Some applications of the WKB method to the wrinkling of bi-annular plates in tension

An application of WKB methods is proposed here for a stretched annular thin plate with piecewise-constant mechanical properties (also known as a bi-annular plate). Unlike the classical scenario involving only a simple annular such plate, in certain cases the neutral stability curve fails to be convex and the critical eigenmodes behave rather differently as the plate becomes progressively thinner (equivalent to ${\mu \to \infty}$ in our notations). On one side of this curve, the corresponding eigenmodes are localised near the inner rim of the annulus, while in the remaining part these functions are concentrated along the interface separating the two annular sub-regions. By using the asymptotic reduction technique proposed by Coman and Haughton in (Acta Mech 185:179–200, 2006), the original fourth-order three-point boundary-value problem is formally reduced to a pair of second-order differential equations coupled through a set of matching conditions at the interface. It is shown that for ${\mu \gg 1}$ the critical eigenvalues for both cases mentioned above can be approximated by solving a couple of simple transcendental equations and that the results predicted compare well with the direct numerical simulations of the original problem.