Journal of Matematics and Statistics 9 (4: 35-39, 3 ISSN: 549-3644 3 doi:.3844/jmssp.3.35.39 Publised Online 9 (4 3 (ttp://www.tescipub.com/jmss.toc EXISTENCE OF SOLUTIONS FOR A CLASS OF ARIATIONAL INEQUALITIES Taalla Frek Department of Matematics, Faculty of Science, Badji Moktar University, Annaba, Algeria Received 3--6; Revised 3--9; Accepted 3--9 ABSTRACT In tis study we considered a deformed elastic solid wit a unilateral contact of a rigid body. We studied te existence, uniqueness and continuity of te deformation of tis solid wit respect to te data. We proved te existence of solutions for a class of variational inequalities. Keywords: ariational Inequalities, Unilateral Contact. INTRODUCTION Several problems in mecanics, pysics, control and tose dealing wit contacts, lead to te study of systems of variational inequalities. Tis model as been studied by Slimane et al. (4; Bernardi et al. (4; Brezis (983; Brezis and Stampaccia (968; Ciarlet (978; Grisvard (985; Haslinger et al. (996; Lions and Stampaccia (967. We consider a solid occupying an open bounded domain Ω of a sufficiently regular boundary Γ = Ω wit unilateral contact wit a rigid obstacle. Teorem. Let P L (ΩR 3 be te resulting of force density. Ten tere exists a unique solution for te variational problem: find u suc tat: Wit: a(u,v = Te bilinear form l(v = Te linear form a ( u,v = l( v, v { ( 3 = v H Ω,R To prove tis teorem we make use of te Lax-Milgram wic is based on proving te continuity and -ellipticity of te bilinear form a(u,v and te continuity of l(v. 35. FORMULATION OF THE CONTACT PROBLEM Here we consider a solid occupying an open bounded domain Ω of a sufficiently regular boundary Γ = Ω. Te solid is supposed to ave: A density on te volume, of force P in Ω Homogenous boundary conditions on Γ Unilateral contact wit a rigid obstacle of equation x 3 = on contact surface Ω c = Ω/Γ. Te displacement is given by: ( u ( x.e3, in Ω c Wit (e, e, e 3 Cartesian base we denote by η te reaction of te obstacle on te solid. Te relations leading to a unilateral contact (witout friction are given by: ( ( u x.e 3, in Ω c η.e 3, in Ω c u x η.e =, in Ω We use te space H ( 3,R 3 c Ω of functions in H (Ω, R 3 equals to zero on Г. Let us introduce te convex subspace K for te autorized displacements, to be defined as:
Taalla Frek / Journal of Matematics and Statistics 9 (4: 35-39, 3 { 3 3 c K = v H Ω,R, v.e, in Ω We consider te following variationnal formulation Find: Suc tat: Wit: ( u, η H ( Ω,R 3 H ( Ω ( P ( 3 e a u,v c η,v = l v, v H Ω,R c η,v = ηvdx Ωc And te reduced problem becomes: Find u K suc tat: Teorem. ( P I ( u l( v u a u,v For any solution (u, η of problem (P e, u is a solution of problem (P I. Let (u,η be a solution of problem (P e and u K, v K and we ave: Problem (P e leads to: We assume tat x = : η,v η,v χ η,u, χ K η,u η,u By replacing v by v-u in line one of problem (P e, we get: Were: a ( u,v u c ( η,v u = l( v u c η,v u = η,v u = η,v + η,u a u, v u l v u, v K 36 Let u be a solution of problem (P I ten (u,η is a solution of (P e : a ( u, v u l( v u, v K By using Green's formula, we get: a ( u, v u η, v u l( v u We assume tat v = i±ϕ, wit ϕ D(ΩR 3, (i.e., ϕ is of a compact support, ten te integral on te contour is zero: a ( u, φ = l ( φ, φ Te integral on a contact area leads to: By assuming tat: η,v u, v K v = v = u η,u = And wit te property of convexity of K, we get: Teorem. χ η,u = χ,u η,u = χ,u For any P H (Ω, R 3, te problem (P e as a unique solution (u,η H (ΩR 3 H (Ω Te existence of te solution u of problem is a direct application of Slimane et al. (4. Let us consider: Remark L( v = a ( u,v l( v In problem (P I, we ave: if v =, ten: l( u a u,u if v = u, ten:
Taalla Frek / Journal of Matematics and Statistics 9 (4: 35-39, 3 a ( u,u l( u L( u = Te Ker of te form (η, v is caracterized by: { ( 3 3 = v H Ω,R, u.e =, in Ω Let v, ten v and -v are in K from te problem (P I and L(u =, we ave: + ( λ + l( v l( u a ( u,v l( v a ( u,u l( u a ( u, v l( v L( u a u, v a u, u b v, l v l u + = We remplace v by -v in L(u to get: a ( u, v l( v L( u a u, v l v a u, v + l v = L is of a compact support in and from te following inf-sup condition: η,v sup β η v H We can prove tat tere exists η H (Ω. Ten (u, η satisfies line one of problem (P e. Te definition of K and L(u =, leads to: χ η,u = χ,u η,u = χ,u, χ K Tis proves te existence of te solution. Let U and U be two solutions of problem (P I. Wit U = u and U = u ten: ( ( a U,W U l W U, W K a U,W U l W U, W K By adding tat W = U and W = U we ave: ( ( ( + ( λ l( U U a ( U,U U l( U U ( ( a U,U U l U U a U,U U b U U, a U,U U l U U 37 ( a U U,U U a U U,U U U U U = U By te inf-sup condition of problem (P e gives us: v H Ω,R, η,v = η,v η = η 3 3. THE DISCRETE PROBLEM We introduce a discrete subspace of suc tat: { 3 = v C Ω,R, v P k v =,on Ω And dim <, terefore tere exists a basis: {ω i, I = to N, we can ten write: v N = β ω i i i= Now, let us construct a closed convex subset K of suc tat K sould be reduced to a finite number of constraints on te β i : K v, v.e ϕ 3 = at every vertex of eac triangle K Ten K K and K. We remark tat problem (P I is equivalent to find u K suc tat: P a u,v u l v u, v K We assume U = u and W = v. Teorem 3. Let U and U k be te solutions of problems (P I and (P, respectively. Let us denote by A L (, te map defined, by a(u, W = (AU, W, ten: U W + M U U = P AU ' U W + U W
Taalla Frek / Journal of Matematics and Statistics 9 (4: 35-39, 3 Wit P is te resultant of te volume force. By te definitions of U and W, we ave: a ( U,U W ( P,U W, W Ka ( U,U W ( P,U W, W K By adding tese inequalities and transposing terms, we obtain: + ( ( ( P,U W a ( U,W a ( U,W a U,U a U,U P,U W + + + Fig.. Mes By subtracting a(u, U + a(u, U from bot sides and grouping terms and by using te continuity and te coercively of te bilinear form a(u, W, we deduce: P AU U W ' ' U U + P AU U W + M U U U W Since: M M U U U W U W Fig.. Isovalue of deplacement u x We obtain: U W + P AU ' M U U = U W + U W W K and W K 4. NUMERICAL RESULTS Consider elastic plate wit te undeformed rectangle sape (, (,. Te body force is te gravity force f and te boundary force g is zero on lower and upper side. On te two vertical sides of te beam are fixed (Fig. -3. 38 Fig. 3. Isovalue of deplacement u y 5. CONCLUSION By starting wit te classical model for a deformed elastic solid wit a unilateral contact of a rigid body, we
Taalla Frek / Journal of Matematics and Statistics 9 (4: 35-39, 3 proved te existence of solutions for a class of variational inequalities. 6. REFERENCES Bernardi, C., Y. Maday and F. Rapetti, 4. Discrétisations ariationnelles De Problèmes Aux Limites Elliptiques. 3rd Edn., Matématiques et Applications, Springer, ISBN-: 3543694, pp: 3. Brezis, H. and G. Stampaccia, 968. Sur la régularité de la solution d'inéquations elliptiques. Bull. Soc. Mat. France, 96: 53-8. Brezis, H., 983. Analyse fonctionnelle, Collection Matématiques Appliquées pour la Maîtrise. Ciarlet, P.G., 978. Te Finite Element Metod for Elliptic Problems. st Edn., Nort-Holland, Amsterdam, New-York, Oxford, ISBN-: 4448587, pp: 53. Grisvard, P., 985. Elliptic Problems in Nonsmoot Domains. st Edn., SIAM, Piladelpia, ISBN-: 69735, pp: 4. Haslinger, J., I. Hlavacek and J. Necas, 996. Numerical metods for unilateral problems in solid mecanics. Handbook Numer. Anal., 4: 33-485. Lions, J.L. and G. Stampaccia, 967. ariational inequalities. Comm. Pure Appl. Mat., : 493-59. DOI:./cpa.363 Slimane, L., A. Bendali and P. Laborde, 4. Mixed formulations for a class of variational inequalities. Model. Mat. Anal. Numer., 38: 77-. DOI:.5/man:49 39