Asymptotic behavior of structures made of curved rods

Abstract. In this paper we study the asymptotic behavior of a structure made of curved rods of thickness 2δ when δ→0. This study is carried on within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of the structure displacements and on the passing to the limit in fixed domains. We show that any displacement of a structure is the sum of an elementary rods-structure displacement (e.r.s.d.) concerning the rods cross sections and a residual one related to the deformation of the cross-section. The e.r.s.d. coincide with rigid body displacements in the junctions. Any e.r.s.d. is given by two functions belonging to H1(S;R3) where S is the skeleton structure (i.e. the set of the rods middle lines). One of this function U is the skeleton displacement, the other R gives the cross-sections rotation. We show that U is the sum of an extensional displacement and an inextensional one. We establish a priori estimates and then we characterize the unfolded limits of the rods-structure displacements. Eventually we pass to the limit in the linearized elasticity system and using all results in [5], 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 coupling the limit of inextensional displacement and the limit of the rods torsion angles. Résumé. Dans cet article on étudie le comportement asymptotique d’une structure formée de poutres courbes d’́epaisseur 2δ quand δ→0. Cette étude se place dans le cadre de l’élasticite ́ linéaire et utilise la méthode de l’éclatement. Cette méthode est basée sur plusieurs décompositions des déplacements et sur le passage a ̀ la limite dans des domaines fixes