On a class of buckling problems in a singularly perturbed domain

We consider the buckling of an annular thin elastic plate when it is subjected to uniform in-plane compressive forces on its outer boundary. This geometrical inhomogeneity means that the pre-buckling stress field is nonconstant and, as a consequence, the resulting variable-coefficient eigenproblem is not solvable in closed form. In the limit when the annulus can be regarded as a disk with a small neighbourhood of its centre removed, singular perturbation techniques are used to construct asymptotic approximations for the critical buckling loads. Our results describe both symmetric and asymmetric buckling patterns and show good agreement with some numerical simulations.