Asymptotic Behaviour of Curved Rods by the Unfolding Method

We consider in this work general curved rods with a circular cross-section of radius δ. Our aim is to study the asymptotic behaviour of such rods as δ→0, in the framework of the linear elasticity according to the unfolding method. It consists in giving some decompositions of the displacements of such rods, and then in passing to the limit in a fixed domain. A first decomposition concerns the elementary displacements of a curved rod which characterize its translations and rotations, and the residual displacements related to the deformation of the cross-section. The second decomposition concerns the displacements of the middle-line of the rod. We prove that such a displacement can be written as the sum of an inextensional displacement and of an extensional one. An extensional displacement will modify the length of the middle-line, while an inextensional displacement will not change this length in a first approximation. We show that the H1−norm of an inextensional displacement is of order 1, while that of an extensional displacement is in general, of order δ. A priori estimates are established and convergence results as δ→0, are given for the displacements. We give their unfolded limits, as well as the unfolded limits of the strain and stress tensors. To prove the convergence of the strain tensor, the introduction of elementary and residual displacements appears as essential. By passing to the limit as δ→0 in the linearized system of the elasticity, we obtain on the one hand, a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem coupling the limit of inextensional displacements and the limit of the angle of torsion. Introduction The rods we consider here are thin curved cylinders built around a middle-line which is a smooth curve in R. To be more precise, given a curve ζ in R, the cross-section of a rod with the middle-line ζ (i.e., its intersection with a normal plane at each point M of ζ), is a disk D(M ; δ) of radius δ. Our work is essentially based on two decompositions of the displacements of the rod. Under the action of any displacement u, the disk D(M ; δ) is submitted on the one hand to a translation and a rotation, that we call elementary displacements of the rod, and, on the other hand, to a displacement that deforms the disk, called residual displacement. The second decomposition concerns the displacements of the middle-line of the rod. We prove that such a displacement can be written as the sum of an inextensional displacement and of an extensional one. As suggested by its name, an extensional displacement will modify the length of the middle-line, while an inextensional displacement will not change this length in a first approximation. If the H−norm of the inextensional displacement is of order 1, that of the extensional displacement is in general, of order δ. The proofs of a priori estimates are based on the method used by Kondratiev and Oleinik [9] and by Cioranescu, Oleinik and Tronel [2] to establish Korn inequalities for frame-type structures and junctions. Once a priori estimates are established, we study the convergence, as δ → 0, of the displacements of a curved rod. We give their unfolded limits, as well as the unfolded limits of the strain and stress tensors. In the next step, we turn our attention to the linearized system of the elasticity written for a family of curved rods. By passing to the limit as δ → 0, we obtain on the one hand, a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem coupling the 1 ha l-0 06 20 20 2, v er si on 1 9 Se p 20 11 Author manuscript, published in "Mathematical Methods in the Applied Sciences 27, 17 (2004) 2081-2110" DOI : 10.1002/mma.546