Uniqueness of solutions in the boundary value problems of bending of thin micropolar plates with surface effects

Abstract

We propose a micropolar thin plate theory that incorporates surface elasticity effects to take into account simultaneously the contribution of high surface-to-volume ratios and the influence of microstructural mechanics at micro/nano scales. Using a direct averaging approach across the plate’s thickness, we derive governing equations that consider the elastic properties of the upper and lower surfaces, treated as two-dimensional micropolar bodies. The boundary value problems are analyzed, with particular attention to solution uniqueness, as a first step toward establishing the well-posedness of the model. A simplified example demonstrates the mathematical formulation and compares the results with classical plate theory. A numerical example highlights the combined effects of surface elasticity and micropolar characteristics as the plate thickness varies, illustrating the influence of these factors on effective bending stiffness. The results reveal that surface and micropolar effects contribute to deformations by varying magnitudes, depending on material properties and plate size.