with no friction and power-like normal response has a unique solution, characterized as a minimizer of the functional VE

V-+J(v)

=

s

~[(u,)+]“‘~+’ +2(v,v) + TC% + 1

ds - (f,v),.

Existence and local uniqueness of solutions to contact problems in elasticity

1767

For il > 1, denote by u(n) the solution with AC, replacing c., i.e. u(J) minimizes the functional

s s

c, “EV+Jl(v)=;u(“,v)+~ Tc% +

1

[(u&]“n+ ‘ds - (f, v)“.

An equivalent characterization of u(J) is obtained by saying that u(n) is the (unique) critical point of the functional Jn (6.10), namely,

c,Cu,(W20, ds = (f, v>v

4um v) + 1

(6.11)

rc

for every VE V. Theorem 6.1: Assume c, > 0 on I,. Then

Proof: It can be easily shown that the functional veV-+p(v)

=

s

rc~&4+lmn+’

ds

is an exterior penalty functional for the constraint set K (6.2). In fact p is weakly lower semicontinuous (since it is convex and differentiable) and satisfies the conditions: p(v)2 0 VVE V and p(v)= 0 if and only if v E K (since c, > 0). It follows from the coercivity and weak lower semicontinuity of the quadratic form a(v,v) and from well-known results (see e.g. [16]), that there exists a subsequence of the family of solutions (u(L)),,, converging weakly in I’, as 1+ + co, to a minimizer of the functional (6.1) on K. Since the minimizer of (6.1) on K is unique it follows by a standard argument that the whole sequence (n(J)), a 1 converges weakly to the unique solution u” of the Signorini’s problem, which is characterized by the variational inequality (u’,v -u”)

2 (f,v - u”),Vv~K

(6.12)

To prove strong convergence it suffices to show that &n,

IWh

= llUOIIY.

Actually, since the coerciveness of the bilinear form a(~, l) over the space V implies that a(~, l) is an equivalent inner product over V, we may limit ourselves to proving ,lima u(u(l), u(1)) = a(#, UO).

(6.13)

From the weak lower semicontinuity of the quadratic form a(v, v) and the weak convergence of u(J) to u one has a(uO,uO)2 ,l&

a(u(l),

u(L)).

(6.14)

On the other hand, letting v = u(L) in (6.11) and observing that (u,,)?’ i > 0 it follows that a(n(~), u(A))G (f, n(W,, which, together with (6.12) at v = 2u” implies a(n(~), n(n)) G (f,n(lz)), + a(uO,nO)- (f,uO),. Hence 1hmm a(@), u(L)) G a(uO,no)

(6.15)

Relations (6.14) and (6.15) show that llimm a(~@),u(n)) exists and coincides with the common value lim a(~@), u(A)) = L 11~~u(u(l), u(L)). The desired equality (6.13) is now immediate ,%++a,

1768

P. RABIER er al.

and the proof is complete. w From Theorem 6.1 and when c, > 0 on l-c (the only physically acceptable assumption!) the solution u” of Signorini’s problem (6.3)-(6.8) then appears as the limiting case of an infinite normal stiffness on l-c. Of course, such an hypothesis is only an approximation of the physical reality which can a priori be made in the aim of simplifying the model. But it is well known that this simplification does not allow in general to consider friction phenomena: by the means of a powerlike normal response, we have shown that friction problems could be taken into account while the solution to the problem with no frictions is “close” to the solution of Signorini’s problem provided that the normal stiffness is “large enough”, a physical reality for highly polished surfaces [17].

REFERENCES [l] R. A. ADAMS, Soboleu Spaces. Academic Press, New York (1975). [Z] N. BACK, M. BURDEKIN and A. COWLEY, Review of the research on fixed and sliding joints, Proc. 13th International Machine Tool Design and Research Conference (Edited bv S. A. TOBIAS and F. KOENIGSBERGER), Macmillan, London, 87 (1973). . ~ [3] L. DEMKOWICZ and J. T. ODEN, On some existence results in contact problems with nonlocal friction. J. Nonlinear Analysis 6(10) (1982). [4] G. DUVAUT, Loi de frottement non locale. 1. de Mdcanique Theorique et Appliquee, Numero special (1982). [S] G. DUVAUT and J. L. LIONS, Inequalities in Mechanics and Physics. Springer-Verlag, New York (1976). [6] I. EKLAND and R. TEMAM, Analyse Conuexe et Probldmes Variationnels. Dunod, Gauthiers-Villars, Paris (1974). [7] R. HUNLICH and J. NAUMAN, On genera1 boundary value problems and duality in linear elasticity, I. Applicace Matematiky, 23(3), 208 (1978). [S] J. JARUSEK, Contact problems with bounded friction, coercive case. Czechoslouak Math. J., 33(108), 337 (1983). [9] N. KIKUCHI and J. T. ODEN, Contact Problems in Elasticity. SIAM, Philadelphia (1985). [lo] M. A. KRASNOSELSKII, Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon

Press, New York (1964). [11] J. KUFNER, 0. JOHN and S. FUCIK, Function Spaces. Noordhoff International Publishing, Leyden (1977). [12] J. L. LIONS, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires. Dunod, Paris (1969). [13] L. A. LYUSTERNIK and V. J. SOBOLEV, Elements of Functional Analysis. Gordon and Breach, New York (1961). [14] J. NECAS, Les Mdthodes Directes en Theorie des Equations Elliptiques. Masson et Cie, Paris (1967). [15] J. NECAS, J. JARUSEK and J. HASLINGER, On the Solution of the Variational Inequality to the Signorini Problem with Small Friction. Bolletino U.M.I., 5, 17-B, 796 (1980). 1161 J. T. ODEN, Qualitative Methods in Nonlinear Mechanics. Prentice Hall, Englewood Cliffs (1985). [17] J. T. ODEN and J. A. C. MARTINS, Models and computational methods for dynamic friction phenomena. Computer Methods Applied Mechanics and Engineering, In press. [18] J. T. ODEN and E. PIRES, Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity. J. Appl. Mech. SO(l), 67 (1983). [19] E. PIRES, Analysis of nonclassical friction laws for contact problems in elastostatics. Ph.D. Dissertation, Univ. of Texas at Austin (1982). [20] E. PIRES and J. T. ODEN, Analysis of contact problems with friction under oscillating loads. Computer Methods in Applied Mechanics and Engineering, 39, 337 (1983). [21] P. RABIER, Definition and properties of a particular notion of convexity. J. Num. Funct. Anal. Opt., 7(4), 279 (1984-85). [22] R. E. SHOWALTER, Hilbert Space Methods for Partial Dijerential Equations. Pitman, San Francisco

(1979). (Received 4 February 1986)