Rippling of long rectangular rubber blocks under bending

Abstract

The problem of finite elastic deformation of a long rectangular rubber block which is deformed in a perturbed cylindrical configuration is examined here, and is motivated from the problem of determining surface rippling that is observed in bent multi-walled carbon nanotubes. The problem of finite elastic bending of a tube is considerably more complicated than the geometrically simpler problem of finite elastic bending of a rectangular block. Accordingly, we examine here the simpler block problem which is assumed to be sufficiently long so that the out of plane end effects may be ignored. The general equations governing plane strain deformations of an isotropic incompressible perfectly elastic Mooney material, which models rubber-like materials, are used to determine small superimposed deformations upon the well-known controllable family for the deformation of rectangular blocks into a sector of a solid bounded by two circular arcs. Traction free boundary conditions are assumed in an average sense along the bounding circular arcs. Physically realistic rippling is found to occur and typical numerical values are used to illustrate the solution graphically.