From wrinkling to global buckling of a ring on a curved substrateR. Lagrange,1F. L´opez Jim´enez,2D. Terwagne,2,*M. Brojan,2,†and P. M. Reis2, 3,‡1Department of Mathematics, Massachusetts Instituteof Technology, Cambridge, MA 02139, USA2Department of Civil and Environmental Engineering,Massachusetts Institute of Technology, Cambridge, MA 02139, USA3Department of Mechanical Engineering,Massachusetts Institute of Technology, Cambridge, MA 02139, USAAbstractWe present a combined analytical approach and numerical study on the stability of a ring boundto an annular elastic substrate, which contains a circular cavity. The system is loaded by depres-surizing the inner cavity.The ring is modeled as an Euler-Bernoulli beam and its equilibriumequations are derived from the mechanical energy which takes into account both stretching andbending contributions. The curvature of the substrate is considered explicitly to model the workdone by its reaction force on the ring. We distinguish two different instabilities: periodic wrinklingof the ring or global buckling of the structure. Our model provides an expression for the criticalpressure, as well as a phase diagram that rationalizes the transition between instability modes.Towards assessing the role of curvature, we compare our results for the critical stress and the wrin-kling wavelength to their planar counterparts. We show that the critical stress is insensitive to thecurvature of the substrate, while the wavelength is only affected due to the permissible discretevalues of the azimuthal wavenumber imposed by the geometry of the problem. Throughout, wecontrast our analytical predictions against finite element simulations.*Current address:Department of Physics, Facult´e des Sciences, Universit´e Libre de Bruxelles (ULB),Bruxelles 1050, Belgium.†Current address: Faculty of Mechanical Engineering, University of Ljubljana, Slovenia.‡Email: [email protected]1arXiv:1509.07392v1[cond-mat.soft]24 Sep 2015
I.INTRODUCTIONWrinkling is a stress-driven mechanical instability that occurs when a stiff and slendersurface layer, bonded to a compliant substrate, is subject to compression. This universal in-stability phenomenon is found in numerous natural and technological/engineering examples,over a wide range of length scales, including: carbon nanotubes , pre-stretched elastomersused in flexible electronics applications , human skin , drying fruit , surface morphol-ogy of the brain  and mountain topographies generated due to tectonic stresses [6, 7].Over the past decade, there has been an upsurge of interest in the study of the mechan-ics of wrinkling, along with a change of paradigm in regarding surface instabilities as anopportunity for functionality, instead of a first step in the route to structural failure [8, 9].