Asymptotics of a non-planar rod in non-linear elasticity

Article type: Research Article

Authors: Pideri, C. | Seppecher, P.

Affiliations: Laboratoire d'Analyse Non Linéaire Appliquée et Modélisation, Université de Toulon et du Var, BP 132-83957, La Garde cedex, France E-mail: seppecher@univ-tln.fr

Abstract: We study the asymptotic behavior of a non-linear elastic material lying in a thin neighborhood of a non-planar line when the diameter of the section tends to zero. We first estimate the rigidity constant in such a domain then we prove the convergence of the three-dimensional model to a one-dimensional model. This convergence is established in the framework of $\varGamma $-convergence. The resulting model is the one classically used in mechanics. It corresponds to a non-extensional line subjected to flexion and torsion. The torsion is an internal parameter which can eventually by eliminated but this elimination leads to a non-local energy. Indeed the non-planar geometry of the line couples the flexion and torsion terms.

Keywords: beam, rod, non-linear elasticity, 3D–1D, [TeX:] $\varGamma $-convergence

Journal: Asymptotic Analysis, vol. 48, no. 1-2, pp. 33-54, 2006