On the Mathematical Structure of Eigen-deformations in a Föppl-von Kármán Bifurcation System

Bifurcations of thin circular elastic plates subjected to uniform normal pressure are explored by taking into account the flexural compliance of the edge restraint. This effect is accounted for by formally reinforcing the outer rim of the plate with a curved beam element, whose net effect is akin to a Hookean spring relating the inclination of the median surface of the plate (with respect to a horizontal plane) and the radial edge moment. The new added feature reflects the imperfect nature of the boundary restraints achieved under realistic physical conditions, and includes as particular cases the usual boundary conditions associated with flexurally simply-supported and clamped plates. It is shown here that in the limit of eigen-deformations with very short wavelengths in the azimuthal direction the two equations in the Föppl-von Kármán bifurcation system remain coupled. However, for edge restraints close to the former type the asymptotic limit of the bifurcation system is described by an Airy-like equation, whereas when the outer rim of the plate is flexurally clamped the Airy-like structure morphs into a standard equation for parabolic cylinder functions. Our singular perturbation arguments are complemented by direct numerical simulations that shed further light on the aforementioned results.