Third-order shear deformable porous metamaterial toroidal shells with arc-type auxetic core: A stability analysis

This study explores the influence of porosity on the stability behavior of shear-deformable auxetic-core sandwich-structured toroidal shell segments (TSSs) with porous carbon nanotube (CNT)-reinforced face sheets supported by a Kerr-type elastic foundation and subjected to external pressure. The CNTs are embedded in a polymer matrix throughout the face sheet thickness, with three distinct porosity distribution patterns examined: uniform, symmetric, and asymmetric. The metamaterial core features an arc-type auxetic design inspired by the traditional reentrant honeycomb structure, with curved ribs that facilitate a smooth transition between adjacent unit cells, reducing stress concentrations. The Kerr elastic foundation is modeled with three parameters: a shear layer in the middle and spring layers at the top and bottom. The governing equations for the TSSs are derived using Reddy's third-order shear deformation theory (TSDT), accounting for von Kármán-type geometric nonlinearity. A three-term deflection solution, based on simply supported boundary conditions, is employed, and the Galerkin method is used to establish the nonlinear load-deflection relationship. The validity of the approach is confirmed through a comparative analysis with existing literature, showing good agreement with theoretical results. Numerical findings provide a detailed analysis of how porosity parameters, including coefficient and distribution type, influence the stability of sandwich TSSs.