Asymptotic analysis of a rod with small bending stiffness

Abstract

In this paper we consider a model problem for the deformation of a rod with small bending stiffness. We show that this problem can be considered as a singular perturbation of the problem in which the rod is replaced by a string with no resistance to bending. We construct an approximate solution to this problem. As the bending stiffness tends to zero this solution tends to the solution of the string problem away from the ends of the rod which are assumed to be clamped. However, as one would expect, there is a boundary layer near each end of the rod. The main point of the paper is to show how to construct the boundary layer corrections.