Nonlinear buckling and symmetry breaking of a soft elastic sheetsliding on a cylindrical substrateNorbert Stoop1and Martin Michael M¨uller2,31Department of Mathematics, Massachusetts Institute of Technology;77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA2Equipe BioPhysStat, LCP-A2MC,Universit´e de Lorraine; 1 boulevard Arago, 57070 Metz, France3Institut Charles Sadron, CNRS-UdS; 23 rue du Loess,BP 84047, 67034 Strasbourg cedex 2, France(Dated: March 18, 2015)AbstractWe consider the axial compression of a thin sheet wrapped around a rigid cylindrical substrate.In contrast to the wrinkling-to-fold transitions exhibited in similar systems, we find that the sheetalways buckles into a single symmetric fold, while periodic solutions are unstable. Upon furthercompression, the solution breaks symmetry and stabilizes into a recumbent fold.Using linearanalysis and numerics, we theoretically predict the buckling force and energy as a function of thecompressive displacement. We compare our theory to experiments employing cylindrical neoprenesheets and find remarkably good agreement.PACS numbers: 46.32.+x, 46.70.-p, 68.60.Bs1arXiv:1503.05030v1[cond-mat.soft]17 Mar 2015
I.INTRODUCTIONWhen you roll up your sleeves to get some work done, you will not pay attention to theintricate folding patterns which form around your arms. However, these patterns are notonly interesting for graphics designers but of eminent importance for biology and technologyalike [1, 2]: one can find them in the twinkling of an eye  as well as in the developmentof organs such as the brain [4, 5], the intestine [6, 7], or the kidney .Technologicalapplications include structures for optics  or microfluidics  to name just a few.One common theme in these examples is that they consist of coupled layers which un-dergo morphological changes in response to external or internal constraints such as a simplecompression or volumetric growth.The materials involved range from swelling hydrogels[11, 12] to supported graphene [13, 14] and many others. A well-studied setup in this con-text consists of a stiff membrane attached to a flat elastic or fluid bulk material [15–17].The interplay between the bending of the sheet and the response of the bulk leads to theformation of wrinkles when the sheet is compressed uniaxially. Beyond a critical compres-sion the wrinkles vanish and localized folds appear in the sheet. Interestingly, the shape ofthe sheet on the fluid can be found analytically [18–20] as long as the sheet does not touchitself .When the substrate is not flat, translational invariance is broken and a whole plethoraof folding patterns can be found [1, 2, 22, 23].In this article we study a particular typeof such a system in a cylindrical geometry.An elastic cylindrical membrane is wrappedaround a solid cylinder of same radius and compressed parallel to the axis of symmetry.