Hygro-electro-elastic analysis of plates and shells via a three-dimensional layer-wise exponential matrix formulation

A fully coupled hygro−electro−elastic (HEE) formulation is developed to investigate the static response of smart plate and shell structures for aerospace applications. The proposed three-dimensional model is based on a system of five coupled second-order differential equations expressed in the orthogonal mixed curvilinear coordinate system (α, β, z). Owing to a unified mathematical framework, spherical shells, cylindrical panels, cylinders, and flat plates can be analyzed under combined hygroscopic, electrical, and mechanical loadings. Simpler geometries are naturally recovered from the general spherical-shell formulation through appropriate assumptions on the curvature radii. The solution strategy combines Navier harmonic expansions along the in-plane directions with the exponential matrix technique through the thickness coordinate, leading to a closed-form three-dimensional solution. The formulation enables the direct evaluation of displacement and stress fields together with moisture content, moisture flux, electric potential, and electric displacement distributions. A layer-wise description is adopted across the thickness, ensuring an accurate representation of local three-dimensional effects for different thickness ratios and stacking sequences while preserving high computational efficiency. The accuracy of the proposed model is first assessed through comparisons with refined two-dimensional coupled HEE formulations available in the literature. Subsequently, a series of benchmark problems involving four structural geometries and three loading scenarios is presented to examine the static HEE behavior of curved smart structures and to highlight the influence of geometry, coupling effects, and loading conditions on the global and local responses. The proposed formulation investigates the mutual interaction among hygrometric, electrical, and mechanical fields in multilayered smart structures subjected to combined environmental and mechanical loading. Such coupling is relevant to aerospace smart structures employing electrically active materials, where moisture diffusion can modify the mechanical response and,consequently, the structural performance under environmental conditions.