ASYMPTOTIC BEHAVIOR OF STRUCTURES MADE OF PLATES

Abstract

The aim of this paper is to study the asymptotic behavior of a structure made of plates of thickness 2δ when δ → 0. This study is carried out within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of displacements of the structure and on the passing to the limit in fixed domains.We begin by studying the displacements of a plate. We show that any displacement is the sum of an elementary displacement concerning the normal lines on the middle surface of the plate and a warping. An elementary displacement is linear with respect to the variable x3. It is written [Formula: see text] where [Formula: see text] is a displacement of the mid-surface of the plate. We show a priori estimates and convergence results when δ → 0. We characterize the limits of the unfolded displacements of a plate as well as the limits of the unfolded strained tensor.Then, we extend these results to structures made of plates. We show that any displacement of a structure is the sum of an elementary displacement of each plate and of a residual displacement. The elementary displacements of the structure (e.p.s.d.) coincide with elementary rod displacements in the junctions. Any e.p.s.d. is given by two functions belonging to H1( S ; ℝ3) where S is the skeleton of the structure (the set formed by the mid-surfaces of the plates constituting the surface). One of these functions, [Formula: see text], is the skeleton displacement. We show that [Formula: see text] is the sum of an extensional displacement and of an inextensional one. The first one characterizes the membrane displacements and the second one is a rigid displacement in the direction of the plates and it characterizes the flexion of the plates.Eventually, we pass to the limit as δ → 0 in the linearized elasticity system. On the one hand, we obtain a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem satisfied by the limit of inextensional displacements.