Asymptotic behavior of structures made of curved rods

TL;DR

This paper analyzes the asymptotic behavior of curved rod structures as their thickness approaches zero, using the unfolding method within linear elasticity to derive limit displacement and rotation fields.

Contribution

It introduces a detailed decomposition of displacements into elementary and residual parts, characterizes their limits, and couples extensional and inextensional behaviors in the asymptotic analysis.

Findings

Displacements split into elementary and residual components.

Limit displacements satisfy specific variational problems.

Coupling of extensional and inextensional displacements established.

Abstract

In this paper we study the asymptotic behavior of a structure made of curved rods of thickness 2\delta when \delta rightarrow 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.