On a bending and torsion asymptotic theory for linear nonhomogeneous anisotropic elastic rods

Abstract

We use asymptotic techniques on a re-scaled three-dimensional linear elasticity model to justify and generalise models describing the elastic behaviour of rods, taking into account the nonhomogeneity and anisotropy of the material they are made of. General bending models for nonhomogeneous anisotropic rods are obtained as the limit of the tridimensional elasticity model when the cross section area tends to zero. We also include convergence results mathematically justifying the method. Next, by identification of second-order terms in the asymptotic expansion we obtain a general theory for transversally nonhomogeneous isotropic rods taking into account Timoshenko and Saint Venant's effects.