The mechanics of tension-driven out-of-plane buckling in auxetic plates: Bridging continuum theory and lattice metamaterials

Abstract

Shape morphing driven by structural instabilities is prevalent in nature and increasingly utilized in advanced technologies across many length scales. Auxetic materials, characterized by their negative Poisson’s ratio and large expansion under tension, offer opportunities for programmability through tailored geometries of their unit cells. In this study, we explore the out-of-plane buckling behaviour of laterally constrained auxetic plates subjected to uniaxial tension using linear buckling analysis, aiming to utilize them as effective media for programmable shape morphing. We derive an analytical model describing the out-of-plane buckling of laterally constrained auxetic plates under tension, validated via finite-element simulations. A systematic parametric analysis explores how key design parameters – Poisson’s ratio, plate aspect ratio, and boundary conditions – influence critical buckling thresholds and mode characteristics. To exploit this mechanical equivalence for design, we reveal congruent governing mechanics with their continuum counterparts, demonstrating that homogenized continuum theory accurately predicts the buckling behaviour of discrete lattice structures when unit-cell resolution meets a critical threshold. Leveraging this equivalence, we develop a machine learning surrogate model to map the geometric parameters of latticed plates to continuum plate theory, enabling rapid predictive design; 3D-printed specimens of latticed auxetic plates are experimentally characterized to validate buckling modes. By inversely computing their equivalent bending stiffness using continuum theory, we bridge the microscale geometry of metamaterials with macroscale mechanical responses. This work establishes a foundational understanding of buckling-induced shape morphing in auxetic plates, unlocking their potential in applications ranging from soft robotics and deployable structures to adaptive surfaces.