Marius Cocu

Elaine Pratt

Michel Raous

CONSTRUCTIVE ASPECTS OF FUNCTIONAL ANALYSIS FOR THE TREATMENT OF FRICTIONAL CONTACT

This paper is a survey of our recent results concerning frictional unilateral contact problems. Mathematical and numerical results obtained over the past few years for both static and quasistatic problems are summarized. Undergoing developments concerning viscoelastic quasistatic problems are outlined.

INTRODUCTION

As most authors in this field, we focused to begin with on the following static formulation of friction expressed in terms of the displacements and not velocities : no displacement occurs if the tangential force does not reach the sliding limit, whereas a natural velocity formulation of friction would imply in such a case that the velocity is zero and therefore that the displacement can be non zero. Such a static formulation does not have a real mechanical meaning except for very specific cases such as for example when the loading is proportional and monotone (i.e. F(t)=(t)F 0 with  monotone). Nevertheless, most of the mathematical results and of the first numerical methods have been developped in terms of this static formulation. In order to avoid the lack of regularity of the contact forces, a regularized problem corresponding to a nonlocal friction law was introduced by

Duvaut [START_REF] Duvaut | Equilibre d'un solide élastique avec contact unilatéral et frottement de Coulomb[END_REF]. In section 2 we comment on the existence and uniqueness results obtained for this problem under the condition that the friction coefficient is small (Cocu [START_REF] Cocu | Existence of solutions of Signorini problems with friction[END_REF]). Some regularity results have also been established for this problem (see ).

Rather than regularizing the friction by a nonlocal law, an alternative was to exploit the regularization obtained through the finite element discretization. In Licht et al. [START_REF] Licht | Remarks on a numerical method for unilateral contact including friction[END_REF],

existence and conditional uniqueness (small friction) of the solution can be found for the discretized problem with P 1 elements, and the convergence of the algorithm used for the numerical computation is proved. Various numerical methods used in the group for Mechanics and Modelling of Contact in the LMA are presented.

Section 3 concerns recent mathematical and numerical results on evolution quasistatic problems in the presence of general loadings. Friction is now formulated in terms of velocity. A difficulty arises when writing the variational formulation which has to deal simultaneously with a displacement condition (unilateral condition) and a velocity relationship (friction). If a displacement or a velocity formulation is chosen, one of the two conditions has to be approximated. This difficulty is overcome by maintaining both variables, displacement and velocity and thus setting the problem as two coupled variational inequalities (see Cocu et al. [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF]). One is an implicit variational inequality dealing essentially with friction and the other one is a variational inequality characterizing the unilateral condition ; they are strongly coupled. An incremental formulation ensues and, if the friction is sufficiently small, uniqueness of the solution of the incremental problem is established and appropriate estimates are given. Existence of a solution to the quasistatic problem is then obtained through the incremental solution. Two numerical methods are derived: one of them is written in terms of the velocities (increments of displacement), the other one is written in terms of the displacements. In the incremental formulation, a sequence of problems, very similar to the static one (but including extra terms), has to be solved : the numerical methods previously developed are then very useful. Finally an extension to elastoviscoplastic and viscoelastic materials is presented; existence and uniqueness results for a quasistatic contact problem with given friction are established.

ELASTOSTATIC CONTACT PROBLEMS

Unilateral contact with nonlocal friction

We consider an elastic body occupying a C 1 -smooth domain

 of R d , d = 2, 3, with boundary  =  1   2   3  where  1  2 and 
3 are open and disjoint parts of  The body is fixed along  1 with mes(  1) > 0 and a force density   is applied on  2 . The solid is in contact with a rigid fixed support on  3 , the displacements on this part of the boundary being restricted by a unilateral condition of non penetration into the support (Signorini's conditions), and submitted to nonlocal friction when contact occurs. On  a volume force density  1 is applied. We suppose that we may neglect the inertial terms, that   belongs to (L 2 ()) d and that   belongs to (L  ( 2 )) d . We denote by  = ( ij ) the stress tensor, e = (e ij ) the strain tensor, u = (u i ) the displacement field and E = (a ijkl ) the elasticity tensor of the material with the usual properties of symmetry and ellipticity. On  3 we use the classical notations for the normal and tangential components of the displacement and stress vectors.

Problem P 1 : Find u such that div  = -  in , (2.1)
 = E e in  () e = grad s u in  ()

u = 0 on   , (2.4) n =  2 on  2 , (2.5) u N    N ² 0, u N  N = 0 on    ()  T   R N and  T   R N  u T = 0  T =  R N    0 u T = - T ,    on    ()
where  is the friction coefficient and R N is a regularization of the normal contact force see Duvaut [START_REF] Duvaut | Equilibre d'un solide élastique avec contact unilatéral et frottement de Coulomb[END_REF] or Demkowicz-Oden [START_REF] Demkowicz | On some existence and uniqueness results in contact problems with nonlocal friction[END_REF].

In order to give a variational formulation we shall adopt the following notations:

V = { v  (H 1 ()) d ; v = 0 a.e. on   }, (2.8)
|| || shall denote the norm induced by ( , ) , the scalar product on V,

K = { v  V; v N ² 0 a.e. on   }, (2.9)
a(u, v) =  ij (u)e ij (v)   , (2.10) j(u,v) =  R N (u)  3  v T , (2.11)
(f, v) =  1 v   +  2 v  2  .
(2.12) We suppose that a ijkl  L  () 1 ² i,j,k,l ² d,   L  (  ) with    a.e. on   and that R:

H -1 /2  3 ( )→ L 2  3
( ) is a compact linear operator, where (see [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF])

H -1/ 2  3 ( ) is the dual space of H 1/2 ( 3 ) = w  3 ;w H 1/2 () , w = 0 on  1       .
The following properties on a(.,.) and j(.,.) hold (see e.g. [START_REF] Duvaut | Les Inéquations en Mécanique et en Physique[END_REF], [START_REF] Cocu | Existence of solutions of Signorini problems with friction[END_REF]) :

m  0 such that u  V a ( u, u)  m u 2 , (2.13) M  0 such that u  V v  V a(u,v)  M u v , (2.14) C  0  (u 1 , u 2 ,v 1 ,v 2 ) V 4 j(u 1 , v 1 ) -j(u 1 , v 2 ) -j(u 2 ,v 1 ) + j(u 2 , v 2 )  ˆ  C u 1 -u 2 v 1 -v 2 , (2.15)
where

ˆ  =  L  ( 3 )
. We may now present the following weak formulation of problem P 1 .

Problem P 2 : Find u  K such that a(u, v-u) + j(u,v) -j(u,u)  (f,v-u)  v K. (2.16)
The proof of the formal equivalence, i.e. if u is sufficiently regular, is straightforward (see e.g. Duvaut -Lions [START_REF] Duvaut | Les Inéquations en Mécanique et en Physique[END_REF] or Demkowicz -Oden [START_REF] Demkowicz | On some existence and uniqueness results in contact problems with nonlocal friction[END_REF]).

The main mathematical difficulty when considering Coulomb's friction law arises from the fact that, in the previous functional framework, the contact forces belong to

H -1/ 2  3 ( )
, so that the compactness condition, which is used in the general theory of variational methods, does not hold. The nonlocal friction law (2.7) enables us to overcome this difficulty; some physical and theoretical arguments in favour of nonlocal friction laws can be found in Oden-Pires [START_REF] Oden | Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity[END_REF].

The following result on this nonlocal formulation of a static frictional contact problem holds (see e.g.Cocu [START_REF] Cocu | Existence of solutions of Signorini problems with friction[END_REF]). The solution of P 2 is also locally regular. The proof of the following regularity result is based on the method of translation (see ).

Theorem 2.2. If  is C 3 -smooth in all x     a ijkl  C 1 (  ) 1 ² i,j,k,l ² d,   C 1 (  3 ) and the regularized stress R N (u)  C 1 (  3 ) with R N (u) ² 0, then for every open set U such that U     3 we have u  [ H 2 (U) ] d .
This regularity property enables us to consider pointwise unilateral contact conditions on   

Regularization due to space discretization

Here the terms due to friction are not regularized by a nonlocal friction law as was the case in the preceeding section. The variational formulation of the problem is as follows.

Problem D 1 : Find uK such that

a(u, v -u) - N (u), v T -u T  (f,v -u) v  K , (2.17)
where .,. denotes the duality pairing on

H 1/2 ( 3 )  H -1/ 2 ( 3 )
.

Not much can be said about problem D 1 from a mathematical point of view without adding extra assumptions, however if we discretize problem D 1 using a finite element method, the discretized problem can be shown to possess interesting properties ( in fact the discretization has a similar regularizing effect on the friction terms as the nonlocal law).

Let us introduce the following notations:

-T h is a regular triangulation of the domain  and

 = T TT h , - X h = w C 0 ( ) ; w TT h  P 1     
 where P 1 is the set of polynomials of degree 1,

- V h = v (X h ) d ; v = 0 on  1   , - K h = v  V h ; v N  0 on  3   , -˜ X h is the space of traces of X h on  3 , - ˜ V h is the space of traces of V h on  3 ,
- h is the restriction to  3 of the interpolation operator associated with X h . Note that  h possesses the two following properties which are due to the use of P 1 finite elements:

(a)  h (v) is positive for all positive continuous v on  3 , (b)

c(h)  h ( v h ) L 2 ( 3 )  c(h) v h v h  ˜ V h .
The following problem is then a discrete formulation of problem D 1 in V h .

Problem D 2 : Find u h  K h such that v K h a(u h ,v - u h ) -F N h (u h ), h ( v T - u T h )  (f, v - u h ) . (2.18)
The extra projection  h ( v ) shall prove to be very constructive for writing the finite dimensional problem presented in the next section. In order to simplify the notations we have supposed here that the friction coefficient  is constant, the general case can be obtained by setting  h  instead of . Existence and uniqueness results are obtained by expressing problem D 2 as a fixed point problem on the sliding limit g. First if B is the set of positive linear mappings on ˜ X h , for all gB the following problem,

Problem D 3 : Find u g h  K h such that v K h a(u g h ,v -u g h ) + g, h ( v T -u gT h )  (f,v -u g h ) (2.19)
has a unique solution. This is because the use of P 1 finite elements implies that the mapping

v → g,  h ( v T )
is convex and therefore problem D 3 is a classical minimization problem of a convex functional with quadratic growth.

Let us now define the mapping T by

T(g) = - F N h (u g h ) for g  B .
T(g) is a positive linear mapping on ˜ X h and the existence of a fixed point of T is equivalent to the existence of a solution u h of problem D 2 . T can fairly easily be shown to possess the following two properties :

(i)  c(h), a positive constant depending on h, such that g 1 ,g 2 B,

T(g 2 ) - T(g 1 )    c(h) g 2 - g 1  where g  = v h  ˜ V h Sup g,v h v h H 1 /2 ( 3 ) (ii)  M >0 such that T(g)   M g  B .
Theorem 2.3. There exists a solution to problem D 2 and this solution is unique if

  1 c(h) .
The existence of a solution to problem D 2 is obtained by applying Brouwer's theorem to the mapping T which is continuous because of (i) on the convex compact H, intersection of B with the ball of centre 0 and radius M ( T(H)  H because of (ii) ).

The uniqueness of the solution to problem D 2 is obtained, under the condition that   1 c(h) , by applying Banach's theorem to the mapping T which is then a contraction because of (i) and the fixed point of T is obtained as the limit of the fixed point iterates g n+1 =T(g n ) with a given g 0 .

Numerical treatment

The previous results are of course constructive and enable us to deduce a converging algorithm for the computation of the solution of (2.18). Using the fixed point method on the sliding limit g to determine the solution ˜ g =  F N , we have to solve a sequence of variational inequalities given by (2.19). The properties of the bilinear form a imply that these problems are equivalent to minimizations of a non-differentiable functional under the constraint induced by the non-penetration condition. The finite dimensional problem is set as follows.

Problem N 1 : For a given G, find

u  K h = K j j=1 M  such that : J(u)  J(v) v  K h (2.20) with J(v) = 1 2 v T Av - F T v + G T v
T , where : -A ij = a(w i ,w j ),

-F i = (f,w i ), - G i = g w i dx  C 
, -w i is a basis function of the finite dimensional set associated with the finite element discretization, -if I c is the set of the subscripts related to a normal component of a displacement of a contact node, K j is defined by :

K j = R if j  I c R - if j  I c   
A is a nn symmetric and positive definite. matrix, v and F are two n-dimensional vectors and v T and G are two m-dimensional vectors, where n is the number of degrees of freedom (twice the number of nodes) and m is the number of tangential components of the contact displacement (number of contact nodes).

For solving problem N 1, various numerical methods can be used (see Raous et al [START_REF] Raous | Numerical methods for frictional contact problems and applications[END_REF],

Lebon-Raous [START_REF] Lebon | Multibody contact problem including friction in structure assembly[END_REF], Barbarin-Raous [START_REF] Raous | Preconditioned conjugate gradient method for a unilateral problem with friction[END_REF]):

-Successive Over-Relaxation with Projection, -Aitken acceleration of Gauss-Seidel, -Projected Conjugate Gradient with preconditioning.

Remark . Other numerical methods have been developed in our group "Mechanics and Modelling of Contact" at the LMA. They are based on other formulations of the original problem :

-mathematical programmming method (Lemke method) based on complementary formulation, -Lagrangian and augmented Lagrangian formulations and associated solvers.

Some of these methods are compared in another paper of this special issue (Chabrand et al [START_REF] Chabrand | Comparison of various numerical methods for solving unilateral contact problems with friction[END_REF]).

QUASISTATIC CONTACT PROBLEMS

A unilateral contact problem with a nonlocal friction law in elasticity

We still consider a unilateral contact problem with a nonlocal friction law as in 2.1 but now the loading is supposed to be time dependent and the evolution to be quasistatic. For such evolution problems we can no longer adopt the static formulation of friction considered in section 2 so that the velocity field is present in the friction law. Two coupled variational inequalities constitute the variational formulation of this quasistatic problem. In this section we shall outline the proof of the existence of a solution to this quasivariational problem that can be found in [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF], [START_REF] Cocu | Existence d'une solution du problème quasi-statique de contact unilatéral avec frottement non local[END_REF].

We use the same notations as in section 2.1. We suppose that the given forces  1 and  2 are sufficiently smooth in space and time, i.e.

 1  W 1,2 (0,T;(L 2 ()) d ) and  2  W 1,2 (0,T;(L 2 ( 2 )) d
) (see e.g.Cazenave -Haraux [START_REF] Cazenave | Introduction aux Problèmes d'Evolution Semi -linéaires[END_REF]) and that we may neglect the inertial terms. The quasistatic problem can be set as follows.

Problem Q 1 : Find u such that for all t[0,T] equations (2.1) -( 2.3) and conditions (2.4) -(2.6) hold together with the initial condition

u(0) = u 0 in  , (3.1)
and with the following friction conditions for all t[0,T]

 T   R N and if  T   R N  Ý u T = 0 on  3 , if  T =  R N    0 Ý u T = - T on  3 .    (3.2)
We denote by W the space W 1,2 (0,T;V) , where V is defined by (2.8), and by . W the norm on W and ((.,.)) the scalar product on L 2 (0,T;V). We suppose that C(0,T;L _ ( 3 )), that t[0,T] (t)³0 a.e. on  3 and that the initial condition u 0 belongs to K and satisfies the following compatibility condition : a(u 0 ,w-u 0 ) + j(u 0 ,w-u 0 ) ³ (f(0),w-u 0 ) wK.

(3.3)
We adopt the following weak formulation of problem Q 1 (see [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF], [START_REF] Cocu | Existence d'une solution du problème quasi-statique de contact unilatéral avec frottement non local[END_REF]).

Problem Q 2 : Find uW 1,2 (0,T;V) such that u(0)=u 0 and for almost all t[0,T], u(t)K and satisfies

a(u( t), v-Ý u (t)) + j(u( t ), v) - j( u( t), Ý u (t))  ( f (t ), v -Ý u (t) ) +  N ( u( t)), v N -Ý u N (t ) vV (3.4)  N (u(t)),z N -u N (t)  0 z  K . (3.5)
If a solution u of problem Q 2 is sufficiently regular it also solves problem Q 1 .

The mapping j(.,.) involved in this variational formulation the following property:

C  0  (u 1 , u 2 ,v 1 ,v 2 ) V 4 j(u 1 , v 1 ) -j(u 1 , v 2 ) -j(u 2 ,v 1 ) + j(u 2 , v 2 )   C u 1 -u 2 v 1 -v 2 with  = Sup t 0,T   (t) L  ( 3 ) .
An incremental formulation is obtained by operating a time discretization of problem Q 2 , taking nN * and setting AEt = T/n , t i = i.t and f i = f(t i ) for i = 0,...,n. We use an implicit scheme and obtain the following sequence ( Q i n ) i=0 ,..., n-1 of variational inequalities defined for a given u 0 K by

Problem Q i n : Find u i+1  K such that a(u i+1 ,v - u i +1 - u i t ) + j(u i +1 , v) - j(u i +1 , u i+1 - u i t )  (f i +1 ,v - u i+1 - u i t ) +  N (u i +1 ), v N - u N i+1 - u N i t v  V, (3.6)
 N (u i +1 ),z N - u N i +1  0 z  K. (3.7)
It can be shown by using its strong formulaton (see [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF]) that problem (

Q i n
) is equivalent to the following problem:

Problem S i n : Find u i+1  K such that a(u i+1 ,w - u i+1 ) + j(u i+1 ,w - u i ) - j(u i +1 , u i+1 - u i )  (f i +1 ,w - u i +1 ) w  K .
Inequalities such as S i n , for a given i, are of the same type as (2.16) and thus possess a unique solution if the friction coefficient is sufficiently small. To be more precise, the friction coefficient must satisfy the following inequality   m / C where m and C have been defined in (2.13) and (2.15) . We shall from now on suppose that the friction coefficient is small enough to ensure that the above inequality is true. We suppose that u 0 satisfies the same condition (3.3) that we have imposed on u 0 , i.e. that u 0 satisfies :

a(u 0 , w -u 0 ) + j(u 0 , w -u 0 )  (f 0 ,w -u 0 ) w  K . ( 3.8)
The following bounds for the incremental solution u i and also for the difference

AEu i =u i+1 -u i can easily be obtained if  C  m : t  0,T   v  K  N (u(t)), v N -u N (t)  0 . ( 3.13)
Equation (3.12) is an integrated form of equation (3.4).

A contact problem with friction for a class of inelastic bodies

The purpose of this is to study the existence and uniqueness of solutions of quasistatic contact problems with given friction (or Tresca's friction law), involving the time derivative of u for a general model of elastoviscoplastic materials with internal state variables ( [START_REF] Necas | Mathematical theory of elastic and elasto plastic bodies[END_REF], [START_REF] Laborde | Approximation en viscoplasticité[END_REF]). This model includes some viscoelastic ones as, for example, the Maxwell's model (see [START_REF] Raous | Contacts unilatéraux avec frottement en viscoélasticité[END_REF], [START_REF] Raous | A periodic problem in viscoelasticity with variable coefficients[END_REF]).

The quasistatic evolution contact problem with given friction for a general model of elastoviscoplastic material with internal state variables has the following classical formulation.

Problem Q 3 : Find the fields of displacements u, inelastic strain tensor e i = (

e ij i
) and internal variables  = (  ,...,   ) such that for all t  [0,T] (2.1), (2.3), (2.4), (2.5) and (3.1) hold together with the following equations and conditions:

 = E e e in  (3.14)
Ý e i =  1 , ( ) in  (3.15)
Ý  =  2 , ( ) in , (3.16)
 T  g and if  T  g  Ý u T = 0 on  3 , if  T = g    0 Ý u T = - T on  3 .    ()
with e(u) = e e +e i , where e = (e ij ) and e e = ( e ij e

) are respectively the total strain and the elastic strain, and

e 0 i  S with S =  = ( ij )   ij  L 2 ()   ij =  ji , 1  i,j  d and   L 2 ()   We suppose that   :R s dxd x R   R s dxd with R s dxd = b= (b ij )  R dxd ; b ij = b ji
1 i, j  d} and   :R s dxd x R   R  satisfy the following Lipschitz conditions:

     such that  b , c  R s dxd ,      R  ,   (b,  ) -  (c,  ) R dxd    (b -c R dxd + - R  ). (3.21)      such that  b , c  R s dxd ,     R  ,   (b, ) -  (c, ) R     (b -c R dxd +  - R  ) ()
Choosing v = Ý w in (3.31) and v = Ý z in (3.32), integrating both sides of these inequalities and adding the resulting inequalities, we obtain, using the initial conditions, ) for all  ³ 0, () and, using (3.28), (3.29) and the properties of      , one can infer (see [START_REF] Cocu | Unilateral contact problems with friction for an elastoviscoplastic material with internal state variables[END_REF]) the existence of C 3 , ' > 0 such that , for all   '

e i () - e i () S,  C 3 ( - C 3 ) -1  - S, . (3.35)
Finally, from (3.34), (3.33), (3.35) and (3.28) it results that there exists " > 0 such that if  " then s L   (0,T;S)  L   (0,T;S) is a contraction and the theorem is proved.

Numerical treatment

For solving the elastic quasistatic problem of section 3.1, the fixed point method on the sliding limit (introduced in section 2 with the problem D 3 ) is now associated to the incremental formulation of problem S

Problem N 2 : Find

u i +1  K j j=1 M  such that J 1 ( u G i +1 )²J 1 ( v )  v  K j j=1 M  (3.36) with J 1 ( v )= 1 2 v T A v -( F i+1 ) T v + ( G i +1 ) T v T - u T i . ( 3

.37)

The same methods as the ones of section 2.3 can be used with a slight modification for the treatment of the friction term. In problem N 2 , it has to be emphasised that the convex set remains the same for every t i+1 because the formulation is written with respect to the displacements. As presented in Cocu et al. [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF] an alternative resolution can be deduced from the incremental formulation : it is set in terms of the velocities (increments of displacement), the friction term remains the same as the one of the static case but the constraint convex set changes at each step. The problem can be expressed in the following way.

Problem N 3 : Find

u G i+1  K j i j=1 M  such that J 2 (u G i+1 )  J 2 (v ) v  K j i j=1 M  (3.38) with J 2 (v ) = 1 2 v T Av -(R i + F i +1 ) T v + (G i +1 ) T v T , (3.39)
where the convex set depends on the solution obtained at the previous step and is given by

K j i = R if j I c  ; ( + v j i )  0   if j I c    R i = F i - Au i
being the previous residue of equilibrium (equal to the contact forces). So, the various algorithms used for solving the static problem can be extended to the treatment of the quasistatic problem thanks to the simple modifications which give the dependance on the loading history.

In viscoelasticity or viscoplasticity, as shown in Raous [START_REF] Raous | Contacts unilatéraux avec frottement en viscoélasticité[END_REF][18] [START_REF] Raous | Fissuration sous contraintes alternnées en viscoélasticité et viscoplasticité[END_REF], the problem can be set as a quasistatic elastic problem coupled with a differential equation by a viscous strain term. Various numerical methods can be used to integrate the differential equation, explicit Runge-Kutta methods such as Euler or Heun, or implicit ones such as -methods.

In linear viscoelasticity, either explicit or implicit methods are easy to implement (see Raous [START_REF] Raous | A periodic problem in viscoelasticity with variable coefficients[END_REF] and Chabrand et al. [START_REF] Chabrand | Residual thermal tempering stresses[END_REF]). A generalization to viscoplasticity is given in Raous [START_REF] Raous | Fissuration sous contraintes alternnées en viscoélasticité et viscoplasticité[END_REF] but explicit methods are prefered in that case.

CONCLUSION

Mathematical tools of functional analysis play a major part in this paper as the static and quasistatic unilateral contact problems with friction are written as implicit variational inequalities, of which the dual form would be quasivariational inequalities.

We have chosen to work directly with the initial formulation of the problem in terms of inequalities rather than adopting other formulations such as the compliance model. The results concerning the quasistatic problems constitute a new advance in the field and may help us in the formulation of dynamic frictional contact problems on which we are actually working.

Theorem 2 . 1 .

 21 The problem P 2 has at least one solution. If  satisfies the inequality ˆ   m / C  where m and C have been defined in (2.13) and (2.15), then problem P 2 has a unique solution.

  , 0) =   (x) in  ()  N = F N on   (3.19)

.

  Then a problem similar to the static discrete problem N 1 is set for each t i+1 . It includes an extra term u T i which characterizes the memory of the system induced by friction.

m - C i = 0,...,n , (3.9)

We define, for all nN * , a function f n : [0,T]V by f n (0) f(0) and f n (t) = f(t i+1 ) for t]t i ,t i+1 ] where t i = i.AEt and AEt = T/n .

We then consider inequalities S i n with f i = f(t i ) for i = 0,...,n and we define the function u n L 2 (0,T;V) by u n (0) = u 0 = u 0 where u 0 K is the initial condition of problem Q 2 and for t]t i ,t i+1 ] by u n (t) = u i+1 solution of S i n i = 0,...,n-1.

Lemma 3.1.

There exists a subsequence ( u n p ) p of (u n ) such that for all t[0,T] (

For the proof, see [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF]. We shall omit the subscript p from now on for simplicity.

Remark. For all t belonging to [0,T] and for all n N * u n (t) belongs to K so that the weak limit u(t) also belongs to K.

In order to show that u is sufficiently regular we shall now define the following sequence of functions :

where n belongs to the subset of N corresponding to the subsequence of (u n ) one has exhibited in the lemma 3.1. The mapping ˜ u n is obviously an element of W 1,2 (0,T;V) . We have the following result.

Lemma 3.2.

There exists a subsequence of ( ˜ u n ) that converges weakly in W 1,2 (0,T;V) and the weak limit of this subsequence is equal to u.

The functions u n and ˜ u n are defined by using u i+1 , solution of S i n , but also solution

), u N i +1 = 0 and as

),v N and by integration on [0,T] we obtain for all vL 2 (0,T;V):

The proof is straightforward as (u n ) converges weakly towards u, (

) converges weakly towards Ý u in L 2 (0,T;V), (f n ) converges strongly towards f, and R is a compact operator.

Lemma 3.4.([5]) We have

(0,T;V) , and the unilateral condition

We suppose also that

)) and that, for simplicity, g  L ( 3 ) is independent of time. In order to set the classical problem in variational form let us define:

where V is defined by (2.8).

We suppose that the initial conditions u 0 and e 0 i satisfy the following compatibility condition:

where a is defined by (2.10).

If U is a Banach space we denote by L   (0,T;U) , ³ 0, the space L  (0, T;U) with the norm y U, = ess supe - t y(t) U ; t  T] } . For any ' , " ³ 0 the norms U,' and U," are equivalent but suitable choices for  will be useful in proving the existence and uniqueness result. We set  = L 2 ()  .

If one knows the evolution of e i on [0,T], then, using Green's formula, the solution u satisfies the following variational inequality for all t  

We can now state the following mixed formulation of problem Q 3 .

for almost all t   with the initial conditions (3.1), (3.17) and (3.18), where W is defined by (3.3).

Let us state the following standard result.

Lemma 3.5. For each   L  (0, T;S) there exist a unique ()  W 1, (0,T;) and a unique e i ()  W 1, (0,T;S) which for almost all t   satisfy the following equations:

with the initial conditions (3.17) and (3.18).

The differential equation (3.29) satisfies the Cauchy -Lipschitz theorem for any given   L  (0, T;S) so that the proof of this lemma is straightforward.

Lemma 3.6. For each   L  (0, T;S) there exists a unique u()  W which satisfies the variational inequality (3.27), for almost all t   where e i = e i () is given by lemma 3.5.

For proof, we observe that for each   L  (0, T;S) we have e i ()  W 1, (0,T;S) so that (3.27) is a variational inequality for which, by using similar arguments as for the Problem Q 2 , we obtain the existence of a solution u()  W. The uniqueness readily follows. (0,T;V) x W 1, (0,T;S) x W 1, (0,T;) .

Proof.. For each   L  (0, T;S) let s() = (s ij ()) be given by

where u() and e i () are defined as above. Thus we define a new map s: L  (0, T;S)  L  (0, T;S) . We will show that s : L   (0,T;S)  L   (0,T;S) is a contraction mapping for a suitable choice of  Then s has a unique fixed point  and (u() e i () ()) is the unique solution of the problem.

Let    L   (0,T;S) . Then z = u() w=u()  W 1,2 (0,T;V) are the solutions of the following variational inequalities for almost all t   a(z,v-Ý z ) +j(v) -j( Ý z ) ³ (f,v-Ý z ) +b(e i (), e(v-Ý z ))   V, (3.31) a(w,v-Ý w ) +j(v) -j( Ý w ) ³ (f,v-Ý w ) +b(e i (), e(v-Ý w ))  v  V. (3.32)