Compressed sandwich structures, comprising two stiff face plates separated by a softer core material, while designed principally as efficient integral structures, can lose this quality when faces buckle locally. Interaction between overall (Euler) buckling and local buckling of one face suggests that failure will localize into the centre. A variational formulation, leading to a pair of nonlinear differential equations subject to integral constraints, describes the post–buckling response. These are solved by a combination of numerical shooting and continuation techniques, such that the response far into the unstable post–buckling regime can be portrayed. Solutions with both linear and nonlinear constitutive core relations are compared with the results of an engineering (body–force) approach, and with those of earlier (periodic) Rayleigh–Ritz analyses. The latter demonstrate the extra destabilization that comes with localization.