Anca Capatina
email: anca.capatina@imar.ro

Marius Cocou
email: cocou@lma.cnrs-mrs.fr

Michel Raous
email: raous@lma.cnrs-mrs.fr

A class of implicit variational inequalities and applications to frictional contact

Keywords: Mathematics Subject Classification (2000). 35K85, 49J40, 65N12, 65N30 Keywords. Evolution problems, implicit variational inequalities, quasistatic frictional contact

This paper deals with the mathematical and numerical analysis of a class of abstract implicit evolution variational inequalities. The results obtained here can be applied to a large variety of quasistatic contact problems in linear elasticity, including unilateral contact or normal compliance conditions with friction. In particular, a quasistatic unilateral contact problem with nonlocal friction is considered. An algorithm is derived and some numerical examples are presented.

Introduction

This paper is concerned with the mathematical analysis and the approximation of a class of abstract implicit evolution variational inequalities which constitutes a generalization of variational inequalities related to various quasistatic contact problems and of some parabolic variational inequalities of second kind, see, e.g., [START_REF] Glowinski | Numerical analysis of variational inequalities[END_REF]- [START_REF] Cocou | Formulation and approximation of quasistatic frictional contact[END_REF] and references therein.

The static contact problems have been extensively studied and their analysis has contributed to considerably developing the mathematical and numerical analysis of elliptic variational inequalities. As is well known, in the case of frictional contact, this static formulation does not have any mechanical significance, except in some very special cases.

Nevertheless, the incremental formulation obtained by performing time discretization of quasistatic or dynamic problems is similar to a sequence of static problems, which facilitates the mathematical and numerical analysis.

In the present work, a new system of evolution variational inequalities, representing a unified approach to quasistatic contact problems in elasticity, is analyzed. In Section 2, using an implicit time discretization scheme and some estimates, strong convergence is obtained and an existence result is established.

In Section 3, numerical analysis is carried out on the general problem. Convergence results are proved using a method based on an semi-discrete internal approximation and an implicit time discretization scheme.

In Section 4, we apply the results of Sections 2 and 3 to a quasistatic unilateral contact problem with nonlocal friction in linear elasticity.

This particular problem concerns the contact between an elastic body and a rigid support but a large variety of contact conditions can be easily analyzed using our general results, as, for example, the unilateral or bilateral contact with nonlocal friction between two elastic bodies or the frictional contact with normal compliance conditions as well as the corresponding frictionless cases.

In Section 5, some algorithms and numerical results are presented and the effects of space and time discretizations are described on a test example.

A general system of evolution inequalities

Let (V, (•, •)) be a real Hilbert space with the associated norm • and let (H, (•, •) H ) be a real Hilbert space with the associated norm • H . Let a : V × V → R be a bilinear and symmetric form on V . We assume that a is V-elliptic and continuous i.e.

a(v, v) ≥ α v 2 , |a(v, w)| ≤ M v w ∀ v, w ∈ V, (2.1)
where α, M are positive constants. Let K be a closed convex cone contained in V with its vertex at 0 and let (K(g)) g∈V be a family of nonempty closed convex subsets of K satisfying the following conditions: 0 ∈ K(0) and

if g n → g in V, v n ∈ K(g n ) and v n ⇀ v in V then v ∈ K(g). (2.2)
We assume that for all g ∈ V there exists an operator β(g, •) :

K(g) → H such that if g n → g in V, v n ∈ K(g n ) and v n ⇀ v in V then β(g n , v n ) → β(g, v) in H (2.3)
and for all g ∈ V , v ∈ K(g),

β(g, v) H ≤ k 1 ( g + v ), (2.4)
where k 1 is a positive constant. For all g ∈ V , let j(g, •, •) : K(g) × V → R be a functional satisfying the following conditions: j(g, v, •) is sub-additive for all g ∈ V, v ∈ K(g), that is (2.5) j(g, v, w 1 + w 2 ) ≤ j(g, v, w 1 ) + j(g, v, w 2 ) ∀ g, w 1,2 ∈ V, v ∈ K(g), j(g, v, •) is positively homogeneous for all g ∈ V, v ∈ K(g),

that is j(g, v, θw) = θj(g, v, w) ∀ g, w ∈ V, v ∈ K(g), θ ≥ 0,

j(0, 0, w) = 0 ∀ w ∈ V, (2.7)
and

|j(g 1 , v 1 , w 2 ) + j(g 2 , v 2 , w 1 ) -j(g 1 , v 1 , w 1 ) -j(g 2 , v 2 , w 2 )| ≤ k 2 ( g 1 -g 2 + β(g 1 -g 2 , v 1 -v 2 ) H ) w 1 -w 2 ∀ g i , w i ∈ V, v i ∈ K(g i ), i = 1, 2, (2.8)
where k 2 is a positive constant. We assume that k 1 and k 2 satisfy the following condition:

k 1 k 2 < α. (2.9)
For all g ∈ V , we consider a functional b(g, •, •) : K(g) × V → R which satisfies the following conditions:

∀ g ∈ V, v ∈ K(g), b(g, v, •) is linear on V, (2.10) if g n → g in V, v n ∈ K(g n ), v n ⇀ v in V and w n → w in V then b(g n , v n , w n ) → b(g, v, w), (2.11)
and

|b(g 1 , v 1 , w) -b(g 2 , v 2 , w)| ≤ k 3 ( g 1 -g 2 + v 1 -v 2 ) w ∀ g i , w ∈ V, v i ∈ K(g i ), i = 1, 2, (2.12)
where k 3 is a positive constant. From the above properties of a, j and K and by a classical argument, it follows that for all g ∈ V, d ∈ K, w ∈ K(g) the elliptic variational inequality

u ∈ K a(u, v -u) + j(g, w, v -d) -j(g, w, u -d) ≥ 0 ∀ v ∈ K
has a unique solution, so that we can define the mapping S g,d : K(g) → K as S g,d (w) = u. We assume that for all g ∈ V, d ∈ K K(g) is stable under S g,d i.e. S g,d (K(g)) ⊂ K(g).

(2.13) For all g ∈ V, d ∈ K, we consider the following problems:

( P )        u ∈ K(g) a(u, v -u) + j(g, u, v -d) -j(g, u, u -d) ≥ b(g, u, v -u) ∀ v ∈ V, b(g, u, z -u) ≥ 0 ∀ z ∈ K, and
( Q) u ∈ K(g) a(u, v -u) + j(g, u, v -d) -j(g, u, u -d) ≥ 0 ∀ v ∈ K.
We assume that if u is a solution of ( Q), then u is a solution of ( P ).

(2.14)

Remark 2.1 If u satisfies ( P ), then u obviously satisfies ( Q).

Let f ∈ W 1,2 (0, T ; V ) be given and u 0 ∈ K(f (0)) be the unique solution of the following implicit elliptic variational inequality (see, e.g., [START_REF] Capatina-Radoslovescu | Internal approximation of quasivariational inequalities[END_REF]):

a(u 0 , w -u 0 ) + j(f (0), u 0 , w) -j(f (0), u 0 , u 0 ) ≥ 0 ∀ w ∈ K.
(2.15) Indeed, using the hypotheses (2.1), (2.4), (2.8) and (2.13), it follows that S f (0),0 : K(f (0)) → K(f (0)) is a contraction if the condition (2.9) holds. Therefore the variational inequality (2.15) has a unique solution.

Remark 2.2 i) Since, by (2.6) j(•, •, 0) = 0, from (2.8) for w 2 = 0, w 1 = w and from (2.4) it follows that

|j(g 2 , v 2 , w) -j(g 1 , v 1 , w)| ≤ k 2 ( g 1 -g 2 + β(g 1 -g 2 , v 1 -v 2 ) H ) w ≤ ((k 1 + 1)k 2 g 1 -g 2 + k 1 k 2 v 1 -v 2 ) w ∀ g i , w ∈ V, v i ∈ K(g i ), i = 1, 2.
(2.16)

ii) As j satisfies (2.5), (2.6), j(g, v, •) is convex and from (2.5), (2.7), (2.16) for

g 2 = g, v 2 = v, g 1 = v 1 = 0, it results that j(g, v, •) is Lipschitz continuous on V for all g ∈ V and for all v ∈ K(g).
iii) Using (2.3) and (2.16), as by (2.4) β(0, 0) = 0, we have the following property if

g n → g in V, v n ∈ K(g n ) and v n ⇀ v in V then j(g n , v n , w) → j(g, v, w) ∀ w ∈ V.
(2.17)

We consider the following evolution system of coupled variational inequalities.

Problem P: Find u ∈ W 1,2 (0, T ; V ) such that (P )              u(0) = u 0 , u(t) ∈ K(f (t)) ∀ t ∈] 0, T [, a(u(t), v -u(t)) + j(f (t), u(t), v) -j(f (t), u(t), u(t)) ≥ b(f (t), u(t), v -u(t)) ∀ v ∈ V a.e. on ]0, T [, b(f (t), u(t), z -u(t)) ≥ 0 ∀ z ∈ K, ∀ t ∈] 0, T [.
In order to prove that there exists a solution to problem P, we consider an incremental formulation obtained by using an implicit time discretization scheme.

For n ∈ N * , we set ∆t := T /n and t i := i ∆t, i = 0, 1, ..., n. If θ is a continuous function of t ∈ [ 0, T ] valued in some vector space, we use the notations θ i := θ(t i ) unless θ = u, and if η i , ∀ i ∈ {0, 1, ..., n}, are elements of some vector space, then we set

∂η i := η i+1 -η i ∆t ∀ i ∈ {0, 1, ..., n -1}.
Let us also denote K i := K(f (t i )), ∀ i ∈ {0, 1, ..., n}, and u 0 := u 0 . We then approximate (P ) using the following sequence of incremental problems (P i n ) i=0,1,...,n-1 .

Problem P i n : Find u i+1 ∈ K i+1 such that

(P i n )        a(u i+1 , v -∂u i ) + j(f i+1 , u i+1 , v) -j(f i+1 , u i+1 , ∂u i ) ≥ b(f i+1 , u i+1 , v -∂u i ) ∀ v ∈ V, b(f i+1 , u i+1 , z -u i+1 ) ≥ 0 ∀ z ∈ K.
Remark 2.3 i) Since K is a cone, the solutions u of (P ) and u i+1 of

(P i n ) obviously satisfy b(f (t), u(t), u(t)) = 0 on [ 0, T ] and b(f i+1 , u i+1 , u i+1 ) = 0, ∀ i ∈ {0, 1, ..., n -1}.
ii) It follows also that b(f (t), u(t), u(t)) = 0 a.e. on [ 0, T ], (2.18) which can be obtained simply by the fact that since u satisfies the second inequality of (P ) we have, for all t ∈] 0, T [ and for all ∆t > 0, sufficiently small,

b(f (t), u(t), u(t + ∆t) -u(t) ∆t ) ≥ 0, and b(f (t), u(t), u(t -∆t) -u(t) -∆t ) ≤ 0.
From (2.14) and Remark 2.1 for g = f i+1 and d = u i , it follows that the problem P i n is equivalent to the following variational inequality: find

u i+1 ∈ K i+1 such that (Q i n ) a(u i+1 , w -u i+1 ) + j(f i+1 , u i+1 , w -u i ) -j(f i+1 , u i+1 , u i+1 -u i ) ≥ 0 ∀ w ∈ K.
From the hypotheses (2.1), (2.4), (2.8), (2.9) and (2.13), it follows that S f i+1 ,u i : K i+1 → K i+1 is a contraction. Therefore (Q i n ) has a unique solution.

A straightforward computation and the fact that f is absolutely continuous give us the following estimates.

Lemma 2.1 Let u i+1 ∈ K i+1 be the solution of (Q i n ), i ∈ {0, 1, ..., n -1}. Then u 0 ≤ C 0 , u i+1 ≤ C 0 , (2.19)
u i+1 -u i ≤ C 1 t i+1 t i ḟ (τ ) dτ, (2.20) n-1 i=0 u i+1 -u i 2 ≤ C 2 1 ∆t T 0 ḟ (τ ) 2 dτ, (2.21)
where

C 0 = (k 1 + 1)k 2 f C([ 0,T ];V ) α -k 1 k 2 , C 1 = (k 1 + 1)k 2 α -k 1 k 2 . (2.22)
Now, if we define

                 u n (0) = ûn (0) = u 0 , f n (0) = f 0 and ∀ i ∈ {0, 1, ..., n -1}, ∀ t ∈]t i , t i+1 ], u n (t) = u i+1 , ûn (t) = u i + (t -t i )∂u i , f n (t) = f i+1 , (2.23)
then the functions u n , f n ∈ L 2 (0, T ; V ) and ûn ∈ W 1,2 (0, T ; V ) satisfy the following incremental formulation for all t ∈ [ 0, T ]:

(P n )            u n (t) ∈ K(f n (t)), a(u n (t), v - d dt ûn (t)) + j(f n (t), u n (t), v) -j(f n (t), u n (t), d dt ûn (t)) ≥ b(f n (t), u n (t), v - d dt ûn (t)) ∀ v ∈ V, b(f n (t), u n (t), z -u n (t)) ≥ 0 ∀ z ∈ K. Remark 2.4 From (2.23) and Remark 2.3 i), it follows that b(f n (t), u n (t), v - d dt ûn (t)) = b(f i+1 , u i+1 , v -∂u i ) ≥ b(f n (t), u n (t), v) ∀ t ∈]t i , t i+1 [ , ∀ v ∈ V.
Lemma 2.2 For all t ∈ [ 0, T ] we have

u n (t) ≤ C 0 , (2.24)
and for all s, t ∈ [ 0, T ], s < t we have

u n (s) -u n (t) ≤ C 1 min{t+∆t,T } s ḟ (τ ) dτ, (2.25) u n -ûn L 2 (0,T ;V ) ≤ C 1 ḟ L 2 (0,T ;V ) ∆t √ 3 , (2.26)
d dt ûn L 2 (0,T ;V ) ≤ C 1 ḟ L 2 (0,T ;V ) . ( 2
u n -ûn L 2 (0,T ;V ) = ∆t √ 3 d dt ûn L 2 (0,T ;V ) = ∆t 3 n-1 i=0 u i+1 -u i 2
and estimate (2.21).

Lemma 2.3 There exists a subsequence of (u n ) n , denoted by (u np ) p , and an element u ∈ W 1,2 (0, T ; V ) such that Proof.

u np (t) ⇀ u(t) in V ∀ t ∈ [ 0, T ], (2.28) ûnp ⇀ u in W 1,2 (0, T ; V ). ( 2
Applying a diagonal process (see, e.g., [START_REF] Cocou | Formulation and approximation of quasistatic frictional contact[END_REF]), it follows from (2.24) and (2.25) that we can extract a subsequence (u

n k ) k of (u n ) n such that u n k (t) ⇀ u(t) in V ∀ t ∈ [ 0, T ], hence u n k ⇀ u in L 2 (0, T ; V ). Thus from (2.26) and
(2.27) we deduce that u ∈ W 1,2 (0, T ; V ) and there exists a subsequence of

(û n ) n such that (2.29) holds. Since f, u ∈ W 1,2 (0, T ; V ) ⊂ C([ 0, T ]; V ) and f n (t) → f (t) in V ∀ t ∈ [ 0, T ], from (2.2) it results that u(t) ∈ K(f (t)) ∀ t ∈ [ 0, T ].
We shall omit the subscript p from now on. In order to prove (2.30) and (2.31), let s ∈] 0, T ]. Using the definitions (2.23) and the properties of a, we obtain

s 0 a(u n (t), d dt ûn (t)) dt ≥ 1 2 a(u n (s), u n (s)) - 1 2 a(u 0 , u 0 ) - t i+1 s a(u n (t), d dt ûn (t)) dt, (2.32)
where i ∈ {0, 1, ..., n -1} is such that s ∈]t i , t i+1 ]. Therefore, by going to the limit, one obtains (2.30).

Next, since by Remark 2.

2 ii) j(f (t), u(t), •) is convex lower semicon- tinuous for all t ∈ [ 0, T ], the mapping v → s 0 j(f (t), u(t), v(t)) dt is convex
and lower semicontinuous on L 2 (0, T ; V ) (see, e.g., [START_REF] Brezis | Problèmes unilatéraux[END_REF]). Thus

lim inf n→∞ s 0 j(f (t), u(t), d dt ûn (t)) dt ≥ s 0 j(f (t), u(t), u(t)) dt. (2.33)
Using (2.16) and (2.27) we have

| s 0 (j(f n (t), u n (t), d dt ûn (t)) -j(f (t), u(t), d dt ûn (t))) dt| ≤ C 1 ḟ L 2 (0,T ;V ) ( s 0 (k 2 β(f n (t) -f (t), u n (t) -u(t)) H +k 2 f n (t) -f (t) ) 2 dt) 1 2
, from which by using (2.3) it follows that

lim n→∞ s 0 (j(f n (t), u n (t), d dt ûn (t)) -j(f (t), u(t), d dt ûn (t))) dt = 0. ( 2
u n (t) → u(t) in V ∀ t ∈ [ 0, T ], (2.35) ûn → u in L 2 (0, T ; V ), (2.36)
d dt ûn ⇀ u in L 2 (0, T ; V ), (2.37)
where u ∈ W 1,2 (0, T ; V ) is a solution of problem P.

Proof.

Let (u n ) n be the subsequence given by lemma 2.3. We shall prove that its limit u is a solution of problem P. Let s ∈ [ 0, T ]. Integrating both sides in the first inequality of (P n ) over [ 0, s ] and passing to the limit, by the relations (2.1), (2.17), Remark 2.4 and lemma 2.3 it follows that for all v ∈ L 2 (0, T ; V )

s 0 a(u(t), v(t) -u(t)) dt + s 0 j(f (t), u(t), v(t))dt - s 0 j(f (t), u(t), u(t))dt ≥ s 0 b(f (t), u(t), v(t))dt.
(2.38)

Now, by writing the inequality corresponding to (Q i n ) satisfied by u n and passing to the limit, we obtain that for all t ∈ [ 0, T ]

a(u(t), v -u(t)) + j(f (t), u(t), v -u(t)) ≥ 0 ∀ v ∈ K, (2.39)
where we have used that

u n (t -∆t) ⇀ u(t) in V ∀ t ∈] 0, T ] and (2.15). From (2.14), taking d = u, it follows that u satisfies b(f (t), u(t), z -u(t)) ≥ 0 ∀ z ∈ K, ∀ t ∈ [ 0, T ]. (2.40)
Finally, by Remark 2.3 and Lebesgue's theorem, we obtain that u is a solution of problem P.

In order to show the strong convergences (2.35), (2.36) (see also [START_REF] Andersson | A quasistatic frictional problem with normal compliance[END_REF]), we first prove that

lim n→∞ s 0 a(u n (t), d dt ûn (t)) dt = s 0 a(u(t), u(t)) dt ∀ s ∈ [ 0, T ]. (2.41) On one hand, taking v = 0 in (P n ), v = 0, v = 2 u(t)
a(u n (s), u n (s)) = a(u(s), u(s)) ∀ s ∈ [ 0, T ].
Therefore, using the V-ellipticity of a and (2.26), we obtain the strong convergences (2.35), (2.36) and the theorem is proved.

Approximation and numerical analysis

This section deals with the discretization of problem P. We shall prove a convergence result for a method based on an internal approximation and a backward difference scheme.

First we consider a semi-discrete approximation of (P ). For a positive parameter h converging to 0, let (V h ) h be a family of finite-dimensional subspaces of V and let (K h ) h be a family of closed convex cones with their vertices at 0 such that K h ⊂ K and (K h ) h is an internal approximation of K (see, e.g., [START_REF] Glowinski | Numerical analysis of variational inequalities[END_REF]), i.e.

∀ v ∈ K, ∃ v h ∈ K h such that v h → v in V. (3.1)
From the properties of a, j and K h , it follows that for all g ∈ V,

d h ∈ K h , w h ∈ K h ∩ K(g), the elliptic variational inequality u h ∈ K h a(u h , v h -u h ) + j(g, w h , v h -d h ) -j(g, w h , u h -d h ) ≥ 0 ∀ v h ∈ K h
has a unique solution. Hence we can define a mapping S h g,d h :

K h ∩K(g) → K h by S h g,d h (w h ) = u h . We suppose that for all g ∈ V, d h ∈ K h S h g,d h (K h ∩ K(g)) ⊂ K h ∩ K(g). (3.2)
For all g ∈ V, d h ∈ K h , we consider the following problems:

( Ph )

       u h ∈ K h ∩ K(g), a(u h , v h -u h ) + j(g, u h , v h -d h ) -j(g, u h , u h -d h ) ≥ b(g, u h , v h -u h ) ∀ v h ∈ V h , b(g, u h , z h -u h ) ≥ 0 ∀ z h ∈ K h ,

and

( Qh )

u h ∈ K h ∩ K(g), a(u h , v h -u h ) + j(g, u h , v h -d h ) -j(g, u h , u h -d h ) ≥ 0 ∀ v h ∈ K h .
We assume that if u h is a solution of ( Qh ), then u h is a solution of ( Ph ).

(3.3)

Remark 3.1 If u h satisfies ( Ph ), then u h obviously satisfies ( Qh ).
Let us now consider the following semi-discrete problem.

Problem P h : Find u h ∈ W 1,2 (0, T ; V h ) such that (P h )              u h (0) = r h u 0 , u h (t) ∈ K h ∩ K(f (t)) ∀ t ∈] 0, T [, a(u h (t), v h -uh (t)) + j(f (t), u h (t), v h ) -j(f (t), u h (t), uh (t)) ≥ b(f (t), u h (t), v h -uh (t)) ∀ v h ∈ V h a.e. on ] 0, T [, b(f (t), u h (t), z h -u h (t)) ≥ 0 ∀ z h ∈ K h , ∀ t ∈] 0, T [, where r h u 0 ∈ K h ∩K(f (0)) is the unique solution of the variational inequality a(r h u 0 , w h -r h u 0 ) + j(f (0), r h u 0 , w h ) -j(f (0), r h u 0 , r h u 0 ) ≥ 0 ∀ w h ∈ K h . (3.4)
The full discretization of (P h ) is obtained by using an implicit scheme as in Section 2 for (P ). For u 0 h := r h u 0 and i ∈ {0, 1, ..., n -1}, we define u i+1 h as the solution of the following problem.

Problem P i hn : Find u i+1 h ∈ K i+1 h such that (P i hn )        a(u i+1 h , v h -∂u i h ) + j(f i+1 , u i+1 h , v h ) -j(f i+1 , u i+1 h , ∂u i h ) ≥ b(f i+1 , u i+1 h , v h -∂u i h ) ∀ v h ∈ V h , b(f i+1 , u i+1 h , z h -u i+1 h ) ≥ 0 ∀ z h ∈ K h , where K i+1 h := K h ∩ K(f i+1 ). By (3.
3) and Remark 3.1 it is easily seen that (P i hn ) is equivalent to the following implicit variational inequality: find

u i+1 h ∈ K i+1 h such that (Q i hn ) a(u i+1 h , w h -u i+1 h ) + j(f i+1 , u i+1 h , w h -u i h ) -j(f i+1 , u i+1 h , u i+1 h -u i h ) ≥ 0 ∀ w h ∈ K h . From (2.1), (2.4), (2.8), (2.9) and (3.2) it follows that S h f i+1 ,u i h : K i+1 h → K i+1 h is a contraction, so that (Q i hn ) has a unique solution. If we define the functions            u hn (0) = ûhn (0) = r h u 0 and ∀ i ∈ {0, 1, ..., n -1}, ∀ t ∈]t i , t i+1 ], u hn (t) = u i+1 h , ûhn (t) = u i h + (t -t i )∂u i h , (3.5)
then we have the analogue to theorem 2.1 in the finite-dimensional case.

Theorem 3.1 Assume that (2.1)-(2.12), (3.2)-(3.4) hold. Then there exists a subsequence of (u hn ) n , still denoted by (u hn ) n , such that

u hn (t) → u h (t) in V ∀ t ∈ [ 0, T ], (3.6)
ûhn ⇀ u h in W 1,2 (0, T ; V ), (3.7)
where u h ∈ W 1,2 (0, T ; V h ) is a solution of (P h ).

We now proceed to find a priori estimates for the solutions of (P h ) which are limits of subsequences of (u hn ) n . Lemma 3.1 Let u h be a solution of (P h ) given by theorem 3.1. Then

u h (t) ≤ C 0 ∀ t ∈ [ 0, T ], (3.8
)

u h (s) -u h (t) ≤ C 1 t s ḟ (τ ) dτ ∀ s, t ∈ [ 0, T ], s < t, (3.9
)

u h W 1,2 (0,T ;V ) ≤ C 2 0 T + C 2 1 ḟ 2 L 2 (0,T ;V ) , (3.10)
C 0 , C 1 being defined by (2.22).

Proof.

Using the same argument as in the proof of lemma 2.2, we obtain

u hn (t) ≤ C 0 ∀ t ∈ [ 0, T ], u hn (s) -u hn (t) ≤ C 1 min{t+∆t, T } s ḟ (τ ) dτ ∀ s, t ∈ [ 0, T ], s < t, ûhn 2 W 1,2 (0,T ;V ) ≤ C 2 0 T + C 2 1 ḟ 2 L 2 (0,T ;V )
. Combining these results with (3.6) and (3.7), the estimates (3.8)-(3.10) follow.

We have the following convergence and existence result.

Theorem 3.2 Under the assumptions (2.1)-(2.15), (3.1)-(3.4), there exists a subsequence of (u h ) h such that

u h (t) → u(t) in V ∀ t ∈ [ 0, T ], (3.11
) uh ⇀ u in L 2 (0, T ; V ), (3.12)
where u ∈ W 1,2 (0, T ; V ) is a solution of (P ).

Proof.

From lemma 3.1 it follows that there exists a subsequence of (u h ) h and an element u ∈ W 1,2 (0, T ; V ) such that

u h (t) ⇀ u(t) in V ∀ t ∈ [ 0, T ], (3.13)
u h ⇀ u in W 1,2 (0, T ; V ). (3.14)
As one can prove that

r h u 0 → u 0 in V, (3.15)
from (3.13) we obtain lim inf Next we prove that u satisfies (2.38). In order to pass to the limit in

h→0 s 0 a(u h (t), uh (t)) dt ≥ 1 2 (lim inf h→0 a(u h (s), u h (s)) -lim h→0 a(r h u 0 , r h u 0 )) ≥ 1 2 (a(u(s), u(s)) -a(u 0 , u 0 )) = s 0 a(u(t), u(t)) dt ∀ s ∈ [ 0, T ]. ( 3
(P h ), for v ∈ V we set v h = π h v, where π h v is the projection of v on V h defined by a(π h v, w h ) = a(v, w h ) ∀ w h ∈ V h . Thus if v ∈ L 2 (0, T ; V ) then π h v ∈ L 2 (0, T ; V h ) and π h v(t) → v(t) in V a.e. on [ 0, T ], which imply lim h→0 s 0 j(f (t), u h (t), π h v(t)) dt = s 0 j(f (t), u(t), v(t)) dt ∀ s ∈ [ 0, T ] and lim h→0 s 0 b(f (t), u h (t), π h v(t)) dt = s 0 b(f (t), u(t), v(t)) dt ∀ s ∈ [ 0, T ].
Since b(f (t), u h (t), uh (t)) = 0 a.e. on [ 0, T ], by integrating (P h ) over [ 0, s ] for v h = π h v and passing to the limit, we obtain that u satisfies (2.38). Now, we prove the strong convergence (3.11). Using the same argument in the proof of theorem 2.1, by taking v = 0, v = 2 u in (2.38), v h = 0, v h = 2 uh (t) in (P h ) and using (3.16), (3.17), we have that for all s ∈ [ 0, T ] Finally we prove that u satisfies (2.40). From (P h ) we deduce that for all t ∈ [ 0, T ]

lim inf h→0 s 0 a(u h (t), uh (t)) dt = s 0 a(u(t), u(t)) dt, (3.18
a(u h (t), v h -u h (t)) + j(f (t), u h (t), v h -u h (t)) ≥ 0 ∀ v h ∈ K h .
(3.20)

Let v ∈ K and, by (3.1), let v h ∈ K h be such that v h → v in V . By passing to the limit in (3.20) and using (3.11), (2.14) we obtain that u ∈ K(f ) and satisfies (2.40). Using theorems 3.1 and 3.2, we obtain the following main approximation result.

Theorem 3.3 Under the assumptions of theorem 3.2, there exists a subsequence of (u hn ) hn such that

u hn (t) → u(t) in V ∀ t ∈ [ 0, T ], (3.21) uhn ⇀ u in L 2 (0, T ; V ), (3.22)
where u ∈ W 1,2 (0, T ; V ) is a solution of (P ).

Applications to contact mechanics 4.1 Nonlocal friction laws and regularizations: a short overview

The classical formulation of the unilateral contact problem in linear elasticity, in the static frictionless case, was given by A. Signorini, who proved also the uniqueness of solution, and the existence of solutions was studied by G. Fichera, J.L. Lions and G. Stampacchia.

If the unilateral (or Signorini) contact conditions with local Coulomb's friction law are considered, some serious physical and mathematical difficulties appear. From the physical point of view, the application of Coulomb's law to continuum mechanics is not obvious, see, e.g., [START_REF] Oden | Contact problems in elastostatics with nonlocal friction laws[END_REF]- [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF] and references therein. Also, the static formulation is not realistic even if it can be useful in solving some intermediate problems like, for example, the incremental ones. Hence the evolution problem, in the quasistatic or the dynamic case, should be considered. From the mathematical point of view, as the normal component of the stress vector on the contact boundary is generally defined in the dual space H -1 2 , this lack of smoothness does not guarantee good convergence properties for the friction functional that appears in the variational formulation of the problem. Thus, there were physical and mathematical arguments for considering nonlocal friction laws by using appropriate regularizations of the pointwise Coulomb friction conditions that preserve the multivalued character of the unilateral contact conditions and of the friction law.

In elastostatics, for a class of frictional contact problems, G. Duvaut [START_REF] Duvaut | Equilibre d'un solide élastique avec contact unilatéral et frottement de Coulomb[END_REF], L. Demkowicz, J.T. Oden [START_REF] Demkowicz | On some existence and uniqueness results in contact problems with nonlocal friction[END_REF] and M. Cocou [START_REF] Cocou | Existence of solutions of Signorini problems with friction[END_REF] considered a mollification of the normal stresses on the contact boundary that enable to prove the existence, and uniqueness if the coefficient of friction is sufficiently small, of the variational solutions.

For the quasistatic unilateral contact problem, the same regularization technique was extended by M. Cocou et al. [START_REF] Cocou | Formulation and approximation of quasistatic frictional contact[END_REF] in order to prove, for a sufficiently small coefficient of friction, the existence of a solution. In that case, the simultaneous presence of the displacement and of the velocity fields is necessary to describe both the Signorini conditions and the physically consistent friction law. As was shown by P. Ballard [START_REF] Ballard | A counter-example to uniqueness in quasistatic elastic contact problems with small friction[END_REF], the uniqueness of the quasistatic solution does not hold, in general, even for an arbitrarily small coefficient of friction.

Another classical approach consisted in a regularization of the unilateral contact conditions by using a normal compliance model, see [START_REF] Oden | Models and computational methods for dynamic friction phenomena[END_REF], [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF] for its description and [START_REF] Andersson | A quasistatic frictional problem with normal compliance[END_REF], [START_REF] Han | Quasistatic contact problems in viscoelasticity and viscoplasticity[END_REF] for existence results in the quasistatic case. The contact law is there described by a single-valued mapping instead a multivalued one and small penetrations are allowed.

A quasistatic contact problem with nonlocal friction

Let us consider a linearly elastic body occupying a bounded domain Ω ⊂ R d , d = 2, 3 with a Lipschitz boundary Γ = Γ 1 ∪ Γ 2 ∪ Γ 3 , where Γ 1 , Γ 2 , Γ 3 are open and disjoint parts of Γ with meas(Γ 1 ) > 0.

We assume that in Ω a body force ϕ 1 is prescribed, on Γ 1 the displacement vector equals zero, a traction ϕ 2 is applied on Γ 2 and that the inertial effects are negligible.

The body is in contact with a rigid fixed support on Γ 3 and the displacements on this part of the boundary satisfy Signorini's contact conditions and are subjected to nonlocal friction when contact occurs.

Let us denote by σ the stress tensor, with the components σ = (σ ij ) , by ε the infinitesimal strain tensor, with the components ε = (ε ij ) , by u the displacement field, with the components u = (u i ) , and by E the elasticity tensor, with the components E = (a ijkl ) satisfying the usual properties of symmetry and ellipticity. where • , • denotes the duality pairing on ([H

1 2 (Γ 3 )] d ) ′ × [H 1 2 (Γ 3 )] d , w ∈ V 0
satisfies w = w a.e. on Γ 3 , and we define the normal component of the stress vector σ N (v) ∈ (H

1 2 (Γ 3 )) ′ by ∀ w ∈ H 1 2 (Γ 3 ) σ N (v), w = Ω σ(v) • ε( w)dx -(L, w), (4.2)
where • , • denotes also the duality pairing on (H

1 2 (Γ 3 )) ′ × H 1 2 (Γ 3 ), w ∈ V 0 satisfies wT = 0 a.e. on Γ 3 , wN = w a.e. on Γ 3 .
It is easy to verify that for all v ∈ S L the definitions of σ(v)n and of σ N (v) do not depend on the choices of w having the above properties, respectively.

For all L ∈ V 0 we introduce the functional

J L : S L × V 0 → R by J L (v, w) = Γ 3 µ|Rσ N (v)||w T |ds ∀ v ∈ S L , w ∈ V 0 , (4.3)
where R : (H

1 2 (Γ 3 )) ′ → L 2 (Γ 3
) is a linear and compact mapping. Now, let L ∈ V 0 be given by the relation

(L, v) = (ϕ 1 , v) [L 2 (Ω)] d + (ϕ 2 , v) [L 2 (Γ 2 )] d ∀ v ∈ V 0 (4.4)
and let u 0 ∈ K 0 satisfying the following compatibility condition:

Ω σ(u 0 ) • ε(w -u 0 )dx + J L(0) (u 0 , w) -J L(0) (u 0 , u 0 ) ≥ (L(0), w -u 0 ) ∀ w ∈ K 0 . (4.5)
A primal variational formulation of P c is as follows, see, e.g., [START_REF] Cocou | Formulation and approximation of quasistatic frictional contact[END_REF].

Problem P 0 : Find u ∈ W 1,2 (0, T ; V 0 ) such that (P 0 )                u(0) = u 0 , u(t) ∈ K 0 ∀ t ∈] 0, T [, Ω σ(u(t)) • ε(v -u(t))dx + J L(t) (u(t), v) -J L(t) (u(t), u(t)) ≥ (L(t), v -u(t)) + σ N (u(t)), v N -uN (t) ∀ v ∈ V 0 a.e. on ] 0, T [, σ N (u(t)), z N -u N (t) ≥ 0 ∀ z ∈ K 0 , ∀ t ∈] 0, T [. Let us define a : V 0 × V 0 → R by a(v, w) = Ω a ijkl ε ij (v)ε kl (w)dx = Ω σ(v) • ε(w)dx ∀ v, w ∈ V 0 . (4.6)
The bilinear form a(• , •) is continuous on V 0 × V 0 and since meas(Γ 1 ) > 0, by Korn's inequality is also V 0 -elliptic i.e. satisfies

∃ M > 0 such that |a(v, w)| ≤ M v w ∀ v, w ∈ V 0 , ∃ α > 0 such that a(v, v) ≥ α v 2 ∀ v ∈ V 0 ,
where

• = • [H 1 (Ω)] d . Let G 1 , G 2 ∈ V 0 and v 1 , v 2 be such that v 1 ∈ S G 1 , v 2 ∈ S G 2 .
Then from the properties of σ N , R and a it follows that the mapping J has the following property: ∃ C, C ′ > 0 such that

|J G 1 (v 1 , w 2 ) + J G 2 (v 2 , w 1 ) -J G 1 (v 1 , w 1 ) -J G 2 (v 2 , w 2 )| ≤ C μ Γ 3 |Rσ N (v 1 ) -Rσ N (v 2 )||w 1 -w 2 |ds ≤ C ′ μ( G 1 -G 2 + M v 1 -v 2 ) w 1 -w 2 (4.7) for all G i , w i ∈ V 0 , v i ∈ S G i , i = 1, 2, where μ = µ L ∞ (Γ 3 ) .
An incremental formulation can be written by using a time discretization of (P 0 ) as in Section 2. Therefore we obtain the following sequence of incremental problems (P i 0,n ) i=0,1,...,n-1 . Problem P i 0,n : Find u i+1 ∈ K 0 such that

(P i 0,n )        a(u i+1 , v -∂u i ) + J L i+1 (u i+1 , v) -J L i+1 (u i+1 , ∂u i ) ≥ (L i+1 , v -∂u i ) + σ N (u i+1 ), v N -∂u i N ∀ v ∈ V 0 , σ N (u i+1 ), z N -u i+1 N ≥ 0 ∀ z ∈ K 0 .
Let us now use similar notations to (2.23). Then u n ∈ L 2 (0, T ; V 0 ) and ûn ∈ W 1,2 (0, T ; V 0 ) satisfy the following incremental problem:

(P 0,n )                    a(u n (t), v - d dt ûn (t)) + J Ln(t) (u n (t), v) -J Ln(t) (u n (t), d dt ûn (t)) ≥ (L n (t), v - d dt ûn (t)) + σ N (u n (t)), v N - d dt ûnN (t) ∀ v ∈ V 0 , ∀ t ∈ [ 0, T ], σ N (u n (t)), z N -u nN (t) ≥ 0 ∀ z ∈ K 0 , ∀ t ∈ [ 0, T ].
We have the following existence and approximation result.

Theorem 4.1 Under the above assumptions and if μ < α C ′ there exists a subsequence

(u np ) p of (u n ) n such that u np (t) → u(t) in V 0 ∀ t ∈ [ 0, T ], ûnp → u in L 2 (0, T ; V 0 ) and d dt ûnp ⇀ u in L 2 (0, T ; V 0 ), as p → ∞, where
u is a solution of (P 0 ).

Proof.

Taking V = V 0 , K = K 0 , H = L 2 (Γ 3 ) and j(L, v, w) = J L (v, w) -(L, w), b(L, v, w) = σ N (v), w N , K(L) = K 0 ∩ S L , β(L, v) = µ|Rσ N (v)| ∀ v ∈ S L , w ∈ V 0 ,
we see that (P 0 ) can be written in the form (P ) with f = L, where L is defined by (4.4).

Using the properties of J and Green's formula, it can be easily seen that the hypotheses of theorem 2.1 are satisfied and the theorem therefore follows.

We can also apply theorems 3.1, 3.2 and 3.3 by choosing an internal approximation as in [START_REF] Capatina-Radoslovescu | Internal approximation of quasivariational inequalities[END_REF].

Using a similar approach that will not be repeated here, one can study the existence and approximation of solutions to the quasistatic unilateral contact problem with nonlocal friction between two linearly elastic bodies.

Clearly, the results presented in Sections 2 and 3 can be equally applied to frictional contact with normal compliance or to bilateral contact problems, see, e.g., [START_REF] Han | Quasistatic contact problems in viscoelasticity and viscoplasticity[END_REF], [START_REF] Shillor | Models and analysis of quasistatic contact[END_REF] for various formulations of these problems. In particular, the set of admissible displacement fields K is a whole vector subspace and b = 0 in all these cases so that their variational formulation consists in a single evolution implicit inequality.

Algorithms and numerical examples 5.1 Description of the algorithm

Let (T h ) h be a set of regular triangulations of Ω such that T h = (Ω j ) j∈J h satisfies the relation Ω = ∪ j∈J h Ω j with Ω k ∩ Ω l = ∅ for all k, l ∈ J h , k = l.
We introduce the following sets:

V 0h := {v h ∈ [C(Ω)] d ; v h|Ω j ∈ [P 1 (Ω j )] d , ∀ j ∈ J h , v h = 0 on Γ 1 }, K 0h := {v h ∈ V 0h ; v hN ≤ 0 a.e. on Γ 3 }, S 0h := {τ h ∈ L 2 (Γ 3 ); τ h|Γ 3,j ∈ P 0 (Γ 3,j ), ∀ j ∈ J h such that Γ 3,j = ∅},
where P k (ω) denotes the space of polynomials of degree lower or equal to k on ω and Γ 3,j = Γ 3 ∩ Ω j .

When the classical finite element approximations (V 0h , K 0h ) of (V 0 , K 0 ) are used for the problem (P i 0,n ), it can be shown, by using a similar argument as in [START_REF] Cocou | Formulation and approximation of quasistatic frictional contact[END_REF] and [START_REF] Raous | Quasistatic Signorini problem with Coulomb friction and coupling to adhesion[END_REF], where the continuous case was considered, that the fully discrete problem can be set under the form of the following implicit variational inequality: find

u i+1 h ∈ K i+1 0h such that ( Qi hn ) a(u i+1 h , w h -u i+1 h ) + J L i+1 (u i+1 h , w h -u i h ) -J L i+1 (u i+1 h , u i+1 h -u i h ) ≥ (L i+1 , w h -u i+1 h ) ∀ w h ∈ K 0h .
In the following µ is supposed to be constant. We choose as a regularization mapping R the projection on the finite-dimensional space S 0h 0 for a given h 0 (see [START_REF] Licht | Remarks on a numerical method for unilateral contact including friction[END_REF]). So within finite element formulation the regularization can be considered as a natural consequence of the discretization. A fixed point method is then introduced on the sliding threshold and ( Qi hn ) is found to be equivalent to the following problem:

(P 1 inc )
at each time step, find the fixed point of the mapping

Φ : S 0h → S 0h defined by Φ(g) = -µRσ N (u i+1 g ) ∀ g ∈ S 0h ,
where u i+1 g is the solution of the problem

(P 2 inc )        for a given g ∈ S 0h , find u i+1 g ∈ K 0h such that a(u i+1 g , v h -u i+1 g ) + p g (v h -u i h ) -p g (u i+1 g -u i h ) ≥ (L i+1 , v h -u i+1 g ) ∀ v h ∈ K 0h with p g (w) := Γ 3 g|w T |ds ∀ w ∈ V 0 .
Now, (P 2 inc ) is equivalent to the following minimization problem under constraint:

(P opt )
For a given g ∈ S 0h , find u i+1 g ∈ K 0h such that

F (u i+1 g ) ≤ F (v h ) ∀ v h ∈ K 0h with F (v) := 1 2 a(v, v) + p g (v -u i h ) -(L i+1 , v) ∀ v ∈ V 0 .
This problem is very similar to a static problem apart from the fact that the known solution u i h of the previous step is now present in the friction term. The influence of the loading history, due to the velocity formulation of the friction condition, is characterized by this extra term. The convex set K 0h remains unchanged from one step to the next. A minimization method is used to solve (P opt ). We consider here the numerical solvers for two-dimensional problems. A Gauss-Seidel method with projection is robust and very easy to implement when dealing with the non differentiable part related to the friction term. The convergence of the algorithm is improved by using an Aitken acceleration procedure (see [START_REF] Raous | Quasistatic Signorini problem with Coulomb friction and coupling to adhesion[END_REF]) which is as efficient as an overelaxation method but does not require an optimum coefficient. Other methods can be found in [START_REF] Raous | Quasistatic Signorini problem with Coulomb friction and coupling to adhesion[END_REF]: mathematical programming (using a complementarity formulation of the problem), conjugate gradient method, etc. The Gauss-Seidel algorithm with Aitken acceleration has been used in the following computations illustrating the behaviour of the solution regarding the space and time discretizations. For three-dimensional problems, the conjugate gradient method with projection and preconditionning given in [START_REF] Raous | Quasistatic Signorini problem with Coulomb friction and coupling to adhesion[END_REF] can be used but the friction law needs to be regularized in that case.

A test example: a block sliding on a plane

The previous algorithm has been succesfully used on many problems, including industrial ones (like, e.g., pressure vessel, composite material, friction and squeal). Here, our aim is to stress the points concerning the convergence regarding the space and the time discretizations that can be observed numerically. The influence of the coefficient of friction is presented and the convergence of the fixed point method is studied. For that purpose, a very simple example has been chosen, the solution of which is sufficiently complex to illustrate this influence of the friction and of the fully discrete schemes on the accuracy of the numerical solution.

The example

This example is a part of a study on the instability phenomena which can be involved in friction problems [START_REF] Martins | Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction[END_REF]. A Polyurethane block (L = 80mm, h = 40mm, E = 5daN/m 2 , ν = 0.48) is firstly pressed on a rigid plane with a normal prescribed displacement ( u 0 N = -0.5 mm) on the top of the block, and a tangential displacement is then imposed on the plane at a constant velocity (see Figure 1 and Figure 2). A plane stress hypothesis is assumed here (the thickness is 9.6 mm). The block beeing elastic, the time variable can be considered as a loading parameter.

The solution

It is worth noting that the solution is strongly affected by the friction and, depending on the coefficient of friction, the shape and the regularity of solutions can be different. To illustrate the influence of the coefficient of friction on the convergence, the changes in the contact forces σ N and σ T /µ on the contact boundary, occurring when the prescribed displacement u 0 T of the plate increases, are shown in Figure 3 and Figure 4 for µ = 0.5 and µ = 1. It can be noted that at µ = 0.5, the sliding zone spreads out from the center of the zone, while at µ = 1 it spreads out from the right edge. In both cases, before complete total sliding occurs, the solution shows the existence of three zones: a separate part (where the forces are zero), a sliding one (where the plot of σ N overlaps that of σ T /µ) and a sticking one (where the plots of σ N and σ T /µ are separate).

Convergence of the algorithm

Fixed point method on the sliding limit The convergence of the fixed point method described by P 1 inc on the sliding threshold is generally observed after less than 10 iterations and it is not very sensitive regarding the mesh size. A diagonal process is used, which means that for the first iterations the prescribed accuracy of the Gauss- Seidel method is low (a rough solution is sufficient to get the contact force) and increase progressively up to ǫ = 5.10 -6 , a convenient convergence test value. This test value saves about 50% of Gauss-Seidel iterations.

Convergence of Gauss-Seidel algorithm with Aitken acceleration

The Aitken acceleration is an heuristic process based on considering 3 terms of the Projected Gauss-Seidel sequence and computing a new sequence based on an assumption of geometrical sequence property. It is combined with the Gauss-Seidel iterations in order to compute a good initial value to a regular Gauss-Seidel process, the convergence of which has been established on the unilateral contact problem with friction.

In that way, the number of iterations needed to treat one loading step and including the fixed point method on the sliding threshold g, is of the same order of magnitude as the number of iterations required when an overrelaxation method with projection is used. This avoids the determination of an optimal relaxation parameter which needs a lot of preliminary computations.

In order to illustrate these comments on the algorithm convergence, we shall present some numerical results. For our example, by considering a mesh with 4193 nodes, including 65 contact nodes, for the step t = 2 where both u 0 T and u 0 N are applied, we have the following results: -convergence on the sliding threshold is obtained within 8 iterations; -convergence of the Gauss-Seidel method with Aitken acceleration (including the previous fixed point iterations) is obtained after 654 iterations, while using an overrelaxation method, which requires a costly preliminary computation of the optimal overrelaxation parameter, the convergence is obtained within 621 iterations.

Effects of space and time discretizations

Various structured meshes were used here and the same P 1 elements (linear interpolation) were used in order to obtain an internal approximation of the convex K 0 . The characteristics of the various meshes are given in Table 1 and the mesh M 2 is given in Figure 5 whereas the other meshes can be easily deduced by cutting every triangle in 3 parts. The comparisons between the various solutions have been conducted on the values of the tangential displacement at the central node of the contact zone. In order to obtain a sliding condition for this node in every case, the solutions have been compared at u 0 T = -1mm when µ = 0.5, u 0 T = -2mm when µ = 1 and u 0 T = -3mm when µ = 1.5.

Mesh name

Effects of time discretization

When the loading is piecewise proportional and monotone, the step size can be theoretically as large as a whole interval. This is globally confirmed on Figure 5: The medium size mesh M 2 tables 2 and 3 in the present example including two intervals. When at least two and respectively three increments are used on each interval the error is less than 0.4 per cent. This accuracy is only slightly improved when a larger number of increments is used. This is not surprising since a more accurate solution is obtained when the intermediate solutions (those obtained before total sliding occurs) are well evaluated, but the difference is slight. That is of great importance for the computations: it allows that small time steps are not required on the intervals where the loading is piecewise proportional and monotone.

The computations presented in tables 2 and 3 were carried out on the mesh M 4 (561 nodes, 33 contact nodes). The value chosen as reference to estimate the error was the solution computed with 24 increments for the normal loading and 36 for the tangential one.

Effects of space discretization and dependency on the coefficient of friction

In Figure 6, the convergence with respect to the mesh size h is shown at various values of the coefficient of friction (µ = 0.5, 1, 1.5). The solution chosen as reference was that computed using the mesh M 7 (33153 nodes, 257 contact nodes). The plotted relative error is a mean value of the errors computed on the sliding displacements of several nodes located in the central zone for the various meshes. Meshes M 0 to M 5 (see Table 1) are considered in Figure 6 and L/h is the number of contact nodes minus one. It has to be noted that no significant influence of the magnitude of the coefficient of friction was observed. The differences observed between the convergence at µ = 0.5 and at µ = 1 or µ = 1.5 were mostly due to the difference between the morphologies of the solutions (see Figure 3 and Figure 4): sliding zone spreading from a center zone when the coefficient of friction is small (µ = 0.5) and from the left side when it is larger (µ = 1, µ = 1.5).

Figure 1 :Figure 2 :

 12 Figure 1: Geometry and prescribed displacements

Figure 3 :

 3 Figure 3: Changes in time in the contact forces σ N (solid line) and σ T /µ (broken line) at µ = 0.5 (sticking, sliding and separation zones)

Figure 4 :

 4 Figure 4: Changes in time in the contact forces σ N (solid line) and σ T /µ (broken line) at µ = 1 (sticking, sliding and separation zones)

5 Figure 6 :

 56 Figure 6: Error depending on the mesh size at µ = 0.5, 1, 1.5

  .[START_REF] Ballard | A counter-example to uniqueness in quasistatic elastic contact problems with small friction[END_REF] Following the argument in the proof of lemma 2.3, by (3.13) and(3.14) we have that for all s ∈ [ 0, T ]

	s	s
	lim inf h→0	j(f (t), u h (t), uh (t)) dt ≥	j(f (t), u(t), u(t)) dt.	(3.17)
	0	0

Table 1 :

 1 Total number of nodes Number of contact nodes Characteristics of the various meshes

	M 0	6	3
	M 1	15	5
	M 2	45	9
	M 3	153	17
	M 4	561	33
	M 5	2145	65
	M 6	8385	129
	M 7	33153	257

Table 2 :

 2 Effects of the number of increments (µ = 0.5)

	Increment number Relative error on
	normal tangential the central node
	1	1	2.1 %
	2	3	0.35 %
	4	6	0.28 %
	12	18	0.01 %

Table 3 :

 3 Effects of the number of increments (µ = 1)

	Increment number Relative error on
	normal tangential the central node
	1	1	1.5 %
	2	3	0.37 %
	4	6	0.28 %
	12	18	0.01%

Acknowledgements

The present research work was supported by a collaboration agreement between the C.N.R.S. and the Romanian Academy.

We shall use the classical decomposition into the normal and tangential components of the displacement vector and stress vector u = u N n + u T with u N = u • n, σn = σ N n + σ T with σ N = (σn) • n, where n is the outward normal unit vector to Γ with the components n = (n i ).

The classical formulation of the quasistatic problem is as follows. Problem P c : Find a displacement field u = u(x, t) which satisfies the initial condition u(0) = u 0 in Ω and for all t ∈] 0, T [, the following equations and boundary conditions:

where µ is the coefficient of friction and Rσ N is a regularization of the normal contact force.

In order to obtain a variational formulation for this problem, we adopt the following hypotheses:

We use the following notations:

For all L ∈ V 0 and v ∈ S L we define the stress vector σ(v)n ∈ ([H


Marius Cocou

Mathieu Schryve

Michel Raous

A dynamic unilateral contact problem with adhesion and friction in viscoelasticity

Keywords: Mathematics Subject Classification (2000). 35K85, 35R35, 49J40, 74A55, 74D05, 74H20 Unilateral contact, adhesion, healing, friction, dynamic problems, viscoelasticity

come

A dynamic unilateral contact problem with adhesion and friction in viscoelasticity

Marius Cocou, Mathieu Schryve, Michel

Introduction

In this paper we consider an interaction law including dynamic unilateral contact, recoverable adhesion and nonlocal friction between two viscoelastic bodies. The adhesion is characterized by the intensity of adhesion, first introduced by M. Frémond (see, e.g., [START_REF] Frémond | Adhérence des solides[END_REF][START_REF] Frémond | Contact with adhesion. Nonsmooth Mechanics and Applications[END_REF]). An interface law for a quasistatic evolution where rebonding is not allowed was originally proposed in [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF] in the framework of continuum thermodynamics. The corresponding quasistatic problem coupling unilateral contact, adhesion and local friction for an elastic body has been analysed in [START_REF] Cocou | Existence results for unilateral quasistatic contact problems with friction and adhesion[END_REF]. This interface law has been used on various applications : fibre-matrix interaction in the context of composite materials [START_REF] Raous | Unilateral contact, friction and adhesion: 3D cracks in composite materials[END_REF], steel-concrete interaction [START_REF] Raous | Model coupling friction and adhesion for steel-concrete interfaces[END_REF] and, more recently, tectonic plates interaction in the context of earthquakes.

The extension to reversible adhesion proposed in the present paper constitutes an approximation of interactions of van der Waals type for rubberglass contact and other specific phenomena as those studied by M. Barquins [START_REF] Barquins | Sliding friction of rubber and Schallamach waves -A review[END_REF] and K.L. Johnson, K. Kendall, A.D. Roberts [START_REF] Johnson | Mechanics of adhesion[END_REF][START_REF] Johnson | Surface energy and the contact of elastic solids[END_REF]. This model, which extends the one considered in [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF][START_REF] Raous | Unilateral contact, friction and adhesion: 3D cracks in composite materials[END_REF], is dedicated to the description of bond damage and can be called a healing model because it allows us to describe both the formation and the rupture of adhesive contacts during the approach of the bodies. Also, in this model bond damage and healing behaviours are rate-dependent. The dissipation effects are related to both viscoelasticity and adhesion.

Dynamic frictional contact problems with normal compliance laws for a viscoelastic body have been considered by J.A.C. Martins and J.T. Oden [START_REF] Martins | Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws[END_REF], K.L. Kuttler [START_REF] Kuttler | Dynamic friction contact problems for general normal and friction laws[END_REF], O. Chau, W. Han and M. Sofonea [START_REF] Chau | A dynamic frictional contact problem with normal damped response[END_REF]. Dynamic unilateral or bilateral contact problems with friction for viscoelastic bodies have been studied in [START_REF] Eck | Unilateral Contact Problems -Variational Methods and Existence Theorems[END_REF][START_REF] Kuttler | Dynamic contact with Signorini's condition and slip rate depending friction[END_REF][START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF] and dynamic frictionless problems with adhesion have been analysed in [START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF] and in the references therein.

In this work we consider a coupled dynamic problem combining reversible adhesion, friction and unilateral contact. In section 2, classical and variational formulations of the dynamic contact problem are presented. The variational formulation is given as an implicit variational inequality coupled with a parabolic variational inequality which describes the evolution of the intensity of adhesion. Also, classical and variational formulations of an auxiliary penalized problem are considered. In section 3, general existence and uniqueness results are proved. In section 4, these abstract results are used to prove the existence and the uniqueness of penalized solutions and the existence of solutions to unilateral contact problem. In section 5, some numerical examples are presented and discussed.

Classical and variational formulations

We consider two viscoelastic bodies, characterized by a Kelvin-Voigt constitutive law, which occupy the reference domains Ω α of R d , d = 2 or 3, with Lipschitz continuous boundaries Γ α = ∂Ω α , α = 1, 2. In this paper we assume the small deformation hypothesis and we use Cartesian coordinate representations.

Let Γ α U , Γ α F and Γ α C be three open disjoint sufficiently smooth parts of Γ α

such that Γ α = Γ α U ∪ Γ α F ∪ Γ α C
and, to simplify the estimates, meas(Γ α U ) > 0, α = 1, 2.

We denote by y α (x α , t) the position at time t ∈ [0, T ], where T > 0, of the material point represented by x α in the reference configuration, by u α (x α , t) := y α (x α , t)x α the displacement vector of x α at time t, with the Cartesian coordinates u α = (u α 1 , ..., u α d ) = (u α , u α d ). Let ε α , with the Cartesian coordinates ε α = (ε ij (u α )), and σ α , with the Cartesian coordinates σ α = σ α ij , be the infinitesimal strain tensor and the stress tensor, respectively, corresponding to Ω α , α = 1, 2.

Excepting in the last section, to simplify notations we assume that the displacement U α = 0 is prescribed on Γ α U × ]0, T [ , α = 1, 2, and that the densities of both bodies are equal to 1. Let f = (f 1 , f 2 ) and F = (F 1 , F 2 ) denote the given body forces in Ω 1 ∪Ω 2 and tractions on Γ 1 F ∪Γ 2 F , respectively. The initial displacements and velocities of the bodies are denoted by

u 0 = (u 1 0 , u 2 0 ), u 1 = (u 1 1 , u 2 1
). The usual summation convention will be used for i, j, k, l = 1, . . . , d.

We suppose that the solids can be in unilateral contact between the potential contact surfaces Γ 1 C and Γ 2 C . We assume also that the surfaces Γ 1

C

and Γ 2 C can be parametrized by two

C 1 functions, ϕ 1 , ϕ 2 , defined on an open subset Ξ of R d-1 such that ϕ 1 (ξ) -ϕ 2 (ξ) ≥ 0 ∀ ξ ∈ Ξ and each Γ α C is the graph of ϕ α on Ξ that is Γ α C = { (ξ, ϕ α (ξ)) ∈ R d ; ξ ∈ Ξ}, α = 1, 2. Let m α : Ξ → R d ,
with m 1 (ξ) := (∇ϕ 1 (ξ), -1), m 2 (ξ) := (-∇ϕ 2 (ξ), 1), ∀ ξ ∈ Ξ, be the outward normal to Γ α C , α = 1, 2. Since the displacements, their derivatives and the gap are assumed small, by using a similar method as the one considered by P. Boieri, F. Gastaldi and D. Kinderlehrer [START_REF] Boieri | Existence, uniqueness, and regularity results for the two-body contact problem[END_REF] (see also [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF]) we obtain the following contact condition at time t on the set Ξ:

0 ≤ ϕ 1 (ξ) -ϕ 2 (ξ) + u 1 d (ξ, ϕ α (ξ), t) -u 2 d (ξ, ϕ α (ξ), t) -∇ϕ 1 (ξ) • u 1 (ξ, ϕ 1 (ξ), t) + ∇ϕ 2 (ξ) • u 2 (ξ, ϕ 2 (ξ), t) ∀ ξ ∈ Ξ, or, using the definition of m 1 , m 2 , m 1 (ξ) • u 1 (ξ, ϕ 1 (ξ), t) + m 2 (ξ) • u 2 (ξ, ϕ 2 (ξ), t) ≤ ϕ 1 (ξ) -ϕ 2 (ξ) ∀ ξ ∈ Ξ. (2.1) Let n α := m α /|m α | denote the unit outward normal vector to Γ α C , α = 1, 2.
The initial normalized gap between the two contact surfaces is defined as

g 0 (ξ) := ϕ 1 (ξ) -ϕ 2 (ξ) 1 + |∇ϕ 1 (ξ)| 2 ∀ ξ ∈ Ξ.
We shall use the following notations for the normal and tangential components of a displacement field v α , α = 1, 2, of the relative displacement corresponding to v := (v 1 , v 2 ), by including the initial gap g 0 in the normal direction, and of the stress vector σ α n α on Γ α C :

v α := v α (ξ, t) = v α (ξ, ϕ α (ξ), t), v α N := v α N (ξ, t) = v α (ξ, ϕ α (ξ), t) • n α (ξ), v α T := v α T (ξ, t) = v α -v α N n α , [v N ] := [v N ](ξ, t) = v 1 N + v 2 N -g 0 , [v T ] := [v T ](ξ, t) = v 1 T -v 2 T , σ α N := σ α N (ξ, t) = (σ α n α ) • n α , σ α T := σ α T (ξ, t) = σ α n α -σ α N n α , (2.2)
for all ξ ∈ Ξ and for all t ∈ [0, T ]. We denote also by

g := -[u N ] = g 0 -u 1 N -u 2
N the gap corresponding to the solution u := (u 1 , u 2 ). Assuming that ∇ϕ 1 (ξ) ≃ ∇ϕ 2 (ξ), it follows that the unilateral contact condition (2.1) at time t can be written as

[u N ] (ξ, t) = -g(ξ, t) ≤ 0 ∀ ξ ∈ Ξ.
(2.3) Following M. Frémond [START_REF] Frémond | Adhérence des solides[END_REF][START_REF] Frémond | Contact with adhesion. Nonsmooth Mechanics and Applications[END_REF], we introduce the internal state variable β, which represents the intensity of adhesion (β = 1 means that the adhesion is total, β = 0 means that there is no adhesion and 0 < β < 1 is the case of partial adhesion). In the following, we will consider only isothermal evolutions.

Classical formulation

Let A α , B α denote two fourth-order tensors, the elasticity tensor and the viscosity tensor corresponding to Ω α , with the components A α = (A α ijkl ) and B α = (B α ijkl ), respectively. We assume that these components satisfy the following classical symmetry and ellipticity conditions:

C ijkl = C jikl = C klij ∈ W 1,∞ (R d ), ∀ i, j, k, l = 1, . . . , d, ∃ α C > 0 such that C ijkl τ ij τ kl ≥ α C τ ij τ ij ∀ τ = (τ ij ) verifying τ ij = τ ji , ∀ i, j = 1, . . . , d, where C ijkl = A α ijkl , C = A α or C ijkl = B α ijkl , C = B α ∀ i, j, k, l = 1, . . . , d, α = 1, 2.
We choose the following state variables (see [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF], [START_REF] Cocou | A variational analysis of a contact interaction problem in viscoelasticity[END_REF]): the infinitesimal strain tensor ε

= (ε 1 , ε 2 ) = (ε(u 1 ), ε(u 2 )) in Ω 1 ∪ Ω 2 , the normal relative displacement [u N ] = u 1 N +u 2 N -g 0 , the tangential relative displacement [u T ] = u 1
Tu 2 T , and the intensity of adhesion β in Ξ. We assume that σ 1 n 1 = -σ 2 n 2 in Ξ, that the normal behaviour is purely elastic for a fixed value of β and that the only dissipative processes on the potential contact surfaces are adhesion and friction.

We define ϑ : R → R a truncation operator as ϑ(s) = -r if s ≤ -r , ϑ(s) = s if |s| < r and ϑ(s) = r if s ≥ r , where r > 0 is a given characteristic length (see, e.g., [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF][START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF]).

We consider the following classical formulation of the dynamic problem coupling adhesion, nonlocal friction and unilateral contact.

Problem P c : Find u = (u 1 , u 2 ) and β such that u(0) = u 0 , u(0) = u 1 , β(0) = β 0 and üα -div σ α (u α , uα ) = f α in Ω α × ]0, T [ , (2.4
)

σ α (u α , uα ) = A α ε(u α ) + B α ε( uα ) in Ω α × ]0, T [ , (2.5
)

u α = 0 on Γ α U × ]0, T [ , σ α n α = F α on Γ α F × ]0, T [ , α = 1, 2, (2.6
)

σ 1 n 1 + σ 2 n 2 = 0 in Ξ × ]0, T [ , (2.7)
[u N ] ≤ 0, σ N + C N [u N ] β 2 ≤ 0, (σ N + C N [u N ] β 2 ) [u N ] = 0 in Ξ × ]0, T [ , (2.8)
| σ T | ≤ µ | (Rσ) N + C N [u N ] β 2 | in Ξ × ]0, T [ and (2.9) | σ T | < µ | (Rσ) N + C N [u N ] β 2 | ⇒ [ uT ] = 0, | σ T | = µ | (Rσ) N + C N [u N ] β 2 | ⇒ ∃ λ ≥ 0 , [ uT ] = -λ σ T , β ∈ [0, 1] in Ξ × ]0, T [ and (2.10) b β ≥ w if β = 0, b β = w -C N ϑ([u N ] 2 ) β if β ∈ ]0, 1[, b β ≤ w -C N ϑ([u N ] 2 ) if β = 1,
where

β 0 ∈ [0, 1] in Ξ, C N > 0, b > 0, w > 0, σ α = σ α (u α , uα ), α = 1, 2, σ N := σ 1 N , σ T := σ 1 T , σ := σ 1
, µ is the coefficient of friction and R is a regularization with good approximation properties which will be presented later.

Note that the healing (rebonding) process is allowed. In the particular case when β = 0, that is the adhesion is totally broken, the classical Signorini's conditions with nonlocal friction are obtained.

Variational formulation

We shall adopt the following notations for some Sobolev spaces and corresponding duality pairings:

H s := [H s (Ω 1 )] d × [H s (Ω 2 )] d ∀ s ∈ R, v, w -s,s = v 1 , w 1 H -s (Ω 1 ),H s (Ω 1 ) + v 2 , w 2
H -s (Ω 2 ),H s (Ω 2 ) ∀ v = (v 1 , v 2 ) ∈ H -s , ∀ w = (w 1 , w 2 ) ∈ H s .
We define the Hilbert spaces (H, |.|) with the associated scalar product denoted by (. , .), (V , . ) with the associated scalar product (of H 1 ) denoted by . , . and the sets K, Λ as follows:

H := H 0 = [L 2 (Ω 1 )] d × [L 2 (Ω 2 )] d , V = V 1 × V 2 ,
where

V α = {v α ∈ [H 1 (Ω α )] d ; v α = 0 a.e. on Γ α U }, α = 1, 2, K = {v = (v 1 , v 2 ) ∈ V ; [v N ] ≤ 0 a.e. in Ξ}, Λ = {λ ∈ L 2 (Ξ) ; λ ∈ [0, 1] a.e. in Ξ}. We assume that F = (F 1 , F 2 ) ∈ W 1,∞ (0, T ; [L 2 (Γ 1 F )] d )×W 1,∞ (0, T ; [L 2 (Γ 2 F )] d ), f = (f 1 , f 2 ) ∈ W 1,∞ (0, T ; [L 2 (Ω 1 )] d ) × W 1,∞ (0, T ; [L 2 (Ω 2 )] d ), u 0 ∈ K, u 1 ∈ V , β 0 ∈ Λ, µ ∈ L ∞ (Ξ), µ ≥ 0 a.e. in Ξ.
Let us define two bilinear, continuous and symmetric mappings a, b on

H 1 × H 1 → R by a(v, w) = a 1 (v 1 , w 1 ) + a 2 (v 2 , w 2 ), b(v, w) = b 1 (v 1 , w 1 ) + b 2 (v 2 , w 2 ) ∀ v = (v 1 , v 2 ), w = (w 1 , w 2 ) ∈ H 1 , where, for α = 1, 2, a α (v α , w α ) = Ω α A α ε(v α ) • ε(w α ) dx, b α (v α , w α ) = Ω α B α ε(v α ) • ε(w α ) dx.
Using the previous hypotheses, we consider L as an element of W 1,∞ (0, T ;

H 1 ) such that ∀ t ∈ [0, T ] L, v = α=1,2 Ω α f α • v α dx + α=1,2 Γ α F F α • v α ds ∀ v = (v 1 , v 2 ) ∈ H 1 . We suppose that R : [L 2 sym (Ω 1 )] d 2 → [H 1 (Ω 1 )] d 2 is a linear and continuous regularization of σ(u 1 , v 1 ) = σ 1 (u 1 , v 1 ), satisfying (Rσ (u 1 0 , u 1 1 )) N = 0 and ∃ C > 0, Rσ (u 1 , v 1 ) [H 1 (Ω 1 )] d 2 ≤ C ( |u 1 | + |v 1 | ) ∀ u 1 , v 1 ∈ V 1 , (2.11)
where |.| denotes also the norm of [L 2 (Ω 1 )] d . A similar type of regularization can be found in [START_REF] Kuttler | Dynamic contact with Signorini's condition and slip rate depending friction[END_REF] and the same regularization was considered in [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF].

We define the following mappings:

c : L ∞ (Ξ) × (H 1 ) 2 → R, c(β, u, v) = Ξ C N ϑ([u N ]) β 2 (v 1 N + v 2 N ) dξ, J : L ∞ (Ξ) × (H 1 ) 3 → R, J(β, u, v, w) = Ξ µ | (Rσ (u 1 , v 1 )) N + C N ϑ([u N ]) β 2 | | [w T ] | dξ ∀ β ∈ L ∞ (Ξ), ∀ u = (u 1 , u 2 ), v = (v 1 , v 2 ), w = (w 1 , w 2 ) ∈ H 1 , γ : H 1 × [L 2 (Ξ)] 2 → R, γ(u, δ, λ) = Ξ C N b ϑ([u N ] 2 ) δ λ dξ ∀ u = (u 1 , u 2 ) ∈ H 1 , ∀ δ, λ ∈ L 2 (Ξ).
We denote by (. , .) Ξ the scalar product in L 2 (Ξ), with the associated norm |.| Ξ , and we set χ := w b . We assume also the following compatibility relation on initial conditions:

∃ l ∈ H such that (l, v) + a(u 0 , v) + b(u 1 , v) + c(β 0 , u 0 , v) = L(0), v ∀ v ∈ V . (2.12)
A variational formulation of the problem P c is the following.

Problem P v : Find u ∈ W 1,2 (0, T ; V ) ∩ C 1 ([0, T ]; H -1/2 ), β ∈ W 1,∞ (0, T ; L ∞ (Ξ)) such that u(0) = u 0 , u(0) = u 1 in Ω 1 ∪ Ω 2 , β(0) = β 0 in Ξ, u(t) ∈ K, β(t) ∈ Λ for all t ∈ ]0, T [ and u(T ), v(T ) -u(T ) -1/2, 1/2 -(u 1 , v(0) -u 0 ) - T 0 ( u, v -u) dt + T 0 {a(u, v -u) + b( u, v -u) + c(β, u, v -u)} dt (2.13) + T 0 {J(β, u, u, v + κ u -u) -J(β, u, u, κ u)} dt ≥ T 0 L, v -u dt ∀ v ∈ L ∞ (0, T ; V ) ∩ W 1,2 (0, T ; H) with v(t) ∈ K a.e. t ∈ ]0, T [, ( β, λ -β) Ξ + γ(u, β, λ -β) ≥ (χ, λ -β) Ξ (2.14) ∀ λ ∈ L 2 (0, T ; L 2 (Ξ)) with λ(t) ∈ Λ a.e. t ∈ ]0, T [,
where κ > 0.

The formal equivalence between the variational system (2.13),(2.14) and the classical problem (2.4)-(2.10) can be easily proved by using Green's formula and an integration by parts.

An auxiliary penalized problem

We consider a penalized contact problem that is a dynamic contact problem with normal compliance law, the solution of which is denoted by

u ε = (u 1 ε , u 2 ε ), β ε
, where ε > 0, verifying the same equations and initial conditions in Ω 1 ∪ Ω 2 , the same boundary conditions as in problem P c , excepting the unilateral contact conditions. The new contact conditions in Ξ × ]0, T [ are

σ 1 N = σ 2 N = - 1 ε [u εN ] + -C N [u εN ] β 2 ε
, where r + = max(r, 0),

σ 1 T (u 1 ε , u1 ε ) = -σ 2 T (u 2 ε , u2 ε ), | σ T | ≤ µ | (Rσ) N + C N [u εN ] β 2 ε | and | σ T | < µ | (Rσ) N + C N [u εN ] β 2 ε | ⇒ [ uεT ] = 0, | σ T | = µ | (Rσ) N + C N [u εN ] β 2 ε | ⇒ ∃ λ ≥ 0 , [ uεT ] = -λ σ T , β ε ∈ [0, 1] and b βε ≥ w if β ε = 0, b βε = w -C N ϑ([u εN ] 2 ) β ε if β ε ∈ ]0, 1[, b βε ≤ w -C N ϑ([u εN ] 2 ) if β ε = 1.

Let us define the mapping p

ε : V × V → R by p ε (v, w) = 1 ε Ξ ϑ([v N ] + )(w 1 N + w 2 N ) dξ ∀ v, w ∈ V . (2.15)
We shall study the following variational formulation of the penalized problem.

Problem P ε : Find u ε ∈ W 2,2 (0, T ; H) ∩ W 1,2 (0, T ; V ) and β ε ∈ W 1,∞ (0, T ; L ∞ (Ξ)) such that u ε (0) = u 0 , uε (0) = u 1 in Ω 1 ∪ Ω 2 , β ε (0) = β 0 in Ξ, β ε (s) ∈ Λ for all s ∈ ]0, T [, and a.e. t ∈ ]0, T [ (ü ε , w -uε ) + a(u ε , w -uε ) + b( uε , w -uε ) + p ε (u ε , w -uε ) (2.16) +c(β ε , u ε , w -uε ) + J(β ε , u ε , uε , w) -J(β ε , u ε , uε , uε ) ≥ L, w -uε ∀ w ∈ V , ( βε , λ -β ε ) Ξ + γ(u ε , β ε , λ -β ε ) ≥ (χ, λ -β ε ) Ξ ∀ λ ∈ Λ.
(2.17)

General existence and uniqueness results

The existence and uniqueness of solutions of problems P ε will be obtained by using the following abstract problem.

Let (H 0 , |.|), (V 0 , . ) and (Π 0 , |.| Π 0 ) be three Hilbert spaces with corresponding scalar products denoted by (. , .), . , . and (. , .) Π 0 , respectively, such that V 0 is dense in H 0 with compact imbedding from V 0 into H 0 and let Λ 0 be a closed convex set in Π 0 . We assume that 0 ∈ Λ 0 and also that Λ 0 is bounded, to simplify the estimates.

We define two bilinear and symmetric forms, a 0 , b 0 :

V 0 × V 0 → R and the mapping γ 0 : V 0 × Π 0 × Π 0 → R such that ∃ m a , m b > 0 a 0 (u, v) ≤ m a u v , b 0 (u, v) ≤ m b u v , (3.1)
∃ A, B > 0 a 0 (v, v) ≥ A v 2 , b 0 (v, v) ≥ B v 2 ∀ u, v ∈ V 0 , (3.2) ∀ u ∈ V 0 γ 0 (u, •, •) is a bilinear and symmetric form, (3.3) ∃ m γ > 0 such that ∀ u 1,2 ∈ V 0 , ∀ δ 1,2 ∈ Λ 0 , ∀ λ ∈ Π 0 , |γ 0 (u 1 , δ 1 , λ) -γ 0 (u 2 , δ 2 , λ)| ≤ m γ ( u 1 -u 2 + |δ 1 -δ 2 | Π 0 ) |λ| Π 0 , (3.4)
γ 0 (u, λ, λ) ≥ 0 ∀ u ∈ V 0 , ∀ λ ∈ Π 0 . (3.5)
Let φ 0 : [0, T ] × Λ 0 × V 3 0 → R and τ 0 : V 0 → R be two mappings satisfying φ 0 (t, λ, •, •, •) and τ 0 are sequentially weakly continuous, (3.6)

φ 0 (t, λ, u, v, w 1 + w 2 ) ≤ φ 0 (t, λ, u, v, w 1 ) + φ 0 (t, λ, u, v, w 2 ), (3.7)
φ 0 (t, λ, u, v, θw) = θ φ 0 (t, λ, u, v, w), (3.8)
φ 0 (0, 0, 0, 0, w) = 0, (3.9)

∃ η 0 > 0 such that |τ 0 (u)| ≤ η 0 u , (3.10)
∀ t ∈ [0, T ], ∀ λ ∈ Λ 0 , ∀ u, v, w, w 1,2 ∈ V 0 , ∀ θ ≥ 0, ∃ η 1 > 0 such that ∀ t 1,2 ∈ [0, T ], ∀ λ 1,2 ∈ Λ 0 , ∀ u 1,2 , v 1,2 , w ∈ V 0 , |φ 0 (t 1 , λ 1 , u 1 , v 1 , w) -φ 0 (t 2 , λ 2 , u 2 , v 2 , w)| ≤ η 1 (|t 1 -t 2 | + |λ 1 -λ 2 | Π 0 + |τ 0 (u 1 -u 2 )| + |v 1 -v 2 |) w , (3.11) ∃ η 2 > 0 such that ∀ t 1,2 ∈ [0, T ], ∀ λ 1,2 ∈ Λ 0 , ∀ u 1,2 , v 1,2 , w 1,2 ∈ V 0 , |φ 0 (t 1 , λ 1 , u 1 , v 1 , w 1 ) -φ 0 (t 1 , λ 1 , u 1 , v 1 , w 2 ) + φ 0 (t 2 , λ 2 , u 2 , v 2 , w 2 ) -φ 0 (t 2 , λ 2 , u 2 , v 2 , w 1 )| ≤ η 2 (|t 1 -t 2 | + |λ 1 -λ 2 | Π 0 + u 1 -u 2 + |v 1 -v 2 |) w 1 -w 2 .
(3.12)

We assume that

L 0 ∈ W 1,∞ (0, T ; V 0 ), χ 0 ∈ W 1,2 (0, T ; Π 0 ), u 0 , u 1 ∈ V 0 , β 0 ∈ Λ 0 and the following compatibility condition: ∃ l 0 ∈ H 0 such that ∀ w ∈ V 0 (l 0 , w) + a 0 (u 0 , w) + b 0 (u 1 , w) + φ 0 (0, β 0 , u 0 , u 1 , w) = L 0 (0), w . (3.13)
We consider the following problem.

Problem Q : Find u ∈ W 2,2 (0, T ; H 0 ) ∩ W 1,2 (0, T ; V 0 ), β ∈ W 1,∞ (0, T ; Π 0 ) such that u(0) = u 0 , u(0) = u 1 , β(0) = β 0 , β(s) ∈ Λ 0 for all s ∈ ]0, T [, and a.e. t ∈ ]0, T [ (ü, v -u) + a 0 (u, v -u) + b 0 ( u, v -u) (3.14) +φ 0 (t, β, u, u, v) -φ 0 (t, β, u, u, u) ≥ L 0 , v -u ∀ v ∈ V 0 , ( β, λ -β) Π 0 + γ 0 (u, β, λ -β) ≥ (χ 0 , λ -β) Π 0 ∀ λ ∈ Λ 0 . (3.15)
The existence and uniqueness of the solution for problem Q will be proved by using a result presented in [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF], an incremental technique and a fixed point argument.

We define the set

X = {λ ∈ C 0 ([0, T ]; Π 0 ) ; λ(0) = β 0 , λ(t) ∈ Λ 0 ∀ t ∈ ]0, T ]},
where the Banach space C 0 ([0, T ]; Π 0 ) is endowed with the norm

λ k = max t∈[0,T ] [exp(-kt) |λ(t)| Π 0 ] for all λ ∈ C 0 ([0, T ]; Π 0 ), k ≥ 0.
Using the Theorem 3.2 of [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF], which for every β ∈ X clearly can be applied with minor modifications to (3.14) with the initial conditions u 0 , u 1 , we have the following result.

Lemma 3.1. For each β ∈ X there exists a unique

u β ∈ W 2,2 (0, T ; H 0 ) ∩ W 1,2 (0, T ; V 0 ) solution of the inequality (3.14) such that u β (0) = u 0 , uβ (0) = u 1 .
Lemma 3.2. Let β 1 , β 2 ∈ X and let u β 1 , u β 2 be the corresponding solutions of (3.14) with the same initial conditions u 0 , u 1 , respectively. Then there exists a constant

C 1 > 0, independent of β 1 , β 2 , u β 1 , u β 2 , such that for all t ∈ [0, T ] | uβ 1 (t) -uβ 2 (t)| 2 + u β 1 (t) -u β 2 (t) 2 ≤ C 1 t 0 |β 1 (s) -β 2 (s)| 2 Π 0 ds. (3.16) Proof. Let u β 1 , u β 2 be the solutions of (3.14) corresponding to β 1 , β 2 ∈ X.
Taking in each inequality v = uβ 2 and v = uβ 1 , respectively, for a.e. s ∈ ]0, T [ we have

(ü β 1 -üβ 2 , uβ 1 -uβ 2 ) + a 0 (u β 1 -u β 2 , uβ 1 -uβ 2 ) + b 0 ( uβ 1 -uβ 2 , uβ 1 -uβ 2 ) ≤ φ 0 (s, β 1 , u β 1 , uβ 1 , uβ 2 ) -φ 0 (s, β 1 , u β 1 , uβ 1 , uβ 1 ) +φ 0 (s, β 2 , u β 2 , uβ 2 , uβ 1 ) -φ 0 (s, β 2 , u β 2 , uβ 2 , uβ 2 ) ≤ η 2 ( |β 1 -β 2 | Π 0 + u β 1 -u β 2 + | uβ 1 -uβ 2 | ) uβ 1 -uβ 2 ,
where the second inequality follows by (3.12). For all t ∈ [0, T ], as the solutions u β 1 , u β 2 verify the same initial conditions, by integrating between 0 and t we obtain

1 2 | uβ 1 (t) -uβ 2 (t)| 2 + 1 2 a 0 (u β 1 (t) -u β 2 (t), u β 1 (t) -u β 2 (t)) + t 0 b 0 ( uβ 1 -uβ 2 , uβ 1 -uβ 2 ) ds ≤ η 2 t 0 |β 1 -β 2 | Π 0 uβ 1 -uβ 2 ds +η 2 t 0 ( u β 1 -u β 2 uβ 1 -uβ 2 + | uβ 1 -uβ 2 | uβ 1 -uβ 2 ) ds.
Using Young's inequalities for the last three terms with an appropriate constant, and V 0 -ellipticity of a 0 , b 0 , it follows that

1 2 | uβ 1 (t) -uβ 2 (t)| 2 + A 2 u β 1 (t) -u β 2 (t) 2 + B 2 t 0 uβ 1 -uβ 2 2 ds ≤ 3η 2 2 2B t 0 |β 1 -β 2 | 2 Π 0 ds + 3η 2 2 2B t 0 ( u β 1 -u β 2 2 + | uβ 1 -uβ 2 | 2 ) ds.
By Gronwall's lemma we obtain the estimate (3.16). Now, for every element u ∈ W 1,2 (0, T ; V 0 ), we consider the inequality (3.15) with the initial condition β 0 , the solution of which is denoted by β u .

The existence and uniqueness results for this parabolic inequality follow by classical references, see, e.g., [START_REF] Brézis | Problèmes unilatéraux[END_REF] or [START_REF] Barbu | Semigroups and Differential Equations in Banach Spaces[END_REF], but we prefer to present a direct proof based on an incremental technique, for the convenience of the reader. Proposition 3.3. For each u ∈ W 1,2 (0, T ; V 0 ) there exists a unique β u ∈ X ∩ W 1,∞ (0, T ; Π 0 ), solution of the inequality (3.15).

Proof. We consider an incremental formulation obtained by using an implicit time discretization scheme for (3.15). For n ∈ N * , we set ∆t := T /n and t i := i ∆t, i = 0, 1, ..., n, and β 0 := β 0 . If ψ is a continuous function of t ∈ [0, T ] valued in some vector space, we use the notations ψ i := ψ(t i ) and if (θ i ) i∈{0,1,...,n} are elements of some vector space, then for all i ∈ {0, 1, ..., n -1} we set ∆θ i := θ i+1 -θ i . We then approximate (3.15) using the sequence of following incremental problems: for i = 0, 1, ..., n -1, find

β i+1 ∈ Λ 0 such that ∆β i ∆t , λ -β i+1 Π 0 + γ 0 (u i+1 , β i+1 , λ -β i+1 ) ≥ (χ i+1 0 , λ -β i+1 ) Π 0 ∀ λ ∈ Λ 0 .
(3.17) As the previous inequality can be written in the equivalent form

(β i+1 , λ -β i+1 ) Π 0 + ∆t γ 0 (u i+1 , β i+1 , λ -β i+1 ) ≥ (∆t χ i+1 0 + β i , λ -β i+1 ) Π 0 ∀ λ ∈ Λ 0 ,
which is an elliptic variational inequality of the first kind that contains the scalar product in Π 0 and γ 0 satisfying (3.3)-(3.5), by a classical result it follows that there exists a unique solution β i+1 of (3.17).

Taking in (3.17) λ = β i , using (3.4) and Cauchy-Schwarz inequality, we have

∆β i ∆t , ∆β i Π 0 ≤ -γ 0 (u i+1 , β i+1 , ∆β i ) + (χ i+1 0 , ∆β i ) Π 0 ≤ m γ ( u i+1 + |β i+1 | Π 0 )|∆β i | Π 0 + |χ i+1 0 | Π 0 |∆β i | Π 0 . Hence, |∆β i | Π 0 ∆t ≤ m γ ( u i+1 + |β i+1 | Π 0 ) + |χ i+1 0 | Π 0 , for i = 0, 1, ..., n -1. (3.
18) Let us define the following functions: and the estimates (3.18), i = 0, 1, ..., n -1, imply the following relation:

β n (0) = βn (0) = β 0 , u n (0) = u 0 , χ n (0) = χ 0 0 and ∀ i ∈ {0, 1, ..., n -1}, ∀ t ∈ ]t i , t i+1 ], β n (t) = β i+1 , βn (t) = β i + (t -t i ) ∆β i ∆t , u n (t) = u i+1 , χ n (t) = χ i+1 0 . (3.19) Then β n ∈ L 2 (0, T ; Π 0 ), βn ∈ W 1,2 (0, T ; Π 0 ), u n → u in L 2 (0, T ; V 0 ), χ n → χ 0 in L 2 (0, T ; Π 0 ) and β n (t) ∈ Λ 0 , u n (t) → u(t) in V 0 , χ n (t) → χ 0 (t) in Π 0 ∀ t ∈ [0, T ]. Since u ∈ W 1,2 (0, T ; V 0 ) ⊂ C 0 ([0, T ]; V 0 ), χ 0 ∈ W 1,2 (0, T ; Π 0 ) ⊂ C 0 ([0, T ]; Π 0 ),
d dt βn (t) Π 0 ≤ m γ ( u n (t) +|β n (t)| Π 0 )+|χ n (t)| Π 0 for a.e. t ∈ ]0, T [. (3.21)
Thus, there exists a constant C 2 > 0 satisfying βn W 1,∞ (0,T ;Π 0 ) ≤ C 2 for all n ∈ N * , so that there exists a subsequence, still denoted by ( βn ) n , and an element

β u ∈ W 1,∞ (0, T ; Π 0 ) such that βn ⇀ * β u in W 1,∞ (0, T ; Π 0 ), (3.22)
and, by (3.20),

β n ⇀ * β u in L ∞ (0, T ; Π 0 ). (3.23)
Applying a diagonal process, see, e.g., [START_REF] Cocou | Formulation and approximation of quasistatic frictional contact[END_REF], it follows from (3.22), (3.20) that, up to a subsequence,

β n (t) ⇀ β u (t) in Π 0 ∀ t ∈ [0, T ], (3.24)
which implies that β u (0) = β 0 and β u (t) ∈ Λ 0 for all t ∈ ]0, T [. We shall prove that the limit β u is a solution of the inequality (3.15). The sequence of inequalities (3.17), i = 0, 1, ..., n -1, is equivalent to the following incremental formulation: β n (t) ∈ Λ 0 for all t ∈ [0, T ] and

d dt βn (t), λ -β n (t) Π 0 + γ 0 (u n (t), β n (t), λ -β n (t)) (3.25) ≥ (χ n (t), λ -β n (t)) Π 0 ∀ λ ∈ Λ 0 , for a.e. t ∈ ]0, T [.
Integrating both sides of (3.25) over [0, T ] it follows that for all λ ∈ L 2 (0, T ; Π 0 ) such that λ(t) ∈ Λ 0 for a.e. t ∈ ]0, T [,

T 0 d dt βn (t), λ(t) -β n (t) Π 0 dt + T 0 γ 0 (u n (t), β n (t), λ(t) -β n (t)) dt (3.26) ≥ T 0 (χ n (t), λ(t) -β n (t)) Π 0 dt.
Using the properties of the corresponding sequences, of the scalar product and of γ 0 , we have

T 0 d dt βn (t), β n (t) Π 0 dt ≥ 1 2 [(β n (T ), β n (T )) Π 0 -(β 0 , β 0 ) Π 0 ], lim n→∞ T 0 [γ 0 (u n (t), β n (t), β n (t)) -γ 0 (u(t), β n (t), β n (t))] dt = 0.
Therefore, by passing to the limit we obtain

Π 0 dt = T 0 ( βu (t), λ(t)) Π 0 dt, lim n→∞ T 0 γ 0 (u n (t), β n (t), λ(t)) dt = lim n→∞ T 0 [γ 0 (u n (t), β n (t), λ(t)) -γ 0 (u(t), β n (t), λ(t))] dt + lim n→∞ T 0 γ 0 (u(t), β n (t), λ(t)) dt = T 0 γ 0 (u(t), β u (t), λ(t)) dt, lim n→∞ T 0 (χ n (t), λ(t) -β n (t)) Π 0 dt = T 0 (χ 0 (t), λ(t) -β u (t)) Π 0 dt,
finally by passing to the limit in (3.26), we obtain that for all λ ∈ L 2 (0, T ; Π 0 ) such that λ(t) ∈ Λ 0 for a.e.

t ∈ ]0, T [ T 0 ( βu (t), λ(t) -β u (t)) Π 0 dt + T 0 γ 0 (u(t), β u (t), λ(t) -β u (t)) dt ≥ T 0 (χ 0 (t), λ(t) -β u (t)) Π 0 dt.
By Lebesgue's theorem, it follows that β u is a solution of the parabolic inequality (3.15). In order to show the uniqueness of β u , let β 1 , β 2 be two solutions of (3.15) corresponding to u ∈ W 1,2 (0, T ; V 0 ). Taking in each inequality λ = β 2 and λ = β 1 , respectively, we derive that for a.e. s ∈ ]0, T [

( β1 (s) -β2 (s), β 1 (s) -β 2 (s)) Π 0 + γ 0 (u(s), β 1 (s) -β 2 (s), β 1 (s) -β 2 (s)) ≤ 0.
Using that β 1 , β 2 satisfy the same initial condition, for all t ∈ ]0, T [ by integrating over [0, t] we have

1 2 |β 1 (t) -β 2 (t)| 2 Π 0 + t 0 γ 0 (u(s), β 1 (s) -β 2 (s), β 1 (s) -β 2 (s)) ds ≤ 0, which implies that β 1 = β 2 .
Lemma 3.4. Let u 1 , u 2 ∈ W 1,2 (0, T ; V 0 ) and let β u 1 , β u 2 ∈ X be the corresponding solutions of (3.15) with the same initial condition β 0 , respectively. Then there exists a constant C 3 > 0, independent of u 1 , u 2 , β u 1 , β u 2 , such that for all t ∈ [0, T ]

|β u 1 (t) -β u 2 (t)| 2 Π 0 ≤ C 3 t 0 u 1 (s) -u 2 (s) 2 ds. (3.27) Proof. Let β u 1 , β u 2 be the solutions of (3.15) corresponding to u 1 , u 2 . Tak- ing in each inequality λ = β u 2 , λ = β u 1 , respectively, for all t ∈ ]0, T [, by
integrating over [0, t], using (3.4) and some elementary inequality we have

1 2 |β u 1 (t) -β u 2 (t)| 2 Π 0 ≤ t 0 [γ 0 (u 2 , β u 2 , β u 1 -β u 2 ) -γ 0 (u 1 , β u 1 , β u 1 -β u 2 )] ds = t 0 [γ 0 (u 2 , β u 2 , β u 1 -β u 2 ) -γ 0 (u 2 , β u 1 , β u 1 -β u 2 )] ds + t 0 [γ 0 (u 2 , β u 1 , β u 1 -β u 2 ) -γ 0 (u 1 , β u 1 , β u 1 -β u 2 )] ds ≤ m γ t 0 |β u 1 -β u 2 | 2 Π 0 ds + m γ t 0 u 1 -u 2 |β u 1 -β u 2 | Π 0 ds ≤ m γ 2 t 0 u 1 (s) -u 2 (s) 2 ds + 3 m γ 2 t 0 |β u 1 (s) -β u 2 (s)| 2 Π 0 ds,
where, to simplify, the variable s was omitted in some relations. By Gronwall's lemma we obtain the estimate (3.27).

Now we can prove the following existence and uniqueness result. Proof. For every β ∈ X let u β ∈ W 2,2 (0, T ; H 0 ) ∩ W 1,2 (0, T ; V 0 ) be the solution of the inequality (3.14) corresponding to β such that u β (0) = u 0 , uβ (0) = u 1 and let β u β ∈ X ∩ W 1,∞ (0, T ; Π 0 ) be the solution of the inequality (3.15) corresponding to u β . We define the mapping T : X → X by ∀ β ∈ X T β = β u β and we will prove that T : X → X has a unique fixed point, which is equally the solution of the problem Q. For all β 1 , β 2 ∈ X, for all t ∈ [0, T ], using (3.27) and (3.16), we have

|T β 1 (t) -T β 2 (t)| 2 Π 0 ≤ C 3 t 0 u β 1 (s) -u β 2 (s) 2 ds ≤ C 1 C 3 t 0 s 0 exp(-2kr) exp(2kr) |β 1 (r) -β 2 (r)| 2 Π 0 dr ds ≤ C 1 C 3 β 1 -β 2 2 k t 0 exp(2ks) 2k ds ≤ C 1 C 3 4k 2 • exp(2kt) β 1 -β 2 2 k .
Then

T β 1 -T β 2 k = max t∈[0,T ] [exp(-kt) |T β 1 (t) -T β 2 (t)| Π 0 ] ≤ √ C 1 C 3 2k β 1 -β 2 k .
Hence, for all

β 1 , β 2 ∈ X T β 1 -T β 2 k ≤ √ C 1 C 3 2k β 1 -β 2 k ,
so that if k is sufficiently large it follows that T is a contraction and its fixed point is the solution of the problem Q.

Approximation and existence of variational solutions

Now, the previous general results are applied to analyse the penalized and the unilateral contact problems.

Existence and uniqueness of penalized solutions

We prove the following existence and uniqueness result for the penalized problem.

Theorem 4.1. There exists a unique solution to the problem P ε .

Proof. We apply Theorem 3.5 to

H 0 = H, V 0 = V , Π 0 = L 2 (Ξ), Λ 0 = Λ, u 0 = u 0 , u 1 = u 1 , a 0 = a, b 0 = b, L 0 = L, (. , .) Π 0 = (. , .) Ξ , |.| Π 0 = |.| Ξ , γ 0 = γ, χ 0 = χ and φ 0 (t, λ, u, v, w) = p ε (u, w) + c(λ, u, w) + J(λ, u, v, w) ∀ t ∈ [0, T ], ∀ λ ∈ L 2 (Ξ), ∀ u, v, w ∈ V .
As meas(Γ α U ) > 0, the ellipticity property of the coefficients A α ijkl , B α ijkl and the Korn's inequality imply that there exist A α , B α > 0 such that

a α (v α , v α ) ≥ A α v α 2 V α , b α (v α , v α ) ≥ B α v α 2 V α ∀ v α ∈ V α , α = 1, 2,
and we obtain

a(v, v) ≥ A v 2 , b(v, v) ≥ B v 2 ∀ v ∈ V , (4.1)
where

A = min(A 1 , A 2 ), B = min(B 1 , B 2 ). For all u 1,2 ∈ V , δ 1,2 ∈ Λ, λ ∈ L 2 (Ξ)
, we have the following relations:

|γ(u 1 , δ 1 , λ) -γ(u 2 , δ 2 , λ)| = C N b Ξ (ϑ([u 1N ] 2 ) δ 1 -ϑ([u 2N ] 2 ) δ 2 ) λ dξ = C N b Ξ [(ϑ([u 1N ]) 2 -ϑ([u 2N ]) 2 ) δ 1 + ϑ([u 2N ] 2 )(δ 1 -δ 2 )] λ dξ ≤ C N b Ξ [2r|ϑ([u 1N ]) -ϑ([u 2N ])| + r 2 |δ 1 -δ 2 |] |λ| dξ ≤ max(2r, r 2 ) C N b Ξ (|[u 1N ] -[u 2N ]| + |δ 1 -δ 2 |) |λ| dξ = max(2r, r 2 ) C N b Ξ (|u 1 1N + u 2 1N -u 1 2N -u 2 2N | + |δ 1 -δ 2 |) |λ| dξ ≤ m γ ( u 1 -u 2 + |δ 1 -δ 2 | Ξ ) |λ| Ξ .
Thus, (

are satisfied. We can also easily verify (

. For all λ ∈ L 2 (Ξ), u ∈ V , the mappings p ε (u, •), c(λ, u, •) are linear on V and the mapping J(λ, u, v, •) is a semi-norm on V , which imply that φ 0 satisfies conditions (3.7), (3.8). The mapping φ 0 equally satisfies (3.6) and (3.9).

We set

τ 0 : V → R, τ 0 (v) = |v 1 N + v 2 N | Ξ + |v| ∀ v ∈ V .
The mapping τ 0 , which clearly verifies (3.10), is weakly continuous on V , since it contains a trace term considered in L 2 (Ξ) and the norm in H. As the function s → (s -g) + is Lipschitz continuous on R, using (3.10) we also have

∃ ηε , η ε > 0 |p ε (u 1 , w 1 ) -p ε (u 1 , w 2 ) + p ε (u 2 , w 2 ) -p ε (u 2 , w 1 )| = |p ε (u 1 , w 1 -w 2 ) -p ε (u 2 , w 1 -w 2 )| ≤ ηε |τ 0 (u 1 -u 2 )| w 1 -w 2 ≤ η ε u 1 -u 2 w 1 -w 2 ∀ u 1,2 , w 1,2 ∈ V , (4.2) and ∃ ηc , η c > 0 |c(λ 1 , u 1 , w 1 ) -c(λ 1 , u 1 , w 2 ) + c(λ 2 , u 2 , w 2 ) -c(λ 2 , u 2 , w 1 )| = |c(λ 1 , u 1 , w 1 -w 2 ) -c(λ 2 , u 2 , w 1 -w 2 )| ≤ ηc (| λ 1 -λ 2 | Ξ + |τ 0 (u 1 -u 2 )|) w 1 -w 2 ≤ η c (| λ 1 -λ 2 | Ξ + u 1 -u 2 ) w 1 -w 2 ∀ λ 1,2 ∈ Λ, u 1,2 , w 1,2 ∈ V . (4.3)
Using the relations (2.11) and (3.10) one can clearly establish the following estimates: ∃ ηJ , η J > 0 such that

|J(λ 1 , u 1 , v 1 , w 1 ) -J(λ 1 , u 1 , v 1 , w 2 ) + J(λ 2 , u 2 , v 2 , w 2 ) -J(λ 2 , u 2 , v 2 , w 1 )| ≤ ηJ (| λ 1 -λ 2 | Ξ + |τ 0 (u 1 -u 2 )| + |v 1 -v 2 |) w 1 -w 2 ≤ η J (| λ 1 -λ 2 | Ξ + u 1 -u 2 + |v 1 -v 2 |) w 1 -w 2 ∀ λ 1,2 ∈ Λ, u 1,2 , v 1,2 , w 1,2 ∈ V . (4.4)
From (4.2)-(4.4) it follows that (3.12) is satisfied and if we set w = w 1 , w 2 = 0, we obtain (3.11). Finally, relation (2.12) implies the validity of (3.13).

Thus, by Theorem 3.5 the problem P ε admits a unique solution.

Existence of a variational solution

We shall use several times the following compactness theorem proved by J. Simon [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF].

Theorem 4.2. Let X, U and Y be three Banach spaces such that X ⊂ U ⊂ Y with compact imbedding X → U . Let F be bounded in L p (0, T ; X), where 1 ≤ p < ∞, and ∂F/∂t = { ḟ ; f ∈ F} be bounded in L 1 (0, T ; Y ). Then F is relatively compact in L p (0, T ; U ).

Let F be bounded in L ∞ (0, T ; X) and ∂F/∂t be bounded in L r (0, T ; Y ), where r > 1. Then F is relatively compact in C 0 ([0, T ]; U ). Theorem 4.3. Under the assumptions of Section 2 there exists a solution of the problem P v .

Proof. First, we establish some estimates on the penalized solutions u ε and β ε which will enable us to pass to the limit in P ε in order to obtain a solution of P v . If we choose w = 0 in (2.16), by integrating from 0 to t ∈ ]0, T [ we have

t 0 (ü ε , uε ) ds + t 0 a(u ε , uε ) ds + t 0 b( uε , uε ) ds + t 0 p ε (u ε , uε ) ds + t 0 c(β ε , u ε , uε ) ds ≤ t 0 L, uε ds.
As a is a symmetric bilinear mapping, g 0 is independent of time and u 0 belongs to K, we obtain for all t ∈ ]0, T [

1 2 | uε (t)| 2 + 1 2 a(u ε (t), u ε (t)) + t 0 b( uε , uε ) ds + 1 2ε |ϑ([u εN (t)] + )| 2 Ξ ≤ t 0 L, uε ds - t 0 c(β ε , u ε , uε ) ds + 1 2 |u 1 | 2 + 1 2 a(u 0 , u 0 ).
Using (4.1), Young's inequality, the properties of the truncation operator ϑ and Gronwall's lemma, it follows that there exists a positive constant M independent of ε such that, for all ε > 0, the following estimates on u ε hold

| uε (t)| ≤ M, u ε (t) ≤ M |[u εN (t)] + | Ξ ≤ M √ ε ∀ t ∈ ]0, T [, T 0 uε 2 ds ≤ M. (4.5)
From (2.16) we obtain for all ϕ = (ϕ

1 , ϕ 2 ) ∈ L 2 (0, T ; [H 1 0 (Ω 1 )] d × [H 1 0 (Ω 2 )] d ) T 0 (ü ε , ϕ) dt + T 0 a(u ε , ϕ) dt + T 0 b( uε , ϕ) dt = α=1,2 T 0 Ω α f α • ϕ α dx dt.
Hence, the term üε is bounded in L 2 (0, T ; H -1 ) by a constant independent of ε. For all v ∈ L ∞ (0, T ; V ) ∩ W 1,2 (0, T ; H) such that v(t) ∈ K for almost every t ∈ ]0, T [, we choose in (2.16) w = uε + 1 κ (vu ε ). Then, integrating with respect to t ∈ ]0, T [ from (2.16), we obtain

T 0 (ü ε , v -u ε ) dt + T 0 a(u ε , v -u ε ) dt + T 0 b( uε , v -u ε ) dt + T 0 p ε (u ε , v -u ε ) dt + T 0 c(β ε , u ε , v -u ε ) dt (4.6) + T 0 {J(β ε , u ε , uε , v + κ uε -u ε ) -J(β ε , u ε , uε , κ uε )} dt ≥ T 0 L, v -u ε dt.
Integrating by parts the acceleration term in (4.6) and by a monotonicity argument for p ε we derive

uε (T ), v(T ) -u ε (T ) -1/2, 1/2 -(u 1 , v(0) -u 0 ) - T 0 ( uε , v -uε ) dt + T 0 {a(u ε , v -u ε ) + b( uε , v -u ε ) + c(β ε , u ε , v -u ε )} dt (4.7) + T 0 {J(β ε , u ε , uε , v + κ uε -u ε ) -J(β ε , u ε , uε , κ uε )} dt ≥ T 0 L, v -u ε dt ∀ v ∈ L ∞ (0, T ; V ) ∩ W 1,2 (0, T ; H) with v(t) ∈ K a.e. t ∈ ]0, T [.
From the estimates (4.5) and the previous estimate on the acceleration, it follows that there exists u such that, up to a subsequence,

u ε ⇀ * u in L ∞ (0, T ; V ), uε ⇀ u in L 2 (0, T ; V ), uε ⇀ * u in L ∞ (0, T ; H), üε ⇀ ü in L 2 (0, T ; H -1 ). (4.8)
As W 1,2 (0, T ; V ) ⊂ C 0 ([0, T ]; V ), it follows that, for all t ∈ [0, T ], (u ε (t)) ε is bounded in V by a constant independent of ε and of t, so that by a diagonal process we can extract a subsequence, still denoted by (u ε ) ε , such that

u ε (t) ⇀ u(t) in V ∀ t ∈ [0, T ]. (4.9)
By (4.8) we can easily pass to the limit in the linear terms of (4.7). To pass to the limit in the nonlinear terms we need the compactness result of Theorem 4.2. As ∂Ω α is Lipschitz continuous, the imbeddings from

V α into H α , from V α into [H 1/2 (Ω α )] d and from H α into [H -1/2 (Ω α )] d are compact, α = 1, 2.
Then we may apply Theorem 4.2 with

F = ( uε ) ε , X = V , U = H, Y = H -1 , p = 2, F = (u ε ) ε , X = V , U = H 1/2 , Y = H, r = 2, F = ( uε ) ε , X = H, U = H -1/2 , Y = H -1 , r = 2,
so that, up to a subsequence, we obtain

uε → u in L 2 (0, T ; H), u ε → u in C 0 ([0, T ]; H 1/2 ), uε → u in C 0 ([0, T ]; H -1/2 ). (4.10) Hence, uε (T ), v(T ) -u ε (T ) -1/2,1/2 → u(T ), v(T ) -u(T ) -1/2,1/2 . (4.11)
The functional v → T 0 a(v, v) dt is convex and continuous on L 2 (0, T ; V ), so it is sequentially weakly lower semicontinuous, which implies that lim inf (4.12)

Since v → b(v, v) is convex and continuous on V , it is sequentially weakly lower semicontinuous on V . Thus, as u ε (T ) ⇀ u(T ) in V , we obtain

lim inf ε→0 T 0 b( uε , u ε ) dt = lim inf ε→0 1 2 b(u ε (T ), u ε (T )) - 1 2 b(u 0 , u 0 ) ≥ 1 2 b(u(T ), u(T )) - 1 2 b(u 0 , u 0 ) = T 0 b( u, u)dt. (4.13)
As the imbedding from H 1/2 (Ξ) into L 2 (Ξ) is compact, by (4.9) we have

[u εN ](t) → [u N ](t) in L 2 (Ξ) ∀ t ∈ [0, T ]. (4.14)
Then from (4.5) 3 we derive

0 = lim ε→0 |[u εN (t)] + | Ξ = |[u N ] + (t)| Ξ ∀ t ∈ [0, T ].
Hence, [u N (t)] ≤ 0 almost everywhere on Ξ and for all t ∈ [0, T ], which implies u(t) ∈ K for all t ∈ [0, T ]. Let β := β u ∈ W 1,∞ (0, T ; L ∞ (Ξ)), with β(s) ∈ Λ for all s ∈ ]0, T [, be the solution, which by Proposition 3.3 exists and is unique, of the parabolic variational inequality

( β, λ -β) Ξ + γ(u, β, λ -β) ≥ (χ, λ -β) Ξ ∀ λ ∈ Λ (4.15)
with the initial condition β 0 . We will verify that (u, β) is a solution of the problem P v . By a similar result as in Lemma 3.4, it follows that there exists a constant M ′ > 0, independent of u ε , u, β ε , β, such that for all t ∈ [0, T ]

|β ε (t) -β(t)| 2 Ξ ≤ M ′ t 0 |[u εN ](s) -[u N ](s)| 2 Ξ ds. (4.16)
Then by (4.5) 2 and (4.14) we have

β ε (t) → β(t) in L 2 (Ξ) ∀ t ∈ [0, T ]. ( 4
.17)

To pass to the limit in the friction term, we apply Theorem 4.2 with F =

( uε ) ε , X = V , U = H 1-ι , where 1/2 > ι > 0, Y = H -1 , p = 2 and we obtain u ε → u in W 1,2 (0, T ; H 1-ι ). As the mapping v → [v T ] defined from H 1-ι into [L 2 (Ξ)] d is compact, we have [u εT ] → [u T ] in W 1,2 (0, T ; [L 2 (Ξ)] d ). ( 4
( βε , λ -β ε ) Ξ dt + T 0 γ(u ε , β ε , λ -β ε ) dt ≥ T 0 (χ, λ -β ε ) Ξ dt (4.19) ∀ λ ∈ L 2 (0, T ; L 2 (Ξ)
) with λ(t) ∈ Λ a.e. t ∈ ]0, T [, so that using (4.14), (4.17) and a weakly lower semicontinuity argument, by passing to the lower limit we obtain

T 0 ( β, λ -β) Ξ dt + T 0 γ(u, β, λ -β) dt ≥ T 0 (χ, λ -β) Ξ dt ∀ λ ∈ L 2 (0, T ; L 2 (Ξ)) with λ(t) ∈ Λ a.e. t ∈ ]0, T [.
By Lebesgue's theorem, it follows that (u, β) verify also the parabolic inequality (2.14).

We note that the abstract results presented in section 3 can be also applied to prove the existence and uniqueness of solutions to dynamic contact problems with normal compliance and adhesion between a viscoelastic body and an obstacle.

Similar results as the previous ones are easily seen to hold for the contact with irreversible adhesion, where the evolution of the intensity of adhesion is governed by a differential equation satisfying the relation (3.27), see, e.g., [START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF].

Numerical examples

In this section, an example is presented to illustrate the mechanical behaviour associated with this interface law and to study the influence of the main mechanical parameters on the solution. The dynamic behaviour of a cylindrical viscoelastic block in adhesive frictional contact (µ = 0.2) under axisymmetric conditions with a stiff sphere is analysed, see Figure 1.

The previous sections concern a dynamic contact problem in viscoelasticity, which constitutes an extension of the model considered in [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF][START_REF] Raous | Unilateral contact, friction and adhesion: 3D cracks in composite materials[END_REF] to the case of recoverable adhesion. In elastodynamics, the discrete problem, the algorithms and the numerical solution were given in [START_REF] Raous | Unilateral contact, friction and adhesion: 3D cracks in composite materials[END_REF][START_REF] Acary | Formulation dynamique d'un modèle de zone cohésive tridimensionnelle couplant endommagement et interface[END_REF] for non recoverable adhesion. They are based on a combination of the Non Smooth Contact Dynamics (NSCD) method, developed by M. Jean and J.J. Moreau [START_REF] Jean | The non smooth contact dynamics method[END_REF][START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF][START_REF] Dubois | LMGC90 -Une plateforme de développement dédiée à la modélisation de problèmes d'interaction[END_REF], and the methods proposed in [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF][START_REF] Raous | Unilateral contact, friction and adhesion: 3D cracks in composite materials[END_REF] for the treatment of the adhesion. This approach is extended here to viscoelasticity and to healing process, see also [START_REF] Raous | Recoverable adhesion and friction[END_REF][START_REF] Schryve | Modèle d'adhésion cicatrisante et applications au contact verre/élastomère[END_REF]. The evolution of the adhesion β is computed by using an implicit method and so a fixed point on β is necessary (the convergence test on the relative variation is taken as 10 -5 ). The convergence of the relaxation method included in the NSCD algorithm is tested on the relative variation of the contact power (test condition : 10 -5 ). The implementation of the model with recoverable adhesion was conducted in the finite element code LMGC90 [START_REF] Dubois | LMGC90 -Une plateforme de développement dédiée à la modélisation de problèmes d'interaction[END_REF].

The finite element discretizations consist of P 1 elements and an initial triangulation with 480 nodes for the cross-section of the block and 270 nodes for the cross-section of the sphere is considered. The corresponding size of minimal length of edge mesh is h c = 0.33 µm. A θ-method is used to solve the dynamic equations and the choice θ = 0.51 gives good stability and a small numerical damping.

The sphere is initially in contact with the viscoelastic block at the point O. A compressive cycle is imposed with a prescribed displacement U 1 (t) = u z (t)e z on the upper part of the sphere, see Figure 2. The viscoelastic con-Figure 2: Imposed displacement on the upper face of the sphere u z . stitutive law is described by the following relations:

σ = σ r + σ ir , ε = 1 + ν E σ r - ν E tr(σ r )I, ε = 1 + ν Ẽ σ ir - ν Ẽ tr(σ ir )I,
where E is Young's modulus, ν is Poisson's ratio and Ẽ, ν are the viscosity coefficients. We considered for the sphere E (1) = 2.10 11 Pa, ν (1) = 0.3, ρ (1) =7800 kg.m -3 , Ẽ(1) = η (1) E (1) with η (1) → 0, and for the block Ẽ(2) = η (2) E (2) with η := η (2) ∈ ]0, 10] s, E (2) = 5.10 6 Pa, ν (2) = ν(2) = 0.48, ρ (2) = 1000 kg.m -3 . This example is used to investigate the dependency of the solution on the variation of the following parameters: -the adhesion parameters C N and w, -the volumic viscosity parameter η, -the mesh size.

Effect of adhesion parameters

We consider here three cases of interfacial behaviour with different choices of the adhesion parameters C N and w as given in Table 1.

For η = 1 s, the evolution of β at the candidate contact point B (see Figure 1) and the evolution of contact radius, characterized by the relation g = 0, in these three cases are presented in Figures 3 and4, respectively.

In case 1, the bonds are weak and brittle. Adhesion has minor effect on the solution and a loss of contact is observed. In case 2, the bonds are strong and brittle. An initial contact is observed (radius 0.9 mm), which is due to the stronger adhesion that acts during the initial step, when only gravity is Adhesion has major effect on the solution and the two bodies remain always in contact even at t=10 s.

Effect of the volumic viscosity

The evolutions of the displacement vector and of the intensity of adhesion in case 1, as presented in Figures 5 and6, show a good convergence of viscoelastic solutions towards an elastic solution as η tends to 0.

Effect of the mesh size

We consider the case 3 with η equal to 1 s. The distributions of the normal and tangential components of the stress vector, σ N and σ T , versus the position r are displayed at t = 8 s in Figure 7 for different meshes, as given in Table 2.

The bodies are in contact (σ N ≤ 0 and g = 0) for r ≤ r C , where r C ≃ 4.5 mm is the contact radius, and the bodies are in adhesion (σ N ≥ 0) for Table 2: The different meshes.

r A > r > r C , where r A ≃ 5.5 mm is the adhesion radius. For r ≥ r A the stress vector is negligible.

Theorem 3 . 5 .

 35 Assume that (3.1)-(3.12) and the compatibility condition (3.13) hold. Then there exists a unique solution of the problem Q.

  ε , u ε ) dt ≥ T 0 a(u, u) dt.

Figure 1 :

 1 Figure 1: Initial configurations, boundary conditions, finite element discretization of the problems and points O(r =0, z =0), A(1.63, 0), B(3.2, 0). Γ 1 C and Γ 2 C are bounded by O, C 1 (10.1, 3.98) and O, C 2 (9.82, 0), respectively (length unit is mm).

  Bonds behaviourw (J.m -2 ) C N (N.m -3 ) b (N.s.m -1

Figure 3 :

 3 Figure 3: Evolution of β at point B in three cases.

Figure 4 :

 4 Figure 4: Evolution of the contact radius (g = 0) in three cases.

Figure 5 :

 5 Figure 5: Evolutions of the displacement at point O for different values of η in case 1.

Figure 6 :

 6 Figure 6: Evolutions of β at point A for different values of η in case 1.

Figure 7 :

 7 Figure 7: Distributions of the normal and tangential interfacial stress σ N and σ T versus position r for different mesh sizes of the interaction surfaces Γ 1 C

  it follows that ( u n (t) ) n∈N * and (|χ n (t)| Π 0 ) n∈N * are bounded by constants independent of t ∈ [0, T ].Also, for all n ∈ N * we have

	|β n (t) -βn (t)| Π 0 ≤	T n	d dt	βn (t)	Π 0	for a.e. t ∈ ]0, T [	(3.20)

Table 1 :

 1 Three cases of interfacial behaviour

Laboratoire de Mécanique et d'Acoustique CNRS, 31 chemin Joseph Aiguier, 13402 Marseille Cedex

and Université de Provence, UFR -MIM, Marseille, France 2 Laboratoire de Mécanique et d'Acoustique CNRS, 31 chemin Joseph Aiguier, 13402 Marseille Cedex 20 and Université de Provence, UFR -MIM, Marseille, France 3 Laboratoire de Mécanique et d'Acoustique CNRS, 31 chemin Joseph Aiguier, 13402 Marseille Cedex 20, France E-mails: cocou, schryve, raous@lma.cnrs-mrs.fr