Structural Optimization and Form-finding of Cylindrical Shells for Targeted Elastic Postbuckling Response

This paper presents a finite element based numerical study on controlling the postbuckling behavior of thin-walled cylindrical shells under axial compression. With the increasing interest of various disciplines for harnessing elastic instabilities in materials and mechanical systems, the postbuckling behavior of thin-walled cylindrical shells may have a new role to design materials and structures at multiple scales with switchable functionalities, morphogenesis, etc. In the design optimization approach presented herein, the mode shapes and their amplitudes are linearly combined to generate initial geometrical designs with predefined imperfections. A nonlinear postbuckling finite element analysis evaluates the design objective function, i.e., the desired postbuckling forcedisplacement path. Single and multi-objective optimization problems are formulated with design variables consisting of shape parameters that scale base eigenvalue shapes. A gradientbased algorithm and numerical sensitivity evaluations are used. Results suggest that an optimized shape for a cylindrical shell can achieve a targeted response in the elastic postbuckling regime with multiple mode transitions and energy dissipation characteristics. The optimization process and the obtained geometry can be potentially used for energy harvesting and other sensing and actuation applications. INTRODUCTION The design optimization of thin-wall axially compressed cylindrical shells with buckling as a constraint [1, 2] typically focuses on maximizing the critical buckling load in the primary static branch because postbuckling behavior has long been regarded as an undesirable phenomenon due to the significant loss of load-carrying capacity and catastrophic failure that follows once it occurs [3]. Yet, a paradigm has shift evolving over the past decade and has turned its attention to utilize such instable behavior for new purposes [4-6]. It is well known that a load drop in the stable load-deformation path relates to the dissipation of strain energy as the shell transitions into a stable post-buckled shape. Several studies on axially compressed cylindrical shells have focused on the response near the first bifurcation point for purposes of residual capacity estimation [7-9], see path (a) in Fig. 1. Physically, mode transitions imply changes in the buckling waves on the cylinder’s surface. Under certain geometric and stiffness constraints multiple buckling events can still occur under increasing axial shortening. The curvature in cylindrical shells provides a natural geometric constraint that allows the attainment of multiple critical points, as shown in path (b) of Fig. 1. While path (b) has a relatively lower stiffness it features multiple bifurcation points due to changes in the deformed geometry after each critical point. Fig. 1. Schematic diagram of the postbuckling behavior of a compressed cylindrical shell.