Wrinkling analysis of a stiff shallow film mounted on a cylindrically curved compliant substrate, Part I: Cylindrical mode

It is well known that when a material bilayer which consists of a stiff thin film perfectly bonded on a compliant substrate develops mechanical instabilities upon the application of in-plane compressive loads. This paper investigates in depth a particular surface instability, namely the one dimensional cylindrical mode, for the case where the initial configuration of the bilayer is cylindrically curved and is acted upon membrane biaxial strains. The investigation of the role of the curvature-induced anisotropy to the geometrical characteristics of the cylindrical wrinkling mode, i.e. the wavelength and the amplitude, is the main focus of this study. To this end, the film is modelled as a thin shallow shell and is analysed within the framework of the general shallow shell theory while the mechanical behaviour of the much softer substrate is investigated through linear elasticity theory for solids. Two cylindrical modes are presented analytically, namely along and across the ridge of the initial curvature, and it has been found that the latter is the favoured wrinkling mode due to the lower total energy of the bilayer in the buckled state. The perturbation method has been employed for the calculation of the displacement components in the compliant substrate and semi-analytical expressions for the wavelengths and the amplitudes are presented which may act as a useful design tool. The results of the present study are in good agreement compared to the results of other studies available in the open literature.