Singular behaviour in a generalized boundary eigenvalue problem for annular plates in tension

A novel application of boundary-layer asymptotic techniques to a generalized linear eigenvalue problem is presented. Our investigation is concerned with a bifurcation equation that governs the formation of wrinkles in thin annular plates subjected to in-plane tensile loading on the inner boundary. If η denotes the ratio of the inner and outer radii of the annulus, then the critical wrinkling load satisfies Λ = Λ_C(η), where the function Λ_C is available only numerically. It is known that there is a critical value ̂η such that Λ → ∞ as η → ̂η but, until now, little has been understood about this singular behaviour. Asymptotic methods enable us to capture accurately and describe the nature of this blow-up phenomenon which we show is sensitive to the forms of the boundary conditions imposed at the edges of the annular plate. Our analytical findings are complemented by a series of comparisons with direct numerical simulations that shed further light on the singular behaviour.