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

Catherine Pideri (1) , Pierre Seppecher (2)

1 ANLA - Laboratoire d'Analyse Non Linéaire Appliquée
2 IMATH - Institut de Mathématiques de Toulon - EA 2134

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 Γ-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.

Mots clés: beam; rod; non-linear elasticity; 3D-1D; Γ-convergence

Domaines: Mécanique [physics.med-ph]

Catherine Pideri, Pierre Seppecher. Asymptotics of a non-planar rod in non-linear elasticity. Asymptotic Analysis, 2006, 48 (1-2), pp.33-54. ⟨hal-00527296⟩