In this paper, we present a mathematical approach of thin curved rods within the framework of linear elasticity. We show that any displacement of a curved rod is the sum of a Bernoulli-Navier displacement and residual displacements (including shearing and warping in the most complete decomposition). We give estimates of the terms of the decompositions with respect to δ and the L 2 norm of the strain tensor. Next, we present different loads applied to a curved rod and give the asymptotic behaviors of the corresponding linearized elasticity problems.

The aim of this paper is to decompose the displacements of thin-walled beams with rectangular cross-section. The decomposition is accompanied by estimates of all its terms with respect to the norm of the strain tensor. Korn's inequality is also given.

In this paper, we show that any displacement of a plate is the sum of a Kirchhoff-Love displacement and two terms, one for shearing and one for warping. Then, the plate is loaded in order to obtain that the bending and shearing contribute the same order of magnitude to the fiber rotations.

The paper is dedicated to the modeling and asymptotic investigation of a linear elasticity problem, in the form of variational inequality, for a textile structure. The textile is made of long and thin fibers crossing each others, forming a periodic squared domain. The domain is clamped only partially and an in plane sliding between the fibers is bounded by a contact function, which is chosen to be loose. We also assume a non-penetration condition for the fibers. Both partial clamp and loose contact arise a domain split, leading to different behaviors in each of the four parts. The homogenization is made via unfolding method, with an additional dimension reduction to further simplify the problem. The four cell problems are inequalities heavily coupled by the outer plane macro-micro constraints, while the macroscopic limit problem results to be an inequality of Leray-Lions type with only macro in plane constraints. On both scales, no uniqueness is expected.