Enhancing nonlinear static stability behavior of axially compressed sandwich composite toroidal shells with a bio-inspired auxetic core

Modern engineering increasingly utilizes complex curved shell structures made from lightweight materials, especially in high-performance fields such as aerospace and aeronautic engineering, where stability under extreme conditions is essential. Developing a new generation of auxetic metamaterials with enhanced mechanical properties drives the need for innovative sandwich structures. Accordingly, this study assesses the influence of a novel bio-inspired butterfly-shaped auxetic core on the stability of axially compressed sandwich toroidal shell segments (TSSs), aiming to improve upon traditional re-entrant auxetic structures. The primary focus is to evaluate how this new auxetic design enhances shell stability, which is crucial for advancing lightweight, high-performance structures. Inspired by butterfly wing structures, the butterfly-shaped core improves stiffness and exhibits a negative Poisson's ratio (NPR), leading to superior stability. The face sheets are reinforced with carbon nanotubes (CNTs) embedded in a polymer matrix, with either uniform (UD) or functionally graded (FG) distributions. A three-parameter model represents the Kerr-type elastic foundation, consisting of a central shear layer and two spring layers. The governing equations are derived using von Kármán shell theory and Stein and McElman approximations, with the Galerkin method applied to establish nonlinear load–deflection relationships under simply supported boundary conditions. Validation against existing studies confirms the model's accuracy. Results show that the butterfly-shaped auxetic core outperforms traditional re-entrant structures in terms of stability, critical buckling loads, and postbuckling behavior. The effects of core unit cell geometry, elastic foundation parameters, shell geometry, and CNT distribution are also examined. These findings provide valuable insights into the design of lightweight metamaterial TSSs with NPR.