A WKB approximation for the divergence-buckling instability of a travelling web

Abstract

A WKB method is proposed to analyse edge-buckling phenomena in a simplified model for axially moving, stretched thin elastic webs. The fourth-order bifurcation equation for this configuration is characterised by the presence of two turning points. Connection matrices for these points are constructed and their implication on determining the critical buckling values are discussed. In particular, we derive a transcendental eigenrelation for localised eigenmodes whose complexity depends on the order of the WKB approximation employed. When secondary boundary-layer effects are ignored, the eigenrelation can be solved with high accuracy using regular perturbation techniques. Although the inclusion of (bending) edge effects makes the eigenrelation less tractable analytically, it can still be efficiently solved with the help of standard numerical methods. Comparisons with direct numerical simulations and previously derived approximations are also presented.