ASYMPTOTIC BEHAVIOR OF STRUCTURES MADE OF CURVED RODS

Author:

GRISO GEORGES1

Affiliation:

1. Laboratoire Jacques-Louis Lions, CNRS & Université Pierre et Marie Curie (Paris VI), Boîte postale 187, 4, Place Jussieu 75252 Paris Cedex 05, France

Abstract

In this paper, we study the asymptotic behavior of a structure made of curved rods of thickness 2δ when δ tends to 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. coincides with rigid body displacements in the junctions. Any e.r.s.d. is given by two functions belonging to [Formula: see text] where [Formula: see text] is the skeleton structure (i.e. the set of rods with middle lines). One of this function [Formula: see text] is the skeleton displacement, the other [Formula: see text] gives the cross section rotation. We show that [Formula: see text] 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 [6], where on 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 rod torsion angles.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Analysis

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