The study of asymptotically fine wrinkling in nonlinear elasticity using a boundary layer analysis

•We study nonlinear elastic wrinkling using a linear-stability analysis.•An analytic postbuckling analysis is performed to determine the magnitude of buckle.•Perform a boundary layer analysis to study the asymptotic behaviour.

Fine sinusoidal wrinkling on the surfaces of mechanically compressed objects has been observed in many contexts over many years. In this paper we investigate such wrinkling through the application of a boundary layer analysis to an elastostatic problem in nonlinear elasticity. We determine the onset of buckling using a linear-stability analysis, and the leading-order postbuckling behaviour through consideration of higher order terms of the energy. The object is assumed to (initially) ‘preserve’ its shape, so that these equations reduce to ordinary differential equations. We then apply a boundary-layer analysis to this problem, determining (in the asymptotic limit of large wavenumbers) the leading order behaviours of the eigenmode, the critical parameter, and the magnitude of the buckle. We find that, to leading order, the shape of the buckle and the time of buckling are independent of the local geometry, however the magnitude of the buckle is dependent on the local geometry. Indeed we find that the magnitude of a buckle with wavenumbers ςγ2(⁎) and ςγ3(⁎) (for fixed γ2(⁎) and γ3(⁎)) has leading asymptotic order ς−3/2λ(2), for an increment λ(2)λ(2) of the critical parameter beyond the critical time of buckling. We provide electronic supplementary material which extends this analysis to that of incompressible elasticity. Finally we confirm the accuracy of our ansatz on a compressed NeoHookean ring.