An explicit analytical description for deep post-buckling behaviors of substrate-free films with complex in-plane biaxial loads

Abstract

Substrate-free films (i.e., films with in-plane and out-of-plane constraints only on some or all edges) can be found everywhere in nature, daily life, and industrial applications. The theory of their buckling and wrinkling behavior has been mature and widely used in engineering. Significant progress has been made in numerical computation of film instability. However, on the one hand, the inherent drawback of numerical calculations is that they cannot provide explicit results, and the relevant mechanism and parametric analysis are limited; on the other hand, the widely used finite element simulation is difficult to compare the energy of different modes, because the mode is always constrained to a certain order in the simulation (they cannot perform the post-buckling of specified higher order modes as the calculation process is guided by the inherent energy minimization mechanism of the computational frameworks and the higher order modes were excluded). Such two aspects constrain the mechanism study of deep post-buckling related behaviors that dependent on accurate and explicit descriptions of high order modes, e.g., the deep post-buckling bifurcation/secondary buckling. As a result, an explicit analytical description for deep post-buckling behaviors of films is still of significant research value. Reviewing classical explicit analytical descriptions, in any case of the explicit descriptions by multiple trigonometric series method, Galerkin method and perturbation method, in essence, at most three terms of double trigonometric series are used to describe the post-buckling behaviors of thin films explicitly, which always introduces significant error when the film enters deep post-buckling stage. To overcome the stated problems, herein, considering a uniform rectangular film model with arbitrary in-plane loads, and based on the Galerkin method, we firstly develop a high-precision explicit analytical description for the deep post-buckling behaviors of films upon complex in-plane biaxial loads, which proposes a new insight on the buckling mode transition upon extremely deep post-buckling or complex biaxial loads, and suggests that the correct solution of the deep post-buckling bifurcation/secondary buckling requires a complete dynamic model. The research results have theoretical significance on improving the instability mechanical system of thin films, and can also provide theoretical basis for precise control of wrinkle morphology and design of film devices corresponding applications.