Size-dependent thermo-mechanical stability of flexomagnetic nano-plates with initial imperfections

Abstract

This study presents a comprehensive thermo-mechanical stability analysis of an imperfect rectangular piezo-flexomagnetic nano-plate. The theoretical model simultaneously incorporates both piezomagnetic and direct flexomagnetic effects, enabling a more comprehensive representation of magneto-mechanical coupling phenomena at the nanoscale. To capture the size-dependent behavior inherent to such nano-structures, the strain-gradient theory is employed through the inclusion of a material length-scale parameter. The governing differential equations and corresponding boundary conditions are derived based on the von Kármán nonlinear strain–displacement relations, classical plate theory, and principle of minimum total potential energy. Closed-form analytical solutions are obtained for critical buckling and post-buckling behavior under mechanical, thermal, and coupled thermo-mechanical loading conditions for both roller and hinge edge supports. The analytical formulation is validated through comparisons with benchmark results reported in the literature. A parametric investigation is conducted to evaluate the effects of key parameters—including flexomagnetic coupling, aspect ratio, boundary conditions, initial geometric imperfection, and thermal loading—on the buckling and post-buckling response of the nano-plate. The numerical results reveal that the influence of the flexomagnetic effect is more pronounced under uniaxial in-plane loading compared to biaxial loading. Additionally, in biaxial loading conditions, the impact of the flexomagnetic property is significantly greater for aspect ratios less than unity. The stability performance of the nano-plate shows consistent improvement due to flexomagnetic effects for both uniaxial and biaxial loading scenarios. Size effects play a critical role in nanoscale structural behavior, as evidenced by the substantial increase in critical buckling load with the length-scale parameter. Geometric imperfections generally lower the critical load, though their impact on the post-buckling response varies with both imperfection magnitude and boundary constraints. Thermal loading demonstrates a more pronounced destabilizing effect compared to purely mechanical loading, particularly in plates with imperfections. Boundary conditions substantially influence the structural response: roller supports offer greater initial load capacity, whereas hinged supports develop enhanced membrane stiffening at larger deformation amplitudes. These findings offer valuable insights for the design and development of smart two-dimensional nano-devices where flexomagnetic coupling can be utilized for enhanced stability control.