Wrinkled Membranes Part Ii: Analytical Models

We present a general analytical model for determining the location and pattern of wrinkles in thin membranes and for making preliminary estimates of their wavelength and amplitude. A rectangular membrane under simple shear and a square membrane subject to corner loads are analysed. In the first problem, our model predicts the wavelength and the wrinkle amplitude to be respectively inversely proportional and directly proportional to the fourth root of the shear angle. Both values are directly proportional to the square root of the height and thickness of the membrane, and are independent of the Young's modulus. In the second problem two wrinkling regimes are identified. The first regime is characterised by radial corner wrinkles and occurs for load ratios less than 1/(√ 2−1); the number of wrinkles is proportional to the fourth root of the radius of the wrinkled region and the magnitude of the corner force, and inversely proportional to the Young's modulus and thickness cubed. The amplitude of these wrinkles is inversely proportional to their number, directly proportional to the square root of the radius of the wrinkled region and the magnitude of the corner force, and inversely proportional to the square root of the Young's modulus and thickness. The second regime occurs for load ratios larger than 1/(√ 2 − 1), and is characterised by a large diagonal wrinkle, plus small radial wrinkles at all four corners. Analytical expressions for the variation of the width and amplitude of the large wrinkle with the load ratio are obtained for this case also. All analytical predictions are compared with experimental and computational results from the other two papers in this series.