Tamtam Proceedings - lamsin

550 Ayadi1. IntroductionConsider a thin plate of thickness 2ɛ occupying a two-dimensional open set ω. Assumethat it is simply supported on the whole of its edge γ and climbed on a part γ 0 whoseLebesgue measure is not zero. Furthermore, the plate is subjected to a one-parameterplane load λ.h on another part γ 1 of its edge.It is well known, if there are any obstacles in the neighborhood of the plate, that thereexists an increasing sequence (λ n ) escaping to the infinity and such that each load λ n .hinvolves an instability of the plate manifested by a great vertical displacement [1, 6, 8, 12].Such a displacement is called buckling mode of the plate corresponding to the bucklingload λ n .h.In this paper, we suppose that the plate is in presence of a rigid fixed plane obstaclethat lies just above it. The paper is organized as follows. The first section is devoted tothe description of the studied unilateral buckling model. It should be mentioned here thatthe problem has been explored since the late seventies. But it was investigated withina general framework and was given a difficult proof [11]. Then, in the second section,the use of a conformal finite element method leads to a discrete nonlinear eigenvalueproblem that is mathematically well posed [9]. The third and principal section is devotedto both construction and convergence justification of the suggested algorithm. In orderto test our algorithm and validate the finite element method proposed, a unit length beamis considered in the forth section. Some numerical results, in both cases: without anyobstacle and in presence of an obstacle, are suggested. First we give the buckling criticalmode, then we give some curves that show the dependence of the approximated criticalload upon the mesh size.2. Mathematical Modelling of Unilateral BucklingWhen taking into account the unilateral contact condition and considering nonlinearelasticity as constitutive law, we obtain a very difficult mathematical problem [4, 6, 10].Nevertheless, we know an obvious solution to the latter. It is the linear elasticity solutionfor which the vertical displacement is null and the plane displacements are solution to thefollowing variational equation:2∑α,β,ν,µ=1∫ωE αβνµ∂u p ν∂x µ∂v α∂x βdω =2∑∫α=1γ 1H α v α dγ for all v ∈ V, (1)where E αβνµ is the membrane stiffness tensor (depending on Young’s modulus and thePoisson’s coefficient), H α = ∫ ɛ−ɛ h αdx 3 and V = {v ∈ H 1 (ω) 2 : v = 0 on γ 0 }.TAMTAM –Tunis– 2005