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

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. Thin elastic objects like beams or plates are of crucial importance in structural design for their low weight and cost. Their properties and the link with the properties of the material they are made of, was a constant subject of study for mechanicians. Approximations for the displacements at small scale which are induced by a global displacement of the structure are well known from the pioneer works of Euler, Bernoulli and Navier. The energy induced by such a global displacement can be estimated and then the behavior of the structure is known. The mathematical justifications of these approximations are more recent. A very wide literature is devoted to this subject. Here we are more particularly interested by the so-called 3D-1D reduction, the limit object is one-dimensional (the reader can refer, for instance, to (2) or (15) for a review of one-dimensional elastic models). Non-planar thin objects like shells or curved rods are also important. Here we restrict our attention to non-planar curved rods: the limit object is a one-dimensional non-planar curve. The mechanical appli- cations are numerous: let us simply mention that helicoidal springs are nothing else but curved elastic rods and that they are widely used in mechanisms. The mathematical literature devoted to curved rods is much more restricted than to the straight ones. This lack seems essentially due to the difficulty for obtaining fine a priori estimates in this complex geometry. First works based on formal expansions are due to Jamal et al. (6) or Sanchez et al. (13). Complete studies have been performed in the last years by Jurak et al. (7), by Griso (5) and by Pideri et al. (14). These studies, like besides the major part of the studies devoted to the straight case, are performed in the framework of linear elasticity. The first reason is mathematical: a fundamental tool was missing