DEFORMATION OF RODS WITH SMALL BENDING STIFFNESS

Abstract

In this paper we continue the study of the deformation of a rod with small bending stiffness begun in two earlier papers. In the theory that we have developed problems are posed as singular perturbations of problems in which the rod is replaced by a string having no resistance to bending. In this paper we apply a general theory of singular perturbations for boundary-value problems for nonlinear systems of ordinary differential equations to obtain our result. This result consists of a construction of an approximate solution to a model problem of the deformation of a rod with small bending stiffness, together with a proof that our solution approximates an exact solution to the problem. Our approximate solution consists of an interior approximation which is a solution of the string problem together with boundary-layer correction terms. The most important point of the paper is the construction of these boundary-layer terms. We also consider a case in which our construction fails together with an example which serves to illuminate this situation.