Inhomogeneities and localised buckling patterns

The aim of this paper is to re-examine the phenomenon of localisation in the classical problem of buckling of the elastica on a linear elastic foundation. Largely speaking, our efforts are concentrated on certain spatial variations of the mechanical characteristics of the foundation. This new feature changes the spectral properties of the linearised bifurcation problem and leads to linear localised eigenmodes; the expressions of these buckling modes are found with the help of a complex WKB approximation. By considering numerical simulations of the full non-linear problem, we then investigate the evolution of the instability patterns in the post-buckling regime. In contrast to other recent studies on similar problems, the approach taken here is conducted upon the assumption of finite length for the system under discussion.