EXISTENCE OF SOLUTIONS FOR A CLASS OF VARIATIONAL INEQUALITIES

Several problems in mechanics, physics, control and  those dealing with contacts, lead to the study of s ystems of variational inequalities. This model has been studied by Slimane et al. (2004); Bernardi et al. (2004); Brezis (1983); Brezis and Stampacchia (1968); Ciarlet (1978); Grisvard (1985) ; Haslinger et al. (1996); Lions and Stampacchia (1967). We consider a solid occupying an open bounded domain Ω of a sufficiently regular boundary Γ = ∂Ω with unilateral contact with a rigid obstacle.


INTRODUCTION
Several problems in mechanics, physics, control and those dealing with contacts, lead to the study of systems of variational inequalities.
We consider a solid occupying an open bounded domain Ω of a sufficiently regular boundary Γ = ∂Ω with unilateral contact with a rigid obstacle.

Theorem 1.1
Let P∈L 2 (ΩR 3 ) be the resulting of force density. Then there exists a unique solution for the variational problem: find u∈V such that: To prove this theorem we make use of the Lax-Milgram which is based on proving the continuity and V-ellipticity of the bilinear form a(u,v) and the continuity of l(v).

FORMULATION OF THE CONTACT PROBLEM
Here we consider a solid occupying an open bounded domain Ω of a sufficiently regular boundary Γ = ∂Ω.
The solid is supposed to have: • A density on the volume, of force P in Ω • Homogenous boundary conditions on Γ • Unilateral contact with a rigid obstacle of equation We use the space Let us introduce the convex subspace K for the authorized displacements, to be defined as: We consider the following variationnal formulation Find: Such that: And the reduced problem becomes: Find u∈K such that: For any solution (u, η) of problem (P e ), u is a solution of problem (P I ).

Proof
Let (u,η) be a solution of problem (P e ) and u∈K, ∀v∈K and we have: We assume that x = 0: By replacing v by v-u in line one of problem (P e ), we get: Let u be a solution of problem (P I ) then (u,η) is a solution of (P e ): By using Green's formula, we get: We assume that v = i±ϕ, with ϕ∈D(ΩR 3 ), (i.e., ϕ is of a compact support), then the integral on the contour is zero: The integral on a contact area leads to: And with the property of convexity of K, we get: , u 0 , u , u ,u 0 χ − η = χ − η = χ ≥

Proof
The existence of the solution u of problem is a direct application of Slimane et al. (2004).
Let us consider:

Remark
In problem (P I ), we have: • if v = 0, then: The Ker of the form (η, v) is characterized by: Let v∈V, then v and -v are in K from the problem (P I ) and L(u) = 0, we have: We remplace v by -v in L(u) to get: L is of a compact support in V and from the following inf-sup condition: We can prove that there exists η∈H −1 (Ω). Then (u, η) satisfies line one of problem (P e ). The definition of K and L(u) = 0, leads to: This proves the existence of the solution. Let U 1 and U 2 be two solutions of problem (P I ). With U 1 = u 1 and U 2 = u 2 then: By adding that W = U 2 and W = U 1 we have: By the inf-sup condition of problem (P e ) gives us: ( )

THE DISCRETE PROBLEM
We introduce a discrete subspace V h of V such that: And dim V h <∞, therefore there exists a basis: {ω i }, I = 1 to N h , we can then write: We assume U = u and W = v.

Theorem 3.1
Let U and U k be the solutions of problems (P I ) and (P h ), respectively. Let us denote by A∈L (V, V') the map defined, by a(U, W) = (AU, W), then:

JMSS
With P is the resultant of the volume force.

Proof
By the definitions of U and W, we have: By adding these inequalities and transposing terms, we obtain: By subtracting a(U, U h )+ a(U h , U) from both sides and grouping terms and by using the continuity and the coercively of the bilinear form a(U, W), we deduce: We obtain:

NUMERICAL RESULTS
Consider elastic plate with the undeformed rectangle shape (0, 10)×(0, 2). The body force is the gravity force f and the boundary force g is zero on lower and upper side. On the two vertical sides of the beam are fixed (Fig. 1-3).

CONCLUSION
By starting with the classical model for a deformed elastic solid with a unilateral contact of a rigid body, we