Decomposition of Curved Rod Displacements: A Mathematical Theory of Linearly Elastic Curved Rods

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. We give estimates of the terms of the decomposition with respect to the thickness of the rod and the 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.