A General Theory and Analytical Solutions for Post-Buckling Behaviors of Thin Sheets

Abstract

A robust and general theory is of great importance to understanding the mechanism of wrinkling, describing its behaviors and guiding the design of thin sheets. Two widely accepted theories, tension-field theory (Wagner, 1929) and thin-film theory (Cerda and Mahadevan, 2003), have been successfully used in predicting the location and pattern of wrinkling and defining critical conditions for its onset, but they have failed to describe the post-buckling behaviors (i.e., bifurcations, increasing wavenumber, and corresponding changes in morphology). In this paper, we propose a new theory of wrinkling that considers the effects of both mechanical and geometrical characteristics of thin sheets on the spatial variation in wrinkles and is valid for the general problems of post-buckling analyses. By circumventing the Föppl–von Kármán equations, the theory offers a compelling complement to thin-film theory and provides analytical details of wrinkles, especially for closed form of post-buckling behaviors. An energy barrier is introduced to assess the configurational changes of wrinkles as they evolve. Three typical examples are selected for validating the robustness of the theory and exploring its implications. More broadly, the present work provides important guidelines for eliminating wrinkles in thin sheet structures.